Metrical Almost Periodicity and Applications to Integro-Differential Equations 9783111233871, 9783111233031

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Marko Kostić Metrical Almost Periodicity and Applications to Integro-Differential Equations

De Gruyter Studies in Mathematics

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Edited by Carsten Carstensen, Berlin, Germany Gavril Farkas, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesztesy, Waco, Texas, USA Niels Jacob, Swansea, United Kingdom Zenghu Li, Beijing, China Guozhen Lu, Storrs, USA Karl-Hermann Neeb, Erlangen, Germany René L. Schilling, Dresden, Germany Volkmar Welker, Marburg, Germany

Volume 95

Marko Kostić

Metrical Almost Periodicity and Applications to Integro-Differential Equations

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Mathematics Subject Classification 2020 Primary: 42A75, 43A60, 35B15; Secondary: 35B10, 34C27, 47D06 Author Prof. Dr. Marko Kostić University of Novi Sad Faculty of Technical Sciences Trg D. Obradovića 6 21125 Novi Sad Serbia [email protected]

ISBN 978-3-11-123303-1 e-ISBN (PDF) 978-3-11-123387-1 e-ISBN (EPUB) 978-3-11-123417-5 ISSN 0179-0986 Library of Congress Control Number: 2023933268 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2023 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Preface The theory of almost periodic functions is still a very active field of research of many authors. The main purpose of this monograph, entitled “Advances in Almost Periodicity” is to continue the research studies presented in my previously published monographs “Selected Topics in Almost Periodicity” [431] and “Almost Periodic and Almost Automorphic Solutions to Integro-Differential Equations” [428]. In this monograph, we continue the analysis of multi-dimensional almost periodic type functions in Banach spaces started in [431]. Before proceeding any further, we would like to note that this is probably the first research monograph, which examines multi-dimensional ρ-almost periodic functions and multi-dimensional ρ-almost periodic functions in general metric, where ρ is an arbitrary binary relation on the underlying Banach space under our consideration. We provide certain applications to the abstract Volterra integro-differential inclusions in Banach spaces, the abstract impulsive differential inclusions in Banach spaces and the classical partial differential equations. We continue our work in the general framework of the Lebesgue spaces with variable coefficients. This monograph is composed of three parts, which are further broken down into chapters, sections, and subsections. The numbering of theorems, propositions, lemmas, corollaries, definitions, etc., is done by chapter and section, as in my previously published monographs [426, 427, 428, 429, 430, 431]. The reference list is sorted in the alphabetical order and contains more than five hundred new titles not quoted in [428] and [431]. The notation of most important function spaces and the index are also made. Concerning the prerequisites, we would like to note that the readers should be familiar with the basics of functional analysis, the integration theory, the (abstract) partial differential equations and the theory of vector-valued almost periodic functions. The monograph could be interesting not only for the experts in the fields of functional analysis, almost periodicity and partial differential equations, but also for the master students and PhD students in mathematics. I would like to thank my family, closest friends, relatives, godfather and colleagues. Special thanks go to Prof. S. Pilipović (Novi Sad, Serbia), as well as to V. Fedorov (Chelyabinsk, Russia), C.-C. Chen (Taichung, Taiwan), W.-S. Du (Kaoshiung, Taiwan), M. Fečkan (Bratislava, Slovakia), V. Kumar (Kamand, India), E. M. A. El-Sayed (Alexandria, Egypt), B. Chaouchi (Khemis Miliana, Algeria), D. Velinov, P. Dimovski, B. Prangoski (Skopje, Macedonia), J. Vindas (Ghent, Belgium), L. Neyt (Trier, Germany), R. Ponce (Talca, Chile), C. Lizama (Santiago, Chile), M. Pinto (Santiago, Chile), A. Chávez (Trujillo, Peru), B. Jovanović (MI SANU, Belgrade), M. T. Khalladi (Adrar, Algeria), A. Rahmani (Adrar, Algeria), M. Hasler (Pointe-à-Pitre, Guadeloupe, France), K. Khalil (Marrakesh, Maroco), P. J. Miana, L. Abadias, J. E. Galé (Zaragoza, Spain), M. Murillo-Arcila, J. A. Conejero, A. Peris, J. Bonet (Valencia, Spain), M. S. Moslehian (Mashhad, Iran), C.-C. Kuo (New Taipei City, Taiwan), V. Valmorin (Pointe-à-Pitre, Guadeloupe, France), D. N. Cheban https://doi.org/10.1515/9783111233871-201

VI � Preface (Chisinau, Moldova), V. Keyantuo (Rio Piedras Campus, Puerto Rico, USA), T. Diagana (Huntsville, USA) and G. M. N’Guérékata (Baltimor, USA). Loznica/Novi Sad April, 2023

Marko Kostić

Introduction Motivated by some earlier results of P. Bohl [144, 145] and E. Esclangon [292, 293, 294], the Danish mathematician H. Bohr, the younger brother of the Nobel Prize-winning physicist N. Bohr, introduced the class of almost periodic functions in 1925 [146]. The theory of almost periodic functions is still very popular, full of open problems, conjectures and possibilities for further expansions. Suppose that F : ℝn → X is a continuous function, where (X, ‖ ⋅ ‖) is a complex Banach space. Then we say that F(⋅) is almost periodic if and only if for each ε > 0 there exists l > 0 such that for each t0 ∈ ℝn there exists τ ∈ B(t0 , l) ≡ {t ∈ ℝn : |t − t0 | ⩽ l} such that 󵄩󵄩 󵄩 󵄩󵄩F(t + τ) − F(t)󵄩󵄩󵄩 ⩽ ε,

t ∈ ℝn ;

here | ⋅ − ⋅ | denotes the Euclidean distance in ℝn . This is equivalent to saying that for any sequence (bk ) in ℝn there exists a subsequence (ak ) of (bk ) such that the sequence of functions (F(⋅+ak )) converges in Cb (ℝn : X), the Banach space of all bounded continuous functions from ℝn into X, equipped with the sup-norm. We know that F(⋅) is almost periodic if and only if there exists a sequence of trigonometric polynomials in ℝn , which converges uniformly to F(⋅). If the function F : ℝn → X is locally p-integrable, where 1 ⩽ p < ∞, then we say that F(⋅) is Stepanov-p-almost periodic if and only if for every ε > 0 there exists l > 0 such that for each t0 ∈ ℝn there exists τ ∈ B(t0 , l) ∩ ℝn with 󵄩󵄩 󵄩 󵄩󵄩F(t + τ + u) − F(t + u)󵄩󵄩󵄩Lp ([0,1]n :X) ⩽ ε,

t ∈ ℝn .

Further on, we say that a locally p-integrable function F : ℝn → X is: (i) equi-Weyl-p-almost periodic if and only if, for every ε > 0, there exist two finite real numbers l > 0 and L > 0 such that for each t0 ∈ ℝn there exists τ ∈ B(t0 , L) ∩ ℝn with sup [l

t∈ℝn

− np 󵄩 󵄩

󵄩 󵄩󵄩F(τ + ⋅) − F(⋅)󵄩󵄩󵄩Lp (t+l[0,1]n :X) ] < ε.

(ii) Weyl-p-almost periodic if and only if, for every ε > 0, there exists a finite real number L > 0 such that for each t0 ∈ ℝn there exists τ ∈ B(t0 , L) ∩ ℝn with lim sup sup [l l→+∞

t∈ℝn

− np 󵄩 󵄩

󵄩 󵄩󵄩F(τ + ⋅) − F(⋅)󵄩󵄩󵄩Lp (t+l[0,1]n :X) ] < ε.

(1)

It is well known that any Bohr almost periodic function is Stepanov-p-almost periodic, as well as that any Stepanov-p-almost periodic function is equi-Weyl-p-almost periodic. The set of all Bohr almost periodic functions (Stepanov-p-almost periodic functions, the equiWeyl-p-almost periodic functions) is a vector space with the usual operations, which is https://doi.org/10.1515/9783111233871-202

VIII � Introduction no longer true for the set of Weyl-p-almost periodic functions; furthermore, it can be simply proved that any p-locally integrable function F : ℝn → X for which the limit lim|t|→+∞ F(t) exists in X is equi-Weyl-p-almost periodic (p ⩾ 1). The class of Weyl-palmost periodic functions contains all above-mentioned; we also know that any equiWeyl-p-almost periodic function is Besicovitch-p-almost periodic, and that there exists a Weyl-p-almost periodic function f : ℝ → ℝ which is not Besicovitch-p-almost periodic [431]. The notion of Besicovitch-p-almost periodicity for a function F : ℝn → X can be p introduced in many equivalent ways; traditionally, if F ∈ Lloc (ℝn : X), then we first define the Besicovitch seminorm ‖F‖ℳp

1 := lim sup[ n (2t) t→+∞

1/p

󵄩 󵄩p ∫ 󵄩󵄩󵄩F(s)󵄩󵄩󵄩 ds]

.

(2)

[−t,t]n

In actual fact, it can be simply proved that ‖⋅‖ℳp is a seminorm on the space ℳp (ℝn : X) p consisting of those Lloc (ℝn : X)-functions F(⋅) for which ‖F‖ℳp < ∞. Denote Kp (ℝn : p n X) := {f ∈ ℳ (ℝ : X) ; ‖F‖ℳp = 0} and Mp (ℝn : X) := ℳp (ℝn : X)/Kp (ℝn : X). The seminorm ‖ ⋅ ‖ℳp on ℳp (ℝn : X) induces the norm ‖ ⋅ ‖M p on M p (ℝn : X) under which M p (ℝn : X) is complete; hence, (M p (ℝn : X), ‖ ⋅ ‖M p ) is a Banach space. It p is said that a function F ∈ Lloc (ℝn : X) is Besicovitch-p-almost periodic if and only if there exists a sequence of trigonometric polynomials (almost periodic functions, equivalently) converging to F(⋅) in (M p (ℝn : X), ‖ ⋅ ‖M p ). The vector space consisting of all Besicovitch-p-almost periodic functions is denoted by Bp (ℝn : X). Clearly, Bp (ℝn : X) is a closed subspace of M p (ℝn : X) and therefore a Banach space itself. Concerning the Banach space M p (ℝn : X), we would like to recall that this space is not separable for any finite exponent p ⩾ 1; see, e. g., [615, Theorem 18] which concerns the one-dimensional case. Albeit very general, the class of all Besicovitch almost periodic functions, i. e., the class of all Besicovitch-1-almost periodic functions, is extremely important, because any Besicovitch almost periodic function F : ℝn → X has a mean value; many other classes of generalized almost periodic functions, which are defined in a much more elementary way, do not possess this feature. For example, the class of almost periodic functions in the Lebesgue measure (the class of m-almost periodic functions, equivalently), which was introduced by S. Stoiński [709] in 1994, does not possess this feature: A Lebesgue measurable function f : ℝ → X is said to be m-almost periodic if and only if for each real numbers ε, η > 0 the set 󵄩 󵄩 {τ ∈ ℝ : sup m({t ∈ [x, x + 1] : 󵄩󵄩󵄩f (t + τ) − f (t)󵄩󵄩󵄩 ⩾ η}) ⩽ ε} x∈ℝ

Introduction

� IX

is relatively dense in ℝ. Arguing as in the scalar-valued case, it readily follows that any Stepanov-p-almost periodic function is m-almost periodic (1 ⩽ p < +∞); this is no longer true for the class of equi-Weyl-almost periodic functions, since it can be easily shown that the function χ[0,1/2] (⋅) is equi-Weyl-1-almost periodic, but not m-almost periodic. It is also well known that if g : ℝ → ℂ is an almost periodic function which has a bounded analytical extension in a strip around the real axis, then the function f : ℝ → ℂ, given by f (x) := 1/g(x), if g(x) ≠ 0 and f (x) := 0, if g(x) = 0, is m-almost periodic. For the sequel, we recall that P. Kasprzaka, A. Nawrocki, and J. Signerska-Rynkowska have analyzed, in [401], the integrate-and-fire models with an almost periodic type input function. In [401, Example 3.3], the authors have constructed a continuous m-almost periodic function t f : ℝ → ℝ such that the mean value M(f ) := limt→+∞ (1/t) ∫0 f (s) ds does not exist. k

More precisely, let Ak := 4k ℤ + 2k , Bk := A2k and ak := (k + 1)2 ⋅ 22 for all k ∈ ℕ. Define the function fk : ℝ → ℝ in the following way: if there does not exist z ∈ Bk such that t ∈ [z + (1/2) − (1/(k + 1)), z + (1/2) + (1/(k + 1))), then we set fk (t) := 0; if z ∈ Bk and t ∈ [z+(1/2)−(1/(k +1)), z+(1/2)), resp. t ∈ [z+(1/2), z+(1/2)+(1/(k +1))) for some k ∈ ℕ, then we set fk (t) := ak t−ak (z+(1/2)−(1/(k+1))), resp. fk (t) := −ak t+ak (z+(1/2)+(1/(k+1))). It has been proved that the function f (t) := ∑∞ k=0 fk (t), t ∈ ℝ is well-defined, continuous, m-almost periodic and does not possess the mean value (it is also well known that there exists a uniformly continuous and bounded Levitan almost periodic function f : ℝ → ℝ which is not m-almost periodic; see, e. g., [577, Example 3.1]). See also the research studies [483] by J. Kwapisz and [692] by J. Signerska-Rynkowska; for more details about the onedimensional m-almost periodic functions and their applications, we refer the reader to the research article [173] by D. Bugajewski, A. Nawrocki and the doctoral dissertation [578] of A. Nawrocki. Further on, the notion of almost automorphy was discovered by the American mathematician S. Bochner in 1955 while he was studying problems related to differential geometry [143]. The study of almost automorphy on (semi-)topological groups starts presumably with the papers of W. A. Veech [739, 740], which were published during the period 1965–1967. For more details about almost automorphic functions on semitopological groups, we refer the reader to [196, Section 4]. Suppose that F : ℝn → X is a continuous function. Then we say that the function F(⋅) is almost automorphic if and only if for every sequence (bk ) in ℝn there exist a subsequence (ak ) of (bk ) and a mapping G : ℝn → X such that lim F(t + ak ) = G(t)

k→∞

and

lim G(t − ak ) = F(t),

k→∞

(3)

pointwisely for t ∈ ℝn . The range of an almost automorphic function F(⋅) is relatively compact in X, and the limit function G(⋅) is bounded on ℝn but not necessarily continuous on ℝn . If the convergence of limits appearing in (3) is uniform on compact subsets of ℝn , then we say that the function F(⋅) is compactly almost automorphic; clearly, any almost periodic function F : ℝn → X is compactly almost automorphic. It is well known

X � Introduction that an almost automorphic function F(⋅) is compactly almost automorphic if and only if F(⋅) is uniformly continuous. Any compactly almost automorphic function F : ℝn → ℝ is Levitan N-almost periodic and admits Fourier series (not necessarily unique) whose Bochner–Fejer sums converge to F(⋅) uniformly on compact subsets of ℝn ; see, e. g., [790, Theorem 2.1] and [789] for the notion and more details. The Stepanov and Weyl classes of multi-dimensional almost automorphic functions have recently been analyzed in [431]. In this monograph, we primarily consider new classes of multi-dimensional almost periodic functions; concerning almost automorphic functions, the only new contribution of ours is contained in Section 3.4, where we analyze the multi-dimensional Besicovitch almost automorphic functions. For more details about almost periodic functions, almost automorphic functions, various generalizations and applications, we refer the reader to the research monographs and articles [99, 125, 196, 259, 263, 313, 317, 344, 428, 495, 797]. In our previous research studies presented in the monographs [428] and [431], we have applied our theoretical results in the analysis of the existence and uniqueness of almost periodic type solutions for various classes of the abstract Volterra integrodifferential equations, the abstract fractional differential equations, and the classical partial differential equations. For example, the obtained results are successfully applied in the analysis of the existence and uniqueness of almost periodic type solutions for the following (fractional) Poisson heat equations 𝜕 [m(x)v(t, x)] = (Δ − b)v(t, x) + f (t, m(x)v(t, x)), t ∈ ℝ, x ∈ Ω; { 𝜕t v(t, x) = 0, (t, x) ∈ [0, ∞) × 𝜕Ω, γ

D [m(x)v(t, x)] = Δv(t, x) + bv(t, x), t ⩾ 0, x ∈ Ω; { { { t v(t, x) = 0, (t, x) ∈ [0, ∞) × 𝜕Ω; { { { x ∈ Ω, {m(x)v(0, x) = u0 (x), and the following fractional semilinear equation with higher-order differential operators in the Hölder space X = C α (Ω): γ

β { {Dt u(t, x) = − ∑ aβ (t, x)D u(t, x) − σu(t, x) + f (t, u(t, x)), |β|⩽2m { { {u(0, x) = u0 (x),

t ⩾ 0, x ∈ Ω; x ∈ Ω,

γ

where Dt denotes the Caputo fractional derivative. As already mentioned in [431], a great number of results concerning the existence and uniqueness of almost periodic (automorphic) type solutions to the abstract (semilinear) fractional differential equations have recently been established by many authors. Basically, we need to analyze the invariance of some types of generalized almost periodicity (automorphy) under the actions of the infinite convolution product t

F(t) := ∫ R(t − s)f (s) ds, −∞

t ∈ ℝ,

(4)

Introduction

� XI

and the finite convolution product t

t 󳨃→ ∫ R(t − s)f (s) ds,

t ⩾ 0;

(5)

0

here, we commonly assume that (R(t))t>0 ⊆ L(X, Y ) is a (degenerate) strongly continuous operator family between the Banach spaces X and Y which has a certain growth order at zero and exponentially (at least, polynomially) decays as t → +∞. We continue to analyze the invariance of generalized almost periodicity (automorphy) under the actions of convolution products in this monograph. Concerning some applications and theoretical results not mentioned in [428] and [431], we would like to emphasize the following: 1. The basic results about the existence and uniqueness of almost periodic solutions of the abstract higher-order differential equations with integer derivatives are briefly summarized in [428] (for more details about the subject, we refer the reader to the monographs [777] by T.-J. Xiao, J. Liang and [427] by M. Kostić). Here, we will only mention that Y. S. Eidelman and I. V. Tikhonov have investigated, in [288], the existence and uniqueness of periodic solutions of the abstract differential equations u(N) (t) = Au(t),

u(t + T) = u(t),

t∈ℝ

(6)

and u(N) (t) = Au(t),

u(j) (0) = u(j) (T),

0 ⩽ j ⩽ N − 1;

here, A is a closed linear operator on a Banach space X and N ∈ ℕ. Concerning the abstract Cauchy problem (6), the authors have proved that it has exactly one classical solution u(t) ≡ 0 if and only if none of the numbers λk = (2πik/T)N , k ∈ ℤ, is an eigenvalue of A; furthermore, it has been proved that any non-trivial classical solution of this problem u(t) can be represented by the series u(t) = ∑ e2πikt/T fk , k∈ℤ

2.

where Afk = (2πik/T)N fk with some elements fk ∈ D(A), k ∈ ℤ. Let us also note that the above series converges uniformly to u(t) on the real line. In [303], M. Fečkan has investigated the existence of nontrivial periodic solutions for the following systems of nonlinear differential equations: uxx − utt + n2 u = εg(t, v), vxx + vtt = εf (u, v),

and

v(0, t) = v(π, t) = 0,

u(0, t) = u(π, t) = 0

(7)

XII � Introduction uxx − utt + n2 u = εg(t, v), vxx + vtt = εf (u, v),

3.

u(x, t + T) = u(x, t) = 0,

u(t, x) = 0,

t ∈ ℝ, x ∈ (0, π)n ;

t ∈ ℝ, x ∈ 𝜕(0, π)n ;

u(x, t + T) = u(x, t),

5.

(8)

under certain reasonable assumptions. The system (7) is important in the study of the Zakharov system for an electromagnetic field in plasma, while the system (8) is a coupled dissipative wave equation. In order to achieve his aims, the author uses the generalized implicit function theorem of Moser’s type (cf. also the research article [613] by H. Petzeltová for further information in this direction). Let us note that A. Nowakowski and A. Rogowski [586] have investigated the timeperiodic solutions of the following nonlinear wave equation in higher dimensions: utt − uxx + F(x, t, u) = 0,

4.

v(x, t + T) = v(x, t) = 0,

t ∈ ℝ, x ∈ (0, π)n .

P. Mironescu and V. D. Radulescu have obtained an interesting result [557, Theorem, p. 654] concerning the existence and uniqueness of periodic solutions of the equation −Δv = v(1 − |v|2 ) in ℝ and ℝ2 . Periodic solutions to the following functional differential equation u′ (t) = Au(t) + F(t, ut ),

t ⩾ 0;

u(0) = φ,

have been sought by P. K. Pavlakos and I. G. Stratis in [605]. The authors have assumed that A is the infinitesimal generator of a strongly continuous semigroup of operators as well as that F(⋅, ⋅) is a nonlinear mapping satisfying certain assumptions; as emphasized, the established results can be used for proving the existence of periodic solutions of the partial functional differential equation ut (x, t) = uxx (x, t) + f (t, u(x, t − r), ux (x, t − r)),

6.

(x, t) ∈ [0, π] × [0, ∞),

accompanied with the initial conditions u(0, t) = u(π, t) = 0 and u(x, t) = φ(x, t), t ∈ [−r, 0], with a suitably chosen function f . Concerning almost periodic type solutions of fractional functional differential equations, we refer the reader to the research article [697] by J. Vanterler da C. Sousa and the list of references quoted therein (cf. also [290, 473, 775]). The existence and uniqueness of time periodic mild solutions for a class of nonautonomous evolution equation with multi-delays have been considered by P. Chen [204]. In this paper, the author has extended some earlier results of J. Zhu, Y. Liu, and Z. Li [816] concerning autonomous evolution equation with multi-delays. The main results of article [204] are Theorem 1.1, Theorem 1.2 and Theorem 1.3, which can be applied in the analysis of time periodic mild solutions for the following nonautonomous evolution equation with multi-delay:

Introduction

� XIII

ut (x, t) = uxx (x, t) + a(t)u(x, t) + g(x, t) + f (u(x, t − τ1 ), u(x, t − τ2 ), . . . , u(x, t − τn )), (x, t) ∈ (0, 1) × ℝ;

u(0, t) = u(1, t) = 0,

7.

t ∈ ℝ,

where a : ℝ → ℝ is a continuously differentiable function which is periodic in the variable t, f : ℝn → ℝ is a continuous nonlinear function, g : [0, 1] × ℝ → ℝ and g(x, t) is periodic in the variable t; τ1 , τ2 , . . . , τn are positive real constants. See also the research articles [318] by A. Fischer and [636] by B. Rachid. W. Shao, F. Zhang and Y. Li have considered, in [681], the existence and uniqueness of 2π-periodic solutions of the nonlinear nth order delay differential equation n−1

x (n) + ∑ aj x (j) + g(t, x(t − τ(t))) = p(t), j=1

8.

where τ, p : ℝ → ℝ and g : ℝ × ℝ → ℝ are continuous functions, τ, p are 2πperiodic, g is 2π-periodic with respect to the first variable, and aj are real constants (1 ⩽ j ⩽ n − 1). Their results complement previously known results concerning the existence of periodic solutions for the higher-order Duffing equations (cf. also [11, 177, 224, 225, 500, 516, 522, 556] for further information about these subjects). It could be also worthwhile to mention that J. W. Macki, P. Nisti and P. Zecca [529] has developed an abstract approach for finding periodic solutions for a class of differential equations with delay which has the form z′ (t) − Az(t) − Bz(t − τ) = F[z(t)],

9.

t ∈ ℝ,

where F : ℝm → ℝm is a continuous mapping, A and B are complex matrices of format m × m, τ > 0 and z : ℝ → ℝm . See also the paper [523] by E. Liz and S. Trofimchuk and the paper [514] by A. Liu and C. Feng. M. Kyed and J. Sauer have investigated the time-periodic solutions to parabolic boundary value problems ut + Au = f

in ℝ × Ω;

Bj u = gj

on ℝ × 𝜕Ω,

(9)

where A is an elliptic operator of order 2m and B1 , . . . , Bm satisfy an appropriate complementing boundary condition. The domain Ω can be the whole Euclidean space, the half Euclidean space, or a bounded domain. The problem (9) is decomposed into an elliptic problem and a purely oscillatory problem; the authors systematically analyzed the purely oscillatory problem in this paper. 10. It is worth noting that V. Keblikas has recently investigated, in his doctoral dissertation [404], the time periodic problems for Navier–Stokes equations in domains with cylindrical outlets to infinity (concerning the periodic solutions of the Navier– Stokes equations and the Kuramoto–Sivashinsky equations, we refer the reader to the doctoral dissertation of M. Cadiot [175] and the list of references quoted in [431]).

XIV � Introduction 11. Concerning the research monographs devoted to the studies of almost periodic type functions and their applications, which have not been cited in [428] and [431], we will quote here the monographs [62] by F. M. Arscott, [211] by V. A. Chulaevsky, [286] by M. S. P. Eastham, [474] by P. Kuchment, and [558] by Yu. A. Mitropolsky, A. M. Samoilenko, and D. I. Martinyuk. The main concern of monograph [62] is to construct and analyze special functions needed for the deeper study of differential equations arising in mathematical physics; especially, the author investigates the Mathieu differential equation, Hille’s type differential equations, Lamé’s differential equation and the ellipsoidal wave equation. In [211], the author has analyzed various classes of almost periodic operators and nonlinear integrable systems. The spectral theory of periodic differential equations has been analyzed in [286]. The author presents numerous interesting examples of ordinary differential equations and their systems which possess Bloch periodic solutions; in particular, the following results have been proved in [286, Theorem 1.5.1]: Consider the system of ordinary differential equations y′ (x) = C(x)y(x), where C(x) is a complex-valued, piecewise continuous matrix of format n × n such that C(x + a) = C(x) for some non-zero real constant a. Then there exists a non-zero complex constant ρ and a non-trivial solution ψ(x) of this system such that ψ(x + a) = ρψ(x) for all x ∈ ℝ. In [474], the reader may find a basic source of information concerning the Floquet theory for partial differential equations. The author basically employs the Fredholm theory to achieve his aims and consider various classes of integro-differential equations, including the abstract hypoelliptic equations on the whole axis, elliptic and parabolic boundary value problems in a cylinder, equations with deviating argument, and invariant differential equations on Riemann symmetric spaces of noncompact type. Here we will only recall that the famous theorem of F. Bloch [138] states that if the equation Δu + qu = 0 has a non-zero bounded solution in ℝn , then this equation has a non-zero bounded Bloch solution. This result was extended by E. Shnoĺ [688], who proved that it suffices to require the existence of a non-zero solution of this equation which has an exponential growth. The general result of this type for scalar periodic elliptic differential operators in ℝn with smooth coefficients has been proved in [474, Theorem 4.3.1]. See also the survey article [475] by P. Kuchment for an overview of results about periodic elliptic operators. In [558], Yu. A. Mitropolsky, A. M. Samoilenko and D. I. Martinyuk have investigated various numerical-analytic methods for some classes of nonlinear systems with aftereffects; in particular, the authors thoroughly investigate the Bubnov–Galerkin method and apply this method in the analysis of the existence and uniqueness of (quasi-)periodic solutions for some classes of systems of integro-differential equations. The reducibility of some classes of linear systems of difference equations with (quasi-)periodic coefficients and the reducibility of some classes of nonlinear systems of differential equations in the vicinity of the toroidal set are also analyzed.

Introduction

� XV

12. Time-periodic solutions to the compressible Hall-magnetohydrodynamic systems have been considered by M. Zheng in [208]. 13. Concerning almost periodic type solutions of the nonlinear evolution equations of first order, we may refer to [10, 236, 355, 358, 372, 378, 625]. It should be particularly emphasized that the existence and uniqueness of Eberlain weakly almost periodic solutions for a class of nonlinear evolution equations of first order, with a family (A(t))t⩾0 of nonlinear m-dissipative operators satisfying A(t + T) = A(t) for all t ⩾ 0 (T > 0), have been analyzed in the doctoral dissertation of J. Kreulich [468] (see also the research articles [469, 470] by the same author, [786] by I. B. Yaacov, T. Tsankov, and [674] by G. Seifert); furthermore, it is worth noting that M. Ôtani [592] has constructed an example of the periodic evolution system governed by the time-dependent subdifferential operators admitting almost periodic orbits, which are not quasiperiodic (cf. also the papers [181, 182, 357], and [588, 589] for further information about the subject). 14. Almost periodic type solutions for certain classes of quasilinear hyperbolic systems of the form Vt + A(x, t, V )Vx + B(x, t, V )V = f (x, t),

x ∈ (0, 1), t ∈ ℝ,

where V : (0, 1) × ℝ → ℝn and f : (0, 1) × ℝ → ℝn are vectorial functions, A, B are real-valued matrices of format n × n whose entries are real-valued functions, have recently been analyzed by I. Kmit, L. Recke, and V. Tkachenko [417, 418]; cf. also the article [631] by P. Qu and the list of references quoted in the above-mentioned papers. 15. The existence and uniqueness of Stepanov-like pseudo almost automorphic solutions for the following neutral partial functional differential equation: u′ (t) = Au(t) + L(ut ) + f (t),

t ∈ ℝ,

where A is a linear operator acting on a Banach space X and a Hille–Yosida type condition holds for A, have recently been studied by M. Maqbul in [542]. The phase space ℬ is a linear space of functions from (−∞, 0] into X satisfying some extra assumptions; for every t ⩾ 0 the history ut ∈ ℬ is defined by ut (s) := u(t + s) for all s ⩽ 0, L is a bounded linear operator from ℬ into X and f : [0, ∞) → X is a continuous function. 16. Concerning the almost periodic type orbits, solenoids in generic Hamiltonian systems and adding machines, we refer the reader to the research articles [103] by H. Bell, K. Meyer and [536] by L. Markus, K. Meyer. Fermi–Ulam ping-pong problems with Bohr almost periodic forcing functions have been analyzed in [480, 670, 812, 813]. It is also worth noting that B. Dorn, V. Keicher and E. Sikolya have analyzed the asymptotic periodicity of recurrent flows in infinite networks

XVI � Introduction [273]. For proving their main results, the authors essentially use the famous Jacobs– Glicksberg–deLeeuw splitting theorem. 17. Let us note that L. Corsi, R. Montalto and M. Procesi have considered, in [232], the existence and uniqueness of almost periodic solutions of the following quasi-linear Airy equation ut + uxxx + Q(u, ux , uxx , uxxx ) = f (t, x),

t ∈ 𝕋 ≡ (ℝ/2πℤ),

where Q is a Hamiltonian, quadratic nonlinearity and f is an analytic forcing term with zero average with respect to the variable x. 18. I. A. Rudakov has analyzed, in [648], the existence of periodic solutions of the following generalized wave equation with nonconstant coefficients: p(x)utt − (p(x)ux )x = g(x, t, u) + f (x, t);

u(x, t + T) = u(x, t),

0 < x < π, t ∈ ℝ,

accompanied with the Dirichlet boundary conditions u(0, t) = u(π, t) = 0 or with the mixed type conditions u(0, t) = u′ (π, t) = 0/u′ (0, t) = u(π, t) = 0. The function p(x) is assumed to be strictly positive and two times continuously differentiable on the segment [0, π]. See also [618, 647, 691] and the lists of references in the above-mentioned papers. 19. Concerning differential equations with piecewise constant argument, we would like to note that R. Yuan [793] has investigated the existence of almost periodic solutions of the following first-order neutral differential equation with piecewise constant argument: d t+1 (y(t) + py(t − 1)) = q(2⌊ ⌋) + g(t, y(t), y(⌊t⌋)), dt 2

(10)

where p ≠ 0 and q ≠ 0, 1 are real constants, and g : ℝ × ℝ × ℝ → ℝ is an almost periodic function with respect to the first variable, uniformly on the compact subsets of ℝ × ℝ with respect to the second variable and the third variable. A continuous function y : ℝ → ℝ is said to be a solution of (10) if and only if the equation (10) is satisfied on the set ℝ ∖ ℤ, the function t 󳨃→ y(t) + py(t − 1), t ∈ ℝ is differentiable on the set ℝ ∖ ℤ and the one-sided derivatives of this function exist on the set ℤ. In order to achieve the main results, the author has first considered the equation d t+1 (y(t) + py(t − 1)) = q(2⌊ ⌋) + f (t), dt 2

(11)

where f : ℝ → ℝ is an almost periodic function. The main results of paper are [793, Theorem 1, Theorem 2], where it has been proved the following: (i) Suppose that p ≠ 0 and q ≠ 0, 1 are real constants, |p| < 1 and the equation

Introduction

� XVII

(1 − q)λ2 + (−p2 + 2pq − q − 1)λ + p2 = 0 has two distinct solutions λ1 , λ2 such that |λ1,2 | ≠ 1. Then for any almost periodic function f (t), the equation (11) has a unique almost periodic solution. Furthermore, if f (t) is ω-periodic and ω = 2n0 /m0 for some mutually prime positive integers n0 and m0 , then the equation (11) has an (m0 ω)-periodic solution. (ii) Suppose that the requirements of (i) hold, as well as that g : ℝ × ℝ × ℝ → ℝ is an almost periodic function with respect to the first variable, uniformly on the compact subsets of ℝ × ℝ with respect to the second variable and the third variable, and there exists a positive real number η > 0 such that 󵄨󵄨 󵄨 󵄨󵄨g(t, x1 , y1 ) − g(t, x2 , y2 )󵄨󵄨󵄨 ⩽ η(|x1 − x2 | + |y1 − y2 |),

t, x1 , x2 , y1 , y2 ∈ ℝ.

Then there exists a positive real number η∗ > 0, such that the equation (10) has a unique almost periodic solution whenever 0 ⩽ η∗ . Furthermore, if g(t, x, y) is ω-periodic in t and ω = 2n0 /m0 for some mutually prime positive integers n0 and m0 , then the equation (10) has an (m0 ω)-periodic solution. See also [12, 13, 547] and other references cited in [793], as well as the papers by M. Miraoui [554, 555]. 20. Almost periodic potentials in higher dimensions have been considered by V. G. Papanicolaou in [603]. Using a probabilistic approach, the author has considered the kernel k(t, x, y) of the semigroup generated by the operator L = (Δ/2) + q, where q(⋅) is almost periodic function defined on ℝn . It has been proved that the function x 󳨃→ k(t, x, x + u) is an almost periodic function with the respect to the variable x, with the frequency module not bigger than that of q. The author has also proved that the kernel k(t, x, y) is uniformly continuous on [a, b] × ℝn × ℝn . 21. Almost periodic optimal control problems and their links with periodic problems have been analyzed by D. Pennequin [609], while the time-periodic solutions of quasilinear parabolic differential equations have been investigated by G. M. Lieberman in [511, 512, 513]. See also the research articles [56, 57, 58, 155, 340, 477, 490, 677]. 22. The existence of a response solution (i. e., quasi-periodic solution with the same frequency as the forcing) for the quasi-periodically forced generalized ill-posed Boussinesq equation: ytt (t, x) = νyxxxx + yxx + (y3 + εf (ωt, x)) , xx

x ∈ [0, π], t ∈ ℝ (ν > 0),

(12)

accompanied with the following hinged boundary conditions y(t, 0) = y(t, π) = yxx (t, 0) = yxx (t, π), has been analyzed by F. Wang, H. Cheng and J. Si in [755]; here, ω = (1, α) with α being any irrational number. For the previous results obtained in this direction,

XVIII � Introduction we recall that many authors have analyzed the existence of quasi-periodic type solutions for the following generalized Boussinesq equation of the form: ytt = νyxxxx + yxx + (g(y))xx

(13)

using the results from the Kolmogorov–Arnold–Moser (KAM) theory. The special case of (13) with the hinged boundary conditions and g(y) = y5 has been analyzed by C. Shang [680] and Y. Shi et al. [686]. See also the classical results by R. de la Llave and his coauthors [253, 254] for the special case g(y) = y2 . Concerning the equation (12), we would like to emphasize that the authors allow that α is arbitrary irrational number, so that the frequency ω = (1, α) need not be of Diophantine or Brjuno type, but it is always a Liouvillean frequency. Here, we would like to recall that a frequency ω = (ω1 , . . . , ωn ) ∈ ℝn is called Diophantine if and only if there exist two real numbers ν > 0 and τ > n − 1 such that for every k = (k1 , . . . , kn ) ∈ ℤn ∖ {0}, we have |⟨k, ω⟩| ⩾ |k|−τ . If ∑ 2−m

m⩾0

23.

24.

25.

26.

max

0 0, such that the double inequality 󵄨󵄨 󵄨 0 < 󵄨󵄨󵄨α − 󵄨󵄨

p 󵄨󵄨󵄨󵄨 1 󵄨< q 󵄨󵄨󵄨 qν

has (at most) finitely many rational solutions p/q; if such a set is empty, then we set ν(α) = +∞ and call α a Liouville number; equivalently, an irrational number α is a

XX � Introduction Liouville number if and only if for every n ∈ ℕ there exist p ∈ ℤ and q ∈ ℕ ∖ {1}, such that 󵄨󵄨 󵄨 0 < 󵄨󵄨󵄨α − 󵄨󵄨

p 󵄨󵄨󵄨󵄨 1 󵄨󵄨 < n . 󵄨 q󵄨 q

We know that ν(α) = 2 if α is an irrational algebraic number, as well as that ν(α) ⩾ 2 if α is an irrational transcendental number; cf. also [172, Theorem 2]. Using the theory of continued fractions, the authors have proved the following: (i) If α is not a Liouville number, then limt→+∞ t −2(ν(α)−1)−ε f (t) = 0. (ii) For any function g : (0, ∞) → (0, ∞) such that limt→+∞ g(t) = +∞, we can always find a Liouville number α ∈ ℝ such that t −2 f (t) = 0. t→+∞ g(t) lim

The organization and main ideas of this monograph can be briefly described as follows. In the preliminary chapter of the monograph, we recollect the basic definitions and results about linear and multilinear operators in Banach spaces, integration in Banach spaces, fractional calculus, fixed point theorems, and Laplace transform; a special subsection is reserved for an overview of the basic definitions and results about the Lebesgue spaces with variable exponents. After that, we divide the material into three parts. The first part of monograph, which is devoted to the study of multi-dimensional ρ-almost periodic type functions and their applications, consists of two chapters, which are further divided into sections and subsections; the second part of monograph, which is devoted to the study of metrically multi-dimensional ρ-almost periodic type functions and their applications, is organized similarly and this part is divided into four chapters. The third part of monograph is devoted to the study of almost periodic type solutions for some new classes of the abstract Volterra integro-differential inclusions. In order to shorten the introductory part, we have decided to explain the main ideas and organization of each section of the monograph within itself. For further information concerning multi-dimensional almost periodic type functions, multi-dimensional almost automorphic type functions and their applications, we refer the reader to our newly published research monograph [431]. We have been forced to quote this monograph multiple times henceforth since some recent results of ours concerning multi-dimensional almost periodic type functions and their generalizations are still not published in the final form. Like any research monograph published so far, this monograph certainly contains some typographical errors and drawbacks. For example, it is well known that onedimensional almost periodic type functions have many more applications than the multi-dimensional almost periodic type functions; although we have presented here several new applications of the multi-dimensional almost periodic type functions not

Introduction

� XXI

considered in [431], some important formulae from the classical theory of partial differential equations and the theory of operator semigroups are multiple times repeated for applications, as certain prototypes, with different classes of almost periodic type functions under our consideration (for the sake of better readability, we have also repeated sometimes the formulae for the infinite convolution like mappings and the classical convolution mappings; it could be interesting to find some other formulas from the classical theory of partial differential equations, which can be employed for our purposes). The author will gratefully acknowledge the receipt of any useful comment, objection, or suggestion of the readers.

Contents Preface � V Introduction � VII Notation � XXIX 1 1.1 1.1.1

Preliminaries � 1 Basic concepts and research tools � 1 Lebesgue spaces with variable exponents Lp(x) � 9

Part I: Multi-dimensional ρ-almost periodicity 2 2.1 2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 2.1.6 2.1.7 2.2 2.2.1 2.2.2 2.2.3 2.3 2.3.1 2.3.2 2.3.3 2.4 2.4.1

Multi-dimensional ρ-almost periodic type functions and their Stepanov generalizations � 17 Multi-dimensional ρ-almost periodic type functions � 19 T -almost periodic type functions in finite-dimensional spaces � 30 𝔻-asymptotically Bohr (ℬ, I ′ , ρ)-almost periodic type functions � 32 (ω, ρ)-periodic functions and (ωj , ρj )j∈ℕn -periodic functions � 37 Liouville type results for anti-periodic linear operators � 40 Applications to the heat equation in ℝn , the fractional diffusion-wave equation in ℝn and the iterated polyharmonic operators in ℝn � 42 New applications in one-dimensional setting � 44 Heteroclinic period blow-up � 45 (ω, ρ)-periodic solutions of abstract integro-differential impulsive equations on Banach space � 47 Nonhomogeneous linear impulsive problem � 50 (ω, ρ)-periodic solutions of integro-differential impulsive problem � 64 (ω, ρ)-BVP for impulsive integro-differential equations of fractional order on Banach space � 69 Stepanov multi-dimensional ρ-almost periodic functions in Lebesgue spaces with variable exponents � 75 Stepanov multi-dimensional Bochner transform, Stepanov distance and Stepanov norm � 75 Stepanov multi-dimensional ρ-almost periodicity � 76 𝔻-asymptotically Stepanov ρ-almost periodic type functions � 85 Some new classes of ρ-almost periodic type functions in ℝn � 88 (S, 𝔻, ℬ)-asymptotically (ω, ρ)-periodic type functions and (S, ℬ)-asymptotically (ωj , ρj , 𝔻j )j∈ℕn -periodic type functions � 88

XXIV � Contents 2.4.2 2.4.3 2.4.4 2.4.5 2.4.6 3 3.1 3.1.1 3.2 3.2.1 3.2.2 3.2.3 3.3 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 3.4 3.4.1 3.4.2 3.5 3.5.1 3.5.2 3.5.3 3.6

Multi-dimensional quasi-asymptotically ρ-almost periodic type functions � 92 Remotely ρ-almost periodic type functions � 93 ρ-Slowly oscillating type functions in ℝn � 95 Applications to the abstract Volterra integro-differential equations � 97 Semi-(ρj , ℬ)j∈ℕn -periodic functions � 102 Multi-dimensional ρ-almost periodic type functions: Weyl, Besicovitch and Doss generalizations � 104 Multi-dimensional Weyl ρ-almost periodic type functions � 104 Applications to the abstract Volterra integro-differential equations � 108 Multi-dimensional Doss ρ-almost periodic type functions � 110 Relationship between Weyl almost periodicity and Doss almost periodicity � 126 Invariance of Doss ρ-almost periodicity under the actions of convolution products � 129 Applications to the abstract Volterra integro-differential equations and partial differential equations � 131 Multi-dimensional Besicovitch almost periodic type functions and applications � 135 Multi-dimensional Besicovitch almost periodic type functions � 139 Multi-dimensional Besicovitch normal type functions � 147 Besicovitch–Doss almost periodicity � 150 On condition (B) � 155 Applications to the abstract Volterra integro-differential equations � 160 Besicovitch multi-dimensional almost automorphic type functions and applications � 171 Multi-dimensional Besicovitch almost automorphy in Lebesgue spaces with variable exponent � 173 Applications to the abstract Volterra integro-differential equations � 182 Multi-dimensional ρ-almost periodic type distributions � 187 ρ-almost periodic type distributions in ℝn � 190 𝔻-asymptotically ρ-almost periodic type distributions in ℝn � 194 Some applications � 199 Notes and appendices � 201

Part II: Metrical ρ-almost periodicity 4 4.1 4.1.1

Metrically ρ-almost periodic type functions and applications � 235 (RX , ℬ, 𝒫 , L)-multi-almost periodic type functions � 235 Generalization of multi-dimensional (Stepanov) almost automorphy � 246

Contents

4.1.2 4.1.3 4.1.4 4.2 4.2.1 4.2.2 4.3 4.3.1 4.3.2 4.3.3 4.3.4 5 5.1 5.1.1 5.1.2 5.1.3 5.1.4 5.2 5.2.1 5.2.2 5.2.3 6 6.1 6.1.1 6.1.2 6.2 6.2.1 6.2.2 6.3 6.3.1 6.4 6.4.1 6.4.2 6.4.3

� XXV

Bohr (ℬ, I ′ , ρ, 𝒫 )-multi-almost periodic type functions � 250 Generalization of multi-dimensional (Stepanov) ρ-almost periodicity � 257 Applications to the abstract Volterra integro-differential equations � 259 Stepanov ρ-almost periodic functions in general metric � 261 Invariance of metrical Stepanov c-almost periodicity under the actions of infinite convolution products � 264 An application to the abstract semilinear fractional differential equations � 266 Weyl ρ-almost periodic functions in general metric � 269 Metrical Weyl distance � 270 Metrical Weyl ρ-almost periodic type functions � 273 Invariance of metrical Weyl ρ-almost periodicity under actions of infinite convolution products � 282 Applications to the abstract Volterra integro-differential equations � 284 Asymptotical ρ-almost periodicity in general metric � 288 Asymptotically ρ-almost periodic type functions in general metric � 288 Metrical (S, 𝔻, ℬ)-asymptotical (ω, ρ)-periodicity and metrical (S, ℬ)-asymptotical (ωj , ρj , 𝔻j )j∈ℕn -periodicity � 288 Metrically ρ-slowly oscillating type functions in ℝn � 291 Metrically quasi-asymptotically ρ-almost periodic type functions � 294 Applications to the abstract Volterra integro-differential equations � 298 Multi-dimensional weighted ergodic components in general metric � 300 Stepanov weighted ergodic components in general metric � 302 Weyl weighted ergodic components in general metric � 307 Weighted pseudo-ergodic components in general metric � 312 Metrical approximations of functions � 318 Metrical approximations: the main concept � 319 Metrical normality and metrical Bohr type definitions � 325 Metrically semi-(cj , ℬ)j∈ℕn -periodic functions � 330 Metrical approximations: Stepanov, Weyl, Besicovitch and Doss concepts � 332 Stepanov and Weyl metrical approximations � 332 Besicovitch and Doss metrical approximations � 340 Further results and applications � 345 Invariance under the actions of infinite convolution products � 346 Metrical almost periodicity: Poisson, Levitan and Bebutov concepts � 351 Levitan (N, c)-almost periodic functions and uniformly Poisson c-stable functions � 361 Multi-dimensional Levitan N-almost periodic functions � 363 Applications to the abstract Volterra integro-differential equations � 365

XXVI � Contents 7 7.1 7.1.1 7.1.2 7.1.3 7.1.4 7.1.5 7.2 7.2.1 7.2.2 7.2.3 7.3

Special classes of metrically almost periodic functions � 371 Hölder ρ-almost periodic type functions in ℝn � 371 Multi-dimensional Hölder ρ-almost periodic type functions � 372 Hölder-α-(𝔽, ℬ)-boundedness and Hölder-α-(𝔽, ℬ, Λ′ , ρ)-continuity � 385 Extensions of Hölder ρ-almost periodic type functions � 391 Invariance of Hölder ρ-almost periodicity under the actions of convolution products � 393 Further applications and examples � 395 Generalized almost periodic functions with values in ordered Banach spaces � 399 Generalized Bohr (ℬ, I1′ , I2′ , ρ1 , ρ2 )-almost periodic type functions � 400 Some subclasses of generalized Bohr (ℬ, I1′ , I2′ , ρ1 , ρ2 , 𝒫 )-almost periodic type functions � 404 Applications to the abstract Volterra integro-differential equations � 408 Notes and appendices to Part II � 417

Part III: Almost periodic type solutions to integro-differential inclusions 8 8.1 8.2 8.2.1 8.2.2 8.3 8.4 8.4.1 9 9.1 9.1.1 9.1.2 9.1.3 9.1.4

Abstract fractional equations with proportional Caputo fractional derivatives � 427 Proportional fractional integrals and proportional Caputo fractional derivatives � 428 Abstract proportional Caputo fractional differential inclusions � 433 Solution operator families for (DFP)ζR and (DFP)ζL � 434 Some applications to the abstract Volterra integro-differential inclusions � 440 Almost periodic type solutions to semilinear proportional Caputo fractional differential equations � 443 Nonexistence of (ω, c)-periodic solutions of (286) and nonexistence of Poisson stable like solutions of (286) � 447 On quasi-periodic properties of proportional fractional integrals � 447 Almost periodic type solutions to abstract impulsive Volterra integro-differential inclusions � 452 Abstract impulsive Volterra integro-differential inclusions � 452 Abstract impulsive differential inclusions of integer order � 454 Abstract impulsive higher-order Cauchy problems � 461 On abstract impulsive fractional differential inclusions � 463 Abstract Volterra integro-differential inclusions with impulsive effects � 466

Contents

9.2 9.2.1 9.2.2 9.2.3 9.2.4 9.2.5 9.2.6 9.2.7 9.2.8 9.2.9

�

XXVII

The existence and uniqueness of almost periodic type solutions � 471 Piecewise continuous almost periodic functions � 472 (ℬ, ρ)-piecewise continuous almost periodic type functions � 474 Relations with Stepanov almost periodic type functions � 479 Composition principles for (ℬ, (tk ))-piecewise continuous almost periodic type functions � 484 Asymptotically almost periodic type solutions of (ACP)1;1 � 487 Asymptotically Weyl almost periodic type solutions of (ACP)1;1 � 488 Besicovitch–Doss almost periodic type solutions of (ACP)1;1 � 492 Almost periodic type solutions of the abstract higher-order impulsive Cauchy problems � 495 Almost periodic type solutions of the abstract Volterra integro-differential inclusions with impulsive effects � 497

Bibliography � 503 Index � 537

Notation – – – – – – – – – – – – – – – – – – – – – – – –

– – – –

ℕ, ℤ, ℚ, ℝ, ℂ: the natural numbers, integers, rationals, reals, complexes. If s ∈ ℝ, then we denote ⌊s⌋ = sup{l ∈ ℤ : s ⩾ l} and ⌈s⌉ = inf{l ∈ ℤ : s ⩽ l}. Re z, Im z: the real and imaginary part of a complex number z ∈ ℂ; |z|: the modul of z, arg(z): the argument of z ∈ ℂ ∖ {0}. Σα = {z ∈ ℂ ∖ {0} : | arg(z)| < α}, α ∈ (0, π]. card(G) or |G|: the cardinality of G. ℕ0 = ℕ ∪ {0}. ℕn = {1, . . . , n}. ℕ0n = {0, 1, . . . , n}. supp(f ): the support of function f (t). ℝn : the real Euclidean space, n ⩾ 2. If α = (α1 , . . . , αn ) ∈ ℕn0 is a multi-index, then we set |α| = α1 + ⋅ ⋅ ⋅ + αn . α α x α = x1 1 ⋅ ⋅ ⋅ xn n for x = (x1 , . . . , xn ) ∈ ℝn and α = (α1 , . . . , αn ) ∈ ℕn0 . α α (α) |α| f := 𝜕 f /𝜕x1 1 ⋅ ⋅ ⋅ 𝜕xn n ; Dα f := (−i)|α| f (α) . Suppose that (X, τ) is a topological space and F ⊆ X. Then the interior, the closure, the boundary, and the complement of F with respect to X are denoted by int(F) (or F ∘ ), F, 𝜕F and F c , respectively. X, Y , Z, E: complex Banach spaces. L(X, Y ): the space of all continuous linear mappings between X and Y , L(X) = L(X, X). X ∗ : the dual space of X. A: a linear operator on X. 𝒜: a multivalued linear operator on X (MLO). If F is a submanifold of X, then we denote by 𝒜|F the part of 𝒜 in F. χΩ (⋅): the characteristic function, defined to be identically one on Ω and zero elsewhere. Γ(⋅): the Gamma function. If α > 0, then we define gα (t) := t α−1 /Γ(α), t > 0; g0 (t) ≡ the Dirac delta distribution. Suppose that 1 ⩽ p < ∞, (X, ‖ ⋅ ‖) is a complex Banach space, and (Ω, ℛ, μ) is a measure space. Then Lp (Ω, X, μ) denotes the space which consists of those strongly μ-measurable functions f : Ω → X such that ‖f ‖p := (∫Ω ‖f (⋅)‖p dμ)1/p is finite; Lp (Ω, μ) ≡ Lp (Ω, ℂ, μ). L∞ (Ω, X, μ): the space which consists of all strongly μ-measurable, essentially bounded functions. ‖f ‖∞ := ess supt∈Ω ‖f (t)‖, the norm of a function f ∈ L∞ (Ω, X, μ). Lp (Ω : X) ≡ Lp (Ω, X) ≡ Lp (Ω, X, μ), if p ∈ [1, ∞] and μ = m is the Lebesgue measure; Lp (Ω) ≡ Lp (Ω : ℂ). p Lloc (Ω : X): the space of Lebesgue measurable functions u(⋅) satisfying that, for every bounded open p p subset Ω′ of Ω, one has u|Ω′ ∈ Lp (Ω′ : X); Lloc (Ω) ≡ Lloc (Ω : ℂ) (1 ⩽ p ⩽ ∞).

–

C k (Ω : X): the space of k-times continuously differentiable functions (k ∈ ℕ0 ) from a non-empty open subset Ω ⊆ ℂ into X; C(Ω : X) ≡ C 0 (Ω : X). 𝒟 = C0∞ (ℝ), ℰ = C ∞ (ℝ) and 𝒮 = 𝒮(ℝ): the Schwartz spaces of test functions. If 0 ≠ Ω ⊆ ℝ, then 𝒟Ω denotes the subspace of 𝒟 consisting of those functions φ ∈ 𝒟 for which supp(φ) ⊆ Ω; 𝒟0 ≡ 𝒟[0,∞) . 𝒟′ = L(𝒟 : ℂ): the space consisting of all scalar-valued distributions. If k ∈ ℕ, p ∈ [1, ∞] and Ω is an open non-empty subset of ℝn , then W k,p (Ω : X) stands for the Sobolev space of those X-valued distributions u ∈ 𝒟′ (Ω : X) such that, for every i ∈ ℕ0k and for every α ∈ ℕn0 with |α| ⩽ k, we have Dα u ∈ Lp (Ω : X). k,p Wloc (Ω : X): the space of those X-valued distributions u ∈ 𝒟′ (Ω : X) such that, for every bounded open

–

ℱ , ℱ −1 : the Fourier transform and its inverse transform, respectively.

– – – – –

subset Ω′ of Ω, we have u|Ω′ ∈ W k,p (Ω′ : X).

https://doi.org/10.1515/9783111233871-203

XXX � Notation

– – – – – – – – – – – – –

L1loc ([0, ∞)), resp. L1loc ([0, τ)): the space of scalar-valued locally integrable functions on [0, ∞), resp. [0, τ). Jtα : the Riemann–Liouville fractional integral of order α > 0. Dαt : the Caputo fractional derivative of order α > 0. γ Dt,+ : the Weyl–Liouville fractional derivative of order γ ∈ (0, 1]. Eα,β (z): the Mittag-Leffler function (α > 0, β ∈ ℝ); Eα (z) ≡ Eα,1 (z). Ψγ (t): the Wright function (0 < γ < 1). If 0 ≠ Ω ⊆ ℝn , then Lp(x) (Ω : X) denotes the Lebesgue space with variable exponent p(x). Let 0 ≠ I, Λ ⊆ ℝn : C0,𝔻,ℬ (I × X : Y ) denotes the vector space consisting of all continuous functions Q : I × X → Y such that, for every B ∈ ℬ, we have limt∈𝔻,|t|→+∞ Q(t; x) = 0, uniformly for x ∈ B. Ω,p(u)

LS,ℬ (Λ × X : Y ): the space of all Stepanov (Ω, p(u))-bounded functions on ℬ. Ω,p(u),ρ

Ω,p(u),ρ

APS ′ (Λ × X : Y ) and APSℬ (Λ × X : Y ): the spaces consisting of all Stepanov (Ω, p(u))-(ℬ, Λ′ , ρ)ℬ,Λ almost periodic functions and Stepanov (Ω, p(u))-(ℬ, ρ)-almost periodic functions, respectively. Ω,p(u),ρ Ω,p(u),ρ URS ′ (Λ × X : Y ) and URSℬ (Λ × X : Y ): the spaces of all Stepanov (Ω, p(u), ρ)-(ℬ, Λ′ )-uniformly ℬ,Λ recurrent functions and Stepanov (Ω, p(u), ρ)-ℬ-uniformly recurrent functions, respectively. Bp (ℝn : X): the space of all Besicovitch-p-almost periodic functions (1 ⩽ p < ∞), and so on. p(t) Let the set I be Lebesgue measurable. Then, the Banach space Lν (I : Y ) is defined through p(t)

Lν (I : Y ) := {u : I → Y ; u(⋅) is measurable and ‖u‖p(t) < ∞}, where p ∈ 𝒫(I), and –

󵄩 󵄩 ‖u‖p(t) := 󵄩󵄩󵄩󵄩u(t)ν(t)󵄩󵄩󵄩󵄩Lp(t) (I:Y ) . Suppose that ν : Λ → (0, ∞) is an arbitrary function satisfying that the function 1/ν(⋅) is locally bounded. Then the vector space C0,ν (Λ : Y ) [Cb,ν (Λ : Y )] consists of all continuous functions u : Λ → Y such that lim|t|→∞,t∈Λ ‖u(t)‖Y ν(t) = 0 [supt∈Λ ‖u(t)‖Y ν(t) < +∞]. Equipped with the norm ‖ ⋅ ‖ := supt∈Λ ‖ ⋅ (t)ν(t)‖Y , C0,ν (Λ : Y ) [Cb,ν (Λ : Y )] is a Banach space.

1 Preliminaries 1.1 Basic concepts and research tools The main purpose of this section is to remind the readers of the basic definitions and results about linear operators, multivalued linear operators, integration in Banach spaces, fractional calculus, fixed point theorems, and Laplace transform; in Section 1.1.1, we consider the Lebesgue spaces with variable exponents Lp(x) . Vector-valued functions, closed operators By (X, ‖ ⋅ ‖) we denote a Banach space over the field of complex numbers. If (Y , ‖ ⋅ ‖Y ) is also a Banach space over the field of complex numbers, then by L(X, Y ) we denote the space consisting of all continuous linear mappings from X into Y ; L(X) ≡ L(X, X). We topologize the spaces L(X, Y ) and X ∗ , the dual space of X, in the usual way. If not explicitly stated otherwise, by I we denote the identity operator on Y . A linear operator A : D(A) → X is said to be closed if and only if the graph of the operator A, defined by GA := {(x, Ax) : x ∈ D(A)}, is a closed subset of X ×X. The null space (kernel) and range of A are denoted by N(A) and R(A), respectively. Let us recall that a linear operator A is called closable if and only if there exists a closed linear operator B such that A ⊆ B. If F is a linear submanifold of X, then we define the part of A in F by D(A|F ) := {x ∈ D(A) ∩ F : Ax ∈ F} and A|F x := Ax, x ∈ D(A|F ). The power An of A is defined inductively (n ∈ ℕ0 ). If α ∈ ℂ ∖ {0}, A and B are linear operators, then we define the operators αA, A + B and AB in the usual way. The Gamma function is denoted by Γ(⋅) and the principal branch is always used to take the powers. Set, for every α > 0, gα (t) := t α−1 /Γ(α),

t > 0,

g0 (t) ≡ the Dirac δ-distribution and 0ζ := 0. Define Σα := {z ∈ ℂ ∖ {0} : | arg(z)| < α}, α ∈ (0, π]. By C(Ω : X) we denote the space of all continuous functions f : Ω → X, where 0 ≠ Ω ⊆ ℂn (n ∈ ℕ); C(Ω) ≡ C(Ω : ℂ). If s ∈ ℝ and n ∈ ℕ, we define ⌊s⌋ := sup{l ∈ ℤ : s ⩾ l}, ⌈s⌉ := inf{l ∈ ℤ : s ⩽ l}, ℕn := {1, . . . , n} and ℕ0n := {0, 1, . . . , n}. If X, Y ≠ 0, then we set YX := {f | f : X → Y}. Suppose that I = ℝ or I = [0, ∞). By Cb (I : X) we denote the vector space consisting of all bounded continuous functions from I into X; the abbreviation C0 (I : X) denotes the vector subspace of Cb (I : X) consisting of those functions f : I → X such that lim|t|→∞ ‖f (t)‖ = 0. By BUC(I : X) we denote the space of all bounded uniformly continuous functions from I to X; Cb (I) ≡ Cb (I : ℂ), C0 (I) ≡ C0 (I : ℂ) and BUC(I) ≡ BUC(I : ℂ). Equipped with the sup-norm, Cb (I : X), C0 (I : X) and BUC(I : X) are Banach spaces. https://doi.org/10.1515/9783111233871-001

2 � 1 Preliminaries Multivalued linear operators and solution operator families subgenerated by them Suppose that X and Y are two Banach spaces. A multivalued map (multimap) 𝒜 : X → P(Y ) is said to be a multivalued linear operator (MLO) if and only if the following holds: (i) D(𝒜) := {x ∈ X : 𝒜x ≠ 0} is a linear subspace of X; (ii) 𝒜x + 𝒜y ⊆ 𝒜(x + y), x, y ∈ D(𝒜) and λ𝒜x ⊆ 𝒜(λx), λ ∈ ℂ, x ∈ D(𝒜). If X = Y , then we say that 𝒜 is an MLO in X. Recall that for every x, y ∈ D(𝒜) and λ, η ∈ ℂ with |λ| + |η| ≠ 0, we have λ𝒜x + η𝒜y = 𝒜(λx + ηy). Also, if 𝒜 is an MLO, then 𝒜0 is a linear submanifold of Y and 𝒜x = f + 𝒜0 for any x ∈ D(𝒜) and f ∈ 𝒜x. Define R(𝒜) := {𝒜x : x ∈ D(𝒜)} and N(𝒜) := 𝒜−1 0 := {x ∈ D(𝒜) : 0 ∈ 𝒜x} (we call that the range and kernel space of 𝒜, respectively). The inverse 𝒜−1 of an MLO is defined by D(𝒜−1 ) := R(𝒜) and 𝒜−1 y := {x ∈ D(𝒜) : y ∈ 𝒜x}. We know that 𝒜−1 is an MLO in X, as well as that N(𝒜−1 ) = 𝒜0 and (𝒜−1 )−1 = 𝒜. If N(𝒜) = {0}, i. e., if 𝒜−1 is single-valued, then 𝒜 is said to be injective. Assuming that 𝒜, ℬ : X → P(Y ) are two MLOs, we define its sum 𝒜 + ℬ by D(𝒜 + ℬ) := D(𝒜) ∩ D(ℬ) and (𝒜 + ℬ)x := 𝒜x + ℬx, x ∈ D(𝒜 + ℬ). Clearly, 𝒜 + ℬ is likewise an MLO. Suppose now that 𝒜 : X → P(Y ) and ℬ : Y → P(Z) are two MLOs, where Z is a complex Banach space. The product of 𝒜 and ℬ is defined by D(ℬ𝒜) := {x ∈ D(𝒜) : D(ℬ) ∩ 𝒜x ≠ 0} and ℬ𝒜x := ℬ(D(ℬ) ∩ 𝒜x). We have that ℬ𝒜 : X → P(Z) is an MLO and (ℬ𝒜)−1 = 𝒜−1 ℬ−1 . The scalar multiplication of an MLO 𝒜 : X → P(Y ) with the number z ∈ ℂ, z𝒜 for short, is defined by D(z𝒜) := D(𝒜) and (z𝒜)(x) := z𝒜x, x ∈ D(𝒜). The integer powers of an MLO 𝒜 : X → P(X) are defined inductively as follows: 𝒜0 =: I; if 𝒜n−1 is defined, set D(𝒜n ) := {x ∈ D(𝒜n−1 ) : D(𝒜) ∩ 𝒜n−1 x ≠ 0}, and n

𝒜 x := (𝒜𝒜

n−1

)x =

⋃

y∈D(𝒜)∩𝒜n−1 x

𝒜y,

x ∈ D(𝒜n ).

Suppose that 𝒜 : X → P(Y ) and ℬ : X → P(Y ) are two MLOs. Then the inclusion 𝒜 ⊆ ℬ is equivalent to saying that D(𝒜) ⊆ D(ℬ) and 𝒜x ⊆ ℬx for all x ∈ D(𝒜). We say that an MLO operator 𝒜 : X → P(Y ) is closed if and only if for any sequences (xn ) in D(𝒜) and (yn ) in Y such that yn ∈ 𝒜xn for all n ∈ ℕ we have that the suppositions limn→∞ xn = x and limn→∞ yn = y imply x ∈ D(𝒜) and y ∈ 𝒜x. Any MLO has a closed linear extension, in contrast to the usually analyzed single-valued linear operators. The following lemma will be used later on (see e. g., [429, Theorem 1.2.3]):

Lemma 1.1.1. Suppose that 𝒜 : X → P(Y ) is a closed MLO, Ω is a locally compact, separable metric space and μ is a locally finite Borel measure defined on Ω. Let f : Ω → X and g : Ω → Y be μ-integrable, and let g(x) ∈ 𝒜f (x), x ∈ Ω. Then ∫Ω f dμ ∈ D(𝒜) and ∫Ω g dμ ∈ 𝒜 ∫Ω f dμ.

1.1 Basic concepts and research tools

� 3

Suppose that 𝒜 is an MLO in X and C ∈ L(X). The C-resolvent set of 𝒜, ρC (𝒜) for short, is defined as the union of those complex numbers λ ∈ ℂ satisfying that (i) R(C) ⊆ R(λ − 𝒜); (ii) (λ − 𝒜)−1 C is a single-valued linear continuous operator on X. The operator λ 󳨃→ (λ − 𝒜)−1 C is said to be the C-resolvent of 𝒜. If C = I, then we say that ρ(𝒜) ≡ ρC (𝒜) is the resolvent set of 𝒜 and the mapping λ 󳨃→ R(λ : 𝒜) ≡ (λ − 𝒜)−1 is called the resolvent of 𝒜 (λ ∈ ρ(𝒜)). For more details regarding the generalized resolvent equations, the analytical properties of C-resolvents of multivalued linear operators and fractional powers of multivalued linear operators, we refer the reader to [429]. We will use condition (P) henceforth: (P) There exist finite constants c, M > 0 and β ∈ (0, 1], such that Ψ := Ψc := {λ ∈ ℂ : Re λ ⩾ −c(|Im λ| + 1)} ⊆ ρ(𝒜) and −β 󵄩󵄩 󵄩 󵄩󵄩R(λ : 𝒜)󵄩󵄩󵄩 ⩽ M(1 + |λ|) ,

λ ∈ Ψ.

If A = 𝒜 is single-valued and satisfies condition (P), then A is said to be almost sectorial; cf. [427, 428] and references cited therein for more details about this important class of operators. Of concern is the following abstract degenerate Volterra inclusion: t

ℬu(t) ⊆ 𝒜 ∫ a(t − s)u(s)ds + ℱ (t),

t ∈ [0, τ),

(14)

0

where 0 < τ ⩽ ∞, a ∈ L1loc ([0, τ)), a ≠ 0, ℱ : [0, τ) → P(Y ), and 𝒜: X → P(Y ), ℬ: X → P(Y ) are two given mappings (possibly non-linear). We need the following notion: Definition 1.1.2 (cf. [429, Definition 3.1.1(i)]). (i) A function u ∈ C([0, τ) : X) is said to be a pre-solution of (14) if and only if (a ∗ u)(t) ∈ D(𝒜) and u(t) ∈ D(ℬ) for t ∈ [0, τ), as well as (14) holds. (ii) A solution of (14) is any pre-solution u(⋅) of (14) satisfying additionally that there exist functions uℬ ∈ C([0, τ) : Y ) and ua,𝒜 ∈ C([0, τ) : Y ) such that uℬ (t) ∈ ℬu(t) t

and ua,𝒜 (t) ∈ 𝒜 ∫0 a(t − s)u(s)ds for t ∈ [0, τ), as well as uℬ (t) ∈ ua,𝒜 (t) + ℱ (t),

t ∈ [0, τ).

(iii) A strong solution of (14) is any function u ∈ C([0, τ) : X) satisfying that there exist two continuous functions uℬ ∈ C([0, τ) : Y ) and u𝒜 ∈ C([0, τ) : Y ) such that uℬ (t) ∈ ℬu(t), u𝒜 (t) ∈ 𝒜u(t) for all t ∈ [0, τ), and

4 � 1 Preliminaries uℬ (t) ∈ (a ∗ u𝒜 )(t) + ℱ (t),

t ∈ [0, τ).

In the remainder of this subsection, we will analyze multivalued linear operators as subgenerators of (a, k)-regularized (C1 , C2 )-existence and uniqueness families and (a, k)regularized C-resolvent families. Unless specified otherwise, we assume that 0 < τ ⩽ ∞, k ∈ C([0, τ)), k ≠ 0, a ∈ L1loc ([0, τ)), a ≠ 0, 𝒜 : Y → P(Y ) is an MLO, C1 ∈ L(X, Y ), C2 ∈ L(Y ) is injective, C ∈ L(Y ) is injective and C 𝒜 ⊆ 𝒜C. We will use the following notion (see, e. g., [429, Definition 3.2.1, Definition 3.2.2]): Definition 1.1.3. (i) It is said that 𝒜 is a subgenerator of a (local, if τ < ∞) mild (a, k)regularized (C1 , C2 )-existence and uniqueness family (R1 (t), R2 (t))t∈[0,τ) ⊆ L(X, Y ) × L(Y ) if and only if the mappings t 󳨃→ R1 (t)y, t ⩾ 0 and t 󳨃→ R2 (t)x, t ∈ [0, τ) are continuous for every fixed x ∈ Y and y ∈ X, as well as the following conditions hold: t

(∫ a(t − s)R1 (s)y ds, R1 (t)y − k(t)C1 y) ∈ 𝒜, t

t ∈ [0, τ), y ∈ X

and

(15)

0

∫ a(t − s)R2 (s)y ds = R2 (t)x − k(t)C2 x,

whenever t ∈ [0, τ) and (x, y) ∈ 𝒜.

(16)

0

(ii) Let (R1 (t))t∈[0,τ) ⊆ L(X, Y ) be strongly continuous. Then it is said that 𝒜 is a subgenerator of a (local, if τ < ∞) mild (a, k)-regularized C1 -existence family (R1 (t))t∈[0,τ) if and only if (15) holds. (iii) Let (R2 (t))t∈[0,τ) ⊆ L(Y ) be strongly continuous. Then it is said that 𝒜 is a subgenerator of a (local, if τ < ∞) mild (a, k)-regularized C2 -uniqueness family (R2 (t))t∈[0,τ) if and only if (16) holds. Definition 1.1.4. Suppose that 0 < τ ⩽ ∞, k ∈ C([0, τ)), k ≠ 0, a ∈ L1loc ([0, τ)), a ≠ 0, 𝒜 : Y → P(Y ) is an MLO, C ∈ L(Y ) is injective, and C 𝒜 ⊆ 𝒜C. Then it is said that a strongly continuous operator family (R(t))t∈[0,τ) ⊆ L(Y ) is an (a, k)-regularized C-resolvent family with a subgenerator 𝒜 if and only if (R(t))t∈[0,τ) is a mild (a, k)-regularized C-uniqueness family having 𝒜 as subgenerator, R(t)C = CR(t) and R(t)𝒜 ⊆ 𝒜R(t) (t ∈ [0, τ)). If τ = ∞, then (R(t))t⩾0 is said to be exponentially bounded (bounded) if and only if there exists ω ∈ ℝ (ω = 0) such that the family {e−ωt R(t) : t ⩾ 0} is bounded; the infimum of such numbers is said to be the exponential type of (R(t))t⩾0 . The above notion can be simply understood for the classes of mild (a, k)-regularized C1 -existence families and mild (a, k)-regularized C2 -uniqueness families. The integral generator of a mild (a, k)-regularized C2 -uniqueness family (R2 (t))t∈[0,τ) (mild (a, k)-regularized (C1 , C2 )-existence and uniqueness family (R1 (t), R2 (t))t∈[0,τ) ) is defined by

1.1 Basic concepts and research tools

� 5

t

𝒜int := {(x, y) ∈ X × X : R2 (t)x − k(t)C2 x = ∫ a(t − s)R2 (s)y ds, t ∈ [0, τ)}; 0

we define the integral generator of an (a, k)-regularized C-resolvent family (R(t))t∈[0,τ) in the same way. For simplicity, we will assume henceforth that any (a, k)-regularized C-resolvent family considered below is likewise a mild (a, k)-regularized C-existence family (subgenerated by 𝒜). We refer the reader to [427, 428, 429] for several simple conditions ensuring this property. Integration in Banach spaces The following elementary definition can be found in many textbooks: Definition 1.1.5. (i) We say that a function f : I → X is simple if and only if there exist k ∈ ℕ, elements zi ∈ X, 1 ⩽ i ⩽ k and Lebesgue measurable subsets Ωk , 1 ⩽ i ⩽ k of I, such that m(Ωi ) < ∞, 1 ⩽ i ⩽ k and k

f (t) = ∑ zi χΩi (t), i=1

t ∈ I.

(17)

(ii) We say that a function f : I → X is measurable if and only if there exists a sequence (fn ) in X I such that for every n ∈ ℕ, fn (⋅) is a simple function and limn→∞ fn (t) = f (t) for a. e. t ∈ I. (iii) Let −∞ < a < b < ∞ and a < τ ⩽ ∞. We say that a function f : [a, b] → X is absolutely continuous if and only if for every ε > 0 there exists a number δ > 0 such that for any finite collection of open subintervals (ai , bi ), 1 ⩽ i ⩽ k of [a, b] with ∑ki=1 (bi − ai ) < δ, we have ∑ki=1 ‖f (bi ) − f (ai )‖ < ε; a function f : [a, τ) → X is said to be absolutely continuous if and only if for every τ0 ∈ (a, τ), the function f|[a,τ0 ] : [a, τ0 ] → X is absolutely continuous. If f : I → X and (fn ) is a sequence of measurable functions such that limn→∞ fn (t) = f (t) for a. e. t ∈ I, then the function f (⋅) is measurable, as well. The Bochner integral of a simple function f : I → X, f (t) = ∑ki=1 zi χΩi (t), t ∈ I is defined by k

∫ f (t) dt := ∑ zi m(Ωi ). I

i=1

The definition of Bochner integral does not depend on the representation (17), as easily shown. We say that a measurable function f : I → X is Bochner integrable if and only if there exists a sequence of simple functions (fn ) in X I such that limn→∞ fn (t) = f (t) for a. e. t ∈ I and

6 � 1 Preliminaries 󵄩 󵄩 lim ∫󵄩󵄩󵄩fn (t) − f (t)󵄩󵄩󵄩 dt = 0;

n→∞

(18)

I

if this is the case, the Bochner integral of f (⋅) is defined by ∫ f (t) dt := lim ∫ fn (t) dt. n→∞

I

I

This definition does not depend on the choice of a sequence of simple functions (fn ) in X I satisfying limn→∞ fn (t) = f (t) for a. e. t ∈ I and (18). We know that f : I → X is Bochner integrable if and only if f (⋅) is measurable and the function t 󳨃→ ‖f (t)‖, t ∈ I is ∞ integrable. For any Bochner integrable function f : [0, ∞) → X, we have ∫0 f (t) dt = τ

limτ→+∞ ∫0 f|[0,τ] (t) dt. The space of all Bochner integrable functions from I into X is designated by L1 (I : X); endowed with the norm ‖f ‖1 := ∫I ‖f (t)‖ dt, L1 (I : X) is a Banach space. It is said that a function f : [0, ∞) → X is locally (Bochner) integrable if and only if f (⋅)|[0,τ] is Bochner integrable for every τ > 0. The space of all locally integrable functions from [0, ∞) into X is denoted by L1loc ([0, ∞) : X). If f : [a, b] → X is Bochner integrable, where t

−∞ < a < b < +∞, then the function F(t) := ∫a f (s) ds, t ∈ [a, b] is absolutely continuous and F ′ (t) = f (t) for a. e. t ∈ [a, b]. We need the following fundamental results:

Theorem 1.1.6. (i) (The dominated convergence theorem) Suppose that (fn ) is a sequence of Bochner integrable functions from X I and that there exists an integrable function g : I → ℝ such that ‖fn (t)‖ ⩽ g(t) for a. e. t ∈ I and n ∈ ℕ. If f : I → X and limn→∞ fn (t) = f (t) for a. e. t ∈ I, then f (⋅) is Bochner integrable, ∫I f (t) dt = limn→∞ ∫I fn (t) dt and limn→∞ ∫I ‖fn (t) − f (t)‖ dt = 0. (ii) (The Fubini theorem) Let I1 and I2 be segments in ℝ and let I = I1 × I2 . Suppose that F : I → X is measurable and ∫I ∫I ‖f (s, t)‖ dt ds < ∞. Then f (⋅, ⋅) is Bochner 1

2

integrable, the repeated integrals ∫I ∫I f (s, t) dt ds and ∫I ∫I f (s, t) ds dt exist and are equal to the integral ∫I f (s, t) ds dt.

1

2

2

1

Suppose now that 1 ⩽ p < ∞ and (Ω, ℛ, μ) is a measure space. By Lp (Ω : X) we denote the space of all strongly μ-measurable functions f : Ω → X such that ‖f ‖p := (∫Ω ‖f (⋅)‖p dμ)1/p is finite. The space L∞ (Ω : X) consisting of all strongly μ-measurable, essentially bounded functions, is a Banach space equipped with the norm ‖f ‖∞ := ess supt∈Ω ‖f (t)‖, f ∈ L∞ (Ω : X). The famous Riesz–Fischer theorem says that (Lp (Ω : X), ‖⋅‖p ) is a Banach space for all p ∈ [1, ∞]; furthermore, (L2 (Ω : X), ‖⋅‖2 ) is a Hilbert space. Let us recall, if limn→∞ fn = f in Lp (Ω : X), then there exists a subsequence (fnk ) of (fn ) such that limk→∞ fnk (t) = f (t) μ-almost everywhere. If the Banach space X is reflexive, then Lp (Ω : X) is reflexive for all p ∈ (1, ∞) and its dual is isometrically isomorphic to Lp/(p−1) (Ω : X).

1.1 Basic concepts and research tools

�

7

p

Let 0 ≠ Ω ⊆ ℝn (n, k ∈ ℕ). The space Lloc (Ω : X) for 1 ⩽ p ⩽ ∞ is defined in the p p usual way (T, τ > 0); Lloc (Ω) ≡ Lloc (Ω : ℂ). If Ω is open, then C k (Ω : X) denotes the space of k-times continuously differentiable functions f : Ω → X. Suppose now that k ∈ ℕ and p ∈ [1, ∞]. Then the Sobolev space W k,p (Ω : X) consists of those X-valued distributions u ∈ 𝒟′ (Ω : X) such that, for every multi-index α ∈ ℕn0 with |α| ⩽ k, we have Dα u ∈ Lp (Ω, X). Here, the derivative Dα is taken in the sense of k,p distributions. By Wloc (Ω : X) we denote the space of those X-valued distributions u ∈ 𝒟′ (Ω : X) such that for every bounded open subset Ω′ of Ω, we have u|Ω′ ∈ W k,p (Ω′ : X). The basic facts about strongly continuous semigroups, integrated semigroups, and C-regularized semigroups may be obtained by consulting the monographs [426] and [429]; for more details about various classes of (a, k)-regularized C-resolvent families and fractional solution operator families appearing in the theory of the abstract (degenerate) Volterra integro-differential equations, we refer the reader to [427, 429] and references cited therein. Fractional calculus and fractional differential equations Before proceeding further, we will only note that the fractional calculus and fractional differential equations are extremely growing fields of research, which have invaluable importance in engineering, physics, chemistry, mechanics, electricity, control theory and many other branches of applied science. For more details about fractional calculus and fractional differential equations, we refer the reader to the monographs cited in [426], [429], and [431]. Suppose that α > 0, m = ⌈α⌉ and I = (0, T) for some T ∈ (0, ∞]. Then the Riemann– Liouville fractional integral Jtα of order α is defined by Jtα f (t) := (gα ∗ f )(t),

f ∈ L1 (I : X), t ∈ I.

The Caputo fractional derivative Dαt u(t) is defined for those functions u ∈ C m−1 ([0, ∞) : m X) for which gm−α ∗ (u − ∑m−1 k=0 uk gk+1 ) ∈ C ([0, ∞) : X), by Dαt u(t) :=

m−1 dm [g ∗ (u − ∑ uk gk+1 )]. dt m m−α k=0

It is worth noticing that the existence of Caputo fractional derivative Dαt u for t ⩾ 0 implies the existence of Caputo fractional derivative Dζt u for t ⩾ 0 and any ζ ∈ (0, α). At some places, we will use a slightly weakened notion of Caputo fractional derivatives, as explicitly emphasized. γ Let us recall that the Weyl–Liouville fractional derivative Dt,+ u(t) of order γ ∈ (0, 1) is defined for those continuous functions u : ℝ → X such that t

t 󳨃→ ∫ g1−γ (t − s)u(s) ds, −∞

t∈ℝ

8 � 1 Preliminaries is a well-defined, continuously differentiable mapping, by t

γ

Dt,+ u(t) :=

d ∫ g1−γ (t − s)u(s) ds, dt

t ∈ ℝ.

−∞

Define D1t,+ u(t) := −(d/dt)u(t). The Mittag-Leffler functions and the Wright functions are of crucial importance in fractional calculus. Let α > 0 and β ∈ ℝ. Then the Mittag-Leffler function Eα,β (z) is defined by zn , Γ(αn + β) n=0 ∞

Eα,β (z) := ∑

z ∈ ℂ;

we define, for short, Eα (z) := Eα,1 (z), z ∈ ℂ. If γ ∈ (0, 1), then we define the Wright function Φγ (⋅) by (−z)n , n!Γ(1 − γ − γn) n=0 ∞

Φγ (z) := ∑

z ∈ ℂ.

Let us recall that Φγ (⋅) is an entire function as well as that: (i) Φγ (t) ⩾ 0, t ⩾ 0, γ

∞

(ii) ∫0 e−λt γst −1−γ Φγ (t −γ s) dt = e−λ s , Re λ > 0, s > 0, and ∞

(iii) ∫0 t r Φγ (t) dt =

Γ(1+r) , Γ(1+γr)

r > −1.

Fixed point theorems The fixed point theory is a rapidly growing field of research. We will recall here the statements of the Banach contraction principle, the Bryant fixed point theorem and the Schauder fixed point theorem; for more details about the fixed point theory, the reader may consult the monographs cited in [431]. Let (E, d) be a metric space. Then we say that T : E → E is a contraction mapping on E if and only if there exists a constant q ∈ [0, 1) such that d(T(x), T(y)) ⩽ qd(x, y) for all x, y ∈ E. For our further work, it will be necessary to recall the following well-known results: Theorem 1.1.7 (The Banach contraction principle, 1922). Let (E, d) be a complete metric space, and let T : E → E be a contraction mapping. Then T admits a unique fixed point x in X (i. e. T(x) = x). Theorem 1.1.8 (The Bryant fixed point theorem, 1968). Let (E, d) be a complete metric space, and let T : E → E satisfy that there is an integer n ∈ ℕ such that T n : E → E is a contraction mapping. Then T has a unique fixed point x in E.

1.1 Basic concepts and research tools

� 9

The first version of the following theorem was conjectured and proven on Banach spaces by J. Schauder in 1930; four years later A. Tychonoff proved the theorem in the case that K is a compact convex subset of a locally convex space: Theorem 1.1.9 (The Schauder fixed point theorem). Suppose that K is a non-empty convex compact subset of a locally convex Hausdorff space V and T is a continuous mapping of K into itself. Then T has at least one fixed point. Laplace transform Let 0 < τ ⩽ ∞ and a ∈ L1loc ([0, τ)). Then we say that the function a(t) is a kernel on [0, τ) t

if and only if for each f ∈ C([0, τ)) the assumption ∫0 a(t − s)f (s) ds = 0, t ∈ [0, τ) implies f (t) = 0, t ∈ [0, τ). We need the following conditions on the kernel k(t): (P1): k(t) is Laplace transformable, i. e., it is locally integrable on [0, ∞) and there exists b ∞ ̃ β ∈ ℝ such that k(λ) := (ℒk)(λ) := limb→∞ ∫0 e−λt k(t)dt := ∫0 e−λt k(t)dt exists for ̃ exists}. all λ ∈ ℂ with Re λ > β. Put abs(k) := inf{Re λ : k(λ) ̃ (P2): k(t) satisfies (P1) and k(λ) ≠ 0, Re λ > β for some β ⩾ abs(k). We say that a function h(⋅) belongs to the class LT − E if there exists an exponentially bounded function f ∈ C([0, ∞) : E) and a real number a > 0 such that h(λ) = (ℒf )(λ), λ > a. For more details about vector-valued Laplace transform, we refer the reader to [54], [429], and [777]. 1.1.1 Lebesgue spaces with variable exponents Lp(x) Concerning the Lebesgue spaces with variable exponents, the research monograph [265] by L. Diening et al. is of invaluable importance. Suppose that 0 ≠ Ω ⊆ ℝ is a nonempty subset. Let M(Ω : X) be the collection of all measurable functions f : Ω → X; M(Ω) := M(Ω : ℝ). By 𝒫 (Ω) we denote the collection of all Lebesgue measurable functions p : Ω → [1, ∞]. For any p ∈ 𝒫 (Ω) and f ∈ M(Ω : X), we define p(x) {t , { { φp(x) (t) := {0, { { {∞,

t ⩾ 0, 1 ⩽ p(x) < ∞, 0 ⩽ t ⩽ 1, p(x) = ∞, t > 1, p(x) = ∞

and 󵄩 󵄩 ρ(f ) := ∫ φp(x) (󵄩󵄩󵄩f (x)󵄩󵄩󵄩) dx. Ω

We define the Lebesgue space Lp(x) (Ω : X) with variable exponent by

(19)

10 � 1 Preliminaries Lp(x) (Ω : X) := {f ∈ M(Ω : X) : lim ρ(λf ) = 0}. λ→0+

Let us recall that this space can be equivalently introduced by: Lp(x) (Ω : X) = {f ∈ M(Ω : X) : there exists λ > 0 such that ρ(λf ) < ∞}; see e. g., [265, p. 73]. For every u ∈ Lp(x) (Ω : X), we introduce the Luxemburg norm of u(⋅) in the following way: ‖u‖p(x) := ‖u‖Lp(x) (Ω:X) := inf{λ > 0 : ρ(f /λ) ⩽ 1}. Equipped with the above norm, Lp(x) (Ω : X) is a Banach space, which coincides with the usual Lebesgue space Lp (Ω : X) in the case that p(x) = p ⩾ 1 is a constant function. For any p ∈ M(Ω), we define p− := ess inf p(x) and x∈Ω

p+ := ess sup p(x). x∈Ω

Put C+ (Ω) := {p ∈ M(Ω) : 1 < p− ⩽ p(x) ⩽ p+ < ∞ for a. e. x ∈ Ω} and D+ (Ω) := {p ∈ M(Ω) : 1 ⩽ p− ⩽ p(x) ⩽ p+ < ∞ for a. e. x ∈ Ω}. If p ∈ D+ (Ω), then we know that the function ρ(⋅) given by (19) is modular in the sense of [265, Definition 2.1.1], as well as that Lp(x) (Ω : X) = {f ∈ M(Ω : X) : for all λ > 0 we have ρ(λf ) < ∞}. Furthermore, if p ∈ C+ (Ω), then Lp(x) (Ω : X) is uniformly convex and therefore reflexive [296]. For the sequel, we need the following lemma (see e. g., [265, Lemma 3.2.20, (3.2.22); Corollary 3.3.4; p. 77] for the scalar-valued case): Lemma 1.1.10. (i) (The Hölder inequality) Let p, q, r ∈ 𝒫 (Ω) such that 1 1 1 = + , q(x) p(x) r(x)

x ∈ Ω.

Then, for every u ∈ Lp(x) (Ω : X) and v ∈ Lr(x) (Ω), we have uv ∈ Lq(x) (Ω : X) and ‖uv‖q(x) ⩽ 2‖u‖p(x) ‖v‖r(x) .

1.1 Basic concepts and research tools

� 11

(ii) Let Ω be of a finite Lebesgue’s measure and let p, q ∈ 𝒫 (Ω) be such that q ⩽ p a. e. on Ω. Then Lp(x) (Ω : X) is continuously embedded in Lq(x) (Ω : X). (iii) Let f ∈ Lp(x) (Ω : X), g ∈ M(Ω : X) and 0 ⩽ ‖g‖ ⩽ ‖f ‖ a. e. on Ω. Then g ∈ Lp(x) (Ω : X) and ‖g‖p(x) ⩽ ‖f ‖p(x) . We will also use the following simple lemma, whose proof can be omitted: Lemma 1.1.11. Suppose that f ∈ Lp(x) (Ω : X) and A ∈ L(X, Y ). Then Af ∈ Lp(x) (Ω : Y ) and ‖Af ‖Lp(x) (Ω:Y ) ⩽ ‖A‖ ⋅ ‖f ‖Lp(x) (Ω:X) . For further information concerning the Lebesgue spaces with variable exponents Lp(x) , we refer the reader to [296, 581] and the list of references given in [265].

�

Part I: Multi-dimensional ρ-almost periodicity

In this part, we investigate various classes of vector-valued ρ-almost periodic type functions in ℝn and ρ-almost periodic type solutions of the abstract (degenerate) Volterra integro-differential equations in Banach spaces. We use the following notation and terminology. Suppose that X, Y , Z and T are given non-empty sets. Let us recall that a binary relation between X into Y is any subset ρ ⊆ X × Y . If ρ ⊆ X × Y and σ ⊆ Z × T with Y ∩ Z ≠ 0, then we define ρ−1 ⊆ Y × X and σ ∘ ρ = σ ∘ ρ ⊆ X × T by ρ−1 := {(y, x) ∈ Y × X : (x, y) ∈ ρ} and σ ∘ ρ := {(x, t) ∈ X × T : ∃y ∈ Y ∩ Z such that (x, y) ∈ ρ and (y, t) ∈ σ}, respectively. As is well known, the domain and range of ρ are defined by D(ρ) := {x ∈ X : ∃y ∈ Y such that (x, y) ∈ X × Y } and R(ρ) := {y ∈ Y : ∃x ∈ X such that (x, y) ∈ X × Y }, respectively; ρ(x) := {y ∈ Y : (x, y) ∈ ρ} (x ∈ X), x ρ y ⇔ (x, y) ∈ ρ. If ρ is a binary relation on X and n ∈ ℕ, then we define ρn inductively; ρ−n := (ρn )−1 and ρ0 := ΔX := {(x, x) : x ∈ X}. Set ρ(X ′ ) := {y : y ∈ ρ(x) for some x ∈ X ′ } (X ′ ⊆ X). For any set A we define its power set P(A) := {B | B ⊆ A}. If t0 ∈ ℝn and ε > 0, then we set B(t0 , ε) := {t ∈ ℝn : |t − t0 | ⩽ ε}, where | ⋅ | denotes the Euclidean norm in ℝn . Set IM := {t ∈ I : |t| ⩾ M} (I ⊆ ℝn ; M > 0). Further on, by (e1 , e2 , . . . , en ) we denote the standard basis of ℝn ; ⟨⋅, ⋅⟩ denotes the usual inner product in ℝn . Define S1 := {z ∈ ℂ : |z| = 1}. If not stated otherwise, we will always assume henceforth that (X, ‖ ⋅ ‖), (Y , ‖ ⋅ ‖Y ) and (Z, ‖ ⋅ ‖Z ) are three complex Banach spaces, n ∈ ℕ, and ℬ is a certain collection of subsets of X satisfying that for each x ∈ X there exists B ∈ ℬ such that x ∈ B.

https://doi.org/10.1515/9783111233871-002

2 Multi-dimensional ρ-almost periodic type functions and their Stepanov generalizations The main aim of this chapter is to introduce and analyze various classes of multidimensional ρ-almost periodic type functions and their Stepanov generalizations. We start by recalling that the notion of a periodic function has recently been reconsidered by E. Alvarez, A. Gómez and M. Pinto [36] as follows: Let I = ℝ or I = [0, ∞). Then a continuous function f : I → X is said to be (ω, c)-periodic (ω > 0, c ∈ ℂ ∖ {0}) if and only if f (t + ω) = cf (t) for all t ∈ I. It is well known that a continuous function f : I → X is (ω, c)-periodic if and only if the function g(⋅) ≡ c−⋅/ω f (⋅) is periodic and g(t + ω) = g(t) for all t ∈ I; here, c−⋅/ω denotes the principal branch of the exponential function (cf. also [15] and [34, 35]). In [410], the authors have recently analyzed various classes of (ω, c)-almost periodic type functions. As already mentioned in [431], the notions of affine-periodicity and pseudo affineperiodicity play an incredible role in the qualitative analysis of solutions for various classes of systems of ordinary differential equations, systems of functional differential equations, and systems of Newtonian equations of motion with friction; cf. [188, 206, 502, 507, 549, 752, 754, 773, 810] for some results obtained in this direction. By a (Q, T) affineperiodic function x : ℝ → ℝn we mean any continuous function x(⋅) which satisfies x(t + T) = Qx(t), t ∈ ℝ, where Q is a regular matrix of format n × n and T > 0. In [188], X. Chang and Y. Li have analyzed the rotating periodic solutions of the second-order dissipative dynamical system: u′′ + cu′ + ∇g(u) + h(u) = e(t),

t ∈ ℝ,

where c > 0 is a constant, g(u) = g(|u|), h ∈ C(ℝn : ℝn ), h(u) = Qh(Q−1 u) for some orthogonal matrix Q ∈ O(n) and e ∈ C(ℝ : ℝn ) satisfies e(t + T) = Qe(t) for all t ∈ ℝ. The authors have proved that the above equation admits a solution of the form u(t + T) = Qu(t), t ∈ ℝ, which is usually called rotating periodic solution. We can also refer the reader to the research article [507] by Y. Li, H. Wang and X. Yang, where the authors have analyzed Fink’s conjecture on affine-periodic solutions and Levinson’s conjecture to Newtonian systems. Among many other themes, the authors have analyzed the following system of Newtonian equations of motion with friction x ′′ + A(t, x)u′ + ∇V (x) + h(u) = e(t),

t ∈ ℝ,

where A : ℝ × ℝm → ℝm,m , V : ℝm → ℝ and e : ℝ → ℝm are continuous, A(t, x) satisfies the local Lipschitz condition with respect to the variable x, V (⋅) is continuously differentiable and A(t + T, x)y ≡ QA(t, Q−1 x)Q−1 y, ∇V (x) ≡ Q∇V (Q−1 x), e(t + T) ≡ Qe(t). We will particularly analyze the accumulation of (Q, T) affine-periodic solutions on the https://doi.org/10.1515/9783111233871-003

18 � 2 Multi-dimensional ρ-almost periodic type functions and their Stepanov generalizations heteroclinic cycles and the phenomenon of heteroclinic/homoclinic period blow-up. See also [23, 308] and references cited therein. The concept of (w, 𝕋)-periodicity for a continuous function f : [0, ∞) → X, where ω > 0 and 𝕋 : X → X is a linear isomorphism, has recently been introduced and analyzed by M. Fečkan, K. Liu, and J. Wang in [307, Definition 2.2] as a generalization of the concept of affine-periodicity to the infinite-dimensional setting; more precisely, a continuous function f : [0, ∞) → X is called (w, 𝕋)-periodic if and only if f (t+ω) = 𝕋f (t) for all t ⩾ 0. In the above-mentioned paper, the existence and uniqueness of (w, 𝕋)-periodic solutions have been investigated for various classes of impulsive evolution equations, linear and semilinear problems by using some results from the theory of strongly continuous semigroups, the Fredholm alternative type theorems and the fixed point theorems. Before proceeding any further, we would like to observe that the notion of (w, 𝕋)periodicity is very general, although it might not seem at first sight. For example, if (T(t))t⩾0 is a strongly continuous non-degenerate semigroup generated by a closed linear operator A in X, then the unique mild solution of the abstract first-order Cauchy problem u′ (t) = Au(t), t ⩾ 0, equipped with the initial condition u(0) = x, is given by u(t) = T(t)x, t ⩾ 0. It is clear that this solution has the property that for each ω > 0 we have u(t + ω) = T(ω)u(t), t ⩾ 0 and u(⋅) is therefore (ω, T(ω))-periodic for each ω > 0; needless to say that, if (T(t))t⩾0 can be extended to a strongly continuous group in X, then the operator T(ω) is a linear isomorphism. This particularly holds if A is a complex matrix of format n × n; then the unique solution of the system of ordinary differential equations u′ (t) = Au(t), t ⩾ 0; u(0) = x, given by u(t) := T(t)x, t ⩾ 0, is (ω, T(ω))-periodic for each real number ω > 0. Further on, in [194], A. Chávez et al. have analyzed various classes of almost periodic functions of the form F : I × X → Y , where (Y , ‖ ⋅ ‖Y ) is a complex Banach space and 0 ≠ I ⊆ ℝn ; the multi-dimensional c-almost periodic type functions have recently been investigated in [410], while the multi-dimensional (ω, c)-almost periodic type functions have recently been investigated in [437]; here, c ∈ ℂ ∖ {0}. For multi-periodic solutions of various classes of ordinary differential equations and partial differential equations, we also refer the reader to [123, 124, 323, 408, 478, 479] and [552, 659, 660, 732, 733, 734]. Especially, we would like to mention the investigations of G. Nadin [567, 568, 569] concerning the space-time periodic reaction-diffusion equations, G. Nadin–L. Rossi [570] concerning transition waves for Fisher-KPP equations, L. Rossi [646] concerning Liouville type results for almost periodic type linear operators (let us only notice here that the Liouville-type results for semilinear elliptic second-order equations in unbounded domains have been also analyzed in the research article [111] by H. Berestycki, F. Hamel and L. Rossi), the investigation of B. Scarpellini [663] concerning the space almost periodic solutions of reaction-diffusion equations, and the recent investigation of R. Xie, Z. Xia, J. Liu [779] concerning the quasi-periodic limit functions, (ω1 , ω2 )-(quasi)-periodic limit functions and their applications, given only in the twodimensional setting.

2.1 Multi-dimensional ρ-almost periodic type functions

�

19

2.1 Multi-dimensional ρ-almost periodic type functions The main aim of this section is to continue some of the above-mentioned research studies by investigating various classes of multi-dimensional ρ-almost periodic type functions F : I × X → Y and multi-dimensional (ω, ρ)-almost periodic type functions F : I × X → Y , where n ∈ ℕ, 0 ≠ I ⊆ ℝn , X and Y are complex Banach spaces and ρ is a binary relation on Y . The introduced notion seems to be very general and we feel it is our duty to say that we have not been able to perceive with forethought all related problems and questions concerning multi-dimensional ρ-almost periodic type functions here (even the one-dimensional setting requires further analyses; see, e. g., [428]). It is our strong belief that our ideas will motivate many other authors to further explore this interesting topic in the near future. The organization and main ideas of this section, which considers Bohr (ℬ, I ′ , ρ)almost periodic functions and (ℬ, I ′ , ρ)-uniformly recurrent type functions, can be briefly described as follows. The basic notion is introduced in Definition 2.1.1 and further analyzed in Proposition 2.1.3, Corollary 2.1.4, Proposition 2.1.6, Remark 2.1.7, and Proposition 2.1.8; this notion is completely new even in the one-dimensional setting. After that, we justify the introduction of notion in Examples 2.1.9–2.1.11. The main structural characterizations of Bohr (ℬ, I ′ , ρ)-almost periodic functions ((ℬ, I ′ , σ)-uniformly recurrent functions) are given in Theorem 2.1.12. After that, we clarify a simple result concerning the compositions of Bohr (ℬ, I ′ , ρ)-almost periodic functions with uniformly continuous functions in Proposition 2.1.13; the supremum formula for (ℬ, I ′ , ρ)-uniformly recurrent type functions is established in Proposition 2.1.14. The convolution invariance of Bohr (ℬ, I ′ , ρ)-almost periodic ((ℬ, I ′ , ρ)-uniformly recurrent) functions is investigated in Theorem 2.1.15; after that, in Theorem 2.1.17, we analyze the invariance of Bohr (ℬ, I ′ , ρ)-almost periodicity and (ℬ, I ′ , σ)-uniform recurrence under the actions of the infinite convolution product t1

t2

tn

t 󳨃→ F(t) := ∫ ∫ ⋅ ⋅ ⋅ ∫ R(t − s)f (s) ds, −∞ −∞

t ∈ ℝn .

(20)

−∞

We also clarify some obvious composition principles necessary for studying the semilinear Cauchy problems. In Section 2.1.1, we further analyze the notion of T-almost periodicity in the case that Y = ℂk is finite-dimensional and T = A is a complex matrix of format k × k, where k ∈ ℕ. We provide many interesting results and observations in the case that A is singular or non-singular; sometimes the geometry of region I ⊆ ℝn is crucial for the validity of these results. In the case that the matrix A is singular, we construct an example of an A-almost periodic function F : [0, ∞) → ℂ2 which is not almost periodic. Section 2.1.2 thoroughly investigates 𝔻-asymptotically Bohr (ℬ, I ′ , ρ)-almost periodic type functions. In this subsection, we prove an extension type theorem for multidimensional T-almost periodic functions, where T ∈ L(Y ) is a linear isomorphism and

20 � 2 Multi-dimensional ρ-almost periodic type functions and their Stepanov generalizations the region I satisfies certain properties (Theorem 2.1.30). The main aim of Section 2.1.3 is to analyze the corresponding problems for (ω, ρ)-periodic functions and (ωj , ρj )j∈ℕn periodic functions. The existence of mean value of multi-dimensional ρ-almost periodic functions is analyzed in Corollary 2.1.5, Corollary 2.1.31 and an interesting problem concerning this issue is proposed at the end of the second subsection. We provide several exemplifying applications to the abstract (degenerate) Volterra integro-differential equations in Banach spaces. Before proceeding any further, the author would like to express his sincere gratitude to Professor G. M. N’Guérékata, who invited him to write and publish a paper for the special issue of Applicable Analysis dedicated to the memory of Professor A. A. Pankov. The monograph “Bounded and Almost Periodic Solutions of Nonlinear Operator Differential Equations”, published by Kluwer Acad. Publ. in 1990, can serve as an excellent source for the readers interested to acquire a basic knowledge about Bohr compactifications and almost periodic functions on topological groups (let us also mention here that the almost periodic functions on locally compact abelian groups have been considered in the research monograph [402] by Y. Katznelson); see also the research monographs [598, 599] for some other scientific research interests of Professor A. A. Pankov. With the exception of Remark 2.1.2, which is new, the material of this section is completely taken from our joint research article [304] with M. Fečkan, M. T. Khalladi, and A. Rahmani. In [441], we have recently introduced and analyzed the following notion with ρ = cI, where I denotes the identity operator on Y : Definition 2.1.1. Suppose that 0 ≠ I ′ ⊆ ℝn , 0 ≠ I ⊆ ℝn , F : I × X → Y is a continuous function, ρ is a binary relation on Y and I + I ′ ⊆ I. Then we say that: (i) F(⋅; ⋅) is Bohr (ℬ, I ′ , ρ)-almost periodic if and only if for every B ∈ ℬ and ε > 0 there exists l > 0 such that for each t0 ∈ I ′ there exists τ ∈ B(t0 , l) ∩ I ′ such that, for every t ∈ I and x ∈ B, there exists an element yt;x ∈ ρ(F(t; x)) such that 󵄩󵄩 󵄩 󵄩󵄩F(t + τ; x) − yt;x 󵄩󵄩󵄩Y ⩽ ε.

(21)

(ii) F(⋅; ⋅) is (ℬ, I ′ , ρ)-uniformly recurrent if and only if for every B ∈ ℬ there exists a sequence (τ k ) in I ′ such that limk→+∞ |τ k | = +∞ and that for every t ∈ I and x ∈ B, there exists an element yt;x ∈ ρ(F(t; x)) such that lim

󵄩 󵄩 sup 󵄩󵄩󵄩F(t + τ k ; x) − yt;x 󵄩󵄩󵄩Y = 0.

k→+∞ t∈I;x∈B

(22)

It is clear that the Bohr (ℬ, I ′ , ρ)-almost periodicity of F(⋅; ⋅) implies the (ℬ, I ′ , ρ)uniform recurrence of F(⋅; ⋅); the converse statement is not true in general [431]. In the case that ρ = T : Y → Y is a single-valued function (not necessarily linear or continuous), then we obtain the most important case for our further investigations, when the function F(⋅; ⋅) is (ℬ, I ′ , T)-almost periodic, resp. (ℬ, I ′ , T)-uniformly recurrent. In the case that X = {0} (I ′ = I), we omit the term “ℬ” (“I ′ ”) from the notation; furthermore, if

2.1 Multi-dimensional ρ-almost periodic type functions

� 21

T = cI for some complex number c ∈ ℂ ∖ {0}, then we also say that the function F(⋅; ⋅) is (ℬ, I ′ , c)-almost periodic, resp. (ℬ, I ′ , c)-uniformly recurrent. Remark 2.1.2. It is worth noting that we require the continuity of function F(⋅; ⋅) in Definition 2.1.1. Concerning this issue, we would like to present the following explanatory example: Suppose that y : ℝ → ℝ is a Bohr almost periodic function, m ∈ ℕ and there exists an integer k0 ∈ ℤ such that y(mk0 ) ≠ y(m(k0 + 1)). Define f (t) := y(m⌊

t+1 ⌋), m

t ∈ ℝ.

Then the function f (⋅) is not continuous on the real line, but for every ε > 0, there exists l > 0 such that any interval I ⊆ ℝ of length ⩾ l contains a point τ such that |f (t+τ)−f (t)| ⩽ ε, t ∈ ℝ. See Example 9.2.7 for more details. In [409, Proposition 2.6, Corollary 2.10, Proposition 2.11], we have considered the question whether a given one-dimensional c-almost periodic function (c-uniformly recurrent function), where c ∈ S1 , is almost periodic (uniformly recurrent). Concerning this question for general binary relations in the multi-dimensional setting, we will state and prove the following result (see also Proposition 2.1.21 and Example 2.1.22 below for the case that the space Y is finite-dimensional and the equality τ + I = I is not satisfied for some points τ ∈ I ′ ): Proposition 2.1.3. Suppose that 0 ≠ I ′ ⊆ ℝn , 0 ≠ I ⊆ ℝn , I + I ′ ⊆ I and the function F : I × X → Y is Bohr (ℬ, I ′ , ρ)-almost periodic ((ℬ, I ′ , ρ)-uniformly recurrent), where ρ is a binary relation on Y satisfying R(F) ⊆ D(ρ) and ρ(y) is a singleton for any y ∈ R(F). If for each τ ∈ I ′ we have τ + I = I, then I + (I ′ − I ′ ) ⊆ I and the function F(⋅; ⋅) is Bohr (ℬ, I ′ − I ′ , I)-almost periodic ((ℬ, I ′ − I ′ , I)-uniformly recurrent). Proof. We will consider only Bohr (ℬ, I ′ , ρ)-almost periodic functions. Let τ ∈ I ′ − I ′ be given. Then there exist points τ1 , τ2 ∈ I ′ such that τ = τ1 − τ2 ; if t ∈ I, then the prescribed assumption τ2 + I = I implies the existence of a point t′ ∈ I such that t = τ2 + t′ . Hence, t + τ = τ1 + t′ ∈ I and therefore I + (I ′ − I ′ ) ⊆ I, as claimed. Further on, let ε > 0 and B ∈ ℬ be given. Then there exists l > 0 such that for each t10 , t20 ∈ I ′ there exist two points τ 1 ∈ B(t10 , l) ∩ I ′ and τ 2 ∈ B(t20 , l) ∩ I ′ such that, for every t ∈ I and x ∈ B, we have 󵄩󵄩 󵄩 󵄩󵄩F(t + τ 1 ; x) − ρ(F(t; x))󵄩󵄩󵄩Y ⩽ ε/2

󵄩 󵄩 and 󵄩󵄩󵄩F(t + τ 2 ; x) − ρ(F(t; x))󵄩󵄩󵄩Y ⩽ ε/2.

This implies 󵄩󵄩 󵄩 󵄩󵄩F(t + τ 1 ; x) − F(t + τ 2 ; x)󵄩󵄩󵄩Y ⩽ ε,

t ∈ I, x ∈ B,

i. e., 󵄩󵄩 󵄩 󵄩󵄩F(v + [τ2 − τ 1 ]; x) − F(v; x)󵄩󵄩󵄩Y ⩽ ε,

v ∈ τ1 + I = I, x ∈ B.

22 � 2 Multi-dimensional ρ-almost periodic type functions and their Stepanov generalizations Since τ2 − τ1 ∈ B(t20 − t10 , 2l) ∩ (I ′ − I ′ ), this simply implies the required conclusion by definition. Corollary 2.1.4. Suppose that 0 ≠ I ′ ⊆ ℝn and the function F : ℝn × X → Y is Bohr (ℬ, I ′ , ρ)-almost periodic ((ℬ, I ′ , ρ)-uniformly recurrent), where ρ is a binary relation on Y satisfying R(F) ⊆ D(ρ) and ρ(y) is a singleton for any y ∈ R(F). Then the function F(⋅; ⋅) is Bohr (ℬ, I ′ − I ′ , I)-almost periodic ((ℬ, I ′ − I ′ , I)-uniformly recurrent). If I = ℝ, then Corollary 2.1.4 enables one to see that any c-uniformly recurrent function F : ℝ → Y , where c ∈ ℂ ∖ {0}, is uniformly recurrent (see [409, Definition 2.3]). Strictly speaking, this is a new result which is not clarified in [409]: more precisely, in [409, Corollary 2.10, Proposition 2.11(ii)], we have proved that any bounded c-uniformly recurrent function F : I → Y , where c ∈ ℂ∖{0} satisfies |c| = 1 and I = ℝ or I = [0, ∞), is uniformly recurrent (if |c| ≠ 1, then the unique c-uniformly recurrent function F : I → Y is the zero function; see [409, Proposition 2.6]). We have the following important corollary of Proposition 2.1.3: Corollary 2.1.5. Suppose that 0 ≠ I ′ ⊆ ℝn , I ′ − I ′ = ℝn and the function F : ℝn → Y is Bohr (I ′ , ρ)-almost periodic, where ρ is a binary relation on Y satisfying R(F) ⊆ D(ρ) and ρ(y) is a singleton for any y ∈ R(F). Then the function F(⋅) has the Bohr–Fourier coefficient ℳλ (F) := lim

T→+∞

1 1 ∫ e−i⟨λ,t⟩ F(t) dt = lim n ∫ e−i⟨λ,t⟩ F(t) dt T→+∞ T (2T)n KT

(23)

LT

for any λ ∈ ℝn , and the set of all points λ ∈ ℝn for which ℳλ (F) ≠ 0 is at most countable; here, for every T > 0, KT := {(t1 , t2 , . . . , tn ) ∈ ℝn : |ti | ⩽ T, i ∈ ℕn } and LT := {(t1 , t2 , . . . , tn ) ∈ [0, ∞)n : ti ⩽ T, i ∈ ℕn }. For our later purposes, it would be very useful to formulate the following result: Proposition 2.1.6. Suppose that 0 ≠ I ′ ⊆ ℝn , 0 ≠ I ⊆ ℝn , I + I ′ ⊆ I, ρ = A is a linear non-injective operator on Y and G : I×X → Y is a Bohr (ℬ, I ′ , A)-almost periodic ((ℬ, I ′ , A)uniformly recurrent) function. Let Q : I × X → N(A) be any continuous function satisfying that for each B ∈ ℬ we have lim|t|→+∞,t∈I+I ′ Q(t; x) = 0, uniformly for x ∈ B. Suppose further that the following condition holds: (D) For every t0 ∈ I ′ , for every l > 0 and for every l′ ⩾ 2l, there exists t′0 ∈ I ′ such that B(t′0 , l) ⊆ B(t0 , 2l′ ) and |t + τ| ⩾ l′ for all t ∈ I and τ ∈ B(t′0 , l). Then the function F = G + Q : I × X → Y is likewise Bohr (ℬ, I ′ , A)-almost periodic ((ℬ, I ′ , A)-uniformly recurrent). Proof. We will consider only Bohr (ℬ, I ′ , A)-almost periodic functions. Let B ∈ ℬ and ε > 0 be fixed. Then there exists l > 0 such that for each t0 ∈ I ′ there exists τ ∈ B(t0 , l)∩I ′ such that for every t ∈ I and x ∈ B, the element yt;x = A(G(t; x)) satisfies (21), with the function F(⋅; ⋅) and the number ε replaced respectively with the function G(⋅; ⋅) and the

2.1 Multi-dimensional ρ-almost periodic type functions

�

23

number ε/2. Our assumption implies the existence of a finite real number l0 (ε) > 0 such that for each t ∈ I + I ′ with |t| ⩾ l0 (ε) and x ∈ B we have ‖Q(t; x)‖Y ⩽ ε/2. Take now l′ := 2 max(l, l0 (ε)). Then the requirements of Definition 2.1.1 are satisfied with the number l replaced therein with the number 2l′ . In actual fact, let a point t0 ∈ I ′ be given. Then we can find a point t′0 ∈ I ′ in accordance with condition (D). Since G(⋅; ⋅) is Bohr (ℬ, I ′ , A)-almost periodic, we have the existence of a point τ ∈ B(t′0 , l) ∩ I ′ such that ‖G(t + τ; x) − AG(t; x)‖Y ⩽ ε/2 for all t ∈ I and x ∈ B. Due to (D), we have τ ∈ B(t0 , 2l′ ) ∩ I ′ and |t + τ| ⩾ l′ for all t ∈ I; hence, ‖Q(t + τ; x)‖Y ⩽ ε/2 for all t ∈ I and x ∈ B, which simply implies the required statement, since R(Q) ⊆ N(A). Remark 2.1.7. Condition (D) is valid in the case that I = [0, ∞) and I ′ = (0, ∞). Then, for every t0 > 0, l > 0 and l′ ⩾ 2l, we can take t0′ := t0 + 2l′ − l. It is worth noting that the statements of [409, Proposition 2.16, Proposition 2.21] can be reformulated for T-almost periodicity, where T ∈ L(Y ); concerning the second statement, we need an additional assumption that T ∈ L(Y ) is a linear isomorphism since, in this case, the estimate (21) implies ‖T −1 F(t + τ; x) − F(t; x)‖Y ⩽ ε ⋅ ‖T −1 ‖L(Y ) for all t ∈ I and x ∈ B. The proofs are very similar to those in which T = I and therefore omitted: Proposition 2.1.8. (i) Suppose that ρ = T ∈ L(Y ), 0 ≠ I ⊆ ℝn , I + I ⊆ I, I is closed, F : I × X → Y is Bohr (ℬ, T)-almost periodic and ℬ is any family of compact subsets of X. If (∀l > 0) (∃t0 ∈ I) (∃k > 0) (∀t ∈ I)(∃t′0 ∈ I)

′ ′′ (∀t′′ 0 ∈ B(t0 , l) ∩ I) t − t0 ∈ B(t0 , kl) ∩ I,

then for each B ∈ ℬ we have that the set {F(t; x) : t ∈ I, x ∈ B} is relatively compact in Y ; in particular, supt∈I;x∈B ‖F(t; x)‖Y < ∞. (ii) Suppose that ρ = T ∈ L(Y ) is a linear isomorphism, 0 ≠ I ⊆ ℝn , I + I ⊆ I, I is closed and F : I × X → Y is Bohr (ℬ, T)-almost periodic, where ℬ is a family consisting of some compact subsets of X. If the following condition holds (∃t0 ∈ I) (∀ε > 0)(∀l > 0) (∃l′ > 0) (∀t′ , t′′ ∈ I)

B(t0 , l) ∩ I ⊆ B(t0 − t′ , l′ ) ∩ B(t0 − t′′ , l′ ),

then for each B ∈ ℬ the function F(⋅; ⋅) is uniformly continuous on I × B. We continue by providing some illustrative examples: Example 2.1.9. There is no need to say that the linearity of operator T is crucial sometimes; suppose, for simplicity, that ρ = T ∈ L(Y ), I = I ′ = [0, ∞) or I = I ′ = ℝ, and X = {0} (see also [409, Proposition 2.9] and [441, Proposition 2.13]). For each t ∈ I, τ ∈ I and l ∈ ℕ, we have

24 � 2 Multi-dimensional ρ-almost periodic type functions and their Stepanov generalizations l−1

F(t + lτ) − T l F(t) = ∑ T j [F(t + (l − j)τ) − TF(t + (l − j − 1)τ)]. j=0

Hence, l−1 󵄩 󵄩 󵄩 󵄩 sup󵄩󵄩󵄩F(t + lτ) − T l F(t)󵄩󵄩󵄩Y ⩽ l(1 + ‖T‖) sup󵄩󵄩󵄩F(t + τ) − TF(t)󵄩󵄩󵄩Y , t∈I

t∈I

which implies that the function F(⋅) is T l -almost periodic (T l -uniformly recurrent), provided that F(⋅) is T-almost periodic (T-uniformly recurrent). In particular, the function F(⋅) is almost periodic (uniformly recurrent), provided that F(⋅) is T-almost periodic (T-uniformly recurrent) and there exists a positive integer l ∈ ℕ such that T l = I. Example 2.1.10. The following example is taken from [431], where we have considered case c = 1, only; this example shows that it is slightly redundant to assume that I ′ ⊆ I in Definition 2.1.1. Suppose that L > 0 is a fixed real number as well as that the function t 󳨃→ (f (t), g(t)), t ∈ ℝ is c-almost periodic. Set I := {(x, y) ∈ ℝ2 : |x − y| ⩾ L}, I ′ := {(τ, τ) : τ ∈ ℝ} and u(x, y) :=

f (x) + g(y) , x−y

(x, y) ∈ I.

Then I + I ′ ⊆ I, but I ′ is not a subset of I. Furthermore, if ε > 0 is given and τ > 0 is a common (ε, c)-period of the functions f (⋅) and g(⋅), then we have: 󵄩󵄩 󵄩 ‖f (x + τ) − cf (x)‖ + ‖g(y + τ) − cg(y)‖ 󵄩󵄩u(x + τ, y + τ) − cu(x, y)󵄩󵄩󵄩 ⩽ |x − y| ⩽ 2ε/L,

(x, y) ∈ I.

This implies that the function u(⋅, ⋅) is Bohr (I ′ , c)-almost periodic. Observe, finally, that under some regularity conditions for the functions f (⋅) and g(⋅), the function u(⋅, ⋅) is a solution of the partial differential equation uxy −

uy ux + = 0. x−y x−y

Example 2.1.11. The conclusions established in [441, Example 2.5, Example 2.12] (cf. also [194, Example 2.13, Example 2.15]) for c-almost periodicity can be formulated for T-almost periodicity, where T ∈ L(Y ), providing certain additional assumptions. For simplicity, let us consider the concrete situation of [441, Example 2.5(i)]. Suppose that Fj : X → Y is a continuous function, for each B ∈ ℬ we have supx∈B ‖Fj (x)‖Y < ∞, and t

t

the Z-valued mapping t 󳨃→ (∫0 f1 (s) ds, . . . , ∫0 fn (s) ds), t ⩾ 0 is bounded and T-almost periodic (1 ⩽ j ⩽ n). Suppose further that the multiplication ⋅ : Z × Y → Y is defined and satisfies:

2.1 Multi-dimensional ρ-almost periodic type functions

�

25

(i) there exists a finite real constant c > 0 such that ‖zy‖Y ⩽ c‖z‖Z ‖y‖Y for all z ∈ Z and y ∈ Y; (ii) z1 y − z2 y = (z1 − z2 )y for all z1 , z2 ∈ Z and y ∈ Y ; (iii) zy1 − zy2 = z(y1 − y2 ) for all z ∈ Z and y1 , y2 ∈ Y . Set n

tj+1

F(t1 , . . . , tn+1 ; x) := ∑ ∫ fj (s) ds ⋅ Fj (x) j=1 t j

for all x ∈ X and tj ⩾ 0, 1 ⩽ j ⩽ n.

Arguing as in [194, Example 2.13(i)], we may deduce that the mapping F : [0, ∞)n+1 ×X → Y is Bohr (ℬ, T)-almost periodic. The use of binary relation ρ in Definition 2.1.1 suggests a very general way of approaching to many known classes of almost periodic functions; before we go any further, we would like to note that this general approach has some obvious unpleasant consequences because, under the general requirements of Definition 2.1.1, any continuous function F(⋅; ⋅) is always Bohr (ℬ, I ′ , ρ)-almost periodic, provided that R(F) × R(F) ⊆ ρ (in particular, it is very redundant to assume any kind of boundedness of function F(⋅; ⋅) in Definition 2.1.1; see [409, Proposition 2.2] and [437, Proposition 2.8] for some particular results obtained in this direction). Therefore, given two sets 0 ≠ I ′ ⊆ ℝn , 0 ≠ I ⊆ ℝn satisfying I + I ′ ⊆ I and a continuous function F : I × X → Y , it is natural to introduce the following non-empty sets: 𝒜I ′ ,I,F := {ρ ⊆ Y × Y ; F(⋅; ⋅) is Bohr (ℬ, I , ρ)-almost periodic} ′

and ℬI ′ ,I,F := {ρ ⊆ Y × Y ; F(⋅; ⋅) is (ℬ, I , ρ)-uniformly recurrent}. ′

Clearly, we have 0 ≠ 𝒜I ′ ,I,F ⊆ ℬI ′ ,I,F ; further on, the assumptions ρ ∈ 𝒜I ′ ,I,F (ρ ∈ ℬI ′ ,I,F ) and ρ ⊆ ρ′ imply ρ′ ∈ 𝒜I ′ ,I,F (ρ′ ∈ ℬI ′ ,I,F ). The set 𝒜I ′ ,I,F (ℬI ′ ,I,F ), equipped with the relation of set inclusion, becomes a partially ordered set; for the sake of brevity, we will not consider the minimal elements and the least elements (if exist) of these partially ordered sets here. The interested reader may try to construct some examples concerning this issue. Now, we will state and prove some parts of the following theorem for Bohr (ℬ, I ′ , ρ)almost periodic type functions: Theorem 2.1.12. Suppose that 0 ≠ I ′ ⊆ ℝn , 0 ≠ I ⊆ ℝn , F : I × X → Y is Bohr (ℬ, I ′ , ρ)almost periodic ((ℬ, I ′ , ρ)-uniformly recurrent), ρ is a binary relation on Y and I + I ′ ⊆ I. Then the following holds:

26 � 2 Multi-dimensional ρ-almost periodic type functions and their Stepanov generalizations (i) Set σ := {(‖y1 ‖Y , ‖y2 ‖Y ) | ∃t ∈ I ∃x ∈ X : y1 = F(t; x) and y2 ∈ ρ(y1 )}. Then the function ‖F(⋅; ⋅)‖Y is Bohr (ℬ, I ′ , σ)-almost periodic ((ℬ, I ′ , σ)-uniformly recurrent). (ii) Suppose that λ ∈ ℂ ∖ {0}. Set ρλ := {λ(y1 , y2 ) | ∃t ∈ I ∃x ∈ X : y1 = F(t; x) and y2 ∈ ρ(y1 )}. Then the function λF(⋅; ⋅) is Bohr (ℬ, I ′ , ρλ )-almost periodic ((ℬ, I ′ , ρλ )-uniformly recurrent). (iii) Suppose a ∈ ℝn and x0 ∈ X. Define G : (I − a) × X → Y by G(t; x) := F(t + a; x + x0 ), t ∈ I − a, x ∈ X, as well as ℬx0 := {−x0 + B : B ∈ ℬ}, Ia′ := I ′ and ρa,x0 := {(y1 , y2 ) | ∃t ∈ I − a ∃x ∈ X : y1 = F(t + a; x + x0 ) and y2 ∈ ρ(y1 )}. Then the function G(⋅; ⋅) is Bohr (ℬx0 , Ia′ , ρa,x0 )-almost periodic ((ℬx0 , Ia′ , ρa,x0 )-uniformly recurrent). (iv) Suppose that a ∈ ℝ ∖ {0} and b ∈ ℂ ∖ {0}. Define the function G : (I/a) × X → Y by G(t; x) := F(at; bx), t ∈ I/a, x ∈ X, as well as ℬb := {b−1 B : B ∈ ℬ}, Ia′ := I ′ /a and ρa,b := {(y1 , y2 ) | ∃t ∈ I/a ∃x ∈ X : y1 = F(at; bx) and y2 ∈ ρ(y1 )}. Then the function G(⋅; ⋅) is Bohr (ℬb , Ia′ , ρa,b )-almost periodic ((ℬb , Ia′ , ρa,b )-uniformly recurrent). (v) Assume that for each B ∈ ℬ there exists εB > 0 such that the sequence (Fk (⋅; ⋅)) of Bohr (ℬ, I ′ , ρ)-almost periodic functions ((ℬ, I ′ , ρ)-uniformly recurrent functions) converges uniformly to a function F(⋅; ⋅) on the set B∘ ∪ ⋃x∈𝜕B B(x, εB ). Then the function F(⋅; ⋅) is Bohr (ℬ, I ′ , ρ)-almost periodic ((ℬ, I ′ , ρ)-uniformly recurrent), provided that D(ρ) is a closed subset of Y and ρ is continuous in the following sense: (Cρ ) For each ε > 0 there exists δ > 0 such that, for every y1 , y2 ∈ Y with ‖y1 −y2 ‖Y < δ, we have ‖z1 − z2 ‖Y < ε/3 for every z1 ∈ ρ(y1 ) and z2 ∈ ρ(y2 ). Proof. The proofs of parts (i)–(iv) are almost trivial and therefore omitted; let us show (v). Due to the proofs of [194, Proposition 2.7, Proposition 2.8], it follows that the function F(⋅; ⋅) is continuous. Fix ε > 0 and B ∈ ℬ. Since D(ρ) is a closed subset of Y and the sequence (Fk (⋅; ⋅)) converges uniformly to a function F(⋅; ⋅) on the set B∘ ∪ ⋃x∈𝜕B B(x, εB ), we have that F(t; x) ∈ D(ρ) for all t ∈ I and x ∈ X. Further on, let a number δ > 0 be chosen in accordance with the continuity of relation ρ. For ε0 ≡ min(ε/3, δ), we can find a positive integer k ∈ ℕ such that ‖Fk (t; x) − F(t; x)‖Y < ε0 . Let t ∈ I and x ∈ B be fixed. Let ykt;x ∈ ρ(Fk (t; x)) be the element determined from the definition of Bohr (ℬ, I ′ , ρ)-almost periodicity of function Fk (⋅; ⋅), with the number ε replaced with the number ε/3 therein. Pick up now an arbitrary element yt;x from ρ(F(t; x)). Then we have (a point τ ∈ I ′ satisfies the requirements in Definition 2.1.1 for the function Fk (⋅; ⋅)): 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 k 󵄩 󵄩󵄩F(t + τ; x) − yt;x 󵄩󵄩󵄩Y ⩽ 󵄩󵄩󵄩F(t + τ; x) − Fk (t + τ; x)󵄩󵄩󵄩Y + 󵄩󵄩󵄩Fk (t + τ; x) − yt;x 󵄩󵄩󵄩Y 󵄩 󵄩 + 󵄩󵄩󵄩yt;x − ykt;x 󵄩󵄩󵄩Y ⩽ ε0 + (ε/3) + (ε/3) ⩽ ε. This completes the proof of (v). The proof of following proposition is simple and therefore omitted: Proposition 2.1.13. Suppose that 0 ≠ I ′ ⊆ ℝn , 0 ≠ I ⊆ ℝn , F : I × X → Y is a continuous function, ρ is a binary relation on Y and I + I ′ ⊆ I. If F : I × X → Y is Bohr (ℬ, I ′ , ρ)-almost

2.1 Multi-dimensional ρ-almost periodic type functions

� 27

periodic/(ℬ, I ′ , ρ)-uniformly recurrent, and ϕ : Y → Z is uniformly continuous on the set R(F) ∪ ρ(R(F)), then ϕ ∘ F : I × X → Z is Bohr (ℬ, I ′ , ϕ ∘ ρ)-almost periodic/(ℬ, I ′ , ϕ ∘ ρ)uniformly recurrent. The supremum formula for (ℬ, I ′ , ρ)-uniformly recurrent functions reads as follows: Proposition 2.1.14. Suppose that 0 ≠ I ′ ⊆ ℝn , 0 ≠ I ⊆ ℝn , I + I ′ ⊆ I and F : I × X → Y is a (ℬ, I ′ , ρ)-uniformly recurrent function, where ρ = T ∈ L(Y ) is a linear isomorphism. Then for each real number a > 0 we have: 󵄩 󵄩 sup 󵄩󵄩󵄩F(t; x)󵄩󵄩󵄩Y ⩽

t∈I,x∈B

sup

t∈I+I ′ ,|t|⩾a,x∈B

󵄩󵄩 −1 󵄩 󵄩󵄩T F(t; x)󵄩󵄩󵄩Y ,

(24)

and for each x ∈ X we have 󵄩 󵄩 sup󵄩󵄩󵄩F(t; x)󵄩󵄩󵄩Y ⩽ t∈I

sup

t∈I+I ′ ,|t|⩾a

󵄩󵄩 −1 󵄩 󵄩󵄩T F(t; x)󵄩󵄩󵄩Y ,

so that the function F(⋅; x) is identically equal to zero, provided that the function F(⋅; ⋅) is (ℬ, I ′ , ρ)-uniformly recurrent and lim|t|→+∞,t∈I+I ′ F(t; x) = 0. Proof. Let a > 0, ε > 0 and B ∈ ℬ be given. Then there exists a sequence (τ k ) in I ′ such that limk→+∞ |τ k | = +∞ and for every t ∈ I and x ∈ B, we have that (22) holds with yt;x = TF(t; x). This implies the existence of an integer k ∈ ℕ such that 󵄩󵄩 󵄩 󵄩󵄩F(t + τ k ; x) − TF(t; x)󵄩󵄩󵄩Y ⩽ ε,

t ∈ I, x ∈ B.

(25)

The operator T is a linear isomorphism, so that (25) immediately implies 󵄩󵄩 −1 󵄩 󵄩 −1 󵄩 󵄩󵄩T F(t + τ k ; x) − F(t; x)󵄩󵄩󵄩Y ⩽ ε ⋅ 󵄩󵄩󵄩T 󵄩󵄩󵄩,

t ∈ I, x ∈ B

and 󵄩󵄩 󵄩 󵄩 −1 󵄩 󵄩 −1 󵄩 󵄩󵄩F(t; x)󵄩󵄩󵄩Y ⩽ 󵄩󵄩󵄩T F(t + τ k ; x)󵄩󵄩󵄩Y + ε ⋅ 󵄩󵄩󵄩T 󵄩󵄩󵄩,

t ∈ I, x ∈ B.

Since ε > 0 was arbitrary, this yields 󵄩󵄩 󵄩 󵄩 −1 󵄩 󵄩󵄩F(t; x)󵄩󵄩󵄩Y ⩽ 󵄩󵄩󵄩T F(t + τ k ; x)󵄩󵄩󵄩Y ,

t ∈ I, x ∈ B

and (24). The remainder of proof, for a fixed element x ∈ X, follows from the same arguments and the existence of a set B ∈ ℬ such that x ∈ B. Regarding the convolution invariance of Bohr (ℬ, I ′ , ρ)-almost periodic ((ℬ, I ′ , ρ)uniformly recurrent) functions, we will clarify the following result:

28 � 2 Multi-dimensional ρ-almost periodic type functions and their Stepanov generalizations Theorem 2.1.15. Suppose that h ∈ L1 (ℝn ) and F : ℝn × X → Y is a continuous function satisfying that for each B ∈ ℬ there exists a finite real number εB > 0 such that supt∈ℝn ,x∈B⋅ ‖F(t, x)‖Y < +∞, where B⋅ ≡ B∘ ∪ ⋃x∈𝜕B B(x, εB ). Suppose further that ρ = A is a closed linear operator on Y satisfying that: (B1) For each t ∈ ℝn and x ∈ B, the function s 󳨃→ AF(t − s; x), s ∈ ℝn is Bochner integrable. Then the function (h ∗ F)(t; x) := ∫ h(σ)F(t − σ; x) dσ,

t ∈ ℝn , x ∈ X

(26)

ℝn

is well defined and for each B ∈ ℬ we have supt∈ℝn ,x∈B⋅ ‖(h∗F)(t; x)‖Y < +∞; furthermore, if F(⋅; ⋅) is Bohr (ℬ, I ′ , A)-almost periodic ((ℬ, I ′ , A)-uniformly recurrent), then the function (h ∗ F)(⋅; ⋅) is Bohr (ℬ, I ′ , A)-almost periodic ((ℬ, I ′ , A)-uniformly recurrent). Proof. We will consider only Bohr (ℬ, I ′ , A)-almost periodic functions. The function (h ∗ F)(⋅; ⋅) is well-defined and supt∈ℝn ,x∈B⋅ ‖(h ∗ F)(t; x)‖Y < +∞ for all B ∈ ℬ. The continuity of this function at the fixed point (t0 ; x0 ) ∈ ℝn × X follows from the existence of a set B ∈ ℬ such that x0 ∈ B, the assumption supt∈ℝn ,x∈B⋅ ‖F(t; x)‖Y < +∞ and the dominated convergence theorem. Let ε > 0 and B ∈ ℬ be given. Then there exists l > 0 such that for each t0 ∈ I ′ there exists τ ∈ B(t0 , l) ∩ I ′ such that for every t ∈ ℝn and x ∈ B, there exists an element yt;x = A(F(t; x)) such that (21) holds. Due to Lemma 1.1.1 and condition (B1), for every t ∈ ℝn and x ∈ B, we have that zt,x := A((h ∗ F)(t; x)) = ∫ℝn h(s)A(F(t − s; x)) ds. Therefore, we have 󵄩󵄩 󵄩 󵄩󵄩(h ∗ F)(t + τ; x) − zt,x 󵄩󵄩󵄩Y 󵄨 󵄨 󵄩 󵄩 ⩽ ∫ 󵄨󵄨󵄨h(σ)󵄨󵄨󵄨 ⋅ 󵄩󵄩󵄩F(t + τ − s; x) − A(F(t − s; x))󵄩󵄩󵄩Y ds ℝn

⩽ ε ⋅ ‖h‖L1 (ℝn ) ,

t ∈ ℝn , x ∈ B,

which completes the proof. Remark 2.1.16. The requirements of Theorem 2.1.15 are satisfied if A ∈ L(X). For the sake of completeness, we will provide all relevant details of the following result, which has numerous important applications in the analysis of the existence and uniqueness of time A-almost periodic solutions for various classes of abstract (degenerate) Volterra integro-differential equations; the statement can be extended for the corresponding Stepanov classes: Theorem 2.1.17. Let 0 ≠ I ′ ⊆ ℝn , let A be a closed linear operator on X, and let (R(t))t>0 ⊆ L(X) be a strongly continuous operator family such that R(t)A ⊆ AR(t) for all t ∈ ℝn and ∫(0,∞)n ‖R(t)‖ dt < ∞. If f : ℝn → X is a bounded (I ′ , A)-almost periodic function, resp.

2.1 Multi-dimensional ρ-almost periodic type functions

�

29

bounded (I ′ , A)-uniformly recurrent function, and the function Af : ℝn → X is well-defined and bounded, then the function F : ℝn → X, given by (20), is well-defined, bounded and (I ′ , A)-almost periodic, resp. well-defined, bounded and (I ′ , A)-uniformly recurrent. Proof. We will consider only (I ′ , A)-almost periodic functions. It is clear that the function F(⋅) is well-defined and bounded since F(t) :=

∫ R(s)f (t − s) ds,

t ∈ ℝn ,

(0,∞)n

as well as ∫(0,∞)n ‖R(t)‖ dt < ∞ and the function f (⋅) is bounded. Similarly, Lemma 1.1.1, our assumptions R(t)A ⊆ AR(t) for all t ∈ ℝn and the boundedness of function Af (⋅) together imply that AF(t) = ∫(0,∞)n R(s)Af (t − s) ds for all t ∈ ℝn . Furthermore, for every τ ∈ I ′ and t ∈ ℝn , we have

󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄩 󵄩󵄩F(t + τ) − AF(t)󵄩󵄩󵄩 = 󵄩󵄩󵄩 ∫ R(s)[f (t + τ − s) − Af (t − s)] ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 n (0,∞)

⩽

󵄩 󵄩 󵄩 󵄩 ∫ 󵄩󵄩󵄩R(s)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩f (t + τ − s) − Af (t − s)󵄩󵄩󵄩 ds.

(0,∞)n

Keeping in mind the corresponding definition of (I ′ , A)-almost periodicity, the above calculation simply completes the proof of theorem. Suppose that F : I × X → Y and G : I × Y → Z are given continuous functions. Then we define the multi-dimensional Nemytskii operator W : I × X → Z by W (t; x) := G(t; F(t; x)),

t ∈ I, x ∈ X.

(27)

The following composition principle slightly generalizes the statement of [441, Theorem 2.19]; keeping in mind the corresponding definitions, the proof is almost the same as the proof of this theorem and therefore omitted: Theorem 2.1.18. Suppose that the functions F : I × X → Y and G : I × Y → Z are continuous, 0 ≠ I ′ ⊆ ℝn , 0 ⊆ I ⊆ ℝn and σ is a binary relation on Z. (i) Suppose further that for every B ∈ ℬ and ε > 0, there exists l > 0 such that for each t0 ∈ I ′ there exists τ ∈ B(t0 , l) ∩ I ′ such that, for every t ∈ I and x ∈ B, there exist elements yt;x ∈ ρ(F(t; x)) and zt;x ∈ σ(W (t; x)) such that (21) holds, as well as that there exists a finite real constant L > 0 such that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩G(t + τ; F(t + τ; x)) − G(t + τ; yt,x )󵄩󵄩󵄩Z ⩽ L󵄩󵄩󵄩F(t + τ; x) − yt;x 󵄩󵄩󵄩Y

(28)

󵄩󵄩 󵄩 󵄩󵄩G(t + τ; yt,x ) − zt;x 󵄩󵄩󵄩Z ⩽ ε.

(29)

and

30 � 2 Multi-dimensional ρ-almost periodic type functions and their Stepanov generalizations Then the function W (⋅; ⋅), given by (27), is Bohr (ℬ, I ′ , σ)-almost periodic. (ii) Suppose that for every B ∈ ℬ, there exists a sequence (τ k ) in I ′ such that limk→+∞ |τ k | = +∞ and that for every t ∈ I and x ∈ B, there exist elements yt;x ∈ ρ(F(t; x)) and zt;x ∈ σ(W (t; x)) such that (22) holds as well as that for each k ∈ ℕ the equations (28)–(29) hold, with the number τ replaced with the number τk therein. Then the function W (⋅; ⋅), given by (27), is (ℬ, I ′ , σ)-uniformly recurrent.

2.1.1 T-almost periodic type functions in finite-dimensional spaces In this subsection, we will clarify some basic results concerning T-almost periodic type functions of form F : I → ℂk , where k ∈ ℕ and I = [0, ∞) or I = ℝ. We will also provide several illustrative examples in this direction. First of all, note that the argumentation contained in the proof of [409, Proposition 2.6] enables one to deduce the following result (let us only point out that if T is a linear isomorphism, then the estimate ‖f (t)‖ ⩽ ‖T −1 ‖ ⋅ ‖Tf (t)‖, t ∈ I can be used): Proposition 2.1.19. Suppose that ρ = T ∈ L(Y ), I = [0, ∞) or I = ℝ, and I ′ = [0, ∞). If the function F : I → Y is (I ′ , T)-uniformly recurrent and F ≠ 0, then ‖T‖L(Y ) ⩾ 1; furthermore, if T is a linear isomorphism, then ‖T −1 ‖L(Y ) ⩾ 1. For the linear continuous operators which are not scalar multiples of the identity operator, case in which ‖T‖L(Y ) > 1 is possible: Example 2.1.20. Suppose that Y := ℂ2 is equipped with the norm ‖(z1 , z2 )‖Y := √|z1 |2 + |z2 |2 (z1 , z2 ∈ ℂ), a ∈ ℂ satisfies |a| > 1, the function u : [0, ∞) → ℂ is almost periodic, F(t) := (u(t), u(t)), t ⩾ 0, and T := [

a a

1−a ]. 1−a

(30)

Then the function F(⋅) is T-almost periodic, but ‖T‖L(Y ) > 1. Furthermore, the assumption that T is a linear isomorphism does not imply ‖T −1 ‖L(Y ) = 1; consider the same pivot space Y , the same function F(⋅) and the matrix 2 T := [ 1

−1 ]. 0

Then we have both ‖T‖L(Y ) > 1 and ‖T −1 ‖L(Y ) > 1. It is clear that Proposition 2.1.19 provides a large class of complex matrices T = A of format k ×k (k ∈ ℕ) such that the only (I ′ , A)-uniformly recurrent function F : [0, ∞) → ℂk is the zero function, actually. Now, we will state and prove the following result:

2.1 Multi-dimensional ρ-almost periodic type functions

�

31

Proposition 2.1.21. Suppose that k ∈ ℕ, T = A = [aij ] is a non-zero complex matrix of format k × k, I = ℝ or I = [0, ∞), I ′ ⊆ ℝ, I + I ′ ⊆ I and the function F : I → ℂk is Bohr (I ′ , A)-almost periodic (bounded (I ′ , A)-uniformly recurrent). If F = (F1 , . . . , Fk ), then there exists a non-trivial linear combination of functions F1 , . . . , Fk which is Bohr (I ′ , I)-almost periodic (bounded (I ′ , I)-uniformly recurrent). Proof. We will prove the statement only for (I ′ , A)-almost periodicity. Let ε > 0 be given. Then there exists a finite real number l > 0 such that, for every t0 ∈ I, there exists a point τ ∈ B(t0 , l) ∩ I ′ such that 󵄨󵄨 󵄨 󵄨󵄨Fi (t + τ) − [ai1 F1 (t) + ⋅ ⋅ ⋅ + aik Fk (t)]󵄨󵄨󵄨 ⩽ ε

for all t ∈ I and i ∈ ℕk .

(31)

Suppose that λ ∈ σp (A)∖{0}, where σp (A) denotes the point spectrum of A; such a number λ exists since A ≠ 0. Then there exists a tuple (α1 , . . . , αk ) ∈ ℝk ∖ {(0, . . . , 0)} such that α1 ai1 + α2 ai2 + ⋅ ⋅ ⋅ + αi (aii − λ) + ⋅ ⋅ ⋅ + αk aik = 0 for all i ∈ ℕk ; (α1 , . . . , αk ) is, in fact, an eigenvector of matrix A, which corresponds to the eigenvalue λ. Multiplying (31) with αi , and adding all obtained inequalities for i = 1, . . . , k, we get that the function u(t) := α1 F1 (t) + ⋅ ⋅ ⋅ + αk Fk (t), t ∈ I satisfies |u(t + τ) − λu(t)| ⩽ ε(|α1 | + ⋅ ⋅ ⋅ + |αk |) for all t ∈ I. If |λ| ≠ 1, then [409, Proposition 2.6] immediately gives u(t) ≡ 0 and the proof is completed. If |λ| = 1, then the result simply follows by applying [409, Corollary 2.10, Proposition 2.11]. Using the conclusion given directly after Corollary 2.1.4, we have that the terms “bounded (I ′ , A)-uniformly recurrent” and “bounded (I ′ , I)-uniformly recurrent” can be replaced with the terms “(I ′ , A)-uniformly recurrent” and “(I ′ , I)-uniformly recurrent” in the case that I = ℝ, when the function F(⋅) is (I ′ − I ′ , I)-almost periodic ((bounded) (I ′ − I ′ , I)-uniformly recurrent). If I = [0, ∞), A is invertible and F(⋅) is uniformly continuous, then the function F(⋅) is ((I ′ − I ′ ) ∩ [0, ∞), I)-almost periodic ((bounded) ((I ′ − I ′ ) ∩ [0, ∞), I)-uniformly recurrent), which follows from an application of Corollary 2.1.4 and Theorem 2.1.30 below. If the matrix A is not invertible and I = [0, ∞), then the case in which any of the functions F1 (⋅), . . . , Fk (⋅) is not (I ′ , I)-almost periodic (bounded (I ′ , I)-uniformly recurrent) can occur, which can be easily shown by using Proposition 2.1.6 (see also Remark 2.1.7), so that there exists an A-almost periodic function (bounded A-uniformly recurrent function) F : [0, ∞) → ℂk , which is not almost periodic (uniformly recurrent): Example 2.1.22. Let Y := ℂ2 , let a, u : [0, ∞) → ℂ and F(⋅) := (u(⋅), u(⋅)) possess the same meaning as in Example 2.1.20. Further on, let I = [0, ∞), I ′ = (0, ∞), and let the matrix T = A be given by (30). Then N(A) = {(α, β) ∈ ℂ2 : αa + β(1 − a) = 0}. Suppose that q = (q1 , q2 ) : [0, ∞) → N(A) is any continuous function tending to zero as the norm of the argument goes to plus infinity. Then Proposition 2.1.6 implies that the function t 󳨃→ (u(t) + q1 (t), u(t) + q2 (t)), t ⩾ 0 is also (I ′ , A)-almost periodic. It is clear that this

32 � 2 Multi-dimensional ρ-almost periodic type functions and their Stepanov generalizations function cannot be almost periodic in the case that some of the functions q1 (⋅) or q2 (⋅) are not identically equal to the zero function.

2.1.2 𝔻-asymptotically Bohr (ℬ, I ′ , ρ)-almost periodic type functions We need the following definition from [194]: Definition 2.1.23. Suppose that 𝔻 ⊆ I ⊆ ℝn and the set 𝔻 is unbounded. By C0,𝔻,ℬ (I ×X : Y ) we denote the vector space consisting of all continuous functions Q : I × X → Y such that, for every B ∈ ℬ, we have limt∈𝔻,|t|→+∞ Q(t; x) = 0, uniformly for x ∈ B. If X = {0}, then we abbreviate C0,𝔻,ℬ (I × X : Y ) to C0,𝔻,ℬ (I : Y ); furthermore, if 𝔻 = I, then we omit the term “𝔻” from the notation. Now we are able to introduce the following notion: Definition 2.1.24. Suppose that 𝔻 ⊆ I ⊆ ℝn , the set 𝔻 is unbounded, 0 ≠ I ′ ⊆ ℝn , 0 ≠ I ⊆ ℝn , F : I × X → Y is a continuous function, ρ is a binary relation on Y and I +I ′ ⊆ I. Then we say that the function F(⋅) is (strongly) 𝔻-asymptotically Bohr (ℬ, I ′ , ρ)almost periodic, resp. (strongly) 𝔻-asymptotically (ℬ, I ′ , ρ)-uniformly recurrent, if and only if there exists a Bohr (ℬ, I ′ , ρ)-almost periodic function, resp. (ℬ, I ′ , ρ)-uniformly recurrent function, (F0 : ℝn × X → Y ) F0 : I × X → Y and a function Q ∈ C0,𝔻,ℬ (I × X : Y ) such that F(t; x) = F0 (t; x) + Q(t; x), t ∈ I, x ∈ X. The functions F0 (⋅; ⋅) and Q(⋅; ⋅) are usually called the principal part of F(⋅; ⋅) and the corrective (ergodic) part of F(⋅; ⋅), respectively. We will not reconsider here [441, Theorem 2.22] for 𝔻-asymptotically Bohr (ℬ, I ′ , ρ)almost periodic functions and 𝔻-asymptotically (ℬ, I ′ , ρ)-uniformly recurrent functions. In the following result, which is applicable to the general binary relations satisfying conditions clarified in Theorem 2.1.12(v), we follow a new approach based on the use of supremum formula (the argumentation contained in the proof of [259, Theorem 4.29] can be used only for the binary relations ρ = T ∈ L(Y ) which are linear isomorphisms; see e. g., [194, Proposition 2.27(ii)] and [441, Proposition 2.24(ii)], where we have also assumed that I ′ = I): Theorem 2.1.25. Suppose that ρ is a binary relation on Y satisfying that D(ρ) is a closed subset of Y and condition (Cρ ). Suppose, further, that for each integer j ∈ ℕ the function Fj (⋅; ⋅) = Gj (⋅; ⋅)+Qj (⋅; ⋅) is I-asymptotically Bohr (ℬ, I ′ , ρ)-almost periodic (I-asymptotically (ℬ, I ′ , ρ)-uniformly recurrent), where Gj (⋅; ⋅) is Bohr (ℬ, I ′ , ρ)-almost periodic ((ℬ, I ′ , ρ)uniformly recurrent) and Qj ∈ C0,I,ℬ (I × X : Y ). Let for each B ∈ ℬ there exist εB > 0 such that the sequence (Fj (⋅; ⋅)) converges uniformly to a function F(⋅; ⋅) on the set B∘ ∪ ⋃x∈𝜕B B(x, εB ), and let Qj ∈ C0,I,ℬ∘ (I × X : Y ), where ℬ∘ ≡ {B∘ : B ∈ ℬ}. If for each natural number m, k ∈ ℕ the function Gk (⋅; ⋅) − Gm (⋅; ⋅) satisfies the following supremum formula:

2.1 Multi-dimensional ρ-almost periodic type functions

� 33

(S1) for every a > 0, we have 󵄩 󵄩 sup 󵄩󵄩󵄩Gk (t; x) − Gm (t; x)󵄩󵄩󵄩Y =

t∈I,x∈B∘

sup

t∈I,|t|⩾a, x∈B∘

󵄩󵄩 󵄩 󵄩󵄩Gk (t; x) − Gm (t; x)󵄩󵄩󵄩Y ,

then F(⋅; ⋅) is I-asymptotically Bohr (ℬ∘ , I ′ , ρ)-almost periodic (I-asymptotically (ℬ∘ , I ′ , ρ)uniformly recurrent). Proof. Let ε > 0 and B ∈ ℬ be given. Then there exists a natural number k0 ∈ ℕ such that for every natural numbers k, m ∈ ℕ with min(k, m) ⩾ k0 , we have 󵄩 󵄩 sup 󵄩󵄩󵄩Fk (t; x) − Fm (t; x)󵄩󵄩󵄩Y < ε/3.

(32)

t∈I, x∈B∘

Let k, m ∈ ℕ with min(k, m) ⩾ k0 be fixed. Then there exists a finite real number ak,m > 0 such that sup

t∈I,|t|⩾ak,m , x∈B∘

󵄩󵄩 󵄩 󵄩󵄩Qk (t; x)󵄩󵄩󵄩Y ⩽ ε/3

and

sup

t∈I,|t|⩾ak,m , x∈B∘

󵄩󵄩 󵄩 󵄩󵄩Qm (t; x)󵄩󵄩󵄩Y ⩽ ε/3.

(33)

Keeping in mind (32)–(33), we get: sup

t∈I,|t|⩾ak,m , x∈B∘

⩽ (ε/3) + ⩽ (ε/3) +

󵄩󵄩 󵄩 󵄩󵄩Gk (t; x) − Gm (t; x)󵄩󵄩󵄩Y sup

t∈I,|t|⩾ak,m ,

x∈B∘

sup

t∈I,|t|⩾ak,m , x∈B∘

󵄩󵄩 󵄩 󵄩󵄩Qk (t; x) − Qm (t; x)󵄩󵄩󵄩Y 󵄩󵄩 󵄩 󵄩󵄩Qk (t; x)󵄩󵄩󵄩Y +

sup

t∈I,|t|⩾ak,m , x∈B∘

󵄩󵄩 󵄩 󵄩󵄩Qm (t; x)󵄩󵄩󵄩Y ⩽ ε.

Using this estimate and the supremum formula (S1), we get that 󵄩 󵄩 sup 󵄩󵄩󵄩Gk (t; x) − Gm (t; x)󵄩󵄩󵄩Y ⩽ ε.

t∈I, x∈B∘

(34)

Therefore, the sequence (Gk (t; x)) is Cauchy and therefore convergent for each t ∈ I and x ∈ X. If we denote by G(⋅; ⋅) the corresponding limit function, then (34) yields 󵄩 󵄩 sup 󵄩󵄩󵄩Gk (t; x) − G(t; x)󵄩󵄩󵄩Y ⩽ ε.

t∈I, x∈B∘

Applying Theorem 2.1.12(v), we get that the function G(⋅; ⋅) is Bohr (ℬ∘ , I ′ , ρ)-almost periodic ((ℬ∘ , I ′ , ρ)-uniformly recurrent). Hence, the sequence (Qk (⋅; ⋅) = Fk (⋅; ⋅)−Gk (⋅; ⋅)) converges to a function Q(⋅; ⋅), uniformly on I × B∘ . This simply implies Q ∈ C0,I,ℬ∘ (I × X : Y ), finishing the proof. Set It := (−∞, t1 ]×(−∞, t2 ]×⋅ ⋅ ⋅×(−∞, tn ] and 𝔻t := It ∩𝔻 for any t = (t1 , t2 , . . . , tn ) ∈ ℝn . The following result extends the statement of [441, Proposition 2.23] in the case that

34 � 2 Multi-dimensional ρ-almost periodic type functions and their Stepanov generalizations X = Y ; the proof follows from Theorem 2.1.15 and the argumentation contained in the proof of [194, Proposition 2.56], where we have assumed I ′ = I: Proposition 2.1.26. Let A be a closed linear operator on X, and let (R(t))t>0 ⊆ L(X) be a strongly continuous operator family such that R(t)A ⊆ AR(t) for all t ∈ ℝn and ∫(0,∞)n ‖R(t)‖ dt < ∞. Suppose, further, that g : ℝn → X is a bounded (I ′ , A)-almost

periodic function, resp. bounded (I ′ , A)-uniformly recurrent function, and the function Ag : ℝn → X is well defined and bounded, q ∈ C0,𝔻 (I : X), f := g + q, lim

|t|→∞,t∈𝔻

󵄩 󵄩 ∫ 󵄩󵄩󵄩R(t − s)󵄩󵄩󵄩 ds = 0

It ∩𝔻c

and for each r > 0 we have lim

󵄩󵄩 󵄩 󵄩󵄩R(t − s)󵄩󵄩󵄩 ds = 0.

∫

|t|→∞,t∈𝔻

𝔻t ∩B(0,r)

Then the function F(t) := ∫ R(t − s)f (s) ds,

t∈I

𝔻t

is strongly 𝔻-asymptotically (I ′ , A)-almost periodic, resp. strongly 𝔻-asymptotically (I ′ , A)-uniformly recurrent; furthermore, the principal part of F(⋅) is bounded (I ′ , A)almost periodic, resp. bounded (I ′ , A)-uniformly recurrent. If 𝔻 = [α1 , ∞) × [α2 , ∞) × ⋅ ⋅ ⋅ × [αn , ∞) for some real numbers α1 , α2 , . . . , αn , then α 𝔻t = [α1 , t1 ] × [α2 , t2 ] × ⋅ ⋅ ⋅ × [αn , tn ]. In this case, the function F(t) = ∫t R(t − s)f (s) ds, t ∈ I is strongly 𝔻-asymptotically (I ′ , A)-almost periodic, resp. strongly 𝔻-asymptotically (I ′ , A)-uniformly recurrent, where we accept the notation α

t1 t2

tn

∫⋅ = ∫∫⋅⋅⋅∫. t

α1 α2

αn

The following definition is also meaningful: Definition 2.1.27. Suppose that 𝔻 ⊆ I ⊆ ℝn and the set 𝔻 is unbounded, as well as 0 ≠ I ′ ⊆ ℝn , 0 ≠ I ⊆ ℝn , F : I × X → Y is a continuous function, I + I ′ ⊆ I and ρ is a binary relation on X. Then we say that: (i) F(⋅; ⋅) is 𝔻-asymptotically Bohr (ℬ, I ′ , ρ)-almost periodic of type 1 if and only if for every B ∈ ℬ and ε > 0 there exist l > 0 and M > 0 such that for each t0 ∈ I ′ there exists τ ∈ B(t0 , l) ∩ I ′ such that for every t ∈ I and x ∈ B with t, t + τ ∈ 𝔻M , there exists an element yt,x ∈ ρ(F(t; x)) such that

2.1 Multi-dimensional ρ-almost periodic type functions

� 35

󵄩󵄩 󵄩 󵄩󵄩F(t + τ; x) − yt,x 󵄩󵄩󵄩Y ⩽ ε.

(35)

(ii) F(⋅; ⋅) is 𝔻-asymptotically (ℬ, I ′ , ρ)-uniformly recurrent of type 1 if and only if for every B ∈ ℬ there exist a sequence (τ k ) in I ′ and a sequence (Mk ) in (0, ∞) such that limk→+∞ |τ k | = limk→+∞ Mk = +∞ and that for every t ∈ I and x ∈ B, there exists an element yt;x ∈ ρ(F(t; x)) such that lim

sup

k→+∞ t,t+τ k ∈𝔻M ;x∈B k

󵄩󵄩 󵄩 󵄩󵄩F(t + τ k ; x) − yt,x 󵄩󵄩󵄩Y = 0.

In the case that X = {0} (I ′ = I), we omit the term “ℬ” (“I ′ ”) from the notation, as before. The proof of the following slight extension of [441, Proposition 2.26] is trivial: Proposition 2.1.28. Suppose that 𝔻 ⊆ I ⊆ ℝn and the set 𝔻 is unbounded, as well as 0 ≠ I ′ ⊆ ℝn , 0 ≠ I ⊆ ℝn , F : I × X → Y is a continuous function, I + I ′ ⊆ I and ρ is a binary relation on X. If F(⋅; ⋅) is 𝔻-asymptotically Bohr (ℬ, I ′ , ρ)-almost periodic, resp. 𝔻-asymptotically (ℬ, I ′ , ρ)-uniformly recurrent, then F(⋅; ⋅) is 𝔻-asymptotically Bohr (ℬ, I ′ , ρ)-almost periodic of type 1, resp. 𝔻-asymptotically (ℬ, I ′ , ρ)-uniformly recurrent of type 1. In the case that ρ = cI, where c ∈ ℂ ∖ {0} satisfies |c| = 1, the converse statement holds provided some additional conditions on the region I ′ = I; see [441, Theorem 2.27]. This result can be further extended to the case in which ρ = T ∈ L(X) is not necessarily a linear isomorphism; more precisely, we have the following: Theorem 2.1.29. Suppose that ρ = T ∈ L(X), 0 ≠ I ⊆ ℝn , I + I = I, I is closed as well as F : I → Y is a uniformly continuous, bounded function which is both I-asymptotically Bohr T-almost periodic function of type 1 and I-asymptotically Bohr I-almost periodic function of type 1. If (∀l > 0) (∀M > 0) (∃t0 ∈ I) (∃k > 0) (∀t ∈ IM+l )(∃t′0 ∈ I)

′ ′′ (∀t′′ 0 ∈ B(t0 , l) ∩ I) t − t0 ∈ B(t0 , kl) ∩ IM ,

there exists L > 0 such that IkL ∖ I(k+1)L ≠ 0 for all k ∈ ℕ and IM + I ⊆ IM for all M > 0, then the function F(⋅) is I-asymptotically Bohr T-almost periodic. Proof. The proof of [441, Theorem 2.27] contains some minor typographical errors and, because of that, we will provide all details of the proof for the sake of completeness. Since we have assumed that F(⋅) is an I-asymptotically Bohr I-almost periodic function of type 1, using [194, Theorem 2.34] it follows that for each sequence (bk ) in I there exist a subsequence (bkl ) of (bk ) and a function F ∗ : I → Y such that liml→+∞ F(t+bkl ) = F ∗ (t), uniformly in t ∈ I. Clearly, for each integer k ∈ ℕ there exist lk > 0 and Mk > 0 such that for each t0 ∈ I there exists τ ∈ B(t0 , l) ∩ I such that (35) holds with T = I, ε = 1/k and 𝔻 = I. Let τ k be any fixed element of I such that |τ k | > Mk + k 2 and (35) holds with T = I,

36 � 2 Multi-dimensional ρ-almost periodic type functions and their Stepanov generalizations ε = 1/k and 𝔻 = I (k ∈ ℕ). Then there exist a subsequence (τ kl ) of (τ k ) and a function F ∗ : I → Y such that lim F(t + τ kl ) = F ∗ (t),

l→+∞

uniformly for t ∈ I.

(36)

The mapping F ∗ (⋅) is clearly continuous and we can prove that F ∗ (⋅) is Bohr T-almost periodic as follows. Let ε > 0 be fixed, and let l > 0 and M > 0 be such that for each t0 ∈ I there exists τ ∈ B(t0 , l) ∩ I such that (35) holds, with 𝔻 = I and the number ε replaced therein by ε/(3(1 + ‖T‖)). Let t ∈ I be fixed, and let l0 ∈ ℕ be such that |t + τ kl | ⩾ M and 0 |t + τ + τ kl | ⩾ M. Then 0

󵄩󵄩 ∗ 󵄩 ∗ 󵄩󵄩F (t + τ) − TF (t)󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ⩽ 󵄩󵄩󵄩F ∗ (t + τ) − F(t + τ + τ kl )󵄩󵄩󵄩 + 󵄩󵄩󵄩F(t + τ + τ kl ) − TF(t + τ kl )󵄩󵄩󵄩 0 0 0 󵄩 󵄩 󵄩 󵄩 + 󵄩󵄩󵄩TF(t + τ kl ) − TF ∗ (t)󵄩󵄩󵄩 ⩽ 2 ⋅ (ε/3) + ‖T‖ ⋅ 󵄩󵄩󵄩F(t + τ kl ) − F ∗ (t)󵄩󵄩󵄩 ⩽ ε, 0 0 which implies the required. The function t 󳨃→ F(t) − F ∗ (t), t ∈ I belongs to the space C0,I (I : Y ) due to (36) and the fact that F : I → Y is an I-asymptotically Bohr I-almost periodic function of type 1. This completes the proof. Concerning the extensions of Bohr (I ′ , ρ)-almost periodic functions and (I ′ , ρ)uniformly recurrent functions, we may deduce the following result, in which we require that ρ = T ∈ L(X) is a linear isomorphism; the proof is almost the same as the corresponding proof of [441, Theorem 2.28] (we only need to replace any appearance of the letters c and c−1 with the letters T and T −1 , respectively, in a part of proof where arg(c)/π ∉ ℚ): Theorem 2.1.30. Suppose that ρ = T ∈ L(Y ) is a linear isomorphism, I ′ ⊆ ℝn , I ⊆ ℝn , I + I ′ ⊆ I, the set I ′ is unbounded, F : I → Y is a uniformly continuous, Bohr (I ′ , T)-almost periodic function, resp. a uniformly continuous, (I ′ , T)-uniformly recurrent function, S ⊆ ℝn is bounded and the following condition holds: ′ (AP-E) For every t′ ∈ ℝn , there exists a finite real number M > 0 such that t′ + IM ⊆ I. Then there exists a uniformly continuous, Bohr (I ′ ∪ S, T)-almost periodic, resp. a unĩ = formly continuous, (I ′ ∪ S, T)-uniformly recurrent, function F̃ : ℝn → Y such that F(t) F(t) for all t ∈ I; furthermore, in T-almost periodic case, the uniqueness of such a func̃ holds provided that ℝn ∖ (I ′ ∪ S) is a bounded set and any Bohr T-almost periodic tion F(⋅) function defined on ℝn is almost automorphic. The general requirements of Theorem 2.1.30 hold provided that (v1 , . . . , vn ) is a basis of ℝn and I = {α1 v1 + ⋅ ⋅ ⋅ + αn vn : αi ⩾ 0 for all i ∈ ℕn }

2.1 Multi-dimensional ρ-almost periodic type functions

�

37

is a convex polyhedral in ℝn and I ′ is its proper convex subpolyhedral. But, in this case, we do not have that ℝn ∖ (I ′ ∪ S) is a bounded set and the question of uniqueness of uniformly continuous, Bohr (I ′ , T)-almost periodic extensions of function F(⋅) to ℝn naturally appears. Keeping in mind Corollary 2.1.5 and Theorem 2.1.30, we can immediately clarify the following result: Corollary 2.1.31. Suppose that ρ = T ∈ L(Y ) is a linear isomorphism, I ′ ⊆ ℝn , [0, ∞)n ⊆ I ⊆ ℝn , I ′ − I ′ = ℝn , I + I ′ ⊆ I, F : I → Y is a uniformly continuous, Bohr (I ′ , T)-almost periodic function, and condition (AP-E) holds. Then the Bohr–Fourier coefficient ℳλ (F), given by the second equality in (23), exists for any λ ∈ ℝn , and the set of all points λ ∈ ℝn for which ℳλ (F) ≠ 0 is at most countable. It is clear that the requirements of Corollary 2.1.31 hold provided that I ′ = I = [0, ∞)n . Concerning the existence of Bohr–Fourier coefficients, we would like to propose the following problem at the end of this subsection:

Problem. Suppose that ρ = T ∈ L(Y ) is not a linear isomorphism, I ′ ⊆ ℝn , [0, ∞)n ⊆ I ⊆ ℝn , I ′ − I ′ = ℝn , I + I ′ ⊆ I and condition (AP-E) holds. Does there exists a (uniformly continuous) Bohr (I ′ , T)-almost periodic function F : I → Y such that the Bohr–Fourier coefficient ℳλ (F), given by the second equality in (23), does not exist for some λ ∈ ℝn ?

Observe only that the function F : [0, ∞) → ℂ2 , constructed in Example 2.1.22, is asymptotically almost periodic and therefore the mean value ℳλ (F) exists for any λ ∈ ℝn . 2.1.3 (ω, ρ)-periodic functions and (ωj , ρj )j∈ℕn -periodic functions In [437], we have recently generalized the notion of Bloch (p, k)-periodicity by proposing a new definition of (ω, c)-periodicity, where ω ∈ ℝn ∖ {0} and c ∈ ℂ ∖ {0}. The notion of (ω, c)-periodicity is a special case of the notion (ω, ρ)-periodicity introduced as follows: Definition 2.1.32. Let ω ∈ ℝn ∖{0}, ρ be a binary relation on X and ω+I ⊆ I. A continuous function F : I → X is said to be (ω, ρ)-periodic if and only if F(t + ω) ∈ ρ(F(t)), t ∈ I. Example 2.1.33. Let C ∈ L(X), and let x ∈ X. A strongly continuous family (T(t))t⩾0 in L(X) is said to be a C-regularized semigroup if and only if T(t + s)C = T(t)T(s) for all t, s ⩾ 0 and T(0) = C. Put u(t) := T(t)x, t ⩾ 0. Then the function u(⋅) is a mild solution of the abstract Cauchy inclusion u′ (t) ∈ 𝒜u(t), t ⩾ 0; u(0) = Cx, where a closed MLO 𝒜 is the integral generator of (T(t))t⩾0 ; see [429, Sections 3.1–3.4] for more details on the subject. It is clear that u(⋅) is (ω, C −1 T(ω))-periodic as well as that the binary relation ρω := C −1 T(ω) is not single-valued in general (ω > 0). Let us recall that an unbounded subset A ⊆ ℕ is called syndetic if and only if there exists a strictly increasing sequence (ak ) of positive integers such that A = {ak : k ∈ ℕ}

38 � 2 Multi-dimensional ρ-almost periodic type functions and their Stepanov generalizations and supk∈ℕ (ak+1 − ak ) < +∞. Keeping in mind this notion, we can state the following simple result: Proposition 2.1.34. Suppose that ω ∈ ℝn ∖ {0}, ρ is a binary relation on X, ω + I ⊆ I and the function F : I → X is (ω, ρ)-periodic. If A and B are syndetic subsets of ℕ, we set σ := ⋃m∈A ρm and I ′ := {mω : m ∈ B}. Then the function F(⋅) is Bohr (I ′ , σ)-almost periodic. Proof. Since ω + I ⊆ I, we inductively get I + I ′ ⊆ I. Since F(t + mω) ∈ ρm (F(t)), t ∈ I, m ∈ ℕ, the result immediately follows from the corresponding definition of Bohr (I ′ , σ)-almost periodicity, the definition of a syndetic set and the definition of binary relation σ. If we additionally assume that kω + I ⊆ I for all positive real numbers k > 0, then the above statement continues to hold with the set I ′ := [0, ∞) ⋅ ω. In connection with the statement of Proposition 2.1.34, it is worth mentioning case in which we have the existence of a positive integer m ∈ ℕ such that ρm ⊆ ρ, which implies that ρkm ⊆ ρ for all k ∈ ℕ. Then we can simply show that the preassumptions of Proposition 2.1.34 imply that the function F(⋅; ⋅) is Bohr (I ′ , ρ)-almost periodic (with the set I ′ := [0, ∞) ⋅ ω, provided that kω + I ⊆ I for all positive real numbers k > 0). It is worth noting that the choice of binary relation σ is important here because, if c ∈ ℂ ∖ {0} and |c| ≠ 1, then the (ω, c)-periodicity of a non-zero function F(⋅) cannot imply its c-almost periodicity due to [409, Proposition 2.7] (ρ = cI). The following definition is also meaningful: Definition 2.1.35. Let ωj ∈ ℝ ∖ {0}, ρj ∈ ℂ ∖ {0} be a binary relation on X and ωj ej + I ⊆ I (1 ⩽ j ⩽ n). A continuous function F : I → X is said to be (ωj , ρj )j∈ℕn -periodic if and only if F(t + ωj ej ) ∈ ρj (F(t)), t ∈ I, j ∈ ℕn . In the case that ρj = cj I for some non-zero complex numbers cj (1 ⩽ j ⩽ n), then we also say that the function F(⋅) is (ωj , cj )j∈ℕn -periodic; furthermore, if cj = 1 for all j ∈ ℕn , then we say that F(⋅) is (ωj )j∈ℕn -periodic. It is clear that if F : I → X is (ωj , ρj )j∈ℕn -periodic, then F(t + mωj ej ) ∈ ρm j (F(t)), t ∈ I, m ∈ ℕ, j ∈ ℕn . This enables one to transfer the statement of Proposition 2.1.34 and conclusions given in the paragraph following it to (ωj , ρj )j∈ℕn -periodic functions; details can be omitted. The interested reader may try to reformulate the statement of Theorem 2.1.12 for (ω, ρ)-periodic functions and (ωj , ρj )j∈ℕn -periodic functions. Further on, in the scalarvalued case, the following holds: If the function F : I → ℂ ∖ {0} is (ω, ρ)-periodic, resp. (ωj , ρj )j∈ℕn -periodic, then the function (1/F)(⋅) is (ω, σ)-periodic, resp. (ωj , σj )j∈ℕn periodic, provided that for every non-zero complex numbers x and y, we have that the assumption (x, y) ∈ ρ implies (1/x, 1/y) ∈ σ, resp. the assumption (x, y) ∈ ρj implies (1/x, 1/y) ∈ σj for all j ∈ ℕn . This can be also reworded for Bohr (ℬ, I ′ , ρ)-almost pe-

2.1 Multi-dimensional ρ-almost periodic type functions

� 39

riodic functions and (ℬ, I ′ , ρ)-uniformly recurrent functions; details can be left to the interested readers. It is straightforward to deduce the following extension of [437, Proposition 2.5]: Proposition 2.1.36. Let ωj ∈ ℝ ∖ {0}, ρj is a binary relation on X and ωj ej + I ⊆ I (1 ⩽ j ⩽ n). If a continuous function F : I → X is (ωj , ρj )j∈ℕn -periodic and σ : ℕn → ℕn is a permutation, then ω + I ⊆ I, where ω := ∑nj=1 ωj ej , and the function F(⋅) is (ω, ρ)-periodic with ρ = ∏nj=1 ρσ(j) . We continue with the following illustrative example: Example 2.1.37. As we have shown in [437, Example 2.2, Example 2.6], there exists a continuous, unbounded function F : ℝn → ℝ which satisfies that F(t+(1, 1, . . . , 1)) = F(t) for all t ∈ ℝn as well as that there do not exist numbers ω1 , ω2 ∈ ℝ ∖ {0} and numbers c1 , c2 ∈ ℂ ∖ {0} such that the function F(⋅) is (ωj , cj )j∈ℕ2 -periodic. The analysis carried out in these examples shows that there do not exist numbers ω1 , ω2 ∈ ℝ ∖ {0}, a number c1 = ρ1 ∈ ℂ ∖ {0} and a binary relation ρ2 on ℂ such that the function F(⋅) is (ωj , ρj )j∈ℕ2 periodic. The reader can simply formulate certain statements concerning the pointwise multiplication of scalar-valued (ω, ρ)-periodic functions ((ωj , ρj )j∈ℕn -periodic functions) and the vector-valued (ω, ρ)-periodic functions ((ωj , ρj )j∈ℕn -periodic functions). Concerning the convolution invariance of spaces consisting of (ω, ρ)-periodic functions ((ωj , ρj )j∈ℕn periodic functions), we will state and prove the following result: Proposition 2.1.38. Suppose that ω ∈ ℝn ∖ {0}, ρ = 𝒜 is a closed MLO on X, and ρj = 𝒜j is a closed MLO on X (1 ⩽ j ⩽ n). Suppose, further, that F : ℝn → X is (ω, 𝒜)-periodic ((ωj , 𝒜j )j∈ℕn -periodic) and bounded. If h ∈ L1 (ℝn ), then the function (h ∗ F)(t) := ∫ h(y)F(t − y) dy,

t ∈ ℝn

ℝn

is (ω, 𝒜)-periodic ((ωj , 𝒜j )j∈ℕn -periodic) and bounded. Proof. Keeping in mind the dominated convergence theorem, the boundedness and the continuity of function F(⋅), we easily get that the function (h ∗ F)(⋅) is well-defined, bounded and continuous. Suppose that the function F(⋅) is (ω, 𝒜)-periodic. Then we have F(t + ω − s) ∈ 𝒜(F(t − s)) and therefore h(s)F(t + ω − s) ∈ 𝒜(h(s)F(t − s)) (t, s ∈ ℝn ). Applying Lemma 1.1.1, we get (h ∗ F)(t + ω) ∈ 𝒜((h ∗ F)(t)) for all t ∈ ℝn , as required. The proof is the same for (ωj , 𝒜j )j∈ℕn -periodic functions. We leave to the interested readers problem of transferring the statements of Theorem 2.1.17 and Theorem 2.1.18 to (ω, ρ)-periodic functions and (ωj , ρj )j∈ℕn -periodic functions. For simplicity, we will not consider here 𝔻-asymptotically (ω, ρ)-periodic functions and 𝔻-asymptotically (ωj , ρj )j∈ℕn -periodic functions, as well.

40 � 2 Multi-dimensional ρ-almost periodic type functions and their Stepanov generalizations In the next four subsections, we will present several important examples and applications of our results to the abstract Volterra integro-differential equations in Banach spaces. We start with the following theme:

2.1.4 Liouville type results for anti-periodic linear operators Our first applications are closely connected with the investigation of L. Rossi [646], which concerns Liouville type results for periodic and almost periodic linear operators. The author has investigated bounded solutions of the partial differential equation Pu := 𝜕t u − aij (x, t)𝜕ij u − bi (x, t)𝜕i u − c(x, t)u = f (x, t),

x ∈ ℝn , t ∈ ℝ;

(37)

all considered coefficients aij (x, t), bi (x, t), c(x, t) are assumed to be real-valued and satisfy the general assumptions stated on [646, p. 2482]. If the coefficients aij (x, t), bi (x, t), c(x, t) do not depend on t, then P can be written as Pu = 𝜕t u − Lu, where L is a linear differential operator in ℝn . Furthermore, if the coefficients aij (x), bi (x), c(x) are (ωj )j∈ℕn -periodic (ωj ∈ ℝ ∖ {0} for all j ∈ ℕn ), then we say that L is (ωj )j∈ℕn -periodic, as well. If this is the case, then some results from the Krein–Rutman theory yield the existence of a unique real number λ, called periodic principal eigenvalue of −L, such that the eigenvalue problem −Lϕ = λϕ in ℝn ;

ϕ(⋅) is periodic, with the same period as L,

admits positive solutions. It is well known that the positive solution ϕ(⋅), which is called the principal eigenfunction, is unique up to a multiplicative constant; by λp (−L) and φp (⋅) we denote the periodic principal eigenvalue and eigenfunction of −L, respectively. In general case, the coefficients aij (x, t), bi (x, t), c(x, t) can be complex-valued. Then it is said that the linear differential operator P is: (i) (ω, c)-periodic (ω ∈ ℝn+1 ∖ {0}, c ∈ ℂ ∖ {0}) if and only if all functions aij (x, t), bi (x, t), c(x, t) are (ω, c)-periodic; (ii) (ωj , cj )j∈ℕn+1 -periodic (ωj ∈ ℝ ∖ {0}, cj ∈ ℂ ∖ {0}) if and only if all functions aij (x, t), bi (x, t), c(x, t) are (ωj , cj )j∈ℕn+1 -periodic. In the case that all coefficients aij (x, t), bi (x, t), c(x, t) are real-valued, we can slightly extend the statement of [646, Lemma 2.1, Theorem 1.1, Theorem 1.3(i)] in the following way, by considering (anti-)periodic case c = ±1 in place of the already considered periodic case c = 1 (with the meaning clear): Lemma 2.1.39. Suppose that the linear differential operator P is (lm em , 1)-periodic and the function f (x, t) is (lm em , c)-periodic, where c = ±1, lm ∈ ℝ ∖ {0} and m ∈ ℕn+1 . If there exists a bounded continuous function v(x, t) satisfying that inf(x,t)∈ℝn+1 v(x, t) > 0

2.1 Multi-dimensional ρ-almost periodic type functions

�

41

and Pv = ϕ for some nonnegative function ϕ ∈ L∞ (ℝn+1 ), then any bounded solution u(x, t) of the equation (37) is (lm em , c)-periodic. Proof. The proof is almost completely the same as the proof of [646, Lemma 2.1] and we will only point out a few important details for case c = −1. First of all, we consider the function ψ(X) := u(X +lm em )+u(X), X ∈ ℝn+1 in place of the already considered function ψ(X) := u(X + lm em ) − u(X), X ∈ ℝn+1 . We will prove that ψ(X) ⩽ 0 for all X ∈ ℝn+1 . If we assume that this is not true, then there exists a sequence (XN ) in ℝn+1 such that ψ(XN )/v(XN ) → k ≡ supX∈ℝn+1 (ψ/v)(X) > 0 as N → +∞. Define ψN := ψ(⋅ + XN ), N ∈ ℕ. Since we have assumed that the linear differential operator P is (lm em , 1)-periodic and the function f (x, t) is (lm em , −1)-periodic, a simple calculation shows that 𝜕t ψN − aij (X + XN )𝜕ij ψN − bi (X + XN )𝜕i ψN − c(X + XN )ψN = 0,

X ∈ ℝn+1 , N ∈ ℕ.

Then we can repeat verbatim all calculations and arguments from the corresponding proof given on p. 2486 of [646]; in the last line of this page, we only need to write ξh + ξh+1 in place of ξh − ξh+1 in order to conclude that limh→+∞ [ξh + ξh+1 ] = −∞, which is a contradiction with the boundedness of sequence (ξh )h∈ℕ . This implies u(X + lm em ) + u(X) ⩽ 0 for all X ∈ ℝn+1 . It is our strong belief that the final part of the proof of the above-mentioned lemma is a little bit incorrect as well as that we can use a simple trick here, working also in the case that c = −1: since P[−u] = −f , we can apply the first part of proof (with the same number lm ; see also [646, p. 2487, l. 2], where the author suggests the use of number −lm ) in order to see that u(X + lm em ) + u(X) ⩾ 0 for all X ∈ ℝn+1 . Therefore, u(X + lm em ) + u(X) = 0 for all X ∈ ℝn+1 , which completes the proof. Theorem 2.1.40. Suppose that the linear differential operator P is (lm em , 1)-periodic and the function f (x, t) is (lm em , c)-periodic, where c = ±1, lm ∈ ℝ∖{0} and m ∈ ℕn+1 . Then any bounded solution u(x, t) of the equation (37) is (lm em , c)-periodic, provided that c(x, t) ⩽ 0 for all x ∈ ℝn and t ∈ ℝ. Proof. It suffices to put c(x, t) ≡ 1 in Lemma 2.1.39. Theorem 2.1.41. Suppose that the linear differential operator L is (ωj )j∈ℕn -periodic (ωj ∈ ℝ ∖ {0} for all j ∈ ℕn ), and P = 𝜕t − L. If the function f (⋅) is (ωj , cj )j∈ℕn+1 -periodic with some ωn+1 ∈ ℝ ∖ {0} and cj = ±1 for all j ∈ ℕn+1 , then any bounded solution u(x, t) of the equation (37) is (ωj , cj )j∈ℕn+1 -periodic. Proof. The proof simply follows from Lemma 2.1.39 and the corresponding part of proof of [646, Theorem 1.3(i)]. We continue by observing that the statements of [646, Theorem 1.3(ii)–(iii)] do not admit satisfactory reformulations for multi-dimensional almost anti-periodic type functions as well as that it would be very tempting to reconsider the statements of [646, Theorem 1.3(i); Theorem 4.1] for multi-dimensional c-almost periodic type functions. For the sake of simplicity, we will not consider here the corresponding analogues of

42 � 2 Multi-dimensional ρ-almost periodic type functions and their Stepanov generalizations Lemma 2.1.39 and Theorem 2.1.40-Theorem 2.1.41 for the Dirichlet and Robin boundary value problems on uniformly smooth domains in ℝn (see [646, Section 5] for more details). 2.1.5 Applications to the heat equation in ℝn , the fractional diffusion-wave equation in ℝn and the iterated polyharmonic operators in ℝn Our results on the convolution invariance of multi-dimensional ρ-almost periodicity can be applied to the Gaussian semigroup in ℝn and the Poisson semigroup in ℝn without serious difficulties. The obtained results can be further extended for general binary relations by assuming some extra conditions, and we will briefly explain this idea here: 1. Let Y := BUC(ℝn ) be the Banach space of bounded uniformly continuous functions F : ℝn → ℂ. Then it is well known that the Gaussian semigroup (G(t)F)(x) := (4πt)−(n/2) ∫ F(x − y)e−

|y|2 4t

t > 0, f ∈ Y , x ∈ ℝn ,

dy,

(38)

ℝn

can be extended to a bounded analytic C0 -semigroup of angle π/2, generated by the Laplacian ΔY acting with its maximal distributional domain in Y ; see e. g. [54, Example 3.7.6] for more details. Suppose that 0 ≠ I ′ ⊆ ℝn , 0 ≠ I ⊆ ℝn , and F(⋅) is bounded Bohr (ℬ, I ′ , ρ)-almost periodic, resp. bounded (ℬ, I ′ , ρ)-uniformly recurrent, where ρ is any binary relation defined on ℂ such that for each t ∈ ℝn the element yt = zt ∈ ρ(F(t)) is chosen such that the equation (21) holds as well as that the mapping s 󳨃→ zt−s , s ∈ ℝn belongs to Y . Let a number t0 > 0 be fixed. We define a new binary relation σ on ℂ by σ := {((G(t0 )F)(x), (4πt0 )−(n/2) ∫ zx−y e−

|y|2 4t

dy) : x ∈ ℝn };

ℝn

let us only note here that σ need not be a function even if ρ is a function. Then the function ℝn ∋ x 󳨃→ u(x, t0 ) ≡ (G(t0 )F)(x) ∈ ℂ is bounded Bohr (ℬ, I ′ , σ)-almost periodic, resp. bounded (ℬ, I ′ , σ)-uniformly recurrent. This holds because for every x, τ ∈ ℝn , we have: 2

󵄨󵄨 󵄨 󵄨 − |y| −(n/2) 󵄨󵄨 ∫ 󵄨󵄨F(x − y + τ) − zx−y 󵄨󵄨󵄨e 4t0 dy 󵄨󵄨u(x + τ, t0 ) − σ(u(x, t0 ))󵄨󵄨󵄨 ⩽ (4πt0 ) ℝn

⩽ (4πt0 )−(n/2) ε ∫ e

|y|2

− 4t

0

dy.

