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March  2008, 1(1): 177-186. doi: 10.3934/dcdss.2008.1.177

Uniqueness of the principal eigenvalue in nonlocal boundary value problems

J. R. L. Webb 1,

1. 

Department of Mathematics, University of Glasgow, Glasgow G12 8QW

Received  August 2006 Revised  August 2007 Published  December 2007

In the study of nonlinear boundary value problems, existence of a positive solution can be shown if the nonlinearity 'crosses' the principal eigenvalue, the eigenvalue corresponding to a positive eigenfunction. It is well known that such an eigenvalue is unique for symmetric problems but it was unclear for general nonlocal boundary conditions. Here some old results due to Krasnosel'skiĭ are applied to show that the nonlocal problems which have been well studied over the last few years do have a unique principal eigenvalue. Some estimates and some comparison results are also given.
Keywords: nonlocal boundary conditions, $u_0$-positive operators, principal eigenvalue..
Mathematics Subject Classification: Primary: 34B18; Secondary: 34B16, 47N2.
Citation: J. R. L. Webb. Uniqueness of the principal eigenvalue in nonlocal boundary value problems. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 177-186. doi: 10.3934/dcdss.2008.1.177
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