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Volume 5, Number 1

January 2010

ISSN:1559-1948 (PRINT), 1559-1956 (ONLINE) EUDOXUS PRESS,LLC

JOURNAL OF APPLIED FUNCTIONAL ANALYSIS \

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SCOPE AND PRICES OF

JOURNAL OF APPLIED FUNCTIONAL ANALYSIS A quartely international publication of EUDOXUS PRESS,LLC ISSN:1559-1948(PRINT),1559-1956(ONLINE) Editor in Chief: George Anastassiou Department of Mathematical Sciences The University of Memphis Memphis, TN 38152,USA E mail: [email protected]

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The purpose of the "Journal of Applied Functional Analysis"(JAFA) is to publish high quality original research articles, survey articles and book reviews from all subareas of Applied Functional Analysis in the broadest form plus from its applications and its connections to other topics of Mathematical Sciences. A sample list of connected mathematical areas with this publication includes but is not restricted to: Approximation Theory, Inequalities, Probability in Analysis, Wavelet Theory, Neural Networks, Fractional Analysis, Applied Functional Analysis and Applications, Signal Theory, Computational Real and Complex Analysis and Measure Theory, Sampling Theory, Semigroups of Operators, Positive Operators, ODEs, PDEs, Difference Equations, Rearrangements, Numerical Functional Analysis, Integral equations, Optimization Theory of all kinds, Operator Theory, Control Theory, Banach Spaces, Evolution Equations, Information Theory, Numerical Analysis, Stochastics, Applied Fourier Analysis, Matrix Theory, Mathematical Physics, Mathematical Geophysics, Fluid Dynamics, Quantum Theory. Interpolation in all forms, Computer Aided Geometric Design, Algorithms, Fuzzyness, Learning Theory, Splines, Mathematical Biology, Nonlinear Functional Analysis, Variational Inequalities, Nonlinear Ergodic Theory, Functional Equations, Function Spaces, Harmonic Analysis, Extrapolation Theory, Fourier Analysis, Inverse Problems, Operator Equations, Image Processing, Nonlinear Operators, Stochastic Processes, Mathematical Finance and Economics, Special Functions, Quadrature, Orthogonal Polynomials, Asymptotics, Symbolic and Umbral Calculus, Integral and Discrete Transforms, Chaos and Bifurcation, Nonlinear Dynamics, Solid Mechanics, Functional Calculus, Chebyshev Systems. Also are included combinations of the above topics. Working with Applied Functional Analysis Methods has become a main trend in recent years, so we can understand better and deeper and solve important problems of our real and scientific world. JAFA is a peer-reviewed International Quarterly Journal published by Eudoxus Press,LLC. We are calling for high quality papers for possible publication. The contributor should send four copies of the contribution to the EDITOR in CHIEF in TEX,LATEX double spaced,in ten point type size. They should be sent BY REGULAR MAIL ONLY, NOT REGISTERED MAIL, AND NO E-MAIL SUBMISSIONS [ See: Instructions to Contributors] Journal of Applied Functional Analysis(JAFA)

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Journal of Applied Functional Analysis Editorial Board Associate Editors Editor in-Chief: George A.Anastassiou Department of Mathematical Sciences The University of Memphis Memphis,TN 38152,USA 901-678-3144 office 901-678-2482 secretary 901-751-3553 home 901-678-2480 Fax [email protected] Approximation Theory,Inequalities,Probability, Wavelet,Neural Networks,Fractional Calculus

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Tel.(+33)(0)3.20.43.42.18 Fax (+33)(0)3.20.43.43.02 [email protected] Approximation Theory, Functional Analysis, Operator Theory. 4) Erik J.Balder Mathematical Institute Universiteit Utrecht P.O.Box 80 010 3508 TA UTRECHT The Netherlands Tel.+31 30 2531458 Fax+31 30 2518394 [email protected] Control Theory, Optimization, Convex Analysis, Measure Theory, Applications to Mathematical Economics and Decision Theory. 5) Carlo Bardaro Dipartimento di Matematica e Informatica Universita di Perugia Via Vanvitelli 1 06123 Perugia, ITALY TEL+390755853822 +390755855034 FAX+390755855024 E-mail [email protected] Web site: http://www.unipg.it/~bardaro/ Functional Analysis and Approximation Theory, Signal Analysis, Measure Theory, Real Analysis. 6) Heinrich Begehr Freie Universitaet Berlin I. Mathematisches Institut, FU Berlin, Arnimallee 3,D 14195 Berlin Germany, Tel. +49-30-83875436, office +49-30-83875374, Secretary Fax +49-30-83875403 [email protected] Complex and Functional Analytic Methods in PDEs, Complex Analysis, History of Mathematics. 7) Fernando Bombal Departamento de Analisis Matematico Universidad Complutense Plaza de Ciencias,3 28040 Madrid, SPAIN Tel. +34 91 394 5020 Fax +34 91 394 4726 [email protected]

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10 JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.5, NO.1, 10-30, COPYRIGHT 2010 EUDOXUS PRESS, LLC

A perturbation result for the layer potentials of general second order differential operators with constant coefficients Matteo Dalla Riva & Massimo Lanza de Cristoforis

Abstract: We consider a hypersurface in Euclidean space Rn parametrized by a diffeomorphism of the boundary of a regular domain in Rn to Rn , and a density function on the hypersurface, which we think as points in suitable Schauder spaces, and a family of second order differential operators with constant coefficients and a corresponding family of fundamental solutions depending on a parameter. Then we investigate the dependence of the corresponding layer potentials, which we also think as points in suitable Schauder spaces, upon variation of the diffeomorphism and of the density and of the parameter, and we show a real analyticity theorem for such a dependence. Keywords: Layer potentials, second order differential operators with constant coefficients, domain perturbation, special nonlinear operators. 2000 Mathematics Subject Classification: 31B10, 47H30.

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Introduction.

As is well known, the potential theoretic method is a powerful tool to analyze boundary value problems for elliptic differential equations and systems and can be used in particular to study boundary perturbation problems (cf. e.g., Fichera [2].) Thus it is clear that it is important to understand the dependence of layer potentials both on variation of the support of integration and on data such as the integral kernel and the density (or moment.) In [5], [6], [7], those authors have considered layer potentials associated to the Laplace equation and to the Helmholtz equation. In this paper, we shall extend the methods of those papers to consider general strongly elliptic operators of second order with complex coefficients, as a preliminary step for a later analysis of the case of elliptic operators of higher order. We fix a bounded open connected subset Ω of Rn with Rn \ clΩ connected, which we consider as a “base domain”. We assume that Ω is of class C m,α for some integer m ≥ 1 and α ∈]0, 1[. Then we consider a class of diffeomorphisms A∂Ω of ∂Ω into Rn . If φ ∈ A∂Ω , the Jordan-Leray separation theorem ensures 1

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that Rn \ φ(∂Ω) has exactly two open connected components, and we denote by I[φ] and E[φ] the bounded and unbounded open connected components of Rn \ φ(∂Ω), respectively. Next we introduce a family of differential operators. Let N denote the number of multi-indexes α ∈ Nn with |α| ≤ 2. For each a ≡ (aα )|α|≤2 ∈ CN , we set (2) a(2) ≡ (alj )l,j=1,...,n a(1) ≡ (aj )j=1,...,n (2)

with alj ≡ ael +ej and aj ≡ aej , where {ej : j = 1, . . . , n} is the canonical basis of Rn . We note that the matrix a(2) is symmetric. Then we set  X   N α E ≡ a ≡ (aα )|α|≤2 ∈ C : aα ξ inf Re >0 . n ξ∈R ,|ξ|=1

|α|=2

Clearly, E coincides with the set of coefficients a ≡ (aα )|α|≤2 such that the differential operator X aα Dα P [a, D] ≡ |α|≤2

is strongly elliptic and has complex coefficients. Then we shall consider the following assumption. Let K be a real Banach space. Let O be an open subset of K. (1.1) Let a(·) be a real analytic map of O to E. Let S(·, ·) be a real analytic map of (Rn \ {0}) × O to C such that S(·, κ) is a fundamental solution of P [a(κ), D] for all κ ∈ O . For all continuous functions f of ∂Ω to C and φ ∈ A∂Ω , one can consider the function f ◦ φ(−1) defined on φ(∂Ω), and it makes sense to consider the simple layer potential Z v[φ, f, κ](ξ) ≡ S (ξ − η, κ) f ◦ φ(−1) (η) dση ∀ξ ∈ Rn . φ(∂Ω)

Then we introduce the function V [φ, f, κ](x) ≡ v[φ, f, κ] ◦ φ(x)

∀x ∈ ∂Ω .

(1.2)

We prove that the map V [·, ·, ·] of (C m,α (∂Ω, Rn ) ∩ A∂Ω ) × C m−1,α (∂Ω) × O to C m,α (∂Ω) which takes (φ, f, κ) to the function V [φ, f, κ] defined in (1.2) is real analytic (see Theorem 5.6.) Then we consider the functions of ∂Ω to C defined by Z Vl [φ, f, κ](x) ≡ ∂ξl S(φ(x) − η, κ)f ◦ φ(−1) (η) dση , (1.3) φ(∂Ω) Z V∗ [φ, f, κ](x) ≡ Dξ S(φ(x) − η, κ)a(2) (κ)νφ (φ(x))f ◦ φ(−1) (η)dση , (1.4) φ(∂Ω)

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RIVA-CRISTOFORIS: A PERTURBATION RESULT...

for all x ∈ ∂Ω, and for all (φ, f, κ) ∈ (C m,α (∂Ω, Rn ) ∩ A∂Ω ) × C m−1,α (∂Ω) × O, and for all l ∈ {1, . . . , n}, and by Z W [φ, f, κ](x) ≡ − Dξ S(φ(x) − η, κ)a(2) (κ)νφ (η)f ◦ φ(−1) (η)dση (1.5) φ(∂Ω) Z − S(φ(x) − η, κ)νφt (η)a(1) (κ)f ◦ φ(−1) (η)dση , φ(∂Ω)

for all x ∈ ∂Ω and for all (φ, f, κ) ∈ (C m,α (∂Ω, Rn )∩A∂Ω )×C m,α (∂Ω)×O. Here ∂ξl S(·, κ) and Dξ S(·, κ) denote the derivative with respect to ξl and the gradient of S(ξ, κ) with respect to the first argument, respectively, and νφ denotes the exterior unit normal field to I[φ]. The functions Vl , V∗ , W are associated to the φ-pull backs on ∂Ω of the derivatives of the simple layer and of the double layer potential and are well known to intervene in the integral equations associated to boundary value problems for the elliptic operator P [a(κ), D]. We prove that Vl and V∗ are real analytic from (C m,α (∂Ω, Rn ) ∩ A∂Ω ) × C m−1,α (∂Ω) × O to C m−1,α (∂Ω) and that W is real analytic from (C m,α (∂Ω, Rn ) ∩ A∂Ω ) × C m,α (∂Ω) × O to C m,α (∂Ω). We note that Potthast [10], [11], [12] has proved a Fr´echet differentiability result, at least in case f is of class C 0,α and when P [a(κ), D] is the Helmholtz operator by exploiting a different method. Our work stems from that of [5] for the Cauchy integral operator, and from that of [6] and [7] for the Laplace and for the Helmholtz operator, respectively. The paper is organized as follows. Section 2 is a section of preliminaries. In section 3, we introduce some basics on elliptic operators and on corresponding layer potentials. In section 4, we introduce an auxiliary boundary value problem. In section 5, we prove our main results.

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Technical preliminaries

We denote the norm on a (real) normed space X by k · kX . Let X and Y be normed spaces. We endow the product space X × Y with the norm defined by k(x, y)kX ×Y ≡ kxkX + kykY for all (x, y) ∈ X × Y, while we use the Euclidean norm for Rn . For standard definitions of Calculus in normed spaces, we refer to Prodi and Ambrosetti [13]. The symbol N denotes the set of natural numbers including 0. Throughout the paper, n ∈ N \ {0, 1} . The inverse function of an invertible function f is denoted f (−1) , as opposed to the reciprocal of a complex-valued function g, or the inverse of a matrix A, which are denoted g −1 and A−1 , respectively. A dot “·” denotes the inner product in Rn , or the matrix product between matrices. Let A be a matrix. Then At denotes the transpose matrix of A and Aij denotes the (i, j) entry of
t A. If A is invertible, we set A−t ≡ A−1 . Let D ⊆ Rn . Then cl D denotes the closure of D and ∂D denotes the boundary of D. For all R > 0, x ∈ Rn , 3

RIVA-CRISTOFORIS: A PERTURBATION RESULT...

xj denotes the j-th coordinate of x, |x| denotes the Euclidean modulus of x in Rn or in C, and Bn (x, R) denotes the ball {y ∈ Rn : |x − y| < R}. Let Ω be an open subset of Rn . The space of m times continuously differentiable complex-valued functions on Ω is denoted by C m (Ω, C), or more simply by C m (Ω). D(Ω) denotes the space of functions of C ∞ (Ω) with compact support. n The dual D0 (Ω) denotes the space of distributions in Ω. Let f ∈ (C m (Ω)) . The s-th component of f is denoted fs , and Df denotes the gradient mas . Let η ≡ (η1 , . . . , ηn ) ∈ Nn , |η| ≡ η1 + . . . + ηn . Then trix ∂f ∂xl η

s,l=1,...,n |η| f denotes ∂xη∂1 ...∂x ηn n 1 η

D f . The subspace of C m (Ω) of those functions f whose derivatives D f of order |η| ≤ m can be extended with continuity to cl Ω is denoted C m (cl Ω). The subspace of C m (cl Ω) whose functions have m-th order derivatives that are H¨ older continuous with exponent α ∈]0, 1] is denoted C m,α (cl Ω) (cf. e.g., Gilbarg and Trudinger [3].) The subspace of C m (cl Ω) of those functions f such that f|cl(Ω∩Bn (0,R)) ∈ C m,α (cl(Ω ∩ Bn (0, R))) for all m,α R ∈]0, +∞[ is denoted Cloc (cl Ω). Let D ⊆ Cn . Then C m,α (cl Ω, D) denotes n {f ∈ (C m,α (cl Ω)) : f (cl Ω) ⊆ D}. Now let Ω be a bounded open subset of Rn . Then C m (cl Ω) and C m,α (cl Ω) are endowed with their usual norm and are well known to be Banach spaces (cf. e.g., Troianiello [15, §1.2.1].) We say that a bounded open subset of Rn is of class C m or of class C m,α , if it is a manifold with boundary imbedded in Rn of class C m or C m,α , respectively (cf. e.g., Gilbarg and Trudinger [3, §6.2].) For standard properties of functions in Schauder spaces, we refer the reader to Gilbarg and Trudinger [3] and to Troianiello [15] (see also [4, §2, Lem. 3.1, 4.26, Thm. 4.28], [6, §2].) If M is a manifold imbedded in Rn of class C m,α , with m ≥ 1, α ∈]0, 1[, one can define the Schauder spaces also on M by exploiting the local parametrizations. In particular, one can consider the spaces C k,α (∂Ω) on ∂Ω for 0 ≤ k ≤ m with Ω a bounded open set of class C m,α , and the trace operator of C k,α (clΩ) to C k,α (∂Ω) is linear and continuous. Moreover, for each R > 0 such that clΩ ⊆ Bn (0, R), there exists a linear and continuous extension operator of C k,α (∂Ω) to C k,α (clΩ), and of C k,α (clΩ) to C k,α (clBn (0, R)) (cf. e.g., Troianiello [15, Thm. 1.3, Lem. 1.5].) We note that throughout the paper “analytic” means “real analytic”. For the definition and properties of analytic operators, we refer to Prodi and Ambrosetti [13, p. 89]. In particular, we mention that the pointwise product in Schauder spaces is bilinear and continuous, and thus analytic, and that the map which takes a nonzero function to its reciprocal, or an invertible matrix of functions to its inverse matrix is real analytic in Schauder spaces (cf. e.g., [6, pp. 141, 142].) Now let Ω be a bounded open connected subset of Rn of class C 1 such that n R \ clΩ is connected. We denote by A∂Ω and by AclΩ the sets of functions of class C 1 (∂Ω, Rn ) and of class C 1 (clΩ, Rn ) which are injective and whose differential is injective at all points x ∈ ∂Ω, and at all points x ∈ clΩ, respectively. One can verify that A∂Ω is open in C 1 (∂Ω, Rn ) and that AclΩ is open in

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RIVA-CRISTOFORIS: A PERTURBATION RESULT...

C 1 (clΩ, Rn ) (cf. [4, Cor. 4.24, Prop. 4.29], [6, Lem. 2.5].) Moreover, if φ ∈ A∂Ω , the Jordan-Leray separation theorem ensures that Rn \ φ(∂Ω) has exactly two open connected components, and we denote by I[φ] and E[φ] the bounded and unbounded open connected components of Rn \ φ(∂Ω), respectively. Then we have the following two Lemmas (cf. [7, §2].) Lemma 2.1 Let m ∈ N \ {0}, α ∈]0, 1[. Let Ω be a bounded open subset of class C m,α of Rn such that both Ω and Rn \ clΩ are connected. Let νΩ denote the outward unit normal field to ∂Ω. Let ω ∈ C m,α (∂Ω, Rn ), |ω(x)| = 1, ω(x) · νΩ (x) > 1/2 for all x ∈ ∂Ω. Then the following statements hold. (i) If φ ∈ C m,α (∂Ω, Rn ) ∩ A∂Ω , then I[φ] is a bounded open connected set of class C m,α and ∂I[φ] = φ(∂Ω) = ∂E[φ]. (ii) There exists δΩ ∈]0, +∞[ such that the sets Ωω,δ Ω+ ω,δ

≡ {x + tω(x) : x ∈ ∂Ω, t ∈] − δ, δ[} , ≡ {x + tω(x) : x ∈ ∂Ω, t ∈] − δ, 0[} ,

Ω− ω,δ

≡

{x + tω(x) : x ∈ ∂Ω, t ∈]0, δ[} ,

are connected and of class C m,α , and ∂Ωω,δ ∂Ω+ ω,δ

= {x + tω(x) : x ∈ ∂Ω, t ∈ {−δ, δ}} , = {x + tω(x) : x ∈ ∂Ω, t ∈ {−δ, 0}} ,

∂Ω− ω,δ

= {x + tω(x) : x ∈ ∂Ω, t ∈ {0, δ}} ,

− n and Ω+ ω,δ ⊆ Ω, Ωω,δ ⊆ R \ clΩ, for all δ ∈]0, δΩ [.

(iii) Let δ ∈]0, δΩ [. If Φ ∈ AclΩω,δ , then φ ≡ Φ|∂Ω ∈ A∂Ω . n o (iv) If δ ∈]0, δΩ [, then the set A0clΩω,δ ≡ Φ ∈ AclΩω,δ : Φ(Ω+ ) ⊆ I[Φ ] is |∂Ω ω,δ 0 open in AclΩω,δ and Φ(Ω− ω,δ ) ⊆ E[Φ|∂Ω ] for all Φ ∈ AclΩω,δ .

(v) If δ ∈]0, δΩ [ and Φ ∈ C m,α (clΩω,δ , Rn ) ∩ A0clΩω,δ , then both Φ(Ω+ ω,δ ) and − m,α Φ(Ωω,δ ) are open sets of class C , and         + − − ∂Φ Ω+ = Φ ∂Ω , ∂Φ Ω = Φ ∂Ω ω,δ ω,δ ω,δ ω,δ . Then we have the following lemma (cf. e.g., [7, Prop. 2.5, 2.6].) Lemma 2.2 Let m, α, Ω, ω, δΩ be as in Lemma 2.1. Let φ0 ∈ C m,α (∂Ω, Rn ) ∩ A∂Ω . Then the following statements hold. (i) There exist δ0 ∈]0, δΩ [ and Φ0 ∈ C m,α (clΩω,δ0 , Rn ) ∩ A0clΩω,δ such that 0 φ0 ≡ Φ0|∂Ω . (ii) Let δ0 , Φ0 be as in (i). Then there exist an open neighborhood W0 of φ0 in C m,α (∂Ω, Rn ) ∩ A∂Ω , and a real analytic operator E0 of C m,α (∂Ω, Rn ) to C m,α (clΩω,δ0 , Rn ) which maps W0 to C m,α (clΩω,δ0 , Rn ) ∩ A0clΩω,δ and 0 such that E0 [φ0 ] = Φ0 and E0 [φ]|∂Ω = φ for all φ ∈ W0 . 5

RIVA-CRISTOFORIS: A PERTURBATION RESULT...

3

15

Some basic properties of elliptic operators and of layer potentials

As is well known, the differential operator P [a, D] has at least a fundamental solution Sa (·) for each a ∈ E. The membership of a in E ensures that P [a, D] is elliptic and that accordingly Sa (·) is real analytic on Rn \{0}, and that any other fundamental solution of P [a, D] differs from Sa (·) by a real analytic function defined on the whole of Rn . We collect in the following statement some known facts on the layer potentials associated to Sa (·). We find convenient to set Ω− ≡ Rn \ clΩ , for all open subsets Ω of Rn . Theorem 3.1 Let α ∈]0, 1[, m ∈ N\{0}. Let Ω be a bounded open subset of Rn of class C m,α . Let a ∈ E. Let Sa be a fundamental solution of P [a, D]. Then the following statements hold. (i) If µ ∈ C 0,α (∂Ω), then the function vSa [∂Ω, µ] of Rn to C defined by Z vSa [∂Ω, µ](ξ) ≡ Sa (ξ − η)µ(η) dση ∀ξ ∈ Rn , ∂Ω

is continuous. (ii) If µ ∈ C m−1,α (∂Ω), then the function vS+a [∂Ω, µ] ≡ vSa [∂Ω, µ]|clΩ belongs to C m,α (clΩ) and the operator which takes µ to vS+a [∂Ω, µ] is continuous from C m−1,α (∂Ω) to C m,α (clΩ). (iii) If µ ∈ C m−1,α (∂Ω), then the function vS−a [∂Ω, µ] ≡ vSa [∂Ω, µ]|clΩ− belongs m,α to Cloc (clΩ− ). If R ∈]0, +∞[ and clΩ ⊆ Bn (0, R), then the operator of m−1,α C (∂Ω) to C m,α (clBn (0, R)\Ω) which takes µ to vS−a [∂Ω, µ]|clBn (0,R)\Ω is continuous. (iv) If µ ∈ C m−1,α (∂Ω), l ∈ {1, . . . , n}, then the integral Z vSa ,l [∂Ω, µ](ξ) ≡ ∂ξl Sa (ξ − η)µ(η) dση

∀ξ ∈ Rn ,

∂Ω

converges in the sense of Lebesgue for all ξ ∈ Rn \ ∂Ω and in the sense of a principal value for all ξ ∈ ∂Ω. (v) Let l ∈ {1, . . . , n}. If µ ∈ C m−1,α (∂Ω), then vSa ,l [∂Ω, µ]|Ω admits a continuous extension vS+a ,l [∂Ω, µ] to clΩ and vS+a ,l [∂Ω, µ] ∈ C m−1,α (clΩ), and vSa ,l [∂Ω, µ]|Ω− admits a continuous extension vS−a ,l [∂Ω, µ] to clΩ− and m−1,α vS−a ,l [∂Ω, µ] ∈ Cloc (clΩ− ), and ∂ ± v [∂Ω, µ](ξ) ∂ξl Sa (νΩ (ξ))l =∓ µ(ξ) + vSa ,l [∂Ω, µ](ξ) , 2νΩ (ξ)t a(2) νΩ (ξ)

vS±a ,l [∂Ω, µ](ξ) =

6

16

RIVA-CRISTOFORIS: A PERTURBATION RESULT...

(DvS±a [∂Ω, µ](ξ))a(2) νΩ (ξ) Z 1 = ∓ µ(ξ) + (DSa (ξ − η))a(2) νΩ (ξ)µ(η) dση 2 ∂Ω for all ξ ∈ ∂Ω. (vi) Let l ∈ {1, . . . , n}. The operator of C m−1,α (∂Ω) to C m−1,α (clΩ) which takes µ to vS+a ,l [∂Ω, µ] is continuous. If R ∈]0, +∞[ and clΩ ⊆ Bn (0, R), then the operator of C m−1,α (∂Ω) to C m−1,α (clBn (0, R) \ Ω) which takes µ to vS−a ,l [∂Ω, µ]|clBn (0,R)\Ω is continuous. (vii) Let wSa [∂Ω, µ, a] be the function of Rn to C defined by Z wSa [∂Ω, µ, a](ξ) ≡ − (DSa (ξ − η))a(2) νΩ (η)µ(η) dση ∂Ω Z t − Sa (ξ − η)νΩ (η)a(1) µ(η) dση ∀ξ ∈ Rn , ∂Ω

for all µ ∈ C 0,α (∂Ω). If µ ∈ C m,α (∂Ω), then the restriction wSa [∂Ω, µ, a]|Ω can be extended uniquely to an element wS+a [∂Ω, µ, a] of C m,α (clΩ) and the restriction wSa [∂Ω, µ, a]|Ω− can be extended uniquely to an element m,α wS−a [∂Ω, µ, a] of Cloc (clΩ− ) and we have wS+a [∂Ω, µ, a] − wS−a [∂Ω, µ, a] = µ (DwS+a [∂Ω, µ, a])a(2) νΩ

−

(DwS−a [∂Ω, µ, a])a(2) νΩ

=0

on ∂Ω , on ∂Ω .

(viii) If µ ∈ C 0,α (∂Ω), then we have wSa [∂Ω, µ, a](ξ)

= −

n X

(2)

alj

j,l=1

Z −

∂ ∂ξl

Z Sa (ξ − η)(νΩ (η))j µ(η) dση ∂Ω

t Sa (ξ − η)νΩ (η)a(1) µ(η) dση

∀ξ ∈ Rn \ ∂Ω .

∂Ω m,α

(ix) If µ ∈ C (∂Ω) and U is an open neighborhood of ∂Ω in Rn and µ ˜ ∈ m C (U ), µ ˜|∂Ω = µ, then the following equality holds ∂ wS [∂Ω, µ, a](ξ) ∂ξr a Z n X (2) ∂ = alj Sa (ξ − η) ∂ξl ∂Ω j,l=1    ∂µ ˜ ∂µ ˜ (η) − (νΩ (η))j (η) dση · (νΩ (η))r ∂ηj ∂ηr  Z  + (DSa (ξ − η))a(1) + a0 Sa (ξ − η) (νΩ (η))r µ(η) dση ∂Ω Z t − ∂ξr Sa (ξ − η)νΩ (η)a(1) µ(η) dση ∀ξ ∈ Rn \ ∂Ω . ∂Ω

7

RIVA-CRISTOFORIS: A PERTURBATION RESULT...

17

(x) The operator of C m,α (∂Ω) to C m,α (clΩ) which takes µ to wS+a [∂Ω, µ, a] is continuous. If R ∈]0, +∞[ and clΩ ⊆ Bn (0, R), then the linear operator of C m,α (∂Ω) to C m,α (clBn (0, R)\Ω) which takes µ to wS−a [∂Ω, µ, a]|clBn (0,R)\Ω is continuous. For a proof and appropriate references of Theorem 3.1 (i)–(vi), we refer to [1]. Statements (vii)–(x) can be proved by exploiting exactly the same classical computations which can be found for example in the proof of [7, Thm. 3.4 (ii), (iii), (iv)]. Next we introduce the following result, which shows that the homogeneous equation P [a, D]u = 0 has a unique solution u in W01,2 (Ω0 ) if the volume of the domain of definition Ω0 is small enough and if a ranges in bounded subsets of E which are away from its boundary. Here W 1,2 (Ω0 ) denotes the Sobolev space of functions of L2 (Ω0 ) which have first order distributional derivatives in L2 (Ω0 ) with its usual norm and W01,2 (Ω0 ) denotes the closure of D(Ω0 ) in W 1,2 (Ω0 ) (cf. e.g., Gilbarg and Trudinger [3, p. 153].) Also, we find convenient to set  X   α −1 aα ξ E(η) ≡ a ≡ (aα )|α|≤2 ∈ E : inf Re > η, max |aα | < η , n ξ∈R ,|ξ|=1

|α|=2

|α|≤2

S for all η ∈]0, 1[. Obviously E = η∈]0,1[ E(η) and each E(η) is open in CN , where N denotes the number of multi-indexes α ∈ Nn with |α| ≤ 2. Lemma 3.2 Let η ∈]0, 1[. Then there exists M (η) ∈]0, +∞[ such that equation P [a, D]u = 0 ,

(3.3)

has the unique weak solution u = 0 in W01,2 (Ω0 ), for all a ∈ E(η) and for all open subsets Ω0 of Rn such that meas(Ω0 ) < M (η). Proof. Let u ∈ W01,2 (Ω0 ) solve (3.3) for some a ∈ E(η). Then we have Z ∀v ∈ W01,2 (Ω0 ) . (Dv)a(2) (Du)t − va(1) (Du)t − a0 uv dx = 0 Ω0

By exploiting the membership of a in E(η), we deduce that Z  Z (2) t Re (Du)a (Du) dx ≥ η |Du|2 dx ∀u ∈ W01,2 (Ω0 ) . Ω0

Ω0

Hence, Z Re

 (Du)a(2) (Du)t − ua(1) (Du)t − a0 |u|2 dx Ω0 Z ≥ η|Du|2 − |a(1) ||Du||u| − |a0 ||u|2 dx Ω0 Z 1 ≥ (η − )|Du|2 − |a(1) |2 |u|2 − |a0 ||u|2 dx , 4 0 Ω 8

(3.4)

18

RIVA-CRISTOFORIS: A PERTURBATION RESULT...

for all  ∈]0, η[. Since Ω0 has finite measure, we know that there exists a constant cP > 0 such that Z Z 2 0 2/n |u| dx ≤ cP (meas(Ω )) |Du|2 dx ∀u ∈ W01,2 (Ω0 ) , (3.5) Ω0

Ω0

for all open subsets Ω0 of Rn of finite measure (cf. e.g., Tartar [14, p. 50]) and thus by taking  = 21 η, inequality (3.5) implies that Z  (2) t (1) t 2 Re (Du)a (Du) − ua (Du) − a0 |u| dx (3.6) Ω0   Z  1 (1) 2 η − |a | + |a0 | cP (M (η))2/n |Du|2 dx ≥ 2 2η Ω0    Z 1 η n 2/n + ≥ − cP (M (η)) |Du|2 dx 2 2η 3 η Ω0 for all open subsets Ω0 of Rn of finite measure less or equal to a constant M (η), which we can choose so small that the term in brackets in the right hand side of (3.6) is positive. Finally, by equation (3.4) and inequality (3.6), we deduce the validity of the lemma (see also (3.5).) 2 Then we have the following immediate consequence of the classical elliptic theory. Theorem 3.7 Let m ∈ N \ {0}, α ∈]0, 1[. Let η ∈]0, 1[. Let M (η) > 0 be as in Lemma 3.2. If Ω0 is a bounded open connected subset of Rn of class C m,α such that meas(Ω0 ) < M (η), and if a ∈ E(η), and if (f, g) ∈ C m−2,α (clΩ0 ) × C m,α (∂Ω0 ), then there exists a unique u ∈ C m,α (clΩ0 ) such that  P [a, D]u = f in Ω0 , (3.8) u=g on ∂Ω0 . Here C −1,α (clΩ) denotes the space of distributions in Ω which equal the divergence of an element of class C 0,α (clΩ, Cn ) endowed with the quotient norm. Proof. As is well known, g is the trace of a function of class C m,α (clΩ0 ). Then the existence of a solution of problem (3.8) follows from that of the corresponding problem for g = 0. Existence of a solution in W01,2 (Ω0 ) follows by the LaxMilgram Lemma and by inequalities (3.5), (3.6). Then by the classical Schauder regularity theory, we deduce that the solution is actually of class C m,α (clΩ0 ) (cf. e.g., Morrey [9, Thm. 6.4.8].) The uniqueness of problem (3.8) follows by Lemma 3.2. 2

9

RIVA-CRISTOFORIS: A PERTURBATION RESULT...

4

19

An auxiliary boundary value problem

For each m, α, Ω, ω, δΩ as in Lemma 2.1, δ ∈]0, δΩ [, Φ ∈ C m,α (clΩω,δ , Rn ) ∩ A0clΩω,δ , a ∈ E, we set m−2,α m,α SΦ ≡ C m−2,α (clΦ(Ω+ (clΦ(Ω− (Φ(∂Ω)) ω,δ )) × C ω,δ )) × C

×C

m−1,α

(Φ(∂Ω)) × C

m,α

(Φ((∂Ω+ ω,δ )

\ ∂Ω)) × C

m,α

(Φ((∂Ω− ω,δ )

(4.1) \ ∂Ω))

and B[a, Φ](v + , v − ) ≡ (Dv + )|Φ(∂Ω) a(2) νΦ|∂Ω − (Dv − )|Φ(∂Ω) a(2) νΦ|∂Ω , m,α for all (v + , v − ) ∈ C m,α (clΦ(Ω+ (clΦ(Ω− ω,δ )) × C ω,δ )). Then we have the following.

Theorem 4.2 Let m, α, Ω, ω, δΩ be as in Lemma 2.1. Let η ∈]0, 1[. Then there exists δη ∈]0, δΩ [ suchthat if δ ∈]0, δη ], and if (a, Φ) belongs to E(η) × 

C m,α (clΩω,δ , Rn ) ∩ A0clΩω,δ , and if |det(DΦ)| ≤ η −1 on clΩω,δ , then the boundary value problem    + + +  P [a, D]v = f in Φ Ω ,     ω,δ     P [a, D]v − = f − in Φ Ω−  ω,δ ,    + − v −v =g on Φ (∂Ω) , (4.3) B[a, Φ](v + , v − ) = g1 on Φ (∂Ω) ,        v + = h+ on Φ (∂Ω+  ω,δ ) \ ∂Ω ,        v − = h− on Φ (∂Ω− ω,δ ) \ ∂Ω , m,α admits a unique solution (v + , v − ) ∈ C m,α (clΦ(Ω+ (clΦ(Ω− ω,δ )) × C ω,δ )) for + − + − each given (f , f , g, g1 , h , h ) in SΦ .

Proof. Let M (η) > 0 be as in Lemma 3.2. We take δη ∈]0, δΩ [ such that η −1 meas(Ωω,δ ) ≤ M (η) for all δ ∈]0, δη ]. Then we also have meas(Φ(Ωω,δ )) ≤ M (η), for all δ ∈]0, δη ]. Now let δ ∈]0, δη ] and (f + , f − , g, g1 , h+ , h− ) ∈ SΦ . We first show existence for (4.3). By Theorem 3.7, there exist v˜+ ∈ C m,α (clΦ(Ω+ ω,δ )) and v˜− ∈ C m,α (clΦ(Ω− )) such that ω,δ     v + = f + in Φ Ω+  ω,δ ,  P [a, D]˜ v˜+ = g on Φ  (∂Ω) ,     v˜+ = h+ on Φ (∂Ω+ ) \ ∂Ω , ω,δ

and

  v− = f −   P [a, D]˜ v˜− = 0    v˜− = h−

  in Φ Ω− ω,δ , on Φ  (∂Ω) ,

 on Φ (∂Ω− ω,δ ) \ ∂Ω . 10

20

RIVA-CRISTOFORIS: A PERTURBATION RESULT...

Next we note that the boundary value problem   P [a, D]u+ = 0     P [a, D]u− = 0   u+ − u− = 0    B[a, Φ](u+ , u− ) = −g1 + B[a, Φ](˜ v + , v˜− )

  in Φ Ω+ ω,δ ,   in Φ Ω− ω,δ , on Φ (∂Ω) , on Φ (∂Ω) ,

m,α has a solution (u+ , u− ) ∈ C m,α (clΦ(Ω+ (clΦ(Ω− ω,δ )) × C ω,δ )). Indeed, we can + − + − take u ≡ vSa [Φ(∂Ω), µ] and u ≡ vSa [Φ(∂Ω), µ] with µ ≡ g1 − B[a, Φ](˜ v + , v˜− ), where Sa is a fundamental solution of P [a, D] (cf. Theorem 3.1.) Then boundary m,α value problem (4.3) has a solution (v + , v − ) ∈ C m,α (clΦ(Ω+ (clΦ(Ω− ω,δ ))×C ω,δ )) if and only if system    + +  P [a, D]V = 0 in Φ Ω ,     ω,δ    − −  P [a, D]V = 0 in Φ Ωω,δ ,     + − V −V =0 on Φ (∂Ω) , (4.4) + − B[a, Φ](V , V ) = 0 on Φ (∂Ω) ,        V + = u+ on Φ (∂Ω+  ω,δ ) \ ∂Ω ,       − −  V =u on Φ (∂Ω− ) \ ∂Ω , ω,δ m,α has a solution (V + , V − ) ∈ C m,α (clΦ(Ω+ (clΦ(Ω− ω,δ )) × C ω,δ )), and in case ± ± ± ± of existence, V = v − v˜ + u . Then we now turn to consider problem (4.4). By a standard argument based on the Green identity for P [a, D] (cf. e.g., Miranda [8, p. 12]), problem (4.4) admits a solution (V + , V − ) if and only if problem  P [a, D]V = 0 in Φ (Ω    ω,δ ) ,   + V =u on Φ (∂Ω+ ω,δ ) \ ∂Ω ,      V = u− on Φ (∂Ω− ) \ ∂Ω , ω,δ

has a solution V ∈ C m,α (clΦ(Ωω,δ )), and in case of existence, V + = V|clΦ(Ω+

ω,δ )

and V − = V|clΦ(Ω− ) . Since the existence for such a system follows by Theorem ω,δ

3.7, the proof of the existence for problem (4.3) is complete. We now turn to consider uniqueness for problem (4.3). Let the pair (v + , v − ) m,α + − + − in C m,α (clΦ(Ω+ (clΦ(Ω− ω,δ )) × C ω,δ )) solve (4.3) with (f , f , g, g1 , h , h ) = 0. Then we define a function v of clΦ(Ωω,δ ) to C by setting v = v + on clΦ(Ω+ ω,δ ) − + − − and v = v on clΦ(Ωω,δ ). Clearly, v satisfies P [a, D]v = 0 in Φ(Ωω,δ ) ∪ Φ(Ωω,δ ) and is continuous on clΦ(Ωω,δ ). Since v + − v − = 0 and B[a, Φ](v + , v − ) = 0 on Φ(∂Ω), a standard argument based on the Green identity for P [a, D] shows that v solves P [a, D]v = 0 in Φ(Ωω,δ ) in the sense of distributions (cf. e.g., Miranda [8, p. 12].) Since v equals 0 on ∂Φ(Ωω,δ ), our choice of δη and Lemma 3.2 imply that v = 0. Hence, (v + , v − ) = (0, 0). 2

11

RIVA-CRISTOFORIS: A PERTURBATION RESULT...

21

We note that if Sa is a fundamental solution of the differential operator P [a, D], then (vS+a [Φ(∂Ω), µ]|clΦ(Ω+ ) , vS−a [Φ(∂Ω), µ]|clΦ(Ω− ) ) is the only soluω,δ

ω,δ

tion of problem (4.3) with f− = 0 , g = 0, g1 = −µ , − − h ≡ vSa [Φ(∂Ω), µ]|Φ((∂Ω− )\∂Ω) .

f+ = 0 , h+ ≡ vS+a [Φ(∂Ω), µ]|Φ((∂Ω+

ω,δ )\∂Ω)

,

ω,δ

Thus problem (4.3) with such data identifies the pair (vS+a [Φ(∂Ω), µ]|clΦ(Ω+ ) , vS−a [Φ(∂Ω), µ]|clΦ(Ω− ) ) . ω,δ

ω,δ

In order to obtain a problem which identifies the pair (vS+a [Φ(∂Ω), µ] ◦ Φ|clΩ+ , vS−a [Φ(∂Ω), µ] ◦ Φ|clΩ− ) , ω,δ

ω,δ

we wish to change the variable in (4.3) with the above data by means of the function Φ. However, we note that if m = 1, then the map Φ is only one time continuously differentiable, while the differential operator P [a, D] is of order 2. Thus we now follow [7] and we introduce the following Lemmas. Lemma 4.5 Let m, m0 ∈ N, m > 0, m ≥ m0 . Let α ∈]0, 1[. Let Ω be an open 0 bounded subset of Rn of class C m,α . Then the operator div of C m ,α (clΩ, Cn ) 0 to C m −1,α (clΩ) is bounded linear continuous open and surjective. Lemma 4.5 follows by the definition of C −1,α (clΩ) as the space of distributions in Ω which equal the divergence of an element of class C 0,α (clΩ, Cn ) endowed with the quotient norm and by standard results in elliptic theory (cf. e.g., Gilbarg and Trudinger [3, Thms. 6.14, 6.19] and Troianiello [15, Thm. 1.3, Lem. 1.5].) Then by Lemma 4.5, we have the following (see also [7, Lem. 4.5] for a proof.) Lemma 4.6 Let m ∈ N, α ∈]0, 1[. Let Ω be an open bounded connected subset of Rn of class C max{1,m},α . The set   Z Y m,α (Ω) ≡ w ∈ C m,α (clΩ, Cn ) : (Dϕ)w dx = 0 ∀ϕ ∈ D (Ω) Ω

is a closed linear subspace of C

m,α

n

(clΩ, C ) and the quotient

Z m,α (Ω) ≡ C m,α (clΩ, Cn ) /Y m,α (Ω) is a Banach space. Moreover, if we denote by ΠΩ the canonical projection f of of C m,α (clΩ, Cn ) onto Z m,α (Ω), there exists a unique homeomorphism div m,α m−1,α f Z (Ω) onto C (clΩ) such that div = div ◦ ΠΩ . Then we have the following lemma, which generalizes the corresponding Lemma of [6], [7].

12

22

RIVA-CRISTOFORIS: A PERTURBATION RESULT...

Lemma 4.7 Let m ∈ N\{0}, α ∈]0, 1[. Let Ω be a bounded open connected subset of Rn of class C m,α . Let A[·, ·, ·] be the map of E ×(C m,α (clΩ, Rn ) ∩ AclΩ )× C m,α (clΩ) to the space C m−1,α (clΩ, Cn ) defined by n o A[a, Φ, u] ≡ (DΦ)−1 a(2) (DΦ)−t (Du)t + (DΦ)−1 a(1) u | det DΦ| ∀(a, Φ, u) ∈ E × (C m,α (clΩ, Rn ) ∩ AclΩ ) × C m,α (clΩ) . Let Q[·, ·, ·] be the map of E ×(C m,α (clΩ, Rn ) ∩ AclΩ )×C m,α (clΩ) to Z m−1,α (Ω) defined by f Q[a, Φ, u] ≡ ΠΩ A[a, Φ, u] + a0 div

(−1)

(u| det DΦ|) ,

(4.8)

for all (a, Φ, u) ∈ E × (C m,α (clΩ, Rn ) ∩ AclΩ ) × C m,α (clΩ). Then we have Q[a, Φ, u] = ΠΩ f

(4.9)

if and only if P [a, D](u ◦ Φ(−1) ) = div

n   o ((DΦ)f ) ◦ Φ(−1) | det D Φ(−1) | ,

(4.10)

in the sense of distributions in Φ (Ω), for all f in C m−1,α (clΩ, Cn ) and for all (a, Φ, u) in E × (C m,α (clΩ, Rn ) ∩ AclΩ ) × C m,α (clΩ). Proof. Since u| det DΦ| ∈ C m−1,α (clΩ), Lemma 4.5 ensures that there exists g ∈ C m,α (clΩ, Cn ) such that f div

(−1)

(u| det DΦ|) = ΠΩ g .

(4.11)

Thus equation (4.9) is equivalent to equation ΠΩ A[a, Φ, u] = ΠΩ (f − a0 g), an equation which we rewrite in the following form   Z Dϕ A[a, Φ, u] + a0 g − f dx = 0 ∀ϕ ∈ D(Ω) . (4.12) Ω

By equality (4.11), we have Z Z ϕu| det DΦ| dx = − (Dϕ)g dx Ω

∀ϕ ∈ D(Ω) .

Ω

Hence, by changing the variables in the integral of equation (4.12), we obtain Z D(ϕ ◦ Φ(−1) )a(2) D(u ◦ Φ(−1) )t + D(ϕ ◦ Φ(−1) )a(1) u (4.13) Φ(Ω)

−a0 (ϕ ◦ Φ(−1) )(u ◦ Φ(−1) ) dy Z   = D(ϕ ◦ Φ(−1) )((DΦ)f ) ◦ Φ(−1) | det D Φ(−1) | dy Φ(Ω)

13

∀ϕ ∈ D(Ω) .

RIVA-CRISTOFORIS: A PERTURBATION RESULT...

23

By exploiting a standard argument based on the convolution with a family of mollifiers, equation (4.13) is easily seen to be equivalent to the same equation with ϕ ◦ Φ(−1) replaced by an arbitrary ψ of D(Φ(Ω)). Hence, equation (4.13) is equivalent to equation (4.10) and thus the proof is complete. 2 We now transplant boundary value problem (4.3), which is defined on the − + − pair of sets (Φ(Ω+ ω,δ ), Φ(Ωω,δ )) to the pair of sets (Ωω,δ , Ωω,δ ) by means of the function Φ. We do so by means of the following. Theorem 4.14 Let m, α, Ω, ω, δΩ be as in Lemma 2.1. Let δ ∈]0, δΩ [. Let T m,α denote the map of E ×(C m,α (clΩω,δ , Rn )∩A0clΩω,δ )×C m,α (clΩ+ (clΩ− ω,δ )×C ω,δ ) to the Banach space     m−1,α m,α Ω− Z ≡ Z m−1,α Ω+ (∂Ω) × C m−1,α (∂Ω) (4.15) ω,δ × Z ω,δ × C     − m,α (∂Ω ×C m,α (∂Ω+ ) \ ∂Ω × C ) \ ∂Ω , ω,δ ω,δ which takes (a, Φ, V + , V − ) to T [a, Φ, V + , V − ] ≡ (Q[a, Φ, V + ], Q[a, Φ, V − ], V + − V − , + − J[a, Φ, V + , V − ], V|(∂Ω , V|(∂Ω + − )\∂Ω ω,δ

(4.16)

ω,δ )\∂Ω

),

where we have set J[a, Φ, V + , V − ] ≡ DV + (DΦ)−1 a(2) n[Φ] − DV − (DΦ)−1 a(2) n[Φ] and n[Φ](x) ≡

(DΦ(x))−t νΩ (x) |(DΦ(x))−t νΩ (x)|

on ∂Ω , (4.17)

∀x ∈ ∂Ω .

Then the following statements hold. (i) Let (a, Φ) ∈ E × (C m,α (clΩω,δ , Rn ) ∩ A0clΩω,δ ), (F + , F − , G, G1 , H + , H − ) ∈ m,α Z. Then a pair (V + , V − ) of C m,α (clΩ+ (clΩ− ω,δ ) × C ω,δ ) satisfies the equation T [a, Φ, V + , V − ] = (F + , F − , G, G1 , H + , H − ) (4.18) if and only if the pair (V + ◦ Φ(−1) , V − ◦ Φ(−1) ) ∈ C m,α (clΦ(Ω+ ω,δ )) × C m,α (clΦ(Ω− )) satisfies problem (4.3) with ω,δ g ≡ G ◦ Φ(−1) ,

g1 ≡ G1 ◦ Φ(−1) ,

(−1)

h± ≡ H ± ◦ Φ|Φ((∂Ω±

ω,δ )\∂Ω)

,

f ± ≡ div{((DΦ)f˜± ) ◦ Φ(−1) |det(DΦ(−1) )|}, n ˜± = F ± . where f˜± ∈ C m−1,α (clΩ± ω,δ , C ) and ΠΩ± f ω,δ

(ii) Let η ∈]0, 1[. Let δη be as in Theorem 4.2. If δ ∈]0, δη ], (a, Φ) ∈ E(η) × (C m,α (clΩω,δ , Rn ) ∩ A0clΩω,δ ), |det(DΦ)| ≤ η −1 , then T [a, Φ, ·, ·] is a linear m,α homeomorphism of C m,α (clΩ+ (clΩ− ω,δ ) × C ω,δ ) onto Z. 14

24

RIVA-CRISTOFORIS: A PERTURBATION RESULT...

Proof. Let (V + , V − ) satisfy equation (4.18). By Lemma 4.6, there exist f˜± as in statement (i). By elementary calculus, we know that n[Φ](x) = νΦ|∂Ω ◦ Φ(x)

∀x ∈ ∂Ω ,

(cf. e.g., [7, Lem. 4.2].) Then Lemma 4.7 and standard calculus imply that (V + ◦ Φ(−1) , V − ◦ Φ(−1) ) satisfies problem (4.3). The proof of the converse is similar. Hence, statement (i) holds. We now prove statement (ii). By continuity of the pointwise product in Schauder spaces and by elementary properties of functions in Schauder spaces (cf. e.g., [4, §2]), the map T [a, Φ, ·, ·] is linear and continuous. Then by the Open Mapping Theorem it suffices to show that T [a, Φ, ·, ·] is a bijection. If (F + , F − , G, G1 , H + , H − ) ∈ Z, then elementary properties of functions in Schauder spaces imply that the sextuple (f + , f − , g, g1 , h+ , h− ) defined as in statement (i) belongs to SΦ (cf. (4.1).) Then Theorem 4.2 ensures that problem (4.3) admits a unique solution (v + , v − ). Then statement (i) ensures that the pair (V + , V − ) ≡ (v + ◦Φ, v − ◦Φ) solves problem (4.18). If (F + , F − , G, G1 , H + , H − ) = 0, then (f + , f − , g, g1 , h+ , h− ) must vanish and accordingly both (v + , v − ) and (v + ◦ Φ, v − ◦ Φ) must vanish. 2 By Theorem 4.2 and by Theorem 4.14 and by Theorem 3.1, we immediately deduce the validity of the following (see also [7, Lem. 4.2] for the form of the area element σn [Φ] below.) Corollary 4.19 Let m, α, Ω, ω, δΩ be as in Lemma 2.1. Let δ ∈]0, δΩ [. Let a ∈ E, Φ ∈ C m,α (clΩω,δ , Rn ) ∩ A0clΩω,δ . Let f ∈ C m−1,α (∂Ω). Let Sa be a fundamental solution of P [a, D]. Then a pair (V + , V − ) ∈ C m,α (clΩ+ ω,δ ) × C m,α (clΩ− ) satisfies the equation ω,δ T [a, Φ, V + , V − ] = (0, 0, 0, −f, h+ , h− ) with Z

+

h (x) ≡

Sa (Φ(x) − Φ(y))f (y)σn [Φ](y) dσy

∀x ∈ (∂Ω+ ω,δ ) \ ∂Ω ,

Sa (Φ(x) − Φ(y))f (y)σn [Φ](y) dσy

∀x ∈ (∂Ω− ω,δ ) \ ∂Ω ,

∂Ω

h− (x) ≡

Z ∂Ω

where σn [Φ] ≡ |det(DΦ)||(DΦ)−t νΩ |, if and only if V+

= vS+a [Φ(∂Ω), f ◦ Φ(−1) ] ◦ Φ|clΩ+ , ω,δ

V

5

−

=

vS−a [Φ(∂Ω), f

◦Φ

(−1)

] ◦ Φ|clΩ− . ω,δ

Real analyticity of layer potentials corresponding to families of fundamental solutions

In this section, we shall prove our main result, which concerns layer potentials associated to families of fundamental solutions of families of elliptic differential 15

RIVA-CRISTOFORIS: A PERTURBATION RESULT...

25

operators of second order. We first introduce the following Lemma, which can be proved by the same argument of [7, Lem. 4.8]. Lemma 5.1 Let m, α, Ω, ω, δΩ be as in Lemma 2.1. Let δ ∈]0, δΩ [. Let assumption (1.1) hold. Then the map Vδ of   C m,α (clΩω,δ , Rn ) ∩ A0clΩω,δ × C m−1,α (∂Ω) × O to C m,α (∂Ωω,δ ) defined by Z Vδ [Φ, f, κ](x) ≡ S(Φ(x) − Φ(y), κ)f (y)σn [Φ](y) dσy

∀x ∈ ∂Ωω,δ ,

∂Ω

  for all (Φ, f, κ) ∈ C m,α (clΩω,δ , Rn ) ∩ A0clΩω,δ × C m−1,α (∂Ω) × O is real analytic. We now introduce some notation. Let m, α, Ω, ω, δΩ be as in Lemma 2.1. If (1.1) holds, we set Z + v [φ, f, κ](ξ) ≡ S (ξ − η, κ) f ◦ φ(−1) (η) dση ∀ξ ∈ clI[φ] , φ(∂Ω) Z v − [φ, f, κ](ξ) ≡ S (ξ − η, κ) f ◦ φ(−1) (η) dση ∀ξ ∈ clE[φ] , φ(∂Ω)

for all (φ, f, κ) ∈ (C m,α (∂Ω, Rn ) ∩ A∂Ω ) × C m−1,α (∂Ω) × O, and n o Uη,δ ≡ Φ ∈ C m,α (clΩω,δ , Rn ) ∩ A0clΩω,δ : |det(DΦ)| < 1/η , O(η) ≡

{κ ∈ O : a(κ) ∈ E(η)} ,

for all η ∈]0, 1[, δ ∈]0, δΩ [. Then we have the following result. Proposition 5.2 Let m, α, Ω, ω, δΩ be as in Lemma 2.1. Let assumption (1.1) hold. Let η ∈]0, 1[. Let δη ∈]0, δΩ [ be as in Theorem 4.2. Let δ ∈ ]0, δη ]. Let V ± [Φ, f, κ] ≡ v ± [Φ|∂Ω , f, κ] ◦ Φ|clΩ± on clΩ± ω,δ for all (Φ, f, κ) ∈ ω,δ

(C m,α (clΩω,δ , Rn ) ∩ A0clΩω,δ ) × C m−1,α (∂Ω) × O. Then the maps of Uη,δ × m,α C m−1,α (∂Ω) × O(η) to C m,α (clΩ+ (clΩ− ω,δ ) and to C ω,δ ), which take (Φ, f, κ) + − to V [Φ, f, κ] and to V [Φ, f, κ] are real analytic, respectively. Proof. First we set X Y

≡ ≡

C m,α (clΩω,δ , Rn ) × C m−1,α (∂Ω) × K , m,α C m,α (clΩ+ (clΩ− ω,δ ) × C ω,δ ) ,

Vη,δ

≡

Uη,δ × C m−1,α (∂Ω) × O(η) .

16

26

RIVA-CRISTOFORIS: A PERTURBATION RESULT...

Then we consider the map Λ of U ≡ Vη,δ × Y to the Banach space Z of (4.15) defined by Λ[Φ, f, κ, V + , V − ] ≡ T [a(κ), Φ, V + , V − ] −(0, 0, 0, −f, V + [Φ, f, κ]|(∂Ω+

ω,δ )\∂Ω

, V − [Φ, f, κ]|(∂Ω−

ω,δ )\∂Ω

)

for all (Φ, f, κ, V + , V − ) ∈ U. By Corollary 4.19, the set of zeros of Λ in U coincides with the graph of the map (V + [·, ·, ·], V − [·, ·, ·]). Thus we can deduce the real analyticity of the operator (V + [·, ·, ·], V − [·, ·, ·]) by showing that we can apply the Implicit Function Theorem for real analytic operators (cf. e.g., Prodi and Ambrosetti [13, Thm. 11.6]) to the equation Λ[Φ, f, κ, V + , V − ] = 0 around the point (Φ1 , f1 , κ1 , V + [Φ1 , f1 , κ1 ], V − [Φ1 , f1 , κ1 ]) for all (Φ1 , f1 , κ1 ) in Vη,δ . The domain U = Vη,δ × Y of Λ is clearly open in X × Y. Since |det(DΦ)| ∈ C m−1,α (clΩω,δ ), the continuity of the imbedding of C 0 (clΩω,δ ) into C −1,α (clΩω,δ ) in case m = 1 (cf. [7, Lem. 4.4]) and Lemma 4.6 imply that the m,α n 0 (clΩ± operator which takes (Φ, V ± ) in (C m,α (clΩ± ω,δ ) ω,δ , R ) ∩ AclΩ± ) × C ω,δ

(−1)

f to div (V ± |det(DΦ)|) in Z m−1,α (Ω± ω,δ ) is bilinear and continuous. Then by the real analyticity of the map which takes an invertible matrix with Schauder functions as entries to its inverse, we conclude that both Q[·, ·, ·] and J[·, ·, ·, ·] are real analytic (cf. (4.8), (4.16), (4.17).) Then Lemma 5.1 and the linearity and continuity of the trace operator on the boundary, imply that Λ is real analytic. Thus it suffices to show that the differential d(V + ,V − ) Λ[Φ1 , f1 , κ1 , V + [Φ1 , f1 , κ1 ], V − [Φ1 , f1 , κ1 ]] is a homeomorphism. Now by standard rules of calculus in Banach space, such a differential coincides with T [a(κ1 ), Φ1 , ·, ·]. Since δ ∈]0, δη ], a(κ1 ) ∈ E(η), Φ1 ∈ Uη,δ , Theorem 4.14 (ii) ensures that T [a(κ1 ), Φ1 , ·, ·] is a linear homeomorphism of Y onto Z, and thus the proof is complete. 2 Corollary 5.3 Let the assumptions of Proposition 5.2 hold. Let W + [Φ, f, κ] − and W − [Φ, f, κ] denote the continuous extensions to clΩ+ ω,δ and to clΩω,δ of the functions Z − Dξ S(Φ(x) − η, κ)a(2) (κ)νΦ|∂Ω (η)f ◦ Φ(−1) (η) dση Φ(∂Ω) Z t − S(Φ(x) − η, κ)νΦ (η)a(1) (κ)f ◦ Φ(−1) (η) dση ∀x ∈ Ω+ ω,δ , |∂Ω Φ(∂Ω) Z − Dξ S(Φ(x) − η, κ)a(2) (κ)νΦ|∂Ω (η)f ◦ Φ(−1) (η) dση Φ(∂Ω) Z t − S(Φ(x) − η, κ)νΦ (η)a(1) (κ)f ◦ Φ(−1) (η) dση ∀x ∈ Ω− ω,δ , |∂Ω Φ(∂Ω)

17

RIVA-CRISTOFORIS: A PERTURBATION RESULT...

  for all (Φ, f, κ) ∈ C m,α (clΩω,δ , Rn ) ∩ A0clΩω,δ × C m,α (∂Ω) × O, respectively. m,α Then the maps of Uη,δ × C m,α (∂Ω) × O(η) to C m,α (clΩ+ (clΩ− ω,δ ) and to C ω,δ ) + − which take (Φ, f, κ) to W [Φ, f, κ] and to W [Φ, f, κ] are real analytic, respectively. + Proof. We proceed as in [6,  Prop.3.11]. We first  consider  W [·, ·, ·]. We observe  + + m,α m−1,α n that the map Γ of C clΩω,δ to C clΩω,δ × C m−1,α clΩ+ ω,δ , C defined by   Γ[g] ≡ (g, ∂x1 g, . . . , ∂xn g) ∀g ∈ C m,α clΩ+ ω,δ ,   is a linear homeomorphism of C m,α clΩ+ ω,δ onto the image space Im Γ, a sub    + m−1,α n space of C m−1,α clΩ+ × C clΩ , C . Thus it suffices to show that ω,δ ω,δ ∂ + s = 1, . . . , n are real analytic ∂xs W  [·, ·, ·] for  + C m−1,α clΩω,δ . Now let R > supx∈Ω∪Ωω,δ |x|.

the nonlinear maps W + [·, ·, ·] and

from Uη,δ ×C m,α (∂Ω)×O(η) to By Troianiello [15, Thm. 1.3, Lem. 1.5], there exists a linear and continuous operator F of C m,α (∂Ω) to C m,α (clBn (0, R)) such that F[f ]|∂Ω = f , for all f ∈ C m,α (∂Ω). By Theorem 3.1 (viii), (ix), we have the following identities W + [Φ, f, κ] (5.4) n X ∂ (2) alj (κ) =− (V + [Φ, nj [Φ]f, κ])((DΦ)−1 )sl − V + [Φ, nt [Φ]a(1) f, κ], ∂xs l,s,j=1

and
∂ W + [Φ, f, κ] (5.5) ∂xs n n n  X X
 ∂ ∂Φr X (2) −1 alj (κ) V + [Φ, Mrj [f, Φ], κ] (DΦ) = ∂xs ∂xt tl t=1 r=1 l,j=1

+

n X r=1

+


∂Φr D V + [Φ, nr [Φ]f, κ] (DΦ)−1 · a(1) (κ) ∂xs

n X ∂Φr r=1

∂xs

a0 (κ)V + [Φ, nr [Φ]f, κ] −

∂ + V [Φ, nt [Φ]a(1) f, κ] , ∂xs

for all s ∈ {1, . . . , n}, where −1 −t Mrj [f, Φ] = (DΦ) · νΩ · # (" n #" n   X X ∂(F[f ])  −1 −1 (DΦ) · (DΦ) (νΩ )l ∂xl lr lj l=1 l=1 " n #" n #)   X X ∂(F[f ])  −1 −1 − (DΦ) (νΩ )l (DΦ) . ∂xl lj lr l=1

l=1

18

27

28

RIVA-CRISTOFORIS: A PERTURBATION RESULT...

Then by the real analyticity of the pointwise product in Schauder spaces and by the real analyticity of the map which takes an invertible matrix with Schauder functions as entries to its inverse, and by the real analyticity of the linear and continuous map F[·] and of the trace operator, and by the real analyticity of V + [·, ·, ·], and by assumption (1.1), and by identities (5.4) and (5.5), we conclude ∂ W + [·, ·, ·] are real analytic from Uη,δ × C m,α (∂Ω) × O(η) that W + [·, ·, ·] and ∂x   s − to C m−1,α clΩ+ ω,δ . Similarly, we can show that W [Φ, f, κ] depends real analytically on (Φ, f, κ). 2 We are now ready to prove our main result. Theorem 5.6 Let m ∈ N \ {0}, α ∈]0, 1[. Let Ω be a bounded open subset of Rn of class C m,α such that both Ω and Rn \ clΩ are connected. Let assumption (1.1) hold. Then the following statements hold. (i) The map V [·, ·, ·] of (C m,α (∂Ω, Rn ) ∩ A∂Ω )×C m−1,α (∂Ω)×O to the space C m,α (∂Ω) defined by (1.2) is real analytic. (ii) The map Vl [·, ·, ·] of (C m,α (∂Ω, Rn ) ∩ A∂Ω )×C m−1,α (∂Ω)×O to the space C m−1,α (∂Ω) defined by (1.3) is real analytic for all l ∈ {1, . . . , n}. (iii) The map V∗ [·, ·, ·] of (C m,α (∂Ω, Rn ) ∩ A∂Ω ) × C m−1,α (∂Ω) × O to the space C m−1,α (∂Ω) defined by (1.4) is real analytic. (iv) The map W [·, ·, ·] of (C m,α (∂Ω, Rn ) ∩ A∂Ω ) × C m,α (∂Ω) × O to the space C m,α (∂Ω) defined by (1.5) is real analytic. Proof. We first consider statements (i)–(iii). It clearly suffices to show that if (φ0 , f0 , κ0 ) ∈ (C m,α (∂Ω, Rn ) ∩ A∂Ω ) × C m−1,α (∂Ω) × O, then the operators of (i)–(iii) are real analytic in a neighborhood of (φ0 , f0 , κ0 ). Now let ω, δ0 , W0 , E0 be as in Lemma 2.2 for φ0 . Possibly shrinking W0 , we can assume that there exists η ∈]0, 1[ such that sup

sup

|det(DE0 [φ](x))| < η −1 ,

κ0 ∈ O(η) .

φ∈W0 x∈clΩω,δ0

Now by definition of the operators in (i)–(iii), and by Theorem 3.1, we have = v + [φ, f, κ] ◦ φ = V + [E0 [φ], f, κ] , nl [E0 [φ]] ∂ + Vl [φ, f, κ] = f+ (v [φ, f, κ]) ◦ φ ∂ξl 2nt [E0 [φ]]a(2) n[E0 [φ]]
nl [E0 [φ]] + −1 f + (DV [E [φ], f, κ]) · (DE [φ]) , = 0 0 l 2nt [E0 [φ]]a(2) n[E0 [φ]]
1 V∗ [φ, f, κ] = f + (Dv + [φ, f, κ]) ◦ φ a(2) (κ)νφ ◦ φ 2 1 = f + (DV + [E0 [φ], f, κ])(DE0 [φ])−1 · a(2) (κ) · n[E0 [φ]] , 2 V [φ, f, κ]

19

RIVA-CRISTOFORIS: A PERTURBATION RESULT...

on ∂Ω for all (φ, f, κ) ∈ W0 × C m−1,α (∂Ω) × O(η) where V + is as in Proposition 5.2 for some arbitrary δ ∈]0, min{δη , δ0 }]. Hence, statements (i)–(iii) follow by Lemma 2.2 and Proposition 5.2. In order to prove statement (iv), we just note that 1 W [φ, f, κ] = − f + W + [E0 [φ], f, κ] 2

on ∂Ω ,

for all (φ, f, κ) ∈ W0 × C m,α (∂Ω) × O(η), and then we argue as above by exploiting Corollary 5.3 instead of Proposition 5.2. 2 Acknowledgement. We acknowledge the support of the research project “Problemi di stabilit` a per operatori differenziali” of the University of Padova, Italy. The authors are indebted to Dr. Paolo Musolino for a number of comments which have improved this paper.

References [1] M. Dalla Riva, A particular family of fundamental solutions for elliptic partial differential operators with constant coefficients, typewritten manuscript, (2009). [2] G. Fichera, Una introduzione alla teoria delle equazioni integrali singolari, Rend. Mat. e Appl., 17, (1958), 82–191. [3] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Springer Verlag, 1983. [4] M. Lanza de Cristoforis, Properties and pathologies of the composition and inversion operators in Schauder spaces, Acc. Naz. delle Sci. detta dei XL, 15, (1991), 93–109. [5] M. Lanza de Cristoforis and L. Preciso, On the analyticity of the Cauchy integral in Schauder spaces, Journal of Integral Equations and Applications, 11, (1999), 363–391. [6] M. Lanza de Cristoforis and L. Rossi, Real analytic dependence of simple and double layer potentials upon perturbation of the support and of the density, J. Integral Equations Appl., 16, 137–174 (2004). [7] M. Lanza de Cristoforis and L. Rossi, Real analytic dependence of simple and double layer potentials for the Helmholtz equation upon perturbation of the support and of the density, Analytic Methods of Analysis and Differential Equations, AMADE 2006, Eds. A.A. Kilbas and S.V. Rogosin, Cambridge Scientific Publishers, Cambridge (UK) (2008), pp. 193–220. [8] C. Miranda, Partial differential equations of elliptic type, Second revised edition, Springer-Verlag, Berlin, 1970. 20

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RIVA-CRISTOFORIS: A PERTURBATION RESULT...

[9] C.B. Jr. Morrey, Multiple integrals in the calculus of variations, SpringerVerlag, New York, 1966. [10] R. Potthast, Domain derivatives in electromagnetic scattering, Math. Methods Appl. Sci., 19 (1996), 1157–1175. [11] R. Potthast, Fr´echet differentiability of boundary integral operators in inverse acoustic scattering, Inverse Problems, 10 (1994), 431–447. [12] R. Potthast, Fr´echet differentiability of the solution to the acoustic Neumann scattering problem with respect to the domain, J. Inverse Ill-Posed Probl., 4 (1996), 67–84. [13] G. Prodi and A. Ambrosetti, Analisi non lineare, Editrice Tecnico Scientifica, Pisa, 1973. [14] L. Tartar, An introduction to Sobolev spaces and interpolation spaces. Lecture Notes of the Unione Matematica Italiana, 3. Springer, Berlin; UMI, Bologna, 2007. [15] G.M. Troianiello, Elliptic differential equations and obstacle problems, Plenum Press, New York and London, 1987. Dipartimento di Matematica Pura ed Applicata, Universit`a degli Studi di Padova, Via Trieste 63, 35121 Padova, Italia.

21

JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.5, NO.1, 31-41, COPYRIGHT 2010 EUDOXUS PRESS, LLC

ORTHONORMAL WAVELET SYSTEMS AND MULTIRESOLUTION ANALYSES RICHARD A. ZALIK

Department of Mathematics and Statistics, Auburn University, AL 36849-5310. email:[email protected] 2000 Mathematics Subject Classification:42C40 Let A ∈ Rd×d be a matrix that preserves the lattice Zd and |a| := det A. In [8], the author studied the properties of wavelet systems in L2 (Rd ) of the form {|a|j/2 ψℓ (Aj t + k); j ∈ Z, k ∈ Zd , 1 ≤ ℓ ≤ m} that are associated with a multiresolution analysis of multiplicity n generated by A. The purpose of the present paper is to extend those results to wavelet systems in L2 (Rd ) of the form {|a|j/2 ψℓ (Aj t + Bk); j ∈ Z, k ∈ Zd , 1 ≤ ℓ ≤ m} that are associated with a multiresolution analysis of multiplicity n generated by A and B, where A ∈ Rd×d is a matrix that preserves the lattice Zd and B ∈ Rd×d is a nonsingular matrix. 1. Introduction In what follows, Z will denote the set of integers, T := [0, 1], and Td will denote the d–dimensional torus. The underlying space will be L2 (Rd ), where d ≥ 1 is an integer and R is the set of real numbers, I will stand for the identity matrix, A, B ∈ Rd×d , a := det A, b := det B, C := (A−1 )T , and D := (B −1 )T . Boldface lowcase letters will denote elements of Rd , which will be represented as column vectors; x · y will stand for the standard dot product of the vectors x and y; ||x||2 := x · x. The inner product of two functions f, g ∈ L2 (Rd ) will be denoted by hf, gi, their bracket product with respect to B by [f, g]B , and the norm of f by ||f ||; thus, Z hf, gi :=

f (t)g(t) dt, p ||f || := hf, f i, Rd

and

[f, g](t) := [f, g]I (t). The Fourier transform of a function f will be denoted by fb. If f ∈ L(Rd ), Z b f (x) := e−i2πx·t f (t) dt. Rd

d

For every j ∈ Z and k ∈ Z the dilation operator DjA and the translation operator TkB are defined on L2 (Rd ) by DjA f (t) := |a|j/2 f (Aj t) and TkB f (t) := f (t + Bk). In particular, Tk f (t) := TkI f (t). 1

31

32

ZALIK: ORTHONORMAL WAVELET SYSTEMS...

A function f will be called BZd –periodic if it is defined on Rd and TkB f = f for every k ∈ Zd . A set S ⊂ L2 (Rd ) is called B shift–invariant if f ∈ S implies that TkB f ∈ S for every k ∈ Zd . If B = I, then we speak of a Zd – periodic function f and of a shift – invariant space S, omitting mention of the matrix I. If f is a BZd –periodic function and b := det B, then (1)

f B (t) := D1B f (t) = |b|1/2 f (Bt)

is Zd –periodic. Let u ⊂ L2 (Rd ); then T B (u) := {TkBu; u ∈ u, k ∈ Zd } and S B (u) := span T B (u), where the closure is in L2 (Rd ). In particular, T (u) := T I (u) and S(u) := S I (u) If u = {u1 , · · · , um } then S B (u) is called a finitely generated B shift–invariant space and the functions uℓ are called the generators of S B (u). In this case we will also use the symbols T B (u1 , · · · , un ) and S B (u1 , · · · , un ) to denote S B (u) and T B (u) respectively. Let H be a (separable) Hilbert space with inner product h·, ·i and norm k · k := h·, ·i1/2 . A sequence F = {fk , k ∈ Z} ⊂ H is called a Riesz sequence if there are constants 0 < A ≤ B such that for every sequence {ck , k ∈ Z} ⊂ ℓ2

2

X

X X

A |ck |2 ≤ c f |ck |2 . k k ≤ B

k∈Zd

k∈Zd k∈Zd F is called a Riesz basis of H if it is a Riesz sequence and its linear span is dense in H. The constants A and B are called (lower and upper) bounds of the Riesz basis. Clearly, every orthonormal basis is a Riesz basis with bounds A = B = 1. The theory of Riesz bases is discussed in, e.g., [3, 7]. Let Λ ⊂ Z and u = {uk ; k ∈ Λ} ⊂ S ⊂ L2 (Rd ). If S is a B shift–invariant space then u is called a basis generator of S, and we say that u provides a basis for S, if for every f ∈ S there are BZd –periodic functions pk , uniquely determined by f (up to a set of measure 0), such that X fb = pk u ck . k∈Λ

If u is a finite set, the uniqueness of the functions pk is equivalent to GB u (x) being nonsingular for almost every x ∈ Td . The theory of basis generators has been extensively developed by De Boor, DeVore, Ron and Shen in [1, 2, 5], under the assumption that B = I. In [8] we applied some of these results to the study of Schauder basis generators, Riesz basis generators and orthonormal basis generators, i.e. sets u such that T (u) is either a Schauder basis, a Riesz basis, or an orthonormal basis of S(u). Note that a Riesz basis generator is a basis generator. In the following section we will extend some of the results of [8] to the case of an arbitrary lattice BZd , where B is nonsingular. 2

ZALIK: ORTHONORMAL WAVELET SYSTEMS...

33

2. Some theorems on Riesz bases of translates and Linear Transformations Given a sequence of functions u := {u1 , · · · , um } in L2 (Rd ) and B ∈ Rd×d , by G [u1 , · · · , um ](x) or GB u (x) we will denote its B Gramian matrix, viz.  m GB bℓ , ubj ]B (x) . u (x) := [u B

ℓ,j=1

In particular,

Gu (x) := GIu (x). If u = {u1 , · · · un } and the functions uB ℓ are defined as in (1), then B uB := {uB 1 , · · · , un }.

