Theory of Fractional Evolution Equations 9783110769272, 9783110769180

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Yong Zhou, Bashir Ahmad, and Ahmed Alsaedi Theory of Fractional Evolution Equations

Fractional Calculus in Applied Sciences and Engineering

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Editor-in Chief Changpin Li Editorial Board Virginia Kiryakova Francesco Mainardi Dragan Spasic Bruce Ian Henry YangQuan Chen

Volume 11

Yong Zhou, Bashir Ahmad, and Ahmed Alsaedi

Theory of Fractional Evolution Equations |

Mathematics Subject Classification 2020 26A33, 34A08, 35R11 Authors Prof. Dr. Yong Zhou Faculty of Information Technology Macau University of Science and Technology Macau 999078 P.R. China Faculty of Mathematics and Computational Science Xiangtan University Hunan 411105 P.R. China [email protected], [email protected]

Prof. Dr. Bashir Ahmad NAAM Research Group Faculty of Science King Abdulaziz University Jeddah 21589 Saudi Arabia [email protected] Prof. Dr. Ahmed Alsaedi NAAM Research Group Faculty of Science King Abdulaziz University Jeddah 21589 Saudi Arabia [email protected]

ISBN 978-3-11-076918-0 e-ISBN (PDF) 978-3-11-076927-2 e-ISBN (EPUB) 978-3-11-076936-4 ISSN 2509-7210 Library of Congress Control Number: 2021950459 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. © 2022 Walter de Gruyter GmbH, Berlin/Boston Cover image: naddi/iStock/thinkstock Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

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To our beloved families

Contents Preface | XI Introduction | XIII 1 1.1 1.2 1.2.1 1.2.2 1.2.3 1.3 1.3.1 1.3.2 1.4 1.4.1 1.4.2 1.4.3 1.4.4 1.5 1.5.1 1.5.2

Preliminaries | 1 Notations, concepts and lemmas | 1 Fractional calculus | 2 Definitions | 2 Properties | 9 Special functions | 12 Semigroups | 13 C0 -semigroup | 13 Almost sectorial operators | 14 Results from analysis | 18 Laplace and Fourier transforms | 18 Weak compactness of sets and operators | 19 Measure of noncompactness | 20 Fixed-point theorems | 23 Stochastic process | 25 Random variables | 25 Stochastic calculus | 27

2 2.1 2.1.1 2.1.2 2.1.3 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5 2.4

Fractional evolution equations of order α ∈ (0, 1) | 29 Initial value problems with Hilfer derivative | 29 Introduction | 29 Definition of mild solutions | 30 Existence of mild solutions | 37 Initial value problems with nondense domain | 44 Introduction | 44 Integral solution to nonhomogeneous Cauchy problem | 45 Integral solution to nonlinear Cauchy problem | 53 An example | 59 Terminal value problems with the Liouville–Weyl derivative | 59 Introduction | 59 Definition of mild solutions | 60 Lemmas | 62 Compact semigroup case | 70 Noncompact semigroup case | 72 Attractivity of evolution equations with almost sectorial operators | 76

VIII | Contents 2.4.1 2.4.2 2.4.3 2.4.4 2.4.5 2.4.6

Introduction | 76 Preliminaries | 76 Auxiliary lemmas | 79 Compact semigroup case | 84 Noncompact semigroup case | 86 Example | 86

3 3.1 3.1.1 3.1.2 3.1.3 3.1.4 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5 3.2.6

Fractional control systems of order α ∈ (0, 1) | 89 Controllability | 89 Introduction | 89 Existence of mild solutions | 90 Controllability results | 97 Examples | 103 Optimal control | 106 Introduction | 106 Preliminaries | 108 Convergence of mild solutions | 110 Convergence of optimal control and cost functional | 121 Error estimates of optimal control and cost functional | 123 Applications | 126

4 4.1 4.1.1 4.1.2 4.1.3 4.1.4 4.1.5 4.2 4.2.1 4.2.2 4.2.3 4.2.4

Fractional evolution equations and inclusions of order α ∈ (1, 2) | 129 Fractional control systems | 129 Introduction | 129 Preliminaries | 131 Definition of mild solutions | 132 Controllability results | 140 An example | 147 Nonlocal evolution inclusions | 150 Introduction | 150 Preliminaries | 151 Existence results | 155 Control problem | 164

5 5.1 5.1.1 5.1.2 5.1.3 5.2 5.2.1 5.2.2

Fractional neutral evolution equations and inclusions | 169 Nonlocal Cauchy problem | 169 Introduction | 169 Existence and uniqueness | 170 An example | 189 Topological properties of solution sets | 192 Introduction | 192 Preliminaries | 193

Contents | IX

5.2.3 5.2.4 5.2.5

Statement of Problem | 194 Topological structure of solution sets | 199 An Example | 212

6 6.1 6.1.1 6.1.2 6.1.3 6.1.4 6.1.5 6.2 6.2.1 6.2.2 6.2.3 6.2.4

Fractional evolution equations on whole real axis | 215 Periodic solutions and S-asymptotically periodic solutions | 215 Introduction | 215 Preliminaries | 216 Linear equations | 222 Nonlinear equations | 228 Examples | 233 Asymptotically almost periodic solutions | 234 Introduction | 234 Preliminaries | 235 Existence | 237 Applications | 249

7 7.1 7.1.1 7.1.2 7.1.3 7.1.4 7.1.5 7.1.6 7.2 7.2.1 7.2.2 7.2.3 7.2.4 7.3 7.3.1 7.3.2 7.3.3 7.3.4

Discrete time fractional evolution equations | 251 Cauchy problem | 251 Introduction | 251 Preliminaries | 254 Resolvent sequences | 257 Inhomogeneous Cauchy problem | 261 Semilinear Cauchy problem | 263 Examples | 268 Stability | 272 Introduction | 272 Preliminaries | 273 Existence of stable solutions | 275 Ulam–Hyers–Rassias stability | 282 Asymptotic almost periodicity | 285 Introduction | 285 Preliminaries | 287 Existence | 296 Applications | 311

Bibliography | 313 Index | 323

Preface Fractional evolution equations provide a unifying framework to investigate wellposedness of complex systems of various types describing the time evolution of concrete systems (like time-fractional diffusion equations and wave equations). This monograph is devoted to the study of the existence, attractivity, stability, periodicity and control for fractional evolution equations. The content of this monograph is based on the research work carried out by the authors and other experts on the topic during the past 5 years. The book contains up-to-date and comprehensive information on the topic. It is useful for researchers working in the areas of pure and applied mathematics, and related disciplines. It may also be used as an excellent source for graduate level advanced courses on evolution equations. We would like to thank Professors M. Fečkan, J.R. Graef, J. Henderson, M. Kirane, V. Kiryakova, C.P. Li, F. Liu, S.K. Ntouyas and J.J. Trujillo for their support. We also express our sincere gratitude to our collaborators and doctoral students for their help. We acknowledge with gratitude the support of the Fundo para o Desenvolvimento das Ciências e da Tecnologia (FDCT) of Macau under Grant 0074/2019/A2. Macau, China and Jeddah, Saudi Arabia, January 2022

https://doi.org/10.1515/9783110769272-201

Yong Zhou Bashir Ahmad Ahmed Alsaedi

Introduction Fractional calculus has attracted the attention of mathematicians and engineers for a long time. The concept of fractional (more precisely, noninteger) differentiation dates back to a famous correspondence between L’Hospital and Leibniz in 1695. Many mathematicians contributed to the development of this branch of mathematical analysis with the pioneer work owing to Euler, Laplace, Abel, Liouville and Riemann. Factional calculus was mainly regarded as a purely theoretical topic for centuries, with a little connection to the practical problems of physics and engineering. During the past three decades, this subject served as an effective modeling methodology for researchers. The tools of fractional calculus are found to be of great interest in improving the mathematical modeling of several phenomena occurring in engineering and sciences such as physics, mechanics, electricity, chemistry, biology, economics, etc. Now the research on fractional differential equations has become an active and popular area of investigation around the world. In the recent years, there has been a significant development in ordinary and partial differential equations involving fractional derivatives; see the monographs of Miller et al. [163], Podlubny [174], Hilfer [124], Kilbas et al. [133], Diethelm [84], Zhou [225, 226] and the references therein. Fractional evolution equations provide a unifying framework to investigate well-posedness of complex systems of various types describing the time evolution of concrete systems. A strong motivation for investigating this class of equations owes to the fact that differential models involving fractional derivative operators provide an excellent instrument for describing memory and hereditary properties of the associated physical phenomena. The fractional-order models of real systems are considered to be more adequate than their classical counterparts, since the description of some systems is more accurate when the fractional derivative is used. The advantages of fractional derivative becomes evident in modeling mechanical and electrical properties of real materials, description of rheological properties of rocks and various other applications. Such models are interesting not only for engineers and physicists but also for pure mathematicians. This monograph is devoted to the theoretical analysis of fractional evolution equations. In particular, we are interested in the existence, attractivity, stability, periodicity and control theory for fractional evolution equations. The development of such a basic theory provides a useful platform for further research concerning the dynamics, numerical analysis and applications of fractional differential equations. This monograph is arranged and organized as follows: In order to make the book self-contained, we devote the first chapter to a description of general information on fractional calculus, special functions, semigroups, Laplace and Fourier transforms, measure of noncompactness, fixed-point theorems and stochastic process. https://doi.org/10.1515/9783110769272-202

XIV | Introduction The second chapter deals with existences of mild solutions, integral solutions and globally attractive solutions for fractional evolution equations of order α ∈ (0, 1). In Section 2.1, we study fractional evolution equations with the Hilfer fractional derivative, which is a generalization of both Riemann–Liuoville and Caputo fractional derivatives. By the noncompact measure method, we obtain some sufficient conditions to ensure the existence of mild solutions. In Section 2.2, we discuss the existence of integral solutions for two classes of fractional-order evolution equations with nondensely defined linear operators. First, we consider the nonhomogeneous fractional-order evolution equation and obtain its integral solution by Laplace transform and probability density function. Subsequently, based on the form of integral solution for nonhomogeneous fractional-order evolution equations, we investigate the existence of integral solution for nonlinear fractional-order evolution equations by the method of noncompact measure. In Section 2.3, the terminal value problem for a class of nonlinear fractional evolution equations with Liouville–Weyl derivative is considered. By using Fourier transform, such a problem is converted into a singular integral equation on an infinite interval. Some sufficient conditions are obtained to ensure the existence of mild solutions when the semigroup is compact as well as noncompact. In Section 2.4, we discuss the question of the attractivity of solutions for fractional evolution equations with almost sectorial operators. We establish sufficient conditions for the existence of globally attractive solutions for the Cauchy problems when the semigroup is compact as well as noncompact. Our results essentially reveal certain characteristics of solutions for fractional evolution equations, which are not possessed by integer-order evolution equations. The third chapter is devoted to the study of fractional control systems of order α ∈ (0, 1). In Section 3.1, we obtain the existence of mild solutions and controllability for fractional evolution inclusions in Banach spaces, avoiding hypotheses of compactness on the semigroup generated by the linear part and any conditions on the multivalued nonlinearity expressed in terms of measures of noncompactness. Finally, two examples are given to illustrate the theoretical results. In Section 3.2, we investigate a class of fractional evolution equations and optimal controls in Hilbert spaces. The strategy of this section is establishing low dimensional approximations for this type of equations by using approximation methods. We derive three kinds of convergence results of mild solutions under appropriate assumptions. Then the convergence result holds for cost functionals as well. Later, error estimates of cost functionals and optimal controls are obtained. Finally, the proposed procedure is illustrated by an example. The fourth chapter is mainly concerned with the investigation of fractional evolution equations and inclusions with order α ∈ (1, 2). First, we obtain some interesting results for mild solutions and controllability to fractional evolution systems with order α ∈ (1, 2) in Banach spaces. By using Laplace transform and a Wright-type function, we deduce a new representation of solution operators and introduce a new concept of mild solutions for the objective equations. Then we proceed to establish a new compact result for the solution operators when the sine family is compact. The control-

Introduction

| XV

lability of mild solutions is also discussed. Second, we study the existence of mild solutions and compactness for the set of mild solutions to a nonlocal problem of fractional evolution inclusions of order α ∈ (1, 2). The main tools of our study include fractional calculus, multivalued analysis, cosine family, measure of noncompactness and a fixed-point theorem. Finally, we demonstrate the application of the obtained results to a control problem. The fifth chapter is devoted to the study of neutral evolution equations and inclusions. In Section 5.1, by using the fractional power of operators and some fixed-point theorems, we discuss a class of fractional neutral evolution equations with nonlocal conditions. We give various criteria on the existence and uniqueness of mild solutions and an example to illustrate the applications of the abstract results. In Section 5.2, we investigate the topological structure for the solution set of Caputo-type neutral fractional stochastic evolution inclusions in Hilbert spaces. We introduce the concept of mild solutions for factional neutral stochastic inclusions and show that the solution set is nonempty compact and Rδ -set, which means that the solution set may not be a singleton, but from the point of view of algebraic topology, it is equivalent to a point in the sense that it has the same homology group as a one-point space. Finally, we illustrate the obtained theory with the aid of an example. In the sixth chapter, we investigate fractional evolution equations with the Liouville–Weyl fractional derivative on whole real axis. In Section 6.1, we give appropriate definitions of mild solutions with the aid of Fourier transform. Then we accurately estimate the spectral radius of resolvent operator and obtain the existence and uniqueness of periodic solutions, S-asymptotically periodic solutions and other types of bounded solutions. In Section 6.2, combining the fixed-point theorem due to Krasnoselskii and a decomposition technique, we give some sufficient conditions to ensure the existence of asymptotically almost periodic mild solutions. An example is also presented as an application to illustrate the feasibility of the abstract result. The seventh chapter deals with discrete-time fractional evolution equations involving the Riemann–Liouville-like difference operator. Based on the relationship between C0 -semigroups and a distinguished class of sequences of operators, we discuss the structure of the solutions for the inhomogenous Cauchy problem of abstract fractional difference equations. The criteria for the existence and uniqueness of solutions for the semilinear Cauchy problem are established. Further, we show the existence of stable solutions for the nonlinear Cauchy problem by means of a fixed-point technique and the compact method. Moreover, we establish the Ulam–Hyers–Rassias stability of the proposed problem. Two examples are presented to explain the main results. In summary, this monograph serves as an excellent source of knowledge for the one who is interested in the theory of fractional differential equations and their applications in science and engineering.

1 Preliminaries 1.1 Notations, concepts and lemmas As usual, ℕ denotes the set of positive integer numbers and ℕ0 the set of nonnegative integer numbers. The set ℝ denotes the real numbers, ℝ+ denotes the set of nonnegative reals and ℝ+ denotes the set of positive reals. Let ℂ be the set of complex numbers. We recall that a vector space X equipped with a norm | ⋅ | is called a normed vector space. A subset E of a normed vector space X is said to be bounded if there exists a number K such that |x| ≤ K for all x ∈ E. A subset E of a normed vector space X is called convex if for any x, y ∈ E, ax + (1 − a)y ∈ E for all a ∈ [0, 1]. A sequence {xn } in a normed vector space X is said to converge to the vector x in X if and only if the sequence {|xn − x|} converges to zero as n → ∞. A sequence {xn } in a normed vector space X is called a Cauchy sequence if for every ε > 0 there exists an N = N(ε) such that for all n, m ≥ N(ε), |xn − xm | < ε. Clearly, a convergent sequence is also a Cauchy sequence, but the converse may not be true. A space X where every Cauchy sequence of elements of X converges to an element of X is called a complete space. A complete normed vector space is said to be a Banach space. Let E be a subset of a Banach space X. A point x ∈ X is said to be a limit point of E if there exists a sequence of vectors in E, which converges to x. We say a subset E is closed if E contains all of its limit points. The union of E and its limit points is called the closure of E and will be denoted by E.̄ Let E, F be normed vector spaces, and E be a subset of X. An operator T : E → F is continuous at a point x ∈ E if and only if for any ε > 0 there is a δ > 0 such that |T x − T y| < ε for all y ∈ E with |x − y| < δ. Further, T is continuous on E, or simply continuous, if it is continuous at all points of E. We say that a subset E of a Banach space X is compact if every sequence of vectors in E contains a subsequence which converges to a vector in E. We say that E is relatively compact in X if every sequence of vectors in E contains a subsequence, which converges to a vector in X, i. e., E is relatively compact in X if Ē is compact. Let J = [a, b] (−∞ < a < b < ∞) be a finite interval of ℝ. We assume that X is a Banach space with the norm | ⋅ |. Denote C(J, X) be the Banach space of all continuous functions from J into X with the norm 󵄨 󵄨 ‖x‖ = sup󵄨󵄨󵄨x(t)󵄨󵄨󵄨, t∈J

where x ∈ C(J, X). The space C n (J, X) (n ∈ ℕ0 ) denotes the set of mappings, which have continuous derivatives up to order n on J, AC(J, X) is the space of functions which are absolutely continuous on J and AC n (J, X) (n ∈ ℕ0 ) is the space of functions f such that f ∈ C n−1 (J, X) and f (n−1) ∈ AC(J, X). In particular, AC 1 (J, X) = AC(J, X). https://doi.org/10.1515/9783110769272-001

2 | 1 Preliminaries Let 1 ≤ p ≤ ∞. The space Lp (J, X) denotes the Banach space of all measurable functions f : J → X, equipped with the norm 1

(∫ |f (t)|p dt) p , ‖f ‖Lp = { J infμ(J)=0 {supt∈J\J ̄ |f (t)|}, ̄

1 ≤ p < ∞, p = ∞.

In particular, L1 (J, X) is the Banach space of measurable functions f : J → X with the norm 󵄨 󵄨 ‖f ‖L = ∫󵄨󵄨󵄨f (t)󵄨󵄨󵄨dt, J

and L∞ (J, X) is the Banach space of measurable functions f : J → X which are bounded, equipped with the norm 󵄨 󵄨 ‖f ‖L∞ = inf{c > 0 | 󵄨󵄨󵄨f (t)󵄨󵄨󵄨 ≤ c, a. e. t ∈ J}. Lemma 1.1 (Hölder inequality). Assume that p, q ≥ 1, and g ∈ Lq (J, X), then for 1 ≤ p ≤ ∞, fg ∈ L1 (J, X) and

1 p

+

1 q

= 1. If f ∈ Lp (J, X),

‖fg‖L ≤ ‖f ‖Lp ‖g‖Lq . Lemma 1.2 (Lebesgue dominated convergence theorem). Let E be a measurable set and let {fn } be a sequence of measurable functions such that limn→∞ fn (x) = f (x) a. e. in E, and for every n ∈ ℕ, |fn (x)| ≤ g(x) a. e. in E, where g is integrable on E. Then lim ∫ fn (x)dx = ∫ f (x)dx.

n→∞

E

E

Finally, we state the Bochner theorem. Lemma 1.3 (Bochner theorem). A measurable function f : (a, b) → X is Bochner integrable if |f | is Lebesgue integrable.

1.2 Fractional calculus 1.2.1 Definitions The gamma function Γ(z) is defined by ∞

Γ(z) = ∫ t z−1 e−t dt 0

(Re(z) > 0),

1.2 Fractional calculus | 3

where t z−1 = e(z−1) log(t) . This integral is convergent for all complex z ∈ ℂ (Re(z) > 0). For this function, the reduction formula Γ(z + 1) = zΓ(z)

(Re(z) > 0)

holds. In particular, if z = n ∈ ℕ0 , then Γ(n + 1) = n!

(n ∈ ℕ0 )

with (as usual) 0! = 1. Let us consider some of the starting points for a discussion of fractional calculus. One development begins with a generalization of repeated integration. Thus if f is locally integrable on (c, ∞), then the n-fold iterated integral is given by t −n c Dt f (t)

sn−1

s1

= ∫ ds1 ∫ ds2 ⋅ ⋅ ⋅ ∫ f (sn )dsn c

=

c

t

c

1 ∫(t − s)n−1 f (s)ds (n − 1)! c

for almost all t with −∞ ≤ c < t < ∞ and n ∈ ℕ. Writing (n − 1)! = Γ(n), an immediate generalization is the integral of f of fractional order α > 0, t

−α c Dt f (t)

1 = ∫(t − s)α−1 f (s)ds (left hand) Γ(α) c

and similarly for −∞ < t < d ≤ ∞, d

−α t Dd f (t)

1 = ∫(s − t)α−1 f (s)ds (right hand) Γ(α) t

both being defined for suitable f . A number of definitions for the fractional derivative have emerged over the years. We refer the reader to Diethelm [84], Hilfer [124], Kilbas, Srivastava and Trujillo [133], Podlubny [174] and Samko, Kilbas and Marichev [179]. In this book, we restrict our attention to the use of the Riemann–Liouville, Caputo fractional derivatives and Hilfer fractional derivatives. In this section, we introduce some basic definitions and properties of the fractional integrals and fractional derivatives, which are used further in this book. The materials in this section are taken from Kilbas, Srivastava and Trujillo [133].

4 | 1 Preliminaries Definition 1.1 (Left and right Riemann–Liouville fractional integrals). Let J = [a, b] (−∞ < a < b < ∞) be a finite interval of ℝ. The left and right Riemann–Liouville −α + fractional integrals a D−α t f (t) and t Db f (t) of order α ∈ ℝ , are defined by t

−α a Dt f (t) =

1 ∫(t − s)α−1 f (s)ds, Γ(α) a

t > a, α > 0

(1.1)

t < b, α > 0,

(1.2)

and b

−α t Db f (t)

1 = ∫(s − t)α−1 f (s)ds, Γ(α) t

respectively, provided the right-hand sides are pointwise defined on [a, b]. When α = n ∈ ℕ, the definitions (1.1) and (1.2) coincide with the nth integrals of the form t

−n a Dt f (t)

1 = ∫(t − s)n−1 f (s)ds (n − 1)!

−n t Db f (t)

1 = ∫(s − t)n−1 f (s)ds. (n − 1)!

a

and b

t

Definition 1.2 (Left and right Riemann–Liouville fractional derivatives). The left and right Riemann–Liouville fractional derivatives a Dαt f (t) and t Dαb f (t) of order α ∈ ℝ+ are defined by α a Dt f (t)

=

dn −(n−α) D f (t) dt n a t t

dn 1 = (∫(t − s)n−α−1 f (s)ds), Γ(n − α) dt n a

t>a

and α n t Db f (t) = (−1)

dn −(n−α) D f (t) dt n t b

b

1 dn = (−1)n n (∫(s − t)n−α−1 f (s)ds), Γ(n − α) dt t

t < b,

1.2 Fractional calculus | 5

respectively, where n = [α] + 1, [α] means the integer part of α. In particular, when α = n ∈ ℕ0 , then 0 a Dt f (t) n a Dt f (t)

= t D0b f (t) = f (t), = f (n) (t)

n t Db f (t)

and

= (−1)n f (n) (t),

where f (n) (t) is the usual derivative of f (t) of order n. If 0 < α < 1, then α a Dt f (t)

t

d 1 (∫(t − s)−α f (s)ds), = Γ(1 − α) dt a

t>a

and α t Db f (t)

b

d 1 (∫(s − t)−α f (s)ds), =− Γ(1 − α) dt t

t < b.

Remark 1.1. If f ∈ C([a, b], ℝN ), it is obvious that Riemann–Liouville fractional integral of order α > 0 exists on [a, b]. On the other hand, following Lemma 2.2 in [133], we know that the Riemann–Liouville fractional derivative of order α ∈ [n − 1, n) exists almost everywhere on [a, b] if f ∈ AC n ([a, b], ℝN ). The left and right Caputo fractional derivatives are defined via above Riemann– Liouville fractional derivatives. Definition 1.3 (Left and right Liouville-Caputo fractional derivatives). The left and right Caputo fractional derivatives CaDαt f (t) and CtDαb f (t) of order α ∈ ℝ+ are defined by n−1

f (k) (a) (t − a)k ) k! k=0

C α aDt f (t)

= a Dαt (f (t) − ∑

C α t Db f (t)

= t Dαb (f (t) − ∑

and n−1

f (k) (b) (b − t)k ), k! k=0

respectively, where n = [α] + 1,

for α ∈ ̸ ℕ0 ;

n = α,

for α ∈ ℕ0 .

In particular, when 0 < α < 1, then C α aDt f (t)

= a Dαt (f (t) − f (a))

(1.3)

6 | 1 Preliminaries and C α t Db f (t)

= t Dαb (f (t) − f (b)).

The Riemann–Liouville fractional derivative and the Caputo fractional derivative are connected with each other by the following relations. Proposition 1.1. (i) If α ∈ ̸ ℕ0 and f (t) is a function for which the Caputo fractional derivatives CaDαt f (t) and CtDαb f (t) of order α ∈ ℝ+ exist together with the Riemann–Liouville fractional derivatives a Dαt f (t) and t Dαb f (t), then n−1

f (k) (a) (t − a)k−α Γ(k − α + 1) k=0

C α aDt f (t)

= a Dαt f (t) − ∑

C α t Db f (t)

= t Dαb f (t) − ∑

and n−1

f (k) (b) (b − t)k−α , Γ(k − α + 1) k=0

where n = [α] + 1. In particular, when 0 < α < 1, we have C α aDt f (t)

= a Dαt f (t) −

f (a) (t − a)−α Γ(1 − α)

C α t Db f (t)

= t Dαb f (t) −

f (b) (b − t)−α . Γ(1 − α)

and

(ii) If α = n ∈ ℕ0 and the usual derivative f (n) (t) of order n exists, then CaDnt f (t) and C n t Db f (t) are represented by C n aDt f (t)

C n t Db f (t)

= f (n) (t) and

= (−1)n f (n) (t).

(1.4)

Proposition 1.2. Let α ∈ ℝ+ and let n be given by (1.3). If f ∈ AC n ([a, b], ℝN ), then the Caputo fractional derivatives CaDαt f (t) and CtDαb f (t) exist almost everywhere on [a, b]. (i) If α ∈ ̸ ℕ0 , CaDαt f (t) and CtDαb f (t) are represented by C α aDt f (t)

t

1 = (∫(t − s)n−α−1 f (n) (s)ds) Γ(n − α) a

1.2 Fractional calculus | 7

and C α t Db f (t)

b

(−1)n (∫(s − t)n−α−1 f (n) (s)ds), Γ(n − α)

=

t

respectively, where n = [α] + 1. In particular, when 0 < α < 1 and f ∈ AC([a, b], ℝN ), C α aDt f (t)

t

=

1 (∫(t − s)−α f ′ (s)ds) Γ(1 − α)

(1.5)

a

and C α t Db f (t)

b

1 (∫(s − t)−α f ′ (s)ds). =− Γ(1 − α)

(1.6)

t

(ii) If α = n ∈ ℕ0 , then CaDαt f (t) and CtDαb f (t) are represented by (1.4). In particular, C 0 aDt f (t)

= CtD0b f (t) = f (t).

Remark 1.2. If f is an abstract function with values in Banach space X, then integrals which appear in the above definitions are taken in Bochner’s sense. μ,ν

Definition 1.4 (Hilfer fractional derivative). The Hilfer fractional derivative a Dt f (t) of order n − 1 < μ < n and 0 ≤ ν ≤ 1 is defined by μ,ν a Dt f (t)

= a Dt

−ν(n−μ)

dn −(1−ν)(n−μ) D f (t), dt n a t

provided the right-hand side is pointwise defined on [a, b]. Remark 1.3. (i) When ν = 0 and n − 1 < μ < n, the Hilfer fractional derivative corresponds to the classical Riemann-Liouville fractional derivative: μ,0 a Dt f (t)

=

dn −(n−μ) μ D f (t) = a Dt f (t). dt n a t

(ii) When ν = 1, n − 1 < μ < n, the Hilfer fractional derivative corresponds to the classical Caputo fractional derivative: μ,1

a Dt f (t) = a Dt

−(n−μ)

dn μ f (t) = CaDt f (t). dt n

The fractional integrals and derivatives, defined on a finite interval [a, b] of ℝ, are naturally extended to whole axis ℝ.

8 | 1 Preliminaries Definition 1.5 (Left and right Liouville fractional integrals on the real axis). The left −α and right Liouville–Weyl fractional integrals −∞ D−α t f (t) and t D+∞ f (t) of order α > 0 on the whole axis ℝ are defined by t

−α −∞ Dt f (t)

1 = ∫ (t − s)α−1 f (s)ds Γ(α)

(1.7)

−∞

and ∞

−α t D+∞ f (t) =

1 ∫ (s − t)α−1 f (s)ds, Γ(α) t

respectively, where t ∈ ℝ and α > 0. Definition 1.6 (Left and right Liouville–Weyl fractional derivatives on the real axis). The left and right Liouville–Weyl fractional derivatives −∞ Dαt f (t) and t Dα+∞ f (t) of order α on the whole axis ℝ are defined by α −∞ Dt f (t)

=

dn ( D−(n−α) f (t)) dt n −∞ t

=

1 dn ( ∫ (t − s)n−α−1 f (s)ds) Γ(n − α) dt n

t

−∞

and α t D+∞ f (t)

= (−1)n

dn ( D−(n−α) f (t)) dt n t +∞ ∞

1 dn = (−1)n n ( ∫ (s − t)n−α−1 f (s)ds), Γ(n − α) dt t

respectively, where n = [α] + 1, α ≥ 0 and t ∈ ℝ. In particular, when α = n ∈ ℕ0 , then 0 −∞ Dt f (t) n −∞ Dt f (t)

= t D0+∞ f (t) = f (t), = f (n) (t)

and

n t D+∞ f (t)

= (−1)n f (n) (t),

where f (n) (t) is the usual derivative of f (t) of order n. If 0 < α < 1 and t ∈ ℝ, then α −∞ Dt f (t)

t

1 d = ( ∫ (t − s)−α f (s)ds) Γ(1 − α) dt −∞

∞

=

α f (t) − f (t − s) ds ∫ Γ(1 − α) sα+1 0

1.2 Fractional calculus | 9

and α t D+∞ f (t) = −

∞

d 1 ( ∫ (s − t)−α f (s)ds) Γ(1 − α) dt t

∞

=

f (t) − f (t + s) α ds. ∫ Γ(1 − α) sα+1 0

Formulas (1.5) and (1.6) can be used for the definition of the Caputo fractional derivatives on the whole axis ℝ. Definition 1.7 (Left and right Caputo fractional derivatives on the real axis). The left C α and right Caputo fractional derivatives −∞ Dt f (t) and Ct Dα+∞ f (t) of order α (with α > 0 and α ∈ ̸ ℕ) on the whole axis ℝ are defined by C α −∞ Dt f (t)

t

=

1 ∫ (t − s)n−α−1 f (n) (s)ds Γ(n − α)

(1.8)

−∞

and C α t D+∞ f (t)

∞

(−1)n = ∫ (s − t)n−α−1 f (n) (s)ds, Γ(n − α)

(1.9)

t

respectively. When 0 < α < 1, the relations (1.8) and (1.9) take the following forms: C α −∞ Dt f (t)

t

=

1 ∫ (t − s)−α f ′ (s)ds Γ(1 − α) −∞

and C α t D+∞ f (t)

∞

1 =− ∫ (s − t)−α f ′ (s)ds. Γ(1 − α) t

1.2.2 Properties We present here some properties of the fractional integral and fractional derivative operators that will be useful throughout this book.

10 | 1 Preliminaries Proposition 1.3. If α ≥ 0 and β > 0, then −α a Dt (t α a Dt (t

Γ(β) (t − a)β+α−1 Γ(β + α) Γ(β) = (t − a)β−α−1 Γ(β − α)

− a)β−1 =

(α > 0),

− a)β−1

(α ≥ 0)

and −α t Db (b α t Db (b

Γ(β) (b − t)β+α−1 Γ(β + α) Γ(β) = (b − t)β−α−1 Γ(β − α)

− t)β−1 =

(α > 0),

− t)β−1

(α ≥ 0).

In particular, if β = 1 and α ≥ 0, then the Riemann–Liouville fractional derivatives of a constant are, in general, not equal to zero: α a Dt 1 =

(t − a)−α , Γ(1 − α)

α t Db 1 =

(b − t)−α . Γ(1 − α)

On the other hand, for j = 1, 2, . . . , [α] + 1, α a Dt (t

− a)α−j = 0,

α t Db (b

− t)α−j = 0.

−α The semigroup property of the fractional integral operators a D−α t and t Db are given by the following result.

Proposition 1.4. If α > 0 and β > 0, then the equations −β −α a Dt (a Dt f (t))

= a Dt

−α−β

f (t)

and

−β −α t Db (t Db f (t))

= t Db

−α−β

f (t)

(1.10)

are satisfied at almost every point t ∈ [a, b] for f ∈ Lp ([a, b], ℝN ) (1 ≤ p < ∞). If α+β > 1, then the relations in (1.10) hold at any point of [a, b]. Proposition 1.5. (i) If α > 0 and f ∈ Lp ([a, b], ℝN ) (1 ≤ p ≤ ∞), then the following equalities α −α a Dt (a Dt f (t))

= f (t) and

α −α t Db (t Db f (t))

= f (t)

(α > 0)

hold almost everywhere on [a, b]. (ii) If α > β > 0, then, for f ∈ Lp ([a, b], ℝN ) (1 ≤ p ≤ ∞), the relations β −α a Dt (a Dt f (t))

= a Dt

−α+β

hold almost everywhere on [a, b].

f (t)

and

β −α t Db (t Db f (t))

= t Db

−α+β

f (t)

1.2 Fractional calculus |

11

In particular, when β = k ∈ ℕ and α > k, then k −α a Dt (a Dt f (t))

= a D−α+k f (t) t

k −α t Db (t Db f (t))

and

= (−1)k t D−α+k f (t). b

p −α p To present the next property, we use the spaces of functions a D−α t (L ) and t Db (L ) defined for α > 0 and 1 ≤ p ≤ ∞ by −α p a Dt (L )

p N = {f : f = a D−α t φ, φ ∈ L ([a, b], ℝ )}

−α p t Db (L )

p N = {f : f = t D−α b ϕ, ϕ ∈ L ([a, b], ℝ )},

and

respectively. The composition of the fractional integral operator a D−α t with the fractional derivative operator a Dαt is given by the following result. Proposition 1.6. Let α > 0, n = [α] + 1 and let fn−α (t) = a D−(n−α) f (t) be the fractional t integral (1.1) of order n − α. p (i) If 1 ≤ p ≤ ∞ and f ∈ a D−α t (L ), then −α α a Dt (a Dt f (t))

= f (t).

(ii) If f ∈ L1 ([a, b], ℝN ) and fn−α ∈ AC n ([a, b], ℝN ), then the equality −α α a Dt (a Dt f (t))

n

= f (t) − ∑ j=1

fn−α (a) (t − a)α−j , Γ(α − j + 1) (n−j)

holds almost everywhere on [a, b]. Proposition 1.7. Let α > 0 and n = [α] + 1. Also, let gn−α (t) = t D−(n−α) g(t) be the fracb tional integral (1.2) of order n − α. p (i) If 1 ≤ p ≤ ∞ and g ∈ t D−α b (L ), then −α α t Db (t Db g(t))

= g(t).

(ii) If g ∈ L1 ([a, b], ℝN ) and gn−α ∈ AC n ([a, b], ℝN ), then the equality −α α t Db (t Db g(t))

n

= g(t) − ∑ j=1

(−1)n−j gn−α (b) (b − t)α−j , Γ(α − j + 1) (n−j)

holds almost everywhere on [a, b]. In particular, if 0 < α < 1, then −α α t Db (t Db g(t))

= g(t) −

g1−α (b) (b − t)α−1 , Γ(α)

12 | 1 Preliminaries where g1−α (t) = t Dα−1 b g(t) while for α = n ∈ ℕ, the following equality holds: n−1

(−1)k g (k) (b) (b − t)k . k! k=0

−n n t Db (t Db g(t)) = g(t) − ∑

Proposition 1.8. Let α > 0 and let y ∈ L∞ ([a, b], ℝN ) or y ∈ C([a, b], ℝN ). Then C α −α a Dt (a Dt y(t))

= y(t) and

C α −α t Db (t Db y(t))

= y(t).

Proposition 1.9. Let α > 0 and let n be given by (1.3). If y ∈ AC n ([a, b], ℝN ) or y ∈ C n ([a, b], ℝN ), then −α C α a Dt (a Dt y(t))

n−1

y(k) (a) (t − a)k k! k=0

= y(t) − ∑

and −α C α t Db (t Db y(t))

n−1

(−1)k y(k) (b) (b − t)k . k! k=0

= y(t) − ∑

In particular, if 0 < α ≤ 1 and y ∈ AC([a, b], ℝN ) or y ∈ C([a, b], ℝN ), then −α C α a Dt (a Dt y(t))

= y(t) − y(a) and

−α C α t Db (t Db y(t))

= y(t) − y(b).

(1.11)

Remark 1.4. If f , g are abstract functions with values in Banach space X, then integrals which appear in the above properties are taken in Bochner’s sense.

1.2.3 Special functions Definition 1.8 ([163, 174]). The generalized Mittag-Leffler function Eα,β is defined by 1 λα−β eλ zk = dλ, ∫ α Γ(αk + β) 2πi λ − z k=0 ∞

Eα,β (z) := ∑

α, β > 0, z ∈ ℂ,

ϒ

where ϒ is a contour which starts and ends at −∞ and encircles the disc |λ| ≤ |z|1/α counter-clockwise. If 0 < α < 1, β > 0, then the asymptotic expansion of Eα,β as |z| → ∞ is given by 1 (1−β)/α z exp(z 1/α ) + εα,β (z), | arg z| ≤ 21 απ, Eα,β (z) = { α εα,β (z), | arg(−z)| < (1 − 21 α)π,

(1.12)

1.3 Semigroups | 13

where N−1

εα,β (z) = − ∑

n=1

z −n + O(|z|−N ), Γ(β − αn)

as |z| → ∞.

For short, set Eα (z) := Eα,1 (z),

eα (z) := Eα,α (z).

Then Mittag-Leffler functions have the following properties. Proposition 1.10. For α ∈ (0, 1) and t ∈ ℝ, (i) Eα (t), eα (t) > 0; (ii) (Eα (t))′ = α1 eα (t); (iii) limt→−∞ Eα (t) = limt→−∞ eα (t) = 0; t (iv) ∫0 Eα,β (ωt α )t β−1 dt = t β Eα,β+1 (ωt α ), α, β > 0, ω ∈ ℝ; α−1 (v) C0Dαt Eα (ωt α ) = ωEα (ωt α ), 0 Dα−1 eα (ωt α )) = Eα (ωt α ), ω ∈ ℂ. t (t Definition 1.9 ([158]). The Wright function Mα is defined by (−z)n n!Γ(−αn + 1 − α) n=0 ∞

Mα (z) := ∑ =

1 ∞ (−z)n Γ(nα) sin(nπα), ∑ π n=1 (n − 1)!

z∈ℂ

with 0 < α < 1. For −1 < r < ∞, λ > 0, the following results hold. Proposition 1.11. (W1) Mα (t) ≥ 0, t > 0; α ∞ α (W2) ∫0 t α+1 Mα ( t1α )e−λt dt = e−λ ;

∞ Γ(1+r) ; Γ(1+αr) ∞ −zt (W4) ∫0 Mα (t)e dt = Eα (−z), z ∈ ∞ (W5) ∫0 αtMα (t)e−zt dt = eα (−z), z

(W3) ∫0 Mα (t)t r dt =

ℂ; ∈ ℂ.

1.3 Semigroups 1.3.1 C0 -semigroup Let X be a Banach space and B(X) be the Banach space of linear bounded operators. Definition 1.10. A semigroup is a one-parameter family {T(t)}t≥0 ⊂ B(X) satisfying the conditions:

14 | 1 Preliminaries (i) T(t)T(s) = T(t + s), for t, s ≥ 0; (ii) T(0) = I. Here, I denotes the identity operator in X. Definition 1.11. A semigroup {T(t)}t≥0 is uniformly continuous if 󵄩 󵄩 lim 󵄩󵄩T(t) − T(0)󵄩󵄩󵄩B(X) = 0,

t→0+󵄩

that is, if 󵄩 󵄩 lim 󵄩󵄩󵄩T(t) − T(s)󵄩󵄩󵄩B(X) = 0.

|t−s|→0

Definition 1.12. We say that the semigroup {T(t)}t≥0 is C0 -semigroup if the map t → T(t)x is strongly continuous, for each x ∈ X, i. e., lim T(t)x = x,

t→0+

∀ x ∈ X.

Definition 1.13. Let T(t) be a C0 -semigroup defined on X. The infinitesimal generator A of T(t) is the linear operator defined by A(x) = lim

t→0+

where D(A) = {x ∈ X : limt→0+

T(t)x−x t

T(t)x − x , t

for x ∈ D(A),

exists in X}.

1.3.2 Almost sectorial operators We first introduce some special functions and classes of functions, which will be used in the following; for more details, we refer to [159, 172]. Let Sμ0 with 0 < μ < π be the open sector {z ∈ ℂ\{0} : | arg z| < μ} and Sμ be its closure, that is, Sμ = {z ∈ ℂ\{0} : | arg z| ≤ μ} ∪ {0}. Denote by D(A) the domain of A, by σ(A) its spectrum, while ρ(A) := ℂ − σ(A) is the resolvent set of A. We state the concept of almost sectorial operators as follows. γ

Definition 1.14 ([172]). Let −1 < γ < 0 and 0 < ω < π/2. By Θω (X), we denote the family of all linear closed operators A : D(A) ⊂ X → X which satisfy:

1.3 Semigroups | 15

(i) σ(A) ⊂ Sω ; (ii) for every ω < μ < π there exists a positive constant Cμ such that 󵄩 󵄩󵄩 γ 󵄩󵄩R(z; A)󵄩󵄩󵄩B(X) ≤ Cμ |z| ,

for all z ∈ ℂ \ Sμ ,

where R(z; A) = (zI − A)−1 , z ∈ ρ(A), which are bounded linear operators the resolvent of A. A linear operator A will be called an almost sectorial operator on X γ if A ∈ Θω (X). γ

Remark 1.5. Let A ∈ Θω (X). Then the definition implies that 0 ∈ ρ(A). Set γ

0

0

γ

0

F0 (Sμ ) = ⋃ Ψs (Sμ ) ⋃ Ψ0 (Sμ ), s k, one easily sees that ψkn ∈ F0 (Sμ0 ). γ Assume that A ∈ Θω (X) with −1 < γ < 0 and 0 < ω < π/2. Following [172], a closed linear operator f → f (A) can be constructed for every f ∈ F (Sμ0 ) via an extended functional calculus. In the following, we give a short overview to this construction. γ For f ∈ F0 (Sμ0 ), via Dunford–Riesz integral, the operator f (A) is defined by f (A) =

1 ∫ f (z)R(z; A)dz, 2πi Γθ

(1.13)

16 | 1 Preliminaries where the integral contour Γθ := {ℝ+ eiθ } ∪ {ℝ+ e−iθ }, is oriented counterclockwise and ω < θ < μ < π. It follows that the integral is absolutely convergent and defines a bounded linear operator on X, and its value does not depend on the choice of θ. Notice in particular that for k, n ∈ ℕ ∪ {0} with n > k, ψkn (A) = Ak (A + 1)−n and the operator ψkn (A) is injective. Notice also that if f ∈ F (Sμ0 ), then there exist γ

k, n ∈ ℕ such that fψkn ∈ F0 (Sμ0 ). Hence, for f ∈ F (Sμ0 ), one can define a closed linear operator, still denoted by f (A), D(f (A)) = {x ∈ X | (fψkn )(A)x ∈ D(A(n−1)k )}, f (A) = (ψkn (A)) (fψkn )(A), −1

and the definition of f (A) does not depend on the choice of k and n. We emphasize γ that f (A) is indeed an extension of the original and the triple (F0 (Sμ0 ), F (Sμ0 ), f (A)) is called an abstract functional calculus on X (see [159]). With respect to this construction we collect some basic properties. For more details, we refer to [172]. Proposition 1.12. The following assertions hold: γ (i) αf (A) + βg(A) = (αf + βg)(A), (fg)(A) = f (A)g(A) for ∀ f , g ∈ F0 (Sμ0 ), α, β ∈ ℂ; (ii) f (A)g(A) ⊂ (fg)(A) for ∀ f , g ∈ F (Sμ0 ), and (iii) f (A)g(A) = (fg)(A), provided that g(A) is bounded or D((fg)(A)) ⊂ D(g(A)).

Since for each β ∈ ℂ, z β ∈ F (Sμ0 ) (z ∈ ℂ \ (−∞, 0], 0 < μ < π), one can define, via γ

the triple (F0 (Sμ0 ), F (Sμ0 ), f (A)), the complex powers of A which are closed by Aβ = z β (A),

β ∈ ℂ.

However, in difference to the case of sectorial operators, having 0 ∈ ρ(A) does not imply that the complex powers A−β with Re(β) > 0 are bounded. The operator A−β belongs to B(X) whenever Re(β) > 1 + γ. So, in this situation, the linear space X β := D(Aβ ), β > 1 + γ, endowed with the graph norm |x|β = |Aβ x|, x ∈ X β , is a Banach space. Next, we turn our attention to the semigroup associated with A. Since given t ∈ 0 S π −ω , e−tz ∈ H ∞ (Sμ0 ) satisfies the conditions (a) and (b) of Lemma 2.13 of [172], the 2

family

T(t) = e−tz (A) =

1 ∫ e−tz R(z; A)dz, 2πi Γθ

here ω < θ < μ
0 such that 󵄩󵄩 󵄩 −γ−1 , 󵄩󵄩T(t)󵄩󵄩󵄩B(X) ≤ C0 t

for all t > 0;

(iv) The range R(T(t)) of T(t), t ∈ S0π −ω is contained in D(A∞ ). Particularly, R(T(t)) ⊂ 2

D(Aβ ) for all β ∈ ℂ with Re(β) > 0, Aβ T(t)x =

1 ∫ z β e−tz R(z; A)xdz, 2πi

for all x ∈ X,

Γθ

and hence there exists a constant C ′ = C ′ (γ, β) > 0 such that 󵄩󵄩 β 󵄩 ′ −γ−Re(β)−1 , 󵄩󵄩A T(t)󵄩󵄩󵄩B(X) ≤ C t (v)

for all t > 0;

If β > 1 + γ, then D(Aβ ) ⊂ ΣT , where ΣT is the continuity set of the semigroup {T(t)}t≥0 , that is, 󵄨 ΣT = {x ∈ X 󵄨󵄨󵄨 lim T(t)x = x}. t→0+

Remark 1.7. We note that the condition (ii) of the proposition does not satisfy for t = 0 or s = 0. The relation between the resolvent operators of A and the semigroup T(t) is characterized by the following.

18 | 1 Preliminaries γ

Proposition 1.14 ([172]). Let A ∈ Θω (X) with −1 < γ < 0 and 0 < ω < π/2. Then for every λ ∈ ℂ with Re(λ) > 0, one has ∞

R(λ; −A) = ∫ e−λt T(t)dt. 0

1.4 Results from analysis 1.4.1 Laplace and Fourier transforms In this subsection, we present definitions and some properties of Laplace and Fourier transforms. Definition 1.15. The Laplace transform of a function f (t) of a real variable t ∈ ℝ+ is defined by ∞

(L f )(s) = L [f (t)](s) = f ̄(s) := ∫ e−st f (t)dt

(s ∈ ℂ).

(1.15)

0

The inverse Laplace transform is given for x ∈ ℝ+ by the formula γ+i∞

1 (L g)(x) = L [g(s)](x) := ∫ esx g(s)ds πi −1

−1

(γ = Re(s)).

(1.16)

γ−i∞

Proposition 1.15. Let f (t) be defined on (0, ∞) and 0 < α < 1. Then Laplace transform of fractional integral and fractional differential operator satisfies: −α ̄ (i) 0D−α t f (s) = s f (s); α (ii) 0Dαt f (s) = s f ̄(s) − (0Dα−1 t f )(0); (iii) CDα f (s) = sα f ̄(s) − sα−1 f (0). 0 t

Definition 1.16. The Fourier transform of a function f (t) of a real variable t ∈ ℝ is defined by ∞

(F f )(w) = F [f (t)](w) = f ̂(w) := ∫ e−it⋅w f (t)dt

(w ∈ ℝ).

(1.17)

−∞

The inverse Fourier transform is given by the formula ∞

1 1 ̂ (F g)(w) = F [g(t)](w) = g(−w) := ∫ eit⋅w g(t)dt 2π π −1

−1

−∞

(w ∈ ℝ).

(1.18)

1.4 Results from analysis | 19

The integrals in (1.17) and (1.18) converge absolutely for functions f , g ∈ L1 (ℝ) and in the norm of the space L2 (ℝ) for f , g ∈ L2 (ℝ). Proposition 1.16. Let f (t) be defined on (−∞, ∞) and 0 < α < 1. Then Fourier transform of the Liouville–Weyl fractional integral and fractional differential operator satisfies: −α ̂ ? (i) −∞ D−α t f (w) = (iw) f (w); −α ̂ (ii) t? D−α ∞ f (w) = (−iw) f (w); α ̂ α ? (iii) D f (w) = (iw) f (w); (iv)

−∞ t α ? t D∞ f (w)

= (−iw)α f ̂(w).

1.4.2 Weak compactness of sets and operators Let X be a real Banach space with norm | ⋅ | and X ∗ be its topological dual, i. e., the vector space of all linear continuous functionals from X to ℝ, which endowed with the dual norm ‖f ‖ = sup|x|≤1 |⟨f , x⟩|, for f ∈ X ∗ , is in its turn a real Banach space, too. Here, ⟨⋅, ⋅⟩ denotes duality product. Thereafter, if x ∈ X and f ∈ X ∗ , ⟨f , x⟩ denotes f (x). We denote by Fin(X ∗ ) the class of all finite subsets in X ∗ . Let F ∈ Fin(X ∗ ). Then the function | ⋅ |F : X → ℝ, defined by 󵄨 󵄨 |x|F = max{󵄨󵄨󵄨⟨f , x⟩󵄨󵄨󵄨 : f ∈ F} for each x ∈ X, is a seminorm on X. The family of seminorms {| ⋅ |F : F ∈ Fin(X ∗ )} denotes the so-called weak topology and X, endowed with this topology, denoted by Xw , is a locally convex topological vector space. A subset A of a Banach space X is called weakly closed if it is closed in the weak w topology. The symbol D denotes the weak closure of D. We will say that {xn } ⊂ X converges weakly to x0 ∈ X, and we write xn ⇀ x0 , if for each f ∈ X ∗ , f (x) → f (x0 ). We recall (see [41]) that a sequence {xn } ⊂ C([0, b], X) weakly converges to an element x ∈ C([0, b], X) if and only if: (i) there exists N > 0 such that, for every n ∈ ℕ+ and t ∈ [0, b], |xn (t)| ≤ N; (ii) for every t ∈ [0, b], xn (t) ⇀ x(t). Definition 1.17. (i) A subset A of a normed space X is said to be (relatively) weakly compact if (the weak closure of) A is compact in the weak topology of X. (ii) A subset A of a Banach space X is weakly sequentially compact if any sequence in A has a subsequence which converges weakly to an element of X.

20 | 1 Preliminaries Definition 1.18. Suppose that X and Y are Banach spaces. A linear operator T from X into Y is weakly compact if T(B) is a relatively weakly compact subset of Y whenever B is a bounded subset of X. We mention also two results that are contained in the so-called Eberlein–Smulian theory. Theorem 1.1 ([130]). Let Ω be a subset of a Banach space X. The following statements are equivalent: (i) Ω is relatively weakly compact; (ii) Ω is relatively weakly sequentially compact. Corollary 1.1 ([130]). Let Ω be a subset of a Banach space X. The following statements are equivalent: (i) Ω is weakly compact; (ii) Ω is weakly sequentially compact. We recall Krein–Smulian theorem and Pettis measurability theorem. Theorem 1.2 ([87]). The convex hull of a weakly compact set in a Banach space X is weakly compact. Theorem 1.3 ([173]). Let (E, Σ) be a measure space, X be a separable Banach space. Then a function f : E → X is measurable if and only if for every x∗ ∈ X ∗ the function x∗ ∘ f : E → ℝ is measurable with respect to Σ and the Borel σ-algebra in ℝ.

1.4.3 Measure of noncompactness We recall here some definitions and properties of measure of noncompactness and condensing maps. For more details, we refer the reader to Banaś and Goebel [33], Deimling [82] and Kamenskii et al. [129]. Definition 1.19. Let Y + be the positive cone of an order Banach space (Y, ≤). A function β defined on the set of all bounded subsets of the Banach space X with values in Y + is called a measure of noncompactness (MNC) on X if β(coΩ) = β(Ω) for all bounded subsets Ω ⊂ X. The MNC β is said to be: (i) Monotone if for all bounded subsets B1 , B2 of X, B1 ⊆ B2 implies β(B1 ) ≤ β(B2 ); (ii) Nonsingular if β({x}∪B) = β(B) for every x ∈ X and every nonempty subset B ⊆ X; (iii) Regular β(B) = 0 if and only if B is relatively compact in X.

1.4 Results from analysis | 21

One of the most important examples of MNC is Hausdorff MNC β defined on each bounded subset B of X by m

β(B) = inf{ε > 0 : B ⊂ ⋃ Bε (xj ) where xj ∈ X}, j=1

where Bε (xj ) is a ball of radius ≤ ε centered at xj , j = 1, 2, . . . , m. Without confusion, Kuratowski MNC β1 defined on each bounded subset B of X by m

β1 (B) = inf{ε > 0 : B ⊂ ⋃ Mj and diam(Mj ) ≤ ε}, j=1

where the diameter of Mj is defined by diam(Mj ) = sup{|x −y| : x, y ∈ Mj }, j = 1, 2, . . . , m. It is well known that Hausdorff MNC β and Kuratowski MNC β1 enjoy the above properties (i)–(iii) and other properties: (iv) β(B1 + B2 ) ≤ β(B1 ) + β(B2 ), where B1 + B2 = {x + y : x ∈ B1 , y ∈ B2 }; (v) β(B1 ∪ B2 ) ≤ max{β(B1 ), β(B2 )}; (vi) β(λB) ≤ |λ|β(B) for any λ ∈ ℝ. In particular, the relationship between Hausdorff MNC β and Kuratowski MNC β1 is given by (vii) β(B) ≤ β1 (B) ≤ 2β(B). In the following, several examples of useful measures of noncompactness in spaces of continuous functions are presented. Example 1.1. We consider general example of MNC in the space of continuous functions C([a, b], X). For Ω ⊂ C([a, b], X) define ϕ(Ω) = sup β(Ω(t)), t∈[a,b]

where β is Hausdorff MNC in X and Ω(t) = {y(t) : y ∈ Ω}. Example 1.2. Consider another useful MNC in the space C([a, b], X). For a bounded Ω ⊂ C([a, b], X), set ν(Ω) = ( sup β(Ω(t)), modC (Ω)); t∈[a,b]

here, the modulus of equicontinuity of the set of functions Ω ⊂ C([a, b], X) has the following form: 󵄨 󵄨 modC (Ω) = lim sup max 󵄨󵄨󵄨x(t1 ) − x(t2 )󵄨󵄨󵄨. δ→0 |t −t |≤δ x∈Ω

1

2

(1.19)

22 | 1 Preliminaries Example 1.3. We consider one more MNC in the space C([a, b], X). For a bounded Ω ⊂ C([a, b], X), set ν(Ω) = max ( sup e−Lt β(D(t)), modC (D)), D∈Δ(Ω) t∈[a,b]

where Δ(Ω) is the collection of all denumerable subsets of Ω, L is a constant, and modC (D) is given in formula (1.19). Let J = [0, b], b ∈ ℝ+ . For any W ⊂ C(J, X), we define t

t

∫ W(s)ds = {∫ u(s)ds : u ∈ W}, 0

for t ∈ [0, b],

0

where W(s) = {u(s) ∈ X : u ∈ W}. We present here some useful properties. Proposition 1.17. If W ⊂ C(J, X) is bounded and equicontinuous, then coW ⊂ C(J, X) is also bounded and equicontinuous. Proposition 1.18 ([114]). If W ⊂ C(J, X) is bounded and equicontinuous, then t → β(W(t)) is continuous on J, and t

β(W) = max β(W(t)), t∈J

t

β(∫ W(s)ds) ≤ ∫ β(W(s))ds, 0

0

for t ∈ [0, b]. Proposition 1.19. If W ⊂ C([0, ∞), X) is bounded and equicontinuous, and limt→∞ |u(t)| = 0 uniformly for u ∈ W, then t → β(W(t)) is continuous on [0, ∞), and β(W) = β(W[0, ∞)) = sup β(W(t)). t∈[0,∞)

Proposition 1.20 ([164]). Let {un }∞ n=1 be a sequence of Bochner integrable functions from ̃ for almost all t ∈ J and every n ≥ 1, where m̃ ∈ L(J, ℝ+ ), then J into X with |un (t)| ≤ m(t) + the function ψ(t) = β({un (t)}∞ n=1 ) belongs to L(J, ℝ ) and satisfies t

t

β({∫ un (s)ds : n ≥ 1}) ≤ 2 ∫ ψ(s)ds. 0

0

1.4 Results from analysis | 23

Proposition 1.21 ([45]). If W is bounded, then for each ε > 0, there is a sequence {un }∞ n=1 ⊂ W, such that β(W) ≤ 2β({un }∞ n=1 ) + ε. Proposition 1.22 ([164]). Let {un }∞ n=1 be a sequence of Bochner integrable functions from ̃ [0, T] into X with |un (t)| ≤ m(t) for almost all t ∈ [0, T] and every n ≥ 1, where m̃ ∈ + L([0, T], ℝ+ ), then the function ψ(t) = β({un (t)}∞ n=1 ) belongs to L([0, T], ℝ ) and satisfies T

T

β({∫ un (s)ds : n ≥ 1}) ≤ ∫ ψ(s)ds. t

t

Lemma 1.4 ([225]). Let Ω be a closed and convex subset of a Banach space X. Suppose the operator T : Ω → Ω is continuous and T (Ω) is bounded. For each bounded subset Ω0 ⊂ Ω, set 1

T (Ω0 ) = T (Ω0 ),

n

T (Ω0 ) = T (co(T

n−1

(Ω0 ))),

n = 2, 3, . . . .

If there exists a constant 0 ≤ k < 1 and a positive integer n0 such that for any bounded subset Ω0 ⊂ Ω, β(T n0 (Ω0 )) ≤ kβ(Ω0 ), then there exists D ⊂ Ω such that β(T (D)) = 0. p Lemma 1.5 ([28]). Let {fn }∞ n=1 ⊂ L (J, X) (p ≥ 1) be an integrable bounded sequence satisfying

β({fn }∞ n=1 ) ≤ γ(t),

a. e., t ∈ J,

where γ(⋅) ∈ L1 (J, ℝ+ ). Then, for each ϵ > 0 there exist a compact set Kϵ ⊆ X, a measurp able set Jϵ ⊂ J with measure less than ϵ, and a sequence of functions {gnϵ }∞ n=1 ⊂ L (J, X) such that {gnϵ (t)}∞ n=1 ⊆ Kϵ , for t ∈ J, and 󵄩󵄩 󵄩 ϵ 󵄩󵄩fn (t) − gn (t)󵄩󵄩󵄩 < 2γ(t) + ϵ,

for each n ≥ 1 and for every t ∈ J − Jϵ .

Lemma 1.6 ([57]). If {Wn }∞ n=1 ⊂ X is a nonempty, decreasing, closed sequence and limn→∞ β(Wn ) = 0, then ⋂∞ n=1 Wn is nonempty and compact. 1.4.4 Fixed-point theorems In this subsection, we present some fixed-point theorems, which will be used in the following chapters.

24 | 1 Preliminaries Definition 1.20. A family F in C(J, X) is called uniformly bounded if there exists a positive constant K such that |f (t)| ≤ K for all t ∈ J and all f ∈ F. Further, F is called equicontinuous, if for every ε > 0 there exists a δ = δ(ε) > 0 such that |f (t1 ) − f (t2 )| < ε for all t1 , t2 ∈ J with |t1 − t2 | < δ and all f ∈ F. Definition 1.21. The map T : D ⊆ X → X is said to be a β-contraction, if there exists a positive constant k < 1, such that β(T (D)) ≤ kβ(D) for any bounded and closed subset D ⊆ X, where β is a measure of noncompactness. Lemma 1.7 (Arzela–Ascoli theorem). If a family F = {f (t)} in C(J, ℝ) is uniformly bounded and equicontinuous on J, then F has a uniformly convergent subsequence {fn (t)}∞ n=1 . If a family F = {f (t)} in C(J, X) is uniformly bounded and equicontinuous on J, and for any t ∗ ∈ J, {f (t ∗ )} is relatively compact, then F has a uniformly convergent subsequence {fn (t)}∞ n=1 . Let 󵄨 󵄨 C0 ([0, ∞), X) = {x ∈ C([0, ∞), X) : lim 󵄨󵄨󵄨x(t)󵄨󵄨󵄨 = 0}, t→∞ with the norm ‖x‖0 = supt∈[0,∞) |x(t)| < ∞. It is obvious that C0 ([0, ∞), X) is a Banach space. We also need the following generalized Arzela–Ascoli theorem. Lemma 1.8 ([115]). The set H ⊂ C0 ([0, ∞); X) is relatively compact if and only if the following conditions hold: (i) for any b > 0, the set H is equicontinuous on [0, b]; (ii) for any t ∈ [0, ∞), H(t) = {u(t) : u ∈ H} is relatively compact in X; (iii) limt→∞ |u(t)| = 0 uniformly for u ∈ H. Theorem 1.4 (Banach contraction mapping principle). Let (X, d) be a complete metric space, and let T : Ω → Ω be a contraction mapping: d(T x, T y) ≤ kd(x, y), where 0 < k < 1, for each x, y ∈ Ω. Then there exists a unique fixed point x of T in Ω, i. e., T x = x. Theorem 1.5 (Darbo–Sadovskii’s fixed-point theorem). If Ω is bounded, closed and convex subset of Banach space X, the continuous mapping T : Ω → Ω is a β-contraction, then the mapping T has at least one fixed point in Ω. Theorem 1.6 (Schauder’s fixed-point theorem). Let Ω be a closed, convex and nonempty subset of a Banach space X. Let T : Ω → Ω be a continuous mapping such that T Ω is a relatively compact subset of X. Then T has at least one fixed point in Ω.

1.5 Stochastic process |

25

Theorem 1.7 (Schaefer’s fixed-point theorem). Let X be a Banach space and let F : X → X be a completely continuous mapping. Then either: (i) the equation x = λFx has a solution for λ = 1, or (ii) the set {x ∈ X : x = λFx for some λ ∈ (0, 1)} is unbounded. Theorem 1.8 (Krasnoselskii’s fixed-point theorem). Let X be a Banach space, let Ω be a bounded, closed, convex subset of X and let S , T be mappings of Ω into X such that S z + T w ∈ Ω for every pair z, w ∈ Ω. If S is a contraction and T is completely continuous, then the equation S z + T z = z has a solution on Ω. Theorem 1.9 (Nonlinear alternative of Leray–Schauder type). Let C be a nonempty, convex subset of X. Let U be a nonempty, open subset of C with 0 ∈ U and F : U → C be a compact and continuous operators. Then either: (i) F has fixed points, or (ii) there exist y ∈ 𝜕U and λ∗ ∈ [0, 1] with y = λ∗ F(y). Theorem 1.10 (O’Regan’s fixed-point theorem). Let U be an open set in a closed, convex set C of X. Assume 0 ∈ U, T(U) is bounded and T : U → C is given by T = T1 + T2 , where T1 : U → X is completely continuous, and T2 : U → X is a nonlinear contraction. Then either: (i) T has a fixed point in U, or (ii) there is a point x ∈ 𝜕U and λ ∈ (0, 1) with x = λT(x). Theorem 1.11 (O’Regan’s fixed-point theorem). Let E be a metrizable, locally, convex, linear, topological space and let Q be a weakly compact, convex subset of E. Suppose G : Q → C(Q) has weakly sequentially closed graph. Then G has a fixed point; here, C(Q) denoted the family of nonempty, closed, convex subsets of Q.

1.5 Stochastic process We present some important concepts and results of stochastic process in this section. The material is taken from Arnold [22], Gawarecki et al. [103] and Prato et al. [77].

1.5.1 Random variables Let Ω be a sample space and F a σ-algebra of the subset of Ω. A function ℙ(⋅) defined on F and taking values in the unit interval [0, 1] is called a probability measure, if: (i) ℙ(Ω) = 1; (ii) ℙ(A) ≥ 0 for all A ∈ F ;

26 | 1 Preliminaries (iii) for an at most countable family {An , n ≥ 1} of mutually disjoint event, we have ℙ{⋃ An } = ∑ ℙ(An ). n≥1

n≥1

The triple (Ω, F , ℙ) is a probability space. Let 𝔽 = (Ft )t≥0 be a family of sub-σ-algebras Ft of σ-algebra F such that Fs ⊂ Ft for 0 ≤ s < t < ∞. The space ℙ𝔽 = (Ω, F , 𝔽, ℙ) is said to be a filtered probability space. We say that a filtration Ft satisfies the usual conditions if F0 contains all ℙ-null sets of F and Ft = ⋂ε>0 Ft+ε for every t ≥ 0. If the last condition is satisfied, we say that a filtration F is right continuous. Let (X, BX ) be measurable space, we mean an (F , BX )-measurable mapping x : Ω → X, i. e., such that x−1 (A) ∈ F for every A ∈ BX , where as usual, BX denotes the Borel σ-algebra on X and x−1 (A) = {ω ∈ Ω : x(ω) ∈ A}. We shall also say that x is a random variable on Ω with values at X. The integral of an integrable random variable x is called its mean value or expectation and is denoted by 𝔼(x) = ∫ x(w)dℙ. Let K and H be separable Hilbert spaces, and Q be either a symmetric nonnegative definite trace-class operator on K or Q = IK , the identity operator on K . In case Q is trace-class, we will always assume that its all eigenvalues λj > 0, j = 1, 2, . . . ; otherwise, we can start with the Hilbert space ker(Q)⊥ instead of K . The associated eigenvectors forming an orthonormal basis (ONB) in K will be denoted by ej . Denote L (K , H ) by all bounded linear operators from K to H . Then the space of Hilbert–Schmidt operators from K to H is defined as ∞

2

L2 (K , H ) = {Φ ∈ L (K , H ) : ∑ |Φei |H < ∞}. i=1

It is well known (see [181]) that L2 (K , H ) equipped with the norm ∞

‖Φ‖L2 (K ,H ) = ∑ |Φei |2H i=1

is a Hilbert space. 1 On the other hand, the space Q 2 K equipped with the scalar product ⟨u, v⟩

∞ 1 Q2

K

=∑ j=1

1 (u, ej )K (v, ej )K λj 1

is a separable Hilbert space with an ONB {λj2 ej }∞ j=1 .

27

1.5 Stochastic process |

1

1

Consider L20 = L2 (Q 2 K , H ), the space of Hilbert–Schmidt operators from Q 2 K 0 to H . If {ẽj }∞ j=1 is an ONB in H , then the Hilbert–Schmidt norm of an operator Φ ∈ L2 is given by

∞

1

2

∞

1

2

‖Φ‖L 0 = ∑ (Φ(λj2 ej ), ẽi )H = ∑ (Φ(Q 2 ej ), ẽi )H 2

i,j=1

i,j=1

1 2 1 1 ∗ 󵄩 󵄩 = 󵄩󵄩󵄩ΦQ 2 󵄩󵄩󵄩L (K ,H ) = tr(ΦQ 2 )(ΦQ 2 ) . 2

1.5.2 Stochastic calculus An X-valued stochastic process (briefly, an X-valued process) indexed by a set I is a

family of X-valued random variables {X(i), i ∈ I} defined on some underlying probability space (Ω, F , ℙ).

Definition 1.22. An X-valued process {X(i), i ∈ I} is called Gaussian, if for all N > 1

and i1 , . . . , iN ∈ I the X N -valued random variable (X(i1 ), . . . , X(iN )) is Gaussian.

Definition 1.23. A real-valued process {W(t), t ∈ [0, T]} is called a Brownian motion, if it enjoys the following properties: (i)

(ii)

W(0) = 0;

W(t) − W(s) is independent of {W(r), r ∈ [0, s]} for 0 ≤ s ≤ t ≤ T;

(iii) W(t) − W(s) is Gaussian with variance (t − s). Definition 1.24.

(i) For an L (H , X)-valued step function of the form Φ(t, ω) = ϕ1 (ω)I[t0 ,t1 ] (t) +

∑ni=2 ϕi (ω)I(ti−1 ,ti ] (t), where 0 = t0 < t1 < ⋅ ⋅ ⋅ < tn = T and ϕi , i = 1, . . . , n, are, respec-

tively, F0 -measurable and Fti -measurable L2 (K , H )-valued random variables such that ϕi (ω) ∈ L (K , H ), i = 1, . . . , n. We define the stochastic integral process t

∫0 Φ(s)dW(s), 0 ≤ t ≤ T, by t

n

∫ Φ(s)dW(s) = ∑ ϕi (W(ti ) − W(ti−1 )). 0

i=1

(ii) A function Φ : [0, T] → L (H , X) is said to be stochastically integrable with respect to the H -cylindrical Brownian motion W if there exists a sequence of finite rank step functions Φn : [0, T] → L (H , X) such that:

(a) for all h ∈ H , we have limn→∞ Φn h = Φh in measure;

28 | 1 Preliminaries (b) there exists an X-valued random variable x such that t

lim ∫ Φn (s)dW(s) = x

n→∞

0

in probability. The stochastic integral of a stochastically integrable function x : [0, T] → L (H , X) is then defined as the limit in probability t

t

∫ Φ(s)dW(s) = lim ∫ Φn (s)dW(s). n→∞

0

0

The relationship t

x(t, ω) = ∫ Φ(s, ω)dW(s, ω) 0

can also be written as dx(t) = Φ(t)dW(t). This is a special so-called stochastic differential. Let us look at a somewhat more general stochastic process of the form t

t

x(t, ω) = x(0, ω) + ∫ f (s, ω)ds + ∫ Φ(s, ω)dW(s, ω); 0

(1.20)

0

t

here, ∫0 f (s, ω)ds is the usual Lebesgue or possibly Riemann integral. Definition 1.25. We shall say that a stochastic process x(t) defined by equation (1.20) possesses the stochastic differential f (t)dt + Φ(t)dW(t) and we shall write dx(t) = f (t)dt + Φ(t)dW(t) = fdt + ΦdW.

2 Fractional evolution equations of order α ∈ (0, 1) 2.1 Initial value problems with Hilfer derivative 2.1.1 Introduction A strong motivation for investigating fractional evolution equations comes from physics. For example, fractional diffusion equations are abstract partial differential equations that involve fractional derivatives in space and time. They are useful to model anomalous diffusion, where a plume of particles spreads in a different manner than the classical diffusion equation predicts. The time-fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order q ∈ (0, 1), namely 𝜕tq u(z, t) = Au(z, t), β

t ≥ 0, z ∈ ℝ. β

β

β

We can take A = 𝜕z 1 , for β1 ∈ (0, 1], or A = 𝜕z + 𝜕z 2 for β2 ∈ (1, 2], where 𝜕tq , 𝜕z 1 , 𝜕z 2 are the fractional derivatives of order q, β1 , β2 respectively. We refer the interested reader to Eidelman and Kochubei [89], EI-Sayed [88], Zhou and He [229] and the references therein for more details. For some fundamental results in the theory of fractional evolution equations involving the Riemann–Liouville fractional derivative and Caputo derivative, we refer the reader to the monographs of Bajlekova [30], Zhou [225, 226] and a series of papers [8, 37, 122, 186] and the references therein. Since the definition of mild solution in integer-order abstract differential equations obtained by variation of constant formulas cannot be generalized directly to fractional-order abstract differential equations, Zhou and Jiao [230] gave a suitable concept on mild solutions by applying Laplace transform and probability density functions for an evolution equation with a Caputo fractional derivative. Using the same method, Zhou et al. [226] gave a suitable concept on mild solutions for an evolution equation with a Riemann–Liouville fractional derivative. By using a sectorial operator, Su et al. [186] gave a definition of mild solution for a fractional differential equation with order 1 < q < 2 and investigated its existence. Agarwal et al. [8] studied the existence and dimension of the set for mild solutions of semilinear fractional differential inclusions. Wang et al. [205] researched the abstract fractional Cauchy problem with almost sectorial operators. On the other hand, Hilfer [124] proposed a generalized Riemann–Liouville fractional derivative, for short, Hilfer fractional derivative, which includes Riemann– Liouville fractional derivative and Caputo fractional derivative. This operator appeared in the theoretical simulation of dielectric relaxation in glass forming materials. The study of initial and boundary value problems for fractional evolution equations with a Hilfer fractional derivative has received great attention from some researchers. In [101], Furati et al. considered an initial value problem for a class of nonlinear https://doi.org/10.1515/9783110769272-002

30 | 2 Fractional evolution equations of order α ∈ (0, 1) fractional differential equations involving Hilfer fractional derivative. In [180], the solution of a fractional diffusion equation with a Hilfer time fractional derivative was obtained in terms of Mittag-Leffler functions and Fox’s H-function. By a noncompact measure method, Gu and Trujillo [112] established some sufficient conditions to ensure the existence of mild solutions for evolution equations with the Hilfer fractional derivative. In this section, we consider a class of evolution equation with the Hilfer fractional derivative μ,ν 0 Dt x(t) = Ax(t) + f (t, x(t)), −(1−μ)(1−ν) x(0) = x0 , 0 Dt

{

t ∈ J ′ = (0, b],

(2.1)

μ,ν

where 0 Dt is the Hilfer fractional derivative, 0 < μ < 1, 0 ≤ ν ≤ 1, the state x(⋅) takes value in a Banach space X with norm | ⋅ |, A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators (i. e., C0 semigroup) {Q(t)}t≥0 in a Banach space X and f : J × X → X are given functions satisfying some assumptions, x0 ∈ X. By using Laplace transform and density function, we first give the definition of mild solution. Then we obtain some sufficient conditions ensuring the existence of mild solution by using a noncompact measure method and the Arzela–Ascoli theorem. This section is based on Gu and Trujillo [112].

2.1.2 Definition of mild solutions In this subsection, we will introduce the definition of a mild solution of system (2.1) and give some assumptions and lemmas, which are useful in this section. Let J = [0, b] and J ′ = (0, b], by C(J, X) and C(J ′ , X) we denote the spaces of all continuous functions from J to X and J ′ to X, respectively. Define Y = {x ∈ C(J ′ , X) : lim t (1−ν)(1−μ) x(t) exists and is finite} t→0

with the norm ‖ ⋅ ‖Y defined by 󵄨 󵄨 ‖x‖Y = sup󵄨󵄨󵄨t (1−ν)(1−μ) x(t)󵄨󵄨󵄨. t∈J ′

Obviously, Y is a Banach space. We also note that: (i) When ν = 1, then Y = C(J, X) and ‖ ⋅ ‖Y = ‖ ⋅ ‖; (ii) Let x(t) = t (ν−1)(1−μ) y(t) for t ∈ J ′ , x ∈ Y if and only if y ∈ C(J, X), and ‖x‖Y = ‖y‖.

2.1 Initial value problems with Hilfer derivative |

31

Let Br (J) = {y ∈ C(J, X) | ‖y‖ ≤ r} and BYr (J ′ ) = {x ∈ Y | ‖x‖Y ≤ r}, then Br and BYr are two bounded, closed and convex subsets of C(J, X) and Y, respectively. Throughout this section, we introduce the following hypotheses: (H1) Q(t) is continuous in the uniform operator topology for t > 0, and {Q(t)}t≥0 is uniformly bounded, i. e., there exists M > 1 such that supt∈[0,+∞) |Q(t)| < M; (H2) for each t ∈ J ′ , the function f (t, ⋅) : X → X is continuous and for each x ∈ X, the function f (⋅, x) : J ′ → X is strongly measurable; (H3) there exists a function m ∈ L(J ′ , ℝ+ ) such that −μ 0 Dt m

∈ C(J ′ , ℝ+ ),

lim t (1−ν)(1−μ) 0 Dt m(t) = 0, −μ

t→0+

and 󵄨󵄨 󵄨 󵄨󵄨f (t, x)󵄨󵄨󵄨 ≤ m(t),

for all x ∈ X and almost all t ∈ J.

It is obvious that if (H3) holds, there exists a constant r > 0 such that t

|x0 | t (1−ν)(1−μ) M( + sup{ ∫(t − s)μ−1 m(s)ds}) ≤ r. Γ(ν(1 − μ) + μ) t∈J Γ(μ) 0

(H4) There exists a constant l > 0 such that for any bounded D ⊆ X, β(f (t, D)) ≤ lt (1−ν)(1−μ) β(D),

for a. e. t ∈ [0, b],

where β is the Hausdorff measure of noncompactness. Lemma 2.1 ([101]). The Cauchy problem (2.1) is equivalent to the integral equation x(t) =

x0 t (ν−1)(1−μ) Γ(ν(1 − μ) + μ) t

+

1 ∫(t − s)μ−1 [Ax + f (s, x(s))]ds, Γ(μ)

t ∈ J′.

0

The Wright function Mμ (θ) is defined by (see Definition 1.9) (−θ)n−1 , n=1 (n − 1)!Γ(1 − μn) ∞

Mμ (θ) = ∑

0 < μ < 1, θ ∈ ℂ,

which satisfies the following equality: ∞

∫ θδ Mμ (θ)dθ = 0

Γ(1 + δ) , Γ(1 + μδ)

for θ ≥ 0.

(2.2)

32 | 2 Fractional evolution equations of order α ∈ (0, 1) Lemma 2.2. If integral equation (2.2) holds, then we have t

x(t) = Sν,μ (t)x0 + ∫ Kμ (t − s)f (s, x(s))ds,

t ∈ J′,

(2.3)

0

where ∞

Kμ (t) = t μ−1 Pμ (t),

Pμ (t) = ∫ μθMμ (θ)Q(t μ θ)dθ,

Sν,μ (t) = 0Dt

−ν(1−μ)

0

Kμ (t).

Proof. Let λ > 0. Applying the Laplace transform ∞

χ(λ) = ∫ e

∞ −λs

x(s)ds

and ω(λ) = ∫ e−λs f (s, x(s))ds

0

0

to (2.2), we have χ(λ) = λ(1−ν)(1−μ)−1 x0 +

1 1 Aχ(λ) + μ ω(λ) λμ λ

= λν(μ−1) (λμ I − A) x0 + (λμ I − A) ω(λ) −1

∞

−1

∞

μ

(2.4)

μ

= λν(μ−1) ∫ e−λ s Q(s)x0 ds + ∫ e−λ s Q(s)ω(λ)ds, 0

0

provided that the integrals in (2.4) exist, where I is the identity operator defined on X. Let ψμ (θ) =

μ M (θ−μ ), θμ+1 μ

whose Laplace transform is given by ∞

μ

∫ e−λθ ψμ (θ)dθ = e−λ ,

where μ ∈ (0, 1).

(2.5)

0

Using (2.4), we have ∞

μ

∞

μ

∫ e−λ s Q(s)x0 ds = ∫ μt μ−1 e−(λt) Q(t μ )x0 dt 0

0 ∞∞

= ∫ ∫ μψμ (θ)e−(λtθ) Q(t μ )t μ−1 x0 dθdt 0 0 ∞∞

= ∫ ∫ μψμ (θ)e−λt Q( 0 0

t μ t μ−1 ) x dθdt θμ θμ 0

(2.6)

2.1 Initial value problems with Hilfer derivative |

∞

∞

= ∫ e−λt [μ ∫ ψμ (θ)Q( 0 ∞

0

33

t μ t μ−1 ) x dθ]dt θμ θμ 0

= ∫ e−λt t μ−1 Pμ (t)x0 dt, 0

and ∞

μ

∫ e−λ s Q(s)ω(λ)ds 0 ∞∞

μ

= ∫ ∫ μt μ−1 e−(λt) Q(t μ )e−λs f (s, x(s))dsdt 0 0 ∞∞∞

= ∫ ∫ ∫ μψμ (θ)e−(λtθ) Q(t μ )e−λs t μ−1 f (s, x(s))dθdsdt 0 0 0 ∞∞∞

= ∫ ∫ ∫ μψμ (θ)e 0 0 0 ∞ −λt

= ∫e

−λ(t+s)

t μ t μ−1 Q( μ ) μ f (s, x(s))dθdsdt θ θ

t ∞

[μ ∫ ∫ ψμ (θ)Q(

0 ∞

t

0

0

0 0

(2.7)

(t − s)μ (t − s)μ−1 ) f (s, x(s))dθds]dt θμ θμ

= ∫ e−λt [∫(t − s)μ−1 Pμ (t − s)f (s, x(s))ds]dt. Since the Laplace inverse transform of λν(μ−1) is L

−1

(λ

ν(μ−1)

t ν(1−μ)−1 , Γ(ν(1−μ))

)={ δ(t),

0 < ν ≤ 1,

ν = 0,

where δ(t) is the Delta function, by (2.4), (2.6) and (2.7), for t ∈ J ′ we obtain x(t) = (L (λ −1

ν(μ−1)

t

) ∗ Kμ (t))x0 + ∫ Kμ (t − s)f (s, x(s))ds t

= (0Dt

−ν(1−μ)

0

Kμ (t))x0 + ∫ Kμ (t − s)f (s, x(s))ds t

0

= Sν,μ (t)x0 + ∫ Kμ (t − s)f (s, x(s))ds. 0

This completes the proof.

(2.8)

34 | 2 Fractional evolution equations of order α ∈ (0, 1) Due to Lemma 2.2, we give the following definition of the mild solution of (2.1). Definition 2.1. By the mild solution of the Cauchy problem (2.1), we mean that the function x ∈ C(J ′ , X), which satisfies t

x(t) = Sν,μ (t)x0 + ∫ Kμ (t − s)f (s, x(s))ds,

t ∈ J′.

0

Remark 2.1. (i) By (2.8), we find ν(1−μ)

D0+ (ii)

Sν,μ (t) = Kμ (t),

t ∈ J′.

When ν = 0, the fractional equation (2.1) degenerates to the classical Riemann– Liouville fractional equation, which has been studied by Zhou et al. in [226]. In there, S0,μ (t) = Kμ (t) = t μ−1 Pμ (t),

t ∈ J′.

(iii) When ν = 1, the fractional equation (2.1) degenerates to the classical Caputo fractional equation, which has been studied by Zhou and Jiao in [230]. In there, S1,μ (t) = Sμ (t),

t ∈ J,

where Sμ (t) is defined in [230]. Proposition 2.1. Under assumption (H1), Pμ (t) is continuous in the uniform operator topology for t > 0. Proof. For any t > 0, h > 0 and x ∈ X, we have 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 μ μ 󵄨󵄨Pμ (t + h)x − Pμ (t)x󵄨󵄨 = 󵄨󵄨 ∫ μθMμ (θ)[Q((t + h) θ) − Q(t θ)]xdθ󵄨󵄨󵄨. 󵄨󵄨 󵄨󵄨 󵄨0 󵄨 Since ∞ 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∫ μθMμ (θ)[Q((t + h)μ θ) − Q(t μ θ)]xdθ󵄨󵄨󵄨 ≤ 2M ∫ μθMμ (θ)dθ|x| = 2M |x|, 󵄨󵄨 󵄨󵄨 Γ(μ) 󵄨󵄨 󵄨󵄨 0 0

then by Lebesgue dominated convergence theorem, we have 󵄨󵄨 󵄨 󵄨󵄨Pμ (t + h)x − Pμ (t)x󵄨󵄨󵄨 → 0

independently of t and x, as h → 0.

Therefore, Pμ (t) is continuous in the uniform operator topology for t > 0. This completes the proof.

2.1 Initial value problems with Hilfer derivative |

35

Proposition 2.2. Under assumption (H1), for any fixed t > 0, {Kμ (t)}t>0 and {Sν,μ (t)}t>0 are linear operators, and for any x ∈ X, μ−1 Mt (ν−1)(μ−1) 󵄨 󵄨 Mt 󵄨 󵄨󵄨 |x| and 󵄨󵄨󵄨Sν,μ (t)x󵄨󵄨󵄨 ≤ |x|. 󵄨󵄨Kμ (t)x󵄨󵄨󵄨 ≤ Γ(μ) Γ(ν(1 − μ) + μ)

Proof. From the equality ∞

∫ θδ Mμ (θ)dθ = 0

Γ(1 + δ) , Γ(1 + μδ)

we know that 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 M 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 μ |x|, 󵄨󵄨Pμ (t)x󵄨󵄨 = 󵄨󵄨 ∫ μθMμ (θ)Q(t θ)xdθ󵄨󵄨󵄨 ≤ 󵄨󵄨 󵄨󵄨 Γ(μ) 󵄨0 󵄨

for t ∈ J and x ∈ X,

then we have μ−1 󵄨󵄨 󵄨 Mt |x|, 󵄨󵄨Kμ (t)x󵄨󵄨󵄨 ≤ Γ(μ)

for t ∈ J ′ and x ∈ X.

For t ∈ J ′ and x ∈ X, 󵄨󵄨 󵄨 󵄨 −ν(1−μ) 󵄨 Kμ (t)x󵄨󵄨󵄨 󵄨󵄨Sν,μ (t)x󵄨󵄨󵄨 = 󵄨󵄨󵄨0Dt t 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 1 = 󵄨󵄨󵄨 ∫(t − s)ν(1−μ)−1 Kμ (s)xds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 Γ(ν(1 − μ)) 󵄨 󵄨 0

t 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 1 󵄨 = 󵄨󵄨 ∫(t − s)ν(1−μ)−1 sμ−1 Pμ (s)xds󵄨󵄨󵄨 󵄨󵄨 Γ(ν(1 − μ)) 󵄨󵄨 󵄨 󵄨 0

󵄨󵄨 󵄨󵄨 (ν−1)(1−μ) 1 󵄨󵄨 t 󵄨󵄨 = 󵄨󵄨󵄨 ∫(1 − s)ν(1−μ)−1 sμ−1 Pμ (ts)xds󵄨󵄨󵄨 󵄨󵄨 Γ(ν(1 − μ)) 󵄨󵄨 󵄨 󵄨 0

(2.9)

1

t (ν−1)(1−μ) M ≤ ∫(1 − s)ν(1−μ)−1 sμ−1 ds|x| Γ(ν(1 − μ))Γ(μ) 0

t M |x|. Γ(ν(1 − μ) + μ) (ν−1)(1−μ)

= This completes the proof.

Proposition 2.3. Under assumption (H1), {Kμ (t)}t>0 and {Sν,μ (t)}t>0 are strongly continuous, which means that, for any x ∈ X and 0 < t ′ < t ′′ ≤ b, we have 󵄨󵄨 ′ ′′ 󵄨 󵄨󵄨Kμ (t )x − Kμ (t )x 󵄨󵄨󵄨 → 0

and

󵄨󵄨 ′ ′′ 󵄨 󵄨󵄨Sν,μ (t )x − Sν,μ (t )x󵄨󵄨󵄨 → 0,

as t ′′ → t ′ .

36 | 2 Fractional evolution equations of order α ∈ (0, 1) Proof. By Proposition 2.1, we know that {Pμ (t)}t>0 is strongly continuous, then we easily obtain {Kμ (t)}t>0 is also strongly continuous. For any x ∈ X and 0 < t1 < t2 ≤ b, we have 󵄨 󵄨󵄨 󵄨󵄨Sν,μ (t2 )x − Sν,μ (t1 )x󵄨󵄨󵄨 1 = Γ(ν(1 − μ)) t t1 󵄨󵄨 󵄨󵄨󵄨 2 󵄨󵄨 󵄨󵄨 ν(1−μ)−1 × 󵄨󵄨∫(t2 − s) Kμ (s)xds − ∫(t1 − s)ν(1−μ)−1 Kμ (s)xds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨0 󵄨 0 1 = Γ(ν(1 − μ)) t1 t 󵄨󵄨 󵄨󵄨󵄨 2 󵄨󵄨 󵄨󵄨 ν(1−μ)−1 μ−1 × 󵄨󵄨∫(t2 − s) s Pμ (s)xds − ∫(t1 − s)ν(1−μ)−1 sμ−1 Pμ (s)xds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨0 0 󵄨󵄨 󵄨󵄨 t2 󵄨󵄨 1 󵄨󵄨󵄨 ν(1−μ)−1 μ−1 ≤ s Pμ (s)xds󵄨󵄨󵄨 󵄨∫(t − s) 󵄨󵄨 Γ(ν(1 − μ)) 󵄨󵄨󵄨󵄨 2 󵄨 t1

󵄨󵄨 t1 󵄨󵄨󵄨 󵄨󵄨 1 󵄨󵄨∫((t2 − s)ν(1−μ)−1 − (t1 − s)ν(1−μ)−1 )sμ−1 Pμ (s)xds󵄨󵄨󵄨 + 󵄨󵄨 󵄨 Γ(ν(1 − μ)) 󵄨󵄨󵄨 󵄨󵄨 ≤

μ−1 Mt1

0

1 (t − t )ν(1−μ) |x| Γ(ν(1 − μ))Γ(μ) ν(1 − μ) 2 1 t 󵄨󵄨 󵄨󵄨󵄨 1 󵄨󵄨 M 󵄨󵄨 ν(1−μ)−1 + − (t1 − s)ν(1−μ)−1 )sμ−1 ds󵄨󵄨󵄨|x|. 󵄨󵄨∫((t2 − s) 󵄨󵄨 󵄨 Γ(ν(1 − μ))Γ(μ) 󵄨󵄨 󵄨 0

Since t1 󵄨󵄨 t1 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨∫((t2 − s)ν(1−μ)−1 − (t1 − s)ν(1−μ)−1 )sμ−1 ds󵄨󵄨󵄨 ≤ 2 ∫(t1 − s)ν(1−μ)−1 sμ−1 ds 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 0 0

exists, then by Lebesgue dominated convergence theorem, we have 󵄨󵄨 󵄨󵄨 t1 󵄨 󵄨󵄨 󵄨󵄨∫((t2 − s)ν(1−μ)−1 − (t1 − s)ν(1−μ)−1 )sμ−1 ds󵄨󵄨󵄨 → 0 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 0

as t2 → t1 .

Consequently, we have 󵄨󵄨 󵄨 󵄨󵄨Sν,μ (t2 )x − Sν,μ (t1 )x 󵄨󵄨󵄨 → 0

as t2 → t1 ,

i. e., {Sν,μ (t)}t>0 is strongly continuous. This completes the proof.

(2.10)

2.1 Initial value problems with Hilfer derivative |

37

2.1.3 Existence of mild solutions For any x ∈ Y, define an operator T as follows: (Tx)(t) = (T1 x)(t) + (T2 x)(t), where t

(T1 x)(t) = Sν,μ (t)x0

and (T2 x)(t) = ∫ Kμ (t − s)f (s, x(s))ds,

for all t ∈ J ′ .

0

By (2.9) and (H3), we have lim t (1−ν)(1−μ) Sν,μ (t)x0

t→0+

= =

1

1 ∫(1 − s)ν(1−μ)−1 sμ−1 x0 ds Γ(ν(1 − μ))Γ(μ) x0 Γ(ν(1 − μ) + μ)

0

(2.11)

and t 󵄨󵄨 󵄨󵄨󵄨 Mt (1−ν)(1−μ) t 󵄨󵄨 (1−ν)(1−μ) 󵄨 󵄨󵄨t ∫ Kμ (t − s)f (s, x(s))ds󵄨󵄨󵄨 ≤ ∫(t − s)μ−1 m(s)ds → 0, 󵄨󵄨 󵄨󵄨󵄨 Γ(μ) 󵄨 󵄨 0 0

as t → 0+. Thus, we can define operator T as follows. For any y ∈ C(J, X), let x(t) = t (ν−1)(1−μ) y(t), (T y)(t) = (T1 y)(t) + (T2 y)(t), t (1−ν)(1−μ) (T1 x)(t),

for t ∈ (0, b],

t (1−ν)(1−μ) (T2 x)(t), (T2 y)(t) = { 0,

for t ∈ (0, b],

(T1 y)(t) = {

x0 , Γ(ν(1−μ)+μ)

for t = 0,

for t = 0.

Obviously, x is a mild solution of (2.1) in Y if and only if the operator equation y = T y has a solution y ∈ C(J, X). In the following, by using the measure of noncompactness and the Arzela–Ascoli theorem, we will obtain some sufficient conditions ensuring the existence of mild solution. Lemma 2.3. Assume that (H1)–(H3) hold, then {T y : y ∈ Br (J)} is equicontinuous.

38 | 2 Fractional evolution equations of order α ∈ (0, 1) Proof. Step I. The set {T1 y : y ∈ Br (J)} is equicontinuous. For any y ∈ Br (J), let x(t) = t (ν−1)(1−μ) y(t), t ∈ (0, b]. For 0 ≤ t1 < t2 ≤ b, we have (1−ν)(1−μ) 󵄨 󵄨 (1−ν)(1−μ) 󵄨󵄨 󵄨 Sν,μ (t2 )x0 − t1 Sν,μ (t1 )x0 󵄨󵄨󵄨 󵄨󵄨(T1 y)(t2 ) − (T1 y)(t1 )󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨t2 (1−ν)(1−μ) 󵄨 (1−ν)(1−μ) 󵄨 Sν,μ (t2 ) − t1 Sν,μ (t1 )󵄨󵄨󵄨|x0 |. ≤ 󵄨󵄨󵄨t2

By (2.11) and Proposition 2.3, we know that t (1−ν)(1−μ) Sν,μ (t) is uniformly continuous on J. Consequently, {T1 y : y ∈ Br (J)} is equicontinuous. Step II. The set {T2 y : y ∈ Br (J)} is equicontinuous. For any y ∈ Br (J), let x(t) = t (ν−1)(1−μ) y(t), t ∈ (0, b], then x ∈ BYr (J ′ ). For t1 = 0, 0 < t2 ≤ b, we get t2 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 (1−ν)(1−μ) 󵄨 ∫ Kμ (t2 − s)f (s, x(s))ds󵄨󵄨󵄨 󵄨󵄨(T2 y)(t2 ) − (T2 y)(0)󵄨󵄨 = 󵄨󵄨t2 󵄨󵄨 󵄨󵄨 󵄨 󵄨 0 t2

M (1−ν)(1−μ) t ≤ ∫(t2 − s)μ−1 m(s)ds Γ(μ) 2 0

→ 0,

as t2 → 0.

