Existence of Positive Solutions for Multipoint Boundary Value Problem with -Laplacian on Time Scales

Abstract

We consider the existence of positive solutions for a class of second-order multi-point boundary value problem with -Laplacian on time scales. By using the well-known Krasnosel'ski's fixed-point theorem, some new existence criteria for positive solutions of the boundary value problem are presented. As an application, an example is given to illustrate the main results.

1. Introduction

The theory of time scales has become a new important mathematical branch since it was introduced by Hilger [1]. Theoretically, the time scales approach not only unifies calculus of differential and difference equations, but also solves other problems that are a mix of stop start and continuous behavior. Practically, the time scales calculus has a tremendous potential for application, for example, Thomas believes that time scales calculus is the best way to understand Thomas models populations of mosquitoes that carry West Nile virus [2]. In addition, Spedding have used this theory to model how students suffering from the eating disorder bulimia are influenced by their college friends; with the theory on time scales, they can model how the number of sufferers changes during the continuous college term as well as during long breaks [2]. By using the theory on time scales we can also study insect population, biology, heat transfer, stock market, epidemic models [2–6], and so forth. At the same time, motivated by the wide application of boundary value problems in physical and applied mathematics, boundary value problems for dynamic equations with p-Laplacian on time scales have received lots of interest [7–16].

In [7], Anderson et al. considered the following three-point boundary value problem with p-Laplacian on time scales:

(1.1)

where , and for some positive constants They established the existence results for at least one positive solution by using a fixed point theorem of cone expansion and compression of functional type.

For the same boundary value problem, He in [8] using a new fixed point theorem due to Avery and Henderson obtained the existence results for at least two positive solutions.

In [9], Sun and Li studied the following one-dimensional p-Laplacian boundary value problem on time scales:

(1.2)

where is a nonnegative rd-continuous function defined in and satisfies that there exists such that is a nonnegative continuous function defined on for some positive constants They established the existence results for at least single, twin, or triple positive solutions of the above problem by using Krasnosel'skii's fixed point theorem, new fixed point theorem due to Avery and Henderson and Leggett-Williams fixed point theorem.

For the Sturm-Liouville-like boundary value problem, in [17] Ji and Ge investigated a class of Sturm-Liouville-like four-point boundary value problem with p-Laplacian:

(1.3)

where By using fixed-point theorem for operators on a cone, they obtained some existence of at least three positive solutions for the above problem. However, to the best of our knowledge, there has not any results concerning the similar problems on time scales.

Motivated by the above works, in this paper we consider the following multi-point boundary value problem on time scales:

(1.4)

where and we denote with

In the following, we denote for convenience. And we list the following hypotheses:

• (C1) is a nonnegative continuous function defined on

• (C2) is rd-continuous with

2. Preliminaries

In this section, we provide some background material to facilitate analysis of problem (1.4).

Let the Banach space is rd-continuous be endowed with the norm and choose the cone defined by

(2.1)

It is easy to see that the solution of BVP (1.4) can be expressed as

(2.2)

If where

(2.3)

we define the operator by

(2.4)

It is easy to see , for and if then is the positive solution of BVP (1.4).

From the definition of for each we have and

In fact,

(2.5)

is continuous and nonincreasing in Moreover, is a monotone increasing continuously differentiable function,

(2.6)

then by the chain rule on time scales, we obtain

(2.7)

so,

For the notational convenience, we denote

(2.8)

Lemma 2.1.

is completely continuous.

Proof.

First, we show that maps bounded set into bounded set.

Assume that is a constant and Note that the continuity of guarantees that there exists such that . So

(2.9)

That is, is uniformly bounded. In addition, it is easy to see

(2.10)

So, by applying Arzela-Ascoli Theorem on time scales, we obtain that is relatively compact.

Second, we will show that is continuous. Suppose that and converges to uniformly on . Hence, is uniformly bounded and equicontinuous on . The Arzela-Ascoli Theorem on time scales tells us that there exists uniformly convergent subsequence in . Let be a subsequence which converges to uniformly on . In addition,

(2.11)

Observe that

(2.12)

Inserting into the above and then letting , we obtain

(2.13)

here we have used the Lebesgues dominated convergence theorem on time scales. 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.

The proof is complete.

Lemma 2.2.

Let then for and for

Proof.

Since , it follows that is nonincreasing. Hence, for ,

(2.14)

from which we have

(2.15)

For

(2.16)

we know

(2.17)

The proof is complete.

Lemma 2.3 ([18]).

Let be a cone in a Banach space Assum that are open subsets of with If

(2.18)

is a completely continuous operator such that either

• (i) or

• (ii)

Then has a fixed point in

3. Main Results

In this section, we present our main results with respect to BVP (1.4).

For the sake of convenience, we define number of zeros in the set , and number of in the set

Clearly, or 2 and there are six possible cases:

• (i)
• (ii)
• (iii)
• (iv)
• (v)
• (vi)

Theorem 3.1.

BVP (1.4) has at least one positive solution in the case and

Proof.

First, we consider the case and Since then there exists such that for where satisfies

(3.1)

If with then

(3.2)

It follows that if then for

Since then there exists such that for where is chosen such that

(3.3)

Set and

If with then

(3.4)

So that

(3.5)

In other words, if then Thus by of Lemma 2.3, it follows that has a fixed point in with .

Now we consider the case and Since there exists such that for , where is such that

(3.6)

If with then we have

(3.7)

Thus, we let so that for

Next consider By definition, there exists such that for , where satisfies

(3.8)

Suppose is bounded, then for all pick

(3.9)

If with then

(3.10)

Now suppose is unbounded. From condition it is easy to know that there exists such that for If with then by using (3.8) we have

(3.11)

Consequently, in either case we take

(3.12)

so that for we have Thus by (ii) of Lemma 2.3, it follows that has a fixed point in with

The proof is complete.

Theorem 3.2.

Suppose , and the following conditions hold,

• (C3): there exists constant such that for where

  

  (3.13)

• (C4): there exists constant such that for where

  

  (3.14)

furthermore, Then BVP (1.4) has at least one positive solution such that lies between and

Proof.

Without loss of generality, we may assume that

Let for any In view of we have

(3.15)

which yields

(3.16)

Now set for we have

(3.17)

Hence by condition we can get

(3.18)

So if we take then

(3.19)

Consequently, in view of (3.16), and (3.19), it follows from Lemma 2.3 that has a fixed point in Moreover, it is a positive solution of (1.4) and

The proof is complete.

For the case or we have the following results.

Theorem 3.3.

Suppose that and hold. Then BVP (1.4) has at least one positive solution.

Proof.

It is easy to see that under the assumptions, the conditions and in Theorem 3.2 are satisfied. So the proof is easy and we omit it here.

Theorem 3.4.

Suppose that and hold. Then BVP (1.4) has at least one positive solution.

Proof.

Since for there exists a sufficiently small such that

(3.20)

Thus, if , then we have

(3.21)

by the similar method, one can get if then

(3.22)

So, if we choose then for we have which yields condition in Theorem 3.2.

Next, by for there exists a sufficiently large such that

(3.23)

where we consider two cases.

Case 1.

Suppose that is bounded, say

(3.24)

In this case, take sufficiently large such that then from (3.24), we know for which yields condition in Theorem 3.2.

Case 2.

Suppose that is unbounded. it is easy to know that there is such that

(3.25)

Since then from (3.23) and (3.25), we get

(3.26)

Thus, the condition of Theorem 3.2 is satisfied.

Hence, from Theorem 3.2, BVP (1.4) has at least one positive solution.

The proof is complete.

From Theorems 3.3 and 3.4, we have the following two results.

Corollary 3.5.

Suppose that and the condition in Theorem 3.2 hold. Then BVP (1.4) has at least one positive solution.

Corollary 3.6.

Suppose that and the condition in Theorem 3.2 hold. Then BVP (1.4) has at least one positive solution.

Theorem 3.7.

Suppose that and hold. Then BVP (1.4) has at least one positive solution.

Proof.

In view of similar to the first part of Theorem 3.1, we have

(3.27)

Since for there exists a sufficiently small such that

(3.28)

Similar to the proof of Theorem 3.2, we obtain

(3.29)

The result is obtained, and the proof is complete.

Theorem 3.8.

Suppose that and hold. Then BVP (1.4) has at least one positive solution.

Proof.

Since similar to the second part of Theorem 3.1, we have for

By similar to the second part of proof of Theorem 3.4, we have where Thus BVP (1.4) has at least one positive solution.

The proof is complete.

From Theorems 3.7 and 3.8, we can get the following corollaries.

Corollary 3.9.

Suppose that and the condition in Theorem 3.2 hold. Then BVP (1.4) has at least one positive solution.

Corollary 3.10.

Suppose that and the condition in Theorem 3.2 hold. Then BVP (1.4) has at least one positive solution.

Theorem 3.11.

Suppose that and the condition of Theorem 3.2 hold. Then BVP (1.4) has at least two positive solutions such that

Proof.

By using the method of proving Theorems 3.1 and 3.2, we can deduce the conclusion easily, so we omit it here.

Theorem 3.12.

Suppose that and the condition of Theorem 3.2 hold. Then BVP (1.4) has at least two positive solutions such that

Proof.

Combining the proofs of Theorems 3.1 and 3.2, the conclusion is easy to see, and we omit it here.

4. Applications and Examples

In this section, we present a simple example to explain our result. When ,

(4.1)

where,

It is easy to see that the condition and are satisfied and

(4.2)

So, by Theorem 3.1, the BVP (4.1) has at least one positive solution.