1. Introduction

In this paper, we will be concerned with the existence of positive solutions for the following boundary value problem on time scales:

where is an increasing homeomorphism and homomorphism and .

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, ().

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

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

Recently, there has been much attention paid to the existence of positive solutions for second-order nonlinear boundary value problems on time scales, for examples, see [1–6] and references therein. At the same time, multipoint nonlinear boundary value problems with -Laplacian operators on time scales have also been studied extensively in the literature, for details, see [4, 5, 7–13] and the references therein. But to the best of our knowledge, few people considered the second-order dynamic equations of increasing homeomorphism and positive homomorphism on time scales.

For the existence problems of positive solutions of boundary value problems on time scales, some authors have obtained many results in the recent years, especially [6, 7, 9, 10, 14, 15] and the references therein. To date few papers have appeared in the literature concerning multipoint boundary value problems for an increasing homeomorphism and homomorphism on time scales.

In [16], Liang and Zhang studied the existence of countably many positive solutions for nonlinear singular boundary value problems:

where is an increasing homeomorphism and positive homomorphism and . By using the fixed point index theory and a new fixedpoint theorem in cones, they obtained countably many positive solutions for problem (1.3).

Very recently, Sang et al. [6] investigated the nonlinear -point BVP on time scales (1.1) and (1.2).

Let

They mainly obtained the following results.

Theorem 1.1. Assume that , , and hold, there exist , such that and suppose that satisfies the following additional conditions: ()()() Then (1.1) and (1.2) has at least two positive solutions and .

Motivated by the above papers, the purpose of our paper is to show the existence of twin or an arbitrary odd number of positive solutions to the BVP (1.1), (1.2). The most important is that the authors would like to point out that there is only the -Laplacian case for increasing homeomorphism and homomorphism, this point was proposed by professor Jeff Webb. This is the main motivation for us to write down the present paper. We also point out that when , (1.1) and (1.2) becomes a boundary value problem of differential equations and just is the problem considered in [15]. Our main results extend and include the main results of [5, 15, 16].

The rest of the paper is arranged as follows. We state some basic time scale definitions and prove several preliminary results in Section 2. Sections 3, 4, and 5 are devoted to the existence of positive solutions of (1.1) and (1.2), with the main tool being the Avery-Henderson and Leggett-Williams fixed point theorems. Finally, in Section 6, we give an example to illustrate our main results.

2. Preliminaries and Some Lemmas

For convenience, we list the following definitions which can be found in [2, 17–19].

Definition 2.1. 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 set .

Definition 2.2. 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 .

Definition 2.3. 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 .

Definition 2.4. 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

Lemma 2.5. For the BVP (2.6) and (2.7) has the unique solution where

Proof. Let be as in (2.8). By [18, Theorem  2.10 (iii)], taking the delta derivative of (2.8), we have moreover, we get taking the nabla derivative of this expression yields . And routine calculations verify that satisfies the boundary value conditions in (2.7), so that given in (2.8) is a solution of (2.6) and (2.7).
It is easy to see that the BVP has only the trivial solution. Thus in (2.8) is the unique solution of (2.6) and (2.7). The proof is complete.

Lemma 2.6. Assume that holds, for and , then the unique solution of (2.6) and (2.7) satisfies

Proof. Let Since then .
According to Lemma 2.5, we get
If , we have So .
Let the norm on be the maximum norm. Then the is a Banach space. Choose the cone defined by
Clearly, for . Define the operator by where It is obvious from Lemma 2.6 that, for .
From the definition of , we claim that for each , and satisfies (1.2) and is the maximum value of on [].
In fact, let
Then it holds Since then . So .
Moreover, is a monotone increasing and continuous function and
then we obtain so, . 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 2.7. If , then for .

Proof. Since it follows that is nonincreasing. Thus, for , from which we have The proof is complete.

In the rest of this section, we provide some background material from the theory of cones in Banach spaces, and we then state several fixed point theorems which we will use later.

Let be a Banach space and a cone in . A map is said to be a nonnegative, continuous, and increasing functional provided that is nonnegative, continuous and satisfies for all and .

Given a nonnegative continuous functional on a cone of a real Banach space , we define, for each , the set .

Lemma 2.8 (see [22]). Let be a cone in a real Banach space . Let and be increasing, nonnegative continuous functionals on , and let be a nonnegative continuous functional on with such that, for some and , for all . Suppose that there exists a completely continuous operator and such that and (i) for all (ii) for all (iii) and for Then, has at least two fixed points, and belonging to satisfying

The following lemma is similar to Lemma 2.8.

Lemma 2.9 (see [23]). Let be a cone in a real Banach space . Let and be increasing, nonnegative continuous functionals on , and let be a nonnegative continuous functional on with such that, for some and , for all . Suppose that there exists a completely continuous operator and such that and (i) for all (ii) for all (iii) and for Then, has at least two fixed points, and belonging to satisfying

Let be given and let be a nonnegative continuous concave functional on the cone . Define the convex sets , by

Finally we state the Leggett-Williams fixed point theorem [3].

Lemma 2.10 (see [3]). Let be a cone in a real Banach space , completely continuous, and a nonnegative continuous concave functional on with for all Suppose that there exist such that (i) and for (ii) for (iii) for with Then, has at least three fixed points, , , satisfying

Now, for the convenience, we introduce the following notations. Let and fixed such that , denote

Define the nonnegative, increasing, and continuous functionals and on by

We observe that, for each ,

In addition, for each , Thus

Finally, we also note that , and .

3. Existence Theorems of Twin Positive Solutions

Theorem 3.1. Assume that there are positive numbers such that Assume further that satisfies the following conditions: (i), (ii), (iii). Then (1.1) and (1.2) has at least two positive solutions and such that

Proof. By the definition of the operator and its properties, it suffices to show that the conditions of Lemma 2.8 hold with respect to .
We first show that if then . Indeed, if , then