(2)

We begin with the following simple but important result: Lemma 1. Let b ∈ L2 (Rd×d be a nonsingular matrix, u ∈ L2 (Rd ), D := (B−1 )T , and let uB (x) be given by (1). Then (a) T B (u) is an orthogonal basis of S B (u) if and only if T (uB ) is an orthogonal basis of S(uB ). (b) T B (u) is a Riesz basis in S B (u) if and only if T (uB ) is a Riesz basis in S(uB ). Moreover, T B (u) and T (uB ) have the same Riesz bounds. Proof. Part (a) follows from a change of variables, whereas part (b) follows from the following computations: X X X ck e2πis·Bk u b(Bs)||, ck TkB (u)|| = || ck e2πix·Bk u b(x)|| = |b|−1/2 || || k∈Zd

k∈Zd

k∈Zd

and X X X ck e2πis·Bk u b(Bs)||. ck e2πiDx·Bk u b(Bx)|| = |b|−1/2 || ck Tk (uB )|| = |b|−1/2 || || k∈Zd

k∈Zd

k∈Zd



The remaining results in this section will follow from Lemma 1 and the corresponding results in [8]. Theorem 2. Let u := {u1 , · · · , un } and v := {v1 , · · · , vm }, and let B ∈ Rd×d be nonsingular. Then (a) If T B (u) and T B (v) are Riesz bases of the same shift–invariant space S ⊂ L2 (Rd ), then n = m. (b) If T B (u) and T B (v) are Riesz sequences such that n = m and S B (u) ⊂ S B (v), then T B (u) is a Riesz basis of S B (v). (c) Let T B (u) and T B (v) be Riesz sequences in L2 (Rd ), and assume that S B (u) is a proper subset of S B (v). Then n < m and there are functions w1 , · · · , wm−n such that T B (w1 , · · · , wm−n ) is an orthonormal basis of the orthogonal complement S B (u)⊥ of S B (u) in S B (v), and T B (u1 , · · · , un , w1 , · · · , wm−n ) is a Riesz basis of S B (v1 , · · · , vm ). 3

34

ZALIK: ORTHONORMAL WAVELET SYSTEMS...

Proof. The assertion follows from [8, Theorem 1] applied to uB and vB .



From the identity B (x) = |b|−1/2 u uc b(Dx),

(3) we obtain

Theorem 3. Let u := {u1 , · · · , un } ∈ L2 (Rd ), h := {h1 , · · · , hm } ∈ L2 (Rd ), let B ∈ Rd×d be a nonsingular matrix, b := det B, D := (B−1 )T , and assume that S B (u) ⊂ S B (h). Then there are DZd –periodic functions qℓ,j (x) such that ubℓ (x) =

m X j=1

qℓ,j (x)hbj (x)

a.e.;

ℓ = 1, · · · , n.

Proof. The hypotheses imply that S(uB ) ⊂ S(hB ) Applying [8, Theorem F] we see that there are Zd –periodic functions pℓ,j (x) such that m X B (x) = B (x) a.e.; uc pℓ,j (x)hc ℓ = 1, · · · , n. j ℓ j=1

Setting qℓ,j (x) := pℓ,j (BT x) and applying (3) to uj and hj , the assertion follows.  The DZd –periodic matrix

 n,m QD (x) := qℓ,j (x)

ℓ,j=1

will be called a transition matrix from the sequence T B (h) to the sequence T B (u). If h is a basis generator of S B (h), then QD (x) is unique (up to a set of measure 0). Lemma 4. Let B ∈ Rd×d be nonsingular and u = {u1 , · · · , um } ⊂ L2 (Rd ). Then T B (u) is an orthonormal sequence in L2 (Rd ) if and only if GD u (x) = |b|I a.e. In particular if n = 1, then T B (u) is an orthonormal sequence in L2 (Rd ) if and only if X |b u(x + Dk)|2 = |b|. k∈Zd

Proof. From (3) we deduce that GuB (x) = |b|−1 GD u (Dx),

(4)

and the assertion follows from, e.g. [8, Lemma D]. d×d

 −1 T

B

Lemma 5. Let B ∈ R be nonsingular, D := (B ) , and assume that T (u) and T B (v1 , · · · , vm ) are orthonormal sequences in L2 (Rd ), and that there are DZd – periodic functions pℓ such that m X u b(x) = pℓ (x)vbℓ (x) a.e. ℓ=1

Then

m X

|pℓ (x)|2 = 1

ℓ=1

4

a.e.

ZALIK: ORTHONORMAL WAVELET SYSTEMS...

Proof. The hypotheses imply that m X u b(Dx) = pℓ (Dx)vbℓ (Dx)

35

a.e.

ℓ=1

Setting qℓ (x) := pℓ (Dx) and applying (3) to u and the functions vℓ we see that m X u bB (x) = qℓ (x)vbℓ B (x) a.e. ℓ=1

Since T (u ) and are orthonormal sequences in L2 (Rd ), and the functions qℓ (x) are Z –periodic, the assertion follows by an application of [8, Lemma E].  B

T (v1B , · · · d

B , vm )

Recall that if u is a finite set of functions such that T (u) is a Riesz basis, then Gu is positive definite for almost every x ∈ Td . Thus the square root of Gu (i.e. the unique positive definite matrix Hu such that Hu2 = Gu ) exists for almost every x ∈ Td . Thus, (4) implies that also the square root of GD u exists. Proposition 6. Let B ∈ Rd×d be nonsingular, and assume that u := {u1 , · · · , um } ⊂ L2 (Rd ) is such that T B (u) is a Riesz sequence. Let  −1/2 RD (x) := GD u (x) and

T 1/2 D T c1 (x), · · · , hc (h R (x)(c u1 (x), · · · , uc m (x)) := |b| m (x)) . B Then T (h) is an orthonormal basis of S (u). B

Proof. Lemma 1 implies that T (uB ) is a Riesz sequence. The hypotheses and (3) imply that B (x), · · · , h c B (x))T = |b|1/2 RD (Dx)(b (hc uB (x), · · · , u bB (x))T . m

1

1

m

But (4) implies that

−1/2

|b|1/2 RD (Dx) = [GuB (x)] . Thus, [8, Proposition G] implies that T (hB ) is an orthonormal basis of S(hB ), and the assertion follows from Lemma 1.  Theorem 7. Let B ∈ Rd×d be nonsingular, let D := (B−1 )T , assume that T B (h) is an orthonormal sequence in L2 (Rd ), that u is a set of functions such that S B (u) ⊂ S B (h), and let QD (x) denote the transition matrix from T B (h) to T B (u). Then (5)

D D ∗ GD u (x) = |b|Q (Dx)(Q (Dx))

a.e.,

and the following statements are equivalent: (a) T B (u) is a Riesz basis of S B (h) with bounds 0 < A ≤ B. (b) r = m and for almost every x ∈ Td
−1 ||GD and || GD || ≤ |b|−1 A−1 . u (x)|| ≤ |b|B u (x) (c) r = m and for almost every x ∈ Td ||QD (x)|| ≤ B 1/2


−1 || QD (x) || ≤ A−1/2 .

and

In particular, T B (u) is an orthonormal basis of S B (h) if and only if r = m and d GD ψ (x) = |b|I for almost every x ∈ T , or, equivalently, if and only if r = m and D Q (x) is a unitary matrix for almost every x ∈ Td . 5

36

ZALIK: ORTHONORMAL WAVELET SYSTEMS...

Proof. The hypotheses imply that T (hB ) is an orthonormal sequence in L2 (Rd ) and that S(uB ) ⊂ S(hB ). If Q(x) denotes the transition matrix from T (hB ) to T (uB ) then [8, Theorem 5] implies that GuB (x) = Q(x) (Q(x))∗

a.e.

By definition,

or, from (3),

c B (x), · · · , u c B (x), · · · , h c B (x))T = Q(x)(h B (x))T (uc m m 1 1

T T c c (c u1 (Dx), · · · , uc m (Dx)) = Q(x)(h1 (Dx), · · · , hm (Dx)) .

This implies that (6)

Q(x) = QD (Dx)

and (5) follows from (4) and (6). Assume now that (a) holds; then T (uB ) is a Riesz basis of S(uB ) with bounds 0 < A ≤ B, and [8, Theorem 5] implies that r = m and for almost every x ∈ T d (7)

||GuB (x)|| ≤ B

and || (GuB (x))

−1

|| ≤ A−1 ,

and (b) follows from (4). If (b) is satisfied, then (4) implies (7), and (c) follows from [8, Theorem 5] and (6). Finally, if (c) is satisfied then (6) implies that ||Q(x)|| ≤ B1/2 and ||(Q(x))−1 || ≤ −1/2 A ; thus [8, Theorem 5] implies that T (uB ) is a Riesz basis of S(hB ) with bounds A and B, and (a) follows from Lemma 1. Let us now prove the last paragraph in the statement of the theorem: T B (u) is an orthonormal basis of S(hB ) if and only if T (uB ) is an orthonormal basis of S(hB ), and [8, Theorem 5] and (4) imply that this is equivalent to GD u (Dx) = |b|I. Finally, (4) and (5) imply that T (uB ) is an orthonormal basis of S(hB ) if and only if QD (x) is unitary.  3. Wavelets A ∈ Rd×d is called a dilation matrix preserving the lattice Zd if AZd ⊂ Zd and all its eigenvalues have modulus greater than 1. These conditions imply that A ∈ Zd×d , and that if a := det A then |a| is an integer larger than 1 (cf. Madych [4]). Assume that A ∈ Rd×d is a dilation matrix preserving the lattice Zd . A coset of AZd is a set of the form j + AZd = {j + Ar; r ∈ Zd }, where j ∈ Zd . An element of a coset is called a representative of the coset. Any pair of cosets are either identical or disjoint, and the union of all disjoint cosets equals Zd . There are exactly |a| disjoint cosets. (cf. Wojtaszczyk [6]). The collection of all disjoint cosets is denoted by Zd /AZd . A set J ⊂ Zd is said to be a full collection of representatives of Zd /AZd if it contains exactly |a| elements and [ (j + AZd ) = Zd . j∈J

6

ZALIK: ORTHONORMAL WAVELET SYSTEMS...

Theorem 8. Let B ∈ Rd×d be a nonsingular matrix, u = {u1 , · · · , un } ⊂ L2 (Rd ), and assume that T B (u) is an orthonormal sequence. Let A be a dilation matrix preserving the lattice Zd , a := det A, m := |a|n, let J be a full collection of representatives of Zd /AZd . For x > 0 define I(x) := [1, x] ∩ Z, and let p = (p1 , p2 ) : I(m) −→ I(n) × J be a bijection. If vℓ (t) := |a|1/2 up1 (ℓ) (At + Bp2 (ℓ)) and v := {v1 , · · · , vm }, then T B (v) is an orthonormal basis of S B (A; u), and every Riesz basis generator of S B (A; u) has exactly m functions. Proof. The hypotheses imply that T (uB ) is an orthonormal basis of S(uB ), and from [8, Theorem 3] we conclude that if wℓ (t) := |a|1/2 uB p1 (ℓ) (At + p2 (ℓ)) and W := {w1 , · · · , wm }, then T (w) is an orthonormal basis of S(A; uB ). Let b := det B and L : S B (A, u) −→ S(A; uB );

Lf := f B .

Since LTkvℓ = Tk wℓ , proceeding as in the proof of Lemma 1 we see that L is an isometry from S(A, uB ) onto S B (A; u), and the assertion follows.  Note. There is a typographical error in the statement of [8, Theorem 3]: The range of the function p described in that theorem is I(n) × J. Let A ∈ Rd×d be a dilation matrix preserving the lattice Zd , and assume that B ∈ Rd×d is nonsingular. A multiresolution analysis (MRA) of multiplicity n in L2 (Rd ) (generated by A and B) is a sequence {Vj ; j ∈ Z} of closed linear subspaces of L2 (Rd ) such that: (i) Vj ⊂ Vj+1 for every j ∈ Z. (ii) For every j ∈ Z, f (t) ∈ Vj if and only if f (At) ∈ Vj+1 . (iii)

S

j∈Z

Vj is dense in L2 (Rd ).

(iv) There are functions u := {u1 , · · · , un } such that T B (u) is an orthonormal basis of V0 . From Proposition 6 we deduce that the condition that T B (u) be an orthonormal basis may be replaced by the condition that T B (u) be a Riesz basis. It follows from the definition of multiresolution analysis that there are DZd – periodic functions pℓ,j ∈ L2 (Td ) such that the functions uℓ satisfy the scaling identity n X ubℓ (AT x) = pℓ,j (x)ubj (x), j, ℓ = 1, · · · , n a.e., j=1

The functions uℓ are called scaling functions for the multiresolution analysis, and the functions pℓ,j are called the low pass filters associated with u. 7

37

38

ZALIK: ORTHONORMAL WAVELET SYSTEMS...

Assume that A is a dilation matrix preserving the lattice Zd and that B ∈ Zd×d is nonsingular. A finite set of functions ψ = {ψ1 , · · · , ψm } ∈ L2 (Rd ) will be called an orthonormal or Riesz wavelet system if the affine sequence [ T B (Aj ; ψ) = {DjA TkB ψℓ ; j ∈ Z, k ∈ Zd , ℓ = 1, · · · , m} j∈Z

is respectively an orthonormal basis or a Riesz basis of L2 (Rd ). If d = 1 we omit the word “system”. If we need to emphasize the connection with the matrices A and B we will say that the wavelet system is generated by A and B. Let ψ := {ψ1 , · · · , ψm } be a Riesz wavelet system in L2 (Rd ) P generated by matrices A and B; for j ∈ Z we define Pj := S B (Aj ; ψ) and Vj := r 0:

x;y jx yj

We give Theorem 1. Let f 2 C N (R) ; N function of compact support x 2 R: Suppose '

0: Put

1; x 2 R; and k 2 Z: Let ' be a bounded 1 P [ a; a] ; a > 0 such that ' (x j) = 1; all j= 1

(Bk f ) (x) = (Dk f ) (x) =

1 X

j 2k

f

j= 1 1 X

kj

' 2k x

(f ) ' 2k x

j ;

j ;

j= 1

where kj (f ) =

n X

j r + k 2k 2 n

wr f

r=0

n 2 N; wr

0;

n P

;

wr = 1: Then

r=0

E1k (x) = (Dk f ) (x) n P

r=1

N X Bk f (i) (x) 2ki ni i! i=1

(Bk f ) (x)

n X r=1

wr r

i

!

wr rN ! 1 f (N ) ; 2krn :

2kN nN N !

(1)

Remark. (i) Given that f (N ) is continuous and bounded or uniformly continuous we have that ! 1 f (N ) ; 2krn < 1; and E1k (x) ! 0; x 2 R; as k ! 1:

2

44

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

(ii) One also has kE1k k1 = Dk f n P

r=1

N X Bk f (i)

Bk f

i=1

n X

2ki ni i!

wr r

r=1

i

!

1

wr rN ! 1 f (N ) ; 2krn :

2kN nN N !

Under the assumptions of (i) we get also kE1k k1 ! 0; as k ! 1: (iii) By (1) we get ! N n X Bk f (i) (x) X i j(Dk f ) (x) (Bk f ) (x)j wr r 2ki ni i! r=1 i=1 + Given that f (i) and

1

n P

r=1

wr rN ! 1 f (N ) ; 2krn :

2kN nN N !

(2)

< 1; for i = 1; : : : ; N; we obtain Bk f (i) (x)

j(Bk f ) (x)

N X f (i) 1 2ki ni i! i=1

(Dk f ) (x)j

+

n P

r=1

n X

wr r

r=1

i

f (i)

1

;

!

wr rN ! 1 f (N ) ; 2krn 2kN nN N !

:

(3)

Clearly we get j(Bk f ) (x)

N X f (i) 1 ! 1 f (N ) ; 21k + 2ki i! 2kN N ! i=1

(Dk f ) (x)j

=: T1

and kBk f

Dk f k1

T1 :

So as k ! 1; we have kBk f Dk f k1 ! 0: Proof of Theorem1. Because f 2 C N (R) ; N n X r=0

wr f

j r + k k 2 2 n

j 2k

=f

n X

+

r=0

wr

+

N X f (i) i=1

Z

(j=2k )+

j=2k

3

j 2k

i!

r 2k n

n X

wr

r=0

f (N ) (t)

(4) 1 we have ri 2ki ni f (N ) j=2k

j 2k

+ 2krn t (N 1)!

N 1

dt:

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

45

Hence we get 1 X

k kj (f ) ' 2 x

j= 1 N X

+

n 1 X X + wr ' 2k x r=0

j 2k

f (i)

j= 1

Z

j

j 2k

f

j= 1

1 P

i=1

1 X

j =

' 2k x

j

n X

2ki i!ni

(j=2k )+

r 2k n

f

' 2k x

wr r

i

r=0

(N )

(t)

f

(N )

j

! j 2k

k

j=2

j=2k

j= 1

+ 2krn t (N 1)!

N 1

dt:

So, we observe that (Dk f ) (x)

N X Bk f (i) (x) 2ki ni i! i=1

(Bk f ) (x)

n X

wr r

r=0

i

!

= R1 ;

where R1 =

n X

wr

r=1

1 X

k

' 2 x

j

Z

(j=2k )+

r 2k n

f

(N )

(t)

f

(N )

j 2k

k

j=2

j=2k

j= 1

+ 2krn t (N 1)!

Set jr

=

Z

(j=2k )+

r 2k n

f

(N )

(t)

f

(N )

j=2

j 2k

k

j=2k

So that

n X

jR1 j

wr

r=1

1 X

' 2k x

j

+ 2krn t (N 1)!

N 1

dt :

jr :

j= 1

Next we see that jr

Z

(j=2k )+

r 2k n

f (N ) (t)

j 2k

f (N ) j=2k

j=2k

Z

(j=2k )+

r 2k n

!1 f

(N )

k

; t

(j=2 )

j 2k

j=2k

! 1 f (N ) ; = ! 1 f (N ) ;

r 2k n r 2k n

Z

(j=2k )+

r 2k n

j 2k

j=2k N r 2k n

N!

+ 2krn t (N 1)!

+ 2krn t (N 1)!

:

That is jr

+ 2krn t (N 1)!

! 1 f (N ) ; 4

r 2k n

N r 2k n

N!

:

N 1

dt

N 1

dt N 1

dt

N 1

dt:

46

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

So we have found that n P

r=1

jR1 j

wr rN ! 1 f (N ) ; 2krn :

2kN nN N !

The proof of the theorem is complete. We continue with Theorem 2. Let f 2 C N (R) ; N

1; x 2 R; and k 2 Z: Let ' be a bounded 1 P [ a; a] ; a > 0 such that ' (x j) = 1; all

function of compact support x 2 R: Suppose '

j= 1

0: Put

(Bk f ) (x) = (Ck f ) (x) =

1 X

j 2k

f

j= 1 1 X

kj

' 2k x

j ;

(f ) ' 2k x

j ;

j= 1

where k kj (f ) = 2

Z

2

2

k

(j+1)

f (t) dt = 2k kj

Z

2

k

f

t+

0

j 2k

dt:

Then E2k (x) = (Ck f ) (x)

(Bk f ) (x)

! 1 f (N ) ; 21k : 2kN (N + 1)!

N X Bk f (i) (x) 2ki (i + 1)! i=1

(5)

Remark. (i) Given that f (N ) is continuous and bounded or uniformly continuous we have that ! 1 f (N ) ; 21k < 1; and E2k (x) ! 0; x 2 R; as k ! 1: (ii) One also has kE2k k1 = Ck f

Bk f

! 1 f (N ) ; 21k : 2kN (N + 1)!

N X Bk f (i) ki 2 (i + 1)! i=1

1

Under the assumptions of (i) we get also kE2k k1 ! 0; as k ! 1: (iii) By (5) we get j(Bk f ) (x)

(Ck f ) (x)j

N X ! 1 f (N ) ; 21k Bk f (i) (x) + : 2ki (i + 1)! 2kN (N + 1)! i=1

5

(6)

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

Given that f (i) j(Bk f ) (x)

47

< 1; for i = 1; : : : ; N; we obtain

1

N X

f (i) 1 ! 1 f (N ) ; 21k + =: T2 ; 2ki (i + 1)! 2kN (N + 1)! i=1

(Ck f ) (x)j

and kBk f

Ck f k1

T2 :

(7)

So as k ! 1; we get kBk f Ck f k1 ! 0: Proof of Theorem2. Because f 2 C N (R) ; N f

t+

j 2k

j 2k

=f

+

Z

+

N X f (i) i=1

j 2k

i!

1 we have

ti

t+(j=2k )

f (N ) (s)

N 1

t + 2jk (N

f (N ) j=2k

j=2k

s 1)!

ds:

Hence we get k kj (f ) = 2

Z

k

2

f

t+

0

k

+2

Z

Z

k

2

j 2k

j 2k

dt = f

+

t+(j=2k )

f

(N )

(s)

f

N X f (i) 2jk 2ki (i + 1)! i=1

(N )

j=2

j=2k

0

N 1

t + 2jk (N

k

s 1)!

!

ds dt:

Hence we get 1 X

kj

(f ) ' 2k x

j =

j= 1

+

1 X

k

k

' 2 x

j 2

Z

1 X

j= 1 2

k

Z

' 2k x

j +

N 1 X X f (i) i=1 j= 1

t+(j=2k )

f

(N )

(s)

f

(N )

j=2

j 2k 2ki

t + 2jk (N

k

j=2k

0

j= 1

j 2k

f

' 2k x (i + 1)!

j

N 1

s 1)!

!

ds dt:

So, we see that (Ck (f )) (x)

N X Bk f (i) (x) = R2 ; 2ki (i + 1)! i=1

(Bk (f )) (x)

where R2 =

1 X

k

' 2 x

k

j 2

Z

2

k

t+(j=2k )

f

(N )

(s)

f

(N )

j=2

k

j=2k

0

j= 1

Z

t + 2jk (N

Set j

(t) =

Z

t+(j=2k )

f

(N )

(s)

f

(N )

j=2k

6

k

j=2

t + 2jk (N

N 1

s 1)!

ds :

N 1

s 1)!

!

ds dt:

48

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

So that

1 X

jR2 j

k

k

' 2 x

j 2

Z

k

2

j

Next we observe that Z t+(j=2k ) f (N ) (s) j (t)

f

(N )

j=2

t+(j=2k )

! 1 f (N ) ; s

t + 2jk (N

(j=2k )

j=2k

Z

! 1 f (N ) ; t

t+(j=2k )

s 1)!

ds

N 1

s 1)!

ds

N 1

t + 2jk (N

s 1)!

! 1 f (N ) ; t

tN ; N!

j=2k

N 1

t + 2jk (N

k

j=2k

Z

(t) dt:

0

j= 1

ds

tN : N!

= ! 1 f (N ) ; t I.e. we get j

(t)

and k

2

Z

k

2

j

(t) dt

2

k

0

Z

2

k

0

k

tN dt N!

! 1 f (N ) ; t

2 1 ! 1 f (N ) ; k N! 2

Z

2

k

tN dt

0

1 2k ! 1 f (N ) ; k 2 = (N + 1)! 2 1 1 ! 1 f (N ) ; k = kN 2 (N + 1)! 2

k(N +1)

:

That is we obtain k

2

Z

2

k

j

(t) dt

0

1 1 ! 1 f (N ) ; k 2kN (N + 1)! 2

It is clear that jR2 j

:

! 1 f (N ) ; 21k : 2kN (N + 1)!

The proof of the theorem is now …nished. We continue with Theorem 3. Let f 2 C N (R) ; N function of compact support

1; x 2 R; and k 2 Z: Let ' be a bounded 1 X [ a; a] ; a > 0 such that ' (x j) = 1; all j= 1

7

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

x 2 R: Suppose '

49

0: Put 1 X

(Ck f ) (x) =

kj

(f ) ' 2k x

j ;

j= 1

where

Z

k kj (f ) = 2

(j+1)

f (t) dt = 2k kj

2

and

k

2

Z

2

k

f

t+

0

1 X

(Dk f ) (x) =

kj

(f ) ' 2k x

j 2k

dt;

j ;

j= 1

where kj

(f ) =

n X

wr f

r=0

n 2 N; wr

0;

n X

j r + k k 2 2 n

;

wr = 1: Then

r=0

E3k (x) = (Ck f ) (x) n X

wr

r=0

h

(Dk f ) (x) r N +1 n

N X Dk f (i) (x) 2ki (i + 1)! i=1 ! i r N +1 n

+ 1

2kN (N + 1)!

! 1 f (N ) ;

i+1

r n

1

1 2k

i+1

( 1)

r n

:

(8)

Remark (i) Given that f (N ) is continuous and bounded or uniformly continuous we have that ! 1 f (N ) ; 21k < 1; and E3k (x) ! 0; x 2 R; as k ! 1: (ii) One also has kE3k k1 = Ck f

Dk f

N X i=1

n X

wr

r=0

h

r N +1 n

Dk f (i) 2ki (i + 1)! r N +1 n

+ 1

2kN (N + 1)!

r n

1 i

!

i+1

i+1

( 1)

! 1 f (N ) ;

1 2k

r n

i+1 1

:

Under the assumption of (i) we get also kE3k k1 ! 0; as k ! 1: (iii) By (8) we get j(Ck f ) (x)

+

(Dk f ) (x)j

N X i=1

n X r=0

wr

h

r N +1 n

Dk f (i) (x) 2ki 1 (i + 1)! r N +1 n

+ 1

2kN (N + 1)! 8

i

!

! 1 f (N ) ;

i+1

1 2k

:

(9)

50

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

Clearly then j(Ck f ) (x)

N X

(Dk f ) (x)j

! 1 f (N ) ; 21k Dk f (i) (x) + : 2ki 1 (i + 1)! 2kN 1 (N + 1)!

i=1

Given that f (i)

(10)

< 1; for i = 1; : : : ; N; we get

1

Dk f (i) (x)

f (i)

1

;

and j(Ck f ) (x)

N X

(Dk f ) (x)j

i=1

2ki

f (i) 1 ! 1 f (N ) ; 21k + =: T3 1 (i + 1)! 2kN 1 (N + 1)!

(11)

(Dk f ) (x)k1

(12)

and k(Ck f ) (x)

T3 :

So as k ! 1; we get kCk f Dk f k1 ! 0: Proof of Theorem 3. Because f 2 C N (R) ; N f

t+

j 2k

j r + k 2k 2 n

=f

+

Z

+

N X f (i)

+ i!

i=1

t+(j=2k)

f (N ) (s)

(j=2k )+

j 2k

r 2k n

1 we have i

r

t

2k n t + 2jk (N

j r + k k 2 2 n

f (N )

r 2k n

N 1

s 1)!

ds:

Hence we get k kj (f ) = 2

Z

2

k

f

j 2k

t+

0

+

N X n X

wr f

(i)

+

r=0

n X

k

wr 2

Z

0

2

k

j r + k 2k 2 n

wr f

r=0

j 2k

+

r 2k n

i!

i=1 r=0

n X

dt =

2k

Z

2

k

t

0

Z

i

r 2k n

t+(j=2k )

(j=2k )+

f r 2k n

9

(N )

(s)

f

(N )

dt j r + k k 2 2 n

t + 2jk (N

N 1

s 1)!

!

ds dt:

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

51

That is we have 1 X

k kj (f ) ' 2 x

1 X

j =

j= 1

kj

j= 1

+

1 X

N X j=

+

i=1 1 X 2

1

k

2ki

t + 2jk (N

j

n X

r n

1

(i + 1)! j

i+1

i+1

( 1)

wr 2k

r=0

Z

k

t+(j=2 )

f (N ) (s)

(j=2k )+

0

j

f (i) ' 2k x

kj

' 2k x

j= 1

Z

(f ) ' 2k x

N 1

s 1)!

r j + k 2k 2 n

f (N )

r 2k n

!

ds dt:

Consequently we get N X Dk f (i) (x) 2ki (i + 1)! i=1

(Ck f ) (x) (Dk f ) (x)

1

r n

i+1

i+1

( 1)

r n

i+1

= R3 ;

where R3 =

1 X

' 2k x

2

n X

wr 2k

r=0

j= 1

Z

j

Z

k

k

t+(j=2 )

f

(j=2k )+

0

(N )

(s)

f

(N )

r 2k n

t + 2jk (N

r j + k 2k 2 n

N 1

s 1)!

ds dt:

Set jr (t) =

Z

t+(j=2k )

f (N ) (s)

(j=2k )+

f (N )

r 2k n

t + 2jk (N

j r + k 2k 2 n

So that jR3 j

1 X

j= 1

' 2k x

j

n X r=0

Next we observe that

10

wr 2k

Z

0

2

k

jr

(t) dt:

N 1

s 1)!

!

ds :

r n

i+1

52

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

r 2k n

(i) Case of Z

jr (t)

t: So we have

t+(j=2k )

f (N ) (s)

(j=2k )+

Z

r 2k n

t+(j=2k )

(j=2k )+

!1 f

!1 f

r 2k n

(N )

(N )

Z

r k 2 n

;t

t+(j=2k )

(j=2k )+

r 2k n

r k 2 n

t + 2jk (N

t + 2jk (N

s 1)!

N! t 2krn N!

1 2k

jr

(t) =

Z

(j=2k )+

(j=2k )+

r 2k n

f

(N )

(s)

f

(N )

r 2k n

f

(N )

Z (j=2k )+ kr 2 n

(s)

f

N r 2k n

t

1 2k

N!

ds

:

! 1 f (N ) ;

r 2k n

r ; k 2 n

! 1 f (N ) ;

1 2k

t t

t

t + 2jk (N 1)!

s

N 1

t + 2jk (N 1)!

t+(j=2k ) r 2k n

t + 2jk (N 1)!

s

s

Z (j=2k )+ kr s 2 n

ds

N

N!

r 2k n

t

N!

N

:

So we derive jr (t)

! 1 f (N ) ;

r 2k n

1 2k

t

N

:

N!

Therefore we have found jr (t)

! 1 f (N ) ; 21k N! 11

t

t + 2jk (N 1)!

s

j r + k k 2 2 n

j r + k k 2 2 n

! 1 f (N ) ;

t+(j=2k )

= !1 f

N 1

j r + k 2k 2 n

(N )

t+(j=2k )

(N )

ds

t: We have

t+(j=2k )

Z

N 1

s 1)!

:

! 1 f (N ) ;

jr (t) r 2k n

ds

N

So we obtain

(ii) Case of

s 1)!

N r 2k n

t

;t

! 1 f (N ) ;

j 2k

! 1 f (N ) ; s

N 1

t + 2jk (N

j r + k k 2 2 n

f (N )

r 2k n

N

:

N 1

ds

N 1

ds

N 1

ds

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

53

Furthermore we see that Z

k

2

jr

Z

! 1 f (N ) ; 21k N!

(t) dt

0

=

dt

2k n

r 2k n

N

r 2k n

0

"

! 1 f (N ) ; 21k = (N + 1)!

N

r

t

"0Z

! 1 f (N ) ; 21k = N!

k

2

t

N +1

r k 2 n

1 2k

+

! 1 f (N ) ; 21k 1 (N + 1)! 2k(N +1)

r n

dt +

Z

N +1

r n

+ 1

r 2k n

t r 2k n

r k 2 n

N +1

k

2

#

N

dt

#

N +1

:

Thus 2k

Z

2

k

jr (t) dt

0

! 1 f (N ) ; 21k 2kN (N + 1)!

r n

N +1

+ 1

N +1

r n

:

Finally we derive that jR3 j

! 1 f (N ) ; 21k 2kN (N + 1)!

!

n X

N +1

r n

wr

r=0

+ 1

r n

N +1

;

proving the theorem. We continue with Theorem 4. Let f 2 C N (R) ; N 1; x 2 R and k 2 Z; also f (i) 1 < 1; i = 1; : : : ; N: Let ' be a bounded function of compact support [ a; a] ; a > 0 1 X such that ' (x j) = 1 all x 2 R: Suppose ' 0 and ' is Lebesgue j= 1

measurable (then

R1

1

' (x) dx = 1): De…ne

'kj (x) := 2k=2 ' 2k x j all k; j 2 Z; Z 1 f; 'kj = f (t) 'kj (t) dt; 1

and

(Ak f ) (x) = =

1 X

j= 1 1 X

j= 1

f; 'kj 'kj (x) Z

also de…ne (Bk f ) (x) =

1

f

1 1 X

u ' (u 2k

f

j= 1

12

j 2k

j) du ' 2k x

' 2k x

j :

j ;

54

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

Then j(Ak f ) (x)

(Bk f ) (x)j

kAk f

Bk f k1

N X f (i) 1 i aN a a + ! f (N ) ; k ; ki i! kN N ! 1 2 2 2 i=1

x 2 R: So as k ! 1; we get kAk f Bk f k1 ! 0: 1 X R1 Proof. By ' 2k x j = 1 we have that 1 ' (u Notice that

(Ak f ) (x)

(13)

j) du = 1:

j= 1

1 X

(Bk f ) (x) = =

j= 1 1 X

=

1

f

Z

j= 1

f

u ' (u 2k

j 2k

' 2k x

f

u ' (u 2k

1

j= 1 Z 1 1 X

j= 1 1 X

j 2k

f

j= 1

Z

j= 1 1 X

=

1 X

f; 'kj 'kj (x)

1

1

f

1

u 2k

' 2k x

j) du ' 2k x

j

j

j

j) du

f

j 2k

f

j 2k

' 2k x

j

j) du ' 2k x

' (u

j :

That is (Ak f ) (x) (Bk f ) (x) =

1 X

j= 1

Z

1

f

1

u 2k

f

j 2k

j) du ' 2k x

' (u

[ a; a] we have that ' (u j) is nonzero when a u j a; a u j + a: Hence Z j+a 1 X j u f ' (u j) du ' 2k x (Ak f ) (x) (Bk f ) (x) = f k k 2 2 j a j= 1

j :

By supp ' that is when j

Next we see that f where R4 =

u 2k Z

f

j 2k

=

N X f (i) i=1

u=2k

f (N ) (t)

f (N )

j=2k

13

j 2k

i!

j 2k

i

(u

j) 2ki u 2k

(N

+ R4 ; t

N 1

1)!

dt:

j :

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

(i) Case of j

55

u: We have Z

jR4 j

u=2k

f (N ) (t)

j=2k

Z

u=2k (N )

!1 f

; t

!1 f

;

= ! 1 f (N ) ; (ii) Case of j

(u

Z

j) 2k

(u

u 2k

u=2k

t

(N

dt

N 1

t

u 2k

N 1

1)!

(N

j=2k

j)

t

(N

j 2k

j=2k (N )

u 2k

j 2k

f (N )

dt

1)! N 1

dt

1)!

N

(u j) : 2kN N !

2k

u: We have Z

jR4 j =

j=2k

f (N ) (t)

f (N )

(N )

(N )

u=2k

Z

j=2k

f

(t)

f

u=2k

Z

j=2k

! 1 f (N ) ;

u=2k

! 1 f (N ) ; = ! 1 f (N ) ;

j

u 2k

j

u=2k

u 2k

(N

(N

1)!

u N 1 2k

t (N

1)!

u N 1 2k

t (N

1)!

u N 1 2k

t

j 2k

j t 2k Z j=2k

u N 1 2k

t

j 2k

1)!

dt

dt

dt

dt

N

(j u) : 2kN N !

So we have proved that jR4 j i.e.

! 1 f (N ) ;

ju

jj 2k

N

ju jj 2kN N !

! 1 f (N ) ;

a 2k

aN ; 2kN N !

aN a ! 1 f (N ) ; k : 2kN N ! 2 Furthermore we observe that jR4 j

Z

j+a

j a

u f k 2

f

j 2k

' (u

Z j+a N X f (i) j=2k j) du = (u 2ki i! j a i=1 Z j+a + R4 ' (u j) du:

i

j) ' (u

j) du

j a

Therefore Z

j+a

j a

f

u 2k

f

j 2k

' (u

j) du

14

N X f (i) 1 i aN a a + ! f (N ) ; k : ki i! kN N ! 1 2 2 2 i=1

56

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

The last proves the theorem. We continue with Theorem 5. Let f 2 C N (R) ; N 1; x 2 R and k 2 Z; also f (i) 1 < 1; i = 1; : : : ; N: Let ' be a bounded function of compact support [ a; a] ; a > 0 1 X such that ' (x j) = 1 all x 2 R: Suppose ' 0 and ' is Lebesgue j= 1

measurable (then

R1

1

' (x) dx = 1): De…ne

'kj (x) := 2k=2 ' 2k x f; 'kj =

Z

1

j

all k; j 2 Z;

f (t) 'kj (t) dt;

1

and 1 X

(Ak f ) (x) = =

f; 'kj 'kj (x)

j= 1 1 X j= 1

Z

1

Also de…ne (Dk f ) (x) =

u ' (u 2k

f

1 1 X

kj

j) du ' 2k x

(f ) ' 2k x

j :

j ;

j= 1

where kj

(f ) =

n X

wr f

r=0

n X

n 2 N; wr

0;

j(Ak f ) (x)

(Dk f ) (x)j

r j + k 2k 2 n

;

wr = 1: Then

r=0

kAk f

Dk f k1

(14)

N N X f (i) 1 (a + 1) (a + 1) i (a + 1) + ! 1 f (N ) ; ki i! kN 2 N !2 2k i=1

So as k ! 1; we get kAk f Dk f k1 ! 0: 1 X R1 Proof. By ' 2k x j = 1 we have that 1 ' (u j= 1

15

j) du = 1:

:

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

57

Notice that (Ak f ) (x)

1 X

(Dk f ) (x) = =

j= 1 1 X

2k=2 f; 'kj Z

1 X

j= 1

=

' 2k x j " n 1 X X

wr

Z

wr

Z

r=0

' 2k x

j

u f k ' (u 2 1

1

j r + k 2k 2 n

wr f

r=0

1

1

j

n X

j) du

u f k 2

j) du

f

j r + k 2k 2 n

j r + k 2k 2 n

f

u 2k

j a

r=0

' 2k x

j r + k 2k 2 n

f

j :

Next we see that f

u 2k

j r + k k 2 2 n

f

=

N X f (i)

j 2k

+ i!

i=1

r 2k n

u

r i n

j 2ki

+ R5 ;

where R5 =

Z

u 2k j 2k

+

f

(N )

(t)

f

(N )

r 2k n

u: We have easily that

jR5 j

! 1 f (N ) ; 21k u j N !2kN

16

u 2k

j r + k k 2 2 n

r n

(i) Case j +

' (u

! #

j) du

#

j :

That is, by the compact support of ' we have " n Z j+a 1 X X (Ak f ) (x) (Dk f ) (x) = wr f j= 1

(f ) ' 2k x

(f ) ' 2k x

u f k ' (u 2 1

r=0

j= 1

kj

1

' 2k x j " n 1 X X

j= 1

=

kj

j= 1

j= 1

=

1 X

f; 'kj 'kj (x)

r n

u

t

(N

j

r n

N 1

1)!

N

:

dt:

' (u

j) du

#

58

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

(ii) Case j + Z

jR5 j =

Z

r n

u: We have that

j 2k

+

r 2k n

j r + k 2k 2 n

f (N )

u 2k j 2k

+

r 2k n

j r + k k 2 2 n

f (N )

u 2k

! 1 f (N ) ; 21k j + N !2kN

r n

u

(N

dt

1)!

u N 1 2k

t

f (N ) (t)

(N

1)!

dt

N

r n

j+

u N 1 2k

t

f (N ) (t)

u

:

So we have proved that ! 1 f (N ) ; 21k j + N !2kN

jR5 j

r n

u

i.e.

j+

N

r n

u

! 1 f (N ) ; a+1 N 2k (a + 1) ; kN N !2

N

jR5 j

a+1 (a + 1) ! 1 f (N ) ; k N !2kN 2

:

Furthermore we observe that Z

j+a

f

j a

u 2k

j r + k 2k 2 n

f

' (u

j) du =

N X f (i) i=1

' (u

j 2k

+

r 2k n

i!2ki j) du +

Z

j+a

u

r n

j

j a

Z

j+a

j a

R5 ' (u

j) du:

Therefore Z

j+a

f

j a

u 2k

j r + k k 2 2 n

f

' (u

j) du

N X f (i) 1 i (a + 1) ki i! 2 i=1 N

+

(a + 1) (a + 1) ! 1 f (N ) ; N !2kN 2k

The last proves the theorem. We continue with Theorem 6. Let f 2 C N (R) ; N 1; x 2 R and k 2 Z; also f (i) 1 < 1; i = 1; : : : ; N: Let ' be a bounded function of compact support [ a; a] ; a > 0 1 X such that ' (x j) = 1 all x 2 R: Suppose ' 0 and ' is Lebesgue j= 1

measurable (then

R1

1

' (x) dx = 1): De…ne

'kj (x) := 2k=2 ' 2k x

17

j

all k; j 2 Z;

:

i

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

Z

f; 'kj = and 1 X

(Ak f ) (x) = =

1

59

f (t) 'kj (t) dt;

1

f; 'kj 'kj (x)

j= 1 1 X

Z

j= 1

1

u ' (u 2k

f

1

Also de…ne

1 X

(Ck f ) (x) =

kj

j) du ' 2k x

(f ) ' 2k x

j :

j ;

j= 1

where kj

(f ) = 2

k

Z

j(Ak f ) (x)

(Ck f ) (x)j

(j+1)

f (t) dt = 2

k

kj

2

Then

k

2

i=1

2

k

f

t+

0

kAk f

N X

Z

Ck f k1

j 2k

(15) N

(i)

f (a + 1) (a + 1) i 1 (a + 1) + kN ! 1 f (N ) ; i!2ki 2 N! 2k

So as k ! 1; we get kAk f Ck f k1 ! 0: 1 X R1 Proof. By ' 2k x j = 1 we have that 1 ' (u Notice that (Ak f ) (x)

1 X

(Ck f ) (x) =

2k=2 f; 'kj

j= 1

=

1 X

j= 1

=

1 X

j= 1

=

1 X

j= 1

=

1 X

j= 1

"

"

"

Z 2k

u f k ' (u 2 1 2

k

0

2k

Z

2

Z

k

0

2

k

kj

1

Z

Z

0

:

j) du = 1:

j= 1

1 X

2

k

Z

(f ) ' 2k x 2k

j) du

1

1 1

f

Z

2

k

f

f

!

j t+ k 2 2k

j) du dt

Z

dt ' 2k x 2

k

f

0

u ' (u 2k

u f k 2

(f ) ' 2k x

j

j

0

u f k ' (u 2 1 1

kj

j= 1

1

Z

1 X

f; 'kj 'kj (x)

j= 1

=

dt:

j) du j t+ k 2

18

f

t+

' (u

j 2k

j t+ k 2 #

j dt ' 2k x

dt ' 2k x #

#

j) du dt ' 2k x

j

j :

j

60

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

That is, by the compact support of ' we have " Z k Z 1 2 j+a X (Ak f ) (x) (Ck f ) (x) = 2k f 0

j= 1

' 2k x

u 2k

j a

j t+ k 2

f

#

' (u

j) du dt

j :

Next we see that f

u 2k

f

R6 =

Z

where

(i) Case t +

u 2k

t+

f

j 2k

u : 2k

jR6 j =

Z

(N )

N X f (i) t + i! i=1

(s)

f

(N )

u : 2k t+

u 2k

j 2k

j 2k

u 2k u 2k

j t+ k 2

i

j 2k

t

N 1

s

(N

+ R6 ;

ds:

1)!

t

u 2k

j 2k

t N!

j N 2k

:

We have that f

f

(N )

u 2k

! 1 f (N ) ; t +

N 1

j t+ k 2

(N )

u 2k

t+

j 2k

We have easily that

! 1 f (N ) ;

j 2k

Z

=

j 2k

jR6 j (ii) Case t +

j 2k

t+

(s)

s 2uk (N 1)!

ds

N 1

j t+ k 2 j 2k

f

(N )

f

(N )

t+

u 2k

(s)

s 2uk (N 1)! u N 2k

j 2k

N!

ds

:

So we have proved that

jR6 j

!1 f

(N )

; t+

(j

t+

u) 2k

(j u) 2k

N

N!

! 1 f (N ) ;

N

(a + 1) 2k

(a + 1) : 2kN N !

I.e. we found that N

jR6 j

(a + 1) (a + 1) ! 1 f (N ) ; kN 2 N! 2k

:

Furthermore we observe that Z

j+a

j a

f

u 2k

f

t+

j 2k

' (u

j) du =

N X f (i) t + i! i=1

' (u

j 2k

j) du +

Z

j+a

j a

Z

j) 2k

i

t

j+a

j a

19

(u

R6 ' (u

j) du:

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

61

Therefore Z

j+a

j a

u f k 2

j t+ k 2

f

' (u

N X f (i) 1 i (a + 1) ki i!2 i=1

j) du

N

+

(a + 1) (a + 1) ! 1 f (N ) ; kN 2 N! 2k

:

The last proves the theorem. We give Theorem 7. Let f 2 C N (R) ; N function of compact support x 2 R: Suppose '

1; x 2 R and k 2 Z: Let ' be a bounded 1 X [ a; a] ; a > 0 such that ' (x j) = 1 all

0: Assume further f (i) (Bk f ) (x) =

1 X

1 X

1

j 2k

f

j= 1

(Ck f ) (x) =

j= 1

kj

< 1; i = 1; : : : ; N: Put

' 2k x

j ;

(f ) ' 2k x

j ;

j= 1

where kj

(f ) = 2

k

Z

2

k

f

t+

0

and

1 X

(Dk f ) (x) =

kj

j 2k

dt;

(f ) ' 2k x

j ;

j= 1

where kj (f ) =

n X

wr f

r=0

n 2 N; wr (i)

0;

n X

j r + k 2k 2 n

;

wr = 1: Then

r=0

j(Bk f ) (x)

(Dk f ) (x)j

kBk f N X i=1

Dk f k1

(i)

f ! 1 f (N ) ; 21k 1 + ; 2ki i! 2kN N !

(16)

(ii) j(Bk f ) (x)

(Ck f ) (x)j

kBk f

N X i=1

Ck f k1

f (i) 1 ! 1 f (N ) ; 21k + ; 2ki (i + 1)! 2kN (N + 1)!

20

(17)

62

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

(iii) j(Ck f ) (x)

(Dk f ) (x)j

kCk f

N X i=1

0:

So as k ! 1; we get kBk f

2ki

Dk f k1

f (i) 1 ! 1 f (N ) ; 21k + : 1 (i + 1)! 2kN 1 (N + 1)!

Dk f k1 ! 0; kBk f

Ck f k1 ! 0; kCk f

(18) Dk f k1 !

Proof. By Theorems 1-3 and especially use of (4), (7) and (12).

2

Estimates for Distances of Fuzzy Wavelet type Operators

We need the following background De…nition (see [7]). Let : R ! [0; 1] with the following properties (i) is normal, i.e., 9x0 2 R; (x0 ) = 1: (ii) ( x + (1 ) y) min f (x) ; (y)g ; 8x; y 2 R; 8 2 [0; 1] ( is called a convex fuzzy subset). (iii) is upper semicontinuous on R; i.e., 8x0 2 R and 8" > 0; 9 neighborhood V (x0 ) : (x) (x0 ) + "; 8x 2 V (x0 ) : (iv) The set supp ( ) is compact in R (where supp ( ) := fx 2 R : (x) > 0g): We call a fuzzy real number. Denote the set of all with RF : E.g., fx0 g 2 RF ; for any x0 2 R; where fx0 g is the characteristic function at x0 : r For 0 < r 1 and 2 RF de…ne [ ] := fx 2 R : (x) rg and 0

[ ] := fx 2 R :

(x)

0g: r

Then it is well known that for each r 2 [0; 1] ; [ ] is a closed and bounded interval of R: For u; v 2 RF and 2 R; we de…ne uniquely the sum u v and the product u by r r r r r [u v] = [u] + [v] ; [ u] = [u] ; 8r 2 [0; 1] ; r

r

where [u] + [v] means the usual addition of two integrals (as subsets of R) and r [u] means the usual product between a scalar and a subset of R (see, e.g., [7]). Notice 1 u = u and it holds u vh = v u;i u=u : If 0 r1 r2 1 (r) (r) (r) (r) (r) (r) r2 r1 r then [u] [u] : Actually [u] = u ; u+ ; where u u+ ; u ; u+ 2 R; 8r 2 [0; 1] : For De…ne

> 0 one has u

(r)

(r)

=(

D : RF

u)

; respectively.

RF ! R+

21

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

by D (u; v) := sup max r2[0;1]

n

u

(r)

v

(r)

(r)

(r)

; u+

v+

h i (r) (r) r where [v] = v ; v+ ; u; v 2 RF : We have that D is a metric on RF : Then (RF ; D) is a complete metric space, see [7] : Let f; g : R ! RF : We de…ne the distance

o

63

;

D (f; g) = sup D (f (x) ; g (x)) : x2R

P Here stands for fuzzy summation and e 0 := f0g 2 RF is the neutral element with respect to ; i.e., u e 0=e 0 u = u; 8u 2 RF :

We need (r) (r) Remark([4]). Here r 2 [0; 1] ; xi ; yi 2 R; i = 1; : : : ; m 2 N: Suppose that (r)

(r)

2 R; for i = 1; : : : ; m:

sup max xi ; yi

r2[0;1]

Then one sees easily that sup max r2[0;1]

m X i=1

(r) xi ;

m X

(r) yi

i=1

!

m X

(r)

(r)

sup max xi ; yi

:

i=1 r2[0;1]

De…nition. Let f : R ! RF ; we de…ne the (…rst) fuzzy modulus of continuity of f by (F ) ! 1 (f; ) = sup D (f (x) ; f (y)) ; > 0: x;y2R jx yj

U We de…ne CF (R) the space of uniformly continuous functions from R ! RF ; also C (R; RF ) the space of fuzzy continuous functions on R: (F )

U (R) : Then ! 1 Proposition 8 ([4]). Let f 2 CF (F )

Proposition 9 ([4]). It holds lim ! 1 U CF

!0

(R) :

(f; ) < 1; any (F )

(f; ) = ! 1

> 0:

(f; 0) = 0; i¤ f 2

Proposition 10 ([4]). Let f : R ! RF be a fuzzy real number valued (F ) (r) (r) function. Assume that ! 1 (f; ) ; ! 1 f ; ; ! 1 f+ ; are …nite; for > 0; all r 2 [0; 1] : Then n o (F ) (r) (r) ! 1 (f; ) = sup max ! 1 f ; ; ! 1 f+ ; : r2[0;1]

22

64

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

U Note. It is clear from Propositions 9, 10 that if f 2 CF (R) ; then f CU (R) (uniformly continuous on R).

(r)

2

De…nition. Let x; y 2 RF : If there exists z 2 RF : x = y z; then we call z the H-di¤erence of x and y; denoted x y: De…nition ([7]). Let T := [x0 ; x0 + ] R; with > 0: A function f : T ! RF is H-di¤erentiable at x 2 T if there exists an f 0 (x) 2 RF such that the limits (with respect to D) lim f (x+h)h f (x) ; lim f (x) fh(x h) exist and are h!0+

h!0+

equal to f 0 (x) : We call f 0 the H-derivative or fuzzy derivative of f at x: Above is assumed that the H-di¤erences f (x + h) f (x); f (x) f (x exist in RF in an neighborhood of x:

h)

We denote by C N (R; RF ) ; N 1; the space of all N times continuously fuzzy di¤erentiable functions from R into RF : We mention Theorem 11 ([11]). Let f : [a; b] R ! RF be H fuzzy di¤erentiable. Let t 2 [a; b] ; 0 r 1: Clearly h i r (r) (r) [f (t)] = (f (t)) ; (f (t))+ R: (r)

Then (f (t))

are di¤erentiable and r

(r)

[f 0 (t)] = (r)

I.e. (f 0 )

= f

(r)

0

(f (t))

0

(r)

; (f (t))+

obtain f (i) (t)

=

:

; 8r 2 [0; 1]:

Remark ([4]). Let f 2 C N (R; RF ) ; N r

0

(r)

(i)

(f (t))

(r)

1: Then by Theorem 11 we (i)

; (f (t))+

; for i = 0; 1; 2; : : : ; N; and

in particular we have that f (i)

(r)

= f

(r)

(i)

;

for any r 2 [0; 1] : (r)

Note. (i) Let f : R ! RF fuzzy continuous, then f : R ! R are continuous, 8r 2 [0; 1] : (r) (ii) Let f 2 C N (R; RF ) ; N 1: Then by Theorem 11, we have f 2 C N (R) ; for any r 2 [0; 1] : We need

23

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

NB De…nition. Denote by CF (R) := ff : R ! RF j such that all fuzzy (i) derivatives f : R ! RF ; i = 0; 1; : : : ; N exist and are fuzzy continuous and furthermore D f (i) ; e 0 < 1; for i = 1; : : : ; N g; N 1:

Notice here that D

f (i) ; e 0 = sup max

f (i)

(r)

;

r2[0;1]

f (r)

= sup max

(i)

1

;

r2[0;1]

f (i) f (r)

1

Notice also that D f (i) ; e 0 < 1; implies

f (i)

(r) 1

(r) + (i) +

1

; i = 1; : : : ; N: 1

< 1; i = 1; : : : ; N; 8r 2 [0; 1] :

We need De…nition ([8], p. 644). Let f : [a; b] ! RF : We say that f is FuzzyRiemann integrable to I 2 RF if for any " > 0; there exists > 0 such that for any division P = f[u; v] ; g of [a; b] with the norms (P ) < ; we have ! X D (v u) f ( ) ; I < ": P

We write I := (F R)

Rb a

f (x) dx:

We mention Rb Theorem 12 ([9]). Let f : [a; b] ! RF be fuzzy continuous. Then (F R) a f (x) dx exists and belongs to RF ; furthermore it holds " #r "Z # Z b

(F R)

b

f (x) dx

a

=

(r)

(f )

a

(r)

(x) dx; (f )+ (x) dx ;

8r 2 [0; 1] : (r) Clearly f : [a; b] ! R are continuous functions. In this section we study the fuzzy corresponding analogs of real wavelet type operators Ak ; Bk ; Ck ; Dk ; k 2 Z; of …rst section. For simplicity we keep the same notation at the fuzzy level. So, depending on the context we understand accordingly whether our operator is real of fuzzy, that is whether is operating on real valued functions or on fuzzy valued functions. We present the next main fuzzy wavelet type result. NB Theorem 13. Let f 2 CF (R) ; N 1; x 2 R; and k 2 Z: Let ' be a bounded real valued function of compact support [ a; a] ; a > 0 such that

24

65

66

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

1 P

' (x

j= 1

j) = 1; all x 2 R: Suppose ' 1 X

(Bk f ) (x) =

j 2k

f

j= 1

(Ck f ) (x) =

1 X

k

2

(F R)

Z

' 2k x

k

2

f

0

j= 1

and

0: Put

1 X

(Dk f ) (x) =

kj

(f )

j t+ k 2

j ;

dt

' 2k x

!

' 2k x

j ;

j ;

j= 1

where kj

n 2 N; wre Then (i)

0;

n P

r e=0

(f ) =

wre = 1:

n X

r e=0

D ((Bk f ) (x) ; (Dk f ) (x))

wre

re j + k 2k 2 n

f

;

D (Bk f; Dk f ) N D X i=1

f (i) ; e 0

2ki i!

(F )

+

!1

f (N ) ; 21k ; 2kN N !

(19)

(ii) D ((Bk f ) (x) ; (Ck f ) (x))

D (Bk f; Ck f ) f (i) ; e 0

N D X i=1

2ki (i + 1)!

(F )

+

!1 f (N ) ; 21k ; 2kN (N + 1)!

+

!1 2kN

(20)

and (iii) D ((Ck f ) (x) ; (Dk f ) (x))

D (Ck f; Dk f ) N D X i=1

2ki

1

f (i) ; e 0

(i + 1)!

(F )

f (N ) ; 21k : 1 (N + 1)!

Note. We see that D f (N ) (x) ; f (N ) (y)

D f (N ) (x) ; e 0 + D f (N ) (y) ; e 0 2D

25

f (N ) ; e 0 < 1:

(21)

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

67

(F )

Thus ! 1 f; 21k < 1; 8k 2 Z: Consequently as k ! 1 we obtain D (Bk f; Dk f ) ; D (Bk f; Ck f ) ; D (Ck f; Dk f ) ! 0 with rates. Proof. (i) We observe the following n (r) (r) D ((Bk f ) (x) ; (Dk f ) (x)) = sup max ((Bk f ) (x)) ((Dk f ) (x)) ; r2[0;1]

(r)

o

(r)

((Bk f ) (x))+ ((Dk f ) (x))+ n (r) = sup max Bk f (x)

Dk f

r2[0;1]

(r)

Bk f+

sup max r2[0;1] (r)

n

sup max r2[0;1] (i)

f+

8 > N

: i=1

r2[0;1]

(r) + 1

f (i)

N X

2ki i!

i=1

f (i)

+ =

1

2kN N !

N X i=1

(r)

2ki i!

1

f (N )

!1 +

N X 1 sup max ki 2 i! r2[0;1] i=1

f (i)

!1

+

> ;

f (i)

; 1

26

+

(N )

;

(r)

;

1 2k

+

1 2k

9 > > = > > ;

;

1

f (N )

; !1 ;

(r)

(N )

2kN N !

(r)

1 1 (F ) ! f (N ) ; k 2kN N ! 1 2

proving the theorem’s (19).

; 21k

f

9 = ; 21k >

(r)

f (N )

1

2kN N !

(r)

;

!1

2kN N !

(r) +

1

(x) ;

(i)

(r) f+

+

(r)

o

o

2ki i!

f (N )

!1

r2[0;1]

f (i) ; e 0 +

1

2kN N !

sup max ! 1

1 D 2ki i!

> > : i=1

(x)

Dk f

Dk f+ 1 8 (r) > > f N

N

;

; 21k

f (i)

;

f (N )

2kN

+

(N )

;

f

(r)

(r)

9 > > =

1 2k

> > ;

;

;

1 2k

(r) +

; !1

1

f (N )

1 1 (F ) !1 f (N ) ; k (N + 1)! 2

proving the theorem’s (20).

27

(N )

; 21k

2kN (N + 1)!

9 = ; 21k >

1

1 sup max ! 1 2kN (N + 1)! r2[0;1]

N X

(r)

(x) ;

;

2kN (N + 1)!

f (N )

2kN (N + 1)!

1 sup max (i + 1)! r2[0;1]

(r)

(i)

2kN (N + 1)!

(r) +

f (N )

!1

+

+

+

o 1

o

(r) f+

!1

!1

1

(r)

> 2ki (i + 1)! > : i=1

2ki (i + 1)!

(r)

(x)

Ck f

Ck f+ 1 8 (r) > > f N

: i=1 2 (i + 1)!

2ki (i + 1)!

2ki

f (i)

(r)

(r)

(17)

(r)

Ck f+

Bk f

Bk f+

N X

(r)

(x)

;

;

(r) +

;

1 2k

;

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

69

(iii) We observe the next D ((Ck f ) (x) ; (Dk f ) (x)) = sup max r2[0;1]

n

(r)

(r)

((Ck f ) (x))

((Dk f ) (x))

(r)

(r)

((Ck f ) (x))+ ((Dk f ) (x))+ n (r) = sup max Ck f (x)

o

Dk f

r2[0;1]

(r)

Ck f+

sup max r2[0;1]

(r)

(x)

n

Dk f+

Ck f

(r)

Dk f

(x) (r)

o 1

(r)

;

(x) ;

;

o (r) Dk f+ 1 8 (i) (N ) (r) (r) > > f !1 f ; 21k N

2ki 1 (i + 1)! 2kN 1 (N + 1)! r2[0;1] > i=1 : 9 (i) (N ) (r) (r) 1 > f !1 f+ ; 2k > N = + X 1 + 2ki 1 (i + 1)! 2kN 1 (N + 1)! > > i=1 ; (r)

Ck f+

= sup max r2[0;1]

2ki

N X

2ki 1

i=1

i=1

+ =

f (i)

N X

2kN

N X i=1

1

8 > N

: i=1 2 (r) + 1

(i + 1)!

+

1

(r) 1

(i + 1)!

!1

+

f (N )

2kN

1

1 sup max (i + 1)! r2[0;1]

2kN

(r) +

1 D (i + 1)!

1

; 21k

(N + 1)!

(N + 1)! > ;

f (i) ; e 0 +

(r)

;

9 = ; 21k > (r)

f (i)

f (i)

;

1

1 sup max ! 1 1 (N + 1)! r2[0;1]

2ki 1

f (N )

!1

f (N )

2kN

proving the theorem’s (21).

28

1

(r)

;

1 2k

(r) +

; !1

1

f (N )

1 1 (F ) !1 f (N ) ; k (N + 1)! 2

(r) +

;

;

1 2k

;

70

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

Above we need that ([5]) (r)

= Bk f

(r)

;

(r)

= Ck f

(r)

; and

(r)

= Dk f

(r)

;

(Bk f ) (Ck f ) (Dk f ) 8r 2 [0; 1]:

Denote by Cb (R; RF ) the space of bounded fuzzy continuous functions on R with respect to metric D: We …nish with the following fuzzy wavelet type main result NB Theorem 14. Let f 2 CF (R) \ Cb (R; RF ) ; N 1; x 2 R; and k 2 Z: Let the scaling function ' (x) a real valued function with supp ' (x) [ a; a] ; 1 P 0 < a < +1; ' is continuous on [ a; a] ; ' (x) 0; such that ' (x j) = 1 j= 1 R1 on R (then 1 ' (x) dx = 1): De…ne 'kj (t) := 2k=2 ' 2k t j all k; j 2 Z; t 2 R Z j+a 2k f; 'kj := (F R) f (t) 'kj (t) dt; j a 2k

and the fuzzy wavelet type operator (Ak f ) (x) =

1 X

f; 'kj

j= 1

'kj (x) ; x 2 R:

The fuzzy wavelet type operators Bk ; Ck ; Dk are as in Theorem 13. Then (i) D ((Ak f ) (x) ; (Bk f ) (x)) N D X i=1

f (i) ; e 0

2ki i!

D (Ak f; Bk f ) ai +

aN a (F ) ! f (N ) ; k ; 2kN N ! 1 2

(22)

(ii) D ((Ak f ) (x) ; (Ck f ) (x)) N D X i=1

f (i) ; e 0

i!2ki

D (Ak f; Ck f ) N

i

(a + 1) +

(a + 1) (a + 1) (F ) ! f (N ) ; 2kN N ! 1 2k

29

;

(23)

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

71

and (iii) D ((Ak f ) (x) ; (Dk f ) (x))

D (Ak f; Dk f ) N D X i=1

f (i) ; e 0

2ki i!

N

i

(a + 1) +

(a + 1) (a + 1) (F ) !1 f (N ) ; kN N !2 2k (24)

Notice that D (Ak f; Bk f ) ; D (Ak f; Ck f ) ; D (Ak f; Dk f ) ! 0 as k ! 1 with rates. Proof. Similar to the proof of Theorem 13. Also notice here (see also [5]) (r) (r) that (Ak f ) = Ak f ; 8r 2 [0; 1] : It is based on (13), (14) and (15). U Note. In [3] we proved, for f 2 CF (R) as k ! 1; we get uniformly that Ak ; Bk ; Ck ; Dk ! I unit operator with rates in the D metric. In the case of Ak we need also f be fuzzy bounded. As related work we mention [2].

References [1] G.A. Anastassiou, Quantitative Approximations, Chapman & Hall/CRC, Boca Raton, New York, 2001. [2] G.A. Anastassiou, Rate of convergence of Fuzzy Neural Network operators, univariate case, J. Fuzzy Mathematics, 10 (2002), no.3, 755-780. [3] G.A. Anastassiou, Fuzzy Wavelet type Operators, Nonlinear Functional Analysis and Appl., Vol. 9, No. 2, 2004, 251-269. [4] G.A. Anastassiou, Fuzzy Approximation by Fuzzy Convolution type Operators, Computers and Math. with Appl., special issue edited by G. Anastassiou, Vol. 48, 2004, 1369-1386. [5] G.A. Anastassiou, High Order Fuzzy Approximation by Fuzzy Wavelet type and Neural Network Operators, Computers and Math. with Appl., special issue edited by G. Anastassiou, Vol. 48, 2004, 1387-1401. [6] G.A. Anastassiou, Quantitative estimates for Distance between Fuzzy Wavelet type operators, Journal of Concrete and Applicable Mathematics, Vol 5, No. 1, 25-52, 2007. [7] Congxin Wu and Zengtai Gong, On Henstock integral of fuzzy number valued functions (I),Fuzzy Sets and Systems, 120, No. 3, 2001, 523-532.

30

:

72

ANASTASSIOU: ESTIMATES FOR WAVELET OPERATORS

[8] S. Gal, Approximation Theory in Fuzzy Setting, Chapter 13 in Handbook of Analytic-Computational Methods in Applied Mathematics, pp. 617-666, edited by G. Anastassiou, Chapman & Hall/CRC, 2000, Boca Raton, New York. [9] R. Goetschel, Jr. and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18 (1986), 31-43. [10] H. Gonska, P. Pitul, and I. Rasa, On di¤ erences of positive linear operators, Carpathian J. Math. 22 (2006), No. 1-2, 65-78. [11] O. Kaleva, Fuzzy di¤ erential equations, Fuzzy Sets and Systems, 24 (1987), 301-317.

31

JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.5, NO.1, 73-84, COPYRIGHT 2010 EUDOXUS PRESS, LLC

General Over-Relaxed A−Proximal Point Algorithms and Applications to Nonlinear Variational Inclusions

Rom U. Verma Department of Mathematics University of Toledo Toledo, Ohio 43606, USA [email protected]

Abstract Based on the notion of A−monotonicity and the general firm nonexpansiveness of the resolvent operator corresponding to A−monotonicity, the convergence analysis of the over-relaxed proximal point algorithm in the context of the approximation solvability of a class of nonlinear variational inclusions is examined. Several results on the general firm nonexpansiveness are also established. The obtained results generalize the results on firm nonexpansiveness. 2000 Mathematics Subject Classifications: 49J40, 47H10, 65B05 Keywords: General firm nonexpansiveness, Variational inclusions, Maximal monotone mapping, A-monotone mapping, Relaxed proximal point algorithm, generalized resolvent operator.

1. Introduction Let X be a real Hilbert space with the norm k · k and the inner product h., .i. We consider the nonlinear inclusion problem: determine a solution to 0 ∈ M (x),

(1)

where M : X → 2X is a set-valued mapping on X. Rockafellar [24] generalized the algorithm of Martinet [17] in the context of convex programming, which is referred to as the proximal point algorithm in the literature. This work includes the general convergence and rate of convergence analysis for solving (1) when M is monotone. Then in [25], Rockafellar has shown as how the proximal point algorithm can be formulated in conjunction with convex programming duality theory to present a general convergence analysis for the multiplier method in convex programming. Eckstein and Bertsekas [6] introduced the relaxed proximal point algorithm and applied to the solvability of inclusion problems of the form (1). Furthermore, they demonstrated that the Douglas-Rachford splitting method [2] for convex programming was, in fact, a special case of the proximal point algorithm. Next, we state the following theorem of Eckstein and Bertsekas [6], which presents the convergence analysis for the relaxed proximal point algorithm.

73

74

VERMA: NONLINEAR VARIATIONAL INCLUSIONS

Theorem 1.1. [6, Theorem 3] Let M : X → 2X be a set-valued maximal monotone mapping on X with 0 ∈ range(M ), and let the sequence {xk } be generated by the iterative algorithm xk+1 = (1 − αk )xk + αk wk ∀ k ≥ 0,

(2)

where wk is such that kwk − (I + ck M )−1 (xk )k ≤ k ∀ k ≥ 0, and the scalar sequences {k }, {αk } and {ck } satisfy Σ∞ k=0 k < ∞, inf αk > 0, sup αk < 2, c = infck > 0. Then the sequence {xk } converges weakly to a zero of M. In [32 − 37, 39], the author introduced and studied the notion of A−monotonicity in the context of solving variational inclusion problems based on the resolvent operator techniques. The notion of A−monotonicity generalizes the general theory of multivalued maximal monotone mappings, including the notion of H−monotonicity introduced by Fang and Huang [8], and provides a general framework to examining variational inclusion problems. Resolvent operator methods have been used in literature for a while and are still being applied to a broad spectrum of problems arising from several fields, such as equilibria problems, optimization and control theory, operations research, and mathematical programming. In this paper, we generalize the relaxed proximal point algorithm to the case of A−monotone mappings, and then we apply it to the approximation solvability of a class of nonlinear inclusion problem involving A− monotone set-valued mappings in a Hilbert space setting. The convergence analysis for the generalized relaxed proximal point algorithm is discussed in detail. Also, several results on the generalized firm nonexpansiveness, Lipschitz continuity of the generalized resolvent operator corresponding to A− monotone mappings are included. The obtained results generalize a number of results on the general maximal monotonicity, resolvent operator technique and firm nonexpansiveness by Eckstein [5], Eckstein and Bertsekas [6], Rockafellar [24, 25], and others. For more literature, we recommend the reader [1-40]. The contents are organized as follows: section 1 deals with a historical development of the proximal point algorithm in conjunction with the maximal monotonicity, and with the approximation solvability of a class of nonlinear inclusion problems based on the proximal point algorithm. Section 2 introduces the notion of A−monotonicity and general firm nonexpansiveness, and then presents a number of results connecting A−monotonicity, general firm nonexpansiveness and generalized resolvent operator, while Section 3 presents the generalized version of the relaxed proximal point algorithm of Eckstein and Bertsekas [6] to the case of A−monotone mappings. Section 4 contains specializations to maximal relaxed monotone mappings and applications. 2. A-Monotonicity and Firm Nonexpansiveness In this section we first explore some basic properties derived from the notion of A− monotonicity. Then we establish some results involving A − monotonicity and the firm nonexpansiveness. Let X denote a real Hilbert space with the norm k · k and inner product < ., . > . Let M : X → 2X be a multivalued mapping on X. We shall denote both the map M and its graph by M, which means, the set {(x, y) : y ∈ M (x)}. This is equivalent to stating that a mapping is any subset M of X × X, and M (x) = {y : (x, y) ∈ M }. If M is single-valued, we shall still use M (x) to represent the unique y such that (x, y) ∈ M rather than the singleton set {y}. This interpretation shall much depend on the context. The domain of a map M is defined (as its projection onto the first argument) by dom(M ) = {x ∈ X : ∃y ∈ X : (x, y) ∈ M } = {x ∈ X : M (x) 6= ∅}.

VERMA: NONLINEAR VARIATIONAL INCLUSIONS

dom(M)=X, shall denote the full domain of M, and the range of M is defined by range(M ) = {y ∈ X : ∃x ∈ X : (x, y) ∈ M }. The inverse M −1 of M is {(y, x) : (x, y) ∈ M }. For a real number ρ and a mapping M, let ρM = {x, ρy) : (x, y) ∈ M }. If L and M are any mappings, we define L + M = {(x, y + z) : (x, y) ∈ L, (x, z) ∈ M }.

Definition 2.1. Let M : X → 2X be a multivalued mapping on X. The map M is said to be: (i) (r)− strongly monotone if there exists a positive constant r such that hu∗ − v∗ , u − vi ≥ rku − vk2 ∀(u, u∗ ), (v, v∗ ) ∈ Graph(M ). (ii) (r)−strongly pseudomonotone if hv∗ , u − vi ≥ 0 implies hu∗ , u − vi ≥ rku − vk2 ∀(u, u∗), (v, v∗ ) ∈ Graph(M ). (iii) pseudomonotone if hv∗ , u − vi ≥ 0 implies hu∗ , u − vi ≥ 0 ∀(u, u∗), (v, v∗ ) ∈ Graph(M ). (iv) (m)−relaxed monotone if there exists a positive constant m such that hu∗ − v∗ , u − vi ≥ (−m)ku − vk2 ∀(u, u∗), (v, v∗ ) ∈ Graph(M ). (v) (c)− cocoercive if there is a positive constant c such that hu∗ − v∗ , u − vi ≥ cku∗ − v∗ k2 ∀(u, u∗), (v, v∗ ) ∈ Graph(M ).

Definition 2.2. A mapping M : X → 2X is said to be maximal (m)− relaxed monotone if (i) M is (m)−relaxed monotone, (ii) For (u, u∗ ) ∈ X × X, and hu∗ − v∗ , u − vi ≥ (−m)ku − vk2 ∀(v, v∗ ) ∈ Graph(M ), we have u∗ ∈ M (u). Definition 2.3. Let M : X → 2X be a mapping on X. The map M is said to be:

75

76

VERMA: NONLINEAR VARIATIONAL INCLUSIONS

(i) Nonexpansive if ku∗ − v∗ k ≤ ku − vk ∀(u, u∗ ), (v, v∗ ) ∈ Graph(M ). (ii) Firmly nonexpansive if ku∗ − v∗ k2 ≤ hu∗ − v∗ , u − vi ∀(u, u∗ ), (v, v∗ ) ∈ Graph(M ). (iii) (c)−firmly nonexpansive if there exists a constant c > 0 such that ku∗ − v∗ k2 ≤ chu∗ − v∗ , u − vi ∀(u, u∗ ), (v, v∗ ) ∈ Graph(M ).

Definition 2.4. (Alternative) Let A : X → X be a single-valued mapping. The map M : X → 2X is said to be A−monotone if (i) M is (m)−relaxed monotone (ii) R(A + ρM ) = X for ρ > 0. Proposition 2.1. Let A : X → X be an (r)-strongly monotone single-valued mapping and let M : X → 2X be an A− monotone mapping. Then M is maximal (m)−relaxed monotone for r 0 0. Proof. It follows from the definition of A− monotonicity of M. 2 3. A−Proximal Point Algorithm and Application This section primarily deals with the relaxed A−proximal point algorithm and its application to approximation solvability of the inclusion problem (1). Several results connecting the generalized A−monotonicity and corresponding resolvent operator are established, which unify the results on the firm expansiveness from Eckstein and Bertsekas [6]. Furthermore, some auxiliary results on A−monotonicity, maximal relaxed monotonicity, and maximal monotonicity are obtained. The solvability of the problem (1) depends on the equivalence between (1) and the problem of finding the fixed point of the associated generalized resolvent operator. Lemma 3.1. Let X be a real Hilbert space, let A : X → X be (r)−strongly monotone, and let M : X → 2X be A− monotone. Then the generalized resolvent operator associated with M and defined by M Jρ,A (u) = (A + ρM )−1 (u) ∀ u ∈ X, 1 is ( r−ρm )−Lipschitz continuous r − ρm > 0.

Lemma 3.2. Let X be a real Hilbert space, let A : X → X be (r)−strongly monotone, and let M : X → 2X be A− monotone. Then the generalized resolvent operator associated with M and defined by M Jρ,A (u) = (A + ρM )−1 (u) ∀ u ∈ X, 1 is ( r−ρm )−firmly nonexpansive for r − ρm > 0. M that Proof. For any u, v ∈ X, it follows from the definition of the resolvent operator Jρ,A

1 M M [u − A(Jρ,A (u))] ∈ M (Jρ,A )(u), ρ and

1 M M [v − A(Jρ,A (v))] ∈ M (Jρ,A )(v). ρ Since M is (m)−relaxed monotone, we have 1 M M hu − v − [A(Jρ,A (u)) − A(Jρ,A (v))], ρ M M Jρ,A (u) − Jρ,A (v)i ≥

M M −mkJρ,A (u) − Jρ,A (v)k2 .

