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

Year 2018, Volume: 1 Issue: 3 - To memory of Prof. RNDr. Beloslav Rieˇcan, DrSc., 293 - 311, 24.10.2018

Khalid Hilal Karim Guida , Lahcen Ibnelazyz Mohamed Oukessou

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

• Benchohra.M , Henderson, J.Ntouyas, SK. Impulsive Differential Equations and inclusions. Hindawi publishing, New York (2006).
• P. Chen, Y. Li, Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions, Results Math, 63 (2013) 731-744.
• 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.
• M.M. El-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos solitons fractals 14(2002)433-440.
• M.M. El-Borai, Semigroups and some nonlinear fractional differential equations, Appl. Math. Comput. 149 (2004) 823-831.
• 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).
• Gou, M., Xue, X. Li. R. Controllability of impulsive evolution inclusions with nonlocal conditions. J. Optim. Theory Appl 120, 255-374 (2004).
• Banas,J Goebel, Measure of noncompactness in banach space Lecture notes in Pure and Applied Mathematics, Vol60, Marcel Dekker, New york (1980).
• R. Hilfer, Apllications of Fractional Calculus in Physics, World Scientific, Singapore(2000).
• 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.
• 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.
• 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.
• V. Lakshmikantham, Theory of fractional functional differential equations, Nonlinear Anal. (2007), doi:10.1016/j.na.2007.09.025.
• V. Lakshmikantham, A.S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. (2007), doi:10.1016/j.na.2007.09.025.
• Lakshmikantham. V, Bainov. D. D., Simeonov. PS. Theory of impulsive Differential Equations. World Scientific, Singapore (1989).
• K. S. Miller, B.Ross, An introduction to the fractional Calulus nad Fractional Differential Equations, Wiley, New York, 1993.
• I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1993.
• A. Pazy, Semigroups of linear Operators and Applications to partial Differential Equations, Springer-Verlag, Berlin 1983.
• 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.
• J. Wang, Y. Zhou, A class of fractional evolution equations and optimal controls, Nonlinear Anal. RWA 12 (2011) 262-272.
• Xi Fu, Xiaoyou Liu, Bowen Lu: On a new class of impulsive fractional evolution equations. Advances in Difference Equations, vol. 2015, 227, 2015.
• Yong Zhou, Basic theory of fractional differential equations, Xiangtan University, China, 2014.