Abstract
In this paper, we study the existence and uniqueness of solutions for the boundary value problem of fractional difference equations
and
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.
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1 Introduction
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 [1], 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 [2], 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. [3]. 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 [4–12].
In recent years, a number of papers on the discrete fractional calculus have appeared, such as [13–30], 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 [13]. They presented in [13] more rules for composing fractional sums and differences. Goodrich studied a two-point fractional boundary value problem in [16], 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 [14] 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 [15].
Atici and Eloe in [14] considered a two-point boundary value problem for the finite fractional difference equation
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.
Goodrich in [18] considered a discrete fractional boundary value problem of the form
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 .
Motivated by all the works above, in this paper, we first aim to study the following boundary value problem:
where , , is continuous and is a continuous functional.
Our second aim is to investigate the boundary value problem of a fractional difference equation with parameter
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.
2 Preliminaries
For the convenience of the readers, we first present some useful definitions and fundamental facts of fractional calculus theory, which can be found in [13, 14].
Definition 2.1 [14]
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 [14]
The ν th fractional sum of a function f, for , is defined by
for . We also define the ν th fractional difference for by , where and is chosen so that .
Lemma 2.1 [13]
If , then for any .
Lemma 2.2 [14]
Let . Then for some with .
Lemma 2.3 [17]
For t and s, for which both and are defined, we find that
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).
Lemma 2.4 Let and . A function y is a solution of the problem
where if and only if , , has the form
where
Proof By Lemma 2.2, we obtain
From (2.2) we get
Noting that
we deduce .
By (2.3) we have
then . We note that
and
it follows that .
From (2.4) we know
which implies that
Consequently, we deduce that has the form
for , where
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. □
Theorem 2.1 The Green function satisfies the following conditions:
-
(i)
for .
-
(ii)
for .
-
(iii)
There exists a positive number such that
The proof of this theorem is similar to that of Theorem 3.2 in [14]. Hence, we omit the proof here.
3 Existence and uniqueness of solution
In this section, we wish to show that under certain conditions, problem (1.1) has at least one solution. We know that problem (1.1) can be recast as an equivalent summation equation. It follows from Lemma 2.4 that y is a solution of (1.1) if and only if y is a fixed point of the operator , where
for . We use this fact to prove the first existence theorem.
Theorem 3.1 Define . Suppose that and are Lipschitz in y. That is, there exist such that , for any functions , defined on . Then if the condition
holds, then problem (1.1) has a unique solution.
Proof We show that T is a contraction mapping. To achieve this, we notice that for given and ,
By an application of Lemma 2.3, we get
Similar to the above inequality, we have
From another application of Lemma 2.1, we obtain
So, putting (3.4)-(3.6) in (3.3), we conclude that
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.
Theorem 3.2 Suppose that there exists a constant such that satisfies the inequality
and satisfies the inequality
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.
Assume that inequalities (3.7) and (3.8) hold for given f and g. For convenience, we let
which is a strictly positive constant. Then we have
As in the proof of Theorem 3.1, we can simplify the expression on the right-hand side of inequality (3.10). Indeed, we know that
On the one hand, from Lemma 2.1 we know is increasing in t, thus we have
On the other hand,
Inserting (3.11)-(3.13) into (3.10), we can obtain
By substituting (3.9) into (3.14), we have
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 [26]
Let ℬ be a Banach space, and let be a cone. Assume that and are two bounded open subsets contained in ℬ such that and . Assume further that is a completely continuous operator. If either
-
(1)
for and for , or
-
(2)
for and for ,
then T has at least one fixed point in .
Define the Banach space ℬ by
with the norm .
For is increasing, we get . Thus, there exists a positive constant such that
In fact, for , is strictly positive, then we let
Take . Then
Denote . Then we have
Define the cone
We define an operator as follows:
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 .
Suppose that f is a nonnegative function. Then, from Theorem 2.1 and (4.1), we have
Thus, .
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;
(F2)
(F3)
Set , for , , , for .
Theorem 4.1 Suppose that conditions (F1) and (F2) hold. If there exist a sufficiently small positive constant δ and a sufficiently large constant such that holds, then for each
boundary value problem (1.2) has at least one positive solution.