(3)

78

VERMA: NONLINEAR VARIATIONAL INCLUSIONS

In light of (3), we have M M (u) − Jρ,A (v)i hu − v, Jρ,A

≥

M M hA(Jρ,A (u)) − A(Jρ,A (v)), M M Jρ,A (u) − Jρ,A (v)i − M M ρmkJρ,A (u) − Jρ,A (v)k2

≥

M M (r − ρm)kJρ,A (u) − Jρ,A (v)k2 . 2

Lemma 3.3. Let X be a real Hilbert space, let A : X → X be (r)−strongly monotone, and let M is firmly nonexpansive for r − ρm > 1. M : X → 2X be A− monotone. Then I − Jρ,A Theorem 3.1. Let X be a real Hilbert space, let A : X → X be (r)−strongly monotone, and let M : X → 2X be A−monotone. Then the following statements are mutually equivalent: (i) An element u ∈ X is a solution to (1). (ii) For an u ∈ X, we have M u = Jρ,A (A(u)),

where M Jρ,A (u) = (A + ρM )−1 (u).

Proof. It follows from the definition of the generalized resolvent operator corresponding to M. Theorem 3.2. Let X be a real Hilbert space, let A : X → X be (r)−strongly monotone and (s)−Lipschitz continuous, and let M : X → 2X be A−monotone. For an arbitrarily chosen element x0, let the sequence {xk } be generated by the relaxed A−proximal point algorithm xk+1 = (1 − αk )xk + αk yk for k ≥ 0 with M kyk − Jρ,A (A(xk ))k ≤ δk kyk − xk k,

where δk → 0, M yk+1 = (1 − αk )xk + αk Jρ,A (A(xk )),

hxk − x∗ , JρMk ,A (A(xk )) − JρMk ,A (A(x∗ ))i ≥ γkJρMk ,A (A(xk )) − JρMk ,A(A(x∗ ))k2, (4) P ∞ M for γ > 0, Jρ,A = (A + ρk M )−1, and sequences {δk }, {αk } and {ρk } satisfy αk > 1, k=0 δk ≤ ∞, and ρk ↑ ρ. Then the sequence {xk } converges linearly to a unique solution x∗ of (1) with rate s s2 s2 1 − α[2(1 − ) − α(1 − (2γ − 1) )] < 1, 2 (r − ρm) (r − ρm)2

VERMA: NONLINEAR VARIATIONAL INCLUSIONS

where α2k + 2αk (1 − αk )γ > 0 and α = lim supk→∞ αk . Proof. Applying Theorem 3.1, x∗ , a solution to (1), satisfies the relaxed proximal point algorithm. It further follows from Theorem 3.1 that any solution to (1) is a fixed point of JρMk ,A oA for all k ≥ 0. Next, using (4), we find the estimate kyk+1 − x∗ k2 = k(1 − αk )xk + αk JρMk ,A (A(xk )) − [(1 − αk )x∗ + αk JρMk ,A (A(x∗ ))k2 =

k(1 − αk )(xk − x∗) − αk (JρMk ,A (A(xk )) − JρMk ,A (A(x∗ )))k2

=

(1 − αk )2kxk − x∗ k2 + 2αk (1 − αk )hxk − x∗, JρMk ,A (A(xk )) − JρMk ,A (A(x∗ ))i + α2k kJρMk ,A (A(xk )) − JρMk ,A (A(x∗ ))k2

≤

(1 − αk )2kxk − x∗ k2 + 2αk (1 − αk )γkJρMk ,A (A(xk )) − JρMk ,A (A(x∗ ))k2 + α2k kJρMk ,A (A(xk )) − JρMk ,A (A(x∗ ))k2

=

(1 − αk )2kxk − x∗ k2 + [α2k + 2αk (1 − αk )γ]kJρMk ,A (A(xk )) − JρMk ,A (A(x∗ ))k2

≤

(1 − αk )2kxk − x∗ k2 + [α2k + 2αk (1 − αk )γ]

=

{(1 − αk )2 + [α2k + 2αk (1 − αk )γ]

=

s2 kxk − x∗k2 (r − ρk m)2

s2 }kxk − x∗ k2 (r − ρk m)2 s2 s2 {1 − αk [2(1 − ) − αk (1 − (2γ − 1) )]}kxk − x∗ k2, 2 (r − ρk m) (r − ρk m)2

where s < r − ρk m, and α2k + 2αk(1 − αk )γ > 0. Therefore, kyk+1 − x∗ k ≤ θk kxk − x∗ k for s < r − ρk m, where α2k + 2αk (1 − αk )γ > 0, and θk =

s

1 − αk [2(1 −

s2 s2 ) − αk (1 − (2γ − 1) )]. 2 (r − ρk m) (r − ρk m)2

Clearly, it follows that

=

kxk+1 − yk+1 k M (A(xk ))]k k(1 − αk )xk + αk yk − [(1 − αk )xk + αk Jρ,A

=

M kαk (yk − Jρ,A (A(xk )))k

≤

αk δk kyk − xk k.

Since xk+1 = (1 − αk )xk + αk yk , it implies that αk (yk − xk ) = xk+1 − xk .

79

80

VERMA: NONLINEAR VARIATIONAL INCLUSIONS

Now we estimate kxk+1 − x∗ k = kyk+1 − x∗ k + kxk+1 − yk+1 k ≤ ≤ = ≤

kyk+1 − x∗ k + kxk+1 − yk+1 k kyk+1 − x∗ k + αk δk kyk − xk k kyk+1 − x∗ k + δk kxk+1 − xk k θk kxk − x∗ k + δk kxk+1 − x∗ k + δk kxk − x∗ k.

(5) (6)

Therefore, we have kxk+1 − x∗ k ≤

θk + δk k kx − x∗ k, 1 − δk

(7)

where

=

θk + δk = lim sup θk lim sup 1 − δk s s2 s2 1 − α[2(1 − ) − α(1 − (2γ − 1) )] < 1. 2 (r − ρm) (r − ρm)2

Finally, to show the uniqueness of the solution, assume that x∗1 and x∗2 are two distinct solutions of (1). By Theorem 3.1, we have x∗1 = JρMk ,A (A(x∗1 )), and x∗2 = JρMk ,A (A(x∗2 )). Since JρMk ,A is ( r−ρ1k m )− Lipschitz continuous and A is (s)− Lipschitz continuous, we arrive at

≤ ≤

kx∗1 − x∗2 k = kJρMk ,A(A(x∗1 )) − JρMk ,A (A(x∗2 ))k 1 kA(x∗1 ) − A(x∗2 )k r − ρk m s kx∗ − x∗2 k. r − ρk m 1

Therefore, we find [1 −

s ]kx∗ − x∗2 k ≤ 0 for s < r − ρk m. r − ρk m 1

It follows from this that kx∗1 − x∗2 k = 0for s < r − ρk m. 2

4. Some Applications In this section, based on results from Sections 2 and 3, we derive some special cases of Theorem 3.2 for the H− monotone mapping – introduced and studied by Fang and Huang [8]. Definition 4.1. Let H : X → X be a single-valued mapping. The map M : X → 2X is said to be H−monotone if

VERMA: NONLINEAR VARIATIONAL INCLUSIONS

(i) M is monotone, (ii) R(H + ρM ) = X for ρ > 0. Lemma 4.1. [8] Let X be a real Hilbert space, let H : X → X be (r)−strongly monotone, and let M : X → 2X be H− monotone. Then the generalized resolvent operator associated with M and defined by M Jρ,H (u) = (H + ρM )−1 (u) ∀ u ∈ X, is ( 1r )−Lipschitz continuous for r > 0. Lemma 4.2. Let X be a real Hilbert space, let H : X → X be (r)−strongly monotone, and let M : X → 2X be H− monotone. Then the generalized resolvent operator associated with M and defined by M Jρ,H (u) = (H + ρM )−1 (u) ∀ u ∈ X, is ( 1r )−firmly nonexpansive for r > 0. Lemma 4.3. Let X be a real Hilbert space, let H : X → X be (r)−strongly monotone, and let M : X → 2X be H −monotone. Then the following statements are mutually equivalent: (i) An element u ∈ X is a solution to (1). (ii) For an u ∈ X, we have M (H (u)). u = Jρ,H

where M Jρ,H (u) = (H + ρM )−1 (u).

Theorem 4.1. Let X be a real Hilbert space, let H : X → X be (r)−strongly monotone and (s)−Lipschitz Continuous, and let M : X → 2X be H−monotone. For an arbitrarily chosen element x0, let the sequence {xk } be generated by the relaxed H−proximal point algorithm xk+1 = (1 − αk )xk + αk yk for k ≥ 0 with M (H(xk ))k ≤ δk kyk − xk k, kyk − Jρ,H

where δk → 0, M (H(xk )), yk+1 = (1 − αk )xk + αk Jρ,H

(8) hxk − x∗ , JρMk ,H (H(xk )) − JρMk ,H (H(x∗ ))i ≥ γkJρMk ,H (H(xk )) − JρMk ,H (H(x∗ ))k2 , P ∞ M for γ > 0, Jρ,H = (H + ρk M )−1, and sequences {δk }, {αk } and {ρk } satisfy αk > 1, k=0 δk ≤ ∞, and ρk ↑ ρ. Then the sequence {xk } converges linearly to a unique solution x∗ of (1) with rate r s2 s2 1 − α∗ [2(1 − 2 ) − α∗ (1 − (2γ − 1) 2 )] < 1, r r

81

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where α∗ = lim supk→∞ αk and s < r Theorem 4.2. Let X be a real Hilbert space, and let M : X → 2X be maximal monotone. For an arbitrarily chosen element x0 , let the sequence {xk } be generated by the relaxed proximal point algorithm xk+1 = (1 − αk )xk + αk yk ) for k ≥ 0 with kyk − JρM (xk )k ≤ δk kyk − xk k, where δk → 0, yk+1 = (1 − αk )xk + αk JρM (xk ), P∞ JρM = (I + ρk M )−1, and sequences {δk }, {αk } and {ρk } satisfy αk > 1, k=0 δk ≤ ∞, and ρk ↑ ρ. Then the sequence {xk } converges linearly to a unique solution x∗ of (1) with rate p 1 − α∗[2 − α∗ ] < 1, where α∗ = lim supk→∞ αk .

References [1] Y. J. Cho, and H. Y. Lan, A new class of generalized nonlinear multi-valued quasi-variationallike inclusions with H− monotone mappings, Mathematical Inequalities & Applications (in press). [2] J. Douglas, and H. H. Rachford, On the numerical solution of heat conduction problems in two and three space variables, Transactions of the American Mathematical Society 82 (1956), 421–439. [3] J. Eckstein, Splitting methods for monotone operators with applications to parallel optimization, Doctoral dissertation, Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, MA, 1989. [4] J. Eckstein, Nonlinear proximal point algorithm using Bregman functions, with applications to convex programming, Mathematics of Operations Research 18 (1993), 203–226. [5] J. Eckstein, Approximation iterations in Bregman-function-based proximal algorithm, Mathematical Programming 83 (1998), 113–123. [6] J. Eckstein, and D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Mathematical Programming 55 (1992), 293– 318. [7] J. Eckstein, and M. C. Ferris, Smooth methods of multipliers for complementarity problems, Mathematical Programming 86 (1999), 65–90.

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[8] Y. P. Fang, and N. J. Huang, H−monotone operators and system of variational inclusions, Communications on Applied Nonlinear Analysis 11 (1)(2004), 93–101. [9] Y. P. Fang, N. J. Huang, and H. B. Thompson, A new system of variational inclusions with (H, η)−monotone operators in Hilbert spaces, Computers and Mathematics with Applications 49 (2-3)(2005), 365–374. [10] M. C. Ferris, Finite termination of the proximal point algorithm, Mathematical Programming 50 (199), 359–366. [11] M. M. Jin, Perturbed algorithm and stability for strongly monotone nonlinear quasi-variational inclusion involving H− accretive operators, Mathematical Inequalities & Applications (in press). [12] M. M. Jin, Iterative algorithm for a new system of nonlinear set-valued variational inclusions involving (H, η)− monotone mappings Journal of Inequalities in Pure and Applied Mathematics (in press). [13] H. Y. Lan, A class of nonlinear (A, η)− monotone operator inclusion problems with relaxed cocoercive mappings, Advances in Nonlinear Variational Inequalities 9(2)(2006), 1–11. [14] H. Y. Lan, J. H. Kim, and Y. J. Cho, On a new class of nonlinear A− monotone multivalued variational inclusions, Journal of Mathematical Analysis and Applications (in press). [15] H. Y. Lan, New resolvent operator technique for a class of general nonlinear (A, η)− accretive equations in Banach spaces, International Journal of Applied Mathematical Sciences (in press). [16] Z. Liu, J. S. Ume, and S. M. Kang, Generalized nonlinear variational-like inequalities in reflexive Banach spaces, Journal of Optimization Theory and Applications 126 (1)(2002), 157–174. [17] B. Martinet, R´egularisation d’in´equations variationnelles par approximations successives. Rev. Francaise Inform. Rech. Oper. Ser. R-3 4 (1970), 154–158. [18] A. Moudafi, Mixed equilibrium problems: Sensitivity analysis and algorithmic aspect,Computers and Mathematics with Applications 44(2002), 1099-1108. [19] J. -S. Pang, Complementarity problems, Handbook of Global Optimization ( edited by R. Horst and P. Pardalos), Kluwer Academic Publishers, Boston, MA, 1995, ppm. 271–338. [20] T. Pennanen, Local convergence of the proximal point algorithm and multiplier methods without monotonicity, Mathematics of Operations Research 27(1)(2002), 170–191. [21] S. M. Robinson, Composition duality and maximal monotonicity, Mathematical Programming 85(199a), 1–13. [22 S. M. Robinson, Linear convergence of epsilon-subgradient descent methods for a class of convex functions, Mathematical Programming 86(199b), 41–50. [23] R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific Journal of Mathematics 33 (1970b), 209–216. [24] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM Journal of Control and Optimization 14 (1976a), 877–898.

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[25] R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Mathematics of Operations Research 1 (1976b), 97–116. [26] R. T. Rockafellar, and R. J-B. Wets,Variational Analysis, Springer-Verlag, Berlin, 1998. [27] M. V. Solodov, and B. F. Svaiter, An inexact hybrid generalized proximal point algorithm and some new results on the theory of Bregman functions, Mathematics of Operations Research 25 (2)(2000), 214–230. [28] P. Tossings,The perturbed proximal point algorithm and some of its applications , Applied Mathematics and Optimization 29 (1994), 125–159. [29] P. Tseng, Applications of a splitting algorithm to decomposition in convex programming and variational inequalities, SIAM Journal of Control and Optimization 29(1991), 119–138. [30] P. Tseng, Alternating projection-proximal methods for convex programming and variational inequalities, SIAM Journal of Optimization 7(1997), 951–965. [31] P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM Journal of Control and Optimization 38(2000), 431–446. [32] R. U. Verma, New class of nonlinear A− monotone mixed variational inclusion problems and resolvent operator technique, Journal of Computational Analysis and Applications 8(3)(2006), 275–285. [33] R. U. Verma, Nonlinear A− monotone variational inclusion systems and the resolvent operator technique, Journal of Applied Functional Analysis 1(1)(2006), 183–190. [34] R. U. Verma, A−monotonicity and its role in nonlinear variational inclusions, Journal of Optimization Theory and Applications 129(3)(2006), 457–467. [35] R. U. Verma, Nonlinear A- monotone mixed variational inclusion problems based on resolvent operator techniques, Mathematical Sciences Research Journal 9(10)(2005), 255–267. [36] R. U. Verma, Approximation-solvability of a class of A− monotone variational inclusion Problems, Journal of the Korean Society for Industrial and Applied Mathematics 8(1)(2004), 55-66. [37] R. U. Verma, A- monotonicity and applications to nonlinear variational inclusion problems, Journal of Applied Mathematics and Stochastic Analysis 17(2)(2004), 193–195. [38] R. U. Verma, A fixed-point theorem involving Lipschitzian generalized pseudocontractions, Proceedings of the Royal Irish Academy 97A(1)(1997), 83–86. [39] R. U. Verma, Approximation solvability of a class of set-valued variational inclusions involving (A, η)− monotone mappings, Journal of Mathematical Analysis and Applications, 337(2008), 969–975. [40] E. Zeidler, Nonlinear Functional Analysis and its Applications II/B, Springer-Verlag, New York, New York, 1990.

JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.5, NO.1, 85-99, COPYRIGHT 2010 EUDOXUS PRESS, LLC

Traffic Flower with n Petals Alexander P. Buslaev Departament of Mathematics The Moscow State Automobile and Road Technical University 125319, Leningradskii pr., Moscow, Russia [email protected] AMS 2000 Mathematics Subject Classification: 34B45 Key words and phrases: System of ordinary differential equations, Stationary solution, stability, critical states Abstract In this paper, some properties of solutions of a system of nonlinear ordinary differentional equationa are obtained. Above system describes a flow on the graph of ”flower with n petals” type when the flow mass is constant.

1 Introduction Let A = (aij ) be a stochastic square matrix n ∗ n, i.e. aij ≥ 0 ∀i, j, n X

aij = 1, i = 1..., n.

(1)

j=1

We assume, that ~ ρ = (ρ1..., ρn) is a vector with non-negative coordinates, 0 ≤ ρi ≤ 1, i = 1..., n, k~ ρkln = ρ1 + ... + ρn = C. 1

(2)

Let f (x) = x (1 − x) χ[0,1] (x) , and χ[0,1] be the characteristic function of a segment [0, 1]. Then the system of ODE    ρ˙ 1 = (a11 − 1)f (ρ1) + a21 f (ρ2) + ... + an1 f (ρn )      

ρ˙ 2 = a12f (ρ1) + (a22 − 1) f (ρ2) + ... + an2 f (ρn )

  ................................................................       ρ˙ = a f (ρ ) + a f (ρ ) + ... + (a − 1) f (ρ ), n 1n 1 2n 2 nn n

(3)

describes traffic on the graph, representing ”flower” - general vertex + a crossroad, where the closed identical edges in quantity of n pieces, (Fig. 1) are united. The function f (x) is

1

85

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BUSLAEV: TRAFFIC FLOWER...

Figure 1: The traffic flower with six petals a classic model of intensity of movement which depends on density and is called as main diagram, [1]. Further in (3) we will consider a value of density on the appropriate petal ρi = ρi (t), 0 ≤ ρi ≤ 1, i = 1, ..., n. Thus the matrix A is a matrix of hashing of flows in the vertex O. The entering flow from i petal is redistributed in the vertex O according to the shares determined in the line i of the matrix A, for this reason the condition (1) is true. Let’s consider further, that in (2) C is a constant, i.e. the flow is closed and, hence, it is described by system (3). The purpose of work consists of research of qualitative properties of flow , i.e. solutions of (3). Thus the stationary point is a solution of (3) ρ ~(t) ≡ ρ(0), satisfying condition (2). The solution ρ ~(t) such that k~ ρkln∞ = max |ρi | = 1 1≤i≤n

(4)

is called critical. According to physics of the system it means that, at least on one edge flow density is maximal and also speed of leaving flow is equal to 0. Hence, the edge ceases to pass a flow, ”the petal falls down ”. Thus, in case , when the system (3) with n petals is in a critical mode it is reduced to a flower with smaller number of petals and, accordingly, smaller mass. There exists different scenarios of a transformation of the matrix A to a stochastic matrix of the smaller size , i.e. different scenarios of movement control in a critical mode. We will postulate the following rules

2

BUSLAEV: TRAFFIC FLOWER...

87

(1) If the edge with number i has critical density, and next (i − 1) and (i + 1) edges are in an operating conditions, then line i of the matrix A shares half-and-half between (i − 1) (mod n) and (i + 1) (mod n), the column i deletes. (2) If the critical condition i comes simultaneously with the next one, then transformation (I) is made above the sum of lines from cluster of critical edges and accordingly columns delete.

2 Flower with Two Petals Let n = 2, a11 = a1, a22 = a2. Then the system (3) becomes    ρ˙ 1 = (a1 − 1) f (ρ1 ) + (1 − a2 ) f (ρ2)  

(5)

ρ˙ 2 = (1 − a1 ) f (ρ1 ) + (a2 − 1) f (ρ2)

with the following conditions    0 ≤ ρ1 ≤ 1, 0 ≤ ρ2 ≤ 1,  

(6)

ρ1 + ρ2 = C.

Figure 2: Double petal flower As ρ2 = C − ρ1, then (5) - (6) are reduced to the problem (see Fig. 2)    ρ˙1 = (a1 − 1)f (ρ1) + (1 − a2)f (ρ2)   0 ≤ ρ1 ≤ 1, 0 ≤ C − ρ1 ≤ 1,

where Cis a parameter, 0 < C < 2. Because of symmetry we will consider, that 1 ≥ a2 ≥ a1 ≥ 0.

3

,

(7)

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BUSLAEV: TRAFFIC FLOWER...

Lemma 2.1. If 0 < a1 = a2 < 1, then for any initial conditions ρ1(0) 6= C/2 (a) if C > 1 then the solution goes into a critical mode in finite time; (b) if 0 < C < 1 then the solution converges to a stationary point ρ1 ≡ C/2. Proof. If 0 ≤ ρ1 ≤ 1, and 0 ≤ C − ρ1 ≤ 1, that ρ˙ 1 = (1 − a2)(f (C − ρ1) − f (ρ1)) = (1 − a2 )(C − 2ρ1)(1 − C), whence everything also follows. Lemma 2.2. Let 0 < a1 < a2 < 1. Then the problem (7) has the unique stationary solution. Proof. If C < 1, then ρ1 ∈ [0, C], and function g(x) = (a1 − 1)f (x) + (1 − a2 )f (C − x) at x = 0 satisfies to an inequality g(0) = (1 − a2)f (C) > 0, and at x = C it is fair g(C) = (a1 − 1)f (C) < 0. Thus, from a continuity g(x) it follows that there is at least one zero. However, as

g(x) = (1 − a1 )f (C − x) + (a1 − a2 )f (C − x) + (a1 − 1)f (x) = = (1 − a1 )(C − 2x)(1 − C) + (a1 − a2 )f (C − x), that, first number of critical points is no more than two, i.e. g(x) is parabola, second by virtue of the above mentioned condition g(0)g(C) < 0 in an interval [0, C] there is only one zero g(x). If C > 1, then x ∈ [C − 1, 1], And on the same reasons there is exactly one critical point. The lemma 2.2 is proved. Theorem 2.1. If 0 < a1 < a2 < 1, than for any initial conditions ρ1(0) such, that g(ρ1(0)) 6= 0 (a) If C > 1 then the solution (7) monotonously turns to a critical mode in finite time; (b) If 0 < C < 1 then the solution (7) monotonously converges to the stationary point ρ1(t) ≡ C1 , g(C1) = 0.

4

BUSLAEV: TRAFFIC FLOWER...

89

Proof. System    ρ˙ 1 = (a1 − 1)ρ1(1 − ρ1) + (1 − a2 )ρ2(1 − ρ2)

  ρ˙ 2 = (−a1 + 1)ρ1(1 − ρ1 ) + (−1 + a2 )ρ2(1 − ρ2)

(8)

has the following vector field (see Fig. 3).

X2 C(0,1)

B(1,1)

A(0,1)

O

X1

Figure 3: The vector field of system (8) The set of stationary points ( Fig. 3) consists of two fragments of the hyperbole described by the equation (a1 − 1)x1(1 − x2 ) + (1 − a2)x2 (1 − x2) = 0 at the square OABC. The direction of the vector field is checked directly. The theorem is proved. More general formulation of the problem for two petals has been considered in [3],[5].

3 General Properties of System (3)Solutions We research a question of stationary points (3). Let ρ ~∗ = (ρ∗1...ρ∗n) be a stationary point, f~∗ = (f (ρ∗1)...f (ρ∗n)), ~ (A − E)f~∗ = 0.

(9)

As det(A − E) = 0, then a nontrivial solution of the system (A − E)~x = ~0, ~ x ∈ Rn exists. As AT is a stochastic matrix, for spectral radiuses ρ(A) and ρ(AT ) the following

5

90

BUSLAEV: TRAFFIC FLOWER...

inequality is true ρ(AT ) = 1 ≤ ρ(A.)

(10)

If A is a double stochastic matrix, i.e. A and AT are stochastic, then under condition of positivity of all elements A the Perron-Frobenius theorem [2] is true. Lemma 3.1. There exists and unique to within normalize a vector f~∗ = (f1∗...fn∗) ∈ Rn+ with positive elements such, that Af~∗ = f~∗ .

(11)

f~∗ = λ∗(1, 1, ...1).

(12)

λ∗ = 0.25.

(13)

f (ρ) = a

(14)

It is obvious, that

Suppose that

The equation

has no more than two solutions, at a < 0 or at a > there exists numbers ρ1, ρ2, 0 < ρ1
0, f~∗ is the solution of (12). We consider the following equation (λf1∗) + fδ−1 (λf2∗) + .... + fδ−1 (λfn∗) = C. ϕ~δ(λ) = fδ−1 n 1 2

(17)

The function ϕ~δ (λ), where ~δ is the fixed set of ±1, ~δ ∈ (±1, ±1....., ±1) is continuous and determined on an interval λ ∈ [0, 1]. As A and AT are stochastic matrixes, then f ∗ = (1/4)(1, 1...1), (λ/4) + fδ−1 (λ/4) + .... + fδ−1 (λ/4), ϕ~δ(λ) = fδ−1 n 1 2

(18)

where δi = 1 or δi = −1. As f−−1 (a) + f+−1 (a) = 1,

6

(19)

BUSLAEV: TRAFFIC FLOWER...

91

then ϕ~δ (λ) = s + (n − 2s) f+−1 (λ/4) ,

(20)

where [~δ] = n − 2s is the signature, excess of quantity of coordinates pluses of a vector ~δ above quantity of minuses (n ≥ 2s) , or ϕ~δ (λ) = s + (n − 2s) f−−1 (λ/4)

(21)

if of quantity of coordinates minuses of a vector ~δ excess above quantity of pluses one. Let us denote ~δ± = (~δ ∪ −~δ). Lemma 3.2. Two-value function ϕ~δ± (λ) = s + (n − 2s) f±−1 (λ/4)

(22)

has the continuous plot from two monotonous components with area of values [s, n − s]. Proof follows from an explicitfunction of f (x). Corollary. For any C ∈ [s, n − s], [~δ] = n − 2s, n ≥ 2s there exists and unique λ ∈ [0, 1], wich satisfies the equation ϕ~δ± (λ) = s + (n − 2s) f±−1 (λ/4) = C.

(23)

As at fixed s, that conterminous on parity with n it is possible to choose Cnn−s ways of distribution of marks ± in a chain from n symbols so, that signature will be equal n − 2s (n > 2s), at C ∈ [s, n−s] there is an appropriate quantity of branches of stationary points. Theorem 3.1.

If A is a double stochastic matrix with positive elements, then for

every s = 0, 1...., [n/2] exists Cnn−s branches of stationary points with a range of definition C ∈ [s, n − s]. For n = 3 this result is present in [6] for partial singular case of an one-dimensional chain.

4 Traffic Stability on Double Petal Flower for One-parametrical Families of State Functions Let’s consider double petal traffic flower with state function , i.e. dependence of flow speed from density of the following type v(x) = (1 − x)γ , 0 ≤ x ≤ 1, i.e., f (x) = f (x, γ) = x(1 − x)γ , 0 ≤ x ≤ 1.

7

(24)

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BUSLAEV: TRAFFIC FLOWER...

α1 2

1

α2

1-α1

1-α2

Figure 4: The dual image of a 2-flower. In case of a stochastic matrix of transition we have α1 = α2 = α. Thus, the equation for density, for example, on the first petal looks like

ρ˙ = −αf (ρ) + αf (C − ρ) = −αf (ρ, γ) + αf (C − ρ, γ) = g(ρ, γ, C)

(25)

with following conditions max(0, C − 1) ≤ ρ ≤ min(1, C).

(26)

At γ = 1 we receive model of paragraph 2, i.e. the solution (25) is stable in and only if C < 1. If γ = 0, that ρ˙ = −αρ + α(C − ρ) = α(C − 2ρ.)

(27)

The equation (27) has a stationary point ρ∗ = C/2,, which is stable for any permitted initial conditions. We research a problem of stability of the equation (25) - (26) at any γ ∈ [0, ∞). Lemma 4.1. (1) Function f (x, γ) = x(1 − x)γ achieves a maximum in the unique critical point f 0(x∗ , γ) = 0, x∗ =

1 1+γ ;

1 ] and also decreases on an interval (2) Function f (x, γ) increases on an interval [0, 1+γ 1 , 1]; [ 1+γ

(3) If 1 < C < 2(1 + γ)−1, then on an interval [max(0, C − function f (x) increases, and function f (C − x) decreases. Proof. It is checked directly.

8

1 1 1+γ ), min( 1+γ , C)]

the

BUSLAEV: TRAFFIC FLOWER...

93

f(x,γ)

f(c-x,γ)

C-1

C

/2

1

C

x

Figure 5: 0 < γ < 1, 1 < C < 2. Lemma 4.2. (1) Function g(x, γ, C) is determined on an interval [max(0, C−1), min(1, C)]; (2) g(x, γ, C) is symmetric concerning the point (C/2, g(C/2, γ, C) = 0), i.e. is odd relative to x = C/2; (3) Let C > 1. If C −

1 1+γ




2 γ

then there is unique and unstable stationary point ρ ≡ C/2 with area

of achive in critical mode [C − 1, C/2), (C/2, 1]. (2) Let 1 < γ < ∞. Then (2.1) If 1 < C < 2, then there is a unique stationary point ρ ≡ C/2 with area of an achive in the critical mode [C − 1, C/2), (C/2, 1]. (2.2) If

2 γ+1

< C < 1, then there is the unstable stationary point ρ ≡ C/2 and two

stable stationary points ρ ≡ ρ∗1 with area of an attraction 0 < ρ < C/2, and ρ ≡ ρ∗2 with area of the attraction C/2 < ρ < C. (2.3) If 0 < C
0, n

(31)

then the solution ρ1 (t) increases until the condition − 2n−1 n ρ1 (t) + C = 0. will not be executed yet. Therefore, if Cn ≥ 1, 2n − 1

(32)

the system (23) passes gets into a critical mode, as (25) in this case it is carried out automatically. If

and if ρ1(0) >

Cn 2n−1

Cn 2n−1 ,

< 1, then by virtue (32) ρ1 (0) < ρ1 (t) %

Cn , 2n − 1

ρ1 (t) &

Cn . 2n − 1

Cn 2n−1

and ρ1(t) increases on t

then

Thus ρ1(t) does not get a critical mode. As the obvious kind of the solution (28) looks like ρ1(t) = that at k ≥ 2 1 ρk (t) = n

Z

(

2n−1 Cn + Ae− n t , 2n − 1

2n−1 Cn + Ae− n t )e−t dt = 2n − 1

11

(33)

96

BUSLAEV: TRAFFIC FLOWER...

=− If A > 0, i.e. ρ1(0) > If ρ1 (0)
1.

5.2 Stochastic Flow As before A = (aij ) is the double stochastic matrix n ∗ n, i.e. aij > 0, ∀i, j

Pn

j=1

aij = 1,

i = 1, ..., n. Therefore ρ ~ = (ρ1, ..., ρn) is the vector with not negative coordinates, 0 ≤ pi ≤ 1, i = 1, ..., n, k~ ρkln = ρ1 + ... + ρn = C.

(35)

1

Let’s consider a system    ρ˙ 1 = (a11 − 1)ρ1 + a21ρ2 + ... + an1 ρn       ρ˙ 2 = a12ρ1 + (a22 − 1) ρ2 + ... + an2 ρn

  ................................................................       ρ˙ = a ρ + a ρ + ... + (a − 1) ρ , n

1n 1

2n 2

nn

(36)

n

Under the condition of the Perron-Frobenius theorem the system (36) has unique stationary point f ∗ = C(1, 1, ...1)/n

(37)

Theorem 5.1. For any allowable initial conditions ρ ~(0), k~ ρkln∞ = max |ρi| < 1, a solution(35)-(36) 1≤i≤n

converges to the stationary (37) for any allowable loading 0 < C < n. Proof. See also [8]. Lemma 5.1. If ~ ρ(t) is the solution of the system (37) with initial conditions, wich satisfies (35) and (36), then ~ = f~∗ . lim ρ(t)

t→∞

12

BUSLAEV: TRAFFIC FLOWER...

97

Lemma 5.2. If ρ ~(t) is the solution of the system (37), then ρ ~(t) is separated from critical modes max |ρi | = 1. 1≤i≤n

6 Optimization and Control on Traffic Flower The traffic flow on a flower is characterized by a vector of density ρ ~ = (ρ1..., ρn) (pic. 4, n = 4) and a intensity vector ~ q = (q1 = f (ρ1)..., qn = f (ρn )).

№1

№4

№2

№3 Figure 6: Traffic 4 - a flower As the length of each edge is equal to unit, as well as weight of each particle of flow, then qi , i = 1..., n is capacity of a flow on an edge with number i, and the value Σi qi = Q is the capacity of a flow on a transport flower at fixed t, and the value Z t1

Q(t)dt = A(t)

t0

is the flow work in a time interval [t0 , t1 ]. One of probable criterias of control is Q(t) → max, kρkl1 = ρ1 (t) + ... + ρn (t) ≡ C i.e. f (ρ1) + ... + f (ρn ) → max, ρ1(t) + ... + ρn (t) ≡ C.

13

(38)

98

BUSLAEV: TRAFFIC FLOWER...

№1

№4

№2

№3 Figure 7: The dual image of a 4-flower. Necessary conditions of extremum of a problem (38) are f 0 (ρ∗i ) = λ, or ρ∗i = 0. If f 0 is monotonously decreases, then within normalized critical points iof the problem (38) looks like ρ∗1 = ρ∗2 = ... = ρ∗m , ρm+1 = ... = ρn = 0. Hence, ρ∗i = C/m, 1 ≤ i ≤ m and we have Qm = mf (C/m) → max, 1 ≤ m ≤ n. We assume that function H(x) = f (x)/x, 0 ≤ x ≤ C. If function H(x) decreases on an interval [0, 1], then a sequence Qm grows on parameter m. So in a case f (x) = x(1 − x) we will receive H(x) = (1 − x), i.e. H 0(x) < 0. So maxm (mf (C/m)) is reached at m = n. Thus, it is true. Theorem 6.1. If f 0 (x) and H(x) monotonously decrease on an interval [0, 1], then the maximal capacity at any allowable loading 0 < C < n it is reached for equaldistributed flow ρ∗i = C/n, i = 1.., n.

14

BUSLAEV: TRAFFIC FLOWER...

7. Referenses [1] Drew D.R. Traffic flow theory and control / NY, McGraw Hill, 1968. [2] Gantmaher F.R. Matrix theory (in Russian), Moscow, Phismathgiz, 1962. [3] Buslaev A.P., Tatashev A.G., Yashina M.V. About flows on graphs/ Vladikavkaz mathematical journal, 2004. V. 6, N.4, P. 4-18. [4] Nazarov A.I. About stability of one system of nonlinear ODE from modelling of traffic flow / Vestnik of the St.-Peterburg Univ., 2006, Ser.1, N 3, P. 35-42 ,(in Russian) [5] Buslaev A.P., Tatashev A.G., Yashina M.V./ Stability of Flows on Networks. /Proceedings of International Conference ”Traffic and Granular Flows -2005”, Springer, 2006, P. 427-435 [6] Lukanin V.N.,Buslaev .P.,Trofimenko Y.V.,Yashina .V. Traffic Flows and Environment, (in Russian). Moscow, Infra-M ,1998 [7] Buslaev A.P., Tatashev A.G., Yashina M.V. Stability of Flows on graph with tree links/ Proceedings of International Conference ”Traffic and Granular Flows -2007”, Springer, 2009 (in print) [8] Van Kampen N.G. Stochastic Processes in Physics and Chemistry / North Holland Physics Publ.,1984

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100 JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.5, NO.1, 100-112, COPYRIGHT 2010 EUDOXUS PRESS, LLC

ON THE RELATIVE CONTROLLABILITY AND MINIMUM ENERGY CONTROL OF SYSTEMS WITH DELAY IN STATE AND CONTROL VARIABLES DAVIES IYAI Department of Mathematics and Computer Science, Rivers State University of Science and Technology, P.M.B 5080, Port Harcourt, Rivers State, Nigeria. Email: [email protected] Abstract This research work establishes necessary and sufficient conditions for relative controllability and minimum control energy of linear time varying systems with delays in state and control variable. The aim is to use the integral equivalence of our system to deduce our controllability matrix and the reachable set. With the aid of the properties of these matrices, we achieve our results by some equivalent relationship. Keywords: Relative controllability, control energy, linear system, controllability matrix, reachable set. 2000 Mathematics Subject Classification: Primary 93B05, Secondary 34H05 1.

INTRODUCTION

The study of controllability of systems, first carried out in details by Kalman [11] has attracted lots of interest in modern research notably from Davies and Jackreece [5], Jackreece and Davies [10], Obukhovskii and Rubbioni [13] etc. because of it’s direct connectivity with variety of mathematical models. Controllability of systems can be encountered in many fields of science, engineering and amongst others environmental management, industrial processes, medicine, biology, economy. This research work is not only interested in the controllability of systems but also in reaching the targets of systems with minimum wastage of control energy. It is evident that successes in life pursuits are predicated upon our ability to direct our energies in reaching the desired target. This enigma has created the necessity for us to sort out for ways of achieving this success with minimum energy control in the shortest possible time. However, the need to steer the state of a system from an initial point to a desired target with minimum energy control poses a challenge despite the pioneering and ambitious work on the optimal control problem of single degree-of-freedom differential system of the form

&x& + f ( x, x& ) = u

(1.1)

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101

by Lee and Markus [2] where they found controllers for the system (1.1) which steers the initial states of the system to the equilibrium time optimally, and then indicatated how these controllers so found for each initial state can be used to construct the feedback controller synthesis. Hermes and Lasalle [3] gave the linear time optimal control problem for the ordinary differential system

x& (t ) = A(t ) x (t ) + B (t )u (t )

(1.2)

by finding a control subject to its constraints in such a way that the solution of (1.2) reaches a continuously moving target in the state space in minimum time. They further gave this minimum energy control required to transfer the system from an initial state to target and defined the energy control on the assumption that the system is controllable. The dividend of these earlier works is in lending focus and clarity of definition to the minimum energy control problem. Recently, Iheagwam [12], Davies [4] working independently gave answers to the following questions that form the crux of the minimum energy control problem. Does a minimum energy control for the pursuit of a moving target in a context described by a differential system exist? What is the form of this energy control? Is it unique? Sebakhy and Bayoumi [9] studied the controllability of linear time-varying systems with delay in control of the form

x& (t ) = A(t ) x (t ) + B (t )u (t ) + C (t )u (t − h)

(1.3)

were they gave an expression for the control required to transfer the system (1.3) from a given state to any desired state using minimum control energy. Klamka [6], using the relative controllability matrix of a linear time varying system with distributed delays in the control of the form

x& (t ) = A(t ) x(t ) +

0

∫ [dH (t , s)]u(t + s)

(1.4)

−h

examined the minimum control energy and derived the control law for the system (1.4). The objective of this research is to extend the work in Klamka [6] by establishing necessary and sufficient condition for the relative controllability and also derive control laws which establishes the minimum energy control required to transfer a class of linear time varying systems with delays in state and control variables given by h

x& (t ) = ∑ Ai x(t − i ) + B u (t ) + i =0

h

∑ D u (t − i ) i

i =1

2

(1.5)

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t ∈ [t 0 , t1 ] , x(t ) = φ (t ) , φ ∈ [− h,0] An example is also given 2. BASIC NOTATIONS AND PRELIMINARIES

n × n matrices and B , Di are n × m

Consider the system (1.5), where x (t ) is an n-vector, Ai are matrices. The control function u (t ) ∈ E line and E

n

m

is a measurable m − vector. Here E = ( −∞ , ∞ ) is the real

is the n-dimensional Euclidean space with norm

⋅ . We let C = (C[− h,0] , E n ) be the

Banach space of continuous functions and the norm of an element

φ in C by

φ = sup φ ( s ) − h≤ s ≤0

We let L1 ([a, b] , E ) be the space of Lebesgue integrable functions taking [ a , b] into E m

φ = ∫ φ ( s) ds , φ ∈ L1 ( [a, b] , E n

n

with

).

b

a

L 2 ([a, b] , E m ) is the space of square integrable functions taking [a, b] → E m If x ∈ C ([a, b] , E ) for any a < b , then for each fixed t ∈ [ a, b] , the symbol xt denotes an element n

of

C given by xt ( s ) = x(t + s ) . For functions u ∈ L 2 ([a, b] , E m ) the symbol u t is similarly defined.

(

In this paper, the control space will be L2 [a, b] , E loc

m

), the space of essentially bounded measurable

m

functions on finite intervals with values in E . The control constraint set will be denoted by

(

)

{

}

m m m U = Lloc , where C = u ∈ E : u j ≤ 1, j = 1,......, m 2 [ a, b] , E

The above conditions on Ai , B and Di ensure that for each initial data (t 0 , φ ) , a unique solution of (1.5) exists through (t 0 , φ ) (see Hale [7], p. 142) which is continuous in (t 0 , φ ) .The solution of system (1.5) at

t = t1 following Sebakhy and Bayoumi [9], Manitius and Olbrot [1] will be given as

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IYAI: ON RELATIVE CONTROLLABILITY...

h

x(t1 , t 0 , φ , u ) = X (t1 , t 0 ) φ (t 0 ) + ∑

t0

∫ X (t , s + i) A φ (s) ds 1

i =1 −i

t1

h

t0

i =1

103

i

+ ∫ X (t1 , s ) B u ( s ) + ∑ Di u ( s − i ) ds or h

x(t1 , t 0 , φ , u ) = ∑

t0

∫ X (t , s + i) A φ (s) ds 1

i −1 −i

h ⎧⎪ + X (t1 , t 0 ) ⎨ φ (t 0 ) + ∑ ⎪⎩ i =1

+

t1 − i

∫

⎫⎪ X ( t , s + i ) D u ( s ) ds ⎬ i t ∫ 0 ⎪⎭ −i t0

h

X (t1 , s ) B + ∑ X (t1 , s + i ) Di u ( s ) ds i =1

t0

+

i

t1

∫ X (t , s) B u (s) ds 1

(2.1)

t1 − i

where X (t , s ) is the fundamental solution of (1.5) which satisfies the equation h ∂ X (t , s ) = ∑ Ai X (t − i, s ) , t > s ∂t i =0

X (t , s ) =

⎧I , ⎨ ⎩0 ,

(2.2)

t=s t s ∂s i =1

(2.3)

Following the methods of Dauer and Gahl [8], we define a matrix function Z by h ⎧ X ( t , s ) B X (t1 , s + i ) Di for t 0 ≤ s < t1 − i + ⎪ 1 ∑ Z (t 0 , l , z ) = ⎨ i =1 ⎪ X (t , s ) B for t − i ≤ s ≤ t 1 1 ⎩ 1

We now give some definition upon which our study hinges.

4

(2.4)

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Definition 2.1 The complete state at time t of system (1.5) is said to be z (t ) = { x(t ) , φ , u t

}.

Definition 2.2 Following Arstein [14], we define relative controllability of system (1.5). That, system (1.5) is relatively controllable on [t 0 , t1 ] , if for every z (t 0 ) and every vector x1 ∈ E , there exists a control n

that the corresponding trajectory of system (1.5) satisfies We now define the

u ∈ B , such

x(t1 ) = x1 .

n × n symmetric and semi positive controllability matrix of our system as t1

W (t 0 , t1 , z ) = ∫ Z (t 0 , l , z ) Z T (t 0 , l , z ) dl t0

where the symbol T denotes the matrix transpose. The controllability matrix satisfies the condition

x 2W (t 0 , t1 , z ) x = x, W (t 0 , t1 , z ) x = x where

2 W ( t 0 ,t1 , z )

≥ 0 for all x ∈ E n

⋅, ⋅ denotes the inner product space in E n

The reachable set of (1.5) at

t1 is given by

⎧⎪ t1 ⎫⎪ R (t 0 , l , z ) = ⎨ ∫ Z (t1 , l , z ) u (l ) dl : u ∈ L2 ⎬ ⎪⎩ t0 ⎪⎭

(2.5)

here R (t 0 , l , z ) is an m × m continuous symmetric matrix defined on [t 0 , t1 ] such that

R − (t 0 , l , z ) exists for all t ∈ [t 0 , t1 ] . We now introduce the following notation for brevity.

W1 (t 0 , t1 , z ) =

∫ [Z (t , l , z ) R (t , l , z )].Z t1

0

-

0

T

(t 0 , l , z )dl

(2.6)

t0

−1

u = R - (t0 , l , z ) Z T (t0 , l , z ).W1 (t0 , t1 , z ) R(t0 , l , z )

(2.7)

3. CONTROLLABILITY RESULTS Here sufficient conditions for the controllability of system (1.5) when formulated and proved.

5

L2 control is assumed are

IYAI: ON RELATIVE CONTROLLABILITY...

105

Theorem 3.1 The following are equivalent (i) W (t 0 , t1 , z ) is non singular for each t (ii) System (1.5) is proper in E

n

for each interval [t 0 , t1 ]

(iii) System (1.5) is relatively controllable on each interval [t 0 , t1 ] Proof: We shall show that (i) ⇒ (ii), (ii) ⇒ (iii) and (iii) ⇒ (i) to complete the proof. Let us show first that (i) ⇒ (ii) m

Define the operator, K : L2 ([t 0 , t1 ], E ) → E

K (u ) =

t1

∫ Z (t

0

n

by

, l , z ) u (l ) dl

(3.1)

t0

where K is a continuous linear operator from one Hilbert space to another. Thus, R ( K ) ⊆ E is a linear n

subspace and its orthogonal complement satisfies the relation

{R( K )}⊥

= N (K ∗ )

(3.2)

∗

∗

where K is the adjoint of K , K : E → U ⊆ L2 by the non-singularity of the controllability matrix n

W (t 0 , t1 , z ) , the symmetric operator KK ∗ = W (t 0 , t1 , z ) is positive definite and hence

{R( K )}⊥ = {0}

(3.3)

N ( K ∗ ) = {0}

That is,

For any c ∈ E , n

(3.4)

u ∈ L2 , < c , Ku > = < K ∗ c, u > t1

t1

t0

t0

< c , K u > = < c , ∫ Z (t 0 , l , z ) u (l) dl > = ∫ c T [Z (t 0 , l , z ) ] u (l) dl Thus, K

∗

is given by c → c

T

[Z (t 0 , l , z ) ];

(3.5)

l ∈ [t 0 , t1 ] . N ( K ∗ ) is therefore the set of all c ∈ E n

such that

c T [Z (t 0 , l , z ) ] = 0

(3.6)

6

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IYAI: ON RELATIVE CONTROLLABILITY...

almost everywhere in [t 0 , t1 ] , since N ( K ) = {0} all such ∗

c are equal to zero, that is c = 0 .This

establishes the properness of system (1.5). (ii) ⇒ (iii) We now show that system (1.5) is relatively controllable on each interval [t 0 , t1 ] . Let c ∈ E , if system (1.5) is proper then c n

T

[Z (t 0 , l , z ) ] = 0

almost everywhere such that l ∈ [t 0 , t1 ] for each t1

∫ c [Z (t T

0

t1 implies c = 0 . Thus

, l , z ) ] u (l )dl = 0

t0

for

u ∈ L2 . It follows that the only vector orthogonal to the set ⎧⎪ t1 ⎪⎫ R(t 0 , l , z ) = ⎨ ∫ c T Z (t 0 , l , z ) u (l ) dl : u ∈ L 2 ⎬ ⎪⎩t0 ⎪⎭

is the zero vector. Hence

{R(t 0 , l , z )}⊥ = {0} That is, R (t 0 , l , z ) = E . This establishes relative controllability on [t 0 , t1 ] of n

system (1.5). (iii) ⇒ (i) We now show that if the system (1.5) is relatively controllable then W (t 0 , t1 , z ) is non-singular. Let us assume for contradiction that W is singular, then, there exists an

VWV T = 0

n vector V ≠ 0 such that (3.7)

Then t1

∫

2

V [ Z (t 0 , l , z )] dl = 0

(3.8)

t0

This implies that 2

V [ Z (t 0 , l , z )] dl = 0 almost everywhere, hence

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IYAI: ON RELATIVE CONTROLLABILITY...

107

V [ Z (t 0 , l , z )] = 0

(3.9)

almost everywhere for l ∈ [t 0 , t1 ] . This contradicts the assumption of properness of the system since

V ≠ 0 and this completes the proof. Corollary 3.1 System (1.5) is relatively controllable on [t 0 , t1 ] if and only

rank W (t 0 , t1 , z ) = n Proof: This is Corollary 1 of Sebaky and Bayoumi [9] Theorem 3.2 System (1.5) is relatively controllable if and only if 0 ∈ int R (t 0 , l , z ) for each t1 > t 0 Proof: If R (t 0 , l , z ) is a closed and convex subset of E , then a point n

y1 on the boundary of

R(t 0 , l , z ) implies that, there is a support plane Π of R (t 0 , l , z ) through y1 , that is c T ( y − y1 ) ≤ 0 for each y ∈ R (t 0 , l , z ) where

c ≠ 0 is an outward normal to Π . If u1 is the control corresponding to

y1 , we have t1

t1

t0

t0

c T ∫ [Z (t 0 , l , z )]u (l )dl ≤ c T ∫ [Z (t 0 , l , z )]u1 (l )dl for each

u ∈ U . Since U is a unit sphere, this last inequality holds for each u ∈ U , if and only if t1

t1

c T ∫ [Z (t 0 , l , z )] u (l )dl ≤ ∫ c T [Z (t 0 , l , z )] u1 (l )dl t0

t0 t1

≤ ∫ c T [Z (t 0 , l , z )] dl t0

8

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IYAI: ON RELATIVE CONTROLLABILITY...

and u1 (t ) = sgn c Z (t 0 , l , z ) as T

were not in the interior of

y1 is on the boundary. Since we always have 0 ∈ R (t 0 , l , z ) . If 0

R(t 0 , l , z ) then 0 is on the boundary, hence, from the foregoing, this implies

t1

0 = ∫ c T [Z (t 0 , l , z )] dl

so that c

T

[Z (t 0 , l , z )] = 0

almost everywhere

t ∈ [t 0 , t1 ] . This by our

t0

definition implies that the system is not proper since c

T

≠ 0 . This completes the proof.

Theorem 3.3 - If system (1.5) is relatively controllable on

[t 0 , t1 ] for each t1 > t 0 , then the domain of

null controllability of system (1.5) contains zero in its interior Proof - Assume that system (1.5) is relatively controllable on

[t 0 , t1 ] , t1 > t 0 then by Theorem 3.2,

0 ∈ int R(t 0 , l , z ) , for each t1 > t 0 . Since x = 0 is a solution of system (1.5) with u = 0 , we have

0 ∈ D . Hence, If 0 ∉ int D , then there exists a sequence φ m ⊆ E n such that φ m → 0 as m → ∞ and no

φm

is in D , that is

φ m ≠ 0 . From (2.1), we have h

0 ≠ x(t1 , t 0 , φ , u ) = X (t1 , t 0 ) φ (t 0 ) + ∑

t0

∫ X (t , s + i) A φ ( s) ds 1

i =1 − i

t1

h

t0

i =1

i

+ ∫ X (t1 , s ) B u ( s ) + ∑ Di u ( s − i ) ds for any

t1 > t 0 and any u ∈ U . Hence, for u = 0 , def

h

z m = x(t1 , t 0 , φ ,0) = X (t1 , t 0 ) φ (t 0 ) + ∑

t0

∫ X (t , s + i) A φ (s) ds ,

i =1 −i

is not in

1

i

R(t 0 , l , z ) for any t1 > t 0 . Therefore the sequence z m ⊆ E n is such that

z m ∉ R(t 0 , l , z ) , z m ≠ 0 , but z m → 0 as m → ∞ . Therefore, 0 ∉ R (t 0 , l , z ) . This is a contradiction and hence proves that 0 ∈ int D . 4. THE MINIMUM ENERGY CONTROL Here we derive an explicit expression for the control that transfers system (1.5) from

t1 and examine the minimum energy control required for this transfer.

9

z (t 0 ) to x1 at time

IYAI: ON RELATIVE CONTROLLABILITY...

109

Theorem 4.1 Let u (t ) be any control which transfer

z (t 0 ) with the initial control u (t0 ) to x1 at time t1 and let

u ∗ (t ) be the control defined by (2.7), then t1

∫

t1

2

u (l )

R ( t0

t0

dl ≥ ∫ u ∗ (l ) ,l , z ) t0

almost everywhere on

2 R ( t 0 ,l , z )

(4.1)

dl

[t 0 , t1 ] and the minimum control energy required for the transfer ( assuming the

transfer is possible ) is given by t1

E (u ∗ ) = ∫ u ∗ (l ) t0

2 R ( t 0 ,l , z )

dl = R(t 0 , l , z )

2

(4.2)

W1−1 ( t 0 ,t1 , z )

Proof: substituting (2.7) into (2.1), it is easy to verify that the control

u ∗ (t ) transfers the complete state

z (t 0 ) to x1 at time t1 . Since u (t ) and u ∗ (t ) transfer z (t 0 ) to x1 at time t1 , we have the following equalities t1

∫ Z (t

t0

t1

0

, l , z )u (l )dl = ∫ Z (t 0 , l , z )u ∗ (l )dl

(4.3)

t0

subtracting both sides and using the inner product gives t1

∫ Z (t

−1

0

, l , z )(u (l ) − u ∗ (l )) dl.W1 (t 0 , t1 , z ) R (t 0 , l , z ) = 0

(4.4)

t0

using (2.7) and the properties of the inner product, we obtain t1

∫

u (l ) − u ∗ (l ), u ∗ (l ) dl = 0

(4.5)

t0

By some easy manipulation and using (4.5) we derive t1

∫

t0

u (l )

2 R ( t0

t1

dl ≥ ∫ u ∗ (l ) ,l , z ) t0

2 R ( t 0 ,l , z )

(4.6)

dl

10

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IYAI: ON RELATIVE CONTROLLABILITY...

The minimal value of the control energy in transferring

is given by

t1

E(u ) = ∫ u (l) ∗

∗

t0

(4.7) Since the matrix

z (t 0 ) to x1 at time t1 by using the control u ∗ (t )

t1

2 R(t0 ,l , z )

2

dl = ∫ R−1 (t 0 , l, z) Z T (t 0 , l, z).W −1 (t 0 , t1 , z)R(t 0 , l, z) dl R(t 0 , l, z) t0

W1 (t1 , t 0 , z ) is symmetric, by the properties of the inner product and by (4.7), we

have t1

[

][

][

]

E (u ) = ∫ R(t 0 , l , z )W1 (t1 , t 0 , z ) . Z (t 0 , l , z ) R −1 (t 0 , l , z ) . Z T (t 0 , l , z )W1 (t 0 , t1 , z ) R(t 0 , l , z ) dl ∗

−1

−1

t0

−1

= R(t 0 , l , z ),W1 (t 0 , t1 , z ) R (t 0 , l , z ) 2

= R(t 0 , l , z ) W −1 ( t This completes the proof.

1

0 ,t1 , z )

5. EXAMPLE Consider the system

x& (t ) = A0 x(t ) + A1 x(t − 1) + Bu (t ) + D1u (t ) + D2 (t − 1)

(5.1)

where

0⎞ ⎛−1 ⎛0 ⎟⎟ , A1 = ⎜⎜ A0 = ⎜⎜ ⎝ 0 − 2⎠ ⎝1 ⎛1 ⎞ D0 = ⎜⎜ ⎟⎟ , ⎝0⎠

0⎞ ⎛ 0⎞ ⎟⎟ , B = ⎜⎜ ⎟⎟ 0⎠ ⎝1 ⎠

⎛0 ⎞ D1 = ⎜⎜ ⎟⎟ ⎝ − 1⎠

To verify relative controllability of (5.1), it is easily seen that, the principal fundamental matrix solution is given by

⎛ e −t X (t , s ) = ⎜⎜ 1−t 1− 2 t ⎝e − e

0 ⎞ ⎟ e − 2t ⎟⎠

11

IYAI: ON RELATIVE CONTROLLABILITY...

and the matrix Z (t 0 , l , z ) as defined in (2.4) will be given by

matrix is given by

t1 ⎛0 W (t 0 , l , z ) = ∫ ⎜⎜ t0 ⎝ 0

111

⎛ 0 ⎞ Z (t 0 , l , z ) = ⎜⎜ −t ⎟⎟ . The controllability ⎝e ⎠

0 ⎞ ⎟ dt , while rank W (t 0 , l , z ) = 1 . This implies that e − 2t ⎟⎠

system (5.1) is relatively controllable by Corollary 3.1. Furthermore, we verify the minimum control energy for system (5.1) as follows; we require the reachable set defined in (2.5) with

u = 1 and is given by

⎧⎪ t1 ⎛ 0 ⎞ ⎫⎪ R(t 0 , l , z ) = ⎨ ∫ ⎜⎜ − 2t ⎟⎟ dt ⎬ ⎪⎩ t0 ⎝ e ⎠ ⎪⎭ Hence, the minimum control energy will be given as

⎛0 ⎞ E (u ) = ∫ ⎜⎜ − 2t ⎟⎟ dt ⎠ t0 ⎝ e ∗

t1

2

CONCLUSION From the sequel, the relative controllability of system (1.5) has been established. Also established is the relationship between relative controllability of the system (1.5) and the domains of relative controllability. This study has been able to show that, if a system is relatively controllable then zero is in the interior of the domain of the reachable set. From the above results, the minimum control energy required to transfer system (1.5) from an initial state to a targeted state within a specified time limit in the state space has also been established.

12

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REFERENCES 1. A. Manitius and A. W. Albrot, Controllability conditions for linear systems with delayed state and control, Arch. Automatic Tele mech. 17119 – 131 (1972). 2. E. B. Lee and L. Markus, Foundations of optimal control theory, John Wiley New Yok 1967. 3. H. Hermes and J. P. Lassale, Functional analysis and time optimal control, Academic Press, New York, 1969. 4. I. Davies optimal control of linear discrete systems with delay in state and control. ABCUS: The Journal of the Mathematical Association of Nigeria 33(2B) 451-461 (2006) 5. I. Davies and P. Jackreece, Controllability and null controllability of linear systems, J. Appl. Sci. Environ. Mgt. 9 31-36 (2005). 6. J. Klamka, Relative controllability and minimum energy control of linear systems with distributed delays in control, IEE Transaction on Automatic Control AC-21 594-595 (1976). 7. J. K. Hale, Theory of functional differential equations, Springer – Verlag, New York, 1977. 8. J. P. Dauer and R. D. Gahl, Controllability of nonlinear delay systems, Journal of Optimization Theory and Application, 21 59 – 70 (1977). 9. O. Sebakhy and M. N. Bayoumi, Controllability of linear time varying systems with delay in the control, Int. Journal of control, 17 127 – 135 (1973). 10. P. Jackreece and I. Davies, Euclidean null controllability of nonlinear infinite delay systems with varying multiple delays in control and implicit derivative, J. Appl. Sci. Environ. Mgt. 9 57-62 (2005). 11. R. E. Kalman Contributions to the theory of optimal control, Bol. Soc. Mat. Mex. 5 102-119 (1960). 12. V. A. Iheagwarm, The Minimum control energy problem for differential systems with distributed delays in control, Journal of the Nigeria Mathematical Society, 21 85-96 (2002), 13. V. Obukhovskii and P. Rubbioni, On a controllability problem for systems governed by semi linear functional differential inclusion in Banach spaces. Journal of the Juliusz Schauder Center, 15 141151 (2000). 14. Z. Arstein, Linear systems with delayed control: A reduction, IEEE Trans. Aut. Control, A C – 27 869 – 879 (1982).

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JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.5, NO.1, 113-120, COPYRIGHT 2010 EUDOXUS PRESS, LLC 113

Global Existence of Solutions for Nonlinear integral equations of second order H. L. Tidke Department of M athematics N orthM aharashtraU niversity, Jalgaon, India, [email protected] ABSTRACT We prove the existence of mild solutions of nonlinear mixed Volterra-Fredholm integrodifferential equations with nonlocal conditions in Banach spaces. The results are obtained by using the theory of strongly continuous cosine family, application of the topological transversality theorem known as Leray-Schauder alternative and rely on a priori bounds of solutions. 1. Introduction Let X be a Banach space with norm k.k. Let B = C([0, b], X) be the Banach space of all continuous functions from [0, b] into X endowed with supremum norm kxkB = sup{kx(t)k : t ∈ [0, b]}. This paper concerns the mixed Volterra-Fredholm integrodifferential equation of the type: Z t Z b
00 x (t) = Ax(t) + f t, x(t), k(t, s, x(s))ds, h(t, s, x(s))ds , t ∈ J = [0, b] (1.1) 0

x(0) + g(x) = x0 ,

0

x0 (0) = η,

(1.2)

where A is an infinitesimal generator of a strongly continuous cosine family {C(t) : t ∈ R} in Banach space X, f : J × X × X × X → X, k, h : J × J × X → X, g : C(J; X) → X are functions, and x0 is a given element of X. The existence, uniqueness and other properties of solutions of these equations (1.1) or special forms with classical conditions have been studied by many authors by using different techniques, see [10, 11, 12, 13, 14, 22]. The work on nonlocal initial value problem(IVP for short) was initiated by Byszewski. In [6, 7] Byszewski using the method of semigroups and the Banach fixed point theorem proved the existence and uniqueness of mild, strong and classical solution of first order IVP. For the importance of nonlocal conditions in different fields, the interesting reader is referred to [1, 2, 3, 4, 9, 20, 21, 24, 25, 26, 27] and the references are given therein. For properties of semigroup theory, we refer the reader to the books [8, 18, 19, 23] The objective of the present paper is to study the global existence of solutions of the equations (1.1)–(1.2). The main tool used in our analysis is based on an application of the topological transversality theorem known as Leray-Schauder alternative, rely on a priori bounds of solutions and the theory of strongly continuous cosine family. The interesting and useful aspect of the method employed here is that it yields simultaneously the global existence of solutions and the maximal interval of existence. The paper is organized as follows. In section 2, we present the preliminaries and hypotheses. Section 3 deals with main result. 1

114

TIDKE: ABOUT NONLINEAR INTEGRAL EQUATIONS

2

2. Preliminaries and Hypotheses Before we state our main result, we list the some preliminaries and hypotheses that will be used in our subsequent discussion. In many cases it is advantageous to treat second order abstract differential equations directly rather than to convert them to first order systems. We will make use of some of the basic ideas from cosine family and it is useful machinery for the study of abstract second order equations. We say that a family {C(t) : t ∈ R} of operators in the space L(X) of bounded linear operators on X is a strongly continuous cosine family if (i) C(0) = I (I is the identity operator); (ii) C(t)x is strongly continuous in t on R for each fixed x ∈ X; (iii) C(t + s) + C(t − s) = 2C(t)C(s) for all t, s ∈ R. The strongly continuous sine family {S(t) : t ∈ R}, associated to the given strongly continuous cosine family {C(t) : t ∈ R}, is defined by Z t S(t)x = C(s)xds, x ∈ X, t ∈ R. 0

The infinitesimal generator A : X → X of a cosine family {C(t) : t ∈ R} is defined by d2 C(t)x|t = 0, dt2

Ax =

x ∈ D(A),

where D(A) = {x ∈ X : C(.)x ∈ C 2 (R, X)}. For more details on strongly continuous cosine and sine families, we refer the reader to the book of Goldstein [19] and to the papers of Fattorini [16, 17] and Travis and Webb [26, 27]. Definition 2.1. A continuous solution x(t) of the integral equation x(t) = C(t)[x0 − g(x)] + S(t)η Z t Z + S(t − s)f s, x(s), 0

s

b

Z


h(s, τ, x(τ ))dτ ds,

k(s, τ, x(τ ))dτ,

0

t∈J

0

is called a mild solution of (1.1)–(1.2) on J. Let us list the following hypotheses: (H1 ) A is the infinitesimal generator of a strongly continuous cosine family {C(t) : t ∈ R} compact for t > 0, and there exists a constant M such that kC(t)kL(X) ≤ M,

for all

t ∈ R.

(H2 ) There exists a constant G such that kg(x)k ≤ G, for x ∈ X. (H3 ) There exists a continuous function p : [0, b] → R+ such that Z t k k(t, s, x(s))dsk ≤ p(t)kxk, 0

for every t, s ∈ [0, b] and x ∈ X. (H4 ) There exists a continuous function q : [0, b] → R+ such that Z b k h(t, s, x(s))dsk ≤ q(t)kxk, 0

for every t, s ∈ [0, b] and x ∈ X.

TIDKE: ABOUT NONLINEAR INTEGRAL EQUATIONS

115

3

(H5 ) There exists a continuous function l : [0, b] → R+ and a continuous increasing function K : R+ → (0, ∞) such that kf (t, x, y, z)k ≤ l(t)K(kxk + kyk + kzk), for every t ∈ [0, b] and x, y, z ∈ X. (H6 ) For each t ∈ [0, b] the function f (t, ., ., .) : [0, b] × X × X × X → X is continuous and for each x, y, z ∈ X the function f (., x, y, z) : [0, b] × X × X × X → X is strongly measurable. (H7 ) For each t, s ∈ [0, b] the functions k(t, s, .), h(t, s, .) : [0, b] × [0, b] × X → X are continuous and for each x ∈ X the functions k(., ., x), h(., ., x) : [0, b]×[0, b]×X → X are strongly measurable. (H8 ) For every positive integer m there exists αm ∈ L1 (0, b) such that kf (t, x, y, z)k ≤ αm (t), for t ∈ [0, b] a. e.

sup kxk≤m,kyk≤m,kzk≤m

In the sequel we will use the following results: Lemma 2.2 ([27]). Let C(t), (resp.S(t)), t ∈ R be a strongly continuous cosine (resp. sine) family on X. Then there exist constants N ∗ ≥ 1and w ≥ 0 such that kC(t)k ≤ N ∗ e|t| , for all t ∈ R, Z t2 ∗ kS(t1 ) − S(t2 )k ≤ N | ew|s| ds|, for all t1 , t2 ∈ R. t1

Lemma 2.3 ([15], p-61). Let S be a convex subset of a normed linear space E and assume 0 ∈ S. Let F : S → S be a completely continuous operator, and let ε(F ) = {x ∈ S : x = λF x for some 0 < λ < 1}. Then either ε(F ) is unbounded or F has a fixed point. 3. Existence of Mild solutions Theorem 3.1. Let g : B → X be a continuous function. Assume that the hypotheses (H1 ) − (H8 ) hold and if b satisfies the following condition Z b Z ∞ ds , (3.1) Mb l(s)[1 + p(s) + q(s)]ds < K(s) 0 c where c = M (kx0 k + G + bkηk). Then problem (1.1)-(1.2) has at least one mild solution on J. Proof. To prove the existence of a mild IVP (1.1)-(1.2), we apply Lemma 2.3. First we establish the priori bounds for the mild solutions of the parameterized problem with parameter λ ∈ (0, 1) such that Z t Z b
00 x (t) = Ax(t) + λf t, x(t), k(t, s, x(s))ds, h(t, s, x(s))ds , (3.2) 0

x(0) + λg(x) = λx0 ,

0

x0 (0) = λη,

(3.3)

and show that the solutions to this system are bounded. First it is not hard to see that system (3.2)-(3.3) has a mild solution satisfying the integral equation x(t) = λC(t)[x0 − g(x)] + λS(t)η Z t Z +λ S(t − s)f s, x(s), 0

0

s

Z k(s, τ, x(τ ))dτ, 0

b


h(s, τ, x(τ ))dτ ds.

116

TIDKE: ABOUT NONLINEAR INTEGRAL EQUATIONS

4

Using hypotheses (H1 ) − (H5 ) and the fact that λ ∈ (0, 1), we have kx(t)k ≤ kλ[C(t)(x0 − g(x))]k + kλS(t)ηk Z t Z s Z b
+ kλ S(t − s)f s, x(s), k(s, τ, x(τ ))dτ, h(s, τ, x(τ ))dτ dsk 0

0

0

≤ M (kx0 k + G) + M bkηk Z b Z s Z t
h(s, τ, x(τ ))dτ kds k(s, τ, x(τ ))dτ, M bkf s, x(s), + 0 0 0 Z t l(s)K(kx(s)k + p(s)kx(s)k + q(s)kx(s)k)ds ≤ M [kx0 k + G + bkηk] + M b 0 Z t ≤ M [kx0 k + G + bkηk] + M b M bl(s)[1 + p(s) + q(s)]K(kx(s)k)ds. (3.4) 0

Define a function u(t) by right-hand side of (3.4). Using the fact that K is continuous increasing function, we obtain Z t u(t) = M [kx0 k + G + bkηk] + M b M bl(s)[1 + p(s) + q(s)]K(kx(s)k)ds. 0

Then kx(t)k ≤ u(t) and u(0) = M [kx0 k + G + bkηk] = c. Therefore, Z t u(t) = c + M b l(s)(1 + p(s) + q(s))K(kx(s)k)ds 0 Z s u(t) ≤ c + M b l(s)(1 + p(s) + q(s))K(u(s))ds. 0

Differentiating the above inequality and using the fact that K is increasing continuous, we get u0 (t) ≤ M bl(t)(1 + p(t) + q(t))K(u(t)) u0 (t) ≤ M bl(t)(1 + p(t) + q(t)). K(u(t))

(3.5)

Integrating (3.5) from 0 to t and using change of variables t → s = u(t) and the condition (3.1), we obtain Z u(t) Z t ds ≤ Mb l(s)(1 + p(s) + q(s))ds u(0) K(s) 0 Z b Z ∞ ds ≤ Mb l(s)(1 + p(s) + q(s))ds < . (3.6) K(s) 0 c From inequality (3.6) and mean value theorem we observe that there exists a constant γ, independent of λ ∈ (0, 1) such that u(t) ≤ γ for t ∈ J and hence kx(t)k ≤ γ for t ∈ J and consequently, we have kxkB = sup{kx(t)k : t ∈ J} ≤ γ. In order to apply Lemma 2.3, we must prove that the operator F : B → B defined for t ∈ J by (F y)(t) = C(t)[x0 − g(y(t))] + S(t)η Z t Z s Z b
+ S(t − s)f s, y(s), k(s, τ, y(τ ))dτ, h(s, τ, y(τ ))dτ ds 0

is completely continuous operator.

0

0

(3.7)

TIDKE: ABOUT NONLINEAR INTEGRAL EQUATIONS

117

5

Let Bm = {y ∈ B : kykB ≤ m} for some m ≥ 1. We first show that F maps Bm into an equicontinuous family of functions with values in X. Let y ∈ Bm and t1 , t2 ∈ J. Then if 0 <  < t1 < t2 ≤ b, we have k(F y)(t2 ) − (F y)(t1 )k = kC(t2 )[x0 − g(y)] + S(t2 )η Z t2 Z + S(t2 − s)f s, y(s), 0

s

Z k(s, τ, y(τ ))dτ,

0

b


h(s, τ, y(τ ))dτ ds

0

− C(t1 )[x0 − g(y)] − S(t1 )η Z b Z s Z t1
h(s, τ, y(τ ))dτ dsk k(s, τ, y(τ ))dτ, S(t1 − s)f s, y(s), − 0

0

0

≤ kC(t2 ) − C(t1 )kL(X) (kx0 k + G) + kS(t2 ) − S(t1 )kL(X) kηk Z t1 − + kS(t2 − s) − S(t1 − s)kL(X) 0

Z

s

Z

b
× kf s, y(s), k(s, τ, y(τ ))dτ, h(s, τ, y(τ ))dτ kds 0 0 Z t1 + kS(t2 − s) − S(t1 − s)kL(X)

Z

s

Z

t1 −

× kf s, y(s), 0

Z

b


h(s, τ, y(τ ))dτ kds

k(s, τ, y(τ ))dτ,

t2

0

Z kS(t2 − s)kL(X) kf s, y(s),

+ t1

s

Z k(s, τ, y(τ ))dτ,

0

b


h(s, τ, y(τ ))dτ kds

0

≤ kC(t2 ) − C(t1 )kL(X) (kx0 k + G) + kS(t2 ) − S(t1 )kL(X) kηk Z t1 − kS(t2 − s) − S(t1 − s)kL(X) αm (s)ds 0 Z t1 + kS(t2 − s) − S(t1 − s)kL(X) αm (s)ds t1 − t2

Z

kS(t2 − s)kL(X) αm (s)ds.

