scholarly journals Continuity of random attractors on a topological space and fractional delayed FitzHugh-Nagumo equations with WZ-noise

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yangrong Li ◽  
Shuang Yang ◽  
Guangqing Long
Keyword(s):  
Topological Space ◽  
Dense Subset ◽  
Complete Metric Space ◽  
Hausdorff Metric ◽  
Density Theorem ◽  
Deterministic Case ◽  
Abstract Result ◽  
General Topological Space ◽  
Random Attractors ◽  
Nagumo Equations

<p style='text-indent:20px;'>We study the continuity of a family of random attractors parameterized in a topological space (perhaps non-metrizable). Under suitable conditions, we prove that there is a residual dense subset <inline-formula><tex-math id="M1">\begin{document}$ \Lambda^* $\end{document}</tex-math></inline-formula> of the parameterized space such that the binary map <inline-formula><tex-math id="M2">\begin{document}$ (\lambda, s)\mapsto A_\lambda(\theta_s \omega) $\end{document}</tex-math></inline-formula> is continuous at all points of <inline-formula><tex-math id="M3">\begin{document}$ \Lambda^*\times \mathbb{R} $\end{document}</tex-math></inline-formula> with respect to the Hausdorff metric. The proofs are based on the generalizations of Baire residual Theorem (by Hoang et al. PAMS, 2015), Baire density Theorem and a convergence theorem of random dynamical systems from a complete metric space to the general topological space, and thus the abstract result, even restricted in the deterministic case, is stronger than those in literature. Finally, we establish the residual dense continuity and full upper semi-continuity of random attractors for the random fractional delayed FitzHugh-Nagumo equation driven by nonlinear Wong-Zakai noise, where the size of noise belongs to the parameterized space <inline-formula><tex-math id="M4">\begin{document}$ (0, \infty] $\end{document}</tex-math></inline-formula> and the infinity of noise means that the equation is deterministic.</p>

scholarly journals Properties of the trajectories of set-valued integrals in banach spaces

1990 ◽  
Vol 41 (2) ◽  
pp. 271-281
Author(s):  
Nikolaos S. Papageorgiou
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Separable Banach Space ◽  
Hausdorff Metric ◽  
Density Theorem ◽  
Mosco Convergence ◽  
Convexity Theorem ◽  
Convergence Theorems ◽  
Convergence Of Sets ◽  
Differentiability Properties

Let F: T → 2x \ {} be a closed-valued multifunction into a separable Banach space X. We define the sets and We prove various convergence theorems for those two sets using the Hausdorff metric and the Kuratowski-Mosco convergence of sets. Then we prove a density theorem of CF and a corresponding convexity theorem for F(·). Finally we study the “differentiability” properties of K(·). Our work extends and improves earlier ones by Artstein, Bridgland, Hermes and Papageorgiou.

scholarly journals On Almost Continuous Mappings and Baire Spaces

1978 ◽  
Vol 21 (2) ◽  
pp. 183-186 ◽  
Author(s):  
Shwu-Yeng T. Lin ◽  
You-Feng Lin
Keyword(s):  
Topological Space ◽  
Dense Subset ◽  
Baire Space ◽  
Real Numbers ◽  
Range Space ◽  
Continuous Mappings ◽  
Almost Continuous

AbstractIt is proved, in particular, that a topological space X is a Baire space if and only if every real valued function f: X →R is almost continuous on a dense subset of X. In fact, in the above characterization of a Baire space, the range space R of real numbers may be generalized to any second countable, Hausdorfï space that contains infinitely many points.

scholarly journals A GENERALIZATION OF SIERPINSKI THEOREM ON UNIQUE DETERMINING OF A SEPARATELY CONTINUOUS FUNCTION

2021 ◽  
Vol 9 (1) ◽  
pp. 250-263
Author(s):  
V. Mykhaylyuk ◽  
O. Karlova
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Dense Subset ◽  
Continuous Functions ◽  
Baire Space ◽  
Regular Space ◽  
Topological Spaces ◽  
Infinite Cardinal ◽  
Urysohn Space ◽  
Completely Regular

