# Super asymptotically nonexpansive actions of semitopological semigroups on
Fréchet and locally convex spaces

Bui Ngoc Muoi and Ngai-Ching Wong  Department of Applied Mathematics, National
Sun Yat-sen University, Kaohsiung, 80424, Taiwan<EMAIL_ADDRESS><EMAIL_ADDRESS>Dedicated to Professor Wataru Takahashi on the occasion
of his 75th birthday

###### Abstract.

Let $\operatorname{LUC}(S)$ be the space of left uniformly continuous
functions on a semitopological semigroup $S$. Suppose that $S$ is right
reversible and $\operatorname{LUC}(S)$ has a left invariant mean. Let $(X,d)$
be a Fréchet space. Let $\tau$ be a locally convex topology of $X$ weaker than
the $d$-topology such that the metric $d$ is $\tau$-lower semicontinuous. Let
$K$ be a $d$–separable and $\tau$–compact convex subset of $X$. We show that
every jointly $\tau$-continuous and super asymptotically $d$–nonexpansive
action $S\times K\mapsto K$ of $S$ has a common fixed point. Similar results
in the locally convex space setting are provided.

###### Key words and phrases:

Semitopological semigroups, left invariant means, reversible semigroups, super
asymptotically nonexpansive actions, fixed points, Fréchet space.

###### 2010 Mathematics Subject Classification:

Primary 47H10; Secondary 47H20, 47H09.

Corresponding author: Ngai-Ching Wong, E-mail<EMAIL_ADDRESS>
##  1\. Introduction

Let $S$ be a _semitopological semigroup_ , i.e., $S$ is a semigroup with a
(Hausdorff) topology such that for each fixed $t\in S$, the mappings $s\mapsto
ts$ and $s\mapsto st$ are continuous. An _action_ of $S$ on a topological
space $K$ is a mapping of $S\times K$ into $K$, denoted by $(s,x)\mapsto s.x$,
such that $(st).x=s.(t.x)$ for all $s,t\in S$ and $x\in K$. We call the action
_separately_ (resp.  _jointly_) _continuous_ if the mapping $(s,x)\mapsto s.x$
is separately (resp. jointly) continuous. A point $x_{0}\in K$ is called a
_common fixed point_ for $S$ if $s.x_{0}=x_{0}$ for all $s\in S$.

In 1976, Karlovitz [9, page 322] proved that if $K$ is a weak* compact convex
subset of $l_{1}$ (which is the Banach dual of $c_{0}$) then every
nonexpansive mapping from $K$ to itself has a fixed point. Indeed, a
commutative family of weak* continuous nonexpansive mappings on a weak*
compact convex subset of a Banach dual space always attains a common fixed
point (see [3, Theorem 3.5]). However, an affine nonexpansive mapping on a
weak* compact convex subset of a general Banach dual space does not always
have a fixed point (see e.g. [21, Example 3.2]). This indicates that we need,
in general, both the topological and the uniform properties to ensure the
existence of a fixed point.

Observe that every (Hausdorff) locally convex space carries a uniform
structure defined by continuous seminorms. More precisely, we write $(X,Q)$
for a locally convex space $X$ with a family $Q$ of selected seminorms
defining its topology. Extending from the Banach space version (see [19]) we
say that an action of a semitopological semigroup $S$ on a subset $K$ of
$(X,Q)$ is

  * •

asymptotically uniformly $Q$–nonexpansive (see [6]) if for $x,y\in K$ there
exists a left ideal $I_{xy}$ of $S$ such that $q(s.x-s.y)\leq q(x-y),\forall
s\in I_{xy},\,\forall q\in Q$;

  * •

asymptotically separately $Q$–nonexpansive if for $x,y\in K$ and $q\in Q$
there exists a left ideal $I^{q}_{xy}$ of $S$ such that $q(s.x-s.y)\leq
q(x-y),\forall s\in I^{q}_{xy}$;

  * •

super asymptotically uniformly $Q$–nonexpansive if for $x\in K$ and $t\in S$
there exists a left ideal $I^{t}_{x}$ of $S$ such that $q(st.x-st.y)\leq
q(x-y),\forall s\in I^{t}_{x},\ \forall y\in K,\,\forall q\in Q$;

  * •

super asymptotically separately $Q$–nonexpansive if for $x\in K$, $t\in S$ and
$q\in Q$ there exists a left ideal $I^{q,t}_{x}$ of $S$ such that
$q(st.x-st.y)\leq q(x-y),\forall s\in I^{q,t}_{x},\,\forall y\in K$.

Note that these uniformity notions depend on the seminorms in $Q$ but not the
topology $Q$ defining.

When $K$ is a metric space with metric $d$, we call the action

  * •

asymptotically $d$–nonexpansive (see [7]) if for each $x,y\in K$, there exists
a left ideal $I_{xy}$ of $S$ such that $d(s.x,s.y)\leq d(x,y)$ for all $s\in
I_{xy}$;

  * •

super asymptotically $d$–nonexpansive if for each $x\in K$ and $t\in S$, there
exists a left ideal $I^{t}_{x}$ of $S$ such that $d(st.x,st.y)\leq d(x,y)$ for
all $y\in K$ and $s\in I^{t}_{x}$.

Let $X$ be a metrizable locally convex space. Let $Q=\\{q_{n}:n=1,2,\ldots\\}$
be a countable family of seminorms defining the topology of $X$, and define a
metric

$d(x,y)=\sum_{n=1}^{\infty}\frac{1}{2^{n}}\frac{q_{n}(x-y)}{1+q_{n}(x-y)}.$
(1.1)

It is easy to see that every asymptotically (resp. super asymptotically)
uniformly $Q$–nonexpansive action on a subset $K$ of $X$ is asymptotically
(resp. super asymptotically) $d$–nonexpansive. Conversely, let $d$ be a
translation invariant metric defining the topology of $X$ such that all open
metric balls are absolutely convex. As shown in Remark 2.1, there is a family
$Q$ of continuous seminorms defining the metric topology of $X$ such that
asymptotically (resp. super asymptotically) $d$–nonexpansive actions on a
subset $K$ of $X$ are exactly those being asymptotically (resp. super
asymptotically) uniformly $Q$–nonexpansive. Consequently, the fixed point
theory for various uniformly $Q$-nonexpansive actions is equivalent to that
for the $d$-nonexpansive ones (see Corollary 2.13).

In Section 2, we consider the case the topological structure and the uniform
structure of $K$ arising from different context. More precisely, let
$(X,\tau)$ be a locally convex space. Assume in addition that there is a
translation invariant metric $d$ on $X$ making $(X,d)$ a Fréchet space.
Moreover, the $\tau$-topology is weaker than the $d$-topology and $d$ is
$\tau$-lower semicontinuous. Following [13], we call $\tau$ a _$d$
–admissible_ topology in this case. For example, the weak (resp. weak*)
topology of a Banach space (resp. Banach dual space) is admissible with
respect to the norm metric.

Let $K$ be a $d$–separable and $\tau$–compact convex subset of $X$. We show in
Theorem 2.2, among other results, that if $S$ is a right reversible
semitopological semigroup such that $\operatorname{LUC}(S)$ has a left
invariant mean (for definitions see Section 2), then every jointly
$\tau$–continuous and super asymptotically $d$–nonexpansive action $S\times
K\mapsto K$ of $S$ on $K$ has a common fixed point. This extends earlier
results of the authors [19], and also supplements other results in literature,
e.g., [1, 10, 12, 22], from the Banach space setting to the Fréchet space
setting.

