Abstract and Applied Analysis

Volume 2014 (2014), Article ID 340159, 11 pages

http://dx.doi.org/10.1155/2014/340159

Research Article

Uniqueness and Existence of Solution for a System of Fractional -Difference Equations

Wen-Xue Zhou1,2 and Hai-zhong Liu¹

¹College of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, China

²School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Received 28 March 2014; Revised 27 May 2014; Accepted 13 June 2014; Published 7 July 2014

Academic Editor: Felix Sadyrbaev

Copyright © 2014 Wen-Xue Zhou and Hai-zhong Liu. 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.

Abstract.

We prove the existence and uniqueness of solution for a system of fractional differential equations. Our results are based on the nonlinear alternative of Leray-Schauder type and Banach’s fixed-point theorem.

1. Introduction

This paper is mainly concerned with the uniqueness and existence of solution for a system of fractional -difference equations given by

where , is the fractional -derivatives of the Caputo type, , ,   ,   ,   , and    are arbitrary real constants, and are given continuous functions.

In the last few years, fractional differential equations (in short FDEs) have been studied extensively. The motivation for those works stems from both the development of the theory of fractional calculus itself and the applications of such constructions in various sciences such as physics, mechanics, chemistry, and engineering. For an extensive collection of such results, we refer the readers to the monographs by Kilbas et al. [1], Miller and Ross [2], Oldham and Spanier [3], Podlubny [4], and Samko et al. [5].

Some basic theory for the initial value problems of fractional differential equations involving Riemann-Liouville differential operator has been discussed by Lakshmikantham and Vatsala ([6–8]), Babakhani and Daftardar-Gejji ([9–11]), Bai [12], and so on. Also, there are some papers which deal with the existence and multiplicity of solutions (or positive solution) for nonlinear FDE of BVPs by using techniques of nonlinear analysis (fixed-point theorems, Leray-Shauder theory, topological degree theory, etc.)—see ([13–18]) and the references therein. The study of a coupled system of fractional order is also very significant because this kind of system can often occur in applications. The reader is referred to the papers ([19–22]) and the references cited therein.

The pioneer work on -difference calculus or quantum calculus dates back to Jackson’s papers ([23, 24]), while a systematic treatment of the subject can be found in [25, 26]. For some recent existence results on -difference equations, see [27–29] and the references therein.

There has also been a growing interest on the subject of discrete fractional equations on the time scale . Some interesting results on the topic can be found in a series of papers [30–38]. Fractional -difference equations have recently attracted the attention of several researchers. For some earlier work on the topic, we refer to [39, 40], whereas some recent work on the existence theory of fractional -difference equations can be found in [41–45]. However, the study of boundary value problems of fractional -difference equations is at its infancy and much of the work on the topic is yet to be done.

From the above works, we can see a fact, although the fractional boundary value problems have been investigated by some authors. To the best of our knowledge, there have been few papers which deal with problem (1) for nonlinear fractional differential equation. Motivated by all the works above, in this paper we discuss problem (1). Using nonlinear alternative of Leray-Schauder type, we will give the existence and uniqueness of solution for a system of fractional differential equations with Riemann-Liouville integral boundary conditions of different order (1).

The paper is organized as follows. In Section 2, we give some preliminary results that will be used in the proof of the main results. In Section 3, we establish the uniqueness and existence of a solution for the nonlinear fractional differential equation boundary value problem (1). In last section, we give two examples to illustrate our work.

2. Preliminaries and Lemmas

In this section, we cite some definitions and fundamental results of the -calculus as well as of the fractional -calculus ([46, 47]). We also give a lemma that will be used in obtaining the main results of the paper.

Let and define [47]

The -analogue of the power is

If is not a positive integer, then

Note that if , then . The -gamma function is defined by

and satisfies (see [47]).

The -derivative of a function is here defined by

and -derivatives of higher order by

The -integral of a function defined in the interval is given by

If and is defined in the interval , its integral from to is defined by

Similarly, as done for derivatives, an operator can be defined, namely, by

The fundamental theorem of calculus applies to these operators and ; that is,

and if is continuous at , then

Basic properties of the two operators can be found in the book that is mentioned in [8]. We now point out three formulas that will be used later ( denotes the derivative with respect to variable ) [43]:

Remark 1. We note that if and , then [43].

