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Existence of positive solutions for a discrete fractional boundary value problem
Advances in Difference Equations volume 2014, Article number: 253 (2014)
This paper is concerned with the existence of positive solutions to a discrete fractional boundary value problem. By using the Krasnosel’skii and Schaefer fixed point theorems, the existence results are established. Additionally, examples are provided to illustrate the effectiveness of the main results.
MSC:26A33, 39A10, 47H07.
Fractional differential equations have received increasing attention within the last ten years or so. The theory of fractional differential equations has been a new important mathematical branch due to its wide applications in different research areas and engineering, such as physics, chemistry, economics, control of dynamical etc. For more details, see [1–9] and the references therein. On the other hand, accompanied with the development of the theory for fractional calculus, fractional difference equations have attracted increasing attention slowly but steadily in the past three years or so. Some research papers have appeared, see [10–19]. For example, Atici and Eloe  analyzed the conjugate discrete fractional boundary value problem (FBVP) with delta derivative:
Goodrich  studied the discrete fractional boundary value problems:
In , Lv discussed the existence of solutions for discrete fractional boundary value problems with a p-Laplacian operator:
They obtained a series of excellent results of discrete fractional boundary value problems. Motivated by the aforementioned works, in this paper we consider a discrete fractional boundary value problem (FBVP):
where , denotes the Riemann-Liouville fractional difference operator, and for any number and each interval I of R, . We appeal to the convention that for any , where y is a function defined on . By using the Krasnosel’skii and Schaefer fixed point theorems, the existence results are established and two examples are also provided to illustrate the effectiveness of the main results.
The rest of the paper is organized as follows. In Section 2, we introduce some lemmas and definitions which will be used later. In Section 3, the existence of positive solutions for the boundary value problem (1.1) is investigated. In Section 4, two examples are provided to illustrate the effectiveness of the main results.
2 Basic definitions and preliminaries
Firstly we present here some necessary definitions and lemmas which are used throughout this paper.
Define for any t and v for which the right-hand side is defined. If is a pole of the gamma function and is not a pole, then .
Definition 2.2 
The v th fractional sum of a function f, for , is defined to be
for . Define the v th fractional difference for by , and satisfies .
Lemma 2.3 
Let t and v be any numbers for which and are defined. Then .
Lemma 2.4 
Assume that . Then
for some , with .
Lemma 2.5 (The nonlinear alternative of Leray and Schauder )
Let E be a Banach space with closed and convex. Let U be a relatively open subset of C with and be a continuous and compact mapping. Then either
the mapping T has a fixed point in ; or
there exist and with .
Lemma 2.6 
Let B be a Banach space and let be a cone. Assume that and are bounded open sets contained in B such that and . Assume further that is a completely continuous operator. If either
for and for ; or
for and for ;
then T has at least one fixed point in .
We state next the structural assumptions that we impose on (1.1).
(H1) Assume that the nonlinearity function is continuous.
(H2) Assume that there exist nonnegative continuous functions , , such that , , .
(H3) Assume that uniformly for .
(H4) Assume that uniformly for .
3 Existence results
In this section, we will establish the existence of at least one positive solution for problem (1.1). At first, we state and prove some preliminary lemmas.
Lemma 3.1 Let be given. Then the unique solution of the discrete fractional boundary value problem
Here, for , is defined by
Proof Suppose that defined on is a solution of (3.1). Using Lemma 2.4, for some constants , we have
By and Definition 2.1, we obtain .
Then, for all , we obtain 
In view of , we have
Substituting the values of and in (3.4), we have
Lemma 3.2 The function given in (3.3) satisfies the following:
Proof First of all, (3.3) implies that . Note that , we know .
Second of all, by (3.3) and the definition of D in (3.6), we obtain
On the other hand,
The proof of Lemma 3.2 is completed. □
Let B be the collection of all functions with the norm .
Define the operator by
In view of the continuity of f, it is easy to know that T is continuous. Furthermore, it is not difficult to verify that T maps bounded sets into bounded sets and equi-continuous sets. Therefore, in the light of the well-known Arzelá-Ascoli theorem, we know that T is a compact operator (see [11, 12]).
Let and set
We have the following theorem.
Theorem 3.3 Assume that (H1) and (H2) hold. Then system (1.1) has at least one positive solution provided that
Proof Let with . If , that is, . From (H1), (H2) and (3.9), we have
which shows that .
Consider the eigenvalue problem
Assume that y is a solution of (3.11), we obtain
It shows that . By Lemma 2.5, T has a fixed point in . The proof is completed. □
We define the cone by
Lemma 3.4 Let T be the operator defined in (3.9) and K be the cone defined in (3.13). Then .
Proof Note that for each , we have
Therefore, it holds that
The conclusion of Lemma 3.4 holds. □
Theorem 3.5 Suppose that conditions (H1), (H3) and (H4) hold. Then problem (1.1) has at least one positive solution.
Proof We have already shown in Lemma 3.4. By condition (H3), we can select sufficiently small so that both and hold for all and , where .
Let . Then, for , we have
It implies that T is a cone contraction on .
On the other hand, from condition (H4), we may select a number such that both and hold for all and , where and . Define , we obtain
whenever , so that T is a cone expansion on .
In summary, we may invoke Lemma 2.6 to deduce the existence of a function such that , where is a positive solution to problem (1.1). The proof is completed. □
Example 4.1 Consider the fractional difference boundary value problem
Set , , , . We have
By a simple computation, we can obtain , , . Therefore, . The conditions of Theorem 3.3 hold, the boundary value problem (4.1) has at least one positive solution.
Example 4.2 Consider the fractional difference boundary value problem
Set , . We have
The conditions of Theorem 3.5 hold, the boundary value problem (4.2) has at least one positive solution.
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This work was jointly supported by the Natural Science Foundation of China under Grants 11471278, the Natural Science Foundation of Hunan Province under Grants 13JJ3120 and 14JJ2133, and the Construct Program of the Key Discipline in Hunan Province. We would like to show our great thanks to the anonymous referee for his/her valuable suggestions and comments, which improve the former version of this paper and make us rewrite the paper in a more clear way.
The authors declare that they have no competing interests.
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
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Wang, J., Xiang, H. & Chen, F. Existence of positive solutions for a discrete fractional boundary value problem. Adv Differ Equ 2014, 253 (2014). https://doi.org/10.1186/1687-1847-2014-253
- positive solution
- fractional boundary value problem