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Witold Jarczyk 《Aequationes Mathematicae》1991,42(1):202-219
Summary The class of all self-mappings of a real compact interval which are iterable, i.e., embeddable in a continuous iteration semigroup, was characterized by M. C. Zdun in his paper [8]. Here we propose and characterize a slightly more general notion called almost iterability which (just as the notion of iterability) is motivated by problems in dynamics and stochastic processes.Dedicated to the memory of Professor Alexander Ostrowski on the occasion of the 100th anniversary of his birth 相似文献
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Summary Using the Isaacs-Zimmermann's theory of iterative roots of functions, we prove a theorem concerning the problemP 250 posed by J. Tabor:Letf: E E be a given mapping. Denote byF the set of all iterative roots off. InF we define the following relation: if and only if is an iterative root of. The relation is obviously reflexive and transitive. The question is: Is it also antisymmetric? If we consider iterative roots of a monotonic function the answer is yes. But in general the question is open.Here we prove that there exists a three-element decomposition {
i
;i = 1, 2, 3} of the setE
E
with blocks i of the same cardinality 2cardE
such that the functions from 1 do not possess any proper iterative root, the quasi-ordering is not antisymmetric onF(f) for anyf 2, and is an ordering onF(f) for anyf 3. Iff is a strictly increasing continuous self-bijection ofE, then the relation is an ordering onF(f) ifff is different from the identity mapping of the setE. 相似文献
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Summary Forf ( C
n() and 0 t x letJ
n
(f, t, x) = (–1)n
f(–x)f
(n)(t) +f(x)f
(n)
(–t). We prove that the only real-analytic functions satisfyingJ
n
(f, t, x) 0 for alln = 0, 1, 2, are the exponential functionsf(x) = c e
x,c, . Further we present a nontrivial class of real-analytic functions satisfying the inequalitiesJ
0
(f, x, x) 0 and
0
x
(x – t)n – 1Jn(f, t, x)dt 0 (n 1). 相似文献
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J. Smítal 《Aequationes Mathematicae》1989,37(2-3):279-281
Summary We construct a non-constant Darboux functionf: R R which is a solution of the Euler's functional equationf(x + f(x)) = f(x) for everyx. This function is a counter-example to a statement given in the literature. 相似文献
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J. Tabor 《Aequationes Mathematicae》1988,35(2-3):164-185
Let 0 < 1. In the paper we consider the following inequality: |f(x + y) – f(x) – f(y)| min{|f(x + y)|, |f(x) + f(y)|}, wheref: R R. Solutions and continuous solutions of this inequality are investigated. They have similar properties as additive functions, e.g. if the solution is bounded above (below) on a set of positive inner Lebesgue measure then it is continuous. Some sufficient condition for this inequality is also given.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday 相似文献
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