+

(3.8)

t1

The right hand side of (3.8) is independent of y ∈ Bm and tends to zero as t2 − t1 → 0 and  sufficiently small, since C(t), S(t) are uniformly continuous for t ∈ J and the compactness of C(t), S(t) for t > 0 imply the continuity in the uniform operator topology (see Lemma 2.2). Thus F Bm is an equicontinuous family of functions with values in X. We next show that F Bm is uniformly bounded. From the definition of operator F and using the hypotheses (H1 ) − (H5 ) and the fact that kykB ≤ m, we obtain k(F y)(t)k = kC(t)[x0 − g(y)] + S(t)η Z t Z s Z b
+ S(t − s)f s, y(s), k(s, τ, y(τ ))dτ, h(s, τ, y(τ ))dτ dsk 0 0 0 Z t ≤c+ kS(t − s)kL(X) αm (s)ds 0 Z t ≤c+ M bαm (s)ds 0

118

TIDKE: ABOUT NONLINEAR INTEGRAL EQUATIONS

6 b

Z ≤ c + Mb

αm (s)ds. 0

This implies that the set {(F y)(t) : kykB ≤ m, 0 ≤ t ≤ b} is uniformly bounded in X and hence F Bm is uniformly bounded. We have already shown that F Bm is an equicontinuous and uniformly bounded collection. To prove that F maps Bm into a precompact set in B, it is sufficient, by ArzelaAscoli’s Theorem, to show that the set {(F y)(t) : y ∈ Bm } is precompact in X for each t ∈ J. Let 0 < t ≤ b be fixed and  real number satisfying 0 <  < t. For y ∈ Bm , we define (F y)(t) = C(t)[x0 − g(x)] + S(t)η Z t− Z + S(t − s)f s, y(s), 0

s

b

Z


h(s, τ, y(τ ))dτ ds.

k(s, τ, y(τ ))dτ,

0

(3.9)

0

Since C(t), S(t) are compact operators, the set Y (t) = {(F y)(t) : y ∈ Bm } is precompact in X, for every , 0 <  < t. Moreover, for every y ∈ Bm , we get Z t k(F y)(t) − (F y)(t)k ≤ kS(t − s)kL(X) t−

Z

s

b

Z

× kf s, y(s), k(s, τ, y(τ ))dτ, 0 Z t αm (s)ds. ≤ Mb


h(s, τ, y(τ ))dτ kds

0

t−

This shows that there exists precompact sets arbitrarily close to the set {(F y)(t) : y ∈ Bm }. Hence the set {(F y)(t) : y ∈ Bm } is precompact in X. It remains to show that F : B → B is continuous. Let {un } be a sequence of elements of B converging to u in B. Then there exists an integer r such that kun k ≤ r for all n and t ∈ J. By hypotheses (H6 ) − (H8 ), we have Z b Z t
h(t, s, un (s))ds k(t, s, un (s))ds, f t, un (t), 0

0

Z → f t, u(t),

t

Z k(t, s, u(s))ds,

0

b

h(t, s, u(s))ds




0

for each t ∈ J, and since Z

t

kf (t, un (t),

b

Z k(t, s, un (s))ds,

h(t, s, un (s))ds)

0

0 t

Z − f (t, u(t),

b

Z

h(t, s, u(s)))dsk ≤ 2αr (t).

k(t, s, u(s))ds, 0

0

Then by by dominated convergence theorem, we have kF un − F ukB = sup k(F un )(t) − (F u)(t)k t∈[0,b]

= sup kC(t)[g(u) − g(un )] t∈J

Z +

t

Z h S(t − s) f s, un (s),

0

s

Z

0

Z − f s, u(s),

s

h(s, τ, un (τ ))dτ 0

Z k(s, τ, u(τ ))dτ,

0

b

k(s, τ, un (τ ))dτ, b

h(s, τ, u(τ ))dτ 0


i

dsk




TIDKE: ABOUT NONLINEAR INTEGRAL EQUATIONS

119

7

≤ kC(t)kL(X) kg(u) − g(un )k Z Z t h kS(t − s)kL(X) k f s, un (s), +

s

Z

Z

s

− f s, u(s),

Z

b

k(s, τ, u(τ ))dτ,


i h(s, τ, u(τ ))dτ kds

0

0

≤ M kg(u) − g(un )k Z Z th + Mb kf s, un (s), 0

s

Z

− f s, u(s),

s




0

Z k(s, τ, u(τ ))dτ,

0

b

h(s, τ, un (τ ))dτ

k(s, τ, un (τ ))dτ,

0

Z




0

0

0

b

h(s, τ, un (τ ))dτ

k(s, τ, un (τ ))dτ,

b


i h(s, τ, u(τ ))dτ k ds → 0.

0

Thus F is continuous. This completes the proof that F is completely continuous operator. Finally, the set ε(F ) = {y ∈ B : y = λF y, λ ∈ (0, 1)} is bounded in B as we proved in the first part. Consequently, by Lemma 2.3, the operator F has a fixed point in B. This means that the IVP (1.1)-(1.2) has a solution.  Remark 3.2. We note that in the special case, if we take (i) M bl(s)[1 + p(s) + q(s)] = 1 in condition (3.1) and the integral on the right side in (3.1) is assumed to diverge, then the solutions of equations (1.1) − (1.2) exist for every b < ∞. Acknowledgments. The work was supported by North Maharashtra University, Jalgaon (INDIA). References [1] K. Balchandran and S. Ilamaran, Existence and uniqueness of mild and strong solutions of a semilinear evolution equation with nonlocal condition, Indian J. Pure Appl. Math., 25(4)(1994), 411-418. [2] K. Balchandran and M. Chandrasekran, Existence of solutions of nonlinear itegrodifferential equation with nonlocal conditions, J. Appl. Stoch. Anal., 10(1996), 279-288. [3] K. Balchandran and M. Chandrasekran, Existence of solutions of delay differential equation with nonlocal condition, Indian J. Pure Appl. Math., 27(5)(1996), 443-449. [4] K. Balchandran and M. Chandrasekran, Nonlcal Cauchy problem for quasilinear integrodifferential equation in Banach spaces, Dyn. Sys. Appl. 8(1)(1999), 35-44. [5] J. Bochenek, An abstract nonlinear second order differental equation, Ann. Pol. Math., 54(1991), 155166. [6] L. Byszewski, Theorems about the existence and uniquess of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 162(1991), 494-505. [7] L. Byszewski, Existence and uniquess of mild and classical solutions of semilinear functional-differential evolution nonlocal Cauchy problem, Select. Problems Math., 6(1995), 25-33. [8] C. Corduneanu, Integral equations and Applications, Cambridge Univ. Press, New York, (1990). [9] J. P. Dauer and K. Balchandran, Existence of solutions for an interodifferential equation with nonlocal in Banach spaces, Libertas Math. 16(1996), 133-143. [10] M. B. Dhakne and G. B. Lamb, Existence result for an abstract nonlinear integrodifferential equation, Ganit: J. Bangladesh Math. Soc., 21(2001), 29-37. [11] M. B. Dhakne and G. B. Lamb, On global existence of solutions of nonlinear integrodifferential equations in Banach spaces, Bull. of the inst. of math. academia sinica, 30(1), (2002), 51-65. [12] M. B. Dhakne and G. B. Lamb, On an abstract nonlinear second order integrodifferential equation, J. Fun. Space Appl., 5(2)(2007), 167-174. [13] M. B. Dhakne and S. D. Kendre, On abstract nonlinear mixed Volterra-Fredholm integrodifferential equations, Communications on Applied Nonlinear Analysis, Vol.1, No.4(2006), 101-111. [14] M. B. Dhakne and S. D. Kendre, On an abstract nonlinear mixed integrodifferential equation, Proceeding of the Second International Conference on Nonlinear Systems (Bulletin of the Marathwada Mathematical Society), Vol.8, No.2(2007), 12-22. [15] J. Dugundji and A. Granas, Fixed Point Theory, Vol. I, Monographie Matematycane, PNW Warsawa, (1982).

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8

[16] H. O. Fattoroni, Ordinary differential equations in linear topological spaces, I, J. Diff. Eq. 5(1968), 72-105. [17] H. O. Fattoroni, Ordinary differential equations in linear topological spaces, II, J. Diff. Eq. 6(1969), 50-70. [18] A. Friedman, A Partial differential equations, Holt Rinehart and Winston, Inc., New York, (1969). [19] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Univ. Press, New York, (1985). [20] S. Ntouyas and P. Tsamatos, Global existence for semilinear evolution equations with nonlocal conditions, J. Math. Anal. Appl., 210(1997), 679-687. [21] S. Ntouyas and P. Ch. Tsamatos, Global existence for second order semilinear ordinary and delay integrodifferential equations with nonlocal conditions, Applic. Anal., 67(1997), 245-257. [22] B. G. Pachpatte Global existence results for certain integrodifferential equations, Demostratio Mathematica, Vol. XXIX,(1986), 23-30. [23] A. Pazy, Semigroup of linear operators and applications to partial differential equations, Springer Verlag, New York, (1983). [24] H. L. Tidke and M. B. Dhakne, On Nonlinear Mixed Volterra-Fredholm of First order Integrodifferential equations, International J. of Math. Sci. and Engg. Appls.(IJMSEA), Vol.2, No.II, (2008), 47-58. [25] H. L. Tidke, Global Existence of Nonlinear Mixed Volterra-Fredholm Integrodifferential Equation with Nonlocal Condition,(Submitted). [26] C. C. Travis and G. F. Webb, Second order differential equations in Banach spaces, Proc. Int. Symp. on Nonlinear Equatioins in Abstract Spaces, Academic Press, New York,(1978), 331-361. [27] C. C. Travis and G. F. Webb, Cosine families and abstract nonlinear second order differential equations, Acta Math. Hungarica, 32(1978), 75-96.

JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.5, NO.1, 121-138, COPYRIGHT 2010 EUDOXUS PRESS, LLC 121

Tensor Product Technique and the Degenerate Homogeneous Abstract Cauchy Problem By Ziqan, A. M. American Arab University-Jenin and Al Horani, M. and Khalil, R. University of Jordan-Amman-Jordan [email protected] [email protected]

Abstract In this paper, we use the technique of tensor product of Banach spaces to study the Abstract Cauchy Problem in Banach spaces. We also introduce semidiagonal operators on Hilbert spaces and discuss the solution of the Cauchy problem for such operators. —————————————————————————————– Key words and Phrases: Tensor product, Banach spaces, degenerate Cauchy problem. AMS classi…cation number. 47D06.

1

Introduction

Let I be the unit interval [0; 1] and C(I) be the Banach space of all real valued continuous functions on I under the sup-norm, and C 1 (I) be the Banach space of all continuously di¤erentiable functions under the norm kf k = kf k1 + kf 0 k1 . If X is a Banach space, then C 1 (I; X) is the Banach space of all

1

122

ZIQAN ET AL: ABOUT ABSTRACT CAUCHY PROBLEM

continuously di¤erentiable functions de…ned on I with values in X, under the norm kuk = kuk1 + ku0 k1 for all u 2 C 1 (I; X) : One of the classical di¤erential equations in Banach spaces is the so called Abstract Cauchy Problem. The general form of such equation is Bu0 (t) = Au(t) + F (t)z where A; B are densely de…ned linear operators on the codomain of the function u , where u is continuously di¤erentiable on I = [0; 1] or [0; 1) with values in the Banach space X: If B 6= I, the identity operator, then the equation is called degenerate. Otherwise, it is called non-degenerate. If f = 0 or z = 0 , then the equation is homogeneous, otherwise it is called nonhomogeneous. The non-degenerate Cauchy problem has been investigated by many authors using di¤erent techniques to solve it. If B = I, f = 0 and A densely de…ned linear operator , the abstract Cauchy problem has been studied extensively. We refer the reader to Pazy, 1983, and the references there in. A. Favini (1979) investigated the degenerate Cauchy problem of parabolic type in Banach space. F. Nneubrander,1994, and Bäumer, B. and F. Nneubrander 1994 used Laplace-Stieltjes transform to obtain existence and uniqueness results for exponentially bounded solutions of the homogeneous degenerate Cauchy problem where A and B are closed operators. A. Lorenzi (2001) used the projection method to derive existence and uniqueness of the solution of …rst-order degenerate abstract Cauchy problem. M. Alhorani (2004) studied the inverse problem of the degenerate abstract Cauchy problem where suitable hypotheses on the involved operators are made to reduce the given problem to a non-degenerate case. Thaller, B. and Thaller, S. 1995, and 1996 studied the Cauchy problem Bu0 = Au; under six assumptions on A and B: In this paper, we use tensor product technique to get solutions of the degenerate homogeneous abstract Cauchy problem. 2

ZIQAN ET AL: ABOUT ABSTRACT CAUCHY PROBLEM

2

123

Basics and Background

Let K be a compact Hausdor¤ space, X be a Banach space. C(K; X) denotes the Banach space of all continuous functions from K into X with the norm kf k1 = sup kf (t)k If X = R we write C(K) for C(K; R): t2K

Let X and Y T : D(T )

be Banach spaces with duals X and Y respectively and

X ! Y be linear. T is called closed operator if its graph G(T ) =

f(x; T x) : x 2 D(T )g is closed in the normed space X on X

Y , where the norm

Y is de…ned by k(x; y)k = kxk + kyk.

. For x 2 X and y 2 Y , we de…ne the map:x y : X ! Y withx y (x ) = hx; x iy: Clearly x We call x X

y is a bounded linear operator and kx

yk = kxk kxk :

y an atom: Let K = fx

y : x 2 X and y 2 Y g : We shall write n P Y for the span of K in L (X ; Y ) : For T = xi yi 2 X Y , de…ne

kT k_ = sup

n P

i=1

i=1

jhxi ; x ihyi ; y ij : x 2 B1 (X ) and y 2 B1 (Y ) :So, kk_ is

just the operator norm on L (X ; Y ) : This is called the injective norm of T . The space (X closure of X

Y; kk_ ) need not be complete. We let X

_

Y denote the

Y in L (X ; Y ) and it is called the completed injective tensor

product of X with Y: One of the nice features of the injective tensor product of normed spaces is the fact that For any compact Hausdor¤ space K, and any Banach space _

X; C(K; X) is isometrically isomorphic to C(K) X: In particular, C(S K) = C(S)

_

C(K); for S; and K are compact metric spaces.

Another norm that one can de…ne is the projective norm. For T 2 n P X Y the projective norm of T is kT k^ = inf kxi k kyi k , where the i=1

in…mum is taken over all representations of T in X

Y: The space X ^

Y with

the(X

Y; kk^ ) need not be complete. So, we let X

Y to be the completion

of (X

Y; kk^ ) :One of the nice features of the projective tensor product of

two Banach spaces is the following result: Let (I; ) be any …nite measure

3

124

ZIQAN ET AL: ABOUT ABSTRACT CAUCHY PROBLEM

space. Then L1 (I; X) is isometrically isomorphic to L1 (I)

^

X: We refer to [4

], for more on tensor products.

3

The Function Space

Through out this paper we let `2 be the classical Hilbert space of square sumP mable sequences which is f(an ) : jan j2 < 1, an 2 Rg:C(I; `2 ) is the space of continuos functions on the compact interval I = [0; 1] with values in `2 :

The subspace of continuously di¤erentiable functions on I will be denoted by 1 P C 1 (I; `2 ): Any function in C(I; `2 ) can be written in the form u(t) = ui (t) i ; i=1

2

where ( i ) is the natural basis of ` :However, we don’t have guarantee that 1 ^ ^ P kui k1 < 1: But we know that C(I) `2 and C 1 (I) `2 are both in i=1

C(I; `2 ): This is because for any two normed spaces X; Y we have X X

_

Y; and C(I; `2 ) = C(I)

_

Y

`2 as was pointed out in section2. So, we can

introduce the following subspace of functions in C(I; `2 ) 1 1 P P W = u 2 C 1 (I; `2 ) : u = ui kui k1 + ui i , kuk1 = i=1

We know that W

^

i=1

is a huge space sinceC 1 (I)

`2

1

0g [ ft 2 I : g(t) < 0g Since E c is open set, it can be written as a countable union of disjoint open 1

intervals, say E c = [ ( i ; i=1

on each ( i ;

i ):Now,

i)

, i = 1; 2; :::; g(t) 6= 0, so

we have g 0 (t) Bx = Ax , 8t 2 ( i ; g(t)

i)

; i = 1; 2; :..

This implies that g 0 (t) = g(t) So g 0 (t)

i g(t)

i

(constant) , on ( i ;

= 0, on ( i ;

i)

g(t) = ci e Now, we claim that ( i ; g(t) = ck e

kt

on (

k;

continuity of g(t) at

k) k,

This implies g = 0 on ( (

k;

k ).

i)

it

i );

; i = 1; 2; :::, and 8t 2 ( i ;

i ); i

= 1; 2; :::

; i = 1; 2; :::are adjacent intervals. Indeed, if

and if on the interval ( we have ck e k;

i = 1; 2; :::

k)

k k

k;

k+1 ); g(t)

= 0; then by the

= 0 which is impossible unless ck = 0:

contradicting the assumption that g(t) 6= 0 on

Hence we can assume that 1

E c = [ ( i; i=1

and therefore g(t) can be written 8 > > c1 e > < g(t) = c2 e > > > : .. .

i+1 )

1t

;t 2 (

1;

2)

2t

;t 2 (

2;

3)

6

.. .

ZIQAN ET AL: ABOUT ABSTRACT CAUCHY PROBLEM

127

Finally, we show that g(t) 6= 0, 8t 2 I: In fact, since g is continuos at t =

k; k

= 2; 3; :::, we have lim ck e t!

that ck e

k

k

= ck+1 e

k+1 k

kt

= lim+ ck+1 e t!

k

= 0. This implies

k

= 0, which is impossible unless ck ; ck+1 both equal g (t) Bx g(t)

zero. Hence g(t) 6= 0 , 8t 2 I. Therefore, g (t) g(t)

k+1 t

= ; 8t 2 I, which implies g(t) = ce

t

= Ax, 8t 2 I: Hence

8t 2 I , where g(0)x = z0 = cx:

Thus, u(t) = e t z0:

Corollary 4.2. For u(t) = g(t)x; g 2 C 1 (I) problem (P1) has a solution. If x 2 = KerA \ KerB then by Theorem 4.1 has a unique solution.

Proof

If x belongs to KerA \ KerB, then Bu0 (t) = Bg 0 (t)x = g (t)Bx = 0 and Au(t) = Ag(t)x = g(t)Ax = 0 Thus u(t) = g(t)x is a solution to (P1) for all g 2 C 1 (I) such that z0 = g(0)x:

5

Solution For The Finite Rank Functions Case

Here we are looking for a solution for (P1) among functions of the form n2 P u(t) = uik (t) ik , where ui 2 C 1 (I) ; i = 1; 2; :::; n. Such functions are ik=n1

called …nite rank functions. First we assume that B = I , so problem (P1) becomes u0 (t) = Au(t)

(P2)

u(0) = z Theorem 5.1. Problem P2 has a unique solution for any densely de…ned linear operator A on `2 : Proof. Let for simplicity u(t) =

n P

ui (t) i : Then u0 (t) =

i=1 n P

i=1

ui (t)

i

=

n P

n P

i=1

ui (t)A

i=1

7

i

ui (t) i : So

......................................(1)

128

ZIQAN ET AL: ABOUT ABSTRACT CAUCHY PROBLEM

Since u0 (t) is a linear combination of : Thus Y = [ 1 ;

2 ; :::; n ]

1 ; :::; n ;

then Au(t) belongs to [ 1 ;

is an invariant subspace of A; and we can consider

the restriction of A on [ 1 ; ^

2 ; :::; n ]; ^

^

A : [ 1;

2 ; :::; n ]

! [ 1;

A has a matrix representation (i.e A = [aij ] where aij = hA i ; inner product of

j

2 ; :::; n ]

2 ; :::; n ]: j i).

Hence

Taking the

with both sides of equation (1) we get n X

ji

ui (t)h i ;

i=1

=

n X

ui (t)hA i ;

i=1

ji

Since { i g is an orthonormal set, we have n P u0j (t) = ui (t)aij .....................................................(2) i=1

Then equation (2) represents a system of n linear di¤erential equations, ^

which can be solved by …nding the spectrum of A; which is a standard procedure. Further, with the initial condition in (P2), the solution is unique. Now, we try to solve (P1) in case B is not the identity, and u(t) is a …nite rank function. Without loss of generality, we look for a solution of the form n P u(t) = ui (t) i . Further, we assume that A; B are densely de…ned operators i=1

2

on ` : Since Bu (t) = Au(t), then A([ 1 ;

2 ; :::; n ])

simplify the notation, we can assume that [ 1 ;

= B([ 1 ;

2 ; :::; n ]

2 ; :::; n ]):

So to

is invariant under both

A and B, and An and Bn are the restriction of A and B to [ 1 ;

2 ; :::; n ]:

With this setup, we can prove the following theorem.

Theorem 5.2. Let Bn = B

[

1 ; 2 ;:::; n ]

be orthogonally diagonalizable lin-

ear operator such that A jker B is invertible. Then, problem (P1) has a unique solution. Proof. Since Bn is diagonalizable, there exists a basis # = f 1 ; :::;

ng

such

that the matrix representation of Bn with respect to # is D = diag( 1 ; :::; where

1 ; :::;

n

are the eigenvalues of B and

1 ; :::; n

n)

are the corresponding

eigenvectors. Further, since B is orthogonally diagonalizable; ( i ) is orthonormal. 8

ZIQAN ET AL: ABOUT ABSTRACT CAUCHY PROBLEM

Now, if

i

129

6= 0 ,8i = 1; :::; n; then Bn is invertible and hence problem (P1)

becomes u0 (t) = Bn 1 An u(t), which has a unique solution by Theorem 5.1. n P 0 Assume i 6= 0 for i = 1; :::; r. Let u(t) = i (t) i : Then Bn u (t) = n P

i=1

0 i (t)Bn i ;

and An u(t) = n P

0 i (t)Bn i

i=1

i=1

n P

i (t)An i i=1 n P

. Hence

i (t)An i .................................................:(3)

=

i=1

Taking the inner product of i with both sides of (3) we get n n P P 0 i (t)hAn i ; j i .....................................(4) i (t)hBn i ; j i = i=1

Let A~ = Ajker B = [hA i ;

i=1

j i]i;j=r+1;

;n

:

From (3.3.5), we have 3 2 2 3 2 3 0 0 7 6 1 36 1 7 76 1 7 2 6 .. 6 7 7 6 7 6 0 76 . 7 4 A1 A2 5 6 : 7 6 : = 6 : 7 ::::::::::(5) 76 : 7 6 7 76 : 7 6 0 ~ A3 A 6 74 6 r 4 : 5 5 5 4 0 n n 0 0 2 3 Ir 0 5 ; we get: Multiplying (5) from left by the matrix4 0 A~ 1 3 2 2 3 2 0 7 6 1 0 3 2 7 6 .. 6 1 6 1 7 6 0 7 . A2 7 6 .. 7 4 A1 6 . 56 6 .. 76 . 7 = 6 74 6 0 5 A~ 1 A3 In r 4 7 6 r 0 5 4 n n 0 0

So, we have 2 6 6 6 4

and

0

1

..

.

0

r

2

32

0 1

3

2

3

2

76 7 6 1 7 6 7 6 .. 7 6 .. 7 6 7 6 . 7 = A1 6 . 7 + A2 6 5 4 5 4 54 0 r

3

6 1 7 6 . 7 0 = A~ 1 A3 6 .. 7 + In 4 5 r

Hence from(6) we get

r

2

6 6 r6 4

r+1

.. . n

3

r+1

.. . n

7 7 7 ::::::::::::::::::(6) 5 9

3 7 7 7 5

3 7 7 7 5

130

ZIQAN ET AL: ABOUT ABSTRACT CAUCHY PROBLEM

2 6 6 6 4

r+1

.. . n

3

2

3

6 1 7 6 . 7 A~ 1 A3 6 .. 7 :::::::::::::::::::::::(7) 4 5

7 7 7= 5

r

Substituting (7) in (5) we get , 2 6 6 6 4

1

.. 0

r )::Hence

2

6 1 7 6 . 7 A2 A~ A3 ) 6 .. 7 4 5 1

r

r

3

1

Case 1. M has

3

3

(t) 7 .. 7 . 7 ; M = D 1 (A1 5 r (t) For such M we have the following cases: 6 6 Let V1 (t) = 6 4

2

6 1 7 6 . 7 A2 A~ 1 A3 ) 6 .. 7 .................(8) 4 5

7 6 6 .. 7 6 . 7 = D 1 (A1 5 4 2

3

0 r

r

0 1

0 1

76 7 7 6 .. 7 7 6 . 7 = (A1 54 5

.

Let D = diag( 1 ; : : : ; 3 2 0 r

32

0

1;

;

r

A2 A~ 1 A3 ). ThenV10 (t) = M V1 (t).

distinct eigenvalues, then the general solution

of the system is of the form r X

V1 (t) =

i Ei e

it

i=1

where Ei is the corresponding eigenvector Case 2. M has

1;

;

k

eigenvalues with multiplicity m1 ;

; mk (m1 +

+ mk = r): For such case we have the following sub-cases Case 2.1. For each p = 1;

; k;

p

has mp linearly independent eigenvec-

tors. Hence the general solution of the system is of the form V1 (t) =

k X

(

p1 Ep1

+

+

pmp Epmp )e

pt

p=1

Case2.2.

For each p = 1;

; k;

eigenvector. Then 10

p

has a single linearly independent

ZIQAN ET AL: ABOUT ABSTRACT CAUCHY PROBLEM

V1 (t) =

k h P

(

p1 Ep1

+

p2 (Ep1 t

+ Ep2 )

mp 1

+

pmp ( p1 Ep1 t

131

i + pmp Epmp ) e

+

p=1

Case 2.3.

pt

:(9)

has 1 < m0p < mp linearly independent eigenvectors, then

p

the solution of the system is of the form V1 (t) = h where

k X

(

p1 Ep1

+

+

pm0p Epm0p )e

pt

+

+

p=1

pm0p +1 (Qp1 t p1 ;

;

nation of Ep1 ;

+ Qp2 ) +

pmp ; p

mp m0p 1

+

pmp ( p1 Qp1 t

+

+

p(mp m0p ) Qp(mp

; k are known constant ,Qp1 is a linear combi-

= 1;

; Epm0p and Qpq = (M

p I)Qpq 1 ;

2

3

c 6 1 7 6 .. 7 For simplicity we may assume that V1 (0) = 6 . 7 ; and we consider the 4 5 cr general solution which is given by (9), since other cases can be treated in a similar way with slight di¤erence in notations. Then 2 3 c1 6 7 k 6 .. 7 P + pmp ( pmp Epmp )::::::::::::::::::::(10) 6 . 7= p1 Ep1 4 5 p=1 cr Let P = [E11 : : E1m1 : : Ek1 : : Ekmk ], then [c1 ;

; c r ]T = P [

11 ;

Now we can choose Epq ; q = 1; (i)kP

1

k

;

1m1 ;

;

k1 ;

; mp ; p = 1;

;

1

(iii)

p1 (mp

mp 1 p1 Ep1 t

1)Ep1 tmp

2

+

+ +

+

pmp Epmp

pmp 1 Epmp 1

1 1

Then from (10) we have k r P P j p1 j + + pmp jci j ::::::::::::::::::::::::(12) p=1

i=1

11

::::::::::::::(11)

; k such that

Proof Case 5.1 (ii) j p j

T kmk ]

i m0p ) ) e

pt

132

ZIQAN ET AL: ABOUT ABSTRACT CAUCHY PROBLEM

If we assume that < (M ) k h X p=1 !

e

k

k X p=1

!

e

k X p=1

p1 Ep1 k

j

+

p1 j

+

mp 1

+

p1 j kEp1 k

j

! < 1, then from (9),(10) and (11) we get

+

pmp ( p1 Ep1 t

+

+

!

e

pmp

r X i=1

+

mp 1 p1 Ep1 t

(

pmp

+

pmp Epmp )

+

+

i

e

pt

pmp Epmp )

jci j

Also, if we di¤erentiate V1 (t) we get V10 (t)

=

k X

p

p=1

k h X

h

(

p1 Ep1

p2 Ep1

+

p2 (Ep1 t

+

pmp (

+ Ep2 )

+

mp 1

pmp ( p1 Ep1 t

1)Ep1 tmp

p1 (mp

2

+

+

+

i + pmp Epmp ) e

pmp

i E ) e pmp 1 1

+

pmp Epmp )

p=1

pt

pt

Thus k X p=1

h j pj j

k h X p=1

2e!

j

k X p=1

p1 j kEp1 k +

p2 j kEp1 k

j

p1 j +

+

+

+

+

pmp

mp p1 Ep1 t

p1 (mp

pmp

2e!

pmp

(

r X i=1

1

+

1)Ep1 tmp

2

+

+

pmp

i

e

Epmp 1

jci j

Therefore u 2 W: Here we give an example as an application of Theorem 5.1 on R2 which can be considered as a …nite dimensional subspace of `2 :

Example 5.1. Let B ,A be two linear operators on R2 such that the conditions in Theorem 5.1 are satis…ed and the matrix representation of B and A are 2 B=4

1 1 1 1

3

2

5; A = 4

2 1 3 0

3

5. 12

pt

1

+ i

e

pt

+

ZIQAN ET AL: ABOUT ABSTRACT CAUCHY PROBLEM

133

2

2

The eigenvalues of B are 2; 0 and the corresponding eigenvectors are 4 p1 2

,4

p1 2

3

5:

Let u(t) =

0 1 (t)B

2 4

1 (t)

p1 2 p1 2

3

2

3

p1 2

4

5+

p1 2

2

5+ 02 (t)B 4

2 (t)

p1 2 p1 2

3

2

p1 2

4

p1 2

5=

2 01 (t) =

1 (t)hA

and 0=

1 (t)hA

Thus

2 4

4 p1 2 p1 2

p1 2 p1 2

3 2

5 ,then

1 (t)A

2 4

2

5;4

3 2 5;4

p1 2 p1 2

p1 2 p1 2

p1 2

3 5

3

:Taking the inner product of (13) with 4 2

p1 2

3

5i +

3

5i +

3

p1 2

5+ 2 (t)A 4

p1 2

p1 2 p1 2

2

3 2 5,4

2 (t)hA

2 (t)hA

2 4

2 4

p1 2 p1 2

p1 2 p1 2

p1 2 p1 2

3

p1 2 p1 2

3

5 ::::::::(13)

5 to get

3 2 5;4

3 2 5;4

p1 2 p1 2

p1 2 p1 2

3

5i

3

5i

2 01 (t) = 3 1 (t) + 2 2 (t) 0 =

Hence u(t) =

6

3 pc e 2 t 2

2 4

1 0

3

5+

3 pc e 2 t 2

2 4

2 (t)

0 1

3

5:

Solutions for In…nite Rank functions

In this section, we are going to solve problems (P1) and (P2) among functions 1 P of the form u(t) = uj (t) j 2 W . We make the following assumptions j=1

Assumption 6.1. Assume that A is a densely de…ned linear operator on

`2 such that [ 1 ;

2;

] is invariant under A and < 13

! < 1, 8 2

(A) :

134

ZIQAN ET AL: ABOUT ABSTRACT CAUCHY PROBLEM

Assumption 6.2. Assume that (jcj j) and (jcj f i;

2;

belong to `1 , where

j j)

g is the set of eigenvalues of A.

We call A : [ 1 ;

2;

] ! `2 diagonal linear operator if the matrix repre-

sentation of A with respect to the basis f jg is diagonal. Theorem 6.1. Let A be a diagonal linear operator, such that Assumption 6:1and 2:2 are satis…ed. Then problem (P2) has a unique solution. 1 1 P P 2 Proof. Since uj u0j j and j converge uniformly in C (I; ` ), then u0 (t) =

1 P

j=1

j=1

j=1

1 X

u0j (t) j

j=1

1 P

So,

j=1

(u0j (t)

uj (t) = cj e

jt

1 P

u0j (t) j . Also, since Au(t) =

uj (t) j )

=

1 X

uj (t)A

j

=

j=1

then,

j

1 X

uj (t)

j j

j=1

= 0: Hence u0j (t)

j

uj (t)A

j=1

uj (t)

where cj = uj (0): Thus u(t) =

1 P

cj e

j jt

= 0 for every j and j:

Now, e

it

= e
0 : j c j , if ! 0 i i

8 1 P > > < e! jci j + j i ci j ,if ! > 0 i=1

> > :

1 P

i=1

jci j + j i ci j ,if !

0

From assumption 6.2, we see that kuk1 < 1 , so u 2 W: De…nition 6.2. A linear operator B de…ned on a Hilbert space H is called semi-diagonal if there exist orthogonal subspaces fWj g1 j=1 , such that 14

ZIQAN ET AL: ABOUT ABSTRACT CAUCHY PROBLEM

135

1. dimWj < 1; 8j 2. B(Wj ) 3. H =

1 j=1

Wj ; 8j Wj

Theorem 6.3. Let A be semi-diagonal linear operator, such that assumptions 6.1 and 6.2 are ful…lled. Then problem (P2) has a unique solution . Proof. since A is semi-diagonal, there exist W1 ; W2 ,...orthogonal subspace of `2 such that for each j dim Wj < 1; A(Wj )

1

Wj and `2 =

j=1

Wj . With out

loss of generality, according to the decomposition of `2 , we may assume that ; nj where n0 = 0; j = 1; 2; ; is the corresponding basis for 1 1 P P Wj . Since uj (t) j and u0j (t) j converge uniformly in C (I; `2 ), we have nj

1 +1

u0 (t) = u(t) =

;

1 P

j=1 1 P

j=1

j=1

u0j (t) j , and by assumption on A; we have Au(t) =

uk (t)

k

k=1

=

nj P

1 P

j=1k=nj

uk (t)

k

nj P

where

k=nj

1 +1

can write u0 (t) = Au(t) as

nj 1 X X j=1 k=nj

u0k (t) k

=

nj 1 X X j=1 k=nj

1 +1

uk (t)

uj (t)A j : Now,

j=1

k

1 +1

uk (t)A

1 P

2 Wj . Hence we

k

1 +1

Since the subspaces W1 ; W2 ,... are orthogonal, A(Wj )

Wj and

nj P

k=nj

Wj we have

nj X

k=nj

u0k (t) k

=

nj X

k=nj

1 +1

uk (t)A

uk (t)

k

1 +1

k

1 +1

This system, together with the initial condition u(0) restricted to Wj ,has a unique solution. So, on each Wj , we have a unique solution wi0 (t). Apply the same argument as in the last part of the proof of Theorem 5.2 for A we conclude that 1 X j=1

and

1 X j=1

!

kuj k1

e

u0j

!

1

nj 1 X X j=1 k=nj

e

1 +1

nj 1 X X j=1 k=nj

15

1 +1

jck j < 1

j k ck j < 1

Wj

= Mj

2

136

ZIQAN ET AL: ABOUT ABSTRACT CAUCHY PROBLEM

Therefore, kuk1 < 1 and so u 2 W . Now we are going to solve problem (P1). But …rst we need to assume the following assumptions on the operators A and B in order to make sure that the desired solution belong to the spaceW Assumption 6.3. A and B are semi-diagonal linear operators with the same decomposition (i.e there exist orthogonal subspaces fWj g1 j=1 such that 1. dimWj < 1; 8j 2. Wj is invariant under A and B for every j 1

3. `2 =

j=1

Wj

Assumption 6.4. Bj = B

is orthogonally diagonalizable such that

Wj

A~j = Aj jker Bj is invertible for every j. From the proof of Theorem 5.2 we have on each Wj Mj = Dj 1 (Aj1 + Aj2 A~j 1 Aj3 ) Assumption 6.5. For each j , < (Mj )

! 0 and T; T > 1, we have ai  S T i min ki ak ai k ; i 2 N (2.2) bj  S T j min kj bk bj k ; j 2 N: As a consequence of (2.1), we have [cf. [7]] 0

2

2

1

+1

1

+1

1

1

0

1

0

1

ai ak  a0 ai k ; i; k 2 N0 bj bk  b0 bj +k ; j; k 2 N0 :

(2.3)

For sequences above, we have the following: De nition 2.1 (i) Denote by H;fa g;a the set of all those in nitly smooth functions (x) such that for some positive constants Cj; and a dependent on  2 i 1 (1 + x ) (x D)j x   1 (x)  Cj; (a + )i ai ; (2.4) holds true for arbitrary  > 0 and i; j 2 N0 . On the other hand, let H;(a );a be the set of all C 1 -functions (x) satisfying (2.4) for all a > 0. i

i

(ii) An in nitly smooth functions (x) 2 H;fb g;b (respectively, H;(b );b ) if for some constant b > 0 (respectively, for all b > 0), there is Cj; > 0 both of which depend on  where i



(1 + x )i(x D)j x 2

1

 

1



(x)  Cj; (a + )i bi ;

i

(2.5)

d. for arbitrary  > 0, where D = dx

(iii) De ne H;;ffab gg;b;a (respectively, H;;((ab ));b;a ) if for some constant C ; > 0 there are constants a > 0 and b > 0 (respectively, for arbitrary a > 0, b > 0) we j

j

i

i

have



(1 + x )i(x D)j x 2

1

 

1



(x)  Cj; (a + )i bj (b + )j ai bj ;

where  and are arbitrary positive constants.

2

(2.6)

160

AL-OMARI: ABOUT TEMPERED ULTRADISTRIBUTIONS

Spaces, so obtained, consist of ultradierentiable functions of rapid descent which are, indeed, subspaces of H; de ned in [6]. Furthermore, H; a ;a, H; b ;b and H;; ab ;b;a are contained in H; fa g;a , H;fb g;b and H;;ffab gg;b;a , respectively. The set of all continuous linear forms on H; a ;a, H; b ;b and H;; ab ;b;a is denoted by H ; a ;a, H ; b ;b and H ;; ab ;b;a , respectively and are tempered ultradistributions of Beurlingtype. Similarly, H ; fa g;a, H ;fb g;b and H ;;ffab gg;b;a are spaces of tempered (temperate) ultradistributions of Roumieu-type. It is interesting to observe that the tempered ultradistributions of Roumieu-type can be characterized as subspaces of the tempered ultra-distributions of Beurling type. Owing to the fact that both types of tempered (slow growth) ultra-distributions assume analysis which is similar, we intend to direct the investigations to ultradistributionsof Roumieu-type. In view of above constructions we assign to H; fa g;a, H;fb g;b and H;;ffab gg;b;a the topologies generated by the respective collections of seminorms ( i)

( j)

j

( i)

i

( j)

j

i

( j)

( i)

( j)

( j)

( i)

( i)

( j) ( i)

j

j

i

i

i

j

j

i

j(1 + x )i(x D)j x

 

j(1 + x )i(x D)j x

 

2

;

j; () = sup

1

(a + )iai

x2(0;1)

2

;

i; () = sup

x2(0;1)

1

; () = sup

;

x2(0;1)

1

(b + )j aj

j(1 + x )i(x D)j x 2

1

1

(x)j

(x)j

 

1

(a + )i(b + )j aibj

for conditions already mentioned.

; i 2 N0

(2.7)

; j 2 N0

(2.8)

(x)j

(2.9)

;

3 Operators for Multiplication and Necessary Conditions De nition: For some positive constants a and b, denote by fa g;a , fb g;b and ffba gg;b;a the set of all those C 1-functions (x), over (0; 1), such that for all positive integers i; j , their respective formulae i

j

j

i







2



2

(x D)i(x)  C (1 + x )iaiai; 1

(x D)j (x)  F (1 + x )j bj bj ; 1

3

(3.1) (3.2)

AL-OMARI: ABOUT TEMPERED ULTRADISTRIBUTIONS



161

(3.3) (x D)i(x)  E (1 + x )iaibj aibj ; hold good where C , F and E are certain positive constants. Spaces fa g;a, fb g;b and ffba gg;b;a and the spaces fa g;a,  fb g;b and ffab gg;a;b in [7] are quitely equivalent and are shown to be multipliers for respective spaces H; fa g;a, H;fb g;b and H;;ffab gg;b;a. For detailed analysis see [ [7],Theorems 11,12,13]. Proposition 1 (a) Let  2 fa g;a and  2 H;fa g;a , then 1

2

j

i

j

j

i

i

j i

j

j

i

i

i

i

(x D)i(x) 2 fa g;a 1

i

. (b) Let  2 fbj g;b and  2 H;fbj g;b , then

(x D)j (x) 2 fb g;b 1

j

. (c) Let  2 ffbajigg;b;a and  2 H;;ffabji gg;b;a , then

(x D)j (x) 2 ffab gg;b;a 1

j

i

.

Proof The proof can be easily established by induction on i, taking into account

that

(x)(x 1 D)(x) = x+ +1 (x 1 D)x+ +1 (x)(x) (x)x+ +1 (x 1 D)x   1 (x);

together with the fact that if  belongs to either H; fa g;a, H;fb g;b or H;;ffab gg;b;a, then x  (x D)i x   (x) belongs to the same space. Details are thus avoided. Therefore, the derived proposition suggests to introduce on fa g;a, fb g;b and ffba gg;b;a the respective seperating collections of seminorms jx   (x)(x D)i(x)j ; i 2 N `;;a () = sup (3.4) j

+ +1

1

j

i

1

i

i

j

j

i

1

;i

x2(0;1)

`;;b ;j () = sup

x2(0;1)

jx

`;;b ;a () = sup

x2(0;1)

ai ai 1 (x)(x 1 D)j (x)j ;j 2 N bj bj

 

jx

1

 

1

(x)(x 1 D)j (x)j ; ai bj ai bj

(3.5) (3.6)

for all  2 fa g;a; fb g;b and ffab gg;b;a, respectively where,  belongs to the respective spaces H; fa g;a, H;fb g;b and H;;ffab gg;b;a. j

j

i

i

j

i

j

i

4

162

AL-OMARI: ABOUT TEMPERED ULTRADISTRIBUTIONS

2 H; fa g;a

and the sequence ai , i = 1; 2; 3;


satisfy (2.6). Then a necessary condition for a C 1 -function (x) to be in fai g;a is

Theorem 1 Let 

i

`;;a ;i () < 1;

for all i 2 N .

Proof: Let  2 H; fa g;a , i; j 2 N and x 2 (0; 1). Leibnitz rule being employed

leads directly to (1 + x )i(x D)j x i



2

1

 

1



()(x)  j r

j r=0

P

!

jx

 

1

r (x)(x

1

j;

D)r (x)

(3.7)

where

r (x) = (1 + x2 )i x   1 (x 1 D)j r x   1 (x): (3.8) The fact that (x) 2 H; fa g;a, implies that x   1(x 1D)j r x   1(x) and (1 + x2)i(x) belong to H; fa g;a. This together with the de nition of the Hankel type integral, we can inferre that  2 H; fa g;a. i

i

Upon multiplying (3.7) by (a +1)ia , we obtain i j(1 + x )i(x D)j x   ()(x)j < (a + )iai i

2

1

1

j

j r

X

r=0

!

jx

 

1

r (x)(x 1 D)r (x)j ; ai ai

where r satis es (3.8) and  > 0. For i > r, we infer from (2.6) that 1 a : ai

0

ar ai

r

(3.9)

(3.10)

Using (3.10) in (3.9) and considering supremum over all x 2 (0; 1), i; r 2 N , yields ;

j; ()

<
j . This completes the proof of the theorem.

Theorem 2 Let  2 H;fb g;b and j 2 N , then the necessary condition for a function (x) to be in fb g;b is to be in nitly smooth and j

j

`;;b ;j () < 1;

for sequences (bj ) having (2.3) imposed.

5

(3.11)

AL-OMARI: ABOUT TEMPERED ULTRADISTRIBUTIONS

163

Proof: Let  2 H;fb g;b and i; j

2 N , then analysis similar to that of the proof of Theorem 2 leads directly to the relation (1 + x )i(x D)j x   ()(x) < (b + )j bj (3.12) j j x   r (x)(x D)r (x)j b j F : j

2

1

1

!

X

1

1

r

r=0

br br

0

bj r bj

r

Once again, letting j and r traverse the set of natural numbers and considering supremum over all x 2 (0; 1) we obtain ;

i; ()

1 (Theorem 5.2) and from H p to H q for 1 < p ≤ q < ∞ (Theorem 6.2). All of them are done under the same additional condition. As their applications, we also obtained some sufficient and necessary conditions for the weighted composition operator to be compact from H p to H q for the above cases. For convenience, we always abbreviate H p (Bn ) to H p . 2. Some Lemmas p Lemma 2.1. Let ϕ is holomorphic self-map of Bn and 0 < p < ∞. For R ψ ∈ H , where any measurable subset E of ∂Bn , denote µψ,ϕ,p (E) = ϕ−1 (E)∩∂Bn |ψ|p dσ. Then Z Z gdµψ,ϕ,p = |ψ|p (g ◦ ϕ)dσ, Bn

∂Bn

where g is an arbitrary measurable positive function in B n . Proof

If g is a measurable simple function defined on B n given by g =

n P

αi χEi ,

i=1

then n X

Z gdµψ,ϕ,p = Bn

αi µψ,ϕ,p (Ei ) =

n X

i=1

Z = ∂Bn

Z

Z αi

|ψ|p dσ

ϕ−1 (Ei )∩∂Bn

i=1

n X αi χϕ−1 (Ei )∩∂Bn )dσ |ψ|p ( i=1

|ψ|p (g ◦ ϕ)dσ.

= ∂Bn

Now, if g is a measurable positive function in B n , then we can take an increasing sequence {gm } of positive and simple functions such that gm (z) → g(z) for all z ∈ B n , it follows that Z Z gm dµψ,ϕ,p → gdµψ,ϕ,p . Bn

On the other hand,

|ψ|p (g

m

Bn

◦ ϕ) is an increasing sequence such that

|ψ(z)|p (gm (ϕ(z)) → |ψ(z)|p (g(ϕ(z)) for all z ∈ B n , so Z

Z

Z

|ψ| (gm ◦ ϕ)dσ →

gm dµψ,ϕ,p = Bn

p

∂Bn

∂Bn

And the conclusion follows by the uniqueness of the limit.

|ψ|p (g ◦ ϕ)dσ.

254

4

Z.S. FANG AND Z.H.ZHOU

Lemma 2.2. (See p116 in [Zhu2]) Suppose 0 < p < ∞ and f ∈ H p . Then |f (z)| ≤ ||f ||p for all z ∈ Bn . (1−|z|2 )n/p Lemma 2.3. Let Ω be a domain in C n , f ∈ H(Ω). If a compact set K and its neighborhood G satisfy K ⊂ G ⊂⊂ Ω and ρ = dist(K, ∂G) > 0, then √ ∂f n (z)| ≤ sup | sup |f (z)|. ρ z∈G z∈K ∂zj Since ρ = dist(K, ∂G) > 0, for any a ∈ K, the polydisc   ρ Pa = (z1 , · · · , zn ) ∈ C n : |zj − aj | < √ , j = 1, · · · , n n is contained in G. Using Cauchy inequality, we have √ √ ∂f n n ∂zj (a) ≤ ρ sup |f (z)| ≤ ρ sup |f (z)|. Proof

z∈G

z∈∂0 Pa

So the Lemma follows. Lemma 2.4. For fixed 0 < δ < 1, let G = {z ∈ Bn : |z| ≤ 1 − δ}. Then lim sup |f (z) − f (rz)| = 0

r→1 z∈G

for any f ∈ H p (Bn ). Proof sup |f (z) − f (rz)| = sup | z∈G

n X

(f (rz1 , rz2 , · · · , rzj−1 , zj , · · · , zn )

z∈G j=1

− f (rz1 , rz2 , · · · , rzj , zj+1 , · · · , zn ))| Z 1 n X ∂f (rz1 , rzj−1 , tzj , zj+1 , · · · , z n )dt| ≤ sup | |zj ∂zj z∈G r j=1

≤ (1 − r)n sup | z∈G

∂f (z)|. ∂zj

δ 2 },

Define G1 = {z ∈ Bn : |z| ≤ 1 − then G ⊂ G1 and dist(G, ∂G1 ) = 2δ . It follows from Lemma 2.3 that √ ∂f 2 n sup | (z)| ≤ sup |f (z)|. δ z∈G1 z∈G ∂zj If p = ∞, then

√ 2(1 − r)n n sup |f (z) − f (rz)| ≤ ||f ||∞ . δ z∈G For 0 < p < ∞, it follows from Lemma 2.2 that √ ||f ||p 2(1 − r)n n sup |f (z) − f (rz)| ≤ sup 2 n/p δ z∈G z∈G1 (1 − |z| ) √ ||f ||p 2(1 − r)n n ≤ sup n/p δ z∈G1 (1 − |z|) √ 2(1 − r)n n ||f ||p ≤ . δ ( 2δ )n/p Let r → 1, the conclusion follows. Lemma 2.5. (See corollary 1.3 in [CowMac]) A sequence in a reflexive functional Banach space converges weakly if and only if it is bounded and converges point-wise.

255

ESSENTIAL NORMS OF WEIGHTED COMPOSITION OPERATORS

5

Lemma 2.6. Assume {fm } is a bounded sequence in H p (Bn )(p > 1), and {fm } converges weakly to 0, then for any compact operator K from H p (Bn ) to Y (Y is a normalized linear space), we have kKfm kY → 0. Proof

This is easily followed by Lemma 2.5 and the property of compact operator. 3. From H ∞ to H q

Case 1. p = ∞, q = 2 It is well known that for any f ∈ H(Bn ), f has homogeneous expansion f (z) = ∞ P P Fs (z), where Fs (z) is the homogeneous polynomial c(α)z α , z α = z α1 · · · z αn , s=0

|α|=s

α = (α1 , · · · , αn ), and |α| = α1 + · · · αn . If f ∈ H 2 (Bn ), then X |c(α)|2 ||z α ||22 , ||f ||22 = α

where ||z α ||22 =

(n − 1)!α! , (n − 1 + |α|)!

α

where { ||zzα ||2 } is an orthonormal basis for H 2 (Bn ), and c(α) = Dα f (0)/α! with α! = α1 ! · · · αn !. If necessary, we refer the reader to see [Rud]. For m a positive integer, define the operators from H 2 (Bn ) to itself: ∞ ∞ X X Rm ( Fs ) = Fs s=0

s=m+1

and Qm = I − Rm . It is easy to show that Rm is compact and kRm k = 1. Lemma 3.1. Wψ,ϕ : H ∞ → H q , 0 < q < ∞ is bounded if and only if ψ ∈ H q . Proof If Wψ,ϕ is bounded, let f = 1, then Wψ,ϕ f = ψf (ϕ) = ψ ∈ H q . Conversely, apparently we have ||Wψ,ϕ f ||q ≤ ||ψ||q ||f ||∞ for any f ∈ H ∞ , that is, ||Wψ,ϕ || ≤ ||ψ||q . Using the same methods as that of Gorkin-MacCluer in [GorM], with minor modifications, we can obtain the following Lemmas 3.2 and 3.3. But for the reader’s convenience, we give still the detail proof for the results. Lemma 3.2. If Wψ,ϕ : H ∞ → H 2 and ψ ∈ H 2 , then kWψ,ϕ ke = lim kRm Wψ,ϕ k. m→∞

Proof On one hand, by hypothesis and Lemma 3.1, we know Wψ,ϕ is bounded, so the compactness of Qm implies that Qm Wψ,ϕ is also compact, kWψ,ϕ ke = k(Rm + Qm )Wψ,ϕ ke = kRm Wψ,ϕ ke ≤ kRm Wψ,ϕ k, it follows that kWψ,ϕ ke ≤ lim inf kRm Wψ,ϕ k. n→∞

On the other hand, let K :

H∞

→

H2

be compact. Since kRm k = 1,

||Wψ,ϕ − K|| ≥ ||Rm (Wψ,ϕ − K)|| = ||Rm Wψ,ϕ − Rm K|| ≥ ||Rm Wψ,ϕ || − ||Rm K||.

256

6

Z.S. FANG AND Z.H.ZHOU

Note that K is compact, the image of the unit ball in H ∞ under K has compact closure in H 2 . Since ||Rm || = 1 and Rm K tends to 0 point-wise in H 2 , Rm K tends to 0 uniformly on the unit ball of H ∞ , that is ||Rm K|| → 0 as n → ∞. It follows that ||Wψ,ϕ ||e ≥ lim sup ||Rm Wψ,ϕ ||, m→∞

this completes the proof. Lemma 3.3. For Wψ,ϕ : H ∞ → H 2 and ψ ∈ H 2 , if k is fixed positive integer and g is any non-constant holomorphic function on Bn with ||g||∞ ≤ 1, then ||Qk Wψ,ϕ (g m )||2 → 0 as m → ∞. Proof

If α is a multi-index with |α| ≤ k, then ||z α ||22 =

(n − 1)!α! ≤ (k!)n ≡ c(n, k). (n − 1 + |α|)!

n

1 ) ⊆ Bn and Cauchy’s estimates, for any holomorphic function F in Bn , we Since D (0, 2n have Dα F (0) ≤ (2n)|α| ||F ||∞,Dn (0, 1 ) 2n α! n 1 where||F ||∞,Dn (0, 1 ) denotes the maximum modulus of F on the polydisc D (0, 2n ). 2n

Since the series coefficients for F are c(α) = ψ · g m ◦ ϕ are bounded above by

Dα F (0) , α!

(2n)|α| ||ψ · g m ◦ ϕ||∞,Dn (0,

we get the series coefficients for 1 ) 2n

.

n

1 ), then s < 1 by hypothesis. This Let c = max|ψ| and s = max|g ◦ ϕ| on D (0, 2n m m implies that ||ψ · g ◦ ϕ||∞,Dn (0, 1 ) ≤ cs , which tends to 0 as m → ∞. For fixed 2n P α α 2 k, ||Qk Wψ,ϕ (g m )||22 = |α|≤k |c(α)|||z ||2 , where c(α) is the coefficients of z in the m expansion of ψ · (g ◦ ϕ) . By the above estimate, we have X ||Qk Wψ,ϕ (g m )||22 ≤ ((2n)k csm )2 c(n, k) ≤ c0 (n, k)s2m . |α|≤k

For fixed k, the last expression tends to 0 as m → ∞. Lemma 3.4. Let  > 0, set E = {ξ ∈ ∂Bn : |ϕ(ξ)| ≥ 1 − } and let Ec denote its complement in ∂Bn , ψ ∈ H 2 . Define an operator K : H ∞ → H 2 by K(f ) = P (χEc ψ ·(f ◦ ϕ)), where P is the orthogonal projection of L2 onto H 2 (where we identify a function in H 2 with its radial limit function). Then K is compact from H p to H 2 , for any 2 < p ≤ ∞. Proof Let {fm } be a sequence from the unit ball of H p . By Lemma 2.4, {fm } is a normal family when 2 < p < ∞, and this is obviously true for p = ∞. So there is a subsequence which converges uniformly on compact subset of Bn , to say f . For simplicity we still denote this subsequence as {fm }. Clearly f ∈ H p . So ||Kfm − Kf ||22 ≤ ||P ||2 ||χEc ψ · (fm ◦ ϕ) − χEc ψ · (f ◦ ϕ)||22 Z ≤ |χEc ψ · (fm ◦ ϕ) − χEc ψ · (f ◦ ϕ)|2 dσ ∂Bn Z = |ψ · (fm ◦ ϕ) − ψ · (f ◦ ϕ)|2 dσ. Ec

Since {fm } are uniformly bounded on Ec and ψ ∈ H 2 , the above expression tends to 0 as n → ∞ by Lebesgue’s dominated convergence theorem. This verifies the compactness of K. Theorem 3.1. For Wψ,ϕ : H ∞ → H 2 and ψ ∈ H 2 , then ||Wψ,ϕ ||e = (µψ,ϕ,2 (ϕ(E)))1/2 , where E = {ξ ∈ ∂Bn : |ϕ∗ (ξ)| = 1}.

257

ESSENTIAL NORMS OF WEIGHTED COMPOSITION OPERATORS

7

Proof we consider the lower estimate first. Let g be a non-constant inner function on Bn and set h = g m for a positive integer m, then Z Z m 2 ∗ ∗ ∗ 2 ||Wψ,ϕ (g )||2 = |ψ · (h ◦ ϕ )| dσ = |h∗ |dµψ,ϕ,2 ∂Bn Bn Z |h∗ |dµψ,ϕ,2 ≥ µψ,ϕ,2 (ϕ(E)) ≥ ϕ(E)

where the last inequality follows by the fact that |h∗ | = 1 a.e [dµ] on ϕ(E), this is true that h is inner and the restriction of µψ,ϕ,2 to ∂Bn is absolutely continuous with respect to σ. In fact, for any measurable subset E of ∂Bn , Z |ψ|2 dσ, µψ,ϕ,2 (E) = ϕ−1 (E)∩∂Bn

by hypothesis of Cϕ , if σ(E) = 0, then σ(ϕ−1 (E)) = 0, and µψ,ϕ,2 (E) = 0 follows. So ||Rk Wψ,ϕ || ≥ ||Rk Wψ,ϕ (g m )|| ≥ ||Wψ,ϕ (g m )|| − ||Qk Wψ,ϕ (g m )|| ≥ µψ,ϕ,2 (ϕ(E)) − ||Qk Wψ,ϕ (g m )||. for all m. Fix k and let m → ∞ and apply Lemma 3.2 we obtain ||Rk Wψ,ϕ || ≥ (µψ,ϕ,2 (ϕ(E)))1/2 for any k. Now let k → ∞, by Lemma 3.1 we have the desired lower estimate on ||Wψ,ϕ ||e . Now we turn to the upper estimate. Take K as in Lemma 3.3, for any g ∈ H ∞ with ||g||∞ = 1, we have ||Wψ,ϕ (g) − K(g)||2 = ||ψ · g ◦ ϕ − P (χEc ψ · (f ◦ ϕ))||2 = ||P (χE ψ · (f ◦ ϕ))||2 ≤ ||χEc ψ · (f ◦ ϕ)||2 Z Z 1 1 2 = ( |ψ · g ◦ ϕ| dσ) 2 ≤ ||g ◦ ϕ||∞ ( |ψ|2 dσ) 2 E E Z 1 ≤ ||g ◦ ϕ||∞ ( |ψ|2 dσ) 2 ϕ−1 (ϕ(E ))∩∂Bn

= ||g ◦ ϕ||∞ (µψ,ϕ,2 (ϕ(E )))1/2 . Let m ↓ 0 and Km the corresponding operator defined by Km (f ) = P (χEcm ψ · (f ◦ ϕ)). For p = ∞ we have ||Wψ,ϕ ||e ≤ ||Wψ,ϕ − Km || ≤ (µψ,ϕ,2 (ϕ(Em )))1/2 for all m, and let m → ∞, as desired. Corollary 3.1. Wψ,ϕ : H ∞ → H 2 is compact if and only if ψ ∈ H 2 and σ(E) = 0. Proof If Wψ,ϕ is compact, it is obviously bounded, it follows from Lemma 3.1 that ψ ∈ H 2 . From Theorem 3.1, the compactness of Wψ,ϕ implies µψ,ϕ,2 (ϕ(E)) = 0, so σ(ϕ−1 (ϕ(E))∩∂Bn ) = 0 (see 5.5.9 in [Rud]), therefore 0 ≤ σ(E) ≤ σ(ϕ−1 (ϕ(E))∩∂Bn ) = 0, σ(E) = 0. On the other hand, if ψ ∈ H 2 , from the proof of theorem 3.1, it follows that Z 1 ||Wψ,ϕ ||e ≤ ( |ψ|2 dσ) 2 E

when  → 0 and since σ(E) = 0, we get ||Wψ,ϕ ||e = 0, so Wψ,ϕ is compact.

258

8

Z.S. FANG AND Z.H.ZHOU

In the above proof, set ψ = 1 ∈ H 2 , then ||W1,ϕ ||e = ||Cϕ ||e ≤ σ(E)1/2 . And if set ψ = 1 in theorem 3.1, then ||Cϕ ||e ≥ (µ1,ϕ,2 (E))1/2 = σ(ϕ−1 (ϕ(E)))1/2 , so σ(ϕ−1 (ϕ(E))) = σ(E), we have the following Corollary Corollary 3.2. (Theorem 1 in[GorM]) Cϕ : H ∞ → H 2 is bounded and ||Cϕ ||e = σ(E)1/2 . Case 2. p = ∞, q 6= 2 Theorem 3.2. Suppose Wψ,ϕ : H ∞ → H q (q > 1), and ψ ∈ H q , then 1 (µψ,ϕ,q (ϕ(E)))1/q ≤ ||Wψ,ϕ ||e ≤ 2(µψ,ϕ,q (ϕ(E)))1/q . 2 Proof We consider upper estimate first. Obviously Wψ,rϕ is compact for any fixed 0 < r < 1. Let E = {ξ ∈ ∂Bn : |ϕ(ξ)| ≥ 1 − } and let Ec denote its complement in ∂Bn . So ||Wψ,ϕ − Wψ,rϕ || =

sup ||(Wψ,ϕ − Wψ,rϕ )f ||q ||f ||∞ =1

Z sup (

=

||f ||∞ =1

∂Bn

Z sup (

=

||f ||∞ =1

||f ||∞ =1

|ψ(f ◦ ϕ) − ψ(f ◦ (rϕ))|q dσ)1/q

E

Z sup (

+

|ψ(f ◦ ϕ) − ψ(f ◦ (rϕ))|q dσ)1/q

|ψ(f ◦ ϕ) − ψ(f ◦ (rϕ))|q dσ)1/q .

Ec

Apply Lemma 2.4, we can choose r sufficiently close to 1 to make the second term less than ||ϕ||q . For the first term, the triangle inequality yields |f ◦ ϕ(ξ) − f ◦ (rϕ)(ξ)| ≤ 2 So, the first term is less than Z Z q 1/q 2( |ψ| dσ) ≤ 2( E

ϕ−1 (ϕ(E ))∩∂Bn

|ψ|q dσ)1/q = 2(µψ,ϕ,q (ϕ(E )))1/q .

Let m ↓ 0, and Em = {ξ ∈ ∂Bn : |ϕ(ξ)| ≥ 1 − m }, then µψ,ϕ,q (ϕ(Em )) → µψ,ϕ,q (ϕ(E)), the upper estimate follows. Now we turn to lower estimate. Let f be a non-constant inner function in Bn , K is any compact operator. For any positive integer m, the sequence {f m } are in the unit ball of H ∞ , So there exists a subsequence {f mk } such that {K(f mk )} converges in norm. Therefore, given  > 0, there exists M such that ||K(f mk ) − K(f ml )||q <  for any k, l > M . Fix k > M , there exists r with 0 < r < 1 such that (ψ(f ◦ ϕ)mk )r (z) = ψ(rz)(f ◦ ϕ(rz))mk satisfies ||(ψ(f ◦ ϕ)mk )r ||q ≥ ||(ψ(f ◦ ϕ)mk )|| − . Thus, for m ≥ M f mk − f ml ||q 2 ≥ (1/2)||(ψ(f ◦ ϕ)mk ) − (ψ(f ◦ ϕ)ml )||q − /2 ≥ (1/2)(||(ψ(f ◦ ϕ)mk )||q − ||(ψ(f ◦ ϕ)ml )||q ) − /2 ≥ (1/2)(||(ψ(f ◦ ϕ)mk )||q − ||(ψ(f ◦ ϕ)ml )r ||q ) − .

||Wψ,ϕ − K|| ≥ ||(Wψ,ϕ − K)

259

ESSENTIAL NORMS OF WEIGHTED COMPOSITION OPERATORS

9

letting l → ∞ and h = f mk , we have ||Wψ,ϕ − K|| ≥ (1/2)(||(ψ(f ◦ ϕ)mk )||q −  Z |ψ ∗ · (h∗ ◦ ϕ∗ )|q dσ)1/q −  = (1/2)( Z∂Bn = (1/2)( |h∗ |q dµψ,ϕ,q )1/q −  ZB n ≥ (1/2)( |h∗ |q dµψ,ϕ,q )1/q −  ϕ(E)

≥ (1/2)(µψ,ϕ,q (ϕ(E)))1/q −  Now letting  → 0 yields the result. Corollary 3.3. Wψ,ϕ : H ∞ → H q is compact if and only if ψ ∈ H q and σ(E) = 0. Proof Combining Lemma 3.1 and Theorem 3.2, the corollary follows. Corollary 3.4. (Theorems 2 and 3 [GorM] Cϕ : H ∞ → H q is bounded and 1 σ(E)1/q ≤ ||Cϕ ||e ≤ 2σ(E)1/q . 2 Proof Let ψ = 1 ∈ H q , then Wψ,ϕ = Cϕ , the corollary follows by Theorem 3.2. 4. From H p to H q for 1 < q < p < ∞ Theorem 4.1. Assume Wψ,ϕ : H p → H q (1 < p < ∞) is bounded, then ||Wψ,ϕ ||e ≥ (µψ,ϕ,q (ϕ(E)))1/q . Proof Let g be a non-constant inner function on Bn and set h = g m for a positive integer m. Then ||g m ||p = 1 for any m, and g m converges weakly to 0 as m → ∞, thus ||Kfw || → 0 for any compact operator from H P to H q when |w| → 1. Like in Theorem 3.1, we have ||Wψ,ϕ − K|| ≥ lim sup ||(Wψ,ϕ − K)(g m )||q m→∞

≥ lim sup ||Wψ,ϕ (g m )||q − lim sup ||K(g m )||q m→∞ m→∞ Z m = lim sup ||Wψ,ϕ (g )||q = lim sup( |ψ ∗ · (h∗ ◦ ϕ∗ )|q dσ)1/q m→∞ m→∞ ∂Bn Z Z = lim sup( |h∗ |dµψ,ϕ,q )1/q ≥ lim sup( |h∗ |dµψ,ϕ,q )1/q m→∞

m→∞

Bn 1/q

≥ (µψ,ϕ,2 (ϕ(E)))

ϕ(E)

.

This ends the proof. Corollary 4.1. Assume Wψ,ϕ : H p → H q , p > 1, 0 < q < ∞ is compact, then σ(E) = 0. Remark 1. We will show that when 0 < p < q < ∞ and Wψ,ϕ : H p → H q is bounded, then µψ,ϕ,q (ϕ(E)) = 0 (see Corollary 6.1), So the above estimate is useless. Theorem 4.2. Suppose 1 < q < p < ∞ and there exists r > q such that Wψ,ϕ : H p → H r (1 < p < ∞) is bounded, then ||Wψ,ϕ ||e ≤ ||P || · ||Wψ,ϕ ||p,r · σ(E) where P is the Szeg¨ o projection of Lq (σ) onto H q .

r−q qr

260

10

Z.S. FANG AND Z.H.ZHOU

Proof

We consider the operator K : H p → H q defined by K(f ) = P (χEc ψ · (f ◦ ϕ)),

where P is the Szeg¨o projection of Lq (σ) onto H q . Like in Lemma 3.3, K is compact operator from H p to H q . So for any g ∈ H p with ||g||p = 1, we have ||Wψ,ϕ (g) − K(g)||q = ||ψ · g ◦ ϕ − P (χEc ψ · (f ◦ ϕ))||q = ||P (χE ψ · (f ◦ ϕ))||q ≤ ||P || · ||χEc ψ · (f ◦ ϕ)||q Z 1 |ψ · g ◦ ϕ|q dσ) q = ||P || · ( ZE 1 ≤ ||P || · ( χEc |ψ · g ◦ ϕ|q dσ) q ∂Bn

≤ ||P || · ||Wψ,ϕ (g)||r σ(E ) ≤ ||P || · ||Wψ,ϕ ||p,r · σ(E )

r−q qr r−q qr

.

Letting  → 0 yields the conclusion. 5. From H p to H ∞ Theorem 5.1. For Wψ,ϕ : H p → H ∞ , and 0 < p < ∞, then Wψ,ϕ is bounded if and |ψ(z)| only if sup (1−|ϕ(z)| 2 )n/p < ∞. z∈Bn

Proof ” ⇒ ” For any w ∈ Bn , define fw (z) = ||fw ||p = 1. So C ≥ ||Wψ,ϕ || =

(1−|w|2 )n/p , (1−)2n/p

sup ||Wψ,ϕ f ||∞ ≥ sup ||Wψ,ϕ fw ||∞ z∈Bn

||f ||p =1

=

and it is easy to check

sup sup |ψ(z)||fw (ϕ(z))| w∈Bn z∈Bn

setting w = ϕ(z), as desired. ”⇐” ||Wψ,ϕ || =

sup ||Wψ,ϕ ||∞ = sup sup |ψ(z)f (ϕ(z))| ||f ||p =1 z∈Bn

||f ||p =1

≤

sup sup |ψ(z)| ||f ||p =1 z∈Bn

||f ||p |ψ(z)| = sup n/p 2 2 n/p (1 − |ϕ(z)| ) z∈Bn (1 − |ϕ(z)| )

Theorem 5.2. For Wψ,ϕ : H p → H ∞ (p > 1), and Wψ,ϕ is bounded, then lim

sup

δ→0 dist(ϕ(z),∂Bn ) |2nq/p Proof Let K be any compact operator from H p to H ∞ . For any w ∈ Bn define (1−|w|2 )n/p fw (z) = (1−) 2n/p , it is easy to check ||fw ||p = 1 and fw converge weakly to 0 as |w| → 1, thus ||Kfw || → 0 when |w| → 1. So for any 0 < δ < 1, ||Wψ,ϕ − K|| ≥ lim sup ||(Wψ,ϕ − K)fw ||q |w|→1

≥ lim sup ||Wψ,ϕ fw ||q − lim sup ||Kfw ||q |w|→1

|w|→1

Z

|ψ(z)|q

= lim sup |w|→1

∂Bn

Z ≥ lim sup |w|→1

Bn

(1 − |w|2 )nq/p dσ(z) |1− < ϕ(z), w > |2nq/p

(1 − |w|2 )nq/p dµψ,ϕ,q (z) |1− < z, w > |2nq/p

The conclusion follows. We cannot give the upper estimate in the above form, but we have the following theorem.

264

14

Z.S. FANG AND Z.H.ZHOU

Theorem 6.3. Assume 1 < p ≤ q < ∞ and Wψ,ϕ : H p → H q is bounded, then Wψ,ϕ : H p → H q is compact if and only if Z (1 − |w|2 )nq/p lim dµψ,ϕ,q (z) = 0. |w|→1 B n |1− < z, w > |2nq/p Proof The necessary condition follows by theorem 6.2. We consider the sufficient condition. By Lemma 6.2, we only have to show µψ,ϕ,q is vanishing pq −Carleson measure. From the proof of (iii) ⇒ (i) in theorem 6.1, for any z ∈ ∂Bn , set |z0 | = 1 − h2 . Suppose Z (1 − |w|2 )nq/p dµψ,ϕ,q (z) = 0. lim |w|→1 B n |1− < z, w > |2nq/p That is , ∀ > 0, ∃1 > r > 0, when |w| > r we have Z (1 − |w|2 )nq/p dµψ,ϕ,q (z)| < . | 2nq/p B n |1− < z, w > | When h < 2(1 − r), for any z ∈ ∂Bn , the corresponding |z0 | > r, so Z (1 − |z0 |2 )nq/p  > dµψ,ϕ,q (w) 2nq/p B n |1− < w, z0 > | Z c dµψ,ϕ,q ≥ nq/p Sh (z) h cµψ,ϕ,q (Sh (z)) ≥ . hnq/p This is true for any z ∈ ∂Bn . So µψ,ϕ,q is vanishing pq − Carleson measure. References [ConH] M.D.Contreras and A G. Her´ andez-D´ıaz, Weighted composition operators in weighted Banach spaces of ananytic functions, J.Austral.Math.Sox.(Serier A) 69(2000), 41-46. [CowMac] C.C.Cowen and B.D.MacCluer,Composition operators on spaces of analytic functions, CRC Press, Boca Raton , FL, 1995. [Goe] T.Goebeler, Composition operators acting between Hardy spaces, Integr. Equ. Oper. Theory 41(2001),389-395. [GorM] P.Gorkin and B.D.MacCluer,Essential norms of composition operators, Integr. Equ. Oper. Theory 48(2004),27-40. [GorMS] P.Gorkin, R. Mortini,and D.Suarez, Homotopic composition operators on H ∞ (BN ), Function space, Edwarssville, IL, 2002, 177-188, Contemporary mathematics, 328, Amer. Math. Soc., Providence, RI, 2003. [Hal] Paul.R.Halmos, Measure theory,Springer-Verlag, GTM 18,1970. [JarR] H.Jarchow and R. Riedl, Factorization of composition operators through Bloch space, Illinois J. Math. 39(1995),431-440. [LS1] L.Luo and J.H.Shi, Compositon operatoers between Hardy spaces on the unit ball, Acta.Math.Sinica 44(2001), 209-216. [LS2] L.Luo and J.H.Shi, Compositon operatoers between the weighted Bergman spaces on bounded symmetric domains of C n , Chinese Journal of Contemporary mathematics 21(2000), 55-64. [Kos] P.Koosis, Introduction to Hp spaces, second edition, Cambridge University Press,Cambridge,1998. [Mac] B.D.MacCluer,Compact compositon operators on H p (BN ), Michigan Math.J. 32(1985), 237-248. [Rud] W.Rudin, Function theory in the unit ball of C n , Springer-Verlag, New York, 1980. [Tja] M. Tjani,Compact Compsotion operators some Mobius invariant Banach spaces, Thesis, Michigan State University,1996. [Shap1] Joel. H. Shapiro,Composition operators and classical function theory, Spriger-Verlag, 1993. [Shap2] L.H.Shapiro, The essential norm of a composition operator, Annals Math. 125(1987), 375-404. [Shap3] J. H. Shapiro, Compact composition operators on spaces of boundary regular holomorphic functions, Proc.Amer.Math.Soc. 100(1987),49-57. [Smi1] W.Smith,Compsotion operators between Bergman and Hardy spaces, Trans.Amer.Math.Soc. 348(1996),2331-2348. [Smi2] W.Smith,Compsotion operators on BMOA, Proc.Amer.Math.Soc. 127(1999),2715-2725.

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ESSENTIAL NORMS OF WEIGHTED COMPOSITION OPERATORS

15

[SmiZ] W.Smith and R. Zhao,Compsotion operators mapping into the Qp spaces, Analysis 17(1999), 239-263. [X] H.M.Xu and T.S.Liu, Weighted compositon operatoers between Hardy spaces on the unit ball, Chin. Quart. J. of. Math, 19(2004),111-119. [Zhe] L. Zheng, The essential norms and spectra of composition operators on H ∞ , Pacific J. Math 203 (2002), 503-510. [Zho] Z.H. Zhou. Composition operators on the Lipschitz space in polydiscs, Sci. China Ser. A 46 (1) (2003), 33-38. [ZC] Z.H.Zhou and Renyu Chen, Weighted composition operators fom F (p, q, s) to Bloch type spaces, International Jounal of Mathematics 19(8)(2008), 899-926. [ZC1] Z. H. Zhou and Renyu Chen, On the composition operators on the Bloch space of several complex variables, Science in China (Series A) 48(Supp.), (2005) 392-399. [ZL] Z.H. Zhou and Yan Liu. The essential norms of composition operators between generalized Bloch spaces in the polydisc and their applications, Journal of Inequalities and Applications 2006(2006), Article ID 90742, 1-22. [ZS1] Z.H. Zhou and J.H. Shi. Compact composition operators on the Bloch space in polydiscs, Science in China (Series A) 44 (2001), 286-291. [ZS2] Z.H. Zhou and J.H. Shi. Composition operators on the Bloch space in polydiscs, Complex Variables 46 (1) (2001), 73-88. [ZS3] Z.H. Zhou and J.H. Shi. Compactness of composition operators on the Bloch space in classical bounded symmetric domains, Michigan Math. J. 50 (2002), 381-405. [Zhu1] K.H.Zhu, Operator theory in function spaces, Marcel DeKKer.New YorK. 1990. [Zhu2] K.h.Zhu, Spaces of holomorphic functions in the unit ball, Springer 2004.

