Some Criteria for the Asymptotic Behavior of a Certain Second Order Nonlinear Perturbed Differential Equation ()
1. Introduction
This paper is concerned with the problem of asymptotic behavior of the second order nonlinear perturbed differential equation
(1)
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.
2. Main Results
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
, (2)
(3)
and
(4)
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
. (5)
Differentiating (5) and making use of (1) and from hypothesis (H1), (H2) and (H3), it follows that
(6)
By using the inequality
(7)
we get
(8)
Integrating this inequality, from 
to
, we get
(9)
Dividing (9) by 
and hence integrating from 
to 
we obtain
(10)
Using (4) we get
(11)
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
. (12)
If there exists a differentiable function 
such that (3) holds and,
(13)
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
(14)
Integrating this inequality from 
to 
we get
(15)
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.
3. Example
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
.