## Recent Developments and Applications on Qualitative Theory of Fractional Equations and Related Topics

View this Special IssueResearch Article | Open Access

# Solvability for a Discrete Fractional Three-Point Boundary Value Problem at Resonance

**Academic Editor:**Jehad Alzabut

#### Abstract

This paper is concerned with the existence of solutions to a discrete three-point boundary value problem at resonance involving the Riemann-Liouville fractional difference of order . Under certain suitable nonlinear growth conditions imposed on the nonlinear term, the existence result is established by using the coincidence degree continuation theorem. Additionally, a representative example is presented to illustrate the effectiveness of the main result.

#### 1. Introduction

For any number , we denote and , for any with , throughout this paper. It is also worth noting that, in what follows, we appeal to the conventions and for a given function defined on and with .

In this paper, we will consider the existence of solutions for the following discrete fractional three-point boundary value problem: where is a real number, denotes the Riemann-Liouville fractional difference of order , , , , and which implies that the problem (1) is at resonance. We note that the problem (1) happens to be at resonance in the sense that the associated linear homogeneous boundary value problem has , , as a nontrivial solution.

The continuous fractional calculus has received increasing attention within the last ten years or so and the theory of fractional differential equations has been a new important mathematical branch due to its demonstrated applications in various fields of science and engineering. For more details, see [1–14] and references therein. Significantly less is known, however, about the discrete fractional calculus, but in recent several years, a lot of papers have appeared; see [15–36]. For example, in [19], Atıcı and Eloe explored a discrete fractional conjugate boundary value problem with the Riemann-Liouville fractional difference. To the best of our knowledge, this is a pioneering work on discussing boundary value problems in discrete fractional calculus. After that, Goodrich studied discrete fractional boundary value problems involving the Riemann-Liouville fractional difference intensively and obtained a series of excellent results; see [20–26]. Bastos et al. in [28, 29] considered the discrete fractional calculus of variations and established the necessary conditions for fractional difference variational problems. Abdeljawad introduced the Caputo fractional difference and developed some useful properties of it in [30]. Ferreira [35] investigated the existence and uniqueness of solutions for some discrete fractional nonresonance boundary value problems of order less than one by using the Banach fixed point theorem.

Although the solvability of fractional boundary value problems has been studied extensively, there are few papers dealing with it under resonance conditions, besides [37–41]. Additionally, as far as we know, the existence of solutions to discrete fractional boundary value problems at resonance has not been studied.

Motivated by the aforementioned results, we will investigate the discrete fractional boundary value problem (1) at resonance and establish some sufficient conditions for the existence of solutions to it by using the coincidence degree theory.

For the sake of convenience, we will always assume that the following conditions hold in this paper:, , and ; is continuous.

The remainder of this paper is organized as follows. Section 2 preliminarily provides some necessary basic knowledge for the theory of discrete fractional calculus and the coincidence degree continuation theorem. In Section 3, the existence result of solutions for problem (1) will be established with the help of the coincidence degree theory. Finally in Section 4, a concrete example is provided to illustrate the possible application of the established analytical result.

#### 2. Preliminaries

Since the theory of discrete fractional calculus is in its infancy to some extent, in order to make this paper self-contained, we begin by presenting here some necessary basic definitions and lemmas about it. For more details, see [15, 16, 19, 34].

*Definition 1 (see [15]). *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 [42]). *The th discrete fractional sum of a function , for , is defined by
Also, we define the trivial sum , .

*Definition 3 (see [15]). *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 the 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 . Also, it is worth reminding the reader that the in (or ) represents an input for the function (or ) and not for the function . For ease of notation, we throughout this paper omit the subscript in and when it does not to lead to domains confusion and general ambiguity.

Lemma 5 (see [19]). *Let . Then , where , , and is the smallest integer greater than or equal to .*

Lemma 6 (see [21]). *Let and such that is well defined. Then
*

Next, we will briefly recall some notations in the frame of Mawhin’s coincidence degree continuous theorem. For more details, see [43].

Let and be two real Banach spaces. Consider an operator equation , where is a linear operator and is a nonlinear operator. The operator will be called a Fredholm operator of index zero if and is closed in . If is a Fredholm operator of index zero, then there exist continuous projectors and such that , and , . It follows that is invertible and its inverse is denoted by .