ℝn

For example, we can consider here the non-linear mapping ρ(z) := zk , z ∈ ℂ (k ∈ ℕ∖{1}). We can similarly clarify the corresponding results for the Poisson semigroup, which is given by

2.1 Multi-dimensional ρ-almost periodic type functions

(T(t)F)(x) :=

t ⋅ dy Γ((n + 1)/2) , ∫ F(x − y) 2 (n+1)/2 π (t + |y|2 )(n+1)/2 n

� 43

t > 0, f ∈ Y , x ∈ ℝn ;

ℝ

see [54, Example 3.7.9] for more details. 2. Suppose that 1 < β < 2. The fractional diffusion-wave equation 𝔻t u(t, x) = Δu(t, x), (β)

t > 0, x ∈ ℝn ,

subjected with the initial conditions u(0, x) = u0 (x),

ut (0, x) = 0,

where 𝔻t u(t, x) is the Caputo–Dzhrbashyan fractional derivative, given by (β)

t

(β) 𝔻t u(t, x)

u (0, x) 1 𝜕2 u(0, x) := − t 1−β , ∫(t − τ)1−β u(τ, x) dτ − t 1−β t Γ(2 − β) 𝜕2 t Γ(2 − β) Γ(1 − β)

t > 0,

0

was studied by many authors (see, e. g., the research article [420] by A. N. Kochubei and references cited therein). A unique solution u(t, x) of this problem is obtained as the convolution with a Green kernel u(t, x) = ∫ G(t, x − ξ)u0 (ξ) dξ = ∫ G(t, ξ)u0 (x − ξ) dξ. ℝn

ℝn

Since the Green kernel G(t, x) satisfies the estimate (cf. the research article [626] by A. V. Pskhu for more details; c > 0 and a > 0 denote the positive real constants here): 󵄨󵄨 󵄨 −nβ/2 γn (|x|t −β/2 )E(|x|t −β/2 ), 󵄨󵄨G(t, x)󵄨󵄨󵄨 ⩽ ct where γn (z) := 1, if n = 1 [γn (z) := | ln z|, if n = 2; γn (z) := z2−n , if n ⩾ 3] and E(z) := exp(−az2/(2−β) ), it can be easily shown the following: If the function x 󳨃→ u0 (x), x ∈ ℝn is bounded, continuous and (I ′ , c)-almost periodic ((I ′ , c)-uniformly recurrent) for some c ∈ S1 , then the solution u(t, x) is bounded, continuous, and (I ′ , c)-almost periodic ((I ′ , c)uniformly recurrent) in the variable x for every fixed value of variable t > 0. 3. Using the Laplace transform of vector-valued distributions, M. Kunzinger, E. A. Nigscha, and N. Ortner [481] have presented the explicit solutions for the Cauchy– Dirichlet problem of the iterated wave operator m

(Δn + 𝜕y2 − 𝜕t2 ) , the iterated Klein–Gordon operator m

(Δ2n+1 + 𝜕y2 − 𝜕t2 − ξ 2 ) ,

44 � 2 Multi-dimensional ρ-almost periodic type functions and their Stepanov generalizations the iterated metaharmonic operator m

(Δn + 𝜕y2 − p2 ) , and the iterated heat operator m

(Δn + 𝜕y2 − 𝜕t ) , in the half-space y > 0. In particular, the authors have shown that under certain logical assumptions, the Cauchy problem 2

(𝜕x21 + ⋅ ⋅ ⋅ + 𝜕x2n + 𝜕y2 ) = 0 u|y=0 = g0 (x),

in (x, y) ∈ ℝn × (0, ∞);

(𝜕uy )|y=0 = g1 (x)

has a unique solution u(x, y) = 2

Γ((n + 3)/2) 3 dξ y ∫ g0 (x − ξ) 2 (n+1)/2 π (|ξ| + y2 )(n+3)/2 n ℝ

Γ((n + 1)/2) 2 dξ + y ∫ g0 (x − ξ) 2 . (n+1)/2 π (|ξ| + y2 )(n+1)/2 n ℝ

Suppose now that the function x 󳨃→ (g0 (x), g1 (x)), x ∈ ℝn is bounded, continuous and (I ′ , c)-almost periodic ((I ′ , c)-uniformly recurrent) for some c ∈ S1 . Then it can be easily shown that the solution u(x, y) is bounded, continuous, and (I ′ , c)-almost periodic ((I ′ , c)-uniformly recurrent) in the variable x for every fixed value of variable y > 0. This example and the example given in the second application of this subsection can be also used to justify the introduction of multi-dimensional ρ-almost periodic functions, with ρ being not a function but a general binary relation; cf. also the first application of this subsection.

2.1.6 New applications in one-dimensional setting The trick employed in the previous subsection can be applied to the remarkable examples [797, Examples 4, 5, 7, 8; pp. 32–34] given in the research monograph of S. Zaidman. Further on, Theorem 2.1.17 and Proposition 2.1.26 can be applied in the qualitative analysis of ρ-almost periodic solutions for various classes of the abstract (degenerate) Volterra integro-differential equations; in the analysis of corresponding semilinear Cauchy problems, applications of Theorem 2.1.18 are necessary. For example, we can incorporate the above mentioned results in the study of existence and uniqueness of asymptotically ρ-almost periodic type solutions of the fractional Poisson heat equation

2.1 Multi-dimensional ρ-almost periodic type functions

� 45

γ

D [m(x)v(t, x)] = (Δ − b)v(t, x) + f (t, x), t ⩾ 0, x ∈ Ω; { { { t v(t, x) = 0, (t, x) ∈ [0, ∞) × 𝜕Ω, { { { x ∈ Ω, {m(x)v(0, x) = u0 (x), and the following fractional Poisson heat equation with Weyl–Liouville derivatives: γ

Dt,+ [m(x)v(t, x)] = (Δ − b)v(t, x) + f (t, x),

t ∈ ℝ, x ∈ Ω;

see also [428, Example 3.10.4] for some possible applications to the differential operators in Hölder spaces. We will omit all related details, because the considerations are almost the same as in [428] and a great number of other research articles.

2.1.7 Heteroclinic period blow-up We first consider a singularly perturbed symmetric ODE with a symmetric heteroclinic cycle connecting hyperbolic equilibria. Then we deal with a conservative ODE with the same heteroclinic structure. We show in both cases there are (Q, T) affine-periodic solutions accumulating on the heteroclinic cycle with T → ∞; this is called as heteroclinic/homoclinic period blow-up. For instance, the Duffing equation ẍ − x + 2x 3 = 0 ̃ with γ(t) ̃ ̇ has a symmetric homoclinic cycle ±γ(t) = (γ(t), γ(t)), γ(t) = sech t, which is accumulated by periodic solutions with periods tending to infinity. Related results are investigated in [736, 737]. Here, Q : ℝn → ℝn is a linear mapping such that Qk = I for some k ∈ ℕ. 1. Symmetric singular systems. Consider the system εu̇ = f (u) + εg(t, u, ε) ≡ fε (u, t),

(39)

for f ∈ C 2 (ℝn : ℝn ), g ∈ C 2 (ℝ × ℝn × ℝ) and g(t, u, ε) is 1-periodic in t. Suppose fε (Qu, t) = Qfε (u, t) for any (t, u, ε). Then Qfε (u, t) = fε (Qu, t + 1). Assume that the boundary layer problem u̇ = f (u) has a solution γ(t) heteroclinic to hyperbolic equilibria p0 , p1 satisfying Qp0 = p1 , i. e., γ(t) → p0 as t → −∞ and γ(t) → p1 as t → +∞. Furthermore, assume that the variational system ̇ = Df (γ(t))u(t) u(t)

(40)

̇ on ℝ up to a multiplicative constant. Then the has the unique bounded solution γ(t) system adjoint to (40) is of the form

46 � 2 Multi-dimensional ρ-almost periodic type functions and their Stepanov generalizations ̇ = −Df (γ(t)) v(t) v(t) ∗

and has unique (up to a multiplicative constant) bounded solution ψ(t) ≠ 0 on ℝ. Assume [95] that a function +∞

M(α) = ∫ ψ∗ (t)g(α, γ(t), 0) dt −∞

has a simple zero at α = α0 . Then there exists ε0 > 0 such that for any ε ∈ (0, ε0 ) and m ≥ 1, the system (39) has an (m, Q) affine-periodic solution; all such solutions accumulate on the symmetric heteroclinic cycle p

⋃ {Qi γ(t) | t ∈ ℝ}. i=1

As an example, we consider εż1 = z2 ,

εż2 = − sin z1 + εh(t)z2 ,

(41)

where h : ℝ → ℝ is 1-periodic and C 2 -smooth. Clearly, (41) is equivalent to the equation ε2 ẍ + sin x − ε2 h(t)ẋ = 0. The boundary layer equation is the pendulum equation ż1 = z2 ,

ż2 = − sin z1 .

(42)

Now we take Q = −I. It is well-known that two hyperbolic equilibria p0 = (−π, 0) and ̇ p1 = (π, 0) of (42) are connected with a heteroclinic solution γ(t) = (θ(t), θ(t)), where −t ′ θ(t) = π − 4 arctan e . If there is an α0 such that h(α0 ) = 0 ≠ h (α0 ), then for any m ∈ ℕ and ε > 0 small, the system (41) has a 2m-periodic solution (z1mε , z2mε ); all such solutions accumulate on the heteroclinic cycle {γ(t) | t ∈ ℝ} ∪ {−γ(t) | t ∈ ℝ} and satisfy −z1mε (t) = z1mε (t + m),

−z2mε (t) = z2mε (t + m).

2. Symmetric conservative systems. We suppose that ẋ = g(x)

(43)

2.2 (ω, ρ)-periodic solutions of abstract integro-differential impulsive equations

� 47

is C 1 -smooth and conservative, i. e., (43) has a C 1 -smooth first integral H : ℝn → ℝ with H(Qu) = H(u) for any u ∈ ℝn . We also suppose that (43) has a heteroclinic orbit γ(t) to hyperbolic equilibria p0 , p1 satisfying Qp0 = p1 . Let us assume that the variational system v̇ = Dg(γ(t))v has the unique bounded ̇ up to a multiplicative constant and DH(γ(0)) ≠ 0. Then there is an ω0 > 0 solution γ(t) such that (43) has, for every ω ≥ ω0 , an (ω, Q) affine-periodic solution; these solutions accumulate on the symmetric heteroclinic cycle p

⋃ {Qi γ(t) | t ∈ ℝ} i=1

as ω → ∞. This result generalizes an accumulation of periodic solutions to homoclinics/heteroclinics in one-degree of freedom conservative equations like in the above-mentioned Duffing equation ẍ − x + 2x 3 = 0 to symmetric higher-degree of freedom conservative systems.

2.2 (ω, ρ)-periodic solutions of abstract integro-differential impulsive equations on Banach space The needs of modern technology and its development cause the growth of the interest in systems with discontinuous trajectories, impulsive automatic control systems, and impulsive computing systems. At present, these kinds of systems become very important and they are intensively developing and broadening the scope of their applications in technical problems, biological phenomena involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, pharmacokinetics, and frequency modulated systems; see [486]. These processes are subject to short-term perturbations whose duration is negligible compared with the duration of the whole process (see [21, 65, 77, 78, 350, 486, 699]). So, logically, the interest for investigating the qualitative features of the solutions of these impulsive systems is enormous. In this section, we consider (ω, ρ)-periodic solutions for a class of (fractional) impulsive differential equations. The impulsive differential equations describe evolution processes characterized by the fact that at certain moments they experience a change of state abruptly, i. e., these processes are subject to short-term perturbations whose duration is negligible compared with the duration of the whole process (see [20, 64, 77, 78, 341, 350, 371, 486, 501, 608, 700, 757, 767]). In [498], the existence and uniqueness of (ω, c)-periodic solutions of impulsive differential systems with coefficient of matrices have been investigated. The authors of [520] have used the fixed point theorems in order to clarify certain results concerning

48 � 2 Multi-dimensional ρ-almost periodic type functions and their Stepanov generalizations the existence and uniqueness of (ω, c)-periodic solutions for nonlinear impulsive differential equations. Further on, the authors of [307] have established certain results on the existence and uniqueness of (ω, 𝕋)-periodic solutions for impulsive linear and semilinear problems. There are many papers on periodic solutions for periodic systems on infinite-dimensional spaces (see [37, 518, 661, 776]); we also refer to [341] and [602], where the integro-differential systems on finite and infinite-dimensional Banach space are investigated. The existence of piecewise continuous mild solutions and optimal control of integro-differential systems is presented in [767]. In [761], the integro-differential impulsive periodic systems on infinite-dimensional spaces are discussed. To our best knowledge, the existence and uniqueness of (ω, c)-periodic solutions for integro-differential systems (c ∈ ℂ, c ≠ 0) have not been extensively studied. As a continuation of the investigations on (ω, 𝕋)-periodic solutions for linear and semilinear problems, and periodic solutions for integro-differential impulsive periodic systems, we consider here (ω, ρ)-periodic solutions of impulsive differential equations as a generalization of the previous concepts (see [15, 37, 77, 228, 307, 309, 428, 498, 519, 602, 638, 661, 760, 761, 762, 776]). The main aim of this section is to present some results concerning the existence and uniqueness of (ω, ρ)-periodic solutions for certain classes of the abstract semilinear integro-differential impulsive equations, considered on the infinite-dimensional pivot Banach space X. In order to stay consistent with the notation used in our recent papers [305, 306], from which we have taken the material of this section, the identity operator on X will be denoted by E here. The author would like to express his sincere gratitude to his friend and colleague D. Velinov, who established the main results of these papers. The organization of section can be briefly described as follows. After recalling some preliminary results and definitions about piecewise continuous (ω, ρ)-periodic functions, we present several results on the solutions of the nonhomogeneous linear impulsive equations and certain useful estimates for the further investigations. After that, we use the Banach fixed point theorem and the Schauder fixed point theorem to prove the existence and uniqueness of the (ω, ρ)-periodic solutions for semilinear integro-differential equations under our considerations. In the last subsection, we consider (ω, ρ)-BVPs for impulsive differential equations of fractional order. The space of bounded piecewise continuous functions f : [0, ∞) → X, denoted by PC([0, ∞) : X), consists of all bounded continuous functions and those bounded functions f : [0, ∞) → X which are continuous at the point zero and for which there exists a strictly increasing sequence (ti ) of positive real numbers without accumulation points, or a finite sequence (ti ) of positive real numbers, such that f ∈ C((ti , ti+1 ] : X), f (ti− ) := f (ti ) and f (ti+ ) exist for any i ∈ ℕ, where the symbols f (ti− ) and f (ti+ ) denote the left limit and the right limit of the function y(t) at the point t = ti , i ∈ ℕ, respectively. Let us recall that PC([0, ∞) : X) is a Banach space endowed with the sup-norm; we similarly define the Banach space PC([0, ω] : X), where ω > 0 is a finite real number. In this section, we assume that ρ : X → X is a linear isomorphism; a measurable function f : [0, ∞) → X is called (ω, ρ)-periodic if and only if there is a real number

2.2 (ω, ρ)-periodic solutions of abstract integro-differential impulsive equations

� 49

ω > 0 such that f (t + ω) = ρf (t) for all t ⩾ 0. By Φω,ρ we denote the set of all piecewise continuous, (ω, ρ)-periodic functions, i. e., Φω,ρ = {f : f ∈ PC([0, ∞) : X) and y(⋅ + ω) = ρy(⋅)}. We continue the investigations started in [307] by studying the (ω, ρ)-periodic solutions of the following abstract integro-differential impulsive equation t

{ { { ̇ = Ay(t) + f (t, y(t), ∫ g(t, s, y(t)) ds), {y(t) { 0 { { { Δy| = B y(t) + d , t=τ k k k {

t ≠ τk , k ∈ ℕ;

(44)

k ∈ ℕ,

where A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators (T(t))t⩾0 , and Bk is a bounded linear operator on X for all k ∈ ℕ. Of concern are the following assumptions: (A1) A is the infinitesimal generator of a strongly continuous semigroup of bounded operators (T(t))t⩾0 in X. The operators Bk , k ∈ ℕ are bounded linear operators and T(t)Bk = Bk T(t) for all k ∈ ℕ, t ⩾ 0. (A2) The constants dk and the time sequence τk > 0 are such that Bk+m = Bk , dk+m = ρdk , τk+m = τk + ω, k ∈ ℕ, for some fixed m = i(0, ω), where by i(0, s) is denoted the number of impulsive points in the segment [0, s]. (A3) ρ : X → X is a linear isomorphism and ρA = Aρ, ρBk = Bk ρ for all k ∈ ℕ. (A4) The operator ρ − T(ω) ∏m k=1 (E + Bk ) is injective. (A5) For all t ⩾ 0 and y ∈ X, we have t+ω

t

f (t + ω, ρy, ρ ∫ g(t, s, y) ds) = ρf (t, y, ∫ g(t, s, y) ds). 0

(A6)

0

For all t ⩾ s ⩾ 0 and y ∈ X, it holds g(t + ω, s, ρy) = ρg(t, s, y).

(A7)

Let f : [0, ∞) × X × X → X and the function t 󳨃→ (t, x, y) be measurable for all (x, y) ∈ X × X. For every ν > 0, there exists Lf (ν) > 0 such that for almost all t ⩾ 0 and all x1 , x2 , y1 , y2 ∈ X with ‖x1 ‖, ‖x2 ‖, ‖y1 ‖, ‖y2 ‖ ⩽ ν, we have 󵄩󵄩 󵄩 󵄩󵄩f (t, x1 , y1 ) − f (t, x2 , y2 )󵄩󵄩󵄩 ⩽ Lf (ν)(‖x1 − x2 ‖ + ‖y1 − y2 ‖). Set D := {(t, s) ∈ [0, ∞) × [0, ∞) : 0 ⩽ s ⩽ t}. The function g : D × X → X is continuous, and for each ν > 0 there exists Lg (ν) > 0 such that for each (t, s) ∈ D and for each x, y ∈ X with ‖x‖, ‖y‖ ⩽ ν, we have 󵄩󵄩 󵄩 󵄩󵄩g(t, s, x) − g(t, s, y)󵄩󵄩󵄩 ⩽ Lg (ν)‖x − y‖.

50 � 2 Multi-dimensional ρ-almost periodic type functions and their Stepanov generalizations (A8)

There are constants α, β ⩾ 0 such that t 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 f (t, y(t), ∫ g(t, s, y(t)) ds)󵄩󵄩󵄩 ⩽ α + β‖y‖, 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 0

for any t ⩾ 0. (A9) Let M ⩾ 1 and γ ∈ ℝ be such that ‖T(t)‖ ⩽ Meγt for all t ⩾ 0. (A10) X is finite-dimensional. 2.2.1 Nonhomogeneous linear impulsive problem Of concern is the following equation t

y′ (t) = f (t, y(t), ∫ g(t, s, y(t)) ds)

(45)

0

accompanied with the conditions t+ω

t

f (t + ω, ρy, ρ ∫ g(t, s, y) ds) = ρf (t, y, ∫ g(t, s, y) ds) 0

(46)

0

and g(t + ω, s, ρy) = ρg(t, s, y).

(47)

We need the following auxiliary lemma: Lemma 2.2.1. Let f be continuous and locally Lipschitzian in last two coordinates. If t+ω

t

f (t + ω, ρy(t), ρ ∫ g(t, s, y(t)) ds) = ρf (t, y(t), ∫ g(t, s, y(t)) ds), 0

t ⩾ 0,

0

and y(t) is a solution of the equation t

y′ (t) = f (t, y(t), ∫ g(t, s, y(t)) ds), 0

then y ∈ Φω,ρ if and only if y(ω) = ρy(0). Proof. Let y ∈ Φω,ρ . By the definition of Φω,ρ , we have y(t + ω) = ρy(t) for all t ⩾ 0. Put t = 0, so y(ω) = ρy(0). Now, let y(ω) = ρy(0). Set x(t) := ρ−1 y(t + ω), t ⩾ 0. Using (46)–(47), we have

2.2 (ω, ρ)-periodic solutions of abstract integro-differential impulsive equations

x ′ (t) = ρ−1 y′ (t + ω)

� 51

t+ω

= ρ f (t + ω, y(t + ω), ∫ g(t + ω, s, y(t + ω)) ds) −1

0

t+ω

= ρ−1 f (t + ω, ρρ−1 y(t + ω), ∫ g(t + ω, s, ρρ−1 y(t + ω)) ds) 0

t+ω

= ρ−1 f (t + ω, ρx(t), ∫ g(t, s, ρx(t)) ds) 0

t

= ρ−1 ρf (t + ω, x(t), ∫ g(t, s, y(t)) ds) t

0

= f (t + ω, x(t), ∫ g(t, s, y(t)) ds). 0

Additionally, x(0) = ρ−1 y(ω) = y(0). Now, both y(t) and x(t), t ⩾ 0, satisfy (45) and y(0) = x(0). From the uniqueness of solutions, we conclude that x(t) = y(t), so y(t + ω) = ρy(t), t ⩾ 0. We need the following lemma: Lemma 2.2.2. Let (A1)–(A3) hold. Then the homogeneous linear impulsive evolution equation {

̇ = Ay(t), y(t)

Δy|t=τk = Bk y(t) + dk ,

t = ̸ τk , k ∈ ℕ k ∈ ℕ,

(48)

has a solution y ∈ Φω,ρ if and only if y(ω) = ρy(0) or i(τi ,ω)

i(0,ω)

(ρ − T(ω)( ∏ (E + Bk ))) y(0) = ∑ T(ω − τi ) ∏ (E + Bk )di . 0 0 such that t

∫ φp1 (u) (2F1 (t)φ(m(K))[m(K)] −t

−1 ‖φ(|h(u

− v)|)‖Lq(v) (u−K)

F(t + diam(K))

) du ⩽ 1,

for any t ⩾ t1 (ε). Then the function (h ∗ F)(⋅; ⋅) is Doss-(p1 , ϕ, F1 , ℬ, Λ′ , A)-almost periodic, resp. Doss-(p1 , ϕ, F1 , ℬ, Λ′ , A)-uniformly recurrent. Proof. We will consider only Doss-(p, ϕ, F, ℬ, Λ′ , A)-almost periodic functions. It is clear that the function (h ∗ F)(⋅; ⋅) is well-defined and supt∈ℝn ,x∈B ‖(h ∗ F)(t; x)‖Y < +∞ for all B ∈ ℬ. Let ε > 0 and B ∈ ℬ be given. Then there exists l > 0 such that for each t0 ∈ Λ′ there exists a point τ ∈ B(t0 , l)∩Λ′ such that, for every t > 0, x ∈ B and ⋅ ∈ Λt , the element y⋅;x = A(F(⋅; x)) satisfies (67). Since A is a closed linear operator and condition (B1) holds, for every t ∈ ℝn and x ∈ B, we have zt,x := A((h ∗ F)(t; x)) = ∫ℝn h(s)A(F(t − s; x)) ds. Let t > 0 and x ∈ B be fixed. The prescribed assumptions together with the well-known Jensen integral inequality and the Hölder inequality (see e. g. [431] and Lemma 1.1.10(i)) imply: 󵄩󵄩 󵄩 󵄩 󵄩󵄩ϕ(󵄩󵄩(h ∗ F)(⋅ + τ; x) − z⋅,x 󵄩󵄩󵄩 )󵄩󵄩󵄩 󵄩󵄩 󵄩 󵄩Y 󵄩󵄩 −1 󵄨 󵄨 󵄩 󵄩 ⩽ ϕ(m(K)[m(K)] ∫󵄨󵄨󵄨h(σ)󵄨󵄨󵄨 ⋅ 󵄩󵄩󵄩F(⋅ + τ − σ; x) − AF(⋅ − σ; x)󵄩󵄩󵄩Y dσ) K

󵄨 󵄨 󵄩 󵄩 ⩽ φ(m(K))[m(K)] ∫ ϕ(󵄨󵄨󵄨h(σ)󵄨󵄨󵄨 ⋅ 󵄩󵄩󵄩F(⋅ + τ − σ; x) − AF(⋅ − σ; x)󵄩󵄩󵄩Y ) dσ −1

K

󵄨 󵄨 󵄩 󵄩 ⩽ φ(m(K))[m(K)] ∫ φ(󵄨󵄨󵄨h(σ)󵄨󵄨󵄨)ϕ(󵄩󵄩󵄩F(⋅ + τ − σ; x) − AF(⋅ − σ; x)󵄩󵄩󵄩Y ) dσ −1

K

= φ(m(K))[m(K)]

−1

󵄨 󵄨 󵄩 󵄩 ∫ φ(󵄨󵄨󵄨h(⋅ − v)󵄨󵄨󵄨)ϕ(󵄩󵄩󵄩F(v + τ; x) − AF(v; x)󵄩󵄩󵄩Y ) dv ⋅−K

󵄩󵄩 󵄩 󵄨 󵄨󵄩 󵄩 󵄩󵄩 ⩽ 2φ(m(K))[m(K)] 󵄩󵄩󵄩φ(󵄨󵄨󵄨h(⋅ − v)󵄨󵄨󵄨)󵄩󵄩󵄩Lq(v) (⋅−K) 󵄩󵄩󵄩ϕ(󵄩󵄩󵄩F(v + τ; x) − AF(v; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 p(v) , 󵄩 󵄩L (⋅−K) −1 󵄩

for any ⋅ ∈ (ℝn )t . Now the final conclusion simply follows as in the proof of [454, Proposition 6] using the corresponding definition of Doss-(p1 , ϕ, F1 , ℬ, Λ′ , A)-almost periodicity and the definition of Luxemburg norm. Unfortunately, the assumption supp(h) ⊆ K for some compact subset K of ℝn is almost inevitable here, so that we cannot so easily apply Proposition 3.2.15 in the qualitative analysis of solutions of the abstract inhomogeneous heat equation in ℝn ; see [194] for more details. The statements of [304, Proposition 2.5, Proposition 2.20] and the con-

3.2 Multi-dimensional Doss ρ-almost periodic type functions

� 125

clusions from [304, Example 2.5], showing that the assumption Λ′ ⊆ Λ is redundant, can be simply formulated in our new context. Concerning the extensions of Doss ρ-almost periodic type functions (see [304, 431] for some results established for various classes of multi-dimensional ρ-almost periodic type functions), we will only note that a very simple argumentation shows that any Doss-(p, ϕ, F, ℬ, Λ′ , ρ)-almost periodic function F : Λ × X → Y can be extended to a ̃ Doss-(p, ϕ, F, ℬ, Λ′ , ρ1 )-almost periodic function F̃ : ℝn × X → Y , defined by F(t) := 0, ̃ t ∉ Λ, F(t) := F(t), t ∈ Λ, with ρ1 := ρ ∪ {(0, 0)}, if the following conditions hold (the corresponding analysis given on [456, p. 16] contains small typographical errors): (i) ϕ(⋅) is locally bounded; (ii) The Lebesgue measure of 𝜕Λ is equal to zero; (iii) For each set B ∈ ℬ, there exists a finite real number t0 > 0 such that for every t ⩾ t0 and τ ∈ Λ′ , we have 󵄩 󵄩 lim sup F(t) sup[ϕ(󵄩󵄩󵄩F(⋅ + τ; x)󵄩󵄩󵄩Y )] t→+∞

x∈B

Lp ((ℝn ∖Λ)t ∩(Λ−τ))

= 0.

We will state only one composition principle for Doss ρ-almost periodic type functions. The following result for one-dimensional Doss (p, c)-almost periodic type functions can be deduced following the lines of the proof of [409, Theorem 2.28]: Proposition 3.2.16. Suppose that 1 ⩽ p < ∞, c ∈ ℂ and F : Λ × X → Y satisfies that there exists a finite real number L > 0 such that 󵄩󵄩 󵄩 󵄩󵄩F(t; x) − F(t; y)󵄩󵄩󵄩Y ⩽ L‖x − y‖,

t ∈ Λ, x, y ∈ X.

(78)

(i) Suppose that f : Λ → X is Doss (p, Λ′ , c)-uniformly recurrent, where Λ′ := {αk : k ∈ ℕ} for some strictly increasing sequence (αk ) of positive reals tending to plus infinity. If lim lim sup

k→+∞ t→+∞

1 t

󵄩󵄩 󵄩󵄩p ∫ 󵄩󵄩󵄩F(s + αk ; cf (s)) − cF(s; f (s))󵄩󵄩󵄩 ds = 0, 󵄩 󵄩

(79)

[−t,t]∩Λ

then the mapping ℱ (t) := F(t; f (t)), t ∈ Λ is Doss (p, Λ′ , c)-uniformly recurrent. (ii) Suppose that f : Λ → X is Doss (p, Λ′ , c)-almost periodic. If for each ε > 0 the set of all positive real numbers τ > 0 such that lim sup t→+∞

1 t

󵄩 󵄩p ∫ 󵄩󵄩󵄩f (s + τ) − cf (s)󵄩󵄩󵄩 ds < ε [−t,t]∩Λ

and lim sup t→+∞

1 t

󵄩󵄩 󵄩󵄩p ∫ 󵄩󵄩󵄩F(s + τ; cf (s)) − cF(s; f (s))󵄩󵄩󵄩 ds < ε, 󵄩 󵄩

[−t,t]∩Λ

126 � 3 Multi-dimensional ρ-almost periodic type functions is relatively dense in [0, ∞), then the mapping ℱ (t) := F(t; f (t)), t ∈ Λ is Doss (p, Λ′ , c)almost periodic. We can similarly analyze the composition principles for multi-dimensional Doss c-almost periodic functions (see also [304] for related results concerning the general class of multi-dimensional ρ-almost periodic functions). In combination with Proposition 3.2.15, this enables one to analyze the existence and uniqueness of bounded, continuous, Doss-(p, c)-almost periodic solutions to the following Hammerstein integral equation of convolution type on ℝn : y(t) = ∫ k(t − s)F(s; y(s)) ds,

t ∈ ℝn ,

ℝn

where the kernel k(⋅) has compact support; see also the issue [194, 4., Section 3]. 3.2.1 Relationship between Weyl almost periodicity and Doss almost periodicity It is worth noting that Proposition 3.2.10 can be formulated for multi-dimensional ρ-almost periodic functions and their Stepanov generalizations. This is very predictable and details can be left to the interested readers. In this subsection, we would like to point out the following, much more important fact with regards to Proposition 3.2.10: It is well known that, in the one-dimensional setting the class of Doss-p-almost periodic functions provides a proper extension of the class of Besicovitch-p-almost periodic functions; see [428] for more details. On the other hand, the class of Weyl-p-almost periodic functions taken in the generalized approach of A. S. Kovanko [463] is not contained in the class of Besicovitch-p-almost periodic functions, as clearly marked in [431]. A very simple observation shows that the class of Doss-p-almost periodic functions extends the class of Weyl-p-almost periodic functions as well, which is defined in the usual way. So, let Λ = ℝ, let f (⋅) be Weyl-p-almost periodic, and let a number ε > 0 be given. Then there exists a finite real number L > 0 such that any interval Λ0 ⊆ Λ of length L contains a point τ ∈ Λ0 such that (1) holds with n = 1; hence, there exists a finite real number l0 (ε, τ) > 0 such that for every l ⩾ l0 (ε, τ) and x ∈ ℝ, we have x+l

󵄩 󵄩p ∫ 󵄩󵄩󵄩f (t + τ) − f (t)󵄩󵄩󵄩 dt ⩽ lεp . x

Plugging x = 0 and x = −l here, we easily get that for each real number l ⩾ l0 (ε, τ), we have: l

󵄩 󵄩p ∫󵄩󵄩󵄩f (s + τ) − f (s)󵄩󵄩󵄩 ds ⩽ 2lεp , −l

which simply implies that f (⋅) is Doss-p-almost periodic.

3.2 Multi-dimensional Doss ρ-almost periodic type functions

�

127

In [431, Theorem 8.3.8], we have particularly proved the following: Suppose that σ ∈ (0, 1), p ∈ [1, ∞), (1 − σ)p < 1 and a > 1 − (1 − σ)p. Define f (x) := |x|σ , x ∈ ℝ. Then the function f (⋅) is Weyl-p-almost periodic, Besicovitch p-unbounded, and has no mean value (see [431] for the notion). As a consequence, we have that a Weyl-p-almost periodic function (Doss-p-almost periodic function) is not necessarily Besicovitch-p-almost periodic; also, a Doss-p-almost periodic function f : ℝ → ℝ has no mean value and can be Besicovitch p-unbounded in general (1 ⩽ p < ∞). The above consideration can be simply extended to the multi-dimensional setting. The usual concept of multi-dimensional Weyl-p-almost periodicity is obtained by plugging p(⋅) ≡ p ∈ [1, ∞), ϕ(x) ≡ x, x ⩾ 0, 𝔽(l, t) ≡ l−n/p , l > 0, t ∈ Λ, Ω = [0, 1]n , Λ′ = Λ = [0, ∞)n or ℝn and ρ = I. The proof of following proposition is quite simple and therefore omitted (we employ almost all of the above-mentioned conditions, but we allow the situation in which Λ′ ≠ Λ and ϕ(x) is not identically equal to x for all x ⩾ 0): Proposition 3.2.17. Suppose that (WM1) holds with Λ = [0, ∞)n or ℝn , p(⋅) ≡ p ∈ [1, ∞), 𝔽(l, t) ≡ 𝔽(l), l > 0, t ∈ Λ, Ω = [0, 1]n , and ρ is single-valued on R(F). Suppose that for each l0 > 0 there exists a finite real number t0 ⩾ l0 such that n/p

(t/l0 )

⩽ 𝔽(l0 )/F(t),

t ⩾ t0 .

(80)

If F ∈ WΩ,Λ′ ,ℬ (Λ × X : Y ), then F(⋅; ⋅) is Doss-(p, ϕ, F, ℬ, Λ′ , ρ)-almost periodic. (p,ϕ,𝔽),ρ

It is worth noting that condition (80) holds in the classical situation 𝔽(l, t) ≡ l−n/p and F(t) ≡ t −n/p (l, t > 0; t ∈ Λ). We continue with the following instructive example: Example 3.2.18. Let ζ ⩾ 1 and 0ζ := 0. Define the complex-valued function ∞ 1 t fζ (t) := ∑ sinζ ( l ), l 2 l=1

t ∈ ℝ.

Then the function fζ (⋅) is Lipschitz continuous and uniformly recurrent. To prove the Lipschitz continuity of function fζ (⋅), it suffices to observe that the function t 󳨃→ sinζ (t), t ∈ ℝ is continuous and 󵄨󵄨 ζ ζ 󵄨 󵄨󵄨sin x − sin y󵄨󵄨󵄨 ⩽ ζ |x − y|,

x, y ∈ ℝ.

(81)

To see that the function fζ (⋅) is uniformly recurrent (cf. [431] for the notion), it suffices to see that for each integer k ∈ ℕ ∖ {1} we have 󵄨󵄨 󵄨 k 󵄨󵄨fζ (t + 2 π) − fζ (t)󵄨󵄨󵄨 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 t + 2k π 󵄨󵄨 1 ζ t 󵄨󵄨 = 󵄨󵄨󵄨∑ [sinζ ( ) − sin ( )] 󵄨󵄨 l l 󵄨󵄨 l 2 2 󵄨󵄨 󵄨l=1

128 � 3 Multi-dimensional ρ-almost periodic type functions 󵄨󵄨k−1 󵄨󵄨 󵄨󵄨 ∞ 󵄨󵄨 k 󵄨󵄨 1 󵄨󵄨 󵄨󵄨 1 󵄨 t + 2k π ζ t 󵄨󵄨 + 󵄨󵄨 ∑ [sinζ ( t + 2 π ) − sinζ ( t )]󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∑ [sinζ ( ) − sin ( )] 󵄨 󵄨 󵄨󵄨 l l l l 󵄨󵄨 󵄨 󵄨 2 2 2 2 󵄨󵄨 󵄨󵄨l=k l 󵄨󵄨 󵄨 l=1 l 󵄨󵄨 ∞ 󵄨 󵄨 󵄨 󵄨󵄨 ∞ 1 󵄨󵄨 󵄨󵄨 1 t + 2k π t t + 2k π t 󵄨󵄨󵄨 󵄨 󵄨 = 󵄨󵄨󵄨 ∑ [sinζ ( ) − sinζ ( l )]󵄨󵄨󵄨 ⩽ ∑ 󵄨󵄨󵄨sinζ ( ) − sinζ ( l )󵄨󵄨󵄨 l l 󵄨󵄨 󵄨 2 2 2 2 󵄨󵄨󵄨 󵄨󵄨󵄨l=k l 󵄨 l=k l 󵄨󵄨 ∞ ζ 2πζ ⩽ ∑ 2k−l π = , t ∈ ℝ, l k l=k where we have applied (81) in the last line of computation. In the case that ζ = 2v for some integer v ∈ ℕ, we have that the function fζ (⋅) is Besicovitch unbounded. This can be inspected as in the proof of [358, Theorem 1.1], with the additional observation that 2k−l π

2k−l π

2 (2v − 1)!! ∫ sin t dt = ∫ sin2 t dt 3 (2v)!! 2v

0

(k ∈ ℕ ∖ {1}, 1 ⩽ l ⩽ k);

0

here, we have used the well-known recurrent formula 2k−l π

2k−l π

2v − 1 ∫ sin t dt = ∫ sin2v−2 t dt, 2v 2v

0

0

which can be deduced with the help of the partial integration. We would like to ask whether the function fζ (⋅) is Besicovitch unbounded in general case and for which functions p ∈ D+ (ℝ) we have that f (⋅) is Doss-p(⋅)-almost periodic, i. e. Doss-(p(⋅), 1)-almost periodic? We continue by observing that the functions of the form ∞

f (t) := ∑ al sinζl ( l=1

t ), bl

t ∈ ℝ,

where ζl ⩾ 0 for all l ∈ ℕ, and (al ) and (bl ) are real sequences such that the above series is absolutely convergent, are still very unexplored within the theory of almost periodic functions. For example, we know that the sequence of partial sums of the series (therefore, the sequence of trigonometric polynomials) ∞ 1 x x 󳨃→ f (x) := ∑ sin , l l l=1

x∈ℝ

is a Cauchy sequence with respect to the Weyl metric W 2 but its sum is not equi-Weyl-2almost periodic; see e. g. [495, p. 247] and [428] for the notion. On the other hand, using the identity sin

x+τ x x τ x τ τ − sin = 2[cos cos − sin sin ] sin , l l l 2l l 2l 2l

x ∈ ℝ, l ∈ ℕ, τ ∈ ℝ,

3.2 Multi-dimensional Doss ρ-almost periodic type functions

� 129

and the argumentation contained in the proof of [442, Theorem 1.1], it readily follows that for each τ ∈ ℝ and p ⩾ 1 we have t

1 󵄨󵄨 󵄨p ∫󵄨f (x + τ) − f (x)󵄨󵄨󵄨 dx = 0. t→+∞ t 󵄨 lim

−t

In particular, the function f (⋅) is Doss-p-almost periodic for any finite exponent p ⩾ 1. We would like to ask whether the function f (⋅) is equi-Weyl-p-almost periodic for some exponent p ∈ [1, 2) or Weyl-p-almost periodic for some finite exponent p ⩾ 1? We close this subsection with the observation that for every finite exponent p ⩾ 1, there exists a (Besicovitch-)Doss-p-almost periodic function f : ℝ → ℝ which is not Weyl-p-almost periodic; see e. g. [42, Example 6.24]. 3.2.2 Invariance of Doss ρ-almost periodicity under the actions of convolution products This subsection investigates the invariance of Doss ρ-almost periodicity under the actions of infinite convolution products (for simplicity, we will not consider here the finite convolution products). From the application point of view, the one-dimensional framework is the most important, and here we will only note that the established results admit straightforward extensions for the infinite convolution product (20) and the finite convolution product tn

t1 t2

t 󳨃→ ∫ ∫ ⋅ ⋅ ⋅ ∫ R(t − s)f (s) ds, α1 α2

αn

t ∈ [0, ∞)n ;

see [194] and [431] for more details. We start by stating the following extension of [454, Theorem 3]: Theorem 3.2.19. Suppose that ψ : (0, ∞) → (0, ∞), φ : [0, ∞) → [0, ∞), ϕ : [0, ∞) → [0, ∞) is a convex monotonically increasing function satisfying ϕ(xy) ⩽ φ(x)ϕ(y) for all x, y ⩾ 0 and p ∈ 𝒫 (ℝ). Suppose further 0 ≠ Λ′ ⊆ ℝ, A is a closed linear operator commuting with R(⋅), f ̌ : ℝ → X is Doss-(p, ϕ, F, Λ′ , A)-almost periodic, resp. Doss-(p, ϕ, F, Λ′ , A)uniformly recurrent, and measurable, F1 : (0, ∞) → (0, ∞), q ∈ 𝒫 (ℝ), 1/p(x) + 1/q(x) = 1, (R(t))t>0 ⊆ L(X) is a strongly continuous operator family and for every real number x ∈ ℝ we have ∞

󵄩 󵄩󵄩 󵄩 ∫ 󵄩󵄩󵄩R(v + x)󵄩󵄩󵄩󵄩󵄩󵄩f ̌(v)󵄩󵄩󵄩 dv < ∞

−x

and

(82)

130 � 3 Multi-dimensional ρ-almost periodic type functions ∞

󵄩 󵄩󵄩 󵄩 ∫ 󵄩󵄩󵄩R(v + x)󵄩󵄩󵄩󵄩󵄩󵄩Af ̌(v)󵄩󵄩󵄩 dv < ∞.

(83)

−x

Suppose further that for each ε > 0 there exist an increasing sequence (am ) of positive real numbers tending to plus infinity and a number t0 (ε) > 0 satisfying that for every t ⩾ t0 (ε), we have t −1 󵄩 󵄩 −1 F1 (t) lim sup[[φ(󵄩󵄩󵄩R(⋅ + x)󵄩󵄩󵄩)]Lq(⋅) [−x,−x+a ] F(t + am ) ]) dx ⩽ 1. ∫ φp(x) (2φ(am )am m→+∞

−t

m

Then the function F : ℝ → X, given by (4), is well-defined and Doss-(p, ϕ, F1 , Λ′ , A)-almost periodic, resp. Doss-(p, ϕ, F1 , Λ′ , A)-uniformly recurrent. ∞ Proof. It is clear that F(x) = ∫−x R(v + x)f ̌(v) dv, x ∈ ℝ; hence, (82) implies that the function F(⋅) is well-defined as well as that the integrals in the definitions of F(x) and F(x+τ)− F(x) converge absolutely (x ∈ ℝ). Furthermore, since A is a closed linear operator com∞ muting with R(⋅), and since we have assumed (83), we have AF(x) = ∫−x R(v + x)Af ̌(v) dv, x ∈ ℝ. The remainder of proof is almost the same as the proof of the corresponding part of [454, Theorem 3], with the distance f ̌(v + τ) − f ̌(v) replaced therein with the distance f ̌(v + τ) − Af ̌(v).

Using a similar argumentation and inspecting carefully the proof of [428, Theorem 2.13.10], we may conclude that the following result holds true: Theorem 3.2.20. Let 0 ≠ Λ′ ⊆ ℝ, 1/p + 1/q = 1 and (R(t))t>0 ⊆ L(X) satisfy β−1 󵄩󵄩 󵄩 Mt , 󵄩󵄩R(t)󵄩󵄩󵄩 ⩽ 1 + tγ

t > 0 for some finite constants γ > β, β ∈ (0, 1], M > 0.

Let A be a closed linear operator commuting with R(⋅), let a function g : ℝ → X be Doss-(p, Λ′ , A)-almost periodic, resp. Doss-(p, Λ′ , A)-uniformly recurrent, and Stepanov p-bounded, and let q(β − 1) > −1 provided that p > 1, resp. β = 1, provided that p = 1. Assume that the function t 󳨃→ Ag(t), t ∈ ℝ is Stepanov p-bounded. Then the function G : ℝ → X, defined through (4) with f = g therein, is bounded, continuous and Doss-(p, Λ′ , A)-almost periodic, resp. Doss-(p, Λ′ , A)-uniformly recurrent. Furthermore, if g(⋅) is Bp -continuous, then G(⋅) is Bp -continuous, as well. Remark 3.2.21. If A = cI for some c ∈ ℂ, then we can consider two different pivot spaces X and Y in Theorem 3.2.19 and Theorem 3.2.20. See also [428, Theorem 2.13.7], where we have used the estimate +∞

󵄩 󵄩 ∫ (1 + t)󵄩󵄩󵄩R(t)󵄩󵄩󵄩 dt < +∞, 0

which cannot be basically satisfied for fractional solution operator families.

3.2 Multi-dimensional Doss ρ-almost periodic type functions

� 131

3.2.3 Applications to the abstract Volterra integro-differential equations and partial differential equations In this subsection, we aim to present some applications of our abstract results to the abstract Volterra integro-differential equations and the partial differential equations. 1. We start by observing that our results about the invariance of Doss ρ-almost periodicity under the actions of convolution products, established in Section 3.2.2, can be applied in the analysis of the existence and uniqueness of Doss-(p, A)-almost periodic solutions in the time variable for various kinds of the abstract (degenerate) Volterra integro-differential equations (see, e. g. [428] for more details). For example, we can apply Theorem 3.2.20 in the analysis of the existence and uniqueness of Doss (p, c)-almost periodic solutions of the following fractional Poisson heat equation with Weyl–Liouville fractional derivatives: γ

Dt,+ [m(x)v(t, x)] = (Δ − b)v(t, x) + f (t, x),

t ∈ ℝ, x ∈ Ω,

where γ ∈ (0, 1), 1 ⩽ p < ∞ and c ∈ ℂ; possible applications can be also given to the higher-order differential operators in Hölder spaces. All this has been seen many times and details can be omitted. 2. It is worth noting that Proposition 3.2.13, Proposition 3.2.16 and Theorem 3.2.20 can be implemented in the analysis of the existence and uniqueness of Doss-(p, Λ′ , c)-uniformly recurrent solutions for various classes of abstract fractional semilinear Cauchy inclusions and equations. Suppose, for instance, that γ ∈ (0, 1), a closed multivalued linear operator 𝒜 on X satisfies all requirements from [428, Subsection 2.9.2] and the solution family Pγ (⋅) is defined as there. Define Rγ (t) := t γ−1 Pγ (t), t > 0. Then we know that ‖Rγ (t)‖ = O(t γ−1 /(1 + t 2γ )), t > 0. Let p ∈ (1, ∞), let 1/p + 1/q = 1, and let q(γ − 1) > −1. Fix now a strictly increasing sequence (αk ) of positive reals tending to plus infinity, and define BCD(αk );c (ℝ : X) := {f : ℝ → X ; f (⋅) is bounded and (p, Λ′ , c)-uniformly recurrent}, where Λ′ := {αk : k ∈ ℕ}. By Proposition 3.2.13(iv), the set BCD(αk );c (ℝ : X) equipped with the metric d(⋅, ⋅) := ‖ ⋅ − ⋅ ‖∞ is a complete metric space. Suppose now that a mapping F : Λ × X → Y satisfies the estimate (79). We say that a continuous function u : ℝ → X is a mild solution of the semilinear Cauchy inclusion γ

Dt,+ u(t) ∈ 𝒜u(t) + F(t; u(t)), t ∈ ℝ, if and only if t

u(t) = ∫ Rγ (t − s)F(s; u(s)) ds, −∞

t ∈ ℝ.

(84)

132 � 3 Multi-dimensional ρ-almost periodic type functions Keeping in mind Proposition 3.2.16 and Theorem 3.2.20, we can simply prove the following analogue of [409, Theorem 3.1]: Theorem 3.2.22. Suppose that the above requirements hold as well as that the function F : ℝ × X → X satisfies that for each bounded subset B of X there exists a finite real constant MB > 0 such that supt∈ℝ supx∈B ‖F(t; x)‖ ⩽ MB . If there exists a finite real number L > 0 such that (78) holds, and there exists an integer k ∈ ℕ such that Mk < 1, where xk

x2

n

−∞ −∞

−∞

i=2

t

󵄩 󵄩 󵄩 󵄩 Mk :=Lk sup ∫ ∫ ⋅ ⋅ ⋅ ∫ 󵄩󵄩󵄩Rγ (t − xk )󵄩󵄩󵄩 ∏󵄩󵄩󵄩Rγ (xi − xi−1 )󵄩󵄩󵄩 dx1 dx2 ⋅ ⋅ ⋅ dxk , t⩾0

then the abstract semilinear fractional Cauchy inclusion (84) has a unique bounded Doss-(p, Λ′ , c)-uniformly recurrent solution which belongs to the space BCD(αk );c (ℝ : X). 3. In this issue, we continue our analysis of the famous d’Alembert formula. Let a > 0; then we know that the regular solution of the wave equation utt = a2 uxx in domain {(x, t) : x ∈ ℝ, t > 0}, equipped with the initial conditions u(x, 0) = f (x) ∈ C 2 (ℝ) and ut (x, 0) = g(x) ∈ C 1 (ℝ), is given by the d’Alembert formula x+at

1 1 u(x, t) = [f (x − at) + f (x + at)] + ∫ g(s) ds, 2 2a x−at

x ∈ ℝ, t > 0.

(85)

Suppose now that the function x 󳨃→ (f (x), g [1] (x)), x ∈ ℝ is Doss-(p, c)-almost periodic ⋅ for some p ∈ [1, ∞) and c ∈ ℂ, where g [1] (⋅) ≡ ∫0 g(s) ds. Clearly, the solution u(x, t) can be extended to the whole real line in the time variable; we will prove that the solution u(x, t) is Doss-(p, c)-almost periodic in (x, t) ∈ ℝ2 . In actual fact, we have (x, t, τ1 , τ2 ∈ ℝ): 󵄨󵄨 󵄨 󵄨󵄨u(x + τ1 , t + τ2 ) − cu(x, t)󵄨󵄨󵄨 󵄨󵄨 1 󵄨󵄨 ⩽ 󵄨󵄨󵄨f ((x − at) + (τ1 − aτ2 )) − cf (x − at)󵄨󵄨󵄨 󵄨 2󵄨 󵄨󵄨 1 󵄨󵄨󵄨 + 󵄨󵄨f ((x + at) + (τ1 + aτ2 )) − cf ([x + at + (τ1 + aτ2 )] − (τ1 + aτ2 ))󵄨󵄨󵄨 󵄨 󵄨 2 󵄨 1 󵄨󵄨󵄨 [1] 󵄨 + 󵄨󵄨g ((x − at) + (τ1 − aτ2 )) − cg [1] (x − at)󵄨󵄨󵄨 󵄨 2a 󵄨 󵄨󵄨 1 󵄨󵄨󵄨 [1] + 󵄨󵄨g ((x + at) − −(τ1 − aτ2 )) − cg [1] (x + at)󵄨󵄨󵄨. 󵄨 2a 󵄨

(86)

l

If τ1 − aτ2 satisfies that lim supl→+∞ (1/l) ∫−l |f (v + τ1 − aτ2 ) − f (v)|p dv ⩽ εp , then there l

exists a finite real number l0 (ε, τ1 , τ2 ) > 0 such that ∫−l |f (v + τ1 − aτ2 ) − f (v)|p dv ⩽ εp l, l ⩾ l0 (ε, τ1 , τ2 ) and therefore 󵄨󵄨 󵄨󵄨p ∫ 󵄨󵄨󵄨f ((x − at) + (τ1 − aτ2 )) − cf (x − at)󵄨󵄨󵄨 dx dt 󵄨 󵄨

|(x,t)|⩽l

3.2 Multi-dimensional Doss ρ-almost periodic type functions

�

133

l l

󵄨󵄨 󵄨󵄨p ⩽ ∫ ∫󵄨󵄨󵄨f ((x − at) + (τ1 − aτ2 )) − cf (x − at)󵄨󵄨󵄨 dx dt 󵄨 󵄨 −l −l l

l

󵄨󵄨 󵄨󵄨p = ∫[∫󵄨󵄨󵄨f ((x − at) + (τ1 − aτ2 )) − cf (x − at)󵄨󵄨󵄨 dt] dx 󵄨 󵄨 −l −l l

=

x+al

󵄨󵄨 󵄨󵄨p 1 ∫[ ∫ 󵄨󵄨󵄨f (v + (τ1 − aτ2 )) − cf (v)󵄨󵄨󵄨 dv] dx 󵄨 󵄨 a −l x−al l

⩽

l(1+a)

󵄨󵄨 󵄨󵄨p 1 ∫[ ∫ 󵄨󵄨󵄨f (v + (τ1 − aτ2 )) − cf (v)󵄨󵄨󵄨 dv] dx 󵄨 󵄨 a −l −l(1+a)

l

⩽

1 p 1 ε l(1 + a) ∫ dx = εp l2 (1 + a), a a

l ⩾ (1 + a)−1 l0 (ε, τ1 , τ2 ),

−l

where we have applied the Fubini theorem in the third line of computation. The remaining three addends in (86) can be estimated similarly, so that the final conclusion simply follows as in the final part of [409, Example 1.2]. 3. In [431], we have recently the existence and uniqueness of c-almost periodic type solutions of the wave equation (62) in ℝ3 , where d > 0, g ∈ C 3 (ℝ3 : ℝ) and h ∈ C 2 (ℝ3 : ℝ). Let us recall that the famous Kirchhoff formula (see, e. g. [653, Theorem 5.4, pp. 277–278]; we will use the same notion and notation) says that the function u(t, x) := =

𝜕 1 [ 𝜕t 4πd 2 t

∫ g(σ) dσ] + 𝜕Bdt (x)

1 4πd 2 t

∫ g(σ) dσ 𝜕Bdt (x)

1 dt ∫ g(x + dtω) dω + ∫ ∇g(x + dtω) ⋅ ω dω 4π 4π 𝜕B1 (0)

t + ∫ h(x + dtω) dω 4π 𝜕B1 (0)

:= u1 (t, x) + u2 (t, x) + u3 (t, x),

𝜕B1 (0)

t ⩾ 0, x ∈ ℝ3 ,

is a unique solution of problem (62) which belongs to the class C 2 ([0, ∞) × ℝ3 ). Let us fix now a number t0 > 0. Then the function x 󳨃→ u(t0 , x), x ∈ ℝ3 is Doss-(1, x, F, Λ′ , c)-almost periodic (Doss-(1, x, F, Λ′ , c)–uniformly recurrent) provided that the functions g(⋅), ∇g(⋅) and h(⋅) are of the same type (0 ≠ Λ′ ⊆ ℝ3 ; c ∈ ℂ). This is a simple consequence of the following computation, given here only for the function u3 (t, ⋅): 󵄨 󵄨 ∫ 󵄨󵄨󵄨u3 (t, x + τ) − cu3 (x, t)󵄨󵄨󵄨 dx |x|⩽l

134 � 3 Multi-dimensional ρ-almost periodic type functions

⩽

t ∫ 4π

󵄨 󵄨 ∫ 󵄨󵄨󵄨h(x + τ + dtω) − ch(x + dtω)󵄨󵄨󵄨 dω dx

t ∫ 4π

󵄨 󵄨 ∫ 󵄨󵄨󵄨h(x + τ + dtω) − ch(x + dtω)󵄨󵄨󵄨 dx dω

|x|⩽l 𝜕B1 (0)

=

𝜕B1 (0) |x|⩽l

tε ⩽ ∫ dω, 4πF(l) 𝜕B1 (0)

provided that l − dt ⩾ l0 (ε, τ), the last being determined from the Doss-(1, x, F, Λ′ , c)almost periodicity of function h(⋅) with a number ε > 0 given in advance. We can similarly analyze the existence and uniqueness of Doss-(1, x, F, Λ′ , c)-almost periodic (Doss-(1, x, F, Λ′ , c)–uniformly recurrent) solutions of the wave equation (63) in ℝ2 , where d > 0, g ∈ C 3 (ℝ2 : ℝ) and h ∈ C 2 (ℝ2 : ℝ). For the wave equation in higher dimensions, we also refer the reader to the book chapter [278] by P. Drábek and G. Holubová; cf. also https://web.stanford.edu/class/math220a/handouts/waveequation3.pdf. We close the section with the observation that we can further extend the notion introduced in Definition 3.2.1 by allowing that the function F(t) depends not only on t > 0 but also on τ ∈ Λ′ . For example, we can consider the following notion (with the exception of assumption F : (0, ∞) → (0, ∞), which is replaced by the assumption F : (0, ∞) × Λ′ → (0, ∞) here, we retain all remaining standing assumptions of ours): Definition 3.2.23. We say that the function F(⋅; ⋅) is Doss-(p, ϕ, F, ℬ, Λ′ , ρ)-almost periodic if and only if, for every B ∈ ℬ and ε > 0, there exists l > 0 such that for each t0 ∈ Λ′ there exists a point τ ∈ B(t0 , l) ∩ Λ′ such that, for every t > 0, x ∈ B and ⋅ ∈ Λt we have the existence of an element y⋅;x ∈ ρ(F(⋅; x)) such that 󵄩 󵄩 lim sup F(t, λ) sup[ϕ(󵄩󵄩󵄩F(⋅ + τ; x) − y⋅;x 󵄩󵄩󵄩Y )] t→+∞

Lp(⋅) (Λt )

x∈B

< ε.

In actual fact, it is very important to assume sometimes that the function F depends also on τ ∈ Λ′ . We will illustrate this fact by considering the second-order partial differential equation Δu = −f , where f ∈ C 2 (ℝ3 ) has a compact support. Let us recall that the Newtonian potential of f (⋅), defined by u(x) :=

f (x − y) 1 dy, ∫ 4π |y|

x ∈ ℝ3 ,

ℝ3

is a unique function belonging to the class C 2 (ℝ3 ), vanishing at infinity and satisfying Δu = −f ; see also [653, Theorem 3.9, pp. 126–127]. In our next application, we will assume that ρ = cI for some c ∈ ℂ, p(⋅) ≡ 1 and ϕ(x) ≡ x, as well as that 0 ≠ Λ′ ⊆ Λ = ℝ3 and there exists a finite real number d > 0 such that

3.3 Multi-dimensional Besicovitch almost periodic type functions and applications

F1 (t, τ) ⩾ d[F(t, τ) + sup ⋅∈Λt

∫ (⋅+τ−K)∖(⋅−K)

dy ], |y|

t > 0, λ ∈ Λ′ .

�

135

(87)

Then we have the following: Theorem 3.2.24. Suppose that f is Doss-(1, ϕ, F, Λ′ , c)-almost periodic and supp(f ) ⊆ K. Then u is Doss-(1, ϕ, F1 , Λ′ , c)-almost periodic. Proof. Let ε > 0 be given, and let τ ∈ Λ′ be as in Definition 3.2.23. Using the Fubini theorem, we have the existence of a finite real number t1 (ε, τ) > 0 such that: 󵄩󵄩 󵄩 󵄩󵄩u(x + τ) − cu(x)󵄩󵄩󵄩L1 (Λ ) t

|f (x + τ − y) − cf (x − y)| 1 ⩽ dy dx ∫∫ 4π |y| Λt ℝ3

=

dy 1 󵄨 󵄨 ∫ [∫󵄨󵄨󵄨f (x + τ − y) − cf (x − y)󵄨󵄨󵄨 dx] 4π |y| ℝ3 Λt

=

dy 1 󵄨 󵄨 ∫ [∫󵄨󵄨󵄨f (x + τ − y) − cf (x − y)󵄨󵄨󵄨 dx] 4π |y| x−K Λt

+

1 4π

∫

dy 󵄨 󵄨 [∫󵄨󵄨󵄨f (x + τ − y) − cf (x − y)󵄨󵄨󵄨 dx] |y|

(x+τ−K)∖(x−K) Λt

1 ε 1 ε ⩽ m(K) + 4π F(t, τ) 4π F(t, τ)

∫

∫

(x+τ−K)∖(x−K) Λt

dy , |y|

t ⩾ t1 (ε, τ).

Keeping in mind the assumption (87) and the notion introduced above, this simply implies the required statement. We can similarly analyze the two-dimensional analogue of this example by considering the logarithmic potential of f (⋅); see also [653, Remark 3.7, p. 128] and [431].

3.3 Multi-dimensional Besicovitch almost periodic type functions and applications In this section, we analyze multi-dimensional Besicovitch almost periodic type functions. We clarify the main structural properties for the introduced classes of Besicovitch almost periodic type functions, explore the notion of Besicovitch–Doss almost periodicity in the multi-dimensional setting, and provide some applications of our results to the abstract Volterra integro-differential equations and the partial differential equations.

136 � 3 Multi-dimensional ρ-almost periodic type functions p

Let us recall that for each F ∈ Lloc (ℝn : X) we set ‖F‖ℳp := lim sup[ t→+∞

1 (2t)n

1/p

󵄩 󵄩p ∫ 󵄩󵄩󵄩F(s)󵄩󵄩󵄩 ds]

.

[−t,t]n

p

Then ‖ ⋅ ‖ℳp is a seminorm on the space ℳp (ℝn : X) consisting of those Lloc (ℝn : X)functions F(⋅) such that ‖F‖ℳp < ∞. Define Kp (ℝn : X) := {f ∈ ℳp (ℝn : X) ; ‖F‖ℳp = 0} and Mp (ℝn : X) := ℳp (ℝn : X)/Kp (ℝn : X). The seminorm ‖ ⋅ ‖ℳp on ℳp (ℝn : X) induces the norm ‖ ⋅ ‖M p on M p (ℝn : X) under which M p (ℝn : X) is complete; as a conclusion, we have that (M p (ℝn : X), ‖ ⋅ ‖M p ) is a p Banach space. We say that a function F ∈ Lloc (ℝn : X) is Besicovitch-p-almost periodic if and only if there exists a sequence of trigonometric polynomials converging to F(⋅) in (M p (ℝn : X), ‖ ⋅ ‖M p ). The vector space consisting of all Besicovitch-p-almost periodic functions is denoted by Bp (ℝn : X). Clearly, Bp (ℝn : X) is a closed subspace of M p (ℝn : X) and therefore a Banach space itself. Concerning the Banach space M p (ℝn : X), we would like to recall that this space is not separable for any finite exponent p ⩾ 1; see, e. g. [615, Theorem 18], which concerns the one-dimensional case. For further information about Besicovitch almost periodic functions, Besicovitch almost automorphic functions and their applications, we refer the reader to [39, 40, 66, 70, 94, 122, 126, 127, 129, 130, 147], [166, 167, 184, 185, 247, 313, 321, 342, 343, 368], [376, 492, 503, 535, 564, 601, 614, 615, 656] and references cited therein; we would like to specially emphasize here the important research monograph [597] by A. A. Pankov. The spatially Besicovitch almost periodic solutions for certain classes of nonlinear second-order elliptic equations, single higher-order hyperbolic equations and nonlinear Schrödinger equations have been investigated in the fifth chapter of this monograph. For the basic source of information about homogenization in algebras with mean value and generalized Besicovitch spaces (the work of J. L. Woukeng and his coauthors), we refer the reader to [431, Part II, Chapter 9, pp. 619–621]. Concerning some other subjects with respect to the Besicovitch classes of almost periodic functions, we would like to note the following: (i) The Sobolev–Besicovitch spaces for traces of almost periodic functions have been investigated by A. M. Bresani and G. Dell’acqua in [159]. (ii) Direct and inverse approximation theorems in the spaces of Besicovitch–Museilak–Orlicz almost periodic functions have been analyzed by S. Chaichenko, A. Shidlich, and T. Shulyk in [187]. (iii) The dual spaces of the Besicovitch almost periodic function spaces have been investigated by E. Følner in [322].

3.3 Multi-dimensional Besicovitch almost periodic type functions and applications

� 137

(iv)

The asymptotic decay to the mean-value of essentially bounded Besicovitch almost periodic solutions to nonlinear anisotropic degenerate parabolic-hyperbolic equations has been analyzed by H. Frid and Y. Li in [328]. (v) The embedding theorems for Besicovitch almost periodic functions have been analyzed by Yu. Kh. Khasanov in [413]. (vi) The Besicovitch almost periodic solutions for fractional-order quaternion-valued neural networks with discrete and distributed delays have been analyzed by Y. Li, M. Huang, and B. Li in [100]. See also the research article [735], where W. Qi and Y. Li have analyzed the almost anti-periodic oscillation excited by external inputs and synchronization of Clifford-valued recurrent neural networks. (vii) The Hausdorff–Young theorem for the Besicovitch–Orlicz space of almost periodic functions has been investigated by M. Morsli and D. Drif in [565]. (viii) The Cauchy problem for scalar conservation laws in the class of Besicovitch almost periodic functions has been considered by E. Yu. Panov in [601]. The concept Besicovitch-p-almost periodicity has not been well explored for the functions of the form F : Λ → X, where 0 ≠ Λ ⊆ ℝn and Λ ≠ ℝn (some particular results in the one-dimensional setting are given in the monograph [428], with Λ = [0, ∞)). This fact has strongly influenced us to write the paper [450], from which the material of this section is picked up. It is worth noting that this is probably the first research article, which examined the existence and uniqueness of Besicovitch-p-almost periodic solutions for certain classes of PDEs on some proper subdomains of ℝn ; even the most simplest examples of quasi-linear partial differential equations of first order considered here vividly exhibit the necessity of further analyses of Besicovitch-p-almost periodic functions which are not defined on the whole Euclidean space ℝn . The organization and main ideas of this section can be briefly described as follows. The main aim of Section 3.3.1 is to introduce and analyze various classes of multidimensional Besicovitch almost periodic type functions. We start this subsection by introducing the class e − (ℬ, ϕ, F) − Bp(⋅) (Λ × X : Y ), where Λ is a general non-empty subset of ℝn , p ∈ 𝒫 (Λ), ϕ : [0, ∞) → [0, ∞) is Lebesgue measurable and F : (0, ∞) → (0, ∞); see Definition 3.3.3, which is crucial for our further work. The class PAP0,p (Λ, ℬ, F, ϕ) of weighted ergodic components, introduced recently in [431, Definition 6.4.13], makes a proper subclass of the class e − (ℬ, ϕ, F) − Bp(⋅) (Λ × X : Y ) since its definition is obtained by plugging the trivial sequence (Pk ≡ 0) of trigonometric polynomials in Definition 3.3.3 (we omit the term “ℬ” from the notation for the functions of the form F : Λ → Y ). Before proceeding any further, we would like to note that there is a large class of PDEs of first order (second order) whose solutions are not Besicovitch almost periodic in ℝn and which belong to the class PAP0,p (Λ, ℬ, F, ϕ), where 0 ≠ Λ ⊊ ℝn . For instance, we have the following: Example 3.3.1. (i) Let a and c be two non-zero real numbers; then the classical C 1 solution of the equation aux + cu = 0 is given by u(x, y) = g(y)e−(c/a)x , (x, y) ∈ ℝ2 .

138 � 3 Multi-dimensional ρ-almost periodic type functions Keeping in mind the notion introduced after Definition 3.2.1, it can be simply shown that any non-trivial solution of this equation cannot be Besicovitch almost periodic in ℝ2 as well as that the solution always belongs to the class PAP0,1 ([0, ∞) × ℝ, F, x) provided that the function g(⋅) is Besicovitch bounded and limt→+∞ F(t) = 0 (see also Example 3.3.4 and Example 3.3.7 below). (ii) The solutions of second-order PDEs on rectangular domains, obtained by the wellknown method of separation of variables, can belong to the space PAP0,p (Λ, ℬ, F, ϕ). Consider, for the illustration purposes, the heat equation ut (x, t) = uxx (x, t) on the domain Λ = [0, 2π] × [0, ∞), equipped with the initial conditions u(x, 0) = f (x) ∈ L1 [0, 2π] and u(0, t) = u(2π, t) = 0. A unique solution of this problem is given by ∞

k2

u(x, t) = ∑ bk sin(kx/2)e− 4 t , k=1

(x, t) ∈ Λ,

where f (x) = ∑k=1 bk sin(kx/2), x ∈ [0, 2π]. Since there exists a finite real constant M > 0 such that |bk | ⩽ M for all k ∈ ℕ, a simple computation shows that u(x, t) ∈ PAP0,1 (Λ, t −ζ , x) for any real number ζ > 0. We provide some structural characterizations of class e −(ℬ, ϕ, F)−Bp(⋅) (Λ×X : Y ) in Proposition 3.3.5, Theorem 3.3.6 and Proposition 3.3.8 (a special attention is paid to the case in which ϕ(x) ≡ x α for some α > 0). A composition principle for multi-dimensional Besicovitch almost periodic functions is clarified in Theorem 3.3.11. In Section 3.3.2, we investigate the notion of multi-dimensional Besicovitch normality. The class of Besicovitch-(R, ℬ, ϕ, F) − Bp(⋅) -normal functions is introduced in Definition 3.3.12 and characterized after that in Proposition 3.3.13, Proposition 3.3.14 and Proposition 3.3.15. In Section 3.3.3, we consider the multi-dimensional analogues of the important research results established by R. Doss in [275, 276]. We pay special attention to the analysis of conditions (A), (A)∞ and (AS); see Theorem 3.3.18, Proposition 3.3.20 and Proposition 3.3.22 for some results obtained in this direction. Section 3.3.4 is focused on the analysis of condition (B); the main results obtained in this part are Proposition 3.3.23 and Proposition 3.3.24. We feel it is our duty to emphasize that the above-mentioned results of R. Doss are primarily intended for the analysis of one-dimensional Besicovitch almost periodic type functions, as well as that we have faced ourselves with many serious problems concerning the multi-dimensional extensions of these results. Some applications to the abstract Volterra integro-differential equations are furnished in Section 3.3.5; it is worth noting that we establish here some new results about the convolution invariance of Besicovitch-p-almost periodicity under the actions of infinite convolution products, and a new result concerning the usually considered convolution invariance of Besicovitch-p-almost periodicity. We provide some new applications in the one-dimensional setting, a new application in the analysis of the existence and uniqueness of Besicovitch-p-almost periodic solutions of the abstract semilinear fractional Cauchy inclusions and the abstract nonautonomous differential equations of first

3.3 Multi-dimensional Besicovitch almost periodic type functions and applications

�

139

order, some new applications to the inhomogeneous heat equation in ℝn and evolution systems generated by the bounded perturbations of the Dirichlet Laplacian. Concerning some open problems proposed, we would like to notice that we introduce the notion of admissibility with respect to the class 𝒞Λ of Besicovitch almost periodic type functions in Definition 3.3.17, and propose after that an open problem concerning the extensions of Besicovitch almost periodic type functions to the whole Euclidean space ℝn ; a similar question has recently been posed for the class of (equi-)Weyl almost periodic type functions in [431]. In addition to the above, we provide numerous illustrative examples and remarks about the multi-dimensional Besicovitch almost periodic type functions under our consideration. For the sequel, we need the following result, which can be deduced in almost the same way as in the proof of [275, Proposition 2]: Lemma 3.3.2. Suppose that the function F : ℝn → X is Bohr almost periodic. If for every a1 ≠ 0, . . . , an ≠ 0 we have (a = (a1 , . . . , an )): 1 k−1 ∑ F(t + (j − 1)a) = 0, k→+∞ k j=0 lim

uniformly in t ∈ ℝn , then F ≡ 0. Although it could be of some importance, we will not discuss here the question whether the statement of Lemma 3.3.2 can be extended to the almost automorphic functions (the uniformly recurrent functions).

3.3.1 Multi-dimensional Besicovitch almost periodic type functions Suppose that Λ is a general non-empty subset of ℝn as well as that p ∈ 𝒫 (Λ), the function ϕ : [0, ∞) → [0, ∞) is Lebesgue measurable and F : (0, ∞) → (0, ∞). Set Λ′′ := {τ ∈ ℝn : τ + Λ ⊆ Λ}. Unless stated otherwise, we assume henceforth that 0 ≠ Ω ⊆ ℝn is a compact set with positive Lebesgue measure, as well as that Λ + lΩ ⊆ Λ for all l > 0, and 0 ≠ Λ′ ⊆ Λ′′ , i. e., Λ + Λ′ ⊆ Λ. In this subsection, we investigate the multidimensional Besicovitch almost periodic type functions, paying a special attention to the class e−(ℬ, ϕ, F)−Bp(⋅) (Λ×X : Y ) and the class of Besicovitch-(R, ℬ, ϕ, F)−Bp(⋅) -normal functions. Recall, a trigonometric polynomial P : Λ × X → Y is any linear combination of functions like (t; x) 󳨃→ ei⟨λ,t⟩ c(x), where c : X → Y is a continuous function; a continuous function F : Λ × X → Y is said to be strongly ℬ-almost periodic if and only if for every B ∈ ℬ we can find a sequence (PkB (⋅; ⋅))k∈ℕ of trigonometric polynomials which converges to F(⋅; ⋅), uniformly on Λ × B. We omit the term “ℬ” from the notation if X = {0}. We are ready to introduce the following notion:

140 � 3 Multi-dimensional ρ-almost periodic type functions Definition 3.3.3. Suppose that F : Λ × X → Y , ϕ : [0, ∞) → [0, ∞) and F : (0, ∞) → (0, ∞). Then we say that the function F(⋅; ⋅) belongs to the class e − (ℬ, ϕ, F) − Bp(⋅) (Λ × X : Y ) if and only if for each set B ∈ ℬ there exists a sequence (Pk (⋅; ⋅)) of trigonometric polynomials such that 󵄩 󵄩 lim lim sup F(t) sup[ϕ(󵄩󵄩󵄩F(t; x) − Pk (t; x)󵄩󵄩󵄩Y )]

k→+∞ t→+∞

x∈B

Lp(t) (Λt )

= 0,

(88)

where we assume that the term in braces belongs to the space Lp(t) (Λt ) for any compact set K. If ϕ(x) ≡ x, then we omit the term “ϕ” from the notation; if X = {0}, then we omit the term “ℬ” from the notation. Immediately from definition, it follows that for every F ∈ e − (ℬ, F) − Bp(⋅) (Λ × X : Y ) p and λ ∈ ℝn , we have ei⟨λ,⋅⟩ F ∈ e−(ℬ, F)−Bp(⋅) (Λ×X : Y ). The Weyl class e−ℬ −WΩ (Λ×X : Y ), introduced in [431, Definition 6.3.18] with p(⋅) ≡ p ∈ [1, ∞) and F(t) ≡ t −n/p , makes a subclass of the class e − (ℬ, F) − Bp(⋅) (Λ × X : Y ), provided some reasonable choices of p compact set Ω; for example, we have that e− ℬ −WΩ (Λ×X : Y ) ⊆ e−(ℬ, F)−Bp(⋅) (Λ×X : Y ) if Ω = [0, 1]n , and Λ = [0, ∞)n or Λ = ℝn . Moreover, we have the following: (i) Equipped with the usual operations, the set e − (ℬ, ϕ, F) − Bp(⋅) (Λ × X : Y ) forms a vector space provided that the function ϕ(⋅) is monotonically increasing and there exists a finite real constant c > 0 such that ϕ(x + y) ⩽ c[ϕ(x) + ϕ(y)] for all x, y ⩾ 0. (ii) For every τ ∈ Λ′′ , x0 ∈ X and F ∈ e − (ℬ, F) − Bp(⋅) (Λ × X : Y ), we have F(⋅ + τ; ⋅ + x0 ) ∈ e−(ℬx0 , F)−Bp(⋅) (Λ×X : Y ) with ℬx0 ≡ {−x0 +B : B ∈ ℬ}, provided that p(⋅) ≡ p ∈ [1, ∞) and there exist two finite real constants cτ > 0 and tτ > 0 such that F(t) ⩽ cτ F(t +|τ|), t ⩾ tτ . (iii) Suppose that the function ϕ(⋅) is monotonically increasing and continuous at the point zero, that there exists a finite real constant c > 0 such that ϕ(x + y) ⩽ c[ϕ(x) + ϕ(y)] for all x, y ⩾ 0, and the mapping t 󳨃→ F(t)[1]Lp(⋅) (Λt ) , t > 0 is bounded at plus infinity. Then F ∈ e − (ℬ, ϕ, F) − Bp(⋅) (Λ × X : Y ) if and only if for each set B ∈ ℬ there exists a sequence (Fk (⋅; ⋅)) of strongly ℬ-almost periodic functions such that (88) holds, with the polynomial Pk (⋅; ⋅) replaced therein with the function Fk (⋅; ⋅). (iv) Let the assumptions of (iii) hold and let there exist a function φ : [0, ∞) → [0, ∞) such that ϕ(xy) ⩽ φ(x)ϕ(y) for all x, y ⩾ 0. Suppose that h : Y → Z is a Lipschitz continuous function and F ∈ e−(ϕ, F)−Bp(⋅) (Λ : Y ). Using [431, Proposition 6.1.11] and the fact that any uniformly continuous Bohr almost periodic function F : Λ → Y , where Λ is a convex polyhedral in ℝn , is strongly almost periodic, we can prove that h ∘ F ∈ e − (ϕ, F) − Bp(⋅) (Λ : Z).

Concerning the notion introduced in Definition 3.3.3, it is clear that the use of constant coefficients p(⋅) ≡ p ∈ [1, ∞) is unquestionably the best. On the other hand, in the introductory part of [431], we have emphasized that, from the theoretical point of view,

3.3 Multi-dimensional Besicovitch almost periodic type functions and applications

� 141

the use of constant coefficients is not adequately enough because many structural results from the theory of generalized almost periodic functions can be further extended using some results from the theory of Lebesgue spaces with variable exponents Lp(x) . Further on, it is clear that Definition 3.3.3 covers some cases that can be freely called pathological; for example, case in which p ∉ D+ (Λ) can be considered: Example 3.3.4. Suppose that Λ = [0, ∞) and F : Λ → ℝ is given by F(t) := 1, if there exists j ∈ ℕ ∖ {1} such that t ∈ [j2 − 1, j2 ], and F(t) := 0, otherwise. Let p(x) ≡ 1 + x 2 , ϕ(x) ≡ x and lim sup F(t) ⋅ inf{λ > 0 : ∑ λ−j t→+∞

2⩽j⩽√t

4

+2j2 −2

⩽ 1} = 0.

Then a simple computation with the Luxemburg norm shows that (88) is satisfied with the trivial sequence (Pk ≡ 0) of trigonometric polynomials, so that F ∈ PAP0,p (Λ, F, ϕ) ⊆ e − (ϕ, F) − Bp(⋅) (Λ : ℂ). The usual notion of Besicovitch-p-almost periodicity (Doss-p-almost periodicity) for the function F : Λ → Y , where 1 ⩽ p < +∞, is obtained by plugging ϕ(x) ≡ x and F(t) ≡ t −n/p (ϕ(x) ≡ x, F(t) ≡ t −n/p and Λ′ = Λ, ρ = I). Further on, we say that a function F : Λ → Y is Besicovitch almost periodic (Doss almost periodic) if and only if F(⋅) is Besicovitch-1-almost periodic (Doss-1-almost periodic). Let us recall that in the usual setting, a Doss almost periodic function f : ℝ → ℂ is not generally Besicovitch almost periodic; for example, A. N. Dabboucy and H. W. Davies have constructed an example of such a function, which has the mean value equal to zero (cf. [235, pp. 352–354] for more details). Now we will state and prove the following result: Proposition 3.3.5. Suppose that ℬ consists of bounded subsets of X, F : Λ × X → Y and, for every fixed element x ∈ X, the function F(⋅; x) is Lebesgue measurable. (i) Suppose that the function ϕ(⋅) is monotonically increasing. If there exists a finite real constant t0 > 0 such that F(t) ⩽ [‖1‖Lp(⋅) (Λt ) ]−1 , t ⩾ t0 , then any function F ∈ e − (ℬ, ϕ, F) − Bp(⋅) (Λ × X : Y ) is Besicovitch-(p, ϕ, F, ℬ)-bounded. (ii) Suppose that ϕ(⋅) is monotonically increasing and continuous at the point t = 0. Let p(⋅) ≡ p ∈ [1, ∞), and let there exist finite real constants c > 0 and t0 > 0 such that for every t ⩾ t0 , we have F(t + 1) ⩾ cF(t) and F(t) ⩽ [m(Λt )]−(1/p) . Then any function F ∈ e − (ℬ, ϕ, F) − Bp (Λ × X : Y ) is Besicovitch-(p, ϕ, F, ℬ, Λ′ , I)-continuous for any set Λ′ ⊆ Λ′′ . (iii) Suppose that p(⋅) ≡ p ∈ [1, ∞), ϕ(⋅) has the same properties as in (ii), as well as that for every real number a > 0 there exist finite real constants ca > 0 and ta > 0 such that for every t ⩾ ta , we have F(t + a) ⩾ ca F(t) and F(t) ⩽ [m(Λt )]−(1/p) . Let F ∈ e − (ℬ, ϕ, F) − Bp (Λ × X : Y ). Then the following holds:

142 � 3 Multi-dimensional ρ-almost periodic type functions (a) The function F(⋅; ⋅) is Doss-(p, ϕ, F, ℬ, Λ, I)-almost periodic, provided that Λ + Λ ⊆ Λ and, for every points (t1 , . . . , tn ) ∈ Λ and (τ1 , . . . , τn ) ∈ Λ, the points (t1 , t2 + τ2 , . . . , tn + τn ), (t1 , t2 , t3 + τ3 , . . . , tn + τn ), . . . , (t1 , t2 , . . . , tn−1 , tn + τn ), also belong to Λ. (b) The function F(⋅; ⋅) is Doss-(p, ϕ, F, ℬ, Λ ∩ Δn , I)-almost periodic, provided that Λ ∩ Δn ≠ 0, Λ+(Λ∩Δn ) ⊆ Λ and that, for every points (t1 , . . . , tn ) ∈ Λ and (τ, . . . , τ) ∈ Λ∩ Δn , the points (t1 , t2 +τ, . . . , tn +τ), (t1 , t2 , t3 +τ, . . . , tn +τ), . . ., (t1 , t2 , . . . , tn−1 , tn +τ), also belong to Λ ∩ Δn . Proof. In order to prove (i), fix a set B ∈ ℬ. Then B is bounded and we have the existence of a trigonometric polynomial Pk (⋅; ⋅) such that 󵄩 󵄩 sup[ϕ(󵄩󵄩󵄩F(⋅; x)󵄩󵄩󵄩Y )]

Lp(⋅) (Λt )

x∈B

󵄩 󵄩 ⩽ sup[ϕ(󵄩󵄩󵄩F(⋅; x) − Pk (⋅; x)󵄩󵄩󵄩Y )] x∈B

Lp(⋅) (Λt )

󵄩 󵄩 ⩽ (ε/2F(t)) + sup[ϕ(󵄩󵄩󵄩Pk (⋅; x)󵄩󵄩󵄩Y )] x∈B

󵄩 󵄩 + sup[ϕ(󵄩󵄩󵄩Pk (⋅; x)󵄩󵄩󵄩Y )] x∈B

Lp(⋅) (Λt )

Lp(⋅) (Λt )

.

Let Pk (⋅; x) = ∑lj=0 ei⟨λl ,⋅⟩ cl (x) for some integer l ∈ ℕ, points λ1 , . . . , λl from ℝn and continuous functions c1 (⋅), . . . , cl (⋅) from X into Y . Then we have the existence of a finite real constant cB > 0 such that (see also Lemma 1.1.10(ii)): 󵄩 󵄩 sup[ϕ(󵄩󵄩󵄩Pk (⋅; x)󵄩󵄩󵄩Y )]

Lp(⋅) (Λt )

x∈B

l

󵄩󵄩 l 󵄩󵄩 󵄩󵄩 󵄩󵄩 ⩽ sup[ϕ(󵄩󵄩󵄩∑ ei⟨λj ,⋅⟩ cj (x)󵄩󵄩󵄩 )] 󵄩 󵄩󵄩 󵄩󵄩j=0 x∈B 󵄩Y Lp(⋅) (Λt )

󵄩 󵄩 ⩽ sup[ϕ(∑󵄩󵄩󵄩cj (x)󵄩󵄩󵄩Y )] x∈B

j=0

Lp(⋅) (Λt )

⩽ [ϕ(cB )]Lp(⋅) (Λ ) ⩽ ϕ(cB )F(t)−1 , t

t ⩾ t0 .

The proof of (ii) is quite similar and follows from the decomposition: 󵄩 󵄩 sup[ϕ(󵄩󵄩󵄩F(⋅ + τ; x) − F(⋅; x)󵄩󵄩󵄩Y )] x∈B

Lp(⋅) (Λt )

󵄩 󵄩 ⩽ sup[ϕ(󵄩󵄩󵄩F(⋅ + τ; x) − Pk (⋅ + τ; x)󵄩󵄩󵄩Y )] x∈B

Lp(⋅) (Λt )

󵄩 󵄩 + sup[ϕ(󵄩󵄩󵄩Pk (⋅ + τ; x) − Pk (⋅; x)󵄩󵄩󵄩Y )]

Lp(⋅) (Λt )

x∈B

󵄩 󵄩 + sup[ϕ(󵄩󵄩󵄩Pk (⋅; x) − F(⋅; x)󵄩󵄩󵄩Y )] x∈B

Lp(⋅) (Λt )

;

let us only note that we need the continuity of ϕ(⋅) at the point t = 0 because, in the final steps of computation, we get a term of form ϕ(cB ∑lj=0 |ei⟨λj ,τ⟩ − 1|), which tends to zero as τ → 0+. The proof of part (a) in (iii) follows from a relatively simple argumentation involving the decomposition used for proving (ii), the given assumptions and the fact that

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�

143

the trigonometric polynomial Pk (⋅; ⋅) is Bohr ℬ-almost periodic due to [431, Proposition 6.1.25(iv)]; the proof of part (b) in (iii) is quite similar because the prescribed assumptions imply that the trigonometric polynomial Pk (⋅; ⋅) is Bohr (ℬ, Λ ∩ Δn )-almost periodic due to [431, Proposition 6.1.25(v)] (cf. also [431, Definition 6.1.9, Definition 6.1.14] for the notion). Suppose that ℬ consists of bounded subsets of X, Λ is unbounded, F, G ∈ e−(ℬ, ϕ, F)− Bp(⋅) (Λ × X : Y ), for every fixed element x ∈ X, the function ϕ(⋅) is monotonically increasing, ϕ(x + y) ⩽ ϕ(x) + ϕ(y) for all x, y ⩾ 0, lim supt→+∞ F(t) = 0, and there exists a finite real constant t0 > 0 such that F(t) ⩽ [‖1‖Lp(⋅) (Λt ) ]−1 , t ⩾ t0 . Due to Proposition 3.3.5(i), we have that the function F(⋅; ⋅) is Besicovitch-(p, ϕ, F, ℬ)-bounded. Let a set B ∈ ℬ be fixed. Then 󵄩 󵄩 dB (F, G) := lim sup F(t) sup[ϕ(󵄩󵄩󵄩F(⋅; x) − G(⋅; x)󵄩󵄩󵄩Y )] t→+∞

Lp(⋅) (Λt )

x∈B

defines a pseudometric on the set e − (ℬ, ϕ, F) − Bp(⋅) (Λ × X : Y ). Using the idea from the original proof of J. Marcinkiewicz [535], we can prove the following theorem (see also [495, pp. 249–252]): Theorem 3.3.6. Let the requirements stated in the previous paragraph hold. Then for every set B ∈ ℬ, the pseudometric space (e − (ℬ, ϕ, F) − Bp(⋅) (Λ × X : Y ), dB ) is complete. The classes with ϕ(x) ≡ x α , α > 0 Without any doubt, the most important case in Definition 3.3.3 and Definition 3.2.1 is that one in which we have ϕ(x) ≡ x α for some real number α > 0. If so, then all requirements necessary for applying Proposition 3.3.5 and the statements stated preceding it hold. The assumptions of Theorem 3.3.6 hold in case α ∈ (0, 1], when we can provide some proper generalizations of the usual notion of Besicovitch-P-almost periodicity. For example, if 1 ⩽ P < +∞, 1 ⩽ p < +∞, αp ∈ (0, 1) and the function F : ℝ → Y is Besicovitch-P-almost periodic, then the Hölder inequality implies that for every trigonometric polynomial P(⋅), we have: t

1/p

1 󵄩 󵄩αp ( ∫󵄩󵄩󵄩F(s) − P(s)󵄩󵄩󵄩Y ds) 2t −t

t

α/P

1 󵄩 󵄩P ⩽ ( ∫󵄩󵄩󵄩F(s) − P(s)󵄩󵄩󵄩Y ds) 2t

,

t > 0,

−t

so that F ∈ e − (x α , t −(1/p) ) − Bp (ℝ : Y ). The converse statement does not hold in general, as the following illustrative example shows: Example 3.3.7 (cf. also Example 3.3.9 below). Let ζ > 1/2 and αζ ∈ (0, 1/2). Define the function F : ℝ → ℝ by F(t) := mζ if t ∈ [m2 , m2 + √|m|) for some m ∈ ℤ, and F(t) := 0, otherwise. Then it can be simply shown that the function F(⋅) is not Besi-

144 � 3 Multi-dimensional ρ-almost periodic type functions covitch bounded and therefore not Besicovitch almost periodic. On the other hand, we have F ∈ PAP0,p (ℝ, t −1 , x α ) ⊆ e − (x α , t −1 ) − B1 (ℝ : ℂ). Let the numbers α > 0 and β > 0 be arbitrary. Using the functions ϕ(x) ≡ x α/p and F(t) ≡ t −β/p in our approach, we can consider the generalized Besicovitch class Bα,β (ℝ : Y ) consisting of those Lebesgue measurable functions F : ℝ → Y such that, for every ε > 0, there exist a trigonometric polynomial P(⋅) and a real number t0 > 0 such that t

󵄩 󵄩α ∫󵄩󵄩󵄩F(s) − P(s)󵄩󵄩󵄩Y ds ⩽ εt β ,

t ⩾ t0 ;

−t

a multi-dimensional generalization can be introduced analogously. The generalized Besicovitch class Bα,β (ℝ : Y ) and its multi-dimensional analogues will be considered a little bit later (let us only observe here that the space Wα , considered by M. A. Picardello [615] in the usual one-dimensional setting with 0 < α ⩽ 1, is nothing else but the space B1,α (ℝ : ℂ)). We will provide the main details of the proof of the following proposition for the sake of completeness: Proposition 3.3.8. Suppose that p, q, r ∈ [1, ∞), 1/r = 1/p + 1/q, F1 (t) ≡ t −n/p , F2 (t) ≡ t −n/q , F(t) ≡ t −n/r , ϕ(x) ≡ x α for some real number α > 0, and any set B of collection ℬ is bounded in X. If F1 ∈ e − (ℬ, ϕ, F1 ) − Bp (Λ × X : ℂ) and F2 ∈ e − (ℬ, ϕ, F2 ) − Bq (Λ × X : Y ), then the function F : Λ × X → Y , given by F(t; x) := F1 (t; x)F2 (t; x), t ∈ Λ, x ∈ X, belongs to the class e − (ℬ, ϕ, F) − Br (Λ × X : Y ). Proof. Let ε > 0 and B ∈ ℬ be given. Then there exist a finite real number t0 > 0, a scalarvalued trigonometric polynomial P1 (⋅; ⋅) and a Y -valued trigonometric polynomial P2 (⋅; ⋅) such that for every x ∈ B, we have 󵄨 󵄨 [ϕ(󵄨󵄨󵄨F1 (⋅; x) − P1 (⋅; x)󵄨󵄨󵄨)] p ⩽ εt n/p , L (Λt )

t ⩾ t0 ,

(89)

and 󵄩 󵄩 [ϕ(󵄩󵄩󵄩F2 (⋅; x) − P2 (⋅; x)󵄩󵄩󵄩Y )] q ⩽ εt n/q , L (Λt )

t ⩾ t0 .

(90)

Clearly, P1 (⋅; ⋅)P2 (⋅; ⋅) is a Y -valued trigonometric polynomial. Applying Proposition 3.3.5(i), we get that the function P1 (⋅; ⋅) is Besicovitch-(p, ϕ, F1 , ℬ)-bounded and the function F2 (⋅; ⋅) is Besicovitch-(q, ϕ, F2 , ℬ)-bounded. Furthermore, it can be simply proved that there exist two finite real numbers M > 0 and t1 > 0 such that, for every t ⩾ t1 , we have: 󵄨 󵄨 F1 (t) sup[ϕ(󵄨󵄨󵄨P1 (⋅; x)󵄨󵄨󵄨)] x∈B

Lp (Λt )

⩽ Const. ⋅ (M + ε)t n/p .

3.3 Multi-dimensional Besicovitch almost periodic type functions and applications

�

145

Keeping in mind that 1/r = 1/p + 1/q, the final conclusion simply follows using these facts, (89)–(90), the existence of a finite real number cα > 0 such that 󵄩 󵄩 ϕ(󵄩󵄩󵄩F1 (⋅; x)F2 (⋅; x) − P1 (⋅; x)P2 (⋅; x)󵄩󵄩󵄩Y ) 󵄨 󵄨 󵄩 󵄩 ⩽ cα [ϕ(󵄨󵄨󵄨F1 (⋅; x) − P1 (⋅; x)󵄨󵄨󵄨) ⋅ ϕ(󵄩󵄩󵄩F2 (⋅; x)󵄩󵄩󵄩Y ) 󵄨 󵄨 󵄩 󵄩 + ϕ(󵄨󵄨󵄨P1 (⋅; x)󵄨󵄨󵄨) ⋅ ϕ(󵄩󵄩󵄩F2 (⋅; x) − P2 (⋅; x)󵄩󵄩󵄩Y )], and the Hölder inequality. We continue by providing the following illustrative application of Proposition 3.3.8: Example 3.3.9. Suppose that 1 ⩽ p1 , . . . , pn , p < +∞ and 1/p = 1/p1 + 1/p2 + ⋅ ⋅ ⋅ + 1/pn . Define the function Fj : ℝ → ℝ by Fj (t) := m1/2pj if t ∈ [m2 , m2 + √|m|) for some m ∈ ℤ, and Fj (t) := 0, otherwise (1 ⩽ j ⩽ n). Then we know that the function Fj (⋅) is Besicovitch-pj almost periodic but not Besicovitch-q-almost periodic if q > pj ; see [122, p. 42] and [42, Example 6.24]. Define F(t) := F1 (t1 ) ⋅ F2 (t2 ) ⋅ ⋅ ⋅ ⋅ ⋅ Fn (tn ), t ∈ ℝn . Applying Proposition 3.3.8 and a simple argumentation, it follows that the function F(⋅) is Besicovitch-p-almost periodic but not Besicovitch-q-almost periodic if q > p. Sometimes we need the value of coefficient p = +∞ in Proposition 3.3.8 and sometimes the usual choice F(t) ≡ t −n/p is wrong if the region Λ is bounded in direction of some real axes: Example 3.3.10. Suppose that 1 ⩽ p < +∞, the function f : [0, 2π] → ℝ is absolutely continuous and the function g : [0, ∞) → Y is Besicovitch-p-almost periodic. Since the Fourier series of function f (⋅) converges uniformly to this function, arguing as in the proof of Proposition 3.3.8, we may conclude that the function F(x, y) := f (x)g(y), (x, y) ∈ Λ ≡ [0, 2π] × [0, ∞) → Y belongs to the class e − (x, t −1/p ) − Bp (Λ : Y ). Further on, the composition principles for one-dimensional Besicovitch-p-almost periodic functions have been analyzed for the first time by M. Ayachi and J. Blot in [70, Lemma 4.1]. In the following theorem, we consider the Besicovitch-p-almost periodicity of the multi-dimensional Nemytskii operator W : ℝn × X → Z, given by (27), where F : ℝn × X → Y and G : ℝn × Y → Z. We follow the ideas from [70] in part (i): Theorem 3.3.11. Suppose that 1 ⩽ p, q < +∞, α > 0, p = αq, F(t) ≡ t −n/p , ϕ(x) ≡ x ζ for some real number ζ > 0, F ∈ e − (ℬ, ϕ, F) − Bp (ℝn × X : Y ), and ℬ is the collection consisting of all bounded subsets of X. (i) Suppose that G : ℝn × Y → Z is Bohr ℬ-almost periodic and there exists a finite real constant a > 0 such that 󵄩󵄩 󵄩 ′ 󵄩 ′ 󵄩α 󵄩󵄩G(t; y) − G(t; y )󵄩󵄩󵄩Z ⩽ a󵄩󵄩󵄩y − y 󵄩󵄩󵄩Y ,

t ∈ ℝn , y, y′ ∈ Y .

(91)

146 � 3 Multi-dimensional ρ-almost periodic type functions Then the function W (⋅; ⋅), given by (27), belongs to the class e − (ℬ, ϕ, t −n/q ) − Bq (ℝn × X : Z). (ii) Define ℬ := { ⋃ F(t; B) ; B ∈ ℬ}. ′

t∈ℝn

q

By e − (ℬ′ , ϕ, t −n/q ) − Ba,α (ℝn × Y : Z) we denote the class of all functions G1 (⋅; ⋅) such that for each set B′ ≡ ∪t∈ℝn F(t; B) ∈ ℬ′ there exists a sequence of Bohr ℬ-almost periodic functions (G1k (⋅; ⋅)) such that (91) holds with the function G(⋅; ⋅) replaced therein by the function G1k (⋅; ⋅) for all k ∈ ℕ, the equation (88) holds with the function F(⋅; ⋅) replaced therein by the function G(⋅; ⋅), the polynomial Pk (⋅; ⋅) replaced therein by the function G1k (⋅; ⋅), the set B replaced therein with the set B′ , and the exponent p(⋅) req placed therein by the constant exponent q. If G ∈ e − (ℬ′ , ϕ, t −n/q ) − Ba,α (ℝn × Y : Z), −n/q q n then W ∈ e − (ℬ, ϕ, t ) − B (ℝ × X : Z). Proof. Let ε > 0 and B ∈ ℬ be given. Then there exist a trigonometric polynomial P(⋅; ⋅) and a finite real number t0 > 0 such that 󵄩 󵄩ζp sup ∫ 󵄩󵄩󵄩F(t; x) − P(t; x)󵄩󵄩󵄩Y dt < εt n , x∈B

t ⩾ t0 .

[−t,t]n

We will first prove (i). Since we have assumed that (91) holds and ℬ is the collection consisting of all bounded subsets of X, the argumentation contained in the proofs of [431, Theorem 6.1.47, Corollary 6.1.48] shows that the function W1 : ℝn × X → Z, given by W1 (t; x) := G(t; P(t; x)), t ∈ ℝn , x ∈ X, is Bohr ℬ-almost periodic. Then the final conclusion simply follows by observing that p = αq, using the next estimate, which holds for any t > 0 and x ∈ B; see (91): 󵄩 󵄩ζq 󵄩 󵄩ζp ∫ 󵄩󵄩󵄩W (t; x) − W1 (t; x)󵄩󵄩󵄩Z dt ⩽ aζq ∫ 󵄩󵄩󵄩F(t; x) − P(t; x)󵄩󵄩󵄩Y dt.

[−t,t]n

[−t,t]n

In order to prove (ii), it suffices to apply (i) and the inequality 󵄩 󵄩 ϕ(󵄩󵄩󵄩G(t; F(t; x)) − Gk (t; P(t; x))󵄩󵄩󵄩Z ) 󵄩 󵄩 󵄩 󵄩 ⩽ 2ζ [ϕ(󵄩󵄩󵄩G(t; F(t; x)) − Gk (t; F(t; x))󵄩󵄩󵄩Z ) + ϕ(󵄩󵄩󵄩Gk (t; F(t; x)) − Gk (t; P(t; x))󵄩󵄩󵄩Z )], where Gk (⋅; ⋅) properly approximates G(⋅; ⋅) in the space e − (ℬ′ , ϕ, t −n/q ) − Bq (ℝn × Y : Z).

3.3 Multi-dimensional Besicovitch almost periodic type functions and applications

� 147

3.3.2 Multi-dimensional Besicovitch normal type functions The notion of a Besicovitch p-normal function f : ℝ → ℂ was introduced by R. Doss in [274] and later reconsidered by the same author in [276]; cf. also [431, Subsection 8.3.2, Definition 8.3.18], where we have recently analyzed the concept Weyl p-almost automorphy (of type 2) without limit functions. In this subsection, we will consider the following notion: Definition 3.3.12. Suppose that R is any collection of sequences in Λ′′ , F : Λ × X → Y , ϕ : [0, ∞) → [0, ∞) and F : (0, ∞) → (0, ∞). Then we say that the function F(⋅; ⋅) is Besicovitch-(R, ℬ, ϕ, F) − Bp(⋅) -normal, if and only if for every set B ∈ ℬ and for every sequence (bk )k∈ℕ in R there exists a subsequence (bkm )m∈ℕ of (bk )k∈ℕ such that, for every ε > 0, there exists an integer m0 ∈ ℕ such that for every integers m, m′ ⩾ m0 , we have 󵄩 󵄩 lim sup F(t) sup[ϕ(󵄩󵄩󵄩F(t + bkm ; x) − F(t + bk ′ ; x)󵄩󵄩󵄩Y )] t→+∞

x∈B

m

Lp(t) (Λt )

< ε.

The usual notion of Besicovitch-p-normality for the function F : Λ → Y , where 1 ⩽ p < +∞, is obtained by plugging ϕ(x) ≡ x and F(t) ≡ t −n/p , with R being the collection of all sequences in Λ′′ . In the sequel, we will occasionally use the following conditions: (I) ϕ(⋅) is monotonically increasing, continuous at the point t = 0, and p(⋅) ≡ p ∈ [1, ∞). (II) There exists c ∈ (0, 1] such that ϕ(x + y) ⩽ c[ϕ(x) + ϕ(y)] for all x, y ⩾ 0, and there exists a function φ : [0, ∞) → [0, ∞) such that ϕ(xy) ⩽ ϕ(x)φ(y) for all x, y ⩾ 0 and D := supm∈ℕ [mφ(1/m)] < +∞. (III) lim supt→+∞ [F(t)m(Λt )1/p ] < +∞ and, for every real number a > 0, we have lim supt→+∞ [F(t)/F(t + a)] ⩽ 1. It is clear that (II) holds provided that ϕ(x) ≡ x α for some real number α ⩾ 1 as well as that (II) does not hold if ϕ(x) ≡ x α for some real number α ∈ (0, 1) because then we have D = +∞. Repeating verbatim the argumentation contained in the proof of [431, Theorem 6.3.19], where we have analyzed the concept Weyl (R, ℬ, p)-normality, the following result can be deduced without any substantial difficulties: Proposition 3.3.13. Suppose that F : Λ × X → Y , F ∈ e − (ℬ, ϕ, F) − Bp(⋅) (Λ × X : Y ), and conditions (I)–(III) hold. Then F(⋅; ⋅) is Besicovitch-(R, ℬ, ϕ, F) − Bp(⋅) -normal. Even in the usual one-dimensional framework, we know that the converse statement of Proposition 3.3.13 is not true in general (see, e. g. [42]) as well as that the usual Besicovitch-p-normality does not imply Besicovitch-p-continuity (see, e. g. [276]). Fur-

148 � 3 Multi-dimensional ρ-almost periodic type functions ther on, let k ∈ ℕ and Fi : Λ × X → Yi (1 ⩽ i ⩽ k). Then we define the function (F1 , . . . , Fk ) : Λ × X → Y1 × ⋅ ⋅ ⋅ × Yk by (F1 , . . . , Fk )(t; x) := (F1 (t; x), . . . , Fk (t; x)),

t ∈ Λ, x ∈ X.

The following result is trivial and its proof is therefore omitted: Proposition 3.3.14. Suppose that k ∈ ℕ, 0 ≠ Λ ⊆ ℝn and we have that, for any sequence which belongs to R, any its subsequence also belongs to R. If the function Fi (⋅; ⋅) is Besicovitch-(R, ℬ, ϕ, F) − Bp(⋅) -normal for 1 ⩽ i ⩽ k, then the function (F1 , . . . , Fk )(⋅; ⋅) is also Besicovitch-(R, ℬ, ϕ, F) − Bp(⋅) -normal. The interested reader may attempt to formulate certain conditions ensuring that the limit function of a sequence of uniformly convergent Besicovitch-(R, ℬ, ϕ, F) − Bp(⋅) normal functions (the functions belonging to the class e−(ℬ, ϕ, F)−Bp(⋅) (Λ×X : Y )) is also Besicovitch-(R, ℬ, ϕ, F) − Bp(⋅) -normal (belongs to the class e − (ℬ, ϕ, F) − Bp(⋅) (Λ × X : Y )); see [431] for many results of this type. Several structural properties of functions belonging to the class e−(ℬ, ϕ, F)−Bp(⋅) (Λ× X : Y ) can be simply reformulated for the class of Besicovitch-(R, ℬ, ϕ, F) − Bp(⋅) -normal functions. For example, we have the following analogue of Proposition 3.3.8: Proposition 3.3.15. Suppose that p, q, r ∈ [1, ∞), 1/r = 1/p + 1/q, F1 (t) ≡ t −n/p , F2 (t) ≡ t −n/q , F(t) ≡ t −n/r , ϕ(x) ≡ x α for some real number α > 0 and, for any sequence which belongs to R, any its subsequence also belongs to R. If the function F1 : Λ × X → ℂ is Besicovitch-(R, ℬ, ϕ, F1 ) − Bp -normal and Besicovitch-(p, ϕ, F1 , ℬ)-bounded as well as the function F2 : Λ×X → Y is Besicovitch-(R, ℬ, ϕ, F2 )−Bq -normal and Besicovitch-(q, ϕ, F2 , ℬ)bounded, then the function F : Λ × X → Y , given by F(t; x) := F1 (t; x)F2 (t; x), t ∈ Λ, x ∈ X, is Besicovitch-(R, ℬ, ϕ, F1 ) − Br -normal. Proof. Let a set B ∈ ℬ and a sequence (bk )k∈ℕ in R be given. Keeping in mind the proof of Proposition 3.3.8 and our assumption that for any sequence which belongs to R any its subsequence also belongs to R, it suffices to show that there exist two sufficiently large real numbers t0 > 0 and M > 0 such that 󵄨 󵄨 sup [ϕ(󵄨󵄨󵄨F1 (⋅ + bk ; x)󵄨󵄨󵄨)] p ⩽ Mt n/p , L (Λt )

x∈B,k∈ℕ

t ⩾ t0

(92)

and 󵄩 󵄩 sup [ϕ(󵄩󵄩󵄩F2 (⋅ + bk ; x)󵄩󵄩󵄩Y )] q ⩽ Mt n/q , L (Λt )

x∈B,k∈ℕ

t ⩾ t0 .

(93)

Since the function F1 : Λ × X → ℂ is Besicovitch-(p, ϕ, F1 , ℬ)-bounded, it can be simply proved that 󵄨 󵄨 sup[ϕ(󵄨󵄨󵄨F1 (⋅ + bk ; x)󵄨󵄨󵄨)] x∈B

Lp (Λt )

⩽ Mt n/p ,

t ⩾ t0 .

3.3 Multi-dimensional Besicovitch almost periodic type functions and applications

�

149

Moreover, we have the existence of a finite real number t0 > 0 and an integer k0 ∈ ℕ such that, for every integers k, k ′ ⩾ k0 , we have 󵄨 󵄨 sup[ϕ(󵄨󵄨󵄨F1 (⋅ + bk ; x) − F1 (⋅ + bk ′ ; x)󵄨󵄨󵄨)]

Lp (Λt )

x∈B

< Mt n/p ,

t ⩾ t0 .

Further on, there exists a finite real constant cα > 0 such that for every integers k, k ′ ⩾ k0 , we have: 󵄨 󵄨 󵄨 󵄨 [ϕ(󵄨󵄨󵄨F1 (⋅ + bk ; x)󵄨󵄨󵄨) − ϕ(󵄨󵄨󵄨F1 (⋅ + bk ′ ; x)󵄨󵄨󵄨)] p L (Λt ) 󵄨󵄨󵄨 󵄨 󵄨 󵄨󵄨󵄨 ⩽ cα [ϕ(󵄨󵄨󵄨󵄨󵄨󵄨F1 (⋅ + bk ; x)󵄨󵄨󵄨 − 󵄨󵄨󵄨F1 (⋅ + bk ′ ; x)󵄨󵄨󵄨󵄨󵄨󵄨)] p 󵄨 󵄨 L (Λt ) 󵄨 󵄨 󵄨 󵄨 ⩽ cα [ϕ(󵄨󵄨F1 (⋅ + bk ; x) − F1 (⋅ + bk ′ ; x)󵄨󵄨)] p ⩽ cα Mt n/p , L (Λt )

t ⩾ t0 , x ∈ B.

This simply implies (92) because 󵄨 󵄨 sup[ϕ(󵄨󵄨󵄨F1 (⋅ + bk ; x)󵄨󵄨󵄨)] x∈B

Lp (Λt )

󵄨 󵄨 ⩽ sup[ϕ(󵄨󵄨󵄨F1 (⋅ + bk0 ; x)󵄨󵄨󵄨)] x∈B

Lp (Λt )

+ cα Mt n/p ,

t ⩾ t0 , k ⩾ k0 .

The estimate (93) can be proved analogously, finishing the proof. Concerning the assumptions on Besicovitch boundedness used in the formulation of Proposition 3.3.15, we have the following: Example 3.3.16. Suppose that p ∈ [1, ∞), σ ∈ (0, 1), F(x) := |x|σ , x ∈ ℝ, and a > 1 − (1 − σ)p > 0. Then we know that the function F(⋅) is not Besicovitch-p-bounded and that for every t ∈ ℝ and ω ∈ ℝ, we have: l

lim l

l→+∞

−a

󵄨 󵄨p ∫󵄨󵄨󵄨|x + t + ω|σ − |x + t|σ 󵄨󵄨󵄨 dx = 0;

(94)

−l

see [431, Theorem 8.3.8] and its proof. Let R denote the collection of all sequences in ℝ and let F(t) ≡ t −a/p . Then the limit equality (94) simply implies that the function F(⋅) is Besicovitch-(R, x, F) − Bp -normal. Hence, the usual Besicovitch-p-normality of a function F(⋅) does not imply its Besicovitch-p-boundedness as well. We close this subsection by introducing the following notion and raising the following issue: Definition 3.3.17. Let 0 ≠ Λ ⊆ ℝn , and let 𝒞Λ = e − (ℬ, ϕ, F) − Bp(⋅) (Λ : Y ) or 𝒞Λ be the class consisting of all Besicovitch-(R, ℬ, ϕ, F) − Bp(⋅) -normal functions. Then we say that the set Λ is admissible with respect to the class 𝒞Λ if and only if for any complex Banach space

150 � 3 Multi-dimensional ρ-almost periodic type functions ̃ = F(t) Y and for any function F : Λ → Y there exists a function F̃ ∈ 𝒞ℝn such that F(t) for all t ∈ Λ. Problem. It is still not known whether the set [0, ∞) ⊆ ℝ is admissible with respect to the class of Besicovitch-p-almost periodic functions, i. e. whether a Besicovitch-p-almost periodic function f : [0, ∞) → Y can be extended to a Besicovitch-p-almost periodic function f ̃ : ℝ → Y defined on the whole real line (1 ⩽ p < ∞). We would like to ask here a more general question: Is it true that a convex polyhedral Λ in ℝn is admissible with respect to the class of multi-dimensional Besicovitch-p-almost periodic functions (Besicovitch-p-normal functions)? 3.3.3 Besicovitch–Doss almost periodicity In this subsection, we discuss and reexamine several structural results established by R. Doss in [275, 276]. We work in the multi-dimensional setting here, considering especially the following conditions: (A) For every B ∈ ℬ and a ∈ Λ′′ , there exists a function FB(a) : Λ × X → Y such that FB(a) (⋅; x) is a-periodic for every fixed element x ∈ B, i. e. FB(a) (t + a; x) = FB(a) (t; x) for all t ∈ Λ, x ∈ B, ‖FB(a) (t; x)‖Y ∈ Lp(t) (Λt ) for all t > 0, x ∈ B, and 󵄩󵄩 k−1 󵄩󵄩 󵄩󵄩 1 󵄩󵄩 (a) 󵄩 lim lim sup F(t) sup[ϕ(󵄩󵄩 ∑ F(t + ja; x) − FB (t; x)󵄩󵄩󵄩 )] = 0. 󵄩 󵄩󵄩 k→+∞ t→+∞ k 󵄩󵄩 j=0 x∈B 󵄩Y Lp(t) (Λt )

(95)

(A)∞ For every B ∈ ℬ and a ∈ Λ′′ , there exists a function FB(a) : Λ × X → Y such that FB(a) (⋅; x) is a-periodic for every fixed element x ∈ B, ‖FB(a) (⋅; x)‖Y ∈ L∞ (ℝn ) for all x ∈ B, and (95) holds. (AS) For every B ∈ ℬ and a = (a1 , a2 , . . . , an ) ∈ Λ′′ such that aj ej ∈ Λ′′ for all j ∈ ℕn , there exists a function FB(a) : Λ × X → Y such that FB(a) (⋅; x) is (aj )j∈ℕn -periodic for

every fixed element x ∈ B, i. e. FB(a) (t + aj ej ; x) = FB(a) (t; x) for all t ∈ Λ, x ∈ B, j ∈ ℕn , ‖FB(a) (t; x)‖Y ∈ Lp(t) (Λt ) for all t > 0, x ∈ B, and (95) holds.

It is clear that (AS) implies (A) as well as that both conditions are equivalent in the onedimensional setting; it is also clear that (A)∞ implies (A). Further on, for every Lebesgue measurable set E ⊆ ℝn and for every Lebesgue measurable function F : ℝn → ℂ, we set ν(E) := lim sup t→+∞

1

t n/p

E

∫ χE (t) dt and

ℳ [F] := lim sup t→+∞

|t|⩽t

If lim

1

t→+∞ t n/p

∫ F(t)χE (t) dt |t|⩽t

1

t n/p

∫ F(t)χE (t) dt. |t|⩽t

3.3 Multi-dimensional Besicovitch almost periodic type functions and applications

� 151

exists in ℂ, then we denote this quantity by ℳE [F]. Suppose now that the function F (a) : ℝn → ℂ is (a1 , a2 , . . . , an )-periodic. Then we can find a sequence of infinitely differentiable functions (φk )k∈ℕ with compact support in S = [0, |a1 |] × ⋅ ⋅ ⋅ × [0, |an |] such that φk → F (a) as k → +∞, in Lp (S). After that, we ̃k (⋅) defined on the whole space ℝn extend φk (⋅) to an (a1 , a2 , . . . , an )-periodic function φ in the usual way. Then it is very simple to prove that 󵄩 ̃k 󵄩󵄩󵄩󵄩Lp ((ℝn ) ) ] = 0; lim lim sup[t −(n/p) 󵄩󵄩󵄩F (a) − φ

k→+∞ t→+∞

(96)

t

cf. also [275, p. 483, l. 7–l. 9]. Keeping this observation in mind, the following result can be deduced, in a plus-minus technical way, following the argumentation contained in the proofs of [275, Proposition 1, Proposition 3, Corollary, Lemma 2] (cf. also Lemma 3.3.2, which is needed for the proof of (ii)): Theorem 3.3.18. (i) Suppose that p ∈ [1, ∞), q ∈ (1, ∞], 1/p + 1/q = 1, the function F : Λ → Y satisfies that ‖F(⋅)‖Y ∈ Lp (Λt ) for all t > 0, as well as that ‖F(⋅ + τ) − F(⋅)‖Y ∈ Lp (Λt ) for all t > 0 and τ ∈ Λ′′ . Suppose, further, that the function F(⋅) is Besicovitch-(p, x, F1 )-bounded and Besicovitch-(p, x, F1 , Λ′′ )-continuous as well as that condition (III) holds, and the set Λ′′ ∩ ℚn is dense in Λ′′ . If the function G : Λ → ℂ satisfies that G(⋅) ∈ Lq (Λt ) for all t > 0, and G(⋅) is Besicovitch-(q, x, F2 )-bounded, then for each sequence (Lm )m∈ℕ there exists a subsequence (Tm )m∈ℕ of (Lm )m∈ℕ such that the function H(τ) := lim F1 (Tm )F2 (Tm ) ∫ G(s)F(s + τ) ds, m→+∞

τ ∈ Λ′′

Λ Tm

is well-defined and bounded. Furthermore, if the function F(⋅) is Doss-(p, x, F1 , Λ′ )almost periodic (Doss-(p, x, F1 , Λ′ )-uniformly recurrent) and Λ′ + Λ′′ ⊆ Λ′′ , then the function H(⋅) is Bohr Λ′ -almost periodic (Λ′ -uniformly recurrent). (ii) Suppose that p ∈ [1, ∞), q ∈ (1, ∞], 1/p + 1/q = 1, the assumptions in (i) hold for the function F : ℝn → ℂ, with Λ = Λ′ = Λ′′ = ℝn , F1 (t) = F2 (t) = t −(n/p) , t ∈ ℝn , and F(⋅) is Doss-(p, x, t −(n/p) )-almost periodic. If condition (A)∞ or (AS) holds with X = {0}, ϕ(x) ≡ x and p(⋅) ≡ p, then, for every real number ε > 0, there exists a finite real number δ > 0 such that, for every Lebesgue measurable set E ⊆ ℝn , the assumption E p ν(E) < δ implies ℳ [|F|Y ] < ε. (iii) Let the assumptions of (ii) hold, and let for each N > 0 the function FN : ℝn → ℂ be defined by FN (t) := F(t), if |F(t)| ⩽ N, and FN (t) := Nei arg(F(t)) , if |F(t)| > N. Then ℝn

limN→+∞ ℳ [F − FN ] = 0. (iv) Suppose that the assumptions in (i) hold for the function F : ℝn → ℂ, with p = 1, Λ = Λ′ = Λ′′ = ℝn , F1 (t) = F2 (t) = t −n , t ∈ ℝn , and F(⋅) is Doss-(1, x, t −n )-almost

152 � 3 Multi-dimensional ρ-almost periodic type functions periodic. If condition (A)∞ or (AS) holds with X = {0}, ϕ(x) ≡ x and p(⋅) ≡ 1, then for n n n each a ∈ ℝn we have that ℳℝ [F] exists in ℂ and ℳℝ [F] = ℳℝ [F (a) ]. Before proceeding further, let us note that it is not clear how we can extend the statement (ii) to the vector-valued functions. It is also worth noting that a serious difficulty in our analysis of the multi-dimensional case presents the fact that it is not clear whether we can further generalize the above-mentioned statement by using condition (A) in place of (A)∞ or (AS). Concerning conditions (A), (A)∞ , (AS) and the equation (96), we would like to present the following example: Example 3.3.19. (i) (see also [431, Example 7.2.2]) Suppose that F0 : {(x, y) ∈ ℝ2 : 0 ⩽ x + y ⩽ 2} → [0, ∞) be any continuous function such that the following conditions hold: (a) F0 (x, y) = F0 (x + 1, y + 1) for every (x, y) ∈ ℝ2 such that x + y = 0; (b) Let Pk = Ak Bk Ck Dk be the rectangle in ℝ2 with vertices Ak = (4k − (2/3), (2/3) − 4k), Bk = (4k − (1/3), (1/3) − 4k), Ck = (4k + (2/3), (4/3) − 4k) and Dk = (4k + (1/3), (5/3) − 4k), for each integer k ∈ ℤ. We have F0 (x, y) ⩾ 2|k| for all integers k ∈ ℤ and (x, y) ∈ Pk . We extend the function F0 (⋅; ⋅) to a continuous (1, 1)-periodic function F : ℝ2 → [0, ∞) in the obvious way. Then the function F(⋅, ⋅) is not Besicovitch almost periodic since it is not Besicovitch bounded; this follows from the next simple estimates: 4k

∫ F(x, y) dx dy ⩾ 16k ∑

√2

l=0

Λ8k √2

3

2l ⩾

16k √2 4k+1 (2 − 1), 3

k ∈ ℕ.

Furthermore, there do not exist a finite real constant M > 0 and an essentially bounded function G : ℝ2 → ℂ such that 󵄨 󵄨 lim sup[t −2 ∫ 󵄨󵄨󵄨F(t) − G(t)󵄨󵄨󵄨 dt] ⩽ M; t→+∞

|t|⩽t

cf. also (96). √ (ii) Set P(x, y) := ei[ 2x+y] + ei[2x+y] , x, y ∈ ℝ, a1 := π(2 + √2), a2 := 2π(1 − √2) and a := (a1 , a2 ). Then there is no continuous (aj )j∈ℕ2 -periodic function F (a1 ,a2 ) (x, y) such that (95) holds with ϕ(x) ≡ x, p(⋅) ≡ 1 and F(t) ≡ t −2 (X = {0}). In actual fact, we have √2a1 + a2 = 2π, 2a1 + a2 = 2π and the validity of (95) would imply lim sup[t −2 t→+∞

󵄨󵄨 √ 󵄨󵄨 ∫ 󵄨󵄨󵄨ei[ 2x+y] + ei[2x+y] − F (a1 ,a2 ) (x, y)󵄨󵄨󵄨 dx dy] = 0. 󵄨 󵄨

(97)

|(x,y)|⩽t

To see that (97) cannot be true, it suffices to observe that there exists a real number √ ε0 := min(x,y)∈[0,a1 ]×[0,−a2 ] |ei[ 2x+y] + ei[2x+y] − F (a1 ,a2 ) (x, y)| > 0 such that

3.3 Multi-dimensional Besicovitch almost periodic type functions and applications

∫ (x,y)∈[0,t]2

�

153

󵄨󵄨 i[√2x+y] 󵄨󵄨 t2 󵄨󵄨e + ei[2x+y] − F (a1 ,a2 ) (x, y)󵄨󵄨󵄨 dx dy ⩾ ε . 󵄨󵄨 󵄨 ⌊a1 ⌋⌊|a2 |⌋ 0

If F : Λ × X → Y , G : Λ × X → Y and B ∈ ℬ, then we set ϕ,F

ℳB (F, G) := lim sup F(t) sup[ϕ(󵄩󵄩󵄩F(t; x) − G(t; x)󵄩󵄩󵄩Y )] t→+∞

x∈B

󵄩

󵄩

Lp(t) (Λt )

.

ϕ,F

Then the quantity ℳB (F, G) always exists in [0, +∞]. The subsequent result states that, under certain reasonable assumptions on the functions ϕ(⋅) and F(⋅), any function F ∈ e − (ℬ, ϕ, F) − Bp(⋅) (Λ × X : Y ) satisfies condition (A): Proposition 3.3.20. Suppose that F : Λ × X → Y , F ∈ e − (ℬ, ϕ, F) − Bp(⋅) (Λ × X : Y ), (I)–(III) and the following conditions hold: (IV) If B ∈ ℬ, a ∈ Λ′′ , and (Fk : Λ × X → Y )k∈ℕ is any sequence of functions which satisfies that Fk (t+a; x) = Fk (t; x), t ∈ Λ, x ∈ B and for each ε > 0 there exists k0 ∈ ℕ such that ϕ,F ℳB (Fk , Fk ′ ) < ε for all integers k, k ′ ⩾ k0 , then there exists a function F : Λ×X → Y ϕ,F

such that F(t + a; x) = F(t; x), t ∈ Λ, x ∈ B and limk→+∞ ℳB (Fk , F) = 0 (here we assume that for each element x ∈ X the functions Fk (⋅; x) and F(⋅; x) are Lebesgue measurable (k ∈ ℕ)). (V) The collection ℬ consists of bounded subsets of X. Then the function F(⋅; ⋅) satisfies condition (A). Proof. We will slightly modify the original argumentation of R. Doss (see [275, pp. 477– 478]). First of all, we will prove that condition (A) holds for any trigonometric polynomial i⟨λl ,⋅⟩ P(⋅; x) = ∑m cl (x), x ∈ X. Let B ∈ ℬ and a ∈ Λ′′ be given; then t + ja ∈ Λ for all l=0 e (a) j ∈ ℕ. Define PB (⋅; x) := ∑l∈La ei⟨λl ,⋅⟩ cl (x), x ∈ X, where La denotes the set of all integers l ∈ [0, m] such that ei⟨λl ,a⟩ = 1. Clearly, the function PB(a) (⋅; x) is a-periodic for every fixed element x ∈ B. Then a simple computation shows that 󵄩󵄩 k−1 󵄩󵄩 󵄩󵄩 1 󵄩󵄩 (a) 󵄩 F(t) sup[ϕ(󵄩󵄩 ∑ P(t + ja; x) − PB (t; x)󵄩󵄩󵄩 )] 󵄩 󵄩󵄩 k 󵄩󵄩 j=0 x∈B 󵄩Y Lp (Λt ) 󵄩󵄩 k−1 󵄩󵄩 󵄩󵄩 1 󵄩󵄩 = F(t) sup[ϕ(󵄩󵄩󵄩 ∑ ∑ ei⟨λl ,t⟩ ei⟨λl ,aj⟩ cl (x)󵄩󵄩󵄩 )] 󵄩󵄩󵄩 k j=0 l∉La 󵄩󵄩󵄩Y Lp (Λ ) x∈B t 󵄩󵄩 󵄩󵄩 i⟨λl ,ak⟩ 󵄩󵄩 1 󵄩 e − 1 󵄩 = F(t) sup[ϕ(󵄩󵄩󵄩 ∑ ei⟨λl ,t⟩ i⟨λ ,a⟩ cl (x)󵄩󵄩󵄩 )] , 󵄩󵄩 k 󵄩󵄩 l e − 1 x∈B 󵄩 l∉La 󵄩Y Lp (Λt )

t > 0.

Using the facts that the function ϕ(⋅) is monotonically increasing and the collection ℬ consists of bounded subsets of X, the above computation yields the existence of a finite real constant cB > 0 such that

154 � 3 Multi-dimensional ρ-almost periodic type functions 󵄩󵄩 k−1 󵄩󵄩 󵄩󵄩 1 󵄩󵄩 c F(t) sup[ϕ(󵄩󵄩󵄩 ∑ P(t + ja; x) − PB(a) (t; x)󵄩󵄩󵄩 )] ⩽ ϕ( B )[F(t)m(Λt )1/p ], 󵄩 󵄩 k 󵄩󵄩 k j=0 󵄩󵄩Y Lp(t) (Λ ) x∈B t

t > 0.

Then the required conclusion follows from the continuity of function ϕ(⋅) at zero and the assumption lim supt→+∞ [F(t)m(Λt )1/p ] < +∞. Before proceeding to the general case, let us note that our assumptions on the function ϕ(⋅), the assumption that for every real number a > 0, we have lim supt→+∞ [F(t)/F(t + a)] ⩽ 1, and a relatively simple argumentation shows that the following holds: (a) Define ϕ,F

ℳB,a (F, G) := lim sup F(t) sup[ϕ(󵄩󵄩󵄩F(t + a; x) − G(t + a; x)󵄩󵄩󵄩Y )] t→+∞

x∈B

󵄩

󵄩

Lp (Λt )

.

Then we have ϕ,F

ϕ,F

ℳB,a (F, G) ⩽ ℳB (F, G). ϕ,F

ϕ,F

(b) ℳB (dF, dG) ⩽ φ(d)ℳB (F, G) for all real numbers d > 0. ϕ,F

ϕ,F

ϕ,F

(c) ℳB (F, G) ⩽ c[ℳB (F, H) + ℳB (H, G)]. (d)

ϕ,F ℳB (F

+ G, H + W ) ⩽

ϕ,F c[ℳB (F, H)

ϕ,F

+ ℳB (G, W )].

Suppose now that there exists a sequence (Pk (⋅; ⋅)) of trigonometric polynomials such ϕ,F that (88) holds, i. e. limk→+∞ ℳB (F, Pk ) = 0. Let ε > 0 be fixed. Using (a) and (d), we get: ϕ,F

ℳB (

F(⋅; ⋅)+F(⋅+a; ⋅)+⋅ ⋅ ⋅+F(⋅+(m − 1)a; ⋅) Pk (⋅; ⋅)+Pk (⋅+a; ⋅)+⋅ ⋅ ⋅ + Pk (⋅+(m − 1)a; ⋅) , ) m m ϕ,F

ϕ,F

ϕ,F

⩽ (c + ⋅ ⋅ ⋅ + cm−1 )φ(1/m)ℳB (F, Pk ) ⩽ mφ(1/m)ℳB (F, Pk ) ⩽ DℳB (F, Pk ),

k, m ∈ ℕ.

(98)

On the other hand, using (c) we get the existence of an integer k0 (ε) ≡ k0 such that: ϕ,F

ℳB (Pk , Pk ′ ) ⩽ 2cε,

k, k ′ ⩾ k0 .

Keeping in mind (98) and this estimate, we get ϕ,F

ℳB (

1 m−1 1 m−1 ∑ Pk (⋅ + ja; ⋅), ∑ P ′ (⋅ + ja; ⋅)) ⩽ 2cDε, m j=0 m j=0 k

k, k ′ ⩾ k0 , m ∈ ℕ.

(99)

(a) If k, k ′ ⩾ k0 , then we find the functions Pk,B and Pk(a) ′ ,B from condition (A); then we can use ϕ,F

(a) (a) (c) and (99) to get the existence of a finite real constant d > 0 such that ℳB (Pk,B , Pk ′ ,B )
0 and x ∈ X. If the assumptions (I)–(III) hold and the function F(⋅; ⋅) satisfies condition (A), then F(⋅; ⋅) is Besicovitch-(p, ϕ, F, ℬ)-bounded. In particular, Proposition 3.3.22 implies that a uniformly recurrent function f : ℝ → ℝ need not satisfy condition (A). In the multi-dimensional setting, it seems very plausible that a trigonometric polynomial P(⋅) does not satisfy condition (AS) in general (see Example 3.3.19(ii)), so that we can freely say that the results established in [275] are primarily intended for the analysis of Besicovitch almost periodic functions of one real variable. We want also to emphasize here that the proof of [275, Proposition 4] contains a small gap since the author has not proved that, for any function f : ℝ → ℂ of class (D) and for any real number λ ∈ ℝ, the function e−iλ⋅ f (⋅) is Doss almost periodic, i. e., satisfies condition [275, 2., p. 477] with p = 1 (this seems to be true, but not specifically proved in the paper). Finally, we would like to mention that it is very likely that the statements of [275, Propositions 4–6] admit extensions to the multi-dimensional setting; hence, it seems very reasonable that a Doss-p-almost periodic function F : ℝn → ℂ which is Besicovitch-p-continuous and satisfies condition (AS) is Besicovitch-p-almost periodic (1 ⩽ p < +∞). This is a very unsatisfactory result in the multi-dimensional setting and we will skip all details with regards to this question here. 3.3.4 On condition (B) In this subsection, we will consider the following condition: (B) Let Ω = [0, 1]n , lΩ ⊆ Λ and Λ + lΩ ⊆ Λ for all l > 0. If F1 : (0, ∞) → (0, ∞), F : (0, ∞) → (0, ∞) and p ∈ 𝒫 (Λ), then condition (B) means that, for every λ ∈ ℝn , we have: 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 lim F1 (l) lim sup F(t) sup󵄩󵄩󵄩[ ∫ − ∫]eiλt F(t; x) dt󵄩󵄩󵄩 = 0. 󵄩󵄩 p(y) l→+∞ t→+∞ 󵄩󵄩 x∈B 󵄩 󵄩 L (Λ :Y ) t y+lΩ lΩ

(100)

156 � 3 Multi-dimensional ρ-almost periodic type functions Let us note that in the original analysis of R. Doss [276], we have p(⋅) ≡ 1, F1 (l) ≡ 1/l, F(t) ≡ 1/t, Λ = ℝ and X = {0}. The situation in which the equation (100) holds for all values of λ ∈ ℝn ∖ {λ0 } but not for the exactly one value λ = λ0 ∈ ℝn is possible; for example, in the one-dimensional setting, we know that the function F : ℝ → ℂ, given by F(t) := e−iλ0 t , t ⩾ 0 and F(t) := −e−iλ0 t , t < 0, satisfies (100) for all values λ ∈ ℝ ∖ {λ0 } but not for λ0 [276]. Concerning condition (B), we will first clarify the following result for the multivariate trigonometric polynomials: Proposition 3.3.23. Suppose that Ω = [0, 1]n , Λ = ℝn and liml→+∞ [ln−1 F1 (l)] = 0. If the collection ℬ consists solely of bounded subsets of X and conditions (I)–(III) hold, then condition (B) holds for any trigonometric polynomial P(⋅; ⋅). i⟨λs ,t⟩ Proof. Let P(t; x) = ∑m cs (x) for some continuous functions cs (⋅). It suffices to s=0 e show that

󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 lim F1 (l) lim sup F(t) sup󵄩󵄩󵄩[ ∫ − ∫]P(t; x) dt󵄩󵄩󵄩 = 0. 󵄩󵄩 p l→+∞ t→+∞ 󵄩󵄩 x∈B 󵄩 󵄩 L (Λ :Y ) t y+lΩ lΩ Let ks denote the number of all non-zero components of vector λs = (λ1s , . . . , λns ). If ks = n, then the term ei⟨λs ,t⟩ cs (x) is meaningless after the integration over the cubes y + lΩ and lΩ. Because of that, we may assume without loss of generality that ks ⩽ n − 1 for all j s ∈ {0, 1, . . . , m} as well as that λs = 0 for all s ∈ {0, 1, . . . , m} and j ∈ {1, . . . , k}, where k ⩽ n − 1. Therefore, we need to prove that 󵄩󵄩 󵄩󵄩 lim F1 (l) lim sup F(t) sup󵄩󵄩󵄩[ ∫ − ∫] l→+∞ t→+∞ 󵄩󵄩 x∈B 󵄩 y+lΩ lΩ m

n

k+1

× ∑ ∑ ei[λs

tk+1 +⋅⋅⋅+λns tn ]

s=0 j=k+1

󵄩󵄩 󵄩󵄩 cs (x) dt1 dt2 . . . dtn 󵄩󵄩󵄩 = 0, 󵄩󵄩 p 󵄩L (Λt :Y )

i. e., that 󵄩󵄩 󵄩󵄩 lim F1 (l)lk lim sup F(t) sup󵄩󵄩󵄩[ ∫ − ∫] l→+∞ t→+∞ 󵄩󵄩 x∈B 󵄩 y+lΩ lΩ m

n

k+1

× ∑ ∑ ei[λs s=0 j=k+1

tk+1 +⋅⋅⋅+λns tn ]

󵄩󵄩 󵄩󵄩 cs (x) dtk+1 dtk+2 . . . dtn 󵄩󵄩󵄩 = 0. 󵄩󵄩 p 󵄩L (Λt :Y )

It is clear that we have the existence of a finite real constant c > 0 such that, for every y ∈ Λt and l > 0, we have:

3.3 Multi-dimensional Besicovitch almost periodic type functions and applications

� 157

󵄩󵄩 󵄩󵄩 m n k+1 n 󵄩󵄩 󵄩 󵄩󵄩[ ∫ − ∫] ∑ ∑ ei[λs tk+1 +⋅⋅⋅+λs tn ] cs (x) dtk+1 dtk+2 . . . dtn 󵄩󵄩󵄩 ⩽ c. 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩 s=0 j=k+1 󵄩Y y+lΩ lΩ Since liml→+∞ [ln−1 F1 (l)] = 0 and lim supt→+∞ F(t)[m(Λt )]1/p < +∞, this simply completes the proof. In the continuation of subsection, we will first consider the one-dimensional setting, show that the necessity in [276, Theorem 1] can be extended to the vector-valued Besicovitch-p-almost periodic functions, where 1 ⩽ p < ∞, and emphasize some difficulties in proving the sufficiency in this theorem in the case that p > 1 (in the existing literature, we have found many open problems regarding Besicovitch-p-almost periodic functions with the exponent p > 1). Keeping in mind the proof of the abovementioned theorem, which works in the vector-valued case, as well as the statements of Proposition 3.3.5(ii) and Proposition 3.3.13, it suffices to show that condition (B) holds for Besicovitch-p-almost periodic functions F : ℝ → Y , with 1 < p < ∞, λ = 0, F1 (l) ≡ 1/l and F1 (t) ≡ 1/t 1/p . Further on, keeping in mind Proposition 3.3.24 below and its proof (condition (101) holds on account of the Hölder inequality), it suffices to show that there exists a finite real constant c > 0 such that, for every real number ε ∈ (0, 1), there exists a real number l0 > 0 such that the assumptions l ⩾ l0 and ℳx,F (F, P) < ε imply the existence of a sufficiently large number tl > 0 such that for every t ⩾ tl , we have l t

p

y+l

t −1 −(1/p)

1/p

󵄩 󵄩 (∫[ ∫ 󵄩󵄩󵄩F(s) − P(s)󵄩󵄩󵄩Y ds] dy)

< cε.

y

−t

Let ε > 0 be given, let l0 = 1 and l ⩾ 1. We will show that we can take c = 5 in the above requirement. First of all, we have the existence of a finite real number t0 > 0 such that t

󵄩 󵄩p ∫󵄩󵄩󵄩F(s) − P(s)󵄩󵄩󵄩Y ds ⩽ εt p ,

t ⩾ t0 .

−t

Then there exists a sufficiently large number tl1 > 0 such that t0 +l

y+l

−(t0 +l)

y

p

1/p

󵄩 󵄩 l−1 t −(1/p) ( ∫ [ ∫ 󵄩󵄩󵄩F(s) − P(s)󵄩󵄩󵄩Y ds] dy)

< ε,

t ⩾ tl1 .

There exists a great similarity in the analysis of estimates for the intervals [−t, −(t0 + l)] and [t0 + l, t] and, because of that, we will only prove that there exists a sufficiently large number tl2 > t0 such that t

l t

−1 −(1/p)

y+l

p

1/p

󵄩 󵄩 ( ∫ [ ∫ 󵄩󵄩󵄩F(s) − P(s)󵄩󵄩󵄩Y ds] dy) t0 +l

y

< 2ε,

t ⩾ tl2 .

158 � 3 Multi-dimensional ρ-almost periodic type functions This follows from the next computation (in the fourth line we can use the inequality appearing on l. 12, p. 134 of [276], which is a consequence of a simple computation with double integrals): t

l t

−1 −(1/p)

p

y+l

1/p

󵄩 󵄩 ( ∫ [ ∫ 󵄩󵄩󵄩F(s) − P(s)󵄩󵄩󵄩Y ds] dy) t0 +l

y

t

⩽l t

−1 −(1/p)

( ∫ [l

1− p1

t0 +l

1/p p

y+l

󵄩 󵄩p ( ∫ 󵄩󵄩󵄩F(s) − P(s)󵄩󵄩󵄩Y ds) y

=l

t

] dy)

1/p

t y+l

−(1/p) −(1/p)

1/p

󵄩 󵄩p ( ∫ ∫ 󵄩󵄩󵄩F(s) − P(s)󵄩󵄩󵄩Y ds dy) t0 +l y

1/p

t+l

󵄩 󵄩p ⩽ l−(1/p) t −(1/p) (l ∫ 󵄩󵄩󵄩F(s) − P(s)󵄩󵄩󵄩Y ds) ⩽ εt

−(1/p)

−t 1/p

[2(t + l)]

< 2ε,

t ⩾ tl2 .

If p > 1, then it is very difficult to show that the validity of condition (B) with F1 (l) ≡ l−1 and F(t) ≡ t −(1/p) for a Besicovitch-p-continuous function F : ℝ → Y implies the validity of condition (A) for F(⋅), even in the scalar-valued case. In actual fact, it is very simple to prove that (B) implies the validity of equation obtained by replacing the term | ⋅ | in the equation [276, (4)] with the term | ⋅ |p . But, if we replace the term (we will consider the scalar-valued case only) c 󵄨󵄨 󵄨󵄨 n−1 n−1 󵄨󵄨 −1 󵄨 󵄨󵄨c ∫[n−1 ∑ f (t + x + kc) − n−1 ∑ f (x + kc)]Km (t) dt 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 k=0 k=0 0

in the equation [276, (*), p. 136] with the term c 󵄨󵄨 󵄨󵄨p n−1 n−1 󵄨󵄨 −1 󵄨 −1 −1 󵄨󵄨c ∫[n ∑ f (t + x + kc) − n ∑ f (x + kc)]Km (t) dt 󵄨󵄨󵄨 , 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 k=0 k=0 0

which can be majorized by ⩽c

−p

p/q c 󵄨󵄨p n−1 n−1 󵄨󵄨 󵄨󵄨 󵄨󵄨 −1 −1 q ∫󵄨󵄨n ∑ f (t + x + kc) − n ∑ f (x + kc)󵄨󵄨󵄨 dt ⋅ (∫ Km (t) dt) 󵄨󵄨 󵄨󵄨 k=0 k=0 󵄨 0󵄨 0 c󵄨

with the help of the Hölder inequality, then it is impossible to control the term c q (∫0 Km (t) dt)p/q as m → +∞. This can be done only in the case that p = 1 because the Fejér kernels

3.3 Multi-dimensional Besicovitch almost periodic type functions and applications

Km (t) = m−1 sin2 (mπt/c)/ sin2 (πt/c),

�

159

t ∈ ℝ (m ∈ ℕ)

are uniformly integrable in a neighborhood of zero with respect to m ∈ ℕ but not uniformly q-integrable in a neighborhood of zero with respect to m ∈ ℕ, if q > 1. Concerning the multi-dimensional setting, we will prove the following result: Proposition 3.3.24. Let Ω = [0, 1]n , Λ = ℝn or Λ = [0, ∞)n , liml→+∞ [ln−1 F1 (l)] = 0, and let 󵄩 󵄩 lim F1 (l) sup ∫󵄩󵄩󵄩F(t; x) − P(t; x)󵄩󵄩󵄩Y dt = 0,

l→+∞

x∈B

B ∈ ℬ.

(101)

lΩ

If there exists a finite real constant c > 0 such that, for every real number ε ∈ (0, 1) and for every set B ∈ ℬ, there exists a real number l0 > 0 such that the assumptions l ⩾ l0 and ℳx,F B (F, P) < ε imply the existence of a sufficiently large number tl > 0 such that for every t ⩾ tl , we have 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 F1 (l)F(t) sup󵄩󵄩󵄩 ∫ (F(t; x) − P(t; x)) dt󵄩󵄩󵄩 < cε, 󵄩󵄩 p 󵄩󵄩 x∈B 󵄩 󵄩 L (Λ :Y ) t y+lΩ

(102)

the collection ℬ consists solely of bounded subsets of X and conditions (I)–(III) hold, then condition (B) holds for any function F ∈ e − (ℬ, F) − Bp (Λ × X : Y ). Proof. Let ε > 0 and B ∈ ℬ be given; we will consider the value λ = 0 in (B) only. It is clear that there exist a trigonometric polynomial P(⋅; ⋅) and a finite real number t0 > 0 such that for every real number t ⩾ t0 , we have 󵄩 󵄩 F(t) sup󵄩󵄩󵄩F(t; x) − P(t; x)󵄩󵄩󵄩Lp (Λ :Y ) < ε. x∈B

t

Then we have (l > 0): 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 F1 (l) lim sup F(t) sup󵄩󵄩[ ∫ − ∫]F(t; x) dt󵄩󵄩󵄩 󵄩 󵄩󵄩 p t→+∞ x∈B 󵄩 󵄩 y+lΩ lΩ 󵄩L (Λt :Y ) 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 ⩽ F1 (l) lim sup F(t) sup󵄩󵄩󵄩[ ∫ − ∫](F(t; x) − P(t; x)) dt󵄩󵄩󵄩 󵄩󵄩 p t→+∞ 󵄩󵄩 x∈B 󵄩 󵄩L (Λt :Y ) y+lΩ lΩ 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 + F1 (l) lim sup F(t) sup󵄩󵄩󵄩[ ∫ − ∫]P(t; x) dt󵄩󵄩󵄩 󵄩 󵄩󵄩 p t→+∞ x∈B 󵄩 󵄩 y+lΩ lΩ 󵄩L (Λt :Y ) 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 ⩽ F1 (l) lim sup F(t) sup󵄩󵄩󵄩 ∫ (F(t; x) − P(t; x)) dt󵄩󵄩󵄩 󵄩󵄩 p t→+∞ 󵄩󵄩 x∈B 󵄩 󵄩L (Λt :Y ) y+lΩ

160 � 3 Multi-dimensional ρ-almost periodic type functions 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 + F1 (l) lim sup F(t) sup󵄩󵄩󵄩∫(F(t; x) − P(t; x)) dt󵄩󵄩󵄩 󵄩 󵄩󵄩 p t→+∞ x∈B 󵄩 󵄩lΩ 󵄩L (Λt :Y ) 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 + F1 (l) lim sup F(t) sup󵄩󵄩󵄩[ ∫ − ∫]P(t; x) dt󵄩󵄩󵄩 . 󵄩󵄩 p t→+∞ 󵄩󵄩 x∈B 󵄩 󵄩 L (Λ :Y ) t y+lΩ lΩ The third term in the last estimate can be majorized as in Proposition 3.3.23. For the second term, we can use the assumption (101) and the inequality lim supt→+∞ F(t)[m(Λt )]1/p < +∞. For the first term, we can use our assumption (102). It is not difficult to prove that for every locally integrable function F : [0, ∞)n → Y , t > 0 and t0 ∈ (0, t), we have the estimate 󵄩 󵄩 󵄩 󵄩 ∫ ∫ 󵄩󵄩󵄩F(s)󵄩󵄩󵄩Y ds dy ⩽ ∫ 󵄩󵄩󵄩F(s)󵄩󵄩󵄩Y ds,

Ω0 y+lΩ

(t+l)Ω

where Ω0 := tΩ ∖ [0, t0 ]n . Using the Hölder inequality and repeating verbatim the argumentation given in the one-dimensional setting, we can prove that the requirements of Proposition 3.3.24 hold with F1 (l) ≡ l−n and F(t) ≡ t −(n/p) . Finally, it seems reasonable to ask whether the validity of condition (B) with p(⋅) ≡ 1 implies, along with the corresponding Besicovitch continuity assumption, the validity of condition (A) in the multi-dimensional setting. We will not consider this question here, nor certain possibilities to extend the results established by A. S. Kovanko [464, 465] to the multi-dimensional setting.

3.3.5 Applications to the abstract Volterra integro-differential equations In this subsection, we will provide several applications of our results to the various classes of abstract Volterra integro-differential equations and the partial differential equations. 1. In this part, we will first prove a new result about the invariance of Besicovitch-palmost periodicity under the actions of infinite convolution product t

t 󳨃→ F(t) := ∫ R(t − s)f (s) ds,

t ∈ ℝ;

(103)

−∞

we will only note here, without going into full details, that this result can be formulated in the multi-dimensional setting as well [431]. We assume that the operator family (R(t))t>0 ⊆ L(X, Y ) satisfies that there exist finite real constants M > 0, β ∈ (0, 1] and γ > β such that

3.3 Multi-dimensional Besicovitch almost periodic type functions and applications

t β−1 󵄩󵄩 󵄩 , 󵄩󵄩R(t)󵄩󵄩󵄩L(X,Y ) ⩽ M 1 + tγ

� 161

t > 0,

(104)

and f (⋅) is Besicovitch-p-almost periodic. The following result is closely connected with the statements of [428, Theorem 2.11.4, Theorem 2.13.10, Theorem 2.13.12]: Proposition 3.3.25. Suppose that the operator family (R(t))t>0 ⊆ L(X, Y ) satisfies (104), as well as that a > 0, α > 0, 1 ⩽ p < +∞, αp ⩾ 1, ap ⩾ 1, αp(β − 1)/(αp − 1) > −1 if αp > 1, and β = 1 if αp = 1. If the function f : ℝ → X is Stepanov-(αp)-bounded, i. e. t+1

󵄩󵄩 󵄩󵄩 󵄩 󵄩αp 󵄩󵄩f 󵄩󵄩Sp := sup ∫ 󵄩󵄩󵄩f (s)󵄩󵄩󵄩 ds < +∞, t∈ℝ

t

and f ∈ e − (x α , t −a ) − Bp (ℝ : X), then the function F(⋅), given by (103), is bounded, continuous and belongs to the class e − (x α , t −a ) − Bp (ℝ : Y ). Proof. Arguing as in the proof of [428, Proposition 2.6.11], we may conclude that the function F(⋅) is well-defined, bounded and continuous. Let (Pk ) be a sequence of trigonometric polynomials such that t

1 󵄩 󵄩αp lim lim sup ∫󵄩󵄩󵄩f (s) − Pk (s)󵄩󵄩󵄩 ds = 0. k→+∞ t→+∞ 2t ap

(105)

−t

t

Applying again [428, Proposition 2.6.11], we get that the function t 󳨃→ Fk (t) ≡ ∫−∞ R(t − s)Pk (s) ds, t ∈ ℝ is almost periodic and we only need to prove that t

lim lim sup

k→+∞ t→+∞

1 󵄩 󵄩αp ∫󵄩󵄩F(s) − Fk (s)󵄩󵄩󵄩 ds = 0. 2t ap 󵄩

(106)

−t

In the remainder of the proof, we will consider case αp > 1 since the consideration is quite similar if αp = 1. Let ζ ∈ (1/(αp), (1/(αp)) + γ − β). Then it is clear that the function s 󳨃→ |s|β−1 (1 + |s|)ζ /(1 + |s|γ ), s ∈ ℝ belongs to the space Lαp/(αp−1) ((−∞, 0)); further on, arguing as in the proof of [428, Theorem 2.11.4], we have that the function s 󳨃→ (1 + |s|)−ζ ‖Pk (s + z) − f (s + z)‖, s ∈ ℝ belongs to the space Lαp ((−∞, 0)) for all k ∈ ℕ and z ∈ ℝ. The estimate (106) follows from the next computation (M1 > 0 and c > 0 are finite real constants): t

1 󵄩 󵄩αp ∫󵄩󵄩F(s) − Fk (s)󵄩󵄩󵄩 ds 2t ap 󵄩 −t

αp t󵄨 0 󵄨󵄨 󵄨󵄨 1 󵄨󵄨 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄨󵄨󵄨 ⩽ ap ∫󵄨󵄨 ∫ 󵄩󵄩R(−z)󵄩󵄩 ⋅ 󵄩󵄩Pk (s + z) − f (s + z)󵄩󵄩 dz󵄨󵄨 ds 󵄨󵄨 󵄨󵄨 2t 󵄨 −t 󵄨−∞

162 � 3 Multi-dimensional ρ-almost periodic type functions t󵄨 0 ζ 󵄨󵄨αp |z|β−1 (1 + |z|) M 󵄨󵄨󵄨󵄨 󵄩󵄩 󵄨󵄨󵄨 −ζ 󵄩 󵄩 ⩽ ap ∫󵄨󵄨 ∫ ⋅ (1 + |z|) 󵄩󵄩Pk (s + z) − f (s + z)󵄩󵄩 dz󵄨󵄨 ds 󵄨󵄨 󵄨󵄨 2t (1 + |z|γ ) 󵄨 −t 󵄨−∞ t

⩽

0

M1 1 󵄩󵄩 󵄩αp ∫ ∫ 󵄩Pk (s + z) − f (s + z)󵄩󵄩󵄩 dz ds 2t ap (1 + |z|αζ )p 󵄩 −t −∞ t t

M1 1 󵄩󵄩 󵄩αp = ap ∫∫ 󵄩󵄩Pk (z) − f (z)󵄩󵄩󵄩 ds dz αζ p 2t (1 + |z − s| ) −t z

+

t

t

M1 1 󵄩󵄩 󵄩αp ∫ ∫ 󵄩Pk (z) − f (z)󵄩󵄩󵄩 ds dz 2t ap (1 + |z − s|αζ )p 󵄩 t

−∞ −t

+∞

M 󵄩 ds 󵄩αp ⩽ ap1 ∫󵄩󵄩󵄩Pk (z) − f (z)󵄩󵄩󵄩 dz ⋅ ∫ t (1 + |s|ζ )αp −t

+

−∞

−3t t

M1 1 󵄩󵄩 󵄩αp ∫ ∫ 󵄩Pk (z) − f (z)󵄩󵄩󵄩 ds dz 2t ap (1 + |z − s|αζ )p 󵄩 −∞ −t 3t t

+

M1 1 󵄩󵄩 󵄩αp ∫∫ 󵄩Pk (z) − f (z)󵄩󵄩󵄩 ds dz 2t ap (1 + |z − s|αζ )p 󵄩 t

−3t −t

+∞

M 󵄩 ds 󵄩αp ⩽ ap1 ∫󵄩󵄩󵄩Pk (z) − f (z)󵄩󵄩󵄩 dz ⋅ ∫ t (1 + |s|ζ )αp −t

+

−∞

3t

+∞

M1 󵄩󵄩 ds 󵄩αp ∫ 󵄩P (z) − f (z)󵄩󵄩󵄩 dz ⋅ ∫ t ap 󵄩 k (1 + |s|ζ )αp −∞

−3t

−3t

+

cM1 t 1 󵄩󵄩 󵄩αp ∫ 󵄩Pk (z) − f (z)󵄩󵄩󵄩 dz, 2t ap (1 + |z/2|αζ )p 󵄩 −∞

involving the Hölder inequality, the Fubini theorem and an elementary change of variables in the double integral; here we use (105) and the fact that −3t

lim ∫

t→+∞

−∞

1 󵄩󵄩 󵄩αp 󵄩󵄩Pk (z) − f (z)󵄩󵄩󵄩 dz = 0. αζ p (1 + |z/2| )

In the following result, the inhomogeneity f (⋅) is not necessarily Stepanov-(αp)bounded: Proposition 3.3.26. Suppose that the operator family (R(t))t>0 ⊆ L(X, Y ) satisfies (104), as well as that a > 0, α > 0, 1 ⩽ p < +∞, αp ⩾ 1, ap ⩾ 1, αp(β − 1)/(αp − 1) > −1 if αp > 1, and β = 1 if αp = 1. If the function f : ℝ → X belongs to the class e − (x α , t −a ) − Bp (ℝ : X)

3.3 Multi-dimensional Besicovitch almost periodic type functions and applications

�

163

and there exists a finite real constant M > 0 such that ‖f (t)‖ ⩽ M(1 + |t|)b , t ∈ ℝ for some real constant b ∈ [0, γ − β), then the function F(⋅), given by (103), is continuous, belongs to the class e − (x α , t −a ) − Bp (ℝ : Y ), and there exists a finite real constant M ′ > 0 such that ‖F(t)‖Y ⩽ M ′ (1 + |t|)b , t ∈ ℝ. Proof. The proof is very similar to the proof of Proposition 3.3.25 and we will only outline the main details. Since b ∈ [0, γ − β), it can be simply shown that the function F(⋅) is welldefined, measurable as well as that there exists a finite real constant M ′ > 0 such that ‖F(t)‖Y ⩽ M ′ (1 + |t|)b , t ∈ ℝ. In order to prove the continuity of function F(⋅) at the fixed point t ∈ ℝ, we take first any integer k ∈ ℕ such that +∞

∫ k

sβ−1 (1 + s)b ε ds ⩽ , 1 + sγ 4M(1 + |t|b )3b

(107)

where a real number ε > 0 is given in advance and the real constant M > 0 has the αp same value as in (104). Since ‖R(⋅)‖L(X,Y ) ∈ Lαp/(αp−1) [0, k] and ‖f ‖ ∈ Lloc [0, k], we can k

apply the Hölder inequality in order to see that the function Fk (⋅) := ∫0 R(s)f (⋅ − s) ds, ⋅ ∈ ℝ is continuous. Take any δ ∈ (0, 1) such that 󵄩󵄩 k 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩∫ R(s)[f (t − s) − f (t ′ − s)] ds󵄩󵄩󵄩 < ε 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩Y 2 0

for |t − t ′ | ⩽ δ.

(108)

A very simple argumentation involving (107) shows that 󵄩󵄩 +∞ 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∫ R(s)[f (t − s) − f (t ′ − s)] ds󵄩󵄩󵄩 ⩽ ε/2, 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩Y k which, along with (108) completes, the proof of continuity of function F(⋅) at the point t. The belonging of function F(⋅) to the class e − (x α , t −1/p ) − Bp (ℝ : Y ) can be proved as above, by taking any real number ζ ∈ ((1/(αp)) + b, (1/(αp)) + γ − β). Remark 3.3.27. Let us quote two important situations in which we do not need the assumption b < γ − β. (i) Suppose that there exist finite real constants M > 0, c > 0 and β ∈ (0, 1] such that 󵄩󵄩 󵄩 −ct β−1 󵄩󵄩R(t)󵄩󵄩󵄩L(X,Y ) ⩽ Me t ,

t > 0.

This estimate appears in the analysis of (degenerate) semigroups of operators satisfying condition (P) and, in particular, in the analysis of semigroups of operators generated by almost sectorial operators (cf. also the important research monograph by A. Favini and A. Yagi [300]).

164 � 3 Multi-dimensional ρ-almost periodic type functions (ii) The existence and uniqueness of (asymptotically) almost periodic solutions of the vectorial parabolic equations on the real hyperbolic manifold M = ℍd (ℝ), where d ⩾ 2, have recently been analyzed by P. T. Xuan, N. T. Van and B. Quoc in [784]. More precisely, the authors have studied the almost periodic solutions of the following abstract parabolic equation 𝜕t u = Lu + BG(t, u),

t∈ℝ

on the Banach space Y (G(TM)), and the asymptotically almost periodic solutions of the following abstract parabolic equation 𝜕t u = Lu + BG(t, u),

t ⩾ 0; u(0, x) = u0 (x) ∈ Y (G(TM)),

where L is the Ebin–Marsden’s Laplace operator, B : X(G(TM)) → Y (G(TM)) is a connection operator between the Banach spaces X(G(TM)), Y (G(TM)) and G(t, u) : ℝ×Y (G(TM)) → X(G(TM)) [G(t, u) : [0, ∞)×Y (G(TM)) → X(G(TM))] is a nonlinear operator satisfying certain assumptions. The estimate 󵄩󵄩 󵄩 −ct β−1 󵄩󵄩R(t)󵄩󵄩󵄩L(X,Y ) ⩽ Me (t + 1),

t > 0,

(109)

for some finite real constants M > 0, c > 0 and β ∈ (0, 1] appears in the analysis of the authors (see the equations [784, (2.5)–(2.6)]). Therefore, Proposition 3.3.26 in combination with the arguments given in the proof of [784, Theorem 3.6] can be applied in the analysis of the existence and uniqueness of Besicovitch almost periodic type solutions of the linear counterparts of the above-mentioned problems. In the analysis of the corresponding semilinear problems, we must additionally employ the composition principle established in Theorem 3.3.11. The estimate (109) also appears in the study of the evolution equations of parabolic type on a non-compact Einstein manifolds [582]. We will only note here that Proposition 3.3.26 in combination with the arguments given in the proof of [582, Theorem 2.9] can be applied in the analysis of the existence and uniqueness of Besicovitch almost periodic type solutions of the problem [582, (2.33)]. It is clear that Proposition 3.3.25 and Proposition 3.3.26 can be applied to a large class of the abstract (degenerate) Volterra integro-differential equations without initial conditions. For example, we can apply this result in the analysis of the existence and uniqueness of Besicovitch-p-almost periodic type solutions of the initial value problem with constant coefficients γ

Dt,+ u(t, x) = ∑ aα f (α) (t, x) + f (t, x),

t ∈ ℝ, x ∈ ℝn

(110)

|α|⩽k γ

in the space Lp (ℝn ), where γ ∈ (0, 1), Dt,+ u(t) denotes the Weyl–Liouville fractional derivative of order γ, 1 ⩽ p < ∞ and some extra assumptions are satisfied. We can also

3.3 Multi-dimensional Besicovitch almost periodic type functions and applications

�

165

consider the existence and uniqueness of Besicovitch-p-almost periodic type solutions of the fractional Poisson heat equation in Lp (ℝn ), and a class of the abstract fractional differential equations with the higher-order elliptic operators in the Hölder spaces [428]. 2. In this part, we analyze the abstract nonautonomous differential equations of first order. First of all, we will remind the readers of some basic definitions from the theory of evolution equations, hyperbolic evolution systems, and Green’s functions (see [533] and the references cited in [428, Section 2.14]). Definition 3.3.28. A family {U(t, s) : t ⩾ s, t, s ∈ ℝ} of bounded linear operators on X is said to be an evolution system if and only if the following holds: (i) U(s, s) = I, U(t, s) = U(t, r)U(r, s) for t ⩾ r ⩾ s and t, r, s ∈ ℝ, (ii) {(τ, s) ∈ ℝ2 : τ > s} ∋ (t, s) 󳨃→ U(t, s)x is continuous for any fixed element x ∈ X. Here, I denotes the identity operator on X. We assume that the family A(⋅) satisfies conditions [428, (H1)–(H2)]; then there exists an evolution system {U(t, s) : t ⩾ s, t, s ∈ ℝ} generated by the family A(t). If Γ(⋅, ⋅) denotes the associated Green’s function, then we know that there exists a finite real constant M > 0 such that 󵄩󵄩 󵄩 −ω|t−s| , 󵄩󵄩Γ(t, s)󵄩󵄩󵄩 ⩽ Me

t, s ∈ ℝ.

(111)

The function +∞

u(t) := ∫ Γ(t, s)f (s) ds,

t∈ℝ

(112)

−∞

is said to be a unique mild solution of the abstract Cauchy problem u′ (t) = A(t)u(t) + f (t),

t ∈ ℝ.

The proof of subsequent theorem can be deduced using the argumentation employed in the proof of [428, Theorem 3.7.1], where we have assumed that p = 1, the argumentation contained in the proof of Proposition 3.3.25, the estimate (111), and the following facts: t (i) If P(⋅) is a trigonometric polynomial, then the functions t 󳨃→ ∫−∞ Γ(t, s)P(s) ds, t ∈ ℝ and t 󳨃→ ∫t Γ(t, s)P(s) ds, t ∈ ℝ are almost periodic; see e. g. the proof of [270, Theorem 2.3]. +∞ +∞ t (ii) We write ∫−∞ Γ(t, s)f (s) ds = ∫t Γ(t, s)f (s) ds + ∫−∞ Γ(t, s)f (s) ds, t ∈ ℝ. The both addends can be considered similarly, while the second addend is identically equal 0 to ∫−∞ Γ(t, t + s)f (t + s) ds, t ∈ ℝ. +∞

Theorem 3.3.29. Suppose that a > 0, α > 0, 1 ⩽ p < +∞, αp ⩾ 1 and ap ⩾ 1. If the function f : ℝ → X is bounded and f ∈ e − (x α , t −a ) − Bp (ℝ : X), then the function u(⋅), given by (112), is bounded, continuous and belongs to the class e − (x α , t −a ) − Bp (ℝ : X).

166 � 3 Multi-dimensional ρ-almost periodic type functions 3. The use of Theorem 3.3.11 is almost mandatory in the analysis of the existence and uniqueness of Besicovitch almost periodic type solutions for some classes of the abstract semilinear Cauchy problems. The first part of this result has a serious unpleasant drawback, because we must impose that the function G : ℝn × Y → Z from its formulation is Bohr ℬ-almost periodic, which automatically leads to the existence and uniqueness of almost periodic solutions of the abstract semilinear Cauchy problems under our consideration, in a certain sense. Here we will present the following illustrative application of Theorem 3.3.11(ii), with ζ = 1, and Proposition 3.3.25. Suppose that the operator family (R(t))t>0 ⊆ L(X) satisfies (104), as well as that 1 ⩽ p, q < +∞, 1/p + 1/q = 1, q(β − 1) > −1 if p > 1, and β = 1 if p = 1. If (R(t))t>0 is a solution operator family which governs solutions of the abstract fractional Cauchy γ inclusion Dt,+ u(t) ∈ 𝒜u(t) + g(t), t ∈ ℝ, where γ ∈ (0, 1), and a closed multivalued linear operator 𝒜 satisfies condition (P), then it is usually said that a continuous function t 󳨃→ u(t), t ∈ ℝ is a mild solution of the abstract semilinear fractional Cauchy inclusion (SCP):

γ

Dt,+ u(t) ∈ 𝒜u(t) + G(t; u(t)),

t∈ℝ

if and only if t

u(t) = ∫ R(t − s)G(s; u(s)) ds,

t ∈ ℝ.

−∞ p

Suppose that G ∈ e − (ℬ, x, t −n/p ) − Ba,1 (ℝ × X : X), where ℬ denotes the family of all bounded subsets of X. Suppose further that there exists a finite real constant a > 0 ∞ such that (91) holds with α = 1, a ∫0 ‖R(s)‖ ds < 1, and supt∈ℝ,x∈B ‖G(t; x)‖ < +∞ for every bounded subset B of X. It can be simply proved that the vector space Cb (ℝ : X) ∩ e − (x, t −1/p ) − Bp (ℝ : X) equipped with the sup-norm is a Banach space. Applying Proposition 3.3.25(ii) and Theorem 3.3.11, we get that the mapping Φ : Cb (ℝ : X) ∩ e − (x, t −1/p ) − Bp (ℝ : X) → Cb (ℝ : X) ∩ e − (x, t −1/p ) − Bp (ℝ : X), given by t (Φu)(t) := ∫−∞ R(t − s)G(s; u(s)) ds, t ∈ ℝ, is a well-defined contraction; therefore, there exists a unique mild solution u(⋅) of the abstract semilinear inclusion (SCP) which belongs to the space Cb (ℝ : X) ∩ e − (x, t −1/p ) − Bp (ℝ : X). 4. Our results about the invariance of Besicovitch-p-almost periodicity under the actions of infinite convolution products can be also formulated for the usual convolution f 󳨃→ F(x) ≡ ∫ h(x − y)f (y) dy,

x ∈ ℝn ,

(113)

ℝn

provided that the function h ∈ L1 (ℝn ) has a certain growth order. Before stating a general result in this direction, we will first consider the inhomogeneous heat equation in ℝn whose solutions are governed by the action of Gaussian semigroup

3.3 Multi-dimensional Besicovitch almost periodic type functions and applications

F 󳨃→ (G(t)F)(x) ≡ (4πt)

−n/2

2

t > 0, x ∈ ℝn .

∫ e−|y| /4t F(x − y) dy,

� 167

(114)

ℝn

Without going into full details, we will only note that the formula (114) makes sense even if the function F(⋅) is polynomially bounded; our basic assumption will be that there exist two finite real numbers b ⩾ 0 and c > 0 such that |F(x)| ⩽ c(1 + |x|)b , x ∈ ℝn as well as that a > 0, α > 0, 1 ⩽ p < +∞, αp ⩾ 1, 1/(αp) + 1/q = 1 and F ∈ e − (x α , t −a ) − Bp (ℝn : ℂ). Let us fix a real number t0 in (114). Then the mapping x 󳨃→ (G(t0 )F)(x), x ∈ ℝn is welldefined and has the same growth as the inhomogeneity f (⋅). Now we will prove that this mapping belongs to the class e − (x α , t −a ) − Bp (ℝn : ℂ) as well. Let ε > 0, let −n/2 󵄩 󵄩󵄩 −|⋅|2 /8t0 󵄩󵄩󵄩αp 󵄩󵄩e 󵄩󵄩 q n , 󵄩 󵄩L (ℝ )

ct0 := (4πt0 ) and let ε0 > 0 be such that

n

2

ε0 ⋅ 2ap ct0 ∫ e−|y| p/8t0 (1 + |y|) dy < ε. ℝn

We know that there exist a trigonometric polynomial P(⋅) and a finite real number t1 > 0 such that 󵄨 󵄨αp ∫ 󵄨󵄨󵄨F(x) − P(x)󵄨󵄨󵄨 dx < ε0 t ap ,

t ⩾ t1 .

(115)

[−t,t]n

Furthermore, we know that the function x 󳨃→ (G(t0 )P)(x), x ∈ ℝn is Bohr almost periodic (see [431, Subsection 6.1.7]) so that the final conclusion simply follows from the next computation involving (115) and the inequality (x + y)ap ⩽ 2ap (x ap + yap ), x, y ⩾ 0 (see also the computation from the proof of Proposition 3.3.25): 1 t ap

󵄨 󵄨αp ∫ 󵄨󵄨󵄨(G(t0 )F)(x) − −(G(t0 )P)(x)󵄨󵄨󵄨 dx

[−t,t]n

⩽ = ⩽ ⩽

ct0

t ap ct0

t ap ct0

t ap ct0

t ap

2 󵄨 󵄨αp ∫ ∫ e−|y| αp/8t0 󵄨󵄨󵄨F(x − y) − P(x − y)󵄨󵄨󵄨 dy dx

[−t,t]n

ℝn

2 󵄨 󵄨αp ∫ e−|y| αp/8t0 ∫ 󵄨󵄨󵄨F(x − y) − P(x − y)󵄨󵄨󵄨 dx dy

ℝn

[−t,t]n

2

∫ e−|y| αp/8t0 ℝn

∫ [−(t+|y|),t+|y|]n

2

󵄨󵄨 󵄨αp 󵄨󵄨F(x) − P(x)󵄨󵄨󵄨 dx dy

∫ e−|y| αp/8t0 ε0 2ap (t ap + |y|ap ) dy, ℝn

t ⩾ t1 .

(116)

168 � 3 Multi-dimensional ρ-almost periodic type functions 2

2

2

The estimate (116) is obtained by writing the term e−|y| /4t0 = e−|y| /8t0 e−|y| /8t0 and applying the Hölder inequality after that. This can be also done in the general case; arguing so, we can prove the following result: Theorem 3.3.30. Suppose that b ⩾ 0, α > 0, a > 0, 1 ⩽ p < +∞, αp ⩾ 1, 1/(αp) + 1/q = 1, f ∈ e − (x α , t −a ) − Bp (ℝn : Y ) and ‖f (x)‖Y ⩽ c(1 + |x|)b , x ∈ ℝn . If there exist two functions h1 : ℝn → ℂ and h2 : ℝn → ℂ such that h = h1 h2 , h1 ∈ Lq (ℝn ) and |h1 (⋅)|α [1 + | ⋅ |]ζ ∈ Lp (ℝn ) with ζ = max(bα, a), then the function F(⋅), given by (113), belongs to the class e − (x α , t −a ) − Bp (ℝn : Y ) and has the same growth order as f (⋅). We can simply apply Theorem 3.3.11 and Theorem 3.3.30 in the analysis of the existence and uniqueness of bounded Besicovitch-p-almost periodic solutions for a class of the semilinear Hammerstein integral equations of convolution type on ℝn ; see [431, p. 362] for more details. It would be very captivating to incorporate Theorem 3.3.30 in the analysis of the existence and uniqueness of Besicovitch-p-almost periodic solutions of the abstract ill-posed Cauchy problems whose solutions are governed by integrated solution operator families or C-regularized solution operator families (cf. [431, pp. 543–545] for more details). In the remaining applications, we will consider the usual case ϕ(x) ≡ x and the class of Besicovitch-p-almost periodic functions. 5. Without going into full details, we would like to note that the argumentation contained in our analysis of [431, Example 3, p. XXXV] enables one to consider the existence and uniqueness of Besicovitch-p-almost periodic type solutions of the wave equation in ℝ2 whose solutions are given by the d’Alembert formula. For example, if the functions f (⋅) and g [1] (⋅) from this example are Besicovitch-p-almost periodic in ℝ, then the solution u(x, t) will be Besicovitch-p-almost periodic in ℝ2 . A similar conclusion can be clarified for the solutions of the wave equation given by the Kirchhoff (Poisson) formula; see [431] for more details. 6. In this issue, we continue our analysis of the evolution systems considered in the final application of [431, Section 6.3, pp. 426–428]. Suppose that Y := Lr (ℝn ) for some r ∈ [1, ∞) and A(t) := Δ + a(t)I, t ⩾ 0, where Δ is the Dirichlet Laplacian on Lr (ℝn ), and a ∈ L∞ ([0, ∞)). Then the evolution system (U(t, s))t⩾s⩾0 ⊆ L(Y ) generated by the family (A(t))t⩾0 exists; this evolution system is given by U(t, t) := I for all t ⩾ 0, and [U(t, s)F](u) := ∫ K(t, s, u, v)F(v) dv,

F ∈ Lr (ℝn ), t > s ⩾ 0,

ℝn

where K(t, s, u, v) := (4π(t − s))

− n2 ∫t a(τ) dτ s

e

We know that, for every τ ∈ ℝn , we have

exp(−

|u − v|2 ), 4(t − s)

t > s, u, v ∈ ℝn .

3.3 Multi-dimensional Besicovitch almost periodic type functions and applications

�

169

t > s ⩾ 0, u, v ∈ ℝn ,

K(t, s, u + τ, v + τ) = K(t, s, u, v),

as well as that, under certain assumptions, a unique mild solution of the abstract Cauchy problem (𝜕/𝜕t)u(t, x) = A(t)u(t, x), t > 0; u(0, x) = F(x) is given by u(t, x) := [U(t, 0)F](x), t ⩾ 0, x ∈ ℝn . Suppose now that F(⋅) ∈ Y is Besicovitch-p-almost periodic for some finite exponent p ⩾ 1. If ε > 0 is given in advance, then we can find a finite real number t0 > 0 and a trigonometric polynomial P(⋅) such that 󵄨 󵄨p ∫ 󵄨󵄨󵄨F(u) − P(u)󵄨󵄨󵄨 du < εt n ,

t ⩾ t0 .

[−t,t]n

The function uP (t, x) := [U(t, 0)P](x), t ⩾ 0, x ∈ ℝn is well-defined, continuous and satisfies that, if τ ∈ ℝn is an ε-almost period of P(⋅), then 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨uP (t, u + τ) − uP (t, u)󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∫ [K(t, 0, u + τ, v) − K(t, 0, u, v)]P(v) dv󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨ℝn 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 = 󵄨󵄨󵄨 ∫ K(t, 0, u + τ, v + τ)P(v + τ) dv − ∫ K(t, 0, u, v)P(v) dv󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨ℝn 󵄨 ℝn 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 = 󵄨󵄨󵄨 ∫ K(t, 0, u, v)[P(v + τ) dv − P(v)] dv󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨ℝn 󵄨 ⩽ ct ∫ e−

|u−v|2 4t

ℝn

⩽ ct ε ∫ e−

󵄨󵄨 󵄨 󵄨󵄨P(v + τ) − P(v)󵄨󵄨󵄨 dv

|u−v|2 4t

dv = ct ε ∫ e−

ℝn

|v|2 4t

dv,

t > 0, u ∈ ℝn ;

ℝn

hence, the function uP (t, ⋅) is almost periodic for every fixed real number t > 0. Writing ∫ K(t, 0, u, v)[F(v) − P(v)] dv = ∫ K(t, 0, u, v − u)[F(v − u) − P(v − u)] dv, ℝn

ℝn 2

2

2

and the term e−|v| /4t as e−|v| /8t ⋅ e−|v| /8t in the corresponding computation after that, we may conclude as before that the function u(t, ⋅) is Besicovitch-p-almost periodic for every fixed real number t > 0. 7. In this part, we will present certain applications of Proposition 3.3.8 in the analysis of the existence and uniqueness of Besicovitch almost periodic solutions for certain classes of PDEs; see, e. g. [653, 787] and references cited therein. We will revisit here the classical theories of quasi-linear partial differential equations of first order and the linear partial differential equations of second order with constant coefficients, considering solutions defined on certain proper subsets Λ of ℝn , where n ⩾ 2.

170 � 3 Multi-dimensional ρ-almost periodic type functions 7.1. It is well known that the general solution of equation ux + uy = u is given by u(x, y) = g(y − x)ex ,

(x, y) ∈ ℝ2 ,

where g : ℝ → ℝ is a continuously differentiable function. Suppose that Λ := (−∞, 0]×ℝ and the function g(⋅) is Besicovitch-q-almost periodic for some finite exponent q > 1. Then there exists a sequence (Pk ) of trigonometric polynomials such that t

󵄨 󵄨q lim lim sup t −1 ∫󵄨󵄨󵄨Pk (s) − g(s)󵄨󵄨󵄨 ds = 0.

k→+∞ t→+∞

−t

Since Qk (x, y) := Pk (x − y), (x, y) ∈ ℝ2 is a sequence of trigonometric polynomials of two variables, the above simply implies along with an elementary argumentation that t t

lim lim sup t

k→+∞ t→+∞

−2

󵄨 󵄨q ∫ ∫󵄨󵄨󵄨Qk (x, y) − g(x − y)󵄨󵄨󵄨 dx dy = 0, −t −t

so that the function G(x, y) := g(x − y), (x, y) ∈ ℝ2 is Besicovitch-q-almost periodic. Since the function (x, y) 󳨃→ ex , (x, y) ∈ Λ is Besicovitch-p-almost periodic for any finite exponent p ⩾ 1, an application of Proposition 3.3.8 yields that the solution u(x, y) is Besicovitch-r-almost periodic on Λ, for any exponent r ∈ [1, q). 7.2. Consider the linear partial differential equation of second order with constant coefficients: Auxx + 2Buxy + Cuyy + 2Dux + 2Euy + Fu = 0,

(117)

where A, B, C, D, E, F are real constants such that B > 0, C > 0, B2 ⩾ AC, E 2 ⩾ CF, B2 > E 2 − CF, and (BE − CD)2 = (B2 − AC)(E 2 − CF). As proposed by J. D. Kečkić in [406] (see also [405]), the general solution u(x, y) of the equation (117) is given by B

u(x, y) = e(− C +

√E 2 −CF C

B

+ e(− C −

)y

f (x + (−

√E 2 −CF C

)y

E 1√ 2 + B − AC)y) C C

g(x + (−

E 1√ 2 − B − AC)y), C C

where f : ℝ → ℝ and g : ℝ → ℝ are arbitrary two times continuously differentiable functions. Suppose that Λ = ℝ×[0, +∞) and the functions f (⋅) and g(⋅) are Besicovitch-qalmost periodic for some finite exponent q > 1. Arguing as above, an application of

3.4 Besicovitch multi-dimensional almost automorphic type functions and applications

�

171

Proposition 3.3.8 yields that the solution u(x, y) of (117) is Besicovitch-r-almost periodic on Λ, for any exponent r ∈ [1, q). We close this section with the observation that it is not clear how one can prove that the solutions considered above are Besicovitch-q-almost periodic.

3.4 Besicovitch multi-dimensional almost automorphic type functions and applications In this section, we analyze various classes of multi-dimensional Besicovitch almost automorphic type functions, working with general Lebesgue spaces with variable exponents. We provide several interpretive examples and applications to the abstract Volterra integro-differential equations. Various classes of multi-dimensional almost automorphic functions have been analyzed by A. Chávez et al. in the research article [196]. This study has recently been continued in [459] and [449], where we have analyzed the Stepanov classes and the Weyl classes of multi-dimensional almost automorphic functions, respectively (let us recall that, in the one-dimensional setting, the notion of Stepanov almost automorphy was introduced by V. Casarino [186] in 2000 and later reconsidered by G. M. N’Guérékata and A. Pankov [345] in 2008, while the notion of Weyl almost automorphy was introduced by S. Abbas [4] in 2012). See also the recent research study [506] by Y. Li and X. Meng, where the authors have analyzed the almost automorphic solutions in distribution sense for a class of quaternion-valued stochastic recurrent neural networks with mixed timevarying delays, and the recent research study [508] by Y. Li, X. Wang, and N. Huo, where the authors have examined the Weyl almost automorphic solutions in distribution sense for a class of Clifford-valued stochastic neural networks with time-varying delays. The main aim of this section is to introduce and analyze the multi-dimensional Besicovitch almost automorphic functions as well as to present certain applications in the analysis of the existence and uniqueness of the Besicovitch almost automorphic type solutions for various classes of the abstract Volterra integro-differential equations and the partial differential equations. Some classes of Besicovitch almost automorphic functions introduced here seem to be new even in the one-dimensional setting; furthermore, the research article [450], from which we have taken the material of this section, is probably the first research article in the existing literature, which seeks for spatially Besicovitch almost automorphic solutions of (abstract) PDEs. The organization and main ideas of this section can be briefly described as follows. The main part is Section 3.4.1, in which we analyze various notions of multi-dimensional Besicovitch almost automorphy in Lebesgue spaces with variable exponent. In Definition 3.4.2, we introduce the notion of Besicovitch-(𝔽, ϕ, p(u), R, ℬ)-multi-almost automorphy and its relatives. Proposition 3.4.3 states that any Besicovitch-(𝔽, ϕ, p(u), R, ℬ)multi-almost automorphic function F(⋅; ⋅) is Besicovitch-(R, ℬ, ϕ, F) − Bp(⋅) -normal, where F(l) ≡ 𝔽(l, 0); Proposition 3.4.6 continues our analysis from [449, Proposition 2.10]. After

172 � 3 Multi-dimensional ρ-almost periodic type functions that, in Definition 3.4.7, we introduce the notions Besicovitch-(𝔽, ϕ, p(u), R, ℬ, Wℬ,R )multi-almost automorphy, the Besicovitch-(𝔽, ϕ, p(u), R, ℬ, Pℬ,R )-multi-almost automorphy and explain how these notions can be introduced for all other classes of functions from Definition 3.4.2. In Proposition 3.4.9, we consider the pointwise products of Besicovitch-(𝔽, ϕ, p(u), R, ℬ, Pℬ,R )-multi-almost automorphic functions with the scalar Besicovitch almost automorphic functions of a similar type. A composition principle for Besicovitch-(𝔽, ϕ, p, R, ℬ)-multi-almost automorphic functions of type 1 is deduced in Theorem 3.4.11. Some applications of our results to the abstract Volterra integro-differential equations and the partial differential equations are provided in Section 3.4.2. It is worth noticing that, in Definition 3.4.12, we introduce the class of Besicovitch-(𝔽, ϕ, p(u), R, w)-multialmost automorphic type functions, in which we aim to control the growth order of limit function F ∗ (⋅) by the weight function w(⋅). This idea seems to be completely new and not explored elsewhere, not even in the one-dimensional setting. The notion introduced in Definition 3.4.12 plays a fundamental role in Proposition 3.4.13, where we investigate the invariance of Besicovitch almost automorphy under the actions of the infinite convolution products, and Theorem 3.4.14, where we investigate the convolution invariance of multi-dimensional Besicovitch almost automorphy. Because of a certain similarity with our previous investigations of the existence and uniqueness of Besicovitch almost periodic solutions of the abstract nonautonomous differential equations of first order and the classical wave equation, we have skipped here some irrelevant details concerning the existence and uniqueness of Besicovitch almost automorphic solutions for these classes of PDEs. In addition to the above, we also provide many useful comments, illustrative examples, and propose some open problems. It is also worth noting that we give some new definitions, observations and examples regarding multi-dimensional Weyl almost automorphic type functions [431]. We need to recall the following definition: Definition 3.4.1 ([196, Definition 2.1]). Suppose that F : ℝn × X → Y is a continuous function. Then we say that the function F(⋅; ⋅) is (R, ℬ)-multi-almost automorphic if and only if for every B ∈ ℬ and for every sequence (bk ) ∈ R there exist a subsequence (bkl ) of (bk ) and a function F ∗ : ℝn × X → Y such that lim F(t + bkl ; x) = F ∗ (t; x)

l→+∞

and lim F ∗ (t − bkl ; x) = F(t; x),

l→+∞

pointwisely for all x ∈ B and t ∈ ℝn . If the convergence in the above-limit equations is uniform with respect to the sets B of collection ℬ, for a fixed number t ∈ ℝn , then we say that the function F(⋅; ⋅) is uniformly (R, ℬ)-multi-almost automorphic.

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3.4.1 Multi-dimensional Besicovitch almost automorphy in Lebesgue spaces with variable exponent The main aim of this subsection is to introduce and analyze various classes of multidimensional Besicovitch almost automorphic functions in Lebesgue spaces with variable exponent. Unless stated otherwise, we will always assume henceforth that Ω := [−1, 1]n ⊆ ℝn , p ∈ 𝒫 (ℝn ) and 𝔽 : (0, ∞) × ℝn → (0, ∞). We start by introducing the following notion: Definition 3.4.2. Suppose that F : ℝn × X → Y . Let for every B ∈ ℬ and (bk = (b1k , b2k , . . . , bnk )) ∈ R there exist a subsequence (bkm = (b1km , b2km , . . . , bnkm )) of (bk ) and a function F ∗ : ℝn × X → Y such that for each x ∈ B, l > 0 and t ∈ ℝn we have ϕ(F(t + u + bkm ; x) − F ∗ (t + u; x)) ∈ Lp(u) (lΩ : Y ), ϕ(F ∗ (t + u − bkm ; x) − F(t + u; x)) ∈ Lp(u) (lΩ : Y ), as well as: (i) 󵄩󵄩 󵄩󵄩 lim lim sup 𝔽(l, t) sup[ϕ(󵄩󵄩󵄩F(t + u + bkm ; x) − F ∗ (t + u; x)󵄩󵄩󵄩 )] m→+∞ l→+∞ 󵄩 󵄩Y x∈B

Lp(u) (lΩ)

= 0 (118)

and 󵄩󵄩 󵄩󵄩 lim lim sup 𝔽(l, t) sup[ϕ(󵄩󵄩󵄩F ∗ (t + u − bkm ; x) − F(t + u; x)󵄩󵄩󵄩 )] m→+∞ l→+∞ 󵄩 󵄩Y x∈B

(ii)

Lp(u) (lΩ)

= 0, (119)

pointwise for all x ∈ B and t ∈ ℝn , or

󵄩󵄩 󵄩󵄩 lim lim inf 𝔽(l, t) sup[ϕ(󵄩󵄩󵄩F(t + u + bkm ; x) − F ∗ (t + u; x)󵄩󵄩󵄩 )] 󵄩 󵄩Y x∈B

=0

󵄩󵄩 󵄩󵄩 lim lim inf 𝔽(l, t) sup[ϕ(󵄩󵄩󵄩F ∗ (t + u − bkm ; x) − F(t + u; x)󵄩󵄩󵄩 )] 󵄩 󵄩Y x∈B

= 0,

m→+∞ l→+∞

Lp(u) (lΩ)

and

m→+∞ l→+∞

(iii)

Lp(u) (lΩ)

pointwise for all x ∈ B and t ∈ ℝn , or

󵄩󵄩 󵄩󵄩 lim lim sup 𝔽(l, t) sup[ϕ(󵄩󵄩󵄩F(t + u + bkm ; x) − F ∗ (t + u; x)󵄩󵄩󵄩 )] 󵄩 󵄩Y l→+∞ m→+∞ x∈B and

Lp(u) (lΩ)

=0

174 � 3 Multi-dimensional ρ-almost periodic type functions

󵄩󵄩 󵄩󵄩 lim lim sup 𝔽(l, t) sup[ϕ(󵄩󵄩󵄩F ∗ (t + u − bkm ; x) − F(t + u; x)󵄩󵄩󵄩 )] 󵄩 󵄩Y l→+∞ m→+∞ x∈B (iv)

Lp(u) (lΩ)

= 0,

pointwise for all x ∈ B and t ∈ ℝn , or

󵄩󵄩 󵄩󵄩 lim lim inf 𝔽(l, t) sup[ϕ(󵄩󵄩󵄩F(t + u + bkm ; x) − F ∗ (t + u; x)󵄩󵄩󵄩 )] m→+∞ 󵄩 󵄩Y l→+∞ x∈B

=0

󵄩󵄩 󵄩󵄩 lim lim inf 𝔽(l, t) sup[ϕ(󵄩󵄩󵄩F ∗ (t + u − bkm ; x) − F(t + u; x)󵄩󵄩󵄩 )] m→+∞ 󵄩 󵄩Y l→+∞ x∈B

= 0,

Lp(u) (lΩ)

and

Lp(u) (lΩ)

pointwise for all x ∈ B and t ∈ ℝn . If (i), resp. [(ii), (iii), (iv)] holds, then we say that the function F(⋅; ⋅) is Besicovitch(𝔽, ϕ, p(u), R, ℬ)-multi-almost automorphic, resp. [weakly Besicovitch-(𝔽, ϕ, p(u), R, ℬ)multi-almost automorphic, Besicovitch-(𝔽, ϕ, p(u), R, ℬ)-multi-almost automorphic of type 1, weakly Besicovitch-(𝔽, ϕ, p(u), R, ℬ)-multi-almost automorphic of type 1]. By 𝔽,ϕ,p(u) 𝔽,ϕ,p(u) 𝔽,ϕ,p(u),1 AAB(R,ℬ) (ℝn × X : Y ), resp. [w − AAB(R,ℬ) (ℝn × X : Y ), AAB(R,ℬ) (ℝn × X : Y ), w − AAB(R,ℬ) (ℝn × X : Y )] we denote the collection of all Besicovitch-(𝔽, ϕ, p(u), R, ℬ)multi-almost automorphic, resp. [weakly Besicovitch-(𝔽, ϕ, p(u), R, ℬ)-multi-almost automorphic, Besicovitch-(𝔽, ϕ, p(u), R, ℬ)-multi-almost automorphic of type 1, weakly Besicovitch-(𝔽, ϕ, p(u), R, ℬ)-multi-almost automorphic of type 1] functions F : ℝn × X → Y. 𝔽,ϕ,p(u),1

Trivially, if the requirements in (i), resp. (iii), of Definition 3.4.2 hold, then the requirements in (ii), resp. (iv), of Definition 3.4.2 hold. The notion introduced in [431, Definition 8.3.17, Definition 8.3.32] is a special case of the notion introduced in Definition 3.4.2. The interested reader may simply clarify some sufficient conditions ensuring that the spaces introduced in Definition 3.4.2 are translation invariant or have a linear vector structure with the usual operations (see also the items [450, (i)–(iv)] clarified at the beginning of the second section). An analog of [450, Proposition 2.13] holds in our new framework. Case ϕ(x) ≡ x, p(u) ≡ p ∈ [1, ∞) and 𝔽(l, t) ≡ l−n/p is the most important, when we say that the function F : ℝn × X → Y is (weakly) Besicovitch p-(R, ℬ)-multi-almost automorphic (of type 1). If, in addition to the above, we have that the collection ℬ consists of bounded subsets of X, then the notion of Besicovitch p-(R, ℬ)-multi-almost automorphy is equivalent with the notion Besicovitch-(R, ℬ) − Bp -normality since, in this case, an extension of the well known result of J. Marcinkiewicz [535] holds (see [450, Theorem 2.5]) and the equations (118)–(119) hold for arbitrary t ∈ ℝn if and only the equations (118)–(119) hold with t = 0 (the value of lim supl→+∞ ⋅ in these equations does

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not depend on t ∈ ℝn ; see the proof of [450, Proposition 3.3]). Furthermore, these two notions are equivalent in the case that ϕ(x) ≡ x α for some α ∈ (0, 1]. If we replace the all operations lim sup and lim inf in Definition 3.4.2 with the classical limits, then we obtain the corresponding notion of Weyl p-almost automorphy (of type 1). If the function F : ℝn × X → Y is Besicovitch p-(R, ℬ)-multi-almost automorphic, for example, and R denotes the collection of all sequences in ℝn , then we omit the term “R” from the notation; furthermore, if X = {0}, then we omit the term “ℬ” from the notation. This particularly means that the function F : ℝn → Y is Besicovitch-p-almost automorphic if and only if F(⋅) is Besicovitch-p-R-multi-almost automorphic with R being the collection of all sequences in ℝn . In [449, Theorem 2.9], we have constructed an example of a Weyl (Besicovitch)-p-almost automorphic function which is not Besicovitch-p-bounded and therefore not Besicovitch-p-almost periodic (p ⩾ 1). Further on, if p = 1, then we say that the function F(⋅) is Besicovitch almost periodic, automorphic, etc. The proof of following result is not difficult and can be omitted: Proposition 3.4.3. Suppose that the function F(⋅; ⋅) is Besicovitch-(𝔽, ϕ, p(u), R, ℬ)-multialmost automorphic, and F(l) ≡ 𝔽(l, 0). Then the function F(⋅; ⋅) is Besicovitch-(R, ℬ, ϕ, F)− Bp(⋅) -normal. We continue by providing the following illustrative examples (the first one is in support of our recent investigation of Weyl almost automorphy [449]): Example 3.4.4 (based on the example of D. Brindle [165, Example 2.2]). Let l∞ denote the Banach space of all bounded numerical sequences, equipped with the sup-norm. Consider the function f : ℝ → l∞ by f (t) := (e−|t|/k )k∈ℕ , t ∈ ℝ. We know that this function is uniformly continuous, bounded, slowly oscillating, and has no mean value, so that f (⋅) is not Besicovitch almost periodic (for the notion and more details, see [431, Example 9.0.20]). On the other hand, we can simply prove that the function f (⋅) is not Stepanov almost automorphic. In actual fact, if we assume the contrary, then the function f (⋅) needs to be almost automorphic (see, e. g. [431, Lemma 2.3.4]), since it is uniformly continuous. This is not the case, because we can use the sequence (bk ≡ k) in the corresponding definition of almost automorphy, with t = 0, in order to conclude that for each ε ∈ (0, e−1 ) there exists an integer k0 ∈ ℕ such that 󵄨󵄨 󵄨󵄨 sup󵄨󵄨󵄨e−l/k − e−m/k 󵄨󵄨󵄨 < ε, 󵄨 󵄨 k∈ℕ

l, m ∈ ℕ.

