Nontrivial Solutions of Systems of Perturbed Hammerstein Integral Equations with Functional Terms

Gennaro Infante

doi:10.3390/math9040330

Abstract

We discuss the solvability of a fairly general class of systems of perturbed Hammerstein integral equations with functional terms that depend on several parameters. The nonlinearities and the functionals are allowed to depend on the components of the system and their derivatives. The results are applicable to systems of nonlocal second order ordinary differential equations subject to functional boundary conditions, this is illustrated in an example. Our approach is based on the classical fixed point index.

Highlights

Nonlocal differential equations have seen recently growing attention by researchers, both in the context of ODEs and PDEs
Goodrich studied the existence of one positive solution of the nonlocal ODE
The approach in [3] relies on classical fixed point index theory applied in the cone of positive continuous functions

Summary

Introduction

Nonlocal differential equations have seen recently growing attention by researchers, both in the context of ODEs and PDEs. Goodrich studied the existence of one positive solution of the nonlocal ODE. The approach in [3] relies on classical fixed point index theory applied in the cone of positive continuous functions. We proceed in a different way; rather than studying a specific boundary value problem (BVP), we provide new results regarding the existence and non-existence of nonzero solutions of the following class of systems of integral equations with functional terms, namely ui(t) = λi ki(t, s) fi(s, u(s), u (s), wi[u]) ds + ∑ ηijγij(t)hij[u], t ∈ [0, 1], (4). We seek solutions of the system (4) in a product of cones of a kind that differs from (3); in particular, we work on products of cones in the space C1[0, 1] where the functions are positive on a subinterval of [0, 1] and are allowed to change sign elsewhere, this follows the line of research initiated by the author and Webb in [8].

We define the operator T as

Reasoning as above yields

Note that f