Extremal Solutions for Coupled Quadratic Fractional Integral Equations Involving the Generalized Q-Function

Abstract

We study a coupled $2\times 2$ system of nonlinear quadratic fractional integral equations in which each component carries its own fractional order $q_i\in(1,2)$ and its own generalized $Q$-function kernel. Using the Banach contraction principle and a monotone iterative technique in a product partially ordered function space, we prove that the system possesses a unique coupled solution and that the successive approximations starting from a coupled lower solution converge uniformly and monotonically to it. A perturbation and limit argument then shows the system has both a coupled maximal solution and a coupled minimal solution, and two comparison theorems demonstrate that any pair of functions satisfying the corresponding integral inequalities is bounded
componentwise above (respectively below) by the maximal (respectively minimal) solution. Three numerical examples with explicit parameter values and graphical illustrations are provided to validate every result.