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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 192509, 12 pages
Existence and Monotone Iteration of Positive Pseudosymmetric Solutions for a Third-Order Four-Point BVP with -Laplacian
Department of Mathematics, Beihua University, Jilin 132013, China
Received 25 February 2013; Revised 27 April 2013; Accepted 28 April 2013
Academic Editor: Guoyin Li
Copyright © 2013 Dan Li et al. 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 existence and monotone iteration of solutions for a third-order four-point boundary value problem with -Laplacian. An existence result of positive, concave, and pseudosymmetric solutions and its monotone iterative scheme are established by using the monotone iterative technique. Meanwhile, as an application of our result, an example is given.
The third-order equations arise in many areas of applied mathematics and physics  and thus have been discussed by many authors and many excellent results were obtained; see [1–31] and the references therein. Recently, wide attention has been paid to the third-order boundary value problems with the -Laplace operator. In fact, the third-order equations involving the -Laplace operator can be seen as a generalized model for various physical, natural or physiological phenomena such as the flow of a thin film of viscous fluid over a solid surface, the solitary wave solution of the Korteweg-de Vries equation or a thyroid-pituitary interaction .
In 2005, Cabada et al.  studied the one-dimensional nonlinear third-order -Laplacian equation with the boundary conditions where is an increasing homeomorphism with . By applying the monotone iterative technique based on suitable antimaximum principles, they obtained the existence of extremal solutions for the problem.
In 2006, using the monotone iterative technique, Zhou and Ma  obtained the existence of positive solutions and established a corresponding iterative scheme for the following third-order -Laplacian problem of the form: In 2007, Wang and Ge  considered third-order differential equation subject to the following integral boundary conditions: The existence result to the problem is obtained by applying the method of upper and lower solutions and Leray-Schauder degree theory.
In 2009, Sun et al.  studied the existence of positive solutions for the following third-order -Laplacian problem: By applying a monotone iterative method, the authors obtained the existence of positive solutions for the problem and established iterative schemes for approximating the solutions.
In 2010, Jin and Lu  considered the following third-order -Laplacian resonant problem of the form: The authors obtained the existence of solutions for the problem by using Mawhin's continuation theorem.
In 2010, by using the fixed point index method, Yang and Yan  established the existence of at least one or at least two positive solutions for the following third-order -Laplacian problem: Motivated by the above works and [32, 33], in this paper, we consider the existence and monotone iteration of positive, pseudosymmetric solutions of the following third-order four-point -Laplacian boundary value problem: subject to boundary conditions where , and be constant. Here is said to be a positive solution of BVP (9), (10) if and only if is the solution of BVP (9), (10) and satisfies for . BVP (9), (10) can model the static deflection of an elastic beam with linear supports at both endpoints.
To the best of our knowledge, the existence results of the pseudosymmetric solutions for the third-order boundary value problem has not been considered.
This work is organized as follows. In Section 2, some notations and preliminaries are introduced. The main results are discussed in Section 3. As applications of our results, an example is given in the last section.
In this section, we give some definitions and lemmas which help to simplify the presentation of our main result.
Definition 1 (see ). Let , . One says that is pseudosymmetric about on , if is symmetric on [,1], that is,
Definition 2. Let , . One says that is pseudo-antisymmetric about on , if is antisymmetric on [,1], that is,
Let the Banach space be endowed with the norm
and define the cone by
For convenience, we consider the following. (H0) is a nonnegative continuous function defined on , on any subinterval of . In addition, and is pseudosymmetric about on . (H1) is continuous and
(H2) for all . (H3) on .
Now, we define an operator as follows: for ,
Obviously under assumptions and , the operator is well defined and it is easy to verify that BVP (9), (10) has a solution if and only if has a fixed point.
The next lemmas are some properties of the operator .
Lemma 3. Assume that , , and hold. Then .
Proof. From the definition of , it is easy to check that is nonnegative on and satisfies (10) for all . Furthermore, since
it follows that is concave on .
Next we prove that is pseudosymmetric about on . In fact, if , then , and it follows that Also since is pseudosymmetric about on , that is, for , then Thus, for all , from , we have Hence is pseudo-antisymmetric about on , and thus is pseudosymmetric about on . Furthermore is pseudosymmetric about on . Thus the function is pseudo-antisymmetric about on , and hence Using the similar technique, we can get From (19), (22), and (23), it follows that If , then . From (24), it follows that This together with (24) implies that In summary, , and then .
The following lemma can be easily verified by a standard argument.
Lemma 4. Assume that , , and hold. Then is completely continuous.
Lemma 5. Assume that , , and hold. Suppose also that there exists such that for , , , Then for , with we have
Proof. First we prove that, for all ,
From assumptions, we have
Since is strictly increasing on , then for all , we have
Thus for ,
Therefore, (30) holds for .
Next we prove that (30) holds for . In fact, if , then , and hence from the fact that and are pseudosymmetric about on , it follows that, for , In summary,
Now, we introduce some notations as follows:
Lemma 6. Assume that , , and hold. Suppose also that there exists such that for , , , Then .
Proof. Define two functionals on as follows: Then If , then From the assumptions, for all , Then, So we have Thus .
3. Main Result
Next we prove that In fact, from it follows that Then (49) holds. Equation (50) can be obtained in a similar way. Thus from (49) and (50), it follows that We note that for , and for , So, Thus, From assumptions, for , and for , Hence from (57) and (58), we have Consequently, from (56) and (59), it follows that From the proof of Lemma 3, we see that is nonnegative, concave, and pseudosymmetric about on , and hence Define as follows: Then is well defined and for , , , In fact, for , For , since , , it follows from (64) and (65) that So from (64)–(67), we have that is, (63) holds when . Assume that (63) holds when , that is, Then from Lemma 5, we obtain So by induction (63) holds.
Since is completely continuous, it follows that is relative compact. Then has a convergent subsequence and such that that is, While from (63) and the fact for each , and on , it follows that Hence, that is, This together with the continuity of and , implies that Also, since we have Furthermore, we have In fact, from and on , we have . Since is concave on , then Consequently from the fact is pseudosymmetric on , we have Let on , then . Set , , , , . Then from Lemma 6, it follows that From Lemma 4, we see that is relative compact, and hence there exists a convergent subsequence and such that that is, Since , then Thus from Lemma 5, By induction, it is easy to see that for , , , From (84)–(89), we see that Therefore, , . By the continuity of and , we have Again from , we have on .
Since every fixed point of in is the solution of BVP (9), (10), then and are two positive, concave and pseudosymmetric solutions of BVP (9), (10). This completes the proof of the theorem.
4. An Example
Next we verify that all conditions of Theorem 7 are satisfied. In fact, obviously the conditions , , , and hold. In addition, for , , , Hence, from Theorem 7, BVP (92) has two positive, concave, and pseudosymmetric solutions and such that where for , where on and for , The first two terms of and three terms of , respectively, are as follows: