This article focuses on deriving the averaging principle for Hilfer fractional stochastic evolution equations (HFSEEs) driven by Lévy noise. We show that the solutions of the averaged equations converge to the corresponding solutions of the original equations, both in the sense of mean square and of probability. Our results enable us to focus on the averaged system rather than the original, more complex one. Given that the existing literature on the averaging principle for Hilfer fractional stochastic differential equations has been established in finite-dimensional spaces, the novelty here is the derivation of the averaging principle for a class of HFSEEs in Hilbert space. Furthermore, an example is allotted to illustrate the feasibility and utility of our results.

In this paper, we investigate the existence uniqueness of mild solutions for a class of ψ-Caputo fractional stochastic evolution equations with varying-time delay driven by fBm, which seems to be the first theoretical result of the ψ-Caputo fractional stochastic evolution equations. Alternative conditions to guarantee the existence uniqueness of mild solutions are obtained using fractional calculus, stochastic analysis, fixed point technique, and noncompact measure method. Moreover, an example is presented to illustrate the effectiveness and feasibility of the obtained abstract results.

This paper concerns the global existence of mild solutions for a class of Hilfer fractional stochastic evolution equations on infinite interval (0, +∞), while the existing work were considered on finite interval. The main difficulties here are how to construct suitable Banach spaces, proper operator relations, and then how to formulate the new criteria to guarantee the global existence of mild solutions on the previous constructed spaces under non-Lipschitz conditions. We mainly rely on the generalized Ascoli–Arzela theorem we established, Wright function, Schauder’s fixed point principle, and Kuratowski’s measure of noncompactness to handle with the infinite interval problems. Moreover, we give two examples to demonstrate the feasibility and utility of our results.