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extremal solutions; monotone map; regular cone; normal cone; quasi-monotone map; reproducing cone; dual cone; differential inequality; monotone iterative technique
In this paper we examine periodic integrodifferential equations in Banach spaces. When the cone is regular, we prove two existence theorems for the extremal solutions in the order interval determined by an upper and a lower solution. Both theorems use only the order structure of the problem and no compactness condition is assumed. In the last section we ask the cone to be only normal but we impose a compactness condition using the ball measure of noncompactness. We obtain the extremal solutions for both the Cauchy and periodic problems in a constructive way, using a monotone iterative technique.
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