In 1932 Sierpi\'nski proved that every real-valued separately continuous function defined on the plane $\mathbb R^2$ is determined uniquely on any everywhere dense subset of $\mathbb R^2$. Namely, if two separately continuous functions coincide of an everywhere dense subset of $\mathbb R^2$, then they are equal at each point of the plane. Piotrowski and Wingler showed that above-mentioned results can be transferred to maps with values in completely regular spaces. They proved that if every separately continuous function $f:X\times Y\to \mathbb R$ is feebly continuous, then for every completely regular space $Z$ every separately continuous map defined on $X\times Y$ with values in $Z$ is determined uniquely on everywhere dense subset of $X\times Y$. Henriksen and Woods proved that for an infinite cardinal $\aleph$, an $\aleph^+$-Baire space $X$ and a topological space $Y$ with countable $\pi$-character every separately continuous function $f:X\times Y\to \mathbb R$ is also determined uniquely on everywhere dense subset of $X\times Y$. Later, Mykhaylyuk proved the same result for a Baire space $X$, a topological space $Y$ with countable $\pi$-character and Urysohn space $Z$. Moreover, it is natural to consider weaker conditions than separate continuity. The results in this direction were obtained by Volodymyr Maslyuchenko and Filipchuk. They proved that if $X$ is a Baire space, $Y$ is a topological space with countable $\pi$-character, $Z$ is Urysohn space, $A\subseteq X\times Y$ is everywhere dense set, $f:X\times Y\to Z$ and $g:X\times Y\to Z$ are weakly horizontally quasi-continuous, continuous with respect to the second variable, equi-feebly continuous wuth respect to the first one and such that $f|_A=g|_A$, then $f=g$. In this paper we generalize all of the results mentioned above. Moreover, we analize classes of topological spaces wich are favorable for Sierpi\'nsi-type theorems.

scholarly journals Modern Dynamical Systems Theory

1974 ◽  
Vol 62 ◽  
pp. 11-18
Author(s):  
L. Markus
Keyword(s):  
Dynamical Systems ◽  
Dynamic Systems ◽  
Vector Fields ◽  
Dense Subset ◽  
Complete Metric Space ◽  
Differentiable Manifold ◽  
Open Dense Subset ◽  
Generic Properties ◽  
Countable Collection ◽  
Generic Subset

In order to analyse generic or typical properties of dynamical systems we consider the space of all C1-vector fields on a fixed differentiable manifold M. In the C1-metric, assuming M is compact, is a complete metric space and a generic subset is an open dense subset or an intersection of a countable collection of such open dense subsets of . Some generic properties (i.e. specifying generic subsets) in are described. For instance, generic dynamic systems have isolated critical points and periodic orbits each of which is hyperbolic. If M is a symplectic manifold we can introduce the space of all Hamiltonian systems and study corresponding generic properties.

scholarly journals The Menger and projective Menger properties of function spaces with the set-open topology

2019 ◽  
Vol 69 (3) ◽  
pp. 699-706 ◽  
Author(s):  
Alexander V. Osipov
Keyword(s):  
Topological Space ◽  
Hausdorff Space ◽  
Dense Subset ◽  
Continuous Functions ◽  
Function Spaces ◽  
List Type ◽  
Finite Sets ◽  
Tychonoff Space ◽  
Menger Space

Abstract For a Tychonoff space X and a family λ of subsets of X, we denote by Cλ(X) the space of all real-valued continuous functions on X with the set-open topology. A Menger space is a topological space in which for every sequence of open covers 𝓤1, 𝓤2, … of the space there are finite sets 𝓕1 ⊂ 𝓤1, 𝓕2 ⊂ 𝓤2, … such that family 𝓕1 ∪ 𝓕2 ∪ … covers the space. In this paper, we study the Menger and projective Menger properties of a Hausdorff space Cλ(X). Our main results state that Cλ(X) is Menger if and only if Cλ(X) is σ-compact; Cp(Y | X) is projective Menger if and only if Cp(Y | X) is σ-pseudocompact where Y is a dense subset of X.

