Research Article | Open Access

# Three-Point Boundary Value Problems of Nonlinear Second-Order -Difference Equations Involving Different Numbers of

**Academic Editor:**Jin L. Kuang

#### Abstract

We study a new class of three-point boundary value
problems of nonlinear second-order *q*-difference equations. Our
problems contain different numbers of *q* in derivatives and
integrals. By using a variety of fixed point theorems (such as Banach’s contraction principle, Boyd and Wong fixed point theorem for nonlinear contractions, Krasnoselskii’s fixed point theorem, and Leray-Schauder nonlinear
alternative) and Leray-Schauder degree theory, some new existence and uniqueness
results are obtained. Illustrative examples
are also presented.

#### 1. Introduction

The -difference calculus or quantum calculus is an old subject that was initially developed by Jackson [1], Carmichael [2], Mason [3], and Adams [4], in the first quarter of 20th century, has been developed over the years, for instance, see [5–14] and the references therein. In fact, -calculus has a rich history, and the details of its basic notions, results, and methods can be found in the text [15]. In recent years, the topic has attracted the attention of several researchers, and a variety of new results can be found in the papers [16–28] and the references cited therein.

In [24], Ahmad et al*.* studied a boundary value problem of nonlinear -difference equations with nonlocal boundary conditions given by
where , , and is a fixed constant. The existence of solutions for problem (1) is shown by means of a variety of fixed point theorems such as Banach's contraction principle, Krasnoselskii’s fixed point theorem, and Leray-Schauder nonlinear alternative.

Yu and Wang [28] considered a boundary value problem with the nonlinear second-order -difference equation, where and is a fixed number. Existence and uniqueness of the solutions are obtained by means of Banach’s contraction principle, Leray-Schauder nonlinear alternative, and Leray-Schauder continuation theorem.

Pongarm et al. [29] considered sequential derivative of nonlinear -difference equation with three-point boundary conditions, where , , , and , are given constants. Existence results are proved based on Banach’s contraction mapping principle, Krasnoselskii’s fixed point theorem, and Leray-Schauder degree theory.

We note that in the above-mentioned papers [24, 28] the -numbers in the equation and the boundary conditions are the same. As far as we know the paper by Pongarm et al. [29] is the first paper which has different values of the -numbers in -derivative and -integral.

In this paper, we discuss the existence of solutions for the following nonlinear -difference equation with three-point integral boundary condition where , , , is a fixed constant, and . Also, , and , are given constants such that .

It is noteworthy that, in the above problem (4), we have three different values of the -numbers, in -derivatives and the -integral. Moreover, we emphasize the fact that, instead the value is usually used in the literature, we use the values of the function and its derivative in an intermediate point .

The aim of this paper is to prove some existence and uniqueness results for the boundary value problem (4). Our results are based on Banach’s contraction mapping principle, nonlinear contraction, Krasnoselskii’s fixed point theorem, Leray-Schauder nonlinear alternative, and Leray-Schauder degree theory.

The rest of the paper is organized as follows. In Section 2, we provide some basic definitions, preliminaries facts, and a lemma, which are used later. The main results are given in Section 3. In the end, Section 4, some results illustrating the results established in this paper are also presented.

#### 2. Preliminaries

Let us recall some basic concepts of -calculus [15, 18].

*Definition 1. *For , one defines the -derivative of a real valued function as

The higher-order -derivatives are given by

For one sets and define, the definite -integral of a function by provided that the series converges.

For , one sets

Note that for , one has , for some ; thus, the definite integral is just a finite sum, so no question about convergence is raised.

One notes that while if is continuous at , then In -calculus, the product rule and integration by parts formula are Further, reversing the order of integration is given by In the limit , the above results correspond to their counterparts in standard calculus.

Lemma 2. *Let and . Then, for any , the boundary value problem,
**
is equivalent to the integral equation
**
where
*

* Proof. *Taking double -integral for (13), we have
By changing the order of -integration, we have
In particular, for , we get
Taking -derivative for (18), for , we obtain
For , we have
Therefore,
Now, using the first condition of (14) with (19), (22), we have
Taking the -integral for (18) from to , we obtain
Substituting in (24) and using the second condition of (14), we get
Solving the system of linear equations (23) and (25) for the unknown constants and , we have
where is defined by (16). Substituting the values of and in (18), we obtain (15). This completes the proof.

