Existence results for fractional q-difference equations with nonlocal q-integral boundary conditions
© Zhao et al.; licensee Springer 2013
Received: 1 December 2012
Accepted: 17 February 2013
Published: 4 March 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.
By applying a fixed point theorem in cones, sufficient conditions for the existence of nontrivial solutions were enunciated.
where is a parameter, and the uniqueness, existence, and nonexistence of positive solutions are considered in terms of different ranges of λ.
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.
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
and satisfies .
provided the sum converges absolutely.
Obviously, if on , then .
where denotes the derivative with respect to the variable t.
where is the smallest integer greater than or equal to α.
Lemma 2.3 Assume that and , then .
Lemma 2.5 ()
Lemma 2.6 ()
Obviously, we have .
In order to define the solution for the problem (1.1), we need the following lemmas.
for some constants . Since , 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 ()
G is a continuous function and for .
- (ii)There exists a positive function such that
The proof is completed. □
3 The main results
Let be a Banach space endowed with the norm . Define the cone by .
It follows from the nonnegativeness and continuity of G and f that the operator satisfies and is completely continuous.
Proof We will prove that under the assumptions (3.2) and (3.3), is a contraction operator for m sufficiently large.
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.
where , with , .
Theorem 3.3 Suppose that there exists such that
Proof We denote . In what follows, we first show that .
Thus, we get .
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)
Then T has at least one fixed point in .
Theorem 3.5 Let be a nonnegative continuous function on . In addition, we assume that
Then the BVP (1.1) has at least one positive solution satisfying .
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 , .
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. □
has a unique positive solution.
Thus Theorem 3.1 implies that the boundary value problem (4.1) has a unique positive solution. □
where , , , . Choosing , , then , .
is continuous and nondecreasing relative to u;
where , , . Choosing , , then , .
So, by Theorem 3.5, the problem (4.3) has at least one positive solution with .
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).
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