Existence of positive solutions of nonlinear fractional q-difference equation with parameter
© Li et al.; licensee Springer 2013
Received: 7 May 2013
Accepted: 7 August 2013
Published: 23 August 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].
where and is a nonnegative continuous function.
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.
where and is 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 , , . 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 .
It is easy to see that . And note that if then .
and 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.
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 operator and 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 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 2.3 
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.
Lemma 2.5 
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 
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 .
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.
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 completed. □
The main results of the paper are as follows.
boundary value problem (1.1)-(1.2) has at least one positive solution. Here we impose if and if .
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. □
boundary value problem (1.1)-(1.2) has at least one positive solution. Here we impose if and if .
We consider two cases.
Thus, (3.15) is also true.
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. □
then boundary value problem (1.1)-(1.2) has a positive solution with .
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.
and so . By Theorem 3.1, boundary value problem (4.1)-(4.2) has a positive solution for each .
It is clear that . By Theorem 3.2, boundary value problem (4.3)-(4.4) has a positive solution for each .
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 .
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).
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