If we plug k = k0 = l here, then we obtain 󵄨󵄨 −1 󵄨 󵄨 󵄨 󵄨󵄨e − e−m/k0 󵄨󵄨󵄨 ⩽ sup󵄨󵄨󵄨e−l/k − e−m/k 󵄨󵄨󵄨 < ε, 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 k∈ℕ which gives a contradiction since the first term tends to e−1 as m → +∞. Moreover, we can simply prove that the function f (⋅) is Weyl-p-almost automorphic for any finite real

176 � 3 Multi-dimensional ρ-almost periodic type functions exponent p ⩾ 1; in actual fact, for every sequence (bk ) in ℝ we can take the same subsequence (bkm ) = (bk ) and the limit function f ∗ ≡ f in the corresponding definition since the function f (⋅) is slowly oscillating and bounded. It can be also proved that the function f (⋅) is Weyl-p-almost automorphic of type 1 (jointly Weyl-p-almost automorphic; see [449, Definition 2.5] for the notion) for any finite real exponent p ⩾ 1 since for any sequence (bk ) tending to plus infinity or minus infinity, we can take the limit function f ∗ ≡ 0 in the corresponding definition (the situation is much simpler for bounded sequences (bk ) when we can take an appropriate translation of function f (⋅) as the limit function f ∗ (⋅)). This example is important because the vector-valued function f (⋅) satisfies that the limit t t limt→+∞ t −1 ∫0 f (s) ds does not exists in l∞ but limt→+∞ t −1 ∫0 ‖f (s)‖ ds = 0. Example 3.4.5. (i) Suppose that ω ∈ ℝn ∖ {0}, ϕ(0) = 0 and a continuous function F : ℝn → Y is ω-periodic, i. e. F(t+ω) = F(t) for all t ∈ ℝn . Let R denote the collection of all sequences in the set ω ⋅ ℤ. Then the function F(⋅) is Besicovitch-(𝔽, ϕ, p(u), R)multi-almost automorphic, resp. [weakly Besicovitch-(𝔽, ϕ, p(u), R)-multi-almost automorphic, Besicovitch-(𝔽, ϕ, p(u), R)-multi-almost automorphic of type 1, weakly Besicovitch-(𝔽, ϕ, p(u), R)-multi-almost automorphic of type 1]. (ii) Let p(⋅) ≡ p ∈ [1, ∞). Then it is very simple to construct an example of an ω-periodic continuous function F : ℝn → Y which is not weakly Besicovitch-(𝔽, ϕ, p, R)-multialmost automorphic of type 1. Suppose, for simplicity, that ϕ(x) ≡ x, n = 2, Y := ℂ and 𝔽(⋅; ⋅) is arbitrary. Suppose, further, that F0 : {(x, y) ∈ ℝ2 : 0 ⩽ x+y ⩽ 2} → [0, ∞) is any continuous function such that F0 (x, y) = F0 (x + 1, y + 1) for every (x, y) ∈ ℝ2 with x + y = 0, as well as that the following condition holds: (⬦) A sequence (ak ) in ℕ and a sequence (rk ) in (0, ∞) satisfy limk→+∞ ak = limk→+∞ rk = +∞, ak + 3rk < ak+1 − 3rk+1 for all k ∈ ℕ, and the value of function F0 (⋅; ⋅) on the projection of the rectangle ak + [−rk , rk ]2 to the strip {(x, y) ∈ ℝ2 : 0 ⩽ x + y ⩽ 2} is greater or equal than k. After that, we extend the function F0 (⋅; ⋅) to a continuous (1, 1)-periodic function defined on the whole space ℝ2 in the usual way. Then it can be simply shown that for each l > 0 we have lim

k→+∞

󵄨 󵄨p ∫ 󵄨󵄨󵄨F(x + ak , y)󵄨󵄨󵄨 dx dy = +∞.

[−l,l]2

This simply implies the required conclusion with R being a collection of all sequences in {(k, 0) : k ∈ ℕ}. The following result, extending [449, Proposition 2.10], can be also formulated in the multi-dimensional setting: p

Proposition 3.4.6. Suppose that p ⩾ 1, σ > 0, 𝔽(l) ≡ l−σ , f ∈ Lloc (ℝ : X) and there exist a strictly increasing sequence (lk ) of positive real numbers tending to plus infinity, a sequence (bk ) of real numbers, and a positive real number ε0 > 0 such that, for every k ∈ ℕ and for every subsequence of (bkm ) of (bk ), we have

3.4 Besicovitch multi-dimensional almost automorphic type functions and applications

�

177

bkm +lk

lim l−σ m→+∞ k

󵄩 󵄩p ∫ 󵄩󵄩󵄩f (x)󵄩󵄩󵄩 dx = +∞.

(120)

bkm −lk

Then the function f (⋅) is not Besicovitch-(𝔽, x, p)-almost automorphic of type 1. Proof. Suppose that the function f (⋅) is Besicovitch-(𝔽, x, p)-almost automorphic of type 1. Let ε > 0 be arbitrary. Then there exist a subsequence (bkm ) of (bk ), a function f ∗ ∈ p Lloc (ℝ : X) and a finite real number l0 > 0 such that, for every l ⩾ l0 , there exists an integer ml ∈ ℕ such that for every integer m ⩾ ml , we have l

󵄩 󵄩p l−σ ∫󵄩󵄩󵄩f (x + bkm ) − f ∗ (x)󵄩󵄩󵄩 dx < ε. −l

Let k ∈ ℕ be such that lk ⩾ l0 . Then, due to (120), we have: lk

ε>

lk−σ

󵄩 󵄩p ∫ 󵄩󵄩󵄩f (x + bkm ) − f ∗ (x)󵄩󵄩󵄩 dx −lk

⩾

lk 󵄩 −σ 1−p lk 2 [ ∫ 󵄩󵄩󵄩f (x

lk

󵄩p 󵄩 󵄩p + bkm )󵄩󵄩󵄩 dx − ∫ 󵄩󵄩󵄩f ∗ (x)󵄩󵄩󵄩 dx]

−lk

−lk

lk +bkm

lk

−lk +bkm

−lk

󵄩 󵄩p 󵄩 󵄩p = lk−σ 21−p [ ∫ 󵄩󵄩󵄩f (x)󵄩󵄩󵄩 dx − ∫ 󵄩󵄩󵄩f ∗ (x)󵄩󵄩󵄩 dx] → +∞,

m → +∞.

We can simply reformulate [449, Example 3.6] in our new framework, as well as the conclusions established in [449, Proposition 3.7, Proposition 3.8]. Further on, the convergence of limits in Definition 3.4.2 is pointwise, for any x ∈ B and t ∈ ℝn . For the sequel, it will be important to note that we can impose further requirements about the convergence of limits in Definition 3.4.2 and consider, in such a way, several new classes of multi-dimensional Besicovitch almost automorphic type functions. For exam𝔽,ϕ,p(u) ple, consider the class AAB(R,ℬ) (ℝn × X : Y ) and assume that for each B ∈ ℬ and (bk = (b1k , b2k , . . . , bnk )) ∈ R we have that WB,(bk ) : B → P(P(ℝn )) and PB,(bk ) ∈ P(P(ℝn ×B)). Then we can introduce the following notion (cf. also [196, Definition 2.2] and the example following it): Definition 3.4.7. We say that a function F : ℝn × X → Y is: (i) Besicovitch-(𝔽, ϕ, p(u), R, ℬ, Wℬ,R )-multi-almost automorphic if and only if (118)– (119) hold pointwisely for all x ∈ B and t ∈ ℝn , as well as that for each x ∈ B the convergence in t is uniform for any element of the collection WB,(bk ) (x);

178 � 3 Multi-dimensional ρ-almost periodic type functions (ii) Besicovitch-(𝔽, ϕ, p(u), R, ℬ, Pℬ,R )-multi-almost automorphic if and only if (118)– (119) hold pointwisely for all x ∈ B and t ∈ ℝn , as well as that the convergence in (118)–(119) is uniform in (t; x) for any set of the collection PB,(bk ) . We similarly define the Wℬ,R -classes and the Pℬ,R -classes of multi-dimensional Besicovitch almost periodic functions from the parts (ii)–(v) of Definition 3.4.2. We can also introduce the corresponding classes of Weyl almost automorphic functions considered in [449]; we only need to replace the operations lim sup and lim inf in Definition 3.4.2 with the usual limits. In connection with Definition 3.4.7 and the above observations, we will present the following example (cf. also [196, Example 5] and [459, Example 2.4]): Example 3.4.8. Suppose that φ : ℝ → ℂ is an almost automorphic function, and (T(t))t∈ℝ ⊆ L(X, Y ) is an operator family which is strongly locally integrable and not strongly continuous at zero. Suppose further that there exist a finite real number M ⩾ 1 and a real number γ ∈ (0, 1) such that M 󵄩󵄩 󵄩 󵄩󵄩T(t)󵄩󵄩󵄩L(X,Y ) ⩽ γ , |t|

t ∈ ℝ ∖ {0},

as well as that R is the collection of all sequences in Δ2 ≡ {(t, t) : t ∈ ℝ} and ℬ is the collection of all bounded subsets of X. Define t

F(t, s; x) := e∫s φ(τ) dτ T(t − s)x,

(t, s) ∈ ℝ2 , x ∈ X,

and assume that for each bounded subset B of X and for each sequence (bk = (bk , bk )) in R the collection PB,(bk ) consists of all sets of form {(t, s) ∈ ℝ2 : |t − s| ⩽ L} × B, where L > 0. Define t

F ∗ (t, s; x) := e∫s φ

∗

(r) dr

T(t − s)x,

(t, s) ∈ ℝ2 , x ∈ X.

If the function φ(⋅) is almost periodic, then it is not difficult to show, with the help of computation established in [196, Example 5], that the function F(⋅, ⋅; ⋅) is Stepanov (Ω, 1)-(R, ℬ, Pℬ,R )-multi-almost automorphic; see [459] for the notion. But, this is no longer possible to be done if the function φ(⋅) is only almost automorphic but not almost periodic. If this is the case, then we can simply prove that the function F(⋅, ⋅; ⋅) is Weyl-(𝔽, x, 1, R, ℬ, Pℬ,R )-multi-almost automorphic of type 1, since for each fixed real number l > 0 we have: lim

m→+∞

󵄩 󵄩 ∫ sup󵄩󵄩󵄩F(t + u1 + bkm , s + u2 + bkm ; x) − F ∗ (t + u1 , s + u2 ; x)󵄩󵄩󵄩Y du1 du2 = 0,

[−l,l]2

x∈B

which follows from an application of the dominated convergence theorem and a simple calculation.

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� 179

Now we will state and prove the following result for the class of Besicovitch(𝔽, ϕ, p(u), R, ℬ, Pℬ,R )-multi-almost automorphic functions. The same result holds for the Besicovitch-(𝔽, ϕ, p(u), R, ℬ, Wℬ,R )-multi-almost automorphic functions; furthermore, we can simply formulate this result for the functions introduced in parts (iii)–(iv) of Definition 3.4.2: Proposition 3.4.9. Suppose that p, q, r ∈ [1, ∞), 1/r = 1/p + 1/q, 𝔽1 (l, t) ≡ l−n/p , 𝔽2 (l, t) ≡ l−n/q , 𝔽(l, t) ≡ l−n/r and ϕ(x) ≡ x α for some real number α > 0. If for each sequence in R any its subsequence also belongs to R, the function F1 : ℝn → ℂ is Besicovitch-(𝔽, ϕ, p, R, ℬ, Pℬ,R )-multi-almost automorphic [weakly Besicovitch-(𝔽1 , ϕ, p, R, ℬ, Pℬ,R )-multi-almost automorphic], F2 : ℝn → Y is Besicovitch-(𝔽, ϕ, q, R, ℬ, Pℬ,R )-multialmost automorphic [weakly Besicovitch-(𝔽1 , ϕ, q, R, ℬ, Pℬ,R )-multi-almost automorphic], and for each set B ∈ ℬ we have the existence of finite real numbers l0 > 0 and mB > 0 such that supt∈ℝn ;x∈B ‖[ϕ(|F1 (t + ⋅)|)]‖Lp (lΩ) ⩽ mB ln/p , l ⩾ l0 and supt∈ℝn ;x∈B ‖[ϕ(‖F2 (t + ⋅)‖Y )]‖Lq (lΩ) ⩽ mB ln/q , l ⩾ l0 [there exists a strictly increasing sequence (lk ) of positive real n/p

numbers tending to plus infinity such that supt∈ℝn ;x∈B;k∈ℕ ‖[ϕ(|F1 (t + ⋅)|)]‖Lp (lk Ω) ⩽ mB lk n/q mB lk ],

and supt∈ℝn ;x∈B;k∈ℕ ‖[ϕ(‖F2 (t + ⋅)‖Y )]‖Lq (lk Ω) ⩽ then the function F : ℝn × X → Y , n given by F(t; x) := F1 (t; x)F2 (t; x), t ∈ ℝ , x ∈ X, is Besicovitch-(𝔽, ϕ, r, R, ℬ, Pℬ,R )-multialmost automorphic [weakly Besicovitch-(𝔽, ϕ, r, R, ℬ, Pℬ,R )-multi-almost automorphic]. Proof. We will consider the Besicovitch-(𝔽, ϕ, p, R, ℬ, Pℬ,R )-multi-almost automorphic functions, only. Let (bk ) ∈ R and B ∈ ℬ be given. Since for every sequence in R any its subsequence also belongs to R, we can extract a subsequence (bkm ) of (bk ) such that 󵄨󵄨 󵄨󵄨 lim lim sup 𝔽1 (l) sup[ϕ(󵄨󵄨󵄨F1 (t + u + bkm ; x) − F1∗ (t + u; x)󵄨󵄨󵄨)] 󵄨 󵄨 x∈B

= 0,

(121)

󵄨󵄨 󵄨󵄨 lim lim sup 𝔽1 (l) sup[ϕ(󵄨󵄨󵄨F1∗ (t + u − bkm ; x) − F1 (t + u; x)󵄨󵄨󵄨)] 󵄨 󵄨 x∈B

= 0,

(122)

m→+∞ l→+∞

m→+∞ l→+∞

Lp (lΩ)

Lp (lΩ)

as well as 󵄩󵄩 󵄩󵄩 lim lim sup 𝔽2 (l) sup[ϕ(󵄩󵄩󵄩F2 (t + u + bkm ; x) − F2∗ (t + u; x)󵄩󵄩󵄩 )] 󵄩 󵄩Y x∈B

= 0,

(123)

󵄩󵄩 󵄩󵄩 lim lim sup 𝔽2 (l) sup[ϕ(󵄩󵄩󵄩F2∗ (t + u − bkm ; x) − F2 (t + u; x)󵄩󵄩󵄩 )] 󵄩 󵄩Y x∈B

= 0.

(124)

m→+∞ l→+∞

Lq (lΩ)

and m→+∞ l→+∞

Lq (lΩ)

Our assumption simply implies that for each set B ∈ ℬ we have the existence of finite real numbers l0′ > 0 and mB′ > 0 such that supt∈ℝn ;x∈B ‖[ϕ(|F1∗ (t + ⋅)|)]‖Lp (lΩ) ⩽ mB′ ln/p , l ⩾ l0′ and supt∈ℝn ;x∈B ‖[ϕ(‖F2∗ (t + ⋅)‖Y )]‖Lq (lΩ) ⩽ mB′ ln/q , l ⩾ l0′ . Keeping in mind these estimates,

180 � 3 Multi-dimensional ρ-almost periodic type functions the equality 1/r = 1/p + 1/q, the required first limit equality follows using (121)–(123), the existence of a finite real number cα > 0 such that 󵄩 󵄩 ϕ(󵄩󵄩󵄩F1 (t + u + bkm ; x)F2 (t + u + bkm ; x) − F1∗ (t + u; x)F2∗ (t + u; x)󵄩󵄩󵄩Y ) 󵄨 󵄨 󵄩 󵄩 ⩽ cα [ϕ(󵄨󵄨󵄨F1 (t + u + bkm ; x) − F1∗ (t + u; x)󵄨󵄨󵄨) ⋅ ϕ(󵄩󵄩󵄩F2 (t + u + bkm ; x)󵄩󵄩󵄩Y ) 󵄨 󵄨 󵄩 󵄩 + ϕ(󵄨󵄨󵄨F1∗ (t + u; x)󵄨󵄨󵄨) ⋅ ϕ(󵄩󵄩󵄩F2 (t + u + bkm ; x) − F2∗ (t + u; x)󵄩󵄩󵄩Y )],

t ∈ ℝn ,

and the Hölder inequality. The second limit equality can be proved analogously, by using (122) and (124). Example 3.4.10. It is worth noting that Proposition 3.4.9 can be applied for the construction of multi-dimensional almost automorphic functions of the form F(t) := F1 (t1 )⋅F2 (t2 )⋅ ⋅ ⋅ ⋅ ⋅ Fn (tn ), t = (t1 , . . . , tn ) ∈ ℝn , where all functions Fj (⋅) are Besicovitch-p-almost automorphic in a certain sense (see also [431, Example 8.1.6] and [450, Example 2.8]). We close this subsection by stating and proving a composition principle for Besicovitch-(𝔽, ϕ, p, R, ℬ)-multi-almost automorphic functions of type 1, which continues our analysis from [196, Theorem 2.20] and [450, Theorem 2.10]. We consider here the Besicovitch-p-almost automorphy of the multi-dimensional Nemytskii operator W : ℝn × X → Z, given by (27), where F : ℝn × X → Y and G : ℝn × Y → Z. Theorem 3.4.11. Suppose that 1 ⩽ p, q < +∞, α > 0, p = αq, F(t) ≡ t −n/p , ϕ(x) ≡ x ζ for some real number ζ > 0, F(⋅; ⋅) is Besicovitch-(𝔽, ϕ, p, R, ℬ)-multi-almost automorphic of type 1 and satisfies that for every B ∈ ℬ and (bk ) ∈ R, the subsequence (bkm ) of (bk ) and

the function F ∗ : ℝn × X → Y from Definition 3.4.2 satisfy F ∗ (t; x) ∈ ⋃s∈ℝn F(s; x), t ∈ ℝn , x ∈ X. Define B′ := ⋃t∈ℝn F(t; B) for each set B ∈ ℬ, and ℬ′ := {B′ : B ∈ ℬ}. Assume that, for every sequence from R, any its subsequence also belongs to R. Then we have the following: (i) Suppose that G : ℝn × Y → Z is uniformly (R, ℬ′ )-almost automorphic and there exists a finite real constant a > 0 such that (91) holds. Then the function W (⋅; ⋅), given by (27), is Besicovitch-(𝔽p/q , ϕ, q, R, ℬ)-multi-almost automorphic of type 1. 𝔽,ϕ,q,1,a,α 𝔽,ϕ,q,1 (ii) By AAB(R,ℬ′ ) (ℝn ×Y : Z) we denote the class of all functions G1 ∈ AAB(R,ℬ′ ) (ℝn ×Y : Z) such that for each set B′ ∈ ℬ′ there exists a sequence of uniformly (R, ℬ′ )-multialmost automorphic functions (G1k (⋅; ⋅)) such that (91) holds with the function G(⋅; ⋅) replaced therein by the function G1k (⋅; ⋅) for all k ∈ ℕ, as well as that for each ε > 0 there exist a sufficiently large real number l0 > 0 and an integer k0 ∈ ℕ such that, for every l ⩾ l0 and k ⩾ k0 , we have 1/q

sup

t∈ℝn ;y∈B

󵄩 󵄩ζq 𝔽(t, l)p/q ( ∫ 󵄩󵄩󵄩Gk (t + u; y) − G(t + u; y)󵄩󵄩󵄩Z du) ′ [−l,l]n

< ε.

3.4 Besicovitch multi-dimensional almost automorphic type functions and applications

� 181

If G ∈ AAB(R,ℬ′ ) (ℝn ×Y : Z), then the function W (⋅; ⋅) is Besicovitch-(𝔽p/q , ϕ, q, R, ℬ)multi-almost automorphic of type 1. 𝔽,ϕ,q,1,a,α

Proof. Let the set B ∈ ℬ and the sequence (bk ) ∈ R be given. By definition, there exist a subsequence (bkm ) of (bk ) and a function F ∗ : ℝn × X → Y such that the requirements of Definition 3.4.2(iii) hold and F ∗ (t; x) ∈ ⋃s∈ℝn F(s; x), t ∈ ℝn , x ∈ X. Since we have assumed that for every sequence from R, any its subsequence also belongs to R, we may assume that the limit function G∗ : ℝn × Y → Z satisfies the corresponding limit equations pointwisely for t ∈ ℝn , uniformly on the set B′ , with the functions F(⋅; ⋅) and F ∗ (⋅; ⋅) replaced therein with the functions G(⋅; ⋅) and G∗ (⋅; ⋅), respectively. Using (91) and the first limit equation for G(⋅; ⋅) and G∗ (⋅; ⋅), we get that 󵄩󵄩 ∗ ∗ ′ 󵄩 α 󵄩󵄩G (t; y) − G (t; y )󵄩󵄩󵄩Z ⩽ a‖x − y‖Y ,

t ∈ ℝn , y, y′ ∈ B′ .

(125)

In order to see that the function W (⋅; ⋅) is Besicovitch-(𝔽p/q , ϕ, q, R, ℬ)-multi-almost automorphic of type 1, we first observe that (here we designate (τ m := bkm ), m ∈ ℕ), for every t ∈ ℝn , x ∈ B and m ∈ ℕ, we have: 󵄩󵄩 󵄩 󵄩󵄩G(t + τ m ; F(t + τ m ; x)) − G∗ (t; F ∗ (t; x))󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩Z 󵄩󵄩 󵄩󵄩 ⩽ 󵄩󵄩󵄩G(t + τ m ; F(t + τ m ; x)) − G(t + τ m ; F ∗ (t; x))󵄩󵄩󵄩 󵄩 󵄩Z 󵄩󵄩 󵄩󵄩 ∗ ∗ ∗ 󵄩 󵄩 + 󵄩󵄩G(t + τ m ; F (t; x)) − G (t; F (t; x))󵄩󵄩 󵄩 󵄩Z 󵄩󵄩 󵄩󵄩α 󵄩󵄩 󵄩󵄩 ∗ ⩽ a󵄩󵄩󵄩F(t + τ m ; x) − F (t; x)󵄩󵄩󵄩 + 󵄩󵄩󵄩G(t + τ m ; F ∗ (t; x)) − G∗ (t; F ∗ (t; x))󵄩󵄩󵄩 . 󵄩 󵄩Y 󵄩 󵄩Z Since x ∈ B and F ∗ (t; x) ∈ B′ for all t ∈ ℝn , we simply deduce the required conclusion from an elementary argumentation involving the fact that, for every fixed real number l > 0, we have: lim

m→+∞

󵄩 󵄩αq ∫ sup󵄩󵄩󵄩G(t + u + τm ; y) − G(t + u; y)󵄩󵄩󵄩Z du = 0,

[−l,l]n

y∈B′

t ∈ ℝn ,

which follows from a simple application of the dominated convergence theorem. Keeping in mind (125) and the estimate 󵄩󵄩 ∗ 󵄩 󵄩󵄩G (t − τl ; F ∗ (t − τl ; x)) − G(t; F(t; x))󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩Z 󵄩󵄩 ∗ 󵄩󵄩 ∗ ∗ ⩽ 󵄩󵄩󵄩G (t − τl ; F (t − τl ; x)) − G (t − τl ; F(t; x))󵄩󵄩󵄩 󵄩 󵄩Z 󵄩󵄩 ∗ 󵄩󵄩 󵄩 󵄩 + 󵄩󵄩G (t − τl ; F(t; x)) − G(t; F(t; x))󵄩󵄩 , l ∈ ℕ, 󵄩 󵄩Z

182 � 3 Multi-dimensional ρ-almost periodic type functions the proof of the second limit equation is quite analogous, finishing the first part of theorem. The second part of theorem follows using the first part of theorem and a simple approximation argument. Before we switch to the next subsection, let us only note that an analogue of Theorem 3.4.11 can be formulated, under certain extra conditions, for the function spaces introduced in Definition 3.4.7; see [196, Theorem 2.20] for more details.

3.4.2 Applications to the abstract Volterra integro-differential equations The main aim of this subsection is to furnish some applications of our results to the abstract Volterra integro-differential equations and the partial differential equations. 1. In this issue, we will first continue our analysis of the invariance of Besicovitch almost periodicity under the actions of infinite convolution product given by (103); as mentioned many times before, this result can be also given in the multi-dimensional setting and applied to a wide class of the abstract (degenerate) Volterra integro-differential equations without initial conditions. For example, we can apply this result in the analysis of the existence and uniqueness of Besicovitch-p-almost automorphic type solutions of the fractional Poisson heat equation in Lp (ℝn ), and a class of the abstract fractional differential equations with the higher-order elliptic operators in the Hölder spaces [428]. Concerning the abstract fractional degenerate equations without initial conditions, the possible applications can be made to the equation α−β

𝜕tα (Bu)(t) = Au(t) + 𝜕t

(Bf )(t),

t ∈ ℝ,

where 𝜕tα denotes the Weyl fractional derivative of order α, A and B are closed linear operators in X, 1 ⩽ β < α < 2, f : ℝ → X is continuous and obeys certain extra features; cf. [189, Chapter 5] for the notion and more details. The semilinear analogues of this problem can be also considered. Before stating our next result, we need to introduce the following notion, which can be constituted in a much more general situation for the classes introduced in Definition 3.4.2: Definition 3.4.12. Suppose that the function F : ℝn → X is Besicovitch-(𝔽, ϕ, p(u), R)multi-almost automorphic [Besicovitch-(𝔽, ϕ, p(u), R, WR )-multi-almost automorphic, Besicovitch-(𝔽, ϕ, p(u), R, PR )-multi-almost automorphic]. Let w : ℝ → (0, ∞). Then we say that F(⋅) is Besicovitch-(𝔽, ϕ, p(u), R, w)-multi-almost automorphic [Besicovitch(𝔽, ϕ, p(u), R, WR , w)-multi-almost automorphic, Besicovitch-(𝔽, ϕ, p(u), R, PR , w)-multialmost automorphic] if and only if for each sequence (bk ) ∈ R the corresponding limit function F ∗ : ℝ → X from Definition 3.4.2 satisfies that there exists a finite real number M > 0 such that ‖F ∗ (t)‖ ⩽ Mw(|t|), t ∈ ℝn .

3.4 Besicovitch multi-dimensional almost automorphic type functions and applications

�

183

The idea of controlling the growth order of limit function F ∗ (⋅) by the weight function w(⋅) seems to be new within the theory of almost automorphic functions, and it is generally applicable in the analysis of many other classes of (generalized) almost automorphic functions known in the existing literature. Now we are ready to formulate the following analogue of [450, Proposition 4.1, Proposition 4.2]: Proposition 3.4.13. Suppose that M > 0, β ∈ (0, 1], γ > β, the operator family (R(t))t>0 ⊆ L(X, Y ) satisfies (104), as well as that a > 0, α > 0, 1 ⩽ p < +∞, αp ⩾ 1, ap ⩾ 1, αp(β − 1)/(αp − 1) > −1 if αp > 1, and β = 1 if αp = 1. If b ∈ [0, γ − β), w(t) := (1 + |t|)b , t ∈ ℝ and the function f : ℝ → X is Besicovitch-(t −a , x α , p, R, w)-multi-almost automorphic [Besicovitch-(t −a , x α , p, R, WR , w)-multi-almost automorphic, Besicovitch-(t −a , x α , p, R, PR , w)-multi-almost automorphic] and there exists a real constant M ′ > 0 such that ‖f (t)‖Y ⩽ M ′ w(t), t ∈ ℝ, then the function F(⋅), given by (103), is continuous, Besicovitch-(t −a , x α , p, R, w)-multi-almost automorphic [Besicovitch-(t −a , x α , p, R, WR , w)-multi-almost automorphic, Besicovitch-(t −a , x α , p, R, PR , w)-multi-almost automorphic] and there exists a finite real constant M ′′ > 0 such that ‖F(t)‖Y ⩽ M ′′ w(t), t ∈ ℝ. Proof. We will consider the class of Besicovitch-(t −a , x α , p, R, w)-multi-almost automorphic functions, only. Since we have assumed that exists a finite real constant M ′ > 0 such that ‖f (t)‖Y ⩽ M ′ w(t), t ∈ ℝ, the function F(⋅) is well-defined and there exists a finite real constant M ′′ > 0 such that ‖F(t)‖Y ⩽ M ′′ w(t), t ∈ ℝ; its continuity can be shown following the argumentation contained in the proof of [450, Proposition 4.2]. Let a sequence (bk ) ∈ R be given. Then there exist a subsequence (bkm ) of (bk ), a function f ∗ : ℝ → X and a finite real constant M > 0 such that ‖f ∗ (t)‖ ⩽ Mw(t), t ∈ ℝ and the equations (118)–(119) hold with the prescribed parameters, and the meaning t clear. Define F ∗ : ℝ → Y by F ∗ (t) := ∫−∞ R(t − s)f ∗ (s) ds, t ∈ ℝ. Then it is clear that F ∗ (⋅) is well-defined as well as that there exists a finite real constant M ′′′ > 0 such that ‖F ∗ (t)‖ ⩽ M ′′′ w(t), t ∈ ℝ. In order to see that the estimate (119) holds for the functions F(⋅) and F ∗ (⋅), take any real number ζ ∈ ((1/(αp)) + b, (1/(αp)) + γ − β). Then we can argue as in the computation carried out in the proof of [450, Proposition 4.1]: l

1 󵄩 󵄩αp ∫󵄩󵄩󵄩F(s + bkm + t) − F ∗ (s + t)󵄩󵄩󵄩 ds ap 2l −l

αp l󵄨 0 󵄨󵄨 󵄨󵄨 1 󵄨󵄨 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄨󵄨󵄨 ∗ ⩽ ap ∫󵄨󵄨 ∫ 󵄩󵄩R(−z)󵄩󵄩 ⋅ 󵄩󵄩F(s + bkm + t + z) − F (s + t + z)󵄩󵄩 dz󵄨󵄨 ds 󵄨󵄨 󵄨󵄨 2l 󵄨 −l 󵄨−∞

l󵄨 0 󵄨󵄨αp M 󵄨󵄨󵄨󵄨 |z|β−1 (1 + |z|)ζ −ζ 󵄩 󵄩󵄩 󵄨󵄨󵄨 ∗ 󵄩 ⩽ ap ∫󵄨󵄨 ∫ ⋅ (1 + |z|) 󵄩󵄩F(s + bkm + t + z) − F (s + t + z)󵄩󵄩 dz󵄨󵄨 ds 󵄨󵄨 󵄨󵄨 2l (1 + |z|γ ) 󵄨 −l 󵄨−∞

184 � 3 Multi-dimensional ρ-almost periodic type functions l

0

M 1 󵄩󵄩 󵄩αp ∗ ⩽ ap1 ∫ ∫ 󵄩󵄩F(s + bkm + t + z) − F (s + t + z)󵄩󵄩󵄩 dz ds αζ p 2l (1 + |z| ) −l −∞ l

l

M 1 󵄩󵄩 󵄩αp ∗ = ap1 ∫ ∫ 󵄩F(bkm + t + z) − F (t + z)󵄩󵄩󵄩 ds dz 2l (1 + |z − s|αζ )p 󵄩 −l z−s

−l l

+

M1 1 󵄩󵄩 󵄩αp ∗ ∫ ∫ 󵄩F(bkm + t + z) − F (t + z)󵄩󵄩󵄩 ds dz 2lap (1 + |z − s|αζ )p 󵄩 l

⩽

−∞ −l

+∞

M1 󵄩󵄩 ds 󵄩αp ∫󵄩F(b + t + z) − F ∗ (t + z)󵄩󵄩󵄩 dz ⋅ ∫ lap 󵄩 km (1 + |s|ζ )αp −l

+

−∞

−3l l

M1 1 󵄩󵄩 󵄩αp ∗ ∫ ∫ 󵄩F(bkm + t + z) − F (t + z)󵄩󵄩󵄩 ds dz 2lap (1 + |z − s|αζ )p 󵄩 −∞ −l 3l l

M 1 󵄩󵄩 󵄩αp ∗ + ap1 ∫ ∫ 󵄩󵄩F(bkm + t + z) − F (t + z)󵄩󵄩󵄩 ds dz αζ p 2l (1 + |z − s| ) l

−3l −l

+∞

M 󵄩 ds 󵄩αp ⩽ ap1 ∫󵄩󵄩󵄩F(bkm + t + z) − F ∗ (t + z)󵄩󵄩󵄩 dz ⋅ ∫ l (1 + |s|ζ )αp −l

−∞

3l

+∞

M ds 󵄩 󵄩αp + ap1 ∫ 󵄩󵄩󵄩F(bkm + t + z) − F ∗ (t + z)󵄩󵄩󵄩 dz ⋅ ∫ l (1 + |s|ζ )αp −∞

−3l

−3l

+

cM1 l 1 󵄩󵄩 󵄩αp ∗ ∫ 󵄩F(bkm + t + z) − F (t + z)󵄩󵄩󵄩 dz, 2lap (1 + |z/2|αζ )p 󵄩 −∞

for any t ∈ ℝ; here we have used the Hölder inequality, the Fubini theorem, and an elementary change of variables in the double integral. The estimate (119) for the functions F(⋅) and F ∗ (⋅) can be proved analogously, finishing the proof. It is worth noting that Proposition 3.4.13 can be reformulated for all other classes of one-dimensional Besicovitch almost automorphic type functions introduced in Definition 3.4.2, but not for multi-dimensional Weyl almost automorphic functions considered in [449], since it is not clear how one can prove the existence of the right limits (classical ones) in the equations from Definition 3.4.2. Using the method proposed in the proofs of [428, Theorem 3.7.1] and Proposition 3.4.13, we can consider the existence and uniqueness of Besicovitch-p-almost automorphic solutions for a class of the abstract nonautonomous differential equations of first order; see also [450, Theorem 4.5]. 2. Concerning the convolution invariance of multi-dimensional Besicovitch almost automorphy, the notion introduced in Definition 3.4.13 plays a crucial role again. We need to control the growth order of limit functions in order to obtain any relevant re-

3.4 Besicovitch multi-dimensional almost automorphic type functions and applications

�

185

sult. The conclusions established in this application can be also formulated for all other classes of functions introduced in Definition 3.4.2. We will consider first the actions of the Gaussian semigroup. Assume that there exist two finite real numbers b ⩾ 0 and c > 0 such that |F(x)| ⩽ c(1 + |x|)b ≡ cw(x), x ∈ ℝn as well as that a > 0, α > 0, 1 ⩽ p < +∞, αp ⩾ 1, 1/(αp) + 1/q = 1 and F(⋅) is Besicovitch-(t −a , x α , p, R, w)-multi-almost automorphic, where R is a general collection of sequences in ℝn . Let us fix a real number t0 in (114). Then the mapping x 󳨃→ (G(t0 )F)(x), x ∈ ℝn is well-defined and has the same growth as the inhomogeneity F(⋅). Writing the 2 2 2 term e−|y| /4t0 = e−|y| /8t0 ⋅ e−|y| /8t0 and applying the Hölder inequality, we may conclude that the function F(⋅) is Besicovitch-(t −a , x α , p, R, w)-multi-almost automorphic; see, e. g. the argumentation given in the fourth application of [450, Section 4]. We can similarly prove the following analogue of [450, Theorem 4.6] (see also [449, Theorem 3.9, Theorem 3.13]): Theorem 3.4.14. Suppose that b ⩾ 0, α > 0, a > 0, 1 ⩽ p < +∞, αp ⩾ 1, 1/(αp) + 1/q = 1, f : ℝn → Y is Besicovitch-(t −a , x α , p, R, w)-multi-almost automorphic [Besicovitch(t −a , x α , p, R, WR , w)-multi-almost automorphic, Besicovitch-(t −a , x α , p, R, PR , w)-multialmost automorphic], where w(t) ≡ (1 + |t|)b , t ∈ ℝ. If there exist two functions h1 : ℝn → ℂ and h2 : ℝn → ℂ such that h = h1 h2 , h1 ∈ Lq (ℝn ) and |h1 (⋅)|α [1+|⋅|]ζ ∈ Lp (ℝn ) with ζ = max(bα, a), then the function F(⋅), given by F(x) ≡ ∫ h(x − y)f (y) dy,

x ∈ ℝn ,

ℝn

is Besicovitch-(t −a , x α , p, R, w)-multi-almost automorphic [Besicovitch-(t −a , x α , p, R, WR , w)-multi-almost automorphic, Besicovitch-(t −a , x α , p, R, PR , w)-multi-almost automorphic]. The notion introduced in Definition 3.4.13 is important if we want to reconsider the fifth application and the sixth application of [450, Section 4]. We will only note that the analysis of the existence and uniqueness of Besicovitch-p-almost automorphic type solutions of the wave equation, whose solutions are given by the d’Alembert formula [the Kirchhoff formula; the Poisson formula] can be carried out in almost the same way as in the almost periodic case, and the same conclusions can be achieved; the analysis of existence and uniqueness of Besicovitch-p-almost automorphic type solutions connected with the use of evolution systems considered in the above-mentioned sixth application of [450, Section 4] and the final application of [431, Section 6.3, pp. 426–428] can be carried out as in the almost periodic case as well. The notion introduced in Definition 3.4.13 is important if we want to reconsider the application from [196, Example 1], given directly before Subsection 1.1 of this paper. More precisely, suppose that A generates a strongly continuous semigroup (T(t))t⩾0 on a Banach space X whose elements are certain complex-valued functions defined on ℝn . Under some assumptions, we have that the function

186 � 3 Multi-dimensional ρ-almost periodic type functions t

u(t, x) = (T(t)u0 )(x) + ∫[T(t − s)f (s)](x) ds,

t ⩾ 0, x ∈ ℝn

0

is a unique classical solution of the abstract Cauchy problem ut (t, x) = Au(t, x) + F(t, x),

t ⩾ 0, x ∈ ℝn ; u(0, x) = u0 (x),

where F(t, x) := [f (t)](x), t ⩾ 0, x ∈ ℝn . In many concrete situations (for example, this holds for the Gaussian semigroup on ℝn ), we have the existence of a kernel (t, y) 󳨃→ E(t, y), t > 0, y ∈ ℝn which is integrable on any set [0, T] × ℝn (T > 0) and satisfies that [T(t)f (s)](x) = ∫ F(s, x − y)E(t, y) dy,

t > 0, s ⩾ 0, x ∈ ℝn .

ℝn

If a real number t0 > 0 is fixed and the above requirement holds, then we have observed, in [194, Example 0.1], that the almost periodic behaviour of function x 󳨃→ ut0 (x) ≡ t

∫00 [T(t0 − s)f (s)](x) ds, x ∈ ℝn depends on the almost periodic behaviour of function F(t, x) in the space variable x. The argumentation given there is applicable not only for almost periodicity, but also for almost automorphy and various generalizations of these concepts, provided that the exponent p(⋅) has a constant value 1. For example, if the function F(t, x) is Besicovitch-(𝔽, x, 1, R, 1)-multi-almost automorphic with respect to the variable x ∈ ℝn , uniformly in the variable t on compact subsets of [0, ∞), the solution ut0 (⋅) will be Besicovitch-(𝔽, x, 1, R, 1)-multi-almost automorphic, as well; see, e. g. the computations carried out on [431, pp. 402–403] for more details. 3. Without going into full details, we will only note that Theorem 3.4.11 can be applied in the analysis of the existence and uniqueness of bounded, continuous, Besicovitch-(𝔽, ϕ, p, R, ℬ)-multi-almost automorphic of type 1, solutions for a various classes of the abstract (fractional) semilinear Cauchy problems; here ℬ = ℬ′ can be chosen to be the collection consisting of all bounded subsets of the Banach space X and p ⩾ 1 is any finite real exponent. See also the third application of [450, Section 4] for more details.

Some conclusions and final remarks Concerning certain drawbacks and possibilities for further investigations of multidimensional Besicovitch almost automorphic type functions, we want to mention first that, besides the above-introduced classes of functions, we can also consider some else. For example, in parts (i) and (ii) of Definition 3.4.2 we can operate with lim infm→+∞ in place of limm→+∞ , while in parts (i) and (ii) of Definition 3.4.2 we can operate with lim infl→+∞ in place of liml→+∞ ; the notion introduced in [431, Definition 8.3.18, Definition 8.3.28] can be extended only if we use the function ϕ(⋅) in the analysis. We have not considered these topics here, as well as the integration and differentiation of multi-dimensional Besicovitch (Weyl) almost automorphic type functions.

3.5 Multi-dimensional ρ-almost periodic type distributions

� 187

Concerning some open problems, we would like to recall first that we have asked, in [449, Question 5.1], whether for a given real exponent p ⩾ 1 we can find a Weyl p-almost automorphic function of type 1 which is not Weyl p-almost automorphic? The same question can be proposed for the Besicovitch almost automorphic type functions. Moreover, in [449, Question 5.3], we have asked whether an almost automorphic function is automatically Weyl-p-almost automorphic? But it is also not clear whether an almost automorphic function is Besicovitch-p-almost automorphic for some finite real exponent p ⩾ 1. Finally, we would like to note that Y. Li, X. Wang and N. Huo have recently analyzed, in [509], the Besicovitch almost automorphic stochastic processes in distribution and provided some applications in the study of the Clifford-valued stochastic neural networks; for some other applications of almost automorphic functions not mentioned in [428] and [431], we also refer the reader to [48, 49, 747, 788].

3.5 Multi-dimensional ρ-almost periodic type distributions In the one-dimensional setting, the notion of a scalar-valued bounded distribution and the notion of a scalar-valued almost periodic distribution have been introduced by L. Schwartz [673]; these notions have been later extended to the vector-valued distributions by I. Cioranescu in [220]. The class of scalar-valued asymptotically almost periodic distributions has been introduced by I. Cioranescu in [219]. It is also worth noting that the notion of a vector-valued asymptotically almost periodic distribution has been analyzed by D. N. Cheban [197] following a different approach (cf. also I. K. Dontvi [272] and A. Halanay, D. Wexler [350]). For more details about the subject, we refer the reader to [92, 152, 153, 154, 156, 461, 703], the recent research studies [157] by C. Bouzar, F. Z. Tchouar, [435] by M. Kostić and the list of references in the recent research monographs [428, 429, 430, 431]. The main aim of this section is to transfer the results established in the recent research article [460] by M. Kostić et al. to the vector-valued ρ-almost periodic distributions T : 𝒟(ℝn ) → X; the material is taken from [452]. To the best knowledge of the author, there is no relevant reference in the existing literature devoted to the study of vector-valued almost periodic distributions which are defined on the space of the test functions 𝒟(ℝn ) if n ⩾ 2 (let us only recall that the space of scalar-valued almost periodic distributions in ℝn has recently been introduced by N. Strungaru and V. Terauds in [717] for the purpose of new investigations in the diffraction theory; cf. also the research article [645] by L. I. Ronkin, who investigated almost periodic distributions in tube domains of ℂn following a completely different approach). Here, we consider ρ-almost periodic type distributions T : 𝒟(ℝn ) → X, where ρ is a general binary relation on X, and slightly extend the results from [460] given in the one-dimensional setting. Furthermore, we analyze vector-valued 𝔻-asymptotically ρ-almost periodic distributions in ℝn

188 � 3 Multi-dimensional ρ-almost periodic type functions and provide some new applications of our theoretical results to the (abstract) partial differential equations; here, 𝔻 is an arbitrary unbounded subset of ℝn . The organization of this section can be briefly summarized as follows. The first original contributions of ours are given in Theorem 2.1.25 and Corollary 3.5.3, where we provide two new results about 𝔻-asymptotically Bohr (I ′ , ρ)-almost periodic functions. After that, we recall the basic definitions and results concerning vector-valued (asymptotically) almost periodic distributions. The main results are given in Section 3.5.1, where we thoroughly analyze ρ-almost periodic type distributions in ℝn . We open this subsection by introducing the notion of an (I ′ , ρ)-almost periodic ((I ′ , ρ)-uniformly recurrent) distribution; see Definition 3.5.4 (0 ≠ I ′ ⊆ ℝn ). The main structural results about these classes of vector-valued distributions are given in Theorem 3.5.6, Theorem 3.5.7, Theorem 3.5.8, and Proposition 3.5.9. The notion of vector-valued distributional ergodic ′ ′ spaces B𝔻,0 (ℝn : X) and B𝔻,0,b (ℝn : X) is introduced in Definition 3.5.10. Some structural results about these spaces are given in Proposition 3.5.11, Proposition 3.5.12, and Proposition 3.5.14; cf. also Remark 3.5.13. Section 3.5.2 investigates 𝔻-asymptotically ρ-almost periodic type distributions in ℝn . The notion of a 𝔻-asymptotically (I ′ , ρ)-almost periodic [𝔻-asymptotically (I ′ , ρ)-uniformly recurrent] distribution and the notion of a 𝔻-asymptotically (𝒟Ω , I ′ , ρ)-almost periodic [𝔻-asymptotically (𝒟Ω , I ′ , ρ)-uniformly recurrent] distribution is introduced in Definition 3.5.15 (0 ≠ Ω ⊆ ℝn ); in Definition 3.5.17, we introduce the corresponding classes of vector-valued distributions of type 1. Some structural results about these spaces are given in Proposition 3.5.16 and the statements (A1)–(A6), where we examine certain possibilities to extend the statement of [460, Theorem 11] to the multi-dimensional setting. Some new applications of our results to the (abstract) partial differential equations are given in Section 3.5.3. Before proceeding further, we would like to notice that many other spaces of (vectorvalued) almost periodic generalized functions have been explored in the existing literature. For the sake of brevity and better exposition, we have decided to analyze the spaces of multi-dimensional (𝔻-asymptotically) almost periodic ultradistributions, hyperfunctions, and Colombeau generalized functions somewhere else; cf. [428, Section 2.15, Section 3.8, Section 3.9] and [431, Subsection 4.1.5, Subsection 4.1.6, pp. 303–309] for more details in this direction. The spaces of (ω, ρ)-almost periodic generalized functions and the spaces of (ωj , ρj )j∈ℕn -almost periodic generalized functions will be briefly considered in the final section of this part. Let us recall, if the set 𝔻 ⊆ ℝn is unbounded, then we define C0,𝔻 (ℝn : X) ≡ {F ∈ C(ℝn : X) ;

lim

|x|→+∞;x∈𝔻

F(x) = 0};

C0,ℝn (ℝn : X) ≡ C0 (ℝn : X). In the remainder of this section, we consider the case in which I = ℝn , only. For the sequel, we need the following slight extension of [304, Theorem 2.23], which can be deduced using the same argumentation as in the proof of the above-mentioned result:

3.5 Multi-dimensional ρ-almost periodic type distributions

�

189

Theorem 3.5.1. Suppose that 𝔻 ⊆ I ⊆ ℝn , the set 𝔻 is unbounded, 0 ≠ I ′ ⊆ ℝn , ρ is a binary relation on Y satisfying that D(ρ) is a closed subset of Y and condition (Cρ ). Suppose further that for each integer j ∈ ℕ the function Fj (⋅; ⋅) = Gj (⋅; ⋅) + Qj (⋅; ⋅) is 𝔻-asymptotically Bohr (ℬ, I ′ , ρ)-almost periodic (𝔻-asymptotically (ℬ, I ′ , ρ)-uniformly recurrent), where Gj (⋅; ⋅) is Bohr (ℬ, I ′ , ρ)-almost periodic ((ℬ, I ′ , ρ)-uniformly recurrent) and Qj ∈ C0,𝔻,ℬ (I × X : Y ). Let for each B ∈ ℬ there exist εB > 0 such that the sequence (Fj (⋅; ⋅)) converges uniformly to a function F(⋅; ⋅) on the set B∘ ∪ ⋃x∈𝜕B B(x, εB ), and let Qj ∈ C0,𝔻,ℬ∘ (I × X : Y ), where ℬ∘ ≡ {B∘ : B ∈ ℬ}. If there exists a finite real number c > 0 such that for each natural numbers m, k ∈ ℕ the function Gk (⋅; ⋅) − Gm (⋅; ⋅) satisfies the following supremum formula: (S)𝔻 for every a > 0, we have 󵄩 󵄩 sup 󵄩󵄩󵄩Gk (t; x) − Gm (t; x)󵄩󵄩󵄩Y ⩽ c ∘

t∈I,x∈B

sup

t∈𝔻,|t|⩾a,x∈B∘

󵄩󵄩 󵄩 󵄩󵄩Gk (t; x) − Gm (t; x)󵄩󵄩󵄩Y ,

then F(⋅; ⋅) is 𝔻-asymptotically Bohr (ℬ∘ , I ′ , ρ)-almost periodic (𝔻-asymptotically (ℬ∘ , I ′ , ρ)-uniformly recurrent). For the sequel, we need the following simple lemma: Lemma 3.5.2. Suppose that F : ℝn → Y is a Bohr almost periodic function, 𝔻 ⊆ ℝn and the set 𝔻 is unbounded. If (∀l > 0) (∀M > 0) (∀t ∈ ℝn ) (∃t0 ∈ ℝn ∖ B(0, M)) (∀τ ∈ B(t0 , l)) t + τ ∈ 𝔻,

(126)

then for each a > 0 we have 󵄩 󵄩 sup 󵄩󵄩󵄩F(t)󵄩󵄩󵄩Y =

t∈ℝn

󵄩 󵄩 sup 󵄩󵄩󵄩F(t)󵄩󵄩󵄩Y .

t∈𝔻,|t|⩾a

(127)

Proof. Clearly, it suffices to show that for each fixed number ε > 0 we have 󵄩 󵄩 󵄩 󵄩 sup 󵄩󵄩󵄩F(t)󵄩󵄩󵄩Y ⩽ ε + sup 󵄩󵄩󵄩F(t)󵄩󵄩󵄩Y .

t∈ℝn

t∈𝔻,|t|⩾a

Take l > 0 from the definition of Bohr almost periodicity of the function F(⋅) and fix a point t ∈ ℝn . Then we choose M = |t|+a+l. Due to (126), there exists t0 ∈ ℝn ∖B(0, M) such that for each τ ∈ B(t0 , l) we have t + τ ∈ 𝔻. To complete the proof, we choose an ε-period τ of the function F(⋅) which belongs to the ball B(t0 , l); then we have |t+τ| ⩾ |t0 |−l−|t| ⩾ a and therefore 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩F(t)󵄩󵄩󵄩Y ⩽ ε + 󵄩󵄩󵄩F(t + τ)󵄩󵄩󵄩Y ⩽ ε + sup 󵄩󵄩󵄩F(t)󵄩󵄩󵄩Y . t∈𝔻,|t|⩾a

Now we are able to formulate the following corollary:

190 � 3 Multi-dimensional ρ-almost periodic type functions Corollary 3.5.3. The conclusion of Theorem 2.1.25 holds provided that X = {0}, 𝔻 ⊆ ℝn , the set 𝔻 is unbounded, ρ : Y → Y is a uniformly continuous function, I ′ − I ′ = ℝn and for each a > 0 we have (127). Proof. Suppose now that 𝔻 ⊆ ℝn , the set 𝔻 is unbounded, ρ : Y → Y is a uniformly continuous function, I ′ −I ′ = ℝn and for each a > 0 we have (127). Then [304, Proposition 2.2] implies that for each k, m ∈ ℕ we have that Gk (⋅) − Gm (⋅) is a Bohr almost periodic function and (Cρ ) is satisfied, so that the final conclusion follows from Lemma 3.5.2. Vector-valued (asymptotically) almost periodic distributions Denote by 𝒟(ℝn ) = 𝒟(ℝn : ℂ) the Schwartz space of all infinitely differentiable functions f : ℝn → ℂ with compact support. By 𝒮 (ℝn ) = 𝒮 (ℝn : ℂ) we denote the Schwartz space of all rapidly decreasing functions with values in ℂ, and by ℰ (ℝn ) = ℰ (ℝn : ℂ) we denote the space of all infinitely differentiable functions with values in ℂ. We define the spaces 𝒟(ℝn : X), ℰ (ℝn : X) and 𝒮 (ℝn : X) in a similar way. The spaces of all linear continuous mappings from 𝒟(ℝn ), 𝒮 (ℝn ) and ℰ (ℝn ) into X are denoted by 𝒟′ (ℝn : X), 𝒮 ′ (ℝn : X) and ℰ ′ (ℝn : X), respectively [673]. If T ∈ 𝒟′ (ℝn : X) and φ ∈ 𝒟(ℝn ), then we define T ∗ φ ∈ ℰ (ℝn : X) by (T ∗ φ)(x) := ⟨T, φ(x − ⋅)⟩. If f : ℝn → X, then we define f ̌ : ℝn → X by f ̌(x) := f (−x), x ∈ ℝn ; for any T ∈ 𝒟′ (ℝn : X), we ̌ φ ∈ 𝒟. If 0 ≠ Ω ⊆ ℝn , then we define define Ť ∈ 𝒟′ (ℝn : X) by ⟨T,̌ φ⟩ := ⟨T, φ⟩, 𝒟Ω (ℝn ) := {φ ∈ 𝒟(ℝn ) : supp(φ) ⊆ Ω}. Let 1 ⩽ p ⩽ ∞. By 𝒟Lp (ℝn : X) we denote the vector space consisting of all infinitely differentiable functions f : ℝn → X such that f (α) ∈ Lp (ℝn : X) for all multi-indices α ∈ ℕn0 . The Fréchet topology on 𝒟Lp (ℝn : X) is induced by the following system of norms ‖f ‖k := ∑|α|⩽k ‖f (α) ‖Lp (ℝn :X) , k ∈ ℕ. If X = ℂ, then the above space is simply denoted by 𝒟Lp (ℝn ). The space of all linear continuous mappings f : 𝒟L1 (ℝn ) → X is denoted by 𝒟L′ 1 (ℝn : X). Endowed with the strong topology, 𝒟L′ 1 (ℝn : X) becomes a complete locally convex space; 𝒟L′ 1 (ℝn : X) is a well known space of bounded X-valued distributions. 3.5.1 ρ-almost periodic type distributions in ℝn We start this subsection by introducing the following notion: Definition 3.5.4. Let T ∈ 𝒟′ (ℝn : X). Then we say that T is an (I ′ , ρ)-almost periodic ((I ′ , ρ)-uniformly recurrent) distribution if and only if T ∗ φ ∈ API ′ ,ρ (ℝn : X) (T ∗ φ ∈ URI ′ ,ρ (ℝn : X)) for all φ ∈ 𝒟(ℝn ). By BI′ ′ ,ρ,AP (X) (BI′ ′ ,ρ,UR (X)) we denote the space of all (I ′ , ρ)-almost periodic ((I ′ , ρ)-uniformly recurrent) distributions.

In the case that ρ = cI for some c ∈ ℂ ∖ {0}, where I denotes the identity operator on X, then we write “c” in place of “ρ”; if this is the case, an application of [441, Corollary 2.11] shows that the non-triviality of T implies |c| = 1.

3.5 Multi-dimensional ρ-almost periodic type distributions

� 191

Remark 3.5.5. In this section, we will not consider the spaces of vector-valued semi-ρperiodic generalized functions; in places of the functions spaces considered in Definition 3.5.4, we can also use some other spaces of generalized almost periodic functions like multi-dimensional almost automorphic functions (cf. [431] for more details). We continue by stating the following extension of [460, Theorem 5]: Theorem 3.5.6. Suppose that ρ = A ∈ L(X), 0 ≠ I ′ ⊆ ℝn , there exist an integer k ∈ ℕ and (I ′ , A)-almost periodic ((I ′ , A)-uniformly recurrent) functions Fj : ℝn → X (0 ⩽ j ⩽ k) such that the function t 󳨃→ F(t) ≡ (F0 (t), . . . , Fk (t)),

t ∈ ℝn

(128)

is (I ′ , Ak+1 )-almost periodic ((I ′ , Ak+1 )-uniformly recurrent), where Ak+1 ∈ L(X k+1 ) is given by Ak+1 (x0 , x1 , . . . , xk ) := (Ax0 , Ax1 , . . . , Axk ), (x0 , x1 , . . . , xk ) ∈ X k+1 . Let αj ∈ ℕn0 for 0 ⩽ j ⩽ k. Define T := ∑kj=0 Fj

(αj )

. Then T ∈ BI′ ′ ,A,AP (X) (T ∈ BI′ ′ ,A,UR (X)).

Proof. We will consider Bohr (I ′ , A)-almost periodic functions, only. It is clear that T ∈ 𝒟′ (X) and for each φ ∈ 𝒟(ℝn ) and t ∈ ℝn we have: k

(T ∗ φ)(t) = ⟨T, φ(t − ⋅)⟩ = ∑ ∫ φ(αj ) (t − v)Fj (v) dv j=0 ℝn

k

= ∑ ∫ φ(αj ) (v)Fj (t − v) dv. j=0 ℝn

Let ε > 0 be given. Then there exists l > 0 such that for each t0 ∈ I ′ there exists τ ∈ B(t0 , l) ∩ I ′ such that for every t ∈ ℝn , we have 󵄩󵄩 󵄩 󵄩󵄩Fj (t + τ) − AFj (t)󵄩󵄩󵄩Y ⩽ ε,

0 ⩽ j ⩽ k.

Then k 󵄩󵄩 󵄩 󵄨󵄨 (α ) 󵄨󵄨 󵄩 󵄩 󵄩󵄩(T ∗ φ)(t + τ) − A(T ∗ φ)(t)󵄩󵄩󵄩 ⩽ ∑ ∫ 󵄨󵄨󵄨φ j (v)󵄨󵄨󵄨 ⋅ 󵄩󵄩󵄩Fj (t + τ − v) − AFj (t − v)󵄩󵄩󵄩 dv 󵄨 󵄨 j=0 ℝn

k 󵄨󵄨 󵄨󵄨 ⩽ ε ∑ ∫ 󵄨󵄨󵄨φ(αj ) (v)󵄨󵄨󵄨 dv, 󵄨 󵄨 j=0 ℝn

φ ∈ 𝒟 , t ∈ ℝn ,

which simply yields the required conclusion. The following extension of [460, Theorem 7] is essentially important in our analysis: Theorem 3.5.7. Let T ∈ BI′ ′ ,ρ,AP (X) (T ∈ BI′ ′ ,ρ,UR (X)), let T be a bounded distribution and let ρ : X → X be a uniformly continuous function. Then there exist an integer p ∈ ℕ and a

192 � 3 Multi-dimensional ρ-almost periodic type functions Bohr (I ′ , ρ)-almost periodic (a bounded (I ′ , ρ)-uniformly recurrent) function F : ℝn → X such that T=

∑

(k0 ,k1 ,...,kn )∈ℕn 0 k0 +k1 +⋅⋅⋅+kn =p

(−1)p−k0

p! 𝜕2(p−k0 ) F k0 !k1 ! ⋅ ⋅ ⋅ ⋅ ⋅ kn ! 𝜕x 2k1 ⋅ ⋅ ⋅ ⋅ ⋅ 𝜕x 2kn n

1

(129)

in the distributional sense. Proof. The proof can be deduced by using the argumentation contained in the proofs of [156, Theorem 1] and the last mentioned result (cf. also [304, Theorem 2.11(v)]); we will only prove here that the fundamental solution G of the differential operator (1 − Δ)p = δ is (2p − n − 1)-continuously differentiable and belongs to the space L1 (ℝn ) with all its derivatives of order ⩽ (2p − n − 1); here, p ∈ ℕ is sufficiently large. Let (ℱ f )(ξ) = (2π)−n/2 ∫ℝn e−ixξ f (x) dx denote the Fourier transform on ℝn and let ℱ −1 denote its inverse transform. Then we have (1 − Δ)p G = δ and therefore 2 p

ℱ ((1 + |x| ) ℱ (G)) = δ. −1

This simply implies G(x) = [ℱ −1 ((1 + |ξ|2 )−p )](x), x ∈ ℝn and G(α) (x) = (2π)−n/2 ∫ ℝn

eixξ (iξ)α dξ , (1 + |ξ|2 )p

x ∈ ℝn ,

for all α ∈ ℕn0 with |α| ⩽ (2p − n − 1). This yields that the function G is (2p − n − 1)continuously differentiable, since for each multi-index α ∈ ℕn0 with |α| ⩽ (2p − n − 1) we have that the function ξ 󳨃→ |ξ|α (1 + |ξ|2 )−p , ξ ∈ ℝn belongs to the space L1 (ℝn ). In order to see that all distributional derivatives G(α) (x) with |α| ⩽ (2p − n − 1) belong to the space L1 (ℝn ), it suffices to show that ℱ G(α) ∈ ℱ L1 (ℝn ) for |α| ⩽ 2p − n − 1. This follows from the existence of a finite real constant c > 0 such that 󵄨󵄨 󵄨󵄨 󵄨󵄨 η 󵄨 󵄨󵄨D [(iξ)α (1 + |ξ|2 )−p ]󵄨󵄨󵄨 ⩽ c(1 + |ξ|)|α|−2p−|η| , 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨

0 ⩽ |η| ⩽ ⌊1 + (n/2)⌋, ξ ∈ ℝn

and [427, Lemma 2.5.1(i)], after choosing a real number a > 0 sufficiently small and r > n/2 therein. Here, Dη is the usual differentiation of order η. Keeping in mind the last two results and their proofs, as well as the proof of [156, Proposition 7], we can simply deduce the following (the third part is an extension of [460, Theorem 9]): Theorem 3.5.8. (i) A vector-valued distribution T ∈ 𝒟′ (ℝn : X) is bounded if and only if the function (T ∗ φ)(⋅) is bounded for all φ ∈ 𝒟(ℝn ). (ii) Let T ∈ 𝒟L′ 1 (ℝn : X). Then the following assertions are equivalent: (a) T ∗ φ ∈ AP(ℝn : X), φ ∈ 𝒟(ℝn ).

3.5 Multi-dimensional ρ-almost periodic type distributions

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193

(b) There exist an integer k ∈ ℕ, almost periodic functions Fj : ℝn → X and multiindices αj ∈ ℕn0 (0 ⩽ j ⩽ k) such that T = ∑kj=0 Fj

(αj )

𝒟L′ 1 (ℝn

in the distributional sense.

(iii) Suppose that T ∈ : X) and ρ = A ∈ L(X). Then the following statements are equivalent: (a) We have T ∈ BI′ ′ ,A,AP (X) (T ∈ BI′ ′ ,A,UR (X)). (b) There exist an integer p ∈ ℕ and an A-almost periodic (a bounded A-uniformly recurrent) function F : ℝn → X such that (129) holds in the distributional sense. (c) There exist an integer k ∈ ℕ and A-almost periodic (bounded A-uniformly recurrent) functions Fj : ℝn → X (0 ⩽ j ⩽ k) such that the function F(⋅), defined through (128), is Ak+1 -almost periodic (Ak+1 -uniformly recurrent) and T = ∑kj=0 Fj j for some multi-indices αj ∈ ℕn0 (0 ⩽ j ⩽ k). (d) There exists a sequence (Tk ) of A-almost periodic functions (bounded A-uniformly recurrent functions) from ℰ (ℝn : X) such that limk→∞ Tk = T in 𝒟L′ 1 (ℝn : X). (α )

We say that a bounded distribution T ∈ 𝒟L′ 1 (ℝn : X) is almost periodic if and only if T satisfies any of two equivalent conditions from part (ii); if this is the case, then it can be easily shown that the restriction of T to the space 𝒮 (ℝn ) is an X-valued tempered ′ distribution (see, e. g. [428] for the one-dimensional setting). By BAP (ℝn : X) we denote the space consisting of all almost periodic distributions. The main structural results for the one-dimensional almost periodic type distributions can be simply clarified for the multi-dimensional almost periodic type distributions. For example, the statements of [460, Proposition 2, Proposition 4, Proposition 5, Corollary 1] as well as the conclusions clarifed on l. 1–21 on p. 196 can be simply formulated in our new framework (see [441] for more details about multi-dimensional c-almost periodic type functions and their applications). In the one-dimensional setting, B. Basit and H. Güenzler have proved, in [92], that the regular distribution induced by a Stepanov-p-almost periodic function F(⋅) is almost periodic. The same statement holds in the multi-dimensional setting, as the next result shows: Proposition 3.5.9. Suppose that F : ℝn → X is a Stepanov-p-almost periodic function, where 1 ⩽ p < +∞, and T(φ) := ∫ φ(x)F(x) dx,

φ ∈ 𝒟(ℝn ).

ℝn

Then T is almost periodic. Proof. Since any Stepanov-p-almost periodic function is Stepanov-1-almost periodic, we may assume that p = 1. We know that the function F(⋅) is Stepanov bounded, i. e., there exists a finite real constant M > 0 such that sup

x∈ℝn

󵄩 󵄩 ∫ 󵄩󵄩󵄩F(y)󵄩󵄩󵄩 dy ⩽ M.

x+[0,1]n

194 � 3 Multi-dimensional ρ-almost periodic type functions It is clear that T ∈ 𝒟′ (ℝn : X). Let φ ∈ 𝒟(ℝn ) be fixed. In order to prove that T is bounded, it suffices to show that the function (T ∗ φ)(⋅) is bounded. This follows from the fact that there exists a positive integer m ∈ ℕ such that 󵄩󵄩 󵄩 󵄩󵄩(T ∗ φ)(x)󵄩󵄩󵄩 ⩽ ‖φ‖∞

∑

k∈ℤn ;|k|⩽m

󵄩 󵄩 ∫ 󵄩󵄩󵄩F(x)󵄩󵄩󵄩 dx ⩽ Mm‖φ‖∞ ,

x ∈ ℝn .

k+[0,1]n

Due to Theorem 3.5.8(ii), it suffices to show that T ∗ φ ∈ AP(ℝn : X). It is clear that the function (T ∗ φ)(⋅) is infinitely differentiable. Now, let ε > 0 be given. Since the function F(⋅) is Stepanov-1-almost periodic, there exists l > 0 such that for each t0 ∈ ℝn there exists τ ∈ B(t0 , l) ∩ ℝn with 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩F(t + τ − u) − F(t − u)󵄩󵄩󵄩L1 ([0,1]n :X) = 󵄩󵄩󵄩F(t + τ + u) − F(t + u)󵄩󵄩󵄩L1 (−[0,1]n :X) 󵄩 󵄩 = 󵄩󵄩󵄩F([t − (1, 1, . . . , 1)] + τ + u) − F([t − (1, 1, . . . , 1)] + u)󵄩󵄩󵄩L1 ([0,1]n :X) ⩽ ε,

t ∈ ℝn .

Then there exists a positive integer m ∈ ℕ such that, for every x ∈ ℝn , we have: 󵄩󵄩 󵄩 󵄩󵄩(T ∗ φ)(x + τ) − (T ∗ φ)(x)󵄩󵄩󵄩 󵄨 󵄨 󵄩 󵄩 ⩽ ∑ ∫ 󵄨󵄨󵄨φ(y)󵄨󵄨󵄨 ⋅ 󵄩󵄩󵄩F(x + τ − y + k) − F(x + τ − y)󵄩󵄩󵄩 dy k∈ℤn ;|k|⩽m [0,1]n

⩽ m‖φ‖∞

∑

󵄩 󵄩 ∫ 󵄩󵄩󵄩F(x + τ − y + k) − F(x + τ − y)󵄩󵄩󵄩 dy ⩽ m‖φ‖∞ ε.

k∈ℤn ;|k|⩽m [0,1]n

This completes the proof. We close this subsection with the observation that for every finite exponent p ⩾ 1, there exists an infinitely differentiable equi-Weyl-p-almost periodic function f : ℝ → ℝ such that the regular distribution induced by this function is not almost periodic [428]. 3.5.2 𝔻-asymptotically ρ-almost periodic type distributions in ℝn In the one-dimensional setting, we have recently analyzed the following spaces of vector-valued bounded distributions vanishing at plus infinity and infinity, respectively (we will use the same notation as in [460]): ′ B+,0 (X) := {Q ∈ 𝒟L′ 1 (X) ; lim ⟨Qh , φ⟩ = 0,

φ ∈ 𝒟}

B+′ (X) := {Q ∈ 𝒟L′ 1 (X) ;

φ ∈ 𝒟},

h→+∞

and lim ⟨Qh , φ⟩ = 0,

|h|→+∞

3.5 Multi-dimensional ρ-almost periodic type distributions

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195

where ⟨Qh , φ⟩ := ⟨Q, φ(⋅ − h)⟩, Q ∈ 𝒟′ (X), h ∈ ℝ. Since for every fixed test function φ ∈ 𝒟 and for every real number h ∈ ℝ we have ̌ + h)⟩, ⟨Q,̌ φ(⋅ − h)⟩ = ⟨Q, φ(− ⋅ −h)⟩ = ⟨Q, φ(⋅

(130)

′ ′ we know that Q ∈ B+′ (X) if and only if Q ∈ B+,0 (X) and Q̌ ∈ B+,0 (X). ′ ′ The spaces B+,0 (X) and B+ (X) are very particular cases of the following spaces of ′ vector-valued bounded distributions (take n = 1, 𝔻 = [0, ∞) for the space B+,0 (X), and ′ n = 1, 𝔻 = ℝ for the space B+ (X)):

Definition 3.5.10. Let 𝔻 ⊆ ℝn and let the set 𝔻 be unbounded. Define ′ B𝔻,0 (ℝn : X) := {Q ∈ 𝒟′ (ℝn : X) :

lim

|h|→+∞;h∈𝔻

⟨Qh , φ⟩ = 0,

φ ∈ 𝒟(ℝn )}

and ′ B𝔻,0,b (ℝn : X) := {Q ∈ 𝒟L′ 1 (ℝn : X) :

lim

|h|→+∞;h∈𝔻

⟨Qh , φ⟩ = 0,

φ ∈ 𝒟(ℝn )}.

′ ′ It is clear that B𝔻,0,b (ℝn : X) ⊆ B𝔻,0 (ℝn : X) as well as that the converse statement is not true in general (for example, the regular distribution determined by the continuous ′ function f (t) = t, t ⩽ 0 and f (t) = 0, t ⩾ 0 belongs to the space B𝔻,0 (ℝ : X) but not to ′ the space B+,0 (X), with 𝔻 = [0, ∞), n = 1 and X = ℂ). But, if 𝔻 = ℝn , then we have the ′ ′ ′ ′ equality B𝔻,0,b (ℝn : X) = B𝔻,0 (ℝn : X); the converse inclusion B𝔻,0 (ℝn : X) ⊆ B𝔻,0,b (ℝn : X) follows from Theorem 3.5.8(i). The use of (130) simply implies the following:

Proposition 3.5.11. Suppose that (a1 , a2 , . . . , an ) ∈ ℝn ∖ {0}, 𝔻+ := {t = (t1 , t2 , . . . , tn ) ∈ ℝn : a1 t1 + a2 t2 + ⋅ ⋅ ⋅ + an tn ⩾ 0} and 𝔻− := {t = (t1 , t2 , . . . , tn ) ∈ ℝn : a1 t1 + a2 t2 + ⋅ ⋅ ⋅ + an tn ⩽ ′ n ′ n ′ 0}. Then Q ∈ Bℝ : X) [Q ∈ Bℝ : X)] if and only if Q ∈ B𝔻 (ℝn : X) n ,0 (ℝ n ,0,b (ℝ + ,0 [Q ∈ B′ (ℝn : X)] and Q̌ ∈ B′ (ℝn : X) [Q̌ ∈ B′ (ℝn : X)]. 𝔻− ,0

𝔻+ ,0,b

𝔻− ,0,b

Keeping in mind the proofs of Theorem 3.5.7, Theorem 3.5.8 and [219, Proposition 1], we are in a position to clarify the following extension of [460, Proposition 1]: Proposition 3.5.12. Suppose that Q ∈ 𝒟L′ 1 (ℝn : X). Then the following statements are equivalent: ′ n (i) Q ∈ Bℝ n ,0 (ℝ : X). (ii) We have Q ∗ φ ∈ C0 (ℝn : X) for all φ ∈ 𝒟(ℝn ). (iii) There exist an integer k ∈ ℕ and functions Fj ∈ C0 (ℝn : X) (0 ⩽ j ⩽ k) such that Q = ∑kj=0 Fj

(αj )

for some multi-indices αj ∈ ℕn0 (0 ⩽ j ⩽ k).

(iv) There exists a sequence (Qk ) in ℰ ′ (ℝn : X), which converges to Q for topology of 𝒟L′ 1 (ℝn : X).

196 � 3 Multi-dimensional ρ-almost periodic type functions Remark 3.5.13. (i) The equivalence of statements (i) and (ii) in Proposition 3.5.12 enables one to introduce many other spaces of vector-valued distributions vanishing at infinity along the set 𝔻. For example, we can use the spaces of weighted ergodic components in ℝn in place of the usually considered space C0 (ℝn : X). See [431] for more details as well as the approach obeyed by B. Basit and H. Güenzler in [92] and [428, Subsection 2.15.1]. (ii) It is worth noting that it is very difficult to state a satisfactory analogue of Proposition 3.5.12 in the case that 𝔻 ≠ ℝn . Here we will only note the following: Suppose that Q ∈ 𝒟L′ 1 (ℝn : X) and consider the following statements: ′ (a) Q ∈ B𝔻,0 (ℝn : X). (b) We have Q ∗ φ ∈ C0,𝔻 (ℝn : X) for all φ ∈ 𝒟(ℝn ). (c) We have Q ∗ φ ∈ C0,𝔻 (ℝn : X) for all φ ∈ 𝒟(ℝn ) such that 𝔻 − supp(φ) ⊆ 𝔻. (d) There exist an integer k ∈ ℕ and functions Fj ∈ C0,𝔻 (ℝn : X) (0 ⩽ j ⩽ k) such that Q = ∑kj=0 Fj j for some multi-indices αj ∈ ℕn0 (0 ⩽ j ⩽ k). It is clear that the limit function of any sequence (Fk )k∈ℕ of functions from C0,𝔻 (ℝn : X) which converges uniformly on the whole space ℝn also belongs to the space C0,𝔻 (ℝn : X); furthermore, for every φ ∈ 𝒟(ℝn ), we have (α )

̌ ⟨Q,̌ φ(⋅ − h)⟩ = (Q ∗ φ)(h),

h ∈ ℝn .

Keeping in mind these observations, the above argumentation shows that (a) is equivalent with (b) and implies (c) and (d), while (d) implies only (c). Furthermore, if Ω is a non-empty subset of ℝn , then we set ′ B𝔻,0,b,Ω (ℝn : X) := {Q ∈ 𝒟L′ 1 (ℝn : X) :

lim

|h|→+∞;h∈𝔻

⟨Qh , φ⟩ = 0,

φ ∈ 𝒟Ω (ℝn )}.

′ Then it can be simply shown that Q ∈ B𝔻,0,b,Ω (ℝn : X) if and only if Q ∗ φ ∈ C0,𝔻 (ℝn : X) for all φ ∈ 𝒟−Ω (ℝn ).

The conclusion established directly after the supremum formula [441, (2.4)] clarifies that any (I ′ , c)-uniformly recurrent function F(⋅) is identically equal to zero provided that lim|t|→+∞ F(t) = 0. Keeping in mind this statement and Proposition 3.5.12, we can simply deduce the following simple result: Proposition 3.5.14. Let 0 ≠ I ′ ⊆ ℝn , c ∈ ℂ ∖ {0}, let T be an (I ′ , c)-almost periodic ((I ′ , c)′ n uniformly recurrent) distribution and let T ≠ 0. Then T ∉ Bℝ n ,0 (ℝ : X). We need the following notion: Definition 3.5.15. Suppose that T ∈ 𝒟′ (ℝn : X) and 0 ≠ Ω ⊆ ℝn . (i) We say that T is a 𝔻-asymptotically (I ′ , ρ)-almost periodic [𝔻-asymptotically (I ′ , ρ)-uniformly recurrent] distribution if and only if the function (T ∗ φ)(⋅) is 𝔻-asymptotically (I ′ , ρ)-almost periodic [𝔻-asymptotically (I ′ , ρ)-uniformly recurrent] for all φ ∈ 𝒟(ℝn ).

3.5 Multi-dimensional ρ-almost periodic type distributions

� 197

(ii) We say that T is a 𝔻-asymptotically (𝒟Ω , I ′ , ρ)-almost periodic [𝔻-asymptotically (𝒟Ω , I ′ , ρ)-uniformly recurrent] distribution if and only if the function (T ∗ φ)(⋅) is 𝔻-asymptotically (I ′ , ρ)-almost periodic [𝔻-asymptotically (I ′ , ρ)-uniformly recurrent] for all φ ∈ 𝒟Ω (ℝn ). It is clear that any 𝔻-asymptotically (I ′ , ρ)-almost periodic [𝔻-asymptotically (I ′ , ρ)uniformly recurrent] distribution is a 𝔻-asymptotically (𝒟Ω , I ′ , ρ)-almost periodic [𝔻asymptotically (𝒟Ω , I ′ , ρ)-uniformly recurrent] distribution for any non-empty subset Ω of ℝn . The converse inclusion can be deduced in the following situation: Proposition 3.5.16. Suppose that 𝔻 = ℝn and for every compact set K there exists a point a ∈ ℝn such that a + K ⊆ Ω. Then any 𝔻-asymptotically (𝒟Ω , I ′ , ρ)-almost periodic [𝔻asymptotically (𝒟Ω , I ′ , ρ)-uniformly recurrent] distribution is a 𝔻-asymptotically (I ′ , ρ)almost periodic [𝔻-asymptotically (I ′ , ρ)-uniformly recurrent] distribution. Proof. Let φ ∈ 𝒟(ℝn ) be fixed and K = supp(φ). Then there exists a point a ∈ ℝn such that a + K ⊆ Ω; moreover, since the convolution mapping is translation invariant, we have (T ∗ φ)a = (T ∗ φ)(⋅ + a). It is clear that supp(φ(⋅ + a)) ⊆ Ω, and therefore we have the existence of a Bohr (I ′ , ρ)-almost periodic [(I ′ , ρ)-uniformly recurrent] function F(⋅) and a function Q ∈ C0 (ℝn : X) such that (T ∗ φ)a (t) = F(t) + Q(t), t ∈ ℝn . This implies (T ∗ φ)(t) = F(t − a) + Q(t − a), t ∈ ℝn . Obviously, Q(⋅ − a) ∈ C0 (ℝn : X); moreover, the function F(⋅ − a) is Bohr (I ′ , ρ)-almost periodic [(I ′ , ρ)-uniformly recurrent], which simply implies the required. Now we will introduce the following notion, which substantially generalizes the notion introduced in [460, Definition 2]: Definition 3.5.17. Suppose that T ∈ 𝒟′ (ℝn : X) and 0 ≠ Ω ⊆ ℝn . (i) We say that T is a 𝔻-asymptotically (I ′ , ρ)-almost periodic distribution of type 1 [𝔻-asymptotically (I ′ , ρ)-uniformly recurrent distribution of type 1] if and only if there exist an (I ′ , ρ)-almost periodic ((I ′ , ρ)-uniformly recurrent) distribution Tap ∈ ′ API ′ ,ρ (ℝn : X) [Tur ∈ URI ′ ,ρ (ℝn : X)] and a distribution Q ∈ B𝔻,0,b (ℝn : X) such that ⟨T, φ⟩ = ⟨Tap , φ⟩ + ⟨Q, φ⟩, φ ∈ 𝒟(ℝn ) [⟨T, φ⟩ = ⟨Tur , φ⟩ + ⟨Q, φ⟩, φ ∈ 𝒟(ℝn )]. (ii) We say that T is a 𝔻-asymptotically (𝒟Ω , I ′ , ρ)-almost periodic distribution of type 1 [𝔻-asymptotically (𝒟Ω , I ′ , ρ)-uniformly recurrent distribution of type 1] if and only if there exist an (I ′ , ρ)-almost periodic ((I ′ , ρ)-uniformly recurrent) distribution ′ Tap ∈ API ′ ,ρ (ℝn : X) [Tur ∈ URI ′ ,ρ (ℝn : X)] and a distribution Q ∈ B𝔻,0,b (ℝn : X) such that ⟨T, φ⟩ = ⟨Tap , φ⟩ + ⟨Q, φ⟩, φ ∈ 𝒟Ω (ℝn ) [⟨T, φ⟩ = ⟨Tur , φ⟩ + ⟨Q, φ⟩, φ ∈ 𝒟Ω (ℝn )]. ′ Remark 3.5.18. (i) For simplicity, we will not use the space B𝔻,0 (ℝn : X) here. (ii) We note that the decompositions in Definition 3.5.17(i) are unique in the case that I ′ = 𝔻 = ℝn and ρ = cI for some c ∈ ℂ ∖ {0}; see Proposition 3.5.14.

198 � 3 Multi-dimensional ρ-almost periodic type functions Concerning the statements considered in [460, Theorem 11, Corollary 2], we will make the following observations in the multi-dimensional setting: (A1) Suppose that 𝔻 ⊆ ℝn , the set 𝔻 is unbounded, ρ : X → X is a uniformly continuous function, I ′ − I ′ = ℝn , for each a > 0 we have (127) and T is a 𝔻-asymptotically (ℝn , ρ)-almost periodic distribution. Since the limit function of any sequence (Fk )k∈ℕ of 𝔻-asymptotically (ℝn , ρ)-almost periodic functions which converges uniformly on ℝn is likewise 𝔻-asymptotically (ℝn , ρ)-almost periodic due to Theorem 3.5.1 and Corollary 3.5.3, the argumentation contained in the proof of Theorem 3.5.8 shows that there exist an integer p ∈ ℕ and a bounded 𝔻-asymptotically (I ′ , ρ)-almost periodic function F : ℝn → X such that (129) holds in the distributional sense. (A2) Suppose that there exist an integer p ∈ ℕ and a bounded 𝔻-asymptotically (I ′ , ρ)almost periodic [𝔻-asymptotically (I ′ , ρ)-uniformly recurrent] function F : ℝn → X such that (129) holds in the distributional sense (III). Then (129) holds in the distributional sense on 𝒟Ω for any non-empty subset Ω of ℝn (III)’. Furthermore, we have the following: Suppose that ρ = A ∈ L(X). Then there exist an integer k ∈ ℕ and bounded 𝔻-asymptotically (I ′ , A)-almost periodic (bounded 𝔻-asymptotically (I ′ , A)-uniformly recurrent) functions Fj : ℝn → X (0 ⩽ j ⩽ k) such that the function F(⋅), given by (128), is (I ′ , Ak+1 )-almost periodic ((I ′ , Ak+1 )-uniformly recurrent), where Ak+1 ∈ L(X k+1 ) is given by Ak+1 (x0 , x1 , . . . , xk ) := (Ax0 , Ax1 , . . . , Axk ), (x0 , x1 , . . . , xk ) ∈ X k+1 ; furthermore, there exists certain multi-indices αj ∈ ℕn0 (0 ⩽ j ⩽ k) such that T = ∑kj=0 Fj j in the distributional sense (IV) (in the distributional sense on 𝒟Ω for any non-empty subset Ω of ℝn (IV)’). (A3) If (IV) [(IV)’] holds, then T is a 𝔻′ -asymptotically (I ′ , ρ)-almost periodic distribution of type 1 (𝔻′ -asymptotically (I ′ , ρ)-uniformly recurrent distribution of type 1) [𝔻′ asymptotically (𝒟Ω , I ′ , ρ)-almost periodic distribution of type 1 (𝔻′ -asymptotically (𝒟Ω , I ′ , ρ)-uniformly recurrent distribution of type 1)], provided that the following condition holds: (C) 𝔻′ is an unbounded subset of ℝn and for each compact set K in ℝn there exists M > 0 such that, for every h ∈ 𝔻′ ∖ B(0, M), we have h + K ⊆ 𝔻. (A4) If T is a 𝔻-asymptotically (𝒟Ω , I ′ , ρ)-almost periodic distribution of type 1 [𝔻asymptotically (𝒟Ω , I ′ , ρ)-uniformly recurrent distribution of type 1], then T is a 𝔻-asymptotically (𝒟−Ω , I ′ , ρ)-almost periodic [𝔻-asymptotically (𝒟−Ω , I ′ , ρ)-uniformly recurrent] distribution. (A5) Let ρ = A ∈ L(X) and let there exist a sequence of 𝔻-asymptotically (I ′ , A)-almost periodic [𝔻-asymptotically (I ′ , A)-almost periodic] functions, which converges to T for topology of 𝒟L′ 1 (ℝn : X). Suppose, further, that the following condition holds: (C)Ω 𝔻′ is an unbounded subset of ℝn , (126) holds with the set 𝔻 replaced therein with the set 𝔻′ , and for each compact set K in ℝn with K ⊆ Ω there exists M > 0 such that, for every h ∈ 𝔻′ ∖ B(0, M), we have h + K ⊆ 𝔻. (α )

3.5 Multi-dimensional ρ-almost periodic type distributions

�

199

Using Corollary 3.5.3, [304, Theorem 2.14] as well as the proofs of [460, Theorem 11] and [156, Proposition 7], it readily follows that T is a 𝔻′ -asymptotically (𝒟Ω , I ′ , ρ)almost periodic [𝔻′ -asymptotically (𝒟Ω , I ′ , ρ)-uniformly recurrent] distribution. (A6) If T is a 𝔻-asymptotically (𝒟Ω , I ′ , ρ)-almost periodic [𝔻-asymptotically (𝒟Ω , I ′ , ρ)uniformly recurrent] distribution and 0 ∈ Ω∘ , then there exists a sequence of 𝔻-asymptotically (I ′ , A)-almost periodic [𝔻-asymptotically (I ′ , A)-uniformly recurrent] functions which converges to T for topology of 𝒟L′ 1 (ℝn : X). This essentially follows from the proof of [156, Proposition 7].

3.5.3 Some applications In [460, Section 3], we have recently analyzed the existence of half-asymptotically c-almost periodic (half-asymptotically semi-c-periodic) solutions of the equation T ′ = AT + G,

T ∈ 𝒟′ (X k ) on [0, ∞),

where G is a half-asymptotically c-almost periodic (half-asymptotically semi-c-periodic) X k -valued distribution, k ∈ ℕ, and A = [aij ]1⩽i,j⩽k is a given complex matrix such that σ(A) ⊆ {z ∈ ℂ : Re z < 0}; in such a way, we have slightly expanded the research study [152] by C. Bouzar and M. T. Khalladi. The main aim of this subsection is to provide some new applications of vector-valued ρ-almost periodic type distributions; we will primarily deal with the regular distributions here. 1. Let a > 0; then a unique regular solution of the wave equation utt = a2 uxx in domain {(x, t) : x ∈ ℝ, t > 0}, equipped with the initial conditions u(x, 0) = f (x) ∈ C 2 (ℝ) and ut (x, 0) = g(x) ∈ C 1 (ℝ), is given by the d’Alembert formula (85). If the functions f (⋅) ⋅ and g [1] (⋅) ≡ ∫0 g(s) ds are almost periodic, then we know that the solution u(x, t) can be extended to the whole real line in time variable and that the solution u(x, t) is almost periodic in ℝ2 ; furthermore, if the functions f (⋅) and g(⋅) are almost periodic, then we know that the solution x 󳨃→ u(x, t0 ), x ∈ ℝ is almost periodic for every fixed time t0 ∈ ℝ. It is important to know what happens if the initial values f (⋅) and g [1] (⋅) [f (⋅) and g(⋅)] are regular almost periodic distributions? Without going into full details, we will consider this issue only in the case that a = 1 and g ≡ 0. First of all, we recall that the function f0 (t) = sin(

1 ), 2 + cos t + cos(√2t)

t∈ℝ

is Stepanov p-almost periodic, infinitely differentiable, bounded and not almost periodic (1 ⩽ p < ∞). Let f (t) := f0′ (t), t ∈ ℝ; then we will prove later that the function f (⋅) is neither Stepanov-p-bounded nor Stepanov-p-almost periodic, for any finite exponent p ⩾ 1. On the other hand, due to Proposition 3.5.9 and the fact that the space of almost periodic distributions is closed under differentiation, the regular distribution T induced

200 � 3 Multi-dimensional ρ-almost periodic type functions by f (⋅) is almost periodic (this example also shows that the converse statement in Proposition 3.5.9 does not hold in general). Since the space of almost periodic distributions is a translation invariant vector space with the usual operations, it readily follows that the solution x 󳨃→ u(x, t0 ), x ∈ ℝ is infinitely differentiable in the usual sense and determines a regular almost periodic distribution for every fixed time t0 ∈ ℝ. On the other hand, it is clear that the solution (x, t) 󳨃→ u(x, t), (x, t) ∈ ℝ2 is infinitely differentiable as well as that the regular distribution T determined by this function is bounded, since for each φ ∈ 𝒟(ℝ2 ) we have that the function (T ∗ φ)(⋅; ⋅) is bounded, as easily approved. Now we will prove that the infinitely differentiable function (T ∗ φ)(⋅; ⋅) is almost periodic so that T is almost periodic distribution in ℝ2 . Let ε > 0 be given; then there exists a finite real constant k > 0 such that for every (v, w) ∈ ℝ2 and τ = (τ1 , τ2 ) ∈ ℝ2 , we have: 󵄨󵄨 󵄨 󵄨󵄨(T ∗ φ)(v + τ1 , w + τ2 ) − (T ∗ φ)(v, w)󵄨󵄨󵄨 󵄨k k 󵄨󵄨 󵄨󵄨 1 󵄨󵄨󵄨󵄨 = 󵄨󵄨 ∫ ∫ [f (v + τ1 − x + w + τ2 − t) − f (v − x + w − t)]φ(x, t) dt dx 󵄨󵄨󵄨 󵄨 󵄨󵄨 2 󵄨󵄨 󵄨 −k −k

󵄨k k 󵄨󵄨 󵄨󵄨 1 󵄨󵄨󵄨󵄨 + 󵄨󵄨 ∫ ∫ [f (v + τ1 − x − w − τ2 + t) − f (v − x − w + t)]φ(x, t) dt dx 󵄨󵄨󵄨 󵄨󵄨 2 󵄨󵄨󵄨 󵄨 −k −k

k k

⩽

1 󵄨 󵄨 󵄨 󵄨 ∫ ∫ 󵄨󵄨󵄨 f0 (v + τ1 − x + w + τ2 − t) − f0 (v − x + w − t)󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨φt (x, t)󵄨󵄨󵄨 dt dx 2 −k −k

k k

+

1 󵄨 󵄨 󵄨 󵄨 ∫ ∫ 󵄨󵄨󵄨 f0 (v + τ1 − x − w − τ2 + t) − f0 (v − x − w + t)󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨φt (x, t)󵄨󵄨󵄨 dt dx 2 −k −k

k k

⩽

‖φt ‖∞ 󵄨 󵄨 ∫ ∫ 󵄨󵄨󵄨 f0 (v + τ1 − x + w + τ2 − t) − f0 (v − x + w − t)󵄨󵄨󵄨 dt dx 2 −k −k

k k

+

‖φt ‖∞ 󵄨 󵄨 ∫ ∫ 󵄨󵄨󵄨 f0 (v + τ1 − x − w − τ2 + t) − f0 (v − x − w + t)󵄨󵄨󵄨 dt dx. 2 −k −k

The set of all points τ = (τ1 , τ2 ) such that τ1 ± τ2 are ε-periods of the function f0̂ (⋅) is relatively dense in ℝ2 ; see [194, Example 1.2] for more details. For such a point τ = (τ1 , τ2 ), the above computation shows that 󵄨󵄨 󵄨 2 󵄨󵄨(T ∗ φ)(v + τ1 , w + τ2 ) − (T ∗ φ)(v, w)󵄨󵄨󵄨 ⩽ 2εk ‖φt ‖∞ ,

(v, w) ∈ ℝ2 ,

which simply implies the required. 2. Our analyses from [194, Example 1.1] and [441, Example 1.1] can be also carried out for the regular c-almost periodic distributions, where c ∈ ℂ ∖ {0}. Let t0 > 0 be fixed. We will use the same notation as in the above-mentioned examples; suppose now that the

3.6 Notes and appendices

201

�

regular distribution determined by the function F(t, x) is c-almost periodic with respect to the variable x ∈ ℝn , uniformly in the variable t on compact subsets of [0, ∞); i. e. for every φ ∈ 𝒟(ℝn ), for every compact set K ⊆ [0, ∞) and for every ε > 0, there exists l > 0 such that, for every t0 ∈ ℝn , there exists τ ∈ B(t0 , l) such that 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∫ [F(t, x + τ − y) − cF(t, x − y)]φ(y) dy󵄨󵄨󵄨 < ε, 󵄨󵄨 󵄨󵄨 󵄨󵄨 n 󵄨󵄨 ℝ

x ∈ ℝn , t ∈ K.

Then the regular distribution determined by the function ut0 (⋅) is likewise c-almost pet

riodic, which follows from the next computation with the function H(x) = ∫00 ∫ℝn F(t0 − s, x − y)E(t0 , y) dy ds, x ∈ ℝn and the fixed test function φ ∈ 𝒟(ℝn ): 󵄨󵄨 󵄨 󵄨󵄨(H ∗ φ)(x + τ) − c(H ∗ φ)(x)󵄨󵄨󵄨 t 󵄨󵄨 󵄨󵄨󵄨 0 󵄨󵄨 󵄨󵄨 ⩽ 󵄨󵄨 ∫ ∫ ∫ [F(t0 − s, x + τ − y − z) − cF(t0 − s, x − y − z)]φ(z)E(t0 , y) dy ds dz󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨ℝn 0 ℝn 󵄨 󵄨󵄨 t0 󵄨󵄨 󵄨󵄨 󵄨󵄨 = 󵄨󵄨󵄨∫ ∫ ∫ [F(t0 − s, x + τ − y − z) − cF(t0 − s, x − y − z)]φ(z)E(t0 , y) dz dy ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 0 ℝn ℝn 󵄨 t0

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 ⩽ ∫ ∫ 󵄨󵄨 ∫ [F(t0 − s, x + τ − y − z) − cF(t0 − s, x − y − z)]φ(z) dz󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨E(t0 , y)󵄨󵄨󵄨 dy ds 󵄨󵄨 󵄨󵄨 󵄨 0 ℝn 󵄨ℝn t0

󵄨 󵄨 ⩽ ε ∫ ∫ 󵄨󵄨󵄨E(t0 , y)󵄨󵄨󵄨 dy ds,

x ∈ ℝn .

0 ℝn

We close this section with the observation that we will not introduce and analyze the Bohr–Fourier coefficients of the multi-dimensional ρ-almost periodic type distributions here; see [428] and [717] for some results obtained in this direction.

3.6 Notes and appendices Traveling fronts and almost periodicity The homogeneous reaction-diffusion equation ut −Δu = u(1−u) has been investigated for the first time in the pioneering articles of R. A. Fisher [319] and A. N. Kolmogorov, I. G. Petrovsky, N. S. Piskunov [421]. This equation has been generalized in many different ways. For example, G. Nadin has explored the existence of pulsating traveling fronts for the equation ut − ∇ ⋅ (A(t, x)∇u) + q(t, x)∇u = f (t, x, u),

(131)

202 � 3 Multi-dimensional ρ-almost periodic type functions where the diffusion matrix A, the advection term q and the reaction term f are periodic in the time variable t and the space variable x, in a certain sense; this equation has an invaluable importance in population genetics, combustion, and population dynamics models [567]. The author has proved the existence of speeds c∗ and c∗∗ such that there exists a pulsating traveling front of speed c for all c ⩾ c∗∗ and that there exists no such front of speed c < c∗ . We also give some spreading properties for front-like initial data. In some particular cases, we have c∗ = c∗∗ and this speed can be profiled using a family of eigenvalues associated with the equation. Concerning the homogeneous counterpart of (131), we would like to note that some solutions are planar fronts, i. e., the homogeneous equation (131) has solutions of the form (t, x) = U(x ⋅ e + ct), where e is a unit vector in ℝn and c is the speed of propagation in the direction −e; see the paper [61] by D. G. Aronson and H. F. Weinberger for more details. In the existing literature, the reader may find many various attempts to define satisfactorily the notion of a front of the equation (131); for example, we can define a front of (131) as a solution of (131) which has the form u(t, x) = ϕ(x ⋅ e + ct, t, x), where (z, t, x) 󳨃→ ϕ(z, t, x) is periodic in t and x, converges to 1 as z → +∞ and to 0 as z → −∞. In [567, Definition 1.2, Definition 1.3], G. Nadin has proposed the following notion: Definition 3.6.1. Suppose that the equation (131) admits two space-time periodic solutions p− and p+ such that p− (t, x) < p+ (t, x) for all (t, x) ∈ ℝ × ℝn . (i) A function u is a pulsating traveling front of speed c in the direction −e that connects p− to p+ if and only if u(t, x) = ϕ(x ⋅ e + ct, t, x), where ϕ ∈ L∞ (ℝ × ℝ × ℝn ), for almost every y ∈ ℝ the function (t, x) 󳨃→ ϕ(y + x ⋅ e + ct, t, x) satisfies (131), the function ϕ is periodic in its second and third variables and satisfies: lim ϕ(z, t, x) = p− (t, x),

z→−∞

and

lim ϕ(z, t, x) = p+ (t, x),

z→+∞

(132)

uniformly in (t, x) ∈ ℝ × ℝn . (ii) A function u is a Lipschitz continuous pulsating traveling front of speed c in the direction −e that connects p− to p+ if and only if u(t, x) = ϕ(x ⋅ e + ct, t, x), where ϕ ∈ W 1,∞ (ℝ × ℝ × ℝn ), for almost every y ∈ ℝ the function (t, x) 󳨃→ ϕ(y + x ⋅ e + ct, t, x) satisfies (131), the function ϕ is periodic in its second and third variables and satisfies (132), uniformly in (t, x) ∈ ℝ × ℝn . As already mentioned, the existence of (Lipschitz continuous) pulsating traveling fronts of speed c in the direction −e is the main objective of paper [567]. The main results of this paper are Theorem 2.1, Theorem 2.3–2.4, Theorem 2.6 and Theorem 2.8. For example, in [567, Theorem 2.1(1)], the author has proved that under certain conditions, for every unit vector e, there exists a minimal speed c∗ such that for all speed c ⩾ c∗ , there exists a pulsating traveling front u of speed c in direction −e that links 0 to p. The existence of positive periodic solutions of the equation (131) has been proved in [568, Theorem 1.1] under certain assumptions; cf. also [568, Corollary 1.4, Theorem 1.6].

3.6 Notes and appendices

� 203

Mention should be also made of the important research papers by H. Berestycki and his co-authors [106, 107, 108, 109, 110, 111, 112, 113], the research monographs [367] by P. Hess, which concerns periodic-parabolic boundary value problems and positivity, [687] by N. Shigesada and K. Kawasaki, which considers some applications in the mathematical biology, as well as the research articles [282] by R. Ducasse, L. Rossi and [332] by J. Garnier, T. Giletti, G. Nadin. Regarding the research articles [682, 683] by W. Shen, we will first recall the following notion of an almost periodic traveling wave solution of the reaction-diffusion equation: ut (x, t) = D(t)uxx (x, t) + f (u(x, t), t),

(x, t) ∈ ℝ × ℝ,

(133)

where D(t) ⩾ D0 > 0 is an almost periodic function, and f (u, t) is two times continuously differentiable function that is almost periodic in t uniformly with respect to u in bounded sets. Definition 3.6.2. By an almost periodic traveling wave solution of (133), we mean any classical solution of this equation which can be expressed as u(x, t) = U(x + c(t), t),

(x, t) ∈ ℝ × ℝ,

where the function U(x, t) is continuously differentiable and almost periodic in t uniformly with respect to x in bounded sets (this function is usually called wave profile), while the function c : ℝ → ℝ is continuously differentiable and satisfies that its first derivative is almost periodic. An almost periodic traveling wave solution u(x, t) is called almost periodic standing wave solution if the function c(t) is bounded. The usual definition of an almost periodic traveling wave solution of (133) is obtained by plugging that the function c(t) is linear: c(t) = ct, t ∈ ℝ, where c ∈ ℝ ∖ {0}. But as explained in [682], this definition is not sufficiently general; consider, for example, the equation ut = uxx + u(u − a)(u − 1),

(134)

where a ∈ (0, 1/2). Then we know that there exists a continuously differentiable function ϕ : ℝ → ℝ such that ϕ(−∞) = 0, ϕ(+∞) = +1 and that the function u(x, t) = ϕ(x + c0 t), where c0 ≠ 0, is a traveling wave solution of (134). But, then the function u(x, t) = ϕ(x + t c0 ∫0 h(s) ds) is an almost periodic traveling wave solution of the equation ut = h(t)uxx + h(t)u(u − a)(u − 1), provided that h(t) ⩾ h0 > 0 is a smooth almost periodic function satisfying that the t function h0 t − ∫0 h(s) ds is unbounded, where h0 is the mean value of h(t).

204 � 3 Multi-dimensional ρ-almost periodic type functions The notion of an almost periodic traveling wave solution which connects two stable almost periodic solutions and the notion of a wave-like solution of (133) is introduced in [682, Definition 4.1, Definition 4.2], respectively. The author has proved, in [682, Theorem 4.1, Theorem 4.2], that any almost periodic traveling wave solution u(x, t) = U(x + c(t), t) of (133) has the property that Ux (x, t) > 0 as well as that any almost periodic traveling wave solution of (133) is wave-like. Decay of almost periodic solutions of conservation laws In this part, we will present the most important ideas and research results obtained by H. Frid in [327]. The main objective of this paper is the asymptotic behaviour of solutions of systems of inviscid or viscous conservation laws in one or several space variables; the author has considered the almost periodicity in the generalized sense of Stepanov and Wiener approaches. More precisely, let us consider the equation d

ut + ∑ (fk (u))x = ∑(akl (u))x x , k=1

k

kl

k l

x ∈ ℝn , t > 0,

(135)

where fk , akl are sufficiently smooth functions, accompanied with the initial condition u(0, x) = u0 (x). The notion of an entropy solution of (135) has been introduced in [327, Definition 3]. The main results of paper are Theorem 1 and Theorem 2, where the author particularly proves the following: Suppose that the solution u(x, t) of problem (135), accompanied with the initial condition u(0, x) = u0 (x), is Stepanov almost periodic in the space variable x ∈ ℝn , locally uniformly in t ⩾ 0. Let lε (t) denote an inclusion interval of u(x, t) with respect to ε > 0, and let uT (x, t) ≡ u(Tx, Tt) for T > 0. If lε (t)/t → 0 as t → +∞, and uT (x, t) is pre-compact in L1loc ([0, ∞) × ℝn ) as t → +∞, then uT tends to the mean value u of u0 as T → +∞, and T

1 󵄨 󵄨 lim ∫ Mx (󵄨󵄨󵄨u(x, t) − u󵄨󵄨󵄨) dt = 0. T→+∞ T 0

With the exception of applications of multi-dimensional Stepanov almost periodic functions presented in our recent research monograph [431], this is a rare paper, which contains some applications of Stepanov almost periodic functions in the multi-dimensional setting. Let us also note that the author provides some applications to a class of inviscid systems in chromatography and isentropic gas dynamics, as well (see [327, Section 4, Section 5]). Concerning behaviour of periodic solutions of viscous conservation laws under certain kinds of perturbations, we would like to recommend the research articles [390, 391, 392, 393] by M. A. Johnson et al. See also [85] and [720].

3.6 Notes and appendices

� 205

The existence and uniqueness of almost periodic type solutions to the impulsive integro-differential equations The existence and uniqueness of the (piecewise continuous) almost periodic type solutions for various classes of impulsive integro-differential equations have been analyzed by numerous authors so far (see, e. g., the research monograph [699] by G. Tr. Stamov for a comprehensive survey of results). In [428, 431] and the previous part of this monograph, we have quoted many research monographs and articles concerning impulsive integro-differential equations and their almost periodic solutions; besides these references, we would like to mention here the important monographs [486] by V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, [655] by A. M. Samoilenko, N. A. Perestyuk, as well as the articles [203, 228, 309, 363, 371, 501, 519, 701, 746, 756, 757, 763]. We will mention here the following classes of impulsive integro-differential equations: 1. Of concern are the following systems of impulsive differential equations with impulsive fixed moments {

̇ = A(t)u(t) + f (t), t ≠ tk , u(t) Δu(tk ) = bk ,

k = ±1, ±2, . . .

(136)

and {

̇ = A(t)u(t) + F(t, u(t)), u(t) Δu(tk ) = Ik (u(tk )),

t ≠ tk ,

k = ±1, ±2, . . . ,

(137)

where A : ℝ → ℝn×n , f : ℝ → ℝn , bk ∈ ℝn , F : ℝ × Ω → ℝn and Ik : Ω → ℝn . The solution of (136) or (137), with respect to the initial condition u(t0+ ) = u0 , t0 ∈ ℝ, u0 ∈ ℝn will be denoted by u(t) = u(t; t0 , u0 ). The homogeneous corresponding version ̇ = A(t)u(t), t ≠ tk is said to be hyperbolic (see, e. g., of the upper systems, given by u(t) [371]) if and only if there exist real constants a, λ > 0 such that for each t ∈ ℝ there exist vector spaces M + (t) and M − (t), whose external direct sum is M + (t) ⊕ M − (t) = ℝn , such that if u0 ∈ M + (t0 ), then for all t ⩾ t0 the inequality 󵄩󵄩 󵄩 −λ(t−t0 ) , 󵄩󵄩u(t; t0 , u0 )󵄩󵄩󵄩 ⩽ a‖u0 ‖e holds, and if u0 ∈ M − (t0 ) then for all t ⩽ t0 the inequality 󵄩󵄩 󵄩 λ(t−t0 ) 󵄩󵄩u(t; t0 , u0 )󵄩󵄩󵄩 ⩽ a‖u0 ‖e holds. The existence and uniqueness of piecewise continuous almost periodic type solutions for the systems (136)–(137) will be considered somewhere else. Of concern is also the following class of impulsive differential equations with delay of Lasota–Wazewska type:

206 � 3 Multi-dimensional ρ-almost periodic type functions n

{ ̇ = −a(t)u(t) + ∑ bi (t)e−gi (t)u(t−h) , {u(t) i=1 { { Δu(t ) = a u(t ) + nk , k k k {

t ≠ tk , k = ±1, ±2, . . . ,

where a(t), b(t), gi (t) are continuous functions defined on the real line, h is a positive real number and ak , nk ∈ ℝ, k = ±1, ±2, . . . . The special case of this model, considered for the first time by M. WazewskaCzyzewska and A. Lasota [766], is that case in which we have that ai (t), bi (t), gi (t), i = 1, 2, . . . , n are positive real constants: ̇ = −au(t) + be−gu(t−h) . u(t) This mathematical model is important in the investigation of the development and survival of the red corpuscles in the living organisms. The analysis is continued by the work of M. R. S. Kulenović and G. Ladas [476] for the equation n

̇ = −au(t) + ∑ bi e−gi u(t−h) , u(t) i=1

where a, bi , gi ⩾ 0. 2. In this issue, we will present the ideas and main results about Favard’s theorem for piecewise continuous almost periodic functions, established in the research article [763] by L. Wang and M. Yu. First of all, the authors have constructed Bochner–Fejér polynomials for piecewise continuous almost periodic functions, which served them to establish Favard’s theorem for piecewise continuous almost periodic functions. This result clarifies when the primitive function of a piecewise continuous almost periodic function is again piecewise continuous almost periodic (see also Chapter 9 of Part III). Using some results from coincidence degree theory (see the monograph [330] by R. E. Gaines and J. L. Mawhin), the authors have examined the existence and uniqueness of piecewise almost periodic solutions for the following impulsive single population model with hereditary effects: N ′ (t) = N(t)[a(t) − b(t)N(t) − d(t)N(t − τ(t))],

t ≠ tk ; N(tk+ ) = (1 + ck )N(tk ).

Almost periodic solutions of Lagrangian systems In this issue, we will briefly describe the results about the existence of various types of almost periodic solutions of Lagrangian systems established by S. F. Zakharin and I. O. Parasyuk in [799]. In this paper, the authors have followed a variational method of finding the almost periodic type solutions to the Lagrangian system x ′′ (t) =

𝜕V (t, x) , 𝜕x

(138)

3.6 Notes and appendices

�

207

with the almost (quasi-) periodic in the time variable t force function V : ℝ × ℝn → ℝ. This method, which has been already known for some periodic systems [221, 531, 587, 634] and almost periodic systems [139, 140], consists in finding the extremals of the functional 1 󵄨 󵄨2 J0 (x(t)) = ℳ[󵄨󵄨󵄨x ′ 󵄨󵄨󵄨 + V (t, x(t))], 2 where ℳf denotes the usual mean value of a function f : ℝ → ℝ. The solving of minimization problem for the functional J0 is essentially based on an extension of functional J0 from the space C (1) (ℝ : ℝn ) of Bohr almost periodic functions with almost periodic derivatives to the space of Besicovitch almost periodic functions which possess generalized Besicovitch almost periodic derivatives. As emphasized by the authors, such spaces are important in the theory of non-linear oscillations of finite dimensional systems and infinite dimensional systems [115, 597, 654]. If r ∈ ℕ0 and 0 ≠ Ω ⊆ ℝn , than the authors have dealt with the space B2 (ℝ : C r (Ω)) consisting of those functions g : ℝ × Ω → ℝ which obey the following properties: (i) If t ∈ ℝ, then the function x 󳨃→ g(t, x), x ∈ ℝn belongs to the space C r (Ω : ℝ); (ii) For very multi-index α ∈ ℕn0 with |α| ⩽ r, the function 𝜕α g(t, x)/𝜕x α is Lebesgue measurable; (iii) The function t 󳨃→ (𝜕α g(t, x)/𝜕x α )1/2 , t ∈ ℝ is locally integrable and, for every ε > 0, there exists a trigonometric polynomial Pεα (t, x) in the time-variable t such that t 󵄨󵄨 𝜕α g(s, x) 󵄨󵄨2 1 󵄨 󵄨 lim sup ∫ max󵄨󵄨󵄨 − Pεα (s, x)󵄨󵄨󵄨 ds < ε. α 󵄨 󵄨󵄨 x∈Ω 2t 𝜕x t→+∞ 󵄨 −t

In [799, Section 3], the authors have analyzed Besicovitch quasiperiodic functions on tori and after that generalized and classical solutions of (138). Concerning the variational method of finding the almost periodic type solutions of partial differential equations, we also refer the reader to [29, 69, 70, 252, 635] and references cited therein. It is also worth mentioning the research report [627] by J. Puig, in which author has considered the problems of reducibility for linear equations with quasi-periodic coefficients. As emphasized therein, the motivation for this topic comes from the study of stability of quasi-periodic motions and preservation of invariant tori in Hamiltonian mechanics. Bloch wave approach to almost periodic homogenization and approximations of effective coefficients In this part, we will inscribe the main details and results established recently by S. S. Ganesh and V. Tewary in [331]. In this paper, the authors have considered periodic approximations of almost periodic functions, Bloch wave approach to almost periodic homogenization and corresponding approximations of effective coefficients.

208 � 3 Multi-dimensional ρ-almost periodic type functions The main objective is the following almost periodic second-order elliptic operator in divergence form: 𝒜u := −

𝜕 𝜕u (a (y) ), 𝜕yk kl 𝜕yk

where summation over repeated indices is assumed and the following holds: (i) The coefficients akl : ℝn → ℝ of the symmetric matrix A = [akl ]1⩽k,l⩽n are almost periodic. (ii) The matrix A is coercive, i. e., there exists a finite real constant α > 0 such that ⟨A(y)v, v⟩ ⩾ α⟨v, v⟩,

v ∈ ℝn , a. e. y ∈ ℝn .

If 0 ≠ Ω ⊆ ℝn is an open set, then the main problem under consideration is the homogenization of the following equation in Ω: 𝒜ε uε := −

𝜕u 𝜕 (aε (x) ε ) = f , 𝜕xk kl 𝜕xk

(139)

ε where f ∈ L2 (Ω), uε ∈ H 1 (Ω) and akl (x) ≡ akl (x/ε) for ε > 0. If the solutions uε converges 1 weakly to a limit u ∈ H (Ω), then it has been proved that u satisfies, in a certain sense, an equation of the form

𝒜 u := − ∗

𝜕 𝜕u ∗ (akl (x) ) = f, 𝜕xk 𝜕yk

∗ with precisely determined coefficients akl (x). The equation (139) cannot be so simply considered by the Bloch wave analysis since the coefficients of matrix A are not necessarily periodic: therefore, the authors have introduced first some periodic approximations to the coefficients akl (x). The first type of periodic approximation of a given almost periodic function F : ℝn → ℝ is simple and mostly used in the paper: for a sufficiently large real number R > 0, we set F R (t) := F(t) for all t ∈ [−Rπ, Rπ)n and after that we extend this function periodically to the whole space ℝn . It is well known that F R (⋅) need not converge to F(⋅) as R → +∞, in L∞ (ℝn ), as well as that F R (⋅) converges to F(⋅) as R → +∞, in L2loc (ℝn ). The authors have asked whether the sequence F R (⋅) converges to F(⋅) as R → +∞, in the Besicovitch space B2 (ℝn )? The second type of periodic approximations of an almost periodic function F : ℝn → ℝ can be attributed to M. A. Shubin [690] and it is based on the approximation of irrationals by rationals with the help of the famous Dirichlet’s approximation theorem. In this paper, the author has shown that there exists a strictly increasing sequence (Tk ) of positive real numbers tending to plus infinity and a sequence of YTk -periodic functions (Pk ) such that limk→+∞ ‖u − Pk ‖L∞ ([−Tk π,Tk π)n ) = 0. Moreover, (Pk ) can be chosen to be smooth trigonometric polynomials, but the sequence (Tk ) cannot be so simply determined.

3.6 Notes and appendices

� 209

After that, the authors analyze periodic approximations of almost periodic operators akl (x) and carry out the corresponding Bloch wave analysis. The authors essentially use the Bloch decomposition of space L2 (ℝn ) in their examinations, and work with the general Besicovitch space B2 (ℝn ); see [331, Subsection 4.1-Subsection 4.6] for more details. The main homogenization result is [331, Theorem 5.1]. See also [30, 60, 223, 600, 685, 694, 695] and references cited therein for more details concerning almost periodic type homogenizations. Periodic solutions to differential variational inequalities In this part, we will refer the reader to some research articles concerning differential variational inequalities and inscribe the main ideas and results of article [46] by N. T. V. Anh. As pointed out by the author of this paper, mechanical impact problems, electrical circuits with ideal diodes, and Coulomb friction problems for contacting bodies or economical dynamics, among many others phenomena from the real world, can be successfully described by differential variational inequalities. Although introduced already in 1984, the first systematic study of differential variational inequalities was conducted by J. S. Pang and D. E. Stewart in 2008 [596]. The study of differential variational inequalities with delay is recent and has some importance in modeling of the real world phenomena as well (let us also mention the research articles [504] by Y. Li, X. Huang and [505] by Y. Li, X. Huang, X. Wang, where the authors have recently analyzed Weyl almost periodic functions on time scales and Weyl almost periodic solutions of dynamic equations with delays, as well as Weyl almost periodic solutions for quaternion-valued shunting inhibitory cellular neural networks with time-varying delays). The main aim of paper [46] is to consider the periodic solutions of the following differential variational inequality with delay: x ′ (t) = A(t)x(t) + h(t, x(t), xt ) + B(t, x(t), xt )ℒu(t), ⟨v − u(t), F(t, x(t)) + G(t, u(t))⟩ ⩾ 0,

t > 0;

v ∈ K, a. e. t > 0,

where x(t) ∈ ℝn , u(t) ∈ K with K being a closed convex subset in ℝm , xt denotes the history of the state function up to time t, A(t) is a system matrix of format n × n whose entries are continuous functions of t, ℒ is a constant complex matrix of format m × m, h : [0, ∞) × ℝn × C([−τ, 0] : ℝn ) → ℝn , B : [0, ∞) × ℝn × C([−τ, 0] : ℝn ) → ℝn×m , F : [0, ∞) × ℝn → ℝm and G : K → ℝm satisfy certain extra assumptions. See also [45, 47, 67] and references cited therein. Some notes about Besicovitch almost periodic functions and Besicovitch–Stepanets almost periodic functions Bohr’s equivalence relation in the spaces of one-dimensional Besicovitch almost periodic functions have recently been investigated by J. M. Sepulcre and T. Vidal in [676]. It is also worth mentioning that A. S. Serdyuk and A. L. Shidlich have investigated the Fourier

210 � 3 Multi-dimensional ρ-almost periodic type functions analysis of one-dimensional Besicovitch–Stepanets functions in [678]; more precisely, the authors have examined the direct and inverse theorems on the approximation of almost periodic functions in these spaces. Let us recall here that the study of direct and inverse approximation theorems is initiated by D. Jackson in [381] and S. N. Bernstein [117]. In some papers from the existing literature, the Jackson-type inequalities have been established with moduli of smoothness of Besicovitch-2-almost periodic functions of arbitrary positive integer order and with generalized moduli of smoothness. In [678], the authors have analyzed the Besicovitch–Stepanets spaces B𝒮 p consisting of those Besicovitch-1-almost periodic functions which satisfy, that the sums of the p-th degrees of absolute values of their Fourier coefficients are finite, where 1 ⩽ p < +∞. The norm of a function in the space B𝒮 p is defined as the norm of a sequence of its Fourier coefficients in the space lp . Let us recall that A. I. Stepanets and his followers have considered the 2π-periodic Lebesgue summable functions in place of the Besicovitch-1-almost periodic functions, denoting such spaces by 𝒮 p . Concerning the multi-dimensional Stepanets spaces, we will only mention here that F. G. Abdullayev et el. [7] have investigated the exact constants in direct and inverse approximations for functions of several variables in the spaces 𝒮 p . Almost periodic solutions of ODEs In this part, we will remind the readers of several important research articles and results concerning the almost periodic type solutions for various classes of ODEs. 1. In connection with almost periodic type solutions of ODEs with delays, we will first note that R. Grafton have analyzed, in [338], a periodicity result for autonomous functional differential equation x ′ (t) = −[f (x(t − 1)) + f (x(t − 2)) + ⋅ ⋅ ⋅ + f (x(t − N + 1))],

t ∈ ℝ,

where F : ℝ → ℝ is a continuous odd function and N ∈ ℕ. This work has stimulated many ongoing research after that; here, we will only note that H. Chen, H. Tang, and J. Sun have investigated, in [201], periodic solutions for a class of second-order delay differential equations x ′′ (t) = f (x(t)) − [f (x(t − 1)) + f (x(t − 2)) + ⋅ ⋅ ⋅ + f (x(t − N + 1))],

t ∈ ℝ,

with the help of the critical point theory and S 1 index theory. See also the research articles by J. L. Kaplan and J. A. Yorke [397, 398, 399]. 2. Concerning variational methods for finding almost periodic type solutions of ODEs, we will only note that J. R. Ward (Jr.) has investigated, in [765], the periodic solutions of the problem u′′ + g(u) = e(t), t ∈ ℝ; u(0) = u(T), u′ (0) = u′ (T), provided that the functions g(⋅) and e(⋅) have certain features. For more details about this topic, we also refer the reader to the monographs [632] by H. P. Rabinowitz and [718] by M. Struwe.

3.6 Notes and appendices

�

211

3. Now we will briefly describe some results obtained by J. Hua [374] with regards to the existence of periodic solutions to the following nonlinear differential equation x ′ (t) = f (x, t)x + g(t, x),

t ∈ ℝ.

(140)

The main results of paper are [374, Theorem 2, Theorem 3], where the author has proved the following: (i) Suppose that the functions f (t, x) and g(t, x) are continuous ω-periodic in the time variable t, uniformly with the respect to the variable x ∈ ℝ as well as that (a) f (t, x) ⩽ −α < 0; (b) |f (t, x) − f (t, y)| ⩽ L1 |x − y|, |g(t, x) − g(t, y)| ⩽ L2 |x − y|, L2 < α; (c) αL2 + L1 supt∈[0,ω] |g(t, 0)| < L22 + α(α − L2 ). Then there exists a unique continuous ω-periodic solution x(t) of (140). Moreover, we have that |x(t)| ⩽ (supt∈[0,ω] |g(t, 0)|)/(α − L2 ) for all t ∈ ℝ and x(t) is uniformly asymptotic stable. (ii) Suppose that the functions f (t, x) and g(t, x) are continuous ω-periodic in the time variable t, uniformly with the respect to the variable x ∈ ℝ as well as that (a) f (t, x) ⩾ α > 0; (b) |f (t, x) − f (t, y)| ⩽ L1 |x − y|, |g(t, x) − g(t, y)| ⩽ L2 |x − y|, L2 < α; (c) αL2 + L1 supt∈[0,ω] |g(t, 0)| < L22 + α(α − L2 ). Then there exists a unique continuous ω-periodic solution x(t) of (140). Moreover, we have that |x(t)| ⩽ (supt∈[0,ω] |g(t, 0)|)/(α − L2 ) for all t ∈ ℝ and x(t) is unstable. After that, the author analyzes the existence and uniqueness of periodic solutions to the following nonlinear Abel-type first-order differential equation: x ′ (t) = a(t)x 3 + b(t)x 2 + c(t)x + d(t),

t ∈ ℝ,

which is a very special case of (140); see also [218, 375, 583] for more details about this important subject. 4. The Sturm–Liouville equations with Besicovitch almost periodic coefficients have been analyzed by A. Dzurnak, A. B. Mingarelli [284] and F.-H. Wong, C.-C. Yeh [769]; see also [324, 416]. Concerning the almost periodic solutions of the Euler–Lagrange type equations with a convex Lagrangian and an almost periodic forcing term, we refer the reader to the research article [141] by J. Blot and references cited therein. 5. The first results concerning the reducibility for linear ordinary differential equations are established by H. Poincaré (the normal form theory). Using KAM theory, which has been developed by Kolmogorov, Arnold, and Moser within the framework of the theory of Hamiltonian systems, many authors have recently investigated the reducibility of finite-dimensional systems. For example, F. H. Meng and W. H. Qiu have examined the reducibility for a class of two-dimensional almost periodic systems with small per-

212 � 3 Multi-dimensional ρ-almost periodic type functions turbation [548]. We also refer the reader to [14, 101, 214, 334, 395, 396, 723, 771, 790], the doctoral dissertation of G. C. O’Brien [164], and other references quoted in [548]. 6. In [543], M. Maqbul has recently investigated the almost periodic solutions for a class of nonlinear Duffing systems with time-varying coefficients and Stepanov-almost periodic forcing terms. The author has first considered the equation u′ (t) = A(t)u(t) + f (t),

t ∈ ℝ,

where f : ℝ → ℝn is an almost periodic function and A(t) is an n × n almost periodic matrix defined on ℝ. After that, the obtained results are applied in the analysis of the existence and uniqueness of almost periodic solutions to the following nonlinear Duffing system u′ (t) = −f1 (t)u(t) + v(t) + F1 (t);

v′ (t) = −f2 (t)v(t) + [α(t) + f2 (t)]u(t) − β(t)um (t − ϕ(t)) + F2 (t),

of ordinary differential equations, where f1 (⋅) is a bounded continuous function on the real line satisfying inft∈ℝ f1 (t) > 0, f2 , α, β, ϕ are almost periodic functions defined on the real line, F1 , F2 are Stepanov almost periodic continuous functions on the real line and m ∈ ℕ ∖ {1}. The obtained results can be incorporated in the study of the existence and uniqueness of positive almost periodic solutions for the following nonlinear Duffing equations with a deviating argument: u′′ (t) + cu′ (t) − au(t) + um (t − ϕ(t)) = ψ(t),

t ∈ ℝ,

which has been previously analyzed by L. Peng and W. Wang in [607]; cf. also [174] and other references cited in [543]. 7. Let us note that M. L. Hbid and R. Qesmi [362] have analyzed the periodic solutions for the following functional differential equation u′ (t) = f (u(t − ντ(t)), ε),

t ∈ ℝ,

under certain assumptions. In order to achieve their aims, the authors have used a perturbation method for retarded differential equations with constant delay, a bifurcation result, and the Poincaré procedure. 8. Suppose that f : [0, T] × ℝn → ℝn is a continuous function, where T > 0 is a given real number. The existence of periodic solutions of the nonlinear system of ODEs x ′ = f (t, x),

t ∈ [0, T],

has been examined by J. Knežević-Miljanović in [419]. The main assumption is that f (t, x) is a function of Kamke type on a certain subset D ⊆ ℝn ; see [419, Theorem 2.1] for more details.

3.6 Notes and appendices

� 213

9. In [179], T. Carballo and D. Cheban have analyzed the almost periodic and asymptotically almost periodic solutions on the positive real line for the Liénard equation x ′ + f (x)x ′ + g(x) = F(t), where F(⋅) is a real-valued almost periodic or asymptotically almost periodic function and g : (a, b) → ℝ is a strictly decreasing function. The authors have examined the related problem for the vectorial Liénard equation. 10. Without going into further details, we will only mention in passing that M. Akhmet, M. Tleubergenova and A. Zhamanshin [26] have recently introduced modulo periodic Poisson stable functions and analyzed after that quasi-linear ordinary differential equations with modulo periodic Poisson stable coefficients. For some other references with regards to these subjects, we also refer the reader to [133, 176, 213, 591, 658, 790] as well as to the trilogy [168, 169, 170] by P. J. Bryant, J. W. Miles, and the doctoral thesis of S. Boudjema [151]. 11. In [279], N. Drisi and B. Es-sebbar have analyzed the existence and uniqueness of almost automorphic solutions to logistic equations with discrete and continuous delay. 12. It is worth noticing that Ju. V. Komlenko and E. L. Tonkov have examined, in [423], the ordinary differential equation of the form n

m

Lu = u(n) (t) + ∑ ∑ akj (t)x (k−1) (t − τj (t)) = 0, k=1 j=0

t ∈ ℝ (τ0 (t) = 0),

(141)

where akj (t) are complex-valued functions of period ω > 0 and τj (t) are real-valued functions of period ω. A complex number λ is said to be multiplier of (141) if and only if this equation has a non-trivial (n − 1)-times continuously differentiable solution u(t) such that u(n) (t) is locally integrable and u(t + ω) ≡ λu(t). The authors have found some sufficient conditions ensuring that a given complex number λ is not a multiplier of the equation (141). See also [379, 422, 610, 611, 726, 727, 728] and the scientific work of I. N. Blinov [133, 134, 135, 136, 137]. 13. Based on the investigation of J. Mallet-Paret and J. A. Yorke [532], K. Geba and W. Marzantowicz [333] have analyzed the global bifurcation of periodic solutions for the following problem u′ (t) = ϕ(u(t), a),

t ∈ ℝ,

where a ∈ ℝ, 0 ≠ Ω ⊆ ℝm is open, ϕ : Ω×ℝ → ℝm and u : ℝ → ℝm . The presented results are obtained using the results from the topological degree theory (cf. also [149, 537, 538, 539, 698, 770] and the research monograph [387] by J. Jezierski and W. Marzantowicz). 14. Concerning the Lienard ordinary differential type equations, we recommend for the reader the references [215, 216, 314, 336, 525, 621, 622]. For periodic solutions of ODEs and their systems, we also refer the reader to [150, 210, 227, 250, 347, 389, 540, 722, 730, 731]. Concerning the existence of positive periodic solutions for certain classes of

214 � 3 Multi-dimensional ρ-almost periodic type functions functional differential equations with periodic delay appearing in population models, we also refer the reader to the research article [325] by D. Franco, E. Liz and P. J. Torres. Periodic solutions of nonlinear wave equations The study of qualitative properties of solutions to nonlinear wave equations is an old mathematical problem. In this part, we will briefly describe the main result of paper [84] by V. Barbu and N. H. Pavel concerning T-periodic solutions for the following nonlinear one dimensional wave equation with x-dependent coefficients: u(x)ytt − (u(x)yx )x + g(y) = f (x, t), y(0, t) = y(π, t) = 0, y(x, t + T) = y(x, t),

0 < x < π, t ∈ ℝ,

0 < x < π, t ∈ ℝ.

The authors have assumed the following conditions: (H1) u ∈ H 2 (0, π), u(x) ⩾ 1 for a. e. x ∈ (0, π), and ess inf[

2

1 u′′ 1 u′ − ( ) ]>0 2 u 4 u

(H2) The function g : ℝ → ℝ is Lipschitz continuous and monotonically increasing. This equation is a nonlinear model for the forced vibrations of a nonhomogeneous string as well as for propagation of waves in non-isotropic media. The main result of this paper is [84, Theorem 3.1], which extends the main result of paper [76] by A. Bahri and H. Brezis. In the formulation of these results, the authors have used condition (H3) as well; we will only note that the main results of research articles [76, 84, 160, 633] are established by assuming that the period T is a rational multiple of π. Concerning the situation in which the period T is an irrational multiple of π, we refer the reader to the paper [544] by P. J. McKenna. Let us also note that P. Krejčí has investigated the hysteresis and periodic solutions of semi-linear and quasi-linear wave equations in [466]. The existence of periodic solutions and asymptotic behavior of a nonlinear heat equation with hysteresis in the source term have been analyzed by J. Kopfová in [425] with the help of a homotopy version of the Leray–Schauder fixed point theorem (cf. also the research articles [574] by M. Nakao, R. Koyanagi and [584] by M. N. Nkashama, M. Willem). Finally, let us note that the (non-)existence of steady periodic solutions of the Prandtl equations was investigated by M. Renardy [644] in 2013 as well as that the (non-)existence of periodic solutions for certain classes of second order semilinear parabolic equations has been analyzed in the research articles [73, 74, 75] by Sh. G. Bagyrov; see also references cited therein.

3.6 Notes and appendices

� 215

On the scientific work of O. Vejvoda In our monograph [431], we have emphasized that the periodic solutions for various classes of ordinary differential equations and partial differential equations have been analyzed by O. Vejvoda and his collaborators during 1960s–1980s; a comprehensive survey of results can be found in the monograph [741] written by O. Vejvoda and L. Herrmann, V. Lovicar as contributors. In this issue, we will briefly describe some results established by O. Vejvoda and his followers (these results have not been particularly mentioned in [431]): 1. In a joint research study with V. Št’astnová [704], the existence of periodic solutions of the problem ut = uxx + cu + g(t, x) + εf (t, x, u, ux , ε);

u(t, 0) = h0 (t) + εχ0 (t, u(t, 0), u(t, π), ε);

u(t, π) = h1 (t) + εχ1 (t, u(t, 0), u(t, π), ε),

where the functions g, f , h0 , h1 , χ0 , χ1 are periodic in the variable t, have been sought. In this paper, the constant c need not be less than 0 and there is no limitation on the growth rate of function f with respect to u or ux . 2. In a joint research article with J. Kurzweil [482], the periodic solutions for the system of ordinary differential equations d y = f (y , . . . , yr , t), dt s s 1

s = 1, 2, . . . , r,

(142)

have been analyzed. More precisely, let for each s = 1, 2, . . . , r and for each fixed real numbers y01 , . . . , y0r , the function fs (y01 , . . . , y0r , t) be of period ω > 0. In his notable book “Lyapunov and Poincare Methods in the Theory of Nonlinear Oscillations” (Leningrad– Moscow, 1949, in Russian), I. G. Malkin has mistakenly stated that, if the solution of this system t 󳨃→ (y1 (t), . . . , yr (t)), t ∈ ℝ exists and each function ys (⋅) is T-periodic, where T > 0, then the number T/ω must be rational. The first counterexample has been constructed by Prof. V. Knihal, who analyzed the ordinary differential equation d2 2 y + y = (y2 + (y′ ) − 1) sin 2πt. 2 dt In [482], the authors have given some results about the periodic solutions of the system (142), provided that the T-periodic solution of this system exists and the number T/ω is irrational. Some results about the almost periodic solutions of system (142) are also given. 3. In his research article [742], O. Vejvoda has examined the existence and stability of the periodic solutions of the second kind for the following mechanical system x ′′ + x = εf (x, x ′ , φ, φ′ , ε),

216 � 3 Multi-dimensional ρ-almost periodic type functions φ′′ = εM(φ′ ) + ε2 g(x, x ′ , φ, φ′ , ε), where the functions f and g are periodic with respect to the third variable. The solution has been sought in the form x = x(t, ε), φ = ω(ε)t + εΦ(t, ε), where the functions x(t, ε) and Φ(t, ε) are (2πN/ω(ε))-periodic for some positive integer N. See also [88, 472, 488, 705, 716, 743, 744] and references cited therein, as well as the research monograph [672] by Š. Schwabig, M. Tvrdý, and O. Vejvoda. For more details about personal life of Otto Vejvoda (1922–2009) we refer the reader to the memorial article [467] by P. Krejčí. Motivated by some earlier works of O. Vejvoda, P. Rabinowitz, J. Hale and some others, H. Brezis [160] has investigated the study of T-periodic solutions for the following nonlinear vibrating string equation: utt − uxx + g(u) = f (x, t), u(x, t) = 0,

0 < x < π, t ∈ ℝ,

x = 0, π; t ∈ ℝ,

u(x, t + T) = u(x, t),

(143)

0 < x < π, t ∈ ℝ,

where g : ℝ → ℝ is a continuous function with g(0) = 0 and f (x, t) is a given T-periodic function in the variable t. As pointed out in [160], the problem (143) may be viewed as an infinite-dimensional Hamiltonian system H 𝜕 p 0 ( ) = ( p ) + ( ), 𝜕t q f −Hq where p = u, q = ut and the Hamiltonian H is defined on the space H01 (0, π) × L2 (0, π) through H(p, q) :=

π

π

π

0

0

0

1 ∫(px )2 dx + ∫ G(p) dx + ∫ q2 dx, 2

with G(⋅) being the first anti-derivative of g; the first systematic study of periodic solutions of the Hamiltonian systems has been carried out by H. Poincaré [620]. The existence of forced vibrations and the existence of free vibrations are two main problems investigated here (cf. also [38, 161, 162] for some other references quoted in [160], the research article of J. K. Hale [351], the research article [119] by M. Berti and P. Bolle, and the research article [120] by M. Berti and R. Montalto). Almost periodic type solutions of stochastic integro-differential equations The author is really not an expert in the field of stochastic integro-differential equations. Besides the references quoted in [428] and [431], we can recommend for the readers the following articles: [99, 178, 246, 348, 349, 407, 497, 509, 510, 563, 738, 764]; see also the references [41, 86, 96, 217, 230, 234], [261, 267, 271, 356, 515, 606, 624, 652, 729] and

3.6 Notes and appendices

�

217

[774, 780, 781, 782, 783, 802, 803] for various results about the existence and uniqueness of almost periodic and almost automorphic type solutions to the (abstract) Volterra integrodifferential equations. Almost periodic sequences and almost periodic solutions of the abstract difference equations Denote by l∞ (ℕ : X) the Banach space of all bounded X-valued sequences equipped with the sup-norm. Let us recall that an X-valued sequence (xn )n∈ℕ is called almost periodic if and only if for each ε > 0 there exists an integer N0 (ε) ∈ ℕ such that among any N0 (ε) consecutive natural numbers there exists at least one natural number τ ∈ ℕ satisfying that ‖xn+τ − xn ‖ ⩽ ε,

n ∈ ℕ;

the number τ is said to be an ε-period of sequence (xn )n∈ℕ . We know that any almost periodic X-valued sequence is bounded. The notion of a Stepanov almost periodic sequence was introduced and analyzed by J. Andres and D. Pennequin in [43]. The authors have actually shown that any Stepanov almost periodic sequence is almost periodic and vice versa, which is completely different from the corresponding statement for the functions. The notion of a Weyl almost periodic sequence was introduced and analyzed by A. Iwanik [380] within the field of topological dynamics. It is worth noting that V. Bergelson et al. [114] have recently considered the notion of scalar-valued Besicovitch (rationally) almost periodic sequences, scalar-valued Weyl (rationally) almost periodic sequences and Weyl rational subsets of ℕ. Their definitions can be simply reconsidered for the vector-valued sequences; see Section 9.2 for more details and some applications given. We have quoted some references about almost periodic type sequences and almost periodic type solutions of difference equations and their systems in [428] and [431]. Here we would like to note that these topics have been also examined in [102, 212, 337, 361, 369, 415, 526, 604, 805, 806]. Besicovitch ρ-almost periodic type functions in ℝn In this part, we will introduce several new classes of multi-dimensional Besicovitch ρ-almost periodic type functions and briefly describe some problems which occurs in the most simplest case when ρ is a certain multiple of the identity operator. We will always assume that ρ1 ⊆ Y × Y and ρ2 ⊆ Y × Y are binary relations, Λ is a general non-empty subset of ℝn as well as that p ∈ 𝒫 (Λ), ϕ : [0, ∞) → [0, ∞) is measurable, F : (0, ∞) → (0, ∞) and the following condition holds (1 ⩽ i ⩽ 3; 1 ⩽ j ⩽ 2): ϕi : [0, ∞) → [0, ∞) is measurable, Fj : (0, ∞) → (0, ∞) and p ∈ 𝒫 (Λ).

218 � 3 Multi-dimensional ρ-almost periodic type functions Set Λ′′ := {τ ∈ ℝn : τ + Λ ⊆ Λ}. In the remainder of this part, we will always assume that 0 ≠ Ω ⊆ ℝn is a compact set with positive Lebesgue measure, Λ + lΩ ⊆ Λ for all l > 0, as well as 0 ≠ Λ′ ⊆ Λ′′ and therefore Λ + Λ′ ⊆ Λ. Now we are ready to introduce the following notion, which seems to be completely new in the purely multi-dimensional case n > 1: Definition 3.6.3. Let F : Λ × X → Y . (i) Suppose that the set A ⊆ Λ′ ⊆ ℝn has no finite accumulation point and δ > 1. Then we say that the set A is Λ′δ -satisfactorily uniform if and only if there exists a finite real number l > 0 such that, for every t0 , t1 ∈ Λ′ , we have δ ⋅ |A|B(t0 ,l) > |A|B(t1 ,l) ;

(144)

here and hereafter, |A|D denotes the number of elements of set A ∩ D, where D ⊆ ℝn . (ii) We say that the function F(⋅; ⋅) is Besicovitch-(p, ϕi , Fj , ℬ, Λ′ , ρ1 , ρ2 , δ)-almost periodic if and only if, for every B ∈ ℬ and ε > 0, there exists a Λ′δ -satisfactorily uniform set A = {τi : i ∈ ℤ} ⊆ ℝn such that for every τ ∈ A, t > 0, x ∈ B and ⋅ ∈ Λt , we have the existence of an element y⋅;x ∈ ρ1 (F(⋅; x)) such that 󵄩 󵄩 lim sup F1 (t) sup[ϕ1 (󵄩󵄩󵄩F(⋅ + τ; x) − y⋅;x 󵄩󵄩󵄩Y )] t→+∞

x∈B

Lp(⋅) (Λt )

< ε,

(145)

as well as, for every l > 0, t > 0, x ∈ B and ⋅ ∈ Λt + lΩ, there exists an element z⋅;x ∈ ρ2 (F(⋅; x)) such that lim sup F2 (t) sup[ϕ2 (lim sup t→+∞

x∈B

k→+∞

1 (2k + 1)

k

󵄩 󵄩 × ∑ [ϕ3 (l−n 󵄩󵄩󵄩F(⋅ + τi ; x) − z⋅;x 󵄩󵄩󵄩Y )] i=−k

Lp(⋅) (y+lΩ:Y )

)]

Lp(y) (Λt )

< ε.

(146)

(iii) We say that the function F(⋅; ⋅) is Besicovitch-(p, ϕ2 , ϕ3 , F2 , ℬ, Λ′ , ρ2 , δ)-almost periodic of type 1 if and only if, for every B ∈ ℬ and ε > 0, there exists a Λ′δ -satisfactorily uniform set A = {τi : i ∈ ℤ} ⊆ ℝn such that for every t > 0, x ∈ B and ⋅ ∈ Λt + Ω, there exists an element z⋅;x ∈ ρ2 (F(⋅; x)) such that (146) holds with l = 1. (iv) Suppose that p(⋅) ≡ p ∈ [1, ∞). Then we say that the function F(⋅; ⋅) is Besicovitch(p, ϕi , Fj , ℬ, Λ′ , ρ1 , ρ2 , δ)-almost periodic of type 2 if and only if, for every B ∈ ℬ and ε > 0, there exists a Λ′δ -satisfactorily uniform set A = {τi : i ∈ ℤ} ⊆ ℝn such that, for every τ ∈ A, t > 0, x ∈ B and ⋅ ∈ Λt , we have the existence of an element

3.6 Notes and appendices

� 219

y⋅;x ∈ ρ1 (F(⋅; x)) such that (145) holds, as well as that, for every l > 0, t > 0, x ∈ B and t ∈ Λt + lΩ, there exists an element zt;x ∈ ρ2 (F(t; x)) such that lim sup F2 (t) sup[∫ ϕ2 (lim sup t→+∞

k

× ∑ [l i=−k

x∈B

−n

Λt

k→+∞

1 (2k + 1)

󵄩 󵄩 p ∫ [ϕ3 (󵄩󵄩󵄩F(t + τi ; x) − zt;x 󵄩󵄩󵄩Y )] dt]) dy]

1/p

< ε.

(147)

y+lΩ

(v) Suppose that p(⋅) ≡ p ∈ [1, ∞). Then we say that the function F(⋅; ⋅) is Besicovitch(p, ϕi , Fj , ℬ, Λ′ , ρ2 , δ)-almost periodic of type 3 if and only if, for every B ∈ ℬ and ε > 0, there exists a Λ′δ -satisfactorily uniform set A = {τi : i ∈ ℤn } ⊆ ℝn such that for every t > 0, x ∈ B and t ∈ Λt + Ω, there exists an element zt;x ∈ ρ2 (F(t; x)) such that (147) holds with l = 1. The notion of a Bohr (Besicovitch) almost periodic set in ℝn can be also introduced and analyzed; see the research article [301] by S. Favorov for more details about this non-trivial problematic as well as the research articles [71, 72, 118, 302, 339, 485, 489, 491, 637, 696, 798] and references cited therein. For more details about quasicrystals, almost periodic type patterns, and irregular sampling, we refer the reader to the research articles [315] by S. Ferri, J. Galindo, C. Gómez, [550] by Y. Meyer, the list of references quoted in these papers, and the monograph [431]. It is clear that (144) implies δ ⋅ infn |A|B(t,l) ⩾ sup |A|B(t,l) . t∈ℝ

t∈ℝn

The usual notion of a satisfactorily uniform set A ⊆ ℝn is obtained by plugging Λ′ = ℝ and δ = 2 in Definition 3.6.3(i); we say that the discrete set A ⊆ ℝn is δ-satisfactorily uniform if and only if A is Λ′δ -satisfactorily uniform with Λ′ = ℝn . Concerning this notion, we would like to observe, probably for the first time in the existing literature, that the use of number δ = 2 is completely meaningless in the structural characterizations of the space of one-dimensional Besicovitch-p-almost periodic functions f : ℝ → ℂ given in [125]. Let us explain this fact in more detail: the statement of [125, Corollary 2], which continues to hold for the vector-valued functions, is essentially used in the proofs of [125, Theorem 7∘ , (IC)B (A), p. 95; Theorem 1∘ , (IC)Br (A), p. 100]. All other technical lemmas and results from [125] continue to hold with δ-satisfactorily uniform sets and here we will only note that, in the concrete situation of [125, Remark, pp. 93–94], we have the following estimate with the use of δ-satisfactorily uniform sets: μ(b) ν(b) δμ(b) ⩽p⩽ ⩽ ⩽ δp. b b b

220 � 3 Multi-dimensional ρ-almost periodic type functions This simply implies that for each number δ′ > δ we have the existence of a real number b′ > b, where b has the same meaning as in the above-mentioned remark, such that the estimate n(t, T) ⩽ δ′ p, 2T

T > b′

holds; see the equation [125, (1), p. 94]. Therefore, we have the following result: Theorem 3.6.4. Suppose that 1 ⩽ p < +∞ and the function f : ℝ → X is locally p-integrable. Then f (⋅) is Besicovitch-p-almost periodic if and only if for each (some) δ > 1 there exists a δ-satisfactorily uniform set A = {τi : i ∈ ℤ} ⊆ ℝ such that 1/p

t

1 󵄩 󵄩p lim sup( ∫󵄩󵄩󵄩f (s + τi ) − f (s)󵄩󵄩󵄩 ds) 2t t→+∞

0, t

x+l

−t

x

k 1 1 󵄩 󵄩p lim sup( ∫[lim sup ∑ l−1 ∫ 󵄩󵄩󵄩f (s + τi ) − f (s)󵄩󵄩󵄩 ds] dx) 2t t→+∞ k→+∞ 2k + 1 i=−k

1/p

< ε.

Concerning the notion of a Λ′δ -satisfactorily uniform set, we will provide the following explanatory examples: Example 3.6.5. (i) If A is a Λ′δ -satisfactorily uniform set, then A − A need not be a Λ′δ -satisfactorily uniform set. In actual fact, the set A − A can have a finite accumulation point, as the following special case with Λ′ = ℝ and δ = 2 shows. Let A := ℤ ∪ {k + k −1 : k ∈ ℤ ∖ {0}}. Then A is a satisfactorily uniform set, but 0 is an accumulation point of the set A − A, as easily approved. (ii) The set ℤn is δ-satisfactorily uniform for each number δ > 1. Given a number δ0 > 1, it could be interesting to construct a set A = {τi : i ∈ ℤ} ⊆ ℝn which is δ-satisfactorily uniform for δ > δ0 but not δ0 -satisfactorily uniform. (iii) In the multi-dimensional setting, it is not completely clear how one can order a discrete set A = {τi : i ∈ ℤ} ⊆ ℝn ; this fact has a series of obvious unpleasant consequences in applications to the partial differential equations. We propose the following unsatisfactory approach: Let A0 consist of those elements of A with the minimal norm. Then A0 can be written as the union of disjoint sets A−s2 , . . . , As1 , where Aj is determined in the following way (−s2 ⩽ j ⩽ s1 ): Let x−s2 < ⋅ ⋅ ⋅ < x−2 < x−1 < 0 ⩽ x0 < x1 < ⋅ ⋅ ⋅ < xs1 , and {x−s2 , . . . , xs1 } is the first projection of set A0 . We first order the elements of A0 whose first projection is x0 by the minimal second coordinate, the minimal third coordinate, etc.; we continue the enumeration with the sets A1 , A2 , . . . , As1 . After that we proceed to the elements of sets A−1 , A−2 , . . . , A−s2 and give them the negative values of indices i by the maximal second coordinate, the

3.6 Notes and appendices

�

221

maximal third coordinate, etc.; and so on and so forth. After ordering the elements with the minimal norm, we proceed to the elements with the second minimal norm and order them as above (we continue the enumeration). It is worth noting that, in the one-dimensional setting, we obtain the enumeration A = {τi : i ∈ ℤ} ⊆ ℝ with τi < τj for i < j. Now we would like to raise the following important issue: Question 3.6.6. Can we prove a multi-dimensional analogue of Theorem 3.6.4? For every space of Besicovitch almost periodic type functions introduced in Definition 3.6.3, we can also introduce the corresponding space of Besicovitch uniformly recurrent type functions: Definition 3.6.7. Let F : Λ × X → Y . (i) We say that the function F(⋅; ⋅) is Besicovitch-(p, ϕi , Fj , ℬ, Λ′ , ρ1 , ρ2 )-uniformly recurrent if and only if, for every B ∈ ℬ, there exists an unbounded set {τi : i ∈ ℕ} ⊆ Λ′ such that for every t > 0, x ∈ B and ⋅ ∈ Λt , we have the existence of an element y⋅;x ∈ ρ1 (F(⋅; x)) such that 󵄩 󵄩 lim lim sup F1 (t) sup[ϕ1 (󵄩󵄩󵄩F(⋅ + τi ; x) − y⋅;x 󵄩󵄩󵄩Y )]

i→+∞ t→+∞

x∈B

Lp(⋅) (Λt )

= 0,

(148)

as well as, for every l > 0, t > 0, x ∈ B and ⋅ ∈ Λt + lΩ, there exists an element z⋅;x ∈ ρ2 (F(⋅; x)) such that lim lim sup F2 (t) sup[ϕ2 (lim sup

i→+∞ t→+∞

x∈B

k→+∞

1 (2k + 1)

k

󵄩 󵄩 × ∑ [ϕ3 (l−n 󵄩󵄩󵄩F(⋅ + τi ; x) − z⋅;x 󵄩󵄩󵄩Y )]

Lp(⋅) (y+lΩ:Y )

i=−k

)]

Lp(y) (Λt )

= 0.

(149)

(ii) We say that the function F(⋅; ⋅) is Besicovitch-(p, ϕ2 , ϕ3 , F2 , ℬ, Λ′ , ρ2 )-uniformly recurrent of type 1 if and only if, for every B ∈ ℬ, there exists an unbounded set {τi : i ∈ ℕ} ⊆ Λ′ such that, for every t > 0, x ∈ B and ⋅ ∈ Λt + Ω, there exists an element z⋅;x ∈ ρ2 (F(⋅; x)) such that (149) holds with l = 1. (iii) Suppose that p(⋅) ≡ p ∈ [1, ∞). Then we say that the function F(⋅; ⋅) is Besicovitch(p, ϕi , Fj , ℬ, Λ′ , ρ1 , ρ2 )-uniformly recurrent of type 2 if and only if, for every B ∈ ℬ, there exists an unbounded set {τi : i ∈ ℕ} ⊆ Λ′ such that, for every τ ∈ A, t > 0, x ∈ B and ⋅ ∈ Λt , we have the existence of an element y⋅;x ∈ ρ1 (F(⋅; x)) such that (148) holds, as well as that, for every l > 0, t > 0, x ∈ B and t ∈ Λt + lΩ, there exists an element zt;x ∈ ρ2 (F(t; x)) such that lim lim sup F2 (t) sup[∫ ϕ2 (lim sup

i→+∞ t→+∞

x∈B

Λt

k→+∞

1 (2k + 1)

222 � 3 Multi-dimensional ρ-almost periodic type functions k

󵄩 p × ∑ [ ∫ [ϕ3 (l 󵄩󵄩F(t + τi ; x) − yt;x 󵄩󵄩󵄩Y )] dt]) dy] i=−k y+lΩ

−n 󵄩 󵄩

1/p

= 0.

(150)

(iv) Suppose that p(⋅) ≡ p ∈ [1, ∞). Then we say that the function F(⋅; ⋅) is Besicovitch(p, ϕi , Fj , ℬ, Λ′ , ρ2 )-uniformly recurrent of type 3 if and only if, for every B ∈ ℬ, there exists an unbounded set {τi : i ∈ ℕ} ⊆ Λ′ such that, for every t > 0, x ∈ B and t ∈ Λt + Ω, there exists an element zt;x ∈ ρ2 (F(t; x)) such that (150) holds with l = 1. Remark 3.6.8. We accept the usual terminology agreements from our previous considerations: If F : Λ → Y , then we omit the term “ℬ” from the notation. Further on, if ρi = ci I for some ci ∈ ℂ, where i = 1, 2, then we also write “ci ” in place of “ρ”; we omit “ci ” from the notation if ci = 1. We also omit the term “Λ′ ” from the notation if Λ′ = Λ. If ϕi (x) ≡ x for i = 1, 2, 3, then we omit it from the notation. The usual notion of Besicovitch-p-almost periodicity of function F : ℝn → Y , where 1 ⩽ p < ∞, is obtained by plugging c1 = c2 = 1, Λ′ = ℝn , ϕi (x) ≡ x for i = 1, 2, 3 and Fj (t) ≡ t−(n/p) for j = 1, 2, in Definition 3.6.3(iv); we define the notion of Besicovitch-p-uniform recurrence in a similar fashion. For some important counterexamples in the theory of one-dimensional Besicovitch-p-almost periodic functions, we refer the reader to [42, Examples 6.24–6.27]. It is clear that the notion introduced here provides a very general approach to the notion of Besicovitch almost periodicity (uniform recurrence) as well as that we work with general subsets Λ of ℝn here. Now we would like to introduce the following notion: Definition 3.6.9. Let 0 ≠ Λ ⊆ ℝn , and let 𝒞Λ denote any class of functions F : Λ → Y introduced in Definition 3.6.3 or Definition 3.6.7. Then we say that the set Λ is admissible with respect to the class 𝒞Λ if and only if for any complex Banach space Y and for any ̃ = F(t) for all t ∈ Λ. function F : Λ → Y there exists a function F̃ ∈ 𝒞ℝn such that F(t) The following question is meaningful: Question 3.6.10. Let (v1 , v2 , . . . , vn ) is a basis of ℝn and let Λ = {α1 v1 + ⋅ ⋅ ⋅ + αn vn : αi ⩾ 0 for all i ∈ ℕn } be a convex polyhedral in ℝn . Further on, let a function F ∈ e − (⋅−n/p ) − Bp (Λ : Y ) be given; cf. Definition (3.3.3) and the paragraph following it for the notion. Is it true that there exists a function F̃ ∈ e − (⋅−n/p ) − Bp (ℝn : Y ) such that F̃ = F on Λ? In support of our investigation of the general case 0 ≠ Λ ⊆ ℝn , we would like to present the following example, as well: Example 3.6.11 (cf. [431, Example 6.1.15]). Suppose that L > 0 is a fixed real number, p ∈ [1, ∞) as well as the functions t 󳨃→ f (t), t ∈ ℝ and t 󳨃→ g(t), t ∈ ℝ are Besicovitch-palmost periodic. Set Λ := {(x, y) ∈ ℝ2 : |x − y| ⩾ L} and Λ′ := {(τ, τ) : τ ∈ ℝ}. We have already mentioned that the function u(x, y) := (f (x)+g(y))/(x −y), (x, y) ∈ Λ, is a solution

3.6 Notes and appendices

� 223

of the partial differential equation uxy − (ux )/(x − y) + (uy )/(x − y) = 0. Further on, if ε > 0 is given, then we can find a satisfactorily uniform set Vε = {τi : i ∈ ℤ} ⊆ ℝ such that, for every i ∈ ℤ, we can find a finite real number t0 > 0 such that, for every real number t ⩾ t0 , we have t

t

−t

−t

󵄨 󵄨p 󵄨 󵄨p ∫󵄨󵄨󵄨 f (x + τi ) − f (x)󵄨󵄨󵄨 dx + ∫󵄨󵄨󵄨g(y + τi ) − g(y)󵄨󵄨󵄨 dy < εt p ,

t ⩾ t0 ,

(151)

and, for every real number l > 0, t

x+l

k 1 󵄨 󵄨p [∫ lim sup ∑ l−1 ∫ 󵄨󵄨󵄨f (s + τi ) − f (s)󵄨󵄨󵄨 ds] dx 2k + 1 k→+∞ i=−k −t

x

t

x+l

+ ∫[lim sup −t

k→+∞

k 1 󵄨 󵄨p ∑ l−1 ∫ 󵄨󵄨󵄨g(s + τi ) − g(s)󵄨󵄨󵄨 ds] dx < εt p , 2k + 1 i=−k x

t ⩾ t0 .

(152)

It is clear that 󵄩󵄩 󵄩 ‖f (x + τ) − f (x)‖ + ‖g(y + τ) − g(y)‖ 󵄩󵄩u(x + τ, y + τ) − u(x, y)󵄩󵄩󵄩 ⩽ |x − y| ⩽

‖f (x + τ) − f (x)‖ + ‖g(y + τ) − g(y)‖ , L

(x, y) ∈ Λ, τ ∈ Vε .

Set Wε := {(τi , τi ) : i ∈ ℤ}. Then Wε is Λ′ -satisfactorily uniform with Λ′ := Δ2 and the function u(x, y) is Besicovitch-(p, ϕi , Fj , Λ′ , ρ1 , ρ2 , δ)-almost periodic of type 2 (ϕi (x) ≡ x, Fj (t) ≡ t (2/p) , ρ1 = ρ2 = I, δ = 2), as a very simple computation involving the Fubini theorem and (151)–(152) shows. Observe, finally, that this example also indicates that the situation in which Λ′ is not a subset of Λ is meaningful. It is clear that we have the following: (i) If F(⋅; ⋅) is Besicovitch-(p, ϕi , Fj , ℬ, Λ′ , ρ1 , ρ2 , δ)-almost periodic, then F(⋅; ⋅) is Besicovitch-(p, ϕi , Fj , ℬ, Λ′ , ρ2 , δ)-almost periodic of type 1. (ii) If F(⋅; ⋅) is Besicovitch-(p, ϕi , Fj , ℬ, Λ′ , ρ1 , ρ2 , δ)-almost periodic of type 2, then F(⋅; ⋅) is Besicovitch-(p, ϕi , Fj , ℬ, Λ′ , ρ2 , δ)-almost periodic of type 3. (iii) If F(⋅; ⋅) is Besicovitch-(p, ϕi , Fj , ℬ, Λ′ , ρ1 , ρ2 , δ)-almost periodic (of type 2), then F(⋅; ⋅) is Besicovitch-(p, ϕi , Fj , ℬ, Λ′ , ρ1 , ρ2 )-uniformly recurrent (of type 2). Furthermore, if F(⋅; ⋅) is Besicovitch-(p, ϕi , Fj , ℬ, Λ′ , ρ2 , δ)-almost periodic of type 1 (of type 3), then F(⋅; ⋅) is Besicovitch-(p, ϕi , Fj , ℬ, Λ′ , ρ2 )-uniformly recurrent of type 1 (of type 3). Now we will prove the following simple result (it is clear that the converse statement can be proved in the case that p = 1, when the corresponding classes coincide):

224 � 3 Multi-dimensional ρ-almost periodic type functions Proposition 3.6.12. Suppose that p(⋅) ≡ p ∈ [1, ∞) and function F : Λ × X → Y is Besicovitch-(p, ϕi , Fj , ℬ, Λ′ , ρ1 , ρ2 , δ)-almost periodic of type 2 (Besicovitch-(p, ϕi , Fj , ℬ, Λ′ , ρ2 , δ)-almost periodic of type 3) [Besicovitch-(p, ϕi , Fj , ℬ, Λ′ , ρ1 , ρ2 , δ)-uniformly recurrent of type 2 (Besicovitch-(p, ϕi , Fj , ℬ, Λ′ , ρ2 , δ)-uniformly recurrent of type 3)]. Let the function ϕ2 (⋅) be monotonically increasing and satisfy that ϕ2 (x) ⩾ [ϕ2 (x 1/p )]p for all x ⩾ 0. Then the function F(⋅; ⋅) is Besicovitch-(p, ϕi , Fj , ℬ, Λ′ , ρ1 , ρ2 , δ)-almost periodic (Besicovitch-(p, ϕi , Fj , ℬ, Λ′ , ρ2 , δ)-almost periodic of type 1) [Besicovitch-(p, ϕi , Fj , ℬ, Λ′ , ρ1 , ρ2 )-uniformly recurrent (Besicovitch-(p, ϕi , Fj , ℬ, Λ′ , ρ2 )-uniformly recurrent of type 1)]. Proof. We will only prove that the Besicovitch-(p, ϕi , Fj , ℬ, Λ′ , ρ1 , ρ2 , δ)-almost periodicity of type 2 implies the Besicovitch-(p, ϕi , Fj , ℬ, Λ′ , ρ1 , ρ2 , δ)-almost periodicity for the function F(⋅; ⋅). Keeping in mind the corresponding definitions, we only need to prove that for every sequence (bi ) of positive real numbers, we have: p

1 1 2k+1 1/p [ϕ2 (lim sup ∑ bi )] ⩽ ϕ2 (lim sup ∑ bi ). k→+∞ 2k + 1 j=1 k→+∞ 2k + 1 j=1 (2k+1)

This simply follows from the fact that the function ϕ2 (⋅) is monotonically increasing, the inequality 2k+1 2k+1 1 1 1/p ∑ bi ⩽ [ ∑ b] (2k + 1) j=1 (2k + 1) j=1 i

1/p

between the exponential means, the equality lim sup[ k→+∞

1 ∑ b] 2k + 1 j=1 i (2k+1)

1/p

= [lim sup k→+∞

1 2k+1 ∑ b] 2k + 1 j=1 i

1/p

,

and our assumption that ϕ2 (x) ⩾ [ϕ2 (x 1/p )]p for all x ⩾ 0. In the subsequent result, we will reconsider [304, Proposition 2.2] for multi-dimensional Besicovitch almost periodic type functions; the best we can do is the following (see also Example 3.6.5): Proposition 3.6.13. Suppose that p(⋅) ≡ p ∈ [1, ∞), 0 ≠ Λ′ ⊆ ℝn , 0 ≠ Λ ⊆ ℝn , Λ + Λ′ ⊆ Λ, Λ′ + (Λ′ − Λ′ ) ⊆ Λ′ , ρ1 is a binary relation on Y satisfying R(F) ⊆ D(ρ1 ) and ρ1 (y) is a singleton for any y ∈ R(F). If for each τ ∈ Λ′ we have τ + Λ = Λ, F(⋅; ⋅) is Besicovitch-(p, ϕi , Fj , ℬ, Λ′ , ρ1 , ρ2 , δ)-almost periodic (of type 2), and there exists a finite real constant c > 0 such that ϕ1 (x + y) ⩽ c[ϕ1 (x) + ϕ1 (y)],

x, y ⩾ 0,

(153)

3.6 Notes and appendices

� 225

then Λ+(Λ′ −Λ′ ) ⊆ Λ and F(⋅; ⋅) is Besicovitch-(p, ϕi , Fj , ℬ, Λ′ −Λ′ , I, ρ2 , δ)-almost periodic (of type 2). The same statement holds for the corresponding classes of Besicovitch uniformly recurrent functions. Proof. We will consider the Besicovitch-(p, ϕi , Fj , ℬ, Λ′ , ρ1 , ρ2 , δ)-almost periodic functions, only. The inclusion Λ + (Λ′ − Λ′ ) ⊆ Λ can be proved as in [304]; suppose that a number ε > 0 and a set B ∈ ℬ are given. Then we know that there exists a Λ′δ -satisfactorily uniform set A = {τi : i ∈ ℤ} ⊆ ℝn such that, for every τ ∈ A, t > 0, x ∈ B and ⋅ ∈ Λt , we have the existence of an element y⋅;x ∈ ρ1 (F(⋅; x)) such that (146) holds as well as, for every l > 0, t > 0, x ∈ B and ⋅ ∈ Λt + lΩ, there exists an element z⋅;x ∈ ρ2 (F(⋅; x)) such that (146) holds. Let j ∈ ℤ be fixed; then a very simple argumentation shows that the set Aj := {τi − τj : i ∈ ℤ} is (Λ′ − Λ′ )-satisfactorily uniform, since we have assumed that Λ′ + (Λ′ − Λ′ ) ⊆ Λ′ , and we only need to show that, for every fixed index i ∈ ℤ, we have: 󵄩 󵄩 lim sup F1 (t) sup[ϕ1 (󵄩󵄩󵄩F(⋅ + τi ; x) − F(⋅ + τj ; x)󵄩󵄩󵄩Y )] t→+∞

Lp (Λt )

x∈B

< Const. ⋅ ε.

(154)

Towards this end, observe that our assumptions imply that there exists a sufficiently large real number t0 (ε) > 0 such that for every t ⩾ t0 (ε) and x ∈ B, we have 󵄩 󵄩 F1 (t)[ϕ1 (󵄩󵄩󵄩F(⋅ + τi ; x) − ρ(F(⋅; x))󵄩󵄩󵄩Y )]

Lp (Λt )

󵄩 󵄩 F1 (t)[ϕ1 (󵄩󵄩󵄩F(⋅ + τj ; x) − ρ(F(⋅; x))󵄩󵄩󵄩Y )]

Lp (Λt )

0 there exists a satisfactorily uniform set A = {τi : i ∈ ℤ} ⊆ ℝ such that

226 � 3 Multi-dimensional ρ-almost periodic type functions 1/p

t

1 󵄨 󵄨p lim sup( ∫󵄨󵄨󵄨f (s + τi ) − cf (s)󵄨󵄨󵄨 ds) 2t t→+∞

0, t

x+l

−t

x

1/p

k 1 1 󵄨 󵄨p lim sup( ∫[lim sup ∑ l−1 ∫ 󵄨󵄨󵄨f (s + τi ) − cf (s)󵄨󵄨󵄨 ds] dx) 2t 2k + 1 t→+∞ k→+∞ i=−k

< ε.

(156)

Then the set mA is also satisfactorily uniform and for each i ∈ ℤ we have: t

󵄨 󵄨p ∫󵄨󵄨󵄨f (s + τi ) − f (s)󵄨󵄨󵄨 ds −t

t

m−1

p

j 󵄨󵄨

󵄨 ⩽ ∫( ∑ |c| 󵄨󵄨f (s + (m − j)τi ) − cf (s + (m − j − 1)τi )󵄨󵄨󵄨) ds −t

j=0

m−1

t+(m−j−1)τi

⩽ cm,p ∑

∫

j=0 −t+(m−j−1)τ

󵄨󵄨 󵄨p 󵄨󵄨f (s + τi ) − cf (s)󵄨󵄨󵄨 ds

i

p

󵄨 󵄨 ⩽ 2mcm,p ε (t + |t + τi | + ⋅ ⋅ ⋅ + 󵄨󵄨󵄨t + (m − 1)τi 󵄨󵄨󵄨) ⩽ 2mcm,p εp (mt +

m2 |τ |) ⩽ Const. ⋅ 2tεp , 2 i

t ⩾ t0 (ε, i),

for some finite real constants cm,p > 0 and t0 (ε, i) > 0. Therefore, (155) holds with the number c replaced by the number 1 therein, which can be simply transferred to the multi-dimensional setting. But, it is not clear how we can deduce an analogous conclusion for the equation (156); arguing as above, we can only prove that there exists a finite real constant cm,p > 0 such that t

x+l

k 1 󵄨 󵄨p ∑ l−1 ∫ 󵄨󵄨󵄨f (s + τi ) − f (s)󵄨󵄨󵄨 ds] dx ∫[lim sup 2k + 1 k→+∞ i=−k x

−t

t

x+l

⩽ cm,p {∫[lim sup k→+∞

−t t

+ ∫[lim sup −t

k→+∞

k 1 󵄨 󵄨p ∑ l−1 ∫ 󵄨󵄨󵄨f (s + τi ) − cf (s)󵄨󵄨󵄨 ds] dx 2k + 1 i=−k

k 1 󵄨 󵄨p ∑ l−1 ∫ 󵄨󵄨󵄨f (s + τi ) − cf (s)󵄨󵄨󵄨 ds] dx 2k + 1 i=−k

t

+ ⋅ ⋅ ⋅ + ∫[lim sup −t

x

x+τi +l

k→+∞

x+τi

k 1 ∑ l−1 2k + 1 i=−k

x+(m−1)τi +l

∫ x+(m−1)τi

󵄨󵄨 󵄨p 󵄨󵄨f (s + τi ) − cf (s)󵄨󵄨󵄨 ds] dx},

which seems to be completely inapplicable for our purposes.

3.6 Notes and appendices

�

227

(ω, ρ)-periodic and (ωj , ρj )j∈ℕn -periodic distributions Periodic generalized functions have been explored in many research articles by now; cf. [431, pp. 599–601] and references cited therein. In this part, we will briefly analyze (ω, ρ)periodic distributions and (ωj , ρj )j∈ℕn -periodic distributions. Let us recall the following notion: (i) Let ω ∈ ℝn ∖ {0} and ρ be a binary relation on X. A continuous function F : ℝn → X is said to be (ω, ρ)-periodic if and only if F(t + ω) ∈ ρ(F(t)), t ∈ ℝn . (ii) Let ωj ∈ ℝ ∖ {0} and ρj be a binary relation on X (1 ⩽ j ⩽ n). A continuous function F : ℝn → X is said to be (ωj , ρj )j∈ℕn -periodic if and only if F(t + ωj ej ) ∈ ρj (F(t)), t ∈ ℝn , j ∈ ℕn . Now we will introduce the following classes of vector-valued distributions: Definition 3.6.15. Suppose that ω ∈ ℝn ∖ {0}, ρ is a binary relation on X [ρj is a binary relation on X (1 ⩽ j ⩽ n)] and T ∈ 𝒟′ (ℝn : X). Then we say that T is an (ω, ρ)-periodic [(ωj , ρj )j∈ℕn -periodic] distribution if and only if, for every φ ∈ 𝒟(ℝn ), the function (T ∗ φ)(⋅) is (ω, ρ)-periodic [(ωj , ρj )j∈ℕn -periodic]. In the case that ρj = cj I for some non-zero complex numbers cj (1 ⩽ j ⩽ n), then we also say that T is (ωj , cj )j∈ℕn -periodic; furthermore, if cj = 1 for all j ∈ ℕn , then we say that T is (ωj )j∈ℕn -periodic. In our previous work, we have introduced and analyzed various classes of (ωj , cj ; rj , 𝕀′j )j∈ℕn -almost periodic type functions. For simplicity, we will not consider here the corresponding classes of vector-valued distributions. The following theorem follows immediately from Definition 3.6.15 and the results already known for (ω, ρ)-periodic [(ωj , ρj )j∈ℕn -periodic] functions: Theorem 3.6.16. (i) Let ωj ∈ ℝ ∖ {0} and ρj be a binary relation on X (1 ⩽ j ⩽ n). If T ∈ n 𝒟′ (ℝn : X) is (ωj , ρj )j∈ℕn -periodic, σ : ℕn → ℕn is a permutation and ω = ∑j=1 ωj ej , then T is (ω, ρ)-periodic with ρ = ∏nj=1 ρσ(j) . (ii) Let ω, a ∈ ℝn ∖{0}, c ∈ ℂ∖{0} and α ∈ ℂ. If T ∈ 𝒟′ (ℝn : X) is (ω, c)-periodic, then Ť is

(−ω, c)-periodic. Moreover, Ta (⋅) := T(⋅ + a) is (ω, c)-periodic and αT is (ω, c)-periodic. (iii) Let ωj ∈ ℝ ∖ {0}, cj ∈ ℂ ∖ {0} and α ∈ ℂ (1 ⩽ j ⩽ n). If T ∈ 𝒟′ (ℝn : X) is (ωj , cj )j∈ℕn periodic, then Ť is (−ωj , cj )j∈ℕn -periodic. Moreover, Ta (⋅) := T(⋅ + a) is (ωj , cj )j∈ℕn periodic and αT is (ωj , cj )j∈ℕn -periodic. Using Theorem 3.5.8(i) and the corresponding result for (ωj , cj )j∈ℕn -periodic functions, we immediately get the following result: Proposition 3.6.17. Suppose that ωj ∈ ℝ ∖ {0}, cj ∈ ℂ ∖ {0} (1 ⩽ j ⩽ n) and T ∈ 𝒟′ (ℝn : X). If T is (ωj , cj )j∈ℕn -periodic and |cj | ⩽ 1 for 1 ⩽ j ⩽ n, then T ∈ 𝒟L′ 1 (ℝn : X). Many other structural characterizations of (ω, ρ)-periodic [(ωj , ρj )j∈ℕn -periodic] distributions can be clarified in this way. The main structural characterizations of (ω, ρ)periodic [(ωj , ρj )j∈ℕn -periodic] distributions are stated in the following result:

228 � 3 Multi-dimensional ρ-almost periodic type functions Theorem 3.6.18. (i) Suppose that ρ = A ∈ L(X), k ∈ ℕ, and the functions Fj : ℝn → X are (ω, ρ)-periodic [(ωj , ρj )j∈ℕn -periodic] for 0 ⩽ j ⩽ k. Let αj ∈ ℕn0 for 0 ⩽ j ⩽ k and T = ∑kj=0 Fj j . Then T is an (ω, ρ)-periodic [(ωj , ρj )j∈ℕn -periodic] distribution. (ii) Let T be a bounded distribution, let T be an (ω, ρ)-periodic [(ωj , ρj )j∈ℕn -periodic] and let ρ : X → X be a continuous function. Then there exist an integer p ∈ ℕ and an (ω, ρ)-periodic [(ωj , ρj )j∈ℕn -periodic] function F : ℝn → X such that (129) holds in the distributional sense. (α )

We can also introduce and analyze 𝔻-asymptotically (ω, ρ)-periodic [(ωj , ρj )j∈ℕn periodic] type distributions. We will skip all details concerning this topic here. On discrete almost periodic Schrödinger operators There is an enormous literature devoted to the study of discrete almost periodic Schrödinger operators on l2 (ℤ), which have the form [Hψ](n) := ψ(n − 1) + ψ(n + 1) + V (n)ψ(n),

n ∈ ℤ, ψ ∈ l2 (ℤ),

where the potential V : ℤ → ℝ is bounded. It is well known that the operator H is bounded and self-adjoint on l2 (ℤ); therefore, its spectrum σ(H) is a compact subset of ℝ. Concerning the spectral analysis of the operator H, the main task is to identify its spectrum and determine the type of the associated spectral measures of H, which are defined in the usual way. The best explored classes of potentials are periodic potentials and random potentials (the ergodic potentials appear in the solid state physics). A random potential is represented by a sequence of independent, identically distributed random variables; on the other hand, a potential V (⋅) is called periodic if and only if there exists p ∈ ℕ such that V (p + n) = V (n) for all n ∈ ℤ. The periodic potentials are special cases of almost periodic potentials, which possess the property that the sequence of translations (Vm (⋅) := V (⋅ − m))m∈ℤ is relatively compact in l∞ (ℤ). Two most important subclasses of almost periodic potentials are limit-periodic potentials or semi-periodic potentials, which are uniform limits of periodic potentials, and quasi-periodic potentials. Let us recall that the one-dimensional quasi-periodic discrete time Schrödinger operators on l2 (ℤ) is given by [Hψ](n) := ψ(n − 1) + ψ(n + 1) + V (nω + θ)ψ(n),

n ∈ ℤ, ψ ∈ l2 (ℤ),

where the n-dimensional torus 𝕋n := (ℝ/ℤ)n is called phase, V : 𝕋n → ℝ, and the rationally independent ω ∈ 𝕋n is called frequency. The famous almost Mathieu operator (AMO), which is defined by [Hλ,α,θ ψ](n) := ψ(n − 1) + ψ(n + 1) + 2λ cos(2π(nα + θ))ψ(n),

n ∈ ℤ, ψ ∈ l2 (ℤ),

is the well-known example of a one-dimensional quasi-periodic discrete time Schrödinger operator on l2 (ℤ). These operators are incredibly important in quantum physics.

3.6 Notes and appendices

� 229

We continue by observing that C. R. de Oliveira and C. Gutierrez have developed, in [255], some techniques for the study of the spectrum of discrete Schrödinger operators [Hλ,α,θ ψ](n) := ψ(n − 1) + ψ(n + 1) + ωn ψ(n),

n ∈ ℤ, ψ ∈ l2 (ℤ),

with ω = (ωn )n∈ℤ being a sequence of real numbers taking a finite number of values. These techniques are used to show the presence of pure singular continuous spectrum for potentials along the shift embedding of some interval exchange transformations. See also the papers [183] by T. O. Carvalho, C. R. de Oliveira and [207] by H. Cheng et al., where the authors have investigated the global rigidity for ultra-differentiable quasi-periodic cocycles and its spectral applications. If V (⋅) is of period p ∈ ℕ, then we know that σ(H) is a finite union of at most p compact intervals in ℝ. Moreover, if V is almost periodic, then σ(H) is usually a Cantor set, and therefore, it is nowhere dense. Many recent investigations explore the structure of σ(H) in the case that V (⋅) is a semi-periodic potential; see, e. g. [68, 238, 239, 241, 316] and references cited therein. Just to quote, the main result of research article [241] by D. Damanik and A. Gorodetski states that, for every limit-periodic potential V and every ε > 0, there is a limit-periodic potential Ṽ such that ‖V − Ṽ ‖∞ < ε and the Schrödinger operator with potential Ṽ has pure point spectrum; see also [241, Theorem 1.2] and [560]. The authors have also extended the Kunz–Souillard method to produce Schrödinger operators with quasi-periodic potentials V (⋅) and pure point spectrum; see also the survey articles [237] by D. Damanik and [792] by J. You. The one-dimensional Schrödinger operators with generalized almost periodic potentials with jump discontinuities and δ-interactions have been analyzed by D. Damanik, M. Zhang and Z. Zhou in [244]; cf. also [242, 243] for some other results established by D. Damanik and his coauthors. Concerning continuous, almost periodic Schrödinger operators, we refer the reader to the landmark survey article [693] by B. Simon, as well as to [148, 240, 287, 291, 312, 562, 566, 768, 804] and references cited therein. In [257], C. R. de Oliveira and M. S. Simsen have investigated the almost periodic orbits of quantum systems and proved that, for periodic time-dependent Hamiltonians, an orbit is almost periodic if and only if the orbit is precompact. As shown by a counterexample, this is no longer true for quasi-periodic time-dependence. Some sufficient conditions ensuring the dynamical stability for the nonautonomous quantum systems under consideration are also presented. Further on, the Toda lattice is proposed by M. Toda [725] in 1967 for modeling of the positions and momenta of a chain: d a (t) = an (t)[bn+1 (t) − bn (t)], dt n d 2 b (t) = 2[an+1 (t) − an2 (t)], n ∈ ℤ, t ∈ ℝ. dt n

(157)

230 � 3 Multi-dimensional ρ-almost periodic type functions The analysis of Toda flow (157) depends heavily on the analysis of the associated Jacobi matrix J(t), which is defined by (J(t)u)n = an−1 (t)un−1 + bn (t)un + an (t)un+1 . For almost periodic type solutions of Toda lattice equations, we refer the reader to the research articles [131, 811] and references cited therein. See also the research article [256] by C. R. de Oliveira and M. S. Simsen, where the authors have constructed an example of a Floquet operator with purely point spectrum and energy instability. Some final comments and results The Wiener–Hopf–Hankel integral operators with piecewise continuous almost periodic symbols and their Fredholm characterizations have been analyzed by G. Bogveradze and L. P. Castro (see [226, pp. 65–74]). See also the research article “Bifurcation of almost periodic solutions in a difference equation” by Y. Hamaya in the edited collection [289], pp. 145–153. The almost periodic solutions for the KdV equation ut − 6uux + uxxx = 0 with almost periodic initial data u(x, 0) = V (x) have recently been analyzed by I. Binder et al. in [131] (cf. also [81, 132, 334, 400, 490]). The existence and uniqueness of almost periodic solutions for a class of singularly perturbed differential equations with piecewise constant argument have been examined by R. Yuan in [795]. Before proceeding to the next chapter of the monograph, let us also note that K. Naito has estimated fractal dimensions of equi-almost periodic attractors in terms of the order of ε-almost period and the coefficient of Hölder’s continuity [572]. Furthermore, the author has given some bounds for the ε-almost periods of quasiperiodic functions using the theory of Diophantine approximations (see, e. g., the research monograph [671] by W. M. Schmidt for more details on the subject). Periodic solutions of the magnetohydrodynamic equations with inhomogeneous boundary condition have been analyzed by I. Kondrashuk et al. in [424]; see also the research article [585] by E. A. Notte, M. D. Rojas and M. A. Rojas. Observe, finally, the time-periodic linear boundary value problems on a finite interval have recently been considered by A. S. Fokas, B. Pelloni, and D. A. Smith in [320].

