Discrete Dynamics in Nature and Society

VolumeΒ 2008Β (2008), Article IDΒ 257680, 19 pages

http://dx.doi.org/10.1155/2008/257680

## Twin Positive Solutions of a Nonlinear -Point Boundary Value Problem for Third-Order -Laplacian Dynamic Equations on Time Scales

^{1}Department of Mathematics, North University of China, Taiyuan 030051, China^{2}Department of Mathematics, School of Science, Tianjin University of Commerce, Tianjin 300134, China

Received 20 April 2008; Accepted 20 June 2008

Academic Editor: BinggenΒ Zhang

Copyright Β© 2008 Wei Han and Guang Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Several existence theorems of twin positive solutions are established for a nonlinear -point boundary value problem of third-order -Laplacian dynamic equations on time scales by using a fixed point theorem. We present two theorems and four corollaries which generalize the results of related literature. As an application, an example to demonstrate our results is given. The obtained conditions are different from some known results.

#### 1. Introduction

A time scale is a nonempty closed subset of . We make the blanket assumption that and are points in . By an interval , we always mean the intersection of the real interval with the given time scale, that is, .

In this paper, we will be concerned with the existence of positive solutions of the -Laplacian dynamic equations on time scales:where is -Laplacian operator; that is, , and

(H_{1})
satisfy and (H_{2}) and there exists such that (H_{3}).We point out that the -derivative and the -derivative in (1.2) and the space in are defined in Section 2.

Recently, there has been much attention paid to the existence of positive solutions for third-order nonlinear boundary value problems of differential equations. For example, see [1β10] and the listed references. Anderson [2] considered the following third-order nonlinear problem:He used the Krasnoselskii and the Leggett and Williams fixed-point theorems to prove the existence of solutions to the nonlinear problem (1.3). Li [6] considered the existence of single and multiple positive solutions to the nonlinear singular third-order two-point boundary value problem:Under various assumptions on and , they established intervals of the parameter which yield the existence of at least two and infinitely many positive solutions of the boundary value problem by using Krasnoselski's fixed-point theorem of cone expansion-compression type. Liu et al. [7] discussed the existence of at least one or two nondecreasing positive solutions for the following singular nonlinear third-order differential equations:Green's function and the fixed-point theorem of cone expansion-compression type are utilized in their paper. In [8], Sun considered the following nonlinear singular third-order three-point boundary value problem:He obtained various results on the existence of single and multiple positive solutions to the boundary value problem (1.6) by using a fixed-point theorem of cone expansion-compression type due to Krasnosel'skii. In [10], Zhou and Ma studied the existence and iteration of positive solutions for the following third-order generalized right-focal boundary value problem with -Laplacian operator:They established a corresponding iterative scheme for (1.7) by using the monotone iterative technique.

On the other hand, the existence of positive solutions for third-order nonlinear boundary value problems of difference equations is also extensively studied by a number of authors (see [1, 3, 5, 9] and the listed references). The present work is motivated by a recent paper [4]. In [4], Henderson and Yin considered the existence of solutions for a third-order boundary value problem on a time-scale equation of the formwhich is uniform for the third-order difference equation and the third-order differential equation.

#### 2. Preliminaries and Lemmas

For convenience, we list the following definitions which can be found in [4, 11β15].

*Definition 2.1. *Let be a time scale. For and ,
define the forward jump operator and the backward jump operator ,
respectively, byfor all .
If , is said to be right-scattered, and if is said to be left-scattered; if , is said to be right-dense, and if is said to be left-dense. If has a right-scattered minimum ,
define ;
otherwise set .
If has a left-scattered maximum ,
define ;
otherwise set .

*Definition 2.2. *For and ,
the delta derivative of at the point is defined to be the number (provided that it exists), with the property
that for each there is a neighborhood of such thatfor all .

For and , the nabla derivative of at is denoted by (provided that it exists), with the property that for each there is a neighborhood of such thatfor all .

