Discrete Dynamics in Nature and Society
Volume 2015 (2015), Article ID 376261, 10 pages
http://dx.doi.org/10.1155/2015/376261

Research Article

Existence and Uniqueness of Solutions for a Discrete Fractional Mixed Type Sum-Difference Equation Boundary Value Problem

Weidong Lv

School of Mathematics and Statistics, Longdong University, Qingyang, Gansu 745000, China

Received 18 July 2015; Revised 6 September 2015; Accepted 6 September 2015

Academic Editor: Douglas R. Anderson

Copyright © 2015 Weidong Lv. 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.

By means of Schauder’s fixed point theorem and contraction mapping principle, we establish the existence and uniqueness of solutions to a boundary value problem for a discrete fractional mixed type sum-difference equation with the nonlinear term dependent on a fractional difference of lower order. Moreover, a suitable choice of a Banach space allows the solutions to be unbounded and two representative examples are presented to illustrate the effectiveness of the main results.

1. Introduction

For , such that is a nonnegative integer, we denote and throughout this paper. It is also worth noting that, in what follows, for any function defined on , we appeal to the convention , when with .

In this paper, we will consider the following discrete fractional mixed type sum-difference equation boundary value problem:

where , and denote the discrete Riemann-Liouville fractional differences of order and , respectively, , , and

where , , and .

Continuous fractional calculus is a generalization of ordinary differentiation and integration on an arbitrary order that can be noninteger. This subject, as old as the problem of ordinary differential calculus, can go back to the times when Leibniz and Newton invented differential calculus. The theory of fractional differential equations has received a lot of attention and now constitutes a new important mathematical branch due to its extensive applications in various fields of science and engineer. For more details, see [1–13] and references therein.

It is well known that discrete analogues of differential equations can be very useful in applications [14], in particular for using computer to simulate the behavior of solutions for certain dynamic equations. However, compared to the long and rich history of continuous fractional calculus, discrete fractional calculus attracted mathematicians and scientists into its fairly new research area in a short period of time. In this time period, the theory of discrete calculus has been developed in many directions parallel to the theory in continuous fractional calculus such as initial value problems and boundary value problems for fractional difference equations, discrete Mittag-Leffler functions, and inequalities with discrete fractional operators; see [15–39] and the references therein. At the same time, in [27], Atıcı and Şengül have shown the usefulness of discrete Gompertz fractional difference equation for tumor growth model, which implies that discrete fractional difference calculus will provide a new excellent tool to model real world phenomena in the future.

Although, among all recently research topics, the branch of discrete finite fractional difference boundary value problems is currently undergoing active investigation [16, 31–38], significantly less is known about discrete infinite fractional difference boundary value problems with the nonlinear term dependent on a fractional difference operator. Here, we should point out that in [39], Lv and Feng, by simple analogy with the ordinary case, introduced some basic definitions of discrete fractional calculus for Banach-valued functions and initially studied a class of discrete infinite fractional mixed type sum-difference equation boundary value problems in abstract spaces by using contracting mapping principle. Furthermore, as far as we know, the theory of discrete fractional mixed type sum-difference equations boundary value problems is still a new research area. So in this paper, we continue to focus on this topic for real-valued functions and provide some sufficient conditions for the existence and uniqueness of solutions to problem (1). Particularly note that problem (1) is not like the problem in [39] and the biggest difference is the nonlinear term in (1) explicitly dependent on the discrete fractional difference operator of lower order. Hence, these differences that cause the main difficulties that we have to deal with in this paper are those of constructing a special Banach space and establishing an appropriate compactness criterion in it.

The outline for the remainder of this paper is as follows. In Section 2, we recall some useful preliminaries for discrete fractional calculus and present the basic space and its compactness criterion for studying problem (1). In Section 3, by employing Schauder’s fixed point theorem and contraction mapping principle, we establish the existence and uniqueness results of problem (1). In Section 4, two concrete examples are provided to illustrate the possible applications of the obtained analytical results.

2. Preliminaries

In this section, we firstly present here some necessary definitions and basic results about discrete fractional calculus.

