Linear and nonlinear, second-order problems with Sturm–Liouville-type, multi-point boundary conditions. (May 2016)

• Record Type: Journal Article
• Authors: Rynne, Bryan P.
• Is Part Of: Nonlinear analysis. Volume 136(2016)
• Page Start: 195
• Page End: 214
• Publication Date: 2016-05
• Subjects: 34B15; Ordinary differential equations -- Nonlinear boundary value problems -- Multi-point boundary conditions -- Linear eigenvalue problem; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.02.015
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider the nonlinear boundary value problem consisting of the equation − u ″ = f ( u ), on ( − 1, 1 ), where f : R → R is continuous, together with general Sturm–Liouville type, multi-point boundary conditions at ± 1 . We obtain Rabinowitz-type global bifurcation results, and then use these to obtain 'nodal' solutions of the problem. We conclude with a nonresonance result for an inhomogeneous form of the problem. These results rely on the spectral properties of the eigenvalue problem consisting of the equation − u ″ = λ u, on ( − 1, 1 ), together with the multi-point boundary conditions. In a previous paper it was shown that, under certain 'optimal' conditions, the basic spectral properties of this eigenvalue problem are similar to those of the standard Sturm–Liouville problem with single-point boundary conditions. In particular, for each integer k ⩾ 0 there exists a unique, simple eigenvalue λ k, whose eigenfunctions have 'oscillation count' equal to k, where the 'oscillation count' was defined in terms of a complicated Prüfer angle construction. Unfortunately, it seems to be difficult to apply the Prüfer angle construction to the nonlinear problem. Accordingly, in this paper we use alternative, non-optimal, oscillation counting methods to obtain the required spectral properties of the linear problem, and these are then applied to the nonlinear problem to yield the results mentioned above.