Abstract and Applied Analysis

Volume 2014 (2014), Article ID 518238, 11 pages

http://dx.doi.org/10.1155/2014/518238

Research Article

Existence and Monotone Iteration of Positive Pseudosymmetric Solutions for a Third-Order Four-Point BVP

Xuezhe Lv, Libo Wang, and Minghe Pei

Department of Mathematics, Beihua University, Jilin 132013, China

Received 19 January 2014; Revised 6 March 2014; Accepted 13 March 2014; Published 8 April 2014

Academic Editor: Bashir Ahmad

Copyright © 2014 Xuezhe Lv 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.

Abstract.

We study the existence and monotone iteration of solutions for a third-order four-point boundary value problem. 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 results, an example is given.

1. Introduction

The third-order equations arise in many areas of applied mathematics and physics, such as the deflection of a curved beam having a constant or varying cross section, three-layer beam, electromagnetic waves, or gravity-driven flows [1], and thus have been studied extensively in the literature; see [1–29] and references therein. Recently, wide attention has been paid to the third-order boundary value problems with nonlocal boundary conditions; see [4, 6–9, 11, 12, 15, 16, 20, 23–30] and references therein.

In 2006, using the monotone iterative technique, Zhou and Ma [30] obtained the existence of positive solutions and established a corresponding iterative scheme for the following third-order -Laplacian problem:

In 2009, Sun et al. [23] 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 2011, Zhang [29] considered the following singular third-order three-point boundary value problem:

The existence and uniqueness of solutions and corresponding iterative scheme to the problem are obtained by applying the cone theory and the Banach contraction mapping principle.

In 2013, Li et al. [7] studied third-order four-point boundary value problem with -Laplacian of the form

By using the monotone iterative technique, the existence result of positive pseudosymmetric solutions and its monotone iterative scheme are established for the problem.

Motivated by above works and [31], in this paper, we consider the existence and monotone iteration of positive pseudosymmetric solutions of the following third-order four-point boundary value problem:

subject to boundary conditions

where and , . Here we say is positive solution of BVP (5), (6), if is the solution of BVP (5), (6) and satisfies for .

To the best of our knowledge, the pseudosymmetric solutions for the second-order boundary value problem have been studied by some authors, see [31–33]. And [7] is the only one concerned with the third-order boundary value problem. We note that the nonlinearity of in our problem contains explicitly and every derivatives of up to order two.

This work is organized as follows. In Section 2, some notations and preliminaries are introduced. The main results are discussed in Section 3. As an application of our results, an example is given in the last section.

2. Preliminary

In this section, we give a definition and some lemmas which help to simplify the presentation of our main result.

Definition 1 (see [33]). Let , . We say is pseudosymmetric about on , if is symmetric on ; that is,

Let the Banach space be endowed with the norm

where ,  . Define a cone by

and by “” denote the induced partial ordering via cone ; that is, for ,  , if and only if .

For convenience, we denote the following.

is a nonnegative continuous function defined on ,   on any subinterval of . In addition, and

    is continuous,

 and, for all ,

    is nonincreasing in , nonincreasing in , and nondecreasing in on .    on .

Now, we define an operator as follows: for ,

Obviously, under assumptions and , the operator is well defined.

Lemma 2. Assume that and hold. Then is a solution of BVP (5), (6) if and only if is a fixed point of .

Proof. At first we show the necessity. Suppose is a solution of BVP (5), (6). Then, integrating (5) and using (6) we infer that

For , integrating (14) on and taking into account , we get

Again integrating (16) on one can obtain

But, from (15) and (16), it follows that

Therefore, for , one has

For , integrating (14) on , we get

Again integrating (20) on one obtains

In particular, we have

In (19), we take , and then

From (22) and (23) one has

Hence from (21) it follows that

Notice that from , , and the fact that , we have

Therefore

This together with (19) implies that is fixed point of .

The sufficiency, by direct computation and using the fact that , follows immediately.

The following 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 (6) for all . Furthermore, since

it follows that is concave on .

Next we prove that is pseudosymmetric about on . In fact, if , then ; it follows that

Note that is pseudosymmetric about on ; that is, for , and then

Thus, for all , from and , we have

Hence, for ,

From (29) and (32), it follows that

If , then . From (33), we have

This together with (33) implies

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. Then is nondecreasing on ; that is, for ,   with .

Proof. Let ,   with . Then . By the definition of , is nonnegative, concave, and pseudosymmetric about on . Therefore

From and the definition of it follows that

We now prove that (37) and (38) hold for . In fact, if , then , and hence, from the fact that   and are pseudosymmetric about on , it follows that, for ,

So

and is concave on .

Finally, we show that is pseudosymmetric about on . To do this we let . We note that ,  , and are pseudosymmetric about on , and thus

In summary, ; that is, .

3. Main Result

Now we establish existence result of positive, concave, and pseudosymmetric solutions and its monotone iterative scheme for BVP (5), (6).

Theorem 6. Assume that , , , and hold. Suppose also that there exist two positive numbers and with such that

where satisfy

Then BVP (5), (6) has positive, concave, and pseudosymmetric solutions ,   with

Proof. We denote . In what follows, we first show that . To do this, let ; then obviously

Also since is concave and pseudosymmetric about on , then is nonincreasing on , , and for . Hence for and achieve the maximum at . Consequently

From , , and (42), it follows that

This together with Lemma 3 implies

Hence , and thus .

Let . Then , and thus . Let ; then . Define iterative sequence as follows:

Since , we have ,  . From Lemma 4, has a convergent subsequence and there exists such that

From the definition of and (42), for , we have

and, for , we have

Thus ,  .

On the other hand, since, for , we have

and, for , we have

it follows that on . Hence is concave on .

Also, since, for ,

then is pseudosymmetric about on . So , and hence from Lemma 5 it follows that ; that is, . By induction, we can show without any difficulty that

that is,

Thus is concave on and pseudosymmetric about on ; consequently

From (50)–(60), it follows that

that is, . Let in (49) to obtain

Also, from , we have . This together with the concavity of implies that

Again using the fact that is pseudosymmetric about on , we have

Hence on . Therefore, from Lemma 2, is a concave pseudosymmetric positive solution of BVP (5), (6).

Let on ; then . Set

Then, from Lemma 3, the sequence is well defined. Since , we have , and thus from Lemma 5 it follows that

By induction we can show that

Similarly to , we can show that there exists such that . Taking limit in (65), we get . Obviously, on . Therefore, from Lemma 2, is a concave pseudosymmetric positive solution of BVP (5), (6). This completes the proof of the theorem.

4. An Example

Consider the following third-order four-point boundary value problem:

Let

Then . It is easy to see that BVP (68) corresponds to BVP (5), (6) when , , , and .

Next we verify that all conditions of Theorem 6 are satisfied. In fact, obviously the conditions , , and hold. In addition, by the definition of , we have

Hence the condition is also satisfied.

Now, we take ,  . Then

and thus

On the other hand, we also have

In summary, all conditions of Theorem 6 are satisfied. Hence, from Theorem 6, BVP (68) has concave pseudosymmetric positive solution ,   with

Conflict of Interests

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

Acknowledgments

The authors are grateful to the anonymous referee for his or her constructive comments and suggestions which led to improvement of the original paper. This work was supported by NSFC (11201008).

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