Research Article | Open Access

Yanbin Sang, Hua Su, Yafeng Xiao, "Several Existence Theorems of Nonlinear -Point BVP for an Increasing Homeomorphism and Homomorphism on Time Scales", *Discrete Dynamics in Nature and Society*, vol. 2008, Article ID 252876, 13 pages, 2008. https://doi.org/10.1155/2008/252876

# Several Existence Theorems of Nonlinear -Point BVP for an Increasing Homeomorphism and Homomorphism on Time Scales

**Academic Editor:**Bing Zhang

#### Abstract

Several existence theorems of positive solutions are established for nonlinear -point boundary value problem for the following dynamic equations on time scales , , , , where is an increasing homeomorphism and homomorphism and . As an application, an example to demonstrate our results is given.

#### 1. Introduction

In this paper, we study the existence of positive solutions of the following dynamic equations on time scales: where is an increasing homeomorphism and homomorphism and .

A projection is called an increasing homeomorphism and homomorphism if the following conditions are satisfied:

(i) if , then , for all ;(ii) is a continuous bijection and its inverse mapping is also continuous;(iii), for all .

We will assume that the following conditions are satisfied throughout this paper:

(H_{1}) satisfy ,
and (H_{2}) and there exists ,
such that (H_{3}).

Recently, there is much attention focused on the existence of positive solutions for second-order, three-point boundary value problem on time scales. On the other hand, three-point and -point boundary value problems with -Laplacian operators on time scales have also been studied extensively, for details, see [1–11] and references therein. But with an increasing homeomorphism and homomorphism, few works were done as far as we know.

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

We would like to mention some results of Anderson et al. [2], He [4, 5], Sun and Li [9], Ma et al. [12], Wang and Hou [13], Wang and Ge [14], which motivate us to consider our problem.

In [2], Anderson et al. considered the following problem: where , and for some positive constants . They established the existence results of at least one positive solution by using a fixed point theorem of cone expansion and compression of functional type.

In [4, 5], He considered the existence of positive solutions of the -Laplacian dynamic equations on time scales: satisfying the boundary conditions or where , and for some positive constants . He obtained the existence of at least double and triple positive solutions of the problem (1.3), (1.4), and (1.5) by using a new double fixed point theorem and triple fixed point theorem, respectively.

In recent papers, Ma et al. [12] have obtained the existence of monotone positive solutions for the following BVP: The main tool is the monotone iterative technique.

In [9], Sun and Li studied the following -Laplacian, -point BVP on time scales: where for . Some new results are obtained for the existence of at least twin or triple positive solutions of the problem (1.7) by applying Avery-Henderson and Leggett-Williams fixed point theorems, respectively.

In [15], Sang and xi investigated the existence of positive solutions of the -Laplacian dynamic equations on time scales: where is -Laplacian operator, that is, Let they mainly obtained the following results.

Theorem 1.1. *Assume (H _{1}), (H_{2}),
and (H_{3}) hold, and assume that one of the following
conditions holds:*

(H

_{4})

*(H*

*there exist**with*such that_{5})

*there exist with such that**Then, (1.8)*

*have a positive solution.*In this paper, we will establish two new theorems of positive solution of (1.8), our work concentrates on the case when the nonlinear term does not satisfy the conditions of Theorem 1.1. At the end of the paper, we will give an example which illustrates that our work is true.

#### 2. Preliminaries and Some Lemmas

For convenience, we list the following definitions which can be found in [16–19].

*Deffinition. *A time scale **T** 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 .

*Deffinition. *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 is the number ,
(provided it exists), with the property that for each ;
there is a neighborhood of such that for all .

*Deffinition. *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 .

*Deffinition. *If ,
then we define the delta integral by If ,
then we define 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:

We can prove the following lemmas by the methods of
[15].

Lemma 2.5. *For ,
the BVP
* (2.6) *
has the unique solution:** where*

Lemma 2.6. *Assume* (H_{1}) *holds,
For and ,
then the unique solution of* (2.6) *satisfies*

Lemma 2.7. *Assume* (H_{1}) *holds, if and ,
then the unique solution of* (2.6) *satisfies** where *

Let the norm on be the maximum norm. Then, the is a Banach space. It is easy to see that the BVP (1.1) has a solution if and only if is a fixed point of the operator equation: where Throughout this paper, we will assume that . Denote where 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 [20], we can obtain that is relatively compact. In view of Lebesgue's dominated convergence theorem on time scales [21], it is easy to prove that is continuous. Hence, is completely continuous.

Lemma 2.8. *Let** for ,
then *

Lemma 2.9 ([22]). *Let be a Banach space, and let be a cone. Assume are open bounded subset 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. Let

For , By Lemma 2.6, and are well defined.

