This paper is concerned with the problem of asymptotic behavior of the second order nonlinear perturbed differential equation

where

.

Throughout the paper according to the results we shall impose the following conditions:

(H₁) Let and there exists a constant

such that and for

(H₂) and there exists a continuous function such that for(H₃) and there exists a continuous function such that for

(H₄) and there exists a continuous function such that for

(H₅) and for every

.

We shall also restrict our attention only to the solution of the differential equation (1.1) which exist on some ray of the form.

The oscillatory behavior of the solution of second order ordinary differential equations including the existence of oscillatory and nonoscillatory solutions has been the subject of intensive investigation. This problem has received the attention of many others. See for example, [1-5]. Since the publication of Hammet’s paper in [6] in 1971, the asymptotic behavior of the solution of the ordinary and functional differential equations has been widely discussed in the literature [2,4,7-9].

In this paper we give sufficient conditions so that for every nonoscillatory solution of (1), we have

. Our results contain the results in [10]

as particular cases.

In this section we prove our main results.

Theorem 2.1. Let conditions (H₁), (H₂), (H₃) and (H₅) hold. If there exists a differentiable function such that

and

where

.

Then for every nonoscillatory solution of equation

(1), we have.

Proof. Let be a nonoscillatory solution of (1). We may assume that for. Define

Differentiating (5) and making use of (1) and from hypothesis (H₁), (H₂) and (H₃), it follows that

By using the inequality

we get

Integrating this inequality, from to, we get

Dividing (9) by and hence integrating from to we obtain

Using (4) we get

If, then there exists a positive constant such that for all and consequently, by (H₅)

which contradicts (11). Thus we must have

. The proof for the case

for is similar and hence is omitted.

We note that when, (H₁) condition can be weakened. Indeed from the proof of Theorem 2.1, the following result can obtain easily.

Theorem 2.2. Let conditions (H₁), (H₂), (H₃) and (H₅) hold. Suppose that

, and for

and

.

Then for every solution of Equation (1), we have.

By taking (H₄) instead of (H₃) we obtain the following result which can be applied for example to the damped equation

.

Theorem 2.3. Let conditions (H₁), (H₂), (H₄) and (H₅) hold. Suppose that

If there exists a differentiable function such that (3) holds and,

where

then for every solution of Equation (1), we have

.

Proof. Let be a nonoscillatory solution of Equation (1). We may assume that for . Differentiating (5) and making use of (1) and from hypothesis (H₁), (H₂), (H₄) and (12), as in the proof of Theorem 2.1, we can obtain easily that

Integrating this inequality from to we get

Dividing (15) by and hence integrating from to we obtain

The rest of of the proof is similar to that of Theorem 2.1 and hence is omitted.

Remark 2.1. Grace and Lalli, consider the following equation

in [10] and give a similar result. But if we compare Theorem 2.3 with Theorem 2 in [10], we observe that they have a condition such as for which impose some restriction on and. In our result we remove this condition, so Theorem 2.3 is weaker then Theorem 2 in [10].

Remark 2.2. To give similar results for the equation

where, , ,

and is a positive real number, still remains as an open problem and will be interesting.

Consider the differential equation of the form

where and are continuous function such that and. All conditions of Theorem 2.1 are satisfied. Then every nonoscillatory solutions of (16) we have. In particular

has a solution and.