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Applying metric (Banach-like) and topological (Schauder-like) fixed-point theorems, the existence of metric and topological fractals is respectively proved as (sub)invariant subsets of the Hutchinson–Barnsley map generated by a multifunction system. Weakly contractive and compact multifunction systems are considered, but systems of more general multifunctions are discussed as well. The notions of hyperspaces and AR-spaces are employed for this goal.

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A multivalued version of the celebrated Sharkovskii theorem is established which is applicable to differential equations and inclusions for obtaining subharmonic periodic solutions. The results in our earlier paper (Set-Valued Anal. 10(1) (2002), 1-14) are completed to a sharp form. A multivalued analogue of the Levinson transformation theory (dissipativity implies the existence of harmonics) is stated. (2000): 34A60, 34C25, 47H04, 58C06. Mathematics Subject Classifications

An extension of the celebrated Sharkovskiȋ cycle coexisting theorem (see [14]) is given for (strongly) admissible multivalued self-maps in the sense of [8], on a Cartesian product of linear continua. Vectors of admissible self-maps have a triangular structure as in [10]. Thus, we make a joint generalization of the results in [2], [5], [6] (a multivalued case), in [10] (a multidimensional case), and in [15] (a linear continuum case). The obtained results can be applied, unlike in the single-valued case, to differential equations and inclusions.