Second-order, three-point, boundary value problems with jumping non-linearities

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Abstract

We consider the non-linear, three-point boundary value problem consisting of the equation (1)- u? = f (u) + h, a.e. on (0, 1), where h ? L1 (0, 1), together with the boundary conditions (2)u (0) = 0, u (1) = a u (?), where ?, a ? (0, 1). The function f : R ? R is continuous, and we assume that the following limits exist and are finite: f± 8 {colon equals} under(lim, s ? ± 8) frac(f (s), s) . We allow f8 ? f- 8-such a non-linearity f is said to be jumping. Related to (1) is the equation (3)- u? = a u+ - b u- + ? u, on (0, 1), where a, b, ? ? R, and u± (x) = max {± u (x), 0} for x ? [0, 1]. The problem (2) and (3) is 'positively homogeneous' and jumping. Values of ? for which (2) and (3) has a non-trivial solution u will be called half-eigenvalues, while the corresponding solutions u will be called half-eigenfunctions. We show that a sequence of half-eigenvalues exists, the corresponding half-eigenfunctions having specified nodal properties, and we obtain certain spectral and degree theoretic properties of the set of half-eigenvalues. These properties lead to solvability and non-solvability results for the problem (1) and (2). The set of half-eigenvalues is closely related to the 'Fucík spectrum' of the problem, and equivalent solvability and non-solvability results for (1) and (2) are obtained from either the half-eigenvalue or the Fucík spectrum approach. © 2007 Elsevier Ltd. All rights reserved.

Original languageEnglish
Pages (from-to)3294-3306
Number of pages13
JournalNonlinear Analysis: Theory, Methods and Applications
Volume68
Issue number11
DOIs
Publication statusPublished - 1 Jun 2008

Keywords

  • Boundary value problems
  • Half-eigenvalues
  • Jumping non-linearities
  • Three-point

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