On a class of second order quasilinear ordinary differential equations
Consider an equation (A) \((|y' |^{\alpha - 1} y')' + q(t) |y |^{\beta - 1} y = 0\), where \(\alpha\) and \(\beta\) are positive constants, and \(q : [a, \infty) \to \mathbb{R}\) is a continuous function. If \(\alpha = 1\) then (A) reduces to the Emden-Fowler type equation (C) \(y'' + q(t) |y |^{\beta - 1} y = 0\). In the first part of the paper the authors discuss the question of global existence and uniqueness of the initial value problem (A), (B) \(y(t_0) = y_0\), \(y'(t_0) = y_1\). Particularly, the following results is proved: If \(q(t)\) is positive on \([a, \infty)\) and \(q(t)\) has either finite positive variation of finite negative variation on any compact subinterval of \([a, \infty)\), then for any \(y_0, y_1\), there exists a unique solution of the initial value problem (A), (B) defined on the entire interval \([a, \infty)\). In the second part of the paper the authors study the problem of oscillation and nonoscillation of solutions of (A). They are concentrated on the case where the function \(q(t)\) may change its sign infinitely often on \([a, \infty)\). It is shown that all proper solutions of (A) are oscillatory if \[ \int^\infty_a q(t)dt = \lim_{T \to \infty} \int^T_a q(t)dt = \infty. \] This result generalizes \textit{P. Waltman}'s oscillation theorem [J. Math. Anal. Appl. 10, 439-441 (1965; Zbl 0131.089)] for the Emden-Fowler equation (C). In the case of conditional integrability of \(q(t)\), i.e. if \[ \int^\infty_a q(t)dt = \lim_{T \to \infty} \int^T_a q(t)dt \quad \text{exists and is finite}, \tag{*} \] the authors suppose \[ Q(t) : = \int^\infty_t q(s)ds \geq 0 \tag{**} \] and obtain a partial generalization of a result of \textit{M. Naito} [Proc. Am. Math. Soc. 109, 769-774 (1990; Zbl 0701.34044)]. A necessary and sufficient condition for (A) to have a minimal (or maximal) nonoscillatory solution are given in the case \((*)\), which generalizes a theorem of \textit{M. Naito} [Trans. Am. Math. Soc. 282, 577-588 (1984; Zbl 0556.34055)]. Finally, assuming \((**)\) all the time and considering separately two cases, \(\alpha < \beta\) and \(\alpha > \beta\), the authors derive the criteria for all proper solutions of (A) to be oscillatory. The result for the case \(\alpha < \beta\) generalize an oscillation theorem of \textit{G. J. Butler} [Ann. Mat. Pura Appl., IV. Ser. 105, 73-92 (1975; Zbl 0307.34029)] for the superlinear Emden-Fowler equation (C).
scientific article; zbMATH DE number 811375