## Sturm-Liouville eigenvalue problems for half-linear ordinary differential equations.(English)Zbl 1006.34025

The authors consider the half-linear Sturm-Liouville eigenvalue problem $\begin{gathered} (p(t)|x'|^{\alpha-1}x')'+\lambda q(t)|x|^{\alpha-1}x=0, \qquad a\leq t\leq b,\tag{HL}\\ Ax(a)-A'x'(a)=0,\qquad Bx(b)+B'x'(b)=0,\end{gathered}$ where $$\alpha>0$$ is a constant, $$p,q$$ are real-valued continuous functions on $$[a,b]$$ with $$p(t)>0$$ and $$\lambda$$ is a real parameter. The function $$q(t)$$ may change its sign in $$[a,b]$$. The main purpose of the paper is to extend the classical Sturm-Liouville eigenvalue problem for the linear equation $$(p(t)x')'+\lambda q(t)x=0$$, which is a special case of the above equation when $$\alpha=1$$, in a natural way to the more general half-linear equation. More precisely, it is proved the following result:
If $$AA'\geq 0$$, $$BB'\geq 0$$, $$A^2+B^2\neq 0$$, and $$q(t)$$ takes both positive value and negative value on $$[a,b]$$, then the totality of eigenvalues of (HL) consists of two sequences $$\{\lambda_n^+\}_{n=0}^\infty$$ and $$\{\lambda_n^-\}_{n=0}^\infty$$ such that $$\dots<\lambda_n^-<\dots<\lambda_1^-<\lambda_0^-<0<\lambda_0^+<\lambda_1^+< \dots<\lambda_n^+<\dots$$ and $$\lim_{n\to\infty}\lambda_n^+=+\infty$$, $$\lim_{n\to\infty}\lambda_n^-=-\infty$$. The eigenfunctions associated with $$\lambda=\lambda_n^+$$ and $$\lambda_n^-$$ have exactly $$n$$ zeros on $$(a,b)$$.
A crucial role in the proof is played by the generalized Prüfer transformation.
Reviewer: Pavel Rehak (Brno)

### MSC:

 34B24 Sturm-Liouville theory 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
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### References:

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