Let denotes the Banach space of all the continuous functions from to endowed with the norm defined by . Define an operator by Observe that the problem (4) has solutions if and only if the operator has fixed points.

For the sake of convenience, we set a constant as

#### 3. Main Results

Now, we are in the position to establish the main results. Our first result is based on Banach’s fixed point theorem.

Theorem 3. *Assume that is a continuous function satisfying the conditions*(H_{1})*, for all and , *(H_{2})*,**
where is a Lipschitz constant, and is defined by (28).**Then, the boundary value problem (4) has a unique solution.*

*Proof. *We transform the boundary value problem (4) into a fixed point problem , where is defined by (27). Assume that , and choose a constant satisfying
Now, we will show that , where . For any , we have
Therefore, .

Next, we will show that is a contraction. For any and for each , we have
Since , is a contraction. Thus, the conclusion of the theorem follows by Banach’s contraction mapping principle. This completes the proof.

Next, we can still deduce the existence and uniqueness of a solution to the boundary value problem (4). We will use nonlinear contraction to accomplish this.

*Definition 4. *Let be a Banach space and let be a mapping. is said to be a nonlinear contraction if there exists a continuous nondecreasing function such that and for all with the following property:

Lemma 5 (Boyd and Wong [30]). *Let be a Banach space and let be a nonlinear contraction. Then, has a unique fixed point in .*

Theorem 6. *Suppose that *()* there exists a continuous function such that
for all and , where
* *and is defined in (16). ** Then, the boundary value problem (4) has a unique solution.*

*Proof. *Let the operator be defined as (27). We define a continuous nondecreasing function by
such that and , for all .

Let . Then, we get
Thus,
This implies that . Hence, is a nonlinear contraction. Therefore, by Lemma 5, the operator has a unique fixed point in , which is a unique solution of problem (4).

The third result is based on the following Krasnoselskii fixed point theorem [31].

Theorem 7. *Let be a bounded closed convex and nonempty subset of a Banach space . Let , be operators such that: *(i)* whenever , *(ii)* is compact and continuous, *(iii)* is a contraction mapping. ** Then, there exists such that .*

Theorem 8. *Assume that and hold. In addition one supposes that: *()*, for all , with . ** If
**
where is given by (28), then the boundary value problem (4) has at least one solution on .*

*Proof. *Setting and choosing a constant
we consider .

In view of Lemma 2, we define the operators and on the ball as
For , by computing directly, we have
Therefore, . Condition (38) implies that is a contraction mapping. Next, we will show that is compact and continuous. Continuity of coupled with the assumption implies that the operator is continuous and uniformly bounded on . We define . For with and , we have
Actually, as , the right-hand side of the above inequality tends to be zero. So, is relatively compact on . Hence, by the Arzelá-Ascoli Theorem, is compact on . Therefore, all the assumptions of Theorem 7 are satisfied, and the conclusion of Theorem 7 implies that the boundary value problem (4) has at least one solution on . This completes the proof.

As the fourth result, we prove the existence of solutions of (4) by using Leray-Schauder nonlinear alternative.

Theorem 9 (Nonlinear Alternative for Single Valued Maps [32]). *Let be a Banach space, a closed convex subset of , an open subset of , and . Suppose that is a continuous, compact (that is, is a relatively compact subset of ) map. Then, either *(i)* has a fixed point in or *(ii)*there is a (the boundary of in ) and with .*

Theorem 10. *Assume that: *()* there exists a continuous nondecreasing function and a function such that
*()* there exists a constant such that
**Then, the boundary value problem (4) has at least one solution on .*

*Proof. *We will show that * maps bounded sets (balls) into bounded sets in *. For a positive number , let be a bounded ball in . Then, for , we have

Consequently,

Next, we will show that * maps bounded sets into equicontinuous sets of **.* Let with and . Then, we have
As , the right-hand side of the above inequality tends to zero independently of . As satisfies the above assumptions; therefore, it follows by the Arzelá-Ascoli theorem that is completely continuous.

Let be a solution. Then, for and following the similar computations as in the first step, we have

Consequently, we have
In view of , there exists such that . Let us set
Note that the operator is continuous and completely continuous. From the choice of , there is no such that for some . Consequently, by the nonlinear alternative of Leray-Schauder type (Theorem 9), we deduce that has a fixed point which is a solution of the problem (4). This completes the proof.