In Section 3, we provide some fixed point theorems for super asymptotically
separately $Q$–nonexpansive actions on a locally convex space $(X,Q)$. In this
setting, both the uniform structure and the topological structure arise from
the same family $Q$ of continuous seminorms of $X$. To end this paper, we
provide examples to demonstrate that the notions of asymptotic
nonexpansiveness and super asymptotic nonexpansiveness can be different, while
they coincide for separately continuous actions of right reversible compact
semigroups.

##  2\. Asymptotically nonexpansive actions on Fréchet spaces

Let $S$ be a semitopological semigroup throughout. We say that $S$ is right
(resp. left) reversible if any two closed left (resp. right) ideals of $S$
intersect. Denote by $l^{\infty}(S)$ (resp. $\operatorname{CB}(S)$) the Banach
space of bounded (resp. bounded and continuous) real-valued functions on $S$
with the supremum norm. For each $s\in S$ and $f\in l^{\infty}(S)$, we denote
by $l_{s}f$ the left translation of $f$ by $s$ where $l_{s}f(t)=f(st)$ for all
$t\in S$. A subspace $X\subseteq l^{\infty}(S)$ that contains all constant
functions is called left translation invariant if $l_{s}(X)\subseteq X$ for
all $s\in S$. A linear functional $m\in X^{*}$ is called a _mean_ if
$\|m\|=m(1)=1$. A mean $m$ is called a left invariant mean, or
$\operatorname{LIM}$ in short, if $m(l_{s}f)=m(f)$ for all $s\in S$ and $f\in
X$. We also have symmetric concepts about right invariant means. An _invariant
mean_ of $X$ is a mean which is both left and right invariant.

Let $\operatorname{LUC}(S)$ be the space of left uniformly continuous
functions on $S$; namely those $f$ for which the map $s\mapsto l_{s}f$ from
$S$ into $\operatorname{CB}(S)$ is norm continuous. The space of right
uniformly continuous functions $\operatorname{RUC(S)}$ is defined
symmetrically. When $S$ is topological group, $\operatorname{LUC(S)}$ and
$\operatorname{RUC(S)}$ coincide (see e.g. [5, Theorem 15.4]). Let
$\operatorname{AP}(S)$ (resp. $\operatorname{WAP}(S)$) be the subspace of
almost periodic (resp. weakly almost periodic) functions in
$\operatorname{CB}(S)$; namely those $f$ for which $\left\\{l_{s}f:s\in
S\right\\}$ is relatively compact in the norm (resp. weak) topology of
$\operatorname{CB}(S)$. All
$\operatorname{LUC}(S),\operatorname{AP}(S),\operatorname{WAP}(S)$ are left
translation invariant subspaces of $\operatorname{CB}(S)$. In general, we have

$\operatorname{AP}(S)\subseteq\operatorname{LUC}(S)\subseteq\operatorname{CB}(S)\quad\text{and}\quad\operatorname{AP}(S)\subseteq\operatorname{WAP}(S)\subseteq\operatorname{CB}(S).$

When $S$ is compact, we have
$\operatorname{AP}(S)=\operatorname{LUC}(S)\subseteq\operatorname{WAP}(S)=\operatorname{CB}(S)$
(see, e.g., [15, page 2952]).

We recall that a locally convex space $X$ is metrizable if and only if there
is a countable family of seminorms defining its topology. In this case, we can
choose a “good” metric $d$ to define the topology of $X$ such that

  * (i)

$d(x+y,x+z)=d(y,z)$ for all $x,y,z$ in $X$,

  * (ii)

$B_{r}(0)=\left\\{x\in X:d(x,0)<r\right\\}$ is open and absolutely convex for
all $r>0$.

The metric topology coincides with the topology defined by the countable
family $Q$ of seminorms

$q_{r}(x)=\inf\left\\{\lambda>0:x\in\lambda\bar{B}_{r}(0)\right\\},\quad
r\in\mathbb{Q}\cap(0,+\infty),$ (2.1)

where $\bar{B}_{r}(0)=\\{x\in X:d(x,0)\leq r\\}$. In this case, we can recover
the metric $d$ by

$d(x,y)=\inf\left\\{r\in\mathbb{Q}\cap(0,+\infty):q_{r}(x-y)\leq 1\right\\},$
(2.2)

when (1.1) gives rise to an equivalent metric instead (see Rudin [20, Theorem
1.24]). Clearly, the norm metric $d(x,y)=\|x-y\|$ of a Banach space is a
“good” metric.

######  Remark 2.1.

Let $d$ be a “good” metric of a Fréchet space $X$, and $Q$ be the countable
family of seminorms defined by (2.1). Then (resp. super) asymptotically
$d$–nonexpansive actions of a semigroup $S$ on a subset $K$ of $X$ are exactly
those being (resp. super) asymptotically uniformly $Q$–nonexpansive.

Indeed, suppose for any $x\in K$ and $t\in S$ there exists a left ideal
$I^{t}_{x}$ of $S$ such that

$d(st.x,st.y)\leq d(x,y),\quad\forall s\in I^{t}_{x},\,\forall y\in K.$

This gives

$x-y\in\bar{B}_{r}(0)\quad\implies\quad
st.x-st.y\in\bar{B}_{r}(0),\quad\forall r\in\mathbb{Q}\cap(0,+\infty).$

In other words,

$q_{r}(x-y)\leq 1\quad\implies\quad q_{r}(st.x-st.y)\leq 1,\quad\forall
r\in\mathbb{Q}\cap(0,+\infty).$

Therefore,

$\displaystyle q_{r}(st.x-st.y)\leq q_{r}(x-y),\quad\forall s\in
I^{t}_{x},\,\forall y\in K,\,\forall q_{r}\in Q.$ (2.3)

Conversely, it follows from (2.2) and (2.3) that any super asymptotically
uniformly $Q$–nonexpansive is super asymptotically $d$–nonexpansive.

The case for asymptotically $d$-nonexpansive actions is similar.

######  Theorem 2.2.

Let $S$ be a right reversible semitopological semigroup such that
$\operatorname{LUC}(S)$ has a left invariant mean. Let $(X,d)$ be a Fréchet
space with a “good” metric $d$. Let $\tau$ be a $d$–admissible locally convex
topology of $X$. Then $S$ has the following fixed point property.

> $\mathbf{(F^{\sup}_{jc,dsep})}$ Every super asymptotically $d$–nonexpansive
> and jointly $\tau$ continuous action of $S$ on a $d$–separable and
> $\tau$–compact convex subset $K$ of $X$ has a common fixed point.

We note that in the setting of Theorem 2.2, the left ideal in the definition
of the super asymptotic $d$–nonexpansiveness can be assumed closed. Indeed,
suppose that for any $x\in Y$, there exists for each $t\in S$ a left ideal
$I^{t}_{x}$ of $S$ such that $d(stx,sty)\leq d(x,y)$ for all $s\in I^{t}_{x}$
and all $y\in Y$. For each $s_{0}\in\overline{I^{t}_{x}}$, there is a net
$\\{s_{\lambda}\\}$ in $I^{t}_{x}$ converging to $s_{0}$. Then
$\\{s_{\lambda}tx\\}$ converges to $s_{0}tx$ in the $\tau$ topology. Since
$\tau$ is $d$–admissible,

$d(s_{0}tx,s_{0}ty)\leq\liminf_{\lambda}d(s_{\lambda}tx,s_{\lambda}ty)\leq
d(x,y).$

Consequently, the said left ideal can be chosen to be $\overline{I^{t}_{x}}$.

