Existence of solutions for discrete fractional boundary value problems with a p-Laplacian operator
© Lv; licensee Springer 2012
Received: 10 May 2012
Accepted: 30 August 2012
Published: 18 September 2012
This paper is concerned with the existence of solutions to a discrete fractional boundary value problem with a p-Laplacian operator. Under certain nonlinear growth conditions of the nonlinearity, the existence result is established by using Schaefer’s fixed point theorem. Additionally, a representative example is presented to illustrate the effectiveness of the main result.
MSC:26A33, 39A05, 39A12.
For any number and each interval I of , we denote and throughout this paper. It is also worth noting that, in what follows, we appeal to the convention that for any , where u is a function defined on .
where , , , , and denote the Caputo fractional differences of order α and β respectively, is a continuous function and is the p-Laplacian operator, that is, , . Obviously, is invertible and its inverse operator is , where is a constant such that .
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 extensive applications in various fields of science, such as physics, mechanics, chemistry, engineering, etc. For more details, see [1–14] and references therein. Although the discrete fractional calculus has seen slower progress, within the recent several years, a lot of papers have appeared; see [15–35]. For example, a recent paper by Atıcı and Eloe  explores 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 [20–25]. Particularly note that Abdeljawad introduced the conception of Caputo fractional difference and presented some useful properties of it in .
p-Laplacian boundary value problems for ordinary differential equations, finite difference equations and dynamic equations have been studied extensively, but there are few papers dealing with the fractional p-Laplacian boundary value problems, besides [36–38], especially for discrete fractional p-Laplacian boundary value problems involving Caputo fractional differences.
Motivated by the aforementioned works, we will consider the existence of solutions to the discrete fractional boundary value problem (1.1) and establish the sufficient conditions for the existence of at least one solution to it by using Schaefer’s fixed point theorem.
The remainder of this paper is organized as follows. Section 2 preliminarily provides some necessary background material for the theory of discrete fractional calculus. In Section 3, the main existence result for problem (1.1) is established with the help of Schaefer’s fixed point theorem. Finally, in Section 4, a concrete example is provided to illustrate the possible application of the established analytical result.
For convenience, we first present here some necessary basic definitions on discrete fractional calculus.
Definition 2.1 
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 2.2 
provided that the right-hand side is well defined. We employ the convention that if t is a pole of the Gamma function and is not a pole, then .
Definition 2.3 
Definition 2.4 
Next, we present here several lemmas which will be important in the sequel.
Lemma 2.1 
where n is the smallest integer greater than or equal to ν.
Therefore, (2.1) also holds for , which implies that (2.1) holds for any positive integer. □
In view of Lemma 2.1 and Lemma 2.2, the following fact is obvious.
where , , and n is the smallest integer greater than or equal to ν.
Finally, we need the following additional lemma that will be used in Section 3 of this paper.
Lemma 2.4 
3 Existence result
In this section, we will establish the existence of at least one solution for problem (1.1). To accomplish this, we first state and prove the following result which is of particular importance in what follows.
for some , .
where , .
The proof is complete. □
To prove the main result, we need Schaefer’s fixed point theorem.
Lemma 3.2 
is bounded. Then Φ has a fixed point in E.
It is easy to verify that the operator is well defined, and the fixed points of the operator are solutions of problem (1.1).
Now, the main result is stated as follows.
Proof The proof will be divided into the following two steps.
Step 1: is completely continuous.
At first, in view of the continuity of f, it is easy to verify that is continuous. Furthermore, it is not difficult to verify that maps bounded sets into bounded sets and equi-continuous sets. Therefore, in the light of the well-known Arzelá-Ascoli theorem, we know that is a compact operator.
Step 2: a priori bounds.
Now, it remains to show that the set S is bounded.
Here we will consider the following two cases: (1) or (2) .
Case 1: Suppose that . Then it is evident that the set S is bounded from (3.5).
Consequently, in both Case 1 and Case 2, we have proved that the set S is bounded.
Then, we can see that satisfies all conditions of Schaefer’s fixed point theorem. Thus, we approach a conclusion that has at least one fixed point which is the solution of problem (1.1). The proof is complete. □
4 An illustrative example
In this section, we will illustrate the possible application of the above established analytical result with a concrete example.
