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Positive solutions of nonlinear boundary value problems for delayed fractional q-difference systems
Advances in Difference Equations volume 2014, Article number: 51 (2014)
In this paper, we investigate the existence and uniqueness of positive solutions to nonlinear boundary value problems for delayed fractional q-difference systems by applying the properties of the Green function and some well-known fixed-point theorems. As applications, some examples are presented to illustrate the main results.
MSC:39A13, 34B18, 34A08.
In the past decades, fractional differential equations have been proved to be valuable tools in the investigation of many phenomena in various fields of science and engineering such as physics, mechanics, chemistry, biology, engineering, etc. Therefore, the subject of fractional differential equations has gained considerable attention by many researchers. Some recent results on fractional boundary value problems can be found in [1–4] and references therein. For example, Ahmad and Nieto  dealt with some existence results for a boundary value problem involving a nonlinear fractional order integrodifferential equation with integral boundary conditions based on a contraction mapping principle and Krasnoselskiii’s fixed-point theorem. Ahmad et al.  investigated the existence and uniqueness of solutions for a class of Caputo-type fractional boundary value problems involving four-point nonlocal Riemann-Liouville integral boundary conditions of different order by means of standard tools of fixed-point theory and Leray-Schauder nonlinear alternative. Ouyang et al.  considered the following nonlinear system of fractional order differential equations with delays:
where is the standard Riemann-Liouville fractional derivative. By using some fixed-point theorems and some properties of the Green function, the existence of positive solutions was obtained.
The q-difference calculus or quantum calculus is an old subject that was initially developed by Jackson [8, 9]; basic definitions and properties of q-difference calculus can be found in the book mentioned in .
The fractional q-difference calculus had its origin in the works by Al-Salam  and Agarwal . Recently, maybe due to the explosion in research within the fractional differential calculus setting, new developments in this theory of fractional q-difference calculus were made; for example, q-analogues of the integral and differential fractional operators properties such as the q-Laplace transform, q-Taylor’s formula, Mittage-Leffler function [13–16], just to mention some.
More recently, boundary value problems of nonlinear fractional q-difference equations have gained popularity and importance. Many researchers pay attention to the existence and multiplicity of solutions or positive solutions for nonlinear boundary value problems of fractional q-difference equations by means of upper and lower solutions method and some fixed-point theorems such as the Krasnoselskii fixed-point theorem, the Leggett-Williams fixed-point theorem, and the Schauder fixed-point theorem; for examples, see [17–21] and the references therein. El-Shahed and Al-Askar  studied the existence of multiple positive solutions to the nonlinear q-fractional boundary value problems by using Guo-Krasnoselskii’s fixed-point theorem in a cone. Graef and Kong  investigated the uniqueness, existence, and nonexistence of positive solutions for the boundary value problem with fractional q-derivatives in terms of different ranges of λ. Ma and Yang  obtained the existence of solutions for multi-point boundary value problems of nonlinear fractional q-difference equations by means of the Banach contraction principle and Krasnoselskii’s fixed-point theorem. Zhao et al.  showed some existence results of positive solutions to nonlocal q-integral boundary value problems of a nonlinear fractional q-derivative equation using the generalized Banach contraction principle, the monotone iterative method, and Krasnoselskii’s fixed-point theorem. Ferreira  and  dealt with the existence of positive solutions to nonlinear q-difference boundary value problems,
respectively. By applying a fixed-point theorem in cones, sufficient conditions for the existence of nontrivial solutions were enunciated.
In , Liang and Zhang discussed the following nonlinear q-fractional three-point boundary value problem:
By using a fixed-point theorem in partially ordered sets, the authors obtained sufficient conditions for the existence and uniqueness of positive and nondecreasing solutions to the above boundary value problem.
In , Ahmad et al. studied the following nonlocal boundary value problems of nonlinear fractional q-difference equations,
where denotes the Caputo fractional q-derivative of order α, and (). The existence of solutions for the problem was shown by applying some well-known tools of fixed-point theory, such as Banach contraction principle, Krasnoselskii’s fixed-point theorem, and Leray-Schauder nonlinear alternative.