Definition 2 (see [40]). Let and let be a function defined on . The fractional -integral of the Riemann-Liouville type is and

Definition 3 (see [48]). The fractional -derivative of the Riemann-Liouville type of order is defined by and

where is the smallest integer greater than or equal to .

Definition 4 (see [48]). The fractional -derivative of the Caputo type of order is defined by

where is the smallest integer greater than or equal to .

Lemma 5. Let and let be a function defined on . Then the next formulas hold:

(1),(2).

Lemma 6 (see [42]). Let and . Then the following equality holds:

Lemma 7 (see [48]). Let and . Then the following equality holds:

For convenience, one introduces the following notations:

From Lemmas 5 and 7, we can obtain the following lemma.

Lemma 8. Let and ; then the unique solution of the linear fractional boundary value problem

is given by

The following lemma is fundamental in the proofs of our main result.

Lemma 9 (see [49]; nonlinear alternative of Leray-Schauder type). Let be a Banach space with closed and convex. Assume that is a relatively open subset of with and is continuous, compact (i.e., is a relatively compact subset of ) map. Then either

(i) has a fixed point in or(ii)there exist and with .

3. Main Results

In this section, we will discuss the uniqueness and existence of solutions for boundary value problem (1).

First of all, we define the Banach space endowed with the norm . For , let ; then is a Banach space.

For convenience, we set

and let . Note

Employing Lemma 8, system (1) can be expressed as

where , , , are given by (21), and , , , are given by (22)

From Lemma 8 in Section 2, we can obtain the following lemma.

Lemma 10. Suppose that and are continuous; then is a solution of BVP (1) if and only if is a solution of the integral equations (24).

Let ; define an operator as

where

then, by Lemma 10, the fixed point of operator coincides with the solution of system (1).

In the first result, we prove uniqueness of solution of the boundary value problem (1) via Banach’s contraction principle.

Theorem 11. Assume that are continuous functions and the following conditions hold:

(H1)there exist two -integrable functions that satisfy

In addition, assume that

where

where and are given by (19) and (22), respectively. Then system (1) has a unique solution.

Proof. Let us set , ,

Define

For , we obtain

Then, for , , we have

In view of (31), we obtain

From the last estimate we deduce that .

By a similar way as done above we have

and .

Therefore, we obtain

From the last estimate we can choose ; then, for every , we have .

In order to show that is a contraction, let , and, for any , we get

which, in view of and (31), implies that

Similarly, we have .

Thus, we have

Since , , therefore, the operator is a contraction. Hence, by Banach’s contraction principle, the operator has a unique fixed point, which is the unique solution of the system (1). This completes the proof.

The second result is based on the nonlinear alternative of Leray-Schauder type (Lemma 9).

Theorem 12. Assume that are continuous functions and the following conditions hold:

(H2)there exist four functions , , and two nondecreasing functions , such that

 where .(H3)There exists a constant such that

 where

Then system (1) has at least one solution on .

Proof. Consider the operator defined by (25). The proof consists of several steps. As a first step, it will be shown that maps bounded sets into bounded sets in . For a positive number , let be bounded set in ; then, for , we have

As before, it can be shown that

Similarly, we have

Thus, maps bounded sets into bounded sets in .

Next, we show that maps bounded sets into equicontinuous sets of . Let with and , where is a bounded set of . Then taking into consideration the inequality , for , we obtain

Clearly, the right-hand side of the above inequality tends to zero independently of as . Thus, it follows by the Arzelá-Ascoli theorem that is completely continuous. Similarly, is completely continuous. Therefore, is completely continuous.

Let us set . Note that the operator is continuous and completely continuous. From the choice of , assume that there is such that , for some . By (H3), we obtain

which is a contradiction. In consequence, by the nonlinear alternative of Leray-Schauder type (Lemma 9), we deduce that has a fixed point which is a solution of the system (1). The proof is complete.

In the sequel we present two examples which illustrate Theorems 11 and 12.

4. Examples

Example 1. Consider the following fractional -difference nonlocal boundary value problem:

In this case, , , , , , , , , , , , , , , , , and are constants to be fixed later on. Moreover, , , , , , and . Consider

Clearly, we have

Choose

Hence all the assumptions of Theorem 11 are satisfied. Therefore, by Theorem 11, the problem (50) has a unique solution.