For 0 < t1 < t2 ≤ b, we have 󵄨󵄨 󵄨 󵄨󵄨(T2 y)(t2 ) − (T2 y)(t1 )󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 t2 󵄨󵄨 󵄨󵄨 (1−ν)(1−μ) ≤ 󵄨󵄨󵄨∫ t2 (t2 − s)μ−1 Pμ (t2 − s)f (s, x(s))ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨t1 󵄨 󵄨󵄨 t1 󵄨󵄨 (1−ν)(1−μ) + 󵄨󵄨󵄨∫ t2 (t2 − s)μ−1 Pμ (t2 − s)f (s, x(s))ds 󵄨󵄨 󵄨0 t1

− ∫ t1

(1−ν)(1−μ)

0

󵄨󵄨 󵄨󵄨 (t1 − s)μ−1 Pμ (t2 − s)f (s, x(s))ds󵄨󵄨󵄨 󵄨󵄨 󵄨

󵄨󵄨 t1 󵄨󵄨 (1−ν)(1−μ) + 󵄨󵄨󵄨∫ t1 (t1 − s)μ−1 Pμ (t2 − s)f (s, x(s))ds 󵄨󵄨 󵄨0 t1

−

≤

(1−ν)(1−μ) (t1 ∫ t1 0

μ−1

− s)

󵄨󵄨 󵄨󵄨 Pμ (t1 − s)f (s, x(s))ds󵄨󵄨󵄨 󵄨󵄨 󵄨

󵄨 t2 󵄨󵄨 󵄨󵄨 M 󵄨󵄨󵄨󵄨 (1−ν)(1−μ) (t2 − s)μ−1 m(s)ds󵄨󵄨󵄨 󵄨󵄨∫ t2 󵄨󵄨 Γ(μ) 󵄨󵄨󵄨 󵄨 t1

2.1 Initial value problems with Hilfer derivative | t1

M (1−ν)(1−μ) (1−ν)(1−μ) (t1 − s)μ−1 − t2 (t2 − s)μ−1 ]m(s)ds + ∫[t1 Γ(μ) 0

󵄨󵄨 t1 󵄨󵄨 󵄨󵄨 󵄨󵄨 (1−ν)(1−μ) μ−1 󵄨 (t1 − s) [Pμ (t2 − s) − Pμ (t1 − s)]f (s, x(s))ds󵄨󵄨󵄨 + 󵄨󵄨∫ t1 󵄨󵄨 󵄨󵄨 󵄨0 󵄨 ≤ I1 + I2 + I3 , where 󵄨 t2 M 󵄨󵄨󵄨󵄨 (1−ν)(1−μ) I1 = (t2 − s)μ−1 m(s)ds 󵄨∫ t Γ(μ) 󵄨󵄨󵄨󵄨 2 0

t1

− ∫ t1

(1−ν)(1−μ)

0

t1

󵄨󵄨 󵄨󵄨 (t1 − s)μ−1 m(s)ds󵄨󵄨󵄨, 󵄨󵄨 󵄨

2M (1−ν)(1−μ) (1−ν)(1−μ) I2 = (t1 − s)μ−1 − t2 (t2 − s)μ−1 ]m(s)ds, ∫[t1 Γ(μ) 0

󵄨 t1

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 (1−ν)(1−μ) (t1 − s)μ−1 [Pμ (t2 − s) − Pμ (t1 − s)]f (s, x(s))ds󵄨󵄨󵄨. I3 = 󵄨󵄨󵄨∫ t1 󵄨󵄨 󵄨󵄨 󵄨0 󵄨 By condition (H3), one can deduce that limt2 →t1 I1 = 0. Noting that (1−ν)(1−μ)

[t1

(t1 − s)μ−1 − t2

(1−ν)(1−μ)

≤ t1

(1−ν)(1−μ)

(t1 − s)μ−1 m(s)

(t2 − s)μ−1 ]m(s)

and t1

∫ t1

(1−ν)(1−μ)

(t1 − s)μ−1 m(s)ds

0

exist, then by Lebesgue dominated convergence theorem, we have t1 (1−ν)(1−μ)

∫[t1 0

(t1 − s)μ−1 − t2

(1−ν)(1−μ)

then one can deduce that limt2 →t1 I2 = 0.

(t2 − s)μ−1 ]m(s)ds → 0,

as t2 → t1 ,

39

40 | 2 Fractional evolution equations of order α ∈ (0, 1) For ε > 0 be enough small, we have t1 −ε

I3 ≤ ∫ t1

(1−ν)(1−μ)

0

󵄨 󵄨󵄨 󵄨 (t1 − s)μ−1 󵄨󵄨󵄨Pμ (t2 − s) − Pμ (t1 − s)󵄨󵄨󵄨󵄨󵄨󵄨f (s, x(s))󵄨󵄨󵄨ds

t1

+ ∫ t1

(1−ν)(1−μ)

t1 −ε

≤

t1 (1−ν)(1−μ) t1 ∫(t1 0

󵄨 󵄨󵄨 󵄨 (t1 − s)μ−1 󵄨󵄨󵄨Pμ (t2 − s) − Pμ (t1 − s)󵄨󵄨󵄨󵄨󵄨󵄨f (s, x(s))󵄨󵄨󵄨ds

󵄨 󵄨 − s)μ−1 m(s)ds sup 󵄨󵄨󵄨Pμ (t2 − s) − Pμ (t1 − s)󵄨󵄨󵄨 s∈[0,t1 −ε]

t1

2M (1−ν)(1−μ) + (t1 − s)μ−1 m(s)ds ∫ t1 Γ(μ) t1 −ε

≤ I31 + I32 + I33 , where I31 = I32

rΓ(μ) 󵄨 󵄨 sup 󵄨󵄨P (t − s) − Pμ (t1 − s)󵄨󵄨󵄨, M s∈[0,t1 −ε]󵄨 μ 2

󵄨 t1 2M 󵄨󵄨󵄨󵄨 (1−ν)(1−μ) (t1 − s)μ−1 m(s)ds = 󵄨∫ t Γ(μ) 󵄨󵄨󵄨󵄨 1 t1 −ε

0

− ∫ (t1 − ε)

(1−ν)(1−μ)

(t1 − ε − s)

0

I33 =

μ−1

󵄨󵄨 󵄨󵄨 m(s)ds󵄨󵄨󵄨, 󵄨󵄨 󵄨

󵄨 t1 −ε 2M 󵄨󵄨󵄨󵄨 (1−ν)(1−μ) (t1 − ε − s)μ−1 󵄨 ∫ [(t1 − ε) Γ(μ) 󵄨󵄨󵄨󵄨 0

− t1

(1−ν)(1−μ)

󵄨󵄨 󵄨󵄨 (t1 − s)μ−1 ]m(s)ds󵄨󵄨󵄨. 󵄨󵄨 󵄨

By Proposition 2.1, it is easy to see that I31 → 0 as t2 → t1 . Similar to the proof that I1 , I2 tend to zero, we get I32 → 0 and I33 → 0 as ε → 0. Thus, I3 tends to zero independently of y ∈ Br (J) as t2 → t1 , ε → 0. Therefore, |(T2 y)(t2 ) − (T2 y)(t1 )| → 0 independently of y ∈ Br (J) as t2 → t1 , which means that {T2 y : y ∈ Br (J)} is equicontinuous. Therefore, {T y : y ∈ B(J)} is equicontinuous. This completes the proof. Lemma 2.4. Assume that (H1)–(H3) hold, then T maps Br (J) into Br (J), and is continuous on Br (J).

2.1 Initial value problems with Hilfer derivative

| 41

Proof. Step I. The operator T maps Br (J) into Br (J). For any y ∈ Br (J), let x(t) = t (ν−1)(1−μ) y(t). Then x ∈ BYr (J ′ ). For t ∈ J, by Proposition 2.2, we have 󵄨󵄨 t 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 (1−ν)(1−μ) 󵄨󵄨 󵄨󵄨 (1−ν)(1−μ) 󵄨󵄨󵄨 󵄨󵄨 t ≤ ( T y)(t) S (t)x + t K (t − s)f (s, x(s))ds ∫ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 μ ν,μ 0 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨0 t

≤

M|x0 | Mt (1−ν)(1−μ) 󵄨 󵄨 + ∫(t − s)μ−1 󵄨󵄨󵄨f (s, x(s))󵄨󵄨󵄨ds Γ(ν(1 − μ) + μ) Γ(μ) 0

t

|x0 | t (1−ν)(1−μ) ≤ M( + sup{ ∫(t − s)μ−1 m(s)ds}) Γ(ν(1 − μ) + μ) t∈J Γ(μ) 0

≤ r. Hence, ‖T y‖ ≤ r, for any y ∈ Br (J). Step II. The operator T is continuous in Br (J). For any ym , y ∈ Br (J), m = 1, 2, . . . , with limm→∞ ym = y, we have lim y (t) m→∞ m

= y(t)

and

lim t (ν−1)(1−μ) ym (t) = t (ν−1)(1−μ) y(t),

m→∞

for t ∈ J ′ .

Then by (H2), we have f (t, xm (t)) = f (t, t (ν−1)(1−μ) ym (t)) → f (t, t (ν−1)(1−μ) y(t)) = f (t, x(t)),

as m → ∞,

where xm (t) = t (ν−1)(1−μ) ym (t) and x(t) = t (ν−1)(1−μ) y(t). On the one hand, using (H3), we get for each t ∈ J ′ , 󵄨 󵄨 (t − s)μ−1 󵄨󵄨󵄨f (s, xm (s)) − f (s, x(s))󵄨󵄨󵄨 ≤ (t − s)μ−1 2m(s),

a. e. in [0, t),

and the function s → (t − s)μ−1 2m(s) is integrable for s ∈ [0, t) and t ∈ J ′ . By the Lebesgue dominated convergence theorem, we have t

󵄨 󵄨 ∫(t − s)μ−1 󵄨󵄨󵄨f (s, xm (s)) − f (s, x(s))󵄨󵄨󵄨ds → 0,

as m → 0.

0

Thus, for t ∈ J, 󵄨󵄨 󵄨 󵄨󵄨(T ym )(t) − (T y)(t)󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 t 󵄨󵄨 (1−ν)(1−μ) 󵄨󵄨󵄨 󵄨 ≤t 󵄨󵄨󵄨∫ Kμ (t − s)(f (s, xm (s)) − f (s, x(s)))ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 0

42 | 2 Fractional evolution equations of order α ∈ (0, 1) t

Mt (1−ν)(1−μ) 󵄨 󵄨 ≤ ∫(t − s)μ−1 󵄨󵄨󵄨f (s, xm (s)) − f (s, x(s))󵄨󵄨󵄨ds Γ(μ) 0

→ 0,

as m → ∞.

Therefore, T ym → T y pointwise on J as m → ∞, by which Lemma 2.3 implies that T ym → T y uniformly on J as m → ∞ and so T is continuous. This completes the proof. Proposition 2.4 ([215]). Suppose β > 0, a(t) is a nonnegative function locally integrable on 0 ≤ t < T (some T ≤ +∞) and g(t) is a nonnegative, nondecreasing, continuous function defined on 0 ≤ t < T, g(t) is bounded, and suppose u(t) is nonnegative and locally integrable on 0 ≤ t < T with t

u(t) ≤ a(t) + g(t) ∫(t − s)β−1 u(s)dt 0

on this interval. Then t

(g(t)Γ(β))n (t − s)β−1 a(s)]ds, Γ(nβ) n=1 ∞

u(t) ≤ a(t) + ∫[ ∑ 0

0 ≤ t < T.

Especially, when a(t) = 0, then u(t) = 0 for all 0 ≤ t < T. Theorem 2.1. Assume that (H1)–(H4) hold, then the Cauchy problem (2.1) has at least one mild solution in BYr (J ′ ). Proof. Let y0 (t) = t (1−ν)(1−μ) Sν,μ (t)x0 for all t ∈ J and ym+1 = T ym , m = 0, 1, 2, . . . . Consider set H = {ym : m = 0, 1, 2, . . .}, and we will prove set H is relatively compact. It follows from Lemmas 2.3 and 2.4 that H is uniformly bounded and equicontinuous on J. Next, we only prove that for any t ∈ J, set H (t) = {ym (t), m = 0, 1, 2, . . .} is relatively compact in X. Under the condition (H4), by the properties of measure of noncompactness and Proposition 1.20, for any t ∈ J we have β(H (t)) = β({ym (t)}m=0 ) = β({y0 (t)} ∪ {ym (t)}m=1 ) = β({ym (t)}m=1 ) ∞

∞

∞

and β({ym (t)}m=1 ) ∞

= β({(T ym )(t)}m=0 ) = β({t

∞

(1−ν)(1−μ)

Sν,μ (t)x0 + t

t (1−ν)(1−μ)

∫ Kμ (t − s)f (s, xm (s))ds}

∞

0

m=0

)

2.1 Initial value problems with Hilfer derivative t

= β({t

(1−ν)(1−μ)

| 43

∞

∫ Kμ (t − s)f (s, xm (s))ds}

m=0

0

)

t

≤

2M (1−ν)(1−μ) ∞ t ∫(t − s)μ−1 β(f (s, {s(ν−1)(1−μ) ym (s)}m=0 ))ds Γ(μ) 0

t

2Ml (1−ν)(1−μ) ∞ ≤ t ∫(t − s)μ−1 s(1−ν)(1−μ) β({s(ν−1)(1−μ) ym (s)}m=0 )ds Γ(μ) 0 t

≤

2Mlt (1−ν)(1−μ) ∞ ∫(t − s)μ−1 β({ym (s)}m=0 )ds, Γ(μ) 0

then t

β(H (t)) ≤

2Mlt (1−ν)(1−μ) ∫(t − s)μ−1 β(H (s))ds. Γ(μ) 0

Therefore, by Propositions 1.18 and 2.4, we obtain that β(H (t)) = 0, then H (t) is

relatively compact. Consequently, it follows from the Arzela–Ascoli theorem that set

H is relatively compact, i. e., there exists a convergent subsequence of {ym }∞ m=0 . With

no confusion, let limm→∞ ym = y∗ ∈ Br (J).

Thus, by continuity of the operator T , we have y∗ = lim ym = lim T ym−1 = T ( lim ym−1 ) = T y∗ , m→∞

m→∞

m→∞

which implies the Cauchy problem (2.1) has at least a mild solution. This completes the proof.

Remark 2.2. The assumption (H2) is replaced by the following assumption: 1

(H2′ ) there exists a constant μ1 ∈ (0, μ) and m ∈ L μ1 (J, ℝ+ ) such that 󵄨󵄨 󵄨 󵄨󵄨f (t, x)󵄨󵄨󵄨 ≤ m(t),

for all x ∈ X and almost all t ∈ J.

Then we have the following corollary. Corollary 2.1. Assume that (H1), (H2′ ), (H3) and (H4) hold, then the Cauchy prob-

lem (2.1) has at least one mild solution in BYr (J ′ ).

Remark 2.3. Obviously, our results can be applied to the evolution equations with the Riemann–Liouville fractional derivative and Caputo fractional derivative.

44 | 2 Fractional evolution equations of order α ∈ (0, 1)

2.2 Initial value problems with nondense domain 2.2.1 Introduction We consider the nonhomogeneous fractional-order evolution equation C q D u(t) = Au(t) + f (t), t ∈ (0, b], {0 t u(0) = u0 ,

(2.12)

and the nonlinear fractional-order evolution equation C q D u(t) = Au(t) + g(t, u(t)), {0 t u(0) = u0 ,

t ∈ (0, b],

(2.13)

where C0Dqt is the Caputo fractional derivative of order 0 < q < 1, the state u(⋅) takes values in a Banach space X with norm | ⋅ |, A : D(A) ⊆ X → X is a nondensely closed linear operator on X, f and g are given functions satisfying appropriate conditions. For the integer-order evolution equation: u′ (t) = Au(t) + f (t, u(t)),

{

u(0) = u0 ,

t ∈ (0, b],

in case A is a Hille–Yosida operator and is densely defined (i. e., D(A) = X), the problem has been extensively studied (see [169]). When A is a Hille–Yosida operator but its domain is nondensely defined, there are many results (see [1, 100, 193, 194] and the references therein). It is noted that Da Prato and Sinestrari are the first to work on equations with nondense domains; see [76]. For nonlinear fractional evolution equation (2.13) with initial data or nonlocal condition, when A is densely defined, there have been many results on the existence of mild solutions (see [141, 170, 205, 230]). In [230], Zhou and Jiao proposed a suitable concept on mild solution by applying a probability density function and Laplace transform, which is widely used now. When A is not densely defined, there have been some investigations (see [166]). Motivated by the above discussion, in this section, we will first give the integral solution for nonhomogeneous fractional evolution equation (2.12) by Laplace transform and probability density function, and subsequently investigate the existence of integral solution for the nonlinear fractional-order evolution equation (2.13) by the Arzela–Ascoli theorem and the measure of noncompactness. In what follows, we do not require the C0 -semigroup (will be given later) to be compact. This section is based on [113].

2.2 Initial value problems with nondense domain

| 45

2.2.2 Integral solution to nonhomogeneous Cauchy problem Let X0 = D(A) and A0 be the part of A in D(A) defined by D(A0 ) = {x ∈ D(A) : Ax ∈ D(A)},

A0 x = Ax.

Proposition 2.5 ([169]). The part A0 of A generates a strongly continuous semigroup (i. e., C0 -semigroup) {Q(t)}t≥0 on X0 . In the forthcoming analysis, we need the following hypotheses. (H1) The linear operator A : D(A) ⊂ X → X satisfies the Hille–Yosida condition, that is, there exist two constant ω ∈ ℝ and M such that (ω, +∞) ⊆ ρ(A) and M 󵄩󵄩 −k 󵄩 , 󵄩󵄩(λI − A) 󵄩󵄩󵄩B(X) ≤ (λ − ω)k

for all λ > ω, k ≥ 1.

(H2) Q(t) is continuous in the uniform operator topology for t > 0, and {Q(t)}t≥0 is uniformly bounded, that is, there exists M > 1 such that supt∈[0,+∞) |Q(t)| < M. Here, we derive the integral solution for the nonhomogeneous fractional-order evolution equation (2.12) with the aid of Laplace transform and the probability density function. Let J = [0, b]. For the Cauchy problem (2.12), it is assumed that u0 ∈ X0 and f : J → X is continuous. Definition 2.2. A function u(t) is said to be an integral solution of (2.12) if: (i) u : J → X is continuous; (ii) 0 D−q t u(t) ∈ D(A) for t ∈ J and (iii) −q u(t) = u0 + A0 D−q t u(t) + 0 Dt f (t),

t ∈ J.

(2.14)

Remark 2.4. If u(t) is an integral solution of (2.12), then u(t) ∈ X0 for t ∈ J. In fact, by −q −(1−q) −q −1 0 Dt u(t) ∈ D(A), we have 0 Dt u(t) = 0 Dt 0 Dt u(t) ∈ D(A) for t ∈ J. Then u(t) = limh→0+

1 t+h ∫ u(s)ds h t

∈ X0 for t ∈ J.

Consider the auxiliary problem C q D u(t) = A0 u(t) + f (t), t ∈ (0, b], {0 t u(0) = u0 .

(2.15)

By Definition 2.2, the integral solution of (2.15) can be written as −q u(t) = u0 + A0 0 D−q t u(t) + 0 Dt f (t)

(2.16)

46 | 2 Fractional evolution equations of order α ∈ (0, 1) for u0 ∈ X0 and t ∈ J. The following lemma gives an equivalent form of (2.16) by means

of Laplace transform.

Lemma 2.5. If f takes values in X0 , then the integral equation (2.16) can be expressed

as

t

u(t) =

(0 D−(1−q) Kq (t))u0 t

+ ∫ Kq (t − s)f (s)ds,

t ∈ J,

(2.17)

0

where Kq (t) = t q−1 Pq (t),

∞

Pq (t) = ∫ qθMq (θ)Q(t q θ)dθ. 0

Proof. Let λ > 0. Applying the Laplace transform, ∞

χ(λ) = ∫ e

∞ −λs

u(s)ds and ω(λ) = ∫ e−λs f (s)ds

0

0

to (2.16), we obtain χ(λ) = λ−1 u0 +

1 0 1 A χ(λ) + q ω(λ) λq λ

= λq−1 (λq I − A0 ) u0 + (λq I − A0 ) ω(λ) −1

∞

∞

q

−1

(2.18)

q

= λq−1 ∫ e−λ s Q(s)u0 ds + ∫ e−λ s Q(s)ω(λ)ds, 0

0

provided that the integrals in (2.18) exist, where I is the identity operator defined on X.

The Laplace transform of ψq (θ) =

q

θq+1

Mq (θ−q ),

is ∞

q

∫ e−λθ ψq (θ)dθ = e−λ , 0

(2.19)

2.2 Initial value problems with nondense domain

| 47

where q ∈ (0, 1). Using (2.19), we have ∞

∫e

−λq s

∞

q

Q(s)u0 ds = ∫ qt q−1 e−(λt) Q(t q )u0 dt

0

0 ∞∞

= ∫ ∫ qψq (θ)e−(λtθ) Q(t q )t q−1 u0 dθdt 0 0 ∞∞

= ∫ ∫ qψq (θ)e−λt Q( 0 0 ∞ −λt

= ∫e

t q t q−1 ) u dθdt θq θq 0

∞

[q ∫ ψq (θ)Q( 0

0 ∞

(2.20)

t q t q−1 ) u dθ]dt θq θq 0

= ∫ e−λt t q−1 Pq (t)u0 dt 0

and ∞

q

∫ e−λ s Q(s)ω(λ)ds 0 ∞∞

q

= ∫ ∫ qt q−1 e−(λt) Q(t q )e−λs f (s)dsdt 0 0 ∞∞∞

= ∫ ∫ ∫ qψq (θ)e−(λtθ) Q(t q )e−λs t q−1 f (s)dθdsdt 0 0 0 ∞∞∞

= ∫ ∫ ∫ qψq (θ)e 0 0 0

−λ(t+s)

t q t q−1 Q( q ) q f (s)dθdsdt θ θ

t ∞

∞

= ∫ e−λt [q ∫ ∫ ψq (θ)Q( 0 ∞

t

0

0

0 0

(t − s)q (t − s)q−1 ) f (s)dθds]dt θq θq

= ∫ e−λt [∫(t − s)q−1 Pq (t − s)f (s)ds]dt.

Since the Laplace inverse transform of λq−1 is L

−1

(λq−1 ) =

t −q = g1−q (t), Γ(1 − q)

(2.21)

48 | 2 Fractional evolution equations of order α ∈ (0, 1) therefore, by (2.18), (2.20) and (2.21), for t ∈ J, we obtain t

u(t) = (L −1 (λq−1 ) ∗ Kq (t))u0 + ∫ Kq (t − s)f (s)ds t

0

(2.22)

= (0 D−(1−q) Kq (t))u0 + ∫ Kq (t − s)f (s)ds. t 0

This completes the proof. Remark 2.5. Let Sq (t) = 0 Dt−(1−q) Kq (t). By the uniqueness of Laplace inverse transform, it is obvious that operators Sq (t) and Pq (t) (obtained here) are the same as the ones given in [225]. In addition, we also obtain the relationship between Sq (t) and Kq (t);

that is, Sq (t) = 0 D−(1−q) Kq (t) for t ≥ 0. So, we can say that {Kq (t)}t≥0 is generated by A0 . t Proposition 2.6 ([225]). With assumption (H2), Pq (t) is continuous in the uniform operator topology for t > 0. Proposition 2.7 ([225]). With assumption (H2), for any fixed t > 0, {Kq (t)}t>0 and {Sq (t)}t>0 are linear operators, and for any x ∈ X0 , q−1 󵄨󵄨 󵄨 Mt |x| 󵄨󵄨Kq (t)x󵄨󵄨󵄨 ≤ Γ(q)

󵄨 󵄨 and 󵄨󵄨󵄨Sq (t)x 󵄨󵄨󵄨 ≤ M|x|.

Proposition 2.8 ([225]). With assumption (H2), {Kq (t)}t>0 and {Sq (t)}t>0 are strongly continuous, that is, for any x ∈ X0 and 0 < t ′ < t ′′ ≤ b, 󵄨󵄨 ′ ′′ 󵄨 󵄨󵄨Kq (t )x − Kq (t )x 󵄨󵄨󵄨 → 0

and

󵄨󵄨 ′ ′′ 󵄨 󵄨󵄨Sq (t )x − Sq (t )x󵄨󵄨󵄨 → 0,

as t ′′ → t ′ .

If we assume that f takes values in X0 , then (2.17) can be written as t

u(t) = Sq (t)u0 + ∫ Kq (t − s) lim Bλ f (s)ds λ→+∞

0

(2.23)

or t

u(t) = Sq (t)u0 + lim ∫ Kq (t − s)Bλ f (s)ds, λ→+∞

(2.24)

0

where Bλ = λ(λI − A)−1 , since limλ→+∞ Bλ x = x for x ∈ X0 . When f takes values in X, but not in X0 , then the limit in (2.24) exists (as we will prove), but the limit in (2.23) will no longer exist. Lemma 2.6. Any solution of integral equation (2.14) with values in X0 is represented by (2.24).

2.2 Initial value problems with nondense domain

| 49

Proof. Let uλ (t) = Bλ u(t),

fλ (t) = Bλ f (t),

uλ = Bλ u0 .

By applying Bλ to (2.14), we have −q uλ (t) = uλ + A0 0 D−q t uλ (t) + 0 Dt fλ (t).

Hence, by Lemma 2.5, we obtain t

uλ (t) = Sq (t)uλ + ∫ Kq (t − s)fλ (s)ds. 0

As u(t), u0 ∈ X0 , we have uλ (t) → u(t),

uλ → u0 ,

Sq (t)uλ → Sq (t)u0 ,

as λ → +∞.

Thus (2.24) holds. This completes the proof. Let us define t

t

Φq (t)x = lim ∫ Kq (t − s)Bλ x ds = lim ∫ Kq (s)Bλ x ds, λ→+∞

λ→+∞

0

(2.25)

0

for x ∈ X and t ≥ 0. Proposition 2.9. For x ∈ X and t ≥ 0, the limit in (2.25) exists and defines a bounded linear operator Φq (t). Proof. Let t

t

0

0

Φ0q (t)x = ∫ Kq (t − s)x ds = ∫ Kq (s)x ds, for x0 ∈ X0 and t ≥ 0. Then the definition Φq (t) = (λI − A)Φ0q (t)(λI − A)−1 , for λ > ω, extends Φ0q (t) from X0 to X. This definition is independent of λ because of the resolvent identity. As Φq (t) maps X into X0 , we have Φq (t)x = lim Bλ Φq (t)x = lim Φ0q (t)Bλ x. λ→+∞

This completes the proof.

λ→+∞

50 | 2 Fractional evolution equations of order α ∈ (0, 1) Proposition 2.10. For x ∈ X0 and t ≥ 0, CDq0+ Φ0q (t)x = Sq (t)x and Sq (t)x = AΦ0q (t)x + x. The proof of the above proposition follows directly from the definitions of Sq (t) and Φ0q (t) for t ≥ 0. Lemma 2.7. (i) For x ∈ X and t ≥ 0, 0 D−q t Φq (t) ∈ D(A) and Φq (t)x = A(0 D−q t Φq (t)x) +

tq x. Γ(1 + q)

(2.26)

(ii) For x ∈ D(A), Φq (t)Ax + x = Sq (t)x.

(2.27)

Proof. (i) For x ∈ X and t ≥ 0, let 0 −1 V(t) = λ0 D−q t Φq (t)(λI − A) x +

tq (λI − A)−1 x − Φ0q (t)(λI − A)−1 x. Γ(1 + q)

Clearly V(0) = 0. By Proposition 2.10, we have C q D0+ V(t) = λΦ0q (t)(λI = λΦ0q (t)(λI = λΦ0q (t)(λI = λΦ0q (t)(λI

− A)−1 x + (λI − A)−1 x − CDq0+ Φ0q (t)(λI − A)−1 x − A)−1 x + (λI − A)−1 x − Sq (t)(λI − A)−1 x

− A)−1 x + (λI − A)−1 x − AΦ0q (t)(λI − A)−1 x − (λI − A)−1 x − A)−1 x − AΦ0q (t)(λI − A)−1 x

= (λI − A)Φ0q (t)(λI − A)−1 x = Φq (t)x.

Then −q V(t) = 0 D−q t Φq (t)x + V(0) = 0 Dt Φq (t)x

and −q (λI − A)V(t) = (λI − A)0 D−q t Φq (t)x = λ0 Dt Φq (t)x +

tq x − Φq (t)x. Γ(1 + q)

Thus Φq (t)x = A(0 D−q t Φq (t)x) +

tq x. Γ(1 + q)

2.2 Initial value problems with nondense domain

| 51

(ii) For x ∈ D(A), it follows by Proposition 2.10 that t

t

Φq (t)Ax = lim ∫ Kq (s)Bλ Ax ds = lim A0 ∫ Kq (s)Bλ x ds λ→+∞

λ→+∞

0

0

= A0 Φ0q (t)x = Sq (t)x − x. This completes the proof.

Theorem 2.2. u(t) is an integral solution of (2.12) if and only if t

u(t) = Sq (t)u0 + lim ∫ Kq (t − s)Bλ f (s)ds λ→+∞

(2.28)

0

for t ∈ J and u0 ∈ X0 . Proof. In view of Lemma 2.6, we only need to show that (2.28) is the integral solution of (2.12). Indeed it is sufficient to prove the theorem for u0 = 0, because it can easily be proved for the special case f = 0. We complete the proof in two steps. Step I. Assume that f is continuously differentiable, then for t ∈ J, we have t

uλ (t) = ∫ Kq (s)Bλ f (s)ds 0

s

t

= ∫ Kq (s)Bλ (f (0) + ∫ f ′ (r)dr)ds 0

0

t

t

s

= ∫ Kq (s)Bλ f (0)ds + ∫ Kq (s)Bλ (∫ f ′ (r)dr)ds 0

t

0

0

= Φ0q (t)Bλ f (0) + ∫ Φ0q (t − r)Bλ f ′ (r)dr. 0

By Lemma 2.7, for t ∈ J, we obtain u(t) = lim uλ (t) λ→+∞

t

= Φq (t)f (0) + ∫ Φq (t − r)f ′ (r)dr 0

= A(0 D−q t Φq (t)f (0)) +

tq f (0) Γ(1 + q)

52 | 2 Fractional evolution equations of order α ∈ (0, 1) t

+ ∫[A(0 D−q t Φq (t − r)) + 0

(t − r)q ′ ]f (r)dr Γ(1 + q)

t

−q ′ = A[0 D−q t Φq (t)f (0) + ∫ 0 Dt Φq (t − r)f (r)dr] 0

q

t

1 t f (0) + + ∫(t − r)q f ′ (r)dr Γ(1 + q) Γ(1 + q) 0

t

−q ′ = A[0 D−q t Φq (t)f (0) + 0 Dt (∫ Φq (t − r)f (r)dr)] 0

t

q

+

t 1 f (0) + ∫(t − r)q f ′ (r)dr Γ(1 + q) Γ(1 + q) 0

=

A(0 D−q t u(t))

+

−q 0 Dt f (t).

Step II. We approximate f by continuously differentiable functions fn such that 󵄨 󵄨 sup󵄨󵄨󵄨f (t) − fn (t)󵄨󵄨󵄨 → 0, t∈J

as n → ∞.

Letting t

un (t) = lim ∫ Kq (s)Bλ fn (s)ds, λ→∞

0

we have −q un (t) = A(0 D−q t un (t)) + 0 Dt fn (t).

Then t 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨un (t) − um (t)󵄨󵄨 = 󵄨󵄨 lim ∫ Kq (s)Bλ [fn (s) − fm (s)]ds󵄨󵄨󵄨 󵄨󵄨λ→∞ 󵄨󵄨 󵄨 󵄨 0 t

MM 󵄨 󵄨 ≤ ∫(t − s)q−1 󵄨󵄨󵄨fn (s) − fm (s)󵄨󵄨󵄨ds Γ(q) 0

MMbq ≤ ‖f − f ‖, Γ(q) n m

(2.29)

2.2 Initial value problems with nondense domain

| 53

which implies that {un } is a Cauchy sequence and its limit, denoted by u(t), exists. Taking limit on both sides of (2.29), we obtain −q u(t) = A(0 D−q t u(t)) + 0 Dt f (t),

for t ∈ J.

Therefore, (2.28) is the integral solution of (2.12). This completes the proof. Remark 2.6. (i) Integrating the last term in (2.28) and using Proposition 2.8, the integral solution (2.28) can be expressed as t

u(t) = Sq (t)u0 +

d ∫ Φq (t − s)f (s)ds. dt 0

(ii)

(λq I − A)−1 x = λ ∫0 e−λt Φq (t)x ds for x ∈ X and λq > ω. In fact, by taking Laplace transform of (2.26), we obtain ∞

L [Φq (t)x] = AL [0 Dt Φq (t)x] + L [ −q

tq x] Γ(1 + q)

= λ−q AL [Φq (t)x] + λ−q−1 x = λ−1 (λq I − A) x. −1

(iii) We can say that A generates the operator {Φq (t)}t≥0 . When q = 1, {Φq (t)}t≥0 degenerates into {S(t)}t≥0 , which is integrated semigroup generated by A in [193]. 2.2.3 Integral solution to nonlinear Cauchy problem In this subsection, we study the existence of the integral solution of the nonlinear fractional evolution equation (2.13). We need the following assumptions: (H3) for each t ∈ J, the function g(t, ⋅) : X → X is continuous and for each x ∈ X, the function g(⋅, x) : J → X is strongly measurable; (H4) there exists a function m ∈ L(J, ℝ+ ) such that −q 0 Dt m(t)

∈ C(J, ℝ+ ),

󵄨󵄨 󵄨 󵄨󵄨g(t, x)󵄨󵄨󵄨 ≤ m(t)

lim 0 D−q t m(t) = 0,

t→0+

for all x ∈ X and almost all t ∈ J;

(H5) there exists a constant l > 0 such that for any bounded D ⊆ X, β(g(t, D)) ≤ lβ(D),

for a. e. t ∈ J,

where β is the Hausdorff measure of noncompactness.

54 | 2 Fractional evolution equations of order α ∈ (0, 1) By Theorem 2.2, it is easy to see that the integral solution of (2.13) is equal to the solution of t

d u(t) = Sq (t)u0 + ∫ Φq (t − s)g(s, u(s))ds dt

(2.30)

0

or t

u(t) = Sq (t)u0 + lim ∫ Kq (t − s)Bλ g(s, u(s))ds. λ→+∞

(2.31)

0

For u ∈ C(J, X0 ), define an operator (T u)(t) = (T1 u)(t) + (T2 u)(t), where t

(T1 u)(t) = Sq (t)u0

and (T2 u)(t) = lim ∫ Kq (t − s)Bλ g(s, u(s))ds, λ→+∞

0

for all t ∈ J. Let Br (J) = {u ∈ C(J, X0 ) : ‖u‖ ≤ r}. Lemma 2.8. Suppose that conditions (H1)–(H4) hold. Then {T u : u ∈ Br (J)} is equicontinuous. Proof. By Proposition 2.8, Sq (t)u0 is uniformly continuous on J. Consequently, {T1 u : u ∈ Br (J)} is equicontinuous. For u ∈ Br (J), taking t1 = 0, 0 < t2 ≤ b, we obtain t2 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 󵄨 󵄨󵄨(T2 u)(t2 ) − (T2 u)(0)󵄨󵄨 = 󵄨󵄨 lim ∫ Kq (t − s)Bλ g(s, u(s))ds󵄨󵄨󵄨 󵄨󵄨λ→+∞ 󵄨󵄨 󵄨 󵄨 0 t2

MM ≤ ∫(t2 − s)q−1 m(s)ds → 0, Γ(q) 0

For 0 < t1 < t2 ≤ b, we have 󵄨󵄨 󵄨 󵄨󵄨(T2 u)(t2 ) − (T2 u)(t1 )󵄨󵄨󵄨

t2 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 q−1 󵄨 ≤ 󵄨󵄨 lim ∫(t2 − s) Pq (t2 − s)Bλ g(s, u(s))ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨λ→+∞ 󵄨 󵄨 t1 t1 󵄨󵄨 󵄨󵄨 󵄨 + 󵄨󵄨 lim ∫(t2 − s)q−1 Pq (t2 − s)Bλ g(s, u(s))ds 󵄨󵄨λ→+∞ 󵄨 0

as t2 → 0.

2.2 Initial value problems with nondense domain |

55

t1

󵄨󵄨 󵄨󵄨 − lim ∫(t1 − s)q−1 Pq (t2 − s)Bλ g(s, u(s))ds󵄨󵄨󵄨 󵄨󵄨 λ→+∞ 󵄨 0

t1 󵄨󵄨 󵄨󵄨 󵄨 + 󵄨󵄨 lim ∫(t1 − s)q−1 Pq (t2 − s)Bλ g(s, u(s))ds 󵄨󵄨λ→+∞ 󵄨 0 t1 󵄨󵄨 󵄨󵄨 − lim ∫(t1 − s)q−1 Pq (t1 − s)Bλ g(s, u(s))ds󵄨󵄨󵄨 󵄨󵄨 λ→+∞ 󵄨 0

󵄨 t2 󵄨󵄨 󵄨󵄨 MM 󵄨󵄨󵄨󵄨 q−1 ≤ 󵄨󵄨∫(t2 − s) m(s)ds󵄨󵄨󵄨 󵄨󵄨 Γ(q) 󵄨󵄨󵄨 󵄨 t1

t1

+

MM ∫[(t1 − s)q−1 − (t2 − s)q−1 ]m(s)ds Γ(q) 0

t1 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 + 󵄨󵄨 lim ∫(t1 − s)q−1 [Pq (t2 − s) − Pq (t1 − s)]Bλ g(s, u(s))ds󵄨󵄨󵄨 󵄨󵄨λ→+∞ 󵄨󵄨 󵄨 󵄨 0

≤ I1 + I2 + I3 , where I1 =

t1 󵄨 t2 󵄨󵄨 󵄨󵄨 MM 󵄨󵄨󵄨󵄨 q−1 q−1 󵄨󵄨∫(t2 − s) m(s)ds − ∫(t1 − s) m(s)ds󵄨󵄨󵄨, 󵄨󵄨 Γ(q) 󵄨󵄨󵄨 󵄨 0

I2 =

t1

0

2MM ∫[(t1 − s)q−1 − (t2 − s)q−1 ]m(s)ds, Γ(q) 0

t1 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 I3 = 󵄨󵄨 lim ∫(t1 − s)q−1 [Pq (t2 − s) − Pq (t1 − s)]Bλ g(s, u(s))ds󵄨󵄨󵄨. 󵄨󵄨λ→+∞ 󵄨󵄨 󵄨 󵄨 0

By condition (H4), one can deduce that limt2 →t1 I1 = 0. Noting that [(t1 − s)q−1 − (t2 − s)q−1 ]m(s) ≤ (t1 − s)q−1 m(s), t

and ∫01 (t1 − s)q−1 m(s)ds exists, it follows by the Lebesgue dominated convergence theorem that t1

∫[(t1 − s)q−1 − (t2 − s)q−1 ]m(s)ds → 0, 0

which implies that limt2 →t1 I2 = 0.

as t2 → t1 ,

56 | 2 Fractional evolution equations of order α ∈ (0, 1) For ε > 0 small enough, by (H4), we have t1 −ε

󵄨 󵄨󵄨 󵄨 I3 ≤ M ∫ (t1 − s)q−1 󵄨󵄨󵄨Pq (t2 − s) − Pq (t1 − s)󵄨󵄨󵄨󵄨󵄨󵄨g(s, u(s))󵄨󵄨󵄨ds 0

t1

󵄨 󵄨󵄨 󵄨 + M ∫ (t1 − s)q−1 󵄨󵄨󵄨Pq (t2 − s) − Pq (t1 − s)󵄨󵄨󵄨󵄨󵄨󵄨g(s, u(s))󵄨󵄨󵄨ds t1 −ε

t1

󵄨 󵄨 ≤ M ∫(t1 − s)q−1 m(s)ds sup 󵄨󵄨󵄨Pq (t2 − s) − Pq (t1 − s)󵄨󵄨󵄨 s∈[0,t1 −ε]

0

t1

2MM + ∫ (t1 − s)q−1 m(s)ds Γ(q) t1 −ε

≤ I31 + I32 + I33 , where I31 = I32

rΓ(q) MM

󵄨 󵄨 sup 󵄨󵄨󵄨Pq (t2 − s) − Pq (t1 − s)󵄨󵄨󵄨,

s∈[0,t1 −ε]

t1 −ε 󵄨 t1 󵄨󵄨 󵄨󵄨 2MM 󵄨󵄨󵄨󵄨 q−1 q−1 = 󵄨󵄨∫(t1 − s) m(s)ds − ∫ (t1 − ε − s) m(s)ds󵄨󵄨󵄨, 󵄨 󵄨󵄨 Γ(q) 󵄨󵄨 󵄨 0

0

t1 −ε

I33 =

2MM ∫ [(t1 − ε − s)q−1 − (t1 − s)q−1 ]m(s)ds. Γ(q) 0

By Proposition 2.6, it follows that I31 → 0 as t2 → t1 . Applying the arguments similar to the ones employed in proving that I1 , I2 tend to zero, we obtain I32 → 0 and I33 → 0 as ε → 0. Thus, I3 tends to zero independently of u ∈ Br (J) as t2 → t1 , ε → 0. Therefore, |(T2 u)(t2 ) − (T2 u)(t1 )| → 0 independently of u ∈ Br (J) as t2 → t1 , which implies that {T2 u : u ∈ Br (J)} is equicontinuous. Therefore, {T u : u ∈ B(J)} is equicontinuous. The proof is complete. Lemma 2.9. Assume that (H1)–(H4) hold. Then T maps Br (J) into Br (J), and is continuous in Br (J). Proof. Claim I. The operator T maps Br (J) into Br (J). Obviously, by (H4), there exists a constant r > 0 such that t

M M(|u0 | + sup{ ∫(t − s)q−1 m(s)ds}) ≤ r. Γ(q) t∈J 0

2.2 Initial value problems with nondense domain

| 57

For any u ∈ Br (J), by Proposition 2.7, we have t 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 lim K (t − s)B g(s, u(s))ds + S (t)u ≤ ( T u)(t) ∫ 󵄨󵄨 󵄨󵄨 q 󵄨󵄨 q λ 0 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 λ→+∞ 󵄨󵄨 󵄨󵄨 0 t

≤ M|u0 | +

MM 󵄨 󵄨 ∫(t − s)q−1 󵄨󵄨󵄨g(s, u(s))󵄨󵄨󵄨ds Γ(q) 0

t

M ≤ M(|u0 | + sup{ ∫(t − s)q−1 m(s)ds}) ≤ r. Γ(q) t∈J 0

Hence, ‖T u‖ ≤ r for any u ∈ Br (J). Claim II. The operator T is continuous in Br (J). For any um , u ∈ Br (J), m = 1, 2, . . . , with limm→∞ um = u, by (H3), we have g(t, um (t)) → g(t, u(t)) as m → ∞, for t ∈ J. On the one hand, using (H4), for each t ∈ J, we obtain 󵄨 󵄨 (t − s)q−1 󵄨󵄨󵄨g(s, um (s)) − g(s, u(s))󵄨󵄨󵄨 ≤ 2(t − s)q−1 m(s),

a. e. in [0, t).

As the function s → 2(t − s)q−1 m(s) is integrable for s ∈ [0, t) and t ∈ J, by the Lebesgue dominated convergence theorem, we obtain t

󵄨 󵄨 ∫(t − s)q−1 󵄨󵄨󵄨g(s, um (s)) − g(s, u(s))󵄨󵄨󵄨ds → 0 as m → ∞. 0

For t ∈ J, we obtain 󵄨󵄨 󵄨 󵄨󵄨(T um )(t) − (T u)(t)󵄨󵄨󵄨 t 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ≤ 󵄨󵄨󵄨 lim ∫ Kq (t − s)Bλ (g(s, um (s)) − g(s, u(s)))ds󵄨󵄨󵄨 󵄨󵄨λ→+∞ 󵄨󵄨 󵄨 󵄨 0 t

MM 󵄨 󵄨 ≤ ∫(t − s)q−1 󵄨󵄨󵄨g(s, um (s)) − g(s, u(s))󵄨󵄨󵄨ds → 0 Γ(q)

as m → ∞.

0

Therefore, T um → T u pointwise on J as m → ∞. Hence, it follows by Lemma 2.8 that T um → T u uniformly on J as m → ∞ and so T is continuous. The proof is complete. Theorem 2.3. Assume that (H1)–(H5) hold. Then the Cauchy problem (2.13) has at least one integral solution in Br (J).

58 | 2 Fractional evolution equations of order α ∈ (0, 1) Proof. Let y0 (t) = Sq (t)u0 for all t ∈ J and ym+1 = T ym , m = 0, 1, 2, . . . . Consider the set

H = {ym : m = 0, 1, 2, . . .}, and show that it is relatively compact.

By Lemmas 2.8 and 2.9, H is uniformly bounded and equicontinuous on J. Next, for any t ∈ J, we just need to show that H (t) = {ym (t), m = 0, 1, 2, . . .} is relatively compact in X0 . By the assumption (H5) together with Proposition 1.20, for any t ∈ J, we have β(H (t)) = β({ym (t)}m=0 ) = β({y0 (t)} ∪ {ym (t)}m=1 ) = β({ym (t)}m=1 ) ∞

∞

∞

and β({ym (t)}m=1 ) = β({(T ym )(t)}m=0 ) ∞

∞

t

∞

= β({Sq (t)u0 + lim ∫ Kq (t − s)Bλ g(s, ym (s))ds} λ→+∞

t

m=0

0 ∞

= β({ lim ∫ Kq (t − s)Bλ g(s, ym (s))ds} λ→+∞ t

≤

m=0

0

)

2MM ∞ ∫(t − s)1−q β(g(s, {ym (s)}m=0 ))ds Γ(q) 0

≤

)

t

2MMl ∞ ∫(t − s)1−q β({ym (s)}m=0 )ds. Γ(q) 0

Thus, t

β(H (t)) ≤

2MMl ∫(t − s)1−q β(H (s))ds. Γ(q) 0

Therefore, by generalized Grownwall’s inequality [215], we infer that β(H (t)) = 0. In consequence, H (t) is relatively compact. Hence, it follows from the Arzela–Ascoli theorem that H is relatively compact. Therefore, there exists a convergent subsequence ∗ of {ym }∞ m=0 . For the sake of clarity, let limm→∞ ym = y ∈ Br (J). Thus, by continuity of the operator T , we have y∗ = lim ym = lim T ym−1 = T ( lim ym−1 ) = T y∗ , m→∞

m→∞

m→∞

which implies the Cauchy problem (2.13) has at least an integral solution.

2.3 Terminal value problems with the Liouville–Weyl derivative

| 59

2.2.4 An example Consider the fractional time partial differential equation: 2

q

𝜕 𝜕 z(t, x) + G(t, z(t, x)), q z(t, x) = { 𝜕x2 { { 𝜕t z(t, 0) = z(t, π) = 0, { { { {z(0, x) = z0 ,

x ∈ [0, π], t ∈ (0, b], 0 < q < 1, t ∈ (0, b],

(2.32)

x ∈ [0, π],

where G : [0, b] × ℝ → ℝ is a given function. Let u(t)(x) = z(t, x),

t ∈ [0, b], x ∈ [0, π],

g(t, u)(x) = G(t, u(x)),

t ∈ [0, b], x ∈ [0, π].

We choose X = C([0, π], ℝ) endowed with the uniform topology and consider the operator A : D(A) ⊂ X → X defined by D(A) = {u ∈ C 2 ([0, π], ℝ) : u(0) = u(π) = 0},

Au = u′′ .

It is well known that the operator A satisfies the Hille–Yosida condition with (0, +∞) ⊂ ρ(A), ‖(λI − A)−1 ‖ ≤ λ1 for λ > 0, and D(A) = {u ∈ X : u(0) = u(π) = 0} ≠ X. We can show that problem (2.13) is an abstract formulation of problem (2.32). Under suitable conditions, Theorem 2.3 implies that problem (2.32) has a unique solution z on [0, b] × [0, π].

2.3 Terminal value problems with the Liouville–Weyl derivative 2.3.1 Introduction Terminal value problems for differential equations nowadays play an essential part in the modeling of numerous phenomena in engineering, physical science and so forth. Terminal value problems emerge naturally in the simulation of techniques that are measured at a later point, eventually after the methodology has started. Terminal value problems of classical ordinary differential equations have been investigated by [6, 184]. The theory of terminal value problems for fractional differential equations has been paid attention quite recently. K. Diethelm [85] established existence, uniqueness and stability results for fractional terminal value problems. Shah and Rehman [183] established existence and uniqueness of solutions of fractional terminal value problems on the infinite interval.

60 | 2 Fractional evolution equations of order α ∈ (0, 1) In this section, we consider the terminal value problem for the fractional evolution equation in an ordered Banach space X, α u(t) + Au(t) = f (t, u(t)), t ∈ [0, ∞), tD { +∞ u(∞) = 0,

(2.33)

where t Dα+∞ is the Liouville–Weyl fractional derivative of order α ∈ (0, 1) which is defined in Definition 1.6, −A : D(A) ⊂ X → X is the infinitesimal generator of a C0 -semigroup {T(t)}t≥0 , f ∈ C([0, ∞), X). In Subsection 2.3.2, by applying Fourier transform, the problem (2.33) is converted into a singular, integral equation on an infinity interval. Then we first give definitions on mild solutions of (2.33). In Subsection 2.3.3, we give some useful lemmas. The Subsections 2.3.4 and 2.3.5 focus on the existence of solutions of the terminal value problem (2.33). By the generalized Arzela–Ascoli theorem, Schauder’s fixed-point theorem, Darbo–Sadovskii’s fixed-point theorem and some analysis skills, some sufficient conditions are obtained to ensure the existence of mild solution in cases that the semigroup is compact as well as noncompact. This section is based on [220].

2.3.2 Definition of mild solutions Let X be a Banach space with the norm | ⋅ |. Set 󵄨 󵄨 󵄨 C0 ([0, ∞), X) = {u󵄨󵄨󵄨u ∈ C([0, ∞), X) and lim 󵄨󵄨󵄨u(t)󵄨󵄨󵄨 = 0}, t→∞ with the norm ‖u‖0 = supt∈[0,∞) |u(t)|. Obviously, (C0 ([0, ∞), X), ‖ ⋅ ‖0 ) is a Banach space. Let B(X) denote the Banach space of all bounded linear operators from X to X with the norm ‖T‖B(X) = sup{|T(u)| : |u| = 1}. Suppose that −A is the infinitesimal generator of a C0 -semigroup {T(t)}t≥0 of uniformly bounded linear operators on X. This means that there exists M ≥ 1 such that 󵄩 󵄩 M = sup 󵄩󵄩󵄩T(t)󵄩󵄩󵄩B(X) < ∞. t∈[0,∞)

Let ∞

V(t) = α ∫ θWα (θ)T(t α θ)dθ,

t ≥ 0,

(2.34)

0

where Wα (θ) is a Wright function, which is defined in Definition 1.9. We have the following results.

2.3 Terminal value problems with the Liouville–Weyl derivative

| 61

Proposition 2.11 ([225]). For any fixed t ≥ 0, V(t) is a linear and bounded operator, i. e., for any u ∈ X, M 󵄨 󵄨󵄨 |u|. 󵄨󵄨V(t)u󵄨󵄨󵄨 ≤ Γ(α) Proposition 2.12 ([225]). {V(t)}t≥0 is strongly continuous, which means that, for any u ∈ X and any 0 ≤ t ′ < t ′′ < ∞, we have 󵄨󵄨 ′′ ′ 󵄨 󵄨󵄨V(t )u − V(t )u󵄨󵄨󵄨 → 0

as t ′′ → t ′ .

Due to the proof in [225], we can easily get the strong continuity of {V(t)}t≥0 on [0, ∞). Here, we omit the proof. Proposition 2.13 ([225]). Assume that the operator T(t), t > 0 is compact. Then V(t), t > 0 is also a compact operator. Proposition 2.14 ([169]). Assume that T(t) (t > 0) is a compact operator. Then T(t) (t > 0) is equicontinuous, i. e., T(t) is continuous in the uniform operator topology for t > 0. Lemma 2.10. Assume that u is a solution of problem (2.33), then u satisfies the following integral equation: ∞

u(t) = ∫ (s − t)α−1 V(s − t)f (s, u(s))ds, t

t ∈ [0, ∞),

where V is defined by (2.34). Proof. Let F u be the Fourier transform of u. That is, ∞

(F u)(λ) = ∫ e−iλt u(t)dt,

for λ ∈ ℝ.

−∞

Thus, (F (t Dα∞ u))(λ) = (−iλ)α (F u)(λ) (see Definition 1.16). Applying Fourier transform to (2.33), we get (−iλ)α (F u)(λ) + A(F u)(λ) = (F f )(λ), Together with (W2) of Proposition 1.11, we obtain (F u)(λ) = ((−iλ)α I + A) (F f )(λ) −1

∞

α

= ∫ e−(−iλ) t T(t)(F f )(λ)dt 0

for λ ∈ ℝ.

62 | 2 Fractional evolution equations of order α ∈ (0, 1) ∞ ∞

α

= ∫ ∫ αt α−1 e−(−iλt) T(t α )f (s, u(s))e−iλs dsdt 0 −∞ ∞ ∞ ∞

=∫ ∫ ∫ 0 −∞ 0 ∞ ∞ ∞

α2

τ2α+1

t α−1 Wα (τ−α )eiλt T(

tα )f (s, u(s))e−iλs dτdsdt τα

= ∫ ∫ ∫ ατt α−1 Wα (τ)T(t α τ)f (s, u(s))e−iλ(s−t) dτdsdt 0 −∞ 0 ∞ ∞∞

= ∫ ∫ ∫ ατt α−1 Wα (τ)T(t α τ)f (s, u(s))e−iλ(s−t) dτdtds −∞ 0 0 ∞ s ∞

= ∫ ∫ ∫ ατ(s − t)α−1 Wα (τ)T((s − t)α τ)f (s, u(s))e−iλt dτdtds −∞ −∞ 0 ∞ ∞∞

= ∫ ∫ ∫ ατ(s − t)α−1 Wα (τ)T((s − t)α τ)f (s, u(s))e−iλt dτdsdt −∞ t 0 ∞ ∞ −iλt

= ∫ e

∫ (s − t)

α−1

t

−∞ ∞

∞

f (s, u(s))( ∫ ατWα (τ)T((s − t)α τ)dτ)dsdt 0

∞

= ∫ e−iλt ( ∫ (s − t)α−1 V(s − t)f (s, u(s))ds)dt, t

−∞

where (−iλ)α I ∈ ρ(−A). By the uniqueness of Fourier transform, we deduce that the assertion of the lemma holds. This completes the proof. Definition 2.3. A function u : [0, ∞) → X is said to be a mild solution of problem (2.33) if ∞

u(t) = ∫ (s − t)α−1 V(t − s)f (s, u(s))ds, t

t ∈ [0, ∞),

where V is given by (2.34).

2.3.3 Lemmas We introduce the following hypotheses: (H0 ) {T(t)}t≥0 is equicontinuous, i. e., T(t) is continuous in the uniform operator topology for t > 0. (H1 ) There exist constants L1 ≥ 0, β1 ∈ (α, 1) and δ > 0 such that |f (t, u)| ≤ L1 t −β1 |u|δ , t ∈ (0, ∞), u ∈ X.

2.3 Terminal value problems with the Liouville–Weyl derivative

| 63

Since 0 < α < β1 < 1, we can choose γ1 > 0 such that 1 − β1 − δγ1 > 0

and α + γ1 − β1 − δγ1 < 0.

Let T1 > 1 be sufficiently large such that ML1 Γ(β1 + δγ1 − α) α+γ1 −β1 −δγ1 T1 < 1. Γ(β1 + δγ1 ) Define a set S1 as follows: 󵄨 󵄨 S1 = {u ∈ C0 ([0, ∞), X) : 󵄨󵄨󵄨t γ1 u(t)󵄨󵄨󵄨 ≤ 1 for t ≥ T1 }. It is easy to see that S1 is a nonempty, convex and closed subset of C0 ([0, ∞), X). Define an operator T : C0 ([0, ∞), X) → C0 ([0, ∞), X) as follows: ∞

α−1

T u(t) = ∫ (s − t) t

V(s − t)f (s, u(s))ds.

(2.35)

Then we can obtain the following lemmas. Lemma 2.11. Assume that (H0 ), (H1 ) hold, then {T u : u ∈ S1 } is equicontinuous on [0, ∞) and limt→∞ |(T u)(t)| = 0 uniformly for u ∈ S1 . Proof. Claim I. It is easy to see that ∀ε(< 1) > 0, there exists T > T1 such that t α−β1 −δγ1 ≤ ε,

for t ≥ T.

For any u ∈ S1 and t2 > t1 ≥ T, we have 󵄨󵄨 󵄨 󵄨󵄨T u(t1 ) − T u(t2 )󵄨󵄨󵄨 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 󵄨󵄨 α−1 󵄨 ≤ 󵄨󵄨 ∫ (s − t1 ) V(s − t1 )f (s, u(s))ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 t1 󵄨 󵄨󵄨 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 + 󵄨󵄨󵄨 ∫ (s − t2 )α−1 V(s − t2 )f (s, u(s))ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨t2 ∞

ML1 󵄨 󵄨δ ≤ ∫ (s − t1 )α−1 s−β1 󵄨󵄨󵄨u(s)󵄨󵄨󵄨 ds Γ(α) t1

∞

+

ML1 󵄨 󵄨δ ∫ (s − t2 )α−1 s−β1 󵄨󵄨󵄨u(s)󵄨󵄨󵄨 ds Γ(α) t2

64 | 2 Fractional evolution equations of order α ∈ (0, 1)

≤

∞

∞

t1

t2

ML1 ML1 ∫ (s − t1 )α−1 s−β1 −δγ1 ds + ∫ (s − t2 )α−1 s−β1 −δγ1 ds Γ(α) Γ(α)

ML1 Γ(β1 + δγ1 − α) α−β1 −δγ1 α−β −δγ ≤ (t1 + t2 1 1 ) Γ(β1 + δγ1 ) 2ML1 Γ(β1 + δγ1 − α) α−β1 −δγ1 T ≤ Γ(β1 + δγ1 ) 2ML1 Γ(β1 + δγ1 − α) ε, ≤ Γ(β1 + δγ1 ) which can deduce that |T u(t1 ) − T u(t2 )| → 0 as t2 → t1 . Let M1 = maxt∈[0,T] |f (t, u(t))|. For 0 = t1 < t2 < T, we have 󵄨󵄨 󵄨 󵄨󵄨T u(0) − T u(t2 )󵄨󵄨󵄨 t2

≤ ∫s

󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 α−1 󵄨󵄨V(s)f (s, u(s))󵄨󵄨ds + 󵄨󵄨󵄨 ∫ (s − t2 ) V(s − t2 )f (s, u(s))ds 󵄨󵄨 󵄨t2

α−1 󵄨󵄨

0

∞ 󵄨󵄨 󵄨󵄨 − ∫ sα−1 V(s)f (s, u(s))ds󵄨󵄨󵄨 󵄨󵄨 󵄨 t2

≤

t2

T

0

t2

MM1 󵄨 󵄨 ∫ sα−1 ds + ∫((s − t2 )α−1 − sα−1 )󵄨󵄨󵄨V(s − t2 )f (s, u(s))󵄨󵄨󵄨ds Γ(α) ∞

󵄨 󵄨 + ∫ ((s − t2 )α−1 − sα−1 )󵄨󵄨󵄨V(s − t2 )f (s, u(s))󵄨󵄨󵄨ds T

T

MM1 t2α 󵄨 󵄨 ≤ + ∫((s − t2 )α−1 − sα−1 )󵄨󵄨󵄨V(s − t2 )f (s, u(s))󵄨󵄨󵄨ds Γ(α + 1) t2

∞

󵄨 󵄨 + ∫ ((s − t2 )α−1 − sα−1 )󵄨󵄨󵄨V(s − t2 )f (s, u(s))󵄨󵄨󵄨ds T

≤

T

MM1 t2α MM1 + ∫((s − t2 )α−1 − sα−1 )ds Γ(α + 1) Γ(α) t2

∞

+

ML1 ∫ ((s − t2 )α−1 − sα−1 )s−β1 −δγ1 ds Γ(α) T

MM1 t2α MM1 ≤ + ((T − t2 )α − T α + t2α ) Γ(α + 1) Γ(α + 1) ∞

∞

T

T

ML1 ML1 + ∫ (s − t2 )α−1 s−β1 −δγ1 ds + ∫ sα−1−β1 −δγ1 ds Γ(α) Γ(α)

2.3 Terminal value problems with the Liouville–Weyl derivative

| 65

2MM1 t2α ML1 ML1 T α−β1 −δγ1 + ∫ (s − T)α−1 s−β1 −δγ1 ds + Γ(α + 1) Γ(α) (β1 + δγ1 − α)Γ(α) ∞

≤

T

2MM1 t2α ML1 Γ(β1 + δγ1 − α)T α−β1 −δγ1 ML1 T α−β1 −δγ1 ≤ + + Γ(α + 1) Γ(β1 + δγ1 ) (β1 + δγ1 − α)Γ(α) 2MM1 t2α ML1 Γ(β1 + δγ1 − α)ε ML1 ε ≤ + + . Γ(α + 1) Γ(β1 + δγ1 ) (β1 + δγ1 − α)Γ(α) It is easy to see that |T u(t1 ) − T u(t2 )| → 0 as t2 → 0. For 0 < t1 < t2 ≤ T, we

have

󵄨󵄨 󵄨 󵄨󵄨T u(t1 ) − T u(t2 )󵄨󵄨󵄨 t2

M 󵄨 󵄨 ≤ ∫(s − t1 )α−1 󵄨󵄨󵄨f (s, u(s))󵄨󵄨󵄨ds Γ(α) t1

∞

M 󵄨 󵄨 + ∫ ((s − t2 )α−1 − (s − t1 )α−1 )󵄨󵄨󵄨f (s, u(s))󵄨󵄨󵄨ds Γ(α) t2

󵄨󵄨 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 + 󵄨󵄨󵄨 ∫ (s − t2 )α−1 (V(s − t2 ) − V(s − t1 ))f (s, u(s))ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨t2 t2

≤

MM1 ∫(s − t1 )α−1 ds Γ(α) t1

T

+

M 󵄨 󵄨 ∫((s − t2 )α−1 − (s − t1 )α−1 )󵄨󵄨󵄨f (s, u(s))󵄨󵄨󵄨ds Γ(α) t2

∞

M 󵄨 󵄨 + ∫ ((s − t2 )α−1 − (s − t1 )α−1 )󵄨󵄨󵄨f (s, u(s))󵄨󵄨󵄨ds Γ(α) T

󵄨󵄨 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 + 󵄨󵄨󵄨 ∫ (s − t2 )α−1 (V(s − t2 ) − V(s − t1 ))f (s, u(s))ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨t2 ≤

MM1 MM1 (t − t )α + ((T − t2 )α − (T − t1 )α + (t2 − t1 )α ) Γ(α + 1) 2 1 Γ(α + 1) ∞

+

ML1 ∫ ((s − t2 )α−1 − (s − t1 )α−1 )s−β1 −δγ1 ds Γ(α) T

󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 󵄨󵄨 + 󵄨󵄨󵄨 ∫ (s − t2 )α−1 (V(s − t2 ) − V(s − t1 ))f (s, u(s))ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨t2 󵄨

66 | 2 Fractional evolution equations of order α ∈ (0, 1)

≤

2MM1 ML1 Γ(β1 + δγ1 − α)ε (t2 − t1 )α + Γ(α + 1) Γ(β1 + δγ1 ) 󵄨󵄨 󵄨󵄨󵄨 ∞ 󵄨󵄨 󵄨 + 󵄨󵄨󵄨 ∫ (s − t2 )α−1 (V(s − t2 ) − V(s − t1 ))f (s, u(s))ds󵄨󵄨󵄨. 󵄨󵄨 󵄨󵄨 󵄨 󵄨t2

For η > 0 small enough, we have 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∫ (s − t2 )α−1 (V(s − t2 ) − V(s − t1 ))f (s, u(s))ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 t2 ∞

󵄩 󵄩 󵄨 󵄨 ≤ ∫ (s − t2 )α−1 󵄩󵄩󵄩V(s − t2 ) − V(s − t1 )󵄩󵄩󵄩B(X) 󵄨󵄨󵄨f (s, u(s))󵄨󵄨󵄨ds T

T

󵄩 󵄩 󵄨 󵄨 + ∫ (s − t2 )α−1 󵄩󵄩󵄩V(s − t2 ) − V(s − t1 )󵄩󵄩󵄩B(X) 󵄨󵄨󵄨f (s, u(s))󵄨󵄨󵄨ds t2 +η

t2 +η

󵄩 󵄩 󵄨 󵄨 + ∫ (s − t2 )α−1 󵄩󵄩󵄩V(s − t2 ) − V(s − t1 )󵄩󵄩󵄩B(X) 󵄨󵄨󵄨f (s, u(s))󵄨󵄨󵄨ds t2

t2 +η

∞

≤

2MM1 2ML1 ∫ (s − t2 )α−1 s−β1 |u|δ ds + ∫ (s − t2 )α−1 ds Γ(α) Γ(α) t2

T

T

󵄨 󵄨 + ∫ (s − t2 )α−1 󵄨󵄨󵄨f (s, u(s))󵄨󵄨󵄨ds t2 +η

󵄩 󵄩 sup 󵄩󵄩󵄩V(s − t2 ) − V(s − t1 )󵄩󵄩󵄩B(X)

s∈[t2 +η,T]

∞

≤

2ML1 2MM1 ηα ∫ (s − T)α−1 s−β1 −δγ1 ds + Γ(α) Γ(α + 1) T

M Tα + 1 α ≤

󵄩 󵄩 sup 󵄩󵄩󵄩V(s − t2 ) − V(s − t1 )󵄩󵄩󵄩B(X)

s∈[t2 +η,T]

2ML1 Γ(β1 + δγ1 − α)ε 2MM1 ηα + Γ(β1 + δγ1 ) Γ(α + 1) α MT 󵄩 󵄩 sup 󵄩󵄩V(s − t2 ) − V(s − t1 )󵄩󵄩󵄩B(X) , + 1 α s∈[t2 +η,T]󵄩

which can deduce that |T u(t1 ) − T u(t2 )| → 0 as t2 → t1 . For any t1 < T < t2 , we obtain 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨T u(t1 ) − T u(t2 )󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨T u(t1 ) − T u(T)󵄨󵄨󵄨 + 󵄨󵄨󵄨T u(T) − T u(t2 )󵄨󵄨󵄨 → 0, as t2 → t1 .

2.3 Terminal value problems with the Liouville–Weyl derivative

| 67

Therefore, combining the above arguments, |(T u)(t2 ) − (T u)(t1 )| tends to zero independently of u ∈ S1 as t2 − t1 → 0, which means that {T u : u ∈ S1 } is equicontinuous. Claim II. By (H1 ), for any u ∈ S1 and t > T, we have 󵄨󵄨 󵄨󵄨 ∞ 󵄨󵄨 󵄨 󵄨󵄨󵄨 󵄨󵄨 α−1 󵄨 󵄨󵄨(T u)(t)󵄨󵄨 = 󵄨󵄨 ∫ (s − t) V(s − t)f (s, u(s))ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨t ∞

ML1 󵄨 󵄨δ ≤ ∫ (s − t)α−1 s−β1 󵄨󵄨󵄨u(s)󵄨󵄨󵄨 ds Γ(α) t ∞

≤

ML1 ∫ (s − t)α−1 s−β1 −δγ1 ds Γ(α) t

ML1 Γ(β1 + δγ1 − α)t α−β1 −δγ1 ≤ Γ(β1 + δγ1 ) → 0,

as t → ∞.

Therefore, we get that limt→∞ |(T u)(t)| = 0 uniformly for u ∈ S1 . Lemma 2.12. Assume that (H0 ) and (H1 ) hold. Then T maps S1 into S1 , and T is continuous in S1 . Proof. Claim I. The operator T maps S1 into S1 . For any u ∈ S1 and t ≥ T1 , we have 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 󵄨󵄨 γ1 󵄨 󵄨󵄨 γ α−1 󵄨󵄨t (T u)(t)󵄨󵄨󵄨 = 󵄨󵄨󵄨t 1 ∫ (s − t) V(s − t)f (s, u(s))ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 t 󵄨 ∞

≤

ML1 t γ1 ∫ (s − t)α−1 s−β1 −δγ1 ds Γ(α) t

ML1 Γ(β1 + δγ1 − α) γ1 +α−β1 −δγ1 ≤ t Γ(β1 + δγ1 ) ML1 Γ(β1 + δγ1 − α) γ1 +α−β1 −δγ1 ≤ T1 Γ(β1 + δγ1 ) ≤1

for T1 large enough.

By Claim II in the proof of Lemma 2.11, it is easy to see that T u ∈ C0 ([0, ∞), X). Due to the above consideration, we obtain that T maps S1 into S1 . Claim II. The operator T is continuous in S1 . For any um , u ∈ S1 , m = 1, 2, . . . , with limm→∞ ‖um − u‖0 = 0, we have lim u (t) m→∞ m

= u(t)

uniformly for t ∈ [0, ∞).

68 | 2 Fractional evolution equations of order α ∈ (0, 1) Then by the continuity of f , we get lim f (t, um (t)) = f (t, u(t)) pointwise for t ∈ [0, ∞).

m→∞

On the one hand, we get for each t ∈ [0, ∞), 󵄨󵄨 󵄨 󵄨󵄨f (s, um (s)) − f (s, u(s))󵄨󵄨󵄨 ≤ F(s), where 2M1 , F(s) = { 2L1 s−β1 −δγ1 ,

s ∈ [0, T],

s ∈ [T, ∞).

In addition, the function ∫t (s − t)α−1 F(s)ds exists. Then for t ∈ [0, T], by Lebesgue’s ∞

dominated convergence theorem, we have

󵄨󵄨 󵄨 󵄨󵄨(T um )(t) − (T u)(t)󵄨󵄨󵄨 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 󵄨󵄨 α−1 󵄨 = 󵄨󵄨 ∫ (s − t) V(s − t)(f (s, um (s)) − f (s, u(s)))ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨t 󵄨 ∞

M 󵄨 󵄨 ≤ ∫ (s − t)α−1 󵄨󵄨󵄨f (s, um (s)) − f (s, u(s))󵄨󵄨󵄨ds Γ(α) t

T

=

M 󵄨 󵄨 ∫(s − t)α−1 󵄨󵄨󵄨f (s, um (s)) − f (s, u(s))󵄨󵄨󵄨ds Γ(α) t

∞

M 󵄨 󵄨 + ∫ (s − t)α−1 󵄨󵄨󵄨f (s, um (s)) − f (s, u(s))󵄨󵄨󵄨ds Γ(α) T

∞

2ML1 Mε ≤ + ∫ (s − t)α−1 s−β1 −δγ1 ds Γ(α) Γ(α) T ∞

≤

2ML1 Mε + ∫ (s − T)α−1 s−β1 −δγ1 ds Γ(α) Γ(α) T

2ML1 Γ(β1 + δγ1 − α) α−β1 −δγ1 Mε ≤ + T Γ(α) Γ(β1 + δγ1 ) 2ML1 Γ(β1 + δγ1 − α)ε Mε ≤ + . Γ(α) Γ(β1 + δγ1 )

2.3 Terminal value problems with the Liouville–Weyl derivative

| 69

For t ∈ [T, ∞), we have 󵄨 󵄨󵄨 󵄨󵄨(T um )(t) − (T u)(t)󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨󵄨 α−1 = 󵄨󵄨 ∫ (s − t) V(s − t)(f (s, um (s)) − f (s, u(s)))ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨t ∞

2ML1 ≤ ∫ (s − t)α−1 s−β1 −δγ1 ds Γ(α) t

2ML1 Γ(β1 + δγ1 − α) α−β1 −δγ1 t ≤ Γ(β1 + δγ1 ) 2ML1 Γ(β1 + δγ1 − α) α−β1 −δγ1 ≤ T Γ(β1 + δγ1 ) 2ML1 Γ(β1 + δγ1 − α)ε ≤ . Γ(β1 + δγ1 ) Therefore, T um → T u pointwise on [0, ∞) as m → ∞. By Lemma 2.11, we can conclude that T um → T u uniformly on [0, ∞) as m → ∞ and so T is continuous. Hereafter, we assume the following condition holds: (H2 ) there exist constants L2 ≥ 0 and β2 ∈ (α, 1) such that |f (t, u)| ≤ L2 t −β2 , t ∈ (0, ∞), u ∈ X. Let γ2 ∈ (0, β2 − α). Then there exists T2 > 1 such that ML2 Γ(β2 − α) γ2 +α−β2 T2 < 1. Γ(β2 ) Define a set S2 as follows: 󵄨 󵄨 S2 = {u ∈ C0 ([0, ∞), X) : 󵄨󵄨󵄨t γ2 u(t)󵄨󵄨󵄨 ≤ 1 for t ≥ T2 }. It is easy to see that S2 is a nonempty, convex and closed subset of C0 ([0, ∞), X). Lemma 2.13. Assume that (H0 ) and (H2 ) hold, then {T u : u ∈ S2 } is equicontinuous on [0, ∞) and limt→∞ |(T u)(t)| = 0 uniformly for u ∈ S2 . Lemma 2.14. Assume that (H0 ) and (H2 ) hold. Then T maps S2 into S2 , and T is continuous in S2 . The proofs of Lemmas 2.13–2.14 are similar to those of Lemmas 2.11–2.12, so we omit them.

70 | 2 Fractional evolution equations of order α ∈ (0, 1) 2.3.4 Compact semigroup case This subsection deals with existence of mild solution for problem (2.33) when the operator −A generates a compact C0 -semigroup {T(t)}t≥0 on X. Theorem 2.4. Assume that {T(t)}t≥0 is compact. Furthermore, assume that (H0 ) and

(H1 ) hold. Then problem (2.33) admits at least one mild solution.

Proof. By Lemma 2.12, it follows that T S1 ⊂ S1 and T is continuous in S1 . Next, we will

show that T is relatively compact. One can infer from Lemma 2.11 that {T u : u ∈ S1 }

is equicontinuous and limt→∞ |T u(t)| = 0 uniformly for u ∈ S1 . It remains to verify that, for any t ∈ [0, ∞), V(t) = {(T u)(t) : u ∈ S1 } is relatively compact in X. For any ε > 0 and η > 0, define an operator Tε,η on S1 by the formula: ∞ ∞

(Tε,η u)(t) = α ∫ ∫ θ(s − t)α−1 Wα (θ)T((s − t)α θ)f (s, u(s))dθds t+ε η

∞ ∞

α

= αT(ε η) ∫ ∫ θ(s − t)α−1 Wα (θ)T((s − t)α θ − εα η)f (s, u(s))dθds. t+ε η

Then from the compactness of T(εα η) (εα η > 0), we obtain the relatively compactness of the set Vε,η (t) = {(Tε,η u)(t) : u ∈ S1 } in X. For t = 0, we have 󵄨󵄨 󵄨 󵄨󵄨(T u)(0) − (Tε,η u)(0)󵄨󵄨󵄨 η 󵄨󵄨 󵄨󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 ≤ α󵄨󵄨 ∫ ∫ θsα−1 Wα (θ)T(sα θ)f (s, u(s))dθds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨0 0 󵄨 󵄨󵄨 󵄨󵄨 ε ∞ 󵄨󵄨 󵄨󵄨 + α󵄨󵄨󵄨∫ ∫ θsα−1 Wα (θ)T(sα θ)f (s, u(s))dθds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨0 η ∞

≤ αM ∫ s 0

η

α−1 󵄨󵄨

󵄨 󵄨󵄨f (s, u(s))󵄨󵄨󵄨ds ∫ θWα (θ)dθ 0

ε

∞

󵄨 󵄨 + αM ∫ sα−1 󵄨󵄨󵄨f (s, u(s))󵄨󵄨󵄨ds ∫ θWα (θ)dθ η

0

T1

∞

η

0

T1

0

≤ αM(M1 ∫ sα−1 ds + L1 ∫ sα−1 s−β1 −δγ1 ds) ∫ θWα (θ)dθ

2.3 Terminal value problems with the Liouville–Weyl derivative ε

∞

0

η

| 71

+ αMM1 ∫ sα−1 ds ∫ θWα (θ)dθ

≤

(MM1 T1α

→ 0,

η

α−β −δγ

αML1 T1 1 1 ) ∫ θWα (θ)dθ + MM1 εα + β1 + δγ1 − α 0

as ε → 0, η → 0.

Let t ∈ (0, ∞) be fixed. We have 󵄨󵄨 󵄨 󵄨󵄨(T u)(t) − (Tε,η u)(t)󵄨󵄨󵄨 󵄨󵄨 ∞ η 󵄨󵄨 󵄨󵄨 󵄨󵄨 ≤ α󵄨󵄨󵄨 ∫ ∫ θ(s − t)α−1 Wα (θ)T((s − t)α θ)f (s, u(s))dθds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨t 0 󵄨

󵄨󵄨 󵄨󵄨 t+ε ∞ 󵄨󵄨 󵄨󵄨 + α󵄨󵄨󵄨 ∫ ∫ θ(s − t)α−1 Wα (θ)T((s − t)α θ)f (s, u(s))dθds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨t η ∞

α−1 −β1 −δγ1

≤ αML1 ∫ (s − t) t

s

η

ds ∫ θWα (θ)dθ 0

t+ε

∞

+ αML1 ∫ (s − t)α−1 s−β1 −δγ1 ds ∫ θWα (θ)dθ η

t

η

ML1 Γ(α + 1)Γ(β1 + δγ1 − α)t α−β1 −δγ1 ≤ ∫ θWα (θ)dθ Γ(β1 + δγ1 ) 1+ εt

0

+ αML1 t α−β1 −δγ1 ∫ (s − 1)α−1 s−β1 −δγ1 ds 1

→ 0,

as ε → 0, η → 0.

Therefore, there exist relatively compact sets arbitrarily close to the set V(t). Hence, the set V(t) is also relatively compact in X. From Lemma 1.8, we deduce that T S1 is relatively compact in C0 ([0, ∞), X). Therefore, from Schauder’s fixedpoint theorem, the operator T has a fixed point, which is the mild solution of problem (2.33). Substituting (H1 ) with (H2 ), we give the following corollary and omit its proof, which is similar to that of Theorem 2.4. Corollary 2.2. Assume that {T(t)}t≥0 is compact. Furthermore, assume that (H0 ) and (H2 ) hold. Then problem (2.33) admits at least one mild solution.

72 | 2 Fractional evolution equations of order α ∈ (0, 1) 2.3.5 Noncompact semigroup case In the case that T(t) is noncompact, we impose the following assumption: (H3 ) there exists a constant ϱ > 0 such that for any bounded D ⊂ X, β(f (t, D)) ≤ ϱβ(D), where β is the Hausdorff measure of noncompactness. Theorem 2.5. Assume that (H0 ), (H1 ) and (H3 ) hold. Then the Cauchy problem (2.33) admits at least one mild solution. Proof. Similar to Corollary 2.2, we only need to prove that T is compact in a subset of S1 . For each bounded subset, S0 ⊂ S1 . For any ε > 0, there exists T > 0 such that t α−β1 −δγ1 ≤ ε,

for t ≥ T.

Set T1 (S0 ) = T (S0 ),

Tn (S0 ) = T (co(Tn−1 (S0 ))),

n = 2, 3, . . . .

∞ Then, by Propositions 1.19–1.22, there is a sequence {u(1) n }n=1 ⊂ S0 such that

β(T1 (S0 (t))) = β(T (S0 (t))) ∞

≤ 2β( ∫ (s − t)α−1 V(s − t)f (s, {u(1) n (s)}n=1 )ds) + ε. ∞

t

For t ∈ [T, ∞), we have 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 󵄨󵄨 α−1 (1) 󵄨 β(T1 (S0 (t))) ≤ 4 sup 󵄨󵄨 ∫ (s − t) V(s − t)f (s, un (s))ds󵄨󵄨󵄨 + ε 󵄨 󵄨󵄨 n∈ℕ+ 󵄨󵄨 󵄨 t ∞

4ML1 ≤ ∫ (s − t)α−1 s−β1 −δγ1 ds + ε Γ(α) t

4ML1 Γ(β1 + δγ1 − α)t α−β1 −δγ1 +ε Γ(β1 + δγ1 ) 4ML1 Γ(β1 + δγ1 − α)ε ≤ + ε. Γ(β1 + δγ1 ) ≤

2.3 Terminal value problems with the Liouville–Weyl derivative

| 73

For t ∈ [0, T], we have β(T1 (S0 (t))) T

≤ 2β(∫(s − t)α−1 V(s − t)f (s, {u(1) n (s)}n=1 )ds) ∞

t

∞

+ 2β( ∫ (s − t)α−1 V(s − t)f (s, {u(1) n (s)}n=1 )ds) + ε ∞

T

T

≤

4M ∞ ∫(s − t)α−1 β(f (s, {u(1) n (s)}n=1 ))ds Γ(α) t

󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 + ε + 4 sup 󵄨󵄨󵄨 ∫ (s − t)α−1 V(s − t)f (s, u(1) (s))ds n 󵄨󵄨 󵄨 n∈ℕ+ 󵄨󵄨 󵄨󵄨 T T

4Mϱβ(S0 ) ≤ ∫(s − t)α−1 ds Γ(α) t

∞

+

4ML1 ∫ (s − T)α−1 s−β1 −δγ1 ds + ε Γ(α) α

T

4MϱT β(S0 ) 4ML1 Γ(β1 + δγ1 − α)T α−β1 −δγ1 + +ε Γ(α + 1) Γ(β1 + δγ1 ) 4MϱT α β(S0 ) 4ML1 Γ(β1 + δγ1 − α)ε + + ε. ≤ Γ(α + 1) Γ(β1 + δγ1 )

≤

Since ε > 0 is arbitrary, for any t ∈ [0, ∞), we have β(T1 (S0 (t))) ≤

4MϱT α β(S ). Γ(α + 1) 0

∞ Using Propositions 1.19–1.22 again, there is a sequence {u(2) n }n=1 ⊂ co(T1 (S0 )) such that for t ∈ [T, ∞), we have

󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 󵄨󵄨 ∞ β(T2 (S0 (t))) ≤ 4 sup 󵄨󵄨󵄨 ∫ (s − t)α−1 V(s − t)f (s, {u(2) (s)}n=1 )ds󵄨󵄨󵄨 + ε n 󵄨 󵄨󵄨 n∈ℕ+ 󵄨󵄨 󵄨 t ∞

4ML1 ≤ ∫ (s − t)α−1 s−β1 −δγ1 ds + ε Γ(α) t

4ML1 Γ(β1 + δγ1 − α)t α−β1 −δγ1 +ε Γ(β1 + δγ1 ) 4ML1 Γ(β1 + δγ1 − α)ε ≤ + ε. Γ(β1 + δγ1 )

≤

74 | 2 Fractional evolution equations of order α ∈ (0, 1) For t ∈ [0, T], we have T

β(T2 (S0 (t))) ≤ 2β(∫(s − t)α−1 V(s − t)f (s, {u(2) n (s)}n=1 )ds) ∞

t

∞

+ 2β( ∫ (s − t)α−1 V(s − t)f (s, {u(2) n (s)}n=1 )ds) + ε ∞

T

T

≤

4M ∞ ∫(s − t)α−1 β(f (s, {u(2) n (s)}n=1 ))ds Γ(α) t

󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 󵄨󵄨 ∞ + 4 sup 󵄨󵄨󵄨 ∫ (s − t)α−1 V(s − t)f (s, {u(2) (s)}n=1 )ds󵄨󵄨󵄨 + ε n 󵄨 󵄨󵄨 n∈ℕ+ 󵄨󵄨 󵄨 T T

≤

4Mϱ ∞ ∫(s − t)α−1 β({u(2) n (s)}n=1 )ds Γ(α) t

∞

4ML1 + ∫ (s − T)α−1 s−β1 −δγ1 ds + ε Γ(α) T

≤

T

(4Mϱ)2 β(S0 ) ∫(s − t)α−1 (T − s)α ds Γ(α)Γ(α + 1) t

4ML1 Γ(β1 + δγ1 − α)T α−β1 −δγ1 + +ε Γ(β1 + δγ1 ) ≤

(4Mϱ)2 T 2α β(S0 ) 4ML1 Γ(β1 + δγ1 − α)ε + + ε. Γ(2α + 1) Γ(β1 + δγ1 )

It can be shown, by mathematical induction, that for every n̄ ∈ ℕ, β(Tn̄ (S0 (t))) ≤

(4Mϱ)n̄ T nᾱ β(S0 ) . ̄ + 1) Γ(nα

Since (4MϱT α )n̄ = 0, ̄ + 1) ̄ n→∞ Γ(nα lim

there exists a positive integer n̂ such that (4Mϱ)n̂ t nα̂ (4MϱT α )n̂ ≤ = k < 1. ̂ + 1) ̂ + 1) Γ(nα Γ(nα

2.3 Terminal value problems with the Liouville–Weyl derivative

| 75

Then β(Tn̂ (S0 (t))) ≤ kβ(S0 ). We know from Proposition 1.17, Tn̂ (S0 (t)) is bounded and equicontinuous. Then, from Proposition 1.17, we have β(Tn̂ (S0 )) = sup β(Tn̂ (S0 (t))). t∈[0,∞)

Hence, β(Tn̂ (S0 )) ≤ kβ(S0 ). Furthermore, by Lemma 1.4, there exists a D ⊂ S1 such that β(T (D)) = 0. Thus, the operator T is an α-contraction in D. By Lemma 2.12, we know that T is continuous. Hence, Theorem 1.5 shows that T has a fixed point in D ⊂ S1 . Therefore, problem (2.33) has a mild solution in S1 . Substituting (H1 ) with (H2 ), we get the corollary which is a similar result of Theorem 2.5. Corollary 2.3. Assume that (H0 ), (H2 ) and (H3 ) hold. Then the Cauchy problem (2.33) admits at least one mild solution. Example 2.1. Let X = L2 ([0, π], ℝ). Consider the following fractional partial differential equation: α 2 { t D y(t, z) + 𝜕z y(t, z) = 𝜕z G(t, y(t, z)), { { ∞ y(t, 0) = y(t, π) = 0, { { { {y(∞, z) = 0,

z ∈ [0, π], t ∈ [0, ∞), t ∈ [0, ∞),

(2.36)

z ∈ [0, π],

where 0 < α < 1, G is a given function. We define an operator A by Av = v′′ with the domain D(A) = {v(⋅) ∈ X : v, v′ absolutely continuous, v′′ ∈ X, v(0) = v(π) = 0}.

Then −A generates a C0 -semigroup {T(t)}t≥0 , which is compact. Thus equation (2.36) can be reformulated in X as α t D u(t) + Au(t) = f (t, u(t)), t ∈ [0, ∞), { ∞ u(∞) = 0,

76 | 2 Fractional evolution equations of order α ∈ (0, 1) where u(t) = y(t, ⋅), that is, u(t)(z) = y(t, z), t ∈ [0, ∞), z ∈ [0, π]. The function f : [0, ∞) × X → X is given by f (t, u(t))(z) = 𝜕z G(t, y(t, z)). Taking L1 > 0, α < β1 < 1, δ ∈ ℝ and f (t, u) = L1 t −β1 sinδ u, the assumption (H1 ) is clearly satisfied. As a result from Corollary 2.3, problem (2.36) has at least one mild solution.

2.4 Attractivity of evolution equations with almost sectorial operators 2.4.1 Introduction Consider the following Cauchy problems of fractional evolution equation with the Caputo derivative: C α D x(t) = Ax(t) + f (t, x(t)), t ∈ [0, ∞), {0 t x(0) = x0 ,

(2.37)

and fractional evolution equation with the Riemann–Liouville derivative α 0Dt x(t) = Ax(t) + f (t, x(t)), −(1−α) x(0) = x0 , 0Dt

{

t ∈ [0, ∞),

(2.38)

where C0Dαt is a Caputo fractional derivative of order α, 0 < α < 1, 0Dαt is a Riemann– Liouville fractional derivative of order α, 0Dt−(1−α) is a Riemann–Liouville fractional integral of order 1 − α, A is an almost sectorial operator on Banach space X, f : [0, ∞) × X → X is a continuous function satisfying some assumptions and x0 is an element of X. In this section, we study the attractivity of solutions for Cauchy problems (2.37) and (2.38). We establish sufficient conditions for the global attractivity for mild solutions of (2.37) and (2.38) in cases that semigroup associated with A is compact as well as noncompact. The obtained results essentially reveal certain characteristics of solutions for fractional evolution equations, which are not possessed by integer order evolution equations. This section is based on [227]. 2.4.2 Preliminaries Let 0 < κ < 1, and let Sμ0 with 0 < μ < π be the open sector {z ∈ ℂ\{0} : | arg z| < μ} and Sμ be its closure, that is, Sμ = {z ∈ ℂ\{0} : | arg z| ≤ μ} ∪ {0}.

2.4 Attractivity of evolution equations with almost sectorial operators | 77

Let X be a Banach space with norm |⋅|. As usual, for a linear operator A, we denote by D(A) the domain of A, by σ(A) its spectrum, while ρ(A) := ℂ − σ(A) is the resolvent set of A, and denote by the family R(z; A) = (zI − A)−1 , z ∈ ρ(A) of bounded linear operators the resolvent of A. Moreover, we denote by B(X) the space of all bounded linear operators from Banach space X to X. As in [58], we state the concept of almost sectorial operators as follows. Definition 2.4. Let 0 < κ < 1 and 0 < ω < π2 . By Θ−κ ω (X), we denote the family of all linear closed operators A : D(A) ⊂ X → X, which satisfy: (i) σ(A) ⊂ Sω = {z ∈ ℂ \ {0} : | arg z| ≤ ω} ∪ {0} and (ii) for every ω < μ < π there exists a constant Cμ such that 󵄩󵄩 󵄩 −κ 󵄩󵄩R(z; A)󵄩󵄩󵄩B(X) ≤ Cμ |z| ,

for all z ∈ ℂ \ Sμ .

(2.39)

A linear operator A will be called an almost sectorial operator on X if A ∈ Θ−κ ω (X). We denote the semigroup associated with A by {Q(t)}t≥0 . For t ∈ S0π −ω , 2

Q(t) = e−tz (A) =

1 ∫ e−tz R(z; A)dz, 2πi Γθ

where the integral contour Γθ = {ℝ+ eiθ } ∪ {ℝ+ e−iθ } is oriented counterclockwise and ω < θ < μ < π2 − | arg t| forms an analytic semigroup of growth order 1 − κ. π Lemma 2.15 ([172]). Let A ∈ Θ−κ ω (X) with 0 < κ < 1 and 0 < ω < 2 . Then the following properties remain true: (i) the functional equation Q(s + t) = Q(s)Q(t) for all s, t ∈ S0π −ω holds; 2

(ii) there is a constant C0 > 0 such that |Q(t)x| ≤ C0 t κ−1 x, for all t > 0.

Denote by Eα,β the generalized Mittag-Leffler special function defined by zk 1 λα−β eλ = dλ, ∫ α Γ(αk + β) 2πi λ − z k=0 ∞

Eα,β (z) := ∑

α, β > 0, z ∈ ℂ,

ϒ

where ϒ is a contour, which starts and ends as −∞ and encircles the disc |λ| ≤ |z|1/α counterclockwise. For short, set Eα (z) := Eα,1 (z),

eα (z) := Eα,α (z).

78 | 2 Fractional evolution equations of order α ∈ (0, 1) Define operator families {Sα (t)}, {Pα (t)} by Sα (t) := Eα (−zt α )(A) =

1 ∫ Eα (−zt α )R(z; A)dz, 2πi Γθ

Pα (t) := eα (−zt α )(A) =

1 ∫ eα (−zt α )R(z; A)dz, 2πi Γθ

where the integral contour Γθ := {ℝ+ eiθ } ∪ {ℝ+ e−iθ }, is oriented counterclockwise. By (W4) and (W5) of Proposition 1.11 and the Fubini theorem, we get Sα (t)x =

1 ∫ Eα (−zt α )R(z; A)xdz 2πi Γθ

∞

α 1 = ∫ Mα (λ) ∫ e−λzt R(z; A)xdzdλ 2πi

0

Γθ

∞

= ∫ Mα (θ)Q(t α θ)xdθ,

for t ∈ S0π −ω , x ∈ X, 2

0

where Mα (θ) is the Wright function. A similar argument shows that ∞

Pα (t)x = ∫ αθMα (θ)Q(t α θ)xdθ,

for t ∈ S0π −ω , x ∈ X. 2

0

Lemma 2.16 ([205]). For each fixed t ∈ S0π −ω , Sα (t) and Pα (t) are linear and bounded 2

operators on X. Moreover, there exist constants C1 > 0 and C2 > 0 such that for all t > 0, 󵄨󵄨 󵄨 −α(1−κ) |x| and 󵄨󵄨Sα (t)x󵄨󵄨󵄨 ≤ C1 t

󵄨 󵄨󵄨 −α(1−κ) |x|. 󵄨󵄨Pα (t)x󵄨󵄨󵄨 ≤ C2 t

Lemma 2.17 ([205]). For t > 0, Sα (t) and Pα (t) are strongly continuous, which means that, for any x ∈ X and t ′ < t ′′ , we have 󵄨󵄨 ′′ ′ 󵄨 󵄨󵄨Sα (t )x − Sα (t )x󵄨󵄨󵄨 → 0,

󵄨󵄨 ′′ ′ 󵄨 ′′ ′ 󵄨󵄨Pα (t )x − Pα (t )x󵄨󵄨󵄨 → 0 as t → t .

Lemma 2.18 ([226]). If Q(t) is compact for every t > 0, then Sα (t), Pα (t) are compact for every t > 0. Lemma 2.19 ([226]). Let η > 1 − κ. For all x ∈ D(Aη ), we have lim Sα (t)x = x

t→0+

and

lim Pα (t)x = x.

t→0+

2.4 Attractivity of evolution equations with almost sectorial operators | 79

Lemma 2.20 ([133]). If a > 0 and b > 0, then t

∫(t − s)a−1 sb−1 ds = 0

Γ(a)Γ(b) a+b−1 t . Γ(a + b)

The following definitions of the mild solution of (2.37) and (2.38) are given in [226]. Definition 2.5 ([226]). By the mild solution of the Cauchy problem (2.37), we mean that the function x ∈ C([0, ∞), X), which satisfies t

x(t) = Sα (t)x0 + ∫(t − s)α−1 Pα (t − s)f (s, x(s))ds,

t > 0.

(2.40)

0

Definition 2.6 ([226]). By the mild solution of the Cauchy problem (2.38), we mean that the function x ∈ C([0, ∞), X), which satisfies x(t) = t

α−1

t

Pα (t)x0 + ∫(t − s)α−1 Pα (t − s)f (s, x(s))ds,

t > 0.

(2.41)

0

Definition 2.7. The solution x(t) of Cauchy problem (2.37) (or (2.38)) is attractive if x(t) → 0 as t → ∞. 2.4.3 Auxiliary lemmas Let 󵄨 󵄨 C0 ([0, ∞), X) = {x ∈ C([0, ∞), X) : lim 󵄨󵄨󵄨x(t)󵄨󵄨󵄨 = 0}, t→∞ with the norm ‖x‖0 = supt∈[0,∞) |x(t)| < ∞. It is obvious that C0 ([0, ∞), X) is a Banach space. For any x ∈ C([0, ∞), X), define an operator F as follows: t

(F x)(t) = Sα (t)x0 + ∫0 (t − s)α−1 Pα (t − s)f (s, x(s))ds, t > 0,

{

(F x)(0) = x0 .

(2.42)

It is clear that x(t) is a mild solution of (2.37) if and only if there exists a fixed-point x∗ , i. e., there exists x∗ (t) such that the operator equation x ∗ (t) = (F x ∗ )(t) holds for all t ∈ (0, ∞). We introduce the following assumption: (H1) |f (t, x)| ≤ Lt −β for t ∈ (0, ∞) and x ∈ C((0, ∞), X), where L ≥ 0, ακ < β < 1.

80 | 2 Fractional evolution equations of order α ∈ (0, 1) Since 0 < ακ < β < 1, we can choose a γ > 0 sufficiently small such that γ − α + ακ < 0

and γ + ακ − β < 0.

Let T > 0 be sufficiently large such that C1 |x0 |T γ−α+ακ +

C2 LΓ(ακ)Γ(1 − β) γ+ακ−β T ≤ 1. Γ(ακ − β + 1)

(2.43)

Define a set Ω as follows: 󵄨 󵄨 Ω = {x(t) : x ∈ C([0, ∞), X), 󵄨󵄨󵄨x(t)󵄨󵄨󵄨 ≤ t −γ for t ≥ T}. It is easy to see that Ω ≠ 0, and Ω is a closed and convex subset of C0 ((0, ∞), X). Lemma 2.21. Assume that (H1) holds. Then {F x : x ∈ Ω} is equicontinuous and limt→∞ |(F x)(t)| = 0 uniformly for x ∈ Ω.

Proof. Since −α(1 − κ) < 0 and ακ − β < 0, there exists a T1 > T such that C1 t −α(1−κ) |x0 |
T1 .

