Existence and Uniqueness of Positive Solutions for a Coupled System of Nonlinear Fractional Differential Equations

Affiliation(s)

College of Sciences, Guangxi University for Nationalities, Nanning 530006, Guangxi Province, P. R. China.

College of Sciences, Guangxi University for Nationalities, Nanning 530006, Guangxi Province, P. R. China.

ABSTRACT

In this paper, we research the existence and uniqueness of positive solutions for a coupled system of fractional differential equations. By means of some standard fixed point principles, some results on the existence and uniqueness of positive solutions for coupled systems are obtained.

Cite this paper

M. Li and Y. Liu, "Existence and Uniqueness of Positive Solutions for a Coupled System of Nonlinear Fractional Differential Equations,"*Open Journal of Applied Sciences*, Vol. 3 No. 1, 2013, pp. 53-61. doi: 10.4236/ojapps.2013.31B1011.

M. Li and Y. Liu, "Existence and Uniqueness of Positive Solutions for a Coupled System of Nonlinear Fractional Differential Equations,"

References

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[12] C. F. Li, X. Z. Luo and Y. Zhou, “Existence of Positive Solutions of the Boundary Value Problem for Nonlinear Fracional Differential Equations,” Computers and Mathematics with Applications, Vol. 59, No. 3, 2010, pp. 1363-1375. doi：10.1016/j.camwa.2009.06.029

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[17] Z. H. Liu and J. H. Sun, “Nonlinear Boundary Value Problems of Fractional Functional Integro-differential Equations,” Computers and Mathematics with Applications, Vol. 64, No. 10, 2012, pp. 3228–3234. doi：10.1016/j.camwa.2012.02.026

[18] Z. H. Liu and J. H. Sun, “Nonlinear Boundary Value Problems of Fractional Differential Systems,” Computers and Mathematics with Applications, Vol. 64, No. 4, 2012, pp. 463-475. doi：10.1016/j.camwa.2011.12.020

[19] R. Ma and L. Xu, “Existence of Positive Solutions of a Nonlinear Fourth-order Boundary Value Problem,” Applied Mathematics Letters, Vol. 23, No. 5, 2010, pp. 537-543. doi：10.1016/j.aml.2010.01.007

[20] K. S. Miller and B. Ross, “An Introduction to the Fractional Calculus and Fractional Differential Equations,” Wiley, New York, 1993.

[21] I. Podlubny, “Fractional Differential Equations,” Academic Press, San Diego, 1999.

[22] D. R. Smart, “Fixed Point Theorems,” Cambridge University Press, 1980.

[23] J. Sabatier, O. P. Agrawal, J. A. T. Machado (Eds.), “Advances in Fractional Calculus:Theoretical Developments and Applications in Physics and Engineering,” Springer, Dordrecht, 2007. doi：10.1007/978-1-4020-6042-7

[24] S. G. Samko, A. A. Kilbas and O. I. Marichev, “Fractional Integral and Derivatives,” Theory and Applications, Gordon and Breach, Yverdon, 1993.

[25] J. H. Sun, Y. L. Liu and G. F. Liu, “Existence of Solutions for Fractional Differential Systems with Antiperiodic Boundary Conditions,” Computers & Mathematics with Applications, Vol. 64, No. 6, 2012, pp. 1557-1566. doi：10.1016/j.camwa.2011.12.083

[26] X. Su, “Boundary Value Problem for a Couple Systems of Nonliear Fractional Differential Equations,” Applied Mathematics Letters, Vol. 22, 2009.

[27] J. H. Wang, H. J. Xiang and Z. G. Liu, “Positive Solution to Nonzero Boundary Values Problem for a Coupled System of Nonlinear Fractional Differential Equations,” International Journal of Differential Equations, Vol. 2010, 2010, p. 12. doi：10.1155/2010/186928

[28] X. Xu, D. Jiang and C. Yuan, “Multiple Positive Solutions for the Boundary Value Problem of a Nonlinear Fractional Differential Equation,” Nonlinear Analysis, Vol. 71, No. 10, 2009, pp. 4676-4688. doi：10.1016/j.na.2009.03.030

[29] X. Yang, Z. L. Wei and W. Dong, “Existence of Positive Solutions for the Boundary Value Problem of Nonlinear Fractional Differential Equations,” Communications in Nonlinear Science and Numerical Simula-tion.

[30] X. Yang, Z. Wei and W. Dong, “Existence of Positive Solutions for the Boundary Value Problem of Nonlineear Fractional Differential Equations,” Communications in Nonlinear Science and Numerical Simulation, Vol. 17, 2012, pp. 85-92.

[31] S. Zhang, “Positive Solutions for Boundary Value Problems of Nonlinear Fractional Differential Equations,” Elec. Journal of Differential Equations, Vol. 36, 2006, pp. 1-12.

