The existence of solutions to a class of boundary value problems with fractional difference equations
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In this paper, we study the existence and uniqueness of solutions for the boundary value problem of fractional difference equations
respectively, where , , is a continuous function and is a continuous functional. We prove the existence and uniqueness of a solution to the first problem by the contraction mapping theorem and the Brouwer theorem. Moreover, we present the existence and nonexistence of a solution to the second problem in terms of the parameter λ by the properties of the Green function and the Guo-Krasnosel’skii theorem. Finally, we present some examples to illustrate the main results.
MSC:34A08, 34B18, 39A12.
Keywordsdiscrete fractional equation boundary value problem existence and uniqueness of solution fixed point theorem eigenvalue
In recent years, fractional differential equations have been of great interest. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in various sciences such as physics, mechanics, chemistry and engineering. Mathematicians have employed this fractional calculus in recent years to model and solve a variety of applied problems. Indeed, as Podlubny outlines in , fractional calculus aids significantly in the fields of viscoelasticity, capacitor theory, electrical circuits, electro-analytical chemistry, neurology, diffusion, control theory and statistics.
The continuous fractional calculus has developed greatly in the last decades. Some of the recent progress in the continuous fractional calculus includes the paper , in which the authors explored a continuous fractional boundary value problem of conjugate type using cone theory, they then deduced the existence of one or more positive solutions. Of particular interest with regard to the present paper is the recent work by Benchohra et al. . In that paper, the authors considered a continuous fractional differential equation with nonlocal conditions. Other recent work in the direction of those articles may be found, for example, in [4, 5, 6, 7, 8, 9, 10, 11, 12].
In recent years, a number of papers on the discrete fractional calculus have appeared, such as [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], which has helped to build up some of the basic theory of this area. For example, Atici and Eloe discussed the properties of the generalized falling function, a corresponding power rule for fractional delta-operators and the commutativity of fractional sums in . They presented in  more rules for composing fractional sums and differences. Goodrich studied a two-point fractional boundary value problem in , which gave the existence results for a certain two-point boundary value problem of right-focal type for a fractional difference equation. At the same time, a number of papers appeared, and these began to build up the theoretical foundations of the discrete fractional calculus. For example, a recent paper by Atici and Eloe  explored some of the theories of a conjugate discrete fractional boundary value problem. Discrete fractional initial value problems were considered in a paper by Atici and Eloe .
where is a real number, is an integer and is continuous. They analyzed the corresponding Green function, provided an application and obtained sufficient conditions for the existence of positive solutions for a two-point boundary value problem for a nonlinear finite fractional difference equation.
where , is a continuous function, is a given functional, and . He established the existence and uniqueness of a solution to this problem by the contraction mapping theorem, the Brouwer fixed point theorem and the Guo-Krasnosel’skii fixed point theorem.
Although the boundary value problem of fractional difference equations has been studied by several authors, the present works are almost all concerned with , very little is known in the literature about a fractional difference equation with .
where , , is continuous and is a continuous functional.
where , , is continuous and is a continuous functional, λ is a positive parameter. We establish some sufficient conditions for the nonexistence and existence of at least one or two positive solutions for the boundary value problem by considering the eigenvalue intervals of the nonlinear fractional differential equation with boundary conditions.
The plan of this paper is as follows. We first give the form of solutions of problem (1.1), second we prove the existence and uniqueness of a solution to problem (1.1) by the contraction mapping theorem and the Brouwer theorem, and then the eigenvalue intervals for the boundary value problem of nonlinear fractional difference equation (1.2) are considered by the properties of the Green function and the Guo-Krasnosel’skii fixed point theorem on cones. Finally we present some examples to illustrate the main results.
Definition 2.1 
We define for any t and ν, for which the right-hand side is defined. We also appeal to the convention that if is a pole of the gamma function and is not a pole, then .
Definition 2.2 
for . We also define the ν th fractional difference for by , where and is chosen so that .
Lemma 2.1 
If , then for any .
Lemma 2.2 
Let . Then for some with .
Lemma 2.3 
Now let us consider a linear boundary value problem, which is important for us to facilitate the analysis of problems (1.1) and (1.2).
we deduce .
it follows that .
This shows that if (2.1)-(2.4) has a solution, then it can be represented by (2.7) and that every function of the form (2.7) is a solution of (2.1)-(2.4), which completes the proof. □
- (iii)There exists a positive number such that
The proof of this theorem is similar to that of Theorem 3.2 in . Hence, we omit the proof here.
3 Existence and uniqueness of solution
for . We use this fact to prove the first existence theorem.
holds, then problem (1.1) has a unique solution.
