Periodic mild solutions of stochastic fractional integro-differential equations with delay and weakly singular kernels in Banach spaces

Abstract

This paper establishes the existence of mean-square $T$-periodic mild solutions for stochastic fractional integro-differential equations with delay and weakly singular kernels in Banach spaces. The model involves a Caputo fractional derivative, a weakly singular kernel, and stochastic perturbations driven by a cylindrical Wiener process. Under suitable periodicity, growth, and continuity conditions, we use resolvent family theory, stochastic analysis, and Krasnoselskii’s fixed point theorem to prove the existence of solutions. An illustrative example demonstrates the results.