OSCILLATION CRITERIA FOR SECOND ORDER NONLINEAR PERTURBED DIFFERENTIAL EQUATIONS

Sufficient conditions for the oscillation of the nonlinear second order differential equation (a(t)x ′) ′ + Q(t, x ′) = P (t, x, x ′) are established where the coefficients are continuous and a(t) is nonnegative.


INTRODUCTION
We are concerned here with the oscillatory behavior of solutions of the following second order nonlinear differential equation: where a : [T 0 , ∞) → R, Q : [T 0 , ∞) × R → R, and P : [T 0 , ∞) × R × R → R are continuous and a(t) > 0. Throughout the paper, we shall restrict our attention only to the solutions of the differential equation (1.1) which exist on some ray of the form [T 0 , ∞).
In this paper we give more general integral criteria to the oscillation of (1.1), which contain the results in [8] as particular cases.
A solution of (1.1) is said to be oscillatory if it has arbitrarily large zeros, and otherwise it is said to be nonoscillatory.If all solutions of (1.1) are oscillatory, (1.1) is called oscillatory.The oscillatory behavior of solutions of second order ordinary differential equation including the existence of oscillatory and nonoscillatory solutions has been the subject of intensive investigations.This problem has received the attention of many authors.Many criteria have been found which involve the average behavior of the integral of the alternating coefficient.Among numerous papers dealing with this subject we refer in particular to [1, 3, to 16 and 19, 20].

MAIN RESULTS
Assume that there exist continuous functions p, q Theorem 1. Suppose that conditions (2.1),(2.2),and (2.3) hold and let ρ be a positive continuously differentiable function on the interval where Proof.Let x be a nonoscillatory solution on an interval [T, ∞), T ≥ T 0 of the differential equation (1.1).Without loss of generality, this solution can be supposed such that x(t) = 0. We assume that x(t) is positive on [T, ∞) (the case x(t) < 0 can be treated similarly and will be omitted).Then Multiplying (2.6) by ρ(t) and integrating from T to t , we obtain (2.7) Where . We use the following notation Then we have by condition (2.2) hence, there exist T 1 ≥ T such that x ′ (t) < 0 for t ≥ T 1 .
Proof.Let x be a nonoscillatory solution on an interval [T, ∞), of the differential equation (1).Without loss of generality, this solution can be supposed such that x(t) > 0. for all t ≥ T. We consider the following three cases for the behavior of x ′ .Case 1: x ′ is oscillatory.Then there exists a sequence So, for some constant M we have By the Schwarz's inequality, we have and hence for every t ≥ T Furthermore, ( and therefore for all t ≥ T and lim sup This contradicts condition (2.20).Case 2: We distinguish two mutually exclusive cases where )ds is finite.In this case, it follows that (2.22) holds for t ≥ T. Once again, we can complete the proof by the procedure of the proof of Case 1. ii ds for all t ≥ T. Put Furthermore, we choose a T 1 ≥ T so that and then for every t ≥ T 1 we have and integrating from T 1 to t, we obtain The last inequality implies for t ≥ T 1 x ′ (t) ≤ − η a(t) , Therefore, we conclude that lim t→∞ x(t) = −∞ .This contradicts the assumption that x(t) > 0. This completes the proof of the theorem.

Remark 1 .
Condition (2.9) implies that ∞ T R(s)ds < ∞ and lim inf t→∞ t T R(s)ds = ∞ T R(s)ds, hence (2.10) takes the form ∞ T R(s)ds ≥ 0 for all large T, Proof.Let x be a nonoscillatory solution on an interval [T ,∞) of the differential equation (1.1 s))ds is infinite.By Condition (2.19), and from (2.22) it follows that there exists a constant µ such that 3then EJQTDE, 2010 No. 25, p. 9 This means that (2.19), (2.20) hold.Thus, from Theorem 4 it follows that, when (2.21) is satisfied, our differential equation is oscillatory.