### On a multivalued version of the Sharkovskii theorem and its application to differential inclusions, III

DOI: http://dx.doi.org/10.12775/TMNA.2003.046

#### Abstract

An extension of the celebrated Sharkovskiĭ cycle coexisting

theorem (see [< i> Coexistence of cycles of a continuous map of a line into itself< /i> ,

Ukrain. Math. J. < b> 16< /b> (1964), 61–71]) is given for (strongly) admissible multivalued

self-maps in the sense of [L. Górniewicz,

< i> Topological Fixed Point Theory of Multivalued Mappings< /i> , Kluwer, Dordrecht,

1999], on a Cartesian product of linear

continua. Vectors of admissible self-maps have a triangular

structure as in [[P. E. Kloeden, < i> On Sharkovsky’s cycle coexisting ordering< /i> ,

Bull. Austral. Math. Soc. < b> 20< /b> (1979), 171–177]. Thus, we make a joint generalization of the

results in [J. Andres, J. Fišer and L. Jüttner,

< i> On a multivalued version of the Sharkovskiĭ

theorem and its application to differential inclusions< /i> ,

Set-Valued Anal. < b> 10< /b> (2002), 1–14],

[J. Andres and L. Jüttner, < i> Period three plays a negative role in a multivalued version

of Sharkovskiĭ’s theorem< /i> , Nonlinear Anal. < b> 51< /b> (2002), 1101–1104],

[J. Andres, L. Jüttner and K. Pastor, < i> On a multivalued version of the Sharkovskiĭ

theorem and its application to differential inclusions II< /i> ] (a multivalued case), in

[P. E. Kloeden, < i> On Sharkovsky’s cycle coexisting ordering< /i> ,

Bull. Austral. Math. Soc. < b> 20< /b> (1979), 171–177] (a

multidimensional case), and in

[H. Schirmer, < i> A topologist’s view of Sharkovskiĭ’s theorem< /i> , Houston, J. Math.

< b> 11< /b> (1985),

385–395] (a linear continuum case).

The obtained results can be applied, unlike in the single-valued

case, to differential equations and inclusions.

theorem (see [< i> Coexistence of cycles of a continuous map of a line into itself< /i> ,

Ukrain. Math. J. < b> 16< /b> (1964), 61–71]) is given for (strongly) admissible multivalued

self-maps in the sense of [L. Górniewicz,

< i> Topological Fixed Point Theory of Multivalued Mappings< /i> , Kluwer, Dordrecht,

1999], on a Cartesian product of linear

continua. Vectors of admissible self-maps have a triangular

structure as in [[P. E. Kloeden, < i> On Sharkovsky’s cycle coexisting ordering< /i> ,

Bull. Austral. Math. Soc. < b> 20< /b> (1979), 171–177]. Thus, we make a joint generalization of the

results in [J. Andres, J. Fišer and L. Jüttner,

< i> On a multivalued version of the Sharkovskiĭ

theorem and its application to differential inclusions< /i> ,

Set-Valued Anal. < b> 10< /b> (2002), 1–14],

[J. Andres and L. Jüttner, < i> Period three plays a negative role in a multivalued version

of Sharkovskiĭ’s theorem< /i> , Nonlinear Anal. < b> 51< /b> (2002), 1101–1104],

[J. Andres, L. Jüttner and K. Pastor, < i> On a multivalued version of the Sharkovskiĭ

theorem and its application to differential inclusions II< /i> ] (a multivalued case), in

[P. E. Kloeden, < i> On Sharkovsky’s cycle coexisting ordering< /i> ,

Bull. Austral. Math. Soc. < b> 20< /b> (1979), 171–177] (a

multidimensional case), and in

[H. Schirmer, < i> A topologist’s view of Sharkovskiĭ’s theorem< /i> , Houston, J. Math.

< b> 11< /b> (1985),

385–395] (a linear continuum case).

The obtained results can be applied, unlike in the single-valued

case, to differential equations and inclusions.

#### Keywords

Sharkovskiĭ theorem; multivalued version; multidimensional version; linear continuum setting; infinitely many orbits; subharmonics of differential systems

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