For any x ∈ Ω, and t1 , t2 > T1 , we get 󵄨󵄨 󵄨 󵄨󵄨(F x)(t2 ) − (F x)(t1 )󵄨󵄨󵄨

t2

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨Sα (t2 )x0 󵄨󵄨󵄨 + 󵄨󵄨󵄨Sα (t1 )x0 󵄨󵄨󵄨 + ∫(t2 − s)α−1 󵄨󵄨󵄨Pα (t2 − s)f (s, x(s))󵄨󵄨󵄨ds 0

t1

󵄨 󵄨 + ∫(t1 − s)α−1 󵄨󵄨󵄨Pα (t1 − s)f (s, x(s))󵄨󵄨󵄨ds 0

≤ C1 t2−α(1−κ) |x0 | + C1 t1−α(1−κ) |x0 | t2

+ C2 L ∫(t2 − s) ≤

0 −α(1−κ) C1 t2 |x0 |

ακ−1 −β

t1

s ds + C2 L ∫(t1 − s)ακ−1 s−β ds 0

+

C1 t1−α(1−κ) |x0 |

C LΓ(ακ)Γ(1 − β) ακ−β C2 LΓ(ακ)Γ(1 − β) ακ−β + 2 t + t < ε. Γ(ακ − β + 1) 2 Γ(ακ − β + 1) 1

2.4 Attractivity of evolution equations with almost sectorial operators | 81

Furthermore, for 0 < t1 < t2 ≤ T1 , by Lemma 2.22 and Lebesgue dominated con-

vergence theorem, we have

󵄨 󵄨󵄨 󵄨󵄨(F x)(t2 ) − (F x)(t1 )󵄨󵄨󵄨

󵄨󵄨 t2 󵄨 󵄨󵄨 󵄨 ≤ 󵄨󵄨󵄨Sα (t2 )x0 − Sα (t1 )x0 󵄨󵄨󵄨 + 󵄨󵄨󵄨∫(t2 − s)α−1 Pα (t2 − s)f (s, x(s))ds 󵄨󵄨 󵄨0 t1 󵄨󵄨 󵄨󵄨 − ∫(t1 − s)α−1 Pα (t1 − s)f (s, x(s))ds󵄨󵄨󵄨 󵄨󵄨 󵄨 0 󵄨󵄨 󵄨󵄨 ≤ 󵄨󵄨Sα (t2 )x0 − Sα (t1 )x0 󵄨󵄨 󵄨󵄨 t1 󵄨󵄨 󵄨󵄨 󵄨󵄨 + 󵄨󵄨󵄨∫((t2 − s)α−1 − (t1 − s)α−1 )Pα (t2 − s)f (s, x(s))ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨0 󵄨

󵄨󵄨 t2 󵄨󵄨 󵄨󵄨 󵄨󵄨 + 󵄨󵄨󵄨∫(t2 − s)α−1 Pα (t2 − s)f (s, x(s))ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨t1 󵄨

󵄨󵄨 t1 󵄨󵄨 󵄨󵄨 󵄨󵄨 + 󵄨󵄨󵄨∫(t1 − s)α−1 (Pα (t2 − s) − Pα (t1 − s))f (s, x(s))ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨0 󵄨 󵄨󵄨 󵄨󵄨 ≤ 󵄨󵄨Sα (t2 )x0 − Sα (t1 )x0 󵄨󵄨 t1

󵄨 󵄨 + C2 sup 󵄨󵄨󵄨f (t, x(t))󵄨󵄨󵄨 ∫[(t1 − s)ακ−1 − (t2 − s)ακ−1 ]ds t∈[0,T1 ]

0

t2

+ C2

󵄨 󵄨 sup 󵄨󵄨󵄨f (t, x(t))󵄨󵄨󵄨 ∫(t2 − s)ακ−1 ds

t∈[0,T1 ]

t1

t1

󵄨 󵄨 + ∫(t1 − s)α−1 󵄨󵄨󵄨(Pα (t2 − s) − Pα (t1 − s))f (s, x(s))󵄨󵄨󵄨ds 0

󵄨 󵄨 ≤ 󵄨󵄨󵄨Sα (t2 )x0 − Sα (t1 )x0 󵄨󵄨󵄨 C 󵄨 󵄨 + 2 sup 󵄨󵄨󵄨f (t, x(t))󵄨󵄨󵄨[(t1ακ − t2ακ + (t2 − t1 )ακ ] ακ t∈[0,T1 ] +

C2 󵄨 󵄨 sup 󵄨󵄨f (t, x(t))󵄨󵄨󵄨(t2 − t1 )ακ ακ t∈[0,T1 ]󵄨 t1

󵄨 󵄨 + ∫(t1 − s)α−1 󵄨󵄨󵄨(Pα (t2 − s) − Pα (t1 − s))f (s, x(s))󵄨󵄨󵄨ds 0

→ 0,

as t2 → t1 .

82 | 2 Fractional evolution equations of order α ∈ (0, 1) If 0 = t1 < t2 ≤ T1 , by Lemma 2.19, we have (F x)(t2 ) → x0 = (F x)(0),

as t2 → 0.

Therefore, combining the above arguments, it is clear that the family of functions {F x : x ∈ Ω} is equicontinuous. It remains to verify that limt→∞ |(F x)(t)| = 0 uniformly for x ∈ Ω. Indeed, we have t

󵄨󵄨 󵄨 󵄨 󵄨 󵄨 α−1 󵄨 󵄨󵄨(F x)(t)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨Sα (t)x0 󵄨󵄨󵄨 + ∫(t − s) 󵄨󵄨󵄨Pα (t − s)f (s, x(s))󵄨󵄨󵄨ds 0

t

≤ C1 |x0 |t −α+ακ + C2 L ∫(t − s)ακ−1 s−β ds 0

≤ C1 |x0 |t −α+ακ + → 0,

(2.44)

C2 LΓ(ακ)Γ(1 − β) ακ−β t Γ(ακ − β + 1)

as t → ∞.

This shows that limt→∞ |(F x)(t)| = 0 uniformly for x ∈ S. The proof is completed. Lemma 2.22. Assume that (H1) holds. Then F maps Ω into Ω and F is continuous in Ω. Proof. Claim I. The operator F maps Ω into Ω. For x ∈ Ω, by Lemma 2.21, we know F x ∈ C([0, ∞), X). On the other hand, by using the condition (H1), we have t

󵄨󵄨 󵄨 󵄨 󵄨 󵄨 α−1 󵄨 󵄨󵄨(F x)(t)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨Sα (t)x0 󵄨󵄨󵄨 + ∫(t − s) 󵄨󵄨󵄨Pα (t − s)f (s, x(s))󵄨󵄨󵄨ds 0

t

γ 󵄨󵄨

󵄨 󵄨 󵄨 = (t 󵄨󵄨Sα (t)x0 󵄨󵄨󵄨 + t γ ∫(t − s)α−1 󵄨󵄨󵄨Pα (t − s)f (s, x(s))󵄨󵄨󵄨ds)t −γ 0

t

≤ (C1 |x0 |t γ−α+ακ + C2 Lt γ ∫(t − s)ακ−1 s−β ds)t −γ 0

≤ (C1 |x0 |t

γ−α+ακ

C LΓ(ακ)Γ(1 − β) γ+ακ−β −γ t )t . + 2 Γ(ακ − β + 1)

From the inequality (2.43), we have C LΓ(ακ)Γ(1 − β) γ+ακ−β −γ 󵄨󵄨 󵄨 γ−α+ακ + 2 T )t 󵄨󵄨(F x)(t)󵄨󵄨󵄨 ≤ (C1 |x0 |T Γ(ακ − β + 1) ≤ t −γ ,

which implies that F Ω ⊂ Ω.

t ≥ T,

2.4 Attractivity of evolution equations with almost sectorial operators | 83

Claim II. The operator F is continuous in Ω. For any xm , x ∈ Ω, m = 1, 2, . . . with limt→∞ xm = x, we will show that F xm → F x, as m → ∞. For ∀ε > 0, there exists a T1 > T such that 2C2 LΓ(ακ)Γ(1 − β) ακ−β < ε. T1 Γ(ακ − β + 1) Then, for t > T1 , we get t

󵄨󵄨 󵄨 󵄨 α−1 󵄨 󵄨󵄨(F xm )(t) − (F x)(t)󵄨󵄨󵄨 ≤ ∫(t − s) 󵄨󵄨󵄨Pα (t − s)(f (s, xm (s)) − f (s, x(s)))󵄨󵄨󵄨ds 0

t

󵄨 󵄨 󵄨 󵄨 ≤ C2 ∫(t − s)ακ−1 (󵄨󵄨󵄨f (s, xm (s))󵄨󵄨󵄨 + 󵄨󵄨󵄨f (s, x(s))󵄨󵄨󵄨)ds 0

t

≤ 2C2 L ∫(t − s)ακ−1 s−β ds 0

2C LΓ(ακ)Γ(1 − β) ακ−β T1 < ε. ≤ 2 Γ(ακ − β + 1) For 0 < t ≤ T1 , we have 󵄨󵄨 󵄨 󵄨󵄨(F xm )(t) − (F x)(t)󵄨󵄨󵄨 t

󵄨 󵄨 ≤ ∫(t − s)α−1 󵄨󵄨󵄨Pα (t − s)(f (s, xm (s)) − f (s, x(s)))󵄨󵄨󵄨ds 0

t

󵄨 󵄨 ≤ C2 ∫(t − s)ακ−1 󵄨󵄨󵄨f (s, xm (s)) − f (s, x(s))󵄨󵄨󵄨ds. 0

Since limm→∞ |f (t, xm (t)) − f (t, x(t))| = 0, by Lebesgue dominated convergence theorem, we have 󵄨󵄨 󵄨 󵄨󵄨(F xm )(t) − (F x)(t)󵄨󵄨󵄨 → 0,

as m → ∞.

Therefore, it is obvious that 󵄩󵄩 󵄩 󵄩󵄩(F xm ) − (F x)󵄩󵄩󵄩 → 0,

as m → ∞,

which implies that the operator F is continuous. This proof is completed.

84 | 2 Fractional evolution equations of order α ∈ (0, 1) 2.4.4 Compact semigroup case In the following, we suppose that the operator Q(t) is compact for t > 0. Theorem 2.6. Assume that (H1) holds. Then the Cauchy problem (2.37) admits at least one attractive solution. Proof. Obviously, x is a mild solution of (2.37) in Ω if and only if x is a fixed point of F in Ω. So it is enough to show that the operator F has a fixed point in Ω. By Lemma 2.22, it follows that F : Ω → Ω is bounded and continuous. Next, it will be shown that F is relatively compact. One can infer from Lemma 2.21 that {F x : x ∈ Ω} is equicontinuous and limt→∞ |F x(t)| = 0 uniformly for x ∈ Ω. It remains to verify that V(t) = {(F x)(t) : x ∈ Ω} is relatively compact in X for any t ∈ [0, ∞). Obviously, V(0) is relatively compact in X. Let t ∈ (0, ∞) be fixed. For ∀ ε > 0 and ∀ η > 0, define an operator Fε,η on Ω as follows: t−ε ∞

(Fε,η x)(t) = Sα (t)x0 + ∫ ∫ αθ(t − s)α−1 Mα (θ) 0 η α

× Q((t − s) θ)f (s, x(s))dθds α

t−ε ∞

= Sα (t)x0 + Q(ε η) ∫ ∫ αθ(t − s)α−1 Mα (θ) α

0 η α

× Q((t − s) θ − ε η)f (s, x(s))dθds. Then from the compactness of Sα (t) and Q(εα η)(εα η > 0), we obtain that the set Vε,η (t) = {(Fε,η x)(t) : x ∈ Ω} is relatively compact in X for all ε ∈ (0, t) and for all η > 0. Moreover, for every x ∈ Ω, we have 󵄨󵄨 󵄨 󵄨󵄨(F x)(t) − (Fε,η x)(t)󵄨󵄨󵄨 η 󵄨󵄨 󵄨󵄨󵄨 t 󵄨󵄨 󵄨󵄨 ≤ 󵄨󵄨∫ ∫ αθ(t − s)α−1 Mα (θ)Q((t − s)α θ)f (s, x(s))dθds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨0 0 󵄨

󵄨󵄨 t ∞ 󵄨󵄨 󵄨󵄨 󵄨󵄨 + 󵄨󵄨󵄨 ∫ ∫ αθ(t − s)α−1 Mα (θ)Q((t − s)α θ)f (s, x(s))dθds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨t−ε η 󵄨 t

≤ αC0 ∫(t − s) 0

η

ακ−1 󵄨󵄨

t

󵄨 󵄨󵄨f (s, x(s))󵄨󵄨󵄨ds ∫ θMα (θ)dθ 0

ακ−1 󵄨󵄨

+ αC0 ∫ (t − s) t−ε

∞

󵄨 󵄨󵄨f (s, x(s))󵄨󵄨󵄨ds ∫ θMα (θ)dθ 0

2.4 Attractivity of evolution equations with almost sectorial operators | 85 t

≤ αC0 L ∫(t − s) 0

η

ακ−1 −β

s ds ∫ θMα (θ)dθ 0

t

∞

+ αC0 L ∫ (t − s)ακ−1 s−β ds ∫ θMα (θ)dθ t−ε

≤

η

0

αC0 LΓ(ακ)Γ(1 − β) ακ−β t ∫ θMα (θ)dθ Γ(ακ − β + 1) 0

1

∞

+ αC0 Lt ακ−β ∫ (1 − s)α−1 s−β ds ∫ θMα (θ)dθ 0

1−ε/t

→ 0,

as ε → 0, η → 0.

Therefore, there exist relatively compact sets arbitrarily close to the set V(t). Hence, the set V(t) is also relatively compact in X. By Lemma 1.8, we deduce that V ⊂ C0 ([0, ∞), X) is relatively compact. Therefore, by Schauder’s fixed-point theorem, (2.37) has a mild solution x ∈ Ω, and x(t) tends to zero as t → ∞. The proof is complete. In the following, we give also sufficient conditions of the existence of globally attractive solutions for (2.38) in case that A is compact. For any x ∈ C([0, ∞), X), define an operator G as follows: t

(G x)(t) = t α−1 Pα (t)x0 + ∫0 (t − s)α−1 Pα (t − s)f (s, x(s))ds, t > 0,

{

(0D−(1−α) G x)(0) = x0 . t

(2.45)

Since 0 < ακ < β < 1, we can choose a γ > 0 sufficiently small such that γ − 1 + ακ < 0

and γ + ακ − β < 0.

Let T > 0 be sufficiently large such that C1 |x0 |T γ−1+ακ +

C2 LΓ(ακ)Γ(1 − β) γ+ακ−β T ≤ 1. Γ(ακ − β + 1)

(2.46)

Define a set Ω as follows: 󵄨 󵄨 Ω = {x(t)|x ∈ C((0, ∞), X), 󵄨󵄨󵄨x(t)󵄨󵄨󵄨 ≤ t −γ for t ≥ T}. It is easy to see that Ω ≠ 0, and Ω is a closed, convex and bounded subset of C0 ((0, ∞), X). By using the similar method as in Subsection 2.4.3, we can prove the following results. Thus, we omit their proofs.

86 | 2 Fractional evolution equations of order α ∈ (0, 1) Lemma 2.23. Assume that (H1) holds. Then {G x : x ∈ Ω} is equicontinuous and limt→∞ |(G x)(t)| = 0 uniformly for x ∈ Ω. Lemma 2.24. Assume that (H1) holds. Then G maps Ω into Ω and G is continuous in Ω. Theorem 2.7. Assume that (H1) holds. Then the Cauchy problem (2.38) admits at least one attractive solution.

2.4.5 Noncompact semigroup case When Q(t) is noncompact, we introduce the following assumption. (H2) There exists a constant k > 0 such that for any bounded set E ⊂ X, β(f (t, E)) ≤ kβ(E), where β is the Hausdorff measure of noncompactness. Theorem 2.8. Assume that (H1) and (H2) hold. Then the Cauchy problem (2.37) admits at least one attractive solution. Proof. By Lemma 2.22, we know that F : Ω → Ω is bounded and continuous. Next, it will be shown that V ⊂ C0 ([0, ∞), X) is relatively compact. By Lemma 2.21, we know that {F x : x ∈ Ω} is equicontinuous and limt→∞ |F x(t)| = 0 uniformly for x ∈ Ω. It remains to verify that for any t ∈ [0, ∞), V(t) = {(F x)(t) : x ∈ Ω} is relatively compact in X. We omit the proof of this step as it is similar to that of Theorem 4.19 in [225]. Therefore, by Darbo–Sadovskii’s fixed-point theorem, (2.37) has a mild solution x ∈ Ω, with x(t) → 0 as t → ∞. This completes the proof. In the following, we give also sufficient conditions of the existence of globally attractive solutions for (2.38) in cases that A is noncompact. Theorem 2.9. Assume that (H1) and (H2) hold. Then the Cauchy problem (2.38) admits at least one attractive solution.

2.4.6 Example Let Ω be a bounded domain in ℝN (N ≥ 0) with boundary 𝜕Ω of class C 4 . Example 2.2. Consider the following fractional initial-boundary value problem: (CDα x)(t, z) = Δx(t, z) + f (t, x(t, z)), a. e. t > 0, z ∈ Ω, { { {0 t x| = 0, t > 0, { { 𝜕Ω { z ∈ Ω, {x(0, z) = x0 (z),

(2.47)

2.4 Attractivity of evolution equations with almost sectorial operators | 87

in the space C l (Ω) (0 < l < 1), where Δ stands for the Laplacian operator with respect to the spatial variable and C0Dαt , representing the regularized Caputo fractional derivative of order α (0 < α < 1). Set à = Δ,

D(A)̃ = {x ∈ C 2+l (Ω) : x = 0 on 𝜕Ω}.

It follows from [158, Example 2.3] that there exist v, ε > 0, such that l

−1 Ã + v ∈ Θ 2π −ε (C l (Ω)). 2

Then problem (2.47) can be written abstractly as (see [205, Example 6.2]) C α D x(t) = Ax(t) + f (t, x), t ∈ [0, ∞), {0 t x(0) = x0 .

(2.48)

Let f (t, x(t)) = t −β sin x(t). Then the assumptions (H1) is clearly satisfied. In consequence, according to Theorem 2.6, the problem (2.47) has at least one attractive mild solution. On the other hand, for the first-order evolution equation x′ (t) = Ax(t) + t −β , 0 < β < 1, t ∈ [0, ∞),

{

x(0) = x0 ,

(2.49)

all solutions of (2.49) do not converge to 0. It means that our results essentially reveal certain characteristics of solutions for fractional evolution equations, which are not possessed by integer-order evolution equations.

3 Fractional control systems of order α ∈ (0, 1) 3.1 Controllability 3.1.1 Introduction Consider fractional semilinear differential inclusions in Banach spaces of the type C q D x(t) ∈ Ax(t) + F(t, x(t)), {0 t x(0) = x0 ,

a. e. t ∈ [0, b], 0 < q ≤ 1,

(3.1)

where C0Dqt is the Caputo fractional derivative of order q, b > 0 is a finite number, A is the infinitesimal generator of a strongly continuous semigroup {T(t)}t≥0 in X, the state x(⋅) takes values in a Banach space X and x0 is an element of the Banach space X, F : [0, b] × X —∘ X is a multivalued map. Further, we investigate the following fractional control system: C q D x(t) ∈ Ax(t) + Bu(t) + F(t, x(t)), a. e. t ∈ [0, b], 0 < q ≤ 1, {0 t x(0) = x0 , 1

(3.2)

where the control function u(⋅) takes its value in L q1 ([0, b]; U) for q1 ∈ (0, q), a Banach space of admissible control functions and U is a Banach space, B : U → X is a bounded linear operator. The existence of mild solutions and controllability problems for various types of nonlinear fractional evolution inclusions in infinite dimensional spaces by using different kinds of approaches have been considered in many recent publications (see, e. g., [94, 145, 202] and the references therein). In most of the existing articles, various fixed-point theorems and measure of noncompactness are employed to obtain the fixed points of the solution operator of the Cauchy problems under the restrictive hypotheses of compactness on the semigroup generated by the linear part and on the nonlinear term. But, in infinite dimensional Banach spaces, the compactness of the associated evolution operator is in contradiction with the controllability of a linear system while using locally Lp -controls, for p > 1. As it was pointed out by [39], it is meaningful to introduce conditions assuring controllability for semilinear equations without requiring the compactness of the semigroup or evolution operator generated by the linear part. In this section, another approach is considered, and it exploits the weak topology of the state space. This tool was introduced to study semilinear differential inclusions associated to boundary value conditions; see [38]. We prove the existence of mild solutions of (3.1) and the controllability results of (3.2) by means of weak topology, avoiding hypotheses of compactness on the semigroup generated by the linear part and any conditions on the multivalued nonlinearity expressed in terms of measures of noncompactness. https://doi.org/10.1515/9783110769272-003

90 | 3 Fractional control systems of order α ∈ (0, 1) The section is organized as follows. In Subsection 3.1.2, we study the existence of mild solutions for (3.1). In Subsection 3.1.3, we prove the controllability for the fractional control system (3.2), and in Subsection 3.1.4 two examples are given to illustrate the obtained theory. This section is based on [234].

3.1.2 Existence of mild solutions Let (X, ‖ ⋅ ‖) be a reflexive Banach space and Xw denote the space X endowed with the w weak topology. For a set D ⊂ X, the symbol D denotes the weak closure of D. We recall that a bounded subset D of a reflexive Banach space X is weakly relatively compact. In the whole section, without generating misunderstanding, we denote by ‖ ⋅ ‖p both the Lp ([0, b]; X)-norm and Lp ([0, b]; ℝ)-norm and by ‖ ⋅ ‖0 the C([0, b]; X)-norm. We study the fractional semilinear differential inclusion (3.1) under the following assumptions: (HA ) the operator A generates a strongly continuous semigroup {T(t)}t≥0 in X, and there exists a constant M1 ≥ 1 such that supt∈[0,b] ‖T(t)‖ ≤ M1 . We assume that the multivalued nonlinearity F : [0, b] × X —∘ X has nonempty convex and weakly compact values and: (H1 ) the multifunction F(⋅, x) : [0, b] —∘ X has a measurable selection for every x ∈ X; (H2 ) the multimap F(t, ⋅) : X —∘ X is weakly sequentially closed for a. e. t ∈ [0, b], i. e., it has a weakly sequentially closed graph; (H3 ) there exists a constant q1 ∈ (0, q) and for every r > 0, there exists a function 1

μr ∈ L q1 ([0, b]; ℝ+ ) such that for each c ∈ X, ‖c‖ ≤ r: 󵄩󵄩 󵄩 󵄩󵄩F(t, c)󵄩󵄩󵄩 = sup{‖x‖ : x ∈ F(t, c)} ≤ μr (t)

for a. e. t ∈ [0, b].

Now, we define the mild solution of fractional evolution inclusion (3.1). Definition 3.1. A continuous function x : [0, b] → X is said to be a mild solution of 1

fractional differential system (3.1) if x(0) = x0 and there exists f ∈ L q1 ([0, b]; X) such that f (t) ∈ F(t, x(t)) on t ∈ [0, b] and x satisfies the following integral equation: t

x(t) = T (t)(x0 ) + ∫(t − s)q−1 S (t − s)f (s)ds, 0

where ∞

q

T (t) = ∫ ξq (θ)T(t θ)dθ, 0

∞

q

S (t) = q ∫ θξq (θ)T(t θ)dθ, 0

3.1 Controllability | 91

ξq (θ) = ϖq (θ) =

1 −1− q1 −1 θ ϖq (θ q ) ≥ 0, q

1 ∞ Γ(nq + 1) sin(nπq), ∑ (−1)n−1 θ−qn−1 π n=1 n!

θ ∈ (0, ∞),

and ξq is a probability density function defined on (0, ∞), that is, ∞

ξq (θ) ≥ 0,

θ ∈ (0, ∞) and

∫ ξq (θ)dθ = 1. 0

Remark 3.1. It is not difficult to verify that for v ∈ [0, 1], ∞

v

∞

∫ θ ξq (θ)dθ = ∫ θ−qv ϖq (θ)dθ = 0

0

Γ(1 + v) . Γ(1 + qv)

The following results will be used in the proof of our main results. Lemma 3.1 ([230]). The operators T and S have the following properties: (i) for any fixed t ≥ 0, T (t) and S (t) are linear and bounded operators, i. e., for any x ∈ X, qM1 󵄩󵄩 󵄩 󵄩 󵄩 ‖x‖; 󵄩󵄩T (t)x󵄩󵄩󵄩 ≤ M1 ‖x‖ and 󵄩󵄩󵄩S (t)x󵄩󵄩󵄩 ≤ Γ(1 + q) (ii) {T (t), t ≥ 0} and {S (t), t ≥ 0} are strongly continuous. Given p ∈ C([0, b]; X), let us denote 1

Λp = {f ∈ L q1 ([0, b]; X) : f (t) ∈ F(t, p(t)) for a. e. t ∈ [0, b]}. The set Λp is always nonempty as Proposition 3.1 below shows. Proposition 3.1. Assume that a multimap F : [0, b]×X —∘X satisfies (H1 ), (H2 ) and (H3 ), the set Λp is nonempty for any p ∈ C([0, b]; X). Proof. Let p ∈ C([0, b]; X). By the uniform continuity of p, there exists a sequence {pn } of step functions, pn : [0, b] → X such that 󵄩 󵄩 sup 󵄩󵄩󵄩pn (t) − p(t)󵄩󵄩󵄩 → 0,

t∈[0,b]

as n → ∞.

(3.3)

Hence, by (H1 ), there exists a sequence of functions {fn } such that fn (t) ∈ F(t, pn (t)) for a. e. t ∈ [0, b] and fn : [0, b] → X is measurable for any n ∈ ℕ. From (3.3), there exists a bounded set E ⊂ X such that pn (t), p(t) ∈ E, for any t ∈ [0, b] and n ∈ ℕ and

92 | 3 Fractional control systems of order α ∈ (0, 1) 1

by (H3 ) there exists ηn ∈ L q1 ([0, b]; ℝ) such that 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩fn (t)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩F(t, pn (t))󵄩󵄩󵄩 ≤ ηn (t),

∀ n ∈ ℕ, and a. e. t ∈ [0, b].

1

Hence, {fn } ⊂ L q1 ([0, b]; X) is bounded and uniformly integrable and {fn (t)} is bounded in X for a. e. t ∈ [0, b]. According to the reflexivity of the space X and by the Dunford– Pettis theorem (see [87, p. 294]), we have the existence of a subsequence, denoted as the sequence, such that 1

fn ⇀ g ∈ L q1 ([0, b]; X). By Mazur’s convexity theorem, we obtain a sequence kn

̃f = ∑ λ f , n n,i n,i i=0

λn,i ≥ 0,

kn

∑ λn,i = 1

i=0

1

such that ̃fn → g in L q1 ([0, b]; X) and, up to subsequence, ̃fn (t) → g(t) for all t ∈ [0, b]. By (H3 ), the multimap F(t, ⋅) is locally weakly compact for a. e. t ∈ [0, b], i. e., for a. e. t and every x ∈ X there is a neighborhood V of x, such that the restriction of F(t, ⋅) to V is weakly compact. Hence, by (H2 ) and the locally weak compactness, we easily get that F(t, ⋅) : Xw —∘ Xw is u. s. c. for a. e. t ∈ [0, b]. Thus, F(t, ⋅) : X —∘ Xw is u. s. c. for a. e. t ∈ [0, b]. To conclude, we have only to prove that g(t) ∈ F(t, p(t)) for a. e. t ∈ [0, b]. Indeed, let N0 with Lebesgue measure zero be such that F(t, ⋅) : X —∘ Xw is u. s. c. fn (t) ∈ F(t, pn (t)) and ̃fn (t) → g(t) for all t ∈ [0, b] \ N0 and n ∈ ℕ. Fix t0 ∉ N0 and assume, by contradiction, that g(t0 ) ∉ F(t0 , p(t0 )). Since F(t0 , p(t0 )) is closed and convex, from the Hahn–Banach theorem there is a weakly, open, convex set V ⊃ F(t0 , p(t0 )) satisfying g(t0 ) ∉ V. Since F(t0 , ⋅) : X —∘ Xw is u. s. c., we can find a neighbourhood U of p(t0 ) such that F(t0 , x) ⊂ V for all x ∈ U. The convergence pn (t0 ) ⇀ p(t0 ) as m → ∞ then implies the existence of n0 ∈ ℕ such that pn (t0 ) ∈ U for all n > n0 . Therefore, g0 (t0 ) ∈ F(t0 , pn (t0 )) ⊂ V for all n > n0 . Since V is convex, we also have that ̃fn (t0 ) ∈ V for all n > n0 and, by the convergence, we arrive at the contradictory conclusion that g(t0 ) ∈ V. We obtain that g(t) ∈ F(t, p(t)) for a. e. t ∈ [0, b]. We define the solution multioperator Γ : C([0, b]; X) —∘ C([0, b]; X) as Γ(p) = {x ∈ C([0, b]; X) : x(t) = T (t)x0 + S(f )(t), f ∈ Λp }, t

where S(f )(t) = ∫0 (t − s)q−1 S (t − s)f (s)ds. We first prove that the operator S is continuous.

(3.4)

3.1 Controllability | 93

For any fn , f ∈ Λp and fn → f (n → ∞), using (H3 ), we get for each t ∈ [0, b], 󵄩 󵄩 (t − s)q−1 󵄩󵄩󵄩fn (s) − f (s)󵄩󵄩󵄩 ≤ 2(t − s)q−1 μr (s),

a. e. s ∈ [0, t). q−q1

t

1−q1 )b 1−q1 ]1−q1 ‖μr ‖ 1 . By the On the other hand, the function ∫0 (t − s)q−1 μr (s)ds ≤ [( q−q 1

Lebesgue dominated convergence theorem, we have

q1

t

󵄩 󵄩 ∫(t − s)q−1 󵄩󵄩󵄩fn (s) − f (s)󵄩󵄩󵄩ds → 0,

as n → ∞.

0

Thus, 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 q−1 󵄩󵄩S(fn ) − S(f )󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩∫(t − s) S (t − s)(fn (s) − f (s))ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩 󵄩 0 t

qM1 󵄩 󵄩 ≤ ∫(t − s)q−1 󵄩󵄩󵄩fn (s) − f (s)󵄩󵄩󵄩ds → 0, Γ(1 + q)

as n → ∞.

0

So, the operator S is continuous. It is easy to verify that the fixed points of the multioperator Γ are mild solutions of fractional differential system (3.1). Fix n ∈ ℕ, consider Qn the closed ball of radius n in C([0, b]; X) centered at the origin and denote by Γn = Γ|Qn : Qn —∘ C([0, b]; X) the restriction of the multioperator Γ on the set Qn . We describe some properties of Γn . Proposition 3.2. The multioperator Γn has a weakly sequentially closed graph. Proof. Let {pm } ⊂ Qn and {xm } ⊂ C([0, b]; X) satisfying xm ∈ Γn (pm ) for all m and pm ⇀ p, xm ⇀ x in C([0, b]; X); we will prove that x ∈ Γn (p). Since pm ∈ Qn for all m and pm (t) ⇀ p(t) for every t ∈ [0, b], it follows that ‖p(t)‖ ≤ lim infm→∞ ‖pm (t)‖ ≤ n for all t (see [47, Proposition III.5]). The fact that xm ∈ Γn (pm ) means that there exists a sequence {fm }, fm ∈ Λpm such that for every t ∈ [0, b], t

xm (t) = T (t)x0 + ∫(t − s)q−1 S (t − s)f (s)ds. 0

We observe that, according to (H3 ), ‖fm (t)‖ ≤ μn (t) for a. e. t and every m, i. e., {fm } is bounded and uniformly integrable and {fm (t)} is bounded in X for a. e. t ∈ [0, b]. Hence, by the reflexivity of the space X and by the Dunford–Pettis theorem (see [87, p. 294]), we have the existence of a subsequence, denoted as the sequence, and a func1

tion g such that fm ⇀ g in L q1 ([0, b]; X).

94 | 3 Fractional control systems of order α ∈ (0, 1) Therefore, we have Sfm ⇀ Sg. Indeed, let e′ : X → ℝ be a linear continuous operator. By the linearity and continuity of the operator S, we have that the operator t

g → e (∫(t − s)q−1 S (t − s)g(s)ds) ′

0 1

is a linear and continuous operator from L q1 ([0, b]; X) to ℝ for all t ∈ [0, b]. Then, from the definition of the weak convergence, we have for every t ∈ [0, b], t

t

0

0

e′ (∫(t − s)q−1 S (t − s)fm (s)ds) → e′ (∫(t − s)q−1 S (t − s)g(s)ds). Thus, t

xm (t) ⇀ T (t)x0 + ∫(t − s)q−1 S (t − s)g(s)ds = x0 (t),

∀ t ∈ [0, b],

0

implying, for the uniqueness of the weak limit in X that x0 (t) = x(t) for all t ∈ [0, b]. Finally, as the reason for the fourth part of Proposition 3.1, it is possible to show that g(t) ∈ F(t, p(t)) for a. e. t ∈ [a, b]. Proposition 3.3. The multioperator Γn is weakly compact. Proof. We first prove that Γn (Qn ) is relatively weakly sequentially compact. Let {pm } ⊂ Qn and {xm } ⊂ C([0, b]; X) satisfying xm ∈ Γn (pm ) for all m. By the definition of the multioperator Γn , there exists a sequence {fm }, fm ∈ Λpm such that t

xm (t) = T (t)x0 + ∫(t − s)q−1 S (t − s)fm (s)ds,

∀ t ∈ [0, b].

0

Further, as the reason for Proposition 3.2, we have that there exists a subsequence, 1

denoted as the sequence, and a function g such that fm ⇀ g in L q1 ([0, b]; X). Therefore, t

xm (t) ⇀ l(t) = T (t)x0 + ∫(t − s)q−1 S (t − s)g(s)ds,

∀ t ∈ [0, b].

0

Furthermore, by the weak convergence of {fm }, by (HA ), we have 1−q1

q−q1 M1 q 1 − q1 󵄩󵄩 󵄩 [( )b 1−q1 ] 󵄩󵄩xm (t)󵄩󵄩󵄩 ≤ M1 ‖x0 ‖ + Γ(1 + q) q − q1

‖μn ‖ 1

q1

3.1 Controllability | 95

for all m ∈ ℕ and t ∈ [0, b]. As the reason for Proposition 3.2, it is then easy to prove that xm ⇀ l in C([0, b]; X). Thus, Γn (Qn ) is relatively weakly compact by Theorem 1.1. Proposition 3.4. The multioperator Γn has convex and weakly compact values. Proof. Fix p ∈ Qn , since F is convex valued, from the linearity of the integral and of the operators T (t) and S (t), it follows that the set Γn (p) is convex. The weak compactness of Γn (p) follows by Propositions 3.2 and 3.3. We are able now to state the main results of this section. Theorem 3.1. Assume that (HA ), (H1 ) and (H2 ) hold. In addition, suppose that 1

(H4 ) there exists a sequence of functions {ωn } ⊂ L q1 ([0, b]; ℝ+ ) such that 󵄩 󵄩 sup 󵄩󵄩󵄩F(t, c)󵄩󵄩󵄩 ≤ ωn (t),

a. e. t ∈ [0, b], n ∈ ℕ

‖c‖≤n

with lim inf n→∞

‖wn ‖ 1 n

q1

= 0.

(3.5)

Then inclusion (3.1) has at least a mild solution. Proof. We show that there exists n ∈ ℕ such that the operator Γn maps the ball Qn into itself. Assume, to the contrary, that there exist sequences {zn }, {yn } such that zn ∈ Qn , 1 yn ∈ Γn (zn ) and yn ∉ Qn , ∀ n ∈ ℕ. Then there exists a sequence {fn } ⊂ L q1 ([0, b]; X), fn (s) ∈ F(s, zn (s)) such that t

yn (t) = T (t)x0 + ∫(t − s)q−1 S (t − s)fn (s)ds,

∀ t ∈ [0, b].

0

As the reason for Proposition 3.3, we have n < ‖yn ‖0 ≤ M1 ‖x0 ‖ +

1−q1

q−q1 1 − q1 M1 q [( )b 1−q1 ] Γ(1 + q) q − q1

‖wn ‖ 1 . q1

Then 1
0 such that L

M1 ‖x0 ‖ +

q−q

1 M1 q [( 1−q1 )b 1−q1 ]1−q1 ‖α‖ 1 Γ(1+q) q−q1 q

1

β(L)

> 1,

(3.7)

then inclusion (3.1) has at least a mild solution. Proof. It is sufficient to prove that the operator Γ maps the ball QL into itself. In fact, given any z ∈ QL , y ∈ Γ(z) and y ∉ QL , we have 1−q1

q−q1 M1 q 1 − q1 ‖y‖0 ≤ M1 ‖x0 ‖ + [( )b 1−q1 ] Γ(1 + q) q − q1

1−q1

q−q1 M1 q 1 − q1 )b 1−q1 ] ≤ M1 ‖x0 ‖ + [( Γ(1 + q) q − q1

q1

b

󵄨 󵄨1 󵄩 󵄩 1 (∫󵄨󵄨󵄨α(η)󵄨󵄨󵄨 q1 (β(󵄩󵄩󵄩z(η)󵄩󵄩󵄩)) q1 dη) 0

‖α‖ 1 β(L) < L, q1

The conclusion then follows by Theorem 1.11, like Theorem 3.1.

3.1.3 Controllability results In this subsection, we deal with the controllability for the fractional semilinear differential inclusions (3.2) in a reflexive Banach space. We assume that: (HB ) The control function u(⋅) takes its value in U, a Banach space of admissible con1

trol functions, where U = L q1 ([a, b]; U) for q1 ∈ (0, q) and U is a Banach space. The operator B : U → X is a bounded linear operator, with ‖B‖ ≤ M2 .

(3.8)

Definition 3.2. A continuous function x : [0, b] → X is said to be a mild solution of 1

system (3.2) if x(0) = x0 and there exists f ∈ L q1 ([0, b]; X) such that f (t) ∈ F(t, x(t)) on t ∈ [0, b] and x satisfies the following integral equation: t

t

0

0

x(t) = T (t)(x0 ) + ∫(t − s)q−1 S (t − s)Bu(s)ds + ∫(t − s)q−1 S (t − s)f (s)ds. We will consider the controllability problem for system (3.2), i. e., we will study conditions, which guarantee the existence of a mild solution to problem (3.2) satisfy-

98 | 3 Fractional control systems of order α ∈ (0, 1) ing x(b) = x1 ,

(3.9)

where x1 ∈ X is a given point. A pair (x, u) consisting of a mild solution x(⋅) to (3.2) 1

satisfying (3.9) and of the corresponding control u(⋅) ∈ L q1 ([0, b]; U) is called a solution of the controllability problem. We assume the standard assumption that the corresponding linear problem (i. e., when F(t, c) ≡ 0) has a solution. More precisely, we suppose that: (HW ) The controllability operator W : U → X given by b

Wu = ∫(b − s)q−1 S (b − s)Bu(s)ds 0

has a bounded inverse which takes values in U \ ker(W) and there exists a positive constant M3 > 0 such that 󵄩󵄩 −1 󵄩󵄩 󵄩󵄩W 󵄩󵄩 ≤ M3 . 1

(3.10) 1

Let q1 ∈ (0, q). We denote with S1 : L q1 ([0, b]; X) → C([0, b]; X) and S2 : L q1 ([0, b]; X) → C([0, b]; X) the following integral operators for ∀ t ∈ [0, b]: t

S1 f (t) = ∫(t − s)q−1 S (t − s)f (s)ds,

(3.11)

0 b

t

S2 f (t) = ∫(t − s)q−1 S (t − s)BW −1 (− ∫(b − η)q−1 S (b − η)f (η)dη)(s)ds,

(3.12)

0

0

and we define the solution multioperator Π : C([0, b]; X) —∘ C([0, b]; X) as Π(p) = {x ∈ C([0, b]; X) : x(t) = T (t)x0 + S1 (f )(t) t

+ ∫(t − s)q−1 S (t − s)BW −1 (x1 − T (b)x0 )(s)ds + S2 f (t), f ∈ Λp }. 0

It is easy to verify that the fixed points of the multioperator Π are mild solutions of fractional differential system (3.2) and (3.9). Proposition 3.5. The operators S1 and S2 defined in (3.11) and (3.12) are linear and continuous.

3.1 Controllability | 99

Proof. The linearity follows from the linearity of the integral operator and of the operators B, W −1 , and we have 󵄩󵄩 󵄩󵄩 t 󵄩󵄩 󵄩 󵄩󵄩󵄩 󵄩󵄩 q−1 󵄩 󵄩󵄩S1 f (t)󵄩󵄩 = 󵄩󵄩∫(t − s) S (t − s)f (s)ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩0 ≤

q−q1 M1 q 1 − q1 [( )b 1−q1 ] Γ(1 + q) q − q1

1−q1

‖f ‖ 1 , q1

∀ t ∈ [0, b].

Moreover, by (3.8) and (3.10), we obtain b 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩 q−1 q−1 −1 󵄩󵄩S2 f (t)󵄩󵄩 = 󵄩󵄩∫(t − s) S (t − s)BW (− ∫(b − η) S (b − η)f (η)dη)(s)ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩0 󵄩 0

≤

t b 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 qM1 M2 ∫(t − s)q−1 󵄩󵄩󵄩W −1 (− ∫(b − η)q−1 S (b − η)f (η)dη)(s)󵄩󵄩󵄩ds 󵄩󵄩 󵄩󵄩 Γ(1 + q) 󵄩 󵄩 0

≤

0 1−q1

1 − q1 qM1 M2 [( )b ] Γ(1 + q) q − q1 b 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 × 󵄩󵄩󵄩W −1 (− ∫(b − η)q−1 S (b − η)f (η)dη)󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 1 󵄩 󵄩 q1 0 q−q1 1−q1

b 󵄩󵄩 󵄩󵄩 󵄩󵄩 M1 q q−1 󵄩󵄩 󵄩󵄩 (b − η) f (η)dη ∫ 󵄩󵄩 󵄩󵄩 Γ(1 + q) 󵄩󵄩 󵄩󵄩 0

1−q1 󵄩 󵄩

≤

q−q1 qM1 M2 M3 1 − q1 [( )b 1−q1 ] Γ(1 + q) q − q1

≤

q−q1 q2 M12 M2 M3 1 − q1 [( )b 1−q1 ] 2 q − q1 Γ (1 + q)

2(1−q1 )

‖f ‖ 1 , q1

for t ∈ [0, b]. Fix n ∈ ℕ, and we denote by Πn = Π|Qn : Qn —∘ C([0, b]; X) the restriction of the multioperator Π on the set Qn . We describe some properties of Πn . Proposition 3.6. The multioperator Πn has a weakly, sequentially, closed graph. Proof. Let {pm } ⊂ Qn and {xm } ⊂ C([0, b]; X) satisfying xm ∈ Πn (pm ) for all m ∈ ℕ and pm ⇀ p, xm ⇀ x in C([0, b]; X); we will prove that x ∈ Πn (p). Since pm ∈ Qn for all m and pm (t) ⇀ p(t) for every t ∈ [0, b], it follows that ‖p(t)‖ ≤ lim infm→∞ ‖pm (t)‖ ≤ n for all t (see [47, Proposition III.5]). The fact that xm ∈ Πn (pm ) means that there exists a sequence {fm }, fm ∈ Λpm such that t

xm (t) = T (t)x0 + S1 fm (t) + ∫(t − s)q−1 S (t − s)BW −1 (x1 − T (b)x0 )(s)ds 0

+ S2 fm (t),

∀ t ∈ [0, b].

100 | 3 Fractional control systems of order α ∈ (0, 1) We observe that, according to (H3 ), ‖fm (t)‖ ≤ μn (t) for a. e. t and every m, i. e., {fm } is bounded and uniformly integrable and {fm (t)} is bounded in X for a. e. t ∈ [0, b]. Hence, by the reflexivity of the space X and by the Dunford–Pettis theorem (see [87, p. 294]), we have the existence of a subsequence, denoted as the sequence, and a func1

tion g such that fm ⇀ g in L q1 ([0, b]; X). In view of the linearity and continuity for Si , we have Si fm ⇀ Si g for i = 1, 2. Thus, t

xm (t) ⇀ T (t)x0 + S1 g(t) + ∫(t − s)q−1 S (t − s)BW −1 (x1 − T (b)x0 )(s)ds 0

+ S2 g(t) = x0 (t),

∀ t ∈ [0, b],

implying, for the uniqueness of the weak limit in X, that x0 (t) = x(t) for all t ∈ [0, b]. Similar to the proof of Proposition 3.2, we can prove that g(t) ∈ F(t, p(t)) for a. e. t ∈ [0, b]. Proposition 3.7. The multioperator Πn is weakly compact. Proof. We first prove that Πn (Qn ) is weakly relatively sequentially compact. Let {pm } ⊂ Qn and {xm } ⊂ C([0, b]; X) satisfying xm ∈ Πn (pm ) for all m ∈ ℕ. By the definition of the multioperator Πn , there exists a sequence {fm }, fm ∈ Λpm such that t

xm (t) = T (t)x0 + S1 fm (t) + ∫(t − s)q−1 S (t − s)BW −1 (x1 − T (b)x0 )(s)ds 0

+ S2 fm (t),

∀ t ∈ [0, b].

Further, as the reason for Proposition 3.6, we have that there exist a subsequence, 1

denoted as the sequence, and a function g such that fm ⇀ g in L q1 ([0, b]; X). Therefore, t

xm (t) ⇀ l(t) = T (t)x0 + S1 g(t) + ∫(t − s)q−1 S (t − s)BW −1 (x1 − T (b)x0 )(s)ds 0

+ S2 g(t),

∀ t ∈ [0, b].

Furthermore, by the weak convergence of {fm }, by (HA ), (3.8), (3.10), and the continuity of the operators S1 and S2 we have q−q1 1 − q1 M1 q 󵄩󵄩 󵄩 [( )b 1−q1 ] 󵄩󵄩xm (t)󵄩󵄩󵄩 ≤ M1 ‖x0 ‖ + Γ(1 + q) q − q1

+

qM1 M2 M3 1 − q1 [( )b Γ(1 + q) q − q1

× (‖x1 ‖ + M1 ‖x0 ‖ +

q−q1 1−q1

1−q1

1−q1

‖μn ‖ 1

q1

]

q−q1 M1 q 1 − q1 [( )b 1−q1 ] Γ(1 + q) q − q1

1−q1

‖μn ‖ 1 ), q1

3.1 Controllability | 101

for all m ∈ ℕ and t ∈ [0, b]. As the reason for Proposition 3.6, it is then easy to prove that xm ⇀ l in C([0, b]; X). Thus, Πn (Qn ) is relatively weakly compact by Theorem 1.1. Proposition 3.8. The multioperator Πn has convex and weakly compact values. Proof. Fix p ∈ Qn , since F is convex valued, from the linearity of the integral and of the operators T (t), S (t), B and W −1 , it follows that the set Πn (p) is convex. The weak compactness of Πn (p) follows by Propositions 3.6 and 3.7. We are able now to state the main results of this subsection. Theorem 3.4. Assume that (HA ), (H1 ), (H2 ), (HB ), (HW ) hold. If 1

(H4 )′ there exists a sequence of functions {ωn } ⊂ L q1 ([0, b]; ℝ+ ) such that 󵄩 󵄩 sup 󵄩󵄩󵄩F(t, c)󵄩󵄩󵄩 ≤ ωn (t),

a. e. t ∈ [0, b], n ∈ ℕ

‖c‖≤n

with lim inf

‖ωn ‖ 1 n

n→∞

q1

= 0,

(3.13)

then controllability problem (3.2) and (3.9) has a solution. Proof. We show that there exists n ∈ ℕ such that the operator Πn maps the ball Qn into itself. Assume, to the contrary, that there exist sequences {zn }, {yn } such that zn ∈ Qn , 1

yn ∈ Πn (zn ) and yn ∉ Qn , ∀ n ∈ ℕ. Then there exists a sequence {fn } ⊂ L q1 ([0, b]; X), fn (s) ∈ F(s, zn (s)) such that t

yn (t) = T (t)x0 + S1 fn (t) + ∫(t − s)q−1 S (t − s)BW −1 (x1 − T (b)x0 )(s)ds + S2 fn (t),

0

∀ t ∈ [0, b].

As the reason for Proposition 3.7, we have q1

b

q1

b

󵄩 󵄨 󵄩1 󵄨1 ‖yn ‖0 ≤ C1 + C2 (∫󵄩󵄩󵄩fn (η)󵄩󵄩󵄩 q1 dη) ≤ C1 + C2 (∫󵄨󵄨󵄨ωn (η)󵄨󵄨󵄨 q1 dη) , 0

0

where C1 = M1 ‖x0 ‖ + C2 =

1−q1

q−q1 qM1 M2 M3 1 − q1 [( )b 1−q1 ] Γ(1 + q) q − q1

q−q1 M1 q 1 − q1 [( )b 1−q1 ] Γ(1 + q) q − q1

1−q1

(1 +

(‖x1 ‖ + M1 ‖x0 ‖),

(3.14) 1−q1

q−q1 qM1 M2 M3 1 − q1 [( )b 1−q1 ] Γ(1 + q) q − q1

).

(3.15)

102 | 3 Fractional control systems of order α ∈ (0, 1) Then 1
0 such that L > 1, C1 + C2 ‖α‖ 1 β(L) q1

where C1 and C2 are the positive constants defined in (3.14) and (3.15), then the controllability problem (3.2) and (3.9) has a solution. Proof. It is sufficient to prove that the operator Π maps the ball QL into itself. In fact, given any z ∈ QL , y ∈ Π(z) and y ∉ QL , we have q1

b

󵄨 󵄨1 󵄩 󵄩 1 ‖y‖0 ≤ C1 + C2 (∫󵄨󵄨󵄨α(η)󵄨󵄨󵄨 q1 (β(󵄩󵄩󵄩z(η)󵄩󵄩󵄩)) q1 dη) ≤ C1 + C2 ‖α‖ 1 β(L) < L. q1 0

The conclusion then follows by Theorem 1.11, like in Theorem 3.4. 3.1.4 Examples As applications, we give two examples to illustrate our theoretical results. Example 3.1. Consider the following fractional differential inclusion of the form: 2

𝜕3 { 2 x(t, y) ∈ xyy (t, y) + P(t, x(t, y)), { { { 𝜕t 3 x(t, 0) = x(t, 1) = 0, { { { { {x(0, y) = 0,

t ∈ J = [0, 1], t ∈ J = [0, 1],

(3.17)

0 < y < 1,

where P : J × X → 2X \ {Ω}. Let X = L2 (0, 1) and define A : D(A) ⊂ X → X by Ax = xyy , where domain D(A) is given by {x ∈ X : x, xy are absolutely continuous, xyy ∈ X, x(t, 0) = x(t, 1) = 0}. The operator A can be written as ∞

Ax = ∑ n2 ⟨x, xn ⟩, n=1

x ∈ D(A),

104 | 3 Fractional control systems of order α ∈ (0, 1) where xn (y) = √2 sin ny, n = 1, 2, . . . are eigenfunctions of A. Moreover, for any x ∈ X we have ∞

2

T(t)x = ∑ e−n t ⟨x, xn ⟩xn . n=1

Clearly, A generates the above strongly continuous semigroup T(t) on Y. Here, T(t) satisfies the hypothesis (HA ). Define x(t)(y) = x(t, y), F(t, x(t))(y) = P(t, x(t, y)). Here, F : J ×X → 2X \{Ω}. With the choice of A and F, system (3.17) can be rewritten as C q D x(t) ∈ Ax(t) + F(t, x(t)), {0 t x(0) = 0.

t ∈ J, q =

2 3

∈ (0, 1),

Assume that F satisfies (H1 ), (H2 ) and one of (H4 )–(H6 ). Thus, all the conditions of Theorem 3.1, 3.2 and 3.3 are satisfied. Hence, system (3.17) has at least a mild solution on J. Example 3.2. Let X = U = L2 (0, 1). Consider the following fractional differential inclusion with control: 3

C 4 { D x(t, y) ∈ xyy (t, y) + F(t, x(t, y)) + Bu(t, y), { { {0 t x(t, 0) = x(t, 1) = 0, { { { { x(0, y) = ϕ(y), {

y ∈ [0, 1], t ∈ J = [0, 1], t ∈ J = [0, 1],

(3.18)

0 ≤ y ≤ 1.

Similarly, A and T(t) are defined as in Example 3.1. Then the operator S (⋅) can be written as ∞

S (t) =

3 3 ∫ θξ 3 (θ)T(t 4 θ)dθ. 4 4

0

Define ∞

Bu = ∑ e

−

n=1

1 1+n2

⟨u, xn ⟩xn ,

and W : U → X as follows: 1

1

Wu := ∫(1 − s)− 4 S (1 − s)Bu(s, y)ds. 0

3.1 Controllability | 105

Since ∞

‖u‖ = √ ∑ ⟨u, xn ⟩2 , n=1

for u ∈ U, we have ∞

‖Bu‖ = √ ∑ e

−

2 1+n2

n=1

∞

⟨u, xn ⟩2 ≤ √ ∑ ⟨u, xn ⟩2 = ‖u‖, n=1

which implies ‖B‖ ≤ 1. Hence, (HB ) holds. Since q =

Then

3 4

>

1

1 2

= q1 , we take U = L q1 (J, U) = L2 (J, U). Next, let u(s, y) = x(y) ∈ U.

1

1

Wu = ∫(1 − s)− 4 0

∞

3 3 ∫ θξ 3 (θ)T((1 − s) 4 θ)Bxdθds 4 4

0

1

− 41

= ∫(1 − s) 0

∞

∞ 3 2 3 4 − 1 ∫ θξ 3 (θ) ∑ e 1+n2 e−n (1−s) θ ⟨x, xn ⟩xn dθds 4 4 n=1 0

∞

∞

= ∫ ξ 3 (θ) ∑ e 0

4

−

1 1+n2

n=1

0

∞ 1

∞

1

3 1 2 3 4 ∫ θ(1 − s)− 4 e−n (1−s) θ ds⟨x, xn ⟩xn dθ 4

= ∫ ξ 3 (θ) ∑ ∫ n−2 e 0 ∞

4

1 1+n2

n=1 0 ∞

= ∫ ξ 3 (θ) ∑ n−2 e 0 ∞

−

4

−

1 1+n2

n=1

= ∑ n−2 e n=1

−

1 1+n2

d −n2 (1−s) 43 θ (e )ds⟨x, xn ⟩xn dθ ds 2

(1 − e−n θ )⟨x, xn ⟩xn dθ

(1 − 𝔼 3 (−n2 ))⟨x, xn ⟩xn , 4

where 2

∞

2

𝔼 3 (−n ) := ∫ e−n θ ξ 3 (θ)dθ 4

0

4

2

is a Mittag-Leffler function. Note that 0 < 1 − e−θ < 1 − e−n θ < 1 for any θ > 0. So

0 < 1 − 𝔼 3 (−1) ≤ 1 − 𝔼 3 (−n2 ) ≤ 1. From the above computations, we know that W is 4

4

106 | 3 Fractional control systems of order α ∈ (0, 1) surjective. So, we may define a right inverse W −1 : X → U by 1

n2 e 1+n2 ⟨x, xn ⟩xn (W x)(t, y) := ∑ , 2 n=1 1 − 𝔼 3 (−n ) ∞

−1

4

for x = ∑∞ n=1 ⟨x, xn ⟩xn . Since ∞

‖x‖D(A) := ‖Ax‖ := √ ∑ n2 ⟨u, xn ⟩2 , n=1

for x ∈ D(A), we derive 1

∞ 4 1+n2 n e ⟨x, xn ⟩2 󵄩󵄩 −1 󵄩 󵄩󵄩(W x)(t, ⋅)󵄩󵄩󵄩 = √ ∑ 2 2 n=1 (1 − 𝔼 3 (−n )) 4

≤

1 2

∞ e √ ∑ n2 ⟨x, xn ⟩2 1 − 𝔼 3 (−1) n=1 4 1

=

e2 ‖x‖ . 1 − 𝔼 3 (−1) D(A) 4

Note that W −1 x is independent of t ∈ J. Consequently, we obtain 1

e2 󵄩󵄩 −1 󵄩󵄩 =: M3 . 󵄩󵄩W 󵄩󵄩 ≤ 1 − 𝔼 3 (−1) 4

Hence, condition (HW ) is satisfied. Next, we suppose that F : J ×X → 2X satisfies (H1 ), (H2 ) and (H4 )′ . Now, the system (3.18) can be abstracted as 3

{C0 Dt4 x(t) ∈ Ax(t) + F(t, x(t)) + Bu(t), { {x(0) = ϕ.

t ∈ J,

Clearly, all the assumptions in Theorem 3.4 are satisfied. Then the system (3.18) is controllable on J.

3.2 Optimal control 3.2.1 Introduction A fractional optimal control problem appears when the performance index and system dynamics are described by fractional differential equations. In particular, a typical

3.2 Optimal control | 107

case is the fractional optimal control of a distributed system. Recently, there has been growing interest in theory analysis for solving the fractional optimal control problems, the main contributions of which were addressed by Antil et al. [17], El-borai et al. [90], Frederico and Torres [99], Rapaić and Jeličić [177], Wang et al. [204] and Wang and Zhou [203]. Meanwhile, special attention has been given to numerical approximations of the related optimization problems. For example, Agrawal [11, 12] presented a general formulation and solution scheme for fractional, optimal, control problems. Alipour and Rostamy [14] used BPs operational matrices to solve time varying fractional optimal control problems. Optimal synergetic control for fractional differential systems was addressed by Djennoune and Bettayeb [86]. Zaky and Machado [217] derived some generalized necessary conditions for distributed-order fractional, optimal control problems and proposed an efficient numerical scheme. Additional papers can be found in Lotfi et al. [154], Meidner and Vexler [162] and references therein. It should be mentioned that the infinite dimensional property of the working space makes optimal control problems associated with fractional differential equations more challenging to treat compared to the fractional differential equations case. Consequently, it is reasonable and significant to formulate low dimensional approximations to moderate the intrinsic computational difficulty. The usage of such approximations is necessary and specifically relevant due mainly to its avoidance of solving the optimal control problem of infinite-dimensional differential equations. Although a significant amount of work has been done with respect to approximations of optimal control problems for integer-order differential equations, there still remains little work to be done on that in fractional-order case. Chekroun, Kroner and Liu [64] consider finite-dimensional approximations of the following initial-value problem: dx = Ax + F(x) + C(u(t)), { dt x(0) = x0 ,

t ∈ (0, T],

where x0 lies in H, and H denotes a separable Hilbert space. In [64], the nonlinear optimal control problems in Hilbert spaces are considered for which the authors derive approximation theorems for Galerkin approximations. The originality of their approach relies on the identification of a set of natural assumptions that allows ones to deal with a broad class of nonlinear evolution equations and cost functionals for which the authors derive convergence of the value functions associated with the optimal control problem of the Galerkin approximations. Motivated by the aforementioned classical works, in this section we study approximations of the following fractional evolution equations: C α D x(t) = Ax(t) + F(x) + C(u(t)), {0 t x(0) = x0 ,

t ∈ (0, T],

(3.19)

108 | 3 Fractional control systems of order α ∈ (0, 1) where C0Dαt is the Caputo fractional derivative of order α ∈ (0, 1), x takes values in a separable Hilbert space H; the operator A : D(A) ⊂ H → H is the infinitesimal generator of a C0 -semigroup {Q(t)}t≥0 ; u takes values in a separable Hilbert space U; the mapping C : U → H satisfies C(0) = 0 and additional assumptions specified later. It is a remarkable fact that the equation (3.19) is the abstract form of a class of fractional partial equations, a typical application of which is 𝜕α

m

α y(t, ξ ) − Δy(t, ξ ) = f (ξ , t, y(t, ξ )) + ∑i=1 ui gi , { { { 𝜕t y(t, ξ ) = 0, { { { {y(0, ξ ) = y0 (ξ ),

in (0, T) × Ω, on (0, T) × 𝜕Ω, in Ω.

Here, we assume that Ω is a bounded domain in ℝn , 1 ≤ n ≤ 3, with a Lipschitz boundary 𝜕Ω, ui are free temporal amplitudes and gi are fixed spatial-shape functions. The section is organized as follows. Subsection 3.2.2 contains some notations, definitions and preliminary results. Three convergence types of this approximation scheme of mild solutions are provided in Subsection 3.2.3. Subsection 3.2.4 is devoted to obtaining the convergence result for cost functionals. In Subsection 3.2.5, we derive error estimates of cost functionals and optimal controls. Finally, an example is given for demonstration. This section is based on [171].

3.2.2 Preliminaries Let H be a separable Hilbert space and let Br (x) denote the closed ball centered at x ∈ H with radius r > 0. In particular, for x = 0, Br (x) can be abbreviated to Br . We will use the notation A ∈ G(M, ω) (M ≥ 1, ω ≥ 0) for an operator A : D(A) ⊂ H → H, which is the infinitesimal generator of a C0 -semigroup {Q(t)}t≥0 satisfying ‖Q(t)‖ ≤ Mewt . Let us state a version of the Trotter–Kato theorem appeared in [169]. Theorem 3.7. Let AN ∈ G(M, ω) and assume: (i) limN→∞ AN x = Ax for every x ∈ D, where D is a dense subset of H. (ii) There exists a λ0 with Re(λ0 ) > ω for which (λ0 I − A)D is dense in H. Then the closure Ā of A is in G(M, ω). Moreover, if {QN (t)}t≥0 and {Q(t)}t≥0 are the C0 -semigroups generated by AN and A,̄ respectively, then lim QN (t)x = Q(t)x,

N→∞

for all t ≥ 0, x ∈ H

(3.20)

and the limit in (3.20) is uniform in t for t in bounded intervals. Now, let q > α1 . We introduce the set of admissible controls: W := Lq ([0, T], U).

(3.21)

3.2 Optimal control | 109

We first introduce an appropriate formulation of mild solutions. Let u ∈ W. The function x : [0, T] → H is called a mild solution of the equation (3.19) if x ∈ C([0, T], H) satisfies t

x(t) = S (t)x0 + ∫(t − s)α−1 P (t − s)[F(x(s)) + C(u(s))]ds,

(3.22)

0

where ∞

∞

α

S (t) = ∫ Mα (θ)Q(t θ)dθ, 0 ∞

α

P (t) = ∫ αθMα (θ)Q(t θ)dθ, 0

k

θ . k!Γ(1 − α(1 + k)) k=0

Mα (θ) = ∑

In what follows, x(t; x0 , u) denotes a mild solution of the equation (3.19) associated with the control u. Lemma 3.2 ([225] [see also [230]]). (i) For any fixed t ∈ [0, T], {S (t)} and {P (t)} are linear and bounded operators, i. e., for any x ∈ H, 󵄩󵄩 󵄩 α 󵄩󵄩S (t)x󵄩󵄩󵄩 ≤ MEα (ωt )‖x‖

󵄩 󵄩 and 󵄩󵄩󵄩P (t)x 󵄩󵄩󵄩 ≤ MEα,α (ωt α )‖x‖,

where tk Γ(αk + 1) k=0 ∞

Eα (t) = ∑

and

tk . Γ(αk + α) k=0 ∞

Eα,α (t) = ∑

(ii) For x ∈ H and t1 , t2 ∈ (0, T], we have 󵄩󵄩 󵄩 󵄩󵄩S (t1 ) − S (t2 )x󵄩󵄩󵄩 → 0,

󵄩󵄩 󵄩 󵄩󵄩P (t1 ) − P (t2 )x󵄩󵄩󵄩 → 0

as t1 → t2 .

Let HN ⊂ H denote the finite dimensional subspace for N ∈ ℤ+ and let PN : H → HN be orthogonal projectors and fulfil the following convergence: 󵄩 󵄩 lim 󵄩󵄩(PN − I)x󵄩󵄩󵄩 = 0,

N→∞󵄩

x ∈ H,

and HN ⊂ D(A),

∀ N ≥ 1.

(3.23)

Set AN := PN APN : H → HN .

(3.24)

110 | 3 Fractional control systems of order α ∈ (0, 1) In particular, we find D(AN ) = H. We construct the corresponding approximations of (3.19) on HN as follows: C α D xN (t) = AN xN (t) + PN F(xN ) + PN C(u(t)), {0 t xN (0) = PN x0 .

t ∈ (0, T],

(3.25)

Now, we give some conditions with respect to the approximation operators AN , which possess the same properties as A. (A1 ) For every N ∈ ℤ+ , the operator eAN t : HN → HN extends to a C0 -semigroup {QN (t)}t≥0 on H. In addition, AN ∈ G(M, ω), that is, 󵄩󵄩 󵄩 ωt 󵄩󵄩QN (t)󵄩󵄩󵄩 ≤ Me ,

N ≥ 1, t ≥ 0.

(3.26)

(A2 ) The following limit is satisfied: lim ‖AN ϕ − Aϕ‖ = 0,

ϕ ∈ D(A).

N→∞

(3.27)

Note that if the operator A is uniformly elliptic, then it is easy to build a family of operators {AN } so that (3.27) holds. Thus, (A1 ) and (A2 ) are verified. Under the conditions (A1 ), for N ∈ ℤ+ and t ∈ [0, T], a mild solution of the approximation equation (3.25) means xN ∈ C([0, T], HN ) fulfilling the following equality: t

xN (t) = SN (t)PN x0 + ∫(t − s)α−1 PN (t − s)[PN F(xN (s)) + PN C(u(s))]ds,

(3.28)

0

where ∞

α

SN (t) = ∫ Mα (θ)QN (t θ)dθ, 0

∞

α

PN (t) = ∫ αθMα (θ)QN (t θ)dθ. 0

Furthermore, the conclusions of Lemma 3.2 are also true for the operators {SN (t)} and {PN (t)}.

3.2.3 Convergence of mild solutions We give the following assumptions: (A3 ) The nonlinear term F : H → H is locally Lipschitz, i. e., for any ball Br ⊂ H there exists a constant LF (Br ) > 0 such that 󵄩󵄩 󵄩 󵄩󵄩F(x1 ) − F(x2 )󵄩󵄩󵄩 ≤ LF (Br )‖x1 − x2 ‖,

∀ x1 , x2 ∈ Br .

3.2 Optimal control | 111

(A4 ) For every x0 ∈ H and each u ∈ W, the equation (3.19) has a unique mild solution x(t; x0 , u) ∈ C([0, T], H). Meanwhile, its approximation equation (3.25) has a unique solution xN (t; PN x0 , u) ∈ C([0, T], HN ) for each N ∈ ℤ+ as well. Furthermore, there exists M1 := M1 (T, x0 , u) such that 󵄩 󵄩󵄩 󵄩󵄩xN (t; PN x0 , u)󵄩󵄩󵄩 ≤ M1 ,

∀ t ∈ [0, T], N ∈ ℤ+ .

(3.29)

We notice that (A4 ) is easily showed by a priori estimate if F and C satisfy the linear growth condition. Remark 3.3. Under the assumption (A4 ), we have x(⋅; x0 , u) ∈ C([0, T], H) for any x0 ∈ H and u ∈ W. Therefore, x(t; x0 , u) is bounded for all t ∈ [0, T]. We can suppose that ‖x(t; x0 , u)‖ ≤ M1 as well. Let us introduce an operator: t

α−1

(Φx)(t) = ∫(t − s)

t

α−1

P (t − s)F(x(s; u))ds + ∫(t − s)

P (t − s)C(u(s))ds.

0

0

We present the property of the operator Φ below. Remark 3.4. {(Φx)(t) : t ∈ [0, T], x ∈ BM1 } is equicontinuous. Such equicontinuity can be derived for fractional evolution equations; see [225, Lemma 4.26]. Next, we present a result that mild solutions of the approximation equation (3.25) are arbitrarily close to that of the equation (3.19) as N tends to ∞. Theorem 3.8. Assume that the operator C : U → H satisfies the following growth condition: 󵄩󵄩 󵄩 󵄩󵄩C(ϖ)󵄩󵄩󵄩 ≤ a‖ϖ‖U + b,

∀ ϖ ∈ H,

(3.30)

where a > 0, b ≥ 0 and q > α1 . In addition, suppose that (A1 ) − (A4 ) are satisfied. Then for each x0 ∈ H and u ∈ W, we have 󵄩 󵄩 lim sup 󵄩󵄩xN (t; PN x0 , u) − x(t; x0 , u)󵄩󵄩󵄩 = 0.

N→∞ t∈[0,T]󵄩

Proof. Let x0 ∈ H and u ∈ W. Put wN (t; u) := x(t; u) − xN (t; u),

for t ∈ [0, T].

112 | 3 Fractional control systems of order α ∈ (0, 1) From (3.22) and (3.28), we immediately calculate that wN (t; u) = S (t)x0 − SN (t)PN x0 t

+ ∫(t − s)α−1 [P (t − s) − PN (t − s)PN ]F(x(s; u))ds 0

t

+ ∫(t − s)α−1 [P (t − s) − PN (t − s)PN ]C(u(s))]ds

(3.31)

0

t

+ ∫(t − s)α−1 PN (t − s)PN [F(x(s; u)) − F(xN (s; u))]ds. 0

For 0 ≤ s ≤ t ≤ T, x0 ∈ H and u ∈ W, set 󵄩 󵄩 rN (s; u) := 󵄩󵄩󵄩x(s; u) − xN (s; u)󵄩󵄩󵄩, 󵄩 󵄩 ζN (t; x0 ) := 󵄩󵄩󵄩[S (t) − SN (t)PN ]x0 󵄩󵄩󵄩, 󵄩 󵄩 dN (s; u) := 󵄩󵄩󵄩[P (t − s) − PN (t − s)PN ]F(x(s; u))󵄩󵄩󵄩, 󵄩 󵄩 d̃ N (s; u) := 󵄩󵄩󵄩[P (t − s) − PN (t − s)PN ]C(u(s))󵄩󵄩󵄩. Using the inequality (3.31), we have the following calculation: t

rN (t; u) ≤ ζN (t; x0 ) + ∫(t − s) 0

α−1

t

dN (s; u)ds + ∫(t − s)α−1 d̃ N (s; u)ds 0

t

󵄩 󵄩 + MLF (BM1 ) ∫(t − s)α−1 Eα,α (ω(t − s)α )󵄩󵄩󵄩x(s; u) − xN (s; u)󵄩󵄩󵄩ds t

0

≤ ζN (t; x0 ) + ∫(t − s) 0

α−1

t

dN (s; u)ds + ∫(t − s)α−1 d̃ N (s; u)ds t

0

+ MLF (BM1 )Eα,α (ωT α ) ∫(t − s)α−1 rN (s; u)ds, 0

here we make use of the locally Lipschitz continuity of F, Remark 3.3 and Lemma 3.2. Then the singular Gronwall’s lemma allows us to obtain that t

rN (t; u) ≤ ηN (t) + κ ∫(t − s)α−1 Eα (κ(t − s))ηN (s)ds, 0

for t ∈ [0, T],

(3.32)

113

3.2 Optimal control |

1

where κ = [MLF (BM1 )Eα,α (ωT α )] α and t

α−1

ηN (t) = ζN (t; x0 ) + ∫(t − s)

t

dN (s; u)ds + ∫(t − s)α−1 d̃ N (s; u)ds.

0

0 t

t

Therefore, it remains to estimate that ζN (t; x0 ), ∫0 (t − s)α−1 dN (s; u)ds and ∫0 (t − s)α−1 d̃ N (s; u)ds tend to 0 uniformly in t as N → ∞. Since 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩[S (t) − SN (t)PN ]x0 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩[S (t) − SN (t)]x0 󵄩󵄩󵄩 + 󵄩󵄩󵄩SN (t)[I − PN ]x0 󵄩󵄩󵄩 ∞

󵄩 󵄩 ≤ ∫ Mα (θ)󵄩󵄩󵄩[Q(t α θ) − QN (t α θ)]x0 󵄩󵄩󵄩dθ

(3.33)

0

󵄩 󵄩 + MEα (ωt α )󵄩󵄩󵄩(I − PN )x0 󵄩󵄩󵄩

and ∞

∞

α 󵄩 󵄩 ∫ Mα (θ)󵄩󵄩󵄩[Q(t α θ) − QN (t α θ)]x0 󵄩󵄩󵄩dθ ≤ 2M‖x0 ‖ ∫ Mα (θ)eωθt dθ

0

0

= 2M‖x0 ‖Eα (ωt α ) < ∞,

for t ∈ [0, T].

Observe that the conditions (A1 ) and (A2 ) allow us to utilize Theorem 3.7, by the

Lebesgue’s dominated convergence theorem and (3.23). This follows that the righthand side of (3.33) tends to zero as N → ∞. It implies that limN→∞ ζN (t; x0 ) = 0 for t ∈ [0, T].

t

For ∫0 (t − s)α−1 dN (s; u)ds, we have t

∫(t − s)α−1 dN (s; u)ds 0

t

α−1

≤ ∫(t − s) 0

t

∞

󵄩 󵄩 ∫ αθMα (θ)󵄩󵄩󵄩[Q((t − s)α θ) − QN ((t − s)α θ)]F(x(s; u))󵄩󵄩󵄩dθds 0

󵄩 󵄩 + ∫(t − s)α−1 󵄩󵄩󵄩PN (s)[I − PN ]F(x(s; u))󵄩󵄩󵄩ds 0

=: I1N (t) + I2N (t).

(3.34)

114 | 3 Fractional control systems of order α ∈ (0, 1) By the fundamental estimations on I1N (t) and I1N (t), we get I1N (t)

t

∞

󵄩 ωθ(t−s)α dθ 󵄩󵄩F(x(s; u))󵄩󵄩󵄩ds ∫ αθMα (θ)e

α−1 󵄩 󵄩

≤ 2M ∫(t − s)

0

0

t

󵄩 󵄩 󵄩 󵄩 ≤ 2M ∫(t − s)α−1 Eα,α (ω(t − s)α )[󵄩󵄩󵄩F(0)󵄩󵄩󵄩 + LF (BM1 )󵄩󵄩󵄩x(s; u)󵄩󵄩󵄩]ds 0

󵄩 󵄩 ≤ 2M[󵄩󵄩󵄩F(0)󵄩󵄩󵄩 + M1 LF (BM1 )]t α Eα,α+1 (ωt α ) < ∞, and t

󵄩 󵄩 I2N (t) ≤ M‖I − PN ‖ ∫(t − s)α−1 Eα,α (ω(t − s)α )󵄩󵄩󵄩F(x(s; u))󵄩󵄩󵄩ds 0

󵄩 󵄩 ≤ M‖I − PN ‖[󵄩󵄩󵄩F(0)󵄩󵄩󵄩 + M1 LF (BM1 )]t α Eα,α+1 (ωt α ) < ∞,

for t ∈ [0, T].

t

Using Theorem 3.7 and (3.23) again, it yields that ∫0 (t − s)α−1 dN (s; u)ds also tends to zero as N → ∞. t Let us estimate ∫0 (t − s)α−1 d̃ N (s; u)ds as N → ∞. In view of the growth condition of C, we obtain for a. e. t ∈ [0, T], t

∫(t − s)α−1 d̃ N (s; u)ds 0

t

󵄩 󵄩 ≤ M(1 + ‖PN ‖) ∫(t − s)α−1 Eα,α (ω(t − s)α )[a󵄩󵄩󵄩u(s)󵄩󵄩󵄩U + b]ds 0

q−1 ≤ M(1 + ‖PN ‖)[a‖u‖W Eα,α (ωT )( ) αq − 1 α

q−1 q

T

αq−1 q

+ bT α Eα,α+1 (ωT α )].

Here, we use u ∈ Lq ([0, T], U). t Similar to the above arguments to evaluate ∫0 (t−s)α−1 dN (s; u)ds, we conclude from the Lebesgue’s dominated convergence theorem that t

lim ∫(t − s)α−1 d̃ N (s; u)ds = 0.

N→∞

0

Hence, limN→∞ ηN (t) = 0 pointwise on t ∈ [0, T]. On the other hand, by Lemma 3.2 and Remark 3.4, it is easy to see that {ζN (t; x0 )}, t t {∫0 (t − s)α−1 dN (s; u)ds} and {∫0 (t − s)α−1 d̃ N (s; u)ds} are equicontinuous on the interval [0, T]. So, we obtain the equicontinuity of ηN (t) on the interval [0, T], thus

3.2 Optimal control | 115

limN→∞ ηN (t) = 0 uniformly on t ∈ [0, T]. This together with (3.32) implies that the desired convergence result is obtained. 3.2.3.1 Local approximation result on u We define a new set Wad ⊂ Lq ([0, T], U) as Wad := {f ∈ Lq ([0, T], U) : f (t) ∈ V for a. e. t ∈ [0, T]},

q>

1 , α

(3.35)

where V is a bounded subset of U. First, we declare a natural property that the remaining part of solutions of the ̃ can be taken arbitrarily small as equation (3.19), i. e., supt∈[0,T] ‖(I − PN )x(t; x0 , u)‖ N → ∞ and uniformly in u,̃ if ũ belongs to a small enough open set of Wad . On the other hand, the proof of the following result also shows that the mapping x(t; x0 , ⋅) is continuous with respect to u. Lemma 3.3. Suppose that (A1 )–(A4 ) hold. If C : U → H is locally Lipschitz, then for every (x0 , u) ∈ H × Wad the mild solution x(t; x0 , u) of (3.19) has the following property: for any ε > 0 there exist N0 ≥ 1 and a neighborhood Ou ⊂ Wad of u such that 󵄩 󵄩 sup sup 󵄩󵄩󵄩(I − PN )x(t; x0 , u)̃ 󵄩󵄩󵄩 < ε,

̃ u t∈[0,T] u∈O

∀ N ≥ N0 .

Proof. Fix u ∈ Wad and r > 0. For any ũ ∈ Br (u), x0 ∈ H and t ∈ [0, T], consider ̃ It follows from (3.22) that ψ(t) := x(t; x0 , u) − x(t; x0 , u). t

󵄩󵄩 󵄩 α−1 α 󵄩 ̃ 󵄩󵄩󵄩󵄩ds 󵄩󵄩ψ(t)󵄩󵄩󵄩 ≤ M ∫(t − s) Eα,α (ω(t − s) )󵄩󵄩󵄩F(x(s; u)) − F(x(s; u)) 0

t

󵄩 󵄩󵄩 ̃ + M ∫(t − s)α−1 Eα,α (ω(t − s)α )󵄩󵄩󵄩C(u(s)) − C(u(s)) 󵄩󵄩ds. 0

In view of Remark 3.3, we can choose M1 > 0 such that x(t; x0 , u)‖ ≤ M1 for all t ∈ [0, T]. On account of ũ ∈ Br (u), we can define 󵄩 󵄩 t ∗ := max{t ∈ [0, T] : 󵄩󵄩󵄩x(t; x0 , u)̃ 󵄩󵄩󵄩 < 2M1 }. First, we show that t ∗ > 0. In fact, since x(t; x0 , u)̃ is a mild solution for any (x0 , u)̃ ∈ H×Wad , then x(⋅; x0 , u)̃ ∈ C([0, T], H). This together with ‖x0 ‖ ≤ M1 (due to ‖x(t; x0 , u)‖ ≤ M1 ) implies that for every ũ ∈ Br (u) there exists tû ̃ > 0 such that 󵄩󵄩 󵄩 󵄩󵄩x(t; x0 , u)̃ 󵄩󵄩󵄩 < 2M1 , and, therefore, t ∗ > 0.

∀ t ∈ [0, tû ̃ ],

116 | 3 Fractional control systems of order α ∈ (0, 1) We denote by B(V) the minimum closed ball including V in U. It follows from the locally Lipschitz property of F and C and the Hölder’s inequality that t

󵄩 󵄩 󵄩󵄩 α α−1 󵄩 󵄩󵄩ψ(t)󵄩󵄩󵄩 ≤ MLF (B2M1 )Eα,α (ωT ) ∫(t − s) 󵄩󵄩󵄩ψ(s)󵄩󵄩󵄩ds 0

t

󵄩 ̃ 󵄩󵄩󵄩󵄩U ds + MLC (B(V))Eα,α (ωT α ) ∫(t − s)α−1 󵄩󵄩󵄩u(s) − u(s) t

0

󵄩 󵄩 ≤ MLF (B2M1 )Eα,α (ωT α ) ∫(t − s)α−1 󵄩󵄩󵄩ψ(s)󵄩󵄩󵄩ds 0

q−1 + MLC (B(V))Eα,α (ωT )( ) αq − 1 α

q−1 q

t

αq−1 q

‖u − u‖̃ W .

Then by the Gronwall’s inequality we conclude that 󵄩󵄩 󵄩 α 󵄩󵄩ψ(t)󵄩󵄩󵄩 ≤ MLC (B(V))Eα,α (ωT ) ×(

q−1 ) αq − 1

q−1 q

(t ∗ )

αq−1 q

α

rEα (MLF (B2M1 )Eα,α (ωT α )Γ(α)(t ∗ ) ),

for t ∈ [0, t ∗ ]. Set M ∗ := MLC (B(V))Eα,α (ωT α ) q−1 ×( ) αq − 1

q−1 q

(t ∗ )

αq−1 q

α

Eα (MLF (B2M1 )Eα,α (ωT α )Γ(α)(t ∗ ) )

M1 and r1 := 2M ∗. We declare that t ∗ = T if r ≤ r1 . On the contrary, if t ∗ < T, we get ‖ψ(t ∗ )‖ ≤ M ∗ r ≤ M1 . This implies that 2

󵄩󵄩 ∗ 󵄩 󵄩 ∗ 󵄩 󵄩 ∗ 󵄩 3M1 , 󵄩󵄩x(t ; x0 , u)̃ 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩x(t ; x0 , u)󵄩󵄩󵄩 + 󵄩󵄩󵄩ψ(t )󵄩󵄩󵄩 ≤ 2 which is in contradiction with the definition of t ∗ . Thus, for each r ∈ (0, r1 ] we know that 󵄩󵄩 󵄩 ∗ 󵄩󵄩ψ(t)󵄩󵄩󵄩 ≤ rM ,

∀ t ∈ [0, T].

Therefore, for any ε > 0, there exists rε > 0 small enough such that ‖ψ(t)‖ ≤ rε M ∗ . So, ‖ψ(t)‖ ≤ ε2 for any t ∈ [0, T]. It means that 󵄩 󵄩 ε sup 󵄩󵄩󵄩x(t; x0 , u) − x(t; x0 , u)̃ 󵄩󵄩󵄩 < . 2 ̃ u∈B(u,r ε ) t∈[0,T] sup

(3.36)

3.2 Optimal control | 117

On the other hand, since (3.23) holds and x(⋅; x0 , u) ∈ C([0, T], H), then for any ε > 0, there exists N0 ∈ Z ∗ such that 󵄩 ε 󵄩 sup 󵄩󵄩󵄩(I − PN )x(t; x0 , u)󵄩󵄩󵄩 < , 2 t∈[0,T]

∀ N ≥ N0 .

Hence, we can conclude that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩(I − PN )x(t; x0 , u)̃ 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩(I − PN )[x(t; x0 , u)̃ − x(t; x0 , u)]󵄩󵄩󵄩 + 󵄩󵄩󵄩(I − PN )x(t; x0 , u)󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≤ (1 + ‖PN ‖)󵄩󵄩󵄩x(t; x0 , u)̃ − x(t; x0 , u)󵄩󵄩󵄩 + 󵄩󵄩󵄩(I − PN )x(t; x0 , u)󵄩󵄩󵄩 < ε.

We choose the open ball 𝔹rε (u) in Wad as Ou . Then the conclusion holds. Theorem 3.9. If the assumptions of Lemma 3.3 are satisfied, then for every (x0 , u) ∈ H × Wad the mild solution x(t; x0 , u) of (3.19) has the following property: for any ε > 0, there exists a neighborhood N0 ≥ 1 and Ou ⊂ Wad of u such that 󵄩 󵄩 sup sup 󵄩󵄩󵄩xN (t; PN x0 , u)̃ − x(t; x0 , u)̃ 󵄩󵄩󵄩 < ε,

̃ u t∈[0,T] u∈O

∀ N ≥ N0 ,

where xN is the solution of the approximation equation (3.25). Proof. According to (3.36), we know that the mild solution of (3.19) depends continuously on the control u ∈ Wad . By the similar argument, (3.36) remains true to the solutions of approximation equation (3.25), which ensures their continuous dependence on u. Therefore, we infer for any ε > 0 and u ∈ Wad , the existence of a neighborhood Ou ⊂ Wad including u such that 󵄩 󵄩 ε sup sup 󵄩󵄩󵄩x(t; x0 , u)̃ − x(t; x0 , u)󵄩󵄩󵄩 < , 3 ε 󵄩󵄩 󵄩 sup sup 󵄩󵄩xN (t; PN x0 , u)̃ − xN (t; PN x0 , u)󵄩󵄩󵄩 < , 3 ̃ u t∈[0,T] u∈O ̃ u t∈[0,T] u∈O

∀ N ≥ 1.

In addition, we know from Theorem 3.8 that there exists N0 > 0 such that 󵄩 󵄩 ε sup 󵄩󵄩󵄩xN (t; PN x0 , u) − x(t; x0 , u)󵄩󵄩󵄩 < , 3

t∈[0,T]

∀ N ≥ N0 .

Therefore, it yields that 󵄩 󵄩 sup sup 󵄩󵄩󵄩xN (t; PN x0 , u)̃ − x(t; x0 , u)̃ 󵄩󵄩󵄩

̃ u t∈[0,T] u∈O

󵄩 󵄩 󵄩 󵄩 ≤ sup sup 󵄩󵄩󵄩xN (t; PN x0 , u)̃ − xN (t; PN x0 , u)󵄩󵄩󵄩 + sup 󵄩󵄩󵄩xN (t; PN x0 , u) − x(t; x0 , u)󵄩󵄩󵄩 ̃ u t∈[0,T] u∈O

t∈[0,T]

118 | 3 Fractional control systems of order α ∈ (0, 1) 󵄩 󵄩 + sup sup 󵄩󵄩󵄩x(t; x0 , u) − x(t; x0 , u)̃ 󵄩󵄩󵄩 ̃ u t∈[0,T] u∈O

< ε. The proof is complete. 3.2.3.2 Uniform approximation Now, we show that the convergence result holds uniformly with respect to u when V is a compact set of U. To achieve our aim, we suppose that (A5 ) V is a compact set of U. (A6 ) For each (x0 , u) ∈ H × Wad , the equation (3.19) and its approximation equation (3.25) have a unique mild solution x(⋅; x0 , u) ∈ C([0, T], H) and xN (⋅; PN x0 , u) ∈ C([0, T], HN ), respectively. In addition, there exists M2 := M2 (T, x0 ) such that 󵄩󵄩 󵄩 󵄩󵄩x(t; x0 , u)󵄩󵄩󵄩 ≤ M2 , 󵄩󵄩 󵄩 󵄩󵄩xN (t; PN x0 , u)󵄩󵄩󵄩 ≤ M2 ,

∀ t ∈ [0, T], u ∈ Wad , N ∈ ℤ+ .

(3.37)

(A7 ) For each (x0 , u) ∈ H × Wad , any mild solution x(t; x0 , u) of (3.19) has the following property: 󵄩 󵄩 lim sup sup 󵄩󵄩(I − PN )x(t; x0 , u)󵄩󵄩󵄩 = 0.

N→∞ u∈Wad t∈[0,T]󵄩

Notice that compared with (A4 ), the condition (A6 ) is stronger since the mild solution x(t; x0 , u) of (3.19) is bounded, and the bound does not rely on u. Lemma 3.4 ([64]). Suppose that (A3 ), (A6 ) and (A7 ) are satisfied. Then 󵄩 󵄩 lim sup sup 󵄩󵄩(I − PN )F(x(t; x0 , u))󵄩󵄩󵄩 = 0.

N→∞ u∈Wad t∈[0,T]󵄩

Theorem 3.10. Let (A1 )–(A3 ) and (A5 )–(A7 ) be satisfied. If C : U → H is a continuous map, then the solution x(t; x0 , u) of (3.19) has the following property for every (x0 , u) ∈ H × Wad : 󵄩 󵄩 lim sup sup 󵄩󵄩xN (t; PN x0 , u) − x(t; x0 , u)󵄩󵄩󵄩 = 0.

N→∞ u∈Wad t∈[0,T]󵄩

Proof. According to the proof of Theorem 3.8 and (3.32), to achieve the conclusion, it remains to prove that the two terms t

󵄩 󵄩 ∫(t − s)α−1 󵄩󵄩󵄩[P (t − s) − PN (t − s)PN ]F(x(s; x0 , u))󵄩󵄩󵄩ds 0

3.2 Optimal control |

119

and t

󵄩 󵄩 ∫(t − s)α−1 󵄩󵄩󵄩[P (t − s) − PN (t − s)PN ]C(u(s))󵄩󵄩󵄩ds 0

tend to 0 as N → ∞ uniformly on t ∈ [0, T] and u ∈ Wad . On the one hand, Lemma 3.4 ensures for arbitrary ε > 0 and t ∈ [0, T] the existence of N0 ∈ ℤ+ such that 󵄩󵄩 󵄩 󵄩󵄩(I − PN )F(x(s; x0 , u))󵄩󵄩󵄩 ≤ ε,

s ∈ [0, t], u ∈ Wad , ∀ N ≥ N0 .

For a particular choice of N0 , we get t

󵄩 󵄩 ∫(t − s)α−1 󵄩󵄩󵄩[P (t − s) − PN (t − s)PN ]F(x(s; x0 , u))󵄩󵄩󵄩ds 0

t

󵄩 󵄩 ≤ ∫(t − s)α−1 󵄩󵄩󵄩[P (t − s) − PN (t − s)PN ]PN0 F(x(s; x0 , u))󵄩󵄩󵄩ds 0

(3.38)

t

󵄩 󵄩 + ∫(t − s)α−1 󵄩󵄩󵄩[P (t − s) − PN (t − s)PN ](I − PN0 )F(x(s; x0 , u))󵄩󵄩󵄩ds. 0

In view of Lemma 3.2, one can find t

󵄩 󵄩 ∫(t − s)α−1 󵄩󵄩󵄩[P (t − s) − PN (t − s)PN ](I − PN0 )F(x(s; x0 , u))󵄩󵄩󵄩ds 0

t

󵄩 󵄩 ≤ 2M ∫(t − s)α−1 Eα,α (ω(t − s)α )󵄩󵄩󵄩(I − PN0 )F(x(s; u))󵄩󵄩󵄩ds

(3.39)

0 α

≤ 2Mt Eα,α+1 (ωt α )ε. On the other hand, we know from (A6 ) and the locally Lipschitz property of F that PN0 F(x(s; x0 , u)) ∈ HN0

󵄩 󵄩 󵄩 󵄩 and 󵄩󵄩󵄩PN0 F(x(s; x0 , u))󵄩󵄩󵄩 ≤ ‖PN0 ‖[󵄩󵄩󵄩F(0)󵄩󵄩󵄩 + LF (BM2 )M2 ]

for all u ∈ Wad and s ∈ [0, t]. Since the dimension of HN0 is finite, we know that the closure of the set Σ(x) = {PN0 F(x(s; x0 , u)) : u ∈ Wad , s ∈ [0, T]}

120 | 3 Fractional control systems of order α ∈ (0, 1) is compact. For each z ∈ Σ(x) and ε > 0, by the same argument employed in (3.34), we have lim PN (t)PN z = P (t)z.

(3.40)

N→∞

It follows from the equicontinuity of {PN (⋅)PN z} that the limit in (3.40) also holds uniformly on [0, T]. Therefore there exists N1 (z) ∈ ℤ+ such that 󵄩 󵄩 ε sup 󵄩󵄩󵄩[P (t) − PN (t)PN ]z 󵄩󵄩󵄩 ≤ , 2

t∈[0,T]

∀ N ≥ N1 (z).

From the continuity of P (t) and PN (t), we can refer the existence of a neighborhood Oz ⊂ HN0 including z such that 󵄩 󵄩 sup 󵄩󵄩󵄩[P (t) − PN (t)PN ]z̃󵄩󵄩󵄩 ≤ ε,

t∈[0,T]

∀ N ≥ N1 (z), z̃ ∈ Oz .

(3.41)

Since the compactness of Σ(x) allows us to withdraw a finite cover of Σ(x), which is taken as such neighborhood Oz and (3.41) is satisfied, then there exists N1 ∈ ℤ+ such that 󵄩 󵄩 sup 󵄩󵄩󵄩[P (t) − PN (t)PN ]z̃󵄩󵄩󵄩 ≤ ε,

t∈[0,T]

∀ N ≥ N1 , z̃ ∈ Σ(x).

For each u ∈ Wad , taking z̃ = PN0 F(x(s; x0 , u)), it follows that 󵄩 󵄩 sup 󵄩󵄩󵄩[P (t − s) − PN (t − s)PN ]PN0 F(x(s; x0 , u))󵄩󵄩󵄩 ≤ ε,

s∈[0,t]

t ∈ [0, T], ∀ N ≥ N1 .

Hence, t

εt α 󵄩 󵄩 . ∫(t − s)α−1 󵄩󵄩󵄩[P (t − s) − PN (t − s)PN ]PN0 F(x(s; x0 , u))󵄩󵄩󵄩ds ≤ α 0

Thus, we know from (3.38) and (3.39) that t

󵄩 󵄩 sup sup ∫(t − s)α−1 󵄩󵄩󵄩[P (t − s) − PN (t − s)PN ]F(x(s; x0 , u))󵄩󵄩󵄩ds

u∈Wad t∈[0,T]

≤ εT α (

0

1 + 2MEα,α+1 (ωT α )). α

We consider now the term t

󵄩 󵄩 sup sup ∫(t − s)α−1 󵄩󵄩󵄩[P (t − s) − PN (t − s)PN ]C(u(s))󵄩󵄩󵄩ds.

u∈Wad t∈[0,T]

0

(3.42)

3.2 Optimal control | 121

The compactness of V ⊂ U and the continuity of C show the compactness of C(V). For u ∈ Wad , since u(t) belongs to V for almost every t ∈ [0, T], from a similar compactness argument as we derived (3.42), there exists a positive integer N2 such that 󵄩 󵄩 sup 󵄩󵄩󵄩[P (t − s) − PN (t − s)PN ]C(u(s))󵄩󵄩󵄩 ≤ ε,

s∈[0,t]

t ∈ [0, T], ∀ N ≥ N2 .

Therefore, for all u ∈ Wad and N ≥ N2 , one can find that t

εT α 󵄩 󵄩 . sup sup ∫(t − s)α−1 󵄩󵄩󵄩[P (t − s) − PN (t − s)PN ]C(u(s))󵄩󵄩󵄩ds ≤ α u∈Wad t∈[0,T] 0

The desired uniform convergence of the two terms is obtained. 3.2.4 Convergence of optimal control and cost functional For the equation (3.19), we consider its cost functional J : H × Wad → ℝ+ as follows: T

J (x0 , u) := ∫[Λ(x(s; x0 , u)) + ϒ(u(s))]ds,

x0 ∈ H.

(3.43)

0

Here, we suppose that (AL ) Λ : H → ℝ+ and ϒ : U → ℝ+ are continuous, and Λ is locally Lipschitz. The corresponding optimal control problem is to find min J (x0 , u) over (x, u) ∈ L2q ([0, T], H) × Wad subject to (3.19) with x(0) = x0 ∈ H. For the approximation equation (3.25), we also consider its cost functional JN : HN × Wad → ℝ+ : T

JN (PN x0 , u) := ∫[Λ(xN (s; PN x0 , u)) + ϒ(u(s))]ds,

x0 ∈ H,

0

and its optimal control problem is to find min JN (PN x0 , u) over (xN , u) ∈ L2q ([0, T], HN ) × Wad subject to (3.25) with xN (0) = PN x0 ∈ HN . Therefore, the optimal control problems of (3.19) and of (3.25) are respectively given by v(x0 ) := inf J (x0 , u), u∈Wad

∀ (t, x0 ) ∈ [0, T] × H,

vN (PN x(0)) := inf JN (PN x0 , u), u∈Wad

∀ (t, PN x0 ) ∈ [0, T] × HN .

Next, we present a convergence result of optimal controls.

(3.44) (3.45)

122 | 3 Fractional control systems of order α ∈ (0, 1) Theorem 3.11. Let the conditions of Theorem 3.10 and (AL ) hold. Suppose that (3.44) (resp., in (3.45)) admits a minimizer u∗ (resp., u∗N ) in Wad for each x0 ∈ H. Then we have 󵄨 󵄨 lim 󵄨󵄨vN (PN x0 ) − v(x0 )󵄨󵄨󵄨 = 0

N→∞󵄨

for any x0 ∈ H.

