Impulsive integro-differential equations on unbounded domain in Banach space. (English) Zbl 0864.45009

The authors treat the initial value problem for a first order impulsive integro-differential equation with an infinite number of impulsive times \(0<t_1<t_2< \cdots <t_k \) on the interval \([0,\infty)\) in a Banach space \(E\): \[ \begin{split} u'=f(t,u,Tu,Su)\;\bigl(t\in [0, \infty) \backslash \cup\{t_k\}\bigr), \\ u(t_k+)- u(t_k-)= I_k \bigl(u(t_k)\bigr) \;(k=1,2,3, \dots), \quad u(0)=u_0, \end{split} \] where \(f\in C([0,\infty) \times E\times E\times E,E)\), \(I_k\in C(E,E)\) \((k=1,2,3, \dots)\), \(u_0\in E\), and \[ Tu(t)= \int^t_0k(t,s)u(s)ds, \quad Su(t)= \int^\infty_0 h(t,s) u(s)ds, \] and further suitable conditions are imposed. Using the Banach fixed point theorem the authors show existence, uniqueness and continuous dependence on the initial values.
Reviewer: J.Voigt (Dresden)


45N05 Abstract integral equations, integral equations in abstract spaces
45G10 Other nonlinear integral equations
45J05 Integro-ordinary differential equations