Department of Mathematics Tianjin Polytechnic University Tianjin 300160 P.R. China. E-mail address: [email protected]

Department of Mathematics Tianjin University Tianjin 300072 P.R. China. E-mail address: [email protected]

266 JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.5, NO.3, 266-289, COPYRIGHT 2010 EUDOXUS PRESS, LLC

THE ABSTRACT WAVELET TRANSFORM JAIME NAVARRO

Universidad Aut´onoma Metropolitana Departamento de Ciencias B´asicas P. O. Box 16-306, M´exico City, 02000 M´exico [email protected] Abstract. Given an abstract locally compact topological group, the abstract wavelet transform is defined so that the inversion formula is used to prove that the image of the wavelet transform is a reproducing kernel Hilbert space, and it is also shown that any function in this space can be reconstructed by its sampled values. Key words and phrases: unitary representation, admissible function, wavelet transform, inversion formula, sampled values. 2000 AMS Mathematics Subject Classification 42A38, 42C40

1. Introduction

The continuous wavelet transform for a given signal f is a time-frequency localization method and it is considered as an alternative of the windowed Fourier transform. The wavelet transform determines better than the windowed Fourier transform the localization of high and low-frequencies w for a specific time t [1]. The continuous wavelet transform with respect to an “admissible” function has been used to detect singularities of functions in the Hilbert space L2 .R/. For example, by using a reconstruction formula given by Grossmann-Morlet-Paul [4], and the locally compact topological group (1)

G D f .a; b/ j a 2 R n f0g; b 2 R g; 2

and a group representation U acting on the Hilbert space L .R/ given by   1 x b U.a; b/h .x/ D p h ; a jaj

(2)

it was proved that for a given function f in L2 .R/, the singularities of the continuous wavelet transform .Lh f /.a; b/ are the singularities of f [7]. 1

267

2

THE ABSTRACT WAVELET TRANSFORM

In two dimensions, the continuous wavelet transform has been used to study singularities of distributions u in S 0 .R2 /. In this case, the wavelet transform of u yields a function on phase space whose high-frequency singularities are precisely the elements in the wave front set of u [9]. For n dimensions [6], one option is to consider a radially symmetric function h. In this case, the group is given by G D f.a; b/ j a > 0 and b 2 Rn g

(3)

and the representation U of G acting on L2 .Rn / is taken as 1 x b .U.a; b/h/.x/ D n h. a /. As before, the singularities of f in L2 .Rn / are the singua2

larities of .Lh f /.a; b/ [8]. In [10], we consider a second option for h, with just two conditions; h in C01 .Rn / and y h.0/ D 0. That is, if we drop the condition that h is radially symmetric, then we need extra parameters besides a in RC and b 2 Rn to define a locally compact topological group G so that the representation U of G acting on L2 .Rn / satisfies the condition that as a ! 0, then .U.g/h/.x/ with g in G concentrates around x D b. In this case, the locally compact topological group is taken as ˚ G D .a; b; / j a > 0; b 2 R2 ; and  2 Œ0; 2 / ; (4) and then, it is proved again that the singularities of f are the singularities of .Lh f /.a; b/.

For a product of two locally compact groups, the wavelet transform as well as its inverse were studied by means of a special homogeneous space [2]. In this paper, we follow the locally compact topological groups point of view, to define the wavelet transform, where our group G is given as the product of n C 1 locally compact topological groups A1 ; A2; : : : ; An ; B, by means of a square integrable, irreducible, and Q unitary representation acting on the Hilbert space L2 . niD1 Ai /, where this representation depends of n C 1 parameters ai 2 Ai , with i D 1; 2; : : : ; n, and b 2 B so that the inversion Q formula is obtained for a given function f 2 L2 . niD1 Ai /. A reproducing kernel Hilbert space on L2 .G/ is given as well as a sampling formula for functions in the image of the wavelet transform. Moreover, we study different characterizations of the wavelet transform by using the Fourier transform, and the convolution of two functions.

2. Notations and definitions Let us begin by defining a homomorphism for an Abelian locally compact topological group. So, consider n C 1 locally compact topological groups A1 ; A2 ; : : : ; An and B where

268

JAIME NAVARRO

A1 ; A2; : : : ; An

are

Abelian. Now, for each b 2 B consider Qn A such that for a D .a ; a ; : : : ; a / 1 2 n 2 A, i D1 i

3

the

map

€b W A ! A, where A D

a ! €b .a/ D .€b .a1 /; €b .a2 /; : : : ; €b .an // is a homeomorphism. Note that for i D 1; 2; : : : ; n the homomorphism € from B into the group of all automorphisms of Ai given by .ai ; b/ ! €b .ai / is continuous on Ai  B to Ai . On the other hand, for a D .a1 ; a2 ; : : : ; an / and a0 D .a10 ; a20 ; : : : ; an0 / in A define aa0 D .a1 ; a2; : : : ; an /.a10 ; a20 ; : : : ; an0 / D .a1 a10 ; a2 a20 ; : : : ; an an0 /: So, A is an Abelian group, where eA D .e1 ; e2 ; : : : ; en / is the identity with ei the identity in Ai , and a 1 D .a1 ; a2; : : : ; an / 1 D .a1 1 ; a2 1 ; : : : ; an 1 / is the inverse with ai 1 the inverse in Ai , for i D 1; 2; : : : ; n. Definition 1. Define G as the product of A and B. That is, consider G D A  B D f.a; b/ j a 2 A; and b 2 Bg : In G define .a; b/.a0 ; b 0 / D .a€b .a0 /; bb 0 /:

(5)

Then, with this product G becomes a group, where eG D .eA ; eB / is the identity (eA is de identity in A and eB is the identity in B), and where .a; b/ 1 D .€b 1 .a 1 /; b 1 / is the inverse. Note also that G D A  B is a locally compact topological group. Then we will denote by dG .a; b/ the left Haar measure on G, the left Haar measure on A by dA .a/ and the left Haar measure in B by dB .b/. Definition 2. Given a group G and a set E, an action of G on E is a map .s; x/ ! sx of G  E ! E such that 1) ex D x for any x in E, and where e is the identity in G 2) s.tx/ D .st/x for any x in E, and where s; t are in G. Then we have the following Lemma. Lemma 1. The function
W G  A ! A given by .a; b/
x D a€b .x/ is an action of G on A where .a; b/ 2 G and x 2 A. Proof. 1) Let eG D .eA ; eB / be the identity in G, and let x be in A. Then eG
x D .eA ; eB /
x D eA €eB .x/ D €eB .x/ D x: 2) Let .a; b/ and .a0 ; b 0 / be in G, and let x be in A. Then       .a; b/ .a0 ; b 0/
x D .a; b/ a0 €b 0 .x/ D a€b a0 €b 0 .x/   D a€b .a0 /€bb 0 .x/ D .a€b .a0 /; bb 0 /
x D .a; b/.a0 ; b 0 /
x:

269

4

THE ABSTRACT WAVELET TRANSFORM

2 Definition 3. Let G be a locally compact topological group. The support of the function f W G ! C denoted by supp.f / is defined as the closure of fx 2 Gjf .x/ ¤ 0g, and C0 .G/ is defined as the set of continuous functions f W G ! C such that supp.f / is compact. Definition 4. Let G be a locally compact topological group, H a closed subgroup of G. A measure  ¤ 0 on G=H is said to be relatively invariant under G if for each s 2 G there is a function  W G ! .0; 1/ such that for every function h 2 C0 .G=H /, we have Z Z h.s 1 z/d.z/ D .s/ h.z/d.z/ G=H

G=H

In this case, the function  W G ! .0; 1/ is a continuous homomorphism such that H .z/ .z/ D  with z 2 H and where H and G are the modular functions on H and G G .z/ respectively. Lemma 2. Let G be a locally compact topological group, and let H be a closed subgroup of H .g/ G. If there is a continuous positive homomorphism  on G satisfying .g/ D  with G .g/ g 2 H , then there exists on G=H a relatively invariant measure  [11]. In our case, for G D A  B, there is a positive continuous homomorphism  W G ! .0; 1/ that satisfies Lemma 2 for h in C0 .A/ [2]. That is, for any h 2 C0 .A/ we have Z Z
1 h .a; b/
x dA .x/ D .a; b/ h.x/dA .x/: (6) A

A

Definition 5. For 1  p < 1 and for a complex valued function defined on the locally compact topological group A, define   Z ˇ Lp .A/ D h W A ! C ˇ jh.x/jp dA .x/ < 1 A

where dA .x/ is the left Haar measure on A.

Formula (6) can be extended for all h 2 L1 .A/ [11]. So, from now on consider  W G ! .0; 1/ satisfying (6) for h 2 L1 .A/. 3. Unitary operators Definition 6. For h 2 L2 .A/ define the following operators 1 .Ja h/.x/ D p hŒ.a; eB / .a; eB / where .a; eB / 2 G; x 2 A, and a 2 A: 1/

2/

.Tb h/.x/ D p

where .eA ; b/ 2 G; x 2 A; and b 2 B:

1 .eA ; b/

hŒ.eA ; b/

1


x;

1


x;

270

JAIME NAVARRO

5

Lemma 3. For the operators Ja and Tb , Ja Ja 0 D Jaa 0 ; where a; a0 2 A: Tb Tb 0 D Tbb 0 ; where b; b 0 2 B: Tb Ja D J€b .a/;Tb ; and Ja Tb D Tb J€b

1/ 2/ 3/

1 .a/

;

where

a 2 A;

and

b 2 B:

Proof. 1) .Ja Ja 0 h/.x/ D ŒJa .Ja 0 h/.x/ D p

1 .a; eB /

.Ja 0 h/Œ.a; eB /

1


x

1 p hŒ.a0 ; eB / 1 .a; eB / 1
x .a; eB / .a0 ; eB /   1 1 h ..a; eB /.a0 ; eB // 1
x D p hŒ.aa0 ; eB / Dp 0 0  Œ.a; eB /.a ; eB / .aa ; eB / Dp

1

1


x D .Jaa 0 h/.x/:

2)

1 .Tb Tb 0 h/.x/ D ŒTb .Tb 0 h/.x/ D p .Tb 0 h/Œ.eA ; b/ 1
x .eA ; b/ 1 1 Dp p hŒ.eA ; b 0 / 1 .eA ; b/ 1
x 0 .eA ; b/ .eA ; b /   1 1 Dp h ..eA ; b/.eA ; b 0 // 1
x D p hŒ.eA ; bb 0 / 0 Œ.eA ; b/.eA ; b / .eA ; bb 0/

1


x D .Tbb 0 h/.x/:

3) On one hand,

1 .Tb Ja h/.x/ D ŒTb .Ja h/.x/ D p .Ja h/Œ.eA ; b/ 1
x .eA ; b/ 1 1 Dp p hŒ.a; eB / 1 .eA ; b/ 1
x .eA ; b/ .a; eB /   1 1 Dp h ..eA ; b/.a; eB // 1
x D p hŒ.€b .a/; b/ Œ.eA ; b/.a; eB / .€b .a/; b/

1


x:

On the other hand,

1

.Tb h/Œ.€b .a/; eB / 1
x .€b .a/; eB / 1 1 Dp p hŒ.eA ; b/ 1 .€b .a/; eB / 1
x .€b .a/; eB / .eA ; b/ 1 Dp hŒ..€b .a/; eB /.eA ; b// 1
x Œ.€b .a/; eB /.eA ; b/ 1 1 hŒ.€b .a/€eB .eA /; eB b/ 1
x D p Dp hŒ.€b .a/; b/ .€b .a/€eB .eA /; eB b/ .€b .a/; b/

.J€b .a/Tb h/.x/ D p

In a similar way, it can be proved that Ja Tb D Tb J€b

1 .a/

:

Lemma 4. For a 2 A and b 2 B, the operators Ja and Tb are unitary operators.

1


x: 

271

6

THE ABSTRACT WAVELET TRANSFORM

Proof. Let h be in L2 .A/. Then by (6), 1)

Z

2

2)

Z

2

jjJa hjj D j.Ja h/.x/j dA .x/ D .Ja h/.x/.Ja h/.x/dA .x/ A A Z 1 1 p h..a; eB / 1
x/ p h..a; eB / 1
x/dA .x/ D .a; eB / .a; eB / A Z 1 D .hh/..a; eB / 1
x/dA .x/ .a; eB / A Z Z 1 D .a; eB / .hh/.x/dA .x/ D jh.x/j2 dA .x/ D jjhjj2: .a; eB / A A

Z Z jjTb hjj2 D j.Tb h/.x/j2 dA .x/ D .Tb h/.x/.Tb h/.x/dA .x/ A A Z 1 1 1 p h..eA ; b/
x/ p h..eA ; b/ 1
x/dA .x/ D .e ; b/ .e ; b/ A A A Z 1 D .hh/..eA ; b/ 1
x/dA .x/ .eA ; b/ A Z Z 1 D jh.x/j2 dA .x/ D jjhjj2 : .eA ; b/ .hh/.x/dA .x/ D .eA ; b/ A A Moreover, since Ja D Ja 1 D Ja 1 ; and Tb D Tb 1 D Tb 1 ; it follows that both, Ja and Tb are unitary.  4. Fourier transform Definition 7. Let G be a locally compact topological Abelian group, and let T D fz 2 C j jzj D 1g. We say that the function  W G ! T is a character on G if  is a continuous homomorphism. Definition 8. Given a locally compact topological Abelian group G, we define the dual group of G as b D f W G ! T j  is a characterg G

b In this case we denote .g/ D hg; i where g 2 G and  2 G: Note

that

b G

is

clearly

an

Abelian

group

under

pointwise

multiplication

.1 2/.g/ D 1.g/2 .g/. Its identity element is the constant function 1 and the inverse element is  1.g/ D .g/ D .g 1 /. The dual group of a locally compact topological Abelian group is used to define an abstract version of the Fourier transform. b ! C Definition 9. Given h 2 L1 .G/, the Fourier transform of h is the function b h W G defined by

b h./ D

Z

h.g/.g/dG .g/;

G

(7)

272

JAIME NAVARRO

7

where the integral is relative to the left Haar measure on G. b the inverse Fourier transform of h is the function Definition 10. For a function h 2 L1 .G/, z h W G ! C defined as Z z h.g/ D h./.g/db ./; (8) G b G b where db ./ is the left Haar measure on G. G b Then for h 2 L1 .G/ and b h 2 L1 .G/,

h.g/ D

Z

Lemma 5. For h 2 C0 .A/ we have 1/ 2/

b G

b h./.g/db ./: G

1 p.a; e / p .1 T h/./ D .e ; b/

.Ja h/./ D

B

A

b

where  2 b A; a 2 A, and b 2 B.

.a/ b h./

b h. ı €b /;

Proof. 1) Note that since Z Z 1 .Ja h/./ D .Ja h/.x/.x/dA .x/ D p h..a; eB / .a; eB / A A

1

1


x/.x/dA .x/;

and by Lemma 1,

.x/ D Œaa

1

€eB .x/ D .a/Œ.a

1

; eB /
x D .a/Œ.a; eB /

1


x;

it follows from (6) that

1

Z

1

1


x/ .a/ ..a; eB / 1
x/dA .x/ .a; eB / Z 1 Dp .a/ .h/..a; eB / 1
x/dA .x/ .a; eB / A Z 1 Dp .a/.a; eB / .h/.x/dA .x/ .a; eB / A Z p p D .a; eB / .a/ h.x/.x/dA .x/ D .a; eB / .a/ b h./:

.Ja h/./ D

A

p

h..a; eB /

A

2) Similarly, since

1

.Tb h/./ D and

Z

A

.Tb h/.x/.x/dA .x/ D

Z

A

1 p h..eA ; b/ .eA ; b/

1


x/.x/dA .x/;

.x/ D  Œ€bb 1 .x/ D  Œ€b .€b 1 .x// D  Œ€b .eA €b 1 .x//     D  €b ..eA ; b 1/
x/ D  €b ..eA ; b/ 1
x/ D . ı €b /..eA ; b/

1


x/;

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8

THE ABSTRACT WAVELET TRANSFORM

it follows from (6) that Z 1 .Tb h/./ D p h..eA ; b/ 1
x/. ı €b /..eA ; b/ 1
x/dA .x/ .eA ; b/ A Z 1 Dp .h
 ı €b /..eA ; b/ 1
x/dA .x/ .eA ; b/ A Z 1 Dp .eA ; b/ .h
 ı €b /.x/dA .x/ .eA ; b/ A Z p p h.x/. ı €b /.x/dA.x/ D .eA ; b/ b h. ı €b /: D .eA ; b/

1

A

Corollary 1. Let h be in L1 .A/. Then for a in A p .Ja Tb h/./ D .a; b/ .a/ b h. ı €b /:



3

Proof. It comes from Lemma 5.



Lemma 6. The function
W G  Ay ! Ay given by Œ.a; b/
.x/ D Œ.a; b/
x is an action of G on Ay where x 2 A. Proof. 1) .eG
/.x/ D Œ.eA ; eB /
.x/ D Œ.eA ; eB /
x D .x/. 2) One one hand,    
.a; b/ .a0 ; b 0 /
 .x/ D ..a; b/
/ .a0 ; b 0 /
x D Œ.a; b/
 a0 €b 0 .x/       D  .a; b/
a0 €b 0 .x/ D  a€b .a0 €b 0 .x// D  a€b .a0 /€bb 0 .x/ : On the other hand,

This proves Lemma 6.

    .a; b/.a0 ; b 0 /
 .x/ D .a€b .a0 /; bb 0 /
 .x/     D  .a€b .a0 /; bb 0 /
x D  a€b .a0 /€bb 0 .x/ :

Corollary 2. Let h be in L1 .A/. Then for a 2 A,

1 .1 T h/./ D .T b h/./ 3 .J T h/./ D .J T

1/ .Ja h/./ D .Ja 1b h/./ 2/ 3/

b

a b

Proof. Note that from Lemma 6,

b

1

a

1

b

b h/./:

1



274

JAIME NAVARRO

9

1) Z p 1 1 1 b b .Ja 1 h/./ D p h..a ; eB /
/ D .a; eB / h.x/Œ.a; eB /
 .x/dA .x/ .a 1 ; eB / A Z Z p p D .a; eB / h.x/Œ.a; eB /
xdA .x/ D .a; eB / h.x/Œa€eB .x/dA .x/ A A Z p p D .a; eB / .a/ h.x/.x/dA .x/ D .a; eB / .a/ b h./ D .Ja h/./:

1

A

2)

Z p 1 1 1 b b .Tb 1 h/./ D p h..eA ; b /
/ D .eA ; b/ h.x/Œ.eA ; b/
 .x/dA .x/ .eA ; b 1/ A Z Z p p D .eA ; b/ h.x/Œ.eA ; b/
xdA .x/ D .eA ; b/ h.x/Œ.eA /€b .x/dA .x/ A ZA p p b h.x/.€b .x//dA .x/ D .eA ; b/ h. ı €b / D .Tb h/./: D .eA ; b/

1

A

3)

3 .J T h/./ D .J Tbh/./ D .J a b

a

1

b

a

1

T b 1b h/./:



5. Unitary representation Definition 11. For .a; b/ in G D A  B, define the n C 1 parameter family of operators U.a; b/ D Ja Tb . Note that U.a; b/ acts on the Hilbert space L2 .A/ by:

.U.a; b/h/.x/ D .Ja Tb h/.x/ D .Ja .Tb h//.x/ D p

1 .a; eB /

 .Tb h/ .a; eB /

  1 p h .eA ; b/ 1 .a; eB / 1
x .a; eB / .eA ; b/    1 1 Dp h ..a; eB /.eA ; b// 1
x D p h .a; b/ ..a; eB /.eA ; b// .a; b/ Dp

1

1

1


x




x :

Lemma 7. U.a; b/ D Ja Tb is a unitary representation of G acting on the Hilbert space L2 .A/. Proof. Note that since the operators Ja and Tb are unitary (Lemma 4), it follows that U.a; b/ is unitary. Now, let us prove that U.a; b/ is a representation of G acting on L2 .A/. On one hand, from Lemma 3,   U .a; b/.a0 ; b 0 / D U.a€b .a0 /; bb 0 / D Ja€b .a 0/ Tbb 0 D Ja J€b .a 0/ Tb Tb 0 D Ja Tb Ja 0 Tb 0 D U.a; b/U.a0 ; b 0 /:

275

10

THE ABSTRACT WAVELET TRANSFORM

On the other hand, since U.eA ; eB / D JeA TeB D I , where I is the identity operator, it follows that U.a; b/ is a representation of G acting on L2 .A/.



Lemma 8. The left Haar measure on G D A  B is given by : d.a; b/ D

1 dA .a/dB .b/: .a; b/

Proof. Let h be in L1 .G/, then by (6) Z

G

 h .a0 ; b 0 /

1

Now, replacing h by

Then Z

G

Z

G

 .a; b/ dA .a/dB .b/ D .a0 ; b 0 /

h 

we have

Z

h.a; b/dA .a/dB .b/: G

  Z h .a0 ; b 0 / 1 .a; b/ h.a; b/ 0 0 dA .a/dB .b/ D .a ; b / dA .a/dB .b/:  Œ.a0 ; b 0 / 1 .a; b/ .a; b/ G

 h .a0 ; b 0 /

That is

1

 .a; b/ Z

G

1 dA .a/dB .b/ D .a; b/

 h .a0 ; b 0 /

1

 .a; b/ d.a; b/ D

Z

h.a; b/

Z

h.a; b/d.a; b/:

G

1 dA .a/dB .b/: .a; b/

G

This shows that d.a; b/ is a left Haar measure on G.



6. Admissibility condition Definition 12. A function h in L2 .A/ is said to be admissible if Z j hh; U.a; b/hi j2 d.a; b/ < 1: G

Lemma 9. Let h be in L1 .A/ \ L2 .A/. If .B/ < 1, then Z Ch  jb h. ı €b /j2 dB .b/ B

y is uniformly bounded for  2 A: Proof. Note that since

it follows that jb h. ı €b /j 

b h. ı €b / D

Z

Z

h.x/. ı €b /.x/dA .x/; A

jh.x/j j.€b .x//jdA .x/ D A

Z

jh.x/jdA .x/ D jjhjj1 : A

Hence, Ch  jjhjj21 .B/ < 1:



Lemma 10. Let h be in L1 .A/ \ L2 .A/ if Z y ı €b /j2 dB .b/ 0 < Ch  jh. B

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JAIME NAVARRO

11

y then h is admissible. is uniformly bounded for  2 A,

Proof. Since 0 < Ch < 1 for any  2 Ay and h is in L2 .A/, it follows that R  jjhjj2 Ch D jjb hjj2 Ch Z  Z  2 2 b b D jh./j dAy./ jh. ı €b /j dB .b/ < 1: Ay

B

Then R can be written as Z Z ˇ ˇ2 ˇb 2 b ˇ RD ˇh./j jh. ı €b /ˇ dAy./dB .b/ B Ay  Z Z ˇ ˇ2 ˇb p ˇ D h. ı €b /ˇ dAy./ ˇh./ .eA ; b/ b B

Ay

1 .eA ; b/

dB .b/:

Then by Corollary 1, RD

Z Z B

Ay

 2 b jh./Tb h./j dAy./

b

2 Z

b

D h T h

b B

D

Z

b

 

S

bb hT h

1

y L2 .A/

.eA ; b/

2

1 dB .b/ .eA ; b/

dB .b/

1

dB .b/ .eA ; b/ 1  ˇˇ 2 Z Z ˇˇ 1 ˇ @ ˇˇ b D hTb h .a/ˇ dA .a/A dB .b/ ˇ .eA ; b/ B Aˇ ! ˇ2 Z Z ˇZ ˇ ˇ 1 ˇ .b ˇ D ˇ y hTb h/. / .a/dAy. /ˇ dA .a/ .e ; b/ dB .b/ A B A A ˇ2 Z Z ˇZ p ˇ ˇ 1 ˇ ˇ b D ˇ y h. / .a; eB / .a/ Tb h. /dAy. /ˇ .a; b/ dA .a/dB .b/ B A A ˇ2 Z ˇZ ˇ ˇ ˇ b ˇ d.a; b/ D h. /.J T h/. /d . / .a/ 1 b ˇ Ay ˇ

B 0

b



L2 .A/

S b

b

b

G

b

Ay

ˇ2 Z ˇD E ˇ ˇ ˇb ˇ d.a; b/ h; J T h a b ˇ 2 y L .A/ ˇ G Z ˇ ˇ ˇhh; Ja Tb hi 2 ˇ2 d.a; b/ D L .A/ ZG D j hh; U.a; b/hi j2 d.a; b/:

1

D

G

That is

Z

j hh; U. b/hi j2 d.a; b/ D Ch jjhjj2 < 1;

(9)

G

which means that h is admissible.



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12

THE ABSTRACT WAVELET TRANSFORM

7. The abstract wavelet transform Definition 13. Given .a; b/ in G D A  B and h admissible in L2 .A/, the abstract wavelet transform with respect to h is defined as the linear operator Lh .a; b/ W L2 .A; dA / ! L2 .G; d.a; b// such that for any f in L2 .A/ we have .Lh f /.a; b/ D hf; U.a; b/hiL2 .A/ : That is, .Lh f /.a; b/ D

Z

f .x/.U.a; b/h/ .x/dA .x/ D A

Z

A

f .x/ p

1 .a; b/

h..a; b/

1


x/dA .x/:

Now, in order to get back the function f from the abstract wavelet transform .Lh f /.a; b/, we will apply the Grossmann-Morlet-Paul theorem [4], where the hypotheses for the representation U.a; b/ are: unitary, irreducible and strongly continuous. In our case, our representation is unitary, so the following lemmas will show that U.a; b/ D Ja Tb is irreducible and strongly continuous. Lemma 11. The representation U.a; b/ of the group G D A  B acting on L2 .A/ is irreducible. Proof. Suppose h 2 L2 .A/ n f0g and suppose f 2 L2 .A/ is such that hf; U.a; b/hi D 0 for all .a; b/ in G. To show the representation U.a; b/ is irreducible [5], we will show that f D 0 in L2 .A/: Under the assumptions we have Z j hf; U.a; b/hi j2 d.a; b/ D 0; G

but by (9), 0D

Z

j hf; U.a; b/hi j2 d.a; b/ D Ch jjf jj2 :

G

Since h is not identically zero, it follows that jjf jj D 0. Thus f D 0.



Definition 14. For a function h W A ! C, define the left and right translations of h by .Ia h/.x/ D h.a

1

x/

and

.Da h/.x/ D h.xa/;

where

a; x 2 A:

Definition 15. For a function h W A ! C, we say that: a) h is left uniformly continuous if jjIa h hjj1 ! 0 as a ! eA b) h is right uniformly continuous if jjDa h hjj1 ! 0 as a ! eA , where jj

jj1 is the uniform norm.

Lemma 12. If h 2 C0 .A/, then h is left and right uniformly continuous [3]. Lemma 13. If h is in C0 .A/, then

278

JAIME NAVARRO

a) jjJa h b) jjTb h

13

hjj1 ! 0 as a ! eA

hjj1 ! 0 as b ! eB :

Proof. a) Note that

ˇ ˇ 1 ˇ j.Ja h/.x/ h.x/j D ˇ p h..a; eB / 1
x/ ˇ .a; eB / ˇ ˇ 1 1 ˇ 1  p jh.a x/ h.x/j C ˇ p ˇ .a; eB / .a; eB /

ˇ ˇ ˇ ˇ 1 ˇ ˇ h.x/ˇ D ˇ p h.a ˇ ˇ .a; eB /

1

x/

ˇ ˇ ˇ 1ˇ jh.x/j: ˇ

ˇ ˇ ˇ h.x/ˇ ˇ

Then, since .a; eB / is continuous at .eA ; eB /, it follows from Lemma 12 that jjJa h

hjj1 ! 0 as a ! eA :

b) Similarly, since ˇ ˇ 1 ˇ h.x/j D ˇ p h..eA ; b/ ˇ .eA ; b/

j.Tb h/.x/  p

1

.eA ; b/

j .h ı €b

1

h ı €eB

/ .x/




1


x/

ˇ ˇ ˇ h.x/ˇ ˇ

ˇ ˇ 1 ˇ .x/j C ˇ p ˇ .eA ; b/

and .eA ; b/ is continuous at .eA ; eB /, it follows that jjTb h

ˇ ˇ ˇ 1ˇ jh.x/j ˇ

hjj1 ! 0 as b ! eB .



Lemma 14. Let h be in L2 .A/. Then 1) jjJa h hjj2 ! 0 as a ! eA 2) jjTb h hjj2 ! 0 as b ! eB : Proof. Let  > 0 be given. Since C0 .A/ is dense in L2 .A/ , there is l 2 C0 .A/ such that jjh l jj2 < 3 . Now, since l is uniformly continuous, it follows from Lemma 13 that there is a neighbor1 hood V of eA such that jjJa l l jj1 < 3 pm.S for all a 2 V and where S D supp.l /. / Then

Z

j.Ja l /.x/ l.x/j2 dA .x/  jjJa l l jj21 m.S /: p Thus, jjJa l l jj2  jjJa l l jj1 m.S /: Hence, jjJa l l jj2 < 3 . Therefore, for a 2 V we have from Lemma 4, jjJa l

l jj22

jjJa h

D

S

hjj2  jjJa h

D jjJa .h

Ja l jj2 C jjJa l

l /jj2 C jjJa l

l jj2 C jjh

l jj2

l jj2 C jjh

l jj2    D jjh l jj2 C jjJa l l jj2 C jjh l jj2 < C C D : 3 3 3 2) In a similar way, it can be proved that jjTb h hjj2 ! 0 as b ! eB : Lemma 15. Let h be in L2 .A/. Then jjU.a; b/h

hjj2 ! 0 as .a; b/ ! .eA ; eB /:



279

14

THE ABSTRACT WAVELET TRANSFORM

Proof. By Lemma 12, jjU.a; b/h

hjj2 D jjJa Tb h

 jjJa .Tb h

hjj2 D jjJa Tb h

h/jj2 C jjJa h

hjj2 D jjTb h

Ja h C Ja h hjj2 C jjJa h

hjj2 hjj2 ! 0

as a ! eA and b ! eB .



Lemma 16. The representation U.a; b/ is strongly continuous. U.a0 ; b 0 /hjj2 ! 0 as .a; b/ ! .a0 ; b 0 / for any h in

Proof. Let us prove that jjU.a; b/h 2 L .A/.

Consider a; a0 in A. Then for b; b 0 in B, and Lemma 3 Ja Tb h D .Ja Tb T.b 0 / 1 J.a 0 / 1 /.Ja 0 Tb 0 h/ D .Ja Tb.b 0 / 1 J.a 0/ 1 /.Ja 0 Tb 0 h/ D Ja .J€

..a b.b0 / 1

0/ 1 /

Tb.b 0/ 1 /.Ja 0 Tb 0 h/ D .Ja€

..a b.b0 / 1

0 / 1/

Tb.b 0 / 1 /.Ja 0 Tb 0 h/:

Then for u D Ja 0 Tb 0 h; Ja 0 Tb 0 h D Ja€

Ja Tb h D Ja€

..a b.b0 / 1

0/ 1 /

D Ja€

..a b.b0 / 1

0/

..a b.b0 / 1

0/ 1 /

Tb.b 0 / 1 u

u

Tb.b 0/ 1 u Ja€ 0 1 ..a 0/ 1 / u C Ja€ 0 1 ..a 0/ b.b / b.b / i  h  u C Ja€ 0 1 ..a 0 / 1/ u u : 1 / Tb.b 0 / 1 u

1/

u

u

b.b /

Then by Lemma 14, jjJa Tb h  jjJa€

Ja 0 Tb 0 hjj2 ..a b.b0 / 1

Thus, jjJa Tb h

 Tb.b 0 / 1 u

ujj2 C jjJa€

D jjTb.b 0/ 1 u as .a.a0 /

0 / 1/

1

; b.b 0 /

1

 u jj2 C jjJa€

..a b.b0 / 1

0/ 1 /

u

..a b.b0 / 1

0/ 1 /

u

ujj2

ujj2 ! 0

/ ! .eA ; eB /:

Ja 0 Tb 0 hjj2 ! 0 as .a; b/ ! .a0 ; b 0 /:



8. Inversion formula Lemma 17. For any f; g in L2 .A/ and an admissible non-zero function h in L2 .A/, we have the following identity in the weak sense Z 1 f D .Lh f /.a; b/ U.a; b/h d.a; b/: Ch G Proof. The representation U.a; b/ is a strongly continuous irreducible unitary representation of the locally compact topological group G D A  B acting on the Hilbert space L2 .A/. So, if there is a non-zero admissible vector h in L2 .A/, then by the Grossmann-Morlet-Paul theorem [4], for f; g in L2 .A/,

280

JAIME NAVARRO

Hence,

Z

15

(10)

hf; U.a; b/hi hg; U.a; b/hid.a; b/ D Ch hf; gi : G

1 f D Ch

in the weak sense.

Z

.Lh f /.a; b/ U.a; b/h d.a; b/ G

 9. Isometry

Lemma 18. Given f; g in L2 .A/, hf; giL2.A/ D

1 hLh f; Lh giL2.G/ Ch

(11)

for any non-zero admissible function h in L2 .A/. Proof. It comes from (10).



Lemma 19. For f; g in L2 .A/, jjf jj2L2.A/ D

1 jjLh f jj2L2.G/: Ch

(12)

Proof. If f D g, then from Lemma 18, hf; f iL2.A/ D

1 hLh f; Lh f iL2 .G/ : Ch

That is, jjf jj2L2.A/ D

1 jjLh f jj2L2.G/: Ch 

Note that from Lemma 19, Z Z 1 jf .a/j2 dA .a/ D j.Lh f /.a; b/j2 d.a; b/: Ch G A 10. Co-variance properties In this section, the co-variance of wavelet transforms is given as well as with respect to admissible functions. Lemma 20. If h in L2 .A/ is admissible, then for f in L2 .A/ and for a0 in A and b 0 in B, 1/

.Lh Tb 0 f /.a; b/ D .Lh f /.€.b 0 / 1 .a/; .b 0 /

2/

.Lh Ja 0 f /.a; b/ D .Lh f /..a0 /

1

1

b/

a; b/:

Proof. From Lemma 3, 1) ˝ ˛ .Lh Tb 0 f /.a; b/ D hTb 0 f; Ja Tb hi D f; T.b 0/ 1 Ja Tb h D E D E D f; J€.b0 / 1 .a/T.b 0 / 1 Tb h D f; J€.b0 / 1 .a/T.b 0 / 1 b h D .Lh f /.€.b 0 / 1 .a/; .b 0 /

1

b/:

281

16

THE ABSTRACT WAVELET TRANSFORM

2) .Lh Ja 0 f /.a; b/ D hJa 0 f; Ja Tb hi ˝ ˛ ˝ ˛ D f; J.a 0/ 1 Ja Tb h D f; J.a 0/ 1 a Tb h D .Lh f /..a0 /

1

a; b/: 

Corollary 3. If h 2 L2 .A/ is admissible, then for f 2 L2 .A/ and for a0 2 A and b 0 2 B, .Lh Ja 0 Tb 0 f /.a; b/ D .Lh f /.€.b 0 / 1 ..a0 /

1

a/; .b 0 /

1

b/:

(13)

Proof. It comes from Lemma 20.



Lemma 21. If f 2 L2 .A/ and h is admissible in L2 .A/, then for a0 2 A and b 0 2 B, we have Ja 0 h and Tb 0 h are admissible. Moreover 1/

.LJa0 h f /.a; b/ D .Lh f /.a€b .a0 /; b/

2/

.LTb0 h f /.a; b/ D .Lh f /.a; bb 0 /:

Proof. First, we will show that Ja 0 and Tb 0 are admissible. It is clear that Ja 0 h and Tb 0 h are in L2 .A/. By Lemma 10, since Z CJa0 h  j.Ja 0 h/. ı €b /j2 dB .b/ Z ˇp B Z ˇ2 ˇ ˇ D h. ı €b /ˇ dB .b/ D .a0 ; eB / jb h. ı €b /j2 dB .b/ < 1; ˇ .a0 ; eB / . ı €b /.a0 / b

1

B

B

it follows that Ja 0 h is admissible.

Also, since Z CTb0 h  j.Tb 0 h/. ı €b /j2 db .b/ B Z ˇp Z ˇ2 ˇ ˇ 0 0 b 0 D .e ; b / h. ı € ı € / d .b/ D .e ; b / jb h. ı €bb 0 /j2 dB .b/ < 1; ˇ A B A b b ˇ

1

B

B

it follows that Tb 0 h is admissible.

Now, note that 1) ˝ ˛ .LJa0 h f /.a; b/ D hf; Ja Tb Ja 0 hi D f; Ja J€b .a 0/ Tb h D .Lh f /.a€b .a0 /; b/;

and 2)

.LTb0 h f /.a; b/ D hf; Ja Tb Tb 0 hi D hf; Ja Tbb 0 hi D .Lh f /.a; bb 0 /:  Corollary 4. If f 2 L2 .A/ and h is admissible in L2 .A/, then .LJa0 Tb0 h f /.a; b/ D .Lh f /.a€b .a0 /; bb 0 /:

(14)

282

JAIME NAVARRO

17

Proof. It comes from Lemma 21.



Corollary 5. If f 2 L2 .A/ and h is admissible in L2 .A/, then .LJa0 Tb0 h Ja 0 Tb 0 f /.a; b/ D .Lh f /.€.b 0 / 1 ..a0 /

1

a€b .a0 //; .b 0 /

1

bb 0 /:

Proof. It comes from (13) and (14).

(15) 

11. Main Results We now give a characterization of the image of the wavelet transform by a reproducing kernel. Note that not every function F .a; b/ in L2 .G/ is the wavelet transform of some function f in L2 .A/. That is, ˚ I m.Lh .a; b// D F .a; b/ j .Lh f /.a; b/ D F .a; b/ for some f 2 L2 .A/

is a proper subspace of L2 .G/. To see this, note that for F .a; b/ D .Lh f /.a; b/, we have F .a; b/ is bounded since jF .a; b/j D j.Lh f /.a; b/j D j hf; U.a; b/hi j  jjf jj2jjhjj2 : Hence, any unbounded and square integrable function F .a; b/ is not in I m.Lh .a; b//. Then we have the following result. Theorem 1. The image of the wavelet transform with respect to an admissible function h in L2 .A/ is the closed subspace of functions F .a; b/ in L2 .G/ that satisfy Z 1 F .a; b/ D F .a0 ; b 0 /K.a; b I a0 ; b 0 /d.a0 ; b 0 / Ch G

where K.a; b I a0 ; b 0 / D ŒLh U.a; b/h.a0 ; b 0 / is the reproducing kernel associated with h. Proof. If F is in I m.Lh .a; b//, there is f 2 L2 .A/ such that .Lh f /.a; b/ D F .a; b/. Then by (11) with g D U.a; b/h, 1 F .a; b/ D .Lh f /.a; b/ D hf; U.a; b/hi D hLh f; Lh gi Ch Z 1 D .Lh f /.a0 ; b 0 / .Lh g/.a0 ; b 0 / d.a0 ; b 0 /: Ch G Now by taking K.a; b I a0 ; b 0 / D .Lh g/.a0 ; b 0 /; we have Z 1 F .a; b/ D F .a0 ; b 0 /K.a; b I a0 ; b 0 /d.a0 ; b 0 /: (16) Ch G This shows that the image of Lh .a; b/ is a reproducing kernel Hilbert space embedded as a close subspace of L2 .G; C1 d.a; b//, where by Corollary 3, h

283

18

THE ABSTRACT WAVELET TRANSFORM

K.a; b I a0 ; b 0 / D ŒLh Ja Tb h .a0 ; b 0 / D .Lh h/.€b

1

.a

1 a0 /; b 1 b 0 /

is the reproducing kernel.



Now, we will develop a sampling formula for functions in the reproducing Hilbert Space .k/ .k/ .k/ I m.Lh .a; b//. Thus, for a.k/ D .a1 ; a2 ; : : : ; an / in A, consider a countable set f.a.k/ ; b .k//gkD1;2;::: in G D AB such that fU.a.k/ ; b .k//hgkD1;2;::: is an orthonormal basis of L2 .A/ where h is admissible in L2 .A/. Then since the wavelet transform is an isometry (11), it follows that fLh U.a.k/ ; b .k/ /hgkD1;2;::: is an orthonormal basis for I m.Lh .a; b//. Then we have the following expression for the L2 norm for Lh U.a; b/h. Lemma 22. For .a; b/ in G and an admissible function h in L2 .A/, set g D U.a; b/h, and for a given countable set f.a.k/ ; b .k//gkD1;2;::: in G, set gk D U.a.k/ ; b .k//h. Then jjLh gjj2L2 .G/ D Ch2

1 X

j.Lh gk /.a; b/j2 :

kD1

Proof.

Since fLh gk gkD1;2;:::

is an orthonormal basis for I m.Lh .a; b// and

.Lh g/.a; b/ 2 I m.Lh .a; b//, it follows that .Lh g/.a; b/ D

1 X

(17)

dk .Lh gk /.a; b/;

kD1

where dk D hLh g; Lh gk i : Note that from (11),

dk D hLh g; Lh gk i D Ch hg; gk i D Ch hgk ; gi

(18)

D Ch hgk ; U.a; b/hi D Ch .Lh gk /.a; b/: Then by (17) and (18), 2

jjLh gjj D hLh g; Lh gi D

D

D

1 X

kD1 1 X

kD1

dk

*

1 X

1 X

dk Lh gk ;

kD1

+

1 X

kD1

jdk j2 D Ch2

1 X

dk Lh gk

kD1

dk Lh gk ; Lh gk D

kD1

dk dk D

*

1 X

+

dk dk hLh gk ; Lh gk i

kD1 1 X

j.Lh gk /.a; b/j2 :

kD1

 Lemma 23. Suppose that h in L2 .A/ is admissible. Then the series representation for Lh g in I m.Lh .a; b//, where g D U.a; b/h is absolutely convergent in G.

284

JAIME NAVARRO

19

Proof. Since fLh gk gkD1;2;::: is an orthonormal basis for I m.Lh .a; b//, it follows from (17) that, .Lh g/.a; b/ D

1 X

dk .Lh gk /.a; b/;

kD1

where dk D Ch .Lh gk /.a; b/. So, the series converges in L2 .G/: Then from Lemma 22, 1 1 ˇ ˇ X X ˇ ˇ jdk .Lh gk /.a; b/j D ˇCh .Lh gk /.a; b/.Lh gk /.a; b/ˇ kD1

kD1

D Ch

1 X

j.Lh gk /.a; b/j2 D Ch

kD1

1 1 jjLh gjj2 D jjLh gjj2 < 1: Ch Ch2

Thus, the series converges absolutely in G.



Now, we will give our second main result. Theorem 2. For a wavelet h in L2 .A/ we have that any F in I m.Lh .a; b// can be reconstructed in terms of its sampled values fF .a.k/ ; b .k//gkD1;2;::: by the following interpolation formula 1 X F .a; b/ D Ch F .a.k/ ; b .k//.Lh gk /.a; b/; (19) kD1

where gk D U.a.k/ ; b .k//h:

Proof. Since fLh gk gkD1;2;::: is an orthonormal basis for I m.Lh .a; b//, it follows that for F in I m.Lh .a; b//, 1 X F .a; b/ D hF; Lh gk iL2 .G/ .Lh gk /.a; b/: kD1

First note that from Lemma 23, this series representation for F is absolutely convergent. Moreover,  1 Z X F .a; b/ D F .a0 ; b 0 /.Lh gk /.a0 ; b 0 /d.a0 ; b 0 / .Lh gk /.a; b/: G

kD1

Then by Theorem 1, F .a; b/ D

D

1 Z X

kD1 1 X

0

0

F .a ; b /K.a G

.k/

;b

.k/

 I a ; b / d.a ; b / .Lh gk /.a; b/ 0

0

0

0

Ch F .a.k/ ; b .k// .Lh gk /.a; b/:

kD1



285

20

THE ABSTRACT WAVELET TRANSFORM

12. Characterization of the wavelet transform In this section, we will see that the wavelet transform can be written in different ways by using the co-variance properties, the Fourier transform, and the convolution of two functions. Lemma 24. Suppose f 2 L2 .A/ and h is admissible in L2 .A/, then 1/ 2/

.Lh f /.a; b/ D .LJa0 h Ja 0 f /.a0 a€b ..a0 /

1

0

/; b/ 0

.Lh f /.a; b/ D .LTb0 h Tb 0 f /.€b 0 .a/; b b.b /

1

/;

where .a; b/ and .a0 ; b 0 / are in G. Proof. It comes from Lemmas 20 and 21.



Corollary 6. Suppose f 2 L2 .A/ and h in L2 .A/ is admissible, then .Lh f /.a; b/ D .LJa0 Tb0 h Ja 0 Tb 0 f /.a0 €b 0 .a/€b ..a0 /

1

/; b 0 b.b 0 /

1

/:

Proof. It comes from Corollary 5.



Lemma 25. If f 2 L2 .A/ and h in L2 .A/ is admissible, then .Lh f /.a; b/ D .Lb fy/.a h

Proof. By Corollary 2, .Lb fy/.a h

1

;b

1

1

;b

1

/:

E E D D h D fy; Ja Tb h D hf; Ja Tb hi D .Lh f /.a; b/: / D fy; Ja 1 Tb 1b

1



Lemma 26. Suppose f and h in L2 .A/ are admissible, then .Lh f /.a; b/ D .Lf h/.a; b/

1

:

Proof. By Definition 13 and Lemma 3,

D E .Lh f /.a; b/ D hf; Ja Tb hi D hTb 1 Ja 1 f; hi D J€ 1 .a 1/ Tb 1 f; h b D E 1 1 D h; J€ 1 .a 1/ Tb 1 f D .Lf h/.€b 1 .a /; b / D .Lf h/.a; b/ 1 : b



Definition 16. If f; g 2 L1 .G/, then the convolution of f and g is defined as the function Z .f  g/.x/ D f .y/g.y 1 x/dG .y/; G

where x; y 2 G Lemma 27. If h 2 C0 .A/ is admissible and f 2 L2 .A/, then h i 1 .Lh f /.a; b/ D p f  .Tb h/ .a/; .a; eB / where  means



.x/ D

.x

1

/.

(20)

286

JAIME NAVARRO

21

Proof. p

1

h i f  .Tb h/ .a/

.a; eB / Z 1 Dp f .x/.Tb h/ .x 1 a/dA .x/ .a; eB / A Z 1 Dp f .x/.Tb h/.a 1 x/dA .x/ .a; eB / A Z 1 Dp f .x/.Tb h/..a; eB / 1
x/dA .x/ .a; eB / A Z 1 1 Dp f .x/ p h..eA ; b/ 1 .a; eB / .a; eB / A .eA ; b/ Z 1 h..a; b/ 1
x/dA .x/ D f .x/ p .a; b/ A

: 1


x/dA .x/

D .Lh f /.a; b/:

 Lemma 28. If f 2 L1 .A/ and g 2 C0 .A/ is admissible, then f  g is admissible. Proof. Note that since f 2 L1 .A/ and g 2 L2 .A/, it follows that f  g 2 L2 .A/ and jjf  gjj2  jjf jj1 jjgjj2 , [3]. Also since f  g D fygy, [11], then Z ˇ Z ˇ ˇ2 ˇ2 ˇ ˇ ˇy ˇ g . ı €b /ˇ dB .b/ Cf g  ˇf  g. ı €b /ˇ dB .b/ D ˇf . ı €b /y B B Z Z 2 2 2 y D jf . ı €b /j jy g . ı €b /j dB .b/  jjf jj1 jy g. ı €b /j2 dB .b/ < 1:

1

1

B

B

 Corollary 7. Suppose h 2 L2 .A/ is admissible.

If f 2 C0 .A/ and g 2 C0 .A/ is

admissible, then .Lh .f  g//.a; b/ D .Lf g h/.a; b/

1

Proof. By Lemma 29, f  g is admissible. Then, the result comes from Lemma 26.

(21) 

13. Example We will give now an example related to the definition of the abstract wavelet transform. According to section 2, let us consider the additive group Rn with identity eRn D .0; 0; : : : ; 0/ and the multiplicative group RC with identity eRC D 1. Now, let us take the homomorphism € from RC to the group of all automorphisms of Rn . That is, for each s 2 RC , the map €s W Rn ! Rn is defined as €s .k1 ; k2; : : : ; kn / D .€s .k1 /; €s .k2 /; : : : ; €s .kn // D .sk1 ; sk2 ; : : : ; skn/, where .k1 ; k2 ; : : : ; kn / 2 Rn . In this case, the product in ˚ G D Rn  RC D .k1 ; k2; : : : ; kn ; s/j.k1 ; k2; : : : ; kn / 2 Rn and s 2 RC

287

22

THE ABSTRACT WAVELET TRANSFORM

is defined as .k1 ; k2 ; : : : ; kn ; s/.k10 ; k20 ; : : : ; kn0 ; s 0 / D .k1 C €s .k10 /; k2 C €s .k20 /; : : : ; kn C €s .kn0 /; ss 0 / D .k1 C sk10 ; k2 C sk20 ; : : : ; kn C skn0 ; ss 0/: Note that with this product, G becomes a locally compact topological group where the identity is eG D .eRn ; eRC / D .0; 0; : : : ; 0; 1/ and .k1 ; k2 ; : : : ; kn ; s/

1

D .€s

1

. k1 /; €s

1

D. s

1

k1 ; s

1

. k2 /; : : : ; €s

k2 ; : : : ; s

1

n

1

kn ; s

. kn /; s 1

n

1

/

/:

n

Moreover, G acts on R with the following action
W G  R ! R given by .k1 ; k2 ; : : : ; kn ; s/
.x1 ; x2; : : : ; xn/ D .k1 C €s .x1 /; k2 C €s .x2 /; : : : ; kn C €s .xn // D .k1 C sx1 ; k2 C sx2; : : : ; kn C sxn /; where .x1 ; x2; : : : ; xn / 2 Rn . On the other hand, the function  W G ! .0; 1/ satisfying Z

Rn

h..k1 ; k2; : : : ; kn ; s/

1


x/dRn .x/ D .k1 ; k2 ; : : : ; kn ; s/

is given by .k1 ; k2 ; : : : ; kn ; s/ D s n , where h 2 L1 .Rn /. Also note that dRn .x/ D dx, and 1

.k1 ; k2 ; : : : ; kn ; s/ D. s

1

k1 C €s

1

1


x D . s

.x1 /; s

1

1

k1 ; s

k2 C €s

1

k2 ; : : : ; s

.x2 /; : : : ; s

. s 1 k1 C s 1 x1 ; s 1k2 C s 1x2 ; : : : ; s   x1 k1 x2 k2 xn kn D ; ;:::; : s s s

1

kn C s

1

Z

Rn

h.x/dRn .x/

kn /
.x1 ; x2; : : : ; xn /

1 1

kn C €s

1

.xn //

xn /

Now following Section 5, for .k1 ; k2 ; : : : ; kn ; s/ 2 G define the family of two operators U.k1 ; k2 ; : : : ; kn ; s/ D J.k1 ;k2 ;:::;kn/ Ts . Then for h 2 L2 .Rn /, .J.k1 ;k2 ;:::;kn / h/.x/ D p 

1 .k1 ; k2 ; : : : ; kn ; 1/

h .k1 ; k2 ; : : : ; kn ; 1/

1 x1 k1 x2 k2 xn kn D p h ; ;:::; 1 1 1 .1/n D h.x1

k1 ; x2

n

k2 ; : : : ; xn



1


x




kn /;

n

where x 2 R and .k1 ; k2 ; : : : ; kn / 2 R , and

.Ts h/.x/ D p

1

h .0; 0; : : : ; 0; s/

1


x




.0; 0; : : : ; 0; s/   x x 1 x1 0 x2 0 xn 0 1 xn  1 2 Dp h ; ;:::; D ; ;:::; ; n h s s s s s s .s/n .s/ 2

288

JAIME NAVARRO

23

where x D .x1 ; x2 ; : : : ; xn / 2 Rn ; and s 2 RC . Then U.k1 ; k2; : : : ; kn ; s/ is a unitary representation of G acting on L2 .Rn / by ŒU.k1 ; k2 ; : : : ; kn ; s/h.x/ D .J.k1 ;k2 ;:::;kn / Ts h/.x/   1 x1 k1 x2 k2 xn kn D .Ts h/.x1 k1 ; x2 k2 ; : : : ; xn kn / D n h ; ;:::; : s s s s2 Moreover since the left Haar measure on RC is dRC D 1s ds, it follows from Lemma 8, that the left Haar measure on G is 1 dRn .k1 ; k2 ; : : : ; kn / dRC .s/ .k1 ; k2 ; : : : ; kn ; s/ 1 1 1 D n d.k1 ; k2 ; : : : ; kn / ds D nC1 d.k1 ; k2 ; : : : ; kn /ds: s s s Thus, the admissibility condition for a radially symmetric function h 2 L2 .Rn / (Lemma 10) becomes d.k1 ; k2 ; : : : ; kn ; s/ D

Ch 

Z

RC

Z ˇ ˇ2 ˇy ˇ ˇh. ı €s /ˇ dRC .s/ D

21 y jh.s/j ds; s RC

where  2 R , and b h.r / D b .jr j/. Then, if we apply the change of variable y D jsj, we get n

Z

1 jb .y/j2 dy: y RC Hence, for .k1 ; k2; : : : ; kn ; s/ 2 G and h admissible in L2 .Rn /, the continuous wavelet Ch 

transform for f 2 L2 .Rn / with respect to h is given by ˝ ˛ .Lh f /.k1 ; k2 ; : : : ; kn ; s/ D f; J.k1;k2 ;:::;kn / Ts h   Z 1 x1 k1 x2 k2 xn kn D f .x1 ; x2 ; : : : ; xn / n h ; ;:::; d.x1 ; x2 ; : : : ; xn/ s s s s2 Rn   Z 1 x k D f .x/ n h dx; s s2 Rn which agrees with the one given in [1]. Also, the inverse formula according to Lemma 17 is given by Z 1 f D .Lh f /.k1 ; k2 ; : : : ; kn ; s/ŒU.k1 ; k2; : : : ; kn ; s/hd.k1 ; k2 ; : : : ; kn ; s/ Ch G in the weak sense.

289

24

THE ABSTRACT WAVELET TRANSFORM

That is, Z Z 1 ds f D hf; U.k1 ; k2 ; : : : ; kn ; s/hi U.k1 ; k2; : : : ; kn ; s/hd.k1 ; k2 ; : : : ; kn / nC1 Ch RC Rn s Z 1 D hf; U.k; s/hi U.k; s/h d.k; s/; Ch G which also agrees with the one given in [8]. References [1] I. Daubechies, Ten Lectures on Wavelets, Siam, Philadelphia, 1992. [2] M. Fashandi, R.B. Kamyabi, A. Niknam, M.A. Pourabdollah, Continuous wavelet transform on a special homogeneous space, J. Math. Phys., Vol. 44, No.9, pp. 4260-4266, (2003). [3] G. Folland, A course in Abstract Harmonic Analysis, CRC Press, 2000. [4] A. Grossmann, J. Morlet, and T. Paul, Transforms associated to square integrable group representations, I. General Results,

J. Math. Phys., 26, pp. 2473-2479, (1985).

[5] C. E. Heil, D.F. Walnut, Continuous and Discrete Wavelet Transforms, Siam Review, Vol 31, No 4, pp. 628 - 666, (1982). [6] R. Murenzi, Wavelet Transforms Associated to the n-Dimensional Euclidean Group with Dilations :Signal in More Than One Dimension ,

J.M. Combes, A. Grossmann, and Ph, Tchamitchian, eds., Springer-

Verlag , Berlin, pp. 239-246, (1989). [7] J. Navarro, The Wavelet Transform of Distributions, Memorias del XXVII Congreso Nacional de la Sociedad Matem´ atica Mexicana, 16, pp. 145-154, (1995). [8] J. Navarro, Use of the Wavelet Transform in Rn to find singularities of functions in L2 .Rn /, Revista Colombiana de Matem´ aticas, Vol. 32, No.2, pp. 93-99, (1998). [9] J. Navarro, Singularities of Distributions Via the Wavelet Transform, Siam J. Math. Anal., Vol. 30 , No 2, pp. 454-467, (1999). [10] J. Navarro, M. A. Alvarez, The wavelet transform with rotations, Sampling Theory in Signal and Image Processing, Vol. 2, pp. 101-106, (2005). [11] H. Reiter, Classical Harmonic Analysis and Locally Compact Groups, Oxford University Press, 2000.

290JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.5, NO.3, 290-301, COPYRIGHT 2010 EUDOXUS PRESS, LLC

On the Green Function of the Operator

(¢B + m6)k

Related to the Bessel Helmholtz Operator and the Bessel Klein-Gordon Operator E. Suntonsinsoungvon and A. Kananthai∗

Department of Mathematics, Chiang Mai University, Chiang Mai, 50200 Thailand. Abstract In this article, we study the Green function of the form (¢B + m6 )k which is iterated k-times and is defined by Ã k 3 !3  p+q p X X (¢B + m6 )k =  Bxi + Bxi  + m6  i=1

i=p+1

where p + q = n is the dimension of R+ n , Bxi =

∂2 ∂x2i

+

2vi ∂ xi ∂xi ,

2vi = 2αi + 1,

αi > − 21 , xi > 0, i = 1, 2, . . . , n, m is a positive real number and k is a positive integer. At first, we study the Green function or elementary solution of the operator (¢B + m6 )k . Moreover, the operator (¢B + m6 )k can be related to the Bessel Helmholtz operator (4B + m2 )k and the Bessel ultra-hyperbolic KleinGordon operator (¤B + m2 )k . After that, we apply such a Green function to solve the solution of the equation (¢B +m6 )k u(x) = f (x) where f is a generalized function and u(x) is an unknown function for x ∈ Rn .

Keywords: Green function, Bessel Helmholtz operator, Bessel Ultra-hyperbolic KleinGordon operator ∗

Corresponding author. E-mail : [email protected]

1

SUNTON-KANANTHAI: ON THE GREEN FUNCTION...

1

291

Introduction

The operator ¢B can be expressed in the form à p !3 à p+q !3 X X ¢B = Bx i + Bx i i=1

=

" p X i=1

·

i=p+1

Bx i +

p+q X i=p+1

# Ã

Bxi 

p X

!2 Bx i

−

i=1

¸ 3 2 = 4B 4B − (4B + ¤B ) (4B − ¤B ) 4 3 1 = 4B ¤2B + 43B 4 4

p X

Bx i

i=1

p+q X

à Bxi +

i=p+1

p+q X

!2  Bxi



i=p+1

(1.1) 2

2vi ∂ ∂ 1 where p + q = n is the dimension of R+ n , Bxi = ∂x2 + xi ∂xi , 2vi = 2αi + 1, αi > − 2 , i xi > 0, i = 1, 2, . . . , n, 4B is the Laplace-Bessel operator which is defined by

4B = Bx1 + Bx2 + · · · + Bxn ,

(1.2)

and ¤ is the Bessel ultra-hyperbolic operator which is defined by ¤B = Bx1 + Bx2 + · · · + Bxp − Bxp+1 − Bxp+2 − · · · − Bxp+q .

(1.3)

Furthermore, Yildirim et al. [8] first introduced the Bessel diamond operator ♦B which is defined by ¡ ¢2 ¡ ¢2 ♦B = Bx1 + Bx2 + · · · + Bxp − Bxp+1 + Bxp+2 + · · · + Bxp+q . (1.4) The Bessel diamond operator can also be expressed in the form ♦B = 4B ¤B = ¤B 4B , from (1.1) we have 1 3 (1.5) ¢B = ♦B ¤B + 43B . 4 4 Later, Bunpog and Kananthai [2] have studied the elementary solution or Green function of the operator (♦B + m4 )k which related to the Bessel Helmholtz operator 4B + m2 and the Bessel Klein-Gordon operator ¤B + m2 and obtained the function ¡ ¢−1 G(x) = (T2k (x) ∗ W2k (x)) ∗ C ∗k (x) (1.6) which is a Green function for the operator Ã k !2 Ã p+q !2 p X X (♦B + m4 )k =  Bxi − Bx i + m 4  , i=1

i=p+1

2

(1.7)

292

SUNTON-KANANTHAI: ON THE GREEN FUNCTION...

where the symbol ∗k denotes the convolution of itself k-times and the symbol ∗ − 1 is an inverse of the convolution algebra, T2k (x) is the elementary solution of the Bessel Helmholtz operator (4B + m2 )k iterated k-times, that is T2k (x) satisfy the equation (4B + m2 )k u(x) = δ(x),

(1.8)

W2k (x) is the elementary solution of the Bessel Klein-Gordon operator (¤B + m2 )k iterated k-times, that is W2k (x) satisfy the equation (¤B + m2 )k u(x) = δ(x)

(1.9)

C(x) = δ(x) − m2 (T2 (x) + W2 (x)) + 2m4 (T2 (u) ∗ W2 (x)) .

(1.10)

and C(x) is defined by

The purpose of this work is to find the elementary solution or Green function of the operator (¢B + m6 )k , that is (¢B + m6 )k G(x) = δ(x),

(1.11)

where G(x) is the Green function, δ is the Direc-delta distribution, m is a positive real number, k is a nonnegative integer and x = (x1 , x2 , . . . , xn ) ∈ R+ n . We then find the solution of the equation (¢B + m6 )k u(x) = f (x) where f is a given generalized function and u(x) is an unknown function.

2

Preliminaries

Before reaching the main results, the following definitions and the basic concepts are needed. At first, the generalized shift operator Txy has the following form [5], Ã n ! Z π Z π Y 2vi −1 y ∗ ··· ϕ(s1 , . . . , sn ) sin θi dθ1 · · · dθn , Tx =Cv 0

0

i=1

Qn Γ(vi +1) ∗ where s2i = x2i + yi2 − 2xi yi cos θi , x, y ∈ R+ n and Cv = i=1 Γ( 1 )Γ(vi ) . We remark that 2 this shift operator is closely connected with the Bessel differential operator [5], d2 ϕ 2vi dϕ d2 ϕ 2vi dϕ + = + , dx2i xi dxi dyi2 yi dyi ϕ(xi , 0) = f (x), ϕyi (xi , 0) = 0, 3

SUNTON-KANANTHAI: ON THE GREEN FUNCTION...

293

y where xi , yi ∈ R+ n for i = 1, 2, . . . , n. The convolution operator determined by the Tx is as follows à n ! Z Y (f ∗ ϕ)(x) = f (y)Txy ϕ(x) yi2vi dy. (2.1) R+ n

i=1

Convolution in (2.1) is known as a B-convolution. We note the following properties of the B-convolution and the generalized shift operator, (a) Txy · 1 = 1. (b) Tx0 · f (x) = f (x). + (c) If f (x), g(x) ∈ C(R+ n ), g(x) is a bounded function for x ∈ Rn and ! Ã n Z Y 2vi |f (x)| xi dx < ∞, R+ n

then

Ã

Z R+ n

Txy f (x)g(y)

i=1

n Y

! yi2vi

Z dy =

i=1

R+ n

f (y)Txy g(x)

à n Y

! yi2vi

dy.

i=1

(d) From (c), we have the following equality for g(x) = 1, Ã n ! Ã n ! Z Z Y Y Txy f (x) yi2vi dy = f (y) yi2vi dy. R+ n

R+ n

i=1

i=1

(e) (f ∗ g)(x) = (g ∗ f )(x). The Fourier-Bessel transformation and its inverse transformation are defined as follows [7], Ã n ! Z Y (FB f )(x) = Cv f (y) Jvi − 1 (xi yi )yi2vi dy, R+ n

2

i=1

(FB−1 f )(x) = (FB f )(−x), Cv =

à n Y i=1

µ

vi − 12

2

1 Γ vi + 2

¶!−1 ,

where Jvi − 1 (xi yi ) is the normalized Bessel function which is the eigenfunction of the 2 Bessel differential operator. There are following equalities for Fourier-Bessel transformation [7], FB δ(x) = 1 and FB (f ∗ g)(x) = FB f (x) · FB g(x). 4

294

SUNTON-KANANTHAI: ON THE GREEN FUNCTION...

Lemma 2.1 There is a following equality for Fourier-Bessel transformation µ ¶ n + 2|v| − α h ³ α ´i−1 α−n−2|v| −α n+2|v|−2α FB (|x| ) = 2 Γ Γ |x| , 2 2 where |v| = v1 + v2 + · · · + vn . Proof. See [7].

¤

k Lemma 2.2 Given the equation 4kB u(x) = δ(x) for x ∈ R+ n , where 4B is the LaplaceBessel operator iterated k-times defined by (1.2). Then u(x) = (−1)k S2k (x) is an elementary solution of the operator 4kB , where

S2k (x) =

2n+2|v|−4k Γ( n+2|v|−2k ) 2 1 Πni=1 2vi − 2 Γ(vi

+

1 )Γ(k) 2

|x|2k−n−2|v| .

Proof. See [8].

(2.2) ¤

Lemma 2.3 Given the equation ¤kB u(x) = δ(x) for x ∈ Γ+ = {x ∈ R+ n : x1 > 0, x2 > 0, · · · , xn > 0 and V > 0}, where ¤kB is the Bessel-ultra-hyperbolic operator iterated k-times defined by (1.3). Then u(x) = R2k (x) is an elementary solution of the operator ¤kB , where 2k−n−2|v| 2 V (2.3) R2k (x) = Kn (2k) for V = x21 + x22 + · · · + x2p − x2p+1 − · · · − x2p+q and Kn (2k) =

π

n+2|v|−1 2

Γ( 2+2k−n−2|v| )Γ( 1−2k )Γ(2k) 2 2

Γ( 2+2k−p−2|v| )Γ( p−2k ) 2 2

Proof. See [8].

¤

Lemma 2.4 The functions S2k (x) and R2k (x) are homogeneous distributions of order (2k − n − 2|v|) for Re(2k) < n + 2|v|. In particular, the B-convolution S2k (x) ∗ R2k (x) exists and is a tempered distribution. Proof. See [8].

¤

Lemma 2.5 (The elementary solution of Bessel Helmholtz operator) Given the equation (4B + m2 )k u(x) = δ(x) for x ∈ R+ n , where 4B is defined by (1.2). Then u(x) = T2k (x, m) is an elementary solution of the operator (4B + m2 )k where ¶ ∞ µ X −k T2k (x, m) = (m2 )r (−1)k+r S2k+2r (x) (2.4) r r=0

for S2k+2r is defined by (2.2). 5

SUNTON-KANANTHAI: ON THE GREEN FUNCTION...

295

Proof. Since the operator 4B is a linearly continuous and have 1 − 1 mapping, it has an inverse. By Lemma 2.2, we obtain ¶ ∞ µ X −k T2k (x, m) = (m2 )r 4−k−r δ(x) = (4B + m2 )−k δ(x), B r r=0

where (4B + m2 )−k is the inverse operator of the operator (4B + m2 )k . By applying the operator (4B + m2 )k to both sides of the above equation, we have (4B + m2 )k T2k (x, m) = (4B + m2 )k (4B + m2 )−k δ(x). Therefore, (4B + m2 )k T2k (x, m) = δ(x). This completes the proof.

¤

Lemma 2.6 (The elementary solution of Bessel Klein-Gordon operator) Given the equation (¤B + m2 )k u(x) = δ(x) for x ∈ R+ n , where ¤B is defined by (1.3). Then u(x) = W2k (x, m) is an elementary solution of the operator (¤B + m2 )k where ¶ ∞ µ X −k (m2 )r R2k+2r (x) (2.5) W2k (x, m) = r r=0

for R2k+2r is defined by (2.3). Proof. The proof of Lemma 2.6 is similar to the proof of lemma 2.5. Lemma 2.7 (The B-convolution of tempered distribution) Let k and r be nonnegative integer. (a) Let T2k (x, m) and T2r (x, m) be defined by (2.4), then T2k (x, m) ∗ T2r (x, m) = T2k+2r (x, m). (b) Let W2k (x, m) and W2r (x, m) be defined by (2.5), then W2k (x, m) ∗ W2r (x, m) = W2k+2r (x, m). (c) Let S2k (x) and S2r (x) be defined by (2.2), then S2k (x) ∗ S2r (x) = S2k+2r (x). (d) Let R2k (x) and R2r (x) be defined by (2.3), then R2k (x) ∗ R2r (x) = R2k+2r (x). 6

¤

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SUNTON-KANANTHAI: ON THE GREEN FUNCTION...

Proof. (a) From the equation (4B +m2 )k+r G(x) = δ(x), we obtain G(x) = T2k+2r (u, m) by Lemma 2.5. For any nonnegative integer r, we write (4B + m2 )k+r G(x) = (4B + m2 )k (4B + m2 )r G(x) = δ(x), then by Lemma 2.5 again we have the following equation (4B + m2 )r G(x) = T2k (u, m). Convoluing both sides of the above equation by W2r (u, m), we obtain T2r (u, m) ∗ (4B + m2 )r G(x) = T2r (u, m) ∗ T2k (u, m) or (4B + m2 )r T2r (u, m) ∗ G(x) = T2r (u, m) ∗ T2k (u, m). Hence, by Lemma 2.5 we have δ(x) ∗ G(x) = T2r (u, m) ∗ T2k (u, m). It follows that G(x) = T2r (u, m) ∗ T2k (u, m). From the fact that G(x) = T2k+2r (u, m), we obtain T2k (u, m) ∗ T2r (u, m) = T2k+2r (u, m). The proof of (b), (c) and (d) are similar to (a).

¤

Lemma 2.8 Let k and r be nonnegative integer. (a) Let S2r (x) and S2r−2k (x) be defined by (2.2), then 4kB S2r (x) = (−1)k S2r−2k (x). (b)Let R2r (x) and R2r−2k (x) be defined by (2.3), then ¤kB R2r (x) = R2r−2k (x). Proof. (a) By Lemma 2.7 (c), we obtain S2k (x) ∗ S2r−2k (x) = δ(x) ∗ S2r (x). By Lemma 2.3 we have S2k (x) ∗ S2r−2k (x) = 4kB (−1)k S2k (x) ∗ S2r (x) = S2k (x) ∗ 4kB (−1)k S2r (x). Therefore, 4kB S2r (x) = (−1)k S2r−2k (x). (b) The proof is similar to (a).

¤ 7

SUNTON-KANANTHAI: ON THE GREEN FUNCTION...

297

Lemma 2.9 Let k be nonnegative integer. (a) Let T2k (x, m) and S−2k (x) be defined by (2.4) and (2.2) respectively, then 4kB T2k (x, m) = T2k (x, m) ∗ (−1)k S−2k (x). (b) Let W2k (x, m) and R−2k (x) be defined by (2.5) and (2.3) respectively, then ¤kB W2k (x, m) = W2k (x, m) ∗ R−2k (x). Proof. (a) From (2.4) we have 4kB T2k (x, m)

¶ ∞ µ X −k = (m2 )r (−1)k+r 4kB S2k+2r (x). r r=0

By Lemma 2.8 (a), we have 4kB T2k (x, m) = T2k (x, m) ∗ (−1)k S−2k (x). (b) The proof is similar to (a).

¤

Lemma 2.10 (The existence of the convolution T6k (x, m) ∗ W4k (x, m)) The convolution T6k (x, m) ∗ W4k (x, m) exists and is a tempered distribution where T6k (x, m) = T2k (x, m) ∗ T2k (x, m) ∗ T2k (x, m) and W4k (x, m) = W2k (x, m) ∗ W2k (x, m) such that T2k (x, m) and W2k (x, m) are defined by (2.4) and (2.5), respectively. Proof. From (2.4) and (2.5), we have T2k (x, m) ∗ W2k (x, m) Ã∞ µ ! Ã∞ µ ! X −k ¶ X −k ¶ = (m2 )r (−1)k+r S2k+2r (x) ∗ (m2 )r R2k+2r (x) r r r=0 r=0 ¶ ¶µ ∞ X ∞ µ X −k −k = (m2 )r+s (−1)k+r S2k+2r (x) ∗ R2k+2s (x). r s r=0 s=0

By Lemma 2.6, the B-convolution of S2k+2r (x)∗R2k+2r (x) exists and is also a tempered distribution. Then T2k (x, m) ∗ W2k (x, m) exists and is also a tempered distribution. Since T2k (x, m), W2k (x, m) and T2k (x, m) ∗ W2k (x, m) exists and is also a tempered distribution, by Donoghue [2, p. 152] we obtain T6k (x, m) ∗ W4k (x, m) exists and is also a tempered distribution. ¤ Lemma 2.11 Let T6 (x, m) and W4 (x, m) be defined by (2.4) and (2.5) with k = 3 and k = 2, respectively. Then 8

298

SUNTON-KANANTHAI: ON THE GREEN FUNCTION...

2

(a) (4B + m2 ) (¤B + m2 ) (T6 (x, m) ∗ W4 (x, m)) = T4 (x, m) 3

(b) (4B + m2 ) (T6 (x, m) ∗ W4 (x, m)) = W4 (x, m) (c) (4B + ¤B ) (T6 (x, m) ∗ W4 (x, m)) = (T6 (x, m) ∗ W4 (x, m)) ∗ (R−2 (x) − S−2 (x)) (d) (4B + ¤B )2 (T6 (x, m) ∗ W4 (x, m)) = (T6 (x, m) ∗ W4 (x, m)) ∗ (S−4 (x) − 2S−2 (x) ∗ R−2 (x) + R−4 (x)) where S−2 (x), S−4 (x) are defined by (2.2) and R−2 (v), R−4 (x) are defined by (2.3). Proof. (a) We have ¡ ¢¡ ¢2 4B + m2 ¤B + m2 (T6 (x, m) ∗ W4 (x, m)) ¡ ¢ ¡ ¢2 = 4B + m2 T2 (x, m) ∗ T4 (x, m) ∗ ¤B + m2 W4 (x, m) = δ(x) ∗ T4 (x, m) ∗ δ(x), by Lemma 2.5 and 2.6, = T4 (x, m). (b) We get ¡ ¢3 ¡ ¢3 4B + m2 (T6 (x, m) ∗ W4 (x, m)) = 4B + m2 T6 (x, m) ∗ W4 (x, m) = δ(x) ∗ W4 (x, m), by Lemma 2.5, = W4 (x, m). (c) We obtain (4B + ¤B ) (T6 (x, m) ∗ W4 (x, m)) = 4B T2 (x, m) ∗ T4 (x, m) ∗ W4 (x, m) + T6 (x, m) ∗ ¤B W2 (x, m) ∗ W2 (x, m) = T6 (x, m) ∗ W4 (x, m) ∗ (−1)S−2 (x) + T6 (x, m) ∗ W4 (x, m) ∗ R−2 (x), by Lemma 2.9 = (T6 (x, m) ∗ W4 (x, m)) ∗ (R−2 (x) − S−2 (x)) . (d) We have (4B + ¤B )2 (T6 (x, m) ∗ W4 (x, m)) ¢ ¡ = 42B + 24B ¤B + ¤2B (T6 (x, m) ∗ W4 (x, m)) = 42B T4 (x, m) ∗ T2 (x, m) ∗ W4 (x, m) + 24B T2 (x, m) ∗ T4 (x, m) ∗ ¤B W2 (x, m) ∗ W2 (x, m) + T6 (x, m) ∗ ¤2B W4 (x, m) = T6 (x, m) ∗ W4 (x, m) ∗ (−1)2 S−4 (x) + 2T6 (x, m) ∗ W4 (x, m) ∗ (−1)S−2 (x) ∗ R−2 (x) + T6 (x, m) ∗ W4 (x, m) ∗ R−4 (x), by Lemma 2.9 = (T6 (x, m) ∗ W4 (x, m)) ∗ (S−4 (x) − 2S−2 (x) ∗ R−2 (x) + R−4 (x)) . ¤ 9

SUNTON-KANANTHAI: ON THE GREEN FUNCTION...

