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Periodic solutions for nonlinear Volterra integrodierential equations in Banach spaces

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Dimitrios A. Kandilakis, Nikolaos S. Papageorgiou

Periodic solutions for nonlinear Volterra integrodierential equations in Banach spaces

Comment.Math.Univ.Carolinae 38,2 (1997) 283-296.

Abstract: In this paper we examine periodic integrodifferential equations in Ba- nach 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 solu- tion. Both theorems use only the order structure of the problem and no compact- ness 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.

Keywords: extremal solutions, monotone map, regular cone, normal cone, quasi- monotone map, reproducing cone, dual cone, differential inequality, monotone iter- ative technique

AMS Subject Classification: 45J05

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