Proof. According to the definition of v(x0 ) and vN (PN x0 ), we get v(x0 ) = J (x0 , u∗ ) ≤ J (x0 , u∗N ),

vN (PN x0 ) = JN (x0 , u∗N ). Then the first inequality leads to T

v(x0 ) ≤ ∫[Λ(x(s; x0 , u∗N )) + ϒ(u∗N (s))]ds. 0

Subtract JN (PN x0 , u∗N ) on both sides of the above inequality. Using (A6 ) and the locally Lipschitz condition of Λ, we estimate T

v(x0 ) − vN (PN x0 ) ≤ ∫[Λ(x(s; x0 , u∗N )) − Λ(xN (s; PN x0 , u∗N ))]ds 0

T

󵄩 󵄩 ≤ LΛ (BM2 ) ∫󵄩󵄩󵄩x(s; x0 , u∗N ) − xN (s; PN x0 , u∗N )󵄩󵄩󵄩ds 0

󵄩 󵄩 ≤ TLΛ (BM2 ) sup 󵄩󵄩󵄩x(s; x0 , u∗N ) − xN (s; PN x0 , u∗N )󵄩󵄩󵄩. s∈[0,T]

A similar argument gives then 󵄩 󵄩 vN (PN x0 ) − v(x0 ) ≤ TLΛ (BM2 ) sup 󵄩󵄩󵄩x(s; x0 , u∗N ) − xN (s; PN x0 , u∗N )󵄩󵄩󵄩. s∈[0,T]

From Theorem 3.10, we draw the desired conclusion. In view of [203], we also obtain the following result which gives a sufficient condition for the existence of optimal pairs. Remark 3.5. Denote L (t, x, u) = Λ(x(t; x0 , u)) + ϒ(u(t)). Assume that (HL ) (i) the functional L : [0, T] × H × Wad → ℝ ∪ {∞} is Borel measurable; (ii) L (t, ⋅, ⋅) is sequentially lower semicontinuous on H × Wad for a. e. t ∈ [0, T]; (iii) L (t, x, ⋅) is convex on Wad for a. e. t ∈ [0, T] and each x ∈ H; (iv) there exist constants c, d ≥ 0 and φ ∈ L1 ([0, T], ℝ) such that q

L (t, x, u) ≥ φ(t) + c‖x‖ + d‖u‖ .

3.2 Optimal control |

123

(H3 )′ There exists a constant b > 0 such that 󵄩 󵄩󵄩 󵄩󵄩F(t, x)󵄩󵄩󵄩 ≤ b(1 + ‖x‖),

∀ x ∈ H.

If (A3 ), (A5 ) and (3.30) hold, then there exists for each x0 ∈ H a minimizer u∗ in Wad of the minimization problem in (3.44).

3.2.5 Error estimates of optimal control and cost functional In this subsection, we investigate some easy and conducive error estimates. To achieve our purpose, we suppose that: (A8 ) (i) The operator A : D(A) ⊂ H → H is self-adjoint. (ii) Assumptions (A6 ) and (AL ) hold. In view of (A8 )(i), we know that A has discrete spectrum {λk } and ek is an eigenfunction of A with the corresponding eigenvalue λk . Let HN := span{ek : k = 1, . . . , N}, N ∈ ℤ+ and we construct the approximation equation (3.25) on HN . Lemma 3.5. Let the assumption (A8 ) be satisfied. Then for any x0 ∈ H and u ∈ Wad , there exists ρ > 0 independent of N such that αq−1 2q 󵄨󵄨 󵄨 󵄩 +1 1− 1 󵄩 T 2q + T 2q )󵄩󵄩󵄩PN⊥ x(⋅; x0 , u)󵄩󵄩󵄩L2q , 󵄨󵄨J (x0 , u) − JN (PN x0 , u)󵄨󵄨󵄨 ≤ LΛ (BM2 )(ρ αq + 2q − 1

where PN⊥ = I − PN . Proof. Since Λ is locally Lipschitz, by the immediate calculation, we get 󵄨󵄨 󵄨 󵄨󵄨J (x0 , u) − JN (PN x0 , u)󵄨󵄨󵄨 T

󵄩 󵄩 ≤ LΛ (BM2 ) ∫󵄩󵄩󵄩x(s; x0 , u) − xN (s; PN x0 , u)󵄩󵄩󵄩ds 0

T

󵄩 󵄩 󵄩 󵄩 ≤ LΛ (BM2 ) ∫󵄩󵄩󵄩PN x(s; x0 , u) − xN (s; PN x0 , u)󵄩󵄩󵄩 + 󵄩󵄩󵄩PN⊥ x(s; x0 , u)󵄩󵄩󵄩ds, 0

where we rewrite x(s; x0 , u) = PN x(s; x0 , u) + PN⊥ x(s; x0 , u). Set ̃ w(s) := PN x(s; x0 , u) − xN (s; PN x0 , u),

(3.46)

124 | 3 Fractional control systems of order α ∈ (0, 1) and apply PN to both sides of (3.19). Then (3.19) converts to the following form: C α D PN x(t) = AN PN x(t) + PN F(PN x + PN⊥ x) + PN C(u(t)), {0 t PN x(0) = PN x0 ∈ HN .

t ∈ (0, T],

It follows from (3.25) that w̃ satisfies the equation C α ̃ = AN w(t) ̃ + PN F(PN x + PN⊥ x) − PN F(xN ), t ∈ (0, T], D w(t) {0 t ̃ w(0) = 0.

(3.47)

We take the inner product of the equation (3.47) with w̃ to obtain ̃ = (AN w(t), ̃ ̃ + (PN F(PN x + PN⊥ x) − PN F(xN ), w). ̃ (C0Dαt w,̃ w) w) According to (A8 )(i), the local Lipschitz continuity of F and Young’s inequality, we obtain 1 C α 󵄩󵄩 2 ̃ 󵄩󵄩󵄩󵄩 ≤ (AN w(t), ̃ ̃ ̃ D 󵄩w(t) w(t)) + (PN (F(PN x + PN⊥ x) − F(xN )), w) 20 t 󵄩 N

󵄩 󵄩2 󵄩 󵄩 ̃ ≤ ∑ λk 󵄩󵄩󵄩w̃ k (t)󵄩󵄩󵄩 + LF (BM2 )󵄩󵄩󵄩PN x + PN⊥ x − xN 󵄩󵄩󵄩‖w‖ k=1

󵄩 ̃ 󵄩󵄩2 ̃ + 󵄩󵄩󵄩󵄩PN⊥ x 󵄩󵄩󵄩󵄩)‖w‖ ̃ ≤ λN 󵄩󵄩󵄩w(t) 󵄩󵄩 + LF (BM2 )(‖w‖

1 3 2 󵄩 ̃ 󵄩󵄩2 ̃ 2 + 󵄩󵄩󵄩󵄩PN⊥ x 󵄩󵄩󵄩󵄩 ) ≤ λN 󵄩󵄩󵄩w(t) 󵄩󵄩 + LF (BM2 )( ‖w‖ 2 2 3 󵄩 ̃ 󵄩󵄩2 LF (BM2 ) 󵄩󵄩 ⊥ 󵄩2 = [λN + LF (BM2 )]󵄩󵄩󵄩w(t) 󵄩󵄩PN x(t; x0 , u)󵄩󵄩󵄩 . 󵄩󵄩 + 2 2

Integrating the above inequality (with order α) and using the Hölder inequality, one can find t

1 󵄩 ̃ 󵄩󵄩2 󵄩󵄩 ̃ 󵄩󵄩2 [2λ + 3LF (BM2 )] ∫(t − s)α−1 󵄩󵄩󵄩w(s) 󵄩󵄩w(t)󵄩󵄩 ≤ 󵄩󵄩 ds Γ(α) N +

≤

LF (BM2 ) Γ(α)

t

0

󵄩 󵄩2 ∫(t − s)α−1 󵄩󵄩󵄩PN⊥ x(s; x0 , u)󵄩󵄩󵄩 ds 0

t

1 󵄩 ̃ 󵄩󵄩2 [2λ + 3LF (BM2 )] ∫(t − s)α−1 󵄩󵄩󵄩w(s) 󵄩󵄩 ds Γ(α) N 0

+

LF (BM2 )

q−1 ( ) Γ(α) αq − 1

q−1 q

t

αq−1 q

t

1 q

󵄩 󵄩2q (∫󵄩󵄩󵄩PN⊥ x(s; x0 , u)󵄩󵄩󵄩 ds) . 0

3.2 Optimal control |

125

A generalized Gronwall’s inequality can give 󵄩󵄩 ̃ 󵄩󵄩2 LF (BM2 ) q − 1 ( ) 󵄩󵄩w(t)󵄩󵄩 ≤ Γ(α) αq − 1

q−1 q

t

t

αq−1 q 1 q

󵄩 󵄩2q × (∫󵄩󵄩󵄩PN⊥ x(s; x0 , u)󵄩󵄩󵄩 ds) Eα ([2λN + 3LF (BM2 )]t α ). 0

Thus, there exists ρ > 0 such that T

󵄩 󵄩 󵄩 󵄩 ∫󵄩󵄩󵄩PN x(s; x0 , u) − xN (s; PN x0 , u)󵄩󵄩󵄩 + 󵄩󵄩󵄩PN⊥ x(s; x0 , u)󵄩󵄩󵄩ds 0

T

≤ ρ∫s 0

αq−1 2q

s

1 2q

󵄩 󵄩2q 󵄩 1− 1 󵄩 (∫󵄩󵄩󵄩PN⊥ x(τ; x0 , u)󵄩󵄩󵄩 dτ) ds + T 2q 󵄩󵄩󵄩PN⊥ x(⋅; x0 , u)󵄩󵄩󵄩L2q 0

αq−1 2q 󵄩 +1 1− 1 󵄩 T 2q + T 2q )󵄩󵄩󵄩PN⊥ x(⋅; x0 , u)󵄩󵄩󵄩L2q . ≤ (ρ αq + 2q − 1

The desired estimation is obtained from (3.47) and the above inequality. Theorem 3.12. Let the assumption (A8 ) be satisfied. Suppose that (3.44) (resp., in (3.45)) admits a minimizer u∗ (resp., u∗N ) in Wad for each x0 ∈ H. Then we have αq−1 2q 󵄨󵄨 󵄨 +1 1− 1 T 2q + T 2q ) 󵄨󵄨v(x0 ) − vN (PN x0 )󵄨󵄨󵄨 ≤ LΛ (BM2 )(ρ αq + 2q − 1 󵄩 󵄩 󵄩 󵄩 × (󵄩󵄩󵄩PN⊥ x(⋅; x0 , u∗ )󵄩󵄩󵄩L2q + 󵄩󵄩󵄩PN⊥ x(⋅; x0 , u∗N )󵄩󵄩󵄩L2q ),

for any x0 ∈ H,

where ρ is the constant in Lemma 3.5. Proof. In view of the definitions of v and vN , we have v(x0 ) − vN (PN x0 ) ≤ J (x0 , u∗N ) − JN (PN x0 , u∗N ),

vN (PN x0 ) − v(x0 ) ≤ JN (PN x0 , u∗ ) − J (x0 , u∗ ), which together with Lemma 3.5 implies that

αq−1 2q 󵄩 +1 1− 1 󵄩 T 2q + T 2q )󵄩󵄩󵄩PN⊥ x(⋅; x0 , u∗N )󵄩󵄩󵄩L2q , αq + 2q − 1 αq−1 2q 󵄩 +1 1− 1 󵄩 T 2q + T 2q )󵄩󵄩󵄩PN⊥ x(⋅; x0 , u∗ )󵄩󵄩󵄩L2q . vN (PN x0 ) − v(x0 ) ≤ LΛ (BM2 )(ρ αq + 2q − 1

v(x0 ) − vN (PN x0 ) ≤ LΛ (BM2 )(ρ

Therefore, the conclusion holds.

126 | 3 Fractional control systems of order α ∈ (0, 1) Corollary 3.1. Let the conditions of Theorem 3.12 be satisfied. In addition, suppose that there exist ς > 0 and some neighborhood O ⊂ Wad of u∗ such that for x0 ∈ H and v ∈ O, we have 󵄩 󵄩 ς󵄩󵄩󵄩u∗ − v󵄩󵄩󵄩Lq ≤ J (x0 , v) − J (x0 , u∗ ). If u∗N ∈ O, then αq−1 1 2q 󵄩󵄩 ∗ +1 1− 1 ∗󵄩 T 2q + T 2q ) 󵄩󵄩u − uN 󵄩󵄩󵄩Lq ≤ LΛ (BM2 )(ρ ς αq + 2q − 1 󵄩 󵄩 󵄩 󵄩 × (󵄩󵄩󵄩PN⊥ x(⋅; x0 , u∗ )󵄩󵄩󵄩L2q + 󵄩󵄩󵄩PN⊥ x(⋅; x0 , u∗N )󵄩󵄩󵄩L2q ).

3.2.6 Applications Here, we assume that Ω is a bounded domain in ℝn (n ≥ 2) with a Lipschitz boundary 𝜕Ω, and we investigate the fractional heat equations: α

𝜕 in (0, T) × Ω, α y(t, ξ ) − Δy(t, ξ ) = f (t, ξ , y(t, ξ )) + C(u(t, ξ )), { { { 𝜕t y(t, ξ ) = 0, on (0, T) × 𝜕Ω, { { { in Ω, {y(0, ξ ) = x0 (ξ ),

(3.48)

α

where 𝜕t𝜕 α is the Caputo fractional partial derivative of order α ∈ (0, 1), the control u : [0, T] × Ω → Ω. Let A : L2 (Ω) → L2 (Ω) be defined by D(A) := H01 (Ω) ∩ H 2 (Ω), Ay = Δy,

y ∈ D(A).

It is known that A is self-adjoint with discrete spectrum, i. e., there exists the eigenfunctions sequence {ek }k∈ℤ+ constituting an orthonormal basis of L2 (Ω) with corresponding λk such that: ⋅ ⋅ ⋅ ≤ λk ≤ ⋅ ⋅ ⋅ ≤ λ2 ≤ λ1 < 0, Aek = λk ek ,

k = 1, 2, . . . , ek ∈ H01 (Ω) ∩ H 2 (Ω).

Then A can generate a C0 -semigroup {T(t)}t≥0 on L2 (Ω) with ‖T(t)‖ ≤ 1. Let H = U = L2 (Ω) and HN = span{ek : k = 1, . . . , N} ⊂ H. The operator PN denotes the orthogonal projector from H to HN . We define the approximations AN : H → HN of the operator A as AN = PN ΔPN .

3.2 Optimal control |

(IF )

127

Assume that F : H → H and C : U → H are both globally Lipschitz and C(0) = 0.

Let V = H 2 (Ω). It is clear to know that the embedding V 󳨅→ U is compact. Therefore, we define Wad := {f ∈ Lq ([0, T], U) : f (t) ∈ V for a. e. t ∈ [0, T]},

q>

1 . α

It is not difficult to prove that the conditions (A1 ), (A2 ), (A6 ) and (A7 ) are satisfied. Then the conclusion of Theorem 3.10 holds. We consider the following cost functional: T

󵄩

󵄩2

󵄩

󵄩2

J (x0 , u) = ∫ ∫󵄩󵄩󵄩y(t, ξ , u)󵄩󵄩󵄩 + 󵄩󵄩󵄩u(t, ξ )󵄩󵄩󵄩 dξdt. 0 Ω

Let Λ = ∫Ω ‖y(t, ξ , u)‖2 dξ . Then Λ is locally Lipschitz. The optimal control problem is to find: min J (x0 , u) over (y, u) ∈ L2q ([0, T] × Ω, H) × Wad subject to (3.48) with y(0, ξ ) = x0 (ξ ) ∈ H.

4 Fractional evolution equations and inclusions of order α ∈ (1, 2) 4.1 Fractional control systems 4.1.1 Introduction We observe that the dependent relation of the mean square displacement of particles with respect to the time in diffusion phenomena behaves like ⟨x 2 (t)⟩ ∼ const⋅t α , α > 0. It is worth noting that when α = 1, it is a normal diffusion, such as the heat equation. When α ≠ 1, it is an anomalous diffusion, where 0 < α < 1 and 1 < α < 2 stand for the subdiffusive phenomenon and superdiffusive phenomenon, respectively. Diffusion equations with fractional derivatives can provide a nice instrument to describe the memory and hereditary properties of anomalous diffusion processes. Meanwhile, the space variable does not scale like the Brownian motion in anomalous diffusion (Lévy processes) as well, a typical example is the fractional Laplacian (−Δ)s , 0 < s < 1, which are the generators of Lévy processes that contain jumps and long-distance interactions. It is readily seen that fractional elliptic operators emerge from nonlinear elasticity, probability and mathematical biology; some excellent papers point out the importance of fractional modeling in the contexts of free boundaries and minimal surfaces, thin obstacle problems and heat equations, etc.. We refer the reader to Caffarelli and Stinga [51], Caffarelli and Sire [50], Bonforte and Sire [44] and the references therein. Let Ω ⊂ ℝn be an open set with Dirichlet boundary conditions. We consider the following fractional wave equation: 𝜕tα x(t, z) = Δx(t, z) + f (t, z),

z ∈ Ω, t > 0,

(4.1)

where 𝜕tα is the Caputo fractional partial derivative of order α ∈ (1, 2), Δ stands for the Laplace operator. Such equation (4.1) is also called superdiffusive equation, when it applies to a viscoelastic model, such as a signaling problem closely related to seismology, there is an intermediate process between diffusion and wave propagation in an evolution process of the propagation of waves in a viscoelastic medium with a creep compliance of power-law type; see Mainardi [157]. Particularly, if we set α → 2 in (4.1), it becomes the standard wave equation that plays an important role in physical fields. Let us consider the usual Hilbert spaces L2 (Ω) and H01 (Ω), in equation (4.1) and let A = Δ on L2 (Ω). One can find from Arendt et al. [21] that operator A generates a bounded strongly continuous cosine family on domain D(A) = {x ∈ H01 (Ω), Ax ∈ L2 (Ω)}. https://doi.org/10.1515/9783110769272-004

130 | 4 Fractional evolution equations and inclusions of order α ∈ (1, 2) According to the above arguments, the fractional wave equation can be rewritten as an abstract fractional evolution equation C α 0Dt x(t)

= Ax(t) + f (t),

t > 0.

From the view point of physics, we can consider some more special problems on fractional evolution equations, which are abstracted from fractional wave equations. On the other hand, the existence of mild solutions for fractional differential and integrodifferential equations of order α ∈ (1, 2) has attracted much attention in recent years. Li et al. [140] considered two fractional evolution problems with the Riemann– Liouville derivative by using the concept of resolvent family. Shu [186] studied the existence and uniqueness of mild solutions for nonlocal fractional differential equations based on the concept of sectorial operator. Li [142] studied the regularity of mild solutions for fractional abstract Cauchy problems by analytic solution operators. Moreover, Kian and Yamamoto [132] studied the existence and Strichartz estimates of solutions for semilinear fractional wave equations by the method of eigenvalue expansion in a bounded domain. Li et al. [143] discussed the existence and uniqueness of fractional abstract Cauchy problems with order α ∈ (1, 2). In addition, there are some interesting and important controllability results on the fractional differential systems introduced in [9] with order α ∈ (0, 1) and in [139] with order α ∈ (1, 2). In this section, we study the controllability of the following fractional evolution systems: C α D x(t) = Ax(t) + f (t, x(t)) + Bu(t), {0 t x(0) = x0 , x′ (0) = x1 ,

t ∈ J = [0, b],

(4.2)

where C0Dαt is the Caputo fractional derivative of order 1 < α < 2, A is the infinitesimal generator of a strongly continuous cosine family {C(t)}t≥0 of uniformly bounded linear operators in Banach space X, f : J × X → X is a given appropriate function satisfying some assumptions, b is a finite positive number, the state x(⋅) takes values in X and the control function u(⋅) is given in a Banach space L2 (J, U) of admissible control functions, where U is also a Banach space. The operator B is a bounded linear operator from U into X and x0 , x1 are elements of space X. For the case of α = 2 and A = 𝜕x2 or A = Δ in (4.2), the study of controllability of wave equation seems to be complete. Extensive related papers can be found in [95, 135, 144, 235] and the references therein. The semigroup theory has been applied to establish the controllability results of wave equation in some literatures, while it is not valid for the case of α ∈ (1, 2). Therefore, the fractional case is not a direct generalization of the integer case. Additionally, compared to integer-order controllers, the fractional-order controllers of fractional-order control models are more flexible [175]. If A is the matrix in finite dimensional space, Balachandran et al. [31] established a set of

4.1 Fractional control systems | 131

sufficient conditions for the controllability of nonlinear fractional dynamical system, and from their results, one can see that controls of fractional dynamical systems are generalizations that include the normal integer-order systems as special cases. This scetion is organized as follows. In Subsection 4.1.2, we introduce some notations and useful concepts for cosine family. In Subsection 4.1.3, we give a new and specific definition on the mild solution for system (4.2) by the Laplace transform, cosine family and Mainardi’s Wright-type function. Furthermore, we establish a compactness result and other properties of solution operators. Subsection 4.1.4 is concerned with the study of the exact controllability for system (4.2). Finally, an example is provided to illustrate the theory of the obtained results. This section is based on [228]. 4.1.2 Preliminaries Throughout this section, let X be a Banach space with the norm | ⋅ |. We denote that Lb (X, Y) is a space of all bounded linear operators from X to Y equipped with the norm ‖⋅‖Lb (X,Y) and let Lb (X) : X → X, for short ‖⋅‖Lb . Let C(J, X) be the space of all continuous functions from J into X equipped with the sup-norm ‖x‖ = supt∈J |x(t)|. The domain and range of an operator A are defined by D (A) and R (A), respectively, if A : X → X is a linear operator, we denote the resolvent set of A by ρ(A) and the resolvent of A by R(λ, A) = (λI − A)−1 ∈ Lb (X). We briefly review the definition and some properties of a cosine family. For more details on strongly continuous cosine and sine families, we refer the reader to the books [21, 105] and the papers [93, 196, 198]. Definition 4.1 ([196]). A one-parameter family {C(t)}t∈ℝ of bounded linear operators mapping the Banach space X into itself is called a strongly continuous cosine family if and only if: (i) C(0) = I; (ii) C(s + t) + C(s − t) = 2C(s)C(t) for all s, t ∈ ℝ; (iii) C(t)x is continuous in t on ℝ for each fixed point x ∈ X. Let {S(t)}t∈ℝ be the sine family associated with the strongly continuous cosine family {C(t)}t∈ℝ which is defined by t

S(t)x = ∫ C(s)xds,

x ∈ X, t ∈ ℝ.

(4.3)

0

by

The infinitesimal generator of cosine family {C(t)}t∈ℝ of the operator A is defined

Ax =

d2 C(0)x, dt 2

for all x ∈ D (A),

132 | 4 Fractional evolution equations and inclusions of order α ∈ (1, 2) where D (A) = {x ∈ X : C(t)x ∈ C 2 (ℝ, X)}. Denote a set E = {x ∈ X : C(t)x ∈ C 1 (ℝ, X)}. It is known that the infinitesimal generator A is a closed, densely-defined operator in X. Lemma 4.1 ([196]). Let {C(t)}t∈ℝ be a strongly continuous cosine family in X. The following are true: (i) there exist constants M ≥ 1 and ω ≥ 0 such that ‖C(t)‖Lb ≤ Meω|t| for all t ∈ ℝ; (ii)

t

‖S(t2 ) − S(t1 )‖Lb ≤ M| ∫t 2 eω|s| ds| for all t1 , t2 ∈ ℝ; 1

(iii) if x ∈ E, then S(t)x ∈ D (A) and

d C(t)x dt

= AS(t)x.

Lemma 4.2 ([198]). Let {C(t)}t∈ℝ be a strongly continuous cosine family in X. Then limt→0 t −1 S(t)x = x for any x ∈ X. Lemma 4.3 ([196]). Let {C(t)}t∈ℝ be a strongly continuous cosine family in X satisfying ‖C(t)‖Lb ≤ Meω|t| , t ∈ ℝ, and let A be the infinitesimal generator of {C(t)}t∈ℝ . Then for Re λ > ω, λ2 ∈ ρ(A) and ∞

∞

λR(λ2 ; A)x = ∫ e−λt C(t)xdt,

R(λ2 ; A)x = ∫ e−λt S(t)xdt,

0

0

for x ∈ X.

4.1.3 Definition of mild solutions Suppose that A is the infinitesimal generator of a strongly continuous cosine family of uniformly bounded linear operators {C(t)}t≥0 in Banach space X, that is, there exists M ≥ 1 such that ‖C(t)‖Lb ≤ M, t ≥ 0. For the sake of convenience in writing, we always α−1

t set q = α2 for α ∈ (1, 2). We denote the kernel function gα (t) := Γ(α) , t > 0, α > 0. We first consider the following linear nonhomogeneous fractional evolution systems: C α D x(t) = Ax(t) + f (t), {0 t x(0) = x0 , x′ (0) = x1 .

t ∈ [0, ∞),

(4.4)

The system (4.4) is equivalent to the following integral equation: t

x(t) = x0 + x1 t +

1 ∫(t − s)α−1 [Ax(s) + f (s)]ds, Γ(α) 0

provided that the integral in (4.5) exists.

t ∈ [0, ∞),

(4.5)

4.1 Fractional control systems | 133

Theorem 4.1. If (4.5) holds, then t

x(t) = Cq (t)x0 + Kq (t)x1 + ∫(t − s)q−1 Pq (t − s)f (s)ds,

for t ∈ [0, ∞),

(4.6)

0

where ∞

Cq (t) = ∫ Mq (θ)C(t q θ)dθ,

t

Kq (t) = ∫ Cq (s)ds,

0 ∞

0

Pq (t) = ∫ qθMq (θ)S(t q θ)dθ. 0

Proof. Let Re λ > 0 and let L denote the Laplace transform by ∞

ν(λ) = L [x(t)](λ) = ∫ e−λs x(s)ds,

∞

μ(λ) = L [f (t)](λ) = ∫ e−λs f (s)ds. 0

0

Let λα ∈ ρ(A). Now applying the Laplace transform to (4.5), we have ν(λ) =

1 1 1 1 x + x + Aν(λ) + α μ(λ) λ 0 λ2 1 λα λ

= λα−1 (λα − A) x0 + λα−2 (λα − A) x1 + (λα − A) μ(λ). −1

−1

−1

(4.7)

By Lemma 4.3, it follows that for t ≥ 0, ∞

α

α 2

α

∞

α 2

ν(λ) = λ 2 −1 ∫ e−λ t C(t)x0 dt + λ−1 λ 2 −1 ∫ e−λ t C(t)x1 dt 0 ∞

+∫e

0 α

−λ 2 t

(4.8)

S(t)μ(λ)dt.

0

Let Φq (θ) =

q

θq+1

Mq (θ−q ),

θ ∈ (0, ∞).

Its Laplace transform is given by ∞

q

∫ e−λθ Φq (θ)dθ = e−λ , 0

1 for q ∈ ( , 1). 2

(4.9)

134 | 4 Fractional evolution equations and inclusions of order α ∈ (1, 2) First, using (4.9), we have ∞

∞

q

q

λq−1 ∫ e−λ t C(t)x0 dt = ∫ q(λt)q−1 e−(λt) C(t q )x0 dt 0

0 ∞

∞

= ∫−

1 d ( ∫ e−λtθ Φq (θ)dθ)C(t q )x0 dt λ dt 0

0 ∞∞

= ∫ ∫ θΦq (θ)e−λtθ C(t q )x0 dθdt 0 0 ∞ −λt

= ∫e

∞

[ ∫ Φq (θ)C(

0

0

(4.10)

tq )x dθ]dt θq 0

∞

= L [ ∫ Mq (θ)C(t q θ)x0 dθ](λ) 0

= L [Cq (t)x0 ](λ). Furthermore, since L [g1 (t)](λ) = λ−1 , by Laplace convolution theorem, we obtain ∞

q

λ−1 λq−1 ∫ e−λ t C(t)x1 dt = L [g1 (t)](λ) ⋅ L [Cq (t)x1 ](λ) 0

(4.11)

= L [(g1 ∗ Cq )(t)x1 ](λ). Similarly, we obtain that ∞

q

∞

q

∫ e−λ t S(t)μ(λ)dt = ∫ qt q−1 e−(λt) S(t q )μ(λ)dt 0

0 ∞∞

= ∫ ∫ qt q−1 Φq (θ)e−λtθ S(t q )μ(λ)dθdt 0 0 ∞ −λt

∞

= ∫e

[∫ q

0

0

t q−1 tq Φ (θ)S( )μ(λ)dθ]dt q θq θq

∞

= L [ ∫ θqt q−1 Mq (θ)S(t q θ)dθ](λ) ⋅ L [f (t)](λ) 0

t

= L [∫(t − s)q−1 Pq (t − s)f (s)ds](λ). 0

(4.12)

4.1 Fractional control systems | 135

Now, by using the uniqueness theorem of Laplace transform, combining (4.10), (4.11) and (4.12), we have the following identity: t

t

0

0

x(t) = Cq (t)x0 + ∫ Cq (s)x1 ds + ∫(t − s)q−1 Pq (t − s)f (s)ds, for t ≥ 0. The proof is completed. Proposition 4.1. If (4.5) holds. Then for t > 0, t q−1 Pq (t)x = 0D−(2q−1) Cq (t)x, t

for x ∈ X,

Cq′ (t)x = t q−1 APq (t)x,

and

for x ∈ E.

Proof. From the proof of Theorem 4.1, for any x ∈ X, we know that for Re λ > 0, λ2q−1 (λ2q − A) x = L [Cq (t)x](λ), −1

and (λ2q − A) x = L [t q−1 Pq (t)x](λ), −1

and moreover by L [g2q−1 (t)](λ) = λ1−2q , we have L [g2q−1 (t)](λ) ⋅ L [Cq (t)x](λ) = L [t

q−1

Pq (t)x](λ),

which can deduce that t q−1 Pq (t)x = 0D−(2q−1) Cq (t)x for x ∈ X by the uniqueness theorem t of Laplace transform. Furthermore, from Lemma 4.1(iii), we have for x ∈ E, Cq′ (t)x

∞

∞

0

0

d = ∫ Mq (θ) C(t q θ)xdθ = t q−1 ∫ qθMq (θ)AS(t q θ)xdθ = t q−1 APq (t)x. dt

The proof is completed. Example 4.1. Let X = ℝ, a > 0, and A : ℝ → ℝ is defined by Ax = −ax, then C(t) = cos(t√a) is a strongly continuous cosine function and the corresponding sine function is given by S(t) = sin(t√a)/√a. Hence, we get that Cq (t) = E2q,1 (−at 2q ), Kq (t) = tE2q,2 (−at 2q ) and Pq (t) = t q E2q,2q (−at 2q ) for t ≥ 0. Indeed, the Euler’s formula eiθ = cos(θ) + i sin(θ), here, i is the imaginary unit and θ ∈ ℝ, implies that cos(θ) = (eiθ + e−iθ )/2, from Proposition 1.11, we have ∞

Cq (t) = ∫ Mq (θ) cos(t q θ√a)dθ 0 ∞

∞

0

0

q q 1 = ( ∫ Mq (θ)eit θ√a dθ + ∫ Mq (θ)e−it θ√a dθ) 2

1 = (Eq,1 (it q √a) + Eq,1 (−it q √a)) 2 = E2q,1 (−at 2q ).

136 | 4 Fractional evolution equations and inclusions of order α ∈ (1, 2) According to (iv) of Proposition 1.10, we get that Kq (t) = tE2q,2 (−at 2q ). Similarly, using sin(θ) = 2i1 (eiθ − e−iθ ), we also have ∞

1 Pq (t) = ∫ qθMq (θ) sin(t q θ√a)dθ √a 0

∞

∞

0

0

q q 1 = ( ∫ qθMq (θ)eit θ√a dθ − ∫ qθMq (θ)e−it θ√a dθ) 2i√a

1 = (Eq,q (it q √a) − Eq,q (−it q √a)) = t q E2q,2q (−at 2q ). 2i√a Remark 4.1. If X = ℝ, a > 0, and A : ℝ → ℝ is defined by Ax = −ax, then we can reduce system (4.4) to C α 0Dt x(t)

= −ax(t) + f (t),

for t ≥ 0,

(4.13)

with initial value conditions x(0) = x0 and x ′ (0) = x1 . From [133, Chapter 3], it follows that (4.13) associated with initial value conditions has the unique solution α

t

α

x(t) = Eα,1 (−at )x0 + tEα,2 (−at )x1 + ∫(t − s)α−1 Eα,α (−a(t − s)α )f (s)ds, 0

which is in coincidence with (4.6) and Example 4.1. In view of Theorem 4.1, we give a new appropriate definition of the mild solutions for the system (4.2). Definition 4.2. For each u ∈ L2 (J, U), a mild solution of the system (4.2), we mean that the function x ∈ C(J, X) satisfies the following integral equation: t

x(t) = Cq (t)x0 + Kq (t)x1 + ∫(t − s)q−1 Pq (t − s)f (s, x(s))ds t

0

+ ∫(t − s)q−1 Pq (t − s)Bu(s)ds, 0

for each t ∈ J. Remark 4.2. For any t ≥ 0, since C(t) and S(t) are linear operators, it is easy to see that Cq (t), Kq (t) and Pq (t) are also linear operators. Lemma 4.4. For any fixed t ≥ 0 and for any x ∈ X, the following estimates are true: 󵄨󵄨 󵄨 󵄨󵄨Cq (t)x󵄨󵄨󵄨 ≤ M|x|,

󵄨󵄨 󵄨 󵄨󵄨Kq (t)x󵄨󵄨󵄨 ≤ M|x|t,

M 󵄨󵄨 󵄨 |x|t q . 󵄨󵄨Pq (t)x 󵄨󵄨󵄨 ≤ Γ(2q)

4.1 Fractional control systems | 137

Proof. For any x ∈ X and fixed t ≥ 0, according to (W3) of Proposition 1.11, we have ∞

󵄨 q 󵄨 󵄨 󵄨󵄨 󵄨󵄨Cq (t)x󵄨󵄨󵄨 ≤ ∫ Mq (θ)󵄨󵄨󵄨C(t θ)x 󵄨󵄨󵄨dθ ≤ M|x|, 0

which deduces immediately that |Kq (t)x| ≤ M|x|t. In addition, by (4.3), we have ∞

∞

tq θ

0

0

󵄨󵄨 󵄨 󵄨 q 󵄨 󵄨 󵄨 󵄨󵄨Pq (t)x󵄨󵄨󵄨 ≤ ∫ qθMq (θ)󵄨󵄨󵄨S(t θ)x 󵄨󵄨󵄨dθ ≤ ∫ qθMq (θ) ∫ 󵄨󵄨󵄨C(τ)x 󵄨󵄨󵄨dτdθ 0

∞

≤ Mq|x|t q ∫ θ2 Mq (θ)dθ = 0

M |x|t q . Γ(2q)

The proof is completed. Lemma 4.5. The operator Cq (t) is strongly continuous, i. e., for every x ∈ X and for any t ′ , t ′′ ≥ 0, we have 󵄨󵄨 ′′ ′ 󵄨 󵄨󵄨Cq (t )x − Cq (t )x󵄨󵄨󵄨 → 0,

as t ′′ → t ′ .

Proof. For every x ∈ X, since {C(t)}t∈ℝ is strongly continuous, i. e., for any ε > 0 and t, s ∈ ℝ, it follows that |C(t)x − C(s)x| → 0 as t − s → 0. Therefore, for any t ′ , t ′′ ≥ 0, we have ξ

󵄨󵄨 󵄨 󵄨 ′′ ′ 󵄨 ′′ q ′ q 󵄨󵄨Cq (t )x − Cq (t )x󵄨󵄨󵄨 ≤ ∫ Mq (θ)󵄨󵄨󵄨(C((t ) θ) − C((t ) θ))x󵄨󵄨󵄨dθ 0

∞

q q 󵄨 󵄨 + ∫ Mq (θ)󵄨󵄨󵄨(C((t ′′ ) θ) − C((t ′ ) θ))x󵄨󵄨󵄨dθ. ξ

From (W3) of Proposition 1.11, for any ε > 0 let us choose ξ sufficiently large such that ∞

∞

ξ

ξ

q q 󵄨 󵄨 ∫ Mq (θ)󵄨󵄨󵄨(C((t ′′ ) θ) − C((t ′ ) θ))x󵄨󵄨󵄨dθ ≤ 2M|x| ∫ Mq (θ)dθ < ε.

Additionally, we have ξ

q q 󵄨 󵄨 ∫ Mq (θ)󵄨󵄨󵄨(C((t ′′ ) θ) − C((t ′ ) θ))x󵄨󵄨󵄨dθ ≤ 2M|x|, 0

138 | 4 Fractional evolution equations and inclusions of order α ∈ (1, 2) which implies that ξ

q q 󵄨 󵄨 ∫ Mq (θ)󵄨󵄨󵄨(C((t ′′ ) θ) − C((t ′ ) θ))x󵄨󵄨󵄨dθ → 0,

as t ′′ → t ′ .

0

Consequently, by the arbitrariness of ε and let t ′′ → t ′ , one has |Cq (t ′′ )x −Cq (t ′ )x| → 0. The proof is completed. Lemma 4.6. The operators Kq (t) and Pq (t) are uniformly continuous, i. e., for any t ′ , t ′′ ≥ 0, we have 󵄩󵄩 ′′ ′ 󵄩 󵄩󵄩Kq (t ) − Kq (t )󵄩󵄩󵄩Lb → 0,

󵄩󵄩 ′′ ′ 󵄩 󵄩󵄩Pq (t ) − Pq (t )󵄩󵄩󵄩Lb → 0,

as t ′′ → t ′ .

Proof. For every x ∈ X, since {C(t)}t∈ℝ is the family of uniformly bounded linear operators, we have 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄨 ′′ ′ 󵄨 ′′ ′ 󵄩 󵄩 󵄩󵄩Kq (t ) − Kq (t )󵄩󵄩Lb = 󵄩󵄩󵄩∫ Cq (s)ds󵄩󵄩󵄩 ≤ M 󵄨󵄨󵄨t − t 󵄨󵄨󵄨 → 0, 󵄩󵄩 󵄩󵄩 󵄩t ′ 󵄩Lb ′′

as t ′′ → t ′ .

In addition, according to Lemma 4.1(ii), we get that ∞

󵄩󵄩 󵄩 󵄩 ′′ ′ 󵄩 ′′ q ′ q 󵄩󵄩Pq (t ) − Pq (t )󵄩󵄩󵄩Lb ≤ ∫ qθMq (θ)󵄩󵄩󵄩(S((t ) θ) − S((t ) θ))󵄩󵄩󵄩Lb dθ 0

≤

M 󵄨󵄨 ′′ q ′ q󵄨 󵄨󵄨(t ) − (t ) 󵄨󵄨󵄨 → 0, Γ(2q)

as t ′′ → t ′ .

The proof is completed. Remark 4.3. Noting that, from Lemma 4.4 and Lemma 4.6, it is easy to see that for any t, s ≥ 0, x ∈ X, lim t q−1 Pq (t)x = 0,

t→0

󵄩 󵄩 and 󵄩󵄩󵄩t q−1 Pq (t) − sq−1 Pq (s)󵄩󵄩󵄩L → 0, b

as t → s.

Theorem 4.2. Assume that S(t) is compact for every t > 0. Then Pq (t) is a compact operator for every t > 0. Proof. For each positive constant r, set Vr = {x ∈ X : |x| ≤ r}. Obviously, Vr is a bounded subset in X. We only need to verify that for any positive constant r and t > 0, the set ∞

U(t) = { ∫ qθMq (θ)S(t q θ)xdθ, x ∈ Vr } 0

is relatively compact in X.

4.1 Fractional control systems | 139

Let t > 0 be fixed. For any δ > 0 and 0 < ε ≤ t, define the subset in X by S(εq δ) Uε,δ (t) = { q ∫ qθMq (θ)S(t q θ − εq δ)xdθ, x ∈ Vr }. ε δ ∞

δ

Clearly, for each fixed t > 0, Uε,δ (t) is well-defined. In fact, by the uniformly bounded-

ness of cosine family and the uniformly convergence of Mainardi’s Wright-type function for θ ∈ (δ, ∞), we obtain that for any x ∈ Vr ,

󵄨󵄨 󵄨󵄨 q ∞ 󵄨󵄨 󵄨󵄨 S(ε δ) q q 󵄨󵄨 󵄨󵄨 qθM (θ)S(t θ − ε δ)xdθ ∫ q 󵄨󵄨 󵄨󵄨 εq δ 󵄨󵄨 󵄨󵄨 δ ∞

≤ M |x| ∫ qθMq (θ)(t q θ + εq δ)dθ 2

δ ∞

≤ 2M 2 |x|t q ∫ qθ2 Mq (θ)dθ ≤ δ

2M 2 |x|t q . Γ(2q)

Hence, the set Uε,δ (t) is relatively compact since S(εq δ) is compact for εq δ > 0. Furthermore, we have

∞ 󵄨󵄨 󵄨󵄨 q ∞ 󵄨󵄨 󵄨󵄨 S(ε δ) q q q 󵄨󵄨 󵄨󵄨 qθM (θ)S(t θ − ε δ)xdθ − qθM (θ)S(t θ)xdθ ∫ ∫ q q 󵄨󵄨 󵄨󵄨󵄨 εq δ 󵄨󵄨 󵄨 0 δ ∞ 󵄨󵄨 󵄨󵄨 q ∞ 󵄨󵄨 󵄨󵄨 S(ε δ) ≤ 󵄨󵄨󵄨 q ∫ qθMq (θ)S(t q θ − εq δ)xdθ − ∫ qθMq (θ)S(t q θ)xdθ󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ε δ 󵄨 󵄨 δ δ ∞ ∞ 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 󵄨 + 󵄨󵄨󵄨 ∫ qθMq (θ)S(t q θ)xdθ − ∫ qθMq (θ)S(t q θ)xdθ󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨δ 󵄨 0 δ 󵄨󵄨 󵄨󵄨 S(εq δ) 󵄨󵄨 󵄨 󵄨 󵄨󵄨 q q q ≤ ∫ qθMq (θ)󵄨󵄨 q S(t θ − ε δ)x − S(t θ)x󵄨󵄨dθ + ∫ qθMq (θ)󵄨󵄨󵄨S(t q θ)x󵄨󵄨󵄨dθ 󵄨󵄨 󵄨󵄨 ε δ ∞

0

δ

=: l1 + l2 . Since 󵄨󵄨 󵄨󵄨 S(εq δ) 󵄨 󵄨 qθMq (θ)󵄨󵄨󵄨 q S(t q θ − εq δ)x − S(t q θ)x 󵄨󵄨󵄨 ≤ 2M 2 t q qθ2 Mq (θ)|x|, 󵄨󵄨 󵄨󵄨 ε δ

140 | 4 Fractional evolution equations and inclusions of order α ∈ (1, 2) and ∫0 qθ2 Mq (θ)dθ = ∞

2q , Γ(1+2q)

we can see that

∞ 󵄨󵄨 S(εq δ) 󵄨󵄨󵄨 󵄨 ∫ qθMq (θ)󵄨󵄨󵄨 q S(t q θ − εq δ)x − S(t q θ)x 󵄨󵄨󵄨dθ 󵄨󵄨 󵄨󵄨 ε δ 0

is uniformly convergence. In addition, due to the strong continuity of the sine family, for θ ∈ (δ, ∞), using Lemma 4.2, we have 󵄨󵄨 S(εq δ) 󵄨󵄨 󵄨󵄨 󵄨 q q q 󵄨󵄨 q S(t θ − ε δ)x − S(t θ)x󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ε δ 󵄨󵄨 󵄨󵄨 S(εq δ) 󵄨 󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨 q S(t q θ − εq δ)x − S(t q θ − εq δ)x󵄨󵄨󵄨 + 󵄨󵄨󵄨S(t q θ − εq δ)x − S(t q θ)x󵄨󵄨󵄨 → 0, 󵄨󵄨 󵄨󵄨 ε δ as δ → 0. Therefore, we have 󵄨󵄨 󵄨󵄨 S(εq δ) 󵄨 󵄨 l1 ≤ ∫ qθMq (θ)󵄨󵄨󵄨 q S(t q θ − εq δ)x − S(t q θ)x󵄨󵄨󵄨dθ → 0, 󵄨󵄨 󵄨󵄨 ε δ ∞

as δ → 0.

0

δ

On the other hand, by ∫0 qθ2 Mq (θ)dθ → 0 as δ → 0, we have δ

l2 ≤ M|x|t q ∫ qθ2 Mq (θ)dθ → 0,

as δ → 0.

0

Hence, there are relatively compact sets arbitrarily close to the set U(t) for every t > 0. Thus, the set U(t) is relatively compact in X for every t > 0. The proof is completed.

4.1.4 Controllability results Definition 4.3. The system (4.2) is said to be controllable on the interval [0, b] if for every x0 , x1 , y ∈ X, there exists a control u ∈ L2 ([0, b], U) such that a mild solution x(⋅) of system (4.2) satisfies x(b) = y. We assume the following hypotheses: (H1) The function f : J × X → X is continuous and there exists a constant lf > 0 such that 󵄨󵄨 󵄨 󵄨󵄨f (t, x) − f (t, y)󵄨󵄨󵄨 ≤ lf |x − y|, and Mf = supt∈J |f (t, 0)|.

for each t ∈ J and for all x, y ∈ X,

4.1 Fractional control systems | 141

(H2) The linear operator B : U → X is bounded, and W : L2 (J, U) → X is the linear operator defined by b

Wu = ∫(b − s)q−1 Pq (b − s)Bu(s)ds, 0

and has a invertible operator W −1 , which takes values in L2 (J, U) \ ker W and there exist two positive constants M1 and M2 such that ‖B‖Lb (U,X) ≤ M1 ,

󵄩󵄩 −1 󵄩󵄩 󵄩󵄩W 󵄩󵄩Lb (X,L2 (J,U)\ker W) ≤ M2 .

For each r > 0, denote a bounded convex and closed set of C(J, X) as follows: Br = {x ∈ C(J, X) : ‖x‖ ≤ r}. Theorem 4.3. Assume that (H1)–(H2) are satisfied. Furthermore, we suppose that the following inequality holds: Mlf b2q

MM1 M2 b2q + 1] < 1. Γ(2q + 1) Γ(2q + 1) [

(4.14)

Then the evolution system (4.2) is controllable on J. Proof. Let ϕ(t) = Cq (t)x0 + Kq (t)x1 for t ∈ J. Using hypothesis (H2) for an arbitrary function x(⋅), we define the control ux (t) by b

ux (t) = W −1 [y − ϕ(b) − ∫(b − s)q−1 Pq (b − s)f (s, x(s))ds](t).

(4.15)

0

We will show that, the operator P : C(J, X) → C(J, X) defined by t

(P x)(t) = ϕ(t) + ∫(t − s)q−1 Pq (t − s)Bux (s)ds t

0

+ ∫(t − s)q−1 Pq (t − s)f (s, x(s))ds, 0

has a fixed point x(⋅) for the control ux . Then x is a mild solution of evolution system (4.2). Clearly, (P x)(b) = y, which means that the control ux steers the system from the initial state x0 and x1 to y in time b. Hence, we need to prove that the operator P has a fixed point.

142 | 4 Fractional evolution equations and inclusions of order α ∈ (1, 2) From the assumptions, we have b

󵄨 󵄨 󵄨 󵄨 󵄨󵄨 q−1 󵄨 󵄨󵄨Bux (t)󵄨󵄨󵄨 ≤ M1 M2 [|y| + 󵄨󵄨󵄨ϕ(b)󵄨󵄨󵄨 + ∫(b − s) 󵄨󵄨󵄨Pq (b − s)f (s, x(s))󵄨󵄨󵄨ds] 0

≤ M1 M2 [|y| + M|x0 | + Mb|x1 | b

+

M 󵄨 󵄨 ∫(b − s)2q−1 󵄨󵄨󵄨f (s, x(s)) − f (s, 0)󵄨󵄨󵄨ds Γ(2q) 0

b

+

M 󵄨 󵄨 ∫(b − s)2q−1 󵄨󵄨󵄨f (s, 0)󵄨󵄨󵄨ds] Γ(2q) 0

≤ M1 M2 [|y| + M|x0 | + Mb|x1 | + (lf r + Mf )

Mb2q ] =: G(r). Γ(2q + 1)

We first show that P maps Br into itself. From the definition of the operator P

and the hypotheses, for x ∈ Br , we obtain t

󵄨󵄨 󵄨 󵄨 󵄨 󵄨 q−1 󵄨 󵄨󵄨(P x)(t)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨ϕ(t)󵄨󵄨󵄨 + ∫(t − s) 󵄨󵄨󵄨Pq (t − s)Bux (s)󵄨󵄨󵄨ds t

0

󵄨 󵄨 + ∫(t − s)q−1 󵄨󵄨󵄨Pq (t − s)f (s, 0)󵄨󵄨󵄨ds 0

t

󵄨 󵄨 + ∫(t − s)q−1 󵄨󵄨󵄨Pq (t − s)(f (s, x(s)) − f (s, 0))󵄨󵄨󵄨ds 0

≤ M|x0 | + Mb|x1 | +

MMf b2q Mlf b2q r Mb2q G(r) + + . Γ(2q + 1) Γ(2q + 1) Γ(2q + 1)

Hence, from inequality (4.14) it follows that ‖P x‖ ≤ r for large r and then P (Br ) ⊆ Br . Now for any v, w ∈ Br , we have t

󵄨󵄨 󵄨 󵄨 q−1 󵄨 󵄨󵄨(P v)(t) − (P w)(t)󵄨󵄨󵄨 ≤ ∫(t − s) 󵄨󵄨󵄨Pq (t − s)(f (s, v(s)) − f (s, w(s)))󵄨󵄨󵄨ds 0

t

+ ∫(t − s) 0

b 󵄨󵄨 −1 q−1 󵄨󵄨󵄨Pq (t − s)BW [∫(b − τ) 󵄨󵄨 0

q−1 󵄨󵄨󵄨

4.1 Fractional control systems | 143

󵄨󵄨 󵄨󵄨 × Pq (b − τ)(f (τ, v(τ)) − f (τ, w(τ)))dτ](s)󵄨󵄨󵄨ds 󵄨󵄨 󵄨 Mlf t 2q M 2 M1 M2 lf t 2q b2q ≤ ‖v − w‖ + ‖v − w‖, Γ(2q + 1) (Γ(2q + 1))2 which implies from (4.14) that ‖P v − P w‖ < ‖v − w‖. Consequently, we deduce that P is a contraction on Br . Hence by the Banach fixed-point theorem, P has a unique

fixed-point x in C(J, X). We thus obtain that x(⋅) is a mild solution of the problem (4.2) and the proof is completed. We use the below condition instead of (H1) to avoid the Lipschitz continuity of f used in Theorem 4.3. (H3) The function f : J × X → X satisfies: (i) f (t, ⋅) : X → X is continuous for a. e. t ∈ J, and f (⋅, x) : J → X is strongly measurable for all x ∈ X; (ii) there exists a function kf (⋅) ∈ L1 (J, ℝ+ ) such that 󵄨󵄨 󵄨 󵄨󵄨f (t, x)󵄨󵄨󵄨 ≤ kf (t)φ(|x|),

for each t ∈ J and for all x ∈ X,

where φ : ℝ+ → ℝ+ is a nondecreasing continuous function satisfying lim inf r→∞

φ(r) = 0. r

Theorem 4.4. Assume that (H2)–(H3) are satisfied, further suppose that (H4) for every t ∈ J and for each r > 0, the set {Pq (t − s)f (s, x), s ∈ [0, t], x ∈ Br } is relatively compact in X. Then the evolution system (4.2) is controllable on J. Proof. We define the operator P on the space C(J, X) as in Theorem 4.3. Clearly, in order to prove the controllability of the problem (4.2), we must show that the operator P has a fixed point by Theorem 1.6. Noting that, from hypotheses (H2)–(H3), we have b

󵄨󵄨 󵄨 󵄨 󵄨 󵄨 q−1 󵄨 󵄨󵄨Bux (t)󵄨󵄨󵄨 ≤ M1 M2 (|y| + 󵄨󵄨󵄨ϕ(b)󵄨󵄨󵄨 + ∫(b − s) 󵄨󵄨󵄨Pq (b − s)f (s, x(s))󵄨󵄨󵄨ds) 0

b

M 󵄨 󵄨 ≤ M1 M2 (|y| + M|x0 | + Mb|x1 | + ∫(b − s)2q−1 kf (s)φ(󵄨󵄨󵄨x(s)󵄨󵄨󵄨)ds), Γ(2q) 0

144 | 4 Fractional evolution equations and inclusions of order α ∈ (1, 2) which implies that 1

‖Bux ‖L2 (J,X) ≤ M3 +

MM1 M2 b2q− 2 φ(r)‖kf ‖L1 (J,ℝ+ ) , Γ(2q)

(4.16)

where M3 = √bM1 M2 (|y| + M|x0 | + Mb|x1 |). Step I. We shall show that P x ∈ Br for every x ∈ Br . To the contrary, if this is not true, then for each r > 0, there exists a function x r ∈ Br , but P x r does not belong to Br , i. e., 󵄨 󵄨 r < 󵄨󵄨󵄨(P xr )(t)󵄨󵄨󵄨,

for some t ∈ J.

In view of Hölder inequality, we obtain that 󵄨󵄨 󵄨 󵄨󵄨(P x)(t)󵄨󵄨󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨ϕ(t)󵄨󵄨󵄨 +

t

t

M M 󵄨 󵄨 󵄨 󵄨 ∫(t − s)2q−1 󵄨󵄨󵄨Bux (s)󵄨󵄨󵄨ds + ∫(t − s)2q−1 󵄨󵄨󵄨f (s, x(s))󵄨󵄨󵄨ds Γ(2q) Γ(2q) 0

0

t

2q− 21

Mt 2q−1 Mt 󵄨 󵄨 ‖Bux ‖L2 (J,X) + ≤ M|x0 | + Mb|x1 | + ∫ kf (s)φ(󵄨󵄨󵄨x(s)󵄨󵄨󵄨)ds Γ(2q) √4q − 1Γ(2q) ≤ M|x0 | + Mb|x1 | +

2q− 21

2q− 21

2

0

MM3 b M1 M2 Mb Mb2q−1 + [( ) + ]φ(r)‖kf ‖L1 (J,ℝ+ ) . Γ(2q) Γ(2q) √4q − 1Γ(2q) √4q − 1

Therefore, it follows that 1

r < M|x0 | + Mb|x1 | +

1

2

MM3 b2q− 2 M1 M2 Mb2q− 2 Mb2q−1 + [( ) + ]φ(r)‖kf ‖L1 (J,ℝ+ ) . Γ(2q) Γ(2q) √4q − 1Γ(2q) √4q − 1

Dividing both sides of the above inequality by r and taking the limit as r → ∞, from (H3), we thus deduce the contraction. Consequently, for each positive number r, P (Br ) ⊆ Br . Step II. We shall show that P is completely continuous. First, we show P is continuous. Let {xn }∞ n=1 be a sequence of Br such that xn → x in Br as n → ∞. By (H3), due to the continuity of f , we have f (s, xn (s)) → f (s, x(s)),

as n → ∞.

and 󵄨 󵄨 (t − s)2q−1 󵄨󵄨󵄨f (s, xn (s)) − f (s, x(s))󵄨󵄨󵄨 ≤ 2φ(r)(t − s)2q−1 kf (s),

for s ∈ [0, t].

4.1 Fractional control systems | 145

Since s 󳨃→ 2φ(r)(t − s)2q−1 kf (s) is integrable for s ∈ [0, t] and t ∈ J, it follows the Lebesgue dominated convergence theorem that 󵄨 󵄨󵄨 󵄨󵄨(P xn )(t) − (P x)(t)󵄨󵄨󵄨 b t 󵄨󵄨 −1 q−1 󵄨󵄨󵄨 ≤ ∫(t − s) 󵄨󵄨Pq (t − s)BW (∫(b − τ)q−1 Pq (b − τ) 󵄨󵄨 󵄨 0 0 󵄨󵄨 󵄨󵄨 × (f (τ, xn (τ)) − f (τ, x(τ)))dτ)(s)󵄨󵄨󵄨ds 󵄨󵄨 󵄨 t

󵄨 󵄨 + ∫(t − s)q−1 󵄨󵄨󵄨Pq (t − s)(f (s, xn (s)) − f (s, x(s)))󵄨󵄨󵄨ds 0

b

≤

M 2 M1 M2 t 2q 󵄨 󵄨 ∫(b − τ)2q−1 󵄨󵄨󵄨f (τ, xn (τ)) − f (τ, x(τ))󵄨󵄨󵄨dτ Γ(2q + 1)Γ(2q) 0

t

+

M 󵄨 󵄨 ∫(t − s)2q−1 󵄨󵄨󵄨f (s, xn (s)) − f (s, x(s))󵄨󵄨󵄨ds → 0, Γ(2q)

as n → ∞.

0

Thus, ‖P xn − P x‖ → 0 as n → ∞. This means that P is continuous. Next, we will verify that P maps bounded sets into equicontinuous sets. Let 0 ≤ t1 < t2 ≤ b, for any x ∈ Br , we get 󵄨󵄨 󵄨 󵄨󵄨(P x)(t2 ) − (P x)(t1 )󵄨󵄨󵄨

t2

󵄨 󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨ϕ(t2 ) − ϕ(t1 )󵄨󵄨󵄨 + ∫󵄨󵄨󵄨(t2 − s)q−1 Pq (t2 − s)[f (s, x(s)) + Bux (s)]󵄨󵄨󵄨ds t1

t1

󵄨 󵄨 + ∫󵄨󵄨󵄨((t2 − s)q−1 Pq (t2 − s) − (t1 − s)q−1 Pq (t2 − s))[f (s, x(s)) + Bux (s)]󵄨󵄨󵄨ds 0

󵄨 󵄨 = 󵄨󵄨󵄨ϕ(t2 ) − ϕ(t1 )󵄨󵄨󵄨 + I1 + I2 . According to Lemma 4.4 and (4.16), we have t2

t2

M M 󵄨 󵄨 󵄨 󵄨 I1 ≤ ∫(t2 − s)2q−1 kf (s)φ(󵄨󵄨󵄨x(s)󵄨󵄨󵄨)ds + ∫(t2 − s)2q−1 󵄨󵄨󵄨Bux (s)󵄨󵄨󵄨ds Γ(2q) Γ(2q) t1

t1

t2

≤

1 M M φ(r)(t2 − t1 )2q−1 ∫ kf (s)ds + (t2 − t1 )2q− 2 ‖Bux ‖L2 (J,X) Γ(2q) √4q − 1Γ(2q)

→ 0,

t1

as t2 → t1 .

146 | 4 Fractional evolution equations and inclusions of order α ∈ (1, 2) Let Tq (t) = t q−1 Pq (t) for t ∈ J, then we know from Remark 4.3 that Tq (t) is a uniformly continuous operator. For I2 , taking ε > 0 small enough, we have t1 −ε

󵄨 󵄨 I2 ≤ ∫ 󵄨󵄨󵄨(Tq (t2 − s) − Tq (t1 − s))[f (s, x(s)) + Bux (s)]󵄨󵄨󵄨ds 0

t1

󵄨 󵄨 + ∫ 󵄨󵄨󵄨(Tq (t2 − s) − Tq (t1 − s))[f (s, x(s)) + Bux (s)]󵄨󵄨󵄨ds t1 −ε

t1

󵄨 󵄨 󵄩 󵄩 ≤ ∫ kf (s)φ(󵄨󵄨󵄨x(s)󵄨󵄨󵄨)ds sup 󵄩󵄩󵄩Tq (t2 − s) − Tq (t1 − s)󵄩󵄩󵄩L b 0

s∈[0,t1 −ε]

t1

󵄨 󵄨 󵄩 󵄩 + ∫󵄨󵄨󵄨Bux (s)󵄨󵄨󵄨ds sup 󵄩󵄩󵄩Tq (t2 − s) − Tq (t1 − s)󵄩󵄩󵄩L b 0

s∈[0,t1 −ε]

t1

2M 󵄨 󵄨 + ∫ kf (s)φ(󵄨󵄨󵄨x(s)󵄨󵄨󵄨)ds(t2 − t1 + ε)2q−1 Γ(2q) t1 −ε t1

2M 󵄨 󵄨 + ∫ 󵄨󵄨󵄨Bux (s)󵄨󵄨󵄨ds(t2 − t1 + ε)2q−1 Γ(2q) t1 −ε

≤ φ(r)‖kf ‖L1 (J,ℝ+ )

󵄩 󵄩 sup 󵄩󵄩󵄩Tq (t2 − s) − Tq (t1 − s)󵄩󵄩󵄩L b

s∈[0,t1 −ε]

+ √t1 ‖Bux ‖L2 (J,X)

󵄩 󵄩 sup 󵄩󵄩󵄩Tq (t2 − s) − Tq (t1 − s)󵄩󵄩󵄩L b

s∈[0,t1 −ε]

t1

2M + φ(r) ∫ kf (s)ds(t2 − t1 + ε)2q−1 Γ(2q) t1 −ε

+

2M ‖Bux ‖L2 (J,X) √ε(t2 − t1 + ε)2q−1 → 0, Γ(2q)

as t2 → t1 , ε → 0.

Thus, I2 tends to zero independently of x ∈ Br as t2 → t1 , ε → 0. As a result, from the strong continuity of operators Cq (t) and Kq (t) obtained in Lemmas 4.5 and 4.6, we obtain immediately that |(P x)(t2 ) − (P x)(t1 )| tends to zero as t2 → t1 independently of x ∈ Br . Thus, P (Br ) is equicontinuous. By the Arzelà–Ascoli theorem, it remains to verify that P maps Br into a relatively compact set in X, that is, we shall show that the set {(P x)(t), x ∈ Br } is relatively compact for t ∈ J. Indeed, it follows from Remark 4.3 and the hypotheses (H2)–(H4) that the set Θ = {(t − s)q−1 Pq (t − s)f (s, x(s)), x ∈ Br , s ∈ [0, t]}

4.1 Fractional control systems | 147

is relatively compact for each t ∈ J. Furthermore, in view of (4.16), for x ∈ Br , by using the mean value theorem for the Bochner integral, we obtain t

∫(t − s)q−1 Pq (t − s)[Bux (s) + f (s, x(s))]ds ∈ t conv Θ,

for all t ∈ J.

0

Therefore, the set {(P x)(t), x ∈ Br } is relatively compact in X for every t ∈ J. In addition, in view of Step II and {(P x)(t), x ∈ Br } is relatively compact in X, we get that P is a completely continuous operator by the Arzelà–Ascoli theorem. Hence, by Step I and using Theorem 1.6, it follows that P has a fixed point x on Br , which is a mild solution of the system (4.2). Thus, the evolution system (4.2) is controllable on [0, b]. The proof is completed. Remark 4.4. Let us mention that (i) if the Banach space X is finite dimensional, the hypothesis (H4) can be reduced to that of the compactness of sine function S(t) for each t > 0; moreover, it is a generalization of paper [31] that we do not apply the MittagLeffler matrix function. (ii) If function f is completely continuous, then hypothesis (H4) holds obviously. (iii) In Theorem 4.4, if replacing hypothesis (H4) by the following hypothesis: (H5) there exists a suitable constant L > 0 such that β(f (t, D)) ≤ Lβ(D),

for any bounded set D ⊂ X and a. e. t ∈ J,

where β stands for the Hausdorff measure of noncompactness, and using the same method as in [226, Theorem 4.9], we can still establish the controllability of evolution system (4.2).

4.1.5 An example Example 4.2. Let Ω ⊂ ℝN be a bounded domain with sufficiently smooth boundary and X = U = L2 (Ω). As an application of our main results, we consider the following fractional partial differential systems: 3

𝜕 2 u(t, z) = Δu(t, z) + γe sin(u(t, z)) + Bu(t, z), { { { t u(t, z) = 0, { { { ′ {u(0, z) = u0 (z), u (0, z) = u1 (z), 3

−t

t ∈ [0, 1], z ∈ Ω,

t ∈ [0, 1], z ∈ 𝜕Ω, z ∈ Ω,

where 𝜕t2 is the Caputo fractional partial derivative, γ is a positive number.

(4.17)

148 | 4 Fractional evolution equations and inclusions of order α ∈ (1, 2) Let J = [0, 1] and let A be the Laplace operator with Dirichlet boundary condition given by A = Δ and 1

2

D (A) = {f ∈ H0 (Ω), Af ∈ L (Ω)}.

Obviously, we have D (A) = H01 (Ω) ∩ H 2 (Ω). It is known that the operator A generates a uniformly bounded strongly continuous cosine family {C(t), t ≥ 0}; see [21, Section 7.2]. In fact, let λn = n2 π 2 and ϕn (z) = √2/π sin(nπz) for every n ∈ ℕ. It is clear that {−λn , ϕn }∞ n=1 is the eigensystem of the operator A, then 0 < λ1 ≤ λ2 ≤ ⋅ ⋅ ⋅ , λn → ∞ as n → ∞, and {ϕn }∞ n=1 form an orthonormal basis of X. Then ∞

Ax = − ∑ λn (x, ϕn )ϕn , n=1

x ∈ D (A),

where (⋅, ⋅) is the inner product in X. It follows that the cosine function given by ∞

C(t)x = ∑ cos(√λn t)(x, ϕn )ϕn , n=1

x ∈ X,

and the sine function is associated with cosine function given by 1 sin(√λn t)(x, ϕn )ϕn , √λ n n=1 ∞

S(t)x = ∑

x ∈ X.

It is easy to check that S(t) is compact for t ≥ 0 and ‖C(t)‖Lb ≤ 1 for all t ≥ 0. Since α = 32 , we know that q = 43 , and then ‖Cq (t)‖Lb ≤ 1 for all t ≥ 0. The control operator B : U → X is defined by ∞

Bu = ∑ cλn (u, ϕn )ϕn , n=1

c > 0,

where un ,

u={

0,

n = 1, 2, . . . , N,

n = N + 1, N + 2, . . . ,

for N ∈ ℕ and W : L2 (J, U) → X as follows: 1

1

Wu = ∫(1 − s)− 4 P 3 (1 − s)Bu(s)ds. 0

4

4.1 Fractional control systems | 149

1

2 2 Since |u| = (∑∞ n=1 (u, ϕn ) ) for u ∈ U, we obtain ∞

|Bu| = ( ∑

n=1

c2 λn2 (u, ϕn )2 )

1 2

≤ cλN |u|,

which implies that there exists a positive constant M1 such that ‖B‖Lb (U,X) ≤ M1 . Let u(s, z) = x(z) ∈ U and let x stand for xn if n = 1, 2, . . . , N or 0 if n = N + 1, . . . . Therefore, we have 1

− 41

Wu = ∫(1 − s) 0

∞

3 3 ∫ θM 3 (θ)S((1 − s) 4 θ)Bxdθds 4 4

0

1

1

= c ∫(1 − s)− 4

N 3 3 ∫ θM 3 (θ) ∑ √λn sin(√λn (1 − s) 4 θ)(x, ϕn )ϕn dθds 4 4 n=1 ∞

0

0 N ∞

= c ∑ ∫ M 3 (θ)(1 − cos(√λn θ))dθ(x, ϕn )ϕn n=1 0 ∞

4

= c ∑ (1 − E 3 ,1 (−λn ))(x, ϕn )ϕn . n=1

2

1 ), then for every n ∈ ℕ, we have −1 < E 3 ,1 (−λn ) ≤ According to [117], let d = E 3 ,1 (− 10 2 2 d < 1, which implies

0 < 1 − d ≤ 1 − E 3 ,1 (−λn ) < 2. 2

Hence, we know that W is surjective. Thus, for any x = ∑∞ n=1 (x, ϕn )ϕn ∈ X, we define an inverse W −1 : X → L2 (J, U) \ ker W by (W −1 x)(t, y) =

1 ∞ (x, ϕn )ϕn . ∑ c n=1 1 − E 3 ,1 (−λn ) 2

Thus, for x ∈ X, it follows that 1 2

∞ (x, ϕn )2 1 󵄨󵄨 −1 󵄨 1 |x|. ) ≤ 󵄨󵄨(W x)(t, ⋅)󵄨󵄨󵄨 = ( ∑ c n=1 (1 − E 3 ,1 (−λn ))2 c(1 − d) 2

Note that W −1 x is independent of t ∈ J. Consequently, we have 1 󵄩󵄩 −1 󵄩󵄩 . 󵄩󵄩W 󵄩󵄩Lb (X,L2 (J,U)\ker W) ≤ c(1 − d) Hence, the condition (H2) holds.

150 | 4 Fractional evolution equations and inclusions of order α ∈ (1, 2) Let f (t, x) = γe−t sin(x(t)). Then 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 −t 󵄨 󵄨󵄨f (t, x) − f (t, y)󵄨󵄨󵄨 ≤ γe 󵄨󵄨󵄨sin(x(t)) − sin(y(t))󵄨󵄨󵄨 ≤ γ 󵄨󵄨󵄨x(t) − y(t)󵄨󵄨󵄨, for any x, y ∈ X, t ∈ [0, 1]. Thus the system (4.17) can be rewritten as the system (4.2) in X. Hence, the condition (H1) is satisfied. Thus, all conditions of Theorem 4.3 are satisfied provided with γ
0, {0 t x(0) + g(x) = x0 , x′ (0) = x1 ,

(4.18)

where C0Dαt f (t) is Caputo fractional derivative of order 1 < α < 2, A is the infinitesimal generator of a strongly continuous cosine family {C(t)}t≥0 of uniformly bounded linear operators in a Banach space X, F : [0, a] × X → X is a multivalued map, g is a given appropriate function and x0 , x1 are elements of space X. Here, we emphasize that the present work is also motivated by an inclusion of the following partial differential model: 𝜕α u(t, z) ∈ 𝜕z2 u(t, z) + F(t, z, u(t, z)), z ∈ [0, π], t ∈ [0, a], { { { t u(t, 0) = u(t, π) = 0, t ∈ [0, a], { { { ′ {u(0, z) + g(u) = u0 (z), u (0, z) = u1 (z), z ∈ [0, π], where 𝜕tα is the Caputo fractional partial derivative. This model includes a class of fractional wave equations that have a memory effect and are not observed in integer-order differential equations, and it indicates the coexistence of finite wave speed and absence of a wavefront; see, e. g., [36]. It is interesting to note that for the case of α = 2, the above fractional partial differential inclusion reduces to a second-order differential inclusion involving one dimensional wave equations with nonlocal initial-boundary conditions. For the case of α = 1 or α ∈ (0, 1) with initial value u1 (z) vanished, the model contains the classical diffusion equations or fractional diffusion equations. In addition, these types of equations can be handled by the method of semigroup theory (see, e. g., [234]) but not cosine families. The rest of this section is organized as follows. In Subsection 4.2.2, we recall some preliminary concepts related to our study. In Subsection 4.2.3, we establish an existence result for mild solutions of problem (4.18) and discuss the compactness of the set of mild solutions. In Subsection 4.2.4, we show the utility of the obtained work by applying it to a control problem. This section is based on [118].

4.2.2 Preliminaries Let X be a Banach space with the norm ‖ ⋅ ‖. Denote by B(X) the space of all bounded linear operators from X to X equipped with the norm ‖ ⋅ ‖B(X) . Let C(J, X) denote the space of all continuous functions from J into X equipped with the usual sup-norm ‖x‖C = supt∈J ‖x(t)‖, where J = [0, a], a > 0. A measurable function f : J → X is Bochner integrable if ‖f ‖ is Lebesgue integrable. Let Lp (J, X) (p ≥ 1) be the Banach

152 | 4 Fractional evolution equations and inclusions of order α ∈ (1, 2) space of measurable functions (defined in the sense of Bochner integral) endowed 1 with the norm ‖f ‖p = (∫J ‖f (t)‖p dt) p . A multivalued map G is called upper semicontinuous (u. s. c.) on X if, for each x∗ ∈ X, the set G(x∗ ) is a nonempty subset of X, and for every open set B ⊆ X such that G(x∗ ) ⊂ B, there exists a neighborhood V of x∗ with the property that G(V(x∗ )) ⊂ B. G is convex valued if G(x) is convex for all x ∈ X. G is closed if its graph ΓG = {(x, y) ∈ X × X : y ∈ G(x)} is a closed subset of the space X × X. The map G is bounded if G(B) is bounded in X for every bounded set B ⊆ X. We say that G is quasicompact if G(B) is relatively compact for every compact subset B of X. Furthermore, if G is quasicompact with nonempty values, then G is u. s. c. if and only if G has a closed graph. If there exists an element x ∈ X such that x ∈ G(x), then G has a fixed point. Let B be a subset of X. Then we define P(X) = {B ⊆ X : B is nonempty},

Pcl (X) = {B ∈ P(X) : B is closed},

Pcp (X) = {B ∈ P(X) : B is compact},

Pcv (X) = {B ∈ P(X) : B is convex},

Pbd (X) = {B ∈ P(X) : B is bounded}, Pcl,cv (X) = Pcl (X) ∩ Pcv (X).

In addition, let co(B) be the convex hull of a subset B ∈ X and co(B) be the closed convex hull in X. A multivalued map G : J → Pcl (X) is said to be measurable if, for each x ∈ X, the function Z : J → ℝ defined by Z(t) = d(x, G(t)) = inf{‖x − z‖ : z ∈ G(t)} is Lebesgue measurable. Let G : J → P(X). A single-valued map f : J → X is called a selection of G if f (t) ∈ G(t) for every t ∈ J. Definition 4.4. A multivalued map F : J × X → P(X) is called L1 -Carathéodory if: (i) the map t 󳨃→ F(t, x) is measurable for each x ∈ X; (ii) the map u 󳨃→ F(t, x) is upper semicontinuous on X for almost all t ∈ J; (iii) for each positive real number r, there exists hr ∈ L1 (J, ℝ+ ) such that 󵄩󵄩 󵄩 󵄩󵄩F(t, x)󵄩󵄩󵄩P(X) = sup{‖v‖ : v(t) ∈ F(t, x)} ≤ hr (t), by

for ‖x‖ ≤ r, for a. e. t ∈ J.

For every Ω ⊂ P(X), the Hausdorff measure of noncompactness (MNC) is defined β(Ω) = inf{ε > 0 : Ω has a finite ε-net},

and the MNC ν is defined as: for a bounded set D ⊂ C(J, X), we define ν(D) = max (sup β(D(t)), modC (D)), D∈Θ(D)

t∈J

4.2 Nonlocal evolution inclusions | 153

where Θ(D) is the collection of all denumerable subsets of D and modC (D) is the modulus of equicontinuity of the set of functions D that has the following form: 󵄩 󵄩 modC (D) = lim sup max 󵄩󵄩󵄩x(t2 ) − x(t1 )󵄩󵄩󵄩. δ→0 |t −t | 0 such that M‖x0 ‖ + MNg1 r + MNg2 + Ma‖x1 ‖ +

Ma2q−1 Ma2q−1 ‖kf ‖1 + ‖k ‖ r ≤ r. Γ(2q) Γ(2q) f 1

(4.19)

Furthermore, we introduce W0 = {x ∈ C(J, X) : ‖x‖C ≤ r} and observe that W0 is a nonempty bounded, closed and convex subset of C(J, X). Let x ∈ W0 and y ∈ P (x), then there exists f ∈ SF,x such that for each t ∈ J and for any x ∈ W0 , we have t

y(t) = Cq (t)(x0 − g(x)) + Kq (t)x1 + ∫(t − s)q−1 Pq (t − s)f (s)ds. 0

By (H3) and (H4), we have 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩y(t)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩Cq (t)󵄩󵄩󵄩B(X) 󵄩󵄩󵄩x0 − g(x)󵄩󵄩󵄩 + 󵄩󵄩󵄩Kq (t)󵄩󵄩󵄩B(X) ‖x1 ‖ t

󵄩 󵄩 + ∫(t − s)q−1 󵄩󵄩󵄩Pq (t − s)f (s)󵄩󵄩󵄩ds 0

t

M 󵄩 󵄩 󵄩 󵄩 ≤ M‖x0 ‖ + M 󵄩󵄩󵄩g(x)󵄩󵄩󵄩 + Mt‖x1 ‖ + ∫(t − s)2q−1 kf (s)(1 + 󵄩󵄩󵄩x(s)󵄩󵄩󵄩)ds Γ(2q) 0

≤ M‖x0 ‖ + MNg1 ‖x‖C + MNg2 + Mt‖x1 ‖ +

Mt 2q−1 Mt 2q−1 ‖kf ‖1 + ‖k ‖ ‖x‖ Γ(2q) Γ(2q) f 1 C

≤ M‖x0 ‖ + MNg1 r + MNg2 + Ma‖x1 ‖ + ≤ r.

Ma2q−1 Ma2q−1 ‖kf ‖1 + ‖k ‖ r Γ(2q) Γ(2q) f 1

Therefore, ‖y‖C ≤ r, which implies that P (W0 ) ⊆ W0 . Define W1 = co(P (W0 )). Clearly, W1 ⊂ C(J, X) is a nonempty, bounded, closed and convex set. Repeating the arguments employed in the previous step, for any x ∈ W1 , y ∈ P (x), it follows that there exists f ∈ SF,x such that for each t ∈ J and for any x ∈ W1 , t

y(t) = Cq (t)(x0 − g(x)) + Kq (t)x1 + ∫(t − s)q−1 Pq (t − s)f (s)ds. 0

4.2 Nonlocal evolution inclusions | 157

By (H3) and (H4) together with Lemma 4.9(ii), we have t

󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 q−1 󵄩 󵄩󵄩y(t)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩Cq (t)(x0 − g(x))󵄩󵄩󵄩 + 󵄩󵄩󵄩Kq (t)x1 󵄩󵄩󵄩 + ∫(t − s) 󵄩󵄩󵄩Pq (t − s)f (s)󵄩󵄩󵄩ds 0

≤ M‖x0 ‖ + MNg1 r + MNg2 + Ma‖x1 ‖ + ≤ r,

Ma2q−1 Ma2q−1 ‖kf ‖1 + ‖k ‖ r Γ(2q) Γ(2q) f 1

which implies that P (W1 ) ⊆ W1 and W1 ⊂ W0 . Next, for every n ≥ 1, we define Wn+1 = co(P (Wn )). From the above proof, it is easy to see that Wn is a nonempty, bounded, closed and convex subset of C(J, X). Furthermore, W2 = co P (W1 ) ⊂ W1 . By induction, we know that the sequence {Wn }∞ n=1 is a decreasing sequence of nonempty, bounded, closed and convex subsets of C(J, X). Furthermore, we set W = ⋂∞ n=1 Wn and note that W is bounded, closed and convex since Wn is bounded, closed and convex for every n ≥ 1. Now we establish that P (W) ⊆ W. Indeed, P (W) ⊆ P (Wn ) ⊆ co(P (Wn )) = Wn+1 for every n ≥ 1. Therefore, P (W) ⊆ ⋂∞ n=2 Wn . On the other hand, Wn ⊂ W1 for every n ≥ 1. Hence, ∞

∞

n=2

n=1

P (W) ⊆ ⋂ Wn = ⋂ Wn = W.

Step 2. The multivalued map P is ν-condensing. Let 𝔹 ⊆ W be such that ν(𝔹) ≤ ν(P (𝔹)).

(4.20)

We will prove below that 𝔹 is a relatively compact set, that is, ν(𝔹) = 0. Let σ(𝔹) = supt∈J β(𝔹(t)) and let ν(P (𝔹)) be achieved on a sequence {yn }∞ n=1 ⊂ P (𝔹), that is, ∞ ∞ ν({yn }∞ n=1 ) = max(σ({yn }n=1 ), modC ({yn }n=1 )).

Then t

yn (t) = Cq (t)(x0 − g(xn )) + Kq (t)x1 + ∫(t − s)q−1 Pq (t − s)fn (s)ds,

t ∈ J,

0

where {xn }∞ n=1 ⊂ 𝔹 and fn ∈ SF,xn for every n ≥ 1. Since g is compact, the set {g(xn ) : n ≥ 1} is relatively compact and combined with the fact that Cq (t), Kq (t) are strongly continuous for t ≥ 0. Hence, for every t ∈ J, we

158 | 4 Fractional evolution equations and inclusions of order α ∈ (1, 2) have ν({Cq (t)(x0 − g(xn )) + Kq (t)x1 , n ≥ 1}) = 0. Therefore, it is enough to estimate that t

ν({∫(t − s)q−1 Pq (t − s)fn (s)ds, n ≥ 1}) = 0. 0

Claim I. We shall show that σ({yn }∞ n=1 ) = 0. For any t ∈ J, using (H4), Proposition 1.20 and Lemma 4.9(ii), we have t

q−1 β({Pq (t − s)fn (s)}n=1 )ds σ({yn }∞ n=1 ) = sup β({yn (t)}n=1 ) ≤ 2 sup ∫(t − s) ∞

∞

t∈J

t∈J

t

≤ sup t∈J

0

2M ∞ ∫(t − s)2q−1 η(s)β({xn (s)}n=1 )ds Γ(2q) 0

t

2M ≤ sup ∫(t − s)2q−1 η(s)dsσ({xn }∞ n=1 ) Γ(2q) t∈J 2q−1

≤

0 a

1 2Ma ∞ ∫ η(s)dsσ({xn }∞ n=1 ) < σ({xn }n=1 ). Γ(2q) 4 0

∞ On the other hand, (4.20) implies that σ({yn }∞ n=1 ) ≥ σ({xn }n=1 ). In consequence, we have that σ({yn }∞ n=1 ) = 0. Claim II. We shall show that modC ({yn }∞ n=1 ) = 0, i. e., the set 𝔹 is equicontinuous. Let ⋅

ỹn (⋅) = ∫(⋅ − s)q−1 Pq (⋅ − s)fn (s)ds. 0

Therefore, it remains to verify that modC ({ỹn }∞ n=1 ) = 0. Then, for any t1 , t2 ∈ J with t1 < t2 , we have t2

󵄩󵄩̃ 󵄩 󵄩 󵄩 q−1 󵄩󵄩yn (t2 ) − ỹn (t1 )󵄩󵄩󵄩 ≤ ∫󵄩󵄩󵄩(t2 − s) Pq (t2 − s)fn (s)󵄩󵄩󵄩ds t1

t1

󵄩 󵄩 + ∫󵄩󵄩󵄩((t2 − s)q−1 Pq (t2 − s) − (t1 − s)q−1 Pq (t1 − s))fn (s)󵄩󵄩󵄩ds 0

= I1 + I2 .

4.2 Nonlocal evolution inclusions | 159

According to Lemma 4.9(ii), we get t2

I1 ≤

M 󵄩 󵄩 ∫(t2 − s)2q−1 kf (s)(1 + 󵄩󵄩󵄩xn (s)󵄩󵄩󵄩)ds Γ(2q) t1

t2

M ≤ (t − t )2q−1 ∫ kf (s)ds(1 + ‖xn ‖C ) → 0, Γ(2q) 2 1

as t2 → t1 .

t1

Let Tq (t) = t q−1 Pq (t) for t ∈ J. Then we know from Lemma 4.9(iii) that Tq (t) is a strongly continuous operator. For I2 , taking ε > 0 to be small enough, we obtain t1 −ε

󵄩 󵄩 I2 ≤ ∫ 󵄩󵄩󵄩(Tq (t2 − s) − Tq (t1 − s))fn (s)󵄩󵄩󵄩ds 0

t1

󵄩 󵄩 + ∫ 󵄩󵄩󵄩(Tq (t2 − s) − Tq (t1 − s))fn (s)󵄩󵄩󵄩ds t1 −ε

t1 −ϵ

󵄩 󵄩 󵄩 󵄩 ≤ ∫ kf (s)(1 + 󵄩󵄩󵄩xn (s)󵄩󵄩󵄩)ds sup 󵄩󵄩󵄩Tq (t2 − s) − Tq (t1 − s)󵄩󵄩󵄩B(X) s∈[0,t1 −ε]

0

t1

Mε2q−1 M(t2 − t1 + ε)2q−1 󵄩 󵄩 +( + ) ∫ kf (s)(1 + 󵄩󵄩󵄩xn (s)󵄩󵄩󵄩)ds Γ(2q) Γ(2q) t1 −ε

󵄩 󵄩 ≤ ‖kf ‖1 (1 + ‖xn ‖C ) sup 󵄩󵄩󵄩Tq (t2 − s) − Tq (t1 − s)󵄩󵄩󵄩B(X) s∈[0,t1 −ε]

t1

Mε2q−1 M(t2 − t1 + ε)2q−1 + )(1 + ‖xn ‖C ) ∫ kf (s)ds +( Γ(2q) Γ(2q) t1 −ε

→ 0,

as t2 → t1 , ε → 0.

Consequently, we have t

modC ({∫(t − s)q−1 Pq (t − s)fn (s)ds, n ≥ 1}) = 0. 0

As a conclusion, it follows that modC ({yn }∞ n=1 ) = 0. Hence, the multivalued map P is ν-condensing. Step 3. The multimap P (x) is convex and compact for each x ∈ W. Part I. The multimap P (x) has convex values for each x ∈ W.

160 | 4 Fractional evolution equations and inclusions of order α ∈ (1, 2) In fact, if y1 , y2 belong to P (x) for each x ∈ W, then there exist f1 , f2 ∈ SF,x such that for each t ∈ J, we have t

yi (t) = Cq (t)(x0 − g(x)) + Kq (t)x1 + ∫(t − s)q−1 Pq (t − s)fi (s)ds,

i = 1, 2.

0

Let θ ∈ [0, 1]. Then, for each t ∈ J, we get (θy1 + (1 − θ)y2 )(t) = Cq (t)(x0 − g(x)) + Kq (t)x1 t

+ ∫(t − s)q−1 Pq (t − s)(θf1 + (1 − θ)f2 )(s)ds. 0

As F has convex values by the definition of SF,x , we deduce that θf1 (s)+(1−θ)f2 (s) ∈ SF,x . Thus, θy1 + (1 − θ)y2 ∈ P (x). Part II. The multimap P has compact values. In view of the foregoing facts, it is enough to show that W is nonempty and compact in C(J, X), that is, by Lemma 1.6, we need to show that lim ν(Wn ) = 0.

(4.21)

n→∞

As in step 2, we can show that modC (Wn ) = 0, that is, Wn is equicontinuous. Hence, it remains to show that σ(Wn ) = 0. By Proposition 1.21, for each ε > 0, there exists a sequence {yk }∞ k=1 in P (Wn−1 ) such that σ(Wn ) = σ(P (Wn )) ≤ 2σ({yk }∞ k=1 ) + ε. Therefore, by Proposition 1.18 and the nonsingularity of σ, it follows that σ(Wn ) ≤ 2σ({yk }∞ k=1 ) + ε = 2 sup β({yk (t)}k=1 ) + ε. ∞

t∈J

(4.22)

Since yk ∈ P (Wn−1 ) (k ≥ 1), there exists xk ∈ Wn−1 such that yk ∈ P (xk ). Hence, from the compactness of g and the strong continuity of Cq (t) and Kq (t) for t ∈ J, there exists fk ∈ SF,xk such that for every t ∈ J, β({yk (t)}k=1 ) ≤ β({Cq (t)(x0 − g({xk }∞ k=1 )) + Kq (t)x1 }) ∞

t

+ β({∫(t − s)q−1 Pq (t − s)fk (s)ds : k ≥ 1}) t

0

= β({∫(t − s)q−1 Pq (t − s)fk (s)ds : k ≥ 1}). 0

4.2 Nonlocal evolution inclusions |

161

By (H4) and Proposition 1.21, for a. e. t ∈ J, we have β({fk (t)}k=1 ) ≤ β(F(t, {xk (t)}k=1 )) ≤ η(t)β({xk (t)}k=1 ) ≤ η(t)σ(Wn−1 ) =: γ(t). ∞

∞

∞

On the other hand, by (H3), for almost all t ∈ J, ‖fk (t)‖ ≤ kf (t)(1 + r) for every k ≥ 1. Hence, fk ∈ L1 (J, X), k ≥ 1. Note that γ(⋅) ∈ L1 (J, ℝ+ ) from (H4). It follows from Lemma 1.5 that there exist a compact Kϵ ⊂ X, a measurable set Jϵ ⊂ J with measure less than ϵ, and a sequence of functions {gkϵ } ⊂ L1 (J, X) such that {gkϵ (s)}∞ k=1 ⊆ Kϵ for all s ∈ J, and 󵄩󵄩 󵄩 ϵ 󵄩󵄩fk (s) − gk (s)󵄩󵄩󵄩 < 2γ(s) + ϵ,

for every k ≥ 1 and every s ∈ Jϵ′ = J − Jϵ .

Then, using Minkowski’s inequality and the property of the MNC, we obtain β({∫(t − s)q−1 Pq (t − s)(fk (s) − gkϵ (s))ds : k ≥ 1}) Jϵ′

≤

2M ∫(t − s)2q−1 β({(fk (s) − gkϵ (s)) : k ≥ 1})ds Γ(2q) Jϵ′

≤

2M 󵄩 󵄩 ∫(t − s)2q−1 sup󵄩󵄩󵄩fk (s) − gkϵ (s)󵄩󵄩󵄩ds Γ(2q) k≥1 Jϵ′

2q−1

≤

2Ma ∫(2γ(s) + ϵ)ds Γ(2q)

(4.23)

Jϵ′

≤ ≤

2Ma2q 4Ma2q−1 ‖γ‖1 + ϵ Γ(2q) Γ(2q)

4Ma2q−1 2Ma2q σ(Wn−1 )‖η‖1 + ϵ, Γ(2q) Γ(2q)

and β({∫(t − s)q−1 Pq (t − s)fk (s)ds : k ≥ 1}) Jϵ

≤

2M ∞ ∫(t − s)2q−1 β({fk (s)}k=1 )ds Γ(2q) Jϵ

≤

2M 󵄩 󵄩 ∫(t − s)2q−1 sup󵄩󵄩󵄩fk (s)󵄩󵄩󵄩ds Γ(2q) k≥1 Jϵ

2Ma2q−1 ≤ (1 + r) ∫ kf (s)ds. Γ(2q) Jϵ

(4.24)

162 | 4 Fractional evolution equations and inclusions of order α ∈ (1, 2) Using (4.23) and (4.24), we have t

β({∫(t − s)q−1 Pq (t − s)fk (s)ds : k ≥ 1}) 0

≤ β({∫(t − s)q−1 Pq (t − s)fk (s)ds : k ≥ 1}) Jϵ′

+ β({∫(t − s)q−1 Pq (t − s)fk (s)ds : k ≥ 1}) Jϵ

≤ β({∫(t − s)q−1 Pq (t − s)(fk (s) − gkϵ (s))ds : k ≥ 1}) Jϵ′

+ β({∫(t − s)q−1 Pq (t − s)gkϵ (s)ds : k ≥ 1}) Jϵ′

+ β({∫(t − s)q−1 Pq (t − s)fk (s)ds : k ≥ 1}) Jϵ

2Ma2q 2Ma2q−1 4Ma σ(Wn−1 )‖η‖1 + ϵ+ (1 + r) ∫ kf (s)ds. Γ(2q) Γ(2q) Γ(2q) 2q−1

≤

Jϵ

As ϵ is arbitrary, for all t ∈ J, we get t

β({∫(t − s)q−1 Pq (t − s)fk (s)ds}) ≤ 0

4Ma2q−1 ‖η‖1 σ(Wn−1 ). Γ(2q)

Therefore, for each t ∈ J, we have β({yk (t)}k=1 ) ≤ ∞

4Ma2q−1 ‖η‖1 σ(Wn−1 ). Γ(2q)

The above inequality together with (4.22) and arbitrariness of ε, we can deduce that σ(Wn ) ≤

8Ma2q−1 ‖η‖1 σ(Wn−1 ). Γ(2q)

Then, by induction, we find that 0 ≤ σ(Wn ) ≤ (

n

8Ma2q−1 ‖η‖1 ) σ(W0 ), Γ(2q)

for all n ≥ 1.

4.2 Nonlocal evolution inclusions |

163

Since this inequality is true for every n ≥ 1, passing to the limit n → ∞ and by (H4), we obtain (4.21). Hence, W = ⋂∞ n=1 Wn is a nonempty compact set of X, and P has compact values in W. Step 4. The values of P are closed. Let xn , x∗ ∈ W with xn → x∗ as n → ∞, yn ∈ P (xn ), and yn → y∗ as n → ∞. We show that y∗ ∈ P (x∗ ). Indeed yn ∈ P (xn ) means that there exists fn ∈ SF,xn such that t

yn (t) = Cq (t)(x0 − g(x)) + Kq (t)x1 + ∫(t − s)q−1 Pq (t − s)fn (s)ds. 0

Next, we must show that there exists f∗ ∈ SF,x∗ such that t

y∗ (t) = Cq (t)(x0 − g(x)) + Kq (t)x1 + ∫(t − s)q−1 Pq (t − s)f∗ (s)ds. 0

Since xn → x∗ and yn ∈ P (xn ), we deduce that 󵄩󵄩 󵄩󵄩(yn (t) − Cq (t)x0 + Cq (t)g(xn ) − Kq (t)x1 ) 󵄩 − (y∗ (t) − Cq (t)x0 + Cq (t)g(x∗ ) − Kq (t)x1 )󵄩󵄩󵄩 → 0,

as n → ∞.

Now, we consider the linear continuous operator 1

t

F : L (J, X) → C(J, X),

f 󳨃→ (F f )(t) = ∫(t − s)q−1 Pq (t − s)f (s)ds. 0

From step 3 and Lemma 4.12, it follows that F ∘ SF is a closed graph operator. Furthermore, in view of the definition of F , we have that (yn (t) − Cq (t)x0 + Cq (t)g(xn ) − Kq (t)x1 ) ∈ F (SF,xn ). In view of the fact that xn → x∗ as n → ∞, repeated application of Lemma 4.12 yields t

(y∗ (t) − Cq (t)x0 + Cq (t)g(x∗ ) − Kq (t)x1 ) = ∫(t − s)q−1 Pq (t − s)f (s)ds 0

for some f ∈ SF,x∗ . Thus, P is a closed multivalued map. Therefore, we deduce that P : W → P(W) is closed and ν-condensing with nonempty, convex and compact values. Thus, all the hypotheses of Lemma 4.7 are satisfied. Hence, there exists at least one fixed point x ∈ W such that x ∈ P (x), which corresponds to a mild solution of problem (4.18).

164 | 4 Fractional evolution equations and inclusions of order α ∈ (1, 2) Theorem 4.6. Suppose that all the assumptions of Theorem 4.5 are satisfied. Then the set of mild solutions of (4.18) is compact in C(J, X). Proof. Note that the set of mild solutions is nonempty by Theorem 4.5. Indeed, letting r > 0 be defined by (4.19), we can get a mild solution in W0 . Now, we show that an arbitrary number of mild solutions of problem (4.18) belongs to W0 . Let x be a mild solution of problem (4.18). Then t

x(t) = Cq (t)(x0 − g(x)) + Kq (t)x1 + ∫(t − s)q−1 Pq (t − s)f (s)ds, 0

where f ∈ SF,x = {f ∈ L1 (J, X) : f (t) ∈ F(t, x(t)), for a. e. t ∈ J}. Using an argument similar to the one used in Step 1 of the proof of Theorem 4.5, we have 󵄩 󵄩 ‖x‖C = sup󵄩󵄩󵄩x(t)󵄩󵄩󵄩 t∈J

t

󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ≤ sup󵄩󵄩󵄩Cq (t)(x0 − g(x))󵄩󵄩󵄩 + sup󵄩󵄩󵄩Kq (t)x1 󵄩󵄩󵄩 + sup ∫(t − s)q−1 󵄩󵄩󵄩Pq (t − s)f (s)󵄩󵄩󵄩ds t∈J

t∈J

t∈J

0

Ma2q−1 Ma2q−1 ‖kf ‖1 + ‖k ‖ r ≤ M‖x0 ‖ + MNg1 r + MNg2 + Ma‖x1 ‖ + Γ(2q) Γ(2q) f 1

≤ r.

This shows that the mild solutions of problem (4.18) are bounded. Thus, the conclusion follows by Lemma 4.8. The proof is completed.

4.2.4 Control problem Let Ω ⊂ ℝN (N = 1, 2, 3) be an open bounded set and X = U = L2 (Ω). Let us consider the following fractional partial differential equation with the constrained control u and a finite multipoint discrete mean condition: 𝜕α y(t, z) = Δy(t, z) + G(t, z, y(t, z), u(t, z)), t ∈ [0, 1], z ∈ Ω, u ∈ U, { { { t y(t, z) = 0, t ∈ [0, 1], z ∈ 𝜕Ω, { { { n ′ {y(0, z) − ∑i=0 ∫Ω m(ξ , z)y(ti , ξ )dξ = 0, y (0, z) = 0, z ∈ Ω, (4.25) where 𝜕tα is the Caputo fractional partial derivative of order α ∈ (1, 2), 0 ≤ t0 < t1 < ⋅ ⋅ ⋅ < tn ≤ 1, m(ξ , z) : Ω × Ω → X is an L2 -Lebesgue integrable function and G : [0, 1] × Ω × X × U → X is a single-valued continuous measurable function. We define x(t) = y(t, ⋅), that is, x(t)(z) = y(t, z), t ∈ J, z ∈ Ω and let J = [0, 1]. The set of constraints function U : J → Pcl,cv (X) is a measurable multivalued map. If u ∈ U,

4.2 Nonlocal evolution inclusions |

165

then it means that u(t) ∈ U(t, x(t)), for a. e. t ∈ J. The function f : J × X × U is given by f (t, x(t), u(t))(z) = G(t, z, y(t, z), u(t, z)). The problem (4.25) is solved if we show that there exists a control function u such that problem (4.25) admits a mild solution. Let the multivalued map be given by F(t, x(t)) = {f (t, x(t), u(t)), u ∈ U}.