3

299

Main results

Theorem 3.1 Given the equation (¢B + m6 )k G(x) = δ(x), (3.1) ¡ ¢∗−1 then G(x) = T6k (x, m) ∗ W4k (x, m) ∗ C ∗k (x) is a Green function for the operator 6 k (¢B + m ) iterated k-times where ¢B is defined by (1.1), δ is the Direc-delta distribution, x = (x1 , x2 , . . . , xn ) ∈ R+ n , k is a nonnegative integer, m is a nonnegative real number and 3 1 3m4 C(x) = T4 (x, m) + W4 (x, m) − (T4 (x, m) ∗ W6 (x, m)) ∗ (R−2 (x) − S−2 (x)) 4 4 2 3m2 (T4 (x, m) ∗ W6 (x, m)) ∗ (S−4 (x) − 2S−2 (x) ∗ R−2 (x) + R−4 (x)) . (3.2) − 4 ¡ ¢∗−1 C ∗k (x) denotes the convolution of C itself k-times, C ∗k (x) denotes the inverse of ∗k ∗k C (x) in the convolution algebra. Moreover C (x) is a tempered distribution. Proof. Since 4 ¢¡ ¢ ¡ ¢ 3¡ 3m2 2 2 2 1 2 3 3m ¢B +m = 4B + m ¤B + m + 4B + m − (4B + ¤B )− (4B + ¤B )2 , 4 4 2 4 6

by (3.1) we have δ(x) = (¢B + m6 )(¢B + m6 )k−1 G(x) · ¢¡ ¢2 1 ¡ ¢3 3m4 3¡ = 4B + m2 ¤B + m2 + 4B + m2 − (4B + ¤B ) 4 4 2 ¸· ¢¡ ¢2 1 ¡ ¢3 3¡ 3m2 2 − (4B + ¤B ) 4B + m2 ¤B + m2 + 4B + m2 4 4 4 ¸k−1 4 2 3m 3m − (4B + ¤B ) − (4B + ¤B )2 G(x). (3.3) 2 4 By Lemma 2.10 with k = 1, we have T6 (x, m) ∗ W4 (x, m) exists and is a tempered distribution. Convolving both sides of (3.3) by T6 (x, m) ∗ W4 (x, m), we obtain ¸ · ¢¡ ¢ ¢ 1¡ 3m4 3m2 3¡ 2 2 2 2 2 3 4B + m ¤B + m + 4B + m − (4B + ¤B ) − (4B + ¤B ) 4 4 2 4 · ¢2 1 ¡ ¢¡ ¢3 3m4 3¡ T6 (x, m) ∗ W4 (x, m) ∗ 4B + m2 ¤B + m2 + 4 B + m2 − (4B + ¤B ) 4 4 2 ¸k−1 3m2 2 − (4B + ¤B ) G(x) = (T6 (x, m) ∗ W4 (x, m)) ∗ δ(x). 4 10

300

SUNTON-KANANTHAI: ON THE GREEN FUNCTION...

By Lemma 2.11, we have · ¢¡ ¢2 1 ¡ ¢3 3m4 3¡ C(x) ∗ 4B + m2 ¤B + m2 + 4B + m2 − (4B + ¤B ) 4 4 2 ¸k−1 3m2 2 − (4B + ¤B ) G(x) = T4 (x, m) ∗ W6 (x, m). 4 Keeping on convolving both sides of the above equation by T6 (x, m) ∗ W4 (x, m) up to k − 1 times, we have C ∗k (x) ∗ G(x) = (T6 (x, m) ∗ W4 (x, m))∗k , where the symbol ∗k denotes the convolution of itself k-times. By Tellez [6], we have (T6 (x, m) ∗ W4 (x, m))∗k = T6k (x, m) ∗ W4k (x, m), and so C ∗k (x) ∗ G(x) = T6k (x, m) ∗ W4k (x, m).

(3.4)

Now, consider the function C ∗k (x), since W4 (x, m), T4 (x, m), T6 (x, m) ∗ W4 (x, m), R−2 (x)−S−2 (x) and S−4 (x)−2S−2 (x)∗R−2 (x)+R−4 (x) are lies in S 0 where S 0 is a space of tempered distribution, C(x) ∈ S 0 . By Donoghue [2, p. 152], we obtain C ∗k (x) ∈ S 0 . 0 0 Since T6k (x, m)∗W4k (x, m) ∈ S 0 , choose S 0 ⊂ DR where DR is the right-side distribution 0 0 which is a subspace of D of distribution. Thus T6k (x, m) ∗ W4k (x, m) ∈ DR , it follows that T6k (x, m)∗W4k (x, m) is an element of the convolution algebra. Hence, by Zemanian [10, p. 150-151], the equation (3.4) has an unique solution ¡ ¢∗−1 G(x) = T6k (x, m) ∗ W4k (x, m) ∗ C ∗k (x) ¡ ¢∗−1 where C ∗k (x) is an inverse of C ∗k (x) in the convolution algebra, G(x) is called the Green function of the operator (¢B + m6 )k . Since T6k (x, m) ∗ W4k (x, m) and ¡ ∗k ¢∗−1 C (x) are tempered distribution, then by Donoghue [2, p. 152], we obtain ¡ ¢∗−1 T6k (x, m) ∗ W4k (x, m) ∗ C ∗k (x) is a tempered distribution. It follows that G(x) is a tempered distribution. ¤ Theorem 3.2 (An application of Green function) Given the equation (¢B + m6 )k u(x) = f (x)

(3.5)

where f is a given generalized function and u(x) is an unknown function, we obtain u(x) = G(x) ∗ f (x) is an unique solution of (3.5) where G(x) is a Green function for (¢B + m6 )k . 11

SUNTON-KANANTHAI: ON THE GREEN FUNCTION...

301

Proof. Convolving both sides of the equation (3.5) by G(x) where G(x) is a Green function for the operator (¢B + m6 )k in Theorem 3.1, we obtain G(x) ∗ f (x) = G(x) ∗ (¢B + m6 )k u(x) = (¢B + m6 )k G(x) ∗ u(x). Applying the Theorem 3.1, we have G(x) ∗ f (x) = δ(x) ∗ u(x) = u(x). Since G(x) is an unique, u(x) is an unique solution of the equation (3.5).

¤

Acknowledgement. The first author is supported by The Royal Golden Jubilee Project Grant No. PHD/0139/2549 and Graduate School, Chiang Mai University, Thailand. The authors would like to thank The Thailand Research Fund for financial support.

References [1] Bateman, Manuscript Project, Higher Transcendental Function, Vol. I, Mc-Graw Hill, New York, 1953. [2] C. Bunpog and A. Kananthai, On the Green Function of the (♦B + m4 )k Operator Related to the Bessel Helmholtz Operator and the Bessel Klein-Gordon Operator, Journal of Applied Functional Analysis, 4 (1) (2009), pp. 10-19. [3] W. F. Donoghue, Distributions and Fourier transform, Academic Press, 1969. [4] A. Kananthai, On the Green function of the diamond operator related to the KleinGordon operator, Bulletin of the Calcutta Mathematical Society, 93(5) (2001), pp. 353-360. [5] B. M. Levitan, Expansion in Fourier series and integrals with Bessel functions (N.S.), Uspeki Mat. Nauka, 2 (42) (1951) pp. 102-143 (in Russian). [6] M. A. Tellez, The convolution product of Wα (u, m) ∗ Wβ (u, m), Mathematicae Notae, 38 (1995-96). [7] H. Yildirim, Riesz Potentials Generated by a Gemeralized Shift Operator, Ph. D. Thesis, Ankara University 1995. ¨ urk, The solution of the n-dimensional [8] H. Yildirim, M. Z. Sarikaya and S. Ozt¨ Bessel diamond operator and the Fourier-Bessel transform of their convolution, Proc. Indian Acad. Sci. (Math. Sci.), 114 (4) (2004), pp. 375-387. [9] A. H. Zemanian, Distribution Theory and Transform Analysis, Mc-Graw Hill, New York, 1964. 12

302JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.5, NO.3, 302-316, COPYRIGHT 2010 EUDOXUS PRESS, LLC

NONLINEAR SINGULAR INTEGRAL EQUATIONS WITH SHIFT IN THE GENERALIZED HÖLDER SPACES M.H.Saleh, S.M.Amer* and D.Sh.Mohammed Dept.of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt. E-mail: [email protected] Abstract. This paper concerns sufficient conditions for the convergence of Newton-Kantorovich majorant method for the solution of a certain class of nonlinear singular integral equations with shift. A criterion for the Noetherity of a correspondence singular integral functional operator of second order with Carleman shift preserving orientation is obtained and the index formula is given. Key words: Nonlinear singular integral equations, Newton-Kantorovich majorant method, Carleman shift, Generalized Hölder spaces. 0. Introduction There is a large Literature on the classical theory of nonlinear singular integral equations (NSIE) (see [4],[6,7],[13],[15,16] and others).The development of the theory of singular integral equations (SIE) naturally stimulated the study of singular integral equations with shift (SIES).The Noether theory of singular integral operators with shift (SIOS) is developed for a closed and open contour (see [9],[10],[11] and others). Existence results and approximate solutions have been studied for (NSIE) and the nonlinear singular integral equations with shift (NSIES) by authors (see [1-3],[5],[8].[12]). In this paper, our aim is to apply the Newton-Kantorovich majorant method to a class of (NSIES) under certain conditions. Consider the following nonlinear singular integral equation with Carleman shift(NSIES)

( (u ) ) (t )  a (t ) u (t )  b (t ) u ( (t ) )  c (t ) u ( 2 (t ) ) 

d (t ) u ( ) e (t ) u ( ) d  d     i L  t  i L    (t )

1 ( , u ( ) ) 2 ( , u ( ) ) 3 ( , u ( ) ) f (t ) u ( ) 1  d    d  0 ,    i L    2 (t ) iL  t    (t )    2 (t )

{

}

(0.1)

is a simple closed Lyapunov contour, dividing the complex plane into interior L where domain D  is unknown function and the homeomorphism D  , u (t ) and exterior domain  : L  L is a shift preserving orientation, satisfying the Carleman condition  3 (t )   ( ( (t ) ) )  t , tL (0.2) The functions   (t )  0  t  L . satisfies usual Hölder condition,   (t ) whose derivative a (t ) , b (t ) , c (t ) , d (t ) , e (t ) and f (t ) belong to the generalized Hölder space   ,m ( L) . Assume that the functions 1 (t , u (t ) ) , 2 (t , u (t ) ) and 3 (t , u (t ) ) have partial derivatives up to (m  1) order, and satisfy the following conditions:

١

SALEH ET AL: NONLINEAR SINGULAR INTEGRAL EQUATIONS...

 k 1 (t1 , u1 )  k 1 (t 2 , u 2 )   c k (r ){ ( t1  t 2 )  u1  u 2 ti u j ti u j

},

 k 2 (t1 , u1 )  k 2 (t 2 , u 2 )   c k (r ){1 ( t1  t 2 )  u1  u 2 ti u j ti u j and  k 3 (t1 , u1 )  k 3 (t 2 , u 2 )   c k (r ){ 2 ( t1  t 2 )  u1  u 2 ti u j ti u j

303

(0.3)

},

(0.4)

},

(0.5)

where  , 1 ,  2  , i  j  k , k  0,1,..., m  1 and c k (r ) , c k (r ) , c k (r ) are positive increasing functions. The functions 1 (t , u (t ) ) , 2 (t , u (t ) ) and 3 (t , u (t ) ) for any   ,m ( L) belong to the space u   , m ( L) , [14].

1. Formulation of the problem Let  : S (u 0 , R )     be a nonlinear operator defined on the closure of a ball

S (u 0 , R)  { u : u   , u  u 0  R } in a Banach space  into a Banach space  . We give new conditions to ensure the convergence of Newton-Kantorovich approximations toward a solution of (u )  0 , under the hypothesis that  is Frechet differentiable in S (u 0 , R) , and that its derivative  satisfies the local Lipschitz condition: (u1 )  (u 2 )  k (r ) u1  u 2 ,

u1 , u 2  S (u 0 , r ) , 0  r  R

where k (r ) is a non-decreasing function on the interval [0,R] and (u1 )  (u 2 ) k (r )  sup : u1 , u 2  S (u 0 , r ) , u1  u 2 . u1  u 2 Define a scalar function  :[0, R ]  [0, ) by

{

}

(1.1)

(1.2)

r

 (r )       (t ) dt  r ,

(1.3)

0

where the function r

 (r )   k (t ) dt ,

(1.4)

0

and

  (u 0 ) 1 (u 0 )

,   (u 0 ) 1 .

(1.5)

Theorem 1.1 [17]. Suppose that the function  has a unique positive root r* in [0, R] and  ( R )  0 . Then the equation (u )  0 has a unique solution u* in S (u 0 , R) and the Newton-Kantorovich approximations u n  u n1  (u n 1 ) 1 (u n 1 ) , n   , (1.6)

٢

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SALEH ET AL: NONLINEAR SINGULAR INTEGRAL EQUATIONS...

are defined for all n  , belong to S (u 0 , r* ) and converge to u* . Moreover the following estimate holds u n 1  u n  rn 1  rn , u*  u n  r*  rn , (1.7) where the sequence (rn ) n  , converges to r* , is defined by the recurrence formula.

 (rn ) , n . (1.8)  (rn ) In this paper, we investigate some sufficient conditions, which ensure that the class of (NSIES) (0.1) verifies the hypotheses of theorem 1.1. r0  0 , rn 1  rn 

2. Some auxiliary results Definition 2.1 defined  the class of all continuous almost increasing function  1) We denote by on (0,  2] such that  (t )  0 , lim  (t )  0 , where  L . is the length of the curve t 0

implies 0  t1  t 2 such that   the class of all functions  m 2) We denote by t1m  (t 2 )  c(m) t 2m  (t1 ) where m is a natural number. 3) We denote by c(L) with the norm L the space of all continuous functions defined on u c ( L )  max u (t ) . t L

4) For a natural number m we define the generalized Hölder space

  , m  { u c( L) : um ( )   ( ( ) ) ,   m } , where  um ( ) is the modulus of continuity of order m of the function u defined as follows: m  m ωum ( )  sup mh (u; x) and mh (u; x)   (1) m i   u ( x  i h) 0 h  ,  0 i 0 i is the m  difference of the function u (x) with step h . 5) We denote by   m the class of all functions defined as follows 

  ( )  ( ) d   m  m1 d  c~ (m)  ( )},    0 ~ where c (m) is a positive constant.

{

  m    m : 

6) for u   , m ( L) we define the norm:

 um ( ) u  ,m  u c  sup .   0  ( ) 7) Consider the following operators on the space   , m ( L) (i ) (W i u ) (t )  u ( i (t ) ) ,

i  0,1,..., m  1

and

٣

W 3  .

(2.1)

(2.2)

SALEH ET AL: NONLINEAR SINGULAR INTEGRAL EQUATIONS...

305

Under the assumption that  (t ) homeomorphically maps L into itself with preservation of orientation and satisfies the carleman condition

 m (t )  t ,  i (t )  t where

, 1 i  m 1 , m 3

 i (t )   ( i 1 (t ) ) ,  0 (t )  t .

(ii ) The inverse operator W 1 is defined by

(W 1 u )(t )  u (  (t )) , where  (t ) is the inverse of  (t ) . (iii ) The singular operator 1 u ( ) d .  i L   t (iv) The complementary projection operators 1   (   S ) , S 2   , 2 where  is the identity operator. ( S u ) (t ) 

(2.3)

(2.4)

Lemma 2.1.Let the function v(t )  ( Su ) (t ) be defined for all t L and has derivatives up to (m  1) order and satisfy the following condition: v ( k ) (t1 )  v ( k ) (t 2 )  l k { ( t1  t 2 ) } ,

(2.5)

where k  0,1,..., m  1 , t1 , t 2  L , u (t )  ,m ( L) , l k is a positive constant and   . v(t )  ,m ( L) . Then

Proof. for m  1 , we have 1h S u (t )  S u (t  h)  S u (t ) ,

from condition (2.5) 1h S u (t )  S u (t  h)  S u (t )  l 0 ( ( ) ) .

Then,

 1S u ( )  l 0 ( ( ) )

(2.6)

for m  2 , we have 2h S u (t )  S u (t  2 h)  2 S u (t  h)  S u (t ) .

From (2.3) we have

1 u (  2 h)  2 u (  h)  u ( ) d .  t  i L Using Lagrange's formula we have 2h S u (t ) 

1

1  S u (t )   iL 2 h

 ( u (  h   h)  u (   h) ) h d 0

 t

٤

d ,

306

SALEH ET AL: NONLINEAR SINGULAR INTEGRAL EQUATIONS...

using condition (2.5) then we have  S2 u ( )  l1  ( ( ) )   ( ( ) ) .

(2.7)

Thus, the lemma is proved at m  1,2 . Now, we prove that the lemma is true for any m by induction we see that 1

mh S u (t )  h  mh1 S u  (t   h ) d .

(2.8)

0

Suppose that this equality is true at m  1,2 . The equality (2.8) is true at m  n  1, that is 1

nh1 S u (t )  h  nh 2 S u  (t   h) d . 0

Now, we prove that (2.8) is true at m  n , thus n 1 h

 S u (t )   ( n h

1 h

1

S u (t ) )   (h   1 h

0

n 1 h

1

S u  (t   h) d )  h  nh1 S u (t   h ) d . 0

Consequently, 1

 S u (t )  h  mh 1 S u (t   h ) d  l m1  m 1 ( ( ) )   ( ( ) ) . m h

0

Hence,

 Smu ( )   ( ( ) ) .

(2.9)

Therefore, from (2.6),(2.7) and (2.9) we have v(t )  ,m ( L) . Thus the lemma is valid. Lemma 2.2.

Let the condition (2.5) be satisfied for the shift operator W , and

u (t )  ,m ( L) . Then the function (W i u ) (t )  , m ( L) . , i  0,1,2 ,..., m  1.

Proof. From Lemma 2.1 and the properties of the shift  (t ) the proof is immediate. Lemma 2.3. [14] Let the function u (t )c( L) and



 0

 um ( ) d   . Then the following 

inequalities: 

 um ( ) S u c  c1 (m) (  d  u  0

c

)

(2.10)

and 

  um ( )  um ( ) m  ( )  c 2 (m) (  d    m 1 d ) ,  0   are valid, where c1 (m) and c 2 (m) are constants. m Su

(2.11)

then the singular operator    m , Lemma 2.4. Let S is bounded operator on the space   ,m ( L) . Proof.

٥

SALEH ET AL: NONLINEAR SINGULAR INTEGRAL EQUATIONS...

307

From lemma 2.1, we have S u (t )  ,m ( L) , for any u    , m ( L) . Hence Su

 Su  ,m

 sup c  0

 Smu ( )  ( )

.

Using (2.10) and (2.11) we have     um ( )  um ( )  um ( ) 1 m S u  ,m  c1 (m) (  d  u c )  c 2 (m) sup ( d    m1 d ) .     0  ( ) 0   0 By using equality (2.1) we have     ( )  ( )  ( ) 1 m S u  ,m  c1 (m) u  ,m (  d  1)  c 2 (m) u  ,m sup ( d    m 1 d ) .     0  ( ) 0 0   Since    m , then S u  ,m   0 u  ,m , (2.12) is a constant, defined as follows  0 where 

 ( ) d  c1 (m)  c 2 (m) c~ (m) .  0 Therefore the singular operator S is bounded in generalized Hö lder space   ,m ( L) .  0  c1 (m) 

Lemma 2.5. The shift operator W is a linear bounded continuously invertible operator on the space   ,m ( L) . Proof. where u (t ) , u~ (t )  ,m ( L) , Since ~ u (t )  (W u ) (t )  u ( (t ) ) . Therefore,  u~m ( )  um ( ) ~ W u  ,m  max u (t )  sup  max u (t )   0 sup tL t L   0  ( )   0  ( ) (2.13)  0 u

 ,m

,

 u~m ( ) . Then the shift m   0  u ( ) operator W is bounded on the space   ,m ( L) . The continuous invertibility of the where  0  max {1,  0 } and  0 is a constant given by  0  sup

operator W on   , m ( L) follows from the properties of the homeomorphism  (t ) . Lemma 2.6. Let the functions 1 ( , u ( ) ) , 2 ( , u ( ) ) and 3 ( , u ( ) ) satisfy the conditions (0.3),(0.4) and (0.5) respectively. Then the operator  (u ) is Frechet differentiable at every fixed point u   , m ( L) . and its derivative is given by:  (u ) h  a (t ) h (t )  b (t ) h ( (t ) )  c (t ) h ( 2 (t ) )  f (t ) h ( ) 1  d    i L    2 (t ) i

{ L

d (t ) h ( ) e (t ) h ( ) d  d     i L  t  i L    (t )

1u ( , u ( ) ) 2 u ( , u ( ) ) 3 u ( , u ( ) )   h( ) d ,  t    (t )    2 (t )

}

٦

(2.14)

308

SALEH ET AL: NONLINEAR SINGULAR INTEGRAL EQUATIONS...

moreover it satisfies Lipschitz condition:  (u1 )   (u 2 )  , m  k ( r ) u1  u 2  , m ,

 u1 , u 2  S (u 0 , r ) and 0  r  R

(2.15)

where k (r ) is local Lipschitz constant given by k (r )  (  0 c1 (r )  1 c1 (r )   2 c1(r ) ) , 1   0  0 and  2  1  0 . Proof. Let u (t ) be a fixed element and h (t ) be an arbitrary element in the space   , m ( L) . Then we have  (u  h)   (u )   (u ) h   (u , h) , where d (t ) h ( ) e (t ) h ( )  (u ) h  a (t ) h (t )  b (t ) h ( (t ) )  c (t ) h ( 2 (t ) )  d  d     i L  t  i L    (t ) 

f (t ) h ( ) 1 d    i L    2 (t ) i

{ L

1u ( , u ( ) ) 2 u ( , u ( ) ) 3 u ( , u ( ) )   h( ) d ,  t    (t )    2 (t )

}

and

1 1  (u, h)     i L t 

1

{  (1   ) 

1 uu

}

( , u ( )   h( ) ) h 2 ( ) d d

0

1 1   i L    (t )

1

{  (1   ) 

2 uu

}

( , u ( )   h( ) ) h 2 ( ) d d

0

1

1 1 (1   ) 3 uu ( , u ( )   h( ) ) h 2 ( ) d d .   i L    2 (t ) 0 From the inequalities (2.12) and (2.13) we have

{



1 

}

h( )

    (t ) d L

 1 h

,m

(2.16)

,

,m

where 1   0  0 is a constant. Therefore we have

 (u, h) h

,m

 (  0 1

,m

 1  2

,m

  2 3

,m

,m

where  2  1  0 , 1

1 ( )   (1   ) 1uu ( , u ( )   h ( ) ) d , 0

1

 2 ( )   (1   ) 2 uu ( , u ( )   h ( ) ) d , 0

and 1

 3 ( )   (1   ) 3 uu ( , u ( )   h ( ) ) d . 0

٧

) h

,m

,

SALEH ET AL: NONLINEAR SINGULAR INTEGRAL EQUATIONS...

Hence we have

h

lim , m

0

 (u, h) h

,m

309

 0.

,m

Thus the Frechet's derivative is given by (2.14).Moreover  satisfies Lipschitz condition, using conditions (0.3) - (0.5) and inequalities (2.13), (2.14) and (2.16) we have  (u1 )   (u 2 )

where

,m

 k ( r ) u1  u 2

,m

,

k (r )  (  0 c1 (r )  1 c1 (r )   2 c1(r ) ) .

Thus the lemma is valid. 3. Criterion of Noetherity for (SIOS): Using equations (2.2), (2.3) and (2.14) we obtain the following (SIES), for the unknown function h (t ) :

 (u 0 ) h  a (t ) h (t )  b (t ) (W h) (t )  c (t ) (W 2 h) (t )  ( d (t )  1u (t , u 0 (t ) ) ) ( S h) (t )  (3.1)

 ( e (t )  2u ( (t ), u 0 ( (t )) ) ) (W S h) (t )  ( f (t )  3 u ( 2 (t ), u 0 ( 2 (t ) ) ) ) (W 2 S h) (t ) 

1 R (t , ) h( ) d  g (t ) ,  i L

where R (t , ) 

1u (t , u 0 (t ) )  1u ( , u 0 ( ) )



2 u ( (t ), u 0 ( (t ) ) )  2 u ( , u 0 ( ) )

 t    (t ) 3 u ( 2 (t ), u 0 ( 2 (t ) ) )  3 u ( , u 0 ( ) ) .     2 (t ) According to the assumption a * (t )  d (t )  1u (t , u 0 (t ) ) , b * (t )  e (t )  2 u ( (t ), u 0 ( (t )) ) ,



(3.2)

c * (t )  f (t )  3 u ( 2 (t ), u 0 ( 2 (t ) ) ) ,

the dominant equation of equation (3.1) can be written in the following operator form:  (u 0 ) h  a (t ) h (t )  b (t ) h ( (t ) )  c (t ) h (  2 (t ) ) 

a * (t ) h ( ) d   i L   t

h ( ) h ( ) b * (t ) c * (t ) d   d  J (t ) ,    i L    (t )  i L    2 (t )

where

 (t )   R (t , ) h( ) d .

J (t )  g (t )   (t ) ,

L

By using equality (2.4) equation (3.3) reduces to the following (SIOS)

٨

(3.3)

310

SALEH ET AL: NONLINEAR SINGULAR INTEGRAL EQUATIONS...

(  h ) (t )  {[ (a (t )  a * (t ) )   (b (t )  b * (t ) ) W  (c (t )  c * (t ) ) W 2 ]    [ (a (t )  a * (t ) )   (b (t )  b * (t ) ) W  (c (t )  c * (t ) ) W 2 ]  } h (t )  J (t ) , or

(  h ) (t )  (      ) h (t )  J (t ) , where   (a (t )  a * (t ) )   (b (t )  b * (t ) ) W  (c (t )  c * (t ) ) W 2 and   (a (t )  a * (t ) )   (b (t )  b * (t ) ) W  (c (t )  c * (t ) ) W 2 . From the theory of singular integral operators with shift [10]. the Noether condition for the operator  is given by:

a (t )  a * (t )

b (t )  b * (t )

c (t )  c * (t )

1 (t )  c ( (t ) )  c * ( (t ) ) a ( (t ) )  a * ( (t ) ) b ( (t ) )  b * ( (t ) )  0 , b ( 2 (t ) )  b * ( 2 (t ) ) c ( 2 (t ) )  c * ( 2 (t ) ) a ( 2 (t ) )  a * ( 2 (t ) ) a (t )  a * (t )

b (t )  b * (t )

c (t )  c * (t )

 2 (t )  c ( (t ) )  c * ( (t ) ) a ( (t ) )  a * ( (t ) ) b ( (t ) )  b * ( (t ) )  0 . b ( 2 (t ) )  b * ( 2 (t ) ) c ( 2 (t ) )  c * ( 2 (t ) ) a ( 2 (t ) )  a * ( 2 (t ) ) Moreover the index formula of the operator  has the form:  (t ) 1 arg 2 ind   L 6 1 (t )

{

(3.4)

}.

(3.5)

(3.6)

4. Solution of linear singular integral equation with shift Now, we prove that  (u 0 ) has inverse. For this aim, we investigate the solvability of the linear singular integral equation  (u 0 ) h  a (t ) h (t )  b (t ) h ( (t ) )  c (t ) h ( 2 (t ) )   

h ( ) 1 d  ( d (t )  1u (t , u 0 (t ) ) )   t  i L

h ( ) 1 d  ( e (t )  2 u ( (t ), u 0 ( (t )) ) )  i   t  ( ) L

(4.1)

h ( ) 1 1 ( f (t )  3 u ( 2 (t ), u 0 ( 2 (t )) ) )  d  R (t , ) h ( ) d  g (t ) ,  i    t  i ( ) 2 L L

by using equations (2.2), (2.3), equation (4.1) takes the following operator form:

(  h ) (t )  a (t ) h (t )  b (t ) (W h ) (t )  c (t ) (W 2 h ) (t )  a * (t ) ( S h) (t )  b * (t ) (W S h) (t )  c * (t ) (W 2 S h) (t )  ( h) (t )  g (t ) .

٩

(4.2)

SALEH ET AL: NONLINEAR SINGULAR INTEGRAL EQUATIONS...

311

To solve equation (4.2), we apply the operators W , W 2 , S ,W S and W 2 S successively to both sides of equation (4.2), hence we obtain the following system: a (t ) h (t )  b (t ) (W h) (t )  c (t ) (W 2 h) (t )  a * (t ) ( S h) (t )  b * (t ) (W S h) (t )   c * (t ) (W 2 S h) (t )  ( h) (t )  g (t ) , c ( (t ) ) h (t )  a ( (t ) ) (W h) (t )  b ( (t ) ) (W 2 h) (t )  c * ( (t ) ) ( S h) (t )   a * ( (t ) ) (W S h) (t )  b * ( (t ) ) (W 2 S h) (t )  (W  h) (t )  (Wg ) (t ) , b ( 2 (t ) ) h (t )  c ( 2 (t ) ) (W h) (t )  a ( 2 (t ) ) (W 2 h) (t )  b * ( 2 (t ) ) ( S h) (t )   c * ( 2 (t ) ) (W S h) (t )  a * ( 2 (t ) ) (W 2 S h) (t )  (W 2  h) (t )  (W 2 g ) (t ) , a * (t ) h (t )  b * (t ) (W h) (t )  c * (t ) (W 2 h) (t )  a (t ) ( S h) (t )  b (t ) (W S h) (t ) 

(4.3)

 c (t ) (W 2 S h) (t )  ( 1 h) (t )  ( S g ) (t ) , c * ( (t ) ) h (t )  a * ( (t ) ) (W h) (t )  b * ( (t ) ) (W 2 h) (t )  c ( (t ) ) ( S h) (t )   a ( (t ) ) (W S h) (t )  b ( (t ) ) (W 2 S h) (t )  ( 2 h) (t )  (W S g ) (t ) , b * ( 2 (t ) ) h (t )  c * ( 2 (t ) ) (W h) (t )  a * ( 2 (t ) ) (W 2 h) (t )  b ( 2 (t ) ) ( S h) (t )   c ( 2 (t ) ) (W S h) (t )  a ( 2 (t ) ) (W 2 S h) (t )  ( 3 h) (t )  (W 2 S g ) (t ) , where

 1  S ( a (t )   b (t ) W  c (t ) W 2  a * (t ) S  b * (t ) W S  c * (t ) W 2 S   )  ( a * (t )   b * (t ) W  c * (t ) W 2  a (t ) S  b (t ) W S  c (t ) W 2 S ) ,  2 W 1 ,  3 W  2 W 2 1 .

No solutions are lost when W , W 2 , S , W S and W 2 S are applied to equation (4.2), hence all solutions of (4.2) are solution of the system (4.3) and conversely. Let D be a closed subspace defined by

D  { ( h,W h,W 2 h, S h,W S h,W 2 S h ) , h  , m } , and let C be the linear operator from D into   m (L) defined by C  (t )  C (t )  (t ) . Moreover if we put

0 0   2 0  0 W W 2  0 0 W W   0 0  0  0 0 0   0 0 0 

0 0 0 1 S 0 0

0 0 0 0 2 SW 2 0

١٠

0   g   h       0   Wg   Wh   W 2g   W2h  0  ,    , , G  0   Sg   Sh  W S g  W Sh  0      W 2 S g  W 2 S h   3 S W     

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SALEH ET AL: NONLINEAR SINGULAR INTEGRAL EQUATIONS...

then the system (4.3) can be written as the following form: C    G ,

D ,

(4.4)

where b c a* b* c*   a   W (a) W (b) W (c*) W (a*) W (b*)   W (c )  W 2 (b) W 2 (c) W 2 (a) W 2 (b*) W 2 (c*) W 2 (a*)  , C  b* c* a b c   a* W c W (a ) W (b)   ( *) W (a*) W (b*) W (c) W 2 (b*) W 2 (c*) W 2 (a*) W 2 (b) W 2 (c) W 2 (a)   

(4.5)

is a matrix of functions from the space   , m ( L) corresponding to the operator C . Theorem 4.1. Let the hypotheses of lemma 2.6 be satisfied and assume that (1) det C (t )  0 ,  t  L (2) C 1   1.

(4.6) (4.7)

Then the operator  (u 0 ) is invertible, moreover

 (u 0 ) 1 

C* 1 C

1



(

det C 1  ), m m2

(4.8)

where m  min det C (t ) , t L

and C * be the adjoint matrix of C . Proof. It is well known that the condition (4.6) is necessary and sufficient for the invertibility moreover equation (4.4) is equivalent to the equation C on D , of the operator   C 1 G  C 1   ,   D . The problem of the invertibility of the operator C   can be reduced to the following fixed point problem     ,    C 1 G  C 1   ,   D , where   1    2  C 1   1   2 . From condition (4.7) and the contraction mapping theorem, it follows that for every G D , the operator  has a unique fixed point. Then the operator C   and therefore (  (u 0 ) ) is invertible and ( C   ) 1  (C (   C 1  ) ) 1  (   C 1  ) 1 C 1 . Thus, we have

١١

SALEH ET AL: NONLINEAR SINGULAR INTEGRAL EQUATIONS...

 (u 0 )

1

1

 (  C

1

) C

1



C 1 1  C 1 

313

,

since C * be the adjoint matrix of C , we have C *G , C 1 G  det C moreover det C C *G 1 1 C 1  max G  C* (   max C * ), G 1 det C G 1 m det C m2 hence, we get C* det C 1  (u 0 ) 1  (  ). m2 1  C 1  m Thus the theorem is proved. Assume that

   ( a (t )

 ,m

u 0 (t )

 ,m

  0 d (t )

  0 b (t )

 ,m

 ,m

u 0 (t )

 ,m

u 0 (t )  ,m  1 e (t )

  0 1 (t , u (t ) )

 ,m

  02 c (t )

 ,m

u 0 (t )

 ,m

 ,m

u 0 (t )

 ,m

  2 f (t )

 1 2 (t , u (t ) )

 ,m



 ,m

u 0 (t )

  2 3 (t , u (t ) )

 ,m



 ,m

),

and



C* 1 C

1



(

det C 1  ). m m2

Therefore, the following theorem is valid. Theorem 4.2. Suppose that the hypotheses of theorem 4.1 are satisfied, moreover the scalar function  (r ) defined by (1.3), (1.4) has a unique positive root r* in [0, R] and  ( R )  0 . Then the equation (0.1) has a unique solution u* in S (u 0 , R) and the Newton-Kantorovich approximations u n  u n1  (u n 1 ) 1 (u n 1 ) , n   ,

belong to S (u 0 , r* ) and satisfy the following estimate

u n 1  u n  rn 1  rn ,

u*  u n  r*  rn ,

where the sequence (rn ) n  , converges to r* , is defined by the recurrence formula.

r0  0 , rn 1  rn 

 (rn ) , n .  (rn )

We will illustrate theorem 4.2 by the following examples.

١٢

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Example (1) Consider the nonlinear function u3 f (u )   u 3 with derivative f  (u )  u 2  1 since f (u1 )  f (u 2 )  u12  u 22  u1  u 2

u1  u 2 .

From inequality (1.2) we take

k (r ) 

sup

u1 , u 2  r

{

f (u1 )  f (u 2 ) u1  u 2

}

: u1 , u 2  S (0, r ) , 0  r  R ,

it is clear that

f (u1 )  f (u 2 ) u1  u 2

 u1  u 2  2 r ,

therefore we get k (r )  2 r and  (r )  r 2 . Obviously, the scalar function (1.3) takes the form: r 1  (r )      t 2 dt  r     r 3  r . 3 0 Consider the scalar equation  (r )  0 we have 3  r 3  r  3( )  0 , (4.9)   equation (4.9) has a unique positive solution r* in [0, R] if and only if q p ( )2  ( )3  0 , 2 3 where  3 p and q  3 ( )   Hence, the equation f (u )  0 has a unique solution u* in S (u 0 , R) and the assumptions of theorem 1.1 are valid.

Example (2) Consider the nonlinear function f (u )  u 3  2 u 2  u  3 with derivative f (u )  3 u 2  4 u  1 since f (u1 )  f (u 2 )  3 (u12  u 22 )  4 (u1  u 2 )  [ 3 u1  u 2  4 ] u1  u 2 . From inequality (1.2) we take

k (r ) 

sup

u1 , u 2  r

{

f (u1 )  f (u 2 ) u1  u 2

it is clear that

١٣

}

: u1 , u 2  S (0, r ) , 0  r  R ,

SALEH ET AL: NONLINEAR SINGULAR INTEGRAL EQUATIONS...

f (u1 )  f (u 2 ) u1  u 2

315

 3 u1  u 2  4  6 r  4 ,

therefore we get k (r )  6 r  4

and

 (r )  3 r 2  4 r .

Obviously, the scalar function (1.3) takes the form

 (r )     r 3  2  r 2  r . Consider the scalar equation  (r )  0 we have 1  r 3  2 r 2  r  ( )  0. (4.10)   2 Let r  y  , then 3 4 1 2  8 y3 (  ) y  (   )  0, (4.11) 3  3   27 equation (4.11) has a unique positive solution r* in [0, R] if and only if q p ( )2  ( )3  0 , 2 3 4 1 2  8 p   (  ) and q    . 3  3   27 Hence, the equation f (u )  0 has a unique solution u* in S (u 0 , R) and the assumptions of theorem 1.1 are valid. References [1] Amer S.M. and Dardery. S.: On the theory of nonlinear singular integral equations with shift in Hölder spaces. Forum Math. 17 (2005), 753-780. [2] Amer S.M. and Dardery. S.; On a class of nonlinear singular integral equations with shift on a closed contour; Applied Mathematics and Computation; 158 (2004) 781791. [3] Amer S. M. and Nagdy A. S.; On the solution of a certain class of nonlinear singular integral and integro-differential equations with carleman shift Chaos, Solitinos and Fractals 12 (2001), 1473-1484. [4] Apple J. ,Carbone A. ,De Pascale E. ,Zabrejko P.P.; Anote on the existence and uniqueness of Hölder solutions of nonlinear singular equations. Z. Anal. Anwendungen 11 (1992) , 377-384. [5] Cianciaruso F.; The Newton-Kantorovich approximations for nonlinear singular integral equations with shift. Journal of Integral Equations and Applications. volume 14 Number 3, (2002), 223-237. [6] Gakhov F.D.: Boundary Value Problems. English Edition: Pergamon Press Ltd, 1966. [7] Guseinov A. I., and Mukhtarov Kh. Sh.: Introduction to the Theory of Nonlinear Singular Integral Equations. Nauka Moscow 1980 (in Russian).

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[8] Junghanns P. and Weber U.: On the solvability of nonlinear singular integral equations. Z. Anal. Anwendungen 12 (1993), 683-698. [9] Kravchenko V. G., Lebre A. B., Litvinchuk G. S. and Texeira F. S.: Fredholm theory for a class of singular integral operators with Carleman shift and unbounded coefficients. Math. Nachr. 172 (1995), 199-210. [10] Kravchenko V. G, and Litvinchuk G. S.: Introduction to the Theory of Singular Integral Operators with shift. Kluwer Academic Publishers, 1994. [11] Litvinchuk G. S.: Boundary Value Problems and Singular Integral Equations with shift. Nauka, Moscow 1977 (in Russian). [12] Nguyen D. T.: On a class of nonlinear singular integral equations with shift on complex curves. Acta Math Vietnam 14 (1989), 75-92. [13] Pogorzelski W.: Integral Equations and their Applications; vol.1. Oxford Pergamon Press and Warszawa PWN, 1966. [14] Saleh M. H.; Basis of Quadrature Method for Nonlinear Singular Integral Equations with Hilbert Kernel in the space   ,k (in Russian), In Az. NIINTI, No. 279 (1984) 1-40. [15] Wegert E.: Nonlinear Boundary Value Problems for Holomorphic Functions and Singular Integral Equations. vol. 65 Akademie-Verlag, 1992. [16] Wolfersdorf L. V.: On the theory of nonlinear singular integral equations of Cauchy type. Math. Method Appl. Sci. 7 (1985), 493-517. [17] Zabrejko P. P. and Nguen D. F.: The majorant method in the theory of NewtonKantorovich approximations and ptak error estimates. Numer. Funct. Anal. Optim. 9 (1987), 671-684.

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JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.5, NO.3, 317-324, COPYRIGHT 2010 EUDOXUS PRESS, LLC317

Some problems of random operator equations in the Z-C-X Space Cheng Li-ying 1, , Zhu Chuan-xi 1 , Tang Chao 2 1. Institute of mathematics, Nanchang University, Nanchang 330031 2. Institute of Public education, Jiangxi Vocational & Technical College of Electricity, Nanchang 330032

Abstract: In this paper, the new concept of

 -cone is introduced, several important inequalities

are proved, and the existences of random solutions of random semi-closed 1-set-contractive operator equations with different boundary conditions are investigated in the Z-C-X space, some new results are obtained. Keywords: Z-C-X Space, random semi-closed 1-set-contractive operator, random operator equation, random solution. AMS(2000): 60H25, 47H10 Chinese Library Classification : O177.91, O211.63

1. Introduction and preliminaries Let (, U ,  ) be a complete probability measure space,  ()  1 ,let E be a separable real Banach space, ( E , B) be a measurable space, where B denotes the

  algebra of generating by

all subset in E , and let X be a closed convex subset of E , and let D be a bounded open set in X ,

D and D the closure and boundary of D in X , respectively. Definition1.

1

Let E be a separable real Banach space, which satisfies the following conditions:

( H 1 ) E be an algebra over the real number field R ,that has （1） E is closed to multiplication, that is, for every x, y  E ,we have x  y  E ； （2）for every   R, x, y  E , we have ( x)  y  x  ( y )   ( x  y ) ；

( H 2 ) E hasn’t nilpotent element, that is, for every x  E , n  N , if x   ,we have x n   . Then E is called the Z-C-X Space. Obviously, because of E is algebra over the real number field R , we obtain: （3）for every  ,   R, x, y  E , we have  x   y  ( )( x  y ) .

x  x x  x , where x  E , n is natural number. In the Z-C-X Space E, let  n

n

 Foundation details: The National Natural Science Foundation of China (10461007 and 10761007); The Provincial Natural Science Foundation of Jiangxi, China (2007GZS2051and 2008GSZ0076).  Corresponding author: E-mail addresses: [email protected].

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CHENG LI-YING ET AL: ABOUT Z-C-X SPACE...

Definition2. Let X be a cone of E , and suppose that linear functional  : X  [0, ) , which satisfies the following conditions: （a）for every x, y  X , if   x  y , then 0   ( x)   ( y ) ； （b） X is closed to multiplication, that is, for every x, y  X ,we have x  y  X . Then X is called the  -cone. In the paper, we suppose that "  " is the derived partial ordering by

 -cone X in E .

By the definition2, we also obtain: for every x, y  X , if   x  y , then 0   ( x)   ( y ) . That is because of when   x  y , we naturally have  ( x)   ( y )  0 . Lemma1.1.

[2 ]

Let X be a closed convex subset of E , and let D be a bounded open set in X , and

  D ,   1 , suppose that A：  D  X is a random semi-closed 1-set-contractive operator, such that A( , x)   x, for every ( , x)    D,    and  is variable. Then the random operator equation A( , x)   x has a random solution in D .

2. Main results Lemma2.1. when  ,   0, p, q  0, p  q  1 , we have (ⅰ) p  q   ( p  q  ) ,where k  1 ; k

k

k

(ⅱ) p  q   ( p  q  ) ,where 0  k  1 . k

k

k

Proof. when    , the both sides of(ⅰ)and(ⅱ)are equal to  ,hence the equalities k

are true. Suppose  

 , and    .Let   p  q , because p  q  1 ,

then     q (    )   , in like matter, we have

   ,that is      .In(ⅰ)the

inequality can be overwritten to p  q   ( p  q )  , k

that is

k

k

q(  k   k )  p(  k   k )

Let f ( x)  qx , g ( x)  px , then we have f '( x)  qkx k

k

(1) k 1

, g '( x)  pkx k 1 .

Using the Lagrange value theorem to them in [  ,  ] and [ ,  ] , respectively, we have

CHENG LI-YING ET AL: ABOUT Z-C-X SPACE...

319

q (  k   k )  qk1k 1 (    )  pqk1k 1 (    ) , p (  k   k )  pk 2k 1 (    )  pqk 2k 1 (    ) , where 0     2    1  . Because k  1 , then k  1  0 , thus 1

k 1

  2k 1 .Therefore, q(  k   k )  p (  k   k ) ,

which proves that the inequality (1)is true. Hence the sign of inequality of(ⅰ)is true. When 0  k  1 , then k  1  0 , we remark the 1

k 1

  2k 1 in the above process of proving,

then we know that the sign of inequality of(ⅱ)is true. Theorem2.1. Let E be the Z-C-X Space, X be a cone of E , and let D be a bounded open set in X , and

  D ,   1 , suppose that A：  D  X is a random semi-closed 1-set-contractive

operator, such that for every ( , x)    D , p, q  0, p  q  1 , which satisfies to one of the following conditions:

(Y1 ) p( A( , x)) k  q(  x) k  ( pA( , x)  q  x) k , where k  1 , (Y1' ) p( A( , x)) k  q(  x) k  ( pA( , x)  q  x)k , where 0  k  1 , Then the random operator equation A( , x)   x has a random solution in D . Proof. By the virtue of Lemma1.1, we only prove

A( , x)   x, for every ( , x)    D,     1 .

(2)

 0    1 and an (0 , x0 )    D such

In fact, suppose (2) is not true, that is there exists a that A(0 , x0 )   0 x0 . Inserting A(0 , x0 )   0 x0 into (Y1 ) , we obtain

p( 0 x0 ) k  q(  x0 ) k  ( p 0 x0  q x0 ) k , k  1 , that is, ( p 0  q  ) x0  ( p 0  q  ) x0 . k

k

k

k

k

Because X is the  -cone, by the definition2, we have

 (( p 0k  q  k ) x0k )   (( p 0  q )k x0k ) , that is

( p 0k  q k ) ( x0k )  ( p 0  q ) k  ( x0k )

(3)

This is because x0  D, x0   , and E is the Z-C-X Space, which hasn’t nilpotent element, hence x0   ,  ( x0 )  0 .By (3) , we have p 0  q   ( p 0  q  ) . n

n

k

k

k

320

CHENG LI-YING ET AL: ABOUT Z-C-X SPACE...

This is contradiction to(ⅰ)in Lemma2.1. Thus, we know that A( , x)   x, for every ( , x)    D,     1 . Then, by the virtue of Lemma1.1, the random operator equation A( , x)   x has a random solution in D . We can use the same method to prove the random operator equation A( , x)   x has a '

random solution in D , when the operator A satisfies to the boundary condition of (Y1 ) . Corollary1. Let E be the Z-C-X Space, X be a cone of E , and let D be a bounded open set in X , and

  D ,   1 , suppose that A：  D  X is a random semi-closed 1-set-contractive

operator, such that for every ( , x)    D ,which satisfies to one of the following conditions:

(Y2 ) 2k 1[( A( , x)) k  (  x) k ]  ( A( , x)   x) k , where k  1 , (Y2' ) 2k 1[( A( , x)) k  (  x) k ]  ( A( , x)   x) k , where 0  k  1 , Then the random operator equation A( , x)   x has a random solution in D . Proof. We only let p  q 

1 in the theorem2.3, and the conclusion is true. 2

k  (2t  1) n  (2t  1) n  (k   )nt , where 0  t  1 , k    0 , n  1 . 2 2 k  n n Proof. Let f (t )  (2t  1)  (2t  1)  (k   )nt , where 0  t  1 , k    0 , n  1 , 2 2 Lemma2.2.

'

then f (t )  kn[(2t  1)

n 1

 1]   n[1  (2t  1)n 1 ] .

When 0  t  1 , n  1 , we have (2t  1)

n 1

 1  0 , 1  (2t  1) n 1  0 ,thus f ' (t )  0 .

Therefore f (t ) is a strictly monotone increasing function in (0,1] . That is, when t  (0,1] , we have

k  k  (2t  1) n  (2t  1)n  (k   )nt  f (0)   (1) n  0 . 2 2 2 2 k  n n That is (2t  1)  (2t  1)  (k   )nt . 2 2 Theorem2.2. Let E be the Z-C-X Space, X be a cone of E , and let D be a bounded open set in X , f (t ) 

and

  D ,   1 , suppose that A：  D  X is a random semi-closed 1-set-contractive

operator, such that

CHENG LI-YING ET AL: ABOUT Z-C-X SPACE...

(Y3 )

321

k  (2  x  A( , x)) n  (2  x  A( , x)) n  (k   )n x( A( , x)) n 1 , 2 2 for every ( , x)    D , n  1 , k    0 ,

Then the random operator equation A( , x)   x has a random solution in D . Proof. By the virtue of Lemma1.1, we only prove

A( , x)   x, for every ( , x)    D,     1 . In fact, suppose (4) is not true, that is there exists a

(4)

 0    1 and an (0 , x0 )    D such

that A(0 , x0 )   0 x0 . Inserting A(0 , x0 )   0 x0 into (Y1 ) , we obtain

k  (2  x0   0 x0 ) n  (2 x0   0 x0 ) n  (k   )n x0 ( 0 x0 ) n 1 , n  1 , k    0 , 2 2 k  n n n n 1 n That is, [ (2    0 )  (2    0 ) ]x0  ( k   ) n 0 x0 2 2 Because X is the  -cone, by the definition2, we have

k   ([ (2   0 )n  (2   0 ) n ]x0n )   ((k   )n 0n 1 x0n ) ,

2 2 k  n n n n 1 n That is, [ (2    0 )  (2    0 ) ] ( x0 )  (k   ) n 0  ( x0 ) 2 2

(5)

This is because x0  D, x0   , and E is the Z-C-X Space, which hasn’t nilpotent element, hence x0   ,  ( x0 )  0 .By (5) , we have n

n

k  (2   0 ) n  (2   0 ) n  (k   )n 0n 1 . 2 2

By dividing  0 ( 0) on the both sides of the inequality, we have n

k     (2  1) n  (2  1) n  (k   )n 2 0 2 0 0 Let

(6)

  t , by  0    1 , we obtain 0  t  1 . 0

Hence (6) is that

k  (2t  1) n  (2t  1) n  (k   )nt . 2 2

This is contradiction to Lemma2.2.

Thus, we know that A( , x)   x, for every ( , x)    D,     1 . Then, by the virtue of Lemma1.1, the random operator equation A( , x)   x has a random

322

CHENG LI-YING ET AL: ABOUT Z-C-X SPACE...

solution in D .

1 2

Lemma2.3. k (t  1)   (t  1)  ( k   ) nt , where   [0, ), k    1, 0  t  1 , n  1 . n

n

1 2

Proof. Let f (t )  k (t  1)   (t  1)  (k   ) nt ,   [0, ), k    1, 0  t  1 , n  1 , n

'

then f (t )  kn(t  1)

n 1

n

  n(t  1) n 1  (k   )n  kn[(t  1) n 1  1]   n[1  (t  1) n 1 ] ,

when 0  t  1 , n  1 ,we have (t  1)

n 1

 1  0,1  (t  1) n 1  0 ,then f ' (t )  0 .

Therefore f (t ) is a strictly monotone increasing function in (0,1] . That is, when t  (0,1] , we have

f (t )  k (t  1) n   (1  t ) n  (k   )nt  f (0)  k   (1) n  0 . That is k (t  1)   (1  t )  ( k   ) nt . n

n

Theorem2.3. Let E be the Z-C-X Space, X be a cone of E , and let D be a bounded open set in X , and

  D ,   1 , suppose that A：  D  X is a random semi-closed 1-set-contractive

operator, if there is

1 2

  [0, ), k    1, n  1 such that

(Y4 ) k (  x  A( , x)) n   (  x  A( , x)) n  (k   )n x( A( , x))n 1 , for every ( , x)    D Then the random operator equation A( , x)   x has a random solution in D . Proof. By the virtue of Lemma1.1, we only prove

A( , x)   x, for every ( , x)    D,     1 . In fact, suppose (7) is not true, that is there exists a

(7)

 0    1 and an (0 , x0 )    D such

that A(0 , x0 )   0 x0 . Inserting A(0 , x0 )   0 x0 into (Y4 ) , we obtain

k (  x0   0 x0 ) n   (  x0   0 x0 ) n  (k   )n x0 ( 0 x0 ) n 1 , n  1 , That is, [k (    0 )   (    0 ) ] x0  ( k   ) n ( 0 ) n

n

n

n 1

x0n ,

Because X is the  -cone, by the definition2, we have

 ([k (    0 ) n   (    0 ) n ]x0n )   ((k   )n ( 0 ) n 1 x0n ),

CHENG LI-YING ET AL: ABOUT Z-C-X SPACE...

That is, [k (    0 )   (    0 ) ] ( x0 )  ( k   )n ( 0 ) n

n

n

n 1

323

 ( x0n )

(8)

This is because x0  D, x0   , and E is the Z-C-X Space, which hasn’t nilpotent element, hence x0   ,  ( x0 )  0 .By (8) , we have k (    0 )   (    0 )  (k   ) n ( 0 ) n

n

n

n

n 1

.

By dividing  0 ( 0) on the both sides of the inequality, we have n

[k (

    1) n   (  1) n ]  (k   )n 0 0 0

Let

  t , by  0    1 , we obtain 0  t  1 . 0

(9)

Hence (9) is that k (t  1)   (t  1)  ( k   ) nt . n

n

This is contradiction to Lemma2.3. Thus, we know that A( , x)   x, for every ( , x)    D,     1 . Then, by the virtue of Lemma1.1, the random operator equation A( , x)   x has a random solution in D . Remark: when E only is a separable real Banach space, the above theorems can be overwritten to the following theorems, correspondingly. Theorem2.4. Let E be the Z-C-X Space, X be a cone of E , and let D be a bounded open set in X , and

  D ,   1 , suppose that A：  D  X is a random semi-closed 1-set-contractive

operator, such that for every ( , x)    D , p, q  0, p  q  1 , which satisfies to one of the following conditions:

(Y5 ) p A( , x)  q  x  pA( , x)  q x , where k  1 , k

k

k

(Y5' ) p A( , x)  q  x  pA( , x)  q  x , where 0  k  1 , k

k

k

Then the random operator equation A( , x)   x has a random solution in D . Theorem2.5. Let E be the Z-C-X Space, X be a cone of E , and let D be a bounded open set in X , and

  D ,   1 , suppose that A：  D  X is a random semi-closed 1-set-contractive

operator, such that

(Y6 )

k  n n 2 x  A( , x)  2 x  A( , x)  (k   )n  x  A( , x) 2 2

n 1

,

for every ( , x)    D , n  1 , k    0 ,

324

CHENG LI-YING ET AL: ABOUT Z-C-X SPACE...

Then the random operator equation A( , x)   x has a random solution in D . Theorem2.6. Let E be the Z-C-X Space, X be a cone of E , and let D be a bounded open set in X , and

  D ,   1 , suppose that A：  D  X is a random semi-closed 1-set-contractive

operator, if there is

1 2

  [0, ), k    1, n  1 such that

(Y7 ) k  x  A( , x)    x  A( , x)  (k   )n  x A( , x) n

n

n1

for every ( , x)    D Then the random operator equation A( , x)   x has a random solution in D . References [1] Chuan-xi Zhu. Some Problems in the Z_C_X Space. Applied Mathematics and Mechanics,2002,23(8): 837-842.(in Chinese) [2] Chuanxi Zhu. Existence of random solution for a class of random operator equations. Acta Mathematica Scientia, 2003, 23A(3): 276-279. (in Chinese) [3] Guozhen Li. On Random fixed point index and some random fixed point theorems of random 1-set-contrac -tive operator. Acta Math. Appl. Sin, 1996,19 (2): 203-212. [4] Chuan-xi Zhu. Generalizations of Krasnoselskii’s Theorem and Petryshyn’s Theorem. Appl. Math. Lett, 2006,19 (7): 628-632 [5] Chuan-xi Zhu, Chunfang Chen. Calculations of random fixed point index. J. Math. Anal. Appl, 2008, 339: 839-844. [6] Chuan-xi Zhu, Zong-ben Xu. Inequalities and solution of an operator equation. Appl. Math. Lett, 2008,21: 607-611. [7] Naseer Shahzad. Random fixed points of set-valued maps, Nonlinear Anal, 2001, 45: 689-692 [8] H.Aumann. On the number of solutions of nonlinear equations in ordered Banach space. J.Funct.Anal, 1972,11(2): 346-384. [9] Lishan Liu. Some random approximations and random fixed point theorems for 1-set-contractive random operators. Proc. Amer. Math. Soc, 1997,125(2): 512-521.

JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.5, NO.3, 325-329, COPYRIGHT 2010 EUDOXUS PRESS, LLC325

Topological Sequence Entropy of Operators on Function Spaces∗ Gengrong Zhang

1,2†

Fanping Zeng3 , and Kesong Yan3

1. Institute of Mathematics, Shantou University, Shantou, 515063, China 2. Institute of Mathematics, Guangxi University, Nanning, 530004, China 3. Department of Mathematics, Liuzhou Teachers College , Liuzhou, 545004, China

Abstract We study sequence entropy of actions on function spaces with the focus on Markov operators on Compact spaces. We defined the natural definition of topological sequence entropy for Markov operators on C(X). Firstly, we prove that the three are equal. Secondly, It is proved that hA (T ) = hA (S) If T f = f ◦ S is an operate generated by a continuous map: S : X → X and A is an increasing integer sequence. Finally, It is proved that If for every continuous f there exists an invariant function ϕf such that lim sup |T n f (x) − ϕf (x)| = 0 then hA (T ) = 0 for every increasing integer sequence A. n→∞ x∈X

Keyword: k-Warsaw circle; Pointwise recurrent; Equicontinuity; AMS(2000) Subject Classification: 58F10, 54H20

1

Introduction Let C1 (X) denote the set of all continuous functions f : X → [0, 1]. In this paper, X is a compact

Hausdorff space and T denotes a Markov operators actin on C(X). For a continuous f let us define ε Uf = {(x, t) ∈ X × [0, 1] : t > f (x) + ε}. ε ε Ufε = Uf .

Given a finite collection F ⊂ C1 (X) we obtain a finite open cover of X × [0, 1] by the formula W ε UFε = Uf f ∈F

If V is a finite open cover of the unit interval then we let W −1 F −1 V = f (V). f ∈F ∗ Project

supported by NSFCs(10661001, 10861002, 60864002), NSF of Guangxi (0649002), NSF of Guangxi Education

Department(200807MS001), NSF of Guangdong(7301276) † Email: [email protected](G.R. Zhang)

1

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ZHANG ET AL: ...ENTROPY OF OPERATORS...

If T f = f ◦ S, where S : X → X is a continuous transformation and A = {0 = a0 < a1 < a2 < · · ·} is n−1 F ak n a integer sequence, we denote FA = T f. k=0

It is easy to prove the following Lemma 1, in which (i) and (ii) can be found in [2]. Lemma 1 Let F,G be finite subsets of C1 (X), V a finite open cover of the unit interval and ε a positive number. Then W (i) UFε ∪G = UFε UGε W (ii) (F ∪ G)−1 (V) = F −1 (V) G)−1 (V); (iii) if T f = f ◦ S, where S : X → X is a continuous n−1 n−1 W W −ai −1 n −1 transformation then UFε n = (S × Id)−ai (UFε ) and (FA ) (V) = S (F (V)). A

i=0

i=0

Recall that, for any open cover U, the symbol N (U) denotes the minimal cardinality of a subcover chosen from U. Definition 1 Let F ⊂ C1 (X) be a finite collection of functions, A = {0 = a0 < a1 < a2 < · · ·} be a integer sequence and ε > 0. We define (i) H1 (F, ε) = log N (UFε ), n , ε), (ii) h1 (T, F, A, ε) = lim sup n1 H1 (FA n→∞

(iii) h1 (T, A) = sup sup h1 (T, F, A, ε). F

ε

Definition 2 Let F ⊂ C1 (X) be a finite collection of functions, A = {0 = a0 < a1 < a2 < · · ·} be a integer sequence and V be a cover of interval [0, 1]. We define (i) H2 (F, V) = log N (F −1 V), n , V), (ii) h2 (T, F, A, V) = lim sup n1 H2 (FA n→∞

(iii)h2 (T, A) = sup sup h2 (T, F, A, V). F

V

Let dF and s(dF , ε) be the same as in [Tomasz] Definition 3 Let F ⊂ C1 (X) be a finite collection of functions, A = {0 = a0 < a1 < a2 < · · ·} be a integer sequence and ε > 0. We define (i) H3 (F, ε) = log s(dF , ε), n (ii) h3 (T, F, A, ε) = lim sup n1 H3 (FA , ε), n→∞

(iii) h3 (T, A) = sup sup h3 (T, F, A, ε). F

ε

In [2], the authors proved the following Theorems A and B: Theorem A For every Markov operate T holds h1 (T ) = h2 (T ) = h3 (T ). Theorem B If T f = f ◦ S is an operate generated by a continuous map: S : X → X, then h1 (T ) is equal to the classic topological entropy of S. In this paper, we defined the topological sequence entropy of a Markov operator T and proved the following Theorems 1, 2 and 3: Theorem 1 For every Markov operate T and integer sequence A = {0 = a0 < a1 < a2 < · · ·} hold h1 (T, A) = h2 (T, A) = h3 (T, A). 2

ZHANG ET AL: ...ENTROPY OF OPERATORS...

Theorem 2 If

327

T f = f ◦ S is an operate generated by a continuous map: S : X → X and

A = {0 = a0 < a1 < a2 < · · ·} is a integer sequence , then hA (T ) is equal to the classic topological entropy hA (S) of S. Theorem 3. If for every continuous f there exists an invariant function ϕf such that lim sup |T n f (x)− n→∞ x∈X

ϕf (x)| = 0 then hA (T ) = 0 for every integer sequence A = {0 = a0 < a1 < a2 < · · ·}.

2

Proofs of Theorems 1, 2 and 3

Theorem 1 For every Markov operate T and integer sequence A = {0 = a0 < a1 < a2 < · · ·} hold h1 (T, A) = h2 (T, A) = h3 (T, A). Proof. Firstly, we prove that h1 (T, A) ≤ h2 (T, A). Choose ε > 0 and let V be a finite open cover of \

interval [0, 1] of sets with diameters not greater than ε. Let WnA = {U × V : U ∈ (FA )−1 (V, v ∈ V). 0 n For each U × V ∈ WnA , let FA = {f ∈ FA : f (x) ≥ infV for each x ∈ U }. It is not difficult T T ε ε to see that U × V ⊂ Uf ∈ UFε n . It follows that WnA is in scribed in UFε n . 0 f ∈FA

A

A

n −F 0 f ∈FA A

\

Thus, N (UFε n ) ≤ N (WnA )leqN ((FA )−1 (V) · N (V). Since N (V) is independent of n. It follows that A

h1 (T, F, A, ε) ≤ h2 (T, F, A, V). Thus, h1 (T, A) ≤ h2 (T, A). Secondly, we will prove that h2 (T, A) ≤ h3 (T, A). et V be a finite open cover of interval [0, 1]. Denote its Lebesgue number by δ and let E be a maximal (dFAn , ε)-separated set in X. It follows from the maximality of E that the collection {B(x, 2δ ) : x ∈ E} of balls constitutes a finite open cover of n and x ∈ E, the interval (f (x) − 2δ , f (x) + 2δ ) is contained in some element X. for every f ∈ FA S −1 S −1 n −1 V (f, x) of V. Hence B(x, 2δ ) = ) (V) and f (f (x) − 2δ , f (x) + 2δ ) ⊂ f (V (f, x)) ∈ (FA n f ∈FA

n −1 ) (V)) N ((FA

≤ N {B(x,

δ 2)

n f ∈FA

: x ∈ E = N (E) = s(d

n FA

,

δ 2 ).

It follows that h2 (T, A) ≤ h3 (T, A).

Now we will show that h3 (T, A) ≤ h1 (T, A). Let D ⊂ X be a (dF , ε)-separated set of maximal cardinality. Put γ =

ε 6

and define Fe = { 12 f + iγ :

Then, by the proof of Theorem 4.2 in [2], we have s(dF , ε) ≤ N (UFγ ). Recall fn = Fen . So we can that T, as a Markov operator, is linear and preserves constants. This implies that F A A

f ∈ F, i ∈ Z, 0 ≤ i ≤

1 2γ }.

n replace F with FA and obtain s(dF \ , ε) ≤ N (U γ \ ). h3 (T, A) ≤ h1 (T, A) holds by taking upper limits A

FA

and superma. The proof of Theorem 1 is completed. In the following we will use the symbol hA (T ) to denote the common value of h1 (T, A), h2 (T, A) and h3 (T, A). Theorem 2 If T f = f ◦ S is an operate generated by a continuous map: S : X → X and A = {0 = a0 < a1 < a2 < · · ·} is a integer sequence , then hA (T ) is equal to the classic topological entropy hA (S) of S. n −1 Proof. Firstly, by Lemma 1(iii), we have (FA ) (V) =

3

n−1 W i=0

S −ai (F −1 (V)). Denote BV = F −1 (V).

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ZHANG ET AL: ...ENTROPY OF OPERATORS...

n −1 Then, (FA ) (V) =

n−1 W

S −ai (B). Therefore,

i=0

n n −1 h2 (T, F, A, V) = lim sup n1 H2 (FA , V) = lim sup n1 log N ((FA ) V) = lim sup n1 log N (BV ). n→∞

n→∞

n→∞

It follows that h2 (T, A) = sup sup n1 h2 (T, F, A, V) = sup sup lim sup n1 log N (BV ) = sup lim sup n1 log N (BV ) ≤ hA (S). F

V

F

V

n→∞

V

n→∞

Let W be a finite open cover of X. Choose a minimal subcover {W1 , W2 , · · · , Wr } ⊂ W. Using the Tietze−Urysohn Theorem(see [4], Theorem 2.1.8), we can construct F = {f1 , f2 , · · · , fr } consisting of T (Wjc ), i.e. x is continuous functions on X such that if x ∈ Wic then fi (x) = 0. Consequently, if x ∈ j6=i 1 covered exclusively by Wi , then fi (x) = 1(i = 1, ..., r). Fix 0 < ε < 2r . Every member of UFε n is of the A n−1 T T T ε 0 Ug , where Ug ∈ Ug . We will prove that each subcover U chosen from UFε n Ug = form n g∈FA

A

k=0 g∈T ak F

n n determines a subcover of WA of the same or smaller cardinality, where WA =

that an element of U 0 satisfies the following condition

n−1 W

S −ai (W). Suppose

i=0

ε For each k < n, there exists gk ∈ T ak F such that Ugk = Ug for any k = 0, 1, · · · , n − 1, since one cam always find gk ∈ T F such that gk (x) ≥ 1r . Thus,

g∈T ak F

1 if an element of U 0 contains (x, 2r ), it satisfies the condition (1) and determines some set

This set contains x, because gk (x) ≥ n−1 T

1 r

n−1 T

fk ∈ W n . W A

k=0

for each k = 0, 1, · · · , n − 1. Since x is arbitrary, the sets of form

fk constitute a cover of X and hence N (W n ) ≤ N ((U )ε n ). It follows that W F A A

k=0

hA (S) = sup lim sup n→∞

W

1 n

n log N (WA )

≤ sup lim sup n1 log N ((U )εF n ) = sup h1 (T, F, A, ε) W

n→∞

A

W

≤ sup sup h1 (T, F, A, ε) F

ε

= h1 (T, A) = hA (T ). The proof of Theorem 2 is completed. Corollary 1. If T f = f ◦ S is an operate generated by a continuous map: S : X → X, where X = [0, 1], then S is chaotic in the sense of Li-York if and only if there is some integer sequence A = {0 = a0 < a1 < a2 < · · ·} such that hA (T ) > 0. Theorem 3. If for every continuous f there exists an invariant function ϕf such that lim sup |T n f (x)− n→∞ x∈X

ϕf (x)| = 0 then hA (T ) = 0 for every integer sequence A = {0 = a0 < a1 < a2 < · · ·}. Proof. The proof of Theorem 3 is analogous to the proof of Theorem 4.7 in [2], so we omitted it. It is clear that Theorem 3 is a promotion of Theorem 4.7 in [2].

4

ZHANG ET AL: ...ENTROPY OF OPERATORS...

329

References [1] Block,L.and Coppel,W.A.

Dynamics in One Dimension. Lecture Notes in Math.,1513, Spinger,

Berlin,1991. [2] D., Tomasz; F., Bartosz Measure-theoretic and topological entropy of operators on function spaces. Ergodic Theory Dynam. Systems 25 (2005), no. 2, 455–481. [3] F. Blanchard, B. Host, and A. Maass, Topological complexity, Ergodic Theory Dynamical Systems, 20 (2000), 641-662. [4] R. Engelking, General Topology, PWN,WarSaw, 1977. [5] D. Kwietniak, P. Oprocha,. Topological entropy and chaos for maps induced on hyperspaces Chaos Solitons Fractals 33 (2007), no 1, 76-86. [6] J. Canovas,. Topological sequence entropy and topological dynamics of interval maps. Dynamics of Cont. Discrt. Impuls. Sys.-Series(A), 14 (2007) no 1, 47-54.

5

330

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334

TABLE OF CONTENTS, JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOLUME 5, NO.3, 2010

Boundedness and stability of solutions to a kind of nonlinear third order differential equations, Cemil Tunç, Muzaffer Ates,,……………………………………………………………………242 Essential norms of weighted composition operators between Hardy spaces in the unit ball, Zhong-Shan Fang, Ze-Hua Zhou,……………………………………………………………….251 The abstract wavelet transform, Jaime Navarro,………………………………………………..266 On the Green Function of the Operator Related to the Bessel Helmholtz Operator and the Bessel Klein-Gordon Operator, E. Suntonsinsoungvon, A. Kananthai,…………………......................290 Nonlinear singular integral equations with shift in the generalized Hölder spaces, M.H.Saleh, S.M.Amer, D.Sh.Mohammed,…………………………………………………………………..302 Some problems of random operator equations in the Z-C-X Space, Cheng Li-ying, Zhu Chuanxi, Tang Chao,…………………………………………………………………………………..317 Topological Sequence Entropy of Operators on Function Spaces, Gengrong Zhang, Fanping Zeng, Kesong Yan,……………………………………………………………………………...325

335

Volume 5,Number 4

October 2010

ISSN:1559-1948 (PRINT), 1559-1956 (ONLINE) EUDOXUS PRESS,LLC

JOURNAL OF APPLIED FUNCTIONAL ANALYSIS

336

SCOPE AND PRICES OF

JOURNAL OF APPLIED FUNCTIONAL ANALYSIS A quartely international publication of EUDOXUS PRESS,LLC ISSN:1559-1948(PRINT),1559-1956(ONLINE) Editor in Chief: George Anastassiou Department of Mathematical Sciences The University of Memphis Memphis, TN 38152,USA E mail: [email protected]

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The purpose of the "Journal of Applied Functional Analysis"(JAFA) is to publish high quality original research articles, survey articles and book reviews from all subareas of Applied Functional Analysis in the broadest form plus from its applications and its connections to other topics of Mathematical Sciences. A sample list of connected mathematical areas with this publication includes but is not restricted to: Approximation Theory, Inequalities, Probability in Analysis, Wavelet Theory, Neural Networks, Fractional Analysis, Applied Functional Analysis and Applications, Signal Theory, Computational Real and Complex Analysis and Measure Theory, Sampling Theory, Semigroups of Operators, Positive Operators, ODEs, PDEs, Difference Equations, Rearrangements, Numerical Functional Analysis, Integral equations, Optimization Theory of all kinds, Operator Theory, Control Theory, Banach Spaces, Evolution Equations, Information Theory, Numerical Analysis, Stochastics, Applied Fourier Analysis, Matrix Theory, Mathematical Physics, Mathematical Geophysics, Fluid Dynamics, Quantum Theory. Interpolation in all forms, Computer Aided Geometric Design, Algorithms, Fuzzyness, Learning Theory, Splines, Mathematical Biology, Nonlinear Functional Analysis, Variational Inequalities, Nonlinear Ergodic Theory, Functional Equations, Function Spaces, Harmonic Analysis, Extrapolation Theory, Fourier Analysis, Inverse Problems, Operator Equations, Image Processing, Nonlinear Operators, Stochastic Processes, Mathematical Finance and Economics, Special Functions, Quadrature, Orthogonal Polynomials, Asymptotics, Symbolic and Umbral Calculus, Integral and Discrete Transforms, Chaos and Bifurcation, Nonlinear Dynamics, Solid Mechanics, Functional Calculus, Chebyshev Systems. Also are included combinations of the above topics. Working with Applied Functional Analysis Methods has become a main trend in recent years, so we can understand better and deeper and solve important problems of our real and scientific world. JAFA is a peer-reviewed International Quarterly Journal published by Eudoxus Press,LLC. We are calling for high quality papers for possible publication. The contributor should send four copies of the contribution to the EDITOR in CHIEF in TEX,LATEX double spaced,in ten point type size. They should be sent BY REGULAR MAIL ONLY, NOT REGISTERED MAIL, AND NO E-MAIL SUBMISSIONS [ See: Instructions to Contributors] Journal of Applied Functional Analysis(JAFA)

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Journal of Applied Functional Analysis Editorial Board Associate Editors Editor in-Chief: George A.Anastassiou Department of Mathematical Sciences The University of Memphis Memphis,TN 38152,USA 901-678-3144 office 901-678-2482 secretary 901-751-3553 home 901-678-2480 Fax [email protected] Approximation Theory,Inequalities,Probability, Wavelet,Neural Networks,Fractional Calculus

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344 JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.5, NO.4, 344-350, COPYRIGHT 2010 EUDOXUS PRESS, LLC

Numerical simulation of the generalized Huxley equation by homotopy analysis method K.M. Hemida and M.S. Mohamed

1

Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt Abstract: In this paper we present the homotopy analysis method (HAM) for obtaining the numerical solution of the generalized Huxley equation. Convergence of the solution and e¤ects for the method are discussed. Comparisons are made among Adomian’s decomposition method (ADM), homotopy perturbation method (HPM), variational iteration method (VIM), the exact solution and the homotopy analysis method. The results reveal that the proposed method is very e¤ective, simple and also suggest that both the HPM and ADM are special case of the HAM. Key words: Nonlinear PDE; Huxley equation; Homotopy analysis method; Homotopy perturbation method; Adomian decomposition method. 1. Introduction Nonlinear partial di¤erential equations (NPDEs) are encountered in such various …elds as physics, chemistry, biology, mathematics and engineering,. Most nonlinear models of real life problems are still very di¢ cult to solve, either numerically or theoretically. The generalized Huxley equation ut

uxx = u(1

u )(u

);

0

x

1; t

0;

(1)

with the initial condition of u (x; 0) = [ + tanh ( 2 2

x)]

1

(2)

describes nerve pulse propagation in nerve …bres and wall motion in liquid crystals. The exact solution of this equation was derived by Wang et al. [1];using nonlinear transformations and is given by

where

=

4(1+ )

u (x; t) = [ + tanh 2 2 p and = 4 (1 + ):

(x +

(1 + ) 2 (1 + )

1

t )] ;

(3)

Many researchers have used various numerical methods to solve equation (1) numerically. Hashim et al. investigated the generalized Huxley equation, using Adomian decomposition method (ADM) [2] and Wazwaz studied the generalized forms of Burgers, Burgers-Kdv and Burgers-Huxley equations [3]. Hashem et al. studied the generalized Burger’s-Huxley equation [4], and Estevez investigated non-classical symmetries and the singular modi…ed Burger’s 1

E-mail: m s

[email protected]

1

HEMIDA-MOHAMED: SOLVING HUXLEY EQUATION...

