Fixed points in uniform spaces

Abstract

We improve Angelov’s fixed point theorems of Φ-contractions and j-nonexpansive maps in uniform spaces and investigate their fixed point sets using the concept of virtual stability. Some interesting examples and an application to the solution of a certain integral equation in locally convex spaces are also given.

1 Introduction

In 1987 [1], Angelov introduced the notion of Φ-contractions on Hausdorff uniform spaces, which simultaneously generalizes the well-known Banach contractions on metric spaces as well as γ-contractions [2] on locally convex spaces, and he proved the existence of their fixed points under various conditions. Later in 1991 [3], he also extended the notion of Φ-contractions to j-nonexpansive maps and gave some conditions to guarantee the existence of their fixed points. However, there is a minor flaw in his proof of Theorem 1 [3] where the surjectivity of the map j is implicitly used without any prior assumption. Additionally, we observe that such a map j can be naturally replaced by a multi-valued map J to obtain a more general, yet interesting, notion of J-nonexpansiveness. Therefore, in this work, we aim to correct and simplify the proof of Theorem 1 [3] as well as extend the notion of j-nonexpansive maps to J-nonexpansive maps and investigate the existence of their fixed points. Then we introduce J-contractions, a special kind of J-nonexpansive maps, that play the similar role as Banach contractions in yielding the uniqueness of fixed points. With the notion of J-contractions, we are able to recover results on Φ-contractions proved in [1] as well as present some new fixed point theorems in which one of them naturally leads to a new existence theorem for the solution of a certain integral equation in locally convex spaces. Finally, we prove that, under a mild condition, J-nonexpansive maps are always virtually stable in the sense of [4] and hence their fixed point sets are retracts of their convergence sets. An example of a virtually stable J-nonexpansive map whose fixed point set is not convex is also given.

2 Fixed point theorems

For any set S, we will use ${\mathcal{P}}^f(S)$ and $|S|$ to denote the set of all nonempty finite subsets of S and the cardinality of S, respectively. Let $(E,\mathcal{A})$ be a Hausdorff uniform space whose uniformity is generated by a saturated family of pseudometrics $\mathcal{A}=\{d_{\alpha }:\alpha \in A\}$ indexed by A, $\mathrm{\varnothing }\ne X\subseteq E$, and $J:A\to {\mathcal{P}}^f(A)$. Interested readers should consult [5] for general topological concepts of uniform spaces, and [6] for the complete development of fixed point theory in uniform spaces that motivates this work. We first give the definition of a J-nonexpansive map as follows:

Definition 2.1 A self-map $T:X\to X$ is said to be J-nonexpansive if for each $\alpha \in A$,

$$d_{\alpha }(Tx,Ty)\le \sum _{\beta \in J(\alpha )}{d}_{\beta }(x,y),$$

for any $x,y\in X$.

Example 2.2 Let $1<p<\mathrm{\infty }$, $E={\ell }_p$ be equipped with the weak topology, and $T:{\ell }_p\to {\ell }_p$ be defined by

$$T(x_1,x_2,\dots )=(\frac{|x_1+x_3|}{3},\frac{|x_2+x_4|}{3},x_3,x_4,\dots ),$$

for any $(x_1,x_2,\dots )\in {\ell }_p$. Then $\mathcal{A}=\{|f|:f\in {\ell }_p^{\ast }\}$, where $|f|(x)=|f(x)|$ for each $x\in {\ell }_p$.

By Theorem 4.6 in [7], we have

$$\begin{array}{rl}|f(Tx-Ty)|\le & |\frac{\parallel f\parallel }{3}(x_1-y_1+x_3-y_3)|+|\frac{\parallel f\parallel }{3}(x_2-y_2+x_4-y_4)|\\ +|\parallel f\parallel (x_1-y_1)|+|\parallel f\parallel (x_2-y_2)|+|f(x-y)|,\end{array}$$

for each $f\in {\ell }_p^{\ast }$, $x=(x_1,x_2,\dots )\in {\ell }_p$ and $y=(y_1,y_2,\dots )\in {\ell }_p$. Here, $\parallel f\parallel =sup\{|f(x)|:x\in X,\parallel x\parallel \le 1\}$.

By letting $J:{\ell }_p^{\ast }\to {\mathcal{P}}^f({\ell }_p^{\ast })$ be defined by $J(f)=\{|f|,|g_1|,|g_2|,|g_3|,|g_4|\}$, for each $f\in {\ell }_p^{\ast }$, where

$$g_1(x)=\frac{\parallel f\parallel }{3}(x_1+x_3),g_2(x)=\frac{\parallel f\parallel }{3}(x_2+x_4),g_3(x)=\parallel f\parallel x_1,g_4(x)=\parallel f\parallel x_2,$$

for each $x=(x_1,x_2,\dots )\in {\ell }_p$, it follows that T is J-nonexpansive.

The above definition of a J-nonexpansive map clearly extends the definition of a j-nonexpansive map in [3]. Before giving general existence criteria for fixed points of J-nonexpansive maps, we need the following notations. For each $\alpha \in A$ and $n\in \mathbf{N}$, we let

$$A_n(\alpha )=\{({\alpha }_1,\dots ,{\alpha }_n):{\alpha }_1\in J(\alpha )\text{ and }{\alpha }_k\in J({\alpha }_{k-1})\text{ for }1<k\le n\}$$

and

$$A(\alpha )=\{({\alpha }_1,{\alpha }_2,\dots ):{\alpha }_1\in J(\alpha )\text{ and }{\alpha }_k\in J({\alpha }_{k-1})\text{ for }k>1\}.$$

When there is no ambiguity, we will denote an element of both $A_n(\alpha )$ and $A(\alpha )$ simply by $({\alpha }_k)$. Notice that for each $\alpha \in A$ and $n\in \mathbf{N}$, the sets $A_n(\alpha )$ and ${\pi }_n(A(\alpha ))$ are finite, where ${\pi }_n$ denotes the n th coordinate projection $({\alpha }_k)\mapsto {\alpha }_n$.

Lemma 2.3 Every J-nonexpansive map is continuous.

Proof Suppose $T:X\to X$ is J-nonexpansive. Let $x\in X$ and $(x_{\gamma })$ be a net in X converging to x. Then for each $\alpha \in A$, we have

$$d_{\alpha }(T{x}_{\gamma },Tx)\le \sum _{\beta \in J(\alpha )}{d}_{\beta }(x_{\gamma },x).$$

Since $(x_{\gamma })$ converges to x, $(d_{\beta }(x_{\gamma },x))$ converges to 0 for any $\beta \in A$, and this proves the continuity of T. □

Theorem 2.4 Let $T:X\to X$ be J-nonexpansive whose $A(\alpha )$ is finite for any $\alpha \in A$. Then T has a fixed point in X if and only if there exists $x_0\in X$ such that

• (i) the sequence $(T^n{x}_0)$ has a convergence subsequence, and

• (ii) for each $\alpha \in A$ and $({\alpha }_k)\in A(\alpha )$, ${lim}_{n\to \mathrm{\infty }}{d}_{{\alpha }_n}(x_0,T{x}_0)=0$.

Proof (⇒): It is obvious by letting $x_0$ be a fixed point of T.

(⇐): Suppose that $(T^{n_i}{x}_0)$ converges to some $z\in X$. Let $\alpha \in A$ and $({\alpha }_k)\in A(\alpha )$. Then ${lim}_{i\to \mathrm{\infty }}{d}_{\alpha }(z,T^{n_i}{x}_0)=0$ and ${lim}_{n\to \mathrm{\infty }}{d}_{{\alpha }_n}(x_0,T{x}_0)=0$. We can choose $N\in \mathbf{N}$ sufficiently large so that $d_{\alpha }(z,T^{n_i}{x}_0)<ϵ$ and $d_{{\alpha }_{n_i}}(x_0,T{x}_0)<ϵ$, for all $i\ge N$. It follows that

$$\begin{array}{rl}{d}_{\alpha }(z,T^{n_i+1}{x}_0)& \le d_{\alpha }(z,T^{n_i}{x}_0)+d_{\alpha }(T^{n_i}{x}_0,T^{n_i}(T{x}_0))\\ \le d_{\alpha }(z,T^{n_i}{x}_0)+\sum _{({\alpha }_k)\in A_{n_i}(\alpha )}{d}_{{\alpha }_{n_i}}(x_0,T{x}_0)\\ \le (1+|A(\alpha )|)ϵ.\end{array}$$

Since α is arbitrary, $(T^{n_i+1}{x}_0)$ converges to z. By the continuity of T, we have $z=Tz$ and hence z is a fixed point of T. □

As a corollary of the previous theorem, we immediately obtain Theorem 1 [3], with a corrected and simplified proof, as follows:

Corollary 2.5 Let $T:X\to X$ be a j-nonexpansive map. If there exists $x_0\in X$ such that

• (i) the sequence $(T^n{x}_0)$ has a convergence subsequence, and

• (ii) for every $\alpha \in A$, ${lim}_{n\to \mathrm{\infty }}{d}_{j^n(\alpha )}(x_0,T{x}_0)=0$,

then T has a fixed point.

Proof The proof follows directly from the previous theorem by considering the map $J:\alpha \mapsto \{j(\alpha )\}$. Notice that $A(\alpha )=\{(j^n(\alpha ))\}$ which is finite. □

We will now consider a special kind of J-nonexpansive maps that resemble Banach contractions in yielding the uniqueness of fixed points. Let Φ denote the family of all functions $\varphi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ satisfying the following conditions:

(Φ1) ϕ is non-decreasing and continuous from the right, and

(Φ2) $\varphi (t)<t$ for any $t>0$.

Notice that $\varphi (0)=0$, and we will call $\varphi \in \mathrm{\Phi }$ subadditive if $\varphi (t_1+t_2)\le \varphi (t_1)+\varphi (t_2)$ for all $t_1,t_2\ge 0$. Also, for a subfamily ${\{{\varphi }_{\alpha }\}}_{\alpha \in A}$ of Φ, $\alpha \in A$, $({\alpha }_k)\in A_n(\alpha )$ and $i\le n$, we let

$${\varphi }_{({\alpha }_k)}^i={\varphi }_{{\alpha }_1}\circ \cdots \circ {\varphi }_{{\alpha }_i}.$$

Definition 2.6 A self-map $T:X\to X$ is said to be a J-contraction if for each $\alpha \in A$, there exists ${\varphi }_{\alpha }\in \mathrm{\Phi }$ such that

$$d_{\alpha }(Tx,Ty)\le \sum _{\beta \in J(\alpha )}{\varphi }_{\alpha }(d_{\beta }(x,y)),$$

for any $x,y\in X$, and ${\varphi }_{\alpha }$ is subadditive whenever $|J(\alpha )|>1$.

Clearly, a Φ-contraction as defined in [1] is a J-contraction and a J-contraction is always J-nonexpansive. A natural example of a J-contraction can be obtained by adding (finitely many) appropriate Φ-contractions as shown in the following example.

Example 2.7 Given two Φ-contractions $T_1:X\to X$ and $T_2:X\to X$ as defined [1]. Then there exist $j_1,j_2:A\to A$, and for each $\alpha \in A$, there exist ${\varphi }_{1,\alpha },{\varphi }_{2,\alpha }\in \mathrm{\Phi }$ such that

$$d_{\alpha }(T_{1}x,T_{1}y)\le {\varphi }_{1,\alpha }(d_{j_1(\alpha )}(x,y))\text{and}{d}_{\alpha }(T_{2}x,T_{2}y)\le {\varphi }_{2,\alpha }(d_{j_2(\alpha )}(x,y)),$$

for any $\alpha \in A$ and $x,y\in X$. If for each $\alpha \in A$, $j_1(\alpha )\ne j_2(\alpha )$ and there is a subadditive ${\varphi }_{3,\alpha }\in \mathrm{\Phi }$ so that ${\varphi }_{1,\alpha }(t)\le {\varphi }_{3,\alpha }(t)$ and ${\varphi }_{2,\alpha }(t)\le {\varphi }_{3,\alpha }(t)$ for any $t\ge 0$, then the map $H=T_1+T_2$ is clearly a J-contraction with respect to $J(\alpha )=\{j_1(\alpha ),j_2(\alpha )\}$ and ${\varphi }_{H,\alpha }={\varphi }_{3,\alpha }$ for any $\alpha \in A$.

Lemma 2.8 If $T:X\to X$ is a J-contraction. Then we have

$$d_{\alpha }(T^{n}x,T^{n}y)\le \sum _{({\alpha }_k)\in A_n(\alpha )}{\varphi }_{\alpha }\circ {\varphi }_{({\alpha }_k)}^{n-1}(d_{{\alpha }_n}(x,y)),$$

for any $\alpha \in A$, $n\ge 2$ and $x,y\in X$.

Proof Recall that ${\varphi }_{\alpha }$ is assumed to be subadditive whenever $|J(\alpha )|>1$. Then, for any $\alpha \in A$, $n\ge 2$ and $x,y\in X$, we clearly have

$$\begin{array}{rl}{d}_{\alpha }(T^{n}x,T^{n}y)\le & \sum _{{\alpha }_1\in J(\alpha )}{\varphi }_{\alpha }\left(d_{{\alpha }_1}(T^{n-1}x,T^{n-1}y)\right)\\ \le & \sum _{{\alpha }_1\in J(\alpha )}{\varphi }_{\alpha }\left(\sum _{{\alpha }_2\in J({\alpha }_1)}{\varphi }_{{\alpha }_1}\left(d_{{\alpha }_2}(T^{n-2}x,T^{n-2}y)\right)\right)\\ \le & \sum _{{\alpha }_1\in J(\alpha )}\sum _{{\alpha }_2\in J({\alpha }_1)}{\varphi }_{\alpha }\circ {\varphi }_{{\alpha }_1}\left(d_{{\alpha }_2}(T^{n-2}x,T^{n-2}y)\right)\\ ⋮\\ \le & \sum _{{\alpha }_1\in J(\alpha )}\sum _{{\alpha }_2\in J({\alpha }_1)}\cdots \sum _{{\alpha }_n\in J({\alpha }_{n-1})}{\varphi }_{\alpha }\circ {\varphi }_{{\alpha }_1}\circ \cdots \circ {\varphi }_{{\alpha }_{n-1}}(d_{{\alpha }_n}(x,y))\\ =& \sum _{({\alpha }_k)\in A_n(\alpha )}{\varphi }_{\alpha }\circ {\varphi }_{({\alpha }_k)}^{n-1}(d_{{\alpha }_n}(x,y)).\end{array}$$

 □

We now obtain some general criteria for the existence of fixed points of J-contractions.

Theorem 2.9 Suppose X is sequentially complete and $T:X\to X$ is a J-contraction whose $A(\alpha )$ is finite for any $\alpha \in A$. If T satisfies the following conditions:

• (i) for each $\alpha \in A$, there exists $c_{\alpha }\in \mathrm{\Phi }$ such that

  $${\varphi }_{{\alpha }_i}(t)\le c_{\alpha }(t),$$

  for any $({\alpha }_k)\in A(\alpha )$, $i\in \mathbf{N}$, $t\ge 0$, and

• (ii) there exists $x_0\in X$ such that for each $\alpha \in A$, $({\alpha }_k)\in A(\alpha )$, $i\in \mathbf{N}$ and $n,m\in \mathbf{N}$, we have

  $$d_{{\alpha }_i}(T^n{x}_0,T^m{x}_0)\le M_{\alpha }(x_0),$$

  for some $M_{\alpha }(x_0)\in \mathbf{R}$,

then T has a fixed point. Moreover, if for each $\alpha \in A$ and $x,y\in X$, there exists $F_{\alpha }(x,y)\in {\mathbf{R}}_0^{+}$ such that

$$d_{{\alpha }_i}(x,y)\le F_{\alpha }(x,y),$$

for all $({\alpha }_k)\in A(\alpha )$ and $i\in \mathbf{N}$, then the fixed point of T is unique.

Proof For each $\alpha \in A$ and $n,m,N\in \mathbf{N}$, since ${\varphi }_{\alpha }$ is non-decreasing, we have

$$\begin{array}{rl}{d}_{\alpha }(T^n{x}_0,T^m{x}_0)& \le \sum _{{\alpha }_1\in J(\alpha )}{\varphi }_{\alpha }\left(d_{{\alpha }_1}(T^{n-1}{x}_0,T^{m-1}{x}_0)\right)\\ \le \sum _{{\alpha }_1\in J(\alpha )}{\varphi }_{\alpha }(sup\{d_{{\alpha }_1}(T^{n-1}{x}_0,T^{m-1}{x}_0):n,m\ge N\}),\end{array}$$

and by letting $h_N^{\alpha }:=sup\{d_{\alpha }(T^n{x}_0,T^m{x}_0):n,m\ge N\}$, it follows that

$$\begin{array}{rl}{h}_N^{\alpha }\le & \sum _{{\alpha }_1\in J(\alpha )}{\varphi }_{\alpha }(sup\{d_{{\alpha }_1}(T^{n-1}{x}_0,T^{m-1}{x}_0):n,m\ge N\})\\ =& \sum _{{\alpha }_1\in J(\alpha )}{\varphi }_{\alpha }\left(h_{N-1}^{{\alpha }_1}\right)\\ \le & \sum _{{\alpha }_1\in J(\alpha )}\sum _{{\alpha }_2\in J({\alpha }_1)}{\varphi }_{\alpha }\left({\varphi }_{{\alpha }_1}\left(h_{N-2}^{{\alpha }_2}\right)\right)\\ ⋮\\ \le & \sum _{({\alpha }_k)\in A_{N-1}(\alpha )}{\varphi }_{\alpha }\circ {\varphi }_{({\alpha }_k)}^{N-1}\left(h_1^{{\alpha }_{N-1}}\right)\\ \le & \sum _{({\alpha }_k)\in A_{N-1}(\alpha )}{c}_{\alpha }^N(M_{\alpha }(x_0))\\ \le & |A(\alpha )|c_{\alpha }^N(M_{\alpha }(x_0)).\end{array}$$

(1)

Also, for a given $t\ge 0$, since $0\le c_{\alpha }^N(t)=c_{\alpha }(c_{\alpha }^{N-1}(t))<c_{\alpha }^{N-1}(t)$, we have ${lim}_{N\to \mathrm{\infty }}{c}_{\alpha }^N(t)=r_{\alpha }$ for some $r_{\alpha }\ge 0$. Since $c_{\alpha }$ is right continuous, we have ${lim}_{N\to \mathrm{\infty }}{c}_{\alpha }(c_{\alpha }^{N-1}(t))=c_{\alpha }(r_{\alpha })$, and hence $c_{\alpha }(r_{\alpha })=r_{\alpha }$. Therefore, $r_{\alpha }=0$. By (1), it follows that ${lim}_{N\to \mathrm{\infty }}{h}_N^{\alpha }=0$. Since α is arbitrary, $(T^k{x}_0)$ is a Cauchy sequence and, by sequential completeness, converges to some $z\in X$. Notice also that z must be a fixed point of T by continuity.

