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

Volume 7, Issue 2, pp 131--137
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### 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 $1 \leq\alpha\leq n - 2$.

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##### 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, 131--137

##### 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):131--137

##### 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): 131--137

### Keywords

• Fractional differential equation
• Schauder fixed point theorem
• Positive solution.

•  47H10
•  34A08
•  34B18.

### 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), 973-1033.

• [2] B. Ahmad, J. J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions , Comput. Math. Appl., 58 (2009), 1838-1843.

• [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), 174-185.

• [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), 271-280.

• [5] B. Ahmad, J. J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. Math. Appl., 58 (2009), 1838-1843.

• [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), 611-621.

• [7] M. Benchohra, B. A. Slimani , Existence and uniqueness of solutions to impulsive fractional differential equations, Electron. J. Differ. Equat., 2009 (10) (2009), 1-11.

• [8] D. Dunninger, H. Wang , Existence and multiplicity of positive solutions for elliptic systems , Nonlinear Anal., 29 (1997), 1051-1060.

• [9] C. S. Goodrich, Existence of a positive solution to systems of differential equations of fractioanl order, Comput. Math. Appl. , 62 (2011), 1251-1268.

• [10] C. S. Goodrich, Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett., 23 (2010), 1050-1055.

• [11] A. Guezane-Lakoud, R. Khaldi, Solvability of a two-point fractional boundary value problem, J. Nonlinear Sci. Appl. , 5 (2012), 64-73.

• [12] V. Kavitha, M. M. Arjunan, Controllability of impulsive quasi-linear fractional mixed volterra-fredholm-type integrodifferential equations in Banach spaces, J. Nonlinear Sci. Appl., 4 (2) (2011), 152-169.

• [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), 281-293.

• [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), 123-131.

• [18] M. ur Rehman, R. A. Khan, Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations, Appl. Math. Lett., 23 (2010), 1038-1044.

• [19] S. R. Sun, Q. P. Li, Y. N. Li , Existence and uniqueness of solutions for a coupled system of multi-term nonlinear fractional differential equations, Comput. Math. Appl., 64 (2012), 3310-3320.

• [20] X. W. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett., 22 (2009), 64-69.

• [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), 974-982.

• [23] T. G. Wang, F. Xie, Existence and uniqueness of fractional differential equations with integral boundary conditions, J. Nonlinear Sci. Appl., 1 (4) (2008), 206-212.

• [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), 1-8.

• [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), 858-866.