�

Part II: Metrical ρ-almost periodicity

In this part, we analyze the metrically ρ-almost periodic type functions and their applications. It is our strong belief that the results presented in this part will serve as the basis of several serious investigations of almost periodicity in special metrics. We will use the same notation and terminology as in the previous part. For our investigation of metrical ρ-almost periodicity, we need to recall the notion of basic weighted function spaces. Suppose first that the set I is Lebesgue measurable, as well as that ν : I → (0, ∞) is a Lebesgue measurable function. We will be dealing with the following Banach space Lp(t) ν (I : Y ) := {u : I → Y ; u(⋅) is measurable and ‖u‖p(t) < ∞}, where p ∈ 𝒫 (I) and 󵄩 󵄩 ‖u‖p(t) := 󵄩󵄩󵄩u(t)ν(t)󵄩󵄩󵄩Lp(t) (I:Y ) . Suppose now that ν : I → (0, ∞) is any function such that the function 1/ν(⋅) is locally bounded. Then the vector space C0,ν (I : Y ) [Cb,ν (I : Y )] consists of all continuous functions u : I → Y satisfying that lim|t|→∞,t∈I ‖u(t)‖Y ν(t) = 0 [supt∈I ‖u(t)‖Y ν(t) < +∞]. When equipped with the norm ‖ ⋅ ‖ := supt∈I ‖ ⋅ (t)ν(t)‖Y , C0,ν (I : Y ) [Cb,ν (I : Y )] is a Banach space.

https://doi.org/10.1515/9783111233871-005

4 Metrically ρ-almost periodic type functions and applications In this chapter, we investigate various classes of metrically ρ-almost periodic type functions and their applications. We start with the analysis of (RX , ℬ, 𝒫 , L)-multi-almost periodic type functions:

4.1 (RX , ℬ, 𝒫, L)-multi-almost periodic type functions In the one-dimensional setting, the classes of bounded almost periodic functions and semi-periodic functions with the Hausdorff metric have been introduced by S. Stoiński [706, 707] and later reconsidered by many other authors, including A. S. Dzafarov, G. M. Gasanov [283] and A. P. Petukhov [612]. In [714], S. Stoiński has introduced and analyzed the class of unbounded almost periodic functions with the Hausdorff metric (cf. also [711]); real-valued functions almost periodic in variation and Lα -almost periodic functions (for α ∈ (0, 1), we obtain the class of Hölder almost periodic functions of order α, while for α = 1 we obtain the class of Lipschitz almost periodic functions) have been analyzed by the same author in [708] and [710], respectively. Let us recall that any function which is almost periodic in variation (any Hölder almost periodic function of order α ∈ (0, 1); any Lipschitz almost periodic function) f : ℝ → ℝ is almost periodic, while the converse statement is not true: Example 4.1.1 ([710, 713]). (i) The sum of functions f1 (⋅) and f2 (⋅), where f1 (t) := sin(√2πt), t ∈ ℝ, f2 (t) := (t − k) sin(π/(t − k)), t ∈ (k, k + 1) and f2 (t) := 0, t = k (k ∈ ℤ), is almost periodic, but not almost periodic in variation. (ii) The sum of functions f1 (⋅) and f2 (⋅), where f1 (t) := arcsin(t − 4k), t ∈ [4k − 1, 4k + 1), f1 (t) := arcsin(−t + 4k + 2), t ∈ [4k + 1, 4k + 3), f2 (t) := arcsin(√2t − 4k), t ∈ [(4k − 1)/√2, (4k + 1)/√2), f2 (t) := arcsin(−√2t + 4k + 2), t ∈ [(4k + 1)/√2, (4k + 3)/√2) (k ∈ ℤ), is almost periodic in variation, but not Lipschitz almost periodic. The above-mentioned research articles can be viewed as a certain predecessor of this work. Here, we investigate multi-dimensional almost periodic type functions in general metric; any notion of almost periodicity considered in the former paragraph as well as the notion of (IC)-almost periodicity, introduced and analyzed by M. Adamczak in [9], is a very special case of the notion Bohr (ℬ, I ′ , ρ, 𝒫 )-almost periodicity introduced in Definition 2.1.1 below (see also the notion of Ca(∞) -almost periodicity introduced and analyzed in [715] as well as [712], where the author has analyzed the class of (VC)(n) -almost periodic functions consisting of all almost periodic functions in variation f : ℝ → ℝ whose first n derivatives are almost periodic functions in variation). In contrast with the above-mentioned research studies, we primarily deal here with the vector-valued functions and the metric induced by the norm of a weighted Lp -space of functions or https://doi.org/10.1515/9783111233871-006

236 � 4 Metrically ρ-almost periodic type functions and applications the norm of a weighted C0 -space of functions. Following this way of looking at things, we generalize several known classes of multi-dimensional (Stepanov) almost periodic type functions and multi-dimensional (Stepanov) almost automorphic type functions [431]. In this section, we will always assume that 0 ≠ I ⊆ ℝn , (Y , ‖ ⋅ ‖Y ) is a complex Banach space, P ⊆ Y I , the space of all functions from I into Y , the zero function belongs to P, and 𝒫 = (P, d) is a metric space; if f ∈ P, then we designate ‖f ‖P := d(f , 0). We can also slightly generalize our notion by requiring that 𝒫 = (P, d) is a pseudometric space. The organization and main ideas of this section, which reconsiders and continues our former research studies A. Chávez et al. [194, 195, 196] and M. Fečkan et al. [304], can be briefly described as follows. Section 4.1 investigates (RX , ℬ, 𝒫 , L)-multi-almost periodic type functions. In Definition 4.1.4 and Definition 4.1.5, we introduce the classes of (strongly) (R, ℬ, 𝒫 , L)-multi-almost periodic functions and (strongly) (RX , ℬ, 𝒫 , L)-multialmost periodic functions. A concrete motivation for the introduction of such function spaces comes from many reasons, and we will only present here the following important example from [431]: Example 4.1.2. As is well known, the Euler equations in ℝn , where n ⩾ 2, describe the motion of perfect incompressible fluids. The essence of problem is to find the unknown functions u = u(x, t) = (u1 (x, t), . . . , un (x, t)) and p = p(x, t) denoting the velocity field and the pressure of the fluid, respectively, such that 𝜕u + (u ⋅ ∇)u + ∇p = 0 in ℝn × (0, T), 𝜕t div u = 0 in ℝn × (0, T),

(158)

u(x, 0) = u0 (x) in ℝn ,

where u0 = u0 (x) = (u01 (x), . . . , u0n (x)) denotes the given initial velocity field. There are many results concerning the well-posedness of (158) in the case that the initial velocity field u0 (x) belongs to some direct product of (fractional) Sobolev spaces. For our observation, it is crucial to remind the readers of the research article [595] by H. C. Pak and Y. J. Park, who investigated the well-posedness of (158) in the case that the initial veloc1 1 ity field u0 (x) belongs to the space B∞,1 (ℝn )n , where B∞,1 (ℝn ) denotes the usual Besov 0 n n space (see, e. g. [662, Definition 2.1]); let the space B∞,1 (ℝ ) be defined in the same way. It is well known that O. Sawada and R. Takada have proved, in [662, Theorem 1.5], that the almost periodicity of function u0 (x) in ℝn implies that the solution u(⋅, t) of (158) is almost periodic in ℝn for all t ∈ [0, T]. In [431, Example 8.1.4], we have analyzed the situation in which R is an arbitrary collection of sequences in ℝn , and u0 (⋅) has the property that for each sequence (bk ) in R there exists a subsequence (bkl ) of (bk ) such that the 0 sequence of translations (u0 (⋅ + bkl )) is convergent in the space B∞,1 (ℝn )n . Then for each sequence (bk ) in R there exists a subsequence (bkl ) of (bk ) such that for every t ∈ [0, T], 0 the sequence of translations (u(⋅ + bkl , t)) is convergent in the space B∞,1 (ℝn )n . Albeit we will not analyze the properties of solutions in more detail here, we would like to note

4.1 (RX , ℬ, 𝒫, L)-multi-almost periodic type functions

� 237

that the condition imposed on the initial value u0 (x) is equivalent with the condition 0 that the function u0 (x) is (R, 𝒫 )-multi-almost periodic, where P := B∞,1 (ℝn )n and the metric d is induced by the norm in P. The convolution invariance of (RX , ℬ, 𝒫 , L)-multi-almost periodicity and the invariance of (R, ℬ, 𝒫 , L)-multi-almost periodicity under the actions of infinite convolution products are investigated in Proposition 4.1.8 and Theorem 4.1.9, respectively. The convergence of sequences of (RX , ℬ, 𝒫 , L)-multi-almost periodic functions in the metric space 𝒫 is investigated in Proposition 4.1.10, while a composition principle in this direction is deduced in Theorem 4.1.12. In Section 4.1.1, we aim to generalize the spaces of multi-dimensional (Stepanov) almost automorphic functions using our approach of metrical almost periodicity; we also present here a completely new characterization of compactly almost automorphic functions. Section 4.1.2 investigates Bohr (ℬ, I ′ , ρ, 𝒫 )-multi-almost periodic type functions. In Definition 4.1.19, we introduce the classes of Bohr (ℬ, I ′ , ρ, 𝒫 )-almost periodic functions and (ℬ, I ′ , ρ, 𝒫 )-uniformly recurrent functions. Before proceeding any further, we would like to recall the following example from the introductory part of article [194], which presents a strong motivational factor for the introduction of such classes of functions: Example 4.1.3. In the homogenization theory, the crucial problem is the asymptotic behaviour of the solutions of the problem inf{∫ f (hx, Du) + ∫ ψx : u(⋅) Lipschitz continuous and u = 0 on 𝜕Ω}, Ω

(159)

Ω

where 0 ≠ Ω ⊆ ℝn is an open bounded set, ψ(⋅) is essentially bounded on Ω, and f : ℝn × ℝn → [0, ∞) satisfies the usual Carathéodory conditions. Under certain conditions, G. de Giorgi has proved, in [251], that the values in (159) converge to inf{∫ f∞ (Du) + ∫ ψx : u(⋅) Lipschitz continuous and u = 0 on 𝜕Ω}, Ω

(160)

Ω

where f∞ : ℝn → [0, ∞) is a convex function defined by f∞ (x) := lim s−n inf{ ∫ f (x, z + Du) : s→∞

(0,s)n

u(⋅) is Lipschitz continuous and u = 0 on 𝜕((0, s)n )}. In his research study [249], R. De Arcangelis has assumed that f (⋅, z) ∈ L1loc (ℝn ) for every z ∈ ℝn , |z| ⩽ f (x, z) for a. e. x ∈ ℝn and every z ∈ ℝn , and, for every z ∈ ℝn , the following

238 � 4 Metrically ρ-almost periodic type functions and applications holds: For every ε > 0, there exists a finite real number Lε > 0 such that, for every x0 ∈ ℝn , there exists τ ∈ x0 + B(0, Lε ) such that 󵄨󵄨 󵄨 󵄨󵄨f (x + τ, z) − f (x, z)󵄨󵄨󵄨 ⩽ ε(1 + f (x, z)),

for a. e. x ∈ ℝn and every z ∈ ℝn .

Then we know that for every open convex set Ω and for every essentially bounded function ψ(⋅) on Ω, the values in (159) converge to the value in (160). The above-mentioned almost type periodicity of function f (x, z) is closely connected (but not completely equivalent) with the notion of Bohr (I ′ , ρ, 𝒫 )-almost periodicity of function f (x, z), where I ′ := {(x1 , x2 , . . . , xn , 0, 0, . . . , 0) ∈ ℝ2n : xi ∈ ℝ (1 ⩽ i ⩽ n)}, ρ is the identity operator on ℂ, 2n P := L∞ : ℂ) with ν(x, z) := [1 + f (x, z)]−1 , x, z ∈ ℝn and d(f , g) := ‖f − g‖P for all ν (ℝ f , g ∈ P. We present some structural results about Bohr (ℬ, I ′ , ρ, 𝒫 )-almost periodic functions and (ℬ, I ′ , ρ, 𝒫 )-uniformly recurrent functions in Theorem 4.1.22 (here we present a completely new characterization of pre-Levitan N-almost periodic functions), Proposition 4.1.25, Proposition 4.1.28, and Proposition 4.1.32; in addition to the above, many illustrative examples are given here. In Section 4.1.3, we explain how our approach can be employed to provide certain generalizations of multi-dimensional (Stepanov) ρ-almost periodic functions. Some applications of our results to the abstract Volterra integro-differential equations are given after that, as well as some conclusions and final remarks about the considered function spaces. We start this section by introducing the following notion (observe that we do not require the continuity of function F(⋅; ⋅) here a priori): Definition 4.1.4. Suppose that 0 ≠ I ⊆ ℝn , F : I × X → Y is a given function and the following holds: If t ∈ I, b ∈ R and l ∈ ℕ, then we have t + b(l) ∈ I.

(161)

Then we say that the function F(⋅; ⋅) is (R, ℬ, 𝒫 , L)-multi-almost periodic, resp. strongly (R, ℬ, 𝒫 , L)-multi-almost periodic in the case that I = ℝn , if and only if for every B ∈ ℬ and for every sequence (bk = (b1k , b2k , . . . , bnk )) ∈ R there exist a subsequence (bkl = (b1kl , b2kl , . . . , bnkl )) of (bk ) and a function F ∗ : I × X → Y such that for every l ∈ ℕ and x ∈ B, we have F(⋅ + (b1kl , . . . , bnkl ); x) − F ∗ (⋅; x) ∈ P and for every B′ ∈ L(B; b), 󵄩󵄩 󵄩󵄩 lim sup󵄩󵄩󵄩F(⋅ + (b1kl , . . . , bnkl ); x) − F ∗ (⋅; x)󵄩󵄩󵄩 = 0, 󵄩 󵄩P l→+∞ x∈B′

(162)

resp. F(⋅ + (b1kl , . . . , bnkl ); x) − F ∗ (⋅; x) ∈ P, F ∗ (⋅ − (b1kl , . . . , bnkl ); x) − F(⋅; x) ∈ P, (162) and, for every B′ ∈ L(B; b), 󵄩󵄩 󵄩󵄩 lim sup󵄩󵄩󵄩F ∗ (⋅ − (b1kl , . . . , bnkl ); x) − F(⋅; x)󵄩󵄩󵄩 = 0. 󵄩P

l→+∞ x∈B′ 󵄩

4.1 (RX , ℬ, 𝒫, L)-multi-almost periodic type functions �

239

Definition 4.1.5. Suppose that 0 ≠ I ⊆ ℝn , F : I × X → Y is a given function, and the following holds: If t ∈ I, (b; x) ∈ RX and l ∈ ℕ, then we have t + b(l) ∈ I.

(163)

Then we say that the function F(⋅; ⋅) is (RX , ℬ, 𝒫 , L)-multi-almost periodic, resp. strongly (RX , ℬ, 𝒫 , L)-multi-almost periodic in the case that I = ℝn , if and only if for every B ∈ ℬ and for every sequence ((b; x)k = ((b1k , b2k , . . . , bnk ); xk )k ) ∈ RX there exist a subsequence ((b; x)kl = ((b1kl , b2kl , . . . , bnkl ); xkl )kl ) of ((b; x)k ) and a function F ∗ : I × X → Y such that

for every l ∈ ℕ and x ∈ B, we have F(⋅ + (b1kl , . . . , bnkl ); x + xkl ) − F ∗ (⋅; x) ∈ P and, for every B′ ∈ L(B; (b; x)), 󵄩󵄩 󵄩󵄩 lim sup󵄩󵄩󵄩F(⋅ + (b1kl , . . . , bnkl ); x + xkl ) − F ∗ (⋅; x)󵄩󵄩󵄩 = 0, 󵄩 󵄩P l→+∞ x∈B′

(164)

resp. F(⋅ + (b1kl , . . . , bnkl ); x + xkl ) − F ∗ (⋅; x) ∈ P, F ∗ (⋅ − (b1kl , . . . , bnkl ); x − xkl ) − F(⋅; x) ∈ P, (164) holds, and, for every B′ ∈ L(B; (b; x)), 󵄩󵄩 󵄩󵄩 lim sup󵄩󵄩󵄩F ∗ (⋅ − (b1kl , . . . , bnkl ); x − xkl ) − F(⋅; x)󵄩󵄩󵄩 = 0. 󵄩 󵄩P l→+∞ x∈B′

(165)

As above, we may simply conclude that the notion introduced in Definition 4.1.4 is a special case of the notion introduced in Definition 4.1.5. We will omit the term “ℬ” from the notation if X = {0}, i. e. if we consider the functions of the form F : I → Y . Remark 4.1.6. (i) In place of sequences, we can consider general nets here (see, e. g. [546, p. 9]); we will not consider such a general notion here. (ii) The usual notions of (R, ℬ)-multi-almost periodicity and (RX , ℬ)-multi-almost periodicity are special cases of the notion introduced in Definition 4.1.4, resp. Definition 4.1.5, with P = l∞ (I : Y ) := {f : I → Y : supt∈I ‖f (t)‖Y < +∞}, d(f , g) := supt∈I ‖f (t) − g(t)‖Y and 𝒫 := (l∞ (I : Y ), d); see also [546, Example 3.4, p. 14]. It suffices to require that B ∈ L(B; b) [B ∈ L(B; (b; x))] for all B ∈ ℬ and b ∈ R [(b; x) ∈ RX ]. (iii) Concerning the supremum formula for (R, ℬ)-multi-almost periodic functions, which has been established in [194, Proposition 2.6], we will only note that its proof is based on the following argument: Suppose that F : I × X → Y is (R, ℬ, 𝒫 , L)multi-almost periodic and (162) holds. If P has a linear vector structure, then for each positive real number ε > 0 we have the existence of a number l0 ∈ ℕ such that 󵄩󵄩 󵄩󵄩 sup󵄩󵄩󵄩F(⋅ + bkl ; x) − F(⋅ + bkl ; x)󵄩󵄩󵄩 ⩽ ε, 0 󵄩 󵄩P x∈B′

l ⩾ l0 .

If t − bkl + bkl ∈ I for all t ∈ I, and ‖f (⋅)‖P = ‖f (⋅ + τ)‖P for all f ∈ P and τ ∈ ℝn with 0 I + τ ⊆ I, then the above implies

240 � 4 Metrically ρ-almost periodic type functions and applications 󵄩󵄩 󵄩󵄩 sup󵄩󵄩󵄩F(⋅; x) − F(⋅ − bkl + bkl ; x)󵄩󵄩󵄩 ⩽ ε, 0 󵄩 󵄩P x∈B′

l ⩾ l0 .

(iv) The usually considered case is that one in which R, resp. RX , is equal to the collection of all sequences in ℝn , resp. ℝn × X. In [431, Example 8.1.3], we have provided an illustrative example showing the importance of case in which R is not equal to the collection of all sequences in ℝn . Suppose that k ∈ ℕ, Fi : I × X → Yi , Pi ⊆ Y I , the zero function belongs to Pi , 𝒫i = (Pi , di ) is a metric space, and Yi is a complex Banach space (1 ⩽ i ⩽ k). We define the function (F1 , . . . , Fk ) : I × X → Y1 × ⋅ ⋅ ⋅ × Yk by (F1 , . . . , Fk )(t; x) := (F1 (t; x), . . . , Fk (t; x)),

t ∈ I, x ∈ X,

as well as the product P := P1 × ⋅ ⋅ ⋅ × Pk of metric spaces P1 , . . . , Pk ; let us recall that the metric d(⋅, ⋅) on P is given by k

d((f1 , . . . , fk ), (g1 , . . . , gk )) := ∑ di (fi , gi ), i=1

fi , gi ∈ Pi (1 ⩽ i ⩽ k).

The proof of the following extension of [194, Proposition 2.4] is simple and therefore omitted. Proposition 4.1.7. (i) Suppose that k ∈ ℕ, 0 ≠ I ⊆ ℝn , (161) holds and for any sequence which belongs to R we have that any its subsequence also belongs to R. If the function Fi (⋅; ⋅) is (R, ℬ, 𝒫i , L)-multi-almost periodic, resp. strongly (R, ℬ, 𝒫i , L)-multialmost periodic in the case that I = ℝn , for 1 ⩽ i ⩽ k, then the function (F1 , . . . , Fk )(⋅; ⋅) is (R, ℬ, 𝒫 , L)-multi-almost periodic, resp. strongly (R, ℬ, 𝒫 , L)-multi-almost periodic. (ii) Suppose that k ∈ ℕ, 0 ≠ I ⊆ ℝn , (163) holds and for any sequence, which belongs to RX we have that any its subsequence also belongs to RX . If the function Fi (⋅; ⋅) is (RX , ℬ, 𝒫i , L)-multi-almost periodic, resp. strongly (RX , ℬ, 𝒫i , L)-multi-almost periodic in the case that I = ℝn , for 1 ⩽ i ⩽ k, then the function (F1 , . . . , Fk )(⋅; ⋅) is (RX , ℬ, 𝒫 , L)multi-almost periodic, resp. strongly (RX , ℬ, 𝒫 , L)-multi-almost periodic. The convolution invariance of (RX , ℬ)-multi-almost periodicity has recently been analyzed in [194, Proposition 2.5]. In general case, the convolution invariance of (RX , ℬ, 𝒫 , L)-multi-almost periodicity, resp. strong (RX , ℬ, 𝒫 , L)-multi-almost periodicity, can be analyzed only if we assume that the metric space 𝒫 has some extra features (the convolution invariance of (R, ℬ, 𝒫 , L)-multi-almost periodicity, resp. strong (R, ℬ, 𝒫 , L)multi-almost periodicity, can be analyzed similarly). Concerning this issue, we will only formulate the following result without proof (see also the proof of Theorem 4.1.9 below and [431] for many similar results of this type): Proposition 4.1.8. Let P := Cb,ν (ℝn : Y ) and d(f , g) := ‖f − g‖Cb,ν (ℝn :Y ) for all f , g ∈ P, and let there exist a positive real number c > 0 such that ν(t) ⩾ c for all t ∈ ℝn . Suppose

4.1 (RX , ℬ, 𝒫, L)-multi-almost periodic type functions

� 241

that h ∈ L1 (ℝn ), the function F(⋅; ⋅) is (RX , ℬ, 𝒫 , L)-multi-almost periodic, resp. strongly (RX , ℬ, 𝒫 , L)-multi-almost periodic, as well as for every B ∈ ℬ, B′ ∈ L(B; (b; x)), l ∈ ℕ and for every sequence ((b; x)k = ((b1k , b2k , . . . , bnk ); xk )k ) ∈ RX , we have that the mapping t 󳨃→ F(t + bkl ; x + xkl ), t ∈ ℝn is bounded, uniformly for x ∈ B′ . If there exists a function w : ℝn → (0, ∞) such that hw ∈ L1 (ℝn ) and ν(x + y) ⩽ ν(x)w(y),

x, y ∈ ℝn ,

(166)

then the function (h ∗ F)(t; x) := ∫ h(σ)F(t − σ; x) dσ,

t ∈ ℝn , x ∈ X

(167)

ℝn

is (RX , ℬ, 𝒫 , L)-multi-almost periodic, resp. strongly (RX , ℬ, 𝒫 , L)-multi-almost periodic. Theorem 4.1.9. Suppose that (R(t))t>0 ⊆ L(X, Y ) is a strongly continuous operator family. Let P := Cb,ν (ℝn : Y ) and d(f , g) := ‖f − g‖Cb,ν (ℝn :Y ) for all f , g ∈ P, let there exist a positive real number c > 0 such that ν(t) ⩾ c for all t ∈ ℝn , and let there exist a function w : ℝn → (0, ∞) such that (166) holds for all x ∈ ℝn , y ∈ [0, ∞)n and ∫(0,∞)n (1 + w(t))‖R(t)‖ dt < ∞. If f : ℝn → X is a bounded, continuous and (R, 𝒫 , L)-multi-almost periodic function, then the function F : ℝn → Y , given by (20), is bounded, continuous and (R, 𝒫 , L)-multi-almost periodic. Proof. Let (bk = (b1k , b2k , . . . , bnk )) ∈ R be given. Then there exist a subsequence (bkl = (b1kl , b2kl , . . . , bnkl )) of (bk ) and a function f ∗ : ℝn → X such that lim [f (⋅ + (b1kl , . . . , bnkl )) − f ∗ (⋅)] = 0

l→+∞

(168)

in P. Since we have assumed that the function f (⋅) is bounded and there exists a positive real number c > 0 such that ν(t) ⩾ c for all t ∈ ℝn , our choice of function space P simply yields that the function f ∗ : ℝn → X is bounded. Let us prove that the function f ∗ (⋅) is continuous at the point t ∈ ℝn . If t′ ∈ ℝn and ε > 0, then we have 󵄩󵄩 ∗ ∗ ′ 󵄩 󵄩󵄩f (t) − f (t )󵄩󵄩󵄩Y 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ⩽ 󵄩󵄩󵄩f ∗ (t) − f (t + bkl )󵄩󵄩󵄩Y + 󵄩󵄩󵄩f (t + bkl ) − f (t′ + bkl )󵄩󵄩󵄩Y + 󵄩󵄩󵄩f (t′ + bkl ) − f ∗ (t′ )󵄩󵄩󵄩Y . The first addend and the third addend are less or equal than ε/3 for some l = l0 ∈ ℕ, since (168) holds and ν(t) ⩾ c for all t ∈ ℝn , while the second addend is less or equal than ε/3 whenever ‖f (t + bkl ) − f (t′ + bkl )‖Y ⩽ δ, which is determined from the continuity 0 0 of function f (⋅) at the point t; this implies the claimed. Moreover, we have F(t) =

∫ R(s)f (t − s) ds [0,∞)n

for all t ∈ ℝn ,

242 � 4 Metrically ρ-almost periodic type functions and applications the function F(⋅) is bounded and continuous due to the dominated convergence theorem. The integral ∫[0,∞)n R(s)f ∗ (t − s) ds is well defined for all t ∈ ℝn , and we have: lim

l→∞

∫ R(s)[f (⋅ + bkl − s) − f ∗ (⋅ − s)] ds = 0

[0,∞)n

in P, because 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∫ R(s)f (t + bk − s) ds − ∫ R(s)f ∗ (t − s) ds󵄩󵄩󵄩 ν(t) 󵄩󵄩 󵄩󵄩 l 󵄩󵄩 󵄩󵄩Y [0,∞)n [0,∞)n ⩽

󵄩 󵄩 󵄩 󵄩 ∫ 󵄩󵄩󵄩R(s)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩f (t + bkl − s) − f ∗ (t − s)󵄩󵄩󵄩ν(t) ds

[0,∞)n

⩽

󵄩 󵄩 󵄩 󵄩 ∫ 󵄩󵄩󵄩R(s)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩f (t + bkl − s) − f ∗ (t − s)󵄩󵄩󵄩ν(t − s)w(s) ds

[0,∞)n

󵄩 󵄩 󵄩 󵄩 ⩽ 󵄩󵄩󵄩f (⋅ + bkl − s) − f ∗ (⋅ − s)󵄩󵄩󵄩P ⋅ ∫ 󵄩󵄩󵄩R(s)󵄩󵄩󵄩w(s) ds,

t ∈ ℝn , l ∈ ℕ.

[0,∞)n

We can similarly prove that lim

l→∞

∫ R(s)[f ∗ (⋅ − bkl − s) − f (⋅ − s)] ds = 0

[0,∞)n

in P, which completes the proof. Before proceeding any further, we feel it is our duty to say that the assumptions in Proposition 4.1.8, resp. Theorem 4.1.9, imply that the function F(⋅; ⋅) under our consideration is (RX , ℬ)-multi-almost periodic, resp. (R, ℬ)-multi-almost periodic. Therefore, we deal here with certain subclasses of (RX , ℬ)-multi-almost periodic functions, resp. (R, ℬ)-multi-almost periodic functions; see also Remark 4.1.18(i) below, as well as Proposition 4.1.30 and Theorem 4.1.31, where we deal with certain subclasses of almost automorphic functions. In [194, Proposition 2.7, Proposition 2.8], we have examined the uniform convergence of (RX , ℬ)-multi-almost periodic functions [(R, ℬ)-multi-almost periodic functions]. The situation is much more complicated with the notion introduced in this section; concerning the above-mentioned issue, we will first note that, in Definition 4.1.4, we can additionally require that for each l ∈ ℕ we also have F(⋅ + (b1kl , . . . , bnkl ); x) ∈ P, F ∗ (⋅; x) ∈ P, resp. F(⋅ + (b1kl , . . . , bnkl ); x) ∈ P, F ∗ (⋅; x) ∈ P, F ∗ (⋅ − (b1kl , . . . , bnkl ); x) ∈ P and F(⋅; x) ∈ P, if I = ℝn and we consider strongly (R, ℬ, 𝒫 , L)-multi-almost periodic functions; if this is the case, then we say that the function F(⋅; ⋅) is (R, ℬ, 𝒫 , L)-multi-almost periodic of type 1, resp. strongly (R, ℬ, 𝒫 , L)-multi-almost periodic of type 1. The class of

4.1 (RX , ℬ, 𝒫, L)-multi-almost periodic type functions �

243

(RX , ℬ, 𝒫 , L)-multi-almost periodic functions of type 1, resp. strongly (RX , ℬ, 𝒫 , L)-multialmost periodic functions of type 1, is introduced analogously. Now we are able to state and prove the following result: Proposition 4.1.10. Suppose that P has a linear vector structure, P is complete, and the metric d is translation invariant in the sense that d(f + g, h + g) = d(f , h) whenever f , h, f +g, h+g ∈ P. Suppose further that, for each integer j ∈ ℕ the function Fj : I ×X → Y is (RX , ℬ, 𝒫 , L)-multi-almost periodic of type 1 as well as that, for every sequence which belongs to RX , any its subsequence also belongs to RX . If F : I × X → Y and for each B ∈ ℬ, (b; x) = ((bk ; xk )) = ((b1k , b2k , . . . , bnk ); xk ) ∈ RX , B′ ∈ L(B; (b; x)) and 󵄩󵄩 󵄩󵄩 sup󵄩󵄩󵄩Fi (⋅ + bkl ; x + xkl ) − F(⋅ + bkl ; x + xkl )󵄩󵄩󵄩 = 0, 󵄩 󵄩P (i,l)→+∞ x∈B′ lim

(169)

then the function F(⋅; ⋅) is (RX , ℬ, 𝒫 , L)-multi-almost periodic of type 1. Proof. Since d is translation invariant, we have ‖f + g‖P ⩽ ‖f ‖P + ‖g‖P for all f , g ∈ P. Let the set B ∈ ℬ and the sequence ((b; x) = ((b1k , b2k , . . . , bnk ); xk )) ∈ RX be given. Let B′ ∈ L(B; (b; x)) be given, as well. Since for each sequence, which belongs to RX we have that any its subsequence also belongs to RX , the diagonal procedure can be used to get the existence of a subsequence ((bkl ; xkl ) = ((b1kl , b2kl , . . . , bnkl ); xkl )) of ((bk ; xk )) such that for each integer j ∈ ℕ there exists a function Fj∗ : I × X → Y such that 󵄩󵄩 󵄩󵄩 lim sup󵄩󵄩󵄩Fj (⋅ + (b1kl , . . . , bnkl ); x + xkl ) − Fj∗ (⋅; x)󵄩󵄩󵄩 = 0. 󵄩P

l→+∞ x∈B′ 󵄩

(170)

Fix a real number ε > 0. Since 󵄩󵄩 󵄩󵄩 sup󵄩󵄩󵄩Fi∗ (⋅; x) − Fj∗ (⋅; x)󵄩󵄩󵄩 󵄩 󵄩P ′ x∈B 󵄩󵄩 ∗ 󵄩󵄩 ⩽ sup󵄩󵄩󵄩Fi (⋅; x) − Fi (⋅ + (b1kl , . . . , bnkl ); x + xkl )󵄩󵄩󵄩 󵄩 󵄩P x∈B′ 󵄩󵄩 󵄩󵄩 + sup󵄩󵄩󵄩Fi (⋅ + (b1kl , . . . , bnkl ); x + xkl ) − Fj (⋅ + (b1kl , . . . , bnkl ); x + xkl )󵄩󵄩󵄩 󵄩 󵄩P ′ x∈B 󵄩󵄩 󵄩 󵄩 + sup󵄩󵄩󵄩Fj (⋅ + (b1kl , . . . , bnkl ); x + xkl ) − Fj∗ (⋅; x)󵄩󵄩󵄩 , 󵄩 󵄩P ′ x∈B and (170) holds, there exists an integer l0 ∈ ℕ such that for all integers l ⩾ l0 we have: 󵄩󵄩 󵄩󵄩 sup󵄩󵄩󵄩Fi∗ (⋅; x) − Fi (⋅ + (b1kl , . . . , bnkl ); x + xkl )󵄩󵄩󵄩 󵄩 󵄩P x∈B′ 󵄩󵄩 󵄩󵄩 + sup󵄩󵄩󵄩Fj (⋅ + (b1kl , . . . , bnkl ); x + xkl ) − Fj∗ (⋅; x)󵄩󵄩󵄩 < 2ε/3. 󵄩 󵄩P ′ x∈B Using (169), we get the existence of integers N(ε) ∈ ℕ and l1 ⩾ l0 such that for all integers l ⩾ l1 and i, j ∈ ℕ with min(i, j) ⩾ N(ε) we have

244 � 4 Metrically ρ-almost periodic type functions and applications 󵄩󵄩 󵄩󵄩 sup󵄩󵄩󵄩Fi (⋅ + (b1kl , . . . , bnkl ); x + xkl ) − Fj (⋅ + (b1kl , . . . , bnkl ); x + xkl )󵄩󵄩󵄩 < ε/3. 󵄩 󵄩P x∈B′ This implies that (Fj∗ (⋅; x)) is a Cauchy sequence in P for each element x ∈ B′ and therefore convergent to a function F ∗ (⋅; x) ∈ P, say; clearly, 󵄩 󵄩 lim sup󵄩󵄩󵄩Fj∗ (⋅; x) − F ∗ (⋅; x)󵄩󵄩󵄩P = 0.

j→+∞ x∈B′

Further on, observe that for each j ∈ ℕ we have: 󵄩󵄩 󵄩󵄩 sup󵄩󵄩󵄩F(⋅ + (b1kl , . . . , bnkl ); x + xkl ) − F ∗ (⋅; x)󵄩󵄩󵄩 󵄩 󵄩P x∈B′ 󵄩󵄩 󵄩󵄩 ⩽ sup󵄩󵄩󵄩F(⋅ + (b1kl , . . . , bnkl ); x + xkl ) − Fj (⋅ + (b1kl , . . . , bnkl ); x + xkl )󵄩󵄩󵄩 󵄩 󵄩P ′ x∈B 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩 󵄩 + sup󵄩󵄩󵄩Fj (⋅ + (b1kl , . . . , bnkl ); x + xkl ) − Fj∗ (⋅; x)󵄩󵄩󵄩 + sup󵄩󵄩󵄩Fj∗ (⋅; x) − F ∗ (⋅; x)󵄩󵄩󵄩 . 󵄩 󵄩 󵄩 󵄩P P x∈B′ x∈B′ There exists a number j0 (ε) ∈ ℕ such that for all integers j ⩾ j0 we have that the third addend in the above estimate is less or greater than ε/3. After that, using condition (169), we can find an integer N ∈ ℕ such that N ⩾ j0 and for every l ∈ ℕ and j ∈ ℕ with min(j, l) ⩾ N, we have 󵄩󵄩 󵄩󵄩 sup󵄩󵄩󵄩F(⋅ + (b1kl , . . . , bnkl ); x + xkl ) − Fj (⋅ + (b1kl , . . . , bnkl ); x + xkl )󵄩󵄩󵄩 < ε/3. 󵄩 󵄩P x∈B′ Take j = N and apply (170) to complete the proof. Remark 4.1.11. Consider the situation of Theorem 4.1.9. If we assume that f : ℝn → X is a bounded, continuous and (R, 𝒫 , L)-multi-almost periodic function of type 1, then the function F : ℝn → Y , given by (20), will be bounded, continuous and (R, 𝒫 , L)-multialmost periodic of type 1. We can simply prove the following: (i) Suppose that c ∈ ℂ, cf ∈ P for all f ∈ P, and there exists a finite real number ϕ(c) > 0 such that ‖cf ‖P ⩽ ϕ(c)‖f ‖P for all f ∈ P. If the function F : I × X → Y is (strongly) (R, ℬ, 𝒫 , L)-multi-almost periodic [(strongly) (RX , ℬ, 𝒫 , L)-multi-almost periodic], then the function cF(⋅; ⋅) is likewise (strongly) (R, ℬ, 𝒫 , L)-multi-almost periodic [(strongly) (RX , ℬ, 𝒫 , L)-multi-almost periodic]. (ii) Suppose that P has a linear vector structure, the metric d is translation invariant and, for every complex number c ∈ ℂ, we have cf ∈ P, f ∈ P and there exists a finite real number ϕ(c) > 0 such that ‖cf ‖P ⩽ ϕ(c)‖f ‖P for all f ∈ P. Suppose also that for each sequence of collection R [RX ] any its subsequence also belongs to R [RX ]. Then the space consisting of all (strongly) (R, ℬ, 𝒫 , L)-multi-almost periodic [(strongly) (RX , ℬ, 𝒫 , L)-multi-almost periodic] functions is a vector space.

4.1 (RX , ℬ, 𝒫, L)-multi-almost periodic type functions �

245

(iii) Suppose that τ ∈ ℝn , τ + I ⊆ I, x0 ∈ X and, for every f ∈ P, we have f (⋅ + τ) ∈ P and the existence of a finite real number cτ > 0 such that ‖f (⋅ + τ)‖P ⩽ cτ ‖f ‖P for all f ∈ P. Define ℬx0 := {−x0 + B : B ∈ ℬ} and, for every B ∈ ℬ and b ∈ R [(b; x) ∈ RX ], L′ (−x0 + B; b) := {−x0 + B′ : B′ ∈ L(B; b)} [L′ (−x0 + B; (b; x)) := {−x0 + B′ : B′ ∈ L(B; (b, x))}]. If the function F : I × X → Y is (strongly) (R, ℬ, 𝒫 , L)-multi-almost periodic [(strongly) (RX , ℬ, 𝒫 , L)-multi-almost periodic], then the function F(⋅ + τ; ⋅ + x0 ) is (strongly) (R, ℬx0 , 𝒫 , L′ )-multi-almost periodic [(strongly) (RX , ℬx0 , 𝒫 , L′ )-multialmost periodic]. The pointwise products of (R, ℬ)-multi-almost periodic functions have been analyzed in [194, Proposition 2.20]; we would like to note that the pointwise products of (strongly) (R, ℬ, 𝒫 , L)-multi-almost periodic functions can be analyzed under certain extra assumptions. Details can be left to the interested readers. Concerning composition principles, we will prove only one result, which corresponds to [194, Theorem 2.47]. Suppose that F : I × X → Y and G : I × Y → Z are given functions. Let 𝒫F = (PF , df ), 𝒫G = (PG , dG ) and 𝒫W = (PW , dW ) be three metric spaces consisting of certain functions from Y I , Z I and Z I , respectively. Theorem 4.1.12. Suppose that F : I × X → Y is (R, ℬ, 𝒫F , L)-multi-almost periodic, resp. strongly (R, ℬ, 𝒫F , L)-multi-almost periodic in the case that I = ℝn , G : I × Y → Z, the metric dW is translation invariant, for any sequence belonging to R any its subsequence belongs to R, as well as the following condition holds: (i) For every b ∈ R, for every B ∈ ℬ, for every functions f : I × X → Y and g : I × X → Y , for every sequences of functions (fl : I × X → Y ), (gl : I × X → Y ) and (hl : I × X → Y ), and for every set B′ ∈ L(B; b), we can find a subsequence b′ of b, a subsequence (fm ) of (fl ), a function G∗ : I × Y → Z and a finite real constant c > 0 such that 󵄩󵄩 󵄩󵄩 lim sup󵄩󵄩󵄩G(⋅ + b′m ; f (⋅; x)) − G∗ (⋅; f (⋅; x))󵄩󵄩󵄩 = 0, 󵄩 󵄩 PG ′ x∈B

m→+∞

(171)

and 󵄩󵄩 󵄩󵄩 󵄩 󵄩 sup󵄩󵄩󵄩G(⋅ + bm ; gm (⋅; x)) − G(⋅ + bm ; g(⋅; x))󵄩󵄩󵄩 ⩽ c sup󵄩󵄩󵄩gm (⋅; x) − g(⋅; x)󵄩󵄩󵄩P , F 󵄩 󵄩 PG ′ ′ x∈B x∈B

(172)

whenever m ∈ ℕ and gm (⋅; x) − g(⋅; x) ∈ PF for all x ∈ B′ , resp. (171) and (172) hold, as well as 󵄩󵄩 󵄩󵄩 lim sup󵄩󵄩󵄩G∗ (⋅ − b′m ; fm (⋅; x)) − G(⋅; fm (⋅; x))󵄩󵄩󵄩 = 0 m→+∞ 󵄩 󵄩PG ′ x∈B and 󵄩󵄩 󵄩󵄩 󵄩 󵄩 sup󵄩󵄩󵄩G(⋅; hm (⋅; x)) − G(⋅; h(⋅; x))󵄩󵄩󵄩 ⩽ c sup󵄩󵄩󵄩hm (⋅; x) − h(⋅; x)󵄩󵄩󵄩P , F 󵄩 󵄩 PG ′ ′ x∈B x∈B whenever m ∈ ℕ and hm (⋅; x) − h(⋅; x) ∈ PF for all x ∈ B′ .

246 � 4 Metrically ρ-almost periodic type functions and applications Then the function W (⋅; ⋅) is (R, ℬ, 𝒫W , L)-multi-almost periodic, resp. strongly (R,ℬ,𝒫W ,L)multi-almost periodic. Proof. We will consider only (R, ℬ, 𝒫F , L)-multi-almost periodic functions. Let the set B ∈ ℬ and the sequence (bk ) ∈ R be given, and let B′ ∈ L(B; b). By definition, there exist a subsequence (bkl ) of (bk ) and a function F ∗ : I × X → Y such that, for every l ∈ ℕ and x ∈ B, we have F(⋅ + bkl ; x) − F ∗ (⋅; x) ∈ P, and (162) holds. Since (bkl ) ∈ R, we can use assumption (i), with f = F ∗ and the sequence of functions (fl (⋅; ⋅) = F(⋅ + bkl ; ⋅)), to find a subsequence (bkl ) of (bkl ) and a function G∗ : I × Y → Z such that conditions (171) m and (172) hold. It suffices to show that 󵄩󵄩 󵄩󵄩 lim 󵄩󵄩G(⋅ + bkl ; F(⋅ + bkl ; x)) − G∗ (⋅; F ∗ (⋅; x))󵄩󵄩󵄩 = 0. m m 󵄩PW

m→+∞󵄩 󵄩

Denote τ m := bkl for all m ∈ ℕ. We have (x ∈ B′ , m ∈ ℕ): m

󵄩󵄩 󵄩 󵄩󵄩G(⋅ + τ m ; F(⋅ + τ m ; x)) − G∗ (⋅; F ∗ (⋅; x))󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩PW 󵄩󵄩 󵄩󵄩 ⩽ 󵄩󵄩󵄩G(⋅ + τ m ; F(⋅ + τ m ; x)) − G(⋅ + τ m ; F ∗ (⋅; x))󵄩󵄩󵄩 󵄩 󵄩 PW 󵄩󵄩 󵄩󵄩 ∗ ∗ ∗ 󵄩 󵄩 + 󵄩󵄩G(⋅ + τ m ; F (⋅; x)) − G (⋅; F (⋅; x))󵄩󵄩 󵄩 󵄩PW 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 ∗ ⩽ c󵄩󵄩󵄩F(⋅ + τ m ; x) − F (⋅; x)󵄩󵄩󵄩 + 󵄩󵄩󵄩G(⋅ + τ m ; F ∗ (⋅; x)) − G∗ (⋅; F ∗ (⋅; x))󵄩󵄩󵄩 , 󵄩 󵄩 PF 󵄩 󵄩PW which tends to zero as m → +∞. This simply completes the proof. Remark 4.1.13. Suppose, additionally, that F : I × X → Y is (R, ℬ, 𝒫F , L)-multi-almost periodic of type 1, resp. strongly (R, ℬ, 𝒫F , L)-multi-almost periodic of type 1 in the case that I = ℝn . If we assume, in condition (i), that we also have G(⋅+b′m ; f (⋅)) ∈ PW for all f ∈ PF , resp. G(⋅; f (⋅)) ∈ PW for all f ∈ PF , then the function W (⋅; ⋅) will be (R, ℬ, 𝒫W , L)-multialmost periodic of type 1, resp. strongly (R, ℬ, 𝒫W , L)-multi-almost periodic of type 1. 4.1.1 Generalization of multi-dimensional (Stepanov) almost automorphy The main aim of this subsection is to explain how we can use the approach obeyed so far to generalize the important classes of multi-dimensional (Stepanov) almost automorphic functions. Moreover, we will provide a completely new characterization of compactly almost automorphic functions by choosing the pivot space P to be a weighted Lp -space. Concerning multi-dimensional almost automorphic functions, we will clarify the following result (for simplicity, we assume here that the corresponding collection L(B; b) [L(B; (b; x))] is equal to {{x} : x ∈ B} for each set B ∈ ℬ and each sequence b ∈ R [(b; x) ∈ RX ]; observe also that the notion from Definition 3.4.1 can be extended using this approach with the function L):

4.1 (RX , ℬ, 𝒫, L)-multi-almost periodic type functions

� 247

Theorem 4.1.14. Suppose that F : ℝn × X → Y is a continuous function, as well as that p ∈ 𝒫 (ℝn ), ν : I → (0, ∞) is a Lebesgue measurable function and L(B; b) = {{x} : x ∈ B} [L(B; (b; x)) = {{x} : x ∈ B}] for each set B ∈ ℬ and each sequence b ∈ R [(b; x) ∈ RX ]. Let p(t) P := Lν (ℝn : Y ) and d(f , g) := ‖f − g‖Lp(t) (ℝn :Y ) for all f , g ∈ P. ν

(i) Suppose that p ∈ D+ (ℝn ), ν ∈ Lp(t) (ℝn ) and the function F(⋅; ⋅) is half-(R, ℬ)-multialmost automorphic, resp. (R, ℬ)-multi-almost automorphic. If the function F(⋅; x) is bounded for every fixed element x ∈ X, then the function F(⋅) is (R, ℬ, 𝒫 , L)-multialmost periodic, resp. strongly (R, ℬ, 𝒫 , L)-multi-almost periodic. (ii) Suppose that p ∈ D+ (ℝn ), ν ∈ Lp(t) (ℝn ) and the function F(⋅; ⋅) is half-(RX , ℬ)-multialmost automorphic, resp. (RX , ℬ)-multi-almost automorphic. If the function F(⋅; ⋅) is bounded, then the function F(⋅) is (RX , ℬ, 𝒫 , L)-multi-almost periodic, resp. strongly (RX , ℬ, 𝒫 , L)-multi-almost periodic. (iii) If the function F(⋅; x) is uniformly continuous for every fixed element x ∈ X, and the function F(⋅) is (R, ℬ, 𝒫 , L)-multi-almost periodic, resp. strongly (R, ℬ, 𝒫 , L)-multialmost periodic, then the function F(⋅; ⋅) is half-(R, ℬ)-multi-almost automorphic, resp. (R, ℬ)-multi-almost automorphic. (iv) If the function F(⋅; ⋅) is uniformly continuous and (RX , ℬ, 𝒫 , L)-multi-almost periodic, resp. strongly (RX , ℬ, 𝒫 , L)-multi-almost periodic, then the function F(⋅; ⋅) is half-(RX , ℬ)-multi-almost automorphic, resp. (RX , ℬ)-multi-almost automorphic. Proof. We will prove only (ii) and (iv). Let B ∈ ℬ, x ∈ B and ((b; x)k = ((b1k , b2k , . . . , bnk ); xk )k ) ∈ RX be given. We know that there exist a subsequence ((b; x)kl = ((b1kl , b2kl , . . . , bnkl ); xkl )kl ) of ((b; x)k ) and a function F ∗ : I × X → Y such that lim F(t + (b1kl , . . . , bnkl ); x + xkl ) = F ∗ (t; x)

(173)

lim F ∗ (t − (b1kl , . . . , bnkl ); x − xkl ) = F(t; x)

(174)

l→+∞

holds, resp. (173) and l→+∞

hold true. This immediately implies that the function F ∗ (⋅; ⋅) is bounded; since we have assumed that p ∈ D+ (ℝn ) and ν ∈ Lp(t) (ℝn : Y ), the dominated convergence theorem (see Lemma 1.1.10(iv)) implies that 󵄩󵄩 󵄩󵄩 lim 󵄩󵄩󵄩[F(t + bkl ; x + xkl ) − F ∗ (t; x)] ⋅ ν(t)󵄩󵄩󵄩 p(t) n = 0, 󵄩L (ℝ :Y ) l→+∞󵄩

(175)

resp. (175) and 󵄩󵄩 󵄩󵄩 lim 󵄩󵄩󵄩[F ∗ (t − bkl ; x − xkl ) − F(t; x)] ⋅ ν(t)󵄩󵄩󵄩 p(t) n = 0 󵄩L (ℝ :Y ) l→+∞󵄩 are true. Therefore, the function F(⋅; ⋅) is (RX , ℬ, 𝒫 , L)-multi-almost periodic, resp. strongly (RX , ℬ, 𝒫 , L)-multi-almost periodic. To prove (iv), fix B ∈ ℬ and ((b; x)k =

248 � 4 Metrically ρ-almost periodic type functions and applications ((b1k , b2k , . . . , bnk ); xk )k ) ∈ RX . We know that there exist a subsequence ((b; x)kl = ((b1kl , b2kl , . . . , bnkl ); xkl )kl ) of ((b; x)k ) and a function F ∗ : I × X → Y such that, for every l ∈ ℕ and x ∈ B, we have F(⋅ + (b1kl , . . . , bnkl ); x + xkl ) − F ∗ (⋅; x) ∈ P and (164) holds for

all x ∈ B, resp. F(⋅+(b1kl , . . . , bnkl ); x +xkl )−F ∗ (⋅; x) ∈ P, F ∗ (⋅−(b1kl , . . . , bnkl ); x −xkl )−F(⋅; x) ∈ P, (164) holds, and (164)–(165) hold for all x ∈ B. Therefore, 󵄩󵄩 󵄩󵄩 lim 󵄩󵄩󵄩[F(t + bkl ; x + xkl ) − F ∗ (t; x)] ⋅ ν(t)󵄩󵄩󵄩 p(t) = 0, 󵄩L (KT :Y )

l→+∞󵄩

(176)

resp. (176) and 󵄩󵄩 󵄩󵄩 lim 󵄩󵄩󵄩[F ∗ (t − bkl ; x − xkl ) − F(t; x)] ⋅ ν(t)󵄩󵄩󵄩 p(t) =0 󵄩L (KT :Y ) l→+∞󵄩 are true for all T ∈ ℕ. Then Lemma 1.1.10(ii) implies that the above equalities hold with the function p(t) replaced with the constant function 1 therein, so that there exists a set N ⊆ ℝn of Lebesgue measure zero such that (173), resp. (173) and (174), hold for all x ∈ B and t ∈ ℝn ∖ N. Using the uniform continuity of function F(⋅; ⋅) and the well known 3 − ε argument, we can simply show that the function F ∗ (⋅; ⋅) is uniformly continuous and the limit equality (173), resp. the limit equalities (173) and (174), hold for all x ∈ B and t ∈ ℝn , since (F(t + bkl ; x + xkl )) and (F ∗ (t − bkl ; x − xkl )) are Cauchy sequences in Y . Corollary 4.1.15. Suppose that the function F : ℝn → Y is measurable, p ∈ D+ (ℝn ), ν ∈ Lp(t) (ℝn ), R is the collection of all sequences in ℝn , 𝒫 and L are defined as above. Then the following holds: (i) If F(⋅) is almost automorphic, then F(⋅) is strongly (R, 𝒫 , L)-multi-almost periodic. (ii) If F(⋅) is uniformly continuous, then F(⋅) is compactly almost automorphic if and only if F(⋅) is strongly (R, 𝒫 , L)-multi-almost periodic. In the sequel, if L is a function as in the previous two statements, we will also simply say that f (⋅) is (strongly) (R, 𝒫 )-multi-almost periodic [(strongly) (RX , 𝒫 )-multi-almost periodic]; it is not clear whether the converse in (i) holds true. Further on, let p1 ∈ D+ (ℝn ), ν ∈ Lp1 (t) (ℝn : ℂ) and L be defined as above. It is worth noting that we have recently analyzed various classes of multi-dimensional Stepanov almost automorphic functions in [431, Section 8.2] as well as that the class of essentially bounded Stepanov (Ω, p(u))-(R, ℬ)-multi-almost automorphic functions, resp., the class of essentially bounded Stepanov (Ω, p(u))-(RX , ℬ)-multi-almost automorphic functions, can be viewed as a special subclass of (R, ℬ, 𝒫1 , L)-multi-almost periodic functions, resp., (RX , ℬ, 𝒫1 , L)-multi-almost periodic functions; here, Ω is any compact subset of ℝn with positive Lebesgue measure and p ∈ D+ (Ω). In order to see this, we need to recall the following definition: Definition 4.1.16. Suppose that the function F : ℝn × X → Y satisfies that the Bochner ̂ x)](u) := F(t + u; x), t ∈ ℝn , x ∈ X, transform F̂ : ℝn × X → Lp(u) (Ω : Y ), defined by [F(t; u ∈ Ω, is well defined and continuous. Then we say that the function F(⋅; ⋅) is:

4.1 (RX , ℬ, 𝒫, L)-multi-almost periodic type functions �

249

(i) Stepanov (Ω, p(u))-(R, ℬ)-multi-almost automorphic if and only if the function F̂ : ℝn × X → Lp(u) (Ω : Y ) is (R, ℬ)-multi-almost automorphic; (iii) Stepanov (Ω, p(u))-(RX , ℬ)-multi-almost automorphic if and only if the function F̂ : ℝn × X → Lp(u) (Ω : Y ) is (RX , ℬ)-multi-almost automorphic. Suppose now that the function is Stepanov (Ω, p(u))-(R, ℬ)-multi-almost automorphic, resp. Stepanov (Ω, p(u))-(RX , ℬ)-multi-almost automorphic, and the function F : ℝn × X → Y satisfies that the function F(⋅; x) is essentially bounded for every fixed element x ∈ X, resp. the function F(⋅; ⋅) is essentially bounded. Then, for every B ∈ ℬ and for every sequence (b)k ∈ R, resp. for every B ∈ ℬ and for every sequence ((b; x)k ) ∈ RX , there exist a subsequence (b)kl ∈ R of (b)k ∈ R, resp. ((b; x)kl ) of ((b; x)k ) and a function G : ℝn × X → Lp(u) (Ω : Y ) such that 󵄩󵄩 󵄩󵄩 lim 󵄩󵄩󵄩F(t + u + (b1kl , . . . , bnkl ); x) − [G(t; x)](u)󵄩󵄩󵄩 p(u) =0 󵄩L (Ω:Y )

(177)

󵄩󵄩 󵄩󵄩 lim 󵄩󵄩󵄩[G(t − u + (b1kl , . . . , bnkl ); x)](u) − F(t + u; x)󵄩󵄩󵄩 p(u) = 0, 󵄩L (Ω:Y )

(178)

󵄩󵄩 󵄩󵄩 lim 󵄩󵄩󵄩F(t + u + (b1kl , . . . , bnkl ); x + xkl ) − [G(t; x)](u)󵄩󵄩󵄩 p(u) =0 󵄩L (Ω:Y )

(179)

󵄩󵄩 󵄩󵄩 lim 󵄩󵄩󵄩[G(t − u + (b1kl , . . . , bnkl ); x − xkl )](u) − F(t + u; x)󵄩󵄩󵄩 p(u) = 0, 󵄩L (Ω:Y )

(180)

l→+∞󵄩

and l→+∞󵄩

resp. l→+∞󵄩

and l→+∞󵄩

hold for all x ∈ B and t ∈ ℝn . The convergence in (177) and (178), resp. (179) and (180), implies the pointwise convergence for all u ∈ Ω ∖ Nt , where m(Nt ) = 0. We can therefore define the limit function F ∗ : ℝn × X → Y by F ∗ (t; x) := [G(t − u; x)](u), t ∈ ℝn , x ∈ X, where u ∈ Ω∖Nt for all t ∈ ℝn . The function F ∗ (⋅; x) is essentially bounded for every fixed element x ∈ X, resp. the function F ∗ (⋅; ⋅) is essentially bounded; using the dominated convergence theorem, we get the following (see also Section 4.1.3 below, where we have obeyed a slightly different approach for the class of multi-dimensional Stepanov almost periodic functions): Theorem 4.1.17. Suppose that Ω is any compact subset of ℝn with positive Lebesgue measure, p ∈ D+ (Ω) and the function F : ℝn × X → Y satisfies that the function F(⋅; x) is essentially bounded for every fixed element x ∈ X, resp. the function F(⋅; ⋅) is essentially bounded. If the function F(⋅; ⋅) is Stepanov (Ω, p(u))-(R, ℬ)-multi-almost automorphic, resp. Stepanov (Ω, p(u))-(RX , ℬ)-multi-almost automorphic, then the function F(⋅; ⋅) is (R, ℬ, 𝒫1 , L)-multi-almost periodic, resp. (RX , ℬ, 𝒫1 , L)-multi-almost periodic, where p1 ∈

250 � 4 Metrically ρ-almost periodic type functions and applications D+ (ℝn ), ν ∈ Lp1 (t) (ℝn : ℂ) is positive, L being defined as in the formulation of Theop (t) rem 4.1.14, P1 := Lν1 (ℝn : Y ) and d(f , g) := ‖f − g‖Lp1 (t) (ℝn :Y ) for all f , g ∈ P1 . ν

Now we will present several useful observations in the case that ν : I → (0, ∞) satisfies that 1/ν(⋅) is locally bounded, P := Cb,ν (I : Y ) and d(f , g) := ‖f − g‖Cb,ν (I:Y ) for all f , g ∈ P: Remark 4.1.18. (i) Suppose that there exists a positive real number c > 0 such that ν(t) ⩾ c for all t ∈ I, and B ∈ L(B; b) for all B ∈ ℬ and b ∈ R [B ∈ L(B; (b; x)) for all B ∈ ℬ and (b; x) ∈ RX ]. Then any (R, ℬ, 𝒫 , L)-multi-almost periodic function [(RX , ℬ, 𝒫 , L)-multi-almost periodic function] is (R, ℬ)-multi-almost periodic [(RX , ℬ)-multi-almost periodic]. (ii) Suppose that there exists a positive real number c > 0 such that ν(t) ⩽ c for all t ∈ I, and B ∈ L(B; b) for all B ∈ ℬ and b ∈ R [B ∈ L(B; (b; x)) for all B ∈ ℬ and (b; x) ∈ RX ]. Then any (R, ℬ)-multi-almost periodic function [(RX , ℬ)-multi-almost periodic function] is (R, 𝒫 , L)-multi-almost periodic [(RX , 𝒫 , L)-multi-almost periodic]. (iii) Suppose that I = ℝn . Then any (R, 𝒫 , L)-multi-almost periodic function [(RX , 𝒫 , L)multi-almost periodic function] is compactly (R, ℬ)-multi-almost automorphic [compactly (RX , ℬ)-multi-almost automorphic]. This almost immediately follows from the corresponding definitions and the fact that the function 1/ν(⋅) is bounded on any compact of ℝn . 4.1.2 Bohr (ℬ, I ′ , ρ, 𝒫 )-multi-almost periodic type functions Arguing as in Remark 4.1.6(ii) above, we may simply conclude that the notion of Bohr (ℬ, I ′ , ρ)-almost periodicity and the notion of (ℬ, I ′ , ρ)-uniform recurrence are very special cases of the following notion: Definition 4.1.19. Suppose that 0 ≠ I ′ ⊆ ℝn , 0 ≠ I ⊆ ℝn , F : I × X → Y is a given function, ρ is a binary relation on Y and I + I ′ ⊆ I. Then we say that: (i) F(⋅; ⋅) is Bohr (ℬ, I ′ , ρ, 𝒫 )-almost periodic if and only if for every B ∈ ℬ and ε > 0 there exists l > 0 such that for each t0 ∈ I ′ there exists τ ∈ B(t0 , l) ∩ I ′ such that, for every t ∈ I and x ∈ B, there exists an element yt;x ∈ ρ(F(t; x)) such that F(⋅ + τ; x) − y⋅;x ∈ P for all x ∈ B, and 󵄩 󵄩 sup󵄩󵄩󵄩F(⋅ + τ; x) − y⋅;x 󵄩󵄩󵄩P ⩽ ε. x∈B

(ii) F(⋅; ⋅) is (ℬ, I , ρ, 𝒫 )-uniformly recurrent if and only if for every B ∈ ℬ there exists a sequence (τ k ) in I ′ such that limk→+∞ |τ k | = +∞ and that, for every t ∈ I and x ∈ B, there exists an element yt;x ∈ ρ(F(t; x)) such that F(⋅ + τ k ; x) − y⋅;x ∈ P for all k ∈ ℕ, x ∈ B and ′

󵄩 󵄩 lim sup󵄩󵄩󵄩F(⋅ + τ k ; x) − y⋅;x 󵄩󵄩󵄩P = 0.

k→+∞ x∈B

4.1 (RX , ℬ, 𝒫, L)-multi-almost periodic type functions

� 251

If I = ℝn , then we can also consider the notions of strong Bohr (ℬ, I ′ , ρ, 𝒫 )-almost periodicity and strong (ℬ, I ′ , ρ, 𝒫 )-uniform recurrence; we will skip all details for simplicity. Further on, we omit the term “ℬ” if X = {0}, the term “I ′ ” if I ′ = I, and the term “ρ” if ρ = I. If ρ = cI, then we also say that the function F(⋅; ⋅) is Bohr (ℬ, I ′ , c, 𝒫 )-almost periodic [Bohr (ℬ, I ′ , c, 𝒫 )-uniformly recurrent]; furthermore, if c = −1, then we say that the function F(⋅; ⋅) is Bohr (ℬ, I ′ , 𝒫 )-almost anti-periodic [Bohr (ℬ, I ′ , 𝒫 )-uniformly anti-recurrent]. We continue with the following example: Example 4.1.20. Suppose that I = [0, ∞) or I = ℝ. A measurable function ν : I → (0, ∞) is said to be an admissible weight function if and only if there exist finite constants M ⩾ 1 ′ and ω ∈ ℝ such that ν(t) ⩽ Meω|t | ν(t+t ′ ) for all t, t ′ ∈ I. Let ν : [0, ∞) → (0, ∞) be an admissible weight function; then it is well known that the function 1/ν(⋅) is locally bounded (see [430] for more details about linear topological dynamics and hypercyclic strongly continuous semigroups on weighted function spaces). Recently, Z. Yin and Y. Wei have considered the weak recurrence of translation operators on weighted Lebesgue spaces and weighted continuous function spaces [791]. In particular, these authors have shown p that the existence of a function f ∈ Y , where Y = Lν ([0, ∞) : ℂ) or Y = C0,ν ([0, ∞) : ℂ), satisfying that there exists a strictly increasing sequence (αk ) of positive reals tending to plus infinity such that 󵄩 󵄩 lim 󵄩󵄩󵄩f (⋅ + αk ) − f (⋅)󵄩󵄩󵄩Y = 0

k→+∞

is equivalent to saying that lim inft→+∞ ν(t) = 0; see also the research article [163] by W. Brian and J. P. Kelly. In our language, this result can be reworded as follows: Suppose that P := Y and d(f , g) := ‖f − g‖Y for all f , g ∈ P. Then there exists a 𝒫 -uniformly recurrent function f ∈ Y if and only if lim inft→+∞ ν(t) = 0. We will analyze the corresponding p result for the general space Y = Lν (I : ℂ) [Y = C0,ν (I : ℂ)] somewhere else. Before going any further, we would like to propose the following issue: Problem. Suppose that F : ℝn → Y is an almost periodic function which is not (ωj )j∈ℕn periodic for any choice of numbers ωj ∈ ℝ ∖ {0} (1 ⩽ j ⩽ n). Does there exist an unbounded function ν : ℝn → (0, ∞) such that F(⋅) is Bohr 𝒫 -almost periodic with P = Cb,ν (ℝn : Y ) and the metric d being induced by the norm of this Banach space? We can simply prove that any multi-dimensional (ω, ρ)-periodic function is Bohr (I ′ , ρ, 𝒫 )-almost periodic with I ′ = {kω : k ∈ ℕ}, provided that there exists a positive integer l ∈ ℕ such that ρl = I. But, there exist some pathological cases in which the spaces of Bohr (ℬ, I ′ , ρ, 𝒫 )-almost periodic functions consist solely of (ω, ρ)-periodic functions. Without going into full details, we will provide only one example regarding this issue:

252 � 4 Metrically ρ-almost periodic type functions and applications Example 4.1.21. Suppose that the function 1/ν(⋅) is locally bounded, P = C0,ν (ℝn : Y ), the metric d is induced by the norm in P, 0 ≠ ω ∈ ℝn , F : ℝn → Y , and the supremum formula 󵄩 󵄩 sup 󵄩󵄩󵄩F(t + ω) − F(t)󵄩󵄩󵄩Y =

t∈ℝn

󵄩 󵄩 sup 󵄩󵄩󵄩F(t + ω) − F(t)󵄩󵄩󵄩Y n

t∈ℝ ;|t|⩾a

holds for all a > 0. If ν : ℝn → (0, ∞) satisfies lim|t|→+∞ ν(t) = +∞, then F(⋅) is ω-periodic (i. e., F(t + ω) = F(t), t ∈ ℝn ) whenever F(⋅ + ω) − F(⋅) ∈ P. In actual fact, if ε > 0 is a fixed number and the last inclusion holds, then there exists a finite real number M > 0 such that the assumption |t| ⩾ M implies ‖F(t + ω) − F(t)‖Y ⩽ ε. Due to the supremum formula, this implies ‖F(t + ω) − F(t)‖Y ⩽ ε for all t ∈ ℝn , so that the claimed assertion follows from the fact that the number ε > 0 was arbitrary. For the sequel, we need the following notion (see Section 6.4 for more details): Suppose that F : ℝn → X is continuous. Then we say that F(⋅) is pre-Levitan N-almost periodic if and only if, for every ε > 0 and N > 0, there exists l > 0 such that for each t0 ∈ ℝn there exists τ ∈ B(t0 , l) ≡ {t ∈ ℝn : |t − t0 | ⩽ l} such that 󵄩󵄩 󵄩 󵄩󵄩F(t + τ) − F(t)󵄩󵄩󵄩 ⩽ ε,

if t ∈ ℝn and |t| ⩽ N;

we also say that τ is a Levitan (ε, N)-period of function F(⋅). In the previous example, we have analyzed the extreme case lim|t|→+∞ ν(t) = +∞. The opposite extreme case lim|t|→+∞ ν(t) = 0 is also important on account of the following: Theorem 4.1.22. (i) Suppose that F : ℝn → Y is continuous, the function 1/ν(⋅) is locally bounded, P = C0,ν (ℝn : Y ) and the metric d is induced by the norm in P. If F(⋅) is Bohr 𝒫 -almost periodic, then F(⋅) is pre-Levitan N-almost periodic and bounded. (ii) Suppose that the requirements of (i) hold as well as that F : ℝn → Y is bounded, the function ν(⋅) is bounded and lim|t|→+∞ ν(t) = 0. Then F(⋅) is pre-Levitan N-almost periodic if and only if F(⋅) is Bohr 𝒫 -almost periodic. Proof. If F(⋅) is Bohr 𝒫 -almost periodic, N > 0 and ε > 0 are given, then we have the existence of a finite real number M > 0 such that 1/ν(t) ⩽ M for |t| ⩽ N. If the requirements in Definition 2.1.1 hold with the numbers ε/M and τ, then it can be simply proved that τ is a Levitan (ε, N)-almost period of function F(⋅). To show that F(⋅) is bounded, take ε = 1 and find a real number l > 0 such that, for every t0 ∈ ℝn , the cube t0 + [0, l]n contains a point τ such that 󵄩󵄩 󵄩 󵄩󵄩F(t + τ) − F(t)󵄩󵄩󵄩Y ν(t) ⩽ 1,

t ∈ ℝn .

Let ‖F(t)‖Y ⩽ M and 1/ν(t) ⩽ L for t ∈ K√nl . This implies 1 󵄩󵄩 󵄩 󵄩 󵄩 ⩽ M + L, 󵄩󵄩F(t + τ)󵄩󵄩󵄩Y ⩽ 󵄩󵄩󵄩F(t)󵄩󵄩󵄩Y + ν(t)

t ∈ K√nl .

(181)

4.1 (RX , ℬ, 𝒫, L)-multi-almost periodic type functions �

253

It can be simply shown that any point t ∈ ℝn can be written as t0 + τ, where t0 ∈ K√nl and τ satisfies (181). Therefore, ‖F(t)‖Y ⩽ M + L for all t ∈ ℝn and (i) is proved. In order to see that (ii) holds, suppose that F(⋅) is pre-Levitan N-almost periodic and ε > 0. Then there exists a finite real number M > 0 such that the assumptions |t| ⩾ M and τ ∈ ℝn imply 󵄩󵄩 󵄩 󵄩󵄩F(t + τ) − F(t)󵄩󵄩󵄩Y ν(t) ⩽ 2‖F‖∞ ν(t) ⩽ ε. If |t| ⩽ M and τ ∈ ℝn is a Levitan (ε/‖ν‖∞ , M)-period of function F(⋅), then we have ‖F(t+ τ) − F(t)‖Y ν(t) ⩽ εν(t)/‖ν‖∞ ⩽ ε; hence, F(⋅) is Bohr 𝒫 -almost periodic. The converse statement follows from (i). Keeping in mind Theorem 4.1.22, it is meaningful to define, for every almost automorphic function F : ℝn → Y , the following space FF := {ν : ℝn → Y ; the function 1/ν(⋅) is locally bounded and the function F(⋅) is Bohr 𝒫 -almost periodic}.

Due to Theorem 4.1.22(ii), we have that all bounded functions vanishing at infinity belong to the space FF . This inclusion can be strict, since for every (ωj )j∈ℕn -periodic function with some numbers ωj ∈ ℝ ∖ {0} (1 ⩽ j ⩽ n), the space FF consists of all functions ν : ℝn → (0, ∞) such that the function 1/ν(⋅) is locally bounded. We proceed with the following illustrative examples: Example 4.1.23. Let us recall that the function t 󳨃→ F(t) ≡ 1/(2+cos t +cos(√2t)), t ∈ ℝ is Levitan N-almost periodic and unbounded. Due to Theorem 4.1.22(i), there is no function ν : ℝ → (0, ∞) such that the function 1/ν(⋅) is locally bounded and F(⋅) is Bohr 𝒫 -almost periodic with the metric space 𝒫 be defined as above. Example 4.1.24. Let F denote the collection of all functions ν : ℝ → (0, ∞) such that the function 1/ν(⋅) is locally bounded; if this is the case, we denote Pν = C0,ν (ℝ : ℂ) and define dν to be the metric induced by the norm in Pν . We introduce the binary relation ∼ on F by: ν1 ∼ ν2 if and only if every Bohr 𝒫ν1 -almost periodic function F : ℝ → ℂ is Bohr 𝒫ν2 -almost periodic and vice versa; clearly, ∼ is an equivalence relation. If there exist two finite real constants c1 > 0 and c2 > 0 such that c1 ν1 (t) ⩽ ν2 (t) ⩽ c2 ν1 (t) for all t ∈ ℝ, then it is clear that ν1 ∼ ν2 (in particular, if there exist two finite real constants c1 > 0 and c2 > 0 such that c1 ⩽ ν(t) ⩽ c2 for all t ∈ ℝ, P = C0,ν (ℝ : ℂ), and the metric d is induced by the norm in P, then the function F(⋅) is almost periodic if and only if the function F(⋅) is Bohr 𝒫 -almost periodic). We can use Theorem 4.1.22 to show that the existence of finite real constants c1 > 0 and c2 > 0 such that c1 ν1 (t) ⩽ ν2 (t) ⩽ c2 ν1 (t), t ∈ ℝ is only sufficient but not necessary for relation ν1 ∼ ν2 to be satisfied. In actual fact, put ν1 (t) := 1/(t 2 + 1) and ν2 (t) := 1/(t 4 + 1) for all t ∈ ℝ. Then ν1 ∼ ν2 due to Theorem 4.1.22(ii) but we cannot find finite real constants c1 > 0 and c2 > 0 such that c1 ν1 (t) ⩽ ν2 (t) ⩽ c2 ν1 (t), t ∈ ℝ.

254 � 4 Metrically ρ-almost periodic type functions and applications The proof of following result is very similar to the proof of Proposition 2.1.3 and therefore omitted: Proposition 4.1.25. Suppose that 0 ≠ I ′ ⊆ ℝn , 0 ≠ I ⊆ ℝn , I + I ′ ⊆ I and the function F : I ×X → Y is Bohr (ℬ, I ′ , ρ, 𝒫 )-almost periodic ((ℬ, I ′ , ρ, 𝒫 )-uniformly recurrent), where ρ is a binary relation on Y satisfying R(F) ⊆ D(ρ) and ρ(y) is a singleton for any y ∈ R(F). If for each τ ∈ I ′ we have τ + I = I, as well as P has a linear vector structure, the metric d is translation invariant and the following condition holds: (P3) 𝒫1 = (P1 , d1 ) is a metric space, c ∈ (0, ∞) and for every f ∈ P and τ ∈ I ′ we have f (⋅ − τ) ∈ P1 and ‖f (⋅ − τ)‖P1 ⩽ c‖f ‖P . Then I + (I ′ − I ′ ) ⊆ I and the function F(⋅; ⋅) is Bohr (ℬ, I ′ − I ′ , I, 𝒫1 )-almost periodic ((ℬ, I ′ − I ′ , I, 𝒫1 )-uniformly recurrent). Corollary 4.1.26. Suppose that 0 ≠ I ′ ⊆ ℝn and the function F : ℝn × X → Y is Bohr (ℬ, I ′ , ρ, 𝒫 )-almost periodic ((ℬ, I ′ , ρ, 𝒫 )-uniformly recurrent), where ρ is a binary relation on Y satisfying R(F) ⊆ D(ρ) and ρ(y) is a singleton for any y ∈ R(F). Suppose that P has a linear vector structure, the metric d is translation invariant, and condition (P3) holds. Then the function F(⋅; ⋅) is Bohr (ℬ, I ′ −I ′ , I, 𝒫1 )-almost periodic ((ℬ, I ′ −I ′ , I, 𝒫1 )-uniformly recurrent). The following example is a slight modification of [304, Example 2.8]: Example 4.1.27. Suppose that ρ = T ∈ L(Y ), I = I ′ = [0, ∞) or I = I ′ = ℝ, and X = {0}. Clearly, for each t ∈ I, τ ∈ I and l ∈ ℕ, we have l−1

F(t + lτ) − T l F(t) = ∑ T j [F(t + (l − j)τ) − TF(t + (l − j − 1)τ)]. j=0

Let for each f ∈ P and τ ∈ I we have f (⋅ + τ) ∈ P, ‖f (⋅ + τ)‖P = ‖f ‖P , Tf ∈ P and ‖Tf ‖P ⩽ cT ‖f ‖P for some finite real constant cT > 0 independent of f ∈ P. This implies that the function F(⋅) is (T l , 𝒫 )-almost periodic ((T l , 𝒫 )-uniformly recurrent), provided that F(⋅) is (T, 𝒫 )-almost periodic ((T, 𝒫 )-uniformly recurrent). In particular, the function F(⋅) is 𝒫 -almost periodic (𝒫 -uniformly recurrent), provided that F(⋅) is (T, 𝒫 )-almost periodic ((T, 𝒫 )-uniformly recurrent) and there exists a positive integer l ∈ ℕ such that T l = I; a similar statement can be formulated for (T, 𝒫 )-almost anti-periodic ((T, 𝒫 )-uniformly anti-recurrent) functions. Consider now the sum f (⋅) of functions f1 (⋅) and f2 (⋅) from Example 4.1.1(i). In our concrete situation, P = {f ∈ Cb (ℝ : ℝ) ; supt∈ℝ V (t; f ) < ∞} and d(f , g) := ‖f − g‖∞ + supt∈ℝ V (t; f − g), f , g ∈ P, where V (t; f ) denotes the total variation of function f (⋅) on the segment [t − 1, t + 1]. If f (⋅) is (c, 𝒫 )-almost periodic for some c ∈ ℂ ∖ {0}, then the function f (⋅) must be c-almost periodic, and therefore, we must have c = ±1 since the function f (⋅) is real-valued [431]. Since the metric space 𝒫 satisfies all requirements from the first part of this example with T = cI, (−1, 𝒫 )-almost periodicity of f (⋅) would imply

4.1 (RX , ℬ, 𝒫, L)-multi-almost periodic type functions �

255

its 𝒫 -almost periodicity, which is not the case. Therefore, there is no c ∈ ℂ ∖ {0} such that f (⋅) is (c, 𝒫 )-almost periodic. We can similarly prove that there is no c ∈ ℂ ∖ {0} such that the sum f (⋅) of functions f1 (⋅) and f2 (⋅) from Example 4.1.1(ii) is Lipschitz c-almost periodic, with the meaning clear. Concerning Bohr (ℬ, I ′ , ρ, 𝒫 )-almost periodic functions in the finite-dimensional spaces, we will only note that the statement of [304, Proposition 2.20] admits a reformulation in our framework provided that P has a linear vector structure and the metric d is translation invariant. The proof of following simple proposition is trivial and therefore omitted: Proposition 4.1.28. Suppose that 0 ≠ I ′ ⊆ ℝn , 0 ≠ I ⊆ ℝn , ρ is a binary relation on Y and I + I ′ ⊆ I. Suppose further that P1 ⊆ Y I , 𝒫1 = (P1 , d1 ) is a metric space, P ⊆ P1 and there exists a finite real constant c > 0 such that ‖f ‖P1 ⩽ c‖f ‖P for all f ∈ P. Then the following holds: (i) If F : I ×X → Y is Bohr (ℬ, I ′ , ρ, 𝒫 )-almost periodic [(ℬ, I ′ , ρ, 𝒫 )-uniformly recurrent], then F(⋅; ⋅) is Bohr (ℬ, I ′ , ρ, 𝒫1 )-almost periodic [(ℬ, I ′ , ρ, 𝒫1 )-uniformly recurrent]. (ii) Let (161) hold. If F : I × X → Y is (R, ℬ, 𝒫 )-multi-almost periodic, resp. strongly (R, ℬ, 𝒫 )-multi-almost periodic in the case that I = ℝn , then F(⋅; ⋅) is (R, ℬ, 𝒫1 )-multialmost periodic, resp. strongly (R, ℬ, 𝒫1 )-multi-almost periodic. (iii) Let (163) hold. If F : I × X → Y is (RX , ℬ, 𝒫 )-multi-almost periodic, resp. strongly (RX , ℬ, 𝒫 )-multi-almost periodic in the case that I = ℝn , then F(⋅; ⋅) is (RX , ℬ, 𝒫1 )multi-almost periodic, resp. strongly (RX , ℬ, 𝒫1 )-multi-almost periodic. We also have the following: (i) Suppose that c ∈ ℂ, cf ∈ P for all f ∈ P, and there exists a finite real number ϕ(c) > 0 such that ‖cf ‖P ⩽ ϕ(c)‖f ‖P for all f ∈ P. If the function F : I × X → Y is Bohr (ℬ, I ′ , ρ, 𝒫 )-almost periodic [(ℬ, I ′ , ρ, 𝒫 )-uniformly recurrent], then the function cF(⋅; ⋅) is Bohr (ℬ, I ′ , ρc , 𝒫 )-almost periodic [(ℬ, I ′ , ρc , 𝒫 )-uniformly recurrent], where ρc := {(y1 , y2 ) ∈ Y × Y : (∃t ∈ I) (∃x ∈ X) y1 = cF(t; x) and y2 ∈ cρ(F(t; x))}. (ii) Suppose that τ ∈ ℝn , τ + I ⊆ I, x0 ∈ X and, for every f ∈ P, we have f (⋅ + τ) ∈ P and the existence of a finite real number cτ > 0 such that ‖f (⋅ + τ)‖P ⩽ cτ ‖f ‖P for all f ∈ P. Define ℬx0 := {−x0 + B : B ∈ ℬ} for every B ∈ ℬ. If the function F : I × X → Y is Bohr (ℬ, I ′ , ρ, 𝒫 )-almost periodic [(ℬ, I ′ , ρ, 𝒫 )-uniformly recurrent], then the function F(⋅ + τ; ⋅ + x0 ) is Bohr (ℬx0 , I ′ , ρ, 𝒫 )-almost periodic [(ℬx0 , I ′ , ρ, 𝒫 )uniformly recurrent]. We can illustrate the notion introduced in this section and the former section by slight modifications of [194, Example 6.13, Example 6.15]. In order to avoid any plagiarism, we will only rearrange [194, Example 6.13(i)] for our new purposes:

256 � 4 Metrically ρ-almost periodic type functions and applications Example 4.1.29. Suppose that Fj : X → Y is a continuous function satisfying that for each B ∈ ℬ we have supx∈B ‖Fj (x)‖Y < ∞. Suppose further that the complex-valued t

t

mapping t 󳨃→ (∫0 f1 (s) ds, . . . , ∫0 fn (s) ds), t ⩾ 0 is 𝒫 -almost periodic (1 ⩽ j ⩽ n), where 1/ν(⋅) is locally bounded function, P := C0,ν ([0, ∞) : ℂn ) and d(f , g) := ‖f − g‖P for all f , g ∈ P. Set n

tj+1

F(t1 , . . . , tn+1 ; x) := ∑ ∫ fj (s) ds ⋅ Fj (x) j=1 t j

for all x ∈ X and tj ⩾ 0, 1 ⩽ j ⩽ n.

Arguing as in the above-mentioned example, we get that for every B ∈ ℬ, t1 , τ1 ∈ [0, ∞); . . . ; tn+1 , τn+1 ∈ [0, ∞) and ε > 0, 󵄩󵄩 󵄩 󵄩󵄩F(t1 + τ1 , . . . , tn+1 + τn+1 ; x) − F(t1 , . . . , tn+1 ; x)󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩Y tj+1 +τj+1 tj+1 tj 󵄨󵄨 󵄨󵄨 tj +τj 󵄨󵄨 n 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 ⩽ M ∑{󵄨󵄨 ∫ fj (s) ds − ∫ fj (s) ds󵄨󵄨 + 󵄨󵄨 ∫ fj (s) ds − ∫ fj (s) ds󵄨󵄨󵄨}, 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 j=1 󵄨 0 󵄨 󵄨 0 󵄨 0 0 󵄩 󵄩 where M = supx∈B,1⩽j⩽n 󵄩󵄩󵄩Fj (x)󵄩󵄩󵄩Y . This implies that the function F(⋅; ⋅) is Bohr (ℬ, 𝒫1 )almost periodic, where P1 := C0,ν1 ([0, ∞)n+1 : Y ) with −1

1 1 ν1 (t1 , . . . , tn+1 ) := [ + ⋅⋅⋅ + ] , ν(t1 ) ν(tn+1 )

tj ⩾ 0 (1 ⩽ j ⩽ n + 1),

and d1 (f , g) := ‖f − g‖P1 for all f , g ∈ P1 . We continue by observing that we can formulate and prove an analogue of [194, Proposition 2.27(i)] and Proposition 4.1.10 for the function spaces introduced in Definition 2.1.1; on the other hand, the statement of [194, Theorem 2.37] which concerns the extensions of multi-dimensional almost periodic functions does not admit a satisfactory reformulation in our new framework. Proposition 4.1.8 and Theorem 4.1.9 can be reformulated as follows (we do not need here the existence of a finite real number c > 0 such that ν(t) ⩾ c for all t ∈ ℝn ): Proposition 4.1.30. Let P := Cb,ν (ℝn : Y ) and d(f , g) := ‖f − g‖Cb,ν (ℝn :Y ) for all f , g ∈ P.

Suppose that c ∈ ℂ ∖ {0} and 0 ≠ I ′ ⊆ ℝn , h ∈ L1 (ℝn ) and F : ℝn × X → Y is a continuous function satisfying that for each B ∈ ℬ there exists a finite real number εB > 0 such that supt∈ℝn ,x∈B⋅ ‖F(t, x)‖Y < +∞, where B⋅ ≡ B∘ ∪ ⋃x∈𝜕B B(x, εB ). Then the function (h ∗ F)(⋅; ⋅), given by (167), is well defined and for each B ∈ ℬ we have supt∈ℝn ,x∈B⋅ ‖(h∗F)(t; x)‖Y < +∞; furthermore, if F(⋅; ⋅) is Bohr (I ′ , c, 𝒫 )-almost periodic, then (h ∗ F)(⋅; ⋅) is Bohr (I ′ , c, 𝒫 )almost periodic.

4.1 (RX , ℬ, 𝒫, L)-multi-almost periodic type functions

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Theorem 4.1.31. Suppose that (R(t))t>0 ⊆ L(X, Y ) is a strongly continuous operator family, c ∈ ℂ ∖ {0} and 0 ≠ I ′ ⊆ ℝn . Let P := Cb,ν (ℝn : Y ) and d(f , g) := ‖f − g‖Cb,ν (ℝn :Y ) for all f , g ∈ P, and let there exist a function w : ℝn → (0, ∞) such that (166) holds for all x ∈ ℝn , y ∈ [0, ∞)n and ∫(0,∞)n (1 + w(t))‖R(t)‖ dt < ∞. If f : ℝn → X is a bounded,

continuous and Bohr (I ′ , c, 𝒫 )-almost periodic function, then the function F : ℝn → Y , given by (20), is bounded, continuous and Bohr (I ′ , c, 𝒫 )-almost periodic. The proof of following proposition is also simple and therefore omitted (we assume the general requirements on the regions I and I ′ , the binary relation ρ and the collections R, RX from the introduced definitions): Proposition 4.1.32. Suppose that 𝒫Z = (PZ , dZ ) is a metric space, ϕ : 𝒫 → 𝒫Z is uniformly continuous, the function ϕ(F1 (⋅)) − ϕ(F2 (⋅)) belongs to PZ whenever F1 , F2 ∈ Y I and F1 − F2 ∈ P, and ρZ is any binary relation on Z such that ϕ(ρ(F(t; x))) ⊆ ρZ (ϕ(F(t; x))) for all t ∈ I and x ∈ X. Suppose further that F : I × X → Y is (strongly) (R, ℬ, 𝒫 , L)multi-almost periodic [(strongly) (RX , ℬ, 𝒫 , L)-multi-almost periodic/Bohr (ℬ, I ′ , ρ, 𝒫 )almost periodic/(ℬ, I ′ , ρ, 𝒫 )-uniformly recurrent]. Then ϕ ∘ F : I × X → Z is (strongly) (R, ℬ, 𝒫Z , L)-multi-almost periodic [(strongly) (RX , ℬ, 𝒫Z , L)-multi-almost periodic/Bohr (ℬ, I ′ , ρZ , 𝒫 )-almost periodic/(ℬ, I ′ , ρZ , 𝒫 )-uniformly recurrent]. 4.1.3 Generalization of multi-dimensional (Stepanov) ρ-almost periodicity In this subsection, we will employ the approach obeyed in the former part of this section to generalize the notion of Bohr (ℬ, I ′ , ρ)-almost periodicity ((ℬ, I ′ , ρ)-uniform recurrence) and the notion of Stepanov (Ω, p(u))-(ℬ, I ′ , ρ)-almost periodicity (Stepanov (Ω, p(u), ρ)-(ℬ, Λ′ )-uniform recurrence). We have the following: (B1) Suppose that 0 ≠ I ′ ⊆ ℝn , 0 ≠ I ⊆ ℝn , ρ is a binary relation on Y , I + I ′ ⊆ I and F : I × X → Y is Bohr (ℬ, I ′ , ρ)-almost periodic ((ℬ, I ′ , ρ)-uniformly recurrent). Suppose further that the set I is Lebesgue measurable as well as that for each x ∈ X any selection of the multi-valued mapping t 󳨃→ ρ(F(t; x)), t ∈ I is Lebesgue measurable. Let a function ν : I → (0, ∞) be Lebesgue measurable, as well. Suppose now that ν ∈ Lp(t) (I : ℂ). Then Lemma 1.1.10(iii) yields that p(t) the Banach space L∞ (I : Y ) is continuously embedded into Lρ (I : Y ); applying Proposition 4.1.28(i), we easily get that the function F(⋅; ⋅) is Bohr (ℬ, I ′ , ρ, 𝒫 )p(t) almost periodic ((ℬ, I ′ , ρ, 𝒫 )-uniformly recurrent), where P := Lν (I : Y ) and d(f , g) := ‖f − g‖Lp(t) (I:Y ) for all f , g ∈ P. We can also use the space C0,ν (I : Y ) ρ

here; see Remark 4.1.18(ii). (B2) Suppose that 0 ≠ I ′ ⊆ ℝn , 0 ≠ I ⊆ ℝn , Ω is a compact subset of ℝn with positive Lebesgue measure, I + Ω ⊆ I and the function F : I × X → Y is Stepanov (Ω, p(u))-(ℬ, I ′ , ρ)-almost periodic. Suppose further that there exist a countable family D and a collection {kd ∈ I : d ∈ D} such that I = ⋃d∈D (kd + Ω) and

258 � 4 Metrically ρ-almost periodic type functions and applications m((kd1 + Ω) ∩ (kd2 + Ω)) = 0 for all d1 , d2 ∈ D such that d1 ≠ d2 . Define p : I → [1, ∞) by p(t) := p(t − kd ) if there exists a unique d ∈ D such that t ∈ kd + Ω; otherwise, we set p(t) := 0. The last assumption implies that this mapping is well defined and measurable; furthermore, a simple argumentation shows that for every x ∈ X, we have: 󵄩󵄩 󵄩 󵄩󵄩[F(t + τ; x) − ρ(F(t; x))] ⋅ ρ(t)󵄩󵄩󵄩Lp(t) ((kd +Ω):Y ) 󵄩 󵄩 = 󵄩󵄩󵄩[F(t + kd + τ; x) − ρ(F(t + kd ; x))] ⋅ ρ(t + kd )󵄩󵄩󵄩Lp(t) (Ω:Y ) .

(182)

We will assume that S := ∑ ess sup ν(t) < +∞.

(183)

d∈D t∈kd +Ω

Fix B ∈ ℬ and ε > 0. Let l > 0 be determined from the Stepanov (Ω, p(u))-(ℬ, I ′ , ρ)almost periodicity of function F(⋅; ⋅); further on, let 󵄩󵄩 󵄩 󵄩󵄩F(t + τ + u; x) − ρ(F(t + u; x))󵄩󵄩󵄩Lp(u) (Ω:Y ) ⩽ ε,

t ∈ I, x ∈ B.

Then, due to (182) and a simple argumentation involving Lemma 1.1.10(iii), we have: 󵄩󵄩 󵄩 󵄩󵄩F(t + τ; x) − ρ(F(t; x))󵄩󵄩󵄩Lp(t) (I:Y ) ν 󵄩 󵄩 ⩽ ∑ 󵄩󵄩󵄩F(t + τ; x) − ρ(F(t; x))󵄩󵄩󵄩Lp(t) ((k +Ω):Y ) d ν d∈D

󵄩 󵄩 = ∑ 󵄩󵄩󵄩[F(t + τ; x) − ρ(F(t; x))] ⋅ ν(t)󵄩󵄩󵄩Lp(t) ((k +Ω):Y ) d d∈D

󵄩 󵄩 = ∑ 󵄩󵄩󵄩[F(t + kd + τ; x) − ρ(F(t + kd ; x))] ⋅ ν(t + kd )󵄩󵄩󵄩Lp(t) (Ω:Y ) d∈D

󵄩 󵄩 ⩽ ∑ 󵄩󵄩󵄩F(t + kd + τ; x) − ρ(F(t + kd ; x))󵄩󵄩󵄩Lp(t) (Ω:Y ) ⋅ ess sup ν(t) t∈kd +Ω

d∈D

⩽ ∑ ε ⋅ ess sup ν(t) ⩽ εS. d∈D

t∈kd +Ω

Since (183) is assumed, the above implies that the function F(⋅; ⋅) is Bohr (ℬ, I ′ , ρ, 𝒫 )p(t) almost periodic, where P := Lν (I : Y ) and d(f , g) := ‖f − g‖Lp(t) (I:Y ) for all f , g ∈ P. ρ

The above conclusion can be also formulated for Stepanov (Ω, p(u), ρ)-(ℬ, I ′ )uniformly recurrent functions [443].

The above particularly shows that the spaces of Stepanov p-almost periodic functions F : ℝn → Y , where 1 ⩽ p < ∞, can be embedded into the corresponding spaces of Bohr 𝒫 -almost periodic functions. For example, in the case of consideration of onedimensional Stepanov p-almost periodic functions, we have Ω = [0, 1], I ′ = ℝ and ρ = I.

4.1 (RX , ℬ, 𝒫, L)-multi-almost periodic type functions �

259

Then we can take ν(t) := 1/(|t|ζ +1), t ∈ ℝ since, in this case, (183) holds true. On the other hand, it is not clear how one can embed the space of all (equi-)Weyl-p-almost periodic functions F : ℝn → Y in some space of Bohr 𝒫 -almost periodic functions. 4.1.4 Applications to the abstract Volterra integro-differential equations The main aim of this subsection is to apply our results to the analysis of existence and uniqueness of metrically almost periodic type solutions for some classes of abstract Volterra integro-differential equations. 1. Let Y be one of the spaces Lp (ℝn ), C0 (ℝn ) or BUC(ℝn ), where 1 ⩽ p < ∞. Consider again the Gaussian semigroup given by (38). Let c ∈ ℂ ∖ {0} and 0 ≠ I ′ ⊆ ℝn , and let F(⋅) be bounded and (R, 𝒫 )-multi-almost periodic (Bohr (I ′ , c, 𝒫 )-almost periodic), where 𝒫 is given in the formulation of Proposition 4.1.8 (Proposition 4.1.30). Applying this result, we get that for each t0 > 0 the function ℝn ∋ x 󳨃→ u(x, t0 ) ≡ (G(t0 )F)(x) ∈ ℂ is likewise bounded and (R, 𝒫 )-multi-almost periodic (Bohr (I ′ , c, 𝒫 )-almost periodic). 2. It is clear that we can use a combination of Theorem 4.1.9 (Theorem 4.1.31) and Theorem 4.1.12 in the analysis of metrically almost periodic solutions in time-variable for a large class of abstract fractional semi-linear inclusions. For instance, of concern is the following abstract semi-linear Cauchy inclusion: γ

Dt,+ u(t) ∈ −𝒜u(t) + f (t, u(t)), t ∈ ℝ,

(184)

γ

where Dt,+ denotes the Weyl–Liouville fractional derivative of order γ ∈ (0, 1), f : ℝ × X → X has certain properties and 𝒜 is a closed multivalued linear operator on X satisfying condition (P); see [428] for the notion and more details. Define Tν (t)x :=

1 ∫(−λ)ν eλt (λ − 𝒜)−1 x dλ, 2πi

x ∈ X, t > 0 (ν > 0),

Γ

where Γ is the upwards oriented curve λ = −c(|η| + 1) + iη (η ∈ ℝ), Tγ,ν (t)x := t

γν

∞

∫ sν Φγ (s)T0 (st γ )x ds, 0

Sγ (t) := Tγ,0 (t)

t > 0, x ∈ X, ν > −β,

and Pγ (t) := γTγ,1 (t)/t γ ,

t > 0.

Define also Rγ (t) := t γ−1 Pγ (t),

t > 0, x ∈ X.

By a mild solution of (184), we mean any X-continuous function u(⋅) such that u(t) = (Λγ u)(t), t ∈ ℝ, where

260 � 4 Metrically ρ-almost periodic type functions and applications t

t 󳨃→ (Λγ u)(t) := ∫ Rγ (t − s)f (s, u(s)) ds,

t ∈ ℝ.

−∞

Suppose now that R denotes the collection of all sequences in [0, ∞), P := Cb,ν (ℝ : X) and d(f , g) := ‖f − g‖Cb,ν (ℝ:X) for all f , g ∈ P. Let there exist a positive real number c > 0 such that ν(t) ⩾ c for all t ∈ ℝ, and let there exist a function w : ℝ → (0, ∞) such that (166) holds for all x ∈ ℝ, y ⩾ 0 and ∫(0,∞) (1 + w(t))‖Rγ (t)‖ dt < ∞. It can be simply verified that the space 𝒳 consisting of all bounded, continuous, (R, 𝒫 )-multialmost periodic functions f : ℝn → X of type 1 is a complete metric space equipped with the metric d(⋅, ⋅) = ‖ ⋅ − ⋅ ‖P (see also Proposition 4.1.10). Suppose, further, that f : ℝ × X → X is continuous, (R, ℬ, 𝒫 )-multi-almost periodic of type 1 with X ∈ ℬ, and satisfies that supt∈ℝ;x∈B ‖f (t, x)‖ < +∞ for each bounded subset B of X. If we assume that there exists a finite real constant L > 0 such that ‖f (t, x) − f (t, y)‖ ⩽ L‖x − y‖ for all t ∈ ℝ ∞ and x, y ∈ X, as well as that L ∫0 w(t)‖Rγ (t)‖ dt < 1, then the mapping Λγ : 𝒳 → 𝒳 is well defined due to Theorem 4.1.9 and Theorem 4.1.12; moreover, this mapping is a contraction and has a unique fixed point theorem according to the Banach contraction principle. Therefore, there exists a unique bounded, continuous solution of inclusion (184) which is (R, 𝒫 )-multi-almost periodic of type 1. In particular, the above conclusions can be incorporated in the study of the following semi-linear fractional Poisson heat equation with Weyl–Liouville fractional derivatives in the space X = Lp (Ω): γ

Dt,+ [m(x)v(t, x)] = (Δ − b)v(t, x) + f (t, v(t, x)),

t ∈ ℝ, x ∈ Ω,

where 1 ⩽ p < ∞, Ω is an open domain in ℝn with smooth boundary, m ∈ L∞ (Ω), m(x) ⩾ 0 for a. e. x ∈ Ω, γ ∈ (0, 1), Δ is the Dirichlet Laplacian and b > 0; possible applications can be also given to the higher-order differential operators in Hölder spaces [428]. 3. The choice of space 𝒫 used in the first application of this subsection enables one to reformulate the conclusions from [194, Example 1.1] in our new context, provided that ∫ℝn |E(t0 , y)|w(y) dy < ∞. This can be incorporated in the analysis of metrically almost periodic solutions to the inhomogeneous heat equation in ℝn . Similarly, with the same choice of space 𝒫 , we can analyze the existence and uniqueness of Bohr (I ′ , c, 𝒫 )-almost periodic ((I ′ , c, 𝒫 )-uniformly recurrent) solutions of the wave equation. In the multi-dimensional setting, we can also consider the Hammerstein semi-linear integral equation of convolution type on ℝn ; see the fourth application in [194, Section 3]. Concerning some subjects not considered in this section, we will only emphasize here that we have recently analyzed, in [194, 195, 196] and [304], various notions of 𝔻-asymptotical periodicity and 𝔻-asymptotical automorphy in the multi-dimensional setting. We will not reconsider these notions for multi-dimensional almost periodicity in general metric.

4.2 Stepanov ρ-almost periodic functions in general metric

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4.2 Stepanov ρ-almost periodic functions in general metric The main aim of this section is to introduce and analyze various notions of metrical Stepanov ρ-almost periodicity. Because of a certain similarity with our previous research studies, we omit the proofs of our structural results to a great extent. We also explain how one can apply our results in the analysis of certain classes of abstract (semilinear) integro-differential equations. The organization and main ideas of this section can be briefly described as follows. Various classes of metrically Stepanov ρ-almost periodic type functions are introduced and analyzed in Section 4.3.2 (see Definitions 4.2.1–4.2.3). Embeddings of spaces of metrically Stepanov ρ-almost periodic functions into the corresponding spaces of metrically equi-Weyl ρ-almost periodic functions are analyzed in Proposition 4.3.9. The convolution invariance of metrical Stepanov almost periodicity is examined in Theorem 4.2.4, while a composition principle in this direction is clarified in Theorem 4.2.5. Section 4.2.1 investigates the invariance of metrical Stepanov c-almost periodicity under the actions of infinite convolution products; the main results of this subsection are Proposition 4.2.7 and Proposition 4.2.8. The final subsection is reserved for an application concerning the existence and uniqueness of metrically almost periodic solutions for a class of the abstract degenerate semilinear fractional differential equations. Assume now that Ω is a fixed compact subset of ℝn with positive Lebesgue measure, 1 ⩽ p < ∞ and Λ is a non-empty subset of ℝn satisfying Λ + Ω ⊆ Λ. In what follows, by Stepanov ∞-boundedness of function F(⋅) we mean its essential boundedness. We will always assume the validity of the following conditions: (SM1-1): 0 ≠ Λ ⊆ ℝn , 0 ≠ Λ′ ⊆ ℝn , 0 ≠ Ω ⊆ ℝn is a Lebesgue measurable set such that m(Ω) > 0, Λ′ + Λ ⊆ Λ, Λ + Ω ⊆ Λ, ϕ : [0, ∞) → [0, ∞) and 𝔽 : Λ → (0, ∞). (SM1-2): For every t ∈ Λ, 𝒫t = (Pt , dt ) is a metric space of functions from ℂt+Ω , resp. Y t+Ω , in the case of consideration of Definition 4.2.1, resp. Definitions 4.2.2–4.2.3, containing the zero function. We set ‖f ‖Pt := dt (f , 0) for all f ∈ Pt,l . We also assume that 𝒫 = (P, d) is a metric space of functions from ℂΛ containing the zero function and set ‖f ‖P := d(f , 0) for all f ∈ P. The argument from Λ will be denoted by ⋅⋅ and the argument from t + Ω will be denoted by ⋅. In case 0 ∈ Ω, the class of multi-dimensional Stepanov ρ-almost periodic functions is a very special case of the following class of functions: Definition 4.2.1. By SΩ,Λ′ ,ℬ t (Λ × X : Y ) we denote the set consisting of all functions F : Λ × X → Y such that, for every ε > 0 and B ∈ ℬ, there exists a finite real number L > 0 such that for each t0 ∈ Λ′ there exists τ ∈ B(t0 , L) ∩ Λ′ such that, for every x ∈ B, the mapping u 󳨃→ Gx (u) ∈ ρ(F(u; x)), u ∈ Ω is well defined, and (ϕ,𝔽,ρ,𝒫 ,𝒫)

󵄩󵄩 󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 sup󵄩󵄩󵄩𝔽(⋅⋅)󵄩󵄩󵄩ϕ(󵄩󵄩󵄩F(τ + ⋅; x) − Gx (⋅)󵄩󵄩󵄩Y )󵄩󵄩󵄩 󵄩󵄩󵄩 < ε. 󵄩 󵄩P⋅⋅ 󵄩󵄩P 󵄩 x∈B 󵄩

(185)

262 � 4 Metrically ρ-almost periodic type functions and applications Definition 4.2.2. By SΩ,Λ′ ,ℬ t 1 (Λ × X : Y ) we denote the set consisting of all functions F : Λ × X → Y such that, for every ε > 0 and B ∈ ℬ, there exists a finite real number L > 0 such that for each t0 ∈ Λ′ there exists τ ∈ B(t0 , L) ∩ Λ′ such that for every x ∈ B, the mapping u 󳨃→ Gx (u) ∈ ρ(F(u; x)), u ∈ Ω is well defined, and (ϕ,𝔽,ρ,𝒫 ,𝒫)

󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 sup󵄩󵄩󵄩𝔽(⋅⋅)ϕ(󵄩󵄩󵄩F(τ + ⋅; x) − Gx (⋅)󵄩󵄩󵄩P )󵄩󵄩󵄩 < ε. ⋅⋅ 󵄩󵄩P 󵄩 x∈B 󵄩 Definition 4.2.3. By SΩ,Λ′ ,ℬ t,l 2 (Λ × X : Y ) we denote the set consisting of all functions F : Λ × X → Y such that, for every ε > 0 and B ∈ ℬ, there exists a finite real number L > 0 such that for each t0 ∈ Λ′ there exists τ ∈ B(t0 , L) ∩ Λ′ such that, for every x ∈ B, the mapping u 󳨃→ Gx (u) ∈ ρ(F(u; x)), u ∈ Ω is well defined, and (ϕ,𝔽,ρ,𝒫 ,𝒫)

󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 sup󵄩󵄩󵄩ϕ(𝔽(⋅⋅)󵄩󵄩󵄩F(τ + ⋅; x) − Gx (⋅)󵄩󵄩󵄩P )󵄩󵄩󵄩 < ε. ⋅⋅ 󵄩 󵄩 󵄩P x∈B 󵄩 For simplicity, we will not consider here the notion of metrical Bochner transform. The notion of Bohr (ℬ, Λ′ , ρ, 𝒫 )-almost periodicity can be viewed as a special case of the notion introduced in the previous three definitions only if we assume some extra conditions on the metric space 𝒫 and the function F(⋅) under our consideration. Denote by AX,Y any of the above introduced classes of function spaces. If F(⋅; ⋅) belongs to AX,Y , c1 ∈ ℝ∖{0}, τ ∈ ℝn , c, c2 ∈ ℂ∖{0} and x0 ∈ X, then it is not difficult to find some sufficient conditions ensuring that the function cF(⋅; ⋅), F(c1 ⋅; c2 ⋅), ‖F(⋅; ⋅)‖Y or F(⋅ + τ; ⋅ + x0 ) also belongs to AX,Y . As in the case of consideration of metrical Weyl ρ-almost periodicity, it is not simple to say when AX,Y will be a vector space, even in the case that ρ = I; we can use the Jensen integral inequality in general measure spaces to clarify certain inclusions about the introduced classes of functions. We can also analyze the uniformly convergent sequences of functions belonging to the introduced spaces of functions. For example, if (ϕ,𝔽,ρ,𝒫 ,𝒫) (Fk (⋅; ⋅)) is a sequence of functions from the space SΩ,Λ′ ,ℬ t (Λ × X : Y ) and there exists a function F : Λ × X → Y such that limk→+∞ Fk (t; x) = F(t; x), uniformly on Λ × B for (ϕ,𝔽,ρ,𝒫 ,𝒫) each set B of collection ℬ, then F ∈ SΩ,Λ′ ,ℬ t (Λ × X : Y ) provided that the following conditions hold true: (i) Pt and P are Banach spaces for all t ∈ Λ; (ii) There exists a finite real constant c > 0 such that ϕ(x + y) ⩽ c[ϕ(x) + ϕ(y)] for all x, y ⩾ 0; (iii) ϕ(⋅) is continuous at zero; (iv) D(ρ) is closed, ρ is single-valued on R(F) and satisfies (Cρ ); (v) We have 𝔽(⋅⋅) ∈ P. Before proceeding further, let us emphasize that it is too much to expect that the functions of the form χK (⋅), where K is a non-empty compact set in ℝn with a positive Lebesgue measure, belong to the spaces of metrical Stepanov ρ-almost periodic func-

4.2 Stepanov ρ-almost periodic functions in general metric

� 263

tions introduced in this section. As is well known, these functions are always equi-Weyl c-almost periodic for all values of complex parameter c; see [431] for more details. In the remainder of section, we will mainly deal with the notion introduced in Definition 4.2.1. The proof of following theorem is almost the same as the proof of [448, Theorem 3.4] and therefore omitted (we can just follow the argumentation given in the proof of this theorem, with l = 1): Theorem 4.2.4. Suppose that c ∈ ℂ ∖ {0}, φ : [0, ∞) → [0, ∞), ϕ : [0, ∞) → [0, ∞) is a convex monotonically increasing function satisfying ϕ(xy) ⩽ φ(x)ϕ(y) for all x, y ⩾ 0, (ϕ,𝔽,cI,𝒫 ,𝒫) h ∈ L1 (ℝn ), Ω = [0, 1]n , F ∈ SΩ,Λ′ ,ℬ t (ℝn ×X : Y ), where p, q ∈ 𝒫 (ℝn ), 1/p(⋅)+1/q(⋅) = 1, n ν : ℝ → (0, ∞) is a Lebesgue measurable function, ν(t) ⩾ d > 0 for some positive real p(⋅) number d > 0, Pt = Lν (t + Ω : ℂ), the metric dt is induced by the norm of this Banach space (t ∈ Λ), P = L∞ (ℝn : ℂ) and for each x ∈ X we have supt∈ℝn ‖F(t; x)‖Y < ∞. If (SM1-1)–(SM1-2) hold, 𝔽1 : (0, ∞) × ℝn → (0, ∞), the following two conditions (i) If 0 ⩽ |f (⋅)| ⩽ |g(⋅)| and f , g ∈ P1 (f , g ∈ Pt1 for some t ∈ ℝn ), then ‖f ‖P1 ⩽ ‖g‖P1 (‖f ‖Pt1 ⩽ ‖g‖Pt1 ); (ii) If ε > 0 and f ∈ P1 (f ∈ Pt1 for some t ∈ ℝn ), then εf ∈ P1 (εf ∈ Pt1 ) and ‖εf ‖P1 ⩽ ε‖f ‖P1 (‖εf ‖Pt1 ⩽ ε‖f ‖Pt1 ),

hold, and there exists a sequence (ak )k∈ℤn of positive real numbers such that ∑k∈ℤn ak = 1 and 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 −1 ν(⋅) 󵄩 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩𝔽1 (⋅⋅)󵄩󵄩󵄩2 ∑ ak [φ(a−1 h(⋅ − v))] [𝔽(⋅ − k)] 󵄩󵄩 󵄩󵄩 ⩽ 1, k 󵄩󵄩 󵄩󵄩 q(v) (⋅−k+Ω) 󵄩󵄩 1 󵄩󵄩 1 L d 󵄩󵄩 󵄩󵄩 k∈ℤn 󵄩P⋅⋅ 󵄩P (ϕ,𝔽 ,cI,𝒫t1 ,𝒫 1 )

1 then h ∗ F ∈ SΩ,Λ′ ,ℬ

(ℝn × X : Y ); here, as before,

(h ∗ F)(t; x) := ∫ h(σ)F(t − σ; x) dσ,

t ∈ ℝn , x ∈ X.

ℝn

The statements of [448, Proposition 3.5, Corollary 3.6, Theorem 3.10] admit straightforward reformulations in our context. Furthermore, the conclusions established in [194, Example 2.13, Example 2.15], showing the importance of case Λ′ ≠ Λ, and the conclusions established in [431, Example 6.1.15], showing the importance of case Λ′ ⊈ Λ, can be formulated in our new framework. Concerning composition principles for the introduced classes of functions, we will only state the following slight extension of [428, Theorem 2.7.1]; the proof is omitted since it follows from the insignificant modification of the argumentation contained in the proofs of [524, Lemma 2.1, Theorem 2.2] (let us also note that an analogue of [428, Theorem 6.2.30] can be formulated for the metrical Stepanov almost periodicity, as well as that we can consider an analogue of [428, Theorem 2.7.2] with the usually considered Lipschitz assumption, when the situation in which p = q can appear):

264 � 4 Metrically ρ-almost periodic type functions and applications Theorem 4.2.5. Suppose that c ∈ ℂ ∖ {0}, Λ = [0, ∞) or Λ = ℝ, 0 ≠ Λ′ ⊆ Λ, ν : Λ → (0, ∞) is a Lebesgue measurable function, F : Λ × X → Y and f : Λ → X. Let there exist a real number r ⩾ max(p, p/(p − 1)) and a Stepanov r-bounded function LF : Λ → [0, ∞) such that the function LF (⋅ + a)ν(⋅) is Stepanov r-bounded for any a ∈ Λ′ , 󵄩󵄩 󵄩 󵄩󵄩F(t; x) − F(t; y)󵄩󵄩󵄩Y ⩽ LF (t)‖x − y‖,

t ∈ Λ, x, y ∈ X,

(186)

and let there exist a set E ⊆ I with m(E) = 0 such that K := {x(t) : t ∈ Λ ∖ E} is relatively compact in X. Suppose further that for every ε > 0 and for every compact set K ⊆ X, there exist two real numbers l > 0 and L > 0 such that any interval Λ0 ⊆ Λ of length L contains a number τ ∈ Λ0 ∩ Λ′ such that t+1

󵄩 󵄩p sup [ ∫ 󵄩󵄩󵄩F(s + τ; cu) − cF(s; u)󵄩󵄩󵄩 νp (s) ds]

t∈Λ,u∈K

1/p

t

⩽ε

and t+1

󵄩 󵄩p sup[ ∫ 󵄩󵄩󵄩f (s + τ) − cf (s)󵄩󵄩󵄩 νp (s) ds] t∈Λ

t

1/p

⩽ ε. q

Then q := pr/(p + r) ∈ [1, p) and F(⋅; f (⋅)) ∈ S[0,1],Λ′t,q q (Λ : Y ), where Pt,q := Lν ([t, t + 1] : ℂ), the metric dt,ν is induced by the norm of this Banach space (t ∈ Λ), Pq := L∞ (Λ : ℂ), and the metric d is induced by the norm of this Banach space. (x,cI,𝒫 ,𝒫 )

Before proceeding to Section 4.2.1, we would like to propose an open problem concerning the following well-known example of H. Bohr and E. Følner [147]: Question 4.2.6. Let a real number P > 1 be fixed. Then there exists a Stepanov P-bounded function F : ℝ → ℝ which is Stepanov 1-almost periodic but not Stepanov P-almost periodic (see [147, Main example 2, pp. 70–73], and [428] for the notion). We would like to ask whether there exists a Lebesgue measurable function ν(⋅) such that (x,1,I,𝒫 ,𝒫) F ∈ SΩ,ℝ t (ℝ : ℂ) with Ω = [0, 1], Pt = LPν ([t, t + 1] : ℂ) for all t ∈ ℝ and P = L∞ (ℝ : ℂ)? 4.2.1 Invariance of metrical Stepanov c-almost periodicity under the actions of infinite convolution products We start this subsection by observing that the statement of [448, Theorem 3.11] admits a straightforward reformulation in our context: ∞ ∞ Proposition 4.2.7. Assume that ϕ(x) ≡ x, P = L∞ ν (ℝ : ℂ), P1 = Lν1 (ℝ : ℂ), Pt = Lσ ([t, t +

1] : ℂ) and Pt1 = L∞ σ1 ([t, t + 1] : ℂ) for all t ∈ ℝ. Assume, further, that c ∈ ℂ ∖ {0},

4.2 Stepanov ρ-almost periodic functions in general metric

� 265

(R(t))t>0 ⊆ L(X, Y ) is a strongly continuous operator family, ν : ℝ → (0, ∞), ν1 : ℝ → (0, ∞), σ : ℝ → (0, ∞) and σ1 : ℝ → (0, ∞) are Lebesgue measurable functions as well as that the spaces P, P1 , Pt and Pt1 are given as above (t ∈ ℝ). Let (SM1-1) hold with Λ = ℝ, let 𝔽1 : ℝ → (0, ∞), and let ∞

sup 𝔽1 (t) sup σ1 (u)ν1 (t) ∫ t∈ℝ

u∈[t,t+1]

0

‖R(s)‖ ds < +∞. ν(t − s)𝔽(t − s)σ(u − s)

(x,𝔽,cI,𝒫t1 ,𝒫 1 )

t If f ∈ SΩ,Λ′ (ℝ : X), then F ∈ SΩ,Λ′ given by (4), is well defined.

(x,𝔽,cI,𝒫 ,𝒫)

(ℝ : Y ), provided that the function F(⋅),

Now we will prove a result, which shows that the notion introduced and analyzed in the previous part of this section is not fully complete in the study of subject under our consideration, especially when we use the weighted function spaces. For the sake of completeness, we will provide all details of the proof of the following result: Proposition 4.2.8. Suppose that c ∈ ℂ ∖ {0}, 1 ⩽ p < ∞, 1/p + 1/q = 1, ν : ℝ → (0, ∞) and σ : ℝ → (0, ∞) are Lebesgue measurable functions, the function 1/νp (⋅) is locally integrable, and (R(t))t>0 ⊆ L(X, Y ) is a strongly continuous operator family satisfying that M := ∑∞ k=0 ‖R(⋅)‖Lqν [k,k+1] < ∞. If f : ℝ → X satisfies 󵄩 󵄩 sup 󵄩󵄩󵄩f (t − ⋅)/ν(⋅)󵄩󵄩󵄩Lp [k,k+1] < +∞,

t∈ℝ,k∈ℤ

(187)

and for every ε > 0 there exists a finite real number l > 0 such that for every t0 ∈ Λ′ , there exists a number τ ∈ [t0 − l, t0 + l] such that 󵄩 󵄩 sup σ(t)󵄩󵄩󵄩[f (⋅ + τ) − cf (⋅)]/ν(t − ⋅)󵄩󵄩󵄩Lp [t−k−1,t−k] < ε,

t∈ℝ,k∈ℤ

(188)

then the function F : ℝ → Y , given by (4), is well-defined, continuous and satisfies that for every ε > 0, there exists a finite real number l > 0 such that for every t0 ∈ Λ′ , there exists a number τ ∈ [t0 − l, t0 + l] such that 󵄩 󵄩 sup σ(t)󵄩󵄩󵄩F(t + τ) − cF(t)󵄩󵄩󵄩Y ⩽ ε. t∈ℝ

(189)

Proof. We will follow the proofs of [428, Proposition 2.6.11, Proposition 3.5.3] with appro∞ priate changes; w. l. o. g., we may assume that X = Y . Clearly, we have G(t) = ∫0 R(s)g(t− s) ds for all t ∈ ℝ, since the last integral is absolutely convergent by the Hölder inequality and condition (187): ∞

∞ k+1

󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ∫ 󵄩󵄩󵄩R(s)󵄩󵄩󵄩󵄩󵄩󵄩f (t − s)󵄩󵄩󵄩 ds = ∑ ∫ [󵄩󵄩󵄩R(s)󵄩󵄩󵄩ν(s)] ⋅ [󵄩󵄩󵄩f (t − s)󵄩󵄩󵄩/ν(s)] ds 0

k=0 k

266 � 4 Metrically ρ-almost periodic type functions and applications ∞

󵄩 󵄩 󵄩 󵄩 ⩽ ∑ 󵄩󵄩󵄩R(⋅)󵄩󵄩󵄩Lq [k,k+1] sup 󵄩󵄩󵄩f (x − ⋅)/ν(⋅)󵄩󵄩󵄩Lp [k,k+1] , ν

t ∈ ℝ.

x∈ℝ,k∈ℤ

k=0

Let a number ε > 0 be fixed. Then there exists a finite real number l > 0 such that, for every t0 ∈ Λ′ , there exists a number τ ∈ [t0 − l, t0 + l] such that (188) holds. Using the Hölder inequality, we get 󵄩 󵄩 σ(t)󵄩󵄩󵄩F(t + τ) − cF(t)󵄩󵄩󵄩 ∞

󵄩 󵄩 󵄩 󵄩 ⩽ σ(t) ∫ 󵄩󵄩󵄩R(r)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩f (t + τ − r) − cf (t − r)󵄩󵄩󵄩 dr 0

∞ k+1

󵄩 󵄩 󵄩 󵄩 = σ(t) ∑ ∫ 󵄩󵄩󵄩R(r)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩f (t + τ − r) − cf (t − r)󵄩󵄩󵄩 dr k=0 k ∞

k+1

󵄩 󵄩 󵄩 󵄩p ⩽ σ(t) ∑ 󵄩󵄩󵄩R(⋅)󵄩󵄩󵄩Lq [k,k+1] ( ∫ 󵄩󵄩󵄩f (t + τ − r) − cf (t − r)󵄩󵄩󵄩 ν−p (r) dr) ν k=0 ∞

k

1/p

1/p

t−k

󵄩 󵄩 󵄩 󵄩p = σ(t) ∑ 󵄩󵄩󵄩R(⋅)󵄩󵄩󵄩Lq [k,k+1] ( ∫ 󵄩󵄩󵄩f (s + τ) − cf (s)󵄩󵄩󵄩 ν−p (t − r) ds) ν k=0

∞

t−k−1

󵄩 󵄩 ⩽ ∑ 󵄩󵄩󵄩R(⋅)󵄩󵄩󵄩Lq [k,k+1] ε = Mε, ν k=0

t ∈ ℝ.

Hence, (189) holds; we only need to prove yet that the function F(⋅) is continuous. Define k+1

Fk (t) := ∫k R(s)f (t − s) ds for all t ∈ ℝ and k ∈ ℕ0 . Since M < +∞ and (187) holds, the Weierstrass criterion implies that ∑k⩾0 Fk (⋅) = F(⋅) uniformly on the real line. The continuity of function Fk (⋅) for a fixed non-negative integer k can be shown using our assumption that the function 1/νp (⋅) is locally integrable, and repeating almost verbatim the argumentation given in the proof of [428, Proposition 3.5.3] (the only thing worth noting is that for given numbers t ∈ ℝ and k ∈ ℕ0 , we can choose a sequence (fl (⋅)) of p test functions converging to the function f (⋅) in the space Lν(t+⋅) [t + k, t + k + 2]).

Before proceeding to the next subsection, we would like to emphasize that the notion of Stepanov (Ω, p(u))-(R, ℬ)-multi-almost periodicity and the notion of Stepanov (Ω, p(u))-(RX , ℬ)-multi-almost periodicity, introduced recently in [431, Definition 6.2.4, Definition 6.2.5], can be further extended following the approach obeyed in this section and the previous one. We will skip all details concerning this topic here.

4.2.2 An application to the abstract semilinear fractional differential equations In this section, we have not considered the notion of metrical Stepanov uniform recurrence by now. This notion is sometimes crucial, because there exist many serious

4.2 Stepanov ρ-almost periodic functions in general metric

� 267

problems in applying Theorem 4.2.5 and Proposition 4.2.7 (Proposition 4.2.8) to the abstract semilinear Cauchy problems; fortunately, these results can be reformulated for the metrically Stepanov uniformly recurrent functions. In this subsection, we will provide an illustrative application of such analogues of Theorem 4.2.5 and Proposition 4.2.8 in the qualitative analysis of metrically Stepanov uniformly recurrent solutions for a class of abstract degenerate semilinear fractional differential equations in the finitedimensional spaces (the bounded uniformly recurrent solutions for this class of semilinear problems have recently been analyzed in [431]). Before doing this, we would like to note that all applications made to the various classes of the abstract (semilinear) Volterra integro-differential equations and inclusions, given in [448, Section 3, Points 1.–2.], can be simply reformulated for the metrical Stepanov almost periodicity. For example, we can analyze the existence and uniqueness of metrically Stepanov almost periodic solutions of the inhomogeneous heat equation in ℝn and the inhomogeneous fractional Poisson heat equation with Weyl–Liouville fractional derivatives. Consider the finite-dimensional space X := ℂn , where n ⩾ 2; let c > 0, and A, B ∈ n,n ℂ (the space of all complex matrices of format n × n). Suppose that the matrix B is not invertible, the degree of complex polynomial P(λ) := det(λB − A), λ ∈ ℂ is equal to n and its roots lie in the region {λ ∈ ℂ : Re λ < −c(| Im λ| + 1)}. We know that these assumptions imply the existence of a positive real constant M > 0 such that condition (P) holds with β = 1, so that the multivalued linear operator 𝒜 = AB−1 generates an exponentially decaying, analytic, strongly continuous degenerate semigroup (T(t))t⩾0 ; cf. [428] and [431] for the notion and more details. Further on, let 0 < γ < 1 and ν > −1. Define Tγ,ν (t)x := t

γν

∞

∫ sν Φγ (s)T(st γ )x ds,

t > 0, x ∈ X,

and

Pγ (t) := γTγ,1 (t)/t γ ,

t > 0.

0

If β ∈ (0, 1], then there exists a finite real constant M1 > 0 such that 󵄩󵄩 󵄩 γ(β−1) , 󵄩󵄩Pγ (t)󵄩󵄩󵄩 ⩽ M1 t

t>0

󵄩 󵄩 and 󵄩󵄩󵄩Pγ (t)󵄩󵄩󵄩 ⩽ M2 t −2γ ,

t ⩾ 1.

Set Rγ (t) := t γ−1 Pγ (t), t > 0. Of concern is the following abstract fractional inclusion γ

⃗ ∈ −𝒜u(t) ⃗ + F(t, u(t)), ⃗ D+ u(t)

t ∈ ℝ,

(190)

γ

where D+ u(t) denotes the Weyl–Liouville fractional derivative of order γ and F : ℝ×X → ⃗ ∈ B−1 u(t), ⃗ X; after the usual substitution v(t) t ∈ ℝ, this inclusion becomes γ

⃗ ⃗ + F(t, Bv(t)), ⃗ D+ [Bv(t)] = −Av(t)

t ∈ ℝ.

We say that a continuous function u : ℝ → X is a mild solution of (190) if and only if

268 � 4 Metrically ρ-almost periodic type functions and applications t

⃗ = ∫ Rγ (t − s)F(s, u(s)) ⃗ u(t) ds,

t ∈ ℝ.

−∞

Fix now a strictly increasing sequence (αk ) of positive real numbers tending to plus infinity as well as a number c ∈ ℂ ∖ {0} and an essentially bounded function ν : ℝ → (0, ∞). By Y(αk ),ν (ℝ : X) we denote the set of all bounded continuous functions u⃗ : ℝ → X such that 󵄩 ⃗ ⃗ 󵄩󵄩󵄩󵄩ν(t)] = 0. lim sup[󵄩󵄩󵄩u(t + αk ) − cu(t)

k→+∞ t∈ℝ

It can be simply proved that the set Y(αk ),ν (ℝ : X) equipped with the metric d(⋅, ⋅) := ‖ ⋅ − ⋅ ‖∞ forms a complete metric space. We have the following result: Theorem 4.2.9. Suppose that p ∈ (1, ∞), 1/p + 1/q = 1 and F : ℝ × X → X is a measurable function. Let there exist a real number r ⩾ max(p, p/(p − 1)) and a Stepanov r-bounded function LF : ℝ → [0, ∞) such that (186) holds. Suppose further that for every compact set K ⊆ X, we have t+1

lim

󵄩 󵄩p sup [ ∫ 󵄩󵄩󵄩F(s + αk ; cu) − cF(s; u)󵄩󵄩󵄩 νp (s) ds]

k→+∞ t∈ℝ,u∈K

t

1/p

= 0.

Suppose that p(γβ − 1) > 1, ∫0 ‖Rγ (s)‖LF (s) ds < 1 and there exists an essentially bounded function φ : ℝ → (0, ∞) such that ν(t − s) ⩽ φ(t)ν(s) for all t, s ∈ ℝ. Then there exists a unique mild solution of problem (190) which belongs to the space Y(αk ),ν (ℝ : X). ∞

Proof. Define Λ′ := {αk : k ∈ ℕ} and ϒ : Y(αk ),ν (ℝ : X) → Y(αk ),ν (ℝ : X) by t

⃗ ⃗ (ϒu)(t) := ∫ Rγ (t − s)F(s, u(s)) ds,

t ∈ ℝ.

−∞

We claim that the mapping ϒ(⋅) is well defined. Suppose that u⃗ ∈ Y(αk ),ν (ℝ : X). Then the set R(u)⃗ = B is bounded and therefore relatively compact in X. Applying an analogue of Theorem 2.2 for metrically Stepanov uniformly recurrent functions, we get that the ⃗ function H(⋅) ≡ F(⋅, u(⋅)) satisfies t+1

󵄩 󵄩q lim sup[ ∫ 󵄩󵄩󵄩H(s + αk ) − cH(s; u)󵄩󵄩󵄩 νq (s) ds] k→+∞ t∈ℝ

t

1/q

= 0.

Since p(γβ − 1) > 1 and there exists an essentially bounded function φ : ℝ → (0, ∞) with the prescribed assumptions, we can repeat verbatim the proof of Proposition 4.2.8, with the functions σ(⋅) and 1/ν(⋅) replaced therein with the function ν(⋅), and the number q

4.3 Weyl ρ-almost periodic functions in general metric

� 269

replaced therein with the number p, in order to see that ϒu⃗ ∈ Y(αk ),ν (ℝ : X); hence, ∞ the mapping ϒ(⋅) is well defined. Due to our assumption ∫0 ‖Rγ (s)‖LF (s) ds < 1, this mapping is a contraction, so that the proof of theorem completes an application of the Banach contraction principle.

4.3 Weyl ρ-almost periodic functions in general metric In this section, we analyze the existence of Weyl distance in general metric and various classes of multi-dimensional Weyl ρ-almost periodic functions in general metric. The main structural properties for the introduced classes of Weyl almost periodic type functions are established. We also provide some applications of our theoretical results to the abstract Volterra integro-differential equations. The organization and main ideas of this section can be briefly summarized as follows. In Section 4.3.1, we generalize the notions of Weyl p-distance and Weyl p-boundedness in the multi-dimensional setting (1 ⩽ p < ∞), introduced and analyzed in [457, Subsection 3.1]. Our main results in this section are Theorem 4.3.1 and Proposition 4.3.5. The central part is Section 4.3.2, where we introduce and analyze several classes of multidimensional Weyl ρ-almost periodic functions in general metric. In almost all established structural results and characterizations, we deal with some special metric spaces, primarily with the Banach spaces of weighted function spaces. The convolution invariance of metrical Weyl almost periodicity has been analyzed in Theorem 4.3.10, while a composition principle for Weyl almost periodic functions in general metric has been clarified in Theorem 4.3.13 (cf. also Propositions 4.3.11–4.3.12). The invariance of metrical Weyl almost periodicity under the actions of infinite convolution products has been investigated in Section 3.3.1. We present some applications of our theoretical results to the abstract Volterra integro-differential equations in Section 4.3.4; in addition to the above, we present several illustrative examples, open problems, and useful remarks. As a certain drawback of this approach, we want also to emphasize that we have not been able to explain how the notion of metrical Weyl ρ-almost periodicity can be used to intermediate the concepts Stepanov ρ-almost periodicity and equi-Weyl ρ-almost periodicity. On the other hand, it is our strong belief that this is only the beginning of several new investigations of metrical Weyl ρ-almost periodic type functions. We have primarily dealt with the weighted function spaces here, so that the possibilities for the expansion of this work certainly exist if we consider some other classes of Banach function spaces and Fréchet function spaces, like the (little) Hölder spaces, the spaces of functions of bounded variation, as well as the spaces of k-times differentiable functions and the spaces of infinitely differentiable functions.

270 � 4 Metrically ρ-almost periodic type functions and applications 4.3.1 Metrical Weyl distance In this subsection, we will generalize the usual notions of Weyl p-distance and Weyl p-boundedness (1 ⩽ p < ∞). We assume here that F : Λ × X → Y and G : Λ × X → Y ; let us consider the following conditions: (D0) 0 ≠ Λ ⊆ ℝn , 0 ≠ Ω ⊆ ℝn is a Lebesgue measurable set such that m(Ω) > 0, and Λ + lΩ ⊆ Λ for all l > 0. (D1) For every l > 0, 𝒫lΩ = (PlΩ , dlΩ ) is a metric space of functions from Y lΩ containing the zero function. We set ‖f ‖PlΩ :=dlΩ (f , 0) for all f ∈ PlΩ . We also assume that 𝒫 = (P, d) is a metric space of functions from [0, ∞)Λ containing the zero function and set ‖f ‖P := d(f , 0) for all f ∈ P. The argument from Λ will be denoted by ⋅⋅ and the argument from t + lΩ will be denoted by ⋅. (D2) F : (0, ∞) → (0, ∞) is a monotonically decreasing function. (D3) For every t ∈ Λ, l > 0 and x ∈ X, we have F(t + ⋅; x) − G(t + ⋅; x) ∈ PlΩ and the mapping t 󳨃→ ‖F(t + ⋅; x) − G(t + ⋅; x)‖PlΩ belongs to P. (D4) If f , g ∈ P and f ⩽ g, then ‖f ‖P ⩽ ‖g‖P . (D5) There exists a function ψ : (0, ∞) → (0, ∞) such that, if f ∈ P and c > 0, then cf ∈ P and ‖cf ‖P ⩽ ψ(c)‖f ‖P . (D6) If 0 < l1 < l2 < +∞, f ∈ Pl2 Ω and f|l1 Ω ∈ Pl1 Ω , then ‖f|l1 Ω ‖Pl Ω ⩽ ‖f ‖Pl Ω . 1 2 (D7) There exist a function G : ℕ × (0, ∞) → (0, ∞) and a function σ : (0, ∞) → (0, ∞) such that for every k ∈ ℕ and l > 0, we have: 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩F(kl)󵄩󵄩󵄩F(⋅ ⋅ +⋅; x) − G(⋅ ⋅ +⋅; x)󵄩󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩 󵄩P(k+1)lΩ 󵄩󵄩󵄩P ⩽ 󵄩󵄩󵄩F(kl)G(k, l)󵄩󵄩F(⋅ ⋅ +⋅; x) − G(⋅ ⋅ +⋅; x)󵄩󵄩PlΩ 󵄩󵄩󵄩P , F(kl)G(k, l) lim ψ( ) = σ(l) and lim σ(l) ⩽ 1. k→+∞ l→+∞ F(l) If conditions (D0), (D1) and (D2) hold, then for each l > 0 and B ∈ ℬ we set 󵄩󵄩 󵄩 󵄩 󵄩󵄩 B dF,𝒫 (F, G) := sup󵄩󵄩󵄩F(l)󵄩󵄩󵄩F(⋅ ⋅ +⋅; x) − G(⋅ ⋅ +⋅; x)󵄩󵄩󵄩P 󵄩󵄩󵄩 . lΩ ,𝒫 lΩ 󵄩P 󵄩 x∈B B Our first result states that liml→+∞ dF,𝒫 (F, G) exists under the above-clarified lΩ ,𝒫 conditions:

Theorem 4.3.1. Suppose that conditions (D0)–(D7) hold and B ∈ ℬ is a fixed set. Then B liml→+∞ dF,𝒫 (F, G) exists in [0, ∞]. lΩ ,𝒫 Proof. Let a number l1 ∈ (0, ∞) be fixed, and let l2 = kl1 + θl1 for some positive integer k ∈ ℕ and a real number θ ∈ [0, 1). Using (D2)–(D4) and (D6), we have: 󵄩󵄩 󵄩 󵄩 󵄩󵄩 sup󵄩󵄩󵄩F(l2 )󵄩󵄩󵄩F(⋅ ⋅ +⋅; x) − G(⋅ ⋅ +⋅; x)󵄩󵄩󵄩P 󵄩󵄩󵄩 l2 Ω 󵄩P 󵄩 x∈B 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 ⩽ sup󵄩󵄩󵄩F(kl1 )󵄩󵄩󵄩F(⋅ ⋅ +⋅; x) − G(⋅ ⋅ +⋅; x)󵄩󵄩󵄩P (k+1)l1 Ω 󵄩 󵄩 󵄩P x∈B

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󵄩󵄩 G(k, l1 ) 󵄩 󵄩 󵄩󵄩 ⩽ sup󵄩󵄩󵄩F(kl1 ) F(l1 )󵄩󵄩󵄩F(⋅ ⋅ +⋅; x) − G(⋅ ⋅ +⋅; x)󵄩󵄩󵄩P 󵄩󵄩󵄩 l1 Ω 󵄩P 󵄩 F(l1 ) x∈B ⩽ ψ(F(kl1 )

G(k, l1 ) B )dF,𝒫l Ω ,𝒫 (F, G). 1 F(l1 )

Employing now the equality in (D7), we get by letting k → ∞ that: B B lim sup dF,𝒫 (F, G) ⩽ σ(l1 )dF,𝒫 (F, G). lΩ ,𝒫 l Ω ,𝒫 1

l→+∞

(191)

Performing after that the operation lim infl1 →+∞ ⋅ at the both sides of the above inequality, and using the assumption that liml1 →+∞ σ(l1 ) ⩽ 1, we obtain B B lim sup dF,𝒫 (F, G) ⩽ lim inf dF,𝒫 (F, G), lΩ ,𝒫 lΩ ,𝒫 l→+∞

l→+∞

(192)

which implies the required. We call the limit W ,B dF,𝒫 (F, G) := lim dF,𝒫lΩ ,𝒫 (F, G) ∗Ω ,𝒫 l→+∞

the generalized (F, {l > 0 : 𝒫lΩ }, 𝒫 )-Weyl distance of functions F(⋅; ⋅) and G(⋅; ⋅). The usual notion of Weyl p-distance of functions F(⋅; ⋅) and G(⋅; ⋅) is obtained by plugging F(l) := l−(n/p) , l > 0, PlΩ := Lp (lΩ : Y ), dlΩ (f , g) := ‖f − g‖PlΩ (f , g ∈ PlΩ ), P = L∞ (Λ : [0, ∞)), d(f , g) := ‖f − g‖P (f , g ∈ P), and G(k, l) := (k + 1)n/p for all k ∈ ℕ and l > 0. We omit the term “B” from the notation if we consider functions of the form F : Λ → Y . Remark 4.3.2. If liml→+∞ σ(l) < 1, then the proof of Theorem 4.3.1 yields B B lim sup dF,𝒫 (F, G) < lim inf dF,𝒫 (F, G) lΩ ,𝒫 lΩ ,𝒫 l→+∞

W ,B in (192); hence, dF,𝒫

∗Ω ,𝒫

l→+∞

(F, G) = 0 for each set B ∈ ℬ. Case liml→+∞ σ(l) < 1 can occur, as 2

the following example indicates: Suppose that σ > 0, Ω := [0, 1], Λ := [0, ∞), F(l) := e−l (l > 0), PlΩ := L∞ ([0, l] : ℂ) (l > 0), ν(t) := e−t , t ⩾ 0, and P := C0,ν ([0, ∞) : [0, ∞)). Then the functions F(⋅) ≡ ⋅σ and G ≡ 0 satisfy (D3); furthermore, the usual Weyl p-distance of these functions equals +∞ for every finite exponent p ∈ [1, ∞). On the other hand, we W have dF,𝒫 (F, G) = 0 since the assumptions (D0)–(D7) are satisfied with the functions ∗Ω ,𝒫 ψ(c) ≡ c (c > 0) and G(k, l) := ekl (k ∈ ℕ, l > 0), when we have liml→+∞ σ(l) = 0 (we should only note here that we can write the interval [0, (k + 1)l] as the union of the intervals [0, l], . . . , [kl, (k + 1)l] and apply the substitution y 󳨃→ y − il for the interval [il, (i + 1)l], where l > 0, k ∈ ℕ and 1 ⩽ i ⩽ k). We will present the following example as well:

272 � 4 Metrically ρ-almost periodic type functions and applications Example 4.3.3. Suppose that condition (D2) holds, ν : [0, ∞) → (0, ∞) is a monotonically decreasing function, Ω = [0, 1], Λ = ℝ, PlΩ := C0,ν ([0, l] : ℂ), P := L∞ (ℝ : [0, ∞)) and functions F : ℝ → ℂ, G : ℝ → ℂ satisfy (D3). Then it can be simply proved that all conditions (D0)–(D6) hold with ψ(c) ≡ c (c > 0) as well as that condition (D7) holds provided that F(l) := 1/ ln(2 + l), l > 0, G(l, k) := (1 + ln(2 + kl))/(ln(2 + l)), k ∈ ℕ, l > 0 and σ(l) ≡ 1 (l > 0). In this case, we have 1 󵄨󵄨 󵄨 󵄨󵄨F(x + y) − G(x + y)󵄨󵄨󵄨ν(y)]. x∈ℝ y∈[0,l] ln(2 + l)

W dF,𝒫 (F, G) = lim [sup sup ∗Ω ,𝒫 l→+∞

W For example, if F(x) = x + 1 and G(x) = x (x ∈ ℝ), then dF,𝒫 (F, G) = 0 but the usual ∗Ω ,𝒫 Weyl p-distance of these functions equals 1 for every finite exponent p ∈ [1, ∞). The interested reader may endeavor to construct some examples in which the Weyl W p-distance of functions F(⋅) and G(⋅) is strictly less than the distance dF,𝒫 (F, G) con∗Ω ,𝒫 sidered above.

Remark 4.3.4. (i) We have not used above the triangle inequalities in 𝒫lΩ and 𝒫 ; the statement of Theorem 4.3.1 also holds if the above spaces are pseudometric, only. (ii) Suppose that τ ∈ ℝn , τ + Λ ⊆ Λ and, for every t ∈ Λ, we have: 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩F(t + τ + ⋅; x) − G(t + τ⋅; x)󵄩󵄩󵄩PlΩ ⩽ 󵄩󵄩󵄩F(t + ⋅; x) − G(t + ⋅; x)󵄩󵄩󵄩PlΩ , resp. 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩F(t + τ + ⋅; x) − G(t + τ⋅; x)󵄩󵄩󵄩PlΩ = 󵄩󵄩󵄩F(t + ⋅; x) − G(t + ⋅; x)󵄩󵄩󵄩PlΩ . Then for each B ∈ ℬ we have W ,B W ,B dF,𝒫 (F(⋅ ⋅ +τ; ⋅), G(⋅ ⋅ +τ; ⋅)) ⩽ dF,𝒫 (F, G), ∗Ω ,𝒫 ∗Ω ,𝒫

resp. W ,B W ,B dF,𝒫 (F(⋅ ⋅ +τ; ⋅), G(⋅ ⋅ +τ; ⋅)) = dF,𝒫 (F, G). ∗Ω ,𝒫 ∗Ω ,𝒫

(iii) Suppose that for each l > 0 the metric space PlΩ has a linear vector structure and the metric dlΩ is translation invariant. Then ‖f + g‖PlΩ ⩽ ‖f ‖PlΩ + ‖g‖PlΩ for all f , g ∈ PlΩ (l > 0). If, additionally, condition (D3) holds for the functions F(⋅; ⋅) and H(⋅; ⋅) in place of functions F(⋅; ⋅) and G(⋅; ⋅), the above implies W ,B W ,B W ,B dF,𝒫 (F, G) ⩽ dF,𝒫 (F, H) + dF,𝒫 (H, G), ∗Ω ,𝒫 ∗Ω ,𝒫 ∗Ω ,𝒫

B ∈ ℬ.

,B Further on, if the requirements of this part hold, then we denote BW F,𝒫 W ,B Λ × X → Y ; dF,𝒫

∗Ω

W ,B ,𝒫 (F, 0) < ∞} (B ∈ ℬ ). Clearly, the set BF,𝒫

W ,B pseudo-distance dF,𝒫

∗Ω

W ,B ,𝒫 (F, G) for any F, G ∈ BF,𝒫

∗Ω ,𝒫

∗Ω ,𝒫

∗Ω ,𝒫

:= {F :

equipped with the

is a pseudometric space.

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Now we would like to state and prove the following extension of [457, Proposition 3.3]: W ,B Proposition 4.3.5. Suppose that (D0)–(D7) hold. Then dF,𝒫 (F, G) < ∞ if and only if ∗Ω ,𝒫 for all (some) l > 0 we have dF,𝒫lΩ ,𝒫 (F, G) < ∞.

Proof. Suppose that for all l > 0 we have dF,𝒫lΩ ,𝒫 (F, G) < ∞; then this clearly holds for W ,B any fixed real number l > 0. If this is the case, then dF,𝒫 ,𝒫 (F, G) < ∞ due to (191). On ∗Ω

W ,B the other hand, if dF,𝒫 (F, G) < ∞, then there exist two finite real numbers M > 0 and ∗Ω ,𝒫 l0 > 0 such that the assumption l2 ⩾ l0 implies dF,𝒫l Ω ,𝒫 (F, G) < M. The final conclusion 2 follows from this estimate and the next computation, which is valid for any l1 ∈ (0, l0 ]; see conditions (D5) and (D6):

󵄩󵄩 󵄩 󵄩 󵄩󵄩 sup󵄩󵄩󵄩F(l1 )󵄩󵄩󵄩F(⋅ ⋅ +⋅; x) − G(⋅ ⋅ +⋅; x)󵄩󵄩󵄩P 󵄩󵄩󵄩 l1 Ω 󵄩P 󵄩 x∈B 󵄩󵄩 󵄩 󵄩 󵄩󵄩 ⩽ sup󵄩󵄩󵄩F(l1 )󵄩󵄩󵄩F(⋅ ⋅ +⋅; x) − G(⋅ ⋅ +⋅; x)󵄩󵄩󵄩P 󵄩󵄩󵄩 l0 Ω 󵄩P 󵄩 x∈B ⩽ ψ(

F(l1 ) 󵄩󵄩 󵄩 󵄩 󵄩󵄩 ) sup󵄩󵄩F(l )󵄩󵄩F(⋅ ⋅ +⋅; x) − G(⋅ ⋅ +⋅; x)󵄩󵄩󵄩P 󵄩󵄩󵄩 . l0 Ω 󵄩P F(l0 ) x∈B 󵄩󵄩 0 󵄩

Keeping in mind Theorem 4.3.1, we have an open door to generalize the notion of Weyl-(R, ℬ, p)-normality, introduced in [457, Definition 3.5], as well as the notion of space e − ℬ − W p (Λ × X : Y ), introduced in [457, Definition 3.7]. Without going into full details, we want only to note here that the statement of [457, Theorem 3.8] can be formulated in our new framework under certain reasonable assumptions.

4.3.2 Metrical Weyl ρ-almost periodic type functions As already mentioned, the multi-dimensional Weyl ρ-almost periodic functions have been analyzed in [443, Section 3]. The main aim of this subsection is to introduce and analyze the multi-dimensional Weyl ρ-almost periodic functions in general metric. In the first concept, we assume the validity of the following conditions: (WM1-1): 0 ≠ Λ ⊆ ℝn , 0 ≠ Λ′ ⊆ ℝn , 0 ≠ Ω ⊆ ℝn is a Lebesgue measurable set such that m(Ω) > 0, p ∈ 𝒫 (Λ), Λ′ + Λ + lΩ ⊆ Λ, Λ + lΩ ⊆ Λ for all l > 0, ϕ : [0, ∞) → [0, ∞) and 𝔽 : (0, ∞) × Λ → (0, ∞). (WM1-2): For every t ∈ Λ and l > 0, 𝒫t,l = (Pt,l , dt,l ) is a metric space of functions from ℂt+lΩ , resp. Y t+lΩ , in the case of consideration of Definition 4.3.6, resp. Definitions 4.3.7–4.3.8, containing the zero function. We set ‖f ‖Pt,l := dt,l (f , 0) for all f ∈ Pt,l . We also assume that 𝒫 = (P, d) is a metric space of functions from ℂΛ containing the zero function and set ‖f ‖P := d(f , 0) for all f ∈ P. The argument from Λ will be denoted by ⋅⋅ and the argument from t + lΩ will be denoted by ⋅.

274 � 4 Metrically ρ-almost periodic type functions and applications In the case that (WM1-1) holds and 0 ∈ Ω, the notion of Bohr (ℬ, Λ′ , ρ)-almost periodicity is a special case of Definition 4.3.6 with the metric space P chosen to be l∞ (Λ : Y ) and the metric space Pt,l chosen to be l∞ (t + lΩ : Y ), for any t ∈ Λ and l > 0. On the other hand, the notion of Bohr (ℬ, Λ′ , ρ, 𝒫 )-almost periodicity can be viewed as a special case of the notion analyzed in this section only if we assume some extra conditions on the metric space 𝒫 and the function F(⋅) under our consideration. For example, it suffices to choose the metric space Pt,l to be l∞ (t + lΩ : Y ), for any t ∈ Λ and l > 0, as well as that the metric space 𝒫 satisfies that the preassumptions 0 ⩽ ‖f (t)‖Y ⩽ ‖g(t)‖Y , t ∈ Λ and f , g ∈ P imply ‖f ‖P ⩽ ‖g‖P and 𝔽(l, t)

sup

u∈t+lΩ;τ∈Λ′

󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩F(u + τ) − Gx (u)󵄩󵄩󵄩Y ⩽ 󵄩󵄩󵄩F(t + τ) − Gx (t)󵄩󵄩󵄩Y ,

whenever t ∈ Λ, l > 0, x ∈ X and Gx (⋅) ∈ ρ(F(⋅; x)); then the assumption that F(⋅; ⋅) is Bohr (ϕ,𝔽,ρ,𝒫 ,𝒫) (ℬ, Λ′ , ρ, 𝒫 )-almost periodic implies that F(⋅; ⋅) belongs to the class (e−)WΩ,Λ′ ,ℬ t,l (Λ × X : Y ) introduced in Definition 4.3.6 below. In the sequel, we will omit the term “ℬ” from the notation if X = {0}, i. e., if we consider functions of the form F : Λ → Y . All classes of multi-dimensional Weyl ρ-almost periodic functions introduced in the first three definitions of [443, Section 3] are special cases of the following classes of functions, with Pt,l = Lp(⋅) (t + lΩ : ℂ), resp. Pt,l = Lp(⋅) (t + lΩ : Y ), the metric dt,l induced by the norm of this Banach space (t ∈ Λ, l > 0), P = L∞ (Λ : ℂ) and the metric d induced by the norm of this Banach space: Definition 4.3.6. (i) By e − WΩ,Λ′ ,ℬ t,l (Λ × X : Y ) we denote the set consisting of all functions F : Λ × X → Y such that, for every ε > 0 and B ∈ ℬ, there exist two finite real numbers l > 0 and L > 0 such that for each t0 ∈ Λ′ there exists τ ∈ B(t0 , L) ∩ Λ′ such that for every x ∈ B, the mapping u 󳨃→ Gx (u) ∈ ρ(F(u; x)), u ∈ Ω is well defined, and (ϕ,𝔽,ρ,𝒫 ,𝒫)

󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩󵄩 󵄩 󵄩 󵄩 sup󵄩󵄩𝔽(l, ⋅⋅)󵄩󵄩ϕ(󵄩󵄩F(τ + ⋅; x) − Gx (⋅)󵄩󵄩Y )󵄩󵄩 󵄩󵄩 < ε. 󵄩 󵄩P⋅⋅,l 󵄩󵄩 󵄩󵄩 x∈B 󵄩 󵄩P (ii) By WΩ,Λ′ ,ℬ t,l (Λ×X : Y ) we denote the set consisting of all functions F : Λ×X → Y such that for every ε > 0 and B ∈ ℬ, there exists a finite real number L > 0 such that for each t0 ∈ Λ′ there exists τ ∈ B(t0 , L) ∩ Λ′ such that, for every x ∈ B, the mapping u 󳨃→ Gx (u) ∈ ρ(F(u; x)), u ∈ Ω is well defined, and (ϕ,𝔽,ρ,𝒫 ,𝒫)

󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 lim sup sup󵄩󵄩󵄩𝔽(l, ⋅⋅)󵄩󵄩󵄩ϕ(󵄩󵄩󵄩F(τ + ⋅; x) − Gx (⋅)󵄩󵄩󵄩Y )󵄩󵄩󵄩 󵄩󵄩󵄩 < ε. 󵄩 󵄩P⋅⋅,l 󵄩󵄩 󵄩󵄩 l→+∞ x∈B 󵄩 󵄩P Definition 4.3.7. (i) By e − WΩ,Λ′ ,ℬ t,l 1 (Λ × X : Y ) we denote the set consisting of all functions F : Λ × X → Y such that, for every ε > 0 and B ∈ ℬ, there exist two finite real numbers l > 0 and L > 0 such that for each t0 ∈ Λ′ there exists τ ∈ B(t0 , L) ∩ Λ′ (ϕ,𝔽,ρ,𝒫 ,𝒫)

4.3 Weyl ρ-almost periodic functions in general metric

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such that for every x ∈ B, the mapping u 󳨃→ Gx (u) ∈ ρ(F(u; x)), u ∈ Ω is well defined, and 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 sup󵄩󵄩󵄩𝔽(l, ⋅⋅)ϕ(󵄩󵄩󵄩F(τ + ⋅; x) − Gx (⋅)󵄩󵄩󵄩P )󵄩󵄩󵄩 < ε. ⋅⋅,l 󵄩 󵄩 󵄩P x∈B 󵄩 (ii) By WΩ,Λ′ ,ℬ t,l 1 (Λ×X : Y ) we denote the set consisting of all functions F : Λ×X → Y such that, for every ε > 0 and B ∈ ℬ, there exists a finite real number L > 0 such that for each t0 ∈ Λ′ there exists τ ∈ B(t0 , L) ∩ Λ′ such that, for every x ∈ B, the mapping u 󳨃→ Gx (u) ∈ ρ(F(u; x)), u ∈ Ω is well defined, and (ϕ,𝔽,ρ,𝒫 ,𝒫)

󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 lim sup sup󵄩󵄩󵄩𝔽(l, ⋅⋅)ϕ(󵄩󵄩󵄩F(τ + ⋅; x) − Gx (⋅)󵄩󵄩󵄩P )󵄩󵄩󵄩 < ε. ⋅⋅,l 󵄩 󵄩 󵄩 󵄩P l→+∞ x∈B Definition 4.3.8. (i) By e − WΩ,Λ′ ,ℬ t,l 2 (Λ × X : Y ) we denote the set consisting of all functions F : Λ × X → Y such that, for every ε > 0 and B ∈ ℬ, there exist two finite real numbers l > 0 and L > 0 such that for each t0 ∈ Λ′ there exists τ ∈ B(t0 , L) ∩ Λ′ such that, for every x ∈ B, the mapping u 󳨃→ Gx (u) ∈ ρ(F(u; x)), u ∈ Ω is well defined, and (ϕ,𝔽,ρ,𝒫 ,𝒫)

󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 sup󵄩󵄩󵄩ϕ(𝔽(l, ⋅⋅)󵄩󵄩󵄩F(τ + ⋅; x) − Gx (⋅)󵄩󵄩󵄩P )󵄩󵄩󵄩 < ε. ⋅⋅,l 󵄩 󵄩 󵄩P x∈B 󵄩 (ii) By WΩ,Λ′ ,ℬ t,l 2 (Λ×X : Y ) we denote the set consisting of all functions F : Λ×X → Y such that, for every ε > 0 and B ∈ ℬ, there exists a finite real number L > 0 such that for each t0 ∈ Λ′ there exists τ ∈ B(t0 , L) ∩ Λ′ such that for every x ∈ B, the mapping u 󳨃→ Gx (u) ∈ ρ(F(u; x)), u ∈ Ω is well defined, and (ϕ,𝔽,ρ,𝒫 ,𝒫)

󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 lim sup sup󵄩󵄩󵄩ϕ(𝔽(l, ⋅⋅)󵄩󵄩󵄩F(τ + ⋅; x) − Gx (⋅)󵄩󵄩󵄩P )󵄩󵄩󵄩 < ε. ⋅⋅,l 󵄩 󵄩 󵄩 󵄩P l→+∞ x∈B Let us denote by AX,Y any of the above introduced classes of function spaces. If F(⋅; ⋅) belongs to AX,Y , c1 ∈ ℝ ∖ {0}, c, c2 ∈ ℂ ∖ {0}, x0 ∈ X, then it is not difficult to find some sufficient conditions ensuring that some of the functions cF(⋅; ⋅), F(c1 ⋅; c2 ⋅), ‖F(⋅; ⋅)‖Y also belongs to AX,Y ; all this has been seen many times before and details can be omitted (see also Lemma 1.1.10(ii)–(iii)). Let us recall from the corresponding analyses of Stepanov classes that it is much more complicated to say when AX,Y will be a vector space, even in the case that ρ = I, as well as that the use of Jensen integral inequality in general measure spaces enables one to clarify certain inclusions about the introduced classes of (ϕ,𝔽,ρ,𝒫 ,𝒫) functions. If (Fk (⋅; ⋅)) is a sequence of functions from the space WΩ,Λ′ ,ℬ t,l (Λ × X : Y ) and there exists a function F : Λ × X → Y such that limk→+∞ Fk (t; x) = F(t; x), uniformly (ϕ,𝔽,ρ,𝒫 ,𝒫) on Λ × B for each set B of collection ℬ, then F ∈ WΩ,Λ′ ,ℬ t,l (Λ × X : Y ) provided that the following holds:

276 � 4 Metrically ρ-almost periodic type functions and applications (i) Pt,l and P are Banach spaces for all t ∈ Λ and l > 0; (ii) There exists a finite real constant c > 0 such that ϕ(x + y) ⩽ c[ϕ(x) + ϕ(y)] for all x, y ⩾ 0; (iii) ϕ(⋅) is continuous at zero; (iv) D(ρ) is closed, ρ is single-valued on R(F) and satisfies (Cρ ); (v) There exists l0 > 0 such that supl⩾l0 ‖𝔽(l, ⋅⋅)‖P < +∞. Concerning embeddings of metrically Stepanov ρ-almost periodic functions into the spaces of metrically equi-Weyl ρ-almost periodic functions, we will clarify only one result here (this result generalizes some already known results in the classical approach; see [431]): Proposition 4.3.9. Suppose that (SM1-1)–(SM1-2) hold, F ∈ SΩ,Λ′ ,ℬ t (Λ × X : Y ), 𝔽1 : (0, ∞) × Λ → (0, ∞) and (WM1-1)–(WM1-2) hold with the metric space 𝒫 and the metric 1 spaces 𝒫t,l replaced therein by the metric space 𝒫1 and the metric spaces 𝒫t,l (t ∈ Λ; l > 0). Suppose, further, that the following conditions hold: (i) The space 𝒫1 is a Banach space; (ii) If f , g ∈ P1 and 0 ⩽ f ⩽ g, then ‖f ‖P1 ⩽ ‖g‖P1 ; (iii) We have that P = L∞ ν (Λ : ℂ) with some Lebesgue measurable function ν(⋅); (iv) There exist two finite real numbers c > 0, M > 0 and a positive integer l ∈ ℕ such that (ϕ,𝔽,ρ,𝒫 ,𝒫)

󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 1 󵄩󵄩𝔽1 (l, ⋅⋅) ∑ 󵄩󵄩 ⩽ M, 󵄩󵄩 ν(⋅ ⋅ +k)𝔽(⋅ ⋅ +k) 󵄩󵄩󵄩󵄩P 󵄩󵄩 n k∈(ℕl−1 ) 1 0

(193)

and, for every t ∈ Λ and t′ ∈ Λ, the assumption f ∈ Pt,l implies f|(t′ +Ω) ∈ Pt′ whenever t′ + Ω ⊆ t + lΩ, and 󵄩󵄩 󵄩 󵄩 󵄩󵄩ϕ(󵄩󵄩F(τ + ⋅; x) − Gx (⋅)󵄩󵄩󵄩 )󵄩󵄩󵄩 ⩽ c 󵄩󵄩 󵄩 󵄩Y 󵄩󵄩P1 t,l 1 (ϕ,𝔽 ,ρ,𝒫t,l ,𝒫1 )

1 Then F ∈ e − WΩ,Λ′ ,ℬ

∑

k∈(ℕl−1 0 )

󵄩󵄩 󵄩 󵄩 󵄩󵄩ϕ(󵄩󵄩F(τ + ⋅; x) − Gx (⋅)󵄩󵄩󵄩 )󵄩󵄩󵄩 . 󵄩󵄩 󵄩 󵄩Y 󵄩󵄩Pt+k n

(194)

(Λ × X : Y ).

Proof. Fix a real number ε > 0 and a set B ∈ ℬ. Then there exists a finite real number L > 0 such that for each t0 ∈ Λ′ there exists τ ∈ B(t0 , L) ∩ Λ′ such that for every x ∈ B, the mapping u 󳨃→ Gx (u) ∈ ρ(F(u; x)), u ∈ Ω is well defined, and (185) holds. Using (194), we get that 󵄩󵄩 󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩𝔽1 (l, ⋅⋅)󵄩󵄩󵄩ϕ(󵄩󵄩󵄩F(τ + ⋅; x) − Gx (⋅)󵄩󵄩󵄩Y )󵄩󵄩󵄩 1 󵄩󵄩󵄩 󵄩󵄩 󵄩 󵄩P⋅⋅,l 󵄩󵄩P1 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 ⩽ 󵄩󵄩󵄩c𝔽1 (l, ⋅⋅) ∑ 󵄩󵄩󵄩ϕ(󵄩󵄩󵄩F(τ + ⋅; x) − Gx (⋅)󵄩󵄩󵄩Y )󵄩󵄩󵄩 󵄩󵄩󵄩 . 󵄩 󵄩P⋅⋅+k 󵄩󵄩P1 󵄩󵄩 l−1 n k∈(ℕ0 )

4.3 Weyl ρ-almost periodic functions in general metric

� 277

n Due to our choice of space P, we have (t ∈ Λ; k ∈ (ℕl−1 0 ) ):

󵄩󵄩 󵄩 󵄩 ε 󵄩󵄩ϕ(󵄩󵄩F(τ + ⋅; x) − Gx (⋅)󵄩󵄩󵄩 )󵄩󵄩󵄩 󵄩󵄩 󵄩 󵄩Y 󵄩󵄩Pt+k ⩽ ν(t + k)𝔽(t + k) . Then the final conclusion simply follows from our assumption that 𝒫1 is a Banach space, (ii) and the estimate (193). Concerning the convolution invariance of metrical Weyl almost periodicity introduced in the above definitions, we will state the following result, closely connected with the statement of [457, Theorem 2.9] (let us only note that an analogue of Theorem 3.3.5 below can be formulated in this context as well): Theorem 4.3.10. Suppose that c ∈ ℂ ∖ {0}, φ : [0, ∞) → [0, ∞), ϕ : [0, ∞) → [0, ∞) is a convex monotonically increasing function satisfying ϕ(xy) ⩽ φ(x)ϕ(y) for all x, y ⩾ (ϕ,𝔽,cI,𝒫 ,𝒫) 0, h ∈ L1 (ℝn ), Ω = [0, 1]n , F ∈ (e−)WΩ,Λ′ ,ℬ t,l (ℝn × X : Y ), where p, q ∈ 𝒫 (ℝn ), n 1/p(⋅) + 1/q(⋅) = 1, ν : ℝ → (0, ∞) is a Lebesgue measurable function, ν(t) ⩾ d > 0 p(⋅) for some positive real number d > 0, Pt,l = Lν (t + lΩ : ℂ), the metric dt,l is induced by the norm of this Banach space (t ∈ Λ, l > 0), P = L∞ (ℝn : ℂ) and for each x ∈ X we have supt∈ℝn ‖F(t; x)‖Y < ∞. If (WM1-1)–(WM1-2) hold, 𝔽1 : (0, ∞) × ℝn → (0, ∞), the following two conditions (i) If 0 ⩽ |f (⋅)| ⩽ |g(⋅)| and f , g ∈ P (f , g ∈ Pt,l for some t ∈ ℝn and l > 0), then ‖f ‖P ⩽ ‖g‖P (‖f ‖Pt,l ⩽ ‖g‖Pt,l ); (ii) If ε > 0 and f ∈ P (f ∈ Pt,l for some t ∈ ℝn and l > 0), then εf ∈ P (εf ∈ Pt,l ) and ‖εf ‖P ⩽ ε‖f ‖P (‖εf ‖Pt,l ⩽ ε‖f ‖Pt,l ), hold, and, for every l > 0, there exists a sequence (ak )k∈lℤn of positive real numbers such that ∑k∈lℤn ak = 1 and 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 −1 ν(⋅) 󵄩 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩𝔽1 (l, ⋅⋅)󵄩󵄩󵄩2 ∑ ak l−n [φ(a−1 ln h(⋅ − v))] [𝔽(l, ⋅ − k)] 󵄩 󵄩 ⩽ 1, k 󵄩󵄩 q(v) 󵄩󵄩󵄩 L (⋅−k+lΩ) d 󵄩󵄩󵄩󵄩P1 󵄩󵄩󵄩󵄩P1 󵄩󵄩 k∈lℤn 󵄩 ⋅⋅,l 1 (ϕ,𝔽 ,cI,𝒫t,l ,𝒫 1 )

1 then h ∗ F ∈ (e−)WΩ,Λ′ ,ℬ

(195)

(ℝn × X : Y ).

Proof. Let ε > 0 and B ∈ ℬ be fixed. Then we know that there exist two finite real numbers l > 0 and L > 0 such that for each t0 ∈ Λ′ there exists τ ∈ B(t0 , L) ∩ Λ′ such that 󵄩 󵄩 sup sup 𝔽(l, t)ϕ(󵄩󵄩󵄩F(τ + u; x) − F(u; x)󵄩󵄩󵄩Y ) x∈B t∈ℝn

p(u)

Lν

(t+lΩ)

< ε.

For this l > 0, we can find a sequence (ak )k∈lℤn of positive real numbers such that ∑k∈lℤn ak = 1 and (195) holds. It is clear that the value of (h ∗ F)(t; x) is well defined for all t ∈ ℝn and x ∈ X since h ∈ L1 (ℝn ) and for each x ∈ X we have supt∈ℝn ‖F(t; x)‖Y < ∞. Arguing as in the proof of the above-mentioned theorem from [457], and using our assumptions on the function ν(⋅), we get that (t ∈ ℝn , l > 0; ⋅ ∈ t + lΩ):

278 � 4 Metrically ρ-almost periodic type functions and applications 󵄩󵄩 󵄩󵄩 ϕ(󵄩󵄩󵄩(h ∗ F)(⋅ + τ+; x) − c(h ∗ F)(⋅; x)󵄩󵄩󵄩 ) 󵄩 󵄩Y ⩽ ∑ 2ak l−n [φ(ak−1 ln h(⋅ − v))] k∈lℤn

Lq(v) (⋅−k+lΩ)

󵄩󵄩 󵄩󵄩 ν(⋅) 󵄩 󵄩 󵄩 󵄩 × 󵄩󵄩󵄩ϕ(󵄩󵄩󵄩F(τ + v; x) − F(v; x)󵄩󵄩󵄩Y )ν(v)󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩Lp(v) (⋅−k+lΩ) d ε ν(⋅) ⩽ ∑ 2ak l−n [φ(ak−1 ln h(⋅ − v))] q(v) , L (⋅−k+lΩ) 𝔽(l, ⋅ − k) d k∈lℤn so that the final conclusion simply follows from our assumptions (i) and (ii). The statement of [457, Theorem 3.4] can be reformulated in our new framework only after assuming certain very restrictive conditions; the situation is quite similar with the statements of [457, Propositions 3.9–3.11, Theorem 3.15]. It is also worth noting that it is almost impossible to satisfactorily reformulate the statement of [457, Theorem 3.12] in our new framework. Because of that, we will skip all related details concerning these issues here. p(⋅) Concerning the situation in which Pt,l = Lν (t + lΩ : ℂ), the metric dt,l,ν being induced by the norm of this Banach space (t ∈ Λ, l > 0), P = L∞ σ (Λ : ℂ) and the metric d being induced by the norm of this Banach space, where ν : Λ → (0, ∞) and σ : Λ → (0, ∞) are Lebesgue measurable functions, we have the following result: Proposition 4.3.11. Let p, q, r ∈ 𝒫 , 1/p(⋅) = 1/q(⋅) + 1/r(⋅), let 𝒫t,l and 𝒫 be defined as above (t ∈ Λ, l > 0), and let F ∈ (e−)WΩ,Λ′ ,ℬ

(ϕ,𝔽,ρ,𝒫t,l ,𝒫)

(Λ × X : Y ). Suppose that 𝒫q = 𝒫 and

νq : Λ → (0, ∞) is a Lebesgue measurable function such that νq /ν ∈ Lr(⋅) (t + lΩ) for all q t ∈ Λ and l > 0. Then F ∈ (e−)WΩ,Λ′ ,ℬ for all t ∈ Λ and l > 0, as well as

(ϕ,𝔽 ,ρ,𝒫t,l,q ,𝒫q )

𝔽q (l, t) :=

q(⋅)

(Λ × X : Y ), where Pt,l,q = Lνq (t + lΩ : ℂ)

𝔽(l, t) , ‖νq /ν‖Lr(⋅) (t+lΩ)

t ∈ Λ, l > 0.

Proof. The proof is easy and almost trivially follows from the corresponding definitions, the prescribed assumptions, and the Hölder inequality (see Lemma 1.1.10(i)): 󵄩󵄩 󵄩 󵄩 󵄩󵄩ϕ(󵄩󵄩F(τ + ⋅; x) − Gx (⋅)󵄩󵄩󵄩 )󵄩󵄩󵄩 󵄩󵄩 󵄩 󵄩Y 󵄩󵄩P⋅⋅,l,q 󵄩󵄩 󵄩 󵄩󵄩 󵄩 = 󵄩󵄩󵄩ϕ(󵄩󵄩󵄩F(τ + ⋅; x) − Gx (⋅)󵄩󵄩󵄩Y )νq (⋅)󵄩󵄩󵄩 q(⋅) 󵄩 󵄩L (t+lΩ:ℂ) 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 ⩽ 2󵄩󵄩ϕ(󵄩󵄩F(τ + ⋅; x) − Gx (⋅)󵄩󵄩Y )ν(⋅)󵄩󵄩󵄩 p(⋅) ‖ν /ν‖ r(⋅) 󵄩 󵄩L (t+lΩ:ℂ) q L (t+lΩ) 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 = 2󵄩󵄩󵄩ϕ(󵄩󵄩󵄩F(τ + ⋅; x) − Gx (⋅)󵄩󵄩󵄩Y )󵄩󵄩󵄩 ‖νq /ν‖Lr(⋅) (t+lΩ) . 󵄩 󵄩P⋅⋅,l We can similarly prove the following corollary:

4.3 Weyl ρ-almost periodic functions in general metric

� 279

Corollary 4.3.12. Let p, q, r ∈ [1, ∞], 1/p = 1/q + 1/r, let 𝒫t,l and 𝒫 be defined as above (ϕ,l−n/p ,ρ,𝒫 ,𝒫)

t,l with σ ≡ ν ≡ 1 (t ∈ Λ, l > 0), and let F ∈ (e−)WΩ,Λ′ ,ℬ (Λ × X : Y ) with Ω = [0, 1]n . Suppose that 𝒫q = 𝒫 and νq : Λ → (0, ∞) is a Stepanov r-bounded function. Then F ∈

(ϕ,l−n/q ,ρ,𝒫t,l,q ,𝒫q )

(e−)WΩ,Λ′ ,ℬ

q

(Λ × X : Y ), where Pt,l,q = Lνq (t + lΩ : ℂ) for all t ∈ Λ and l > 0.

The composition principles for Weyl almost periodic type functions have been investigated for the first time by F. Bedouhene et al. in [100]. In connection with this issue, we would like to emphasize that the analysis of existence and uniqueness of Weyl almost periodic type solutions of the abstract semilinear Cauchy problems requires further non-trivial examinations (as is well known, the spaces of equi-Weyl almost periodic functions are not complete with respect to the Weyl norm). In [431, Subsection 2.2.2], we have continued the analysis from [100] and prove two new composition principles for Weyl almost periodic functions in the one-dimensional setting [431, Theorem 2.2.37, Theorem 2.2.39]. Albeit these results can be also formulated in the multi-dimensional setting, we want to note here that we can further refine these results using the concept of metrical Weyl almost periodicity. For example, the statement of [431, Theorem 2.2.37] can be extended in the following way; the proof is omitted since it follows from a careful inspection of the proof of this result: Theorem 4.3.13. Suppose that c ∈ ℂ∖{0}, Λ = [0, ∞) or Λ = ℝ, 0 ≠ Λ′ ⊆ Λ, ν : Λ → (0, ∞) is a Lebesgue measurable function, F : Λ → X × Y and f : Λ → X. Let there exist a real number r ⩾ max(p, p/(p − 1)) and a Stepanov r-bounded function LF : Λ → [0, ∞) such that the function LF (⋅)ν(⋅) is also Stepanov r-bounded, 󵄩󵄩 󵄩 󵄩󵄩F(t; x) − F(t; y)󵄩󵄩󵄩Y ⩽ LF (t)‖x − y‖,

t ∈ Λ, x, y ∈ X,

and let there exist a set E ⊆ I with m(E) = 0 such that K := {x(t) : t ∈ Λ ∖ E} is relatively compact in X. Suppose further that for every ε > 0 and for every compact set K ⊆ X, there exist two real numbers l > 0 and L > 0 such that any interval Λ0 ⊆ Λ of length L contains a number τ ∈ Λ0 ∩ Λ′ such that t+l

1 󵄩 󵄩p sup [ ∫ 󵄩󵄩󵄩F(s + τ; cu) − cF(s; u)󵄩󵄩󵄩 νp (s) ds] l t∈Λ,u∈K

1/p

t

⩽ε

(196)

and t+l

1 󵄩 󵄩p sup[ ∫ 󵄩󵄩󵄩f (s + τ) − cf (s)󵄩󵄩󵄩 νpq (s) ds] t∈Λ l t

1/p

⩽ ε,

(197)

resp., there exists a finite number L > 0 such that any interval Λ0 ⊆ Λ of length L contains a number τ ∈ Λ0 ∩ Λ′ satisfying that there exists a number l(ε, τ) > 0 so that (196)–(197) hold for all numbers l ⩾ l(ε, τ). Then q := pr/(p + r) ∈ [1, p) and

280 � 4 Metrically ρ-almost periodic type functions and applications (x,l−1/q ,cI,𝒫t,l,q ,𝒫q )

F(⋅; f (⋅)) ∈ e − W[0,1],Λ′ q Lν (t

(x,l−1/q ,cI,𝒫t,l,q ,𝒫q )

(Λ : Y ), resp. F(⋅; f (⋅)) ∈ W[0,1],Λ′

(Λ : Y ), where

Pt,l,q := + lΩ : ℂ), the metric dt,l,ν is induced by the norm of this Banach space (t ∈ Λ, l > 0), Pq := L∞ (Λ : ℂ), and the metric d is induced by the norm of this Banach space.

We will not consider here the corresponding notion of metrical Weyl ρ-uniform recurrence; in this approach, we can simply consider the existence and uniqueness of metrically Weyl ρ-uniformly recurrent solutions for some classes of the abstract semilinear fractional Cauchy inclusions in the finite-dimensional spaces (Theorem 4.3.13 can be formulated in this framework; cf. [431] for related applications and more details about multi-dimensional Weyl uniformly recurrent type functions). We continue with the following well known example, whose one-dimensional analogue has been analyzed for the first time by J. Stryja in his master thesis [719]: Example 4.3.14. (i) Arguing similarly as in [440, Example 2.12] and [457, Example 2.7], we can prove that for each compact set K ⊆ ℝn with positive Lebesgue measure, for each number σ > 0 and for each function p ∈ 𝒫 (ℝn ), the function F(⋅) := χK (⋅) (x,l−σ ,cI,𝒫 ,𝒫)

p(⋅)

belongs to the space e − W[0,1]n ,ℝn t,l (ℝn : ℂ), where Pt,l = Lν (t + lΩ : ℂ), the n metric dt,l,ν is induced by the norm of this Banach space (t ∈ ℝn , l > 0), P = L∞ σ (ℝ : n ℂ), the metric d is induced by the norm of this Banach space, ν : ℝ → (0, ∞) and σ : ℝn → (0, ∞) are essentially bounded functions. (ii) Arguing similarly as in [440, Example 2.13] and [457, Example 2.8], we can prove that for each number σ > (n − 1)/p and for each finite exponent p ∈ [1, ∞), the function (x,l−σ ,I,𝒫 ,𝒫)

F(⋅) := χ[0,∞)n (⋅) belongs to the space W[0,1]n ,ℝn t,l (ℝn : ℂ), where Pt,l , dt,l,ν , P, d, ν(⋅) and σ(⋅) possess the same meaning as above, with p(u) ≡ p. The situation is tremendously different in the case of consideration of operator cI, where c ∈ ℂ ∖ {0, 1}, since in this case there do not exist a real number σ ∈ ((n − 1)/p, n/p] and functions ν(⋅) and σ(⋅) bounded from below by positive real constants such that the (x,l−σ ,cI,𝒫 ,𝒫)

function F(⋅) belongs to the space W[0,1]n ,ℝn t,l (ℝn : ℂ); see [439] for more details. (iii) The function F(⋅) from the second issue of this example is essentially bounded, and x,l−σ ,I,𝒫 ,𝒫

we know that there is no real number σ > 0 such that F ∈ e − W[0,1]n ,ℝn t,l (ℝn : ℂ), where P = l∞ (Λ : ℂ) and Pt,l = l∞ (t+lΩ : ℂ), for any t ∈ Λ and l > 0; see [457]. On the other hand, for any essentially bounded function G : ℝn → ℂ, we can always find a finite real number σ > 0 and a Lebesgue measurable function ν : ℝn → (0, ∞) x,l−σ ,cI,𝒫t,l ,𝒫

such that G ∈ e − W[0,1]n ,ℝn p Lν (t+lΩ

(ℝn : ℂ) for all c ∈ ℂ, where P = l∞ (ℝn : ℂ) and

Pt,l = : ℂ), for some p ∈ [1, ∞) and any t ∈ Λ, l > 0. Consider, for the purpose of illustration, the one-dimensional case and the Heaviside function G(⋅) = χ[0,∞) (⋅). Take ν(t) := |t|a , t ≠ 0, with some a ∈ (−1, 0) such that ap > −1. Using the inequality |f (x + τ) − cf (x)| ⩽ (1 + |c|)‖f ‖∞ , x ∈ ℝ, a very simple computation shows that x,l−σ ,cI,𝒫 ,𝒫

G ∈ e − W[0,1],ℝ t,l (ℝ : ℂ) for all c ∈ ℂ and σ > ap + 1. (iv) In connection with the last conclusion, we would like to note that a similar statement can be formulated for Stepanov p-bounded functions (1 ⩽ p < ∞). Consider, for sim-

4.3 Weyl ρ-almost periodic functions in general metric

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plicity, the one-dimensional setting and a Stepanov p-bounded function F : ℝ → Y . Define ν(t) := (

1 ) 1 + |t|−ap

1/p

1/p

=(

|t|ap ) 1 + |t|ap

,

t ∈ ℝ,

x,l−σ ,cI,𝒫 ,𝒫

where a ∈ (−1, 0) and ap > −1. Then F ∈ e − W[0,1]n ,ℝn t,l (ℝn : ℂ) for all c ∈ ℂ and σ > ap + 1. In order to see this, observe first that there exists a finite real constant cp > 0 such that, for every real numbers l > 0, t ∈ ℝ and τ ∈ ℝ, we have: 1/p

t+l

󵄩 󵄩p l−σ ( ∫ 󵄩󵄩󵄩F(s + τ) − cF(s)󵄩󵄩󵄩Y νp (s) ds) t

1/p

t+l

󵄩 󵄩p = l ( ∫ 󵄩󵄩󵄩F(s + τ) − cF(s)󵄩󵄩󵄩Y |s|ap /(1 + |s|ap ) ds) −σ

t

|s|ap |s|ap + ⋅ ⋅ ⋅ + max ]. s∈[t,t+1] 1 + |s|ap s∈[t+⌊l⌋,t+l] 1 + |s|ap

⩽ cp l−σ ‖F‖Sp [ max

After that, we can examine separately the following cases: t > 0, t ⩽ 0, t + l ⩽ 0, t ⩽ 0 and t + l > 0. The consideration in all these cases is similar, and we will only continue the above computation provided that t > 0. Since the function t 󳨃→ νp (t), t ⩾ 0 is monotonically decreasing, we have |s|ap |s|ap + ⋅ ⋅ ⋅ + max ] s∈[t,t+1] 1 + |s|ap s∈[t+⌊l⌋,t+l] 1 + |s|ap

[ max ⩽

|t|ap |t + ⌊l⌋|ap + ⋅ ⋅ ⋅ + 1 + |t|ap 1 + |t + ⌊l⌋|ap

⩽ 1 + (1ap + 2ap + ⋅ ⋅ ⋅ + ⌊l⌋ap ) ⩽ Const. ⋅ (1 + lap+1 ),

l ⩾ 1.

This simply implies the required. (v) Let us recall that, for any real number P > 1, H. Bohr and E. Følner have constructed a Stepanov S P -bounded function F : ℝ → ℝ which is Stepanov p-almost periodic for any exponent p ∈ [1, P) but not equi-Weyl-P-almost periodic (see [147, Main example 3, pp. 83–91]). Suppose further that the space 𝒫 , the function ν(⋅) and the space 𝒫t,l are defined as in (iv) with the number p replaced therein with the number P. x,l−σ ,cI,𝒫t,l ,𝒫

Due to our conclusion established in (iv), we have F ∈ e − W[0,1],ℝ all c ∈ ℂ and σ > aP + 1.

(ℝ : ℂ) for

In [194, Example 2.13, Example 2.15], we have exhibited the importance of case Λ ≠ Λ; these examples can be formulated in our new framework as well. Furthermore, in [431, Example 6.1.15], we have exhibited the importance of case Λ′ ⊈ Λ; this example can be also formulated in our new framework. From the high similarity with our ′

282 � 4 Metrically ρ-almost periodic type functions and applications previous research studies, we will skip all details with regards to the above-mentioned examples. Concerning the first-mentioned issue, we will present the following new illustrative example in support of our assumption Λ′ ≠ Λ: Example 4.3.15. Suppose that ψ ∈ L2 (ℝn : ℂ). As is well known, the short-time Fourier transform Vψ : L2 (ℝn : ℂ) → L2 (ℝ2n : ℂ) is defined by (Vψ f )(x, ξ) := ∫ ψ(t − x)e−2πiξt f (t) dt,

f ∈ L2 (ℝn : ℂ).

ℝn

̌ belongs to Suppose now that f ∈ L1 (ℝn : ℂ) ∩ L2 (ℝn : ℂ), as well as that the function ψ(⋅) x,𝔽,cI,𝒫t,l ,𝒫 n 1 n the class (e−)W[0,1]n ,Λ′ (ℝ : ℂ), where c ∈ ℂ∖{0}, 𝒫t,l = L (t+l[0, 1] : ℂ) for all t ∈ ℝn ∞ n and l > 0, P = L (ℝ : ℂ), and the function 𝔽(l, t) ≡ 𝔽(l) does not depend on the second x,𝔽 ,cI,𝒫 1 ,𝒫 1

argument. Then the function (Vψ f )(⋅, ⋅) belongs to the class (e−)W[0,1]12n ,(Λ′t,l×{0}) (ℝ2n : ℂ), where 𝒫t,l = L1 (t + l[0, 1]2n : ℂ) for all t ∈ ℝ2n and l > 0, P = L∞ (ℝ2n : ℂ) and 𝔽1 (l) = l−n 𝔽(l) for all l > 0. This is a consequence of the following computation (y ∈ ℝn , l > 0, τ ∈ Λ′ satisfies the obvious requirements, Ω = [0, 1]2n ): 󵄨 󵄨 ∫ 󵄨󵄨󵄨(Vψ f )(x + τ, ξ) − c(Vψ f )(x, ξ)󵄨󵄨󵄨 dx dξ

y+lΩ

󵄨 󵄨 󵄨 󵄨 ⩽ ∫ ∫ 󵄨󵄨󵄨ψ(t − x − τ) − cψ(t − x)󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨f (t)󵄨󵄨󵄨 dt dx dξ y+lΩ ℝn

󵄨 󵄨 󵄨 󵄨 = ∫ ∫ 󵄨󵄨󵄨ψ(t − x − τ) − cψ(t − x)󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨f (t)󵄨󵄨󵄨 dx dξ dt ℝn y+lΩ

󵄨 ̌ ̌ − t)󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨f (t)󵄨󵄨󵄨 dx dξ dt = ∫ ∫ 󵄨󵄨󵄨ψ(x + τ − t) − cψ(x 󵄨 󵄨 󵄨 ℝn y+lΩ

=∫

󵄨 ̌ ̌ 󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨f (t)󵄨󵄨󵄨 dz dξ dt + τ) − cψ(z) ∫ 󵄨󵄨󵄨ψ(z 󵄨 󵄨 󵄨

ℝn y−t+lΩ n

⩽

εl 󵄨 󵄨 ∫ 󵄨󵄨f (t)󵄨󵄨󵄨 dt. 𝔽(l) 󵄨 ℝn

4.3.3 Invariance of metrical Weyl ρ-almost periodicity under actions of infinite convolution products We start this subsection with the observation that the proofs of [431, Theorem 3.1.46] and Theorem 4.3.10 can be used to clarify certain results on the invariance of some special types of metrical Weyl ρ-almost periodicity under the actions of infinite convolution products. From application point of view, the most important is one-dimensional setting:

4.3 Weyl ρ-almost periodic functions in general metric

� 283

Theorem 4.3.16. Suppose that c ∈ ℂ ∖ {0}, φ : [0, ∞) → [0, ∞), ϕ : [0, ∞) → [0, ∞) is a convex monotonically increasing function satisfying ϕ(xy) ⩽ φ(x)ϕ(y) for all x, y ⩾ 0, (R(t))t>0 ⊆ L(X, Y ) is a strongly continuous operator family, Ω = [0, 1], (ϕ,𝔽,cI,𝒫t,l ,𝒫) f ̌ ∈ (e−)W ′ (ℝ × X : Y ), where p, q ∈ 𝒫 (ℝ), 1/p(⋅) + 1/q(⋅) = 1, ν : ℝ → [d, ∞) Ω,Λ

p(⋅)

for some positive real number d > 0, Pt,l = Lν (t + lΩ : ℂ), the metric dt,l is induced by the norm of this Banach space (t ∈ Λ, l > 0), and P = L∞ (ℝ : ℂ). If for every real numbers x, τ ∈ ℝ we have (82), (WM1-1)–(WM1-2) hold, 𝔽1 : (0, ∞) × ℝn → (0, ∞), the assumptions (i)–(ii) from the formulation of Theorem 4.3.10 hold and, for every l > 0, there exists a sequence (ak )k∈ℕ0 of positive real numbers such that ∑∞ k=0 ak = 1 and 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 ∞ 󵄩󵄩 󵄩 −1 ν(⋅) 󵄩 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩𝔽1 (l, ⋅⋅)󵄩󵄩󵄩2l−1 ∑ φ(ak l−1 )[φ(󵄩󵄩󵄩R(v + ⋅)󵄩󵄩󵄩)] [𝔽(l, − ⋅ +kl)] 󵄩 󵄩 ⩽ 1, 󵄩󵄩 󵄩󵄩 󵄩 󵄩 Lq(v) [−⋅+kl,−⋅+(k+1)l] d 󵄩󵄩󵄩󵄩P1 󵄩󵄩󵄩󵄩P1 󵄩󵄩 󵄩󵄩 k=0 ⋅⋅,l then the function F(⋅), given by (4), is well defined and belongs to the class 1 (ϕ,𝔽 ,cI,𝒫t,l ,𝒫 1 )

(e−)WΩ,Λ′ 1

(ℝ : Y ).

Proof. Clearly, we have F(x) = ∫−x R(v + x)f ̌(v) dv, x ∈ ℝ, so that the estimate (82) yields that the function F(⋅) is well-defined and the integral in definition of F(x) converges absolutely (x ∈ ℝ). Let ε > 0 be fixed. Then there exist two finite real numbers l > 0 and L > 0 such that any subinterval [t0 − L, t0 + L] ⊆ ℝ, where t0 ∈ Λ′ , contains a point τ ∈ Λ′ such that ∞

󵄩 󵄩 sup[𝔽(l, t)[ϕ(󵄩󵄩󵄩f ̌(⋅ + τ) − f ̌(⋅)󵄩󵄩󵄩)Lp(⋅) [t,t+l] ]] ⩽ ε. t∈ℝ

(198)

ν

Using (198), we can repeat verbatim the argumentation contained in the proof of [431, Theorem 3.1.46] in order to see that for each real number x ∈ ℝ the following holds: 󵄩 󵄩 ϕ(󵄩󵄩󵄩F(x + τ) − cF(x)󵄩󵄩󵄩) ∞

󵄩 󵄩 ⩽ 2l−1 ∑ φ(ak l−1 )[φ(󵄩󵄩󵄩R(v + x)󵄩󵄩󵄩)] k=0

Lq(v) [−x+kl,−x+(k+1)l]

[𝔽(l, −x + kl)]

−1 ν(x)

d

.

