Existence of Almost Periodic Solutions for a Class of Abstract Impulsive Differential Equations

Abstract

We study the existence of piecewise almost periodic solutions for a class of abstract impulsive semilinear differential equations.

Piecewise asymptotically almost periodic solution ofneutral Volterra integro-differential equations with impulsiveeffects

“…For more details about this topic, one can see [6,7,9,16,23,24,26], where the authors have given an important overview about the theory of impulsive differential and integro-differential equations. On the other hand, the asymptotic properties of solutions of impulsive differential equations have been studied from different points, such as almost periodicity [8,17,18,20,21,29], almost automorphy [1,30], asymptotic stability [19,28], asymptotic equivalence [5], and oscillation [15]. However, the existence and uniqueness of a piecewise asymptotically almost periodic mild solution for neutral Volterra integro-differential equations with impulsive effects in the form (1.1) is an untreated topic in the literature and this fact is the motivation of the present work.…”

Existence and stability of almost periodic solutions for impulsive differential equations

“…Obviously, if f AP T (R × X, X) and for each compact set K ⊆ X, f(t, x) : R × X X is uniformly continuous in x K uniformly in t R, then {f(⋅, x) : x K} is a uniformly piecewise almost periodic family Lemma 2.2 [18] Let j AP T (R, X), then the range of j, R(j), is a relatively compact subset of X. Lemma 2.3 If a set B ⊆ AP T (R, X) is relatively compact, then ∪ f B R ( f ) is a relatively compact set in X.…”

“…Lemma 2.5 [18] Assume that f AP T (R, X), T AP(L(X)), the sequence {x i : i Z} is almost periodic, and t j i , j Z, are equipotentially almost periodic. Then for each ε > 0 there exist relatively dense sets ε,f ,x i ,T of R and Q ε,f ,x i ,T of Z satisfying:…”

“…There are many articles [9][10][11][12][13][14][15][16] about existence of solutions, periodic solutions and stability for impulsive differential equations. But, only a few articles [17,18] have studied the existence of almost periodic solutions to abstract impulsive differential equations in Banach space.…”

“…In [18], by using Contraction mapping principle, Henriquez et al studied the existence of almost periodic solutions to the following impulsive differential equation:…”