Obviously, problem (4.1) satisfies all assumptions of Theorem 3.1. Hence, we can conclude that problem (4.1) has at least one solution.
The author would like to express his thanks to the referees for their helpful comments and suggestions. This work was supported by the Longdong University Grant XYZK-1010 and XYZK-1007.
- Samko S, Kilbas A, Marichev O: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Yverdon; 1993.MATHGoogle Scholar
- Glöckle W, Nonnenmacher T: A fractional calculus approach to self-similar protein dynamics. Biophys. J. 1995, 68(1):46–53. 10.1016/S0006-3495(95)80157-8View ArticleGoogle Scholar
- Metzler R, Schick W, Kilian H, Nonnenmacher T: Relaxation in filled polymers: a fractional calculus approach. J. Chem. Phys. 1995, 103(16):7180–7186. 10.1063/1.470346View ArticleGoogle Scholar
- Podlubny I Mathematics in Science and Engineering 198. In Fractional Differential Equations. Academic Press, San Diego; 1999.Google Scholar
- Hilfer R: Applications of Fractional Calculus in Physics. World Scientific, Singapore; 2000.View ArticleMATHGoogle Scholar
- Kilbas A, Trujillo J: Differential equations of fractional order: methods, results and problems-I. Appl. Anal. 2001, 78(1–2):153–192. 10.1080/00036810108840931MathSciNetView ArticleMATHGoogle Scholar
- Kilbas A, Trujillo J: Differential equations of fractional order: methods, results and problems-II. Appl. Anal. 2002, 81(2):435–493. 10.1080/0003681021000022032MathSciNetView ArticleMATHGoogle Scholar
- Sabatier J, Agrawal O, Machado Tenreiro J: Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht; 2007.View ArticleMATHGoogle Scholar
- Frederico G, Torres D: Fractional conservation laws in optimal control theory. Nonlinear Dyn. 2008, 53(3):215–222. 10.1007/s11071-007-9309-zMathSciNetView ArticleMATHGoogle Scholar
- Zhang S: The existence of a positive solution for a nonlinear fractional differential equation. J. Math. Anal. Appl. 2000, 252(2):804–812. 10.1006/jmaa.2000.7123MathSciNetView ArticleMATHGoogle Scholar
- Zhang S: Existence of positive solution for some class of nonlinear fractional differential equations. J. Math. Anal. Appl. 2003, 278(1):136–148. 10.1016/S0022-247X(02)00583-8MathSciNetView ArticleMATHGoogle Scholar
- Bai Z, Lü H: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 2005, 311(2):495–505. 10.1016/j.jmaa.2005.02.052MathSciNetView ArticleMATHGoogle Scholar
- Bai C: Triple positive solutions for a boundary value problem of nonlinear fractional differential equation. Electron. J. Qual. Theory Differ. Equ. 2008, 2008(24):1–10.View ArticleMathSciNetGoogle Scholar
- Xu X, Jiang D, Yuan C: Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation. Nonlinear Anal. TMA 2009, 71(10):4678–4688.MathSciNetGoogle Scholar
- Atıcı F, Eloe P: A transform method in discrete fractional calculus. Int. J. Differ. Equ. 2007, 2(2):165–176.MathSciNetGoogle Scholar
- Atıcı F, Eloe P: Initial value problems in discrete fractional calculus. Proc. Am. Math. Soc. 2009, 137(3):981–989.MathSciNetMATHGoogle Scholar
- Atıcı F, Eloe P: Discrete fractional calculus with the nabla operator. Electron. J. Qual. Theory Differ. Equ. 2009, 2009(3):1–12.MathSciNetMATHGoogle Scholar
- Atıcı F, Şengül S: Modeling with fractional difference equations. J. Math. Anal. Appl. 2010, 369(1):1–9. 10.1016/j.jmaa.2010.02.009MathSciNetView ArticleMATHGoogle Scholar
- Atıcı F, Eloe P: Two-point boundary value problems for finite fractional difference equations. J. Differ. Equ. Appl. 2011, 17(4):445–456. 10.1080/10236190903029241View ArticleMathSciNetMATHGoogle Scholar
