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

(11)

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

(12)

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

(13)

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

(14)

Let be nonempty subset of

(15)

(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, ;

(16)

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

  1. (i)

    If for , then

  2. (ii)

    If for , then

Theorem 1.2 (see [15]).

Let be a cone in the real Banach space . Set

(17)

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:

  1. (i)

    and for all

  2. (ii)

    for

  3. (iii)

    for with

Then has at least three fixed points in satisfying

(18)

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. (1)

    is nonnegative on .

  2. (2)

    as , where is defined as

    (21)
  3. (3)

    as , where is defined as

    (22)
  4. (4)

    is -differentiable, is -differentiable on and is continuous.

  5. (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

(23)

under the initial conditions

(24)

Set

(25)

Since the Wronskian of any two solutions of (2.3) is independent of , evaluating at and using the initial conditions (2.4) yield

(26)

Using the initial conditions (2.4), we can deduce from (2.3) for and , the following equations:

(27)
(28)

(See [8].)

Lemma 2.2 (see [8]).

Under the conditions (H1) and the first part of (H4) the solutions and possess the following properties:

(29)

Let be the Green function for the boundary value problem:

(210)

given by

(211)

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

  1. (i)

    for

  2. (ii)

    for and

where

(212)

in which

(213)

Proof.

for , and , for . Besides, is nondecreasing and is nonincreasing, for . Therefore, we have

(214)

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 ,

(215)

and we have for ,

(216)

Let be endowed with maximum norm for , and let be a cone defined by

(217)

where is as in (2.12).

Suppose that is a solution of (1.3), then it can be written as

(218)

where

(219)

Throughout this paper we assume that is the solution of (1.3) with . Clearly, can be expressed as follows:

(220)

where

(221)

Let be a solution of (1.3) and . Noting that for , we have

(222)

where

(223)

Define an operator as follows:

(224)

where

(225)

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

(226)

Thus, .

Lemma 2.5.

is completely continuous.

Lemma 2.6.

If

(227)

for all , then there exist such that , for and , for .

Proof.

Choose such that

(228)

By using the first equality of (2.27), we can choose such that

(229)

If , then for , we have

(230)

Therefore we get

(231)

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

(232)

Here we choose satisfying (2.28). For , we have definition of that

(233)

It follows from (2.32) that

(234)

This shows that

(235)

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):

(31)

where is large enough such that

(32)

and is small enough such that

(33)

where is the eigenfunction related to the smallest eigenvalue of the eigenvalue problem:

(34)

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

(35)

where

(36)

Then is a completely continuous operator. One has from Lemma 2.6 that there exist such that

(37)
(38)

Define by then is a completely continuous operator. By the first equality in (H6) and the definition of , there are and such that

(39)

We now prove that for all and . In fact, if there exists and such that , then satisfies the equation

(310)

and the boundary conditions

(311)

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

(312)

Combining (3.9) and (3.12), we get

(313)

We also have

(314)

Equations (3.13) and (3.14) lead to

(315)

This is impossible. Thus for and . By (3.7) and the homotopy invariance of the fixed-point index (see [11]), we get that

(316)

On the other hand, according to the second inequality of (H6), there exist and such that

(317)

We define

(318)

then it follows that

(319)

Define by , then is a completely continuous operator. We claim that there exists such that

(320)

In fact, if for some and , then

(321)
(322)

where Combining (3.21) with (3.22), we have

(323)

Let Then we get

(324)

Consequently, by the homotopy invariance of the fixed-point index, we have

(325)

where is zero operator. Use (3.16) and (3.25) to conclude that

(326)

Hence, has a fixed point in .

Let . Since for and .

(H7)

(327)

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

(328)

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

(329)

Multiplying the last equation by integrating from to , by (3.28), we obtain

(330)

then we have

(331)

Equations (3.30) and (3.31) lead to

(332)

This is impossible. By homotopy invariance of the fixed-point index, we get that

(333)

Define by , then is a completely continuous operator. By the second inequality in (H7), and definition of , there exist and such that

(334)

We define

(335)

then, it is obvious that

(336)

We claim that there exists such that

(337)

In fact, if for some and , then using (3.36), it is analogous to the argument of (3.13) and (3.14) that

(338)

Equation (3.38) leads to Let . Then we get

(339)

Consequently, by (3.8) and the homotopy invariance of the fixed-point index, we have

(340)

In view of (3.33) and (3.40), we obtain

(341)

Therefore, has a fixed point in . The proof is completed.

Corollary 3.3.

Using the following (H8) or (H9) instead of (H6) or (H7), the conclusions of Theorems 3.1 and 3.2 are true. For ,

(H8)

(342)

(H9)

(343)

4. Existence of Three Positive Solutions

In this section, using Theorem 1.2 (the Leggett-Williams fixed-point theorem) we prove the existence of at least three positive solutions to the BVP (1.3).

Define the continuous concave functional to be , and the constants

(41)
(42)

Theorem 4.1.

Suppose there exists constants such that

(D1) for

(D2) for

(D3) one of the following is satisfied:

  1. (a)
  2. (b)

    there exists a constant such that for and ,

where , , and are as defined in (2.12), (4.1), (4.2), respectively. Then the boundary value problem (1.3) has at least three positive solutions and satisfying

(43)

Proof.

The technique here similar to that used in [5] Again the cone , the operator is the same as in the previous sections. For all we have If , then and the condition (a) of (D3) imply that

(44)

Thus there exist a and such that if , then . For , we have for all , for all Pick any

(45)

Then implies that

(46)

Thus

The condition (b) of (D3) implies that there exists a positive number such that for and . If , then

(47)

Thus Consequently, the assumption (D3) holds, then there exist a number such that and

The remaining conditions of Theorem 1.2 will now be shown to be satisfied.

By (D1) and argument above, we can get that Hence, condition (ii) of Theorem 1.2 is satisfied.

We now consider condition (i) of Theorem 1.2. Choose for , where . Then and so that . For , we have , . Combining with (D2), we get

(48)

for . Thus, we have

(49)

As a result, yields

Lastly, we consider Theorem 1.2(iii). Recall that . If and , then

(410)

Thus, all conditions of Theorem 1.2 are satisfied. It implies that the TPBVP (1.3) has at least three positive solutions with

(411)

5. Examples

Example 5.1.

Let Consider the BVP:

(51)

Then and

(52)

Since , . It is clear that (H1)–(H5) and (H8) are satisfied. Thus, by Corollary 3.3, the BVP (5.1) has at least one positive solution.

Example 5.2.

Let us introduce an example to illustrate the usage of Theorem 4.1. Let

(53)

Consider the TPBVP:

(54)

Then and

(55)

The Green function of the BVP (5.4) has the form

(56)

Clearly, is continuous and increasing . We can also see that . By (2.12), (4.1), and (4.2), we get , and .

Now we check that (D1), (D2), and (b) of (D3) are satisfied. To verify (D1), as , we take , then

(57)

and (D1) holds. Note that , when we set ,

(58)

holds. It means that (D2) is satisfied. Let , we have

(59)

from , so that (b) of (D3) is met. Summing up, there exist constants and satisfying

(510)

Thus, by Theorem 4.1, the TPBVP (5.4) has at least three positive solutions with

(511)