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Existence of positive solutions of nonlinear fractional q-difference equation with parameter
Advances in Difference Equations volume 2013, Article number: 260 (2013)
In this paper, we study the boundary value problem of a class of nonlinear fractional q-difference equations with parameter involving the Riemann-Liouville fractional derivative. By means of a fixed point theorem in cones, some positive solutions are obtained. As applications, some examples are presented to illustrate our main results.
MSC:39A13, 34B18, 34A08.
The q-difference calculus is an interesting and old subject that many researchers devote their time to studying. The q-difference calculus or quantum calculus were first developed by Jackson [1, 2], while basic definitions and properties can be found in the papers [3, 4]. The q-difference calculus describes many phenomena in various fields of science and engineering .
The q-difference calculus is a necessary part of discrete mathematics. More recently, there has been much research activity concerning the fractional q-difference calculus [7–15]. Relevant theory about fractional q-difference calculus has been established , such as q-analogues of integral and difference fractional operators properties as Mittag-Leffler function , q-Laplace transform, q-Taylor’s formula [18, 19], 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 has received a considerable interest of many mathematicians and from many aspects, theoretical and practical. Specifically, q-difference equations have been widely used in mathematical physical problems, dynamical system and quantum models , q-analogues of mathematical physical problems including heat and wave equations , sampling theory of signal analysis [22, 23]. 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 [24–26], an interest has been observed in studying boundary value problems of fractional q-difference equations, especially about the existence of solutions for boundary value problems [3, 4, 27, 28].
In 2010, Ferreita  considered the existence of nontrivial solutions to the fractional q-difference equation
subjected to the boundary conditions
where and is a nonnegative continuous function.
In 2011, Ferreita  went on studying the existence of positive solutions to the fractional q-difference equation
subjected to the boundary conditions
where and is a nonnegative continuous function. By constructing a special cone and using Krasnosel’skii fixed point theorem, some existence results of positive solutions were obtained.
In 2011, El-Shahed and Al-Askar  studied the existence of a positive solution for a boundary value problem of the nonlinear factional q-difference equation
with the boundary conditions
where and is fractional q-derivative of Caputo type.
In 2012, Liang and Zhang  studied the existence and uniqueness of positive solutions for 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.
To the best of our knowledge, there are few papers that consider the boundary value of nonlinear fractional q-difference equations with parameters. Theories and applications seem to be just being initiated. In this paper we investigate the existence of solutions for the following two-point boundary value problem of nonlinear fractional q-difference equations
subject to the boundary conditions
where , , . We prove the existence of positive solutions for boundary value problem (1.1)-(1.2) by utilizing a fixed point theorem in cones. Several existence results for positive solutions in terms of different values of the parameter λ are obtained. This work is motivated by papers [25, 28].
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 positive solutions for boundary value problem (1.1)-(1.2) by a fixed point theorem in cones. Moreover, some examples are given to illustrate our main results.
In the following section, we collect some definitions and lemmas about fractional q-integral and fractional q-derivative which are referred to in .
Let and define
The q-analogue of the power function with is
More generally, if , then
It is easy to see that . And 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
If and f is defined on the interval , its q-integral from a to b is defined by
Similarly as done for derivatives, an operator can be defined as
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.
Lemma 2.1 (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 fundamental theorem of calculus applies to these operators and , i.e.,
and if f is continuous at , then
Basic properties of q-integral operator and q-differential operator can be found in the book .
We now point 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 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:
Lemma 2.3 
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
is the Green function of boundary value problem (2.1)-(2.2).
The following properties of the Green function play important roles in this paper.
Lemma 2.6 
Function G defined above satisfies the following conditions:
3 Main results
We are now in a position to state and prove our main results in this paper.
Let the Banach space be endowed with the norm . Let τ be a real constant with and define the cone by .
Suppose that u is a solution of boundary value problem (1.1)-(1.2). Then
Define the operator by
Then we have the following results.
Lemma 3.1 is completely continuous.
Proof It is easy to see that the operator is continuous in view of continuity of G and f.
By Lemmas 2.1 and 2.6, we have
Now, let be bounded, i.e., there exists a positive constant such that for all . Let . Then, for , from Lemmas 2.1 and 2.6, we have
Hence, is bounded.
On the other hand, for any given , setting
then for each , and , one has , that is to say, is equicontinuous. In fact,
Now we rearrange the above equation as follows, and from the properties of q-integral, we get
Now, we estimate :
for , , ;
for , ;
for , from the mean value theorem of differentiation, we have .
Thus, we have that
By means of the Arzela-Ascoli theorem, is completely continuous. The proof is completed. □
For convenience, we define
The main results of the paper are as follows.
Theorem 3.1 If holds, then for each
boundary value problem (1.1)-(1.2) has at least one positive solution. Here we impose if and if .
Proof Let λ satisfy (3.2) and be such that
By the definition of , we can know that there exists such that
so if with , then by (2.3) and (3.4), we have
Hence, if we choose , then
Let be such that
If with , then by (2.3) and (3.6) we have
Thus, if we set
Now, from (3.5), (3.9) and Lemma 2.4, we conclude that has a fixed point with , and it is clear that u is a positive solution of (1.1)-(1.2). The proof is completed. □
Theorem 3.2 If holds, then for each
boundary value problem (1.1)-(1.2) has at least one positive solution. Here we impose if and if .
Proof Let λ satisfy (3.10) and be given such that
From the definition of , we can see that there exists such that
Further, if , , then the flowing is similar to the second part of Theorem 3.1:
We can obtain that . Thus, if we choose , then
Next, we may choose such that
We consider two cases.
Case 1. Suppose that f is bounded. Then there exists some such that for . Define . Then if with , we have
Case 2. Suppose f is unbounded. Then there exists some such that
Let with . Then by (2.3) and (3.14) we get
Thus, (3.15) is also true.
In both Cases 1 and 2, if we set , where , then
Now that we have obtained (3.13) and (3.17), it follows from Lemma 2.4 that has a fixed point with . It is clear that u is a positive solution of (1.1)-(1.2). The proof is completed. □
Theorem 3.3 If there exist such that
then boundary value problem (1.1)-(1.2) has a positive solution with .
Proof Choose . Then, for , we have
For another thing, choose , then, for , we have
Now that we have obtained (3.18) and (3.19), it follows from Lemma 2.4 that has a fixed point with . It is clear that u is a positive solution of (1.1)-(1.2). The proof is completed. □
In this section, we present some examples to illustrate our main results.
Example 4.1 Consider the following boundary value problem:
Let , and . Then
and so . By Theorem 3.1, boundary value problem (4.1)-(4.2) has a positive solution for each .
Example 4.2 Consider the following boundary value problem:
Let , and . Then , ,
It is clear that . By Theorem 3.2, boundary value problem (4.3)-(4.4) has a positive solution for each .
Example 4.3 We can still consider the example that has been given in Example 4.2,
Here , , . Take . Then
Set , with . Then , and
Thus all the conditions in Theorem 3.3 hold. Hence, by Theorem 3.3, boundary value problem (4.5)-(4.6) has a positive solution with .
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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 (11071143), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119) and supported by Shandong Provincial Natural Science Foundation (ZR2012AM009, ZR2011AL007), also supported by Natural Science Foundation of Educational Department of Shandong Province (J11LA01).
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.