Global solutions for abstract impulsive differential equations. (English) Zbl 1183.34083

The authors consider an impulse differential equation of the sort \[ u'(t)= Au(t)+ f(t,u(t)),\quad t\in\mathbb{R},\quad t\neq t_i,\quad i\in F, \]
\[ \Delta u(t_i)= I_i(u(t_i)),\quad i\in F, \] where \(A\) is the infinitesimal generator of a hyperbolic \(C_0\)-semigroup of bounded linear operators \((T^{(t)})_{t\geq 0}\) on a Banach space \[ (X,\|.\|),\;f: \mathbb{R}\times X\to X,\;I_i: X\to X,\quad i\in F \] are continuous functions, \(F\subset\mathbb{Z}\), \(\{t_i: i\in F\}\) is a discrete set of fixed real numbers \(t_i< t_j\) for \(i< j\). For \(t\in [a,b]\), \(t\neq tn_i\), \(i= 1,\dots, m\) a definition for a mild solution is given. Necessary and sufficient conditions for the boundedness of their solution are given.


34G99 Differential equations in abstract spaces
34A37 Ordinary differential equations with impulses
34D09 Dichotomy, trichotomy of solutions to ordinary differential equations
47D06 One-parameter semigroups and linear evolution equations
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