The proof of Theorem 2.2 needs several lemmas. The first one borrows from the
proof of [6, Theorem 3.1].

######  Lemma 2.3.

Let $S$ be a right reversible semitopological semigroup. Assume $S\times K\to
K$ is a separately continuous action of $S$ on a compact convex subset $K$ of
a locally convex space. Then there exists a subset $L_{0}$ of $K$ which is
minimal with respect to being nonempty, compact, convex and satisfying the
following conditions.

  1. $\mathbf{(\star 1)}$

There exists a collection $\Lambda=\left\\{\Lambda_{i}:i\in I\right\\}$ of
closed subsets of $K$ such that $L_{0}=\bigcap\Lambda$, and

  2. $\mathbf{(\star 2)}$

for each $x\in L_{0}$ there is a left ideal $J_{i}\subseteq S$ such that
$J_{i}.x\subseteq\Lambda_{i}$ for each $i\in I$.

Furthermore, $L_{0}$ contains a subset $Y$ that is minimal with respect to
being nonempty, closed and $S$-invariant, i.e. $s.Y\subset Y$ for all $s\in
S$.

######  Lemma 2.4 (Muoi and Wong [19, Lemma 2.3]).

The subset $Y$ of $K$ in Lemma 2.3 is $S$-preserving, i.e., $s.Y=Y$ for all
$s\in S$, if we suppose further that $\operatorname{LUC}(S)$ has a
$\operatorname{LIM}$ and the action is jointly continuous on $K$.

###### Proof.

For completeness, we sketch here the argument in the proof of [19, Lemma 2.3].
For each pair of $y\in Y$ and $f\in C(Y)$, define a function $R_{y}f\in
l^{\infty}(S)$ by $R_{y}f(s)=f(s.y)$. Following the proof of [16, Theorem 1],
with the joint continuity of the action we can show that
$R_{y}f\in\operatorname{LUC}(S)$.

Let $m$ be a $\operatorname{LIM}$ of $\operatorname{LUC}(S)$. Define a linear
functional $\psi$ on $C(Y)$ by $\psi(f)=m(R_{y}f)$. Observe

$|\psi(f)|\leq\|R_{y}f\|=\sup_{s\in S}|f(s.y)|\leq\|f\|_{C(Y)},\quad\forall
f\in C(Y).$

Since $\psi(1)=1$, we have $\|\psi\|=1$. For each $t\in S$, notice that

$\psi(l_{t}f)=m(R_{y}(l_{t}f))=m(l_{t}(R_{y}f))=m(R_{y}f)=\psi(f).$

Thus, $\psi$ is a left invariant mean of $C(Y)$. Let $\mu$ be the probability
measure on $Y$ corresponding to $\psi$. Define $Y_{0}$ to be the support of
$\mu$, i.e.,

$Y_{0}=\operatorname{supp}(\mu)=\bigcap\left\\{F\subseteq Y:F\ \text{is closed
and}\ \mu(F)=1\right\\}.$

We are going to see that $s.Y_{0}=Y_{0}$ for all $s\in S$. For every Borel
subset $A$ of $Y$ and $s\in S$, define $L_{s}(x)=s.x$ and
$L_{s}^{-1}A=\left\\{x\in Y:s.x\in A\right\\}$. Since $\psi$ is left
invariant, we have

$\mu(A)=\int_{Y}\mathbf{1}_{A}(x)\,d\mu=\int_{Y}\mathbf{1}_{A}(s.x)\,d\mu=\int_{Y}\mathbf{1}_{L_{s}^{-1}A}(x)\,d\mu=\mu(L_{s}^{-1}A).$

Because $L_{s}^{-1}Y_{0}$ is closed and $\mu(L_{s}^{-1}Y_{0})=\mu(Y_{0})=1$,
we have $Y_{0}\subseteq L_{s}^{-1}Y_{0}$ or $sY_{0}\subseteq Y_{0}$. On the
other hand, $\mu(sY_{0})=\mu(L_{s}^{-1}(sY_{0}))\geq\mu(Y_{0})=1$. This
implies $Y_{0}\subseteq sY_{0}$, and thus $sY_{0}=Y_{0}$. By the minimality of
$Y$, we conclude that $Y=Y_{0}$ is $S$-preserving. ∎

######  Lemma 2.5.

Let $(X,d)$ be a Fréchet space with a $d$–admissible locally convex topology
$\tau$. Let $S$ be a right reversible semitopological semigroup. Assume that
an action of $S$ on a $d$–separable and $\tau$–compact subset $Y$ of $X$ is
separately $\tau$ continuous and super asymptotically $d$–nonexpansive.
Suppose $Y$ is minimal with respect to being $\tau$–closed and $S$-invariant.
Suppose further that there exists a nonempty $\tau$–closed subset $F$ of $Y$
such that $F\subset sF$ for all $s\in S$. Then $F$ is $d$–compact. Especially,
if $Y$ is $S$–preserving then $Y$ is $d$–compact.

###### Proof.

We follow an idea from the proof of [13, Lemma 3.1] in which nonexpansive
actions are considered instead. Let $Z$ be the set of all points of continuity
of the identity mapping from $(Y,\tau)$ to $(Y,d)$. By [18, Corollary 1.3],
$Z$ is a dense $G_{\delta}$ subset of $(Y,\tau)$. Let $U=\\{x\in
X:d(x,0)<\varepsilon\\}$ be a $d$–neighborhood of 0 where $\varepsilon>0$. For
each $z\in Z$, there exists a $\tau$–neighborhood $V$ of $0$ such that
$(z+V)\cap Y\subset(z+U)\cap Y$. We can choose a $\tau$–neighborhood $V_{1}$
of $0$ such that $V_{1}+V_{1}\subseteq V$. Since the $d$–topology is stronger
than $\tau$, we have $V_{1}$ contains a $d$–open neighborhood $U_{1}$ of $0$.
Assume $U_{1}=\\{x\in X:d(x,0)<\delta\\}$ for some $\delta>0$. Since $Y$ is
$d$–separable, there exists a sequence $\\{x_{i}:i\in\mathbb{N}\\}$ in $Y$
such that

$Y=\bigcup\left\\{(x_{i}+U_{1})\cap Y:i\in\mathbb{N}\right\\}.$

Since the action is super asymptotically $d$–nonexpansive, for each given
$r_{0}\in S$, there exists a left ideal $I_{1}=I_{x_{1}}^{r_{0}}$ of $S$ such
that $d(sr_{0}.x_{1},sr_{0}.y)\leq d(x_{1},y)$ for all $s\in I_{1}$ and $y\in
Y$. Since $I_{1}r_{0}.x_{1}$ is $S$-invariant in $Y$, by the minimality of
$Y$, its $\tau$–closure must be exactly $Y$. Thus, there exists a $s_{1}\in
I_{1}$ such that $s_{1}r_{0}.x_{1}\in(z+V_{1})\cap Y$. Let $r_{1}=s_{1}r_{0}$,
we have $r_{1}.x_{1}\in(z+V_{1})\cap Y$ and $d(r_{1}.x_{1},r_{1}.y)\leq
d(x_{1},y)$ for all $y\in Y$.

