Abstract and Applied Analysis

Volume 2014 (2014), Article ID 181585, 8 pages

http://dx.doi.org/10.1155/2014/181585

Research Article

Existence of Positive Solutions for a Nonlinear Higher-Order Multipoint Boundary Value Problem

Min Zhao and Yongping Sun

College of Electron and Information, Zhejiang University of Media and Communications, Hangzhou, Zhejiang 310018, China

Received 24 January 2014; Accepted 19 March 2014; Published 10 April 2014

Academic Editor: Naseer Shahzad

Copyright © 2014 Min Zhao and Yongping Sun. 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 existence of positive solutions for a nonlinear higher-order multipoint boundary value problem. By applying a monotone iterative method, some existence results of positive solutions are obtained. The main result is illustrated with an example.

1. Introduction

We consider the following nonlinear higher-order differential equation:

with the multipoint boundary conditions

Throughout this paper, we assume that the following conditions are satisfied:

(H1), , are fixed integers, , () with ;(H2) is nonnegative and ;(H3) is continuous.

In this paper, by a positive solution of problems (1) and (2), we mean a function satisfying the differential equation (1) and the boundary conditions (2) with for all .

The multipoint boundary value problems for ordinary differential equations arise in a variety of different areas of applied mathematics and physics. In recent years, the existence and multiplicity of solutions of nonlinear higher-order differential equations with various multipoint boundary conditions have been studied extensively by numerous researchers using a variety of methods and techniques. For example, Graef and Yang [1] studied a higher-order multipoint boundary value problem

where and are integers, for , and , , , is a parameter. Some existence and nonexistence results of positive solutions were obtained by using Krasnosel’skii’s fixed point theorem. In [2], by applying fixed point index theory, Pang et al. studied the expression and properties of Green's function and obtained the existence of positive solutions for th-order -point boundary value problems

where , (), with . Guo et al. in [3] imposed growth conditions on the nonlinearity which yield the existence of multiple positive solutions by using the Leggett-Williams fixed point theorem. Li and Wei in [4] and Yang and Wei in [5] improved and generalized the results of [2] by using different methods. Graef et al. [6] considered an th-order multipoint boundary value problem

where and are integers, is a parameter, , with on , for , and for . Sufficient conditions were obtained for the existence of one solution and two solutions of the problem for different values of . The analysis mainly relies on the lower and upper solution method and topological degree theory. The results extended and improved some recent work in the literature. In a recent paper [7], we study problems (1) and (2) with by a fixed point theorem of cone expansion and compression of functional type according to Avery et al. [8]. For other existence results of positive solutions for higher-order multipoint problems, for a small sample of such work, we refer the reader to Ahmad and Ntouyas [9], Anderson et al. [10], Davis et al. [11], Du et al. [12, 13], Eloe and Henderson [14], Fu and Du [15], Graef et al. [16, 17], Henderson and Luca [18], Ji and Guo [19], Jiang [20], Liu et al. [21], Liu and Ge [22], Liu et al. [23], Palamides [24], Su and Wang [25], Zhang et al. [26], and Zhang [27] and the references therein.

We noticed that the main tools employed in above-mentioned works are various fixed point theorems, such as Krasnosel'skii, Leggett-Williams, and Avery and Peterson. Recently, the monotone iterative method has been successfully employed to prove the existence of positive solutions of nonlinear boundary value problems for ordinary differential equations. For example, Ma et al. [28] proved the existence of positive solutions of some multipoint -Laplacian boundary value problems via monotone iterative method. Ma and Yang [29] obtained the existence of positive solutions and established two corresponding iterative schemes for a third-order three-point boundary value problem with increasing homeomorphism and positive homomorphism. Sun and Ge [30] applied monotone iterative procedure to prove the existence of positive pseudosymmetric solutions for a three-point second-order -Laplacian boundary value problem. Sun et al. [31] proved the existence of positive solutions for some fourth-order two-point boundary value problems via monotone iterative technique. Yao [32, 33] obtained a successively iterative scheme of positive solution of Lidstone boundary value problem and a beam equation with nonhomogeneous boundary condition, respectively. In this paper, we will study the existence and iteration of positive solutions for problems (1) and (2) by using the monotone iterative method. The monotone iterative scheme can be developed into a computational algorithm for numerical solutions.