This simply implies the required statement, since the assumptions (i)–(ii) from the formulation of Theorem 4.3.10 hold true. ∞ In the following result, we will assume that ϕ(x) ≡ x, P = L∞ ν (ℝ : ℂ), P1 = Lν1 (ℝ : ℂ),

1 ∞ Pt,l = L∞ σ ([t, t + l] : ℂ) and Pt,l = Lσ1 ([t, t + l] : ℂ) for all t ∈ ℝ and l > 0:

Theorem 4.3.17. Suppose that c ∈ ℂ ∖ {0}, (R(t))t>0 ⊆ L(X, Y ) is a strongly continuous operator family, ν : ℝ → (0, ∞), ν1 : ℝ → (0, ∞), σ : ℝ → (0, ∞) and σ1 : ℝ → (0, ∞) 1 are Lebesgue measurable functions as well as that the spaces P, P1 , Pt,l and Pt,l are given as above (t ∈ ℝ, l > 0). Let (WM1-1) hold with Λ = ℝ, let 𝔽1 : (0, ∞) × ℝ → (0, ∞), and let

284 � 4 Metrically ρ-almost periodic type functions and applications ∞

sup 𝔽1 (l, t) sup σ1 (u)ν1 (t) ∫

t∈ℝ;l>0

u∈[t,t+l]

0

‖R(s)‖ ds < +∞. ν(t − s)𝔽(l, t − s)σ(u − s) 1 (x,𝔽,cI,𝒫t,l ,𝒫 1 )

t,l If f ∈ (e−)WΩ,Λ′ (ℝ : X), then F ∈ (e−)WΩ,Λ′ function F(⋅) given by (4) is well defined.

(x,𝔽,cI,𝒫 ,𝒫)

(199)

(ℝ : Y ), provided that the

Proof. Let ε > 0 be fixed. Then there exist two finite real numbers l > 0 and L > 0 such that for each t0 ∈ Λ′ any subinterval [t0 − L, t0 + L] ⊆ ℝ contains a point τ ∈ Λ′ such that ε 󵄩󵄩 󵄩 , 󵄩󵄩f (u + τ) − cf (u)󵄩󵄩󵄩 ⩽ ν(t)𝔽(l, t)σ(u)

provided t ∈ ℝ and u ∈ [t, t + l].

(200)

Using (200) and our assumption that the function F(⋅) given by (4) is well defined, we simply get that for every t ∈ ℝ and for every u ∈ [t, t + l], we have: ∞

󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩F(u + τ) − cF(u)󵄩󵄩󵄩Y ⩽ ∫ 󵄩󵄩󵄩R(s)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩f (u + τ − s) − cf (u − s)󵄩󵄩󵄩 ds 0

∞

⩽ε∫ 0

‖R(s)‖ ds. ν(t − s)𝔽(l, t − s)σ(u − s)

1 Using this estimate, the assumption (199) and the definition of space Pt,l , the required statement follows.

We close this subsection with the following example: Example 4.3.18. Suppose that the functions 𝔽(⋅; ⋅) and 𝔽1 (⋅; ⋅) do not depend on the second argument, as well as that ν ≡ ν1 and σ ≡ σ1 . Suppose, further, that there exist functions wν : [0, ∞) → (0, ∞) and wσ : [0, ∞) → (0, ∞) such that ν(t) ⩽ ν(t − s)wν (s) and

σ(t) ⩽ σ(t − s)wσ (s)

for all t ∈ ℝ and s ⩾ 0.

Then the estimate (199) is satisfied. In particular, all these requirements hold if ν(⋅) and σ(⋅) are admissible weight functions and the operator family (R(t))t>0 ⊆ L(X, Y ) has a certain exponential decay at plus infinity; cf. [446] for the notion.

4.3.4 Applications to the abstract Volterra integro-differential equations In this subsection, we will present the following applications to the abstract Volterra integro-differential inclusions: 1. It is clear that our results on the invariance of metrical Weyl almost periodicity under the actions of infinite convolution products (see Section 4.3.3) can be used in the analysis of the existence and uniqueness of metrical Weyl almost periodic solutions for

4.3 Weyl ρ-almost periodic functions in general metric

� 285

various classes of the abstract (degenerate) fractional differential equations without initial conditions. We will only note here that Theorem 4.3.16 and Theorem 4.3.17 can be used in the qualitative analysis of metrical Weyl almost periodic solutions of the following inhomogeneous fractional Poisson heat equation with Weyl–Liouville fractional derivatives in the space X = Lp (Ω): γ

Dt,+ [m(x)v(t, x)] = (Δ − b)v(t, x) + f (t),

t ∈ ℝ, x ∈ Ω,

where 1 ⩽ p < ∞, Ω is an open domain in ℝn with smooth boundary, m ∈ L∞ (Ω), m(x) ⩾ 0 for a. e. x ∈ Ω, γ ∈ (0, 1), Δ is the Dirichlet Laplacian and b > 0. 2. In order to avoid repeating, we will only note that our results about d’Alembert formula (see the first application given in [457, Section 1]) can be reformulated and 2 1 slightly extended in our new framework, provided that P = L∞ σ (ℝ : ℂ), Pt,l = Lν (t + l[0, 1/a]2 : ℂ) for all t ∈ ℝ2 and l > 0, where the functions σ(⋅) and ν(⋅) satisfy the general 3 1 3 requirements clarified before. Choosing P = L∞ σ (ℝ : ℂ) and Pt,l = Lν (t + l[0, 1] : ℂ) for all t ∈ ℝ3 and l > 0, we can slightly extend the statement of [443, Theorem 7] concerning the solutions of wave equation in ℝ3 given by the convolution with the Newton potential, provided that the function ν(⋅) has the property that there exists a function φ : ℝ3 → [0, ∞) such that ν(x + y) ⩽ ν(x)φ(y) for all x, y ∈ ℝ3 (assuming the same condition for all x, y ∈ ℝn , we can simply reformulate the conclusions obtained in the second part of [457, Application 2., Section 4] in our new framework, with 𝒫t,l = L1ν (t + l[0, 1]n : ℂ) for all t ∈ ℝn and l > 0; with the help of Theorem 4.3.10, we can simply reformulate the conclusions obtained in the first part of [457, Application 2., Section 4] concerning the heat equation in ℝn , as well). We can similarly analyze the existence and uniqueness of metrically Weyl almost periodic solutions of wave equation in ℝ2 given by the convolution with the logarithmic potential, as well as the existence and uniqueness of metrical Weyl almost periodic solutions of the wave equations in ℝ3 (ℝ2 ) given by the Kirchhoff formula (Poisson formula); see [431] for more details. 3. In the third application of [457, Section 4], we have considered the evolution system generated by the family (A(t) ≡ Δ + a(t)I)t⩾0 in the space Y := Lr (ℝn ), where r ∈ [1, ∞), Δ is the Dirichlet Laplacian on Lr (ℝn ), I is the identity operator on Lr (ℝn ) and a ∈ L∞ ([0, ∞)). Recall that U(t, t) := I for all t ⩾ 0 and [U(t, s)F](u) := ∫ K(t, s, u, v)F(v) dv,

F ∈ Lr (ℝn ), t > s ⩾ 0,

ℝn

where K(t, s, u, v) is given by − n2 ∫t a(τ) dτ s

K(t, s, u, v) := (4π(t − s))

e

exp(−

|x − y|2 ), 4(t − s)

t > s, u, v ∈ ℝn .

286 � 4 Metrically ρ-almost periodic type functions and applications We know that, under certain assumptions, a unique mild solution of the abstract Cauchy problem (𝜕/𝜕t)u(t, x) = A(t)u(t, x), t > 0; u(0, x) = F(x) is given by u(t, x) := [U(t, 0)F](x), t ⩾ 0, x ∈ ℝn ; see also the sixth application given in Section 3.3.5. Suppose now that ν : ℝn → (0, ∞) is a Lebesgue measurable function, σ : ℝn → (0, ∞) is a Lebesgue measurable function satisfying that the mapping 1/σ(⋅) is locally bounded, and there exist two finite real constants c1 > 0 and c2 > 0 such that ν(t) ⩾ c1 (x,𝔽,I,𝒫 ,𝒫) for a. e. t ∈ ℝn and σ(t) ⩽ c2 for a. e. t ∈ ℝn . Let F ∈ Lr (ℝn ) ∩ (e−)W[0,1]n ,Λ′ t,l (ℝn : ℂ), p

where 1 ⩽ p < ∞, 0 ≠ Λ′ ⊆ ℝn , P = L∞ (ℝn : ℂ), Pt,l = Lν (t + l[0, 1]n : ℂ) for all t ∈ ℝn and l > 0, and the function 𝔽(l, t) ≡ 𝔽(l) does not depend on t. Let 1/p + 1/q = 1; arguing as in [457], we get that for every t > 0, l > 0 and u, τ ∈ ℝn , there exists a finite real constant ct > 0 such that: 󵄩󵄩 |u−⋅|2 1 󵄩󵄩 ε ε 󵄨󵄨 󵄨 󵄩 󵄩󵄩󵄩 := ct G(l, u). ∑ 󵄩󵄩󵄩e− 4t 󵄨󵄨u(t, u + τ) − u(t, u)󵄨󵄨󵄨 ⩽ ct 󵄩 󵄩 󵄩 q n F(l) k∈lℤn 󵄩 ν(⋅) 󵄩L (k+l[0,1] ) 𝔽(l) Fix a number t > 0 and a new exponent p′ ∈ [1, ∞). Setting 𝔽1 (l, t) :=

𝔽(l) ′ ′ , (∫t+l[0,1]n G(l, u)p du)1/p

l > 0,

we can simply prove that the mapping x 󳨃→ u(t, x), x ∈ ℝn belongs to the class (x,𝔽 ,I,𝒫 1 ,𝒫 1 )

1 (e−)W[0,1]n1 ,Λ′ t,l (ℝn : ℂ), where Pt,l = Lp (t + l[0, 1]n : ℂ) for all t ∈ ℝn and l > 0, ∞ n P = Lσ (ℝ : ℂ), and the metrics dt,l and d are induced by the norms in these Banach spaces (t ∈ ℝn , l > 0). We close this section with the observation that we can also follow the second concept, in which we assume the validity of condition (WM2). The notion analyzed in this section can be also reconsidered as follows (the last three definitions of [443, Section 3] can be extended in such a way): ′

Definition 4.3.19. (i) By e − WΩ,Λ′ ,ℬ (Λ × X : Y ) we denote the set consisting of all functions F : Λ × X → Y such that for every ε > 0 and B ∈ ℬ, there exist two finite real numbers l > 0 and L > 0 such that for each t0 ∈ Λ′ there exists τ ∈ B(t0 , L) ∩ Λ′ such that for every t ∈ Λ and x ∈ B, we have Gx,t,l (u) ∈ ρ(F(t + lu; x)) and [p(u),ϕ,𝔽,ρ,𝒫]

󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 sup󵄩󵄩󵄩ln 𝔽(l, ⋅)ϕ(󵄩󵄩󵄩F(⋅ + τ + lu; x) − Gx,⋅,l (u)󵄩󵄩󵄩Y ) p(u) 󵄩󵄩󵄩 < ε. 󵄩 󵄩󵄩P L (Ω) x∈B 󵄩 (ii) By WΩ,Λ′ ,ℬ (Λ×X : Y ) we denote the set consisting of all functions F : Λ×X → Y such that for every ε > 0 and B ∈ ℬ, there exists a finite real number L > 0 such that for each t0 ∈ Λ′ there exists τ ∈ B(t0 , L) ∩ Λ′ such that, for every l > 0, t ∈ Λ and x ∈ B, we have Gx,t,l (u) ∈ ρ(F(t + lu; x)) and [p(u),ϕ,𝔽,ρ,𝒫]

4.3 Weyl ρ-almost periodic functions in general metric

� 287

󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩 lim sup sup󵄩󵄩󵄩ln 𝔽(l, ⋅)ϕ(󵄩󵄩󵄩F(⋅ + τ + lu; x) − Gx,⋅,l (u)󵄩󵄩󵄩Y ) p(u) 󵄩 < ε. 󵄩󵄩P L (Ω:Y ) 󵄩 󵄩 l→+∞ x∈B 󵄩 1 Definition 4.3.20. (i) By e − WΩ,Λ′ ,ℬ (Λ × X : Y ) we denote the set consisting of all functions F : Λ × X → Y such that for every ε > 0 and B ∈ ℬ, there exist two finite real numbers l > 0 and L > 0 such that for each t0 ∈ Λ′ there exists τ ∈ B(t0 , L) ∩ Λ′ such that, for every t ∈ Λ and x ∈ B, the mapping u 󳨃→ Gx,t,l (u) ∈ ρ(F(t + lu; x)), u ∈ Ω is well defined, belongs to the space Lp(u) (Ω : Y ) and

[p(u),ϕ,𝔽,ρ,𝒫]

󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 sup󵄩󵄩󵄩ln 𝔽(l, ⋅)ϕ(󵄩󵄩󵄩F(⋅ + τ + lu; x) − Gx,⋅,l (u)󵄩󵄩󵄩Lp(u) (Ω:Y ) )󵄩󵄩󵄩 < ε. 󵄩 󵄩󵄩P 󵄩 x∈B 1 (ii) By WΩ,Λ′ ,ℬ (Λ×X : Y ) we denote the set consisting of all functions F : Λ×X → Y such that, for every ε > 0 and B ∈ ℬ, there exists a finite real number L > 0 such that for each t0 ∈ Λ′ there exists τ ∈ B(t0 , L) ∩ Λ′ such that for every l > 0, t ∈ Λ and x ∈ B, the mapping u 󳨃→ Gx,t,l (u) ∈ ρ(F(t + lu; x)), u ∈ Ω is well defined, belongs to the space Lp(u) (Ω : Y ) and

[p(u),ϕ,𝔽,ρ,𝒫]

󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 lim sup sup󵄩󵄩󵄩ln 𝔽(l, ⋅)ϕ(󵄩󵄩󵄩F(⋅ + τ + lu; x) − Gx,⋅,l (u)󵄩󵄩󵄩Lp(u) (Ω:Y ) )󵄩󵄩󵄩 < ε. 󵄩 󵄩󵄩P l→+∞ x∈B 󵄩 2 Definition 4.3.21. (i) By e − WΩ,Λ′ ,ℬ (Λ × X : Y ) we denote the set consisting of all functions F : Λ × X → Y such that, for every ε > 0 and B ∈ ℬ, there exist two finite real numbers l > 0 and L > 0 such that for each t0 ∈ Λ′ there exists τ ∈ B(t0 , L) ∩ Λ′ such that for every t ∈ Λ and x ∈ B, the mapping u 󳨃→ Gx,t,l (u) ∈ ρ(F(t + lu; x)), u ∈ Ω is well defined, belongs to the space Lp(u) (Ω : Y ) and

[p(u),ϕ,𝔽,ρ𝒫]

󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 sup󵄩󵄩󵄩ϕ(ln 𝔽(l, ⋅)󵄩󵄩󵄩F(⋅ + τ + lu; x) − ρ(F(⋅ + lu; x))󵄩󵄩󵄩Lp(u) (Ω:Y ) )󵄩󵄩󵄩 < ε. 󵄩󵄩P 󵄩 x∈B 󵄩 2 (ii) By WΩ,Λ′ ,ℬ (Λ×X : Y ) we denote the set consisting of all functions F : Λ×X → Y such that, for every ε > 0 and B ∈ ℬ, there exists a finite real number L > 0 such that for each t0 ∈ Λ′ there exists τ ∈ B(t0 , L) ∩ Λ′ such that for every l > 0, t ∈ Λ and x ∈ B, the mapping u 󳨃→ Gx,t,l (u) ∈ ρ(F(t + lu; x)), u ∈ Ω belongs to the space Lp(u) (Ω : Y ) and

[p(u),ϕ,𝔽,ρ,𝒫]

󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 lim sup sup󵄩󵄩󵄩ϕ(ln 𝔽(l, ⋅)󵄩󵄩󵄩F(⋅ + τ + lu; x) − ρ(F(⋅ + lu; x))󵄩󵄩󵄩Lp(u) (Ω:Y ) )󵄩󵄩󵄩 < ε. 󵄩 󵄩󵄩P l→+∞ x∈B 󵄩

5 Asymptotical ρ-almost periodicity in general metric The main aim of this chapter is to investigate various classes of multi-dimensional asymptotically ρ-almost periodic type functions in general metric and various classes of weighted ergodic components in general metric. We continue our work in the multidimensional setting, employing the notion and results from the general theory of Lebesgue spaces with variable exponent.

5.1 Asymptotically ρ-almost periodic type functions in general metric In this section, we analyze the following classes of multi-dimensional asymptotically ρ-almost periodic type functions: (i) the class of metrically (S, 𝔻, ℬ)-asymptotically (ω, ρ)-periodic functions; (ii) the class of metrically (S, ℬ)-asymptotically (ωj , ρj , 𝔻j )j∈ℕn -periodic functions; (iii) the class of metrically ρ-slowly oscillating type functions; (iv) the class of 𝔻-asymptotically Bohr (ℬ, I ′ , ρ, 𝒫 )-almost periodic functions of type 1; (v) the class of 𝔻-quasi-asymptotically (ℬ, I ′ , ρ, 𝒫 )-almost periodic functions; (vi) the class of 𝔻-quasi-asymptotically (ℬ, I ′ , ρ, 𝒫 )-uniformly recurrent functions. The organization of section can be briefly described as follows. Section 5.1.1 investigates the notion of metrical (S, 𝔻, ℬ)-asymptotical (ω, ρ)-periodicity and the notion of metrical (S, ℬ)-asymptotical (ωj , ρj , 𝔻j )j∈ℕn -periodicity. Section 5.1.2 is devoted to the study of metrically ρ-slowly oscillating type functions in ℝn , and Section 5.1.3 is devoted to the study of metrically quasi-asymptotically ρ-almost periodic type functions. We present some applications of our theoretical results to the abstract Volterra integro-differential equations in the final subsection. 5.1.1 Metrical (S, 𝔻, ℬ)-asymptotical (ω, ρ)-periodicity and metrical (S, ℬ)-asymptotical (ωj , ρj , 𝔻j )j∈ℕn -periodicity This subsection investigates the notions of metrical (S, 𝔻, ℬ)-asymptotical (ω, ρ)-periodicity and metrical (S, ℬ)-asymptotical (ωj , ρj , 𝔻j )j∈ℕn -periodicity. In the following two definitions, we extend the notion introduced in [431, Definition 7.3.1, Definition 7.3.2] and [444, Definition 2.1, Definition 2.2] (in case of consideration of the first-mentioned definition, set only, for every set B ∈ ℬ, PB := C0 (𝔻 × B : Y ) := {F : 𝔻 × B → Y ; lim|t|→+∞,t∈𝔻 supx∈B ‖F(t; x)‖Y = 0} and dB (F, G) := supx∈B supt∈𝔻 ‖F(t; x)‖Y for all F, G ∈ PB ; similarly we can analyze the notion from the second-mentioned definition): https://doi.org/10.1515/9783111233871-007

5.1 Asymptotically ρ-almost periodic type functions in general metric

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289

Definition 5.1.1. Let ρ be a binary relation on Y , ω ∈ ℝn ∖ {0}, ω + I ⊆ I, 𝔻 ⊆ I ⊆ ℝn and the set 𝔻 be unbounded. Suppose further that for each set B ∈ ℬ, 𝒫B = (PB , dB ) is a metric space of functions from [0, ∞)𝔻×B containing the zero function. A function F : I × X → Y is said to be (S, 𝔻, ℬ, 𝒫ℬ )-asymptotically (ω, ρ)-periodic if and only if for each B ∈ ℬ there exists a function G : 𝔻 × B → Y such that G(⋅; ⋅) ∈ ρ(F(⋅; ⋅)) and 󵄩󵄩 󵄩 󵄩󵄩F(⋅ + ω; ⋅) − G(⋅; ⋅)󵄩󵄩󵄩Y ∈ PB . Definition 5.1.2. Let ρj be a binary relation on Y , ωj ∈ ℝ ∖ {0}, ωj ej + I ⊆ I, 𝔻j ⊆ I ⊆ ℝn and the set 𝔻j be unbounded (1 ⩽ j ⩽ n). Suppose further that for each j ∈ ℕn j

j

j

and for each set B ∈ ℬ, 𝒫B = (PB , dB ) is a metric space of functions from [0, ∞)𝔻j ×B containing the zero function. A function F : I × X → Y is said to be (S, ℬ)-asymptotically j (ωj , ρj , 𝔻j , 𝒫ℬ )j∈ℕn -periodic if and only if for each j ∈ ℕn and for each B ∈ ℬ there exists a function Gj : 𝔻j × B → Y such that Gj (⋅; ⋅) ∈ ρj (F(⋅; ⋅)) and j 󵄩󵄩 󵄩 󵄩󵄩F(⋅ + ω; ⋅) − Gj (⋅; ⋅)󵄩󵄩󵄩Y ∈ PB .

Before proceeding further, we would like to note that, in the case of consideration of functions of the form F : I → Y , we have X = {0} and ℬ = {X}, so that we work de facto with the metric space 𝒫B = 𝒫 = (PB , dB ) = (P, d) consisting of certain functions from [0, ∞)𝔻 and functions G : 𝔻 → Y ; see e. g., Definition 5.1.1. The following result extends the statement of [444, Proposition 2.5] and particularly shows that the notion introduced in Definition 2.4.1 is more general, in a certain sense, than the notion introduced in Definition 5.1.2: Proposition 5.1.3. Let ωj ∈ ℝ ∖ {0}, Tj ∈ L(X), ωj ej + I ⊆ I, 𝔻j ⊆ I ⊆ ℝn and the set 𝔻j j

be unbounded (1 ⩽ j ⩽ n). If F : I × X → X is (S, ℬ)-asymptotically (ωj , Tj , 𝔻j , 𝒫ℬ )j∈ℕn periodic and the set 𝔻, consisting of all tuples t ∈ 𝔻n such that t + ∑ni=j+1 ωi ei ∈ 𝔻j for all j ∈ ℕn−1 , is unbounded in ℝn , then the function F(⋅; ⋅) is (S, 𝔻, ℬ)-asymptotically (ω, T, 𝒫 )periodic, where ω := ∑nj=1 ωj ej and T := ∏nj=1 Tj , provided that for each set B ∈ ℬ there exists a finite real constant cB > 0 such that: 󵄩󵄩 󵄩󵄩 n n 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩F(⋅; ⋅)󵄩󵄩󵄩P ⩽ cB [󵄩󵄩󵄩F(⋅ + ∑ ωi ei ; ⋅) − T1 F(⋅ + ∑ ωi ei ; ⋅)󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 1 i=1 i=2 󵄩 󵄩PB 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 n n 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 + 󵄩󵄩󵄩F(⋅+ ∑ ωi ei ; ⋅)−T2 F(⋅ + ∑ ωi ei ; ⋅)󵄩󵄩󵄩 + ⋅ ⋅ ⋅ + 󵄩󵄩󵄩F(⋅ + ωn en ; ⋅)−Tn F(⋅; ⋅)󵄩󵄩󵄩 ]. 󵄩󵄩 󵄩󵄩 2 󵄩󵄩 󵄩󵄩 n i=2 i=3 󵄩 󵄩 PB 󵄩 󵄩PB Proof. The proof is an almost direct consequence of the corresponding definitions, the prescribed assumption and the following calculation (t ∈ 𝔻): 󵄩󵄩 󵄩 󵄩󵄩F(t + ω; x) − TF(t; x)󵄩󵄩󵄩 󵄩 󵄩 = 󵄩󵄩󵄩F(t1 + ω1 , . . . , tn + ωn ; x) − T1 ⋅ ⋅ ⋅ Tn F(t1 , . . . , tn ; x)󵄩󵄩󵄩

290 � 5 Asymptotical ρ-almost periodicity in general metric 󵄩 󵄩 ⩽ 󵄩󵄩󵄩F(t1 + ω1 , t2 + ω2 , . . . , tn + ωn ; x) − T1 F(t1 , t2 + ω2 , . . . , tn + ωn ; x)󵄩󵄩󵄩 󵄩 󵄩 + ‖T1 ‖ ⋅ 󵄩󵄩󵄩F(t1 , t2 + ω2 , . . . , tn + ωn ; x) − T2 ⋅ ⋅ ⋅ Tn F(t1 , . . . , tn ; x)󵄩󵄩󵄩 󵄩 󵄩 ⩽ 󵄩󵄩󵄩F(t1 + ω1 , t2 + ω2 , . . . , tn + ωn ; x) − T1 F(t1 , t2 + ω2 , . . . , tn + ωn ; x)󵄩󵄩󵄩 󵄩 󵄩 + ‖T1 ‖ ⋅ [󵄩󵄩󵄩F(t1 , t2 + ω2 , . . . , tn + ωn ; x) − T2 F(t1 , t2 , . . . , tn + ωn ; x)󵄩󵄩󵄩 󵄩 󵄩 + ‖T2 ‖ ⋅ 󵄩󵄩󵄩F(t1 , t2 , . . . , tn + ωn ; x) − T3 ⋅ ⋅ ⋅ Tn F(t1 , t2 , . . . , tn ; x)󵄩󵄩󵄩] ⩽ ⋅⋅⋅. In the following proposition, we examine the convolution invariance of function spaces introduced in this subsection: Proposition 5.1.4. Suppose that h ∈ L1 (ℝn ) and F : ℝn × X → Y is a function which satisfies that for each set B ∈ ℬ we have supt∈ℝn ,x∈B ‖F(t, x)‖Y < +∞. Suppose further that ρ = A is a closed linear operator on Y satisfying that: (D) For each t ∈ ℝn and x ∈ X, the function s 󳨃→ AF(t−s; x), s ∈ ℝn is essentially bounded; for each B ∈ ℬ, the function s 󳨃→ supx∈B ‖AF(s; x)‖Y , s ∈ ℝn is bounded. Then the function (h ∗ F)(t; x) := ∫ h(σ)F(t − σ; x) dσ,

t ∈ ℝn , x ∈ X

ℝn

is well defined and for each set B ∈ ℬ we have supt∈ℝn ,x∈B ‖(h ∗ F)(t; x)‖Y < +∞. Furthermore, the following holds: (i) Suppose that 𝔻 = ℝn , ν ∈ L∞ (ℝn : (0, ∞)) and there exists a function φ : ℝn → (0, ∞) such that ν(x) ⩽ ν(y)φ(x − y) for all x, y ∈ ℝn and hφ ∈ L1 (ℝn ). Suppose also n that, for every set B ∈ ℬ, we have PB = C0,ν (ℝn × B : (0, ∞)) or PB = L∞ ν (ℝ × B : n (0, ∞)). If the function F(⋅; ⋅) is (S, ℝ , ℬ)-asymptotically (ω, A, 𝒫ℬ )-periodic, then the function (h ∗ F)(⋅; ⋅) is likewise (S, ℝn , ℬ)-asymptotically (ω, A, 𝒫ℬ )-periodic. (ii) Suppose that 𝔻j = ℝn and the function νj ∈ L∞ (ℝn : (0, ∞)) satisfies that there exists a function φj : ℝn → (0, ∞) such that νj (x) ⩽ νj (y)φj (x −y) for all x, y ∈ ℝn and hφj ∈ j

L1 (ℝn ), for all j ∈ ℕn . Suppose also that for every set B ∈ ℬ and j ∈ ℕn , we have PB = j

n C0,νj (ℝn ×B : (0, ∞)) or PB = L∞ νj (ℝ ×B : (0, ∞)). If for each j ∈ ℕn condition (D) holds with the closed linear operator A replaced therein with the closed linear operator Aj , j

and the function F(⋅; ⋅) is (S, ℬ)-asymptotically (ωj , Aj , ℝn , 𝒫ℬ )j∈ℕn -periodic, then the j

function (h ∗ F)(⋅; ⋅) is likewise (S, ℬ)-asymptotically (ωj , Aj , ℝn , 𝒫ℬ )j∈ℕn -periodic.

Proof. The proof is very similar to the proof of [439, Theorem 2.6], and we will only provide the main details of proof for the issue (i). It can be easily proved that the function (h ∗ F)(⋅; ⋅) is well defined as well as that supt∈ℝn ,x∈B ‖(h ∗ F)(t; x)‖Y < +∞ for all B ∈ ℬ. Fix a real number ε > 0 and a set B ∈ ℬ. Then there exists a sufficiently large real

5.1 Asymptotically ρ-almost periodic type functions in general metric

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291

number M1 > 0 such that ‖F(t + ω; x) − AF(t; x)‖Y ν(t) < ε, provided |t| > M1 and x ∈ B. Since A is closed and condition (D) holds, for every t ∈ ℝn and x ∈ B, the value of Gx (t) := A((h ∗ F)(t; x)) = ∫ℝn h(s)A(F(t − s; x)) ds is well defined. Due to the second part of condition (D), the essential boundedness of function ν(⋅) and the fact that hφ ∈ L1 (ℝn ), we know that there exists a finite constant cB ⩾ 1 such that 󵄩󵄩 󵄩 󵄩󵄩(h ∗ F)(t + ω; x) − Gx (t)󵄩󵄩󵄩Y ν(t) 󵄨 󵄨 󵄩 󵄩 ⩽ ∫ 󵄨󵄨󵄨h(σ)󵄨󵄨󵄨 ⋅ 󵄩󵄩󵄩F(t + ω − σ; x) − AF(t − σ; x)󵄩󵄩󵄩Y ν(t) dσ ℝn

=

󵄨 󵄨 󵄩 󵄩 ∫ 󵄨󵄨󵄨h(t − σ)󵄨󵄨󵄨 ⋅ 󵄩󵄩󵄩F(σ + ω; x) − AF(σ; x)󵄩󵄩󵄩Y ν(t) dσ |σ|⩽M1

󵄨 󵄨 󵄩 󵄩 + ∫ 󵄨󵄨󵄨h(t − σ)󵄨󵄨󵄨 ⋅ 󵄩󵄩󵄩F(σ + ω; x) − AF(σ; x)󵄩󵄩󵄩Y ν(t) dσ |σ|⩾M1

⩽

󵄨 󵄨 󵄩 󵄩 ∫ 󵄨󵄨󵄨h(t − σ)󵄨󵄨󵄨 ⋅ 󵄩󵄩󵄩F(σ + ω; x) − AF(σ; x)󵄩󵄩󵄩Y ν(t) dσ |σ|⩽M1

󵄨 󵄨 󵄩 󵄩 + ∫ 󵄨󵄨󵄨h(t − σ)󵄨󵄨󵄨φ(t − σ) ⋅ 󵄩󵄩󵄩F(σ + ω; x) − AF(σ; x)󵄩󵄩󵄩Y ν(σ) dσ |σ|⩾M1

󵄨 󵄨 ⩽ cB ‖ν‖∞ ∫ 󵄨󵄨󵄨h(t − σ)󵄨󵄨󵄨 dσ + ε‖φh‖L1 (ℝn ) . |σ|⩽M1

To complete the proof, it suffices to observe that there exists a finite real number M2 > 0 such that ∫|σ|⩾M |h(σ)| dσ < ε. 2

We would like to note that an analogue of Proposition 5.1.4, with the same choice of metric spaces, can be formulated for metrically quasi-asymptotically ρ-almost periodic type functions in ℝn and metrically ρ-slowly oscillating type functions in ℝn . Furthermore, the composition principle clarified in [444, Theorem 4.5] can be straightforwardly reformulated for metrically ρ-slowly oscillating type functions in ℝn , providing the same choice of metric spaces as in Proposition 5.1.4.

5.1.2 Metrically ρ-slowly oscillating type functions in ℝn In the following two definitions, we extend the notion recently introduced in [444, Definition 4.1, Definition 4.2]: Definition 5.1.5. Let ρ be a binary relation on Y , 𝔻 ⊆ I ⊆ ℝn and the set 𝔻 be unbounded. Suppose further that for each set B ∈ ℬ, 𝒫B = (PB , dB ) is a metric space of functions from [0, ∞)𝔻×B containing the zero function. Set

292 � 5 Asymptotical ρ-almost periodicity in general metric AI := {ω ∈ ℝn ∖ {0} : ω + I ⊆ I}. A function F : I × X → Y is said to be (𝔻, ℬ, ρ, 𝒫ℬ )-slowly oscillating if and only if for each ω ∈ AI the function F(⋅; ⋅) is (S, 𝔻, ℬ, 𝒫ℬ )-asymptotically (ω, ρ)-periodic. Definition 5.1.6. Let ρj be a binary relation on Y , 𝔻j ⊆ I ⊆ ℝn and the set 𝔻j be unbounded (1 ⩽ j ⩽ n). Suppose, further, that for each j ∈ ℕn and for each set B ∈ ℬ, j j j 𝒫B = (PB , dB ) is a metric space of functions from [0, ∞)𝔻j ×B containing the zero function. Set n

BI := {(ω1 , . . . , ωn ) ∈ (ℝ ∖ {0}) : ωj ej + I ⊆ I for all j ∈ ℕn }. j

A function F : I × X → Y is said to be (ℬ, (ωj , ρj , 𝔻j , 𝒫ℬ )j∈ℕn )-slowly oscillating if and only if for each tuple (ω1 , . . . , ωn ) ∈ BI the function F(⋅; ⋅) is (S, ℬ)-asymptotically j (ωj , ρj , 𝔻j , 𝒫ℬ )j∈ℕn -periodic. There is no need to say that the terms “(S, 𝔻, ℬ, 𝒫ℬ )-asymptotically (ω, ρ)-periodic” and “(𝔻, ℬ, ρ, 𝒫ℬ )-slowly oscillating” are not ideal but suitable in the situation in which we use the weighted C0 -spaces of functions. It is also worth noting that in the usual approach developed by D. Sarason [657] for the functions of form F : [0, ∞) → ℂ, the boundedness and continuity of function F(⋅) are assumed a priori; we do not use these assumptions here. Further on, in the usual approach, any slowly oscillating function F : [0, ∞) → ℂ is uniformly continuous. We have expanded this result in [458, Proposition 2.3]; it is clear that we cannot expect the uniform continuity of functions introduced in Definition 5.1.5 and Definition 5.1.6. We continue by providing two examples: Example 5.1.7. Let X := c0 (ℂ) be the Banach space of all numerical sequences tending to zero, equipped with the sup-norm. Consider the function f (t) := (

4k 2 t 2 ) , (t 2 + k 2 )2 k∈ℕ

t ⩾ 0.

Then we know (see, e. g. [438, Example 2.6]) that the function f (⋅) is uniformly continuous, the range of f (⋅) is not relatively compact in X and for every positive real number τ > 0, we have τ2 󵄩󵄩 󵄩 1 󵄩󵄩f (t + τ) − f (t)󵄩󵄩󵄩 ⩽ 4 + 4 2 , t t

t > 0.

This implies that the function f (⋅) is 𝒫 -slowly oscillating with P = C0,ν ([0, ∞) : X) and ν(t) = (1 + t)ζ , where ζ ∈ (0, 2). Example 5.1.8. Suppose that F : ℝn → ℝ is continuously differentiable and there exist finite real numbers σ ∈ ℝ and M ⩾ 1 such that |∇F(x)| ⩽ M|x|σ , |x| ⩾ 1. Suppose

5.1 Asymptotically ρ-almost periodic type functions in general metric

293

�

further that ν : ℝn → (0, ∞) is any continuous function such that lim|x|→+∞ |x|σ ν(x) = 0. Applying the Lagrange mean value theorem, we get that, for every x, τ ∈ ℝn ∖ {0}, there exists a number c ∈ (0, 1) such that: 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨f (x + τ) − f (x)󵄨󵄨󵄨ν(x) ⩽ 󵄨󵄨󵄨∇f (x + (1 − c)τ)󵄨󵄨󵄨 ⋅ |τ| ⋅ ν(x) |x + (1 − c)τ|σ 󵄨 󵄨σ ⩽ M|τ| ⋅ 󵄨󵄨󵄨x + (1 − c)τ 󵄨󵄨󵄨 ⋅ ν(x) = M|τ| ⋅ [|x|σ ν(x)]. |x|σ Along with an elementary argumentation, this computation shows that for every τ ∈ ℝn , we have lim|x|→+∞ |f (x+τ)−f (x)|ν(x) = 0. Hence, the function F(⋅) is 𝒫 -slowly oscillating with P = C0,ν (ℝn : ℝ) and the metric d induced by the norm of this Banach space. In [458, Proposition 2.2], we have recently observed that it is not so logical to study the class of (𝔻, ℬ, c)-slowly oscillating functions by replacing the term ‖F(t + ω; x) − F(t; x)‖Y in the usual definition by the term ‖F(t + ω; x) − cF(t; x)‖Y , where c ∈ ℂ ∖ {0}. This result can be further generalized as follows: Proposition 5.1.9. Let c ∈ ℂ ∖ {0}, 0 ≠ I ⊆ ℝn , 𝔻 ⊆ I ⊆ ℝn and the set 𝔻 be unbounded. Suppose that ν : I → (0, ∞) is a function which satisfies that there exists a function φ : AI → (0, ∞) such that ν(t) ⩽ ν(t + ω)φ(ω) for all t ∈ 𝔻 and ω ∈ AI , as well as that AI ⊆ 2AI and ω′ + 𝔻 ⊆ 𝔻 for all ω′ ∈ AI /2. Let PB := C0,𝔻,ν (I × B : (0, ∞)) for all B ∈ ℬ. Then the following holds: (i) If a function F : I × X → Y is (𝔻, ℬ, cI, 𝒫ℬ )-slowly oscillating, then for each set B ∈ ℬ we have lim

󵄩 󵄩 sup󵄩󵄩󵄩F(t; x)󵄩󵄩󵄩Y ν(t) = 0.

|t|→+∞,t∈𝔻 x∈B

(201)

(ii) If, in addition to the above, we have ω + 𝔻 ⊆ 𝔻 for all ω ∈ AI , then a function F : I × X → Y is (𝔻, ℬ, cI, 𝒫ℬ )-slowly oscillating if and only if for each set B ∈ ℬ we have (201). Proof. In order to prove (i), suppose that ω′ ∈ AI and B ∈ ℬ; then there exists ω ∈ AI such that ω′ = 2ω. We have (t ∈ I; x ∈ B): [F(t + ω′ ; x) − c2 F(t; x)]ν(t) = [F(t + 2ω; x) − c2 F(t; x)]ν(t)

= [F(t + 2ω; x) − cF(t + ω; x)]ν(t) + c[F(t + ω; x) − cF(t; x)]ν(t),

which implies 󵄩󵄩 󵄩 󵄩󵄩F(t + ω′ ; x) − c2 F(t; x)󵄩󵄩󵄩 ν(t) 󵄩󵄩 󵄩󵄩Y 󵄩󵄩 󵄩 󵄩 󵄩 ⩽ 󵄩󵄩F(t + 2ω; x) − cF(t + ω; x)󵄩󵄩󵄩Y ν(t + ω)φ(ω) + |c| ⋅ 󵄩󵄩󵄩F(t + ω; x) − cF(t; x)󵄩󵄩󵄩Y ν(t). Our assumption (AI /2) + 𝔻 ⊆ 𝔻 implies t + ω ∈ 𝔻, t ∈ 𝔻 and

294 � 5 Asymptotical ρ-almost periodicity in general metric lim

|t|→+∞,t∈𝔻

󵄩󵄩 󵄩 ′ 2 󵄩󵄩F(t + ω ; x) − c F(t; x)󵄩󵄩󵄩Y ν(t) = 0,

uniformly in x ∈ B.

Subtracting the term in the above equality and the corresponding term from the definition of (𝔻, ℬ, cI, 𝒫ℬ )-slowly oscillating property, with the number ω replaced therein with the number ω′ , we obtain lim

󵄩󵄩 2 󵄩 󵄩󵄩(c − c) ⋅ F(t; x)󵄩󵄩󵄩Y ν(t) = 0,

uniformly in x ∈ B.

|t|→+∞,t∈𝔻

This immediately implies (201) since c ≠ 1. In order to prove (ii), it suffices to apply (i) and observe that the assumption ω + 𝔻 ⊆ 𝔻 for all ω ∈ AI implies lim

|t|→+∞,t∈𝔻

󵄩󵄩 󵄩 󵄩󵄩F(t + ω; x)󵄩󵄩󵄩Y ν(t) = 0,

uniformly in x ∈ B,

since ν(t) ⩽ ν(t + ω)φ(ω) for all t ∈ 𝔻 and ω ∈ AI . 5.1.3 Metrically quasi-asymptotically ρ-almost periodic type functions This subsection investigates various classes of metrically quasi-asymptotically ρ-almost periodic type functions. We will always assume the validity of the following condition: (QAAP-1) Suppose that 𝔻 ⊆ I ⊆ ℝn , 0 ≠ I ′ ⊆ ℝn , 0 ≠ I ⊆ ℝn and I + I ′ ⊆ I. If τ ∈ I ′ and M > 0, then we define 𝔻M := {t ∈ 𝔻 : |t| ⩾ M} and Iτ,M := {t ∈ I : t, t + τ ∈ 𝔻M }; further on, we assume that 𝒫τ,M = (Pτ,M , dτ,M ) is a metric space, where Pτ,M ⊆ Y Iτ,M and Pτ,M contains the zero function. We set ‖f ‖Pτ,M := dτ,M (0, f ) for all f ∈ Pτ,M . Now we are able to introduce the following notion: Definition 5.1.10. Suppose that (QAAP-1) holds as well as that F : I × X → Y is a given function. Then we say that: (i) F(⋅; ⋅) is 𝔻-asymptotically Bohr (ℬ, I ′ , ρ, 𝒫 )-almost periodic of type 1 if and only if for every B ∈ ℬ and ε > 0 there exist finite real numbers l > 0 and M > 0 such that for each t0 ∈ I ′ there exists τ ∈ B(t0 , l) ∩ I ′ such that, for every x ∈ B, there exists a function Gx ∈ Y Iτ,M such that Gx (t) ∈ ρ(F(t; x)) for all t ∈ Iτ,M , x ∈ B and 󵄩 󵄩 sup󵄩󵄩󵄩F(⋅ + τ; x) − Gx (⋅)󵄩󵄩󵄩P x∈B

τ,M

⩽ ε.

5.1 Asymptotically ρ-almost periodic type functions in general metric

�

295

(ii) F(⋅; ⋅) is 𝔻-quasi-asymptotically (ℬ, I ′ , ρ, 𝒫 )-almost periodic if and only if for every B ∈ ℬ and ε > 0 there exists l > 0 such that for each t0 ∈ I ′ there exists τ ∈ B(t0 , l)∩I ′ such that there exists a finite real number M ≡ M(ε, τ) > 0 such that, for every x ∈ B, there exists a function Gx ∈ Y Iτ,M such that Gx (t) ∈ ρ(F(t; x)) for all t ∈ Iτ,M , x ∈ B and 󵄩 󵄩 sup󵄩󵄩󵄩F(⋅ + τ; x) − Gx (⋅)󵄩󵄩󵄩P x∈B

τ,M

⩽ ε.

(iii) F(⋅; ⋅) is 𝔻-quasi-asymptotically (ℬ, I ′ , ρ, 𝒫 )-uniformly recurrent (or, equivalently, 𝔻-asymptotically (ℬ, I ′ , ρ, 𝒫 )-uniformly recurrent of type 1) if and only if for every B ∈ ℬ there exist a sequence (τ k ) in I ′ and a sequence (Mk ) in (0, ∞) such that limk→+∞ |τ k | = limk→+∞ Mk = +∞ and, for every x ∈ B, there exists a function Gx ∈ Y Iτk ,Mk such that Gx (t) ∈ ρ(F(t; x)) for all t ∈ Iτ,M , x ∈ B and 󵄩 󵄩 lim sup󵄩󵄩󵄩F(⋅ + τ k ; x) − Gx (⋅)󵄩󵄩󵄩P = 0. τk ,Mk

k→+∞ x∈B

If I ′ = I, then we also say that F(⋅; ⋅) is 𝔻-asymptotically Bohr (ℬ, ρ, 𝒫 )-almost periodic of type 1 (𝔻-quasi-asymptotically (ℬ, ρ, 𝒫 )-almost periodic, 𝔻-quasi-asymptotically (ℬ, ρ, 𝒫 )-uniformly recurrent); furthermore, if X ∈ ℬ, then it is also said that F(⋅; ⋅) is 𝔻-asymptotically (I ′ , ρ, 𝒫 )-almost periodic of type 1 (𝔻-quasi-asymptotically (I ′ , ρ, 𝒫 )almost periodic, 𝔻-quasi-asymptotically (I ′ , ρ, 𝒫 )-uniformly recurrent). If I ′ = I and X ∈ ℬ, then we also say that F(⋅; ⋅) is 𝔻-asymptotically (ρ, 𝒫 )-almost periodic of type 1 (𝔻-quasi-asymptotically (ρ, 𝒫 )-almost periodic, 𝔻-quasi-asymptotically (ρ, 𝒫 )-uniformly recurrent). We remove the prefix “𝔻-” in the case that 𝔻 = I, remove the prefix “(ℬ, )” in the case that X ∈ ℬ and remove the prefix “ρ-” if ρ = I, the identity operator on Y . It is worth noting that we do not assume the continuity of a function F(⋅; ⋅) here. The notion of 𝔻-quasi-asymptotical Bohr (ℬ, I ′ , ρ)-almost periodicity and the notion of 𝔻-quasi-asymptotical (ℬ, I ′ , ρ)-uniform recurrence, introduced and analyzed in [444], are obtained by plugging that Pτ,M = L∞ (Iτ,M : Y ) for all τ ∈ I ′ and M > 0; similarly, the notion of 𝔻-asymptotical Bohr (ℬ, I ′ , ρ)-almost periodicity of type 1 and the notion of 𝔻-asymptotical (ℬ, I ′ , ρ)-uniform recurrence of type 1, introduced and analyzed in [304], are obtained in the same way; see also [431, Definition 6.1.33, Definition 7.1.23, Definition 7.3.14]. Remark 5.1.11. The notion introduced in the former part of this section can be understood in a more general setting. By that we primarily mean that all considered metric spaces can be pseudometric spaces as well as that X and Y can be general non-empty sets, only. Let us consider in more detail part (i) of Definition 5.1.10; then it suffices to assume that I, I ′ , X and Y are non-empty sets, (I ′ , d ′ ) is a pseudometric space [then for each t0 ∈ I ′ the inclusion τ ∈ B(t0 , l) ∩ I ′ means τ ∈ I ′ and d ′ (t0 , τ) ⩽ l], the operation ⊕ : I × I ′ → I is defined [then ⋅ + τ means ⋅ ⊕ τ], 𝒫τ,M = (Pτ,M , dτ,M ) is a pseudo-

296 � 5 Asymptotical ρ-almost periodicity in general metric metric space, where P ⊆ Y Iτ,M contains a zero function, and (Y , ⊖) is a grupoid [then F(⋅ + τ; x) − Gx (⋅) ∈ Pτ,M means F(⋅ ⊕ τ; x) ⊖ Gx (⋅) ∈ Pτ,M ], for any τ ∈ I ′ and M > 0. Suppose that (QAAP-1) holds and F : I × X → Y is a given function. Then it is clear that the following holds: (i) If F(⋅; ⋅) is 𝔻-asymptotically Bohr (ℬ, I ′ , ρ, 𝒫 )-almost periodic of type 1, then F(⋅; ⋅) is 𝔻-quasi-asymptotically (ℬ, I ′ , ρ, 𝒫 )-almost periodic. (ii) If F(⋅; ⋅) is 𝔻-quasi-asymptotically (ℬ, I ′ , ρ, 𝒫 )-almost periodic, then F(⋅; ⋅) is 𝔻-quasiasymptotically (ℬ, I ′ , ρ, 𝒫 )-uniformly recurrent. For simplicity, we will not consider here the Stepanov generalizations of the notion introduced in Definition 5.1.10; see [431, Subsection 6.2.3, Subsection 6.2.5, Subsection 7.3.4] for some results obtained in the case that ρ = I. Recall also that the notion of Stepanov quasi-asymptotical almost periodicity intermediates the concepts Stepanov asymptotical almost periodicity and Weyl almost periodicity, considered in the general approach of A. S. Kovanko [463]. Concerning this issue, we would like to note that an analogue of [438, Proposition 2.12] and its multi-dimensional extensions can be proved in metrical framework, provided that the metric spaces under our consideration are weighted Lp spaces. Details can be left to the interested readers. Furthermore, in [444, Subsection 3.1], we have recently analyzed various classes of remotely ρ-almost periodic type functions. The notion of metrical remote ρ-almost periodicity can be introduced and analyzed, as well; for simplicity, we will skip all related details concerning this topic here. We continue by providing an illustrative example: Example 5.1.12. Suppose that I = ℝ, ν : ℝ → (0, ∞) is any function satisfying that the function 1/ν(⋅) is locally bounded, and f (⋅) is any scalar-valued continuous function such that f (t) = 1 for all t ⩾ 0 and f (t) = 0 for all t ⩽ −1. Then f (⋅) is not equi-Weyl-p-almost periodic for any finite exponent p ⩾ 1, but f (⋅) is quasi-asymptotically 𝒫 -almost periodic, where Pτ,M = C0,ν (ℝ : ℝ) for every τ ∈ ℝ and M > 0. Denote by AX,Y any of the function spaces introduced in the former part of this paper. If F(⋅; ⋅) belongs to AX,Y , c1 ∈ ℝ ∖ {0}, τ ∈ ℝn , c, c2 ∈ ℂ ∖ {0} and x0 ∈ X, then it is not ̌ ⋅), difficult to clarify certain sufficient conditions ensuring that the function cF(⋅; ⋅), F(⋅; F(c1 ⋅; c2 ⋅), ‖F(⋅; ⋅)‖Y or F(⋅ + τ; ⋅ + x0 ) also belongs to AX,Y . In some cases, it is almost trivial to say when AX,Y will be a vector space. Concerning the uniformly convergent sequences of functions belonging to some of the above-introduced function spaces, we must impose some restrictive conditions on the metric spaces under our considerations in order to obtain any relevant. For example, suppose that (Fk (⋅; ⋅) : I × X → Y ) is a sequence of functions and there exists a function F : I × X → Y such that limk→+∞ Fk (t; x) = F(t; x), uniformly on I × B for each set B of collection ℬ. Concerning the binary relation ρ on Y , we assume that D(ρ) is closed, ρ is single-valued on R(F) and continuous on D(ρ) in the usual sense, i. e. condition (Cρ ) holds; we assume the same conditions for the sequence (ρj )j∈ℕn of binary relations on Y , if considered. Then we have the following:

5.1 Asymptotically ρ-almost periodic type functions in general metric

�

297

(i) Suppose that for each positive integer k ∈ ℕ the function Fk (⋅; ⋅) is 𝔻-asymptotically Bohr (ℬ, I ′ , ρ, 𝒫 )-almost periodic of type 1, respectively, 𝔻-quasi-asymptotically (ℬ, I ′ , ρ, 𝒫 )-almost periodic [𝔻-quasi-asymptotically (ℬ, I ′ , ρ, 𝒫 )-uniformly recurrent]. Suppose further that for each τ ∈ I ′ and M > 0 we have that Pτ,M = C0,ν (Iτ,M : ∞ Y ) or Pτ,M = L∞ ν (Iτ,M : Y ), where ν ∈ L (I : (0, ∞)). Then F(⋅; ⋅) is 𝔻-asymptotically Bohr (ℬ, I ′ , ρ, 𝒫 )-almost periodic of type 1, respectively, 𝔻-quasi-asymptotically (ℬ, I ′ , ρ, 𝒫 )-almost periodic [𝔻-quasi-asymptotically (ℬ, I ′ , ρ, 𝒫 )-uniformly recurrent]. (ii) Suppose that for each integer k ∈ ℕ the function Fk (⋅; ⋅) is (S, 𝔻, ℬ, 𝒫ℬ )-asymptotically (ω, ρ)-periodic [(𝔻, ℬ, ρ, 𝒫ℬ )-slowly oscillating]. Suppose, further, that for each B ∈ ℬ we have that PB = C0,ν (𝔻 × B : [0, ∞)) or PB = L∞ ν (𝔻 × B : [0, ∞)), where ν ∈ L∞ (I : (0, ∞)). Then F(⋅; ⋅) is (S, 𝔻, ℬ, 𝒫ℬ )-asymptotically (ω, ρ)-periodic [(𝔻, ℬ, ρ, 𝒫ℬ )-slowly oscillating]. Observe only that, in this place, we have C0,ν (𝔻 × B : Y ) := {F : 𝔻 × B → Y ; lim|t|→+∞,t∈𝔻 supx∈B ‖F(t; x)‖Y ν(t) = 0} and dB (F, G) := supx∈B supt∈𝔻 ‖F(t; x)‖Y ν(t) for all F, G ∈ C0,ν (𝔻 × B : Y ); we define L∞ ν (𝔻 × B : Y ) similarly. (iii) Suppose that for each positive integer k ∈ ℕ the function Fk (⋅; ⋅) is (S, ℬ)-asympj j totically (ωj , ρj , 𝔻j , 𝒫ℬ )j∈ℕn -periodic [(ℬ, (ωj , ρj , 𝔻j , 𝒫ℬ )j∈ℕn )-slowly oscillating]. Suppose further that for each B ∈ ℬ we have that PB = C0,ν (𝔻 × B : [0, ∞)) ∞ or PB = L∞ ν (𝔻 × B : [0, ∞)), where ν ∈ L (I : (0, ∞)). Then F(⋅; ⋅) is (S, ℬ )j j asymptotically (ωj , ρj , 𝔻j , 𝒫ℬ )j∈ℕn -periodic [(ℬ, (ωj , ρj , 𝔻j , 𝒫ℬ )j∈ℕn )-slowly oscillating]. The following result generalizes [439, Proposition 3.4(i)] and [444, Proposition 3.2]: Proposition 5.1.13. Let ω ∈ I ∖ {0}, T ∈ L(X), ‖T‖ ⩽ 1, ω + I ⊆ I, ω + 𝔻 ⊆ 𝔻 and 𝔻 ⊆ I ⊆ ℝn . Set I ′ := ω ⋅ ℕ. Suppose that ν : I → (0, ∞), PB = C0,ν (𝔻 × B : [0, ∞)) ∞ (PB = L∞ ν (𝔻 × B : [0, ∞))) for any B ∈ ℬ , and Pτ,M = C0,ν (Iτ,M : Y ) (Pτ,M = Lν (Iτ,M : Y )) for any τ ∈ I ′ and M > 0. If a function F : I × X → Y is (S, 𝔻, ℬ, 𝒫ℬ )-asymptotically (ω, ρ)periodic, then the function F(⋅; ⋅) is 𝔻-quasi-asymptotically (ℬ, I ′ , ρ, 𝒫 )-almost periodic, provided that for each k ∈ ℕ we have k−1

sup ∑

t∈𝔻 j=0

ν(t) < +∞. ν(t + jω)

(202)

Proof. The proof essentially follows from the argumentation contained in the proof of [439, Proposition 3.4(i)] and our assumption (202). We will only note here that the assumption t ∈ 𝔻 implies t + kω ∈ 𝔻 for each fixed integer k ∈ ℕ, so that for each x ∈ B we have: 󵄩󵄩 󵄩 󵄩󵄩F(t + kω; x) − TF(t; x)󵄩󵄩󵄩Y ν(t)

298 � 5 Asymptotical ρ-almost periodicity in general metric k−1

󵄩 󵄩 ⩽ ∑ ‖T‖k−1−j 󵄩󵄩󵄩F(t + (j + 1)ω; x) − TF(t + jω; x)󵄩󵄩󵄩Y ν(t) j=0

k−1

ν(t) 󵄩 󵄩 ⩽ ∑ 󵄩󵄩󵄩F(t + (j + 1)ω; x) − TF(t + jω; x)󵄩󵄩󵄩Y ν(t + jω) . ν(t + jω) j=0

Concerning the invariance of metrical c-quasi-asymptotical almost periodicity under the actions of convolution products, where c ∈ ℂ ∖ {0}, we will only state the following slight extensions of [438, Proposition 3.1, Proposition 3.2] without proofs and say that these results can be also established in the multi-dimensional setting (we can also consider the Banach space L∞ ν (I : (0, ∞)) here): Proposition 5.1.14. Suppose that c ∈ ℂ ∖ {0}, (R(t))t>0 ⊆ L(X, Y ) is a strongly continuous ∞ operator family and ∫0 ‖R(s)‖ ds < ∞. (i) Suppose that ν : [0, ∞) → (0, ∞) is an essentially bounded function which satisfies that the function 1/ν(⋅) is locally bounded, as well as that there exists a function φ : [0, ∞) → (0, ∞) such that ν(t) ⩽ ν(t − s)φ(s) for all t, s ⩾ 0 and ∞ ∫0 φ(s)‖R(s)‖ ds < ∞. If f : [0, ∞) → X is bounded and (cI, 𝒫 )-quasi-asymptotically almost periodic, then the function F(⋅), defined through (5), is likewise bounded and (cI, 𝒫 )-quasi-asymptotically almost periodic. (ii) Suppose that ν : ℝ → (0, ∞) is an essentially bounded function which satisfies that the function 1/ν(⋅) is locally bounded, as well as that there exists a function φ : [0, ∞) → (0, ∞) such that ν(t) ⩽ ν(t − s)φ(s) for all t ∈ ℝ, s ⩾ 0 and ∞ ∫0 φ(s)‖R(s)‖ ds < ∞. If f : ℝ → X is bounded and (cI, 𝒫 )-quasi-asymptotically almost periodic, then the function F(t), defined through (4), is likewise bounded and (cI, 𝒫 )-quasi-asymptotically almost periodic. It would be very captivating to extend the statements of [194, Theorem 2.34] and [304, Theorem 2.27] for some special kinds of metrically (asymptotically) almost periodic type functions (see also [431, Theorem 7.1.25] for the case in which ρ = cI for some c ∈ ℂ ∖ {0}). The same observation can be given for the statements of [431, Theorem 6.1.40, Proposition 7.3.15, Proposition 7.3.17, Proposition 7.3.18].

5.1.4 Applications to the abstract Volterra integro-differential equations In this subsection, we will present the following applications: 1. It is clear that Proposition 5.1.4 can be applied in the analysis of inhomogeneous heat equation in ℝn . In this part, we will apply Proposition 5.1.4 in the analysis of certain classes of the abstract ill-posed Cauchy problems; we will consider the integrated semigroups here (see [431] and [458] for more details about this kind of applications). Suppose that k ∈ ℕ, aα ∈ ℂ, 0 ⩽ |α| ⩽ k, aα ≠ 0 for some α with |α| = k, P(x) = ∑|α|⩽k aα i|α| x α , x ∈ ℝn , P(⋅) is an elliptic polynomial, i. e. there exist C > 0 and L > 0 such that |P(x)| ⩾

5.1 Asymptotically ρ-almost periodic type functions in general metric

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299

C|x|k , |x| ⩾ L, ω := supx∈ℝn Re(P(x)) < ∞, and X := BUC(ℝn ), the space of bounded uniformly continuous functions f : ℝn → ℂ equipped with the sup-norm. Define P(D) := ∑ aα f (α)

and

Dom(P(D)) := {f ∈ X : P(D)f ∈ X distributionally},

|α|⩽k

and assume that nX > n/2. It is well known that the operator P(D) generates an exponentially bounded r-times integrated semigroup (Sr (t))t⩾0 in X for any r > nX (see [54, 428] and references cited therein for more details about fractionally integrated semigroups). Furthermore, it is well known that for each t ⩾ 0 there exists a function ft ∈ L1 (ℝn ) such that [Sr (t)f ](x) := (ft ∗ f )(x),

x ∈ ℝn , f ∈ X.

Let a number t0 ⩾ 0 be fixed. Assume that A : BUC(ℝn ) → BUC(ℝn ) is given by (Af )(x) := m(x)f (x), x ∈ ℝn , f ∈ BUC(ℝn ), where m ∈ L∞ (ℝn ). Let ν ∈ L∞ (ℝn : (0, ∞)) and let there exist a function φ : ℝn → (0, ∞) such that ν(x) ⩽ ν(y)φ(x − y) for all x, y ∈ ℝn n and hφ ∈ L1 (ℝn ). Assume also that, for every set B ∈ ℬ, we have PB = L∞ ν (ℝ × B : n (0, ∞)) and the function f (⋅) is (S, ℝ , ℬ)-asymptotically (ω, A, 𝒫ℬ )-periodic. Applying Proposition 5.1.4, we get that the function [Sr (t)f ](⋅) is likewise (S, ℝn , ℬ)-asymptotically (ω, A, 𝒫ℬ )-periodic. We can simply incorporate the obtained result in the analysis of the corresponding ill-posed Cauchy problems. 2. Because of a great similarity with our previous research studies, we will only note here that the function spaces introduced in this paper can be important in the qualitative analysis of solutions of the inhomogeneous wave equation in ℝ3 , ℝ2 and ℝ. 3. It is clear that Proposition 5.1.14 can be applied in the analysis of the existence and uniqueness of metrically c-quasi-asymptotically almost periodic solutions for various classes of the abstract (degenerate) inhomogeneous Cauchy problems [428, 431]; for example, we can apply this result in the qualitative analysis of the following fractional equation with higher order differential operators in the Hölder space X = C α (Ω): γ

β { {Dt u(t, x) = − ∑ aβ (t, x)D u(t, x) − σu(t, x) + f (t, x), t ⩾ 0, x ∈ Ω; |β|⩽2m { { u(0, x) = u (x), x ∈ Ω; 0 {

see [428, Example 3.10.4] for the notion and more details. Since the composition principle clarified in [444, Theorem 4.5] can be reformulated for metrically ρ-slowly oscillating type functions in ℝn , providing the same special choices of metric spaces, we are in a position to analyze the metrically (S, 𝔻, ℬ)-asymptotically (ω, ρ)-periodic solutions, e. g., for the class of semilinear Hammerstein integral equations of convolution type in ℝn [194, 444]. We can also analyze metrically (S, 𝔻, ℬ)-asymptotically (ω, ρ)-periodic type solutions for certain classes of the abstract semilinear fractional Cauchy problems [428, 431].

300 � 5 Asymptotical ρ-almost periodicity in general metric

5.2 Multi-dimensional weighted ergodic components in general metric The notion of asymptotical almost periodicity was introduced by A. S. Kovanko in his pioneering paper [462] (1929). Twelve years later, in 1941, the notion of asymptotical almost periodicity was rediscovered, in a slightly different form we are using now, by the famous French mathematician M. Fréchet [326]; both authors, A. S. Kovanko and M. Fréchet, worked in the one-dimensional setting. Suppose that f : [0, ∞) → X is a bounded continuous function; then we say that the function f (⋅) is asymptotically almost periodic if and only if for each ε > 0 there exist two finite real numbers l > 0 and M > 0 such that for each real number t0 ⩾ 0 there exists a number τ ∈ [t0 , t0 + l] such that ‖f (t + τ) − f (t)‖ ⩽ ε, t ⩾ M. Due to the important research results obtained by W. M. Ruess and W. H. Summers in [649, 650, 651], we know that a bounded continuous function f : [0, ∞) → X is asymptotically almost periodic if and only if there exist a uniquely determined almost periodic function g : ℝ → X and a continuous function ϕ : [0, ∞) → X vanishing at plus infinity such that f (t) = g(t) + ϕ(t) for all t ⩾ 0. The functions g(⋅) and ϕ(⋅) are usually called the principal and corrective (ergodic) terms of the function f (⋅), respectively. The results of W. M. Ruess and W. H. Summers have strongly influenced many mathematicians working in the field of asymptotically almost periodic type functions and their applications (see, e. g. the interesting research article [471] by M. Krukowski and B. Przeradzki, where the authors have applied certain compactness results to the scalarvalued integral equations). Let us recall that, in [453], we have introduced the following notion: Let 𝒳Λ denote any space of (Stepanov, Weyl, Besicovitch) almost periodic functions F : Λ × X → Y , and let 𝒬Λ denote any space of weighted ergodic spaces introduced and analyzed in [453]. Then we say that a function F : Λ × X → Y is asymptotically (𝒳Λ , 𝒬Λ )-almost periodic if and only if there exist a function G(⋅; ⋅) ∈ 𝒳Λ and a function Q ∈ 𝒬Λ such that F(t; x) = G(t; x) + Q(t; x) for all t ∈ Λ and x ∈ X; if, moreover, there ex̃ x) = G(t; x) for all t ∈ Λ and x ∈ X, then we say that ists a function G̃ ∈ 𝒳ℝn such that G(t; the function F(⋅; ⋅) is strongly asymptotically (𝒳Λ , 𝒬Λ )-almost periodic. Unfortunately, the uniqueness of decomposition F = G + Q in the above definition cannot be expected in the general framework; as is well known, this fact has some obvious unpleasant consequences concerning applications to the semilinear Cauchy problems (let us only note here that J. Zhang, T.-J. Xiao and J. Liang have introduced the notion of modular norm to overcome this difficulty; see [801] for more details given in the one-dimensional setting). We have systematically analyzed the spaces of multi-dimensional weighted ergodic components Q(⋅; ⋅) in our recent research article [453] (a joint work with B. Chaouchi and W.-S. Du). Especially, in this paper, we have examined the multi-dimensional pseudoalmost periodic functions and components for the first time in the existing literature (in the one-dimensional setting, the notion of pseudo-almost periodicity was introduced in the doctoral dissertation of C. Zhang [800], in 1992). Furthermore, in [453], we have extended the notion of double-weighted pseudo ergodic components, introduced by T. Di-

5.2 Multi-dimensional weighted ergodic components in general metric

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agana in [260] (2011); see also [5, 142, 231, 453, 801] and references cited therein for more details about the subject. The main aim of this section is to further extend and generalize the notion introduced in [453] following the approach obeyed in our recent investigation of metrical almost periodicity and its applications [446]. More precisely, we consider here the weighted Stepanov ergodic spaces in general metric, the weighted Weyl ergodic spaces in general metric, and the weighted pseudo-ergodic spaces in general metric. The introduced spaces of weighted ergodic components seem to be new and not considered elsewhere even in the one-dimensional setting. We also provide many illustrative examples, theoretical results, and applications to the abstract Volterra integro-differential equations. The organization of section is very similar to that of [453] and can be briefly described as follows. In Section 5.2.1, we introduce and analyze multi-dimensional Stepanov weighted ergodic components in general metric. The introduced classes of Stepanov weighted ergodic components are very general; for example, the notion of Stepanov p(u)-boundedness is a very special case of the notion introduced in this section (cf. [431] for the notion). The invariance of Stepanov weighted ergodicity under the actions of infinite convolution products is analyzed in Theorem 5.2.4. In this subsection, we also provide an interesting application in the study of asymptotically almost periodic type solutions at minus infinity for a class of abstract Volterra integro-differential equations (see Example 5.2.7); see also the important research article [299] by R. Farwig and Y. Taniuchi, where the authors have investigated the Navier–Stokes equations in unbounded domains, and [453]. To the best knowledge of the author, the study of Weyl ergodic components has been initiated in [436, Subsection 4.1], where we have only considered the one-dimensional components; the notion from [436] has been extended and generalized to the multi-dimensional setting in [453], where we have considered various classes of Weyl weighted ergodic components in ℝn . The main aim of Section 5.2.2 is to further generalize the notion from [453] by investigating the multi-dimensional Weyl weighted ergodic components in general metric. Here we reconsider and extend some statements from [436] to the multi-dimensional setting (see, e. g., Proposition 5.2.12) and examine, among some other questions, the invariance of metrical Weyl weighted ergodicity under the actions of infinite convolution products (see Theorem 5.2.14). Section 5.2.3 is devoted to the study of multi-dimensional weighted pseudo-ergodic components in general metric; here we also present some applications to the partial differential equations. We use the standard notation throughout the section. As before, we assume here that ℬ is a collection of non-empty subsets of X satisfying that for each x ∈ X there exists B ∈ ℬ such that x ∈ B. If T > 0 and Q : Λ × X → Y , then we set ΛT := {λ ∈ Λ : |λ| ⩽ T} and ̌ x) := Q(−t; x), t ∈ −Λ, x ∈ X. Q̌ : −Λ × X → Y by Q(t;

302 � 5 Asymptotical ρ-almost periodicity in general metric 5.2.1 Stepanov weighted ergodic components in general metric In this subsection, we will always assume that 0 ≠ Λ ⊆ ℝn and Ω is a compact subset of ℝn with a positive Lebesgue measure. We start the subsection by introducing the following notion, which extends the notion from [431, Definition 6.4.1] with the special choice of metric spaces PB := C0 (𝔻 × B : [0, ∞)) := {F : 𝔻 × B → [0, ∞); lim|t|→+∞,t∈𝔻 supx∈B ‖F(t; x)‖Y = 0} and dB (F, G) := supx∈B supt∈𝔻 ‖F(t; x) − G(t; x)‖Y for all F, G ∈ PB (B ∈ ℬ): Definition 5.2.1. Suppose that 𝔻 ⊆ Λ ⊆ ℝn , Λ + Ω ⊆ Λ, p ∈ 𝒫 (Ω) and the set 𝔻 is unbounded. Suppose further that for each set B ∈ ℬ, 𝒫B = (PB , dB ) is a metric space of functions from [0, ∞)𝔻×B containing the zero function, as well as that G : Λ → (0, ∞) and ϕ : [0, ∞) → [0, ∞). Then we say that: Ω,p(u),ϕ,G (i) a function Q : Λ × X → Y belongs to the space S0,𝔻,ℬ,𝒫 (Λ × X : Y ) if and only if for ℬ

every t ∈ 𝔻 and x ∈ X, we have that ϕ(‖Q(t + u; x)‖Y ) ∈ Lp(u) (Ω) as well as that, for every B ∈ ℬ, we have 󵄩 󵄩 G(⋅)[ϕ(󵄩󵄩󵄩Q(⋅ + u; ⋅)󵄩󵄩󵄩Y )]

Lp(u) (Ω)

∈ PB ;

Ω,p(u),ϕ,G,1

(ii) a function Q : Λ × X → Y belongs to the space S0,𝔻,ℬ,𝒫 for every t ∈ 𝔻 and x ∈ X, we have that Q(t + u; x) ∈ L every B ∈ ℬ, we have

ℬ

p(u)

(Λ × X : Y ) if and only if

(Ω : Y ) as well as that, for

󵄩 󵄩 G(⋅)ϕ(󵄩󵄩󵄩Q(⋅ + u; ⋅)󵄩󵄩󵄩Lp(u) (Ω:Y ) ) ∈ PB ; Ω,p(u),ϕ,G,2

(iii) a function Q : Λ × X → Y belongs to the space S0,𝔻,ℬ,𝒫

ℬ

(Λ × X : Y ) if and only if

for every t ∈ 𝔻 and x ∈ X, we have that Q(t + u; x) ∈ Lp(u) (Ω : Y ) as well as that, for every B ∈ ℬ, we have 󵄩 󵄩 ϕ(G(⋅)󵄩󵄩󵄩Q(⋅ + u; ⋅)󵄩󵄩󵄩Lp(u) (Ω:Y ) ) ∈ PB . In the case of consideration of functions of the form F : I → Y , we have X = {0} and

ℬ = {X}, so that the metric space 𝒫B = 𝒫 = (PB , dB ) = (P, d) consists of certain functions

from [0, ∞)𝔻 and functions G : 𝔻 → Y .

Remark 5.2.2. (i) For simplicity, we use the same symbol ⋅ for the arguments t ∈ 𝔻 and x ∈ B (the meaning is clear). (ii) In [431, Definition 6.4.1], we have used a slightly redundant assumption that for every t ∈ Λ and x ∈ X, we have ϕ(‖Q(t + u; x)‖Y ) ∈ Lp(u) (Ω), resp. Q(t + u; x) ∈ Lp(u) (Ω : Y ). We relax the notion here by assuming that this condition holds for every t ∈ 𝔻 and x ∈ X.

5.2 Multi-dimensional weighted ergodic components in general metric

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Under certain reasonable assumptions, the weighted ergodic function spaces introduced above are translation invariant. The interested reader can make an effort to formulate this precisely. The following example shows that the assumption 𝔻 ≠ Λ is sometimes crucial: Example 5.2.3. Suppose that 0 < α < β < π/2, n = 2, Λ := [0, ∞)2 , 𝔻 := {t = (t1 , t2 ) ∈ Λ : arg(t1 + it2 ) ∈ [α, β], t12 + t22 ⩾ 1} and G(t) ≡ 1, ϕ(x) ≡ x, Ω := [0, 1]2 , p ∈ 𝒫 (Ω). Suppose further that P := C0 (𝔻 : [0, ∞)) := {F : 𝔻 → [0, ∞); lim|t|→+∞,t∈𝔻 ‖F(t)‖Y = 0} and d(F, G) := supt∈𝔻 ‖F(t; x) − G(t; x)‖Y for all F, G ∈ P. Define Q(t) := 0 if t = (t1 , t2 ) for some non-negative real numbers t1 , t2 ⩾ 0 such that arg(t1 + it2 ) ∈ [arctan(sin α/(1 + cos α)), arctan(tan β + (1/ cos β))], and Q(t) := 1, otherwise. Then we Ω,p(u),ϕ,G Ω,p(u),ϕ,G have Q ∈ S0,𝔻,𝒫 (Λ : ℂ) but not Q ∈ S0,Λ,𝒫 (Λ : ℂ). Towards this end, we essentially need to show that for every real numbers r ⩾ 1 and γ ∈ [α, β], as well as for every tuple (x, y) ∈ Ω, we have arctan(sin α/(1 + cos α)) ⩽ arg(reiγ + x + iy) ⩽ arctan(tan β + (1/ cos β)).

(203)

In order to prove the right hand side of composed inequality (203), observe that we have: r sin γ + y r sin γ + 1 ⩽ arctan r cos γ + x r cos γ r sin β + 1 ⩽ arctan ⩽ arctan(tan β + (1/ cos β)); r cos β

arg(reiγ + x + iy) = arctan

the right hand side of composed inequality (203) follows from the next computation: r sin γ + y r sin γ ⩾ arctan r cos γ + x r cos γ + 1 r sin α sin α ⩾ arctan ⩾ arctan , r cos α + 1 cos α + 1

arg(reiγ + x + iy) = arctan

where we have used the fact that the function r 󳨃→ increasing.

r sin α , r cos α+1

r ⩾ 1 is monotonically

In the following metrical analogue of [431, Theorem 6.4.2(i)], we consider the invariance of a weighted ergodicity introduced in Definition 5.2.1(i) under the actions of infinite convolution products; a similar result can be formulated in many other ways and for the function spaces introduced in parts (ii) and (iii) of the last-mentioned definition (in [431, Theorem 6.4.2(i)], we have used the space PB = C0 (𝔻 × B : [0, ∞))): Theorem 5.2.4. Suppose that φ : [0, ∞) → [0, ∞), ϕ : [0, ∞) → [0, ∞) is a convex monotonically increasing function satisfying ϕ(xy) ⩽ φ(x)ϕ(y) for all x, y ⩾ 0, Ω = [0, 1]n , p, q ∈ 𝒫 (Ω), 1/p(⋅) + 1/q(⋅) = 1, (ak )k∈ℕn0 is a sequence of positive real numbers such that [0,1]n ,p(u),ϕ,G

∑k∈ℕn ak = 1, Q ∈ S0,[0,∞)n ,ℬ,𝒫 (ℝn × X : Y ), where for each set B ∈ ℬ we have PB := 0

ℬ

304 � 5 Asymptotical ρ-almost periodicity in general metric L∞ (𝔻 × B : [0, ∞)). Let the operator function (R(t))t>0 ⊆ L(Y , Z) be strongly continuous, let the function t1

t

t2

tn

̌ x) ds = ∫ ∫ ⋅ ⋅ ⋅ ∫ R(t − s) Q(s; ̌ x) ds Q1 (t; x) ≡ ∫ R(t − s)Q(s; −∞

−∞ −∞

(204)

−∞

be well defined for all t ∈ ℝn , x ∈ X, and measurable in t ∈ ℝn for every fixed x ∈ X. Suppose, further, that 𝔻1 := −[1, ∞)n , r ∈ 𝒫 (Ω), as well as that for each real number M > 0 and set B ∈ ℬ we have 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 1 󵄩 󵄩󵄩 󵄩󵄩 MG1 (t)󵄩󵄩󵄩 ∑ ak φ(ak−1 )󵄩󵄩󵄩φ(󵄩󵄩󵄩R(s + k)󵄩󵄩󵄩)󵄩󵄩󵄩 q(s) ∈ PB1 ; 󵄩󵄩 n 󵄩 󵄩L (Ω) G(k − t − u) 󵄩󵄩󵄩 r(u) 󵄩k∈ℕ0 󵄩L (Ω)

(205)

here we assume that the function in (205) depends only on the first argument t ∈ 𝔻1 . If the assumptions 0 ⩽ f ⩽ g and g ∈ PB1 imply f ∈ PB1 for each set B ∈ ℬ, then Q1 ∈ [0,1]n ,r(u),ϕ,G

S0,−[1,∞)n ,ℬ,𝒫11 (ℝn × X : Z). ℬ

Proof. Let a set B ∈ ℬ be given. Then we know that there exists a real number M > 0 such that for every t ∈ [0, ∞)n and x ∈ B, we have 󵄩 󵄩 [ϕ(󵄩󵄩󵄩Q(t + u; x)󵄩󵄩󵄩Y )] p(u) < M/G(t). L (Ω:Y ) Since ϕ(⋅) is convex, ∑k∈ℕn ak = 1 and the function ϕ(⋅) is both convex and monotonically 0 increasing, the computation from the proof of [431, Theorem 6.4.2(i)] (we use the Jensen integral inequality and the Hölder inequality here) shows that for every t ∈ −[1, ∞)n , u ∈ Ω and x ∈ B, we have: 󵄩 󵄩 ϕ(󵄩󵄩󵄩Q1 (t + u; x)󵄩󵄩󵄩Z ) ⩽ ϕ( ∑ ak k∈ℕn0

∫ k+[0,1]n

⩽ ∑ ak ϕ( ∫ k∈ℕn0

⩽ ∑ ak k∈ℕn0

k+[0,1]n

∫ k+[0,1]n

⩽ ∑ ak φ(ak−1 ) k∈ℕn0

󵄩 󵄩 󵄩 ̌ 󵄩 ak−1 󵄩󵄩󵄩R(s)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩Q(t + u − s; x)󵄩󵄩󵄩Y ds) 󵄩 󵄩 󵄩 ̌ 󵄩 ak−1 󵄩󵄩󵄩R(s)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩Q(t + u − s; x)󵄩󵄩󵄩Y ds)

󵄩 󵄩 󵄩 ̌ 󵄩 ϕ(ak−1 󵄩󵄩󵄩R(s)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩Q(t + u − s; x)󵄩󵄩󵄩Y ) ds ∫ k+[0,1]n

󵄩 󵄩 󵄩 ̌ 󵄩 φ(󵄩󵄩󵄩R(s)󵄩󵄩󵄩) ⋅ ϕ(󵄩󵄩󵄩Q(t + u − s; x)󵄩󵄩󵄩Y ) ds

󵄩 󵄩 󵄩 ̌ 󵄩 = ∑ ak φ(ak−1 ) ∫ φ(󵄩󵄩󵄩R(s + k)󵄩󵄩󵄩) ⋅ ϕ(󵄩󵄩󵄩Q(t + u − s − k; x)󵄩󵄩󵄩Y ) ds k∈ℕn0

[0,1]n

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5.2 Multi-dimensional weighted ergodic components in general metric

󵄩 󵄩 󵄩 󵄩 = ∑ ak φ(ak−1 ) ∫ φ(󵄩󵄩󵄩R(s + k)󵄩󵄩󵄩) ⋅ ϕ(󵄩󵄩󵄩Q(−t − u + s + k; x)󵄩󵄩󵄩Y ) ds k∈ℕn0

[0,1]n

󵄩󵄩 󵄩 1 󵄩 󵄩󵄩 ⩽ M ∑ ak φ(ak−1 )󵄩󵄩󵄩φ(󵄩󵄩󵄩R(s + k)󵄩󵄩󵄩)󵄩󵄩󵄩 q(s) , 󵄩 󵄩L (Ω) G(k − t − u) n k∈ℕ0

since these assumptions imply −t − u + k + s ∈ [0, ∞)n for all k ∈ ℕn0 and s ∈ Ω. Clearly, for every t ∈ −[1, ∞)n and x ∈ X, the mapping u 󳨃→ ϕ(‖Q1 (t + u; x)‖Z ), u ∈ Ω belongs to the space Lr(u) (Ω) since it is measurable and (205) is assumed. The final result follows by applying (205) and the assumption following it. Remark 5.2.5. In the formulation of Theorem 5.2.4, we have assumed that 0 ⩽ f ⩽ g and g ∈ PB1 imply f ∈ PB1 for each set B ∈ ℬ. In some situations, the above condition is not satisfactory and we must additionally involve here the continuity or measurability of function f , with the meaning clear. This concretely occurs if we consider the metric space PB1 := Lr(t) (𝔻1 × B : [0, ∞)) ≡ {F : 𝔻1 × B → [0, ∞); supx∈B ‖F(t; x)‖Lr(t) (𝔻1 :Y ) < +∞} with dB1 (F, G) := supx∈B ‖F(t; x) − G(t; x)‖Lr(t) (𝔻1 :Y ) for all F, G ∈ PB1 (B ∈ ℬ), where r ∈ 𝒫 (𝔻1 ). We continue by providing an example for illustrative purposes: Example 5.2.6 (See also [431, Example 2.3.6], where we have made small typos by considering the particular case p = 1 at some places; see, e. g. the equation (206) below). Assume that 1 ⩽ p, r < ∞ and Ω = [0, 1]. Let (an )n∈ℤ be a strictly increasing sequence in Λ = ℝ such that limn→+∞ an = +∞ and limn→−∞ an = −∞. Assume, further, that (bn )n∈ℤ is a real sequence and the function f : ℝ → ℝ defined by f (t) := bn if t ∈ [an , an+1 ) for some n ∈ ℤ, which can be written as a countable sums of step functions. Define the functions P1 : ℝ → ℤ and P2 : ℝ → ℤ by P1 (t) := n if and only if t ∈ [an , an+1 ) and P2 (t) := m if and only if t + 1 ∈ [am , am+1 ) (t ∈ ℝ). Then we know that [0,1],p,x,1 the function f (⋅) is S p -bounded, i. e. f ∈ S0,ℝ,𝒫 (ℝ : ℂ) with P := L∞ (ℝ : [0, ∞)) and the metric d(⋅; ⋅) induced by the norm of this Banach space if and only if p sup[bP (t) 1 t∈ℝ

p

⋅ (aP1 (t)+1 − t) +

P2 (t)−1

∑

j=P1 (t)+1

p bj

p

p

⋅ (aj+1 − aj ) + (t + 1 − aP2 (t) ) ⋅

p bP (t) ] 2

1/p

< ∞. (206)

On the other hand, f ∈ S0,ℝ,𝒫 (ℝ : ℂ) with Pr := Lr (ℝ : [0, ∞)) and the metric dr (⋅; ⋅) r induced by the norm of this Banach space if and only if [0,1],p,x,G

∞

p

p

P2 (t)−1

∫ Gr (t)[bP (t) ⋅ (aP1 (t)+1 − t) + ∑ 1

−∞

j=P1 (t)+1

p

p

p

r/p

p

bj ⋅ (aj+1 − aj ) +(t + 1 − aP2 (t) ) ⋅ bP (t) ] 2

dt < ∞.

The statement of [431, Corollary 6.4.3] can be simply reformulated in our new framework, in many different ways, and this can be applied in the analysis of the existence

306 � 5 Asymptotical ρ-almost periodicity in general metric and uniqueness of metrically Stepanov asymptotically almost periodic type solutions at minus infinity for a large class of the abstract Volterra integro-differential equations. For example, we have the following: Example 5.2.7. Suppose that 1 ⩽ p < ∞, 1/p + 1/q = 1, (R(t))t>0 ⊆ L(X, Y ) is a strongly continuous operator family, and there exists a finite real number ω < 0 such that ‖R(t)‖ ⩽ Meωt for all t > 0 (this in particular implies ∑∞ k=0 ‖R(⋅)‖Lq [k,k+1] < ∞). Suppose further that f : ℝ → X is Stepanov-p-almost periodic, q : ℝ → X is measurable, σ ∈ (−∞, 0] and t+1

σ

󵄩 󵄩p sup[(1 + |t|) ( ∫ 󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds) t∈ℝ

1/p

] < +∞.

t

Consider the function G : ℝ → Y given by t

̌ G(t) := ∫ R(t − s)[f (s) + q(s)] ds,

t ∈ ℝ.

−∞ t

Applying [428, Proposition 2.6.11], we get that the function t 󳨃→ q1 (t) ≡ ∫−∞ R(t −s)f (s) ds, t ∈ ℝ is almost periodic. Consider now the function t

∞

̌ ds = ∫ R(s)q(s − t) ds, t 󳨃→ ∫ R(t − s)q(s)

t ∈ ℝ.

0

−∞

This function is well defined and measurable; furthermore, the Hölder inequality and the prescribed assumptions imply that there exists a finite real constant Mσ > 0 such that for every t ∈ ℝ, we have ∞

󵄩󵄩 󵄩 󵄩 ωs 󵄩 󵄩󵄩q1 (t)󵄩󵄩󵄩Y ⩽ M ∫ e 󵄩󵄩󵄩q(s − t)󵄩󵄩󵄩 ds 0 ∞

󵄩 󵄩 󵄩 󵄩 ⩽ M ∑ 󵄩󵄩󵄩eωs 󵄩󵄩󵄩Lq [k,k+1] 󵄩󵄩󵄩q(s − t)󵄩󵄩󵄩Lp [k,k+1] k=0

∞

1/p

t−k

󵄩 󵄩p ⩽ Me ∑ e ( ∫ 󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds) k=0

ωk

t−k−1 ∞

⩽ Me ⋅ Const. ⋅ ∑ eωk [(1 + |k − t|)

−σ

k=0

∞

+ (1 + |k − t − 1|) ] −σ

⩽ Me ⋅ Const. ⋅ 2−σ−1 ∑ eωk [3 + 2|t|−σ + 2k −σ ] ⩽ Mσ [1 + |t|−σ ]. k=0

5.2 Multi-dimensional weighted ergodic components in general metric

� 307

Suppose now that G1 : ℝ → (0, ∞) and ν : ℝ → (0, ∞) are two measurable functions as −σ well as that P1 := Ls(u) ∈ P1 and ν (ℝ : (0, ∞)) for some function s ∈ 𝒫 ([0, 1]), G1 (⋅)(1+|⋅|) r(u) ≡ r ∈ [1, ∞). Using Theorem 5.2.4 (see also Remark 5.2.5) and a simple computation similar to the computation carried out above, we get that the right-hand side of (205) [0,1],r,x,G1 can be estimated as Const. ⋅ (1 + |t|)−σ so that q1 ∈ S0,(−∞,−1],𝒫 1. It is worth noting that the growth estimate of operator family (R(t))t>0 ⊆ L(X, Y ) considered above is not satisfactory if we consider the abstract degenerate (fractional) solution operator families generated by the multivalued linear operators satisfying condition (P). In this case, the growth order ‖R(t)‖ ⩽ Meωt t β−1 , t > 0, for some finite real numbers M > 0 and β ∈ (0, 1], or the growth order ‖R(t)‖ ⩽ Mt β−1 /(1 + t γ ), t > 0, for some finite real numbers M > 0, β ∈ (0, 1] and γ > β, appears. Suppose that q(β − 1) > −1, provided that p > 1, and β = 1, provided that p = 1. If we consider the growth order ‖R(t)‖ ⩽ Meωt t β−1 , t > 0, everything remains the same; in the case of consideration of growth order ‖R(t)‖ ⩽ Mt β−1 /(1 + t γ ), t > 0, a relatively simple argumentation shows that we only need to assume additionally that γ > β − σ. It is clear how we can incorporate the obtained result in the analysis of the existence and uniqueness of metrically Stepanov asymptotically almost periodic type solutions at minus infinity for the fractional Poisson heat equation with Weyl–Liouville fractional derivatives in the space X = Lr (Ω), for instance. 5.2.2 Weyl weighted ergodic components in general metric In this subsection, we analyze Weyl weighted ergodic components in general metric. We start by introducing the following notion, which generalizes the corresponding notion of Weyl weighted ergodic components introduced in [431, Definitions 6.4.4–6.4.6]: Definition 5.2.8. Suppose that 𝔻 ⊆ Λ ⊆ ℝn , the set 𝔻 is unbounded, p ∈ 𝒫 (Λ) as well as 𝔻 + Λ + lΩ ⊆ Λ

for all l > 0,

(207)

ϕ : [0, ∞) → [0, ∞) and 𝔽 : (0, ∞) × Λ → (0, ∞). (i) Suppose, further, that for each set B ∈ ℬ we have that 𝒫B = (PB , dB ) is a metric space consisting of functions from [0, ∞)(0,∞)×B , containing the zero function. By p(u),ϕ,𝔽 e − W0,𝔻,ℬ,𝒫 (Λ × X : Y ) we denote the collection consisting of all functions Q : ℬ Λ × X → Y such that for every t ∈ 𝔻, s ∈ Λ, l > 0 and x ∈ X, we have that ϕ(‖Q(t + u; x)‖Y ) ∈ Lp(u) (s + lΩ) as well as that, for every B ∈ ℬ, we have 󵄩 󵄩 lim sup sup[𝔽(⋅, t)[ϕ(󵄩󵄩󵄩Q(t + u; ⋅)󵄩󵄩󵄩Y )]

|t|→+∞,t∈𝔻 s∈Λ

Lp(u) (s+⋅Ω)

] ∈ PB .

(ii) Suppose further that for each set B ∈ ℬ we have that 𝒫B = (PB , dB ) is a metric space consisting of functions from [0, ∞)𝔻×B , containing the zero function. By

308 � 5 Asymptotical ρ-almost periodicity in general metric p(u),ϕ,𝔽

W0,𝔻,ℬ,𝒫 (Λ × X : Y ) we denote the collection consisting of all functions Q : Λ × X → ℬ Y such that for every t ∈ 𝔻, s ∈ Λ, l > 0 and x ∈ X, we have that ϕ(‖Q(t + u; x)‖Y ) ∈ Lp(u) (s + lΩ) as well as that for every B ∈ ℬ, we have 󵄩 󵄩 lim sup sup[𝔽(l, ⋅)[ϕ(󵄩󵄩󵄩Q(⋅ + u; ⋅)󵄩󵄩󵄩Y )] l→+∞

s∈Λ

Lp(u) (s+lΩ)

] ∈ PB .

Definition 5.2.9. Suppose that 𝔻 ⊆ Λ ⊆ ℝn , the set 𝔻 is unbounded, p ∈ 𝒫 (Λ), as well as (207) holds, ϕ : [0, ∞) → [0, ∞) and 𝔽 : (0, ∞) × Λ → (0, ∞). (i) Suppose further that for each set B ∈ ℬ we have that 𝒫B = (PB , dB ) is a metric space consisting of functions from [0, ∞)(0,∞)×B , containing the zero function. By p(u),ϕ,𝔽,1 e − W0,𝔻,ℬ,𝒫 (Λ × X : Y ) we denote the collection consisting of all functions Q : ℬ Λ × X → Y such that for every t ∈ 𝔻, s ∈ Λ, l > 0 and x ∈ X, we have that Q(t + u; x) ∈ Lp(u) (s + lΩ : Y ) as well as that for every B ∈ ℬ, we have 󵄩 󵄩 lim sup sup[𝔽(⋅, t)ϕ[󵄩󵄩󵄩Q(t + u; ⋅)󵄩󵄩󵄩Lp(u) (s+⋅Ω:Y ) ]] ∈ PB .

|t|→+∞,t∈𝔻 s∈Λ

(ii) Suppose further that for each set B ∈ ℬ we have that 𝒫B = (PB , dB ) is a metric space consisting of functions from [0, ∞)𝔻×B , containing the zero function. By p(u),ϕ,𝔽,1 W0,𝔻,ℬ,𝒫 (Λ × X : Y ) we denote the collection consisting of all functions Q : Λ × X → ℬ Y such that for every t ∈ 𝔻, s ∈ Λ, l > 0 and x ∈ X, we have that Q(t + u; x) ∈ Lp(u) (s + lΩ : Y ) as well as that, for every B ∈ ℬ, we have 󵄩 󵄩 lim sup sup[𝔽(l, ⋅)ϕ[󵄩󵄩󵄩Q(⋅ + u; x)󵄩󵄩󵄩Lp(u) (s+⋅Ω:Y ) ]] ∈ PB . l→+∞

s∈Λ

Definition 5.2.10. Suppose that 𝔻 ⊆ Λ ⊆ ℝn , the set 𝔻 is unbounded, p ∈ 𝒫 (Λ), as well as (207) holds, ϕ : [0, ∞) → [0, ∞) and 𝔽 : (0, ∞) × Λ → (0, ∞). (i) Suppose, further, that for each set B ∈ ℬ we have that 𝒫B = (PB , dB ) is a metric space consisting of functions from [0, ∞)(0,∞)×B , containing the zero function. By p(u),ϕ,𝔽,2 e − W0,𝔻,ℬ,𝒫 (Λ × X : Y ) we denote the collection consisting of all functions Q : ℬ Λ × X → Y such that for every t ∈ 𝔻, s ∈ Λ, l > 0 and x ∈ X, we have that Q(t + u; x) ∈ Lp(u) (s + lΩ : Y ) as well as that for every B ∈ ℬ, we have 󵄩 󵄩 lim sup sup ϕ[𝔽(⋅, t)󵄩󵄩󵄩Q(t + u; ⋅)󵄩󵄩󵄩Lp(u) (s+⋅Ω:Y ) ] ∈ PB .

|t|→+∞,t∈𝔻 s∈Λ

(ii) Suppose further that for each set B ∈ ℬ we have that 𝒫B = (PB , dB ) is a metric space consisting of functions from [0, ∞)𝔻×B , containing the zero function. By p(u),ϕ,𝔽,2 W0,𝔻,ℬ,𝒫 (Λ × X : Y ) we denote the collection consisting of all functions Q : Λ × X → ℬ Y such that, for every t ∈ 𝔻, s ∈ Λ, l > 0 and x ∈ X, we have that Q(t + u; x) ∈ Lp(u) (s + lΩ : Y ) as well as that for every B ∈ ℬ, we have 󵄩 󵄩 lim sup sup ϕ[𝔽(l, ⋅)󵄩󵄩󵄩Q(⋅ + u; ⋅)󵄩󵄩󵄩Lp(u) (s+lΩ:Y ) ] ∈ PB . l→+∞

s∈Λ

5.2 Multi-dimensional weighted ergodic components in general metric

� 309

Remark 5.2.11. (i) For simplicity, we use the same symbol ⋅ for the arguments l > 0 (t ∈ 𝔻) and x ∈ B (as in the previous subsection, the meaning is clear); see also Definition 5.2.15 below. (ii) In [431, Definitions 6.4.4–6.4.6], we have assumed that Λ + Λ ⊆ Λ and Λ + lΩ ⊆ Λ; these conditions imply (207). Moreover, in the above-mentioned definitions, we have assumed that for every t ∈ Λ, s ∈ Λ, l > 0 and x ∈ X, we have ϕ(‖Q(t + u; x)‖Y ) ∈ Lp(u) (s + lΩ) or Q(t + u; x) ∈ Lp(u) (s + lΩ : Y ); here, we relax this assumption by requiring that the above inclusion holds for all t ∈ 𝔻, s ∈ Λ, l > 0 and x ∈ X. (iii) If we consider some special metric spaces, then the use of Jensen integral inequality in general measure spaces enables one to clarify some sufficient conditions under 1,ϕ,𝔽 1,ϕ,𝔽,1 which a function Q ∈ (e−)W0,𝔻,ℬ,𝒫 (ℝn × X : Y ) [Q ∈ (e−)W0,𝔻,ℬ,𝒫 (ℝn × X : Y )] 1,ϕ ,𝔽 ,1

ℬ

1,ϕ ,𝔽 ,2

ℬ

1 1 1 1 belongs to the space (e−)W0,𝔻,ℬ,𝒫 (ℝn ×X : Y ) [(e−)W0,𝔻,ℬ,𝒫 (ℝn ×X : Y )]. A similar ℬ ℬ comment can be given for the weighted ergodic spaces considered in the previous subsection and the subsequent subsection. (iv) If we impose some rational conditions on the metric spaces under our consideration, then we can simply clarify some embedding results between the same classes of metrically (equi)-Weyl multi-dimensional ergodic components (see also Lemma 1.1.10(ii)–(iii)). We can also clarify certain results about the translation invariance of spaces of metrically (equi)-Weyl multi-dimensional ergodic components (cf. also [431, conditions (D)–(D)’]).

Under certain conditions, the ergodic function spaces introduced in Definition 5.2.1 can be embedded into the ergodic function spaces introduced in this subsection. Without going into full details, we will only note that the statement of [431, Proposition 6.4.7] can be straightforwardly reformulated, with condition [431, (6.105)] and the corresponding conditions from the formulations of the second part of this proposition and the third part of this proposition staying the same, if we employ the spaces L∞ ([0, ∞)n × B : [0, ∞)) and L∞ ((0, ∞) × B : [0, ∞)) in our analysis. It is worth noticing that, if a p-locally integrable function q : [0, ∞) → Y is equiWeyl-p-vanishing, then q(⋅) is Weyl-p-vanishing as well (cf. [436, Definition 4.4] for the notion); here 1 ⩽ p < ∞. It is clear that we cannot so simply reconsider this statement for the metrically Weyl ergodic components introduced in this section. Concerning this issue, we will state and prove the following result (the possibility for work also exists if we consider weighted Lp -spaces here): Proposition 5.2.12. Suppose that 1 ⩽ p < ∞, Λ = 𝔻 = [0, ∞)n , ϕ(x) ≡ x, F : (0, ∞) → (0, ∞) satisfies that the function l 󳨃→ ln/p F(l), l ⩾ 1 is monotonically decreasing and Q ∈ p,x,F p,x,F e−W0,[0,∞)n ,𝒫 ([0, ∞)n : Y ), where P = C0 ((0, ∞) : [0, ∞)). Then Q ∈ W0,[0,∞)n ,𝒫 ([0, ∞)n : 1 Y ), where P1 = C0 ([0, ∞)n : [0, ∞)). Proof. Let ε > 0 be given. Then we know that there exists a real number l0 = l0 (ε) ⩾ 1 such that for each real number l ⩾ l0 there exists a real number tl > 0 such that for each t ∈ [0, ∞)n with |t| ⩾ tl we have

310 � 5 Asymptotical ρ-almost periodicity in general metric 1/p

F(l) sup ( s∈[0,∞)n

󵄩󵄩 󵄩p 󵄩󵄩Q(u)󵄩󵄩󵄩Y du)

∫

s+t+[0,l]n

⩽ 2−(n/p) ε.

(208)

Suppose that t0 := tl0 , t ∈ [0, ∞)n , |t| ⩾ t0 , lt = l0 and l > lt . It suffices to show that 1/p

F(l) sup ( s∈[0,∞)n

∫

s+t+[0,l]n

󵄩󵄩 󵄩p 󵄩󵄩Q(u)󵄩󵄩󵄩Y du)

⩽ ε.

This simply follows from the next computation (k = ⌊l/l0 ⌋) involving (208): 1/p

(

∫

s+t+[0,l]n

󵄩󵄩 󵄩p 󵄩󵄩Q(u)󵄩󵄩󵄩Y du)

⩽(

∫

s+t+[0,(k+1)l0 ]n n

󵄩󵄩 󵄩p 󵄩󵄩Q(u)󵄩󵄩󵄩Y du)

⩽ ((k + 1) [2 ⩽ 2−(n/p) ε2n/p

−(n/p)

ln/p n/p l0

p

1/p

1/p

ε/F(l0 )] )

⩽ 2−(n/p) ε(1 +

l ) l0

n/p

1 F(l0 )

1 ε ⩽ , F(l0 ) F(l)

where the last estimate follows form our assumption that the function l 󳨃→ ln/p F(l), l ⩾ 1 is monotonically decreasing. We continue by providing some examples: Example 5.2.13. (i) (see [431, Example 6.4.9]) Suppose that k1 , k2 , . . . , kn ∈ ℕ0 , Λk1 ,k2 ,...,kn := (k1 , k1 + 1) × (k2 , k2 + 1) × ⋅ ⋅ ⋅ × (kn , kn + 1) and Λ = 𝔻 := [0, ∞)n (X = {0}). Define the function Q : Λ → [0, ∞) by Q(t) := Qk1 ,k2 ,...,kn (t) for t ∈ Λk1 ,k2 ,...,kn , where Qk1 ,k2 ,...,kn (t) := 0 if there exists ki (1 ⩽ i ⩽ n) such that ki ∉ {n2 : n ∈ ℕ0 }, and Qk1 ,k2 ,...,kn (t) := 1, otherwise. If there do not exist integers k1 , k2 , . . . , kn ∈ ℕ0 such that t ∈ Λk1 ,k2 ,...,kn , we define Q(t) := 0. Following the analysis from the abovep,x,𝔽

mentioned example, we simply get that Q ∈ e − W0,𝔻,𝒫 (Λ : ℂ), provided that 𝔽(l; t) := (1 + l)−σ for some real number σ < (−1)/r (p(u) ≡ p ∈ [1, ∞), r ∈ [1, ∞)) and P = Lr ((0, ∞) : [0, ∞)). (ii) (see [431, Example 6.4.10]) Suppose that k1 , k2 , . . . , kn ∈ ℤ, Λk1 ,k2 ,...,kn := (k1 , k1 + 1) × (k2 , k2 + 1) × ⋅ ⋅ ⋅ × (kn , kn + 1) and Λ = 𝔻 := ℝn (X = {0}). Define the function Q : Λ → [0, ∞) by Q(t) := Qk1 ,k2 ,...,kn (t) for t ∈ Λk1 ,k2 ,...,kn , where Qk1 ,k2 ,...,kn (t) := 0 if there exists ki (1 ⩽ i ⩽ n) such that ki ∉ {n2 : n ∈ ℕ0 }, and Qk1 ,k2 ,...,kn (t) := √|k1 | ⋅ ⋅ ⋅ ⋅ ⋅ |kn |, otherwise. If there do not exist integers k1 , k2 , . . . , kn ∈ ℤ such that t ∈ Λk1 ,k2 ,...,kn , we define Q(t) := 0. Suppose p ∈ [1, ∞); following the analysis from the abovementioned example, we may conclude the following: (a) If 𝔽(l; t) does not depend on t, then there does not exist a collection 𝒫 of metric p,x,𝔽 spaces with the required properties such that Q ∈ e − W0,𝔻,𝒫 (Λ : ℂ).

5.2 Multi-dimensional weighted ergodic components in general metric

� 311

p,x,𝔽

(b) If 𝔽(l; t) := l−σ for some real number σ > (n/p) + (n/2), then Q ∈ W0,𝔻,𝒫 (Λ : ℂ) for any collection 𝒫 of metric spaces with the required properties. (iii) Suppose that Λ = 𝔻 := [0, ∞)n , Q(t1 , . . . , tn ) := ∏ni=1 arctan ti , t1 , . . . , tn ⩾ 0 and P := C0 ((0, ∞) : [0, ∞)). Then a very simple argumentation (see also Example 5.2.16(iii) p,ϕ,𝔽 below) shows that Q ∈ e − W0,𝔻,𝒫 (Λ : ℂ), provided that liml→+∞ ln/p F(l) = 0. Now we will continue our analysis from [431, Theorem 6.4.11(i)] (we can similarly reconsider the second part of this theorem and the third part of this theorem): Theorem 5.2.14. Suppose that the operator family (R(t))t>0 ⊆ L(Y , Z) is strongly continuous, Q : ℝn ×X → Y satisfies that its restriction QR (⋅; ⋅) to the set [0, ∞)n ×X belongs to the p(u),ϕ,𝔽 p(u),ϕ,𝔽 space e − W0,[0,∞)n ,ℬ,𝒫 ([0, ∞)n × X : Y ), resp. to the space W0,[0,∞)n ,ℬ,𝒫 ([0, ∞)n × X : Y ), ℬ ℬ where PB = L∞ ((0, ∞) × ℬ : [0, ∞)), resp. PB = L∞ ([0, ∞)n × ℬ : [0, ∞)), for each set B of collection ℬ. Suppose that φ : [0, ∞) → [0, ∞), ϕ : [0, ∞) → [0, ∞) is a convex monotonically increasing function satisfying ϕ(xy) ⩽ φ(x)ϕ(y) for all x, y ⩾ 0, (ak,l )k∈ℕn0 is a sequence of positive real numbers such that ∑k∈lℕn ak,l = 1 for all l > 0, 0 and the value Q1 (t; x), given by (204), is well defined for all t ∈ −[0, ∞)n and x ∈ X. Let 𝔽1 : (0, ∞) × (−[0, ∞)n ) → (0, ∞) satisfy that for each set B of collection ℬ we have M

󵄩󵄩 󵄩󵄩 −1 sup 󵄩󵄩󵄩 ∑ ak,⋅ ⋅−n φ(⋅n ak,⋅ ) 󵄩󵄩k∈⋅ℕn |t|→+∞,t∈−[0,∞)n s∈−[0,∞)n 󵄩 0 󵄩󵄩 󵄩󵄩 𝔽1 (⋅, t) 󵄩 󵄩 󵄩󵄩 × [φ(󵄩󵄩󵄩R(v + k)󵄩󵄩󵄩)]Lq(v) (⋅[0,1]n ) ∈ PB 𝔽(⋅, k − t − u) 󵄩󵄩󵄩󵄩Lr(u) (s−⋅[0,1]n ) lim sup

for all M > 0,

(209)

resp. M lim sup l→+∞

󵄩󵄩 󵄩󵄩 −1 sup 󵄩󵄩󵄩 ∑ ak,l l−n φ(ln ak,l ) 󵄩 n n 󵄩 s∈−[0,∞) 󵄩k∈lℕ 0

󵄩 󵄩 × [φ(󵄩󵄩󵄩R(v + k)󵄩󵄩󵄩)]Lq(v) (l[0,1]n )

󵄩󵄩 󵄩󵄩 𝔽1 (l, ⋅) 󵄩󵄩 ∈ PB 𝔽(l, k − ⋅ − u) 󵄩󵄩󵄩󵄩Lr(u) (s−l[0,1]n )

for all M > 0,

where we assume that the function in (205) depends only on the first argument. If the assumptions 0 ⩽ f ⩽ g and g ∈ PB1 imply f ∈ PB1 for each set B ∈ ℬ, r ∈ 𝒫 (ℝn ), then r(u),ϕ,𝔽 r(u),ϕ,𝔽 Q1 ∈ e − W0,−[0,∞)1n ,ℬ,𝒫 1 ((−[0, ∞)n ) × X : Z), resp. Q1 ∈ W0,−[0,∞)1n ,ℬ,𝒫 1 ((−[0, ∞)n ) × X : Z). ℬ

ℬ

p(u),ϕ,𝔽 W0,[0,∞)n ,ℬ,𝒫 ([0, ∞)n ℬ

Proof. We will consider the class e − × X : Y ), only. Let B ∈ ℬ be given. Then we know that there exist two finite real numbers M > 0 and M1 > 0 such that for each t ∈ [0, ∞)n with |t| > M1 and for each s ∈ [0, ∞)n we have 𝔽(l, t)[ϕ(‖Q(t + u; x)‖Y )]Lp(u) (s+lΩ) ⩽ M, l > 0, x ∈ B. Assume that t ∈ −[0, ∞)n , |t| > M1 , l > 0 and s ∈ −[0, ∞)n . Repeating verbatim the argumentation contained in the proof of [431, Theorem 6.4.11(i)], we obtain (our computation involves the Jensen integral inequality, the Hölder inequality, and the assumptions on the function ϕ(⋅)):

312 � 5 Asymptotical ρ-almost periodicity in general metric 󵄩 󵄩 ϕ(󵄩󵄩󵄩Q1 (t + u; x)󵄩󵄩󵄩Z ) ⩽ ϕ( ∑

k∈l[0,1]n

⩽

= ⩽ ⩽

∑

k∈l[0,1]n

∑

k∈l[0,1]n

∑

k∈l[0,1]n

∑

k∈l[0,1]n

⩽2

ak,l

−1 󵄩 ̌ + u − v; x)󵄩󵄩󵄩 dv) 󵄩󵄩R(v)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩Q(t ak,l 󵄩 󵄩 󵄩 󵄩Y

∫ k+l[0,1]n

ak,l ϕ(

∫

k+l[0,1]n

ak,l ϕ(l−n

−1 󵄩 ̌ + u − v; x)󵄩󵄩󵄩 dv) 󵄩󵄩R(v)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩Q(t ak,l 󵄩 󵄩 󵄩 󵄩Y

∫ k+l[0,1]n

−1 n ak,l l−n φ(ak,l l )

󵄩 󵄩 ̌ 󵄩 −1 n 󵄩 ak,l l 󵄩󵄩󵄩R(v)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩Q(t + u − v; x)󵄩󵄩󵄩Y dv) ∫

k+l[0,1]n −1 n ak,l l−n φ(ak,l l )

∑

k∈l[0,1]n

∫ k+l[0,1]n

󵄩 󵄩 󵄩 ̌ 󵄩 ϕ(󵄩󵄩󵄩R(v)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩Q(t + u − v; x)󵄩󵄩󵄩Y ) dv 󵄩 󵄩 󵄩 ̌ 󵄩 φ(󵄩󵄩󵄩R(v)󵄩󵄩󵄩) ⋅ ϕ(󵄩󵄩󵄩Q(t + u − v; x)󵄩󵄩󵄩Y ) dv

󵄩 󵄩 −1 n ak,l l−n φ(ak,l l )[φ(󵄩󵄩󵄩R(v + k)󵄩󵄩󵄩)]Lq(v) (l[0,1]n )

󵄩 󵄩 × [ϕ(󵄩󵄩󵄩Q(−t − u + k + v; x)󵄩󵄩󵄩Y )]Lp(v) (l[0,1]n ) ⩽2

∑

k∈l[0,1]n

󵄩 󵄩 −1 n ak,l l−n φ(ak,l l )[φ(󵄩󵄩󵄩R(v + k)󵄩󵄩󵄩)]Lq(v) (l[0,1]n )

M , 𝔽(l, k − t − u)

for all u ∈ s − l[0, 1]n and k ∈ lℕn0 . This simply completes the proof along with our assumption (209). Because of a great similarity with our previous research of multi-dimensional weighted ergodic components, we will not reconsider here the statement of [431, Corollary 6.4.12] and related applications in the study of the existence and uniqueness of metrically (equi-)Weyl asymptotically almost periodic type solutions at minus infinity for certain classes of the abstract Volterra integro-differential equations.

5.2.3 Weighted pseudo-ergodic components in general metric In this subsection, we will extend and generalize the notion of pseudo-ergodic spaces 1 2 PAP0,p (Λ, ℬ, F, ϕ), PAP0,p (Λ, ℬ, F, ϕ, ψ) and PAP0,p (Λ, ℬ, F, ϕ, ψ), which have recently been analyzed in [453]. We will always assume here that 𝒫B = (PB , dB ) is a metric space consisting of functions from [0, ∞)(0,∞)×B , containing the zero function (B ∈ ℬ). Definition 5.2.15. Suppose that ϕ : [0, ∞) → [0, ∞), ψ : [0, ∞) → [0, ∞), F : (0, ∞) → (0, ∞) and p ∈ 𝒫 (Λ), where 0 ≠ Λ ⊆ ℝn . For every finite number T > 0, by pT (⋅) we denote the restriction of function p(⋅) to ΛT . Define PAP0,p,𝒫ℬ (Λ, ℬ, F, ϕ)

5.2 Multi-dimensional weighted ergodic components in general metric

� 313

:= {Q : Λ × X → Y ; ϕ(‖Q(t; x)‖Y ) ∈ LpT (t) (ΛT ), T > 0, x ∈ X and for each set B ∈ ℬ we have F(⋅)[ϕ(‖Q(t; ⋅)‖Y )]Lp⋅ (t) (Λ ) ∈ PB }, ⋅

1 PAP0,p,𝒫 (Λ, ℬ, F, ϕ, ψ) ℬ

:= {Q : Λ × X → Y ; ψ(‖Q(t; x)‖Y ) ∈ LpT (t) (ΛT ), T > 0, x ∈ X and for each set B ∈ ℬ we have F(⋅)ϕ([ψ(‖Q(t; ⋅)‖Y )]Lp⋅ (t) (Λ ) ) ∈ PB } ⋅

and 2 PAP0,p,𝒫 (Λ, ℬ, F, ϕ, ψ) ℬ

:= {Q : Λ × X → Y ; ψ(‖Q(t; x)‖Y ) ∈ LpT (t) (ΛT ), T > 0, x ∈ X and for each set B ∈ ℬ we have ϕ(F(⋅)[ψ(‖Q(t; ⋅)‖Y )]Lp⋅ (t) (Λ ) ) ∈ PB }. ⋅

If X = {0}, then we omit the term “ℬ” from the notation and write “𝒫 ” in place of “𝒫ℬ ”. We continue by providing some examples: Example 5.2.16. (i) (see also [431, Example 6.4.15]) Let 1/q(t) = 1/p(t) + 1/r(t) for all t ∈ ℝn , where p, q, r ∈ 𝒫 (ℝn ), Q ∈ PAP0,p,𝒫ℬ (Λ, ℬ, F, ϕ), F1 : (0, ∞) → (0, ∞), and F1 (T)[1]LrT (t) (ΛT ) F(T)

⩽ M,

T > 0,

for some finite constant M > 0. Using the argumentation contained in the analysis carried out in the above-mentioned example, we get that Q ∈ PAP0,q,𝒫ℬ (Λ, ℬ, F1 , ϕ), provided that the set PB is closed under pointwise multiplication with bounded functions (B ∈ ℬ). (ii) Suppose that s > 1, Λ := [0, ∞), p(u) := 1 + u (u ⩾ 0), ϕ(x) ≡ x, Q(t) := 0, if there does not exist an integer k ∈ ℕ such that t ∈ [k s , k s + 1], and Q(t) := 1, otherwise; then p ∈ 𝒫 ([0, ∞)), p ∉ D+ ([0, ∞)) and the function Q(⋅) is well defined because for each integer k ∈ ℕ we have (k + 1)s > 1 + k s , as easily approved. Suppose that F ∈ P and P is closed under pointwise multiplication with positive constants; then we have Q ∈ PAP0,p,𝒫 ([0, ∞), F, x). In order to prove this, it suffices to show that for each real number T > 0 we have ‖Q‖LpT (t) ([0,T]) ⩽ (1 + √5)/2. In actual fact, if T

λ ⩾ (1 + √5)/2, then λ2 − λ − 1 ⩾ 0 and it suffices to show that ∫0 (|Q(t)|/λ)p(t) dt ⩽ 1; this follows from the next computation:

314 � 5 Asymptotical ρ-almost periodicity in general metric T

s ⌊T 1/s ⌋ k +1

⌊T 1/s ⌋

s 󵄨 󵄨 p(t) ∫(󵄨󵄨󵄨Q(t)󵄨󵄨󵄨/λ) dt = ∑ ∫ λ−1−t dt ⩽ ∑ λ−2−k

0

k=1

⌊T

1/s

ks

⌋

⩽ ∑ λ−2−k ⩽ k=1

k=1

1 ⩽ 1. λ(λ − 1)

(iii) Suppose that Λ := [0, ∞)n , Q(t1 , . . . , tn ) := ∏ni=1 arctan ti , t1 , . . . , tn ⩾ 0, and P := C0 ((0, ∞) : [0, ∞)). If limT→+∞ T n F(T) = 0, then we have Q ∈ PAP0,1,𝒫 ([0, ∞)n , F, x). This simply follows from the Fubini theorem and a relatively simple argumentation. The translation invariance of spaces introduced in Definition 5.2.15 has been analyzed in [431, Theorem 6.4.17]. We will skip all details concerning this topic for brevity. Further on, the invariance of weighted pseudo-ergodicity under the actions of infinite convolution products has been analyzed in [431, Theorem 6.4.18]. This result can be reconsidered for the metrical weighted pseudo-ergodicity only if we employ some special classes of metric spaces in our analysis, like the space L∞ ((0, ∞) × B : [0, ∞)) or its modifications; see also Theorem 5.2.4 and Theorem 5.2.14. It is also worth noting that the method proposed in the proof of the last-mentioned theorem can be employed to deduce certain results about the convolution invariance of weighted pseudo-ergodic spaces, introduced in this section. The main idea is to consider the decomposition of space ℝn into the orthants I1 = [0, ∞)n , . . . , I2n = −[0, ∞)n of ℝn as well as certain substitutions of variables and the integration over the first orthant I1 after all; details can be left to the interested readers. Such results can be applied in the qualitative analysis of asymptotically almost periodic type solutions of the inhomogeneous heat equation in ℝn , provided that the corresponding ergodic component belongs to some weighted pseudo-ergodic space introduced in Definition 5.2.15. The uniformly convergent sequences of metrically weighted pseudo-ergodic components can be considered, as well as the uniformly convergent sequences of metrically Stepanov (Weyl) weighted ergodic components. Sometimes the ordinary convergence is sufficient to ensure certain features of the limit function, as the following simple result shows (we will only note here that the same result if we consider variable exponents p ∈ D+ (Λ) and r ∈ D+ ((0, ∞)) since the dominated convergence theorem holds in this framework; see, e. g., [265] and [446, Lemma 1.4(iv)]): Proposition 5.2.17. Suppose that ϕ(⋅) is continuous, 1 ⩽ p < +∞, 1 ⩽ r < +∞, Qk ∈ PAP0,r,𝒫 (Λ, F, ϕ) for all k ∈ ℕ and limk→+∞ Qk (t) = Q(t) for a. e. t ∈ Λ. Suppose that Λ is a Lebesgue measurable set and the following conditions hold: (i) There exists a function G ∈ Lr ((0, ∞)) such that for every k ∈ ℕ, we have |F(T)(∫Λ ϕ(‖Qk (t)‖Y )p dt)1/p | ⩽ G(T) for a. e. T ∈ (0, ∞). T (ii) There exists a Lebesgue measurable set A ⊆ (0, ∞) such that for every T ∈ (0, ∞) ∖ A, there exists a function g ∈ Lp (ΛT ) such that ϕ(‖Qk (t)‖Y ) ⩽ g(t) for a. e. t ∈ ΛT .

5.2 Multi-dimensional weighted ergodic components in general metric

� 315

Proof. Our assumptions simply imply with the help of dominated convergence theorem that for every T ∈ (0, ∞) ∖ A, we have limk→+∞ ϕ(‖Qk (t)‖Y ) = ϕ(‖Q(t)‖Y ) in Lp (ΛT ). Keeping in mind condition (i), we can apply the dominated convergence theorem again in order to see that 1/p

󵄩 󵄩 p lim F(T)(∫ ϕ(󵄩󵄩󵄩Qk (t)󵄩󵄩󵄩Y ) dt)

k→+∞

ΛT

󵄩 󵄩 p = F(T)(∫ ϕ(󵄩󵄩󵄩Q(t)󵄩󵄩󵄩Y ) dt)

1/p

ΛT

in Lr ((0, ∞)). This completes the proof. We can simply formulate many simple results concerning the composition principles for metrically weighted pseudo-ergodic components and metrical Stepanov (Weyl) weighted ergodic components. We close the section by providing some examples, applications, and concluding remarks about the introduced notion. In the first example, we revisit the d’Alembert formula [453]: Example 5.2.18. Let a > 0. Then it is well known that the regular solution of the wave equation utt = a2 uxx in domain {(x, t) : x ∈ ℝ, t > 0}, equipped with the initial conditions u(x, 0) = f (x) ∈ C 2 (ℝ) and ut (x, 0) = g(x) ∈ C 1 (ℝ), is given by the d’Alembert formula (85). Clearly, the solution u(⋅; ⋅) can be extended to the whole plane. Suppose now that qi ∈ PAP0,1,𝒫 (ℝ, F, x) for i = 1, 2, as well as that q1 = f and q2 = g [1] , the first anti-derivative of function g. Then the ergodic component u = uq belongs to the space PAP0,1,𝒫 1 (ℝ2 , F1 , x), provided that the following conditions hold: (i) For each real constant c > 0 and function f ∈ P, the function f (c⋅) also belongs to P; (ii) The set P is closed under pointwise multiplication with bounded functions; (iii) For each real constant c > 0 the function F1 (⋅) ⋅ /F(c⋅) is bounded on (0, ∞). In order to see this, we only need to follow the corresponding definitions and the next computation from [453]: 󵄨 󵄨 ∫ 󵄨󵄨󵄨q1 (x − at) + q1 (x + at)󵄨󵄨󵄨 dx dt |(x,t)|⩽T

⩽

󵄨󵄨 󵄨 󵄨󵄨q1 (x − at) + q1 (x + at)󵄨󵄨󵄨 dx dt

∫ |x|⩽T,|t|⩽T T T

󵄨 󵄨 󵄨 󵄨 ⩽ ∫ ∫ [󵄨󵄨󵄨q1 (x − at)󵄨󵄨󵄨 + 󵄨󵄨󵄨q1 (x + at)󵄨󵄨󵄨] dt dx −T −T T (a+1)T

⩽2∫

󵄨 󵄨 ∫ 󵄨󵄨󵄨q1 (v)󵄨󵄨󵄨 dv dx = 2T

−T −(a+1)T

(a+1)T

󵄨 󵄨 ∫ 󵄨󵄨󵄨q1 (v)󵄨󵄨󵄨 dv, −(a+1)T

T > 0.

316 � 5 Asymptotical ρ-almost periodicity in general metric In the second example, we consider regular solutions of the backward wave equation in the upper half-plane: Example 5.2.19 (see, e. g. [797, Example 8, p. 33]). A unique regular solution of the backward wave equation uxx + utt = 0 in the upper half-plane {(x, t) : x ∈ ℝ, t > 0}, which is bounded continuous on {(x, t) : x ∈ ℝ, t ⩾ 0} and satisfies u(x, 0) = f (x), x ∈ ℝ, is given by +∞

u(x, t) =

1 t f (x − y) dy, ∫ 2 π t + y2

x ∈ ℝ, t > 0.

−∞

Let a number t > 0 be fixed. Then we know that the almost periodicity of forcing term f (⋅) implies the almost periodicity of solution x 󳨃→ u(x, t), x ∈ ℝ. Suppose now that f = q ∈ PAP0,1,𝒫 (ℝ, F, x), where the metric space P is closed under the pointwise multiplication with positive constants. Suppose also that there exists a function φ : ℝ → [0, ∞) such that |q(x − y)| ⩽ φ(y)|q(x)| for all x, y ∈ ℝ and +∞

∫ −∞

t2

t φ(y) dy < +∞. + y2

Then the mapping x 󳨃→ u(x, t), x ∈ ℝ belongs to the class PAP0,1,𝒫 (ℝ, F, x) as well. In order to prove this, observe only that for each real number T > 0 we have: T

󵄨 󵄨 F(T) ∫ 󵄨󵄨󵄨u(x, t)󵄨󵄨󵄨 dx −T T +∞

F(T) t 󵄨󵄨 󵄨 ⩽ ∫ ∫ 2 󵄨q(x − y)󵄨󵄨󵄨 dy dx π t + y2 󵄨 −T −∞ +∞ T

=

F(T) t 󵄨󵄨 󵄨 ∫ ∫ 2 󵄨q(x − y)󵄨󵄨󵄨 dx dy π t + y2 󵄨 −∞ −T +∞ T

⩽

tφ(y) 󵄨󵄨 F(T) 󵄨 ∫ ∫ 2 󵄨q(x)󵄨󵄨󵄨 dx dy π t + y2 󵄨 −∞ −T

+∞

T

−∞

−T

1 t 󵄨 󵄨 =[ ∫ 2 φ(y) dy] ⋅ [F(T) ∫ 󵄨󵄨󵄨q(x)󵄨󵄨󵄨 dx]. π t + y2 Finally, we would like to emphasize that we can further extend our notion by considering general topological spaces of functions here (observe that, in all definitions, PB can be just a topological space of functions). As certain drawbacks of our work, we would like to mention the following:

5.2 Multi-dimensional weighted ergodic components in general metric

� 317

(i) We have mainly considered the metrically weighted ergodic components introduced in Definition 5.2.8 as well as the first parts of Definition 5.2.1 and Definition 5.2.15. We can similarly analyze the metrically weighted ergodic components from Definition 5.2.9, Definition 5.2.10, as well as the second parts and the third parts of Definition 5.2.1 and Definition 5.2.15. (ii) For simplicity, we have mainly considered case in which ϕ(x) ≡ x. For some applications given in the general case and case in which ϕ(x) ≡ x α for some real number α ⩾ 1, we refer the reader to [431]. (iii) We have not considered the invariance of metrical weighted ergodicity under the actions of finite convolution products as well as semilinear Cauchy problems.

6 Metrical approximations of functions The main aim of this chapter is to consider the metrical approximations of functions F : Λ × X → Y by trigonometric polynomials and ρ-periodic type functions, where 0 ≠ Λ ⊆ ℝn , X and Y are complex Banach spaces and ρ is a general binary relation on Y (it would be very difficult to summarize here all relevant results concerning function spaces obtained by the closures of the set of trigonometric polynomials in certain norms; see, e. g. the investigations of A. Oliaro, L. Rodino, P. Wahlberg [590], and M. A. Shubin [689] for some non-trivial results established in this direction). We also investigate the generalized Stepanov, Weyl, Besicovitch, and Doss approaches to the metrical approximations of functions, and provide certain applications to the abstract Volterra integro-differential equations. Besides many other novelties of this work, we would like to emphasize here that [190], the research article from which we have taken the material of this chapter, is probably the first in the existing literature, which examines the notion of metrical semi-periodicity, even in the one-dimensional framework. The organization, main ideas, and scheme of this chapter can be briefly described as follows. After collecting some preliminary material, we introduce the notion of strong (ϕ, 𝔽, ℬ, 𝒫 )-almost periodicity (semi-(ϕ, ρ, 𝔽, ℬ, 𝒫 )-periodicity, semi-(ϕ, ρj , 𝔽, ℬ, 𝒫 )j∈ℕn periodicity) in Definition 6.1.1. The main purpose of Proposition 6.1.3 is to consider the compositions of functions introduced here with the Lipschitz type functions. After that, in Example 6.1.4, we provide many engaging examples justifying the introduction of function spaces under our consideration (although the main aim of this chapter is to create the abstract theory of metrical approximations of functions by trigonometric polynomials and ρ-periodic type functions, there are many places where we consider some special pseudometric spaces and specify our general notion; see e. g. Example 6.2.3 and Example 6.2.4 below). Section 6.1.1 is devoted to the study of metrical normality and metrical Bohr type definitions. The notion of (ϕ, R, ℬ, 𝔽, 𝒫 )-normality is introduced in Definition 6.1.5 and later analyzed in Proposition 6.1.6. It is worth noting that the notion of (ϕ, R, ℬ, 𝔽, 𝒫 )normality generalizes many other notions of normality known in the existing literature (see, e. g. [42, Definitions 3.3, 4.2, 4.5, 5.17] for the one-dimensional setting, and [431, Subsection 6.3.1] for the multi-dimensional setting). Definition 6.1.8 introduces the notions of the Bohr (ϕ, 𝔽, ℬ, Λ′ , ρ, 𝒫 )-almost periodicity and the (ϕ, 𝔽, ℬ, Λ′ , ρ, 𝒫 )-uniform recurrence. The main aim of Proposition 6.1.9 is to indicate that the notion of (ϕ, R, ℬ, ϕ, 𝔽, 𝒫 )normality is very general as well as that any function belonging to the introduced func𝒫 tion space e−(ℬ, ϕ, 𝔽)−B𝒫⋅ (Λ×X : Y ) [e−(ℬ, ϕ, 𝔽)j∈ℕn −BI ⋅ (Λ×X : Y )] is (ϕ, R, ℬ, ϕ, 𝔽, 𝒫 )normal under certain logical assumptions (cf. also Proposition 6.1.10, where we reconsider the statements of [446, Proposition 3.7, Corollary 3.8] in our new framework). Without going into full description of introduced definitions and established results in Section 6.2, we will only emphasize that we investigate the generalized Stepanov, Weyl, Besicovitch, and Doss concepts to the metrical approximations of functions here. The Stepanov and Weyl concepts are investigated in Section 6.2.1, while the Besicovitch https://doi.org/10.1515/9783111233871-008

6.1 Metrical approximations: the main concept

�

319

and Doss concepts are investigated in Section 6.2.2. Several theoretical results about the convolution invariance of function spaces introduced here, and many new applications to the abstract Volterra integro-differential equations are given in Section 6.3 (the actions of infinite convolution products are specifically analyzed in Section 6.3.1). Albeit it may look a little bit redundant, we have decided to repeat certain conditions on the function ϕ(⋅) sometimes for the sake of better readability. Before proceeding to our work, let us also recommend for the readers the research monograph [285] by V. K. Dzyadyk, I. A. Shevchuk for more details about the uniform approximation of functions by polynomials, and the research monograph [59] by G. I. Arkhipov, V. N. Chubarikov, A. A. Karatsuba for more details about the trigonometric sums in number theory and analysis.

6.1 Metrical approximations: the main concept In this section, we assume that 0 ≠ Λ ⊆ ℝn , ϕ : [0, ∞) → [0, ∞), 𝔽 : Λ → (0, ∞), P ⊆ [0, ∞)Λ , the space of all functions from Λ into [0, ∞), the zero function belongs to P, and 𝒫 = (P, d) is a pseudometric space. We start by introducing the following notion: Definition 6.1.1. Suppose that 0 ≠ Λ ⊆ ℝn and F : Λ × X → Y . Then we say that F(⋅; ⋅) is strongly (ϕ,𝔽,ℬ,𝒫 )-almost periodic (semi-(ϕ,ρ,𝔽,ℬ,𝒫 )-periodic, semi-(ϕ, ρj , 𝔽, ℬ, 𝒫 )j∈ℕn periodic) if and only if for each B ∈ ℬ there exists a sequence (PkB (t; x)) of trigonometric polynomials (ρ-periodic functions, (ρj )j∈ℕn -periodic functions) such that 󵄩󵄩 󵄩 󵄩 󵄩󵄩 lim sup󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩PkB (⋅; x) − F(⋅; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 = 0. 󵄩P

k→+∞ x∈B 󵄩

(210)

The usual notion of a strongly ℬ-almost periodic function F : Λ → Y (semi-(ρj )j∈ℕn periodic function F : Λ → Y ) is obtained by plugging 𝔽(⋅) ≡ 1, ϕ(x) ≡ x and P = Cb (Λ : Y ). The choice of function 𝔽(⋅) ≠ 1 is sometimes important; for example, we have the following: sin it Example 6.1.2. The function F(t) = ∑∞ i=1 i is strongly (x, 𝔽, 𝒫 )-almost periodic with 𝔽(t) ≡ | sin(t/2)|, t ∉ 2ℤπ, 𝔽(t) := 1, otherwise, and P = Cb (ℝ); see [137, Lemma 3].

Our first result reads as follows: Proposition 6.1.3. Suppose that 0 ≠ Λ ⊆ ℝn , F : Λ × X → Y , h : Y → Z is Lipschitz continuous, ϕ(⋅) is monotonically increasing and there exists a function φ : [0, ∞) → [0, ∞) such that ϕ(xy) ⩽ φ(x)ϕ(y) for all x, y ⩾ 0. Let the assumption (C1)’ hold, where: (C1)’ If f ∈ P, then d ′ f ∈ P for all reals d ′ ⩾ 0, and there exists a finite real constant d > 0 such that ‖d ′ f ‖P ⩽ d(1 + d ′ )‖f ‖P for all reals d ′ ⩾ 0 and all functions f ∈ P. Then we have the following:

320 � 6 Metrical approximations of functions (i) Suppose that F(⋅; ⋅) is semi-(ϕ, ρ, 𝔽, ℬ, 𝒫 )-periodic (semi-(ϕ, ρj , 𝔽, ℬ, 𝒫 )j∈ℕn -periodic), and h ∘ ρ ⊆ ρ ∘ h (h ∘ ρj ⊆ ρj ∘ h for 1 ⩽ j ⩽ n). Then the function h ∘ F : Λ × X → Y is likewise semi-(ϕ, ρ, 𝔽, ℬ, 𝒫 )-periodic (semi-(ϕ, ρj , 𝔽, ℬ, 𝒫 )j∈ℕn -periodic). (ii) Suppose that X = {0}, there exists a finite real constant c > 0 such that ϕ(x + y) ⩽ c[φ(x) + φ(y)] for all x, y ⩾ 0, ϕ(⋅) is continuous at the point zero, 𝔽∈P

and

lim ‖ε𝔽(⋅)‖P = 0,

ε→0+

(211)

the function F(⋅) is strongly (ϕ, 𝔽, ℬ, 𝒫 )-almost periodic and the assumption (C2)’ holds, where: (C2)’ There exists a finite real constant e > 0 such that the assumptions f , g ∈ P and 0 ⩽ w ⩽ d ′ [f + g] for some finite real constant d ′ > 0 imply w ∈ P and ‖w‖P ⩽ e(1 + d ′ )[‖f ‖P + ‖g‖P ]. Then the function (h ∘ F)(⋅) is likewise strongly (ϕ, 𝔽, ℬ, 𝒫 )-almost periodic. Proof. In order to prove (i), observe first that if P : Λ × X → Y is (ω, ρ)-periodic ((ωj , ρj )j∈ℕn -periodic), then the function h ∘ P : Λ × X → Z is likewise (ω, ρ)-periodic ((ωj , ρj )j∈ℕn -periodic), since we have assumed that h∘ρ ⊆ ρ∘h (h∘ρj ⊆ ρj ∘h for 1 ⩽ j ⩽ n). Let L > 0 be the Lipschitzian constant of mapping h(⋅). Then the required statement simply follows from our assumptions on the function ϕ(⋅), the pseudometric space P and the subsequent computation (the set B ∈ ℬ is given in advance): 󵄩󵄩 󵄩 󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩(h ∘ PB )(⋅; x) − (h ∘ F(⋅; x))󵄩󵄩󵄩 )󵄩󵄩󵄩 k 󵄩󵄩 󵄩 󵄩Y 󵄩󵄩P 󵄩󵄩 󵄩 󵄩 󵄩󵄩 ⩽ 󵄩󵄩󵄩𝔽(⋅)ϕ(L󵄩󵄩󵄩PkB (⋅; x) − F(⋅; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 󵄩 󵄩P 󵄩󵄩 󵄩󵄩 B 󵄩󵄩 󵄩󵄩󵄩 󵄩 ⩽ 󵄩󵄩𝔽(⋅)φ(L)ϕ(󵄩󵄩Pk (⋅; x) − F(⋅; x)󵄩󵄩Y )󵄩󵄩 󵄩 󵄩P 󵄩󵄩 󵄩󵄩 B 󵄩 󵄩󵄩 󵄩 ⩽ d(1 + φ(L))󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩Pk (⋅; x) − F(⋅; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 . 󵄩 󵄩P To prove (ii), observe first that the function (h ∘ P)(⋅) is strongly almost periodic for every trigonometric polynomial P(⋅); hence, there exists a sequence (Pk ) of trigonometric polynomials such that limk→∞ ‖Pk − (h ∘ P)‖∞ = 0 [431]. After that, we can apply the argumentation used for proving (i), the additional assumptions given, and the next computation: 󵄩󵄩 󵄩 󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩Pk (⋅) − (h ∘ F)(⋅)󵄩󵄩󵄩 )󵄩󵄩󵄩 󵄩󵄩 󵄩 󵄩Y 󵄩󵄩P 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 ⩽ e(1 + c)[󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩Pk (⋅) − (h ∘ P)(⋅)󵄩󵄩󵄩Y )󵄩󵄩󵄩 + 󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩(h ∘ P)(⋅) − (h ∘ F)(⋅)󵄩󵄩󵄩Y )󵄩󵄩󵄩 ] 󵄩 󵄩P 󵄩 󵄩P 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 ⩽ e(1 + c)[󵄩󵄩󵄩𝔽(⋅)ϕ(ε′ )󵄩󵄩󵄩 + d(1 + φ(L))󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩P(⋅) − F(⋅)󵄩󵄩󵄩Y )󵄩󵄩󵄩 ]; 󵄩 󵄩P 󵄩 󵄩P

6.1 Metrical approximations: the main concept

�

321

here, we first choose a finite real number ε0 > 0 such that ‖x𝔽(⋅)‖P < ε/(2e(1 + c)) for 0 < x < ε0 (cf. (211)), and after that we choose a finite real number ε′ > 0 such that ϕ(ε′ ) < ε0 due to the continuity of function ϕ(⋅) at the point zero. Without going into further details, we will only note here that Proposition 6.1.3 can be reformulated for all other classes of functions analyzed in this section. Condition (C1)’ holds if P is a Banach space or P is a Fréchet space and d(⋅; ⋅) is a metric induced by the fundamental system of increasing seminorms which defines the topology of P; note also that condition (211) does not hold in general metric spaces P: for example, if d(⋅; ⋅) is a discrete unit metric on P, defined by d(f , g) := 0 if and only if f = g, and d(f , g) := 1, otherwise, then (211) does not hold. With this choice of metric space P, we have that a function F(⋅; ⋅) is strongly (x, 1, ℬ, 𝒫 )-almost periodic (semi-(x, ρ, 1, ℬ, 𝒫 )periodic, semi-(x, ρj , 1, ℬ, 𝒫 )j∈ℕn -periodic) if and only if F(⋅; ⋅) is a trigonometric polynomial (ρ-periodic function, (ρj )j∈ℕn -periodic function). We continue by providing several examples: Example 6.1.4. (i) The spaces of Bohr almost periodic functions (semi-(cj )j∈ℕn -periodic functions; cf. [439, Subsection 2.1]) F : ℝn → Y can be constricted if we use the pseudometric spaces 𝒫 such that 𝒫 is continuously embedded into the space Cb (ℝn : Y ). The obvious choice is the space Cb,ν (ℝn : Y ), where the function 1/ν(⋅) is locally bounded, and ν(x) ⩾ c, x ∈ ℝn for some positive real number c > 0. Concerning the choice of this space, we will present the following example (cf. also [590] and [431]). If 0 ≠ Ω ⊆ ℝn , then the space of all Gevrey functions of order s ⩾ 1, denoted by Gs (Ω), is defined as a collection of all infinitely differentiable functions F : ℝn → ℂ such that for each compact set K ⊆ ℝn there exists a finite real constant CK > 0 such that 󵄨󵄨 α 󵄨 1+|α| s 󵄨󵄨D F(t)󵄨󵄨󵄨 ⩽ CK α! for all t ∈ K and α ∈ ℕn0 . It is natural to ask whether an almost periodic function F : ℝn → ℂ, which belongs to the space Gs (Ω) obeys the property of the existence of a global real constant C > 0 such that 󵄨󵄨 α 󵄨 1+|α| s α! 󵄨󵄨D F(t)󵄨󵄨󵄨 ⩽ C for all t ∈ ℝn and α ∈ ℕn0 ? An instructive counterexample in the one-dimensional setting with s > 1, is given in [590, Example 2.1], showing that this is not true in general. We will reexamine this example for our purposes. Set gs (x) := exp(−x 1/(1−s) ), x > 0, gs (x) := 0, x ⩽ 0, ψs (x) := gs (x)gs (1 − x), x ∈ ℝ, ψs,n (x) := ψs (nx), x ∈ ℝ and φs,n (x) := ∑k∈ℤ ψs (x − 2n (2k + 1)), x ∈ ℝ (n ∈ ℕ). It has been shown that the function ∞

Fs (x) := ∑ k −1/4 φs,k (x), k=1

x∈ℝ

322 � 6 Metrical approximations of functions is well defined, as well as that the above series is uniformly convergent in the variable x ∈ ℝ, so that the function Fs (⋅) is actually, semi-periodic, since the function φs,n (⋅) is of period 2n+1 (n ∈ ℕ). Moreover, it has been shown that for each x ∈ ℝ and N ∈ ℕ we have the existence of an integer n0 > N such that ∑ n−1/4 φs,k (x) = n0−1/4 φs,n0 (x) ⩽ n0−1/4 ⩽ (1 + N)−1/4 .

k>N

(212)

Let (xn0 ) be any strictly increasing sequence tending to plus infinity and satisfying that xn0 ∈ ⋃k∈ℤ ([0, n0−1 ]+2n0 (1+2k)). Define the function ν : ℝ → (0, ∞) by ν(x) := 1, if x ∉ {xn0 : n0 ∈ ℕ} and ν(xn0 ) := n01/8 . The use of (212) implies that the function Fs (⋅) is semi-(x, I, 1, 𝒫 )-periodic with P = Cb,ν (ℝ : [0, ∞)). It is worth noting that we can also use the metric spaces 𝒫 equipped with the distance of the form d(f , g) = ‖f − g‖∞ + d1 (f , g),

f , g ∈ P,

(213)

where P is a certain subspace of the space Cb (ℝn : [0, ∞)) and d1 (⋅; ⋅) is a pseudometric on P. In the one-dimensional setting, this approach has been used in many research articles of S. Stoiński and his followers (see, e. g., [707, 708, 709, 710, 711, 712, 713]). Concerning this problematic, we will first present here an example of a p-semi-anti-periodic function in variation (1 ⩽ p < +∞) based on our analysis from [431, Example 4.2.9] and the investigations carried out in [708] and [713]. We already know that the function eix/(2m+1) , m2 m=1 ∞

f (x) := ∑

x∈ℝ

(214)

is not periodic and f (⋅) is semi-anti-periodic (n = 1, c1 = −1), because it is a uniform limit of [π ⋅ (2N + 1)!!]-anti-periodic functions N

eix/(2m+1) , m2 m=1

fN (x) := ∑

x ∈ ℝ (N ∈ ℕ);

cf. also [439, Example 2.11] for the multi-dimensional analogue of this example. Now we will prove that f (⋅) is p-semi-anti-periodic function in variation (1 ⩽ p < +∞), i. e., semi-(x, −I, 1, 𝒫 )-periodic with P being the subspace BVp (ℝ : [0, ∞)) of Cb (ℝ : [0, ∞)) consisting of those functions h(⋅) for which the p-variation of h(⋅) on the interval [t − 1, t + 1], defined by s−1

1/p

󵄨 󵄨p Vp (h; t) := sup(∑󵄨󵄨󵄨h(ui+1 ) − h(ui )󵄨󵄨󵄨 ) Φ

i=0

,

6.1 Metrical approximations: the main concept

�

323

is finite for any t ∈ ℝ (the supremum is taken over all finite partitions Φ = {u0 , . . . , us }, t − 1 = u0 < u1 < ⋅ ⋅ ⋅ < us = t + 1, of the interval [t − 1, t + 1]) and supt∈ℝ Vp (f ; t) < +∞. We equip the space BVp (ℝ : [0, ∞)) with the metric 󵄨 󵄨 d(f , g) := sup(󵄨󵄨󵄨f (t) − g(t)󵄨󵄨󵄨 + Vp (f − g; t)),

f , g ∈ BVp (ℝ : [0, ∞));

t∈ℝ

(215)

cf. also (213). Using the partial sums fN (⋅), it suffices to show that 󵄨󵄨 ∞ 󵄨 󵄨󵄨 ei⋅/(2m+1) 󵄨󵄨󵄨󵄨 lim sup Vp (󵄨󵄨󵄨 ∑ 󵄨; t) = 0. 󵄨󵄨m=N+1 m2 󵄨󵄨󵄨 N→+∞ t∈ℝ 󵄨 󵄨

(216)

Let t ∈ ℝ be fixed, and let Φ = {u0 , . . . , us }, t − 1 = u0 < u1 < ⋅ ⋅ ⋅ < us = t + 1, be any finite partition of the interval [t − 1, t + 1]. Then we have: s−1󵄨󵄨󵄨󵄨󵄨󵄨

e 󵄨󵄨 sup(∑󵄨󵄨󵄨󵄨󵄨󵄨 ∑ 󵄨󵄨󵄨󵄨m=N+1 Φ i=0󵄨󵄨 ∞

󵄨󵄨 ∞ 󵄨󵄨p 1/p eiui /(2m+1) 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 − 󵄨󵄨 ∑ 󵄨󵄨 ) 󵄨󵄨 󵄨󵄨m=N+1 m2 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨 󵄨

iui+1 /(2m+1) 󵄨󵄨󵄨

s−1󵄨󵄨󵄨

∞

s−1󵄨󵄨󵄨

∞

m2

e 󵄨 ⩽ sup(∑󵄨󵄨󵄨 ∑ 󵄨 Φ i=0󵄨󵄨m=N+1

󵄨p 1/p eiui /(2m+1) 󵄨󵄨󵄨󵄨 − ∑ 󵄨 ) m2 󵄨󵄨󵄨󵄨 m=N+1

iui+1 /(2m+1)

|e 󵄨 ⩽ sup(∑󵄨󵄨󵄨 ∑ 󵄨 Φ i=0󵄨󵄨m=N+1

∞

m2

p

p

s−1󵄨󵄨󵄨

󵄨 ∞ |ui+1 − ui | 󵄨󵄨󵄨󵄨 󵄨 ⩽ sup(∑󵄨󵄨󵄨 ∑ 󵄨 ) 󵄨 m2 (2m + 1) 󵄨󵄨󵄨󵄨 Φ i=0󵄨󵄨m=N+1 s−1

1/p

p

⩽ sup(∑ |ui+1 − ui | ) Φ

i=0

s−1

⩽ sup ∑ |ui+1 − ui | ∑

m=N+1

1

∞ m=N+1

m2 (2m

1

m2 (2m

1 → 0, 2 (2m + 1) m m=N+1

=2 ∑

1/p

∑

∞

Φ i=0 ∞

1/p

󵄨 − eiui /(2m+1) | 󵄨󵄨󵄨󵄨 󵄨󵄨 ) 󵄨󵄨 m2 󵄨

iui+1 /(2m+1)

+ 1)

+ 1)

N → +∞.

(ii) If we remove the term |f (t)−g(t)| in (215), then we obtain the concept of slow p-semiperiodicity in variation (1 ⩽ p < +∞); see also [711, pp. 563–564, Theorem 6]. For the sequel, let us recall that A. Haraux and P. Souplet have proved, in [358, Theorem 1.1], that the function 1 t sin2 ( m ), m 2 m=1 ∞

f (t) := ∑

t ∈ ℝ,

(217)

is not Besicovitch-p-almost periodic for any finite exponent p ⩾ 1, as well as that f (⋅) is uniformly recurrent and uniformly continuous; see [428] for the notion and

324 � 6 Metrical approximations of functions more details. Now we will prove that the function f (⋅) is slowly p-semi-periodic in variation (1 ⩽ p < +∞). In our approach, we consider the pseudometric space 𝒫 := (P, d1 ), where P = BVp (ℝ : [0, ∞)) and d1 (f , g) := supt∈ℝ Vp (f −g; t), f , g ∈ P; then we aim to show that the function f (⋅) is (x, 1, 𝒫 )-semi-periodic; but this simply follows from the equality 󵄨󵄨 ∞ 󵄨 󵄨󵄨 sin2 (⋅/2m ) 󵄨󵄨󵄨󵄨 lim sup Vp (󵄨󵄨󵄨 ∑ 󵄨󵄨; t) = 0, 󵄨󵄨m=N+1 󵄨󵄨 N→+∞ t∈ℝ m 󵄨 󵄨 which can be proved as in part (i), for the equality (216). (iii) The spaces of Bohr almost periodic functions (semi-(cj )j∈ℕn -periodic functions) F : ℝn → Y can be extended if we use the pseudometric spaces 𝒫 such that the space Cb (ℝn : Y ) is continuously embedded into 𝒫 . The obvious choice is the space Cb,ν (ℝn : Y ), where the function 1/ν(⋅) is locally bounded and ν(x) ⩽ c, x ∈ ℝn for some finite real number c > 0. But, here we can also use the metric space P consisting of all continuous functions from ℝn into [0, ∞), equipped with the distance 󵄨 󵄨 d(f , g) := sup 󵄨󵄨󵄨arctan(f (x)) − arctan(g(x))󵄨󵄨󵄨, n

f , g ∈ P.

x∈ℝ

Concerning the use of this metric space, we would like to note first that 𝒫 is not complete, since the sequence (fk (⋅) := tan((π/2) − [cos2 ⋅] − (1/k)))

k∈ℕ

of periodic functions is a Cauchy sequence in 𝒫 but not convergent. Denote by 𝒫 = (P, d) the completion of 𝒫 = (P, d). Then the limit function f (⋅) of sequence (fk (⋅)) has the form f (x) = tan((π/2)−[cos2 x]) = cot(cos2 x), x ∉ (π/2)+ℤπ and f (x) = +∞, if x ∈ (π/2) + ℤπ. Therefore, the function f (⋅) is not locally integrable, since f (x) ∼

1 1 ∼ , cos2 x ((π/2) + kπ − x)2

x → (π/2) + kπ (k ∈ ℤ),

which implies that we cannot expect that the function f (⋅) is Besicovitch (Weyl, Stepanov) almost periodic in the usual sense [431]. On the other hand, it is clear that we can extend the notion introduced in Definition 6.1.1 by replacing the limit equality (210) by 󵄩󵄩 󵄩 󵄩 󵄩󵄩 lim sup󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩PkB (⋅; x) − F(⋅; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 = 0. 󵄩P k→+∞ x∈B 󵄩

(218)

Then the function f (⋅) considered above will be (x, 1, 𝒫 )-semi-periodic, with the meaning clear. We will further analyze the approximation equality (218) in our

6.1 Metrical approximations: the main concept

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forthcoming research studies; this concept seems to be very important because it allows one to consider, maybe for the first time in existing literature, the generalized almost periodicity of functions f (⋅) which are not locally integrable. (iv) The use of a complete metric space P = C(ℝn ) is completely irrelevant in our analysis (the use of function spaces P = C k (ℝn ), where k ∈ ℕ, P = C ∞ (ℝn ) and some ultradistributional analogues of these spaces is much more important but we will not consider here this topic). More precisely, let P be equipped with the metric ∞

d(f , g) := ∑ 2−k k=1

supx∈[−k,k]n |f (x) − g(x)|

1 + supx∈[−k,k]n |f (x) − g(x)|

,

f , g ∈ P,

and let ϕ(x) ≡ x, 𝔽 ≡ 1. Then a continuous function F : ℝn → Y is (x, 1, 𝒫 )-almost periodic if and only if there exists a sequence (Pk ) of trigonometric polynomials which converges uniformly to F(⋅) on any compact subset of ℝn . Using the vectorvalued version of the Weierstrass approximation theorem, we can simply prove that any continuous function F : ℝn → Y is (x, 1, 𝒫 )-almost periodic; in actual fact, for every integer k ∈ ℕ, we can find a trigonometric polynomial Pk : ℝn → Y such that sup|t|⩽k ‖Pk (t) − F(t)‖Y ⩽ 1/k. Then (Pk ) converges uniformly to F(⋅) on any compact subset of ℝn . 6.1.1 Metrical normality and metrical Bohr type definitions We refer the reader to the survey article [42] by J. Andres, A. M. Bersani, R. F. Grande and the monograph [431] for more details about various classes of normal-type functions in the theory of almost periodic functions. We start this subsection by introducing the following notion, which generalizes the usual notion of Bohr normality (see, e. g. [42, Definition 2.6] for the one-dimensional setting): Definition 6.1.5. Suppose that R is any collection of sequences in Λ′′ , F : Λ × X → Y , ϕ : [0, ∞) → [0, ∞) and 𝔽 : Λ → (0, ∞). Then we say that the function F(⋅; ⋅) is (ϕ, R, ℬ, 𝔽, 𝒫 )normal, if and only if, for every set B ∈ ℬ and for every sequence (bk )k∈ℕ in R there exists a subsequence (bkm )m∈ℕ of (bk )k∈ℕ such that, for every ε > 0, there exists an integer m0 ∈ ℕ such that, for every integers m, m′ ⩾ m0 , we have 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 sup󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩F(⋅ + bkm ; x) − F(⋅ + bk ′ ; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 < ε. m 󵄩󵄩P 󵄩 x∈B 󵄩

(219)

The usual notion of (R, ℬ, 𝒫 )-normality is obtained by plugging 𝔽(⋅) ≡ 1, ϕ(x) ≡ x, R being a collection of sequences in Λ′′ , and P = L∞ (Λ); see also [446, Definition 2.1], where we have assumed that (219) holds for every set B′ of a collection L(B; b) of certain subsets of B. The notion introduced in [446, Definition 2.1, Definition 2.2] can be further strengthened by using the general functions 𝔽(⋅) and ϕ(x) therein.

326 � 6 Metrical approximations of functions The uniformly convergent sequences of almost periodic type functions and the behaviour of its limit function have been examined in many structural results established so far. In this subsection, we will clarify only one result concerning this issue: Proposition 6.1.6. Suppose that R is any collection of sequences in Λ′′ , Fj : Λ × X → Y , ϕ : [0, ∞) → [0, ∞), 𝔽 : Λ → (0, ∞) and the function Fj (⋅; ⋅) is (ϕ, R, ℬ, 𝔽, 𝒫 )-normal for all j ∈ ℕ. If F : Λ × X → Y and, for every set B ∈ ℬ and for every sequence (bk )k∈ℕ in R, we have 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 sup󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩Fj (⋅ + bj ; x) − F(⋅ + bk ; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 = 0, 󵄩󵄩P (j,k)→+∞ x∈B 󵄩 󵄩 lim

then the function F(⋅; ⋅) is likewise (ϕ, R, ℬ, 𝔽, 𝒫 )-normal, provided that: (i) The function ϕ(⋅) is monotonically increasing and there exists a finite real constant c > 0 such that ϕ(x + y) ⩽ c[ϕ(x) + ϕ(y)] for all x, y ⩾ 0. (ii) Condition (C3)’ holds, where: (C3)’ There exists a finite real constant f > 0 such that the assumptions f , g, h ∈ P and 0 ⩽ w ⩽ d ′ [f + g + h] for some finite real constant d ′ > 0 imply w ∈ P and ‖w‖P ⩽ f (1 + d ′ )[‖f ‖P + ‖g‖P + ‖h‖P ]. Proof. Without loss of generality, we may assume that c = f = 1. Let a real number ε > 0, a set B ∈ ℬ and a sequence (bk )k∈ℕ in R be given. Then there exists a natural number N ∈ ℕ such that the assumption min(j, k) ⩾ N implies 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 sup󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩Fj (⋅ + bj ; x) − F(⋅ + bk ; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 < ε/3. 󵄩󵄩P 󵄩 x∈B 󵄩 After that, we find a subsequence (bkm )m∈ℕ of (bk )k∈ℕ and a natural number m0 ∈ ℕ such that for every integers m, m′ ⩾ m0 , we have: 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 sup󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩FN (⋅ + bkm ; x) − FN (⋅ + bk ′ ; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 < ε/3. m 󵄩󵄩P 󵄩 x∈B 󵄩 Then the final conclusion simply follows from the last two estimates, conditions (i)–(ii) and the next decomposition (m, m′ ⩾ m0 ): 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 sup󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩F(⋅ + bkm ; x) − F(⋅ + bk ′ ; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 m 󵄩󵄩P 󵄩 x∈B 󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 ⩽ sup󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩F(⋅ + bkm ; x) − FN (⋅ + bkm ; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 󵄩󵄩P 󵄩 x∈B 󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 + sup󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩FN (⋅ + bkm ; x) − FN (⋅ + bk ′ ; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 m 󵄩󵄩P 󵄩 x∈B 󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 + sup󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩FN (⋅ + bk ′ ; x) − F(⋅ + bk ′ ; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 . m m 󵄩 󵄩󵄩P x∈B 󵄩

6.1 Metrical approximations: the main concept

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327

Remark 6.1.7. It is clear that condition (ii) from the formulations of Proposition 6.1.3 and Proposition 6.1.6 is a little bit inappropriate because it cannot detect a continuity or a measurability of function w(⋅). Despite of this, the proofs of Proposition 6.1.3, Proposition 6.1.6, and many other statements clarified below still work in concrete situations, so that our results are applicable if P is a solid Banach space, for example (let us recall that the space X consisting of continuous (Lebesgue measurable) functions is called solid if the assumptions f ∈ X and |g(x)| ⩽ |f (x)| a. e. for some continuous (Lebesgue measurable) function g implies g ∈ X and ‖g‖ ⩽ ‖f ‖). We need the following extension of [446, Definition 3.1], where we have 𝔽(⋅) ≡ 1 and ϕ(x) ≡ x: Definition 6.1.8. Suppose that 0 ≠ Λ′ ⊆ ℝn , 0 ≠ Λ ⊆ ℝn , F : Λ × X → Y is a given function, ρ is a binary relation on Y , 𝔽 : Λ → (0, ∞) and Λ′ ⊆ Λ′′ . Then we say that: (i) F(⋅; ⋅) is Bohr (ϕ, 𝔽, ℬ, Λ′ , ρ, 𝒫 )-almost periodic if and only if for every B ∈ ℬ and ε > 0 there exists l > 0 such that for each t0 ∈ Λ′ there exists τ ∈ B(t0 , l) ∩ Λ′ such that for every t ∈ Λ and x ∈ B, there exists an element yt;x ∈ ρ(F(t; x)) such that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 sup󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩F(⋅ + τ; x) − y⋅;x 󵄩󵄩󵄩Y )󵄩󵄩󵄩 ⩽ ε. 󵄩󵄩P 󵄩 x∈B 󵄩 (ii) F(⋅; ⋅) is (ϕ, 𝔽, ℬ, Λ′ , ρ, 𝒫 )-uniformly recurrent if and only if for every B ∈ ℬ there exists a sequence (τ k ) in Λ′ such that limk→+∞ |τ k | = +∞ and that for every t ∈ Λ and x ∈ B, there exists an element yt;x ∈ ρ(F(t; x)) such that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 lim sup󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩F(⋅ + τ k ; x) − y⋅;x 󵄩󵄩󵄩Y )󵄩󵄩󵄩 = 0. 󵄩󵄩P k→+∞ x∈B 󵄩 󵄩 For our purposes, the situation in which ρ = I will be the most important (the consideration of notion of strong (ϕ, ρ, 𝔽, ℬ, 𝒫 )-almost periodicity with a general binary relation ρ on Y is a bit misleading; cf. Definition 6.1.1). We continue by clarifying the following result: Proposition 6.1.9. (i) Suppose that R is any collection of sequences in Λ′′ , F : Λ×X → Y , ϕ : [0, ∞) → [0, ∞) and 𝔽 : Λ → (0, ∞). Let the following conditions hold: (a) The function ϕ(⋅) is monotonically increasing, continuous at the point zero, and there exists a finite real constant c > 0 such that ϕ(x + y) ⩽ c[ϕ(x) + ϕ(y)] for all x, y ⩾ 0. (b) Condition (C3)’ holds. (c) 𝔽(⋅)ϕ(‖P(⋅; x)‖Y ) ∈ P for any trigonometric polynomial (periodic function) P(⋅; ⋅) and x ∈ X. (d) There exists a finite real constant g > 0 such that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩P(⋅; x)󵄩󵄩󵄩 )󵄩󵄩󵄩 ⩽ g 󵄩󵄩󵄩ϕ(󵄩󵄩󵄩P(⋅; x)󵄩󵄩󵄩 )󵄩󵄩󵄩 , 󵄩󵄩 󵄩 󵄩Y 󵄩󵄩P 󵄩Y 󵄩󵄩∞ 󵄩󵄩 󵄩

328 � 6 Metrical approximations of functions for any trigonometric polynomial (periodic function) P(⋅; ⋅) and x ∈ X. (e) There exists a finite real constant h > 0 such that for every x ∈ X and τ ∈ Λ′′ , the assumption 𝔽(⋅)ϕ(‖H(⋅; x)‖Y ) ∈ P implies 𝔽(⋅)ϕ(‖H(⋅ + τ; x)‖Y ) ∈ P and 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩H(⋅ + τ; x)󵄩󵄩󵄩 )󵄩󵄩󵄩 ⩽ h󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩H(⋅; x)󵄩󵄩󵄩 )󵄩󵄩󵄩 . 󵄩󵄩 󵄩 󵄩Y 󵄩󵄩P 󵄩󵄩 󵄩 󵄩Y 󵄩󵄩P (f) Any set B of collection ℬ is bounded. 𝒫 If F ∈ e − (ℬ, ϕ, 𝔽) − B𝒫⋅ (Λ × X : Y ) [e − (ℬ, ϕ, 𝔽)j∈ℕn − BI ⋅ (Λ × X : Y )], then the function F(⋅; ⋅) is (ϕ, R, ℬ, ϕ, 𝔽, 𝒫 )-normal. (ii) Suppose that 0 ≠ Λ′ ⊆ ℝn , 0 ≠ Λ ⊆ ℝn , F : Λ × X → Y is a given function and 𝒫 Λ′ ⊆ Λ′′ . If F ∈ e − (ℬ, ϕ, 𝔽) − B𝒫⋅ (Λ × X : Y ) [e − (ℬ, ϕ, 𝔽)j∈ℕn − BI ⋅ (Λ × X : Y )] and the assumptions (a)–(f) given in the formulation of (i) hold, then the function F(⋅; ⋅) is Bohr (ϕ, 𝔽, ℬ, Λ′ , I, 𝒫 )-almost periodic. Proof. We will prove (i) for the function F ∈ e − (ℬ, ϕ, 𝔽) − B𝒫⋅ (Λ × X : Y ). Let a real number ε > 0, a set B ∈ ℬ, and a sequence (bk )k∈ℕ in R be fixed. Then we can find a trigonometric polynomial P(⋅; ⋅) such that 󵄩󵄩 󵄩 󵄩 󵄩󵄩 sup󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩P(⋅; x) − F(⋅; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 < ε/3. 󵄩 󵄩P x∈B After that we can use condition (e) in order to see that 󵄩󵄩 󵄩 󵄩 󵄩󵄩 sup󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩P(⋅ + τ; x) − F(⋅ + τ; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 < hε/3 󵄩 󵄩P x∈B

(220)

for all τ ∈ Λ′′ . Since B is bounded, the Bochner criterion for almost periodic functions in ℝn ensures, that there exist a subsequence (bkm )m∈ℕ of (bk )k∈ℕ and a natural number m0 ∈ ℕ such that, for every positive integers m′ , m′′ ⩾ m0 , we have 󵄩󵄩 󵄩󵄩 sup󵄩󵄩󵄩P(t + bk ′ ; x) − P(t + bk ′′ ; x)󵄩󵄩󵄩 < ε/3, m m 󵄩 󵄩Y x∈B

t ∈ ℝn .

Employing (c)–(d), we get: 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 sup󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩P(t + bk ′ ; x) − P(t + bk ′′ ; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 ⩽ gϕ(ε/3), m m 󵄩󵄩P 󵄩 x∈B 󵄩

m′ , m′′ ⩾ m0 .

Then the final conclusion simply follows from conditions (a)–(b), the estimate (220), and the next decomposition 󵄩󵄩 󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩F(⋅ + bkm′ ; x) − F(⋅ + bkm′′ ; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩P 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 ⩽ c′ [󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩F(⋅ + bk ′ ; x) − P(⋅ + bk ′ ; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 m m 󵄩󵄩 󵄩󵄩P

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�

329

󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 + 󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩P(⋅ + bk ′ ; x) − P(⋅ + bk ′′ ; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 m m 󵄩󵄩 󵄩󵄩P 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 + 󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩P(⋅ + bk ′′ ; x) − F(⋅ + bk ′′ ; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 ], m m 󵄩󵄩 󵄩󵄩P with a certain positive real constant c′ > 0. The proof of (ii) can be deduced similarly and therefore omitted. The conclusions established in [446, Example 3.9] can be simply formulated for Bohr (ϕ, 𝔽, ℬ, Λ′ , ρ, 𝒫 )-almost periodic functions and (ϕ, 𝔽, ℬ, Λ′ , ρ, 𝒫 )-uniformly recurrent functions. Furthermore, the statements of [446, Proposition 3.7, Corollary 3.8] can be extended in the following way: Proposition 6.1.10. Suppose that 0 ≠ Λ′ ⊆ ℝn , 0 ≠ Λ ⊆ ℝn , Λ + Λ′ ⊆ Λ and the function F : Λ×X → Y is Bohr (ϕ, 𝔽, ℬ, Λ′ , ρ, 𝒫 )-almost periodic ((ϕ, 𝔽, ℬ, Λ′ , ρ, 𝒫 )-uniformly recurrent), where ρ is a binary relation on Y satisfying R(F) ⊆ D(ρ) and ρ(y) is a singleton for any y ∈ R(F). Suppose that for each τ ∈ Λ′ we have τ + Λ = Λ, the function ϕ(⋅) is monotonically increasing, there exists a finite real constant c > 0 such that ϕ(x +y) ⩽ c[ϕ(x)+ϕ(y)] for all x, y ⩾ 0, and the following conditions hold: (P4) 𝒫1 = (P1 , d1 ) is a pseudometric space, c′ ∈ (0, ∞) and for every f ∈ P and τ ∈ Λ′ we have f (⋅ − τ) ∈ P1 and ‖f (⋅ − τ)‖P1 ⩽ c′ ‖f ‖P . (i) We have 𝔾 : Λ → (0, ∞) and 𝔾(⋅) ⩽ infτ∈Λ′ 𝔽(⋅ − τ). (ii) Condition (C2)’ holds. (iii) Condition (C0) holds, where: (C0) There exists a finite real constant e > 0 such that the assumptions g ∈ P1 and 0 ⩽ f ⩽ g imply ‖f ‖P1 ⩽ e‖g‖P1 . Then Λ+(Λ′ −Λ′ ) ⊆ Λ and the function F(⋅; ⋅) is Bohr (ϕ, 𝔾, ℬ, Λ′ −Λ′ , I, 𝒫1 )-almost periodic ((ϕ, 𝔾, ℬ, Λ′ − Λ′ , I, 𝒫1 )-uniformly recurrent). Proof. We will consider only Bohr (ϕ, 𝔽, ℬ, Λ′ , I, 𝒫 )-almost periodic functions. The inclusion Λ + (Λ′ − Λ′ ) ⊆ Λ can be simply verified. Let a real number ε > 0 and a set B ∈ ℬ be given. Then there exists l > 0 such that for each t10 , t20 ∈ Λ′ there exist two points τ 1 ∈ B(t10 , l) ∩ Λ′ and τ 2 ∈ B(t20 , l) ∩ Λ′ such that for every x ∈ B, we have 󵄩󵄩 󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩F(⋅ + τ 1 ; x) − ρ(F(⋅; x))󵄩󵄩󵄩Y )󵄩󵄩󵄩 ⩽ ε/2 󵄩󵄩 󵄩󵄩P and 󵄩󵄩 󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩F(⋅ + τ 2 ; x) − ρ(F(⋅; x))󵄩󵄩󵄩Y )󵄩󵄩󵄩 ⩽ ε/2. 󵄩󵄩 󵄩󵄩P Our assumptions on the function ϕ(⋅) and condition (ii) simply imply:

330 � 6 Metrical approximations of functions 󵄩󵄩 󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 ′ 󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩F(⋅ + τ 1 ; x) − F(⋅ + τ 2 ; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 ⩽ c ε, 󵄩󵄩 󵄩󵄩P

x ∈ B,

with a certain positive real constant c′ > 0. Using (P4) and translation for the vector −τ2 , we get 󵄩󵄩 󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 ′′ 󵄩󵄩𝔽(⋅ − τ2 )ϕ(󵄩󵄩󵄩F(⋅ + [τ2 − τ 1 ]; x) − F(⋅; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 ⩽ c ε, 󵄩󵄩 󵄩󵄩P1

x ∈ B,

with a certain positive real constant c′′ > 0. Clearly, τ2 − τ1 ∈ B(t20 − t10 , 2l) ∩ (Λ′ − Λ′ ); keeping in mind the assumptions (i) and (iii), the last inclusion simply implies the required.

6.1.2 Metrically semi-(cj , ℬ)j∈ℕn -periodic functions Let us recall that the notion of semi-periodicity (sometimes also called limit-periodicity) has been thoroughly analyzed by J. Andres and D. Pennequin in [44]. The main aim of this subsection is to continue our recent analysis of semi-(cj , ℬ)j∈ℕn -periodic functions carried out in [439, Subsection 2.1]; for simplicity, we will not consider general binary relations here, and we will assume that cj ∈ S1 for all j ∈ ℕn . We start by introducing the following metrical analogue of [439, Definition 2.9]: Definition 6.1.11. Suppose that 0 ≠ Λ ⊆ ℝn and F : Λ × X → Y . Then we say that F(⋅; ⋅) is semi-(ϕ, cj , 𝔽, ℬ, 𝒫 )j∈ℕn -periodic of type 1 if and only if for each B ∈ ℬ and ε > 0 there exist non-zero real numbers ωj such that ωj ej ∈ Λ′′ for all j ∈ ℕn and 󵄩󵄩 󵄩 󵄩 󵄩󵄩 sup󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩F(⋅ + mωj ej ; x) − cjm F(⋅; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 < ε, 󵄩 󵄩P x∈B

m ∈ ℕ, j ∈ ℕn .

(221)

In the metrical framework, the use of (221) with m ∈ ℕ is not satisfactorily enough if we use the (pseudo-)metric spaces 𝒫 different from l∞ (Λ). The following definition suggests the use of (221) with m ∈ ℤ and the region Λ = ℝn : Definition 6.1.12. Suppose that F : ℝn × X → Y . Then we say that F(⋅; ⋅) is semi(ϕ, cj , 𝔽, ℬ, 𝒫 )j∈ℕn -periodic of type 2 if and only if for each B ∈ ℬ and ε > 0 there exist non-zero real numbers ωj such that (221) holds for all m ∈ ℤ and j ∈ ℕn . In what follows, we investigate the relationship between the notions of semi(ϕ, cj , 𝔽, ℬ, 𝒫 )j∈ℕn -periodicity and the semi-(ϕ, cj , 𝔽, ℬ, 𝒫 )j∈ℕn -periodicity of type 1 (2); cf. also [439, Theorem 2.10]: Proposition 6.1.13. Suppose that condition (C3)’ holds, ϕ(0) = 0 and there exists a finite real constant c > 0 such that ϕ(x + y) ⩽ c[ϕ(x) + ϕ(y)] for all x, y ⩾ 0. If for every x ∈ X and τ ∈ Λ′′ , the assumption 𝔽(⋅)ϕ(‖G(⋅; x)‖Y ) ∈ P implies 𝔽(⋅)ϕ(‖G(⋅ + τ; x)‖Y ) ∈ P and

6.1 Metrical approximations: the main concept

󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩G(⋅ + τ; x)󵄩󵄩󵄩 )󵄩󵄩󵄩 ⩽ 󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩G(⋅; x)󵄩󵄩󵄩 )󵄩󵄩󵄩 , 󵄩󵄩 󵄩 󵄩Y 󵄩󵄩P 󵄩󵄩 󵄩 󵄩Y 󵄩󵄩P

�

331 (222)

and the function F : Λ × X → Y is semi-(ϕ, cj , 𝔽, ℬ, 𝒫 )j∈ℕn -periodic (Λ = ℝn ), then F(⋅; ⋅) is semi-(ϕ, cj , 𝔽, ℬ, 𝒫 )j∈ℕn -periodic of type 1 (type 2). Proof. We will consider the semi-(ϕ, cj , 𝔽, ℬ, 𝒫 )j∈ℕn -periodic functions of type 1, only. Let ε > 0 and B ∈ ℬ be fixed. Then there exist (cj )j∈ℕn -periodic function P(⋅; ⋅) and non-zero real numbers ωj such that ωj ej ∈ Λ′′ for all j ∈ ℕn and 󵄩󵄩 󵄩 󵄩 󵄩󵄩 sup󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩P(⋅; x) − F(⋅; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 < ε/2. 󵄩 󵄩P x∈B

(223)

Using the prescribed assumptions, we have the existence of a finite real constant c′ > 0 such that: 󵄩󵄩 󵄩 󵄩 󵄩󵄩 sup󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩F(⋅ + mωj ej ; x) − cjm F(⋅; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 󵄩 󵄩P x∈B 󵄩󵄩 󵄩 󵄩 󵄩󵄩 ⩽ c′ [sup󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩F(⋅ + mωj ej ; x) − P(⋅ + mωj ej ; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 󵄩 󵄩P x∈B 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 + sup󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩P(⋅ + mωj ej ; x) − cjm P(⋅; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 󵄩P x∈B 󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 + sup󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩cjm P(⋅; x) − cjm F(⋅; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 ] 󵄩 󵄩P x∈B 󵄩󵄩 󵄩 󵄩 󵄩󵄩 = c′ [sup󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩F(⋅ + mωj ej ; x) − P(⋅ + mωj ej ; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 󵄩P x∈B 󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 + sup󵄩󵄩󵄩𝔽(⋅)ϕ(󵄩󵄩󵄩P(⋅; x) − F(⋅; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 ], 󵄩P x∈B 󵄩

m ∈ ℕ, j ∈ ℕn .

Then the estimate (221) simply follows using the assumption (222) and the estimate (223). Remark 6.1.14. It is worth noting that in the usually considered situation, when ϕ(x) ≡ x and 𝔽(⋅) ≡ 1, the assumptions of Proposition 6.1.13 are satisfied for the p-semi-cperiodic functions F : ℝ → Y in variation (1 ⩽ p < +∞) since the metric d(⋅; ⋅) introduced in (215) is translation invariant, i. e., for every τ ∈ ℝ we have d(f , g) = d(f (⋅ + τ), g(⋅ + τ)), with the meaning clear. The converse of Proposition 6.1.13 cannot be so simply formulated in the metrical framework because the authors of [44] have used, in the proof of [44, Lemma 1], the linearizations of periodic functions which approximates F(⋅). Concerning this issue, we will only formulate the following research result without proof because it is very similar to the proof of [44, Theorem 1]:

332 � 6 Metrical approximations of functions Proposition 6.1.15. Suppose that P = Cb,ν (ℝn : [0, ∞)), the function F : ℝn → Y is continuous and semi-(ϕ, cj , 1, 𝒫 )j∈ℕn -periodic of type 2, where the function ν(⋅) is bounded from above, the function ϕ(⋅) is monotonically increasing and there exists a finite real constant c > 0 such that ϕ(x + y) ⩽ c[ϕ(x) + ϕ(y)] for all x, y ⩾ 0. Then the function F(⋅) is semi-(ϕ, cj , 1, 𝒫 )j∈ℕn -periodic.

6.2 Metrical approximations: Stepanov, Weyl, Besicovitch and Doss concepts In this section, we will consider the generalized Stepanov, Weyl, Besicovitch and, Doss metrical approximations by trigonometric polynomials and ρ-almost periodic type functions.

6.2.1 Stepanov and Weyl metrical approximations In this part, we will assume that condition (S2) holds, where: (S2) Let Ω be any compact subset of ℝn with positive Lebesgue measure such that Λ + Ω ⊆ Λ. We assume that PΩ ⊆ [0, ∞)Ω , the zero function belongs to PΩ , and 𝒫Ω = (PΩ , dΩ ) is a pseudometric space. Let P ⊆ [0, ∞)Λ , let the zero function belong to P, and let 𝒫 = (P, d) be a pseudometric space. By ⋅ and ⋅⋅ we denote the arguments from Ω and Λ, respectively. We will use a similar notation in the analysis of Weyl metrical approximations. We will first introduce the following notion: Definition 6.2.1. Suppose that (S2) holds, ϕ : [0, ∞) → [0, ∞) and 𝔽 : Ω × Λ → (0, ∞). Then we say that the function F(⋅; ⋅) belongs to the class e − (𝔽, ℬ) − S 𝒫Ω (Λ × X : Y ) 𝒫 𝒫 [e − (𝔽, ℬ) − Sρ Ω (Λ × X : Y ); e − (𝔽, ℬ)j∈ℕn − Sρj Ω (Λ × X : Y )] if and only if for every B ∈ ℬ and for every ε > 0 there exists a trigonometric polynomial [ρ-periodic function; (ρj )j∈ℕn -periodic function] P(⋅; ⋅) such that 󵄩󵄩󵄩 󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 sup󵄩󵄩󵄩󵄩󵄩󵄩𝔽(⋅, ⋅⋅)ϕ(󵄩󵄩󵄩P(⋅⋅ + ⋅; x) − F(⋅⋅ + ⋅; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 󵄩󵄩󵄩 < ε. 󵄩PΩ 󵄩󵄩P 󵄩󵄩 x∈B 󵄩 The most important subclass of the general class introduced above is obtained by plugging 𝔽(⋅, ⋅⋅) ≡ 1, P = L∞ (Λ) and PΩ = Lp (Ω), where 1 ⩽ p < ∞. Besides the notion introduced in Definition 6.2.1, we will consider the following ones: Definition 6.2.2. Suppose that (S) holds, ϕ : [0, ∞) → [0, ∞), 𝔽 : Ω × Λ → (0, ∞) and F : Λ × X → Y.

6.2 Metrical approximations: Stepanov, Weyl, Besicovitch and Doss concepts

�

333

(i) Suppose that R is any collection of sequences in Λ′′ . Then we say that the function F(⋅; ⋅) is Stepanov (ϕ, R, ℬ, ϕ, 𝔽, 𝒫 )-normal, if and only if, for every set B ∈ ℬ and for every sequence (bk )k∈ℕ in R there exists a subsequence (bkm )m∈ℕ of (bk )k∈ℕ such that, for every ε > 0, there exists an integer m0 ∈ ℕ such that for every integers m, m′ ⩾ m0 , we have 󵄩󵄩󵄩 󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 sup󵄩󵄩󵄩󵄩󵄩󵄩𝔽(⋅, ⋅⋅)ϕ(󵄩󵄩󵄩F(⋅ + ⋅⋅ + bkm ; x) − F(⋅ + ⋅⋅ + bk ′ ; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 󵄩󵄩󵄩 < ε. m 󵄩PΩ 󵄩󵄩P 󵄩󵄩 x∈B 󵄩 (ii) By SΩ,Λ′ ,ℬ Ω (Λ × X : Y ) we denote the set consisting of all functions F : Λ × X → Y such that for every ε > 0 and B ∈ ℬ, there exists a finite real number L > 0 such that for each t0 ∈ Λ′ there exists τ ∈ B(t0 , L) ∩ Λ′ such that for every x ∈ B, the mapping u 󳨃→ Gx (u) ∈ ρ(F(u; x)), u ∈ Ω + Λ is well defined, and (ϕ,𝔽,ρ,𝒫 ,𝒫)

󵄩󵄩󵄩 󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 sup󵄩󵄩󵄩󵄩󵄩󵄩𝔽(⋅; ⋅⋅)ϕ(󵄩󵄩󵄩F(τ + ⋅ + ⋅⋅; x) − Gx (⋅ + ⋅⋅)󵄩󵄩󵄩Y )󵄩󵄩󵄩 󵄩󵄩󵄩 < ε. 󵄩PΩ 󵄩󵄩P 󵄩󵄩 x∈B 󵄩 The notion introduced in Definition 6.2.1 is stronger than the notion introduced in Definition 6.2.2; the interested reader may simply clarify some sufficient conditions un𝒫 der which a function F ∈ e − (𝔽, ℬ) − S 𝒫Ω (Λ × X : Y ) [F ∈ e − (𝔽, ℬ)j∈ℕn − SI Ω (Λ × X : Y )] is

Stepanov (ϕ, R, ℬ, ϕ, 𝔽, 𝒫Ω , 𝒫 )-normal or belongs to the class SΩ,Λ′ ,ℬ Ω (Λ×X : Y ). In the usual context, we have already discussed this question in Proposition 6.1.9 (a uniformly recurrent analogue of the notion introduced in Definition 6.2.2(ii) can be also analyzed). In Definition 6.2.2(ii), we do not explicitly use the notion of multi-dimensional Bochner transform [431]. In the usual setting, this approach is commonly used and it is a very special case of the general approach obeyed for the introduction of function spaces in Definition 6.2.1 and Definition 6.2.2. We have the following: (ϕ,𝔽,ρ,𝒫 ,𝒫)

p

Example 6.2.3. Suppose that 1 ⩽ p < ∞, c ∈ S1 and f ∈ Lloc (ℝ : Y ). The one-dimensional Bochner transform f ̂ : ℝ → Lp ([0, 1] : Y ) is defined by [f ̂(t)](s) := f (t +s), t ∈ ℝ, s ∈ [0, 1]; a function f (⋅) is said to be Stepanov-(p, c)-semi-periodic if and only if the function f ̂(⋅) is semi-c-periodic [428]. Suppose now that Y = ℝ, the function f (⋅) is semi-c-periodic and can be analytically extended to a strip around the real axis. Then the function sign(f (⋅)) is Stepanov-(p, c)semi-periodic. In actual fact, there exists a sequence (fk ) of c-periodic functions which converges uniformly to the function f (⋅) on the real line. Then fk̂ : ℝ → Lp ([0, 1] : ℝ) is c-periodic and the required conclusion simply follows if we prove that for every ε > 0, there exists an integer k0 ∈ ℕ such that for every k ⩾ k0 , we have t+1

󵄨 󵄨p ∫ 󵄨󵄨󵄨sign(fk (s)) − sign(f (s))󵄨󵄨󵄨 ds ⩽ ε, t

t ∈ ℝ.

(224)

334 � 6 Metrical approximations of functions By the proof of [495, Theorem 5.3.1], we have that there exists a sufficiently small real number ε0 > 0 such that m({x ∈ [t, t + 1] : |f (x)| ⩽ ε0 }) ⩽ 2−p ε for all t ∈ ℝ. Let k0 ∈ ℕ be such that |fk (t) − f (t)| ⩽ ε0 /2 for all k ⩾ k0 and t ∈ ℝ. If |f (s)| ⩾ ε0 , then we have |fk (s)| ⩾ ε0 /2 and sign(fk (s)) = sign(f (s)) for all k ⩾ k0 (s ∈ ℝ). Therefore, t+1

󵄨 󵄨p ∫ 󵄨󵄨󵄨sign(fk (s)) − sign(f (s))󵄨󵄨󵄨 ds t

=

󵄨󵄨 󵄨p 󵄨󵄨sign(fk (s)) − sign(f (s))󵄨󵄨󵄨 ds

∫ {x∈[t,t+1]:|f (x)|⩽ε0 }

󵄨 󵄨 ⩽ 2p m({x ∈ [t, t + 1] : 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 ⩽ ε0 }) ⩽ ε, so that (224) holds. We continue our exposition with the following example: Example 6.2.4. Suppose that 1 ⩽ p < ∞. It is well known that the function f (t) := sin(

1 ), 2 + cos t + cos(√2t)

t∈ℝ

is Stepanov-p-almost periodic (S p -almost periodic); see [428]. Now we will prove that the function f (⋅) is not Lipschitz S p -almost periodic, i. e., that the Bochner transform f ̂ : ℝ → Lp ([0, 1] : ℂ) is not Lipschitz almost periodic (see [710] for the notion and more details). Suppose the contrary; then for each ε > 0 there exists a relatively dense set R ⊆ ℝ such that for every τ ∈ R and t ∈ ℝ, we have p 1/p 1 1 1 1 󵄨󵄨 [sin ζ (s+x) − sin ζ (s+y) ] − [sin ζ (s+x+τ) − sin ζ (s+y+τ) ] 󵄨󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ds) ⩽ ε, lim sup (∫󵄨󵄨 󵄨󵄨 󵄨󵄨 δ→0+ x =y;x,y∈[t−δ,t+δ] x−y 󵄨󵄨 ̸ 󵄨 0 (225) 1󵄨

where we have put ζ (t) := 2 + cos t + cos(√2t), t ∈ ℝ. If τ ∈ R and t ∈ ℝ, then we can insert x = t and y = t + δ in (225) in order to see that p 1/p 1 1 1 1 󵄨󵄨 [sin ζ (s+t) − sin ζ (s+t+δ) ] − [sin ζ (s+t+τ) − sin ζ (s+t+δ+τ) ] 󵄨󵄨󵄨󵄨 󵄨󵄨 󵄨 lim (∫󵄨󵄨 󵄨󵄨 ds) ⩽ ε. 󵄨󵄨 󵄨󵄨 δ→0+ δ 󵄨 0󵄨 1󵄨

Applying the Lagrange mean value theorem, we have lim

δ→0+

1 1 sin ζ (s+t) − sin ζ (s+t+δ)

δ

=−

ζ ′ (s + t) 1 cos , ζ (s + t) ζ 2 (s + t)

s ∈ [0, 1], t ∈ ℝ.

