Generalized Fixed-Point Principles for Integral Equations and Their Quadrature Approximations

We introduce a generalized fixed-point framework based on $d$-$\psi$-$\varepsilon$-contractions and weak $d$-$\psi$-$\varepsilon$-contractions, where the contraction metric and the metric in which the space is complete may be different and the comparison function belongs to the class $\Psi_0(\varepsilon)$. This setting extends several classical fixed-point principles and yields existence, uniqueness, localization, and convergence results for Picard iterations. The theory is applied to nonlinear integral equations and to their quadrature approximations. By introducing suitable invariant sets and a \(\psi\)-\(\varepsilon\)-max inequality, existence and uniqueness results are obtained under assumptions that are substantially different from classical Lipschitz-type conditions. The developed framework is further extended to quadrature integral equations generated by numerical integration formulas. Sufficient conditions are established for the existence and uniqueness of solutions as well as for the convergence of the associated Picard sequences. The theoretical results guarantee convergence of the discrete iterations under appropriate assumptions, thereby providing a direct connection between fixed-point theory and numerical computation. Several examples, including Chandrasekhar-type integral equations and nonlinear weighted integral equations with singular kernels, are presented to illustrate the applicability and effectiveness of the proposed approach. Numerical experiments confirm the theoretical findings and demonstrate the accuracy of the resulting quadrature-Picard schemes.