Existence and uniqueness of discrete weighted pseudo S-asymptotically $$\omega $$ -periodic solution to abstract semilinear superdiffusive difference equation

Abstract

In this paper, we establish sufficient conditions in order to guarantee the existence and uniqueness of discrete weighted pseudo S-asymptotically \(\omega \)-periodic solution to the semilinear fractional difference equation

$$\begin{aligned} {\left\{ \begin{array}{ll} _C\nabla ^{\alpha } u^n=Au^n+g^n(u^n), \quad n\ge 2,\\ u^0=x_0 \in X, \quad u^1=x_1\in X, \\ \end{array}\right. } \end{aligned}$$

where \(1<\alpha <2,\) A is a closed linear operator in a Banach space X which generates an \((\alpha ,\beta )\)-resolvent sequence \(\{S^n_{\alpha ,\beta }\}_{n\in \mathbb N_0}\subset \mathcal {B}(X)\) and \(g:\mathbb N_0\times X\rightarrow X\) a discrete weighted pseudo S-asymptotically \(\omega \)-periodic function satisfying suitable Lipschitz type conditions in the spatial variable (local and global), based in fixed point Theorems. In order to achieve this objective, we prove invariance by convolution and principle of superposition for a class of suitables function spaces.