2. Basic Lemmas

In this section, we present two lemmas, related to the following higher-order differential equation with multipoint boundary conditions:

Lemma 1. Let be a given function; then the solution of problems (6) and (7) is given by

where

Proof. The solution of (6) is

for some (). Noting that the conditions are , we obtain . Consequently, the general solution of problems (6) and (7) is

Therefore, by (14), we have

which implies that

By the condition , (16) and (17), we deduce

Substituting (18) into (14), we obtain the unique solution of problems (6) and (7) as

The proof is completed.

Lemma 2. Green’s function defined by (9) has the following properties:

(i) is continuous on , ;(ii), for all , ;(iii), for all .

Proof. The statement (i) is obvious. For the proof of the statement (ii), we note that, for all , if , from definition, it is clear that for . If , from (10), we obtain that

For any , if , from (12), it is obvious that . If , from (12), we have

which implies that

By (20) and (22), we get

On the other hand, by (10) and (12), we find that

Therefore,

In view of (23) and (25), we have the assertion.

Now, we prove the statement (iii). In fact, from the statement (ii), we know that for any . Thus, is nondecreasing with respect to for any . Consequently,

If , then, from (10), we have

Also, if , from (10), we have

Thus, from (27) and (28), we obtain

which together with (9) and (11) implies

Inequations (26) and (30) show that the statement (iii) is true. Then, the proof is completed.

3. Main Results

In this section, we consider the existence of positive solutions for problems (1) and (2) by using the monotone iterative method. In the sequel, for any , we define . Let be a Banach space with the norm

We define a cone by

and an integral operator by

Obviously, the fixed points of are nonnegative solutions of problems (1) and (2). Applying Ascoli-Arzelà theorem and a standard argument, we can prove that is completely continuous.

For any , it flows from Lemma 2 (iii) that

which implies that

On the other hand, by Lemma 2 (iii), we have

which together with (35) implies

In addition, it follows from Lemma 2 (ii) that

Therefore, (37) and (38) indicate that .

For convenience, we introduce the following notation:

The conditions (H1) and (H2) indicate that is well defined and .

Theorem 3. Suppose (H1), (H2), and (H3) hold. Assume that and there exists constant , such that

(H4), for , , ;(H5).

Then, problems (1) and (2) possess at least two positive solutions and , such that

(i) and , where , ;(ii) and , where .

Proof. We define . In what follows, we first prove . In fact, if , then ; thus

By assumptions (H4) and (H5), we have

Thus, by the definition of and Lemma 2 (ii), for , we get

Then, (42) shows that ; thus we get .

Let ; then it is evident that . Let (). The fact that implies that (). Since is completely continuous, we assert that the sequence has a convergent subsequence such that .

Since , we have

So, by (H4), one has

Thus, by the induction, we have

Hence, . Applying the continuity of and taking the limit in , we get .

Let , , and (). Then, . Since , we have (). Since is completely continuous, we assert that the sequence has a convergent subsequence such that .

Since , by Lemma 2 (ii) and (H4) and (H5), for , we have

Thus, we obtain that

So, by Lemma 2 (ii) and (H4), we have

By the induction, we have

Hence, . Applying the continuity of and , we get .

Furthermore, assumption implies that the zero function is not a solution of problems (1) and (2); thus , . The definition of the cone follows that we have , , . Thus, and are positive solutions of problems (1) and (2). The proof is completed.

Remark 4. The iterative sequences in Theorem 3 start off with the zero function and a known simple function, respectively.