Now suppose that for each $x,y\in X$ and $\alpha \in A$, there exists $F_{\alpha }(x,y)\in {\mathbf{R}}_0^{+}$ such that $d_{{\alpha }_i}(x,y)\le F_{\alpha }(x,y)$ for all $({\alpha }_k)\in A(\alpha )$ and $i\in \mathbf{N}$. If x, y are fixed points of T, then by Lemma 2.8, we have for each $\alpha \in A$ and $n\in \mathbf{N}$,

$$\begin{array}{rl}{d}_{\alpha }(x,y)& =d_{\alpha }(T^{n}x,T^{n}y)\\ \le \sum _{({\alpha }_k)\in A_n(\alpha )}{\varphi }_{\alpha }\circ {\varphi }_{({\alpha }_k)}^{n-1}(d_{{\alpha }_n}(x,y))\\ \le \sum _{({\alpha }_k)\in A_n(\alpha )}{c}_{\alpha }^n(d_{{\alpha }_n}(x,y))\\ \le |A(\alpha )|c_{\alpha }^n(F_{\alpha }(x,y)).\end{array}$$

Since ${lim}_{n\to \mathrm{\infty }}{c}_{\alpha }^n(F_{\alpha }(x,y))=0$, we must have $x=y$. □

As a corollary of the previous theorem, we immediately obtain Theorem 1 in [1] as follows.

Corollary 2.10 Suppose X is a bounded and sequentially complete subset of E and $T:X\to X$ is Φ-contraction. If

• (i) for each $\alpha \in A$, there exists $c_{\alpha }\in \mathrm{\Phi }$ such that ${\varphi }_{j^n(\alpha )}(t)\le c_{\alpha }(t)$ for all $n\in \mathbf{N}$ and $t\ge 0$,

• (ii) for each $n\in \mathbf{N}$, $sup\{d_{j^n(\alpha )}(x,y):x,y\in X\}\le p(\alpha ):=sup\{d_{\alpha }(x,y):x,y\in X\}$,

then there exists a unique fixed point $x\in X$ of T.

Proof For each $x_0,x,y\in X$, $\alpha \in A$, $({\alpha }_k)\in A(\alpha )$ and $i,m,n\in \mathbf{N}$, by letting $J(\alpha )=\{j(\alpha )\}$ and $M_{\alpha }(x_0)=p(\alpha )=F_{\alpha }(x,y)$, we have $A(\alpha )=\{(\alpha ,j(\alpha ),j^2(\alpha ),\dots ,j^k(\alpha ),\dots )\}$, $d_{{\alpha }_i}(T^m{x}_0,T^n{x}_0)=d_{j^i(\alpha )}(T^m{x}_0,T^n{x}_0)\le M_{\alpha }(x_0)$ and $d_{{\alpha }_i}(x,y)\le F_{\alpha }(x,y)$. Hence, by Theorem 2.9, T has a unique fixed point. □

Theorem 2.11 Suppose X is sequentially complete and $T:X\to X$ is a self-map satisfying: for each $\alpha \in A$ and $k\in \mathbf{N}$, there exist ${\varphi }_{\alpha ,k}\in \mathrm{\Phi }$, a finite set $D_{\alpha ,k}$ and a map $P_{\alpha ,k}:D_{\alpha ,k}\to A$ such that

$$d_{\alpha }(T^{k}x,T^{k}y)\le \sum _{\gamma \in D_{\alpha ,k}}{\varphi }_{\alpha ,k}(d_{P_{\alpha ,k}(\gamma )}(x,y)),$$

for any $x,y\in X$.

• 1. If there exists $x_0\in X$ such that for each $\alpha \in A$ there exists $M_{\alpha }(x_0)\in {\mathbf{R}}_0^{+}$ so that ${\sum }_{k\in \mathbf{N}}|D_{\alpha ,k}|{\varphi }_{\alpha ,k}(M_{\alpha }(x_0))<\mathrm{\infty }$ and

  $$d_{P_{\alpha ,k}(\gamma )}(x_0,T{x}_0)\le M_{\alpha }(x_0),$$

for all $k\in \mathbf{N}$ and $\gamma \in D_{\alpha ,k}$, then T has a fixed point in X.

• 2. If for each $\alpha \in A$ and $x,y\in X$, there exists $F_{\alpha }(x,y)\in {\mathbf{R}}_0^{+}$ such that ${\sum }_{k\in \mathbf{N}}|D_{\alpha ,k}|{\varphi }_{\alpha ,k}(F_{\alpha }(x,y))<\mathrm{\infty }$ and

  $$d_{P_{\alpha ,k}(\gamma )}(x,y)\le F_{\alpha }(x,y),$$

for all $k\in \mathbf{N}$ and $\gamma \in D_{\alpha ,k}$, then T has a unique fixed point in X and, for any $x\in X$, the sequence $(T^{n}x)$ converges to the fixed point of T.

Proof First notice that T is clearly a J-contraction.

• 1. For any $\alpha \in A$ and $m>n\in \mathbf{N}$, we have

  $$\begin{array}{rl}{d}_{\alpha }(T^n{x}_0,T^m{x}_0)& \le \sum _{n\le i<m}{d}_{\alpha }(T^i{x}_0,T^{i+1}{x}_0)\\ \le \sum _{n\le i<m}\sum _{\gamma \in D_{\alpha },i}{\varphi }_{\alpha ,i}(d_{P_{\alpha ,i}(\gamma )}(x_0,T{x}_0))\\ \le \sum _{n\le i<m}|D_{\alpha ,i}|{\varphi }_{\alpha ,i}(M_{\alpha }(x_0)).\end{array}$$

Also, since ${\sum }_{k\in \mathbf{N}}|D_{\alpha ,k}|{\varphi }_{\alpha ,k}(M_{\alpha }(x_0))<\mathrm{\infty }$, $(T^k{x}_0)$ is a Cauchy sequence and converges to a fixed point of T by the sequential completeness of X and the continuity of T.

• 2. For any $x\in X$, $\alpha \in A$ and $m>n\in \mathbf{N}$, we have

  $$\begin{array}{rl}{d}_{\alpha }(T^{n}x,T^{m}x)& \le \sum _{n\le i<m}{d}_{\alpha }(T^{i}x,T^{i+1}x)\\ \le \sum _{n\le i<m}\sum _{\gamma \in D_{\alpha },i}{\varphi }_{\alpha ,i}(d_{P_{\alpha ,i}(\gamma )}(x,Tx))\\ \le \sum _{n\le i<m}|D_{\alpha ,i}|{\varphi }_{\alpha ,i}(F_{\alpha }(x,Tx)).\end{array}$$

Also, since ${\sum }_{k\in \mathbf{N}}|D_{\alpha ,k}|{\varphi }_{\alpha ,k}(F_{\alpha }(x,Tx))<\mathrm{\infty }$, $(T^{k}x)$ is a Cauchy sequence and converges to a fixed point of T by the sequential completeness of X and the continuity of T.

