Fixed point theorems for left amenable semigroups of non-Lipschitzian mappings in Banach spaces

Abstract

In this paper, we provide the existence and convergence theorems of fixed points for left amenable semigroups of asymptotically nonexpansive type mappings in general Banach spaces, which extend and improve many recent results in this area.

MSC:47H09, 47H10, 47H20.

1 Introduction

Let E be a Banach space and C a nonempty bounded closed convex subset of E. A mapping T on C is said to be nonexpansive if $\parallel Tx-Ty\parallel \le \parallel x-y\parallel$ for all $x,y\in C$. A well-known result of Browder [1] asserts that if E is uniformly convex, then every nonexpansive mapping on C has a fixed point. Kirk [2], Belluce and Kirk [3] extended this result to the case that X has a normal structure or Opial’s property. Goebel and Kirk [4] proved that if E is a uniformly convex Banach space, then every asymptotically nonexpansive mapping on C has a fixed point.

As is well known, not every semigroup of nonexpansive mappings on a subset of a Banach space has a fixed point [5]. The existence and convergence of fixed points for semigroups of various mappings have been studied extensively [6–10]. Recently, Suzuki and Takahashi [8], Takahashi and Zembayashi [9], Zhu and Li [10] proved the existence theorems of fixed points for semigroups $\mathrm{\Im }=\{T(t):t\ge 0\}$ of nonexpansive, asymptotically nonexpansive and asymptotically nonexpansive type mappings, respectively. For instance, in [9], Takahashi and Zembayashi proved the following theorem:

Theorem 1.1 [9]

Let C be a nonempty compact convex subset of a Banach space E and $\mathrm{\Im }=\{T(t):t\ge 0\}$ be a semigroup of asymptotically nonexpansive mappings on C, then the set of common fixed points $F(\mathrm{\Im })$ of ℑ is nonempty.

Many results are known in the case that the semigroup G is commutative, amenable or reversible [11–24]. In the case of an amenable semigroup, the first result was established by Takahashi [21] where he proved:

Theorem 1.2 [21]

Let C be a nonempty compact convex subset of a Banach space E. Let $\mathrm{\Im }=\{T(t):t\in G\}$ be an amenable semigroup of nonexpansive mappings on C. Then C contains a common fixed point for ℑ.

Theorem 1.2 was proved for a commutative semigroup by DeMarr [11]. Later in [13], Lau showed that the fixed point property is equivalent to the existence of a left invariant mean on $AP(\mathrm{\Im })$, the space of almost periodic functions on the semigroup ℑ. It should be pointed out that if ℑ is left reversible, then $AP(\mathrm{\Im })$ always has a left invariant mean [13], but the converse is false [14]. And in [16], Lau, Miyake and Takahashi gave the following existence theorem:

Theorem 1.3 [16]

Let C be a nonempty weakly compact convex subset of a Banach space E. Let G be a left reversible semigroup (with identity) and $\mathrm{\Im }=\{T(t):t\in G\}$ be a semigroup of nonexpansive mappings on C. Let X be a left invariant ℑ-stable subspace of $l^{\mathrm{\infty }}(G)$ containing 1, and μ be a left invariant mean on X. Then $F(\mathrm{\Im })=F(T_{\mu })\cap C_a$, where $C_a$ denotes the set of almost periodic elements in C, i.e., all $x\in C$ such that $\{T(s)x:s\in G\}$ is relatively compact in the norm topology of E. Further, if C is compact, then the set $F(\mathrm{\Im })$ is nonempty.

In [20], Saeidi extended Theorem 1.3 to the case for left reversible semigroups of asymptotically nonexpansive mappings. Inspired and motivated by [8–10, 16, 20, 21, 23], we investigate the existence and convergence of fixed points for left amenable semigroups of asymptotically nonexpansive type mappings in Banach spaces. We first provide the existence theorem of fixed points for left amenable semigroups of asymptotically nonexpansive type mappings in Banach spaces. Utilizing this result, we obtain a strong convergence theorem of iterative sequences for left amenable semigroups of asymptotically nonexpansive type mappings. The results obtained in this paper extend and improve many recent results in [8–10, 16, 20, 23].

