Positive solutions of nonlinear boundary value problems for delayed fractional q-difference systems
© Yuan and Yang; licensee Springer. 2014
Received: 18 October 2013
Accepted: 14 January 2014
Published: 31 January 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.
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.
respectively. By applying a fixed-point theorem in cones, sufficient conditions for the existence of nontrivial solutions were enunciated.
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.
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.
The existence results were obtained by applying some well-known fixed-point theorems.
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.
and satisfies .
We note that if and , then .
Definition 2.1 ()
Definition 2.2 ()
where m is the smallest integer greater than or equal to α.
Definition 2.3 ()
where m is the smallest integer greater than or equal to α.
Theorem 2.5 ()
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.
T has a fixed point in ,
there exist and with .
for and for , or
for and for .
Then T has a fixed point in .
3 Existence of positive solutions
In this section, we always assume that .
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.
and for all ;
for all with .
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).
and this finishes the proof of (b). □
Now, we are ready to present the main results.
then (1.1) has a unique positive solution.
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.
then (1.1) has a unique positive solution.
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.
Therefore, for , which implies that T is continuous.
Immediately, we can easily see that TA is a bounded subset of Ω.
Now the equicontinuity of T on B follows easily from the fact that is continuous and hence uniformly continuous on .
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.
for , , and
for , ,
where , . Then (1.1) has at least a positive solution.
that is, 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.
It follows from Theorem 3.3 that (4.1) has a unique positive solution on .
It follows from Theorem 3.5 that (4.2) has at least one positive solution on .
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.
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