- Research
- Open Access
Existence and approximation of solution for a nonlinear second-order three-point boundary value problem
- Xian-Ci Zhong^{1}Email author,
- Qiong-Ao Huang and
- Han-Mei Wei
- Received: 19 April 2016
- Accepted: 13 September 2016
- Published: 21 September 2016
Abstract
A nonlinear second-order ordinary differential equation with four cases of three-point boundary value conditions is studied by investigating the existence and approximation of solutions. First, the integration method is proposed to transform the considered boundary value problems into Hammerstein integral equations. Second, the existence of solutions for the obtained Hammerstein integral equations is analyzed by using the Schauder fixed point theorem. The contraction mapping theorem in Banach spaces is further used to address the uniqueness of solutions. Third, the approximate solution of Hammerstein integral equations is constructed by using a new numerical method, and its convergence and error estimate are analyzed. Finally, some numerical examples are addressed to verify the given theorems and methods.
Keywords
- three-point boundary value problems
- Hammerstein integral equation
- existence
- approximate solution
- convergence and error estimate
MSC
- 34B10
- 45B05
1 Introduction
It is noted that when \(g(x)=0\), the existence of solutions for equation (1) with various boundary value conditions has been studied widely. For example, the method of lower and upper solutions is developed by Ma [9] and the multiplicity solutions for a three-point boundary value problem at resonance were given. Xu [10, 11] considered the singular three-point and m-point boundary value problems, respectively. The multiplicity results and existence of positive solutions were analyzed by using a fixed point index theory. Yao [12] investigated the existence of positive solutions for a second-order three-point boundary value problem and a successive iteration method was given for computing the solutions. Nieto [13] studied the existence of solution for a second-order nonlinear ordinary differential equation with three-point boundary value conditions at resonance. In addition, the problems of nonlinear second-order ordinary differential equations with m-point and integral boundary value conditions were further investigated in [14–19] and so on. On the other hand, when \(\psi(x,\varphi(x))=a(x)F(x,\varphi(x))\) and \(\psi (x,\varphi(x))\) degenerates to \(a(x)\psi(\varphi(x))\), the multipoint boundary value problems of nonlinear second-order ordinary differential equations were dealt with in [20–27]. Recently, the special linear case of \(\psi(x,\varphi(x))=q(x)\varphi(x)\) has been studied in [28] and an approximate solution has been given. The case of \(g(x)=0\) in equation (1) with impulse three-point boundary value conditions have been studied in [29], where the existence conditions for obtaining a nontrivial solution have been given.
As shown in the above-mentioned work, the existence of the solutions for second-order multipoint boundary value problems is always focused on. Moreover, the approximate solutions of boundary value problems are very important in engineering applications. A monotone iterative technique was developed for the approximate solution of a second-order three-point boundary value problem in [30]. This paper generally focuses on the nonlinear second-order ordinary differential equation with four cases of three-point boundary value conditions in (1)-(5). A general method is proposed to transform the nonlinear three-point boundary value problems into nonlinear Hammerstein integral equations. The existence and uniqueness of solutions for the obtained Hammerstein integral equations are considered by using the Schauder fixed point theorem and the contraction mapping theorem, respectively. A new numerical method is further proposed to construct the approximate solutions of Hammerstein integral equations. Some numerical examples are computed to show the effectiveness of the proposed methods.
2 Hammerstein integral equations
In this section, we will transform the nonlinear second-order ordinary differential equation (1) with three-point boundary value conditions in (2)-(5) into Hammerstein integral equations. Then the existence and uniqueness of solutions for the obtained Hammerstein integral equations will be investigated.
2.1 Transformations
In the following, we apply the integration method to get the following four theorems.
Theorem 1
Proof
After some computations, one verifies that (14) and (15) can be rewritten as indicated in (9). □
Furthermore, one can obtain the following theorem and the proof has been omitted for saving space.
Theorem 2
In the following, by considering the boundary value conditions in the cases of III and IV, we give Theorems 3 and 4, respectively.
Theorem 3
Proof
Theorem 4
The proof can be completed similar to that of Theorem 3.
It is seen from Theorems 1-4 that the nonlinear second-order three-point boundary value problems have been transformed into Hammerstein integral equations. We remark that the integration method is uniform and enough to transform any nonlocal boundary value problem of ordinary differential equations into an integral equation [19, 31]. In the end, as a check, we consider the special case of \(g(x)=\alpha=\beta=0\) and obtain the boundary value problems as those in [32]. Based on Theorems 1-4, the solutions and Green’s functions of the nonlinear three-point boundary value problems in [32] can be determined easily.