If is an open bounded subset of and , the operator will be called -compact on if is bounded and is compact.

Now, we present the coincidence degree continuation theorem as follows, which will be used in the sequel to establish the existence of solutions for problem (1).

Theorem 7 (see [43]). *Let be a Fredholm operator of index zero and let be -compact on . Assume that the following conditions are satisfied:*(i)* for every ;*(ii)* for every ;*(iii)*, where is a projection such that .**Then the operator equation has at least one solution in .*

Finally, we wish to fix our framework for the study of problem (1). First of all, we denote and , and it is clear that and are two Banach spaces when equipped with the usual maximum norm; that is, for any and , and . Next, we define the linear operator by with and the nonlinear operator as Then the problem (1) is equivalent to an operator equation

#### 3. Main Results

In this section, we will establish the existence of at least one solution for the problem (1). To accomplish this, we firstly present here several lemmas which will be used in the sequel.

For convenience, we define the operator by and by Lemma 6, we can find that .

Lemma 8. *If holds, then
**
where and are defined by (8) and (12), respectively.*

*Proof . *At first, in view of Lemma 5 and , we can easily verify that (13) holds. Next, we prove that (14) also holds.

For any , then there exists a function such that . Based on Lemma 5, we have
where , . From conditions and , we can easily obtain that .

Conversely, for any with , if we set , then , and it is easy to verify that . Moreover, by the relation , we have , which lead to . So we get that (14) holds. The proof is complete.

Lemma 9. *If holds, then defined by (8) is a Fredholm operator of index zero.*

*Proof . *In order to show that is a Fredholm operator of index zero, firstly, we consider the following operator defined by
where is defined by (12). Evidently, and is a continuous linear projector. In fact, for any , we have
that is to say, is idempotent. Hence, is a projector.

From the definition of and (14), it is easy to see that leads to , and if , we can get , which implies that . So, we derive .

For any , set . Since and , we have . Moreover, take . Then can be written as , , for . On the other hand, since , by (14), we can get , which implies that . So, we have and .

Now, since and is closed in , is a Fredholm operator of index zero. The proof is complete.

Let be defined by It is clear that is a linear continuous projector and Also, proceeding as the proof of Lemma 9, we can show that

Define operator by where .

From the definitions of and , it is easy to see that the inverse of is . In fact, if , then we have Also, if , by Lemma 5, we have where . Then it follows from and that which implies that . Consequently, we have for . So, .

Lemma 10. *Suppose that holds. If is an open bounded subset and , then is -compact on .*

*Proof. *By the continuity of , we can verify that and are bounded. So we get that is compact. Therefore is -compact on . The proof is complete.

To establish the main result, we need the following conditions.There exist two nonnegative functions with such that There exists a constant such that for each satisfying , .There exists a constant such that for any , , if , then either or

Theorem 11. *If hold, then the problem (1) has at least one solution in .*

*Proof. *This proof will be divided into four main steps. Now let us prove the steps one by one.*Step **1*. Set and prove that is bounded in .

For , then and , so . By (14), we have . From , there exists a constant such that . Since , by Lemma 5, we have
Considering , we get
Consequently, by (29), (30), Lemma 6, , and the monotonicity of functions and on , we have
So, by the fact that in and (31), we can derive that is bounded.*Step **2*. Set and prove that is bounded in .

For any , there exists a constant such that , , and . So it follows from (14) that . By virtue of and the fact that is decreasing for on , we can derive that , which implies that is bounded in . *Step **3*. Set and prove that is bounded in , where is a linear isomorphism defined by
For any , there exists such that
If , then . Hence . By , we get . If , then . For , if , then, from , we can obtain that
Therefore, we have
which is a contradiction. So, is bounded in . *Step **4*. Let be a bounded open set such that and prove that
It follows from Lemma 10 that is -compact on . Then by Steps 1 and 2, we have (i) for every ;(ii) for every .

