  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.
