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

Abstract

The prime aim of the present paper is to continue developing the theory of tempered fractional integrals and derivatives of a function with respect to another function. This theory combines the tempered fractional calculus with the $\Psi$-fractional calculus, both of which have found applications in topics including continuous time random walks. After studying the basic theory of the $\Psi$-tempered operators, we prove mean value theorems and Taylor's theorems for both Riemann--Liouville type and Caputo type cases of these operators. Furthermore, we study some nonlinear fractional differential equations involving $\Psi$-tempered derivatives, proving existence-uniqueness theorems by using the Banach contraction principle, and proving stability results by using Gr\"onwall type inequalities.