Research Article | Open Access

# Triple Positive Solutions for Third-Order -Point Boundary Value Problems on Time Scales

**Academic Editor:**Leonid Shaikhet

#### Abstract

We study the following third-order -point boundary value problems on time scales , , , , , where is an increasing homeomorphism and homomorphism and , . We obtain the existence of three positive solutions by using fixed-point theorem in cones. The conclusions in this paper essentially extend and improve the known results.

#### 1. Introduction

The theory of time scales was initiated by Hilger [1] as a mean of unifying and extending theories from differential and difference equations. The study of time scales has led to several important applications in the study of insect population models, neural networks, heat transfer, and epidemic models; see, for example [2–6]. Recently, the boundary value problems with -Laplacian operator have also been discussed extensively in the literature, for example, see [7–15].

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

In [16], Anderson considered the following third-order nonlinear boundary value problem (BVP): He used the Krasnoselskii and Leggett-Williams fixed-point theorems to prove the existence of solutions to the nonlinear boundary value problem.

In [9, 10], He considered the existence of positive solutions of the -Laplacian dynamic equations on time scales satisfying the boundary conditions or where . He obtained the existence of at least double and triple positive solutions of the problems by using a new double fixed point theorem and triple fixed point theorem, respectively.

In [15], Zhou and Ma firstly 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 the problem by using the monotone iterative technique.

However, to the best of our knowledge, little work has been done on the existence of positive solutions for the increasing homeomorphism and positive homomorphism operator on time scales. So the goal of the present paper is to improve and generate -Laplacian operator and establish some criteria for the existence of multiple positive solutions for the following third-order -point boundary value problems on time scales where is an increasing homeomorphism and homomorphism and , and satisfy

(), , () is continuous, and there exits such that , where .A projection is called an increasing homeomorphism and homomorphism, if the following conditions are satisfied:

(i)if , then ;(ii) is continuous bijection and its inverse mapping is also continuous;(iii)#### 2. Preliminaries and Lemmas

For convenience, we list the following definitions which can be found in [1–5].

*Definition 2.1. *A time scale is a nonempty closed subset of real numbers . For and , define the forward jump operator and backward jump operator , respectively, by
for 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 it exists) with the property that for each , there is a neighborhood of such that
for all .

For and , the nabla derivative of at , denoted by (provided it exists) with the property that for each , there is a neighborhood of such that
for 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 we define the delta integral by
If , then we define the nabla integral by

*Definition 2.5. *Let be a real Banach space over . A nonempty closed set is said to be a cone provided that

(i), implies ; (ii) implies .

*Definition 2.6. *Given a cone in a real Banach space , a functional is said to be increasing on , provided , for all with .

*Definition 2.7. *Given a cone in a real Banach space , we define for each the set

*Definition 2.8. *A map is called 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 called nonnegative continuous concave functional on a cone of a real Banach space if is continuous and
for all and .

Let be nonnegative continuous convex functionals on , let be a nonnegative continuous concave functional on , and let be a nonnegative continuous functional on . For nonnegative real numbers , and we define the following convex set:

Theorem 2.9 ([17]). *Let be a cone in a real Banach space . Let and be nonnegative continuous convex functionals on , let be a nonnegative continuous concave functional on , and let be a nonnegative continuous functional on satisfying for , such that for some positive numbers and , and for all . Suppose that is a completely continuous operator and there exist nonnegative numbers , and with such that*

(i)* and for ;*(ii)* for with ;*(iii)* and for with .**Then has at least three fixed points satisfying
*

Theorem 2.10 ([18]). *Let be a bounded closed convex subset of a Banach space . Assume that are disjoint closed convex subsets of and are nonempty open subsets of with and . Suppose that is completely continuous and the following conditions hold:*

(i)*;*(ii)* has no fixed points in .**Then has at least three points such that , and .*

Lemma 2.11. *If condition holds, then for , the boundary value problem (BVP)
**
has the unique solution
*

*Proof. *By caculating, we can easily get (2.12). So we omit it.

