ON A CLASS OF CAUCHY-TYPE PROBLEMS ASSOCIATED WITH WEIGHTED FRACTIONAL DERIVATIVES

Abstract

This paper investigates the existence and uniqueness of mild solutions for a class of nonlocal Cauchy–type problems involving integro–differential equations with weighted $ \varsigma $–Caputo fractional derivatives. The equations incorporate generalized Volterra and Fredholm integral operators with singular kernels of order $ \ell \in (0, 1) $, within the framework of an ordered Banach space. The main results are established by combining semigroup theory, monotone iterative techniques, upper and lower solution methods, Kuratowski's measure of noncompactness, generalized Gronwall inequalities, and fixed point theorems. Continuous dependence of mild solutions on initial data is also proved. Two concrete examples are provided to illustrate the applicability of the abstract results.