Half-eigenvalues of periodic Sturm–Liouville problems (original) (raw)

We consider the nonlinear Sturm-Liouville problem Àðpu 0 Þ 0 þ qu ¼ au þ À bu À þ lu; in ð0; 2pÞ; ð1Þ uð0Þ ¼ uð2pÞ; ðpuÞ 0 ð0Þ ¼ ðpuÞ 0 ð2pÞ; ð2Þ where 1=p; qAL 1 ð0; 2pÞ; with p40 a.e. on ð0; 2pÞ; a; bAL 1 ð0; 2pÞ; l is a real parameter, and u 7 ðtÞ ¼ maxf7uðtÞ; 0g for tA½0; 2p: Values of l for which (1)-(2) has a non-trivial solution u will be called half-eigenvalues while the corresponding solutions u will be called halfeigenfunctions. 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 Àðpu 0 Þ 0 þ qu ¼ f ðt; uÞ þ h; in ð0; 2pÞ ð 3Þ (together with (2)), where hAL 1 ð0; 2pÞ; f : ½0;