(4.26)

Then the set of mild solutions of control problem (4.25) with the right-hand side given by (4.26) coincides with the set of mild solutions of problem (4.18). Let A be the Laplace operator with Dirichlet boundary conditions defined by A = Δ with 2

1

2

D (A) = {v ∈ L (Ω) : v ∈ H0 (Ω) ∩ H (Ω)}.

Let {−λk , ϕk }∞ k=1 be the eigensystem of the operator A. Then 0 < λ1 ≤ λ2 ≤ ⋅ ⋅ ⋅ , λk → ∞ as k → ∞, and {ϕk }∞ k=1 forms an orthonormal basis of X. Furthermore, ∞

Ax = − ∑ λk (x, ϕk )ϕk , k=1

x ∈ D (A),

where (⋅, ⋅) is the inner product in X. It is known that the operator A generates a strongly continuous uniformly bounded cosine family (see, e. g., [36]), which in this case is defined by ∞

C(t)x = ∑ cos(√λk t)(x, ϕk )ϕk , k=1

x ∈ X,

and then ‖C(t)‖B(X) ≤ 1 for every t ∈ ℝ. Hence, (H1) holds. Taking α = 32 , we have q = 43 . Let g : C(J, X) → X be given by g(x)(z) = n ∑i=0 Kg x(ti )(z) with Kg v(z) = ∫Ω m(ξ , z)v(ξ )dξ for v ∈ X, z ∈ Ω (noting that Kg : X → X is completely continuous). Thus, the assumption in (H5) holds true. With the choice of operator A, the problem (4.25) can be reformulated in X as the following nonlocal control problem: C α Dt x(t)

{

= Ax(t) + f (t, x(t), u(t)),

x(0) = g(x),

x (0) = 0.

t ∈ J, u ∈ U,

′

(4.27)

Next, the results obtained in Subsection 4.2.3 can be applied to the following problem of fractional evolution inclusions: C α D x(t) ∈ Ax(t) + F(t, x(t)), {0 t x(0) = g(x), x′ (0) = 0.

t ∈ J,

Theorem 4.7. Assume that the following conditions hold:

(4.28)

166 | 4 Fractional evolution equations and inclusions of order α ∈ (1, 2) (H6) U : J → Pcl,cv (X) is a measurable multivalued map; (H7) the function f : J × X × X → X is L1 -Carathéodory, linear in the third argument and there exists a function kf (⋅) ∈ L1 (J, ℝ+ ) satisfying ‖kf ‖1 < √π(1 − n‖m‖)/2 such that ‖f (t, x, y)‖ ≤ kf (t)(1 + ‖x‖) for almost all t ∈ J and all x ∈ X; (H8) there exists a function η(⋅) ∈ L1 (J, ℝ+ ) satisfying ‖η‖1 < √π/16 such that β(f (t, D, U(t, D))) ≤ η(t)β(D), for every bounded subset D ⊂ C(J, X). Then the control problem (4.25) has at least one mild solution. In addition, the set of mild solutions is compact. Proof. From (H6) and (H7), the map t 󳨃→ F(t, ⋅) is obviously a measurable multivalued map, and then F(⋅, ⋅) ∈ Pcv,cl (X). Now, we show that the selection set of F is not empty. Since U is a measurable multivalued map, it follows by Lemma 4.10 that there exists a sequence of measurable selections {un }∞ n=1 ⊂ U such that U(t) = ⋃{un (t), n ≥ 1}

for every t ∈ J.

Let vn (t) = f (t, x(t), un (t)) for n ≥ 1 and t ∈ J. In view of the continuity of f , vn is thus measurable. Hence, {vn (t), n ≥ 1} ⊆ F(t, x(t)). Conversely, if f (t, x(t), u(t)) ∈ F(t, x(t)) for any u ∈ U, then there exists a subsequence in U, which will be still denoted by {un }∞ n=1 , such that un → u as n → ∞. It follows from the continuity of f that f (t, x(t), un (t)) → f (t, x(t), u(t)) as n → ∞. Hence, f (t, x(t), u(t)) ∈ {vn (t), n ≥ 1}. This means that F(t, x(t)) = ⋃{vn (t), n ≥ 1}. Consequently, from Lemma 4.10 F(⋅, x) is measurable. Next, we show that the map x 󳨃→ F(⋅, x) is an u. s. c. multivalued map by means of contradiction. First, we suppose that F is not u. s. c. at some point x0 ∈ Ω. Then there exists an open neighborhood W ⊆ X such that F(t, x0 ) ⊂ W, and for every open neighborhood V ⊆ Ω of x0 there exists x1 ∈ V such that F(t, x1 ) ⊄ W. Let Vn = {x ∈ Ω, ‖x − x0 ‖
0 and C = C([−r, 0], E) be the space of continuous functions from [−r, 0] into E. For any element z ∈ C , define the norm ‖z‖∗ = supϑ∈[−r,0] |z(ϑ)|. Consider the nonlocal Cauchy problem of the following form: C q D [x(t) − h(t, xt )] + Ax(t) = f (t, xt ), {0 t x0 (ϑ) + (g(xt1 , . . . , xtn ))(ϑ) = φ(ϑ),

t ∈ (0, a],

(5.1)

ϑ ∈ [−r, 0],

where C0Dqt is Caputo fractional derivative of order 0 < q < 1, 0 < t1 < ⋅ ⋅ ⋅ < tn ≤ a, a > 0, −A is the infinitesimal generator of an analytic semigroup {T(t)}t≥0 of operators on E, f , h : [0, ∞) × C → E and g : C n → C are given functions satisfying some assumptions, φ ∈ C and define xt by xt (ϑ) = x(t + ϑ), for ϑ ∈ [−r, 0]. A strong motivation for investigating the nonlocal Cauchy problem (5.1) comes from physics. For example, fractional diffusion equations are abstract partial differential equations that involve fractional derivatives in space and time. They are useful to model anomalous diffusion, where a plume of particles spreads in a different manner than the classical diffusion equation predicts. The time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order q ∈ (0, 1), namely 𝜕tq u(z, t) = Au(z, t), β

t ≥ 0, z ∈ ℝ. β

β

β

We can take A = 𝜕z 1 , for β1 ∈ (0, 1], or A = 𝜕z + 𝜕z 2 for β2 ∈ (1, 2], where 𝜕tq , 𝜕z 1 , 𝜕z 2 are the fractional derivatives of order q, β1 , β2 , respectively. We refer the interested reader to [29, 161, 182, 218] and the references therein for more details. The nonlocal condition can be applied in physics with better effect than the classical initial condition x(ϑ) = φ(ϑ), ϑ ∈ [−r, 0]. For example, g(xt1 , . . . , xtn ) can be written as n

(g(xt1 , . . . , xtn ))(ϑ) = ∑ci xti (ϑ), i=1

https://doi.org/10.1515/9783110769272-005

170 | 5 Fractional neutral evolution equations and inclusions where ci (i = 1, 2, . . . , n) are given constants and 0 < t1 < ⋅ ⋅ ⋅ < tn ≤ a. Nonlocal conditions were initiated by Byszewski [48] when he proved the existence and uniqueness of mild and classical solutions of nonlocal Cauchy problems. As remarked by Byszewski and Lakshmikantham [49], the nonlocal condition can be more useful than the standard initial condition to describe some physical phenomena. In Subsection 5.1.2, we establish criteria on existence and uniqueness of mild solutions for the nonlocal Cauchy problem (5.1) by considering an integral equation, which is given in terms of probability density and semigroup. The methods of the functional analysis concerning to an analytic semigroup of operators and some fixed-point theorems are applied effectively. In Subsection 5.1.3, we give also an example to illustrate the applications of the abstract results. This section is based on [231].

5.1.2 Existence and uniqueness Throughout this section, let −A be the infinitesimal generator of an analytic semigroup {T(t)}t≥0 of uniformly bounded linear operators on E. Let 0 ∈ ρ(A), where ρ(A) is the resolvent set of A. Then for 0 < η ≤ 1, it is possible to define the fractional power Aη as a closed linear operator on its domain D(Aη ). For analytic semigroup {T(t)}t≥0 , the following properties will be used: (i) there is an M ≥ 1 such that M :=

󵄨 󵄨 sup 󵄨󵄨󵄨T(t)󵄨󵄨󵄨 < ∞,

t∈[0,+∞)

(5.2)

(ii) for any η ∈ (0, 1], there exists a positive constant Cη such that 󵄨󵄨 η 󵄨 Cη 󵄨󵄨A T(t)󵄨󵄨󵄨 ≤ η , t

0 < t ≤ a.

(5.3)

We rewrite the nonlocal Cauchy problem (5.1) in the equivalent integral equation x(t) = φ(0) − (g(xt1 , . . . , xtn ))(0) − h(0, x0 ) + h(t, xt ) { { { t 1 + Γ(q) t ∈ [0, a], ∫0 (t − s)q−1 [−Ax(s) + f (s, xs )]ds, { { { ϑ ∈ [−r, 0], {x0 (ϑ) + (g(xt1 , . . . , xtn ))(ϑ) = φ(ϑ),

(5.4)

provided that the integral in (5.4) exists. Before giving the definition of mild solution of (5.1), we first prove the following lemma.

5.1 Nonlocal Cauchy problem

| 171

Lemma 5.1. If (5.4) holds, then we have x(t) = ∫0 ϕq (θ)T(t q θ)[φ(0) − (g(xt1 , . . . , xtn ))(0) − h(0, x0 )]dθ { { { { { { + h(t, xt ) { { { t ∞ + q ∫0 ∫0 θ(t − s)q−1 ϕq (θ)AT((t − s)q θ)h(s, xs )dθds { { { t ∞ { { + q ∫0 ∫0 θ(t − s)q−1 ϕq (θ)T((t − s)q θ)f (s, xs )dθds, { { { { {x0 (ϑ) + (g(xt1 , . . . , xtn ))(ϑ) = φ(ϑ), ∞

(5.5) t ∈ [0, a],

ϑ ∈ [−r, 0],

where ϕq is a probability density function defined on (0, ∞), that is, ∞

ϕq (θ) ≥ 0,

θ ∈ (0, ∞) and

∫ ϕq (θ)dθ = 1. 0

Proof. Let λ > 0. Applying Laplace transforms, ∞

ν(λ) = ∫ e

∞ −λs

x(s)ds,

χ(λ) = ∫ e−λs h(s, xs )ds,

0

0

and ∞

ω(λ) = ∫ e−λs f (s, xs )ds 0

to (5.4), we have ν(λ) =

1 1 1 [φ(0) − (g(xt1 , . . . , xtn ))(0) − h(0, x0 )] + χ(λ) − q Aν(λ) + q ω(λ) λ λ λ

= λq−1 (λq I + A) [φ(0) − (g(xt1 , . . . , xtn ))(0) − h(0, x0 )] −1

+ λq (λq I + A) χ(λ) + (λq I + A) ω(λ) −1

=λ

q−1

q

∞

−1

q

∫ e−λ s T(s)[φ(0) − (g(xt1 , . . . , xtn ))(0) − h(0, x0 )]ds 0 ∞

+λ ∫e

−λq s

0

∞

q

T(s)χ(λ)ds + ∫ e−λ s T(s)ω(λ)ds, 0

where I is the identity operator defined on E. Consider the one-sided stable probability density ψq (θ) =

Γ(nq + 1) 1 ∞ sin(nπq), ∑ (−1)n−1 θ−qn−1 π n=1 n!

θ ∈ (0, ∞),

(5.6)

172 | 5 Fractional neutral evolution equations and inclusions whose Laplace transform is given by ∞

q

∫ e−λθ ψq (θ)dθ = e−λ ,

where q ∈ (0, 1).

(5.7)

0

Using (5.7), we get ∞

q

λq−1 ∫ e−λ s T(s)[φ(0) − (g(xt1 , . . . , xtn ))(0) − h(0, x0 )]ds 0 ∞

q

= ∫ q(λt)q−1 e−(λt) T(t q )[φ(0) − (g(xt1 , . . . , xtn ))(0) − h(0, x0 )]dt 0 ∞

= ∫−

1 d −(λt)q [e ]T(t q )[φ(0) − (g(xt1 , . . . , xtn ))(0) − h(0, x0 )]dt λ dt

(5.8)

0 ∞∞

= ∫ ∫ θψq (θ)e−λtθ T(t q )[φ(0) − (g(xt1 , . . . , xtn ))(0) − h(0, x0 )]dθdt 0 0 ∞ −λt

= ∫e

∞

[ ∫ ψq (θ)T( 0

0 ∞

tq )[φ(0) − (g(xt1 , . . . , xtn ))(0) − h(0, x0 )]dθ]dt, θq

q

∫ e−λ s T(s)ω(λ)ds 0 ∞∞

q

= ∫ ∫ qt q−1 e−(λt) T(t q )e−λs f (s, xs )dsdt 0 0 ∞∞∞

= ∫ ∫ ∫ qψq (θ)e−(λtθ) T(t q )e−λs t q−1 f (s, xs )dθdsdt 0 0 0 ∞∞∞

= ∫ ∫ ∫ qψq (θ)e−λ(t+s) T( 0 0 0 ∞ −λt

= ∫e 0

t ∞

[q ∫ ∫ ψq (θ)T( 0 0

t q t q−1 ) f (s, xs )dθdsdt θq θq

(t − s)q−1 (t − s)q )f (s, x ) dθds]dt, s θq θq

and ∞

q

λq ∫ e−λ s T(s)χ(λ)ds 0 ∞∞

q

= ∫ ∫ qλq t q−1 e−(λt) T(t q )e−λs h(s, xs )dsdt 0 0

(5.9)

5.1 Nonlocal Cauchy problem

∞ ∞

| 173

q

= ∫ [ ∫ −T(t q )e−λs h(s, xs )ds]de−(λt) 0

0

= (e

−(λt)q

∞

q

∫ −T(t )e

−λs

0 ∞∞

󵄨󵄨∞ 󵄨󵄨 h(s, xs )ds)󵄨󵄨󵄨 󵄨󵄨 󵄨t=0

q

+ ∫ ∫ qt q−1 e−(λt) AT(t q )e−λs h(s, xs )dsdt 0 0

t ∞

∞

= ∫ e−λt [h(t, xt ) + q ∫ ∫ ψq (θ)AT( 0

0 0

(t − s)q (t − s)q−1 )h(s, xs ) dθds]dt. q θ θq

According to (5.6), (5.8)–(5.10), we have ∞

∞

ν(λ) = ∫ e−λt [ ∫ ψq (θ)T( 0

0

tq )[φ(0) − (g(xt1 , . . . , xtn ))(0) − h(0, x0 )]dθ θq

t ∞

+ h(t, xt ) + q ∫ ∫ ψq (θ)AT( t ∞

0 0

+ q ∫ ∫ ψq (θ)T( 0 0

(t − s)q (t − s)q−1 )h(s, x ) dθds s θq θq

(t − s)q (t − s)q−1 )f (s, xs ) dθds]dt. q θ θq

Now we can invert the last Laplace transform to get ∞

x(t) = ∫ ψq (θ)T( 0

tq )[φ(0) − (g(xt1 , . . . , xtn ))(0) − h(0, x0 )]dθ + h(t, xt ) θq

t ∞

+ q ∫ ∫ ψq (θ)AT( 0 0

t ∞

+ q ∫ ∫ ψq (θ)T( 0 0

(t − s)q (t − s)q−1 )h(s, xs ) dθds q θ θq

(t − s)q (t − s)q−1 )f (s, x ) dθds s θq θq

∞

= ∫ ϕq (θ)T(t q θ)[φ(0) − (g(xt1 , . . . , xtn ))(0) − h(0, x0 )]dθ + h(t, xt ) 0

t ∞

+ q ∫ ∫ θ(t − s)q−1 ϕq (θ)AT((t − s)q θ)h(s, xs )dθds 0 0

t ∞

+ q ∫ ∫ θ(t − s)q−1 ϕq (θ)T((t − s)q θ)f (s, xs )dθds, 0 0

(5.10)

174 | 5 Fractional neutral evolution equations and inclusions where ϕq (θ) = q1 θ−1−1/q ψq (θ−1/q ) is the probability density function defined on (0, ∞). This completes the proof. For any x ∈ E, define operators {Sq (t)}t≥0 and {Tq (t)}t≥0 by ∞

Sq (t)x = ∫ ϕq (θ)T(t q θ)xdθ

∞

and

Tq (t)x = q ∫ θϕq (θ)T(t q θ)xdθ.

0

0

Due to Lemma 5.1, we give the following definition of the mild solution of (5.1). Definition 5.1. By the mild solution of the nonlocal Cauchy problem (5.1), we mean that the function x ∈ C([−r, a], E), which satisfies x(t) = Sq (t)[φ(0) − (g(xt1 , . . . , xtn ))(0) − h(0, x0 )] + h(t, xt ) { { { { t { { + ∫0 (t − s)q−1 ATq (t − s)h(s, xs )ds t { { + ∫0 (t − s)q−1 Tq (t − s)f (s, xs )ds, { { { { {x0 (ϑ) + (g(xt1 , . . . , xtn ))(ϑ) = φ(ϑ),

t ∈ [0, a],

ϑ ∈ [−r, 0].

Before stating and proving the main results, we introduce the following hypotheses: (H1 ) T(t) is a compact operator for every t > 0, (H2 ) for almost all t ∈ [0, a], the function f (t, ⋅) : C → E is continuous and for each z ∈ C , the function f (⋅, z) : [0, a] → E is strongly measurable, 1

(H3 ) there exist a constant q1 ∈ [0, q) and m ∈ L q1 ([0, a], ℝ+ ) such that |f (t, z)| ≤ m(t) for all z ∈ C and almost all t ∈ [0, a], (H4 ) there exists a constant L > 0 such that ‖g(xt1 , . . . , xtn ) − g(yt1 , . . . , ytn )‖∗ ≤ L‖x − y‖, for x, y ∈ C([−r, a], E), (H5 ) h : [0, a] × C → E is a continuous function and there exist a constant β ∈ (0, 1) and H, H1 > 0 such that h ∈ D(Aβ ) and for any z, y ∈ C , t ∈ [0, a], the function Aβ h(⋅, z) is strongly measurable and Aβ h(t, ⋅) satisfies the Lipschitz condition 󵄨󵄨 β 󵄨 β 󵄨󵄨A h(t, z) − A h(t, y)󵄨󵄨󵄨 ≤ H‖z − y‖∗

(5.11)

󵄨󵄨 β 󵄨 󵄨󵄨A h(t, z)󵄨󵄨󵄨 ≤ H1 (‖z‖∗ + 1).

(5.12)

and the inequality

Remark 5.1. The condition (H3 ) can be replaced by the condition 1

(H3 )′ there exist a constant q1 ∈ [0, q) and mk ∈ L q1 ([0, a], ℝ+ ) such that |f (t, z)| ≤ mk (t) for all z ∈ C , ‖z‖∗ ≤ k and almost all t ∈ [0, a], where k is any positive constant. Since (H3 )′ would not be an essential generalization, we only consider (H3 ) throughout the following text.

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We prove the following lemmas relative to operators {Sq (t)}t≥0 and {Tq (t)}t≥0 be-

fore we proceed further.

Lemma 5.2. For any fixed t ≥ 0, Sq (t) and Tq (t) are linear and bounded operators. Proof. For any fixed t ≥ 0, since T(t) is linear operator, it is easy to see that Sq (t) and Tq (t) are also linear operators.

For ξ ∈ [0, 1], direct calculation gives that ξ

Γ(1 + q ) 1 . ∫ ξ ψq (θ)dθ = Γ(1 + ξ ) θ ∞

0

Then we have ∞

ξ

∞

∫ θ ϕq (θ)dθ = ∫ 0

0

Γ(1 + ξ ) 1 ψq (θ)dθ = . Γ(1 + qξ ) θqξ

(5.13)

In the case ξ = 1, we have ∞

∞

∫ θϕq (θ)dθ = ∫ 0

0

1 1 ψ (θ)dθ = . θq q Γ(1 + q)

For any x ∈ E, according to (5.2) and (5.13), we have 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 q 󵄨󵄨Sq (t)x󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∫ ϕq (θ)T(t θ)xdθ󵄨󵄨󵄨 ≤ M|x| 󵄨󵄨 󵄨󵄨󵄨 󵄨 0 and 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 qM 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 q |x|. 󵄨󵄨Tq (t)x󵄨󵄨 = 󵄨󵄨q ∫ θϕq (θ)T(t θ)xdθ󵄨󵄨󵄨 ≤ 󵄨󵄨 󵄨󵄨 Γ(1 + q) 󵄨 0 󵄨 This completes the proof. Lemma 5.3. Operators {Sq (t)}t≥0 and {Tq (t)}t≥0 are strongly continuous, which means that for ∀x ∈ E and 0 ≤ t ′ < t ′′ ≤ a, we have 󵄨󵄨 ′′ ′ 󵄨 󵄨󵄨Sq (t )x − Sq (t )x󵄨󵄨󵄨 → 0

and

󵄨󵄨 ′′ ′ 󵄨 󵄨󵄨Tq (t )x − Tq (t )x 󵄨󵄨󵄨 → 0

as t ′ → t ′′ .

176 | 5 Fractional neutral evolution equations and inclusions Proof. For any x ∈ E and 0 ≤ t ′ < t ′′ ≤ a, we get that 󵄨󵄨 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 󵄨󵄨 ′′ q ′ q ′′ ′ 󵄨󵄨 󵄨 󵄨󵄨Tq (t )x − Tq (t )x 󵄨󵄨 = 󵄨󵄨q ∫ θϕq (θ)[T((t ) θ) − T((t ) θ)]xdθ󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 0 ∞

q q 󵄨 󵄨 ≤ qM ∫ θϕq (θ)󵄨󵄨󵄨[T((t ′′ ) θ − (t ′ ) θ) − I]x 󵄨󵄨󵄨dθ. 0

According to the strong continuity of {T(t)}t≥0 and (5.13), we know that |Tq (t ′′ )x − Tq (t ′ )x| tends to zero as t ′′ −t ′ → 0, which means that {Tq (t)}t≥0 is strongly continuous. Using the similar method, we can also obtain that {Sq (t)}t≥0 is also strongly continuous, and this completes the proof. Lemma 5.4. If the assumption (H1 ) is satisfied, then Sq (t) and Tq (t) are also compact operators for every t > 0. Proof. For each positive constant k, set Yk = {x ∈ E : |x| ≤ k}. Then Yk is clearly a bounded subset in E. We only need to prove that for any positive constant k and t > 0, the sets ∞

V1 (t) = { ∫ ϕq (θ)T(t q θ)xdθ, x ∈ Yk }

∞

and V2 (t) = {q ∫ θϕq (θ)T(t q θ)xdθ, x ∈ Yk } 0

0

are relatively compact in E. Let t > 0 be fixed. For ∀ δ > 0, define the subset in E by ∞

V (t) = { ∫ ϕq (θ)T(t q θ)xdθ, x ∈ Yk }. δ

δ

Then for any x ∈ Yk , we have ∞

q

q

∞

∫ ϕq (θ)T(t θ)xdθ = T(t δ) ∫ ϕq (θ)T(t q θ − t q δ)xdθ. δ

δ

From the compactness of T(t q δ) (t q δ > 0), we obtain that the set V δ (t) is relatively compact in E for ∀ δ > 0. Moreover, for every x ∈ Yk , we have ∞ δ 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 δ 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨 ∫ ϕq (θ)T(t q θ)xdθ − ∫ ϕq (θ)T(t q θ)xdθ󵄨󵄨󵄨 = 󵄨󵄨󵄨∫ ϕq (θ)T(t q θ)xdθ󵄨󵄨󵄨 ≤ Mk ∫ ϕq (θ)dθ. 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 0 0 0 δ

Therefore, there are relatively compact sets arbitrarily close to the set V1 (t), t > 0. Hence, the set V1 (t), t > 0 is also relatively compact in E. Similarly, we can conclude that the set V2 (t), t > 0 is relatively compact in E. The proof is complete.

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Lemma 5.5. For any x ∈ E, β ∈ (0, 1) and η ∈ (0, 1], we have ATq (t)x = A1−β Tq (t)Aβ x,

0 ≤ t ≤ a,

and Γ(2 − η) 󵄨 qCη 󵄨󵄨 η , 󵄨󵄨A Tq (t)󵄨󵄨󵄨 ≤ qη t Γ(1 + q(1 − η))

0 < t ≤ a.

Proof. For any x ∈ E, β ∈ (0, 1) and η ∈ (0, 1], we have ∞

ATq (t)x = q ∫ θϕq (θ)AT(t q θ)xdθ 0 ∞

= q ∫ θϕq (θ)A1−β T(t q θ)Aβ xdθ 0 1−β

=A

Tq (t)Aβ x.

By (5.3) and (5.13), we get 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 󵄨󵄨 η 󵄨󵄨 󵄨󵄨󵄨 η q 󵄨󵄨A Tq (t)x󵄨󵄨 = 󵄨󵄨q ∫ θϕq (θ)A T(t θ)xdθ󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 0 󵄨 ∞

≤ q ∫ θϕq (θ) 0

= =

qCη |x| t qη

qCη t qη

Cη |x|dθ q (t θ)η

∞

∫ θ1−η ϕq (θ)dθ 0

Γ(2 − η) |x|, Γ(1 + q(1 − η))

0 < t ≤ a.

The proof is complete. For each positive constant k, let Bk = {x ∈ C([−r, a], E) : ‖x‖ ≤ k}. Then Bk is clearly a bounded, closed and convex subset in C([−r, a], E). The following existence results for the nonlocal Cauchy problem (5.1) are based on Krasnoselskii’s fixed-point theorem. Theorem 5.1. If (H1 )–(H5 ) are satisfied, then the nonlocal Cauchy problem (5.1) has a mild solution provided that: Γ(1+β)C1−β H1 qβ (i) ML + (M + 1)|A−β |H1 + βΓ(1+qβ) a < 1, (ii) ML + (M + 1)|A−β |H +

Γ(1+β)C1−β H βΓ(1+qβ)

aqβ < 1.

178 | 5 Fractional neutral evolution equations and inclusions Proof. Define the function υ ∈ C([−r, a], E) such that |υ(t)| ≡ 0, t ∈ [−r, a]. For any positive constant k and x ∈ Bk , in view of (5.12), Lemma 5.2 and the condition (H4 ), for t ∈ [0, a], it follows that 󵄨 󵄨󵄨 󵄨󵄨Sq (t)[φ(0) − (g(xt1 , . . . , xtn ))(0) − h(0, x0 )]󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ≤ M(󵄨󵄨󵄨φ(0)󵄨󵄨󵄨 + L‖x − υ‖ + 󵄨󵄨󵄨g(υt1 , . . . , υtn )(0)󵄨󵄨󵄨 + 󵄨󵄨󵄨A−β Aβ h(0, x0 )󵄨󵄨󵄨) 󵄨 󵄨 󵄩 󵄩 ≤ M[‖φ‖∗ + Lk + 󵄩󵄩󵄩g(υt1 , . . . , υtn )󵄩󵄩󵄩∗ + 󵄨󵄨󵄨A−β 󵄨󵄨󵄨H1 (k + 1)].

(5.14)

For any positive constant k and x ∈ Bk , since xt is continuous in t, according to (H2 ), f (t, xt ) is a measurable function on [0, a]. Direct calculation gives that (t − s)q−1 ∈ 1

L 1−q1 [0, t] for t ∈ [0, a] and q1 ∈ [0, q). Let b=

q−1 ∈ (−1, 0), 1 − q1

M1 = ‖m‖

1

L q1 [0,a]

.

By using Lemma 1.1 (Hölder inequality) and (H3 ), for t ∈ [0, a], we obtain t

t

0

0

q−1 󵄨 󵄨 ∫󵄨󵄨󵄨(t − s)q−1 f (s, xs )󵄨󵄨󵄨ds ≤ (∫(t − s) 1−q1 ds)

1−q1

‖m‖

1

L q1 [0,t]

M1 ≤ a(1+b)(1−q1 ) . (1 + b)1−q1

(5.15)

In view of Lemma 5.2 and (5.15), we get t

󵄨 󵄨 ∫󵄨󵄨󵄨(t − s)q−1 Tq (t − s)f (s, xs )󵄨󵄨󵄨ds 0

t

(5.16)

qM 󵄨 󵄨 ≤ ∫󵄨󵄨(t − s)q−1 f (s, xs )󵄨󵄨󵄨ds Γ(1 + q) 󵄨 0

qM1 M a(1+b)(1−q1 ) , ≤ Γ(1 + q)(1 + b)1−q1

for t ∈ [0, a].

Thus, |(t − s)q−1 Tq (t − s)f (s, xs )| is Lebesgue integrable with respect to s ∈ [0, t] for all t ∈ [0, a]. From Lemma 1.3 (Bochner’s theorem), it follows that (t − s)q−1 Tq (t − s)f (s, xs ) is Bochner integrable with respect to s ∈ [0, t] for all t ∈ [0, a]. Since h ∈ D(Aβ ), the function Aβ h(⋅, z) is strongly measurable for any z ∈ C , t ∈ [0, a], and Aβ h(t, ⋅) satisfies the Lipschitz condition, then Aβ h(s, xs ) is strongly measurable on [0, a]. In addition, in view of the fact that {T(t)}t≥0 is an analytic semigroup, then for t ∈ (0, a] and θ ∈ (0, ∞), the operator function s → (t − s)q−1 AT((t − s)q θ) is continuous in the uniform operator topology in [0, t), and thus (t−s)q−1 ATq (t−s)h(s, xs ) is continuous in [0, t). Applying (5.12) and Lemma 5.5, for any x ∈ Bk , t ∈ [0, a], the

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| 179

following relation holds: t

󵄨 󵄨 ∫󵄨󵄨󵄨(t − s)q−1 ATq (t − s)h(s, xs )󵄨󵄨󵄨ds 0

t

󵄨 󵄨 = ∫󵄨󵄨󵄨(t − s)q−1 A1−β Tq (t − s)Aβ h(s, xs )󵄨󵄨󵄨ds 0

t

≤ ∫(t − s)q−1 0

qΓ(1 + β)C1−β

Γ(1 + qβ)(t −

s)q(1−β)

(5.17)

H1 (k + 1)ds

t

qΓ(1 + β) = C H (k + 1) ∫(t − s)qβ−1 ds Γ(1 + qβ) 1−β 1 0

Γ(1 + β) ≤ C H (k + 1)aqβ . βΓ(1 + qβ) 1−β 1 Thus, |(t − s)q−1 ATq (t − s)h(s, xs )| is Lebesgue integrable with respect to s ∈ [0, t] for all t ∈ [0, a]. From Lemma 1.3 (Bochner’s theorem), it follows that (t−s)q−1 ATq (t−s)h(s, xs ) is Bochner integrable with respect to s ∈ [0, t] for all t ∈ [0, a]. For each positive k, define two operators F1 and F2 on Bk as follows: (F1 x)(t) = Sq (t)[φ(0) − (g(xt1 , . . . , xtn ))(0) − h(0, x0 )] + h(t, xt ) { { { t + ∫0 (t − s)q−1 ATq (t − s)h(s, xs )ds, { { { {(F1 x)(ϑ) = φ(ϑ) − (g(xt1 , . . . , xtn ))(ϑ),

t ∈ [0, a],

ϑ ∈ [−r, 0],

and t

(F2 x)(t) = ∫0 (t − s)q−1 Tq (t − s)f (s, xs )ds, { (F2 x)(ϑ) = 0,

t ∈ [0, a],

ϑ ∈ [−r, 0],

where x ∈ Bk . Obviously, x is a mild solution of (5.1) if and only if the operator equation x = F1 x + F2 x has a solution x ∈ Bk . Therefore, the existence of a mild solution of (5.1) is equivalent to determining a positive constant k0 , such that F1 + F2 has a fixed point on Bk0 . In fact, in view of (i) of Theorem 5.1, by choosing k0 such that 󵄩 󵄩 󵄨 󵄨 󵄨 󵄨 k0 = M[‖φ‖∗ + Lk0 + 󵄩󵄩󵄩g(υt1 , . . . , υtn )󵄩󵄩󵄩∗ + 󵄨󵄨󵄨A−β 󵄨󵄨󵄨H1 (k0 + 1)] + 󵄨󵄨󵄨A−β 󵄨󵄨󵄨H1 (k0 + 1) +

Γ(1 + β)C1−β H1 (k0 + 1) qβ qM1 M a(1+b)(1−q1 ) + a , 1−q 1 βΓ(1 + qβ) Γ(1 + q)(1 + b)

(5.18)

180 | 5 Fractional neutral evolution equations and inclusions we can prove that F1 + F2 has a fixed point on Bk0 . Our proof will be divided into three steps. Step I. We shall show that F1 x + F2 y ∈ Bk0 whenever x, y ∈ Bk0 . For any fixed y ∈ Bk0 and 0 ≤ t ′ < t ′′ ≤ a, we get that 󵄨󵄨 ′′ ′ 󵄨 󵄨󵄨(F2 y)(t ) − (F2 y)(t )󵄨󵄨󵄨

t′ 󵄨󵄨 󵄨󵄨 t ′′ 󵄨󵄨 󵄨󵄨 ′′ q−1 q−1 ′′ 󵄨 = 󵄨󵄨∫(t − s) Tq (t − s)f (s, ys )ds − ∫(t ′ − s) Tq (t ′ − s)f (s, ys )ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨0 0

󵄨󵄨 t ′′ 󵄨󵄨 󵄨󵄨 󵄨󵄨 q−1 ≤ 󵄨󵄨󵄨∫(t ′′ − s) Tq (t ′′ − s)f (s, ys )ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨t ′ 󵄨 t′ 󵄨󵄨 t ′ 󵄨󵄨 󵄨󵄨 ′′ 󵄨󵄨 q−1 q−1 ′′ + 󵄨󵄨󵄨∫(t − s) Tq (t − s)f (s, ys )ds − ∫(t ′ − s) Tq (t ′′ − s)f (s, ys )ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨0 󵄨 0 t′ 󵄨󵄨 t ′ 󵄨󵄨 󵄨󵄨 ′ 󵄨󵄨 q−1 q−1 ′′ + 󵄨󵄨󵄨∫(t − s) Tq (t − s)f (s, ys )ds − ∫(t ′ − s) Tq (t ′ − s)f (s, ys )ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨0 󵄨 0

󵄨󵄨 t ′′ 󵄨󵄨 󵄨󵄨 󵄨󵄨 q−1 = 󵄨󵄨󵄨∫(t ′′ − s) Tq (t ′′ − s)f (s, ys )ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨t ′ 󵄨 󵄨󵄨 t ′ 󵄨󵄨 󵄨󵄨 󵄨󵄨 q−1 q−1 ′′ ′ ′′ 󵄨 + 󵄨󵄨∫[(t − s) − (t − s) ]Tq (t − s)f (s, ys )ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨0 󵄨 󵄨󵄨 t ′ 󵄨󵄨 󵄨󵄨 ′ 󵄨󵄨 q−1 ′′ ′ 󵄨 + 󵄨󵄨∫(t − s) [Tq (t − s) − Tq (t − s)]f (s, ys )ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨0 󵄨 = I1 + I2 + I3 , where 󵄨󵄨 t ′′ 󵄨󵄨 󵄨󵄨 󵄨󵄨 q−1 I1 = 󵄨󵄨󵄨∫(t ′′ − s) Tq (t ′′ − s)f (s, ys )ds󵄨󵄨󵄨, 󵄨󵄨 󵄨󵄨 󵄨t ′ 󵄨 󵄨󵄨 t ′ 󵄨󵄨 󵄨󵄨 󵄨󵄨 q−1 q−1 I2 = 󵄨󵄨󵄨∫[(t ′′ − s) − (t ′ − s) ]Tq (t ′′ − s)f (s, ys )ds󵄨󵄨󵄨, 󵄨󵄨 󵄨󵄨 󵄨0 󵄨 󵄨󵄨 t ′ 󵄨󵄨 󵄨󵄨 󵄨󵄨 q−1 I3 = 󵄨󵄨󵄨∫(t ′ − s) [Tq (t ′′ − s) − Tq (t ′ − s)]f (s, ys )ds󵄨󵄨󵄨. 󵄨󵄨 󵄨󵄨 󵄨0 󵄨

5.1 Nonlocal Cauchy problem

| 181

Therefore, for every y ∈ Bk0 , (F2 y)(t) is continuous in t ∈ [−r, a]. Using the similar argument and (5.12), we can conclude that for every x ∈ Bk0 , (F1 x)(t) is also continuous in t ∈ [−r, a]. For every pair x, y ∈ Bk0 and t ∈ [0, a], by using the similar methods as we did in (5.14)–(5.17) and noting that (5.18), we have 󵄨 󵄨󵄨 󵄨󵄨(F1 x)(t) + (F2 y)(t)󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨Sq (t)[φ(0) − (g(xt1 , . . . , xtn ))(0) − h(0, x0 )]󵄨󵄨󵄨 + 󵄨󵄨󵄨h(t, xt )󵄨󵄨󵄨

󵄨󵄨 t 󵄨󵄨 󵄨󵄨 t 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 + 󵄨󵄨󵄨∫(t − s)q−1 ATq (t − s)h(s, xs )ds󵄨󵄨󵄨 + 󵄨󵄨󵄨∫(t − s)q−1 Tq (t − s)f (s, ys )ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨0 󵄨 󵄨 󵄨0 󵄩󵄩 󵄩󵄩 󵄨󵄨 −β 󵄨󵄨 󵄨󵄨 −β 󵄨󵄨 ≤ M[‖φ‖∗ + Lk0 + 󵄩󵄩g(υt1 , . . . , υtn )󵄩󵄩∗ + 󵄨󵄨A 󵄨󵄨H1 (k0 + 1)] + 󵄨󵄨A 󵄨󵄨H1 (k0 + 1) Γ(1 + β)C1−β H1 (k0 + 1) qβ qM1 M a(1+b)(1−q1 ) + a + 1−q 1 βΓ(1 + qβ) Γ(1 + q)(1 + b)

(5.19)

= k0 .

Noting that M ≥ 1, we have 󵄨󵄨 󵄨 󵄩 󵄩 󵄨󵄨(F1 x)(ϑ) + (F2 y)(ϑ)󵄨󵄨󵄨 ≤ M[‖φ‖∗ + Lk0 + 󵄩󵄩󵄩g(υt1 , . . . , υtn )󵄩󵄩󵄩∗ ] ≤ k0 ,

ϑ ∈ [−r, 0].

Hence, ‖F1 x + F2 y‖ ≤ k0 for every pair x, y ∈ Bk0 . Step II. The operator F1 is a contraction on Bk0 . For any x, y ∈ Bk0 and t ∈ [0, a], according to (5.11), (H4 ), Lemma 5.2 and Lemma 5.5, we have 󵄨󵄨 󵄨 󵄨󵄨(F1 x)(t) − (F1 y)(t)󵄨󵄨󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨Sq (t)[(g(xt1 , . . . , xtn ))(0) − (g(yt1 , . . . , ytn ))(0)]󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 + 󵄨󵄨󵄨Sq (t)[h(0, x0 ) − h(0, y0 )]󵄨󵄨󵄨 + 󵄨󵄨󵄨h(t, xt ) − h(t, yt )󵄨󵄨󵄨 t

󵄨 󵄨 + ∫(t − s)q−1 󵄨󵄨󵄨A1−β Tq (t − s)[Aβ h(s, xs ) − Aβ h(s, ys )]󵄨󵄨󵄨ds 0

Γ(1 + β)C1−β H qβ 󵄨 󵄨 a ‖x − y‖ ≤ ML‖x − y‖ + (M + 1)󵄨󵄨󵄨A−β 󵄨󵄨󵄨H‖x − y‖ + βΓ(1 + qβ) Γ(1 + β)C1−β H qβ 󵄨 󵄨 = (ML + (M + 1)󵄨󵄨󵄨A−β 󵄨󵄨󵄨H + a )‖x − y‖. βΓ(1 + qβ) Noting that M ≥ 1, we have 󵄨󵄨 󵄨 󵄨󵄨(F1 x)(ϑ) − (F1 y)(ϑ)󵄨󵄨󵄨 ≤ ML‖x − y‖,

ϑ ∈ [−r, 0],

182 | 5 Fractional neutral evolution equations and inclusions which implies that Γ(1 + β)C1−β H qβ 󵄨 󵄨 ‖F1 x − F1 y‖ ≤ (ML + (M + 1)󵄨󵄨󵄨A−β 󵄨󵄨󵄨H + a )‖x − y‖. βΓ(1 + qβ) According to (ii) of Theorem 5.1, we get that F1 is a contraction. Step III. The operator F2 is a completely continuous operator. First, we will prove that F2 is continuous on Bk0 . Let {x n } ⊆ Bk0 with x n → x on Bk0 . Then by (H2 ) and the fact that xtn → xt for t ∈ [0, a], we have f (s, xsn ) → f (s, xs ),

a. e. t ∈ [0, a] as n → ∞.

Noting that |f (s, xsn ) − f (s, xs )| ≤ 2m(s), by the dominated convergence theorem, we have 󵄨󵄨 t 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 n q−1 n 󵄨󵄨 (F x )(t) − (F x)(t) = (t − s) T (t − s)[f (s, x ) − f (s, x )]ds ∫ 󵄨󵄨 2 󵄨󵄨 󵄨󵄨 2 q s s 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨0 t

≤

qM 󵄨 󵄨 ∫(t − s)q−1 󵄨󵄨󵄨f (s, xsn ) − f (s, xs )󵄨󵄨󵄨ds → 0, Γ(1 + q)

as n → ∞,

0

which implies that F2 is continuous. Next, we will show that {F2 x, x ∈ Bk0 } is relatively compact. It suffices to show that the family of functions {F2 x, x ∈ Bk0 } is uniformly bounded and equicontinuous, and for any t ∈ [0, a], {(F2 x)(t), x ∈ Bk0 } is relatively compact in E. For any x ∈ Bk0 , we have ‖F2 x‖ ≤ k0 , which means that {F2 x, x ∈ Bk0 } is uniformly bounded. In the following, we will show that {F2 x, x ∈ Bk0 } is a family of equicontinuous functions. For any x ∈ Bk0 and 0 ≤ t ′ < t ′′ ≤ a, we get that |(F2 x)(t ′′ ) − (F2 x)(t ′ )| ≤ I1 + I2 + I3 , where I1 , I2 and I3 are defined as in Step I. By using the analogous argument performed in (5.15) and (5.16), we can conclude that I1 ≤

qM1 M(t ′′ − t ′ )(1+b)(1−q1 ) , Γ(1 + q)(1 + b)1−q1

1−q1

t′

qM q−1 q−1 1 I2 ≤ (∫((t ′ − s) − (t ′′ − s) ) 1−q1 ds) Γ(1 + q) 0

t′

1−q1

qM1 M b b ≤ (∫((t ′ − s) − (t ′′ − s) )ds) Γ(1 + q) 0

‖m‖

1

L q1 [0,t ′ ]

5.1 Nonlocal Cauchy problem

| 183

qM1 M 1+b 1+b 1+b 1−q ((t ′ ) − (t ′′ ) + (t ′′ − t ′ ) ) 1 1−q 1 Γ(1 + q)(1 + b) qM1 M (1+b)(1−q1 ) (t ′′ − t ′ ) . ≤ Γ(1 + q)(1 + b)1−q1 =

For t ′ = 0, 0 < t ′′ ≤ a, it is easy to see that I3 = 0. For t ′ > 0 and ε > 0 small

enough, we have

t ′ −ε

I3 ≤ ∫ (t ′ − s)

q−1 󵄨

󵄨󵄨Tq (t ′′ − s) − Tq (t ′ − s)󵄨󵄨󵄨󵄨󵄨󵄨f (s, xs )󵄨󵄨󵄨ds 󵄨 󵄨󵄨 󵄨

0 t′

q−1 󵄨

+ ∫ (t ′ − s) t ′ −ε

≤

󵄨󵄨Tq (t ′′ − s) − Tq (t ′ − s)󵄨󵄨󵄨󵄨󵄨󵄨f (s, xs )󵄨󵄨󵄨ds 󵄨 󵄨󵄨 󵄨

M1 ((t ′ )1+b − ε1+b )(1−q1 ) (1 + b)1−q1 +

󵄨 󵄨 sup 󵄨󵄨󵄨Tq (t ′′ − s) − Tq (t ′ − s)󵄨󵄨󵄨

s∈[0,t ′ −ε]

2qM1 M ε(1+b)(1−q1 ) . Γ(1 + q)(1 + b)1−q1

The hypothesis (H1 ) and Lemma 5.4 imply the continuity of Tq (t) (t > 0) in t in

the uniform operator topology, it is easy to see that I3 tends to zero independently of

x ∈ Bk0 as t ′′ − t ′ → 0, ε → 0. Thus, |(F2 x)(t ′ ) − (F2 x)(t ′′ )| tends to zero independently

of x ∈ Bk0 as t ′′ − t ′ → 0, which means that {F2 x, x ∈ Bk0 } is equicontinuous.

It remains to prove that for any t ∈ [−r, a], V(t) = {(F2 x)(t), x ∈ Bk0 } is relatively

compact in E.

Obviously, for t ∈ [−r, 0], V(t) is relatively compact in E. Let 0 < t ≤ a be fixed. For

∀ ε ∈ (0, t) and ∀ δ > 0, define an operator Fε,δ on Bk0 by the formula t−ε ∞

(Fε,δ x)(t) = q ∫ ∫ θ(t − s)q−1 ϕq (θ)T((t − s)q θ)f (s, xs )dθds 0 δ

t−ε ∞

= q ∫ ∫ θ(t − s)q−1 ϕq (θ)[T(εq δ)T((t − s)q θ − εq δ)]f (s, xs )dθds 0 δ

t−ε ∞

= T(εq δ)q ∫ ∫ θ(t − s)q−1 ϕq (θ)T((t − s)q θ − εq δ)f (s, xs )dθds, 0 δ

where x ∈ Bk0 . Then from the compactness of T(εq δ) (εq δ > 0), we obtain that the set

Vε,δ (t) = {(Fε,δ x)(t), x ∈ Bk0 } is relatively compact in E for ∀ ε ∈ (0, t) and ∀ δ > 0.

184 | 5 Fractional neutral evolution equations and inclusions Moreover, for every x ∈ Bk0 , we have 󵄨 󵄨󵄨 󵄨󵄨(F2 x)(t) − (Fε,δ x)(t)󵄨󵄨󵄨 󵄨󵄨 t δ 󵄨󵄨 = q󵄨󵄨󵄨∫ ∫ θ(t − s)q−1 ϕq (θ)T((t − s)q θ)f (s, xs )dθds 󵄨󵄨 󵄨0 0 t ∞

+ ∫ ∫ θ(t − s)q−1 ϕq (θ)T((t − s)q θ)f (s, xs )dθds 0 δ

t−ε ∞

󵄨󵄨 󵄨󵄨 − ∫ ∫ θ(t − s)q−1 ϕq (θ)T((t − s)q θ)f (s, xs )dθds󵄨󵄨󵄨 󵄨󵄨 󵄨 0 δ 󵄨󵄨 t δ 󵄨󵄨 󵄨󵄨 󵄨󵄨 q−1 q 󵄨 ≤ q󵄨󵄨∫ ∫ θ(t − s) ϕq (θ)T((t − s) θ)f (s, xs )dθds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨0 0 󵄨

󵄨󵄨 t ∞ 󵄨󵄨 󵄨󵄨 󵄨󵄨 + q󵄨󵄨󵄨 ∫ ∫ θ(t − s)q−1 ϕq (θ)T((t − s)q θ)f (s, xs )dθds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨t−ε δ 󵄨 t

≤ qM(∫(t − s) 0

q−1 1−q1

t

+ qM( ∫ (t − s)

1−q1

ds) q−1 1−q1

δ

‖m‖

1 L q1

[0,t]

1−q1

ds)

t−ε

∫ θϕq (θ)dθ 0 ∞

‖m‖

1

L q1 [t−ε,t]

∫ θϕq (θ)dθ 0

δ

≤

qM1 Ma(1+b)(1−q1 ) qM1 M ε(1+b)(1−q1 ) . ∫ θϕq (θ)dθ + (1 + b)1−q1 Γ(1 + q)(1 + b)1−q1 0

Therefore, there are relatively compact sets arbitrarily close to the set V(t), t > 0.

Hence, the set V(t), t > 0 is also relatively compact in E.

Therefore, {F2 x, x ∈ Bk0 } is relatively compact by the Arzela–Ascoli theorem. Thus,

the continuity of F2 and relatively compactness of {F2 x, x ∈ Bk0 } imply that F2 is a completely continuous operator. Hence, Krasnoselskii’s fixed-point theorem shows that F1 + F2 has a fixed point on Bk0 . Therefore, the nonlocal Cauchy problem (5.1) has a mild solution. The proof is complete.

In the following, we give an existence result in the case where (H4 ) is not satisfied.

We need the following assumption.

(H4 )′ The function g is completely continuous, and there exist positive constants L1 , L1 ′ such that ‖g(xt1 , . . . , xtn )‖∗ ≤ L1 ‖x‖ + L1 ′ for all x ∈ C([−r, a], E).

5.1 Nonlocal Cauchy problem

| 185

Theorem 5.2. If assumptions (H1 )–(H3 ), (H4 )′ and (H5 ) are satisfied, then the nonlocal Cauchy problem (5.1) has a mild solution provided that: Γ(1+β)C1−β H1 qβ a < 1, (i) ML1 + (M + 1)|A−β |H1 + βΓ(1+qβ) (ii) (M + 1)|A−β |H +

Γ(1+β)C1−β H βΓ(1+qβ)

aqβ < 1.

Proof. For any positive constant k and x ∈ Bk , according to (5.12), (H4 )′ and Lemma 5.2,

it follows that

󵄨󵄨 󵄨 󵄨󵄨Sq (t)[φ(0) − (g(xt1 , . . . , xtn ))(0) − h(0, x0 )]󵄨󵄨󵄨 󵄨 󵄨 ≤ M[‖φ‖∗ + L1 k + L1 ′ + 󵄨󵄨󵄨A−β 󵄨󵄨󵄨H1 (k + 1)].

(5.20)

Therefore, the function Sq (t)[φ(0) − (g(xt1 , . . . , xtn ))(0) − h(0, x0 )] exists. According to (5.15)–(5.17), for x ∈ Bk , the functions (t − s)q−1 Tq (t − s)f (s, xs ) and (t − s)q−1 ATq (t − s)h(s, xs ) are Bochner integrable with respect to s ∈ [0, t] for all t ∈ [0, a].

For each positive k, define two operators G1 and G2 on Bk as follows: t

(G1 x)(t) = h(t, xt ) − Sq (t)h(0, x0 ) + ∫0 (t − s)q−1 ATq (t − s)h(s, xs )ds, t ∈ [0, a],

{

(G1 x)(ϑ) = 0,

ϑ ∈ [−r, 0],

and (G2 x)(t) = Sq (t)[φ(0) − (g(xt1 , . . . , xtn ))(0)] { { { t + ∫0 (t − s)q−1 Tq (t − s)f (s, xs )ds, { { { {(G2 x)(ϑ) = φ(ϑ) − (g(xt1 , . . . , xtn ))(ϑ),

t ∈ [0, a],

ϑ ∈ [−r, 0],

where x ∈ Bk . In view of (i) of Theorem 5.2, we can choose k1 such that 󵄨 󵄨 󵄨 󵄨 k1 = M[‖φ‖∗ + L1 k1 + L′1 + 󵄨󵄨󵄨A−β 󵄨󵄨󵄨H1 (k1 + 1)] + 󵄨󵄨󵄨A−β 󵄨󵄨󵄨H1 (k1 + 1) +

Γ(1 + β)C1−β H1 (k1 + 1) qβ qM1 M a . a(1+b)(1−q1 ) + 1−q βΓ(1 + qβ) Γ(1 + q)(1 + b) 1

(5.21)

In the following, we will prove that F has a fixed point on Bk1 . Our proof will be

divided into three steps.

Step I. We shall show that G1 x + G2 y ∈ Bk1 whenever x, y ∈ Bk1 .

Obviously, for every pair x, y ∈ Bk1 , (G1 x)(t) and (G2 y)(t) are continuous in t ∈

[−r, a]. For every pair x, y ∈ Bk1 and t ∈ [0, a], by using (5.20), (5.21) and similar meth-

186 | 5 Fractional neutral evolution equations and inclusions ods as we did in (5.15)–(5.17), we have 󵄨 󵄨󵄨 󵄨󵄨(G1 x)(t) + (G2 y)(t)󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨Sq (t)[φ(0) − (g(xt1 , . . . , xtn ))(0) − h(0, x0 )]󵄨󵄨󵄨 + 󵄨󵄨󵄨h(t, xt )󵄨󵄨󵄨

󵄨󵄨 t 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨󵄨 t 󵄨󵄨 󵄨󵄨 󵄨 󵄨 q−1 󵄨 + 󵄨󵄨∫(t − s) ATq (t − s)h(s, xs )ds󵄨󵄨󵄨 + 󵄨󵄨󵄨∫(t − s)q−1 Tq (t − s)f (s, xs )ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨0 󵄨0 󵄨 󵄨 󵄨 󵄨 󵄨 ≤ M[‖φ‖∗ + L1 k1 + L′1 + 󵄨󵄨󵄨A−β 󵄨󵄨󵄨H1 (k1 + 1)] + 󵄨󵄨󵄨A−β 󵄨󵄨󵄨H1 (k1 + 1) Γ(1 + β)C1−β H1 (k1 + 1) qβ qM1 M + a a(1+b)(1−q1 ) + βΓ(1 + qβ) Γ(1 + q)(1 + b)1−q1

(5.22)

= k1 .

Noting that M ≥ 1, we have 󵄨󵄨 󵄨 ′ 󵄨󵄨(G1 x)(ϑ) + (G2 y)(ϑ)󵄨󵄨󵄨 ≤ M[‖φ‖∗ + L1 k1 + L1 ] ≤ k1 ,

ϑ ∈ [−r, 0].

Hence, ‖G1 x + G2 y‖ ≤ k1 for every pair x, y ∈ Bk1 . Step II. The operator G1 is a contraction on Bk1 . For any x, y ∈ Bk1 and t ∈ [0, a], according to (5.11), we have 󵄨󵄨 󵄨 󵄨󵄨(G1 x)(t) − (G1 y)(t)󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨Sq (t)[h(0, x0 ) − h(0, y0 )]󵄨󵄨󵄨 + 󵄨󵄨󵄨h(t, xt ) − h(t, yt )󵄨󵄨󵄨 t

󵄨 󵄨 + ∫(t − s)q−1 󵄨󵄨󵄨A1−β Tq (t − s)[Aβ h(s, xs ) − Aβ h(t, ys )]󵄨󵄨󵄨ds 0

Γ(1 + β)C1−β H qβ 󵄨 󵄨 a ‖x − y‖ ≤ (M + 1)󵄨󵄨󵄨A−β 󵄨󵄨󵄨H‖x − y‖ + βΓ(1 + qβ) Γ(1 + β)C1−β H qβ 󵄨 󵄨 ≤ ((M + 1)󵄨󵄨󵄨A−β 󵄨󵄨󵄨H + a )‖x − y‖, βΓ(1 + qβ) which implies Γ(1 + β)C1−β H qβ 󵄨 󵄨 ‖G1 x − G1 y‖ ≤ ((M + 1)󵄨󵄨󵄨A−β 󵄨󵄨󵄨H + a )‖x − y‖. βΓ(1 + qβ) According to (ii) of Theorem 5.2, we get that G1 is a contraction. Step III. The operator G2 is a completely continuous operator. First, we will show that {G2 x, x ∈ Bk1 } is equicontinuous. For any x ∈ Bk1 and 0 ≤ t ′ < t ′′ ≤ a, we get that 󵄨󵄨 󵄨 󵄨 ′′ ′ 󵄨 ′′ ′ 󵄨󵄨(G2 x)(t ) − (G2 x)(t )󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨[Sq (t ) − Sq (t )][φ(0) − (g(xt1 , . . . , xtn ))(0)]󵄨󵄨󵄨 + I1 + I2 + I3 ,

5.1 Nonlocal Cauchy problem

| 187

where I1 , I2 and I3 are defined as in the proof of Theorem 5.1. According to the strong continuity of {Sq (t)}t≥0 and (H4 )′ , we know that |(G2 x)(t ′ ) − (G2 x)(t ′′ )| tends to zero independently of x ∈ Bk1 as t ′′ − t ′ → 0, which means that {G2 x, x ∈ Bk1 } is equicontinuous. It remains to prove that for t ∈ [−r, a], the set {(G2 x)(t), x ∈ Bk1 } is relatively compact in E. Obviously, for t ∈ [−r, 0], the set {(G2 x)(t), x ∈ Bk1 } is relatively compact in E by (H4 )′ . According to the argument of Theorem 5.1, we only need to prove that for any t ∈ (0, a], the set V ′ (t) = {Sq (t)[φ(0) − (g(xt1 , . . . , xtn ))(0)], x ∈ Bk1 } is relatively compact in E. In fact, the set V ′ (t), t > 0 is relatively compact in E according to (H1 ) and Lemma 5.4. Moreover, {G2 x, x ∈ Bk1 } is uniformly bounded by (5.22). Therefore, {G2 x, x ∈ Bk1 } is relatively compact by the Arzela–Ascoli theorem. Using the similar argument as that we did in the proof of Theorem 5.1, we know that G2 is continuous on Bk1 by (H2 ) and (H4 )′ . Thus, G2 is a completely continuous operator. Hence, Krasnoselskii’s fixed-point theorem shows that G1 + G2 has a fixed point on Bk1 , which means that the nonlocal Cauchy problem (5.1) has a mild solution. The proof is complete. The following existence and uniqueness result for the nonlocal Cauchy problem (5.1) is based on Banach contraction principle. We will need the following assumptions: (H6 ) f (t, xt ) is strongly measurable for any x ∈ C([−r, a], Bk ) and almost all t ∈ [0, a], 1

(H7 ) there exist a constant q2 ∈ [0, q) and ρ ∈ L q2 ([0, a], ℝ+ ) such that for any x, y ∈ C([−r, a], Bk ), we have |f (t, xt ) − f (t, yt )| ≤ ρ(t)‖x − y‖, t ∈ [0, a], where k is a positive constant. Theorem 5.3. Assume that (H3 )–(H7 ) are satisfied. If (i) of Theorem 5.1 holds, then the nonlocal Cauchy problem (5.1) has a unique mild solution provided that Γ(1 + β)C1−β Haqβ qM2 Ma(1+b )(1−q2 ) 󵄨 󵄨 ML + (M + 1)󵄨󵄨󵄨A−β 󵄨󵄨󵄨H + + < 1, βΓ(1 + qβ) Γ(1 + q)(1 + b′ )1−q2 ′

where b′ =

q−1 1−q2

∈ (−1, 0), M2 = ‖ρ‖

(5.23)

1

L q2 [0,a].

Proof. It is easy to see that Sq (t)[φ(0) − (g(xt1 , . . . , xtn ))(0) − h(0, x0 )] exists, (t − s)q−1 ATq (t − s)h(s, xs ) and (t − s)q−1 Tq (t − s)f (s, xs ) are Bochner integrable with respect to s ∈ [0, t] for all t ∈ [0, a]. For x ∈ Bk , define the operator F on Bk by (Fx)(t) = Sq (t)[φ(0) − (g(xt1 , . . . , xtn ))(0) − h(0, x0 )] + h(t, xt ) { { { { t { { + ∫0 (t − s)q−1 ATq (t − s)h(s, xs )ds t { { + ∫0 (t − s)q−1 Tq (t − s)f (s, xs )ds, t ∈ [0, a], { { { { ϑ ∈ [−r, 0]. {(Fx)(ϑ) = −(g(xt1 , . . . , xtn ))(ϑ) + φ(ϑ),

188 | 5 Fractional neutral evolution equations and inclusions Obviously, it is sufficient to prove that F has a unique fixed point on Bk0 , where k0

is defined as in (5.18).

According to (5.19), we know that F is an operator from Bk0 into itself. For any

x, y ∈ Bk0 and t ∈ [0, a], according to (H4 ), (H5 ) and (H7 ), we have 󵄨 󵄨󵄨 󵄨󵄨(Fx)(t) − (Fy)(t)󵄨󵄨󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨Sq (t)[(g(xt1 , . . . , xtn ))(0) − (g(yt1 , . . . , ytn ))(0)]󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 + 󵄨󵄨󵄨Sq (t)[h(0, x0 ) − h(0, y0 )]󵄨󵄨󵄨 + 󵄨󵄨󵄨h(t, xt ) − h(t, yt )󵄨󵄨󵄨 t

󵄨 󵄨 + ∫(t − s)q−1 󵄨󵄨󵄨A1−β Tq (t − s)[Aβ h(s, xs ) − Aβ h(s, ys )]󵄨󵄨󵄨ds 0

t

󵄨 󵄨 + ∫(t − s)q−1 󵄨󵄨󵄨Tq (t − s)[f (s, xs ) − f (s, ys )]󵄨󵄨󵄨ds 0

Γ(1 + β)C1−β H qβ 󵄨 󵄨 ≤ ML‖x − y‖ + (M + 1)󵄨󵄨󵄨A−β 󵄨󵄨󵄨H‖x − y‖ + a ‖x − y‖ βΓ(1 + qβ) t

qM + ∫(t − s)q−1 ρ(s)‖x − y‖ds Γ(1 + q) 0

Γ(1 + β)C1−β H qβ 󵄨 󵄨 ≤ ML‖x − y‖ + (M + 1)󵄨󵄨󵄨A−β 󵄨󵄨󵄨H‖x − y‖ + a ‖x − y‖ βΓ(1 + qβ) t

+

q−1 qM (∫(t − s) 1−q2 ds) Γ(1 + q)

1−q2

‖ρ‖

0

1

L q2 [0,t]

‖x − y‖

Γ(1 + β)C1−β Haqβ qM2 Ma(1+b )(1−q2 ) 󵄨 󵄨 )‖x − y‖, ≤ (ML + (M + 1)󵄨󵄨󵄨A−β 󵄨󵄨󵄨H + + βΓ(1 + qβ) Γ(1 + q)(1 + b′ )1−q2 ′

which means that F is a contraction according to (5.23). By applying the Banach contraction principle, we know that F has a unique fixed point on Bk0 . The proof is com-

plete.

Theorem 5.4. Assume that assumptions (H1 ), (H3 ), (H4 )′ and (H5 )–(H7 ) are satisfied.

If (i) of Theorem 5.2 holds, then the nonlocal Cauchy problem (5.1) has a mild solution provided that

󵄨 󵄨 (M + 1)󵄨󵄨󵄨A−β 󵄨󵄨󵄨H +

Γ(1 + β)C1−β Haqβ βΓ(1 + qβ)

qM2 Ma(1+b )(1−q2 ) < 1. Γ(1 + q)(1 + b′ )1−q2 ′

+

5.1 Nonlocal Cauchy problem

| 189

Proof. In fact, for each positive k, define two operators U1 and U2 on Bk as follows: (U1 x)(t) = −Sq (t)h(0, x0 ) + h(t, xt ) { { { { t { { + ∫0 (t − s)q−1 ATq (t − s)h(s, xs )ds t { { + ∫0 (t − s)q−1 Tq (t − s)f (s, xs )ds, { { { { {(U1 x)(ϑ) = 0,

t ∈ [0, a],

ϑ ∈ [−r, 0],

and {

(U2 x)(t) = Sq (t)[φ(0) − (g(xt1 , . . . , xtn ))(0)], t ∈ [0, a], (U2 x)(ϑ) = φ(ϑ) − (g(xt1 , . . . , xtn ))(ϑ),

ϑ ∈ [−r, 0].

According to the arguments above, we can easily get that U1 x + U2 y ∈ Bk1 whenever x, y ∈ Bk1 , where k1 is defined as in (5.21). Furthermore, we can obtain that U1 is a contraction on Bk1 according to (H3 ) and (H5 )–(H7 ) and U2 is a completely continuous operator according to (H1 ) and (H4 )′ . Hence, Krasnoselskii’s fixed-point theorem shows that U1 + U2 has a fixed point on Bk1 , which means that the nonlocal Cauchy problem (5.1) has a mild solution and this completes the proof.

5.1.3 An example Let E = L2 ([0, π], ℝ). Consider the following fractional partial differential equations: π

𝜕q (u(t, z) − ∫0 U(z, y)ut (ϑ, y)dy) = 𝜕z2 u(t, z) + 𝜕z G(t, ut (ϑ, z)), { { { t u(t, 0) = u(t, π) = 0, { { { π n {u(ϑ, z) + ∑i=0 ∫0 k(z, y)uti (ϑ, y)dy = (φ(ϑ))(z),

t ∈ (0, a], t ∈ [0, a],

(5.24)

ϑ ∈ [−r, 0],

where 𝜕tq is Caputo fractional partial derivative of order 0 < q < 1, a > 0, z ∈ [0, π], G is a given function, n is a positive integer, 0 < t0 < t1 < ⋅ ⋅ ⋅ < tn < a, φ ∈ C([−r, 0], E), that is, φ(ϑ) ∈ E = L2 ([0, π], ℝ), k(z, y) ∈ L2 ([0, π] × [0, π], ℝ) and ut (ϑ, z) = u(t + ϑ, z), t ∈ [0, a], ϑ ∈ [−r, 0]. We define an operator A by Av = −v′′ with the domain D(A) = {v(⋅) ∈ E : v, v′ absolutely continuous, v′′ ∈ E, v(0) = v(π) = 0}. Then −A generates a strongly continuous semigroup {T(t)}t≥0 which is compact, analytic and self-adjoint. Furthermore, −A has a discrete spectrum, the eigenvalues are −n2 , n ∈ ℕ, with corresponding normalized eigenvectors un (z) = (2/π)1/2 sin(nz). We also use the following properties: −n2 t (i)′ for each v ∈ E, T(t)v = ∑∞ ⟨v, un ⟩un . In particular, T(⋅) is a uniformly stable n=1 e semigroup and ‖T(t)‖L2 [0,π] ≤ e−t ,

190 | 5 Fractional neutral evolution equations and inclusions 1

1

−2 for each v ∈ E, A− 2 v = ∑∞ n=1 1/n⟨v, un ⟩un . In particular, ‖A ‖L2 [0,π] = 1,

(ii)′

1

(iii)′ the operator A 2 is given by

∞

1

A 2 v = ∑ n⟨v, un ⟩un n=1

1

on the space D(A 2 ) = {v(⋅) ∈ E, ∑∞ n=1 n⟨v, un ⟩un ∈ E}. Clearly, (5.2), (5.3) and (H1 ) are satisfied.

The system (5.24) can be reformulated as the following nonlocal Cauchy problem

in E:

C q D (xt − h(t, xt )) + Ax(t) = f (t, xt ), {0 t x0 (ϑ) + (g(xt1 , . . . , xtn ))(ϑ) = φ(ϑ),

t ∈ (0, a],

ϑ ∈ [−r, 0],

where xt = ut (ϑ, ⋅), that is, (x(t + ϑ))(z) = u(t + ϑ, z), t ∈ [0, a], z ∈ [0, π], ϑ ∈ [−r, 0]. The function h : [0, a] × C → E is given by

π

(h(t, xt ))(z) = ∫ U(z, y)ut (ϑ, y)dy. 0 π

Let (Uh v)(z) = ∫0 U(z, y)v(y)dy, for v ∈ E = L2 ([0, π], ℝ), z ∈ [0, π]. The function f : [0, a] × C → E is given by

(f (t, xt ))(z) = 𝜕z G(t, ut (ϑ, z)), and the function g : C n → C is given by n

(g(xt1 , . . . , xtn ))(ϑ) = ∑ Kg xti (ϑ), i=0

π

where (Kg v)(z) = ∫0 k(z, y)v(y)dy, for v ∈ E = L2 ([0, π], ℝ), z ∈ [0, π]. 1 We can take q = 1/2 and f (t, xt ) = t 1/3 sin xt , then (H2 ), (H3 ), (H6 ) and (H7 ) are π

π

satisfied. Furthermore, assume that L = L1 = (n + 1)[∫0 ∫0 k 2 (z, y)dydz]1/2 . Then (H4 )

and (H4 )′ are satisfied (noting that Kg : E → E is completely continuous). In fact, for

5.1 Nonlocal Cauchy problem

| 191

v1 , v2 ∈ E, we have π

2

π

1/2

‖Kg v1 − Kg v2 ‖L2 [0,π] = (∫(∫ k(z, y)(v1 (y) − v2 (y))dy) dz) 0

0

π

π

π

2

2

1/2

≤ (∫(∫ k (z, y)dy ∫[v1 (y) − v2 (y)] dy)dz) 0

0

π π

0 2

1/2

= (∫ ∫ k (z, y)dydz) ‖v1 − v2 ‖L2 [0,π] . 0 0

Moreover, we assume that the following conditions hold: (a) the function U(z, y), z, y ∈ [0, π] is measurable and π π

∫ ∫ U 2 (z, y)dydz < ∞, 0 0

(b) the function 𝜕z U(z, y) is measurable, U(0, y) = U(π, y) = 0, and let π π

2

1 2

H = (∫ ∫(𝜕z U(z, y)) dydz) < ∞. 0 0

From (a) it is clear that Uh is a bounded linear operator on E. Furthermore, Uh (v) ∈ 1 1 D(A 2 ), and ‖A 2 Uh ‖L2 [0,π] < ∞. In fact, from the definition of Uh and (b) it follows that π

π

⟨Uh (v), un ⟩ = ∫ un (z)(∫ U(z, y)v(y)dy)dz 0

=

1 2

0

1 2 ( ) ⟨U(v), cos(nz)⟩, n π

where U is defined by π

(U(v))(z) = ∫ 𝜕z U(z, y)v(y)dy. 0

From (b), we know that U : E → E is a bounded linear operator with ‖U‖L2 [0,π] ≤ H. 1

Hence, we can write ‖A 2 Uh (v)‖L2 [0,π] = ‖U(v)‖L2 [0,π] , which implies that (5.12) holds. Obviously, (5.11) holds according to (b). Hence, according to Theorems 5.1 and 5.2, system (5.24) has a mild solution provided that (i) and (ii) of Theorems 5.1 and 5.2 hold. From Theorem 5.3, system (5.24)

192 | 5 Fractional neutral evolution equations and inclusions admits a unique mild solution provided that (i) of Theorem 5.3 and (5.23) hold. From Theorem 5.4, system (5.24) has a mild solution provided that the inequalities in Theorem 5.4 hold.

5.2 Topological properties of solution sets 5.2.1 Introduction Stochastic differential inclusions play an important role in characterizing many social, physical, biological and engineering problems, see, e. g., Gawarecki and Mandrekar [103], Kisielewicz [134], and Prato and Zabczyk [77]. Neutral stochastic differential equations and inclusions can be used to describe some systems such as aeroelasticity, the lossless transmission lines, stabilization of lumped control systems and theory of heat conduction in materials with fading memory when noise or stochastic perturbation are taken into account, for details; see Hernández and Henriquez [123], Mahmudov [156] and the references therein. Toufik [195] obtained the existence of mild solutions for the fractional stochastic evolution inclusions. Zhou [226] derived topological properties of solution sets for fractional stochastic evolution inclusions. An important aspect of such structure is the Rδ -property, i. e., an Rδ -set is acyclic (in particular, nonempty, compact and connected) and may not be a singleton but, from the point of view of algebraic topology, it is equivalent to a point in the sense that it has the same homology group as one point space. There has been a great interest in the study of topological structure of solution sets, for instance, see Andres and Pavlačková [16], Bothe [45], Bressan and Wang [46], Chen et al. [66], De Blasi and Myjak [81], Deimling [83], Gabor and Quincampoix [102], Górniewicz and Pruszko [110], Staicu [188], and Wang et al. [206, 208] and the references cited therein. In this section, we consider the following problem of fractional stochastic evolution inclusions in Hilbert spaces: q

C D [x(t) − h(t, xt )] ∈ Ax(t) + Σ(t, xt ) dW(t) , dt {0 t x(t) = ϕ(t), q

t ∈ [0, b],

t ∈ [−τ, 0],

(5.25)

where C0Dt is Caputo fractional derivative of order q ∈ ( 21 , 1), A is the infinitesimal generator of a strongly continuous semigroup {T(t) : t ≥ 0} in a Hilbert space H with inner product (⋅, ⋅) and norm | ⋅ |, h : J × C([−τ, 0], H) → H, Σ : J × C([−τ, 0], H) —∘ B(K, H) is a nonempty, bounded, closed and convex multimap, {W(t) : t ≥ 0} is a given K-valued Brownian motion or Wiener process with a finite trace nuclear covariance operator Q ≥ 0. Here, C([−τ, 0], H) is the space of all continuous functions from [−τ, 0] to H equipped with the norm ‖c‖2∗ = supθ∈[−τ,0] E|c(θ)|2 , K is a Hilbert space with inner product (⋅, ⋅)K and norm | ⋅ |K , B(K, H) denotes the Banach space of all bounded linear operators from K to H.

5.2 Topological properties of solution sets | 193

The study of inclusions (5.25) constitutes an important area of research. However, this topic is relatively less developed and needs to be explored further [13]. To the best of our knowledge, the investigation of topological properties of the solution set for (5.25) is yet to addressed. In this section, our objective is to establish that the solution set for the inclusions (5.25) is a nonempty compact Rδ -set. The rest of the section is organized as follows. Subsection 5.2.2 contains some preliminary material on differential inclusions and the notation to be followed in the sequel, while Subsection 5.2.3 presents the concept of mild solutions for factional neutral stochastic inclusions. In Subsection 5.2.4, we show that the solution set for the inclusions problem (5.25) is nonempty compact and discuss its Rδ -structure. Finally, we illustrate the theory with the aid of an example. This section is based on [233].

5.2.2 Preliminaries Let H, K be real separable Hilbert spaces, and (Ω, F , ℙ) be a complete probability space equipped with a normal filtration Ft , t ∈ [0, b] satisfying the usual conditions (that is, right continuous and F0 contains all ℙ-null sets). We consider Q-Wiener process on (Ω, F , ℙ) with the linear bounded covariance operator Q satisfying the condition Tr(Q) < ∞. We assume that there exist a complete orthonormal system {en }n≥1 on K, a bounded sequence of nonnegative real numbers {λn } such that Qen = λn en , n = 1, 2, . . . and a sequence {Wn }n≥1 of independent Brownian motions such that ∞

(W(t), e)K = ∑ √λn (en , e)K Wn (t), n=1

e ∈ K, t ∈ [0, b],

and Ft = Ftω , where Ftω is the sigma algebra generated by {W(s) : 0 ≤ s ≤ t}. Let 1

1

L02 = L2 (Q 2 K; H) be the space of all Hilbert–Schmidt operators from Q 2 K to H with the inner product (Ψ, ϒ)L0 = tr[ΨQϒ∗ ]. 2

Denote by L2 (Ω; H) the Banach space of all Fb -measurable square integrable random variables with values in H with the norm ‖ ⋅ ‖. Let C [−τ, b] be a subspace of all continuous H-valued stochastic processes x ∈ C([−τ, b]; L2 (Ω; H)) endowed with the norm 1

󵄨 󵄨2 2 ‖x‖C [−τ,b] = ( sup E 󵄨󵄨󵄨x(t)󵄨󵄨󵄨 ) . t∈[−τ,b]

We assume that A is the infinitesimal generator of an analytic semigroup {T(t) : t ≥ 0} of uniformly bounded linear operators on H. Let 0 ∈ ρ(A), where ρ(A) is the resolvent set of A. Under these conditions, it is possible to define the fractional power Aβ , 0 < β ≤ 1, as a closed linear operator on its domain D(Aβ ). For analytic semigroup {T(t) : t ≥ 0}, the following properties will be used:

194 | 5 Fractional neutral evolution equations and inclusions (i) there exists M ≥ 1 such that M := supt≥0 ‖T(t)‖ < ∞; (ii) for any β ∈ (0, 1], there exists a positive constant Cβ such that 󵄩 Cβ 󵄩󵄩 β 󵄩󵄩A T(t)󵄩󵄩󵄩 ≤ β , t

0 < t ≤ b.

Let P(H) stand for the collection of all nonempty subsets of H. As usual, we denote Pcp (H) = {D ∈ P(H) : compact}, Pcl,cv (H) = {D ∈ P(H) : closed and convex}, Pcp,cv (H) = {D ∈ P(H) : compact and convex}, co(D) (resp., co(D)) be the convex hull (resp., convex closed hull in D) of a subset D. Definition 5.2. A subset D of a metric space is called an Rδ -set if there exists a decreasing sequence {Dn } of compact and contractible sets such that ∞

D = ⋂ Dn . n=1

We need the following known results in the forthcoming analysis. Lemma 5.6 ([129]). Let H be a Hilbert space and φ : H → P(H) be a closed quasicompact multimap with compact values. Then φ is u. s. c.. Lemma 5.7 ([45]). Let φ : D ⊂ H —∘ P(H) be a multimap with weakly compact convex values. Then φ is weakly u. s. c. if and only if {xn } ⊂ D with xn → x0 ∈ D and yn ∈ φ(xn ) implies yn ⇀ y0 ∈ φ(x0 ), up to a subsequence. Theorem 5.5 ([232]). Let D be a bounded convex closed subset of Banach space X. Let φ1 : D → X be a single-valued map and φ2 : D —∘ Pcp,cv (X) be a multimap such that φ1 (x) + φ2 (x) ∈ P(D) for x ∈ D. In addition, it is assumed that: (a) φ1 is a contraction with the contraction constant k < 21 , and (b) φ2 is u. s. c. and compact.

Then the fixed-point set Fix(φ1 + φ2 ) := {x : x ∈ φ1 (x) + φ2 (x)} is a nonempty compact set.

5.2.3 Statement of Problem To study the fractional stochastic evolution inclusions (5.25), we assume that (H1 ) the function h : [0, b]×C([−τ, 0]; H) → H is continuous and there exist a constant 1 β ∈ ( 2q , 1) and d, d1 > 0 with √2d(‖A−β ‖2 +

Mβ2

2qβ−1

b2qβ )
0 and (t, c) ∈ [0, b] \ E × C([−τ, 0]; H), there exists N > 0 such that for all n ≥ N, ⟨x ∗ , Σn (t, c)⟩ ⊂ ⟨x ∗ , Σ(t, c)⟩ + (−ϵ, ϵ); (iv) Σn (t, ⋅) : C([−τ, 0]; H) → Pcl,cv (L02 ) is continuous for a. e. t ∈ [0, b] with respect to Hausdorff metric for each n ≥ 1; (v) for each n ≥ 1, there exists a selection σ̃ n : [0, b] × C([−τ, 0]; H) → L02 of Σn such that σ̃ n (⋅, c) is measurable for each C([−τ, 0]; H) and for any compact subset D ⊂ H

196 | 5 Fractional neutral evolution equations and inclusions there exist constants CV > 0 and δ > 0 for which the estimate 󵄩2 󵄩 E 󵄩󵄩󵄩σ̃ n (t, c1 ) − σ̃ n (t, c2 )󵄩󵄩󵄩L0 ≤ CV α(t)‖c1 − c2 ‖2∗ 2

holds for a. e. t ∈ [0, b] and each c1 , c2 ∈ V with V := D + Bδ (0); (vi) Σn verifies the condition (H2 )(i) with Σn instead of Σ for each n ≥ 1. Let us first introduce two families of operators on H: ∞

Sq (t) = ∫ ξq (θ)T(t q θ)dθ,

for t ≥ 0,

0 ∞

Kq (t) = ∫ qθξq (θ)T(t q θ)dθ,

for t ≥ 0,

0

where ξq (θ) =

1 ∞ Γ(1 + qn) sin(nπq), ∑ (−θ)n−1 πq n=1 n!

θ ∈ (0, +∞).

Definition 5.3. A stochastic process x ∈ C [−τ, b] is said to be a mild solution of the problem (5.25) if x(t) = ϕ(t) for t ∈ [−τ, 0] and there exists σ(t) ∈ L2 ([0, b]; L02 ) such that σ(t) ∈ Σ(t, xt ) for a. e. t ∈ [0, b], and x satisfies the following integral equation: t

x(t) = Sq (t)[ϕ(0) − h(0, ϕ)] + h(t, xt ) + ∫(t − s)q−1 AKq (t − s)h(s, xs )ds 0

t

+ ∫(t − s)q−1 Kq (t − s)σ(s)dW(s),

t ∈ [0, b].

0

Lemma 5.10 ([225]). The operators Sq (t) and Kq (t) have the following properties: (i) for each fixed t ≥ 0, Sq (t) and Kq (t) are linear and bounded operators, i. e., for any x ∈ H, 󵄨󵄨 󵄨 󵄨󵄨Sq (t)x󵄨󵄨󵄨 ≤ M|x| and

󵄨󵄨 󵄨 M|x| ; 󵄨󵄨Kq (t)x 󵄨󵄨󵄨 ≤ Γ(q)

(ii) {Sq (t) : t ≥ 0} and {Kq (t) : t ≥ 0} are strongly continuous; (iii) {Sq (t) : t > 0} is compact if {T(t) : t > 0} is compact;

5.2 Topological properties of solution sets | 197

(iv) for any x ∈ H, β ∈ (0, 1), we have AKq (t)x = A1−β Kq (t)Aβ x and Mβ 󵄩 󵄩󵄩 1−β 󵄩󵄩A Kq (t)󵄩󵄩󵄩 ≤ q(1−β) , t qC1−β Γ(1+β) . Γ(1+qβ)

where Mβ =

Remark 5.2. For any x ∈ C [−τ, b], define a solution multioperator F : C [−τ, b] → P(C [−τ, b]) as follows: F = F1 (x) + F2 (x),

where t

−Sq (t)h(0, ϕ) + h(t, xt ) + ∫0 (t − s)q−1 AKq (t − s)h(s, xs )ds,

F1 (x)(t) = {

0,

S(σ)(t),

F2 (x)(t) = {y(t) ∈ C [−τ, b], y(t) = {

ϕ(t),

σ ∈ SelΣ (x), t ∈ [0, b],

t ∈ [−τ, 0],

t ∈ [0, b],

t ∈ [−τ, 0],

},

and the operator S : L2 ([0, b]; L02 ) → C [−τ, b] is defined by t

S(σ) = Sq (t)ϕ(0) + ∫(t − s)q−1 Kq (t − s)σ(s)dW(s). 0

Observe that the fixed points of the multioperator F are mild solutions of the problem (5.25). Lemma 5.11. Let D be a bounded set of C [−τ, b]. If (H1 ) holds, then {Φ(x)(t) : x ∈ D} is equicontinuous on [0, b], where t

Φ(x)(t) = ∫(t − s)q−1 AKq (t − s)h(s, xs )ds,

t ∈ [0, b].

0

Proof. For each x ∈ D with 0 ≤ t1 < t2 ≤ b, we obtain 󵄨 󵄨2 E 󵄨󵄨󵄨Φ(x)(t2 ) − Φ(x)(t1 )󵄨󵄨󵄨 t2

󵄨 󵄨2 ≤ 3(t2 − t1 ) ∫(t2 − s)2(q−1) E 󵄨󵄨󵄨A1−β Kq (t2 − s)Aβ h(s, xs )󵄨󵄨󵄨 ds t1

t1

2 󵄨 󵄨2 + 3t1 ∫((t2 − s)q−1 − (t1 − s)q−1 ) E 󵄨󵄨󵄨A1−β Kq (t2 − s)Aβ h(s, xs )󵄨󵄨󵄨 ds 0

198 | 5 Fractional neutral evolution equations and inclusions t1

󵄨2 󵄨 + 3t1 ∫(t1 − s)2(q−1) E 󵄨󵄨󵄨[A1−β Kq (t2 − s) − A1−β Kq (t1 − s)]Aβ h(s, xs )󵄨󵄨󵄨 ds 0

=: J1 (t1 , t2 ) + J2 (t1 , t2 ) + J3 (t1 , t2 ). After a fundamental calculation, one can estimate each term as follows: J1 (t1 , t2 ) ≤

3d1 Mβ2 (t2

t2

− t1 ) ∫(t2 − s)2(qβ−1) (1 + ‖xs ‖2∗ )ds t1

≤ 3d1 Mβ2 (1 + ‖x‖2C [−τ,b] ) J2 (t1 , t2 ) ≤

t1 2 3d1 Mβ t1 ∫((t2 0

≤

3d1 Mβ2 t1 (1

+

(t2 − t1 )2qβ , 2qβ − 1 2

− s)q−1 − (t1 − s)q−1 ) (t2 − s)2q(β−1) (1 + ‖xs ‖2∗ )ds

t1 2 ‖x‖C [−τ,b] ) ∫((t1

− s)2(q−1) − (t2 − s)2(q−1) )(t2 − s)2q(β−1) ds,

0 t1 −δ

J3 (t1 , t2 ) ≤ 3t1 d1

󵄩2 󵄩 sup 󵄩󵄩󵄩A1−β Kq (t2 − s) − A1−β Kq (t1 − s)󵄩󵄩󵄩 ∫ (t1 − s)2(q−1) (1 + ‖xs ‖2∗ )ds

s∈[0,t1 −δ]

0

t1

+ 6d1 Mβ2 t1 ∫ (t1 − s)2(q−1) [(t2 − s)2q(β−1) + (t1 − s)2q(β−1) ](1 + ‖xs ‖2∗ )ds t1 −δ

≤ 3t1 d1 (1 + ‖x‖2C [−τ,b] ) ×

t12q−1 − δ2q−1 2q − 1

󵄩 󵄩2 sup 󵄩󵄩󵄩A1−β Kq (t2 − s) − A1−β Kq (t1 − s)󵄩󵄩󵄩

s∈[0,t1 −δ]

+ 12d1 Mβ2 t1 (1 + ‖x‖2C [−τ,b] )

δ2qβ−1 . 2qβ − 1

Therefore, for t2 −t1 sufficiently small, J1 (t1 , t2 ) and J3 (t1 , t2 ) tend to zero by Lemma 5.10. For J2 (t1 , t2 ), notice that t1

∫((t1 − s)2(q−1) − (t2 − s)2(q−1) )(t2 − s)2q(β−1) ds 0

t1

≤ ∫(t2 − s)2(qβ−1) + (t1 − s)2(q−1) (t2 − s)2q(β−1) ds 0

t1

≤ ∫(t2 − s)2(qβ−1) + (t1 − s)2(qβ−1) ds < ∞. 0

5.2 Topological properties of solution sets | 199

Using Lebesgue’s dominated convergence theorem, one can deduce that J2 (t1 , t2 ) → 0 as t2 − t1 → 0. Hence, we obtain the result.

5.2.4 Topological structure of solution sets In this subsection we study the topological properties of solution set. For computational convenience, set 2−2q1

4M 2 Tr(Q) 1 − q1 ) ( q − q1 Γ2 (q)

Λ=

,

2 2qβ 󵄩 󵄩2 Mβ b d̃ = 4d1 (󵄩󵄩󵄩A−β 󵄩󵄩󵄩 + ) + Λb2(q−q1 ) ‖α‖ 1 . 2q1 −1 2qβ − 1

The following compactness characterizations of the solution set to the problem (5.25) will be useful. 1

Lemma 5.12. Suppose that T(t) is compact for t > 0 and there exists γ ∈ L 2q1 −1 ([0, b]; ℝ+ ) such that 󵄩 󵄩2 E 󵄩󵄩󵄩Σ(t, c)󵄩󵄩󵄩L0 ≤ γ(t) 2

for a. e. t ∈ [0, b] and c ∈ C([−τ, 0]; H).

Then the multimap F2 is compact in C [−τ, b]. Proof. Let D be a bounded set of C [−τ, b]. For each t ∈ [−τ, b], it will be shown that Δ(t) = {F2 (x)(t) : x ∈ D} is relatively compact in L2 (Ω; H). Obviously, for t ∈ [−τ, 0], Δ(t) = {ϕ(t)} is relatively compact in L2 (Ω; H). Let t ∈ [0, b] be fixed, for x ∈ D and y(t) ∈ Δ(t), there exists σ ∈ SelΣ (x) such that t

y(t) = Sq (t)[ϕ(0) − h(0, ϕ)] + ∫(t − s)q−1 Kq (t − s)σ(s)dW(s). 0

Let t ∈ [0, b] be arbitrary and ε > 0 be small enough. Define the operator Ψε : Δ(t) → L2 (Ω; H) by Ψε y(t) = Sq (t)[ϕ(0) − h(0, ϕ)] t−ε ∞

+ T(εq δ) ∫ ∫ qθ(t − s)q−1 ξq (θ)T((t − s)q θ − εq δ)σ(s)dθdW(s). 0 δ

200 | 5 Fractional neutral evolution equations and inclusions Using the compactness of T(t) for t > 0, we deduce that the set {Ψε y(t) : y ∈ Δ(t)} is relatively compact in L2 (Ω; H) for every ε, 0 < ε < t. Moreover, we have

2

󵄩󵄩 󵄩󵄩 t δ 󵄩󵄩 󵄩󵄩 󵄨2 󵄨󵄨 󵄨 E 󵄨󵄨Ψε y(t) − y(t)󵄨󵄨 ≤ 2E 󵄩󵄩󵄩∫ ∫ qθ(t − s)q−1 ξq (θ)T((t − s)q θ)σ(s)dθdW(s)󵄩󵄩󵄩 󵄩󵄩 0 󵄩󵄩 󵄩L2 󵄩0 0

󵄩󵄩2 󵄩󵄩 t ∞ 󵄩󵄩 󵄩󵄩 + 2E 󵄩󵄩󵄩 ∫ ∫ qθ(t − s)q−1 ξq (θ)T((t − s)q θ)σ(s)dθdW(s)󵄩󵄩󵄩 󵄩󵄩 0 󵄩󵄩 󵄩L2 󵄩t−ε δ 2 t

δ

󵄩 󵄩2 ≤ 2 Tr(Q)(M ∫ qθξq (θ)dθ) ∫(t − s)2(q−1) E 󵄩󵄩󵄩σ(s)󵄩󵄩󵄩L0 ds 0

2

0 2 t

∞

󵄩 󵄩2 + 2 Tr(Q)(M ∫ qθξq (θ)dθ) ∫ (t − s)2(q−1) E 󵄩󵄩󵄩σ(s)󵄩󵄩󵄩L0 ds δ

2 t

δ

2

t−ε

≤ 2 Tr(Q)(M ∫ qθξq (θ)dθ) ∫(t − s)2(q−1) γ(s)ds 0

+

0

t

2M 2 Tr(Q) ∫ (t − s)2(q−1) γ(s)ds Γ2 (q) t−ε

2

δ

≤ 2 Tr(Q)(M ∫ qθξq (θ)dθ) b2(q−q1 ) ( 0 2

2−2q1

2M Tr(Q) 1 − q1 ( ) + q − q1 Γ2 (q) → 0,

ε

2(q−q1 )

2−2q1

1 − q1 ) q − q1 t

(∫ γ t−ε

1 2q1 −1

‖γ‖

1 2q1 −1

2q1 −1

(s)ds)

as ε → 0.

Thus, ‖Ψε y(t) − y(t)‖C → 0, which shows that there is a relatively compact set arbi-

trarily close to the set Δ(t). Thus, the set Δ(t) is also relatively compact in L2 (Ω; H) for

each t ∈ [0, b]. Hence, Δ(t) = {Γ2 (x)(t) : x ∈ D} is relatively compact in L2 (Ω; H) for each t ∈ [−τ, b].

Next, we verify that the set {F2 (x)(t) : x ∈ D} is equicontinuous on (0, b]. For each

y ∈ F2 (x) and 0 < t1 < t2 ≤ b, we obtain

󵄨 󵄨2 󵄨 󵄨2 E 󵄨󵄨󵄨y(t2 ) − y(t1 )󵄨󵄨󵄨 ≤ 4E 󵄨󵄨󵄨(Sq (t2 ) − Sq (t1 ))[ϕ(0) − h(0, ϕ)]󵄨󵄨󵄨

󵄩󵄩 t2 󵄩󵄩2 󵄩󵄩 󵄩󵄩 + 4E 󵄩󵄩󵄩∫(t2 − s)q−1 Kq (t2 − s)σ(s)dW(s)󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 0 󵄩t1 󵄩L2

5.2 Topological properties of solution sets | 201 2 󵄩󵄩 t1 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 q−1 q−1 󵄩 + 4E 󵄩󵄩∫((t2 − s) − (t1 − s) )Kq (t2 − s)σ(s)dW(s)󵄩󵄩 󵄩󵄩 0 󵄩󵄩 󵄩L2 󵄩0

󵄩󵄩2 󵄩󵄩 t1 󵄩󵄩 󵄩󵄩 q−1 󵄩 + 4E 󵄩󵄩∫(t1 − s) [Kq (t2 − s) − Kq (t1 − s)]σ(s)dW(s)󵄩󵄩󵄩 󵄩󵄩 0 󵄩󵄩 󵄩L2 󵄩0 =: I1 (t1 , t2 ) + I2 (t1 , t2 ) + I3 (t1 , t2 ) + I4 (t1 , t2 ). As before, one can obtain the following estimates: 󵄩 󵄩2 I1 (t1 , t2 ) ≤ 4󵄩󵄩󵄩Sq (t2 ) − Sq (t1 )󵄩󵄩󵄩 E|x0 |2 , t2

4M 2 Tr(Q) I2 (t1 , t2 ) ≤ ∫(t2 − s)2(q−1) γ(s)ds Γ2 (q) t1

≤ Λ‖γ‖

1 2q1 −1

(t2 − t1 )2(q−q1 ) , t1

4M 2 Tr(Q) 2 I3 (t1 , t2 ) ≤ ∫[(t2 − s)q−1 − (t1 − s)q−1 ] γ(s)ds 2 Γ (q) 0

≤ Λ‖γ‖

1

q−q1 1−q1

2q1 −1

(t1

q−q1 1−q1

− t2

q−q1

2(1−q1 )

+ (t2 − t1 ) 1−q1 )

,

t1 −δ

󵄩 󵄩2 I4 (t1 , t2 ) ≤ 8 Tr(Q) sup 󵄩󵄩󵄩Kq (t2 − s) − Kq (t1 − s)󵄩󵄩󵄩 ∫ (t1 − s)2(q−1) γ(s)ds s∈[0,t1 −δ]

0

t1

+

8M 2 Tr(Q) ∫ (t1 − s)2(q−1) γ(s)ds Γ2 (q) t1 −δ

2−2q1

󵄩 󵄩2 1 − q1 ) ≤ 8 Tr(Q) sup 󵄩󵄩󵄩Kq (t2 − s) − Kq (t1 − s)󵄩󵄩󵄩 ( q−q s∈[0,t1 −δ]

+ 2Λδ

2(q−q1 )

‖γ‖

1

1 2q1 −1

‖γ‖

1 2q1 −1

2(q−q1 )

t1

.