Similarly, there exists a left ideal $I_{2}=I_{x_{2}}^{r_{1}}$ of $S$ such
that $d(sr_{1}.x_{2},sr_{1}.y)\leq d(x_{2},y)$ for all $s\in I_{2}$ and all
$y\in Y$. There exists $s_{2}\in I_{2}$ such that
$s_{2}r_{1}.x_{2}\in(z+V_{1})\cap Y$. Let $r_{2}:=s_{2}r_{1}$, we have
$r_{2}.x_{2}\in(z+V_{1})\cap Y$ and $d(r_{2}.x_{2},r_{2}.y)\leq d(x_{2},y)$
for all $y\in Y$. By induction, we can choose a sequence
$\left\\{r_{i}:i\in\mathbb{N}\right\\}$ in $S$ such that

$\displaystyle r_{i}.x_{i}\in(z+V_{1})\cap Y,\quad r_{i}=s_{i}s_{i-1}\cdots
s_{1}r_{0},$ and $\displaystyle d(r_{i}.x_{i},r_{i}.y)\leq
d(x_{i},y),\quad\forall y\in Y\mbox{ and }i\geq 1.$

For each $y\in(x_{i}+U_{1})\cap Y$, we have
$r_{i}y=(r_{i}y-r_{i}x_{i})+r_{i}x_{i}$ where $r_{i}x_{i}\in(z+V_{1})\cap Y$
and $d(r_{i}x_{i},r_{i}y)\leq d(x_{i},y)=d(x_{i}-y,0)<\delta$. Thus,

$r_{i}((x_{i}+U_{1})\cap Y)\subseteq(z+V_{1}+U_{1})\cap Y\subseteq(z+V)\cap
Y.$

We rewrite the action $r.x$ in the form of $L_{r}x$. Then $(x_{i}+U_{1})\cap
Y\subseteq L_{r_{i}}^{-1}((z+V)\cap Y)$, where $L_{r_{i}}^{-1}((z+V)\cap Y)$
is $\tau$–open by the separate $\tau$–continuity of the action. By the
$\tau$–compactness, we can cover $Y$ by finitely many such open sets. Let

$Y=\bigcup_{i=1}^{n}L_{r_{i}}^{-1}((z+V)\cap Y).$

It follows from the super asymptotic $d$–nonexpansiveness of the action that
there exist closed left ideals $J_{i}=J_{z}^{t_{i}}$, where
$t_{i}=s_{n+1}s_{n}\cdots s_{i+1}$ and $i=1,\ldots,n$, such that
$d(st_{i}.z,st_{i}.y)\leq d(z,y)$ for all $y\in Y$ and $s\in J_{i}$. Since $S$
is right reversible, there exists $t_{0}\in\cap_{i=1}^{n}J_{i}$. Consequently,

$\displaystyle d(t_{0}t_{i}.z,t_{0}t_{i}.y)\leq d(z,y)\quad\text{for all}\
y\in Y\mbox{ and }\ i=1,\ldots,n.$ (2.4)

By the assumption, $F\subset sF$ for all $s\in S$, we have

$\begin{array}[]{rl}F\subset L_{t_{0}}L_{r_{n+1}}F\subset
L_{t_{0}}L_{r_{n+1}}Y&=L_{t_{0}}L_{r_{n+1}}\left\\{\bigcup_{i=1}^{n}L_{r_{i}}^{-1}((z+V)\cap
Y)\right\\}\\\ &\subseteq\bigcup_{i=1}^{n}\left\\{L_{t_{0}}L_{s_{n+1}\cdots
s_{i+1}}((z+U)\cap Y)\right\\}\\\
&=\bigcup_{i=1}^{n}\left\\{L_{t_{0}t_{i}}((z+U)\cap Y)\right\\}\\\
&\subseteq\bigcup_{i=1}^{n}\left\\{(L_{t_{0}t_{i}}z+U)\cap Y)\right\\}\\\
&\subseteq\bigcup_{i=1}^{n}\left\\{L_{t_{0}t_{i}}z+U\right\\}.\end{array}$

The second last inclusion above follows from (2.4). This proves that the
$d$–closed subset $F$ can be covered by finitely many translates of any given
$d$–neighborhood $U$ of $0$. In other words, $F$ is totally bounded. Since
$(X,d)$ is complete, $F$ is $d$–compact. ∎

We provide below a metric version of DeMarr’s Lemma [4, Lemma 1].

######  Lemma 2.6.

Let $(X,d)$ be a metrizable locally convex space with a “good” metric $d$. Let
$Y$ be a compact subset of $X$ containing more than one point. Then there
exists a point $u$ in the convex hull $\operatorname{conv}(Y)$ of $Y$ such
that

$\sup\left\\{d(u,y):y\in Y\right\\}<\sup\left\\{d(x,y):x,y\in Y\right\\}.$

###### Proof.

Let $r=\mbox{diam}(Y)=\sup\left\\{d(x,y):x,y\in Y\right\\}$. Then
$x-y\in\bar{B}_{r}(0)$ for all $x,y\in Y$. For each
$\lambda\in\mathbb{Q}\cap(0,+\infty)$, the metric balls $B_{\lambda}(0)$ is
open, convex and balance. Thus

$\displaystyle B_{\lambda}(0)$ $\displaystyle=\left\\{x\in
X:d(x,0)<\lambda\right\\}$ $\displaystyle=\left\\{x\in
X:q_{\lambda}(x)<1\right\\}=\left\\{x\in X:q_{\lambda}(\lambda
x)<\lambda\right\\},$

where the seminorm $q_{\lambda}$ is defined in (2.1). This implies

$d(x,0)=q_{\lambda}(\lambda x)\ \text{for each}\
\lambda\in\mathbb{Q}\cap(0,+\infty)\ \text{and}\ x\in B_{\lambda}(0).$

Choose a $\lambda\in\mathbb{Q}\cap(0,+\infty)$ such that $\lambda>r$, we have
$d(x,y)=p_{\lambda}(\lambda(x-y))$ for all $x,y\in Y$. Since $Y$ is compact,
there exists a finite subset $M=\\{x_{1},x_{2},\ldots,x_{n}\\}\subset Y$ which
is maximal with respect to being that $d(x_{i},x_{j})=r$ for all $i\neq j$.

Let
$u=\frac{1}{n}\sum_{i=1}^{n}x_{i}\in\operatorname{conv}(Y)\subseteq\bar{B}_{r}(0)$
and $y_{0}\in Y$ such that $r_{0}=d(u,y_{0})=\max_{y\in Y}d(u,y)$. Suppose on
the contrary that $r_{0}=r$. Then

$r=d(u,y_{0})=p_{\lambda}(\lambda(u-y_{0}))\leq\frac{1}{n}\sum_{i=1}^{n}p_{\lambda}(\lambda(x_{i}-y_{0}))=\frac{1}{n}\sum_{i=1}^{n}d(x_{i},y_{0})\leq
r.$

This drives $d(x_{i},y_{0})=r$ for all $i=1,\ldots,n$. By the maximality of
$M$, we have $y_{0}=x_{i_{0}}$ for some $i_{0}\in\\{1,\ldots,n\\}$. This
conflicts with the fact that $d(x_{i_{0}},y_{0})=r>0$. ∎

Following the idea in the proofs of [6, Theorem 3.1] and [2, Theorem 4.2], we
are able to prove our main result.

