Research Article | Open Access

# Positive Solutions for Third-Order -Laplacian Functional Dynamic Equations on Time Scales

**Academic Editor:**Hui-Sheng Ding

#### Abstract

We study the following third-order -Laplacian functional dynamic equation on time scales: , , , , , and . By applying the Five-Functional Fixed Point Theorem, the existence criteria of three positive solutions are established.

#### 1. Introduction

Recently, much attention has been paid to the existence of positive solutions for the boundary value problems with -Laplacian operator on time scales; for example, see [1–22] and the references therein. But, to the best of our knowledge, there is not much concerning -Laplacian functional dynamic equations on time scales [6, 12–14, 19, 21, 22], especially for the third-order -Laplacian functional dynamic equations on time scales [14, 22].

In [14], Song and Gao were concerned with the existence of positive solutions for the -Laplacian functional dynamic equation on time scales: where and is -Laplacian operator; that is, , , , , and is continuous; is left dense continuous (i.e., ) and does not vanish identically on any closed subinterval of , where denotes the set of all left dense continuous functions from to ; is continuous and ; is continuous, for all ; is continuous and satisfies the condition that there are such that The existence of two positive solutions to problem (1) was obtained by using a double fixed point theorem due to Avery et al. [23] in a cone.

In [22], Wang and Guan considered the existence of positive solutions to problem (1) by applying the well-known Leggett-Williams Fixed Point Theorem.

Motivated by [14, 22], we will show that problem (1) has at least three positive solutions by means of the Five-Functional Fixed Point Theorem [24] (which is a generalization of the Leggett-Williams Fixed Point Theorem [25]). It is worth noting that the Five-Functional Fixed Point Theorem is used extensively in yielding three solutions for BVPs of differential equations, difference equations, and/or dynamic equations on time scales; see [6, 26, 27] and references therein.

Throughout this work we assume knowledge of time scales and time-scale notation, first introduced by Hilger [28]. For more on time scales, please see the texts by Bohner and Peterson [29, 30].

In the remainder of this section, we state the following theorem, which is crucial to our proof.

Let , , be nonnegative, continuous, and convex functionals on and let , be nonnegative, continuous, and concave functionals on . Then, for nonnegative real numbers , , , , and , we define the convex sets

Theorem 1 (see [24]). *Let be a cone in a real Banach space . Suppose there exist positive numbers and ; nonnegative, continuous, and concave functionals and on ; and nonnegative, continuous, and convex functionals , , and on , with
**
for all . Suppose
**
is completely continuous and there exist nonnegative numbers , , , , with such that*(i)* and for ;*(ii)* and for ;*(iii)* for with ;*(iv)* for with .**Then has at least three fixed points such that
*

#### 2. Existence of Three Positive Solutions

We note that is a solution of BVP (1) if and only if

Let be endowed with , so is a Banach space. Define cone by

For each , extend to with for .

Define by We seek a point, , of in the cone . Define

Then denotes a positive solution of BVP (1).

We have the following results.

Lemma 2. *Let , and then*(1)* is completely continuous;*(2)* for ;*(3)* is decreasing ;*(4)* for and .*

*Proof. *(1)–(3) are Lemma 3.1 of [14]. It is easy to conclude that (4) is satisfied by the concavity of .

Let be fixed such that , and set

Throughout this paper, we assume and .

We define the nonnegative, continuous, and concave functionals , and the nonnegative, continuous, and convex functionals , , on the cone , respectively, as

We observe that for each .

In addition, by Lemma 2, we have . Hence for all .

For convenience, we define

We now state growth conditions on so that BVP (1) has at least three positive solutions.

Theorem 3. *Let , , and suppose that satisfies the following conditions: **, if , uniformly in , and , if , ;**, if , uniformly in ;**, if , uniformly in , and , if , .**Then BVP (1) has at least three positive solutions of the form
**
where , , and with .*

*Proof. *Let , and then , and consequently, for . Since , so , and this implies

From , we have

Therefore

We now turn to property (i) of Theorem 1. Choosing , , it follows that
which shows that , and, for , we have

From , we have

We conclude that (i) of Theorem 1 is satisfied.