Finally, we prove that problem (4) has at least one solution on by using Leray-Schauder degree theory.

Theorem 11. *Let be a continuous function. Assume that: *()* there exist constants , where is given by (28) and such that for all , . ** Then, the boundary value problem (4) has at least one solution.*

*Proof. *Let us define an operator as (27). We wish to prove that there exists at least one solution of the fixed point equation
We define a ball , with a constant radius given by
Then, it is sufficient to show that satisfies
Now, we set
Then, by the Arzelá-Ascoli theorem, we conclude that a continuous map defined by is completely continuous. If (53) holds, then the following Leray-Schauder degrees are well defined. From the homotopy invariance of topological degree, it follows that
where denotes the unit operator. By the nonzero property of Leray-Schauder degree, for at least one . Let us assume that for some . Then, for all , we obtain
Taking norm and solving it for , this yields
Let , then (53) holds. This completes the proof

#### 4. Examples

In this section, we illustrate our main results with some examples. Let us consider the following boundary value problem of nonlinear second-order -difference equations with three-point boundary conditions Here, we have , , , , , , and . We find that (a) Let be a continuous function given by Since, , then is satisfied with . We can find that Hence, by Theorem 3, problem (58) with given by (60) has a unique solution on .(b) If is a continuous function given by Choosing , we find that Here, Therefore, by Theorem 6, the problem (58) with given by (62) has a unique solution on .(c) Consider a continuous function given by We can show that with and . By Theorem 11, the problem (58) with the given by (65) has at least one solution on .

#### Acknowledgment

This research of T. Sitthiwirattham and J. Tariboon is supported by King Mongkut’s University of Technology North Bangkok, Thailand. Sotiris K. Ntouyas is a Member of Nonlinear Analysis and Applied Mathematics (NAAM), Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.

