Comptes Rendus
no. 1
Ordinary differential equations
Topological properties of solution sets for partial functional evolution inclusions
[Propriétés topologiques des ensembles de solutions d'inclusions fonctionnelles partielles d'évolution]

Présenté par : the Editorial Board

Yong Zhou1, 2 ; Li Peng1
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.

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 Rδ-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.

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 Rδ-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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2016.11.011
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 
@article{CRMATH_2017__355_1_45_0,
     author = {Yong Zhou and Li Peng},
     title = {Topological properties of solution sets for partial functional evolution inclusions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {45--64},
     publisher = {Elsevier},
     volume = {355},
     number = {1},
     year = {2017},
     doi = {10.1016/j.crma.2016.11.011},
     language = {en},
}
TY  - JOUR
AU  - Yong Zhou
AU  - Li Peng
TI  - Topological properties of solution sets for partial functional evolution inclusions
JO  - Comptes Rendus. Mathématique
PY  - 2017
SP  - 45
EP  - 64
VL  - 355
IS  - 1
PB  - Elsevier
DO  - 10.1016/j.crma.2016.11.011
LA  - en
ID  - CRMATH_2017__355_1_45_0
ER  - 
%0 Journal Article
%A Yong Zhou
%A Li Peng
%T Topological properties of solution sets for partial functional evolution inclusions
%J Comptes Rendus. Mathématique
%D 2017
%P 45-64
%V 355
%N 1
%I Elsevier
%R 10.1016/j.crma.2016.11.011
%G en
%F CRMATH_2017__355_1_45_0
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/

[1] J. Andres; G. Gabor; L. Górniewicz Topological structure of solution sets to multi-valued asymptotic problems, Z. Anal. Anwend., Volume 19 (2000), pp. 35-60

[2] J. Andres; M. Pavlačková Topological structure of solution sets to asymptotic boundary value problems, J. Differ. Equ., Volume 248 (2010), pp. 127-150

[3] N. Aronszajn Le correspondant topologique de l'unicité dans la théorie des équations différentielles, Ann. Math. (2), Volume 43 (1942) no. 4, pp. 730-738

[4] A. Bakowska; G. Gabor Topological structure of solution sets to differential problems in Fréchet spaces, Ann. Pol. Math., Volume 95 (2009), pp. 17-36

[5] M. Benchohra; S. Abbas Advanced Functional Evolution Equations and Inclusions, Springer, 2015

[6] D. Bothe Multi-valued perturbations of m-accretive differential inclusions, Isr. J. Math., Volume 108 (1998), pp. 109-138

[7] A. Bressan; Z.P. Wang Classical solutions to differential inclusions with totally disconnected right-hand side, J. Differ. Equ., Volume 246 (2009), pp. 629-640

[8] D.H. Chen; R.N. Wang; Y. Zhou Nonlinear evolution inclusions: topological characterizations of solution sets and applications, J. Funct. Anal., Volume 265 (2013), pp. 2039-2073

[9] K. Deimling Multivalued Differential Equations, de Gruyter, Berlin, 1992

[10] J. Diestel; W.M. Ruess; W. Schachermayer Weak compactness in L1(μ;X), Proc. Amer. Math. Soc., Volume 118 (1993), pp. 447-453

[11] T. Donchev; E. Farkhi; B.S. Mordukhovich Discrete approximations, relaxation, and optimization of one-sided Lipschitzian differential inclusions in Hilbert spaces, J. Differ. Equ., Volume 243 (2007), pp. 301-328

[12] G. Gabor; A. Grudzka Structure of the solution set to impulsive functional differential inclusions on the half-line, Nonlinear Differ. Equ. Appl., Volume 19 (2012), pp. 609-627

[13] G. Gabor; A. Grudzka Erratum to: structure of the solution set to impulsive functional differential inclusions on the half-line, Nonlinear Differ. Equ. Appl., Volume 22 (2015), pp. 175-183

[14] G. Gabor; M. Quincampoix On existence of solutions to differential equations or inclusions remaining in a prescribed closed subset of a finite-dimensional space, J. Differ. Equ., Volume 185 (2002), pp. 483-512

[15] S.C. Hu; N.S. Papageorgiou On the topological regularity of the solution set of differential inclusions with constraints, J. Differ. Equ., Volume 107 (1994), pp. 280-289

[16] M. Kamenskii; V. Obukhovskii; P. Zecca Condensing Multi-valued Maps and Semilinear Differential Inclusions in Banach Spaces, Walter de Gruyter, Berlin, New York, 2001

[17] S.K. Ntouyas; D. O'Regan Existence results for semilinear neutral functional differential inclusions via analytic semigroups, Acta Appl. Math., Volume 98 (2007), pp. 223-253

[18] A. Pazy Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983

[19] V. Staicu On the solution sets to nonconvex differential inclusions of evolution type, Discrete Contin. Dyn. Syst., Volume 2 (1998), pp. 244-252

[20] V. Staicu On the solution sets to differential inclusions on unbounded interval, Proc. Edinb. Math. Soc., Volume 43 (2000), pp. 475-484

[21] I.I. Vrabie Compactness Methods for Nonlinear Evolutions, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 75, Longman and John Wiley & Sons, 1995

[22] R.N. Wang; Q.H. Ma; Y. Zhou Topological theory of non-autonomous parabolic evolution inclusions on a noncompact interval and applications, Math. Ann., Volume 362 (2014), pp. 173-203

[23] J.H. Wu Theory and Applications of Partial Functional Differential Equations, vol. 119, Springer Science & Business Media, 2012

Cité par Sources :

Commentaires - Politique