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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 $u'(t)= Au(t)+ f(t,u(t)),\quad 0< t< T_0,\quad t\neq t_i,\quad u(0)= u_0,$ $\Delta u(t_i)= I_i(u(t_i)),\quad i= 1,2,\dots,\;0<t_1< t_2<\cdots< T_0,$ 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