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The generalized quasilinearization method is applied in this paper to a telegraph system with periodic boundary conditions. We consider the case in which the forcing function

It is well known that the telegraph equation can be used to model many problems such as fluid mechanics, thermodynamics, and elastic mechanics; examples can be found in [

The convergence of the solution plays an important role in the development of the qualitative theory, and the higher-order convergence of solutions is also very important in practical applications. Quasilinearization is an efficient method for constructing approximate solutions of a variety of nonlinear problems. It was introduced by Bellman and Kalaba [

In this paper, we attempt to extend this generalized quasilinearization method to a telegraph system by assuming that

For convenience, we give some notions and lemmas.

Let

Let

First, we consider the linear equation

Let

We assume the following conditions throughout this paper.

When

Assume that

Then, (

If

Let

Now, one has the following linear telegraph system:

We have the following result.

Assume that conditions

Then, system (

In this paper, we consider the nonlinear telegraph system with doubly periodic boundary conditions:

Let system (

Here and in what follows, the inequalities related to upper and lower solutions are in the distribution sense.

The function

If the above inequality is reversed, the function

Now, we will give two important lemmas which are necessary in our further discussion.

Assume that conditions

The functions

The function

for

Let

We first show that

For this purpose, we set

To prove

The process can be continued successively to obtain

In this section, we apply the method of generalized quasilinearization for a telegraph system. We obtain that the convergence of the sequences of successive approximations is of order

Assume that conditions

The Fréchet derivatives

Then, there exist monotone sequences

In view of condition

Let us first consider the following systems:

Initially, we show that

Next, we must show that

Assume now that

Now, we show that

Furthermore, we need to prove that

To prove the uniform convergence, we can see easily that the sequence

Finally, we have to show that the convergence of

Similarly, we can get

Assume that conditions

The function

Then, there exist monotone sequences

It can be noted from condition

Consider the following systems:

We can prove as in Theorem

Now, let us show that

Now, we prove that

Thus, using mathematical induction, one can obtain

Employing the Ascoli-Arzela theorem, both sequences

At last, we show that the convergence of the sequences of

For the convergence of

To prove the convergence of

Assume that conditions

The function

Then, there exist monotone sequences

In order to construct monotone sequences

Assume that conditions

The function

Then, there exist monotone sequences

To construct monotone sequences

Now, we give an example to illustrate the result in the previous section.

Consider the following telegraph system with periodic boundary conditions:

One can see easily that

The successive approximations

Firstly, we construct an increasing sequence of lower solutions which converges uniformly to the solution

Using the solution

Similarly, we can find all successive approximations

Next, we shall construct a decreasing sequence of upper solutions which will converge uniformly to the exact solution. Since

According to the solution

Proceeding as before, we can find all successive approximations

This illustrates the application of the procedure proven above for approximately obtaining the solution.

The authors declare that they have no conflicts of interest.

All the authors completed the paper together and all of them read and approved the final manuscript.

This paper is supported by the National Natural Science Foundation of China (11771115, 11271106).