Oscillations in second order differential equations with alternating coefficients

Convergence of oscillatory solutions of $$(a(t)\psi (x(t))x \cdot (t)) \cdot + p(t)x \cdot (t) + q(t)f(x(t)) = 0, \left( { \cdot = \frac{d}{{dt}}} \right)$$ is discussed. The results obtained extend some of the results of Graef, Kitamura, Kusano, Onose, Spikes and Singh.     

Generic oscillations of second order delay differential equations with impulses

In this paper, we will study generic oscillation and generic nonoscillation of second order impulsive delay differential equations. Some necessary and sufficient conditions and sufficient conditions are obtained for both phenomena based on the root of characteristic equation.     

Nonoscillation criteria for quasilinear second order differential equations

Some nonoscillation criteria for quasilinear second order differential equations are obtained. These results generalize the classical results of Hille, Wintner and Opial and recent results of Elbert, Yan, Del Pino as well as Takasi and Yoshida.     

Oscillations of certain second order nonlinear differential equations

A second order nonlinear differential equation

Positive solutions for quasilinear second order differential equation

It is well known that the Krasnoselskii's fixed point theorem is very very important. It was extensively used for studying the boundary value problems. In this article, the Krasnoselskii's fixed point theorem is extended. The new fixed point theorem is obtained. The second order quasilinear differential equation (Φ (y′))′+a(t)f(t,y,y′)=0,, 0<t<1 subject to mixed boundary condition is studied, where f is a nonnegative continuous function, Φ (v)= |v|^p-2 v, p>1. We show the existence of at least one positive solution by using the new fixed point theorem in cone.     

Oscillation of second order quasilinear neutral delay differential equations

In this paper, by employing Riccati transformation technique, some new sufficient conditions for the oscillation criteria are given for the second order quasilinear neutral delay differential equations with delayed argument in the form $$\bigl(r(t)\bigl|z'(t)\bigr|^{\alpha-1}z'(t)\bigr)'+q(t)f\bigl(x\bigl(\sigma(t)\bigr)\bigr)=0,\quad t\geq t_0,$$ where z(t)=x(t)?p(t)x(??(t)), 0??p(t)??p<1, lim _t???? p(t)=p ₁<1, q(t)>0, ??>0. Two examples are considered to illustrate the main results.     

Oscillation theorems for second order quasilinear perturbed differential equations

Abstract. New oscillation criteria for the second order perturbed differential equation are pre-sented. The special case of the results includes the corresponding results in previous papers,extends and unifies a number of known results.     

On the asymptotic behavior of nonoscillatory solutions of second order quasilinear ordinary differential equations

In this paper second order quasilinear ordinary differential equations are considered, and a necessary and sufficient condition for the existence of a slowly growing positive solution is established. Moreover, the precise asymptotic forms as t→∞ of slowly growing positive solutions and slowly decaying positive solutions are obtained.     

Anti-periodic solutions for second order differential equations

By using the bilinear form and the Leray-Schauder principle, we prove the new anti-periodic existence results for second order differential equations.     

Asymptotic behavior of solutions of second order quasilinear differential equations with delay depending on the unknown function

The asymptotic behavior of the nonoscillatory solutions of quasilinear differential equations of second order with delay depending on the unknown function is considered. The main results given by [Bainov et al. (J. Comput. Appl. Math. 91 (1998) 87–96) and Wong (Funkcial. Ekvac. 11 (1968) 207–234)] are improved and generalized.     

Unbounded solutions of differential equations of second order

In this paper, the existence of unbounded solutions for the following nonlinear asymmetric oscillator