(226)

6.2 Metrical approximations: Stepanov, Weyl, Besicovitch and Doss concepts

335

�

Keeping this equality and (226) in mind, the dominated convergence theorem implies that 1/p 1󵄨 󵄨p 󵄨󵄨 ζ ′ (s + t + τ) 1 ζ ′ (s + t) 1 󵄨󵄨󵄨󵄨 󵄨 sup(∫󵄨󵄨󵄨 2 cos − 2 cos ds) ⩽ ε, 󵄨 󵄨󵄨 ζ (s + t + τ) ζ (s + t + τ) ζ (s + t) ζ (s + t) 󵄨󵄨󵄨󵄨 t∈ℝ 󵄨 0

which implies that the function g(t) :=

ζ ′ (t) 1 sin t + √2 sin(√2t) 1 cos = cos , ζ (t) (2 + cos t + cos(√2t))2 ζ 2 (t) 2 + cos t + cos(√2t)

t∈ℝ

is S p -almost periodic. This cannot be true because the function g(⋅) is not Stepanov (p-)bounded, i. e. t+2π

󵄨 󵄨 sup ∫ 󵄨󵄨󵄨g(s)󵄨󵄨󵄨 ds = +∞. t∈ℝ

t

Let us prove this. Applying the substitutions v = ζ (t) and u = 1/v after that, it readily follows that for each t ∈ ℝ we have: 1 2+cos t+cos(√2t)

t+2π

󵄨󵄨 󵄨󵄨 󵄨 󵄨 ∫ 󵄨󵄨󵄨g(s)󵄨󵄨󵄨 ds = 󵄨󵄨󵄨 󵄨󵄨 󵄨 t

∫ 1 2+cos(t+2π)+cos(√2(t+2π))

󵄨󵄨 󵄨󵄨 | cos u| du󵄨󵄨󵄨. 󵄨󵄨 󵄨

Therefore, there exist two finite real constants c > 0 and c1 > 0 such that t+2π

󵄨 󵄨 ∫ 󵄨󵄨󵄨g(s)󵄨󵄨󵄨 ds ⩾ c⌊ t

|

1 2+cos t+cos(√2t)

−

1 | 2+cos(t+2π)+cos(√2(t+2π))

π/2

⌋

= c⌊

4| sin(√2π)| ⋅ | sin(√2(t + π))| ⌋ π ⋅ [2 + cos t + cos(√2t)] ⋅ [2 + cos(t + 2π) + cos(√2(t + 2π))]

⩾ c⌊

| sin(√2π)| ⋅ | sin(√2(t + π))| | sin(√2(t + π))| , ⌋ ⩾ c1 π ⋅ [2 + cos t + cos(√2t)] 2 + cos t + cos(√2t)

t → +∞. (227)

We proceed by using some arguments given in the remarkable paper [576] by A. Nawrocki. Let Q2m be the odd natural number from the proof of [576, Theorem 4]; then we have limm→+∞ Q2m = +∞ and x2 1 ⩾ 2 m4 , π 2 + cos xm + cos(√2xm )

m ∈ ℕ,

(228)

336 � 6 Metrical approximations of functions where we have put xm := Q2m π (m ∈ ℕ). Suppose that √2(Q2m + 1)π = km π + am , where km ∈ ℕ and |am | ⩽ π/2 (m ∈ ℕ). Since 󵄨󵄨 km 󵄨󵄨󵄨󵄨 |am | 󵄨󵄨√ , 󵄨󵄨 2 − 󵄨󵄨 = 󵄨󵄨 󵄨 Q2m + 1 󵄨 (Q2m + 1)π

m ∈ ℕ,

the Liouville theorem (see, e. g. [576, Theorem 1]) implies the existence of a finite real number d > 0 such that |am | ⩾ dπ/(Q2m + 1) for all m ∈ ℕ. Then we can apply the Jensen inequality in order to see that 2d 2d 󵄨󵄨 󵄨 2 = , 󵄨󵄨sin(am )󵄨󵄨󵄨 ⩾ |am | ⩾ π 1 + Q2m 1 + (xm /π)

m ∈ ℕ.

(229)

Keeping in mind (228)–(229), we get lim sup m→+∞

| sin(√2(xm + π))| = +∞, 2 + cos xm + cos(√2xm )

contradicting (227). Finally, we would like to ask whether the function f (⋅) is S p -almost periodic in variation, i. e., whether the function f ̂(⋅) is almost periodic in variation. It is well known that any uniformly continuous Stepanov-p-almost periodic function F : ℝn → Y is almost periodic [431]. The interested reader may try to prove a metrical analogue of this result. Before proceeding to the study of Weyl metrical approximations, we will state and prove the following simple result: Proposition 6.2.5. Suppose that F : Λ × X → ℂ, c ∈ ℂ ∖ {0} and the following conditions hold: (i) The function ϕ(⋅) is monotonically increasing and there exists a function φ : [0, ∞) → [0, ∞) such that ϕ(xy) ⩽ ϕ(x)φ(y) for all x, y ⩾ 0. (ii) For every set B ∈ ℬ, there exists a finite real constant cB > 0 such that |F(t; x)| ⩾ cB , t ∈ Λ, x ∈ B. (iii) The assumptions 0 ⩽ f ⩽ g and g ∈ PΩ (g ∈ P) imply f ∈ PΩ (f ∈ P). (iv) Condition (C1)’ holds for PΩ and P. Then we have the following: (a) If F(⋅; ⋅) is Stepanov (ϕ, R, ℬ, ϕ, 𝔽, 𝒫 )-normal, then the function (1/F)(⋅; ⋅) is Stepanov (ϕ, R, ℬ, ϕ, 𝔽, 𝒫 )-normal. (b) If F ∈ SΩ,Λ′ ,ℬ

(ϕ,𝔽,cI,𝒫Ω ,𝒫)

(ϕ,𝔽,c−1 I,𝒫Ω ,𝒫)

(Λ × X : Y ), then (1/F) ∈ SΩ,Λ′ ,ℬ

(Λ × X : Y ).

Proof. We will prove only (a). Due to (ii), we have |F(t; x)| ≠ 0, t ∈ Λ, x ∈ X. Let a set B ∈ ℬ and a sequence (bk )k∈ℕ in R be given. Then we know that there exists a subsequence (bkm )m∈ℕ of (bk )k∈ℕ such that, for every ε > 0, there exists an integer m0 ∈ ℕ such

6.2 Metrical approximations: Stepanov, Weyl, Besicovitch and Doss concepts

�

337

that, for every integers m, m′ ⩾ m0 , we have (230). Denote by dΩ > 0 and d > 0 the corresponding constants from condition (C1)’ for the spaces PΩ and P, respectively. Then we have: 󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩󵄩 󵄨 󵄨 󵄩󵄩 󵄩󵄩 sup󵄩󵄩󵄩󵄩󵄩󵄩𝔽(⋅, ⋅⋅)ϕ(󵄨󵄨󵄨(1/F)(⋅ + ⋅⋅ + bkm ; x) − (1/F)(⋅ + ⋅⋅ + bk ′ ; x)󵄨󵄨󵄨)󵄩󵄩󵄩 󵄩󵄩󵄩 m 󵄩PΩ 󵄩󵄩 󵄩󵄩󵄩 x∈B 󵄩 󵄩P 󵄩󵄩󵄩󵄩 󵄩 󵄩󵄩 󵄩 |F(⋅ + ⋅⋅ + b ; x) − F(⋅ + ⋅⋅ + b ; x)| 󵄩󵄩 󵄩󵄩 km km ′ 󵄩󵄩󵄩󵄩 = sup󵄩󵄩󵄩󵄩󵄩󵄩𝔽(⋅, ⋅⋅)ϕ( )󵄩󵄩󵄩 󵄩󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 |F(⋅ + ⋅⋅ + b ; x) ⋅ F(⋅ + ⋅⋅ + b ; x)| x∈B 󵄩 km k ′ 󵄩󵄩󵄩 󵄩P 󵄩P m

Ω

󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩󵄩 󵄨 󵄨 󵄩󵄩 󵄩󵄩 ⩽ sup󵄩󵄩󵄩󵄩󵄩󵄩𝔽(⋅, ⋅⋅)φ(cB−2 )ϕ(󵄨󵄨󵄨F(⋅ + ⋅⋅ + bkm ; x) − F(⋅ + ⋅⋅ + bk ′ ; x)󵄨󵄨󵄨)󵄩󵄩󵄩 󵄩󵄩󵄩 m 󵄩PΩ 󵄩󵄩 󵄩󵄩󵄩 x∈B 󵄩 󵄩P 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄨 󵄨 󵄩󵄩 󵄩󵄩 ⩽ sup󵄩󵄩󵄩dΩ (1 + φ(cB−2 ))󵄩󵄩󵄩𝔽(⋅, ⋅⋅)ϕ(󵄨󵄨󵄨F(⋅ + ⋅⋅ + bkm ; x) − F(⋅ + ⋅⋅ + bk ′ ; x)󵄨󵄨󵄨)󵄩󵄩󵄩 󵄩󵄩󵄩 m 󵄩 󵄩 󵄩PΩ 󵄩󵄩 x∈B 󵄩 󵄩 󵄩P 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄨 󵄨 󵄩󵄩 󵄩󵄩 ⩽ d(1 + dΩ (1 + φ(cB−2 ))) sup󵄩󵄩󵄩󵄩󵄩󵄩𝔽(⋅, ⋅⋅)ϕ(󵄨󵄨󵄨F(⋅ + ⋅⋅ + bkm ; x) − F(⋅ + ⋅⋅ + bk ′ ; x)󵄨󵄨󵄨)󵄩󵄩󵄩 󵄩󵄩󵄩 , m 󵄩PΩ 󵄩󵄩 󵄩󵄩󵄩 x∈B 󵄩 󵄩P

which simply implies the required. 2. Weyl metrical approximations. In this part, we will assume that condition (L) holds, where: (L) Let Ω be any compact subset of ℝn with positive Lebesgue measure such that Λ+Ω ⊆ Λ. We assume that for each real number l > 0 we have that Pl ⊆ [0, ∞)lΩ , the zero function belongs to Pl , and 𝒫l = (Pl , dl ) is a pseudometric space. Let P ⊆ [0, ∞)Λ , let the zero function belong to P, and let 𝒫 = (P, d) be a pseudometric space. Now we are ready to introduce the following notion: Definition 6.2.6. Suppose that (L) holds, ϕ : [0, ∞) → [0, ∞) and 𝔽 : (0, ∞) × (∪l>0 lΩ) × Λ → (0, ∞). Then we say that the function F(⋅; ⋅) belongs to the class e−(𝔽, ϕ, ℬ)−W 𝒫⋅ (Λ× 𝒫 𝒫 X : Y ) [e − (𝔽, ϕ, ℬ) − Wρ ⋅ (Λ × X : Y ); e − (𝔽, ϕ, ℬ)j∈ℕn − Wρj ⋅ (Λ × X : Y )] if and only if for every B ∈ ℬ and for every ε > 0 there exist a real number l0 > 0 and a trigonometric polynomial [ρ-periodic function; (ρj )j∈ℕn -periodic function] P(⋅; ⋅) such that 󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 sup󵄩󵄩󵄩󵄩󵄩󵄩𝔽(l, ⋅, ⋅⋅)ϕ(󵄩󵄩󵄩P(⋅⋅ + ⋅; x) − F(⋅⋅ + ⋅; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 󵄩󵄩󵄩 < ε, 󵄩Pl 󵄩󵄩 󵄩󵄩󵄩 x∈B 󵄩 󵄩P

l ⩾ l0 .

The usual notion of class e − ℬ − W 𝒫⋅ (Λ × X : Y ), introduced and analyzed in [431, Subsection 6.3.1], is obtained by plugging 𝔽(l, ⋅, ⋅⋅) ≡ l−n/p , P = L∞ (Λ) and Pl = Lp (lΩ) for 𝒫 𝒫⋅ all l > 0. The classes e − (𝔽, ϕ, ℬ) − Wω,ρ⋅ (Λ × X : Y ) and e − (𝔽, ϕ, ℬ)j∈ℕn − Wωj ,ρ (Λ × X : j Y ) have not been considered elsewhere even in the one-dimensional setting, with this choice of metric spaces.

338 � 6 Metrical approximations of functions Besides the notion introduced in Definition 6.2.6, we will consider the following ones (cf. also Definition 6.2.2): Definition 6.2.7. Suppose that (L) holds, ϕ : [0, ∞) → [0, ∞) and 𝔽 : (0, ∞) × (∪l>0 lΩ) × Λ → (0, ∞). (i) Suppose that R is any collection of sequences in Λ′′ . Then we say that the function F(⋅; ⋅) is Weyl (ϕ, R, ℬ, ϕ, 𝔽, 𝒫 )-normal, if and only if for every set B ∈ ℬ and for every sequence (bk )k∈ℕ in R there exists a subsequence (bkm )m∈ℕ of (bk )k∈ℕ such that, for every ε > 0, there exists an integer m0 ∈ ℕ such that for every integers m, m′ ⩾ m0 , we have 󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 lim sup sup󵄩󵄩󵄩󵄩󵄩󵄩𝔽(l, ⋅, ⋅⋅)ϕ(󵄩󵄩󵄩F(⋅ + ⋅⋅ + bkm ; x) − F(⋅ + ⋅⋅ + bk ′ ; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 󵄩󵄩󵄩 < ε. (230) m 󵄩Pl 󵄩󵄩 󵄩󵄩󵄩 l→+∞ x∈B 󵄩 󵄩P (ii) By e−WΩ,Λ′ ,ℬ Ω (Λ×X : Y ) we denote the set consisting of all functions F : Λ×X → Y such that, for every ε > 0 and B ∈ ℬ, there exist two finite real numbers l > 0 and L > 0 such that for each t0 ∈ Λ′ there exists τ ∈ B(t0 , L) ∩ Λ′ such that for every x ∈ B, the mapping u 󳨃→ Gx (u) ∈ ρ(F(u; x)), u ∈ (∪l>0 lΩ) + Λ is well defined, and (ϕ,𝔽,ρ,𝒫 ,𝒫)

󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 sup󵄩󵄩󵄩󵄩󵄩󵄩𝔽(l, ⋅, ⋅⋅)ϕ(󵄩󵄩󵄩F(τ + ⋅ + ⋅⋅; x) − Gx (⋅ + ⋅⋅)󵄩󵄩󵄩Y )󵄩󵄩󵄩 󵄩󵄩󵄩 < ε. 󵄩Pl 󵄩󵄩 󵄩󵄩󵄩 x∈B 󵄩 󵄩P (iii) By WΩ,Λ′ ,ℬ Ω (Λ×X : Y ) we denote the set consisting of all functions F : Λ×X → Y such that, for every ε > 0 and B ∈ ℬ, there exists a finite real number L > 0 such that for each t0 ∈ Λ′ there exists τ ∈ B(t0 , L) ∩ Λ′ such that for every x ∈ B, the mapping u 󳨃→ Gx (u) ∈ ρ(F(u; x)), u ∈ (∪l>0 lΩ) + Λ is well defined, and (ϕ,𝔽,ρ,𝒫 ,𝒫)

󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 lim sup sup󵄩󵄩󵄩󵄩󵄩󵄩𝔽(l, ⋅, ⋅⋅)ϕ(󵄩󵄩󵄩F(τ + ⋅ + ⋅⋅; x) − Gx (⋅ + ⋅⋅)󵄩󵄩󵄩Y )󵄩󵄩󵄩 󵄩󵄩󵄩 < ε. 󵄩Pl 󵄩󵄩 󵄩󵄩󵄩 l→+∞ x∈B 󵄩 󵄩P The notion introduced in Definition 6.2.6 is stronger than the notion introduced in Definition 6.2.7; the interested reader may simply clarify some reasons justifying this (cf. also Proposition 6.1.9). As already emphasized for the Stepanov metrical approximations, a uniformly recurrent analogue of the notion introduced in Definition 6.2.7(ii)–(iii) can be also analyzed. Concerning the linear structure of introduced function spaces, we will only note the following: Suppose that the function ϕ(⋅) is monotonically increasing and there exists a finite real constant c > 0 such that ϕ(x + y) ⩽ c[ϕ(x) + ϕ(y)] for all x, y ⩾ 0. Then we have the following: (i) Equipped with the usual operations, the set of all strongly (ϕ, 𝔽, ℬ, 𝒫 )-almost periodic functions forms a vector space, provided that condition (C2)’ holds. (ii) Equipped with the usual operations, the set e − (𝔽, ℬ) − S 𝒫Ω (Λ × X : Y ) forms a vector space provided that condition (C2)’ holds and condition (C2)’ holds for the pseudometric space PΩ .

6.2 Metrical approximations: Stepanov, Weyl, Besicovitch and Doss concepts

�

339

(iii) Equipped with the usual operations, the set e − (𝔽, ϕ, ℬ) − W 𝒫⋅ (Λ × X : Y ) forms a vector space provided that condition (C2)’ holds and condition (C2)’ holds for the pseudometric space PlΩ , for any l > 0. The interested reader may try to reformulate the statements of [448, Proposition 3.5, Corollary 3.6] in our new context. Concerning the embeddings of Stepanov classes introduced in this paper into the equi-Weyl classes of functions introduced in this paper, we will clarify the following result only: Proposition 6.2.8. Suppose that Λ = ℝn and the following conditions hold: (i) P1 is a Banach space and condition (C0) holds for P1 . p (ii) PΩ = Lν (Ω : [0, ∞)) for some p ∈ [1, ∞) and a Lebesgue measurable function ν : Ω → (0, ∞); P = C0,w (Λ : [0, ∞)) for some function w : Λ → (0, ∞) such that the function 1/w(⋅) is locally bounded. (iii) There exists a real number l0 > 0 such that, for every l ⩾ l0 , we have 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 1 󵄩󵄩 ∑ 󵄩󵄩 ⩽ Const. 󵄩󵄩 󵄩 󵄩󵄩k∈(ℕ⌈l⌉−1 )n w(⋅⋅ + k) 󵄩󵄩󵄩P1 0 (iv) There exists a real number l0 > 0 such that, for every l ⩾ l0 , we have 󵄩󵄩 󵄩 󵄩󵄩𝔽1 (l, ⋅, ⋅⋅)W (⋅, ⋅⋅; x)󵄩󵄩󵄩P1 ⩽ l

∑

k∈(ℕ⌈l⌉−1 )n 0

󵄩󵄩 󵄩 󵄩󵄩𝔽(⋅, ⋅⋅ + k)W (⋅, ⋅⋅ + k; x)󵄩󵄩󵄩PΩ ,

provided that the above terms are well-defined. 1

If F ∈ e − (𝔽, ℬ) − S 𝒫Ω (Λ × X : Y ), then F ∈ e − (𝔽1 , ϕ, ℬ) − W 𝒫⋅ (Λ × X : Y ). Proof. The proof simply follows from the prescribed assumptions and the next computation (the meaning of inequality in the fourth line of computation is clear from the context): 󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 sup󵄩󵄩󵄩󵄩󵄩󵄩𝔽1 (l, ⋅, ⋅⋅)ϕ(󵄩󵄩󵄩P(⋅⋅ + ⋅; x) − F(⋅⋅ + ⋅; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 󵄩󵄩󵄩 󵄩Pl1 󵄩󵄩 1 󵄩󵄩󵄩 x∈B 󵄩 󵄩P 󵄩󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 ⩽ j sup󵄩󵄩󵄩󵄩󵄩󵄩 ∑ 󵄩󵄩󵄩𝔽(⋅, ⋅⋅ + k)ϕ(󵄩󵄩󵄩P(⋅⋅ + k + ⋅; x) − F(⋅⋅ + k + ⋅; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩󵄩 󵄩 󵄩󵄩PΩ 󵄩󵄩 1 x∈B 󵄩 󵄩 k∈(ℕ⌈l⌉−1 󵄩P )n 0

1/p 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄨󵄨 󵄨󵄨p p 󵄩 󵄩 󵄩 = j sup󵄩󵄩󵄩 ∑ (∫󵄨󵄨󵄨𝔽(u, ⋅⋅ + k)ϕ(󵄩󵄩󵄩P(⋅⋅ + k + u; x) − F(⋅⋅ + k + u; x)󵄩󵄩󵄩Y )󵄨󵄨󵄨 ν (u)du) 󵄩󵄩󵄩 󵄨 󵄩󵄩 1 󵄩󵄩k∈(ℕ⌈l⌉−1 )n 󵄨 x∈B 󵄩 󵄩P Ω 0 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 ε 󵄩󵄩 ⩽ j ⋅ ε ⋅ Const. ⩽ j sup󵄩󵄩󵄩 ∑ 󵄩󵄩 󵄩 w(⋅⋅ + k) 󵄩󵄩P1 x∈B 󵄩 n 󵄩k∈(ℕ⌈l⌉−1 ) 0

340 � 6 Metrical approximations of functions 6.2.2 Besicovitch and Doss metrical approximations In this subsection, we will assume that condition (BD) holds, where: (BD) For each real number t > 0 we have Pt ⊆ [0, ∞)Λt , the zero function belongs to Pt , and 𝒫t = (Pt , dt ) is a pseudometric space. We are ready to introduce the following notion: Definition 6.2.9. Suppose that F : Λ × X → Y , ϕ : [0, ∞) → [0, ∞) and 𝔽 : (0, ∞) × Λ → (0, ∞). Then we say that the function F(⋅; ⋅) belongs to the class e−(ℬ, ϕ, 𝔽)−B𝒫⋅ (Λ×X : Y ) 𝒫 𝒫 [e − (ℬ, ϕ, 𝔽) − Bρ ⋅ (Λ × X : Y ); e − (ℬ, ϕ, 𝔽)j∈ℕn − Bρj ⋅ (Λ × X : Y )] if and only if for each set B ∈ ℬ there exists a sequence (Pk (⋅; ⋅)) of trigonometric polynomials [ρ-periodic functions; (ρj )j∈ℕn -periodic functions] such that 󵄩󵄩 󵄩 󵄩 󵄩󵄩 lim lim sup sup󵄩󵄩󵄩𝔽(t, ⋅)ϕ(󵄩󵄩󵄩F(⋅; x) − Pk (⋅; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 = 0. 󵄩Pt k→+∞ t→+∞ x∈B 󵄩

(231)

If ϕ(x) ≡ x, then we omit the term “ϕ” from the notation; if X = {0}, then we omit the term “ℬ” from the notation. Immediately from definition, it follows that for every F ∈ e−(ℬ, ϕ, 𝔽)−B𝒫⋅ (Λ×X : Y ) and λ ∈ ℝn , we have ei⟨λ,⋅⟩ F ∈ e − (ℬ, ϕ, 𝔽) − B𝒫⋅ (Λ × X : Y ); this also holds for the corresponding classes introduced in Definition 6.1.1, Definition 6.2.1 and Definition 6.2.6. The class e − (ℬ, ϕ, 𝔽) − Bp(⋅) (Λ × X : Y ) considered in [450] (although some results established in this subsection and the subsequent section generalize the corresponding results from Section 3.3, we will refer to the easily accessible article [450] here) is nothing else but the class e − (ℬ, ϕ, 𝔽) − B𝒫⋅ (Λ × X : Y ) with 𝔽(t, ⋅) ≡ 𝔽(t) and Pt = Lp(⋅) (Λt ) for all t > 0. The 𝒫 𝒫 classes e − (ℬ, ϕ, 𝔽) − Bω,ρ⋅ (Λ × X : Y ) and e − (ℬ, ϕ, 𝔽)j∈ℕn − Bωj⋅,ρj (Λ × X : Y ) have not been considered elsewhere even in the one-dimensional setting, with this choice of metric spaces. We continue this part by extending the notion of Besicovitch-(R, ℬ, ϕ, 𝔽) − Bp(⋅) normality: Definition 6.2.10. Suppose that R is any collection of sequences in Λ′′ , F : Λ × X → Y , ϕ : [0, ∞) → [0, ∞) and 𝔽 : (0, ∞) → (0, ∞). Then we say that the function F(⋅; ⋅) is Besicovitch (ϕ, R, ℬ, ϕ, 𝔽, 𝒫 )-normal, if and only if for every set B ∈ ℬ and for every sequence (bk )k∈ℕ in R there exists a subsequence (bkm )m∈ℕ of (bk )k∈ℕ such that, for every ε > 0, there exists an integer m0 ∈ ℕ such that for every integers m, m′ ⩾ m0 , we have 󵄩󵄩 󵄩 󵄩 󵄩󵄩 lim sup sup󵄩󵄩󵄩𝔽(t, ⋅)ϕ(󵄩󵄩󵄩F(⋅ + bkm ; x) − F(⋅ + bk ′ ; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 < ε. m 󵄩 󵄩Pt t→+∞ x∈B The interested reader may try to clarify some assumptions under which any func𝒫 tion e − (𝔽, ϕ, ℬ) − W 𝒫⋅ (Λ × X : Y ) [e − (𝔽, ϕ, ℬ)j∈ℕn − WI ⋅ (Λ × X : Y )] is Besicovitch

6.2 Metrical approximations: Stepanov, Weyl, Besicovitch and Doss concepts

�

341

(ϕ, R, ℬ, ϕ, 𝔽, 𝒫 )-normal; as is well known, the converse statement does not hold (cf. [450, Proposition 2.12, Example 2.15] and Proposition 6.1.9). A metrical analogue of [450, Proposition 2.13] can be formulated without any substantial difficulties; for simplicity, we will not consider here any Bohr analogue of the notion introduced in Definition 6.2.9 and Definition 6.2.10. The following notion generalizes the notion introduced in Definition 3.2.1: Definition 6.2.11. Let ϕ : [0, ∞) → [0, ∞) and 𝔽 : (0, ∞) × Λ → (0, ∞). (i) Suppose that the function F : Λ × X → Y satisfies 𝔽(t; ⋅)ϕ(‖F(⋅; x)‖Y ) ∈ Pt for all t > 0 and x ∈ X. Then we say that the function F(⋅; ⋅) is Besicovitch-(𝒫 , ϕ, 𝔽, ℬ)-bounded if and only if, for every B ∈ ℬ, there exists a finite real number MB > 0 such that 󵄩󵄩 󵄩 󵄩 󵄩󵄩 lim sup sup󵄩󵄩󵄩𝔽(t; ⋅)ϕ(󵄩󵄩󵄩F(⋅; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 ⩽ MB . 󵄩 Pt t→+∞ x∈B 󵄩 (ii) Suppose that the function F : Λ×X → Y satisfies that 𝔽(t; ⋅)ϕ(‖F(⋅+τ; x)−y⋅;x ‖Y ) ∈ Pt for all t > 0, x ∈ X, τ ∈ Λ′ and y⋅;x ∈ ρ(F(⋅; x)). (a) We say that the function F : Λ × X → Y is Besicovitch-(𝒫 , ϕ, 𝔽, ℬ, Λ′ , ρ)continuous if and only if for every B ∈ ℬ as well as for every t > 0, x ∈ B and ⋅ ∈ Λt , we have the existence of an element y⋅;x ∈ ρ(F(⋅; x)) such that lim

τ→0,τ∈Λ′

󵄩󵄩 󵄩 󵄩 󵄩󵄩 lim sup sup󵄩󵄩󵄩𝔽(t; ⋅)ϕ(󵄩󵄩󵄩F(⋅ + τ; x) − y⋅;x 󵄩󵄩󵄩Y )󵄩󵄩󵄩 = 0. 󵄩Pt t→+∞ x∈B 󵄩

(b) We say that the function F(⋅; ⋅) is Doss-(𝒫 , ϕ, 𝔽, ℬ, Λ′ , ρ)-almost periodic if and only if, for every B ∈ ℬ and ε > 0, there exists l > 0 such that for each t0 ∈ Λ′ there exists a point τ ∈ B(t0 , l) ∩ Λ′ such that, for every t > 0, x ∈ B and ⋅ ∈ Λt , we have the existence of an element y⋅;x ∈ ρ(F(⋅; x)) such that 󵄩󵄩 󵄩 󵄩 󵄩󵄩 lim sup sup󵄩󵄩󵄩𝔽(t; ⋅)ϕ(󵄩󵄩󵄩F(⋅ + τ; x) − y⋅;x 󵄩󵄩󵄩Y )󵄩󵄩󵄩 < ε. 󵄩 󵄩 Pt t→+∞ x∈B (c) We say that the function F(⋅; ⋅) is Doss-(𝒫 , ϕ, 𝔽, ℬ, Λ′ , ρ)-uniformly recurrent if and only if, for every B ∈ ℬ, there exists a sequence (τk ) ∈ Λ′ such that for every t > 0, x ∈ B and ⋅ ∈ Λt , we have the existence of an element y⋅;x ∈ ρ(F(⋅; x)) such that 󵄩󵄩 󵄩 󵄩 󵄩󵄩 lim lim sup sup󵄩󵄩󵄩𝔽(t; ⋅)ϕ(󵄩󵄩󵄩F(⋅ + τk ; x) − y⋅;x 󵄩󵄩󵄩Y )󵄩󵄩󵄩 = 0. 󵄩Pt k→+∞ t→+∞ x∈B 󵄩 Let us note that the statement of Proposition 6.1.10 can be simply reformulated for Doss-(𝒫 , ϕ, 𝔽, ℬ, Λ′ , ρ)-almost periodic type functions. Furthermore, the next result can be deduced following the lines of proof of [450, Proposition 2.4] (for simplicity, we assume here that the function 𝔽 does not depend on the second argument):

342 � 6 Metrical approximations of functions Proposition 6.2.12. Suppose that ℬ consists of bounded subsets of X, F : Λ × X → Y and, for every fixed element x ∈ X, the function F(⋅; x) is Lebesgue measurable. Suppose, further, that the function ϕ(⋅) is monotonically increasing and there exists a finite real constant c > 0 such that ϕ(x + y) ⩽ c[ϕ(x) + ϕ(y)] for all x, y ⩾ 0. (i) Let the following conditions hold: (a) For every t > 0, Pt contains all positive constants, and for every real number d > 0, there exist two real numbers d ′ > 0 and t0 > 0 such that, for every t ⩾ t0 , we have ‖d‖Pt ⩽ d ′ . (b) There exist two real numbers t1 > 0 and M > 0 such that 𝔽(t)‖1‖Pt ⩽ M, t ⩾ t1 . (c) There exists a real number t2 > 0 such that condition (C2)’ holds for any pseudometric space Pt , t ⩾ t2 . Then, any function F ∈ e−(ℬ, ϕ, 𝔽)−B𝒫⋅ (Λ×X : Y ) is Besicovitch-(𝒫 , ϕ, 𝔽, ℬ)-bounded. (ii) Suppose that ϕ(⋅) is continuous at the point t = 0. Let the following conditions hold: (a) For every t > 0, the pseudometric space Pt contains all positive constants and limε→0+ ‖ε‖Pt = 0. (b) There exist two real numbers t1 > 0 and c > 0 such that condition (C0) holds for Pt , t ⩾ t1 . (c) There exist two real numbers t2 > 0 such that condition (C3)’ holds for any pseudometric space Pt , t ⩾ t2 . (d) There exist two finite real constants c > 0 and t2 > 0 such that, for every t ⩾ t2 , the assumption F(⋅; ⋅) ∈ Pt implies F(⋅ + τ; ⋅) ∈ Pt for all τ ∈ Λ′ with |τ| ⩽ 1, and ‖F(⋅ + τ; x)‖Pt ⩽ c‖F(⋅; x)‖Pt for all t ⩾ t2 , x ∈ X and τ ∈ Λ′ with |τ| ⩽ 1. Then, any function F ∈ e − (ℬ, ϕ, 𝔽) − B𝒫⋅ (Λ × X : Y ) is Besicovitch-(𝒫 , ϕ, 𝔽, ℬ, Λ′ , I)continuous for any set Λ′ ⊆ Λ′′ . (iii) Suppose that ϕ(⋅) is continuous at the point t = 0, conditions (a)–(c) of (ii) hold and condition (d) of (ii) holds for every τ ∈ Λ′ = Λ. Let F ∈ e − (ℬ, ϕ, 𝔽) − B𝒫⋅ (Λ × X : Y ). Then the following holds: (a) The function F(⋅; ⋅) is Doss-(𝒫 , ϕ, 𝔽, ℬ, Λ, I)-almost periodic, provided that Λ + Λ ⊆ Λ and, for every points (t1 , . . . , tn ) ∈ Λ and (τ1 , . . . , τn ) ∈ Λ, the points (t1 , t2 + τ2 , . . . , tn + τn ), (t1 , t2 , t3 + τ3 , . . . , tn + τn ), . . . , (t1 , t2 , . . . , tn−1 , tn + τn ), also belong to Λ. (b) The function F(⋅; ⋅) is Doss-(𝒫 , ϕ, 𝔽, ℬ, Λ ∩ Δn , I)-almost periodic, provided that Λ∩Δn ≠ 0, Λ+(Λ∩Δn ) ⊆ Λ and that, for every points (t1 , . . . , tn ) ∈ Λ and (τ, . . . , τ) ∈ Λ∩Δn , the points (t1 , t2 +τ, . . . , tn +τ), (t1 , t2 , t3 +τ, . . . , tn +τ), . . . , (t1 , t2 , . . . , tn−1 , tn + τ), also belong to Λ ∩ Δn . Suppose that ℬ consists of bounded subsets of X, F, G : Λ × X → Y , the function ϕ(⋅) is monotonically increasing, ϕ(0) = 0 and ϕ(x + y) ⩽ ϕ(x) + ϕ(y) for all x, y ⩾ 0. Let conditions (a)–(b) given in the formulation of Proposition 6.2.12(i), and let condition (c) holds with d = 1, for the both functions F and G. Equipped with the usual operations, the set e − (ℬ, ϕ, 𝔽) − B𝒫⋅ (Λ × X : Y ) forms a vector space on account of condition (c).

6.2 Metrical approximations: Stepanov, Weyl, Besicovitch and Doss concepts

�

343

Therefore, the functions F(⋅; ⋅) and G(⋅; ⋅) are Besicovitch-(𝒫 , ϕ, 𝔽, ℬ)-bounded. Let a set B ∈ ℬ be fixed. Then 󵄩󵄩 󵄩 󵄩 󵄩󵄩 dB (F, G) := lim sup sup󵄩󵄩󵄩𝔽(t; ⋅)ϕ(󵄩󵄩󵄩F(⋅; x) − G(⋅; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 󵄩 󵄩 Pt t→+∞ x∈B defines a pseudometric on the set e − (ℬ, ϕ, 𝔽) − B𝒫⋅ (Λ × X : Y ). We will not consider here some sufficient conditions ensuring that the pseudometric space e−(ℬ, ϕ, 𝔽)−B𝒫⋅ (Λ×X : Y ) is complete. In [456, Subsection 2.1], we have investigated the relationship between the Doss almost periodiicty and Weyl almost periodicity and proved especially that every Weyl-palmost periodic function f : ℝ → Y is Doss-p-almost periodic (1 ⩽ p < +∞). The multidimensional analogue of this statement has been clarified in [456, Proposition 8] and we will only note that it could be interesting to formulate this result in the metrical framework. We continue our work by emphasizing the following facts: (BE1) The reader can simply clarify some natural conditions under which the introduced spaces of functions are translation invariant. Suppose, for example, that 𝒫 τ ∈ Λ′′ , x0 ∈ X and F ∈ e − (ℬ, ϕ, 𝔽) − B𝒫⋅ (Λ × X : Y ) [e − (ℬ, ϕ, 𝔽) − Bρ ⋅ (Λ × X : Y ); e − (ℬ, ϕ, 𝔽)j∈ℕn − Bρj ⋅ (Λ × X : Y )]. Then we have F(⋅ + τ; ⋅ + x0 ) ∈ e − (ℬx0 , ϕ, 𝔽) − 𝒫

B𝒫⋅ (Λ × X : Y ) [e − (ℬ, ϕ, 𝔽) − Bρ ⋅ (Λ × X : Y ); e − (ℬ, ϕ, 𝔽)j∈ℕn − Bρj ⋅ (Λ × X : Y )] with ℬx0 ≡ {−x0 + B : B ∈ ℬ}, provided that there exists a finite real constant cτ > 0 such that for every x ∈ B and t > 0, the assumption 𝔽(t + |τ|, ⋅)ϕ(‖H(⋅, x)‖Y ) ∈ Pt+|τ| implies 𝔽(t, ⋅)ϕ(‖H(⋅ + τ, x)‖Y ) ∈ Pt and 𝒫

𝒫

󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩𝔽(t, ⋅)ϕ(󵄩󵄩󵄩H(⋅ + τ, x)󵄩󵄩󵄩 )󵄩󵄩󵄩 ⩽ cτ 󵄩󵄩󵄩𝔽(t + |τ|, ⋅)ϕ(󵄩󵄩󵄩H(⋅, x)󵄩󵄩󵄩 )󵄩󵄩󵄩 󵄩󵄩 󵄩 󵄩Y 󵄩󵄩Pt 󵄩󵄩 󵄩 󵄩Y 󵄩󵄩Pt+|τ| . (BE2) Suppose that the function ϕ(⋅) is monotonically increasing, continuous at the point zero, there exists a finite real constant c > 0 such that ϕ(x + y) ⩽ c[ϕ(x) + ϕ(y)] for all x, y ⩾ 0, the mapping t 󳨃→ ‖𝔽(t, ⋅)‖Pt , t > 0 is bounded at plus infinity, and for each t > 0 we have that Pt has a linear vector structure and contains all positive constants. Then F ∈ e−(ℬ, ϕ, 𝔽)−B𝒫⋅ (Λ×X : Y ) if and only if for each set B ∈ ℬ there exists a sequence (Fk (⋅; ⋅)) of strongly ℬ-almost periodic functions such that (231) holds with the polynomial Pk (⋅; ⋅) replaced therein with the function Fk (⋅; ⋅). We can similarly consider the corresponding question for the classes introduced in Definition 6.1.1, Definition 6.2.1 and Definition 6.2.6. The pointwise multiplication of multi-dimensional Besicovitch almost periodic type functions has recently been analyzed in [450, Proposition 2.7, Proposition 2.14]. Here we will clarify only one result concerning pointwise multiplication of functions obtained as metrical approximations by trigonometric polynomials or ρ-periodic type functions

344 � 6 Metrical approximations of functions (the proof is very similar to the proof of the above-mentioned statement and therefore omitted): Proposition 6.2.13. Suppose that the following conditions hold: (i) There exists a finite real constant c > 0 such that ϕ(x + y) ⩽ c[ϕ(x) + ϕ(y)] for all x, y ⩾ 0, and there exists a function φ : [0, ∞) → [0, ∞) such that ϕ(xy) ⩽ φ(x)ϕ(y) for all x, y ⩾ 0. (ii) Condition (BD) holds with the pseudometric spaces 𝒫t1 = (Pt1 , dt1 ) and 𝒫t2 = (Pt2 , dt2 ) as well as the assumptions f ∈ Pt1 and g ∈ Pt2 imply f ⋅ g ∈ Pt (t > 0) and the existence of a finite real constant k > 0 such that ‖f ⋅ g‖Pt ⩽ k‖f ‖Pt1 ⋅ ‖g‖Pt2 ,

t > 0, f ∈ Pt1 , g ∈ Pt2 .

(232)

(iii) Condition (C0) holds for Pt , provided that t is sufficiently large. (iv) For every set B ∈ ℬ, there exists a real number t0 > 0 such that, for every x ∈ B, the function 𝔽1 (t, ⋅)φ(|P(⋅; x)|) belongs to Pt1 for any t > 0 and any trigonometric polynomial [c-periodic function; (cj )j∈ℕn -periodic function], the function 𝔽2 (t, ⋅)φ(‖G(⋅; x)‖Y ) belongs to Pt2 for any t > 0, and 󵄩󵄩 󵄨 󵄨 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 lim sup[sup󵄩󵄩󵄩𝔽1 (t, ⋅)φ(󵄨󵄨󵄨P(⋅; x)󵄨󵄨󵄨)󵄩󵄩󵄩 1 + sup󵄩󵄩󵄩𝔽2 (t, ⋅)φ(󵄩󵄩󵄩G(⋅; x)󵄩󵄩󵄩Y )󵄩󵄩󵄩 2 ] = 0, 󵄩 󵄩 󵄩 󵄩Pt P t t→+∞ x∈B x∈B for any trigonometric polynomial [c-periodic function; (cj )j∈ℕn -periodic function] P(⋅; ⋅). 𝒫1

1

If F ∈ e − (ℬ, ϕ, 𝔽1 ) − B𝒫⋅ (Λ × X : ℂ) [e − (ℬ, ϕ, 𝔽1 ) − Bc I⋅ (Λ × X : ℂ); e − (ℬ, ϕ, 𝔽1 )j∈ℕn − 𝒫1

1

2

𝒫2

B j ⋅ (Λ × X : ℂ)] and G ∈ e − (ℬ, ϕ, 𝔽2 ) − B𝒫⋅ (Λ × X : Y ) [e − (ℬ, ϕ, 𝔽2 ) − Bc ⋅I (Λ × X : c1 I

2

𝒫⋅2

Y ); e − (ℬ, ϕ, 𝔽2 )j∈ℕn − B j (Λ × X : Y )], then F ⋅ G ∈ e − (ℬ, ϕ, 𝔽) − B (Λ × X : Y ) c2 I

𝒫⋅

[e − (ℬ, ϕ, 𝔽) − Bc I⋅ (Λ × X : Y ); e − (ℬ, ϕ, 𝔽)j∈ℕn − Bc I⋅ (Λ × X : Y )] with F = F1 F2 ; here 𝒫 1

j

j

c1 , c2 ∈ ℂ ∖ {0} and c1 , c2 ∈ ℂ ∖ {0} for all j ∈ ℕn .

𝒫 j

In connection with Proposition 6.2.13, let us notice that the existence and uniqueness of Besicovitch-p-almost periodic solutions for certain classes of PDEs on some proper subdomains of ℝn have recently been analyzed in [450]. The interested reader may try to provide some applications of Proposition 6.2.13 in the analysis of the existence and uniqueness of metrical Besicovitch almost periodic solutions for certain classes of PDEs; see, e. g., the seventh application from [450, Section 4], where the Besicovitch-p-almost periodicity of solutions is clarified for the equations depending on two variables, on the sectors of form Λ = (−∞, 0] × ℝ or Λ = [0, ∞) × ℝ. Let us also emphasize that the equation (232) can be viewed as the abstract Hölder inequality in pseudometric spaces. We close this section with the observation that we will not analyze here the metrical analogues of conditions (A), (AS), (A)∞ and (B) here.

6.3 Further results and applications

� 345

6.3 Further results and applications Concerning the convolution invariance of function spaces introduced in this chapter so far, we will only present here a few comments and a concrete application in the study of the existence and uniqueness of metrically Besicovitch almost periodic solutions of the heat equation in ℝn ; the theoretical analysis is very similar to the analysis of the invariance under the actions of infinite convolution products carried out in the next subsection (see [446, 447, 448] for some results obtained recently in this direction). It is well known that a unique solution of the heat equation ut (t, x) = uxx (t, x), t ⩾ 0, x ∈ ℝn ; u(0, x) = F(x), x ∈ ℝn is given by the action of Gaussian semigroup given by (38). Suppose now that there exist two finite real numbers b ⩾ 0 and c > 0 such that |F(x)| ⩽ c(1 + |x|)b , x ∈ ℝn as well as that a > 0, α > 0, 1 ⩽ p < +∞, αp ⩾ 1, and p 1/(αp)+1/q = 1. Suppose further that ν ∈ Lloc (ℝn ), there exist finite real numbers M0 > 0 and t0 > 0 such that ∫ νp (s) ds ⩽ M0 t ap ,

t ⩾ t0 ,

(233)

[−t,t]n

as well as there exists a function φ : ℝn → [0, ∞) such that ν(x) ⩽ ν(x − y)φ(y) for all x, y ∈ ℝn and 2

∫ e−c|y| (1 + |y|ap )φp (y) dy < +∞, ℝn p

for any c > 0. Let for each t > 0 we have Pt := Lν ([−t, t]n ), and let dt be the metric induced by the norm of this Banach space. Let us fix a real number t0 in (38), and let F ∈ e − (x α , t −a ) − B𝒫⋅ (ℝn : ℂ). Then the mapping x 󳨃→ (G(t0 )F)(x), x ∈ ℝn is welldefined and has the same growth as f (⋅); let us prove that this mapping belongs to the class e − (x α , t −a ) − B𝒫⋅ (ℝn : ℂ) as well. Let ε > 0 be given in advance. Set, as in [450], −n/2 󵄩 󵄩󵄩 −|⋅|2 /8t0 󵄩󵄩󵄩αp 󵄩󵄩e 󵄩󵄩 q n . 󵄩 󵄩L (ℝ )

ct0 := (4πt0 )

By our assumption, there exist a trigonometric polynomial P(⋅) and a finite real number t1 > 0 such that 󵄨 󵄨αp ∫ 󵄨󵄨󵄨F(x) − P(x)󵄨󵄨󵄨 νp (x) dx < ε0 t ap ,

t ⩾ t1 .

[−t,t]n

The function x 󳨃→ (G(t0 )P)(x), x ∈ ℝn is Bohr almost periodic and the required conclusion simply follows from the next computation: 1 t ap

󵄨 󵄨αp ∫ 󵄨󵄨󵄨(G(t0 )F)(x) − (G(t0 )P)(x)󵄨󵄨󵄨 νp (x) dx

[−t,t]n

346 � 6 Metrical approximations of functions

⩽ = ⩽

⩽ ⩽

ct0

t ap ct0

t ap ct0

t ap ct0

t ap ct0

t ap

󵄨 󵄨αp ∫ ∫ e−|y| αp/8t0 󵄨󵄨󵄨F(x − y) − P(x − y)󵄨󵄨󵄨 dy ⋅ νp (x) dx 2

[−t,t]n ℝn

2 󵄨 󵄨αp ∫ e−|y| αp/8t0 ∫ 󵄨󵄨󵄨F(x − y) − P(x − y)󵄨󵄨󵄨 νp (x) dx dy

ℝn

[−t,t]n

2

∫ e−|y| αp/8t0 ℝn

∫ [−t+|y|,t+|y|]n

2

∫ e−|y| αp/8t0 [ ℝn

󵄨󵄨 󵄨αp p 󵄨󵄨F(x) − P(x)󵄨󵄨󵄨 ν (x) dx dy

∫

[−t+|y|,t+|y|]n

󵄨󵄨 󵄨αp p p 󵄨󵄨F(x) − P(x)󵄨󵄨󵄨 ν (x − y) dx]φ (y) dy

2

∫ e−|y| αp/8t0 ε0 2ap (t ap + |y|ap )φp (y) dy,

t ⩾ t1 .

(234)

ℝn

The argumentation employed for proving the estimate (234) can be used to deduce some results about the invariance of certain types of metrical Besicovitch almost periodicity under the actions of the usual convolution product f 󳨃→ F(x) ≡ ∫ h(x − y)f (y) dy,

x ∈ ℝn ,

ℝn

provided that the function h ∈ L1 (ℝn ) has a certain growth order. For instance, an extension of [450, Theorem 4.6] can be proved in this context. Furthermore, using a similar idea as above, we can consider the existence and uniqueness of metrically Besicovitch almost periodic solutions for some special classes of evolution equations of first order; see, e. g. the sixth application from [450, Section 4] for more details.

6.3.1 Invariance under the actions of infinite convolution products In this subsection, we will present a few results concerning the invariance of introduced classes of functions under the actions of infinite convolution product (103); for simplicity, we will not consider the multi-dimensional case here. Our first result considers (c, Λ′ )-almost periodic functions in variation; we will use the following notion: Suppose that 0 ≠ Λ′ ⊆ ℝ and c ∈ ℂ, |c| = 1. A continuous function f : ℝ → Y is called (c, Λ′ )-almost periodic in variation if and only if for each ε > 0 there exists a finite real number l > 0 such that, for every t0 ∈ Λ′ , there exists a number τ ∈ [t0 − l, t0 + l] such that 󵄨 󵄨 sup(󵄨󵄨󵄨f (t + τ) − cf (t)󵄨󵄨󵄨 + V1 (f (⋅ + τ) − cf (⋅); t)) < ε. t∈ℝ

We are ready to state the following simple result:

(235)

6.3 Further results and applications

�

347

Proposition 6.3.1. Suppose that (R(t))t>0 ⊆ L(X, Y ) is a strongly continuous operator ∞ family and ∫0 ‖R(t)‖ dt < +∞. If the function f : ℝ → X is (c, Λ′ )-almost periodic in variation (0 ≠ Λ′ ⊆ ℝ; c ∈ ℂ, |c| = 1), then the function F(⋅), given by (103), is likewise (c, Λ′ )-almost periodic in variation. Proof. Let ε > 0 be given. Then there exists a finite real number l > 0 such that, for every t0 ∈ Λ′ , there exists a number τ ∈ [t0 − l, t0 + l] such that (235) holds (cf. also [431, Theorem 6.1.53]). We need to show that (235) holds with the function f (⋅) replaced by the function F(⋅) therein. But, keeping in mind the elementary definitions, it is very simple to prove that 󵄩 󵄩 sup(󵄩󵄩󵄩f (t + τ) − cf (t)󵄩󵄩󵄩 + V1 (f (⋅ + τ) − cf (⋅); t)) t∈ℝ

∞

󵄩 󵄩 ⩽ sup(󵄩󵄩󵄩F(t + τ) − cF(t)󵄩󵄩󵄩Y + V1 (F(⋅ + τ) − cF(⋅); t)) ⋅ ∫ ‖R(s)‖ ds. t∈ℝ

0

This implies the required result. We continue by stating the following analogue of [446, Theorem 3.13]: Proposition 6.3.2. Suppose that (R(s))s>0 ⊆ L(X, Y ) is a strongly continuous operator family, c ∈ ℂ ∖ {0} and 0 ≠ Λ′ ⊆ ℝ. Let P := Cb,ν (ℝ : [0, ∞)) and d(f , g) := ‖f − g‖Cb,ν (ℝ:[0,∞)) for all f , g ∈ P, and let there exist a function φ : ℝ → [0, ∞) continuous at the point zero and satisfying that ϕ(xy) ⩽ ϕ(x)φ(y) for all x, y ∈ ℝ. Suppose further that ‖R(s)‖ ds < ∞ for all x ∈ ℝ. If f : ℝ → X is a bounded, ∫(0,∞) ‖R(s)‖ ds < ∞ and ∫(0,∞) ν(x−s)

continuous and Bohr (ϕ, 𝔽, Λ′ , c, 𝒫 )-almost periodic function, then the function F : ℝ → Y , given by (103), is bounded, continuous, and Bohr (ϕ, 𝔽1 , Λ′ , c, 𝒫1 )-almost periodic, provided that P1 is a Banach space, ∞

󵄩 󵄩 𝔽1 (⋅)ϕ( ∫ 󵄩󵄩󵄩R(s)󵄩󵄩󵄩 0

ds ) ∈ P1 ν(⋅ − s)

(236)

and condition (C0) holds for P1 . Proof. Since ∫0 ‖R(s)‖ ds < ∞, a very simple argumentation from our previous research studies shows that the function F(⋅) is bounded and continuous. Let us show that F(⋅) is Bohr (ϕ, 𝔽1 , Λ′ , c, 𝒫1 )-almost periodic. Let ε > 0 be arbitrary and let τ ∈ Λ′ be a corresponding ε-period of the function f (⋅). Then the final conclusion simply follows ∞ ‖R(s)‖ from the next computation involving the conditions ∫0 ν(x−s) ds < ∞ for all x ∈ ℝ, (236) and the continuity of function φ(⋅) at the point zero: ∞

󵄩󵄩 󵄩󵄩 ∞ 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩𝔽1 (⋅)ϕ(󵄩󵄩󵄩 ∫ R(s)[f (⋅ + τ − s) − cf (⋅ − s)] ds󵄩󵄩󵄩 )󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩Y 󵄩󵄩P1 0

348 � 6 Metrical approximations of functions ∞ 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ⩽ 󵄩󵄩󵄩𝔽1 (⋅)ϕ( ∫ 󵄩󵄩󵄩R(s)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩f (⋅ + τ − s) − cf (⋅ − s)󵄩󵄩󵄩 ds)󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩P1 0 ∞ 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 ε 󵄩 󵄩 ⩽ 󵄩󵄩󵄩𝔽1 (⋅)ϕ( ∫ 󵄩󵄩󵄩R(s)󵄩󵄩󵄩 ⋅ ds)󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 ν(⋅ − s) 󵄩 󵄩 P1 0 ∞ 󵄩󵄩󵄩 󵄩󵄩󵄩 ds 󵄩 󵄩 󵄩 󵄩 ⩽ 󵄩󵄩󵄩𝔽1 (⋅)φ(ε)ϕ( ∫ 󵄩󵄩󵄩R(s)󵄩󵄩󵄩 ⋅ )󵄩󵄩󵄩 . 󵄩󵄩 󵄩󵄩 ν(⋅ − s) 󵄩 󵄩 P1 0

Now we will assume that the operator family (R(t))t>0 ⊆ L(X, Y ) satisfies that there exist finite real constants M > 0, β ∈ (0, 1] and γ > β such that (104) holds. The following result is a slight extension of [450, Proposition 4.1]: Proposition 6.3.3. Suppose that the operator family (R(t))t>0 ⊆ L(X, Y ) satisfies (104), as well as that a > 0, α > 0, 1 ⩽ p < +∞, αp ⩾ 1, ap ⩾ 1, αp(β − 1)/(αp − 1) > −1 if αp > 1, and β = 1 if αp = 1. Suppose further that the function f : ℝ → X is Stepanov-(αp)-bounded, i. e. t+1

󵄩 󵄩αp ‖f ‖Sp := sup ∫ 󵄩󵄩󵄩f (s)󵄩󵄩󵄩 ds < +∞, t∈ℝ

t

as well as that the function ν : ℝ → (0, ∞) is monotonically decreasing, Stepanov-pbounded, and satisfies that the function f (⋅)ν1/α (⋅) is Stepanov-(αp)-bounded as well as that there exist finite real numbers M0 > 0 and t0 > 0 such that (233) holds with n = 1. Let p for each t > 0 we have Pt := Lν ([−t, t]), and let dt be the metric induced by the norm of α −a this Banach space. If f ∈ e − (x , t ) − B𝒫⋅ (ℝ : X), then the function F(⋅), given by (103), is bounded, continuous and belongs to the class e − (x α , t −a ) − B𝒫⋅ (ℝ : Y ). Proof. The proof of [428, Proposition 2.6.11] indicates that the function F(⋅) is welldefined, bounded, and continuous. Let (Pk ) be a sequence of trigonometric polynomials such that t

lim lim sup

k→+∞ t→+∞

1 󵄩 󵄩αp ∫󵄩󵄩f (s) − Pk (s)󵄩󵄩󵄩 νp (s) ds = 0. 2t ap 󵄩 −t

t

The function t 󳨃→ Fk (t) ≡ ∫−∞ R(t − s)Pk (s) ds, t ∈ ℝ is almost periodic due to the abovementioned proposition. Since there exist finite real numbers M > 0 and t0 > 0 such that (233) is true with n = 1, the conclusions established in (BE2) hold. Hence, we need to prove that t

1 󵄩 󵄩αp lim lim sup ∫󵄩󵄩󵄩F(s) − Fk (s)󵄩󵄩󵄩 νp (s) ds = 0. k→+∞ t→+∞ 2t ap −t

(237)

6.3 Further results and applications

� 349

In the remainder of the proof, we will only consider case αp > 1. Suppose that ζ ∈ (1/(αp), (1/(αp)) + γ − β). Then the function s 󳨃→ |s|β−1 (1 + |s|)ζ /(1 + |s|γ ), s ∈ ℝ belongs to the space Lαp/(αp−1) ((−∞, 0)); further on, since we have assumed that the function ν(⋅) is Stepanov-p-bounded and the functions f (⋅), f (⋅)ν1/α (⋅) are Stepanov-(αp)-bounded, the argumentation contained in the proof of [428, Theorem 2.11.4] shows that the function s 󳨃→ (1 + |s|)−ζ ‖Pk (s + z) − f (s + z)‖ν1/α (s + z), s ∈ ℝ belongs to the space Lαp ((−∞, 0)) for all k ∈ ℕ and z ∈ ℝ. We have (M1 > 0 is a finite real constant, t > 0): t

1 󵄩 󵄩αp ∫󵄩󵄩F(s) − Fk (s)󵄩󵄩󵄩 νp (s) ds 2t ap 󵄩 −t

αp t󵄨 0 󵄨󵄨 󵄨󵄨 1 󵄨󵄨 󵄩󵄩 󵄩󵄩 󵄨󵄨󵄨 p ⩽ ap ∫󵄨󵄨 ∫ ‖R(−z)‖ ⋅ 󵄩󵄩Pk (s + z) − f (s + z)󵄩󵄩 dz󵄨󵄨 ν (s) ds 󵄨󵄨 󵄨󵄨 2t 󵄨 −t 󵄨−∞

⩽

t󵄨 0 ζ 󵄨󵄨αp 󵄨󵄨 |z|β−1 (1 + |z|) M 󵄨󵄨󵄨󵄨 󵄩󵄩 1/α −ζ 󵄩 󵄩 󵄨󵄨 ds ⋅ (1 + |z|) P (s + z) − f (s + z) ν (s + z) dz ∫ ∫ 󵄨 󵄩󵄩 k 󵄩󵄩 󵄨󵄨 2t ap 󵄨󵄨󵄨󵄨 (1 + |z|γ ) 󵄨󵄨 −t −∞

⩽

M1 1 󵄩󵄩 󵄩αp p ∫ ∫ 󵄩Pk (s + z) − f (s + z)󵄩󵄩󵄩 ν (s + z) dz ds. 2t ap (1 + |z|αζ )p 󵄩

t

0

−t −∞

The estimate (237) then follows from the remainder of the long computation carried out in the proof of [450, Proposition 4.1]. In the following slight extension of [450, Proposition 4.2], the inhomogeneity f (⋅) is not necessarily Stepanov-(αp)-bounded and the weight ν(⋅) is not necessarily Stepanov-p-bounded. The proof is almost the same as the proof of Proposition 6.3.3 and the above-mentioned results from [450]: Proposition 6.3.4. Suppose that the operator family (R(t))t>0 ⊆ L(X, Y ) satisfies (104), as well as that a > 0, α > 0, 1 ⩽ p < +∞, αp ⩾ 1, ap ⩾ 1, αp(β − 1)/(αp − 1) > −1 if αp > 1, and β = 1 if αp = 1. Suppose, further, that there exists a finite real constant M > 0 such that ‖f (t)‖ ⩽ M(1 + |t|)b , t ∈ ℝ for some real constant b ∈ [0, γ − β), f (⋅) is Lebesgue measurable, the function ν : ℝ → (0, ∞) is monotonically decreasing, there exists a finite real constant M ′ > 0 such that ‖ν(t)‖ ⩽ M ′ (1 + |t|)b/α , t ∈ ℝ and there exist finite real numbers M0 > 0 and t0 > 0 such that (233) holds with n = 1. Let for each t > 0 we have p Pt := Lν ([−t, t]), and let dt be the metric induced by the norm of this Banach space. If f ∈ e − (x α , t −a ) − B𝒫⋅ (ℝ : X), then the function F(⋅), given by (103), is continuous, belongs to the class e − (x α , t −a ) − B𝒫⋅ (ℝ : Y ), and there exists a finite real constant M ′ > 0 such that ‖F(t)‖Y ⩽ M ′ (1 + |t|)b , t ∈ ℝ. As mentioned in a great number of our recent research articles, Proposition 6.3.1, Proposition 6.3.3 and Proposition 6.3.4 can be applied to a large class of the abstract (degenerate) Volterra integro-differential equations without initial conditions. Here we

350 � 6 Metrical approximations of functions will only note that we can apply these results in the analysis of the existence and uniqueness of metrically Besicovitch-p-almost periodic type solutions of the initial value probγ lem (110) in the space Lp (ℝn ), where γ ∈ (0, 1), Dt,+ u(t) denotes the Weyl–Liouville fractional derivative of order γ and 1 ⩽ p < ∞. The statement of [450, Theorem 4.5], which concerns the existence and uniqueness of Besicovitch-p-almost periodic solutions of the abstract nonautonomous differential equations of first order, can be simply reformulated in our new context, with the use of p the same pivot Banach spaces Pt = Lν ([−t, t]) for all t > 0. For simplicity, we will not consider here the invariance of various classes of metrically Stepanov almost periodic type functions and metrically Weyl almost periodic type functions under the actions of infinite convolution products; see [447, Subsection 2.1] and [448, Subsection 3.1] for some results obtained in this direction. Before proceeding further, we would like to emphasize that, in our definitions given in Section 6.1, we have taken the norm in Y of certain terms and assumed that P ⊆ [0, ∞)Λ . Without going into full details, we will only note here that we can also consider the following notion: Assume that 0 ≠ Λ ⊆ ℝn , ϕY : Y → Y , 𝔽 : Λ → (0, ∞), PY ⊆ Y Λ , the zero function belongs to PY , and 𝒫Y = (PY , dY ) is a pseudometric space. (i) We say that the function F(⋅; ⋅) is strongly (ϕY , 𝔽, ℬ, 𝒫Y )-almost periodic (semi(ϕY , ρ, 𝔽, ℬ, 𝒫Y )-periodic, semi-(ϕY , ρj , 𝔽, ℬ, 𝒫Y )j∈ℕn -periodic) if and only if for each B ∈ ℬ there exists a sequence (PkB (t; x)) of trigonometric polynomials (ρ-periodic functions, (ρj )j∈ℕn -periodic functions) such that 󵄩󵄩 󵄩󵄩 lim sup󵄩󵄩󵄩𝔽(⋅)ϕY (PkB (⋅; x) − F(⋅; x))󵄩󵄩󵄩 = 0. 󵄩 PY

k→+∞ x∈B 󵄩

(ii) We say that the function F(⋅; ⋅) is Bohr (ϕY , 𝔽, ℬ, Λ′ , ρ, 𝒫Y )-almost periodic if and only if for every B ∈ ℬ and ε > 0 there exists l > 0 such that for each t0 ∈ Λ′ there exists τ ∈ B(t0 , l) ∩ Λ′ such that, for every t ∈ Λ and x ∈ B, there exists an element yt;x ∈ ρ(F(t; x)) such that 󵄩󵄩 󵄩󵄩 sup󵄩󵄩󵄩𝔽(⋅)ϕY (F(⋅ + τ; x) − y⋅;x )󵄩󵄩󵄩 ⩽ ε. 󵄩 󵄩PY x∈B (iii) We say that the function F(⋅; ⋅) is (ϕY , 𝔽, ℬ, Λ′ , ρ, 𝒫Y )-uniformly recurrent if and only if for every B ∈ ℬ there exists a sequence (τ k ) in Λ′ such that limk→+∞ |τ k | = +∞ and that for every t ∈ Λ and x ∈ B, there exists an element yt;x ∈ ρ(F(t; x)) such that 󵄩󵄩 󵄩󵄩 lim sup󵄩󵄩󵄩𝔽(⋅)ϕY (F(⋅ + τ k ; x) − y⋅;x )󵄩󵄩󵄩 = 0. 󵄩 󵄩 PY k→+∞ x∈B It seems very plausible that many structural results from Section 6.1 can be reformulated for these classes of functions. At the end of section, we will only mention some topics not considered so far:

6.4 Metrical almost periodicity: Poisson, Levitan and Bebutov concepts

�

351

1. We have not considered extensions of functions obtained as metrical approximations by trigonometric polynomials and ρ-periodic type functions. 2. There are several different ways to introduce the notion of a function of bounded variation in multiple dimensions; for more details about this important subject, we refer the reader to the master thesis [158] by S. Breneis and references cited therein. We will analyze multi-dimensional (metrical) almost periodic type functions in variation somewhere else; cf. also Section 7.3 for further information in this direction. 3. In our analysis, we do not require that the set Λ is unbounded. If the set Λ is bounded, the situation is completely without control and a series of further investigations can be carried out. 4. Let us recall that the composition principles for the metrically Stepanov c-almost periodic type functions and the metrically Weyl c-almost periodic type functions have been considered in [447, Theorem 2.6] and [448, Theorem 3.7], respectively; a composition principle for Besicovitch-p-almost periodic type functions has been deduced in [450, Theorem 2.10] following the approach of M. Ayachi and J. Blot [70, Lemma 4.1]. It is also worth noting that the notion of a (c, Λ′ )-uniformly recurrent function in p-variation can be also introduced (1 ⩽ p < +∞); then an analogue of the composition principle stated in [409, Theorem 2.28] can be proved under certain very restrictive assumptions (the interested reader may try to reconsider the results established in our recent research study [409] by M. T. Khalladi et al. for c-almost periodic type functions in p-variation and Hölder c-almost periodic type functions, which can defined in a similar fashion). We will not analyze the composition principles for the introduced classes of functions and related applications to the semilinear Cauchy problems here.

6.4 Metrical almost periodicity: Poisson, Levitan and Bebutov concepts Let us recall that a continuous function F : ℝn → X is said to be uniformly recurrent (uniformly Poisson stable) if and only if there exists a sequence (τk ) in ℝn such that limk→+∞ |τk | = +∞ and limk→+∞ F(t + τk ) = F(t), uniformly in t ∈ ℝn (uniformly in t ∈ K, for any compact subset K ⊆ ℝn ). Any almost periodic function is uniformly recurrent and any uniformly recurrent function is uniformly Poisson stable, while the converse statements are not true in general. The class of almost periodic functions can be generalized following the approach of B. M. Levitan (see, e. g., [496] for the one-dimensional setting): Suppose that F : ℝn → X is a continuous function, N > 0 and ε > 0. Then a point τ ∈ ℝn is said to be an ε, N-almost period of function F(⋅) if and only if 󵄩󵄩 󵄩 󵄩󵄩F(t + τ) − F(t)󵄩󵄩󵄩 ⩽ ε

for all t ∈ ℝn with |t| ⩽ N;

denote by E(ε, N) the set consisting of all ε, N-almost periods of function F(⋅). We say that a continuous function F(⋅) is Levitan pre-almost periodic if and only if for each

352 � 6 Metrical approximations of functions N > 0 and ε > 0, there exists a finite real number l > 0 such that for each t0 ∈ ℝn there exists τ ∈ B(t0 , l) ∩ E(ε, N), i. e., the set E(ε, N) is relatively dense in ℝn for each N > 0 and ε > 0. It is worth noting that B. Ya. Levin has shown, in [494], that the sum of two Levitan pre-almost periodic functions f : ℝ → ℝ and g : ℝ → ℝ need not be Levitan pre-almost periodic in general. In the definition of a Levitan (N-)almost periodic function F : ℝn → X, we additionally require that, for every real numbers N > 0 and ε > 0, there exist a finite real number η > 0 and a relatively dense set Eη;N of (η, N)-almost periods of F(⋅) such that Eη;N ± Eη;N ⊆ E(ε, N); cf. [496, condition (2), p. 54, l. 3] and the corresponding footnote for more details concerning this issue in the onedimensional setting. Due to R. Yuan’s result [794, Theorem 3.1], we know that a bounded uniformly continuous function f : ℝ → X is compactly almost automorphic if and only if f (⋅) is Levitan N-almost periodic (cf. also B. Basit [90], A. Reich [643] and references cited therein for more details about the relationship between the almost automorphic functions and the Levitan N-almost periodic functions on topological groups). The notion of a recurrent function in the continuous Bebutov system [98] is based on the use of topology of uniform convergence on compact sets (cf. also Subsection 2.3.9 in the monograph [121] by G. Bertotti and I. D. Mayergoyz, the paper [245] by L. I. Danilov and references cited therein for further information in this direction). A uniformly recurrent function is also called pseudo-periodic by H. Bohr, which has been accepted by many other authors later on; a recurrent function in the continuous Bebutov system is also called (uniformly) Poisson-stable motion by M. V. Bebutov. The Levitan almost periodic solutions and the uniformly Poisson stable solutions for various classes of (abstract) differential equations have been sought in many research articles by now; see, e. g. [26, 28, 197, 198, 521, 528, 577, 667, 668, 669] and references cited therein. Concerning the Poisson stability of motions of dynamical systems and solutions of differential equations, we would like to specifically mention the research monographs [664, 665] by B. A. Shcherbakov. It is also worth noticing that T. Caraballo and D. Cheban have analyzed the existence and uniqueness of Levitan/Bohr almost periodic (almost automorphic) solutions of the second-order monotone differential equations in [180]. The Poisson stability of motions for monotone nonautonomous dynamical systems and the Poisson stable solutions for certain classes of monotone nonautonomous differential equations have been analyzed by D. Cheban and Z. Liu in [199]; cf. also [200, 664, 665, 666, 669, 675, 684] and references cited therein for related results obtained within the theory of dynamical systems. Finally, let us note that the interpolation by Levitan almost periodic functions was considered by S. Hartman in [360] (1974), while the difference property for perturbations of vector-valued Levitan almost periodic functions was considered by B. Basit and H. Günzler in [91]; see also M. G. Lyubarskii [528]. The main aim of this section is to continue the analysis raised in the previous section by investigating the Levitan and Bebutov metrical approximations of functions F : Λ × X → Y by trigonometric polynomials and ρ-periodic type functions. We analyze here various classes of multi-dimensional Levitan almost periodic functions in general metric and multi-dimensional Bebutov uniformly recurrent functions in general metric.

6.4 Metrical almost periodicity: Poisson, Levitan and Bebutov concepts

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353

The organization and main ideas of section can be briefly described as follows. We introduce the basic function spaces of metrically almost periodic functions in the sense of Levitan/Bebutov approach in the second subsection; cf. Definition 6.4.1 and Definitions 6.4.4–6.4.5. The main structural results established in this subsection are Proposition 6.4.6, Proposition 6.4.9 and Proposition 6.4.11; cf. also Example 6.4.7, Example 6.4.8 and Example 6.4.10. Levitan (N, c)-almost periodic functions and uniformly Poisson c-stable functions [multi-dimensional Levitan N-almost periodic functions] are specifically analyzed in Section 6.4.1 [Section 6.4.2]. In Section 6.4.2, we introduce the spaces of strongly Levitan N-almost periodic functions of type 1, the Levitan N-almost periodic functions of type 1 and the strongly Levitan N-almost periodic functions. We show that these spaces have the linear vector structure and propose an open problem whether the Levitan N-almost periodic functions form a vector space with the usual operations if n ⩾ 2. By a simple counterexample, we show that the Bogolybov theorem (see, e. g. [496, pp. 55–57]) cannot be straightforwardly extended to the higher-dimensional case n ⩾ 2; cf. also Proposition 6.4.15 and Example 6.4.16. In Section 6.4.3, we present several applications of our theoretical results to the abstract Volterra integro-differential equations. We investigate the invariance of Levitan N-almost like periodicity under the actions of the infinite convolution products and certain applications to the abstract Cauchy problems without initial conditions; we also investigate here the convolution invariance of certain kinds of multi-dimensional Levitan N-almost periodic type functions and reconsider several results established recently by A. Nawrocki in [577]. We continue our analysis of the wave equation in ℝn and provide certain comments and final remarks about the introduced spaces of Levitan N-almost periodic type functions. We propose several open problems to our readers, providing also a great number of important references concerning the subjects under our considerations. The material is taken from [191]. In this section, we assume that 0 ≠ Λ ⊆ ℝn and ϕ : [0, ∞) → [0, ∞). If 0 ≠ K ⊆ ℝn is a compact set and K ∩ Λ ≠ 0, then we assume that the function 𝔽K : K ∩ Λ → [0, ∞) and the pseudometric space 𝒫K = (PK , dK ), where PK ⊆ [0, ∞)K∩Λ , are given. Define ‖g‖PK := dK (g, 0) for any g ∈ PK . The following notion plays an important role in our analysis: Definition 6.4.1. Suppose that 0 ≠ Λ ⊆ ℝn and F : Λ × X → Y . Then we say that F(⋅; ⋅) is Levitan strongly (ϕ, 𝔽K , ℬ, 𝒫K )-almost periodic (Levitan semi-(ϕ, ρ, 𝔽K , ℬ, 𝒫K )-periodic, Levitan semi-(ϕ, ρj , 𝔽K , ℬ, 𝒫K )j∈ℕn -periodic) if and only if for each B ∈ ℬ and for each non-empty compact set K ⊆ ℝn such that K ∩ Λ ≠ 0 there exists a sequence (PkB,K : Λ × X → Y ) of trigonometric polynomials (ρ-periodic functions (PkB,K : Λ × X → Y ), (ρj )j∈ℕn periodic functions (PkB,K : Λ × X → Y )) such that 𝔽K (⋅)[ϕ(‖PkB,K (⋅; x) − F(⋅; x)‖Y )]|K∩Λ ∈ PK for all x ∈ X, and 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩 lim sup󵄩󵄩󵄩𝔽K (⋅)[ϕ(󵄩󵄩󵄩PkB,K (⋅; x) − F(⋅; x)󵄩󵄩󵄩Y )] 󵄩 = 0. 󵄩󵄩PK |K∩Λ 󵄩 k→+∞ x∈B 󵄩 󵄩

354 � 6 Metrical approximations of functions As in our former research studies, we omit the term “ϕ” if ϕ(x) ≡ x, “ρ” if ρ = I, the term “ℬ” if X = {0} and ℬ = {X}. We will also omit the term “𝔽K ” if 𝔽K ≡ 1 for each non-empty compact set K ⊆ ℝn such that K ∩ Λ ≠ 0 (in the case that ϕ(x) ≡ x, ρ = I, PK = Cb (K ∩ Λ) and 𝔽K ≡ 1 for each non-empty compact set K ⊆ ℝn such that K ∩ Λ ≠ 0, the notion of Levitan strong (ϕ, 𝔽K , ℬ, 𝒫K )-almost periodicity is completely regardless; for example, any continuous function F : ℝn → Y is Levitan strongly (x, 𝔽K , 𝒫K )-almost periodic due to the Weierstrass approximation theorem). The following result can be proved as the corresponding result from the previous section: Proposition 6.4.2. Suppose that 0 ≠ Λ ⊆ ℝn , F : Λ × X → Y , h : Y → Z is Lipschitz continuous, ϕ(⋅) is monotonically increasing and there exists a function φ : [0, ∞) → [0, ∞) such that ϕ(xy) ⩽ φ(x)ϕ(y) for all x, y ⩾ 0. Let the assumptions (C0-K)–(C1-K) hold for every non-empty compact set K ⊆ ℝn with K ∩ Λ ≠ 0, where: (C0-K) The assumptions 0 ⩽ f ⩽ g and g ∈ PK imply f ∈ PK and ‖f ‖PK ⩽ ‖g‖PK . (C1-K) If f ∈ PK , then d ′ f ∈ PK for all reals d ′ ⩾ 0 and there exists a finite real constant dK > 0 such that ‖d ′ f ‖PK ⩽ dK (1 + d ′ )‖f ‖PK for all reals d ′ ⩾ 0 and all functions f ∈ PK . Then we have the following: (i) Suppose that F(⋅; ⋅) is Levitan semi-(ϕ, ρ, 𝔽K , ℬ, 𝒫K )-periodic (Levitan semi-(ϕ, ρj , 𝔽K , ℬ, 𝒫K )j∈ℕn -periodic) and h ∘ ρ ⊆ ρ ∘ h (h ∘ ρj ⊆ ρj ∘ h for 1 ⩽ j ⩽ n). Then the function h ∘ F : Λ × X → Y is likewise Levitan semi-(ϕ, ρ, 𝔽K , ℬ, 𝒫K )-periodic (Levitan semi-(ϕ, ρj , 𝔽K , ℬ, 𝒫K )j∈ℕn -periodic). (ii) Suppose that X = {0}, there exists a finite real constant c > 0 such that ϕ(x + y) ⩽ c[φ(x) + φ(y)] for all x, y ⩾ 0, ϕ(⋅) is continuous at the point zero and for every nonempty compact set K ⊆ ℝn with K ∩ Λ ≠ 0, we have 𝔽K ∈ PK

and

󵄩 󵄩 lim 󵄩󵄩ε𝔽K (⋅)󵄩󵄩󵄩P = 0.

ε→0+󵄩

K

Suppose further that the function F(⋅) is Levitan strongly (ϕ, 𝔽K , ℬ, 𝒫K )-almost periodic and the assumption (C2-K) holds, where: (C2-K) There exists a finite real constant eK > 0 such that the assumptions f , g ∈ PK and 0 ⩽ w ⩽ d ′ [f + g] for some finite real constant d ′ > 0 imply w ∈ PK and ‖w‖PK ⩽ eK (1 + d ′ )[‖f ‖PK + ‖g‖PK ]. Then the function (h ∘ F)(⋅) is likewise Levitan strongly (ϕ, 𝔽K , ℬ, 𝒫K )-almost periodic. We continue by providing the following illustrative example: Example 6.4.3. Let us consider again the function f (⋅) given by (217). Let ε0 > 0 be a fixed real number; using the elementary inequality | sin t| ⩽ |t|, t ∈ ℝ, it readily follows that the function f (⋅) is Levitan semi-(x, I, 𝔽K , 𝒫K )-semi-periodic with 𝔽[−N,N] ≡ N −2−ε0 and 𝒫[−N,N] := C[−N, N] for N > 0.

6.4 Metrical almost periodicity: Poisson, Levitan and Bebutov concepts

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355

n Similarly, any continuous function F : ℝn → ℂ given by F(t) = ∑∞ m=1 Pm (t), t ∈ ℝ , where Pm (⋅) is a trigonometric polynomial (m ∈ ℕ), is Levitan semi-(x, I, 𝔽K , 𝒫K )-semiperiodic with the constant function 𝔽[−N,N]n ≡ cN > 0 appropriately chosen and 𝒫[−N,N]n := C([−N, N]n ) for N > 0, provided that there exist a summable sequence of non-negative real numbers (am ) and a continuous function G : ℝn → ℂ such that n |F(t)| ⩽ |G(t)| ⋅ ∑∞ m=1 am , t ∈ ℝ .

We continue by introducing the following notion: Definition 6.4.4. Suppose that R is any collection of sequences in Λ′′ , F : Λ × X → Y and ϕ : [0, ∞) → [0, ∞). Then we say that the function F(⋅; ⋅) is Levitan (ϕ, R, ℬ, 𝔽K , 𝒫K )normal if and only if for every set B ∈ ℬ, for every non-empty compact set K ⊆ ℝn with K ∩ Λ ≠ 0 and for every sequence (bk )k∈ℕ in R, there exists a subsequence (bkm )m∈ℕ of (bk )k∈ℕ such that for every ε > 0, there exists an integer m0 ∈ ℕ such that, for every integers m, m′ ⩾ m0 , we have 𝔽K (⋅)[ϕ(‖F(⋅ + bkm ; x) − F(⋅ + bk ′ ; x)‖Y )]|K∩Λ ∈ PK for all m x ∈ X, and 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩 sup󵄩󵄩󵄩𝔽K (⋅)[ϕ(󵄩󵄩󵄩F(⋅ + bkm ; x) − F(⋅ + bk ′ ; x)󵄩󵄩󵄩Y )] 󵄩 < ε. m 󵄩󵄩PK |K∩Λ 󵄩 󵄩 x∈B 󵄩 Definition 6.4.5. Suppose that 0 ≠ Λ′ ⊆ ℝn , 0 ≠ Λ ⊆ ℝn , F : Λ × X → Y is a given function, ρ is a binary relation on Y and Λ′ ⊆ Λ′′ . Then we say that: (i) F(⋅; ⋅) is Levitan (ϕ, 𝔽K , ℬ, Λ′ , ρ, 𝒫K )-almost periodic if and only if for every B ∈ ℬ, ε > 0 and for every non-empty compact set K ⊆ ℝn with K ∩ Λ ≠ 0, there exists l > 0 such that for each t0 ∈ Λ′ there exists τ ∈ B(t0 , l) ∩ Λ′ such that, for every t ∈ Λ and x ∈ B, there exists an element yt;x ∈ ρ(F(t; x)) such that 𝔽K (⋅)[ϕ(‖F(⋅ + τ; x) − y⋅;x ‖Y )]|K∩Λ ∈ PK for all x ∈ X, and 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩 sup󵄩󵄩󵄩𝔽K (⋅)[ϕ(󵄩󵄩󵄩F(⋅ + τ; x) − y⋅;x 󵄩󵄩󵄩Y )] 󵄩 ⩽ ε. 󵄩󵄩PK |K∩Λ 󵄩 󵄩 x∈B 󵄩 (ii) F(⋅; ⋅) is Bebutov (ϕ, 𝔽K , ℬ, Λ′ , ρ, 𝒫K )-uniformly recurrent, if and only if for every B ∈ ℬ and for every non-empty compact set K ⊆ ℝn with K ∩ Λ ≠ 0, there exists a sequence (τ k ) in Λ′ such that limk→+∞ |τ k | = +∞ and that for every t ∈ Λ and x ∈ B, there exists an element yt;x ∈ ρ(F(t; x)) such that 𝔽K (⋅)[ϕ(‖F(⋅ + τ k ; x) − y⋅;x ‖Y )]|K∩Λ ∈ PK for all x ∈ X, and 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩 lim sup󵄩󵄩󵄩𝔽K (⋅)[ϕ(󵄩󵄩󵄩F(⋅ + τ k ; x) − y⋅;x 󵄩󵄩󵄩Y )] 󵄩 = 0. 󵄩󵄩PK |K∩Λ 󵄩 k→+∞ x∈B 󵄩 󵄩 If the sequence (τ k ) in Λ′ is independent of the choice of a non-empty compact set K ⊆ ℝn with K ∩Λ ≠ 0, for a set B ∈ ℬ given in advance, then we say that the function F(⋅; ⋅) is Bebutov (ϕ, 𝔽K , ℬ, Λ′ , ρ, 𝒫K )-uniformly recurrent of type 1.

356 � 6 Metrical approximations of functions In any normal situation, a Bebutov (ϕ, 𝔽K , ℬ, Λ′ , ρ, 𝒫K )-uniformly recurrent function is already Bebutov (ϕ, 𝔽K , ℬ, Λ′ , ρ, 𝒫K )-uniformly recurrent of type 1: Proposition 6.4.6. Suppose that 0 ≠ Λ′ ⊆ ℝn and 0 ≠ Λ ⊆ ℝn are unbounded sets, F : Λ × X → Y is a given function, ρ is a binary relation on Y and Λ′ ⊆ Λ′′ . If F(⋅; ⋅) is Bebutov (ϕ, 𝔽K , ℬ, Λ′ , ρ, 𝒫K )-uniformly recurrent, then F(⋅; ⋅) is Bebutov (ϕ, 𝔽K , ℬ, Λ′ , ρ, 𝒫K )uniformly recurrent of type 1, provided that for each compact set K ⊆ ℝn such that K ∩ Λ ≠ 0, there exists a finite real constant cK > 0 such that for every x ∈ X, τ ∈ Λ′ and for every compact set K ′ ⊆ ℝn which contains K, we have 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩𝔽K (⋅)[ϕ(󵄩󵄩󵄩F(⋅ + τ; x) − y⋅;x 󵄩󵄩󵄩Y )] 󵄩 󵄩󵄩 󵄩󵄩PK |K∩Λ 󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ⩽ cK 󵄩󵄩󵄩𝔽K ′ (⋅)[ϕ(󵄩󵄩󵄩F(⋅ + τ; x) − y⋅;x 󵄩󵄩󵄩Y )] ′ 󵄩󵄩󵄩 , 󵄩󵄩 |K ∩Λ 󵄩 󵄩PK ′

y⋅;x ∈ ρ(F(⋅; x)).

(238)

Proof. Let the set B ∈ ℬ be given. We know that there exists a natural number N0 ∈ ℕ such that [−N, N]n ∩ Λ ≠ 0 for every natural number N ⩾ N0 . Choose a point τN in Λ′ such that |τ N | ⩾ N and 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩 sup󵄩󵄩󵄩𝔽[−N,N]n (⋅)[ϕ(󵄩󵄩󵄩F(⋅ + τ N ; x) − y⋅;x 󵄩󵄩󵄩Y )] ⩽ 1/N. 󵄩󵄩 n 󵄩 󵄩󵄩P |[−N,N] ∩Λ x∈B 󵄩 [−N,N]n Then the sequence (τN ) in Λ′ satisfies the desired requirements since we have assumed (238). The notion introduced in the previous section is a special case of the notion introduced in the previous three definitions, provided that for each non-empty compact set K ⊆ ℝn with K ∩ Λ ≠ 0 we have the existence of a finite real constant cK > 0 such that d(f|K∩Λ , g|K∩Λ ) ⩽ cK d(f , g) for all f , g ∈ P, 𝔽K = 𝔽|K∩Λ , and PK = {f|K∩Λ ; f ∈ P}; here and hereafter, 𝒫 = (P, d) is a pseudometric space and P ⊆ [0, ∞)Y contains the zero function. Usually, we plug 𝔽K (⋅) ≡ 1, ϕ(x) ≡ x, ρ = I and PK = C(Λ ∩ K : Y ) in Definition 6.4.4 and Definition 6.4.5. For some examples of the one-dimensional uniformly Poisson stable functions, we refer the reader to [666] and the research monograph [258, pp. 219–223] by J. de Vries (the function constructed by B. A. Shcherbakov in [666] is uniformly recurrent, in fact, which simply follows from an application of [666, Lemma, p. 324]). Any compactly almost automorphic function f : ℝ → X is bounded, uniformly continuous and Levitan N-almost periodic, which simply implies that F(⋅) is uniformly Poisson stable; on the other hand, we know that there exists a compactly almost automorphic function f : ℝ → ℝ, which is not (asymptotically) uniformly recurrent (see, e. g., [431, Example 2.4.35]). Therefore, the class of compactly almost automorphic functions seems to be ideal for finding certain functions which are uniformly Poisson stable but not uniformly recurrent (the function analyzed in Example 6.4.3 is uniformly recurrent but not Stepanov almost automorphic; cf. [431] for the notion). Another class of functions which

6.4 Metrical almost periodicity: Poisson, Levitan and Bebutov concepts

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are always uniformly Poisson stable, but not necessarily uniformly recurrent can be simply constructed by taking the compositions of the uniformly recurrent functions, which do not have relatively compact range with the functions, which are only continuous but not uniformly continuous. We continue with the following example: Example 6.4.7. Define f : [0, ∞) → c0 by f (t) := (

4k 2 t 2 ) , + k 2 )2 k∈ℕ

(t 2

t ⩾ 0.

Besides many other features, we know that the function f (⋅) is bounded, uniformly continuous, quasi-asymptotically almost periodic, and not almost automorphic; see [429, Example 3.11.14] and Example 5.1.7. Here we will prove that the function f (⋅) is not uniformly Poisson stable. Let us assume the contrary, and let 0 < ε < inf

k∈ℕ (2k 2

4k 4 ; + 2k + 1)2

(239)

then there exists a strictly increasing sequence (τk ) of positive real numbers such that |f (τk ) − f (0)| ⩽ ε, k ⩾ k0 for some positive integer k0 ∈ ℕ. This implies sup k∈ℕ

4k 2 τk2

(τk2 + k 2 )2

⩽ ε,

k ⩾ k0 .

Let τk ∈ [lk , lk + 1) for some lk ∈ ℕ (k ⩾ k0 ). Then the previous estimate implies ε ⩾ sup k∈ℕ

4k 2 τk2

(τk2 + k 2 )2

⩾

4lk2 lk2

((lk + 1)2 + lk2 )2

> ε,

which is a contradiction; see (239). In the following example, the considered metric space PK is different from C(Λ ∩ K : Y ): Example 6.4.8. Suppose that (T(t)) ⊆ L(X, Y ) is a strongly continuous operator family and ℬ denotes the collection of all bounded subsets of X. Define t

F(t, s; x) := e∫s φ(τ) dτ T(t − s)x,

(t, s) ∈ ℝ2 , x ∈ X.

Then our analysis from [431, Example 8.1.5] shows the following: If the function φ(⋅) is bounded and Levitan (x, R, 1, 𝒫K )-normal, where R denotes the collection of all sequences in ℝ and PK = L1 (ℝ) for each compact set K ⊆ ℝ, then the function F(⋅, ⋅; ⋅) is Levitan (x, R1 , ℬ, 1, 𝒫1,K )-normal, where R1 denotes the collection of all sequences in {(x, x) : x ∈ ℝ} and P1,K = C(ℝ) for each compact set K ⊆ ℝ2 .

358 � 6 Metrical approximations of functions The usually considered spaces of Levitan N-almost periodic functions and uniformly Poisson stable functions are translation invariant; the basic properties of multidimensional ρ-almost periodic functions clarified in [304, Proposition 2.11] can be formulated in our new setting, as well. Using the same argumentation as before, we can deduce the following result: Proposition 6.4.9. Suppose that R is any collection of sequences in Λ′′ , Fj : Λ×X → Y and the function Fj (⋅; ⋅) is Levitan (ϕ, R, ℬ, 𝔽K , 𝒫K )-normal for all j ∈ ℕ. If F : Λ × X → Y and, for every set B ∈ ℬ, for every sequence (bk )k∈ℕ in R and for every non-empty compact set K ⊆ ℝn with K ∩ Λ ≠ 0, we have 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 sup󵄩󵄩󵄩𝔽K (⋅)[ϕ(󵄩󵄩󵄩Fj (⋅ + bj ; x) − F(⋅ + bk ; x)󵄩󵄩󵄩Y )] 󵄩󵄩󵄩 = 0, 󵄩󵄩PK K∩Λ (j,k)→+∞ x∈B 󵄩 󵄩 lim

then the function F(⋅; ⋅) is likewise Levitan (ϕ, R, ℬ, 𝔽K , 𝒫K )-normal, provided that: (i) The function ϕ(⋅) is monotonically increasing and there exists a finite real constant c > 0 such that ϕ(x + y) ⩽ c[ϕ(x) + ϕ(y)] for all x, y ⩾ 0. (ii) Condition (C3-K) holds, where: (C3-K) For every non-empty compact set K ⊆ ℝn with K ∩ Λ ≠ 0, there exists a finite real constant fK > 0 such that the assumptions f , g, h ∈ PK and 0 ⩽ w ⩽ d ′ [f + g + h] for some finite real constant d ′ > 0 imply w ∈ PK and ‖w‖PK ⩽ fK (1 + d ′ )[‖f ‖PK + ‖g‖PK + ‖h‖PK ]. Without going into full details, we will only note here that the supremum formula can be formulated in our new framework; for example, if a function F : ℝn → X is uniformly Poisson stable, then for each positive real number a > 0 we have supt∈ℝn ‖F(t)‖ = supt∈ℝn ;|t|⩾a ‖F(t)‖ ∈ [0, ∞]; cf. also [431, Proposition 2.4.13, Proposition 6.1.6, Proposition 8.1.15] for the case that ρ = I, and [304, Proposition 2.13] for the case in which ρ = T ∈ L(Y ) is a linear isomorphism (the statement of [304, Proposition 2.20] concerning multi-dimensional ρ-almost periodic functions in the finite-dimensional spaces can be formulated in our new setting, as well). We continue with some examples: Example 6.4.10. (i) The spaces of Levitan N-almost periodic functions (Bebutov uniformly recurrent functions) F : ℝn → Y can be constricted if we use the pseudometric spaces 𝒫K such that 𝒫K is continuously embedded into the space C(K : Y ) for every non-empty compact set K ⊆ ℝn . For instance, the function f (t) := 1/(2 + cos t + cos(√2t)), t ∈ ℝ is unbounded, continuous, and Levitan (x, 1, 𝒫K )-almost periodic, where 𝒫K = C(K); see [496, pp. 58–59] for more details. Therefore, the function f (⋅) is uniformly Poisson stable, as well; before going any further, we would like to ask whether the function f (⋅) is uniformly recurrent or Besicovitch p-bounded for some finite exponent p ⩾ 1 (cf. [431] for the notion)?

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Here, we can also use the metric spaces 𝒫K equipped with the distance of the form 󵄩 󵄩 d(f|K , g|K ) := sup󵄩󵄩󵄩f (t) − g(t)󵄩󵄩󵄩 + d1 (f|K , g|K ), t∈K

f , g ∈ P,

where P is a subspace of the space Cb (ℝn : [0, ∞)), d1 (⋅; ⋅) is a pseudometric on PK , 𝔽K = 𝔽|K∩Λ and PK = {f|K∩Λ ; f ∈ P}. (ii) The spaces of Levitan N-almost periodic functions (Bebutov uniformly recurrent functions) F : ℝn → Y can be extended if we use the pseudometric spaces 𝒫K such that the space C(K : Y ) is continuously embedded into 𝒫K for every non-empty compact set K ⊆ ℝn . The use of the incomplete metric space P consisting of all continuous functions from ℝn into [0, ∞), equipped with the distance d(f , g) := 󵄨 󵄨 supx∈ℝn 󵄨󵄨󵄨arctan(f (x)) − arctan(g(x))󵄨󵄨󵄨, f , g ∈ P, allows one to consider the generalized almost periodicity of functions f (⋅) which are not locally integrable. We can similarly analyze the generalized Levitan N-almost periodicity and the generalized Bebutov uniform recurrence of the functions f (⋅) which are not locally integrable by replacing the term ‖ ⋅ ‖PK by ‖ ⋅ ‖P , where 𝒫K = (PK , dK ) is the completion of the K metric space 𝒫K . (iii) Define the function f : ℝ → ℝ by f (x) := n3n+1 sin(2πx) if x ∈ [3n , 3n +1]+2⋅3n+1 ℤ for some n ∈ ℕ, and f (x) := 0, otherwise. Then f (⋅) is clearly unbounded; moreover, we know that f (⋅) is Levitan N-almost periodic (see [577, Lemma 2.18, Example 2.19]). (iv) Suppose that (αk ) is a fixed sequence of positive real numbers such that limk→+∞ αk = +∞. If fi : ℝ → ℂ is a bounded, uniformly Poisson stable function such that limk→+∞ fi (⋅ + αk ) = fi (⋅), uniformly on compacts of ℝ (1 ⩽ i ⩽ n), then the function F(⋅), given by F(t1 , . . . , tn ) := f1 (t1 ) ⋅ ⋅ ⋅ ⋅ ⋅ fn (tn ),

t = (t1 , . . . , tn ) ∈ ℝn ,

is bounded, uniformly Poisson stable and satisfies limk→+∞ F(⋅ + βk ) = F(⋅), uniformly on compacts of ℝn , where βk = (αk , . . . , αk ) for all k ∈ ℕ. (v) The introduced notion has the meaning even if ρ = 0 ∈ L(Y ). In this case, we can simply construct a Bebutov (x, 1, 0, 𝒫K )-uniformly recurrent function f : ℝ → ℝ, which does not vanish as |t| → +∞; here, PK = C(K) for each non-empty compact subset K of ℝ. (vi) We can simply justify the introduction of our concepts with the set Λ′ being not equal to the set Λ or some of its proper subsets. In the usually considered situation, the notion introduced in Definition 6.4.1 is more specific than the notion introduced in Definition 6.4.4 and Definition 6.4.5. Concerning this issue, we will clarify the following result: Proposition 6.4.11. (i) Suppose that R is any collection of sequences in Λ′′ , F : Λ×X → Y and ϕ : [0, ∞) → [0, ∞). Let the following conditions hold:

360 � 6 Metrical approximations of functions (a) The function ϕ(⋅) is monotonically increasing, continuous at the point zero, and there exists a finite real constant c > 0 such that ϕ(x + y) ⩽ c[ϕ(x) + ϕ(y)] for all x, y ⩾ 0. (b) Condition (C3-K) holds. (c) For every non-empty compact set K ⊆ ℝn with K ∩ Λ ≠ 0, we have that 𝔽K (⋅)ϕ(‖P(⋅; x)‖Y ) ∈ PK for any trigonometric polynomial (periodic function) P(⋅; ⋅) and x ∈ X. (d) For every non-empty compact set K ⊆ ℝn with K ∩ Λ ≠ 0, there exists a finite real constant gK > 0 such that 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩𝔽K (⋅)[ϕ(󵄩󵄩󵄩P(⋅; x)󵄩󵄩󵄩 )] 󵄩 󵄩󵄩 󵄩 󵄩Y |K∩Λ 󵄩󵄩󵄩PK ⩽ gK sup ϕ(󵄩󵄩P(t; x)󵄩󵄩Y ), t∈K∩Λ for any trigonometric polynomial (periodic function) P(⋅; ⋅) and x ∈ X. (e) For every non-empty compact set K ⊆ ℝn with K ∩ Λ ≠ 0, there exists a finite real constant hK > 0 such that for every x ∈ X and τ ∈ Λ′′ , the assumptions H : Λ × X → Y and 𝔽K (⋅)[ϕ(‖H(⋅; x)‖Y )]|K∩Λ ∈ PK imply 𝔽K (⋅)[ϕ(‖H(⋅ + τ; x)‖Y )]|K∩Λ ∈ PK and 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩𝔽K (⋅)[ϕ(󵄩󵄩󵄩H(⋅ + τ; x)󵄩󵄩󵄩 )] 󵄩 󵄩 󵄩 󵄩󵄩 󵄩 󵄩Y |K∩Λ 󵄩󵄩󵄩PK ⩽ hK 󵄩󵄩󵄩𝔽K (⋅)[ϕ(󵄩󵄩H(⋅; x)󵄩󵄩Y )]|K∩Λ 󵄩󵄩󵄩PK . (f) Any set B of collection ℬ is bounded. If the function F(⋅; ⋅) is Levitan strongly (ϕ, 𝔽K , ℬ, 𝒫K )-almost periodic (Levitan semi(ϕ, I, 𝔽K , ℬ, 𝒫K )j∈ℕn -periodic), then the function F(⋅; ⋅) is Levitan (ϕ, R, ℬ, ϕ, 𝔽K , 𝒫K )normal. (ii) Suppose that 0 ≠ Λ′ ⊆ ℝn , 0 ≠ Λ ⊆ ℝn , F : Λ × X → Y is a given function and Λ′ ⊆ Λ′′ . If the function F(⋅; ⋅) is Levitan strongly (ϕ, 𝔽K , ℬ, 𝒫K )-almost periodic (Levitan semi-(ϕ, I, 𝔽K , ℬ, 𝒫K )j∈ℕn -periodic), then the function F(⋅; ⋅) is Levitan (ϕ, 𝔽K , ℬ, Λ′ , I, 𝒫K )-almost periodic, provided that the assumptions (a)–(f) given in the formulation of (i) hold. Proof. We will include all details of the proof of (i) for the sake of completeness, considering the class of Levitan strongly (ϕ, 𝔽K , ℬ, 𝒫K )-almost periodic functions, only. Let F(⋅; ⋅) be such a function, let ε > 0, B ∈ ℬ, and let (bk )k∈ℕ belong to R. Suppose further that K is a non-empty compact set of ℝn such that K ∩ Λ ≠ 0. Then there exists a trigonometric polynomial PkB,K (⋅; ⋅) such that 𝔽K (⋅)[ϕ(‖PkB,K (⋅; x) − F(⋅; x)‖Y )]|K∩Λ ∈ PK for all x ∈ X, and 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩 sup󵄩󵄩󵄩𝔽K (⋅)[ϕ(󵄩󵄩󵄩PkB,K (⋅; x) − F(⋅; x)󵄩󵄩󵄩Y )] 󵄩 < ε/3. 󵄩󵄩PK |K∩Λ 󵄩 󵄩 x∈B 󵄩 Using (e), we get: 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩 sup󵄩󵄩󵄩𝔽K (⋅)[ϕ(󵄩󵄩󵄩PkB,K (⋅ + τ; x) − F(⋅ + τ; x)󵄩󵄩󵄩Y )] 󵄩 < hK ε/3 󵄩󵄩PK |K∩Λ 󵄩 󵄩 x∈B 󵄩

(240)

6.4 Metrical almost periodicity: Poisson, Levitan and Bebutov concepts

�

361

for all τ ∈ Λ′′ . The set B is bounded due to (f), so that the Bochner criterion for almost periodic functions in ℝn ensures that there exist a subsequence (bkm )m∈ℕ of (bk )k∈ℕ and a natural number m0 ∈ ℕ such that, for every positive integers m′ , m′′ ⩾ m0 , we have 󵄩󵄩 󵄩󵄩 sup󵄩󵄩󵄩PkB,K (t + bk ′ ; x) − PkB,K (t + bk ′′ ; x)󵄩󵄩󵄩 < ε/3, m m 󵄩Y x∈B 󵄩

t ∈ ℝn .

Using (c)–(d), we obtain: 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 sup󵄩󵄩󵄩𝔽K (⋅)[ϕ(󵄩󵄩󵄩PkB,K (⋅ + bk ′ ; x) − PkB,K (⋅ + bk ′′ ; x)󵄩󵄩󵄩 )] 󵄩 ⩽ gK ϕ(ε/3), m m 󵄩 󵄩Y |K∩Λ 󵄩󵄩󵄩PK 󵄩 x∈B 󵄩 for all positive integers m′ , m′′ ⩾ m0 . Then the final conclusion follows from conditions (a)–(b), the estimate (240) and the following decomposition: 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩𝔽K (⋅)[ϕ(󵄩󵄩󵄩F(⋅ + bkm′ ; x) − F(⋅ + bkm′′ ; x)󵄩󵄩󵄩Y )] 󵄩 󵄩󵄩 󵄩󵄩PK |K∩Λ 󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩 ⩽ cK′ [󵄩󵄩󵄩𝔽K (⋅)[ϕ(󵄩󵄩󵄩F(⋅ + bk ′ ; x) − P(⋅ + bk ′ ; x)󵄩󵄩󵄩Y )] 󵄩 m m 󵄩󵄩 󵄩󵄩PK |K∩Λ 󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝔽K (⋅)[ϕ(󵄩󵄩󵄩P(⋅ + bk ′ ; x) − P(⋅ + bk ′′ ; x)󵄩󵄩󵄩Y )] m m 󵄩󵄩 󵄩󵄩PK |K∩Λ 󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩 + 󵄩󵄩󵄩𝔽K (⋅)[ϕ(󵄩󵄩󵄩P(⋅ + bk ′′ ; x) − F(⋅ + bk ′′ ; x)󵄩󵄩󵄩Y )] 󵄩 ], m m 󵄩󵄩 󵄩󵄩PK |K∩Λ 󵄩 holding with a certain positive real constant cK′ > 0. The method proposed in the proof of [431, Theorem 6.1.37] does not work for Levitan N-almost periodic functions and uniformly Poisson stable functions, and because of that we will omit here all details regarding the extensions of Levitan N-almost periodic functions and the extensions of uniformly Poisson stable functions. The Levitan and Bebutov classes of metrical semi-(cj , ℬ)j∈ℕn -periodic functions can be considered similarly as before; we leave all details concerning this topic to the interested readers.

6.4.1 Levitan (N, c)-almost periodic functions and uniformly Poisson c-stable functions In this subsection, we will provide some examples and open questions concerning the Levitan (N, c)-almost periodic functions and the uniformly Poisson c-stable functions, where c ∈ ℂ ∖ {0}; for simplicity, we will always consider here the case in which Λ = ℝn , ϕ(x) ≡ x, 𝔽K ≡ 1 and PK = C(K) for each non-empty compact set K ⊆ ℝn . Let ρ = cI; then any Levitan (ϕ, 𝔽K , ℬ, Λ′ , ρ, 𝒫K )-almost periodic function is simply called

362 � 6 Metrical approximations of functions Levitan (N, Λ′ , c)-almost periodic and any Bebutov (ϕ, 𝔽K , ℬ, Λ′ , ρ, 𝒫K )-uniformly recurrent function is simply called uniformly Poisson (Λ′ , c)-stable [we omit the term “Λ′ ” from the notation if Λ′ = ℝn ]. The proofs of [431, Proposition 4.2.14, Proposition 7.1.13] do not work for the Levitan/Bebutov concepts; because of that, we would like to ask the following: Problem. Suppose that c ∈ ℂ ∖ {0, 1}. Can we find a bounded continuous function F : ℝn → ℂ, which is Levitan (N, c)-almost periodic (uniformly c-Poisson stable) but not Levitan (N, c2 )-almost periodic (uniformly c2 -Poisson stable)? The following important counterexample shows that there exists a non-trivial uniformly Poisson c-stable function F : ℝ → ℝ for any complex number c ≠ 0; this counterexample is based on the conclusions established in [22, Lemma 3.5] and particularly shows that the requirements of [431, Proposition 7.1.9] do not imply c = ±1 for uniformly Poisson c-stable functions: Example 6.4.12. Consider the function φ : ℝ → [0, ∞) given by ∞

φ(t) := ∑ sin2 ( k=1

πt ), 2k

t ∈ ℝ.

In the above-mentioned lemma, E. Ait Dads, B. Es-sebbar and L. Lhachimi have proved that the function φ(⋅) is continuous and lim φ(t + 2l ) = lim φ(t − 2l ) = φ(t) + φ(1),

l→+∞

l→+∞

(241)

pointwisely on ℝ. We will first prove that the function φ(⋅) is Lipschitz continuous as well as that the convergence in (241) is uniform on compact subsets of ℝ. The Lipschitz continuity of function φ(⋅) simply follows from the fact that the function sin2 ⋅ is Lipschitz with the corresponding Lipschitz constant L = 1 and the Lagrange mean value theorem, which shows that |φ(x) − φ(y)| ⩽ π|x − y|, x, y ∈ ℝ. Let K = [a, b] ⊆ ℝ be a compact set. Then we have: l

φ(t + 2l ) = ∑ sin2 ( k=1

∞ πt πt π ) + sin2 ( k+l + k ), ∑ k 2 2 2 k=1

t ∈ ℝ.

Using this equality, the above-mentioned fact that the function sin2 ⋅ is Lipschitz continuous with the corresponding Lipschitz constant L = 1 and the Langrange mean value theorem, we simply get that the convergence in (241) is uniform in t ∈ K since: ∞

∑ sin2 (

k=l+1

and

∞ πt π 2 (|a| + |b|)2 )⩽ ∑ , k 2 4k k=l+1

k, l ∈ ℕ

6.4 Metrical almost periodicity: Poisson, Levitan and Bebutov concepts

󵄨󵄨 ∞ 󵄨󵄨 ∞ 󵄨󵄨 󵄨 󵄨󵄨 ∑ sin2 ( πt + π ) − ∑ sin2 ( π )󵄨󵄨󵄨 ⩽ π(|a| + |b|) ∑ 2−k , 󵄨󵄨 k+l k k 2 2 2 󵄨󵄨󵄨󵄨 󵄨󵄨k=1 k=1 k=l+1

�

363

k, l ∈ ℕ;

here we have also used the elementary inequality | sin t| ⩽ |t|, t ∈ ℝ. Therefore, we have that the function φ(⋅) is Bebutov (x, 1, {2l : l ∈ ℕ}, ρ, 𝒫K )-uniformly recurrent, where PK = C(K) for each non-empty compact subset K of ℝ, D(ρ) := [0, ∞) and ρ(φ(t)) := φ(t)+ φ(1) for all t ∈ ℝ (note that the function φ(⋅) is surjective since it is continuous, φ(0) = 0 and the function φ(⋅) is Besicovitch unbounded, which follows from the inequality φ(t) ⩾ f (πt), t ∈ ℝ with the function f (⋅) being defined through (217)). Suppose now that c ∈ ℂ ∖ {0} is fixed and consider any non-trivial continuous (φ(1), c)-periodic function p : ℝ → ℂ, i. e., any non-trivial continuous function p : ℝ → ℂ such that p(t + φ(1)) = cp(t) for all t ∈ ℝ. Then an elementary argumentation involving the equation (241) and the uniform continuity of function p(⋅) on the interval [−a, a], where a = maxt∈K (φ(t) + φ(1) + 1), shows that the function (p ∘ φ)(⋅) is uniformly Poisson c-stable, non-trivial and liml→+∞ (p ∘ φ)(⋅ + 2l ) = (p ∘ φ)(⋅), uniformly on compacts of ℝ. Let us finally consider the case c = 1 and the function p(t) := sin(2πφ(t)/φ(1)), t ∈ ℝ. Then p(⋅) is of period φ(1), and we know that the function (p ∘ φ)(⋅) is uniformly Poisson stable. At the end of this example, we would like to ask whether the function (p ∘ φ)(⋅) is uniformly recurrent (cf. also [22, Proposition 3.6])? Before proceeding to the next subsection, we would like to ask whether we can construct a continuous Levitan (N, c)-almost periodic function f : ℝ → ℝ for any complex number c ≠ 0? 6.4.2 Multi-dimensional Levitan N-almost periodic functions Let us recall that a continuous function F : ℝn → X is almost periodic if and only if for each finite real number ε > 0 there exist relatively dense sets E j in ℝ (1 ⩽ j ⩽ n) such that the set E ≡ ∏nj=1 E j is contained in the set of all ε-almost periods of F(⋅); see, e. g., [431, Introduction, pp. 31–32]. Motivated by this fact, we will first introduce the following notion (let us only note here that we can similarly reconsider the notion of pre-Levitan almost periodicity): Definition 6.4.13. Suppose that F : ℝn → X is a continuous function. Then we say that the function F(⋅) is strongly Levitan N-almost periodic if and only if F(⋅) is pre-Levitan almost periodic and, for every real numbers N > 0 and ε > 0, there exist a finite real j number η > 0 and the relatively dense sets Eη;N in ℝ (1 ⩽ j ⩽ n) such that the set j

Eη;N ≡ ∏nj=1 Eη;N consists solely of (η, N)-almost periods of F(⋅) and Eη;N ± Eη;N ⊆ E(ε, N). It is clear that any strongly Levitan N-almost periodic function F : ℝn → X is Levitan N-almost periodic as well as that the both notions coincide in the one-dimensional setting.

364 � 6 Metrical approximations of functions We continue this section by introducing the following notion, which has been proposed for the first time by V. A. Marchenko in [534] (1950, the one-dimensional setting): Definition 6.4.14. Suppose that F : ℝn → X is a continuous function. Then we say that F(⋅) is Levitan N-almost periodic of type 1 if and only if for each j ∈ ℕn there exists a j real sequence (λk ) such that, for every N > 0 and ε > 0, there exist numbers k ∈ ℕ and j

δ > 0 such that any point τ = (τ1 , τ2 , . . . , τn ) ∈ ℝn satisfying |λl τj | ⩽ δ (mod 2π) for all l ∈ ℕk and j ∈ ℕn is an ε, N-almost period of the function F(⋅). Now we will clarify the following result: Proposition 6.4.15. Consider the next statements: (i) F(⋅) is Levitan N-almost periodic of type 1. (ii) F(⋅) is strongly Levitan N-almost periodic. (iii) F(⋅) is Levitan N-almost periodic. Then (i) ⇔ (ii) and (ii) ⇒ (iii). Proof. Using Kronecker’s theorem, the Bogolyubov theorem, and the argumentation given on [495, pp. 54–55, 57], we can simply prove, as in the one-dimensional setting, that the statements (i) and (ii) are equivalent. The statement (ii) ⇒ (iii) is trivial.

Further on, we can simply show that the Levitan N-almost periodic functions of type 1 form a vector space with the usual operations. Due to Proposition 6.4.15, it follows that the strongly Levitan N-almost periodic functions form a vector space with the usual operations. Concerning this result, it is logical to ask whether the Levitan N-almost periodic functions form a vector space with the usual operations as well as whether the all statements from the formulation of Theorem 6.4.15 are mutually equivalent, i. e. whether the implication (iii) ⇒ (i) is true? Concerning the last question, we would like to notice that the arguments contained on p. 57 of [495] can be repeated once more in order to see that this is true, provided that the following generalization of the Bogolyubov theorem holds true in the higher-dimensional setting: (SB) Suppose that A is a relatively dense set in ℝn and δ > 0. Then there exist a real number η > 0 and a finite set {ω1 , . . . , ωk } in ℝn such that 󵄨 󵄨 {τ ∈ ℝn : 󵄨󵄨󵄨eiωl τ − 1󵄨󵄨󵄨 < η for all l = 1, . . . ., k} ⊆ A − A + A − A + Bδ . Unfortunately, the statement (SB) does not hold in the case that n ⩾ 2, as the following trivial counterexample shows: Example 6.4.16. Suppose that A = ℤn , n ⩾ 2 and δ ∈ (0, 1/4); then A is relatively dense in ℝn . If (SB) is true, then there exist a sufficiently small number η0 > 0 and two real numbers ω1 and ω2 (take k = 1 in (SB)) such that W = {τ ∈ ℝn : |ω1 τ1 + ⋅ ⋅ ⋅ + ωn τn | ⩽ η0 (mod 2π)} ⊆ A − A + A − A + Bδ = A + Bδ . If ω1 = ω2 = 0, then W = ℝn and the

6.4 Metrical almost periodicity: Poisson, Levitan and Bebutov concepts

�

365

contradiction is obvious; otherwise, W is a hyperplane in ℝn and this clearly implies that W cannot be contained in A + Bδ = (A − A) + ⋅ ⋅ ⋅ + (A − A) + Bδ , where we have exactly l addends in the above sum with the number l ∈ ℕ ∖ {1} being arbitrarily chosen. Therefore, it is also important to ask whether the Bogolyubov theorem can be satisfactorily formulated in the higher-dimensional setting? We close this subsection with the observation that we will not reconsider the approximation theorem for multi-dimensional Levitan N-almost periodic functions and the uniqueness theorem for multi-dimensional Levitan N-almost periodic functions here; see [496, pp. 60–62] for the corresponding results established in the onedimensional setting.

6.4.3 Applications to the abstract Volterra integro-differential equations In this subsection, we will present several new theoretical results and their applications to the abstract Volterra integro-differential equations and the classical partial differential equations. 1. Let us recall once more that, in the one-dimensional setting, the analysis of existence and uniqueness of almost periodic type solutions for various classes of the abstract Volterra integro-differential equations without initial conditions leans heavily on the analysis of the qualitative features of the infinite convolution product (4). In [24, 25, 26, 27], M. Akhmet et al. have proposed a new method in the analysis of the existence and uniqueness of unpredictable solutions of linear differential and discrete equations, which is called the method of included intervals by authors. The proof of the following result is almost entirely based on the method of included intervals (cf. also [664, 665], where B. A. Shcherbakov have developed the so-called comparability method by character of recurrence, which is not so easy for comprehension and applications): Proposition 6.4.17. Let 0 ≠ Λ′ ⊆ ℝ, let A be a closed linear operator on X, and let (R(t))t>0 ⊆ L(X) be a strongly continuous operator family such that R(t)A ⊆ AR(t) for ∞ all t > 0 and ∫0 ‖R(t)‖ dt < ∞. Let 𝒫K = C(K) for each non-empty compact set K of ℝ. If f : ℝ → X is a bounded, continuous and Levitan (x, Λ′ , A, 𝒫K )-almost periodic, resp. bounded, continuous and Bebutov (x, Λ′ , A, 𝒫K )-uniformly recurrent, and the function Af : ℝ → X is well defined and bounded, then the function F : ℝ → X, given by (4), is bounded, continuous and Levitan (x, Λ′ , A, 𝒫K )-almost periodic, resp. bounded, continuous and Bebutov (x, Λ′ , A, 𝒫K )-uniformly recurrent. Proof. We will consider only Bebutov (x, Λ′ , A, 𝒫K )-uniformly recurrent functions. It is clear that the function F(⋅) is well-defined and bounded since ∞

F(t) := ∫ R(s)f (t − s) ds, 0

t ∈ ℝ,

366 � 6 Metrical approximations of functions as well as ∫0 ‖R(t)‖ dt < ∞ and the function f (⋅) is bounded. The continuity of F(⋅) simply follows from the dominated convergence theorem. Since R(t)A ⊆ AR(t) for all ∞ t > 0 and the function Af (⋅) is bounded, we have that AF(t) = ∫0 R(s)Af (t − s) ds for all t ∈ ℝ. Let ε > 0, let −∞ < a < b < +∞, and let K = [a, b]. Further on, let −∞ < c < a be such that ∞

∞

󵄩 󵄩 (‖f ‖∞ + ‖Af ‖∞ ) ⋅ ∫ 󵄩󵄩󵄩R(s)󵄩󵄩󵄩 ds < ε/2.

(242)

a−c

Then there exists a sequence (τ k ) in Λ′ such that limk→+∞ |τ k | = +∞ and ∞

󵄩 󵄩 󵄩 󵄩 ( ∫ 󵄩󵄩󵄩R(t)󵄩󵄩󵄩 dt) ⋅ sup 󵄩󵄩󵄩f (t + τk ) − Af (t)󵄩󵄩󵄩 < ε/2. t∈[c,b]

0

(243)

The final conclusion simply follows from the estimates (242)–(243) and the next computation, which holds for every k ∈ ℕ and t ∈ K: 󵄩󵄩 󵄩 󵄩󵄩F(t + τk ) − AF(t)󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩 t 󵄩󵄩 󵄩 = 󵄩󵄩󵄩 ∫ R(t − s)[f (s + τk ) − Af (s)] ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩−∞ 󵄩 c

t

󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ⩽ ∫ 󵄩󵄩󵄩R(t − s)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩f (s + τk ) − Af (s)󵄩󵄩󵄩 ds + ∫󵄩󵄩󵄩R(t − s)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩f (s + τk ) − Af (s)󵄩󵄩󵄩 ds c

−∞

∞

󵄩 󵄩 ⩽ (‖f ‖∞ + ‖Af ‖∞ ) ⋅ ∫ 󵄩󵄩󵄩R(s)󵄩󵄩󵄩 ds c

a−c

󵄩 󵄩 󵄩 󵄩 + (∫󵄩󵄩󵄩R(t − s)󵄩󵄩󵄩 ds) ⋅ sup 󵄩󵄩󵄩f (s + τk ) − Af (s)󵄩󵄩󵄩 ⩽ (ε/2) + (ε/2) = ε. t

s∈[c,b]

Remark 6.4.18. (i) Suppose that νK : K → (0, ∞) satisfies that the function 1/νK (⋅) is bounded. Then the use of weighted function space Cb,νK (K) seems to be discutable here, since the argumentation contained in the proof given above indicates that the function νK (⋅) has to be bounded, as well, unless we assume certain very restrictive conditions. (ii) The statement of Proposition 6.4.17 can be transferred to the multi-dimensional setting without any serious difficulty; cf. [431] for more details. (iii) For simplicity, let us consider here the class of Levitan (x, ℝ, I, 𝒫K )-almost periodic functions. If we additionally suppose that for each N > 0 and ε > 0 there exists a relatively dense set Eη;N of (η, N)-almost periods of f (⋅) such that Eη;N ± Eη;N ⊆ E(ε, N), then the same property holds for the function F(⋅).

6.4 Metrical almost periodicity: Poisson, Levitan and Bebutov concepts

�

367

As mentioned many times before, our results about the convolution invariance of generalized metrical Levitan almost periodicity and generalized metrical Bebutov uniform recurrence under the actions of the infinite convolution product (4) can be successfully applied in the qualitative analysis of solutions for a large class of the abstract (degenerate) Volterra integro-differential equations without initial conditions; for example, we can consider the almost periodic type solutions for the following Volterra integro-differential equation: t

u′ (t) = Au(t) + ∫ a(t − s)Au(s) + f (t),

t ∈ ℝ,

(244)

−∞

where A is a closed linear operator in X, a(t) = αgη (t)e−βt , t > 0 and f : ℝ → X is a continuous function obeying certain properies (α ≠ 0, β > 0, η ⩾ 1); cf. [189, Chapter 3] for more details about the subject. We can also consider the semilinear analogues of (244). Concerning the existence and uniqueness of almost periodic solutions of the equation t

u (t) = Au(t) + ∫ B(t − s)u(s) + f (t), ′

t ∈ ℝ,

−∞

as well as the existence and uniqueness of asymptotically almost periodic solutions of the equation t

u′ (t) = Au(t) + ∫ B(t − s)u(s) + f (t),

t ⩾ 0; u(0) = x,

0

where A and B(⋅) are closed linear operator in X, we may refer, e. g. to [366] and [431, pp. 230–232] and references cited therein. 2. Now we will continue our investigations by examining the convolution invariance of Levitan almost like periodicity. In actual fact, we would like to observe that the method of included intervals enables one to simply deduce the following result: Proposition 6.4.19. Let 0 ≠ Λ′ ⊆ ℝn , let A be a closed linear operator on X, h ∈ L1 (ℝn ), and let 𝒫K = C(K) for each non-empty compact set K of ℝ. If f : ℝn → X is bounded, continuous, and Levitan (x, Λ′ , A, 𝒫K )-almost periodic, resp. bounded, continuous, and Bebutov (x, Λ′ , A, 𝒫K )-uniformly recurrent, and the function Af : ℝn → X is well defined and bounded, then the function H : ℝn → X, given by H(t) := (h ∗ f )(t) := ∫ h(t − s)f (s) ds,

t ∈ ℝn ,

ℝn

is bounded, continuous and Levitan (x, Λ′ , A, 𝒫K )-almost periodic, resp. bounded, continuous and Bebutov (x, Λ′ , A, 𝒫K )-uniformly recurrent.

368 � 6 Metrical approximations of functions Remark 6.4.20. (i) Unfortunately, we cannot formulate a satisfactory analogue of Proposition 6.4.19 in the case that there exist two finite real numbers b > 0 and c > 0 such that ‖f (t)‖ ⩽ c(1 + |t|)b , t ∈ ℝn and h(⋅)[1 + | ⋅ |]b ∈ L1 (ℝn ), when the function H(⋅) is clearly well-defined. (ii) Let us consider the class of Levitan (x, ℝ, I, 𝒫K )-almost periodic functions. If we suppose, additionally, that for each N > 0 and ε > 0 there exists a relatively dense set Eη;N of (η, N)-almost periods of f (⋅) such that Eη;N ± Eη;N ⊆ E(ε, N), then the same property holds for the function H(⋅). Taken together with the statement of Proposition 6.4.19, this provides a proper extension of the well known result [171, Theorem 13], proved for the first time by D. Bugajewski, X. Gan and P. Kasprzak. It is clear how we can incorporate Proposition 6.4.19 in the analysis of the existence and uniqueness of the Levitan almost periodic type solutions for some classes of (fractional) partial differential equations. 3. We can similarly consider the existence and uniqueness of Levitan almost periodic type solutions of the inhomogeneous abstract Cauchy problems considered in [194, Example 1.1], the backward wave equation considered in [797, Example 8, p. 33], and provide certain applications to the abstract ill-posed Cauchy problems considered on [431, pp. 543–545]. 4. Concerning semilinear Cauchy problems, we will only note that the composition principle clarified in [431, Theorem 7.1.18] can be simply formulated for the uniformly Poisson c-stable functions. Keeping in mind Proposition 6.4.19, we can simply consider the existence and uniqueness of the uniformly Poisson c-stable solutions for the class of semilinear Hammerstein integral equations of convolution type on ℝn ; see [431, p. 470]. Without going into full details, we will only note that the statement of [171, Theorem 12] and some composition principles clarified in [431, Subsection 6.1.5] can be formulated in our new framework. 5. We would like to notice that the convolution invariance of Levitan N-almost periodicity has recently been considered by A. Nawrocki in [577, Theorem 4.2, Theorem 4.4, Theorem 4.8]. Concerning these statements, clarified in the one-dimensional setting, we would like to make the following comments: (i) Theorem 4.2 can be formulated for the uniformly Poisson stable (Levitan N-almost periodic) functions f : ℝ → Y . Then the convolution f ∗g exists and it is a uniformly Poisson stable (Levitan N-almost periodic) function. (ii) Theorem 4.4 can be formulated for the Stepanov bounded, uniformly Poisson stable (Levitan N-almost periodic) functions f : ℝ → Y . The resulting convolution f ∗ gλ exists and it is a bounded, uniformly Poisson stable (Levitan N-almost periodic) function. (iii) Concerning Theorem 4.8, we will only note that the Stepanov unboundedness condition [577, (4.2)] is not suitable for work with the uniformly Poisson stable functions. Let us assume, in place of this condition, that for each real number ω > 0 and for each strictly increasing sequence (αk ) of positive integers we have

6.4 Metrical almost periodicity: Poisson, Levitan and Bebutov concepts

lim sup k→+∞

(αk +1)ω

∫

αk ω

�

369

f (t) dt = +∞.

Then the resulting convolution f ∗ gλ exists and it is a bounded, uniformly Poisson stable function. The statement can be also formulated for the vector-valued functions. In [577, Section 5], the author has applied the above results (cf. also [577, Corollaries 4.11–4.14]) in the analysis of the existence and uniqueness of Levitan N-almost periodic solutions to the linear ordinary differential equation of first order y′ (x) = λy(x) + f (x),

x ∈ ℝ.

The statements of [577, Lemma 5.5, Theorem 5.10(i)] can be formulated for the uniformly Poisson stable solutions of this equation. 6. We can continue our recent investigations of the wave equation in ℝn , where n ⩽ 3. We will only consider here the wave equation utt = a2 uxx in domain {(x, t) : x ∈ ℝ, t > 0}, equipped with the initial conditions u(x, 0) = f (x) ∈ C 2 (ℝ) and ut (x, 0) = g(x) ∈ C 1 (ℝ). As is well known, its unique regular solution is given by the d’Alembert formula (85). Our consideration from [194, Example 1.2] shows that the solution u(x, t) can be extended to the whole real line in time variable and that u(x, t) will be uniformly ⋅ c-Poisson stable in (x, t) ∈ ℝ2 , provided that the function t 󳨃→ (f (⋅), ∫0 g(s) ds), t ∈ ℝ is uniformly c-Poisson stable, with the meaning clear (c ∈ ℂ ∖ {0}). We would like to emphasize that it is not clear how one can prove that the solution u(x, t) will be Levi⋅ tan N-almost periodic in (x, t) ∈ ℝ2 , provided that the functions f (⋅) and ∫0 g(s) ds are Levitan N-almost periodic. Finally, we would like to mention some topics not considered in our former work: 1. The Levitan and Bebutov approaches to the metrical approximations by trigonometric polynomials and ρ-periodic type functions can be further generalized using the approaches of Stepanov, Weyl and Besicovitch. Many structural results established in the previous section can be reformulated in this context. 2. It is well known that every Levitan N-almost periodic function f : ℝ → ℂ can be represented by the uniform limit of ratios of scalar-valued almost periodic functions (see A. G. Baskakov [93]). We will not discuss here the question whether this result continues to hold for multi-dimensional Levitan N-almost periodic functions. 3. In this section, we have not analyzed 𝔻-asymptotically Levitan (Bebutov) almost periodic type functions in general metric. Concerning this issue, we will only note that we have recently provided some new results about the existence and uniqueness of 𝔻-asymptotically almost automorphic solutions of the inhomogeneous wave equation utt (t, x) − d 2 Δx u(t, x) = f (t, x), u(0, x) = g(x),

ut (0, x) = h(x),

x ∈ ℝ2 , t > 0;

370 � 6 Metrical approximations of functions where d > 0, f (t, x) is continuously differentiable in the variable t ∈ ℝ and continuous in the variable x ∈ ℝ, g ∈ C 2 (ℝ2 : ℝ) and h ∈ C 1 (ℝ2 : ℝ). It is well known that the unique solution of this problem is given by x+at

1 1 u(x, t) = [g(x − at) + g(x + at)] + ∫ h(s) ds 2 2a x−at

+

t

x+d(t−s)

0

x−d(t−s)

1 ∫[ 2d

∫

f (r, s) dr] ds,

x ∈ ℝ, t > 0,

so that the conclusions established on [431, pp. 541–542] can be also formulated for 𝔻-asymptotically uniformly Poisson stable solutions (defined in the obvious way), for example. Details can be left to the interested readers.

7 Special classes of metrically almost periodic functions 7.1 Hölder ρ-almost periodic type functions in ℝn The main aim of this section is to explore several new classes of multi-dimensional Hölder ρ-almost periodic type functions [445]. Here we use the special (pseudo-)metric spaces analyzed by S. Stoiński [710] within the framework of the theory of onedimensional scalar-valued almost periodic functions; cf. also [707, 708, 711, 713]. As before, the considered functions are of the form F : Λ × X → Y , where 0 ≠ Λ ⊆ ℝn , X and Y are complex Banach spaces. At the very beginning, we would like to emphasize that the situation in which Y ≠ ℝ is not simple, because in this setting the Lagrange mean value theorem does not allow a complete generalization (see, e. g. the formulations of Theorem 7.1.15(ii) and Theorem 7.1.16(ii), which do not admit the converses in the vector-valued case; for further information on the subject we refer the reader to the research articles [541] by J. Matkowski, [545] by R. McLeod, [796] by P. P. Zabreiko and references cited therein). Nevertheless, we can freely say that we have basically achieved our aims, substantially expanding the investigation [710] carried out by S. Stoiński. Several new results, observations and illustrative examples about onedimensional Hölder ρ-almost periodic type functions are given. It is also worth noting that we have proved two extension type theorems for the Hölder-α-almost periodic functions and presented certain results about the invariance of Hölder-α-almost periodicity under the actions of the infinite convolution products, providing also certain applications to the abstract Volterra integro-differential equations and the classical PDEs. The organization and main ideas of this section can be briefly described as follows. Multi-dimensional Hölder ρ-almost periodic type functions are thoroughly analyzed in Section 7.1.1. Suppose that 0 < α ⩽ 1; in Definition 7.1.3, we introduce the notion of a strongly (𝔽, ℬ)-Hölder-α-almost periodic [of type 1] (semi-(ρ, 𝔽, ℬ)-Hölder-α-periodic [of type 1], semi-(ρj , 𝔽, ℬ)j∈ℕn -Hölder-α-periodic [of type 1]) function F : Λ × X → Y . The corresponding classes of Lipschitz almost periodic type functions are obtained by plugging α = 1. In Definition 7.1.4, we introduce the classes of (𝔽, ℬ, Λ′ , ρ)-Hölder-α-almost periodic functions [of type 1] and (𝔽, ℬ, Λ′ , ρ)-Hölder-α-uniformly recurrent functions [of type 1]. Further on, in Definition 7.1.7, we introduce the notion of a strongly uniformly (𝔽, ℬ)-Hölder-α-almost periodic function [of type 1] (uniformly semi-(ρ, 𝔽, ℬ)-Hölder-αperiodic function [of type 1], uniformly semi-(ρj , 𝔽, ℬ)j∈ℕn -Hölder-α-periodic function [of type 1]); cf. also Definition 7.1.8. This concept is new even in the one-dimensional setting and can be constituted for the function spaces introduced in Definition 7.1.4. The notion introduced in Definition 7.1.7 is extremely important for us because the later investigation of the invariance of the Hölder-α-almost periodicity under the actions of convolution products, which are crucial for applications (see Section 7.1.4), shows that https://doi.org/10.1515/9783111233871-009

372 � 7 Special classes of metrically almost periodic functions the established results cannot be clarified for the corresponding notion introduced in [710], Definition 7.1.3 and Definition 7.1.4. The main results about the introduced function spaces are given in Theorem 7.1.6, Theorem 7.1.15, Theorem 7.1.16, Corollary 7.1.17, Corollary 7.1.18, and Theorem 7.1.24; for example, in Corollary 7.1.17(i) we prove that a continuously differentiable function f : ℝ → ℝ is Lipschitz almost periodic if and only if f (⋅) is uniformly Lipschitz almost periodic if and only if the functions f (⋅) and f ′ (⋅) are almost periodic in the usual sense. Theorem 7.1.22 clarifies an interesting sufficient condition for an absolutely continuous function f : ℝ → Y to be Lipschitz almost periodic. Without going into full details, we will only note that the notions of the Hölder-α-(𝔽, ℬ)-boundedness and the Hölder-α-(𝔽, ℬ, Λ′ , ρ)-continuity are examined in Section 7.1.2, as well as that the extension type theorems for the Hölder ρ-almost periodic type functions are examined in Section 7.1.3. Some applications of our results to the abstract Volterra integro-differential equations are given in the already mentioned Section 7.1.4 and Section 7.1.5. In addition to the above, several comments and interpretive examples are given in Remark 7.1.13, Remark 7.1.29, Remark 7.1.30, Examples 7.1.10–7.1.12 and Example 7.1.20, for instance; some open problems are proposed, as well. In this section, we will not consider the Hölder ρ-almost periodic type functions on time scales and related applications to the (semilinear) dynamic differential equations; concerning these subjects, we refer the readers to the research articles [748, 749, 750] by C. Wang, R. P. Agarwal, and D. O’Regan, as well as the research articles [750, 751] by C. Wang et al. For the sequel, we need to state the following analogue of [194, Proposition 2.18] (the proof is very similar to the proof of the above-mentioned result and therefore omitted): Proposition 7.1.1. Suppose that k ∈ ℕ, the function Fi : ℝn × X → Yi is Bohr ℬ-almost periodic and satisfies that supx∈B;t∈ℝn ‖Fi (t; x)‖Y < +∞ for all B ∈ ℬ and 1 ⩽ i ⩽ k. Then the function (F1 , . . . , Fk )(⋅; ⋅) is also Bohr ℬ-almost periodic. In this section, we assume that 0 ≠ Λ ⊆ ℝn is a closed set which coincides with the set of its accumulation points, F : Λ × X → Y is a continuous mapping and 0 < α ⩽ 1. 7.1.1 Multi-dimensional Hölder ρ-almost periodic type functions First, if B ∈ ℬ and t ∈ Λ, then we define Lα,B (t, δ; F) := sup

sup

x∈B u1 ,u2 ∈Λ∩B(t,δ);u1 =u ̸ 2

‖F(u1 ; x) − F(u2 ; x)‖Y . |u1 − u2 |α

We say that a point t ∈ Λ is an α-regular point of function F(⋅; ⋅) if and only if for every set B ∈ ℬ there exists a finite real number δ > 0 such that Lα,B (t, δ; F) < +∞. After that, 1,ℬ we define P0,α (Λ × X : Y ) as the set of all functions F : Λ × X → Y satisfying that each

7.1 Hölder ρ-almost periodic type functions in ℝn

�

373

1,ℬ point t ∈ Λ is an α-regular point of function F(⋅; ⋅). If F ∈ P0,α (Λ × X : Y ) and B ∈ ℬ, then we define

Lα,B (t; F) := lim Lα,B (t, δ; F), δ→0+

t ∈ Λ.

1,ℬ ℬ Finally, by P0,α (Λ × X : Y ) we denote the set of all functions F ∈ P0,α (Λ × X : Y ) satisfying that for each set B ∈ ℬ the function t 󳨃→ Lα,B (t, 1; F), t ∈ Λ is locally bounded. It is simply 1,ℬ ℬ shown that P0,α (Λ×X : Y ) and P0,α (Λ×X : Y ) are vector spaces with the usual operations as well as that these spaces are translation invariant in the usual sense. If X = {0} and ℬ ℬ = {X}, then we abbreviate P0,α (Λ × X : Y ) to P0,α (Λ : Y ); similarly, if X = {0} and ℬ = {X}, then we abbreviate Lα,B (t, δ; F) to Lα (t, δ; F), and Lα,B (t; F) to Lα (t; F).

Remark 7.1.2. Without going into full details, we will only note that, in our definitions, ℬ we require that a Hölder-α-almost periodic function belongs to the space P0,α (Λ × X : Y ), which is a slightly redundant condition (we essentially employ this condition only in the proof of Proposition 7.1.27 below). This condition can be slightly relaxed by assuming that for each set B ∈ ℬ there exists a finite real number δ ∈ (0, 1] such that the function t 󳨃→ Lα,B (t, δ; F), t ∈ Λ is locally bounded. It can be simply proved that the space 𝒫α = (Pα , dα ) is a metric space if Pα = Cb (Λ : Y ) ∩ P0,α (Λ : Y ) and 󵄩 󵄩 dα (f , g) := sup(󵄩󵄩󵄩f (t) − g(t)󵄩󵄩󵄩Y + lim Lα (t, δ; f − g)); δ→0+

t∈Λ

here Cb (Λ : Y ) denotes the Banach space of all bounded continuous functions f : Λ → Y equipped with the sup-norm. Moreover, it can be simply proved that the space 𝒫α1 = (Pα1 , dα1 ) is a pseudometric space if Pα1 = P0,α (Λ : Y ) and dα (f , g) := sup lim Lα (t, δ; f − g); t∈Λ δ→0+

this space is not a metric space, because we have dα (x 2 , 0) = 0 for all α ∈ (0, 1] with Λ = ℝ, for example. Now we are able to consider the following special cases of the notion introduced in [190, Section 4]; for simplicity, we consider here the situation in which ϕ(x) ≡ x: Definition 7.1.3. We say that F(⋅; ⋅) is strongly (𝔽, ℬ)-Hölder-α-almost periodic (semi(ρ, 𝔽, ℬ)-Hölder-α-periodic, semi-(ρj , 𝔽, ℬ)j∈ℕn -Hölder-α-periodic) if and only if for each B ∈ ℬ there exists a sequence (PkB (t; x)) of trigonometric polynomials (ρ-periodic functions, (ρj )j∈ℕn -periodic functions) such that 󵄩󵄩 󵄩󵄩 lim sup󵄩󵄩󵄩𝔽(⋅)(PkB (⋅; x) − F(⋅; x))󵄩󵄩󵄩 = 0. 󵄩Pα

k→+∞ x∈B 󵄩

(245)

374 � 7 Special classes of metrically almost periodic functions Furthermore, if we replace the metric space Pα with the pseudometric space Pα1 in (245), then we obtain the notion of a strongly (𝔽, ℬ)-Hölder-α-almost periodic function of type 1 (semi-(ρ, 𝔽, ℬ)-Hölder-α-almost periodic function of type 1, semi-(ρj , 𝔽, ℬ)j∈ℕn -Hölder-αalmost periodic function of type 1). The corresponding classes of Lipschitz almost periodic type functions (of type 1) are obtained with α = 1; for example, a strongly (𝔽, ℬ)-Hölder-1-almost periodic function is also called strongly (𝔽, ℬ)-Lipschitz-almost periodic. Definition 7.1.4. Suppose that 0 ≠ Λ′ ⊆ ℝn , F : Λ × X → Y is a given function, ρ is a binary relation on Y , 𝔽 : Λ → (0, ∞) and Λ′ ⊆ Λ′′ . Then we say that: (i) F(⋅; ⋅) is (𝔽, ℬ, Λ′ , ρ)-Hölder-α-almost periodic if and only if for every B ∈ ℬ and ε > 0 there exists l > 0 such that for each t0 ∈ Λ′ there exists τ ∈ B(t0 , l) ∩ Λ′ such that, for every t ∈ Λ and x ∈ B, there exists an element yt;x ∈ ρ(F(t; x)) such that 󵄩󵄩 󵄩󵄩 sup󵄩󵄩󵄩𝔽(⋅)(F(⋅ + τ; x) − y⋅;x )󵄩󵄩󵄩 ⩽ ε. 󵄩 Pα x∈B 󵄩

(246)

(ii) F(⋅; ⋅) is (𝔽, ℬ, Λ′ , ρ)-Lipschitz-uniformly recurrent if and only if for every B ∈ ℬ there exists a sequence (τ k ) in Λ′ such that limk→+∞ |τ k | = +∞ and that for every t ∈ Λ and x ∈ B, there exists an element yt;x ∈ ρ(F(t; x)) such that 󵄩󵄩 󵄩󵄩 lim sup󵄩󵄩󵄩𝔽(⋅)(F(⋅ + τ k ; x) − y⋅;x )󵄩󵄩󵄩 = 0. 󵄩Pα

k→+∞ x∈B 󵄩

The corresponding classes of (𝔽, ℬ, Λ′ , ρ)-Hölder-α-almost periodic functions of type 1 and (𝔽, ℬ, Λ′ , ρ)-Hölder-α-uniformly recurrent functions of type 1 are introduced similarly as in Definition 7.1.3; the corresponding Lipschitz classes are obtained by plugging α = 1. In the sequel, we remove the term “𝔽” [“ρ”; “Λ′ ”] from the notation if 𝔽 ≡ 1 [ρ = I; Λ = Λ]; furthermore, we write “c” in place of “ρ” if ρ = cI for some c ∈ ℂ ∖ {0}. It is clear that any strongly (𝔽, ℬ)-Hölder-α-almost periodic function is strongly (𝔽, ℬ)-Hölder-α-almost periodic of type 1 as well as that the assumption 0 < α′ ⩽ α′′ ⩽ 1 implies that any (𝔽, ℬ)-Hölder-α′′ -almost periodic function (of type 1) is (𝔽, ℬ)-Hölder-α′ almost periodic (of type 1); this also holds for all other classes of functions introduced above. Since the inequality ′

󵄨󵄨 󵄨 󵄨󵄨[‖a‖Y − ‖b‖Y ] − [‖c‖Y − ‖d‖Y ]󵄨󵄨󵄨 ⩽ 󵄩󵄩󵄩[a − b] − [c − d]󵄩󵄩󵄩 󵄨󵄨 󵄨󵄨 󵄩 󵄩Y and its reverse inequality do not hold for arbitrary elements a, b, c, d of the Banach space Y , we cannot expect that the (𝔽, ℬ, Λ′ , ρ)-Hölder-α-almost periodicity (of type 1) of function F : Λ×X → Y , for example, implies the (𝔽, ℬ, Λ′ , ρ)-Hölder-α-almost periodicity (of type 1) of function ‖F‖Y : Λ × X → ℂ. It could be interesting to construct, if exists, a

7.1 Hölder ρ-almost periodic type functions in ℝn

�

375

Lipschitz almost periodic function f : ℝ → ℝ such that the function |f |(⋅) is not Lipschitz almost periodic. We continue with the following observation: Remark 7.1.5. With the notion introduced in [708] and [710], S. Stoiński has proved that any Lipschitz almost periodic function is almost periodic in variation. We feel it is our duty to say that the proof of this result, which has not been stated as a proposition or a theorem, is a little bit misleading and incomplete. For example, the notion of number δn′ in the equation [710, (1), p. 690] is not clear as well as the way how this equation implies the inequality stated on the fifth line on [710, p. 691], with the term 1 + δn′ appeared. Fortunately, the result is correct and we can use the Heine–Borel theorem here. In actual fact, due to the definition of an Lipschitz almost periodic function f : ℝ → ℝ, for every real number ε > 0 we have the existence of a relatively dense set R ⊆ ℝ such that for every τ ∈ R, we have |f (t + τ) − f (t)| ⩽ ε/2, t ∈ ℝ and lim

sup

δ→0+ u ,u ∈[t−δ,t+δ];u =u 1 2 1 ̸ 2

|[f (u2 + τ) − f (u2 )] − [f (u1 + τ) − f (u1 )]| ⩽ ε/2, |u1 − u2 |

t ∈ ℝ.

(247)

The equation (247) means that for every τ ∈ R and t ∈ ℝ, there exists a finite real number δt > 0 such that sup

u1 ,u2 ∈[t−δt ,t+δt ];u1 =u ̸ 2

|[f (u2 + τ) − f (u2 )] − [f (u1 + τ) − f (u1 )]| ⩽ ε/2. |u1 − u2 |

Due to the Heine–Borel theorem, for every τ ∈ R and t ∈ ℝ, there exists a finite real number δ > 0 such that the previous estimate holds for every distinct points u1 , u2 ∈ [t − 1, t + 1] such that |u1 − u2 | < δ. If P is any partition of the interval [t − 1, t + 1], then we can refine it by adding some points such that the mesh of a newly formed partition P′ is strictly less than δ; then it can be simply proved that V[t−1,t+1] (fτ − f ) ⩽ 2 × (ε/2) = ε. The consideration from the previous remark is simply generalized in our next result, stated here only for the class of one-dimensional (ℬ, Λ′ , ρ)-Lipschitz-almost periodic functions: Theorem 7.1.6. Suppose that Λ = [a, ∞) for some a ∈ ℝ or Λ = ℝ, F : Λ × X → Y and ℬ consists solely of compact subsets of X. If the function F(⋅; ⋅) is (ℬ, Λ′ , ρ)-Lipschitz-almost periodic, then F(⋅; ⋅) is (ℬ, Λ′ , ρ)-almost periodic in variation, which means that for each B ∈ ℬ and ε > 0 there exists l > 0 such that for each t0 ∈ Λ′ there exists τ ∈ B(t0 , l) ∩ Λ′ such that, for every t ∈ Λ and x ∈ B, there exists an element yt;x ∈ ρ(F(t; x)) such that 󵄩 󵄩 sup sup(󵄩󵄩󵄩F(⋅ + τ; x) − y⋅;x 󵄩󵄩󵄩Y + V[t−1,t+1]∩Λ (F(⋅ + τ; x) − y⋅;x )) ⩽ ε. x∈B t∈Λ

The considerations from Remark 7.1.5 and Theorem 7.1.6 stimulate us to introduce the following stronger versions of Definition 7.1.3 as well:

376 � 7 Special classes of metrically almost periodic functions Definition 7.1.7. We say that F(⋅; ⋅) is strongly uniformly (𝔽, ℬ)-Hölder-α-almost periodic (of type 1) [uniformly semi-(ρ, 𝔽, ℬ)-Hölder-α-periodic (of type 1), uniformly semi-(ρj , 𝔽, ℬ)j∈ℕn -Hölder-α-periodic (of type 1)] if and only if for each B ∈ ℬ and ε > 0 there exist a real number δ > 0 and a trigonometric polynomial (ρ-periodic function, (ρj )j∈ℕn -periodic function) (PB (t; x)) such that 󵄩 󵄩 sup sup 𝔽(t)[󵄩󵄩󵄩F(t; x) − PB (t; x)󵄩󵄩󵄩Y x∈B t∈Λ

+

sup

u1 ,u2 ∈Λ∩B(t,δ);u1 =u ̸ 2

(sup sup 𝔽(t)[ x∈B t∈Λ

‖[PB − F](u1 ; x) − [PB − F](u2 ; x)‖Y ] 0 there exists a trigonometric polynomial (ρ-periodic function, (ρj )j∈ℕn -periodic function) (PB (t; x)) such that 󵄩 󵄩 sup sup 𝔽(t)[󵄩󵄩󵄩F(t; x) − PB (t; x)󵄩󵄩󵄩Y + x∈B t∈Λ

(sup 𝔽(t)[ x∈B

sup

u1 ,u2 ∈Λ;u1 =u ̸ 2

sup

u1 ,u2 ∈Λ;u1 =u ̸ 2

‖[PB − F](u1 ; x) − [PB − F](u2 ; x)‖Y ] 0, Λ := {(x, y) ∈ ℝ2 : |x − y| ⩾ L} and Λ′ := {(τ, τ) : τ ∈ ℝ}. Suppose further that f : ℝ → ℝ and g : ℝ → ℝ are continuous periodic functions of period τ0 > 0. Define u(x, y) := (f (x)+g(y))/(x −y), (x, y) ∈ Λ. Then it can be easily shown that the function u(x, y) is Λ-uniformly Λ′ -Hölder-α-almost periodic for each α ∈ (0, 1]. We continue by providing a few other illustrative examples:

7.1 Hölder ρ-almost periodic type functions in ℝn

� 377

Example 7.1.10. Suppose that p(t) := sin t + sin(√2t), t ∈ ℝ. Although we will deduce a much more general result in Corollary 7.1.18, we will directly prove here that p(⋅) is uniformly Lipschitz almost periodic, i. e. that for each ε > 0 there exist finite real numbers l > 0 and δ > 0 such that for every t0 ∈ ℝ, there exists τ ∈ [t0 − l, t0 + l] such that Au1 ,u2 ,τ 󵄨󵄨 󵄨󵄨 sup(󵄨󵄨󵄨[sin(t + τ) + sin(√2(t + τ))] − [sin t + sin(√2t)]󵄨󵄨󵄨 + sup ) ⩽ ε, 󵄨 󵄨 |u t∈ℝ u1 ,u2 ∈B(t,δ);u1 =u ̸ 2 1 − u2 | where 󵄨󵄨 Au1 ,u2 ,τ := 󵄨󵄨󵄨[sin(u1 + τ) + sin(√2(u1 + τ))] − [sin u1 + sin(√2u1 )] 󵄨 󵄨󵄨 − ([sin(u2 + τ) + sin(√2(u2 + τ))] − [sin u2 + sin(√2u2 )])󵄨󵄨󵄨. 󵄨 Let τ be an (ε/10)-period of the function t 󳨃→ (sin t + sin √2t, cos t, cos √2t), t ∈ ℝ, and let δ > 0 be chosen such that, for every two distinct points u1 , u2 ∈ ℝ with |u1 − u2 | ⩽ δ, we have: 󵄨󵄨 sin(u + τ) − sin(u + τ) 󵄨󵄨 󵄨󵄨 󵄨 1 2 − cos(u2 + τ)󵄨󵄨󵄨 ⩽ ε/8, 󵄨󵄨 󵄨󵄨 󵄨󵄨 u1 − u2 󵄨󵄨 sin √2(u + τ) − sin √2(u + τ) 󵄨󵄨 󵄨󵄨 󵄨 1 2 − 2 cos(√2(u2 + τ))󵄨󵄨󵄨 ⩽ ε/8, 󵄨󵄨 󵄨󵄨 󵄨󵄨 u1 − u2 󵄨󵄨 sin(u ) − sin(u ) 󵄨󵄨 1 2 󵄨󵄨󵄨 󵄨󵄨󵄨 ⩽ ε/8 − cos(u ) 2 󵄨󵄨 󵄨󵄨 u1 − u2 󵄨 󵄨 and 󵄨󵄨 sin(√2u ) − sin(√2u ) 󵄨󵄨 󵄨󵄨 󵄨 1 2 − 2 cos(√2u2 )󵄨󵄨󵄨 ⩽ ε/8; 󵄨󵄨 󵄨󵄨 󵄨󵄨 u1 − u2 see [710, l. -1, p. 691]. Then, for every two distinct points u1 , u2 ∈ ℝ with |u1 − u2 | ⩽ δ, we have 󵄨󵄨 A 󵄨󵄨 󵄨󵄨 u1 ,u2 ,τ 󵄨󵄨 󵄨󵄨 󵄨 √ √ 󵄨󵄨 u − u − [cos(u2 + τ) + 2 cos( 2(u2 + τ)) − cos u2 − 2 cos( 2u2 )]󵄨󵄨󵄨 ⩽ ε/2, 󵄨󵄨 1 󵄨󵄨 2 which simply implies the required conclusion. It is worth noting that the function p(⋅) is uniformly Lipschitz almost anti-periodic as well, which can be proved in a similar fashion by finding a relatively dense set R ⊆ ℝ such that for every τ ∈ R, we have | sin(t + τ) + sin t| ⩽ ε, | sin(√2(t + τ)) + sin(√2t)| ⩽ ε, | cos(t + τ) + cos t| ⩽ ε and | cos(√2(t +τ))+cos(√2t)| ⩽ ε for all t ∈ ℝ. This follows from the existence of a relatively dense set R ⊆ ℝ such that for every τ ∈ R, we have | cos(τ/2)| + | cos(√2τ/2)| ⩽ ε/2, the equality sin(t +τ)+sin t = 2 sin(t +(τ/2)) cos(τ/2), the equality cos(t +τ)+cos t = 2 cos(t + (τ/2)) cos(τ/2) and the corresponding equalities for the terms sin(√2(t + τ)) + sin(√2t)

378 � 7 Special classes of metrically almost periodic functions and cos(√2(t + τ)) + cos(√2t); before proceeding further, we will only note that we can similarly prove that the vector-valued function t 󳨃→ (sin(α1 t), sin(α2 t), . . . , sin(αk t), cos(β1 t), cos(β2 t), . . . , cos(βj t)),

t ∈ ℝ,

where k, j ∈ ℕ and αi , βj ∈ ℝ ∖ {0}, is almost anti-periodic; cf. [428] for the notion and more details. Example 7.1.11. Suppose that 0 < α < β < 1. A very simple method for the construction of Hölder-α-almost periodic functions which are not Lipschitz almost periodic is presented in [710, Theorem 9]. Concerning this result, we will only note here that the same arguments can be used for the construction of a uniformly Hölder-α-almost antiperiodic function f : ℝ → ℝ which is not Lipschitz almost periodic. Put, for example, f (t) := | sin t + sin(√2t)|β , t ∈ ℝ. This function is almost anti-periodic and a very simple argumentation shows that the function f (⋅) enjoys the required properties. Example 7.1.12. The function f (⋅), given by (214), is not periodic and f (⋅) is semi-antiperiodic (n = 1, c1 = −1). We already know that f (⋅) is a p-semi-anti-periodic function in variation (1 ⩽ p < +∞). In this part, we would like to emphasize that the function f (⋅) is ℝ-uniformly semi-anti-Lipschitz-periodic, i. e., ℝ-uniformly semi-(−I, 𝒫1 )-periodic as well. This essentially follows from the estimate 󵄨󵄨 ∞ 󵄨 ∞ iu1 /(2m+1) 󵄨󵄨 − eiu2 /(2m+1) 󵄨󵄨󵄨󵄨 1 󵄨󵄨 ∑ e ⩽ ∑ 󵄨 󵄨󵄨 󵄨 2 (u − u ) 2 (2m + 1) 󵄨 m m 󵄨󵄨m=N+1 󵄨󵄨 m=N+1 1 2

(N ∈ ℕ; u1 , u2 ∈ ℝ).

Let us also note that we can similarly prove that the function f (⋅), given by (217), is ℝ-uniformly semi-anti-Lipschitz-periodic of type 1 as well. We continue with the following important observation: Remark 7.1.13. Suppose that 0 < α′ < α′′ ⩽ 1 and the function F : Λ → Y is locally Hölder α′′ -continuous, i. e. for each compact set K ⊆ ℝn we have the existence of a finite ′′ real constant cK > 0 such that ‖F(u1 ) − F(u2 )‖Y ⩽ cK |u1 − u2 |α for all u1 , u2 ∈ K ∩ Λ. Then for each point t ∈ Λ we have limδ→0+ Lα (t, δ; F) = 0, the space 𝒫α′ coincides with Cb (Λ : Y ), and the function F : Λ → Y is Hölder-α′ -almost periodic (Hölder-α′ -uniformly recurrent) if and only if F(⋅) is Bohr almost periodic (uniformly recurrent) in the usual sense. In particular, if the function F : ℝn → Y is continuously differentiable and 0 < α′ < 1, then F(⋅) is Hölder-α′ -almost periodic (Hölder-α′ -uniformly recurrent) if and only if F(⋅) is Bohr almost periodic (uniformly recurrent) in the usual sense. Example 7.1.14. We would like to note that we will not consider the Stepanov, Weyl and Besicovitch generalizations of Hölder-α-almost periodic type functions here. In connection with this subject, we would like to note that we have already shown that the Stepanov-p-almost periodic function

7.1 Hölder ρ-almost periodic type functions in ℝn

f (t) := sin(

1 ), 2 + cos t + cos(√2t)

�

379

t∈ℝ

is not Lipschitz S p -almost periodic. Here we would like to note that the function f ̂(⋅) is Hölder-α-almost periodic for every exponent α ∈ (0, 1) since f ̂(⋅) is continuously differentiable, as easily shown, and almost periodic in the usual sense (see Remark 7.1.13). In our next result, we consider the situation in which the function F : ℝn → Y is continuously differentiable and α′′ = 1 (cf. [710, Theorem 8] for a similar statement given in the one-dimensional setting): Theorem 7.1.15. Suppose that the function F : ℝn × X → Y is continuously differentiable with respect to the first variable, uniformly on the individual sets B of the collection ℬ. Then the following holds: (i) If Y = ℝ, n = 1 and F(⋅; ⋅) is (ℬ, Λ′ )-Lipschitz-almost periodic ((ℬ, Λ′ )-Lipschitzuniformly recurrent), then the function ((F(⋅; ⋅), F ′ (⋅; ⋅)) is Bohr (ℬ, Λ′ )-almost periodic ((ℬ, Λ′ )-uniformly recurrent). (ii) If Λ′ = ℝn , the functions F(⋅; ⋅), Fx1 (⋅; ⋅), . . . , Fxn (⋅; ⋅) are Bohr ℬ-almost periodic and ℬ is any family of compact subsets of X, then the function F(⋅; ⋅) is uniformly ℬ-Lipschitzalmost periodic. (iii) If Λ′ = ℝn , the functions F(⋅; ⋅), Fx1 (⋅; ⋅), . . . , Fxn (⋅; ⋅) are Bohr ℬ-almost periodic and bounded on the individual sets of the collection ℬ (the function (F1 , . . . , Fk )(⋅; ⋅) is ℬ-uniformly recurrent and bounded on the individual sets of the collection ℬ), then the function F(⋅) is ℬ-Lipschitz-almost periodic (ℬ-Lipschitz-uniformly recurrent). Proof. We will consider the almost periodic classes of functions here, only. Suppose that the function F(⋅; ⋅) is (ℬ, Λ′ )-Lipschitz-almost periodic, Y = ℝ and n = 1. Let ε > 0 and B ∈ ℬ be given. Then there exists l > 0 such that for each t0 ∈ Λ′ there exists τ ∈ B(t0 , l) ∩ Λ′ such that for every t ∈ ℝ and x ∈ B, (246) holds with the number ε replaced by the number ε/2 therein and the element yt;x = F(t; x); i. e. 󵄨 󵄨 sup sup(󵄨󵄨󵄨F(t + τ; x) − F(t; x)󵄨󵄨󵄨 + lim Lα,B (t, δ; Fτ (⋅; x) − F(⋅; x))) ⩽ ε/2, x∈B t∈ℝ

δ→0+

(248)

where Fτ (⋅) := F(⋅ + τ). Fix an element x ∈ B, a point t ∈ ℝ, a real number δ > 0 and the distinct points u1 , u2 ∈ B(t, δ). Applying the Lagrange mean value theorem, we get the existence of a number c ∈ (0, 1) such that 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨(Fτ − F)(u1 ; x) − (Fτ − F)(u2 ; x)󵄨󵄨󵄨 ⩾ 󵄨󵄨󵄨(Fτ − F)′ ((1 − c)u1 + cu2 ; x)󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨u1 − u2 󵄨󵄨󵄨. 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 Keeping in mind (248)-(249), we get 󵄨 󵄨 sup sup(󵄨󵄨󵄨F(t + τ; x) − F(t; x)󵄨󵄨󵄨 x∈B t∈ℝ

(249)

380 � 7 Special classes of metrically almost periodic functions

+ lim sup

sup

δ→0+ u1 ,u2 ∈B(t,δ);u1 =u ̸ 2

󵄨󵄨 󵄨󵄨 ′ inf 󵄨󵄨󵄨(Fτ − F) ((1 − c)u1 + cu2 ; x)󵄨󵄨󵄨) ⩽ ε/2. 󵄨 c∈(0,1)󵄨

(250)

After that, due to our assumption on the continuous differentiability of the function F : ℝ × X → Y , we can find a sufficiently small real number δτ > 0 such that for every δ ∈ (0, δτ ), we have: sup

u1 ,u2 ∈B(t,δ);u1 =u ̸ 2

󵄨󵄨 󵄨󵄨 ′ ′ sup 󵄨󵄨󵄨(Fτ − F) ((1 − c)u1 + cu2 ; x) − (Fτ − F) (t; x)󵄨󵄨󵄨 ⩽ ε/2. 󵄨 c∈(0,1)󵄨

Employing (250), the above yields 󵄨 󵄨 󵄨 󵄨 sup sup(󵄨󵄨󵄨F(t + τ; x) − F(t; x)󵄨󵄨󵄨 + 󵄨󵄨󵄨(Fτ − F)′ (t; x)󵄨󵄨󵄨) ⩽ ε. x∈B t∈ℝ

This simply implies that the function (F(⋅; ⋅), F ′ (⋅; ⋅)) is Bohr (ℬ, Λ′ )-almost periodic, as claimed. For (ii), let us fix again a real number ε > 0 and a set B ∈ ℬ. Due to [194, Proposition 2.21], we have that the function F : ℝn × X → Y is uniformly continuously differentiable with respect to the first variable, uniformly on the individual sets B of the collection ℬ; that is, for each set B ∈ ℬ and for every real number ε > 0, there exists a finite real number δ > 0 such that for every t, t′ ∈ ℝn with |t − t′ | ⩽ δ, we have: n

󵄩 󵄩 󵄩 󵄩 sup(󵄩󵄩󵄩F(t; x) − F(t′ ; x)󵄩󵄩󵄩Y + ∑󵄩󵄩󵄩Fxi (t; x) − Fxi (t′ ; x)󵄩󵄩󵄩Y ) < ε. x∈B

i=1

(251)

Since ℬ consists of compact subsets of X and Λ′ = ℝn , we know that (see [194, Proposition 2.18]) there exists l > 0 such that for each t0 ∈ ℝn there exists τ ∈ B(t0 , l) ∩ ℝn such that for every t ∈ ℝn and x ∈ B, we have 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩F(t + τ; x) − F(t; x)󵄩󵄩󵄩Y + 󵄩󵄩󵄩Fx1 (t + τ; x) − Fx1 (t; x)󵄩󵄩󵄩Y + ⋅ ⋅ ⋅ + 󵄩󵄩󵄩Fxn (t + τ; x) − Fxn (t; x)󵄩󵄩󵄩Y ⩽ ε. Fix again an element x ∈ B, a point t ∈ ℝn , a real number δ > 0 and the distinct points u1 , u2 ∈ B(t, δ). Applying the Lagrange mean value theorem to the vector-valued function [0, 1] ∋ t 󳨃→ (Fτ − F)((1 − t)u1 + tu2 ) ∈ Y (see, e. g. [545, Theorem 5]), we get the existence of a number c ∈ (0, 1) such that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩(Fτ − F)(u1 ; x) − (Fτ − F)(u2 ; x)󵄩󵄩󵄩 ⩽ 󵄩󵄩󵄩[∇(Fτ − F)]((1 − c)u1 + cu2 ; x)󵄩󵄩󵄩 ⋅ |u1 − u2 |. 󵄩󵄩 󵄩󵄩Y 󵄩󵄩 󵄩󵄩Y n Due to (251), we get that: lim

sup

δ→0+ u ,u ∈B(t,δ);u =u 1 2 1 ̸ 2

󵄩󵄩 ‖(Fτ − F)(u1 ; x) − (Fτ − F)(u2 ; x)‖Y 󵄩󵄩󵄩 ⩽ 󵄩󵄩[∇(Fτ − F)](t; x)󵄩󵄩󵄩 n , 󵄩 󵄩Y |u1 − u2 |

7.1 Hölder ρ-almost periodic type functions in ℝn

�

381

ℬ uniformly in t ∈ ℝn . We can similarly prove that F ∈ P0,1 (ℝn × X : Y ), which simply implies the required. The proof of (iii) can be deduced in the same way as above by appealing to Proposition 7.1.1 in place of [194, Proposition 2.18].

We can similarly prove the following (see also the notion introduced in Definition 7.1.4; for a better sound, we have abbreviated here and hereafter the term “uniformly ℬ-Lipschitz-uniformly recurrent” to “uniformly ℬ-Lipschitz-recurrent”): Theorem 7.1.16. Suppose that the function F : ℝn × X → Y is uniformly continuously differentiable with respect to the first variable, uniformly on the individual sets B of the collection ℬ. Then the following holds: (i) If Y = ℝ, n = 1 and F(⋅; ⋅) is uniformly (ℬ, Λ′ )-Lipschitz-almost periodic (uniformly (ℬ, Λ′ )-Lipschitz-recurrent), then the function (F(⋅; ⋅), F ′ (⋅; ⋅)) is uniformly Bohr (ℬ, Λ′ )almost periodic (uniformly (ℬ, Λ′ )-recurrent). (ii) If Λ′ = ℝn , the functions F(⋅; ⋅), Fx1 (⋅; ⋅), . . . , Fxn (⋅; ⋅) are Bohr ℬ-almost periodic and bounded on the individual sets of the collection ℬ (the function (F1 , . . . , Fk )(⋅; ⋅) is ℬ-uniformly recurrent and bounded on the individual sets of the collection ℬ), then the function F(⋅) is uniformly ℬ-Lipschitz-almost periodic (uniformly ℬ-Lipschitzrecurrent). The following corollaries of Theorem 7.1.15 and Theorem 7.1.16 are immediate: Corollary 7.1.17. Suppose that the function f : ℝ → ℝ is continuously differentiable. (i) Then f (⋅) is Lipschitz almost periodic if and only if f (⋅) is uniformly Lipschitz almost periodic if and only if the functions f (⋅) and f ′ (⋅) are almost periodic. (ii) Then f (⋅) is Lipschitz uniformly recurrent if and only if the function (f (⋅), f ′ (⋅)) is uniformly recurrent. (iii) Suppose that the function f : ℝ → ℝ and its first derivative f ′ : ℝ → ℝ are uniformly continuous. Then f (⋅) is Lipschitz uniformly recurrent if and only if f (⋅) is uniformly Lipschitz uniformly recurrent if and only if the function (f (⋅), f ′ (⋅)) is uniformly recurrent. Corollary 7.1.18. Suppose that P : ℝn × X → Y is a trigonometric polynomial and ℬ consists of bounded subsets of X. Then P(⋅; ⋅) is uniformly Lipschitz almost periodic. It seems very plausible that Corollary 7.1.17(i) has not been exploited to the full extent by S. Stoiński in his research study [710] since the statements of [710, Theorem 5, Theorem 10] almost immediately follows from a combination of Corollary 7.1.17 and some elementary results about the almost periodic functions (the first statement holds in the vector-valued case, while the second statement holds in the vector-valued case provided that Y does not contain an isomorphic copy of the space c0 ; see also [428, Theorem 2.1.1(vi)]). Furthermore, Corollary 7.1.17 yields that the assumptions of [710, Theorem 3] imply that the both functions x(⋅) and x ′ (⋅) under the consideration of the author

382 � 7 Special classes of metrically almost periodic functions are bounded and uniformly continuous, so that [710, Theorem 3] is an immediate consequence of [710, Theorem 2]. The both results, [710, Theorem 2] and [710, Theorem 3], give sufficient conditions for the uniform Lipschitz continuity of mappings under consideration. Example 7.1.19. The sum of functions f1 (⋅) and f2 (⋅), where f1 (t) := arcsin(t − 4k), t ∈ [4k − 1, 4k + 1), f1 (t) := arcsin(−t + 4k + 2), t ∈ [4k + 1, 4k + 3), f2 (t) := arcsin(√2t − 4k), t ∈ [(4k − 1)/√2, (4k + 1)/√2), f2 (t) := arcsin(−√2t + 4k + 2), t ∈ [(4k + 1)/√2, (4k + 3)/√2) (k ∈ ℤ), is almost periodic in variation, but not Lipschitz almost periodic [710]. Now we will prove that the function f (t) := f1 (t) + f2 (t), t ∈ ℝ is not Hölder-α-almost periodic for any α ∈ (1/2, 1] as well as that the function f (⋅) is uniformly Hölder-α-almost periodic for α ∈ (0, 1/2). Suppose first α ∈ (1/2, 1]. Then it suffices to show that for each real number δ > 0 we have |(π/2) + arcsin u1 | = +∞. |1 + u1 |α u1 ∈(−1,−1+δ] sup

(252)

Applying the Lagrange mean value, we have the existence of a finite real number ξ ∈ (−1, −1 + δ) such that |(π/2) + arcsin u1 | (−1)/2 (−1)/2 = |1 + u1 |1−α (1 − ξ 2 ) ⩾ |1 + u1 |1−α (1 − u12 ) |1 + u1 |α ∼ 2(−1)/2 |1 + u1 |1−2α → +∞,

δ → 0+.

This implies (252). Suppose now that α ∈ (0, 1/2). Then the required conclusion simply follows from the fact that the function f (⋅) is almost periodic and the equalities lim

sup

|f1 (u1 ) − f1 (u2 )| = 0, |u1 − u2 |α

lim

sup

|f2 (u1 ) − f2 (u2 )| = 0, |u1 − u2 |α

δ→0+ u ,u ∈B(t,δ);u =u 1 2 1 ̸ 2 δ→0+ u ,u ∈B(t,δ);u =u 1 2 1 ̸ 2

(253)

which hold uniformly in t ∈ ℝ. We will prove only (253). First of all, this equality is clearly true if the function f (⋅) is continuously differentiable in a neighborhood of the point t ∈ ℝ since we can apply the Lagrange mean value theorem then. Therefore, the only problem is that one in which we have t = ±1 + 4k for some integer k ∈ ℤ; without loss of generality, we may assume that t = −1. We essentially need to show that lim

sup

δ→0+ u ,u ∈[−1,−1+δ];u =u 1 2 1 ̸ 2

|f1 (u1 ) − f1 (u2 )| = 0, |u1 − u2 |α

uniformly in t ∈ ℝ, since the consideration is similar if −1 − δ ⩽ u1 < u2 ⩽ −1; if −1 − δ ⩽ u2 ⩽ −1 ⩽ u1 ⩽ −1 + δ, then we pass to the above case because f1 (u2 ) = arcsin(−2 − u2 )

7.1 Hölder ρ-almost periodic type functions in ℝn

� 383

and |u1 − u2 | ⩾ |u1 + u2 + 2|. Using the substitutions v1 = arcsin u1 and v2 = arcsin u2 , it suffices to show that for each real number δ ∈ (0, 1) we have lim

sup

δ→0+ v ,v ∈[(π/2)−δ,π/2];v 0 there exists a finite real number l > 0 such that for each t0 ∈ Λ there exists a point τ ∈ B(t0 , l) ∩ Λ such that 1

󵄨󵄨 󵄨󵄨 ′ ∫󵄨󵄨󵄨(fτ − f ) ((1 − s)u1 + su2 )󵄨󵄨󵄨 ds) < ε. 󵄨 δ→0+ u ,u ∈B(t,δ) 󵄨

󵄨 󵄨 sup(󵄨󵄨󵄨f (t + τ) − f (t)󵄨󵄨󵄨 + lim t∈Λ

sup

1

2

0

Then the function f (⋅) is Lipschitz almost periodic. Remark 7.1.23. If the function f : Λ → Y is absolutely continuous and Y has the Radon– Nikodym property, then f (⋅) is differentiable a. e.; see [54, Proposition 1.2.3, Proposition 1.2.4, Definition 1.2.5]. Albeit clarified in a little bit different framework, the (pseudo)metric spaces used for the introduction of the multi-dimensional Hölder ρ-almost periodic type functions satisfy the general requirements of [190, Proposition 2.2, Proposition 2.9] with ϕ(x) ≡ x. Therefore, we can clarify the following statements (the reformulations can be also given

7.1 Hölder ρ-almost periodic type functions in ℝn

� 385

for the corresponding classes of Hölder ρ-almost periodic type functions of type 1 and Lipschitz ρ-almost periodic type functions; cf. also [710, Theorem 6]): Theorem 7.1.24. (i) Suppose that 0 ≠ Λ ⊆ ℝn , F : Λ × X → Y and h : Y → Z is Lipschitz continuous. Then we have the following: (a) Suppose that F(⋅; ⋅) is semi-(ρ, 𝔽, ℬ)-Hölder-α-almost periodic (semi-(ρj , 𝔽, ℬ)j∈ℕn Hölder-α-almost periodic), and h ∘ ρ ⊆ ρ ∘ h (h ∘ ρj ⊆ ρj ∘ h for 1 ⩽ j ⩽ n). Then the function h ∘ F : Λ × X → Y is likewise semi-(ρ, 𝔽, ℬ)-Hölder-α-almost periodic (semi-(ρj , 𝔽, ℬ)j∈ℕn -Hölder-α-almost periodic). (b) Suppose that X = {0} and the function F(⋅) is strongly (𝔽, ℬ)-Hölder-α-almost periodic. Then the function (h ∘ F)(⋅) is likewise strongly (𝔽, ℬ)-Hölder-α-almost periodic. (ii) Suppose that 0 ≠ Λ′ ⊆ ℝn , 0 ≠ Λ ⊆ ℝn , Λ + Λ′ ⊆ Λ and the function F : Λ × X → Y is (𝔽, ℬ, Λ′ , ρ)-Hölder-α-almost periodic ((𝔽, ℬ, Λ′ , ρ)-Hölder-α-uniformly recurrent), where ρ is a binary relation on Y satisfying R(F) ⊆ D(ρ) and ρ(y) is a singleton for any y ∈ R(F). Suppose that for each τ ∈ Λ′ we have τ + Λ = Λ, 𝔾 : Λ → (0, ∞) and 𝔾(⋅) ⩽ infτ∈Λ′ 𝔽(⋅ − τ). Then Λ + (Λ′ − Λ′ ) ⊆ Λ and the function F(⋅; ⋅) is (𝔾, ℬ, Λ′ − Λ′ , I)Hölder-α-almost periodic ((𝔾, ℬ, Λ′ − Λ′ , I)-Hölder-α-uniformly recurrent). The statement of [190, Proposition 2.11] can be also restated for our purposes. Before proceeding to the next subsection, we would like to recall that B. Basit [89] has proved that there exists a complex-valued almost periodic function f : ℝ2 → ℂ such that the x function F : ℝ2 → ℂ, defined by F(x, y) := ∫0 f (t, y) dt, (x, y) ∈ ℝ2 , is bounded, but not almost periodic. This result has recently been reconsidered by S. M. A. Alsulami [33, Theorem 2.2] and A. Chávez et al. [194, Theorem 2.44, Corollary 2.45] under certain restrictive assumptions. Here we will not reconsider [710, Theorem 10] and the above-mentioned results for multi-dimensional Lipschitz almost periodic type functions.

7.1.2 Hölder-α-(𝔽, ℬ)-boundedness and Hölder-α-(𝔽, ℬ, Λ′ , ρ)-continuity We start this subsection by introducing the following notion: Definition 7.1.25. Suppose that F : Λ × X → Y is a given function and 𝔽 : Λ → (0, ∞). Then we say that the function F(⋅; ⋅) is Hölder-α-(𝔽, ℬ)-bounded if and only if for each set B ∈ ℬ we have 󵄩 󵄩 sup sup 𝔽(t)[󵄩󵄩󵄩F(t; x)󵄩󵄩󵄩Y + lim x∈B t∈Λ

sup

δ→0+ u ,u ∈B(t,δ)∩Λ;u =u 1 2 1 ̸ 2

‖F(u2 ; x) − F(u1 ; x)‖Y ] < +∞; |u1 − u2 |α

(254)

furthermore, we say that the function F(⋅; ⋅) is uniformly Hölder-α-(𝔽, ℬ)-bounded if and only if for each set B ∈ ℬ there exists δ0 ∈ (0, 1] such that, for every δ ∈ (0, δ0 ], we have:

386 � 7 Special classes of metrically almost periodic functions

󵄩 󵄩 sup sup 𝔽(t)[󵄩󵄩󵄩F(t; x)󵄩󵄩󵄩Y + x∈B t∈Λ

sup

u1 ,u2 ∈B(t,δ)∩Λ;u1 =u ̸ 2

‖F(u2 ; x) − F(u1 ; x)‖Y ] < +∞. |u1 − u2 |α

Definition 7.1.26. Suppose that 0 ≠ Λ′ ⊆ ℝn , F : Λ × X → Y is a given function, ρ is a binary relation on Y , 𝔽 : Λ → (0, ∞) and Λ′ ⊆ Λ′′ . Then we say that the function F(⋅; ⋅) is Hölder-α-(𝔽, ℬ, Λ′ , ρ)-continuous if and only if for every t ∈ Λ and x ∈ X there exists an element yt;x ∈ ρ(F(t; x)) such that for every set B ∈ ℬ, we have lim

󵄩 󵄩 sup sup 𝔽(t)[󵄩󵄩󵄩F(t + τ; x) − yt;x 󵄩󵄩󵄩Y

τ→0,τ∈Λ′ x∈B t∈Λ

+ lim

sup

‖[F(u2 + τ; x) − yu2 ;x ] − [F(u1 + τ; x) − yu1 ;x ]‖Y |u1 − u2 |α

δ→0+ u ,u ∈B(t,δ)∩Λ;u =u 1 2 1 ̸ 2

] = 0;

(255)

furthermore, we say that the function F(⋅; ⋅) is uniformly Hölder-α-(𝔽, ℬ, Λ′ , ρ)-continuous if and only if for every t ∈ Λ and x ∈ X there exists an element yt;x ∈ ρ(F(t; x)) such that for every set B ∈ ℬ, there exists δ0 ∈ (0, 1] such that, for every δ ∈ (0, δ0 ], we have: lim

󵄩 󵄩 sup sup 𝔽(t)[󵄩󵄩󵄩F(t + τ; x) − yt;x 󵄩󵄩󵄩Y

τ→0,τ∈Λ′ x∈B t∈Λ

+

sup

‖[F(u2 + τ; x) − yu2 ;x ] − [F(u1 + τ; x) − yu1 ;x ]‖Y |u1 − u2 |α

u1 ,u2 ∈B(t,δ)∩Λ;u1 =u ̸ 2

] = 0.

The notion of (uniform) Hölder-α-(𝔽, ℬ)-boundedness of type 1 and the notion of (uniform) Hölder-α-(𝔽, ℬ, Λ′ , ρ)-continuity of type 1 are obtained by removing the term ‖F(t; x)‖Y in (254) and the term ‖F(t + τ; x) − yt;x ‖Y in (255), respectively (of course, we assume that all terms in these equations are well-defined). Now we will state the following generalization of [710, Theorem 1], which can be applied in case that Λ = [0, ∞)n or Λ = ℝn : Proposition 7.1.27. Suppose that F ∈ P0,α (Λ × X : Y ), ρ = T ∈ L(Y ), the function 𝔽 : Λ → (0, ∞) satisfies that the function 1/𝔽 : Λ → (0, ∞) is locally bounded, and the following condition holds: (∀l > 0) (∃t0 ∈ Λ) (∃k > 0) (∀t ∈ Λ)(∃t′0 ∈ Λ)

′ ′′ (∀t′′ 0 ∈ B(t0 , l) ∩ Λ) t − t0 ∈ B(t0 , kl) ∩ Λ.

(256)

Then we have: (i) Suppose that 𝔽(⋅) is bounded, the function F(⋅; ⋅) is (uniformly) (𝔽, ℬ, T)-Hölder-αalmost periodic, and for every compact K ⊆ ℝn and set B ∈ ℬ, we have 󵄩 󵄩 sup sup 󵄩󵄩󵄩F(t; x)󵄩󵄩󵄩Y < +∞. x∈B t∈K∩Λ

7.1 Hölder ρ-almost periodic type functions in ℝn

�

387

Then the function F(⋅; ⋅) is (uniformly) Hölder-α-(𝔽, ℬ)-bounded. (ii) If the function F(⋅; ⋅) is (uniformly) (𝔽, ℬ, T)-Hölder-α-almost periodic of type 1, then the function F(⋅; ⋅) is (uniformly) Hölder-α-(𝔽, ℬ)-bounded of type 1. Proof. We will prove only (i), for the class of (𝔽, ℬ, T)-Hölder-α-almost periodic functions. Let B ∈ ℬ and ε = 1; we follow the proof of [194, Proposition 2.16] with appropriate changes. There exists a finite real number l > 0 such that for each s ∈ Λ there exists τ ∈ B(s, l) ∩ Λ such that (246) holds. Let t0 ∈ Λ and k > 0 be such that (256) holds. By the prescribed assumption, the set {F(s; x) : s ∈ B(t0 , kl) ∩ Λ, x ∈ B} is bounded in Y . Let t ∈ Λ be fixed, and let M := sups∈B(t0 ,kl)∩Λ, x∈B ‖F(s; x)‖Y . By our assumption, there exists ′ ′′ t′0 ∈ Λ such that for every t′′ 0 ∈ B(t0 , l) ∩ Λ, we have t ∈ t0 + [B(t0 , kl) ∩ Λ]. Furthermore, there exist τ ∈ B(t′0 , l) ∩ Λ and δ ∈ (0, 1) such that 󵄩 󵄩 𝔽(s)[󵄩󵄩󵄩F(s + τ; x) − TF(s; x)󵄩󵄩󵄩Y + lim

sup

δ→0+ u ,u ∈B(s,δ)∩Λ;u =u 1 2 1 ̸ 2

‖[F(u2 + τ; x) − TF(u2 ; x)] − [F(u1 + τ; x) − TF(u1 ; x)]‖Y ] ⩽ 1, |u1 − u2 |α

for any s ∈ Λ and x ∈ B. Clearly, s = t − τ ∈ B(t0 , kl) ∩ Λ, so that the last estimate implies that for every x ∈ B, we have: 󵄩 󵄩 𝔽(s)[󵄩󵄩󵄩F(t; x) − TF(s; x)󵄩󵄩󵄩Y + lim

sup

δ→0+ u ,u ∈B(t,δ)∩Λ;u =u 1 2 1 ̸ 2

‖[F(u2 ; x) − TF(u2 − τ; x)] − [F(u1 ; x) − TF(u1 − τ; x)]‖Y ] ⩽ 1. |u1 − u2 |α

This implies 𝔽(t)‖F(t; x)‖Y ⩽ ‖𝔽‖∞ ((1/𝔽(s)) + ‖T‖M) and sup

u1 ,u2 ∈B(t,δ)∩Λ;u1 =u ̸ 2

⩽

‖F(u1 ; x) − F(u2 ; x)‖Y |u1 − u2 |α

‖F(u1 ; x) − F(u2 ; x)‖Y 1 + ‖T‖ sup , 𝔽(s) |u1 − u2 |α u1 ,u2 ∈B(s,δ)∩Λ;u1 =u ̸ 2

so that the required conclusion simply follows from our assumptions that the function 𝔽 : Λ → (0, ∞) is bounded, the function 1/𝔽 : Λ → (0, ∞) is locally bounded and the assumption that F ∈ P0,α (Λ × X : Y ). Some results about the one-dimensional Hölder-α-continuous functions are given in [710, Theorem 2, Theorem 3, Theorem 4]. Our first contribution generalizes [710, Theorem 2] and has no direct connection with any kind of the almost periodicity (the assumption Λ = ℝn or Λ = [0, ∞)n has been made for the sake of convenience):

388 � 7 Special classes of metrically almost periodic functions 1,ℬ Theorem 7.1.28. Suppose that Λ = ℝn or Λ = [0, ∞)n , and the function F ∈ P0,α (Λ×X : Y ) satisfies that for each fixed element x ∈ X the function F(⋅; x) is continuously differentiable as well as that the function F(⋅; ⋅) is uniformly continuous on Λ × B for each set B ∈ ℬ. Suppose, further, that for each set B ∈ ℬ the following holds: 1. If α = 1, the functions Fx1 (⋅; ⋅), . . . , Fxn (⋅; ⋅) are uniformly continuous on Λ × B. 2. If α ∈ (0, 1), the functions Fx1 (⋅; ⋅), . . . , Fxn (⋅; ⋅) are uniformly continuous or bounded on Λ × B.

Then the function F(⋅; ⋅) is uniformly Hölder-α-(1, ℬ, Λ, I)-continuous. Proof. Let a set B ∈ ℬ be fixed. For a fixed element x ∈ B and points u1 , u2 , τ ∈ Λ, we consider the function H : [0, 1] → Y given by H(t) := F((1 − t)u1 + tu2 + τ; x) − F((1 − t)u1 + tu2 ; x), t ∈ [0, 1]. Applying [545, Corollary, p. 202], we get the existence of an integer m ∈ ℕ, the real numbers λk ∈ [0, 1] and the real numbers ck ∈ (0, 1) such that λ1 + ⋅ ⋅ ⋅ + λk = 1 and [F(u2 + τ; x) − F(u2 ; x)] − [F(u1 + τ; x) − F(u1 ; x)] m

n

k=1

k=1

= ∑ λk ∑ (u1k − u2k ) ⋅ (Fxk ((1 − ck )u1 + ck u2 + τ; x) − Fxk ((1 − ck )u1 + ck u2 ; x)). Applying the Cauchy–Schwartz inequality, the above implies: ‖[F(u2 + τ; x) − F(u2 ; x)] − [F(u1 + τ; x) − F(u1 ; x)]‖Y |u1 − u2 |α m

n

k=1

k=1

󵄩 󵄩2 ⩽ ∑ λk |u1 − u2 |1−α √ ∑ 󵄩󵄩󵄩Fxk ((1 − ck )u1 + ck u2 + τ; x) − Fxk ((1 − ck )u1 + ck u2 ; x)󵄩󵄩󵄩Y . (257)

This simply yields the required conclusion provided that α ∈ (0, 1) and the functions Fx1 (⋅; ⋅), . . . , Fxn (⋅; ⋅) are bounded on Λ×B. If the functions Fx1 (⋅; ⋅), . . . , Fxn (⋅; ⋅) are uniformly continuous on Λ × B, then for each δ ∈ [0, 1/2] we have |u1 − u2 |1−α ⩽ 1, provided that there exists t ∈ Λ such that u1 , u2 ∈ B(t, δ), so that the final conclusion follows from the decomposition n

2 √ ∑ 󵄩󵄩󵄩󵄩Fxk ((1 − ck )u1 + ck u2 + τ; x) − Fxk ((1 − ck )u1 + ck u2 ; x)󵄩󵄩󵄩󵄩Y k=1

n

󵄩 󵄩2 ⩽ √2√ ∑ 󵄩󵄩󵄩Fxk ((1 − ck )u1 + ck u2 + τ; x) − Fxk (t + τ; x)󵄩󵄩󵄩Y k=1

n

󵄩 󵄩2 + √2√ ∑ 󵄩󵄩󵄩Fxk (t + τ; x) − F(t; x)󵄩󵄩󵄩Y k=1

7.1 Hölder ρ-almost periodic type functions in ℝn

� 389

n

󵄩 󵄩2 + √2√ ∑ 󵄩󵄩󵄩Fxk ((1 − ck )u1 + ck u2 ; x) − F(t; x)󵄩󵄩󵄩Y k=1

and an elementary argumentation. 1,ℬ Remark 7.1.29. Suppose that Λ = ℝn or Λ = [0, ∞)n , and the function F ∈ P0,α (Λ × X : Y ) satisfies that for each fixed element x ∈ X the function F(⋅; x) is continuously differentiable. Suppose further that for each set B ∈ ℬ there exists a finite real number cB > 0 such that n

󵄩 󵄩 ∑󵄩󵄩󵄩Fxi (t; x) − Fxi (s; x)󵄩󵄩󵄩Y ⩽ cB |t − s|α , i=1

t, s ∈ Λ, x ∈ B.

(258)

Then for each set B ∈ ℬ the mapping τ 󳨃→ sup

sup

x∈B u1 ,u2 ∈Λ;u1 =u ̸ 2

‖[F(u2 + τ; x) − F(u2 ; x)] − [F(u1 + τ; x) − F(u1 ; x)]‖Y , |u1 − u2 |α

λ∈Λ

is Lipschitz continuous at zero. In order to show this, fix an element x ∈ B and points u1 , u2 , τ ∈ Λ. Let us consider after that the function G : [0, 1] → Y 2 given by G(t) := (F(u2 +tτ; x), F(u1 +tτ; x)), t ∈ [0, 1]. Applying [545, Corollary, p. 202], we get the existence of an integer m ∈ ℕ, the real numbers λk ∈ [0, 1] and the real numbers ck ∈ (0, 1) such that λ1 + ⋅ ⋅ ⋅ + λk = 1 and m

(F(u2 + τ; x)−F(u2 ; x), F(u1 + τ; x) − F(u1 ; x)) = ∑ λk ( k=1

∑nk=1 τk Fxk (u2 + ck τ; x) ∑nk=1 τk Fxk (u1 + ck τ; x)

).

