On tempered Hilfer fractional derivatives with respect to functions and the associated fractional differential equations

Abstract

We investigate the Hilfer-type operator within the topic of tempered fractional calculus with respect to functions. This operator, the tempered Psi-Hilfer derivative, is defined for the first time here, and its fundamental properties are studied, such as composition properties, function space mappings, and other functional analysis properties. We also consider fractional differential equations involving these operators, and establish existence, uniqueness, well-posedness, and stability results for such equations under suitable conditions.