Abstract

Several existence theorems of positive solutions are established for nonlinear -point boundary value problem for the following dynamic equations on time scales , , , , where is an increasing homeomorphism and homomorphism and . As an application, an example to demonstrate our results is given.

1. Introduction

In this paper, we study the existence of positive solutions of the following dynamic equations on time scales: where is an increasing homeomorphism and homomorphism and .

A projection is called an increasing homeomorphism and homomorphism if the following conditions are satisfied:

(i) if , then , for all ;(ii) is a continuous bijection and its inverse mapping is also continuous;(iii), for all .

We will assume that the following conditions are satisfied throughout this paper:

(H₁) satisfy , and (H₂) and there exists , such that (H₃).

Recently, there is much attention focused on the existence of positive solutions for second-order, three-point boundary value problem on time scales. On the other hand, three-point and -point boundary value problems with -Laplacian operators on time scales have also been studied extensively, for details, see [1–11] and references therein. But with an increasing homeomorphism and homomorphism, few works were done as far as we know.

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

We would like to mention some results of Anderson et al. [2], He [4, 5], Sun and Li [9], Ma et al. [12], Wang and Hou [13], Wang and Ge [14], which motivate us to consider our problem.

In [2], Anderson et al. considered the following problem: where , and for some positive constants . They established the existence results of at least one positive solution by using a fixed point theorem of cone expansion and compression of functional type.

In [4, 5], He considered the existence of positive solutions of the -Laplacian dynamic equations on time scales: satisfying the boundary conditions or where , and for some positive constants . He obtained the existence of at least double and triple positive solutions of the problem (1.3), (1.4), and (1.5) by using a new double fixed point theorem and triple fixed point theorem, respectively.

In recent papers, Ma et al. [12] have obtained the existence of monotone positive solutions for the following BVP: The main tool is the monotone iterative technique.

In [9], Sun and Li studied the following -Laplacian, -point BVP on time scales: where for . Some new results are obtained for the existence of at least twin or triple positive solutions of the problem (1.7) by applying Avery-Henderson and Leggett-Williams fixed point theorems, respectively.

In [15], Sang and xi investigated the existence of positive solutions of the -Laplacian dynamic equations on time scales: where is -Laplacian operator, that is, Let they mainly obtained the following results.

Theorem 1.1. Assume (H₁), (H₂), and (H₃) hold, and assume that one of the following conditions holds:
(H₄ )there exist with such that(H₅ )there exist with such that Then, (1.8) have a positive solution.

In this paper, we will establish two new theorems of positive solution of (1.8), our work concentrates on the case when the nonlinear term does not satisfy the conditions of Theorem 1.1. At the end of the paper, we will give an example which illustrates that our work is true.

2. Preliminaries and Some Lemmas

For convenience, we list the following definitions which can be found in [16–19].

Deffinition. A time scale T 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, set .

Deffinition. 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 the number , (provided it exists), with the property that for each ; there is a neighborhood of such that for all .

Deffinition. A function is left-dense continuous (i.e., continuous) if is continuous at each left-dense point in , and its right-sided limit exists at each right-dense point in .

Deffinition. If , then we define the delta integral by If , then we define the nabla integral by To prove the main results in this paper, we will employ several lemmas. These lemmas are based on the linear BVP:
We can prove the following lemmas by the methods of [15].

Lemma 2.5. For , the BVP (2.6) has the unique solution: where

Lemma 2.6. Assume (H₁) holds, For and , then the unique solution of (2.6) satisfies

Lemma 2.7. Assume (H₁) holds, if and , then the unique solution of (2.6) satisfies where

Let the norm on be the maximum norm. Then, the is a Banach space. It is easy to see that the BVP (1.1) has a solution if and only if is a fixed point of the operator equation: where Throughout this paper, we will assume that . Denote where is the same as in Lemma 2.7. It is obvious that is a cone in . By Lemma 2.7, . So by applying Arzela-Ascoli theorem on time scales [20], we can obtain that is relatively compact. In view of Lebesgue's dominated convergence theorem on time scales [21], it is easy to prove that is continuous. Hence, is completely continuous.

Lemma 2.8. Let for , then

Lemma 2.9 ([22]). Let be a Banach space, and let be a cone. Assume are open bounded subset of with and let be a completely continuous operator such that
(i) and ; or(ii), and Then, has a fixed point in

Now, we introduce the following notations. Let

For , By Lemma 2.6, and are well defined.

3. Existence Theorems of Positive Solution

Theorem 3.1. Assume (H₁), (H₂), (H₃) hold, and assume that the following conditions hold:
(A₁) and (A₂)(A₃)there exist , such thatThen, the problems (1.1) have at least one positive solution.

Theorem 3.2. Assume (H₁), (H₂), (H₃) hold, and assume that the following conditions hold:
(B₁) and (B₂);(B₃)there exist , such thatThen, the problems (1.1) have at least one positive solution.

Proof of Theorem 3.1. By , then we can get that there exist such that If , then . By condition (A₃), we have
Let so that
Therefore, It follows that Noticing , we have Therefore, there exist such that . It implies that .
On the other hand, by , we can get that there exist , and such that then .
Let . By Lemma 2.8 and condition , applying Lemma 2.8, it follows that It follows that Noticing , we get

Therefore, there exists with such that . It implies that for .
By Lemma 2.9, we assert that the operator has one fixed point such that . Therefore, is positive solution of the problems (1.1).