Existence and Uniqueness of Solutions to Impulsive Fractional Integro-Differential Equations with Nonlocal Conditions

Affiliation(s)

Department of Mathematics and Computational Science, Hengyang Normal University, Hengyang, China.

Department of Mathematics and Computational Science, Hengyang Normal University, Hengyang, China.

ABSTRACT

In this article, by using Schaefer fixed point theorem, we establish sufficient conditions for the existence and uniqueness of solutions for a class of impulsive integro-differential equations with nonlocal conditions involving the Caputo fractional derivative.

In this article, by using Schaefer fixed point theorem, we establish sufficient conditions for the existence and uniqueness of solutions for a class of impulsive integro-differential equations with nonlocal conditions involving the Caputo fractional derivative.

KEYWORDS

Caputo Fractional Derivative; Impulses; Nonlocal Conditions; Existence; Uniqueness; Fixed Point

Caputo Fractional Derivative; Impulses; Nonlocal Conditions; Existence; Uniqueness; Fixed Point

Cite this paper

Z. Gao, L. Yang and G. Liu, "Existence and Uniqueness of Solutions to Impulsive Fractional Integro-Differential Equations with Nonlocal Conditions,"*Applied Mathematics*, Vol. 4 No. 6, 2013, pp. 859-863. doi: 10.4236/am.2013.46118.

Z. Gao, L. Yang and G. Liu, "Existence and Uniqueness of Solutions to Impulsive Fractional Integro-Differential Equations with Nonlocal Conditions,"

References

[1] [1] J. A. Tenreiro Machado, V. Kiryakova and F. Mainardi, “Recent History of Fractional Calculus,” Communications in Nonlinear Science and Numerical Simulation, Vol. 16, No. 3, 2011, pp. 1140-1153.

[2] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, “Theory and Applications of Fractional Differential Equations,” North-Holland Mathematics Studies, Vol. 204, 2006. doi:10.1016/S0304-0208(06)80001-0

[3] K. Diethelm, “The Analysis of Fractional Differential Equations,” Springer-Verlag, Berlin, Heidelberg, 2010. doi:10.1007/978-3-642-14574-2

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

[5] I. Podlubny, “Fractional Differential Equations,” Academic Press, San Diego, New York, London, 1999.

[6] S. G. Samko, A. A. Kilbas and O. I. Marichev, “Fractional Integral and Derivatives,” Gordon and Breach Science Publisher, London, 1993.

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

[8] V. Lakshmikantham, S. Leela and J. Vasundhara Devi, “Theory of Fractional Dynamic Systems,” Cambridge Academic Publishers, Cambridge, 2009.

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

[10] B. Ahmad and J. J. Nieto, “Existence of Solution for Non-Local Boundary Value Problems of Higher-Order Nonlinear Fractional Differential Equations,” Abstract and Applied Analysis, Vol. 2009, 2009, pp. 1-9. doi:10.1155/2009/494720

[11] A. A. Kilbas and S. A. Marzan, “Nonlinear Differential Equations with the Caputo Fractional Derivative in the Space of Continuously Differentiable Functions,” Differential Equations, Vol. 41, No. 1, 2005, pp. 84-89. doi:10.1007/s10625-005-0137-y

[12] V. D. Milman and A. D. Myshkis, “On the Stability of Motion in the Presence of Impulses (Russian),” Siberial Mathematical Journal, Vol. 1, No. 2, 1960, pp. 233-237.

[13] A. M. Samoilenko and N. A. Perestyuk, “Differential Equations with Impulses,” Viska Scola, Kiev, 1987 (in Russian).

[14] V. Lakshmikantham, D. D. Baino and P. S. Simeonov, “Theory of Impulsive Differential Equations,” World Scientific Publishing Corporation, Singapore City, 1989. doi:10.1142/0906

[15] D. D. Baino and P. S. Simeonov, “Systems with Impulsive Effects,” Horwood, Chichister, 1989.

[16] D. D. Baino and P. S. Simeonov, “Impulsive Differential Equations: Periodic Solutions and Its applications,” Longman Scientific and Technical Group, England, 1993.

[17] D. D. Baino and V. C. Covachev, “Impulsive Differential Equations with a Small Perturbations,” World Scientific, New Jersey, 1994. doi:10.1142/2058

[18] M. Benchohra, J. Henderson and S. K. Ntonyas, “Impulsive Differential Equations and Inclusions,” Hindawi Publishing Corporation, New York, 2006. doi:10.1155/9789775945501

[19] R. P. Agarwal, M. Benchohra and B. A. Salimani, “Existence Results for Differential Equations with Fractional Order and Impulses,” Memoir on Differential Equations and Mathematical Physics, Vol. 44, 2008, pp. 1-21.

[20] M. Benchohra and B. A. Salimani, “Existence and Uniqueness of Solutions to Impulsive Fractional Differential Equations,” Electronic Journal of Differential Equations, Vol. 2009, No. 10, 2009, pp. 1-11.

[21] M. Fecken, Y. Zhong and J. Wang, “On the Concept and existence of Solutions for Impulsive Fractional Differential Equations,” Communications in Non-Linear Science and numerical Simulation, Vol. 17, No. 7, 2012, pp. 3050-3060. doi:10.1016/j.cnsns.2011.11.017

[22] L. Byszewski and V. Lakshmikantham, “Theorem about the Existence and Uniqueness of a Solution of a Nonlocal Abstract Cauchy Problem in a Banach Space,” Journal of Applied Analysis, Vol. 40, 1991, pp. 11-19. doi:10.1080/00036819008839989

[23] L. Byszewski, “Theorems about Existence and Uniqueness of Solutions of a Semilinear Evolution Nonlocal Cauchy Problem,” Journal of Mathematical Analysis and Applications, Vol. 162, No. 2, 1991, pp. 494-505. doi:10.1016/0022-247X(91)90164-U

[24] L. Byszewski, “Existence and Uniqueness of Mild and Classical Solutions of Semilinear Functional-Differential Evolution Nonlocal Cauchy Problem,” Selected Problems of Mathematics, 50th Anniversary Cracow University of Technology, No. 6, Cracow University of Technology, Krakow, 1995, pp. 25-33.

[25] J. X. Sun, “Nonlinear Functional Analysis and Its Application,” Science Press, Beijing, 2008.

[1] [1] J. A. Tenreiro Machado, V. Kiryakova and F. Mainardi, “Recent History of Fractional Calculus,” Communications in Nonlinear Science and Numerical Simulation, Vol. 16, No. 3, 2011, pp. 1140-1153.

[2] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, “Theory and Applications of Fractional Differential Equations,” North-Holland Mathematics Studies, Vol. 204, 2006. doi:10.1016/S0304-0208(06)80001-0

[3] K. Diethelm, “The Analysis of Fractional Differential Equations,” Springer-Verlag, Berlin, Heidelberg, 2010. doi:10.1007/978-3-642-14574-2

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

[5] I. Podlubny, “Fractional Differential Equations,” Academic Press, San Diego, New York, London, 1999.

[6] S. G. Samko, A. A. Kilbas and O. I. Marichev, “Fractional Integral and Derivatives,” Gordon and Breach Science Publisher, London, 1993.

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

[8] V. Lakshmikantham, S. Leela and J. Vasundhara Devi, “Theory of Fractional Dynamic Systems,” Cambridge Academic Publishers, Cambridge, 2009.

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

[10] B. Ahmad and J. J. Nieto, “Existence of Solution for Non-Local Boundary Value Problems of Higher-Order Nonlinear Fractional Differential Equations,” Abstract and Applied Analysis, Vol. 2009, 2009, pp. 1-9. doi:10.1155/2009/494720

[11] A. A. Kilbas and S. A. Marzan, “Nonlinear Differential Equations with the Caputo Fractional Derivative in the Space of Continuously Differentiable Functions,” Differential Equations, Vol. 41, No. 1, 2005, pp. 84-89. doi:10.1007/s10625-005-0137-y

[12] V. D. Milman and A. D. Myshkis, “On the Stability of Motion in the Presence of Impulses (Russian),” Siberial Mathematical Journal, Vol. 1, No. 2, 1960, pp. 233-237.

[13] A. M. Samoilenko and N. A. Perestyuk, “Differential Equations with Impulses,” Viska Scola, Kiev, 1987 (in Russian).

[14] V. Lakshmikantham, D. D. Baino and P. S. Simeonov, “Theory of Impulsive Differential Equations,” World Scientific Publishing Corporation, Singapore City, 1989. doi:10.1142/0906

[15] D. D. Baino and P. S. Simeonov, “Systems with Impulsive Effects,” Horwood, Chichister, 1989.

[16] D. D. Baino and P. S. Simeonov, “Impulsive Differential Equations: Periodic Solutions and Its applications,” Longman Scientific and Technical Group, England, 1993.

[17] D. D. Baino and V. C. Covachev, “Impulsive Differential Equations with a Small Perturbations,” World Scientific, New Jersey, 1994. doi:10.1142/2058

[18] M. Benchohra, J. Henderson and S. K. Ntonyas, “Impulsive Differential Equations and Inclusions,” Hindawi Publishing Corporation, New York, 2006. doi:10.1155/9789775945501

[19] R. P. Agarwal, M. Benchohra and B. A. Salimani, “Existence Results for Differential Equations with Fractional Order and Impulses,” Memoir on Differential Equations and Mathematical Physics, Vol. 44, 2008, pp. 1-21.

[20] M. Benchohra and B. A. Salimani, “Existence and Uniqueness of Solutions to Impulsive Fractional Differential Equations,” Electronic Journal of Differential Equations, Vol. 2009, No. 10, 2009, pp. 1-11.

[21] M. Fecken, Y. Zhong and J. Wang, “On the Concept and existence of Solutions for Impulsive Fractional Differential Equations,” Communications in Non-Linear Science and numerical Simulation, Vol. 17, No. 7, 2012, pp. 3050-3060. doi:10.1016/j.cnsns.2011.11.017

[22] L. Byszewski and V. Lakshmikantham, “Theorem about the Existence and Uniqueness of a Solution of a Nonlocal Abstract Cauchy Problem in a Banach Space,” Journal of Applied Analysis, Vol. 40, 1991, pp. 11-19. doi:10.1080/00036819008839989

[23] L. Byszewski, “Theorems about Existence and Uniqueness of Solutions of a Semilinear Evolution Nonlocal Cauchy Problem,” Journal of Mathematical Analysis and Applications, Vol. 162, No. 2, 1991, pp. 494-505. doi:10.1016/0022-247X(91)90164-U

[24] L. Byszewski, “Existence and Uniqueness of Mild and Classical Solutions of Semilinear Functional-Differential Evolution Nonlocal Cauchy Problem,” Selected Problems of Mathematics, 50th Anniversary Cracow University of Technology, No. 6, Cracow University of Technology, Krakow, 1995, pp. 25-33.

[25] J. X. Sun, “Nonlinear Functional Analysis and Its Application,” Science Press, Beijing, 2008.