Remark 5. We can easily get that and are the maximal and minimal solution of problems (1) and (2) in . Of course, may happen and then problems (1) and (2) have only one solution in .

4. Example

Example 1. Consider the fourth-order four-point boundary value problem

In this problem, , , , , , , , , and . It is obvious that (H1)–(H3) hold. By direct calculation, we get

Choose ; then it is easy to check that (H4) and (H5) hold. Thus, all the conditions of Theorem 3 are satisfied. By Theorem 3, problem (50) has two positive solutions and , such that , , , and .

The two iterative sequences are as follows:

The second and third terms of the two schemes are as follows:

Conflict of Interests

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

Acknowledgments

The authors would like to express their thanks to the anonymous referee for her/his valuable suggestions and comments. This research was partially supported by the Natural Science Foundation of Zhejiang Province of China (LY12A01012).

References

J. R. Graef and B. Yang, “Positive solutions to a multi-point higher order boundary value problem,” Journal of Mathematical Analysis and Applications, vol. 316, no. 2, pp. 409–421, 2006.

C. Pang, W. Dong, and Z. Wei, “Green's function and positive solutions of th order -point boundary value problem,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1231–1239, 2006.

Y. Guo, Y. Ji, and J. Zhang, “Three positive solutions for a nonlinear th-order -point boundary value problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 11, pp. 3485–3492, 2008.

Y. Li and Z. Wei, “Multiple positive solutions for th order multipoint boundary value problem,” Boundary Value Problems, vol. 2010, Article ID 708376, 13 pages, 2010.

J. Yang and Z. Wei, “Positive solutions of th order -point boundary value problem,” Applied Mathematics and Computation, vol. 202, no. 2, pp. 715–720, 2008.

J. R. Graef, L. Kong, and B. Yang, “Existence of solutions for a higher order multi-point boundary value problem,” Results in Mathematics, vol. 53, no. 1-2, pp. 77–101, 2009.

Y. Sun, Q. Sun, and X. Zhang, “Existence and nonexistence of positive solutions for a higher-order three-point boundary value problem,” Abstract and Applied Analysis, vol. 2014, Article ID 513051, 7 pages, 2014.

R. Avery, J. Henderson, and D. O'Regan, “Functional compression-expansion fixed point theorem,” Electronic Journal of Differential Equations, vol. 2008, 12 pages, 2008.

B. Ahmad and S. K. Ntouyas, “A study of higher-order nonlinear ordinary differential equations with four-point nonlocal integral boundary conditions,” Journal of Applied Mathematics and Computing, vol. 39, no. 1-2, pp. 97–108, 2012.

D. R. Anderson, T. O. Anderson, and M. M. Kleber, “Green's function and existence of solutions for a functional focal differential equation,” Electronic Journal of Differential Equations, vol. 2006, no. 12, 14 pages, 2006.

J. M. Davis, P. W. Eloe, and J. Henderson, “Triple positive solutions for multipoint conjugate boundary value problems,” Georgian Mathematical Journal, vol. 6, no. 5, pp. 415–420, 1999.

Z. Du, W. Liu, and X. Lin, “Multiple solutions to a three-point boundary value problem for higher-order ordinary differential equations,” Journal of Mathematical Analysis and Applications, vol. 335, no. 2, pp. 1207–1218, 2007.

Z. Du, X. Lin, and W. Ge, “Some higher-order multi-point boundary value problem at resonance,” Journal of Computational and Applied Mathematics, vol. 177, no. 1, pp. 55–65, 2005.

P. W. Eloe and J. Henderson, “Positive solutions and nonlinear multipoint conjugate eigenvalue problems,” Electronic Journal of Differential Equations, vol. 1997, no. 3, 11 pages, 1997.

Z. Fu and Z. Du, “Existence of solution for a class of th-order multi-point boundary value problem,” Journal of Applied Mathematics and Computing, vol. 33, no. 1-2, pp. 423–435, 2010.

