Oscillation Theorems for a Class of Nonlinear Second Order Differential Equations with Damping

Xiaojing Wang,Guohua Song

doi:10.4236/apm.2013.31a032

Abstract

The oscillatory behavior of solutions of a class of second order nonlinear differential equations with damping is studied and some new sufficient conditions are obtained by using the refined integral averaging technique. Some well known results in the literature are extended. Moreover, two examples are given to illustrate the theoretical analysis.

Highlights

In this paper, we are concerned with the oscillatory behavior of solutions of the second-order nonlinear differential equations with damping r t x t k x t p t k x t (1.1) C R, R .In what follows with respect to Equation (1.1), we shall assume that there are positive constants c, c1, c2, 1 and 2 satisfying (A1) r t 0 and xf x 0 for all x 0 ;(A2) 0 c x t c1 for all x ; (A3) (A4)q t and k 0 and y c2yk y g x t for ; all y R; (A5) f x
The oscillatory behavior of solutions of a class of second order nonlinear differential equations with damping is studied and some new sufficient conditions are obtained by using the refined integral averaging technique
We are concerned with the oscillatory behavior of solutions of the second-order nonlinear differential equations with damping

Summary

Introduction

We are concerned with the oscillatory behavior of solutions of the second-order nonlinear differential equations with damping r t x t k x t p t k x t (1.1). A solution of Equation (1.1) is said to be oscillatory if it has arbitrarily large zeros, otherwise it is said to be nonoscillatory. In 2011, Wang [22] established some oscillation criteria for Equation (1.1) firstly, some new sharper results are obtained in the present paper. An important method in the study of oscillatory behaviour for Equations (1.1)-(1.4) is the averaging technique which comes from the classical results of Wintner [19] and Hartman [18]. Our results strengthen and improve the recent results of [1] and [21,22]

The Main Results

H t 1c1 s r

H 2 1c1r s s s