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Existence of Positive Solutions for Multipoint Boundary Value Problem with Laplacian on Time Scales
Advances in Difference Equations volume 2009, Article number: 312058 (2009)
Abstract
We consider the existence of positive solutions for a class of secondorder multipoint boundary value problem with Laplacian on time scales. By using the wellknown Krasnosel'ski's fixedpoint 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 pLaplacian on time scales have received lots of interest [7–16].
In [7], Anderson et al. considered the following threepoint boundary value problem with pLaplacian on time scales:
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 onedimensional pLaplacian boundary value problem on time scales:
where is a nonnegative rdcontinuous 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 LeggettWilliams fixed point theorem.
For the SturmLiouvillelike boundary value problem, in [17] Ji and Ge investigated a class of SturmLiouvillelike fourpoint boundary value problem with pLaplacian:
where By using fixedpoint 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 multipoint boundary value problem on time scales:
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 rdcontinuous with
2. Preliminaries
In this section, we provide some background material to facilitate analysis of problem (1.4).
Let the Banach space is rdcontinuous be endowed with the norm and choose the cone defined by
It is easy to see that the solution of BVP (1.4) can be expressed as
If where
we define the operator by
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,
is continuous and nonincreasing in Moreover, is a monotone increasing continuously differentiable function,
then by the chain rule on time scales, we obtain
so,
For the notational convenience, we denote
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
That is, is uniformly bounded. In addition, it is easy to see
So, by applying ArzelaAscoli 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 ArzelaAscoli Theorem on time scales tells us that there exists uniformly convergent subsequence in . Let be a subsequence which converges to uniformly on . In addition,
Observe that
Inserting into the above and then letting , we obtain
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 ,
from which we have
For
we know
The proof is complete.
Lemma 2.3 ([18]).
Let be a cone in a Banach space Assum that are open subsets of with If
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
If with then
It follows that if then for
Since then there exists such that for where is chosen such that
Set and
If with then
So that
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
If with then we have
Thus, we let so that for
Next consider By definition, there exists such that for , where satisfies
Suppose is bounded, then for all pick
If with then
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
Consequently, in either case we take
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
which yields
Now set for we have
Hence by condition we can get
So if we take then
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
Thus, if , then we have
by the similar method, one can get if then
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
where we consider two cases.
Case 1.
Suppose that is bounded, say
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
Since then from (3.23) and (3.25), we get
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
Since for there exists a sufficiently small such that
Similar to the proof of Theorem 3.2, we obtain
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 ,
where,
It is easy to see that the condition and are satisfied and
So, by Theorem 3.1, the BVP (4.1) has at least one positive solution.
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Acknowledgments
This research is supported by the Natural Science Foundation of China (60774004), China Postdoctoral Science Foundation Funded Project (20080441126), Shandong Postdoctoral Funded Project (200802018), the Natural Science Foundation of Shandong (Y2007A27, Y2008A28), and the Fund of Doctoral Program Research of University of Jinan (B0621, XBS0843).
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Keywords
 Banach Space
 Eating Disorder
 West Nile Virus
 Fixed Point Theorem
 Epidemic Model