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Zbl 1183.34083
Hernández M., Eduardo; Aki, Sueli M.Tanaka
Global solutions for abstract impulsive differential equations.
(English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 3-4, A, 1280-1290 (2010). ISSN 0362-546X

The authors consider an impulse differential equation of the sort $$u'(t)= Au(t)+ f(t,u(t)),\quad t\in\bbfR,\quad t\ne t_i,\quad i\in F,$$ $$\Delta u(t_i)= I_i(u(t_i)),\quad i\in F,$$ where $A$ is the infinitesimal generator of a hyperbolic $C_0$-semigroup of bounded linear operators $(T^{(t)})_{t\ge 0}$ on a Banach space $$(X,\Vert.\Vert),\ f: \bbfR\times X\to X,\ I_i: X\to X,\quad i\in F$$ are continuous functions, $F\subset\bbfZ$, $\{t_i: i\in F\}$ is a discrete set of fixed real numbers $t_i< t_j$ for $i< j$. For $t\in [a,b]$, $t\ne tn_i$, $i= 1,\dots, m$ a definition for a mild solution is given. Necessary and sufficient conditions for the boundedness of their solution are given.
MSC 2000:
*34G99 ODE in abstract spaces
34A37 Differential equations with impulses
34D09 Dichotomy, trichotomy
47D06 One-parameter semigroups and linear evolution equations

Keywords: exponential dichotomy; $C_{0}$-semigroup; impulsive system

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