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We present two qualitative results concerning the solutions of the following equation:

During the last twenty years, the theory of stochastic differential equations has successfully attracted considerable attentions of scholars; for example, see [

Stochastic delay differential equation, also known as stochastic functional differential equation, is a natural generalization of stochastic ordinary differential equation by allowing the coefficients to depend on the past values. The Razumikhin argument, generalized Itô formula, and Euler-Maruyama formula play an important role in studying stochastic differential equations. Unfortunately, it is generally not possible to give explicit expressions for the solutions to stochastic differential equations. Therefore, most of the papers are interested in being able to characterize at least qualitatively the behaviour of the solutions. Thus, Lyapunov theory is a powerful tool for qualitative analysis of stochastic differential equations, since the advantage of this method can judge the behaviour of systems without any prior knowledge of the explicit solutions, while the greatest disadvantage of the Lyapunov approach is that no universal method has been given, which enables us to find a Lyapunov function or determine that no such function exists.

It is worth mentioning that there are few results on the stability and boundedness of solutions for first-order stochastic delay differential equations; for example, see [

In 2004, Kolmanovskii and Shaikhet [

Later, in 2006, Rodkina and Basin [

On the other hand, the corresponding problem for the stability and boundedness of solutions of second-order stochastic delay differential equations was studied far less often. So our main aim in this paper is to establish new results on the stability and boundedness for solutions of second-order stochastic delay differential equation of the type

Consider the following nonautonomous

In this section, for the stability result, we impose

Hence, the stochastic delay differential equation (

The zero solution of the stochastic differential equation is said to be stochastically stable or stable in probability, if for every pair of

The zero solution of the stochastic differential equation is said to be stochastically asymptotically stable, if it is stochastically stable, and moreover for every

Let

It should be emphasized that the operator

To prove our main stability result, we shall introduce the following theorems.

Assume that there exist

Then, (

Furthermore,

Assume that there exist

Now we present the main stability result of (

Further to the basic assumptions imposed on

In fact, (

Define the Lyapunov functional

From (

Thus, by using the inequality

Let us choose

Now, in view of (

Since

Then, there exists a positive constant

Also, since

It follows that, by using the inequality

Therefore, we obtain

Then, there exists a positive constant

Thus, from the results (

In the next section, we shall state and prove our main second result on boundedness of (

Consider the

In [

A solution

We assume that for any solution

A special case of the general condition (

Here, we will present the main theorems used in our main boundedness result.

Assume there exists a function

Assume there exists a function

(1) Assume that the hypotheses of Theorem

(2) Assume that the hypotheses of Theorem

The following theorem is the second main result for (

Let the conditions (i) and (ii) of Theorem

Equation (

Consider the Lyapunov functional

From (

Also, from (

Therefore, from (

If we take

Therefore, if

Thus, condition (ii) of Theorem

Then, we have

Hence, the conditions (

In this section, we display two examples to illustrate the application of the results we obtained in the previous sections.

As an application of Theorem

The equivalent system of (

From (

It is obvious that

Thus,

Let us choose

Since

Then, there exists a positive constant

Also, by using the fact that

So there exists a positive constant

Thus, from the results (

As an application of Theorem

The equivalent system of (

From (

It is obvious that

Thus,

Since

If we take

The authors declare that there is no conflict of interests regarding the publication of this paper.