Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-12T20:32:16.838Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  25 January 2007

Gennaro Infante
Affiliation:
Dipartimento di Matematica, Università della Calabria, 87036 Arcavacata di Rende, Cosenza, Italy ([email protected])
J. R. L. Webb
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK ([email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2006