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Nonlinear impulsive evolution equations. (English) Zbl 0932.34067

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)),0<t<T 0 ,tt i ,u(0)=u 0 ,
Δu(t i )=I i (u(t i )),i=1,2,,0<t 1 <t 2 <<T 0 ,

in a Banach space X, where A is the generator of a strongly continuous semigroup, Δu(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:
34G20Nonlinear ODE in abstract spaces
34A37Differential equations with impulses