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• W2964347153 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 Prufer angle construction. Unfortunately, it seems to be difficult to apply the Prufer 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." @default.
• W2964347153 created "2019-07-30" @default.
• W2964347153 creator A5067895351 @default.
• W2964347153 date "2016-05-01" @default.
• W2964347153 modified "2023-10-01" @default.
• W2964347153 title "Linear and nonlinear, second-order problems with Sturm–Liouville-type, multi-point boundary conditions" @default.
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• W2964347153 doi "https://doi.org/10.1016/j.na.2016.02.015" @default.
• W2964347153 hasPublicationYear "2016" @default.
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