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:
where and is the fractional q-derivatives of the Caputo type.
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.
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:
where is the fractional q-derivative of the Caputo type, and .
Recently, in , the authors investigate the following singular semipositone integral boundary value problem for fractional q-derivatives equation:
where , is the q-derivative of Riemann-Liouville type of order α, is continuous and semipositone, and may be singular at .
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].
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 q-derivatives of higher order by
The q-integral of a function f defined in the interval is given by
If and f is defined in the interval , then its integral from a to b is defined by
Similar to that 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 2.4Let andfbe a function defined on . Then the following formulas hold:
Lemma 2.5 ()
Let andnbe 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 , and
In order to define the solution for the problem (1.1), we need the following lemmas.
Lemma 2.7For given , the unique solution of the boundary value problem
subject to the boundary conditions
is given by
Proof Since , we put . In view of Definition 2.1 and Lemma 2.4, we see that
Then it follows from Lemma 2.5 that the solution of (2.7) and (2.8) is given by
for some constants . From , we have .
Differentiating both sides of (2.11) and with the help of (2.4) and (2.6), we obtain,
Then by the boundary condition , we get . Using the boundary condition , we get
Hence, we have
This completes the proof of the lemma. □
Lemma 2.8The function defined by (2.10) satisfies the following conditions:
(i) , and for all .
(ii) for all .
Proof We start by defining the following two functions:
Obviously, . Now , and for
Moreover, for , it follows from (2.4) and Lemma 2.3 that
which implies that is an increasing function with respect to t. It is clear that is increasing in t. Therefore, is an increasing function of t for all , and so .
When , then
Finally, we prove part (ii). When , we have
If , then we have
which implies that part (ii) holds. This completes the proof of the lemma. □
Remark 2.9 If we let , then
According to , we may take , .
3 The main results
Let be a Banach space endowed with the norm . Define the cone by .
Define the operator as follows:
Theorem 3.1Assume that is continuous and there exists a nonnegative function such that
Then the BVP (1.1) has a unique positive solution for any , provided
If, in addition, on , then the conclusion is true for .
Proof We will show that under the assumptions (3.2) and (3.3), is a contraction operator for m sufficiently large.
By the definition of , for , we have
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. □
Remark 3.2 When is a constant, the condition (3.2) reduces to a Lipschitz condition.
Our next existence results is based on Krasnoselskii’s fixed-point theorem .
Lemma 3.3 (Krasnoselskii’s)
LetEbe a Banach space, and let be a cone. Assume , are open subsets ofEwith , and let be a completely continuous operator such that, either
(1) , and , , or
(2) , and , .
ThenThas at least one fixed point in .
Define a cone by
Obviously, K is a cone of nonnegative functions in X.
Lemma 3.4The operator is completely continuous.
Proof Firstly, we prove that . By (2.9) and Lemma 2.8, we have
On the other hand,
Hence, we have .
Next, we show that T is uniformly bounded. For fixed , consider a bounded subset of K defined by , and let . Then for , we get
which implies that is bounded.
Finally, we show that T is equicontinuous. For all , setting
For any , we can prove that if and , then
In fact, we have
If , then
If , , then
By means of Arzela-Ascoli theorem, is completely continuous.
For the sake of convenience, we introduce the following weight functions:
Theorem 3.5Suppose that there exists two positive numbers such that one of the following conditions is satisfied
( ) , ;
( ) , .
Then the BVP (1.1) has at least one positive solution , such that for . If, in addition, on , then the conclusion is true for .
Proof Because the proofs are similar, we prove only the case ( ). Denote . Then for any , we get , , , and , . By assumption ( ), we have
In view of (2.9) and Lemma 2.8, we have
On the other hand, define . For any and , we have and , . Thus,
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. □
Theorem 3.6Suppose that there exists three positive numbers such that one of the following conditions is satisfied
( ) , , ;
( ) , , .
Then the BVP (1.1) has at least two positive solutions such that for . If, in addition, on , then the conclusion is true for .
Proof We prove only the case ( ). Since is continuous and , there exist two positive numbers , such that and , . Thus, it follows from the assumption ( ) that
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. □
Denote the integer part of m by . Generally, we have the following theorem.
Theorem 3.7Suppose that there exists positive numbers such that one of the following conditions is satisfied:
( ) , , ;
( ) , , .
Then the BVP (1.1) has at leastmpositive solutions , , such that for . If, in addition, on , then the conclusion is true for .
Example 4.1 The fractional q-difference boundary value problem
has a unique positive solution for any .
Proof In this case, , , , , . Let
and . It is easy to prove that
A simple computation showed
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 for any . □
Example 4.2 Consider the following fractional boundary value problem:
where , , , . Choosing , then .
By calculation, we get . By Lemma 2.6, Lemma 2.8 and with the aid of a computer, we obtain that
Let . Take , , then , and satisfies
So, by Theorem 3.5, the problem (4.2) has one positive solution such that for .
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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