Perturbative oscillation criteria and Hardy-type inequalities. (English) Zbl 0903.34030

Gesztesy, F.; Ünal, M.

Math. Nachr. 189, 121-144 (1998).

The authors compare oscillation properties of solutions to Sturm-Liouville equations \(\tau_0 \psi_0= \lambda \psi_0\) and \(\tau\psi =\lambda \psi\), where \(\tau_0\) is of the type \[ \tau_0= -{d\over dx} p_0(x) {d\over dx} +q_0(x) \] and its perturbation \(\tau\) is of the form \(\tau= \tau_0+ q(x)\), \(x\in (a,b)\), \(-\infty\leq a<b \leq\infty\).
Under certain conditions on \(q(x)\), if \((\tau_0 -\lambda_0)\) is nonoscillatory near \(b\) (resp. \(a)\) for some \(\lambda_0 \in \mathbb{R}\) (with \(\psi_0 (\lambda_0,x)\) denoting a positive solution to \(\tau_0 \psi= \lambda_0 \psi)\), the authors establish two theorems for that \((\tau- \lambda_0)\) be nonoscillatory or be oscillatory near \(b\) (resp. \(a)\).
The special case \(p_0= \psi_0=1\), \(q_0= \lambda_0 =0\) in the first theorem stands for the original oscillation criterion by A. Kneser [Math. Ann. Qd 42, 409-435 (1893; JFM 25.0533.01)] and in the second one represents a generalization of Kneser’s result due to H. Weber (1912).
Finally, making use of certain types of factorizations of general Sturm-Liouville differential expressions on \((a,b)\), the authors prove a natural generalization of Hardy’s inequality.

Reviewer: N.Hayek (La Laguna)

MSC:

34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34B24	Sturm-Liouville theory

Keywords:

Kneser’s oscillation criterion; Sturm-Liouville equations; JFM 25.0533.01; Hardy’s inequality

Citations:

JFM 25.0533.01

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