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On Nonlinear Boundary Conditions Involving Decomposable Linear Functionals

Part of: Special classes of linear operators Integral, integro-differential, and pseudodifferential operators Boundary value problems Nonlinear operators and their properties

Published online by Cambridge University Press:  27 October 2014

Christopher S. Goodrich
Christopher S. Goodrich*
Affiliation:
Department of Mathematics, Creighton Preparatory School, Omaha, NE 68114, USA Department of Mathematics, University of Rhode Island, Kingston, RI 02881, USA, ([email protected])
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Abstract

In this paper we consider the existence of a positive solution to boundary-value problems with non-local nonlinear boundary conditions, the archetypical example being −y″(t) = λf(t,y(t)), t ∈ (0, 1), y(0) = H(φ(y)), y(1) = 0. Here, H is a nonlinear function, λ > 0 is a parameter and φ is a linear functional that is realized as a Lebesgue—Stieltjes integral with signed measure. By requiring φ to decompose in a certain way, we show that this problem has at least one positive solution for each λ ∈ (0, λ0), for a number λ0 > 0 that is explicitly computable. We also give a separate result that holds for all λ > 0.

Keywords

positive solutionnonlinear boundary conditionnon-local boundary conditionfixed point indexeigenvalue

MSC classification

Primary: 34B09: Boundary eigenvalue problems 34B10: Nonlocal and multipoint boundary value problems 34B15: Nonlinear boundary value problems 34B18: Positive solutions of nonlinear boundary value problems 47H07: Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
Secondary: 47B40: Spectral operators, decomposable operators, well-bounded operators, etc. 47G10: Integral operators 47H11: Degree theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 58 , Issue 2 , June 2015 , pp. 421 - 439
Copyright
Copyright © Edinburgh Mathematical Society 2014 

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