Iterative Methods for Nonlinear Equations or Systems and Their Applications 2014View this Special Issue
Iteration for a Third-Order Three-Point BVP with Sign-Changing Green's Function
We are concerned with the following third-order three-point boundary value problem: , where and . Although corresponding Green's function is sign-changing, we still obtain the existence of monotone positive solution under some suitable conditions on by applying iterative method. An example is also included to illustrate the main results obtained.
Third-order differential equations arise from a variety of different areas of applied mathematics and physics, for example, in the deflection of a curved beam having a constant or varying cross-section, a three-layer beam, electromagnetic waves or gravity driven flows, and so on .
Recently, the existence of single or multiple positive solutions to some third-order three-point boundary value problems (BVPs) has received much attention from many authors; see [2–7] and the references therein. However, all the above-mentioned papers are achieved when corresponding Green’s functions are nonnegative, which is a very important condition. A natural question is that whether we can obtain the existence of positive solutions to some third-order three-point BVPs when corresponding Green’s functions are sign-changing.
In 2008, Palamides and Smyrlis  studied the existence of at least one positive solution to the singular third-order three-point BVP with an indefinitely signed Green’s function where . Their technique was a combination of the Guo-Krasnoselskii fixed point theorem [9, 10] and properties of the corresponding vector field.
Very recently, for the third-order three-point BVP with sign-changing Green’s function Sun and Zhao proved the existence of single or multiple positive solutions when by using the Guo-Krasnoselskii and Leggett-Williams fixed point theorems in [11, 12] and obtained the existence of a positive solution when by using iterative technique in .
In 2013, Li et al.  established the existence of at least one positive solution to the following third-order three-point BVP with sign-changing Green’s function: where and . The main tool used was the Guo-Krasnoselskii fixed point theorem [9, 10].
It is worth mentioning that there are other types of works on sign-changing Green’s functions which prove the existence of sign-changing solutions, positive in some cases; see Infante and Webb’s papers [15–17].
Motivated greatly by the above-mentioned works, in this paper, we will study the BVP (3) by applying iterative method. Throughout this paper, we always assume that and . Although corresponding Green’s function is sign-changing, we still obtain the existence of monotone positive solution for the BVP (3) under some suitable conditions on . Moreover, our iterative scheme starts off with zero function, which implies that the iterative scheme is feasible.
2. Main Results
For any , we consider the BVP
So, if we let , then
Let Banach space be equipped with the norm and Then is a cone in . Note that this induces an order relation “” in by defining if and only if .
In the remainder of this paper, we always assume that and satisfies the following two conditions:for each , the mapping is decreasing;for each , the mapping is increasing.
Now, we define an operator as follows: Obviously, if is a fixed point of in , then is a nonnegative and decreasing solution of the BVP (3).
Lemma 1. is completely continuous.
Proof. Let . Then, for , we have
which together with and implies that
For , we have
which together with and shows that
So, is decreasing on . At the same time, since , we know that is nonnegative on . This indicates that . Furthermore, although is not continuous, it follows from known textbook results, for example, see , that is completely continuous.
Theorem 2. Assume that for and there exist two positive constants and such that the following conditions are satisfied:;, , .
If we construct an iterative sequence , , where for , then converges to in and is a decreasing and positive solution of the BVP (3).
Proof. Let . Then we may assert that .
In fact, if , then it follows from that which shows that . So, .
Now, we prove that converges to in and is a decreasing and positive solution of the BVP (3).
Indeed, in view of and , we have , . Since the set is bounded and is completely continuous, we know that the set is relatively compact. In what follows, we prove that is monotone by induction. First, it is obvious that , which shows that . Next, we assume that . Then is decreasing and , . So, it follows from that So, This together with implies that In view of (19) and (20), we know that , which indicates that . Thus, we have shown that , . Since is relatively compact and monotone, there exists such that , which together with the continuity of and the fact that implies that . This indicates that is a decreasing and nonnegative solution of the BVP (3). Moreover, since for , we know that zero function is not a solution of the BVP (3), which shows that is a positive solution of the BVP (3).
3. An Example
Consider the BVP
If we let , , and , , then all the hypotheses of Theorem 2 are fulfilled with and . Therefore, it follows from Theorem 2 that the BVP (21) has a decreasing and positive solution. Moreover, the iterative scheme is for and The first, second, third, and fourth terms of this scheme are as follows:
In , only the existence of at least one positive solution to the BVP (3) was obtained when and . In this paper, when and , we have successfully constructed an iterative sequence, whose limit is just a desired monotone positive solution of the BVP (3). Moreover, our iterative scheme starts off with zero function, which implies that the iterative scheme is feasible.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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