1. Introduction

Due to the unification of the theory of differential and difference equations, there have been many investigations working on the existence of positive solutions to boundary value problems for dynamic equations on time scales. Also there is much attention paid to the study of multipoint boundary value problem with -Laplacian; see [1–10].

For convenience, throughout this paper we denote as the -Laplacian operator, that is, , . , where .

In [11], the author discussed the positive solutions of a -point boundary value problem for a second-order dynamic equation on a time scale

(1.1)

where ,  , and with . And he got the existence of at least two positive solutions of the above problem by means of a fixed point theorem in a cone.

Zhao and Ge [9] considered the following multi-point boundary value problem with one-dimensional -Laplacian:

(1.2)

where , , , , , . By using a fixed point theorem in a cone, they obtained the existence of at least one, two, or three positive solutions under some sufficient conditions.

Motivated by the above results, in this paper, we investigate the nonlocal boundary value problem with -Laplacian

(1.3)

where and .

For convenience, we list the following hypotheses:

  1. (H1)

    , , , , ;

  2. (H2)

    and is not identically zero on any compact subinterval of ;

  3. (H3)

    and is not identically zero on any compact subinterval of , also it satisfies

    (1.4)

By using the Four functionals fixed point theorem and Five functionals fixed point theorem, we obtain the existence criteria of at least one positive solution and three positive solutions for the BVP (1.3). As an application, an example is worked out finally. The remainder of this paper is organized as follows. Section 2 is devoted to some preliminary discussions. We give and prove our main results in Section 3.

2. Preliminaries

The basic definitions and notations on time scales can be found in [12, 13]. In the following, we will provide some background materials on the theory of cones in Banach spaces. For more details, please refer to [14, 15].

Definition 2.1.

Let be a Banach space. A nonempty, closed set is said to be a cone provided that the following hypotheses are satisfied:

  1. (1)

    if , , then

  2. (2)

    if , , then

Every cone induces a partial ordering "" on defined by if and only if

Definition 2.2.

A map is said to be a nonnegative continuous concave functional on a cone of a real Banach space if is continuous and for all and . Similarly, we say that the map is a nonnegative continuous convex functional on a cone of a real Banach space if is continuous and for all and .

Let and be nonnegative continuous concave functionals on , and let and be nonnegative continuous convex functionals on ; then for positive numbers and , define the sets

(2.1)

The following lemma can be found in [16].

Lemma 2.3 (four functionals fixed point theorem).

If P is a cone in a real Banach space E, and are nonnegative continuous concave functionals on , and are nonnegative continuous convex functionals on , and there exist nonnegative positive numbers and , such that

(2.2)

is a completely continuous operator, and is a bounded set. If

  1. (i)
  2. (ii)

    , for all with and

  3. (iii)

    , for all with

  4. (iv)

    , for all with and ,

  5. (v)

    , for all with

then has a fixed point in .

We are now in a position to present the Five functionals fixed point theorem (see [17]). Let be nonnegative continuous convex functionals on and nonnegative continuous concave functionals on . For nonnegative numbers and define the following convex sets:

(2.3)

Lemma 2.4 (five functionals fixed point theorem).

Let be a cone in a real Banach space . Suppose that there exist nonnegative numbers and , nonnegative continuous concave functionals and on , and nonnegative continuous convex functionals and on , with

(2.4)

Suppose that is completely continuous and there exist nonnegative numbers with such that

  1. (i)

    and for

  2. (ii)

    and for

  3. (iii)

    for with

  4. (iv)

    for with

then A has at least three fixed points such that

(2.5)

Consider the Banach space equipped with the norm . Suppose , with . For the sake of convenience, we take the notations

(2.6)

Define a cone

(2.7)

and an operator by

(2.8)

Lemma 2.5.

.

Proof.

For , ,

(2.9)

From the definition of , it is clear that

(2.10)

is continuous, and is the maximum value of on .

Let , then is continuous, is delta differentiable on , and is continuously differentiable. Moreover is monotonically increasing for and . Then by the chain rule [12, Theorem , page 31], we obtain

(2.11)

where is in the interval . So, . This completes the proof.

3. Main Results and an Example

Theorem 3.1.

Assume that (H1), (H2), and (H3) hold, if there exist constants , , , with , , and suppose that satisfies the following conditions:

(A1) for all

(A2) for all

then the BVP (1.3) has a fixed point such that

(3.1)

Define maps

(3.2)

and let , and be defined by (2.1).

In order to complete the proof of Theorem 3.1, we first need to prove the following lemma.

Lemma 3.2.

is bounded and is completely continuous.

Proof.

For all , , which means that is a bounded set.

According to Lemma 2.5, it is clear that .

In view of the continuity of , there exists a constant such that , for all , . Consider

(3.3)

which means that is uniformly bounded.

In addition, for all , we have

(3.4)

Applying the Arzelà-Ascoli theorem on time scales [18], one can show that is relatively compact.

Now we prove that is continuous. Let be a sequence in which converges to uniformly on . Because is relatively compact, the sequence admits a subsequence converging to uniformly on . In addition,

(3.5)

Observe that

(3.6)

Hence, by the Lebesgue's dominated convergence theorem on time scales [19], insert into the above equality and then let , we obtain

(3.7)

From the definition of , we know that on . This shows that each subsequence of uniformly converges to . Therefore the sequence uniformly converges to . This means that is continuous at . So, is continuous on since is arbitrary. Thus, is completely continuous. This completes the proof.

Proof of Theorem 3.1.

Let

(3.8)

Clearly, . By direct calculation,

(3.9)

So, which means that (i) in Lemma 2.3 is satisfied.

For all with and , we have , and for all with and , we obtain that . Hence, (ii) and (iv) in Lemma 2.3 are fulfilled.

For any with

(3.10)

and for all with ,

(3.11)

Thus (iii) and (v) in Lemma 2.3 hold true. So, by Lemma 2.3, the BVP (1.3) has a fixed point in . This completes the proof.

Theorem 3.3.

Assume that (H1), (H2), and (H3) hold. If there exist constants , with , , , , , further suppose that satisfies the following conditions:

(B1) for all

(B2) for all

then the BVP (1.3) has at least three positive solutions , and such that

(3.12)

Proof.

Define these maps

(3.13)

and let , , and be defined by (2.3). It is clear that

(3.14)

Using similar methods as those in Lemma 3.2, we obtain that is completely continuous. Thus, we only need to show that . Let , then

(3.15)

which implies that .

Let and

(3.16)

we can verify that . By calculation,

(3.17)

So, , , , which means that and are not empty.

For ,

(3.18)

and for ,

(3.19)

Thus (i) and (ii) in Lemma 2.4 hold.

On the other hand, for with , we have . And for with , we can obtain Thus, (iii) and (iv) in Lemma 2.4 hold.

So, by Lemma 2.4, we obtain that the BVP (1.3) has at least three positive solutions such that

(3.20)

This completes the proof.

Remark 3.4.

Let , , , we can find that the conditions of Theorem 3.1 are contained in Theorem 3.3.

Example 3.5.

Let , , consider the following eight-point BVP:

(3.21)

where , , , , , , , , , , , , , for all , and

(3.22)

where is continuous, , and .

Set , , by calculation,

(3.23)

and let , , , , . Clearly, we can verify that the conditions in Theorem 3.3 are fulfilled. Thus, by Theorem 3.3, the BVP (3.21) has at least three positive solutions , and such that

(3.24)

Remark 3.6.

If we let , , , , we can also verify that the conditions in Theorem 3.1 are satisfied.