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).