###### Proof of Theorem 2.2.

By Lemmas 2.4 and 2.5, the nonempty $S$-preserving set $Y$ given in Lemma 2.3
is a $d$–compact subset of the Fréchet space $X$. Consequently, the
$d$–topology agrees with $\tau$ on $Y$. If $Y$ contains exactly one point then
we are done. Otherwise, let

$r=\mbox{diam}(Y)=\sup\left\\{d(x,y):x,y\in Y\right\\}.$

By Lemma 2.6, there exists a $u\in\operatorname{conv}(Y)$ such that

$r_{0}=\sup\left\\{d(u,y):y\in Y\right\\}<\sup\left\\{d(x,y):x,y\in
Y\right\\}=r.$

Let $0<\varepsilon<r-r_{0}$. For each $\Lambda\in\left\\{\Lambda_{i}:i\in
I\right\\}$ in Lemma 2.3, we set

$\displaystyle N_{\varepsilon,\Lambda}$ $\displaystyle=\bigcap_{y\in
Y}\\{x\in\Lambda:d(x,y)\leq r_{0}+\varepsilon\\}$ and $\displaystyle N_{0}$
$\displaystyle=\bigcap\left\\{N_{\varepsilon,\Lambda_{i}}:\ i\in
I\right\\}=L_{0}\cap\bigcap_{y\in Y}\bar{B}[y,r_{0}+\varepsilon],$

where $\bar{B}[y,\delta]$ denotes the closed ball centered at $y$ of radius
$\delta$.

We show that $N_{0}$ satisfies conditions $\mathbf{(\star 1)}$ and
$\mathbf{(\star 2)}$. Indeed, every $N_{\varepsilon,\Lambda_{i}}$ is
$d$–compact. Thus $N_{0}$ is a $d$–compact subset of $L_{0}$, and contains
$u$. For each $x\in N_{0}$ and $i\in I$, there exists a left ideal $I\subseteq
S$ such that $I.x\subseteq\Lambda_{i}$. By the super asymptotic
$d$–nonexpansiveness of the action, for each $t\in S$ there exists a left
ideal $I_{x}^{t}$ such that $d(st.y,st.x)\leq d(y,x)$ for all $y\in K$ and
$s\in I_{x}^{t}$. By the right reversibility of $S$, there exists a
$t_{0}\in\overline{I}\cap\overline{I_{x}^{t}t}$. Since $\Lambda_{i}$ is
$\tau$–closed, $St_{0}.x\subseteq\Lambda_{i}$. Consider a net $s_{\lambda}\in
I_{x}^{t}$ such that $s_{\lambda}t\to t_{0}$. From
$d(ss_{\lambda}t.y,ss_{\lambda}t.x)\leq d(y,x)\leq r_{0}+\varepsilon$ for all
$\lambda$, $y\in Y$ and $s\in S$, we have $d(st_{0}.y,st_{0}.x)\leq d(y,x)\leq
r_{0}+\varepsilon$. Since $Y\subset st_{0}.Y$, we have
$d(y^{\prime},st_{0}.x)\leq r_{0}+\varepsilon$ for all $y^{\prime}\in Y$. In
other words, there exists a left ideal $J=St_{0}$ of $S$ such that
$J.x\subseteq N_{\varepsilon,\Lambda_{i}}$. Consequently, the nonempty,
$\tau$–compact, convex subset $N_{0}$ satisfies conditions $\mathbf{(\star
1)}$ and $\mathbf{(\star 2)}$.

By the minimality of $L_{0}$, we have $Y\subseteq
L_{0}=N_{0}\subseteq\bigcap_{y\in Y}\bar{B}[y,r_{0}+\varepsilon]$. This gives
us a contradiction that $\operatorname{diam}(Y)\leq r_{0}+\epsilon<r$.
Therefore, $Y$ contains a unique point and it is the common fixed point for
the action of $S$ on $K$. ∎

When the semigroup $S$ is right reversible, the following proposition of us in
[19] is an extension of a result by Lau and Zhang [15, Theorem 6.2]. Their
result holds for norm nonexpansive and jointly weak* continuous actions on a
weak* compact convex subset of a Banach dual space. Here, in this paper, we
have a new proof. In fact, it is a direct consequence of Theorem 2.2, by
noting that for a Banach space (resp. a Banach dual space) the weak (resp.
weak*) topology is $\|\cdot\|$–admissible.

######  Proposition 2.7 (Muoi and Wong [19, Theorem 2.5]).

Let $S$ be a right reversible semitopological semigroup. Assume that
$\operatorname{LUC}(S)$ has a $\operatorname{LIM}$. Then $S$ has the following
fixed point property.

> $\mathbf{(F^{sup}_{jw^{*}c,Nsep})}$ Every super asymptotically nonexpansive
> and jointly weak* continuous action of $S$ on a weak* compact convex and
> norm separable subset of a Banach dual space has a common fixed point.

It is known that every finite Radon measure on a weakly compact subset of a
Banach space has a norm separable support (see, e.g., [8, Theorem 4.3, page
256]). This implies that the subset $Y$ in Lemma 2.4 is norm separable, when
$(X,\|\cdot\|)$ is a Banach space and $\tau$ is the weak topology of $X$.
Consequently, Theorem 2.2 holds also for a nonseparable subset $K$ in this
case. This provides a new proof of the following result.

######  Proposition 2.8 (Muoi and Wong [19, Theorem 2.1]).

Let $S$ be a right reversible semitopological semigroup. Assume that
$\operatorname{LUC}(S)$ has a left invariant mean. Then $S$ has the following
fixed point property.

> $\mathbf{(F^{\sup}_{jwc})}$ Every super asymptotically nonexpansive and
> jointly weakly continuous action of $S$ on a weakly compact convex subset of
> a Banach space has a common fixed point.

Recall that for a discrete semitopological semigroup $S$, the condition that
$\operatorname{LUC}(S)$ has a $\operatorname{LIM}$ is strictly stronger than
that $S$ is left reversible (see [14, page 2549]), while in general it might
not be the case.

######  Lemma 2.9 (Lau and Zhang [15, Lemma 3.4]).

Let S be left reversible semitopological semigroup. Consider an action
$S\times Y\mapsto Y$ of $S$ on a compact subset $Y$ of a locally convex space
$X$. Then

  * (i)

there is a closed subset $F$ of $Y$ such that $F\subset sF$ for all $s\in S$
if the action is separately continuous;

  * (ii)

there is a closed subset $F$ of $Y$ such that $sF=F$ for all $s\in S$ if the
action is jointly continuous.

###### Proof.

We give a different proof than [15, Lemma 3.4], since our approach is more in
line with the reasoning in this paper. Following an idea in [17], see also [6,
Lemma 4], let $F=\bigcap\\{sY:s\in S\\}$ where each $sY$ is compact. Consider
any finite collection $\\{s_{1}Y,s_{2}Y,\ldots,s_{n}Y\\}$. By the left
reversibility of $S$, there is a $t\in S$ such that
$t\in\bigcap_{i=1}^{n}\overline{s_{i}S}$. We have

$\bigcap_{i=1}^{n}s_{i}Y\supset\bigcap_{i=1}^{n}\overline{s_{i}(SY)}\supset\bigcap_{i=1}^{n}\overline{s_{i}S}Y\supset
tY\neq\emptyset.$ (2.5)

It follows that $F$ is nonempty.

We claim that $F\subset sF$ for all $s\in S$. We need to prove that $y\in sF$
whenever $y\in F$ and $s\in S$. Consider any finite collection
$\\{s_{1}Y,s_{2}Y,\ldots,s_{n}Y\\}$, from (2.5) we have

$\left(L_{s}^{-1}\\{y\\}\right)\cap\bigcap_{i=1}^{n}s_{i}Y\supset\left(L_{s}^{-1}\\{y\\}\right)\bigcap
tY\neq\emptyset,$

since $y\in F\subset stY$ then there is a $x\in Y$ such that $y=stx$, hence
$tx\in L_{s}^{-1}\\{y\\}$. By the finite intersection property,
$L_{s}^{-1}\\{y\\}\bigcap F\neq\emptyset$. Consequently, $y\in sF$. That
proves our claim.

