# Nontrivial solutions of systems of Hammerstein integral equations with first derivative dependence

## TL;DR

This paper establishes new fixed point index results for systems of Hammerstein integral equations with first derivative dependence, addressing existence, non-existence, and multiplicity of solutions, including positive solutions for third order ODEs with nonlocal boundary conditions.

## Contribution

It introduces novel fixed point index techniques for Hammerstein systems with derivative dependence and applies them to boundary value problems for third order ODEs.

## Key findings

- Proved conditions for existence of nontrivial solutions.
- Identified criteria for non-existence of solutions.
- Provided examples demonstrating applicability.

## Abstract

By means of classical fixed point index, we prove new results on the existence, non-existence, localization and multiplicity of nontrivial solutions for systems of Hammerstein integral equations where the nonlinearities are allowed to depend on the first derivative. As a byproduct of our theory we discuss the existence of positive solutions of a system of third order ODEs subject to nonlocal boundary conditions. Some examples are provided in order to illustrate the applicability of the theoretical results.