We next address (ii) of Theorem 1. If we take , , then

From this we know that . If , then

From , we have

Now we show that (iii) of Theorem 1 is satisfied. If and , then

Finally, if and , then from (4) of Lemma 2 we have
which shows that condition (iv) of Theorem 1 is fulfilled.

Thus, all the conditions of Theorem 1 are satisfied. Hence, has at least three fixed points , , satisfying

Let
which are three positive solutions of BVP (1).

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors thank the referees and the editors for their helpful comments and suggestions. Research was supported by the Postdoctoral Fund in China (Grant no. 2013M531717), the Excellent Young Teacher Training Program of Lanzhou University of Technology (Grant no. Q200907), and the Natural Science Foundation of Gansu Province of China (Grant no. 1310RJYA080).

#### References

- D. 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 | MathSciNet - D. R. Anderson, “Existence of solutions for a first-order $p$-Laplacian BVP on time scales,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 69, no. 12, pp. 4521–4525, 2008. View at: Publisher Site | Google Scholar | MathSciNet - L. Bian, X. He, and H. Sun, “Multiple positive solutions of $m$-point BVPs for third-order $p$-Laplacian dynamic equations on time scales,”
*Advances in Difference Equations*, vol. 2009, Article ID 262857, 12 pages, 2009. View at: Publisher Site | Google Scholar | MathSciNet - A. Cabada, “Existence results for $p$-Laplacian boundary value problems on time scales,”
*Advances in Difference Equations*, vol. 11, Article ID 21819, 2006. View at: Google Scholar | MathSciNet - A. Cabada, “Discontinuous functional $p$-Laplacian boundary value problems on time scales,”
*International Journal of Difference Equations*, vol. 2, no. 1, pp. 51–60, 2007. View at: Google Scholar | MathSciNet - W. Guan, “Three positive solutions for $p$-Laplacian functional dynamic equations on time scales,”
*Electronic Journal of Qualitative Theory of Differential Equations*, No. 28, 7 pages, 2008. View at: Google Scholar | Zentralblatt MATH | MathSciNet - W. Han and S. Kang, “Multiple positive solutions of nonlinear third-order {BVP} for a class of $p$-Laplacian dynamic equations on time scales,”
*Mathematical and Computer Modelling*, vol. 49, no. 3-4, pp. 527–535, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z. M. 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 | 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 - Z. M. He and Z. Long, “Three positive solutions of three-point boundary value problems for
*p*-Laplacian dynamic equations on time scales,”*Nonlinear Analysis: Theory, Methods & Applications*, vol. 69, no. 2, pp. 569–578, 2008. View at: Publisher Site | Google Scholar - S. Hong, “Triple positive solutions of three-point boundary value problems for $p$-Laplacian dynamic equations on time scales,”
*Journal of Computational and Applied Mathematics*, vol. 206, no. 2, pp. 967–976, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. Song and C. Xiao, “Positive solutions for $P$-Laplacian functional dynamic equations on time scales,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 66, no. 9, pp. 1989–1998, 2007. View at: Publisher Site | Google Scholar | MathSciNet - C. X. Song and P. X. Weng, “Multiple positive solutions for $p$-Laplacian functional dynamic equations on time scales,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 68, no. 1, pp. 208–215, 2008. View at: Publisher Site | Google Scholar | MathSciNet - C. Song and X. Gao, “Positive solutions for third-order
*p*-Laplacian functional dynamic equations on time scales,”*Boundary Value Problems*, vol. 2011, Article ID 279752, 2011. View at: Publisher Site | Google Scholar - Y. Su, W. Li, and H. Sun, “Positive solutions of singular $p$-Laplacian {BVP}s with sign changing nonlinearity on time scales,”
