Journal of Function Spaces
Volume 2017 (2017), Article ID 1982568, 10 pages
https://doi.org/10.1155/2017/1982568

Research Article

Rapid Convergence for Telegraph Systems with Periodic Boundary Conditions

Peiguang Wang¹ and Xiang Liu²

¹College of Electronic and Information Engineering, Hebei University, Baoding 071002, China
²College of Mathematics and Information Science, Hebei University, Baoding 071002, China

Correspondence should be addressed to Peiguang Wang; pgwang@hbu.edu.cn

Received 4 July 2017; Revised 7 October 2017; Accepted 19 October 2017; Published 13 November 2017

Academic Editor: Xinguang Zhang

Copyright © 2017 Peiguang Wang and Xiang Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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 satisfies the following condition: exists and is quasimonotone nondecreasing or nonincreasing. We develop nonlinear iterates of order which will be different with being even or odd. Finally, we develop two sequences which converge to the solution of the telegraph system and the convergence is of order .

1. Introduction

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 [1–4]. Recently, the maximum principle for the telegraph equation has been studied by many authors. Ortega and Robles-Perez [5] established a maximum principle for periodic solutions of the telegraph equation

with doubly periodic boundary condition. Mawhin et al. [6] gave a maximum principle for bounded solutions of the telegraph equation which are -periodic with respect to and bounded over with respect to . Li [7] built a maximum principle for time-periodic solutions of the telegraph equation

which are bounded on for and -periodic for . Wang and An [8] gave the maximum principle for the linear telegraph system

and they obtained a monotone sequence of approximate solutions converging uniformly and quadratically to a solution of the semilinear telegraph system.

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 [9] and generalized by Lakshmikantham and Vatsala [10]. Up till now, there are some higher-order convergence results for a variety of differential equations under different assumptions. In [11], Mohapatra et al. applied the generalized quasilinearization method for first-order initial value problems and obtained the th order of convergence on conditions that the th derivatives and of the functions and exist and are continuous and and are -hyperconvex or -hyperconcave in . Cabada et al. [12] used the generalized quasilinearization method for first-order boundary value problems to obtain the th order of convergence by assuming that exists and is continuous. Afterwards, the results of th-order convergence, such as second-order boundary value problems [13], impulsive differential equations [14], reaction diffusion equations [15], parabolic integrodifferential equations [16], nonlinear boundary value problems [17], and singular differential systems [18], were also obtained under weaker assumptions; namely, exists and is nondecreasing or nonincreasing in and one-sided Lipschitzian in for . The nonlinear iterate is of order and it is different when is even or odd. To the best of our knowledge, no results on the rapid convergence of solutions for a nonlinear telegraph system can be found.

In this paper, we attempt to extend this generalized quasilinearization method to a telegraph system by assuming that exists and is quasimonotone nondecreasing or nonincreasing. We obtain two sequences which converge to the solution of the telegraph system and the convergence is of order ( being even or odd) for .

2. Preliminaries

For convenience, we give some notions and lemmas.

Let be the torus defined as

where is a real number set and is an integer set. Doubly -periodic functions will be regarded as functions defined on . and denote the Lebesgue integrable function space and continuous function space on . The norms can be defined as and .

Let denote the space of distribution on and represent a Banach space .

First, we consider the linear equation

where , , and for .

Let be the differential operator,

acting on functions on . From the discussion in [5, 7], we know that if has the resolvent

where is the unique solution of (5), and the restriction of on or is compact. In particular, is a completely continuous operator. Furthermore, for , the Green function of the differential operator is explicitly expressed; we can refer to Lemma in [5]. According to the definition of , we have

We assume the following conditions throughout this paper.

, for , and ., , .

When , we have the following unique existence and positive estimate result.

Lemma 1 (see [7]). Assume that and is the Banach space .
Then, (5) has a unique solution is a linear bounded operator with the following properties:

(i) is a completely continuous operator.(ii)If , , has the positive estimate

Let be a Banach space ordered by the positive cone and equipped with norm and be a bounded linear operator. By maximum principle to the following system:

we mean that implies since is a solution of (10).