345

and Burger’s-Huxley equation [5]. Hashemi et al. solved the generalized Huxley equation by He’s homotopy perturbation method (HPM) [6]. Batiha et al. solved the generalized Huxley equation by variational iteration method (VIM) [7]. In this paper, we solve the generalized Huxley equation by Homotopy analysis method (HAM) [8-14]. The results are compared with the available exact solutions and those were obtained by the (ADM) [2], homotopy perturbation method (HPM) [6] and variational iteration method (VIM) [7]. 2. The homotopy analysis method (HAM) We apply the HAM [8-14] to Huxley equation with initial conditions. We consider the following di¤erential equation (4)

N [u(x; t)] = 0;

where N is a nonlinear operator for this problem, x and t denote independent variables, u(x; t) is an unknown function. By means of the HAM, one …rst construct zero-order deformation equation (1

q)L( (x; t; q)

u0 (x; t)) = qhH(t)N [ (x; t; q)];

(5)

where q 2 [0; 1] is the embedding parameter, h 6= 0 is an auxiliary parameter, H(t) 6= 0 is an auxiliary function, L is an auxiliary linear operator, u0 (x; t) is an initial guess. Obviously, when q = 0 and q = 1, it holds

(x; t; 0) = u0 (x; t);

(x; t; 1) = u(x; t):

(6)

Liao [8-14] expanded (x; t; q) in Taylor series with respect to the embedding parameter q, as follows: (x; t; q) = u0 (x; t) +

1 X

um (x; t)q m ;

(7)

m=1

where

1 @ m (x; t; q) um (x; t) = jq=0 m! @ mq

(8)

Assume that the auxiliary linear operator, the initial guess, the auxiliary parameter h and the auxiliary function H(t) are selected such that the series (7) is convergent at q = 1, then we have from (7) u(x; t) = u0 (x; t) +

1 X

um (x; t):

(9)

m=1

Let us de…ne the vector

u! n (t) = fu0 (x; t); u1 (x; t); u2 (x; t); :::::; un (x; t)g:

2

(10)

346

HEMIDA-MOHAMED: SOLVING HUXLEY EQUATION...

Di¤erentiating (5) m times with respect to q, then setting q = 0 and dividing then by m!, we have the mth-order deformation equation {m um 1 (x; t)) = hH(t)Rm (u! m 1 );

$(um (x; t) where

Rm (u! m 1) =

1 (m

and {m =

(11)

@ m 1 N [ (x; t; q)] jq=0 ; 1)! @m 1q

(

0 m

(12)

1;

(13)

1 m > 1:

The mth-order deformation Eq. (11) is linear and thus can be easily solved, especially by means of symbolic computation software such as Mathematica, Maple, MathLab. 3. Analysis of the method by the HAM To solve Eq. (1) with an initial condition (2) by means of HAM, we choose the linear operator @ (x; t; q) ; @t with property L[c] = 0; where c is a constant. We de…ne a nonlinear operator as L[ (x; t; q)] =

N [ (x; t; q)] =

@ (x; t; q) @ 2 (x; t; q) @t @x2

(1+ )

+1

(x; t; q)+

2 +1

(x; t; q)+

(14)

(x; t; q): (15)

We construct the zeroth-order deformation equation (1

q)L( (x; t; q)

u0 (x; t)) = qhH(t)N [ (x; t; q)]:

For q = 0 and q = 1, we can write (x; t; 0) = u0 (x; t) = u(x; 0); (16)

(x; t; 1) = u(x; t): Thus, we obtain the mth-order deformation equations {m um 1 (x; t)) = hH(t)Rm (u! m 1 );

$(um (x; t) where Rm (u! m 1) =

@

m 1 (x; t; q)

@2

@t +

m 1 (x; t; q) @x2

+

2 +1 m 1 (x; t; q)

(1 + )

+1 m 1 (x; t; q)

(17)

m 1 (x; t; q):

In order to obey both the rule of solution expression and the rule of the coe¢ cient ergodicity [12], the auxiliary function can be determined uniquely H(t) = 1: 3

HEMIDA-MOHAMED: SOLVING HUXLEY EQUATION...

347

Now the solution of the mth-order deformation equations (17) for m

1 become

um (x; t) = {m um 1 (x; t) + h$ 1 Rm (u! m 1 ):

(18)

So, a few terms of series solution are as follows: 1

h

u1 (x; t) =

2

1

x)) (2(

((1 + tanh ( ) ( 1+2 ) 2

+2(

1

+2(

1

) (

2(

1

) ( 1+2 ) 2

2(

1

1+

)

1

) ( )

tanh2 (

(2(

(2(

1

1

) ( )

1

) ( )

) ( 1+2 ) 2

1

) (1)

) (1)

1

2(

2

x)) )2

) ( 1+2 ) 2

x)

) (

1+

) 1

((1 + tanh ( 1

((1 + tanh (

tanh (

2

x)) )

1

x) + 2(

) ( 1+2 ) 2 1

1

(2(

tanh2 (

1+

2(

x)

((1 + tanh (

+ 2( 1

1

(19)

x)] ;

u0 (x; t) = [ + tanh ( 2 2

x)) )

2 2

(20)

)t):

According to the HAM, we can conclude that (21)

u (x; t) = u0 (x; t) + u1 (x; t) + ::::::

Therefore, substituting the values of u0 (x; t) and u1 (x; t) from Eqs. (19), (20) into. Eq. (21) yields: h

u (x; t) =

When h =

2

1

) ( 1+2 ) 2

+2(

1

+2(

1

) (

2(

1

) ( 1+2 ) 2

2(

1

1+

1

) ( )

)

tanh2 (

(2(

(2(

1

x)) (2(

((1 + tanh (

1

1

) ( )

1

) ( )

1+

2(

x) 1

) (1)

(2(

1

tanh2 (

x)) ) ) (1)

x)) )2

((1 + tanh (

2

tanh ( 2(

1

2

) ( 1+2 ) 2

x)

) (

1+

) 1

x)) )

((1 + tanh ( 1

1

x) + 2(

) ( 1+2 ) 2 1

((1 + tanh (

+ 2( 1

) ( 1+2 ) 2

2 2

)t) + [ + tanh ( 2 2

(22) 1

x)] :

1; we obtain

u (x; t) =

1

2 ((1

1

x)) (2(

+ tanh ( ) ( 1+2 ) 2

+2(

1

+2(

1

) (

2(

1

) ( 1+2 ) 2

2(

1

1+

1

) ( )

)

tanh2 (

(2(

(2(

1

1

) ( )

+ 2( 1

1

) ( )

1

) ( 1+2 ) 2

1+

2(

x)

((1 + tanh ( 1

) (1)

(2(

((1 + tanh (

tanh2 (

1

x) + 2(

) ( 1+2 ) 2 1

x)) ) ) (1)

2

x)) )2

2

2(

1

x)

) (

1+

) 1

x)) )

2 2

)t) + [ + tanh ( 2 2

which the same as the solution obtained by [6]. Then we …nd at h = u (x; t)HAM = u (x; t)ADM = u (x; t)HPM : 4

) ( 1+2 ) 2

tanh (

((1 + tanh ( 1

1

1;

(23) 1

x)] ;

348

HEMIDA-MOHAMED: SOLVING HUXLEY EQUATION...

4. Numerical results and discussion We shall illustrate the accuracy and e¢ ciency of HAM applied to Eq. (1) compared to the ADM [2], HPM[6] and VIM [7]. For this purpose, we consider the same parameter values for the generalized Huxley equation (1) as considered speci…cally in [2], we take and

= 1,

= 0:001

= 1; 2; 3. We present in Tables 1–3, the values of exact solution, …ve-term approximate

of ADM , 1-iteration VIM and 2-term of HAM. Table 1 numerical solutions for x

t

Exact

= 1,

= 0:001 and h = 0:6

ADM [2]

=1

HPM [6]

VIM [7]

0.1 0.05 5.000302E-4 5.000052E-4 5.000052E-4 5.000052E 04

HAM 5.000100E-4

0.1

5.000427E-4 4.999927E-4 4.999927E-4 4.999927E 04

5.000030E-4

1

5.002676E-4 4.997678E-4 4.997678E-4 4.997678E 04

4.998680E-4

0.05 5.001009E-4 5.000759E-4 5.000759E-4 5.000759E 04

5.000810E-4

0.1

5.001134E-4 5.000634E-4 5.000634E-4 5.000634E 04

5.000730E-4

1

5.003383E-4 4.998385E-4 4.998385E-4 4.998385E 04

4.999380E-4

0.05 5.001716E-4 5.001466E-4 5.001466E-4 5.001466E 04

5.001520E-4

0.1

5.001841E-4 5.001341E-4 5.001341E-4 5.001341E 04

5.001440E-4

1 5.004090E-4 4.999092E-4 4.999092E-4 4.999092E 04 Table 2 numerical solutions for = 1, = 0:001 and = 2 h = 0:6

5.000090E-4

0.5

0.9

x

t

Exact

ADM [2]

HPM [6]

VIM [7]

0.1 0.05 2.236188E-2 2.236077E-2 2.236077E-2 2.236077E -2

0.5

2.236100E-2

0.1

2.236244E-2 2.236021E-2 2.236021E-2 2.236023E -2

2.236070E-2

1

2.237250E-2 2.235015E-2 2.235015E-2 2.235015E -2

2.223546E-2

0.05 2.236447E-2 2.236335E-2 2.236335E-2

2.236335E-2

2.236360E-2

0.1

2.236279E-2

2.236320E-2

2.237508E-2 2.235273E-2 2.235273E-2 2.235273E -2

2.235720E-2

0.05 2.236705E-2 2.236593E-2 2.236593E-2 2.236593E -2

2.236620E-2

0.1

2.236761E-2 2.236537E-2 2.236537E-2 2.236537E -2

2.236580E-2

1

2.237766E-2 2.235531E-2 2.235531E-2 2.235531E -2

2.235980E-2

1 0.9

HAM

2.236502E-2 2.236279E-2 2.236279E-2

5

HEMIDA-MOHAMED: SOLVING HUXLEY EQUATION...

Table 3 numerical solutions for x

t

Exact

= 1,

= 0:001 and h = 0:6

ADM [2]

=3

HPM [6]

0.1 0.05 7.937402E-2 7.937005E-2 7.937005E-2

0.5

VIM [7]

HAM

7.937005E-2

7.937080E-2

0.1

7.937600E-2 7.936807E-2 7.936807E-2 7.936807E -2

7.936970E-2

1

7.941169E-2 7.933234E-2 7.933234E-2 7.933236E -2

7.934820E-2

0.05 7.938196E-2 7.937799E-2 7.937799E-2

7.937799E-2

7.937880E-2

0.1

7.937601E-2

7.937760E-2

7.941962E-2 7.934029E-2 7.934029E-2 7.934031E -2

7.935620E-2

0.05

7.93899E-2

7.938989E-2 7.938989E-2 7.938592E -2

7.938670E-2

0.1

7.939187E-2 7.939187E-2 7.939187E-2 7.938394E -2

7.938550E-2

1 0.9

349

7.938394E-2 7.937601E-2 7.937601E-2

1 7.942754E-2 7.934823E-2 7.934823E-2 7.934825E -2 7.936410E-2 The results clearly show that HAM is more e¢ cient than the ADM, HPM, and VIM. The HAM avoids the needs for calculating the Adomian polynomials which can be di¢ cult in some cases. 5. Conclusion In this paper, we propose HAM to solve the generalized Huxley equation. The solution is also, given by ADM, HPM and VIM. We reveal the relationship between HAM and other methods, That is ADM and HPM are special case of HAM for this problem. Compared with ADM, HPM and VIM, this illustrative problem shows that HAM has the following advantages. The HAM contains a certain auxiliary parameter h which provides us with a simple way to adjust and control the convergence region and rate of convergence of the series solution.

References [1] X.Y.Wang, Z. S. Zhu, Y. K. Lu, Solitary wave solutions of the generalized Burgers-Huxley equation, Phys. Lett. A. 23(1990)271-274. [2] I. Hashim, M. S. M. Noorani, and B. Batiha, “A note on the Adomian decomposition method for the generalized Huxley equation,” Applied Mathematics and Computation, 181(2)(2006)1439–1445. [3] A.M. Wazwaz, Travelling wave solutions of generalized forms of Burgers-Kdv and BurgersHuxley equations, App. Math. Comput. 169(2005)639-656.

6

350

HEMIDA-MOHAMED: SOLVING HUXLEY EQUATION...

[4] I. Hashim, M.S.M. Noorani, M.R. said, Solving the generalized burgers-Huxley equation using the Adomian decomoposition method, Math. Comput. Modell. 43(2006)1404-1411. [5] P.G Estevez, Non-classical symmetries and the singular modi…ed Burger’s and Burger’sHuxley equation, Phys. Lett. A 27(1994)2113-2127. [6] S. H. Hashemi, H. R. Mohammadi Daniali, D. D. Ganji, Numerical simulation of the generalized Huxley equation by He’s homotopy perturbation method, Appl. Math. and Comp. 92(2007)157-161. [7] B. Batiha, M.S.M. Noorani, I. Hashim, Numerical simulation of the generalized Huxley equation by He’s variational iteration method , Appl. Math. and Comp.186(2007)13221325. [8] SJ Liao, The proposed homotopy analysis technique for the solution of nonlinear problem. Ph.D thesis, Shanghai Jiao Tong University;1992. [9] SJ Liao, An approximate solution technique which does not depend upon small parameters: a special example. Int. J. Nonlinear Mech. 30(1995)371-380. [10] SJ Liao, An approximate solution technique which does not depend upon small parameters (II): an application in ‡uid mechanics, Int. J. Nonlinear Mech. 32(1997)815-822. [11] SJ Liao, An explicit, totally analytic approximation of Blasius viscous ‡ow problems, Int. J. Nonlinear Mech. 34(4)(1999)759-778. [12] SJ Liao, Beyond perturbation: introduction to the homotopy analysis method,CRC Press, Boca Raton: Chapman& Hall, 2003. [13] SJ Liao, On the homotopy analysis method for nonlinear problems, Appl Math Comput.147(2004)499-513. [14] SJ Liao, Notes on the homotopy analysis method: Some de…ntions and theorems, Commun Nonlinear Sci Numer Simulat. 14(2009)983-997.

7

JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.5, NO.4, 351-356, COPYRIGHT 2010 EUDOXUS PRESS, LLC 351

BASIS PROPERTY IN Lp (0; 1) OF THE ROOT FUNCTIONS CORRESPONDING TO A BOUNDARY-VALUE PROBLEM H. MENKEN AND KH. R. MAMEDOV Abstract. The non-self adjoint Sturm-Liouville operators with periodic and anti-periodic boundary conditions are studied. The basis property in the space Lp (0; 1) (p > 1 ) of the root functions of these operators is proved. For the basisness in Lp (0; 1) the inequality in the F. Riesz’s theorem is used.

1. Introduction Let us consider di¤erential operators generated by the di¤erential equation (1.1)

`(y)

y 00 + q(x)y = y

and the periodic boundary conditions (1.2)

y 0 (0) = y 0 (1);

y(0) = y(1);

or the anti-periodic boundary conditions (1.3)

y(0) =

y 0 (0) =

y(1);

y 0 (1);

where the potential q(x) is an complex-valued function de…ned on [0; 1]. We note that the boundary conditions (1.2) and (1.3) in Birkho¤ classi…cation are regular, but not strongly regular (see [14], p.71). In this work we study the basis properties of the root functions for boundary value problem (1.1), (1.2) and (1.1), (1.3) which correspond to periodic and antiperiodic problems. This problem is important for the study of Sturm-Liouville operators with periodic complex potential on the whole real line, which is called the Hill operator, and it is of independent interest. A recent progress in the study of the Hill operator is presented in a paper by Djakov and Mityagin [2] and references therein. It is well known that the basisness of the root functions of linear di¤erential operators depends on regularity of boundary conditions generating the given differential operators. The basisness in the space L2 (0; 1) of the root functions of a linear di¤erential operator of order n with strongly regular boundary conditions was shown by Mikhailov (1962) [13], Kesel’man (1964) [7], and in the monagraph of Dunford and Schwartz [3]. Some examples of di¤erential operators with regular but not strongly regular boundary conditions whose root functions do not form a basis in L2 (0; 1) were given by Kesel’man (1964) [7] and Walker (1977) [17]. 2000 Mathematics Subject Classi…cation. 34L10, 34B24, 47E05. Key words and phrases. Sturm-Liouville operator, F. Riesz’s inequality, basisness of root functions, not strongly regular boundary conditions. 1

352

2

H. M ENKEN AND KH. R. M AM EDOV

Ionkin (1976) [4] studied one non-classical heat conduction problem in homogeneous rod, which is reduced to following boundary-value problem by method of partition y 00 + y = 0; y 0 (0) y 0 (1) = 0; y(0) = 0 whose boundary conditions are regular, but not strongly regular. All the eigenvalues of this problem starting with the second one are double, the general number of associated functions is in…nite. However, it was established that the chosen specially system of root functions forms an unconditional basis in L2 (0; 1). Shkalikov (1979, 1982) [15, 16] proved that the eigenfunctions and associated functions of an ordinary di¤erential operator of 2n order with not strongly regular boundary conditions do not form a Riesz basis, they form a basis with parentheses. Kerimov and Mamedov (1998) [6] proved that the root functions of the boundary value problems (1.1), (1.2) and (1.1), (1.3) form a Riesz basis in L2 (0; 1) when q(x) 2 C (4) [0; 1] is complex-valued functions satisfying the condition q(0) 6= q(1). Kurbanov (2006) [8] ( by using the results in [6]) obtained the basis property in the space Lp (0; 1) of the root functions of the boundary value problems(1.1), (1.2) and (1.1), (1.3). Dernek and Veliev (2005) [1] and Makin (2006) [9] obtained some results on the basis properties of the root functions in terms of the Fourier coe¢ cient of the potential q(x). Moreover, some theorems for determining whether the root functions form a Riesz basis in L2 (0; 1) or not were given in [9]. Makin (2006) [10] established a classi…cation on the boundary conditions for a second order di¤erential equation under which the root functions form a Riesz basis. Mamedov and Menken (2008) [11] proved that the root functions of the boundary problems (1.1), (1.2) and (1.1), (1.3) form a Riesz basis in L2 (0; 1) when q(x) 2 C (4) [0; 1] satisfying the conditions q(0) = q(1) and q 0 (0) 6= q 0 (1). In the present work under the same conditions we prove the basis property in the space Lp (0; 1) (1 < p < 1) of the root functions the boundary problems (1.1), (1.2) and (1.1), (1.3). The basis property in Lp (0; 1) of the root functions depends on the inequality at the following theorem in the book of Kashin and Saakyan (see [5], Section I). 1

Theorem 1. [5] A system 'j j=1 is a basis in the Banach space X if only if the following conditions are satis…ed: 1 j=1 1 'j j=1

a) 'j

is complete in X,

b) is minimal, c) there exists a number M > 0 such that for each f 2 X, the inequality N X

(f;

j=1

where the sequence

1 j j=1

j )'j

M kxk ; N = 1; 2;

is the biorthogonal adjoint system to 'j

1 j=1

:

We use also the inequality in F. Riesz theorem which was given in [18]. We need the following asymptotic formulas which were obtained in [11, 12]. Lemma 1. Let q(x) 2 C (4) [0; 1], q(0) = q(1) and q 0 (0) 6= q 0 (1). Then the following assertions hold:

353

BASIS PPROPERTY IN Lp (0; 1) OF THE ROOT FUNCTIONS

3

a) All eigenvalues of the boundary-value problem (1.1), (1.2), starting from some number, are simple and form two in…nite sequences k;1 ; k;2 ; k = N; N + 1; ; where N is a positive integer and R1

q 0 (1) q 0 (0)+ q 2 (t)dt k;1

(1.4)

=

(2k )2

0

(4k )2

R

+ O( k13 ) (k

N );

+ O( k13 ) (k

N );

1

q 0 (1) q 0 (0) k;2

=

2

(2k ) +

(4k

q 2 (t)dt

0 )2

and the corresponding eigenfunctions are of the form (1.5)

yk;1 (x)

(1.6)

yk;2 (x)

1 sin 2k x + O( ) (k k 1 = cos 2k x + O( ) (k k

=

N ); N );

b) All eigenvalues of the boundary value problem (1.1), (1.3), starting from some number, are simple and form two in…nite sequence k;1 ; k;2 ; k = N; N + 1; ; where N is a positive integer and

k;1

k;2

= =

q 0 (1) q 0 (0)

2

[(2k + 1) ] +

R1

q 2 (t)dt

0

[2(2k+1) ]2

R1

+ O( k13 ) (k

N );

+ O( k13 ) (k

N );

q 0 (1) q 0 (0)+ q 2 (t)dt

2

0

[(2k + 1) ]

[2(2k+1) ]2

and the corresponding eigenfunctions are of the form (1.7)

yk;1 (x)

=

(1.8)

yk;2 (x)

=

1 sin(2k + 1) x + O( ) (k k 1 cos(2k + 1) x + O( ) (k k

N ); N ):

Theorem 2. [11] Assume that the conditions of Lemma 1 are satis…ed. Then, the system of the root functions of the boundary problem (1.1), (1.2) forms a Riesz basis in L2 (0; 1). It can be easily showed that the system of the root functions of the boundary problem (1.1), (1.2) has a biorthogonal system consisting of the root functions of the adjoint operator ` (v) = v 00 + q(x)v; v(1) = v(0); v 0 (1) = v 0 (0); and the eigenfunctions of the adjoint operator have of the form (1.9)

k;1 (x)

1 = 2 sin 2k x + O( ); k

(1.10)

k;2 (x)

1 = 2 cos 2k x + O( ): k

354

4

H. M ENKEN AND KH. R. M AM EDOV

2. Main Results In the present work we obtain the following main results. Theorem 3. Let q(x) 2 C (4) [0; 1], q(0) = q(1) and q 0 (0) 6= q 0 (1). Then, the system of the root functions of the boundary problem (1.1), (1.2) forms a basis in the space Lp (0; 1) (1 < p < 1): Proof. Let 1 < p < 2 be …xed. According the Theorem 2, the system of the root 1 1 1 functions fyk gk=1 = fyk;1 (x); yk;2 (x)gk=1 is a basis in L2 (0; 1) and f n gn=1 = 1 1 f k;1 (x); k;2 (x)gk=1 is biorthogonal to the systemfyk gk=1 : Thus, this system is complete in Lp (0; 1): For basisness in Lp (0; 1) of the system it is su¢ cient to show to existence of a constant M > 0 such that N X

(2.1)

(f;

M kf kp ; (N = 1; 2;

n )yn

n=1

p

);

for all f 2 Lp (0; 1); where k kp denotes the norm in Lp (0; 1): (see Theorem 1). p p Let 1 (x) = 1, 2n+1 (x) = 2 sin 2n x; 2n (x) = 2 cos 2n x; (n = 1; 2; ): The asymptotic formulas 1 yn (x) = p 2

(2.2)

n (x)

1 + O( ); n

n (x)

=

p

2

n (x)

1 + O( ) n 1

1

are also valid for su¢ ciently large n where f n (x)gn=1 = 1 (x); 2n (x); 2n+1 (x) n=1 is a basis in the space Lp (0; 1): This follows immediately from (1.5), (1.6) and (1.9), (1.10). By (2.2) N X

(f;

N X

n )yn

n=1

(f;

n)

n

n=1

p

1 (f; O( )) n n=1

(f;

n )O(

n=1

p

N X

(2.3)

N X

+

+

n p

1 ) n

+ p

N X

1 1 (f; O( ))O( ) n n n=1

: p

We shall now prove that all the summands on the right side of (2.3) are bounded from above by constant kf kp : 1 Since f n gn=1 is a basis of Lp (0; 1); applying Theorem 1 , we have N X

(2.4)

(f;

n) n

n=1

C kf kp ; C = const (N = 1; 2;

p

)

for all f 2 Lp (0; 1): Thus, by the inequality in F. Riesz theorem (see [18], p.154, the inequality (2.10)) N X

n=1

(2.5)

(f;

n )O(

1 ) n

C

N X

(f;

n)

n=1

p

C

N X

n=1

j(f;

1 n q n )j

! q1

N X 1 np n=1

! p1

C kf kp ;

355

BASIS PPROPERTY IN Lp (0; 1) OF THE ROOT FUNCTIONS

where

1 p

+

1 q

= 1: Using the Parseval’s equality we have

N X

1 (f; O( )) n n=1 (2.6)

(2.7)

5

n p

N X

1 (f; O( )) n n=1 C kf k1

N X

1 1 (f; O( ))O( ) n n n=1

p

2

N X 1 2 n n=1

C kf k1

=

n

! 21

N X 1 n2 n=1

N X

n=1

1 (f; O( )) n

2

! 12

C kf kp ; ! 12

C kf kp :

Using the inequalities (2.4)-(2.6) in the estimate (2.3) we have (2.1), i.e. the basis1 ness of the system fyn (x)gn=1 in the space Lp (0; 1) at 1 < p < 2 is proved. 1 Now let 2 < p < 1: It is clear that the adjoint system f n (x)gn=1 is a basis of the space Lp (0; 1): Consequently, this system is complete in the space Lq (0; 1); where p1 + 1q = 1: Note that 1 < q < 2: By means of absolute analogous discussion used above the basisness in Lq (0; 1) 1 of the system f n (x)gn=1 is proved. Hence it follows the basisness in Lp (0; 1) 1 (2 < p < 1) of the system of fyn (x)gn=1 : Thus, the theorem is proved. Similarly, the following result is proved for the boundary problem (1.1), (1.3). Theorem 4. Let q(x) 2 C (4) [0; 1], q(0) = q(1) and q 0 (0) 6= q 0 (1). Then, the system of the root functions of the boundary problem (1.1), (1.3) forms a basis in the space Lp (0; 1) (1 < p < 1): Moreover, if p = 2 this system is a Riesz basis in L2 (0; 1): We note that the smoothness condition on q(x) follows from the asymptotic formulas which were obtained in [11, 12]. This smoothness condition may be reduced, but it does not play important role in the proof of the basis property. References [1] N. Dernek and O. A. Veliev, On the Riesz basisness of the root functions of the nonself-adjoint Sturm-Liouville operators, Israel Journal of Mathematics, 145, 113-123 (2005). [2] P. Djakov and B. S. Mitjagin, Instability Zones of Periodic 1-dimensional Schrodinger and Dirac Operators, Uspekhi Mat. Nauk, 61:4, 77-182 (2006). English Transl. in Russian Math. Surves, 61:4, 663-776 (2006). [3] N. Dunford and J. T. Schwartz, Linear Operators, Prt.3 Spectral Operators, Wiley, New York, 1971. [4] N. I. Ionkin, The solution of a boundary-value problem in heat conduction with a nonclassical boundary condition, Di¤ er. Equations, V.13, N. 2, 294-304 (1977). [5] B. S. Kashin and A. A. Saakyan, Orthogonal Series, American Mathematical Society, 1989. [6] N. B. Kerimov and Kh. R. Mamedov, On the Riesz basis property of the root functions in certain regular boundary value problems, Math. Notes, V. 64, N.4, 483-487 (1998). [7] G. M. Kesel’man, On the unconditional convergence of expansions in the eigenfunctions of some di¤erential operators, Izv. Vyssh. Uchebn. Zaved. Mat. [Soviet Math. (Iz. VUZ)], N.2, 82-93 (1964). [8] V. M. Kurbanov, A theorem on equivalent bases for a di¤erential operator, Dokl. Akad. Nauk, 406, N. 1, 17-20 (2006). [9] A. S. Makin, Convergence of expansions in the root functions of periodic boundary value problems, Doklady Math., V.73, N. 1, 71-76 (2006).

356

6

H. M ENKEN AND KH. R. M AM EDOV

[10] A. S. Makin, On spectral decompositions corresponding to non-self-adjoint Sturm-Lioville operators, Doklady Math., V.73, N. 1, 15-18 (2006). [11] Kh. R. Mamedov and H. Menken, On the basisness in L2 (0; 1) of the root functions in not strongly regular boundary value problems, European Journal of Pure and Applied Math., Vol. 1, No. 2, 51-60 (2008). [12] Kh. R. Mamedov and H. Menken, Asymptotic formulas for eigenvalues and eigenfunctions of a nonself-adjoint Sturm-Liouville operator, Further Progress in Analysis (H. G. W. Begehr, A. O. Çelebi and R. P. Gilbert, eds.), World Scienti…c Publishing, 2009, pp.798-806. [13] V. P. Mikhailov, On Riesz bases in L2 (0; 1), Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.], 144, N.5, 981-984 (1962). [14] M. A. Naimark, Linear Di¤ erential Operators, Part I, Frederick Ungar Pub. Co., New York, 1967. [15] A. A. Shkalikov, On the Riesz basis property of the root vectors of ordinary di¤erential operators, Russian Math. Surveys, Vol. 34, 5, 249-250 (1979). [16] A. A. Shkalikov, On the basis property of the eigenfunctions of ordinary di¤erential operators with integral boundary conditions, Vestnik Moscow University, Ser. Mat. Mekh., Vol 37, 6, 12-21 (1982). [17] P. W. Walker, A nonspectral Birkho¤-regular di¤erential operators, Proc. of American Math. Soc., V. 66, N.1, 187-188 (1977). [18] A. Zygmund, Trigonometric Series, Vol. 2, Cambridge Univ. Press, Cambridge, 1959; Russian transl., Mir, Moskow, 1965. (Hamza Menken) Mersin University, Science and Arts Faculty, Mathematics Department, 33343 Ciftlikkoy Campus, Mersin-Turkey E-mail address : [email protected] (Khanlar R. Mamedov) Mersin University, Science and Arts Faculty, Mathematics Department, 33343 Ciftlikkoy Campus, Mersin-Turkey E-mail address : [email protected]

JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.5, NO.4, 357-369, COPYRIGHT 2010 EUDOXUS PRESS, LLC357

Dimensions of bivariate C 1 cubic spline spaces over unconstricted triangulations with valence six Huan-Wen Liu and Na Yi Faculty of Mathematics and Computer Sciences, Guangxi University for Nationalities, Nanning, Guangxi 530006, P.R. China [email protected], [email protected] Abstract. In this paper, we consider the open problem of dimensions of bivariate C 1 cubic spline spaces. Firstly, by introducing two new operators, element-3 and element-4, in constructing a triangulation, a kind of unconstricted triangulations with valence 6 is defined, which is an extension of the unconstricted triangulation introduced by Farin in [Dimensions of spline spaces over unconstricted triangulations, J. Comput. Appl. Math., 192(2006), pp.320-327] and some well-known triangulations such as the Morgan-Scott triangulation and the Robbins triangulation are therefore included. Then, by using the technique of minimal determining set, the dimension of bivariate C 1 cubic spline spaces over the unconstricted triangulation with valence six is determined and the Lagrange interpolation on all vertices in the unconstricted triangulation with valence 6 is considered. Key words: bivariate C 1 cubic spline space, unconstricted triangulations with valence six, dimension, Lagrange interpolation AMS Subject Classifications. 65D07, 41A15, 41A63

1

Introduction Let ∆ be a regular triangulation of a simply connected polygonal domain Ω in R2 , i.e., ∆ is a set of

closed triangles whose union coincides with Ω such that the intersection of any two triangles in ∆ is either empty, a common edge or a vertex. Let V(∆), VI (∆), VB (∆), E(∆), E I (∆), E B (∆) and F(∆) denote the sets of vertices, interior vertices, boundary vertices, edges, interior edges, boundary edges and triangles in ∆, respectively, and we omit the triangulation and simply use the notations V, VI , VB , E, E I , E B and F if statements apply to any triangulation or there is otherwise no confusion. For two given integers n and r with 0 ≤ r ≤ n − 1, the space of bivariate splines of degree n and smoothness order r with respect to ∆ is defined by S nr (∆) = {s ∈ C r (Ω) : s|T ∈ Pn , ∀T ∈ F} ,

1

(1.1)

358

LIU-YI: BIVARIATE CUBIC SPLINE SPACES...

where Pn is the space of bivariate polynomials of total degree being at most n. The dimension problem of bivariate spline spaces was initiated with a conjecture by Strang [12]. The first result was given by Morgan and Scott [9] for bivariate spline space S n1 (∆) with n ≥ 5. Later, Schumaker [11] gave a lower bound formula for the dimension of the spaces S nr (∆). Alfeld and Schumaker [3] and Wang and Lu [13] independently proved that Schumaker’s lower bound is in fact the dimension of S nr (∆) for n ≥ 4r + 1. By carefully working with the smoothness conditions in terms of B-net representation of spline functions, Hong [8] determined the dimension for the space S nr (∆) when n ≥ 3r + 2. Alfeld, Piper and Schumaker [2] also extended Morgan and Scott’s results [9] to the space S 41 (∆). Alfeld and Schumaker [4] r determined the dimension of S 3r+1 (∆) for a nondegenerate triangulation.

However, when n ≤ 3r, the dimension of S nr (∆) is poorly understood as it may depend on the geometric shape of ∆, see [10]. Especially, the dimension of S 31 (∆) is still open and the following two conjectures provide extremely challenging research problems, see [1]. Conjecture 1.1. Let σ(∆) be the number of singular vertices in triangulation ∆, then dim S 31 (∆) = 2|VI | + 3|VB | + 1 + σ,

(1.2)

where the so-called singular vertex v is the intersection point of the two diagonals of a quadrilateral. Conjecture 1.2. Given a triangulation ∆ with its all vertices being vi (i = 1, ..., |V|) and numbers zi , i = 1, 2, ..., |V|, there exists a function s ∈ S 31 (∆) such that s(vi ) = zi , i = 1, ..., |V|.

(1.3)

Billera [5] and Whiteley [14] gave the generic dimension of S n1 (∆) for n ≥ 2, where the so-called generic dimension is such that if dim S n1 (∆) does not equal it then there is an arbitrary small perturbation in the location of the vertices that will cause dim S n1 (∆) to equal the generic value, also see [1]. According to [5], the generic dimension of S 31 (∆) is 2|VI | + 3|VB | + 1. This means that the related generic triangulation does not contain any singular vertex. Recently, Farin [7] introduced a flap-and-pair manner to construct a triangulation and determined the dimension of S 31 (∆) over a kind of special triangulations, called the unconstricted triangulations. In this paper, by defining two kinds of operators in triangulation construction, called element-3 and element-4, we shall first extend the unconstricted triangulations to a kind of new triangulations, called unconstricted triangulations with valence 6. The new triangulations include some well-known difficult cases such as the Morgan-Scott triangulation and the Robbins triangulation. Then by using the technique of minimal determining set, dimensions of bivariate C 1 cubic spline spaces over the unconstricted triangulations with valence 6 are given. At the end of this paper, the Conjecture 1.2 is also partially answered.

LIU-YI: BIVARIATE CUBIC SPLINE SPACES...

2

359

Preliminaries Let T := [u, v, w] be a triangle in ∆. As we know, every polynomial p ∈ P3 defined on T can be written

uniquely in the Bernstein-B´ezier (B-net) representation as follows X

p(x, y) = p(α, β, γ) =

cTijk

i+ j+k=3

3! i j k αβ γ , i! j!k!

(2.4)

where (α, β, γ) are the barycentric coordinates of (x, y) with respect to T and {cTijk }i+ j+k=3 are the BernsteinB´ezier (B-net) coefficients of p(α, β, γ). Clearly, each cTijk associates with the domain point ξiTjk

! i j k := , , , 3 3 3

thus the B-net coefficients of p on T can be indexed with the set n o DT := ξiTjk

i+ j+k=3

,

and the spline space S 30 (∆) is in one-to-one correspondence with the set of domain points D∆ :=

[

DT .

T ∈∆

Following [3], for any vertex v ∈ V, we define the m-th ring around v to be the set Rm (v) := {domain points which are distance m from v}.

(2.5)

A related concept is the m-th disk around v defined by Dm (v) :=

m [

Ri (v).

(2.6)

i=0

For each ξ ∈ D∆ , let λξ be the linear functional such that for any spline s ∈ S 30 (∆), λξ s = the B-net coefficient cξ of s associated with domain point ξ.

(2.7)

A subset M ⊆ D∆ is said to be a determining set for S 31 (∆) if for ∀s ∈ S 31 (∆), we have λξ s = 0 for all ξ ∈ M, implies s ≡ 0.

(2.8)

Furthermore, M is called a minimal determining set for S if there is no other determining set for S with the cardinality being smaller than |M|. Obviously, if M is a determining set for S , we then have dim S ≤ |M|, and furthermore, if M is a minimal determining set for S , then dim S = |M|, see [3]. Definition 2.1. Suppose e j−1 , e j , e j+1 are three consecutive edges attached to a vertex v, then edge e j is called to be degenerate at v if two edges e j−1 and e j+1 are collinear, otherwise e j is called to be nondegenerate at v.

360

LIU-YI: BIVARIATE CUBIC SPLINE SPACES...

The following result on C 1 smoothness of s(x, y) across two adjacent triangles is a special case of the general smoothness condition given in [6]. Lemma 2.1. Let T (1) = [v0 , v1 , v2 ], T (2) = [v0 , v1 , v3 ] be two adjacent triangles in ∆ sharing a common edge [v0 , v1 ]. Suppose s(x, y) agrees with p1 (x, y) ∈ P3 in T (1) and with p2 (x, y) ∈ P3 in T (2) . Assume that p1 (x, y) =

X i+ j+k=3

c(1) i jk

3! i j k αβγ, i! j!k! 1 1 1

X

p2 (x, y) =

c(2) i jk

i+ j+k=3

3! i j k αβγ, i! j!k! 2 2 2

(2.9)

where (α1 , β1 , γ1 ) and (α2 , β2 , γ2 ) are barycentric coordinates of (x, y) with respect to triangles T (1) and T (2) , S respectively. Then s(x, y) ∈ C 1 (T (1) T (2) ) if and only if the following smoothness conditions (1) c(2) i, j,0 = ci, j,0 , i + j = 3,

(2.10)

(1) (1) (1) c(2) i, j,1 = αci+1, j,0 + βci, j+1,0 + γci, j,1 , i + j = 2,

(2.11)

hold, where (α, β, γ) are the barycentric coordinates of v3 with respect to triangle T (1) . Corollary 2.1. Under the condition of Lemma 2.1, if the common edge [v0 , v1 ] is degenerate at v0 , then the above Eq.(2.11) degenerates into (1) (1) c(2) i, j,1 = αci+1, j,0 + γci, j,1 , i + j = 2.

3

(2.12)

The unconstricted triangulations Given a triangulation ∆, let v ∈ V, the number d of all edges emanating from v is called the valence of

v, denoted by val(v). Definition 3.1. A subtriangulation ∆0 of a triangulation ∆ is a triangulation satisfying T ∈ F(∆0 ) =⇒ T ∈ F(∆).

(3.13)

Definition 3.2. For a triangulation ∆, if val(v) ≥ d, ∀v ∈ Vb (∆), then we call ∆ a constricted triangulation with valence d. Definition 3.3. Given a triangulation ∆, if it does not contain any constricted subtriangulation with valence d, then we call ∆ an unconstricted triangulation with valence d. The set of all unconstricted triangulations with valence d is denoted by Ad . It is clear that A0 = A1 = A2 = ∅,

(3.14)

A3 ⊂ A4 ⊂ · · · ⊂ Ak ⊂ Ak+1 ⊂ · · · .

(3.15)

According to above Definition 3.3, the unconstricted triangulation ∆ introduced by Farin in [7] is in fact an unconstricted triangulation with valence 4, i.e., ∆ ∈ A4 . In this paper, we are interested in the

LIU-YI: BIVARIATE CUBIC SPLINE SPACES...

unconstricted triangulation with valence 6, i.e., ∆ ∈ A6 . In addition, it is easy to check by using the following Theorem 3.1 that both the Morgan-Scott triangulation and the Robbins triangulation (see Figure 1) are in A6 but A4 .

Figure 1: Left: the Morgan-Scott triangulation, Right: the Robbins triangulation. To give the construction manner of a triangulation ∆ ∈ A6 , we first recall the two definitions of a flap and a pair of triangles, which are introduced in [7]. Definition 3.4. The triangle formed by one boundary edge of ∆ and a point outside ∆ is called a flap to ∆. The point outside ∆ is called the expansion vertex of the flap. Definition 3.5. The two triangles formed by two adjacent boundary edges of ∆ and a point outside ∆ is called a pair of triangles to ∆. The point outside ∆ is called the expansion vertex of the pair of triangles. Besides, we need introduce two new concepts. Definition 3.6. A set of three triangles formed by a point outside ∆ and three adjacent boundary edges of ∆ is called an element-3 to ∆ . The point outside ∆ is called the expansion vertex of the element-3. Definition 3.7. A set of four triangles formed by a point outside ∆ and four adjacent boundary edges of ∆ is called an element-4 to ∆. The point outside ∆ is called the expansion vertex of the element-4. Examples of both element-3 and element-4 are shown by the dash lines in Figure 2.

Figure 2: Left: an element-3, Right: an element-4. It is noted that, for a given triangulation ∆, it is difficult to judge whether ∆ ∈ A6 or not by using Definitions 3.1 and 3.2. To make the judgement easier, we have Theorem 3.1. If ∆ ∈ A6 , then ∆ can be constructed in the following manner.

361

362

LIU-YI: BIVARIATE CUBIC SPLINE SPACES...

Start: one triangle. Recursive step: assuming a subtriangulation ∆ has been constructed, we may extend the triangulation ∆ to a new triangulation ∆0 using four operations on boundary vertices: 1) Adding a flap; 2) Adding a pair of triangles; 3) Adding an element-3; 4) Adding an element-4. Proof. Assume that ∆ ∈ A6 . Set ∆(0) := ∆ and let v0 be a boundary vertex such that it has the minimal valence among all the boundary vertices in ∆. It is clear that 2 ≤ val(v0 ) ≤ 5. Then we go to step 1. Step 1. If val(v0 ) = 2, we remove the flap in which v0 is the expansion vertex. It is clear that the resulting triangulation is still an unconstricted triangulation with valence 6, otherwise, the original triangulation ∆ would not be an unconstricted triangulation with valence 6, which is in contradiction with the assumption. By repeating the procedure of removing a flap, the triangulation ∆(0) becomes a new triangulation which is either a single triangle (if the procedure of removing a flap continues, then nothing would be left) or a triangulation with the valences of all boundary vertices being greater than 2. In the latter case, we denote the resulting triangulation by ∆(0) again and let v0 still be a boundary vertex such that it has the minimal valence among all the boundary vertices in ∆(0) . It is clear that 3 ≤ val(v0 ) ≤ 5. Then we go to step 2. Step 2. If val(v0 ) = 3, we remove the pair of triangles in which v0 is the expansion vertex. It is clear that the resulting triangulation is still an unconstricted triangulation with valence 6. By repeating the procedures of removing a flap and a pair of triangles, the triangulation ∆(0) becomes a new triangulation which is either a single triangle or a triangulation with the valences of all boundary vertices being greater than 3. In the latter case, we denote the resulting triangulation by ∆(0) again and let v0 still be a boundary vertex such that it has the minimal valence among all the boundary vertices in ∆(0) . It is clear that 4 ≤ val(v0 ) ≤ 5. Then we go to step 3. Step 3. If val(v0 ) = 4, we remove the element-3 in which v0 is the expansion vertex. It is clear that the resulting triangulation is still an unconstricted triangulation with valence 6. By repeating the procedures of removing a flap, a pair of triangles and an element-3, the triangulation ∆(0) becomes a new triangulation which is either a single triangle or a triangulation with the valences of all boundary vertices being greater than 5. In the latter case, we denote the resulting triangulation by ∆(0) again and let v0 still be a boundary vertex such that it has the minimal valence among all the boundary vertices in ∆(0) . It is clear that val(v0 ) = 5. Then we go to step 4. Step 4. We remove the element-4 in which v0 is the expansion vertex. The resulting triangulation is still an unconstricted triangulation with valence 6. By repeating the procedures of removing a flap, a pair of triangles, an element-3 and an element-4, the triangulation ∆(0) becomes a new triangulation which is a

LIU-YI: BIVARIATE CUBIC SPLINE SPACES...

363

single triangle. By reversing the above procedures of removing a flap, a pair of triangles, an element-3 and an element4, we obtain the construction procedures of adding a flap, a pair of triangles, an element-3 and an element-4. That is to say, any triangulation ∆ ∈ A6 can be constructed by recursively adding a flap, a pair of triangles, an element-3 and an element-4 only. As an example, the construction procedure of the Robbins triangulation is shown in Figure 3.

Figure 3: The construction of the Robbins triangulation, from left to right: adding three flaps, adding a pair of triangles, adding an element-3 and adding an element-4.

4

dim S 31 (∆) over the unconstricted triangulation with valence six In [7], Farin proved that the conjecture (1.2) is correct in the case of the unconstricted triangulation

with valence four, but in his paper the singular vertex is excluded. In this section, we shall prove that the conjecture (1.2) is still correct when the triangulation ∆ is in A6 with some singular vertices being also included. Theorem 4.1. Let ∆ be an unconstricted triangulation with valence 6, i.e., ∆ ∈ A6 . Assume that there is no degenerate edge at any nonsingular vertex in ∆, then dim S 31 (∆) = 2|VI | + 3|VB | + 1 + σ.

(4.16)

Proof. It is clear that Eq.(4.16) holds for ∆ being one single triangle. For an inductive proof, assume that Eq. (4.16) holds for a subtriangulation ∆(k) of ∆. Let ∆(k+1) be the triangulation after we add a flap, a pair of triangles, an element-3 or an element-4 to ∆(k) . S S 1) When ∆(k+1) = ∆(k) a flap = ∆(k) [v, v1 , v2 ], where v is the expansion point of the added flap, it is clear that     VI ∆(k+1) = VI ∆(k) ,

        VB ∆(k+1) = VB ∆(k) + 1, σ ∆(k+1) = σ ∆(k) .

(4.17)

By using Lemma 2.1, all the B-net coefficients associated with the domain points in D∆(k+1) \ D1 (v) can be   determined by the B-net coefficients associated with the minimal determining set for S 31 ∆(k) , see Figure

364

LIU-YI: BIVARIATE CUBIC SPLINE SPACES...

  4. Hence a determining set for S 31 ∆(k+1) can be constructed by adding all three domain points in the first   disk around v, D1 (v), to the minimal determining set for the space S 31 ∆(k) . This means     dim S 31 ∆(k+1) ≤ dim S 31 ∆(k) + 3       = 2 VI ∆(k) + 3 VB ∆(k) + 1 + σ ∆(k) + 3       = 2 VI ∆(k+1) + 3 VB ∆(k+1) + 1 + σ ∆(k+1) .

(4.18)

Eq. (4.18) together with Schumaker’s lower bound [11] leads to         dim S 31 ∆(k+1) = 2 VI ∆(k+1) + 3 VB ∆(k+1) + 1 + σ ∆(k+1) .

(4.19)

This is consistent with Eq. (4.16).

v v1 v2

Figure 4: The variation of the minimal determining set when a flap is added. 2) When ∆(k+1) = ∆(k)

S S S a pair of triangles = ∆(k) [v, v1 , v2 ] [v, v2 , v3 ], where v is the expansion

point of the added pair of triangles, it is clear that     VI ∆(k+1) = VI ∆(k) + 1,

    VB ∆(k+1) = VB ∆(k) .

(4.20)

By using Lemma 2.1, all the B-net coefficients associated with the domain points in D∆(k+1) \ D1 (v) can be   determined by the B-net coefficients associated with the minimal determining set for S 31 ∆(k) .   The construction of a determining set for S 31 ∆(k+1) depends on whether the new interior vertex v2 is     singular or not. If v2 is nonsingular, i.e., σ ∆(k+1) = σ ∆(k) , see Figure 5(1), then the edge [v, v2 ] is T nondegenerate at v2 , hence the B-net coefficient associated with the domain point in R1 (v) [v, v2 ] can be   further determined by using Lemma 2.1, and a determining set for S 31 ∆(k+1) can be constructed by adding   two other domain points in D1 (v) to the minimal determining set for the space S 31 ∆(k) . If v2 is singular,     see Figure 5(2), i.e., σ ∆(k+1) = σ ∆(k) + 1, then the edge [v, v2 ] is degenerate at v2 . By using Corollary T 2.1, the B-net coefficient associated with the domain point in R1 (v) [v, v2 ] cannot be determined, so three   domain points in D1 (v) have to be added to the minimal determining set for the space S 31 ∆(k) to form a

LIU-YI: BIVARIATE CUBIC SPLINE SPACES...

365

  determining set for S 31 ∆(k+1) . Hence, we have dim S 31



(k+1)

∆



       1 (k) (k+1)   = σ ∆(k)  dim S 3 ∆  + 2, if σ ∆    ≤   dim S 31 ∆(k) + 3, if σ ∆(k+1) = σ ∆(k) +1             VI ∆(k) + 3 VB ∆(k) + 1 + σ ∆(k) + 2, if σ ∆(k+1) = σ ∆(k)  2     =                2 VI ∆(k) + 3 VB ∆(k) + 1 + σ ∆(k) + 3, if σ ∆(k+1) = σ ∆(k) +1       = 2 VI ∆(k+1) + 3 VB ∆(k+1) + 1 + σ ∆(k+1) . (4.21)

Eq. (4.21) together with Schumaker’s lower bound [11] leads to         dim S 31 ∆(k+1) = 2 VI ∆(k+1) + 3 VB ∆(k+1) + 1 + σ ∆(k+1) .

(4.22)

v v v1

v1

v2

v3

v2

v3 (1)

(2)

Figure 5: The variation of the minimal determining set when a pair of triangles is added. 3) When ∆(k+1) = ∆(k)

S

{an element-3} = ∆(k)

S S S [v, v1 , v2 ] [v, v2 , v3 ] [v, v3 , v4 ], where v is the expan-

sion point of the added element-3, it is clear that     VI ∆(k+1) = VI ∆(k) + 2,

    VB ∆(k+1) = VB ∆(k) − 1.

(4.23)

By using Lemma 2.1, all the B-net coefficients associated with the domain points in D∆(k+1) \ D1 (v) can be   determined by the B-net coefficients associated with the minimal determining set for S 31 ∆(k) .     On one hand, if both v2 and v3 are nonsingular, i.e., σ ∆(k+1) = σ ∆(k) , see Figure 6(1), then edges [v, v2 ] and [v, v3 ] are nondegenerate at v2 and v3 , respectively, therefore by using Lemma 2.1, two B-net T S coefficients associated with two domain points in R1 (v) ([v, v2 ] [v, v3 ]) can be further determined, and   a determining set for S 31 ∆(k+1) can be constructed by adding the expansion vertex v to the minimal deter  mining set for the space S 31 ∆(k) . On the other hand, if one of two vertices v2 and v3 , for example v2 , is singular, see Figure 6(2), then    σ ∆(k+1) = σ ∆(k) + 1, and the edge [v, v2 ] is degenerate at v2 but the edge [v, v3 ] is nondegenerate 

366

LIU-YI: BIVARIATE CUBIC SPLINE SPACES...

at v3 , hence only the B-net coefficient associated with the domain point in R1 (v)

T [v, v3 ] can be further

determined, so two more domain points in D1 (v) have to be added. We actually have        1 (k) (k+1)     = σ ∆(k)  dim S 3 ∆  + 1, if σ ∆ 1 (k+1)    dim S 3 ∆ ≤   dim S 31 ∆(k) + 2, if σ ∆(k+1) = σ ∆(k) +1             VI ∆(k) + 3 VB ∆(k) + 1 + σ ∆(k) + 1, if σ ∆(k+1) = σ ∆(k)  2     =                2 VI ∆(k) + 3 VB ∆(k) + 1 + σ ∆(k) + 2, if σ ∆(k+1) = σ ∆(k) +1       (4.24) = 2 VI ∆(k+1) + 3 VB ∆(k+1) + 1 + σ ∆(k+1) . Eq. (4.24) together with Schumaker’s lower bound [11] leads to         dim S 31 ∆(k+1) = 2 VI ∆(k+1) + 3 VB ∆(k+1) + 1 + σ ∆(k+1) .

(4.25)

v

v

v4

v2 v3

v1

v1

(1)

v4 v2

v3

(2)

Figure 6: The variation of the minimal determining set when an element-3 is added. 4) When ∆(k+1) = ∆(k)

S

{an element-4} = ∆(k)

S S S S [v, v1 , v2 ] [v, v2 , v3 ] [v, v3 , v4 ] [v, v4 , v5 ], where v

is the expansion point of the added element-4, it is clear that     VI ∆(k+1) = VI ∆(k) + 3,

    VB ∆(k+1) = VB ∆(k) − 2.

(4.26)

By using Lemma 2.1, all the B-net coefficients associated with the domain points in D∆(k+1) \ D1 (v) can   be determined by the B-net coefficients associated with the minimal determining set for S 31 ∆(k) . The   construction of a determining set for S 31 ∆(k+1) depends on the following cases. i) All the three interior vertices v2 , v3 and v4 are nonsingular. Thus edges [v, v2 ], [v, v3 ] and [v, v4 ] are     nondegenerate at v2 , v3 and v4 , respectively, see Figure 7(1). In this case, σ ∆(k+1) = σ ∆(k) , and three T S S B-net coefficients associated with three domain points in R1 (v) ([v, v2 ] [v, v3 ] [v, v4 ]) can be further   determined, therefore a determining set for S 31 ∆(k+1) can be the same to the minimal determining set for   the space S 31 ∆(k) .

LIU-YI: BIVARIATE CUBIC SPLINE SPACES...

367

ii) One of the three interior vertices v2 , v3 and v4 is singular, for example say v2 , see Figure 7(2). Thus     edges [v, v3 ] and [v, v4 ] are nondegenerate at v3 and v4 , respectively. In this case, σ ∆(k+1) = σ ∆(k) + 1, T S and two B-net coefficients associated with domain points in R1 (v) ([v, v3 ] [v, v4 ]) can be further deter  mined, therefore a determining set for S 31 ∆(k+1) can be constructed by adding the expansion vertex v to   the minimal determining set for the space S 31 ∆(k) . iii) Two of the three interior vertices v2 , v3 and v4 are singular. Obviously, v3 cannot be singular, i.e.,     the edge [v, v3 ] is nondegenerate at v3 , see Figure 7(3). In this case, σ ∆(k+1) = σ ∆(k) + 2, and the T B-net coefficient associated with the domain point in R1 (v) [v, v3 ] can be further determined, therefore   a determining set for S 31 ∆(k+1) can be constructed by adding other two domain points in D1 (v) to the   minimal determining set for the space S 31 ∆(k) . Hence, we have 

dim S 31 ∆(k+1)



         dim S 31 ∆(k) , if σ ∆(k+1) = σ ∆(k)           ≤ dim S 31 ∆(k) + 1, if σ ∆(k+1) = σ ∆(k) +1            dim S 1 ∆(k) + 2, if σ ∆(k+1) = σ ∆(k) +2 3           (k)  VI ∆ + 3 VB ∆(k) + 1 + σ ∆(k) ,   2 if σ ∆(k+1) = σ ∆(k)                    (k)  VB ∆(k) + 1 + σ ∆(k) + 1, if σ ∆(k+1) = σ ∆(k) +1 = + 3 2 V ∆ I                      2 VI ∆(k) + 3 VB ∆(k) + 1 + σ ∆(k) + 2, if σ ∆(k+1) = σ ∆(k) +2       = 2 VI ∆(k+1) + 3 VB ∆(k+1) + 1 + σ ∆(k+1) . (4.27)

Eq. (4.27) together with Schumaker’s lower bound [11] leads to         dim S 31 ∆(k+1) = 2 VI ∆(k+1) + 3 VB ∆(k+1) + 1 + σ ∆(k+1) .

(4.28)

This completes the proof of the theorem.

v

v

v

v5

v1 v2

v3

v1 v4

v2

v3 v4

v1 v5

v5

v2 v3

(1)

(2)

(3)

Figure 7: The variation of the minimal determining set when an element-4 is added.

v4

368

LIU-YI: BIVARIATE CUBIC SPLINE SPACES...

From the proof of Theorem 4.1, it can be seen that when a flap or a pair of triangles or an element-3 is added in every expansion step from ∆(k) to ∆(k+1) , the expansion vertex can be always chosen as a domain point in the minimal determining set for S 31 (∆(k+1) ); however, when an element-4 is added, the expansion vertex can be also chosen as a domain point in the minimal determining set for S 31 (∆(k+1) ) only if there is at least one singular vertex among the three added interior vertices. Hence we actually have proved the existence of the Lagrange interpolation by S 31 (∆) on all the vertices in ∆ and have partially answered the Conjecture 1.2. Theorem 4.2. Let ∆ ∈ A6 such that there is at least one singular vertex among the three added interior vertices when every element-4 is added in the construction of ∆. If there is no degenerate edge at any nonsingular vertex in ∆, then for arbitrary numbers zi , i = 1, 2, ..., |V(∆)|, there exists a function s ∈ S 31 (∆) to satisfy that s(vi ) = zi , i = 1, ..., |V(∆)|,

(4.29)

where vi , i = 1, ..., |V(∆)|, are all the vertices in ∆.

Acknowledgement The work is supported by the Natural Science Foundation of China (No. 10462001), Guangxi Natural Science Foundation (No. 0575029), Guangxi Shi-Bai-Qian Scholars Program (No. 2001224) and Hunan Key Laboratory for Computation and Simulation in Science and Engineering.

References [1] P. ALFELD, Bivariate spline spaces and minimal determining sets, J. Comp. Appl. Math., 119 (2000), pp. 13-27. [2] P. ALFELD, B. PIPER AND L. L. SCHUMAKER, An explicit basis for C 1 quartic bivariate splines, SIAM J. Numer. Anal., 24 (1987), pp. 891-911. [3] P. ALFELD AND L. L. SCHUMAKER, The dimension of spline spaces of smoothness r for d ≥ 4r+1, Constr. Approx., 3 (1987), pp. 189-197. [4] P. ALFELD AND L. L. SCHUMAKER, On the dimension of bivariate spline spaces of smoothness r and degree d = 3r + 1, Numer. Math., 57 (1990), pp. 651-661. [5] L.J. BILLERA, Homology of smooth splines: generic triangulations and a conjecture of Strang, Trans. Am. Math. Soc., 310 (1988), 325-340.

LIU-YI: BIVARIATE CUBIC SPLINE SPACES...

[6] G. FARIN, Triangular Bernstein-B´ezier patches, Comput. Aided Geom. Des., 3 (1986), pp. 83-128. [7] G. FARIN, Dimensions of spline spaces over unconstricted triangulations, J. Comput. Appl. Math., 192 (2006), pp. 320-327. [8] D. HONG, Spaces of bivariate spline functions over triangulations, Approx. Theory Appl., 7 (1991), pp. 56-75. [9] J. MORGAN AND R. SCOTT, A nodal basis for C 1 piecewise polynomials of degree n ≥ 5, Math. Comp., 29 (1975), pp. 736-740. [10] J. MORGAN AND R. SCOTT, The dimension of piecewise polynomial, manuscript, (1975). [11] L.L. SCHUMAKER, On the dimension of spaces of piecewise polynomials in two variables, in Multivariable Approximation Theory, W. Schempp and K. Zeller, eds., Birkh¨auser, Basel, 1979, pp. 396412. [12] G. STRANG, Piecewise polynomials and the finite elements method, Bull. Amer. Math. Soc., 79 (1973), pp. 1129-1137. [13] R.H. WANG AND X.G. LU, On dimensions of spaces of bivariate splines with triangulations, Science in China (Series A), 32 (1989), pp. 674-684. [14] W. Whiteley, The combinatorics of bivariate splines, in Applied Geometry and Discrete Mathematics, V. Klee Festschrift, P. Gritzmann and B. Sturmfels, eds., AMS Press, Providence, 1991, pp. 567-708.

369

370 JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.5, NO.4, 370-376, COPYRIGHT 2010 EUDOXUS PRESS, LLC

Existence of solutions to a model of long range di¤usion involving ‡ux By Manar A .Qudah and Marwan S. Abualrub E- mail : [email protected] and [email protected] Department of Mathematics / University of Jordan / P.O.Box 11942 / Amman - Jordan 2000 Mathematics Subject Classi…cation code: 92B99 Keywords : long range di¤usion, insect dispersal. Abstract A model for insect dispersal has been considered, existence and uniqueness of solutions to the long range di¤usion involving ‡ux for such model has been shown in Lp,q space. 1. Introduction The dynamics of population has been described using mathematical models which have been very successful in giving good e¤ect in the study of animal and human populations. Fife [4] considered reaction and di¤usion systems which are distributed in 3-dimensional spaces or on a surface rather than on a line. Abualrub [1] studied di¤usion in two dimensional spaces for which di¤usion is more realistic and applicable in life. Also he talked about long range di¤usion with population pressure in Plankton-Herbivore populations.In this paper we include long range di¤usion involving ‡ux for insect population and then talk about the existence and uniqueness of solutions to our model in the Lp,q space. But we are going to …nd the required p and q in similar approach used in [2] . 2. Long Range Di¤usion Involving Flux Here we consider long range di¤usion to a modi…ed insect dispersal model in two dimensions as follows ut

D

(2)

u=

1u

+

2u

2

+

3 ux

+

4

u

+1

u (x; 0) = f (x) ;where u = u (x; t) is the insect population density and x 2 R2 . Here 1

(1) (2)

QUDAH-ABUALRUB: LONG RANGE DIFFUSION...

371

represents the laplacian operator and (2)

=

2 X

@4 : 2 2 i;j=1 @xi @xj

(3)

ut is the rate of change of the insect population density, D (2) u is the long range di¤usion term; where D is a small constant, and ; 4 are positive constants. But u2 is the interaction between the Males and Females of the insect population, ux is the instantaneous ‡ux in the x direction due to molecular di¤usion. (u +1 ) is the regular di¤usion of the insect population. We now want to discuss existence and uniqueness of solutions to equation (1) togother with condition (2) in the Lp,q space which is the function space consisting of Lebesgue measurable functions u (x; t) such that kukp;q < 1; where k kp,q is the norm in Lp,q , p is taken in the t variable, and q is taken in the x variable. We want to …nd the appropriate values for p and q in the next section. In addition, we will consider large values of time since we are talking about long range di¤usion. 3. Existence and Uniqueness of Solutions First of all, we have to prove the following Lemma for the initial data: Lemma 1 Assume that u satis…es equations (1) and (2). If f (x) 2 LP (R2 ), and jK (x,t)j ;where

D 1 jxj+t 4

2

; t > 0; and D is a constant. Then, K

f 2 L3q

represents the convolution in space only.

PROOF. As we did in [1]; we will use the following estimate for the Kernel; namely D jK (x,t)j ; t > 0: (4) 1 2 jxj + t 4 Now, if f (x) 2 LP (R2 )we have K

f

Z

Df (y) dy 2

1

jx

R2

yj + t 4

We …rst take the p norm in t; namely kK

f kp

Z

Df (y) dy jx

R2

2

1

yj + t 4

2 p

372

QUDAH-ABUALRUB: LONG RANGE DIFFUSION...

Applying Minkowski’s integral inequality on the right hand side of the above inequality to obtain

kK

f kp

D

Z

0

R2

D

jf (y)j B @

Z

R2

=D

Z

R2

Z

R+

0

B jf (y)j @

dt jx

C

2p A dy

1

jx

yj + t 4 1

yj + t 4

11

p

1

jf (y)j dy

jx

yj + t

1 4

1

2p

C 4 A dy

4 P

2

;where is a constant. We now take the q norm in x of the above inequality to obtain kK

f kp;q

Z

D

jf (y)j dy

R2

jx

1

yj + t 4

2

4 P

q

The right hand side of the above inequality is less than or equal to constant kf kq , 4 = 1q p2 (using the Benedek-Panzone Potential Theorem [3] ;see if p1 = 1q 2p Appendix). This implies that p = 3q and hence K f 2 L3q ; this concludes the proof for the initial data. Theorem 2 The solution u(x; t) of (1) and (2) exists and it is unique in the space L3 , for > 12 ; whenever the initial data f (x) is small enough in the norm of its space and if e 1 t ux (x; t) 2 L3 ; : PROOF. We begin by eliminating the linear term u(x; t) = e

1t

1u

of (1), so we let

w(x; t):

Then we get, wt

D

(2)

w=

2e

1t

w2 +

3 wx

+

4

w(x; 0) = f (x); where x 2 R2 :

w

+1

;

(5)

(6)

We may assume population pressure in the …rst and second terms in the right hand side of equation (5).Therefore, we write 2

= c1 w and 3

3

= c2 (wx )

QUDAH-ABUALRUB: LONG RANGE DIFFUSION...

373

then (5) becomes: wt

(2)

D

1t

w = c1 e

w

+2

+ c2 (wx )

+1

+

4

+1

w

(7)

;

w(x; 0) = f (x); where x 2 R2 :

(8)

Now w(x; t) can be obtained by solving the integral equation: w=

Zt Z

K (x

h

y; t

) c1 e

0 R2

1t

w

+2

+ c2 (wx )

+1

+

w

4

+1

i

dyd +

Z

K (x

R2

(9) ;where K is the fundamental solution to the homogeneous problem of (7), in two dimensions,namely K(x; t) =

1 e 2 t

jxj2 2t

; jxj = x21 + x22

1 2

; and x 2 R2 :

Also K can be approximated by (4).We will now rewrite (9) simply as h

w=K

c1 e

1t

w

+2

+ c2 (wx )

+1

+

4

w

+1

i

+K

f

(10)

;where represents the convolution in space and time; and w(x; t) is a weak solution of (9) provided that the integrals in (10) exist in the Lebesque sense. Using integration by parts on the term K (w +1 ) ; and set wx = H in 4 (10), we obtain: h

w=K

1t

c1 e

w

+2

+ c2 H

+1

i

+

4

w

+1

+K

f

(11)

;where =

K=

2 X @2K

@x2i Now for the …rst, second, third and fourth terms on the right hand side of (11), we shall use exponents r, s, p,q, respectively, when considering the Lp norm. For the …rst term in (11) we have: i =1

D

jKj

1

jxj + t 4

2

D

=

1

jxj + t 4

2+4 4

So, 1 = q

+2 r

4 = 2+4

+2 r 4

2 ; where 1 < 3

r 3 < +2 2

(12)

y; t) f (y)dy

374

QUDAH-ABUALRUB: LONG RANGE DIFFUSION...

Setting r = q we get: 3 ( + 1) 2 + 2 < 32 ( + 1) < 23 ( + 2), which gives r=

Using (12) ; (13) we have

(13) > 1:

For the second term in (11) ; we have: 1 = q

+1 s

1 = 2+1

+1 s

1 ; where 1 < 3

s 12 :

For the third term in (11) ; we have: j j = j Kj D 1

jxj + t 4 =

4

D 1

jxj + t 4

4+2 2

So, 1 +1 = q p Setting p = q we get:

2 = 4+2

+1 p

1 ; where 1 < 3

p : 2

(18)

Now to get a contraction mapping Lp R 2

R+

! Lp R 2 5

R+

QUDAH-ABUALRUB: LONG RANGE DIFFUSION...

375

in (11), we have to equate all exponents in K f 2 L3q ,(13) ; (15) and (17). That is 3 3q = ( + 1) = 3 = 3 : 2 In view of (13) ; (15) and (17) and the above relationship we arrive at: p = 3 ;q = ;r = 3 ;

(19)

=

Consequently, the following relationship exists between =2

and ; namely

1

Hence, 1 > : 2 Now, it is enough to show the uniqueness of the solution. Our mapping in (11) will be: kT ( )k3 C ( ) k( )k3 + k( )k3 : w(x; t) 2 L3

;

; where

;where C is a constant depends on . That is if we apply the mapping T to (11) we have: T (w) = K G + 4 w +1 + K f (20) ;where G = c1 e

1t

w

+2

+ c2 H

+1

;

then kT (w)k3

C ( ) kwk3 +1 + kgk3 ;

(21)

;where g is an auxiliary function which is the sum of the …rst and the third terms in the right hand side of equation (20).We are going now to compare (21) with the following mapping: y= x

+1

where ; are positive constants and x a linear function and it is convex. For

(22)

+ ; 0. Now x

+1

increases faster than

= 0 we have only one non-zero root of (22) because the graph of

y= x

+1

and y = x will intersect in only one non-zero point.

For the same reason if 0 < < (where is su¢ ciently small) ; we have two roots, say xf1 ; xf2 . Let xf1 be the smallest root, therefore if xf1 is small enough then the mapping T will be a contraction mapping which maps the ball of radius xf1 into itself. This implies that the solution to the equation w = T (w), in (20) ; exists and its unique in the ball of radius xf1 .Here xf1 depends on the size of the initial data. This completes the proof of Theorem 2. 6

376

QUDAH-ABUALRUB: LONG RANGE DIFFUSION...

Remark 3 We have shown in Theorem 2 that must be greater than 12 to guarantee the existence and uniquness of solutions for equation(1) togother with condition (2). Let us now take = 1: Therefore, equation (1) becomes: ut

D

(2)

u=

1u

+

2u

2

+

3 ux

+

4

u2

(23)

We will work on (23) in future research. Appendix: Benedek-Panzone Potential Theorem: Let X=En (the nth dimensional Euclidean space), and = ( 1 ; 2 ; ::::; n ) be an n-tuple of real numbers, 1 = ; 1 < P < 1 ; then 0 < i < 1: If P and Q are such that P1 Q f

jxj

n

P

Q

c kf kP holds for every f 2 L ; where

c( ; P ):

=

n X

i and

c =

i=1

References: [1] Abualrub, M.S.: Long range di¤usion-reaction model on population dynamics. Documenta Mathematica (J. of German Mathematician Union). Vol. 3, pp. 331-340, Dec. 1998. [2] Abualrub, M.S.: Existence and uniqueness of Solutions to a di¤usive predator-prey model. Journal of Applied Functional Analysis,Vol.4, No.1: pp. 107-111. Copyright 2009 Eudoxus press, LLC. [3] Benedek, A and Panzone, R.: The space Lp with mixed norm. Duke Math.J.,Vol. 28: pp. 301–324, 1961. [4] Fife, P.C. Stationary patterns for reaction-di¤usion equations. Pitman Research Notes in mathematics, eds., W. E. Fitzgibbon,III and H. W. Walker.Vol. 14: pp.81-121, 1977.