2 Preliminaries

Let C be a nonempty bounded subset of a Banach space E. Let G be a semitopological semigroup, i.e., G is a semigroup with a Hausdorff topology such that for $s\in G$ the mappings $s\mapsto st$ and $s\mapsto ts$ from G to G are continuous. Let $\mathrm{\Im }=\{T(t):t\in G\}$ be a continuous representation of G on C, i.e., $T(ts)x=T(t)T(s)x$, $t,s\in G$, $x\in C$ and the mapping $(t,x)\mapsto T(t)x$ from $G\times C$ into C is continuous when $G\times C$ has the product topology. Recall that ℑ is said to be

• (1) nonexpansive if for all $x,y\in C$ and $t\in G$,

  $$\parallel T(t)x-T(t)y\parallel \le \parallel x-y\parallel ;$$

• (2) asymptotically nonexpansive [25–27] if there exists a function $k:G\mapsto [0,+\mathrm{\infty })$ with ${inf}_{s\in G}{sup}_{t\in G}k(ts)\le 1$ such that for all $x,y\in C$ and $t\in G$,

  $$\parallel T(t)x-T(t)y\parallel \le k(t)\parallel x-y\parallel ;$$

• (3) asymptotically nonexpansive type [25–27] if for each $x\in C$, there exists a function $r(\cdot ,x):G\mapsto [0,+\mathrm{\infty })$ with ${inf}_{s\in G}{sup}_{t\in G}r(ts,x)=0$ such that for all $x,y\in C$ and $t\in G$,

  $$\parallel T(t)x-T(t)y\parallel \le \parallel x-y\parallel +r(t,x).$$

It is easily seen that $(1)\Rightarrow (2)\Rightarrow (3)$ and that both inclusions are proper [25–27].

Let $l^{\mathrm{\infty }}(G)$ be the Banach space of all bounded real valued functions on G with the supremum norm. Then, for each $s\in G$ and $f\in l^{\mathrm{\infty }}(G)$, we can define $l_{s}f$ in $l^{\mathrm{\infty }}(G)$ by $(l_{s}f)(t)=f(st)$ for all $t\in G$. Let X be a subspace of $l^{\mathrm{\infty }}(G)$ containing 1 and $X^{\ast }$ be its dual space. An element $\mu \in X^{\ast }$ is called a mean on X if $\parallel \mu \parallel =\mu (1)=1$. We always denote the value of μ at $f\in X$ by ${\mu }_t〈f(t)〉=\mu (f)$. Let X be left invariant, i.e., $l_s(X)\subset X$ for all $s\in G$. A mean μ on X is said to be left invariant if $\mu (l_{s}f)=\mu (f)$ for all $s\in G$ and $f\in X$. Further, X is called left amenable if X has a left invariant mean. In this case, we also say that G is a left amenable semigroup. Recall that a semigroup G is called left reversible if any two closed right ideals of G have nonvoid intersection. In this case, $(G,\le )$ is a directed system when the binary relation ≤ on G is defined by $s\le t$ if and only if $\{s\}\cup \overline{sG}\supseteq \{t\}\cup \overline{tG}$, $s,t\in G$. As is well known, the class of left reversible semigroups includes all commutative semigroups and if a semigroup G is left amenable, then G is left reversible. But the converse is false [28].

Let $\mathrm{\Im }=\{T(t):t\in G\}$ be an asymptotically nonexpansive type semigroup on C. Let $F(\mathrm{\Im })$ denote the set of all fixed points of ℑ, i.e., $F(\mathrm{\Im })=\{x\in C:T(s)x=x,\mathrm{\forall }s\in G\}$. A subspace X of $l^{\mathrm{\infty }}(G)$ is called ℑ-stable if functions $s\mapsto 〈T(s)x,x^{\ast }〉$ and $s\mapsto \parallel T(s)x-y\parallel$ on G are in X for all $x,y\in C$ and $x^{\ast }\in E^{\ast }$. We know that if μ is a mean on X and if for each $x^{\ast }\in E^{\ast }$ the function $s\mapsto 〈T(s)x,x^{\ast }〉$ is contained in X and C is weakly compact, then there exists a unique point $x_0$ of E such that ${\mu }_s〈T(s)x,x^{\ast }〉=〈x_0,x^{\ast }〉$ for all $x^{\ast }\in E^{\ast }$. Such a point $x_0$ is always denoted by $T_{\mu }x$. Obviously, $T_{\mu }x=x$ for each $x\in F(\mathrm{\Im })$.

3 Main results

Lemma 3.1 Let C be a nonempty weakly compact convex subset of a Banach space E. Let G be a left reversible semigroup and $\mathrm{\Im }=\{T(t):t\in G\}$ be a continuous representation of G as asymptotically nonexpansive type mappings on C, with the condition ${lim\hspace{0.17em}sup}_{s\in G}r(s,x)=0$ for all $x\in C$. Let X be a left invariant ℑ-stable subspace of $l^{\mathrm{\infty }}(G)$ containing 1, and μ be a left invariant mean on X. Then $F(\mathrm{\Im })=F(T_{\mu })\cap C_a$.

