---

Discontinuity, Nonlinearity, and Complexity 3(2) (2014) 133--145 | DOI:10.5890/DNC.2014.06.003

Xianghu Liu; Yanfang Li

School of Mathematics and Computer Science, Zunyi Normal College,563002, Zunyi,China

Download Full Text PDF

 

Abstract

This paper is concerned with the existence and uniqueness of mild solution of some fractional impulsive equations withfinite delay. Firstly,we introduce the fractional calculus,Gronwall inequality, leray-schauder’sfixed point theorem,Secondly with the help of them, the sufficient condition for the existence and uniqueness of solutions is presented. Finally we give an example to illustrate our main results.

Acknowledgments

The authors thanks the referees for their careful reading of the manuscript and insightful comments, which help to improve the quality of the paper. We would also like to acknowledge the valuable comments and suggestions from the editors, which vastly contributed to improve the presentation of the paper.

References

1. [1]	Miller, K.S. and Ross, B. (1993), An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York.

2. [2]	Hu, S. and Papageorgiou, N.S. (1997), Handbook of multivalued Analysis Theory, Kluwer Academic Publishers, Dordrecht Boston, London.

3. [3]	Kilbas, A.A., Hari, M., Srivastava, J., and Trujillo, Juan (2006), Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam.

4. [4]	Lakshmikantham, V., Leela, S. and Vasundhara Devi, J. (2009), Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge.

5. [5]	Diethelm, Kai (2010), The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin & Heidelberg.

6. [6]	Wang, J.R., Zhou, Y. andMedved, Mi. (2012),On the Solvability and Optimal Controls of Fractional Integrodifferential Evolution Systems with Infinite Delay, Journal of Optimization Theory and Applications, 152, 31-50.

7. [7]	Wang, J.R., Zhou, Y., and Wei, W. (2011), A class of fractional delay nonlinear integrodifferential controlled systems in Banach space, Communications in Nonlinear Science and Numerical Simulation, 16, 4049-4059.

8. [8]	Shu, X.-B., Lai, Y.Z., and Chen, Y.M. (2011) ,The existence ofmild solutions for impulsive fractional partial differential equations, Nonlinear Analysis, 74 ,2003-2011.

9. [9]	Benchohra, M. and Henderson, J. (2008), Existence results for fractional order functional differential equations with infinite delay, Journal of Mathematical Analysis and Applications , 338, 1340-1350.

10. [10]	Balachandran, K. and Zhou, Y. (2012), Relative controllability of fractional dynamical with delays in control, Communications in Nonlinear Science and Numerical Simulation , 17, 3508-3520.

11. [11]	Cuevas, C., Lizama, C. (2008), Almost automorphic solutions to a class of semilinear fractional differential equations, Applied Mathematics Letters, 21, 1315-1319.

12. [12]	Bazhlekova, E. (2001), Fractional Evolution Equations in Banach Spaces, Ph.D Thesis, Eindhoven University of Technology.

13. [13]	Xue, L., Xiong, J.J. (2011), Existence and uniqueness of mild solutions for abstract delay fractional differential equations, Computers & Mathematics with Applications, 62, 1398-1404.

14. [14]	Hernández, E.,O'Regan, D., and Balachandran, K. (2010), On recent developments in the theory of abstract differential equations with fractional derivatives, Nonlinear Analysis, 73, 3462-3471.

15. [15]	Mophou, G.M., N'Guérékata, G.M. (2010), Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay, Applied Mathematics and Computation, 216, 61-69.

16. [16]	Wang, J.R., Lv, L.L., and Zhou, Y. (2012), New concepts and results in stability of fractional differential equations, Communications in Nonlinear Science and Numerical Simulation, 17, 2530-2538.

17. [17]	Mophou, G.M. (2010), Existence and uniqueness of mild solution to impulsive fractional differential equations, Nonlinear Analysis: Theory, Methods & Applications, 72, 1604-1615.

18. [18]	Hale, J., Kato, J. (1978), Phase space for retarded equations with infinte delay, Funkcialaj, 21, 11-41.

19. [19]	Wang, J.R., Fečkan, M., and Zhou, Y. (2011), On the new concept of solutions and existence results for impulsive fractional evolution equations, Dynamics of PDE, 8(4), 345-361.

20. [20]	Wang, J.R., Fečkan, M., and Zhou, Y. (2013), Relaxed controls for nonlinear fractional impulsive fvolution equations, Journal of Optimization Theory and Applications, in press.

21. [21]	Ye, H.P., Gao, J.M., and Ding, Y.S. (2007), A generalized Gronwall inequality and its application to a fractional differential equation, Journal of Mathematical Analysis and Applications, 328, 1075-1081.

22. [22]	Zhou,Y. and Jiao, F. (2010), Existence of mild solutions for fractional neutral evolution equations, Computers & Mathematics with Applications, 59, 1063-1077.

23. [23]	Balder, E. (1987), Necessary and sufficient conditions for L1-strong-weak lower semicontinuity of integral functional, Nonlinear Analysis: Real World Applications, 11, 1399-1404.

24. [24]	Finner, H. (2007),A generalization of Hölder's inequality and some probability inequalities, the Annals of Probobity, 1893-1901.

25. [25]	Granas, A. and Dugundji, J. (2003), Fixed Point Theory, Springer, New York.