Abstract
We consider the nonlinear equation−u′′=f(u)+h,on(−1,1),where f:R→R and h:[−1,1]→R are continuous, together with general SturmLiouville type, multipoint boundary conditions at ±1. We will obtain existence of solutions of this boundary value problem under certain `nonresonance' conditions, and also Rabinowitztype 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′′=λu,on(−1,1),together with the multipoint 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 SturmLiouville problem with singlepoint boundary conditions. In particular, for each integer k≥0 there exists a unique, simple eigenvalue λk, whose eigenfunctions have `oscillation count' equal to k, where the `oscillation count' was defined in terms of a complicated Pr\"ufer angle construction. Unfortunately, it seems to be difficult to apply the Pr\"ufer angle construction to the nonlinear problem. Accordingly, in this paper we use alternative, nonoptimal, 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.
Original language  English 

Pages (fromto)  195–214 
Number of pages  20 
Journal  Nonlinear Analysis: Theory, Methods and Applications 
Volume  136 
Early online date  9 Mar 2016 
DOIs  
Publication status  Published  May 2016 
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Bryan Patrick Rynne
 School of Mathematical & Computer Sciences  Professor
 School of Mathematical & Computer Sciences, Mathematics  Professor
Person: Academic (Research & Teaching)