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

Abstract

517.988.6 The conditions for the existence of solutions of nonlinear differential equations in a space of functions bounded on the axis are established by using local linear approximations of these equations. Main Object of InvestigationLet E be a finite-dimensional Banach space with norm ⋅ E , let X and Y be arbitrary Banach spaces, and let L X Y ( , ) be a Banach space of linear continuous operators A acting from the space,We now consider differential operators F and G acting from the space C 1 (R , E) into the spaceR and defined by the formulaswhere the mappings f : R × E → E and g : E → E are continuous and, for any number r > 0,In view of the requirements imposed on the mappings f and g, the operators F and G are bounded. Recall that the operator C : X → Y is bounded if it maps every bounded set into a bounded set [1].

Ограниченные И Периодические Решения Нелинейных Дифференциально-Функциональных Уравнений

“…[1]- [4]). Здесь в общем случае изложена идея этого мето-да, показано, что он применим к исследованию широкого класса нелинейных дифференциально-функциональных уравнений, и приведены примеры нахож-дения условий разрешимости нелинейных уравнений в пространствах ограни-ченных непрерывных на оси функций и периодических функций.…”