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1

Ergodic and fixed point theorems for sequences and nonlinear mappings in a Hilbert space

Rouhani B. D.

Demonstratio Mathematica

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2018

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Vol. 51, nr 1

27--36

EN

In this paper, we introduce the notion of 2-generalized hybrid sequences, extending the notion of nonexpansive and hybrid sequences introduced and studied in our previous work [Djafari Rouhani B., Ergodic theorems for nonexpansive sequences in Hilbert spaces and related problems, Ph.D. thesis, Yale University, 1981; and other published in J. Math. Anal. Appl., 1990, 2002, and 2014; Nonlinear Anal., 1997, 2002, and 2004], and prove ergodic and convergence theorems for such sequences in a Hilbert space H. Subsequently, we apply our results to prove new fixed point theorems for 2-generalized hybrid mappings, first introduced in [Maruyama T., Takahashi W., Yao M., Fixed point and mean ergodic theorems for new nonlinear mappings in Hilbert spaces, J. Nonlinear Convex Anal., 2011, 12, 185-197] and further studied in [Lin L.-J., Takahashi W., Attractive point theorems and ergodic theorems for nonlinear mappings in Hilbert spaces, Taiwanese J. Math., 2012, 16, 1763-1779], defined on arbitrary nonempty subsets of H.

2

Ergodic theory approach to chaos: remarks and computational aspects

Mitkowski P. J., Mitkowski W.

International Journal of Applied Mathematics and Computer Science

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2012

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Vol. 22, no. 2

259-267

EN

We discuss basic notions of the ergodic theory approach to chaos. Based on simple examples we show some characteristic features of ergodic and mixing behaviour. Then we investigate an infinite dimensional model (delay differential equation) of erythropoiesis (red blood cell production process) formulated by Lasota. We show its computational analysis on the previously presented theory and examples. Our calculations suggest that the infinite dimensional model considered possesses an attractor of a nonsimple structure, supporting an invariant mixing measure. This observation verifies Lasota's conjecture concerning nontrivial ergodic properties of the model.

3

On positive definite and stationary sequences with respect to polynomial hypergroups

Lasser R.

Journal of Applied Analysis

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2011

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Vol. 17, nr 2

207-230

EN

We study bounded positive definite double sequences which are stationary with respect to a polynomial hypergroup structure generated by (Rn(t)) ∈nNo. Connected with bounded positive definite and Rn-stationary double sequences is an Rn-stationary sequence of elements in a Hilbert space. We derive an ergodic theorem for such Rn-stationary sequences and we give a complete characterization of the space of multipliers defined by such an Rn-stationary sequence. Further we give examples of bounded positive definite double sequences.

4

Ergodic theorems for Markov chains represented by iterated function systems

Stenflo O.

Bulletin of the Polish Academy of Sciences. Mathematics

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2001

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Vol. 49, no 1

27-43

EN

We consider Markov chains represented in the form Xn+1 = f (Xn, In), where {In} is a sequence of independent, identically distributed (i.i.d.) random variables, and where f is a measurable function. Any Markov chain {Xn} on a Polish state space may be represented in this form i.e. can be considered as arising from an iterated function system (IFS). A distributional ergodic theorem, including rates of convergence in the Kantorovich distance is proved for Markov chains under the condition that an IFS representation is "stochastically contractive" and "stochastically bounded". We apply this result to prove our main theorem giving upper bounds for distances between invariant probability measures for iterated function systems. We also give some examples indicating how ergodic theorems for Markov chains may be proved by finding contractive IFS representations. These ideas are applied to some Markov chains arising from iterated function systems with place dependent probabilities.