Fixed point of some Markov operator of Frobenius-Perron type generated by a random family of point-transformations in ℝd
Existence of fixed point of a Frobenius-Perron type operator P : L1 ⟶ L1 generated by a family {φy}y∈Y of nonsingular Markov maps defined on a σ-finite measure space (I, Σ, m) is studied. Two fairly general conditions are established and it is proved that they imply for any g ∈ G = {f ∈ L1 : f ≥ 0,...
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doaj-f46c52fcfa514f858137b396299a2b0d2021-09-06T19:39:56ZengDe GruyterAdvances in Nonlinear Analysis2191-94962191-950X2021-01-0110197298110.1515/anona-2020-0163anona-2020-0163Fixed point of some Markov operator of Frobenius-Perron type generated by a random family of point-transformations in ℝdBugiel Peter0Wędrychowicz Stanisław1Rzepka Beata2Faculty of Mathematics and Computer Science, Jagiellonian University Cracow (Kraków), Cracow, PolandDepartment of Nonlinear Analysis, Ignacy Łukasiewicz Rzeszów University of Technology, al. Powstańców Warszawy 8, 35-959, Rzeszów, PolandDepartment of Nonlinear Analysis, Ignacy Łukasiewicz Rzeszów University of Technology, al. Powstańców Warszawy 8, 35-959, Rzeszów, PolandExistence of fixed point of a Frobenius-Perron type operator P : L1 ⟶ L1 generated by a family {φy}y∈Y of nonsingular Markov maps defined on a σ-finite measure space (I, Σ, m) is studied. Two fairly general conditions are established and it is proved that they imply for any g ∈ G = {f ∈ L1 : f ≥ 0, and ∥f∥ = 1}, the convergence (in the norm of L1) of the sequence {Pjg}j=1∞$\begin{array}{} \{P^{j}g\}_{j = 1}^{\infty} \end{array} $ to a unique fixed point g0. The general result is applied to a family of C1+α-smooth Markov maps in ℝd.https://doi.org/10.1515/anona-2020-0163markov operatormarkov mapsfixed pointradon-nikodym derivativefrobenius-perron operator37a30 |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Bugiel Peter Wędrychowicz Stanisław Rzepka Beata |
spellingShingle |
Bugiel Peter Wędrychowicz Stanisław Rzepka Beata Fixed point of some Markov operator of Frobenius-Perron type generated by a random family of point-transformations in ℝd Advances in Nonlinear Analysis markov operator markov maps fixed point radon-nikodym derivative frobenius-perron operator 37a30 |
author_facet |
Bugiel Peter Wędrychowicz Stanisław Rzepka Beata |
author_sort |
Bugiel Peter |
title |
Fixed point of some Markov operator of Frobenius-Perron type generated by a random family of point-transformations in ℝd |
title_short |
Fixed point of some Markov operator of Frobenius-Perron type generated by a random family of point-transformations in ℝd |
title_full |
Fixed point of some Markov operator of Frobenius-Perron type generated by a random family of point-transformations in ℝd |
title_fullStr |
Fixed point of some Markov operator of Frobenius-Perron type generated by a random family of point-transformations in ℝd |
title_full_unstemmed |
Fixed point of some Markov operator of Frobenius-Perron type generated by a random family of point-transformations in ℝd |
title_sort |
fixed point of some markov operator of frobenius-perron type generated by a random family of point-transformations in ℝd |
publisher |
De Gruyter |
series |
Advances in Nonlinear Analysis |
issn |
2191-9496 2191-950X |
publishDate |
2021-01-01 |
description |
Existence of fixed point of a Frobenius-Perron type operator P : L1 ⟶ L1 generated by a family {φy}y∈Y of nonsingular Markov maps defined on a σ-finite measure space (I, Σ, m) is studied. Two fairly general conditions are established and it is proved that they imply for any g ∈ G = {f ∈ L1 : f ≥ 0, and ∥f∥ = 1}, the convergence (in the norm of L1) of the sequence {Pjg}j=1∞$\begin{array}{}
\{P^{j}g\}_{j = 1}^{\infty}
\end{array} $ to a unique fixed point g0. The general result is applied to a family of C1+α-smooth Markov maps in ℝd. |
topic |
markov operator markov maps fixed point radon-nikodym derivative frobenius-perron operator 37a30 |
url |
https://doi.org/10.1515/anona-2020-0163 |
work_keys_str_mv |
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1717769738604511232 |