When the action is jointly continuous, we show that $F$ is $S$-invariant. We
need to prove that $a.x\in F$ whenever $x\in F$. To see this, for any $b\in
S$, let $e\in S$ such that $e\in\overline{aS}\cap\overline{bS}$. There exist
nets $\\{c_{\lambda}\\}_{\lambda}$ and $\\{d_{\lambda}\\}_{\lambda}$ in $S$
such that $\\{ac_{\lambda}\\}_{\lambda}$, $\\{bd_{\lambda}\\}_{\lambda}$
converge to $e$. Since $x\in F=\bigcap\\{sY:s\in S\\}$, there is a
$x_{\lambda}\in Y$ such that $x=c_{\lambda}x_{\lambda}$ for every $\lambda$.
By the weak compactness of $Y$, we can assume $\\{x_{\lambda}\\}$ converges to
some $x_{0}\in Y$. Therefore,

$\displaystyle a.x=(ac_{\lambda})x_{\lambda}\rightarrow e.x_{0}$ (2.6)

by the joint continuity of the action. This implies $a.x=e.x_{0}$. Since
$bd_{\lambda}x_{0}\rightarrow e.x_{0}$, we have
$a.x=e.x_{0}\in\overline{bS}Y\subset\overline{bSY}\subset\overline{bY}=bY$,
since $bY$ is compact. Consequently, $sF=F$ for all $s\in S$. ∎

The following result supplements Theorem 2.2. The key point in its proof is
that we can bypass Lemma 2.4.

######  Theorem 2.10.

Let $S$ be a reversible semitopological semigroup and $(X,d)$ be a Fréchet
space with a “good” metric $d$. Let $\tau$ is a $d$–admissible locally convex
topology of $X$. Then $S$ has the following fixed point property.

> $\mathbf{(F^{\sup}_{sc,sep})}$ Every super asymptotically $d$–nonexpansive
> and separately $\tau$ continuous action of $S$ on a $d$–separable and
> $\tau$–compact convex subset $K$ of $X$ has a common fixed point.

###### Proof.

By Lemma 2.3, there is a subset $L_{0}$ of $K$ which is minimal with respect
to being nonempty, $\tau$–compact, convex and satisfying conditions
$\mathbf{(\star 1)}$ and $\mathbf{(\star 2)}$. Moreover, $L_{0}$ contains a
subset $Y$ that is minimal with respect to being nonempty, $\tau$–compact and
$S$-invariant. By Lemmas 2.9 and 2.5, there is a $d$–compact subset $F$ of $Y$
such that $F\subset sF$ for all $s\in S$. The remaining part follows similarly
as in the proof of Theorem 2.2 where the set $Y$ is replaced by its
$d$–compact subset $F$. ∎

######  Theorem 2.11.

Let $S$ be a right reversible semitopological semigroup and $(X,d)$ be a
Fréchet space with a “good” metric $d$. Let $\tau$ be a $d$–admissible locally
convex topology on $X$.

  * (i)

Assume $\operatorname{AP}(S)$ has a $\operatorname{LIM}$. Then every super
asymptotically $d$–nonexpansive, separately $\tau$ continuous, and $\tau$
equicontinuous action of $S$ on a $d$–separable and $\tau$–compact convex
subset $K$ of $X$ has a common fixed point.

  * (ii)

Assume $\operatorname{WAP}(S)$ has a $\operatorname{LIM}$. Then every super
asymptotically $d$–nonexpansive, separately $\tau$ continuous, and $\tau$
quasi-equicontinuous action of $S$ on a $d$–separable and $\tau$–compact
convex subset $K$ of $X$ has a common fixed point.

###### Proof.

These are direct consequences of [11, Lemma 3.1], [14, Theorem 3.4], and the
proof of Theorem 2.2. ∎

######  Corollary 2.12.

Let $S$ be a semitopological semigroup as well as a normal space. Let $(X,d)$
be a Fréchet space with a “good” metric $d$. Let $\tau$ be a $d$–admissible
locally convex topology on $X$. Assume that $\operatorname{CB}(S)$ has an
invariant mean. Then $S$ has the fixed point property
$\mathbf{F^{\sup}_{sc,sep}}$.

###### Proof.

It is known that if $S$ is normal and $CB(S)$ has a right invariant mean then
$S$ is right reversible. The assertion follows similarly as in proving Theorem
2.10. ∎

We now discuss $Q$–nonexpansive actions. Let $(X,Q)$ be a Fréchet space in
which $Q=\\{q_{n}:n\in\mathbb{N}\\}$ is a countable family of seminorms
defining the metric topology. A locally convex topology $\tau$ on $X$ is said
to be _$Q$ –admissible_ ([13]) if $\tau$ is weaker than the $Q$–topology while
every seminorm $q_{n}$ in $Q$ is $\tau$–lowersemicontinuous.

Note that one cannot use (1.1) to define a metric and apply Theorem 2.2 to get
the following result, as the metric so defined might not be “good”.

######  Corollary 2.13.

Let $S$ be a right reversible semitopological semigroup. Let $(X,Q)$ be a
Fréchet space with a $Q$–admissible locally convex topology $\tau$. Assume
that $\operatorname{LUC}(S)$ has a left invariant mean. Then $S$ has the
following fixed point property.

> Every super asymptotically uniformly $Q$–nonexpansive and jointly $\tau$
> continuous action of $S$ on a $\tau$–compact convex and $Q$–separable subset
> $K$ of $X$ has a common fixed point.

###### Proof.

Let $Q=\\{q_{n}:n=1,2,\ldots\\}$. Without loss of generality, by summing and
scaling, we can assume that $4q_{n}(x)\leq q_{n+1}(x)$ for all $x\in X$ and
$n=1,2,\ldots$. Then the absolutely convex open sets $V_{n}=\\{x\in
X:q_{n}(x)<1\\}$, $n=1,2,\ldots$, constitute a local basis of zero of $X$,
such that $V_{n+1}+V_{n+1}+V_{n+1}+V_{n+1}\subseteq V_{n}$ for $n=1,2,\ldots$.
Let $D$ be the set of rational numbers $r$ in $(0,1)$ such that
$r=\sum_{n}c_{n}(r)2^{-n}$, where the binary digit $c_{n}(r)$ assumes either
$0$ or $1$, and among them only finitely many $c_{n}(r)$ are $1$. Let
$A(r)=c_{1}(r)V_{1}+c_{2}(r)V_{2}+\cdots$ for any rational number $r\in D$ and
$A(r)=X$ for $r\geq 1$. Following the proof of [20, Theorem 1.24], we can
define a “good” metric $d$ of $X$ defining its topology such that the open
metric balls $B_{\delta}(0)=\\{x\in X:d(0,x)<\delta\\}=\bigcup\\{A(r):r\in
D,0<r<\delta\\}$. It is not difficult to see that $Q$–admissible locally
convex topologies of $X$ are exactly those being $d$-admissible, and super
asymptotically (resp. asymptotically) uniformly $Q$–nonexpansive actions on
any subset $K$ of $X$ are exactly those being super asymptotically (resp.
asymptotically) $d$–nonexpansive. Consequently, we can apply Theorem 2.2. ∎

However, we have an even better version of Corollary 2.13 in the following,
which works for super asymptotically _separately_ $Q$–nonexpansive actions.

