# Time-Scale-Dependent Criteria for the Existence of Positive Solutions to -Laplacian Multipoint Boundary Value Problem

- Wenyong Zhong
^{1}and - Wei Lin
^{2}Email author

**2010**:746106

https://doi.org/10.1155/2010/746106

© W. Zhong andW. Lin. 2010

**Received: **1 May 2010

**Accepted: **30 July 2010

**Published: **15 August 2010

## Abstract

By virtue of the Avery-Henderson fixed point theorem and the five functionals fixed point theorem, we analytically establish several sufficient criteria for the existence of at least two or three positive solutions in the -Laplacian dynamic equations on time scales with a particular kind of -Laplacian and -point boundary value condition. It is this kind of boundary value condition that leads the established criteria to be dependent on the time scales. Also we provide a representative and nontrivial example to illustrate a possible application of the analytical results established. We believe that the established analytical results and the example together guarantee the reliability of numerical computation of those -Laplacian and -point boundary value problems on time scales.

## 1. Introduction

The investigation of dynamic equations on time scales, originally attributed to Stefan Hilger's seminal work [1, 2] two decades ago, is now undergoing a rapid development. It not only unifies the existing results and principles for both differential equations and difference equations with constant time stepsize but also invites novel and nontrivial discussions and theories for hybrid equations on various types of time scales [3–11]. On the other hand, along with the significant development of the theories, practical applications of dynamic equations on time scales in mathematical modeling of those real processes and phenomena, such as the population dynamics, the economic evolutions, the chemical kinetics, and the neural signal processing, have been becoming richer and richer [12, 13].

As one of the focal topics in the research of dynamic equations on time scales, the study of boundary value problems for some specific dynamic equations on time scales recently has elicited a great deal of attention from mathematical community [14–33]. In particular, a series of works have been presented to discuss the existence of positive solutions in the boundary value problems for the second-order equations on time scales [14–21]. More recently, some analytical criteria have been established for the existence of positive solutions in some specific boundary value problems for the -Laplacian dynamic equations on time scales [22, 33].

Here and throughout, is supposed to be a time scale; that is, is any nonempty closed subset of real numbers in with order and topological structure defined in a canonical way. The closed interval in is defined as . Accordingly, the open interval and the half-open interval could be defined, respectively. In addition, it is assumed that , , , , and for some positive constants and . Moreover, is supposed to be the -Laplacian operator, that is, and , in which and . With these configurations and with the aid of the Avery-Henderson fixed point theorem [34], He established the criteria for the existence of at least two positive solutions in (1.1) fulfilling the boundary value conditions (1.2).

In the light of the five functionals fixed point theorem, they established the criteria for the existence of at least three solutions for the dynamic equation (1.1) either with conditions (1.4) or with conditions (1.5).

Here, , , , and . Indeed, Yaslan analytically established the conditions for the existence of at least two or three positive solutions in these boundary value problems by virtue of the Avery-Henderson fixed point theorem and the Leggett-Williams fixed point theorem [36]. It is worthwhile to mention that these theoretical results are novel even for some special cases on time scales, such as the conventional difference equations with fixed time stepsize and the ordinary differential equations.

It is clear that, together with conditions (1.10) and the above hypotheses ( )–( ), the dynamic equation (1.9) not only covers the corresponding boundary value problems in the literature, but even nontrivially generalizes these problems to a much wider class of boundary value problems on time scales. Also it is valuable to mention that condition (1.12) in hypothesis ( ) is necessarily relevant to the graininess operator around the time instant . Such kind of condition has not been required in the literature, to the best of authors' knowledge. Thus, this paper analytically establishes some new and time-scale-dependent criteria for the existence of at least double or triple positive solutions in the boundary value problems (1.9) and (1.10) by virtue of the Avery-Henderson fixed point theorem and the five functionals fixed point theorem. Indeed, these obtained criteria significantly extend the results existing in [26–28].

The remainder of the paper is organized as follows. Section 2 preliminarily provides some lemmas which are crucial to the following discussion. Section 3 analytically establishes the criteria for the existence of at least two positive solutions in the boundary value problems (1.9) and (1.10) with the aid of the Avery-Henderson fixed point theorem. Section 4 gives some sufficient conditions for the existence of at least three positive solutions by means of the five functionals fixed point theorem. More importantly, Section 5 provides a representative and nontrivial example to illustrate a possible application of the obtained analytical results on dynamic equations on time scales. Finally, the paper is closed with some concluding remarks.

## 2. Preliminaries

In this section, we intend to provide several lemmas which are crucial to the proof of the main results in this paper. However, for concision, we omit the introduction of those elementary notations and definitions, which can be found in [11, 12, 33] and references therein.

Lemma 2.1.

Proof.

