This paper studies the integro-differential equations of Hilfer fractional (HF) neutral stochastic evolution on an infinite interval with almost sectorial operators and their attractive solutions. We use semigroup theory, stochastic analysis, compactness methods, and the measure of noncompactness (MNC) as the foundation for our methodologies. We establish the existence and attractivity theorems for mild solutions by considering the fact that the almost sectorial operator is both compact and noncompact. Example that highlight the key findings are also provided.

In this article, we implemented the idea of a fuzzy interval-valued function with the well-known generalized fuzzy fractional operators, associated with different types of convexities and preinvexities. We developed the Prabhakar fuzzy fractional operators using the fuzzy interval-valued function. We presented the novel extensions of Hermite-Hadamard fuzzy-type and trapezoidal fuzzy-type inequalities, based on the $h$-Godunova-Levin convex and $h$-Godunova preinvex fuzzy interval-valued functions.

In this research, we investigate the existence of solution for a class of coupled fractional impulsive hybrid integro-differential equations with hybrid boundary conditions. Our primary tools for this analysis are the Banach contraction mapping principle (BCMP) and Schaefer's fixed point theorem. This study ended with two applied examples to facilitate understanding of the theoretical results obtained.