Oscillation Results for Emden–Fowler Type Differential Equations

AbstractThe third order nonlinear differential equationx‴+a(t)x′+b(t)f(x)=0, (∗)is considered. We present oscillation and nonoscillation criteria which extend and improve previous results existing in the literature, in particular some results recently stated by M. Greguš and M. Greguš, Jr., (J. Math. Anal. Appl.181, 1994, 575–585). In addition, contributions to the classification of solutions are given. The techniques used are based on a transformation which reduces (∗) to a suitable disconjugate form. To this aim auxiliary results on the asymptotic behavior of solutions of a second order linear differential equation associated to (∗) are stated. They are presented in an independent form because they may be applied also to simplify and improve other qualitative problems concerning differential equations with quasiderivatives

The asymptotic nature of a class of second order nonlinear system

In this paper, we obtain some results on the nonoscillatory behaviour of the system (1), which contains as particular cases, some well known systems. By negation, oscillation criteria are derived for these systems. In the last section we present some examples and remarks, and various well known oscillation criteria are obtained

Oscillatory solutions of Emden-Fowler type differential equation

The paper deals with the coexistence between the oscillatory dynamics and the nonoscillatory one for a generalized super-linear Emden–Fowler differential equation. In particular, the coexistence of infinitely many oscillatory solutions with unbounded positive solutions are proved. The asymptotics of the unbounded positive solutions are described as well

Decaying Regularly Varying Solutions of Third-order Differential Equations with a Singular Nonlinearity

summary:This paper is concerned with asymptotic analysis of strongly decaying solutions of the third-order singular differential equation $x^{\prime \prime \prime }+q(t)x^{-\gamma }=0$, by means of regularly varying functions, where $\gamma$ is a positive constant and $q$ is a positive continuous function on $[a,\infty )$. It is shown that if $q$ is a regularly varying function, then it is possible to establish necessary and sufficient conditions for the existence of slowly varying solutions and regularly varying solutions of (A) which decrease to $0$ as $t\rightarrow \infty$ and to acquire precise information about the asymptotic behavior at infinity of these solutions. The main tool is the Schauder–Tychonoff fixed point theorem combined with the basic theory of regular variation