To this end, it is not hard to check that satisfies (2.2), which implies that is a solution of the problems (2.1).

which consequently leads to the completion of the proof, that is, specified in (2.2) is the unique solution of the problems (2.1).

Lemma 2.2.

Proof.

which completes the proof.

Naturally, we denote that and that . With these settings, we have the following properties.

Lemma 2.3.

If then for any , for any pair of with .

for any . Then, through a standard argument [33], it is not hard to validate the following properties on this map.

Lemma 2.4.

Assume that the hypotheses are all fulfilled. Then, , and is completely continuous.

## 3. At Least Two Positive Solutions in Boundary Value Problems

In this section, we aim to adopt the well-known Avery-Henderson fixed point theorem to prove the existence of at least two positive solutions in the boundary value problems (1.9) and (1.10). For the sake of self-containment, we first state the Avery-Henderson fixed point theorem as follows.

Theorem 3.1 (see [34]).

and for all , for all , and and for all . Then, the operator has at least two fixed points, denoted by and , belonging to and satisfying with and with .

Hence, we are in a position to obtain the following results.

Theorem 3.2.

Proof.

for each . Also, notice that for and . Furthermore, from Lemma 2.4, it follows that the operator is completely continuous.

In what follows, we are to verify that all the conditions of Theorem 3.1 are satisfied with respect to the operator .

Thus, condition (i) in Theorem 3.1 is satisfied.

which consequently implies the validity of condition (ii) in Theorem 3.1.

Indeed, the validity of condition (iii) in Theorem 3.1 is verified.

According to Theorem 3.1, we consequently approach the conclusion that the boundary value problems (1.9) and (1.10) possess at least two positive solutions, denoted by and , satisfying with and with , respectively.

## 4. At Least Three Positive Solutions in Boundary Value Problems

In this section, we are to prove the existence of at least three positive solutions in the boundary value problems (1.9) and (1.10) by using the five functionals fixed point theorem which is attributed to Avery [35].

Theorem 4.1 (see [35]).

- (i)
- (ii)
- (iii)
- (iv)

Then the operator admits at least three fixed points , , satisfying , , and with , respectively.

With this theorem, we are now in a position to establish the following result on the existence of at least three solutions in the boundary value problems (1.9) and (1.10).

Theorem 4.2.

Proof.

for . Next, we intend to verify that all the conditions in Theorem 4.1 hold with respect to the operator .

This, with Lemma 2.4, clearly manifests that the operator is completely continuous.

This definitely verifies the validity of condition in Theorem 4.1.

Accordingly, the validity of condition ( ) in Theorem 4.1 is verified.

Therefore, both conditions ( ) and ( ) in Theorem 4.1 are satisfied. Consequently, by virtue of Theorem 4.1, the boundary value problems (1.9) and (1.10) have at least three positive solutions circumscribed on satisfying , , and with .

## 5. A Specific Example

In this section, we provide a representative and nontrivial example to clearly illustrate the feasibility of the time-scale-dependent results of dynamic equations with boundary value conditions that are obtained in the preceding section.

has at least three positive solutions defined on satisfying , , and with .

## 6. Concluding Remarks

In this paper, some novel and time-scale-dependent sufficient conditions are established for the existence of multiple positive solutions in a specific kind of boundary value problems on time scales. This kind of boundary value problems not only includes the problems discussed in the literature but also is adapted to more general cases. The well-known Avery-Henderson fixed point theorem and the five functionals fixed point theorem are adopted in the arguments.

It is valuable to mention that the writing form of the ending point of the interval on time scales should be accurately specified in dealing with different kind of boundary value conditions. Any inaccurate expression may lead to a problematic or incomplete discussion. Also it is noted that some other fixed point theorems and degree theories may be adapted to dealing with various boundary value problems on time scales. In addition, future directions for further generalization of the boundary value problem on time scales may include the generalization of the -Laplacian operator to increasing homeomorphism and homeomorphism, which has been investigated in [39] for the nonlinear boundary value of ordinary differential equations; the allowance of the function to change sign, which has been discussed in [31] and needs more detailed and rigorous investigations.

## Declarations

### Acknowledgments

This paper was supported by the NNSF of China (Grants nos. 10501008 and 60874121) and by the Rising-Star Program Foundation of Shanghai, China (Grant no. 07QA14002). The authors are grateful to the referee and editors for their very helpful suggestions and comments.

