Abstract
In this paper, we study a class of fractional qdifference 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 qdifference equation; nonhomogeneous boundary value problem; positive solution; multiplicity1 Introduction
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 [310] and the references therein.
The early work on qdifference calculus or quantum calculus dates back to Jackson’s papers [11], basic definitions and properties of quantum calculus can be found in the book [12]. For some recent existence results on qdifference equations, we refer to [1315] and the references therein.
The fractional qdifference calculus had its origin in the works by AlSalam [16] and Agarwal [17]. 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 qdifference calculus [1822]. Specifically, fractional qdifference equations have attracted the attentions of several researchers. Some recent work on the existence theory of fractional qdifference equations can be found in [20,2331]. However, the study of boundary value problems for nonlinear fractional qdifference equations is still in the initial stage and many aspects of this topic need to be explored.
By using a fixedpoint theorem in a cone, M. ElShahed and F. AlAskar [25] were concerned with the existence of positive solutions to nonlinear qdifference equation:
where and is the fractional qderivatives of the Caputo type.
In [27], Graef and Kong investigated the boundary value problem with fractional qderivatives
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 fixedpoint theorem, and the LeraySchauder nonlinear alternative, Ahmad, Ntouyas and Purnaras [29] studied the existence of solution for the following nonlinear fractional qdifference equation with nonlocal boundary conditions:
where is the fractional qderivative of the Caputo type, and .
Recently, in [32], the authors investigate the following singular semipositone integral boundary value problem for fractional qderivatives equation:
where , is the qderivative of RiemannLiouville 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 qcalculus equations has tremendous potential for applications. In this paper, we will deal with the following nonhomogeneous boundary value problem with fractional qderivatives:
where , , , , and λ is a parameter, is the qderivative of RiemannLiouville 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 fixedpoint 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 qcalculus and lemmas
For the convenience of the reader, below we cite some definitions and fundamental results on qcalculus as well as the fractional qcalculus. The presentation here can be found in, for example, [12,18,20,22].
The qanalogue of the power function with is
Clearly, if , then . The qgamma function is defined by
The qderivative of a function f is defined by
and qderivatives of higher order by
The qintegral 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:
and
where denotes the derivative with respect to the variable t.
Definition 2.1 Let and f be a function defined on . The fractional qintegral of RiemannLiouville type is and
Definition 2.2 The fractional qderivative of the RiemannLiouville type of order is defined by and
where is the smallest integer greater than or equal to α.
Lemma 2.3 ([20])
Lemma 2.4Letandfbe a function defined on. Then the following formulas hold:
Lemma 2.5 ([20])
Letandnbe a positive integer. Then the following equality holds:
Lemma 2.6 ([22])
Let, , the following is valid:
Particularly, for , , using qintegration by parts, we have
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
where
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,
and
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 functiondefined by (2.10) satisfies the following conditions:
Proof We start by defining the following two functions:
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 .
Finally, we prove part (ii). When , we have
which implies that part (ii) holds. This completes the proof of the lemma. □
According to [20], 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 thatis continuous and there exists a nonnegative functionsuch 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
Consequently,
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 fixedpoint theorem [33].
Lemma 3.3 (Krasnoselskii’s)
LetEbe a Banach space, and letbe a cone. Assume, are open subsets ofEwith, and letbe a completely continuous operator such that, either
ThenThas at least one fixed point in.
Obviously, K is a cone of nonnegative functions in X.
Lemma 3.4The operatoris completely continuous.
Proof Firstly, we prove that . By (2.9) and Lemma 2.8, we have
On the other hand,
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
where
For any , we can prove that if and , then
In fact, we have
By means of ArzelaAscoli theorem, is completely continuous.
For the sake of convenience, we introduce the following weight functions:
and set
□
Theorem 3.5Suppose that there exists two positive numberssuch that one of the following conditions is satisfied
Then the BVP (1.1) has at least one positive solution, such thatfor. 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,
It follows
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 numberssuch that one of the following conditions is satisfied
Then the BVP (1.1) has at least two positive solutionssuch thatfor. 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 existspositive numberssuch that one of the following conditions is satisfied:
Then the BVP (1.1) has at leastmpositive solutions, , such thatfor. If, in addition, on, then the conclusion is true for.
4 Examples
Example 4.1 The fractional qdifference 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
and
which implies that
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
and
Let . Take , , then , and satisfies
So, by Theorem 3.5, the problem (4.2) has one positive solution such that for .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Acknowledgements
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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