Taking into account (258), the above implies: m ‖[F(u1 + τ; x) − F(u1 ; x)] − [F(u2 + τ; x) − F(u2 ; x)]‖Y ⩽ |τ|nc λk = |τ|ncB . ∑ B |u1 − u2 |α k=1

This simply yields the required conclusion. Remark 7.1.30. Consider the situation of Theorem 7.1.28 for a function f : ℝ → ℝ. We have assumed that the functions f (⋅) and f ′ (⋅) are uniformly continuous, but we have not assumed, as in the formulation of [710, Theorem 2], that the function f ′ (⋅) is bounded. But the boundedness of the function f ′ (⋅) follows from the Lagrange mean value theorem, the uniform continuity of the functions f (⋅), f ′ (⋅) and an elementary argumentation, so that [710, Theorem 2] actually holds for any uniformly continuous function f ∈ X01 such that its first derivative f ′ (⋅) is uniformly continuous (we use the same notation as in [710] here). Albeit we will not use this result henceforth, we will also remind the readers of the statement of the well-known Barbalat’s lemma, saying that any uniformly continuous +∞ function f : ℝ → Y such that ∫−∞ ‖f (x)‖Y dx < +∞ vanishes at ±∞.

390 � 7 Special classes of metrically almost periodic functions Now we will extend [710, Theorem 3] in the following way: Theorem 7.1.31. Suppose that ℬ consists of compact subsets of X, the function F : ℝn × X → Y satisfies that for each fixed element x ∈ X the function F(⋅; x) is continuously differentiable, as well as that for each ε > 0, t ∈ ℝn and B ∈ ℬ there exists δ > 0 such that supx∈B ‖∇F(t; x)−∇F(s; x)‖Y n < ε for every s ∈ ℝn with |t−s| ⩽ δ. Suppose further that the function F(⋅; ⋅) is (uniformly) (1, ℬ, T)-Lipschitz almost periodic, where T ∈ L(Y ) is a linear isomorphism. Then the function F(⋅; ⋅) is (uniformly) Hölder-α-(1, ℬ, Λ, I)-continuous. Proof. We will consider only the class of uniformly (1, ℬ, T)-Lipschitz almost periodic functions. Let a set B ∈ ℬ and a real number ε > 0 be given. Then there exist two finite real numbers l ⩾ 2 and δ0 > 0 such that for every t0 ∈ ℝn , we have the existence of a point ν ∈ B(t0 , l) such that 󵄩󵄩 󵄩 󵄩󵄩F(t + ν; x) − TF(t; x)󵄩󵄩󵄩Y ‖[F(u1 + ν; x) − TF(u1 ; x)] − [F(u2 + ν; x) − TF(u2 ; x)]‖Y + sup < ε, (259) |u1 − u2 | u1 ,u2 ∈B(t,δ);u1 =u ̸ 2 for every t ∈ ℝn , x ∈ B and δ ∈ (0, δ0 ]. Furthermore, there exists an absolute Const. such that, for every fixed point t ∈ ℝn , we have the existence of a number ν ∈ B(−t, l) such that for every point τ ∈ ℝn with a sufficiently small norm, we have: 󵄩󵄩 󵄩 󵄩󵄩F(t + τ; x) − F(t; x)󵄩󵄩󵄩Y 󵄩 󵄩 󵄩 󵄩 ⩽ 󵄩󵄩󵄩F(t + τ; x) − T −1 F(t + τ + ν; x)󵄩󵄩󵄩Y + 󵄩󵄩󵄩T −1 F(t + τ + ν; x) − T −1 F(t + ν; x)󵄩󵄩󵄩Y 󵄩 󵄩 󵄩 󵄩 + 󵄩󵄩󵄩T −1 F(t + ν; x) − F(t; x)󵄩󵄩󵄩Y ⩽ 2ε‖T‖ + ‖T −1 ‖ ⋅ 󵄩󵄩󵄩F(t + τ + ν; x) − F(t + ν; x)󵄩󵄩󵄩Y ⩽ Const. ⋅ ε, x ∈ B, because F(⋅; ⋅) is continuous on ℝn × B due to the mean value theorem [545, Corollary, p. 202] and ℬ consists of compact subsets of X. By (257), for every compact subset K ⊆ ℝn , the mapping τ 󳨃→ sup

sup

t∈K,x∈B u1 ,u2 ∈B(t,δ);u1 =u ̸ 2

‖[T −1 F(u1 + τ; x) − T −1 F(u1 ; x)] − [T −1 F(u2 + τ; x) − T −1 F(u2 ; x)]‖Y |u1 − u2 | is continuous at the point zero. The final conclusion simply follows from this fact, the uniform (1, ℬ, T)-Lipschitz almost periodicity of F(⋅; ⋅) [multiply the second addend in (259) with T −1 ], as well as the decomposition of the term [F(u1 + ν; x) − F(u1 ; x)] − [F(u2 + ν; x) − F(u2 ; x)] |u1 − u2 | into three terms:

7.1 Hölder ρ-almost periodic type functions in ℝn

�

391

[F(u1 + τ; x) − T −1 F(u1 + τ + ν; x)] − [F(u2 + τ; x) − T −1 F(u2 + τ + ν; x)] , |u1 − u2 |

[T −1 F(u1 + τ + ν; x) − T −1 F(u1 + ν; x)] − [T −1 F(u2 + τ + ν; x) − T −1 F(u2 + ν; x)] |u1 − u2 | and [F(u1 ; x) − T −1 F(u1 + ν; x)] − [F(u2 ; x) − T −1 F(u2 + ν; x)] . |u1 − u2 | An effort should be made to generalize the result of [710, Theorem 4] to the multidimensional functions with values in Banach spaces.

7.1.3 Extensions of Hölder ρ-almost periodic type functions To the best knowledge of the author, the first research results about the extensions of almost periodic functions in topological groups and semigroups were given by J. F. Berglund [116] in 1970. Eight years later, in 1978, H. Bart and S. Goldberg [87] proved that for every almost periodic function f : [0, ∞) → X, there exists a unique almost periodic function Ef : ℝ → X such that Ef (t) = f (t) for all t ⩾ 0. In a series of our recent research studies, we have considered the extensions of multi-dimensional almost periodic type functions using the argumentation contained in the proof of [650, Theorem 3.4], the important theoretical result established by W. M. Ruess and W. H. Summers. In this subsection, we will state two research results about the extensions of multidimensional, uniformly Hölder ρ-almost periodic type functions. The first result reads as follows: Theorem 7.1.32. Suppose that Λ′ ⊆ Λ ⊆ ℝn , Λ+Λ′ ⊆ Λ, the set Λ′ is unbounded, F : Λ → Y is uniformly continuous and uniformly Λ′ -Hölder-α-almost periodic, resp. uniformly continuous and Λ′ -Hölder-α-uniformly recurrent, S ⊆ ℝn is bounded and the following condition holds: (AP-E) For every t′ ∈ ℝn , there exists a finite real number M > 0 such that t′ + {s ∈ Λ′ : |s| ⩾ M} ⊆ Λ. Define ΩS := [(Λ′ ∪ (−Λ′ )) + (Λ′ ∪ (−Λ′ ))] ∪ S. Then there exists a uniformly continuous and uniformly ΩS -Hölder-α-almost periodic, resp. uniformly continuous and uniformly ̃ = F(t) for all t ∈ Λ; ΩS -Hölder-α-uniformly recurrent, function F̃ : ℝn → Y such that F(t) ̃ holds provided furthermore, in almost periodic case, the uniqueness of such a function F(⋅) n that ℝ ∖ ΩS is a bounded set. Proof. We will deduce the result only for uniformly continuous and uniformly Λ′ Hölder-α-almost periodic functions; without loss of generality we may assume that S = 0. If k ∈ ℕ, then there exists a point τk ∈ Λ′ such that ‖F(t + τ k ) − F(t)‖Y ⩽ 1/k

392 � 7 Special classes of metrically almost periodic functions for all t ∈ Λ and k ∈ ℕ; since the set Λ′ is unbounded, we may assume without loss of generality that limk→+∞ |τ k | = +∞. Hence, we have limk→+∞ F(t + τ k ) = F(t), uniformly for t ∈ Λ. If t′ ∈ ℝn , then there exists a finite real number M > 0 such that ̃ ′ ) exists (see the proof t′ + {s ∈ Λ′ : |s| ⩾ M} ⊆ Λ, and the limit limk→+∞ F(t′ + τ k ) := F(t ̃ ̃ of [194, Theorem 2.36]). We know that the function F(⋅) is uniformly continuous and F(⋅) is Bohr ΩS -almost periodic. Furthermore, it can be simply proved that F̃ ∈ P0,α (Λ : Y ). Let ε > 0 be a given real number. Then we know that there exist two finite real numbers l > 0 and δ0 ∈ (0, 1] such that, for every t0 ∈ Λ′ , there exists a point τ ∈ B(t0 , l) ∩ Λ′ such that for every δ ∈ (0, δ0 ], we have: 󵄩 󵄩 sup(󵄩󵄩󵄩F(t + τ) − F(t)󵄩󵄩󵄩Y t∈Λ

+

sup

u1 ,u2 ∈B(t,δ)∩Λ;u1 =u ̸ 2

‖[F(u1 + τ) − F(u1 )] − [F(u2 + τ) − F(u2 )]‖Y ) < ε. |u1 − u2 |α

(260)

We need to prove that for every τ ∈ [(Λ′ ∪ (−Λ′ )) + (Λ′ ∪ (−Λ′ ))] and δ ∈ (0, δ0 ], we have: 󵄩 ̃ ′ ̃ ′ )󵄩󵄩󵄩 sup (󵄩󵄩󵄩F(t + τ) − F(t 󵄩Y

t′ ∈ℝn

+

sup

u1 ,u2 ∈B(t′ ,δ)∩Λ;u1 =u ̸ 2

̃ 1 + τ) − F(u ̃ 1 )] − [F(u ̃ 2 + τ) − F(u ̃ 2 )]‖Y ‖[F(u ) < Const. ⋅ ε. (261) α |u1 − u2 |

̃ ′ + τ) − F(t ̃ ′ )‖Y and now we We already know [194] that this holds for the first term ‖F(t will prove the same inequality for the second term. In actual fact, it suffices to prove that (261) holds for every τ ∈ Λ′ because it can be easily shown that the validity of (261) for τ ∈ Λ′ (τ = τ1 ∈ Λ′ and τ = τ2 ∈ Λ′ ) implies the validity of (261) for −τ (τ1 + τ2 ). This follows from (260), since for every t′ ∈ ℝn , we have:

u1 ,u2

sup

∈B(t′ ,δ)∩Λ;u

=

̸ 2 1 =u

̃ 1 + τ) − F(u ̃ 1 )] − [F(u ̃ 2 + τ) − F(u ̃ 2 )]‖Y ‖[F(u |u1 − u2 |α

sup

u1 ,u2 ∈B(t′ ,δ)∩Λ;u1 =u ̸ 2

‖ limk→+∞ ([F(u1 + τ + τk ) − F(u1 + τk )] − [F(u2 + τ + τk ) − F(u2 + τk )])‖Y |u1 − u2 |α ⩽

sup

u1 ,u2 ∈B(t′ ,δ)∩Λ;u1 =u ̸ 2

lim sup k→+∞

⩽ lim sup

‖[F(u1 + τ + τk ) − F(u1 + τk )] − [F(u2 + τ + τk ) − F(u2 + τk )]‖Y |u1 − u2 |α sup

k→+∞ u1 ,u2 ∈B(t′ ,δ)∩Λ;u1 =u ̸ 2

‖[F(u1 + τ + τk ) − F(u1 + τk )] − [F(u2 + τ + τk ) − F(u2 + τk )]‖Y ⩽ ε. |u1 − u2 |α

7.1 Hölder ρ-almost periodic type functions in ℝn

� 393

Using the proofs of [441, Theorem 2.28] and [304, Theorem 2.28], we can similarly deduce the following result: Theorem 7.1.33. Suppose that ρ = T ∈ L(Y ) is a linear isomorphism, Λ′ ⊆ ℝn , Λ ⊆ ℝn , Λ + Λ′ ⊆ Λ, the set Λ′ is unbounded, F : Λ → Y is a uniformly continuous and uniformly (Λ′ , T)-Hölder-α-almost periodic, resp. uniformly continuous and (Λ′ , T)-Hölder-αuniformly recurrent, S ⊆ ℝn is bounded and condition (AP-E) holds. Then there exists a uniformly continuous and uniformly (Λ′ ∪ S, T)-Hölder-α-almost periodic, resp. uniformly continuous and (Λ′ ∪ S, T)-Hölder-α-uniformly recurrent, function F̃ : ℝn → Y such that ̃ = F(t) for all t ∈ Λ. F(t) In particular, if (v1 , . . . , vn ) is a basis of ℝn , Λ = {α1 v1 + ⋅ ⋅ ⋅ + αn vn : αi ⩾ 0 for all i ∈ ℕn } is a convex polyhedral in ℝn and Λ′ is its proper convex subpolyhedral, then Theorem 7.1.32 is applicable and we have Ω = ℝn ; hence, in this case, there exists a unique uniformly Λ′ -Hölder-α-almost periodic extension of the function F : Λ → Y to the whole Euclidean space, with the meaning clear. A similar statement can be proved in the case of consideration of Theorem 7.1.33, but then we cannot prove the uniqueness of such an extension. Now we will provide some applications of our results in the analysis of the existence and uniqueness of the Hölder-α-almost periodic type solutions to the various classes of the abstract Volterra integro-differential equations. The only new theoretical result presented here is Proposition 7.1.34.

7.1.4 Invariance of Hölder ρ-almost periodicity under the actions of convolution products In this subsection, we consider the invariance of Hölder ρ-almost periodicity under the actions of infinite convolution product (103). For simplicity, we will not consider the multi-dimensional case here. The following result is very similar to [190, Proposition 4.1]: Proposition 7.1.34. Suppose that (R(t))t>0 ⊆ L(X, Y ) is a strongly continuous operator ∞ family and ∫0 ‖R(t)‖ dt < +∞. If the function f : ℝ → X is uniformly (Λ′ , c)-Hölder-αalmost periodic (uniformly (Λ′ , c)-Hölder-α-uniformly recurrent), where 0 ≠ Λ′ ⊆ ℝ and c ∈ ℂ, |c| = 1, then the function F(⋅), given by (103), is likewise uniformly (Λ′ , c)-Hölder-αalmost periodic (uniformly (Λ′ , c)-Hölder-α-recurrent). Proof. We will consider (Λ′ , c)-Hölder-α-almost periodic functions, only. Let ε > 0 be given. Then there exist two finite real numbers l > 0 and δ > 0 such that for each t0 ∈ Λ′ , there exists a number τ ∈ [t0 − l, t0 + l] such that

394 � 7 Special classes of metrically almost periodic functions

󵄩 󵄩 Q := sup(󵄩󵄩󵄩f (t + τ) − cf (t)󵄩󵄩󵄩 + t∈ℝ

sup

u1 ,u2 ∈B(t,δ);u1 =u ̸ 2

‖[fτ − cf ](u1 ) − [fτ − cf ](u2 )‖ ) < ε. (262) |u1 − u2 |α

We need to show that (262) holds with the function f (⋅) replaced by the function F(⋅) therein. This follows from the next simple estimate ∞

󵄩 󵄩 sup(󵄩󵄩󵄩F(t + τ) − cF(t)󵄩󵄩󵄩 + t∈ℝ

sup

u1 ,u2 ∈B(t,δ);u1 =u ̸ 2

‖[Fτ − cF](u1 ) − [Fτ − cF](u2 )‖ 󵄩 󵄩 ) ⩽ Q ⋅ ∫ 󵄩󵄩󵄩R(s)󵄩󵄩󵄩 ds. |u1 − u2 |α 0

Before proceeding further, let us only note that Proposition 7.1.34 can be formulated for some other classes of functions considered so far, like ℝ-uniformly (Λ′ , c)-Hölder-αalmost periodic functions (ℝ-uniformly (Λ′ , c)-Hölder-α-uniformly recurrent functions). As is well known, Proposition 7.1.34 and its relatives can be successfully applied to a large class of the abstract (degenerate) Volterra integro-differential equations without initial conditions. The inheritance of the uniform (Λ′ , ρ)-Hölder-α-almost periodicity (the uniform ′ (Λ , ρ)-Hölder-α-uniformly recurrence) under the actions of the usual convolution product on ℝn can be analyzed in a similar manner; we will briefly explain this fact in a concrete situation concerning the heat equation in ℝn . It is well known that the solutions of this equation are governed by the action of Gaussian semigroup (38) acting on the space Y := BUC(ℝn ), for example, consisting of all bounded uniformly continuous functions F : ℝn → ℂ equipped with the sup-norm. Suppose that 0 ≠ Λ′ ⊆ ℝn and F(⋅) is bounded, uniformly (Λ′ , ρ)-Hölder-α-almost periodic, resp. bounded, uniformly (Λ′ , ρ)-Hölder-α-recurrent, where ρ is any binary relation defined on ℂ such that for each t ∈ ℝn the element yt = zt ∈ ρ(F(t)) is chosen such that the mapping s 󳨃→ zt−s , s ∈ ℝn belongs to Y and 󵄨 󵄨 sup (󵄨󵄨󵄨F(x + τ) − yx 󵄨󵄨󵄨 +

x∈ℝn

sup

|[F(u1 + τ) − yu1 ] − [F(u2 + τ) − yu2 ]| |u1 − u2 |α

u1 ,u2 ∈B(x,δ);u1 =u ̸ 2

) < +ε. (263)

Let us fix a number t0 > 0. We define a new binary relation σ on ℂ by σ := {((G(t0 )F)(x), (4πt0 )−(n/2) ∫ zx−y e−

|y|2 4t

dy) : x ∈ ℝn }.

ℝn

Then the function ℝn ∋ x 󳨃→ u(x, t0 ) ≡ (G(t0 )F)(x) ∈ ℂ is bounded, uniformly (Λ′ , ρ)Hölder-α-almost periodic, resp. bounded, uniformly (Λ′ , ρ)-Hölder-α-recurrent. This essentially follows from the next simple estimates, holding for every x ∈ ℝn and δ > 0:

� 395

7.1 Hölder ρ-almost periodic type functions in ℝn |y|2 󵄨󵄨 󵄨 󵄨󵄨u(x + τ, t0 ) − σ(u(x, t0 ))󵄨󵄨󵄨 ⩽ (4πt0 )−(n/2) ∫ 󵄨󵄨󵄨F(x − y + τ) − zx−y 󵄨󵄨󵄨e− 4t0 dy 󵄨󵄨 󵄨󵄨 󵄨 󵄨

ℝn

⩽ (4πt0 )−(n/2) ε ∫ e

|y|2

− 4t

0

dy

ℝn

and sup

u1 ,u2 ∈B(x,δ);u1 =u ̸ 2

|[u(u1 + τ, t0 ) − σ(u(u1 , t0 ))] − [u(u2 + τ, t0 ) − σ(u(u2 , t0 ))]| |u1 − u2 |α

⩽ (4πt0 )−(n/2) ×

sup

u1 ,u2 ∈B(x,δ);u1 =u ̸ 2

⩽ (4πt0 )−(n/2) ε ∫ e

∫

|[F(u1 − y + τ) − zu1 −y ] − [F(u2 − y + τ) − zu2 −y ]| |u1 − u2 |α

ℝn

|y|2

− 4t

0

e

|y|2

− 4t

0

dy

dy;

ℝn

see (263). Let us finally note that we can consider here the non-linear mapping ρ(z) := zk , z ∈ ℂ (k ∈ ℕ ∖ {1}). 7.1.5 Further applications and examples In this subsection, we will provide some new applications and examples concerning the existence and uniqueness of the Hölder-α-almost periodic type solutions to the abstract Volterra integro-differential equations and the classical partial differential equations. 1. In this issue, we would like to observe that the analysis carried out in [194, Example 1.1] can be simply used in our new framework. Suppose that a closed linear operator A generates a strongly continuous semigroup (T(t))t⩾0 on a Banach space X whose elements are certain complex-valued functions defined on ℝn . Then we know that, under some assumptions, the function t

u(t, x) = (T(t)u0 )(x) + ∫[T(t − s)f (s)](x) ds,

t ⩾ 0, x ∈ ℝn

(264)

0

is a unique classical solution of the abstract Cauchy problem ut (t, x) = Au(t, x) + F(t, x),

t ⩾ 0, x ∈ ℝn ; u(0, x) = u0 (x),

where F(t, x) := [f (t)](x), t ⩾ 0, x ∈ ℝn . There exist many concrete situations (especially, this holds for the Gaussian semigroup on ℝn ) where there exists a kernel (t, y) 󳨃→ E(t, y), t > 0, y ∈ ℝn which is integrable on any set [0, T] × ℝn (T > 0) and satisfies

396 � 7 Special classes of metrically almost periodic functions [T(t)f (s)](x) = ∫ F(s, x − y)E(t, y) dy,

t > 0, s ⩾ 0, x ∈ ℝn .

ℝn

Let it be the case, and let c ∈ ℂ, |c| = 1. After fixing a positive real number t0 > 0, we consider the inhomogeneous part in the equation (264). We would like to notice that the t Hölder-(α, c)-almost periodic behaviour of function x 󳨃→ ut0 (x) ≡ ∫00 [T(t0 − s)f (s)](x) ds, n x ∈ ℝ depends on the Hölder-(α, c)-almost behaviour of function F(t, x) in the space variable x. Our assumption will be that the function F(t, x) is uniformly Hölder-(α, c)almost periodic with respect to the variable x ∈ ℝn , uniformly in the variable t on compact subsets of [0, ∞); that is, for every ε > 0, there exist two real numbers l > 0 and δ0 > 0 such that, for every point t ∈ ℝn , there exists a point τ ∈ B(t0 , l) such that, for every δ ∈ (0, δ0 ], we have: 󵄨 󵄨 sup sup (󵄨󵄨󵄨F(s, x + τ) − cF(s, x)󵄨󵄨󵄨

s∈[0,t0 ] x∈ℝn

+

sup

u1 ,u2 ∈B(x,δ);u1 =u ̸ 2

|[F(s, u1 + τ) − cF(s, u1 )] − [F(s, u2 + τ) − cF(s, u2 )]| ) < ε. |u1 − u2 |α

Then the function ut0 (⋅) is likewise uniformly Hölder-(α, c)-almost periodic, which follows from the estimate of the term |ut0 (x + τ) − cut0 (x)| in [194] and the following estimates: |[ut0 (u1 + τ) − ut0 (u1 )] − [ut0 (u2 + τ) − ut0 (u2 )]|

sup

u1 ,u2 ∈B(x,δ);u1 =u ̸ 2 t0

⩽∫∫ 0 ℝn t0

|u1 − u2 |α

|[F(s, u1 +τ − y) − cF(s, u1 − y)]−[F(s, u2 + τ − y) − cF(s, u2 − y)]| 󵄨󵄨 󵄨 󵄨󵄨E(t0 , y)󵄨󵄨󵄨 dy ds |u1 − u2 |α

󵄨 󵄨 ⩽ ε ∫ ∫ 󵄨󵄨󵄨E(t0 , y)󵄨󵄨󵄨 dy ds. 0 ℝn

2. In this example, we continue our analysis of the d’Alembert formula from [441, Example 1.2]. Let us suppose that the functions x 󳨃→ f (x), x ∈ ℝ, x 󳨃→ f ′ (x), x ∈ ℝ, ⋅ x 󳨃→ g(x), x ∈ ℝ and x 󳨃→ g [1] (x), x ∈ ℝ are almost periodic, where g [1] (⋅) ≡ ∫0 g(s) ds, i. e., the functions x 󳨃→ f (x), x ∈ ℝ and x 󳨃→ g [1] (x), x ∈ ℝ are Lipschitz almost periodic (see Corollary 7.1.17(i)). Then we know that the solution u(x, t) can be extended to the whole real line in the time variable and that this solution is almost periodic in (x, t) ∈ ℝ2 . A simple calculation gives 1 1 ux (x, t) = [f ′ (x − at) + f ′ (x + at)] + [g(x + at) − g(x − at)], 2 2a and

(x, t) ∈ ℝ2

7.1 Hölder ρ-almost periodic type functions in ℝn

1 1 ut (x, t) = [−f ′ (x − at) + f ′ (x + at)] + [g(x + at) + g(x − at)], 2 2

�

397

(x, t) ∈ ℝ2 .

Arguing in the same way as in the above-mentioned example, we can show that the functions ux (x, t) and ut (x, t) are almost periodic in (x, t) ∈ ℝ2 . By Theorem 7.1.16(ii), we have that the solution u(x, t) is uniformly Lipschitz almost periodic in (x, t) ∈ ℝ2 . 3. Let us reconsider the semilinear Hammerstein integral equation (64). Suppose that k ∈ W 1,1 (ℝn ), the usual Sobolev space of real-valued functions of order 1, and there exists a constant L > 0 such that L(‖k‖L1 (ℝn ) + ‖kt1 ‖L1 (ℝn ) + ⋅ ⋅ ⋅ + ‖ktn ‖L1 (ℝn ) ) < 1,

(265)

and the function F : ℝn+1 → ℝ is continuously differentiable with n+1

󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩F(t; x) − F(t; y)󵄩󵄩󵄩Y + ∑ 󵄩󵄩󵄩Ftk (t; x) − Ftk (t; y)󵄩󵄩󵄩Y ⩽ L|x − y|, k=1

t ∈ ℝn , x, y ∈ ℝ.

Suppose that the functions F(t; x), Ft1 (t; x), . . . , Ftn+1 (t; x) are ℬ-almost periodic with respect to the variable t, where ℬ is the collection of all compact subsets of ℝ, as well as that supt∈ℝn ,x∈B |F(t; x)| < +∞ for every bounded subset B ⊆ ℝ. Consider the Banach space X = {F : ℝn → ℝ : F, Ft1 , . . . , Ftn are Bohr almost periodic}, equipped with the norm n

‖F‖ := ‖F‖∞ + ∑ ‖Fti ‖∞ . i=1

Let g, y ∈ X. Then it can be simply proved that the function t 󳨃→ F(t; y(t)), t ∈ ℝn is continuously differentiable, as well as that (𝜕/𝜕ti )F(t; y(t)) = Fti (t; y(t)) + Ftn+1 (t; y(t)) ⋅ Fti (t), t ∈ ℝn (i ∈ ℕn ). Using the argumentation contained in the proofs of [194, Proposition 2.19, Theorem 2.46, Corollary 2.47], we can simply show that the function t 󳨃→ (𝜕/𝜕ti )F(t; y(t)), t ∈ ℝn is Bohr almost periodic (i ∈ ℕn ). Therefore, the mapping X ∋ y 󳨃→ g(⋅) + ∫ k(⋅ − s)F(s; y(s)) ds ∈ X ℝn

is well defined. This mapping is contraction due to the assumption (265), so that the Banach contraction principle implies that the equation (64) has a unique almost periodic solution which belongs to the space X. Let us mention some subjects not considered here: 1. The L1 -convergence of a sequence (fk ) of the Hölder-αk -almost periodic functions, where (αk ) is a strictly increasing sequence in (0, 1) converging to 1, to a Lipschitz almost

398 � 7 Special classes of metrically almost periodic functions periodic function f (⋅) has been examined in [710, Theorem 7]. This result seems to be incorrectly proved since the equation [710, (3), l. -1, p. 693] is definitely wrong (the use of translation number τ is also not correct here). Because of that, we will not reconsider this result in the multi-dimensional setting. 2. We will not consider here the multi-dimensional 𝔻-asymptotically Hölder-αalmost periodic type functions. 3. For simplicity, we have not considered the corresponding classes of Hölder-αnormal type functions. Concerning this problematic, we would like to notice that the following notion can be analyzed: Definition 7.1.35. Suppose that R is any collection of sequences in Λ′′ , F : Λ × X → Y and 𝔽 : Λ → (0, ∞). Then we say that the function F(⋅; ⋅) is (R, ℬ, 𝔽)-Hölder-α-normal, if and only if, for every set B ∈ ℬ and for every sequence (bk )k∈ℕ in R there exists a subsequence (bkm )m∈ℕ of (bk )k∈ℕ such that for every ε > 0, there exists an integer m0 ∈ ℕ such that, for every integers m, m′ ⩾ m0 , we have 󵄩󵄩 󵄩󵄩 󵄩 󵄩 sup󵄩󵄩󵄩𝔽(⋅)(F(⋅ + bkm ; x) − F(⋅ + bk ′ ; x))󵄩󵄩󵄩 < ε. m 󵄩󵄩Pα 󵄩 x∈B 󵄩 The corresponding classes of (R, ℬ, 𝔽)-Hölder-α-normal functions of type 1 and (R, ℬ, 𝔽)Lipschitz-normal functions (of type 1) are introduced similarly as in Definition 7.1.3. Let us only note that the following result is a special case of [190, Proposition 2.5] with ϕ(x) ≡ x: Proposition 7.1.36. Suppose that 0 ≠ Λ ⊆ ℝn , F : Λ × X → Y , R is any collection of sequences in Λ′′ , Fj : Λ × X → Y , 𝔽 : Λ → (0, ∞) and the function Fj (⋅; ⋅) is (R, ℬ, 𝔽)Hölder-α-normal for all j ∈ ℕ. If F : Λ × X → Y and, for every set B ∈ ℬ and for every sequence (bk )k∈ℕ in R, we have 󵄩󵄩 󵄩󵄩 󵄩 󵄩 sup󵄩󵄩󵄩𝔽(⋅)(Fj (⋅ + bj ; x) − F(⋅ + bk ; x))󵄩󵄩󵄩 = 0, 󵄩󵄩Pα (j,k)→+∞ x∈B 󵄩 󵄩 lim

then the function F(⋅; ⋅) is likewise (R, ℬ, 𝔽)-Hölder-α-normal. The statement of [710, Theorem 6] can be also extended in our new setting. Finally, we would like to raise the following issues: 1. Is it possible to construct a Lipschitz almost periodic function F : ℝn → ℝ which is not uniformly Lipschitz almost periodic? 2. Is it possible to construct a continuously differentiable vector-valued Lipschitz almost periodic function F : ℝn → Y which satisfies that the function Fxi (⋅) is not almost periodic for some integer i ∈ ℕn ?

7.2 Generalized almost periodic functions with values in ordered Banach spaces

� 399

7.2 Generalized almost periodic functions with values in ordered Banach spaces The class of generalized almost periodic functions F : ℝn → ℝ was introduced by Y. Meyer in 2012 (cf. [550, Definition 2.1]; in the initial analysis, Y. Meyer considered the space of all Besicovitch bounded functions defined on ℝn , equipped with the Besicovitch pseudometric). This class, the class of generalized almost periodic measures, and the class of mean-periodic functions provide a solid base for the investigations of quasicrystals, irregular sampling, and modal sets in ℝn . Recently, D. Lenz, T. Spindeler, and N. Strungaru have introduced the notion of a real-valued generalized almost periodic function on a class of locally compact topologically groups (cf. [492, Definition 4.17]). The organization and main ideas of this section can be briefly described as follows. We first recall the basic facts about ordered Banach spaces and reconsider the notion of almost periodicity for the functions with values in real Banach spaces. Our main structural results are given in Section 7.2.1, where we propose a rather general approach with the use of pseudometric spaces. The notion of a generalized Bohr (ℬ, I1′ , I2′ , ρ1 , ρ2 , 𝒫 )almost periodic function [generalized (ℬ, I1′ , I2′ , ρ1 , ρ2 , 𝒫 )-uniformly recurrent function] F : I × X → Y , where 0 ≠ I ⊆ ℝn , ρi is a binary relation on Y , I + Ii′ ⊆ I, i = 1, 2 and Y is an ordered Banach space, is introduced in Definition 7.2.1. Some structural results concerning the introduced classes of function spaces are given without proofs in Proposition 7.2.2, Proposition 7.2.3, Theorem 7.2.4 and Proposition 7.2.5. In Example 7.2.6, we present an example of an unbounded Stepanov almost periodic function f : ℝ → ℝ which is not generalized almost periodic and ask whether an essentially bounded Stepanov-p-almost periodic function, where 1 ⩽ p < ∞, must be generalized almost periodic. In Section 7.2.2, which is crucially important in our study, we analyze some important subclasses of generalized Bohr (ℬ, I1′ , I2′ , ρ1 , ρ2 , 𝒫 )-almost periodic type functions. Here we also provide many useful comments and remarks about the results established by Y. Meyer in [550] and D. Lenz, T. Spindeler, N. Strungaru in [492]. The strong motivational fact for the genesis of paper [451], from which the material of this section is taken, presents the fact that the generalized almost periodic type solutions of ordinary differential equations and partial differential equations have not been sought by now. Some applications of our theoretical results to the abstract Volterra integro-differential equations in order Banach spaces are given; in order to apply our theoretical results, it is crucial to know whether a closed linear operator A under our consideration is the generator of a positive strongly continuous semigroup or a positive fractional solution operator family. Ordered Banach spaces Suppose that (Y , ‖⋅‖Y ) is a real Banach space. By a positive cone in Y , we mean any closed subset Y+ of Y such that Y+ + Y+ ⊆ Y+ and λY+ ⊆ Y+ for all λ > 0. The order relation on Y is defined by x ⩾ y if and only if x − y ∈ Y+ . The cone Y+ is said to be generating if

400 � 7 Special classes of metrically almost periodic functions and only if Y = Y+ − Y+ and the cone Y+ is said to be normal if and only if there is some finite constant c > 0 such that x ⩽ y ⩽ z always implies ‖y‖Y ⩽ c max(‖x‖Y , ‖z‖Y ). For example, if Y = ℝn is equipped with the Euclidean norm, then the cone Y = [0, ∞)n is both normal and generating (unless stated otherwise, we will always use this cone in the case of consideration of the Euclidean space Y = ℝn ). An operator T ∈ L(Y ) is called positive if and only if T(Y+ ) ⊆ Y+ , which will be simply denoted by T ⩾ 0. Suppose now that X is a real Banach space. Then the standard complexification Xℂ of a real Banach space X is the Banach space Xℂ = (X × X, ‖(⋅, ⋅)‖ℂ ), where ‖(x, y)‖ℂ := supθ∈[0,2π] ‖ cos(θ)x + sin(θ)x‖, X × X is a vector space with the usual addition of pairs and multiplication of pairs by complex scalars, defined through (a + ib)(x, y) := (ax − by, bx + ay), for any a, b ∈ ℝ and x, y ∈ X. It is trivial to show that a continuous function F : ℝn → X is almost periodic if and only if the function F̃ : ℝn → X × X, given by ̃ F(t) := (F(t), 0), t ∈ ℝn , is almost periodic. Hence, a continuous function F(⋅) is almost periodic if and only if for any sequence (bk ) in ℝn there exists a subsequence (ak ) of (bk ) such that (F(⋅ + ak )) converges in Cb (ℝn : X) if and only if there exists a linear combination of functions like t 󳨃→ cos(⟨λ, t⟩), t ∈ ℝn and t 󳨃→ sin(⟨λ, t⟩), t ∈ ℝn which converges uniformly to F(⋅) on the whole space ℝn . Finally, the mean value M(F) defined by M(F) := lim

T→+∞

1 ∫ F(t) dt (2T)n s+KT

exists in X and it does not depend on s ∈ ℝn (many other structural properties of almost periodic functions with values in real Banach spaces can be clarified in this way; see also [431, Introduction]).

7.2.1 Generalized Bohr (ℬ, I1′ , I2′ , ρ1 , ρ2 )-almost periodic type functions In this subsection, we will always assume that (X, ‖ ⋅ ‖) is a real or complex Banach space as well as that (Y , ‖ ⋅ ‖Y ) is an ordered Banach space with the fixed positive cone Y+ . Furthermore, we will always assume that 0 ≠ I ′ ⊆ ℝn , 0 ≠ I ⊆ ℝn and ℬ is a certain family of non-empty subsets of X satisfying that for each x ∈ X there exists B ∈ ℬ such that x ∈ B. By 𝒫 = (P, d) we denote a fixed pseudometric space such that P ⊆ Y I and the zero function belongs to P; we write ‖f ‖P := d(f , 0), f ∈ P. Now we are ready to introduce the following notion: Definition 7.2.1. Suppose that F : I × X → Y is a given function, ρi is a binary relation on Y and I + Ii′ ⊆ I, i = 1, 2. Then we say that F(⋅; ⋅) is: (i) generalized Bohr (ℬ, I1′ , I2′ , ρ1 , ρ2 , 𝒫 )-almost periodic if and only if for every ε > 0 and B ∈ ℬ there exist functions Hε : I × X → Y and Gε : I × X → Y such that Hε (⋅; ⋅) is Bohr (ℬ, I1′ , ρ1 )-almost periodic, Gε (⋅; ⋅) is Bohr (ℬ, I2′ , ρ2 )-almost periodic,

7.2 Generalized almost periodic functions with values in ordered Banach spaces

Gε (t; x) ⩽ F(t; x) ⩽ Hε (t; x),

t ∈ I, x ∈ X,

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Hε (⋅; x) − Gε (⋅; x) ∈ P for all x ∈ B and supx∈B ‖Hε (⋅; x) − Gε (⋅; x)‖P ⩽ ε; (ii) generalized (ℬ, I1′ , I2′ , ρ1 , ρ2 , 𝒫 )-uniformly recurrent if and only if for every ε > 0 and B ∈ ℬ there exist functions Hε : I × X → Y and Gε : I × X → Y such that Hε (⋅; ⋅) is (ℬ, I1′ , ρ1 )-uniformly recurrent, Gε (⋅; ⋅) is (ℬ, I2′ , ρ2 )-uniformly recurrent, (266) holds, Hε (⋅; x) − Gε (⋅; x) ∈ P for all x ∈ B and supx∈B ‖Hε (⋅; x) − Gε (⋅; x)‖P ⩽ ε. In the sequel, we omit the term “ℬ” if X = {0}. Keeping in mind [431, Proposition 6.1.19], which ensures that any tuple of Bohr ℬ-almost periodic functions is also Bohr ℬ-almost periodic provided that the collection ℬ consists of some compact subsets of X (cf. Definition 2.1.1 with I ′ = I = ℝn and ρ = I), the proof of the following proposition is rather technical and therefore omitted: Proposition 7.2.2. Suppose that m ∈ ℕ, ρi is a binary relation on Y , I + Ii′ ⊆ I, i = 1, 2 and Fj : I × X → Y is a given function for any j ∈ ℕm . We define the cone Y+m on the Banach m space Y m by Y+m := (Y+ )m , the relations ρm i on Y by ′ ′ ′ (y1 , . . . , ym ) ρm i (y1 , . . . , ym ) ⇔ yj ρi yj ,

j ∈ ℕm (i = 1, 2),

and the pseudometric space 𝒫m = (Pm , dm ) in the following way: If W : I → Y m , then W ∈ Pm if and only if all component functions W1 , . . . , Wm of W belong to P. The pseudometric dm on Pm is given by m

dm (V , W ) := ∑ d(Vj , Wj ), j=1

V , W ∈ Pm .

Set F := (F1 , . . . , Fm ). Then the following holds: m (i) If F(⋅; ⋅) is a generalized Bohr (ℬ, I1′ , I2′ , ρm 1 , ρ2 , 𝒫m )-almost periodic function (gener′ ′ m m alized (ℬ, I1 , I2 , ρ1 , ρ2 , 𝒫m )-uniformly recurrent function), then Fj (⋅; ⋅) is generalized Bohr (ℬ, I1′ , I2′ , ρ1 , ρ2 , 𝒫 )-almost periodic (generalized (ℬ, I1′ , I2′ , ρ1 , ρ2 , 𝒫 )-uniformly recurrent) for all j ∈ ℕm . (ii) The converse statement in (i) holds provided that ℬ consists of some compact subsets of X, Ii′ = I = ℝn and ρi = I for i = 1, 2. An analogue of [550, Theorem 2.7], concerning the pointwise multiplication of generalized almost periodic functions, can be formulated in the case that the requirements of (ii) hold (cf. also the statements of [550, Corollary 2.13, Lemma 2.14, Corollary 2.15], which will not be reconsidered here). Further on, the statements of [304, Proposition 2.2, Corollary 2.3, Proposition 2.5] can be straightforwardly reformulated for the function spaces introduced in Definition 7.2.1. This is also the case with the statements of [304, Proposition 2.7, Theorem 2.11(i)–(iv), Proposition 2.12]:

402 � 7 Special classes of metrically almost periodic functions Proposition 7.2.3. Suppose that ρi = Ti ∈ L(Y ), 0 ≠ I ⊆ ℝn , Ii′ = I is closed, i = 1, 2, ℬ consists of some compact subsets of X and F : I × X → Y is a generalized Bohr (ℬ, I1′ , I2′ , ρ1 , ρ2 , 𝒫 )-almost periodic function. Assume that the cone Y+ has the property that ±f ⩽ g

implies ‖f ‖Y ⩽ ‖g‖Y .

If, additionally, (∀l > 0) (∃t0 ∈ I) (∃k > 0) (∀t ∈ I)(∃t′0 ∈ I)

′ ′′ (∀t′′ 0 ∈ B(t0 , l) ∩ I) t − t0 ∈ B(t0 , kl) ∩ I,

then for each B ∈ ℬ we have supt∈I;x∈B ‖F(t; x)‖Y < ∞. Theorem 7.2.4. Suppose that 0 ≠ Ii′ ⊆ ℝn , 0 ≠ I ⊆ ℝn , F : I × X → Y is generalized Bohr (ℬ, I1′ , I2′ , ρ1 , ρ2 , 𝒫 )-almost periodic (generalized (ℬ, I1′ , I2′ , ρ1 , ρ2 , 𝒫 )-uniformly recurrent), ρ is a binary relation on Y and I + Ii′ ⊆ I for i = 1, 2. Then the following holds: (i) Set σi := {(‖y1 ‖Y , ‖y2 ‖Y ) | ∃t ∈ I ∃x ∈ X : y1 = F(t; x) and y2 ∈ ρi (y1 )}, i = 1, 2. Then the function ‖F(⋅; ⋅)‖Y is generalized Bohr (ℬ, I1′ , I2′ , σ1 , σ2 , 𝒫1 )-almost periodic (generalized (ℬ, I1′ , I2′ , σ1 , σ2 , 𝒫1 )-unifo-rmly recurrent), where 𝒫1 = (P1 , d1 ) is a pseudometric space such that P1 := {‖f (⋅)‖Y − ‖g(⋅)‖Y : f , g ∈ P} and there exists a finite real constant c > 0 such that ‖‖f (⋅)‖Y − ‖g(⋅)‖Y ‖P1 ⩽ c‖f − g‖P for all f , g ∈ P. (ii) Suppose that λ ∈ ℝ ∖ {0}. Set ρλ,i := {λ(y1 , y2 ) | ∃t ∈ I ∃x ∈ X : y1 = F(t; x) and y2 ∈ ρi (y1 )}, i = 1, 2. Then the function λF(⋅; ⋅) is generalized Bohr (ℬ, I1′ , I2′ , ρλ,1 , ρλ,2 , 𝒫 )almost periodic (generalized (ℬ, I1′ , I2′ , ρλ,1 , ρλ,2 , 𝒫 )-uniformly recurrent), provided that for each η ∈ ℝ ∖ {0} there exists a finite real constant cη > 0 such that for every f ∈ P, we have ηf ∈ P and ‖ηf ‖P ⩽ cη ‖f ‖P . (iii) Suppose a ∈ ℝn and x0 ∈ X. Define G : (I − a) × X → Y by G(t; x) := F(t + a; x + ′ x0 ), t ∈ I − a, x ∈ X, as well as ℬx0 := {−x0 + B : B ∈ ℬ}, Ia,i := Ii′ and ρa,x0 ,i := {(y1 , y2 ) | ∃t ∈ I − a ∃x ∈ X : y1 = F(t + a; x + x0 ) and y2 ∈ ρi (y1 )}, i = 1, 2. Then ′ ′ the function G(⋅; ⋅) is generalized Bohr (ℬ, Ia,1 , Ia,2 , ρa,x0 ,1 , ρa,x0 ,2 , 𝒫a )-almost periodic ′ ′ (generalized (ℬ, I1 , I2 , ρ1 , ρ2 , 𝒫a )-uniformly recurrent), provided that 𝒫a = (Pa , da ) is a pseudometric space such that Pa ⊆ Y I−a and there exists a finite real constant ca > 0 such that for every f ∈ P, we have f (a + ⋅) ∈ Pa and ‖f (a + ⋅)‖Pa ⩽ ca ‖f ‖P . (iv) Suppose that a, b ∈ ℝ ∖ {0}. Define the function G : (I/a) × X → Y by G(t; x) := ′ F(at; bx), t ∈ I/a, x ∈ X, as well as ℬb := {b−1 B : B ∈ ℬ}, Ia,i := Ii′ /a and ρa,b,i := {(y1 , y2 ) | ∃t ∈ I/a ∃x ∈ X : y1 = F(at; bx) and y2 ∈ ρi (y1 )}, i = 1, 2. Then the func′ ′ tion G(⋅; ⋅) is generalized Bohr (ℬ, Ia,1 , Ia,2 , ρa,b,1 , ρa,b,2 , 𝒫a )-almost periodic (general′ ′ ized (ℬ, Ia,1 , Ia,2 , ρa,b,1 , ρa,b,2 , 𝒫a )-uniformly recurrent), provided, that 𝒫a = (Pa , da ) is a pseudometric space such that Pa ⊆ Y I/a and there exists a finite real constant ca > 0 such that for every f ∈ P, we have f (a⋅) ∈ Pa and ‖f (a⋅)‖Pa ⩽ ca ‖f ‖P .

7.2 Generalized almost periodic functions with values in ordered Banach spaces

�

403

Proposition 7.2.5. Suppose that 0 ≠ Ii′ ⊆ ℝn , 0 ≠ I ⊆ ℝn , F : I × X → Y is a generalized Bohr (ℬ, I1′ , I2′ , ρ1 , ρ2 , 𝒫 )-almost periodic (generalized (ℬ, I1′ , I2′ , ρ1 , ρ2 , 𝒫 )-uniformly recurrent) function, ρ is a binary relation on Y and I + Ii′ ⊆ I, i = 1, 2. If ϕ : Y → Z is uniformly continuous on the set R(F) ∪ ρ(R(F)), where Z is an ordered Banach space with the cone Z+ , then the function ϕ∘F : I ×X → Z is generalized Bohr (ℬ, I1′ , I2′ , ϕ∘ρ1 , ϕ∘ρ2 , 𝒫Z )-almost periodic (generalized (ℬ, I1′ , I2′ , ϕ ∘ ρ1 , ϕ ∘ ρ2 , 𝒫Z )-uniformly recurrent), where 𝒫Z = (PZ , dZ ) is a pseudometric space, provided that the following conditions hold: (i) x ⩽ y in Y implies ϕ(x) ⩽ ϕ(z) in Z; (ii) There exists a real constant c > 0 such that the assumption f − g ∈ P implies ϕ(f ) − ϕ(g) ∈ PZ and ‖ϕ(f ) − ϕ(g)‖PZ ⩽ c‖f − g‖P . As an immediate consequence of Proposition 7.2.3, we have that an unbounded Stepanov-p-almost periodic function f : ℝ → ℝ cannot be generalized almost periodic in the usual sense (let us only note that Prof. Y. Meyer has mistakenly stated, without a corresponding proof, that the class of generalized almost periodic functions introduced in [550, Definition 2.1], sits between the classes of equi-Weyl-1-almost periodic functions and Besicovitch-1-almost periodic functions); we will construct an example of such a function, with p = 1, following the analysis of Z. Hu and A. B. Mingarelli from [373, Section 5]: Example 7.2.6. Assume that α, β ∈ ℝ and αβ−1 is a well-defined irrational number. Then we know that the functions fα,β (t) = sin(

1 ), 2 + cos αt + cos βt

t∈ℝ

1 ), 2 + cos αt + cos βt

t∈ℝ

and gα,β (t) = cos(

are Stepanov p-almost periodic, infinitely differentiable, bounded, but not almost periodic (1 ⩽ p < ∞); we would like to ask is it true that the function fα,β (⋅) [gα,β (⋅)] is generalized almost periodic in the usual sense (more generally, is it true that an essentially bounded Stepanov p-almost periodic function is generalized almost periodic in the usual sense)? In [373, Section 5], Z. Hu and A. B. Mingarelli have constructed a continuously differentiable, Stepanov-1-almost periodic function f : ℝ → ℝ, which is not almost periodic; this example is important because the constructed function is not bounded: In actual fact, the formula given on [373, p. 735, l. 2–l. 5] shows that for each natural numbers n, k ∈ ℕ we have fn ((2k + 1)n) ⩾ 1/2. This readily implies: ∞

f ((2k + 1)!!) = ∑ fl ((2k + 1)!!) l=0

⩾ f1 ((2k + 1)!!) + f3 ((2k + 1)!!) + ⋅ ⋅ ⋅ + f2k+1 ((2k + 1)!!) ⩾ (k + 1)/2,

k ∈ ℕ.

404 � 7 Special classes of metrically almost periodic functions Using the corresponding results from [304], we can simply deduce some composition principles for Bohr (ℬ, I1′ , I2′ , ρ1 , ρ2 )-almost periodic type functions. Suppose that F : I × X → Y and G : I × Y → Z are given functions, where Z is also an ordered Banach space with the cone Z+ . We will only observe here that the statement of [304, Proposition 2.17] can be straightforwardly reformulated for our new purposes; using the same notion and notation as in the formulation of this result, and following the method obeyed for formulating Theorem 7.2.4 and Proposition 7.2.5, we have that the multi-dimensional Nemytskii operator W (⋅; ⋅), given by W (t; x) := G(t; F(t; x)), t ∈ I, x ∈ X, will be generalized Bohr (ℬ, I1′ , I2′ , σ1 , σ2 , 𝒫Z )-almost periodic (generalized Bohr (ℬ, I1′ , I2′ , σ1 , σ2 , 𝒫Z )-uniformly recurrent), provided some obvious assumptions and the following conditions on the pseudometric space 𝒫Z : (i) x ⩽ y in Y implies G(t; x) ⩽ G(t; z), t ∈ I in Z; (ii) There exists a real constant c > 0 such that the assumption f − g ∈ P implies G(⋅; f (⋅)) − G(⋅; g(⋅)) ∈ PZ and ‖G(⋅; f (⋅)) − G(⋅; g(⋅))‖PZ ⩽ c‖f − g‖P . In this section, we will not consider the extensions of generalized Bohr (ℬ,I1′ ,I2′ , ρ1 , ρ2 , 𝒫 )almost periodic type functions; cf. [304, Subsection 2.2] for more details concerning this subject. The interested reader may also try to formulate an analogue of [550, Theorem 2.5] in our framework as well as to reconsider [431, Example 6.1.13, Example 6.1.15, Example 6.1.16] for generalized almost periodic type functions. 7.2.2 Some subclasses of generalized Bohr (ℬ, I1′ , I2′ , ρ1 , ρ2 , 𝒫 )-almost periodic type functions It is clear that the notion introduced in Definition 7.2.1 is rather general as well as that this notion can be further specialized by using some special classes of pseudometric spaces 𝒫 . Some important examples of pseudometric spaces 𝒫 , which can be used in Definition 7.2.1 are given below: 1. Let P be a vector space consisting of all continuous functions F : ℝn → Y for which M(F) exists in Y . We equip P with the metric 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 1 󵄩 d(F, G) := 󵄩󵄩 lim ∫ [F(t) − G(t)] dt 󵄩󵄩󵄩 , 󵄩󵄩T→+∞ (2T)n 󵄩󵄩 󵄩 󵄩Y KT

F, G ∈ P.

We can use this space in the case that I = ℝn , Ii′ = ℝn and ρi : Y → Y are functions (i = 1, 2). Then we know that any Bohr (ℬ, I ′ , ρi )-almost periodic function F(⋅; ⋅) is Bohr (ℬ, ℝn , I)-almost periodic so that the mean value of function F(⋅; x) exists for all x ∈ X; see [304]. Furthermore, if the collection ℬ consists of some compact subsets of X, I = Ii′ = ℝn and ρi = I for i = 1, 2, then the collection of all generalized Bohr (ℬ, I1′ , I2′ , ρ1 , ρ2 , 𝒫 )-almost periodic functions form a vector space with the usual operations (cf. also [431, Proposition 6.1.19]); equipped with the sup-norm, this space is one of Banach’s and we know

7.2 Generalized almost periodic functions with values in ordered Banach spaces

�

405

that in the one-dimensional setting the regular distribution determined by a locally integrable function belonging to this space need not be an almost periodic distribution (cf. [550, p. 12] for a counterexample of this type). p 2. If F ∈ Lloc (ℝn : Y ), then we define the Besicovitch seminorm ‖⋅‖ℳp through (2). Let p us recall that ‖ ⋅ ‖ℳp is a seminorm on the space ℳp (ℝn : Y ) consisting of those Lloc (ℝn : p n Y )-functions F(⋅) for which ‖F‖ℳp < ∞. Let P = ℳ (ℝ : Y ) and d(F, G) := ‖F − G‖ℳp , F, G ∈ P. We can always use this space; in the analyses carried out in [550] and [492], we always have p = 1 (see also Proposition 7.2.12 below and [431, Subsection 6.4.3], where we have recently analyzed the spaces of generalized weighted pseudo-ergodic components in ℝn ; these spaces can be used for consideration of some special pseudometric spaces 𝒫 in Definition 7.2.1, which generalize the Besicovitch spaces analyzed above). Suppose now that m ∈ ℕ and a function F : ℝn → ℝm is generalized Bohr (ℝn , ℝn , I, I, 𝒫 )-almost periodic with the pseudometric space 𝒫 being chosen by employing the usual Besicovitch seminorm. Then a simple analysis shows that the function F(⋅) is Besicovitch-p-almost periodic in the usual sense. We feel it is our duty to say that the proof of [492, Proposition 4.19] is wrong, since the authors use the inequality ‖f − g‖w,1 ⩽ M(h − g) in the proof; in fact, we have ‖f − g‖w,1 ⩾ M(h − g) and anything reasonable could tell us why the function F(⋅) has to be equi-Weyl-p-almost periodic in the usual sense; cf. also [492, Proposition 4.24]. We continue by providing the following examples: Example 7.2.7. Due to [550, Proposition 2.27], for any non-negative continuous compactly supported function φ : ℝ → ℝ such that φ(0) = 1 we have that the function 2 √ f (x) := ∑∞ k=1 φ(x − k − k 2), x ∈ ℝ is not generalized almost periodic, as well as that f (⋅) is bounded and has the mean value equal to zero (see also [550, Proposition 2.4]). Here we will only observe that the function f (⋅) is equi-Weyl-p-almost periodic for any exponent p ⩾ 1 since the constant sequence of zero polynomials converges to f (⋅) in the Weyl-p-norm, i. e. x+l

1 p lim sup[ ∫ [f (t)] dt] l→+∞ x∈ℝ l x

1/p

= 0.

In order to see this, it suffices to observe that the assumption x ⩽ 1 + k 2 + k √2 ⩽ −1 + m2 + m√2 ⩽ x + l for some k, m ∈ ℕ, x ∈ ℝ and l > 0 implies (m − k)(m + k + √2) ⩽ l + 2, m − k ⩽ √l + 2 and therefore x+l

p

1

p

∫ [f (t)] dt ⩽ √l + 2 ∫ [φ(t)] dt, x

−1

which simply yields the required; here we have assumed that the support of φ(⋅) is contained in [−1, 1]. Without going into full details, we will only emphasize here that this

406 � 7 Special classes of metrically almost periodic functions approach can be used for the construction of a substantially large class of equi-Weyl-palmost periodic functions that are not Stepanov-p-almost periodic (p ⩾ 1). For example, the function considered above is not Stepanov-p-almost periodic (p ⩾ 1) since it is bounded, uniformly continuous and not almost periodic in the usual sense (the uniform continuity of function f (⋅) follows from the uniform continuity of function φ(⋅) on the 2 √ interval [−2, 2] and the fact that the series ∑∞ k=1 φ(x − k − k 2) contains at most two terms for every real number x ∈ ℝ). Example 7.2.8. We know that the function F(⋅) = χ[0,∞)n (⋅) is not equi-Weyl-p-almost periodic for any exponent p ⩾ 1 as well as that F(⋅) is Weyl-p-almost periodic for any exponent p ⩾ 1; see [431] for a slightly stronger result. Using the mean value theorem, we may simply conclude that the function F(⋅) is not Besicovitch-p-almost periodic for any exponent p ⩾ 1 as well; therefore, the function F(⋅) is not generalized almost periodic in the usual sense. We can also prove this fact in the following way: Suppose the contrary and take any number ε ∈ (0, 1). Then there exist two almost periodic functions Gε (⋅) and Hε (⋅) such that Gε (t) ⩽ F(t) ⩽ Hε (t) for all t ∈ ℝn and lim

t→+∞

1 (2t)n

∫ [Hε (t) − Gε (t)] dt = lim

1

t→+∞ t n

[−t,t]n

∫ [Hε (t) − Gε (t)] dt ⩽ ε.

(267)

[0,t]n

Since the function Gε (⋅) is almost automorphic, we can find a strictly increasing sequence (kl ) of positive integers and a function Gε∗ (⋅) such that liml→+∞ Gε (t − (kl , kl , . . . , kl )) = Gε∗ (t) and liml→+∞ Gε∗ (t + (kl , kl , . . . , kl )) = Gε (t) for all t ∈ ℝn . The first limit equality in combination with the inequality Gε (t) ⩽ F(t) ⩽ 0, t ∈ −(0, ∞)n implies that Gε∗ (t) ⩽ 0 for all t ∈ ℝn . Using the second limit equality, we get that Gε (t) ⩽ 0 for all t ∈ ℝn . This is in contradiction with (267) because then we would have 1

ε ⩾ lim

t→+∞ t n

∫ [Hε (t) − Gε (t)] dt ⩾ lim

1

t→+∞ t n

[0,t]n

∫ Hε (t) dt ⩾ lim

1

t→+∞ t n

[0,t]n

∫ 1 dt = 1. [0,t]n

3. Suppose that Ω = [−1, 1]n , F : ℝn → Y and G : ℝn → Y are two functions satisfying p that F(t+⋅)−G(t+⋅) ∈ Lp (lΩ : Y ) for all t ∈ ℝn and l > 0. The Stepanov distance DS (F, G) lΩ

p

[the Weyl distance DW (F, G)] of functions F(⋅) and G(⋅) is defined as before (cf. [431] for more details). Let l > 0 and let P be the vector space of all p-locally integrable functions p p F : ℝn → Y such that DS (F, 0) < +∞, resp. DW (F, 0) < +∞, equipped with the distance p

lΩ

p

d(F, G) := DS (F, G) for all F, G ∈ P, resp. d(F, G) := DW (F, G) for all F, G ∈ P. Ω Our next contribution reads as follows:

Proposition 7.2.9. Suppose that a function F : ℝn → ℝm is generalized Bohr (ℝn , ℝn , I, I, 𝒫 )-almost periodic with the pseudometric space 𝒫 being chosen in this way (m ∈ ℕ). Then the function F(⋅) is Stepanov-p-almost periodic, resp. equi-Weyl-p-almost periodic. Proof. Without loss of generality, we may consider the one-dimensional setting m = n = 1 because the proof in the higher-dimensional setting is quite similar; we will consider

7.2 Generalized almost periodic functions with values in ordered Banach spaces

� 407

the class of equi-Weyl-p-almost periodic functions, only. Let ε > 0 be given. Then we can find two almost periodic functions hε : ℝ → ℝ and gε : ℝ → ℝ such that gε (t) ⩽ f (t) ⩽ hε (t), t ∈ ℝ, hε − gε ∈ P and ‖hε − gε ‖P ⩽ ε. Hence, we can find a finite real number l0 > 0 such that for every l ⩾ l0 , we have x+l

1 󵄨 󵄨p sup ∫ 󵄨󵄨h (t) − gε (t)󵄨󵄨󵄨 dt < ε. l x∈ℝ 󵄨 ε

(268)

x

Let τ ∈ ℝ be a common ε-period of functions hε (⋅) and gε (⋅). Then we have gε (x + τ) − hε (x) ⩽ f (x + τ) − f (x) ⩽ hε (x + τ) − gε (x),

x ∈ ℝ.

Therefore, we have: x+l

[

1 󵄨󵄨 󵄨p ∫ 󵄨f (t + τ) − f (t)󵄨󵄨󵄨 dt] l 󵄨 x

1/p

x+l

x+l

2p−1 󵄨󵄨 2p−1 󵄨󵄨 󵄨p 󵄨p ⩽[ ∫ 󵄨󵄨hε (t + τ) − gε (t)󵄨󵄨󵄨 dt + ∫ 󵄨󵄨gε (t + τ) − hε (t)󵄨󵄨󵄨 dt] l l x

⩽

1/p

x

x+l

4(p−1)/p 󵄨 󵄨p [ ∫ 󵄨󵄨󵄨hε (t + τ) − hε (t)󵄨󵄨󵄨 dt l1/p x+l

x

x+l

1/p

󵄨 󵄨p 󵄨 󵄨p + 2 ∫ 󵄨󵄨󵄨hε (t) − gε (t)󵄨󵄨󵄨 dt + ∫ 󵄨󵄨󵄨gε (t + τ) − gε (t)󵄨󵄨󵄨 dt] x

x

,

which simply implies the required statement taking into account the estimate (268). Remark 7.2.10. The statement of above proposition continues to hold for general vector-valued functions F : ℝn → Y , provided that the cone Y+ satisfies that there exists a finite real number c > 0 such that the assumption x ⩽ y ⩽ z in Y implies ‖y‖Y ⩽ c(‖x‖Y + ‖z‖Y ). We can simply prove the analogues of [550, Corollary 2.6, Corollary 2.8] for this choice of pseudometric space 𝒫 and introduce the Fourier coefficients in the same way as in [550, Definition 2.9]. The statement of [550, Lemma 2.10] can be extended using the metric space P consisting of all p-locally integrable functions F : ℝn → Y such that p DW (F, 0) < +∞, equipped with the Weyl distance (the proof is almost the same and follows from a simple computation after choosing the sequence of 2/εp -periodic functions in each variable), so that any measurable function with a compact support F : ℝn → ℝ is generalized Bohr (ℝn , ℝn , I, I, 𝒫 )-almost periodic and, due to Proposition 7.2.9, equiWeyl-p-almost periodic. Furthermore, the same property holds for any continuous function F : ℝn → ℝ for which the finite limit lim|t|→+∞ F(t) exists. In order to see this, we

408 � 7 Special classes of metrically almost periodic functions can assume without loss of generality that the above limit is equal to zero as well as that 0 ⩽ F(t) ⩽ 1 for all t ∈ ℝn . After that, we use the sequence (Fk (⋅) ≡ F(⋅)χ[−k,k]n (⋅))k∈ℕ, which satisfies that for each ε > 0 there exists k0 ∈ ℕ such that |Fk (t) − F(t)| ⩽ ε/3, t ∈ ℝn , k ⩾ k0 . For the function Fk (⋅) we can find two almost periodic functions Gk,ε (⋅) and Hk,ε (⋅) such that the Weyl distance of its difference is less or equal than ε/3 and Gk,ε (⋅) ⩽ F(⋅) ⩽ Hk,ε (⋅). Then, clearly, Gk,ε (⋅) − (ε/3) ⩽ Fk (⋅) − (ε/3) ⩽ F(⋅) ⩽ Fk (⋅) + (ε/3) ⩽ Hk,ε (⋅) + (ε/3), and the Weyl distance of functions Gk,ε (⋅) + (ε/3) and Hk,ε (⋅) + (ε/3) is less or equal than ε, which yields the required. The proof given above also implies that the function F(⋅) for which the finite limit lim|t|→+∞ F(t) exists must be semi-(x, I, 1, 𝒫 )j∈ℕn -periodic. Before proceeding any further, we will only mention in passing that the generalized spaces of Stepanov weighted ergodic components and the generalized spaces of Weyl weighted ergodic components (cf. [431, Subsection 6.4.1, Subsection 6.4.2]) can be also used for consideration of some special pseudometric spaces 𝒫 in Definition 7.2.1. 4. Suppose that the set I is Lebesgue measurable and ν : I → (0, ∞) is a Lebesgue p measurable function. Then the use of Banach space P := Lν (I : Y ) where 1 ⩽ p < +∞, can be also meaningful, as well as the use of the Banach space P := Cb,ν (I : Y ) where ν : I → (0, ∞) is an arbitrary function satisfying that the function 1/ν(⋅) is locally bounded.

7.2.3 Applications to the abstract Volterra integro-differential equations In this subsection, we will provide certain applications of our theoretical results to the various classes of the abstract Volterra integro-differential equations. We start with the analysis of the invariance of generalized Bohr (ℬ, I1′ , I2′ , ρ1 , ρ2 )-almost periodicity under the actions of infinite convolution products (103); for simplicity, we consider the onedimensional setting only. 1. Abstract fractional Cauchy problems without initial conditions. We need the following notion: Definition 7.2.11. Suppose that X is an ordered Banach space with the fixed cone X+ and (R(t))t>0 ⊆ L(X, Y ) is a strongly continuous operator family. Then we say that (R(t))t>0 ⊆ L(X, Y ) is positive if and only if, for every t > 0, we have R(t) ⩾ 0, i. e. R(t)[X+ ] ⊆ Y+ . The first result of this subsection reads as follows: Proposition 7.2.12. Suppose that X = Y , (R(t))t>0 ⊆ L(X) is a positive strongly continuous operator family and there exist finite real constants M > 0, β ∈ (0, 1] and γ > β such that

7.2 Generalized almost periodic functions with values in ordered Banach spaces

t β−1 󵄩󵄩 󵄩 , 󵄩󵄩R(t)󵄩󵄩󵄩L(X) ⩽ M 1 + tγ

t > 0.

�

409

(269)

Let a > 0, α > 0, 1 ⩽ p < +∞, αp ⩾ 1, ap ⩾ 1, αp(β − 1)/(αp − 1) > −1 if αp > 1, and β = 1 if αp = 1. Suppose also that f : ℝ → X is a bounded, generalized Bohr (I1′ , I2′ , ρ1 , ρ2 , 𝒫 )almost periodic function, where ρi = Ti ∈ L(X), R(t)Ti ⊆ Ti R(t), t > 0, Ii′ − Ii′ = ℝ (i = 1, 2), P is a collection of all (αp)-locally integrable functions u : ℝ → Y such that t αp lim supt→+∞ [t −ap ∫−t ‖u(s)‖Y ds] < +∞, and d(u, v) := lim sup[ t→+∞

1

t ap

t

󵄩 󵄩αp ∫󵄩󵄩󵄩u(s) − v(s)󵄩󵄩󵄩Y ds],

u, v ∈ P.

−t

Then the function F(⋅), given by (103), is bounded and generalized Bohr (I1′ , I2′ , ρ1 , ρ2 , 𝒫 )almost periodic. Proof. Since ∫0 R(t) dt < +∞ and f (⋅) is bounded, it is clear that the function F(⋅), given by (103), is measurable and bounded. By our definition, for every ε > 0, there exist functions hε : ℝ → X and gε : ℝ → X such that hε (⋅) is Bohr (I1′ , T1 )-almost periodic, gε (⋅) is Bohr (I2′ , T2 )-almost periodic, gε (t) ⩽ f (t) ⩽ hε (t), t ∈ ℝ, hε − gε ∈ P and ‖hε − gε ‖P ⩽ ε. Since we have assumed that ρi = Ti ∈ L(X), R(t)Ti ⊆ Ti R(t), t > 0 and Ii′ − Ii′ = ℝ (i = 1, 2), the following holds [304]: (a) The functions hε (⋅) and gε (⋅) are almost periodic and therefore bounded. t (b) The function t 󳨃→ Gε (t) ≡ ∫−∞ R(t − s)gε (s) ds, t ∈ ℝ is bounded and (I1′ , T1 )-almost ∞

t

periodic, while the function t 󳨃→ Hε (t) ≡ ∫−∞ R(t − s)hε (s) ds, t ∈ ℝ is bounded and Bohr (I2′ , T2 )-almost periodic.

Now we will prove that Gε (t) ⩽ F(t) ⩽ Hε (t),

t ∈ ℝ.

t

Let t ∈ ℝ be fixed. It suffices to prove that ∫−∞ R(t − s)[f (s) − gε (s)] ds ∈ Y+ = X+ . But, we have f (s) − gε (s) ∈ X+ for all s ∈ ℝ and therefore R(t − s)[f (s) − gε (s)] ∈ X+ for all s ∈ ℝ. t′

Since X+ is a closed subset of X, we only need to show that ∫a R(t −s)[f (s)−gε (s)] ds ∈ X+ for a < t ′ < t. This follows from the definition of the Riemann integral along with the inclusions X+ + X+ ⊆ X+ and λX+ ⊆ X+ for all λ > 0. It remains to be proved that Hε − Gε ∈ P and ‖Hε − Gε ‖P ⩽ Const. ⋅ ε. The first inclusion simply follows from the boundedness and the continuity of functions Hε (⋅) and Gε (⋅), along with the assumption ap ⩾ 1. The inequality ‖Hε − Gε ‖P ⩽ Const. ⋅ ε can be deduced using the computation from the proof of [450, Proposition 10], with the term Pk (⋅) − f (⋅) replaced therein with the term hε (⋅) − gε (⋅).

410 � 7 Special classes of metrically almost periodic functions Remark 7.2.13. (i) The situation X ≠ Y can occur if ρi = ci IX for i = 1, 2, where IX is the identity operator on X. Then the resulting function will be bounded and generalized Bohr (I1′ , I2′ , c1 I, c2 I, 𝒫 )-almost periodic. See also [450, Proposition 11, Remark 2]. ∞ (ii) It is clear that the estimate (269) does not imply ∫0 (1 + t)‖R(t)‖ dt < +∞. The last estimate holds for the exponentially decaying strongly continuous operator families and we will only mention in passing here that the computation given in the proof of [434, Theorem 3.1, pp. 196–197] can be used for proving an analogue of Proposition 7.2.12 with the exponent p = 1. (iii) We can use the same pseudometric space as in the formulation of Proposition 7.2.12 for providing some applications to the inhomogeneous heat equation in ℝn whose solutions are governed by the action of Gaussian semigroup. For example, if the function F : ℝn → ℝ is a generalized Bohr (c1 , c2 , 𝒫 )-almost periodic (ρ1 = c1 I, ρ2 = c2 I, I1′ = I2′ = ℝn ; c1 , c2 = ±1), then the solution (G(t0 )F)(x) := (4πt0 )−(n/2) ∫ F(x − y)e−

|y|2 4t

dy,

x ∈ ℝn

ℝn

is likewise generalized Bohr (c1 , c2 , 𝒫 )-almost periodic; see the computation directly before the formulation of [450, Theorem 4.3] (a similar approach shows that the use of general binary relations ρ1 and ρ2 is meaningful in our analysis; cf. [304] for more details). The convolution invariance of generalized almost periodicity introduced in Definition 7.2.1 can be analyzed similarly as in the proofs of this result and Proposition 7.2.12 (a possible application can be given to the fractional transport equation considered recently by N.-D. Li, R. Liu and M. Li in [499, Example 3]). (iv) Suppose that 1/p + 1/q = 1, q(β − 1) > −1 if p > 1, resp., β = 1 if p = 1. Let P be a collection of all p-locally integrable functions u : ℝ → Y such that x+l

p

lim supl→+∞ [l−1 supx∈ℝ ∫x−l ‖u(s)‖Y ds] < +∞, equipped with the Weyl distance x+l

1 󵄩 󵄩p d(u, v) := lim [ sup ∫ 󵄩󵄩󵄩u(s) − v(s)󵄩󵄩󵄩Y ds] l→+∞ l x∈ℝ

1/p

,

u, v ∈ P.

x−l

Then the function F(⋅), given by (103), is bounded and generalized Bohr (I1′ , I2′ , ρ1 , ρ2 , 𝒫 )-almost periodic, provided that the function f (⋅) enjoys the same properties. This follows from the proof of Proposition 7.2.12 and the computation contained in the proof of [428, Theorem 2.11.4], with the term g(s + t + τ) − g(s + t) replaced therein with the term hε (t + s) − gε (t + s). It is clear that Proposition 7.2.12 can be applied to the abstract (degenerate) Volterra integro-differential equations without initial conditions, provided that the solution operator family (R(t))t>0 is positive and satisfies (269). We will provide some examples of this type:

7.2 Generalized almost periodic functions with values in ordered Banach spaces

� 411

Example 7.2.14. Suppose that 1 < p < +∞ and X := Lp (ℝn : ℝ), X+ := Lp (ℝn : [0, ∞)). Then we know that there exists a positive real number ω > 0 such that the operator A := Δ + V (x) − ω, defined in the usual way, generates a positive, exponentially decaying strongly continuous semigroup (T(t))t⩾0 on X, provided that V ⩾ 0, V ∈ Lr (ℝn : ℝ) and r ⩾ max(p, n/2) if p ≠ n/2, and r > n/2 if p = n/2 (for more details, see [55, pp. 114–115] and [499, Example 8, (93)]). Let γ ∈ (0, 1) and let Rγ (t)f := γt

γ−1

∞

∫ sΦγ (s)T(st γ )f ds,

t > 0, f ∈ X,

0

where Φγ (⋅) denotes the Wright function. Since T(t) ⩾ 0 for all t ⩾ 0 and Φγ (t) ⩾ 0 for all t ⩾ 0, it follows that Rγ (t) ⩾ 0 for all t > 0; furthermore, using [428, Lemma 2.9.1], it follows that there exists a finite real constant M > 0 such that ‖Rγ (t)‖ ⩽ Mt γ−1 /(1 + t 2γ ), t > 0. On the other hand, we know that a unique solution of the abstract fractional differential equation (184) with 𝒜 = A is given by t

u(t) = ∫ Rγ (t − s)f (s) ds,

t ∈ ℝ;

−∞

for simplicity, we will not consider the semilinear analogues of (184) here (let us only note that the statement of [409, Theorem 3.1] can be formulated for the generalized uniformly recurrent type solutions of the abstract semilinear fractional differential equations under consideration). Now it is clear how we can apply Proposition 7.2.12 in the analysis of the existence and uniqueness of bounded, generalized Bohr (I1′ , I2′ , ρ1 , ρ2 , 𝒫 )-almost periodic solutions of problem (184). We can similarly apply Proposition 7.2.12 in the analysis of the existence and uniqueness of bounded, generalized Bohr (I1′ , I2′ , ρ1 , ρ2 , 𝒫 )-almost periodic solutions of the abstract fractional differential equations with the Weyl–Liouville fractional derivatives in which the operator A is replaced by its fractional analogue Aβ := −(−Δ)β + V (x) − ω for some b ∈ (0, 1); see [499, Remark 26] for more details. We continue our exposition by noticing that the resolvent positive operators considered by W. Arendt in [55, Example 3.2, Example 3.3] can be also used to provide certain applications of Proposition 7.2.12; without going into full details (cf. also [499, Example 7]), we recall that the operator (Af )(x) := −f ′ (x) + (α/x)f (x), x ∈ (0, 1], acting with domain D(A) := {f ∈ C 1 [0, 1] : f ′ (0) = f (0) = 0} in the Banach space C0 (0, 1] := {f ∈ C[0, 1] : f (0) = 0}, is resolvent positive, does not generate a strongly continuous semigroup, and generates an exponentially bounded, analytic θ-times resolvent family for any number θ ∈ (0, 1); cf. [499] for the notion. Furthermore, ρ(A) = ℂ and x

[R(λ : A)f ](x) = ∫( 0

α

x ) f (x − t)e−λt dt, x−t

λ ∈ ℂ.

We will only mention in passing that the last formula shows that the operator −A is extremely ill-posed since it cannot be even the generator of a hyperfunction semigroup

412 � 7 Special classes of metrically almost periodic functions (cf. S. Ōuchi [593]). The reason is quite simple; we have the estimate ‖R(λ : −A)‖ ⩾ eλ/2 /2, λ > 0, which can be proved by considering the function f (x) := x, x ∈ [0, 1] whose norm is equal to one. On the other hand, we have 1

[R(λ : A)f ](1) ⩾ ∫(1 − t) 0

α−1 −λt

e

1

dt ⩾ ∫ e−λt ⩾ e−λ/2 /2,

λ < 0.

1/2

We close this example with the observation that almost anything relevant has been said about the positive solutions of the abstract degenerate integro-differential equations and the positive solutions of the (fractional) Poisson heat type equations (cf. the research monographs [300] and [429] for more details about the subject). Example 7.2.15. Due to our considerations from [428, Subsection 2.9.2], an application of Proposition 7.2.12 to the abstract fractional differential equations is possible in any situation in which the operator A generates a positive strongly continuous semigroup on a Banach space X; then we can consider the operator A − ω for a sufficiently large real number ω > 0. For some important examples of differential operators generating positive strongly continuous semigroups in ordered Banach spaces, we refer the reader to [32, 53, 83, 559, 571, 580]; concerning the systems of ordinary differential equations, let us only mention that Proposition 7.2.12 is applicable provided that A ∈ ℝn,n is a real Metzler matrix of format n × n, i. e., every off-diagonal entry of A is non-negative (this is equivalent to saying that the matrix exponential (etA )t⩾0 generated by A is positive), and the spectrum of A is contained in the left half-plane {λ ∈ ℂ : Re λ < 0}. 2. Asymptotically Bohr (ℬ, I1′ , I2′ , ρ1 , ρ2 , 𝒫 )-almost periodic type functions and related applications. In this part, we have not considered the generalized 𝔻-asymptotically Bohr (ℬ, I1′ , I2′ , ρ1 , ρ2 , 𝒫 )-almost periodic type functions by now. The notion of Meyer almost periodicity has recently been introduced in [550, Definition 4.22] and this notion can be reconsidered in many different ways: it is essential to explore the classes of functions which can be written as sums of a generalized Bohr (ℬ, I1′ , I2′ , ρ1 , ρ2 , 𝒫 )-almost periodic type function and a multi-dimensional weighted ergodic component in general metric. In this section, we will only explain how the method and pseudometric proposed in the proof of Proposition 7.2.12 can be useful in the analysis of the existence and uniqueness of asymptotically generalized almost periodic solutions for a class of fractional relaxation inclusions with Caputo derivatives. Consider again the situation in which the operator A generates a positive, exponentially decaying strongly continuous semigroup (T(t))t⩾0 on X. We define the operator family (Rγ (t))t⩾0 as in Example 7.2.14; define also the operator family (Sγ (t))t⩾0 by ∞

Sγ (t)x := ∫ Φγ (s)T(st γ )x ds, 0

t > 0, x ∈ X.

7.2 Generalized almost periodic functions with values in ordered Banach spaces

�

413

Of concern is the following abstract semilinear Cauchy problem γ

(DFP)f ,γ : Dt u(t) = Au(t) + f (t), t > 0; u(0) = x0 , γ

where the Caputo fractional derivative Dt u(t) is defined as before. It is well known that a unique classical solution of problem (DFP)f ,γ is given by the formula t

u(t) = Sγ (t)x0 + ∫ Rγ (t − s)f (s) ds,

t ⩾ 0;

0

see [428] for more details. We know that limt→+∞ Sγ (t)x0 = 0 as well as that the term t

∫0 Rγ (t − s)f (s) ds can be written by (for simplicity we will not perturb the function f (⋅) by a weighted ergodic component here): t

t

+∞

∫ Rγ (t − s)f (s) ds = ∫ Rγ (t − s)f (s) ds + ∫ Rγ (s)f (t − s) ds, 0

t

−∞

t ⩾ 0.

(270)

Suppose now that the function f (⋅) satisfies the requirements of Proposition 7.2.12. Then we know that the first addend in the above sum is bounded and generalized Bohr (I1′ , I2′ , ρ1 , ρ2 , 𝒫 )-almost periodic; the second addend in the above sum vanishes as t → +∞ since +∞ 󵄩󵄩 +∞ 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∫ Rγ (s)f (t − s) ds󵄩󵄩󵄩 ⩽ ‖f ‖∞ ∫ 󵄩󵄩󵄩Rγ (s)󵄩󵄩󵄩 ds → 0, 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩Y t t

t → +∞.

Summa summarum, there exists a bounded, generalized Bohr (I1′ , I2′ , ρ1 , ρ2 , 𝒫 )-almost periodic function h : ℝ → X and a function q ∈ C0 ([0, ∞) : X) such that u(t) = h(t)+q(t), t ⩾ 0. In the case that X = Ii′ = ℝ and ρi = I for i = 1, 2, this means that u(⋅) is, in fact, a generalized almost periodic function in the usual sense, since we know that any function q : [0, ∞) → ℝ vanishing at plus infinity is generalized almost periodic; for example, if the function f (⋅) is generalized almost periodic and A = aI, where a < 0, then the solution u(t) of (DFP)f ,γ is likewise generalized almost periodic. There is only a few relevant references concerning the positive solutions of the (abstract) linear Volterra integro-differential equations (see, e. g. [222, 298, 377, 551, 573]). Here we will only note that Proposition 7.2.12 is applicable in the analysis of generalized almost periodic solutions of the following linear Volterra integral equation t

X(t) = F(t) − ∫ a(t − s)X(s) ds,

t ⩾ 0.

(271)

0

It is well known that, under certain assumptions, a unique solution of (271) is given by

414 � 7 Special classes of metrically almost periodic functions t

X(t) = F(t) − ∫ k(t − s)F(s) ds,

t ⩾ 0,

0

where the kernel k(t) is the solution of the linear equation t

k(t) = a(t) − ∫ a(t − s)k(s) ds,

t ⩾ 0.

(272)

0

By a well-known result of R. K. Miller [551, Theorem 2, Lemma 2], we know the following: If a ∈ L1 (0, 1), a(t) is positive, continuous, non-increasing on the interval (0, ∞) and ln a(t) is convex on the interval (0, ∞), then there exists a unique, locally integrable solution k(t) of (272), which is continuous for t > 0 and continuous at t = 0+ if a(0+) < +∞; moreover, we have: (i) 0 ⩽ k(t) ⩽ a(t), t > 0; ∞ (ii) ∫0 k(t) dt ⩽ 1; (iii) k(t) is not locally identical to the zero function. So, if additionally assume that there exists a finite real constant M > 0 such that a(t) ⩽ M/(1 + t γ ), t > 0, then the same estimate holds for the function k(t); see, e. g. https://www.desmos.com/calculator/0pzgylmkwu for case in which γ = 3/2. By the proof of Proposition 7.2.12 and decomposition (270), it follows that the generalized almost periodicity of forcing term F(⋅) implies the generalized almost periodicity of solution X(⋅). Concerning the vector-valued setting, we would like to emphasize that it is not clear how one can prove that a function q : [0, ∞) → Y vanishing at plus infinity is generalized almost periodic (this can be possibly done in some concrete situations). 3. Generalized almost periodic solutions of wave equation in ℝn ; examples of S. Zaidman. The classical solutions of the inhomogeneous wave equation in dimension 1, 2 and 3 are given by the d’Alembert formula, the Poisson formula and the Kirchhoff formula; see [431] for more details about the subject. It is worth noting that the generalized almost periodicity of the initial data implies the generalized almost periodicity of solution of the wave equation. We will illustrate this fact by considering the d’Alembert formula, only. In [194, Example 1.2], we have proved the following: if the functions f (⋅) and ⋅ g [1] (⋅) ≡ ∫0 g(s) ds are almost periodic, then the solution u(x, t) is almost periodic in ℝ2 . Using a simple computation involving the Fubini theorem and this fact, it readily follows that the solution u(x, t) will be generalized almost periodic in ℝ2 provided that the initial data are generalized almost periodic in ℝ. The situation is a little bit complicated with the c-almost periodic solutions since the collection of all c-almost periodic functions is not a vector space with the usual operations. Besides the condition that the

7.2 Generalized almost periodic functions with values in ordered Banach spaces

�

415

function (f (⋅), g [1] (⋅)) is c-almost periodic, provided that g(⋅) is a non-trivial function, we must also use a stronger version of Definition 7.2.1 to obtain that the solution u(x, t) is generalized c-almost periodic in ℝ2 . Generally speaking, in Definition 7.2.1 we must impose conditions that I = I1′ = I2′ = ℝn , ρi = cI and the function (Hε (⋅; ⋅), Gε (⋅; ⋅)) is Bohr (ℬ, ℝn , cI)-almost periodic (concerning [194, Example 1.1], we will only note here that the possible applications can be given only in the case that the corresponding kernel E(t, y) has a certain growth order; see also the computation before the formulation of [450, Theorem 4.3]). Consider now the situation in which the functions f (⋅) and g(⋅) are almost periodic and the number t0 > 0 is fixed. Then we know (see, e. g. S. Zaidman [797, Example 5, p. 32]; we assume here a = 1 for simplicity) that the solution u(x, t0 ) is almost periodic in ℝ. Let us assume now that the functions f (⋅) and g(⋅) are generalized Bohr (ℝ, ℝ, I, I, 𝒫 )-almost periodic, with the pseudometric space 𝒫 being the same as in the formulation of Proposition 7.2.12. Then the solution u(x, t0 ) will be generalized Bohr (ℝ, ℝ, I, I, 𝒫 )-almost periodic as well. Towards this end, fix a positive real number ε > 0. Then there exist almost periodic functions hε (⋅), gε (⋅), Hε (⋅) and Gε (⋅) such that gε (x) ⩽ f (x) ⩽ hε (x) and Gε (x) ⩽ g(x) ⩽ Hε (x) as well as a sufficiently large real number t0 > 0 such that, for every t ⩾ t0 , we have 1

t ap

t

t

−t

−t

1 󵄨 󵄨 󵄨αp 󵄨αp ∫󵄨󵄨󵄨hε (x) − gε (x)󵄨󵄨󵄨 dx + ap ∫󵄨󵄨󵄨Hε (x) − Gε (x)󵄨󵄨󵄨 dx ⩽ ε. t

(273)

Then x+t0

1 1 Gu,ε (x) ≡ [gε (x − t0 ) + gε (x + t0 )] + ∫ Gε (s) ds ⩽ u(x, t0 ) 2 2 x−t

x+t0

1 1 ⩽ Hu,ε (x) ≡ [hε (x − t0 ) + hε (x + t0 )] + ∫ Hε (s) ds, 2 2 x−t0

x∈ℝ

and there exists cp > 0 such that 1

t ap

t

󵄨 󵄨αp ∫󵄨󵄨󵄨Hu,ε (x) − Gu,ε (x)󵄨󵄨󵄨 dx

−t

⩽

cp

t ap

t

t

cp 󵄨 󵄨 󵄨αp 󵄨αp ∫󵄨󵄨󵄨hε (x + t0 ) − gε (x + t0 )󵄨󵄨󵄨 dx + ap ∫󵄨󵄨󵄨hε (x − t0 ) − gε (x − t0 )󵄨󵄨󵄨 dx t

−t

−t

󵄨󵄨αp cp 󵄨󵄨󵄨 󵄨 󵄨󵄨 󵄨 + ap ∫󵄨󵄨󵄨 ∫ 󵄨󵄨󵄨Hε (s) − Gε (s)󵄨󵄨󵄨 ds󵄨󵄨󵄨 dx. 󵄨󵄨 󵄨󵄨 t 󵄨 −t 󵄨x−t0 t 󵄨 x+t0

416 � 7 Special classes of metrically almost periodic functions The first two terms are easily controlled on account of (273). For the third addend, we assume first that αp = 1. Then we have cp

t ap

t x+t0

󵄨 󵄨 ∫ ∫ 󵄨󵄨󵄨Hε (s) − Gε (s)󵄨󵄨󵄨 ds dx

−t x−t0

⩽

cp

t+t0 s−t0

t ap

+

cp 󵄨 󵄨 ∫ ∫ 󵄨󵄨󵄨Hε (s) − Gε (s)󵄨󵄨󵄨 dx ds + ap t

t−t0

cp

t ap

t

−t+t0 s+t0

󵄨 󵄨 ∫ 󵄨󵄨󵄨Hε (s) − Gε (s)󵄨󵄨󵄨 dx ds

∫

−(t+t0 ) −t

t−t0 s+t0

∫

󵄨 󵄨 ∫ 󵄨󵄨󵄨Hε (s) − Gε (s)󵄨󵄨󵄨 dx ds,

−t+t0 s−t0

which simply implies the required conclusion after a simple calculation involving (273). x+t If αp > 1, then we first apply the Hölder inequality for the term ∫x−t 0 |Hε (s) − Gε (s)| ds 0 with exponents 1/αp and 1/αq, where 1/αp + 1/αq = 1, and repeat verbatim the computation given in the case that αp = 1. Consider finally the backward wave equation uxx + utt = 0 in the upper half-space {(x, t) : x ∈ ℝ, t > 0}. Then a unique regular solution u(x, t) of this equation which is bounded and continuous on {(x, t) : x ∈ ℝ, t ⩾ 0}, subjected with the initial condition u(x, 0) = f (x), is given by +∞

1 t u(x, t) = ∫ 2 2 f (x − s) ds, π t +s

x ∈ ℝ, t > 0.

−∞

Let t0 > 0 be fixed. Then we know that the almost periodicity of f (⋅) implies the almost periodicity of solution u(x, t) in the variable x; see, e. g. [797, Example 8, p. 33]. We also have that the generalized almost periodicity of f (⋅) implies the generalized almost periodicity of solution u(x, t) in the variable x. This follows from the above fact and the next simple computation (the meaning of functions hε (⋅) and gε (⋅) is clear): t +∞

t 1 ∫ ∫ 2 0 2 [hε (x − s) − gε (x − s)] ds dx πt t0 + s −t −∞

+∞ t

=

t 1 ∫ ∫ 2 0 2 [hε (x − s) − gε (x − s)] dx ds πt t0 + s −∞ −t +∞

t+|s|

−∞

−(t+|s|)

t 1 ⩽ ∫ 2 0 2 πt t0 + s

∫ [hε (x) − gε (x)] dx ds,

x ∈ ℝ, t > 0.

Let us finally mention some topics not considered by now:

7.3 Notes and appendices to Part II

� 417

1. It is well known that a composition of the generalized almost periodic function F : ℝn → ℝ with a Riemann integrable function is likewise generalized almost periodic. We will not reconsider here the statements of [550, Lemma 2.17, Theorem 2.18, Lemma 2.20, Theorem 2.23] in our framework. 2. It is clear that Example 7.2.6 indicates that the use of Stepanov generalizations of Bohr (ℬ, I ′ , ρ)-almost periodic functions can serve one to extend the notion introduced in Definition 7.2.1. We will skip all details concerning this concept as well as all details concerning the existence and uniqueness of generalized almost periodic type solutions of the abstract nonautonomous differential equations of first order. 3. It is also clear that the notion introduced in Definition 7.2.1 can be reconsidered by replacing the classes of Bohr (ℬ, I1′ , ρ1 )-almost periodic functions and Bohr (ℬ, I2′ , ρ2 )almost periodic functions with various classes of almost automorphic type functions, Levitan N-almost periodic functions and (ρj )j∈ℕ -periodic type functions; see also Example 7.2.7 and Example 7.2.8 given above.

7.3 Notes and appendices to Part II In this section, we will provide several useful comments and notes about special classes of metrically almost periodic type functions, considered in the second part of this monograph. Multi-dimensional functions bounded in variation and multi-dimensional almost periodic functions in variation In this part, we will first remind the readers of several traditional ways for the introduction of multivariate functions bounded in variation (cf. [158] for more details). Let us consider a rectangle R := [a1 , b1 ] × ⋅ ⋅ ⋅ × [an , bn ] ⊆ ℝn and a function F : R → ℝ. Define Δhk (F, x) := F(x1 , . . . , xk + hk , . . . , xn ) − F(x1 , . . . , xk , . . . , xn ) and, recursively, Δh1 h2 ...hk (F, x) := Δhk (Δh1 ...hk−1 (F, x)). Further on, let Πk be a collection of finite ordered families πk of points tk1 < tk2 < ⋅ ⋅ ⋅ < N +1 tk k ∈ [ak , bk ]. For every partition πk , we denote by hki the difference tki+1 − tki . The Vitali variation of F(⋅), introduced by G. Vitali in 1908 [745], is defined to be the supremum over (π1 , . . . , πn ) ∈ Π1 × ⋅ ⋅ ⋅ × Πn of the sums Nn N1 󵄨󵄨 󵄨󵄨 i ∑ ⋅ ⋅ ⋅ ∑ 󵄨󵄨󵄨Δhi1 ...hin (F, (x11 , . . . , xnin ))󵄨󵄨󵄨 . 󵄨 1 n 󵄨

i1 =1

in =1

(274)

418 � 7 Special classes of metrically almost periodic functions If the Vitali variation is finite, then we say that the function F(⋅) has bounded (finite) Vitali variation. It is well known that F(⋅) has finite Vitali variation if and only if F(⋅) can be written as difference of two functions for which all the sums of type (274) are non-negative. The next definition of a multi-dimensional function bounded in variation was proposed in the two-dimensional setting by G. H. Hardy in 1905 [359] (see also [346]). Denote by H̃ n (F) the Vitali variation of F(⋅). Further on, let α, ᾱ be pair of ordered subsets, which gives a partition of ℕn . For each such pair and for each (y1 , . . . , ys ) ∈ [aα1 , bα1 ] × ⋅ ⋅ ⋅ × [aαs , bαs ], y

Fα (⋅) denotes the function of n − s variables z1 , . . . , zn−s given by F(x1 , . . . , xn ) where xαi = yi and xᾱj = zj . The Hardy variation of F(⋅) is defined by VH (F) := sup sup H̃ n−s (fαy ). y

α

If VH (F) < ∞, then we say that the function F(⋅) has bounded (finite) Hardy variation. If a function has bounded Hardy variation, then it also has bounded Arzelà variation (see C. Arzelà [63] (1905) and [346], p. 543 for more details). Denote by Γ the class of continuous functions γ = (γ1 , . . . , γn ) : [0, 1] → R such that each component γj (⋅) is increasing. Let Π be the family of (N + 1)-tuples of points 0 ⩽ t1 < ⋅ ⋅ ⋅ < tN+1 ⩽ 1. Then the Arzelà variation of a function f : R → ℝ is defined through N

󵄨 󵄨 VA (F) := sup TV (f ∘ γ) = sup(sup{∑󵄨󵄨󵄨F(γ(ti+1 )) − F(γ(ti ))󵄨󵄨󵄨 : (t1 , . . . , tN+1 ) ∈ Π}), γ∈Γ

γ∈Γ

i=1

where TV (F ∘γ) is the classical total variation of the real-valued function F ∘γ(⋅). It is well known that a function F(⋅) has finite Arzelà variation if and only if F(⋅) can be written as the difference of two functions F + − F − with the property that F ± (x1 , . . . , xn ) ⩽ F ± (y1 , . . . , yn ) if xi ⩽ yi . The following concept has been proposed by J. Pierpont in [616]. Let Πm k be {ak = < ⋅ ⋅ ⋅ < akm+1 = bk } the equidistant partition of the segment [ak , bk ]. These partitions m generate a partition Σm of the rectangle R into 2m closed rectangles Rm 1 , . . . , R2m having equal side lengths. Then the Pierpont variation of a function F : R → ℝ is defined by ak0

sup m

1

mn−1

2m

∑ ω(F, Rm i ), i=1

where ω(F, E) denotes the oscillation of the function F(⋅) over the set E: ω(F, E) := sup F − inf F. E

E

7.3 Notes and appendices to Part II

�

419

If the Pierpont variation of F(⋅) is finite, then we say that F(⋅) has bounded (finite) Pierpont variation. Let us recall that, if a function F(⋅) has bounded Arzela variation, then F(⋅) has also bounded Pierpont variation. The modern definition of a multivariate function of bounded variation is based on the distributional techniques and goes as follows (see [295, Section 5.1] for more details): Let 0 ≠ Ω ⊆ ℝn be open. The total variation of a function u ∈ L1 (Ω) is defined through V (u; Ω) := sup{∫ u(x)(div ψ)(x) dx : ψ ∈ 𝒟Ω (ℝn ), ‖ψ‖∞ ⩽ 1}, Ω

where 𝒟Ω denotes the set consisting of all infinitely differentiable functions with support contained in Ω. A function u ∈ L1 (Ω) is said to be a function of bounded variation if and only if V (u; Ω) < +∞. The above-mentioned concepts can be used to introduce and analyze various classes of multi-dimensional almost periodic functions in variation. For example, if we accept the last concept, then we can consider the following special class of metrically almost periodic functions: Definition 7.3.1. Suppose that 0 ≠ I ′ ⊆ ℝn and F : ℝn → ℝ is a locally integrable function. Then we say that the function F(⋅; ⋅) is Bohr I ′ -almost periodic in variation if and only if for every ε > 0 there exists l > 0 such that for each t0 ∈ I ′ there exists τ ∈ B(t0 , l) ∩ I ′ such that for every t ∈ ℝn , we have: 󵄨󵄨 󵄨 󵄨󵄨F(t + τ) − F(t)󵄨󵄨󵄨 + V (F(⋅ + τ) − F(⋅); L(t; 1)) ⩽ ε. The use of partial integration shows, for instance, that any continuously differentiable, almost periodic function F : ℝn → ℝ whose all partial derivatives of first order are also almost periodic, is necessarily Bohr ℝn -almost periodic in variation. We close this part with the observation that this result and all above-considered concepts can be simply generalized to the vector-valued setting. Almost periodic functions in the Hausdorff metric As we have already mentioned, the classes of bounded almost periodic functions and semi-periodic functions with the Hausdorff metric have been introduced by S. Stoiński [706, 707]. This class has been later reconsidered by many other authors, including A. S. Dzafarov, G. M. Gasanov [283], and A. P. Petukhov [612]. Let (X, d) be a metric space and A1 , A2 ⊆ X. The Hausdorff distance between the sets A1 and A2 is defined through H(A1 , A2 ) := inf{ε > 0 : A1 ⊆ U(A2 , ε) and A2 ⊆ U(A1 , ε)}. The Hausdorff distance is usually considered in the Euclidean space X = ℝ2 equipped with the metric d((x1 , y1 ), (x2 , y2 )) := max(|x1 − x2 |, |y1 − y2 |), (xi , yi ) ∈ ℝ2 , i = 1, 2. The

420 � 7 Special classes of metrically almost periodic functions smallest set F(f ), closed and convex with respect to the y-axis, that contains the graph of a (multivalued) function f : ℝ → ℝ is called the complemented graph of the function f (⋅). The Hausdorff distance between f (⋅) and g(⋅) is defined by H(f , g) := H(F(f ), F(g)). Unless stated otherwise, we assume that the graphs of all functions under our examinations are complemented and the functions are bounded. A bounded (multivalued, in general) function f : ℝ → ℝ is called H-almost periodic if and only if for every ε > 0, there is a relatively dense set of points τ ∈ ℝ such that H(f , fτ ) < ε, where fτ (⋅) := f (⋅ + τ). Denote by ℍ the class of H-almost periodic functions. Then ℍ extends the class of Bohr almost periodic functions f : ℝ → ℝ, as easily approved (we must say that this generalization is a rather different compared with the generalizations of Stepanov, Weyl and Besicovitch). For simplicity, we will consider only the single-valued functions henceforth. In what follows, we will briefly recollect the main results obtained in the research study [612] by A. P. Petukhov: (i) Any H-almost periodic function f : ℝ → ℝ can be approximated with arbitrary accuracy by trigonometric polynomials in the Hausdorff metric. (ii) The space ℍ equipped with the Hausdorff distance H is complete. (iii) A subset ℳ ⊆ ℍ is relatively compact in (ℍ, H) if and only if ℳ is uniformly bounded and equi-H-almost periodic. (iv) A bounded function f : ℝ → ℝ is H-almost periodic if and only if the sequence of translations {fτ : τ ∈ ℝ} is relatively compact in (ℍ, H). (v) If a bounded function f : ℝ → ℝ is H-almost periodic and a bounded function g : ℝ → ℝ is Bohr almost periodic, then f + g ∈ ℍ and fg ∈ ℍ. See also [711, Theorem 1, Theorem 2] and [711, Theorem 4, Theorem 5]. (vi) A uniformly continuous H-almost periodic function f : ℝ → ℝ is Bohr almost periodic. (vii) A continuous H-almost periodic function f : ℝ → ℝ is Levitan N-almost periodic. (viii) There exists a continuous H-almost periodic function f : ℝ → ℝ which is not Bohr almost periodic. (ix) There exists a Levitan N-almost periodic function f : ℝ → ℝ which is not H-almost periodic. (x) A sum of two continuous H-almost periodic functions f : ℝ → ℝ and g : ℝ → ℝ is not H-almost periodic, in general. The class of unbounded almost periodic functions with the Hausdorff metric has been introduced by S. Stoiński in [714]; cf. also [711]. Let us only note that the author has shown, in [714, Example 2.8], that the sum of functions f : ℝ → ℝ and g : ℝ → ℝ, where f (t) := tan t, t ∈ ⋃k∈ℤ (kπ − π/2, kπ + π/2), f (t) := 0, t ∈ {kπ ± π/2 : k ∈ ℤ}, otherwise; g(t) := tan(√2t), t ∈ ⋃k∈ℤ (kπ √2/2 − √2π/4, kπ √2/2 + √2π/4), g(t) := 0, t ∈ {kπ √2/2 ± √2π/4 : k ∈ ℤ}, otherwise, is an unbounded H-almost periodic function.

7.3 Notes and appendices to Part II

� 421

Recently, D. Ji, L. Yang and J. Zhang [388] have introduced a new class of almost periodic time scales called Hausdorff almost periodic time scales by using the Hausdorff distance and proposed a generalization of almost periodic functions on time scales so. We will not consider here the multi-dimensional almost periodic type functions in the Hausdorff metric and the corresponding classes of functions on the Hausdorff almost periodic time scales. On some results of W. Chojnacki Suppose that f : ℝ → ℝ is a continuous function. Due to an old result of H. Bohr, we know that the function t 󳨃→ exp(if (t)), t ∈ ℝ is almost periodic if and only if there exists a real constant a ∈ ℝ such that the function t 󳨃→ f (t) + at, t ∈ ℝ is almost periodic. The following extensions of this result have been proved by W. Chojnacki in [209, Theorem 1, Theorem 2]: Theorem 7.3.2. (i) Suppose that f : ℝ → ℝ is a uniformly continuous function satisfying that, for every t ∈ ℝ, there exists at ∈ ℝ such that the function x 󳨃→ f (x + t) − f (x) + at x, x ∈ ℝ is almost periodic. Then there exists a real constant a ∈ ℝ such that the function t 󳨃→ f (t) + at, t ∈ ℝ is almost periodic if and only if the function t 󳨃→ exp(if (t)), t ∈ ℝ is equi-Weyl-1-almost periodic. (ii) Suppose that f : ℝ → ℝ is a Stepanov-1-almost periodic function. Then the function t t 󳨃→ exp(i ∫0 f (s) ds), t ∈ ℝ is equi-Weyl-1-almost periodic if and only if, it is almost t

periodic, which implies that the function t 󳨃→ ∫0 f (s) ds − M(f )t, t ∈ ℝ is almost periodic.

In the same paper, the author has introduced the notion of so-called E r almost periodicity for a function f : ℝ → ℝ and shown, using a probabilistic approach, that the statement of Theorem 7.3.2 is no longer true if the equi-Weyl-1-almost periodicity is replaced by E 2 almost periodicity. It is worth noting that an E r almost periodic function f : ℝ → ℝ is Besicovitch-p-almost periodic for any finite exponents 1 ⩽ p, r < ∞ as well as that the notion of E r almost periodicity is a special case of the notion introduced in Definition 6.1.1 with the metric space P ⊆ [0, ∞)ℝ consisting of all Lebesgue measurable functions f : ℝ → ℝ for which there exists a finite constant λ > 0 such that t lim supt→+∞ (2t)−1 ∫−t exp(|f (s)/λ|p ) ds ⩽ 2, equipped with the pseudo-distance t

󵄨 󵄨 p d(f , g) := inf{λ > 0 : lim sup(2t) ∫ exp([󵄨󵄨󵄨f (s) − g(s)󵄨󵄨󵄨/λ] ) ds ⩽ 2}. −1

t→+∞

−t

In [209, Lemma, p. 53], the author has shown that for any uniformly continuous equiWeyl-1-almost periodic function f : ℝ → ℝ its mean value M(f ) is the uniform limit of convex combinations of translates of f (⋅). We will illustrate Theorem 7.3.2(i) with the following example:

422 � 7 Special classes of metrically almost periodic functions Example 7.3.3. Consider the function F(t) := f (t) + bt 2 , t ∈ ℝ, where f (⋅) is given by (217) and b ∈ ℝ. Due to [431, Theorem 2.4.2], we know that the requirements of Theorem 7.3.2(i) hold with at = −2bt. Since the function t 󳨃→ F(t) + at, t ∈ ℝ is not Besicovitch bounded, as easily approved, it follows that the function t 󳨃→ exp(iF(t)), t ∈ ℝ is not equi-Weyl-1-almost periodic. Before proceeding further, we would like to emphasize that it would be very difficult to state a satisfactory analogue of Theorem 7.3.2 in the multi-dimensional setting.

�

Part III: Almost periodic type solutions to integro-differential inclusions

In this part, we will continue our analysis of almost periodic type solutions of the abstract (degenerate) Volterra integro-differential equations in Banach spaces. We will consider the existence and uniqueness of almost periodic type solutions for various classes of the abstract fractional inclusions with proportional Caputo fractional derivatives and the abstract impulsive Volterra integro-differential inclusions, using the same terminology as in Part I and Part II.

https://doi.org/10.1515/9783111233871-010

8 Abstract fractional equations with proportional Caputo fractional derivatives In this chapter, we consider the almost periodic type solutions for the abstract fractional equations with proportional Caputo fractional derivatives. The proportional Caputo fractional derivative is a relatively new type of fractional derivative extending the classical Caputo fractional derivative. In 2017, F. Jarad et al. [383] (see also [382]) introduced this notion which has many advantages when compared with the notion of the classical Caputo fractional derivative; this concept clearly enables one to consider some broader applications in modeling of various phenomena appearing in natural and technical sciences; see, e. g. [1, 2, 3, 17, 370, 411, 412, 679]. This chapter provides several new concepts in this direction and consider various classes of abstract (degenerate) fractional solution operator families connected with the use of proportional Caputo fractional derivatives. We also analyze the existence and uniqueness of almost periodic type solutions for various classes of proportional Caputo fractional differential inclusions in Banach spaces, as well as the quasi-periodic properties of the proportional fractional integrals (see also [18] and [31]). For the sequel, we need to recall the following notion from [431]: Let ω > 0 and c ∈ ℂ\{0}. Then a continuous function f : [0, ∞) → X is said to be (ω, c)-almost periodic [S-asymptotically (ω, c)-periodic] if and only if the function c−⋅/ω f (⋅) is almost periodic [limt→+∞ ‖f (t + ω) − cf (t)‖ = 0]. If |c| = 1, then we have: t 󵄩 󵄩 󵄩 t+ω 󵄩 lim 󵄩󵄩f (t + ω) − cf (t)󵄩󵄩󵄩 = |c| lim 󵄩󵄩󵄩c− ω f (t + ω) − c− ω f (t)󵄩󵄩󵄩, t→+∞

t→+∞󵄩

so that a continuous function f : [0, ∞) → X is S-asymptotically (ω, c)-periodic if and only if the function c−⋅/ω f (⋅) is S-asymptotically ω-periodic, that is, S-asymptotically (ω, c)-periodic with c = 1. Furthermore, if |c| ⩾ 1, then we have that the function c−⋅/ω f (⋅) is S-asymptotically ω-periodic if the function f (⋅) is S-asymptotically (ω, c)-periodic. Although we will not use this fact in the sequel, let us only note that, if |c| ⩾ 1 and a continuous function f : [0, ∞) → X is both, S-asymptotically (ω, c)-periodic and (ω, c)almost periodic, then f (⋅) is necessarily (ω, c)-periodic function; this can be shown with the help of substitution c−(⋅/ω) f (⋅), [51, Proposition 2] and the corresponding statement with c = 1. Let I = [0, ∞) and Φω,c := {u ∈ Cb (I : X) ; u(ω) = cu(0)}. It is clear that any (ω, c)-periodic function u : I → X belongs to the space Φω,c if |c| ⩽ 1 as well as that the converse statement is far from being generally true; furthermore, Φω,c is a closed linear subspace of Cb (I : X), and therefore, a Banach space itself when equipped with the sup-norm. The structure and main ideas of this chapter can be briefly described as follows. Section 8.1 investigates the proportional fractional integrals and the proportional Caputo fractional derivatives of vector-valued functions. The main theoretical results are given in Section 8.2, where we study the abstract proportional fractional differential inclusions https://doi.org/10.1515/9783111233871-011

428 � 8 Abstract fractional equations with proportional Caputo fractional derivatives in Banach spaces (for simplicity, we will not consider the solution operator families in locally convex spaces here; see [429] for more details about this topic). Solution operator families for the abstract fractional Cauchy problems (DFP)ζR and (DFP)ζL are investigated in Section 8.2.1; a few relevant applications to the abstract Volterra integro-differential inclusions are given in Section 8.2.2. Further on, some results about the existence and uniqueness of (ω, c)-periodic type solutions for some special classes of the semilinear proportional Caputo fractional differential equations are given in Section 8.3. In Section 8.4.1, we consider the quasi-periodic properties of proportional fractional integrals and observe that the quasi-periodic properties of Riemann–Liouville (Caputo) fractional derivatives established in the research studies [50, 51] by I. Area, J. Losada and J. J. Nieto hold for the vector-valued functions. We apply these results in the continuation of Section 8.4, where we prove the nonexistence of (ω, c)-periodic solutions of the semilinear fractional Cauchy problem (50) and the nonexistence of Poisson stable like solutions of the same problem. In this chapter, we will consider the proportional fractional integrals, the proportional Caputo fractional derivatives of order α ∈ (0, 1), and the corresponding abstract fractional relaxation inclusions with the proportional Caputo fractional derivatives only. The proportional Caputo fractional derivatives of order α > 1, and the corresponding abstract fractional oscillation inclusions will be considered somewhere else. By (E, ‖ ⋅ ‖) we denote a complex Banach space (it is our strong belief that using the same symbol for the norms of X and E will not cause any confusion henceforth).; with a little abuse of notation, the symbol I denotes the identity operator on E. The material of this chapter is taken from our recent research article [641] (a joint work with A. Rahmani, W.-S. Du, M. T. Khalladi, and D. Velinov).

8.1 Proportional fractional integrals and proportional Caputo fractional derivatives First of all, we recall the definition of proportional fractional integral of a locally integrable function u : [0, ∞) → X (see [383]): (0 I

α,ζ

t

ζ −1 1 (t−s) u)(t) := α (t − s)α−1 u(s) ds, ∫e ζ ζ Γ(α)

t ⩾ 0, α ⩾ 0, ζ ∈ (0, 1].

0

If the function u : [0, ∞) → X is differentiable and its first derivative is locally integrable, then we define its proportional Caputo fractional derivative by (c0 Dα,ζ u)(t) := (0 I 1−α,ζ (D1,ζ u))(t) := for t ⩾ 0, α ∈ (0, 1), ζ ∈ (0, 1],

t

ζ −1 1 (t−s) (t − s)−α (D1,ζ u)(s) ds, ∫e ζ 1−α ζ Γ(1 − α)

0

8.1 Proportional fractional integrals and proportional Caputo fractional derivatives

�

429

where (D1,ζ u)(t) := (Dζ u)(t) := (1 − ζ )u(t) + ζu′ (t). For ζ = 1, the proportional Caputo fractional derivative is reduced to the classical Caputo fractional derivative. Additionally, for ζ ∈ (0, 1] and α ∈ (0, 1), we note that (0 I α,ζ (c0 Dα,ζ u))(t) = u(t) − u(0)e

ζ −1 t ζ

t⩾0

,

(275)

and (c0 Dα,ζ (0 I α,ζ u))(t) = u(t),

t ⩾ 0.

(276)

For our further work, it will be important to observe that the proportional Caputo fractional derivatives of order α can be defined even for a continuous (locally integrable) function u : [0, ∞) → X (for example, it is not so satisfactory to consider the wellposedness of the fractional relaxation problem (DFP)ζR below for the continuously differentiable functions; see Definition 8.2.1). In actual fact, if the function u : [0, ∞) → X is differentiable and its first derivative is locally integrable, then we have: t

(c0 Dα,ζ u)(t) := ζ α−1 (1 − ζ ) ∫ e t

+ ζα ∫ e

ζ −1 (t−s) ζ

g1−α (t − s)u(s) ds

0

ζ −1 (t−s) ζ

g1−α (t − s)u′ (s) ds := I(t) + II(t),

t ⩾ 0.

0

The term I(t) is clearly definable for any continuous (locally integrable) function u : [0, ∞) → X. Concerning the term II(t), we manipulate as follows: Clearly, e

1−ζ ζ

t

II(t) = ζ α (g1−α ∗ e

1−ζ ζ

u (⋅))(t),

⋅ ′

t ⩾ 0,

so that the partial integration implies (gα ∗ e

1−ζ ζ

⋅

II(⋅))(t) = ζ α (g1 ∗ e α

= ζ (e

1−ζ ζ

1−ζ ζ

u (⋅))(t)

⋅ ′

⋅

⋅

1−ζ 1−ζ s u(⋅) − u(0) − ∫ e ζ u(s) ds)(t), ζ

t ⩾ 0.

0

Convoluting with g1−α (⋅), we get that for every t ⩾ 0, (g1 ∗ e

1−ζ ζ

⋅

II(⋅))(t) = ζ α [g1−α ∗ (e

1−ζ ζ

⋅

⋅

u(⋅) − u(0) −

1−ζ 1−ζ s ∫ e ζ u(s) ds)](t), ζ

0

430 � 8 Abstract fractional equations with proportional Caputo fractional derivatives so that α

II(t) = ζ e

ζ −1 t ζ

⋅

1−ζ 1−ζ d 1−ζ s [g ∗ (e ζ ⋅ u(⋅) − u(0) − ∫ e ζ u(s) ds)](t), dt 1−α ζ

t ⩾ 0.

(277)

0

Although the equation (277) can be used to provide the definition of the proportional Caputo fractional derivatives of order α for any locally integrable function u(⋅), we will restrict ourselves to the following notion: Definition 8.1.1. Let T ∈ (0, ∞]. (i) Suppose that u ∈ C([0, T) : X). The proportional Caputo fractional derivative Dα,ζ t u(t) is defined provided g1−α ∗(e by t

α−1 Dα,ζ (1 − ζ ) ∫ e t u(t) := ζ

ζ −1 (t−s) ζ

1−ζ ζ

⋅

u(⋅)−u(0)− 1−ζ ∫0 e ζ ⋅

1−ζ ζ

s

u(s) ds) ∈ C 1 ([0, T) : X),

g1−α (t − s)u(s) ds

0

+ ζ αe

ζ −1 t ζ

⋅

1−ζ 1−ζ d 1−ζ s [g1−α ∗(e ζ ⋅ u(⋅)−u(0)− ∫ e ζ u(s) ds)](t), dt ζ

t ∈ [0, T).

0

(ii) We define the proportional Caputo fractional derivative Dα,ζ t,w u(t) for those functions u : [0, T) → X for which u|(0,T) (⋅) ∈ C((0, T) : X), e 1

L ((0, T) : X) and g1−α ∗ (e Dα,ζ t,w u(t)

:= ζ

α−1

t

(1 − ζ ) ∫ e

1−ζ ζ

⋅

u(⋅) − u(0) −

ζ −1 (t−s) ζ

1−ζ ζ

⋅ ∫0 e

1−ζ ζ

⋅

1−ζ ζ

s

u(⋅) − u(0) − 1−ζ ∫0 e ζ ⋅

1−ζ ζ

s

u(s) ds ∈

1,1

u(s) ds) ∈ W ((0, T) : X), by

g1−α (t − s)u(s) ds

0 α

+ζ e

ζ −1 t ζ

⋅

1−ζ 1−ζ d 1−ζ s [g1−α ∗ (e ζ ⋅ u(⋅)−u(0)− ∫ e ζ u(s) ds)](t), dt ζ

t ∈ (0, T).

0

For ζ = 1, our proportional Caputo fractional derivatives reduce to Dαt u(t) and Furthermore, suppose that 1 > α > β > 0; then, immediately from Defini-

Dαt,w u(t).

α,ζ tion 8.1.1, it follows that the existence of fractional derivative Dα,ζ t u(t) (Dt,w u(t)) implies β,ζ

β,ζ

the existence of fractional derivative Dt u(t) (Dt,w u(t)). Remark 8.1.2. Denote A(t) = [g1−α ∗ (e 1−ζ

1−ζ ζ

1−ζ 1−ζ ⋅ ζ s ∫0 e u(s) ds)](t), t ∈ [0, T) ζ 1−ζ t 1−ζ [0, T). Since ∫0 e ζ s u(s) ds = (g1 ∗ e ζ ⋅ u(⋅))(t), 1

⋅

u(⋅) − u(0) −

and B(t) = [g1−α ∗ (e ζ ⋅ u(⋅) − u(0))](t), t ∈ t ⩾ 0, we have the following: Suppose that u ∈ C([0, T) : X), resp. u ∈ L ([0, T) : X). Then A ∈ C 1 ([0, T) : X), resp. A ∈ W 1,1 ((0, T) : X), if and only if B ∈ C 1 ([0, T) : X), resp. B ∈ W 1,1 ((0, T) : X). Observe also that, for a given function u|(0,T) (⋅) ∈ C((0, T) : X),

8.1 Proportional fractional integrals and proportional Caputo fractional derivatives

there exists only one value u(0) such that g1−α ∗ (e

1−ζ ζ

⋅

1,1

W ((0, T) : X).

u(⋅) − u(0) −

1−ζ ζ

∫0 e ⋅

1−ζ ζ

431

�

s

u(s) ds) ∈

It is worth noting that Definition 8.1.1(ii) can be used to compute the value of the fractional derivative Dα,ζ t,w ga (t) under certain assumptions: Example 8.1.3. Suppose that 0 < α < a < 1 and 0 < ζ < 1. Define u(t) := ga (t) for t > 0 1−ζ

and u(0) := 0. In order to compute X(t) = (d/dt)[g1−α ∗ e ζ ⋅ u(⋅)](t), t > 0, we integrate this equality, convolve the obtained equality with gα (t) and differentiate the obtained equality after this; it follows that e transform, we get ̃ X(λ) =

1−ζ ζ

t

ga (t) = (gα ∗ X)(t), t > 0. Applying the Laplace

λα (λ +

, ζ −1 a ) ζ

1−ζ , ζ

Re λ >

so that X(t) is locally integrable on [0, ∞) and, more precisely, a X(t) = t a−α−1 ℰ1,a−α (−at),

t > 0,

where γ

Γ(k + γ) zk , Γ(γ)Γ(αk + β) k! k=0 ∞

ℰα,β (z) = ∑

z∈ℂ

is the generalized Mittag-Leffler function [785]. Therefore, t

α−1 Dα,ζ (1 − ζ ) ∫ e t,w u(t) = ζ

ζ −1 (t−s) ζ

g1−α (t − s)ga (s) ds

0

α

+ζ e

ζ −1 t ζ

a t a−α−1 ℰ1,a−α (−at)

− ζ αe

ζ −1 t ζ

1−ζ 1−ζ (g1−α ∗ e ζ ⋅ ga (⋅))(t), ζ

t > 0.

It is predictable that the equalities (275)–(276) continue to hold in our new framework: Proposition 8.1.4. Suppose that u ∈ C([0, T) : X), resp. u ∈ L1loc ([0, T) : X). Then the following holds: α,ζ (i) If Dα,ζ t u(t), resp. Dt,w u(t) is well-defined, then we have [0 I α,ζ (Dα,ζ t,w u)](t) = u(t) − u(0)e

ζ −1 t ζ

,

for t ∈ [0, T), resp., for a. e. t ∈ (0, T).

(ii) We have α,ζ [Dα,ζ t,w (0 I u)](t) = u(t),

for t ∈ [0, T), resp., for a. e. t ∈ (0, T).

(278)

432 � 8 Abstract fractional equations with proportional Caputo fractional derivatives Proof. The proofs of both statements follows from the relative simple but tedious computations; for the sake of completeness, we will prove only (ii). By our definitions, we need to show that ζ α−1 (1 − ζ )(e + ζ αe −

ζ −1 t ζ

ζ −1 ⋅ ζ

g1−α (⋅) ∗ ζ −α e

ζ −1 ⋅ ζ

gα (⋅) ∗ u)(t)

1−ζ ζ −1 d [g1−α ∗ [(e ζ ⋅ ζ −α e ζ ⋅ gα (⋅) ∗ u) dt

1−ζ ζ −1 1−ζ (g1 ∗ (e ζ ⋅ ζ −α e ζ ⋅ gα (⋅) ∗ u))]](t) = u(t), ζ

for t ∈ [0, T), resp., for a. e. t ∈ (0, T). Since (g1−α ∗ gα )(t) = g1 (t), t > 0, it can be simply proved that ζ α−1 (1 − ζ )(e

ζ −1 ⋅ ζ

g1−α (⋅) ∗ ζ −α e

ζ −1 ⋅ ζ

gα (⋅) ∗ u)(t) =

1 − ζ ζ ζ−1 ⋅ (e ∗ u)(t), ζ

for t ∈ [0, T), resp., for a. e. t ∈ (0, T). Since e

1−ζ ζ

t

(e

ζ −1 ⋅ ζ

gα (⋅) ∗ u)(t) = (gα ∗ [e

1−ζ ζ

⋅

u(⋅)])(t),

for t ∈ [0, T), resp., for a. e. t ∈ (0, T), the previous equality and a simple calculation shows that we need to deduce that (e

ζ −1 ⋅ ζ

∗ u)(t) = e

ζ −1 t ζ

(e

1−ζ ζ

⋅

∗ u)(t),

for t ∈ [0, T), resp., for a. e. t ∈ (0, T). But, this is a trivial equality. Using the operational properties of the Laplace transform (see, e. g. [54, Section 1.6, p. 36]), we can simply prove that the Laplace transform of the proportional Caputo fractional derivative Dα,ζ t,w u(t) can be computed by 1−ζ ? α−1 Dα,ζ (1 − ζ )(λ + ) t,w u(λ) = ζ ζ + ζ α (λ +

α−1

̃ u(λ)

α

̃ 1−ζ u(0) 1 − ζ u(λ) ̃ − ) [u(λ) − ], 1−ζ ζ ζ λ+ ζ λ + 1−ζ ζ

Re λ > max(0, abs(u)), (279)

provided that the function u(t) satisfies (P1). Remark 8.1.5. Observe that we cannot generally expect the validity of the equation (279) for Re λ > max( ζ ζ−1 , abs(u)) since we need to apply [54, Corollary 1.6.6] here.

8.2 Abstract proportional Caputo fractional differential inclusions

� 433

8.2 Abstract proportional Caputo fractional differential inclusions Of concern are the following proportional Caputo fractional differential inclusions: (DFP)ζR : Dα,ζ t Bu(t) ∈ 𝒜u(t) + ℱ (t),

α ∈ (0, 1), ζ ∈ (0, 1], t ⩾ 0,

(DFP)ζL : ℬDα,ζ t u(t) ∈ 𝒜u(t) + ℱ (t),

α ∈ (0, 1), ζ ∈ (0, 1], t ⩾ 0,

{

Bu(0) = Bu0 ,

and {

u(0) = u0 ,

where ℱ : [0, ∞) → P(E), 𝒜: X → P(E) and ℬ: X → P(E) are given mappings (possibly non-linear), and B : D(B) ⊆ X → E is a single-valued operator. In the following definition, we extend the notion introduced recently in [429, Definition 3.1.1(ii)–(iii)], where we have considered the case ζ = 1: Definition 8.2.1. (i) (a) By a p-solution of (DFP)ζR , we mean any X-valued function t 󳨃→ u(t), t ⩾ 0 such that the term t 󳨃→ Dα,ζ t Bu(t), t ⩾ 0 is well-defined, u(t) ∈ D(𝒜) for t ⩾ 0, and the requirements of (DFP)ζR hold. (b) A pre-solution of (DFP)ζR is any p-solution of (DFP)ζR that is continuous for t ⩾ 0. (c) A solution of (DFP)ζR is any pre-solution u(⋅) of (DFP)ζR such that there exists a function u𝒜 ∈ C([0, ∞) : E) with u𝒜 (t) ∈ 𝒜u(t) for t ⩾ 0, and Dα,ζ t Bu(t) ∈ u𝒜 (t) + ℱ (t), t ⩾ 0. (ii) (a) By a pre-solution of (DFP)ζL , we mean any continuous X-valued function t 󳨃→ u(t), t ⩾ 0 such that the term t 󳨃→ Dα,ζ t u(t), t ⩾ 0 is well defined and continuous, α,ζ as well as that Dt u(t) ∈ D(ℬ) and u(t) ∈ D(𝒜) for t ⩾ 0, and (DFP)ζL holds. (b) A solution of (DFP)ζL is any pre-solution u(⋅) of (DFP)ζL such that there exist functions uα,ℬ ∈ C([0, ∞) : E) and u𝒜 ∈ C([0, ∞) : E) such that uα,ℬ (t) ∈ ℬDα,ζ t u(t) and u𝒜 (t) ∈ 𝒜u(t) for t ⩾ 0, as well as that uα,ℬ (t) ∈ u𝒜 (t) + ℱ (t), t ⩾ 0. Similarly as above, we can introduce the notion of a (pre-)solution of the problems (DFP)ζR and (DFP)ζL on any finite interval [0, τ) or [0, τ], where 0 < τ < ∞. We assume henceforth that 𝒜 and ℬ are multivalued linear operators. Before proceeding further, we will only notice that we cannot consider the abstract Cauchy problems (DFP)ζR or (DFP)ζL in full generality, by passing to the multivalued linear operators ℬ−1 𝒜 or 𝒜ℬ−1 (see also [429, Remark 3.1.2] for more details concerning this issue with ζ = 1) as well as that we will revisit the Ljubich’s uniqueness criterium [429, Theorem 3.1.6] for the abstract Cauchy problems with proportional Caputo fractional derivatives somewhere else.

434 � 8 Abstract fractional equations with proportional Caputo fractional derivatives 8.2.1 Solution operator families for (DFP)ζR and (DFP)ζL In this subsection, we analyze various types of solution operator families for the abstract fractional Cauchy problems (DFP)ζR and (DFP)ζL with ℬ = B = I. For the beginning, let us consider the abstract proportional Caputo inclusion (DFP)ζR with the function ℱ (t) = f (t) being single-valued, X = E and the initial value u0 replaced therein with the initial value Cu0 , where C ∈ L(E) is injective. Applying (278) and Lemma 1.1.1, we get u(t) − e

ζ −1 t ζ

t

Cu0 ∈ ζ

−α

𝒜∫e 0

ζ −1 (t−s) ζ

t

+ ζ −α ∫ e

gα (t − s)u(s) ds

ζ −1 (t−s) ζ

gα (t − s)f (s) ds,

t ⩾ 0.

0

If f ≡ 0, the above justifies the introduction of the following solution operator families for (DFP)ζR : Definition 8.2.2. Suppose that a(t) and k(t) are given by a(t) := ζ −α e

ζ −1 t ζ

gα (t),

t>0

and

k(t) := (e

ζ −1 ⋅ ζ

∗ k0 )(t),

t ⩾ 0,

(280)

where k0 (t) is the Dirac delta distribution δ(t) or k0 ∈ L1loc ([0, ∞)) [recall that δ̃ = 1]. Then a mild (α, ζ , k0 , C1 )-existence family (R1 (t))t∈[0,τ) ⊆ L(X, E) (a mild (α, ζ , k0 , C2 )uniqueness family (R2 (t))t∈[0,τ) ⊆ L(E); an (α, ζ , k0 , C)-resolvent family (R(t))t∈[0,τ) ⊆ L(E)) subgenerated by 𝒜 is nothing else but a mild (a, k)-regularized C1 -existence family subgenerated by 𝒜 (mild (a, k)-regularized C2 -uniqueness family subgenerated by 𝒜; (a, k)-regularized C-resolvent family subgenerated by 𝒜). The integral generator of a mild (α, ζ , k0 , C2 )-uniqueness family (R2 (t))t∈[0,τ) ⊆ L(E) [an (α, ζ , k0 , C)-resolvent family (R(t))t∈[0,τ) ⊆ L(E)] is defined to be the integral generator of the corresponding mild (a, k)-regularized C2 -uniqueness family [(a, k)-regularized C-resolvent family]. Observe here that the Titchmarsh convolution theorem yields that k(t) is not identically equal to the zero function, as well as that a very simple argumentation shows that the function k(t) is continuous for t ⩾ 0. In Definition 8.2.2, we assume that the operators C and C2 are injective; for more details about the situation in which some of these operators is possible non-injective, we refer the reader to [429, Subsection 3.2.2]. Immediately from Definition 8.2.2, it follows that we can apply [429, Proposition 3.2.3, Proposition 3.2.8, Theorem 3.2.9] and the equation [429, (274)] to deduce several important structural properties of the introduced solution operator families for the problems (DFP)ζR and (DFP)ζL ; the properties of subgenerators of solution operator families for these problems can be clarified following the corresponding analysis from [429, Section 3.2]. An application of [429, Proposition 3.2.13] is possible provided that

8.2 Abstract proportional Caputo fractional differential inclusions

� 435

k0 (t) is the Dirac delta distribution; we will only state here the following particular consequences of [429, Theorem 3.2.4, Theorem 3.2.5]: Theorem 8.2.3. Suppose 𝒜 is a closed MLO in X, C1 ∈ L(X, E), C2 ∈ L(E), C2 is injective, the kernels a(t) and k(t) are given through (280), and ω ⩾ max(0, abs(|k|)). (i) Let (R1 (t))t⩾0 be strongly continuous, and let the family {e−ωt R1 (t) : t ⩾ 0} be equicontinuous. Then (R1 (t))t⩾0 is a mild (α, ζ , k0 , C1 )-existence family with a subgenerator 𝒜 if and only if for every λ ∈ ℂ with Re λ > ω and k̃0 (λ) ≠ 0, we have R(C1 ) ⊆ R(I − ζ −α (λ − ((ζ − 1)/ζ ))−α 𝒜) and ∞

C y ζ −1 k̃0 (λ) 1 ζ −1 ∈ (I − ζ −α (λ − ) 𝒜) ∫ e−λt R1 (t)y dt, ζ λ− ζ −α

y ∈ X.

0

(ii) Let (R2 (t))t⩾0 be strongly continuous, and let the family {e−ωt R2 (t) : t ⩾ 0} be equicontinuous. Then (R2 (t))t⩾0 is a mild (α, ζ , k0 , C1 )-uniqueness family with a subgenerator 𝒜 if and only if for every λ ∈ ℂ with Re λ > ω and k̃0 (λ) ≠ 0, the operator I − ζ −α (λ − ((ζ − 1)/ζ ))−α 𝒜 is injective and ∞

Cx k̃0 (λ) 2 ζ −1 = ∫ e−λt [R2 (t)x − (a ∗ R2 )(t)y] dt, λ− ζ

whenever (x, y) ∈ 𝒜.

0

Theorem 8.2.4. Let (R(t))t⩾0 ⊆ L(E) be a strongly continuous operator family such that there exists ω ⩾ max(0, abs(|k|)) satisfying that the family {e−ωt R(t) : t ⩾ 0} is bounded. Suppose that 𝒜 is a closed MLO in E and C 𝒜 ⊆ 𝒜C. (i) Assume that 𝒜 is a subgenerator of the global (α, ζ , k0 , C1 )-resolvent family (R(t))t⩾0 satisfying (15) for all x = y ∈ E. Then, for every λ ∈ ℂ with Re λ > ω and k̃0 (λ) ≠ 0, the operator I − ζ −α (λ − ((ζ − 1)/ζ ))−α 𝒜 is injective, R(C) ⊆ R(I − ζ −α (λ − ((ζ − 1)/ζ ))−α 𝒜), as well as ζ α k̃0 (λ)(λ

ζ −1 − ) ζ

α−1

α

−1

ζ −1 [ζ (λ − ) − 𝒜] Cx ζ α

∞

= ∫ e−λt R(t)x dt, 0

{ζ α (λ −

α

x ∈ E, Re λ > ω0 , k̃0 (λ) ≠ 0,

ζ −1 ) : Re λ > ω, k̃0 (λ) ≠ 0} ⊆ ρC (𝒜) ζ

(281) (282)

and R(s)R(t) = R(t)R(s), t, s ⩾ 0. (ii) Assume (281)–(282). Then 𝒜 is a subgenerator of the global (α, ζ , k0 , C1 )-resolvent family (R(t))t⩾0 satisfying (15) for all x = y ∈ E and R(s)R(t) = R(t)R(s), t, s ⩾ 0.

436 � 8 Abstract fractional equations with proportional Caputo fractional derivatives Keeping in mind the last two statements, we can simply clarify the complex characterization theorem and the real characterization theorem for the generation of solution operator families for the problems (DFP)ζR and (DFP)ζL ; see also [429, Theorem 3.2.10, Theorem 3.2.12]. Differential and analytical properties of solution operator families for the problems (DFP)ζR and (DFP)ζL can be clarified following the corresponding analysis from [429, Subsection 3.2.1]; for the sequel, we need the following notion: Definition 8.2.5. (i) Suppose that 𝒜 is an MLO in X. Let θ ∈ (0, π], and let (R(t))t⩾0 be an (α, ζ , k0 , C)-resolvent family (R(t))t∈[0,τ) ⊆ L(E) subgenerated by 𝒜. Then it is said that (R(t))t⩾0 is an analytic (α, ζ , k0 , C)-resolvent family of angle θ, if and only if there exists a function R : Σθ → L(E) which satisfies that for every x ∈ E, the mapping z 󳨃→ R(z)x, z ∈ Σθ is analytic, as well as that: (a) R(t) = R(t), t > 0 and (b) limz→0,z∈Σγ R(z)x = R(0)x for all γ ∈ (0, θ) and x ∈ E. (ii) Let (R(t))t⩾0 be an analytic (α, ζ , k0 , C)-resolvent family of angle θ ∈ (0, π]. Then it is said that (R(t))t⩾0 is an exponentially bounded, analytic (α, ζ , k0 , C)-resolvent family of angle θ, resp. bounded analytic (α, ζ , k0 , C)-resolvent family of angle θ, if and only if for every γ ∈ (0, θ), there exists ωγ ⩾ 0, resp. ωγ = 0, such that the family {e−ωγ Re z R(z) : z ∈ Σγ } ⊆ L(E) is bounded. We will identify R(⋅) and R(⋅) henceforth. Basically, the following result is the first original result of this section; it can be simply formulated for the class of analytic (α, ζ , k0 , C)-resolvent families, as well: Theorem 8.2.6. Suppose that the kernels a(t) and k(t) are given by (280), (R1 (t))t∈[0,τ) ⊆ L(X, E) [(R2 (t))t∈[0,τ) ⊆ L(E); (R(t))t∈[0,τ) ⊆ L(E)] and k1 (t) := e(1−ζ )t/ζ k(t), t ⩾ 0. Then (R1 (t))t∈[0,τ) [(R2 (t))t∈[0,τ) ; (R(t))t∈[0,τ) ] is a mild (gα , k1 )-regularized C1 -existence family [a mild (gα , k1 )-regularized C2 -uniqueness family; a (gα , k1 )-regularized C-resolvent family] subgenerated by ζ −α 𝒜 if and only if (e(ζ −1)t/ζ R1 (t))t∈[0,τ) [(e(ζ −1)t/ζ R2 (t))t∈[0,τ) ; (e(ζ −1)t/ζ R(t))t∈[0,τ) ] is a mild (α, ζ , k0 , C1 )-existence family [a mild (α, ζ , k0 , C2 )-uniqueness family (R2 (t))t∈[0,τ) ; an (α, ζ , k0 , C)-resolvent family (R(t))t∈[0,τ) ] subgenerated by 𝒜. Proof. The proof is very simple and we will present it only for the mild (α, ζ , k0 , C1 )existence families. We know that for every y ∈ X and t ∈ [0, τ), we have t

(∫ gα (t − s)R1 (s)y ds, R1 (t)y − e

1−ζ ζ

t

k(t)C1 y) ∈ ζ −α 𝒜.

0

But this is equivalent to saying that for every y ∈ X and t ∈ [0, τ), we have t

(∫ e

ζ −1 t ζ

gα (t − s)R1 (s)y ds, e

ζ −1 t ζ

R1 (t)y − k(t)C1 y) ∈ ζ −α 𝒜,

0

i. e. that for every y ∈ X and t ∈ [0, τ), we have

8.2 Abstract proportional Caputo fractional differential inclusions t

(∫ ζ −α e

ζ −1 (t−s) ζ

gα (t − s)e

ζ −1 s ζ

R1 (s)y ds, e

ζ −1 t ζ

� 437

R1 (t)y − k(t)C1 y) ∈ 𝒜.

0

This simply implies the required statement. Remark 8.2.7. Keeping in mind Theorem 8.2.6, we can also consider some applications of degenerate (a, k)-regularized C-resolvent families from [429, Section 2.2, Subsection 2.3.3] to the abstract fractional Cauchy inclusions with the proportional fractional Caputo derivatives. We will skip all details concerning this issue here. Now we will state and prove the following analogue of [429, Proposition 3.2.15(i)]: Proposition 8.2.8. Suppose that the kernels a(t) and k(t) are given through (280), a closed MLO 𝒜 is a subgenerator of an (α, ζ , k0 , C)-regularized resolvent family (R(t))t∈[0,τ) , C −1 f ∈ C([0, τ) : E), u0 ∈ E, b(t) := ζ α (e

ζ −1 ⋅ ζ

g1−α ∗ k0 )(t),

t ∈ (0, τ),

(283)

and t

u(t) := R(t)u0 + ∫ R(t − s)C −1 f (s) ds,

t ∈ [0, τ).

(284)

0

Then u(t) is a unique solution of the abstract Cauchy inclusion (14) with ℬ ≡ I and ℱ (t) = k(t)Cu0 + (k ∗ f )(t), t ∈ [0, τ). If, moreover, u0 ∈ D(𝒜) and there exists a function f𝒜 ∈ C([0, τ) : E) such that f𝒜 (t) ∈ 𝒜C −1 f (t), t ∈ [0, τ), then u(t) is a unique strong solution of (14) with ℬ ≡ I and ℱ (t) = k(t)Cu0 + (k ∗ f )(t), t ∈ [0, τ). Proof. The uniqueness of solutions follows from [429, Proposition 3.2.8(ii)]. To prove the existence of solutions, observe first that u ∈ C([0, τ) : E) and a ∗ b = k. Due to our standing assumptions, we have R(t)u0 − k(t)Cu0 ∈ 𝒜(a ∗ R)(t)u0 , t ∈ [0, τ). Moreover, Lemma 1.1.1 implies that (R ∗ C −1 f )(t) ∈ 𝒜(a ∗ R ∗ C −1 f )(t),

t ∈ [0, τ).

This simply implies the first statement with uℬ = u and ua,𝒜 = u − ℱ ; cf. Definition 1.1.2(ii). Suppose now that u0 ∈ D(𝒜) and there exists a function f𝒜 ∈ C([0, τ) : E) with the prescribed properties. Then there exists v0 ∈ E such that (u0 , v0 ) ∈ 𝒜 and therefore (R(t)u0 , R(t)v0 ) ∈ 𝒜 for all t ∈ [0, τ). Then u(t) is a strong solution of (14) with ℬ ≡ I and ℱ (t) = k(t)Cu0 + (k ∗ f )(t), t ∈ [0, τ) since we can take the function t

u𝒜 (t) = R(t)v0 + ∫ R(t − s)f𝒜 (s) ds,

t ∈ [0, τ)

0

in the third part of Definition 1.1.2 (we need to apply Lemma 1.1.1 once more here).

438 � 8 Abstract fractional equations with proportional Caputo fractional derivatives Remark 8.2.9. We feel it is our duty to say that we have made a small mistake by stating that the solutions of fractional Cauchy problems constructed in [427, Proposition 2.1.32] and [429, Proposition 3.2.15] are continuously differentiable on the interval (0, τ), which is not true in general. Consider, for example, the situation of [427, Proposition 2.1.32(i)]. Let E := l1 , the Banach space of norm-summable numerical sequences (xk ) equipped with the norm ‖(xk )‖ := ∑∞ k=1 |xk |, let 0 < α < 1, and let a closed linear operator Aα on iαπ/2 E be defined through D(Aα ) := {(xk ) ∈ E : ∑∞ kxk ), k=1 k|xk | < +∞} and Aα (xk ) := (e (xk ) ∈ D(Aα ). Then E. Bazhlekova has proved, in [97, Example 2.24], that the operator Aα generates a bounded (gα , I)-regularized resolvent family (Sα (t))t⩾0 , given by Sα (t)(xk ) := (Eα (eiαπ/2 kt α )xk ),

t ⩾ 0, (xk ) ∈ E,

as well as that the operator Aα + I does not generate an exponentially bounded (gα , I)regularized resolvent family; here, Eα (z) denotes the Mittag-Leffler function (see [97] and [427] for the notion used below). Suppose now that (xk ) ∈ D(Aα ) and (1−α)/α |xk | = +∞. If the mapping t 󳨃→ Sα (t)(xk ), t > 0 is differentiable at some ∑∞ k=1 k point t > 0, then we must have d S (t)(xk ) = (t α−1 Eα,α (eiαπ/2 kt α )xk ). dt α Due to the asymptotic expansion formula for the Mittag-Leffler functions (see, e. g., [427, Theorem 1.3.1, (17)–(19)]), we have 󵄨󵄨 α−1 󵄨 󵄨 󵄨 󵄨󵄨t Eα,α (eiαπ/2 kt α )󵄨󵄨󵄨 ∼ 1 t α−1 󵄨󵄨󵄨(eiαπ/2 kt α )(1−α)/α 󵄨󵄨󵄨 ∼ 1 k (1−α)/α , 󵄨󵄨 󵄨󵄨 α 󵄨󵄨 󵄨󵄨 α (1−α)/α as k → +∞. This would imply ∑∞ |xk | < +∞, which is a contradiction. Therek=1 k fore, the mapping t 󳨃→ Sα (t)(xk ), t > 0 is not differentiable at any point t > 0.

The interested reader may try to formulate some analogues of [429, Proposition 3.2.15(ii)] in our new framework; we continue by stating the following result: Theorem 8.2.10. Suppose that the requirements of Proposition 8.2.8 hold with k0 (t) = δ(t) being the Dirac delta distribution, u0 ∈ D(𝒜) and there exists a function f𝒜 ∈ C([0, τ) : E) such that f𝒜 (t) ∈ 𝒜C −1 f (t), t ∈ [0, τ). Then the function u(t), given by (284), is a solution of the abstract fractional Cauchy problem (DFP)ζR with B = I and ℱ (t) = (b∗f )(t), t ∈ [0, τ), where the function b(t) is given by (283). Proof. By Definition 8.2.1(i), it suffices to prove that the fractional derivative Dα,ζ t u(t) is well defined as well as that Dα,ζ t u(t) = R(t)v0 + (R ∗ f𝒜 )(t) + (b ∗ f )(t),

t ∈ [0, τ).

(285)

Keeping in mind our consideration from Remark 8.1.2, we have that Dα,ζ t u(t) is well defined if and only if

8.2 Abstract proportional Caputo fractional differential inclusions

g1−α ∗ [e

1−ζ ζ

⋅

(R(⋅)u0 + (R ∗ C −1 f )(⋅)) − Cu0 ](⋅) ∈ C 1 ([0, τ) : E).

Due to Theorem 8.2.6, we have that (e family with a subgenerator ζ t ∫0 gα (t

ζ that −α

− s)e

1−ζ ζ

s

� 439

−α

1−ζ ζ

t

R(t))t∈[0,τ) is a (gα , k1 )-regularized C-resolvent 1−ζ ζ

𝒜. Since k1 (0) = k(0) = 1, we have e

t

R(t)u0 − Cu0 =

R(s)v0 ds, where v0 ∈ 𝒜u0 is chosen arbitrarily. This simply implies g1−α ∗ e

1−ζ ζ

⋅

[R(⋅)u0 − Cu0 ](⋅) ∈ C 1 ([0, τ) : E);

therefore, we need to prove that g1−α ∗ e

1−ζ ζ

⋅

(R ∗ C −1 f )(⋅) ∈ C 1 ([0, τ) : E).

We have [g1−α ∗ e

1−ζ ζ

⋅

(R ∗ C −1 f )(⋅)](t)

t

1−ζ ζ

= ∫ g1−α (t − s)e 0

t

s

= ∫ g1−α (t − s) ∫ e 0

s

s

s−r

∫[k(s − r)f (r) + ζ −α ∫ gα (s − r − v)e 0

1−ζ ζ

ζ −1 (s−r−v) ζ

R(v)f𝒜 (r) dv] dr ds

0 r

f (r) dr ds + ζ −α (g1−α ∗ gα ∗ [e

1−ζ ζ

⋅

R(⋅)] ∗ [e

1−ζ ζ

⋅

f𝒜 ])(t)

0

t

= ∫ g2−α (t − s)e

1−ζ ζ

s

f (s) ds + ζ −α (g1 ∗ [e

1−ζ ζ

⋅

R(⋅)] ∗ [e

1−ζ ζ

⋅

f𝒜 ])(t),

t ∈ [0, τ),

0

where we have applied the partial integration for the first addend in the last equality. This yields 1−ζ d [g ∗ e ζ ⋅ (R ∗ C −1 f )(⋅)](t) dt 1−α

t

= ∫ g2−α (t − s)e

1−ζ ζ

s

f (s) ds + ζ −α ([e

1−ζ ζ

⋅

R(⋅)] ∗ [e

1−ζ ζ

⋅

f𝒜 ])(t),

t ∈ [0, τ),

0

which implies that the fractional derivative Dα,ζ t u(t) is well defined. Since the both sides of the equation (285) are well defined, its equality is equivalent with the corresponding equality obtained by convoluting the both sides of (285) with a(t). This is equivalent to saying that u(t) is a strong solution of the associated Volterra inclusion (14), as easily approved, so that an application of Proposition 8.2.8 completes the proof.

440 � 8 Abstract fractional equations with proportional Caputo fractional derivatives 8.2.2 Some applications to the abstract Volterra integro-differential inclusions It is clear that Theorem 8.2.6, Proposition 8.2.8 and Theorem 8.2.10 can be applied to the various classes of the abstract Volterra integro-differential inclusions and the abstract fractional Cauchy inclusions with the proportional Caputo fractional derivatives: 1. (cf. [429, Example 3.2.23].) Suppose that 0 < α < 1, the closed linear operators A and B satisfy the condition [300, (3.14)] with α = 1 and some real constants 0 < β ⩽ 1, γ ∈ ℝ and c, C > 0 (in our notation, we have A = L and B = M). Then we know that the multivalued linear operator AB−1 generates an exponentially bounded, analytic (g1 , g1+σ )-regularized I-resolvent family of angle Σarcctan(1/c) , provided that σ > 1 − β. The subordination principle then implies (see, e. g., [432, Theorem 3.9] and [429]) that AB−1 generates an exponentially bounded, analytic (gα , g1+ασ )-regularized I-resolvent family of angle θ ⩾ min((π/2), (π/2)(α−1 − 1)) for any σ > 1 − β; the value of this angle can be probably increased using the argumentation contained in the proof of [432, Theorem 3.10], but we will not discuss this question here. For example, we can consider the well-posedness of the abstract Volterra integral inclusions associated with the following Poisson heat equation in the space E = Lp (Ω): 𝜕 [m(x)v(t, x)] = Δv − bv, t ⩾ 0, x ∈ Ω; { { { 𝜕t (P)b : {v(t, x) = 0, (t, x) ∈ [0, ∞) × 𝜕Ω, { { x ∈ Ω, {m(x)v(0, x) = u0 (x),

where Ω is a bounded domain in ℝn , b > 0, m(x) ⩾ 0 a. e. x ∈ Ω, m ∈ L∞ (Ω) and 1 < p < ∞. Let B be the multiplication in Lp (Ω) with m(x), and let A = Δ − b act with the Dirichlet boundary conditions; these operators satisfy the general requirements of this example described above. Set 𝒜 := ζ α AB−1 . Then Theorem 8.2.6 implies that the multivalued linear operator 𝒜 generates an exponentially bounded, analytic (α, ζ , k0 , I)-resolvent family of angle θ ⩾ min((π/2), (π/2)(α−1 − 1)) for any σ > 1 − β; ζ −1

ζ −1

ζ −1

here k0 (t) = e ζ t gασ (t), t > 0, b(t) = ζ α e ζ t g1−α+ασ (t), t > 0 and k(t) = e ζ t g1+ασ (t), t > 0. Then Proposition 8.2.8 is applicable for any u0 ∈ E and f ∈ C([0, τ) : E). We can similarly apply Proposition 8.2.8 to the certain classes of the abstract Volterra integral inclusions with the almost sectorial operators (cf. [427, 428, 429] and references cited therein). 2. (cf. [427, Section 2.5].) In this part, we will briefly explain how we can apply our theoretical results in the analysis of the abstract fractional Cauchy problems with the proportional Caputo fractional derivatives and the abstract differential operators generating fractional resolvent families (cf. also [429, Subsection 3.10.1]). Suppose that the requirements of [427, Theorem 2.5.3] are satisfied with α ∈ (0, 1). Then there exists an injective operator C ∈ L(E) such that the single-valued linear operator P(A) generates an exponentially bounded (gα , C)-regularized resolvent family. In our concrete situation, we have that k0 (t) = δ(t) is the Dirac delta distribution and b(t) = ζ α e

ζ −1 t ζ

g1−α (t),

8.2 Abstract proportional Caputo fractional differential inclusions

� 441

t > 0. Due to Theorem 8.2.6, the operator ζ α P(A) generates an exponentially bounded (α, ζ , δ, C)-regularized resolvent family. If u0 ∈ D(P(A)) and the function t 󳨃→ C −1 f (t), t ∈ [0, τ) is continuous, then Theorem 8.2.10 implies that there exists a unique solution of the abstract fractional Cauchy problem (DFP)ζR with B = I, 𝒜 = ζ α P(A) and ℱ (t) = (b ∗ f )(t), t ∈ [0, τ). As in many research studies carried out so far, it is clear how we can use this result in the analysis of the abstract fractional Cauchy problems in Lp -spaces with the proportional Caputo fractional derivatives and the abstract differential (non-elliptic, in general) differential operators in Lp -spaces with the constant coefficients; for example, we can consider the well-posedness of the following abstract fractional Cauchy problem: i(2−α)π

α 2 {Dα,ζ Δu(t, x) + ζ α (e t u(t, x) = ζ e { u(0) = (I − Δ)−γ u0 , {

ζ −1 ⋅ ζ

∗ f )(t),

α ∈ (0, 1), ζ ∈ (0, 1], t ⩾ 0,

provided that 1 < p < ∞ and γ ⩾ n|(1/p) − (1/2)|/α (see also [427, Example 2.5.6]). 3. Concerning the classes of mild (α, ζ , k0 , C1 )-existence families and the mild (α, ζ , k0 , C2 )-uniqueness families, we will only emphasize, that Theorem 8.2.6 and Theorem 8.2.10 can be successfully applied in the analysis of the fractional heat equation with the proportional Caputo fractional derivative in the space E := {f ∈ C(ℝ) : 2 lim|x|→∞ ex f (x) = 0}; see the final part of [427, Section 2.8] for more details. In our new framework, some other applications can be given using [427, Theorem 2.3.1(ii), Theorem 2.3.3, Remark 2.5.4(ii), Example 2.6.39]. 4. Suppose that k0 (t) = δ(t), the Dirac Delta distribution, and consider the situation of Theorem 8.2.6; then k1 (t) ≡ 1. If the corresponding (gα , 1)-regularized C-resolvent family (R1 (t))t⩾0 subgenerated by ζ −α 𝒜 exists and satisfies that ‖R1 (t)‖ ⩽ Meωt , t ⩾ 0 for some real numbers M > 0 and ω ∈ [0, (1 − ζ )/ζ ), then the corresponding (α, ζ , δ, C)regularized resolvent family (R(t))t⩾0 subgenerated by 𝒜 is exponentially decaying. Therefore, Proposition 8.2.8 and Theorem 8.2.10 can be successfully applied in the analysis of the existence and uniqueness of asymptotically almost periodic (automorphic) type solutions of the corresponding abstract Volterra integral inclusions and the abstract fractional Cauchy inclusions with the proportional Caputo fractional derivatives; see, e. g. [16, Lemma 2.13], [248, Lemma 4.1] and [428, Propositions 2.6.13, 2.7.5, 2.11.10, 3.5.4]. 5. As a consequence of Theorem 8.2.10, we have that the solution of the abstract fractional inclusion (DFP)ζR with B = I can be expected, in the usually considered situation, only if the function ℱ (t) belongs to the range of convolution transform b ∗ ⋅, where the function b(t) is given by (283). It is also clear that Theorem 8.2.10 gives us rise to define, under certain logical assumptions, the mild solution of the abstract semilinear fractional Cauchy inclusion (α ∈ (0, 1), ζ ∈ (0, 1]) t

(DFP)ζR,s : Dα,ζ t u(t) ∈ 𝒜u(t) + ∫0 b(t − s)f (s, u(s)) ds, t ∈ [0, τ), { u(0) = Cu0 ,

442 � 8 Abstract fractional equations with proportional Caputo fractional derivatives as any continuous function u : [0, τ) → E such that t

u(t) = R(t)u0 + ∫ R(t − s)C −1 f (s, u(s)) ds,

t ∈ [0, τ).

0

Concerning the abstract Cauchy inclusion (DFP)ζR , we would like to note the following, t

as well: Suppose that F(t) = ∫0 b(t − s)C −1 f (s) ds, t ⩾ 0 (τ = +∞). If we want to find a t

strongly continuous operator family (P(t))t⩾0 ⊆ L(E) such that ∫0 R(t − s)C −1 f (s) ds = t

∫0 P(t − s)C −1 F(s) ds, t ⩾ 0, then we can apply the Laplace transform and Theorem 8.2.4. After a simple computation, we get: α

α

−1

−1

ζ −1 ζ −1 ̃ P(λ) = [ζ α (λ − ) − 𝒜] Cx = ζ −α [(λ − ) − ζ −α 𝒜] Cx, ζ ζ for Re λ > 0 sufficiently large. Hence, if (W (t))t⩾0 ⊆ L(E) and α

(λ − ζ

∞ −α

𝒜) Cx = ∫ e −1

−λt

W (t)x dt,

x ∈ E, Re λ > 0 sufficiently large,

0 ζ −1

then we have P(t) = ζ −α e ζ t W (t), t ⩾ 0. This implies the following: 5.1. If the multivalued linear operator ζ −α 𝒜 satisfies the condition [429, (PW), p. 366], then our consideration from [429, Section 3.5] implies that the mild solution of the abstract semilinear fractional Cauchy inclusion (α ∈ (0, 1), ζ ∈ (0, 1]) (DFP)ζF,s : Dα,ζ t u(t) ∈ 𝒜u(t) + F(t, u(t)), { u(0) = Cu0 ,

t ∈ [0, τ),

should be defined as any continuous function u : [0, τ) → E such that t

u(t) = R(t)u0 + αζ −α ∫ e 0

η−1 (t−s) ζ

∞

(t − s)−2α ∫ rΦα (r(t − s)−α )T(r)C −1 F(s, u(s)) dr ds, 0

t ∈ [0, τ), where (T(t))t>0 denotes the (degenerate) semigroup with a removable singularity at zero, generated by ζ −α 𝒜, and Φα (⋅) denotes the Wright function. 5.2. Suppose now that P1 (x) and P2 (x) are non-zero complex polynomials in n variables, 0 < α < 1, iAj , 1 ⩽ j ⩽ n are commuting generators of bounded C0 -groups on a Banach space E, and A := (A1 , . . . , An ). Consider the following semilinear fractional degenerate abstract Cauchy problem

8.3 Almost periodic type solutions to semilinear proportional Caputo fractional

(DFP)ζα : {

� 443

Dα,ζ t P2 (A)u(t) = P1 (A)u(t) + F(t, u(t)), t ⩾ 0, u(0) = Cu0 .

If (R(t))t⩾0 is the corresponding (gα , C)-regularized resolvent family for the problem (DFP)ζα with ζ = 1 (in the sense of considerations made in [429, Section 2]), and (P(t))t⩾0 is a global (α, α, ζ −α P1 (A), P2 (A), C)-resolvent family in the sense of [429, Definition 2.2.28], then the above consideration indicates that the mild solution of the abstract semilinear Cauchy inclusion (DFP)ζα should be defined as any continuous function u : [0, τ) → E such that t

u(t) = R(t)u0 + ∫ P(t − s)C −1 F(s, u(s)) ds,

t ∈ [0, τ).

0

Hence, we can consider the existence and uniqueness of the asymptotically almost periodic (automorphic) type solutions of the corresponding abstract fractional Cauchy inclusions (DFP)ζR,s , (DFP)ζF,s and (DFP)ζα in a rather technical way; cf. the fourth application of this subsection, [429, Theorem 2.2.29, Theorem 2.2.30] and the consideration from [428, Section 2.9] for more details.

8.3 Almost periodic type solutions to semilinear proportional Caputo fractional differential equations Let I = [0, ∞), α ∈ (0, 1) and ζ ∈ (0, 1]. In this section, we continue the investigation of R. Agarwal, S. Hristova, and D. O’Regan [17] by studying the almost periodic type solutions of the following proportional Caputo fractional differential equation with 𝒜 ≡ 0: {

c α,ζ 0 D u(t)

= f (t, u(t)),

u(0) = u0 .

α ∈ (0, 1), ζ ∈ (0, 1], t ⩾ 0,

(286)

By a mild solution of (286) we mean any continuous function u : [0, ∞) → X such that (see also [17, Lemma 3]): u(t) = u0 e

ζ −1 t ζ

t

+

ζ −1 1 (t−s) (t − s)α−1 f (s, u(s)) ds, ∫e ζ α ζ Γ(α)

t ⩾ 0.

(287)

0

We will occasionally use the following conditions: (CP1) The function f : I × X → X is continuous, ω

∫ f (ω − s, x)e 0

ζ −1 s ζ

sα−1 ds = 0,

x ∈ X,

(288)

444 � 8 Abstract fractional equations with proportional Caputo fractional derivatives and for every bounded subset B of X, we have supt⩾0;x∈B ‖f (t, x)‖ < +∞. (CP2) There exists a finite real constant Lf > 0 such that 󵄩󵄩 󵄩 󵄩󵄩f (t, u) − f (t, v)󵄩󵄩󵄩 ⩽ Lf ‖u − v‖ for all t ⩾ 0 and u, v ∈ X. (CP3) c = e

ζ −1 ω ζ

.

Then, we have the following: Proposition 8.3.1. Let (CP1)–(CP3) hold, and let Lf < (1 − ζ )α . Then there exists a unique solution u ∈ Φω,c of (286). Proof. The solution of (286) is given by (287). We define the operator T : Φω,c → Φω,c by (Tu)(t) := u0 e

ζ −1 t ζ

t

ζ −1 1 (t−s) + α (t − s)α−1 f (s, u(s)) ds, ∫e ζ ζ Γ(α)

t ⩾ 0.

0

Let us show that T is well defined. So, let u ∈ Φω,c . Then a straightforward computation involving conditions (288) and (C3) shows that (Tu)(ω) = c(Tu)(0). We have that Tu(⋅) is a bounded function, since we have assumed that for every bounded subset B of X, we have supt⩾0;x∈B ‖f (t, x)‖ < +∞ and ∞

[Γ(α)]

−1

∫e

ζ −1 (t−s) ζ

(t − s)α−1 ds = [(1 − ζ )/ζ ]

−α

< +∞;

0

moreover, it is elementary to prove that Tu(⋅) is a continuous function. Therefore, T is well defined; since we have assumed that Lf < (1 − ζ )α , it can be simply shown that the mapping T is a contraction, which implies that the equation (286) has a unique solution u ∈ Φω,c due to the Banach contraction principle. We continue by providing the following illustrative example: Example 8.3.2. Suppose that X = ℂ and f (t, x) = f1 (t)f2 (x), where f1 ∈ Cb ([0, ∞) : X) and f2 (⋅) is Lipschitz continuous on ℝ with the Lipschitz constant Lf2 > 0 satisfying ‖f1 ‖∞ Lf2 < (1 − ζ )α . If ω

∫ f1 (ω − s)e

ζ −1 s ζ

sα−1 ds = 0,

0

then Proposition 8.3.1 can be simply applied. It is clear that the proportional fractional integrals, the proportional Caputo fractional derivatives and the abstract Cauchy fractional problem (286) can be considered for the functions defined on the finite interval [0, ω]. Define Ψω,c := {u ∈ C[0, ω] : u(ω) = cu(0)}; then Ψω,c is a Banach space when equipped with the sup-norm.

8.3 Almost periodic type solutions to semilinear proportional Caputo fractional

� 445

For the sequel, we need the following version of the Schauder fixed point theorem; cf. [814]: Theorem 8.3.3. Let X be a Banach space and Ω ⊆ X be a convex, closed and bounded set. If T : Ω → Ω is a continuous operator such that T(Ω) is pre-compact, then T has at least one fixed point in Ω. Consider the following conditions: (CP1)’ The function f : [0, ω] × X → X is continuous and ω

∫ f (ω − s, x)e

ζ −1 s ζ

sα−1 ds = 0,

x ∈ X.

0

(CP4) There exist real constants C1 , C2 > 0 such that 󵄩󵄩 󵄩 󵄩󵄩f (t, u)󵄩󵄩󵄩 ⩽ C1 ‖u‖ + C2

for all t ∈ [0, ω] and u ∈ X.

In the subsequent result, we apply the Schauder fixed point theorem to prove the existence of a solution u ∈ Ψω,c of the problem (286) on [0, ω]: Theorem 8.3.4. Let the conditions (CP1)’ and (CP3)–(CP4) hold, and let ω

Iω := ∫ e

ζ −1 s ζ

sα−1 ds.

0

If C1 Iω < ζ α Γ(α), then the proportional Caputo fractional differential equation (286) has at least one solution u ∈ Ψω,c on [0, ω]. Proof. Define Bm := {u ∈ Ψω,c : ‖u‖ ⩽ m}, where m(1 −

C1 Iω CI ) > ‖u0 ‖ + α 2 ω . ζ α Γ(α) ζ Γ(α)

Let T be the operator defined as in the proof of Proposition 8.3.1. For every t ∈ [0, ω] and u ∈ Bm , we have t

ζ −1 1 󵄩󵄩 󵄩 󵄩 󵄩 (t−s) (t − s)α−1 󵄩󵄩󵄩f (s, u(s))󵄩󵄩󵄩 ds ∫e ζ 󵄩󵄩(Tu)(t)󵄩󵄩󵄩 ⩽ ‖u0 ‖ + α ζ Γ(α)

0

t

ζ −1 1 󵄩 󵄩 (t−s) ⩽ ‖u0 ‖ + α (t − s)α−1 (C1 󵄩󵄩󵄩u(s)󵄩󵄩󵄩 + C2 ) ds ∫e ζ ζ Γ(α)

0

446 � 8 Abstract fractional equations with proportional Caputo fractional derivatives 1

⩽ ‖u0 ‖ +

ζ α Γ(α)

(C1 ‖u‖ + C2 )Iω < m.

Thus, ‖Tu‖ ⩽ m and T(Bm ) ⊆ Bm . Now we will prove that the operator T is continuous on Bm . Let (uk ) be a sequence in Bm such that uk → u on Bm , when k → ∞. Since f (t, x) is a continuous function, we have f (s, uk (s)) → f (s, u(s)), when k → ∞. Hence, e

ζ −1 (t−s) ζ

(t − s)α−1 f (s, uk (s)) → e

ζ −1 (t−s) ζ

(t − s)α−1 f (s, u(s)),

as k → ∞.

By (CP4), we have t

ζ −1 󵄩󵄩 ζ −1 (t−s) 󵄩󵄩 (t − s)α−1 f (s, uk (s)) − e ζ (t−s) (t − s)α−1 f (s, u(s))󵄩󵄩󵄩 ds ∫ 󵄩󵄩󵄩e ζ 󵄩 󵄩

0

t

⩽ 2(C1 m + C2 ) ∫ e

ζ −1 (t−s) ζ

(t − s)α−1 ds ⩽ 2(C1 m + C2 )Iω < +∞.

0

Now, by the Lebesgue dominated convergence theorem, we have t

ζ −1 󵄩󵄩 ζ −1 (t−s) 󵄩󵄩 (t − s)α−1 f (s, uk (s)) − e ζ (t−s) (t − s)α−1 f (s, u(s))󵄩󵄩󵄩 ds → 0, ∫ 󵄩󵄩󵄩e ζ 󵄩 󵄩

0

as k → ∞. This simply implies that the operator T is continuous on Bm . Next, we prove that the operator T is pre-compact. For any 0 ⩽ s1 ⩽ s2 and u ∈ Bm , we have 󵄩󵄩 󵄩 󵄩󵄩(Tu)(s1 ) − (Tu)(s2 )󵄩󵄩󵄩 ζ −1 󵄨 󵄨󵄨 ζ −1 󵄨 ⩽ ‖u0 ‖ ⋅ 󵄨󵄨󵄨e ζ s1 − e ζ s2 󵄨󵄨󵄨 󵄨 󵄨 s1 󵄩󵄩 ζ −1 󵄩󵄩 1 (s −s) + 󵄩󵄩󵄩 α ∫ e ζ 1 (s1 − s)α−1 f (s, u(s)) ds 󵄩󵄩 ζ Γ(α) 󵄩 0

s2 󵄩󵄩 ζ −1 󵄩󵄩 1 (s −s) − α ∫ e ζ 2 (s2 − s)α−1 f (s, u(s)) ds󵄩󵄩󵄩 󵄩󵄩 ζ Γ(α) 󵄩 0

ζ −1 󵄨 󵄨󵄨 ζ −1 󵄨 ⩽ ‖u0 ‖ ⋅ 󵄨󵄨󵄨e ζ s1 − e ζ s2 󵄨󵄨󵄨 󵄨 󵄨

s1

ζ −1 ζ −1 1 󵄩 󵄩 (s −s) (s −s) + α ∫ (e ζ 1 (s1 − s)α−1 − e ζ 2 (s2 − s)α−1 )󵄩󵄩󵄩f (s, u(s))󵄩󵄩󵄩 ds ζ Γ(α)

0 s2

+

ζ −1 1 󵄩 󵄩 (s −s) ∫ e ζ 2 (s2 − s)α−1 󵄩󵄩󵄩f (s, u(s))󵄩󵄩󵄩 ds ζ α Γ(α)

s1

8.4 Nonexistence of (ω, c)-periodic solutions of (286) and nonexistence

� 447

ζ −1 󵄨 󵄨󵄨 ζ −1 󵄨 ⩽ ‖u0 ‖ ⋅ 󵄨󵄨󵄨e ζ s1 − e ζ s2 󵄨󵄨󵄨 󵄨 󵄨

s1

ζ −1 ζ −1 C m + C2 (s −s) (s −s) + 1α ∫ (e ζ 1 (s1 − s)α−1 − e ζ 2 (s2 − s)α−1 ) ds ζ Γ(α)

0 s2

+

ζ −1 C1 m + C2 (s −s) ∫ e ζ 2 (s2 − s)α−1 ds ζ α Γ(α)

s1

ζ −1 󵄨 󵄨󵄨 ζ −1 󵄨 ⩽ ‖u0 ‖ ⋅ 󵄨󵄨󵄨e ζ s1 − e ζ s2 󵄨󵄨󵄨 󵄨 󵄨

s1

ζ −1 ζ −1 C m + C2 (s −s) (s −s) + 1α ∫ (e ζ 1 (s1 − s)α−1 − e ζ 2 (s2 − s)α−1 ) ds ζ Γ(α)

0

C m + C2 α + α1 (s − s ) → 0, ζ Γ(α + 1) 2 1 as s1 → s2 , independently of u ∈ Bm ; here we can apply the dominated convergence theorem for the second addend. Therefore, T(Bm ) is equicontinuous. Since T(Bm ) is uniformly bounded, the Arzelá–Ascoli theorem (see, e. g. [487]) implies that T(Bm ) is precompact. Using Schauder’s fixed point theorem, we finally get that the proportional Caputo fractional differential equation (286) has at least one solution u ∈ Ψω,c .

8.4 Nonexistence of (ω, c)-periodic solutions of (286) and nonexistence of Poisson stable like solutions of (286) In this section, we consider the nonexistence of (ω, c)-periodic solutions of (286) and the nonexistence of Poisson stable like solutions of (286). Before doing this, we will continue and slightly extend the results of I. Area, J. Losada, and J. J. Nieto [50, 51] concerning the quasi-periodic properties of the Riemann–Liouville fractional integrals (see also I. Area, J. Losada, J. J. Nieto [52], and J. M. Jonnalagadda [394] for the discrete analogues). For simplicity, we will not thoroughly analyze here the quasi-periodic properties of proportional Caputo fractional derivatives.

8.4.1 On quasi-periodic properties of proportional fractional integrals Let us consider first the statement of Proposition 8.3.1 with f (t, x) ≡ f (t), t ⩾ 0. If (C3) holds, then the existence of a unique nonzero (ω, c)-periodic solution of (286) can be expected only if t+ω

∫ f (t + ω − s)e t

ζ −1 s ζ

sα−1 ds = 0,

t ⩾ 0,

448 � 8 Abstract fractional equations with proportional Caputo fractional derivatives i. e. ω

∫ f (ω − s)e

ζ −1 (t+s) ζ

(t + s)α−1 ds = 0,

t ⩾ 0,

(289)

0

which is a very restrictive assumption. In [50, Theorem 1], the authors have proved that the validity of (289) with ζ = 1 and f ∈ L1loc ([0, ∞) : ℝ) being a nonzero ω-periodic function, implies that the Riemann–Liouville integral Jtα f (t) = (0 I α,1 f )(⋅) cannot be an ω-periodic function for any α ∈ (0, 1); moreover, the authors have proved, in [50, Section 4], that (0 I α,1 f )(⋅) cannot be ω′ -periodic for any α ∈ (0, 1) and ω′ > 0 (see also [50, Corollary 2] for a fractional derivative analogue of the first-mentioned result). Therefore, it is logical to ask whether these results continue to hold for an arbitrary value of parameter ζ ∈ (0, 1). Before considering this issue, we would like to state and prove a new theoretical result about the quasi-periodic properties of the Riemann–Liouville fractional integrals of essentially bounded ω-periodic functions. Suppose that α ∈ (0, 1), ω > 0 and f : [0, ∞) → X is a non-zero essentially bounded ω-periodic function. Then [51, Lemma 3] continues to hold for f (⋅), as it can be simply verified, so that the function Jtα f (⋅) is S-asymptotically ω-periodic (cf. [365] for the notion). If we suppose that the function Jtα f (⋅) is Poisson stable, this would imply by [365, Lemma 3.1] that the function Jtα f (⋅) is ω-periodic. This will be used in the proof of the following proper extension of [51, Theorem 9], which has been formulated in a slightly different manner as Theorem 2.3.48 of [431]: Theorem 8.4.1. Suppose that α ∈ (0, 1), ω > 0 and f : [0, ∞) → X is a non-zero essentially bounded ω-periodic function. Then Jtα f (⋅) cannot be Poisson stable (a restriction of an almost automorphic function to the non-negative real line). Proof. We will first consider the Poisson stable functions. Suppose that Jtα f (⋅) is Poisson stable and x ∗ ∈ X ∗ is an arbitrary functional. Let ⟨x ∗ , f (⋅)⟩ = a(⋅) + ib(⋅), where a(⋅) and b(⋅) are real-valued functions. Then the function Jtα ⟨x ∗ , f (⋅)⟩ = Jtα a(⋅) + iJtα b(⋅) is Poisson stable because Jtα ⟨x ∗ , f (⋅)⟩ = ⟨x ∗ , Jtα f (⋅)⟩, which further implies that the functions Jtα a(⋅) and Jtα b(⋅) are Poisson stable. Since a(⋅) and b(⋅) are essentially bounded functions of period ω, the above discussion implies that Jtα a(⋅) and Jtα b(⋅) are periodic functions of period ω. Then we can apply [50, Theorem 1] in order to see that a(⋅) ≡ b(⋅) ≡ 0. This implies ⟨x ∗ , f (⋅)⟩ ≡ 0 and therefore f (⋅) ≡ 0 since x ∗ was arbitrary. Suppose now that Jtα f (⋅) is a restriction of an almost automorphic function to the non-negative real line. For the remainder of the proof, it is essential to observe that the statements of [51, Lemma 1, Theorem 5] hold not only for continuous periodic functions but also for essentially bounded periodic functions (let us only note here that the functions φn (t) and Φn (t) defined at the beginning of the proof of [51, Theorem 5] are continuous for any essentially bounded T-periodic function f (t), as a very simply computation shows). Keeping in mind our assumptions, we obtain now that the functions Jtα a(⋅) and Jtα b(⋅) are asymptotically ω-periodic on the non-negative real line so that there

8.4 Nonexistence of (ω, c)-periodic solutions of (286) and nonexistence � 449

exist two continuous ω-periodic functions ga,b : ℝ → ℝ and two continuous functions ψa,b : [0, ∞) → ℝ vanishing at plus infinity so that Jtα a(t) = ga (t) + ψa (t), t ⩾ 0 and Jtα b(t) = gb (t) + ψb (t), t ⩾ 0. This is impossible because the almost automorphic function Jtα a(⋅) − ga (⋅) [Jtα b(⋅) − gb (⋅)] cannot vanish at plus infinity on account of the supremum formula [428, Lemma 3.9.9]. Therefore, a(⋅) ≡ b(⋅) ≡ 0, ⟨x ∗ , f (⋅)⟩ ≡ 0 and f (⋅) ≡ 0. Applying the trick used in the first part of the proof and the well known fact that a weakly bounded set in a locally convex space is bounded, we may conclude that the statements of [50, Theorem 1, Corollary 2] and [51, Lemma 2, Lemma 3; Proposition 1, Proposition 2; Theorem 2, Theorem 3, Theorem 4, Theorem 8] hold in the vector-valued case (concerning the above-mentioned statements from [51], it seems very plausible that the continuity of function f (⋅) in their formulations can be replaced with the essential boundedness). It is clear that [51, Corollary 1] cannot be reformulated even for the complex-valued functions, and regarding the main structural results established in [50, 51], it remains to be considered whether the statements of [51, Theorem 5, Theorem 6, Theorem 7] hold in the vector-valued case. We will analyze this question somewhere else. Let us come back now to the question proposed at the end of the first paragraph of this subsection. First of all, we will prove that the statements of [50, Theorem 1] and Theorem 8.4.1 can be simply transferred to the proportional Caputo fractional integrals. More precisely, we have the following: Theorem 8.4.2. Let ζ ∈ (0, 1), α ∈ (0, 1) and (C3) hold. (i) Suppose that f : [0, ∞) → ℝ is a non-zero locally integrable (ω, c)-periodic function. Then the function (0 I α,ζ f )(⋅) cannot be (ω, c)-periodic. (ii) Suppose that f : [0, ∞) → X is a non-zero essentially bounded (ω, c)-periodic func1−ζ

tion. Then the function e ζ ⋅ (0 I α,ζ f )(⋅) cannot be Poisson stable (a restriction of an almost automorphic function to the non-negative real line). (iii) Suppose that f : [0, ∞) → X is a non-zero essentially bounded (ω, c)-periodic function. Then the function e

1−ζ ζ

⋅

(0 I α,ζ f )(⋅) is S-asymptotically ω-periodic. 1−ζ

Proof. We will prove only (i). The function g(t) := e ζ t f (t), t ⩾ 0 is nonzero, locally integrable and ω-periodic, as easily shown. An application of [50, Theorem 1] shows that t

the function t 󳨃→ ∫0 gα (t − s)e e

ζ −1 t ζ

1−ζ t s ∫0 gα (t −s)e ζ f (s) ds, t

1−ζ ζ

s

f (s) ds, t ⩾ 0 cannot be ω-periodic, i. e. the function t 󳨃→

⩾ 0 cannot be (ω, c)-periodic. This implies the required.

Remark 8.4.3. We cannot use (iii) in order to prove that the function (0 I α,ζ f )(⋅) is S-asymptotically (ω, c)-periodic since |c| < 1. Using the same substitution, we can simply transfer the statements of [51, Theorem 2, Theorem 5, Theorem 8] and Theorem 8.4.1 for the proportional fractional integrals. Details can be left to the interested readers.

450 � 8 Abstract fractional equations with proportional Caputo fractional derivatives Remark 8.4.4. It is clear that Dαt [Const.] = 0 for any α ∈ (0, 1) so that we must replace the word “nonzero” in the formulation of [51, Corollary 2] with the word “nonconstant” in order to retain its validity. We will use this fact later on. In what follows, we continue the general analysis of Section 8.4. Let (C3) hold. We are going to prove that the solution u(t) of the fractional Cauchy problem (286), which is given by (287), cannot be (ω, c)-periodic, provided that u(t) is not a constant multiple of the function e

ζ −1 t ζ

, as well as the function f (t, x) is continuous and satisfies f (t + ω, cx) = cf (t, x),

t ⩾ 0, x ∈ X.

(290)

Suppose the contrary and consider the function v(t) := e(1−ζ )t/ζ u(t), t ⩾ 0. Then v(t) is a nonconstant ω-periodic function since v(t + ω) = e(1−ζ )t/ζ e(1−ζ )ω/ζ u(t + ω) = e(1−ζ )t/ζ e(1−ζ )ω/ζ cu(t) = v(t),

t ⩾ 0;

moreover, we have: t

v(t) = u0 + ∫ gα (t − s)[ζ −α e

1−ζ ζ

s

f (s, u(s))] ds,

t ⩾ 0.

0

This implies Dαt v(t) =

1−ζ d [g ∗ (v(⋅) − u0 )] = ζ −α e ζ t f (t, u(t)), dt 1−α

t ⩾ 0.

On the other hand, using (290) we have f (t + ω, u(t + ω)) = f (t + ω, cu(t)) = cf (t, u(t)),

t ⩾ 0,

so that the mapping t 󳨃→ f (t, u(t)), t ⩾ 0 is (ω, c)-periodic, and consequently, the mapping 1−ζ

t 󳨃→ ζ −α e ζ t f (t, u(t)), t ⩾ 0 is ω-periodic. This contradicts [50, Corollary 2] (see also Remark 8.4.4) and yields the required conclusion. We can similarly prove that the function e(1−ζ )⋅/ζ u(⋅) cannot be Poisson stable (a restriction of an almost automorphic function to the non-negative real line), which strongly justifies the consideration of our results from Section 8.3. Before proceeding further, let us note the following: 1. Of concern is the following generalization of (289): ω

∫ a(t + s)f (ω − s) ds = 0,

t ⩾ 0,

(291)

0

with a ∈ L1loc ([0, ∞) : ℂ) and f ∈ L1loc ([0, ω) : X). It could be of importance to find some sufficient conditions on the kernel a(t) ensuring that the assumption (291) implies

8.4 Nonexistence of (ω, c)-periodic solutions of (286) and nonexistence

� 451

f (t) = 0 for a. e. t ∈ [0, ω]. For general non-constant kernels a(t), this is not true, as the following simple counterexample shows: Example 8.4.5. Suppose that a(t) = f (t) = 1 for 0 ⩽ t < ω/2 and a(t) = f (t) = 0 for ω/2 ⩽ t ⩽ ω. Then (291) holds but it is not true that f (t) = 0 for a. e. t ∈ [0, ω]; moreover, 0 ∈ supp(a) ∩ supp(f ). 2. In this section, we have not considered the Caputo fractional proportional derivatives with respect to another functions and the abstract fractional inclusions with this type of fractional derivatives. For more details about the subject, we refer the reader to the research articles [384, 385, 484] and the list of references quoted therein. 3. The Hilfer generalized proportional fractional derivatives have been also introduced and analyzed in the existing literature (see, e. g. the paper [19] by I. Ahmed et al). Concerning the Hadamard proportional fractional integral inequalities, we can recommend for the reader [575, 639, 640] and references cited therein. Before proceeding to the final chapter of the monograph, we would like to note that we will not consider here the corresponding classes of multi-dimensional C (k) -almost periodic type functions and their applications (k ∈ ℕ); see [428, 431] and references quoted therein for more details about this subject in the one-dimensional setting.

9 Almost periodic type solutions to abstract impulsive Volterra integro-differential inclusions The main aim of this chapter is to consider the existence and uniqueness of almost periodic type solutions for various classes of the abstract impulsive Volterra integrodifferential inclusions.

9.1 Abstract impulsive Volterra integro-differential inclusions The fractional calculus and the fractional differential equations play a significant role in modeling of complex systems in various disciplines in science and engineering; see, e. g. [97, 266, 414, 427, 619] and references cited therein for more details about the subject. On the other hand, the theory of impulsive differential equations has significant growth in popularity because of its huge potential of applicability in various fields of pure and applied science. For example, the impulsive differential equations are used for modeling processes exhibiting changes at certain moments, negligible compared with the duration of the whole process; these types of processes cannot be described using the classical theory of integer or fractional differential equations. For further information concerning the theory of impulsive differential equations, we refer the reader to [77, 78, 297, 350, 486, 699, 756, 760, 808, 809, 810, 811, 812, 813, 814, 815] and references cited therein. The main purpose of this section is to provide certain applications of (a, k)-regularized C-resolvent families to the abstract impulsive Volterra integro-differential inclusions in Banach spaces ((a, k)-regularized C-resolvent families in sequentially complete locally convex spaces can be also considered but we will skip all details regarding this topic here). In the existing theory of the abstract impulsive Volterra integro-differential equations, it has been commonly used that the linear operator A under consideration is single-valued and generates a strongly continuous semigroup, cosine operator function or fractional resolvent operator family (for some results concerning applications of the almost sectorial operators, we refer the reader to the research articles [352] and [642]). This is probably the first research monograph which considers the use of C-regularized solution operator families (even global non-degenerate C-regularized semigroups) or multivalued linear operators in the theory of the abstract impulsive Volterra integrodifferential equations (some applications of once integrated semigroups on weakly compactly generated Banach spaces have recently been provided by I. Benedetti, V. Obukhovskii, and V. Taddei in [105], where the authors have investigated the solvability of the impulsive Cauchy problem for integro-differential inclusions with non-densely defined linear operators). Concerning the abstract impulsive degenerate differential equations with Caputo fractional derivatives, we would like to mention that T. D. Ke and C. T. Kinh have recently analyzed in [403] the existence and stability of solutions for a class of degenerate impulsive fractional differential equations using the subordination https://doi.org/10.1515/9783111233871-012

9.1 Abstract impulsive Volterra integro-differential inclusions

� 453

principles and degenerate semigroups of operators (cf. also [778] and Definition 9.1.15 below with a(t) ≡ k(t) ≡ 1 and C = I). The organization of this section can be briefly described as follows. Section 9.1.1 investigates the C-wellposedness of the abstract impulsive differential inclusions of integer order (see Theorem 9.1.2 and Corollary 9.1.3 for some results obtained in this direction). In Section 9.1.2, we investigate the C-wellposedness of the abstract degenerate impulsive higher-order Cauchy problem (ACP)n . In the existing literature, we have not been able to locate any research article concerning the use of (ultra-)distribution semigroups ((ultra-)distribution cosine functions) to the abstract impulsive differential equations of first-order (second-order) or the C-wellposedness of the abstract impulsive higher-order Cauchy problem (ACP)n , even if this problem is non-degenerate, i. e. solvable with the respect to the highest derivative. Motivated by this fact, we present several illustrative applications in Example 9.1.4. In Example 9.1.5, we consider the abstract impulsive differential equations of first order (second order) with the multivalued linear operators 𝒜 satisfying the condition (P); the main importance of Example 9.1.6 is to present some applications of entire (gn , C)-regularized resolvent families in the study of the abstract impulsive Cauchy problem (ACP)n;1 clarified below as well as to initiate the study of the abstract Volterra integro-differential equations with impulsive effects in the complex plane (the angular domains of the complex plane). Before proceeding further with the organization of section, we would like to say that a great number of structural results about the existence and uniqueness of the (abstract) impulsive fractional differential equations have been incorrectly stated in the existing literature because the authors have used the completely wrong formulae for the forms of solutions (see, e. g. the discussions carried out in the research articles [310, 311, 758, 759, 808, 809]). In Section 9.1.3, we will say just a few words about the abstract impulsive fractional differential inclusions with Caputo derivatives or Riemann– Liouville derivatives. The main aim of this subsection is a very simple result, Theorem 9.1.9, which indicates the incorrectness of many structural results published so far in the existing literature. As a simple consequence of the second result of Section 9.1.3, Theorem 9.1.11, we have that it is very difficult to study the existence and uniqueness of the piecewise continuous solutions to the abstract fractional differential inclusions with the Riemann–Liouville derivatives of order α ∈ (0, 1). Section 9.1.4 investigates the abstract Volterra integro-differential inclusions with impulsive effects. The main result of this section is Theorem 9.1.13, which particularly shows that it is much better to analyze the well-posedness of the abstract Volterra integro-differential inclusions with the kernel a(t) = gα (t) than the well-posedness of the abstract impulsive fractional differential inclusions with Caputo derivatives (Riemann–Liouville derivatives). In Theorem 9.1.17, we provide certain applications of a special class of the exponentially bounded (a, k)-regularized C-resolvent families to the abstract impulsive degenerate Volterra equation (301). In order to better exhibit the main aims and ideas of this section, we will consider here the very simple forms of impulsive effects. The material of this section is taken from [280].