Proof If $F(T_{\mu })\cap C_a$ is empty, then so is $F(\mathrm{\Im })$ as $F(\mathrm{\Im })\subset F(T_{\mu })\cap C_a$. Let $z\in F(T_{\mu })\cap C_a$ and define $d={\mu }_s\parallel T(s)z-z\parallel$, then for all $t\in G$, we have

$$\begin{array}{rcl}\parallel T(t)z-z\parallel & =& \parallel T(t)z-T_{\mu }z\parallel =sup\{\left|〈T(t)z-T_{\mu }z,x^{\ast }〉\right|:x^{\ast }\in E^{\ast },\parallel x^{\ast }\parallel =1\}\\ =& sup\{\left|{\mu }_s〈T(t)z-T(s)z,x^{\ast }〉\right|:x^{\ast }\in E^{\ast },\parallel x^{\ast }\parallel =1\}\\ \le & sup\{{\mu }_s\parallel T(t)z-T(s)z\parallel \cdot \parallel x^{\ast }\parallel :x^{\ast }\in E^{\ast },\parallel x^{\ast }\parallel =1\}\\ =& {\mu }_s\parallel T(t)z-T(s)z\parallel ={\mu }_s\parallel T(t)z-T(ts)z\parallel \text{(by }\mu \text{-left invariant)}\\ \le & {\mu }_s\parallel T(s)z-z\parallel +r(t,z)=d+r(t,z),\end{array}$$

i.e., for all $t\in G$,

$$\parallel T(t)z-z\parallel \le d+r(t,z).$$

(3.1)

Next, we shall show $d=0$. In fact, if $d>0$, then for each $t\in G$,

$$d={\mu }_s\parallel T(s)z-z\parallel ={\mu }_s\parallel T(ts)z-z\parallel \le \underset{s\in G}{sup}\parallel T(ts)z-z\parallel ,$$

i.e.,

$$\underset{s\in G}{sup}\parallel T(ts)z-z\parallel \ge d,\mathrm{\forall }t\in G.$$

(3.2)

By ${lim\hspace{0.17em}sup}_{s\in G}r(s,z)=0$, then for any $n\in N$, there exists $s_n\in G$ such that

$$\underset{t\ge s_n}{sup}r(t,z)<\frac{1}{4n}.$$

(3.3)

It follows from (3.2) that we can choose a cluster point $u_1$ of the net $\{T(s)z:s\in G\}$ in the set C with $\parallel u_1-z\parallel \ge d$ and there exists $t_n^{(1)}\in G$ satisfying $t_n^{(1)}\ge s_n$ and $\parallel T(t_n^{(1)})z-u_1\parallel <\frac{1}{4n}$. Combining it with (3.1) and (3.3), we get

$$\begin{array}{rcl}\parallel u_1-z\parallel & \le & \parallel u_1-T\left(t_n^{(1)}\right)z\parallel +\parallel T\left(t_n^{(1)}\right)z-z\parallel \\ \le & \frac{1}{4n}+d+r(t_n^{(1)},z)\le d+\frac{1}{2n}\text{(by }{t}_n^{(1)}\ge s_n\text{)}.\end{array}$$

Hence $\parallel u_1-z\parallel \le d$ and so $\parallel u_1-z\parallel =d$. It follows from (3.1) and (3.3) that

$$\begin{array}{rcl}\parallel T\left(t_n^{(1)}{s}_{n}s\right)z-u_1\parallel & \le & \parallel T\left(t_n^{(1)}{s}_{n}s\right)z-T\left(t_n^{(1)}\right)z\parallel +\parallel T\left(t_n^{(1)}\right)z-u_1\parallel \\ \le & \parallel T(s_{n}s)z-z\parallel +r(t_n^{(1)},z)+\parallel T\left(t_n^{(1)}\right)z-u_1\parallel \\ \le & d+r(s_{n}s,z)+\frac{1}{2n}\le d+\frac{3}{4n}\end{array}$$

(3.4)

for all $s\in G$. Noting

$$\begin{array}{rcl}d& =& \parallel u_1-z\parallel =\parallel u_1-T_{\mu }z\parallel \\ =& sup\{\left|〈u_1-T_{\mu }z,x^{\ast }〉\right|:x^{\ast }\in E^{\ast },\parallel x^{\ast }\parallel =1\}\\ =& sup\{\left|{\mu }_s〈u_1-T(s)z,x^{\ast }〉\right|:x^{\ast }\in E^{\ast },\parallel x^{\ast }\parallel =1\}\\ \le & {\mu }_s\parallel u_1-T(s)z\parallel ,\end{array}$$