#### 3. Existence Theorems of Positive Solution

Theorem 3.1. *Assume* (H_{1}), (H_{2}), (H_{3})
*hold, and assume that the following conditions
hold:*

(A_{1})* and *(A

_{2})(A

_{3})

*there exist , such that*

*Then, the problems*(1.1)*have at least one positive solution.*Theorem 3.2. *Assume* (H_{1}), (H_{2}), (H_{3})
*hold, and assume that the following conditions
hold:*

(B_{1})* and *(B

_{2})

*;*(B

_{3})

*there exist , such that*

*Then, the problems*(1.1)*have at least one positive solution.**Proof of Theorem 3.1. *By ,
then we can get that there exist such that If ,
then .
By condition (A_{3}),
we have

Let so that

Therefore, It follows that Noticing ,
we have Therefore,
there exist such that .
It implies that .

On the other hand, by ,
we can get that there exist ,
and such that then .

Let .
By Lemma 2.8 and condition ,
applying Lemma 2.8, it follows that It follows that Noticing ,
we get

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

By Lemma 2.9, we assert that the operator has one fixed point such that .
Therefore, is positive solution of the problems (1.1).

*Proof of Theorem 3.2. *By ,
then we can get that there exist such that For ,
then .
By condition (B_{3}),
we have

Let so that Denote satisfying Therefore, It follows that Noticing ,
we have Therefore,
there exists such that .
It implies that .

On the other hand, by ,
we can get that there exist ,
and such that then .

Let .
By Lemma 2.8 and condition (B_{3}),
applying Lemma 2.8, it follows that It follows that Noticing ,
we can get Therefore,
there exists
with such that .
It implies that for .

By Lemma 2.9, we assert that the operator has one fixed point such that .
Therefore, is positive solution of the problems (1.1).

#### 4. Example

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

Let . Consider the following BVP: where It is easy to check that is continuous. In this case, , it follows from a direct calculation that Choose it is easy to check that It follows that satisfies the conditions of Theorem 3.1, then problems (1.1) have at least one positive solution.

#### References

- D. R. Anderson, “Solutions to second-order three-point problems on time scales,”
*Journal of Difference Equations and Applications*, vol. 8, no. 8, pp. 673–688, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - D. R. Anderson, R. Avery, and J. Henderson, “Existence of solutions for a one dimensional $p$-Laplacian
on time-scales,”
*Journal of Difference Equations and Applications*, vol. 10, no. 10, pp. 889–896, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. J. DaCunha, J. M. Davis, and P. K. Singh, “Existence results for singular three point boundary value problems on time scales,”
*Journal of Mathematical Analysis and Applications*, vol. 295, no. 2, pp. 378–391, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z. He, “Double positive solutions of three-point boundary value problems for $p$-Laplacian dynamic equations on time scales,”
*Journal of Computational and Applied Mathematics*, vol. 182, no. 2, pp. 304–315, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z. He and X. Jiang, “Triple positive solutions of boundary value problems for $p$-Laplacian dynamic equations on time scales,”
*Journal of Mathematical Analysis and Applications*, vol. 321, no. 2, pp. 911–920, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. R. Kaufmann, “Positive solutions of a three-point boundary-value problem on a time scale,”
*Electronic Journal of Differential Equations*, vol. 2003, no. 82, pp. 1–11, 2003. View at: Google Scholar | Zentralblatt MATH | MathSciNet - H. Luo and Q. Ma, “Positive solutions to a generalized second-order three-point boundary-value problem on time scales,”
*Electronic Journal of Differential Equations*, vol. 2005, no. 17, pp. 1–14, 2005. View at: Google Scholar | Zentralblatt MATH | MathSciNet - H. Su, Z. Wei, and F. Xu, “The existence of countably many positive solutions for a system of nonlinear singular boundary value problems with the $p$-Laplacian operator,”
*Journal of Mathematical Analysis and Applications*, vol. 325, no. 1, pp. 319–332, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H.-R. Sun and W.-T. Li, “Multiple positive solutions for $p$-Laplacian $m$-point boundary value problems on time scales,”
*Applied Mathematics and Computation*, vol. 182, no. 1, pp. 478–491, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H.-R. Sun and W.-T. Li, “Positive solutions for nonlinear three-point boundary value problems on time scales,”
*Journal of Mathematical Analysis and Applications*, vol. 299, no. 2, pp. 508–524, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H.-R. Sun and W.-T. Li, “Positive solutions for nonlinear $m$-point boundary value problems on time scales,”
*Acta Mathematica Sinica*, vol. 49, no. 2, pp. 369–380, 2006 (Chinese). View at: Google Scholar | Zentralblatt MATH | MathSciNet - D.-X. Ma, Z.-J. Du, and W.-G. Ge, “Existence and iteration of monotone positive solutions for multipoint boundary value problem with $p$-Laplacian operator,”
*Computers & Mathematics with Applications*, vol. 50, no. 5-6, pp. 729–739, 2005. View at: Google Scholar | Zentralblatt MATH | MathSciNet - Y. Wang and C. Hou, “Existence of multiple positive solutions for one-dimensional
$p$-Laplacian,”
*Journal of Mathematical Analysis and Applications*, vol. 315, no. 1, pp. 144–153, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Wang and W. Ge, “Positive solutions for multipoint boundary value problems with a one-dimensional $p$-Laplacian,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 66, no. 6, pp. 1246–1256, 2007. View at: Google Scholar | Zentralblatt MATH | MathSciNet - Y. Sang and H. Xi, “Positive solutions of nonlinear