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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:
[1] Wong, Sturmian Theory of Ordinary and Partial Differential Equations (1971)
[2] Swanson, Comparison and Oscillation Theory of Linear Differential Equations (1968) · Zbl 0191.09904
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[4] Li, Proc. Roy. Soc. Edinburgh Sect. 125A pp 133– (1995) · Zbl 0831.26011
[5] Hartman, Ordinary Differential Equations (1964) · Zbl 0125.32102
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