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Oscillation theory for indefinite Sturm-Liouville problems with eigenparameter-dependent boundary conditions. (English) Zbl 0894.34018

The authors extend their theory [Proc. Edinb. Math. Soc., II. Ser. 37, No. 1, 57-72 (1993; Zbl 0791.34023); Proc. R. Soc. Edinb., Sect. A 125, 1205-1218 (1995; Zbl 0852.34024)] of eigenvalue problems

-(pq ' ) ' +qy=λryinI=(0,1)(1)
(λa j +b j )y(j)=(λc j +d j )(py ' )(j),j=0,1,

where λ denotes the eigenparameter, a j , b j , c j , d j are constants, and p, q, r are functions in I with 1/pL 1 (I, + ), qL 1 (I,), rL (I,). New techniques based on 2-parameter embedding [see the authors, Appl. Anal. 29, No. 1/2, 107-142 (1988; Zbl 0683.47011)] are used to treat problems (1) which are “right semidefinite”, “left definite” or “indefinite”, as described previously, op. cit. For example, r(x) can have both signs on sets of positive measure in I and a j d j -b j c j can have either sign. In the right semidefinite case, the oscillation of eigenfunctions resembles that in classic Sturmian theory for all but finitely many eigenvalues λ i , but with an index shift, i.e., for some integers j 0 and k, λ j+k has an eigenfunction with j zeros in I for all j>j 0 . Analogues of this are proved in the other cases and a detailed illustration is included.

34B24Sturm-Liouville theory
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory