Boundary value problems for fractional q-difference equations with nonlocal conditions
© Li et al.; licensee Springer. 2014
Received: 9 November 2013
Accepted: 16 January 2014
Published: 4 February 2014
In this paper, we study the boundary value problem of a fractional q-difference equation with nonlocal conditions involving the fractional q-derivative of the Caputo type, and the nonlinear term contains a fractional q-derivative of Caputo type. By means of Bananch’s contraction mapping principle and Schaefer’s fixed-point theorem, some existence results for the solutions are obtained. Finally, examples are presented to illustrate our main results.
MSC:39A13, 34B18, 34A08.
The q-difference calculus is an interesting and old subject. The q-difference calculus or quantum calculus was first developed by Jackson [1, 2], while basic definitions and properties can be found in the papers [3, 4]. The origin of the fractional q-difference calculus can be traced back to the work in [5, 6] by Al-Salam and by Agarwal. The q-difference calculus describes many phenomena in various fields of science and engineering .
The q-difference calculus is an important part of discrete mathematics. More recently, the fractional q-difference calculus has caused wide attention of scholars, many researchers devoted themselves to studying it [7–14]. The relevant theory of fractional q-difference calculus has been established , such as q-analogues of the fractional integral and the difference operators properties as Mitlagel Leffler function , q-Laplace transform, q-Taylor’s formula [17, 18], just to mention some. It is not only the requirements of the fractional q-difference calculus theory but also its the broad application.
Apart from this old history of q-difference equations, the subject received a considerable interest of many mathematicians and from many aspects, theoretical and practical. So specifically, q-difference equations have been widely used in mathematical physical problems, for dynamical system and quantum models , for q-analogues of mathematical physical problems including heat and wave equations , for sampling theory of signal analysis [21, 22]. What is more, the fractional q-difference calculus plays an important role in quantum calculus.
As generalizations of integer order q-difference, fractional q-difference can describe physical phenomena much better and more accurately. Perhaps due to the development of fractional differential equations [23–25], an interest has been aroused in studying boundary value problems of fractional q-difference equations, especially as regards the existence of solutions for boundary value problems [3, 4, 26–33].
where and is a nonnegative continuous function.
where and is a fractional q-derivative of Caputo type.
where . By using a fixed-point theorem in partially ordered sets, they got some sufficient conditions for the existence and uniqueness of positive solutions to the above boundary problem.
where and is the fractional q-derivative of Caputo type. By virtue of Mönch’s fixed-point theorem and the technique of measure of weak noncompactness and got some conditions of positive solutions.
where is a parameter, is the q-derivative of Rieman-Liouville type of order α. By using the generalized Banach contraction principle, the monotone iterative method and Krasnoselskii’s fixed-point theorem, some existence results of positive solutions to the above boundary value problems have been enunciated.
where , , . They proved the existence of positive solutions for boundary value problems by utilizing a fixed-point theorem in cones. Several existence results for positive solutions in terms of different values of the parameter λ were obtained.
where is the Caputo fractional derivative, is a continuous function, is a continuous function, and .
where , , , and , , and y is a continuous functional. The condition of is representative and general; the conditions of in  and in  can be viewed as two special cases. We will prove the existence of solutions for the boundary value problem (1.1)-(1.2) by utilizing Banach’s contraction mapping principle and Schaefer’s fixed-point theorem. Several existence results for the solutions are obtained. This work is motivated by papers [28, 34].
The paper is organized as follows. In Section 2, we introduce some definitions of q-fractional integral and differential operator together with some basic properties and lemmas to prove our main results. In Section 3, we investigate the existence of solutions for the boundary value problem (1.1)-(1.2) by the Banach contraction mapping principle and Schaefer’s fixed-point theorem. Moreover, some examples are given to illustrate our main results.
It is easy to see that . And note that, if , then .
and it satisfies .
From the definition of q-integral and the properties of series, we can get the following results concerning q-integral, which are helpful in the proofs of our main results.
- (1)If f and g are q-integral on the interval , , , then
If is q-integral on the interval , then .
If f and g are q-integral on the interval , for all , then .
The basic properties of the q-integral operator and the q-differential operator can be found in the book .
Remark 2.1 We note that if and , then .
Definition 2.1 
Definition 2.2 
where is the smallest integer greater than or equal to α.
Next, we list some properties of the q-derivative and the q-integral that are already known in the literature.
Lemma 2.2 
Lemma 2.3 
Lemma 2.4 
F has a fixed point in ; or
there exist and with .
The next result is important in the sequel.
which completes the proof. □
3 Main results
We are now in a position to state and prove our main results in this paper.
Let be the Banach space endowed with the norm . Define the closed subset by .
Obviously, the fixed points of the operator F are solutions of the boundary value problem (1.1)-(1.2).
Then we have the following results.
Theorem 3.1 Assume that
(H1) there exists a constant such that , for each and all ;
(H2) there exists a constant such that , for each ;
(H3) , where and are defined as (3.8) and (3.9).
Then the boundary value problem (1.1)-(1.2) has at least one positive solution.
Note that , so and are convergent, which imply that and are convergent. Thus is convergent. Hence is convergent.
Consequently F is a contraction map as . As a consequence of Banach’s fixed-point theorem, we deduce that F has a fixed point which is a solution of the problem (1.1)-(1.2). The proof is completed. □
Theorem 3.2 Assume and are continuous, and is a continuous functional. Suppose the following conditions are satisfied:
(H4) there exists a continuous function with on ;
(H5) there exists , with .
Then the boundary value problem (1.1)-(1.2) has a solution.
Proof We will prove the result by using Schaefer’s fixed-point theorem and divide the proof into four steps.
First, set , then , we show is continuous.
which shows is uniform bounded.
for , , ;
for , ;
- (3)for . From the mean value theorem of differentiation, we have . Thus, we have
Therefore, is equicontinuous. By means of the Arzela-Ascoli theorem, is a relatively compact set in K, then the operator is completely continuous.
In the fourth step, we have a priori bounds.
where r is defined as (H5). It follows that , that is, there is no such that for . As a consequence of Lemma 2.4, F has a fixed point which is a solution of the boundary value problem (1.1)-(1.2), and the proof is completed. □
In this section, we present some examples to illustrate our main results.
Hence, by Theorem 3.1, the boundary value problem (4.1)-(4.2) has a solution.
and we get with . Choosing , it is clear that . By Theorem 3.2, the boundary value problem (4.3)-(4.4) has a solution.
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 (61374074), the Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119) and supported by the Shandong Provincial Natural Science Foundation (ZR2012AM009, ZR2013AL003). It is also supported by the Graduate Innovation Foundation of University of Jinan (YCX13013).
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