Initial value problems of second order nonlinear impulsive integro-differential equations in Banach spaces. (Q1396259)

Let \(E\) be a real Banach space and \(J = [0, a]\) with \(a>0\). The authors consider the initial value problem for nonlinear second order impulsive integro-differential equations of the type \[ \frac{d^2u}{dt^2}=f(t, u, du/dt, Tu)\quad (t\in J), \] where \(f\in C[J\times E^2, E]\), \[ (Tu)(t)= \int_0^t k(t,s)u(s)\,ds \] where \(k\) is a non-negative scalar-valued kernel. By using a new comparison theorem and the monotone iterative technique, the existence of minimal and maximal solutions of the initial value problem, and some other existence results are obtained.