On the Oscillatory Behavior of Certain Third Order Nonlinear Differential Equation

Abstract

. In this paper we shall study some oscillatory and nonoscillatory properties of solutions of a nonlinear third order differential equation, using the results and methods of the linear differential equation of the third order. The aim of this paper is to study the oscillatory or nonoscillatory properties of solutions of the nonlinear differential equation (1) u 000 + q(t)u 0 + p(t) h(u) = 0 where q 0 (t) and p(t) are continuous function of t 2 (a; 1), \Gamma1 ! a ! 1; h(u) is continuous function of u 2 (\Gamma1; 1) and (i) h(u)u ? 0 for u 6= 0, (ii) lim u!0 h(u) u = \Theta, 0 \Theta ! 1. In this paper, a solution of equation (1) we will understand a nontrivial solution of (1) defined on the interval [T; 1], T ? a. A nontrivial solution of (1) is said to be oscillatory if it has zeros for arbitrarily large values of (the independent variable) t. Otherwise a solution is called nonoscillatory. The object of generalization are the results of the paper [1] concerning oscillatory ..