EXISTENCE RESULTS FOR AN IMPULSIVE FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS WITH A NON-COMPACT SEMIGROUP

Abstract

In this paper we study a fractional dierential equations problem with not instantaneous impulses involving a non-compact semigroup. We present some concepts and facts about the strongly continuous semigroup and the measure of noncompactness. After that we give an existence theorem of our problem using a condensing operator and the measure of noncompactness.

Keywords

• Fractional impulsive dierential equations,measure of non- compactness,Darbo-Sadovskii's fixed point theorem

References

1. Benchohra.M , Henderson, J.Ntouyas, SK. Impulsive Differential Equations and inclusions. Hindawi publishing, New York (2006).
2. P. Chen, Y. Li, Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions, Results Math, 63 (2013) 731-744.
3. P. Chen, X. Zhang, Y. Li. Existence of mild solutions to partial differential equations with non-instantenous impulses. Electronic Journal of Differential Equations, vol. 2016 (2016), No. 241, pp. 1-11.
4. M.M. El-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos solitons fractals 14(2002)433-440.
5. M.M. El-Borai, Semigroups and some nonlinear fractional differential equations, Appl. Math. Comput. 149 (2004) 823-831.
6. Peter L. Falb, Infinite Dimensional control problems: On the closure of the set of attainable states for linear systems, Mathematical Analysis and Application 9, 12-22 (1964).
7. Gou, M., Xue, X. Li. R. Controllability of impulsive evolution inclusions with nonlocal conditions. J. Optim. Theory Appl 120, 255-374 (2004).
8. Banas,J Goebel, Measure of noncompactness in banach space Lecture notes in Pure and Applied Mathematics, Vol60, Marcel Dekker, New york (1980).

9. R. Hilfer, Apllications of Fractional Calculus in Physics, World Scientific, Singapore(2000).
10. H.P Heinz, on the behaviour of measure of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal. 7 (1983) 1351-1371.
11. A. A. Kilbas, H.M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol, 204, Elsevier, Amsterdam, 2006.
12. P. Kumar, D. Pandey, D. bahuguna. On a new class of abstract impulsive functional differential equations of fractional order, J. Nonlinear Sci. Appl. 7 (2014). 102-114.
13. V. Lakshmikantham, Theory of fractional functional differential equations, Nonlinear Anal. (2007), doi:10.1016/j.na.2007.09.025.
14. V. Lakshmikantham, A.S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. (2007), doi:10.1016/j.na.2007.09.025.
15. Lakshmikantham. V, Bainov. D. D., Simeonov. PS. Theory of impulsive Differential Equations. World Scientific, Singapore (1989).
16. K. S. Miller, B.Ross, An introduction to the fractional Calulus nad Fractional Differential Equations, Wiley, New York, 1993.
17. I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1993.
18. A. Pazy, Semigroups of linear Operators and Applications to partial Differential Equations, Springer-Verlag, Berlin 1983.
19. J. Wang, M. Feckan and Y. Zhou, On the new concept of solutions and existence results for impulsive fractional evolution equations, Dyn. Partial Differ. Equ. 8 (2011), 345-361.
20. J. Wang, Y. Zhou, A class of fractional evolution equations and optimal controls, Nonlinear Anal. RWA 12 (2011) 262-272.
21. Xi Fu, Xiaoyou Liu, Bowen Lu: On a new class of impulsive fractional evolution equations. Advances in Difference Equations, vol. 2015, 227, 2015.
22. Yong Zhou, Basic theory of fractional differential equations, Xiangtan University, China, 2014.