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Existence of solutions for fractional q-difference equation with mixed nonlinear boundary conditions
Advances in Difference Equations volume 2014, Article number: 326 (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].
In 2012, Liang and Zhang  studied the three-point boundary problem of fractional q-differences,
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.
In 2013, Zhou and Liu  studied the existence results for fractional q-difference equations with nonlocal q-integral boundary conditions,
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.
In 2013, Goodrich  proved that the nonlocal boundary value problem with mixed nonlinear boundary conditions
has at least one positive solution by imposing some relatively mild structural conditions on f, , and φ.
To the best of our knowledge, very few authors consider the boundary value problem of fractional q-difference equations with mixed nonlinear boundary conditions. Theories and applications seem to be just initiated. This paper will fill up the gap. Here, motivated by , we will consider the boundary value problem of the nonlinear fractional q-difference equations
subject to the boundary conditions
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.
Let and define
The q-analog of the power function with is
More generally, if , then
It is easy to see that . Note that, if then .
The q-gamma function is defined by
and satisfies .
The q-derivative of a function f is here defined by
and q-derivatives of higher order by
The q-integral of a function f defined on the interval is given by
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.
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 .
We now present out three formulas that will be used later ( denotes the derivative with respect to variable i)
Remark 2.1 We note that if and , then .
Definition 2.1 
Let and f be a function defined on . The fractional q-integral of the Riemann-Liouville type is defined by and
Definition 2.2 
The fractional q-derivative of the Riemann-Liouville type of order is defined by and
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.
Let and f be a function defined on . Then the following formulas hold:
Let and p be a positive integer. Then the following equality holds:
Lemma 2.4 
Let X be a Banach space and be a cone. Suppose that and are bounded open sets contained in X such that . Suppose further that is a completely continuous operator. If either
for and for , or
for and for , then S has at least one fixed point in .
The next result is important in the sequel.
Lemma 2.5 Let be a given function. Then the boundary value problem
has a unique solution
Proof In view of Definition 2.2 and Lemma 2.2, we deduce
It follows from Lemma 2.3,
where are some constants to be determined. By the boundary condition , we get . Now if , then differentiating both sides of (2.4) j times for , we obtain
From the boundary conditions , for , it is easy to know . Thus, (2.4) reduces to
Differentiating both sides of (2.5), we obtain
Using the boundary condition , we have
Remark 2.2 
Let . Then and for .
The following properties of the Green’s function play important roles in this paper.
Lemma 2.6 
The function G defined as (2.3) satisfies the following properties:
and for all ;
for all with .
3 Main results
Let the Banach space be endowed with the norm . Let τ be a real constant with and define the cone by
where , obviously, .
Define the operator by
In order to get the integrated and rigorous theory, we make the following assumptions.
(H1) There are constants such that the functional φ satisfies the inequality
(H2) For each given , there are and such that
(H3) There exists a function satisfying the growth condition
for some . For each given and , there is such that
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.
By Lemmas 2.1 and 2.6, we have
where . Thus, .
Now let be bounded, i.e., there exists a positive constant such that for all . By the continuity of , , and φ, we easily see that and are bounded, so there exist constants and such that and . Let . Then for , from Lemma 2.6, we have
we rearrange (3.3) as follows:
Hence, is bounded.
On the other hand, for any given , there exists small enough, such that holds for each and with , that is to say, is equicontinuous. In fact,
Now, we estimate (we discuss in the same way, the proof here is omitted):
for , , ;
for , ;
for , from the mean value theorem of differentiation, we have .
Thus, we have
By means of the Arzela-Ascoli theorem, is completely continuous. The proof is complete. □
Theorem 3.1 Assume that the nonlinearity splits in the sense that , for continuous functions and such that and . Suppose conditions (H1)-(H3) and
hold, where is a closed subinterval. Then the boundary value problem (1.1)-(1.2) has at least one solution.
Proof Begin by selecting the number such that
Now, there exists a number such that for . Then take the open set
By Remark 2.2, Lemma 2.6, and (3.6), for , we find
Hence, , that is, F is a cone expansion on .
On the other hand, we consider two cases:
Case 1. Suppose that g is bounded for . We may find sufficiently large such that
Condition (3.5) implies the existence of such that
Now if , then by (H1) we get . According to condition (H2), it follows that
for all with .
Next, since is a closed subinterval, we may select such that . Then for each ,
Thus combining condition (H3) we see that
Take . Set .
From (3.8) we may assume without loss of generality that
Then by (3.9), (3.10), (3.12), and (3.13), for each , we have
Case 2. Suppose that g is unbounded at +∞. By condition , there exists a number such that for we find that , where meets
Noting that g is unbounded at +∞, we may find such that
Now, put . Then for each we find that by (3.9), (3.10), (3.12), (3.15), and (3.16),
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.
Corollary 3.1 Assume is defined as in Theorem 3.1 and (H1)-(H3) hold. If , . Then the boundary value problem
has at least one solution.
Corollary 3.2 Assume is defined as in Theorem 3.1, . Then φ is linear functional, is a closed subinterval, just need , where m is Lebesgue measure. If (H 1)-(H 3) hold, in addition
then the boundary value problem (3.20)-(3.21) with integral boundary conditions
has at least one solution.
Corollary 3.3 Assume is defined as in Theorem 3.1, . Then φ is linear functional, just need , is a closed subinterval. If (H1)-(H3) hold, moreover,
then the boundary value problem (3.22)-(3.23) with multi-point boundary conditions
has at least one solution.
In this section, we will give an example to expound our main results.
Example 4.1 Consider the following boundary value problem:
here , with and for , is a closed subinterval, , and satisfies (H3).
Since and , then . Choosing and , we have
For each , setting , it is clear that
so (H2) holds.
If only the given function satisfies , for continuous functions and such that and , in addition,
then, by Theorem 3.1, the boundary value problem (4.1)-(4.2) has at least one solution.
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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).
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.