Proceedings of the Edinburgh Mathematical Society (Series 2)

Table of Contents - October 2006 - Volume 49, Issue 03  

Research Article

NONLINEAR NON-LOCAL BOUNDARY-VALUE PROBLEMS AND PERTURBED HAMMERSTEIN INTEGRAL EQUATIONS

Gennaro Infantea1 and J. R. L. Webba2

a1 Dipartimento di Matematica, Università della Calabria, 87036 Arcavacata di Rende, Cosenza, Italy (g.infante@unical.it)

a2 Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK (jrlw@maths.gla.ac.uk)

Abstract

Motivated by some non-local boundary-value problems (BVPs) that arise in heat-flow problems, we establish new results for the existence of non-zero solutions of integral equations of the form

$$ u(t)=\gamma(t)\alpha[u]+\int_{G}k(t,s)f(s,u(s))\,\mathrm{d}s, $$

where $G$ is a compact set in $\mathbb{R}^{n}$. Here $\alpha[u]$ is a positive functional and $f$ is positive, while $k$ and $\gamma$ may change sign, so positive solutions need not exist. We prove the existence of multiple non-zero solutions of the BVPs under suitable conditions. We show that solutions of the BVPs lose positivity as a parameter decreases. For a certain parameter range not all solutions can be positive, but for one of the boundary conditions we consider we show that there are positive solutions for certain types of nonlinearity. We also prove a uniqueness result.

(Online publication January 25 2007)

(Received April 14 2005)

Keywords

  • Primary 34B18;
  • Secondary 34B10;
  • 47H10;
  • 47H30;
  • fixed-point index;
  • cone;
  • positive solution