Existence of asymptotically almost automorphic solutions to some abstract partial neutral integro-differential equations

Pseudo-almost periodic solutions for first-order neutral differential equations

“…Moreover, because Q(t) = 1 + 2 sin 200t is oscillating on R, one can see that all results in Refs. [1][2][3][4][5][6][7][8][9][10][11] cannot be applied to illustrate that all solutions for (4.1) converge exponentially to x * (t). We all know that the pseudo-almost periodic functions contain almost periodic functions, thus, the derived results are still novel if we reduce all time-varying delays and coefficients of (1.2) to periodic functions or almost periodic functions.…”

“…x(t) -P(t)x t -τ 1 (t) = -Q(t)x(t) + f t, x t -τ 2 (t) , (1.2) sufficient conditions for the existence of pseudo-almost periodic (mild) solutions are obtained in [10,11]. On the other hand, the global exponential stability of pseudo-almost periodic solutions plays a key role in characterizing the dynamical behavior of biological and ecological dynamical systems since the exponential convergence rate can be unveiled [12][13][14][15][16][17][18][19][20].…”

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.…”

A Study of Controllability of Impulsive Neutral Evolution Integro-Differential Equations with State-Dependent Delay in Banach Spaces

“…Using the method of semigroups, various types of solutions of semilinear evolution equations have been discussed by Pazy [9]. The theory of neutral differential equations in Banach spaces has been studied by several authors [10][11][12][13][14][15].…”