Definition 1 (see [30]). For any and , the falling factorial function is defined as

provided that the right-hand side is well defined. We appeal to the convention that if is a pole of the Gamma function and is not a pole, then .

Definition 2 (see [40]). The th discrete fractional sum of a function , for , is defined by

Also, we define the trivial sum .

Definition 3 (see [30]). The th discrete Riemann-Liouville fractional difference of a function , for , is defined by

where is the smallest integer greater than or equal to and is the th order forward difference operator. If , then .

Remark 4. From Definitions 2 and 3, it is easy to see that maps functions defined on to functions defined on and maps functions defined on to functions defined on , where is the smallest integer greater than or equal to . For ease of notation, we throughout this paper omit the subscript in and when it is not to lead to domains confusion and general ambiguity.

Lemma 5 (see [30]). Let and . Then

Lemma 6 (see [31]). Let and with . Then

for , , .

Lemma 7 (see [15]). Let and be given. Then, for , ,

Lemma 8 (see [28]). Let , and . Then

Next, we define the space,

equiped with the norm

Furthermore, using the linear functional analysis theory, we can easily verify that is a Banach space, and then we present the following compactness criterion in it.

Lemma 9. Let be a bounded set. If for any given , there exists a positive integer such that

whenever , , and ; then is relatively compact in .

Proof. Evidently, it is sufficient to prove that is totally bounded. In what follows we divide this proof into two steps.
Step  1. Let us consider the case .
Denote by the restriction of on . Then , equipped with the norm , is a finite dimension Banach space. So we know that is relatively compact from the boundness of ; hence is totally bounded; namely, for any , there exist finitely many ball , , , such that

where .
Similarly, denote . Then is also a Banach space with the norm and it can be covered by finitely many balls ; that is,

where .
Step  2. Define
Let us consider the case . It is obvious that . Now, let us take ; then can be covered by the balls , where

In fact, for any , the argument in Step 1 implies that there exist and such that , . Hence, for and , we have

For arbitrary , (12) and (17) yield that

and for any , (13) and (18) ensure that

Relations (17)–(20) show that . Therefore, is totally bounded and this lemma is proved.

3. Main Result

In this section, we will establish the existence and uniqueness of solutions for problem (1) by using Schauder’s fixed point theorem and contraction mapping principle. For the sake of convenience and to abbreviate our presentation, for any function , we denote

in the sequel discussion and list here the following conditions:

() and < .()There exist functions , , with

 and such that

 for ., and there exist nonnegative numbers , and a function with such that

 for .

Lemma 10. If and hold, then, for any ,

Proof. For any , , using , , and the monotonicity of , produces

Summating both sides of (26), we can get (25). The proof is completed.

Lemma 11. If and hold, then the unique solution of problem (1) is

where

Proof. If satisfies the equation of problem (1), then Lemma 6 implies that

for some , . By , we get .
Therefore,

By virtue of Lemmas 5, 7, and 8, we have

Using the condition in (31), we obtain

Now, substitution of into (30) gives

where is defined by (28). The proof is completed.

Remark 12. From the expression of , we can easily find that and for .

For any , define an operator by

and due to Lemma 10 and Remark 12, we have

On the other hand, by virtue of Lemmas 5, 7, 8, and 10, we get

which hold for . So (35) and (37) imply that is well defined and bounded. Furthermore, from Lemma 11, we can transform problem (1) into an operator equation and it is clear to see that is a solution of problem (1) which is equivalent to a fixed point of .

Remark 13. Setting in , we have

for , which implies that condition is stronger than . So under assumptions and , the operator defined by (34) is also well defined.

Now, we are in the position to give the main results of this work.

Theorem 14. Assume that is continuous, and suppose that conditions and hold. Then problem (1) has at least one solution .

