ftp ejde.math.txstate.edu (login: ftp) OSCILLATION OF SECOND-ORDER NONLINEAR DIFFERENTIAL EQUATIONS WITH A DAMPING TERM

We present new oscillation criteria for the second order nonlinear differential equation with damping term of the form (r(t) (x)f(ux)) · +p(t)'(g(x),r(t) (x)f(ux)) +q(t)g(x) = 0, where p, q, r : [to,1) ! R and , g, f : R ! R are continuous, r(t) > 0, p(t) � 0 and (x) > 0, xg(x) > 0 for x 6 0, uf(u) > 0 for u 6 0. Our results generalize and extend some known oscillation criteria in the literature. The relevance of our results is illustrated with a number of examples.


Introduction
This paper concerned with oscillation of the solutions to the second-order nonlinear differential equation with damping term: r(t)x (t) + p(t)x (t) + q(t)g(x(t)) = 0, t ≥ t 0 , where q and p are continuous functions defined on the interval [t 0 , ∞), t 0 > 0 and r(t) > 0 for t ≥ t 0 > 0, g is a continuous function for x ∈ (−∞, ∞), continuously differentiable and satisfies xg(x) > 0, g (x) ≥ k > 0 for all x = 0. (1.2) Equation (1.1) is said to be superlinear if du < ∞ for > 0, (1.3) and sublinear if du < ∞ for > 0. (1.4) We restrict our attention to those solutions of (1.1) which exist on some half line [t x , ∞) and satisfy sup{|x(t)| : t > T } > 0 for any T ≥ t x .We make a standing hypothesis that (1.1) does possess such solutions.A solution of (1.1) is said to be oscillatory if it has arbitrarily large zeros; otherwise it is non-oscillatory.The equation itself is called oscillatory if all its solutions are oscillatory.
In the previous two decades, there has been increasing interest in obtaining sufficient conditions for the oscillation and non-oscillation of solutions of different classes of second order differential equations, see for example [4,5,6,7,8,11,12,14,15,16,17,19,23,24,25,26,27,28,29] and the references therein.a lot of work has been done on the following particular cases of (1.1): An important tool in the study of oscillatory behavior of solutions of these equations is the averaging technique which goes back as far as the classical result of Wintner [32] which proved that (1.5) is oscillatory if Hartman [14] proved that that the limit in (1.8) cannot be replaced by the limit supremum and proved that the condition implies that every solution of (1.5) oscillates.Kamenev [15] improved Wintner's result by proving that the condition for some integer n > 1 is sufficient for the oscillation of (1.5).
Yan [36] proved that if lim sup for some integer n > 1 and there exists a function φ on [t 0 , ∞) satisfying then every solution of equation (1.5) oscillates.Philos [24] further improved Kamenev's result by proving the following: Suppose there exist continuous functions H, h : , see [17].For further results on the oscillation of superlinear and sublinear equations, we refer the reader to [6,7,8,31].
In the study of the differential equation (1.7), many criteria for oscillation exist which involve the behavior of the integral of q however the common restrictions that on the functions q, g and r are required; see for example [4,5,11,12].The presence of the damping term in (1.1) calls for a modified approach to the study of the oscillatory properties of solutions, see for example the paper by Saker, Peng and Agarwal [28], Li, Wang and Yan [23] and the references therein.They cited most of the oscillation results for (1.1) when p(t) and q(t) are positive functions.
Recently, Rogovechenko et al. [27] considered (1.1) and established some sufficient conditions for oscillations.They posed the following open problem: It would be very important to obtain general oscillation criteria for nonlinear differential equations with damping term without requiring additional sign conditions on the coefficients p(t) and q(t).
In this paper, we consider (1.1) and by using a Riccati transformation technique, we establish some oscillation criteria of Kamanev and Philos type with no sign conditions on p(t) and q(t).Our results in this paper are the affirmative answer to the question posed by Rogovechenko et al. [27] and improve and complement the results established by Sun [29].

Main Results
In this section, we will use the Riccati technique to establish sufficient conditions for oscillation of (1.1).Comparisons between our results and the previously known are presented and some examples illustrate the main results.
Proof.Suppose to the contrary that (1.1) possesses a non-oscillatory solution x on an interval [T, ∞), T ≥ t 0 .Without loss of generality, we shall assume that x(t) > 0 for all t ≥ T (the case x(t) < 0 can be treated similarly and will be omitted).Let w(t) be defined by the Riccati transformation This and (1.1) imply for t ≥ T that We consider the following two cases: Case 1. the integral ds is finite.Then there exists a positive constant N such that Now, from (2.4), t t0 ρ(s)q(s)ds where c 1 = w(T ) + T t0 ρ(s)q(s)ds.By Bonnet's theorem since r(t)ρ (t) − ρ(t)p(t) is non-increasing, for a fixed t ≥ T , there exists ξ ∈ du .
Thus, for t ≥ T , t t0 ρ(s)q(s)ds Therefore, taking into account (2.5) and (2.7), we conclude that where Thus, for every t ≥ T , ρ(u)q(u)du ds is infinite.From (2.4), taking into account (2.6) and (2.7), for every t ≥ T we obtain where A = c 1 + k 1 .By the condition (2.2), from (2.8), it follows that for some constant B, −w(t) ≥ B + t T g (x(s))w 2 (s) ρ(s)r(s) ds for every t ≥ T. (2.9) We can consider a T 2 ≥ T such that Then (2.9) ensures that w(t) is negative on [T 2 , ∞).Now, (2.9) gives and consequently for all t ≥ T 2 , log Hence, So, (2.9) yields where C = Cg(x(T 2 )) > 0. Thus, we have ds for all t ≥ T 2 which leads to lim t→∞ x(t) = −∞, which is a contradiction.This completes the proof.
Proof.Suppose to the contrary that (1.1) possesses a non-oscillatory solution x on an interval [T, ∞), T ≥ t 0 .Without loss of generality, we shall assume that x(t) > 0 for all t ≥ T ≥ t 0 (the case x(t) < 0 treated similarly and will be omitted).Again we define w(t) as in Theorem 2.1.and prove that (1.1) holds then we have equation (2.4).First, we claim that  Set ∞ (2.15) Then, we have ) Then, taking a limit superior on both sides, we obtain a contradiction to the condition (2.14).Thus it must be the case I = ∞.As in the proof of Theorem 2.1 (Case 2) we arrive at the contradiction lim t→∞ x(t) = −∞.This completes the proof.