Lemma 2.12. *If condition holds, then for , the boundary value problem (BVP)
**
has the unique solution
**
where , is the inverse function to .*

*Proof. *Integrating both sides of equation in (2.13) on , we have
So,
By boundary value condition , we have
By (2.15) and (2.17) we know
This together with Lemma 2.11 implies that
where . The proof is complete.

Lemma 2.13. *Let condition hold. If and , then the unique solution of (2.13) satisfies
*

*Proof. *By , we can know that the graph of is concave down on and is nonincreasing on . This together with the assumption that the boundary condition is implies that for . This implies that
So we only prove By condition we have
The proof is completed.

#### 3. Triple Positive Solutions

In this section, some existence results of positive solutions to BVP (1.6) are established by imposing some conditions on and defining a suitable Banach space and a cone.

Let be endowed with the ordering if for all and is defined as usual by maximum norm. Clearly, it follows that is a Banach space.

We define a cone by Let and fix such that and define the nonnegative continuous convex functionals and , the nonnegative continuous concave functional , and the nonnegative continuous functional on the cone by For notational convenience, denote

Lemma 3.1 ([9]). *If , then*

(1)* for all ;*(2)* for with .**Define an operator by
**
where . Then, is a solution of boundary value problem (1.6) if and only if is a fixed point of operator . Obviously, for one has for . In addition, for and , this implies . With standard argument one may show that is completely continuous.*

Theorem 3.2. *Suppose conditions and hold, and there exist positive numbers such that*

()*;*()*;*()*.**Then, the BVP (1.6) has at least three positive solutions satisfying
*

*Proof. *Based on Lemma 3.1, it is clear that for and , there are and . Furthermore, and therefore .

Take , then . By means of () one derives
Thus .

Set and , it follows that
which means

For , we have for . By condition () we have
So, (i) of Theorem 2.9 is fulfilled.

If and , then due to of Lemma 3.1
Therefore, (ii) of Theorem 2.9 is fulfilled.

Take and , then , it then follows from () that
As a result, all the conditions of Theorem 2.9 are verified. This completes the proof.

Theorem 3.3. *Suppose that conditions () and () hold. Let and assume that the following conditions are satisfied:*

()*, ;*()* there exists a number such that , ;*()*, .**Then, the BVP (1.6) has at least three positive solutions , and such that
**
Where for real number , is continuous, , for .*

*Proof. *We first show that if condition () holds. If , then , which implies . We have
This implies that .

Next, condition () indicates that there exists such that . Now we let
where is the interior of . Then we have . Moreover, means . Thus has no fixed point in .

To show and has no fixed point in , set , following the definition of , we can know , for . Condition () then gives rise to , which in turn produces
Combining the above two inequalities one achieves , for . That is, . So and has no fixed point in . Therefore, all conditions of Theorem 2.10 are fulfilled, and the BVP (1.6) has at least three positive solutions , and such that

#### 4. Some Examples

In the section, we present some simple examples to explain our results. We only study the case .

*Example 4.1. *Consider the following third-order three-point boundary value problem:
where .

We choose , by computing we can know . Let , then . Obviously, . We define a nonlinearity as follows: Then, by the definition of , we have

(i);(ii);(iii).By Theorem 3.2, BVP (4.1) has at least three positive solutions.

*Example 4.2. *Consider the following third-order three-point boundary value problem:
where .

By computing, we can know . Let , then . Obviously, . We define a nonlinearity as follows: Then, by the definition of , we have

(i);(ii)and there exists such that ;(iii).By Theorem 3.3, BVP (4.3) has at least three positive solutions.

*Remark 4.3. *Consider following nonlinear m-point boundary value problem:
where
and satisfy the conditions () and (). It is clear that is an increasing homeomorphism and homomorphism and . Because -Laplacian operators are odd, they do not apply to our example. Hence we generalize boundary value problem with -Laplacian operator, and the results [8–11, 13–15] do not apply to the example.

*Remark 4.4. *In a similar way, we can get the corresponding results for the following boundary value problem:

#### Acknowledgment

The project was supported by the National Natural Science Foundation of China (10471075).

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#### Copyright

Copyright © 2009 Jian Liu and Fuyi Xu. 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.