(3.5)

we obtain

$$\begin{array}{r}{\mu }_s(\parallel T\left(t_n^{(1)}{s}_{n}s\right)z-z\parallel +\parallel T\left(t_n^{(1)}{s}_{n}s\right)z-u_1\parallel )\\ ={\mu }_s\parallel T\left(t_n^{(1)}{s}_{n}s\right)z-z\parallel +{\mu }_s\parallel T\left(t_n^{(1)}{s}_{n}s\right)z-u_1\parallel \\ ={\mu }_s\parallel T(s)z-z\parallel +{\mu }_s\parallel T(s)z-u_1\parallel \ge 2d.\end{array}$$

This implies that

$$\underset{s\in G}{sup}[\parallel T\left(t_n^{(1)}{s}_{n}s\right)z-z\parallel +\parallel T\left(t_n^{(1)}{s}_{n}s\right)z-u_1\parallel ]\ge 2d.$$

Thus there exists $s_n^{(1)}\in G$ such that

$$\parallel T\left(t_n^{(1)}{s}_n{s}_n^{(1)}\right)z-z\parallel +\parallel T\left(t_n^{(1)}{s}_n{s}_n^{(1)}\right)z-u_1\parallel \ge 2d-\frac{1}{4n}.$$

(3.6)

Since $\{T(s)z:s\in G\}$ is a relatively compact set, $\{T(t_n^{(1)}{s}_n{s}_n^{(1)})z\}$, as a subset of $\{T(s)z:s\in G\}$, has a strong convergent subsequence. Without loss of generality, we can assume that $T(t_n^{(1)}{s}_n{s}_n^{(1)})z\to u_2\in C$. Setting $t_n^{(2)}=t_n^{(1)}{s}_n{s}_n^{(1)}$, then $t_n^{(2)}\ge t_n^{(1)}\ge s_n$, $T(t_n^{(2)})z\to u_2$ and by (3.6),

$$\parallel u_2-z\parallel +\parallel u_2-u_1\parallel \ge 2d.$$

(3.7)

On the other hand,

$$\begin{array}{rcl}\parallel u_2-z\parallel & \le & \parallel u_2-T\left(t_n^{(2)}\right)z\parallel +\parallel T\left(t_n^{(2)}\right)z-z\parallel \\ \le & \parallel u_2-T\left(t_n^{(2)}\right)z\parallel +d+r(t_n^{(2)},z)\text{(by (3.1))}\\ \le & \parallel u_2-T\left(t_n^{(2)}\right)z\parallel +d+\frac{1}{4n}\text{(by (3.3))}\end{array}$$

and

$$\begin{array}{rcl}\parallel u_2-u_1\parallel & \le & \parallel u_2-T\left(t_n^{(2)}\right)z\parallel +\parallel T\left(t_n^{(2)}\right)z-u_1\parallel \\ =& \parallel u_2-T\left(t_n^{(2)}\right)z\parallel +\parallel T\left(t_n^{(1)}{s}_n{s}_n^{(1)}\right)z-u_1\parallel \\ \le & \parallel u_2-T\left(t_n^{(2)}\right)z\parallel +d+\frac{3}{4n}\text{(by (3.4))}.\end{array}$$

Thus we can conclude $\parallel u_2-z\parallel \le d$ and $\parallel u_2-u_1\parallel \le d$. So by (3.7),

$$\parallel u_2-z\parallel =\parallel u_2-u_1\parallel =d.$$

Similar to the proof of (3.5), we can prove ${\mu }_s\parallel u_2-T(s)z\parallel \ge d$ and

$$\begin{array}{r}{\mu }_s(\parallel T\left(t_n^{(2)}{s}_{n}s\right)z-z\parallel +\parallel T\left(t_n^{(2)}{s}_{n}s\right)z-u_1\parallel +\parallel T\left(t_n^{(2)}{s}_{n}s\right)z-u_2\parallel )\\ ={\mu }_s\parallel T\left(t_n^{(2)}{s}_{n}s\right)z-z\parallel +{\mu }_s\parallel T\left(t_n^{(2)}{s}_{n}s\right)z-u_1\parallel +{\mu }_s\parallel T\left(t_n^{(2)}{s}_{n}s\right)z-u_2\parallel \\ ={\mu }_s\parallel T(s)z-z\parallel +{\mu }_s\parallel T(s)z-u_1\parallel +{\mu }_s\parallel T(s)z-u_2\parallel \ge 3d.\end{array}$$

This means

$$\underset{s\in G}{sup}(\parallel T\left(t_n^{(2)}{s}_{n}s\right)z-z\parallel +\parallel T\left(t_n^{(2)}{s}_{n}s\right)z-u_1\parallel +\parallel T\left(t_n^{(2)}{s}_{n}s\right)z-u_2\parallel )\ge 3d.$$