Therefore, for t2 − t1 sufficiently small, the right-hand side of each term tends to zero by Lemma 5.10. The equicontinuity of the cases t1 < t2 ≤ 0 and t1 ≤ 0 ≤ t2 is obvious. Thus, an application of the Arzela–Ascoli theorem justifies that {F2 (x) : x ∈ D} is relatively compact in C [−τ, b]. Hence, F2 is compact in C [−τ, b]. This completes the proof. Let a ∈ [0, b). We consider the singular integral equation of the form for t

φ(t) + h(t, xt ) + ∫a (t − s)q−1 AKq (t − s)h(s, xs )ds { { { t ̃ xs )dW(s), x(t) = { + ∫ (t − s)q−1 Kq (t − s)σ(s, t ∈ [a, b], a { { ̃ t ∈ [−τ, a], {φ(t),

(5.26)

202 | 5 Fractional neutral evolution equations and inclusions ̃ − h(a, φ). ̃ Similar where φ ∈ C([a, b]; H) and φ̃ ∈ C([−τ, a]; H) are such that φ(a) = φ(a) to the proof of [208, Lemma 3.2], we can get the following lemma. 1

̃ c) be L 2q1 −1 -integrable for every c ∈ C([a − τ, a]; H). Lemma 5.13. Let q1 ∈ [ 21 , q), σ(⋅, Assume that {T(t) : t > 0} is compact. In addition, suppose that: 1 ) (i) for any compact subset K ⊂ H, there exist δ > 0 and LK ∈ Lp ([a, b]; ℝ+ ) (p > 2q−1 such that 2 󵄩 ̃ ̃ c2 )󵄩󵄩󵄩󵄩L0 ≤ LK (t)‖c1 − c2 ‖2∗ , E 󵄩󵄩󵄩σ(t, c1 ) − σ(t, 2

for a. e. t ∈ [a, b] and each c1 , c2 ∈ Bδ (K); 1

̃ c)‖2L0 ≤ γ1 (t)(c′ + ‖c‖2∗ ) for a. e. (ii) there exists γ1 (t) ∈ L 2q1 −1 ([a, b]; ℝ+ ) such that E‖σ(t, 2

t ∈ [a, b] and every c ∈ C([a − τ, a]; H), where c′ is arbitrary, but fixed.

If 4d1 ‖A−β ‖2 < 1, then the integral equation (5.26) admits a unique solution for every φ̃ ∈ C([−t, a]; H). Moreover, the solution of (5.26) depends continuously on φ̃ and φ. Proof. Step 1. A priori estimate. Assume that x is a solution of (5.26). Then t

󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 E 󵄨󵄨󵄨x(t)󵄨󵄨󵄨 ≤ 4E 󵄨󵄨󵄨A−β Aβ h(t, xt )󵄨󵄨󵄨 + 4(t − a) ∫(t − s)2(q−1) E 󵄨󵄨󵄨A1−β Kq (t − s)Aβ h(s, xs )󵄨󵄨󵄨 ds a

t

2 󵄩 󵄨 󵄨2 ̃ xs )󵄩󵄩󵄩󵄩L0 ds + 4E 󵄨󵄨󵄨φ(t)󵄨󵄨󵄨 + 4 Tr(Q) ∫(t − s)2(q−1) E 󵄩󵄩󵄩Kq (t − s)σ(s, 2

a

t

󵄩 󵄩2 ≤ 4d1 󵄩󵄩󵄩A−β 󵄩󵄩󵄩 (1 + ‖xt ‖2∗ ) + 4d1 Mβ2 (t − a) ∫(t − s)2(qβ−1) (1 + ‖xs ‖2∗ )ds 2

t

a

󵄨 󵄨2 4M Tr(Q) + 4 max E 󵄨󵄨󵄨φ(t)󵄨󵄨󵄨 + ∫(t − s)2(q−1) γ1 (s)(c′ + ‖xs ‖2∗ )ds [a,b] Γ2 (q) a

t

󵄩 󵄩2 ≤ 4d1 󵄩󵄩󵄩A−β 󵄩󵄩󵄩 (1 + ‖x‖2C [a−τ,t] ) + 4(b − a)d1 Mβ2 ∫(t − s)2(qβ−1) (1 + ‖x‖2C [a−τ,s] )ds t

a

2 󵄨 󵄨2 4M Tr(Q) + 4 max E 󵄨󵄨󵄨φ(t)󵄨󵄨󵄨 + ∫(t − s)2(q−1) γ1 (s)(c′ + ‖x‖2C [a−τ,s] )ds [a,b] Γ2 (q) a

̃ for t ∈ [a, b], and notice that |x(t)| = |φ(t)| for t ∈ [−τ, a]. Let t ∗ ∈ [a − τ, t] be such that ∗ ‖x(t )‖ = ‖x‖C [a−τ,t] . Furthermore, we have ‖x‖2C [a−τ,t] = ‖x‖2C [a−τ,t ∗ ]

2 2qβ 1 󵄩󵄩 −β 󵄩󵄩2 4d1 Mβ (b − a) ̃ + 4 max E 󵄨󵄨󵄨φ(t)󵄨󵄨󵄨2 ≤ [4d1 󵄩󵄩A 󵄩󵄩 + +Λ 󵄨 󵄨 2qβ − 1 [a,b] 1 − 4d1 ‖A−β ‖2

5.2 Topological properties of solution sets | 203 t∗

2(qβ−1)

+ ∫(4(b − a)d1 Mβ2 (t ∗ − s) a

+

4M 2 Tr(Q) ∗ 2(q−1) (t − s) γ1 (s)) 2 Γ (q)

× ‖x‖C [a−τ,s] ds], ̃ = c′ Λ(b − a)2(q−q1 ) ‖γ1 ‖ 1 . By Gronwall’s inequality, there exists M̄ > 0 such with Λ 2q1 −1 ̄ that ‖x‖C [−τ,b] ≤ M. Step 2. Local existence. Let φ ∈ C([a, b]; H) and φ̃ ∈ C([−τ, b]; H) be fixed. From 4d1 ‖A−β ‖2 < 1, we can find one ξ arbitrarily close to a such that 2 2qβ 󵄩 󵄩2 Mβ (ξ − a) 4d1 (󵄩󵄩󵄩A−β 󵄩󵄩󵄩 + ) + Λ(ξ − a)2(q−q1 ) ‖γ1 ‖ 2q 1−1 < 1. 2qβ − 1 L 1 [a,ξ ]

Then, for such an ξ , we can choose ρ satisfying

ρ≥

4d1 (‖A−β ‖2 +

Mβ2 (ξ −a)2qβ ) 2qβ−1

+ 4 max[a,ξ ] E|φ(t)|2 + c′ Λ(ξ − a)2(q−q1 ) ‖γ1 ‖

1 − 4d1 (‖A−β ‖2 +

Mβ2 (ξ −a)2qβ 2qβ−1

1

L 2q1 −1 [a,ξ ]

) − Λ(ξ − a)2(q−q1 ) ‖γ1 ‖

1 L 2q1 −1

[a,ξ ]

that is, 2qβ 2 󵄨 󵄩󵄩 −β 󵄩󵄩2 Mβ (ξ − a) 󵄨2 ) + 4 max E 󵄨󵄨󵄨φ(t)󵄨󵄨󵄨 4d1 (1 + ρ)(󵄩󵄩A 󵄩󵄩 + 2qβ − 1 [a,ξ ]

+ Λ(ξ − a)2(q−q1 ) (c′ + ρ)‖γ1 ‖

1

L 2q1 −1 [a,ξ ]

≤ ρ.

Let us define 󵄨 󵄨2 ̃ for t ∈ [−τ, a]}, Bρ (ξ ) = {x ∈ C [−τ, ξ ] : sup E 󵄨󵄨󵄨x(t)󵄨󵄨󵄨 ≤ ρ and x(t) = φ(t) t∈[−τ,ξ ]

and introduce an operator Ξ on Bρ (ξ ) as follows: Ξx(t) = Ξ1 x(t) + Ξ2 x(t), where t

h(t, xt ) + ∫a (t − s)q−1 AKq (t − s)h(s, xs )ds,

Ξ1 x(t) = {

0,

t ∈ [a, b],

t ∈ [−τ, a],

and t

̃ xs )dW(s), t ∈ [a, b], φ(t) + ∫a (t − s)q−1 Kq (t − s)σ(s,

Ξ2 x(t) = {

̃ φ(t),

t ∈ [−τ, a].

,

204 | 5 Fractional neutral evolution equations and inclusions For x ∈ Bρ (ξ ), we have 󵄨2 󵄨 E 󵄨󵄨󵄨Ξ1 x(t) + Ξ2 x(t)󵄨󵄨󵄨

t

󵄩 󵄩2 ≤ 4󵄩󵄩󵄩A−β 󵄩󵄩󵄩 d1 (1 + ‖x‖2C [a−τ,t] ) + 4(ξ − a)d1 Mβ2 ∫(t − s)2(qβ−1) (1 + ‖x‖2C [a−τ,s] )ds a

t

2

󵄨 󵄨2 4M Tr(Q) + 4 max E 󵄨󵄨󵄨φ(t)󵄨󵄨󵄨 + ∫(t − s)2(q−1) γ1 (s)(c′ + ‖x‖2C [a−τ,s] )ds [a,ξ ] Γ2 (q) 󵄩 󵄩2 ≤ 4d1 (1 + ρ)(󵄩󵄩󵄩A−β 󵄩󵄩󵄩 + 2(q−q1 )

+ Λ(ξ − a)

Mβ2 (ξ

a 2qβ

− a)

2qβ − 1

(c + ρ)‖γ1 ‖ ′

󵄨 󵄨2 ) + 4 max E 󵄨󵄨󵄨φ(t)󵄨󵄨󵄨 [a,ξ ]

1

L 2q1 −1 [a,ξ ]

≤ ρ, for t ∈ [a, ξ ]. Obviously, Ξ maps Bρ (ξ ) into itself. For any x, y ∈ Bρ (ξ ) and t ∈ [a, ξ ], we have 󵄨 󵄨2 E 󵄨󵄨󵄨Ξ1 x(t) − Ξ1 y(t)󵄨󵄨󵄨

t

󵄨 󵄨2 󵄨 󵄨2 ≤ 2E 󵄨󵄨󵄨h(t, xt ) − h(t, yt )󵄨󵄨󵄨 + 2(t − a) ∫(t − s)2(q−1) E 󵄨󵄨󵄨AKq (t − s)[h(s, xs ) − h(s, ys )]󵄨󵄨󵄨 ds a

t

󵄨 󵄨2 󵄨 󵄨2 ≤ 2E 󵄨󵄨󵄨A−β Aβ [h(t, xt ) − h(t, yt )]󵄨󵄨󵄨 + 2(t − a)Mβ2 ∫(t − s)2(qβ−1) E 󵄨󵄨󵄨Aβ [h(s, xs ) − h(s, ys )]󵄨󵄨󵄨 ds a

t

󵄩 󵄩2 ≤ 2d󵄩󵄩󵄩A−β 󵄩󵄩󵄩 ‖xt − yt ‖2∗ + 2d(t − a)Mβ2 ∫(t − s)2(qβ−1) ‖xs − ys ‖2∗ ds a

t

󵄩 󵄩2 ≤ 2d󵄩󵄩󵄩A−β 󵄩󵄩󵄩 ‖x − y‖2C [a−τ,t] + 2d(t − a)Mβ2 ∫(t − s)2(qβ−1) ‖x − y‖2C [a−τ,s] ds 󵄩 󵄩2 ≤ 2d(󵄩󵄩󵄩A−β 󵄩󵄩󵄩 +

Mβ2

2qβ − 1

a

(ξ − a)2qβ )‖x − y‖2C [a−τ,ξ ] .

Noting that Ξ1 x(t) = 0 for t ∈ [−τ, a], we get

‖Ξ1 x − Ξ1 y‖C [a−τ,ξ ]

Mβ2 󵄩󵄩 −β 󵄩󵄩2 √ ≤ 2d(󵄩󵄩A 󵄩󵄩 + ξ 2qβ )‖x − y‖C [a−τ,ξ ] , 2qβ − 1

which shows that Ξ1 is a contraction.

5.2 Topological properties of solution sets | 205

Next, we will prove that Ξ2 is continuous on Bρ (ξ ). Let x n , x ∈ Bρ (ξ ) with x n → x on Bρ (ξ ). Noting that t

2(q−1)

∫(t − s)

t

󵄩 󵄩2 󵄩 󵄩2 LK (s)󵄩󵄩󵄩xsn − xs 󵄩󵄩󵄩∗ ds ≤ 2 ∫(t − s)2(q−1) LK (s)(󵄩󵄩󵄩xsn 󵄩󵄩󵄩∗ + ‖xs ‖2∗ )ds a

a

t

≤ 4ρ ∫(t − s)2(q−1) LK (s)ds < ∞, a

by (i) and the fact that xtn → xt for t ∈ [a, ξ ], it follows from Lebesgue’s dominated convergence theorem that t

2 2 󵄨 󵄨2 4M Tr(Q) 󵄩 ̃ ̃ xs )󵄩󵄩󵄩󵄩L0 ds E 󵄨󵄨󵄨Ξ2 xn (t) − Ξ2 x(t)󵄨󵄨󵄨 ≤ xsn ) − σ(s, ∫(t − s)2(q−1) E 󵄩󵄩󵄩σ(s, 2 Γ2 (q) a

t

2

4M Tr(Q) 󵄩 󵄩2 ≤ ∫(t − s)2(q−1) LK (s)󵄩󵄩󵄩xsn − xs 󵄩󵄩󵄩∗ ds 2 Γ (q) a

→ 0,

as n → ∞.

Moreover, from the proof of Lemma 5.12 we see that Ξ2 is a compact operator. Thus, Ξ2 is a completely continuous operator. Hence, Krasnoselskii’s fixed-point theorem shows that there is a fixed point of Ξ, denoted by x, which is a local solution to equation (5.26). Step 3. Uniqueness. In fact, let y be another local solution of equation (5.26). According to condition (i), we obtain t

󵄨 󵄨2 󵄩 󵄩2 E 󵄨󵄨󵄨x(t) − y(t)󵄨󵄨󵄨 ≤ 3d󵄩󵄩󵄩A−β 󵄩󵄩󵄩 ‖x − y‖2C [a−τ,t] + 3d(t − a)Mβ2 ∫(t − s)2(qβ−1) ‖x − y‖2C [a−τ,s] ds a

t

+

3M 2 Tr(Q) ∫(t − s)2(q−1) LK (s)‖x − y‖2C [a−τ,s] ds Γ2 (q) a

for t ∈ [a, ξ ], and E|x(t) − y(t)|2 = 0 for t ∈ [−τ, a]. Let t ̃ ∈ [a − τ, t] be such that ̃ = ‖x − y‖C [a−τ,t] . Thus, we obtain ‖x(t)̃ − y(t)‖ ‖x − y‖2C [a−τ,t] = ‖x − y‖2C [a−τ,t]̃ t̃

1 ≤ ∫[3d(ξ − a)Mβ2 (t ̃ − s)2(qβ−1) 1 − 3d‖A−β ‖2 a

3M 2 Tr(Q) ̃ + (t − s)2(q−1) LK (s)]‖x − y‖2C [a−τ,s] ds. Γ2 (q)

206 | 5 Fractional neutral evolution equations and inclusions Applying Gronwall’s inequality, we get ‖x − y‖2C [a−τ,t] = 0, which implies x(t) = y(t) for t ∈ [−τ, ξ ]. Step 4. Continuation. In the sequel, the operator Ξ is treated as a mapping from C [a, b] to C [a, b]. Define an operator Ψ : [a, b] × C [a, b] → C [a, b],

y(s), s ∈ [a, t],

Ψ(t, y)(s) = {

y(t),

s ∈ [t, b].

Let J = {t ∈ [a, b] : yt ∈ C [a, b], yt = Ψ(t, Ξ(yt ))}. Then it follows from x ξ = Ψ(ξ , Ξ(xξ )) and xξ = Ψ(t, x) that J ≠ 0 and [a, t] ⊂ J for all t ∈ J. Letting t0 = sup J, there exists a sequence {tn } ⊂ J such that tn ≤ tn+1

for n ∈ ℕ,

lim t n→∞ n

= t0 .

By the continuity of φ and the assumption (H1 ), and following the argument employed in Lemmas 5.11 and 5.12, we conclude that E|x tm (t0 )−x tn (t0 )|2 = E|xtm (tm )−x tn (tn )|2 → 0 as n → ∞. Accordingly, the limn→∞ xtn (t0 ) exists. Consider the function xtn (s),

xt0 (s) = {

tn

s ∈ [a, tn ],

limn→∞ x (t0 ), s ∈ [t0 , b].

Clearly, equicontinuity of the family {xtn } implies that x t0 is continuous. Using Lebesgue’s dominated convergence theorem again, we have x t0 (t0 ) = lim xtn (tn ) n→∞

t

tn

= lim [φ(tn ) + h(tn , xtnn ) + ∫(tn − s)q−1 AKq (tn − s)h(s, xstn )ds n→∞

a

tn

̃ xstn )dW(s)] + ∫(tn − s)q−1 Kq (tn − s)σ(s, a

t

t0

= φ(t0 ) + h(t0 , xt00 ) + ∫(t0 − s)q−1 AKq (t0 − s)h(s, xst0 )ds t0

a

̃ xst0 )dW(s). + ∫(t0 − s)q−1 Kq (t0 − s)σ(s, a

Thus, we find that x t0 = Ψ(t0 , Ξ(xt0 )), which yields t0 ∈ J.

5.2 Topological properties of solution sets | 207

Next, we show that t0 = b. If this is not true, then t0 < b. Let us set t0

t0

a

a

̃ xs )dW(s), ̂ = φ(t) + ∫(t − s)q−1 AKq (t − s)h(s, xst0 )ds + ∫(t − s)q−1 Kq (t − s)σ(s, φ(t) with φ̂ ∈ C [t0 , b] and consider the following integral equation: t

̂ + h(t, xt ) + ∫(t − s)q−1 AKq (t − s)h(s, xs )ds x(t) = φ(t) t0

t

̃ xs )dW(s). + ∫(t − s)q−1 Kq (t − s)σ(s, t0

Applying the earlier arguments, one can obtain that z ∈ C [t0 , t0 + ξ ′ ]. Let x t0 +ξ (s) equal to xt0 (s) for s ∈ [a, t0 ], equal to z(s) for s ∈ [t0 , t0 + ξ ′ ] and equal to z(t0 + ξ ′ ) for ′ s ∈ [t0 + ξ ′ , b]. Then it is clear that xt0 +ξ (s) ∈ C [a, b]. Moreover, ′

x

t0 +ξ ′

(t) = φ(t) +

t +ξ ′ h(t, xt0 )

t

+ ∫(t − s)q−1 AKq (t − s)h(s, xst0 +ξ )ds

t

′

a

̃ xst0 +ξ )dW(s) for t ∈ [a, t0 + ξ ′ ]. + ∫(t − s)q−1 Kq (t − s)σ(s, ′

a

This shows t0 + ξ ′ ∈ J, which is a contradiction. Finally, let φn → φ0 in C([a, b]; H) and φ̃ n → φ̃ 0 in C([−τ, a]; H) as n → ∞, and xn be the solution of (5.26) with the perturbation φn , i. e., n

n

x (t) = φ (t) + t

h(t, xtn )

t

+ ∫(t − s)q−1 AKq (t − s)h(s, xsn )ds a

(5.27)

̃ xsn )dW(s) + ∫(t − s)q−1 Kq (t − s)σ(s, a

for t ∈ [a, b] and xn (t) = φ̃ n (t) for t ∈ [−τ, a]. It is clear that limn→∞ x n exists in C [−τ, a]. From Lemma 5.12, it follows that the set t

̃ xsn )dW(s) : n ≥ 1} {∫(t − s)q−1 Kq (t − s)σ(s, a

208 | 5 Fractional neutral evolution equations and inclusions is relatively compact in C [a, b]. This implies that the family n

{x (t) −

h(t, xtn )

t

− ∫(t − s)q−1 AKq (t − s)h(s, xsn )ds : n ≥ 1} a

is relatively compact in C [a, b]. Next, we only need to prove that limn→∞ x n exists in C [a, b]. On the contrary, if limn→∞ x n does not exist in C [a, b], then for any n ∈ ℕ, we have n1 , n2 with n1 , n2 > n such that ‖x n1 − x n2 ‖C [a,b] > ε0 (ε0 > 0 is a constant), i. e., there exists t ∗ such that 󵄨 󵄨2 󵄩 󵄩2 E 󵄨󵄨󵄨xn1 (t ∗ ) − xn2 (t ∗ )󵄨󵄨󵄨 = 󵄩󵄩󵄩x n1 − x n2 󵄩󵄩󵄩C [a,b] > ε02 . Let n

n

u (t) = x (t) −

h(t, xtn )

t

− ∫(t − s)q−1 AKq (t − s)h(s, xsn )ds. a

Using (H2 ), we have 󵄨 󵄨2 3E 󵄨󵄨󵄨un1 (t ∗ ) − un2 (t ∗ )󵄨󵄨󵄨 n n 󵄨2 󵄨 󵄨2 󵄨 ≥ E 󵄨󵄨󵄨xn1 (t ∗ ) − xn2 (t ∗ )󵄨󵄨󵄨 − 3E 󵄨󵄨󵄨h(t ∗ , xt ∗1 ) − h(t ∗ , xt ∗2 )󵄨󵄨󵄨 󵄨󵄨 t ∗ 󵄨󵄨2 󵄨󵄨 󵄨󵄨 q−1 − 3E 󵄨󵄨󵄨∫(t ∗ − s) AKq (t ∗ − s)[h(s, xsn1 ) − h(s, xsn2 )]ds󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨a 󵄨 2 n2 󵄩 󵄨󵄨 n1 ∗ 󵄩󵄩 −β 󵄩󵄩󵄩󵄩 n1 n2 ∗ 󵄨󵄨2 ≥ E 󵄨󵄨x (t ) − x (t )󵄨󵄨 − 3d󵄩󵄩A 󵄩󵄩󵄩󵄩xt ∗ − xt ∗ 󵄩󵄩󵄩∗ t∗

β−1 󵄩 n1 󵄩󵄩x 󵄩 s

− 3dMβ2 (t ∗ − a) ∫(t ∗ − s) a

󵄩2 − xsn2 󵄩󵄩󵄩∗ ds

2 ∗ β 󵄩2 󵄨󵄨 n1 ∗ 󵄩󵄩 −β 󵄩󵄩 Mβ (t − a) 󵄩󵄩 n1 n2 ∗ 󵄨󵄨2 )󵄩󵄩x − x n2 󵄩󵄩󵄩C [a,b] ≥ E 󵄨󵄨x (t ) − x (t )󵄨󵄨 − 3d(󵄩󵄩A 󵄩󵄩 + β

2 β 󵄩 󵄩 Mβ b )]ε02 , ≥ [1 − 3d(󵄩󵄩󵄩A−β 󵄩󵄩󵄩 + β

which contradicts the compactness of un in C [a, b]. Hence, {xn } converges in C [−τ, b], the limit is denoted by x. Therefore, taking the limit in (5.27) as n → ∞, one finds again by (H2 ) and Lebesgue’s dominated convergence theorem that x is the solution of (5.26) with the perturbation φ0 . This completes the proof. Next, we present an approximation result. We do not provide the proof as it is similar to that of [207, Lemma 2.4].

5.2 Topological properties of solution sets | 209

Lemma 5.14. Let {T(t) : t > 0} be compact and (H1 ) holds. Suppose that there exist two sequences {σn } ⊂ L2 ([0, b]; L02 ) and {x n } ⊂ C [−τ, b] such that limn→∞ σn = σ weakly in L2 ([0, b]; L02 ) and limn→∞ xn = x in C [−τ, b], where x n is a mild solution of the stochastic problem q

C , D [x n (t) − h(t, xtn )] = Ax n (t) + σn (t) dW(t) dt {0 n t x (t) = ϕ(t),

t ∈ [0, b],

t ∈ [−τ, 0].

Then x is a mild solution of the limit problem q

C D [x(t) − h(t, xt )] = Ax(t) + σ(t) dW(t) , dt {0 t x(t) = ϕ(t),

t ∈ [0, b],

t ∈ [−τ, 0].

Theorem 5.6. Let the conditions (H1 ) and (H2 ) be satisfied. In addition, assume that T(t) is compact for t > 0. If d̃ < 1, then the solution set of the inclusion problem (5.25) is a nonempty compact subset of C [−τ, b] for each ϕ ∈ C([−τ, 0]; H). Proof. Let us fix R>

‖ϕ‖2∗ + 8M 2 [d1 ‖A−β ‖2 (1 + ‖ϕ‖2∗ ) + E|ϕ(0)|2 ] + d̃ 1 − d̃

and consider BR (b) = {x ∈ C [−τ, b] :

󵄨 󵄨2 sup E 󵄨󵄨󵄨x(t)󵄨󵄨󵄨 ≤ R}.

t∈[−τ,b]

Clearly, BR (b) is closed and convex subset of C [−τ, b]. We first show that F (BR (b)) ⊂ BR (b). Indeed, taking x ∈ BR (b) and y(t) ∈ F (x), there exists σ ∈ SelΣ (x) such that t

󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 E 󵄨󵄨󵄨y(t)󵄨󵄨󵄨 ≤ 4E 󵄨󵄨󵄨A−β Aβ h(t, xt )󵄨󵄨󵄨 + 4t ∫(t − s)2(q−1) E 󵄨󵄨󵄨A1−β Kq (t − s)Aβ h(s, xs )󵄨󵄨󵄨 ds 0

t

󵄨 󵄨2 󵄩 󵄩2 + 4E 󵄨󵄨󵄨Sq (t)[ϕ(0) − h(0, ϕ)]󵄨󵄨󵄨 + 4 Tr(Q) ∫(t − s)2(q−1) E 󵄩󵄩󵄩Kq (t − s)σ(s)󵄩󵄩󵄩L0 ds 2 t

0

󵄩 󵄩2 ≤ 4d1 󵄩󵄩󵄩A−β 󵄩󵄩󵄩 (1 + ‖x‖2C [−τ,t] ) + 4d1 Mβ2 t ∫(t − s)2(qβ−1) (1 + ‖x‖2C [−τ,s] )ds 0

󵄩 󵄩2 󵄨 󵄨2 + 8M [d1 󵄩󵄩󵄩A−β 󵄩󵄩󵄩 (1 + ‖ϕ‖2∗ ) + E 󵄨󵄨󵄨ϕ(0)󵄨󵄨󵄨 ] 2

t

4M 2 Tr(Q) + ∫(t − s)2(q−1) γ1 (s)(1 + ‖x‖2C [−τ,s] )ds Γ2 (q) 0

210 | 5 Fractional neutral evolution equations and inclusions 2 2qβ 󵄨2 󵄩 󵄩2 Mβ b 󵄩 󵄩2 󵄨 ≤ 4d1 (1 + R)(󵄩󵄩󵄩A−β 󵄩󵄩󵄩 + ) + 8M 2 [d1 󵄩󵄩󵄩A−β 󵄩󵄩󵄩 (1 + ‖ϕ‖2∗ ) + E 󵄨󵄨󵄨ϕ(0)󵄨󵄨󵄨 ] 2qβ − 1

+ Λb2(q−q1 ) (1 + R)‖α‖

1 2q1 −1

for t ∈ [0, b]. With y(t) = ϕ(t) for t ∈ [−τ, 0], we have 󵄨2 󵄨 󵄩 󵄩2 󵄨2 󵄨 ̃ + R) ≤ R E 󵄨󵄨󵄨y(t)󵄨󵄨󵄨 ≤ ‖ϕ‖2∗ + 8M 2 [d1 󵄩󵄩󵄩A−β 󵄩󵄩󵄩 (1 + ‖ϕ‖2∗ ) + E 󵄨󵄨󵄨ϕ(0)󵄨󵄨󵄨 ] + d(1 for t ∈ [−τ, b]. Letting x, y ∈ C [−τ, b] and applying the argument employed in Lemma 5.13, we obtain 󵄨 󵄨2 󵄨 󵄨2 E 󵄨󵄨󵄨F1 x(t) − F1 y(t)󵄨󵄨󵄨 ≤ 2E 󵄨󵄨󵄨A−β Aβ [h(t, xt ) − h(t, yt )]󵄨󵄨󵄨 t

󵄨 󵄨2 + 2tMβ2 ∫(t − s)2(qβ−1) E 󵄨󵄨󵄨Aβ [h(s, xs ) − h(s, ys )]󵄨󵄨󵄨 ds 0

󵄩 󵄩2 ≤ 2d(󵄩󵄩󵄩A−β 󵄩󵄩󵄩 +

Mβ2

2qβ − 1

b2qβ )‖x − y‖2C [−τ,b] .

As F1 x(t) = 0 for t ∈ [−τ, 0], we have 󵄩 󵄩2 ‖F1 x − F1 y‖2C [−τ,b] ≤ 2d(󵄩󵄩󵄩A−β 󵄩󵄩󵄩 +

Mβ2

2qβ − 1

b2qβ )‖x − y‖2C [−τ,b] .

This shows that F1 is a contraction, since √2d(‖A−β ‖2 +

Mβ2

2qβ−1

b2qβ ) < 21 .

It follows from Lemma 5.8 that SelΣ is weakly u. s. c. with convex and weakly compact values. Moreover, using Lemmas 5.12, 5.14 and an argument similar to the one described in Zhou [226], we find that F2 : C [0, b] → P(C [0, b]) is quasicompact and closed (and therefore has closed values). This implies that F2 is u. s. c. due to Lemma 5.6 and, therefore, has compact values. Hence, the operators F1 and F2 satisfy all conditions of Theorem 5.5, and consequently the fixed-points set of the operator F1 + F2 is a nonempty compact subset of C [−τ, b]. In the next result, we denote by Θ(ϕ) the set of all mild solutions to inclusion problem (5.25). Theorem 5.7. Let the hypotheses of Theorem 5.6 be satisfied. Then the solution set of the inclusion problem (5.25) is an Rδ -set. Proof. Consider the fractional stochastic evolution inclusion q

C D [x(t) − h(t, xt )] ∈ Ax(t) + Σn (t) dW(t) , dt {0 t x(t) = ϕ(t),

t ∈ [0, b],

t ∈ [−τ, 0],

(5.28)

5.2 Topological properties of solution sets | 211

where Σn : [0, b] × C([−τ, 0]; H) → Pcl,cv (L02 ) are the multivalued functions already established in Lemma 5.9. Let Θn (ϕ) be the set of all mild solutions of the inclusion

problem (5.28).

From Lemma 5.9(ii) and (vi), it follows that {Σn } verifies the conditions (H2 ) for

each n ≥ 1. Then, from Lemma 5.8, we find that SelΣn is nonempty weakly u. s. c. with

convex and weakly compact values. As shown earlier, the solution set of the inclusion problem (5.28) is nonempty and compact in C [−τ, b] for each n ≥ 1.

Now we show that Θn (ϕ) is contractible for all n ≥ 1. To do this, let x ∈ Θn (ϕ) and

σ̃ n be the selection of Σn , n ≥ 1. For any λ ∈ [0, 1), we are concerned with the existence and uniqueness of solutions to the integral equation λb

y(t) = Sq (t)[ϕ(0) − h(0, ϕ)] + h(t, yt ) + ∫ (t − s)q−1 AKq (t − s)h(s, xs )ds 0 λb

+ ∫ (t − s)

q−1

t

x

Kq (t − s)σ (s)dW(s) + ∫ (t − s)q−1 AKq (t − s)h(s, ys )ds

0

(5.29)

λb

t

+ ∫ (t − s)q−1 Kq (t − s)σ̃ n (s)dW(s),

for t ∈ [λb, b],

λb

where σ x ∈ SelΣ (x) and y(t) = x(t) for t ∈ [λb − τ, λb]. Since the functions σ̃ n satisfy the

conditions of Lemma 5.13 due to Lemma 5.9(ii) and (v), it follows by Lemma 5.13 that

the problem (5.29) has a unique solution on [λb, b], denoted by y(t, λb, x(λb)). Moreover, y(t, λb, x(λb)) depends continuously on (λ, x).

Next, we define a function h1 : [0, 1] × Θn (ϕ) → Θn (ϕ) as x(t), h1 (λ, x) = { y(t, λb, x),

t ∈ [−τ, λb], t ∈ (λb, b],

and observe that h1 is well-defined and continuous. Also, it is clear that h1 (0, x) = y(t, 0, ϕ),

and h1 (1, x) = x.

Thus, Θn (ϕ) is contractible for each n ≥ 1.

Finally, by applying the arguments used in Zhou [226], we find that Θ(ϕ) =

⋂n≥1 Θn (ϕ). In consequence, we conclude that Θ(ϕ) is a compact Rδ -set. The proof is completed.

212 | 5 Fractional neutral evolution equations and inclusions 5.2.5 An Example Setting H = L2 ([0, π]; ℝ+ ) and K = ℝ, we consider the fractional partial differential inclusions of neutral type given by π

𝜕tq (z(t, ξ ) − ∫0 U(ξ , y)zt (θ, y)dy) { { { 2 { ) { { ∈ 𝜕 z(t,ξ + G(t, zt (θ, ξ )) dW(t) , dt 𝜕ξ 2 { {z(t, 0) = z(t, π) = 0, { { { { {z(θ, ξ ) = ϕ(θ)(ξ ),

t ∈ [0, 1], ξ ∈ [0, π], t ∈ [0, 1],

(5.30)

θ ∈ [−τ, 0], ξ ∈ [0, π],

where 𝜕tq is Caputo fractional partial derivative of order q ∈ ( 21 , 1], W(t) is a standard one-dimensional Wiener process defined on a stochastic basis (Ω, F , ℙ), ϕ(θ) ∈ H and zt (θ, ξ ) = z(t + θ, ξ ), t ∈ [0, 1], θ ∈ [−τ, 0]. Define an operator A : D(A) ⊂ H → H by D(A) = {z ∈ H : z, Az =

𝜕2 z . 𝜕ξ 2

𝜕z 𝜕2 z ∈ H, z(0) = z(π) = 0}, are absolutely continuous, 𝜕ξ 𝜕ξ 2

Then ∞

Az = ∑ n2 (z, zn )zn , n=1

where zn (t) = √ π2 sin(nt), n = 1, 2, . . . constitute the orthogonal basis of eigenvectors of A. It is well known that A generates a compact analytic semigroup {T(t) : t ≥ 0} in H (see Pazy [169]). Then the system (5.30) can be reformulated as C q 0Dt [x(t)

− h(t, xt )] ∈ Ax(t) + Σ(t, xt ) dW(t) , dt { x(t) = ϕ(t),

t ∈ [0, 1],

t ∈ [−τ, 0],

where x(t)(ξ ) = z(t, ξ ), xt (θ, ξ ) = zt (θ, ξ ), Σ(t, xt )(ξ ) = G(t, zt (θ, ξ )). Let the function h(t, xt ) : [0, 1] × C([−τ, 0]; H) → H be defined by π

h(t, xt ) = ∫ U(ξ , y)zt (θ, y)dy. 0

Moreover, we assume that the following conditions hold:

5.2 Topological properties of solution sets | 213

(h1 ) the function U(ξ , y) is measurable and π π

∫ ∫ U 2 (ξ , y)dydξ < ∞; 0 0

(h2 ) the function 𝜕ξ U(ξ , y) is measurable, U(0, y) = U(π, y) = 0, and let π π

2

1 2

H = (∫ ∫(𝜕ξ U(ξ , y)) dydξ ) < ∞. 0 0

Clearly, (H1 ) is satisfied. Let Σ(t, c) = [f1 (t, c), f2 (t, c)]. Now, we assume that fi : [0, 1] × C([−τ, 0]; H) → ℝ, i = 1, 2, satisfy (F1 ) f1 is l. s. c. and g2 is u. s. c.; (F2 ) f1 (t, c) ≤ f2 (t, c) for each (t, c) ∈ [0, 1] × C([−τ, 0]; H); (F3 ) there exist α1 , α2 ∈ L∞ ([0, 1]; ℝ+ ) such that 󵄩󵄩 󵄩2 2 󵄩󵄩fi (t, c)󵄩󵄩󵄩L0 ≤ α1 (t)‖c‖∗ + α2 (t), 2

i = 1, 2,

for each (t, c) ∈ [0, 1] × C([−τ, 0]; H). In view of assumptions (F1 )–(F3 ), it readily follows that the multivalued function Σ(⋅, ⋅) : [0, 1] × C([−τ, 0]; H) → P(L02 ) satisfies (H2 ). Thus, all the assumptions of Theorems 5.6 and 5.7 are satisfied and our conclusions can be applied to the inclusion problem (5.30).

6 Fractional evolution equations on whole real axis 6.1 Periodic solutions and S-asymptotically periodic solutions 6.1.1 Introduction As a matter of fact, periodic motion is a very important and special phenomena not only in natural science but also in social science such as climate, food supplement, insecticide population, sustainable development. Several authors have showed that for some fractional-order systems the solutions do not show any periodic behavior if the lower terminal of the derivative is finite; see [19, 131, 191, 201, 214]. Let β ∈ (0, ∞) \ ℕ and n = [β] + 1. If u : (0, ∞) → ℝ is a nonconstant ω-periodic function of class C n , [19, 131] tell us that Dβ u cannot be ω-periodic function, where Dβ is understood as one of the fractional derivatives (Caputo, Riemann–Liouville or Grunwald–Letnikov) with the lower terminal finite. Nevertheless, the authors also point out in [131] that fractional order derivative of other form, such as Liouville–Weyl fractional derivatives defined for periodic function [125], perhaps preserves periodicity. As indicated in [125, 174], the Liouville–Weyl derivative coincides with the Caputo, Riemann–Liouville or Grunwald–Letnikov derivative with β lower limit −∞, which is denoted by −∞ Dt . There is essential difference between finity β

and infinity. As in, for instance, [131, 174], C0Dt sin s = s1−β E2,2−β (−s2 ) for β ∈ (0, 1), while β

β −∞ Dt

sin s = sin(s + π2 β) for β ∈ (−1, ∞), where C0Dt is the Caputo fractional derivative with order β and the lower terminal 0, and Ea,b (z) denotes the two parameter MittagLeffler function. It is obvious that the Liouville–Weyl fractional derivative is suitable for the study of periodic solutions to differential equations. On the other hand, many phenomena are not strictly periodic, therefore, many other generalized periodic cases need to be studied, such as almost periodic, asymptotically almost periodic, S-asymptotically periodic, asymptotically periodic, pseudoperiodic, pseudo-almost periodic. As the advantages of fractional derivatives, such as the memorability and heredity, many papers concern these types of solutions for fractional differential equations. Since S-asymptotically periodic functions in Banach space were first studied by Henríquez et al. [121], there are some papers about S-asymptotically periodic solutions for fractional equations, one can refer to [73, 74, 185]. For almost periodic solutions, asymptotically almost periodic and other types of bounded solutions to fractional differential equations, one can refer to [15, 151, 176, 185, 201]. Ponce [176] studied the existence and uniqueness of bounded solutions for the semilinear fractional integrodifferential equation α −∞ Dt u(t)

t

= Au(t) + ∫ a(t − s)Au(s)ds + f (t, u(t)), −∞

https://doi.org/10.1515/9783110769272-006

t ∈ ℝ,

(6.1)

216 | 6 Fractional evolution equations on whole real axis where A is a closed linear operator defined on a Banach space X, −∞ Dαt is a Weyl fractional derivative of order α > 0 with the lower limit −∞, a ∈ L1 (ℝ+ ) is a scalar-valued kernel and f : ℝ×X → X satisfies some Lipschitz type conditions. Assume that A is the generator of an α-resolvent family {Sα (t)}t≥0 , which is uniformly integrable. The mild solutions of (6.1) were given by t

u(t) = ∫ Sα (t − s)f (s, u(s))ds,

t ∈ ℝ.

−∞

By the Banach contraction principle, existence and uniqueness results of almost periodic, asymptotically almost periodic and other types of bounded are established. In addition, Lizama [151] gave some sufficient conditions ensuring the existence and uniqueness of bounded solutions to a fractional semilinear equation of order 1 < α < 2. In this section, we study the fractional evolution equations in an ordered Banach space X, α −∞ Dt u(t)

+ Au(t) = f (t, u(t)),

t ∈ ℝ,

(6.2)

where −∞ Dαt is the Liouville–Weyl fractional derivative of order α ∈ (0, 1), −A : D(A) ⊂ X → X is the infinitesimal generator of a C0 -semigroup {T(t)}t≥0 . Applying Fourier transform, we give reasonable definitions of mild solutions of (6.2). Then the existence and uniqueness results for the corresponding linear fractional evolution equations are established, and the spectral radius of resolvent operator is accurately estimated. Finally, some sufficient conditions are established for the existence and uniqueness of periodic solutions, S-asymptotically periodic solutions and other types of bounded solutions when f : ℝ × X → X satisfies some ordered or Lipschitz conditions. This section organized as follows. In Subsection 6.1.2, we apply the Fourier transform to (6.2) itself and obtain the expression of the solution. Naturally, we may define that expression as a mild solution under suitable conditions. In Subsection 6.1.3, the existence and uniqueness results for the linear equations are obtained. In Subsection 6.1.4, we consider the existence and uniqueness of the mild solutions for (6.2). In Subsection 6.1.5, we also give two examples to apply the abstract results. This section is based on [168].

6.1.2 Preliminaries This subsection is devoted to some preliminary facts needed in the sequel. Let X be an ordered Banach space with norm ‖ ⋅ ‖ and partial order ≤, whose positive cone P :=

6.1 Periodic solutions and S-asymptotically periodic solutions | 217

{y ∈ X | y ≥ θ} (θ is the zero element of X) is normal with normal constant N. Denote by L (X) the space of all linear bounded operator on Banach space X, with the norm

‖ ⋅ ‖L (X) . The notation Cb (X) stands for the Banach space of all bounded continuous functions from ℝ into X equipped with the sup norm ‖ ⋅ ‖∞ , i. e.,

󵄩 󵄩 󵄨 Cb (X) := {u : ℝ → X 󵄨󵄨󵄨 u is continuous, and ‖u‖∞ = sup󵄩󵄩󵄩u(t)󵄩󵄩󵄩}. t∈ℝ

For u, v ∈ Cb (X), u ≤ v if u(t) ≤ v(t) for all t ∈ ℝ. The space Pω (X) stands for the

subspace of Cb (X) consisting of all X-valued continuous ω-periodic functions. Set 󵄩 󵄩 SAPω (X) = {f ∈ Cb (X) | there exists ω > 0 such that 󵄩󵄩󵄩f (t + ω) − f (t)󵄩󵄩󵄩 → 0 as t → ∞}.

These functions in SAPω (X) are called S-asymptotically ω-periodic (see [121]). We note

that Pω (X) and SAPω (X) are Banach spaces (see [121]), and Pω (X) ⊂ SAPω (X).

A function f ∈ Cb (X), if for any ϵ > 0 there is a real number ω = ω(ϵ) > 0 and

τ = τ(ε) in arbitrary interval of length ω(ε) such that ‖f (t+τ)−f (t)‖∞ < ϵ for all t ∈ ℝ, is

said to be almost periodic (in the sense of Bohr). We denote by AP(X) the set of all these

functions. The space of almost automorphic functions (resp. compact almost auto-

morphic functions) will be written as AA(X) (resp., AAc (X)). The bounded continuous

function f ∈ AA(X) (resp., f ∈ AAc (X)) ⇔ for every sequence {s′n }n∈ℕ there is a subse-

quence {sn }n∈ℕ ⊂ {s′n }n∈ℕ such that g(t) := limn→∞ f (t + sn ) and f (t) = limn→∞ g(t − sn )

for each t ∈ ℝ (resp., uniformly on compact subsets of ℝ). Clearly, the compact almost automorphic function f above is continuous on ℝ.

For convenience, we set C0 (X) := {g ∈ Cb (X) | lim|t|→∞ ‖g(t)‖ = 0}. We regard

the direct sum of Pω (X) and C0 (X) as the space of asymptotically periodic functions

APω (X), the direct sum of AP(X) and C0 (X) as the space of asymptotically almost peri-

odic functions AAP(X), the direct sum of AAc (X) and C0 (X) as the space of asymptotically compact almost automorphic functions, the direct sum of AA(X) and C0 (X) as the space of asymptotically almost automorphic functions. Then we set

T 󵄨󵄨 1 󵄨󵄨 󵄩 󵄩 P0 (X) := {f ∈ Cb (X) 󵄨󵄨 lim ∫ 󵄩󵄩f (s)󵄩󵄩󵄩ds = 0}, 󵄨󵄨 T→∞ 2T 󵄩 −T

and regard the direct sum of Pω (X) and P0 (X) as the space of pseudo-periodic func-

tions, the direct sum of AP(X) and P0 (X) as the space of pseudo-almost periodic func-

218 | 6 Fractional evolution equations on whole real axis tions, the direct sum of AAc (X) and P0 (X) as the space of pseudo-compact almost automorphic functions, the direct sum of AA(X) and P0 (X) as the space of pseudo-almost automorphic functions. The relationship of the above different classes of subspaces is presented in [151, p. 805]: AA(X)

⊂

∪

AAc (X)

⊂

AAAc (X)

⊂

AAP(X)

⊂

APω (X)

PAAc (X) ∪

⊂

∪

⊂

PAA(X) ∪

∪

∪

Pω (X)

⊂

∪

∪

AP(X)

AAA(X)

PAP(X) ∪

⊂

PPω (X)

∩

SAPω (X) Denote by M (ℝ, X) or simply M (X) the following function spaces: M (X) := {Pω (X), AP(X), AAc (X), AA(X), APω (X), AAP(X), AAAc (X),

AAA(X), PPω (X), PAP(X), PAAc (X), PAA(X), SAPω (X)},

and M (ℝ × X, X) the space of all functions g : ℝ × X → X satisfying g(⋅, v) ∈ M (X) uniformly for each v in any bounded subset V of X. For fixed Ω(X) ∈ M (X)∪{C0 (X), P0 (X)}, then for u ∈ Ω(X) and f ∈ Cb (ℝ × X, X), the following conditions could ensure that f (⋅, u(⋅)) ∈ Ω(X): (A1 ) f (t, ⋅) is uniformly continuous with respect to t on ℝ for each bounded subset of X. More precisely, given ϵ > 0 and K ⊂ X, there exists δ > 0 such that u1 , u2 ∈ K and ‖u1 − u2 ‖ < δ imply that ‖f (t, u1 ) − f (t, u2 )‖ < ϵ; (A2 ) for all bounded subset U ⊂ X, {f (t, u) | t ∈ ℝ, u ∈ U} is bounded; (A3 ) if f = f1 +f2 , where f1 ∈ Φ(X) = {AAc (X), AA(X), Pω (X), AP(X)} and f2 ∈ {C0 (X), P0 }\ {0}, f1 (t, ⋅) is uniformly continuous with respect to t on ℝ for each bounded subset of X. Then the results in Table 6.1 hold; see [151, Remark 3.4 and p. 810]. Lemma 6.1 ([151]). Let Ω(X) ∈ M (X) and f ∈ Ω(ℝ × X, X) ∈ M (ℝ × X, X) be given and fixed. Assume that there exists a constant Lf > 0 such that 󵄩󵄩 󵄩 󵄩󵄩f (t, u1 ) − f (t, u2 )󵄩󵄩󵄩 ≤ Lf ‖u1 − u2 ‖ for all t ∈ ℝ and u1 , u2 ∈ X. Let u ∈ Ω(X), then f (⋅, u(⋅)) ∈ Ω(X).

6.1 Periodic solutions and S-asymptotically periodic solutions | 219 Table 6.1: Conditions on f (⋅, u(⋅)) : Ω(X ) → Ω(X ). Ω(X ) Pw AP AAc AA APω AAP AAAc AAA PAP PPω PAA PAAc SAPω C0 P0

(A1 )

(A2 )

(A3 )

∙ ∙ ∙ ∙

∙ ∙ ∙ ∙ ∙ ∙

∙ ∙ ∙ ∙ ∙ ∙

∙ ∙ ∙ ∙ ∙

Now we recall the definitions and properties of operator semigroups, for details see [169]. Assume that −A is the infinitesimal generator of a C0 -semigroup {T(t)}t≥0 . If there are M ≥ 0 and ν ∈ ℝ such that ‖T(t)‖L (X) ≤ Meνt , then ∞

(λI + A) x = ∫ e−λt T(t)xdt, −1

Re λ > ν, x ∈ X.

(6.3)

0

A C0 -semigroup {T(t)}t≥0 is called exponentially stable if there exist constants M > 0 and δ > 0 such that 󵄩󵄩 󵄩 −δt 󵄩󵄩T(t)󵄩󵄩󵄩L (X) ≤ Me ,

t ≥ 0.

(6.4)

The growth bound of the semigroup {S(t)}t≥0 is defined as 󵄩 󵄩 ν0 = inf{ρ ∈ ℝ | ∃M1 > 0, 󵄩󵄩󵄩S(t)󵄩󵄩󵄩L (X) ≤ M1 eρt , ∀ t ≥ 0}.

(6.5)

Furthermore, ν0 could also be expressed by ν0 = lim sup t→+∞

ln ‖S(t)‖L (X) . t

(6.6)

For C0 -semigroup {T(t)}t≥0 , if there exists a constant M > 0 such that 󵄩󵄩 󵄩 󵄩󵄩T(t)󵄩󵄩󵄩L (X) ≤ M,

t ≥ 0,

(6.7)

220 | 6 Fractional evolution equations on whole real axis then it is called uniformly bounded. A C0 -semigroup {T(t)}t≥0 is called compact if T(t) is compact for t > 0. The positive C0 -semigroup {T(t)}t≥0 is C0 -semigroup {T(t)}t≥0 satisfying T(t)y ≥ θ for all y ≥ θ and t ≥ 0. For the positive operators semigroup, one can refer to [34]. Let ∞

V(t) = α ∫ θWα (θ)T(t α θ)dθ,

t ≥ 0,

(6.8)

0

where Wα (θ) is Wright function which is defined in Definition 1.9 and {T(t)}t≥0 is a C0 -semigroup. We have the following results. Lemma 6.2. (i) Assume that {T(t)}t≥0 is a uniformly bounded C0 -semigroup and satisfies (6.7). Then for any fixed t ≥ 0, V(t) is a linear and bounded operator, i. e., for any x ∈ X, we get M 󵄩󵄩 󵄩 ‖x‖. 󵄩󵄩V(t)x 󵄩󵄩󵄩 ≤ Γ(α)

(6.9)

(ii) If {T(t)}t≥0 is a C0 -semigroup, then {V(t)}t≥0 is strongly continuous. (iii) If {T(t)}t≥0 is a positive C0 -semigroup, then V(t) is positive for t ≥ 0. (iv) If {T(t)}t≥0 is exponentially stable and satisfies (6.4), then 󵄩󵄩 󵄩 α 󵄩󵄩V(t)󵄩󵄩󵄩L (X) ≤ MEα,α (−δt ),

for t ≥ 0.

Proof. For the proof of (i)–(ii), we can see [225]. By Proposition 1.11, we obtain (iii). In view of Proposition 1.11, we have 󵄩󵄩 ∞ 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 α 󵄩󵄩V(t)󵄩󵄩L (X) = 󵄩󵄩󵄩α ∫ θWα (θ)T(t θ)dθ󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 0 󵄩L (X) ∞

α

≤ α ∫ θWα (θ)Me−δt θ dθ 0

= MEα,α (−δt α ). Then (iv) holds. Next, we consider the following linear abstract fractional evolution equation: α −∞ Dt u(t)

+ Au(t) = h(t),

t ∈ ℝ,

(6.10)

where −A : D(A) ⊂ X → X generates a C0 -semigroup {T(t)}t≥0 of operators on Banach space X, h : ℝ → X is continuous.

6.1 Periodic solutions and S-asymptotically periodic solutions | 221

For convenience, we assume the following condition: t (H) u ∈ C(ℝ, X), ∫−∞ g1−α (t − s)u(s)ds ∈ C 1 (ℝ, X), u(t) ∈ D(A) for t ∈ ℝ, Au ∈ L1 (ℝ, X) and u satisfies (6.10), where t −α

g1−α (t) = { Γ(1−α) 0,

,

t > 0, t ≤ 0.

Lemma 6.3. Assume that −A generates an exponentially stable C0 -semigroup {T(t)}t≥0 . If u : ℝ → X is a function satisfying equation (6.10) and assumption (H), then u satisfies the following integral equation: t

u(t) = ∫ (t − s)α−1 V(t − s)h(s)ds,

t ∈ ℝ,

−∞

where V is defined by (6.8). Proof. Let F u be the Fourier transform of u. That is, ∞

(F u)(λ) = ∫ e−iλt u(t)dt,

for λ ∈ ℝ.

−∞

Thus (F (−∞Dαt u))(λ) = (iλ)α (F u)(λ) (see Definition 1.16). Applying Fourier transform to (6.10), we get (iλ)α (F u)(λ) + A(F u)(λ) = (F h)(λ),

for λ ∈ ℝ.

In view of (6.3) and Proposition 1.11, we have (F u)(λ) = ((iλ)α I + A) (F h)(λ) −1

∞

α

= ∫ e−(iλ) t T(t)(F h)(λ)dt 0 ∞ ∞

α

= ∫ ∫ αt α−1 e−(iλt) T(t α )h(s)e−iλs dsdt 0 −∞ ∞ ∞ ∞

=∫ ∫ ∫ 0 −∞ 0 ∞ ∞ ∞

α2

τ2α+1

t α−1 Wα (τ−α )e−iλt T(

tα )h(s)eiλs dτdsdt τα

= ∫ ∫ ∫ ατt α−1 Wα (τ)T(t α τ)h(s)eiλ(−t+s) dτdsdt 0 −∞ 0

222 | 6 Fractional evolution equations on whole real axis t

∞

= ∫ e

−iλt

−∞

∫ (t − s)

α−1

∞

h(s)( ∫ ατWα (τ)T((t − s)α τ)dτ)dsdt 0

−∞

∞

t

−∞

−∞

= ∫ e−iλt ( ∫ (t − s)α−1 V(t − s)h(s)ds)dt. By the uniqueness of Fourier transform, we deduce that the assertion of lemma holds. This completes the proof. Definition 6.1. A function u : ℝ → X is said to be a mild solution of problem (6.10) if t

u(t) = ∫ (t − s)α−1 V(t − s)h(s)ds,

t ∈ ℝ,

−∞

where V is given by (6.8).

6.1.3 Linear equations Theorem 6.1. If {T(t)}t≥0 is exponentially stable and satisfies (6.4), h belongs to one of M (X), and t

(Rh)(t) = ∫ (t − s)α−1 V(t − s)h(s)ds,

(6.11)

−∞

where V is defined by (6.8), then Rh and h belong to the same space. Proof. Pω (X): If h ∈ Pω (X), then s+ω

(Rh)(s + ω) = ∫ (s + ω − τ)α−1 V(s + ω − τ)h(τ)dτ −∞ s

= ∫ (s − t)α−1 V(s − t)h(t + ω)dt −∞ s

= ∫ (s − t)α−1 V(s − t)h(t)dt = (Rh)(s). −∞

Therefore, Rh ∈ Pω (X). AP(X): By the hypotheses, for any ϵ > 0, we can find a real number l = l(ϵ) > 0 for any interval of length l(ε), and there exists a number m = m(ε) in this interval such that ‖h(t + m) − h(t)‖∞ < ϵ for all t ∈ ℝ. From Proposition 1.10, Lemma 6.2(iv) and

6.1 Periodic solutions and S-asymptotically periodic solutions | 223

Eα (0) = 1, we have 󵄩 󵄩 sup󵄩󵄩󵄩(Rh)(t + m) − (Rh)(t)󵄩󵄩󵄩 t∈ℝ

󵄩󵄩 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 = sup󵄩󵄩󵄩 ∫ (t − s)α−1 V(t − s)[h(s + m) − h(s)]ds󵄩󵄩󵄩 󵄩󵄩 t∈ℝ 󵄩 󵄩󵄩−∞ 󵄩 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩󵄩󵄩 α−1 ≤ sup󵄩󵄩h(t + m) − h(m)󵄩󵄩󵄩󵄩 ∫ (t − s) V(t − s)ds󵄩󵄩󵄩 󵄩 󵄩󵄩 t∈ℝ 󵄩󵄩−∞ 󵄩L (X) t

≤ Mϵ ∫ (t − s)α−1 Eα,α (−δ(t − s)α )ds −∞

=

󵄨󵄨t Mϵ Mϵ 󵄨 Eα (−δ(t − s)α )󵄨󵄨󵄨 , = 󵄨󵄨−∞ δ δ

and Rh and h are all almost periodic. AAc (X): Since h ∈ AAc (X), there is {tn }n∈ℕ and v ∈ Cb (X) such that h(t + tn ) → v(t) and v(t − tn ) → h(t) as t → ∞, uniformly on compact subsets of ℝ: t+tn

(Rh)(t + tn ) = ∫ (t + tn − s)α−1 V(t + tn − s)h(s)ds (6.12)

−∞ t

= ∫ (t − τ)α−1 V(t − τ)h(τ + tn )dτ, −∞

therefore, by Lebesgue dominated convergence theorem, when n → ∞ we get (Rh)(t + t tn ) → z(t) = ∫−∞ (t − τ)α−1 V(t − τ)v(τ)dτ for all t ∈ ℝ. Furthermore, for a compact set K = [−a, a] and ϵ > 0, by Proposition 1.10 and Lemma 6.2, we choose Mϵ > 0 and Nϵ ∈ ℕ such that ∞ 󵄨󵄨∞ M M 󵄨 󵄩 󵄩 ∫ sα−1 󵄩󵄩󵄩V(s)󵄩󵄩󵄩L (X) ds ≤ − Eα (−δsα )󵄨󵄨󵄨 = Eα (−δMϵα ) ≤ ϵ, 󵄨󵄨Mϵ δ δ

Mϵ

󵄩󵄩 󵄩 󵄩󵄩h(τ + tn ) − v(τ)󵄩󵄩󵄩 ≤ ϵ,

n ≥ Nϵ , τ ∈ [−M, M],

where M = Mϵ + a. For t ∈ K, by Proposition 1.10 and Lemma 6.2, we estimate t

󵄩󵄩 󵄩 󵄩 󵄩 󵄩 α−1 󵄩 󵄩󵄩(Rh)(t + tn ) − z(t)󵄩󵄩󵄩 ≤ ∫ (t − s) 󵄩󵄩󵄩V(t − s)󵄩󵄩󵄩L (X) 󵄩󵄩󵄩h(s + tn ) − v(s)󵄩󵄩󵄩ds −∞

−M

󵄩 󵄩 󵄩 󵄩 ≤ ∫ (t − s)α−1 󵄩󵄩󵄩V(t − s)󵄩󵄩󵄩L (X) 󵄩󵄩󵄩h(s + tn ) − v(s)󵄩󵄩󵄩ds −∞

224 | 6 Fractional evolution equations on whole real axis t

󵄩 󵄩 󵄩 󵄩 + ∫ (t − s)α−1 󵄩󵄩󵄩V(t − s)󵄩󵄩󵄩L (X) 󵄩󵄩󵄩h(s + tn ) − v(s)󵄩󵄩󵄩ds −M ∞

󵄩 󵄩 ≤ (‖h‖∞ + ‖v‖∞ ) ∫ sα−1 󵄩󵄩󵄩V(s)󵄩󵄩󵄩L (X) ds t+M

∞

󵄩 󵄩 + ϵ ∫ sα−1 󵄩󵄩󵄩V(s)󵄩󵄩󵄩L (X) ds 0

≤ ϵ(‖h‖∞ + ‖v‖∞ ) +

ϵM , δ

which implies that the convergence is irrelevant to t ∈ K. Similarly, we can prove that z(t − tn ) → (Rh)(t) if n → ∞ uniformly for t on compact subsets of ℝ. The case of the space AA(X) is similar, too. AA(X): Set h ∈ AA(X). For all sequence {tn′ } ⊂ ℝ, there is {tn } of {tn′ } such that lim h(t + tn ) = v(t),

t ∈ ℝ,

lim v(t − tn ) = h(t),

t ∈ ℝ.

n→∞

and n→∞

Since 󵄩󵄩 󵄩 M 󵄩󵄩(Rh)(t + tn )󵄩󵄩󵄩 ≤ ‖h‖∞ , δ by (6.12), Proposition 1.10 and Lemma 6.2, for any t ≥ s ∈ ℝ we get V(t − s)h(s + tn ) → V(t − s)v(s) as n → ∞. For any t ∈ ℝ, Lebesgue dominated convergence theorem t implies that (Rh)(t + tn ) → z(t) = ∫−∞ (t − s)α−1 V(t − s)v(s)ds as n → ∞. It is similar that z(t − tn ) → (Rh)(t),

n → ∞, t ∈ ℝ.

SAPω (X): Assume that h ∈ SAPω (X). For any ϵ > 0, there exists Tϵ > 0 such that ‖h(t + ω) − h(t)‖ < ϵ for t ≥ Tϵ . Then by Proposition 1.10 and Lemma 6.2, we get 󵄩󵄩 󵄩 󵄩󵄩(Rh)(t + ω) − (Rh)(t)󵄩󵄩󵄩 t 󵄩󵄩 t+ω 󵄩󵄩 󵄩󵄩 󵄩󵄩 = 󵄩󵄩󵄩 ∫ (t + ω − s)α−1 V(t + ω − s)h(s)ds − ∫ (t − s)α−1 V(t − s)h(s)ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩−∞ 󵄩 −∞ t

󵄩 󵄩 ≤ ∫ 󵄩󵄩󵄩(t − τ)α−1 V(t − τ)[h(τ + ω) − h(τ)]󵄩󵄩󵄩dτ −∞

6.1 Periodic solutions and S-asymptotically periodic solutions | 225 Tϵ

t

α−1 󵄩 󵄩

󵄩 󵄩 α−1 󵄩 󵄩󵄩V(t − τ)󵄩󵄩󵄩L (X) dτ + ϵ ∫(t − τ) 󵄩󵄩󵄩V(t − τ)󵄩󵄩󵄩L (X) dτ

≤ 2‖h‖∞ ∫ (t − τ)

Tϵ

−∞ Tϵ

≤ 2‖h‖∞ M ∫ (t − τ)α−1 Eα,α (−δ(t − τ)α )dτ −∞ t

+ ϵM ∫(t − τ)α−1 Eα,α (−δ(t − τ)α )dτ Tϵ

= =

󵄨󵄨Tϵ 󵄨󵄨t 2‖h‖∞ M ϵM 󵄨 󵄨 Eα (−δ(t − τ)α )󵄨󵄨󵄨 + Eα (−δ(t − τ)α )󵄨󵄨󵄨 󵄨󵄨−∞ 󵄨󵄨Tϵ δ δ 2‖h‖∞ M − ϵM ϵM Eα (−δ(t − Tϵ )α ) + , δ δ

for t ≥ Tϵ . It follows that ‖(Rh)(t + ω) − (Rh)(t)‖ → 0 as t → ∞. Thus, Rh ∈ SAPω (X).

We next consider the asymptotic property of the solutions. For w ∈ C0 (X) and

ϵ > 0, we have ‖w(s)‖ < ϵ for some T > 0 and |s| > T. Then Proposition 1.10 and Lemma 6.2 imply that

T

󵄩󵄩 󵄩 󵄩 󵄩 󵄩 α−1 󵄩 󵄩󵄩(Rw)(t)󵄩󵄩󵄩 ≤ ∫ (t − s) 󵄩󵄩󵄩V(t − s)󵄩󵄩󵄩L (X) 󵄩󵄩󵄩w(s)󵄩󵄩󵄩ds −∞

t

󵄩 󵄩 + ϵ ∫(t − s)α−1 󵄩󵄩󵄩V(t − s)󵄩󵄩󵄩L (X) ds T

‖w‖∞ M ϵM Eα (−δ(t − T)α ) + (1 − Eα (−δ(t − T)α )), ≤ δ δ which implies that (Rw)(t) → 0 as t → ∞. Naturally, we can also get the results for the spaces APω (X), AAP(X), AAAc (X) and AAA(X). Set w ∈ P0 (X). For L > 0, we get L

L

t

1 󵄩󵄩 1 󵄩 󵄩 󵄩 󵄩 󵄩 ∫ 󵄩󵄩(Rw)(t)󵄩󵄩󵄩dt ≤ ∫ ( ∫ (t − s)α−1 󵄩󵄩󵄩V(t − s)󵄩󵄩󵄩L (X) 󵄩󵄩󵄩w(s)󵄩󵄩󵄩ds)dt 2L 2L −L −∞

−L

L

∞

−L

0

1 󵄩 󵄩 󵄩 󵄩 ≤ ∫ ( ∫ sα−1 󵄩󵄩󵄩V(s)󵄩󵄩󵄩L (X) 󵄩󵄩󵄩w(t − s)󵄩󵄩󵄩ds)dt 2L ∞

L

0

−L

1 󵄩 󵄩 󵄩 󵄩 = ∫ sα−1 󵄩󵄩󵄩V(s)󵄩󵄩󵄩L (X) ( ∫ 󵄩󵄩󵄩w(t − s)󵄩󵄩󵄩dt)ds. 2L

226 | 6 Fractional evolution equations on whole real axis We can find that the set P0 (X) is translation-invariant and get L

1 󵄩󵄩 󵄩 ∫ 󵄩(Rw)(t)󵄩󵄩󵄩dt → 0 2L 󵄩

as L → ∞,

−L

by Lebesgue dominated convergence theorem. Then PPω (X), PAP(X), PAAc (X) and PAA(X) have the maximal regularity property under the convolution defined by (6.11). Theorem 6.2. Assume that h ∈ Ω(X) ∈ M (X), −A generates an exponentially stable C0 -semigroup {T(t)}t≥0 and satisfies (6.4). Then linear fractional evolution equation (6.10) possesses a unique mild solution u := Rh ∈ Ω(X), and ‖Rh‖∞ ≤

M ‖h‖∞ . δ

(6.13)

Proof. In view of Definition 6.1 and Theorem 6.1, Rh is a mild solution of (6.10) and Rh ∈ Ω(X). By Proposition 1.10 and Lemma 6.2, we have t

󵄩󵄩 󵄩 󵄩 α−1 󵄩 󵄩󵄩(Rh)(t)󵄩󵄩󵄩 ≤ ∫ (t − s) 󵄩󵄩󵄩V(t − s)h(s)󵄩󵄩󵄩ds −∞

t

≤ M‖h‖∞ ∫ (t − s)α−1 Eα,α (−δ(t − s)α )ds −∞

=

󵄨󵄨t M M 󵄨 ‖h‖∞ Eα (−δ(t − s)α )󵄨󵄨󵄨 = ‖h‖∞ , 󵄨 δ δ 󵄨−∞

where V is defined by (6.8). Then we obtain ‖Rh‖∞ ≤

M ‖h‖∞ . δ

Remark 6.1. Equation (6.13) is an optimal estimation. In fact, for X = ℝ, the periodic solution of equation −∞ Dαt x + γx = 1 is x = γ1 . Corollary 6.1. Let h ∈ Ω(X) ∈ M (X). Assume that −A generates a uniformly bounded C0 -semigroup {T(t)}t≥0 and satisfies (6.7). If Re λ > 0, then linear fractional evolution equation α −∞ Dt u(t)

+ Au(t) + λu(t) = h(t),

t ∈ ℝ,

(6.14)

has a unique mild solution u := Rλ h, and ‖Rλ h‖∞ ≤

M ‖h‖ . Re λ ∞

(6.15)

6.1 Periodic solutions and S-asymptotically periodic solutions | 227

Proof. −(A + λI) generates a C0 -semigroup {S(t)}t≥0 , and S(t) = e−λt T(t). Then ‖S(t)‖L (X) = e− Re λt ‖T(t)‖L (X) ≤ Me− Re λt , so S(t) is exponentially stable for Re λ > 0. The conclusion follows by Theorem 6.2. Theorem 6.3. Let h ∈ Ω(X) ∈ M (X). Assume that −A generates an exponentially stable C0 -semigroup {T(t)}t≥0 , i. e., the growth bound ln ‖T(t)‖L (X) < 0. t

ν0 = lim sup t→+∞

Then linear fractional evolution equation (6.10) has a unique mild solution u := Rh ∈ Ω(X), R : Ω(X) → Ω(X) is bounded linear, and spectral radius r(R) ≤ |ν1 | . 0

Proof. By Theorem 6.2 and (i) of Lemma 6.2, we obtain that (6.10) has a unique mild solution u := Rh ∈ Ω(X), and R : Ω(X) → Ω(X) is a bounded linear. For all ν ∈ (0, |ν0 |), there exists M1 ≥ 1 such that 󵄩󵄩 󵄩 −νt 󵄩󵄩T(t)󵄩󵄩󵄩L (X) ≤ M1 e ,

t ≥ 0.

Define a new norm | ⋅ | in X as 󵄩 󵄩 |u| = sup󵄩󵄩󵄩eνt T(t)u󵄩󵄩󵄩. t≥0

Since ‖u‖ ≤ |u| ≤ M1 ‖u‖, then | ⋅ | is equivalent to ‖ ⋅ ‖. The norm of T(t) in X0 := (X, | ⋅ |) is denoted by |T(t)|L (X0 ) . Then for t ≥ 0, we have 󵄨󵄨 󵄨 󵄩 νs 󵄩 󵄨󵄨T(t)u󵄨󵄨󵄨 = sup󵄩󵄩󵄩e T(s)T(t)u󵄩󵄩󵄩 s≥0

󵄩 󵄩 = e−νt sup󵄩󵄩󵄩eν(s+t) T(s + t)u󵄩󵄩󵄩 s≥0

󵄩 󵄩 = e−νt sup󵄩󵄩󵄩eνη T(η)u󵄩󵄩󵄩 η≥t

≤e

−νt

|u|.

This implies that |T(t)|L (X0 ) ≤ e−νt . Proposition 1.11 implies that 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 α 󵄨󵄨V(t)󵄨󵄨L (X0 ) = 󵄨󵄨󵄨α ∫ θWα (θ)T(t θ)dθ󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 0 󵄨L (X0 ) ∞

α

≤ α ∫ θWα (θ)e−νt θ dθ 0

= Eα,α (−νt α ),

(6.16)

228 | 6 Fractional evolution equations on whole real axis where V is defined by (6.8). Since h ∈ Ω(X), we have |h|∞ := supt∈ℝ |h(t)| < ∞. By (6.16) and Proposition 1.10, we have 󵄨󵄨 󵄨󵄨 t 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 α−1 󵄨󵄨 (t − s) V(t − s)h(s)ds = (Rh)(t) ∫ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨−∞ t

≤ |h|∞ ∫ (t − s)α−1 Eα,α (−ν(t − s)α )ds −∞

󵄨t |h|∞ α 󵄨󵄨󵄨 = E (−ν(t − s) )󵄨󵄨 󵄨󵄨−∞ ν α |h|∞ , for t ≥ 0, = ν which implies that |Rh|∞ ≤ ν∞ . Then |R|L (X0 ) ≤ ν1 , and spectral radius r(R) ≤ ν1 . Since ν is any number in (0, |ν0 |), we obtain that r(R) ≤ |ν1 | . |h|

0

Remark 6.2. Similar to [167], if {T(t)}t≥0 is a compact and positive analytic semigroup and P is a regeneration cone, then ν0 = −λ1 , where λ1 is the first eigenvalue of A. Corollary 6.2. Assume that h ∈ Ω(X) and positive cone P ∈ X is a regeneration cone, compact and positive C0 -semigroup {T(t)}t≥0 is generated by −A, whose first eigenvalue λ1 = inf{Re λ | λ ∈ σ(A)} > 0.

(6.17)

Then (6.10) possesses a unique mild solution u := Rh ∈ Ω(X), R : Ω(X) → Ω(X) is positive and bounded linear, and the corresponding spectral radius r(R) = λ1 . 1

Proof. The proof is similar to that in [167].

6.1.4 Nonlinear equations Theorem 6.4. Let X be an ordered Banach space, whose positive cone P is normal with normal constant N. Assume that −A generates a positive C0 -semigroup {T(t)}t≥0 , Ω(X) ∈ M (X), f ∈ Ω(ℝ × X, X) ∈ M (ℝ × X, X) and satisfies the results shown in Table 6.1, f (t, θ) ≥ θ for ∀ t ∈ ℝ, and the following assumptions hold: (H1 ) for κ > 0, there is C = C(κ) > 0 that f (t, u2 ) − f (t, u1 ) ≥ −C(u2 − u1 ), where t ∈ ℝ, θ ≤ u1 ≤ u2 , ‖u1 ‖, ‖u2 ‖ ≤ κ;

6.1 Periodic solutions and S-asymptotically periodic solutions | 229

(H2 ) there exists L < −ν0 , such that f (t, u2 ) − f (t, u1 ) ≤ L(u2 − u1 ), where t ∈ ℝ, θ ≤ u1 ≤ u2 . Then equation (6.2) has a unique positive mild solution u ∈ Ω(X). Proof. Let h0 (t) = f (t, θ), then h0 ∈ Ω(X), h0 ≥ θ. Next, we consider linear equation α −∞ Dt u(t)

+ (A − LI)u(t) = h0 (t),

t ∈ ℝ.

(6.18)

We know that a positive C0 -semigroup eLt T(t) could be generated by −(A − LI), whose growth bound is L + ν0 < 0. From Theorem 6.3, linear equation (6.18) has a unique positive mild solution w0 ∈ Ω(X). If κ0 = N‖w0 ‖∞ + 1 and C is the constant in (H1 ). Without loss of generality, we may assume that C > max{ν0 , −L}. In the following part, we consider linear equation: α −∞ Dt u(t)

+ (A + CI)u(t) = h(t),

t ∈ ℝ.

(6.19)

The operator −(A + CI) is the generator of a positive C0 -semigroup T1 (t) = e−Ct T(t) with growth bound −C + ν0 < 0. From Theorem 6.3, for h ∈ Ω(X), linear equation (6.19) has a unique mild solution u := Q1 h, and Q1 : Ω(X) → Ω(X) is a positive bounded 1 linear operator, and spectral radius r(Q1 ) ≤ C−ν . 0 Set F(u) = f (t, u) + Cu, then by (ii) of Lemma 6.2, Lemma 6.1 and Theorem 6.1, it follows that F(θ) = h0 ≥ θ and Q1 F : Ω(X) → Ω(X) is continuous. By (H1 ), F is incremental on [θ, w0 ]. Set v0 ≡ θ, we can construct the sequences vn = (Q1 ∘ F)(vn−1 ),

wn = (Q1 ∘ F)(wn−1 ),

n = 1, 2, . . . .

(6.20)

By (6.19), we have that Q1 (h0 + Lw0 + Cw0 ) is another mild solution of (6.18). Since the mild solution of (6.18) is unique, we have w0 = Q1 (h0 + Lw0 + Cw0 ).

(6.21)

Let u1 = θ, u2 = w0 (t) in (H2 ), then f (t, w0 ) ≤ h0 (t) + Lw0 (t),

θ ≤ F(θ) ≤ F(w0 ) ≤ h0 + Lw0 + Cw0 .

By (6.21) and (6.22), and the definition and the positivity of Q1 , we obtain Q1 θ = θ = v0 ≤ v1 ≤ w1 ≤ w0 .

(6.22)

230 | 6 Fractional evolution equations on whole real axis Since Q1 ∘ F is an increasing operator on [θ, w0 ], in view of (6.20) we can show that θ ≤ v1 ≤ ⋅ ⋅ ⋅ ≤ vn ≤ ⋅ ⋅ ⋅ ≤ wn ≤ ⋅ ⋅ ⋅ ≤ w1 ≤ w0 .

(6.23)

Therefore, θ ≤ wn − vn = Q1 (F(wn−1 ) − F(vn−1 ))

= Q1 (f (⋅, wn−1 ) − f (⋅, vn−1 ) + C(wn−1 − vn−1 ))

≤ (C + L)Q1 (wn−1 − vn−1 ). By induction,

θ ≤ wn − vn ≤ (C + L)n Qn1 (w0 − v0 ) = (C + L)n Qn1 (w0 ). Since the cone P is normal, then we get 󵄩 󵄩 󵄩 󵄩 ‖wn − vn ‖∞ ≤ N(C + L)n 󵄩󵄩󵄩Qn1 (w0 )󵄩󵄩󵄩∞ ≤ N(C + L)n 󵄩󵄩󵄩Qn1 󵄩󵄩󵄩L (X) ‖w0 ‖∞ .

(6.24)

Moreover, 0 < C + L < C − ν0 for some ε > 0, so it follows that C + L + ε < C − ν0 . By 1

n = r(Q1 ) ≤ the Gelfand formula, limn→∞ ‖Qn1 ‖L (X)

‖Qn1 ‖L (X)

≤

1 (C+L+ε)n

1 . C−ν0

Then there exists N0 such that

for n ≥ N0 . Equation (6.24) implies that

‖wn − vn ‖∞ ≤ N‖w0 ‖∞ (

n

C+L ) → 0, C+L+ε

as n → ∞.

(6.25)

Combining (6.23) and (6.25), by the nested interval method, there is a unique u∗ ∈ ⋂∞ n=1 [vn , wn ] such that lim v n→∞ n

= lim wn = u∗ . n→∞

Since operator Q1 ∘ F is continuous, by (6.20) we have u∗ = (Q1 ∘ F)(u∗ ). It follows from the definition of Q1 and (6.23) that u∗ is a positive mild solution of (6.19) for h(t) = f (t, u∗ (t)) + Cu∗ (t). Hence, u∗ is a positive mild solution of (6.2). Finally, we prove the uniqueness. If u1 , u2 are the positive mild solutions of (6.2), substitute u1 and u2 for w0 , respectively, then wn = (Q1 ∘ F)(ui ) = ui (i = 1, 2). Equation (6.25) implies that ‖ui − vn ‖∞ → 0,

as n → ∞, i = 1, 2.

Thus, u1 = u2 = limn→∞ vn , (6.2) has a unique positive mild solution u ∈ Ω(X).

6.1 Periodic solutions and S-asymptotically periodic solutions | 231

From Theorem 6.4 and Remark 6.2, we obtain the following results. Corollary 6.3. Let X be an ordered Banach space, whose positive cone P is a regeneration cone. Assume that −A generates a compact and positive C0 -semigroup {T(t)}t≥0 , Ω(X) ∈ M (X), f ∈ Ω(ℝ × X, X) ∈ M (ℝ × X, X) and satisfies the results shown in Table 6.1, f (t, θ) ≥ θ for ∀ t ∈ ℝ, (H1 ) and the following condition are satisfied: (H2 )′ there exists L < λ1 (where λ1 is the first eigenvalue of A) such that f (t, u2 ) − f (t, u1 ) ≤ L(u2 − u1 ), for any t ∈ ℝ, θ ≤ u1 ≤ u2 . Then equation (6.2) has a unique positive mild solution u ∈ Ω(X). Theorem 6.5. Assume that −A generates an exponentially stable C0 -semigroup {T(t)}t≥0 and satisfies (6.4), Ω(X) ∈ M (X), f ∈ Ω(ℝ × X, X) ∈ M (ℝ × X, X) and satisfies the results shown in Table 6.1. If the following condition is satisfied: (H3 ) f (t, u) is Lipschitz continuous with respect to u, that is, there exists a constant L ≥ 0 such that 󵄩󵄩 󵄩 󵄩󵄩f (t, u2 ) − f (t, u1 )󵄩󵄩󵄩 ≤ L‖u2 − u1 ‖, for all t ∈ ℝ and u1 , u2 ∈ X, then equation (6.2) has a unique mild solution u ∈ Ω(X) for δ > ML. Proof. We define operator Q by t

(Qu)(t) := ∫ (t − s)α−1 V(t − s)f (s, u(s))ds,

t ∈ ℝ.

−∞

From (ii) of Lemma 6.2, Lemma 6.1 and Theorem 6.1, it follows that Q : Ω(X) → Ω(X) is continuous. By (H3 ), Proposition 1.10 and Lemma 6.2, for all t ∈ ℝ, u1 , u2 ∈ Ω(X), we get t

󵄩󵄩 󵄩 󵄩 α−1 󵄩 󵄩󵄩(Qu2 )(t) − (Qu1 )(t)󵄩󵄩󵄩 ≤ ∫ (t − s) 󵄩󵄩󵄩V(t − s)[f (s, u2 (s)) − f (s, u1 (s))]󵄩󵄩󵄩ds −∞

t

≤ ML‖u2 − u1 ‖∞ ∫ (t − s)α−1 Eα,α (−δ(t − s)α )ds −∞

󵄨󵄨t ML 󵄨 ‖u2 − u1 ‖∞ Eα (−δ(t − s)α )󵄨󵄨󵄨 󵄨󵄨−∞ δ ML ‖u2 − u1 ‖∞ . = δ =

232 | 6 Fractional evolution equations on whole real axis Thus, ‖Qu2 − Qu1 ‖∞ ≤

ML ‖u2 − u1 ‖∞ . δ

When δ > ML, Q is a contraction in Ω(X). Using Banach fixed-point theorem, we have that Q has a unique fixed point in Ω(X). This completes the proof. Corollary 6.4. Assume that −A generates a uniformly bounded C0 -semigroup {T(t)}t≥0 and satisfies (6.7), Ω(X) ∈ M (X), f ∈ Ω(ℝ × X, X) ∈ M (ℝ × X, X) and satisfies the results shown in Table 6.1. If f satisfies (H3 ), then equation (6.2) has a unique mild solution u ∈ Ω(X) for Re λ > ML. Proof. Similar to the proof in Corollary 6.1, we know that −(A + λI) generates an exponentially stable C0 -semigroup {S(t)}t≥0 . By Theorem 6.5, equation (6.2) possesses a unique mild solution for Re λ > ML. Theorem 6.6. Assume that −A generates an exponentially stable C0 -semigroup {T(t)}t≥0 and satisfies (6.4), Ω(X) ∈ M (X), f ∈ Ω(ℝ × X, X) ∈ M (ℝ × X, X) and satisfies the results shown in Table 6.1. Assume that the following condition is satisfied: (H4 ) f (t, u) is local Lipschitz continuous in u: for all r > 0, there exists L(r) > 0 such that 󵄩󵄩 󵄩 󵄩󵄩f (t, u2 ) − f (t, u1 )󵄩󵄩󵄩 ≤ L(r)‖u2 − u1 ‖, for t ∈ ℝ, ‖u1 ‖, ‖u2 ‖ ≤ r. Denote f0 = f (t, θ), then for δ > ML(r)+ Mr ‖f0 ‖∞ equation (6.2) has only one mild solution in B(θ, r) = {u ∈ Ω(X) | ‖u‖∞ < r}. Proof. Let F(u) = f (t, u), we know that the mild solution of equation (6.2) is the fixed point of R ∘ F in B(θ, r). For any u ∈ B(θ, r), by (6.13) and (H4 ), we have 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩R(F(u))󵄩󵄩󵄩∞ ≤ 󵄩󵄩󵄩R(F(θ))󵄩󵄩󵄩∞ + 󵄩󵄩󵄩R(F(u) − F(θ))󵄩󵄩󵄩∞ M‖f0 ‖∞ M‖F(u) − F(θ)‖∞ + ≤ δ δ M‖f0 ‖∞ + ML(r)r ≤ < r, δ where δ > ML(r)+ Mr ‖f0 ‖∞ . By (ii) of Lemma 6.2, Lemma 6.1 and Theorem 6.1, we obtain that R ∘ F : B(θ, r) → B(θ, r) is continuous. On the other hand, for ∀ u1 , u2 ∈ B, by (H4 ), we get ML(r) 󵄩󵄩 󵄩 󵄩 󵄩 ‖u2 − u1 ‖∞ . 󵄩󵄩R(F(u2 )) − R(F(u1 ))󵄩󵄩󵄩∞ ≤ 󵄩󵄩󵄩R(F(u2 ) − F(u1 ))󵄩󵄩󵄩∞ ≤ δ For δ > ML(r) +

M ‖f ‖ , r 0 ∞

we have

ML(r) δ

< 1. Thus, R is a contraction in Ω(X).

6.1 Periodic solutions and S-asymptotically periodic solutions | 233

Banach fixed-point theorem implies that there is only one ũ ∈ B(θ, r) such that ̃ = u.̃ Therefore, equation (6.2) has only one mild solution in B(θ, r). R(F(u)) Corollary 6.5. Assume that −A generates a uniformly bounded C0 -semigroup {T(t)}t≥0 and satisfies (6.7), Ω(X) ∈ M (X), f ∈ Ω(ℝ × X, X) ∈ M (ℝ × X, X), and satisfies the results shown in Table 6.1 and (H4 ). Set f0 = f (t, θ), then equation (6.2) has only one mild solution in B(θ, r) = {u ∈ Ω(X) | ‖u‖∞ < r} for Re λ > ML(r) + Mr ‖f0 ‖∞ . Proof. Similar to the proof in Corollary 6.1, we know that −(A + λI) generates an exponentially stable C0 -semigroup {S(t)}t≥0 . By Theorem 6.6, equation (6.2) has only one mild solution in B(θ, r) for Re λ > ML(r) + Mr ‖f0 ‖∞ . 6.1.5 Examples ̄ we discuss briefly the existence of mild solutions of the Example 6.1. Let X = C0 (Ω), following fractional parabolic partial differential equation 1

{−∞ Dt2 u − Δu = g(x, t, u(x, t)), { {u|𝜕Ω = 0,

(x, t) ∈ Ω × ℝN ,

(6.26)

1

where −∞Dt2 is Liouville–Weyl fractional partial derivative of order 21 with the lower limit −∞, Ω ⊂ ℝN is bounded and boundary 𝜕Ω is sufficiently smooth, Δ is the Laplace operator. Theorem 6.7. Let g(⋅, t, u(⋅, t)) ∈ Ω(X) ∈ M (X), and g(x, t, 0) ≥ 0. Assume that g has continuous partial derivatives for u in arbitrary bounded domain, and the supremum of gu (x, t, u) is smaller than λ1 , where λ1 is the first eigenvalue of Laplace operator −Δ with u|𝜕Ω = 0. Then equation (6.26) possesses only one positive mild solution u(⋅, t) ∈ Ω(X). Proof. Let K = C0+ (Ω)̄ := {f ∈ C(Ω,̄ ℝ+ ) | f |𝜕Ω = 0}, then K is a positive cone in X. Define operator A in X as follows: D(A) = {u | u ∈ X, Δu ∈ X},

Au = −Δu.

In view of [169], −A generates a compact and analytic semigroup {T(t)}t≥0 . Thus, equation (6.26) can be formulated as the abstract fractional partial differential equation (6.2), where f (t, u) := g(⋅, t, u(⋅, t)). By the maximum principle of parabolic equations, {T(t)}t≥0 is a positive C0 -semigroup. It is easy to see that f satisfies (H1 ) and (H2 )′ . By Corollary 6.3, (6.26) has only one positive mild solution u(⋅, t) ∈ Ω(X). Example 6.2. Consider the problem 1

2

{−∞ Dt2 u(x, t) = 𝜕 u(x,t) + b(t) sin u(x, t), 𝜕x2 { {u(0, t) = u(2π, t) = 0,

(x, t) ∈ [0, 2π] × ℝ,

(6.27)

234 | 6 Fractional evolution equations on whole real axis 1

where −∞Dt2 is Liouville–Weyl fractional partial derivative of order limit −∞, b ∈ Cb (ℝ, ℝ). Proof. Set X = L2 [0, 2π], A :=

𝜕2 𝜕x2

1 2

with the lower

and

D(A) = {g ∈ H 2 [0, 2π] | g(0) = g(2π) = 0}. For f (t, u) = b(t) sin u for u ∈ X, t ∈ ℝ, it follows that f ∈ Ω(X) ∈ M (X) for any u ∈ X, with π

󵄩󵄩 󵄩2 󵄨 󵄨2 󵄨 󵄨2 2 2 󵄩󵄩f (t, u1 ) − f (t, u2 )󵄩󵄩󵄩2 ≤ ∫󵄨󵄨󵄨b(t)󵄨󵄨󵄨 󵄨󵄨󵄨sin u1 (s) − sin u2 (s)󵄨󵄨󵄨 ds ≤ ‖b‖∞ ‖u1 − u2 ‖2 , 0

for u1 , u2 ∈ X. In consequence, (6.27) has only one mild solution u(⋅, t) ∈ Ω(X) (by Theorem 6.5).

6.2 Asymptotically almost periodic solutions 6.2.1 Introduction The theory of almost periodic functions was introduced in the literature around 1924–1926 with the pioneering works of the Danish mathematician, Bohr [42, 43]. Loosely speaking, almost periodic functions are those functions which come arbitrarily close to being periodic when one looks over long enough time scales, they play an important role in describing the phenomena that are similar to the periodic oscillations, which can be observed frequently in many fields, such as celestial mechanics, nonlinear vibration, electromagnetic theory, plasma physics, engineering, ecosphere and so on [40, 197, 199]. As a natural extension of almost periodicity, the concept of asymptotic almost periodicity, which was the central issue to be discussed in this section, was introduced in the literature [97, 98] by Fréchet in the early 1940s. Since then, the theory of asymptotically almost periodic functions and their various extensions have attracted a great deal of attention of many mathematicians due to both their mathematical interest and significance as well as applications in physics, mathematical biology, control theory and so forth. In particular, asymptotically almost periodic functions have been utilized to study various ordinary differential equations, partial differential equations, functional differential equations, integrodifferential equations as well as stochastic differential equations (see, for instance, [20, 56, 75, 80, 178, 219, 224] and the references therein), and due to their significance and applications in control theory, mathematical biology and physics, etc., the study of asymptotically almost periodic solutions to various differential equations becomes an attractive topic in the qualitative theory of differential equations.

6.2 Asymptotically almost periodic solutions | 235

The study of almost periodic type solutions to fractional differential equations was initiated by Araya and Lizama [18]. In their work, they investigated the existence and uniqueness of an almost automorphic mild solution of the semilinear fractional differential equation α 0 Dt x(t)

= Ax(t) + F(t, x(t)),

t ∈ ℝ, 1 < α < 2,

when A is a generator of an α-resolvent family and 0 Dαt is the Riemann–Liouville fractional derivative. For more on almost periodic type solutions to fractional differential equations, one can refer to [7, 62, 63, 71, 151, 165, 168, 212] and the references therein. In Section 6.1, we established some sufficient conditions for the existence and uniqueness of periodic solutions and S-asymptotically periodic solutions when F(t, x) satisfies some ordered or Lipschitz conditions. In this section, the nonlinearity F(t, x) does not have to satisfy a Lipschitz condition with respect to x (see Remark 6.3). In particular, as application and to illustrate the feasibility of the abstract result, we will examine some sufficient conditions for the existence of asymptotically almost periodic mild solutions to the fractional partial differential equation given by 𝜕tα u(t, x) = 𝜕x2 u(t, x) + μ(sin t + sin √2t) sin u(t, x) + νe−|t| u(t, x) sin u2 (t, x), t ∈ ℝ, x ∈ [0, π],

with Dirichlet boundary conditions u(t, 0) = u(t, π) = 0, t ∈ ℝ, where μ and ν are constants. The rest of this section is organized as follows. In Subsection 6.2.2, some concepts, the related notations and some useful lemmas are introduced. In Subsection 6.2.3, we present some criteria ensuring the existence of asymptotically almost periodic mild solutions. An example is given to illustrate the feasibility of the abstract result in Subsection 6.2.4. This section is based on [53].

6.2.2 Preliminaries This subsection is concerned with some notations, definitions, lemmas and preliminary facts which are used in what follows. From now on, ℕ, ℤ, ℝ and ℂ stand for the set of natural numbers, integral numbers, real numbers and complex numbers, respectively. Let (X, ‖ ⋅ ‖), (Y, ‖ ⋅ ‖Y ) be two Banach spaces, BC(ℝ, X) (resp., BC(ℝ × Y, X)) is the space of all X-valued bounded continuous functions (resp., jointly bounded continuous functions F : ℝ × Y → X). Furthermore, C0 (ℝ, X) (resp., C0 (ℝ × Y, X)) is the closed subspace of BC(ℝ, X) (resp., BC(ℝ × Y, X)) consisting of functions vanishing at infinity (vanishing at infinity uni-

236 | 6 Fractional evolution equations on whole real axis formly in any compact subset of Y, and in other words, 󵄩 󵄩 lim 󵄩󵄩󵄩g(t, x)󵄩󵄩󵄩 = 0

|t|→+∞

uniformly for x ∈ 𝕂,

where 𝕂 is an any compact subset of Y). Let also 𝕃(X) be the Banach space of all bounded linear operators from X into itself endowed with the norm: ‖T‖𝕃(X) = sup{‖Tx‖ : x ∈ X, ‖x‖ = 1}. First, we recall some basic definitions and results on almost periodic and asymptotically almost periodic functions. Definition 6.2 ([42, 43]). A continuous function F : ℝ → X is said to be (Bohr) almost periodic in t ∈ ℝ if for every ε > 0, there exists l(ε) > 0 such that every interval of length l(ε) contains a number τ with the property that 󵄩󵄩 󵄩 󵄩󵄩F(t + τ) − F(t)󵄩󵄩󵄩 < ε,

for every t ∈ ℝ.

The number τ is called an ε-translation number of F(t) and the collection of those functions is denoted by AP(ℝ, X). Lemma 6.4 ([219]). The space AP(ℝ, X) is a Banach space with the norm ‖F‖∞ = supt∈ℝ ‖F(t)‖. Definition 6.3 ([219]). A function F : ℝ × Y → X is said to be almost periodic if F(t, x) is almost periodic in t ∈ ℝ uniformly for x ∈ 𝕂, where 𝕂 is any compact subset of Y. The collection of those functions is denoted by AP(ℝ × Y, X). Lemma 6.5 ([219]). Let F(t, x) ∈ AP(ℝ×X, X) and φ(t) ∈ AP(ℝ, X), then Φ(t) = F(t, φ(t)) belongs to AP(ℝ, X). Definition 6.4 ([97, 98]). A continuous function F : ℝ → X is said to be asymptotically almost periodic if it can be decomposed as F(t) = G(t) + Φ(t), where G(t) ∈ AP(ℝ, X),

Φ(t) ∈ C0 (ℝ, X).

Denote by AAP(ℝ, X) the set of all such functions. Lemma 6.6 ([219]). The space AAP(ℝ, X) is also a Banach space with the norm ‖ ⋅ ‖∞ . Definition 6.5 ([219]). A function F : ℝ × Y → X is said to be asymptotically almost periodic if it can be decomposed as F(t, x) = G(t, x) + Φ(t, x), where G(t, x) ∈ AP(ℝ × Y, X),

Φ(t, x) ∈ C0 (ℝ × Y, X).

Denote by AAP(ℝ × Y, X) the set of all such functions.

6.2 Asymptotically almost periodic solutions | 237

6.2.3 Existence In this subsection, we study the existence of asymptotically almost periodic mild solutions for the fractional evolution equations in a Banach space X of the form α −∞Dt x(t)

+ Ax(t) = F(t, x(t)),

t ∈ ℝ,

(6.28)

where the fractional derivative is understood in the Liouville–Weyl sense, 0 < α < 1, −A is the infinitesimal generator of a C0 -semigroup on X and F : ℝ × X → X is a given function to be specified later. Let ∞

V(t) = α ∫ θWα (θ)T(t α θ)dθ,

t ≥ 0,

(6.29)

0

where Wα (θ) is Wright function which is defined in Definition 1.9 and {T(t)}t≥0 is a C0 -semigroup. Definition 6.6. A function x : ℝ → X is said to be a mild solution to equation (6.28) if t

x(t) = ∫ (t − s)α−1 V(t − s)F(s, x(s))ds,

t ∈ ℝ,

−∞

where V(t) is given by (6.29). In the proof of our result, we need the following auxiliary results. Lemma 6.7. Assume that {T(t)}t≥0 is exponentially stable satisfying (6.4). Given Y(t) ∈ AP(ℝ, X). Let t

Φ1 (t) := ∫ (t − s)α−1 V(t − s)Y(s)ds,

t ∈ ℝ.