[32] Y. G. Zhao, S. R. Sun, Z. L. Han and M. Zhang, “Positive Solutions for Boundary Value Problems of Nonlinear Fractional Differential Equations,” Applied Mathematics and Computations, Vol. 2, No. 17, 2011, pp. 6950-6958. doi：10.1016/j.amc.2011.01.103

[33] Y. Zhao, S. Sun, Z. Han and Q. Li, “Positive Solutions to Boundary Value Problems of Nonlinear Fractional Differential Equation,” Abstract and Aplied Analysis, Vol. 2011, 2011, p. 16. doi：10.1155/2011/390543

[34] Y. Zhou and F. Jiao, “Nonlocal Cauchy Problem for Fractional Evolution Equations,” Nonlinear Analysis, Vol. 11, 2010, pp. 4465-4475.

[1] R. P. Agarwal, D. O’Regan and S. StaneK, “Positive Solutions of the Boundary Value Problem for Nonlinear Fractional Differential Equations,” Journal of Mathatical Analysis and Application, Vol. 371, No. 1, 2010, pp. 57-68. doi：10.1016/j.jmaa.2010.04.034

[2] B. Ahmad and J. J. Nieto, “Existence Results for a Couple System of Nonlinear Fractional Differential Equations with Three-Point Boundary Conditions,” Computers& Mathematics with Applications, Vol. 58, No. 9, 2009, pp. 1838-1843. doi：10.1016/j.camwa.2009.07.091

[3] C. Z. Bai and J. X. Fang, “The Existence of a Positive Solution for a Singular Coupled Systems of Nonlinear Fractional Differential Equations,” Applied Mathematics and Computation, Vol. 150, No. 3, 2004, pp. 611-621. doi：10.1016/S0096-3003(03)00294-7

[4] Z. B. Bai and H. S. Lu, “Positive Solutions of Boundary Value Problems of Nonlinear Fractional Differential Equation,” Journal of Mathematical Analysis and Applications, Vol. 311, No. 2, 2005, pp. 495-505. doi：10.1016/j.jmaa.2005.02.052

[5] R. A. C. Ferreira, “Positive Solutions for a Class of Boundary Value Problems with Fractional Q-Differences,” Computer & Mathematics with Applications, Vol. 61, 2011, No. 2, pp. 367-373. doi：10.1016/j.camwa.2010.11.012

[6] M. Feckan, Y. Zhou and J. R. Wang, “On the Concept and Existence of Solution for Impulsive Fractional Differential Equations,” Communications in Nonlinear Science and Numerical Simulation, Vol. 17, No. 7, 2012, pp. 3050-3060. doi：10.1016/j.cnsns.2011.11.017

[7] C. S. Goodrich, “Existence of a Positive Solution to a System of Discrete Fractional Boundary Value Problems,” Allied Mathematics and Computation, Vol. 217, No. 9, 2011, pp. 4740-4753. doi：10.1016/j.amc.2010.11.029

[8] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, “Theory and Applications of Fractional Differential Equations,” in: North-Holland Mathematics Studies, Vol. 204, Elservier Science B.V., Amsterdam, 2006.

[9] V. Lakshmikantham, S. Leela and J. Vasundhara Devi, “Theory of Fractional Dynamic Systems,” Camb. Acad. Publ., Cambridge, 2009.

[10] V. Lakshmikantham and A. S. Vatsala, “Basic Theory of Fractional Differntial Equations,” Nonlinear Analysis, Vol. 69, No. 8, 2009, pp. 2677-2682. doi：10.1016/j.na.2007.08.042

[11] V. Lakshmikantham, “Theory of Fractional Functional Diffrential Equations,” Non-linear Analysis, Vol. 69, No. 10, 2008, pp. 3337-3343. doi：10.1016/j.na.2007.09.025

[12] C. F. Li, X. Z. Luo and Y. Zhou, “Existence of Positive Solutions of the Boundary Value Problem for Nonlinear Fracional Differential Equations,” Computers and Mathematics with Applications, Vol. 59, No. 3, 2010, pp. 1363-1375. doi：10.1016/j.camwa.2009.06.029

[13] Z. H. Liu and J. H. Han, “Integral Boundary Value Problems for Fractional Order Integro-differential Equations,” Dynamic Systems and Applica-tions, Vol. 21, 2012, pp. 535-548.

