Nonlinear impulsive evolution equations. (English) Zbl 0932.34067

Liu, James H.

Dyn. Contin. Discrete Impulsive Syst. 6, No. 1, 77-85 (1999).

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.

MSC:

34G20	Nonlinear differential equations in abstract spaces
34A37	Ordinary differential equations with impulses

Keywords:

existence; uniqueness; mild and classical solutions; nonlinear impulsive evolution equations