## 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

### Keywords:

oscillation; nonoscillation

Zbl 0889.34029
Full Text:

### References:

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