A Novel Study on Tempered $({\upkappa},\uppsi)$-Hilfer Fractional Operators

Abstract

The objective of this paper is to introduce new definitions of the tempered (κ,ψ)-fractional operators and establish their various properties. Our research is primarily focused on applying these newly proposed operators to investigate the existence, uniqueness, and κ-Mittag-Leffler-Ulam-Hyers stability of a specific class of boundary value problems involving implicit nonlinear fractional differential equations and tempered (κ,ψ)-Hilfer fractional derivatives. To accomplish this, we make use of the fixed point theorem of Banach and a generalization of the well-known Gronwall inequality. Additionally, we provide illustrative examples to demonstrate the practical effectiveness of our main findings.

Study on a Nonlocal Fractional Coupled System Involving (k,ψ)-Hilfer Derivatives and (k,ψ)-Riemann–Liouville Integral Operators

“…By using Banach's fixed-point theorem, the existence of a unique solution was proven. For some recent results on (k, ψ)-Hilfer fractional derivative operators of orders in (0, 1], see [29,30] and references cited therein. Boundary value problems of the (k, ψ)-Hilfer fractional derivative operator of orders in (1,2] were initiated in [31] by studying the problem…”