Thus there exists $s_n^{(2)}\in G$ such that

$$\parallel T\left(t_n^{(2)}{s}_n{s}_n^{(2)}\right)z-z\parallel +\parallel T\left(t_n^{(2)}{s}_n{s}_n^{(2)}\right)z-u_1\parallel +\parallel T\left(t_n^{(2)}{s}_n{s}_n^{(2)}\right)z-u_2\parallel \ge 3d-\frac{1}{n}.$$

Therefore,

$$\begin{array}{r}\parallel T\left(t_n^{(2)}{s}_n{s}_n^{(2)}\right)z-z\parallel \le d+r(t_n^{(2)}{s}_n{s}_n^{(2)},z)\le d+\frac{1}{4n},\\ \parallel T\left(t_n^{(2)}{s}_n{s}_n^{(2)}\right)z-u_2\parallel \le \parallel T\left(t_n^{(2)}{s}_n{s}_n^{(2)}\right)z-T\left(t_n^{(2)}\right)z\parallel +\parallel T\left(t_n^{(2)}\right)z-u_2\parallel \\ \phantom{\parallel T\left(t_n^{(2)}{s}_n{s}_n^{(2)}\right)z-u_2\parallel }\le \parallel T\left(s_n{s}_n^{(2)}\right)z-z\parallel +r(t_n^{(2)},z)+\parallel T\left(t_n^{(2)}\right)z-u_2\parallel \\ \phantom{\parallel T\left(t_n^{(2)}{s}_n{s}_n^{(2)}\right)z-u_2\parallel }\le d+r(s_n{s}_n^{(2)},z)+r(t_n^{(2)},z)+\parallel T\left(t_n^{(2)}\right)z-u_2\parallel \\ \phantom{\parallel T\left(t_n^{(2)}{s}_n{s}_n^{(2)}\right)z-u_2\parallel }\le d+\frac{1}{2n}+\parallel T\left(t_n^{(2)}\right)z-u_2\parallel \end{array}$$

and

$$\begin{array}{rcl}\parallel T\left(t_n^{(2)}{s}_n{s}_n^{(2)}\right)z-u_1\parallel & \le & \parallel T\left(t_n^{(2)}{s}_n{s}_n^{(2)}\right)z-T\left(t_n^{(1)}\right)z\parallel +\parallel T\left(t_n^{(1)}\right)z-u_1\parallel \\ \le & \parallel T\left(t_n^{(1)}{s}_n{s}_n^{(1)}{s}_n{s}_n^{(2)}\right)z-T\left(t_n^{(1)}\right)z\parallel +\parallel T\left(t_n^{(1)}\right)z-u_1\parallel \\ \le & \parallel T\left(s_n{s}_n^{(1)}{s}_n{s}_n^{(2)}\right)z-z\parallel +r(t_n^{(1)},z)+\parallel T\left(t_n^{(1)}\right)z-u_1\parallel \\ \le & d+r(s_n{s}_n^{(1)}{s}_n{s}_n^{(2)},z)+r(t_n^{(1)},z)+\parallel T\left(t_n^{(1)}\right)z-u_1\parallel \\ \le & d+\frac{1}{2n}+\parallel T\left(t_n^{(1)}\right)z-u_1\parallel .\end{array}$$

Since $\{T(t_n^{(2)}{s}_n{s}_n^{(2)})z\}$ has a strong convergent subsequence, without loss of generality, we can assume that $T(t_n^{(2)}{s}_n{s}_n^{(2)})z\to u_3\in C$. Setting $t_n^{(3)}=t_n^{(2)}{s}_n{s}_n^{(2)}$, then $t_n^{(3)}\ge t_n^{(2)}$, $T(t_n^{(3)})z\to u_3$,

$$\parallel u_3-z\parallel \le d,\parallel u_3-u_2\parallel \le d,\parallel u_3-u_1\parallel \le d$$

and

$$\parallel u_3-z\parallel +\parallel u_3-u_1\parallel +\parallel u_3-u_2\parallel \ge 3d.$$

Thus we have found $u_3\in C$ such that

$$\parallel u_3-u_1\parallel =\parallel u_3-u_2\parallel =\parallel u_3-z\parallel =d.$$

Now, by mathematical induction, we can find a sequence $\{u_i\}\subset C$ satisfying

$$\parallel u_i-z\parallel =d,\parallel u_i-u_j\parallel =d(\mathrm{\forall }i,j\in N,i\ne j).$$

Since $T(t_n^{(i)})z\to u_i$, we can seek out $t_{n_i}^{(i)}\in G$ with $\parallel T(t_{n_i}^{(i)})z-u_i\parallel \le \frac{d}{4}$. Thus

$$\parallel T\left(t_{n_i}^{(i)}\right)z-T\left(t_{n_j}^{(j)}\right)z\parallel \ge \frac{d}{2}(\mathrm{\forall }i,j\in N,i\ne j),$$

which is a contradiction with the relative compactness of $\{T(t_{n_i}^{(i)})z:i\in N\}$. Therefore, we can conclude $d=0$.