[14] Z. H. Liu and X. W. Li, “Existence and Uniqueness of Solutions for the Nonlinear Impulsive Fractional Differential equations,” Communications in Nonlinear Science and Numerical Simulation, Vol. 18, No. 6, 2013, pp. 1362-1373. doi：10.1016/j.cnsns.2012.10.010

[15] Z. H. Liu and X. W. Li, “On the Controllability of Impulsive Fractional Evolution Inclusions in Banach Spaces,” Journal of Optimization Theory and Applications, Vol. 156, No. 1, 2013, pp. 167–182. doi：10.1007/s10957-012-0236-x

[16] Z. H. Liu and L. Lu, “A Class of BVPs for Nonlinear Fractional Differential Equations with P-Laplacian Operator,” E. J. Qualitative Theory of Diffe-rential Equations, No. 70, 2012, pp. 1-16.

[17] Z. H. Liu and J. H. Sun, “Nonlinear Boundary Value Problems of Fractional Functional Integro-differential Equations,” Computers and Mathematics with Applications, Vol. 64, No. 10, 2012, pp. 3228–3234. doi：10.1016/j.camwa.2012.02.026

[18] Z. H. Liu and J. H. Sun, “Nonlinear Boundary Value Problems of Fractional Differential Systems,” Computers and Mathematics with Applications, Vol. 64, No. 4, 2012, pp. 463-475. doi：10.1016/j.camwa.2011.12.020

[19] R. Ma and L. Xu, “Existence of Positive Solutions of a Nonlinear Fourth-order Boundary Value Problem,” Applied Mathematics Letters, Vol. 23, No. 5, 2010, pp. 537-543. doi：10.1016/j.aml.2010.01.007

[20] K. S. Miller and B. Ross, “An Introduction to the Fractional Calculus and Fractional Differential Equations,” Wiley, New York, 1993.

[21] I. Podlubny, “Fractional Differential Equations,” Academic Press, San Diego, 1999.

[22] D. R. Smart, “Fixed Point Theorems,” Cambridge University Press, 1980.

[23] J. Sabatier, O. P. Agrawal, J. A. T. Machado (Eds.), “Advances in Fractional Calculus:Theoretical Developments and Applications in Physics and Engineering,” Springer, Dordrecht, 2007. doi：10.1007/978-1-4020-6042-7

[24] S. G. Samko, A. A. Kilbas and O. I. Marichev, “Fractional Integral and Derivatives,” Theory and Applications, Gordon and Breach, Yverdon, 1993.

[25] J. H. Sun, Y. L. Liu and G. F. Liu, “Existence of Solutions for Fractional Differential Systems with Antiperiodic Boundary Conditions,” Computers & Mathematics with Applications, Vol. 64, No. 6, 2012, pp. 1557-1566. doi：10.1016/j.camwa.2011.12.083

[26] X. Su, “Boundary Value Problem for a Couple Systems of Nonliear Fractional Differential Equations,” Applied Mathematics Letters, Vol. 22, 2009.

[27] J. H. Wang, H. J. Xiang and Z. G. Liu, “Positive Solution to Nonzero Boundary Values Problem for a Coupled System of Nonlinear Fractional Differential Equations,” International Journal of Differential Equations, Vol. 2010, 2010, p. 12. doi：10.1155/2010/186928

[28] X. Xu, D. Jiang and C. Yuan, “Multiple Positive Solutions for the Boundary Value Problem of a Nonlinear Fractional Differential Equation,” Nonlinear Analysis, Vol. 71, No. 10, 2009, pp. 4676-4688. doi：10.1016/j.na.2009.03.030

[29] X. Yang, Z. L. Wei and W. Dong, “Existence of Positive Solutions for the Boundary Value Problem of Nonlinear Fractional Differential Equations,” Communications in Nonlinear Science and Numerical Simula-tion.

[30] X. Yang, Z. Wei and W. Dong, “Existence of Positive Solutions for the Boundary Value Problem of Nonlineear Fractional Differential Equations,” Communications in Nonlinear Science and Numerical Simulation, Vol. 17, 2012, pp. 85-92.

[31] S. Zhang, “Positive Solutions for Boundary Value Problems of Nonlinear Fractional Differential Equations,” Elec. Journal of Differential Equations, Vol. 36, 2006, pp. 1-12.

[32] Y. G. Zhao, S. R. Sun, Z. L. Han and M. Zhang, “Positive Solutions for Boundary Value Problems of Nonlinear Fractional Differential Equations,” Applied Mathematics and Computations, Vol. 2, No. 17, 2011, pp. 6950-6958. doi：10.1016/j.amc.2011.01.103

[33] Y. Zhao, S. Sun, Z. Han and Q. Li, “Positive Solutions to Boundary Value Problems of Nonlinear Fractional Differential Equation,” Abstract and Aplied Analysis, Vol. 2011, 2011, p. 16. doi：10.1155/2011/390543

[34] Y. Zhou and F. Jiao, “Nonlocal Cauchy Problem for Fractional Evolution Equations,” Nonlinear Analysis, Vol. 11, 2010, pp. 4465-4475.