Applying fixed point techniques for solving tripled system of quantum integral equations with numerical results

Abstract

This study investigates the demanding problem of proving the existence of solutions for a tripled system that couples quantum integral equations with quadratic integral equations. To tackle the intrinsic nonlinear and noncompact features of the model, we employ Petryshyn’s fixed-point theorem, a significant generalization of Darbo’s theorem formulated within the framework of measures of noncompactness. Based on this approach, we derive rigorous and verifiable existence conditions applicable to a broad class of quantum integral systems. The theoretical findings are supported by a comprehensive illustrative example that confirms the validity of the proposed criteria. In addition, we develop a constructive collocation method founded on barycentric interpolation and Jackson quadrature for q-integrals, and we verify the required assumptions within the same example. Numerical experiments are finally presented to confirm the practical applicability of the existence results and to demonstrate the accuracy, stability, and robustness of the proposed discretization scheme.