Existence and uniqueness of nontrivial collocation solutions of implicitly linear homogeneous Volterra integral equations

TL;DR

This paper investigates the existence and uniqueness of nontrivial collocation solutions for nonlinear homogeneous Volterra-Hammerstein integral equations, providing theoretical conditions and illustrative examples.

Contribution

It offers new theoretical results on existence and uniqueness of collocation solutions for a class of nonlinear integral equations with non-Lipschitz nonlinearities.

Findings

Conditions for existence and uniqueness established

Examples illustrating the applicability of the methods

Counterexamples showing limitations of previous cases

Abstract

We analyze collocation methods for nonlinear homogeneous Volterra-Hammerstein integral equations with non-Lipschitz nonlinearity. We present different kinds of existence and uniqueness of nontrivial collocation solutions and we give conditions for such existence and uniqueness in some cases. Finally we illustrate these methods with an example of a collocation problem, and we give some examples of collocation problems that do not fit in the cases studied previously.