Then condition (3.2) holds. We find that (1.1) has a unique solution, which completes the proof of the theorem. □
By weakening the conditions imposed on and , we can still obtain the existence of a solution to (1.1). We apply the Brouwer theorem to accomplish this.
Then (1.1) has at least one solution satisfying for all .
Proof Consider the Banach space . T is defined as (3.1). It is obvious that T is a continuous operator. Therefore, our main objective is to show that . That is, whenever , it follows that . Once this is established, we use the Brouwer theorem to deduce the conclusion.
Thus, from (3.15) we deduce that . Consequently, it follows at once by the Brouwer theorem that there exists a fixed point of the map T, say with . So, this function is a solution of (1.1) and satisfies the bound for each . And this completes the proof of the theorem. □
4 Existence of a positive solution
In this section, we show the existence of positive solutions for boundary value problem (1.2).
Lemma 4.1 
for and for , or
for and for ,
then T has at least one fixed point in .
with the norm .
It is easy to see from Lemma 2.4 that y is a solution of (1.2) if and only if y is a fixed point of .
Lemma 4.2 is completely continuous.
Proof Note that is a summation operator on a discrete finite set, so is trivially completely continuous. □
For convenience, we define:
(F1) , where h is a positive function, g is a nonnegative functional;
Set , for , , , for .
boundary value problem (1.2) has at least one positive solution.
Now, from (4.6), (4.9) and Lemma 4.1, we have has a fixed point with . Then the theorem is proved. □
boundary value problem (1.2) has at least one positive solution.
Next, we consider two cases for the construction of .
From (4.13), (4.19) and Lemma 4.1, we get has a fixed point with . This completes the proof. □
In this section, we give some sufficient conditions for the nonexistence of a positive solution to boundary value problem (1.2).
We state the following hypotheses that will be used in what follows.
Theorem 5.1 Assume that (F1) and (F4) hold. If and , then there exists such that for all , boundary value problem (1.2) has no positive solution.
which is a contradiction. Therefore, (1.2) has no positive solution. This completes the proof. □
Theorem 5.2 Assume that (F1) and (F5) hold. If and , then there exists such that for all , boundary value problem (1.2) has no positive solution.
which is a contradiction. Thus, (1.2) has no positive solution. The proof is completed. □
In this section, we present some examples to illustrate the main results.
Therefore, from Theorem 3.1 we deduce that problem (6.1) has a unique solution.
The Banach space .
It is clear that , . So, f and g satisfy the conditions. Thus, by Theorem 3.2 we deduce that problem (6.2) has at least one solution.
thus by Theorem 4.1 we have that boundary value problem (6.3) has at least one positive solution for each .
then by Theorem 4.2 we deduce that boundary value problem (6.4) has at least one positive solution for each .
Therefore, by Theorem 5.1 we deduce that (6.5) has no positive solution for .
Hence by Theorem 5.2 we deduce that (6.6) has no positive solution for .
The existence and uniqueness of a solution to a class of boundary value problems for a fractional difference equation with are studied by the contraction mapping theorem.
The existence of a solution to a class of boundary value problems for a fractional difference equation with is studied by the Brouwer fixed point theorem.
The eigenvalue intervals of a boundary value problem for a class nonlinear fractional difference equations with are investigated by the Guo-Krasnosel’skii fixed point theorem.
The nonexistence of a positive solution boundary value problem for a class nonlinear fractional difference equations with is considered in terms of parameter.
The methods used to prove the existence results are standard and the same; however, their exposition in the framework of problems (1.1) and (1.2) is new.
The major difference is that the equations have different fractional order. The order is in this paper and in  and . The higher order leads the comparable process to being more difficult and complex.
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (11071143), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119) and supported by Shandong Provincial Natural Science Foundation (ZR2012AM009), also supported by the Natural Science Foundation of Educational Department of Shandong Province (J11LA01).
- 1.Podlubny I Mathematics in Science and Engineering. In Fractional Differential Equations. Academic Press, New York; 1999.Google Scholar
- 7.Zhao Y, Sun S, Han Z, Li Q: Positive solutions to boundary value problems of nonlinear fractional differential equations. Abstr. Appl. Anal. 2011., 2011: Article ID 390543Google Scholar
- 9.Pan Y, Han Z: Existence of solutions for a coupled system of boundary value problem of nonlinear fractional differential equations. 1. Proceedings of the 5th International Congress on Mathematical Biology 2011, 109–114.Google Scholar
- 21.Pan Y, Han Z, Sun S, Zhao Y: The existence of solutions to a system of discrete fractional boundary value problems. Abstr. Appl. Anal. 2012. 10.1155/2012/707631Google Scholar
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