# On the Residual Continuity of Global Attractors

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## Abstract

**:**

## 1. Introduction

## 2. Residual Continuity & Dense Continuity

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

- (i)
- a set of first category has an empty interior;
- (ii)
- a residual set is dense;
- (iii)
- a countable intersection of dense open sets is dense.

**Lemma**

**2.**

**Proof.**

**Theorem**

**2**

**Residual continuity ⇔ dense continuity**)

**.**Let $f:\Lambda \to \mathcal{E}$ be a function between complete metric spaces Λ and $\mathcal{E}$. Then, the set ${D}^{c}$ of continuity points of f is residual if and only if it is dense in Λ.

**Proof.**

**Lemma**

**3**

**.**Let I be a compact metric space and Y a metric space. Suppose that ${f}_{n}:I\to Y$, $n\in \mathbb{N}$, is a family of continuous map. If ${f}_{n}$ converges pointwise to f, i.e., ${f}_{n}\left(\epsilon \right)\to f\left(\epsilon \right)$ for each $\epsilon \in I$, then the points of continuity of f form a residual subset of I.

## 3. Almost Uniform Continuity in Lebesgue Measure Sense

**Lemma**

**4.**

**Lemma**

**5**

**Egroff’s theorem**). If a sequence of measurable functions ${g}_{n}$ defined on a measure space $(\Omega ,\mathcal{F},m)$ converges to g pointwise on a set E of finite measure, then for each $\epsilon >0$, there is a set $F\subset E$ with $m\left(F\right)<\epsilon $ such that ${g}_{n}$ converges to g uniformly on $E\backslash F$.

**Lemma**

**6.**

**Proof.**

**Theorem**

**3.**

- (i)
- ${f}_{n}\to f$ uniformly on ${C}_{\epsilon}$;
- (ii)
- the restriction ${f|}_{{C}_{\epsilon}}$ of f to ${C}_{\epsilon}$ is uniformly continuous.

**Proof.**

## 4. Residual Continuity and Almost Uniform Continuity of Global Attractors

**Lemma**

**7.**

- (i)
- ${S}_{\lambda}$ has a global attractor ${\mathcal{A}}_{\lambda}$ for every $\lambda \in \Lambda $;
- (ii)
- There is a bounded subset D of X such that ${\mathcal{A}}_{\lambda}\subset D$ for every $\lambda \in \Lambda $; and
- (iii)
- For $t>0$, ${S}_{\lambda}(t,x)$ is continuous in Λ uniformly for x in bounded subsets of X.

**Theorem**

**4.**

- (i)
- ${S}_{\lambda}$ has a global attractor ${\mathcal{A}}_{\lambda}$ for every $\lambda \in \Lambda $;
- (ii)
- There is a bounded subset D of X such that ${\mathcal{A}}_{\lambda}\subset D$ for every $\lambda \in \Lambda $; and
- (iii)
- For $t>0$, ${S}_{\lambda}(t,x)$ is continuous in λ uniformly for x in bounded subsets of X.

- (i)
- for any $\epsilon >0$, there is a closed measurable subset ${C}_{\epsilon}\subset \Lambda $ with Lebesgue measure $m\left({C}_{\epsilon}\right)>m(\Lambda )-\epsilon $ such that the family of global attractors ${\left\{{\mathcal{A}}_{\lambda}\right\}}_{\lambda \in {C}_{\epsilon}}$ is uniformly continuous on ${C}_{\epsilon}$;
- (ii)
- if D is a uniformly absorbing set (i.e., for any bounded set $E\subset X$, ${\cup}_{\lambda \in \Lambda}{S}_{\lambda}(t,E)\subset D$ for t large enough), then for the closed set ${C}_{\epsilon}\subset \Lambda $ defined above, the global attractors ${\left\{{\mathcal{A}}_{\lambda}\right\}}_{\lambda \in {C}_{\epsilon}}$ are equi-attracting: for any bounded set $E\subset X$,$$\underset{\lambda \in {C}_{\epsilon}}{sup}dist({S}_{\lambda}(t,E),{\mathcal{A}}_{\lambda})\to 0,\phantom{\rule{1.em}{0ex}}as\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4pt}{0ex}}t\to \infty .$$

**Proof.**

**Example**

**1.**

## 5. Conclusive Comments

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Wang, X.; Cui, H.
On the Residual Continuity of Global Attractors. *Mathematics* **2022**, *10*, 1444.
https://doi.org/10.3390/math10091444

**AMA Style**

Wang X, Cui H.
On the Residual Continuity of Global Attractors. *Mathematics*. 2022; 10(9):1444.
https://doi.org/10.3390/math10091444

**Chicago/Turabian Style**

Wang, Xingxing, and Hongyong Cui.
2022. "On the Residual Continuity of Global Attractors" *Mathematics* 10, no. 9: 1444.
https://doi.org/10.3390/math10091444