A positive fixed point theorem with applications to systems of Hammerstein integral equations

Alberto Cabada,José Ángel Cid,Gennaro Infante

doi:10.1186/s13661-014-0254-8

Abstract

Abstract We present new criteria on the existence of fixed points that combine some monotonicity assumptions with the classical fixed point index theory. As an illustrative application, we use our theoretical results to prove the existence of positive solutions for systems of nonlinear Hammerstein integral equations. An example is also presented to show the applicability of our results. MSC: 47H10, 34B10, 34B18, 45G15, 47H30.

Highlights

1 Introduction In this manuscript we pursue the line of research developed in the recent papers [ – ] in order to deal with fixed point theorems on cones that mix monotonicity assumptions and conditions in one boundary, instead of imposing conditions on two boundaries as in the celebrated cone compression/expansion fixed point theorem of Krasnosel’skiı
In [ ] Cid et al, in order to show the existence of positive solutions of the fourth-order boundary value problem (BVP)
We prove that the system ( . ) has a solution for every λ, λ >

Summary

Introduction

In this manuscript we pursue the line of research developed in the recent papers [ – ] in order to deal with fixed point theorems on cones that mix monotonicity assumptions and conditions in one boundary, instead of imposing conditions on two boundaries as in the celebrated cone compression/expansion fixed point theorem of Krasnosel’skiı. In the proposition we recall the main properties of the fixed point index of a completely continuous operator relative to a cone, for more details see [ , ]. We state our first result on the existence of non-trivial fixed points. Let X be a real Banach space, K a normal cone with normal constant d ≥ and nonempty interior (i.e. solid) and T : K → K a completely continuous operator. By assumption, iK (T, V ) = we get the existence of a non-trivial fixed point x belonging to the set V \ K R (when KR ⊂ V ) or to the KR \ V (when V ⊂ KR). This result together with ( ) gives the existence of a non-zero fixed point with the desired localization property. Author details 1Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela, 15782, Spain. 2Departamento de Matemáticas, Universidade de Vigo, Pabellón 3, Campus de Ourense, Ourense, 32004, Spain. 3Dipartimento di Matematica e Informatica, Università della Calabria, Arcavacata di Rende, Cosenza, 87036, Italy

Methods