In the following, we shall show $z\in F(\mathrm{\Im })$. Indeed, for any $h\in G$, $T(h):C\to C$ is continuous at z, then for all $\epsilon >0$, there exists a $\delta >0$ ($\delta <\epsilon$) such that for all $x\in C$ with $\parallel x-z\parallel <\delta$,

$$\parallel T(h)x-T(h)z\parallel <\epsilon .$$

By (3.1) and the definition of $r(\cdot ,z)$, we can get

$$\underset{s\in G}{inf}\underset{t\in G}{sup}\parallel T(ts)z-z\parallel \le d+\underset{s\in G}{inf}\underset{t\in G}{sup}r(ts,z)=0$$

and so we can find a $s_{\delta }\in G$ such that ${sup}_{t\in G}\parallel T(t{s}_{\delta })z-z\parallel <\delta$, i.e., for all $t\in G$,

$$\parallel T(t{s}_{\delta })z-z\parallel <\delta .$$

Hence

$$\begin{array}{rcl}\parallel T(h)z-z\parallel & \le & \parallel T(h)z-T(h)T(t{s}_{\delta })z\parallel +\parallel T(h)T(t{s}_{\delta })z-z\parallel \\ =& \parallel T(h)z-T(h)T(t{s}_{\delta })z\parallel +\parallel T(ht{s}_{\delta })z-z\parallel \\ <& \epsilon +\delta <2\epsilon .\end{array}$$

Since $\epsilon >0$ is arbitrary, we get $z\in F(\mathrm{\Im })$. This completes the proof. □

Now we can give the existence theorem of fixed points for left amenable semigroups of non-Lipschitzian mappings in Banach spaces.

Theorem 3.1 Let C be a nonempty compact convex subset of a Banach space E. Let G be a left reversible semigroup and $\mathrm{\Im }=\{T(t):t\in G\}$ be a continuous representation of G as asymptotically nonexpansive type mappings on C, with the condition ${lim\hspace{0.17em}sup}_{s\in G}r(s,x)=0$ for all $x\in C$. Let X be a left invariant ℑ-stable subspace of $l^{\mathrm{\infty }}(G)$ containing 1, and μ be a left invariant mean on X. Then the set $F(\mathrm{\Im })$ is nonempty.

Proof For all $x,y\in C$ and $t\in G$, we have

$$\begin{array}{rcl}\parallel T_{\mu }x-T_{\mu }y\parallel & =& sup\{\left|〈T_{\mu }x-T_{\mu }y,x^{\ast }〉\right|:x^{\ast }\in E^{\ast },\parallel x^{\ast }\parallel =1\}\\ =& sup\{\left|{\mu }_s〈T(s)x-T(s)y,x^{\ast }〉\right|:x^{\ast }\in E^{\ast },\parallel x^{\ast }\parallel =1\}\\ \le & sup\{{\mu }_s\parallel T(s)x-T(s)y\parallel \cdot \parallel x^{\ast }\parallel :x^{\ast }\in E^{\ast },\parallel x^{\ast }\parallel =1\}\\ =& {\mu }_s\parallel T(s)x-T(s)y\parallel ={\mu }_s\parallel T(ts)x-T(ts)y\parallel \\ \le & \underset{s\in G}{sup}\parallel T(ts)x-T(ts)y\parallel \le \parallel x-y\parallel +\underset{s\in G}{sup}r(ts,z),\end{array}$$

and so by ${lim\hspace{0.17em}sup}_{s\in G}r(s,z)=0$, we get $\parallel T_{\mu }x-T_{\mu }y\parallel \le \parallel x-y\parallel$, i.e., $T_{\mu }$ is a nonexpansive mapping from C into itself. Since a nonexpansive mapping of a compact convex subset of a Banach space into itself has a fixed point [29], $T_{\mu }$ has a fixed point z. By Lemma 3.1, $z\in F(\mathrm{\Im })$. This completes the proof. □

Remark 3.1 Theorem 3.1 is an extension of the main results in [8–10, 16, 20, 23] to the case for left amenable semigroups of asymptotically nonexpansive type mappings in Banach spaces.

