Elements of Mathematical Theory of Evolutionary Equations in Banach Spaces (World Scientific Nonlinear Science Series a)

World Scientific, 2013. — 408 p. — ISBN: 9789814434829. 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. Contents: Reducibility Problems for Difference Equations. Invariant Tori of Difference Equations in the Space M. Periodic Solutions of Difference Equations. Extention of Solutions. Countable-Point Boundary-Value Problems for Nonlinear Differential Equations. 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 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 -- P. 4 Of Cover. 1. Reducibility Problems For Difference Equations -- 2. Invariant Tori Of Difference Equations In The Space M -- 3. Periodic Solutions Of Difference Equations. Extension Of Solutions -- 4. Countable-point Boundary-value Problems For Nonlinear Differential Equations. Anatoly M. Samoilenko, National Academy Of Sciences, Ukraine; Yuriy V. Teplinsky, Kamyanets-podislsky National University, Ukraine. Includes Bibliographical References And Index. Preface; Contents; 1. Reducibility problems for difference equations; 1.1 On analogs of the Erugin and Floquet-Lyapunov theorems for equations in the space; 1.2 Linear equations in the space defined on tori; 1.3 Nonlinear almost periodic equations deFIned on an infinite-dimensional torus; 1.4 Reduction of a discrete dynamical system in the space Rq to the canonical form in a neighborhood of its invariant set; 1.5 Investigation of a discrete dynamical system defined in an abstract Banach space in a neighborhood of its invariant set; 2. Invariant tori of difference equations in the space 3.2 Periodic solutions of nonlinear difference equations of the first order in an abstract Banach space3.3 Periodic solutions of nonlinear difference equations of the second order; 3.4 Asymptotic periodicity of solutions of a linear equation in a complex Banach space; 3.5 Extension "to the left" of solutions of nonlinear degenerate difference equations; 4. Countable-point boundary-value problems for nonlinear differential equations; 4.1 Boundary-value problem on the semiaxis; 4.2 Boundary-value problems on an interval; 4.3 Reduction to a finite-dimensional multipoint case 2.1 Sufficient conditions of existence of a continuous invariant torus2.2 On the differentiability of an invariant torus with respect to the angular variable and the parameter in the coordinate wise meaning; 2.3 Truncation method in studying the smoothness of invariant tori; 2.4 Case of linear and quasilinear systems defined on the infinite-dimensional tori; 2.5 On the existence of the invariant tori of nonlinear systems; 2.6 Differentiability of the invariant tori of nonlinear systems in the Frechet meaning 2.7 Conditions of existence of the Green-Samoilenko function for a linear system defined on the set × T . Reduction of the problem of construction of the invariant torus of this system to an analogous problem in the space Rs × Tm2.8 On the existence of the smooth bounded semi-invariant manifold of a degenerate nonlinear system; 3. Periodic solutions of difference equations. Extension of solutions; 3.1 On the periodic solutions of linear and quasilinear equations with periodic coefficients in the space 4.4 Another means of the reduction. Conditions of commutativity of the limiting transitions (4.42) and (4.43)4.5 Boundary-value problems for differential equations unsolvable with respect to the derivative; 4.6 Reduction to a finite-dimensional multipoint problem; Bibliography; Index