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Zbl 0932.34067
Liu, James H.
Nonlinear impulsive evolution equations.
(English)
[J] Dyn. Contin. Discrete Impulsive Syst. 6, No.1, 77-85 (1999). ISSN 1201-3390

Summary: The author studies existence and uniqueness of mild and classical solutions to nonlinear impulsive evolution equations $$u'(t)= Au(t)+ f(t,u(t)),\quad 0< t< T_0,\quad t\ne t_i,\quad u(0)= u_0,$$ $$\Delta u(t_i)= I_i(u(t_i)),\quad i= 1,2,\dots,\ 0<t_1< t_2<\cdots< T_0,$$ 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 2000:
*34G20 Nonlinear ODE in abstract spaces
34A37 Differential equations with impulses

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

Cited in: Zbl 1006.35055

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