−∞

Then Φ1 (t) ∈ AP(ℝ, X). Proof. Firstly, from (iv) of Lemma 6.2, Proposition 1.10 and Eα (0) = 1 it follows that 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩 α−1 α−1 󵄩 󵄩 󵄩󵄩 Φ (t) = (t − s) V(t − s)Y(s)ds ≤ ‖Y‖ (t − s) V(t − s)ds ∫ ∫ 󵄩󵄩 1 󵄩󵄩 󵄩󵄩 󵄩󵄩 ∞󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩−∞ 󵄩󵄩𝕃(X) 󵄩−∞ 󵄩 t

≤ M‖Y‖∞ ∫ (t − s)α−1 Eα,α (−δ(t − s)α )ds = −∞

M‖Y‖∞ = , δ

󵄨󵄨t M 󵄨 ‖Y‖∞ Eα (−δ(t − s)α )󵄨󵄨󵄨 󵄨󵄨−∞ δ

238 | 6 Fractional evolution equations on whole real axis which implies Φ1 (t) is well-defined and continuous on ℝ. Since Y(t) ∈ AP(ℝ, X), then for every ε > 0 there exists l(ε) > 0 such that every interval of length l(ε) contains a number τ with the property that 󵄩 󵄩󵄩 󵄩󵄩Y(s + τ) − Y(s)󵄩󵄩󵄩 < ε,

for every s ∈ ℝ,

which together with (iv) of Lemma 6.2, Proposition 1.10 and Eα (0) = 1, implies that 󵄩󵄩 󵄩 󵄩󵄩Φ1 (t + τ) − Φ1 (t)󵄩󵄩󵄩 t 󵄩󵄩 󵄩󵄩󵄩 t+τ 󵄩󵄩 󵄩󵄩 α−1 = 󵄩󵄩 ∫ (t + τ − s) V(t + τ − s)Y(s)ds − ∫ (t − s)α−1 V(t − s)Y(s)ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩−∞ 󵄩 −∞ t 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 󵄩󵄩 = 󵄩󵄩󵄩 ∫ (t − s)α−1 V(t − s)Y(s + τ)ds − ∫ (t − s)α−1 V(t − s)Y(s)ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩−∞ 󵄩 −∞ t t 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 ≤ ∫ 󵄩󵄩󵄩(t − s)α−1 V(t − s)[Y(s + τ) − Y(s)]󵄩󵄩󵄩ds ≤ ε󵄩󵄩󵄩 ∫ (t − s)α−1 V(t − s)ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩−∞ 󵄩𝕃(X) −∞ t

≤ Mε ∫ (t − s)α−1 Eα,α (−δ(t − s)α )ds = −∞

󵄨󵄨t Mε 1 󵄨 MεEα (−δ(t − s)α )󵄨󵄨󵄨 , = 󵄨󵄨−∞ δ δ

which implies Φ1 (t) ∈ AP(ℝ, X). Lemma 6.8. Assume that {T(t)}t≥0 is exponentially stable satisfying (6.4). Given Z(t) ∈ C0 (ℝ, X). Let t

Φ2 (t) := ∫ (t − s)α−1 V(t − s)Z(s)ds,

t ∈ ℝ.

−∞

Then Φ2 (t) ∈ C0 (ℝ, X). Proof. First, similar to the proof of Lemma 6.7, it is easy to see that Φ2 (t) is well-defined and continuous on ℝ. Since Z(t) ∈ C0 (ℝ, X), one can choose an N1 > 0, such that ‖Z(t)‖ < ε for all t > N1 . This together with (iv) of Lemma 6.2, Proposition 1.10 and Eα (0) = 1, enables us to conclude that for all t > N1 , 󵄩󵄩 N1 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩󵄩 t 󵄩󵄩 󵄩 󵄩󵄩󵄩 󵄩󵄩 󵄩 󵄩 α−1 α−1 󵄩 󵄩󵄩Φ2 (t)󵄩󵄩 ≤ 󵄩󵄩 ∫ (t − s) V(t − s)Z(s)ds󵄩󵄩󵄩 + 󵄩󵄩󵄩 ∫ (t − s) V(t − s)Z(s)ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩−∞ 󵄩 󵄩N1 󵄩 󵄩󵄩 N1 󵄩󵄩 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 ≤ ‖Z‖∞ 󵄩󵄩󵄩 ∫ (t − s)α−1 V(t − s)ds󵄩󵄩󵄩 + ε󵄩󵄩󵄩 ∫ (t − s)α−1 V(t − s)ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩−∞ 󵄩𝕃(X) 󵄩N1 󵄩𝕃(X)

6.2 Asymptotically almost periodic solutions | 239 N1

≤ M‖Z‖∞ ∫ (t − s)α−1 Eα,α (−δ(t − s)α )ds −∞ t

+ Mε ∫ (t − s)α−1 Eα,α (−δ(t − s)α )ds N1

󵄨󵄨N1 󵄨󵄨t 1 1 󵄨 󵄨 M‖Z‖∞ Eα (−δ(t − s)α )󵄨󵄨󵄨 + MεEα (−δ(t − s)α )󵄨󵄨󵄨 󵄨󵄨−∞ δ 󵄨󵄨N1 δ M Mε ≤ ‖Z‖∞ Eα (−δ(t − N1 )α ) + , δ δ =

which together with (iii) of Proposition 1.10, implies that limt→+∞ ‖Φ2 (t)‖ = 0. On the other hand, from Z(t) ∈ C0 (ℝ, X) it follows that there exists an N2 > 0 such that ‖Z(t)‖ < ε for all t < −N2 . This together with (iv) of Lemma 6.2, Proposition 1.10 and Eα (0) = 1, enables us to conclude that for all t < −N2 , 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 α−1 α−1 󵄩󵄩Φ2 (t)󵄩󵄩󵄩 = 󵄩󵄩󵄩 ∫ (t − s) V(t − s)Z(s)ds󵄩󵄩󵄩 ≤ ε󵄩󵄩󵄩 ∫ (t − s) V(t − s)ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩−∞ 󵄩 󵄩−∞ 󵄩𝕃(X) t

≤ Mε ∫ (t − s)α−1 Eα,α (−δ(t − s)α )ds = −∞

󵄨󵄨t Mε 1 󵄨 MεEα (−δ(t − s)α )󵄨󵄨󵄨 , = 󵄨󵄨−∞ δ δ

which implies that limt→−∞ ‖Φ2 (t)‖ = 0. Now we are in position to state and prove our main result. To prove the result, let us introduce the following assumptions: (H1 ) F(t, x) = F1 (t, x) + F2 (t, x) ∈ AAP(ℝ × X, X) with F1 (t, x) ∈ AP(ℝ × X, X),

F2 (t, x) ∈ C0 (ℝ × X, X)

and there exists a constant L > 0, such that 󵄩󵄩 󵄩 󵄩󵄩F1 (t, x) − F1 (t, y)󵄩󵄩󵄩 ≤ L‖x − y‖,

for all t ∈ ℝ, x, y ∈ X.

(6.30)

(H2 ) There exist a function β(t) ∈ C0 (ℝ, ℝ+ ) and a nondecreasing function Φ : ℝ+ → ℝ+ , such that for all t ∈ ℝ and x ∈ X with ‖x‖ ≤ r, 󵄩󵄩 󵄩 󵄩󵄩F2 (t, x)󵄩󵄩󵄩 ≤ β(t)Φ(r)

and

lim inf r→+∞

Φ(r) = ρ1 . r

(6.31)

Remark 6.3. Assuming that F(t, x) satisfies the assumption (H1 ), it is noted that F(t, x) does not have to meet the Lipschitz continuity with respect to x. Such class of asymptotically almost periodic functions F(t, x) is more complicated than those with Lipschitz continuity and little is known about them.

240 | 6 Fractional evolution equations on whole real axis Lemma 6.9. Given F(t, x) = F1 (t, x) + F2 (t, x) ∈ AAP(ℝ × X, X) with F1 (t, x) ∈ AP(ℝ × X, X),

F2 (t, x) ∈ C0 (ℝ × X, X).

Then it yields that 󵄩 󵄩 󵄩 󵄩 sup󵄩󵄩󵄩F1 (t, x) − F1 (t, y)󵄩󵄩󵄩 ≤ sup󵄩󵄩󵄩F(t, x) − F(t, y)󵄩󵄩󵄩, t∈ℝ

t∈ℝ

x, y ∈ X.

(6.32)

Proof. To show this result, it suffices to verify that {F1 (t, x) − F1 (t, y) : t ∈ ℝ} ⊂ {F(t, x) − F(t, y) : t ∈ ℝ},

x, y ∈ X.

If this is not the case, then, for fixed x, y ∈ X, there exist some t0 ∈ ℝ and ε > 0, such that 󵄩󵄩 󵄩 󵄩󵄩(F1 (t0 , x) − F1 (t0 , y)) − (F(t, x) − F(t, y))󵄩󵄩󵄩 ≥ 3ε,

for all t ∈ ℝ.

It is clear that limt→+∞ ‖F2 (t, x)−F2 (t, y)‖ = 0, which implies that there exists a positive number T such that for all t ≥ T, 󵄩󵄩 󵄩 󵄩󵄩F2 (t, x) − F2 (t, y)󵄩󵄩󵄩 < ε.

(6.33)

Since F1 (t, x) ∈ AP(ℝ × X, X), one can take l = l(ε) > 0, such that [T, T + l] of length l contains at least a τ with the properties 󵄩󵄩 󵄩 󵄩󵄩F1 (t0 + τ, x) − F1 (t0 , x)󵄩󵄩󵄩 < ε,

󵄩󵄩 󵄩 󵄩󵄩F1 (t0 + τ, y) − F1 (t0 , y)󵄩󵄩󵄩 < ε,

which enable us to find that 󵄩󵄩 󵄩 󵄩󵄩F2 (t0 + τ, x) − F2 (t0 + τ, y)󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≥ 󵄩󵄩󵄩F(t0 + τ, x) − F(t0 + τ, y) − F1 (t0 , x) + F1 (t0 , y)󵄩󵄩󵄩 − 󵄩󵄩󵄩F1 (t0 + τ, x) − F1 (t0 , x)󵄩󵄩󵄩 󵄩 󵄩 − 󵄩󵄩󵄩F1 (t0 + τ, y) − F1 (t0 , y)󵄩󵄩󵄩 ≥ ε, which contradicts (6.33), completing the proof. Let β(t) be the function involved in assumption (H2 ). Define t

σ(t) := ∫ β(s)(t − s)α−1 Eα,α (−δ(t − s)α )ds, −∞

Lemma 6.10. The function σ(t) ∈ C0 (ℝ, ℝ+ ).

t ∈ ℝ.

6.2 Asymptotically almost periodic solutions | 241

Proof. Since β(t) ∈ C0 (ℝ, ℝ+ ), one can choose a T1 > 0 such that ‖β(t)‖ < ε for all t > T1 . This together with Proposition 1.10 and Eα (0) = 1, enables us to conclude that for all t > T1 , T1

σ(t) ≤ ∫ β(s)(t − s)α−1 Eα,α (−δ(t − s)α )ds −∞ t

+ ∫ β(s)(t − s)α−1 Eα,α (−δ(t − s)α )ds T1 T1

≤ ‖β‖∞ ∫ (t − s)α−1 Eα,α (−δ(t − s)α )ds −∞ t

+ ε ∫(t − s)α−1 Eα,α (−δ(t − s)α )ds T1

󵄨󵄨T1 󵄨󵄨t 1 ε 󵄨 󵄨 ‖β‖∞ Eα (−δ(t − s)α )󵄨󵄨󵄨 + Eα (−δ(t − s)α )󵄨󵄨󵄨 󵄨󵄨−∞ δ 󵄨󵄨T1 δ ε 1 ≤ ‖β‖∞ Eα (−δ(t − T1 )α ) + , δ δ

=

which together with Proposition 1.10, implies limt→+∞ σ(t) = 0. On the other hand, from β(t) ∈ C0 (ℝ, ℝ+ ) it follows that there exists a T2 > 0 such that ‖β(t)‖ < ε for all t < −T2 . This together with Proposition 1.10 and Eα (0) = 1, enables us to conclude that for all t < −T2 , t

σ(t) ≤ ε ∫ (t − s)α−1 Eα,α (−δ(t − s)α )ds = −∞

󵄨󵄨t ε ε 󵄨 Eα (−δ(t − s)α )󵄨󵄨󵄨 = , 󵄨󵄨−∞ δ δ

which implies limt→−∞ σ(t) = 0. Theorem 6.8. Assume that −A generates an exponentially stable C0 -semigroup {T(t)}t≥0 satisfying (6.4). Let F : ℝ × X → X satisfy the hypotheses (H1 ) and (H2 ). Put ρ2 := supt∈ℝ σ(t). Then equation (6.28) has at least one asymptotically almost periodic mild solution whenever MLδ−1 + Mρ1 ρ2 < 1.

(6.34)

Proof. Consider the coupled system of integral equations t

v(t) = ∫−∞ (t − s)α−1 V(t − s)F1 (s, v(s))ds, t ∈ ℝ, { { { t ω(t) = ∫−∞ (t − s)α−1 V(t − s)[F1 (s, v(s) + ω(s)) − F1 (s, v(s))]ds { { { t + ∫−∞ (t − s)α−1 V(t − s)F2 (s, v(s) + ω(s))ds, t ∈ ℝ. {

(6.35)

242 | 6 Fractional evolution equations on whole real axis If (v(t), ω(t)) ∈ AP(ℝ, X)×C0 (ℝ, X) is a solution to system (6.35), then x(t) := v(t)+ω(t) ∈ AAP(ℝ, X) and it is a solution to the integral equation t

x(t) = ∫ (t − s)α−1 V(t − s)F(s, x(s))ds,

t ∈ ℝ,

−∞

i. e., x(t) is an asymptotically almost periodic mild solution to equation (6.28). Hence, the problem has shifted to show that system (6.35) has at least a solution in AP(ℝ, X) ×

C0 (ℝ, X).

Define a mapping Λ on AP(ℝ, X) by t

(Λv)(t) = ∫ (t − s)α−1 V(t − s)F1 (s, v(s))ds,

t ∈ ℝ.

−∞

First, since the function s → F1 (s, v(s)) is bounded on ℝ and 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 α−1 󵄩󵄩(Λv)(t)󵄩󵄩󵄩 = 󵄩󵄩󵄩 ∫ (t − s) V(t − s)F1 (s, v(s))ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩−∞ 󵄩 t 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩 󵄩 ≤ ‖F1 ‖∞ 󵄩󵄩󵄩 ∫ (t − s)α−1 V(t − s)ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩−∞ 󵄩𝕃(X) t

≤ M‖F1 ‖∞ ∫ (t − s)α−1 Eα,α (−δ(t − s)α )ds −∞

󵄨󵄨t M 󵄨 ‖F1 ‖∞ Eα (−δ(t − s)α )󵄨󵄨󵄨 󵄨󵄨−∞ δ M‖F1 ‖∞ , = δ =

which implies that (Λv)(t) exists. Moreover, from F1 (t, x) ∈ AP(ℝ × X, X) satisfying (6.30), together with Lemma 6.5, it follows that F1 (⋅, v(⋅)) ∈ AP(ℝ, X),

for every v(⋅) ∈ AP(ℝ, X).

This, together with Lemma 6.7, implies that Λ is well-defined and maps AP(ℝ, X) into itself.

In the sequel, we verify Λ is continuous.

6.2 Asymptotically almost periodic solutions | 243

Let vn (t), v(t) be in AP(ℝ, X) with vn (t) → v(t) as n → ∞, then one has 󵄩󵄩 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩 α−1 󵄩󵄩(Λvn )(t) − (Λv)(t)󵄩󵄩 = 󵄩󵄩 ∫ (t − s) V(t − s)[F1 (s, vn (s)) − F1 (s, v(s))]ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩−∞ t

󵄩 󵄩 󵄩 󵄩 ≤ L ∫ (t − s)α−1 󵄩󵄩󵄩V(t − s)󵄩󵄩󵄩𝕃(X) 󵄩󵄩󵄩νn (s) − ν(s)󵄩󵄩󵄩ds −∞

t

≤ ML‖vn − v‖∞ ∫ (t − s)α−1 Eα,α (−δ(t − s)α )ds −∞

ML‖vn − v‖∞ 1 󵄨t = ML‖vn − v‖∞ Eα (−δ(t − s)α )󵄨󵄨󵄨−∞ = . δ δ Therefore, as n → ∞, Λvn → Λv, hence Λ is continuous. Next, we prove that Λ is a contraction on AP(ℝ, X) and has a unique fixed point v(t) ∈ AP(ℝ, X). Let v1 (t), v2 (t) be in AP(ℝ, X), similar to the above proof of the continuity of Λ, one has 󵄩󵄩 󵄩 ML ‖v − v2 ‖∞ , 󵄩󵄩(Λv1 )(t) − (Λv2 )(t)󵄩󵄩󵄩 ≤ δ 1 which implies ML 󵄩󵄩 󵄩 ‖v − v2 ‖∞ . 󵄩󵄩Λv1 − Λv2 󵄩󵄩󵄩∞ ≤ δ 1 This together with (6.34), proves that Λ is a contraction on AP(ℝ, X). Thus, the Banach’s fixed-point theorem implies that Λ has a unique fixed point v(t) ∈ AP(ℝ, X). For the above v(t), define Γ := Γ1 + Γ2 on C0 (ℝ, X) as follows: t

1

(Γ ω)(t) = ∫ (t − s)α−1 V(t − s)[F1 (s, v(s) + ω(s)) − F1 (s, v(s))]ds,

t ∈ ℝ,

−∞ t

2

(Γ ω)(t) = ∫ (t − s)α−1 V(t − s)F2 (s, v(s) + ω(s))ds,

t ∈ ℝ.

−∞

First, from (6.30) it follows that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩F1 (s, v(s) + ω(s)) − F1 (s, v(s))󵄩󵄩󵄩 ≤ L󵄩󵄩󵄩ω(s)󵄩󵄩󵄩,

for all s ∈ ℝ, ω(s) ∈ X,

which implies that F1 (⋅, v(⋅) + ω(⋅)) − F1 (⋅, v(⋅)) ∈ C0 (ℝ, X),

for every ω(⋅) ∈ C0 (ℝ, X).

244 | 6 Fractional evolution equations on whole real axis According to (6.31), one has 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩F2 (s, v(s) + ω(s))󵄩󵄩󵄩 ≤ β(s)Φ(r + sup󵄩󵄩󵄩v(s)󵄩󵄩󵄩) s∈ℝ

for all s ∈ ℝ and ω(s) ∈ X with ‖ω(s)‖ ≤ r, then F2 (⋅, v(⋅) + ω(⋅)) ∈ C0 (ℝ, X) as β(⋅) ∈ C0 (ℝ, ℝ+ ). Those, together with Lemma 6.8, yield that Γ is well-defined and maps C0 (ℝ, X) into itself. To complete the proof, it suffices to prove that Γ has at least one fixed point in C0 (ℝ, X). Set Ωr := {ω(t) ∈ C0 (ℝ, X) : ‖ω‖∞ ≤ r}. In view of (6.31) and (6.34), it is not difficult to see that there exists a constant k0 > 0, such that ML 󵄩 󵄩 k + Mρ2 Φ(k0 + sup󵄩󵄩󵄩v(s)󵄩󵄩󵄩) ≤ k0 . δ 0 s∈ℝ This enables us to conclude that for any t ∈ ℝ and ω1 (t), ω2 (t) ∈ Ωk0 , 󵄩󵄩 1 󵄩 2 󵄩󵄩(Γ ω1 )(t) + (Γ ω2 )(t)󵄩󵄩󵄩 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 󵄩󵄩 ≤ 󵄩󵄩󵄩 ∫ (t − s)α−1 V(t − s)[F1 (s, v(s) + ω1 (s)) − F1 (s, v(s))]ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩−∞ 󵄩 t 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩 󵄩 + 󵄩󵄩󵄩 ∫ (t − s)α−1 V(t − s)F2 (s, v(s) + ω2 (s))ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩−∞ 󵄩 t

󵄩 󵄩 ≤ ∫ 󵄩󵄩󵄩(t − s)α−1 V(t − s)[F1 (s, v(s) + ω1 (s)) − F1 (s, v(s))]󵄩󵄩󵄩ds −∞

t

󵄩 󵄩 + ∫ 󵄩󵄩󵄩(t − s)α−1 V(t − s)F2 (s, v(s) + ω2 (s))󵄩󵄩󵄩ds −∞ t

󵄩 󵄩 󵄩 󵄩 ≤ L ∫ (t − s)α−1 󵄩󵄩󵄩V(t − s)󵄩󵄩󵄩𝕃(X) 󵄩󵄩󵄩ω1 (s)󵄩󵄩󵄩ds −∞

t

󵄩 󵄩 󵄩 󵄩 + Φ(‖ω2 ‖∞ + sup󵄩󵄩󵄩v(s)󵄩󵄩󵄩) ∫ β(s)(t − s)α−1 󵄩󵄩󵄩V(t − s)󵄩󵄩󵄩𝕃(X) ds s∈ℝ

t

−∞

󵄩 󵄩 ≤ L‖ω1 ‖∞ ∫ (t − s)α−1 󵄩󵄩󵄩V(t − s)󵄩󵄩󵄩𝕃(X) ds −∞

6.2 Asymptotically almost periodic solutions | 245 t

󵄩 󵄩 + MΦ(‖ω2 ‖∞ + sup󵄩󵄩󵄩v(s)󵄩󵄩󵄩) ∫ β(s)(t − s)α−1 Eα,α (−δ(t − s)α )ds s∈ℝ

−∞

t

󵄩 󵄩 ≤ ML‖ω1 ‖∞ ∫ (t − s)α−1 Eα,α (−δ(t − s)α )ds + Mσ(t)Φ(‖ω2 ‖∞ + sup󵄩󵄩󵄩v(s)󵄩󵄩󵄩) s∈ℝ

−∞

1 󵄩 󵄩 ML‖ω1 ‖∞ Eα (−δ(t − s)α )|t−∞ + Mρ2 Φ(‖ω2 ‖∞ + sup󵄩󵄩󵄩v(s)󵄩󵄩󵄩) δ s∈ℝ ML‖ω1 ‖∞ 󵄩󵄩 󵄩󵄩 = + Mρ2 Φ(‖ω2 ‖∞ + sup󵄩󵄩v(s)󵄩󵄩) δ s∈ℝ ML 󵄩 󵄩 k + Mρ2 Φ(k0 + sup󵄩󵄩󵄩v(s)󵄩󵄩󵄩) ≤ δ 0 s∈ℝ ≤

≤ k0 , which implies that (Γ1 ω1 )(t) + (Γ2 ω2 )(t) ∈ Ωk0 . Thus, Γ maps Ωk0 into itself. In the following, we show that Γ1 is a contraction on Ωk0 . For any ω1 (t), ω2 (t) ∈ Ωk0 , from (6.30) it follows that 󵄩󵄩 󵄩 󵄩󵄩[F1 (s, v(s) + ω1 (s)) − F1 (s, v(s))] − [F1 (s, v(s) + ω2 (s)) − F1 (s, v(s))]󵄩󵄩󵄩 󵄩 󵄩 ≤ L󵄩󵄩󵄩ω1 (s) − ω2 (s)󵄩󵄩󵄩. Thus, 󵄩󵄩 1 󵄩 1 󵄩󵄩(Γ ω1 )(t) − (Γ ω2 )(t)󵄩󵄩󵄩 󵄩󵄩󵄩 t 󵄩 = 󵄩󵄩󵄩 ∫ (t − s)α−1 V(t − s)[(F1 (s, v(s) + ω1 (s)) − F1 (s, v(s))) 󵄩󵄩 󵄩−∞ 󵄩󵄩 󵄩󵄩 − (F1 (s, v(s) + ω2 (s)) − F1 (s, v(s)))]ds󵄩󵄩󵄩 󵄩󵄩 󵄩 t

󵄩 󵄩 󵄩 󵄩 ≤ L ∫ (t − s)α−1 󵄩󵄩󵄩V(t − s)󵄩󵄩󵄩𝕃(X) 󵄩󵄩󵄩ω1 (s) − ω2 (s)󵄩󵄩󵄩ds −∞

t

≤ ML‖ω1 − ω2 ‖∞ ∫ (t − s)α−1 Eα,α (−δ(t − s)α )ds −∞

ML‖ω1 − ω2 ‖∞ 1 , = ML‖ω1 − ω2 ‖∞ Eα (−δ(t − s)α )|t−∞ = δ δ which implies ML 󵄩󵄩 1 󵄩 1 ‖ω1 − ω2 ‖∞ . 󵄩󵄩(Γ ω1 )(t) − (Γ ω2 )(t)󵄩󵄩󵄩∞ ≤ δ Thus, in view of (6.34), one obtains the conclusion.

246 | 6 Fractional evolution equations on whole real axis From our assumption, it is clear that Γ2 is a continuous mapping from Ωk0 to Ωk0 . Thus, in order to apply the well-known Krasnoselskii’s fixed-point theorem (see Theorem 1.8) to obtain a fixed point of Γ, one needs to verify that Γ2 is completely continuous on Ωk0 . Given ε > 0. Let {ωk }+∞ k=1 ⊂ Ωk0 with ωk → ω0 in C0 (ℝ, X) as k → +∞. Since σ(t) ∈ C0 (ℝ, ℝ+ ), which follows from Lemma 6.10, one may choose a t1 > 0 big enough such that for all t ≥ t1 , Φ(k0 + ‖v‖∞ )σ(t)
0 such that for any k ≥ N, t1

ε 󵄩 󵄩 M ∫ 󵄩󵄩󵄩(t − s)α−1 Eα,α (−δ(t − s)α )[F2 (s, v(s) + ωk (s)) − F2 (s, v(s) + ω0 (s))]󵄩󵄩󵄩ds ≤ . 3 −∞

Thus, when k ≥ N, 󵄩󵄩 2 󵄩 2 󵄩󵄩(Γ ωk )(t) − (Γ ω0 )(t)󵄩󵄩󵄩 󵄩󵄩 t 󵄩󵄩 = 󵄩󵄩󵄩 ∫ (t − s)α−1 V(t − s)F2 (s, v(s) + ωk (s))ds 󵄩󵄩 󵄩−∞

t 󵄩󵄩 󵄩󵄩 − ∫ (t − s)α−1 V(t − s)F2 (s, v(s) + ω0 (s))ds󵄩󵄩󵄩 󵄩󵄩 󵄩 −∞ t1

󵄩 󵄩 ≤ ∫ 󵄩󵄩󵄩(t − s)α−1 V(t − s)[F2 (s, v(s) + ωk (s)) − F2 (s, v(s) + ω0 (s))]󵄩󵄩󵄩ds −∞

max{t,t1 }

+

∫

t1 t1

󵄩󵄩 󵄩 α−1 󵄩󵄩(t − s) V(t − s)[F2 (s, v(s) + ωk (s)) − F2 (s, v(s) + ω0 (s))]󵄩󵄩󵄩ds

󵄩 󵄩 ≤ M ∫ 󵄩󵄩󵄩(t − s)α−1 Eα,α (−δ(t − s)α )[F2 (s, v(s) + ωk (s)) − F2 (s, v(s) + ω0 (s))]󵄩󵄩󵄩ds −∞

6.2 Asymptotically almost periodic solutions | 247 max{t,t1 }

+ 2MΦ(k0 + ‖v‖∞ )

∫

t1

β(s)(t − s)α−1 Eα,α (−δ(t − s)α )ds

t1

󵄩 󵄩 ≤ M ∫ 󵄩󵄩󵄩(t − s)α−1 Eα,α (−δ(t − s)α )[F2 (s, v(s) + ωk (s)) − F2 (s, v(s) + ω0 (s))]󵄩󵄩󵄩ds −∞

+ 2MΦ(k0 + ‖v‖∞ )σ(t) ε 2ε ≤ + = ε. 3 3 Accordingly, Γ2 is continuous on Ωk0 . In the sequel, we consider the compactness of Γ2 . Set Br (X) for the closed ball with center at 0 and radius r in X, Δ = Γ2 (Ωk0 ) and z(t) = Γ2 (u(t)) for u(t) ∈ Ωk0 . First, for all ω(t) ∈ Ωk0 and t ∈ ℝ, 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 󵄩󵄩 2 󵄩󵄩 󵄩󵄩󵄩 α−1 󵄩󵄩 (Γ ω)(t) = (t − s) V(t − s)F (s, v(s) + ω(s))ds ∫ 󵄩󵄩 󵄩󵄩 󵄩󵄩 2 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩−∞ t

󵄩 󵄩 ≤ MΦ(k0 + sup󵄩󵄩󵄩v(s)󵄩󵄩󵄩) ∫ β(s)(t − s)α−1 Eα,α (−δ(t − s)α )ds s∈ℝ

−∞

󵄩 󵄩 = Mσ(t)Φ(k0 + sup󵄩󵄩󵄩v(s)󵄩󵄩󵄩), s∈ℝ

in view of σ(t) ∈ C0 (ℝ, ℝ+ ), which follows from Lemma 6.10, one concludes that lim (Γ2 ω)(t) = 0

|t|→+∞

uniformly for ω(t) ∈ Ωk0 .

As 2

t

(Γ ω)(t) = ∫ (t − s)α−1 V(t − s)F2 (s, v(s) + ω(s))ds −∞ +∞

= ∫ τα−1 V(τ)F2 (t − τ, v(t − τ) + ω(t − τ))dτ. 0

Hence, for given ε0 > 0, one can choose a ξ > 0, such that 󵄩󵄩 +∞ 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∫ τα−1 V(τ)F2 (t − τ, v(t − τ) + ω(t − τ))dτ󵄩󵄩󵄩 < ε0 . 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 ξ

248 | 6 Fractional evolution equations on whole real axis Thus, we get z(t) ∈ ξ c({τα−1 V(τ)F2 (λ, v(λ) + ω(λ)) : 0 ≤ τ ≤ ξ , t − ξ ≤ λ ≤ ξ , ‖ω‖∞ ≤ r}) + Bε0 (X),

where c(K) denotes the convex hull of K. Using that V(⋅) is strongly continuous, which follows from Lemma 6.2, we infer that K = {τα−1 V(τ)F2 (λ, v(λ) + ω(λ)) : 0 ≤ τ ≤ ξ , t − ξ ≤ λ ≤ ξ , ‖ω‖∞ ≤ r} is a relatively compact set, and Δ ⊂ ξ c(K) + Bε0 (X), which implies that Δ is a relatively compact subset of X. Next, we verify the equicontinuity of the set {(Γ2 ω)(t) : ω(t) ∈ Ωk0 }. Let k > 0 be small enough and t1 , t2 ∈ ℝ, ω(t) ∈ Ωk0 . Then by (6.31), we have 󵄩󵄩 2 󵄩 2 󵄩󵄩(Γ ω)(t2 ) − (Γ ω)(t1 )󵄩󵄩󵄩 t 󵄩󵄩󵄩 2 󵄩󵄩 = 󵄩󵄩 ∫ (t2 − s)α−1 V(t2 − s)F2 (s, v(s) + ω(s))ds 󵄩󵄩 󵄩−∞ t1

α−1

− ∫ (t1 − s) −∞ t2

󵄩󵄩 󵄩󵄩 V(t1 − s)F2 (s, v(s) + ω(s))ds󵄩󵄩󵄩 󵄩󵄩 󵄩

󵄩 󵄩 ≤ ∫󵄩󵄩󵄩(t2 − s)α−1 V(t2 − s)F2 (s, v(s) + ω(s))󵄩󵄩󵄩ds t1

t1 −k

󵄩 󵄩 + ∫ 󵄩󵄩󵄩[(t2 − s)α−1 V(t2 − s) − (t1 − s)α−1 V(t1 − s)]F2 (s, v(s) + ω(s))󵄩󵄩󵄩ds −∞ t1

󵄩 󵄩 + ∫ 󵄩󵄩󵄩[(t2 − s)α−1 V(t2 − s) − (t1 − s)α−1 V(t1 − s)]F2 (s, v(s) + ω(s))󵄩󵄩󵄩ds t1 −k

t2

≤ MΦ(k0 + ‖v‖∞ ) ∫ β(s)(t2 − s)α−1 Eα,α (−δ(t2 − s)α )ds + Φ(k0 + ‖v‖∞ ) t1

t1 −k

×

sup

󵄩 󵄩󵄩 α−1 α−1 󵄩󵄩[(t2 − s) V(t2 − s) − (t2 − s) V(t1 − s)]󵄩󵄩󵄩 ∫ β(s)ds

s∈[−∞,t1 −k]

−∞

t1

+ MΦ(k0 + ‖v‖∞ ) ∫ (β(s)(t2 − s)α−1 Eα,α (−δ(t2 − s)α ) t1 −k

6.2 Asymptotically almost periodic solutions | 249

+ β(s)(t1 − s)α−1 Eα,α (−δ(t1 − s)α ))ds

→0

as t2 − t1 → 0, k → 0,

which implies the equicontinuity of the set {(Γ2 ω)(t) : ω(t) ∈ Ωk0 }. Now an application of Lemma 1.8 justifies the compactness of Γ2 . Finally, from the Krasnoselskii’s fixed-point theorem (see Theorem 1.8) it follows that Γ has at least one fixed point in Ωk0 . This proves that system (6.35) has at least one solution in AP(ℝ, X) × C0 (ℝ, X). 6.2.4 Applications In this subsection, we give an example to illustrate the feasibility of the above abstract result. Consider the following fractional partial differential equation with Dirichlet boundary conditions of the form: 𝜕tα u(t, x) = 𝜕x2 u(t, x) { { { { { { + μ(sin t + sin √2t) sin u(t, x) { { + νe−|t| u(t, x) sin u2 (t, x), t ∈ ℝ, x ∈ [0, π], { { { { {u(t, 0) = u(t, π) = 0, t ∈ ℝ,

(6.36)

where μ and ν are positive constants. Take X = L2 [0, π] with norm ‖ ⋅ ‖ and define A : D(A) ⊂ X → X given by Ax =

𝜕2 x(ξ ) 𝜕ξ 2

with the domain D(A) = {x(⋅) ∈ X : x′′ ∈ X, x′ ∈ X is absolutely

continuous on [0, π], x(0) = x(π) = 0}.

It is well known that A is self-adjoint, with compact resolvent and is the infinitesimal generator of an analytic semigroup {T(t)}t≥0 satisfying ‖T(t)‖ ≤ e−t for t > 0. Let F1 (t, x(ξ )) := μ(sin t + sin √2t) sin x(ξ ),

F2 (t, x(ξ )) := νe−|t| x(ξ ) sin x 2 (ξ ).

Then it is easy to verify that F1 , F2 : ℝ × X → X are continuous, F1 (t, x) ∈ AP(ℝ × X, X) satisfying 󵄩󵄩 󵄩 󵄩󵄩F1 (t, x) − F1 (t, y)󵄩󵄩󵄩 ≤ 2μ‖x − y‖,

for all t ∈ ℝ, x, y ∈ X,

250 | 6 Fractional evolution equations on whole real axis and 󵄩 󵄩󵄩 −|t| 󵄩󵄩F2 (t, x)󵄩󵄩󵄩 ≤ νe ‖x‖,

for all t ∈ ℝ, x ∈ X,

which implies F2 (t, x) ∈ C0 (ℝ × X, X). Furthermore, F(t, x) = F1 (t, x) + F2 (t, x) ∈ AAP(ℝ × X, X). Thus, (6.36) can be reformulated as the abstract problem (6.28) and the assumptions (H1 ) and (H2 ) hold with L = 2μ,

Φ(r) = r,

β(t) = νe−|t| ,

ρ1 = 1,

ρ2 ≤ ν.

Then from Theorem 6.8 it follows that equation (6.36) has at least one asymptotically almost periodic mild solution whenever 2μ + ν < 1.

7 Discrete time fractional evolution equations 7.1 Cauchy problem 7.1.1 Introduction In recent years, discrete fractional calculus has received increasing interest by many mathematicians. Gray and Zhang [111] developed a fractional calculus for the discrete nabla difference operator. Atici and Eloe [24, 25] developed the delta-type fractional sums/differences with Riemann–Liouville-like operators and began the study of initial value problems. The existence, uniqueness and positivity of solutions for the discrete fractional boundary value problems have been studied by Goodrich [106, 107], as well as monotonicity properties [78]. The modeling with fractional difference equations began to be studied by Atici and Sengül [26]. Ferreira [96] proved a discrete fractional Gronwall inequality. More recently, Wu and Baleanu [209] studied the discrete fractional logistic map and its chaos, notably with the same definition of fractional delta operator that we will consider in this section. For an overview, one can see the recent monograph by Goodrich and Peterson [109]. In spite of the extensive research in this area, there are still some outstanding problems regarding fractional difference equations. In particular, further research of abstract fractional difference equations with unbounded linear operators remains to be done. Some models among these equations are closely connected with numerical methods for integropartial differential equations that are intermediate between diffusion and wave equations [70] and evolution equations with memory [155]. On the other hand, the mixed models that come from time discretization of partial differential equations occur not only in traffic dynamics, but also in the theory of probability and in the theory of the chain processes of chemistry and radioactivity [35]. Thus, the modeling by means of abstract fractional difference equations provide a new viewpoint and should give new insights about time-discrete behavior because we are taking into account memory effects of the materials that are implicitly present in the mathematical modeling. Besides, in [210] the authors have pointed out that discrete fractional models have some new degrees of freedom, which can be used to capture the hidden aspects of real world phenomena with memory effects. Thus, it is worthwhile to study the behavior of abstract fractional difference equations from not only a purely mathematical but also an applied perspective. As for the abstract fractional difference equations, some results were presented for the first time in [146, 147]. Since then, some developments have been made motivated by these researches [3–5, 128, 137, 138, 148–150, 152]. In [146], by using an operator theoretical method, the author was successful to completely characterize the maximal regularity of solutions for a class of discrete time evolution equations. In [147], Lizama https://doi.org/10.1515/9783110769272-007

252 | 7 Discrete time fractional evolution equations considered the existence and stability for fractional difference equations C α

{

Δ u(n) = Au(n + 1),

u(0) = u0 ∈ X,

n ∈ ℕ0 ,

(7.1)

where CΔα is the Caputo-like fractional difference operator of order 0 < α ≤ 1, and A is a closed linear operator with domain D(A) defined on a Banach space X, ℕ0 = {0, 1, 2, . . .}. It was the first time that the problem (7.1) showed a strong link between the fractional differential and fractional difference operators. Further, by applying the properties of Poisson transformation, a concept that was introduced in [147], as well as methods of operator theory, some results of continuous fractional evolution equations can be generalized to situations involving abstract fractional difference equations. In [2], Abadias and Lizama studied the existence and uniqueness of almost automorphic solutions for nonlinear partial difference-differential equations modeled in abstract form as Δα u(n) = Au(n + 1) + f (n, u(n)),

n ∈ ℤ,

for 0 < α ≤ 1 where A is the generator of a C0 -semigroup defined on a Banach space X and Δα denotes the fractional difference in Weyl-like sense. Some recent works of fractional models with the Grünwald–Letnikov fractional difference can be found in [148, 190] and the references therein, these models can serve as a new microstructural basis for the fractional nonlocal continuum mechanics and physics. In this section, we consider the existence of solutions for nonlinear abstract fractional difference equations Δα u(n) = Au(n + 1) + f (n, u(n)),

{

u(0) = u0 ∈ X,

n ∈ ℕ0 ,

(7.2)

where Δα is the Riemann–Liouville-like fractional difference operator of order 0 < α ≤ 1, f : ℕ0 → X, A is the infinitesimal generator of a bounded C0 -semigroup {Q(t)}t≥0 with domain D(A) defined on a Banach space X. It is important to remark that such problem has only recently been studied in [152] but with A bounded. Therefore, our main contribution in this section is a significant advance in the study of the Cauchy problem (7.2) with unbounded operator A, typically differential operators like the Laplacian, allowing in this way to the analysis of mixed fractional differencedifferential equations, using tools of operator theory. One reason to consider in this section the Riemann–Liouville-like fractional difference operator arises in the recent paper [147] where is proved that Δα is linked with the Riemann–Liouville fractional differential operator Dα by means of the Poisson transformation ℙ. More precisely, the following identity is true: Δα ∘ ℙ = ℙ ∘ Dα . See [4, Theorem 4.2 and Theorem 4.5] for an up-to-date revision of the main properties of

7.1 Cauchy problem |

253

the Poisson transformation and a proof of the above mentioned identity. A second reason is that several fractional difference operators appearing in the current literature are, in fact, related with the operator Δα . For instance, we have the identity C α Δ u(n) = Δα u(n) − k1−α (n + 1)u(0) where 0 < α < 1 and k1−α is defined in (7.3). See [152, Theorem 2.4]. A second example is the identity Δα ∘ τa = τa+1−α ∘ Δαa valid for 0 < α < 1 and a ∈ ℝ, where τa denotes the translation operator and Δαa is the fractional difference operator as defined by Atici and Eloe [25]. This remarkable relationship, known as a transference principle, has been recently proved [108]. We remark that, in contrast with the above mentioned difference operators, the operator Δα enjoys many good properties that enables the handle of abstract fractional difference equations in a simpler way. The main properties are that Δα is a well-defined operator in the vector-valued sequence space s(ℕ0 , X), and that behaves nicely under (finite) convolution [4]. Consequently, many useful tools, like the z-transform, are directly available. This section is organized as follows. In Subsection 7.1.2, we introduce the main tools needed to this work. In Subsection 7.1.3 we prove that, under the assumption that A generates a bounded C0 -semigroup {Q(t)}t≥0 , there exists a sequence of bounded and linear operators {Sα (n)}n∈ℕ0 that is related with the semigroup {Q(t)}t≥0 by means of the subordination formula: ∞∞

Sα (n)x = ∫ ∫ pn (t)fs,α (t)Q(s)x dsdt,

n ∈ ℕ0 , x ∈ X,

0 0

where fs,α is the Lévy distribution and pn is the Poisson distribution. This result improves [2, Theorem 3.5]. We note that an important property of the sequence of operators {Sα (n)}n∈ℕ0 is its automatic regularity: Sα (n)x ∈ D(A) for all n ∈ ℕ0 . Using this remarkable fact, we show in Subsection 7.1.4 that u : ℕ0 → [D(A)] verifies (7.2) if and only if u satisfies u(0) = u0 ∈ D(A) and n−1

u(n) = Sα (n)(I − A)u0 + ∑ Sα (n − 1 − j)f (j, u(j)), j=0

n ∈ ℕ.

In Subsection 7.1.5, we present our main findings on the existence of solutions for abstract fractional semilinear difference equations modeled as (7.2). We prove that if f satisfies a Lipschitz-type condition, then for an initial condition u0 ∈ D(A) the problem (7.2) has a unique solution in the vector valued space of sequences l∞ (ℕ0 ; X). Using a different argument involving the compactness of the semigroup Q(t), we show that if f is merely bounded in the first variable and sublinear in the second one, then the problem (7.2) with initial condition u(0) = 0 has at least one solution in the fnsspace of sequences lf∞ (ℕ; X). Note that such vector-valued Banach space was only recently introduced in the literature by Lizama and Velasco [152]. Finally, in Subsec-

254 | 7 Discrete time fractional evolution equations tion 7.1.6, we provide some simple examples to illustrate our main findings. This section is based on [119].

7.1.2 Preliminaries Let X be a Banach space with norm ‖⋅‖, B (X) be the space of bounded linear operators from X into X endowed with the norm ‖Q‖B(X) = sup{‖Q(x)‖ : ‖x‖ = 1}, where x ∈ X and Q ∈ B (X). We denote by s(ℕ0 ; X) the vectorial space consisting of all vector-valued sequences u : ℕ0 → X. In this context, the forward Euler operator Δ : s(ℕ0 ; X) → s(ℕ0 ; X) is defined by Δu(n) := u(n + 1) − u(n),

n ∈ ℕ0 .

Recall that the finite convolution ∗ of two sequences u, v ∈ s(ℕ0 ; X) is defined as follows: n

(u ∗ v)(n) := ∑ u(n − j)v(j), j=0

n ∈ ℕ0 .

In addition, for α > 0, we consider the scalar sequence {kα (n)}n∈ℕ0 defined by kα (n) =

Γ(n + α) , Γ(α)Γ(n + 1)

n ∈ ℕ0 .

(7.3)

This scalar sequence was introduced by Lizama in [146, 147] in the context of fractional differences. It has several important properties. For instance, the semigroup property n

(kα ∗ kβ )(n) = ∑ kα (n − j)kβ (j) = kα+β (n), j=0

n ∈ ℕ0 , α > 0, β > 0.

It is easy to see that for all n ∈ ℕ0 and for any α ∈ (0, 1], kα (n) ∈ (0, 1] and kα (n) is a non-increasing sequence. Moreover, by [236, Vol. I, p. 77 (1.18)] we have kα (n) =

nα−1 1 (1 + O( )), Γ(α) n

n ∈ ℕ, α > 0.

(7.4)

Definition 7.1 ([147]). Let α > 0, the αth-order fractional sum operator is defined by n

n Γ(n − j + α) u(j) = ∑ kα (n − j)u(j), Γ(α)Γ(n − j + 1) j=0 j=0

Δ−α u(n) = ∑

n ∈ ℕ0 .

7.1 Cauchy problem |

255

Definition 7.2 ([147]). Let α > 0, the αth-order fractional difference operator (in the sense of Riemann–Liouville-like) is defined by Δα u(n) := Δm ∘ Δ−(m−α) u(n),

n ∈ ℕ0 ,

where m − 1 < α < m, m = [α] + 1. Lemma 7.1 ([146]). Let α ∈ (0, 1), a : ℕ0 → ℂ and b : ℕ0 → X be given. Then Δα (a ∗ b)(n) = (a ∗ Δα b)(n) + b(0)a(n + 1),

n ∈ ℕ0 .

We recall the Mittag-Leffler function as follow: μα−β eμ 1 zn = dμ, ∫ α Γ(αn + β) 2πi μ − z n=0 ∞

Eα,β (z) = ∑

z, β ∈ ℂ, Re(α) > 0,

(7.5)

C

where C is a contour, which starts and ends at −∞ and encircles the disk |μ| ≤ |z|1/α counterclockwise. Its Laplace transform is given by ∞

∫ e−λt t β−1 Eα,β (±ωt α )dt = 0

λα−β , λα ∓ ω

ω ∈ ℂ, Re(λ) > |ω|1/α .

(7.6)

We will need the following function, called stable Lévy process, which was introduced by Yosida [216]: σ+i∞

α 1 ft,α (λ) = ∫ ezλ−tz dz, 2πi

σ > 0, t > 0, λ ≥ 0, 0 < α < 1,

(7.7)

σ−i∞

where the branch of z α is taken such that Re(z α ) > 0 for Re(z) > 0. This branch is single-valued in the z-plane cut along the negative real axis. We denote the kernel function gα (t) :=

t α−1 , Γ(α)

t > 0, α > 0,

and in case α = 0, we set g0 (t) = δ(t), the Dirac measure concentrated at the origin. It is well known that this function plays a central role in the theory of fractional calculus. We will need the following result that gives insight on the relationship between fractional powers, the kernel function gα , the Mittag-Leffler function and stable Lévy processes. Proposition 7.1. The following properties hold: α ∞ (i) ∫0 e−λa ft,α (λ)dλ = e−ta , t > 0, a > 0. (ii) ft,α (λ) ≥ 0, λ > 0.

256 | 7 Discrete time fractional evolution equations (iii) ∫0 ft,α (λ)dλ = 1. ∞

(iv) ∫0 fs,α (t)ds = gα (t), t > 0. ∞

(v)

∫0 e−λs fs,α (t)ds = t α−1 Eα,α, (−λt α ), λ ∈ ℂ, t > 0. ∞

Proof. The proof of (i)–(iii) can be found in [216, pp. 260–262]. Next, we shall show that (iv) holds. Since the function fs,α (t) is nonnegative for t > 0, by applying (7.7) and the uniqueness of the inverse Laplace transform, we have ∞

∞ σ+i∞

σ+i∞

0

0 σ−i∞

σ−i∞

α 1 1 1 ∫ fs,α (t)ds = ∫ ∫ ezt e−sz dzds = ∫ α ezt dz = gα (t). 2πi 2πi z

Next, we prove that (v) is true. In fact, for all t > 0, in view of (7.5) and λ ∈ ℂ, by applying (7.7) again, we have ∞ σ+i∞

∞

∫ e−λs fs,α (t)ds = 0

α 1 ∫ ∫ ezt−sz e−λs dzds 2πi

0 σ−i∞ σ+i∞

1 ezt = dz ∫ α 2πi z +λ σ−i∞

=t

α−1

1 2πi

σ ′ +i∞

∫ σ ′ −i∞

zα

ez dz = t α−1 Eα,α (−λt α ). + λt α

Consequently, we obtain the desired results. Finally, for each n ∈ ℕ0 , we recall that the Poisson distribution is defined by pn (t) := e−t

tn , n!

t ≥ 0.

One of the most interesting properties is associated with their infinite integral ∞

∫ pn (t)dt = 1,

n ∈ ℕ0 .

0

Moreover, we recall that the Poisson transformation for a continuous function u(t), t ∈ [0, ∞) is defined by ∞

u(n) := ∫ pn (t)u(t)dt, 0

n ∈ ℕ0 .

(7.8)

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257

This definition was introduced by Lizama [147]. As pointed out in [147], the Poisson transformation reveals a strong relation between gα and kα , and consequently the fractional operators Dαt and Δα . 7.1.3 Resolvent sequences Our basic assumption in this section is that the operator A in (7.2) is the infinitesimal generator of a bounded C0 -semigroup {Q(t)}t≥0 . This means that there is a constant M ≥ 1 such that M = supt∈[0,∞) ‖Q(t)‖B(X) < ∞. It is well known from [169, p. 19, Theorem 5.2(i)] that A is closed and D(A) is dense in X. Next, we use the notion of α-resolvent sequence of bounded and linear operators which was introduced by Abadias and Lizama [2] and it is an important tool to deal with abstract fractional difference equations. Definition 7.3. Let α > 0 and A be a closed linear operator with domain D(A) defined on a Banach space X. An operator-valued sequence {Sα (n)}n∈ℕ0 ⊂ B (X) is called an α-resolvent sequence generated by A if it satisfies the following conditions: (i) Sα (n)x ∈ D(A) for all x ∈ X and ASα (n)x = Sα (n)Ax, for all n ∈ ℕ0 and x ∈ D(A); (ii) Sα (n)x = kα (n)x + A(kα ∗ Sα )(n)x, for all n ∈ ℕ0 and x ∈ X. The main properties of α-resolvent sequences are contained in the following result. Lemma 7.2 ([2]). Let {Sα (n)}n∈ℕ0 be an α-resolvent sequence generated by A. Then: (i) 1 ∈ ρ(A). (ii) For all x ∈ X we have that Sα (0)x = (I − A)−1 x and there exists a scalar sequence {βα,n (j)}n,j∈ℕ such that n

Sα (n)x = ∑ βα,n (j)(I − A)−(j+1) , j=1

n ∈ ℕ.

(iii) For all x ∈ X we have that Sα (0)x ∈ D(A) and Sα (n)x ∈ D(A2 ) for all n ∈ ℕ. Our first result is an improvement of [2, Theorem 3.5] where the exponential stability of the semigroup is assumed. Theorem 7.1. Let 0 < α ≤ 1 and A be the generator of a bounded C0 -semigroup {Q(t)}t≥0 defined on a Banach space X. Then A generates an α-resolvent sequence {Sα (n)}n∈ℕ0 given by ∞∞

Sα (n)x = ∫ ∫ pn (t)fs,α (t)Q(s)x dsdt, 0 0

n ∈ ℕ0 , x ∈ X.

(7.9)

258 | 7 Discrete time fractional evolution equations Proof. From above assumption, we know that there exists M ≥ 1 such that 󵄩 󵄩󵄩 󵄩󵄩Q(t)󵄩󵄩󵄩B(X) ≤ M,

for t ≥ 0.

Define ∞

Qα (t)x := ∫ fs,α (t)Q(s)xds,

t > 0, x ∈ X.

(7.10)

0

Firstly, we observe that Qα (t) is well defined for t > 0. Indeed, in view of (7.7), Proposition 7.1(ii) and (iv), for any x ∈ X, we have ∞

∞

󵄩 󵄩 󵄩󵄩 󵄩 󵄩󵄩Qα (t)x󵄩󵄩󵄩 ≤ ∫ fs,α (t)󵄩󵄩󵄩Q(s)x󵄩󵄩󵄩ds ≤ M‖x‖ ∫ fs,α (t)ds ≤ M‖x‖gα (t),

(7.11)

0

0

and hence, we obtain ∞

∫e

∞

M‖x‖ M‖x‖ 󵄩 , ∫ e− Re(λ)t t α−1 dt ≤ 󵄩󵄩Qα (t)x󵄩󵄩󵄩dt ≤ Γ(α) [Re(λ)]α

− Re(λ)t 󵄩 󵄩

0

0

for Re(λ) > 0, x ∈ X. Consequently, Qα (t) is Laplace transformable and, using Fubini’s theorem, we obtain ∞

∞ ∞

̂ (λ)x := ∫ e−λt Q (t)xdt = ∫ ( ∫ e−λt f (t)dt)Q(s)xds. Q α α s,α 0

0

0

Therefore, by (i) in Proposition 7.1 and a well-known property on the Laplace transformation of C0 -semigroups, we have ∞

α

α

̂ (λ)x = (λα I − A) ∫ e−λ s Q(s)xds = x, (λ I − A)Q α

x ∈ X,

0

and ∞

α

̂ (λ)(λα I − A)x = ∫ e−λ s Q(s)(λα I − A)xds = x, Q α 0

It shows that A commutes with Qα (t) on D(A) and ̂ (λ)x = 1 x + A 1 Q ̂ (λ)x, Q α λα λα α

x ∈ X.

x ∈ D(A).

7.1 Cauchy problem |

259

By the inversion of the Laplace transform, we obtain the identity t

Qα (t)x = gα (t)x + A ∫ gα (t − s)Qα (s)xds,

x ∈ X,

(7.12)

0

and since A is closed, we also get t

Qα (t)x = gα (t)x + ∫ gα (t − s)Qα (s)Axds,

x ∈ D(A).

0

Since A commutes with Qα (t) and A is closed, by applying Poisson transformation into (7.12), similarly to the remaining proof of [2, Theorem 3.5], one can see that n

Sα (n)x = kα (n)x + A ∑ kα (n − j)Sα (j)x, j=0

x ∈ X,

which implies that Definition 7.3(i)–(ii) are satisfied. The proof is completed. The above theorem has two important consequences that we give as corollaries. Corollary 7.1. Let 0 < α ≤ 1 and A be the generator of a bounded C0 -semigroup {Q(t)}t≥0 defined on a Banach space X. Then 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩Sα (n)x󵄩󵄩󵄩 ≤ kα (n) sup󵄩󵄩󵄩Q(t)󵄩󵄩󵄩B(X) ‖x‖, t≥0

for n ∈ ℕ0 , x ∈ X,

where {Sα (n)}n∈ℕ0 is defined by (7.9). Proof. From Theorem 7.1, we know that A generates an α-resolvent sequence {Sα (n)}n∈ℕ0 . Then for M := supt≥0 ‖Q(t)‖B(X) we have ∞∞

󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩Sα (n)x󵄩󵄩󵄩 ≤ ∫ ∫ pn (t)fs,α (t)󵄩󵄩󵄩Q(s)x󵄩󵄩󵄩 dsdt 0 0 ∞

≤ M ∫ e−t 0

tn g (t)dt‖x‖ n! α

= Mkα (n)‖x‖. The second important consequence can be described as follows. Corollary 7.2. Let 0 < α ≤ 1 and A be the generator of a bounded and compact C0 -semigroup {Q(t)}t>0 defined on a Banach space X. Then A generates a compact α-resolvent sequence {Sα (n)}n∈ℕ0 which is defined in (7.9).

260 | 7 Discrete time fractional evolution equations Proof. Let ε > 0 be given. We define an operator sequence Sαε as follows: Sαε (n)x

∞∞

:= ∫ ∫ pn (t)fs,α (t)Q(s)x dsdt 0 ε

∞∞

= Q(ε) ∫ ∫ e−t 0 ε

tn f (t)Q(s − ε)x dsdt, n! s,α

for x ∈ X, where in the second identity we use the semigroup property. Then, from the compactness of Q(ε) (ε > 0), and from Corollary 7.1, we obtain that the set Vε = {Sαε (n)x : n ∈ ℕ0 } is relatively compact in X for ε > 0. Moreover, for any x ∈ X, we have 󵄩󵄩 󵄩󵄩 ∞ ε n 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩 ε −t t fs,α (t)Q(s)x dsdt 󵄩󵄩󵄩 󵄩󵄩Sα (n)x − Sα (n)x󵄩󵄩 = 󵄩󵄩 ∫ ∫ e 󵄩󵄩 󵄩󵄩 n! 󵄩 󵄩0 0 ε ∞

≤ M‖x‖ ∫ ∫ e−t 0 0

Since et > have

tn n!

tn f (t) dtds. n! s,α

n

and fs,α (t) ≥ 0 for t > 0, then e−t tn! < 1 and by (iii) in Proposition 7.1 we ∞

∞

∫ e−t 0

tn f (t)dt < ∫ fs,α (t)dt = 1, n! s,α

s > 0.

0

Therefore, we obtain ε ∞

∫ ∫ e−t 0 0

tn f (t) dtds < ϵ. n! s,α

Consequently, 󵄩󵄩 󵄩 ε 󵄩󵄩Sα (n)x − Sα (n)x󵄩󵄩󵄩 → 0,

as ε → 0.

Hence, there are relatively compact sets arbitrarily close to the set V = {Sα (n)x : n ∈ ℕ0 } for x ∈ X. Thus, the set V is also relatively compact in X. It means that the α-resolvent sequence {Sα (n)}n∈ℕ0 is compact. Observe that the conclusion of compactness for Sα (n) also follows from the representation in Lemma 7.2(ii), namely, assuming that (I − A)−1 is a compact operator. In this way, no assumption on A for the generation of a compact C0 -semigroup is needed.

7.1 Cauchy problem

| 261

7.1.4 Inhomogeneous Cauchy problem In this subsection, we consider the inhomogeneous linear abstract fractional difference equations on a Banach space X given by Δα u(n) = Au(n + 1) + f (n),

{

u(0) = u0 ∈ X,

n ∈ ℕ0 ,

(7.13)

where 0 < α ≤ 1 and A is the generator of a bounded C0 -semigroup {Q(t)}t≥0 . Note that, in particular, (I − A)−1 exists, because λ ∈ ρ(A) for all Re(λ) > 0 by the Hille–Yosida theorem [169]. Definition 7.4. Let f ∈ s(ℕ0 ; X) be given. We say that u ∈ s(ℕ0 ; X) is a strong solution of (7.13) if u(n) ∈ D(A) for all n ∈ ℕ0 and u(n) satisfies (7.13). In what follows, we always denote by {Sα (n)}n∈ℕ0 the α-resolvent sequence generated by A which is defined in (7.9). It is interesting to observe that, in contrast with the continuous case, no additional regularity on the range of the sequence f is needed. Theorem 7.2. Let 0 < α < 1 and f ∈ s(ℕ0 ; X). Then (7.13) admits a strong solution u ∈ s(ℕ0 ; D(A)) if and only if u satisfies u(0) = u0 ∈ D(A) and n−1

u(n) = Sα (n)(I − A)u0 + ∑ Sα (n − 1 − j)f (j), j=0

n ∈ ℕ.

(7.14)

Proof. Sufficiency. Suppose (7.14) holds. By (iii) in Lemma 7.2 we have u(n) ∈ D(A) for all n ∈ ℕ0 . Next, we prove that the following identity holds: Δα Sα (n)x = ASα (n + 1)x,

n ∈ ℕ0 , x ∈ X.

Indeed, convolving the proposed identity of (ii) in Definition 7.3 by k1−α , we have (k1−α ∗ Sα )(n)x = (k1−α ∗ kα )(n)x + A(k1−α ∗ kα ∗ Sα )(n)x,

n ∈ ℕ0 .

Applying the semigroup property of the kernels kα , and the relationship between fractional sum and convolution, the above equality is equivalent to the following expression: n

Δ−(1−α) Sα (n)x = k1 (n)x + A ∑ k1 (n − j)Sα (j)x, j=0

n ∈ ℕ0 .

262 | 7 Discrete time fractional evolution equations Therefore, using k1 (j) = 1 and Δk1 (j) = 0 for all j ∈ ℕ0 , we get n

Δα Sα (n)x = ΔΔ−(1−α) Sα (n)x = Δk1 (n)x + AΔ ∑ k1 (n − j)Sα (j)x j=0

n+1

n

j=0

j=0

= A ∑ Sα (j)x − A ∑ Sα (j)x = ASα (n + 1)x, for all n ∈ ℕ0 and all x ∈ X, and then the claim is proved. Next, we apply the operator Δα into (7.14), it yields n−1

Δα u(n) = Δα Sα (n)(I − A)u0 + Δα ∑ Sα (n − 1 − j)f (j), j=0

n ∈ ℕ.

In addition, by Lemma 7.1, we have n−1

Δα (Sα ∗ f )(n − 1) = ∑ Δα Sα (j)f (n − 1 − j) + Sα (0)f (n) j=0 n

= ∑ Δα Sα (j − 1)f (n − j) + Sα (0)f (n) j=1 n

= ∑ ASα (j)f (n − j) + Sα (0)f (n) j=1 n

= ∑ ASα (j)f (n − j) − ASα (0)f (n) + Sα (0)f (n) j=0

= A(Sα ∗ f )(n) + f (n), where we applied the fact that 1 ∈ ρ(A) and the identity Sα (0)x = (I −A)−1 x for all x ∈ X, see Lemma 7.2. Therefore, it follows that Δα u(n) = Δα Sα (n)(I − A)u0 + Δα (Sα ∗ f )(n − 1)

= ASα (n + 1)(I − A)u0 + Δα (Sα ∗ f )(n − 1)

= Au(n + 1) − A(Sα ∗ f )(n) + Δα (Sα ∗ f )(n − 1)

= Au(n + 1) − A(Sα ∗ f )(n) + A(Sα ∗ f )(n) + f (n) = Au(n + 1) + f (n).

Necessity. By hypothesis, u(0) = u0 ∈ D(A). Using the fact that Δα Sα (n) = ASα (n+1) and Lemma 7.1 we obtain Δα (Sα ∗ u)(n − 1) = A(Sα ∗ u)(n) + u(n),

n ∈ ℕ,

(7.15)

7.1 Cauchy problem

| 263

and again by Lemma 7.1, Δα (Sα ∗ u)(n − 1) = (Sα ∗ Δα u)(n − 1) + Sα (n)u(0),

n ∈ ℕ.

(7.16)

Therefore, if u is a solution of (7.13), then (Sα ∗ Δα u)(n − 1) n−1

= ∑ Sα (n − 1 − j)Δα u(j) j=0

n−1

n−1

j=0

j=0

= ∑ Sα (n − 1 − j)Au(j + 1) + ∑ Sα (n − 1 − j)f (j) n

n−1

j=0

j=0

(7.17)

= A ∑ Sα (n − j)u(j) − ASα (n)u(0) + ∑ Sα (n − 1 − j)f (j) = A(Sα ∗ u)(n) − ASα (n)u(0) + (Sα ∗ f )(n − 1). Putting (7.15) and (7.16) into (7.17), we obtain u(n) = Sα (n)(I − A)u(0) + (Sα ∗ f )(n − 1),

n ∈ ℕ.

The proof is completed.

7.1.5 Semilinear Cauchy problem In this subsection, we study the existence of solutions for the following nonlinear abstract fractional difference equation: Δα u(n) = Au(n + 1) + f (n, u(n)),

{

u(0) = u0 ,

n ∈ ℕ0 ,

(7.18)

where α > 0, A is the generator of a bounded C0 -semigroup and f : ℕ0 × X → X is given. We introduce the next definition of solutions. Definition 7.5. Let 0 < α ≤ 1 and A be the generator of an α-resolvent sequence {Sα (n)}n∈ℕ0 . We say that u ∈ s(ℕ0 ; [D(A)]) is a solution of (7.18) if u satisfies u(0) = u0 ∈ D(A) and n−1

u(n) = Sα (n)(I − A)u0 + ∑ Sα (n − 1 − j)f (j, u(j)), j=0

n ∈ ℕ.

(7.19)

264 | 7 Discrete time fractional evolution equations According to Theorem 7.2, this definition is consistent with true solutions of (7.18). Also note that u(n) ∈ D(A) for all n ∈ ℕ0 because we always have Sα (n)x ∈ D(A) for all x ∈ X and n ∈ ℕ0 ; see Lemma 7.2. First, we consider the problem (7.18) on the vector-valued Banach space of sequences l∞ (ℕ0 ; X), which consists of the following set: 󵄩 󵄩 l∞ (ℕ0 ; X) := {u : ℕ0 → X, sup 󵄩󵄩󵄩u(n)󵄩󵄩󵄩 < ∞}, n∈ℕ0

endowed with the norm ‖u‖∞ = supn∈ℕ0 ‖u(n)‖. In order to state our first existence result, we will need the following hypotheses: (H1) A is the generator of a bounded C0 -semigroup {Q(t)}t≥0 with M := supt≥0 ‖Q(t)‖B(X) and α-resolvent sequence defined in (7.9) for 0 < α < 1; (H2) f (n, 0) ≡ 0, and there exist β ∈ [α, 1) and 0 < L < M1 , such that 󵄩󵄩 󵄩 󵄩󵄩f (n, x) − f (n, y)󵄩󵄩󵄩 ≤ Lk1−β (n)‖x − y‖,

for any x, y ∈ X, n ∈ ℕ0 .

Noting that, by (7.4), the hypothesis (H2) implies f (n, x) → 0 as n → ∞ for all x ∈ X. The next is the first main theorem of this subsection, concerning bounded solutions of problem (7.18). Theorem 7.3. Assume that A satisfies (H1) and f satisfies (H2). Then for any u0 ∈ D(A), the problem (7.18) has a unique solution u in l∞ (ℕ0 ; X). Proof. Let us define the map P : l∞ (ℕ0 ; X) → l∞ (ℕ0 ; X) as follows: n−1

(Pu)(n) := Sα (n)(I − A)u0 + ∑ Sα (n − 1 − j)f (j, u(j)), j=0

n ∈ ℕ,

and (Pu)(0) = u0 . We first show that P is well-defined. In fact, let u ∈ l∞ (ℕ0 ; X) be given, it follows from (H2) that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩f (n, u(n))󵄩󵄩󵄩 ≤ Lk1−β (n)󵄩󵄩󵄩u(n)󵄩󵄩󵄩,

for any u ∈ l∞ (ℕ0 ; X), n ∈ ℕ0 .

By (H2), Corollary 7.1 and observing that 1 + α − β ∈ (0, 1] for 0 < α ≤ β < 1, we obtain n−1

󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩(Pu)(n)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩Sα (n)(I − A)u0 󵄩󵄩󵄩 + ∑ 󵄩󵄩󵄩Sα (n − 1 − j)f (j, u(j))󵄩󵄩󵄩 j=0

n−1

󵄩 󵄩 󵄩 󵄩 ≤ Mkα (n)󵄩󵄩󵄩(I − A)u0 󵄩󵄩󵄩 + ∑ 󵄩󵄩󵄩Sα (n − 1 − j)(f (j, u(j)) − f (j, 0))󵄩󵄩󵄩 j=0

n−1

󵄩 󵄩 󵄩 󵄩 ≤ M 󵄩󵄩󵄩(I − A)u0 󵄩󵄩󵄩 + ML ∑ kα (n − 1 − j)k1−β (j)󵄩󵄩󵄩u(j)󵄩󵄩󵄩 j=0

7.1 Cauchy problem

| 265

󵄩 󵄩 ≤ M 󵄩󵄩󵄩(I − A)u0 󵄩󵄩󵄩 + ML‖u‖∞ k1+α−β (n − 1) 󵄩 󵄩 ≤ M 󵄩󵄩󵄩(I − A)u0 󵄩󵄩󵄩 + ML‖u‖∞ , n ∈ ℕ, where we used the fact that ks (n) < ks (0) = 1 because ks (n) is a nonincreasing sequence for each 0 < s ≤ 1 and n ∈ ℕ. It implies that P is well-defined. For any u, v ∈ l∞ (ℕ0 ; X), by (H2), we get n−1

󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩(Pu)(n) − (Pv)(n)󵄩󵄩󵄩 ≤ M ∑ kα (n − 1 − j)󵄩󵄩󵄩f (j, u(j)) − f (j, v(j))󵄩󵄩󵄩 j=0

n−1

󵄩 󵄩 ≤ ML ∑ kα (n − 1 − j)k1−β (j)󵄩󵄩󵄩u(j) − v(j)󵄩󵄩󵄩 j=0

≤ MLk1+α−β (n − 1)‖u − v‖∞ ≤ ML‖u − v‖∞ , for all n ∈ ℕ. Hence, it follows that ‖Pu − Pv‖∞ ≤ ML‖u − v‖∞ . In view of ML < 1, we conclude the result by the Banach fixed-point theorem. In order to study solutions that behave like nn! at infinity, we consider the fnsspace of vector-valued sequences lf∞ (ℕ; X) introduced in [152] and defined by lf∞ (ℕ; X) := {u : ℕ → X, sup n∈ℕ

endowed with their natural norm ‖u‖f = supn∈ℕ 1 sequence nn! ∑n−1 j=0 jj! has the following properties: sup n∈ℕ

1 n−1 5 ∑ jj! = nn! j=0 18

and

‖u(n)‖ < ∞}, nn! ‖u(n)‖ . nn!

From [152], we note that the

1 n−1 n! − 1 = 0. ∑ jj! = lim n→∞ nn! n→∞ nn! j=0 lim

(7.20)

We will need the following lemma. Lemma 7.3 ([152]). Let U ⊂ lf∞ (ℕ; X) be such that

(a) the set Hn (U) = { u(n) : u ∈ U} is relatively compact in X, for all n ∈ ℕ. nn! ‖u(n)‖ (b) limn→∞ supu∈U nn! = 0, that is, for each ε > 0, there is N > 0 such that for each n ≥ N and for all u ∈ U. Then U is relatively compact in lf∞ (ℕ; X).

‖u(n)‖ nn!

< ε,

For a given function g : ℕ0 × X → X, the Nemytskii operator Ng : lf∞ (ℕ; X) → lf∞ (ℕ; X) (with g restricted to ℕ) is defined by Ng (u)(n) := g(n, u(n)),

n ∈ ℕ.

266 | 7 Discrete time fractional evolution equations In order to obtain our second main result, we will need the following assumptions: (H3) A is the generator of a compact C0 -semigroup {Q(t)}t>0 and α-resolvent sequence defined in (7.9) for 0 < α < 1. (H4) There exist a positive sequence a(⋅) ∈ l∞ (ℕ0 ) and a function ψ : ℝ+ → ℝ+ , with ψ(r) ≤ r for r ∈ ℝ+ such that ‖g(n, x)‖ ≤ a(n)ψ(‖x‖), for all n ∈ ℕ0 and x ∈ X. (H5) The Nemytskii operator Ng is continuous in lf∞ (ℕ; X). Remark 7.1. In particular, for example, suppose that the function g : ℕ0 × X → X is defined by g(n, x) := cos(n)x. It is easy to check that function g satisfies (H4) and (H5). Theorem 7.4. Assume that A satisfies (H3) and g satisfies (H4)–(H5). Then the problem (7.18) with u0 = 0 has at least one solution u ∈ lf∞ (ℕ; X) provided with ‖a‖∞ ≤ 18/(5 supt≥0 ‖Q(t)‖B(X) ). Proof. Let us define the map P : lf∞ (ℕ; X) → lf∞ (ℕ; X) as follows: n−1

(P u)(n) := Sα (n − 1)g(0, u0 ) + ∑ Sα (n − 1 − j)g(j, u(j)), j=1

n ∈ ℕ.

Since u0 = 0 and by hypothesis (H4), we have g(n, 0) = 0 for all n ∈ ℕ0 . Then we can rewrite P as n−1

(P u)(n) = ∑ Sα (n − 1 − j)g(j, u(j)), j=0

n ∈ ℕ,

where we understand u as their extension to ℕ0 by u(0) = 0. First, we show that P is well- defined. Let u ∈ lf∞ (ℕ; X) be given. For each n ∈ ℕ, by Corollary 7.1 and (H4), we have n−1

󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩(P u)(n)󵄩󵄩󵄩 ≤ ∑ 󵄩󵄩󵄩Sα (n − 1 − j)g(j, u(j))󵄩󵄩󵄩 j=0

n−1

󵄩 󵄩 󵄩 󵄩 ≤ sup󵄩󵄩󵄩Q(t)󵄩󵄩󵄩B(X) ∑ kα (n − 1 − j)a(j)ψ(󵄩󵄩󵄩u(j)󵄩󵄩󵄩) t≥0

j=0

n−1

󵄩 󵄩 󵄩 󵄩 ≤ sup󵄩󵄩󵄩Q(t)󵄩󵄩󵄩B(X) ‖a‖∞ ∑ 󵄩󵄩󵄩u(j)󵄩󵄩󵄩 t≥0

j=0

n−1

󵄩 󵄩 ≤ sup󵄩󵄩󵄩Q(t)󵄩󵄩󵄩B(X) ‖a‖∞ ‖u‖f ∑ jj!, t≥0

j=0

where we used the fact that kα (n) < kα (0) = 1 because kα (n) is a nonincreasing sequence for 0 < α ≤ 1 and n ∈ ℕ. Denote M := supt≥0 ‖Q(t)‖B(X) . Then, by (7.20) we

7.1 Cauchy problem

| 267

obtain 1 n−1 ‖(P u)(n)‖ 5 ≤ M‖a‖∞ ‖u‖f ∑ jj! ≤ M‖a‖∞ ‖u‖f . nn! nn! j=0 18

(7.21)

This proves that P is well-defined. Now, we show that P is continuous. Let {um }∞ m=1 ⊂ lf∞ (ℕ; X) be a sequence such that um → u as m → ∞ in the norm of lf∞ (ℕ; X). Then n−1

󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩(P um )(n) − (P u)(n)󵄩󵄩󵄩 ≤ M ∑ kα (n − 1 − j)󵄩󵄩󵄩g(j, um (j)) − g(j, u(j))󵄩󵄩󵄩 j=0

n−1

󵄩 󵄩 ≤ M ∑ 󵄩󵄩󵄩g(j, um (j)) − g(j, u(j))󵄩󵄩󵄩 j=0

n−1

󵄩 󵄩 ≤ M 󵄩󵄩󵄩Ng (um ) − Ng (u)󵄩󵄩󵄩f ∑ jj!. j=0

Therefore, for all n ∈ ℕ, we have n−1 ‖(P um )(n) − (P u)(n)‖ 󵄩 󵄩 1 ≤ M 󵄩󵄩󵄩Ng (um ) − Ng (u)󵄩󵄩󵄩f ∑ jj! nn! nn! j=0

≤

5 󵄩󵄩 󵄩 M 󵄩N (u ) − Ng (u)󵄩󵄩󵄩f → 0, 18 󵄩 g m

as m → ∞,

which means that ‖P um − P u‖f → 0 as m → ∞. Therefore, P is continuous. Since Q(t) is compact for t > 0, then from Corollary 7.2, we know that the sequence of operators {Sα (n)}n∈ℕ0 is compact. Let r > 0 be given. We define a set by Sr := {ω ∈ lf (ℕ; X) | ‖ω‖f ≤ r}. ∞

Clearly, Sr is a nonempty, bounded, closed and convex subset of lf∞ (ℕ; X). In view of (7.21) and (H4), we can deduce that P maps Sr into itself. Thus, it remains to show that P is a compact operator. In order to prove that U := PSr is relatively compact, we will use Lemma 7.3. We check that the conditions in this lemma are satisfied. (a) Let v = P u for any u ∈ Sr . We have v(n) = (P u)(n) n−1

= ∑ Sα (n − 1 − k)g(k, u(k)) k=0 n−1

= ∑ Sα (k)g(n − 1 − k, u(n − 1 − k)), k=0

n ∈ ℕ,

268 | 7 Discrete time fractional evolution equations and then 1 1 n−1 v(n) = ( ∑ Sα (k)g(n − 1 − k, u(n − 1 − k))). nn! n! n k=0 Therefore,

v(n) nn!

∈

1 n!

co(Kn ), where co(Kn ) denotes the convex hull of Kn for the set

n−1

Kn = ⋃ {Sα (k)g(ξ , x) : ξ ∈ {0, 1, 2, . . . , n − 1}, ‖x‖f ≤ r}, k=0

n ∈ ℕ.

On the one hand, for every m ∈ ℕ0 and σ > 0, the set {g(k, x) : 0 ≤ k ≤ m, ‖x‖f ≤ σ} is bounded because from condition (H4) we have ‖g(k, x)‖ ≤ a(k)ψ(‖x(k)‖) ≤ mm!‖a‖∞ σ for all 0 ≤ k ≤ m and ‖x‖f ≤ σ. Consequently, the set {Sα (n)g(k, x) : 0 ≤ k ≤ m, ‖x‖f ≤ σ} is relatively compact in X for all n ∈ ℕ0 from the fact that {Sα (n)}n∈ℕ0 is compact. Then it follows that each set Kn is relatively compact. From the inclusions, Hn (U) = {

1 1 v(n) : v ∈ U} ⊆ co(Kn ) ⊆ co(Kn ), nn! n! n!

we conclude that the set Hn (U) is relatively compact in X, for all n ∈ ℕ. (b) Let u ∈ Sr and v = P u. Using condition (H4), for each n ∈ ℕ we have 1 n−1󵄩󵄩 ‖v(n)‖ 󵄩 ≤ ∑ 󵄩S (n − 1 − j)g(j, u(j))󵄩󵄩󵄩 nn! nn! j=0 󵄩 α ≤

n−1 M 󵄩 󵄩 ‖a‖∞ ∑ kα (n − 1 − j)󵄩󵄩󵄩u(j)󵄩󵄩󵄩 nn! j=0

≤ M‖u‖f ‖a‖∞

1 n−1 1 n−1 ∑ jj! ≤ Mr‖a‖∞ ∑ jj!. nn! j=0 nn! j=0

Then (7.20) implies that limn→∞ ‖v(n)‖ = 0 independently of u ∈ Sr . Therefore, U = nn! PSr is relatively compact in lf∞ (ℕ; X) from Lemma 7.3, and by applying the continuity of operator P , we conclude that P is a completely continuous operator. Thus, the Schauder’s fixed point theorem shows that P has at least one fixed point u ∈ lf∞ (ℕ; X). The proof is completed.

7.1.6 Examples Example 7.1. We can describe the heat flow in a ring of length one with a temperature dependent “source” at discrete time n ∈ ℕ by the following evolution equation (see

7.1 Cauchy problem

| 269

[169, p. 234] and references therein): 2

𝜕 u(n, z) − u(n − 1, z) = 𝜕z 0 < z < 1, 2 u(n, z) + G(u(n, z)), { { { ′ ′ u(n − 1, 0) = u(n − 1, 1), u (n − 1, 0) = u (n − 1, 1), { z z { { {u(0, z) = u0 (z),

(7.22)

where G is a given function. We rewrite this model as an abstract difference equation. As a natural Banach space, we choose X = Cp ([0, 1]) the space of all continuous real valued periodic functions having period 1 with the norm ‖u‖ = max0≤z≤1 |u(z)|. The space X consists therefore of continuous functions on [0, 1] satisfying u(0) = u(1). On this Banach space X, we define an operator A by Av = v′′ with its domain D(A) = {v : v, v′ , v′′ ∈ X, v(0) = v(1)}. Then, by [169, Chapter 8, Lemma 2.1], the operator A generates a C0 -semigroup {Q(t)}t≥0 which is compact, bounded and analytic on X. With those definitions, equation (7.22) is a particular case of the abstract difference equation Δα x(n) = Ax(n + 1) + f (n, x(n)),

n ∈ ℕ0 ,

(7.23)

where 0 < α ≤ 1, x : ℕ0 → X is defined by x(n)(z) := u(n, z) and the function f : ℕ0 × X → X is given by f (n, x(n))(z) = G(u(n, z)). In particular, if f = 0, then the solution of (7.23) with initial condition x(0) = x0 and α = 1 is given by x(n) = (I − A)−n x0 ,

n ∈ ℕ.

Hence, by Theorem 7.1, we conclude that for each 0 < α ≤ 1, the operator A generates an α-resolvent sequence Sα (n). By Corollary 7.2, the operator A generates a compact α-resolvent sequence Sα (n). Example 7.2. Let Ω ⊂ ℝn be a bounded open set and X = L2 (Ω). We consider the following discrete abstract Cauchy problem: α

2

Δ u(n, z) = ∇ u(n + 1, z) + f (n, z), { { { u(n, z) = 0, { { { {u(0, z) = u0 (z),

n ∈ ℕ0 , z ∈ Ω,

n ∈ ℕ0 , z ∈ 𝜕Ω, z ∈ Ω,

where f : ℝ+ × X → X and ∇2 is the Laplacian operator.

(7.24)

270 | 7 Discrete time fractional evolution equations Now, let A = ∇2 be the Laplacian operator with Dirichlet boundary conditions and D(A) = {v ∈ H01 (Ω) ∩ H 2 (Ω), Av ∈ L2 (Ω)}. ∞ We denote by {−λl , ϕl }∞ l=1 the eigensystem of the operator A, where {λl }l=1 denotes the set of eigenvalues satisfying 0 < λ1 ≤ λ2 ≤ ⋅ ⋅ ⋅ ≤ λl ≤ ⋅ ⋅ ⋅, and λl → ∞ as l → ∞, and {ϕl }∞ l=1 denotes the corresponding eigenfunctions. It is well known that ϕl can be normalized so that {ϕl }∞ l=1 is an orthonormal basis of X. Hence, it yields ∞

Au = − ∑ λl (u, ϕl )ϕl , l=1

u ∈ D(A),

where (⋅, ⋅) is the inner product in X. It is clear that the operator A generates a C0 -semigoup {Q(t)}t>0 which is compact, bounded, analytic and explicitly given by ∞

Q(t)u = ∑ e−λl t (u, ϕl )ϕl , l=1

u ∈ D(A).

Hence, by applying Theorem 7.1, (v) in Proposition 7.1 and Corollary 7.2, we can obtain a subordinated, discrete compact α-resolvent family {Sα (n)}n∈ℕ0 as follows: ∞∞

∞

0 0 ∞

l=1

Sα (n)u = ∫ ∫ pn (t) ∑ fs,α (t)e−λl s (u, ϕl )ϕl dsdt ∞

= ∫ pn (t) ∑ t 0

l=1

α−1

(7.25) α

Eα,α (−λl t )(u, ϕl )ϕl dt.

By applying [236, Theorem 5.1] and (7.6), we have ∞

∫ pn (t)t α−1 Eα,α (−λl t α )dt = 0

=

(−1)n (n) (L (t α−1 Eα,α (−λl t α ))) (1) n! 󵄨 (−1)n α −1 (n) 󵄨󵄨 ((s + λl ) ) 󵄨󵄨󵄨 , 󵄨󵄨s=1 n!

where L denotes the Laplace transform. Let βα,0 (0) := 1 and βα,n (0) := 0 for n ∈ ℕ, then by (ii) in Lemma 7.2, there exists a scalar sequence {βα,n (j)}n,j∈ℕ0 such that n 󵄨 (−1)n α −1 (n) 󵄨󵄨 ((s + λl ) ) 󵄨󵄨󵄨 = ∑ βα,n (j)(1 + λl )−(j+1) , 󵄨󵄨s=1 n! j=0

n ∈ ℕ0 .

7.1 Cauchy problem |

271

Thus, we have ∞ n

Sα (n)u = ∑ ∑ βα,n (j)(1 + λl )−(j+1) (u, ϕl )ϕl , l=1 j=0

u ∈ D(A), n ∈ ℕ0 .

(7.26)

Therefore, we obtain that (7.24) possesses a solution and its explicit form is given by ∞ n

u(n) = ∑ ∑ βα,n (j)(1 + λl )−j (u0 , ϕl )ϕl l=1 j=0

n−1 ∞ n−1−m

+ ∑ ∑ ∑ βα,n−1−m (j)(1 + λl )−(j+1) (f (m, ⋅), ϕl )ϕl , m=0 l=1

j=0

n ∈ ℕ.

Example 7.3. For any λ ∈ ℂ, |λ| < 1 and 0 < α ≤ 1, we consider the following equations: Δα u(n) = −λu(n + 1) + f (n, u(n)),

(7.27)

{

u(0) = u0 .

It is clear that Re(λ) > 0 is the generator of the bounded C0 -semigroup Q(t) = e−λt for t ≥ 0. Hence, by applying Theorem 7.1, (v) in Proposition 7.1 and the definition of Mittag-Leffler function, we obtain the discrete α-resolvent family {Sα (n)}n∈ℕ0 as follows: ∞∞

Sα (n) = ∫ ∫ pn (t)fs,α (t)e−λs dsdt 0 0 ∞

∞

0

i=0

= ∫ pn (t)t α−1 Eα,α (−λt α )dt = ∑ (−λ)i

Γ(αi + α + n) . Γ(αi + α)Γ(n + 1)

Then, by the definition of kα (n), the explicit form of the solution of the equation (7.27) is given by ∞

u(n) = ∑ (−λ)i (1 − λ)kαi+α (n)u0 i=0

∞ n−1

+ ∑ ∑ (−λ)i kαi+α (n − j − 1)f (j, u(j)), i=0 j=0

n ∈ ℕ.

Example 7.4. We consider the following nonlinear abstract fractional difference equations: Δ0.5 x(n) = Ax(n + 1) + 0.1k0.4 (n) sin(x(n)),

{

x(0) = u0 .

n ∈ ℕ0 ,

(7.28)

272 | 7 Discrete time fractional evolution equations If A is the generator of a contraction C0 -semigroup {Q(t)}t≥0 , then by Theorem 7.1, we conclude that for each 0 < α ≤ 1, the operator A generates an α-resolvent sequence Sα (n) with M = 1. Hence, (H1) holds. Denote α = 0.5 and f (n, x(n)) = 0.1k0.4 (n) sin(x(n)) for n ∈ ℕ0 . Note that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩f (n, x(n)) − f (n, y(n))󵄩󵄩󵄩 = 0.1k0.4 (n)󵄩󵄩󵄩sin(x(n)) − sin(y(n))󵄩󵄩󵄩 󵄩󵄩󵄩 x(n) + y(n) x(n) − y(n) 󵄩󵄩󵄩󵄩 = 0.2k0.4 (n)󵄩󵄩󵄩cos( ) sin( )󵄩󵄩 2 2 󵄩󵄩 󵄩󵄩 󵄩 󵄩 ≤ 0.1k0.4 (n)󵄩󵄩󵄩x(n) − y(n)󵄩󵄩󵄩, which implies that (H2) holds. Following Theorem 7.3, we obtain that for any u0 ∈ D(A) there exists a unique solution of (7.28) in l∞ (ℕ0 ; X). If A is the generator of a compact C0 -semigroup {Q(t)}t>0 , then by Corollary 7.2, we conclude that for each 0 < α ≤ 1, the operator A generates a compact α-resolvent sequence Sα (n). Then (H3) holds. Denote α = 0.5 and g(n, x(n)) = k0.8 (n) sin(x(n)) for n ∈ ℕ0 . Consider the Nemystkii operator Ng (u) : ℕ → X defined by Ng (u)(n) := g(n, u(n)). Obviously, 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩g(n, x(n))󵄩󵄩󵄩 = a(n)󵄩󵄩󵄩sin(x(n))󵄩󵄩󵄩 ≤ a(n)󵄩󵄩󵄩x(n)󵄩󵄩󵄩, where a(n) = k0.8 (n). Then (H4) and (H5) hold. Thus, following Theorem 7.4, we obtain the existence of solutions of Δ0.5 x(n) = Ax(n + 1) + k0.8 (n) sin(x(n)),

{

x(0) = 0,

n ∈ ℕ0 ,

in the space lf∞ (ℕ; X).

7.2 Stability 7.2.1 Introduction Compared with the continuous fractional differential models, some scholars found that the discrete time fractional differential equations (or called fractional difference equations) can still capture certain hidden aspects of real world phenomena with memory effects, further it will appear some new interesting properties, which are different from the continuous case. Cermák, Gyori and Nechvátal [60] investigated the stability behaviors of discrete fractional systems. Wu and Baleanu [209, 210] considered the discrete fractional logistic map and its chaos whose point out there some new degrees of freedom in discrete fractional models. Abadias and Mianab [5] generalized the algebraic structure of Cesàro sums in the discrete fractional operators setting, several subjects of interest in harmonic and functional analysis are displayed. Goodrich

7.2 Stability | 273

and Lizama [108] showed the positivity, monotonicity and convexity of functions under a different definition for fractional delta operator (see Definition 7.2), which can be derived by a transference principle from known fractional difference operators [25]. In this section, we study the following nonlinear abstract fractional difference equations: Δα u(n) = Au(n + 1) + f (n, u(n)),

{

u(0) = u0 ,

n ∈ ℕ0 ,

(7.29)

where Δα is the Riemann–Liouville-like fractional difference operator of order 0 < α ≤ 1, f : ℕ0 → X, A is the infinitesimal generator of a bounded C0 -semigroup {T(t)}t≥0 with domain D(A) defined on a Banach space X, ℕ0 = {0, 1, 2, . . .}. It is important to remark that such problem has been studied in [152] with A bounded. When f = 0 in (7.29), [147] considered the existence and stability for abstract difference equations with Caputo-like fractional difference operator by means of operator theory. Also in [119], the authors derived a structure of the solutions for the inhomogenous Cauchy problem of abstract fractional difference equations (7.29) and further investigated the existence results to the proposed problem. There are some studies involving the fractional difference operators of Riemann– Liouville and/or Caputo-like, which focus on the stability results of Ulam–Hyers and Ulam–Hyers–Rassias in finite interval of discrete points, Chen and Zhou [67] considered the Ulam–Hyers stability of solutions for a discrete fractional boundary value problem, Chen, Bohner and Jia [65] studied the Ulam–Hyers stability of a initial value problem of Caputo-like fractional difference equations. Obviously, it’s not a very captivating situation that it does not take the whole values on infinite interval of discrete points, for this purpose, we will consider the case of whole values on ℕ in the current section. It should be noted that the Ulam–Hyers stability may not exist on ℕ of problem (7.29) due to the fact that the series ∑∞ j=0 kα (j) is divergent for α > 0. This section is organized as follows: in Subsection 7.2.2, we introduce some important preliminary definitions and results. In Subsection 7.2.3, we consider a nonlinear discrete time abstract fractional differential equation modeled as (7.29), by using a different argument involving the compactness of the semigroup T(t) associated with f satisfying growth type condition in the second variable, we obtain an existence criterion of stable solutions. In Subsection 7.2.4, the Ulam–Hyers–Rassias stability is also established. This section is based on [120].

7.2.2 Preliminaries Let X be a Banach space with norm ‖ ⋅ ‖, B (X) stands for the space of bounded linear operators from X into X with the norm ‖⋅‖B := ‖⋅‖B(X) . We also consider the essentially

274 | 7 Discrete time fractional evolution equations bounded vector-valued Banach space of sequences l∞ (ℕ0 ; X), which is defined by 󵄩 󵄩 l∞ (ℕ0 ; X) := {u : ℕ0 → X, sup 󵄩󵄩󵄩u(n)󵄩󵄩󵄩 < ∞}, n∈ℕ0

endowed with the norm ‖u‖∞ = supn∈ℕ0 ‖u(n)‖.

Our basic assumption is that the operator A in (7.29) is the infinitesimal generator

of a bounded C0 -semigroup {T(t)}t≥0 , which means that there exists a constant M ≥ 1

such that M = supt∈[0,∞) ‖T(t)‖B < ∞. It is well known from [169, p. 19, Theorem 5.2(i)] that A is closed and the domain D(A) of operator A is dense in X. The norm of D(A)

is given by a graph norm defined as ‖x‖A = ‖x‖ + ‖Ax‖ for any x ∈ D(A). Next, we

introduce the following notion of α-resolvent sequence that is an important tool to

deal with abstract fractional difference equations.

Definition 7.6. Let α > 0 and let A be a closed linear operator with domain D(A) de-

fined on a Banach space X. An operator-valued sequence {Sα (n)}n∈ℕ0 ⊂ B (X) is called

an α-resolvent sequence generated by A if it satisfies the following conditions: (i) Sα (n)x ∈ D(A) and ASα (n)x = Sα (n)Ax, for all n ∈ ℕ0 and x ∈ D(A);

(ii) Sα (n)x = kα (n)x + A(kα ∗ Sα )(n)x, for all n ∈ ℕ0 and x ∈ X.

The main properties of α-resolvent sequences are contained in the following re-

sults.

Lemma 7.4 ([2]). Let ρ(A) be the resolvent set of operator A and let {Sα (n)}n∈ℕ0 be an

α-resolvent sequence generated by A. Then:

(i) 1 ∈ ρ(A), and for all x ∈ X we have that Sα (0)x = (I − A)−1 x.

(ii) For all x ∈ X we have that Sα (0)x ∈ D(A) and Sα (n)x ∈ D(A2 ) for all n ∈ ℕ. We next get a strong relationship between C0 -semigroup in the setting of frac-

tional version and α-resolvent sequence, one can find that the Poisson distribution

also acts as a bridge between the discrete and continuous theories; for more details, see [108, 147].

Lemma 7.5 ([119]). Let 0 < α ≤ 1 and let A be the generator of a bounded C0 -semigroup

{T(t)}t≥0 defined on a Banach space X. Then A generates an α-resolvent sequence {Sα (n)}n∈ℕ0 given by

∞∞

Sα (n)x = ∫ ∫ pn (t)fs,α (t)T(s)xdsdt, 0 0

n ∈ ℕ0 , x ∈ X,

(7.30)

7.2 Stability | 275

where pn (t) := e−t t n /n! (n ∈ ℕ0 , t ≥ 0) is the Poisson distribution and function fα,s (t) denotes the stable Lévy process given by σ+i∞

α 1 fs,α (t) = ∫ ezt−sz dz, 2πi

σ > 0, s > 0, t ≥ 0, 0 < α < 1,

σ−i∞

in which the branch of z α is taken such that Re(z α ) > 0 for Re(z) > 0. Lemma 7.6 ([119]). Let 0 < α ≤ 1 and let A be the generator of a bounded C0 -semigroup {T(t)}t≥0 defined on a Banach space X. Then 󵄩󵄩 󵄩 󵄩󵄩Sα (n)x󵄩󵄩󵄩 ≤ Mkα (n)‖x‖,

for n ∈ ℕ0 , x ∈ X.

Furthermore, if A is the generator of a compact C0 -semigroup {T(t)}t>0 . Then A generates a compact α-resolvent sequence {Sα (n)}n∈ℕ0 . Remark 7.2. Noting that if X is a finite dimensional space, we see that the semigroup T(t) can be rewritten by eAt , and if A is the generator of a C0 -semigroup {eAt }t≥0 with respect to ‖A‖B ≤ 1. Then, A generates an α-resolvent sequence if and only if {Sα (n)}n∈ℕ0 is given by ∞

Sα (n) = ∑ Aj kαj+α (n). j=0

In fact, by the properties of stable Lévy process (iii) and Mittag-Leffler functions, we get for n ∈ ℕ0 , ∞∞

∞

∫ ∫ pn (t)fs,α (t)e dsdt = ∫ pn (t)t α−1 Eα,α, (At α )dt As

0 0

0 ∞

= ∑ Aj kαj+α (n). j=0

Conversely, it is easy to check that Sα (n) satisfies the Definition 7.6 for every n ∈ ℕ0 . 7.2.3 Existence of stable solutions In this subsection, we study the existence of stable solutions for the following nonlinear discrete time abstract fractional differential equation (7.29). For this purpose, we introduce the next definition of solutions, which can be seen in [119]. Definition 7.7. Let 0 < α < 1 and A be the generator of an α-resolvent sequence {Sα (n)}n∈ℕ0 . We say that u ∈ l∞ (ℕ0 ; D(A)) is a solution of (7.29) if u satisfies u(0) =

276 | 7 Discrete time fractional evolution equations u0 ∈ D(A) and n−1

u(n) = Sα (n)(I − A)u0 + ∑ Sα (n − 1 − j)f (j, u(j)), j=0

n ∈ ℕ.

According to Lemma 7.5, this definition is consistent with true solutions of (7.29). From Lemma 7.4, it follows that Sα (n)x ∈ D(A) for all x ∈ X and n ∈ ℕ0 and u(n) ∈ D(A) for all n ∈ ℕ0 . In order to use the Schauder’s fixed-point theorem, we need the next compactness result. Lemma 7.7. Let U ⊂ l∞ (ℕ; X) satisfy (a) the set Hn (U) = {u(n) : u ∈ U} is relatively compact in X, for all n ∈ ℕ. (b) limn→∞ supu∈U ‖u(n)‖ = 0, that is, for each ε > 0, there is a N > 0 such that ‖u(n)‖ < ε, for each n ≥ N and for all u ∈ U. Then U is relatively compact in l∞ (ℕ; X). Proof. Let {um }∞ m=1 be a sequence in U, then by (a), for any given n ∈ ℕ, there exists a ∞ convergent subsequence {umk }∞ k=1 ⊂ {um }m=1 such that limk→∞ umk (n) = u(n), i. e., for ∗ each ε > 0, there exists a constant N = N ∗ (n, ε) > 0, such that 󵄩󵄩 󵄩 󵄩󵄩umk (n) − u(n)󵄩󵄩󵄩 < ε,

for k > N ∗ .

