SOLVABILITY AND APPROXIMATION OF NONLINEAR FUNCTIONAL MIXED VOLTERRA–FREDHOLM EQUATION IN BANACH SPACE

Abstract

This study probes into the existence of a unique solution and the numerical approximation of a nonlinear functional Volterra–Fredholm integral equations of the mixed type and second kind. Based on the Lipschitz constants of the functional and kernel, a Bielecki’s norm is defined and used to modify a distance inequality on a constructed self-map. The map is shown to be contractive, thereby establishing solvability. The problem is then approximated by collocating at discrete points and use of a composite multidimensional numerical quadrature approximation. A new Grönwall-type inequality is proposed, and used, to prove the second order of convergence of the numerical scheme. Numerical experiments are provided to verify the theoretical results.