Existence Results for Nonlinear Fractional Difference Equation
© Fulai Chen et al. 2011
Received: 27 September 2010
Accepted: 12 December 2010
Published: 20 December 2010
This paper is concerned with the initial value problem to a nonlinear fractional difference equation with the Caputo like difference operator. By means of some fixed point theorems, global and local existence results of solutions are obtained. An example is also provided to illustrate our main result.
Fractional differential equation has received increasing attention during recent years since fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes . However, there are few literature to develop the theory of the analogues fractional finite difference equation [2–6]. Atici and Eloe  developed the commutativity properties of the fractional sum and the fractional difference operators, and discussed the uniqueness of a solution for a nonlinear fractional difference equation with the Riemann-Liouville like discrete fractional difference operator. To the best of our knowledge, this is a pioneering work on discussing initial value problems (IVP for short) in discrete fractional calculus. Anastassiou  defined a Caputo like discrete fractional difference and compared it to the Riemann-Liouville fractional discrete analog.
For convenience of numerical calculations, the fractional differential equation is generally discretized to corresponding difference one which makes that the research about fractional difference equations becomes important. Following the definition of Caputo like difference operator defined in , here we investigate the existence and uniqueness of solutions for the IVP (1.1). A merit of this IVP with Caputo like difference operator is that its initial condition is the same form as one of the integer-order difference equation.
2. Preliminaries and Lemmas
We start with some necessary definitions from discrete fractional calculus theory and preliminary results so that this paper is self-contained.
Here is defined for mod (1) and is defined for mod (1); in particular, maps functions defined on to functions defined on , where . In addition, . Atici and Eloe  pointed out that this definition of the th fractional sum is the development of the theory of the fractional calculus on time scales .
Definition 2.2 (see ).
Theorem 2.3 (see ).
Lemma 2.6 (see ).
The following fixed point theorems will be used in the text.
Theorem 2.7 (Leray-Schauder alternative theorem ).
Theorem 2.8 (Schauder fixed point theorem ).
Theorem 2.9 (Ascoli-Arzela theorem ).
Let be a Banach space, and is a function family of continuous mappings . If is uniformly bounded and equicontinuous, and for any , the set is relatively compact, then there exists a uniformly convergent function sequence in .
Lemma 2.10 (Mazur Lemma ).
3. Local Existence and Uniqueness
Since is convex and compact, we know that . Hence, for any , the set ( ) is relatively compact. From Theorem 2.9, every contains a uniformly convergent subsequence ( ) on which means that the set is relatively compact. Since is a bounded, equicontinuous and relatively compact set, we have that is completely continuous.
4. Global Uniqueness
Example 5.1 is similar to Example 3.1 in  in which the difference operator is in the Riemann-Liouville like discrete sense. Compared with the solution of Example 3.1 in  defined on , where , the solution of Example 5.1 in this paper is defined on . This difference makes that fractional difference equation with the Caputo like difference operator is more similar to classical integer-order difference equation.
This work was supported by the Natural Science Foundation of China (10971173), the Scientific Research Foundation of Hunan Provincial Education Department (09B096), the Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province, and the Construct Program of the Key Discipline in Hunan Province.
- Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.MATHGoogle Scholar
- Atici FM, Eloe PW: Initial value problems in discrete fractional calculus. Proceedings of the American Mathematical Society 2009,137(3):981-989.MathSciNetView ArticleMATHGoogle Scholar
- Atici FM, Eloe PW: A transform method in discrete fractional calculus. International Journal of Difference Equations 2007,2(2):165-176.MathSciNetGoogle Scholar
- Anastassiou GA: Discrete fractional calculus and inequalities. http://arxiv.org/abs/0911.3370
- Atıcı FM, Eloe PW: Discrete fractional calculus with the nabla operator. Electronic Journal of Qualitative Theory of Differential Equations 2009,2009(3):1-12.MathSciNetMATHGoogle Scholar
- Atıcı FM, Şengül S: Modeling with fractional difference equations. Journal of Mathematical Analysis and Applications 2010,369(1):1-9. 10.1016/j.jmaa.2010.02.009MathSciNetView ArticleMATHGoogle Scholar
- Bohner M, Peterson A: Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleMATHGoogle Scholar
- Granas A, Guenther RB, Lee JW: Some general existence principles in the Carathéodory theory of nonlinear differential systems. Journal de Mathématiques Pures et Appliquées 1991,70(2):153-196.MathSciNetMATHGoogle Scholar
- Hale J: Theory of Functional Differential Equations, Applied Mathematical Sciences. Volume 3. 2nd edition. Springer, New York, NY, USA; 1977:x+365.View ArticleGoogle Scholar
- Lakshmikantham V, Leela S: Nonlinear Differential Equations in Abstract Spaces, International Series in Nonlinear Mathematics: Theory, Methods and Applications. Volume 2. Pergamon Press, Oxford, UK; 1981:x+258.Google Scholar
- Rudin W: Functional Analysis, McGraw-Hill Series in Higher Mathematics. McGraw-Hill, New York, NY, USA; 1973:xiii+397.Google Scholar
This article is published under license to BioMed Central Ltd. 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.