## Authors’ Affiliations

## References

- Aulbach B, Hilger S:
**Linear dynamic processes with inhomogeneous time scale.**In*Nonlinear Dynamics and Quantum Dynamical Systems (Gaussig, 1990), Mathematics Research*.*Volume 59*. Akademie, Berlin, Germany; 1990:9-20.Google Scholar - Hilger S:
**Analysis on measure chains—a unified approach to continuous and discrete calculus.***Results in Mathematics*1990,**18**(1-2):18-56.MathSciNetView ArticleMATHGoogle Scholar - Agarwal RP, Bohner M:
**Basic calculus on time scales and some of its applications.***Results in Mathematics*1999,**35**(1-2):3-22.MathSciNetView ArticleMATHGoogle Scholar - Agarwal RP, Bohner M, Rehak P:
**Half-linear dynamic equations.**In*Nonlinear Analysis and Applications: To V. Lakshmikantham on His 80th Birthday. Vol. 1, 2*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:1-57.View ArticleGoogle Scholar - Lakshmikantham V, Sivasundaram S, Kaymakcalan B:
*Dynamic Systems on Measure Chains, Mathematics and Its Applications*.*Volume 370*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1996:x+285.View ArticleMATHGoogle Scholar - Hoffacker J, Tisdell CC:
**Stability and instability for dynamic equations on time scales.***Computers & Mathematics with Applications*2005,**49**(9-10):1327-1334. 10.1016/j.camwa.2005.01.016MathSciNetView ArticleMATHGoogle Scholar - Zhong W, Lin W, Ruan J:
**The generalized invariance principle for dynamic equations on time scales.***Applied Mathematics and Computation*2007,**184**(2):557-565. 10.1016/j.amc.2006.06.056MathSciNetView ArticleMATHGoogle Scholar - Wu H, Zhou Z:
**Stability for first order delay dynamic equations on time scales.***Computers & Mathematics with Applications*2007,**53**(12):1820-1831. 10.1016/j.camwa.2006.09.011MathSciNetView ArticleMATHGoogle Scholar - Otero-Espinar V, Vivero DR:
**Uniqueness and existence results for initial value problems on time scales through a reciprocal problem and applications.***Computers & Mathematics with Applications*2009,**58**(4):700-710. 10.1016/j.camwa.2009.02.030MathSciNetView ArticleMATHGoogle Scholar - Anderson DR, Wong PJY:
**Positive solutions for second-order semipositone problems on time scales.***Computers & Mathematics with Applications*2009,**58**(2):281-291. 10.1016/j.camwa.2009.02.033MathSciNetView ArticleMATHGoogle Scholar - Bohner M, Peterson A (Eds):
*Advances in Dynamic Equations on Time Scales*. Birkhäuser, Boston, Mass, USA; 2003:xii+348.MATHGoogle Scholar - Bohner M, Peterson A:
*Dynamic Equations on Time Scales. An Introduction with Applications*. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleMATHGoogle Scholar - Atici FM, Biles DC, Lebedinsky A:
**An application of time scales to economics.***Mathematical and Computer Modelling*2006,**43**(7-8):718-726. 10.1016/j.mcm.2005.08.014MathSciNetView ArticleMATHGoogle Scholar - Akin E:
**Boundary value problems for a differential equation on a measure chain.***Panamerican Mathematical Journal*2000,**10**(3):17-30.MathSciNetMATHGoogle Scholar - Agarwal RP, O'Regan D:
**Triple solutions to boundary value problems on time scales.***Applied Mathematics Letters*2000,**13**(4):7-11. 10.1016/S0893-9659(99)00200-1MathSciNetView ArticleMATHGoogle Scholar - Agarwal RP, O'Regan D:
**Nonlinear boundary value problems on time scales.***Nonlinear Analysis: Theory, Methods & Applications*2001,**44**(4):527-535. 10.1016/S0362-546X(99)00290-4MathSciNetView ArticleMATHGoogle Scholar - Anderson DR:
**Solutions to second-order three-point problems on time scales.***Journal of Difference Equations and Applications*2002,**8**(8):673-688. 10.1080/1023619021000000717MathSciNetView ArticleMATHGoogle Scholar - Anderson DR: Nonlinear triple-point problems on time scales. Electronic Journal of Differential Equations 2004, (47):-12.Google Scholar
- Kaufmann ER: Positive solutions of a three-point boundary-value problem on a time scale. Electronic Journal of Differential Equations 2003, (82):-11.Google Scholar
- DaCunha JJ, Davis JM, Singh PK:
**Existence results for singular three point boundary value problems on time scales.***Journal of Mathematical Analysis and Applications*2004,**295**(2):378-391. 10.1016/j.jmaa.2004.02.049MathSciNetView ArticleMATHGoogle Scholar - He Z:
**Existence of two solutions of****-point boundary value problem for second order dynamic equations on time scales.***Journal of Mathematical Analysis and Applications*2004,**296**(1):97-109. 10.1016/j.jmaa.2004.03.051MathSciNetView ArticleMATHGoogle Scholar - Sun H-R, Li W-T:
**Multiple positive solutions for****-Laplacian****-point boundary value problems on time scales.***Applied Mathematics and Computation*2006,**182**(1):478-491. 10.1016/j.amc.2006.04.009MathSciNetView ArticleMATHGoogle Scholar - Sun H-R, Li W-T:
**Existence theory for positive solutions to one-dimensional****-Laplacian boundary value problems on time scales.***Journal of Differential Equations*2007,**240**(2):217-248. 10.1016/j.jde.2007.06.004MathSciNetView ArticleMATHGoogle Scholar - Su Y-H, Li W-T:
**Triple positive solutions of****-point BVPs for****-Laplacian dynamic equations on time scales.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(11):3811-3820. 10.1016/j.na.2007.10.018MathSciNetView ArticleMATHGoogle Scholar - He Z:
**Double positive solutions of three-point boundary value problems for****-Laplacian dynamic equations on time scales.***Journal of Computational and Applied Mathematics*2005,**182**(2):304-315. 10.1016/j.cam.2004.12.012MathSciNetView ArticleMATHGoogle Scholar - He Z, Li L:
**Multiple positive solutions for the one-dimensional****-Laplacian dynamic equations on time scales.***Mathematical and Computer Modelling*2007,**45**(1-2):68-79. 10.1016/j.mcm.2006.03.021MathSciNetView ArticleMATHGoogle Scholar - Yaslan İ:
**Multiple positive solutions for nonlinear three-point boundary value problems on time scales.***Computers & Mathematics with Applications*2008,**55**(8):1861-1869. 10.1016/j.camwa.2007.07.005MathSciNetView ArticleMATHGoogle Scholar - Yaslan İ:
**Existence of positive solutions for nonlinear three-point problems on time scales.***Journal of Computational and Applied Mathematics*2007,**206**(2):888-897. 10.1016/j.cam.2006.08.033MathSciNetView ArticleMATHGoogle Scholar - Anderson DR, Karaca IY:
**Higher-order three-point boundary value problem on time scales.***Computers & Mathematics with Applications*2008,**56**(9):2429-2443. 10.1016/j.camwa.2008.05.018MathSciNetView ArticleMATHGoogle Scholar - Anderson DR, Avery R, Henderson J:
**Existence of solutions for a one dimensional****-Laplacian on time-scales.***Journal of Difference Equations and Applications*2004,**10**(10):889-896. 10.1080/10236190410001731416MathSciNetView ArticleMATHGoogle Scholar - Sang Y, Su H, Xu F:
**Positive solutions of nonlinear****-point BVP for an increasing homeomorphism and homomorphism with sign changing nonlinearity on time scales.***Computers & Mathematics with Applications*2009,**58**(2):216-226. 10.1016/j.camwa.2009.03.106MathSciNetView ArticleMATHGoogle Scholar - Sun H-R:
**Triple positive solutions for****-Laplacian****-point boundary value problem on time scales.***Computers & Mathematics with Applications*2009,**58**(9):1736-1741. 10.1016/j.camwa.2009.07.083MathSciNetView ArticleMATHGoogle Scholar - Zhong W, Lin W:
**Existence of positive solutions in generalized boundary value problem for****-Laplacian dynamic equations on time scales.***Advances in Difference Equations*2009,**2009:**-19.Google Scholar - Avery RI, Henderson J:
**Two positive fixed points of nonlinear operators on ordered Banach spaces.***Communications on Applied Nonlinear Analysis*2001,**8**(1):27-36.MathSciNetMATHGoogle Scholar - Avery RI:
**A generalization of the Leggett-Williams fixed point theorem.***Mathematical Sciences Research Hot-Line*1999,**3**(7):9-14.MathSciNetMATHGoogle Scholar - Leggett RW, Williams LR:
**Multiple positive fixed points of nonlinear operators on ordered Banach spaces.***Indiana University Mathematics Journal*1979,**28**(4):673-688. 10.1512/iumj.1979.28.28046MathSciNetView ArticleMATHGoogle Scholar - Guseinov GSh:
**Integration on time scales.***Journal of Mathematical Analysis and Applications*2003,**285**(1):107-127. 10.1016/S0022-247X(03)00361-5MathSciNetView ArticleMATHGoogle Scholar - Cabada A, Vivero DR:
**Expression of the Lebesgue****-integral on time scales as a usual Lebesgue integral: application to the calculus of****-antiderivatives.***Mathematical and Computer Modelling*2006,**43**(1-2):194-207. 10.1016/j.mcm.2005.09.028MathSciNetView ArticleMATHGoogle Scholar - Bereanu C, Mawhin J:
**Existence and multiplicity results for some nonlinear problems with singular****-Laplacian.***Journal of Differential Equations*2007,**243**(2):536-557. 10.1016/j.jde.2007.05.014MathSciNetView ArticleMATHGoogle Scholar

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