Example 2. Consider the following fractional boundary value problem:

In this case, , , , , , , , , , , , , , , , , and are constants to be fixed later on. Moreover, , , , , , and . Clearly

Clearly , , , , , and . Consequently, , , , , and conditions (43) imply that . Thus, all the assumptions of Theorem 12 are satisfied. Therefore, the conclusion of Theorem 12 applies to problems (54).

Conflict of Interests

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

Acknowledgments

This research was supported by the National Natural Science Foundation of China (11161027, 11262009, and 11226132), the Scientific Research Projects in Colleges and Universities of Gansu Province of China (2013A-043), the Fundamental Research Funds for the Gansu Universities (213061), the Fundamental Research Funds for the Gansu Universities (212084), the Youth Science Foundation of Lanzhou Jiaotong University (2012019), and the National Natural Science Foundation of China (11226132). The authors are thankful to the referees for their careful reading of the paper and insightful comments.

References

A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006.

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.

K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, London, UK, 1974.

I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.

S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1993.

V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis, vol. 69, no. 8, pp. 2677–2682, 2008.

V. Lakshmikantham and A. S. Vatsala, “General uniqueness and monotone iterative technique for fractional differential equations,” Applied Mathematics Letters, vol. 21, no. 8, pp. 828–834, 2008.

V. Lakshmikantham, “Theory of fractional functional differential equations,” Nonlinear Analysis, vol. 69, no. 10, pp. 3337–3343, 2008.

A. Babakhani and V. Daftardar-Gejji, “Existence of positive solutions of nonlinear fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 278, no. 2, pp. 434–442, 2003.

A. Babakhani and V. Daftardar-Gejji, “Existence of positive solutions for -term non-autonomous fractional differential equations,” Positivity, vol. 9, no. 2, pp. 193–206, 2005.

A. Babakhani and V. Daftardar-Gejji, “Existence of positive solutions for multi-term non-autonomous fractional differential equations with polynomial coefficients,” Abstract and Applied Analysis, vol. 2009, Article ID 768920, 12 pages, 2009.

C. Bai, “Positive solutions for nonlinear fractional differential equations with coefficient that changes sign,” Nonlinear Analysis, vol. 64, no. 4, pp. 677–685, 2006.

R. P. Agarwal, M. Benchohra, and S. Hamani, “Boundary value problems for differential inclusions with fractional order,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 181–196, 2008.

W. Zhou and Y. Chu, “Existence of solutions for fractional differential equations with multi-point boundary conditions,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 3, pp. 1142–1148, 2012.

W. Zhou, J. Peng, and Y. Chu, “Multiple positive solutions for nonlinear semipositone fractional differential equations,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 850871, 10 pages, 2012.

W. Zhou and H. Liu, “Existence of weak solutions for nonlinear fractional differential inclusion with nonseparated boundary conditions,” Journal of Applied Mathematics, vol. 2012, Article ID 530624, 13 pages, 2012.

W. Zhou, Y. Chu, and D. Baleanu, “Uniqueness and existence of positive solutions for a multi-point boundary value problem of singular fractional differential equations,” Advances in Diff erence Equations, vol. 2013, no. 114, 2013.

W. Zhou and H. Liu, “Existence solutions for boundary value problem of nonlinear fractional -difference equations,” Advances in Difference Equations, vol. 2013, article 113, 12 pages, 2013.

B. Ahmad and J. J. Nieto, “Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions,” Computers & Mathematics with Applications, vol. 58, no. 9, pp. 1838–1843, 2009.

X. Su, “Boundary value problem for a coupled system of nonlinear fractional differential equations,” Applied Mathematics Letters, vol. 22, no. 1, pp. 64–69, 2009.

J. Sun, Y. Liu, and G. Liu, “Existence of solutions for fractional differential systems with antiperiodic boundary conditions,” Computers & Mathematics with Applications, vol. 64, no. 6, pp. 1557–1566, 2012.

J. Wang, H. Xiang, and Z. Liu, “Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations,” International Journal of Differential Equations, vol. 2010, Article ID 186928, 12 pages, 2010.

F. H. Jackson, “-difference equations,” American Journal of Mathematics, vol. 32, no. 4, pp. 305–314, 1910.

F. H. Jackson, “On q-functions and a certain difference operator,” Transactions of the Royal Society of Edinburgh, vol. 46, no. 2, pp. 253–281, 1909.

