Existence of solutions for delayed $\psi$-Hilfer hybrid fractional integro-differential inclusions via Dhage'sfixed point theorem in Banach algebras

Abstract

In this paper, we investigate the existence of solutions for a class of delayed $\psi$-Hilfer hybrid fractional integro-differential inclusions. The problem involves the $\psi$-Hilfer fractional derivative of order $\varrho\in(0,1)$ and type $\vartheta\in[0,1]$, applied to a hybrid quotient structure whose numerator contains multiple $\psi$-fractional integral perturbations of a history-dependent nonlinearity, and whose denominator is a nonvanishing function coupling the present state, the history segment, and a $\psi$-fractional integral of the state. The right-hand side is a multivalued $L^{1}$-Carathéodory map, allowing for inclusion-type dynamics and uncertainty. The analysis is carried out in the Banach algebra $C(\mathcal{J},\mathbb{R})$ equipped with the supremum norm. By transforming the problem into an equivalent operator inclusion of the form $z\in\mathcal{A}z\,\mathcal{B}z+\mathcal{C}z$ and applying a hybrid fixed point theorem for three operators in a Banach algebra due to Dhage, we establish sufficient conditions for the existence of at least one solution. The main result unifies and extends several recent contributions on hybrid fractional differential inclusions, Hilfer fractional systems with delay, and $\psi$-fractional equations. Two illustrative examples, one with $\psi(\xi)=\xi$ (recovering the Riemann--Liouville case) and one with $\psi(\xi)=\ln(1+\xi)$ (the Hadamard-type case), are provided to demonstrate the applicability of the theoretical results.