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- W2290636706 abstract "We consider the nonlinear equation $$-u'' = f(u) + h , quad text{on} quad (-1,1),$$ where $f : {mathbb R} to {mathbb R}$ and $h : [-1,1] to {mathbb R}$ are continuous, together with general Sturm-Liouville type, multi-point boundary conditions at $pm 1$. We will obtain existence of solutions of this boundary value problem under certain `nonresonance' conditions, and also Rabinowitz-type global bifurcation results, which yield nodal solutions of the problem. These results rely on the spectral properties of the eigenvalue problem consisting of the equation $$-u'' = lambda u, quad text{on} quad (-1,1),$$ 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 $k geq 0$ there exists a unique, simple eigenvalue $lambda_k$, 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.
- W2290636706 created "2016-06-24" @default.
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- W2290636706 date "2015-09-21" @default.
- W2290636706 modified "2023-09-27" @default.
- W2290636706 title "Linear and nonlinear, second-order problems with Sturm-Liouville-type, multi-point boundary conditions" @default.
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