Elements Of Mathematical Theory Of Evolutionary Equations In Banach Spaces

by Anatoliy M Samoilenko, Yuriy Teplinsky

Overview

Evolutionary equations are studied in abstract Banach spaces and in spaces of bounded number sequences. For linear and nonlinear difference equations, which are defined on finite-dimensional and infinite-dimensional tori, the problem of reducibility is solved, in particular, in neighborhoods of their invariant sets, and the basics for a theory of invariant tori and bounded semi-invariant manifolds are established. Also considered are the questions on existence and approximate construction of periodic solutions for difference equations in infinite-dimensional spaces and the problem of extendibility of the solutions in degenerate cases. For nonlinear differential equations in spaces of bounded number sequences, new results are obtained in the theory of countable-point boundary-value problems.The book contains new mathematical results that will be useful towards advances in nonlinear mechanics and theoretical physics.

Product Details

ISBN-13:	9789814434829
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	06/28/2013
Series:	World Scientific Series On Nonlinear Science Series A , #86
Pages:	408
Product dimensions:	5.90(w) x 9.10(h) x 1.10(d)

Table of Contents

Preface v

1 Reducibility problems for difference equations 1

1.1 On analogs of the Erugin and Floquet-Lyapunov theorems for equations in the space m 2

1.2 Linear equations in the space m defined on tori 9

1.3 Nonlinear almost periodic equations denned on an infinite-dimensional torus 18

1.4 Reduction of a discrete dynamical system in the space R^q to the canonical form in a neighborhood of its invariant set 29

1.5 Investigation of a discrete dynamical system defined in an abstract Banach space in a neighborhood of its invariant set 42

2 Invariant tori of difference equations in the space m 61

2.1 Sufficient conditions of existence of a continuous invariant torus 62

2.2 On the differentiability of an invariant torus with respect to the angular variable and the parameter in the coordinate-wise meaning 75

2.3 Truncation method in studying the smoothness of invariant tori 89

2.4 Case of linear and quasilinear systems defined on the infinite-dimensional tori 119

2.5 On the existence of the invariant tori of nonlinear systems 137

2.6 Differentiability of the invariant tori of nonlinear systems in the Fréchet meaning 149

2.7 Conditions of existence of the Green-Samoilenko function for a linear system defined on the set m × T_∞ Reduction of the problem of construction of the invariant torus of this system to an analogous problem in the space R^s × T_m 166

2.8 On the existence of the smooth bounded semi-invariant manifold of a degenerate nonlinear system 172

3 Periodic solutions of difference equations. Extension of solutions 193

3.1 On the periodic solutions of linear and quasilinear equations with periodic coefficients in the space m 193

3.2 Periodic solutions of nonlinear difference equations of the first order in an abstract Banach space 216

3.3 Periodic solutions of nonlinear difference equations of the second order 229

3.4 Asymptotic periodicity of solutions of a linear equation in a complex Banach space 253

3.5 Extension "to the left" of solutions of nonlinear degenerate difference equations 260

4 Countable-point boundary-value problems for nonlinear differential equations 289

4.1 Boundary-value problem on the semiaxis 289

4.2 Boundary-value problems on an interval 310

4.3 Reduction to a finite-dimensional multipoint case 315

4.4 Another means of the reduction. Conditions of commutativity of the limiting transitions (4.42) and (4.43) 335

4.5 Boundary-value problems for differential equations unsolvable with respect to the derivative 344

4.6 Reduction to a finite-dimensional multipoint problem 364

Bibliography 385

Index 397