In this paper, we study a class of fractional q-difference equations with nonhomogeneous boundary conditions. By applying the classical tools from functional analysis, sufficient conditions for the existence of single and multiple positive solutions to the boundary value problem are obtained in term of the explicit intervals for the nonhomogeneous term. In addition, some examples to illustrate our results are given.
MSC: 34A08, 34B18, 39A13.
Keywords:fractional q-difference equation; nonhomogeneous boundary value problem; positive solution; multiplicity
Fractional differential equations have attracted considerable interest because of its demonstrated applications in various fields of science and engineering including fluid flow, rheology, diffusive transport akin to diffusion, electrical networks, probability [1,2]. Many researchers have studied the existence of solutions (or positive solutions) to fractional boundary value problems; for example, see [3-10] and the references therein.
The early work on q-difference calculus or quantum calculus dates back to Jackson’s papers , basic definitions and properties of quantum calculus can be found in the book . For some recent existence results on q-difference equations, we refer to [13-15] and the references therein.
The fractional q-difference calculus had its origin in the works by Al-Salam  and Agarwal . More recently, there seems to be new interest in the study of this subject and many new developments were made in this theory of fractional q-difference calculus [18-22]. Specifically, fractional q-difference equations have attracted the attentions of several researchers. Some recent work on the existence theory of fractional q-difference equations can be found in [20,23-31]. However, the study of boundary value problems for nonlinear fractional q-difference equations is still in the initial stage and many aspects of this topic need to be explored.
By using a fixed-point theorem in a cone, M. El-Shahed and F. Al-Askar  were concerned with the existence of positive solutions to nonlinear q-difference equation:
In , Graef and Kong investigated the boundary value problem with fractional q-derivatives
By applying the Banach contraction principle, Krasnoselskii’s fixed-point theorem, and the Leray-Schauder nonlinear alternative, Ahmad, Ntouyas and Purnaras  studied the existence of solution for the following nonlinear fractional q-difference equation with nonlocal boundary conditions:
Recently, in , the authors investigate the following singular semipositone integral boundary value problem for fractional q-derivatives equation:
Since finding positive solutions of boundary value problems is interest in various fields of sciences, fractional q-calculus equations has tremendous potential for applications. In this paper, we will deal with the following nonhomogeneous boundary value problem with fractional q-derivatives:
where , , , , and λ is a parameter, is the q-derivative of Riemann-Liouville type of order α, is continuous. 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 and Krasnoselskii’s fixed-point theorem, the uniqueness, existence, and multiplicity of positive solution to the BVP (1.1) are obtained in term of the explicit intervals for the nonhomogeneous term. Our results are different from those of [25,27].
2 Preliminaries on q-calculus and lemmas
For the convenience of the reader, below we cite some definitions and fundamental results on q-calculus as well as the fractional q-calculus. The presentation here can be found in, for example, [12,18,20,22].
The q-derivative of a function f is defined by
and q-derivatives of higher order by
The following formulas will be used later, namely, the integration by parts formula:
Lemma 2.3 ()
Lemma 2.5 ()
Lemma 2.6 ()
In order to define the solution for the problem (1.1), we need the following lemmas.
subject to the boundary conditions
is given by
Differentiating both sides of (2.11) and with the help of (2.4) and (2.6), we obtain,
Hence, we have
This completes the proof of the lemma. □
Proof We start by defining the following two functions:
which implies that part (ii) holds. This completes the proof of the lemma. □
According to , we may take , .
3 The main results
By introduction, we get
From the condition (3.3), we have
for m sufficiently large. So, we get
Hence, it follows from the generalized Banach contraction principle that the BVP (1.1) has a unique positive solution for any . If , then the condition on and Lemma 2.8 imply that in . This completes the proof of the theorem. □
Our next existence results is based on Krasnoselskii’s fixed-point theorem .
Lemma 3.3 (Krasnoselskii’s)
Obviously, K is a cone of nonnegative functions in X.
On the other hand,
In fact, we have
For the sake of convenience, we introduce the following weight functions:
In view of (2.9) and Lemma 2.8, we have
By Lemma 3.3, the operator T has at least one fixed point , and . Since , , then, the solution is positive for . As in the proof of Theorem 3.1, is a positive solution for . This completes the proof of the theorem. □
From Theorem 3.5, the operator T has two fixed point , with . Therefore, the BVP (1.1) has at least two positive solutions for . As in the proof of Theorem 3.1, , are two positive solutions for . This completes the proof of the theorem. □
Example 4.1 The fractional q-difference boundary value problem
A simple computation showed
which implies that
Example 4.2 Consider the following fractional boundary value problem:
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.
Dedicated to Professor Hari M Srivastava.
The authors are highly grateful for the referees’ careful reading and comments on this paper. The research is supported by the National Natural Science Foundation of China (Grant No. 11271372, 11201138); it is also supported by the Hunan Provincial Natural Science Foundation of China (Grant No. 13JJ3106, 12JJ2004), and the Scientific Research Fund of Hunan Provincial Education Department (Grant No. 12B034).
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