PSEUDO S-ASYMPTOTICALLY (ω,c)-PERIODIC SEQUENTIAL SOLUTIONS TO SOME SEMILINEAR DIFFERENCE EQUATIONS IN BANACH SPACES

Abstract

The main purpose of this paper is to investigate some existence results for pseudo $S$-asymptotically ($\omega ,c$)-periodic sequential solutions to a semilinear difference equation of convolution type and a semilinear Weyl-like fractional difference equation in Banach spaces. For this purpose, we first give the definition of the pseudo $S$-asymptotically $(\omega ,c)$-periodic sequence and prove the completeness, convolution and superposition theorems for such a sequence in abstract spaces. We show some existence and uniqueness of pseudo $S$-asymptotically $(\omega ,c)$-periodic sequential solutions under some different Lipschitz type conditions of the nonlinear force term with its second variable. We also consider the existence of pseudo $S$-asymptotically $(\omega ,c)$-periodic sequential solutions under a non-Lipschitz growth condition.