7

JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.5, NO.4, 377-388, COPYRIGHT 2010 EUDOXUS PRESS, LLC 377

On Best N Simultaneous Approximation Eyad Abu-Sirhan Math. Department, Ta…la Technical University, Tafeela - Jordan E-mail: [email protected] Abstract Let X and Y be Banach Spaces, L(X; Y ) be the space of bounded linear operators from X into Y , and X Y be the tensor product of X and Y with a uniform cross norm . Given a natural number n, let N be a monotonus norm on Rn : A subspace G is called N simultaneuosly proximinal if for every x1 ; x2 ; :::; xn 2 X there exists y 2 G such that : N (k x1 y k; k x2 y k; :::; k xn y k)

N (k x1 g k; k x2 g k; :::; k xn g k);

for all g 2 G: In this paper, we discuss simultaneous approximation in L(X; Y ) and X Y:

Key Words : Simultaneuos approximation Classi…cation Numbers : Primiry : 41A28, secondary : 41A65

1

Introduction

Throughout this section, (X; k k) and (Y; k k) are Banach Spaces, X denotes the dual of X and L(X; Y ) is the Banach space of all bounded linear operator from X into Y , endowed with usual norm k f k= sup k f (x) k; for every f 2 L(X; Y ): we denote X

_

Y and X

^

kxk=1

Y to the injective and the projective ^

tensor products of X and Y respectively, X Y is a subspace of the space of the nuclear operators from X into Y; [ 8]. We wish to consider here the problem of simultaneous approximation in L(X; Y ) in the sense of Fathi, Hussien, and Khalil [10]. Let G be a closed subspace of Y: Given n points y1 ; y2 ; :::; yn in Y , there are several ways of simultaneuosly approximatig them by an element g in G (see [10], [2]), throughout this paper we will use the de…nition introduced by Fathi, Hussien, and Khalil in [10]. Namely, we say that a norm N in Rn is monotonus if for every t = (ti )1 i n , s = (si )1 i n 2 Rn such that j ti j j si j for i = 1; 2; :::; n we have N (t) N (s): Notice that all the usual norms in Rn are monotonus. 1

378

SIRHAN: SIMULTANEOUS APPROXIMATION

De…nition 1.1. ([3,De…nition 1]). we say that g0 2 G is a best N simultaneuos approximation from G of vectors y1 ; y2 ; :::; yn 2 Y if N (k y1

g0 k; k y2

g0 k; :::; k yn

g0 k)

N (k y1

g k; k y2

g k; :::; k yn

g k)

for all g 2 G: If every n-tuple of vectors y1 ; y2 ; :::; yn 2 Y admits a best N simultaneuos approximation from G; then G is said to be N simultaneuosly proximinal in Y: Of course, for n = 1 the preceding concepts are just best approximation and proximinality. For an example of a Banach space which has a proper subspace that is proximinal but not N simultaneously prximinal see [6]. The problem of best N simultaneuos approximation had been studied by some authers : Tijani Pakhrou [14], studied the problem on L1 ( ; Y ) ; the Banach space of all essensially bounded Bochner integrable Y valued functions on a …nite measure space ( ; ; ) : Fathi, Hussien, and Khalil [10], studied the problem on Lp (I; Y ) ; the Banach space of all p integrable Y valued functions on the interval I = [0; 1] with lebesque measure: J. Mendoza and Tijani [6] ; studied the problem on L1 ( ; Y ) ; the Banach space of all Bochner integrable Y valued functions on a …nite measure space ( ; ; ) : Little works have been done L(X; Y ). E. Abu-Sirhan and R. Khalil [5], studied the problem on L (X; Y ) in a particular case : N is an `1 norm on R2 ; N (a; b) = jaj + jbj ; a; b 2 R: The object of this paper is to study the subspase of L(X; Y ) which are N simultaneuosly proximinal in L(X; Y ): In this paper we will show that if G is a re‡exive subspace of Y , then L(X; G) is N simultaneuosly proximinal in L(X; Y ): Result on other spaces, Tensor produt spaces; L1 ( ; Y ) ; and L1 ( ; Y ) are presented.

2

Preliminary

The uncomplete tensor product of two Banach spaces X and Y is the set of P all …nite sums of the form xi yi withP xi 2 X and yi 2 Y: An equivalence relation is introduce by stipulating that xi yi is (equivalent to) 0 when P hf; xi i yi for all f 2 X , X is the dual space of X: A norm on X Y is termed a cross-norm if (x y) = kxk kyk for all x 2 X and all y 2 Y: A cross-norm is said to be uniform cross-norn if X X Axi Byi kAk kBk xi yi for any bounded linear operators A and B: The completion of the normed linear space X Y with a cross-norm is denoted by X Y: The projective norm is a special uniform cross-norm de…ned by the equation nX X kzk^ = inf kxi k kyi k : xi 2 X; yi 2 Y; z = xi 2

o yi ;

SIRHAN: SIMULTANEOUS APPROXIMATION

379

^

and the completion of X Y under such norm is denoted by X Y: The injective norm is a special uniform cross-norm de…ned by the equation X X xi yi = sup hf; xi i yi ; _

f 2X

and the completion of X Y under such norm is denoted by X Banach spaces X and Y; we have the following equation X

_

Y

L(X; Y ) =

X

^

Y

_

Y: for given

:

P The ideti…cations made here are as follows. With an element i i in X Y (uncompleted tensor product) we associate an operator A 2 L(X; Y ) whose P de…ning equation is Ax = h i ; xi i : With an arbitrary B in L(X; Y ) we ^ P P associate a funcional in X Y by putting ( xi yi ) = hBxi ; yi i : The weak

ality of X

^

topology in L(X; Y ) is the weak topology induced by the du-

Y with

X

^

Y

: Convergence of a net A to A in this topology

means hA x; yi = hAx; yi for all x 2 X and y 2 Y: It is known, [7], that L (`p ; X) is isometrically isomorphic to n o X p `p (X) = (xn ) : kxn k < 1 :

Where p is the conjugate of p: Hence L `1 ; X = `1 (X) : For any measure space (S; ; ) and any Banach space X; it is known that L1 ( ; X) = L1 ( ) space X; we have

^

X: In particular `1 (X) = `1

`1 (X ) = L `1 ; X

`1

=

^

^

X: Thus, for any Banach

= `1 (X)

X

:

_

Remark 2.1 For a Banach Space X, `1 X can be identi…ed with a closed subspace of `1 (X) : The identi…cation as follows : 1 Pn (i) With each element z = i=1 Fi xi in `1 X , Fi = ak 2 `1 , we associate an element Fz in `1 (X) to be de…ning by Fz =

n X

(i)

ak xi

i=1

3

1 k=1

:

k=1

380

SIRHAN: SIMULTANEOUS APPROXIMATION

Observe that kzk_

=

sup 2X ;k k=1

=

sup

k

n i=1

sup

2X ;k k=1 k

=

sup k

=

sup 2X ;k k=1

sup k

(xi ) Fi k1 (i)

n i=1

(xi ) ak

n i=1

(xi ) ak

(i)

(i) n i=1 xi ak

= kFz k : Thus the linear map z 7! Fz ; after being extended by continuity is an isom-

etry of `1

3

_

X into `1 (X) :

Main Result

Throughout this section, n is agiven natural number and N is a monotonous norm in Rn : First, we present the following two lemmas needed to prove our main result. Lemma 3.1. Let X be a Banach Space. If G is a re‡ixive subspace of a Banach Y , then L(X; G) is w closed in L(X; Y ): Proof. Now, we have L (X; G) = L (X; G ) So, let (F ) w

topology on (X

L (X; Y

) = (X

^

Y ) :

L (X; G) be a net converging to F 2 (X ^

^

Y ) in the

Y ) : Let x 2 X and y 2 Y : Then lim F (x

y ) = F (x

y ):

Thus lim y (F (x)) = y (F (x)) : Since y 2 Y was arbitrary, then (F (x)) G = G is a net converging to F (x) in the w topology on Y : Since G is w closed in Y and x 2 X was arbitrary; then F (x) 2 G for all x 2 X and F 2 L (X; G) : Lemma 3.2. ([14, Lemma 2.2]). Let X be a Banach space and let A If A is w closed, then A is N -simultaneuosly proximinal in X .

4

X :

SIRHAN: SIMULTANEOUS APPROXIMATION

381

Theorem 3.3. Let X be a Banach space: If G is a re‡exive subspace of a Banach Y , then L(X; G) is N -simultaneuosly proximinal in L(X; Y ): Proof. Let f1 ; f2 ; :::; fn 2 L (X; Y ) L (X; Y ) : Since L (X; G) is w closed in L (X; Y ) (Lemma 3.1), then it is N -simultaneuosly proximinal in L (X; Y ) (Lemma 3.2): Therefore, there exists g0 2 L (X; G) ; a best N simultaneous approximation from L (X; G) of the vectors f1 ; f2 ; :::; fn 2 L (X; Y ) : Of course, this means that L(X; G) is N -simultaneuosly proximinal in L(X; Y ): Theorem 3.4. Let X be a Banach space G be a closed subspace of a Banach Y . If L(X; G) is N -simultaneuosly proximinal in L(X; Y ); then G is N simultaneuosly proximinal in Y: Proof. Let y1 ; y2 ; :::; yn 2 Y: Let x0 2 X, kx0 k = 1: Choose x 2 X ; kx k = 1; such that < x0 ; x >= 1: The elements x yi : X ! Y; i = 1; 2; :::; n; de…ned by x yi (x) = x (x) yi , are in L(X; Y ): Since L(X; G) is N -simultaneously proximinal in L(X; Y ); there exists g0 2 L(X; G) such that N

kx

y1

g0 k ; kx ; kx yn

y2 g0 k

g0 k ; :::

kx

N

y1

gk ; kx ; kx yn

y2 gk

for all g 2 L(X; G): In particular this inequality is valid if we choose g = x for some z 2 G: So, N

kx

y1

g0 k ; kx ; kx yn

y2 g0 k

g0 k ; :::

N

kx

y1

gk ; :::

;

z;

x zk ; kx y2 x :::; kx yn x zk

zk ;

kx k N (ky1 zk ; ky2 zk ; :::; kyn zk) = N (ky1 zk ; ky2 zk ; :::; kyn zk):

Hence, for any x 2 X; kxk = 1; we have N

kx

y1 (x)

g0 (x)k ; kx ; kx yn (x)

y2 (x) g0 (x)k

Choose x to equal x0 : However, (x N (ky1

g0 (x0 )k ; ky2

g0 (x0 )k ; :::; kyn

g0 (x)k ; :::

N (ky1

zk ; ky2

zk ; :::; kyn

yi )(x0 ) = yi ; i = 1; 2; :::; n: Thus g0 (x0 )k)

N (ky1

zk ; ky2

zk ; :::; kyn

for all z 2 G: Consequently, g0 (x0 ) is a best N simultaneous approximation of the vectors y1 ; y2 ; :::; yn 2 Y: Hence G is N simultaneously proximinal in Y:

5

zk):

zk):

382

SIRHAN: SIMULTANEOUS APPROXIMATION

Theorem 3.5. Let X and Y be Banach spaces. If P is a projecton on X; Q is a projecton on Y; and Q is the adjoint operator of Q; then the following subspaces are N simultaneously proximinal in L (X; Y ) : 1. M = fAP : A 2 L (X; Y )g . 2. N = fQ B : B 2 L (X; Y )g : 3. W = fAP + Q B : A; B 2 L (X; Y )g : Proof. The proof follows from Lemma 3.2 and the fact that the subspaces N; M; and W are w closed in L (X; Y ) ; [8] : Defnition 3.6. A subspace G of a Banach space X is called p-summand, 1 p < 1; if there exists a closed subspace W X; such that X = G W , p p 1 and for x = g + w; one has kxk = (kgk + kwk ) p : We write X = G p W: If G is 1-summand in X, then the projection P : X ! G; P (g + w) = g; is called an L1 projection of X onto G; then G is called 1 complemented in X: In case kxk = maxfkgk ; kwkg; we call G an 1 summand. We refer to [1] for more on contractive projections. Lemma 3.7. Let X and Y be Banach spaces. If G is 1-summand of X; X=G

1

W; then L (X; Y ) = L (G; Y )

1

L (W; Y ) :

Proof. For f1 + f2 2 L (G; Y ) 1 L (W; Y ) ; de…ne f : G f (g + w) = f1 (g) + f2 (w) : Now, de…ne : L (G; Y ) by

L (W; Y ) ! L (G

1

(f1 + f2 ) = f . Let g + w 2 G kf (g + w)k

1

1

W !Y

by

W; Y )

W with kg + wk = 1; then

kf1 (g)k + kf2 (w)k kf1 k kgk + kf2 k kwk max fkf1 k ; kf2 kg (kgk + kwk) = max fkf1 k ; kf2 kg kg + wk = kf1 + f2 k :

Since g + w 2 X; kg + wk = 1 was arbitrary, then kf k convers inequality, kf k =

1

sup kgk+kwk=1

kf1 (g) + f2 (w)k

sup kf1 (g)k = kf1 k :

kgk=1

6

kf1 + f2 k : For the

SIRHAN: SIMULTANEOUS APPROXIMATION

383

Similarly, kf k kf2 k ; and hence kf k max fkf1 k ; kf2 kg : Thus kf k = max fkf1 k ; kf2 kg : Then one can easly check that is onto isometry. Theorem 3.8. Let X and Y be Banach spaces. If G is 1-summand of X; then L (G; Y ) is N simultaneously proximinal in L (X; Y ) : Proof. Assume that X = G 1 W: Let P1 : X ! G be the L1 Projection. Using Lemma 3.7, one can easly show that M = fAP1 : A 2 L (X; Y )g is isometrically isomorphic to L (G; Y ) : The result follows from Theorem 3.5. Lemma 3.9. Let X be a re‡exive Banach space with the approximation prop^

erty, and G be a re‡exive subspace of a Banach space Y: Then X G is w ^

in X

Y

closed

:

Proof. Since X is re‡exive, then X has the Radon Nikodym property, [4]. Hence by assumption on X, X point of X

^

G in X

^

Y

^

Y

= (X

_

Y ) [4 ]. Let S be a w

limit

: Then, we can assume the existence of a net (S ) in

^

X G such that limS = S in the w y 2 Y : Then, limS (x

topology on X

y )

= S(x

^

Y

. Let x 2 X and

y );

lim hy ; S (x )i = hy ; S(x )i : Since y 2 Y is an arbitrary, then (S (x )) is a net in G = G that converging to S (x ) in the w topology on Y : Since G is w closed in Y ; then S (x ) 2 G: Since x 2 X was an arbitrary, then S (x ) 2 G for all x 2 X and S 2 X

^

G: Thus X

^

G is w

closed in X

^

Y

:

Theorem 3.10. Let X be a re‡exive Banach space with the approximation property, n be a natural number, N be a monotonous norm in Rn , and G be a re‡exive subspace of a Banach space Y: Then X proximinal in X

^

X Y

^

Y

X

^

X

^

Y

: Since X

^

G is w

closed in

^

(Lemma

(Lemma 3.9), then it is N -simultaneuosly proximinal in X Y

3.2): Therefore, there exists g0 2 X from X

G is N simultaneously

Y:

Proof. Let f1 ; f2 ; :::; fn 2 X ^

^

^

^

G; a best N simultaneous approximation

G of the vectors f1 ; f2 ; :::; fn 2 X

^

G is N -simultaneuosly proximinal in in X 7

Y: Of course, this means that ^

Y:

384

SIRHAN: SIMULTANEOUS APPROXIMATION

Lemma 3.11. Let G be a w L(`1 ; G) is w closed in `1 (X )

closed subspace of a dual space X: Then :

Proof. Let (xn ) 2 `1 (X ) be a w limit piont of L(`1 ; G) = `1 (G). Then, we can assume the existence of a sequence (Hn ) in `1 (G) such that limHn = (xn ) in the w topology on `1 (X ) Let (sn ) 2 `1 (X ); then

1

n

: Assume that Hn = (hnm )m=1 for all n:

lim h(sn ) ; Hm i = h(sn ) ; (xn )i ;

lim

m!1 1 P

m!1

n=1

hhm n ; sn i

=

1 P

n=1

hsn ; xn i :

Take (sn ) = (0; 0; :::; xnth ; 0; :::), then lim x (hm n ) = x (xn ) and this is true m!1

w

! xn for all n: Since G is w for all x 2 X . Thus hm n for all n; then (xn ) 2 `1 (G): This ends the proof.

closed, then xn 2 G

Theorem 3.12. Let G be a w closed subspace of a dual space X, n be a natural number, and N be a monotonous norm in Rn : Then L(`1 ; G) is N simultaneously proximinal in `1 (X): Proof. By Lemma 3.2 and Lemma 3.11, L(`1 ; G) is N simultaneously proximinal in `1 (X ) : Since L(`1 ; G) = `1 (G)

`1 (X)

`1 (X )

;

then L(`1 ; G) is N simultaneously proximinal in `1 (X): From Remark (2:1) ; we deduce the following `1

_

G

`1

_

`1 (X)

X

`1 (X )

:

Thus we have the following result. Theorem 3.13. Let G be a closed subspace of a Banach space X: If `1 is w 1

`

_

1

closed in ` (X )

1

; then `

_

_

G

G is N simultaneously proximinal in

X:

Proof. The result follows from Lemma(3:2) : In [14] it is shown that if G is a re‡exive subspace of a Banach space Y , then L1 ( ; G) is N simultaneously proximinal in L1 ( ; Y ) : As a result on L1 ( ; X), we have : 8

SIRHAN: SIMULTANEOUS APPROXIMATION

385

Theorem 3.14. If L1 ( ; G) is N simultaneously proximinal in L1 ( ; Y ) ; then G is N simultaneously proximinal in Y: Proof. Let y1 ; y2 ; :::; yn 2 Y: Let 1 be the constant function on : Then 1 yi : ! Y; i = 1; 2; :::; n; de…ned by 1 yi (s) = yi , are elements in L1 ( ; Y ) : Since L1 ( ; G) is N -simultaneously proximinal in L1 ( ; Y ) ; there exists g0 2 L1 ( ; G) such that N

k1

y1

g0 k ; k1 ; k1 y2

y2 g0 k ; ::: g0 k

k1

N

y1

gk ; k1 ; k1 y2

y2 gk

gk ; :::

;

for all g 2 L1 ( ; G) : In particular this inequality is valid if we choose g = 1 z; for some z 2 G: So N

k1

y1

g0 k ; k1 ; k1 y2

There exists s0 2 N

k1

y1 (s0 )

y2 g0 k ; ::: g0 k

= N (ky1

y1

1 zk ; k1 y2 1 ; k1 yn 1 zk

zk ; ky2

zk ; :::; kyn

zk ; :::

zk):

such that

g0 (s0 )k ; k1 ; k1 yn (s0 )

N

k1

N

ky1

y2 (s0 ) g0 (s0 )k ; ::: g0 (s0 )k

N

ky1

g0 (s0 )k ; ky2 g0 (s0 )k ; ::: ; kyn g0 (s0 )k

N

ky1

zk ; ky2 zk ; ::: ; kyn zk

zk ; ky2 zk ; ::: ; kyn zk

Since z 2 G was arbitrary, then g0 (s0 ) 2 G is a best N simultaneous approximation for y1 ; y2 ; :::; yn 2 Y and G is N simultaneously proximinal in Y:

Theorem 3.15: Let G be a closed subspace of the Banach space Y: If X

_

G is

_

N simultaneously proximinal in X Y; then G is N simultaneously proximinal in Y: Proof. Let y1 ; y2 ; :::; yn 2 Y: Let x 2 X; kxk = 1: Choose x 2 X ; kx k = 1; such that < x; x >= 1: The elements x _

y1 , x

Since X G is N -simultaneously proximinal in X such that N

kx

y1

z0 k ; kx kx yn

y2 z0 k ; :::; z0 k

N

_

y2 ; :::; x

_

kx

yn are in X

Y; there exists z0 2 X y1

zk ; kx kx yn

y2 zk

Y:

_

G

zk ; :::;

for all z 2 X G: In particular this inequality is valid if we choose z = x for some g 2 G: So 9

_

g;

;

; :

386

SIRHAN: SIMULTANEOUS APPROXIMATION

kx

N

y1

z0 k ; kx kx yn

y2 z0 k ; :::; z0 k

N

kx

kxk N

y1 ky1

x gk ; kx y2 x ; kx yn x gk

gk ; ky2 gk ; ::: ; kyn gk

gk ; :::

:

Hence, for any v 2 X ; kv k = 1; we have k(x

N

y1 )(v )

z0 (v )k ; k(x ; k(x yn )(v )

y2 )(v ) z0 (v )k

Choose v to equal x : However, (x ky1

N

z0 (v )k ; ky2 z0 (v )k ; ::: ; kyn z0 (v )k

z0 (v )k ; :::

N

ky1

gk ; ky2 gk ; ::: ; kyn gk

yi )(x ) = yi ; i = 1; 2; :::; n: Thus

N

ky1

gk ; ky2 gk ; ::: ; kyn gk

;

for all g 2 G: Consequently, z0 (x ) is a best N -simultaneous approximation of y1 ; y2 ; :::; yn 2 Y: Hence G is N -simultaneously proximinal in Y: Let S be a compact Hausdor¤ space, X be a Banach space. we denote C (S; X) to the Banach space of all X valued contuous functions on S equipped with supremum norm. If X = R; then we write C(S) instead of C(S; R): Corollary 2.16. If C(S; G) is N -simultaneously proximinal in C(S; X); then G is N -simultaneously proximinal in X: Proof. The proof follows from Theorem 2.15 and the fact that C(S; X) = C(S)

_

X; [8].

^

Theorem 3.17. If L1 ( ) G is N simultaneously proximinal in L1 ( ) then G is N simultaneously proximinal in Y:

^

Y;

Proof. The proof is similar to that given in Theorem 3.15. In [10] it is shown that if G is a re‡exive subspace of a Banach space Y , then L1 ( ; G) is N simultaneously proximinal in L1 ( ; Y ) : As a result on L1 ( ; Y ) ; we have : Theorem 3.18. If L1 ( ; G) is N simultaneously proximinal in L1 ( ; Y ); then G is N simultaneously proximinal in Y:

10

:

SIRHAN: SIMULTANEOUS APPROXIMATION

387

Proof. The proof follows from Theorem 3.17 and the fact that L1 ( ; Y ) = L1 ( )

^

Y; [8].

Let S; T be compact Hausdor¤ spaces, then C(S Now we have the following result

T ) = C(S)

_

C(T ); [8].

_

Theorem 3.19. If G is a subspace of C (S) such that G C(T ) is N simultaneously proximinal in C(S T ), then G is N simultaneously proximinal in C(S): _

Proof. Assume that G C(T ) is N simultaneously proximinal in C(S T ): 0 Let x1 ; x2 ; :::; xn 2 C (S) : Put xi (s; t) = xi (s) for all (s; t) 2 S T and i = 0 0 0 1; 2; :::; n: Let z be a best N simultaneous approximation for x1 ; x2 ; :::; xn from G

_

C(T ): Note that for any g 2 G; 1 0 0 0 x1 z ; x2 z ; ::: A N@ 0 ; xn z

0

N@

0

x1

= N (kx1

Let N (kx1

0

g

1 ; x2 0

; xn gk ; kx2

g

g

1 ; :::

1

gk ; :::; kxn

gk) :

1 A

2 T and put h (s) = z (s; t) : Then h 2 G and hk ; kx2

hk ; :::; kxn

hk)

N (kx1

gk ; kx2

gk ; :::; kxn

gk)

for all g 2 G: Hence h is a best N simultaneous approximation for x1 ; x2 ; :::; xn from G:

References [1] Alfsen, E. M. and E¤ros, E. M. Structure in Real Banach Spaces. Ann. of Math. 96, 98-173 (1972). [2] C. Benitez, M. Fernandez, L. Soriano, Suitable Norm for Simultaneous Approximation, Extracta Mathematica, Vol.11, No.3, 485-488 (1996). [3] Chong, Li. and Watson, G. A. On best simultaneous approximation. J. Approx. Theory. 91, 332-348 (1997). [4] Diestel, J., and Uhl, J. J. Vector measures. The American Mathematical Society, Providence, R. I. 1977. [5] E. Abu-Sirhan, Roshdi Khalil, Simultaneos Approximation In Operator and Tensor Product Spaces, Journal of Applied Functional Analysis, Vol 4, No. 1, 112-121. (2009).

11

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SIRHAN: SIMULTANEOUS APPROXIMATION

[6] J. Mendoza, Tijani Pakhrou, Best simultaneous approximation in L1 ( ; X);J. Approx.Theory. 145, 212-220 (2007). [7] J. S. Cohen, Absolutely P -Summing, P -Nuclear Operators and their Conjugates, Math Ann, 201 ; 177-200 (1973). [8] Light, W. A., and Cheney. Approximation Theory in Tensor Product spaces. Lecture Notes in Mathematics. 1169, Singer-Verlag, New York. 1985. [9] Mach, J. Best simultaneous approximation of bounded functions with values in certain Banach spaces. Math. Ann. 240, 157-164 (1979). [10] Saidi, F. Hussein, D. and Khalil, R. Best simultaneous approximation in Lp (I; X): J. Approx. Theory. 116, 369-379 (2002). [11] Shany, B. N. and Singh, S. P. On best simultaneous approximation in Banach spaces. J. Approx. Theory. 35, 222-224 (1982). [12] Tanimoto, S. Characterization of best simulataneous approximation. J. Approx. Theory. 59, 359-361 (1989). [13] Tanimoto, S. On best simulataneous approximation. Math. Jap. 48, 275-279 (1998). [14] Tijani Pakhrou, Best Simultaneous approximation in L1 ( ; X); Math. Nachr. 281, No. 3, 396-401, (2008). [15] W. Deeb and R. Khlil, Best approximation in L(X,Y), Math. Proc. Cambridge Philos. Soc. 104, 527-531 (1988). [16] Watson, G. A. A characterization of best simulataneous approximations. J. Approx.Theory. 75, 175-182 (1993).

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JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.5, NO.4, 389-407, COPYRIGHT 2010 EUDOXUS PRESS, LLC 389

A Rank-One Fitting Method for Solving Symmetric Nonlinear Equations ∗ Gonglin Yuan † , Xiangrong Li† Abstract. In this paper, a rank-one updated method for solving symmetric nonlinear equations is proposed. This method possesses some features: (1) The updated matrix is positive definite whatever line search technique is used; (2) The search direction is descent for the norm function; Under reasonable conditions, we establish its global convergence. Numerical results show that the presented method is competitive to the normal BFGS method for the test problems. Key Words. rank-one update; global convergence; nonlinear equations. AMS subject classifications. 90C26.

1.

Introduction

Consider the following system of nonlinear equations: F (x) = 0, x ∈ ℜn ,

(1.1)

where F : ℜn → ℜn is continuously differentiable and the Jaconbian ∇F (x) of F (x) is symmetric for all x ∈ ℜn . Let θ be the norm function defined by θ(x) = 21 kF (x)k2 . Then the nonlinear equation (1.1) is equivalent to the following global optimization problem min θ(x), x ∈ ℜn . ∗

(1.2)

Foundation item: National Natural Science Foundation of China(10761001) and the Scientific Research Foundation of Guangxi University (Grant No. X081082). † College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi, 530004, P.R. China. E-mail: [email protected].

1

390

YUAN-LI: SOLVING SYMMETRIC NONLINEAR EQUATIONS...

The following iterative method is used for solving (1.1) xk+1 = xk + αk dk ,

(1.3)

where xk is the current iterative point, dk is a search direction, and αk is a positive step-size. It is well known that there are many methods [9, 11, 23, 24, 26, 29] for the unconstrained optimization problems minx∈ℜn f (x) (UOP), where the BFGS method is one of the most effective quasi-Newton methods [2, 3, 4, 6, 28, 30, 31]. These years, lots of modified BFGS methods (see [13, 14, 17, 19, 32]) have been proposed for UOP. Especially, many efficient attempts have been made to modify the usual quasi-Newton methods using both the gradient and function values information (e.g. [18, 32]). Lately, in order to get a higher order accuracy in approximating the second curvature of the objective function, Wei, Yu, Yuan, and Lian [19] proposed a new BFGS-type method for UOP, and the reported numerical results showed that the average performance is better than that of the standard BFGS method. The superlinear convergence of this modified has been established for uniformly convex function. Its global convergence is established by Wei, Li, and Qi [18]. Motivated by their ideas, Yuan and Wei [31] presented a modified BFGS method which can ensure that the update matrices are positive definite for the general convex functions. Moreover, the global convergence is proved for the general convex functions. For general functions, it is now known that the BFGS method may fail for non-convex functions with inexact line search [4], and Mascarenhas [15] showed that the nonconvergence of the standard BFGS method even with exact line search. In order to obtain a global convergence of BFGS method without convexity assumption on the objective function, Li and Fukushima [13, 14] made a slight modification to the standard BFGS method. Different from above techniques, Xu [20] presented a rank-one fitting algorithm for UOP, and the numerical examples are very interesting. Motivated by their ideas, we give a new rank-one fitting algorithm for (1.1) which possesses the global conver2

YUAN-LI: SOLVING SYMMETRIC NONLINEAR EQUATIONS...

gence. The method can ensure that the updated matrices are positive definite without carrying out any line search, the search direction is descent for the normal function, and the numerical results is competitive to the BFGS method. For nonlinear equations, the global convergence is due to Griewank [10] for Broyden’s rank one method. Fan [8], Yuan [21], Yuan, Lu and Wei [27], and Zhang [33] presented the trust region algorithms for nonlinear equations. Zhu [34] gave a family of nonmonotone backtracking inexact quasi-Newton algorithms for solving smooth nonlinear equations. In particular, a Gauss-Newton-based BFGS method is proposed by Li and Fukushima [12] for solving symmetric nonlinear equations, and the modified methods [22, 25] are studied. We all know that the BFGS method for solving (1.1) is to generate a sequence of iterates {xk } by letting xk+1 = xk + αk dk , where αk is the steplength and dk is a solution of the system of linear equation Bk dk + F (xk ) = 0,

(1.4)

where Bk is generated by BFGS update formula Bk+1 = Bk −

Bk sk sTk Bk yk ykT + T , sTk Bk sk sk y k

(1.5)

where sk = xk+1 − xk , and yk = F (xk+1 ) − F (xk ). In the following, we briefly review some line search technique to obtain the stepsize αk . Brown and Saad [1] proposed the following line search method: θ(xk + αk dk ) − θ(xk ) ≤ σαk ∇θ(xk )T dk ,

(1.6)

where θ(xk )T dk = F (xk )T ∇F (xk )dk , σ ∈ (0, 1), αk = r ik , r ∈ (0, 1), and ik is the smallest nonnegative integer i such that (1.6). Based on this technique, Zhu [34] gave the nonmonotone line search technique: θ(xk + αk dk ) − θ(xl(k) ) ≤ σαk ∇θ(xk )T dk , 3

391

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YUAN-LI: SOLVING SYMMETRIC NONLINEAR EQUATIONS...

kθ(xl(k) )k = max0≤j≤m(k) {kθ(xk−j )k}, m(0) = 0, m(k) = min{m(k − 1) + 1, M}, k ≥ 1, and M is a nonnegative integer. From these two techniques, it is easy to see that the Jacobian matric ∇F (xk ) must be computed at every iteration, which will increase the workload especially for large-scale problems or this matric is expensive. Considering these points, Yuan and Lu [25] presented a new backtracking inexact technique to obtain the stepsize αk : kF (xk + αk dk )k2 ≤ kF (xk )k2 + δαk2 F (xk )T dk ,

(1.7)

where δ ∈ (0, 1) is a constant, and dk is a solution of the system of linear equations (1.4). Li and Fukashima [12] give a line search technique to determine a positive step-size αk satisfying kF (xk +αk dk )k2 −kF (xk )k2 ≤ −δ1 kαk F (xk )k2 −δ2 kαk dk k2 +εk kF (xk )k2 , (1.8) where δ1 and δ2 are positive constants, and {εk } is a positive sequence such that ∞ X

εk < ∞.

(1.9)

k=0

The formula (1.8) means that {F (xk )} is norm descent when k is sufficiently large. In this paper, we also use the formula (1.8) as line search to find the step-size αk . Normally, the update matrix is defined by formula (1.5). Is there another way to determine the update formula? Accordingly the search direction dk is determined by the way. In this paper, we give a positive answer. The updated matrix Bk is generated by the following rank-one updated formulas Bk+1 = Bk + vk vkT , (1.10) Hk+1

Hk vk vkT Hk , = Hk − 1 + vkT Hk vk

(1.11)

where, as k = 0, B0 is the given symmetric positive definite matrix, Bk−1 = Hk , and vk = δ0 αk F (xk ), here δ0 is a positive constant. Then we 4

YUAN-LI: SOLVING SYMMETRIC NONLINEAR EQUATIONS...

use the following formula to get the search direction, Bk d + qk (αk−1 ) = 0, where qk (αk−1) =

F (xk + αk−1 Fk ) − F (xk ) , αk−1

(1.12)

(1.13)

Bk follows (1.10), αk−1 is the steplength used at the previous iteration, and the equation (1.13) is inspired by [12]. Throughout the paper, we use these notations: k · k is the Euclidean norm, and F (xk ) and F (xk+1 ) are replaced by Fk and Fk+1 , respectively. This paper is organized as follows. In the next section, the algorithm is stated. The global convergence convergence are established in Section 3. The numerical results are reported in Section 4.

2.

The algorithm

In this section, we state our new algorithm based on formulas (1.3), (1.8), (1.10), (1.11) and (1.12) for solving (1.1). Rank-One Updated Algorithm (ROUA). Step 0: Choose an initial point x0 ∈ ℜn , constants r ∈ (0, 1), 0 < δ0 , δ1 , δ2 < 1, α−1 > 0, a positive sequence {εk } satisfying (1.9), symmetric positive definite matrices B0 and B0−1 = H0 . Let: k = 0; Step 1: If kFk k = 0, stop. Otherwise, solving linear equations (1.12) to get dk ; Step 2: Find a αk is the largest number of {1, r, r 2 , r 3 , · · ·} such that (1.8); Step 3: Let the next iterative point be xk+1 = xk + αk dk ; Step 4: Update Bk+1 and Hk+1 by the formula (1.10) and (1.11) respectively; Step 5: Set k := k + 1. Go to step 1. In this paper, we also give the normal BFGS method for solving (1.1), and the algorithm which has the same conditions to ROUA is stated as 5

393

394

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follows. BFGS Algorithm(BFGSA). In ROUA, the step 4 is replaced by: Update Bk+1 by the formula (1.5). Remark 1. (a) By the step 0 of ROUA, there should exist constants λ1 ≥ λ0 > 0 such that 1 1 kdk2 ≥ dT H0 d ≥ kdk2 , ∀ d ∈ Rn . λ0 λ1 (2.1) (b) By the step 4 of ROUA, it is easy to deduce that the updated matrices are symmetric. λ1 kdk2 ≥ dT B0 d ≥ λ0 kdk2,

3.

Convergence Analysis

In this section, we establish the global convergence of ROUA. Let Ω be the level set defined by ε

Ω = {x|kF (x)k ≤ e 2 kF (x0 )k},

(3.1)

where ε is a constant satisfying ∞ X

εk ≤ ε.

(3.2)

k=0

Then the following lemma is satisfied (see [12]), here we also give its proof. Lemma 3.1 Let {xk } be generated by ROUA. Consider the line search (1.8). Then {xk } ⊂ Ω, moreover, {kF (x)k} converges. Proof. By line search (1.8), we have 1

kFk+1 k ≤ (1 + εk ) 2 kFk k ≤ (1 + εk )kFk k.

6

YUAN-LI: SOLVING SYMMETRIC NONLINEAR EQUATIONS...

Since εk satisfies (3.2), from Lemma 3.3 in [5], we conclude that {kFk k} converges. Moreover, we have for all k 1

kFk+1 k ≤ (1 + εk ) 2 kFk k ≤ ··· ≤

k Y

1

(1 + εi ) 2 kF0 k

i=0 k k+1 1 X (1 + εi )] 2 ≤ kF (x0 )k[ k + 1 i−0 k k+1 1 X εi ] 2 k + 1 i−0 ε k+1 ≤ kF (x0 )k[1 + ] 2 k+1 ε ≤ e 2 kF (x0 )k,

= kF (x0 )k[1 +

which implies that {xk } ⊂ Ω. The proof is complete. In order to get the global convergence, the following assumptions are needed [12, 22]. Assumption A (i) F is continuously differentiable on an open convex set Ω1 containing Ω. (ii) The Jaconbian of F is symmetric, bounded and uniformly nonsingular on Ω1 , i.e., there exist constants M ≥ m > 0 such that k∇F (x)k ≤ M,

∀x ∈ Ω1 ,

(3.3)

and k∇F (x)dk ≥ mkdk, ∀x ∈ Ω1 , d ∈ ℜn .

(3.4)

Remark 2. Assumption A(ii) implies that mkdk ≤ k∇F (x)dk ≤ Mkdk, ∀ x ∈ Ω1 , d ∈ ℜn , mkx − yk ≤ kF (x) − F (y)k ≤ Mkx − yk, ∀ x, y ∈ Ω1 . 7

(3.5) (3.6)

395

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In particular, for all x ∈ Ω1 , we have mkx − x∗ k ≤ kF (x)k = kF (x) − F (x∗ )k ≤ Mkx − x∗ k,

(3.7)

where x∗ stands for the unique solution of (1.1) in Ω1 . Lemma 3.2 Let Assumption A hold and {αk , dk , xk+1 , Fk } be generated by ROUA. Then we have ∞ X

kαk Fk k2 < ∞,

(3.8)

∞ X

kαk dk k2 < ∞.

(3.9)

k=0

and

k=0

Proof. By the line search (1.8), we get δ1 kαk Fk k2 + δ2 kαk dk k2 ≤ kFk k2 − kFk+1k2 + εk kgk k2 .

(3.10)

Since {εk } satisfies (3.2) and {kFk k} is bounded, we obtain (3.8) and (3.9) by summing these inequalities (3.10) for k from 0 to ∞. The proof is complete. Lemma 3.3 Let Assumption A hold. Consider ROUA. Then {kBk k} converges, and for any d ∈ ℜn , then there exist constants m0 and M0 such that M0 kdk2 ≥ dT Bk d ≥ m0 kdk2 , f or all k, ∀ d ∈ Rn ,

(3.11)

1 1 kdk2 ≥ dT Hk d ≥ kdk2 , f or all k, ∀ d ∈ Rn , m0 M0

(3.12)

and

which means that the updated matrices are all positive by ROUA.

8

YUAN-LI: SOLVING SYMMETRIC NONLINEAR EQUATIONS...

Proof. By the updated formula (1.10), we have kBk+1k = kBk + vk vkT k ≤ kBk k + kvk k2 = kBk k + δ02 kαk Fk k2 ≤ kB0 k + δ02

k X

kαi Fi k2 .

(3.13)

i=0

P

By (3.8), we know that ki=0 kαi Fi k2 is convergent. Then we can deduce that {kBk k} is convergent. So there exists a constant M0 such that kBk k ≤ M0 , f or all k.

(3.14)

Accordingly, we get the left side of (3.11). Then, we prove the right side of (3.11). By ROUA, we know that the initial matrix B0 is symmetric positive, then we have (2.1). Using (1.10), for k ≥ 1, we have dT Bk d = dT Bk−1 d + dT vk vkT d = dT Bk−1 d + (dT vk )2 ≥ dT Bk−1 d ≥ ··· ≥ dT B0 d ≥ λ0 kdk2 ,

(3.15)

let m0 = λ0 , thus we get (3.11). By (3.11) and the Remark 1(b), we can deduce that the updated matrices are all symmetric and positive definite. Consider Hk = Bk−1 , we obtain (3.12) immediately. The proof is complete. Since Bk is positive definite, then dk which is determined by (1.12) has the unique solution. Lemma 3.4 Let Assumption A hold. If xk is not a stationary point of (1.2), then there exists a constant α′ > 0 depending on k such that when αk−1 ∈ (0, α′ ), and the unique solution d(αk−1 ) of (1.12) such that ∇θ(xk )d(αk−1 ) < 0. 9

(3.16)

397

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YUAN-LI: SOLVING SYMMETRIC NONLINEAR EQUATIONS...

Proof. By (1.13), we can deduce that lim qk (αk−1 ) = ∇F (xk )F (xk ).

αk−1 →0

(3.17)

From (1.12), we get lim

αk−1 →0+

∇θ(xk )d(αk−1 ) = −

lim

αk−1 →0+

F (xk )T ∇F (xk )Bk−1 qk (αk−1 )

= −F (xk )T ∇F (xk )Bk−1 ∇F (xk )F (xk ).(3.18) Since xk is not a stationary point of (1.2), we have ∇F (xk )F (xk ) 6= 0. By ∇F (xk ) is symmetric and Bk is positive, we obtain (3.16). The proof is complete. By (3.11) and (3.14), we have kqk (αk−1 )k = kBk dk k ≤ M0 kdk k, kdk k ≤

1 kqk (αk−1)k. m0

(3.19)

Now we establish the global convergence theorem of ROUA. Theorem 3.1 Let Assumption A hold and {αk , dk , xk+1 , gk+1} be generated by ROUA. Then the sequence {xk } converges to the unique solution x∗ of (1.1) in sense of lim kFk k = 0. (3.20) k→∞

Proof. By Lemma 3.1, we know that {kFk k} converges. By Lemma 3.2, we get lim kαk Fk k = 0, (3.21) k→∞

then, we have lim kFk k = 0

(3.22)

lim αk = 0.

(3.23)

k→∞

or k→∞

10

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399

Therefore, we only discuss the case of (3.23). In this case, for all k sufficiently large and αk′ = αrk , by (1.8), we obtain kF (xk + αk′ dk )k2 − kF (xk )k2 > −δ1 kαk′ F (xk )k2 − δ2 kαk′ dk k2 + εk kF (xk )k2 > −δ1 kαk′ F (xk )k2 − δ2 kαk′ dk k2 .

(3.24)

By Lemma 3.1, we know that {xk } ⊂ Ω is bounded. Considering (3.19), it is easy to deduce that {qk (αk−1 )} and {dk } are bounded. Let {xk } and {dk (α)} converge to x∗ and d∗ , respectively. Then we have lim qk (αk−1) = ∇θ(x∗ ).

k→∞

(3.25)

Let both sides of (3.24) be divided by αk′ and take limits as k → ∞, we obtain θ(x∗ )d∗ ≥ 0. (3.26) By (3.11) and (1.12), we have 0 = dTk Bk dk + qk (αk−1)T dk ≥ m0 kdk k2 + qk (αk−1 )T dk .

(3.27)

As k → ∞, taking limits in both of (3.27) yields ∇θ(x∗ )d∗ ≤ −m0 kd∗ k2 . This together with (3.26) implies d∗ = 0. From (3.19), we have limk→∞ qk (αk−1 ) = 0, which together with (3.25), we obtain ∇θ(x∗ ) = 0.

(3.28)

By ∇θ(x∗ ) = F (x∗ )∇F (x∗ ) and using that ∇F (x∗ ) is nonsingular, we have F (x∗ ) = 0. This implies (3.20). The proof is complete.

4.

Numerical Results

In this section, we report results of some numerical experiments with ROUA and BFGSA. 11

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The discretized two-point boundary value problem is similar to the problem in [16] F (x) = Ax +

1 T (x) = 0, (n + 1)2

where A is the n × n tridiagonal matrix given by 

8 −1  8 −1  −1   −1 8 −1  A= .. ..  . .  ..  . 



.. . .. . −1

     ,    −1 

8

and T (x) = (T1 (x), T2 (x), ..., Tn (x))T with Ti (x) = sinxi −1, i = 1, 2, ..., n. In the experiments, the parameters in ROUA and BFGSA were chosen as r = 0.1, δ0 = 10−4 , δ1 = δ2 = 10−3 , εk = k12 , and k is the number of iteration. The program was coded in MATLAB 6.5.1. We stopped the iteration when the condition kF (x)k ≤ 10−6 was satisfied. The detailed numerical results are listed on the web site http : //210.36.18.9 : 8018/publication.asp?id = 34337. Dolan and Mor´e [7] gave a new tool to analyze the efficiency of Algorithms. They introduced the notion of a performance profile as a means to evaluate and compare the performance of the set of solvers S on a test set P. Assuming that there exist ns solvers and np problems, for each problem p and solver s, they defined tp,s = computing time (the number of function evaluations or others) required to solve problem p by solver s. Requiring a baseline for comparisons, they compared the performance on problem p by solver s with the best performance by any solver on this 12

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401

problem; that is, using the performance ratio rp,s =

tp,s . min{tp,s : s ∈ S}

Suppose that a parameter rM ≥ rp,s for all p, s is chosen, and rp,s = rM if and only if solver s does not solve problem p. Performance profiles of ROUA and BFGSA methods(NI). Figure 1 1 0.9 0.8

Pp:r(p,s) 0

such that x∈C ( [σ − h, σ + a], E n ), x0 = φ , D(t ) xt is continuously differentiable on (σ ,σ + a) and (2.2) is satisfied on (σ , σ + a) .

3

IYAI: MINIMUM CONTROL ENERGY PROBLEM...

411

We shall consider control systems of the form 0

d D(t ) xt = L(t , x, u ) + ∫ A(θ ) x(t + θ )dθ dt −∞

(2.3)

its linear base control system d D(t ) xt = L(t , x, u ) dt

(2.4)

and its free system 0

d D(t ) xt = L( t , x,0 ) + ∫ A(θ ) x(t + θ )dθ dt −∞

(2.5)

where D(t ) xt = x(t ) − Ax(t − 1) , L( t , x, u ) = Gx(t ) + Bx(t − 1) + Fu (t ) + Hu (t − h) . A , B , G are n × n matrices, F , H are n × m matrices. A(θ ) is an n × n matrix

whose elements are square integrable on ( − ∞, 0 ] . Let X (t ) be the unique n × n constant matrix function with the following properties a) X (t ) = 0 , for t < 0 b) X (t ) = I the identity matrix c) X (t ) − AX (t − 1) is continuous on [0, ∞) d) X (t ) satisfies X& (t ) − Ax& (t − 1) = L(t , x,0) for t ∈(0, ∞) − S 2 , where S 2 is the set of non-negative integers. Then a unique solution of (2.4) exist on [1, t ] satisfying xL (t , u ) = φ (t ) for t ∈ [ 0,1] and by Gahl [11], this solution is given by

4

412

IYAI: MINIMUM CONTROL ENERGY PROBLEM...

t

xL (t , u ) = X (t − 1)φ (1) − X (t − 2)φ (1) + ∫ X (t − s − 1) [ Aφ ( s ) + Bφ ( s )]ds 1

t

(2.6)

+ ∫ X (t − s )[ Fu ( s ) + Hu ( s − h)]ds 1

for all t ∈ [1, t1 ]. xL (t , u ) is a continuous function which satisfies (2.4) on [1, t1 ] except for a finite number of points which are contained in the set S3 = S 2 {t : t = t1 ± h + I ; t ≠ k or h ≠ I , for k ∈ S 2 }

Define the matrix functions Z by Z (t , s ) = X (t − s ) F + X (t − s − h) H

(2.7)

Then it follows immediately that t

xL (t , u ) = xL (t ,0) + ∫ Z (t , s )u ( s )ds

(2.8)

1

Since X (t ) is continuous and bounded on [a , b] − S 2 ∂ Z (t , s) = X& (t − s ) F + X& (t − s − h) H ∂t

is continuous and bounded on [a , b] − S 2 . In a similar manner, any solution of system (2.3) following the methods of Gahl [11] and Sinha [1] will be given by t

0

1

−∞

x(t , φ , u ) = x L (t , u ) + ∫ X (t − s) ∫ A(θ ) x(t + θ )dθds

Or t

x(t , φ , u ) = x L (t ,0) + ∫ Z (t , s)u ( s)ds + 1

t

0

1

−∞

∫ X (t − s) ∫ A(θ ) x(t + θ )dθds

In this paper, the control space will be

5

(2.9)

IYAI: MINIMUM CONTROL ENERGY PROBLEM...

(

Lloc [0 , ∞), E m 2

)

413

the space of essentially bounded measurable functions on finite

intervals with values in E m . The control constraint set will be denoted by

(

U = Lloc [0, ∞) , C m 2

)

{

where C m = u ∈ E m : u j ≤1, j =1,2 ,..., m

}

We now give some definitions upon which our study hinges. Definition 2.1 The complete state of system (2.3) at time t is given by y (t ) = {x(t ), φ (t ), u (t )} Definition 2.2 System (2.3) is said to be relatively controllable on J if for any function φ ∈ C , and each x1 ∈ E n , there is an admissible control u ∈ U such that the solution x(t , φ , u ) of (2.3) satisfies xt0 ( ⋅, φ , u ) = φ , x(t1 , φ , u ) = x1 .

Definition 2.3 The reachable set of (2.3) is a subset of E m given by ⎧t ⎫ P (t1 , t 0 ) = ⎨∫ Z (t , s ) u ( s )ds : u ∈ L2 [1, t1 ], E m ⎬ ⎩1 ⎭

(

)

If the controls are in L 2 ( [1, t1 ], C m ) , we define the constraint reachable set by ⎧t ⎫ R (t1 , t 0 ) = ⎨∫ Z ( t , s ) u ( s )ds : u ∈ L2 [1, t1 ], C m ⎬ ⎩1 ⎭

(

)

Note that P(t1 , t 0 ) is a subset of E m which is symmetric about zero. Definition 2.4 The attainable set (2.3) is given by A(t ) = {x(t , φ , u ) : u ∈ U }

6

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IYAI: MINIMUM CONTROL ENERGY PROBLEM...

Definition 2.5 The target set for system (2.3) is given by T (t ) = {x(t , φ , u )} : t1 > t 0 , u ∈ U Corollary 2.1 Consider system (2.5), with all its assumptions. If there exists v>0 such that A(θ ) ≤ M exp(vθ ) ≤ M , θ ∈ (−∞,0] and if 0 ⎧⎪ ⎡ ⎤ ⎫⎪ B(λ ) = ⎨Re λ ≥ 0 , det ⎢λ ( I − Ae − λ ) − G − Be − λ + ∫ exp(λθ )[ A(θ )dθ ] = 0 ⎥ ⎬ = φ ⎪⎩ −∞ ⎣ ⎦ ⎪⎭

Then the solutions of (2.5) is uniformly asymptotically stable such that xt (t 0 , φ ) ≤ K φ exp [− α (t − s ) ], t ≥ t 0 for some α > 0 , K > 0

Proof: The proof can be observed from Sinha [1] and Onwuatu [9] RELATIONSHIP BETWEEN THE REACHABLE SET AND THE ATTAINABLE SET Here we establish the relationship between the two set functions reachable and attainable sets. However, we shall first establish a relationship between the sets, to enable us see that once the properties have been proved for one set then they are applicable to the other. From equation (2.9) A(t ) = X (t − s )[φ (t 0 ) + R(t1 , t 0 )] for u ∈ U , t ∈ [t0 , t1 ]

This means that the attainable set is the transition of the reachable set through η ∈ E n , where η = [φ (t0 ) + R(t1 , t0 )]

We shall then use the attainable set to establish that the two set functions posses the following properties: convexity, closedness and compactness

7

IYAI: MINIMUM CONTROL ENERGY PROBLEM...

415

Theorem 2.1 The attainable set A(t ) is convex and compact Proof: The convexity of A(t ) follows trivially from the convexity of the constrained set U. We now show that, A(t ) is compact. Assume S to be a convex, compact subset of the space of continuous functions C then x(t , S ,0) is bounded. Also, since Z (t , s ) is integrably bounded, by the analyticity of X

and an

assumption that F and H are of bounded variation, coupled with the fact that u ∈ U , the attainable set A(t ) is bounded in E n . From the weak compactness

argument and the assumed compactness property on S , A(t ) is closed in E n (Chukwu [3]). Having established boundedness and closedness properties for A(t ) , we concluded that A(t ) is compact in E n . The convexity and compactness

of the reachable set R(t1 , t 0 ) follows from the convexity and compactness of the attainable set. Also the convexity and compactness of the target set T (t ) follows trivially from the convexity and compactness of the constrained control set U . Corollary 2.2 The reachable set R(t1 , t 0 ) is a continuous set function on [t0 , ∞) to the metric space of compact subsets of E n . Proof: For t ≥ 0, we set t0 = 0 and let Z (t , u ) = Z (t , s ) u (t ) so that, t

Z(t,u) − Z(t 0 ,u) =

∫ Z (t , s) u(s) ds 1

since u is admissible, we have t

Z(t,u) − Z(t 0 ,u ) ≤ ∫ Z (t , s ) ds 1

by the definition of metric d ,

8

416

IYAI: MINIMUM CONTROL ENERGY PROBLEM...

t

d ( R(t ), R(t 0 )) ≤

∫

Z (t , s ) ds since

1

t

∫

Z (t , s ) ds is absolutely continuous, the

1

reachable set is continuous, and by the transition property of both attainable set and the reachable set we conclude that, the attainable set A(t ) is also continuous. 3. CONTROLLABILITY RESULTS Here we state and prove theorems that summarizes our result on the controllability of system (2.3) Definition 3.1 System (2.3) is controllable if A(t ) ∩ T (t ) ≠ ψ . for t ∈ [t0 , t1 ]. We now introduce computational criteria for system (2.3) following the methods of (Gabasov and Kirillova [12]; Chukwu [4]) to check when the system (2.3) is relatively controllable by introducing the following notation Qk ( s ) = GQk −1 ( s ) + BQk −1 ( s − 1) + AQk ( s − 1), k = 0, 1, 2, L; s = 0, 1, 2, L ⎧I , Qk ( s ) = ⎨ ⎩0,

k = 0, s = 0 k = 0, s < 0 or k < 0, s = 0

and Ω(t1 ) = {Qk ( s ) F , Qk ( s) H , k = 0, 1, L n − 1, s ∈ [t 0 , t1 ] }

We define the rank of Ω(t1 ) as the rank of the block matrix composed of all matrices from the set Ω(t1 ) . Theorem 3.1 (Chukwu [4]; Klamka [8], for every t1 ∈ (0, ∞) the following conditions are equivalent: (i) If c T Z (t , s) = 0 for t ∈ J and c ∈ E n , then c = 0 9

IYAI: MINIMUM CONTROL ENERGY PROBLEM...

(ii)

rank Ω(t1 ) = n

(iii)

The system (2.3) without constraints on the control is (globally) relatively controllable in J

Remark 3.1 To prove the above theorem, we use the fact that Z (t , s) = X (t − s) F + X (t − s − h) H and proceeds as in the cited proofs. The proof of conditions (i) and (ii) follows the same pattern as that shown in Chukwu [4], whereas that of condition (iii) follows the same pattern as that shown in the monograph Klamka [8]. The next theorems are direct consequences of Theorem 3.1 and Corollary 2.1 Theorem 3.2 The system (2.3) is completely controllable in the time interval [t0 , t1 ] if: (i) the system (2.5) is uniformly asymptotically stable (ii) the (n × n) - dimensional controllability matrix W (t1 ) of the system t

(2.3) given by W (t1 ) = ∫ Z (t , s ) Z T (t , s )ds satisfies rank W (t1 ) = n and 1

W −1 (t1 ) ∈ E n×n , Z T (t , s ) ∈ E m×n for every t ∈ [1, t1 ] and (x1 − x(t , 0) ) ∈ E n

Proof: Let the assumptions of Theorem 3.1 be satisfied, and the solutions of system (2.5) uniformly asymptotically stable. We let

(

)

y 0 = ( x(0), x0 ) ∈ E n × L2 [t 0 , t1 ], E n be any initial complete state of the system (2.3)

and x1 ∈ E n be any vector. We shall prove that the admissible control function

(

)

u ∈ L2 [t 0 , t1 ], E m of the form

10

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u (t ) = Z T (t , s)W −1 (t1 )( x − x(t , 0) )

(3.1)

for t ∈ [ t 0 , t1 ] steers the system (2.3) from the initial state y 0 to the state x(t1 , u ) = x1 . Substituting (3.1) into (2.9) for t = t1 , we get t

(

)

x(t1 , u ) = x L (t1 , 0) + ∫ Z (t , s ) Z T (t , s) × W −1 (t1 )( x1 − x(t1 , 0) ) ds 1

= x(t1 , 0) + W (t1 )W −1 (t1 )( x1 − x(t1 , 0) ) = x1

Since y 0 and x1 were arbitrary, the system (2.3) is completely controllable in the interval [t 0 , t1 ] . 4. THE MINIMUM CONTROL ENERGY The minimum control energy problem is best understood in the context of capture problem or rescue effort (see Chukwu [3] and references therein). Emphasis here is on the minimum control energy to reach the target or intercept it. However the need to first determine the existence of control for pursuit is evident. If the intersection of the attainable set and the reachable set is non-empty, the target could be reached using appropriate control energy. The next theorem establishes the existence of such control efforts required to capture a moving target- either moving point function or a compact set function Theorem 4.1 In system (2.3), A(t ) ∩ T (t ) ≠ ψ , if and only if, there exists an admissible control such that weapon for the capture of a target satisfies system (2.3) on the interval [t0 , t1 ].

Proof: Suppose the state y (t ) of the weapon for the capture or rescue mission satisfies the given system then, y (t ) ∈ T (t ). We are to show that there exist x(t , φ , u ) ∈ A(t ) such that y (t ) = x(t , φ , u ) for some u . Let {u n } be a sequence in U ,

since U is compact, lim u n = u. Now x(t ,φ , un ) ∈ A(t ) and from (2.9). n →∞

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t1

t1

0

1

−∞

x(t1 , φ , u n ) = x L (t1 ,0) + ∫ Z (t1 , s) u ( s )ds + ∫ X (t1 − s ) ∫ A(θ ) x(t + θ )dθ ds n

1

419

(4.1)

taking limits on both sides of (4.1), we have t1

t1

0

1

−∞

lim x(t1 , φ , u n ) = x L (t1 ,0) + ∫ Z (t1 , s) lim u n ( s )ds + ∫ X (t1 − s ) ∫ A(θ ) x(t + θ )dθ ds (4.2) n→∞

n →∞

1

(4.2) implies t1

t1

0

1

−∞

lim x(t1 , φ , u n ) = x L (t1 ,0) + ∫ Z (t1 , s )u ( s )ds + ∫ X (t1 − s ) ∫ A(θ ) x(t + θ )dθ ds n

n→∞

1

= x(t , φ , u )

Since A(t ) is compact, lim x(t1 , φ , u n ) ∈ A(t ). That is, x(t ,φ , u ) ∈ A(t ). Thus, there n→∞

exists u ∈ U such that x(t ,φ , u ) = y (t ) . Since y (t ) ∈ T (t ) and A(t ) , A(t ) ∩ A(t ) ≠ ψ or A(t ) ∩ T (t ) ≠ ψ . Conversely, suppose A(t ) ∩ T (t ) ≠ ψ . There is, y (t ) ∈ A(t ) such that y (t ) ∈ T (t ) . This implies that y (t ) = x(t ,φ , u ) being an element of the attainable set. Thus, establishing that, the state of the weapon for capture of a target satisfies systems (2.3). This completes the proof. EXISTENCE OF MINIMUM CONTROL ENERGY Theorem 4.2 In systems (2.3), suppose the system is controllable using a admissible control at time t, then there exists a minimum control energy Proof: Let y (t ) ∈ T (t ) and by the controllability of (4.1), y (t ) ∈ A(t ) , since A(t ) is the translation of the reachable set through η , y (t ) ∈ R(t1 , t 0 ) for t1 > t 0 . Let t ∗ = inf{t , y (t ) ∈ R(t1 , t0 )}. Now, 0 ≤ t ∗ ≤ t1 and there is a non-increasing sequence of

time tn converging to the minimum time t ∗ and a sequence of controls un ∈ U . Let y (tn ) = z (t , u ) ∈ R(t1 , t0 ). Also y (t ∗ ) − z (t ∗ , u n ) ≤ y (t ∗ ) − y (t n ) + y (t n ) − z (t ∗ , u n ) ≤ y (t ∗ ) − y (t n ) + z (t n , u n ) − z (t ∗ , u ∗ )

12

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tn

≤ y (t ∗ ) − y (t n ) + ∫ ∗ z ( s ) ds t

By continuity of y (t ) and the integrability of z (t ) , it follows that z (t ∗ , un ) → W (t ∗ ) as n → ∞ where W (t ∗ ) = z (t ∗ , u ∗ ) for some u ∗ ∈ U . Since ∗

R (t1 , un ) contains z (t ∗ , un ) for each n and R (t1 , t 0 ) is closed then W (t ∗ ) = z (t ∗ , u ∗ ) ∈ R(t1 , t0 ) for some u ∗ ∈ U and by definition of t ∗ , u ∗ is the required

minimum control energy. This establishes the existence of minimum control energy for system (2.3). FORM OF THE MINIMUM CONTROL ENERGY We shall in this section present the form of the minimum control energy for system (2.3), and this can be seen in the following theorem. Theorem 4.3 In system (2.3) u ∗ is the minimum control energy if and only if u ∗ is of the form. u ∗ (t ) = sgn[k T Z (t , s )] where k ∈ E n

Proof: Suppose u ∗ is the minimum control energy for system (2.3) then it maximizes the rate of change of Z (t , u ) = Z (t , s ) u (t ) in the direction of k , that is, we want to minimize k T Z (t , s ) since u (t ) are admissible controls, that is, they are constrained to lie in a unit sphere, we have k T Z (t , s )u (t ) ≤ k T Z (t , s) ≤ k T Z (t , s ) sgn[k T Z (t , s )] ≤ k T Z (t , s )u ∗ (t )

This shows that the minimum control u ∗ has the form u ∗ = sgn [k T Z (t , s )] . Conversely, let u ∗ = sgn[k T Z (t , s )] then for admissible controls u ∈ U

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t1

t1

t1

1

1

[

]

k T ∫ Z (t1 , s ) u ( s ) ds ≤ ∫ k T Z (t1 , s ) sgn ∫ k T Z (t1 , s ) ds 1

t1

[

]

≤ ∫ k T Z (t1 , s ) ds 1

t1

≤ ∫ k T [Z (t1 , s )]u ∗ ( s )ds 1

This shows that u ∗ maximizes k T [ Z (t , s)] over all admissible controls u , hence it is the minimum control energy for system (2.3). REALIZATION FROM THEOREM 4.3 Let t∗

t

z = z (t , u ) = ∫ Z (t , s ) u ( s ) ds and z = z (t , u ) = ∫ Z (t , s )u ( s ) ds ∗

∗

∗

∗

1

1

From the result in Theorem 4.3, k T z ≤ k T z ∗ ⇒ k T ( z − z ∗ ) ≤ 0 , for each z ∈ R(t1 , t 0 ) . Since the reachable set is closed, convex subset of E n , there is a support plane η of R(t1 , t 0 ) through z ∗ with k ≠ 0 an outward normal to η at z ∗ and hence z ∗ is in the boundary of R(t1 , t 0 ) . Thus, showing that, if u ∗ be the minimum control energy then the target is on the boundary of the reachable set. The above realization is now stated below as a theorem. Theorem 4.4 Let u ∗ be the minimum control energy for system (2.3), with t ∗ the minimum time, then the target x(t ∗ ) = x(t ∗ ,φ , u ∗ ) is in the boundary of the attainable set A(t ) i.e. y (t ) ∈ ∂A(t ) ( ∂ symbolizes boundary). Proof: Suppose u ∗ is the minimum control energy, then ∗

x(t ∗ , φ , u ∗ ) = X (t ∗ − s)[η + z ∗ ] , z ∗ ∈ R(t1 , t 0 ). Therefore x(t ∗ , φ , u ∗ ) ∈ A(t ∗ ) suppose

x(t ∗ , φ , u ∗ ) is not on the boundary of A(t ∗ ) then x(t ∗ , φ , u ∗ ) is in the interior of A(t ∗ ); t ∗ > t 0 . Therefore, there is a ball, B ( x(t ∗ , φ , u ∗ ), r ) ∈ A(t ) . Because A(t ) is a

continuous set function of t, we can preserve the above inclusion for t near t ∗ . if

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we reduce the size of the ball B( x(t ∗ , u ∗ ), r ) ; that is, if there is an ε > 0 such that r⎞ ⎛ B⎜ x(t ∗ , φ , u ∗ ), ⎟ ⊂ A(t1 ) for t ∗ − ε ≤ t1 ≤ t ∗ . Thus x(t ∗ , φ , u ∗ ) ∈ A(t1 ) for 2⎠ ⎝ t ∗ − ε ≤ t1 ≤ t ∗ . This contradicts the optimality of t ∗ and u ∗ as the minimum control

energy. The contradiction, however, proves that x(t ∗ , φ , u ∗ ) is on the boundary of the attainable set i.e. x(t ∗ , φ , u ∗ ) ∈ ∂ A(t ∗ ) . UNIQUENESS OF MINIMUM CONTROL ENERGY Theorem 4.5 Consider systems (2.3) with its basic assumption, if u ∗ is the minimum control, then it is unique Proof: Let u ∗ and v ∗ be minimum controls for system (2.3), then both u ∗ and v ∗ maximize k T [ Z (t , s )] for t ∈ J over all admissible controls u , and so we have k

T

t∗

t∗

1

1

∫ Z (t , s)u (s) ds ≤ ∫

k T Z (t , s ) u ∗ ( s ) ds

(4.3)

k T Z (t , s ) v ∗ ( s ) ds

(4.4)

Also using v ∗ we have k

T

t∗

t∗

1

1

∫ Z (t , s)u (s) ds ≤ ∫

Clearly, ⎫⎪ t ⎧⎪ t max ⎨k T ∫ [ Z (t , s )] u ( s ) ds ⎬ = ∫ k T Z (t , s ) u ∗ ( s ) ds −1≤u ≤1 ⎪⎭ 1 ⎪⎩ 1 ∗

∗

(4.5)

implies ⎫⎪ t ⎧⎪ t max ⎨k T ∫ [ Z (t , s )] u ( s) ds ⎬ = ∫ k T Z (t , s ) v ∗ ( s ) ds −1≤u ≤1 ⎪⎭ 1 ⎪⎩ 1 ∗

∗

(4.6)

Subtracting equation (4.5) from equation (4.6) gives v∗ (t ) − u ∗ (t ) = 0 , for all t ∈ J implies v ∗ = u ∗ , proving the uniqueness of the minimum control energy.

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423

Theorem 4.6 Consider systems (2.3) with its basic assumption. Suppose (i) (2.5) is uniformly asymptotically stable (ii)

rank Ω(t1 ) = n

(iii)

u ∗ (t ) = sgn[k T Z (t , s )]

Then there exist a unique minimum control energy that drives the system to target. Proof: Immediately from Theorems 3.1, 4.3 and 4.5 5. EXAMPLE Here, we give numerical examples to illustrate the theoretical analysis. Consider the neutral system d (x(t ) − A−1 x(t − 1) ) = A1 x(t − 1) + A0 x(t ) + B0 u (t ) dt 0

+ B1u (t − 1) + C 0 ∫ exp(vθ ) x(t + θ )dθ

(5.1)

−∞

where ⎡0 A−1 = ⎢ ⎣1

1⎤ , 0⎥⎦

⎡− 1 A0 = ⎢ ⎣ 1

1⎤ , − 2⎥⎦

⎡0 A1 = ⎢ ⎣0

⎡1 ⎤ ⎡0 ⎤ ⎡0 B0 = ⎢ ⎥ , B1 = ⎢ ⎥ , C 0 = ⎢ ⎣0 ⎦ ⎣1 ⎦ ⎣0

3⎤ , − 1⎥⎦

0⎤ − 1⎥⎦

The characteristic root of the homogenous equation 0

x& (t ) = A−1 x& (t − 1) + A1 x(t − 1) + A0 x(t ) + ∫ exp(vθ ) x(t + θ )dθ −∞

is 0

λ2 + 3λ + 1 + (3λ − λ2 ) e − 2λ + (2 − 3λ ) e − λ + (λ + 1) ∫ exp[(λ + ν )θ ]dθ = 0 −∞

16

(5.2)

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and every root of (5.2) has negative real part. Hence by Corollary 2.1, the system (5.1) with u = 0 is uniformly asymptotically stable. Furthermore, we shall show that condition (ii) of Theorem 3.1 is satisfied i.e. rank Ω(t1 ) = n , n = 2 . We require all matrices belonging to the set Ω(t1 ) : ⎡1 ⎤ Q0 (0) = IB0 = ⎢ ⎥ and Q0 ( s ) = 0 for s < 0 ⎣0 ⎦ ⎡0 ⎤ ⎡1 ⎤ ⎡− 1⎤ Q0 (1) = A−1 B0 = ⎢ ⎥ , Q0 (2) = A−21 B0 = ⎢ ⎥ , Q1 (0) = A0 B0 = ⎢ ⎥ ⎣1 ⎦ ⎣0 ⎦ ⎣ 1⎦ ⎡ 1⎤ Q1 (1) = A0 A−1 B0 + A1 B0 + A−1 A0 B0 = ⎢ ⎥ ⎣− 3⎦ ⎡ 2⎤ Q1 (2) = A0 A−21 B0 + A1 A−1 B0 + A−1 B0 = ⎢ ⎥ ⎣1 ⎦ ⎡0 ⎤ Q0 (0) = IB1 = ⎢ ⎥ and Q0 ( s ) = 0 for s < 0 ⎣1 ⎦ ⎡1 ⎤ ⎡0 ⎤ ⎡ 1⎤ Q0 (1) = A−1 B1 = ⎢ ⎥ , Q0 (2) = A−21 B1 = ⎢ ⎥ , Q1 (0) = A0 B1 = ⎢ ⎥ ⎣0 ⎦ ⎣1 ⎦ ⎣ − 2⎦ ⎡0 ⎤ Q1 (1) = A0 A−1 B1 + A1 B1 + A−1 A0 B1 = B1 = ⎢ ⎥ ⎣1 ⎦ ⎡ 2⎤ Q1 (2) = A0 A−21 B1 + A1 A−1 B1 + A−1 B1 = ⎢ ⎥ ⎣ − 2⎦ Ω(t1 ) = {Q0 (0), Q0 (1), Q0 (2), Q1 (0), Q1 (1), Q1 (2), Q0 (0), Q0 (1), Q0 (2), Q1 (0), Q1 (1), Q1 (2)}, n=2

⎡1 0 1 − 1 2 2 0 1 0 1 0 2 ⎤ rank Ω(t1 ) = rank ⎢ ⎥=2 ⎣0 1 0 1 − 3 1 1 0 1 − 2 1 − 2 ⎦

17

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425

From the sequel, condition (ii) of Theorem 3.1 is satisfied in any interval [t 0 , t1 ] and since the system (5.1) with u = 0 is uniformly asymptotically stable, we conclude that system (5.1) is completely controllable. To see the minimum control energy, when complete controllability is established, it is easily verified (Chukwu [5], Pp 60-64) that, the principal fundamental matrix solution of (5.1) (with u = 0 ) is given by ⎛ −1 ⎜ te X (t ) = ⎜ ⎜ − 1 + 9t ⎜ ⎝ 6

− 1 + 9t ⎞ ⎟ 6 ⎟ ⎟ te − 2 ⎟ ⎠

Then by Theorem 4.3, the minimum control energy is of the form u ∗ (t ) = sgn[k T Z (t , s )]

[

]

= sgn k T ( X (t − s ) B + X (t − s − h) B1 ) ⎧ ⎛ −1 ⎜ te ⎪⎪ u ∗ (t ) = sgn ⎨(k1 , k 2 )⎜ ⎜ − 1 + 9t ⎪ ⎜ ⎝ 6 ⎩⎪

− 1 + 9t 6

⎛ ⎞ −1 ⎜ te ⎟1 ⎟⎛⎜ ⎞⎟ + (k , k )⎜ 1 2 ⎜ ⎟ ⎜ − 1 + 9t − 2 ⎟⎝ 0 ⎠ te ⎟ ⎜ ⎝ 6 ⎠

(5.3) ⎞ ⎫ ⎟ 0 ⎪ ⎟⎛⎜ ⎞⎟⎪⎬ ⎟⎜1 ⎟ te − 2 ⎟⎝ ⎠⎪ ⎠ ⎭⎪

− 1 + 9t 6

k k ⎞ ⎛ = sgn ⎜ (6te −1 + 9t − 1) 1 + (6te − 2 + 9t − 1) 2 ⎟ 6 6⎠ ⎝

Hence, the minimum control energy that drives the system state to the target is unique and is given by the equation (5.3). CONCLUSION We have established the relationship that exists between the minimum control energy problem and the complete controllability of systems (2.3). Namely, if system (2.3) is completely controllable on J then there exists minimum control energy for the system. The establishment of the existence, form, and uniqueness of this control energy for infinite neutral differential systems is one of the major results of this research.

18

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REFERENCE 1. A. S. C. Sinha, Null controllability of nonlinear infinite delay systems with restrained control, Int. J. Control 42 (3), 735-741, (1985) 2. E. N. Chukwu, Control in W2(1) of nonlinear interconnected systems of neutral type, J. Austral. Math .Soc. Series B 36 286 – 312, (1994). 3. E. N. Chukwu, The time optimal control theory of linear differential equation of the neutral type, Comput., Math, Appl. Great Britain, 16 851-866, (1988). 4. E. N. Chukwu, Euclidean controllability of linear delay systems with limited controls. IEE Trans. Automat. Contr. 24 (6), 798-800, (1979) 5 E. N. Chukwu, Stability and time optimal control of hereditary systems with application to the economic dynamics of the US, (2nd Edition) Series on Advances in Mathematics for Applied Science), (2002). 6. I. Davies, Euclidean null controllability of infinite neutral differential systems, ANZIAM J. 48 285-293, (2006) 7. I. Davies, Minimum Control energy problem for infinite delay systems, A.M.S.E. 44 (1), 58-68, (2007) 8. J. Klamka, Controllability of control systems. War-saw: Warsaw University of Technology Press, Polish (1990). 9. J. U. Onwuatu, Null Controllability of nonlinear infinite neutral system, KYBERNETIKA, 29 (4), 325-336, (1993). 10. K. Balachandran and E. R. Anandhi, Controllability of neutral functional integrodifferential infinite delay systems in Banach Spaces, Taiwanese Journal of Mathematics, 8 (4), 689 – 702, (2004). 11. R. D. Gahl, Controllability of nonlinear system of neutral type, Journal of Mathematical Analysis and Applications 63 33 – 42, (1978). 12. R. Gabasov and F. Kirillova, The qualitative theory of optimal processes. Marcel Dekker, New York (1976).

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13. R. G. Underwood and E. N. Chukwu Null controllability of nonlinear neutral differential equations. Journal of Mathematical Analysis and Applications 129 326-345, (1988) 14. X. Fu, Controllability of neutral functional differential systems in abstract space, J. Applied Mathematics and Computation, 141 281-296, (2003)

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TABLE OF CONTENTS, JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOLUME 5, NO.4, 2010

Numerical simulation of the generalized Huxley equation by homotopy analysis method, K.M. Hemida, M.S. Mohamed,……………………………………………………………………….344 Basis property in Lp(0,1) of the root functions corresponding to a boundary-value problem, H. Menken, Kh. R. Mamedov,……………………………………………………………………..351 Dimensions of bivariate C1 cubic spline spaces over unconstricted triangulations with valence six, Huan-Wen Liu, Na Yi,……………………………………………………………………..357 Existence of solutions to a model of long range diffusion involving flux, Manar A .Qudah, Marwan S. Abualrub,…………………………………………………………………………...370 On Best N-Simultaneous Approximation, Eyad Abu-Sirhan,………………………………….377 A Rank-One Fitting Method for Solving Symmetric Nonlinear Equations, Gonglin Yuan, Xiangrong Li,…………………………………………………………………………………...389 Minimum control energy problem for infinite neutral differential systems, Davies Iyai,……...408