At last, we prove that condition (iii) of Theorem 7 is satisfied. Let
According to the arguments in Step 3, we have
and therefore, via the homotopy property of degree, we get that
which implies that condition (iii) of Theorem 7 is satisfied. Then by Theorem 7, we can conclude that has at least one solution in ; that is, (1) has at least one solution in . The proof is completed.

#### 4. An Illustrative Example

In this section, we will illustrate the possible application of the above established analytical result with a concrete example.

*Example 1. *Consider the following discrete fractional boundary value problem:
where
It is obvious that is continuous. Corresponding to problem (1), there exist , , , , , and
Therefore, the problem (40) is at resonance.

Choosing , , , then we have Hence, the condition in Theorem 11 holds.

Let . If holds for any , then we can easily verify that which implies that condition of Theorem 11 holds.

Furthermore, we can choose , to show that the condition of Theorem 11 holds. In fact, for any satisfying , we can get that So, by the fact that for and (45), we can derive that, for any satisfying , which implies that (27) in of Theorem 11 holds. Therefore, all conditions of Theorem 11 hold. Hence, we can conclude that problem (40) has at least one solution.

#### Conflict of Interests

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

#### Acknowledgments

The author sincerely thanks the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original paper. The research is supported by the Longdong University Grant XYZK-1402.

