Existence of Positive Solutions to a Boundary Value Problem for a Delayed Nonlinear Fractional Differential System
© Zigen Ouyang et al. 2011
Received: 14 November 2010
Accepted: 24 February 2011
Published: 14 March 2011
Though boundary value problems for fractional differential equations have been extensively studied, most of the studies focus on scalar equations and the fractional order between 1 and 2. On the other hand, delay is natural in practical systems. However, not much has been done for fractional differential equations with delays. Therefore, in this paper, we consider a boundary value problem of a general delayed nonlinear fractional system. With the help of some fixed point theorems and the properties of the Green function, we establish several sets of sufficient conditions on the existence of positive solutions. The obtained results extend and include some existing ones and are illustrated with some examples for their feasibility.
In the past decades, fractional differential equations have been intensively studied. This is due to the rapid development of the theory of fractional differential equations itself and the applications of such construction in various sciences such as physics, mechanics, chemistry, and engineering [1, 2]. For the basic theory of fractional differential equations, we refer the readers to [3–7].
Recently, many researchers have devoted their attention to studying the existence of (positive) solutions of boundary value problems for differential equations with fractional order [8–23]. We mention that the fractional order involved is generally in with the exception that in [12, 23] and in [8, 17]. Though there have been extensive study on systems of fractional differential equations, not much has been done for boundary value problems for systems of fractional differential equations [18–20].
On the other hand, we know that delay arises naturally in practical systems due to the transmission of signal or the mechanical transmission. Though theory of ordinary differential equations with delays is mature, not much has been done for fractional differential equations with delays [24–31].
where is the standard Riemann-Liouville fractional derivative of order for some integer , for , for , and is a nonlinear function from to . The purpose is to establish sufficient conditions on the existence of positive solutions to (1.1) by using some fixed point theorems and some properties of the Green function. By a positive solution to (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 differential equations when for all and . Therefore, the obtained results generalize and include some existing ones.
The remaining part of this paper is organized as follows. In Section 2, we introduce some basics of fractional derivative and the fixed point theorems which will be used in Section 3 to establish the existence of positive solutions. To conclude the paper, the feasibility of some of the results is illustrated with concrete examples in Section 4.
We first introduce some basic definitions of fractional derivative for the readers' convenience.
provided that the integral exists on , where is the Gamma function.
provided that the right-hand side is pointwise defined on , where .
It is well known that if then . Furthermore, if and then for .
The following results on fractional integral and fractional derivative will be needed in establishing our main results.
Lemma 2.3 (see ).
where , .
Lemma 2.4 (see ).
for some , .
Now, we cite the fixed point theorems to be used in Section 3.
Lemma 2.5 (the Banach contraction mapping theorem ).
Let be a complete metric space and let be a contraction mapping. Then has a unique fixed point.
Let be a closed and convex subset of a Banach space . Assume that is a relatively open subset of with and is completely continuous. Then at least one of the following two properties holds:
(i) has a fixed point in ;
(ii)there exists and with .
Let be a cone in a Banach space . Assume that and are open subsets of with and . Suppose that is a completely continuous operator such that either
(i) for and for
(ii) for and for .
Then has a fixed point in .
3. Existence of Positive Solutions
In this section, we always assume that .
This completes the proof.
The following two results give some properties of the Green functions .
For is continuous on and for .
Note that and for . It follows that and hence for .
Therefore, for and the proof is complete.
If , then for .
If , then for .
- (i)Obviously, for . Now, for , we have(3.14)
- (ii)Again, one can easily see that for . When , we have in this case that(3.15)
which implies that for . To summarize, we have proved (ii) and this completes the proof.
Now, we are ready to present the main results.
then (1.1) has a unique positive solution.
This, combined with Lemma 3.3 and (3.17) and (3.18), immediately implies that is a contraction. Therefore, the proof is complete with the help of Lemmas 3.1 and 2.5.
The following result can be proved in the same spirit as that for Theorem 3.4.
then (1.1) has a unique positive solution.
then (1.1) has at least one positive solution.
Let and be defined by (3.19) and (3.20), respectively. We first show that is completely continuous through the following three steps.
which implies that is continuous.
Immediately, we can easily see that is a bounded subset of .
Now the equicontituity of on follows easily from the fact that is continuous and hence uniformly continuous on .
Similarly, we can have if . To summarize, , a contradiction to . This proves the claim. Applying Lemma 2.6, we know that has a fixed point in , which is a positive solution to (1.1) by Lemma 3.1. Therefore, the proof is complete.
As a consequence of Theorem 3.6, we have the following.
If all , , are bounded, then (1.1) has at least one positive solution.
Suppose that there exist and positive constants with such that
(ii) , for ,
where . Then (1.1) has at least a positive solution.
Therefore, we have verified condition (ii) of Lemma 2.7. It follows that 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 3.4 that (4.1) has a unique positive solution on .
By now we have verified all the assumptions of Theorem 3.8. Therefore, (4.6) has at least one positive solution satisfying .
Supported partially by the Doctor Foundation of University of South China under Grant no. 5-XQD-2006-9, the Foundation of Science and Technology Department of Hunan Province under Grant no. 2009RS3019 and the Subject Lead Foundation of University of South China no. 2007XQD13. Research was partially supported by the Natural Science and Engineering Re-search Council of Canada (NSERC) and the Early Researcher Award (ERA) Pro-gram of Ontario.
- Debnath L: Recent applications of fractional calculus to science and engineering. International Journal of Mathematics and Mathematical Sciences 2003, (54):3413-3442.View ArticleGoogle Scholar
- Sabatier J, Agrawal OP, Tenreiro Machado JA: Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht, The Netherlands; 2007.View ArticleGoogle Scholar
- Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies. Volume 204. Elsevier Science, Amsterdam, The Netherlands; 2006:xvi+523.Google Scholar
- Lakshmikantham V: Theory of fractional functional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(10):3337-3343. 10.1016/j.na.2007.09.025View ArticleMathSciNetMATHGoogle Scholar
- Lakshmikantham V, Leela S, Vasundhara Devi J: Theory of Fractional Dynamic Systems. Cambridge Scientific, Cambridge, UK; 2009.MATHGoogle Scholar
- Lakshmikantham V, Vatsala AS: Basic theory of fractional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(8):2677-2682. 10.1016/j.na.2007.08.042View ArticleMathSciNetMATHGoogle Scholar
- Podlubny I: Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.Google Scholar
- Bai C: Triple positive solutions for a boundary value problem of nonlinear fractional differential equation. Electronic Journal of Qualitative Theory of Differential Equations 2008, (24):-10.Google Scholar
- Bai Z: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Analysis: Theory, Methods & Applications 2010, 72(2):916-924. 10.1016/j.na.2009.07.033View ArticleMathSciNetMATHGoogle Scholar
- Bai Z, Lü H: Positive solutions for boundary value problem of nonlinear fractional differential equation. Journal of Mathematical Analysis and Applications 2005, 311(2):495-505. 10.1016/j.jmaa.2005.02.052View ArticleMathSciNetMATHGoogle Scholar
- Benchohra M, Hamani S, Ntouyas SK: Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(7-8):2391-2396. 10.1016/j.na.2009.01.073View ArticleMathSciNetMATHGoogle Scholar
- El-Shahed M: Positive solutions for boundary value problem of nonlinear fractional differential equation. Abstract and Applied Analysis 2007, 2007:-8.Google Scholar
- Jafari H, Daftardar-Gejji V: Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method. Applied Mathematics and Computation 2006, 180(2):700-706. 10.1016/j.amc.2006.01.007View ArticleMathSciNetMATHGoogle Scholar
- Kaufmann ER, Mboumi E: Positive solutions of a boundary value problem for a nonlinear fractional differential equation. Electronic Journal of Qualitative Theory of Differential Equations 2008, (3):-11.MathSciNetGoogle Scholar
- Kosmatov N: A singular boundary value problem for nonlinear differential equations of fractional order. Journal of Applied Mathematics and Computing 2009, 29(1-2):125-135. 10.1007/s12190-008-0104-xView ArticleMathSciNetMATHGoogle Scholar
- Li CF, Luo XN, Zhou Y: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. Computers & Mathematics with Applications 2010, 59(3):1363-1375.View ArticleMathSciNetMATHGoogle Scholar
- Liang S, Zhang J: Positive solutions for boundary value problems of nonlinear fractional differential equation. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(11):5545-5550. 10.1016/j.na.2009.04.045View ArticleMathSciNetMATHGoogle Scholar
- Su X: Boundary value problem for a coupled system of nonlinear fractional differential equations. Applied Mathematics Letters 2009, 22(1):64-69. 10.1016/j.aml.2008.03.001View ArticleMathSciNetMATHGoogle Scholar
- Wang J, Xiang H, Liu Z: Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations. International Journal of Differential Equations 2010, 2010:-12.Google Scholar
- Yang A, Ge W: Positive solutions for boundary value problems of N -dimension nonlinear fractional differential system. Boundary Value Problems 2008, 2008:-15.Google Scholar
- Zhang S: Existence of solution for a boundary value problem of fractional order. Acta Mathematica Scientia B 2006, 26(2):220-228. 10.1016/S0252-9602(06)60044-1View ArticleMATHGoogle Scholar
- Zhang S: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electronic Journal of Differential Equations 2006, (36):-12.Google Scholar
- Zhao Y, Sun S, Han Z, Li Q: The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations. Communications in Nonlinear Science and Numerical Simulation 2011, 16(4):2086-2097. 10.1016/j.cnsns.2010.08.017View ArticleMathSciNetMATHGoogle Scholar
- Babakhani A: Positive solutions for system of nonlinear fractional differential equations in two dimensions with delay. Abstract and Applied Analysis 2010, 2010:-16.Google Scholar
- Babakhani A, Enteghami E: Existence of positive solutions for multiterm fractional differential equations of finite delay with polynomial coefficients. Abstract and Applied Analysis 2009, 2009:-12.Google Scholar
- Benchohra M, Henderson J, Ntouyas SK, Ouahab A: Existence results for fractional order functional differential equations with infinite delay. Journal of Mathematical Analysis and Applications 2008, 338(2):1340-1350. 10.1016/j.jmaa.2007.06.021View ArticleMathSciNetMATHGoogle Scholar
- Deng W, Li C, Lü J: Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dynamics 2007, 48(4):409-416. 10.1007/s11071-006-9094-0View ArticleMathSciNetMATHGoogle Scholar
- Hu L, Ren Y, Sakthivel R: Existence and uniqueness of mild solutions for semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays. Semigroup Forum 2009, 79(3):507-514. 10.1007/s00233-009-9164-yView ArticleMathSciNetMATHGoogle Scholar
- Maraaba TA, Jarad F, Baleanu D: On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives. Science in China. Series A 2008, 51(10):1775-1786. 10.1007/s11425-008-0068-1View ArticleMathSciNetMATHGoogle Scholar
- Mophou GM, N'Guérékata GM: A note on a semilinear fractional differential equation of neutral type with infinite delay. Advances in Difference Equations 2010, 2010:-8.Google Scholar
- Zhang X: Some results of linear fractional order time-delay system. Applied Mathematics and Computation 2008, 197(1):407-411. 10.1016/j.amc.2007.07.069View ArticleMathSciNetMATHGoogle Scholar
- Podlubny I: Fractional Differential Equations. Academic Press, San Diego, Calif, USA; 1993.Google Scholar
- Agarwal RP, Meehan M, O'Regan D: Fixed Point Theory and Applications, Cambridge Tracts in Mathematics. Volume 141. Cambridge University Press, Cambridge, UK; 2001:x+170.View ArticleGoogle Scholar
- Granas A, Guenther RB, Lee JW: Some general existence principles in the Carathéodory theory of nonlinear differential systems. Journal de Mathématiques Pures et Appliquées 1991, 70(2):153-196.MathSciNetMATHGoogle Scholar
- Krasnosel'skii MA: Topological Methods in the Theory of Nonlinear Integral Equations. The Macmillan, New York, NY, USA; 1964:xi + 395.Google Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.