#### References

- F. H. Jackson, “On
*q*-difference equations,”*American Journal of Mathematics*, vol. 32, no. 4, pp. 305–314, 1910. View at: Publisher Site | Google Scholar - R. D. Carmichael, “The general theory of linear
*q*-difference equations,”*American Journal of Mathematics*, vol. 34, no. 2, pp. 147–168, 1912. View at: Publisher Site | Google Scholar - T. E. Mason, “On properties of the solutions of linear
*q*-difference equations with entire function coefficients,”*American Journal of Mathematics*, vol. 37, no. 4, pp. 439–444, 1915. View at: Publisher Site | Google Scholar - C. R. Adams, “On the linear ordinary
*q*-difference equation,”*American Mathematics II*, vol. 30, pp. 195–205, 1929. View at: Google Scholar - W. J. Trjitzinsky, “Analytic theory of linear
*q*-difference equations,”*Acta Mathematica*, vol. 61, no. 1, pp. 1–38, 1933. View at: Publisher Site | Google Scholar | Zentralblatt MATH - T. Ernst, “A new notation for
*q*-calculus and a new*q*-Taylor formula,”*U.U.D.M. Report*1999:25, Department of Mathematics, Uppsala University, Uppsala, Sweden, 1999. View at: Google Scholar - R. J. Finkelstein, “
*q*-Field theory,”*Letters in Mathematical Physics*, vol. 34, no. 2, pp. 169–176, 1995. View at: Publisher Site | Google Scholar - R. J. Finkelstein, “
*q*-deformation of the Lorentz group,”*Journal of Mathematical Physics*, vol. 37, no. 2, pp. 953–964, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH - R. Floreanini and L. Vinet, “Automorphisms of the
*q*-oscillator algebra and basic orthogonal polynomials,”*Physics Letters A*, vol. 180, no. 6, pp. 393–401, 1993. View at: Publisher Site | Google Scholar - R. Floreanini and L. Vinet, “Symmetries of the
*q*-difference heat equation,”*Letters in Mathematical Physics*, vol. 32, no. 1, pp. 37–44, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH - R. Floreanini and L. Vinet, “
*q*-Gamma and*q*-beta functions in quantum algebra representation theory,”*Journal of Computational and Applied Mathematics*, vol. 68, no. 1-2, pp. 57–68, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH - P. G. O. Freund and A. V. Zabrodin, “The spectral problem for the
*q*-Knizhnik-Zamolodchikov equation and continuous*q*-Jacobi polynomials,”*Communications in Mathematical Physics*, vol. 173, no. 1, pp. 17–42, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH - G. Gasper and M. Rahman,
*Basic Hypergeometric Series*, Cambridge University Press, Cambridge, UK, 1990. - G. N. Han and J. Zeng, “On a
*q*-sequence that generalizes the median Genocchi numbers,”*Annales des Sciences Mathématiques du Québec*, vol. 23, pp. 63–72, 1999. View at: Google Scholar - V. Kac and P. Cheung,
*Quantum Calculus*, Springer, New York, NY, USA, 2002. - G. Bangerezako, “Variational
*q*-calculus,”*Journal of Mathematical Analysis and Applications*, vol. 289, no. 2, pp. 650–665, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH - A. Dobrogowska and A. Odzijewicz, “Second order
*q*-difference equations solvable by factorization method,”*Journal of Computational and Applied Mathematics*, vol. 193, no. 1, pp. 319–346, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH - G. Gasper and M. Rahman, “Some systems of multivariable orthogonal
*q*-Racah polynomials,”*Ramanujan Journal*, vol. 13, no. 1–3, pp. 389–405, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - M. E. H. Ismail and P. Simeonov, “
*q*-difference operators for orthogonal polynomials,”*Journal of Computational and Applied Mathematics*, vol. 233, no. 3, pp. 749–761, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - M. Bohner and G. S. Guseinov, “The
*h*-Laplace and*q*-Laplace transforms,”*Journal of Mathematical Analysis and Applications*, vol. 365, no. 1, pp. 75–92, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - M. El-Shahed and H. A. Hassan, “Positive solutions of
*q*-difference equation,”*Proceedings of the American Mathematical Society*, vol. 138, no. 5, pp. 1733–1738, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - B. Ahmad, “Boundary-value problems for nonlinear third-order
*q*-difference equations,”*Electronic Journal of Differential Equations*, vol. 94, pp. 1–7, 2011. View at: Google Scholar - B. Ahmad, A. Alsaedi, and S. K. Ntouyas, “A study of second-order
*q*-difference equations with boundary conditions,”*Advances in Difference Equations*, vol. 2012, p. 35, 2012. View at: Publisher Site | Google Scholar - B. Ahmad, S. K. Ntouyas, and I. K. Purnaras, “Existence results for nonlinear
*q*-difference equations with nonlocal boundary conditions,”*Communications on Applied Nonlinear Analysis*, vol. 19, pp. 59–72, 2012. View at: Google Scholar - B. Ahmad and J. J. Nieto, “On nonlocal boundary value problems of nonlinear
*q*-difference equations,”*Advances in Difference Equations*, vol. 2012, p. 81, 2012. View at: Publisher Site | Google Scholar - B. Ahmad and S. K. Ntouyas, “Boundary value problems for
*q*-difference inclusions,”*Abstract and Applied Analysis*, vol. 2011, Article ID 292860, 15 pages, 2011. View at: Publisher Site | Google Scholar - W. Liu and H. Zhou, “Existence solutions for boundary value problem of nonlinear fractional
*q*-difference equations,”*Advances in Difference Equations*, vol. 2013, 113 pages, 2013. View at: Publisher Site | Google Scholar - C. Yu and J. Wang, “Existence of solutions for nonlinear second-order
*q*-difference equations with first-order*q*-derivatives,”*Advances in Difference Equations*, vol. 2013, p. 124, 2013. View at: Publisher Site | Google Scholar - N. Pongarm, S. Asawasamrit, and J. Tariboon, “Sequential derivatives of nonlinear
*q*-difference equations with three-point*q*-integral boundary conditions,”*Journal of Applied Mathematics*, vol. 2013, Article ID 605169, 9 pages, 2013. View at: Publisher Site | Google Scholar - D. W. Boyd and J. S. W. Wong, “On nonlinear contractions,”
*Proceedings of the American Mathematical Society*, vol. 20, pp. 458–464, 1969. View at: Publisher Site | Google Scholar - M. A. Krasnoselskii, “Two remarks on the method of successive approximations,”
*Uspekhi Matematicheskikh Nauk*, vol. 10, pp. 123–127, 1955. View at: Google Scholar - A. Granas and J. Dugundji,
*Fixed Point Theory*, Springer, New York, NY, USA, 2003.

#### Copyright

Copyright © 2013 Thanin Sitthiwirattham et al. 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.