Asymptotically almost automorphic solutions for some integrodifferential equations with nonlocal initial conditions

Abstract

Of concern is a class of abstract semilinear integrodifferential equations with nonlocal initial conditions. Under some suitable hypotheses, we establish some new theorems about the existence of asymptotically almost automorphic solutions to the integrodifferential equations. Moreover, an example is given to illustrate our results.

Almost automorphic solutions of nonautonomous evolution equations

“…On the other hand, almost automorphic functions and their applications have attracted more and more attention (cf. [2,3,6,[10][11][12]14,16,17,[19][20][21][22]27,28] and the references cited therein). Recall that almost automorphic functions, which were introduced by Bochner [1], turn out to be an important generalization of almost periodic functions.…”

The EXISTENCE OF S-Asymptotically Ω-Periodic MILD SOLUTIONS FOR SOME DIFFERENTIAL EQUATION WITH NONLOCAL CONDITIONS

“…(1). The Cauchy problems with nonlocal conditions have been investigated in many papers ( [1,2,10,15]). …”

“…Hui-Sheng Ding et al [10] investigated the existence of asymptotically almost automorphic solutions for some integrodifferential equations with nonlocal initial conditions…”

Existence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equations

“…Since then, this notion has found several developments and has been generalized into different directions. Until now, the asymptotically almost automorphic functions as well as the asymptotically almost automorphic solutions for differential systems have been investigated by many mathematicians; see [19] by Bugajewski and N'Guérékata, [20] by Diagana, Hernández, and dos Santos, and [21] by Ding, Xiao, and Liang for the asymptotically almost automorphic solutions to integrodifferential equations, see [22] by Zhao, Chang, and N'Guérékata for the asymptotically almost automorphic solutions to the nonlinear delay integral equations, and see [23] by Chang and Tang and [24] by Zhao, Chang, and Nieto for the asymptotically almost automorphic solutions to stochastic differential equations, and the existence of asymptotically almost automorphic solutions has become one of the most attractive topics in the qualitative theory of differential equations due to its significance and applications in physics, mathematical biology, control theory, and so on. We refer the reader to the monographs of N'Guérékata [25] for the recently theory and applications of asymptotically almost automorphic functions.…”