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Hernández M., Eduardo; Aki, Sueli M. Tanaka

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

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 3-4, 1280-1290 (2010).
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.
Reviewer: Stepan Kostadinov (Plovdiv)

Cited in 8 Documents

MSC:

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

Keywords:

exponential dichotomy; \(C_{0}\)-semigroup; impulsive system
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\textit{E. Hernández M.} and \textit{S. M. T. Aki}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 3--4, 1280--1290 (2010; Zbl 1183.34083)
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References:

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