*Definition 2.3. *A function is left-dense continuous (i.e., -continuous) if is continuous at each left-dense point in ,
and its right-sided limit exists at each right-dense point in .

*Definition 2.4. *If ,
then one defines the delta integral by
If ,
then one defines the nabla integral by

To prove the main results in this paper, we will employ several lemmas. These lemmas are based on the linear BVP

Lemma 2.5. *If and ,
then for the BVP* (2.6)-(2.7) *has the unique solution**where *

*Proof. *(i) Let
be a solution, then we will show that (2.8)
holds. By taking the nabla integral of problem (2.6) on ,
we havethenBy taking the nabla integral of
(2.11) on ,
we can getBy taking the delta integral of
(2.12) on ,
we can getSimilarly, let on (2.10), then we have ;
let on (2.10), then we have
Let on (2.12), then we haveLet on (2.13), then we haveSimilarly, let on (2.13), then we haveBy the boundary condition (2.7),
we can getSolving
(2.19), we getBy the boundary condition (2.7),
we can obtainSubstituting (2.20) in the above
expression, one has

(ii) We show that the
function given in (2.8) is a solution.

Let be as in (2.8). By [12, Theorem 2.10(iii)] and
taking the delta derivative of (2.8), we havemoreover, we get

Taking the nabla derivative of
this expression yields .
Also, routine calculation verifies that satisfies the boundary value conditions in
(2.7) so that given in (2.8) is a solution of (2.6) and
(2.7). The proof is complete.

Lemma 2.6. *Assume*β
β
*holds. For and ,
the unique solution of* (2.6) *and* (2.7) *satisfies*

*Proof. *LetSincethen .

According to Lemma 2.5, we getIf ,
we have
So .

Lemma 2.7. *Assume*β
β*holds. If and ,
then the unique solution of* (2.6) *and* (2.7) *satisfies**where *

*Proof. *
It is easy to
check that ;
this implies thatIt is easy to see that for any with .
Hence, is a decreasing function on .
This means that the graph of is concave down on .

For each ,
we havethat is,so thatWith the boundary condition ,
we haveThis completes the proof.

Let the norm on be the maximum norm. Then, the is a Banach space. It is easy to see that BVP (1.1)-(1.2) has a solution if and only if is a fixed point of the operatorwhereDenotewhere is the same as in Lemma 2.7. It is obvious that is a cone in . By Lemma 2.7, . So by applying Arzela-Ascoli theorem on time scales [16], we can obtain that is relatively compact. In view of Lebesgue's dominated convergence theorem on time scales [13], it is easy to prove that is continuous. Hence, is completely continuous.

Lemma 2.8. * is completely continuous.*

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

Assume is a constant and .
Note that the continuity of guarantees that there is such that for .
SoThat is, is uniformly bounded.

In addition, notice that for any ,
we haveSo, by applying Arzela-Ascoli
theorem on time scales [16], we obtain that is relatively compact.

Finally, we prove that is continuous. Suppose that and converges to uniformly on Hence, is uniformly bounded and equicontinuous on The Arzela-Ascoli theorem on time scales [16]
tells us that there exists uniformly convergent subsequence in Let be a subsequence which converges to uniformly on In addition,Observe the expression of and then letting we obtainwhere Here, we have used the Lebesgue
dominated convergence theorem on time scales [13]. 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 proof is
complete.

Lemma 2.9. *Let**For , *

*Proof. *
Sincethen .
For all ,
we haveIn fact, let ;
taking the nabla derivative of this expression, we haveHence, is a nondecreasing function on .
That is,For all ,By (2.51), for ,
we have

Lemma 2.10 (See [17]). *Let be a Banach space, and let be a cone. Assume are open bounded
subsets of with and let** be a completely continuous operator such that *

(i)* and ,
or*(ii)*,
and **Then, has a fixed point in *

Now, we introduce the following notations. LetFor , and ,by Lemma 2.6, where and are well defined.

