Topological properties of solution sets for partial functional evolution inclusions

Yong Zhou; Li Peng

doi:10.1016/j.crma.2016.11.011

Ordinary differential equations

Topological properties of solution sets for partial functional evolution inclusions

Presented by: the Editorial Board

Yong Zhou ^1,²; Li Peng ¹

¹ Faculty of Mathematics and Computational Science, Xiangtan University, Hunan 411105, PR China
² Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Comptes Rendus. Mathématique, Volume 355 (2017) no. 1, pp. 45-64.

Abstract
Résumé

This paper deals with functional evolution inclusions of neutral type in Banach space when the semigroup is compact as well as noncompact. The topological properties of the solution set is investigated. It is shown that the solution set is nonempty, compact and an $R_{δ}$ -set which means that the solution set may not be a singleton but, from the point of view of algebraic topology, it is equivalent to a point, in the sense that it has the same homology group as one-point space. As a sample of application, we consider a partial differential inclusion at end of the paper.

Cette Note traite des inclusions fonctionnelles d'évolution de type neutre dans les espaces de Banach, aussi bien lorsque le semi-groupe est compact que lorsqu'il est non compact. Nous étudions les propriétés topologiques de l'ensemble des solutions. Nous montrons que cet ensemble est non vide, compact, et qu'il est un $R_{δ}$ -ensemble. Ceci signifie qu'il peut ne pas être réduit à un point, mais qu'il est équivalent, pour la topologie algébrique, à un espace réduit à un point. Plus précisément, l'ensemble des solutions a les mêmes groupes d'homologie qu'un ensemble réduit à un point. Comme exemple d'application, nous considérons enfin une inclusion différentielle partielle.

Received: 2016-05-06
Accepted: 2016-11-28
Published online: 2016-12-13

DOI: 10.1016/j.crma.2016.11.011

Author's affiliations:

Yong Zhou ^{1, 2}; Li Peng ¹

¹ Faculty of Mathematics and Computational Science, Xiangtan University, Hunan 411105, PR China
² Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

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     title = {Topological properties of solution sets for partial functional evolution inclusions},
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     pages = {45--64},
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Yong Zhou; Li Peng. Topological properties of solution sets for partial functional evolution inclusions. Comptes Rendus. Mathématique, Volume 355 (2017) no. 1, pp. 45-64. doi : 10.1016/j.crma.2016.11.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.11.011/

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