J. R. Graef, J. Henderson, and B. Yang, “Positive solutions of a nonlinear higher order boundary-value problem,” Electronic Journal of Differential Equations, vol. 2007, no. 45, 10 pages, 2007.

J. R. Graef, L. Kong, and Q. Kong, “Existence of three solutions for a higher-order boundary-value problem,” Electronic Journal of Differential Equations: Conference, vol. 2009, no. 17, 10 pages, 2009.

J. Henderson and R. Luca, “On a system of higher-order multi-point boundary value problems,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 2012, no. 49, 14 pages, 2012.

Y. Ji and Y. Guo, “The existence of countably many positive solutions for some nonlinear th order -point boundary value problems,” Journal of Computational and Applied Mathematics, vol. 232, no. 2, pp. 187–200, 2009.

W. Jiang, “Multiple positive solutions for th-order -point boundary value problems with all derivatives,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 5, pp. 1064–1072, 2008.

L. Liu, B. Liu, and Y. Wu, “Nontrivial solutions for higher-order -point boundary value problem with a sign-changing nonlinear term,” Applied Mathematics and Computation, vol. 217, no. 8, pp. 3792–3800, 2010.

Y. Liu and W. Ge, “Solvability of a (p, n-p)-type multi-point boundary-value problem for higher-order differential equations,” Electronic Journal of Differential Equations, vol. 2003, no. 120, 19 pages, 2003.

X.-J. Liu, W.-H. Jiang, and Y.-P. Guo, “Multi-point boundary value problems for higher order differential equations,” Applied Mathematics E: Notes, vol. 4, pp. 106–113, 2004.

P. K. Palamides, “Multi point boundary-value problems at resonance for -order differential equations: positive and monotone solutions,” Electronic Journal of Differential Equations, vol. 2004, no. 25, 14 pages, 2004.

H. Su and X. Wang, “Positive solutions to singular semipositone -point -order boundary value problems,” Journal of Applied Mathematics and Computing, vol. 36, no. 1-2, pp. 187–200, 2011.

M. Zhang, Y. Yin, and Z. Wei, “Positive solution of singular higher-order -point boundary value problem with nonlinearity that changes sign,” Applied Mathematics and Computation, vol. 201, no. 1-2, pp. 678–687, 2008.

X. Zhang, “Eigenvalue of higher-order semipositone multi-point boundary value problems with derivatives,” Applied Mathematics and Computation, vol. 201, no. 1-2, pp. 361–370, 2008.

D.-X. Ma, Z.-J. Du, and W.-G. Ge, “Existence and iteration of monotone positive solutions for multipoint boundary value problem with -Laplacian operator,” Computers & Mathematics with Applications, vol. 50, no. 5-6, pp. 729–739, 2005.

D.-X. Ma and X.-Z. Yang, “Existence and iteration of positive solutions to third order three-point BVP with increasing homeomorphism and positive homomorphism,” The Rocky Mountain Journal of Mathematics, vol. 43, no. 2, pp. 539–550, 2013.

B. Sun and W. Ge, “Successive iteration and positive pseudo-symmetric solutions for a three-point second-order -Laplacian boundary value problems,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1772–1779, 2007.

Y. Sun, X. Zhang, and M. Zhao, “Successive iteration of positive solutions for fourth-order two-point boundary value problems,” Abstract and Applied Analysis, vol. 2013, Article ID 621315, 8 pages, 2013.

Q. Yao, “Monotone iterative technique and positive solutions of Lidstone boundary value problems,” Applied Mathematics and Computation, vol. 138, no. 1, pp. 1–9, 2003.

Q. Yao, “Successively iterative technique of a classical elastic beam equation with Carathéodory nonlinearity,” Acta Applicandae Mathematicae, vol. 108, no. 2, pp. 385–394, 2009.