2.2 Existence and uniqueness of solution
For a nonlinear equation, the fixed point theorems are always used to address the existence and uniqueness of solutions [33–35]. Here since the considered three-point boundary value problems have been transformed into the Hammerstein integral equations, it is natural to study the existence and uniqueness of solutions for the obtained Hammerstein integral equations. Moreover, it is seen that the existence and uniqueness of solutions for Hammerstein integral equations have been investigated widely such as those in the book [36] and the recent results on \(\mathbf{L}_{1}\) spaces [37]. In the present paper, for the obtained Hammerstein integral equations, we will use the Schauder fixed point theorem to address the existence of the solutions, and apply the Banach fixed point theorem to investigate the uniqueness of the solutions. The obtained results are related to the considered nonlinear boundary value problems of ordinary differential equations.
Theorem 5
Proof
By using the Schauder fixed point theorem, there exists at least a point \(\varphi\in S\) such that \(\mathbf{T}\varphi=\varphi\). □
In addition, we can change some conditions in Theorem 5 to obtain the following corollary.
Corollary 1
Furthermore, strengthening the conditions of the nonlinear term \(\psi (x,\varphi(x))\), we have the uniqueness theorem of solution in Banach spaces.
Theorem 6
Proof
When \(LC_{i}<1\), T is a contraction operator. According to the fixed point theorem in Banach spaces, one can see that \(\mathbf{T}\varphi=\varphi\) has a unique solution in \(\mathbf{L}_{2}[a,b]\). □
3 Approximation of the solution
In practical applications, of much interest is how to obtain the solutions except for the existence of the solutions. However, the closed-form solutions of the Hammerstein integral equations in (9), (17), (19), and (25) cannot be determined easily due to the complexity of the kernels. Thus it is interesting to obtain numerical solutions of Hammerstein integral equations and many methods have been proposed [8, 39–46]. Moreover, it is noted that a simple Taylor-series expansion method has been proposed in [47] and modified in [48, 49] for numerically solving linear Fredholm integral equations of the second kind. Recently, by using the idea of piecewise approximation, the simple Taylor-series expansion method has been further modified in [28]. Here the proposed method in [28] is further extended and applied to solve the nonlinear integral equation of Hammerstein type. The convergence and error estimate of the approximate solution will be made. Moreover, it is seen from Theorems 5 and 6 that a solution in \(\mathbf{L}^{2}[a,b]\) is only determined by using the given conditions. Based on the proposed numerical method, the solution \(\varphi(x)\) should have more smoothing property and here it is assumed \(\varphi(x)\in\mathbf{C}^{n+1}[a,b]\) (\(n\geq0\)). Indeed, the case of \(\varphi(x)\in\mathbf{C}^{n+1}[a,b]\) (\(n\geq0\)) is important in practical applications. Two examples will be given in Section 4 to verify the extended numerical method by comparing a difference format.
3.1 Constructing the approximate solution
Theorem 7
Proof
As shown in [28], one can see from equation (40) that the approximate solution has two parameters. The proposed method is based on the discretization points \(x_{q}\) (\(q=0,1,\ldots, m-1\)), which is different from the simple Taylor-series expansion method in [47–49]. The effectiveness and advantage of the new method will be shown in the given numerical examples of Section 4. Furthermore, in order to give the approximate solution in (40), the convergence of the nonlinear system (39) is requisite. Under some conditions, the nonlinear system (39) is convergent and it will be proved in the next subsection about the error estimate of the approximate solution.
3.2 Convergence and error estimate
From the viewpoint of mathematical theory and practical applications, the convergence and error estimate of the approximate solution are all important. For the approximation method, we have the following theorem.
Theorem 8
Proof
It is found from (50) that when \(n\rightarrow+\infty\) or \(m\rightarrow+\infty\) (i.e. \(h\rightarrow0\)), one always has \(\|\varphi(x)-\varphi_{m,n}(x)\|_{\infty}\rightarrow0\). This completes the proof. □
As shown in Theorem 8, one can choose a pair of feasible values for m (i.e. h) and n to obtain a good approximation of the exact solution. The above observations will be further verified by using the numerical examples in the next section.
4 Numerical results
In order to show the effectiveness of the proposed methods, we give two numerical examples corresponding to cases I and IV, respectively. The existence and uniqueness of the solution will be considered, and the approximate solution will be calculated numerically. All the computations are made by using the programming language of MATLAB (R2014).