Now, since for each $\alpha \in A$, $k\in \mathbf{N}$ and $x,y\in F(T)$,

$$\begin{array}{rl}{d}_{\alpha }(x,y)& =d_{\alpha }(T^{k}x,T^{k}y)\\ \le \sum _{\gamma \in D_{\alpha ,k}}{\varphi }_{\alpha ,k}(d_{P_{\alpha ,k}(\gamma )}(x,y))\\ \le \sum _{\gamma \in D_{\alpha ,k}}{\varphi }_{\alpha ,k}(F_{\alpha }(x,y))\\ =|D_{\alpha ,k}|{\varphi }_{\alpha ,k}(F_{\alpha }(x,y)),\end{array}$$

and ${lim}_{k\to \mathrm{\infty }}|D_{\alpha ,k}|{\varphi }_{\alpha ,k}(F_{\alpha }(x,y))=0$, we have the uniqueness. □

Corollary 2.12 (Theorem 5 in [1])

Let us suppose

• (i) for each $\alpha \in A$ and $n>0$, there exist ${\varphi }_{\alpha ,n}\in \mathrm{\Phi }$ and $j(\alpha ,n)\in A$ such that

  $$d_{\alpha }(T^{n}x,T^{n}y)\le {\varphi }_{\alpha ,n}(d_{j(\alpha ,n)}(x,y)),$$

  for any $x,y\in X$,

• (ii) there exists $x_0\in X$ such that $d_{j(\alpha ,n)}(x_0,T{x}_0)\le p(\alpha )<\mathrm{\infty }$ ($n=1,2,\dots$), ${\sum }_n{\varphi }_{\alpha ,n}(p(\alpha ))<\mathrm{\infty }$ and $j:A\times \mathbf{N}\to A$.

Then T has at least one fixed point in X.

Proof By letting $D_{\alpha ,k}=\{j(\alpha ,k)\}$ for any $\alpha \in A$ and $k\in \mathbf{N}$ and $P_{\alpha ,k}={\pi }_k{|}_{D_{\alpha ,k}}$. Then for each $i\in \mathbf{N}$, we have $|D_{\alpha ,i}|=1$ and $M_{\alpha }(x_0)=p(\alpha )$. By Theorem 2.11(2), T has a fixed point. □

Theorem 2.13 Suppose X is sequentially complete and $T:X\to X$ is a J-contraction whose $A(\alpha )$ is finite for each $\alpha \in A$. If, for each $\alpha \in A$, there exists $c_{\alpha }\in \mathrm{\Phi }$ satisfying:

• (i) $c_{\alpha }(t)/t$ is non-decreasing in t,

• (ii) ${\varphi }_{{\alpha }_n}(t)\le c_{\alpha }(t)$ for any $({\alpha }_k)\in A(\alpha )$, $n\in \mathbf{N}$ and $t\in [0,\mathrm{\infty })$, and

• (iii) there exist $x_0\in X$ and $M_{\alpha }(x_0)\in {\mathbf{R}}^{+}$ such that $d_{{\alpha }_n}(x_0,T{x}_0)\le M_{\alpha }(x_0)$ for any $({\alpha }_k)\in A(\alpha )$ and $n\in \mathbf{N}$,

then T has a fixed point in X.

Proof Let $D_{\alpha ,i}=A_i(\alpha )$, $P_{\alpha ,i}(({\alpha }_k))={\alpha }_i$, and ${\varphi }_{\alpha ,i}(t)=c_{\alpha }^i(t)$ for any $i\in \mathbf{N}$, $\alpha \in A$, $({\alpha }_k)\in A_i(\alpha )$, and $t\in [0,\mathrm{\infty })$. Then for any $\alpha \in A$ and $x,y\in X$, we have, by Lemma 2.8,

$$\begin{array}{rl}{d}_{\alpha }(T^{i}x,T^{i}y)& \le \sum _{({\alpha }_k)\in A_i(\alpha )}{\varphi }_{\alpha }\circ {\varphi }_{({\alpha }_k)}^{i-1}(d_{{\alpha }_i}(x,y))\\ \le \sum _{({\alpha }_k)\in A_i(\alpha )}{c}_{\alpha }^i(d_{{\alpha }_i}(x,y))\\ =\sum _{({\alpha }_k)\in D_{\alpha ,i}}{\varphi }_{\alpha ,i}(d_{P_{\alpha ,i}(({\alpha }_k))}(x,y)).\end{array}$$

Since

$$\frac{|D_{\alpha ,i+1}|{\varphi }_{\alpha ,i+1}(M_{\alpha }(x_0))}{|D_{\alpha ,i}|{\varphi }_{\alpha ,i}(M_{\alpha }(x_0))}=\frac{|A_{i+1}(\alpha )|c_{\alpha }^{i+1}(M_{\alpha }(x_0))}{|A_i(\alpha )|c_{\alpha }^i(M_{\alpha }(x_0))}\le \frac{c_{\alpha }(c_{\alpha }^i(M_{\alpha }(x_0)))}{c_{\alpha }^i(M_{\alpha }(x_0))}\le \frac{c_{\alpha }(M_{\alpha }(x_0))}{M_{\alpha }(x_0)}<1,$$

for any $i\in \mathbf{N}$, we have ${\sum }_{i\in \mathbf{N}}|D_{\alpha ,i}|{\varphi }_{\alpha ,i}(M_{\alpha }(x_0))<\mathrm{\infty }$. Then by Theorem 2.11(1), T has a fixed point. □

Corollary 2.14 (Theorem 2 in [1])

Let us suppose

• (i) the operator $T:X\to X$ is a Φ-contraction,

• (ii) for each $\alpha \in A$ there exists a Φ-function $c_{\alpha }$ such that ${\varphi }_{j^n(\alpha )}(t)\le c_{\alpha }(t)$ for all $n\in \mathbf{N}$ and $c_{\alpha }(t)/t$ is non-decreasing,

• (iii) there exists an element $x_0\in X$ such that $d_{j^n(\alpha )}(x_0,T{x}_0)\le p(\alpha )<\mathrm{\infty }$ ($n=1,2,\dots$).

Then T has at least one fixed point in X.