454 � 9 Almost periodic type solutions to abstract impulsive Volterra integro-differential inclusions Let T > 0. Then the space of X-valued piecewise continuous functions on [0, T] is defined by PC([0, T] : X) ≡ {u : [0, T] → X : u ∈ C((ti , ti+1 ] : X),

u(ti −) = u(ti ) and u(ti +) exist for any i ∈ ℕ0l },

where 0 ≡ t0 < t1 < t2 < ⋅ ⋅ ⋅ < tl < T ≡ tl+1 and the symbols u(ti −) and u(ti +) denote the left and the right limits of the function u(t) at the point t = ti , i ∈ ℕ0l−1 , respectively. Let us recall that PC([0, T] : X) is a Banach space endowed with the norm ‖u‖ := max{supt∈[0,T) ‖u(t+)‖, supt∈(0,T] ‖u(t−)‖}. The space of X-valued piecewise continuous functions on [0, ∞), denoted by PC([0, ∞) : X), if defined as the union of those functions f : [0, ∞) → X such that the discontinuities of f (⋅) form a discrete set and that for each T > 0 we have f|[0,T] (⋅) ∈ PC([0, T] : X). Unless stated otherwise, we will always assume henceforth that the operator C ∈ L(X) is injective.

9.1.1 Abstract impulsive differential inclusions of integer order The main aim of this subsection is to analyze the abstract impulsive differential inclusions of integer order. The applications of C-regularized solution operator families are crucial in the case that the order of equation is greater or equal than three. For the beginning, let us consider the following abstract impulsive higher-order Cauchy inclusion

(ACP)n;1

u(n) (t) ∈ 𝒜u(t) + f (t), t ∈ [0, T] ∖ {t1 , . . . , tl }, { { { (j) (j) (j) k : {(Δu )(tk ) = u (tk +) − u (tk −) = Cyj , k ∈ ℕl , j ∈ ℕ0n−1 , { { (j) j ∈ ℕ0n−1 , {u (0) = Cuj ,

where 𝒜 is an MLO in X. We will use the following concepts of solutions: Definition 9.1.1. (i) By a pre-solution of (ACP)n;1 on [0, T] we mean any function u(⋅) which is n-times continuously differentiable on the intervals [0, t1 ), (t1 , t2 ), (t2 , t3 ), . . . , (tl , T], the right derivatives limt→ti + u(j) (t) exist for 0 ⩽ j ⩽ n and 1 ⩽ i ⩽ l, the left derivatives limt→ti − u(j) (t) exist for 0 ⩽ j ⩽ n and 1 ⩽ i ⩽ l + 1, and the requirements of (ACP)n;1 hold. A solution of (ACP)n;1 on [0, T] is any pre-solution u(t) of (ACP)n;1 on [0, T] which additionally satisfies that there exists a function u𝒜 : [0, T] → X such that u𝒜 (t) ∈ 𝒜u(t) for t ∈ [0, T] ∖ {t1 , t2 , . . . , tl }, u(n) (t) = u𝒜 (t) + f (t) for t ∈ [0, T] ∖ {t1 , t2 , . . . , tl }, the right limits limt→ti + u𝒜 (t) exist for 1 ⩽ i ⩽ l and the left limits limt→ti − u𝒜 (t) exist for 1 ⩽ i ⩽ l + 1.

9.1 Abstract impulsive Volterra integro-differential inclusions

� 455

(ii) Suppose that 0 ≡ t0 < t1 < ⋅ ⋅ ⋅ < tl < tl+1 < ⋅ ⋅ ⋅ < +∞ and the sequence (tl )l has no accumulation point. By a (pre-)solution of (ACP)n;1 on [0, ∞) we mean any function u(⋅) which satisfies that, for every l ∈ ℕ and T ∈ (tl , tl+1 ), the function u|[0,T] (⋅) is a (pre-)solution of (ACP)n;1 on [0, T]. The main result about the well-posedness of the problem (ACP)n;1 reads as follows: Theorem 9.1.2. Suppose that 𝒜 is a closed subgenerator of a local (gn , C)-regularized resolvent family (R(t))t∈[0,τ) , where τ > T and n ∈ ℕ. Suppose that the functions C −1 f (⋅) and f𝒜 (⋅) are continuous on the set [0, T] ∖ {t1 , . . . , tl }, f𝒜 (t) ∈ 𝒜C −1 f (t) for all t ∈ [0, T] ∖ {t1 , . . . , tl }, as well as the right limits and the left limits of the functions C −1 f (⋅) and f𝒜 (⋅) exist at any point of the set {t1 , . . . , tl }. Define n−1 t

u(t) := R(t)u0 + ∑ ∫ gj (t − s)R(s)uj ds t t−s

j=1 0

+ ∫ ∫ gn−1 (t − s − r)R(r)(C −1 f )(s) dr ds + ω(t),

t ∈ [0, T],

(292)

0 0

where 0, t ∈ [0, t1 ], { { { k t−tp p ω(t) := {∑p=1 R(t − tp )yp0 + ∑kp=1 ∑n−1 gj (t − tp − s)R(s)yj ds, j=1 ∫0 { { 0 { if t ∈ (tk , tk+1 ] for some k ∈ ℕl−1 .

(293)

Then the function u(t) is a unique solution of the problem (ACP)n;1 , provided that j u0 , . . . , ul ∈ D(𝒜) and yk ∈ D(𝒜) for all k ∈ ℕl and j ∈ ℕ0n−1 . Proof. The uniqueness of solutions can be simply proved with the help of [429, Proposition 3.2.8(ii)]. Next, we will show that the function n−1 t

t t−s

uh (t) := R(t)u0 + ∑ ∫ gj (t − s)R(s)uj ds + ∫ ∫ gn−1 (t − s − r)R(r)(C −1 f )(s) dr ds j=1 0

0 0

is n-times continuously differentiable on [0, T], uh(n) (t) ∈ 𝒜uh (t) + f (t), t ∈ [0, T] and t

uh (0) = Cuj , j ∈ ℕ0n−1 . If (x, y) ∈ 𝒜, then we have R(t)x − Cx = ∫0 gn (t − s)R(s)y ds, t ∈ [0, T] and therefore (j)

(n)

R (t)x = R(t)y,

t ∈ [0, T];

(j)

t

R (t)x = ∫ gn−j (t − s)R(s)y ds, 0

t ∈ [0, T], j ∈ ℕn−1 0 . (294)

456 � 9 Almost periodic type solutions to abstract impulsive Volterra integro-differential inclusions Let (uj , vj ) ∈ 𝒜, j ∈ ℕ0n−1 ; since t t−s

dj ( j ∫ ∫ gn−1 (t − s − r)R(r)(C −1 f )(s) dr ds) = 0, dt t=0

j ∈ ℕ0n−1 ,

0 0

the above equalities simply imply uh (0) = Cuj , j ∈ ℕ0n−1 and (j)

n−1 t

u(n) (t) = R(t)(v0 ) + ∑ ∫ gn−j (t − s)R(s)vj ds t

+ lim ∫ h→0

0

j=1 0

R(t + h − s) − R(t − s) −1 (C f )(s) ds h t+h

1 ∫ R(t + h − s)(C −1 f )(s) ds h→0 h

+ lim

t

n−1 t

= R(t)(v0 ) + ∑ ∫ gn−j (t − s)R(s)vj ds t

+ lim ∫ h→0

0

+ f (t),

j=1 0

R(t + h − s) − R(t − s) −1 (C f )(s) ds h t ∈ [0, T] ∖ {t1 , . . . , tl }.

(295)

Since t−s

−1

R(t − s)(C f )(s) − f (s) = ∫ gn (t − s − r)R(r)f𝒜 (s) dr,

0 ⩽ s ⩽ t,

0

the dominated convergence theorem shows that t

lim ∫

h→0

0

t

R(t + h − s) − R(t − s) −1 (C f )(s) ds = ∫ R(t − s)f𝒜 (s) ds, h

t ∈ [0, T] ∖ {t1 , . . . , tl }.

0

Keeping in mind that R(t)𝒜 ⊆ 𝒜R(t) for t ∈ [0, T], Lemma 1.1.1, the last equality and (295) together imply that the function uh (⋅) is n-times continuously differentiable on [0, T] and uh(n) (t) ∈ 𝒜uh (t) + f (t), t ∈ [0, T]. Therefore, it suffices to show that the function ω(⋅) is a solution of the problem ω (t) ∈ 𝒜ω(t), { { { (Δω(j) )(tk ) = Cykj , { { { (j) {ω (0) = 0, (n)

t ∈ [0, T] ∖ {t1 , . . . , tl }, k ∈ ℕl , j ∈ ℕ0n−1 , j ∈ ℕ0n−1 .

9.1 Abstract impulsive Volterra integro-differential inclusions

�

457

The third equality is obvious since ω(t) = 0 for t ∈ [0, t1 ]. Let (yk , zk ) ∈ 𝒜 for all k ∈ ℕl as well as (ykj , zkj ) ∈ 𝒜 for all k ∈ ℕl and j ∈ ℕ0n−1 ; the second equality simply follows from (294). To verify the first equality, we observe that (n)

k

ω (t) = ∑ R(t − p=1

p tp )z0

k n−1

t−tp

+ ∑ ∑ ∫ gj (t − tp − s)R(s)zkj ds p=1 j=1 0

for all t ∈ (tk , tk+1 ] (k ∈ ℕ0l−1 ). Since R(t)𝒜 ⊆ 𝒜R(t) for t ∈ [0, T], Lemma 1.1.1 yields that ω(n) (t) ∈ 𝒜ω(t) for all t ∈ [0, T] ∖ {t1 , . . . , tl }. This completes the proof of theorem. Corollary 9.1.3. Suppose that 𝒜 is a closed subgenerator of a global (gn , C)-regularized resolvent family (R(t))t⩾0 , where n ∈ ℕ. Suppose further that 0 < t1 < ⋅ ⋅ ⋅ < tl < ⋅ ⋅ ⋅ < +∞, the sequence (tl )l has no accumulation point, the functions C −1 f (⋅) and f𝒜 (⋅) are continuous on the set [0, T] ∖ {t1 , . . . , tl , . . .}, f𝒜 (t) ∈ 𝒜C −1 f (t) for all t ∈ [0, T] ∖ {t1 , . . . , tl , . . .}, as well as the right limits and the left limits of the functions C −1 f (⋅) and f𝒜 (⋅) exist at any point of the set {t1 , . . . , tl , . . .}. Define the functions u(t) and ω(t) for t ∈ [0, T] by (292) and (293), respectively. Then the function u(t) is a unique solution of the problem (ACP)n;1 for t ∈ [0, T] ∖ {t1 , . . . , tl , . . .}, provided that u0 , . . . , ul , . . . ∈ D(𝒜) and ykj ∈ D(𝒜) for all k ∈ ℕ and j ∈ ℕ0n−1 .

Now we will provide the following illustrative applications of Theorem 9.1.2 and Corollary 9.1.3: Example 9.1.4. (i) Let 𝒜 = A be a closed single-valued linear operator and λ ∈ ρ(A). Then it is well known that A is the integral generator of a distribution semigroup (distribution cosine function) if and only if for each τ > 0 there exists n ∈ ℕ such that A is the integral generator of a local (λ − A)−n -regularized semigroup ((λ − A)−n regularized cosine function) on [0, τ); furthermore, there exists an injective operator C ∈ L(X) such that A is the integral generator of a global C-regularized semigroup (C-regularized cosine function); cf. [426] for the notion and more details. Therefore, Theorem 9.1.2 and Corollary 9.1.3 can be successfully applied in the case that n = 1 (n = 2). (ii) Suppose that the sequence (Mp ) of positive real numbers satisfies M0 = 1, (M.1), (M.2) and (M.3) as well as that a closed linear operator A generates a regular ultradistribution semigroup (regular ultradistribution cosine function) of (Mp )-class; then there exists an injective operator C ∈ L(X) such that A is the integral generator of a global C-regularized semigroup (C-regularized cosine function); cf. [426, Sections 3.5–3.6] for the notion and more details. Consequently, Theorem 9.1.2 and Corollary 9.1.3 can be successfully applied in the case that n = 1 (n = 2). For some important examples of (differential) operators generating ultradistribution semigroups (ultradistribution cosine functions), we refer the reader to [426, Example 3.5.18, Example 3.5.23, Example 3.5.30(ii), Example 3.5.39].

458 � 9 Almost periodic type solutions to abstract impulsive Volterra integro-differential inclusions (iii) Suppose that k ∈ ℕ, aα ∈ ℂ, 0 ⩽ |α| ⩽ k, aα ≠ 0 for some α with |α| = k, P(x) = ∑|α|⩽k aα i|α| x α , x ∈ ℝn , ω := supx∈ℝn Re(P(x)) < +∞ (condition [427, (W), p. 68] holds), and X is one of the spaces Lp (ℝn ) (1 ⩽ p ⩽ ∞), C0 (ℝn ), Cb (ℝn ), BUC(ℝn ). Define P(D) := ∑ aα f (α) |α|⩽k

and

D(P(D)) := {f ∈ E : P(D)f ∈ E distributionally}.

Then it is well known that the operator P(D) generates an exponentially bounded C-regularized semigroup (C-regularized cosine function) with an appropriately chosen regularizing operator C ∈ L(X), so that Corollary 9.1.3 can be successfully applied in the case that n = 1 (n = 2). Example 9.1.5. (i) We can analyze the well-posedness of the abstract impulsive inclusion (ACP)1;1 for the multivalued linear operators 𝒜 satisfying condition (P). Then the degenerate semigroup (T(t))t>0 generated by 𝒜 has an integrable singularity at zero but we can still apply the method obeyed in the proof of Theorem 9.1.2 if the function f (t) satisfies the requirements of [300, Theorem 3.7] and there exist vectors z1 , . . . , zk , . . . from the continuity set of the semigroup (T(t))t>0 such that zk ∈ 𝒜yk , k = 1, . . . , l, . . . The established conclusion can be simply applied in the analysis of the following abstract impulsive Poisson heat equation in the space X = Lp (Ω): d [m(x)v(t, x)] = (Δ − b)v(t, x) + f (t, x), t ⩾ 0, x ∈ Ω; { dt { { { { {v(t, x) = 0, (t, x) ∈ [0, ∞) × 𝜕Ω, { { k ∈ ℕ, {m(x)v(tk +, x) − m(x)v(tk −, x) = fk (x), { { { x ∈ Ω, {m(x)v(0, x) = u0 (x),

under certain logical assumptions; keeping in mind the consideration carried out in [428, Example 3.10.4], we can also provide certain applications of the almost sectorial operators to the abstract impulsive differential equations of first order in Hölder spaces. (ii) Suppose that A, B and C are closed linear operators in X, D(B) ⊆ D(A) ∩ D(C), B−1 ∈ L(X) and the conditions [300, (6.4)–(6.5)] are satisfied with some numbers c > 0 and 0 < β ⩽ α = 1; cf. also [428, Example 3.10.10]. In [300, Chapter VI], the following second-order differential equation without impulsive conditions d (Cu′ (t)) + Bu′ (t) + Au(t) = f (t), dt

t > 0;

u(0) = u0 ,

Cu′ (0) = Cu1

has been analyzed by the usual converting into the first-order matricial system d Mz(t) = Lz(t) + F(t), dt where

t > 0;

Mz(0) = Mz0 ,

9.1 Abstract impulsive Volterra integro-differential inclusions

M=[

I O

O ], C

L=[

O −A

I ], −B

u z0 = [ 0 ] u1

and

F(t) = [

� 459

0 ] (t > 0). f (t)

The argumentation contained in the proof of [300, Theorem 6.1] shows that the operator (L[D(B)]×X − ωM[D(B)]×X )(M[D(B)]×X )−1 satisfies the condition (P) for a sufficiently large number ω > 0, in the pivot space [D(B)] × X. Hence, this MLO generates a degenerate semigroup (T(t))t>0 in [D(B)] × X, having an integrable singularity at zero and exponentially decaying growth rate at infinity. Then we can apply [300, Theorem 3.8, Theorem 3.9] in the analysis of the existence and uniqueness of solutions to the abstract degenerate Cauchy problem without impulsive conditions: d Mz(t) = (L − ωM)z(t) + F(t), dt

t > 0;

Mz(0) = Mz0 .

Furthermore, we can apply Corollary 9.1.3 with n = 1, u(t) = Mz(t), t ⩾ 0 and 𝒜 = (L − ωM)M −1 in the analysis of the existence and uniqueness to the piecewise continuously differentiable solutions of the following second-order impulsive differential equation: d (Cu′ (t)) + (2ωC + B)u′ (t) + (A + ωB + ω2 C)u(t) = f (t), { { { dt u(tk +) − u(tk −) = yk , C[u′ (tk +) + ωu(tk +)] − C[u′ (tk −) + ωu(tk −)] = zk , { { { ′ {u(0) = u0 , C[u (0) + ωu0 ] = Cu1 .

t > 0; k ∈ ℕ,

As is well known, we can simply incorporate this result in the analysis of the existence and uniqueness of piecewise continuously differentiable solutions to the following damped Poisson-wave type equation in the spaces X := H −1 (Ω) or X := Lp (Ω): 𝜕 (m(x) 𝜕u ) + (2ωm(x) − Δ) 𝜕u + (A(x; D) − ωΔ + ω2 m(x))u(x, t) = f (x, t), { { 𝜕t 𝜕t 𝜕t { { { { { t ⩾ 0, x ∈ Ω; u = 𝜕u/𝜕t = 0, (x, t) ∈ 𝜕Ω × [0, ∞), { { u(x, tk +) − u(x, tk −) = yk (x), k ∈ ℕ, { { { { { C[(𝜕u/𝜕t)(x, tk +) + ωu(x, tk +)] − C[(𝜕u/𝜕t)(x, tk −) + ωu(x, tk −)] = zk (x), { { { { {u(0, x) = u0 (x), m(x)[(𝜕u/𝜕t)(x, 0) + ωu0 ] = m(x)u1 (x), x ∈ Ω,

k ∈ ℕ,

where Ω ⊆ ℝn is a bounded open domain with smooth boundary, 1 < p < ∞ and some extra assumptions are satisfied; see [300, Example 6.1] for further information. We continue this subsection by providing an illustrative application of Corollary 9.1.3 with n ⩾ 3: Example 9.1.6. Let (E, ‖ ⋅ ‖) be a complex Banach space, s ∈ ℕ and let iAj , 1 ⩽ j ⩽ n be commuting generators of bounded C0 -groups on E. Further on, let P(x) = ∑|η|⩽d Pη x η (Pη ∈ Mm , x ∈ ℝs ) be a polynomial matrix; cf. [427] for the notion and the notation. Then, due to [427, Theorem 2.3.3], we know that there exists an injective operator C ∈ L(E) with dense range such that the operator P(A), defined in the usual way, is the integral

460 � 9 Almost periodic type solutions to abstract impulsive Volterra integro-differential inclusions generator of a global (gn , Cm )-regularized resolvent family (Wn (t))t⩾0 on E m , where Cm = CIm,m and Im,m denotes the identity matrix of format m × m. Furthermore, the mapping t 󳨃→ Wn (t), t ⩾ 0 can be extended to the whole complex plane and the following holds: (i) R(Wn (z)) ⊆ D∞ (P(A)), z ∈ ℂ and z

P(A) ∫ gn (z − s)Wn (s)x⃗ ds = Wn (z)x⃗ − Cm x,⃗

z ∈ ℂ, x⃗ ∈ E m .

0

(iii) The mapping z 󳨃→ Wn (z), z ∈ ℂ is entire. Suppose now that m = 1, p11 (x) = ∑|α|⩽d aα x α , x ∈ ℝs (aα ∈ ℂ) and E is a function space on which translations are uniformly bounded and strongly continuous. In this case, P(A) is just the operator ∑|α|⩽d aα i|α| (𝜕/𝜕x)α with its maximal distributional domain; for example, let E = Lp (ℝs ), where 1 ⩽ p < +∞. Let us also assume that 0 < t1 < ⋅ ⋅ ⋅ < tl < ⋅ ⋅ ⋅ < +∞ and the sequence (tl )l has no accumulation point. 1. Using Corollary 9.1.3 and the above result, we obtain that there exists a dense subset E0,n of Lp (ℝs ) such that the following abstract impulsive Cauchy problem: n

d |α| (𝜕/𝜕x)α u(t, x), t ⩾ 0, x ∈ ℝs , n u(t, x) = ∑|α|⩽d aα i { { { dt j j d d k j ∈ ℕ0n−1 , k ∈ ℕ, x ∈ ℝs , j u(tk +, x) − dt j u(tk −, x) = fj (x), { { { dt𝜕l x ∈ ℝs , l = 0, 1, . . . , n − 1, { 𝜕tl u(t, x)|t=0 = fl (x),

has a unique solution provided fl (⋅) ∈ E0,n , l ∈ ℕ0n−1 and fjk (⋅) ∈ E0,n , j ∈ ℕ0n−1 , k ∈ ℕ. 2. Let zk = tk + isk , where sk ∈ ℝ (k ∈ ℕ). Define n−1 z

uh (z) := Wn (z)f0 + ∑ ∫ gj (z − s)Wn (s)fj ds + ω(z), j=1 0

z ∈ ℂ,

where 0, if Re z ⩽ t1 , { { { l z k ω(z) := {∑k=1 Wn (z − tk )fjk + ∑n−1 j=1 ∫0 gj (z − s)Wn (s − tk )fj ds, { { 0 { if Re z ∈ (tk , tk+1 ] for some k ∈ ℕl−1 . Then the function u(t) := uh (z)+ω(z), z ∈ ℂ is a unique solution of the following abstract impulsive Cauchy problem in the complex plane: n

d u(z, x) = ∑|α|⩽d aα i|α| (𝜕/𝜕x)α u(z, x), z ∈ ℂ, x ∈ ℝs , { dzn { { j j { { {limz→zk , Re z>tk d j u(z, x) − limz→zk , Re z 0, m = ⌈α⌉ and I = (0, T) for some T ∈ (0, ∞]. The Caputo fractional derivative Dαt u(t) is traditionally defined for those functions u ∈ C m−1 ([0, ∞) : m E) for which gm−α ∗ (u − ∑m−1 k=0 uk gk+1 ) ∈ C ([0, ∞) : E), by Dαt u(t) :=

m−1 dm [gm−α ∗ (u − ∑ uk gk+1 )]; m dt k=0

(296)

here and hereafter, uk = u(k) (0) for 0 ⩽ k ⩽ m − 1. In our striving to investigate the abstract impulsive integro-differential equations with Caputo fractional derivatives, we need to slightly weaken the assumption u ∈ C m−1 ([0, ∞) : E) in the above definition. We propose the following notion: Definition 9.1.7. A function u : [0, T] → X belongs to the space Aα ([0, T] : X) if and only if there exist l ∈ ℕ and points t0 ≡ 0 < t1 < t2 < ⋅ ⋅ ⋅ < tl < T ≡ tl+1 such that the following conditions hold: (i) The function u(⋅) is (m − 1)-times continuously differentiable on the intervals [0, t1 ), (t1 , t2 ), (t2 , t3 ), . . . , (tl , T]; (ii) The right derivatives limt→ti + u(j) (t) exist for 0 ⩽ j ⩽ m − 1 and 1 ⩽ i ⩽ l; the left derivatives limt→ti − u(j) (t) exist for 0 ⩽ j ⩽ m − 1 and 1 ⩽ i ⩽ l + 1. It is clear that the assumption u ∈ Aα ([0, T] : X) implies that the function u(j) (⋅) is essentially bounded on the segment [0, T] for 0 ⩽ j ⩽ m−1. In the following definition, we will introduce the following generalization of the Caputo fractional derivative Dαt u(t); the notion of Sobolev space W m,1 ((0, T) : X) is taken in the sense of [97]: Definition 9.1.8. (i) Suppose that u ∈ Aα ([0, T] : X). Then the Caputo fractional derivam,1 tive Dαt u(t) is defined if and only if gm−α ∗ (u − ∑m−1 ((0, T) : X), k=0 uk gk+1 ) ∈ W by (296). (ii) Suppose that u ∈ L1loc ([0, ∞) : X). Then the Caputo fractional derivative Dαt u(t) is defined for t ⩾ 0 if and only if for each T > 0 we have u|[0,T] ∈ Aα ([0, T] : X) and the Caputo fractional derivative Dαt u(t) is defined on the segment [0, T].

464 � 9 Almost periodic type solutions to abstract impulsive Volterra integro-differential inclusions m,1 The assumption gm−α ∗ (u − ∑m−1 ((0, T) : X) is almost mandak=0 uk gk+1 ) ∈ W tory in any reasonable definition of the Caputo fractional derivative Dαt u(t). Unfortunately, this assumption has several very unpleasant consequences if we want to study the well-posedness of the abstract impulsive fractional differential inclusions with Caputo derivatives following some methods proposed in the existing literature. More precisely, we have the following result:

Theorem 9.1.9. Suppose that 𝒜 is the integral generator of a local (gα , k)-uniqueness C2 resolvent family, where τ > T, α ∈ (0, ∞) ∖ ℕ and k(t) is a kernel. Suppose that ω ∈ Aα ([0, T] : X) and {

Dαt ω(t) ∈ 𝒜ω(t), ω (0) = 0, (j)

t ∈ [0, T] ∖ {t1 , t2 , . . . , tl }; 0 ⩽ j ⩽ m − 1.

Then ω(t) = 0 for all t ∈ [0, T]. Proof. Since ω(j) (0) = 0, 0 ⩽ j ⩽ m−1 and ω(⋅) is (m−1)-times continuously differentiable on [0, t1 ], we have that the function (gm−α ∗ ω)(⋅) is likewise (m − 1)-times continuously differentiable on [0, t1 ] and (gm−α ∗ ω)(j) (0) = 0, 0 ⩽ j ⩽ m − 1. Applying the partial integration and the above equalities, we get: (gα ∗ Dαt ω)(t) = (gα ∗

dm d [gm−α ∗ ω])(t) = (gα+1−m ∗ [gm−α ∗ ω])(t), m dt dt

t ∈ [0, T].

Since the function (gm−α−m ∗ ω)(⋅) is absolutely continuous on [0, T], the last equality implies (g1 ∗ ω)(t) = (gα+1 ∗ Dαt ω)(t),

t ∈ [0, T].

Hence, (g1 ∗ ω)(t) ∈ (gα ∗ 𝒜[g1 ∗ ω])(t),

t ∈ [0, T].

Then an application of [429, Theorem 3.2.8(ii)] gives (g1 ∗ ω)(t) = 0, t ∈ [0, T]. Since ω ∈ Aα ([0, T] : X), the above yields ω(t) = 0 for all t ∈ [0, T]. Remark 9.1.10. Suppose that a piecewise continuous function u : [0, T] → X is (m − 1)times continuously differentiable on the interval [0, t1 ). Then we can define the Caputo fractional derivative Dαt u(t) in the same way as in Definition 9.1.8. Then we have Dαt u(t) = Dαt u(t) provided that u(j) (0) = 0, 0 ⩽ j ⩽ m − 1; here, Dαt u(t) denotes the Riemann– Liouville fractional derivative of function u(t) defined by the equations [97, (1.11)–(1.12), p. 10]. Even in this situation, the proof of Theorem 9.1.9 shows that ω(t) = 0 for all t ∈ [0, T].

9.1 Abstract impulsive Volterra integro-differential inclusions

� 465

Before going any further, we would like to observe that Prof. M. Fečkan, Y. Zhou, and J. R. Wang have noticed, in the concluding remark of research article [311], that we must use certain generalizations of the Caputo fractional derivatives (the Riemann–Liouville fractional derivative or some other types of fractional derivatives) in order to study the well-posedness of the abstract impulsive fractional Cauchy problems. It is our strong belief that this is the only correct way for the investigations of the abstract impulsive fractional Cauchy problems and that anything else is completely misleading and wrong. In a series of recent research studies, many authors have investigated the existence and uniqueness of the almost periodic type solutions for various classes of the abstract fractional differential inclusions with the Riemann–Liouville derivatives of order α ∈ (0, 1); see, e. g. the references quoted in [428]. We define here the Riemann– Liouville fractional derivative Dαt,+ u(t) in a very general manner: Dαt,+ u(t) is defined for those locally integrable functions u : ℝ → X such that for almost every t ∈ ℝ, there exists δt > 0 such that the function s 󳨃→ g1−α (t − s)u(s), s ∈ (t − δt , t + δt ) belongs to the space W 1,1 ((t − δt , t + δt ) : X), by Dαt,+ u(t)

t

d := ∫ g1−α (t − s)u(s) ds. dt −∞

If u ∈ L1 (ℝ : X), then the Fourier transform of the functions u(t) and Dαt,+ u(t) can be defined and we have: (ℱ Dαt,+ u)(λ) = (iλ)α (ℱ u)(λ),

λ ∈ ℝ;

(297)

see, e. g. [414, Property 2.15, p. 90]. Now we are ready to state and prove the following simple result: Theorem 9.1.11. Suppose that 𝒜 is a closed MLO which satisfies that, for almost every λ ∈ ℝ, the multivalued linear operator 𝒜λ = (iλ)α − 𝒜 is injective. Suppose that ω ∈ L1 (ℝ : X), (tl )l∈ℤ is a sequence without accumulation points and Dαt,+ ω(t) ∈ 𝒜ω(t),

t ∈ ℝ ∖ {. . . , t−l , . . . , t0 , . . . , tl , . . .}.

(298)

Then ω(t) = 0 for a. e. t ∈ ℝ. Proof. Since 𝒜 is closed and ω ∈ L1 (ℝ : X), the use of (297)–(298) implies [(iλ)α − 𝒜](ℱ ω)(λ) = 0 for all λ ∈ ℝ. Our assumption and the Riemann–Lebesgue lemma together imply that (ℱ ω)(λ) = 0, λ ∈ ℝ, and therefore, ω(t) = 0 for a. e. t ∈ ℝ. It is clear that Theorem 9.1.11 indicates that it is very difficult to study the existence and uniqueness of piecewise continuous solutions of the abstract fractional differential inclusion Dαt,+ u(t) ∈ 𝒜u(t) + f (t),

t ∈ ℝ ∖ {. . . , t−l , . . . , t0 , . . . , tl , . . .}.

466 � 9 Almost periodic type solutions to abstract impulsive Volterra integro-differential inclusions Almost nothing can be said if 𝒜 is a subgenerator of an exponentially bounded (gβ , gζ )regularized C-resolvent family for some β ∈ (α, 1] and ζ ⩾ 1 since in this case the multivalued linear operator 𝒜λ = (iλ)α − 𝒜 is injective for every λ ∈ ℝ. 9.1.4 Abstract Volterra integro-differential inclusions with impulsive effects In this subsection, we consider the well-posedness of the following abstract impulsive Volterra integro-differential inclusion: t

ℬu(t) ⊆ 𝒜 ∫ a(t − s)u(s)ds + ℱ (t),

t ∈ [0, T] ∖ {t1 , . . . , tl };

0

(Δu)(tm ) = Cym ,

m = 1, . . . , l,

(299)

where 0 ≡ t0 < t1 < ⋅ ⋅ ⋅ < tl < T ≡ tl+1 < ∞, a ∈ L1loc ([0, T]), a ≠ 0, ℱ : [0, T] → P(E), and 𝒜: X → P(E), ℬ: X → P(E) are two given mappings, as well as the well-posedness of the following abstract impulsive Volterra integro-differential inclusion: t

ℬu(t) ⊆ 𝒜 ∫ a(t − s)u(s)ds + ℱ (t),

t ∈ [0, ∞) ∖ {t1 , . . . , tl , . . .};

0

(Δu)(tl ) = Cyl ,

l ⩾ 1,

(300)

where 0 ≡ t0 < t1 < ⋅ ⋅ ⋅ < tl < tl+1 < ⋅ ⋅ ⋅ < +∞, the sequence (tl )l has no accumulation point, a ∈ L1loc ([0, ∞)), a ≠ 0, ℱ : [0, ∞) → P(E), and 𝒜: X → P(E), ℬ: X → P(E) are two given mappings. We will use the following notion: Definition 9.1.12. (i) A function u ∈ 𝒫 C([0, T] : X) is said to be a pre-solution of (299) on [0, T] if and only if u(⋅) is continuous on the set [0, T]∖{t1 , . . . , tl }, (a∗u)(t) ∈ D(𝒜) and u(t) ∈ D(ℬ) for t ∈ [0, T] ∖ {t1 , . . . , tl }, as well as (299) holds. (ii) A solution of (299) on [0, T] is any pre-solution u(⋅) of (299) on [0, T] satisfying additionally that there exist functions uℬ ∈ 𝒫 C([0, T] : E) and ua,𝒜 ∈ 𝒫 C([0, T] : E), continuous on the set [0, T] ∖ {t1 , . . . , tl }, such that uℬ (t) ∈ ℬu(t) and ua,𝒜 (t) ∈ t

𝒜 ∫0 a(t − s)u(s)ds for t ∈ [0, T] ∖ {t1 , . . . , tl }, as well as

uℬ (t) ∈ ua,𝒜 (t) + ℱ (t),

t ∈ [0, T] ∖ {t1 , . . . , tl }.

(iii) A strong solution of (299) on [0, T] is any function u ∈ 𝒫 C([0, T] : X), continuous on the set [0, T]∖{t1 , . . . , tl }, satisfying that there exist two functions uℬ ∈ 𝒫 C([0, T] : E) and u𝒜 ∈ 𝒫 C([0, T] : E), continuous on the set [0, T] ∖ {t1 , . . . , tl }, such that uℬ (t) ∈ ℬu(t), u𝒜 (t) ∈ 𝒜u(t) for all t ∈ [0, T] ∖ {t1 , . . . , tl }, and

9.1 Abstract impulsive Volterra integro-differential inclusions

uℬ (t) ∈ (a ∗ u𝒜 )(t) + ℱ (t),

�

467

t ∈ [0, T] ∖ {t1 , . . . , tl }.

(iv) Suppose that 0 ≡ t0 < t1 < ⋅ ⋅ ⋅ < tl < tl+1 < ⋅ ⋅ ⋅ < +∞ and the sequence (tl )l has no accumulation point. By a (pre-)solution [solution, strong solution] of (300) we mean any function u(⋅), which satisfies that for every l ∈ ℕ and T ∈ (tl , tl+1 ), the function u|[0,T] (⋅) is a (pre-)solution [solution, strong solution] of (299) on [0, T]. The following important results can be simply deduced: (i) If 𝒜 and ℬ are two MLOs and 𝒜 is closed, then any strong solution of (299) on [0, T] [(300)] is already a solution of (299) on [0, T] [(300)]. (ii) If ℬ = I, a(t) and k(t) are kernels and 𝒜 is a closed subgenerator of a mild (a, k)regularized C2 -uniqueness family, then any pre-solution of (299) on [0, T] [(300)] is unique (see [429, Proposition 3.2.8(ii)]). The following essential result can be simply reformulated in the global setting: Theorem 9.1.13. (i) Suppose a(t) and k(t) are kernels, k(0) = 1, C2 ∈ L(X) and 𝒜 is a closed subgenerator of a mild (a, k)-regularized C2 -uniqueness family (R2 (t))t∈[0,τ) ⊆ L(X), where τ > T. Define ℱ (t) := 0 for t ∈ [0, t1 ] and ℱ (t) := ∑m s=1 k(t − ts )C2 ys if t ∈ (tm , tm+1 ] for some integer m ∈ ℕl . Define also u(t) := 0 for t ∈ [0, t1 ] and u(t) := ∑ls=1 R2 (t − ts )ys if t ∈ (tm , tm+1 ] for some integer m ∈ ℕl . If y1 , . . . , yl ∈ D(𝒜), then u(t) is a unique strong solution of problem (299) on [0, T], with the operator C replaced therein with the operator C2 . (ii) Suppose that a(t) and k(t) are kernels, k(0) = 1, C1 ∈ L(X, E) and 𝒜 is a closed subgenerator of a mild (a, k)-regularized C1 -existence family (R1 (t))t∈[0,τ) ⊆ L(X, E) such that R1 (0) = C1 , where τ > T. Define ℱ (t) and u(t) in the same way as above, with the operator C2 replaced therein with the operator C1 and the elements y1 , . . . , yl ∈ X. Then u(t) is a solution of problem (299) on [0, T], with the operator C replaced therein with the operator C1 . Proof. We will prove only (i). The uniqueness of strong solutions of problem (299) on [0, T] follows from the above observations. Let zm ∈ 𝒜ym , m = 1, . . . , l; then we have t R2 (t)ym − k(t)C2 ym = ∫0 a(t − s)R2 (s)zm ds for all t ∈ [0, τ) and m = 1, . . . , l so that t−t

R2 (t − tm )ym − k(t − tm )Cym = ∫0 m a(t − s − tm )R2 (s)zm ds for all t ∈ [0, τ) and m = 1, . . . , l. Adding these equalities for s = 1, . . . , m, we simply obtain that u(t) = (a ∗ u𝒜 )(t) + ℱ (t) on (tm , tm+1 ], where m ∈ ℕl is fixed and u𝒜 (t) is defined in the same way as u(t) with the elements y1 , . . . , yl replaced therein with the elements z1 , . . . , zl (we only want to notice that it is necessary to divide the segment of the integration [0, t] into the segments [0, t1 ], [t1 , t2 ], . . . , [tm−1 , tm ] and [tm , t]). Since k(0) = 1, we have R2 (0)x = C2 x for all x ∈ D(𝒜), which simply completes the proof of theorem. It is clear that Theorem 9.1.13 can be applied to a wide class of the abstract impulsive Volterra integro-differential inclusions; see the research monograph [623, 426, 429] and references cited therein for fairly complete information about the subject. Concerning

468 � 9 Almost periodic type solutions to abstract impulsive Volterra integro-differential inclusions the use of C-regularized solution operator families (integrated solution operator families and convoluted solution operator families with k(0) = 0 cannot be used for providing certain applications of Theorem 9.1.13), we may refer, e. g. to [426, Example 2.1.9, Example 2.1.10(ii); Section 2.5] and [429, Theorem 3.2.21; see also pp. 323–324]; for the sake of completeness, we will present the following illustrative applications of Theorem 9.1.13 only: Example 9.1.14. (i) Suppose that Ω is a bounded domain in ℝn , b > 0, m(x) ⩾ 0 a. e. x ∈ Ω, m ∈ L∞ (Ω) and 1 < p < ∞. Let B be the multiplication in Lp (Ω) with m(x), and let A = Δ − b act with the Dirichlet boundary conditions. Then our analysis from [429, Example 3.2.23] shows that there exists an operator C1 ∈ L(Lp (Ω)) such that the MLO 𝒜 = −AB−1 is a subgenerator of an entire (g1 , g1 )-regularized C1 -existence family. Consider now the following degenerate Volterra integral equation associated to the abstract backward Poisson heat equation in the space X = Lp (Ω): t

m(x)v(t, x) = u0 (x) + ∫0 (−Δ + b)v(s, x) ds, { { { { { { t ∈ [0, ∞) ∖ {t1 , . . . , tl , . . .}, x ∈ Ω; { { { (PR): {v(t, x) = 0, (t, x) ∈ [0, ∞) × 𝜕Ω, { { { { m(x)v(tl +, x) − m(x)v(tl −, x) = C1 [f1 (x) + ⋅ ⋅ ⋅ + fl (x)], { { { { {x ∈ Ω, t ∈ (tl , tl+1 ] (l ∈ ℕ), where t1 < t2 < ⋅ ⋅ ⋅ < tl < ⋅ ⋅ ⋅ , the sequence (tl ) has no accumulation point and fl ∈ X for all l ⩾ 1. Then Theorem 9.1.13 and its proof yield that there exists a global solution of the problem (PR) which can be extended to an analytic function defined on the set ℂ ∖ {t1 , . . . , tl , . . .}; note that we can also apply Theorem 9.1.2 here by assuming that {t1 , . . . , tl , . . .} is a complex sequence obeying certain properties, as well as that it is still not clear how we can consider the well-posedness of the fractional analogue of problem (PR), obtained by replacing the first equation of (PR) with the equation t m(x)v(t, x) = u0 (x) + ∫0 gα (t − s)(−Δ + b)v(s, x) ds for some number α ∈ (0, 1). (ii) Our analysis from [429, Example 3.10.7] and the second equality in [97, (1.21)] shows that we can similarly analyze the following degenerate Volterra integral equation closely connected with the inverse generator problem and the abstract backward Poisson heat equation in the space X = Lp (Ω): t

d (Δ − b)v(t, x) = (Δ − b)v(0, x) + t( ds [(Δ − b)v(s, x)]) + ∫0 gα (t − s)m(s)v(s, x) ds, { { s=0 { { { { t ∈ [0, ∞) ∖ {t1 , . . . , tl , . . .}, x ∈ Ω; { { { v(t, x) = 0, (t, x) ∈ [0, ∞) × 𝜕Ω, { { { { { { {(Δ − b)v(tl +, x) − (Δ − b)v(tl −, x) = C1 [f1 (x) + ⋅ ⋅ ⋅ + fl (x)], { { { x ∈ Ω, t ∈ (tl , tl+1 ] (l ∈ ℕ),

where t1 < t2 < ⋅ ⋅ ⋅ < tl < ⋅ ⋅ ⋅ , the sequence (tl ) has no accumulation point fl ∈ X for all l ⩾ 1 and a number α ∈ (1, 2) satisfies an extra assumption.

9.1 Abstract impulsive Volterra integro-differential inclusions

� 469

We continue this subsection with the observation that the theory of abstract degenerate Volterra integro-differential equations is rather non-trivial as well as that the use of multivalued linear operators is not sufficiently adequate to cover all related problems within this theory. Consider now the following problem: t

Bu(t) = f (t) + ∫ a(t − s)Au(s)ds,

t ∈ [0, T] ∖ {t1 , . . . , tl };

0

B(Δu)(tm ) = CBym ,

m = 1, . . . , l,

(301)

where 0 < T < ∞, t 󳨃→ f (t), t ∈ [0, T] is a Lebesgue measurable mapping with values in X, a ∈ L1loc ([0, T]) and A, B are closed linear operators with domain and range contained in X. We refer the reader to [429, Definition 2.2.1] for the notion of a mild (strong) solution of (301) without impulsive effects; we similarly define the notion of a mild (strong) solution of (301) with impulsive effects. The notion of an exponentially bounded (a, k)-regularized C-resolvent family for (301) has recently been introduced in [429, Definition 2.2.2]: Definition 9.1.15. Suppose that the functions a(t) and k(t) satisfy (P1), as well as that R(t): D(B) → E is a linear mapping (t ⩾ 0). Let C ∈ L(E) be injective, and let CA ⊆ AC. Then the operator family (R(t))t⩾0 is said to be an exponentially bounded (a, k)regularized C-resolvent family for (301) if and only if there exists ω ⩾ max(0, abs(a), abs(k)) such that the following holds: (i) The mapping t 󳨃→ R(t)x, t ⩾ 0 is continuous for every fixed element x ∈ D(B). (ii) There exist M ⩾ 1 and ω ⩾ 0 such that ‖R(t)‖ ⩽ Meωt , t ⩾ 0. ̃ ̃ (iii) For every λ ∈ ℂ with Re λ > ω and k(λ) ≠ 0, the operator B − a(λ)A is injective, ̃ C(R(B)) ⊆ R(B − a(λ)A) and ∞

−1 ̃ ̃ k(λ)(B − a(λ)A) CBx = ∫ e−λt R(t)x dt,

x ∈ D(B).

0

We need the following lemma (cf. [429, Theorem 2.2.8]): Lemma 9.1.16. Let the functions |a|(t) and k(t) satisfy (P1), and let (R(t))t⩾0 be an exponentially bounded (a, k)-regularized C-resolvent family for (301), satisfying (ii) of Definition 9.1.15 with ω ⩾ max(0, abs(|a|), abs(k)). (i) Suppose that v0 ∈ D(B) and the following condition holds: (i.1) for every x ∈ D(B), there exist a number ω0 > ω and a function h(λ; x) ∈ LT − E −1 ̃ ̃ ̃ such that h(λ; x) = k(λ)B(B − a(λ)A) CBx provided Re λ > ω0 and k(λ) ≠ 0. Then the function u(t) = R(t)v0 , t ⩾ 0 is a mild solution of (301) with f (t) = k(t)CBv0 , t ⩾ 0 and without impulsive effects. The uniqueness of mild solutions holds if we suppose additionally that CB ⊆ BC and the function k(t) satisfies (P2). (ii) Suppose that v0 ∈ D(A) ∩ D(B), CB ⊆ BC, and the following condition holds:

470 � 9 Almost periodic type solutions to abstract impulsive Volterra integro-differential inclusions (ii.1) for every x ∈ E, there exist a number ω1 > ω and a function h(λ; x) ∈ LT − E −1 ̃ ̃ ̃ such that h(λ; x) = k(λ)B(B − a(λ)A) Cx provided Re λ > ω1 and k(λ) ≠ 0. Then the function u(t) = R(t)v0 , t ⩾ 0 is a strong solution of (301) with f (t) = k(t)CBv0 , t ⩾ 0 and without impulsive effects. The uniqueness of strong solutions holds if we suppose additionally that the function k(t) satisfies (P2). Let us note that the requirements of Lemma 9.1.16(i) imply that A(a ∗ R)(t)v0 = BR(t)v0 − k(t)CBv0 ,

t ⩾ 0, v0 ∈ D(B).

(302)

Keeping in mind (302), Lemma 9.1.16 and the argumentation contained in the proof of Theorem 9.1.13, we may simply deduce the following result: Theorem 9.1.17. (i) Suppose that the requirements of Lemma 9.1.16(i) hold and k(0) = 1. Define ℱ (t) := 0 for t ∈ [0, t1 ] and ℱ (t) := ∑m s=1 k(t − ts )CBys if t ∈ (tm , tm+1 ] for some integer m ∈ ℕl . Define also u(t) := 0 for t ∈ [0, t1 ] and u(t) := ∑m s=1 R(t − ts )ys if t ∈ (tm , tm+1 ] for some integer m ∈ ℕl . If y1 , . . . , yl ∈ D(B), then u(t) is a mild solution of problem (301) on [0, T]. The uniqueness of mild solutions of problem (301) holds if we suppose additionally that CB ⊆ BC and the function k(t) satisfies (P2). (ii) Suppose that the requirements of Lemma 9.1.16(ii) hold and k(0) = 1. Define ℱ (t) and u(t) as above. If y1 , . . . , yl ∈ D(A) ∩ D(B), then u(t) is a strong solution of problem (301) on [0, T]. The uniqueness of strong solutions of problem (301) holds if we suppose additionally that the function k(t) satisfies (P2). An application of Theorem 9.1.17 can be given to the impulsive degenerate Volterra integral equations associated with the following degenerate fractional Cauchy problem in X = Lp (ℝ2 ), for example: π

2l

Dα [uxx + uxy + uyy − u] = e−iα 2 [(−1)l+1 𝜕x𝜕 2α u + uyy ], t ⩾ 0, (P5): { t u(0, x) = ϕ(x); ut (0, x) = 0 if α ⩾ 1; see [429, Example 2.2.27(i)] for more details. We can similarly provide certain applications of exponentially bounded (a, k)-regularized C-resolvent families generated by a pair of closed linear operators A, B to the abstract impulsive degenerate Volterra integral equations; see [429, Subsection 2.3.3] for more details. We will not consider here the abstract degenerate multi-term Volterra integrodifferential equations with impulsive effects; the interested reader may try to reconsider the problems from [429, Example 2.3.43, Example 2.3.48] by adding certain impulsive effects therein. For the theory of the abstract degenerate Cauchy problems, we also refer the reader to the research monograph [617] by M. V. Plekhanova, V. E. Fedorov and the references quoted therein. Before going further, we would like to quote a few important topics not considered so far:

9.2 The existence and uniqueness of almost periodic type solutions

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1. It seems very plausible that we can similarly analyze the well-posedness of the abstract incomplete Cauchy inclusions with impulsive effects; for more details about the subject, we refer the reader to [429, Section 2.7, Section 3.9]. 2. We have not considered here the abstract (degenerate) impulsive Volterra integrodifferential equations of non-scalar type; cf. [426, pp. 54–56] and [429, Section 2.9] for further information in this direction. 3. Let us finally mention that we have not considered here the abstract impulsive Volterra integro-differential inclusions on the line as well as the existence and uniqueness of discontinuous almost periodic (automorphic) type solutions for certain classes of the abstract impulsive Cauchy problems on the line; see, e. g. [426, pp. 51–53] for more details about this subject in the non-degenerate case.

9.2 The existence and uniqueness of almost periodic type solutions The main aim of this section is to reconsider the notion of a piecewise continuous almost periodic function, which has been thoroughly analyzed in the research monographs [350] by A. Halanay, D. Wexler and [655] by A. M. Samoilenko, N. A. Perestyuk, by introducing and systematically investigating the classes of (pre-)(ℬ, ρ, (tk ))-piecewise continuous almost periodic functions and (pre-)(ℬ, ρ, (tk ))-piecewise continuous uniformly recurrent functions [281]. We also aim to continue the research study carried out in [280] by investigating the almost periodic type solutions for various classes of the abstract impulsive Volterra integro-differential inclusions. We consider here the functions of the form F : I × X → E, where (X, ‖ ⋅ ‖) and (E, ‖ ⋅ ‖E ) are complex Banach spaces and I = ℝ or I = [0, ∞). In the existing literature, it has been commonly assumed that the sequence (tk ) of possible first kind discontinuities of function f (⋅) under our consideration is a Wexler sequence: by using certain results about Stepanov almost periodic type functions, we show that this condition is sometimes rather superfluous and almost completely irrelevant. Before proceeding any further, we would like to note that this is probably the first research monograph, which investigates the existence and uniqueness of the uniformly recurrent type solutions, the Weyl almost periodic type solutions, and the Besicovitch–Doss almost periodic type solutions to the abstract impulsive Volterra integro-differential equations, and probably the first research monograph, which investigates the almost periodic type solutions for certain classes of the abstract higher-order impulsive Cauchy problems. Furthermore, it should be mentioned that we introduce here, for the first time in the existing literature, the class of Weyl-palmost periodic sequences in the sense of general approach of A. S. Kovanko [463], the class of Doss-p-almost periodic sequences (1 ⩽ p < ∞) and analyze their applications in the study of the existence and uniqueness of the Weyl-p-almost periodic solutions (Doss-p-almost periodic solutions) for certain kinds of the abstract impulsive Volterra integro-differential equations.

472 � 9 Almost periodic type solutions to abstract impulsive Volterra integro-differential inclusions The organization and main ideas of this section can be briefly summarized as follows. The main aim of Section 9.2.1 is to recollect the basic facts about the class of piecewise continuous almost periodic functions, which has been commonly used in the existing literature. We extend the notion of piecewise continuous almost periodicity in Section 9.2.2, where we introduce and analyze various classes of (ℬ, ρ)-piecewise continuous almost periodic type functions. More precisely, in Definition 9.2.4, we introduce the classes of (pre-)(ℬ, ρ, (tk ))-piecewise continuous almost periodic functions and (pre-)(ℬ, ρ, (tk ))-piecewise continuous uniformly recurrent functions. The assumption that the corresponding sequence (tk ) of possible discontinuities is a Wexler sequence is almost completely irrelevant in our analysis, but the quasi-uniformly continuity condition (QUC) from Definition 9.2.4 plays an important role in our study. In Examples 9.2.6–9.2.7, we present two illustrative examples of real-valued functions, which are (tk )-piecewise continuous almost periodic (cf. also Remark 9.2.9, Remark 9.2.10 and Remark 9.2.12 for some useful observations about the function spaces introduced in Definition 9.2.4). Several structural characterizations for the introduced classes of piecewise continuous almost periodic type functions have been proved in Proposition 9.2.8, Proposition 9.2.11, and Proposition 9.2.13; it should be specifically emphasized that the supremum-formula holds for certain classes of pre-(ℬ, T, (tk ))-piecewise continuous uniformly recurrent functions (cf. Proposition 9.2.15). In Section 9.2.3, we continue the analysis of L. Qi and R. Yuan from their remarkable paper [629] concerning the relations between the piecewise continuous almost periodic functions and the Stepanov almost periodic type functions. We improve some structural results obtained in [629] by removing the assumption that (tk ) is a Wexler sequence. Our main results in this subsection are Theorem 9.2.16 and Theorem 9.2.18; some consequences of these results are presented in Theorem 9.2.19, Theorem 9.2.20, Proposition 9.2.21, Proposition 9.2.22 and Theorem 9.2.23. Composition principles for (ℬ, (tk ))piecewise continuous almost periodic type functions are investigated in Section 9.2.4. Further on, Section 9.2.5 is devoted to the study of asymptotically almost periodic type solutions of the abstract impulsive differential Cauchy problem (ACP)1;1 , asymptotically Weyl almost periodic type solutions of (ACP)1;1 are sought in Section 9.2.6, the Besicovitch almost periodic type solutions of (ACP)1;1 are sought in Section 9.2.7 and the almost periodic type solutions of the abstract higher-order impulsive Cauchy problems are sought in Section 9.2.8 (let us only mention that the separation condition infk∈ℕ (tk+1 − tk ) > 0 on the corresponding sequence (tk ) of possible discontinuities is not employed in some results). In addition to the above, we present many illustrative examples and open problems.

9.2.1 Piecewise continuous almost periodic functions The piecewise continuous almost periodic type solutions for various classes of impulsive integro-differential equations have been analyzed by numerous authors so far (see, e. g.

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the research monograph [699] by G. Tr. Stamov for a comprehensive survey of results). In this subsection, we analyze the piecewise continuous almost periodic type functions. Let us recall that an X-valued sequence (xk )k∈ℤ [(xk )k∈ℕ ] is called (Bohr) almost periodic if and only if, for every ε > 0, there exists a natural number K0 (ε) such that among any K0 (ε) consecutive integers in ℤ [ℕ], there exists at least one integer τ ∈ ℤ [τ ∈ ℕ] satisfying that ‖xk+τ − xk ‖ ⩽ ε,

k ∈ ℤ [k ∈ ℕ];

as in the case of functions, this number is said to be an ε-period of sequence (xk ). Any almost periodic X-valued sequence is bounded. The equivalent concept of Bochner almost periodicity of X-valued sequences can be introduced, as well; see, e. g. [655, Theorem 70, pp. 185–186] and its important consequences [655, Theorems 71–73, pp. 186–188]. It is well known that a sequence (xk )k∈ℤ in X is almost periodic if and only if there exists an almost periodic function f : ℝ → X such that xk = f (k) for all k ∈ ℤ; see e. g., the proof of [264, Theorem 2] given in the scalar-valued case. It is not difficult to prove that for every almost periodic sequence (xk )k∈ℕ in X, there exists a unique almost periodic sequence (xk̃ )k∈ℤ in X such that xk̃ = xk for all k ∈ ℕ, so that a sequence (xk )k∈ℕ in X is almost periodic if and only if there exists an almost periodic function f : [0, ∞) → X such that xk = f (k) for all k ∈ ℕ. Unless stated otherwise, we will always assume henceforth that (tk )k∈ℤ [(tk )k∈ℕ ] is a sequence in ℝ [in (0, ∞)] such that δ0 := infk∈ℤ (tk+1 −tk ) > 0 [δ0 := infk∈ℕ (tk+1 −tk ) > 0]. j Set tk := tk+j − tk , j, k ∈ ℤ [j, k ∈ ℕ]. We need the following definitions: j

j

Definition 9.2.1. The family of sequences (tk )k∈ℤ [(tk )k∈ℕ ], j ∈ ℤ [j ∈ ℕ] is called equipotentially almost periodic if and only if, for every ε > 0, there exists a relatively dense set Qε in ℝ [in [0, ∞)] such that for each τ ∈ Qε there exists an integer q ∈ ℤ [q ∈ ℕ] such that |ti+q − ti − τ| < ε for all i ∈ ℤ [i ∈ ℕ]. Definition 9.2.2. The sequence (tk )k∈ℤ [(tk )k∈ℕ ] is said to be uniformly almost periodic if and only if, for every ε > 0, there exists a relatively dense set Qε in ℤ [in ℕ] such that 󵄨󵄨 j 󵄨 󵄨󵄨t − t j 󵄨󵄨󵄨 < ε, 󵄨󵄨 i+q i 󵄨󵄨

i, j ∈ ℤ [i, j ∈ ℕ], q ∈ Qε .

We know that, if the sequence (tk )k∈ℤ [(tk )k∈ℕ ] is uniformly almost periodic, then j j the family of sequences (tk )k∈ℤ [(tk )k∈ℕ ], j ∈ ℤ [j ∈ ℕ] is equipotentially almost periodic. See also [655, p. 377] and [629, Lemma 2.12]; let us also note that the family of sequences j (tk )k∈ℤ , j ∈ ℤ is equipotentially almost periodic if and only if there exist a unique nonzero real number ζ and an almost periodic sequence (ak )k∈ℤ such that tk = ζk + ak for all k ∈ ℤ. It seems very plausible that a similar statement holds for an equipotentially almost periodic sequence (tk )k∈ℕ . The usual definition of a piecewise continuous almost periodic function goes as follows (see [350] and [655] for more details about the subject):

474 � 9 Almost periodic type solutions to abstract impulsive Volterra integro-differential inclusions Definition 9.2.3. Suppose that the function f : ℝ → X [f : [0, ∞) → X] is piecewise continuous with the possible first kind discontinuities at the points of a fixed sequence (tk )k∈ℤ [(tk )k∈ℕ ]. Then we say that the function f (⋅) is (tk )-piecewise continuous almost periodic if and only if the following conditions are fulfilled: j j (i) The family of sequences (tk )k∈ℤ [(tk )k∈ℕ ], j ∈ ℤ [j ∈ ℕ] is equipotentially almost periodic, i. e. (tk ) is a Wexler sequence. (ii) For every ε > 0, there exists δ > 0 such that, if the points t1 and t2 belong to (ti , ti+1 ) for some i ∈ ℤ [i ∈ ℕ0 ; t0 ≡ 0] and |t1 − t2 | < δ, then ‖f (t1 ) − f (t2 )‖ < ε. (iii) For every ε > 0, there exists a relatively dense set S in ℝ [in [0, ∞)] such that, if τ ∈ S, then ‖f (t + τ) − f (t)‖ < ε for all t ∈ ℝ such that |t − tk | > ε, k ∈ ℤ [k ∈ ℕ]. Such a point τ is called an ε-almost period of f (⋅). j

For example, let the family of sequences (tk )k∈ℤ , j ∈ ℤ be equipotentially almost periodic. Then we know that the function f : ℝ → ℝ, defined by f (t) := μi if t ∈ (ti , ti+1 ] for some i ∈ ℤ, is (tk )-piecewise continuous almost periodic provided that the sequence (μi )i∈ℤ is almost periodic (cf. [655, pp. 202–203] for the proof of the above fact). For further information about piecewise continuous almost periodic functions and their applications, we refer the reader to the research articles [364] by H. R. Henríquez, B. de Andrade, M. Rabelo, [628] by L. Qi, R. Yuan, [724] by V. Tkachenko and references cited therein. Before proceeding with our original contributions about piecewise continuous almost periodic type functions, it would be worthwhile to mention that J. Xia has considered, in [772], the class of piecewise continuous almost periodic functions following a completely different approach (cf. also the research article [264] by L. Díaz and R. Naulin).

9.2.2 (ℬ, ρ)-piecewise continuous almost periodic type functions We start this subsection by introducing the following notion: Definition 9.2.4. Suppose that ρ is a binary relation on E and the function F : ℝ×X → E [F : [0, ∞) × X → E] satisfies that for every x ∈ X, the function t 󳨃→ F(t; x) is piecewise continuous with the possible first kind discontinuities at the points of a fixed sequence (tk )k∈ℤ [(tk )k∈ℕ ]. Then we say that the function F(⋅) is: (i) pre-(ℬ, ρ, (tk ))-piecewise continuous almost periodic if and only if, for every ε > 0 and B ∈ ℬ, there exists a relatively dense set S in ℝ [in [0, ∞)] such that, if τ ∈ S, x ∈ B and t ∈ ℝ satisfies |t − tk | > ε for all k ∈ ℤ [k ∈ ℕ], then there exists yt,x ∈ ρ(F(t; x)) such that ‖F(t + τ; x) − yt,x ‖ < ε. (ii) (ℬ, ρ, (tk ))-piecewise continuous almost periodic if and only if F(⋅; ⋅) is pre-(ℬ, ρ, (tk ))piecewise continuous almost periodic, the condition (i) from Definition 9.2.3 holds and (QUC) holds, where:

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(QUC) For every ε > 0 and B ∈ ℬ, there exists δ > 0 such that, if x ∈ B and the points t1 and t2 belong to (ti , ti+1 ) for some i ∈ ℤ [i ∈ ℕ0 ; t0 ≡ 0] and |t1 − t2 | < δ, then ‖F(t1 ; x) − F(t2 ; x)‖ < ε. (iii) pre-(ℬ, ρ, (tk ))-piecewise continuous uniformly recurrent if and only if there exists a strictly increasing sequence (αl ) of positive real numbers tending to plus infinity and satisfying that for every ε > 0 and B ∈ ℬ, there exists an integer l0 ∈ ℕ such that, if x ∈ B, l ⩾ l0 and t ∈ ℝ satisfies |t − tk | > ε for all k ∈ ℤ [k ∈ ℕ], then there exists yt,x ∈ ρ(F(t; x)) such that ‖F(t + αl ; x) − yt,x ‖ < ε. (iv) (ℬ, ρ, (tk ))-piecewise continuous uniformly recurrent if and only if F(⋅; ⋅) is pre(ℬ, ρ, (tk ))-piecewise continuous uniformly recurrent and the condition (QUC) holds. We say that the function F(⋅; ⋅) is (pre-)(ℬ, ρ)-piecewise continuous almost periodic [(pre-)(ℬ, ρ)-piecewise continuous uniformly recurrent] if and only if F(⋅; ⋅) is (pre-)(ℬ, ρ, (tk ))-piecewise continuous almost periodic [(pre-)(ℬ, ρ, (tk ))-piecewise continuous uniformly recurrent] for a certain sequence (tk )k∈ℤ [(tk )k∈ℕ ] obeying our general requirements. If ρ = cI for some c ∈ ℂ, then we also say that F(⋅; ⋅) is (pre-)piecewise continuous c-almost periodic [(pre-)piecewise continuous c-uniformly recurrent]; furthermore, if c = −1, then we also say that F(⋅; ⋅) is (pre-)piecewise continuous almost anti-periodic [(pre-)piecewise continuous uniformly anti-recurrent]. We omit the term “ℬ” from the notation if X = {0} and omit the term “c” from the notation if c = 1. Remark 9.2.5. In the notion introduced in Definition 9.2.4(i), we can also require that the inequality ‖F(t + τ; x) − yt,x ‖ < ε holds provided that |t − tk | > M(ε) for all k ∈ ℤ [k ∈ ℕ], where M : (0, ∞) → [0, ∞) satisfies lim infε→0+ M(ε) = 0. This notion is really not interesting because a very simple argumentation shows that a function F : I ×X → E obeys this condition if and only if F(⋅; ⋅) is pre-(ℬ, ρ, (tk ))-piecewise continuous almost periodic. The same holds in the case of consideration of parts (ii), (iii) and (iv) of Definition 9.2.4, so that we will always assume henceforth that M(ε) ≡ ε. Before proceeding any further, we would like to present the following examples: Example 9.2.6. Suppose that c ∈ S1 , ω > 0, t1 ∈ (0, ω] and tk = t1 + (k − 1)ω, k ∈ ℤ [t0 = 0 and tk = t1 + (k − 1)ω, k ⩾ 2]. Suppose further that the function F1 : (t1 , t1 + ω] × X → E satisfies that, for every x ∈ X, F1 (t1 + ω; x) ≠ cF1 (t1 ; x) as well as that limt→t1 + F1 (t; x) exists in E. Then we can extend the function F1 (⋅; ⋅) to a function F : ℝ × X → E [F : [0, ∞) × X → E] such that, for every x ∈ X, the function F(⋅; x) is piecewise continuous, has the possible first kind discontinuities at the points of sequence (tk )k∈ℤ [(tk )k∈ℕ ] and F(t + ω; x) = cF(t; x) for all x ∈ X and t ∈ ⋃k∈ℤ (tk , tk+1 ) [t ∈ ⋃k∈ℕ0 (tk , tk+1 )]. Since the

set of all integers k ∈ ℤ [k ∈ ℕ] such that ck = c is relatively dense in ℤ [ℕ], with the meaning clear, a very simple argumentation shows that the function F(⋅; ⋅) is (c, (tk ))piecewise continuous almost periodic. For example, if P(t) is an (anti-)periodic non-zero trigonometric polynomial with real values, then the piecewise continuous function f0 (⋅)

476 � 9 Almost periodic type solutions to abstract impulsive Volterra integro-differential inclusions determined by the function f (t) := sign(P(t)), t ∈ ℝ is (tk )-piecewise continuous almost (anti-)periodic. For our later purposes, we will denote the collection of all functions F : I × X → E constructed in this way by PPCω,c;(tk ) (I × X : E). Example 9.2.7. Suppose that m ∈ ℕ, y : ℝ → ℝ is a Bohr almost periodic function, and there exists an integer k0 ∈ ℤ such that y(mk0 ) ≠ y(m(k0 + 1)). Define f (t) := y(m⌊

t+1 ⌋), m

t ∈ ℝ.

Then we have f (t) = y(mk) if y ∈ [mk − 1, mk − 1 + m) for some integer k ∈ ℤ so that our assumption implies that the function f (⋅) is not continuous on the real line. On the other hand, for every ε > 0 there exists l > 0 such that any interval I ⊆ ℝ of length ⩾ l contains a point τ such that |f (t + τ) − f (t)| ⩽ ε, t ∈ ℝ. Towards this end, let us recall that for a given ε > 0 in advance, we can always find l > 0 such that any interval I ⊆ ℝ of length ⩾ l contains an integer τ such that |y(t + τ) − y(t)| ⩽ ε/m, t ∈ ℝ; the last estimate simply implies |y(t + mτ) − y(t)| ⩽ ε, t ∈ ℝ so that actually, we can always find a number l′ = lm > 0 such that any interval I ⊆ ℝ of length ⩾ lm contains an integer mτ, where τ ∈ ℤ, such that |y(t + mτ) − y(t)| ⩽ ε, t ∈ ℝ. Let it be the case; then we have 󵄨󵄨 󵄨 t+1 t + 1 󵄨󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 ⌋ + mτ) − y(m⌊ ⌋)󵄨󵄨 ⩽ ε, 󵄨󵄨f (t + mτ) − f (t)󵄨󵄨 = 󵄨󵄨y(m⌊ 󵄨󵄨 󵄨󵄨 m m 󵄨 󵄨

t ∈ ℝ.

Since the function f ̌(⋅) ≡ f (−⋅) is continuous from the left side, the condition (i) from Definition 9.2.3 holds, and (QUC) holds, it readily follows that the function f ̌(⋅) is (tk )piecewise continuous almost periodic. The proof of the following extension of [655, Theorem 77] is simple and therefore omitted: Proposition 9.2.8. Suppose that ρ is a binary relation on E and the function F : I × X → E is (pre-)(ℬ, ρ, (tk ))-piecewise continuous almost periodic [(pre-)(ℬ, ρ, (tk ))-piecewise continuous uniformly recurrent]. If (Z, ‖ ⋅ ‖Z ) is a complex Banach space, ψ : E → Z is uniformly continuous on the set ρ(F(I×B))∪F(I×B) for each set B ∈ ℬ and σ = {(ψ(y1 ), ψ(y2 )) : y1 ρy2 }, then the function ψ ∘ F : I × X → Z is (pre-)(ℬ, σ, (tk ))-piecewise continuous almost periodic [(pre-)(ℬ, σ, (tk ))-piecewise continuous uniformly recurrent]. We continue by providing several useful observations: Remark 9.2.9. (i) Condition (QUC) can be relaxed by assuming that for every ε > 0 and B ∈ ℬ, there exists δ > 0 such that, if x ∈ B and the points t1 and t2 belong to the set ⋃k∈ℤ [tk +ε, tk+1 −ε] [⋃k∈ℕ [tk +ε, tk+1 −ε]] and |t1 −t2 | < δ, then ‖F(t1 ; x)−F(t2 ; x)‖ < ε; cf. also [655, Definition 7, p. 390] for this approach. We feel it is our duty to say that

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the condition (QUC) is primarily intended for the analysis of (ℬ, I, (tk ))-piecewise continuous almost periodic type functions and that some problems naturally occur if ρ ≠ I. (ii) The introduction of class of (pre-)(ℬ, ρ)-piecewise continuous uniformly recurrent functions is strongly justified by the fact that the definition of a piecewise continuous almost periodic function is a bit restrictive due to condition (i). In actual fact, this condition does not allow one to consider the existence and uniqueness of the piecewise continuous solutions for a large class of the abstract impulsive Cauchy problems in which the corresponding sequence of the first kind discontinuities (tk ) is not of linear growth as k → +∞; for example, we cannot consider the case t±k = k 2 for all k ∈ ℤ, which is very legitimate from the point of view of the theory of the abstract impulsive Cauchy problems. Remark 9.2.10. It is clear that any Bohr (ℬ, ρ)-almost periodic [Bohr (ℬ, ρ)-uniformly recurrent] function F(⋅; ⋅) is pre-(ℬ, ρ, (tk ))-piecewise continuous almost periodic [pre-(ℬ, ρ, (tk ))-piecewise continuous uniformly recurrent] for any sequence (tk ) satisfying our general assumptions as well as that any Bohr (ℬ, ρ)-almost periodic [Bohr (ℬ, ρ)-uniformly recurrent] function F(⋅; ⋅) which is uniformly continuous on the set I × B for each B ∈ ℬ is (ℬ, ρ, (tk ))-piecewise continuous almost periodic [(ℬ, ρ, (tk ))piecewise continuous uniformly recurrent] for any sequence (tk ) satisfying our general assumptions. In the almost periodic case, the statements of [304, Proposition 2.2, Proposition 2.7(ii)] show that this condition holds true if ℬ is a family consisting of some compact subsets of X, I = ℝ, R(F) ⊆ D(ρ) and ρ is single-valued on R(F). The subsequent structural result is a generalization of [364, Lemma 2.6]: Proposition 9.2.11. Suppose that F : I × X → E is pre-(ℬ, T, (tk ))-piecewise continuous almost periodic, where ρ = T ∈ L(E) is a linear isomorphism and the condition (QUC) holds. If B ∈ ℬ is a compact subset of X, then the set {F(t; x) : t ∈ I, x ∈ B} is relatively compact in E. Proof. We will basically consider the case in which I = ℝ and explain the essential change in the case that I = [0, ∞). Since ρ = T ∈ L(E) is a linear isomorphism, it suffices to show that the set T({F(t; x) : t ∈ ℝ, x ∈ B}) is relatively compact in E. Let ε > 0 be given. Then there exists δ > 0 such that, if x ∈ B and the points t1 and t2 belong to (ti , ti+1 ) for some i ∈ ℤ and |t1 − t2 | < δ, then ‖F(t1 ; x) − F(t2 ; x)‖ < ε/2. Let δ1 ∈ (0, min{δ, ε/4}). After that, we find l > 0 such that, for every t0 ∈ ℝ, the interval [t0 , t0 + l] contains a point τ ∈ I such that ‖F(t + τ; x) − TF(t; x)‖ ⩽ δ1 for all t ∈ ℝ such that |t − tk | ⩾ δ1 for all j ∈ ℤ. Fix now a point t ∈ ℝ and consider the interval I = [−t, l − t] (if I = [0, ∞), then for each point t ⩾ l we can consider the interval [t − l, t] and a corresponding (δ1 , T)-almost period τ belonging to this set). The set F([0, l] × B) is compact in E; furthermore, if τ ∈ I and the above conditions are satisfied, then we easily obtain the existence of an integer m ∈ ℕ, the points s1 , . . . , sm ∈ ℝ and the elements x1 , . . . , xm ∈ B such that

478 � 9 Almost periodic type solutions to abstract impulsive Volterra integro-differential inclusions TF(t; x) ∈ B(F(t + τ; x), δ1 ) ⊆

⋃

y∈F([0,l]×B)

B(y, δ1 )

⊆ L(F(s1 ; x1 ), ε/2) ∪ ⋅ ⋅ ⋅ ∪ L(F(sm ; xm ), ε/2),

x ∈ B,

provided that |t − tk | ⩾ δ1 for all j ∈ ℤ. If |t − tk | < δ1 for some j ∈ ℤ, then there exists an element t ′ ∈ [tj + δ1 , tj+1 − δ1 ] ∪ [tj−1 + δ1 , tj − δ1 ] such that ‖F(t; x) − F(t ′ ; x)‖ ⩽ ε/2, x ∈ B, which simply completes the proof of theorem. Remark 9.2.12. (i) It is also worth noting that Proposition 9.2.11 provides a proper generalization of [629, Lemma 3.3] as well as that this lemma holds even if the corresponding sequence (τj )j∈ℤ from its formulation is not a Wexler sequence. (ii) We know that there exists a continuous Stepanov-1-almost periodic function f : ℝ → ℝ which is not bounded; therefore, f (⋅) cannot be piecewise continuous almost periodic due to Proposition 9.2.11. We continue by stating the following results; the proofs are rather technical and therefore omitted (the statements of [304, Theorem 2.11(ii)–(iv)] can be also simply reformulated in our new framework): Proposition 9.2.13. Suppose that ρ is a binary relation on E which satisfies that D(ρ) is a closed subset of X and (Cρ ) holds. Suppose further that for each m ∈ ℕ the function Fm : I × X → E satisfies that, for every x ∈ X, the function t 󳨃→ Fm (t; x) is piecewise continuous with the possible first kind discontinuities at the points of a fixed sequence (tk ). Let F : I × X → E and let limm→∞ Fm (t; x) = F(t; x), uniformly on I × B for every fixed set B ∈ ℬ. Then for every x ∈ X, the function t 󳨃→ F(t; x) is piecewise continuous with the possible first kind discontinuities at the points of sequence (tk ) and the following holds: If for each m ∈ ℕ the function Fm (⋅; ⋅) is pre-(ℬ, ρ, (tk ))-piecewise continuous almost periodic [(ℬ, ρ, (tk ))-piecewise continuous almost periodic; pre-(ℬ, ρ, (tk ))-piecewise continuous uniformly recurrent/(ℬ, ρ, (tk ))-piecewise continuous uniformly recurrent], then the function F(⋅; ⋅) is pre-(ℬ, ρ, (tk ))-piecewise continuous almost periodic [(ℬ, ρ, (tk ))piecewise continuous almost periodic; pre-(ℬ, ρ, (tk ))-piecewise continuous uniformly recurrent/(ℬ, ρ, (tk ))-piecewise continuous uniformly recurrent]. Furthermore, if the functions Fm (⋅; ⋅) satisfy condition (QUC), then the function F(⋅; ⋅) satisfies the same condition. Proposition 9.2.14. Suppose that the function F : I × X → E is pre-(ℬ, ρ, (tk ))-piecewise continuous almost periodic [(ℬ, ρ, (tk ))-piecewise continuous almost periodic; pre-(ℬ, ρ, (tk ))-piecewise continuous uniformly recurrent/(ℬ, ρ, (tk ))-piecewise continuous uniformly recurrent]. Then the function ‖F(⋅; ⋅)‖ is pre-(ℬ, σ, (tk ))-piecewise continuous almost periodic [(ℬ, σ, (tk ))-piecewise continuous almost periodic; pre-(ℬ, σ, (tk ))piecewise continuous uniformly recurrent/(ℬ, σ, (tk ))-piecewise continuous uniformly recurrent], where σ := {(‖y1 ‖, ‖y2 ‖) | ∃t ∈ I ∃x ∈ X : y1 = F(t; x) and y2 ∈ ρ(y1 )}. It is worth noting that the supremum-formula holds in our framework:

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Proposition 9.2.15. Suppose that ρ = T ∈ L(E) is a linear isomorphism and F : I × X → E is a pre-(ℬ, T, (tk ))-piecewise continuous, uniformly recurrent function. Then for each a ∈ I and B ∈ ℬ we have 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 sup 󵄩󵄩󵄩F(t; x)󵄩󵄩󵄩 ⩽ 󵄩󵄩󵄩T −1 󵄩󵄩󵄩 sup 󵄩󵄩󵄩F(t; x)󵄩󵄩󵄩.

t∈I;x∈B

t⩾a;x∈B

Proof. It suffices to show that for each fixed number ε > 0 we have 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 sup 󵄩󵄩󵄩F(t; x)󵄩󵄩󵄩 ⩽ 󵄩󵄩󵄩T −1 󵄩󵄩󵄩 sup 󵄩󵄩󵄩F(t; x)󵄩󵄩󵄩 + ε.

t∈I;x∈B

t⩾a;x∈B

Clearly, there exists a sequence (δk ) in (0, ε) such that limk→+∞ δk = 0. After that, we find a strictly increasing sequence (αk ) of positive real numbers tending to plus infinity such that ‖F(t + αk ; x) − TF(t; x)‖ ⩽ δk provided that |t − ti | > δk for all i ∈ ℤ [i ∈ ℕ]. If t ∉ {ti : i ∈ ℤ} [t ∉ {ti : i ∈ ℕ}], then there exists k ∈ ℕ such that |t − ti | > δk for some i ∈ ℤ [i ∈ ℕ]. Hence, we have ‖TF(t; x)‖ ⩽ ‖F(t + αk ; x)‖ + δk ⩽ ‖F(t + αk ; x)‖ + ε and ‖F(t; x)‖ ⩽ ‖T −1 ‖(‖F(t + αk ; x)‖ + ε). The final conclusion follows from the fact that for every x ∈ B, the function F(⋅; x) is continuous from the left side. The statements of [409, Proposition 2.17, Proposition 2.18] continue to hold in our new framework and we have the following: (i) If f : I → ℝ is a pre-c-piecewise continuous uniformly recurrent function, then c = ±1; furthermore, if f (t) ⩾ 0 for all t ∈ I, then c = 1. (ii) If f : I → E is a pre-c-piecewise continuous uniformly recurrent function, then limt→+∞ f (t) ≠ 0. Let us recall that for every almost periodic function f : [0, ∞) → E, there exists a unique almost periodic function g : ℝ → E such that g(t) = f (t) for all t ⩾ 0. We close this section with the observation that is not clear whether we can state a satisfactory analogue of this result for certain subclasses of (pre-)(ℬ, M, ρ)-piecewise continuous almost periodic functions.

9.2.3 Relations with Stepanov almost periodic type functions As observed by S. I. Trofimchuk on [655, p. 389], a piecewise continuous almost periodic function f : I → X is Stepanov almost periodic under additional conditions that are not restrictive, and that our interest in the spaces of piecewise continuous almost periodic functions comes from the fact that these spaces have the stronger topologies than the spaces of Stepanov almost periodic functions. The main purpose of following result is to indicate that any piecewise continuous almost periodic function f : I → X in the sense of Definition 9.2.3 is immediately Stepanov-p-almost periodic for any finite exponent p ⩾ 1, as well as that a much more general result holds true (cf. also [655, Lemma 58, p. 400]):

480 � 9 Almost periodic type solutions to abstract impulsive Volterra integro-differential inclusions Theorem 9.2.16. Suppose that ρ = T ∈ L(E), 1 ⩽ p < +∞, F : I × X → E is pre-(ℬ, T, (tk ))piecewise continuous almost periodic [pre-(ℬ, T, (tk ))-piecewise continuous uniformly recurrent] and, for every B ∈ ℬ, ‖f ‖∞,B ≡ supt∈I,x∈B ‖F(t; x)‖ < +∞. Then the function F(⋅; ⋅) is Stepanov-(ℬ, p, T)-almost periodic [Stepanov-(ℬ, p, T)-uniformly recurrent] for any finite exponent p ⩾ 1. Proof. Without loss of generality, we may assume that I = ℝ and T = cI for some c ∈ ℂ; we will consider only pre-(ℬ, c)-piecewise continuous almost periodic functions. Fix a number ε > 0 and a set B ∈ ℬ. Let a point x ∈ ℝ be also fixed and let the interval [x, x + 1] contains the possible first kind discontinuities at the points {tm , . . . , tm+k } ⊆ [x, x + 1]. Then k ⩽ ⌈1/δ0 ⌉ and we have the existence of a sufficiently small real number ε0 > 0 such that p

ε0 + 2(

1 p + 1)((1 + |c|)‖f ‖∞,B ) ε0 ⩽ εp . δ0

(303)

Let S be a relatively dense set in ℝ such that, if τ ∈ S and b ∈ B, then ‖F(t+τ; x)−cF(t; x)‖ < ε0 for all t ∈ ℝ such that |t − tk | > ε0 , k ∈ ℤ. The function t 󳨃→ F(t + τ; b) − cF(t; b), t ∈ [x, x + 1] is less or equal than ε0 if t ∈ [x, tm − ε0 ] ∪ (tm + ε0 , tm+1 − ε0 ] ∪ ⋅ ⋅ ⋅ ∪ (tm+k , x + 1]; otherwise, we have ‖F(t + τ; b) − cF(t; b)‖ ⩽ (1 + |c|)‖f ‖∞,B . This implies x+1

p p 󵄩 󵄩p ∫ 󵄩󵄩󵄩F(t + τ; b) − cF(t; b)󵄩󵄩󵄩 dt ⩽ ε0 + 2⌈1/δ0 ⌉((1 + |c|)‖f ‖∞,B ) ε0 ,

b ∈ B.

x

Taking into account (303) and a simple computation, we get that x+1

󵄩 󵄩p ∫ 󵄩󵄩󵄩F(t + τ; b) − cF(t; b)󵄩󵄩󵄩 dt ⩽ εp ,

b ∈ B.

x

This simply completes the proof of theorem. Remark 9.2.17. (i) The condition δ0 > 0 is crucial for the proof of Theorem 9.2.16 to work. In [655, Appendix A.5], S. I. Trofimchuk has also analyzed the class of piecewise continuous almost periodic functions in the situation that infk∈ℤ (tk+1 − tk ) = 0 [infk∈ℕ (tk+1 − tk ) = 0]. If we allow the last condition, then a piecewise continuous almost periodic function need not be Besicovitch bounded (Besicovitch almost periodic); cf. [428] for the notion, and [655, p. 400] for a counterexample of this type. (ii) Albeit sometimes inevitable, the condition δ0 > 0 is a little bit redundant. For example, if P(t) is a non-periodic trigonometric polynomial with real values, then we know that the function f (t) := sign(P(t)), t ∈ ℝ is Stepanov-p-almost periodic for any exponent p ⩾ 1; see [428, Example 2.2.3]. Clearly, the zeros of P(⋅) are the points of discontinuity of the piecewise continuous function f0 (⋅) determined by f (⋅). But,

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since P(⋅) is not periodic, its zeros cannot be separated; to illustrate this, let us consider the polynomial P(t) := sin t + sin(√2t), t ∈ ℝ. Any zero of P(⋅) is of the form ′ tk = 2kπ/(1 + √2) for some k ∈ ℤ or tm = (2m + 1)π/(1 − √2) for some m ∈ ℤ. It can be ′ simply proved that tk ≠ tm for all k, m ∈ ℤ as well as that for each ε > 0 there exist two strictly increasing sequences (ak )k∈ℕ and (bk )k∈ℕ of positive integers such that |tak − tb′ k | < ε for all k ∈ ℕ; see, e. g. [576, Definition 2, Theorem 2, Remark 1]. This implies the required. (iii) In the formulation of Theorem 9.2.16, we have assumed immediately that c ∈ S1 . The proof also works in the case that c ∉ S1 but then the obtained conclusion in combination with [409, Proposition 2.6] shows that f ≡ 0. (iv) Keeping in mind Theorem 9.2.16, we can extend the statements of [364, Lemma 3.4, Theorem 3.7, Corollary 3.8] for any Stepanov-p-almost periodic inhomogeneity f (⋅) with the exponent p > 1; see also [428, Theorem 2.14.6] for case p = 1. In the case that (tk )k∈ℤ is a Wexler sequence, L. Qi and R. Yuan have proved that a piecewise continuous function f : ℝ → X, which satisfies the condition (QUC) is (tk )-piecewise continuous almost periodic if and only if the function f (⋅) is Stepanov-palmost periodic for every (some) exponent p ⩾ 1; see [629, Theorem 3.2]. Taken together with the statements of Proposition 9.2.11 and Theorem 9.2.16, the subsequent result provides a proper generalization of [629, Theorem 3.2]. Here we do not assume necessarily that (tk )k∈ℤ is a Wexler sequence and follow the idea from the proof of [655, Lemma 59, pp. 401–402], which is slightly incorrect since it is not clear how we can directly deduce the estimate |x(t + z′n ) − x(t + zn )| ⩾ ε/4 for all t ∈ [0, κ] or a similar estimate |x(t + z′n ) − x(t + zn )| ⩾ ε/4 for all t ∈ [−κ, 0]; see [655, l. 1, p. 402] and observe that, in the above-described situation, we can have zn = tj + ε and z′n = tp − ε for some integers j, n, p so that the quasi-uniform continuity argument cannot be directly applied here: Theorem 9.2.18. Suppose that ρ = T ∈ L(E), 1 ⩽ p < +∞ and F : I × X → E is a Stepanov-(ℬ, p, T)-almost periodic function. If the condition (QUC) holds, then F(⋅; ⋅) is pre-(ℬ, T, (tk ))-piecewise continuous almost periodic. Proof. For the sake of convenience, we will assume that I = ℝ, T = cI for some c ∈ S1 and X = {0}. Let ε > 0 be given; then there exists δ ∈ (0, min{ε/2, δ0 /4}) such that, if the points t1 and t2 belong to the same interval (ti , ti+1 ) of the continuity of function f (⋅) and |t1 − t2 | < δ, then ‖f (t1 ) − f (t2 )‖ < ε/4. Let ηk ∈ (0, εδ1/p /4) for all k ∈ ℕ and let limk→+∞ ηk = 0. We claim that there exists k0 ∈ ℕ such that, for every τ ∈ ℝ t+1

p

with ∫t ‖f (s + τ) − cf (s)‖p ds ⩽ ηk , t ∈ ℝ, we have ‖f (t + τ) − cf (t)‖ ⩽ ε for all t ∉ 0 ⋃l∈ℤ (tl − ε, tl + ε). If we assume the contrary, then for each k ∈ ℕ there exist points t+1

p

sk ∉ ⋃l∈ℤ (tl − ε, tl + ε) and τk ∈ ℝ such that ∫t ‖f (s + τk ) − cf (s)‖p ds ⩽ ηk , t ∈ ℝ and ‖f (sk + τk ) − cf (sk )‖ > ε. Using the continuity of function f (⋅) from the left side, for each k ∈ ℕ there exist points sk′ ∉ ⋃l∈ℤ (tl − (3ε/4), tl + (3ε/4)) and τk ∈ ℝ such that t+1

p

∫t ‖f (s + τk ) − cf (s)‖p ds ⩽ ηk , t ∈ ℝ, ‖f (sk′ + τk ) − cf (sk′ )‖ > 3ε/4 and sk′ + τk ∉ {tl : l ∈ ℤ}. Since δ < ε/2, it follows that, for every k ∈ ℕ, the interval (sk′ − δ, sk′ + δ) belongs to the

482 � 9 Almost periodic type solutions to abstract impulsive Volterra integro-differential inclusions same interval (tj , tj+1 ) of continuity of function f (⋅), for some j ∈ ℤ. On the other hand, at least one of the intervals (sk′ + τk , sk′ + τk + δ) and (sk′ + τk − δ, sk′ + τk ) belongs to the same interval (tp , tp+1 ) of continuity of function f (⋅), for some p ∈ ℤ. If the integer k ∈ ℕ is fixed, then we may assume without loss of generality that the above holds for the interval (sk′ + τk , sk′ + τk + δ); since |c| = 1, this readily implies: 󵄩󵄩 󵄩 󵄩󵄩[f (s + s′ + τk ) − cf (s + s′ )] − [f (s′ + τk ) − cf (s′ )]󵄩󵄩󵄩 k k k k 󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 ′ ′ ′ ′ 󵄩 ⩽ 󵄩󵄩f (s + sk + τk ) − f (sk + τk )󵄩󵄩 + 󵄩󵄩f (s + sk ) − f (sk )󵄩󵄩󵄩 ⩽ ε/2,

a. e. s ∈ [0, δ].

Hence, for every k ∈ ℕ, we have: 󵄩󵄩 ′ ′ 󵄩 󵄩󵄩f (s + sk + τk ) − cf (s + sk )󵄩󵄩󵄩 ⩾ ε/4,

a. e. s ∈ [0, δ],

and p ηk

tk +1

󵄩 󵄩p ⩾ ∫ 󵄩󵄩󵄩f (s + sk′ + τk ) − cf (s + sk′ )󵄩󵄩󵄩 ds tk

tk +δ

󵄩 󵄩p ⩾ ∫ 󵄩󵄩󵄩f (s + sk′ + τk ) − cf (s + sk′ )󵄩󵄩󵄩 ds ⩾ (ε/4)p δ, tk

which is a contradiction. This simply completes the proof of theorem. The argumentation contained in the proof of [629, Theorem 3.8] can be applied even if (tk )k∈ℤ is not a Wexler sequence. Keeping in mind this fact as well as Proposition 9.2.11, Theorem 9.2.16, Theorem 9.2.18 and [431, Theorem 6.2.21], we can extend [629, Theorem 3.8] in the following way: Theorem 9.2.19. Suppose that F : I × X → E is pre-(ℬ, (tk ))-piecewise continuous almost periodic, the condition (QUC) holds and any set B of the collection ℬ is a compact subset of X. Then F(⋅; ⋅) is Bohr ℬ-almost periodic if and only if F(⋅; ⋅) is continuous. We proceed with some applications of Theorem 9.2.16 and Theorem 9.2.18; our first result improves the statement of [655, Lemma 31, pp. 204–206]: Theorem 9.2.20. Suppose that Fi : I × X → E is a pre-(ℬ, (tk ))-piecewise continuous almost periodic function (i = 1, 2) and every set B of collection ℬ is compact in X. If the condition (QUC) holds for the functions F1 (⋅; ⋅) and F2 (⋅; ⋅), then the functions (F1 , F2 )(⋅; ⋅) and αF1 ± βF2 are pre-(ℬ, (tk ))-piecewise continuous almost periodic and satisfy the condition (QUC). Proof. Due to Proposition 9.2.11 and Theorem 9.2.16, we have that the functions F1 (⋅; ⋅) and F2 (⋅; ⋅) are Stepanov-(ℬ, p)-almost periodic. An application of [431, Proposition 6.2.17] shows that the function (F1 , F2 )(⋅; ⋅) is Stepanov-(ℬ, p)-almost periodic so that the function (F1 , F2 )(⋅; ⋅) is pre-(ℬ, (tk ))-piecewise continuous almost periodic by Theorem 9.2.18.

9.2 The existence and uniqueness of almost periodic type solutions

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This clearly implies that the function αF1 ± βF2 is pre-(ℬ, (tk ))-piecewise continuous almost periodic as well. The condition (QUC) clearly holds for the both functions. Observe that we have not assumed above that (tk )k∈ℤ is a Wexler sequence; in particular, if f1 (⋅) and f2 (⋅) are (tk )-piecewise continuous almost periodic functions and the requirements of Theorem 9.2.20 hold, then for each number ε > 0 there exists a relatively dense set of their common ε-almost periods, with the meaning clear. Keeping in mind Proposition 9.2.13, Proposition 9.2.15, and Theorem 9.2.20, we can simply prove an analogue of [304, Theorem 2.23] for (pre-)(ℬ, (tk ))-piecewise continuous almost periodic functions. Further on, as a simple application of Theorem 9.2.20, we have the following: Proposition 9.2.21. Suppose that F1 : I × X → ℂ and F2 : I × X → E are pre-(ℬ, (tk ))piecewise continuous almost periodic functions and every set B of collection ℬ is compact in X. If the condition (QUC) holds for the functions F1 (⋅; ⋅) and F2 (⋅; ⋅), then the function (F1 ⋅ F2 )(⋅; ⋅) is pre-(ℬ, (tk ))-piecewise continuous almost periodic; moreover, the function (F1−1 ⋅ F2 )(⋅; ⋅) is pre-(ℬ, (tk ))-piecewise continuous almost periodic, provided that for each set B ∈ ℬ we have inft∈I;x∈B |F1 (t; x)| > 0. The next result follows from the argumentation contained in the proofs of Theorem 9.2.16, Theorem 9.2.18, and the corresponding result for the Stepanov-p-almost periodic functions: Proposition 9.2.22. Suppose that F : I → E is a pre-piecewise continuous almost periodic function and the condition (QUC) holds. Let ε > 0 be fixed. Then for each number δ ∈ ℝ∖{0} there exists a relatively dense set S of integers such that the set δ ⋅ S consists solely of the ε-almost periods of F(⋅). The Favard type theorems for piecewise continuous almost periodic functions have been considered in the research article [763] by L. Wang and M. Yu. Let us only mention that the authors have clarified, in [763, Theorem 2.3], a sufficient condition for the primitive function of a scalar-valued piecewise continuous almost periodic function to be almost periodic; observe, however, that the established result is very unsatisfactory from the application point of view. On the other hand, using Proposition 9.2.11, Theorem 9.2.16, and the Bohl–Bohr–Amerio theorem (see, e. g. [496, p. 80]), we can clarify the following simple result on the integration of piecewise continuous almost periodic type functions: Theorem 9.2.23. Suppose that F : I → E is a pre-piecewise continuous almost perit odic function and E is uniformly convex. If the function t 󳨃→ F [1] (t) ≡ ∫0 F(s) ds, t ∈ I is bounded, then F [1] (⋅) is almost periodic.

The statement of [304, Proposition 2.2] admits a satisfactory reformulation in our new framework, provided that ρ = T ∈ L(E) is a linear isomorphism; in order to see

484 � 9 Almost periodic type solutions to abstract impulsive Volterra integro-differential inclusions this, we can combine Proposition 9.2.11, Theorem 9.2.16, Theorem 9.2.18, and [443, Theorem 1(i)]. Before proceeding to the next subsection with the observation that the statements of [409, Proposition 2.9, Corollary 2.10, Proposition 2.11] admit satisfactory reformulations in our new context, as well. For example, we can combine Proposition 9.2.11, Theorem 9.2.16, Theorem 9.2.18, and [443, Proposition 2] in order to see that the following generalization of [409, Proposition 2.9] holds true: Proposition 9.2.24. Suppose that ρ = T ∈ L(E) is a linear isomorphism and F : I × X → E is (pre-)(ℬ, T, (tk ))-piecewise continuous almost periodic [(pre-)(ℬ, T, (tk ))-piecewise continuous uniformly recurrent]. Then for each l ∈ ℕ the function F(⋅; ⋅) is (pre-)(ℬ, T l , (tk ))piecewise continuous almost periodic [(pre-)(ℬ, T l , (tk ))-piecewise continuous uniformly recurrent].

9.2.4 Composition principles for (ℬ, (tk ))-piecewise continuous almost periodic type functions In this subsection, we will prove two composition theorems for (ℬ, (tk ))-piecewise continuous almost periodic type functions. In order to achieve our aims, we employ the relations between the (ℬ, (tk ))-piecewise continuous almost periodic type functions and the Stepanov almost periodic type functions. Our first result reads as follows: Theorem 9.2.25. Suppose that (Z, ‖ ⋅ ‖Z ) is a complex Banach space, F : ℝ × X → E is a pre-(ℬ, (tk ))-piecewise continuous almost periodic function and G : ℝ × E → Z is a pre-(ℬ′ , (tk ))-piecewise continuous almost periodic function, where ℬ is a collection of all compact subsets of X and ℬ′ is a collection of all compact subsets of E. If the condition (QUC) holds for the functions F(⋅; ⋅) and G(⋅; ⋅), and there exists L > 0 such that 󵄩󵄩 󵄩 󵄩󵄩G(t; x) − G(t; y)󵄩󵄩󵄩Z ⩽ L‖x − y‖,

t ∈ ℝ, x, y ∈ E,

(304)

then the function W : ℝ × X → Z, given by W (t; x) := G(t; F(t; x)), t ∈ ℝ, x ∈ X, is a pre-(ℬ, (tk ))-piecewise continuous almost periodic function and the condition (QUC) holds for W (⋅; ⋅). Proof. Fix ε > 0 and B ∈ ℬ. Then Proposition 9.2.11 implies that the set B′ := F(ℝ × B) is relatively compact in E. Let δ1 > 0 be chosen from the condition (QUC) for the function F(⋅; ⋅), the number ε/2L and the set B; further on, let δ2 > 0 be chosen from the condition (QUC) for the function G(⋅; ⋅), the number ε/2 and the set B′ . Define δ := min{δ1 , δ2 }. Let t ′ , t ′′ ∈ (tk , tk+1 ) for some k ∈ ℤ, and let |t ′ − t ′′ | < δ. Then 󵄩󵄩 ′ 󵄩 ′ ′′ ′′ 󵄩󵄩G(t ; F(t ; x)) − G(t ; F(t ; x))󵄩󵄩󵄩Z 󵄩 󵄩 󵄩 󵄩 ⩽ 󵄩󵄩󵄩G(t ′ ; F(t ′ ; x)) − G(t ′ ; F(t ′′ ; x))󵄩󵄩󵄩Z + 󵄩󵄩󵄩G(t ′ ; F(t ′′ ; x)) − G(t ′′ ; F(t ′′ ; x))󵄩󵄩󵄩Z

9.2 The existence and uniqueness of almost periodic type solutions

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󵄩 󵄩 󵄩 󵄩 ⩽ L󵄩󵄩󵄩F(t ′ ; x) − F(t ′′ ; x)󵄩󵄩󵄩 + sup󵄩󵄩󵄩G(t ′ ; y) − G(t ′′ ; y)󵄩󵄩󵄩Z y∈B′

⩽ L(ε/2L) + (ε/2) = ε. Therefore, the condition (QUC) holds for W (⋅; ⋅); using a similar argumentation, we can show that for each x ∈ X the mapping t 󳨃→ W (t; x), t ∈ ℝ is continuous from the left side, with the possible first kind of discontinuities at the points of the sequence (tk )k∈ℤ . Consider now the functions FB : ℝ → l∞ (B : E) and GB : ℝ → l∞ (B′ : Z) defined through [FB (t)](x) := F(t; x), t ∈ ℝ, x ∈ B and [GB′ (t)](y) := G(t; y), t ∈ ℝ, y ∈ B′ , where l∞ (B : E) denotes the Banach space of all essentially bounded functions from B into E, equipped with the sup-norm. Due to Proposition 9.2.11, these mappings are well-defined. Using a simple argumentation involving the condition (QUC) for the functions F(⋅; ⋅) and G(⋅; ⋅), it follows that the functions FB (⋅) and GB (⋅) are pre-(tk )-piecewise continuous almost periodic and the condition (QUC) holds for them. Applying Theorem 9.2.16, Theorem 9.2.18 and [431, Proposition 6.2.17], we get that there exists a common set D of (tk )-almost periods for these functions, with the meaning clear. If τ ∈ D and |t − tk | > ε for some k ∈ ℤ, then we have 󵄩󵄩 󵄩 󵄩󵄩G(t + τ; F(t + τ; x)) − G(t; F(t; x))󵄩󵄩󵄩Z 󵄩 󵄩 󵄩 󵄩 ⩽ 󵄩󵄩󵄩G(t + τ; F(t + τ; x)) − G(t + τ; F(t; x))󵄩󵄩󵄩Z + 󵄩󵄩󵄩G(t + τ; F(t; x)) − G(t; F(t; x))󵄩󵄩󵄩Z 󵄩 󵄩 󵄩 󵄩 ⩽ L󵄩󵄩󵄩F(t + τ; x) − F(t; x)󵄩󵄩󵄩 + sup󵄩󵄩󵄩G(t + τ; y) − G(t; y)󵄩󵄩󵄩Z . ′ y∈B

This simply completes the proof of theorem. Our second structural result simply follows from Theorem 9.2.25 and the argumentation contained in the proof of [431, Theorem 6.1.50] (cf. also [304, Theorem 2.17] and [431, Subsection 6.1.5] for similar results): Theorem 9.2.26. Suppose that (Z, ‖ ⋅ ‖Z ) is a complex Banach space, F0 : ℝ × X → E is a pre-(ℬ, (tk ))-piecewise continuous almost periodic function, G1 : ℝ × E → Z is a pre-(ℬ′ , (tk ))-piecewise continuous almost periodic function, where ℬ is a collection of all compact subsets of X and ℬ′ is a collection of all compact subsets of E. Suppose, further, that the condition (QUC) holds for the functions F0 (⋅; ⋅) and G1 (⋅; ⋅), there exists L > 0 such that (304) holds with the function G(⋅; ⋅) replaced therein with the function G1 (⋅; ⋅), and the function Q0 : [0, ∞) × X → E [Q1 : [0, ∞) × E → Z] satisfies that for each set B ∈ ℬ [B′ ∈ ℬ′ ] we have limt→+∞ supx∈B ‖Q0 (t; x)‖ = 0 [limt→+∞ supy∈B′ ‖Q1 (t; y)‖Z = 0]. Then the function W : ℝ × X → Z, given by W (t; x) := [G1 + Q1 ](t; [F0 + Q0 ](t; x)), t ∈ ℝ, x ∈ X, is strongly asymptotically pre-(ℬ, (tk ))-piecewise continuous almost periodic in the sense that there exists a pre-(ℬ, (tk ))-piecewise continuous almost periodic function W2 : ℝ×X → E obeying the condition (QUC), and a function Q2 : [0, ∞)×X → E satisfying that for each set B ∈ ℬ we have limt→+∞ supx∈B ‖Q2 (t; x)‖ = 0 and W (t; x) = W2 (t; x) + Q2 (t; x) for all t ⩾ 0 and x ∈ X.

486 � 9 Almost periodic type solutions to abstract impulsive Volterra integro-differential inclusions Concerning the composition principles for piecewise pseudo almost periodic type functions, we refer the reader to [517, Section 3] for some results established by J. Liu and C. Zhang. Before going any further, we would like to present the following simple application of Theorem 9.2.25 and some useful observations concerning this result: Example 9.2.27. Let 𝒳 denote the set of all pre-(tk )-piecewise continuous almost periodic functions f : ℝ → X satisfying the condition (QUC). Then Proposition 9.2.13 and Theorem 9.2.20 together imply that (𝒳 , ‖ ⋅ ‖∞ ) is a complex Banach space. Consider now Theorem 9.2.25 with E = Z = X and ℬ = ℬ′ being the collection of all compact subsets of X. Consider further the following simple equation u(t) = u0 + G(t; u(t)),

t ∈ ℝ,

(305)

where u0 : ℝ → X is pre-(tk )-piecewise continuous almost periodic and satisfies the condition (QUC) as well as G : ℝ × X → X is pre-(ℬ, (tk ))-piecewise continuous almost periodic and satisfies the condition (QUC). Suppose that there exists L ∈ (0, 1) such that (304) holds. Then the mapping u ∋ 𝒳 󳨃→ u0 + G(⋅; u(⋅)) ∈ 𝒳 is well-defined due to Proposition 9.2.10, Theorem 9.2.20 and Theorem 9.2.25. Moreover, this mapping is a contraction, and therefore, there exists a unique function u ∈ 𝒳 satisfying (305). For example, we can take X = ℂ and G(t; x) = f1 (t)g1 (x) + ⋅ ⋅ ⋅ + fk (t)gk (x),

t ∈ ℝ, x ∈ ℂ (k ∈ ℕ),

where the functions Fj (⋅) are pre-(tk )-piecewise continuous almost periodic and satisfy the condition (QUC), the functions gj (⋅) are bounded, Lipschitz continuous with constants Lj > 0 and 󵄩 󵄩 󵄩 󵄩 L1 󵄩󵄩󵄩f1 󵄩󵄩󵄩∞ + ⋅ ⋅ ⋅ + Lk 󵄩󵄩󵄩fk 󵄩󵄩󵄩∞ < 1. On the other hand, it is very difficult to apply Theorem 9.2.25 to the abstract semilinear integro-differential equations of the form t

u(t) = u0 + ∫ R(t − s)G(s; u(s)) ds,

t ∈ ℝ,

(306)

−∞ +∞

if the operator family (R(t))t>0 ⊆ L(X) satisfies ∫0 ‖R(t)‖ dt < +∞ and condition that the mapping t 󳨃→ R(t)x, t > 0 is (piecewise-)continuous for every element x ∈ X. Then it t is expected that the mapping t 󳨃→ ∫−∞ R(t − s)G(s; u(s)) ds, t ∈ ℝ is Bohr almost periodic in the usual sense, so that we can always use a more general assumption that G : ℝ×X → X is Stepanov-p-ℬ-almost periodic for some p ⩾ 1 and apply the composition theorems for Stepanov almost periodic type functions [428, 431] combined with some result of type [428, Proposition 2.6.11]; cf. also Remark 9.2.17(iv). We will not discuss here the well-

9.2 The existence and uniqueness of almost periodic type solutions

� 487

posedness of problem (306) in the case that the mapping t 󳨃→ R(t)x, t > 0 is only Lebesgue +∞ measurable (x ∈ X) and ∫0 ‖R(t)‖ dt < +∞. In the subsequent subsections, we will analyze almost periodic type solutions to the abstract impulsive differential inclusions of integer order. We start with the observation that is not so simple to analyze the existence and uniqueness of (ω, c)-periodic solutions of the abstract impulsive Volterra integro-differential inclusions on bounded domains, unless some very restrictive assumptions are satisfied. Concerning this topic, which has recently been analyzed by some authors, we will only provide the following simple application of Theorem 9.1.2 with n = 1. Let ω = tk+1 − tk > 0 for all integers k ∈ ℕn , let f (t) ≡ 0 and let the (local) C-regularized semigroup with subgenerator 𝒜 satisfy R(t + ω) = cR(t) for all t ∈ [0, T − ω]. If u0 ∈ D(𝒜), then the solution u(t) of problem (ACP)1;1 satisfies u(T) = cu(0) if and only if cu0 = cl+1 u0 + [cl y10 + ⋅ ⋅ ⋅ + cyl0 ]; if c = 1, this simply means that y10 + ⋅ ⋅ ⋅ + yl0 = 0. 9.2.5 Asymptotically almost periodic type solutions of (ACP)1;1 In this subsection, we will use the following function spaces: If ω ∈ ℝ, then Cω ([0, ∞) : X) stands for the space of all continuous functions f : [0, ∞) → X such that the function t 󳨃→ e−ωt ‖f (t)‖, t ⩾ 0 is bounded; the space PCω ([0, ∞) : X) stands for the space of all piecewise continuous functions f : [0, ∞) → X such that the function t 󳨃→ e−ωt ‖f (t)‖, t ⩾ 0 is bounded. For simplicity, we set yk ≡ yk0 . Suppose that 𝒜 is the integral generator of a global exponentially decaying C-regularized semigroup (T(t))t⩾0 on X; therefore, there exist finite real constants ω < 0 and M ⩾ 1 such that ‖T(t)‖ ⩽ Meωt , t ⩾ 0. Suppose, further, that the functions C −1 f (⋅) and f𝒜 (⋅) satisfy all requirements from Corollary 9.1.3 with n = 1. 1. In this part, we will only assume that the sequence (tk ) has no accumulation point; the separation condition δ0 > 0 is completely regardless. If ∑k⩾1 e−ωtk ‖yk ‖ < +∞, then the function ω(⋅) defined in the proof of Theorem 9.1.2 belongs to the space PCω ([0, ∞) : X) since we have (cf. also [428, Remark 2.6.14(i)]): ∞

󵄩󵄩 󵄩 ωt −ωt 󵄩󵄩ω(t)󵄩󵄩󵄩 ⩽ Me ∑ e k ‖yk ‖ → 0 k=1

as t → +∞;

(307)

we will not further discuss here the sufficient conditions ensuring that the function ω(⋅) belongs to some space of the weighted ergodic components in ℝ (cf. [431, Section 6.4] for more details about these spaces in the multi-dimensional setting). Concerning the function C −1 f (⋅), we can assume that there exists a bounded Stepanov-p-almost periodic function g : ℝ → X and a function q ∈ PCω′ ([0, ∞) : X), for some ω′ ∈ ℝ

488 � 9 Almost periodic type solutions to abstract impulsive Volterra integro-differential inclusions and p ∈ [1, ∞), such that C −1 f (t) = g(t) + q(t) for all t ⩾ 0; see e. g., the proofs of [428, Proposition 2.6.11, Proposition 2.6.13]. A similar conclusion can be given in the case that there exist a bounded Stepanov-p-almost periodic function g : ℝ → X and ′ a function e−ω ⋅ q(⋅) ∈ PAPT0 ([0, ∞) : X), for some ω′ ∈ ℝ and p ∈ [1, ∞), such that C −1 f (t) = g(t) + q(t) for all t ⩾ 0; see [517, p. 3 and Definition 2.7] for the notion, the argumentation contained in the proof of [428, Lemma 2.12.3], and the decomposition used in the proof of [428, Proposition 2.6.13]. We can also use Stepanov-(p, c)-almost periodic functions here. 2. Suppose now, in place of condition ∑k⩾1 e−ωtk ‖yk ‖ < +∞, that (yk )k∈ℕ is an almost j periodic sequence as well as that the family of sequences (tk )k∈ℕ , j ∈ ℕ is equipotentially almost periodic. Then the argumentation contained in the proofs of [364, Lemma 3.4, Lemma 3.6, Theorem 3.7] shows that the function ω(⋅) is piecewise continuous almost periodic. 3. In this issue, we are seeking for the uniformly recurrent analogues of the conclusions established in the previous issue. Suppose that (τm ) is a strictly increasing sequence of positive real numbers such that limm→+∞ τm = +∞ and (ql ) is a strictly increasing sequence of positive integers. Let, for every ε > 0 and m ∈ ℕ, there exist integers s1 ∈ ℕ and s2 ∈ ℕ such that for every l ⩾ s1 and j ∈ ℕ, we have ‖yj+ql − yj ‖ + |tj+qs − tj − τm | ⩽ ε; see also [655, Lemma 35]. If the sequence (yk )k∈ℕ is bounded [the 2 sequences (yk )k∈ℕ and (Ayk )k∈ℕ are bounded], then Proposition 9.2.13 in combination with the argumentation contained in the proofs of [364, Lemma 3.6, Theorem 3.7] shows that the function ω(⋅) is (pre-)piecewise continuous uniformly recurrent; see also the statement (S2) from the proof of Theorem 9.2.28 below. For a concrete example, consider again the bounded Lipschitz continuous function ∞

1 tπ sin2 ( m ), m 2 m=1

f (t) := ∑

t ∈ ℝ;

then limk→+∞ f (t + 2k ) = f (t), uniformly in t ∈ ℝ. Take yj := f (j), tj := j for all j ∈ ℕ and τm := 2m for all m ∈ ℕ. Then the above requirements are satisfied.

9.2.6 Asymptotically Weyl almost periodic type solutions of (ACP)1;1 p

Suppose that 1 ⩽ p < ∞ and f ∈ Lloc (I : E); the notion of (equi-)Weyl-p-almost periodicity of f (⋅) is well known. In order to study the existence and uniqueness of asymptotically (equi-)Weyl-p-almost periodic solutions of the problem (ACP)1;1 , we will use the following conditions: (ew-M1) For every ε > 0, there exist s ∈ ℕ and L > 0 such that every interval I ′ ⊆ [0, ∞) of length L contains a point τ ∈ I ′ which satisfies, that there exists an integer qτ ∈ ℕ such that |ti+qτ − ti − τ| < ε for all i ∈ ℕ and

9.2 The existence and uniqueness of almost periodic type solutions

1 󵄩 󵄩p sup[ ∑󵄩󵄩󵄩yj+qτ − yj 󵄩󵄩󵄩 ] s |J|=s j∈J

� 489

1/p

< ε,

(308)

where the supremum is taken over all segments J ⊆ ℕ of length s. (w-M1) For every ε > 0, there exists L > 0 such that every interval I ′ ⊆ [0, ∞) of length L contains a point τ ∈ I ′ which satisfies that there exist an integer qτ ∈ ℕ and an integer sτ ∈ ℕ such that |ti+qτ − ti − τ| < ε for all integers i ∈ ℕ and (308) holds for all integers s ⩾ sτ . Condition (ew-M1), respectively, condition (w-M1), implies that the family of sequences j (tk )k∈ℕ , j ∈ ℕ is equipotentially almost periodic as well as that the sequence (xk )k∈ℕ is equi-Weyl-p-almost periodic, respectively, Weyl-p-almost periodic, in the following sense: (e-M1) For every ε > 0, there exist s ∈ ℕ and L > 0 such that every interval I ′ ⊆ [0, ∞) of length L contains a point τ ∈ I ′ ∩ ℕ which satisfies that (308) holds with the number qτ replaced therein with the number τ. (M1) For every ε > 0, there exists L > 0 such that every interval I ′ ⊆ [0, ∞) of length L contains a point τ ∈ I ′ ∩ ℕ which satisfies that there exists an integer sτ ∈ ℕ such that (308) holds for all integers s ⩾ sτ , with the number qτ replaced therein with the number τ. In the existing literature, the class of equi-Weyl-1-almost periodic sequences has been commonly used so far (see, e. g. the research articles [114] by V. Bergelson et al., [277] by T. Downarowicz, A. Iwanik, and [380] by A. Iwanik). The class of Weyl-p-almost periodic sequences seems to be not considered elsewhere, even in the scalar-valued case. Before going further, let us mention that is clear that condition (ew-M1) implies (w-M1) as well as that condition (e-M1) implies (M1). Concerning the existence and uniqueness of asymptotically Weyl almost periodic solutions of problem (ACP)1;1 , we will state and prove the following result: Theorem 9.2.28. Suppose that (ew-M1), resp., (w-M1) holds, the functions (C −1 f )(⋅) and fA (⋅) satisfy all requirements of Corollary 9.1.3 with n = 1, u0 ∈ D(A) and yk ≡ y0k ∈ D(A) for all k ∈ ℕ. Suppose, further, that (yk ) and (Ayk ) are bounded sequences, q ∈ PC0 ([0, ∞) : X), the function g : ℝ → X is (equi-)Weyl-p-almost periodic and bounded as well as (C −1 f )(t) = g(t) + q(t) for all t ⩾ 0. Then there exist a bounded continuous (equi-)Weyl-p-almost periodic function G1 : ℝ → X, a bounded piecewise continuous (equi-)Weyl-p-almost periodic function G2 : [0, ∞) → X and a function Q1 ∈ C0 ([0, ∞) : X) such that the unique solution u(t) of problem (ACP)1;1 satisfies u(t) = G1 (t) + G2 (t) + Q1 (t) for all t ⩾ 0. Proof. Keeping in mind [428, Theorem 2.11.4] and the proof of [428, Proposition 2.6.13], it readily follows that there exist a bounded (equi-)Weyl-p-almost periodic function G1 : t ℝ → X and a function Q1 ∈ C0 ([0, ∞) : X) such that T(t)u0 + ∫0 T(t − s)(C −1 f )(s) ds =

490 � 9 Almost periodic type solutions to abstract impulsive Volterra integro-differential inclusions G1 (t) + Q1 (t) for all t ⩾ 0; cf. the formulation of Theorem 9.1.2 with R(t) ≡ T(t). It remains to be proved that the function ω(⋅) from the formulation of Theorem 9.1.2 is bounded, piecewise continuous and (equi-)Weyl-p-almost periodic. Keeping in mind the argumentation contained in the proof of [364, Theorem 3.7], our assumption that (yk ) is a bounded sequence, and the fact that the statement of [428, Proposition 2.3.5] continues to hold for the sequences of piecewise continuous bounded functions, it suffices to show that the function ω1 (⋅), defined by ω1 (t) := 0 if 0 ⩽ t ⩽ t1 and ω1 (t) := T(t − tk )yk , if tk < t ⩽ tk+1 for some integer k ∈ ℕ, is (equi-)Weyl-p-almost periodic. The consideration is similar for the both classes of functions and we may assume, without loss of generality, that condition (ew-M1) holds. Since (Ayk ) is a bounded sequence, we have t T(t)yk − T(s)yk = [T(t)yk − Cyk ] − [T(s)yk − Cyk ] = ∫s T(r)Ayk dr for all t, s ⩾ 0, and therefore, the following statement holds: (S2) For every ε > 0, there exists δ ∈ (0, ε) such that, if t, s ⩾ 0 and |t − s| < δ, then ‖T(t)yk − T(s)yk ‖ ⩽ ε/3 for all k ∈ ℕ. Let ε > 0 be given. Then we know that there exist s ∈ ℕ, as large as we want, and L > 0 such that every interval I ′ ⊆ [0, ∞) of length L contains a point τ ∈ I ′ which satisfies that there exists an integer qτ ∈ ℕ such that |ti+qτ −ti −τ| < δ for all i ∈ ℕ and (308) holds. Suppose now that t > 0, |t − ti | > ε, |t − ti+1 | > ε and ti < t < ti+1 for some integer i ∈ ℕ. Then the argumentation contained in the proof of [364, Lemma 3.6], with ε′ = β = ε, shows that ti+qτ < t + τ < ti+qτ +1 . Therefore, since (S2) holds and |ti+qτ − ti − τ| < δ, we have: 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩ω1 (t + τ) − ω1 (t)󵄩󵄩󵄩 = 󵄩󵄩󵄩T(t + τ − ti+qτ )yi+qτ − T(t − ti )yi 󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ⩽ 󵄩󵄩󵄩T(t + τ − ti+qτ )yi+qτ − T(t − ti )yi+qτ 󵄩󵄩󵄩 + 󵄩󵄩󵄩T(t − ti )(yi+qτ − yi )󵄩󵄩󵄩 󵄩 󵄩 ⩽ (ε/3) + M 󵄩󵄩󵄩yi+qτ − yi 󵄩󵄩󵄩. Suppose now that x ⩾ 0, [x, x + l] ⊆ [tr , tr+m ], x ⩽ tr+1 and x + l ⩾ tr+m−1 for some integers r, m ∈ ℕ0 . Since the separation condition δ0 > 0 holds, we have l ⩾ (m − 2)δ0 and therefore m ⩽ ⌊l/δ0 ⌋ + 2. Hence, there exist absolute real constants M1 > 0 and M2 > 0 such that: x+l

1 󵄩 󵄩p [ ∫ 󵄩󵄩󵄩ω1 (t + τ) − ω1 (t)󵄩󵄩󵄩 dt] l x

1/p

tr +ε

tr+1 −ε

tr

tr +ε

1 󵄩 󵄩p 󵄩 󵄩p ⩽ [ ( ∫ 󵄩󵄩󵄩ω1 (t + τ) − ω1 (t)󵄩󵄩󵄩 dt + ∫ 󵄩󵄩󵄩ω1 (t + τ) − ω1 (t)󵄩󵄩󵄩 dt l tr+1 +ε

󵄩 󵄩p + ∫ 󵄩󵄩󵄩ω1 (t + τ) − ω1 (t)󵄩󵄩󵄩 dt + ⋅ ⋅ ⋅)] tr+1 −ε

1/p

9.2 The existence and uniqueness of almost periodic type solutions

1 󵄩 󵄩p ⩽ M1 [ (ε + (εp + 󵄩󵄩󵄩yr+qτ − yr 󵄩󵄩󵄩 ) + ε + ⋅ ⋅ ⋅)] l

� 491

1/p

1/p

r+m−1 1 󵄩 󵄩p ⩽ M1 [ ((m + 1)ε + (m + 1)εp + ∑ 󵄩󵄩󵄩yw+qτ − yw 󵄩󵄩󵄩 )] l w=r

1/p

r+m−1 1 󵄩 󵄩p ⩽ M1 [ ((⌊l/δ0 ⌋ + 3)ε + (⌊l/δ0 ⌋ + 3)εp + ∑ 󵄩󵄩󵄩yw+qτ − yw 󵄩󵄩󵄩 )] l w=r

⩽ M2 (ε

1/p

⩽ M2 (ε

1/p

1 r+m−1󵄩 󵄩p + ε) + M2 [ ∑ 󵄩󵄩󵄩yw+qτ − yw 󵄩󵄩󵄩 ] l w=r 1 + ε) + M2 [ l

r+⌊l/δ0 ⌋+1

∑

w=r

1/p

󵄩󵄩 󵄩p 󵄩󵄩yw+qτ − yw 󵄩󵄩󵄩 ]

1/p

.

Due to our assumption (308), the above calculation shows that we can take l = δ0 (s−2) in the corresponding definition of equi-Weyl-p-almost periodicity. The proof of the theorem is thereby complete. Remark 9.2.29. If we replace the conjunction of condition (ew-M1), resp. (w-M1), and condition that (Ayk ) is a bounded sequence, by condition that ∑k⩾1 e−ωtk ‖yk ‖ < +∞, then the above argumentation and (307) together imply that there exist a bounded, continuous (equi-)Weyl-p-almost periodic function G1 : ℝ → X and a function Q1 ∈ PC0 ([0, ∞) : X) such that the unique solution u(t) of problem (ACP)1;1 satisfies u(t) = G1 (t) + Q1 (t) for all t ⩾ 0. Here, we can only assume that the sequence (tk ) has no accumulation point; the separation condition δ0 > 0 is completely regardless. Now we would like to present the following simple example in which Theorem 9.2.28 can be applied (X = ℂ): Example 9.2.30. (i) Suppose that ti = i for all i ∈ ℕ, m ∈ ℕ∖{1}, yk = 0 for 1 ⩽ k ⩽ m−1 and yk = 1 for all k ⩾ m. Then it is trivial to show that (ew-M1) holds with L > k + 1 and s ⩾ (k −1)ε−p ; on the other hand, it is clear that (yk )k∈ℕ is not an almost periodic sequence. (ii) Suppose that ti = i for all i ∈ ℕ, σ ∈ (0, 1), p ⩾ 1, (1 − σ)p < 1 and yk = k σ for all k ∈ ℕ. Then the sequence (yk )k∈ℕ is not equi-Weyl-p-almost periodic (p ⩾ 1); on the other hand, (yk )k∈ℕ is Weyl-p-almost periodic for any exponent p ⩾ 1. Towards this end, it suffices to observe that there exists a finite constant cσ,p > 0 such that for every τ, sτ , l ∈ ℕ, we have l+sτ −1

p

∑ [(j + τ)σ − jσ ] ⩽ cσ,p τ p sτ1−(1−σ)p ; j=l

see the proof of [431, Theorem 7.3.8, case 3, p. 566]. The requirements of Theorem 9.2.28 hold with condition (w-M1) being satisfied.

492 � 9 Almost periodic type solutions to abstract impulsive Volterra integro-differential inclusions We close this subsection with the observation that we can similarly analyze the existence and uniqueness of (equi-)Weyl-p-almost periodic solutions for a class of the abstract impulsive nonautonomous differential equation of the form [364, (1.1)].

9.2.7 Besicovitch–Doss almost periodic type solutions of (ACP)1;1 Let us recall the following notion: Suppose that 1 ⩽ p < +∞, F : I → X, ϕ : [0, ∞) → [0, ∞) and F : (0, ∞) → (0, ∞). Then we say that the function F(⋅) belongs to the class e − (ϕ, F)−Bp (I : X) if and only if there exists a sequence (Pk (⋅)) of trigonometric polynomials such that 󵄩 󵄩 lim lim sup F(t)[ϕ(󵄩󵄩󵄩F(⋅) − Pk (⋅)󵄩󵄩󵄩)]

Lp ([−t,t]∩I)

k→+∞ t→+∞

= 0,

where we assume that the term in braces belongs to the space Lp ([−t, t] ∩ I) for all t > 0. If ϕ(x) ≡ x, then we omit the term “ϕ” from the notation. The usual notion is obtained by plugging ϕ(x) ≡ x and F(t) ≡ t −1/p , when we say that the function F(⋅) is Besicovitch-palmost periodic. As an immediate consequence of [125, Proposition 10] and our previous considerations, we have the following result (cf. also Remark 9.2.29; we only assume here that the sequence (tk ) has no accumulation point): Proposition 9.2.31. Suppose that the functions (C −1 f )(⋅) and fA (⋅) satisfy all requirements of Corollary 9.1.3 with n = 1, u0 ∈ D(A) and yk ≡ y0k ∈ D(A) for all k ∈ ℕ. Suppose further that ∑k⩾1 e−ωtk ‖yk ‖ < +∞, q ∈ PC0 ([0, ∞) : X), α > 0, a > 0, αp ⩾ 1, ap ⩾ 1, the function g : ℝ → X is bounded and belongs to the class e − (x α , t −a ) − Bp (ℝ : X) as well as (C −1 f )(t) = g(t) + q(t) for all t ⩾ 0. Then there exist a bounded continuous function G1 : ℝ → X belonging to the class e − (x α , t −a ) − Bp (ℝ : X) and a function Q1 ∈ PC0 ([0, ∞) : X) such that the unique solution u(t) of problem (ACP)1;1 satisfies u(t) = G1 (t) + Q1 (t) for all t ⩾ 0. Remark 9.2.32. Is should be noticed that the solution u(⋅) = G1 (⋅) + Q1 (⋅) belongs to the class e − (x α , t −a ) − Bp (ℝ : X), as well. In order to see this, it suffices to observe that e − (x α , t −a ) − Bp (ℝ : X) is a vector space (see the statement [125, (i), p. 4221]) and Q1 ∈ e − (x α , t −a ) − Bp (ℝ : X), which follows from a relatively simple computation with the sequence (Pk ≡ 0) in the corresponding definition and our assumption ap ⩾ 1. For the sequel, we need to recall the following notion: Let 1 ⩽ p < ∞ and let f ∈ p Lloc (I : X). Then it is said that f (⋅) is Doss p-almost periodic if and only if, for every ε > 0, the set of numbers τ ∈ I for which l

1 󵄩 󵄩p lim sup[ ∫󵄩󵄩󵄩f (s + τ) − f (s)󵄩󵄩󵄩 ds] 2l l→+∞ −l

1/p

< ε,

9.2 The existence and uniqueness of almost periodic type solutions

� 493

in the case that I = ℝ, resp., l

1 󵄩 󵄩p lim sup[ ∫󵄩󵄩󵄩f (s + τ) − f (s)󵄩󵄩󵄩 ds] l l→+∞

1/p

< ε,

0

in the case that I = [0, ∞), is relatively dense in I. Now we would like to reexamine the statement of Theorem 9.2.28 for Doss-p-almost periodic solutions. In order to do that, we need to introduce the following condition: (ed-M1) For every ε > 0, there exists L > 0 such that every interval I ′ ⊆ [0, ∞) of length L contains a point τ ∈ I ′ which satisfies that there exists an integer qτ ∈ ℕ such that |ti+qτ − ti − τ| < ε for all i ∈ ℕ and 1 s󵄩 󵄩p lim sup[ ∑󵄩󵄩󵄩yj+qτ − yj 󵄩󵄩󵄩 ] s j=1 s→+∞

1/p

< ε.

(309)

j

Condition (ed-M1) implies that the family of sequences (tk )k∈ℕ , j ∈ ℕ is equipotentially almost periodic as well as that the sequence (yk )k∈ℕ is Doss-p-almost periodic in the following sense: (d-M1) For every ε > 0, there exists L > 0 such that every interval I ′ ⊆ [0, ∞) of length L contains a point τ ∈ I ′ ∩ ℕ which satisfies that (309) holds with the number qτ replaced therein with the number τ. Before stating our next result, we would like to note that the class of Doss-p-almost periodic sequences has not been defined in the existing literature so far, even in the scalarvalued setting. Theorem 9.2.33. Suppose that (ed-M1) holds, the functions (C −1 f )(⋅) and fA (⋅) satisfy all requirements of Corollary 9.1.3 with n = 1, u0 ∈ D(A) and yk ≡ y0k ∈ D(A) for all k ∈ ℕ. Suppose, further, that (yk ) and (Ayk ) are bounded sequences, q ∈ PC0 ([0, ∞) : X), the function g : ℝ → X is Doss-p-almost periodic and bounded as well as (C −1 f )(t) = g(t)+q(t) for all t ⩾ 0. Then there exist a bounded continuous Doss-p-almost periodic function G1 : ℝ → X, a bounded piecewise continuous Doss-p-almost periodic function G2 : [0, ∞) → X and a function Q1 ∈ C0 ([0, ∞) : X) such that the unique solution u(t) of problem (ACP)1;1 satisfies u(t) = G1 (t) + G2 (t) + Q1 (t) for all t ⩾ 0. Proof. We will only outline the main details of the proof (see also the proof of Theorem 9.2.28). In place of [428, Theorem 2.11.4], we can use [428, Theorem 2.13.10]. If condition (ed-M1) holds in place of condition (ew-M1), then we can use the same arguments as in the proof of Theorem 9.2.28, with x = 0 and r = 0. The remainder of the proof is the same.

494 � 9 Almost periodic type solutions to abstract impulsive Volterra integro-differential inclusions Remark 9.2.34. Due to [428, Proposition 2.13.6], we have that the function Q1 (⋅) is also Doss-p-almost periodic. The pioneering results about Besicovitch-p-almost periodic sequences have been given in [126, 128]. The first systematic study of scalar-valued Besicovitch-p-almost periodic sequences has been carried out by A. Bellow and V. Losert in [104, Section 3]; cf. also V. Bergelson et al. [114]. In the following definition, we introduce the vector-valued version of [104, Definition 3.2]: Definition 9.2.35. Suppose that 1 ⩽ p < +∞ and (yk )k∈ℕ is a sequence in X. Then we say that (yk )k∈ℕ is Besicovitch-p-almost periodic if and only if for every ε > 0 there exists a trigonometric polynomial P(⋅) such that 1 s 󵄩 󵄩p lim sup[ ∑ 󵄩󵄩󵄩yk − P(k)󵄩󵄩󵄩 ] s k=1 s→+∞

1/p

< ε.

It can be simply shown that any Besicovitch-p-almost periodic sequence (yk )k∈ℕ is Besicovitch-p-bounded, i. e. lim sup k→+∞

1 k 󵄩󵄩 󵄩󵄩p ∑󵄩y 󵄩 < +∞; k l=1󵄩 l 󵄩

see, e. g. the proof of [125, Proposition 1(i)] for the continuous analogue of this statement. Therefore, the sequence (k σ )k∈ℕ , considered in Example 9.2.30(ii), is not Besicovitch-palmost periodic since 1 k σp ∑ l ∼ (1 + σp)−1 k σp k l=1

as k → +∞.

Here we will not reconsider the statements established in [104] for the vector-valued Besicovitch-p-almost periodic sequences. Before proceeding further, we would like to address the following questions: (Q1) Is it possible to state a satisfactory analogue of Theorem 9.2.28 and Theorem 9.2.33 for Besicovitch-p-almost periodic solutions of problem (ACP)1;1 (1 ⩽ p < ∞)? j (Q2) Suppose that the family of sequences (tk )k∈ℕ , j ∈ ℕ is equipotentially almost periodic, as well as that the sequence (yk )k∈ℕ is equi-Weyl-p-almost periodic (1 ⩽ p < ∞). Is it true that (ew-M1) holds true? (Q3) Suppose that the sequence (yk )k∈ℕ is (equi-)Weyl-p-almost periodic [Doss-p-almost periodic/Besicovitch-p-almost periodic] (1 ⩽ p < ∞). Is it true that there exists a unique (equi-)Weyl-p-almost periodic [Doss-p-almost periodic/Besicovitch-palmost periodic] sequence (yk̃ )k∈ℤ (defined in the obvious way) such that yk̃ = yk for all k ∈ ℕ?

9.2 The existence and uniqueness of almost periodic type solutions

� 495

(Q4) Is it true that the sequence (yk )k∈ℤ [(yk )k∈ℕ ] is (equi-)Weyl-p-almost periodic [Doss-p-almost periodic/Besicovitch-p-almost periodic] (1 ⩽ p < ∞) if and only if there exists a continuous (equi-)Weyl-p-almost periodic [Doss-p-almost periodic/ Besicovitch-p-almost periodic] function f : ℝ → X [f : [0, ∞) → X] such that yk = f (k) for all k ∈ ℤ [k ∈ ℕ]? In connection with the problem (Q2), see also [655, Lemma 35] and observe that we cannot expect the affirmative answer in the case of consideration of Weyl-p-almost periodic sequences and Doss-p-almost periodic sequences; cf. [431] for more details. It can be very simply shown that the class of Doss-p-almost periodic sequences is the most general, since it contains all Weyl-p-almost periodic sequences and all Besicovitch-p-almost periodic sequences; it is also worth noting that the class of equi-Weyl-p-almost periodic sequences is contained in the class of Besicovitch-p-almost periodic sequences, which is no longer true for the class of Weyl-p-almost periodic sequences. A simple example of a Besicovitch-p-almost periodic sequence, which is not Weyl-p-almost periodic is given as follows: If 1 ⩽ p < +∞ and k ∈ [m2 , m2 + √m) for some integer m ∈ ℕ, then we define yk := m1/2p ; then the sequence (yk )k∈ℕ enjoys the above-mentioned properties (see also [42, Example 6.24], [122, p. 42] and [125, Example 4]). We will examine in more detail the classes of (equi-)Weyl-p-almost periodic sequences, Doss-p-almost periodic sequences and Besicovitch-p-almost periodic sequences somewhere else (1 ⩽ p < ∞).

9.2.8 Almost periodic type solutions of the abstract higher-order impulsive Cauchy problems We will first explain how the results established in the previous three subsections can be used in the analysis of the existence and uniqueness of almost periodic type solutions for certain classes of the abstract higher-order impulsive (degenerate) Cauchy problems. Here, our idea is to convert these problems into the equivalent abstract impulsive (degenerate) Cauchy problems of first order on the product spaces. Suppose, for example, that the operator A generates a strongly continuous semigroup on X as well as that B is a closed densely defined operator on X with D(A) ⊆ D(B). Applying [579, Theorem 3], we get that there exists a real number ω > 0 such that, for every λ ∈ ℂ with Re λ > ω, the matricial operator D := [

A−λ B

I ] −λ

generates an exponentially decaying strongly continuous semigroup (T(t))t⩾0 on X × X. Suppose further that u1 ∈ D(A), u2 ∈ X, and t 󳨃→ f1,2 (t), t ⩾ 0 are continuously differentiable and asymptotically almost periodic. Using Corollary 9.1.3 and a simple computation with the variation of parameters formula, we can reformulate all established

496 � 9 Almost periodic type solutions to abstract impulsive Volterra integro-differential inclusions conclusions from the previous three subsections in the analysis of the abstract impulsive Cauchy problem u′′ (t) − (A − 2λ)u′ (t) − [λ(A − λ) + B]u(t) = f1′ (t) + λf1 (t) + f2 (t), (Δu)(tk ) = u(0) = u1 ,

yk0 ;

′

(Δu )(tk ) = ′

yk1

+ (A −

t ⩾ 0;

λ)yk0 ;

u (0) = u2 + (A − λ)u1 + f1 (0).

Without going into full details, we will only emphasize that the certain classes of the abstract degenerate second-order Cauchy problems with impulsive effects can be also analyzed in a similar manner, by reduction to the abstract degenerate first-order Cauchy problems with impulsive effects on product spaces; see, e. g. [280, Example 2.5(ii)], where we have analyzed the well-posedness of the damped Poisson-wave type equations in Lp spaces (it would be very enticing to apply the same method to the abstract higher-order Cauchy problems considered in [433]). In the existing literature, the authors have found many interesting criteria ensuring that a strongly continuous operator family (R(t))t⩾0 ⊆ L(X) is (asymptotically) almost periodic, i. e. the mapping t 󳨃→ R(t)x, t ⩾ 0 is (asymptotically) almost periodic for every fixed element x ∈ X; cf. [428] for more details. Such operator families can be important in the analysis of the existence and uniqueness of the almost periodic type solutions for certain classes of the abstract impulsive Volterra integro-differential inclusions, as the following illustrative example indicates: Example 9.2.36. The existence and uniqueness of almost periodic solutions for a class of the complete second-order Cauchy problems have been considered by T.-J. Xiao and J. Liang [777, Section 7.1.2] under the assumption that the corresponding problem is strongly well-posed. Specifically, the authors have analyzed the abstract Cauchy problem u′′ (t) + (aA0 + bI)u′ (t) + (cA0 + dI)u(t) = 0,

t ⩾ 0,

where a, b, c, d ∈ ℂ and the operator A0 is a closed linear operator with domain and range contained in a Banach space X. In the case that X := L2 [0, 1] and A0 is the Dirichlet Laplacian, the authors have shown that the both propagator families, (S0 (t))t⩾0 and (S1 (t))t⩾0 , are almost periodic. If so, then we can consider the piecewise continuous almost periodic solutions of the following abstract impulsive Cauchy problem u′′ (t) + (aA0 + bI)u′ (t) + (cA0 + dI)u(t) = 0, { { { { { { t ∈ [0, ∞) ∖ {t1 , . . . , tl , . . .}, (ACP)2 : { (j) {(Δu )(tk ) = u(j) (tk +) − u(j) (tk −) = yk , k ∈ ℕ, j = 0, 1; { j { { { (j) u (0) = u , j = 0, 1, j {

9.2 The existence and uniqueness of almost periodic type solutions

� 497

where 0 < t1 < ⋅ ⋅ ⋅ < tl < ⋅ ⋅ ⋅ < +∞ and the sequence (tl )l has no accumulation point. Due to our consideration from [280], the function u(t) = S0 (t)u0 + S1 (t)u1 + ω(t), t ⩾ 0, where ω(t) := ∑1j=0 [Sj (t−t1 )y1j +⋅ ⋅ ⋅+Sj (t−tk )ykj ] if t ∈ (tk , tk+1 ] for some k ∈ ℕ, is a unique solution of (ACP)2 . Arguing as in Example 9.2.39 below, we may conclude that the assumptions ∑k⩾1 ‖yk0 ‖ < +∞ and ∑k⩾1 ‖yk1 ‖ < +∞ imply that there exist an almost periodic function f : [0, ∞) → X and a function q ∈ PC0 ([0, ∞) : X) such that u(t) = f (t) + q(t) for all t ⩾ 0. 9.2.9 Almost periodic type solutions of the abstract Volterra integro-differential inclusions with impulsive effects We start with the observation that, in the global version of Theorem 9.1.13, we do not need the separation condition δ0 > 0 on the sequence (tk ). For the sequel, we need to recall the following special consequences of [429, Proposition 3.1.15(i)]: (i) Suppose that α ∈ (0, 1), u0 ∈ D(𝒜) as well as C −1 f , f𝒜 ∈ C([0, ∞) : X), f𝒜 (t) ∈ 𝒜C −1 f (t), t ⩾ 0 and 𝒜 is a closed subgenerator of a (gα , C)-regularized resolvent family (R(t))t⩾0 . Then the function u(t) := R(t)x + (R ∗ C −1 f )(t), t ⩾ 0 is a unique solution of the following abstract fractional Cauchy inclusion: 1

u ∈ C ((0, ∞) : X) ∩ C([0, ∞) : X), { { { α D u(t) ∈ 𝒜u(t) + (g1−α ∗ f )(t), t ⩾ 0, { { t { {u(0) = Cu0 . (ii) Suppose that α ∈ (1, 2), u0 ∈ D(𝒜) as well as C −1 f , f𝒜 ∈ C([0, ∞) : X), f𝒜 (t) ∈ 𝒜C −1 f (t), t ⩾ 0 and 𝒜 is a closed subgenerator of a (gα , C)-regularized resolvent family (R(t))t⩾0 . Set v(t) := (g2−α ∗ f )(t), t ⩾ 0. If v ∈ C 1 ([0, ∞) : X), then the function u(t) := R(t)x + (R ∗ C −1 f )(t), t ⩾ 0 is a unique solution of the following abstract fractional Cauchy inclusion: u ∈ C 2 ((0, ∞) : X) ∩ C 1 ([0, ∞) : X), { { { α d Dt u(t) ∈ 𝒜u(t) + dt (g2−α ∗ f )(t), t ⩾ 0, { { { ′ {u(0) = Cu0 , u (0) = 0. Keeping in mind this result, Theorem 9.1.13 and the second equality in [97, (1.21)], we have the following: Theorem 9.2.37. Suppose that u0 ∈ D(𝒜), C −1 f , f𝒜 ∈ C([0, ∞) : X), f𝒜 (t) ∈ 𝒜C −1 f (t), t ⩾ 0 and 𝒜 is a closed subgenerator of a global (gα , C)-regularized resolvent family (R(t))t⩾0 ⊆ L(X). Define ℱ0 (t) := 0 for t ∈ [0, t1 ] and ℱ0 (t) := ∑m s=1 Cys if t ∈ (tm , tm+1 ] for some integer m ∈ ℕ. Define also ω(t) := 0 for t ∈ [0, t1 ] and ω(t) := ∑ls=1 R(t − ts )ys if t ∈ (tm , tm+1 ] for some integer m ∈ ℕ, and assume that y1 , . . . , yl , . . . ∈ D(𝒜).

498 � 9 Almost periodic type solutions to abstract impulsive Volterra integro-differential inclusions t

(i) Suppose that α ∈ (0, 1). Then the function u(t) := R(t)u0 + ∫0 R(t − s)(C −1 f )(s) ds + ω(t), t ⩾ 0 is piecewise continuous and it is a unique strong solution of problem (299) on t [0, ∞) with ℬ = I, a(t) = gα (t) and ℱ (t) = Cu0 + ∫0 f (s) ds + ℱ0 (t), t ⩾ 0. (ii) Suppose that α ∈ (1, 2) and the mapping t 󳨃→ (g2−α ∗ f )(t), t ⩾ 0 is continuously t differentiable. Then the function u(t) := R(t)u0 + ∫0 R(t − s)(C −1 f )(s) ds + ω(t), t ⩾ 0 is piecewise continuous and it is a unique strong solution of problem (299) on [0, ∞) t with ℬ = I, a(t) = gα (t) and ℱ (t) = Cu0 + ∫0 f (s) ds + ℱ0 (t), t ⩾ 0. Concerning the existence and uniqueness of the asymptotically almost periodic type t solutions of problem (299) on [0, ∞) with ℬ = I and ℱ (t) = Cu0 + ∫0 f (s) ds + ℱ0 (t), t ⩾ 0, we will present the following extremely important situation in which all conclusions established in the previous four subsections continue to hold. 1. Suppose that A is a densely defined, closed linear operator, 1 < α < 2, M > 0, ω < 0, Ω ≡ (ω + {λ ∈ ℂ ∖ {0} : arg(−λ) < θ})c ⊆ ρC (A) for some number θ ∈ [0, π(1 − (α/2))), ‖(λ − A)−1 C‖ ⩽ M/|λ − ω|, λ ∈ Ω and the mapping λ 󳨃→ (λ − A)−1 C is analytic in an open neighborhood of the set Ω. Then we know that the operator A is a subgenerator of a global (gα , C)-regularized resolvent family (R(t))t⩾0 satisfying that there exists M ′ > 0 such that ‖R(t)‖ ⩽ M ′ /(1 + |ω|t α ), t ⩾ 0; see e. g., the proof of E. Cuesta’s result [233, Theorem 1] and [428, Section 3.4] for many important generalizations of this result. Suppose now that all requirements of Theorem 9.2.37 hold, the sequence (tk ) has no accumulation point (the separation condition δ0 > 0 is completely regardless here) and ∑k⩾1 (tkα + 1)‖yk ‖ < +∞. Then the function ω(⋅) defined in the proof of Theorem 9.2.37(ii) belongs to the space PCα;1 ([0, ∞) : X) ≡ {f ∈ PC([0, ∞) : X) ; ⋅α ‖f (⋅)‖ ∈ L∞ ([0, ∞) : X)}, since for each k ∈ ℕ and t ∈ (tk , tk+1 ] we have: k k (t − tl )α + tlα tα 󵄩󵄩 α 󵄩 ′ ′ α−1 ‖y ‖] ⩽ M 2 [ ‖y ‖] ∑ 󵄩󵄩t ω(t)󵄩󵄩󵄩 ⩽ M ∑[ 1 + |ω|(t − t )α l 1 + |ω|(t − t )α l l

l=1

l

l=1

k

∞

l=1

l=1

⩽ M ′ 2α−1 ∑[((1/|ω|) + tlα )‖yl ‖] ⩽ M ′ 2α−1 ∑[((1/|ω|) + tlα )‖yl ‖].

(310)

Concerning the function C −1 f (⋅), we can assume that there exists a bounded Stepanov-palmost periodic function g : ℝ → X and a function q ∈ PC0 ([0, ∞) : X), for some p ∈ [1, ∞), such that C −1 f (t) = g(t) + q(t) for all t ⩾ 0. Then the function t 󳨃→ u(t) − ω(t), t ⩾ 0 will be asymptotically almost periodic in the usual sense; see e. g., [428, Proposition 2.6.11, Remark 2.6.12, Proposition 2.6.13]. All other results established in the previous four subsections continue to hold, as marked above. After a careful inspection of the proofs of [364, Lemma 3.6, Theorem 3.7], it suffices to observe that the uniform convergence in the corresponding part of the proof of [364, Theorem 3.7; cf. (3.36), p. 14] is a consequence of the following simple computation, where we assume that the sequence (yk ) is bounded:

9.2 The existence and uniqueness of almost periodic type solutions

1 󵄩󵄩 󵄩 󵄩󵄩R(ti−k − t)yi−k 󵄩󵄩󵄩 ⩽ N sup 1 + |ω|(t − t i∈ℕ

⩽ N sup i∈ℕ

i−k )

� 499

α

1 1 ⩽N , 1 + |ω|(t − ti )α + |ω|(ti − ti−k )α 1 + |ω|(kδ0 )α

for some finite real constant N > 0. In conclusion, we have the following: if (yk )k∈ℕ is an almost periodic sequence, the separation condition δ0 > 0 holds and the family of j sequences (tk )k∈ℕ , j ∈ ℕ is equipotentially almost periodic, then the function ω(⋅) is piecewise continuous almost periodic. It should be noticed that the obtained results can be applied to the abstract (noncoercive) differential operators in Lp -spaces; cf. [426, Section 2.5] for further information in this direction. Remark 9.2.38. Consider now the situation in which γ ∈ (0, 1), u0 ∈ D(𝒜) and 𝒜 satisfies condition (P); cf. also [428, Subsection 2.9.1]. If we consider the subordinated resolvent families (Sγ (t))t>0 and (Rγ (t))t>0 from [428], then the function uh (t) := Sγ (t)x + (Rγ ∗ f )(t), t ⩾ 0 is a unique solution of the following abstract fractional Cauchy inclusion (under certain reasonable assumptions): {

Dαt uh (t) ∈ 𝒜uh (t) + f (t),

t > 0,

uh (0) = u0 .

Keeping in mind the second equality in [97, (1.21)] and the initial condition u(0) = u0 , we simply get that the function uh (⋅) is a unique strong solution of the associated Volterra inclusion uh (t) ∈ u0 + (gα ∗ f )(t) + 𝒜(gα ∗ uh )(t),

t ⩾ 0.

Suppose now that ℱ0 (t) := 0 for t ∈ [0, t1 ] and ℱ0 (t) := ∑m s=1 ys if t ∈ (tm , tm+1 ] for some integer m ∈ ℕ, as well as that y1 , . . . , yl , . . . ∈ D(𝒜). Define ω(t) := 0 for t ∈ [0, t1 ] and ω(t) := ∑m s=1 Sγ (t − ts )ys if t ∈ (tm , tm+1 ] for some integer m ∈ ℕ, and assume that there exist vectors z1 , . . . , zl , . . . from the continuity set of the resolvent operator family (Sγ (t))t>0 such that zl ∈ 𝒜yl for all l ∈ ℕ; cf. also [280, Example 2.5(i)]. Then the function ω(t) is a unique strong solution of the abstract impulsive Volterra inclusion ω(t) ∈ ℱ0 (t) + 𝒜(gα ∗ ω)(t),

t ∈ [0, ∞) ∖ {t1 , t2 , . . . , tl , . . .}.

Therefore, the function u(t) := uh (t) + ω(t), t ⩾ 0 is a unique strong solution of the abstract impulsive Volterra inclusion u(t) ∈ u0 + (gα ∗ f )(t) + ℱ0 (t) + 𝒜(gα ∗ u)(t),

t ∈ [0, ∞) ∖ {t1 , t2 , . . . , tl , . . .}.

(311)

Concerning the existence and uniqueness of asymptotically almost periodic solutions of (311), the situation is far from being simple because the operator family (Sγ (t))t>0

500 � 9 Almost periodic type solutions to abstract impulsive Volterra integro-differential inclusions has an integrable singularity at zero: we must impose certain extra assumptions in order for our proofs to work. This can be simply done for the analogues of the equations [364, (3.27)–(3.28)] but, unfortunately, this is almost impossible to be done for the equation [364, (3.36)] since the series ∑k⩾1 (kδ0 )−γ diverges. Even the computation carried out in (310) cannot be so simply reconsidered in our new situation. We continue by providing the following instructive example: Example 9.2.39. Let α ∈ (0, 2) and θ = π − πα/2, and let us consider the following fractional Cauchy problem Dαt u(t, x) = eiθ uxx (t, x),

0 < x < 1, t ⩾ 0;

cf. also [97, Example 2.20]. Suppose that X := L2 [0, 1] and A := eiθ Δ, where Δ denotes the Laplacian equipped with the Dirichlet boundary conditions. Then we known that A is the integral generator of an asymptotically almost periodic (gα , I)-resolvent family (R(t))t⩾0 as well as that (R(t))t⩾0 is not almost periodic if α ≠ 1; cf. [428, Example 2.6.4]. Suppose now that u0 , y1 , y2 , . . . , yl , . . . ∈ D(A) and ∑k⩾1 ‖yk ‖ < +∞. Define the function ω(⋅) as in the formulation of Theorem 9.1.13(i), with k(t) ≡ 1 and C2 = I. Then it can be simply shown that the function u(t) := R(t)u0 + ω(t), t ⩾ 0 is a unique strong solution of the abstract Volterra equation (299) with ℬ = C = I, 𝒜 = A, a(t) ≡ gα (t) and ℱ (t) ≡ u0 + ℱ0 (t), where ℱ0 (t) = y1 + ⋅ ⋅ ⋅ + yk if tk < t ⩽ tk+1 for some k ∈ ℕ0 . By our assumption and the already mentioned result about the extension of almost periodic functions [87], we know that for each k ∈ ℕ there exist an almost periodic function Rkap : ℝ → X and a function Q ∈ C0 ([0, ∞) : X) such that R(t −tk )yk = Rkap (t −tk )+Q(t −tk ) for all t ⩾ tk . Define Fk : [0, ∞) → X and Qk : [0, ∞) → X by Fk (t) := Rkap (t − tk ), t ⩾ 0,

Qk (t) := −Rkap (t − tk ) for t ∈ [0, tk ] and Qk (t) := Q(t − tk ) for t > tk . It can be simply shown that Fk (⋅) is almost periodic, Qk ∈ PC0 ([0, ∞) : X), ‖Fk (⋅)‖∞ ⩽ ‖R(⋅)‖∞ ⋅ ‖yk ‖, ‖Qk (⋅)‖∞ ⩽ 3‖R(⋅)‖∞ ⋅ ‖yk ‖ as well as that χ[0,tk ] (t)R(t − tk )yk = Fk (t) + Qk (t) for all t ⩾ 0 and k ∈ ℕ; cf. also [259, Lemma 4.28, Theorem 4.29]. Since we have assumed that ∑k⩾1 ‖yk ‖ < +∞, the Weierstrass criterion implies that the series ∑k⩾1 Fk (t) =: F(t), t ⩾ 0 and ∑k⩾1 Qk (t) =: Q(t), t ⩾ 0 are uniformly convergent. Since PC0 ([0, ∞) : X) is a Banach space, it readily follows that Q ∈ PC0 ([0, ∞) : X); on the other hand, it is clear that the function F(⋅) is almost periodic. Hence, the solution u(t) is piecewise continuous asymptotically almost periodic, since u(t) = F(t) + Q(t) for all t ⩾ 0. Observe that we can similarly analyze, with the help of [280, Theorem 4.6] and the foregoing arguments, the existence and uniqueness of asymptotically almost periodic type solutions of problem (301). Let us finally mention a few important topics not considered here: 1. As already mentioned, the Levitan and Bebutov classes of almost periodic type functions can be further generalized using the approaches of Stepanov, Weyl, and Besicovitch. The notion of a Stepanov 1-Levitan N-almost periodic function f : ℝ → X has

9.2 The existence and uniqueness of almost periodic type solutions

� 501

been already introduced in [655, Definition 12, p. 402] and the analogues of [655, Lemmas 58–59] for Stepanov 1-Levitan N-almost periodic functions have been clarified in [655, Lemma 60]. We will only note here that we can similarly define the notion of a Stepanov p-Levitan N-almost periodic function f : I → X, where 1 ⩽ p < +∞, and prove an analogue of Theorem 9.2.16 for Stepanov p-Levitan N-almost periodic functions. 2. The notion of a Bochner spatially almost automorphic sequence (tk )k∈ℤ has recently been introduced by L. Qi and R. Yuan in [630, Definition 3.1]. Any Wekler sequence (tk )k∈ℤ is Bochner spatially almost automorphic, while the converse statement is not true in general. The authors have generalized the notion of piecewise continuous almost periodicity by introducing and examining the classes of Bohr, Bochner, and Levitan piecewise continuous almost automorphic functions; in [630, Theorem 4.8], the authors have proved that these classes coincide, actually. We will consider piecewise continuous almost automorphic type functions and piecewise continuous almost automorphic solutions to the abstract impulsive Volterra integro-differential equations somewhere else (cf. also the research article [268] by W. Dimbour and V. Valmorin for the notion of S-almost automorphy; for discontinuous almost automorphic functions and their applications, we also refer the reader to the research articles by A. Chávez, S. Castillo, M. Pinto [192, 193] and S. Abbas, L. Mahto [6, 530]). 3. The following notion is also meaningful (cf. Example 9.2.6): A function F : I × X → E is said to be semi-(ℬ, ω, c, (tk ))-piecewise continuous periodic if and only if there exists a sequence Fk : I × X → E in PPCω,c;(tk ) (I × X : E) which converges uniformly to the function F(⋅; ⋅) on I × B for each set B ∈ ℬ. The function F : I × X → E is said to be semi-(ℬ, ω, c)-piecewise continuous periodic if and only if F(⋅; ⋅) is semi-(ℬ, ω, c, (tk ))piecewise continuous periodic for some sequence (tk ) constructed in the same way as in Example 9.2.6; finally, we say that the function F : I × X → E is semi-(ℬ, c)-piecewise continuous periodic if and only if F(⋅; ⋅) is semi-(ℬ, ω, c)-piecewise continuous periodic for some number ω > 0. Besides the class of semi-(ℬ, ω, c)-piecewise continuous periodic functions, we can also analyze many other classes of piecewise continuous almost periodic type functions like (S, ℬ)-asymptotically (ω, ρ)-periodic functions, quasi-asymptotically (ℬ, ρ)-almost periodic (uniformly recurrent) functions, (ℬ, ρ)-slowly oscillating functions, and remotely (ℬ, ρ)-almost periodic (uniformly recurrent) functions. Details and results will be given somewhere else.

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Index (α, ζ, k0 , C)-resolvent family 434 – analytic 436 – bounded, analytic 436 – exponentially bounded, analytic 436 (a, k)-regularized (C1 , C2 )-existence and uniqueness family 4 (a, k)-regularized C1 -existence family 4 (a, k)-regularized C2 -uniqueness family 4 p-solution 433 S 1 index theory 210 abstract Cauchy problem – fractional degenerate 442 – (SCP) 166 abstract degenerate Volterra inclusion 3 abstract hypoelliptic equations on the whole axis XIV abstract ill-posed Cauchy problems 298 admissibility with respect to the class 𝒞Λ 149 admissible weight function 251 almost periodic potentials with jump discontinuities 229 almost periodic sequence 473 – Besicovitch 217 – Stepanov 217 – Weyl 217 almost periodic standing wave solution 203 almost periodic traveling wave solution 203 Arzelà variation 418 Barbalat’s lemma 389 Besicovitch-p-almost periodic sequence 494 Bloch decomposition of L2 (ℝn ) 208 Bloch solutions XIV Bochner–Fejér polynomials 206 Bubnov–Galerkin method XIV Cantor set 229 coincidence degree theory 206 comparability method by character of recurrence 365 complexification of real Banach space 400 condition – (A)∞ 150 – (MD − B)S 100 – (Cρ;1 ) 80 – (A) 150 https://doi.org/10.1515/9783111233871-014

– (A1) 49 – (A2) 49 – (A3) 49 – (A4) 49 – (A5) 49 – (A6) 49 – (A7) 49 – (A8) 49 – (A9) 49 – (A10) 49 – (AP-E) 36, 391 – (AS) 150 – (B) 155 – (B)’ 83 – (B2) 89 – (BD) 340 – (C) 198 – (C)Ω 198 – (C0) 329 – (C0-K) 354 – (C1) 69 – (C1)’ 319 – (C1-K) 354 – (C2) 69 – (C2)’ 319 – (C2-K) 354 – (C3) 69 – (C3)’ 326 – (C3-K) 358 – (C4) 69 – (C5) 69 – (C6) 69 – (CP1) 443 – (CP1)’ 445 – (CP2) 443 – (CP3) 443 – (CP4) 445 – (D) 22 – (D0) 270 – (D1) 270 – (D2) 270 – (D3) 270 – (D4) 270 – (D5) 270 – (D6) 270 – (D7) 270 – (L) 337

538 � Index

– (P) 3 – (P1) 9 – (P2) 9 – (P3) 254 – (P4) 329 – (QAAP-1) 294 – (QUC) 475 – (S) 332 – (SM1-1) 261 – (SM1-2) 261 – (WM1) 104 – (WM1-1) 273 – (WM1-2) 273 – (WM2) 105 – (ed-M1) 493 – (ew-M1) 488 – (w-M1) 488 cone 399 – generating 399 – normal 399 continuous linear mapping 1 convex polyhedral in ℝn 37 critical point theory 210 delta distribution 1 differential variational inequalities 209 distribution cosine function 457 distribution semigroup 457 Doss-p-almost periodic sequence 493 dual space 1 ellipsoidal wave equation XIV elliptic and parabolic boundary value problems in a cylinder XIV equation – Camassa–Holm XIX – damped Poisson-wave 459 – Duffing 45 – Euler–Lagrange XIX, 211 – fractional diffusion-wave 43 – fractional Poisson heat 440 – homogeneous reaction-diffusion 201 – Liénard 213 – magnetohydrodynamic 217 – nonlinear Abel-type first-order 211 – nonlinear vibrating string 216 – reaction-diffusion 203 – semilinear elliptic 18 – semilinear Poisson heat 44

– Sturm–Liouville 211 (equi-)Weyl-p-almost periodic sequence 489 equipotentially almost periodic family 473 Euler equations in ℝn 236 evolution system 165 exponential type 4 Favard’s theorem for piecewise continuous almost periodic functions 206 Fejér kernel 158 fractal dimensions of equi-almost periodic attractors 217 fractional calculus 7 fractional derivative – Caputo 7, 463 – Caputo–Dzhrbashyan 43 – proportional Caputo 429, 430 – Weyl–Liouville 7 fractional differential equations 7 function – (ω, c)-periodic 17 – (w, 𝕋)-periodic 18 – (S, 𝔻, ℬ)-asymptotically (ω, ρ)-periodic 88 – (S, 𝔻, ℬ, 𝒫ℬ )-asymptotically (ω, ρ)-periodic 289 – (S, ℬ)-asymptotically (ωj , ρj , 𝔻j )j∈ℕn -periodic 88 j

– (S, ℬ)-asymptotically (ωj , ρj , 𝔻j , 𝒫ℬ )j∈ℕn -periodic 289 – (ϕ, 𝔽, ℬ, Λ′ , ρ, 𝒫)-uniformly recurrent 327 – (ϕ, R, ℬ, 𝔽, 𝒫)-normal 325 – (ϕ, R, ℬ, 𝔽K , 𝒫K )-normal 355 – (ϕY , 𝔽, ℬ, Λ′ , ρ, 𝒫Y )-uniformly recurrent 350 – (ω, ρ)-periodic 37 – (ωj , ρj )j∈ℕn -periodic 38 – (𝔻, ℬ, ρ)-slowly oscillating 95 – (𝔻, ℬ, ρ, 𝒫ℬ )-slowly oscillating 292 – (𝔽, ℬ, Λ′ , ρ)-Hölder-α-almost periodic 374 – (𝔽, ℬ, Λ′ , ρ)-Hölder-α-almost periodic of type 1 374 – (𝔽, ℬ, Λ′ , ρ)-Hölder-α-uniformly recurrent 374 – (𝔽, ℬ, Λ′ , ρ)-Hölder-α-uniformly recurrent of type 1 374 – (𝔽, ℬ, Λ′ , ρ)-Lipschitz-almost periodic 374 – (𝔽, ℬ, Λ′ , ρ)-Lipschitz-uniformly recurrent 374 j – (ℬ, (ωj , ρj , 𝔻j , 𝒫ℬ )j∈ℕn )-slowly oscillating 292 – (ℬ, (𝔻j , ρj )j∈ℕn )-slowly oscillating 95 – (ℬ, I ′ , ρ)-uniformly recurrent 20 – (ℬ, I ′ , ρ, 𝒫)-uniformly recurrent 250 – (ℬ, ρ, (tk ))-piecewise continuous almost periodic 474

Index

– (ℬ, ρ, (tk ))-piecewise continuous uniformly recurrent 474 – (R, ℬ, 𝔽)-Hölder-α-normal 398 – (R, ℬ, 𝔽)-Hölder-α-normal of type 1 398 – (R, ℬ, 𝔽)-Lipschitz-normal 398 – (R, ℬ, 𝔽)-Lipschitz-normal of type 1 398 – (R, ℬ, 𝒫)-multi-almost periodic 238 – (RX , ℬ, 𝒫)-multi-almost periodic 239 – H-almost periodic 420 – PAP0,p,𝒫ℬ (Λ, ℬ, F, ϕ) 312 2 – PAP0,p,𝒫 (Λ, ℬ, F, ϕ, ψ) 312 ℬ – PAP0,p (Λ, ℬ, F, ϕ) 312 – [S, Ω, ℬ, 𝔻, p, ϕ, F]-asymptotically (ωj , ρj , 𝔻j )j∈ℕn -periodic 100 – Λ-uniformly semi-(ρ, 𝔽, ℬ)-Hölder-α-periodic (of type 1) 376 – Λ-uniformly semi-(ρj , 𝔽, ℬ)j∈ℕn -Hölder-α-periodic (of type 1) 376 – p-semi-anti-periodic function in variation 322 – 𝔻-asymptotically (ℬ, I ′ , ρ)-uniformly recurrent of type 1 35 – 𝔻-asymptotically Bohr (ℬ, I ′ , ρ)-almost periodic of type 1 35 – 𝔻-asymptotically Bohr (ℬ, I ′ , ρ, 𝒫)-almost periodic of type 1 294 – 𝔻-asymptotically Stepanov (Ω, p(u))-(ℬ, Λ′ )-almost periodic of type 1 86 – 𝔻-asymptotically Stepanov (Ω, p(u))-(ℬ, Λ′ )-uniformly recurrent of type 1 86 – 𝔻-quasi-asymptotically (ℬ, I ′ , ρ, 𝒫)-uniformly recurrent 295 – 𝔻-quasi-asymptotically (ℬ, I ′ , c)-uniformly recurrent 92 – 𝔻-quasi-asymptotically Bohr (ℬ, I ′ , ρ, 𝒫)-almost periodic 295 – 𝔻-quasi-asymptotically Bohr (ℬ, I ′ , c)-almost periodic 92 – 𝔻-remotely (ℬ, I ′ , ρ)-almost periodic 94 – 𝔻-remotely (ℬ, I ′ , ρ)-almost periodic of type 1 94 – 𝔻-remotely (ℬ, I ′ , ρ)-uniformly recurrent 94 – 𝔻-remotely (ℬ, I ′ , ρ)-uniformly recurrent of type 1 94 – (compactly) (R, ℬ)-multi-almost automorphic 172 – (strongly) 𝔻-asymptotically (ℬ, I ′ , ρ)-uniformly recurrent 32 – (strongly) 𝔻-asymptotically Bohr (ℬ, I ′ , ρ)-almost periodic 32

� 539

– (strongly) 𝔻-asymptotically Stepanov (Ω, p(u))-(ℬ, Λ′ , ρ)-uniformly recurrent 85 – (strongly) 𝔻-asymptotically Stepanov (Ω, p(u))-(ℬ, Λ′ , ρ)-almost periodic 85 – asymptotically (𝒳Λ , 𝒬Λ )-almost periodic 300 – Bebutov (ϕ, 𝔽K , ℬ, Λ′ , ρ, 𝒫K )-uniformly recurrent 355 – Bebutov (ϕ, 𝔽K , ℬ, Λ′ , ρ, 𝒫K )-uniformly recurrent of type 1 355 – Besicovitch (ϕ, R, ℬ, ϕ, 𝔽, 𝒫)-normal 340 – Besicovitch-(p, ϕ, F, ℬ)-bounded 111 – Besicovitch-(p, ϕ, F, ℬ, Λ′ , ρ)-continuous 111 – Besicovitch-(p, ϕ2 , ϕ3 , F2 , ℬ, Λ′ , ρ2 )-uniformly recurrent of type 1 221 – Besicovitch-(p, ϕ2 , ϕ3 , F2 , ℬ, Λ′ , ρ2 , δ)-almost periodic of type 1 218 – Besicovitch-(p, ϕi , Fj , ℬ, Λ′ , ρ1 , ρ2 )-uniformly recurrent 221 – Besicovitch-(p, ϕi , Fj , ℬ, Λ′ , ρ1 , ρ2 )-uniformly recurrent of type 2 221 – Besicovitch-(p, ϕi , Fj , ℬ, Λ′ , ρ1 , ρ2 , δ)-almost periodic 218 – Besicovitch-(p, ϕi , Fj , ℬ, Λ′ , ρ1 , ρ2 , δ)-almost periodic of type 2 218 – Besicovitch-(p, ϕi , Fj , ℬ, Λ′ , ρ2 , δ)-almost periodic of type 3 218 – Besicovitch-(𝔽, ϕ, p(u), R, PR , w)-multi-almost automorphic 182 – Besicovitch-(𝔽, ϕ, p(u), R, w)-multi-almost automorphic 182 – Besicovitch-(𝔽, ϕ, p(u), R, WR , w)-multi-almost automorphic 182 – Besicovitch-(𝔽, ϕ, p(u), R, ℬ)-multi-almost automorphic 173 – Besicovitch-(𝔽, ϕ, p(u), R, ℬ, Wℬ,R )-multi-almost automorphic 177 – Besicovitch-(𝔽, ϕ, p(u), R, ℬ, Pℬ,R )-multi-almost automorphic 177 – Besicovitch-(𝔽, ϕ, p(u), R, ℬ)-multi-almost automorphic of type 1 173, 174 – Besicovitch-(𝒫, ϕ, 𝔽, ℬ)-bounded 341 – Besicovitch-(𝒫, ϕ, 𝔽, ℬ, Λ′ , ρ)-continuous 341 – Besicovitch-(R, ℬ, ϕ, F) − Bp(⋅) -normal 147 – Besicovitch-p-almost periodic 136 – Besicovitch–Museilak–Orlicz almost periodic 136 – Besicovitch–Stepanets almost periodic 209 – Bohr (ϕ, 𝔽, ℬ, Λ′ , ρ, 𝒫)-almost periodic 327 – Bohr (ϕY , 𝔽, ℬ, Λ′ , ρ, 𝒫Y )-almost periodic 350 – Bohr (ℬ, I ′ , ρ, 𝒫)-almost periodic 250

540 � Index

– Bohr (ℬ, I ′ , ρ)-almost periodic 20 – Bohr I ′ -almost periodic in variation 419 – Doss-(p, ϕ, F, ℬ, Λ′ , ρ)-almost periodic 111, 134 – Doss-(p, ϕ, F, ℬ, Λ′ , ρ)-uniformly recurrent 111 – Doss-(𝒫, ϕ, 𝔽, ℬ, Λ′ , ρ)-almost periodic 341 – Doss-(𝒫, ϕ, 𝔽, ℬ, Λ′ , ρ)-uniformly recurrent 341 – Gamma 1 – generalized (ℬ, I1′ , I2′ , ρ1 , ρ2 , 𝒫)-uniformly recurrent 400 – generalized Bohr (ℬ, I1′ , I2′ , ρ1 , ρ2 , 𝒫)-almost periodic 400 – generalized Mittag-Leffler 431 – Laplace transformable 9 – Levitan (ϕ, 𝔽K , ℬ, Λ′ , ρ, 𝒫K )-almost periodic 355 – Levitan N-almost periodic 252 – Levitan semi-(ϕ, ρ, 𝔽K , ℬ, 𝒫K )-periodic 354 – Levitan strongly (ϕ, 𝔽K , ℬ, 𝒫K )-almost periodic 354 – Levitan semi-(ϕ, ρj , 𝔽K , ℬ, 𝒫K )j∈ℕn -periodic 354 – Lipschitz S p -almost periodic 334, 378 – Mittag-Leffler 8 – modulo periodic Poisson stable 213 – of Kamke type 212 – piecewise continuous almost periodic 474 – pre-(ℬ, ρ, (tk ))-piecewise continuous almost periodic 474 – pre-(ℬ, ρ, (tk ))-piecewise continuous uniformly recurrent 474 – semi-(ϕ, ρ, 𝔽, ℬ, 𝒫)-periodic 319 – semi-(ϕ, ρj , 𝔽, ℬ, 𝒫)j∈ℕn -periodic 319 – semi-(ϕ, cj , 𝔽, ℬ, 𝒫)j∈ℕn -periodic of type 1 330 – semi-(ϕ, cj , 𝔽, ℬ, 𝒫)j∈ℕn -periodic of type 2 330 – semi-(ϕY , ρ, 𝔽, ℬ, 𝒫Y )-periodic 350 – semi-(ϕY , ρj , 𝔽, ℬ, 𝒫Y )j∈ℕn -periodic 350 – semi-(ρ, 𝔽, ℬ)-Hölder-α-almost periodic 374 – semi-(ρ, 𝔽, ℬ)-Hölder-α-almost periodic of type 1 374 – semi-(ρ, 𝔽, ℬ)-Lipschitz-almost periodic 374 – semi-(ρ, 𝔽, ℬ)-Lipschitz-almost periodic of type 1 374 – semi-(ρj , 𝔽, ℬ)j∈ℕn -Hölder-α-almost periodic 374 – semi-(ρj , 𝔽, ℬ)j∈ℕn -Hölder-α-almost periodic of type 1 374 – semi-(ρj , 𝔽, ℬ)j∈ℕn -Lipschitz-almost periodic 374 – semi-(ρj , 𝔽, ℬ)j∈ℕn -Lipschitz-almost periodic of type 1 374 – semi-(ρj , ℬ)j∈ℕn -periodic 102 – slowly p-semi-periodic in variation 323 – Stepanov (Ω, p(u))-(ℬ, Λ′ )-uniformly recurrent 77

– Stepanov (Ω, p(u))-ℬ-almost periodic 76 – Stepanov (Ω, p(u))-ℬ-uniformly recurrent 77 – Stepanov (Ω, p(u))-bounded 75 – Stepanov (Ω, p(u))-bounded on ℬ 75 – Stepanov (ϕ, R, ℬ, ϕ, 𝔽, 𝒫)-normal 333 – Stepanov [S, Ω, ℬ, 𝔻, p, ϕ, F]-asymptotically (ω, ρ)-periodic 100 – Stepanov (compactly) (Ω, p(u))-(RX , ℬ)-multi-almost automorphic 249 – Stepanov-[Ω, ℬ, Λ′ , 𝔻, p, ϕ, F, ρ]-quasiasymptotically almost periodic 100 – Stepanov-[Ω, ℬ, Λ′ , 𝔻, p, ϕ, F, ρ]-quasiasymptotically uniformly recurrent 100 – strongly (ϕ, 𝔽, ℬ, 𝒫)-almost periodic 319 – strongly (ϕY , 𝔽, ℬ, 𝒫Y )-almost periodic 350 – strongly (𝔽, ℬ)-Hölder-α-almost periodic 374 – strongly (𝔽, ℬ)-Hölder-α-almost periodic of type 1 374 – strongly (𝔽, ℬ)-Lipschitz-almost periodic 374 – strongly (𝔽, ℬ)-Lipschitz-almost periodic of type 1 374 – strongly (R, ℬ, 𝒫)-multi-almost periodic 238 – strongly (RX , ℬ, 𝒫)-multi-almost periodic 239 – strongly Λ-uniformly (𝔽, ℬ)-Hölder-α-almost periodic (of type 1) 376 – strongly asymptotically (𝒳Λ , 𝒬Λ )-almost periodic 300 – strongly uniformly (𝔽, ℬ)-Hölder-α-almost periodic (of type 1) 376 – unbounded H-almost periodic 420 – uniformly Poisson stable 351 – uniformly recurrent 351 – uniformly semi-(ρ, 𝔽, ℬ)-Hölder-α-periodic (of type 1) 376 – uniformly semi-(ρj , 𝔽, ℬ)j∈ℕn -Hölder-α-periodic (of type 1) 376 – weakly Besicovitch-(𝔽, ϕ, p(u), R, ℬ)-multi-almost automorphic 173 – weakly Besicovitch-(𝔽, ϕ, p(u), R, ℬ)-multi-almost automorphic of type 1 173, 174 – Weyl (ϕ, R, ℬ, ϕ, 𝔽, 𝒫)-normal 338 – Wright 8 Gaussian semigroup 42, 98, 394 Green’s function 165

Index �

Hamiltonian system 216 Hardy variation 418 Hausdorff almost periodic time scale 421 Hausdorff distance 419 heteroclinic period blow-up 45 Hille differential equation XIV Hölder inequality 10 Hölder space X homogenization theory 237 hyperbolic evolution system 165 hyperfunction semigroup 412 integral generator 5 invariant differential equations on Riemann symmetric spaces of non-compact type XIV inverse Laplace transform 9 KAM theory 211 Kirchhoff formula 133 Lagrangian system 206 Lamé differential equation XIV Lebesgue spaces with variable exponent 9 linear partial differential equations of second order 169 Lipschitz continuous pulsating traveling front of speed c in the direction −e that connects p− to p+ 202 local C-regularized cosine function 457 local C-regularized semigroup 457 logarithmic potential 109 logistic equation with discrete and continuous delay 213 Luxemburg norm 10 matrix – coercive 208 measure – locally finite Borel 2 method of included intervals 365 metric space – completion 325 metrical Weyl distance 270 Metzler matrix 412 mild (α, ζ, k0 , C1 )-existence family 434 mild (α, ζ, k0 , C2 )-uniqueness family 434 multi-dimensional Bochner transform 75 multi-dimensional Nemytskii operator 29 multivalued linear operator – closed 2

541

– integer powers 2 – inverse 2 – kernel 2 – MLO 2 – product 2 – sum 2 multivariate function of bounded variation 419 neural networks 137 Newtonian potential 108, 134 non-compact Einstein manifolds 164 nonlinear anisotropic degenerate parabolic-hyperbolic equations 137 nonlinear Duffing equation with deviating argument 212 nonlinear Duffing system 212 nonlinear heat equation 214 nonlinear integrable systems XIV nonlinear systems with aftereffects XIV nonlinear wave equation 213 operator – almost Mathieu 228 – closed 1 – Ebin–Marsden’s Laplace 164 – Laplacian 42 – linear 1 – resolvent positive 411 ordered Banach space 399 part of operator 1 periodic approximations of almost periodic functions 208 Pierpont variation 419 planar fronts 202 Poisson formula 134 Poisson semigroup 42 positive strongly continuous operator family 408 positive strongly continuous semigroup 411 potential – ergodic 228 – limit-periodic 228 – periodic 228 – quasi-periodic 228 – random 228 pre-solution 3, 433, 466 problem – (DFP)ζα 442 – (P5) 470

542 � Index

– (PR) 468 proportional fractional integral 429 pulsating traveling front 201 pulsating traveling front of speed c in the direction −e that connects p− to p+ 202 quasi-linear partial differential equations of first order 169 range 1 real hyperbolic manifolds 163 removable singularity at zero 3, 259 resolvent set 1 Riemann–Liouville fractional integral 7, 447 scalar conservation laws 137 second-order elliptic operator in divergence form 208 second-order PDEs on rectangular domains 137 semilinear Hammerstein integral equation of convolution type on ℝn 100 set – admissible with respect to the class 𝒞Λ 150, 222 – Λ′δ -satisfactorily uniform 218 – syndetic 15 short-time Fourier transform 282 Sobolev–Besicovitch spaces for traces of almost periodic functions 136 solid Banach space 327 solution 3, 433, 466 – mild 469 – strong 3, 466, 470 space 𝔽,ϕ,p(u) – AAB(R,ℬ) (ℝn × X : Y ) 174 ′ – B𝔻,0,b,Ω (ℝn : X) 196 – PC([0, ∞) : X) 454 Ω,p(u),ρ – APSℬ (Λ × X : Y ) 77 – BCD(αn );c (ℝ : X) 131 – BUR(αn );T ,uc 110 – BVp (ℝ : [0, ∞)) 324 – B2 (ℝ : C r (Ω)) 207 ′ – BAP (ℝn : X) 193 ′ – B𝔻,0,b (ℝn : X) 195 ′ – B𝔻,0 (ℝn : X) 195 – C0,𝔻,ℬ (I × X : Y ) 32 – PC([0, T ] : X) 454 (ϕ,𝔽,ρ,𝒫 ,𝒫) – S ′ Ω (Λ × X : Y ) 333

–S

Ω,Λ ,ℬ (ϕ,𝔽,ρ,𝒫t,l ,𝒫)1 Ω,Λ′ ,ℬ

(Λ × X : Y ) 262

–S –S

(ϕ,𝔽,ρ,𝒫t,l ,𝒫)2

Ω,Λ′ ,ℬ (ϕ,𝔽,ρ,𝒫t,l ,𝒫) Ω,Λ′ ,ℬ Ω,p(u),ϕ,G,1

(Λ × X : Y ) 262

(Λ × X : Y ) 261

– S0,𝔻,ℬ,𝒫 (Λ × X : Y ) 302 ℬ

Ω,p(u),ϕ,G,2

– S0,𝔻,ℬ,𝒫 – – – – – –

ℬ

(Λ × X : Y ) 302

Ω,p(u),ϕ,G S0,𝔻,ℬ,𝒫 (Λ × X : Y ) 302 B Ω,p(u) S0,𝔻,ℬ (Λ × X : Y ) 85 Ω,p(u) S0,𝔻 (Λ : Y ) 85 Ω,p(u) URSℬ (Λ × X : Y ) 77 ()p(u),ϕ,𝔽)2 W ′ (Λ × X : Y ) 105 Ω,Λ ,ℬ (ϕ,𝔽,ρ,𝒫Ω ,𝒫) W ′ (Λ × X : Y ) 338 Ω,Λ ,ℬ (ϕ,𝔽,ρ,𝒫t,l ,𝒫)1

–W –W –W

Ω,Λ′ ,ℬ (ϕ,𝔽,ρ,𝒫t,l ,𝒫)2 Ω,Λ′ ,ℬ (ϕ,𝔽,ρ,𝒫t,l ,𝒫)

(Λ × X : Y ) 275 (Λ × X : Y ) 275

(Λ × X : Y ) 274

Ω,Λ′ ,ℬ (p(u),ϕ,𝔽)1 (Λ × X : Y ) 105 Ω,Λ′ ,ℬ (p(u),ϕ,𝔽) W ′ (Λ × X : Y ) 104 Ω,Λ ,ℬ [p(u),ϕ,𝔽,𝒫]1 W ′ (Λ × X : Y ) 287 Ω,Λ ,ℬ [p(u),ϕ,𝔽,𝒫]2 W ′ (Λ × X : Y ) 287 Ω,Λ ,ℬ [p(u),ϕ,𝔽,𝒫] W ′ (Λ × X : Y ) 286 Ω,Λ ,ℬ [p(u),ϕ,𝔽]2 W ′ (Λ × X : Y ) 107 Ω,Λ ,ℬ [p(u),ϕ,𝔽] W ′ (Λ × X : Y ) 106 Ω,Λ ,ℬ p(u),ϕ,𝔽,1 W0,𝔻,ℬ,𝒫 (Λ × X : Y ) 308 ℬ p(u),ϕ,𝔽,1 W ′ (Λ × X : Y ) 106 Ω,Λ ,ℬ p(u),ϕ,𝔽,2 W0,𝔻,ℬ,𝒫 (Λ × X : Y ) 308 ℬ p(u),ϕ,𝔽 W0,𝔻,ℬ,𝒫 (Λ × X : Y ) 308 ℬ n Diffm ap (ℝ ) XIX 𝒫⋅

–W – – – – – – – – – – –

– e − (𝔽, ϕ, ℬ) − W (Λ × X : Y ) 337 𝒫 – e − (𝔽, ϕ, ℬ) − Wρ ⋅ (Λ × X : Y ) 337

– e − (𝔽, ϕ, ℬ)j∈ℕn − Wρj ⋅ (Λ × X : Y ) 337 𝒫

– e − (𝔽, ℬ) − S 𝒫Ω (Λ × X : Y ) 332 𝒫 – e − (𝔽, ℬ) − Sρ Ω (Λ × X : Y ) 332

– e − (𝔽, ℬ)j∈ℕn − Sρj Ω (Λ × X : Y ) 332 𝒫

– e − (ℬ, ϕ, 𝔽) − B (Λ × X : Y ) 340 𝒫 – e − (ℬ, ϕ, 𝔽) − Bρ ⋅ (Λ × X : Y ) 340 𝒫⋅

– e − (ℬ, ϕ, 𝔽)j∈ℕn − Bρj ⋅ (Λ × X : Y ) 340 𝒫

– e − (ℬ, ϕ, F) − Bp(⋅) (Λ × X : Y ) 139 (ϕ,𝔽,ρ,𝒫Ω ,𝒫) –e−W ′ (Λ × X : Y ) 338 –e−W –e−W –e−W

Ω,Λ ,ℬ (ϕ,𝔽,ρ,𝒫t,l ,𝒫)1

Ω,Λ′ ,ℬ (ϕ,𝔽,ρ,𝒫t,l ,𝒫)2 Ω,Λ′ ,ℬ (ϕ,𝔽,ρ,𝒫t,l ,𝒫) Ω,Λ′ ,ℬ

(Λ × X : Y ) 275 (Λ × X : Y ) 275

(Λ × X : Y ) 274

Index

(p(u),ϕ,𝔽)1 (Λ × X : Y ) 105 Ω,Λ′ ,ℬ (p(u),ϕ,𝔽)2 W ′ (Λ × X : Y ) 105 Ω,Λ ,ℬ (p(u),ϕ,𝔽) W ′ (Λ × X : Y ) 104 Ω,Λ ,ℬ [p(u),ϕ,𝔽,𝒫]1 W ′ (Λ × X : Y ) 287 Ω,Λ ,ℬ [p(u),ϕ,𝔽,𝒫]2 W ′ (Λ × X : Y ) 287 Ω,Λ ,ℬ [p(u),ϕ,𝔽,𝒫] W ′ (Λ × X : Y ) 286 Ω,Λ ,ℬ [p(u),ϕ,𝔽]1 W ′ (Λ × X : Y ) 106 Ω,Λ ,ℬ [p(u),ϕ,𝔽]2 W ′ (Λ × X : Y ) 106 Ω,Λ ,ℬ [p(u),ϕ,𝔽] W ′ (Λ × X : Y ) 106 Ω,Λ ,ℬ p(u),ϕ,𝔽,1 W0,𝔻,ℬ,𝒫 (Λ × X : Y ) 308 ℬ p(u),ϕ,𝔽,2 W0,𝔻,ℬ,𝒫 (Λ × X : Y ) 308 ℬ p(u),ϕ,𝔽 W0,𝔻,ℬ,𝒫 (Λ × X : Y ) 307 ℬ p

–e−W –e− –e− –e− –e− –e− –e− –e− –e− –e− –e− –e−

– e − ℬ − W (Λ × X : Y ) 337 𝔽,ϕ,p(u) – w − AAB(R,ℬ) (ℝn × X : Y ) 174 – Φω,ρ 49 – 𝒟L′1 (ℝn : X) 190 – Besicovitch–Orlicz 137 – Schwartz XXIX – separable metric 2 – uniformly convex 483 spectrum 1 Stepanov distance 75 subgenerator 4 supremum formula (S)𝔻 189 symmetric conservative systems 46 symmetric heteroclinic cycle 46 symmetric singular systems 45 systems of inviscid or viscous conservation laws 204 theorem – Bohl–Bohr–Amerio 483 – dominated convergence 6 – Fubini 6 – Hausdorff–Young 137 – Scauder 445

� 543

– Schauder 9 – Titchmarsh convolution 434 traveling fronts 201 – pulsating 201 ultra-differentiable quasi-periodic cocycle 229 ultradistribution cosine function 457 ultradistribution semigroup 457 uniformly almost periodic set of sequences 473 vector-valued – Laplace transform 435 – Sobolev space 7 vector-valued distribution – (ω, ρ)-periodic 227 – (ωj , ρj )j∈ℕn -periodic 227 – ρ-almost periodic 190 – ρ-uniformly recurrent 190 – 𝔻-asymptotically (I ′ , ρ)-almost periodic distribution 197 – 𝔻-asymptotically (I ′ , ρ)-almost periodic distribution of type 1 197 – 𝔻-asymptotically (I ′ , ρ)-uniformly recurrent distribution 197 – 𝔻-asymptotically (I ′ , ρ)-uniformly recurrent distribution of type 1 197 – 𝔻-asymptotically (𝒟Ω , I ′ , ρ)-almost periodic distribution 197 – 𝔻-asymptotically (𝒟Ω , I ′ , ρ)-almost periodic distribution of type 1 197 – 𝔻-asymptotically (𝒟Ω , I ′ , ρ)-uniformly recurrent distribution 197 – 𝔻-asymptotically (𝒟Ω , I ′ , ρ)-uniformly recurrent distribution of type 1 197 – almost periodic 193 – bounded 190 Vitali variation 418 Wexler sequence 474

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