#### 3. Main Results

Theorem 3.1. *Assume*β
, , and
β*hold, and assume that the following conditions
hold:*

(A_{1})*, and *(A_{2})*;*(A_{3})* there exist and such that *(A

_{4})

*there exists such that*

*Then, problem*(1.1)-(1.2)*has at least two positive solutions satisfying .*Theorem 3.2. *Assume*β
, ,
and
β*hold, and assume that the following conditions
hold:*

(B_{1})*, and *(B_{2})*; *(B_{3})* there exist and such that *(B

_{4})

*there exists such that*

*Then, problem*(1.1)-(1.2)*has at least two positive solutions satisfying .**Proof of Theorem 3.1. *Letthen there exist and such thatIf ,
then .
By condition ,
we haveso thatTherefore,It follows thatNoticing ,
we haveTherefore, there exist such that .
It implies that .

If and ,
then .
Similar to the above argument, noticing that ,
we can get .
Therefore, there exist such that .
It implies that .

On the other hand, since is continuous, by condition ,
there exist such thatIf ,
then .
Applying Lemma 2.9, it follows that
In the same way, we
can prove
that if ,
then .

Now, we consider the operator on and , respectively. By Lemma 2.10, we assert that the operator has two fixed points such that and . Therefore, , are positive solutions of problem (1.1)-(1.2).

*Proof of Theorem 3.2. *Letthen there exist and such that If ,
then .
By Lemma 2.9 and condition ,
we haveIt follows thatNoticing ,
we getTherefore, there
exists
with such that .
It implies that for .

If and ,
then .
Similar to the above argument, noticing that ,
we can get .

Therefore, there exist
with such that .
It implies that for .

By condition ,
we can see that there exist such thatIf ,
then , ,
and .
It follows thatSimilarly, if ,
then .

Now, we study the operator on and , respectively. By Lemma 2.10, we assert that the operator has two fixed points such that and . Therefore, , are positive solutions of problem (1.1)-(1.2).

#### 4. Further Discussion

If the conditions of Theorems 3.1 and 3.2 are weakened, we will get the existence of single positive solution of problem (1.1)-(1.2).

Corollary 4.1. *Assume*β
, ,
and
β*hold, and assume that the following conditions
hold:*

(C_{1})*,
and *(C_{2})*;*(C_{3})* there exist and such that *(C

_{4})

*there exists such that*

*Then, problem*(1.1)-(1.2)*has at least one positive solution.*Corollary 4.2. *Assume*β
, ,
and
β*hold, and assume that the following conditions
hold:*

(D_{1})*, and *(D_{2})*;*(C_{3})* there exist and such that *(D

_{4})

*there exists such that*

*Then, problem*(1.1)-(1.2)*has at least one positive solution.*Corollary 4.3. *Assume*β
, ,
and
β *hold, and assume that the following conditions
hold:*

(E_{1})*, and *(E_{2})*;*(E_{3})* there exist and such that *(E

_{4})

*there exists such that*

*Then, problem*(1.1)-(1.2)*has at least one positive solution.*Corollary 4.4. *Assume*β
, ,
and
β*hold, and assume that the following conditions
hold:*

(F_{1})*, and *(F_{2})*;*(F_{3})* there exist and such that *(F

_{4})

*there exists such that*

*Then, problem*(1.1)-(1.2)*has at least one positive solution.*The proof of the above results is similar to those of Theorems 3.1 and 3.2; thus we omit it.

#### 5. Some Examples

In this section, we present a simple example to explain our results. We only study the case .

Let . Consider the following BVP: whereIt is easy to check that is continuous. In this case, , and , and it follows from a direct calculation thatWe haveChoosing , and , it is easy to check thatIt follows that satisfies the conditions of Theorem 3.1; then problem (5.1) has at least two positive solutions.

#### Acknowledgment

Project supported by the National Natural Science Foundation of China (Grant NO. 10471040).

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