######  Theorem 2.14.

Let $S$ be a right reversible semitopological semigroup. Let $(X,Q)$ be a
Fréchet space with a $Q$–admissible locally convex topology $\tau$. Assume
that $\operatorname{LUC}(S)$ has a left invariant mean. Then $S$ has the
following fixed point property.

> $\mathbf{(F^{\sup}_{jc,sQsep})}$ Every super asymptotically separately
> $Q$–nonexpansive and jointly $\tau$ continuous action of $S$ on a
> $\tau$–compact convex and $Q$–separable subset $K$ of $X$ has a common fixed
> point.

###### Proof.

We adapt the proof of Theorem 2.2 to the locally convex space setting. By
Lemma 2.3, there is a subset $L_{0}$ of $K$ which is minimal with respect to
being nonempty, compact, convex and satisfying conditions $\mathbf{(\star 1)}$
and $\mathbf{(\star 2)}$. Moreover, $L_{0}$ contains a subset $Y$ that is
minimal with respect to being nonempty, compact and $S$-invariant. Following
Lemma 2.4, we see that $Y$ is $S-$preserving. With similar arguments as in
Lemma 2.5, we have $Y$ is $Q$-compact.

If $Y$ contains exactly one point then we are done. Otherwise, there exists a
seminorm $q$ in $Q$ such that

$r=\mbox{diam}_{q}(Y):=\sup\left\\{q(x-y):x,y\in Y\right\\}>0.$

Following the arguments in the proof of Lemma 2.6, noting also [4, Lemma 1],
we see that there exists a $u\in\operatorname{conv}(Y)$ such that

$r_{0}=\sup\left\\{q(u-y):y\in Y\right\\}<\sup\left\\{q(x-y):x,y\in
Y\right\\}=r.$

Let $0<\varepsilon<r-r_{0}$. For each $\Lambda\in\left\\{\Lambda_{i}:i\in
I\right\\}$ in Lemma 2.3, we set

$\displaystyle N_{\varepsilon,\Lambda}$ $\displaystyle=\bigcap_{y\in
Y}\\{x\in\Lambda:q(x-y)\leq r_{0}+\varepsilon\\}$ and $\displaystyle N_{0}$
$\displaystyle=\bigcap\left\\{N_{\varepsilon,\Lambda_{i}}:\ i\in
I\right\\}=L_{0}\cap\bigcap_{y\in Y}\bar{B}_{q}[y,r_{0}+\varepsilon],$

where $\bar{B}_{q}[y,\delta]$ denotes the $q$–closed ball centered at $y$ of
radius $\delta$. Following the arguments in the last part of the proof of
Theorem 2.2, we see that $N_{0}$ is nonempty, compact, convex and satisfies
conditions $\mathbf{(\star 1)}$ and $\mathbf{(\star 2)}$.

By the minimality of $L_{0}$, we have $Y\subseteq
L_{0}=N_{0}\subseteq\bigcap_{y\in Y}\bar{B}_{q}[y,r_{0}+\varepsilon]$. This
gives us a contradiction that $\operatorname{diam}_{q}(Y)\leq
r_{0}+\epsilon<r$. Therefore, $Y$ contains a unique point and it is the common
fixed point for the action of $S$ on $K$. ∎

In a similar manner, we will get the $Q$–nonexpansive versions of Theorems
2.10 and 2.11, and Corollary 2.12.

######  Theorem 2.15.

Let $S$ be a right reversible semitopological semigroup. Let $(X,Q)$ be a
Fréchet space with a $Q$–admissible locally convex topology $\tau$.

  * (i)

Assume $S$ is reversible. Then every super asymptotically separately
$Q$–nonexpansive and separately $\tau$ continuous action of $S$ on a
$Q$–separable and $\tau$–compact convex subset $K$ of $X$ has a common fixed
point.

  * (ii)

Assume $\operatorname{AP}(S)$ has a $\operatorname{LIM}$. Then every
separately $\tau$–continuous, $\tau$–equicontinuous and super asymptotically
separately $Q$–nonexpansive action of $S$ on a $\tau$–compact convex subset
$K$ of $X$ has a common fixed point.

  * (iii)

Assume $\operatorname{WAP}(S)$ has a $\operatorname{LIM}$. Then every
$\tau$–separately continuous, $\tau$–quasi-equicontinuous and super
asymptotically separately $Q$–nonexpansive action of $S$ on a $\tau-$compact
convex subset $K$ of $X$ has a common fixed point.

##  3\. Fixed point properties on locally convex spaces

In this section, we consider super asymptotically separately $Q$–nonexpansive
actions of a semitopological semigroup $S$ on a compact convex set $K$ in a
general locally convex space $(X,Q)$, in which $Q$ is a family of seminorms
defining the topology. Note that we do not assume the metrizability or the
completeness of $X$, and we do not assume the separability of $K$ either.

We note that the results in [6, 14, 13], though stated for asymptotically
uniformly $Q$–nonexpansive actions, hold indeed with the same proofs for
asymptotically _separately_ $Q$–nonexpansive actions. However, the following
example tells us that the various separately $Q$–nonexpansiveness are strictly
weaker than their uniform versions.

######  Example 3.1.

Let $X$ be the Fréchet space of all scalar sequences $x=(x_{n})$ equipped with
the topology of coordinate-wise convergence; namely, it is the topology
defined by the countable family $Q$ of seminorms $q_{n}(x)=|x_{n}|$ for
$n\in\mathbb{N}$. Let $K$ be the compact convex subset of $X$ defined by

$K=\\{(x_{n})\in X:|x_{n}|\leq 1\mbox{ for all }n\in\mathbb{N}\\}.$

Let $\mathbb{N}_{0}$ be the additive discrete semigroup of nonnegative
integers acting on $K$ by right shifts:

$(k.x)_{j}=0\mbox{ for all }j\leq k,\quad\mbox{ and }(k.x)_{j}=x_{j-k}\mbox{
elsewhere}.$

Note that any (necessarily two-sided) ideal $J$ of $\mathbb{N}_{0}$ assumes
the form $J=s+\mathbb{N}_{0}$ for $s=\min J$.

This action is not asymptotically uniformly $Q$–nonexpansive. Indeed, consider
any $x=(x_{n}),y=(y_{n})$ in $K$ such that $|x_{n}-y_{n}|$ is strictly
decreasing. For any left ideal $J=s+\mathbb{N}_{0}$ of $\mathbb{N}_{0}$,
choosing $n>k>s$ we have

$q_{n}(k.x-k.y)=|x_{n-k}-y_{n-k}|>|x_{n}-y_{n}|=q_{n}(x-y).$

Thus the action is not asymptotically uniformly $Q$–nonexpansive.

On the other hand, this action is super asymptotically separately
$Q$–nonexpansive. Indeed, for any seminorm $q_{n}$, we choose the ideal
$J=n+\mathbb{N}_{0}$. Then for any $x$ in $K$ and any $t\in\mathbb{N}_{0}$, we
have

$q_{n}((s+t).x-(s+t).y)=|((s+t).x)_{n}-((s+t).y)_{n}|=0\leq
q_{n}(x-y),\quad\forall s\in J,\,\forall y\in K.$

######  Theorem 3.2.

Let $S$ be a right reversible semitopological semigroup and $(X,Q)$ be a
locally convex space.

