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This paper is devoted to presenting an averaging principle for stochastic pantograph equations. Under suitable non-Lipschitz conditions, the solutions to stochastic pantograph equations can be approximated by solutions to averaged stochastic systems in the mean-square sense and probability. At last, an example is given to demonstrate the feasibility of obtained results. Moreover, our results have generalized significantly some previous ones.

Pantograph equations [

On the contrary, the averaging principle as a powerful method has been largely applied in stochastic differential systems, and its main role is to strike a balance between complex models that are more realistic and simpler models that are more amenable to analysis and simulation. Referring to the averaging principle, we are indispensable to recall some excellent articles [

To the best of authors’ knowledge, there is no paper which has considered the approximation theorem as an averaging principle for stochastic pantograph equations. To fill this gap, in this paper, we are intended to study the stochastic pantograph equations as follows:

The highlights and major contributions of this paper are reflected in the subsequent key aspects:

We first attempt to investigate the property of solutions for a class of stochastic pantograph equations by the averaging principle under the non-Lipschitz conditions. Comparing with some previous literatures [

The previous literatures [

The arrangement of the rest paper is as follows. In Section

In this section, we will give some necessary definition and hypothesis conditions, which will be used in later sections.

An

For all

For any other solution

Next, we list the necessary hypothesis conditions, which will be used to prove the main results in next section.

(H1) (non-Lipschitz condition): there exists a function

This section is devoted to present an averaging principle for stochastic pantograph equations. First, we consider the standard form of the system (

To obtain the averaging principle for stochastic pantograph equations, we also need following condition: there exist measurable functions

(H2): for any

By the above preparation, we now consider the original solution

Assume that conditions (H1) and (H2) hold. Then, for a given arbitrarily small number

For any

Taking mathematical expectation on above equation and applying the elementary inequality, we have

Using the elementary inequality again, we have

From (H1) and Cauchy–Schwarz inequality, we have

Also, from (H2) and Cauchy–Schwarz inequality, we have

Analogously, by using the elementary inequality, we also have

From (H1) and Burkholder–Davis–Gundy inequality, we have

Also, from (H2) and Burkholder–Davis–Gundy inequality, we have

According to the estimation of

The proof is completed.

Next, we will present the properties of the uniform convergence in probability between (

Assume that conditions (H1) and (H2) hold. Then, for a given arbitrarily small number

Given

The proof is completed.

If

Consider the following stochastic pantograph equation:

Let

It is clear to see all conditions (H1) and (H2) hold for the functions defined in systems (

In this paper, we mainly discuss an averaging principle for stochastic pantograph equations at the first time. Under suitable non-Lipschitz conditions, we obtain that the solutions to stochastic pantograph equations can be approximated by solutions to averaged stochastic systems in the mean-square sense and probability. Then, an example is given to demonstrate the feasibility of obtained results.

In future work, the interesting extension of our study would be to discuss the averaging principle for the impulsive stochastic pantograph equations with time delays and the fractional stochastic pantograph equations.

The data in this study were mainly collected via discussion during our class. Readers wishing to access these data can do so by contacting the corresponding author.

The authors declare that they have no conflicts of interest.

This work was supported by the NSFC (Grant nos. 12071105 and 11571088) and the Natural Science Special Research Fund Project of Guizhou University (Grant no. 202002).