Existence of Hilfer Fractional Stochastic Differential Equations with Nonlocal Conditions and Delay via Almost Sectorial Operators

Abstract

In this article, we examine the existence of Hilfer fractional (HF) stochastic differential systems with nonlocal conditions and delay via almost sectorial operators. The major methods depend on the semigroup of operators method and the Mo¨nch fixed-point technique via the measure of noncompactness, and the fundamental theory of fractional calculus. Finally, to clarify our key points, we provide an application.

Study of (k,Θ)-Hilfer fractional differential and inclusion systems on the glucose graph

“…In 2000, Hilfer proposed the generalized Riemann-Liouville fractional derivative for short Hilfer fractional derivative [14] , including Riemann-Liouville fractional derivative and Caputo fractional derivative. Subsequently, fractional differential equations with Hilfer fractional derivatives have been studied by many authors and are widely used in physics [15] , [16] , [17] , [18] , [19] , [20] , [21] , [22] . For example, in 2023, Sivasankar and Udhayakumar [20] studied a new set of sufficient conditions for the existence of mild solutions for Hilfer fractional neutral stochastic evolution equations via almost sectorial operators with delay.…”

Existence of Mild Solutions for Nonlocal Evolution Equations with the Hilfer Derivatives

“…Basic fractional calculus theory, Bohnenblust-Karlin's fixed point theorem, and stochastic analysis were utilized. Sivasankar et al [31] utilized basic fractional calculus theory, semigroup method, and M€ onch fixed point theorem with the measure of noncompactness to investigate the existence of fractional stochastic differential systems with nonlocal conditions and delay with almost sectorial operators involving the Hilfer derivatives. In [32], Bedi et al investigated the controllability and stability of fractional evolution equations with the Hilfer derivatives and the results obtained by means of propagation family theory, noncompactness calculation methods, and fixed point theory.…”

Total Controllability for a Class of Fractional Hybrid Neutral Evolution Equations with Non-Instantaneous Impulses

“…For instance, non-local conditions could be used to describe the effect of a pollutant on a distant ecosystem. Fractional differential equations with non-local conditions are powerful tools for understanding and predicting the behavior of complex systems [12,13].…”