an:01371513
Zbl 0932.34067
Liu, James H.
Nonlinear impulsive evolution equations
EN
Dyn. Contin. Discrete Impulsive Syst. 6, No. 1, 77-85 (1999).
00057198
1999
j
34G20 34A37
existence; uniqueness; mild and classical solutions; nonlinear impulsive evolution equations
Summary: The author studies existence and uniqueness of mild and classical solutions to nonlinear impulsive evolution equations
\[
u'(t)= Au(t)+ f(t,u(t)),\quad 0< t< T_0,\quad t\neq t_i,\quad u(0)= u_0,
\]
\[
\Delta u(t_i)= I_i(u(t_i)),\quad i= 1,2,\dots,\;0<t_1< t_2<\cdots< T_0,
\]
in a Banach space \(X\), where \(A\) is the generator of a strongly continuous semigroup, \(\Delta u(t_i)= u(t^+_i)- u(t^-_i)\) and \(I_i\)'s are some operators. The impulsive conditions can be used to model more physical phenomena than the traditional initial value problems \(u(0)= u_0\). The author applies the semigroup theory to study existence and uniqueness of the mild solutions, and to show that the mild solutions give rise to classical solutions if \(f\) is continuously differentiable.