#### 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. View at: MathSciNet - W. G. Glockle and T. F. Nonnenmacher, “A fractional calculus approach to self-similar protein dynamics,”
*Biophysical Journal*, vol. 68, no. 1, pp. 46–53, 1995. View at: Publisher Site | Google Scholar - R. Metzler, W. Schick, H. 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. View at: Publisher Site | Google Scholar - I. Podlubny,
*Fractional Differential Equations*, vol. 198 of*Mathematics in Science and Engineering*, Academic Press, San Diego, Calif, USA, 1999. View at: MathSciNet - R. Hilfer,
*Applications of Fractional Calculus in Physics*, World Scientific, Singapore, 2000. View at: Publisher Site | MathSciNet - 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. View at: Publisher Site | Google Scholar | MathSciNet - 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. View at: Publisher Site | Google Scholar | MathSciNet - J. Sabatier, O. P. Agrawal, and J. A. Tenreiro Machado,
*Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering*, Springer, Dordrecht, The Netherlands, 2007. View at: Publisher Site | MathSciNet - G. S. F. Frederico and D. F. M. Torres, “Fractional conservation laws in optimal control theory,”
*Nonlinear Dynamics*, vol. 53, no. 3, pp. 215–222, 2008. View at: Publisher Site | Google Scholar | MathSciNet - 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. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Zhang, “Existence of positive solution for some class of nonlinear fractional differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 278, no. 1, pp. 136–148, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z. Bai and H. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,”
*Journal of Mathematical Analysis and Applications*, vol. 311, no. 2, pp. 495–505, 2005. View at: Publisher Site | Google Scholar | MathSciNet - C. Bai, “Triple positive solutions for a boundary value problem of nonlinear fractional differential equation,”
*Electronic Journal of Qualitative Theory of Differential Equations*, no. 24, 10 pages, 2008. View at: Google Scholar | MathSciNet - 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. View at: Publisher Site | Google Scholar | MathSciNet - F. M. Atıcı and P. W. Eloe, “A transform method in discrete fractional calculus,”
*International Journal of Difference Equations*, vol. 2, no. 2, pp. 165–176, 2007. View at: Google Scholar | MathSciNet - F. M. Atıcı 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. View at: Publisher Site | Google Scholar | MathSciNet - F. M. Atıcı and P. Eloe, “Discrete fractional calculus with the nabla operator,”
*Electronic Journal of Qualitative Theory of Differential Equations*, no. 3, pp. 1–12, 2009. View at: Google Scholar - 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. View at: Publisher Site | Google Scholar | MathSciNet - 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. View at: Publisher Site | Google Scholar | MathSciNet - 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. View at: Google Scholar | MathSciNet - C. S. Goodrich, “Continuity of solutions to discrete fractional initial value problems,”
*Computers & Mathematics with Applications*, vol. 59, no. 11, pp. 3489–3499, 2010. View at: Publisher Site | Google Scholar | MathSciNet - C. S. Goodrich, “Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions,”
*Computers & Mathematics with Applications*, vol. 61, no. 2, pp. 191–202, 2011. View at: Publisher Site | Google Scholar | MathSciNet - 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. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. S. Goodrich, “On a discrete fractional three-point boundary value problem,”
*Journal of Difference Equations and Applications*, vol. 18, no. 3, pp. 397–415, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. S. Goodrich, “On discrete sequential fractional boundary value problems,”
*Journal of Mathematical Analysis and Applications*, vol. 385, no. 1, pp. 111–124, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. S. Goodrich, “On a first-order semipositone discrete fractional boundary value problem,”
*Archiv der Mathematik*, vol. 99, no. 6, pp. 509–518, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. Chen, X. Luo, and Y. Zhou, “Existence results for nonlinear fractional difference equation,”
*Advances in Difference Equations*, vol. 2011, Article ID 713201, 12 pages, 2011. View at: Publisher Site | Google Scholar | MathSciNet - 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. View at: Publisher Site | Google Scholar | Zentralblatt MATH - N. R. O. Bastos, R. A. C. Ferreira, and D. F. M. Torres, “Necessary optimality conditions for fractional difference problems of the calculus of variations,”
*Discrete and Continuous Dynamical Systems*, vol. 29, no. 2, pp. 417–437, 2011. View at: Publisher Site | Google Scholar | MathSciNet - T. Abdeljawad, “On Riemann and Caputo fractional differences,”
*Computers & Mathematics with Applications*, vol. 62, no. 3, pp. 1602–1611, 2011. View at: Publisher Site | Google Scholar | MathSciNet - T. Abdeljawad, “Dual identities in fractional difference calculus within Riemann,”
*Advances in Difference Equations*, vol. 2013, article 36, 2013. View at: Publisher Site | Google Scholar | MathSciNet - 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. View at: Publisher Site | Google Scholar | MathSciNet - 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. View at: Publisher Site | Google Scholar | MathSciNet - M. Holm, “Sum and difference compositions in discrete fractional calculus,”
*Cubo*, vol. 13, no. 3, pp. 153–184, 2011. View at: Publisher Site | Google Scholar | MathSciNet - R. A. C. Ferreira, “Existence and uniqueness of solution to some discrete fractional boundary value problems of order less than one,”
*Journal of Difference Equations and Applications*, vol. 19, no. 5, pp. 712–718, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. A. C. Ferreira and C. S. Goodrich, “Positive solution for a discrete fractional periodic boundary value problem,”
*Dynamics of Continuous, Discrete & Impulsive Systems*, vol. 19, no. 5, pp. 545–557, 2012. View at: Google Scholar | MathSciNet - Z. Bai and Y. Zhang, “Solvability of fractional three-point boundary value problems with nonlinear growth,”
*Applied Mathematics and Computation*, vol. 218, no. 5, pp. 1719–1725, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Zhang, Z. Bai, and T. Feng, “Existence results for a coupled system of nonlinear fractional three-point boundary value problems at resonance,”
*Computers & Mathematics with Applications*, vol. 61, no. 4, pp. 1032–1047, 2011. View at: Publisher Site | Google Scholar | MathSciNet - W. Jiang, “The existence of solutions to boundary value problems of fractional differential equations at resonance,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 74, no. 5, pp. 1987–1994, 2011. View at: Publisher Site | Google Scholar | MathSciNet - W. Jiang, “Solvability for a coupled system of fractional differential equations at resonance,”
*Nonlinear Analysis: Real World Applications*, vol. 13, no. 5, pp. 2285–2292, 2012. View at: Publisher Site | Google Scholar | MathSciNet - Y. Chen and X. Tang, “Solvability of sequential fractional order multi-point boundary value problems at resonance,”
*Applied Mathematics and Computation*, vol. 218, no. 14, pp. 7638–7648, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K. S. Miller and B. Ross, “Fractional difference calculus,” in
*Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, Nihon University, Koriyama, Japan*, Ellis Horwood Ser. Math. Appl., pp. 139–152, Horwood, Chichester, UK, 1989. View at: Google Scholar - J. Mawhin,
*Topological Degree Methods in Nonlinear Boundary Value Problems*, NSFCBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1979.

#### Copyright

Copyright © 2014 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.