Nontrivial solutions of Hammerstein integral equations with reflections

Alberto Cabada,Gennaro Infante,Fernando Adrián Fernández Tojo

doi:10.1186/1687-2770-2013-86

Abstract

Using the theory of fixed point index, we establish new results for the existence of nonzero solutions of Hammerstein integral equations with reflections. We apply our results to a first-order periodic boundary value problem with reflections. MSC: Primary 34K10; secondary 34B15; 34K13

Highlights

In a recent paper, Cabada and Tojo [ ] studied, by means of methods and results present in [, ], the first-order operator u (t) + ωu(–t) coupled with periodic boundary value conditions, describing the eigenvalues of the operator and providing the expression of the associated Green’s function in the nonresonant case
In [ ], the authors provide the range of values of the real parameter ω for which the Green’s function has constant sign and apply these results to prove the existence of constant sign solutions for the nonlinear periodic problem with reflection of the argument u (t) = h t, u(t), u(–t), t ∈ [–T, T], u(–T) = u(T)
The methodology, analogous to the one utilized by Torres [ ] in the case of ordinary differential equations, consists of two steps

Summary

Introduction

Cabada and Tojo [ ] studied, by means of methods and results present in [ , ], the first-order operator u (t) + ωu(–t) coupled with periodic boundary value conditions, describing the eigenvalues of the operator and providing the expression of the associated Green’s function in the nonresonant case. We continue the study of [ ] and we prove new results regarding the existence of nontrivial solutions of Hammerstein integral equations with reflections of the form u(t) = k(t, s)g(s)f s, u(s), u(–s) ds, t ∈ [–T, T],

Results

Conclusion