Existence of solutions for fractional q-difference equation with mixed nonlinear boundary conditions
© Li et al.; licensee Springer. 2014
Received: 10 September 2014
Accepted: 9 December 2014
Published: 19 December 2014
In this paper, we study the boundary value problem for a class of nonlinear fractional q-difference equation with mixed nonlinear conditions involving the fractional q-derivative of Riemann-Liouuville type. By means of the Guo-Krasnosel’skii fixed point theorem on cones, some results concerning the existence of solutions are obtained. Finally, examples are presented to illustrate our main results.
MSC:39A13, 34B18, 34A08.
Fractional calculus is a generalization of integer order calculus [1, 2]. It has been used by many researchers to adequately describe the evolution of a variety of engineering, economical, physical, and biological processes . There are a large number of papers dealing with the continuous fractional calculus. Among all the topics, boundary value problems for fractional differential equations have attracted considerable attention [4–6]. However, the discrete fractional calculus has seen slower progress, it is still a relatively new and emerging area of mathematics. Some efforts have also been made to develop the theory of discrete fractional calculus in various directions. For some recent works, see . Of particular note is that Atici and Sengül have shown the usefulness of fractional difference equations in tumor growth modeling in .
The early works about q-difference calculus or quantum calculus were first developed by Jackson [9, 10], while basic definitions and properties can be found in the monograph by Kac and Cheung . q-Difference equations have been widely used in mathematical physical problems, dynamical system and quantum models , heat and wave equations , and sampling theory of signal analysis .
As an important part of discrete mathematics, more recently, some researchers devoted their attention to the study of the fractional q-difference calculus, they developed the q-analogs of fractional integral and difference operators properties, the q-Mittag-Leffler function , q-Laplace transform, q-Taylor’s formula, etc. . The origin of the fractional q-difference calculus can be traced back to the works in [17, 18] by Al-Salam and Agarwal. A book on this subject by Annaby and Mansour  summarizes and organizes much of the q-fractional calculus and equations.
As is well known, the aim of finding solutions to boundary value problems is of main importance in various fields of applied mathematics. Recently, there seems to be a new interest in the study of the boundary value problems for fractional q-difference equations [20–27].
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 is a parameter, is the q-derivative of Riemann-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 were enunciated.
has at least one positive solution by imposing some relatively mild structural conditions on f, , and φ.
where , , are integers, is continuous and φ is a linear functional, here is a closed subinterval, and are real-valued, continuous functions. We are interested in the existence of solutions for the boundary value problem (1.1)-(1.2) by utilizing a fixed point theorem on cones.
We should mention that the above boundary conditions are rather general and contain many common cases such as separated boundary conditions, integral boundary conditions, multi-point boundary conditions, etc., by choosing different , , and ϕ. Our results generalize and improve some results on the existence of solutions for fractional q-difference equations. Moreover, problems studied in  and  can be regarded as our special cases.
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 fixed point theorem on cones. Moreover, some examples are given to illustrate our main results.
It is easy to see that . Note that, if then .
and satisfies .
From the definition of q-integral and the properties of series, we can get the following results on 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 .
Basic properties of q-integral and q-differential operators can be found in the book .
Remark 2.1 We note that if and , then .
Definition 2.1 
Definition 2.2 
where p is the smallest integer greater than or equal to α.
Next, we list some properties about q-derivative and q-integral that are already known in the literature.
Lemma 2.4 
for and for , or
for and for , then S has at least one fixed point in .
The next result is important in the sequel.
Remark 2.2 
Let . Then and for .
The following properties of the Green’s function play important roles in this paper.
Lemma 2.6 
and for all ;
for all with .
3 Main results
where , obviously, .
In order to get the integrated and rigorous theory, we make the following assumptions.
The following result plays an important role in the coming discussion.
Lemma 3.1 is completely continuous.
Proof It is easy to see that the operator F is continuous in view of the continuity of G and f.
where . Thus, .
Hence, is bounded.
for , , ;
for , ;
for , from the mean value theorem of differentiation, we have .
By means of the Arzela-Ascoli theorem, is completely continuous. The proof is complete. □
hold, where is a closed subinterval. Then the boundary value problem (1.1)-(1.2) has at least one solution.
Hence, , that is, F is a cone expansion on .
On the other hand, we consider two cases:
for all with .
Take . Set .
To summarize, we conclude from (3.14) and (3.17) that F is a cone compression on .
With the help of Lemma 2.4 we can now deduce the existence of function such that . Hence, the problem (1.1)-(1.2) has at least one solution. The proof is completed. □
Next, by choosing suitable forms of and φ, we present the corresponding boundary value problems with separated boundary conditions, integral boundary conditions and multi-point boundary conditions as corollaries of Theorem 3.1 to illustrate the universality and generalization of our results.
has at least one solution.
has at least one solution.
has at least one solution.
In this section, we will give an example to expound our main results.
here , with and for , is a closed subinterval, , and satisfies (H3).
so (H2) holds.
then, by Theorem 3.1, the boundary value problem (4.1)-(4.2) has at least one solution.
The authors sincerely thank the reviewers for their valuable suggestions and useful comments, which have led to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (61374074), Shandong Provincial Natural Science Foundation (ZR2012AM009, ZR2013AL003), also supported by Graduate Innovation Foundation of University of Jinan (YCX13013).
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