Electron. J. Diff. Eqns., Vol. 2005(2005), No. 111, pp. 1-8.

Existence and uniqueness of mild and classical solutions of impulsive evolution equations

Annamalai Anguraj, Mani Mallika Arjunan

Abstract:
We consider the non-linear impulsive evolution equation
$$\displaylines{ u'(t)=Au(t)+f(t,u(t),Tu(t),Su(t)), \quad 0<t<T_0, \; t\neq t_i,\cr u(0) =u_0,\cr \Delta u(t_i) =I_i(u(t_i)),\quad i=1,2,3,\dots,p. }$$
in a Banach space $ X$, where $ A $ is the infinitesimal generator of a $C_0 $ semigroup. We study the existence and uniqueness of the mild solutions of the evolution equation by using semigroup theory and then show that the mild solutions give rise to a classical solutions.

Submitted June 15, 2005. Published October 17, 2005.
Math Subject Classifications: 34A37, 34G60, 34G20.
Key Words: Semigroups; evolution equations; impulsive conditions.

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Annamalai Anguraj
Department of Mathematics
P.S.G. College of Arts and Science
Coimbatore - 641 014, Tamilnadu, India
email: angurajpsg@yahoo.com
Mani Mallika Arjunan
Department of Mathematics
P.S.G. College of Arts and Science
Coimbatore- 641 014, Tamilnadu, India
email: arjunphd07@yahoo.co.in

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