In , Alsaedi et al. were concerned with the following nonlinear fractional q-difference equations with nonlocal integral boundary conditions:
The existence results were obtained by applying some well-known fixed-point theorems.
Motivated by the above works, in this paper, we consider the following system of nonlinear fractional q-difference equations with delays:
where is the fractional q-derivative of the Riemann-Liouville type, for some , for , for , and is a nonlinear function from to . The purpose of this paper is to establish sufficient conditions on the existence of positive solutions for fractional q-difference system (1.1) by using some properties of the Green function and some fixed-point theorems such as the Banach contraction principle, Krasnoselskii’s fixed-point theorem, and the Leray-Schauder nonlinear alternative. By a positive solution for the fractional q-difference system (1.1) we mean a mapping with positive components on such that (1.1) is satisfied. Obviously, (1.1) includes the usual system of fractional q-difference equations when for all i and j. Therefore, the obtained results generalize and include some existing ones.
For convenience of the reader, we present some necessary definitions and lemmas of fractional q-calculus theory to facilitate analysis of problem (1.1). These details can be found in the recent literature; see  and references therein.
Let and define
The q-analogue of the power with is
More generally, if , then
Note that, if , then . The q-gamma function is defined by
and satisfies .
The q-derivative of a function f is here 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 , its integral from a to b is defined by
Similarly to what is done for derivatives, an operator can be defined, namely,
The fundamental theorem of calculus applies to these operators and , i.e.,
and if f is continuous at , then
Basic properties of the two operators can be found in the book . We now point out three formulas that will be used later ( denotes the derivative with respect to variable i)
We note that if and , then .
Definition 2.1 ()
Let and f be function defined on . The fractional q-integral of the Riemann-Liouville type is and
Definition 2.2 ()
The fractional q-derivative of the Riemann-Liouville type of order is defined by and
where m is the smallest integer greater than or equal to α.
Definition 2.3 ()
The fractional q-derivative of the Caputo type of order is defined by
where m is the smallest integer greater than or equal to α.
Let and f be a function defined on . Then the following formulas hold:
Theorem 2.5 ()
Let and p be a positive integer. Then the following equality holds:
Theorem 2.6 (Banach contraction mapping theorem )
Let M be a complete metric space and let be a contraction mapping. Then T has a unique fixed point.
Let C be a closed and convex subset of a Banach space X. Assume that U is a relatively open subset of C with and is completely continuous. Then at least one of the following two properties holds:
T has a fixed point in ,
there exist and with .
Let P be a cone in a Banach space X. Assume that and are open subsets of X with and . Suppose that is a completely continuous operator such that either
for and for , or
for and for .
Then T has a fixed point in .
3 Existence of positive solutions
Throughout this paper, we let . Then is a Banach space, where
In this section, we always assume that .
Lemma 3.1 Fractional q-difference systems (1.1) is equivalent to the following system of q-integral equations:
for , where
Proof It is easy to see that if satisfies (3.1), then it also satisfies (1.1). So, assume that is a solution to (1.1). In view of Lemma 2.4 and Theorem 2.5, integrating both sides of the first equation of (1.1) of order with respect to t, we can see that
for , . It follows that
for , . Combining with the boundary conditions in (1.1), this yields
Similarly, one can obtain , . Also we have
Then it follows from (3.3) and the boundary condition that
Therefore, for ,
where is defined as in (3.2). The proof is completed. □
Some properties of the Green functions needed in the sequel are now stated and proved.
Lemma 3.2 Function defined above satisfies the following conditions:
and for all ;
for all with .
Proof We start by defining the two functions
It is clear that and . On the other hand, for
Therefore, . Moreover, for fixed ,
i.e., is an increasing function of t. Obviously, is increasing in t, therefore is an increasing function of t for fixed . This concludes the proof of (a).
Suppose now that . Then we have
On the other hand, if , then we have
and this finishes the proof of (b). □
Now, we are ready to present the main results.