Proof. In what follows, we divide this proof into three steps.
Step  1. Choose

and let

Then, for any , by (35), (37), and the fact , we can verify that , which implies .
Step  2. Let be s subset of . We employ Lemma 9 to verify that is relatively compact.
In view of Lemma 10 and the boundness of , there exists such that

By (34) and (36), we have

Observing (42), together with and the conditions of Lemma 9, we only need to show that, for any , there exists sufficiently large positive integer such that, for any ,

and for any with ,

Relation (41) yields that there exists a positive number such that

On the other hand, from the monotonicity of and , there exist such that, for any and ,

Now taking , by virtue of (41), (45), and (46), we get

Moreover, from (45), we have

which holds for any with and arbitrary . Moreover, it follows from (47) and (48) that (43) and (44) hold. Consequently, by Lemma 9, is relatively compact.
Step  3. is a continuous operator.
Let such that as . Then by , for any there exists a positive integer such that

On the other hand, from the continuity of , we know that there exists such that, for any and ,

Therefore, for and , by (49)-(50) and Remark 12, we can obtain that

Meanwhile, for and , applying (49)-(50) again, we can easily verify that

Then, by virtue of (51) and (52), we conclude that as , which asserts the continuity of .
Therefore, by Schauder’s fixed point theorem, we obtain that problem (1) has at least one solution in and the proof is finished.

Theorem 15. Suppose that conditions and hold. Then problem (1) has a unique solution .

Proof. For any , in view of and Remark 12, we have

On the other hand, by (36) and using again, we have

So, from (53), (54) and the facts that and when , we know that is a contraction mapping. By means of Banach contraction mapping principle, we get that has a unique fixed point in ; that is, problem (1) has a unique solution. This completes the proof.

4. Examples

In this section, we will illustrate the possible applications of the above established analytical results with the following two concrete examples.

Example 1. Consider the discrete fractional difference boundary value problem:

Conclusion. Problem (55) has at least one solution .

Proof. Corresponding to problem (1), we have ,

.
From the expression of , it is easy to see that is continuous. Furthermore, we can verify that

So condition is satisfied.
On the other hand, by using a simple inequality

we have

and therefore

where

By directly calculation, we have

Thus, condition holds. So, by Theorem 14, our conclusion follows.

Example 2. Consider the following problem:

Conclusion. Problem (63) has a unique solution .

Proof. It is easy to see that problem (63) is the form of problem (1), where ,

for .
Choosing , , , and , then we can verify that , , , , and

holds for any , .
Clearly, all conditions of Theorem 15 are fulfilled. Therefore, we can conclude that problem (63) has a unique solution.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the Longdong University Grant XYZK-1402 and the Science Research Project of Gansu University 2015A-149.

References

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

W. G. Glöckle and T. F. Nonnenmacher, “A fractional calculus approach to self-similar protein dynamics,” Biophysical Journal, vol. 68, no. 1, pp. 46–53, 1995.

R. Metzler, W. Schick, H.-G. Kilian, and T. F. Nonnenmacher, “Relaxation in filled polymers: a fractional calculus approach,” The Journal of Chemical Physics, vol. 103, no. 16, pp. 7180–7186, 1995.

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

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Company, Singapore, 2000.

A. A. Kilbas and J. J. Trujillo, “Differential equations of fractional order: methods, results and problems-I,” Applicable Analysis, vol. 78, no. 1-2, pp. 153–192, 2001.

A. A. Kilbas and J. J. Trujillo, “Differential equations of fractional order: methods, results and problems-II,” Applicable Analysis, vol. 81, no. 2, pp. 435–493, 2002.

J. Sabatier, O. Agrawal, and J. T. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, 2007.

G. S. Frederico and D. F. Torres, “Fractional conservation laws in optimal control theory,” Nonlinear Dynamics, vol. 53, no. 3, pp. 215–222, 2008.

S. Zhang, “The existence of a positive solution for a nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 252, no. 2, pp. 804–812, 2000.

X. Xu, D. Jiang, and C. Yuan, “Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 10, pp. 4676–4688, 2009.

F. Jiao and Y. Zhou, “Existence of solutions for a class of fractional boundary value problems via critical point theory,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1181–1199, 2011.

A. Di Matteo and A. Pirrotta, “Generalized differential transform method for nonlinear boundary value problem of fractional order,” Communications in Nonlinear Science and Numerical Simulation, vol. 29, no. 1–3, pp. 88–101, 2015.

W. G. Kelley and A. C. Peterson, Difference Equations: An Introduction with Applications, Academic Press, New York, NY, USA, 1991.

