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Perturbative oscillation criteria and Hardy-type inequalities. (English) Zbl 0903.34030
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 $\tau= \tau_0+ q(x)$, $x\in (a,b)$, $-\infty\le a<b \le\infty$. Under certain conditions on $q(x)$, if $(\tau_0 -\lambda_0)$ is nonoscillatory near $b$ (resp. $a)$ for some $\lambda_0 \in \bbfR$ (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 {\it 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.

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