Half-eigenvalues of periodic Sturm–Liouville problems

AbstractWe consider the nonlinear Sturm–Liouville problem (1)−(pu′)′+qu=au+−bu−+λu,in(0,2π),(2)u(0)=u(2π),(pu)′(0)=(pu)′(2π),where 1/p,q∈L1(0,2π), with p>0 a.e. on (0,2π), a,b∈L1(0,2π), λ is a real parameter, and u±(t)=max{±u(t),0} for t∈[0,2π]. Values of λ 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 ΣH.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 ΣH. These properties yield results on the existence and non-existence of solutions of the problem (3)−(pu′)′+qu=f(t,u)+h,in(0,2π)(together with (2)), where h∈L1(0,2π), f:[0,2π]×R→R is a Carathéodory function, and the limitsa(t)≔limξ→∞f(t,ξ)/ξ,b(t)≔limξ→−∞f(t,ξ)/ξ,exist for a.e. t∈[0,2π].When the functions a and b are constant the set ΣH is closely related to the ‘Fučı́k 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