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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 308413, 7 pages
Quasilinearization for the Boundary Value Problem of Second-Order Singular Differential System
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.
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.
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 ). Recently, Abd-Ellateef Kamar et al. investigated the first-order singular systems of differential equations with initial value problem . In , 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  for linear singular systems and the comparison result , we investigate two monotone iterative sequences which converge uniformly and quadratically to the solution of the problem.
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 ). 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  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
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.
All authors completed the paper together. All authors read and approved the final paper.
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).
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