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Existence results for fractional q-difference equations with nonlocal q-integral boundary conditions
Advances in Difference Equations volume 2013, Article number: 48 (2013)
In this paper, we discuss the existence of positive solutions for nonlocal q-integral boundary value problems of fractional q-difference equations. By applying the generalized Banach contraction principle, the monotone iterative method, and Krasnoselskii’s fixed point theorem, some existence results of positive solutions are obtained. In addition, some examples to illustrate our results are given.
MSC:39A13, 34B18, 34A08.
Studies on q-difference equations appeared already at the beginning of the twentieth century in intensive works especially by Jackson , Carmichael  and other authors such as Poincare, Picard, Ramanujan. Up to date, q-difference equations have evolved into a multidisciplinary subject; for example, see [3–6] and the references therein. For some recent work on q-difference equations, we refer the reader to the papers [7–20], and basic definitions and properties of q-difference calculus can be found in the book . On the other hand, fractional differential equations have gained importance due to their numerous applications in many fields of science and engineering including fluid flow, rheology, diffusive transport akin to diffusion, electrical networks, probability, etc. For details, see [22, 23]. Many researchers studied the existence of solutions to fractional boundary value problems; see, for example, [24–33] and the references therein.
The fractional q-difference calculus had its origin in the works by Al-Salam  and Agarwal . More recently, perhaps due to the explosion in research within the fractional differential calculus setting, new developments in this theory of fractional q-difference calculus were made, specifically, q-analogues of the integral and differential fractional operators properties such as the Mittag-Leffler function, the q-Laplace transform, and q-Taylor’s formula [12, 21, 36, 37], just to mention some.
However, the theory of boundary value problems for nonlinear q-difference equations is still in the initial stage and many aspects of this theory need to be explored. Recently, there have been some paper considering the existence of solutions to boundary value problems of fractional q-difference equations, for example, [10, 16–20, 38] and the references therein.
In , Ferreira considered the Dirichlet type nonlinear q-difference boundary value problem
By applying a fixed point theorem in cones, sufficient conditions for the existence of nontrivial solutions were enunciated.
In , Graef and Kong investigated the boundary value problem with fractional q-derivatives
where is a parameter, and the uniqueness, existence, and nonexistence of positive solutions are considered in terms of different ranges of λ.
Furthermore, Ahmad, Ntouyas, and Purnaras  studied the following nonlinear fractional q-difference equation with nonlocal boundary conditions:
where is the fractional q-derivative of the Caputo type and . The existence of solutions for the problem is shown by applying some well-known tools of fixed point theory such as Banach’s contraction principle, Krasnoselskii’s fixed point theorem, and the Leray-Schauder nonlinear alternative.
In this paper, we deal with the following nonlocal q-integral boundary value problem of nonlinear fractional q-derivatives equation:
where , , , , and is a parameter, is the q-derivative of Riemann-Liouville type of order α, is continuous, in which . To the authors’ knowledge, no one has studied the existence of positive solutions for the fractional q-difference boundary value problem (1.1). In the present work, we gave the corresponding Green’s function of the boundary value problem (1.1) and its properties. 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 are enunciated.
2 Preliminaries on q-calculus and lemmas
Let and define
The q-analogue of the power function with is
More generally, if , then
Clearly, if , then . The q-gamma function is defined by
and satisfies .
The q-derivative of a function f is defined by
and the q-derivatives of higher order by
The q-integral of a function f defined in the interval is given by
provided the sum converges absolutely.
If and f is defined in the interval , then its integral from a to b is defined by
Obviously, if on , then .
Similar as done for derivatives, an operator is given by
The fundamental theorem of calculus applies to these operators and , i.e.,
and if f is continuous at , then
The following formulas will be used later, namely, the integration by parts formula
where denotes the derivative with respect to the variable t.
Definition 2.1 Let and f be a function defined on . The fractional q-integral of Riemann-Liouville type is and
Definition 2.2 The fractional q-derivative of the Riemann-Liouville type of order is defined by and
where is the smallest integer greater than or equal to α.
Lemma 2.3 Assume that and , then .
Lemma 2.4 Let and f be a function defined on . Then the following formulas hold:
Lemma 2.5 ()
Let and n be a positive integer. Then the following equality holds:
Lemma 2.6 ()
Let , , the following is valid:
Particularly, for , , using q-integration by parts, we have
Obviously, we have .
In order to define the solution for the problem (1.1), we need the following lemmas.
Lemma 2.7 Let . Then, for a given , the unique solution of the boundary value problem
subject to the boundary condition
is given by
Proof Since , we take . In view of Definition 2.1 and Lemma 2.4, we have
Then it follows from Lemma 2.5 that the solution of (2.7) and (2.8) is given by
for some constants . Since , we have .
