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Oscillation and nonoscillation criteria for second order linear differential equations. (English) Zbl 0912.34032

The authors derive oscillation and nonoscillation criteria for second-order linear differential equations \[ (p(t)u')'+q(t)u=0, \tag{*} \] where \(p,q\) are real-valued functions and \(p(t)>0\). The results are essentially based on the following idea. The transformation of the dependent variable \(u=h(t)w\), \(h(t)\neq 0\), transforms (*) into the equation \[ (P(t)w')'+Q(t)w=0 \tag{**} \] with \(P=ph^2\), \(Q=h[(ph')'+qh]\). In particular, if (*) is rewritten into the form \[ (p(t)u')'+\tilde q(t)u+(q(t)-\tilde q(t))u=0, \] where \(\tilde q\) is such that the equation \((pu')'+\tilde qu=0\) is nonoscillatory and \(h\) is its (nonzero) solution, then the transformation \(u=hw\) converts (*) into (**) with \(Q=(q-\tilde q)h^2\). The above given transformation coupled with some (classical) oscillation and nonoscillation criteria are applied not directly to (*) (as in the classical setting) but to (**).
This idea has been recently used by the reviewer (in a slightly modified form) when investigatigating oscillatory properties of the higher-order Sturm-Liouville equation \((p(t)y^{(n)})^{(n)}+q(t)y=0\) [Math. Nachr. 188, 49-68 (1997; Zbl 0889.34029)] and of the half-linear second-order equation \((p(t)| y'| ^{\alpha-1}y')'+q(t)| y| ^{\alpha-1}y=0\) [O. Došlý, Hiroshima Math. J. 28, 507-521 (1998)].
Reviewer: O.Došlý (Brno)

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations

Citations:

Zbl 0889.34029
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References:

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