- Goodrich C: Solutions to a discrete right-focal fractional boundary value problem. Int. J. Differ. Equ. 2010, 5(2):195–216.MathSciNetGoogle Scholar
- Goodrich C: Continuity of solutions to discrete fractional initial value problems. Comput. Math. Appl. 2010, 59(11):3489–3499. 10.1016/j.camwa.2010.03.040MathSciNetView ArticleMATHGoogle Scholar
- Goodrich C: Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions. Comput. Math. Appl. 2011, 61(2):191–202. 10.1016/j.camwa.2010.10.041MathSciNetView ArticleMATHGoogle Scholar
- Goodrich C: Existence of a positive solution to a system of discrete fractional boundary value problems. Appl. Math. Comput. 2011, 217(9):4740–4753. 10.1016/j.amc.2010.11.029MathSciNetView ArticleMATHGoogle Scholar
- Goodrich C: On a discrete fractional three-point boundary value problem. J. Differ. Equ. Appl. 2012, 18(3):397–415. 10.1080/10236198.2010.503240MathSciNetView ArticleMATHGoogle Scholar
- Goodrich C: On discrete sequential fractional boundary value problems. J. Math. Anal. Appl. 2012, 385(1):111–124. 10.1016/j.jmaa.2011.06.022MathSciNetView ArticleMATHGoogle Scholar
- Chen F, Luo X, Zhou Y: Existence results for nonlinear fractional difference equation. Adv. Differ. Equ. 2011., 2011: Article ID 713201Google Scholar
- Bastos N, Ferreira R, Torres D: Discrete-time fractional variational problems. Signal Process. 2011, 91(3):513–524. 10.1016/j.sigpro.2010.05.001View ArticleMATHGoogle Scholar
- Abdeljawad T: On Riemann and Caputo fractional differences. Comput. Math. Appl. 2011, 62(3):1602–1611. 10.1016/j.camwa.2011.03.036MathSciNetView ArticleMATHGoogle Scholar
- Chen F: Fixed points and asymptotic stability of nonlinear fractional difference equations. Electron. J. Qual. Theory Differ. Equ. 2011, 2011(39):1–18.MathSciNetGoogle Scholar
- Ferreira R: A discrete fractional Gronwall inequality. Proc. Am. Math. Soc. 2012, 140(5):1605–1612. 10.1090/S0002-9939-2012-11533-3View ArticleMathSciNetMATHGoogle Scholar
- Ferreira R, Torres D: Fractional h -difference equations arising from the calculus of variations. Appl. Anal. Discrete Math. 2011, 5(1):110–121. 10.2298/AADM110131002FMathSciNetView ArticleMATHGoogle Scholar
- Holm M: Sum and difference compositions in discrete fractional calculus. CUBO 2011, 13(3):153–184. 10.4067/S0719-06462011000300009MathSciNetView ArticleMATHGoogle Scholar
- Abdeljawad T, Jarad F, Baleanu D: A semigroup-like property for discrete Mittag-Leffler functions. Adv. Differ. Equ. 2012., 2012: Article ID 72Google Scholar
- Jarad F, Abdeljawad T, Baleanu D, Bicen K: On the stability of some discrete fractional non-autonomous systems. Abstr. Appl. Anal. 2012., 2012: Article ID 476581Google Scholar
- Abdeljawad T, Baleanu D: Fractional differences and integration by parts. J. Comput. Anal. Appl. 2011, 13(3):574–582.MathSciNetMATHGoogle Scholar
- Wang J, Xiang H, Liu Z: Positive solutions for three-point boundary value problems of nonlinear fractional differential equations with p -Laplacian. Far East J. Appl. Math. 2009, 37(1):33–47.MathSciNetMATHGoogle Scholar
- Wang J, Xiang H: Upper and lower solutions method for a class of singular fractional boundary value problems with p -Laplacian operator. Abstr. Appl. Anal. 2010., 2010: Article ID 971824Google Scholar
- Cai G: Positive solutions for boundary value problem for fractional differential equation with p -Laplacian operator. Bound. Value Probl. 2012., 2012: Article ID 18Google Scholar
- Miller KS, Ross B: Fractional difference calculus. Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications 1989, 139–152.Google Scholar
- Lloyd N: Degree Theory. Cambridge University Press, Cambridge; 1978.MATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.