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

Abstract

An extension of the celebrated Sharkovskiĭ cycle coexisting theorem (see [Coexistence of cycles of a continuous map of a line into itself, Ukrain. Math. J. 16 (1964), 61–71]) is given for (strongly) admissible multivalued self-maps in the sense of [L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, Kluwer, Dordrecht, 1999], on a Cartesian product of linear continua. Vectors of admissible self-maps have a triangular structure as in [P. E. Kloeden, On Sharkovsky’s cycle coexisting ordering, Bull. Austral. Math. Soc. 20 (1979), 171–177]. Thus, we make a joint generalization of the results in [J. Andres, J. Fišer and L. Jüttner, On a multivalued version of the Sharkovskiĭ theorem and its application to differential inclusions, Set-Valued Anal. 10 (2002), 1–14], [J. Andres and L. Jüttner, Period three plays a negative role in a multivalued version of Sharkovskiĭ’s theorem, Nonlinear Anal. 51 (2002), 1101–1104], [J. Andres, L. Jüttner and K. Pastor, On a multivalued version of the Sharkovskiĭ theorem and its application to differential inclusions II] (a multivalued case), in [P. E. Kloeden, On Sharkovsky’s cycle coexisting ordering, Bull. Austral. Math. Soc. 20 (1979), 171–177] (a multidimensional case), and in [H. Schirmer, A topologist’s view of Sharkovskiĭ’s theorem, Houston, J. Math. 11 (1985), 385–395] (a linear continuum case). The obtained results can be applied, unlike in the single-valued case, to differential equations and inclusions.