*Mathematical and Computer Modelling*, vol. 48, no. 5-6, pp. 845–858, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. Sun, L. Tang, and Y. Wang, “Eigenvalue problem for $p$-Laplacian three-point boundary value problems on time scales,”
*Journal of Mathematical Analysis and Applications*, vol. 331, no. 1, pp. 248–262, 2007. View at: Publisher Site | Google Scholar | MathSciNet - H. Sun and W. Li, “Existence theory for positive solutions to one-dimensional $p$-Laplacian boundary value problems on time scales,”
*Journal of Differential Equations*, vol. 240, no. 2, pp. 217–248, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - D. B. Wang, “Existence, multiplicity and infinite solvability of positive solutions for $p$-Laplacian dynamic equations on time scales,”
*Electronic Journal of Differential Equations*, no. 96, 10 pages, 2006. View at: Google Scholar | MathSciNet - D. B. Wang, “Three positive solutions for $p$-Laplacian functional dynamic equations on time scales,”
*Electronic Journal of Differential Equations*, vol. 2007, no. 95, pp. 1–9, 2007. View at: Google Scholar | MathSciNet - D. B. Wang, “Three positive solutions of three-point boundary value problems for $p$-Laplacian dynamic equations on time scales,”
*Nonlinear Analysis: Theory, Methods &Applications*, vol. 68, no. 8, pp. 2172–2180, 2008. View at: Publisher Site | Google Scholar | MathSciNet - D. Wang and W. Guan, “Multiple positive solutions for $p$-Laplacian functional dynamic equations on time scales,”
*Taiwanese Journal of Mathematics*, vol. 12, no. 9, pp. 2327–2340, 2008. View at: Google Scholar | MathSciNet - D. B. Wang and W. Guan, “Multiple positive solutions for third-order
*p*-Laplacian functional dynamic equations on time scales,”*Advances in Difference Equations*, vol. 2014, article 145, 2014. View at: Publisher Site | Google Scholar - R. I. Avery, C. J. Chyan, and J. Henderson, “Twin solutions of boundary value problems for ordinary differential equations and finite difference equations,”
*Computers & Mathematics with Applications*, vol. 42, no. 3–5, pp. 695–704, 2001. View at: Publisher Site | Google Scholar | MathSciNet - R. Avery and J. Henderson, “Existence of three positive pseudo-symmetric solutions for a one-dimensional $p$-Laplacian,”
*Journal of Mathematical Analysis and Applications*, vol. 277, no. 2, pp. 395–404, 2003. View at: Publisher Site | Google Scholar | MathSciNet - R. W. Leggett and L. R. Williams, “Multiple positive fixed points of nonlinear operators on ordered Banach spaces,”
*Indiana University Mathematics Journal*, vol. 28, no. 4, pp. 673–688, 1979. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Li and J. Shen, “Existence of three positive solutions for boundary value problems with $p$-Laplacian,”
*Journal of Mathematical Analysis and Applications*, vol. 311, no. 2, pp. 457–465, 2005. View at: Publisher Site | Google Scholar | MathSciNet - D. Wang and W. Guan, “Three positive solutions of boundary value problems for $P$-Laplacian difference equations,”
*Computers & Mathematics with Applications*, vol. 55, no. 9, pp. 1943–1949, 2008. View at: Publisher Site | Google Scholar | MathSciNet - S. Hilger, “Analysis on measure chains: a unified approach to continuous and discrete calculus,”
*Results in Mathematics*, vol. 18, no. 1-2, pp. 18–56, 1990. View at: Publisher Site | Google Scholar | MathSciNet - M. Bohner and A. Peterson,
*Dynamic Equations on Time Scales*, An Introduction with Applications, Birkhäuser, Boston, Mass, USA, 2001. View at: Publisher Site | MathSciNet - M. Bohner and A. Peterson,
*Advances in Dynamic Equations on Time Scales*, Birkhäuser, Boston, Mass, USA, 2003.

#### Copyright

Copyright © 2014 Wen Guan and Da-Bin Wang. 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.