Positive Solutions of Hybrid Nonlinear Integro-Fractional Differential Equations via Dhage's Fixed Point Theorem

Abstract

This paper investigates the existence, uniqueness, and monotonicity of positive solutions for a new class of hybrid nonlinear fractional integro-differential equations involving the Caputo–Fabrizio derivative. Unlike the classical Caputo operator, the Caputo–Fabrizio derivative is characterized by a non-singular exponential kernel, which leads to improved analytical tractability and better modeling of memory effects. Using Dhage's fixed point theorem, sufficient conditions for the well-posedness of the problem are derived. A concrete example is provided to demonstrate the effectiveness of the theoretical results, showing the relevance of the Caputo–Fabrizio approach in fractional modeling.