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Asymptotic behavior of nonoscillatory solutions of second-order nonlinear differential equations. (English) Zbl 1082.34042
Ladde, G.S.(ed.) et al., Dynamic systems and applications. Volume 4. Proceedings of the 4th international conference, Morehouse College, Atlanta, GA, USA, May 21–24, 2003. Atlanta, GA: Dynamic Publishers (ISBN 1-890888-00-1/hbk). 312-319 (2004).
The asymptotic behaviour of nonoscillatory solutions of the general second-order nonlinear differential equation \[ u''+f(t,u,u')=0, \quad t\geq 1, \] is investigated. It is supposed that the real function \(f(t,u,v)\) is continuous and satisfies \[ | f(t,u,v)| \leq h_1 \biggl(t,\frac{| u| }{t}\biggr)+h_2(t,| v| ), \] where the functions \(h_1(t,s)\) and \(h_2(t,s)\) are continuous, nonnegative and monotone nondecreasing in \(s\). It is assumed that there exists a constant \(c>0\) such that \[ \int_1^{\infty}[h_1(t,c)+h_2(t,c)]\,dt<\infty. \] Under these conditions, the authors prove the existence of asymptotically linear solutions of equation (1). Due to the general nature of the hypotheses, solutions exist only locally near \(+\infty\) in the sense that the interval of the existence cannot be given a priori. Some examples are presented in order to illustrate the applicability of the theory developed in the paper.
For the entire collection see [Zbl 1054.34001].

34D05 Asymptotic properties of solutions to ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations