Application of Schauder fixed point theorem to a coupled system of differential equations of fractional order

1304
Downloads

1532
Views
Authors
Mengru Hao
 School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, P.R. China.
Chengbo Zhai
 School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, P.R. China.
Abstract
In this paper, by using Schauder fixed point theorem, we study the existence of at least one positive solution
to a coupled system of fractional boundary value problems given by
\[
\begin{cases}
D^{\nu_1}_{0^+}y_1(t) = \lambda_1a_1(t)f(t, y_1(t), y_2(t)) + e_1(t),\\
D^{\nu_2}_{0^+}y_2(t) = \lambda_2a_2(t)g(t, y_1(t), y_2(t)) + e_2(t),
\end{cases}
\]
where \(\nu_1,\nu_2\in (n  1; n]\) for \(n > 3\) and \(n \in N\), subject to the boundary conditions \(y^(i)_1 (0) = 0 = y^(i)_2 (0)\), for
\(0 \leq i \leq n  2\), and \([D^{\alpha}_{0^+}y_1(t)]_{t=1}=0=[D^{\alpha}_{0^+}y_2(t)]_{t=1}\), for \(1 \leq\alpha\leq n  2\).
Share and Cite
ISRP Style
Mengru Hao, Chengbo Zhai, Application of Schauder fixed point theorem to a coupled system of differential equations of fractional order, Journal of Nonlinear Sciences and Applications, 7 (2014), no. 2, 131137
AMA Style
Hao Mengru, Zhai Chengbo, Application of Schauder fixed point theorem to a coupled system of differential equations of fractional order. J. Nonlinear Sci. Appl. (2014); 7(2):131137
Chicago/Turabian Style
Hao, Mengru, Zhai, Chengbo. "Application of Schauder fixed point theorem to a coupled system of differential equations of fractional order." Journal of Nonlinear Sciences and Applications, 7, no. 2 (2014): 131137
Keywords
 Fractional differential equation
 Schauder fixed point theorem
 Positive solution.
MSC
References

[1]
R. P. Agarwal, M. Benchohra, S. Hamani , A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta. Appl. Math., 109 (2010), 9731033.

[2]
B. Ahmad, J. J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with threepoint boundary conditions , Comput. Math. Appl., 58 (2009), 18381843.

[3]
A. Anguraj, M. C. Ranjini, Existence of mild solutions of random impulsive functional differential equations with almost sectorial operators, J. Nonlinear Sci. Appl., 5 (2012), 174185.

[4]
A. Anguraj, M. L. Maheswari , Existence of solutions for fractional impulsive neutral functional infinite delay integrodifferential equations with nonlocal conditions, J. Nonlinear Sci. Appl., 5 (2012), 271280.

[5]
B. Ahmad, J. J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with threepoint boundary conditions, Comput. Math. Appl., 58 (2009), 18381843.

[6]
C. Z. Bai, J. X. Fang, The existence of a positive solution for a singular coupled system of a nonlinear fractional differential equations, Appl. Math. Comput., 150 (3) (2004), 611621.

[7]
M. Benchohra, B. A. Slimani , Existence and uniqueness of solutions to impulsive fractional differential equations, Electron. J. Differ. Equat., 2009 (10) (2009), 111.

[8]
D. Dunninger, H. Wang , Existence and multiplicity of positive solutions for elliptic systems , Nonlinear Anal., 29 (1997), 10511060.

[9]
C. S. Goodrich, Existence of a positive solution to systems of differential equations of fractioanl order, Comput. Math. Appl. , 62 (2011), 12511268.

[10]
C. S. Goodrich, Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett., 23 (2010), 10501055.

[11]
A. GuezaneLakoud, R. Khaldi, Solvability of a twopoint fractional boundary value problem, J. Nonlinear Sci. Appl. , 5 (2012), 6473.

[12]
V. Kavitha, M. M. Arjunan, Controllability of impulsive quasilinear fractional mixed volterrafredholmtype integrodifferential equations in Banach spaces, J. Nonlinear Sci. Appl., 4 (2) (2011), 152169.

[13]
A. A. Kilbas, H. H. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science B. V., Amsterdam (2006)

[14]
Y. J. Liu, H. P. Shi , Existence of unbounded positive solutions for BVPs of singular fractional differential equations, J. Nonlinear Sci. Appl., 5 (2012), 281293.

[15]
K. S. Miller, B. Ross, An introduction to the fractional calculus and differential equations, John Wiley, New York (1993)

[16]
I. Podlubny, Fractional differential equations, Academic Press, New York (1999)

[17]
T. T. Qiu, Z. B. Bai , Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Nonlinear Sci. Appl. , 1 (3) (2008), 123131.

[18]
M. ur Rehman, R. A. Khan, Existence and uniqueness of solutions for multipoint boundary value problems for fractional differential equations, Appl. Math. Lett., 23 (2010), 10381044.

[19]
S. R. Sun, Q. P. Li, Y. N. Li , Existence and uniqueness of solutions for a coupled system of multiterm nonlinear fractional differential equations, Comput. Math. Appl., 64 (2012), 33103320.

[20]
X. W. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett., 22 (2009), 6469.

[21]
V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, HEP (2010)

[22]
G. Wang, L. Zhang, G. Song , Systems of first order impulsive functional differential equations with deviating arguments and nonlinear boundary conditions, Nonlinear Anal.: TMA, 74 (2011), 974982.

[23]
T. G. Wang, F. Xie, Existence and uniqueness of fractional differential equations with integral boundary conditions, J. Nonlinear Sci. Appl., 1 (4) (2008), 206212.

[24]
C. Yang, C. B. Zhai, Uniqueness of positive solutions for a fractional differential equation via a fixed point theorem of a sum operator, Electron. J. Differ. Equat., 70 (2012), 18.

[25]
C. B. Zhai, W. P. Yan, C. Yang, Sum operator method for the existence and uniqueness of positive solutions to RiemannCLiouville fractional differential equation boundary value problems, Commun. Nonlinear Sci. Numer. Simulat., 18 (2013), 858866.