scholarly journals Fractal Dimension of a Random Invariant Set and Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Gang Wang ◽  
Yanbin Tang
Keyword(s):  
Fractal Dimension ◽  
Parabolic Equation ◽  
Degenerate Parabolic Equation ◽  
Invariant Set ◽  
Invariant Sets ◽  
Abstract Result ◽  
Random Attractors ◽  
Finite Fractal Dimension ◽  
Degenerate Parabolic ◽  
Random Invariant Set

We prove an abstract result on random invariant sets of finite fractal dimension. Then this result is applied to a stochastic semilinear degenerate parabolic equation and an upper bound is obtained for the random attractors of fractal dimension.

A Combined Criterion for Existence and Continuity of Random Attractors for Stochastic Lattice Dynamical Systems

2017 ◽  
Vol 27 (02) ◽  
pp. 1750019 ◽  
Author(s):  
Anhui Gu ◽  
Yangrong Li
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Random Systems ◽  
Sufficient Criterion ◽  
Abstract Result ◽  
Lattice Dynamical System ◽  
Random Attractors ◽  
Multiplicative White Noise ◽  
Lattice Dynamical Systems ◽  
Set Up

The paper is devoted to establishing a combination of sufficient criterion for the existence and upper semi-continuity of random attractors for stochastic lattice dynamical systems. By relying on a family of random systems itself, we first set up the abstract result when it is convergent, uniformly absorbing and uniformly random when asymptotically null in the phase space. Then we apply the results to the second-order lattice dynamical system driven by multiplicative white noise. It is indicated that the criterion depending on the dynamical system itself seems more applicable than the existing ones to lattice differential models.

scholarly journals A continuity property related to an index of non-separability and its applications

1992 ◽  
Vol 46 (1) ◽  
pp. 67-79 ◽  
Author(s):  
Warren B. Moors
Keyword(s):  
Banach Space ◽  
Topological Space ◽  
Convex Function ◽  
Compact Subset ◽  
Metric Space ◽  
Dense Subset ◽  
Countable Family ◽  
Set Valued Mapping ◽  
Continuous Convex Function ◽  
Fréchet Differentiable

For a set E in a metric space X the index of non-separability is β(E) = inf{r > 0: E is covered by a countable-family of balls of radius less than r}.Now, for a set-valued mapping Φ from a topological space A into subsets of a metric space X we say that Φ is β upper semi-continuous at t ∈ A if given ε > 0 there exists a neighbourhood U of t such that β(Φ(U)) < ε. In this paper we show that if the subdifferential mapping of a continuous convex function Φ is β upper semi-continuous on a dense subset of its domain then Φ is Fréchet differentiable on a dense Gδ subset of its domain. We also show that a Banach space is Asplund if and only if every weak* compact subset has weak* slices whose index of non-separability is arbitrarily small.

Limiting dynamics for stochastic FitzHugh–Nagumo equations on large domains

2019 ◽  
Vol 19 (05) ◽  
pp. 1950037 ◽  
Author(s):  
Yangrong Li ◽  
Fuzhi Li
Keyword(s):  
Bounded Domain ◽  
Unbounded Domain ◽  
Bounded Domains ◽  
Limit Set ◽  
Coupled Equations ◽  
Lower Semi ◽  
The Family ◽  
Random Attractors ◽  
Nagumo Equations ◽  
Theoretical Results

This paper is devoted to the convergence of bi-spatial random attractors as a family of bounded domains is extended to be unbounded. Some criteria in terms of expansion and restriction are provided to ensure that the unbounded-domain attractor is approximated by the family of bounded-domain attractors in both upper and lower semi-continuity senses. The theoretical results are applied to show that the stochastic FitzHugh–Nagumo coupled equations have an attractor in [Formula: see text]-times Lebesgue space irrespective of whether the domain is bounded or unbounded. Furthermore, we prove that the family of bounded-domain attractors continuously converges to the unbounded-domain attractor, and the latter can be constructed by the metric-limit set of all bounded-domain attractors.

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