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On Semipositone Non-Local Boundary-Value Problems with Nonlinear or Affine Boundary Conditions

Part of: Special classes of linear operators Boundary value problems Nonlinear operators and their properties

Published online by Cambridge University Press:  02 November 2016

Christopher S. Goodrich
Christopher S. Goodrich*
Affiliation:
Department of Mathematics, Creighton Preparatory School, Omaha, NE 68114, USA (cgood@prep.creighton.edu)
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Abstract

We consider the boundary-value problem

where H: [0,+∞) → ℝ and f : [0, 1] × ℝ → ℝ are continuous and λ > 0 is a parameter. We show that if H satisfies a boundedness condition on a specified compact set, then this, together with an assumption that H is either affine or superlinear at +∞, implies existence of at least one positive solution to the problem, even in the case where we impose no growth conditions on f. Finally, since it can hold that f(t, y) < 0 for all (t, y) ∈ [0, 1]×ℝ, the semipositone problem is included as a special case of the existence result.

Keywords

positive solutionnonlinear boundary conditionnon-local boundary conditionfixed-point indexboundary-value problem

MSC classification

Secondary: 34B09: Boundary eigenvalue problems 34B10: Nonlocal and multipoint boundary value problems 34B18: Positive solutions of nonlinear boundary value problems 47B40: Spectral operators, decomposable operators, well-bounded operators, etc. 47H11: Degree theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 3 , August 2017 , pp. 635 - 649
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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