Recall that for each $s\in G$, we define a point evaluation ${\delta }_s$ on X by ${\delta }_s(f)=f(s)$ for every $f\in X$. A convex combination of point evaluation is called a finite mean on G. If λ is a finite mean on G, say $\lambda ={\mathrm{\Sigma }}_{i=1}^n{a}_i{\delta }_{s_i}$, where $s_i\in G$, $a_i\ge 0$, $i=1,2,\dots ,n$, and ${\mathrm{\Sigma }}_{i=1}^n{a}_i=1$, then $\lambda (t)〈T(t)x,x^{\ast }〉={\mathrm{\Sigma }}_{i=1}^n{a}_i〈T(s_i)x,x^{\ast }〉=〈{\mathrm{\Sigma }}_{i=1}^n{a}_{i}T(s_i)x,x^{\ast }〉$ for all $x^{\ast }\in E^{\ast }$. For convenience, we denote it by $\lambda (t)〈T(t)x〉={\mathrm{\Sigma }}_{i=1}^n{a}_{i}T(s_i)x$. A net $\{{\lambda }_{\alpha }:\alpha \in I\}$ of finite means on G is said to be strongly left regular if

$$\underset{\alpha \in I}{lim}\parallel {\lambda }_{\alpha }-l_s^{\ast }{\lambda }_{\alpha }\parallel =0$$

for all $s\in G$, where A is a directed system and $l_s^{\ast }$ is the conjugate operator of $l_s$.

Corollary 3.1 Let C be a nonempty compact convex subset of a Banach space E. Let G be a left reversible semigroup and $\mathrm{\Im }=\{T(t):t\in G\}$ be a continuous representation of G as asymptotically nonexpansive type mappings on C, with the condition ${lim\hspace{0.17em}sup}_{s\in G}r(s,x)=0$ for all $x\in C$. Let X be a left invariant ℑ-stable subspace of $l^{\mathrm{\infty }}(G)$ containing 1 and $\{{\lambda }_{\alpha }:\alpha \in I\}$ be a net of strongly left regular finite means on G. If $z\in C$ satisfies

$$\underset{\alpha \in I}{lim\hspace{0.17em}inf}\parallel {\lambda }_{\alpha }(t)〈T(t)z〉-z\parallel =0,$$

then $z\in F(\mathrm{\Im })$.

Proof Since ${lim\hspace{0.17em}inf}_{\alpha \in I}\parallel {\lambda }_{\alpha }(t)〈T(t)z〉-z\parallel =0$ and $\{{\lambda }_{\alpha }:\alpha \in I\}\subset D^{\ast }$, we can find a subnet $\{{\lambda }_{{\alpha }_{\beta }}:\beta \in A\}$ of $\{{\lambda }_{\alpha }:\alpha \in I\}$ such that ${lim}_{\beta \in A}{\lambda }_{{\alpha }_{\beta }}(t)〈T(t)z〉=z$ and ${\omega }^{\ast }-{lim}_{\beta \in A}{\lambda }_{{\alpha }_{\beta }}=\mu$, where A is a directed system. Hence μ is a left invariant mean on X (see [30]) and $T_{\mu }z=z$, which implies $z\in F(\mathrm{\Im })$. This completes the proof. □

Remark 3.2 Corollary 3.1 is an extension of the main results in [8–10, 23].

Next we shall prove the strong convergence theorem for the iterative sequences of left reversible semigroups of asymptotically nonexpansive type mappings. We need a lemma which plays a crucial role in the proof of Theorem 3.2.

Lemma 3.2 [30]

Let $z_n$ and $w_n$ be bounded sequences in a Banach space X and let ${\alpha }_n$ be a sequence in $(0,1)$ with $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\alpha }_n<1$. Suppose that $z_{n+1}={\alpha }_n{w}_n+(1-{\alpha }_n)z_n$ for all $n\in N$ and

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}(\parallel w_n-w_{n+k}\parallel -\parallel z_n-z_{n+k}\parallel )\le 0$$

for all $k\in N$. Then ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}\parallel w_n-z_n\parallel =0$.

Theorem 3.2 Let C be a nonempty compact convex subset of a Banach space X and G be a left reversible semigroup. Let $\mathrm{\Im }=\{T(t):t\in G\}$ be a continuous representation of G as asymptotically nonexpansive type mappings on C, with the condition ${lim\hspace{0.17em}sup}_{s\in G}r(s,x)=0$ for all $x\in C$. Let X be a left invariant ℑ-stable subspace of $l^{\mathrm{\infty }}(G)$ containing 1, and μ be a left invariant mean on X. Let $x_1\in C$ and define a sequence $\{x_n\}$ in C by

$$x_{n+1}={\alpha }_n{T}_{\mu }{x}_n+(1-{\alpha }_n)x_n,$$

for all $n\in N$, where ${\alpha }_n\subset [0,1]$ satisfies $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\alpha }_n<1$. Then $x_n$ converges strongly to a fixed point $z\in F(\mathrm{\Im })$.

