Periodic solutions for nonlinear Volterra integrodifferential equations in Banach spaces

Dimitrios A. Kandilakis; Nikolaos S. Papageorgiou

Commentationes Mathematicae Universitatis Carolinae (1997)

  • Volume: 38, Issue: 2, page 283-296
  • ISSN: 0010-2628

Abstract

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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.

How to cite

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Kandilakis, Dimitrios A., and Papageorgiou, Nikolaos S.. "Periodic solutions for nonlinear Volterra integrodifferential equations in Banach spaces." Commentationes Mathematicae Universitatis Carolinae 38.2 (1997): 283-296. <http://eudml.org/doc/248086>.

@article{Kandilakis1997,
abstract = {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.},
author = {Kandilakis, Dimitrios A., Papageorgiou, Nikolaos S.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {extremal solutions; monotone map; regular cone; normal cone; quasi-monotone map; reproducing cone; dual cone; differential inequality; monotone iterative technique; extremal solutions; monotone map; regular cone; normal cone; quasi-monotone map; reproducing cone; dual cone; differential inequality; monotone iterative technique; periodic integro-differential equations; Banach spaces},
language = {eng},
number = {2},
pages = {283-296},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Periodic solutions for nonlinear Volterra integrodifferential equations in Banach spaces},
url = {http://eudml.org/doc/248086},
volume = {38},
year = {1997},
}

TY - JOUR
AU - Kandilakis, Dimitrios A.
AU - Papageorgiou, Nikolaos S.
TI - Periodic solutions for nonlinear Volterra integrodifferential equations in Banach spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1997
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 38
IS - 2
SP - 283
EP - 296
AB - 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.
LA - eng
KW - extremal solutions; monotone map; regular cone; normal cone; quasi-monotone map; reproducing cone; dual cone; differential inequality; monotone iterative technique; extremal solutions; monotone map; regular cone; normal cone; quasi-monotone map; reproducing cone; dual cone; differential inequality; monotone iterative technique; periodic integro-differential equations; Banach spaces
UR - http://eudml.org/doc/248086
ER -

References

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  13. Lakshmikantham V., Leela S., An Introduction to Nonlinear Differential Equations in Abstract Spaces, Pergamon Press, Oxford, 1980. MR0616449
  14. Lakshmikantham V., Leela S., On the method of upper and lower solutions in abstract cones, Annales Polonici Math. XLII (1983), 159-164. (1983) Zbl0544.34056MR0728078
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