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.

The study explores controllability results for fuzzy fractional differential equations involving the Hilfer–Katugampola fractional derivative, a generalization of the Riemann–Liouville and Hadamard fractional derivatives. It establishes existence conditions for mild solutions by applying fractional calculus, semigroup theory, the Laplace transform, and Sadovskii’s fixed point theorem. Additionally, an example is included to illustrate the practical applications of the key findings.

The paper investigates the existence of a mild solution for a fuzzy fractional differential equation involving the Caputo–Katugampola fractional derivative. The main results are established using Mönch’s fixed point theorem, along with essential tools like semigroup theory, fractional calculus, and the measure of noncompactness. Finally, our theoretical results are applied by providing an interesting example.