- Open Access
Boundary value problems for fractional difference equations with three-point fractional sum boundary conditions
© Sitthiwirattham et al.; licensee Springer. 2013
- Received: 8 May 2013
- Accepted: 17 September 2013
- Published: 7 November 2013
In this paper, we consider a discrete fractional boundary value problem of the form
where , , and is a continuous function. Existence and uniqueness of the solutions are proved by using the contraction mapping theorem, the nonlinear contraction theorem and Schaefer’s fixed point theorem. Some illustrative examples are also presented.
- fractional difference equations
- boundary value problem
- fixed point theorems
Fractional calculus is an emerging field recently drawing attention from both theoretical and applied disciplines. Fractional order differential equations play a vital role in describing many phenomena related to physics, chemistry, mechanics, control systems, flow in porous media, electrical networks, mathematical biology and viscoelasticity. For a reader interested in the systematic development of the topic, we refer to the books [1–3]. A variety of results on initial and boundary value problems of fractional differential equations and inclusions can easily be found in the literature on the topic. For some recent results, we can refer to [4–12] and references cited therein.
Discrete fractional calculus and fractional difference equations represent a very new area for researchers. Some real-world phenomena are being studied with the help of discrete fractional operators. A good account of papers dealing with discrete fractional boundary value problems can be found in [13–26] and references cited therein.
where , is a continuous function and is a given function. Existence and uniqueness of solutions are obtained by the contraction mapping theorem, the Brouwer theorem and the Guo-Krasnoselskii theorem.
where , so that , , so that , is continuous and nonnegative for , and is a given function. Existence and uniqueness of solutions are obtained by the contraction mapping theorem and the Brouwer theorem.
where , , and is a continuous function.
The plan of this paper is as follows. In Section 2 we recall some definitions and basic lemmas. Also, we derive a representation for the solution to (1.3) by converting the problem to an equivalent summation equation. In Section 3, using this representation, we prove existence and uniqueness of the solutions of boundary value problem (1.3) by the help of the contraction mapping theorem, the nonlinear contraction theorem and Schaefer’s fixed point theorem. Some illustrative examples are presented in Section 4.
In this section, we introduce notations, definitions and lemmas which are used in the main results.
Definition 2.1 We define the generalized falling function by , for any t and α, for which the right-hand side is defined. If is a pole of the gamma function and is not a pole, then .
for and . We also define the α th fractional difference for by , where and is chosen so that .
Lemma 2.1 Let t and α be any numbers for whichandare defined. Then.
for some, with.
To define the solution of boundary value problem (1.3), we need the following lemma which deals with linear variant of boundary value problem (1.3) and gives a representation of the solution.
Substituting a constant into (2.6), we obtain (2.3). □
for , where is defined by (2.4). It is easy to see that problem (1.3) has solutions if and only if the operator F has fixed points.
then problem (1.3) has a unique solution in.
Therefore, F is a contraction. Hence, by the Banach fixed point theorem, we get that F has a fixed point which is a unique solution of problem (1.3) on . □
Next, we can still deduce the existence of a solution to (1.3). We shall use nonlinear contraction to accomplish this.
Lemma 3.1 (Boyd and Wong )
Let E be a Banach space and letbe a nonlinear contraction. Then F has a unique fixed point in E.
and Λ is defined in (2.4).
Then boundary value problem (1.3) has a unique solution.
such that and for all .
for . From (3.4), it follows that . Hence F is a nonlinear contraction. Therefore, by Lemma 3.1, the operator F has a unique fixed point in , which is a unique solution of problem (1.3). □
The following result is based on Schaefer’s fixed point theorem.
Theorem 3.3 Suppose that there exists a constantsuch thatfor eachand all.
Then problem (1.3) has at least one solution on.
Proof We shall use Schaefer’s fixed point theorem to prove that the operator F defined by (3.1) has a fixed point. We divide the proof into four steps.
Since f is a continuous function, we have as . This means that F is continuous.
where Ω is defined by (3.3).
This means that the set is an equicontinuous set. As a consequence of Steps I to III together with the Arzelá-Ascoli theorem, we get that is completely continuous.
This shows that the set E is bounded. As a consequence of Schaefer’s fixed point theorem, we conclude that F has a fixed point which is a solution of problem (1.3). □
In this section, in order to illustrate our result, we consider some examples.
Hence, by Theorem 3.1, boundary value problem (4.1)-(4.3) has a unique solution.
Here , , , , , , . It is clear that for . Thus, we conclude from Theorem 3.3 that (4.4)-(4.6) has at least one solution.
The third author is a member of Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
The present paper was done while J Tariboon and T Sitthiwirattham visited the Department of Mathematics of the University of Ioannina, Greece. It is a pleasure for them to thank Professor SK Ntouyas for his warm hospitality. This research of J Tariboon and T Sitthiwirattham is supported by King Mongkut’s University of Technology North Bangkok, Thailand.
- Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.Google Scholar
- Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies 204. In Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.Google Scholar
- Baleanu D, Diethelm K, Scalas E, Trujillo JJ Series on Complexity, Nonlinearity and Chaos. In Fractional Calculus Models and Numerical Methods. World Scientific, Boston; 2012.Google Scholar
- Guezane-Lakoud A, Khaldi R: Solvability of a three-point fractional nonlinear boundary value problem. Differ. Equ. Dyn. Syst. 2012, 20: 395-403. 10.1007/s12591-012-0125-7MathSciNetView ArticleGoogle Scholar
- Guezane-Lakoud A, Khaldi R: Positive solution to a higher order fractional boundary value problem with fractional integral condition. Rom. J. Math. Comput. Sci. 2012, 2: 41-54.MathSciNetGoogle Scholar
- Kaufmann E: Existence and nonexistence of positive solutions for a nonlinear fractional boundary value problem. Discrete Contin. Dyn. Syst. 2009, 2009: 416-423. suppl.MathSciNetGoogle Scholar
- Wang J, Xiang H, Liu Z: Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations. Int. J. Differ. Equ. 2010., 2010: Article ID 186928Google Scholar
- Bai Z: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. 2010, 72: 916-924. 10.1016/j.na.2009.07.033MathSciNetView ArticleGoogle Scholar
- Sudsutad W, Tariboon J: Boundary value problems for fractional differential equations with three-point fractional integral boundary conditions. Adv. Differ. Equ. 2012., 2012: Article ID 93Google Scholar
- Ntouyas SK: Boundary value problems for nonlinear fractional differential equations and inclusions with nonlocal and fractional integral boundary conditions. Opusc. Math. 2013, 33: 117-138. 10.7494/OpMath.2013.33.1.117MathSciNetView ArticleGoogle Scholar
- Guezane-Lakoud A, Khaldi R: Solvability of a fractional boundary value problem with fractional integral condition. Nonlinear Anal. 2012, 75: 2692-2700. 10.1016/j.na.2011.11.014MathSciNetView ArticleGoogle Scholar
- Ahmad B, Ntouyas SK, Assolani A: Caputo type fractional differential equations with nonlocal Riemann-Liouville integral boundary conditions. J. Appl. Math. Comput. 2013, 41: 339-350. 10.1007/s12190-012-0610-8MathSciNetView ArticleGoogle Scholar
- Baleanu D, Mustafa OG, O’Regan D: On a fractional differential equation with infinitely many solutions. Adv. Differ. Equ. 2012., 2012: Article ID 145Google Scholar
- Liu Y, Ahmad B, Agarwal RP: Existence of solutions for a coupled system of nonlinear fractional differential equations with fractional boundary conditions on the half line. Adv. Differ. Equ. 2013., 2013: Article ID 46Google Scholar
- Ahmad B, Ntouyas SK, Alsaedi A: A study of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multi-strip boundary conditions. Math. Probl. Eng. 2013., 2013: Article ID 320415Google Scholar
- Zhang L, Ahmad B, Wang G, Agarwal RP: Nonlinear fractional integro-differential equations on unbounded domains in a Banach space. J. Comput. Appl. Math. 2013, 249: 51-56.MathSciNetView ArticleGoogle Scholar
- Ahmadian A, Suleiman M, Salahshour S, Baleanu D: A Jacobi operational matrix for solving a fuzzy linear fractional differential equation. Adv. Differ. Equ. 2013., 2013: Article ID 104Google Scholar
- Ferreira R: Existence and uniqueness of solution to some discrete fractional boundary value problems of order less than one. J. Differ. Equ. Appl. 2013, 19: 712-718. 10.1080/10236198.2012.682577View ArticleGoogle Scholar
- Ahmad B, Nieto JJ: Basic theory of nonlinear third-order q -difference equations and inclusions. Math. Model. Anal. 2013, 18: 122-135. 10.3846/13926292.2013.760012MathSciNetView ArticleGoogle Scholar
- Pongarm N, Asawasamrit S, Tariboon J: Sequential derivatives of nonlinear q -difference equations with three-point q -integral boundary conditions. J. Appl. Math. 2013., 2013: Article ID 605169Google Scholar
- Atici F, Eloe P: Two-point boundary value problems for finite fractional difference equations. J. Differ. Equ. Appl. 2011, 17: 445-456. 10.1080/10236190903029241MathSciNetView ArticleGoogle Scholar
- Goodrich CS: Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions. Comput. Math. Appl. 2011, 61: 191-202. 10.1016/j.camwa.2010.10.041MathSciNetView ArticleGoogle Scholar
- Goodrich CS: Existence of a positive solution to a system of discrete fractional boundary value problems. Appl. Math. Comput. 2011, 217: 4740-4753. 10.1016/j.amc.2010.11.029MathSciNetView ArticleGoogle Scholar
- Goodrich CS: On a discrete fractional three-point boundary value problem. J. Differ. Equ. Appl. 2012, 18: 397-415. 10.1080/10236198.2010.503240MathSciNetView ArticleGoogle Scholar
- Abdeljawad T: On Riemann and Caputo fractional differences. Comput. Math. Appl. 2011, 62: 1602-1611. 10.1016/j.camwa.2011.03.036MathSciNetView ArticleGoogle Scholar
- Pan Y, Han Z, Sun S, Huang Z: The existence and uniqueness of solutions to boundary value problems of fractional difference equations. Math. Sci. 2012., 6: Article ID 7Google Scholar
- Boyd DW, Wong JSW: On nonlinear contractions. Proc. Am. Math. Soc. 1969, 20: 458-464. 10.1090/S0002-9939-1969-0239559-9MathSciNetView ArticleGoogle 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.