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,...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2021-01-01
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| Series: | Advances in Nonlinear Analysis |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/anona-2020-0163 |
| Summary: | 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. |
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| ISSN: | 2191-9496 2191-950X |