Proof By letting $J(\alpha )=\{j(\alpha )\}$ for any $\alpha \in A$ and $M_{\alpha }(x_0)=p(\alpha )$. Then $|A(\alpha )|=1$, and, by Theorem 2.13, T has a fixed point. □

Example 2.15 Given a sequentially complete locally convex space X, and two Φ-contractions $T_1,T_2:X\to X$; i.e., there exist $j_1,j_2:A\to A$, and for each $\alpha \in A$, there exist ${\varphi }_{1,\alpha },{\varphi }_{2,\alpha }\in \mathrm{\Phi }$ such that

$$d_{\alpha }(T_{1}x,T_{1}y)\le {\varphi }_{1,\alpha }(d_{j_1(\alpha )}(x,y))\text{and}{d}_{\alpha }(T_{2}x,T_{2}y)\le {\varphi }_{2,\alpha }(d_{j_2(\alpha )}(x,y)),$$

for any $\alpha \in A$ and $x,y\in X$. Suppose further that

• (i) $j_1^{n+1}=j_2^n\circ j_1$ and $j_1^n\circ j_2=j_2^{n+1}$ for any $n\in \mathbf{N}$,

• (ii) for each $\alpha \in A$, ${\varphi }_{1,\alpha }(t)=c_1(\alpha )t$ and ${\varphi }_{2,\alpha }(t)=c_2(\alpha )t$ for some $c_1(\alpha )+c_2(\alpha )\in (0,1)$, and

• (iii) there exists $x_0\in X$ such that $d_{j_1^n(\alpha )}(x_0,T_1{x}_0)\le p_1(x_0,\alpha )<\mathrm{\infty }$ and $d_{j_2^n(\alpha )}(x_0,T_2{x}_0)\le p_2(x_0,\alpha )<\mathrm{\infty }$ for any $\alpha \in A$ and $n=1,2,\dots$.

Then $H=\frac{T_1+T_2}{2}$ is a J-contraction with $J(\alpha )=\{j_1(\alpha ),j_2(\alpha )\}$ and ${\varphi }_{H,\alpha }(t)=(c_1(\alpha )+c_2(\alpha ))t$. Also, by (i) and (iii), we have $|A(\alpha )|=2<\mathrm{\infty }$ and

$$d_{{\alpha }_n}(x_0,H{x}_0)\le \frac{d_{{\alpha }_n}(x_0,T_1{x}_0)+d_{{\alpha }_n}(x_0,T_2{x}_0)}{2}\le \frac{p_1(x_0,\alpha )+p_2(x_0,\alpha )}{2}.$$

Hence, H satisfies all conditions in Theorem 2.13, and it has a fixed point in X. Notice that H may not be a Φ-contraction, by choosing $j_1$ and $j_2$ so that $d_{j_1(\alpha )}+d_{j_2(\alpha )}\notin \mathcal{A}$ for some $\alpha \in A$, and hence Theorem 2 in [1] cannot be applied.

We now end this section by giving an application to the solution of a certain integral equation in locally convex spaces.

Example 2.16 Following terminologies in [8], let X be an $S$-space topologized by the family of seminorms $\{{|\cdot |}_{\alpha }:\alpha \in A\}$ and $C([0,T];X)$ the space of all continuous functions from $[0,T]$ into X topologized by the family of seminorms $\{{\parallel \cdot \parallel }_{\alpha }:\alpha \in A\}$, where ${\parallel x\parallel }_{\alpha }:={sup}_{t\in [0,T]}{|x(t)|}_{\alpha }$ for any $x\in C([0,T];X)$. Let $L(X)$ denote the set of all continuous linear operators on X,

$$L_0(X)=\{l\in L(X):\mathrm{\forall }\alpha \in A,\mathrm{\exists }M(\alpha )>0,\mathrm{\forall }x\in X,{|lx|}_{\alpha }\le M(\alpha ){|x|}_{\alpha }\},$$

and let ${\{S(t)\}}_{t\ge 0}$ be a $C_0$-semigroup on X such that $S:[0,\mathrm{\infty })\to L_0(X)$ is locally bounded.

Now, we replace H3 and H5 in [8] by conditions (N1), (N2) and (N3) as follows:

(N1) $B:C([0,T];X)\to C([0,T];X)$ is an operator such that there exists $J_B:A\to {\mathcal{P}}^f(A)$ so that for any $\alpha \in A$, there is $k_{\alpha ,B}\in L_{\mathrm{loc}}^1([0,T];[0,\mathrm{\infty }))$ such that

$${|Bx(t)-By(t)|}_{\alpha }\le k_{\alpha ,B}(t)\sum _{\beta \in J_B(\alpha )}{|x(t)-y(t)|}_{\beta },$$

for any $x,y\in C([0,T];X)$.

(N2) $f:[0,T]\times X\times X\to X$ is continuous and there exist $J_f:A\to {\mathcal{P}}^f(A)$ and $K_f\in L_{\mathrm{loc}}^1([0,T];[0,\mathrm{\infty }))$ such that for each $\alpha \in A$,

$${|f(t,u_1,v_1)-f(t,u_2,v_2)|}_{\alpha }\le K_f(t)(\sum _{\beta \in J_f(\alpha )}{|u_1-u_2|}_{\beta }+{|v_1-v_2|}_{\alpha }),$$

for any $t\in [0,T]$ and $u_1,u_2,v_1,v_2\in X$,

(N3) $K_f\cdot k_{\alpha ,B}\in L_{\mathrm{loc}}^1([0,T];[0,\mathrm{\infty }))$.

Consider the integral equation

$$x(t)=S(t)x_0+{\int }_0^{t}S(t-s)f(s,x(s),Bx(s))ds;t\in [0,T]$$

(2)

whose solution is closely related to the mild solution of the differential equation

$$\frac{dx}{dt}=ax+f(t,x(t),Bx(t)),$$

where a denotes the infinitesimal generator of ${\{S(t)\}}_{t\ge 0}$.

