Half-eigenvalues of periodic Sturm-Liouville problems

We consider the nonlinear Sturm-Liouville problem -(pu')'+qu=au⁺-bu^-+?u, in (0,2p), u(0)=u(2p), (pu)'(0)=(pu)'(2p), where 1/p, q?L₁(0,2p), with p>0 a.e. on (0,2p), a, b?L₁(0,2p), ? is a real parameter, and u^±(t)=max{±u(t),0} for t?[0,2p]. Values of ? for which (1)-(2) has a non-trivial solution u will be called half-eigenvalues while the corresponding solutions u will be called half-eigenfunctions. The set of half-eigenvalues will be denoted by S_H. We show that a sequence of half-eigenvalues exists, the corresponding half-eigenfunctions having certain nodal properties, and we obtain certain spectral and degree theoretic properties associated with S_H. These properties yield results on the existence and non-existence of solutions of the problem -(pu')'+qu=f(t,u)+h, in (0,2p) (together with (2)), where h?L₁(0,2p), f:[0,2p]× R?R is a Carathéodory function, and the limits a(t):=lim_??8 f(t,?)/?, b(t):=lim_??-8 f(t,?)/?, exist for a.e. t?[0,2p]. When the functions a and b are constant the set S_H is closely related to the 'Fucík spectrum' of the problem, and equivalent solvability results are obtained using the two approaches. However, when a and b are not constant the half-eigenvalue approach yields stronger results. © 2004 Elsevier Inc. All rights reserved.