Nonoscillation criteria for second-order nonlinear differential equations.
The author deals with oscillatory properties of the half-linear second-order differential equation \[ (p(t)| y'| ^{\alpha-1}y')'+q(t)| y| ^{\alpha-1}y=0,\quad \alpha>0, \tag{*} \] where \(p,q\) are continuous functions and \(p(t)>0\). First, the author states known facts that the linear Sturmian comparison theorem extends to (*) and that nonoscillation of (*) is equivalent to the solvability of the Riccati inequality \[ v'+\frac{\alpha | v| ^{\frac{\alpha+1}{\alpha}}}{p^{\frac{1}{\alpha}}(t)} +q(t)\leq 0. \tag{**} \] Then, he looks for the function \(v\) satisfying (**) in the form \(v(t)=ct^{-d}\) and obtains (rather trivial) conditions for the nonoscillation of (*). Note that the paper also contains the following statement (given without proof): Suppose that \(R(t)=\int_t^\infty q(s)\,ds\) exists and \(R(t)>-\infty\). Then equation \text{ (*)} is nonoscillatory if and only if there exists a function \(v\) satisfying for large \(t\) the integral inequality \[ \int_t^\infty \frac{\alpha | v| ^{\frac{\alpha+1}{\alpha}}(s)}{p^{\frac{1}{\alpha}}(s)} \,ds + R(t)\leq v(t). \tag{***} \] Moreover, suppose that \(P(t)\geq p(t)>0\) and \(\int_t^\infty q(s)\leq \int_t^\infty Q(s)\,ds\). If the equation \[ (P(t)| y'| ^{\alpha-1}y')'+Q(t)| y| ^{\alpha-1}y=0,\quad \alpha>0, \tag{****} \] is nonoscillatory, then \text{ (*)}is also nonoscillatory. It would be interesting to see the proof of this statement, in particular, of the statement that nonoscillation of (*) implies the existence of a function \(v\) satisfying (***) for which the first integral in (***) is convergent. The existence of such a functions is known (by the half-linear version of the Hartman-Wintner theorem) only if \(\int^\infty p^{-\frac{1}{\alpha}}(t)\,dt=\infty\). Moreover, something is wrong in the second part of the theorem as the example of the generalized Euler equations (*), (****) with \(p(t)=P(t)\equiv 1\), \(q(t)= [ (\frac{\alpha}{\alpha+1} )^{\alpha+1}+ \varepsilon]t^{-(\alpha+1)}\), \(\varepsilon >0\), \(Q(t)= (\frac{\alpha}{\alpha+1} )^{\alpha+1}\) shows. Here, the assumptions of the statement are satisfied, but (****) is nonoscillatory (\(y=t^{\frac{\alpha}{\alpha+1}}\) is its nonoscillatory solution) while (*) is oscillatory [see e.g., \textit{Á. Elbert}, Stud. Sci. Math. Hung. 19, 447--464 (1984; Zbl 0629.34066)].
scientific article; zbMATH DE number 1861155
scientific article