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Periodic boundary value problems for a class of second-order impulsive integro-differential equations in Banach spaces. (English) Zbl 0889.45016

The 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.

##### MSC:
 45N05 Abstract integral equations, integral equations in abstract spaces 45J05 Integro-ordinary differential equations