Theorem 3.3 Suppose that there exist functions , , such that
for , . If
then (1.1) has a unique positive solution.
It is easy to see that Ω is a complete metric space. Define an operator T on Ω by
Because of the continuity of G and f, it follows easily from Lemma 3.2 that T maps Ω into itself. To finish the proof, we only need to show that T is a contraction. Indeed, for , by (3.4), we have
This, combined with Theorem 2.6 and (3.5), immediately implies that is a contraction. Therefore, the proof is complete with the help of Lemma 3.1 and Theorem 2.6. □
The following result can be proved in the same spirit as that for Theorem 3.3.
Theorem 3.4 Suppose that there exist functions , , and nonnegative constants such that and
for , . If
then (1.1) has a unique positive solution.
Theorem 3.5 Suppose that there exist nonnegative real-valued functions , , such that
for almost every and all . If
then (1.1) has at least one positive solution.
Proof Let Ω and be defined by (3.6) and (3.7), respectively. We first show that T is completely continuous through the following three steps.
Step 1. Show that is continuous. Let be a sequence in Ω such that . Then is bounded in . Since f is continuous, it is uniformly continuous on any compact set. In particular, for any , there exists a positive integer such that
for and , . Then, for and , , we have
Therefore, for , which implies that T is continuous.
Step 2. Show that T maps bounded sets of Ω into bounded sets. Let A be a bounded subset of Ω. Then is bounded. Since f is continuous, there exists an such that
It follows that, for , and ,
Immediately, we can easily see that TA is a bounded subset of Ω.
Step 3. Show that T maps bounded sets of Ω into equicontinuous sets. Let B be a bounded subset of Ω. Similarly as in Step 2, there exists such that
Then, for any and and ,
Now the equicontinuity of T on B follows easily from the fact that is continuous and hence uniformly continuous on .
Now we have shown that T is completely continuous. To apply Theorem 2.7, let
Fix and define . We claim that there is no such that for some . Otherwise, assume that there exist and such that . Then
Therefore, , a contradiction to . This proves the claim. Applying Theorem 2.7, we know that T has a fixed point in , which is a positive solution to (1.1) by Lemma 3.1. Therefore, the proof is complete. □
Corollary 3.6 If all , , are bounded, then (1.1) has at least one positive solution.
To state the last result of this section, we introduce
Theorem 3.7 Suppose that there exist and positive constants with such that
for , , and
for , ,
where , . Then (1.1) has at least a positive solution.
Proof Let Ω be defined by (3.5) and , . Obviously, Ω is a cone in E. From the proof of Theorem 3.5, we know that the operator T defined by (3.6) is completely continuous on Ω. For any , it follows from Theorem 2.8 and condition (b) that
that is, for .
On the other hand, for any , it follows from Lemma 3.2 and condition (a) that, for ,
that is, for . Therefore, we have verified condition (b) of Theorem 2.8. It follows that T has a fixed point in , which is a positive solution to (1.1). This completes the proof. □
4 Some examples
In this section, we demonstrate the feasibility of some of the results obtained in Section 3.
Example 4.1 Consider the following fractional q-difference system:
Here , , , , , ,
One can easily see that (3.4) is satisfied with
and hence 
It follows from Theorem 3.3 that (4.1) has a unique positive solution on .
Example 4.2 Consider the following fractional q-difference system:
Here , , , ,
One can easily see that (3.8) is satisfied. Moreover,
and hence 
It follows from Theorem 3.5 that (4.2) has at least one positive solution on .
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The authors sincerely thank the editor and reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript.
The authors declare that they have no competing interests.
The authors contributed to each part of this work equally and read and approved the final version of the manuscript.
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Yuan, Q., Yang, W. Positive solutions of nonlinear boundary value problems for delayed fractional q-difference systems. Adv Differ Equ 2014, 51 (2014). https://doi.org/10.1186/1687-1847-2014-51
- fractional q-difference systems
- boundary value problems
- positive solutions
- fixed-point theorems