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Sturmian comparison theorem for half-linear second-order differential equations. (English) Zbl 0873.34020
The half-linear differential equation ${\textstyle {\frac{d}{dt}}} \{r(t)\varphi(u'(t))\}+ c(t)\varphi(u(t))=0 \tag{E}$ is considered. Here $$c\in C[0,\infty)$$, $$r\in C^1([0,\infty), (0,\infty))$$ and $$\varphi:\mathbb{R}\to\mathbb{R}$$ is defined by $$\varphi(s)=|s|^{p-2}s$$, where $$p>1$$ is a fixed number. The authors continue the study initiated by A. Elbert [Colloq. Math. Soc. Janos Bolyai 30, 153-180 (1981; Zbl 0511.34006)]. They prove: Let $$r(t)$$, $$r_1(t)$$, $$C^1([a,b], (0,\infty))$$ and $$c(t),c_1(t)\in C[a,b]$$. If the boundary value problem ${\textstyle {\frac{d}{dt}}} \{r_1(t)\varphi(\omega'(t))\}+ c_1(t)\varphi(\omega(t))=0, \qquad \omega(a)= \omega(b)=0$ has a solution with $$\omega(t)\neq 0$$ on $$(a,b)$$ satisfying $\int_a^b \{(r-r_1)|\omega'|^p- (c-c_1)|\omega|^p\}dt\leq 0$ then every solution $$u(t)$$ of (E) must have a zero in $$(a,b)$$ unless $$u(t)$$ and $$\omega(t)$$ are proportional. Some oscillation and nonoscillation criteria are given, too, as applications.

##### MSC:
 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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##### References:
  Wong, Sturmian Theory of Ordinary and Partial Differential Equations (1971)  Swanson, Comparison and Oscillation Theory of Linear Differential Equations (1968) · Zbl 0191.09904  Elbert, Qualitative Theory of Differential Equations 30 pp 153– (1979)  Li, Proc. Roy. Soc. Edinburgh Sect. 125A pp 133– (1995) · Zbl 0831.26011  Hartman, Ordinary Differential Equations (1964) · Zbl 0125.32102  Singh, J. Math. Phys. Sci. 4 pp 363– (1974)
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