Pseudo $S$-Asymptotically Bloch Type Periodicity with Applications to Some Evolution Equations

Abstract

This paper is mainly focused upon the pseudo S -asymptotically Bloch type periodicity and its applications. Firstly, a new notion of pseudo S -asymptotically Bloch type periodic functions is introduced, and some fundamental properties on pseudo S -asymptotically Bloch type periodic functions are established. Then, the notion and properties of weighted pseudo S -asymptotically Bloch type periodic functions are similarly presented. Finally, the obtained results are applied to investigate the existence and uniqueness of pseudo S -asymptotically Bloch type periodic mild solutions for some semi-linear evolution equations in Banach spaces.

Pseudo $ S $-asymptotically Bloch type periodic solutions to a damped evolution equation

“…The aim of this paper is to investigate the existence of pseudo S-asymptotically Bloch type periodic mild solutions to (1) by introducing pseudo S-asymptotically Bloch type periodic function on R + . Our results can be considered as a useful supplement to the works [1,2,10,11]. It should be pointed out that few results are available yet for the existence of pseudo S-asymptotically ω-anti-periodic solutions to (1) since the notion of pseudo S-asymptotically ω-anti-periodic function on the nonnegative real axis is not specially mentioned up to now.…”

“…The concept of asymptotically Bloch type periodic function was also presented in [18], which can be seen as the Bloch type periodic function with a perturbation vanishing at infinity and a natural generalization of asymptotically ω-periodic function (see [19]). Recently, Chang and Wei in [10] introduced the notion of pseudo S-asymptotically Bloch type periodic function on the real line and investigated the existence and uniqueness of pseudo S-asymptotically Bloch type periodic mild solutions to some semilinear evolution equations. It is noted that pseudo S-asymptotically Bloch type periodicity can be regarded as a generalization of S or pseudo S-asymptotic ω-periodicity which has been considerably studied, see for instance [13,20,22,23,24,26,27] and references cited therein.…”

“…It is noted that pseudo S-asymptotically Bloch type periodicity can be regarded as a generalization of S or pseudo S-asymptotic ω-periodicity which has been considerably studied, see for instance [13,20,22,23,24,26,27] and references cited therein. In particular, the concept of pseudo S-asymptotically ω-anti-periodic function on the real line as a special case was pointed out in [10].…”

“…We recall some properties on P SABP ω,k (R + , X) which can be analogously attained from [10]. Definition 2.1.…”

Weighted pseudo S-asymptotically Bloch type periodic solutions for a class of mean field stochastic fractional evolution equations

“…Some publications have also explored the effects of perturbations on Bloch periodic functions, by defining some quasi-Bloch periodicity concepts. For instance, [18,20,32] studied asymptotically Bloch periodic functions and their applications, while [31][32][33][34] investigated (pseudo) S-asymptotically Bloch periodic functions and their applications. These quasi-Bloch periodic functions are extensions of the corresponding asymptotically ω-periodic and (pseudo) S-asymptotically ω-periodic functions in the deterministic case.…”