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

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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.

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