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.


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


Zbl 0511.34006
Full Text: DOI


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