Example 1
The absolute errors of the approximate and exact solutions for Example 1
x | \(\boldsymbol{\varphi_{m,n}(x)}\) : The present method | \(\boldsymbol{\varphi_{h}(x)}\) : The difference method | ||||
---|---|---|---|---|---|---|
(2,2) | (2,4) | (4,2) | (4,4) | h = 0.1 | h = 0.05 | |
0.10 | 1.8921e − 3 | 1.3455e − 4 | 4.0629e − 4 | 8.1435e − 6 | 1.7128e − 4 | 4.2829e − 5 |
0.20 | 3.7502e − 3 | 2.6842e − 4 | 7.8165e − 4 | 1.5896e − 5 | 3.0443e − 4 | 7.6114e − 5 |
0.30 | 5.4142e − 3 | 3.9569e − 4 | 1.0209e − 3 | 2.0120e − 5 | 3.9356e − 4 | 9.8358e − 5 |
0.40 | 6.4733e − 3 | 4.8748e − 4 | 1.2069e − 3 | 2.3084e − 5 | 4.3201e − 4 | 1.0798e − 4 |
0.50 | 6.1339e − 3 | 4.5086e − 4 | 1.2227e − 3 | 2.3171e − 5 | 4.1224e − 4 | 1.0301e − 4 |
0.60 | 4.6732e − 3 | 2.8768e − 4 | 1.0183e − 3 | 1.7900e − 5 | 3.2672e − 4 | 8.1720e − 5 |
0.70 | 3.0518e − 3 | 1.2016e − 4 | 6.9688e − 4 | 1.1472e − 5 | 1.6693e − 4 | 4.1672e − 5 |
0.80 | 7.6299e − 4 | 6.2865e − 5 | 9.8850e − 4 | 3.8926e − 6 | 7.3872e − 5 | 1.8509e − 5 |
0.90 | 3.5063e − 3 | 3.2838e − 4 | 2.0438e − 3 | 2.2064e − 5 | 4.0280e − 4 | 1.0080e − 4 |
1.00 | 1.2268e − 2 | 9.0172e − 4 | 3.5652e − 3 | 4.6343e − 5 | 8.2447e − 4 | 2.0603e − 4 |
Example 2
The absolute errors of the approximate and exact solutions for Example 2
x | \(\boldsymbol{\varphi_{m,n}(x)}\) : The present method | \(\boldsymbol{\varphi_{h}(x)}\) : The difference method | ||||
---|---|---|---|---|---|---|
(2,1) | (2,2) | (4,1) | (4,2) | h = 0.1 | h = 0.05 | |
0.00 | 1.8390e − 3 | 9.9247e − 4 | 1.0502e − 3 | 1.6925e − 4 | 4.9821e − 2 | 2.5019e − 2 |
0.10 | 3.7770e − 3 | 1.2501e − 3 | 1.5834e − 3 | 2.2288e − 4 | 4.5369e − 2 | 2.2780e − 2 |
0.20 | 5.6959e − 3 | 1.5063e − 3 | 2.0998e − 3 | 2.7613e − 4 | 4.0870e − 2 | 2.0517e − 2 |
0.30 | 7.5115e − 3 | 1.7571e − 3 | 2.5446e − 3 | 3.2734e − 4 | 3.6285e − 2 | 1.8212e − 2 |
0.40 | 9.0093e − 3 | 1.9876e − 3 | 2.9475e − 3 | 3.7623e − 4 | 3.1581e − 2 | 1.5848e − 2 |
0.50 | 9.7699e − 3 | 2.1512e − 3 | 3.2074e − 3 | 4.1161e − 4 | 2.6730e − 2 | 1.3411e − 2 |
0.60 | 9.9119e − 3 | 2.2428e − 3 | 3.2926e − 3 | 4.2675e − 4 | 2.1712e − 2 | 1.0891e − 2 |
0.70 | 9.8280e − 3 | 2.3012e − 3 | 3.2046e − 3 | 4.2358e − 4 | 1.6516e − 2 | 8.2834e − 3 |
0.80 | 9.0100e − 3 | 2.2273e − 3 | 2.5986e − 3 | 3.3970e − 4 | 1.1146e − 2 | 5.5885e − 3 |
0.90 | 6.3775e − 3 | 1.7078e − 3 | 1.7315e − 3 | 2.2813e − 4 | 5.6224e − 3 | 2.8184e − 3 |
5 Conclusions
Four cases of nonlinear second-order three-point boundary value problems have been investigated and they are transformed into the Hammerstein integral equations by using the integration method. Based on the Schauder fixed point theorem, the sufficient conditions for the existence of the solutions have been given. The uniqueness of the solutions has been considered by using the Banach fixed point theorem. Furthermore, we have constructed the approximate solution of Hammerstein integral equations by applying a novel numerical method, which depends on the values of two parameters. The convergence and error estimate of the approximate solution have been made, and they show that one can get a good approximation of the exact solution by choosing a pair of the parameters. Two examples have been carried out numerically and the obtained results have revealed that the proposed methods are effective. In the future, the proposed method will be extended to solve nonlinear second-order differential equations with various nonlocal boundary value conditions.
Declarations
Acknowledgements
The authors would like to thank the Associate Editor and the reviewers for the valuable suggestions in improving the paper. The work was supported by the National Natural Science Foundation of China (No. 11362002), the Natural Science Foundation of Guangxi (No. 2016GXNSFAA380012), the Innovation Project of Guangxi Graduate Education (No. YCSZ2015030), the project of outstanding young teachers’ training in higher education institutions of Guangxi, and the project of Guangxi Colleges and Universities Key Laboratory of Mathematics and Its Applications.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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