We now define an operator G on $C_{x_0}([0,T];X)=\{x\in C([0,T];X):x(0)=x_0\}$ by

$$(Gx)(t)=S(t)x_0+{\int }_0^{t}S(t-s)f(s,x(s),Bx(s))ds,$$

for any $x\in C_{x_0}([0,T];X)$. Following the proof of Theorem 3 in [8] and for each $t>0$, $S(t)\in L_0(X)$, then we can show that, for any $\alpha \in A$, there exists $M(\alpha )>0$ such that

$${\parallel Gx-Gy\parallel }_{\alpha }\le H_{\alpha }(\sum _{\beta \in J_f(\alpha )}{\parallel x-y\parallel }_{\beta }+\sum _{\beta \in J_B(\alpha )}{\parallel x-y\parallel }_{\beta }),$$

where $H_{\alpha }=max\{M(\alpha ){\int }_0^T{K}_f(s)ds,M(\alpha ){\int }_0^T{K}_f(s)k_{\alpha ,B}(s)ds\}$. It is easy to see that if for each $\alpha \in A$, $H_{\alpha }\in (0,1)$ and $J_f(\alpha )\cap J_B(\alpha )=\mathrm{\varnothing }$, then G is a J-contraction with $J_G(\alpha )=J_f(\alpha )\cup J_B(\alpha )$.

In particular, if we assume further that for each $\alpha \in A$, $J_f(\alpha )=\{\alpha \}$, $|J_B(\alpha )|=1$ such that $J_B\circ J_B=J_B$ and $H_{\alpha }=H_{J_B(\alpha )}<\frac{1}{2}$. Then for any $k\in \mathbf{N}$ and $x,y\in C_{x_0}([0,T];X)$, we have

$$\begin{array}{rl}{\parallel G^{k}x-G^{k}y\parallel }_{\alpha }& \le H_{\alpha }^k{\parallel x-y\parallel }_{\alpha }+\left(\sum _{i=1}^k{(2H_{J_B(\alpha )})}^{k-i}{H}_{\alpha }^i\right){\parallel x-y\parallel }_{J_B(\alpha )}\\ =H_{\alpha }^k{\parallel x-y\parallel }_{\alpha }+\left(\sum _{i=1}^k{2}^{k-i}{H}_{\alpha }^k\right){\parallel x-y\parallel }_{J_B(\alpha )}\\ \le 2^{k-1}{H}_{\alpha }^k({\parallel x-y\parallel }_{\alpha }+\sum _{i=1}^k{\parallel x-y\parallel }_{J_B(\alpha )}).\end{array}$$

Now, by letting ${\varphi }_{\alpha ,k}(t)=2^{k-1}{H}_{\alpha }^{k}t$, $D_{\alpha ,k}=\{(1,\alpha ),(1,J_B(\alpha ))(2,J_B(\alpha )),\dots ,(k,J_B(\alpha ))\}$, $P_{\alpha ,k}(\gamma )={\pi }_2(\gamma )$, and $F_{\alpha }(x,y)=max\{{\parallel x-y\parallel }_{\alpha },{\parallel x-y\parallel }_{J_B(\alpha )}\}$, we have

• (i) ${\parallel x-y\parallel }_{P_{\alpha ,k}(\gamma )}\le F_{\alpha }(x,y)$ for any $x,y\in C_{x_0}([0,T];X)$, $k\in \mathbf{N}$, $\alpha \in A$, and $\gamma \in D_{\alpha ,k}$,

• (ii) ${\sum }_{k\in \mathbf{N}}|D_{\alpha ,k}|{\varphi }_{\alpha ,k}(F_{\alpha }(x,y))={\sum }_{k\in \mathbf{N}}\frac{k+1}{2}{(2H_{\alpha })}^k{F}_{\alpha }(x,y)<\mathrm{\infty }$ for any $x,y\in C_{x_0}([0,T];X)$ and $\alpha \in A$.

Therefore, by Theorem 2.11(2), G has a unique fixed point, so the integral equation (2) has a unique solution.

3 Fixed point sets

In this section, we will show that, under a mild condition, a J-nonexpansive map is always virtually stable. This immediately gives a connection between the fixed point set and the convergence set of a J-nonexpansive map. Recall that a continuous self-map $T:X\to X$, whose fixed point set $F(T)$ is nonempty, on a Hausdorff space X is said to be virtually stable [4] if for each $x\in F(T)$ and each neighborhood U of x, there exist a neighborhood V of x and an increasing sequence $(k_n)$ of positive integers such that $T^{k_n}(V)\subseteq U$ for all $n\in \mathbf{N}$. When the sequence $(k_n)$ is independent of the point x and the neighborhood U, we simply call T a uniformly virtually stable map with respect to $(k_n)$. For example, a (quasi-) nonexpansive self-map, whose fixed point set is nonempty, on a metric space is always uniformly virtually stable with respect to the sequence $(n)$ of all natural numbers. An important feature of a virtually stable map is the connection between its fixed point set and its convergence set as given in the following theorem.

Theorem 3.1 ([4], Theorem 2.6)

Suppose X is a regular space. If $T:X\to X$ is virtually stable, then $F(T)$ is a retract of $C(T)$, where $C(T)$ is the (Picard) convergence set of T defined as follows:

$$C(T)=\{x\in X:\mathit{\text{the sequence }}\left(T^{n}x\right)\mathit{\text{ converges}}\}.$$

As in the previous section, let $(E,\mathcal{A})$ be a Hausdorff uniform space whose uniformity is generated by a saturated family of pseudometrics $\mathcal{A}=\{d_{\alpha }:\alpha \in A\}$ indexed by A and $\mathrm{\varnothing }\ne X\subseteq E$. The following theorem gives a general criterion for a self-map on X to be virtually stable.

Theorem 3.2 Let $T:X\to X$ be a self-map whose fixed point set $F(T)$ is nonempty, and which satisfies the following conditions:

• (i) for each $\alpha \in A$ and $k\in \mathbf{N}$, there exist a finite set $D_{\alpha ,k}$ and a map $P_{\alpha ,k}:D_{\alpha ,k}\to A$ such that

  $$d_{\alpha }(T^{k}x,T^{k}y)\le \sum _{\gamma \in D_{\alpha ,k}}{d}_{P_{\alpha ,k}(\gamma )}(x,y),$$

  for any $x,y\in X$,

• (ii) there exists $N\in \mathbf{N}$ such that $|D_{\alpha ,n}|\le |D_{\alpha ,N}|$ and $P_{\alpha ,n}(D_{\alpha ,n})\subseteq P_{\alpha ,N}(D_{\alpha ,N})$ for any $n\ge N$ and $\alpha \in A$.

Then T is uniformly virtually stable with respect to the sequence of all natural numbers.

