# Fixed points in uniform spaces

- Phichet Chaoha
^{1}Email author and - Sittichoke Songsa-ard
^{1}

**2014**:134

https://doi.org/10.1186/1687-1812-2014-134

© Chaoha and Songsa-ard; licensee Springer. 2014

**Received: **23 January 2014

**Accepted: **23 May 2014

**Published: **2 June 2014

## 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.

### Keywords

fixed points Φ-contractions uniform spaces## 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 $\alpha \in A$,

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

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 $x\in {\ell}_{p}$.

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, $\parallel f\parallel =sup\{|f(x)|:x\in X,\parallel x\parallel \le 1\}$.

for each $x=({x}_{1},{x}_{2},\dots )\in {\ell}_{p}$, it follows that *T* is *J*-nonexpansive.

*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

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

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*.

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 $t>0$.

**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

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

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*

*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

□

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*

*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

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.

*x*,

*y*are fixed points of

*T*, then by Lemma 2.8, we have for each $\alpha \in A$ and $n\in \mathbf{N}$,

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*

*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$\begin{array}{rl}{d}_{\alpha}({T}^{n}{x}_{0},{T}^{m}{x}_{0})& \le \sum _{n\le i<m}{d}_{\alpha}({T}^{i}{x}_{0},{T}^{i+1}{x}_{0})\\ \le \sum _{n\le i<m}\sum _{\gamma \in {D}_{\alpha},i}{\varphi}_{\alpha ,i}({d}_{{P}_{\alpha ,i}(\gamma )}({x}_{0},T{x}_{0}))\\ \le \sum _{n\le i<m}|{D}_{\alpha ,i}|{\varphi}_{\alpha ,i}({M}_{\alpha}({x}_{0})).\end{array}$

*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*.

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,

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

- (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 $.

*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

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*,

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:

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

for any $t\in [0,T]$ and ${u}_{1},{u}_{2},{v}_{1},{v}_{2}\in X$,

(N3) ${K}_{f}\cdot {k}_{\alpha ,B}\in {L}_{\mathrm{loc}}^{1}([0,T];[0,\mathrm{\infty}))$.

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

*G*on ${C}_{{x}_{0}}([0,T];X)=\{x\in C([0,T];X):x(0)={x}_{0}\}$ by

where ${H}_{\alpha}=max\{M(\alpha ){\int}_{0}^{T}{K}_{f}(s)\phantom{\rule{0.2em}{0ex}}ds,M(\alpha ){\int}_{0}^{T}{K}_{f}(s){k}_{\alpha ,B}(s)\phantom{\rule{0.2em}{0ex}}ds\}$. 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 ${J}_{G}(\alpha )={J}_{f}(\alpha )\cup {J}_{B}(\alpha )$.

- (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)
${\sum}_{k\in \mathbf{N}}|{D}_{\alpha ,k}|{\varphi}_{\alpha ,k}({F}_{\alpha}(x,y))={\sum}_{k\in \mathbf{N}}\frac{k+1}{2}{(2{H}_{\alpha})}^{k}{F}_{\alpha}(x,y)<\mathrm{\infty}$ 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*:

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 $U={\bigcap}_{i=1}^{m}\{w\in X:{d}_{{\alpha}_{i}}(w,z)<\u03f5\}$ for some $\u03f5>0$ and ${\alpha}_{1},\dots ,{\alpha}_{m}\in A$. For each $n\in \mathbf{N}$, let

*X*, $y\in V$, $l\in \mathbf{N}$ and $i\in \{1,\dots ,m\}$. It follows that

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

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

for each $f\in {\ell}_{2}$, $n\in \mathbf{N}$, $x=({x}_{1},{x}_{2},\dots )$ and $y=({y}_{1},{y}_{2},\dots )\in {\ell}_{2}$.

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.

## Declarations

### Acknowledgements

The authors are grateful to the anonymous referee(s) for their valuable comments and suggestions for improving this manuscript. This research is (partially) supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.

## Authors’ Affiliations

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