- Open Access
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.
- fractional q-difference systems
- boundary value problems
- positive solutions
- fixed-point theorems
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 .
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. □
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.
- Agarwal RP, Ahmad B: Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions. Comput. Math. Appl. 2011, 62: 1200-1214. 10.1016/j.camwa.2011.03.001MathSciNetView ArticleGoogle Scholar
- Agarwal RP, O’Regan D, Staněk S: Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. J. Math. Anal. Appl. 2010, 371: 57-68. 10.1016/j.jmaa.2010.04.034MathSciNetView ArticleGoogle Scholar
- Agarwal RP, Lakshmikantham V, Nieto JJ: On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal. 2010, 72: 2859-2862. 10.1016/j.na.2009.11.029MathSciNetView ArticleGoogle Scholar
- Ahmad B, Nieto JJ, Pimentel J: Some boundary value problems of fractional differential equations and inclusions. Comput. Math. Appl. 2011, 62: 1238-1250. 10.1016/j.camwa.2011.02.035MathSciNetView ArticleGoogle Scholar
- Ahmad B, Nieto JJ: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Bound. Value Probl. 2009., 2009: Article ID 708576Google Scholar
- Ahmad B, Ntouyas S, Assolami A: Caputo type fractional differential equations with nonlocal Riemann-Liouville integral boundary conditions. J. Appl. Math. Comput. 2013. 10.1007/s12190-012-0610-8Google Scholar
- Ouyang Z, Chen Y, Zou S: Existence of positive solutions to a boundary value problem for a delayed nonlinear fractional differential system. Bound. Value Probl. 2011., 2011: Article ID 475126Google Scholar
- Jackson FH: On q -functions and a certain difference operator. Trans. R. Soc. Edinb. 1908, 46: 253-281.View ArticleGoogle Scholar
- Jackson FH: On q -definite integrals. Q. J. Pure Appl. Math. 1910, 41: 193-203.Google Scholar
- Kac V, Cheung P: Quantum Calculus. Springer, New York; 2002.View ArticleGoogle Scholar
- Al-Salam WA: Some fractional q -integrals and q -derivatives. Proc. Edinb. Math. Soc. 1966/1967, 15(2):135-140. 10.1017/S0013091500011469MathSciNetView ArticleGoogle Scholar
- Agarwal RP: Certain fractional q -integrals and q -derivatives. Proc. Camb. Philos. Soc. 1969, 66: 365-370. 10.1017/S0305004100045060View ArticleGoogle Scholar
- Abdeljawad T, Benli B, Baleanu D: A generalized q -Mittag-Leffler function by q -Caputo fractional linear equations. Abstr. Appl. Anal. 2012., 2012: Article ID 546062Google Scholar
- Rajković PM, Marinković SD, Stanković MS: On q -analogues of Caputo derivative and Mittag-Leffler function. Fract. Calc. Appl. Anal. 2007, 10: 359-373.MathSciNetGoogle Scholar
- Rajković PM, Marinković SD, Stanković MS: Fractional integrals and derivatives in q -calculus. Appl. Anal. Discrete Math. 2007, 1: 311-323. 10.2298/AADM0701311RMathSciNetView ArticleGoogle Scholar
- Atici FM, Eloe PW: Fractional q -calculus on a time scale. J. Nonlinear Math. Phys. 2007, 14: 333-344.MathSciNetView ArticleGoogle Scholar
- El-Shahed M, Al-Askar FM: On the existence and uniqueness of solutions for q -fractional boundary value problem. Int. J. Math. Anal. 2011, 5: 1619-1630.Google Scholar
- Zhao Y, Chen H, Zhang Q: Existence results for fractional q -difference equations with nonlocal q -integral boundary conditions. Adv. Differ. Equ. 2013., 2013: Article ID 48Google Scholar
- Zhao Y, Chen H, Zhang Q: Existence and multiplicity of positive solutions for nonhomogeneous boundary value problems with fractional q -derivative. Bound. Value Probl. 2013., 2013: Article ID 103Google Scholar
- Yang W: Positive solutions for boundary value problems involving nonlinear fractional q -difference equations. Differ. Equ. Appl. 2013, 5: 205-219.MathSciNetGoogle Scholar
- Yang W: Positive solution for fractional q -difference boundary value problems with ϕ -Laplacian operator. Bull. Malays. Math. Soc. 2013, 36(4):1195-1203.Google Scholar
- El-Shahed M, Al-Askar FM: Positive solutions for boundary value problem of nonlinear fractional q -difference equation. ISRN Math. Anal. 2011., 2011: Article ID 385459Google Scholar
- Graef JR, Kong L: Positive solutions for a class of higher order boundary value problems with fractional q -derivatives. Appl. Math. Comput. 2012, 218: 9682-9689. 10.1016/j.amc.2012.03.006MathSciNetView ArticleGoogle Scholar
- Ma J, Yang J: Existence of solutions for multi-point boundary value problem of fractional q -difference equation. Electron. J. Qual. Theory Differ. Equ. 2011, 92: 1-10.MathSciNetView ArticleGoogle Scholar
- Zhao Y, Ye G, Chen H: Multiple positive solutions of a singular semipositone integral boundary value problem for fractional q -derivatives equation. Abstr. Appl. Anal. 2013., 2013: Article ID 643571Google Scholar
- Ferreira RAC: Nontrivial solutions for fractional q -difference boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2010, 70: 1-10.View ArticleGoogle Scholar
- Ferreira RAC: Positive solutions for a class of boundary value problems with fractional q -differences. Comput. Math. Appl. 2011, 61: 367-373. 10.1016/j.camwa.2010.11.012MathSciNetView ArticleGoogle Scholar
- Liang S, Zhang J: Existence and uniqueness of positive solutions for three-point boundary value problem with fractional q -differences. J. Appl. Math. Comput. 2012, 40: 277-288. 10.1007/s12190-012-0551-2MathSciNetView ArticleGoogle Scholar
- Ahmad B, Ntouyas S, Purnaras I: Existence results for nonlocal boundary value problems of nonlinear fractional q -difference equations. Adv. Differ. Equ. 2012., 2012: Article ID 140Google Scholar
- Alsaedi A, Ahmad B, Al-Hutami H: A study of nonlinear fractional q -difference equations with nonlocal integral boundary conditions. Abstr. Appl. Anal. 2013., 2013: Article ID 410505Google Scholar
- Agarwal RP, Meehan M, O’Regan D Cambridge Tracts in Mathematics 141. In Fixed Point Theory and Applications. Cambridge University Press, Cambridge; 2001.View ArticleGoogle Scholar
- Li C, Luo X, Zhou Y: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. Comput. Math. Appl. 2010, 59: 1363-1375. 10.1016/j.camwa.2009.06.029MathSciNetView ArticleGoogle Scholar
- Granas A, Guenther RB, Lee JW: Some general existence principles in the Carathéodory theory of nonlinear differential systems. J. Math. Pures Appl. 1991, 70: 153-196.MathSciNetGoogle Scholar
- Krasnosel’skii MA: Topological Methods in the Theory of Nonlinear Integral Equations. Macmillan Co., New York; 1964.Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.