Also set-valued functions do not like iterative roots

Jarczyk, Witold; Zhang, Weinian

doi:10.4171/em/57

Cited by 18 publications

(24 citation statements)

References 14 publications

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“…Let F c (X) stands for the set of all setvalued functions f : X → 2 X such that card f (c) > 1 and f (x) is a singleton whenever x ∈ X\{c}. Many other theorems on this topic were proved by Powierża, Jarczyk, Jarczyk, Li and Zhang (see [50,51,[110][111][112][113]). They considered mainly the case where the values of G consist of one or two elements.…”

Section: Set-valued Iterative Roots Of Bijectionsmentioning

confidence: 99%

“…Moreover, they obtained several results indicating that the smallest set-valued iterative root of a given order does not exist. In papers [51,82] it is shown that the phenomenon of the lack of iterative roots appears also for some set-valued functions with exactly one value not being a singleton. Even assumptions such as continuity or strict monotonicity on the single-valued parts of such a set-valued function does not guarantee the existence of its square roots.…”

Section: Set-valued Iterative Roots Of Bijectionsmentioning

confidence: 99%

See 1 more Smart Citation

Recent results on iteration theory: iteration groups and semigroups in the real case

Zdun

Solarz

2013

Aequat. Math.

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Abstract. In this survey paper we present some recent results in iteration theory. Mainly, we focus on the problems concerning real iteration groups (flows) and semigroups (semiflows) such as existence, regularity and embeddability. We also discuss some issues associated to the problem of embeddability, i.e. iterative roots and approximate iterability. The topics of this paper are: (1) measurable iteration semigroups; (2) embedding of diffeomorphisms in regular iteration semigroups in the space R n ; (3) iteration groups of fixed point free homeomorphisms on the plane; (4) embedding of interval homeomorphisms with two fixed points in a regular iteration group; (5) commuting functions and embeddability; (6) iterative roots; (7) the structure of iteration groups on an interval; (8) iteration groups of homeomorphisms of the circle; (9) approximately iterable functions; (10) set-valued iteration semigroups; (11) iterations of mean-type mappings; (12) Hayers-Ulam stability of the translation equation. Most of the results presented here were obtained by means of functional equations. We indicate the relations between iteration theory and functional equations.

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Section: Set-valued Iterative Roots Of Bijectionsmentioning

confidence: 99%

Section: Set-valued Iterative Roots Of Bijectionsmentioning

confidence: 99%

Recent results on iteration theory: iteration groups and semigroups in the real case

Zdun

Solarz

2013

Aequat. Math.

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“…Encountering difficulties in finding iterative roots, ones also made efforts (see e.g. [6,7,15,19]) to find multivalued iterative roots, which have at least one set-valued point (or called jump simply). In general, the composition G • F of multifunctions F : X → 2 Y and G : Y → 2 Z is defined by (G • F )(x) = G(F (x)), where the image F (A) of a set A ⊂ X is defined by F (A) := ∪ x∈A F (x).…”

Section: Introductionmentioning

confidence: 99%

Iterative roots of exclusive multifunctions

Liu,

Li,

Zhang

2020

Preprint

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In this paper we investigate iterative roots of strictly monotone upper semi-continuous multifunctions having finitely many jumps. Known results are concerning roots of order 2 for multifunctions of exact one jump. For the general investigation, we introduce a concept 'intensity' to formulate the growth of jumps under iteration and find a class of strictly monotone and upper semi-continuous multifunctions of intensity 1, called exclusive multifunctions, each of which has an absorbing interval. Then we use the absorbing interval to construct iterative roots of order n for those exclusive multifunctions.

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“…Many advances have been made to iterative roots for various kinds of maps, e.g., continuous strictly monotone self-maps of intervals [3,6], homeomorphisms of the circle [28,29,34], series and transseries [4], set-valued functions [9,18,24,25], high dimensional mappings [15][16][17]23]. Many researchers are also devoted to some properties of iterative roots, for instance, differentiability [11,13,33,38], approximation [39], stability [40,41], category and measure [2,27].…”

Section: Introductionmentioning

confidence: 99%