M. Holm, “Sum and difference compositions in discrete fractional calculus,” Cubo, vol. 13, no. 3, pp. 153–184, 2011.

M. Holm, “Solutions to a discrete, nonlinear, (N − 1; 1) fractional boundary value problem,” International Journal of Dynamical Systems and Differential Equations, vol. 3, no. 1-2, pp. 267–287, 2011.

G. A. Anastassiou, “Nabla discrete fractional calculus and nabla inequalities,” Mathematical and Computer Modelling, vol. 51, no. 5-6, pp. 562–571, 2010.

T. Abdeljawad, D. Baleanu, F. Jarad, and R. P. Agarwal, “Fractional sums and differences with binomial coefficients,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 104173, 6 pages, 2013.

N. Bastos and D. Torres, “Combined Delta-Nabla sum operator in discrete fractional calculus,” Communications in Fractional Calculus, vol. 1, pp. 41–47, 2010.

J. Čermák, T. Kisela, and L. Nechvátal, “Discrete Mittag-Leffler functions in linear fractional difference equations,” Abstract and Applied Analysis, vol. 2011, Article ID 565067, 21 pages, 2011.

T. Abdeljawad, “On Riemann and Caputo fractional differences,” Computers and Mathematics with Applications, vol. 62, no. 3, pp. 1602–1611, 2011.

T. Abdeljawad, “On delta and nabla Caputo fractional differences and dual identities,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 406910, 12 pages, 2013.

R. Dahal and C. S. Goodrich, “A monotonicity result for discrete fractional difference operators,” Archiv der Mathematik, vol. 102, no. 3, pp. 293–299, 2014.

Q. Feng, “Some new generalized Gronwall-Bellman type discrete fractional inequalities,” Applied Mathematics and Computation, vol. 259, pp. 403–411, 2015.

R. A. Ferreira, “A discrete fractional Gronwall inequality,” Proceedings of the American Mathematical Society, vol. 140, no. 5, pp. 1605–1612, 2012.

G.-C. Wu and D. Baleanu, “Discrete fractional logistic map and its chaos,” Nonlinear Dynamics, vol. 75, no. 1-2, pp. 283–287, 2014.

F. M. Atıcı and S. Şengül, “Modeling with fractional difference equations,” Journal of Mathematical Analysis and Applications, vol. 369, no. 1, pp. 1–9, 2010.

F. M. Atici and P. W. Eloe, “Initial value problems in discrete fractional calculus,” Proceedings of the American Mathematical Society, vol. 137, no. 3, pp. 981–989, 2009.

I. K. Dassios and D. I. Baleanu, “Duality of singular linear systems of fractional nabla difference equations,” Applied Mathematical Modelling, vol. 39, no. 14, pp. 4180–4195, 2015.

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.

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.

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

Y. Pan, Z. Han, S. Sun, and C. Hou, “The existence of solutions to a class of boundary value problems with fractional difference equations,” Advances in Difference Equations, vol. 2013, article 275, 20 pages, 2013.

F. Chen and Y. Zhou, “Existence and Ulam stability of solutions for discrete fractional boundary value problem,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 459161, 7 pages, 2013.

Y. Chen and X. Tang, “The difference between a class of discrete fractional and integer order boundary value problems,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 12, pp. 4057–4067, 2014.

W. Lv, “Existence of solutions for discrete fractional boundary value problems with a p-laplacian operator,” Advances in Difference Equations, vol. 2012, article 163, 2012.

W. Lv, “Solvability for discrete fractional boundary value problems with a p-laplacian operator,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 679290, 8 pages, 2013.

W. Lv, “Solvability for a discrete fractional three-point boundary value problem at resonance,” Abstract and Applied Analysis, vol. 2014, Article ID 601092, 7 pages, 2014.

W. Lv and J. Feng, “Nonlinear discrete fractional mixed type sum-difference equation boundary value problems in Banach spaces,” Advances in Difference Equations, vol. 2014, article 184, 12 pages, 2014.

K. Miller and B. Ross, “Fractional difference calculus,” in Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, pp. 139–152, Nihon University, Koriyama, Japan, 1989.