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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 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:

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:

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:

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:

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

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 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,

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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Zhang, M., Sun, S. & Han, Z. Existence of Positive Solutions for Multipoint Boundary Value Problem with -Laplacian on Time Scales.
Adv Differ Equ 2009, 312058 (2009). https://doi.org/10.1155/2009/312058
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DOI: https://doi.org/10.1155/2009/312058
Keywords
- Banach Space
- Eating Disorder
- West Nile Virus
- Fixed Point Theorem
- Epidemic Model