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

Kusano, Takaŝi; Naito, Manabu

Rocky Mt. J. Math. 31, No. 3, 1039-1054 (2001).

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

Keywords:

half-linear differential equation; Sturm-Liouville eigenvalue problem; generalized Prüfer transformation

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