From the assumption (b), for each ε > 0, there exists a constant N ′ > 0, such that for n ≥ N ′, 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 sup 󵄩󵄩󵄩umk (n) − umj (n)󵄩󵄩󵄩 ≤ sup 󵄩󵄩󵄩umk (n)󵄩󵄩󵄩 + sup 󵄩󵄩󵄩umj (n)󵄩󵄩󵄩 ′ ′ n≥N

n≥N ′

n≥N

< ε/2 + ε/2 = ε. Let N ′ be fixed. For each 1 ≤ n < N ′ , then for j, k > N = N(N ′ , ε), we have 󵄩 󵄩 sup 󵄩󵄩󵄩umk (n) − umj (n)󵄩󵄩󵄩

1≤n 0, then from Lemma 7.6, we know that the sequence of operators {Sα (n)}n∈ℕ0 is compact. Let r > 0 be given. We define a set by Sr := {ω ∈ l (ℕ; D(A)) : ‖ω‖∞ ≤ r}. ∞

Clearly, Sr is a bounded, closed and convex subset of l∞ (ℕ; D(A)). In view of (H2), we can deduce that P maps Sr into itself. Thus, it remains to show that P is a compact operator. In order to prove that U := PSr is relatively compact, we will use Lemma 7.7. We check that the conditions in this lemma are satisfied, and we check that u satisfies all assumptions. (a) Let v = P u for any u ∈ Sr . We have n−1

vε (n) = (P ε u)(n) = ∑ Sαε (j)f (n − 1 − j, u(n − 1 − j)), j=0

n ∈ ℕ,

where ∞∞

Sαε (j)x := ∫ ∫ pj (t)fs,α (t)T(s)x dsdt 0 ε

∞∞

= T(ε) ∫ ∫ pj (t)fs,α (t)T(s − ε)x dsdt, 0 ε

x ∈ X,

7.2 Stability | 279

where we use the semigroup property of T(t). Hence, it remains to prove that (Q ∗ f )(n − 1) is bounded, where ∞∞

Q (j)x = ∫ ∫ pj (t)fs,α (t)T(s − ε)x dsdt. 0 ε

In fact, noting the properties of stable Lévy process (ii), we have the following identity: ∞∞

∫ ∫ pj (t)fs,α (t)dsdt = kα (j),

j ∈ ℕ0 ,

0 0

it is easy to check 󵄩󵄩n−1 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∑ Q (j)f (n − 1 − j, u(n − 1 − j))󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 j=0 󵄩󵄩 n−1 ∞ ∞

󵄩 󵄩 ≤ ∑ ∫ ∫ pj (t)fs,α (t)󵄩󵄩󵄩T(s − ε)f (n − 1 − j, u(n − 1 − j))󵄩󵄩󵄩dsdt j=0 0 ε

n−1

≤ MLf ∑ kα (j)k1−β (n − 1 − j)r j=0

≤ MLf r. For any fixed n̂ ∈ ℕ and for all n ≥ n̂ + 1, since n−1

∑ kα (j)k1−β (n − 1 − j) ≤ k1+α−β (n − 1),

j=n̂

and by (7.4) we have k1+α−β (n − 1) =

1 1 (n − 1)α−β [1 + O( )], Γ(1 + α − β) n−1

for n large enough, it follows that for any δ > 0 there is n∗ ∈ ℕ large enough such that n∗ + 1 ≤ n for n large enough and n−1

∑ kα (j)k1−β (n − 1 − j)
0 such that ‖u∗ ‖∞ ≤ C and n−1

u∗ (n) = Sα (n)(I − A)u0 + ∑ Sα (n − 1 − j)f (j, u∗ (j)), j=0

n ∈ ℕ,

7.2 Stability | 281

moreover, in view of (7.4) we have n−1

󵄩 󵄩 󵄩 󵄩󵄩 ∗ 󵄩󵄩 󵄩󵄩 ∗ 󵄩󵄩u (n)󵄩󵄩 ≤ 󵄩󵄩Sα (n)(I − A)u0 󵄩󵄩󵄩 + ∑ 󵄩󵄩󵄩Sα (n − 1 − j)f (j, u (j))󵄩󵄩󵄩 j=0

n−1

󵄩 󵄩 󵄩 󵄩 ≤ Mkα (n)󵄩󵄩󵄩(I − A)u0 󵄩󵄩󵄩 + MLf ∑ kα (n − 1 − j)k1−β (j)󵄩󵄩󵄩u∗ (j)󵄩󵄩󵄩 j=0

≤ 2Mkα (n)‖u0 ‖A + MLf Ck1+α−β (n − 1) → 0,

as n → ∞.

Thus, u∗ is a stable solution. The proof is completed. Remark 7.3. Theorem 7.5 shows that it is not necessary to use the Lipschitz condition to establish the existence for problem (7.29), and this is a general result of the paper [119]. Example 7.5. Let Ω = [0, π] and X = L2 (Ω). We consider the following discrete abstract Cauchy problem: 2

d Δα u(n, z) = dz 2 u(n + 1, z) + k1−β (n)u(n, z), { { { u(n, 0) = u(n, π) = 0, { { { {u(0, z) = 0,

n ∈ ℕ0 , z ∈ Ω, n ∈ ℕ0 ,

(7.31)

z ∈ Ω,

where Δα is the Riemann–Liouville-like fractional difference operator of order 0 < α < β < 1. Let us consider the operator A : D(A) ⊆ X → X defined by D(A) = {v ∈ X : v′ , v′′ ∈ X, v(0) = v(π) = 0},

Av = v′′ .

Clearly, A is closed densely defined in X and it is well known that A generates a compact, uniformly bounded and analytic C0 -semigroup {T(t)}t>0 . Furthermore, A has a discrete spectrum with eigenvalues of the form −m2 , m ∈ ℕ, and corresponding normalized eigenfunctions given by ϕm (z) = √2/π sin(mz). In addition, {ϕm }m∈ℕ is an orthogonal basis for X, and ∞

2

T(t)u = ∑ e−m t (u, ϕm )ϕm , m=1

u ∈ D(A).

Hence, by applying Lemmas 7.5–7.6, we get the discrete compact α-resolvent family {Sα (n)}n∈ℕ0 as follows: ∞ ∞

Sα (n)u = ∑ ∫ pn (t)t α−1 Eα,α (−m2 t α )dt(u, ϕm )ϕm . m=1 0

282 | 7 Discrete time fractional evolution equations Let ∞

Sm (n) = ∫ pn (t)t

α−1

Eα,α (−m2 t α )dt,

0

since the inequalities |Eα,α (−m2 t α )| ≤ 1/Γ(α) for all m ∈ ℕ, t ∈ ℝ+ and |Sm (n)| ≤ kα (n) for n ∈ ℕ0 , it follows that Sm (n) tend to zero as n → ∞ for all m ∈ ℕ. Thus, we have ∞

Sα (n)u = ∑ Sm (n)(u, ϕm )ϕm , m=1

n ∈ ℕ0 .

Therefore, let f (n, u(n)) = k1−β (n)u(n). The problem (7.31) possesses a stable solution by Theorem 7.5 and its expression form is given by ∞ n−1

u(n) = ∑ ∑ Sm (n − 1 − j)(f (j, u(j)), ϕm )ϕm , m=1 j=0

n ∈ ℕ.

7.2.4 Ulam–Hyers–Rassias stability In this subsection, we obtain the Ulam–Hyers–Rassias stability for problem (7.29). We now introduce the following adaptation definition of Ulam–Hyers–Rassias stability for the discrete form of fractional differential equation. Definition 7.8. If u(n) satisfies 󵄩󵄩 α 󵄩 󵄩󵄩Δ u(n) − Au(n + 1) − f (n, u(n))󵄩󵄩󵄩 ≤ ϑ(n),

n ∈ ℕ0 ,

(7.32)

where ϑ(n) ≥ 0 for all n ∈ ℕ0 , and there exist a solution v(n) of the problem (7.29) and a constant C > 0 independent of u(n) and v(n) with 󵄩󵄩 󵄩 󵄩󵄩u(n) − v(n)󵄩󵄩󵄩 ≤ Cϑ(n),

n ∈ ℕ0 ,

for all n ∈ ℕ0 , then problem (7.29) is called the Ulam–Hyers–Rassias stability. In particular, if ϑ(n) is substituted for a constant in the above inequalities, then problem (7.29) is called the Ulam–Hyers stability. Remark 7.4. Obviously, v solves (7.32) if and only if there exists g : ℕ0 → X satisfying 󵄩󵄩 󵄩 󵄩󵄩g(n)󵄩󵄩󵄩 ≤ ϑ(n),

n ∈ ℕ0 ,

such that Δα v(n) = Av(n + 1) + f (n, v(n)) + g(n),

n ∈ ℕ0 .

7.2 Stability | 283

Furthermore, if v ∈ l∞ (ℕ0 , X) is a solution of inequality (7.32), then 󵄩󵄩 󵄩󵄩 n−1 󵄩 󵄩󵄩 󵄩󵄩v(n) − Sα (n)(I − A)v(0) − ∑ Sα (n − 1 − j)f (j, v(j))󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 j=0 n−1

󵄩 󵄩 ≤ ∑ 󵄩󵄩󵄩Sα (n − 1 − j)󵄩󵄩󵄩B ϑ(j). j=0

Remark 7.5. It is hard to get the Ulam–Hyers stability of problem (7.29), because if we substitute the sequence ϑ(n) for a constant, then from Remark 7.4, we see that 󵄩󵄩 󵄩󵄩 n−1 n−1 󵄩󵄩 󵄩 󵄩󵄩v(n) − Sα (n)(I − A)v(0) − ∑ Sα (n − 1 − j)f (j, v(j))󵄩󵄩󵄩 ≤ C ∑ kα (j), 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 j=0 j=0 in which ∑∞ j=0 kα (j) is divergent according to the Raabe’s discriminant, hence the above inequality does not make sense and we cannot find a suitable stability in the sense of Ulam–Hyers. (H4) There exists a nonnegative sequence L(n), such that 󵄩󵄩 󵄩 󵄩󵄩f (n, x) − f (n, y)󵄩󵄩󵄩 ≤ L(n)‖x − y‖,

for any x, y ∈ X, n ∈ ℕ0 ,

with respect to series ∑∞ j=0 L(j) converges absolutely. Theorem 7.6. Assume that (H4) holds. Let ϑ(n) : ℕ0 → ℝ+ be an increasing sequence ∞ such that ∑n−1 j=0 ϑ(j) ≤ ϑ(n) and let u ∈ l (ℕ0 , D(A)) be a solution of inequality (7.32) then problem (7.29) is Ulam–Hyers–Rassias stable. Proof. Let v ∈ l∞ (ℕ0 , D(A)) be a solution of inequality (7.32). By Remark 7.4, from the property of 0 < kα (n) ≤ 1 for α ∈ (0, 1), n ∈ ℕ0 , we have 󵄩󵄩 󵄩󵄩 n−1 󵄩󵄩 󵄩 󵄩󵄩v(n) − Sα (n)(I − A)v(0) − ∑ Sα (n − 1 − j)f (j, v(j))󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 j=0 n−1

󵄩 󵄩 ≤ ∑ 󵄩󵄩󵄩Sα (n − 1 − j)󵄩󵄩󵄩B ϑ(j) j=0

n−1

≤ M ∑ kα (n − 1 − j)ϑ(j) ≤ Mϑ(n). j=0

Let us denote by u ∈ l∞ (ℕ0 , D(A)) the unique solution of the Cauchy problem Δα u(n) = Au(n + 1) + f (n, u(n)),

{

u(0) = v(0).

n ∈ ℕ0 ,

284 | 7 Discrete time fractional evolution equations The solution u of above equation satisfies u(n) = Sα (n)(I − A)v(0) + (Sα ∗ f )(n − 1, u(n − 1)), therefore, it follows that n−1

󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩u(n) − v(n)󵄩󵄩󵄩 ≤ Mϑ(n) + ∑ 󵄩󵄩󵄩Sα (n − 1 − j)(f (j, u(j)) − f (j, v(j)))󵄩󵄩󵄩 j=0

n−1

≤ Mϑ(n) + M ∑ kα (n − 1 − j)L(j)‖u(j) − v(j))‖ j=0

n−1

󵄩 󵄩 ≤ Mϑ(n) + M ∑ L(j)󵄩󵄩󵄩u(j) − v(j)󵄩󵄩󵄩. j=0

On the other hand, let b(n) = ‖u(n) − v(n)‖, from 0 ≤ b(n) ≤ a(n) + ∑n−1 j=0 L(j)b(j) with respect to an increasing sequence a(n) for all n ∈ ℕ0 , we get n−1

b(n) ≤ a(n) ∏(1 + L(j)), j=1

n ∈ {2, 3, . . .} =: ℕ2 .

In fact, in view of b(0) = 0, we have for n = 1 that b(1) ≤ a(1); for n = 2, we get that b(2) ≤ a(2) + a(1)L(1) ≤ a(2)(1 + L(1)). Assume that it’s true for some n = k ∈ ℕ2 . If n = k + 1, then the induction implies k

b(k + 1) ≤ a(k) + ∑ L(j)b(j) j=0

j−1

k

≤ a(k) + ∑ L(j)a(j) ∏(1 + L(i)) + L(1)b(1) j=2

i=1

k

j−1

j=2

i=1

≤ a(k)[1 + ∑ L(j) ∏(1 + L(i))] + L(1)a(1) 1

k−1

j=1

j=1

≤ a(k)[1 + L(1) + L(2) ∏(1 + L(j)) + ⋅ ⋅ ⋅ + L(k) ∏(1 + L(j))], which implies the desired inequality. Since ∑∞ j=1 L(j) is convergent absolutely, it follows that ∏∞ (1+L(j)) is convergent absolutely and then there exists a constant M∗ > 0 such j=1 that ∞

∏(1 + L(j)) ≤ M∗ . j=1

7.3 Asymptotic almost periodicity |

285

Thus, let a(n) = Mϑ(n), there exists a constant C := MM∗ > 0 such that 󵄩 󵄩󵄩 󵄩󵄩u(n) − v(n)󵄩󵄩󵄩 ≤ Cϑ(n),

n ∈ ℕ0 .

Therefore, we conclude the desired result. The proof is completed. Example 7.6. For any 0 < λ < 1 and 0 < α < 1, let us consider the following fractional difference equation: Δα u(n) = −λu(n + 1) + νg(n) sin(u(n)),

n ∈ ℕ0 ,

(7.33)

where g(n) is a bounded sequence on l∞ (ℕ0 ) with n2 |g(n)| ≤ 1, parameter ν > 0. Clearly, λ > 0 is the generator of the exponentially bounded C0 -semigroup T(t) = e−λt for t ≥ 0. Hence, (H1) holds. Let f (n, u) = νg(n) sin(u), it is easy to check the condition (H4) and if ϑ(n) = 2n for all n ∈ ℕ0 and the inequality 󵄩󵄩 α 󵄩 󵄩󵄩Δ u(n) − Au(n + 1) − f (n, u(n))󵄩󵄩󵄩 ≤ ϑ(n),

n ∈ ℕ0

holds, then (7.33) is the Ulam–Hyers–Rassias stable by Theorem 7.6.

7.3 Asymptotic almost periodicity 7.3.1 Introduction The original notion of discrete almost-periodic sequence was proposed by the famous mathematician Walther in 1928 [200] and the theory as well as applications of discrete almost-periodic sequences was further developed by Halanay [116] and Corduneanu [69]. Discrete almost-periodic sequence plays a key role in characterizing the phenomena that are similar to the periodic phenomena. As an important development of the well-known discrete almost-periodic sequence, the notion of a discrete asymptotically almost-periodic sequence, which is the key issue, will be investigated in this work and was introduced in the works of Fan [91] in 1943 based on the Fréchet concept from [97, 98]. Since then, the investigation of the existence of solutions with asymptotically almost periodicity has become one of the important as well as attractive subjects in the theory as well as applications of difference equations due to both the intensive development of the theorem of difference equations itself and the applications in various sciences such as computer science, chemistry, physics, engineering and so forth. For some meaningful and interesting works in this area, we refer readers to Thana [192] for periodic evolution equations, Agarwal, Cuevas and Dantas [10] for difference equations of Volterra type, Song [187] for nonlinear delay difference equations of Volterra type, Cuevas and Pinto [72] as well as Campo, Pinto and Vidal [52] for functional difference equations, Matkowski [160] for functional equations and the references therein.

286 | 7 Discrete time fractional evolution equations On the other hand, the tools of fractional calculus are found to be of great utility in studying various scientific processes and systems. This has been mainly due to the ability of fractional-order operators to describe long-memory effects of underlying processes. In particular, models described by fractional differential equations have gained significant importance and there has been a great interest in developing the theory and applications of fractional differential equations. For examples and details, we refer the reader to a series of recent research articles and the references therein. Recently, fractional difference equations have gained more interest by many researchers. Besides their interest in terms of theory, the study of fractional difference equations has great importance in the aspect applications [108]. The study of modeling with fractional difference equations began with the works of Atici and Sengül [26] and then Atici and Eloe [23, 25], Dassios and Baleanu [79], Goodrich [106, 107] studied the initial value problems, boundary value problems and positivity as well as monotonicity properties [78] of solutions to fractional difference equations, Baleanu and Wu et al. [32, 211] and Lizama [147] investigated the stability of fractional difference equations and Wu and Baleanu [209] studied the logistic map with fractional difference and its chaos. Especially in [2], Abadias and Lizama considered nonlinear partial difference-differential equations in Banach space X forms like △α x(n) = Ax(n + 1) + F(n, x(n)),

n ∈ ℤ,

(7.34)

where α ∈ (0, 1), the operator A generates a C0 -semigroup defined on X, △α denotes the Weyl-like fractional difference operator, F(n, x) : ℤ × X → X is a function satisfying some Lipschitz-type assumptions. By using the operator theoretical method and the Banach fixed-point theorem, they proved the existence and uniqueness of solutions with almost automorphy to equation (7.34). Stimulated by the works of Abadias and Lizama [2], Cao and Zhou [55] established some criteria for the existence and uniqueness of mild solutions with asymptotical almost-periodicity to equation (7.34) with the nonlinear perturbation F(n, x) satisfying a Lipschitz assumption or locally Lipschitz assumption. According to our knowledge, up to now, much less is known about the existence of mild solution with asymptotical almost periodicity to equation (7.34) when the nonlinear perturbation F(n, x) loses the Lipschitz assumption with respect to x. Thus in this work, we will try to fill this gap. Combining a decomposition technique with the Krasnoselskii’s fixed-point theorem, we establish some new existence theorems of mild solutions with asymptotic almost periodicity to equation (7.34). In our results, F(n, x) does not have to satisfy a Lipschitz assumption or locally Lipschitz assumption with respect to x (see Remark 7.7) and F(n, x) with Lipschitz assumption becomes a special case with our conditions (see Remark 7.9). Thus, our results generalize some related conclusions on this topic. In particular, as an application, we prove the existence and

7.3 Asymptotic almost periodicity | 287

uniqueness of mild solutions with asymptotically almost periodicity to the fractional difference scalar equation forms like △α x(n) = λx(n + 1) + F(n, x(n)),

n ∈ ℤ,

where α ∈ (0, 1), λ is a complex number satisfying Re(λ) < 0, F : ℤ × ℂ → ℂ is a function to be specified later. An outline of this section is as follows. We introduce some basic concepts and recall some preliminaries in Subsection 7.3.2. Subsection 7.3.3 is concerned with some new existence theorems for mild solutions with asymptotically almost periodicity to equation (7.34). The last subsection deal with an example to validate the applications of our theoretical results. This section is based on [54].

7.3.2 Preliminaries In this subsection, we introduce some basic concepts and recall some preliminaries. Let ℕ, ℤ, ℤ+ , ℝ, ℝ+ , ℂ be the sets of all natural numbers, integral numbers, positive integral numbers, real numbers, positive real numbers and complex numbers, respectively. For a Banach space (X, ‖ ⋅ ‖), let Bρ (X) = {x ∈ X : ‖x‖ ≤ ρ}. Define s(ℤ, X) by a set consisting of sequences f : ℤ → X. In particularly, l∞ (ℤ, X) is a set consisting of sequences l∞ (ℤ, X) := {f : ℤ → X | f is bounded on ℤ}. The space l∞ (ℤ, X) is a Banach space under the norm 󵄩 󵄩 ‖x‖d := sup󵄩󵄩󵄩x(n)󵄩󵄩󵄩. n∈ℤ

The set AAP0 (ℤ, X) is one consisting of sequences 󵄨 󵄩 󵄩 AAP0 (ℤ, X) := {f (n) ∈ l∞ (ℤ, X) 󵄨󵄨󵄨 lim 󵄩󵄩󵄩f (n)󵄩󵄩󵄩 = 0}. |n|→+∞ Obviously, AAP0 (ℤ, X) is also a Banach space under the norm ‖x‖d . Let l(ℤ, X) be a set consisting of sequences +∞ 󵄨󵄨 󵄨 󵄩 󵄩 l(ℤ, X) := {f : ℤ → X 󵄨󵄨󵄨 ‖f ‖l = ∑ 󵄩󵄩󵄩f (n)󵄩󵄩󵄩 < +∞}. 󵄨󵄨 n=−∞

288 | 7 Discrete time fractional evolution equations Moreover, when X = ℝ, we write l(ℤ) for short. Denote lρ (ℤ, X) a set consisting of sequences +∞ 󵄨󵄨 󵄩 󵄩 󵄨 lρ (ℤ, X) := {f : ℤ → X 󵄨󵄨󵄨 ‖f ‖lρ = ∑ 󵄩󵄩󵄩f (n)󵄩󵄩󵄩ρ(n) < +∞, 󵄨󵄨 n=−∞

ρ : ℤ → ℝ+ is a positive sequence weight}. The space BC(ℝ, X) is one of all X-valued bounded continuous functions and C0 (ℝ, X) is the closed subspace of BC(ℝ, X) consisting of functions vanishing at infinity. Let (Y, ‖ ⋅ ‖Y ) be another Banach space, l∞ (ℤ × Y, X) is a set consisting of functions 󵄨 l∞ (ℤ × Y, X) := {G : ℤ × Y → X 󵄨󵄨󵄨 G is bounded on ℤ × Y and G(n, ⋅) is continuous on Y for each fixed n ∈ ℤ}. The space l∞ (ℤ × Y, X) is a Banach space under the norm 󵄨󵄨 󵄨󵄨 󵄩 󵄩 󵄨󵄨‖G‖󵄨󵄨 := sup 󵄩󵄩󵄩G(n, x)󵄩󵄩󵄩. n∈ℤ,x∈Y

By AAP0 (ℤ × Y, X) we define a set consisting of functions AAP0 (ℤ × Y, X) 󵄨 󵄩 󵄩 := {G(n, x) ∈ l∞ (ℤ × Y, X) 󵄨󵄨󵄨 lim 󵄩󵄩󵄩G(n, x)󵄩󵄩󵄩 = 0 uniformly for x ∈ Y}. |n|→+∞ Let 𝕃(X, Y) be the collection of all bounded linear operators from X to Y. Under the uniform operator topology, ‖T‖𝕃(X,Y) := sup{‖Tf ‖Y : f ∈ X, ‖f ‖ = 1}, we denote by 𝕃(X) = 𝕃(X, X). For A ∈ 𝕃(X), let ρ(A) be the resolvent of A and D(A) be the domain of A. First, we recall the definitions and related properties on discrete almost-periodic sequences as well as discrete asymptotically almost-periodic sequences. Following Bohr, Walther has formulated the notion of discrete almost-periodic sequence. Definition 7.9 ([68, 200]). Let {f (n)}n∈ℤ be a discrete sequence with values in X. If for each ε > 0, the collection 󵄩 󵄩 T(f , ε) := {k ∈ ℤ : 󵄩󵄩󵄩f (n + k) − f (n)󵄩󵄩󵄩 < ε for every n ∈ ℤ} is relatively dense in ℤ, that is for any ε > 0, there is an integer N = N(ε) > 0, such that there exists at least one integer k ∈ Δ, where Δ is any collection consisting of N

7.3 Asymptotic almost periodicity | 289

consecutive integers, satisfying 󵄩 󵄩󵄩 󵄩󵄩f (n + k) − f (n)󵄩󵄩󵄩 < ε,

n ∈ ℤ,

then {f (n)}n∈ℤ is said to be a discrete almost-periodic sequence. The integer k ∈ T(k, ε), with the property in Definition 7.9, is said to be an ε-translation number of the sequence {f (n)}n∈ℤ . By AP(ℤ, X) we denote the collection of such sequences. The following notion of a normal process is needed to formulate an important property of the discrete almost-periodic processes. Definition 7.10 ([68, 223]). A discrete sequence {f (n)}n∈ℤ is called a normal process, if for any sequence {α(k)} ∈ ℤ, there is a subsequence {β(k)} ⊂ {α(k)}, for which {f (n + β(k))} converges uniformly with respect to n ∈ ℤ, as k → ∞. That is to say, for any ε > 0, there exist an integer K(ε) > 0 and a discrete process {f ̄(n)}n∈ℤ such that 󵄩󵄩 󵄩 󵄩󵄩f (n + β(k)) − f ̄(n)󵄩󵄩󵄩 < ε,

for k ≥ K(ε), n ∈ ℤ.

Lemma 7.8 ([68]). A discrete process is almost-periodic if and only if it is normal. Definition 7.11 ([42, 43]). A continuous function f : ℝ → X is said to be (Bohr) almostperiodic in t ∈ ℝ if for each ε > 0, the collection 󵄩 󵄩 T(f , ε) := {τ ∈ ℝ : 󵄩󵄩󵄩f (t + τ) − f (t)󵄩󵄩󵄩 < ε, for every t ∈ ℝ} is relatively dense in ℝ; that is for every ε > 0 there exists l(ε) > 0 such that every interval of length l(ε) contains a number τ with the property that 󵄩󵄩 󵄩 󵄩󵄩f (t + τ) − f (t)󵄩󵄩󵄩 < ε,

for every t ∈ ℝ.

The number τ is called an ε-translation number of f (t) and the collection of those functions is denoted by AP(ℝ, X). Lemma 7.9 ([68, 221, 222]). (I) For any almost-periodic sequence {f (n)}n∈ℤ , there is a function g(t), t ∈ ℝ which is almost-periodic satisfying g(n) = f (n) for n ∈ ℤ. (II) For any almost-periodic function g(t), t ∈ ℝ, {g(n)}n∈ℤ is an almost-periodic sequence. Remark 7.6 ([52]). The discretization of a periodic function may not lead to a periodic sequence. For instance, {cos(k)}, k = 1, 2, 3, . . . , is not a periodic sequence, it is an almost-periodic sequence. Lemma 7.10 ([68, 221, 222]). The sequence {f (n)}n∈ℤ is bounded if {f (n)}n∈ℤ is an almost-periodic sequence.

290 | 7 Discrete time fractional evolution equations Lemma 7.11 ([68, 153]). Under the norm ‖ ⋅ ‖d , AP(ℤ, X) forms a Banach space. Definition 7.12 ([2]). An operator-valued sequence {T(n)}n∈ℕ ⊂ 𝕃(X) is summable if +∞

󵄩 󵄩 ‖T‖1 := ∑ 󵄩󵄩󵄩T(n)󵄩󵄩󵄩𝕃(X) < ∞. n=0

The following lemma, which comes from Gohberg and Feldman [104], is the essential property to study almost periodicity and asymptotic almost periodicity of difference equations. Lemma 7.12 ([104]). Assume {T(n)}n∈ℕ is a summable sequence. Then for any discrete sequence {f (n)}n∈ℤ , which is almost-periodic, the sequence {g(n)}n∈ℤ defined by +∞

g(n) = ∑ T(k)f (n − k), k=0

n∈ℤ

is also an almost-periodic sequence. Definition 7.13 ([187]). Let G : ℤ × Y → X and Ω be any compact set in Y. If for any ε > 0, the collection 󵄩 󵄩 T(G, ε, Ω) := {k ∈ ℤ : 󵄩󵄩󵄩G(n + k, x) − G(n, x)󵄩󵄩󵄩 < ε for each n ∈ ℤ and x ∈ Ω} is relatively dense in ℤ, that is for any ε > 0, there is an integer N = N(ε, Ω) such that there exists at least one integer k ∈ Δ, where Δ is any collection consisting of N consecutive integers, satisfying 󵄩󵄩 󵄩 󵄩󵄩G(n + k, x) − G(n, x)󵄩󵄩󵄩 < ε,

∀ n ∈ ℤ, x ∈ Ω,

then G(n, x) is said to be almost-periodic in n ∈ ℤ uniformly for x ∈ Y. The integer k ∈ T(G, ε, Ω), with the property in Definition 7.12, is said to be the ε-translation number of G(n, x). By AP(ℤ × Y, X), we denote the collection of such functions. Lemma 7.13 ([52, 187]). Let Ω be any compact set in Y and assume G ∈ AP(ℤ × Y, X). Then G(n, ⋅) is continuous on Ω uniformly for n ∈ ℤ, that is for any ε > 0, there is a constant δ > 0 such that 󵄩󵄩 󵄩 󵄩󵄩G(n, x) − G(n, y)󵄩󵄩󵄩 < ε,

∀ n ∈ ℤ, x, y ∈ Ω with ‖x − y‖ < δ,

and G(ℤ × Ω) is relatively compact in X.

7.3 Asymptotic almost periodicity | 291

Lemma 7.14. The following statements are equivalent: (i) G ∈ AP(ℤ × Y, X); (ii) G(⋅, f ) ∈ AP(ℤ, X) for each f ∈ Y and G is continuous in f ∈ Ω uniformly in n ∈ ℤ, i. e., for ∀ε > 0, ∃δ > 0, s. t. 󵄩 󵄩󵄩 󵄩󵄩G(n, f ) − G(n, g)󵄩󵄩󵄩 < ε,

∀n ∈ ℤ, f , g ∈ Ω with ‖f − g‖Y < δ,

where Ω is any compact set in Y. Proof. The proof of (i) ⇒ (ii) follows from Lemma 7.13. In the following, we prove (ii) ⇒ (i). In fact, if G is continuous in f ∈ Ω uniformly in n ∈ ℤ, then G is uniformly continuous in f on Ω, thus ∀ε > 0, ∃δ = δ(ε/3) > 0, s. t. 󵄩󵄩 󵄩 ε 󵄩󵄩G(n, f ) − G(n, g)󵄩󵄩󵄩 < , 3

∀n ∈ ℤ, f , g ∈ Ω with ‖f − g‖Y < δ.

Since Ω is any compact set in Y, then there are f1 , f2 , . . . , fn ∈ Ω so that for each f ∈ Ω, ‖f − fi ‖ < δ for some i. It follows from G(n, fi ) ∈ AP(ℤ, X) that for the above ε, there is an integer N = N(ε, Ω) such that there exists at least one integer k ∈ Δ, where Δ is any collection consisting of N consecutive integers, satisfying 󵄩󵄩 󵄩 ε 󵄩󵄩G(n + k, fi ) − G(n, fi )󵄩󵄩󵄩 < , 3

∀n ∈ ℤ.

Thus for each f ∈ Ω, choose some fi such that ‖f − fi ‖ < δ, then 󵄩󵄩 󵄩 󵄩󵄩G(n + k, f ) − G(n, f )󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩G(n + k, f ) − G(n + k, fi )󵄩󵄩󵄩 + 󵄩󵄩󵄩G(n + k, fi ) − G(n, fi )󵄩󵄩󵄩 + 󵄩󵄩󵄩G(n, fi ) − G(n, f )󵄩󵄩󵄩 ε ε ε < + + = ε, 3 3 3 which yields G ∈ AP(ℤ × Y, X). Lemma 7.15 ([187]). Assume f ∈ AP(ℤ, X). Then for any integer sequence {αk }, there are a subsequence {βk } ⊂ {αk } and a function g : ℤ → X satisfying f (n + βk ) → g(n) uniformly on ℤ as k → ∞. Moreover, g ∈ AP(ℤ, X). Lemma 7.16 ([187]). Assume G ∈ AP(ℤ × Y, X) and Ω is any compact set in Y. Then for any integer sequence {αk }, there are a subsequence {βk } ⊂ {αk } and a function H : ℤ × Y → X satisfying G(n + βk , x) → H(n, x) uniformly on ℤ × Ω as k → ∞. Moreover, H ∈ AP(ℤ × Y, X).

292 | 7 Discrete time fractional evolution equations We now give a composition theorem of discrete almost-periodic functions. Lemma 7.17. Assume G ∈ AP(ℤ × Y, X) satisfying G(n, ⋅) is continuous in each bounded subset of Y uniformly in n ∈ ℤ, i. e., for ∀ε > 0 and any bounded subset K of Y, ∃δ = δ(ε, K) > 0 s. t. 󵄩 󵄩󵄩 󵄩󵄩G(n, f ) − G(n, g)󵄩󵄩󵄩 ≤ ε,

∀n ∈ ℤ, f , g ∈ K with ‖f − g‖Y ≤ δ.

Suppose f : ℤ → Y is a discrete almost-periodic sequence. Then {G(n, f (n))}n∈ℤ belongs to AP(ℤ, X). Proof. From G ∈ AP(ℤ × Y, X) and f ∈ AP(ℤ, Y), together with Lemmas 7.15 and 7.16, it follows that for any integer sequence {αk }, there exist a subsequence {βk } ⊂ {αk } and two functions H : ℤ × Y → X, g : ℤ → Y satisfying G(n + βk , x) → H(n, x) uniformly on ℤ × Ω as k → ∞ and f (n + βk ) → g(n) uniformly on ℤ as k → ∞. Moreover, H ∈ AP(ℤ × Y, X) and g ∈ AP(ℤ, Y). It follows from f ∈ AP(ℤ, Y) and g ∈ AP(ℤ, Y), together with Lemma 7.10, that there exists a bounded subset K of Y such that f (n) ∈ K and g(n) ∈ K for each n ∈ ℤ. As G(n, x) is continuous on K uniformly for n ∈ ℤ, thus for any ε > 0, one can find a constant δ = δ( ε2 ) > 0 satisfying 󵄩󵄩 󵄩 ε 󵄩󵄩G(n, f ) − G(n, g)󵄩󵄩󵄩 < , 2

∀ n ∈ ℤ, f , g ∈ K with ‖f − g‖Y < δ.

Moreover, for every compact set Ω ⊂ Y, there exists an N = N( ε2 ) such that for all k > N, 󵄩󵄩 󵄩 ε 󵄩󵄩G(n + βk , f ) − H(n, f )󵄩󵄩󵄩 < , ∀n ∈ ℤ, x ∈ Ω, 2 󵄩󵄩 󵄩󵄩 󵄩󵄩f (n + βk ) − g(n)󵄩󵄩 < δ, ∀n ∈ ℤ. Thus for all k > N, 󵄩󵄩 󵄩 󵄩󵄩G(n + βk , f (n + βk )) − H(n, f (n))󵄩󵄩󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩G(n + βk , f (n + βk )) − G(n + βk , f (n))󵄩󵄩󵄩 󵄩 󵄩 + 󵄩󵄩󵄩G(n + βk , f (n)) − H(n, f (n))󵄩󵄩󵄩 ε ε < + = ε, 2 2

7.3 Asymptotic almost periodicity | 293

which implies {G(n, f (n))}n∈ℤ is a normal process. Thus, G(n, f (n)) is almost-periodic, which follows from Lemma 7.8. Let G(n, f ) : ℤ × X → X satisfy a Lipschitz assumption in f ∈ X uniformly in n ∈ ℤ, we obtain the following result as an immediate consequence of the previous lemma, Lemma 7.17. Lemma 7.18. Assume G(n, f ) ∈ AP(ℤ × X, X) and satisfies a Lipschitz assumption in f ∈ X uniformly for n ∈ ℤ, i. e., 󵄩󵄩 󵄩 󵄩󵄩G(n, f ) − G(n, g)󵄩󵄩󵄩 ≤ L‖f − g‖,

for ∀x, y ∈ X, n ∈ ℤ and a positive constant L. (7.35)

Then, the conclusion of Lemma 7.17 is true. Lemma 7.18 admits a new version with local Lipschitz assumption on the function G. Lemma 7.19. Assume G(n, f ) ∈ AP(ℤ × X, X) and satisfies a local Lipschitz condition in f ∈ X uniformly for n ∈ ℤ, i. e., for each r > 0 and ∀x, y ∈ X with ‖x‖ ≤ r, ‖y‖ ≤ r, 󵄩󵄩 󵄩 󵄩󵄩G(n, x) − G(n, y)󵄩󵄩󵄩 ≤ LG (r)‖x − y‖,

∀n ∈ ℤ,

(7.36)

where LG : ℝ+ → ℝ+ is a function. Then the conclusion of Lemma 7.17 is true. As an important development of the well-known discrete almost-periodic sequence, the notion of discrete asymptotically almost-periodic sequence, which is based upon the Fréchet concept from [97, 98], was introduced in the literature [91] by Fan. Definition 7.14 ([91, 187]). If a sequence f (n) = g(n) + h(n) with g(n) ∈ AP(ℤ, X) and h(n) ∈ AAP0 (ℤ, X), then the sequence f : ℤ → X is said to be asymptotically almostperiodic. The sequences {g(n)}n∈ℤ is called the almost-periodic component of {f (n)}n∈ℤ and {h(n)}n∈ℤ is called the ergodic perturbation of {f (n)}n∈ℤ . By AAP(ℤ, X) we denote the collection of such sequences. Definition 7.15 ([97, 98]). A continuous function f : ℝ → X is said to be asymptotically almost-periodic if it can be decomposed as f (t) = g(t) + h(t), where g(t) ∈ AP(ℝ, X),

h(t) ∈ C0 (ℝ, X).

Lemma 7.20 ([221, 222]). (I) For any discrete asymptotically almost-periodic sequence {f (n)}n∈ℤ , there is a function g(t), t ∈ ℝ which is asymptotically almost-periodic satisfying g(n) = f (n) for n ∈ ℤ.

294 | 7 Discrete time fractional evolution equations (II) For any asymptotically almost-periodic function g(t), t ∈ ℝ, {g(n)}n∈ℤ is a discrete asymptotically almost-periodic sequence. Lemma 7.21 ([221, 222]). The decomposition of an asymptotically almost-periodic sequence {f (n)}n∈ℤ , f (n) = g(n) + h(n) with g(n) ∈ AP(ℤ, X) and h(n) ∈ AAP0 (ℤ, X), is unique. Lemma 7.22 ([153]). Under the norm ‖ ⋅ ‖d , AAP(ℤ, X) also forms a Banach space. Lemma 7.23. Assume g(n) is the almost-periodic component of the sequence f (n) ∈ AAP(ℤ, X). Then g(ℤ) ⊂ f (ℤ). Proof. Denote by h(n) = f (n) − g(n), the ergodic perturbation of f (n). If g(ℤ) is not contained in f (ℤ), then there are ε0 > 0 and n0 ∈ ℤ such that 󵄩 󵄩 inf 󵄩󵄩f (n) − g(n0 )󵄩󵄩󵄩 ≥ ε0 .

n∈ℤ󵄩

Let us take k ∈ T(g,

ε0 ). 2

Then

󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩h(n0 + k)󵄩󵄩󵄩 = 󵄩󵄩󵄩f (n0 + k) − g(n0 + k)󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ε ≥ 󵄩󵄩󵄩f (n0 + k) − g(n0 )󵄩󵄩󵄩 − 󵄩󵄩󵄩g(n0 + k) − g(n0 )󵄩󵄩󵄩 ≥ 0 , 2 which is a contradiction with h(n) ∈ AAP0 (ℤ, X). Definition 7.16 ([52, 153]). A function G(n, x) : ℤ×Y → X is said to be a discrete asymptotically almost-periodic function in n ∈ ℤ for each x ∈ Y if G(n, x) = H(n, x) + W(n, x) with H(n, x) ∈ AP(ℤ × Y, X) and W(n, x) ∈ AAP0 (ℤ × Y, X). By AAP(ℤ × Y, X) we denote the collection of such functions. Lemma 7.24. Assume G(n, x) ∈ AAP(ℤ × Y, X) and Ω is any compact set in Y. Then G(n, x) is bounded on ℤ × Ω. Proof. As G(n, x) ∈ AAP(ℤ × Y, X), then G(n, x) = H(n, x) + W(n, x) with H(n, x) ∈ AP(ℤ × Y, X) and W(n, x) ∈ AAP0 (ℤ × Y, X). From Lemma 7.13, it follows that H(n, x) is bounded on ℤ × Ω. On the other hand, as W(n, x) ∈ AAP0 (ℤ × Y, X), then W(n, x) is bounded on ℤ × Y. Thus, G(n, x) = H(n, x) + W(n, x) is bounded on ℤ × Ω.

7.3 Asymptotic almost periodicity | 295

Definition 7.17 ([2]). Assume α > 0 and A is a closed-linear operator with domain D(A) ⊂ X. An operator sequence {Sα (n)}n∈ℕ ⊂ 𝕃(X) is said to be a discrete α-resolvent family generated by A if for any n ∈ ℕ and x ∈ D(A), Sα (n)Ax = ASα (n)x,

Sα (n)x = kα (n)x + A(kα ∗ Sα )(n)x.

Definition 7.18 ([2]). For any α > 0, let ρ(n) = |n|α−1 , n ∈ ℤ and f ∈ lρ (ℤ, X) be a sequence. The fractional sum of f is given by n

△−α f (n) := ∑ kα (n − j)f (j), j=−∞

n ∈ ℤ.

Definition 7.19 ([2]). For any α > 0, let ρ(n) = |n|α−1 , n ∈ ℤ and f ∈ lρ (ℤ, X) be a sequence. The fractional difference of f is given by △α f (n) := △p △−(p−α) f (n),

n ∈ ℤ,

where p = [α] + 1, [⋅] is the largest integer function. Lemma 7.25. A set D ⊂ C0 (ℤ, X) is relatively compact if: (1) lim|n|→+∞ x(n) = 0 uniformly for x ∈ D, (2) the set D(n) := {x(n) : x ∈ D} is relatively compact in X for every n ∈ ℤ. Proof. For any n1 , n2 ∈ ℤ with n1 < n2 , set D(n1 , n2 ) = {(x(n1 ), x(n1 + 1), . . . , x(n2 )) | x ∈ D}. For any x∗ = (x(n1 ), x(n1 + 1), . . . , x(n2 )), y∗ = (y(n1 ), y(n1 + 1), . . . , y(n2 )) ∈ D(n1 , n2 ), define 󵄩 󵄩 ρ(x∗ , y∗ ) = max{󵄩󵄩󵄩x(m) − y(m)󵄩󵄩󵄩 | n1 ≤ m ≤ n2 }, it is clear that (D(n1 , n2 ), ρ) is a metric space. Let {xk∗ } be a sequence in D(n1 , n2 ) with xk∗ = (xk (n1 ), xk (n1 + 1), . . . , xk (n2 )), it follows from the condition (2) that {xk (n1 )} is ∗ relatively compact in X. Then there exists subsequence {x1,k } ⊂ {xk∗ } such that {x1,k (n1 )} is convergent in X. Then from the condition (2), it follows that {x1,k (n1 + 1)} is relatively ∗ ∗ compact in X; thus, there exists subsequence {x2,k } ⊂ {x1,k } such that {x2,k (n1 + 1)} is convergent in X. Note that {x2,k (n1 )} is also convergent in X. Repeating like this, we ∗ can obtain a subsequence {xk,j } such that {xk,j (m)} (n1 ≤ m ≤ n2 ) is convergent in X, ∗ i. e., {xk,j } is relatively compact in D(n1 , n2 ). Then {xk∗ } is relatively compact in D(n1 , n2 ). Since {xk∗ } is arbitrary, then (D(n1 , n2 ), ρ) is relatively compact. From the condition (1), it follows that for any ε > o, there exists n0 ∈ ℤ+ such that 󵄩󵄩 󵄩 󵄩󵄩x(n)󵄩󵄩󵄩 < ε,

for all |n| > n0 , x ∈ D.

296 | 7 Discrete time fractional evolution equations As (D(−n0 , n0 ), ρ) is relatively compact, then it has a finite ε-net A. Let ̃ = {(. . . , 0, x(−n0 ), . . . , x(n0 ), 0, . . . | (x(−n0 ), . . . , x(n0 )) ∈ A}. A For any x ∈ D, assume that x∗ ∈ D(−n0 , n0 ) is a corresponding point of x and taking y∗ ∈ A such that ρ(x∗ , y∗ ) < ε. Assume that ỹ ∗ is a corresponding point of y∗ in ̃ then A, 󵄩󵄩 ∗󵄩 󵄩󵄩x − ỹ 󵄩󵄩󵄩d < ε. ̃ is a finite ε-net of D, thus D is totally bounded; this together This implies that A with C0 (ℤ, X) is a complete space and yields that D is relatively compact. 7.3.3 Existence In this subsection, we will state and prove conditions for the existence of mild solutions with asymptotically almost periodicity of the nonhomogeneous nonlinear difference equations of fractional-order given by △α x(n) = Ax(n + 1) + F(n, x(n)),

n ∈ ℤ,

(7.37)

where α ∈ (0, 1), the operator A generates a C0 -semigroup on X, △α denotes the Weyllike fractional difference operator, F(n, x) : ℤ × X → X is a function to be specified later. In [2], Abadias and Lizama obtained the following remarkable result. Lemma 7.26 ([2]). Assume A generates a C0 -semigroup {T(t)}t≥0 on X, which is exponentially stable, i. e., 󵄩󵄩 󵄩 −δt 󵄩󵄩T(t)󵄩󵄩󵄩𝕃(X) ≤ Me ,

for ∀ t > 0 and some constants M > 0, δ > 0.

(7.38)

Then there exists a discrete α-resolvent family {Sα (n)}n∈ℕ generated by A, which is given by ∞∞

Sα (n)x := ∫ ∫ e−t 0 0

tn f (t)T(s)xdsdt, n! s,α

n ∈ ℕ, x ∈ X,

where fs,α (t) is a function given by σ+i∞

α 1 fs,α (t) = ∫ ezt−sz dz, 2πi

σ−i∞

σ > 0, s > 0, t ≥ 0, 0 < α < 1.

7.3 Asymptotic almost periodicity | 297

Moreover, {Sα (n)}n∈ℕ satisfies ∞

n

󵄩 󵄩󵄩 −t t α−1 t Eα,α (−δt α )dt, 󵄩󵄩Sα (n)󵄩󵄩󵄩𝕃(X) ≤ M ∫ e n! 0

and furthermore {Sα (n)}n∈ℕ is summable with ∞ 󵄩 1 󵄩 ‖Sα ‖1 = ∑ 󵄩󵄩󵄩Sα (n)󵄩󵄩󵄩 ≤ . δ k=0

The following definition of mild solutions to equation (7.37), which is given in [2], is essential for us. Definition 7.20 ([2]). Assume {Sα (n)}n∈ℕ ⊂ 𝕃(X) is a discrete α-resolvent family generated by A. A sequence x ∈ s(ℤ, X) is said to be a mild solution of equation (7.37) if for each n ∈ ℤ, m → Sα (m)F(n − 1 − m, x(n − 1 − m)) is summable on ℕ and x satisfies n−1

x(n) = ∑ Sα (n − 1 − k)F(k, x(k)), k=−∞

n ∈ ℤ.

(7.39)

The following auxiliary result plays a key role in the proofs of our main results. Lemma 7.27. Let {S(n)}n∈ℕ be summable. Given sequence F(n) ∈ AP(ℤ, X), G(n) ∈ AAP0 (ℤ, X). Let n−1

Φ(n) := ∑ S(n − 1 − k)F(k), k=−∞

n−1

Ψ(n) := ∑ S(n − 1 − k)G(k), k=−∞

n ∈ ℤ.

Then Φ(n) ∈ AP(ℤ, X) and Ψ(n) ∈ AAP0 (ℤ, X). Proof. From F(n) ∈ AP(ℤ, X), combining with Lemma 7.10, it follows that F(n) is bounded on X. In addition, note that n−1

+∞

󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩 󵄩󵄩Φ(n)󵄩󵄩󵄩 ≤ ∑ 󵄩󵄩󵄩S(n − 1 − k)󵄩󵄩󵄩󵄩󵄩󵄩F(k)󵄩󵄩󵄩 = ∑ 󵄩󵄩󵄩S(k)󵄩󵄩󵄩󵄩󵄩󵄩F(n − 1 − k)󵄩󵄩󵄩 k=−∞

k=0

+∞

󵄩 󵄩 ≤ ‖F‖d ∑ 󵄩󵄩󵄩S(k)󵄩󵄩󵄩 ≤ ‖F‖d ‖S‖1 < +∞, k=0

hence Φ(n) is well-defined. Similarly, Ψ(n) is also well-defined. From Lemma 7.12, together with n−1

+∞

k=−∞

k=0

Φ(n) = ∑ S(n − 1 − k)F(k) = ∑ S(k)F(n − 1 − k), it follows that Φ(n) ∈ AP(ℤ, X).

298 | 7 Discrete time fractional evolution equations As G(n) ∈ AAP0 (ℤ, X), then for ∀ε > 0, ∃N = N(ε) > 0 s. t., 󵄩 󵄩󵄩 󵄩󵄩G(k)󵄩󵄩󵄩 ≤ ε,

∀k > N.

Thus, 󵄩󵄩 󵄩󵄩 n−1 n−1 󵄩󵄩 󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩Ψ(n)󵄩󵄩󵄩 = 󵄩󵄩󵄩 ∑ S(n − 1 − k)G(k)󵄩󵄩󵄩 ≤ ∑ 󵄩󵄩󵄩S(n − 1 − k)󵄩󵄩󵄩󵄩󵄩󵄩G(k)󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 k=−∞ 󵄩k=−∞ +∞

+∞

k=0

k=0

󵄩 󵄩 󵄩 󵄩󵄩 󵄩 = ∑ 󵄩󵄩󵄩S(k)󵄩󵄩󵄩󵄩󵄩󵄩G(n − 1 − k)󵄩󵄩󵄩 ≤ ε ∑ 󵄩󵄩󵄩S(k)󵄩󵄩󵄩 ≤ ε‖S‖1 ,

∀n > N + k + 1,

which implies Ψ(n) ∈ AAP0 (ℤ, X). Now, we state and prove our main results. The following assumptions are required: (H1 ) The C0 -semigroup {T(t)}t≥0 generated by A on X is exponentially stable. (H2 ) The function F(n, x) = F1 (n, x) + F2 (n, x) ∈ AAP(ℤ × X, X) with F1 (n, x) ∈ AP(ℤ × X, X),

F2 (n, x) ∈ C0 (ℤ × X, X)

and it is bounded on ℤ × X. Moreover, 󵄩󵄩 󵄩 󵄩󵄩F1 (n, x) − F1 (n, y)󵄩󵄩󵄩 ≤ L‖x − y‖,

for ∀n ∈ ℤ, x, y ∈ X and a positive constant L. (7.40) (H3 ) There is a function β(n) ∈ C0 (ℤ, ℝ+ ) satisfying +∞

Tn (t) = ∑ β(n − 1 − k)e−t k=0

tk ∈ C0 (ℤ, ℝ+ ) k!

uniformly for t ∈ ℝ+ ,

and a nondecreasing function Φ : ℝ+ → ℝ+ satisfying 󵄩󵄩 󵄩 󵄩󵄩F2 (n, x)󵄩󵄩󵄩 ≤ β(n)Φ(r) and Φ(r) = ρ1 , for ∀n ∈ ℤ, x ∈ X with ‖x‖ ≤ r. lim inf r→+∞ r

(7.41)

(H4 ) For all n1 , n2 ∈ ℤ, n1 ≤ n2 and η > 0, the collection {F2 (n, x) : n1 ≤ n ≤ n2 , ‖x‖ ≤ η} is relatively compact in X. Remark 7.7. Note that in (H2 ), F(n, x) does not need to meet the Lipschitz assumption with respect to x. Such classes of functions F(n, x) are more complicated than those with the Lipschitz assumption (see Remark 7.8). Remark 7.8. In Lemma 7.24, we prove that a discrete asymptotically almost-periodic function F(n, x) : ℤ × X → X is bounded only on ℤ × Ω for any compact set Ω in X,

7.3 Asymptotic almost periodicity | 299

so the condition boundedness in (H2 ) does not conflict with the condition F(n, x) ∈ AAP(ℤ × X, X). First, we give some important lemmas. Lemma 7.28. Let G(n, f ) = G1 (n, f ) + G2 (n, f ) ∈ AAP(ℤ × X, X) with G1 (n, f ) ∈ AP(ℤ × X, X),

G2 (n, f ) ∈ C0 (ℤ × X, X).

Then we have 󵄩 󵄩 󵄩 󵄩 sup󵄩󵄩󵄩G1 (n, f ) − G1 (n, g)󵄩󵄩󵄩 ≤ sup󵄩󵄩󵄩G(n, f ) − G(n, g)󵄩󵄩󵄩, n∈ℤ

n∈ℤ

f , g ∈ X.

(7.42)

Proof. To obtain this result, it suffices to show {G1 (n, f ) − G1 (n, g) : n ∈ ℤ} ⊂ {G(n, f ) − G(n, g) : n ∈ ℤ},

f , g ∈ X.

If this is not true, then there are n0 ∈ ℤ and ε > 0 s. t., 󵄩󵄩 󵄩 󵄩󵄩(G1 (n0 , f ) − G1 (n0 , g)) − (G(n, f ) − G(n, g))󵄩󵄩󵄩 ≥ 3ε,

∀n ∈ ℤ and fixed x, y ∈ X.

As G2 (n, f ) ∈ C0 (ℤ × X, X), then 󵄩󵄩

lim 󵄩G (n, x) n→+∞󵄩 2

󵄩 − G2 (n, y)󵄩󵄩󵄩 = 0,

this yields that ∃N > 0 s. t. 󵄩󵄩 󵄩 󵄩󵄩G2 (n, f ) − G2 (n, g)󵄩󵄩󵄩 < ε,

∀n ≥ N.

(7.43)

From G1 (n, x) ∈ AP(ℤ × X, X), it follows that for the above ε and every compact Ω ⊂ X, there is an integer N = N(ε) > 0, such that there exists at least one integer k ∈ Δ, where Δ is any collection consisting of N consecutive integers, satisfying 󵄩󵄩 󵄩 󵄩󵄩G1 (n0 + k, f ) − G1 (n0 , f )󵄩󵄩󵄩 < ε,

󵄩󵄩 󵄩 󵄩󵄩G1 (n0 + k, g) − G1 (n0 , g)󵄩󵄩󵄩 < ε.

Then 󵄩󵄩 󵄩 󵄩󵄩G2 (n0 + k, f ) − G2 (n0 + k, g)󵄩󵄩󵄩 󵄩 󵄩 ≥ 󵄩󵄩󵄩G(n0 + k, f ) − G(n0 + k, g) − G1 (n0 , f ) + G1 (n0 , g)󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 − 󵄩󵄩󵄩G1 (n0 + k, f ) − G1 (n0 , f )󵄩󵄩󵄩 − 󵄩󵄩󵄩G1 (n0 + k, g) − G1 (n0 , g)󵄩󵄩󵄩 ≥ ε, which contradicts equation (7.43).

300 | 7 Discrete time fractional evolution equations Remark 7.9. From equation (7.42) it follows that if G(n, x) meets the Lipschitz assumption of equation (7.35), then G1 (n, x) satisfies equation (7.40). Then Lipschitz assumptions become a special case of our condition. Note that in [2, 55], a Lipschitz assumption (equation (7.35)) or a locally Lipschitz assumption (equation (7.36)) for F(n, x) of equation (7.37) is needed. Thus our condition in (H2 ) is weaker than those of [2, 55] and our results generalize some related conclusions of [2, 55]. Let β(n) be the sequence in (H3 ). Set n−1 ∞

σ(n) := M ∑ ∫ β(k)e−t k=−∞ 0

t n−1−k t α−1 Eα,α (−δt α )dt (n − 1 − k)!

+∞ ∞

= M ∑ ∫ β(n − 1 − k)e−t k=0 0 ∞

t k α−1 t Eα,α (−δt α )dt k!

= M ∫ Tn (t)t α−1 Eα,α (−δt α )dt,

n ∈ ℤ.

0

Lemma 7.29. The function σ(n) ∈ C0 (ℤ, ℝ+ ). Proof. From Tn (t) ∈ C0 (ℤ, ℝ+ ) uniformly for t ∈ ℝ+ , it follows that for any ε > 0, one can choose an N1 > 0 such that |Tn (t)| < ε, for all |n| > N1 . This combining with Proposition 1.10 and Eα (0) = 1, implies that ∞

σ(n) = M ∫ Tn (t)t α−1 Eα,α (−δt α )dt 0 ∞

≤ Mε ∫ t α−1 Eα,α (−δt α )dt 0

󵄨󵄨 Mε 󵄨 Eα (−δt α )󵄨󵄨󵄨 󵄨󵄨0 δ Mε for ∀|n| > N1 , = δ +∞

=−

which implies lim|n|→+∞ σ(n) = 0. First, we state and prove conditions for the existence of mild solutions with asymptotically almost periodicity of equation (7.37) when the almost-periodic component F1 of F satisfies equation (7.40). Theorem 7.7. Let (H1 )–(H4 ) hold. Put ρ2 := supn∈ℤ σ(n). Then there is a mild solution of equation (7.37) whenever L‖Sα ‖1 + ρ1 ρ2 < 1. Moreover, the mild solution is asymptotically almost-periodic.

(7.44)

7.3 Asymptotic almost periodicity |

301

Proof. We divide into five steps to complete the proof. Step 1. Let Λ be a mapping defined on AP(ℤ, X) given by n−1

(Λv)(n) = ∑ Sα (n − 1 − k)F1 (k, v(k)), k=−∞

n ∈ ℤ.

There is a unique fixed point v(n) ∈ AP(ℤ, X) of Λ. First, according to (H1 ), there is a discrete α-resolvent family {Sα (n)}n∈ℕ ⊂ 𝕃(X) generated by A, which is summable (by Lemma 7.26) and, combining with the discrete function k → F1 (k, v(k)), is bounded on ℤ, which follows from Lemma 7.10, and one has n−1

󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩 󵄩󵄩(Λv)(n)󵄩󵄩󵄩 ≤ ∑ 󵄩󵄩󵄩Sα (n − 1 − k)󵄩󵄩󵄩󵄩󵄩󵄩F1 (k, v(k))󵄩󵄩󵄩 k=−∞ +∞

󵄩 󵄩󵄩 󵄩 = ∑ 󵄩󵄩󵄩Sα (k)󵄩󵄩󵄩󵄩󵄩󵄩F1 (n − 1 − k, v(n − 1 − k))󵄩󵄩󵄩 k=0

+∞

󵄨 󵄨 󵄩 󵄩 ≤ 󵄨󵄨󵄨‖F1 ‖󵄨󵄨󵄨 ∑ 󵄩󵄩󵄩Sα (k)󵄩󵄩󵄩 k=0

󵄨 󵄨 = 󵄨󵄨󵄨‖F1 ‖󵄨󵄨󵄨‖Sα ‖1 < +∞,

for all n ∈ ℤ,

which yields that (Λv)(n) exists. Furthermore from F1 (n, x) ∈ AP(ℤ × X, X) satisfying equation (7.40), combined with Lemmas 7.10 and 7.18, it follows that F1 (⋅, v(⋅)) ∈ AP(ℤ, X),

∀v(⋅) ∈ AP(ℤ, X).

This, combined with Lemma 7.9, yields that Λ is well-defined. Then we show Λ is continuous. Let vj (n), v(n) ∈ AP(ℤ, X) satisfy vj (n) → v(n) as j → ∞, then 󵄩󵄩 n−1 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩(Λvj )(n) − (Λv)(n)󵄩󵄩󵄩 = 󵄩󵄩󵄩 ∑ Sα (n − 1 − k)[F1 (k, vj (k)) − F1 (k, v(k))]󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩k=−∞ 󵄩 n−1

󵄩 󵄩󵄩 󵄩 ≤ ∑ 󵄩󵄩󵄩Sα (n − 1 − k)󵄩󵄩󵄩󵄩󵄩󵄩F1 (k, vj (k)) − F1 (k, v(k))󵄩󵄩󵄩 k=−∞

n−1

󵄩 󵄩󵄩 󵄩 ≤ L ∑ 󵄩󵄩󵄩Sα (n − 1 − k)󵄩󵄩󵄩󵄩󵄩󵄩vj (k) − v(k)󵄩󵄩󵄩 k=−∞

n−1

󵄩 󵄩 ≤ L‖vj − v‖d ∑ 󵄩󵄩󵄩Sα (n − 1 − k)󵄩󵄩󵄩 k=−∞

= L‖Sα ‖1 ‖vj − v‖d .

302 | 7 Discrete time fractional evolution equations Therefore, as j → ∞, Λvj → Λv, which yields that Λ is continuous. Finally, we show there is a unique fixed point v(n) ∈ AP(ℤ, X) of Λ. Let v1 (n), v2 (n) ∈ AP(ℤ, X), similar to the proof of the continuity of Λ, and we have 󵄩 󵄩󵄩 󵄩󵄩(Λv1 )(n) − (Λv2 )(n)󵄩󵄩󵄩 ≤ L‖Sα ‖1 ‖v1 − v2 ‖d , which implies ‖Λv1 − Λv2 ‖d ≤ L‖Sα ‖1 ‖v1 − v2 ‖∞ . From equation (7.44), it follows that Λ is a contraction on AP(ℤ, X). Thus, there is a unique fixed point v(n) ∈ AP(ℤ, X) of Λ. Step 2. Set Ωr := {ω(n) ∈ C0 (ℤ, X) : ‖ω‖d ≤ r}. For the above v(n), define Γ := Γ1 + Γ2 on C0 (ℤ, X) as n−1

(Γ1 ω)(n) = ∑ Sα (n − 1 − k)[F1 (k, v(k) + ω(k)) − F1 (k, v(k))], k=−∞ n−1

2

(Γ ω)(n) = ∑ Sα (n − 1 − k)F2 (k, v(k) + ω(k)), k=−∞

n ∈ ℤ, (7.45)

n ∈ ℤ.

There is a constant k0 s. t. Γ maps Ωk0 into itself. First, according to equation (7.40), one has 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩F1 (k, v(k) + ω(k)) − F1 (k, v(k))󵄩󵄩󵄩 ≤ L󵄩󵄩󵄩ω(k)󵄩󵄩󵄩,

∀k ∈ ℤ, ω(k) ∈ X,

which yields F1 (⋅, v(⋅) + ω(⋅)) − F1 (⋅, v(⋅)) ∈ C0 (ℤ, X),

∀ω(⋅) ∈ C0 (ℤ, X).

According to equation (7.41), one has 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩F2 (k, v(k) + ω(k))󵄩󵄩󵄩 ≤ β(k)Φ(r + sup󵄩󵄩󵄩v(k)󵄩󵄩󵄩), k∈ℤ

󵄩 󵄩 ∀k ∈ ℤ, ω(k) ∈ X with 󵄩󵄩󵄩ω(k)󵄩󵄩󵄩 ≤ r.

Then F2 (⋅, v(⋅) + ω(⋅)) ∈ C0 (ℤ, X) as β(⋅) ∈ C0 (ℤ, ℝ+ ). Thus Γ is well-defined and maps C0 (ℤ, X) into itself, which follows from Lemma 7.27. From equations (7.41) and (7.44), it easily follows that there is a constant k0 > 0 s. t., 󵄩 󵄩 L‖Sα ‖1 k0 + ρ2 Φ(k0 + sup󵄩󵄩󵄩v(k)󵄩󵄩󵄩) ≤ k0 . k∈ℤ

7.3 Asymptotic almost periodicity | 303

Then for any n ∈ ℤ and ω1 (n), ω2 (n) ∈ Ωk0 , one has 󵄩 󵄩󵄩 1 2 󵄩󵄩(Γ ω1 )(n) + (Γ ω2 )(n)󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 n−1 󵄩󵄩 󵄩󵄩 ≤ 󵄩󵄩󵄩 ∑ Sα (n − 1 − k)[F1 (k, v(k) + ω1 (k)) − F1 (k, v(k))]󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩k=−∞ 󵄩󵄩 󵄩󵄩 n−1 󵄩󵄩 󵄩󵄩 + 󵄩󵄩󵄩 ∑ Sα (n − 1 − k)F2 (k, v(k) + ω2 (k))󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩k=−∞ n−1

󵄩 󵄩 ≤ ∑ 󵄩󵄩󵄩Sα (n − 1 − k)[F1 (k, v(k) + ω1 (k)) − F1 (k, v(k))]󵄩󵄩󵄩 k=−∞

n−1

󵄩 󵄩 + ∑ 󵄩󵄩󵄩Sα (n − 1 − k)F2 (k, v(k) + ω2 (k))󵄩󵄩󵄩 k=−∞ n−1

󵄩 󵄩󵄩 󵄩 ≤ L ∑ 󵄩󵄩󵄩Sα (n − 1 − k)󵄩󵄩󵄩󵄩󵄩󵄩ω1 (k)󵄩󵄩󵄩 k=−∞

n−1

󵄩 󵄩 󵄩 󵄩 + Φ(‖ω2 ‖d + sup󵄩󵄩󵄩v(k)󵄩󵄩󵄩) ∑ 󵄩󵄩󵄩β(k)Sα (n − 1 − k)󵄩󵄩󵄩 k∈ℤ

k=−∞

n−1

󵄩 󵄩󵄩 󵄩 ≤ L ∑ 󵄩󵄩󵄩Sα (n − 1 − k)󵄩󵄩󵄩󵄩󵄩󵄩ω1 (k)󵄩󵄩󵄩 k=−∞

∞

n−1 t n−1−k 󵄩 󵄩 + Φ(‖ω2 ‖d + sup󵄩󵄩󵄩v(k)󵄩󵄩󵄩)M ∑ ∫ β(k)e−t t α−1 Eα,α (−δt α )dt (n − 1 − k)! k∈ℤ

k=−∞ 0

n−1

󵄩 󵄩 󵄩 󵄩 ≤ L‖ω1 ‖d ∑ 󵄩󵄩󵄩Sα (n − 1 − k)󵄩󵄩󵄩 + σ(n)Φ(‖ω2 ‖d + sup󵄩󵄩󵄩v(k)󵄩󵄩󵄩) k∈ℤ

k=−∞

󵄩 󵄩 ≤ L‖Sα ‖1 ‖ω1 ‖d + ρ2 Φ(‖ω2 ‖d + sup󵄩󵄩󵄩v(k)󵄩󵄩󵄩) k∈ℤ

󵄩 󵄩 ≤ L‖Sα ‖1 k0 + ρ2 Φ(k0 + sup󵄩󵄩󵄩v(k)󵄩󵄩󵄩) ≤ k0 . k∈ℤ

This indicates that (Γ1 ω1 )(n) + (Γ2 ω2 )(n) ∈ Ωk0 . Thus, Γ maps Ωk0 into itself. Step 3. The mapping Γ1 is a contraction on Ωk0 . Let ω1 (n), ω2 (n) ∈ Ωk0 , according to equation (7.40), one has 󵄩󵄩 󵄩 󵄩󵄩[F1 (k, v(k) + ω1 (k)) − F1 (k, v(k))] − [F1 (k, v(k) + ω2 (k)) − F1 (k, v(k))]󵄩󵄩󵄩 󵄩 󵄩 ≤ L󵄩󵄩󵄩ω1 (k) − ω2 (k)󵄩󵄩󵄩.

304 | 7 Discrete time fractional evolution equations Thus, 󵄩 󵄩󵄩 1 1 󵄩󵄩(Γ ω1 )(n) − (Γ ω2 )(n)󵄩󵄩󵄩 󵄩󵄩 n−1 󵄩󵄩 = 󵄩󵄩󵄩 ∑ Sα (n − 1 − k)[(F1 (k, v(k) + ω1 (k)) − F1 (k, v(k))) 󵄩󵄩 󵄩k=−∞ 󵄩󵄩 󵄩󵄩 − (F1 (k, v(k) + ω2 (k)) − F1 (k, v(k)))]󵄩󵄩󵄩 󵄩󵄩 󵄩 n−1

󵄩 󵄩󵄩 󵄩 ≤ L ∑ 󵄩󵄩󵄩Sα (n − 1 − k)󵄩󵄩󵄩󵄩󵄩󵄩ω1 (k) − ω2 (k)󵄩󵄩󵄩 k=−∞

n−1

󵄩 󵄩 ≤ L‖ω1 − ω2 ‖d ∑ 󵄩󵄩󵄩Sα (n − 1 − k)󵄩󵄩󵄩 k=−∞

= L‖Sα ‖1 ‖ω1 − ω2 ‖d , which indicates 󵄩󵄩 1 󵄩 1 󵄩󵄩Γ ω1 − Γ ω2 󵄩󵄩󵄩d ≤ L‖Sα ‖1 ‖ω1 − ω2 ‖d . Thus Γ1 is a contraction on Ωk0 by equation (7.44). Step 4. The mapping Γ2 is completely continuous on Ωk0 . First, Γ2 is continuous on Ωk0 . + In fact, ∀ε > 0, let {ωj }+∞ j=1 ⊂ Ωk0 with ωj → ω0 as j → +∞. From σ(n) ∈ C0 (ℤ, ℝ ) (from Lemma 7.29), one may choose an n1 > 0 big enough s. t., Φ(k0 + ‖v‖d )σ(n)
0 such that for any j ≥ N, n1 −1

󵄩 󵄩 ε ∑ 󵄩󵄩󵄩Sα (n − 1 − k)[F2 (k, v(k) + ωj (k)) − F2 (k, v(k) + ω0 (k))]󵄩󵄩󵄩 ≤ . 3 k=−∞

7.3 Asymptotic almost periodicity |

305

Thus, when j ≥ N, 󵄩 󵄩󵄩 2 2 󵄩󵄩(Γ ωj )(n) − (Γ ω0 )(n)󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 n−1 n−1 󵄩󵄩 󵄩󵄩 = 󵄩󵄩󵄩 ∑ Sα (n − 1 − k)F2 (k, v(k) + ωj (k)) − ∑ Sα (n − 1 − k)F2 (k, v(k) + ω0 (k))󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 k=−∞ 󵄩 󵄩k=−∞ n1 −1

󵄩 󵄩 ≤ ∑ 󵄩󵄩󵄩Sα (n − 1 − k)[F2 (k, v(k) + ωj (k)) − F2 (k, v(k) + ω0 (k))]󵄩󵄩󵄩 k=−∞

max{n,n1 }

+

∑ n1

󵄩󵄩 󵄩 󵄩󵄩Sα (n − 1 − k)[F2 (k, v(k) + ωj (k)) − F2 (k, v(k) + ω0 (k))]󵄩󵄩󵄩

n1 −1

󵄩 󵄩 ≤ ∑ 󵄩󵄩󵄩Sα (n − 1 − k)[F2 (k, v(k) + ωj (k)) − F2 (k, v(k) + ω0 (k))]󵄩󵄩󵄩 k=−∞

+ 2Φ(k0 + ‖v‖d )

max{n,n1 }

∑ n1

󵄩󵄩 󵄩 󵄩󵄩β(k)Sα (n − 1 − k)󵄩󵄩󵄩

n1 −1

󵄩 󵄩 ≤ ∑ 󵄩󵄩󵄩Sα (n − 1 − k)[F2 (k, v(k) + ωj (k)) − F2 (k, v(k) + ω0 (k))]󵄩󵄩󵄩 k=−∞

+ 2Φ(k0 + ‖v‖d )σ(n − 1) ε 2ε ≤ + = ε. 3 3 Accordingly, Γ2 is continuous on Ωk0 .

Then we consider the compactness of Γ2 .

Let Δ = Γ2 (Ωk0 ) and z(n) = Γ2 (u(n)) for u(n) ∈ Ωk0 . First, for all ω(n) ∈ Ωk0 and

n ∈ ℤ,

󵄩󵄩 n−1 󵄩󵄩 󵄩󵄩 󵄩󵄩 2 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩(Γ ω)(n)󵄩󵄩 = 󵄩󵄩 ∑ Sα (n − 1 − k)F2 (k, v(k) + ω(k))󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩k=−∞ 󵄩 n−1

󵄩 󵄩 󵄩 󵄩 ≤ Φ(k0 + sup󵄩󵄩󵄩v(k)󵄩󵄩󵄩) ∑ 󵄩󵄩󵄩β(k)Sα (n − 1 − k)󵄩󵄩󵄩 k∈ℤ

k=−∞

󵄩 󵄩 ≤ σ(n)Φ(k0 + sup󵄩󵄩󵄩v(k)󵄩󵄩󵄩), k∈ℤ

in view of σ(n) ∈ C0 (ℤ, ℝ+ ), which follows from Lemma 7.29, one has lim (Γ2 ω)(n) = 0

|n|→+∞

uniformly for ω(n) ∈ Ωk0 .

306 | 7 Discrete time fractional evolution equations From n−1

(Γ2 ω)(n) = ∑ Sα (n − 1 − k)F2 (k, v(k) + ω(k)) k=−∞ +∞

= ∑ Sα (k)F2 (n − 1 − k, v(n − 1 − k) + ω(n − 1 − k)), k=0

it follows that for given ε0 > 0, one can choose an n2 > 0 such that 󵄩󵄩 +∞ 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∑ Sα (k)F2 (n − 1 − k, v(n − 1 − k) + ω(n − 1 − k))󵄩󵄩󵄩 < ε0 . 󵄩󵄩 󵄩󵄩 󵄩󵄩k=n2 󵄩󵄩 Thus, we get z(n) ∈ n2 c({Sα (τ)F2 (λ, v(λ) + ω(λ)) : 0 ≤ τ ≤ n2 , n − n2 − 1 ≤ λ ≤ n − 1, ‖ω‖d ≤ r}) + Bε0 (X),

where c(K) is the convex hull of K. Using (H4 ), the collection K = {Sα (τ)F2 (λ, v(λ) + ω(λ)) : 0 ≤ τ ≤ n2 , n − n2 − 1 ≤ λ ≤ n − 1, ‖ω‖d ≤ r} is relatively compact, combing with Δ ⊂ n2 c(K) + Bε0 (X), one obtains that Δ is a relatively compact subset of X. Thus, Γ2 is compact by Lemma 7.25, which further implies Γ2 is completely continuous on Ωk0 . Step 5. Show equation (7.37) has at least one mild solution, which is asymptotically almost-periodic. First, from the results of step 4, combining with the results of steps 2 and 3 as well as the Krasnoselskii’s fixed-point theorem (see Theorem 1.8), it follows that Γ has at least one fixed point ω(n) ∈ Ωk0 ; furthermore, ω(n) ∈ C0 (ℤ, X). Then consider the following coupled system: v(n) = ∑n−1 n ∈ ℤ, { k=−∞ Sα (n − 1 − k)F1 (k, v(k)), { { n−1 ω(n) = ∑k=−∞ Sα (n − 1 − k)[F1 (k, v(k) + ω(k)) − F1 (k, v(k))] { { { + ∑n−1 n ∈ ℤ. k=−∞ Sα (n − 1 − k)F2 (k, v(k) + ω(k)), {

(7.46)

Combing the unique fixed point v(n) ∈ AP(ℤ, X) in step 1 with the fixed point ω(n) ∈ C0 (ℤ, X) in Step 4, it follows that (v(n), ω(n)) ∈ AP(ℤ, X) × C0 (ℤ, X) is a solution to System (7.46); then x(n) := v(n)+ω(n) ∈ AAP(ℤ, X) and it is a solution to the equation n−1

x(t) = ∑ Sα (n − 1 − k)F(k, x(k)), k=−∞

n ∈ ℤ,

7.3 Asymptotic almost periodicity |

307

that is x(n) is a mild solution to equation (7.37), which is asymptotically almostperiodic. In the following, we state and prove conditions for the existence of mild solutions with asymptotically almost-periodicity of equation (7.37) when the almost-periodic component F1 satisfies a locally Lipschitz assumption (H′2 ) F(n, x) = F1 (n, x) + F2 (n, x) ∈ AAP(ℤ × X, X) with F1 (n, x) ∈ AP(ℤ × X, X),

F2 (n, x) ∈ C0 (ℤ × X, X)

and it is bounded on ℤ × X. Moreover, for each r > 0, 󵄩󵄩 󵄩 󵄩󵄩F1 (n, x) − F1 (n, y)󵄩󵄩󵄩 ≤ L(r)‖x − y‖,

∀n ∈ ℤ, x, y ∈ X with ‖x‖ ≤ r, ‖y‖ ≤ r, (7.47)

where L : ℝ+ → ℝ+ is a nondecreasing function. Theorem 7.8. Assume that (H1 ), (H′2 ), (H3 ) and (H4 ) hold and if there exist r0 > 0, r1 > 0 such that ‖Sα ‖1 (L(r0 + r1 ) +

1 󵄩 󵄩 sup󵄩󵄩F (k, 0)󵄩󵄩󵄩) < 1. min{r0 , r1 } k∈ℤ 󵄩 1

Put ρ2 := supn∈ℤ σ(n). Then there is a mild solution of equation (7.37) whenever (7.48)

L(r0 + r1 )‖Sα ‖1 + ρ1 ρ2 < 1. Moreover, the mild solution is asymptotically almost-periodic. Proof. We also divide into five steps to complete the proof. Step 1. Define a mapping Λ : Br0 (AAP(ℤ, X)) → Br0 (AAP(ℤ, X)) by n−1

(Λv)(n) = ∑ Sα (n − 1 − k)F1 (k, v(k)), k=−∞

n ∈ ℤ.

Λ has a unique fixed point v(n) ∈ Br0 (AAP(ℤ, X)). First, according to (H1 ), there is a discrete α-resolvent family {Sα (n)}n∈ℕ ⊂ 𝕃(X) generated by A, which is summable (by Lemma 7.26) and, combined with the discrete function k → F1 (k, v(k)), is bounded on ℤ, which follows from Lemma 7.10, and one

308 | 7 Discrete time fractional evolution equations has n−1

󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩(Λv)(n)󵄩󵄩󵄩 ≤ ∑ 󵄩󵄩󵄩Sα (n − 1 − k)󵄩󵄩󵄩󵄩󵄩󵄩F1 (k, v(k))󵄩󵄩󵄩 k=−∞ +∞

󵄩 󵄩󵄩 󵄩 = ∑ 󵄩󵄩󵄩Sα (k)󵄩󵄩󵄩󵄩󵄩󵄩F1 (n − 1 − k, v(n − 1 − k))󵄩󵄩󵄩 k=0

+∞

󵄩 󵄨 󵄩 󵄨 ≤ 󵄨󵄨󵄨‖F1 ‖󵄨󵄨󵄨 ∑ 󵄩󵄩󵄩Sα (k)󵄩󵄩󵄩 k=0

󵄨 󵄨 = 󵄨󵄨󵄨‖F1 ‖󵄨󵄨󵄨‖Sα ‖1 < +∞,

for all n ∈ ℤ,

which yields that (Λv)(n) exists. Furthermore from F1 (n, x) ∈ AP(ℤ × X, X) satisfying equation (7.47), combined with Lemmas 7.10 and 7.19, it follows that F1 (⋅, v(⋅)) ∈ AP(ℤ, X),

for every v(⋅) ∈ AP(ℤ, X).

Furthermore it follows from Lemma 7.27 that (Λv)(n) ∈ AP(ℤ, X). Let v ∈ Br0 (AP(ℤ, X)), and one has 󵄩󵄩 n−1 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩(Λv)(n)󵄩󵄩 = 󵄩󵄩 ∑ Sα (n − 1 − k)F1 (k, v(k))󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩k=−∞ 󵄩 n−1

󵄩 󵄩󵄩 󵄩 ≤ ∑ 󵄩󵄩󵄩Sα (n − 1 − k)󵄩󵄩󵄩󵄩󵄩󵄩F1 (k, v(k)) − F1 (k, 0)󵄩󵄩󵄩 k=−∞

n−1

󵄩 󵄩󵄩 󵄩 + ∑ 󵄩󵄩󵄩Sα (n − 1 − k)󵄩󵄩󵄩󵄩󵄩󵄩F1 (k, 0)󵄩󵄩󵄩 k=−∞

n−1

󵄩 󵄩󵄩 󵄩 󵄩 󵄩 ≤ L(r0 ) ∑ 󵄩󵄩󵄩Sα (n − 1 − k)󵄩󵄩󵄩󵄩󵄩󵄩v(k)󵄩󵄩󵄩 + ‖Sα ‖1 sup󵄩󵄩󵄩F1 (k, 0)󵄩󵄩󵄩 k=−∞

k∈ℤ

supk∈ℤ ‖F1 (k, 0)‖ )r0 r0 supk∈ℤ ‖F1 (k, 0)‖ ≤ ‖Sα ‖1 (L(r0 + r1 ) + )r0 < r0 . min{r0 , r1 }

≤ ‖Sα ‖1 (L(r0 ) +

Hence, Λv ∈ Br0 (AP(ℤ, X)), which yields that Λ is well defined. Then we show Λ is continuous. Let vj (n), v(n) ∈ AP(ℤ, X) satisfy vj (n) → v(n) as j → ∞, then one has 󵄩󵄩 󵄩󵄩 n−1 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩(Λvj )(n) − (Λv)(n)󵄩󵄩󵄩 = 󵄩󵄩󵄩 ∑ Sα (n − 1 − k)[F1 (k, vj (k)) − F1 (k, v(k))]󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩k=−∞ 󵄩 n−1

󵄩 󵄩󵄩 󵄩 ≤ ∑ 󵄩󵄩󵄩Sα (n − 1 − k)󵄩󵄩󵄩󵄩󵄩󵄩F1 (k, vj (k)) − F1 (k, v(k))󵄩󵄩󵄩 k=−∞

7.3 Asymptotic almost periodicity | 309 n−1

󵄩 󵄩󵄩 󵄩 ≤ L(r0 ) ∑ 󵄩󵄩󵄩Sα (n − 1 − k)󵄩󵄩󵄩󵄩󵄩󵄩vj (k) − v(k)󵄩󵄩󵄩 k=−∞

n−1

󵄩 󵄩 ≤ L(r0 )‖vj − v‖d ∑ 󵄩󵄩󵄩Sα (n − 1 − k)󵄩󵄩󵄩 k=−∞

= L(r0 )‖Sα ‖1 ‖vj − v‖d .

has

Therefore, as j → ∞, Λvj → Λv, hence Λ is continuous. Finally, we show there is a unique fixed point v(n) ∈ AP(ℤ, X) of Λ. Let v1 (n), v2 (n) ∈ Br0 (AP(ℤ, X)) and, similar to the proof of the continuity of Λ, one 󵄩󵄩 󵄩 󵄩󵄩(Λv1 )(n) − (Λv2 )(n)󵄩󵄩󵄩 ≤ ‖Sα ‖1 L(r0 )‖v1 − v2 ‖d ,

which implies 󵄩󵄩 󵄩 󵄩󵄩Λv1 − Λv2 󵄩󵄩󵄩d ≤ ‖Sα ‖1 L(r0 )‖v1 − v2 ‖d . From equation (7.48), it follows that Λ is a contraction on AP(ℤ, X). Thus, there is a unique fixed point v(n) ∈ AP(ℤ, X) of Λ and ‖v(n)‖ ≤ r0 . Step 2. Set Ωr := {ω(n) ∈ C0 (ℤ, X) : ‖ω‖d ≤ r}. For the above v(n) ∈ Br0 (AAP(ℤ, X)), define Γ := Γ1 + Γ2 on C0 (ℤ, X) as equation (7.45). There exists a constant r1 s. t. Γ maps Ωr1 into itself. First, as ω(⋅) ∈ C0 (ℤ, X), then ∃r > 0 s. t. ‖ω(k)‖ ≤ r, ∀k ∈ ℤ. This combined with equation (7.47) implies 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩F1 (k, v(k) + ω(k)) − F1 (k, v(k))󵄩󵄩󵄩 ≤ L(r0 + r)󵄩󵄩󵄩ω(k)󵄩󵄩󵄩, which yields F1 (⋅, v(⋅) + ω(⋅)) − F1 (⋅, v(⋅)) ∈ C0 (ℤ, X). According to equation (7.41), one has 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩F2 (k, v(k) + ω(k))󵄩󵄩󵄩 ≤ β(k)Φ(r + sup󵄩󵄩󵄩v(k)󵄩󵄩󵄩), k∈ℤ

󵄩 󵄩 ∀k ∈ ℤ, ω(k) ∈ X with 󵄩󵄩󵄩ω(k)󵄩󵄩󵄩 ≤ r,

which yields F2 (⋅, v(⋅) + ω(⋅)) ∈ C0 (ℤ, X) as β(⋅) ∈ C0 (ℤ, ℝ+ ). Thus, Γ is well-defined and maps C0 (ℤ, X) into itself, which follows from Lemma 7.27.

310 | 7 Discrete time fractional evolution equations

s. t.,

From equations (7.41) and (7.48), it easily follows that there is a constant r1 > 0 󵄩 󵄩 L(r0 + r1 )‖Sα ‖1 r1 + ρ2 Φ(r1 + sup󵄩󵄩󵄩v(k)󵄩󵄩󵄩) ≤ r1 . k∈ℤ

Then for any n ∈ ℤ and ω1 (n), ω2 (n) ∈ Ωr1 , one has 󵄩󵄩 1 󵄩 2 󵄩󵄩(Γ ω1 )(n) + (Γ ω2 )(n)󵄩󵄩󵄩 󵄩󵄩 n−1 󵄩󵄩 󵄩󵄩 󵄩󵄩 ≤ 󵄩󵄩󵄩 ∑ Sα (n − 1 − k)[F1 (k, v(k) + ω1 (k)) − F1 (k, v(k))]󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩k=−∞ 󵄩 󵄩󵄩 n−1 󵄩󵄩 󵄩󵄩 󵄩󵄩 + 󵄩󵄩󵄩 ∑ Sα (n − 1 − k)F2 (k, v(k) + ω2 (k))󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩k=−∞ 󵄩 n−1

󵄩 󵄩 ≤ ∑ 󵄩󵄩󵄩Sα (n − 1 − k)[F1 (k, v(k) + ω1 (k)) − F1 (k, v(k))]󵄩󵄩󵄩 k=−∞

n−1

󵄩 󵄩 + ∑ 󵄩󵄩󵄩Sα (n − 1 − k)F2 (k, v(k) + ω2 (k))󵄩󵄩󵄩 k=−∞

n−1

󵄩 󵄩󵄩 󵄩 ≤ L(r0 + r1 ) ∑ 󵄩󵄩󵄩Sα (n − 1 − k)󵄩󵄩󵄩󵄩󵄩󵄩ω1 (k)󵄩󵄩󵄩 k=−∞

n−1

󵄩 󵄩 󵄩 󵄩 + Φ(‖ω2 ‖d + sup󵄩󵄩󵄩v(k)󵄩󵄩󵄩) ∑ 󵄩󵄩󵄩β(k)Sα (n − 1 − k)󵄩󵄩󵄩 k∈ℤ

k=−∞

n−1

󵄩 󵄩󵄩 󵄩 ≤ L(r0 + r1 ) ∑ 󵄩󵄩󵄩Sα (n − 1 − k)󵄩󵄩󵄩󵄩󵄩󵄩ω1 (k)󵄩󵄩󵄩 k=−∞

n−1 ∞

󵄩 󵄩 + Φ(‖ω2 ‖d + sup󵄩󵄩󵄩v(k)󵄩󵄩󵄩)M ∑ ∫ β(k)e−t k∈ℤ

n−1

k=−∞ 0

t n−1−k t α−1 Eα,α (−δt α )dt (n − 1 − k)!

󵄩 󵄩 󵄩 󵄩 ≤ L(r0 + r1 )‖ω1 ‖d ∑ 󵄩󵄩󵄩Sα (n − 1 − k)󵄩󵄩󵄩 + σ(n)Φ(‖ω2 ‖d + sup󵄩󵄩󵄩v(k)󵄩󵄩󵄩) k∈ℤ

k=−∞

󵄩 󵄩 ≤ L(r0 + r1 )‖Sα ‖1 ‖ω1 ‖d + ρ2 Φ(‖ω2 ‖d + sup󵄩󵄩󵄩v(k)󵄩󵄩󵄩) k∈ℤ

󵄩 󵄩 ≤ L(r0 + r1 )‖Sα ‖1 r1 + ρ2 Φ(r1 + sup󵄩󵄩󵄩v(k)󵄩󵄩󵄩) ≤ r1 , k∈ℤ

this indicates that (Γ1 ω1 )(n) + (Γ2 ω2 )(n) ∈ Ωr1 . Thus Γ maps Ωr1 into itself. Step 3. The mapping Γ1 is a contraction on Ωr1 .

7.3 Asymptotic almost periodicity | 311

Let ω1 (n), ω2 (n) ∈ Ωr1 and, from equation (7.40), one has 󵄩 󵄩󵄩 󵄩󵄩[F1 (k, v(k) + ω1 (k)) − F1 (k, v(k))] − [F1 (k, v(k) + ω2 (k)) − F1 (k, v(k))]󵄩󵄩󵄩 󵄩 󵄩 ≤ L(r0 + r1 )󵄩󵄩󵄩ω1 (k) − ω2 (k)󵄩󵄩󵄩. Thus, 󵄩󵄩 n−1 󵄩 󵄩󵄩 󵄩󵄩 1 1 󵄩󵄩(Γ ω1 )(n) − (Γ ω2 )(n)󵄩󵄩󵄩 = 󵄩󵄩󵄩 ∑ Sα (n − 1 − k)[(F1 (k, v(k) + ω1 (k)) − F1 (k, v(k))) 󵄩󵄩 󵄩k=−∞ 󵄩󵄩 󵄩󵄩 − (F1 (k, v(k) + ω2 (k)) − F1 (k, v(k)))]󵄩󵄩󵄩 󵄩󵄩 󵄩 n−1

󵄩 󵄩󵄩 󵄩 ≤ L(r0 + r1 ) ∑ 󵄩󵄩󵄩Sα (n − 1 − k)󵄩󵄩󵄩󵄩󵄩󵄩ω1 (k) − ω2 (k)󵄩󵄩󵄩 k=−∞

n−1

󵄩 󵄩 ≤ L(r0 + r1 )‖ω1 − ω2 ‖d ∑ 󵄩󵄩󵄩Sα (n − 1 − k)󵄩󵄩󵄩 k=−∞

= L(r0 + r1 )‖Sα ‖1 ‖ω1 − ω2 ‖d , which implies 󵄩󵄩 1 󵄩 1 󵄩󵄩Γ ω1 − Γ ω2 󵄩󵄩󵄩d ≤ L(r0 + r1 )‖Sα ‖1 ‖ω1 − ω2 ‖d . Thus Γ1 is a contraction on Ωr1 by equation (7.48). Step 4. The mapping Γ2 is completely continuous on Ωk0 . The proof of this step is the same as step 4 of Theorem 7.7. Step 5. Equation (7.37) has at least one mild solution which is asymptotically almost-periodic. The proof of this step is the same as step 5 of Theorem 7.7. 7.3.4 Applications In this subsection, an example is provided to demonstrate the effectiveness of our abstract results. Consider the different scalar equation of fractional-order forms like △α x(n) = λx(n + 1) + F(n, x(n)),

n ∈ ℤ,

(7.49)

where α ∈ (0, 1), λ is a complex number satisfying Re(λ) < 0 and F : ℤ × ℂ → ℂ is a function. It is clear that there is an exponentially stable C0 -semigroup generated by λ given by T(t) = eλt ,

∀t ≥ 0.

312 | 7 Discrete time fractional evolution equations Thus (H1 ) holds. From Abadias and Lizama [2], it follows that there is a discrete α-resolvent family {Sα (n)} (n ∈ ℕ) generated by λ, which is summable and given by ∞

Sα (n) = ∫ e−t 0

t n α−1 t Eα,α (−λt α )dt. n!

Let F2 (n, x) := νe−|n| x sin x 2 .

F1 (n, x) := μ(sin n + sin √2n) sin x,

Then, according to Lemma 7.20, one has F1 (n, x) ∈ AP(ℤ × ℂ, ℂ) satisfying 󵄩󵄩 󵄩 󵄩󵄩F1 (n, x) − F1 (n, y)󵄩󵄩󵄩 ≤ 2μ‖x − y‖,

∀n ∈ ℤ, x, y ∈ ℂ,

and 󵄩󵄩 󵄩 −|n| 󵄩󵄩F2 (n, x)󵄩󵄩󵄩 ≤ νe ‖x‖,

∀n ∈ ℤ, x ∈ ℂ.

This indicates F2 (n, x) ∈ C0 (ℝ × ℂ, ℂ). Furthermore, F(n, x) = F1 (n, x) + F2 (n, x) ∈ AAP(ℝ × ℂ, ℂ). Thus (H2 )–(H4 ) hold with L = 2μ,

Φ(r) = r,

β(n) = νe−|n| ,

ρ1 = 1,

ρ2 ≤ ν.

From Theorem 7.7, it follows that there exists at least one mild solution to equation (7.37) whenever 2μ + ν < 1. Moreover, it is asymptotically almost-periodic.

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Index almost sectorial operator 14, 15, 29, 76, 77 analytic semigroup 16, 77, 169, 170, 178, 193, 212, 228, 233, 249 asymptotically periodic solution 215, 216, 235 attractivity 76 attractive solution 84–86 Caputo fractional derivative 3, 5–7, 9, 29, 43, 44, 76, 87, 89, 108, 130, 151, 169, 192, 215 Cauchy problem 29, 31, 34, 42, 43, 45, 53, 57, 58, 72, 75, 76, 79, 84, 86, 89, 130, 150, 169, 170, 174, 177, 184, 185, 187, 188, 189, 190, 251, 252, 261, 263, 269, 273, 281, 283 C0 -semigroup 13, 14, 44, 45, 60, 70, 75, 108, 110, 126, 154, 216, 219–221, 226–229, 231–233, 237, 241, 252, 253, 257–261, 263, 264, 266, 269, 271–275, 277, 281, 285, 286, 296, 298, 311 compact semigroup 70, 84 controllability 89, 90, 97, 98, 101–103, 130, 131, 140, 143, 147, 150 control problem 106, 107, 121, 127, 151, 164, 165, 166, 167 discrete time evolution equations 251 existence 29, 30, 37, 44, 53, 59, 60, 70, 85, 86, 89, 90, 92, 93, 97, 100, 116, 117, 119, 120, 122, 130, 150, 151, 154, 155, 170, 177, 179, 184, 187, 192, 203, 211, 215, 216, 233, 235, 237, 251–253, 263, 264, 272, 273, 275, 281, 285–287, 296, 300, 307 fixed point theorem 23–25, 60, 85, 86, 89, 143, 170, 177, 184, 187, 189, 205, 232, 233, 243, 246, 249, 265, 268, 276, 280, 286, 306 Fourier transform 18, 19, 60–62, 216, 221, 222 fractional calculus 2, 3, 251, 255, 286 fractional evolution equations 29, 76, 87, 107, 111, 129, 130, 150, 215, 216, 237, 251, 252 fractional evolution inclusions 89, 150, 165 Hilfer fractional derivative 3, 7, 29, 30

initial value problems 29, 44, 251, 286 integral solution 44, 45, 51, 53, 54, 57, 58 Laplace transform 18, 29, 30, 32, 44, 45, 46, 53, 131, 133, 135, 154, 171–173, 255, 256, 258, 259, 270 Liouville–Weyl fractional derivative 8, 60, 215, 216 measure of noncompactness 20, 24, 31, 37, 42, 43, 53, 72, 86, 89, 147, 152 mild solutions 29, 30, 37, 44, 60, 76, 89, 90, 93, 98, 108–111, 130, 132, 136, 150, 151, 154, 156, 164–167, 170, 192, 193, 197, 210, 211, 216, 230, 233, 235, 237, 286, 287, 296, 297, 300, 307 Mittag-Leffler function 12, 13, 30, 105, 255, 271, 275 multivalued map 89, 151, 152, 154, 155, 163–167 neutral evolution equations 169 noncompact semigroup 72, 86 periodic solutions 215, 216, 234, 235 Riemann–Liuoville fractional derivative 4–7, 10, 29, 43, 76, 235 Riemann–Liouville fractional integral 4, 5, 76 S-asymptotically periodic solution 215, 216, 235 solution sets 192, 199 stability 59, 252, 257, 272, 273, 282, 283, 286 stochastic process 25, 27, 28, 193, 196 terminal value problems 59 topological structure 192, 199 uniqueness 48, 59, 62, 94, 100, 130, 135, 170, 187, 205, 211, 215, 216, 222, 230, 235, 251, 252, 256, 286, 287 Ulam–Hyers–Rassias stability 273, 282 Wright function 13, 31, 60, 78, 150, 220, 237

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