Abstract and Applied Analysis

Volume 2013 (2013), Article ID 308413, 7 pages

http://dx.doi.org/10.1155/2013/308413

Research Article

Quasilinearization for the Boundary Value Problem of Second-Order Singular Differential System

Peiguang Wang¹ and Tiantian Kong²

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

²College of Mathematics and Computer Science, Hebei University, Baoding 071002, China

Received 23 August 2013; Accepted 25 November 2013

Academic Editor: Yong Hong Wu

Copyright © 2013 Peiguang Wang and Tiantian Kong. 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.

We study the boundary value problems of second-order singular differential equations. At first, we reduce the BVPs to initial value problems of first-order singular integrodifferential equations and then we employ the quasilinearization method in studying the IVPs and obtain two monotone iterative sequences, which converge uniformly and quadratically to the unique solution of the IVPs. Finally, we get the similar result for the given BVPs.

1. Introduction

It is well known that quasilinearization method is a powerful tool for proving the existence of approximate solutions of nonlinear systems and the converge quadratically to the unique solution of the given problems (see [1]). Recently, Abd-Ellateef Kamar et al. investigated the first-order singular systems of differential equations with initial value problem [2]. In [3], Wang and Liu developed monotone iterative technique combined with the method of upper and lower solutions for studying the second-order singular systems with boundary value problems (BVPs).

In this paper, we extend quasilinearization method to study the second-order singular systems with the boundary conditions:

where is a singular matrix, , , and and are two constant vectors. By using the existence result [4] for linear singular systems and the comparison result [5], we investigate two monotone iterative sequences which converge uniformly and quadratically to the solution of the problem.

2. Preliminaries

Consider the following initial value problem:

where is a singular matrix, , , is an increasing operator, and is a constant vector.

In order to obtain two monotone sequences, we introduce an existence result for the corresponding linear singular systems and a comparison result.

The existence of the solution of the linear initial value problem of the form

is well known and is given by the following lemma.

Lemma 1 (see [4]). Assume that the nonhomogeneous linear system (3) exists and if

(i)there exists a such that exists,(ii) is the solution of  , where

and are the Drazin inverses of , respectively, then the solution is given by

Now one gives the following assumptions for convenience.

Let and be matrices such that exists and is nonnegative for some . Suppose further that exist and are nonnegative, such that

where , , and is a diagonal square matrix with and .There exist with on , such that

All second-order derivatives of exist and are bounded, is convex in for each , is convex in for each , is nonincreasing in for each , and is nonincreasing in for each .Moreover .

To obtain the results, one needs the following comparison theorem.

Lemma 2. Let such that and satisfy assumptions and . Then implies on .

The proof is similar to [5] and one omits it.

3. Main Results

Firstly, we develop the following result which is important for the final result.

Theorem 3. Suppose that assumptions hold and, for ,

(i), where .

Then there exist two monotone sequences and , which converge uniformly on to the unique solution of problem (2) and the convergence rate is quadratic.

Proof. From we find that , and

for all and . The convexity of in implies that

for , , and .

Now consider the following linear problems. For ,

For (10), set

Then, we obtain that . Furthermore, we can get

Hence and are lower and upper solutions of (12), respectively. Thus (12) has a solution on , and we have . Similarly, set

Then we obtain that (14) has a solution on , and we have .

Now we claim that

Let . We get that

Noticing that , we get , on .

Now, assume that, for , (10) and (11) admit solutions and , respectively, such that

Then setting in (10) and (11), we observe that the assumptions in Theorem 3 are satisfied. Thus, there exist solutions and for (10) and (11), respectively, and we now will show that the following relation

holds. Firstly, we can easily know that and .

To prove that , consider . Then

We know that . Hence, from Lemma 2, we deduce that , on . Thus, we have monotone sequences such that

Now, employing Ascoli-Arzela's theorem we conclude that the sequences converge uniformly and monotonically to the unique solution of (2) on .

To show that the convergence rate is quadratic, we begin with . Then

where

Set . Then

where , , , and .

Next, set . Then

where and .

Furthermore, we have that

where .

Using Lemma 2, we show that on , where is the solution of

Thus, using Lemma 1, the solution of the previously mentioned equation is given as

After taking suitable estimates, we obtain

where .

Set . Then we can get

in which

Set . Then we get that

where and .

Similarly, set . Then

where , and .

Then we get that

where and . Thus we have that

Furthermore, we obtain after taking suitable estimates

where and .

Hence we proved that the convergence rate is quadratic.

Next, we consider singular differential systems BVPs and prove the following main result.

Theorem 4. Let assumptions hold. Suppose further the following.

There exist with and on and

For , , where .

Then there exist monotone sequences and which converge uniformly on to the unique solution of (1) and the convergence rate is quadratic.

Proof. Using the transformation , we have that

BVPs (1) can be transformed into IVPs of singular system:

Let and . By using assumption , we get that

and .

Noticing that , we have that

Then from , and

A similar argument shows that

By Theorem 3, there exist monotone sequences such that

and the convergence rate is quadratic. Again set and . Then

and also the convergence rate is quadratic. Noticing that

we can obtain the similar result that converges quadratically to the solution of (1). The proof is complete.

Conflict of Interests

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

Authors’ Contribution

All authors completed the paper together. All authors read and approved the final paper.

Acknowledgments

The authors would like to thank the reviewers for their valuable suggestions and comments. This paper is supported by the National Natural Science Foundation of China (11271106) and the Natural Science Foundation of Hebei Province of China (A2013201232).

References

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W. Wang and Y. Q. Liu, “Monotone iterative for boundary value problem of second order singular dierential system,” Journal of Systems Science and Systems Engineering, vol. 4, no. 4, pp. 266–272, 1995.

S. L. Campbell, Singular Systems of Differential Equations, Pitman Advanced Publishing Program, London, UK, 1980.

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