Proof It follows from

$$\begin{array}{rcl}\parallel T_{\mu }{x}_{n+1}-x_{n+1}\parallel & \le & \parallel T_{\mu }{x}_{n+1}-T_{\mu }{x}_n\parallel +\parallel T_{\mu }{x}_n-x_{n+1}\parallel \\ =& \parallel T_{\mu }{x}_{n+1}-T_{\mu }{x}_n\parallel +(1-{\alpha }_n)\parallel T_{\mu }{x}_n-x_n\parallel \\ =& \parallel T_{\mu }{x}_n-x_n\parallel +\parallel T_{\mu }{x}_{n+1}-T_{\mu }{x}_n\parallel -\parallel x_{n+1}-x_n\parallel \\ \le & \parallel T_{\mu }{x}_n-x_n\parallel \end{array}$$

that ${lim}_{n\to \mathrm{\infty }}\parallel T_{\mu }{x}_n-x_n\parallel$ exists. By Lemma 3.1, we get ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}\parallel T_{\mu }{x}_n-x_n\parallel =0$ and so ${lim}_{n\to \mathrm{\infty }}\parallel T_{\mu }{x}_n-x_n\parallel =0$. Since C is compact, there exists a subsequence $\{x_{n_k}\}\subset \{x_n\}$ such that $x_{n_k}\to z\in C$. Hence, z is a fixed point of $T_{\mu }$. By Lemma 3.1, we have $z\in F(\mathrm{\Im })$ and

$$\begin{array}{rcl}\parallel x_{n+1}-z\parallel & =& \parallel {\alpha }_n{T}_{\mu }{x}_n+(1-{\alpha }_n)x_n-z\parallel \\ \le & {\alpha }_n\parallel T_{\mu }{x}_n-z\parallel +(1-{\alpha }_n)\parallel x_n-z\parallel \\ \le & \parallel x_n-z\parallel .\end{array}$$

Hence ${lim}_{n\to \mathrm{\infty }}\parallel x_n-z\parallel$ exists. Thus ${lim}_{n\to \mathrm{\infty }}\parallel x_n-z\parallel ={lim}_{k\to \mathrm{\infty }}\parallel x_{n_k}-z\parallel =0$, which implies that $x_n$ converges strongly to $z\in F(\mathrm{\Im })$. This completes the proof. □

In the following, we shall give an example of a semigroup which is asymptotically nonexpansive type but not asymptotically nonexpansive on a compact set.

Example 3.1 [27]

Let Δ be the Cantor ternary set. Define the Cantor ternary function

$$\tau (x)=\{\begin{array}{cc}{\sum }_{n=1}^{+\mathrm{\infty }}\frac{b_n}{2^n},\hfill & x={\sum }_{n=1}^{+\mathrm{\infty }}\frac{2b_n}{3^n}\in \mathrm{\Delta }(b_n=0,1),\hfill \\ sup\{\tau (y),y\le x,y\in \mathrm{\Delta }\},\hfill & x\in [0,1]\mathrm{\setminus }\mathrm{\Delta }\hfill \end{array}$$

then $\tau :[0,1]\to [0,1]$ is a continuous and increasing but not absolutely continuous function with $\tau (0)=0$, $\tau (\frac{1}{2})=\frac{1}{2}$ (see [31]). Since a Lipschitzian function is absolutely continuous, τ is non-Lipschitzian. For all $t>0$, we define $T(t):[0,1]\to [0,1]$ by

$$T(t)x=\{\begin{array}{cc}\frac{x}{2^t},\hfill & 0\le x\le \frac{1}{2},\hfill \\ \frac{\tau (1-x)}{2^t},\hfill & \frac{1}{2}<x\le 1.\hfill \end{array}$$

Then $T(t)$ is continuous but not Lipschitzian continuous (since τ is non-Lipschitzian) and for all $x,y\in [0,1]$, $|T(t)x|\le \frac{1}{2^{t+1}}$,

$$|T(t)x-T(t)y|\le \frac{1}{2^t}\le |x-y|+\frac{1}{2^t}.$$

Therefore, we can conclude that the semigroup $\mathrm{\Im }=\{T(t):t>0\}$ is asymptotically nonexpansive type but not an asymptotically nonexpansive on $[0,1]$. Also, 0 is a fixed point of ℑ.