Abstract

We investigate the existence and iteration of positive solutions for the following third-order -Laplacian dynamic equations on time scales: where is -Laplacian operator; that is, , and By applying the monotone iterative technique and without the assumption of the existence of lower and upper solutions, we not only obtain the existence of positive solutions for the problem, but also establish iterative schemes for approximating the solutions.

1. Introduction

A time scale is a nonempty closed subset of . We make the blanket assumption that are points in . By an interval , we always mean the intersection of the real interval with the given time scale; that is, .

In recent years, dynamic equations on time scales have found a considerable interest and attracted by many researchers; see, for example, [111]. The reason seems to be twofold. Theoretically, dynamic equations on time scales not only can unify differential and difference equations [12], but also have displayed much more complicated dynamics [13, 14]. Moreover, the study of time scales has led to several important applications in the study of insect population models, neural networks, stock market, heat transfer, economic, wound healing, and epidemic models.

Recently, there is much attention paid to the existence of positive solutions for second-order nonlinear boundary value problems on time scales; see [2, 6, 15, 16] and the references therein. On the one hand, higher-order nonlinear boundary value problems have been studied extensively; for details, see [4, 10, 17, 18] and references therein. On the other hand, the boundary value problems with -Laplacian operator have also been discussed extensively in the literature; for example, see [3, 79, 11, 1921]. However, to the best of our knowledge, there are not many results concerning the third-order -Laplacian dynamic equations on time scales.

In [22], Yang and Yan studied the following third-order Sturm-Liouville boundary value with -Laplacian: By using the fixed point index method, they established the existence of at least one or two positive solutions for the third-order Sturm-Liouville boundary value problem with -Laplacian.

In [8], Han and Kang were concerned with the existence of multiple positive solutions of third-order -Laplacian dynamic equation on time scales: where , . By using the fixed point index method, they obtained the existence of multiple positive solutions for singular nonlinear boundary value problem.

In this paper, we are concerned with the existence and iteration of positive solutions to the following third-order -Laplacian dynamic equation on time scales:where is -Laplacian operator; that is, for with and .

Throughout this paper, we assume that the following conditions are satisfied:(H1).(H2) is a nonnegative continuous function defined on , and on any subinterval of . In addition, . Here, a positive solution of (3) and (4) means a solution satisfying .(H3) is continuous, and there exist constants , such that, for any , ,(H4) and , and let

Remark 1. Inequalities (5) and (6) are equivalent to the following inequalities, respectively:

This paper is organized as follows. In Section 2, we state some basic time scale definitions and several preliminary results. Section 3 is devoted to the iteration and existence of positive solutions of for the third-order Sturm-Liouville boundary value problem with -Laplacian on time scales (3)-(4). Finally, in Section 4, we give an example to demonstrate our main result.

2. Preliminaries

For convenience, we list the following definitions which can be found in [3, 5, 1315, 23].

Definition 2. A time scale is a nonempty closed subset of real numbers . For and , define the forward jump operator and backward jump operator , respectively, by for all . If , is said to be right scattered, and if , is said to be left scattered; if , is said to be right dense, and if , is said to be left dense. If has a right scattered minimum , define ; otherwise set . If has a left scattered maximum , define ; otherwise .

Throughout this paper, we make the blanket assumption that are points in and define the interval in :Other types of intervals are defined similarly.

Definition 3. For and , the delta derivative of at the point is defined to be the number (provided it exists) with the property that, for each , there is a neighborhood of such that for all .
For and , the nabla derivative of at is denoted by (provided it exists) with the property that, for each , there is a neighborhood of such that for all .

Definition 4. A function is left-dense-continuous (i.e., ld-continuous), if is continuous at each left-dense point in and its right-sided limit exists at each right-dense point in . It is well known that if is ld-continuous, then there is a function such that . In this case, it is defined that

Definition 5. If , then we define the delta integral by If , then we define the nabla integral by

If on , then we say that is concave down on .

By a function being a positive solution of (3)-(4), we mean that is positive on and satisfies (3)-(4). Let be Green’s function of with respect to boundary condition (4); then where .

This implies that satisfiesIn fact, on the one hand, from Green’s function, if is kept fixed then is a decreasing function of for and increasing for . This leads to the conclusion that has a global maximum at , so , for . On the other hand, we can easily know that is nonnegative from the following two equalities:

Let the Banach space be equipped with the norm

Definition 6. Let be a real Banach space. Let be a nonempty, convex closed set in . We say that is a cone it it satisfies the following properties:(i) for ;(ii) implies ( denotes the null element of ).

Define the cone by We define operator byand then all the fixed points of operator are the solutions for BVP (3)-(4).

Remark 7. From the definition of the operator , we can easily get the following properties:

Lemma 8. Assume that (H1)–(H3) hold; then the operator is completely continuous.

Proof. From (H1)–(H3) and the definition of , we deduce that, for any , there is . By (24), we get which implies that is concave on . Thus, we have .
Now we show that is completely continuous. The continuity of is obvious. We prove is compact. Let be a bounded set. Then, there exists such that . We can easily prove that is bounded and equicontinuous; that is, is a relatively compact set according to the Ascoli-Arzela theorem. Thus is completely continuous.

For convenience we use the following notations. Let

3. The Main Results

We now give our results for the existence of positive solutions for BVP (3) and (4).

Theorem 9. Assume that (H1)–(H4) hold and there exists a positive constant such that (H5) for any , , ,(H6), where , , are defined by (5), (6), and (7), respectively. Then boundary value (3) and (4) has two positive, concave solutions and such that wherewhere , is defined by (22). The iterative schemes in this theorem are , , and , , .

Proof. We define . Then, in what follows, we will prove that . If , then ; we have So by (8), (9) and (H5), we haveIn fact, Then, by (18), (30) and (H6), we have and from (23) and (24), we haveThus, we obtained that . So, we have shown that .
Let , then It is easy to get so, .
Let , then from the above expression, we have . We define It follows from , that . Since is completely continuous, we can assert that is a sequentially compact set.
Since we obtain By (22) and (H5), we have Hence by induction we have Thus, there exists such that . Applying the continuity of and , we get which implies that is a nonnegative concave solution of the BVP (3) and (4).
On the other hand, let , then . Let , then . We define . Since , we have . Since is completely continuous, we can assert that is a sequentially compact set.
Since , we have So, By an induction argument similar to the above we obtain Hence, there exists such that . Applying the continuity of and , we obtain which implies that is a nonnegative solution of the BVP (3) and (4). From the condition (H4), we know that the zero function is not the solution of the BVP (3) and (4). Thus, , and by the concavity of we have Next, noting , it follows from (H5) and (22) that , and by induction, we have which implies that . Thus we have This means that is a positive concave solution of the BVP (3) and (4). Thus the BVP (3) and (4) has two positive and concave solutions such that , and, from the above proof, we know that the iterative sequences hold. The proof is completed.

4. Example

Now, we present an example to illustrate the main result: Theorem 9.

Let . We consider the following third-order Sturm-Liouville boundary value problem with -Laplacian on time scales:where , , , , and . Let where, for any and , , we have which implies that (H3) holds.

On the other hand, it is clear that (H1), (H2), and (H4) are satisfied and Since , we have so .

Take ; then which implies that (H6) holds. For , it is clear that (H5) holds. So, by Theorem 9, boundary value problem (46) has two positive and concave solutions and such that where , and where .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.