T. Ernst, “A new notation for q-calculus and a new q-Taylor formula,” U.U.D.M. Report 1999: 25, Department of Mathematics, Up-psala University, 1999.

T. Ernst, “The history of qi calculus and a new method,” UUDM Report 2000, Department of Mathematics, Uppsala University, Uppsala, Sweden, 2000.

B. Ahmad, “Boundary-value problems for nonlinear third-order -difference equations,” Electronic Journal of Differential Equations, vol. 94, pp. 1–7, 2011.

B. Ahmad and S. K. Ntouyas, “Boundary value problems for -difference inclusions,” Abstract and Applied Analysis, Art. ID 292860, 15 pages, 2011.

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, article 35, 2012.

F. M. Atici and P. W. Eloe, “A transform method in discrete fractional calculus,” International Journal of Difference Equations, vol. 2, no. 2, pp. 165–176, 2007.

C. S. Goodrich, “Solutions to a discrete right-focal fractional boundary value problem,” International Journal of Difference Equations, vol. 5, no. 2, pp. 195–216, 2010.

C. S. Goodrich, “Continuity of solutions to discrete fractional initial value problems,” Computers & Mathematics with Applications, vol. 59, no. 11, pp. 3489–3499, 2010.

F. M. Atici and P. W. Eloe, “Linear systems of fractional nabla difference equations,” The Rocky Mountain Journal of Mathematics, vol. 41, no. 2, pp. 353–370, 2011.

F. M. Atıcı and P. W. Eloe, “Two-point boundary value problems for finite fractional difference equations,” Journal of Difference Equations and Applications, vol. 17, no. 4, pp. 445–456, 2011.

N. R. O. Bastos, R. A. C. Ferreira, and D. F. M. Torres, “Discrete-time fractional variational problems,” Signal Processing, vol. 91, no. 3, pp. 513–524, 2011.

N. R. O. Bastos, D. Mozyrska, and D. F. M. Torres, “Fractional derivatives and integrals on time scales via the inverse generalized Laplace transform,” International Journal of Mathematics and Computation, vol. 11, no. J11, pp. 1–9, 2011.

C. S. Goodrich, “Existence of a positive solution to a system of discrete fractional boundary value problems,” Applied Mathematics and Computation, vol. 217, no. 9, pp. 4740–4753, 2011.

C. S. Goodrich, “On discrete sequential fractional boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 385, no. 1, pp. 111–124, 2012.

W. A. Al-Salam, “Some fractional {$q$}-integrals and {$q$}-derivatives,” Proceedings of the Edinburgh Mathematical Society II, vol. 15, no. 2, pp. 135–140, 1967.

R. Agarwal, “Certain fractional q-integrals and q-derivatives,” Proceedings of the Cambridge Philosophical Society, vol. 66, no. 2, pp. 365–370, 1969.

M. El-Shahed and F. M. Al-Askar, “Positive solutions for boundary value problem of nonlinear fractional -difference equation,” ISRN Mathematical Analysis, vol. 2011, Article ID 385459, 12 pages, 2011.

R. A. C. Ferreira, “Positive solutions for a class of boundary value problems with fractional -differences,” Computers & Mathematics with Applications, vol. 61, no. 2, pp. 367–373, 2011.

R. A. C. Ferreira, “Nontrivial solutions for fractional -difference boundary value problems,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 70, 10 pages, 2010.

J. R. Graef and L. Kong, “Positive solutions for a class of higher order boundary value problems with fractional -derivatives,” Applied Mathematics and Computation, vol. 218, no. 19, pp. 9682–9689, 2012.

J. Ma and J. Yang, “Existence of solutions for multi-point boundary value problem of fractional -difference equation,” Electronic Journal of Qualitative Theory of Differential Equations, No. 92, 10 pages, 2011.

G. Gasper and M. Rahman, Basic Hypergeometric Series, vol. 35 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, UK, 1990.

V. Kac and P. Cheung, Quantum Calculus, Springer, New York, NY, USA, 2002.

P. M. Rajkovic, S. D. Marinkovic, and M. S. Stankovic, “On -analogues of Caputo derivative and Mittag-Leffler function,” Fractional Calculus & Applied Analysis, vol. 10, no. 4, pp. 359–373, 2007.

A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York, NY, USA, 2005.