Proof Let $z\in F(T)$ and let U be a neighborhood of z. We may assume that $U={\bigcap }_{i=1}^m\{w\in X:d_{{\alpha }_i}(w,z)<ϵ\}$ for some $ϵ>0$ and ${\alpha }_1,\dots ,{\alpha }_m\in A$. For each $n\in \mathbf{N}$, let

$$V_n=\bigcap _{i=1}^m\bigcap _{\gamma \in D_{{\alpha }_i,n}}\{w\in X:d_{P_{{\alpha }_i,n}(\gamma )}(w,z)<\frac{ϵ}{|D_{{\alpha }_i,n}|}\}.$$

By (ii), there exists $N\in \mathbf{N}$ such that $|D_{{\alpha }_i,n}|\le |D_{{\alpha }_i,N}|$ and $P_{{\alpha }_i,n}(D_{{\alpha }_i,n})\subseteq P_{{\alpha }_i,N}(D_{{\alpha }_i,N})$ for any $n\ge N$ and $i=1,\dots ,m$. Let $V=V_1\cap V_2\cap \cdots \cap V_N$ which is clearly a nonempty open subset of X, $y\in V$, $l\in \mathbf{N}$ and $i\in \{1,\dots ,m\}$. It follows that

$$d_{{\alpha }_i}(T^{l}y,z)=d_{{\alpha }_i}(T^{l}y,T^{l}z)\le \sum _{\gamma \in D_{{\alpha }_i,l}}{d}_{P_{{\alpha }_i,l}(\gamma )}(y,z).$$

If $l<N$, then

$$d_{{\alpha }_i}(T^{l}y,z)<\sum _{\gamma \in D_{{\alpha }_i,l}}\frac{ϵ}{|D_{{\alpha }_i,l}|}=ϵ.$$

If $l\ge N$, since $P_{{\alpha }_i,l}(\gamma )\in P_{{\alpha }_i,l}(D_{{\alpha }_i,l})\subseteq P_{{\alpha }_i,N}(D_{{\alpha }_i,N})$, we have $d_{P_{{\alpha }_i,l}(\gamma )}(y,z)<\frac{ϵ}{|D_{{\alpha }_i,N}|}$ for each $\gamma \in D_{{\alpha }_i,l}$, and hence

$$d_{{\alpha }_i}(T^{l}y,z)<\sum _{\gamma \in D_{{\alpha }_i,l}}\frac{ϵ}{|D_{{\alpha }_i,N}|}=\frac{ϵ|D_{{\alpha }_i,l}|}{|D_{{\alpha }_i,N}|}\le ϵ.$$

Hence, T is uniformly virtually stable with respect to the sequence of all natural numbers. □

Corollary 3.3 Suppose that T is J-nonexpansive with $F(T)\ne \mathrm{\varnothing }$. If there exists $N\in \mathbf{N}$ such that $|A_n(\alpha )|\le |A_N(\alpha )|$ and ${\pi }_n(A_n(\alpha ))\subseteq {\pi }_N(A_N(\alpha ))$ for any $n\ge N$ and $\alpha \in A$, then T is uniformly virtually stable with respect to the sequence of all natural numbers.

Proof By letting $D_{\alpha ,n}=A_n(\alpha )$ and $P_{\alpha ,n}={\pi }_n{|}_{A_n(\alpha )}$ for any $n\in \mathbf{N}$ and $\alpha \in A$, we have

$$d_{\alpha }(T^{l}x,T^{l}y)\le \sum _{\gamma \in D_{\alpha ,l}}{d}_{P_{\alpha ,l}(\gamma )}(x,y),$$

for any $x,y\in X$. The result then follows from Theorem 3.2. □

Example 3.4 Let $E={\ell }_2$ equipped with the weak topology and $T:{\ell }_2\to {\ell }_2$ be defined by

$$T(x_1,x_2,\dots )=(\frac{|x_1+x_3|}{3},\frac{|x_2+x_4|}{3},x_3,x_4,\dots ),$$

for any $(x_1,x_2,\dots )\in {\ell }_2$. Then $\mathcal{A}=\{|f|:f\in {\ell }_2\}$, and by Lemma 4.5 and Theorem 4.6 in [7], we have

$$\begin{array}{r}\left|f(T^{n}x-T^{n}y)\right|\\ \le 2\parallel f\parallel [\frac{\sqrt{2}}{9}(|x_1-y_1+x_3-y_3|+|x_2-y_2+x_4-y_4|)\\ +\frac{\sqrt{2}(|x_1-y_1|+|x_2-y_2|+|x_1-y_1+x_3-y_3|+|x_2-y_2+x_4-y_4|)}{9-6\sqrt{2}}]\\ +\parallel f\parallel (\frac{1}{3}|x_1-y_1|+|x_1-y_1+x_3-y_3|+\frac{1}{3}|x_2-y_2|+|x_2-y_2+x_4-y_4|)\\ +\parallel f\parallel |x_1-y_1|+\parallel f\parallel |x_2-y_2|+|f(x-y)|,\end{array}$$

for each $f\in {\ell }_2$, $n\in \mathbf{N}$, $x=(x_1,x_2,\dots )$ and $y=(y_1,y_2,\dots )\in {\ell }_2$.

By letting $J:{\ell }_2\to \mathcal{P}({\ell }_2)$ be defined by $J(f)=\{|f|,|g_1|,|g_2|,|g_3|,|g_4|\}$ for each $f\in {\ell }_2$, where

$$\begin{array}{c}{g}_1(x)=\parallel f\parallel (\frac{2\sqrt{2}}{9}+\frac{2\sqrt{2}}{9-6\sqrt{2}}+1)(x_1+x_3),\hfill \\ g_2(x)=\parallel f\parallel (\frac{2\sqrt{2}}{9}+\frac{2\sqrt{2}}{9-6\sqrt{2}}+1)(x_2+x_4),\hfill \\ g_3(x)=\parallel f\parallel (\frac{2\sqrt{2}}{9-6\sqrt{2}}+\frac{4}{3})x_1,g_4(x)=\parallel f\parallel (\frac{2\sqrt{2}}{9-6\sqrt{2}}+\frac{4}{3})x_2,\hfill \end{array}$$

for each $x=(x_1,x_2,\dots )\in {\ell }_2$, it follows that T is J-nonexpansive.

Notice that $(0,0,\dots )$ is a fixed point of T, and for each $f\in {\ell }_2$ and $n,m\in \mathbf{N}$, ${\pi }_n(A(|f|))={\pi }_m(A(|f|))$. Then, by Theorem 3.2, T is virtually stable and hence the fixed point set of T is a retract of the convergence set of T. Moreover, the fixed point set is not convex because $x=(1,1,2,2,0,\dots )$ and $y=(1,1,-4,-4,0,\dots )$ are fixed points of T, while the convex combination $\frac{1}{2}x+\frac{1}{2}y=(1,1,-1,-1,0,\dots )$ is not.