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Zbl 1059.47057
Andres, Jan; Pastor, Karel
On a multivalued version of the Sharkovskiĭ theorem and its application to differential inclusions. III. (English)
[J] Topol. Methods Nonlinear Anal. 22, No. 2, 369-386 (2003). ISSN 1230-3429

This paper presents a multivalued version of the Sharkovskiĭ cycle coexisting theorem. The authors consider strongly admissible multivalued self-maps in the sense of {\it L. Górniewicz} [``Topological fixed point theory of multivalued mappings'' (Mathematics and its Applications 495, Kluwer Academic Publishers, Dordrecht) (1999; Zbl 0937.55001)] on a Cartesian product of linear continua. Vectors of admissible maps have a triangular structure as in {\it P. E. Kloeden} [Bull. Aust. Math. Soc. 20, 171--177 (1979; Zbl 0465.58022)]. The multivalued case was considered before in two papers with the same title as this one, {\it J. Andres, J. Fiser} and {\it L. Jütner} [Set-Valued Anal. 10, 1--14 (2002; Zbl 1082.37048)], and another one, submitted for publication, by {\it J. Andres, L. Jütner} and {\it K. Pastor}. The paper {\it J. Andres} and {\it L. Jütner} [Nonlinear Anal., Theory Methods Appl. 51, 1101--1104 (2002; Zbl 1015.37032)] also contains a discussion about the multivalued version of the Sharkovskiĭ theorem. A generalization in the multidimensional case is given in the cited paper of Kloeden, while the linear continuum case is considered by {\it H. Schirmer} [Houston J. Math. 11, 385-395 (1985; Zbl 0606.54031)]. The results of the present paper are a joint generalization of all these ones. \par The final paragraph contains applications to differential equations and inclusions.
[Adriana Buică (Cluj-Napoca)]

MSC 2000:: *47H04 Set-valued operators
37E05 Maps of the interval
37C25 Fixed points, periodic points, fixed-point index theory
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
58C30 Fixed point theorems on manifolds
34A60 ODE with multivalued right-hand sides
47N20 Appl. of operator theory to differential and integral equations

Keywords: Sharkovskij cycle coexistence theorem; linear continuum setting; infinitely many orbits; subharmonics of differential systems

Citations: Zbl 0937.55001; Zbl 0465.58022; Zbl 1015.37032; Zbl 0606.54031; Zbl 1082.37048

Cited in: Zbl 1128.37025

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