1. Introduction

In a recent paper [1], by applying a fixed-point index theorem in cones, Jiang and Weng studied the existence of positive solutions for the boundary value problems described by second-order functional differential equations of the form Aykut [2] applied a cone fixed-point index theorem in cones and obtained sufficient conditions for the existence of positive solutions of the boundary value problems of functional difference equations of the form

In this article, we are interested in proving the existence and multiplicity of positive solutions for the boundary value problems of a second-order functional dynamic equation of the form Throughout this paper we let be any time scale (nonempty closed subset of ) and be a subset of such that and for is not right scattered and left dense at the same time.

Some preliminary definitions and theorems on time scales can be found in books [3, 4] which are excellent references for calculus of time scales.

We will assume that the following conditions are satisfied.

(H1) (H2) is continuous with respect to and for , where denotes the set of nonnegative real numbers. (H3) defined on satisfies Let be nonempty subset of (H4) if then ; for , where denotes the set of all positively regressive and rd-continuous functions. (H5) and are defined on and , respectively, where ; furthermore, ;

There have been a number of works concerning of at least one and multiple positive solutions for boundary value problems recent years. Some authors have studied the problem for ordinary differential equations, while others have studied the problem for difference equations, while still others have considered the problem for dynamic equations on a time scale [5–10]. However there are fewer research for the existence of positive solutions of the boundary value problems of functional differential, difference, and dynamic equations [1, 2, 11–13].

Our problem is a dynamic analog of the BVPs (1.1) and (1.2). But it is more general than them. Moreover, conditions for the existence of at least one positive solution were studied for the BVPs (1.1) and (1.2). In this paper, we investigate the conditions for the existence of at least one or three positive solutions for the BVP (1.3). The key tools in our approach are the following fixed-point index theorem [14], and Leggett-Williams fixed-point theorem [15].

Theorem 1.1 (see [14]). Let be Banach space and be a cone in . Let , and define . Assume is a completely continuous operator such that for
(i) If for , then (ii) If for , then

Theorem 1.2 (see [15]). Let be a cone in the real Banach space . Set Suppose that is a completely continuous operator and is a nonnegative continuous concave functional on with for all . If there exists such that the following conditions hold:
(i) and for all (ii) for (iii) for with Then has at least three fixed points in satisfying

2. Preliminaries

First, we give the following definitions of solution and positive solution of BVP (1.3).

Definition 2.1. We say a function is a solution of BVP (1.3) if it satisfies the following.
(1) is nonnegative on . (2) as , where is defined as (3) as , where is defined as (4) is -differentiable, is -differentiable on and is continuous. (5) for

Furthermore, a solution of (1.3) is called a positive solution if for

Denote by and the solutions of the corresponding homogeneous equation under the initial conditions Set Since the Wronskian of any two solutions of (2.3) is independent of , evaluating at and using the initial conditions (2.4) yield Using the initial conditions (2.4), we can deduce from (2.3) for and , the following equations: (See [8].)

Lemma 2.2 (see [8]). Under the conditions (H1) and the first part of (H4) the solutions and possess the following properties:

Let be the Green function for the boundary value problem: given by where and are given in (2.7) and (2.8), respectively. It is obvious from (2.6), (H1) and (H4), that holds.

Lemma 2.3. Assume the conditions (H1) and (H4) are satisfied. Then (i) for (ii) for and where in which

Proof. for , and , for . Besides, is nondecreasing and is nonincreasing, for . Therefore, we have So statement (i) of the lemma is proved. If for a given then statement (ii) of the lemma is obvious for such values. Now, and . Consequently, , for all such Let us take any . Then we have for , and we have for ,
Let be endowed with maximum norm for , and let be a cone defined by where is as in (2.12).
Suppose that is a solution of (1.3), then it can be written as where
Throughout this paper we assume that is the solution of (1.3) with . Clearly, can be expressed as follows: where
Let be a solution of (1.3) and . Noting that for , we have where
Define an operator as follows: where
It is easy to derive that is a positive solution of BVP (1.3) if and only if is a nontrivial fixed point of , where be defined as before.

Lemma 2.4.

Proof. For , we have . Moreover, we have from definition of that and , for and , respectively. Thus, where . It follows from the definition and Lemma 2.3 that Thus, .

Lemma 2.5. is completely continuous.

Lemma 2.6. If for all , then there exist such that , for and , for .

Proof. Choose such that By using the first equality of (2.27), we can choose such that If , then for , we have Therefore we get Thus, we have from Theorem 1.1, , for . On the other hand, the second equality of (2.27) implies for every , there is an , such that Here we choose satisfying (2.28). For , we have definition of that It follows from (2.32) that This shows that Thus, by Theorem 1.1, we conclude that for . The proof is therefore complete.

3. Existence of One Positive Solution

In this section, we investigate the conditions for the existence of at least one positive solution of the BVP (1.3).

In the next theorem, we will also assume that the following condition on .

(H6): where is large enough such that and is small enough such that where is the eigenfunction related to the smallest eigenvalue of the eigenvalue problem:

Theorem 3.1. If (H1)–(H6) are satisfied, then the BVP (1.3) has at least one positive solution.

Proof. Fix and let for . Then, satisfies (2.27). Define by
where Then is a completely continuous operator. One has from Lemma 2.6 that there exist such that Define by then is a completely continuous operator. By the first equality in (H6) and the definition of , there are and such that We now prove that for all and . In fact, if there exists and such that , then satisfies the equation and the boundary conditions Multiplying both sides of (3.10) by , then integrating from to , and using integration by parts in the left-hand side two times, we obtain Combining (3.9) and (3.12), we get We also have Equations (3.13) and (3.14) lead to This is impossible. Thus for and . By (3.7) and the homotopy invariance of the fixed-point index (see [11]), we get that On the other hand, according to the second inequality of (H6), there exist and such that We define then it follows that Define by , then is a completely continuous operator. We claim that there exists such that In fact, if for some and , then where Combining (3.21) with (3.22), we have Let Then we get Consequently, by the homotopy invariance of the fixed-point index, we have where is zero operator. Use (3.16) and (3.25) to conclude that Hence, has a fixed point in .
Let . Since for and .
(H7)

Theorem 3.2. If (H1)–(H5) and (H7) are satisfied, then the BVP (1.3) has at least one positive solution.

Proof. Define by , then is a completely continuous operator. By the first inequality in (H7), there exist and such that We claim that for and . In fact, if there exist and such that , then satisfies the boundary condition (3.11). Since , we have . Then we have Multiplying the last equation by integrating from to , by (3.28), we obtain then we have Equations (3.30) and (3.31) lead to This is impossible. By homotopy invariance of the fixed-point index, we get that Define by , then is a completely continuous operator. By the second inequality in (H7), and definition of , there exist