  1. (i)

Assume $\operatorname{LUC}(S)$ has a $\operatorname{LIM}$. Then every jointly
continuous super asymptotically separately $Q$–nonexpansive action of $S$ on a
compact convex subset $K$ of $X$ has a common fixed point.

  2. (ii)

Assume $\operatorname{AP}(S)$ has a $\operatorname{LIM}$. Then every
separately continuous, equicontinuous and super asymptotically separately
$Q$–nonexpansive action of $S$ on a compact convex subset $K$ of $X$ has a
common fixed point.

  3. (iii)

Assume $\operatorname{WAP}(S)$ has a $\operatorname{LIM}$. Then every
separately continuous, quasi-equicontinuous and super asymptotically
separately $Q$–nonexpansive action of $S$ on a compact convex subset $K$ of
$X$ has a common fixed point.

###### Proof.

The assertions follow from arguments similar to those in the proofs of
Theorems 2.14 and 2.15. Note that we do not need Lemma 2.5, while its
conclusion holds automatically as the $\tau$–topology coincides with the
$Q$–topology in the current setting. ∎

The following result supplements Theorem 2.10 in the general locally convex
space setting.

######  Theorem 3.3.

Let $S$ be a reversible semitopological semigroup and $(X,Q)$ be a locally
convex space. Then $S$ has the following fixed point property.

> $\mathbf{(F^{\sup}_{sc})}$ Every super asymptotically separately
> $Q$–nonexpansive and separately continuous action of $S$ on a compact convex
> subset $K$ of $X$ has a common fixed point.

######  Remark 3.4.

We do not have the “two topology $Q-\tau$ version” of the above results ready.
The difficulty is that we need to assume $(X,Q)$ to be metrizable to utilize
[18, Corollary 1.3] in proving Lemma 2.5. Without Lemma 2.5 we do not know
whether the $\tau$-compact set $Y$ is also $Q$–compact. Thus we are not able
to apply Lemma 2.6, or [4, Lemma 1], to conclude the existence of a common
fixed point of the action.

Finally, we show that the notions of asymptotic nonexpansiveness, super
asymptotic nonexpansiveness and nonexpansiveness are strictly different.

######  Example 3.5 (Based on [6, Example]).

Let $K=\left\\{(r,\theta):0\leq r\leq 1,0\leq\theta<2\pi\right\\}$ be the
closed unit disk in $\mathbb{R}^{2}$ in polar coordinates and the usual
Euclidean norm. Define continuous mappings $f,g$ from $K$ into $K$ such that

$f(r,\theta)=(r/2,\theta)\quad\text{and}\quad g(r,\theta)=(r,2\theta\
({\operatorname{mod}2\pi})).$

Let $S$ be the discrete semigroup generated by $f$ and $g$ under composition.
Then

$S=\left\\{f^{n}g^{m}:(m,n)\in\mathbb{N}_{0}\times\mathbb{N}_{0}\setminus\left\\{(0,0)\right\\}\right\\},$

where $\mathbb{N}_{0}=\\{0,1,2,\ldots\\}$. Consider the action of $S$ on $K$
given by

$f^{n}g^{m}(r,\theta)=(\frac{r}{2^{n}},2^{m}\theta\;(\operatorname{mod}2\pi)).$

This action is asymptotically nonexpansive but not super asymptotically
nonexpansive (see [19, Example 2.13]).

######  Example 3.6 (Based on [1, Example 3.3(ii)]).

Let $K$ be the closed unit disk in $\mathbb{R}^{2}$. Let $f$ be any continuous
but not nonexpansive function from $[-1,1]$ into $[-1,1]$ such that $f(0)=0$.
Consider the map $T:K\to K$ defined by $T(x_{1},x_{2})=(f(x_{2}),0)$. Then $T$
is not nonexpansive and $T^{n}=0$ for all $n\geq 2$. Define an action of the
reversible discrete additive semigroup $\mathbb{N}$ on $M$ by
$n.(x_{1},x_{2})=T^{n}(x_{1},x_{2})$. It is plain that this action is super
asymptotically nonexpansive but not nonexpansive.

We see, however, in the following that asymptotically nonexpansive actions of
a right reversible compact semitopological semigroup $S$ are automatically
super asymptotically nonexpansive.

######  Proposition 3.7 (Based on [19, Proposition 2.12(iii)]).

Let $S$ be a right reversible compact semitopological semigroup. Let $K$ be a
subset of a locally convex space $(X,Q)$. Then every separately continuous and
asymptotically separately (resp. uniformly) $Q$–nonexpansive action of $S$ on
$K$ is super asymptotically separately (resp. uniformly) $Q$–nonexpansive.

###### Proof.

Fix an $x\in K$, $t\in S$ and a seminorm $q\in Q$. For each $y\in K$ there
exists a left ideal $I_{y}$ of $S$ such that $q(s.x-s.y)\leq q(x-y)$ for all
$s\in I_{y}$. Since the action is separately continuous, we can assume $I_{y}$
is closed. For each finite family of closed left ideals
$\left\\{I_{y_{1}},\ldots,I_{y_{n}}\right\\}$ of $S$, it follows from the
right reversibility of $S$ that $\bigcap_{i=1}^{n}I_{y_{i}}\neq\emptyset$. It
follows from the compactness of $S$ that $I=\bigcap_{y\in K}I_{y}$ is a
nonempty closed left ideal of $S$. Clearly, $q(sx-sy)\leq q(x-y)$ for all
$s\in I$ and $y\in K$. Since $S$ is compact, $St$ is a closed left ideal of
$S$, and thus $St\cap I\neq\emptyset$. Then $I^{t}=\left\\{s\in S:st\in
I\right\\}$ is a nonempty left ideal of $S$. We have $q(st.x-st.y)\leq q(x-y)$
for all $s\in I^{t}$ and $y\in K$. In other words, the action is super
asymptotically separately $Q$-nonexpansive. Finally, the uniform version
follows similarly. ∎

We end this paper with an open problem for a converse of Theorem 2.2, 2.11 and
2.14.

######  Question 3.8.

Let $S$ be a right reversible semitopological semigroup. Does the fixed point
property stated in Theorems 2.2, 2.11 or 2.14 imply that
$\operatorname{LUC(S)}$, $\operatorname{AP(S)}$ or $\operatorname{WAP(S)}$ has
a $\operatorname{LIM}$, respectively?.

It is shown in [15, Proposition 6.5] that $\operatorname{AP(S)}$ has a
$\operatorname{LIM}$ if $S$ has the following fixed point property.

> $\mathbf{(F^{ne}_{jw*,sep})}$ Every norm nonexpansive and jointly weak*
> continuous action of $S$ on a nonempty norm separable and weak* compact
> convex subset of a Banach dual space has a common fixed point.

Note that the fixed point property $\mathbf{(F^{\sup}_{jc,dsep})}$ in Theorem
2.2 is stronger than $\mathbf{(F^{ne}_{jw*,sep})}$. Therefore,
$\mathbf{(F^{\sup}_{jc,dsep})}$ implies the existence of a
$\operatorname{LIM}$ of $\operatorname{AP(S)}$. In particular, the converse of
Theorem 2.2 holds when $S$ is compact, since $\operatorname{AP(S)}$ and
$\operatorname{LUC(S)}$ coincide in this case.

## Acknowledgement

This research is supported by the Taiwan MOST grant 108-2115-M-110-004-MY2.

The authors would like to thank the referee for encouraging comments and
useful suggestions.

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