Cauchy Problem Involving Hybrid Proportional–Caputo Derivative in Banach Spaces

Abstract

This paper studies a class of nonlinear Cauchy problems driven by hybrid proportional-Caputo fractional derivatives in Banach spaces. The nonlinear source term is assumed to satisfy combined Lipschitz and Carathéodory conditions. By employing Weissinger's fixed point theorem together with a fixed point principle for convex-power condensing operators, new quantitative results on existence and uniqueness are established. To illustrate the applicability of the theoretical findings, two examples involving infinite-dimensional fractional systems in spaces of tempered sequences are presented. Moreover, previously known results are recovered as special cases of the main theorems developed in this work.