Now, one has the following linear telegraph system:

Here and in what follows, by a doubly periodic solution of (11), we mean that satisfies (11) in the distribution sense; that is,

We have the following result.

Lemma 2 (see [5]). Assume that conditions and hold, and

, .

Then, system (11) has at least one solution in and satisfies the maximum principle.

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

Let system (13) be equivalent to

where

Remark 3. Here and in what follows, the inequalities related to upper and lower solutions are in the distribution sense.

Definition 4. The function is called a lower solution of system (14) if it satisfies

If the above inequality is reversed, the function is called an upper solution of system (14).
Now, we will give two important lemmas which are necessary in our further discussion.

Lemma 5. Assume that conditions hold.

The functions are lower and upper solutions of system (14) with .The function satisfies the inequality

 for , where

Then, system (14) has a solution that satisfies .

Proof. Let , be the solutions of the following linear system:

which exist because of Lemma 2. According to the iterative schemes (19), we obtain the sequences and which were generated by the initial conditions and , respectively. The regularity of solutions of the linear operator implies that and are continuous if .
We first show that on .
For this purpose, we set . Using condition , we obtain

By Lemma 2, we have ; that is, on . Similarly, letting , we can find that

Using Lemma 2, we get , showing that on .
To prove , set . From condition , we have

As before, this implies that on . Thus, we conclude that

The process can be continued successively to obtain

In particular, the sequence is nondecreasing and converges pointwise to a function that satisfies . A passage to the limit based on the Dominated Convergence Theorem shows that is a solution of

that is, is a solution of (14). Again, the regularity theory allows us to conclude that belongs to .

3. Generalized Quasilinearization Method

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 where is or .

Theorem 6. Assume that conditions , , , and hold:

  , .The Fréchet derivatives exist and are continuous satisfying for , , and is a positive matrix.

Then, there exist monotone sequences and , which converge uniformly to the solution of system (14) in , and the convergence is of order ; that is, there exist positive constants and such that

hold, where , .

Proof. In view of condition , we have

for , , and

for , , where

for , , .
Let us first consider the following systems:

Initially, we show that and are lower and upper solutions of (30), respectively. Let , in (30) and (31). Condition and inequality (27) lead to

We show that and are lower and upper solutions of (30), respectively. Hence, we obtain by Lemma 5 that there exists a solution of (30) such that on . We next prove that and are lower and upper solutions of (31), respectively. From inequalities (27) and (28), we get

which show that and are lower and upper solutions of (31), respectively. As before, by Lemma 5, we can conclude that system (31) has a solution that satisfies on .
Next, we must show that and are lower and upper solutions of system (14), respectively. To prove this, we can apply inequality (27) to obtain

It proves that is a lower solution of (14). Similarly, using inequality (28), we see that

which implies that is an upper solution of (14). It therefore follows that on .
Assume now that and are lower and upper solutions of (14), respectively, and on . We need to show that on , where and are solutions of (30) and (31), respectively, with and .
Now, we show that and are lower and upper solutions of (30), respectively. Let and in (30) and (31). Utilizing the assumption that and are lower and upper solutions of (14), respectively, and inequality (27), we get

which show that and are lower and upper solutions of (30), respectively. Hence, we can now apply Lemma 5 to deduce that there exists a solution of (30) such that on . Using the same technique, we can show that there exists a solution of (31) such that on .
Furthermore, we need to prove that and are lower and upper solutions of (14). For this purpose, by using inequality (27), we get

This proves that is a lower solution of (14). Similar reasoning gives that is an upper solution of (14). Thus, we conclude that on . Therefore, the method of mathematical induction can be applied to obtain

Since and are lower and upper solutions of (14), respectively, and all the assumptions of Lemma 5 hold, we can conclude that there exists a solution of (14) such that . Hence, we have

To prove the uniform convergence, we can see easily that the sequence is uniformly bounded and equicontinuous on . Hence, employing Ascoli-Arzela’s theorem, we have that there exists a subsequence which converges uniformly to . Moreover, is a nondecreasing sequence; it follows that converges to . By the Dominated Convergence Theorem, it is easy to verify that is a solution of (14) on in distribution sense. Similarly, convergence holds for .
Finally, we have to show that the convergence of and is of order . For that, we consider

Now, using the mean value theorem together with condition , we obtain

where . An application of Lemma 2 yields that on , and is the solution of

By applying Lemma 1, we obtain that

On the other hand, we obtain

From the above discussion, we arrive at

where is a positive constant.
Similarly, we can get

Lemma 2 shows that on , where is the solution of

We get from Lemma 1 that

Furthermore, we obtain

It follows that

where is a positive constant. The proof is complete.

