Periodic boundary value problems for a class of second-order impulsive integro-differential equations in Banach spaces.

*(English)*Zbl 0889.45016The paper deals with a periodic boundary value problem for a second order nonlinear integro-differential equation with impulses at fixed moments in a Banach space.

The concepts of lower and upper solutions for this problem are introduced and then the monotone iterative method is used to construct a couple of monotone sequences converging to the extremal solutions of the problem in a sector defined by a lower solution ${u}_{0}$ and an upper solution ${v}_{0}$, with ${u}_{0}\le {v}_{0}$.

For it, the authors first study a periodic boundary value problem for a linear impulsive integro-differential equation and prove a comparison result.

Reviewer: Eduardo Liz (Vigo)

##### MSC:

45N05 | Abstract integral equations, integral equations in abstract spaces |

45J05 | Integro-ordinary differential equations |