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Fixed points in uniform spaces
Fixed Point Theory and Applications volume 2014, Article number: 134 (2014)
Abstract
We improve Angelov’s fixed point theorems of Φcontractions and jnonexpansive 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 Φcontractions on Hausdorff uniform spaces, which simultaneously generalizes the wellknown Banach contractions on metric spaces as well as γcontractions [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 Φcontractions to jnonexpansive 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 multivalued map J to obtain a more general, yet interesting, notion of Jnonexpansiveness. Therefore, in this work, we aim to correct and simplify the proof of Theorem 1 [3] as well as extend the notion of jnonexpansive maps to Jnonexpansive maps and investigate the existence of their fixed points. Then we introduce Jcontractions, a special kind of Jnonexpansive maps, that play the similar role as Banach contractions in yielding the uniqueness of fixed points. With the notion of Jcontractions, we are able to recover results on Φcontractions 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, Jnonexpansive 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 Jnonexpansive 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 Jnonexpansive map as follows:
Definition 2.1 A selfmap T:X\to X is said to be Jnonexpansive 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}.
By Theorem 4.6 in [7], we have
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
for each x=({x}_{1},{x}_{2},\dots )\in {\ell}_{p}, it follows that T is Jnonexpansive.
The above definition of a Jnonexpansive map clearly extends the definition of a jnonexpansive map in [3]. Before giving general existence criteria for fixed points of Jnonexpansive maps, we need the following notations. For each \alpha \in A and n\in \mathbf{N}, we let
and
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 Jnonexpansive map is continuous.
Proof Suppose T:X\to X is Jnonexpansive. 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 Jnonexpansive 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})<\u03f5 and {d}_{{\alpha}_{{n}_{i}}}({x}_{0},T{x}_{0})<\u03f5, for all i\ge N. It follows that
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 jnonexpansive 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 Jnonexpansive 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 nondecreasing 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
Definition 2.6 A selfmap T:X\to X is said to be a Jcontraction 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 Φcontraction as defined in [1] is a Jcontraction and a Jcontraction is always Jnonexpansive. A natural example of a Jcontraction can be obtained by adding (finitely many) appropriate Φcontractions as shown in the following example.
Example 2.7 Given two Φcontractions {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 Jcontraction 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 Jcontraction. 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 Jcontractions.
Theorem 2.9 Suppose X is sequentially complete and T:X\to X is a Jcontraction 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 nondecreasing, we have
and by letting {h}_{N}^{\alpha}:=sup\{{d}_{\alpha}({T}^{n}{x}_{0},{T}^{m}{x}_{0}):n,m\ge N\}, it follows that
Also, for a given t\ge 0, since 0\le {c}_{\alpha}^{N}(t)={c}_{\alpha}({c}_{\alpha}^{N1}(t))<{c}_{\alpha}^{N1}(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}^{N1}(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},
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 Φcontraction. 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 selfmap 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 Jcontraction.

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),
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 Jcontraction 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 nondecreasing 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,
Since
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 Φcontraction,

(ii)
for each \alpha \in A there exists a Φfunction {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 nondecreasing,

(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 Φcontractions {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
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 Jcontraction 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 Φcontraction, 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 Sspace 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:
(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
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,
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
whose solution is closely related to the mild solution of the differential equation
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
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
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 Jcontraction 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
Now, by letting {\varphi}_{\alpha ,k}(t)={2}^{k1}{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 xy\parallel}_{\alpha},{\parallel xy\parallel}_{{J}_{B}(\alpha )}\}, we have

(i)
{\parallel xy\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 Jnonexpansive map is always virtually stable. This immediately gives a connection between the fixed point set and the convergence set of a Jnonexpansive map. Recall that a continuous selfmap 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 selfmap, 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 selfmap on X to be virtually stable.
Theorem 3.2 Let T:X\to X be a selfmap 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
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
If l<N, then
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{\u03f5}{{D}_{{\alpha}_{i},N}} for each \gamma \in {D}_{{\alpha}_{i},l}, and hence
Hence, T is uniformly virtually stable with respect to the sequence of all natural numbers. □
Corollary 3.3 Suppose that T is Jnonexpansive 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 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
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
for each x=({x}_{1},{x}_{2},\dots )\in {\ell}_{2}, it follows that T is Jnonexpansive.
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.
References
Angelov VG: Fixed point theorems in uniform spaces and applications. Czechoslov. Math. J. 1987, 37: 19–33. 10.1007/BF01597873
Cain GL, Nashed MZ: Fixed points and stability for a sum of two operators in locally convex spaces. Pac. J. Math. 1971, 39(3):581–592. 10.2140/pjm.1971.39.581
Angelov VG: J Nonexpansive mappings in uniform spaces and applications. Bull. Aust. Math. Soc. 1991, 43: 331–339. 10.1017/S0004972700029130
Chaoha P, Atiponrat W: Virtually stable maps and their fixed point sets. J. Math. Anal. Appl. 2009, 359: 536–542. 10.1016/j.jmaa.2009.06.015
Weil A: Sur les Espaces à Structure Uniforme et la Topologie Générale. Hermann, Paris; 1938.
Angelov VG: Fixed Points in Uniform Spaces and Applications. Cluj University Press, ClujNapoca; 2009.
Chaoha P, Songsaard S: Fixed points of functionally Lipschitzian maps. J. Nonlinear Convex Anal. 2014, 15(4):665–679.
Chonwerayuth A, Termwuttipong I, Chaoha P: Piecewise continuous mild solutions of a system governed by impulsive differential equations in locally convex spaces. Science Asia 2011, 37: 360–369. 10.2306/scienceasia15131874.2011.37.360
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.
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Chaoha, P., Songsaard, S. Fixed points in uniform spaces. Fixed Point Theory Appl 2014, 134 (2014). https://doi.org/10.1186/168718122014134
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DOI: https://doi.org/10.1186/168718122014134
Keywords
 fixed points
 Φcontractions
 uniform spaces