Theorem 7. Assume that conditions , , , , and hold:

The function , the Fréchet derivatives exist and are continuous satisfying for , , and is a positive matrix.

Then, there exist monotone sequences and , which converge uniformly to the solution of system (14) in , and the convergence is of order ; that is, there exist positive constants and such that

hold, where , .

Proof. It can be noted from condition that

for , , and

for , .
Consider the following systems:

We can prove as in Theorem 6 that there exists a solution of (54) such that on with .
Now, let us show that and are lower and upper solutions of (55), respectively. By setting and in (55), it follows from inequalities (52) and (53) that

This shows that and are lower and upper solutions of (55), respectively. It follows from Lemma 5 that there exists a solution of (55) such that on .
Now, we prove that and are lower and upper solutions of system (14). Utilizing inequality (52), we find that

Hence, is a lower solution of (14). In a similar way, condition together with inequality (53) gives

This proves that is an upper solution of (14). As a result, it follows that on .
Thus, using mathematical induction, one can obtain

where and are solutions of (54) and (55), respectively, with and .
Employing the Ascoli-Arzela theorem, both sequences and converge uniformly to the solution of system (14).
At last, we show that the convergence of the sequences of and is . For this purpose, we consider

For the convergence of , we can prove Theorem 6 to obtain

where is a positive constant.
To prove the convergence of , using the mean value theorem and condition , we arrive at

where . Lemma 2 shows that on , where is the solution of

Employing Lemma 1, it follows that

Therefore, we conclude that

where is a positive constant. The proof is complete.

Theorem 8. Assume that conditions , , , , and hold:

The function , the Fréchet derivatives exist and are continuous satisfying for , , and is a positive matrix.

Then, there exist monotone sequences and , which converge uniformly to the solution of system (14) in , and the convergence is of order ; that is, there exist positive constants and such that

hold, where , .

Proof. In order to construct monotone sequences and which converge uniformly to the solution of (14), we need to consider the following systems:

Since the proof follows on the same line as in Theorem 6, we omit the proof.

Theorem 9. Assume that conditions , , , , and hold:

The function , the Fréchet derivatives exist and are continuous satisfying for , , and is a positive matrix.

Then, there exist monotone sequences and , which converge uniformly to the solution of system (14) in , and the convergence is of order ; that is, there exist positive constants and such that

hold, where , .

Proof. To construct monotone sequences and which converge uniformly to the solution of (14), we consider the following systems:

We omit the proof, since it follows the same way as in Theorem 6.

4. Example

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

Example 1. Consider the following telegraph system with periodic boundary conditions:

where

One can see easily that is a solution of the telegraph system (70). Taking , , it is easy to verify that and are lower and upper solutions of the telegraph system (70), respectively, and , , , and ; that is, the conditions of Theorem 7 are satisfied. According to Theorem 7, we can construct two monotone sequences and .

The successive approximations and are solutions of the telegraph systems

Firstly, we construct an increasing sequence of lower solutions which converges uniformly to the solution . In view of the fact that , the telegraph system (72) reduces to

Then, the telegraph system (74) has a solution .

Using the solution , the telegraph system (72) reduces to the following:

The telegraph system (75) has an exact solution .

Similarly, we can find all successive approximations , which converge uniformly to the solution of the telegraph system (70).

Next, we shall construct a decreasing sequence of upper solutions which will converge uniformly to the exact solution. Since , then the telegraph system (73) can be written as

The function is a solution of the telegraph system (76).

According to the solution , the telegraph system (73) reduces to the following:

The telegraph system (77) has a solution .

Proceeding as before, we can find all successive approximations , which converge uniformly to the solution of the telegraph system (70).

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

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

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

Acknowledgments

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

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