Using the Riemann-Liouville integral of order β for (2.13), we have
where we have used Lemma 2.4 and Lemma 2.6. Using the boundary condition , we get
Hence, we have
This completes the proof of the lemma. □
Remark 2.8 For the special case where , Lemma 2.7 has been obtained by Ferreira .
Lemma 2.9 ()
The function defined by (2.11) satisfies the following properties:
Lemma 2.10 The function defined by (2.10) satisfies the following properties:
G is a continuous function and for .
There exists a positive function such that
Proof It is easy to prove that the statement (i) holds. On the other hand, we note that defined by (2.11) is decreasing with respect to t for and increasing with respect to t for . Hence, we have
The proof is completed. □
3 The main results
Let be a Banach space endowed with the norm . Define the cone by .
Define the operator as follows:
It follows from the nonnegativeness and continuity of G and f that the operator satisfies and is completely continuous.
Theorem 3.1 Suppose that is continuous and there exists a function such that
Then the BVP (1.1) has a unique positive solution provided
Proof We will prove that under the assumptions (3.2) and (3.3), is a contraction operator for m sufficiently large.
By (2.10), (2.11), and (2.12), for , we obtain the estimate
By introduction, we have
According to (3.3), we can choose m sufficiently large such that
Hence, it follows from the generalized Banach contraction principle that the BVP (1.1) has a unique positive solution. □
Remark 3.2 When is a constant, the condition (3.2) reduces to a Lipschitz condition.
For the sake of convenience, we set
where , with , .
Theorem 3.3 Suppose that there exists such that
() is continuous and nondecreasing relative to u, and
Then the BVP (1.1) has one positive solution satisfying
Proof We denote . In what follows, we first show that .
Let ; then . By assumption (), we have
Hence, for any ,
Thus, we get .
Let , ; then . Let ; then . We denote , . According to , we have , . Since T is completely continuous, we assert that has a convergent subsequence and there exists such that . Since , then . According to the definition of T and (), we have
By introduction, we have for , . Thus, there exists such that . From the continuity of T and , we have . The proof is completed. □
Our next existence result is based on Krasnoselskii’s fixed point theorem .
Lemma 3.4 (Krasnoselskii’s)
Let E be a Banach space, and let be a cone. Assume , are open subsets of E with , and let be a completely continuous operator such that
Then T has at least one fixed point in .
Theorem 3.5 Let be a nonnegative continuous function on . In addition, we assume that
(H1) There exists a positive constant such that
where , with , , and
(H2) There exists a positive constant with such that
Then the BVP (1.1) has at least one positive solution satisfying .
Proof By Lemma 2.9, we obtain that . Let . For any , according to (H1) and the definitions of and , we obtain
Let . For any , by (H2) and Lemma 2.10, we have
Now, an application of Lemma 3.4 concludes the proof. □
Theorem 3.6 Assume that there exist positive numbers such that
(H3) for and for , ; or
(H4) for and for , ,
where , , κ are given in (H1).
Then the BVP (1.1) has at least n positive solutions with , .
Proof Suppose that the condition (H3) holds. According to the continuity of f, for every pair , there exists with such that
It follows from Theorem 3.5 that every pair presents a positive solution of the BVP (1.1) such that , .
When the condition (H4) holds, the proofs are similar to those in the case (H3). The proof is completed. □
The fractional boundary value problem
has a unique positive solution.
Proof In this case, , , , . Let
and . It is easy to prove that
By simple calculation, we get
which implies that
Obviously, for any , we have
Thus Theorem 3.1 implies that the boundary value problem (4.1) has a unique positive solution. □
Consider the following fractional boundary value problem:
where , , , . Choosing , , then , .
A simple computation showed . By Lemma 2.6 and with the aid of a computer, we obtain that
Let . Take , , then satisfies
is continuous and nondecreasing relative to u;
So, by Theorem 3.3, the problem (4.2) has one positive solution satisfying
Consider the following fractional boundary value problem:
where , , . Choosing , , then , .
By calculation, we get . By Lemma 2.6 and with the aid of a computer, we obtain that
Let . Take and , . Then satisfies
So, by Theorem 3.5, the problem (4.3) has at least one positive solution with .
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Dedicated to Professor Hari M Srivastava.
The authors are highly grateful for the referee’s careful reading and comments on this paper. The research is supported by the National Natural Science Foundation of China (11271372, 11201138); it is also supported by the Hunan Provincial Natural Science Foundation of China (12JJ2004, 13JJ3106), and the Scientific Research Fund of Hunan Provincial Education Department (12B034).
The authors declare that they have no competing interests.
Each of the authors, YZ, HC, and QZ, contributed to each part of this study equally and read and approved the final version of the manuscript.