## 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)

### MSC:

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