Conditions for the existence of bounded solutions of nonlinear differential and functional differential equations

Abstract

Let E be a finite-dimensional Banach space, let C 0 .R; E/ be a Banach space of functions continuous and bounded on R and taking values in E; let KW C 0 .R; E/ ! C 0 .R; E/ be a c-continuous bounded mapping, let AW E ! E be a linear continuous mapping, and let h 2 C 0 .R; E/: We establish conditions for the existence of bounded solutions of the nonlinear equation Main Functional Spaces and c-Continuous OperatorsLet R be the set of all real numbers, let C be the set of all complex numbers, let E be a real or complex Banach space with norm k k E ; and let L.E; E/ be the Banach algebra of all linear continuous operators AW E ! E with the normBy C 0 .R; E/ we denote the Banach space of functions x D x.t/ continuous and bounded on R and taking values in E with the normNote that kak E D jaj if E D R or E D C: Let C 1 .R; E/ denote the Banach space of functions x 2 C 0 .R; E/ each derivative of which is an element of the space C 0 .R; E/ with the normWe say that a sequence .x n / n 1 of elements of the space C 0 .R; E/ converges locally to an element x 2 C 0 .R; E/ and write x n loc: C 0 .R;E / ! x as n ! 1 if this sequence is bounded and, for every number p > 0; one has National University of Water Management and Nature Resources Use, Rivne, Ukraine.

Method of local linear approximation in the theory of bounded solutions of nonlinear differential equations

“…In [21][22][23], we study the conditions of existence of bounded solutions of this equation and their properties. For Eq.…”

“…Then, for any number H > 0, there exist numbers r > 0 and k ∈ { } R\ 0 such that f x ( ) satisfies inequality (22), where K is the operator defined by equality (19).…”

Systems that generate solutions with a small period

“…Условие вида (4) использовано в работе В. Е. Слюсарчука [5] при изучении вопроса о существовании ограниченных решений скалярного уравнения. The problem of existence of periodic solutions is one of the traditional problems of the theory of differential equations.…”

Systems that generate solutions with small period

“…Из (I.1) и (II.1) следует, что на множестве Σ определен диффеоморфизм Пуанкаре T (ξ) = x(ω, 0, ξ, H) системы (5).…”

“…Хорошо известно, что в этом случае x(t, 0, ξ 0 (H), H)ω-периодическое решение системы (5). Из (I.2) следует, что график этого решения лежит в R × Σ 1 .…”