Integration of vector-valued pseudo-almost periodic functions

Abstract

Abstract. A necessary and sufficient condition is given to show that the indefinite integral of a vector-valued pseudo-almost periodic function is again pseudo-almost periodic. Then we use this result to answer a question about weakly almost periodic functions.Throughout this paper, X denotes a Banach space and $a stands for [a, oo) when a G R and for R when a = -oo ; W($a, X) denotes the space of all bounded continuous functions from JJa to X. Also, m denotes Lebesgue measure on M.Let / G f (R, X). The translate of / by 5 G R is the function RJ(t) = f(t + s),teR.f is called (weakly) almost periodic if the set {Ruf} = {Rsf : s G R} is (weakly) relatively compact in %?(R, X) [3]. Denote by ( Ws/PÇk, X)) jf&(R,X) all such functions.In case X = C, we will omit X from our notation and write, for example, W{Ja) for &{Ja,X).

Translation invariance and related properties of μ-pseudo almost automorphic (periodic) functions with application

“…Then these concepts are generalized in various ways, say, pseudo almost periodicity (Zhang [1][2][3]), weighted pseudo almost periodicity (Diagana [4,5]), pseudo almost automorphy (Liang, Xiao and Zhang [6,7]), weighted pseudo almost automorphy (Blot et al [8]), etc. These concepts have been widely used in the investigation of ordinary differential equations, partial differential equations, functional differential equations and fractional differential equations.…”

Existence of Weighted Pseudo Almost Periodic Solutions for some Partial Differential Equations with Delay

“…The notion of pseudo-almost periodicity was introduced in the literature in the early 90s by C. Zhang [18][19][20], as a natural generalization of the classical almost periodicity in the sense of Bohr. Since then, the existence of pseudo-almost periodic solutions to differential equations, partial differential equations, and functional differential equations has been of a great interest to several authors and hence generated various contributions: E. Aitdads, O. Arino, and K. Ezzinbi [3][4][5] obtained sufficient condition for existence of pseudo almost periodic solutions of some delay differential equations, and other contributions upon pseudo almost periodic solutions to various differential equations have recently been made in T. Diagana, E.M. Hernàndez, G.M.…”

A new composition theorem for $S^p$-weighted pseudo almost periodic functions and applications to semilinear differential equations

“…Here f is an appropriate function to be specified later. The concept of pseudo almost periodicity was introduced by Zhang [1][2][3] in the early nineties. It is a natural generalization of the classical almost periodicity in the sense of Bochner.…”