On Oscillation of Nonlinear Differential Equations of Second Order

Abstract

In this paper, we present some new sufficient conditions for the oscillation of all solutions of the second order non-linear ordinary differential equation of the form

ç r(t) x(t)÷ + q(t)F(g(x(t)), r(t) x(t)) = H (t, x(t))

where q and r are continuous functions on the interval [t0 , ¥), t0 ³ 0 , r(t) is a positive function, g is continuously differentiable function on the real line R except possibly at 0 with xg(x) > 0 and g ¢(x) ³ k > 0

for all x ¹ 0, F is a continuous function on RxR with u F(u, v) > 0 for all u ¹ 0 and F(lu, lv) = lF(u, v)

for any l Î (0, ¥) and H is a continuous function on [t0 , ¥)×R with H (t, x(t)) g(x(t)) £ p(t) for all x ¹ 0 and t ³ t0 . The oscillatory behavior of ordinary differential equations has been extensively studied by many authors, see for examples [1-14] and the references therein. This research work which is obtained using Riccati Technique, extends and improves many of the known results of oscillation in the literatures such as our oscillation results extend result of Wong and Yeh[14], result of Philos[9], result of Onose[8], result of Philos and Purnaras[10], result of E. M. Elabbasy[3], results of Greaf, Rankin and Spikes[5], results of Grace and Lalli[4] and results of Moussadek Rmail[7] and some other previous results. We illustrate our oscillation results and the improvement over other known oscillation conditions by examples, numerically are solved in MATLAB.