Proof By condition (F2), there exists such that
So, for with , by (4.3) and (4.4), we have, for all ,
Thus, if we choose , then (4.5) implies that
Similarly, by condition (F2), we can find and a sufficiently large constant such that
And then we set and . Then and imply
thus
Therefore, for given , by Theorem 2.1, (4.3) and (4.7), we have
Hence, from (4.8) we have
Now, from (4.6), (4.9) and Lemma 4.1, we have has a fixed point with . Then the theorem is proved. □
Theorem 4.2 Assume that conditions (F1) and (F3) hold. If there exists a sufficiently large constant such that holds, then for each
boundary value problem (1.2) has at least one positive solution.
Proof By condition (F3), there exist and a sufficiently large constant such that
Let . Then, for ,
Then (4.12) implies that
Next, we consider two cases for the construction of .
Case 1. Suppose that g is bounded. Then there exists some such that
From (4.10) we know
Thus, from (4.14) and (4.15), we get
Case 2. Suppose that g is unbounded. From (4.10) we know , so for a sufficiently small constant . Then, by condition (F3), there exists some such that
Choose such that and for , . Define . Now we set , then . Thus, for , we have
Then, in both Case 1 and Case 2, we have
From (4.13), (4.19) and Lemma 4.1, we get has a fixed point with . This completes the proof. □
5 Nonexistence
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.
(F4)
(F5)
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.
Proof Since and , there exist positive numbers , , and such that and
Let . Then we have
Suppose that is a positive solution of (1.2). Then we show that this leads to a contradiction for . Since , for ,
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.
Proof Since and , we can get that there exist positive numbers , , , such that , and
Let . Then
Assume that is a positive solution of (1.2). We show that this leads to a contradiction for . Since , for , thus
which is a contradiction. Thus, (1.2) has no positive solution. The proof is completed. □
6 Example
In this section, we present some examples to illustrate the main results.
Example 6.1 Suppose that , . Let and . Then (1.1) becomes
In this case, let , . Inequality (3.2) is
Therefore, from Theorem 3.1 we deduce that problem (6.1) has a unique solution.
Example 6.2 Suppose that , and . Let and . Then problem (1.1) is
The Banach space .
We note that
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.
Example 6.3 Suppose that , . Let for , for . Take , . Then , and problem (1.2) becomes
By routine numerical calculations, we have
and
Then
thus . So, the conditions of Theorem 4.1 are satisfied. Since
thus by Theorem 4.1 we have that boundary value problem (6.3) has at least one positive solution for each .
Example 6.4 Suppose that , . Let for , for . Take . Then , and problem (1.2) becomes
By routine numerical calculations, we have
and
Then
thus . So, the conditions of Theorem 4.2 are satisfied. Since
then by Theorem 4.2 we deduce that boundary value problem (6.4) has at least one positive solution for each .
Example 6.5 Suppose that , . Let for , for . Take . Then , and problem (1.2) becomes
Thus, we have
By calculation,
so
Therefore, by Theorem 5.1 we deduce that (6.5) has no positive solution for .
Example 6.6 Suppose that , . Let for , for . Take . Then , and problem (1.2) becomes
Thus, we have
By calculation,
By the definition of [14], we know
and then
Hence by Theorem 5.2 we deduce that (6.6) has no positive solution for .
7 Conclusion
This paper is an extension of [14] and [18]. The main contributions of this paper include:
-
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.
In contrast to [14] and [18], the similarities and differences are as follows:
-
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 [14] and [18]. The higher order leads the comparable process to being more difficult and complex.
-
Nonlocal boundary conditions are considered in this paper and [18], Dirichlet boundary conditions are considered in [14].
-
Both the existence and nonexistence are considered in this paper, but only the existence is considered in [14] and [18].
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Acknowledgements
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).
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Pan, Y., Han, Z., Sun, S. et al. The existence of solutions to a class of boundary value problems with fractional difference equations. Adv Differ Equ 2013, 275 (2013). https://doi.org/10.1186/1687-1847-2013-275
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DOI: https://doi.org/10.1186/1687-1847-2013-275