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Positive solutions for boundary value problems of fractional difference equations depending on parameters
Advances in Difference Equations volume 2013, Article number: 376 (2013)
We use the Krasnosel’skii fixed point theorem to obtain the sufficient conditions of the existence of two positive solutions for the boundary value problem of fractional difference equations depending on parameters.
MSC:26A33, 39A05, 39A12.
In this paper, we consider the boundary value problems of fractional difference equations depending on parameters of the form
where , , , are continuous functions. For each j, we have that () are given functions. We point out that fractional difference equations have been extensively studied in recent years. Systems of discrete fractional boundary value problems are also popular. In , the authors discussed the existence of positive solutions for coupled systems of nonlinear fractional difference equations:
where , , , . In , Goodrich studied the following pair of discrete fractional boundary value problems:
where , , , it is the same as  when . Goodrich obtained the existence of at least one positive solution to this problem by means of the Krasnosel’skii theorem for cones. We shall deduce the existence of at least two positive solutions to problem (1.1)-(1.2) in this paper. These results extend the results of .
The paper is organized as follows. In Section 2, we present basic definitions and demonstrate some lemmas in order to prove our main results. In Section 3, we establish some results for the existence of at least two solutions to problem (1.1)-(1.2), and we present an example to illustrate our main results.
Definition 2.1 
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 
The ν th fractional sum of a function f defined on the set , for , is defined to be
where . We also define the ν th fractional difference for by
where and .
Lemma 2.3 
Let t and ν be any numbers for which and are defined. Then
Lemma 2.4 
Let . Then
for some , with .
Lemma 2.5 
Let and be given. The unique solution of the FBVP
is given by
where is defined by
Lemma 2.6 
The Green’s function given in Lemma 2.5 satisfies:
for each ;
for each ;
there exists a number such that
First of all, we let represent the Banach space of all maps from into ℝ when equipped with the usual maximum norm . Then, we put . By equipping χ with the norm
it follows that is a Banach space.
Now consider the operator defined by
where we define by
Theorem 2.7 Let and be given for , where stands for the continuous functions on . Then is a solution of discrete FBVP (1.1)-(1.2) if and only if is a fixed point of S.
Proof From Lemma 2.4, we find that a general solution to problem (1.1)-(1.2) is
From boundary condition (1.2), we get
On the other hand, applying boundary condition (1.2) to implies that
Finally, we can get that
The opposite direction is obvious, so it is omitted. Consequently, we get that is a solution of (1.1)-(1.2) if and only if is a fixed point of S, as desired. □
Lemma 2.8 The function is strictly decreasing in for . In addition,
On the other hand, the function is strictly increasing in for . In addition,
Proof Note that for every ,
So, the first claim about holds. On the other hand,
It follows that
It may be shown in a similar way that satisfies the properties given in the statement of this lemma. We omit the details. □
Corollary 1 Let . There are constants such that , for , where is the usual maximum norm.
Theorem 2.9 
Let ℬ be a Banach space, and let be a cone in ℬ. Assume that and are open subsets of ℬ with . Assume, further, that
is a completely continuous operator. If either
, , , ; or
, , , .
Then T has a fixed point in .
3 Main results
In this section, we present the theorems for the existence of at least two positive solutions to problem (1.1)-(1.2). In the sequel, we let
We now present the conditions that we presume in the sequel.
(L1) for , .
(L2) for , .
(L3) for , .
(L4) for , .
(G1) The functionals , are linear. In particular, we assume that
for , .
(G2) For , there are
for each , and in addition
(G3) We have that each of , , , and is nonnegative for .
Let . In the sequel, we shall also make use of the cone
where is a constant defined by
where , come from Corollary 1 and is associated by Lemma 2.6(iii) to , .
Lemma 3.1 Let S be the operator defined as in (2.2). Then .
Proof By means of (G1), we get
By assumptions (G2) and (G3) together with the nonnegativity of and the fact that , we can get . By means of the same method, we obtain for .
On the other hand, we show that
for . In fact, by Lemma 2.6(iii), we have
where , .
Let . Then we obtain
Finally, by the definitions (), it is clear that
So, we conclude that . This completes the proof. □
Lemma 3.2 Suppose that conditions (G1)-(G3) hold, and there exist two different positive numbers , , , such that
Then the operator S has a fixed point such that
Proof Let . Then, for any and , we have
On the other hand, for any and , we have
By the use of Theorem 2.9, there exists such that , the proof is complete. □
Theorem 3.3 Suppose that conditions (L1), (L2) and (G1)-(G3) hold. Then, for every , problem (1.1)-(1.2) has at least two positive solutions, where
Proof Define the function
we have that . In view of (L1), we see that , that is, , and
In view of (L2), we see further that . Thus, there exists such that , . For any , by the intermediate value theorem, there exist two points , such that . Thus, we have
On the other hand, in view of (L1) and (L2), we see that there exist , such that
where γ is defined by (3.5). An application of Lemma 3.2 leads to two distinct solutions of (1.1)-(1.2) which satisfy
The proof is complete. □
Theorem 3.4 Suppose that (L3), (L4) and (G1)-(G3) hold. Then, for any , equation (1.1)-(1.2) has at least two positive solutions, where
and γ is defined by (3.5).
The proof is similar to Theorem 3.3 and hence omitted.
We now present an example to illustrate the sorts of boundary conditions that can be treated by Theorem 3.3.
Example 3.1 Consider the following boundary value problems:
where , , , we take
, and is defined on the time scale , is defined on the time scale .
It is easy to get that (F1), (F2) hold. On the other hand, (G1) holds. Now, we see that (G2), (G3) hold. In fact,
By using a similar method, we get
Hence, (G2) holds.
Finally, we numerically calculate that
Similarly, we have
We obtain that each of , , and is nonnegative for . So, condition (G3) holds. Namely, , and , , , satisfy the conditions of Theorem 3.3.
A computation shows that , . Then, for every (), problem (3.10)-(3.12) has at least two positive solutions.
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The authors are very grateful to the reviewers for their valuable suggestions and useful comments, which led to an improvement of this paper. Project supported by the National Natural Science Foundation of China (Grant No. 11271235) and Shanxi Province (2008011002-1) and Shanxi Datong University Institute (2009-Y-15, 2010-B-01, 2013K5) and the Development Foundation of Higher Education Department of Shanxi Province (20101109, 20111020).
The authors declare that they have no competing interests.
SK conceived of the study, and participated in its design and coordination. XZ drafted the manuscript. HC participated in the design of the study and the sequence correction. All authors read and approved the final manuscript.
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Kang, S., Zhao, X. & Chen, H. Positive solutions for boundary value problems of fractional difference equations depending on parameters. Adv Differ Equ 2013, 376 (2013). https://doi.org/10.1186/1687-1847-2013-376
- discrete boundary value problem
- fixed point theory
- positive solution