Pseudo almost periodic and automorphic mild solutions to nonautonomous neutral partial evolution equations

Abstract

In this work, we present a new concept of Stepanov weighted pseudo almost periodic and automorphic functions which is more generale than the classical one, and we obtain a new existence result of µ-pseudo almost periodic and µ-pseudo almost automorphic mild solutions for some nonautonomous evolution equations with Stepanov µ-pseudo almost periodic terms. An example is shown to illustrate our results.

Stepanov pseudo‐almost periodic functions and applications

“…In the case where f is uniformly Lipschitzian with respect to the second variable and 𝜇-pseudo-almost periodic (resp. Stepanov 𝜇-pseudo-almost periodic) in t, it was shown (see previous studies [5][6][7][8]11,13,14,19,23,24 ) that the output (the corresponding solution) u is 𝜇-pseudo-almost periodic. Extensively, the existence and uniqueness results established so far are obtained in view of some composition results and the Banach strict contraction principle.…”

“…In the literature, we found several works devoted to the existence and uniqueness of 𝜇-pseudo-almost periodic solutions to semilinear evolution equations and inclusions; we quote previous studies. [4][5][6][7][8][11][12][13][14][15][16][17][18][19][20][21][22][23] In that case, a solution (at least in a mild sense) is usually represented by an integral operator (this holds under a certain decay of the associated linear system). More specifically, for given an integral operator solution, namely, (u)(t) = ∫ R (t, s)𝑓 (s, u(s))ds, t ∈ R, (1.1) where the input parameters, ((t, s)) t≥s which represents the Green function (resp.…”

“…In the literature, we found several works devoted to the existence and uniqueness of μ ‐pseudo‐almost periodic solutions to semilinear evolution equations and inclusions; we quote previous studies 4–8,11–23 . In that case, a solution (at least in a mild sense) is usually represented by an integral operator (this holds under a certain decay of the associated linear system).…”

A discussion on boundary controllability of nonlocal impulsive neutral integrodifferential evolution equations

“…El-Khalil et al 34 established results for regularity and existence of solutions for the non-autonomous nondense control system in abstract space having finite delays. Akdad et al 35 presented existence results for the Stepanov 𝜇-pseudo almost automorphic and periodic mild solution. Zhou and Jiao, Vijayakumar and Udhayakumar, Hernández and O'Regan, and Henriquez [36][37][38][39] investigated existence and uniqueness for certain classes of the fractional system by employing the fixed point approach, fractional theories, and measure of noncompactness, etc.…”