## 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:

 [1] Hille, Non - Oscillation Theorems, Trans. Amer. Math. Soc. 64 pp 234– (1948) · Zbl 0031.35402 [2] Kusano , T. Naito , Y. [3] Kusano, Nonoscillation Theorems for a Class of Quasilinear Differential Equations of Second Order, J. Math. Anal. Appl. 189 pp 115– (1995) · Zbl 0823.34039 [4] Kwong, Integral Inequalities and Second Order Linear Oscillation, J. Diff. Equs. 45 pp 16– (1982) · Zbl 0498.34022 [5] Leighton, The Detection of the Oscillation of Solutions of a Second Order Linear Differential Equation, Duke J. Math. 17 pp 57– (1950) · Zbl 0036.06101 [6] Li , H. J. Yeh , C. C. [7] Swanson, Comparison and Oscillation Theory of Linear Differential Equations (1968) · Zbl 0191.09904 [8] Willett, On the Oscillatory Behavior of the Solutions of Second Order Linear Differential Equations, Ann. Polon. Math. 21 pp 175– (1969) · Zbl 0174.13701 [9] Wintner, On the Non - Existence of Conjugate Points, Amer. J. Math. 73 pp 368– (1951) · Zbl 0043.08703
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.