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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 P f (S) and |S| to denote the set of all nonempty finite subsets of S and the cardinality of S, respectively. Let (E,A) be a Hausdorff uniform space whose uniformity is generated by a saturated family of pseudometrics A={ d α :αA} indexed by A, XE, and J:A 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:XX is said to be J-nonexpansive if for each αA,

d α (Tx,Ty) β J ( α ) d β (x,y),

for any x,yX.

Example 2.2 Let 1<p<, E= p be equipped with the weak topology, and T: p p be defined by

T( x 1 , x 2 ,)= ( | x 1 + x 3 | 3 , | x 2 + x 4 | 3 , x 3 , x 4 , ) ,

for any ( x 1 , x 2 ,) p . Then A={|f|:f p }, where |f|(x)=|f(x)| for each x p .

By Theorem 4.6 in [7], we have

| f ( T x T y ) | | f 3 ( x 1 y 1 + x 3 y 3 ) | + | f 3 ( x 2 y 2 + x 4 y 4 ) | + | f ( x 1 y 1 ) | + | f ( x 2 y 2 ) | + | f ( x y ) | ,

for each f p , x=( x 1 , x 2 ,) p and y=( y 1 , y 2 ,) p . Here, f=sup{|f(x)|:xX,x1}.

By letting J: p P f ( p ) be defined by J(f)={|f|,| g 1 |,| g 2 |,| g 3 |,| g 4 |}, for each f p , where

g 1 (x)= f 3 ( x 1 + x 3 ), g 2 (x)= f 3 ( x 2 + x 4 ), g 3 (x)=f x 1 , g 4 (x)=f x 2 ,

for each x=( x 1 , x 2 ,) 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 αA and nN, we let

A n (α)= { ( α 1 , , α n ) : α 1 J ( α )  and  α k J ( α k 1 )  for  1 < k n }

and

A(α)= { ( α 1 , α 2 , ) : α 1 J ( α )  and  α k J ( α k 1 )  for  k > 1 } .

When there is no ambiguity, we will denote an element of both A n (α) and A(α) simply by ( α k ). Notice that for each αA and nN, the sets A n (α) and π n (A(α)) are finite, where π n denotes the n th coordinate projection ( α k ) α n .

Lemma 2.3 Every J-nonexpansive map is continuous.

Proof Suppose T:XX is J-nonexpansive. Let xX and ( x γ ) be a net in X converging to x. Then for each αA, we have

d α (T x γ ,Tx) β J ( α ) d β ( x γ ,x).

Since ( x γ ) converges to x, ( d β ( x γ ,x)) converges to 0 for any βA, and this proves the continuity of T. □

Theorem 2.4 Let T:XX be J-nonexpansive whose A(α) is finite for any αA. Then T has a fixed point in X if and only if there exists x 0 X such that

  1. (i)

    the sequence ( T n x 0 ) has a convergence subsequence, and

  2. (ii)

    for each αA and ( α k )A(α), lim n d α 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 zX. Let αA and ( α k )A(α). Then lim i d α (z, T n i x 0 )=0 and lim n d α n ( x 0 ,T x 0 )=0. We can choose NN sufficiently large so that d α (z, T n i x 0 )<ϵ and d α n i ( x 0 ,T x 0 )<ϵ, for all iN. It follows that

d α ( z , T n i + 1 x 0 ) d α ( z , T n i x 0 ) + d α ( T n i x 0 , T n i ( T x 0 ) ) d α ( z , T n i x 0 ) + ( α k ) A n i ( α ) d α n i ( x 0 , T x 0 ) ( 1 + | A ( α ) | ) ϵ .

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:XX be a j-nonexpansive map. If there exists x 0 X such that

  1. (i)

    the sequence ( T n x 0 ) has a convergence subsequence, and

  2. (ii)

    for every αA, lim n d j n ( α ) ( 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:α{j(α)}. Notice that A(α)={( j n (α))} 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 ϕ:[0,)[0,) satisfying the following conditions:

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

(Φ2) ϕ(t)<t for any t>0.

Notice that ϕ(0)=0, and we will call ϕΦ subadditive if ϕ( t 1 + t 2 )ϕ( t 1 )+ϕ( t 2 ) for all t 1 , t 2 0. Also, for a subfamily { ϕ α } α A of Φ, αA, ( α k ) A n (α) and in, we let

ϕ ( α k ) i = ϕ α 1 ϕ α i .

Definition 2.6 A self-map T:XX is said to be a J-contraction if for each αA, there exists ϕ α Φ such that

d α (Tx,Ty) β J ( α ) ϕ α ( d β ( x , y ) ) ,

for any x,yX, and ϕ α is subadditive whenever |J(α)|>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 :XX and T 2 :XX as defined [1]. Then there exist j 1 , j 2 :AA, and for each αA, there exist ϕ 1 , α , ϕ 2 , α Φ such that

d α ( T 1 x, T 1 y) ϕ 1 , α ( d j 1 ( α ) ( x , y ) ) and d α ( T 2 x, T 2 y) ϕ 2 , α ( d j 2 ( α ) ( x , y ) ) ,

for any αA and x,yX. If for each αA, j 1 (α) j 2 (α) and there is a subadditive ϕ 3 , α Φ so that ϕ 1 , α (t) ϕ 3 , α (t) and ϕ 2 , α (t) ϕ 3 , α (t) for any t0, then the map H= T 1 + T 2 is clearly a J-contraction with respect to J(α)={ j 1 (α), j 2 (α)} and ϕ H , α = ϕ 3 , α for any αA.

Lemma 2.8 If T:XX is a J-contraction. Then we have

d α ( T n x , T n y ) ( α k ) A n ( α ) ϕ α ϕ ( α k ) n 1 ( d α n ( x , y ) ) ,

for any αA, n2 and x,yX.

Proof Recall that ϕ α is assumed to be subadditive whenever |J(α)|>1. Then, for any αA, n2 and x,yX, we clearly have

d α ( T n x , T n y ) α 1 J ( α ) ϕ α ( d α 1 ( T n 1 x , T n 1 y ) ) α 1 J ( α ) ϕ α ( α 2 J ( α 1 ) ϕ α 1 ( d α 2 ( T n 2 x , T n 2 y ) ) ) α 1 J ( α ) α 2 J ( α 1 ) ϕ α ϕ α 1 ( d α 2 ( T n 2 x , T n 2 y ) ) α 1 J ( α ) α 2 J ( α 1 ) α n J ( α n 1 ) ϕ α ϕ α 1 ϕ α n 1 ( d α n ( x , y ) ) = ( α k ) A n ( α ) ϕ α ϕ ( α k ) n 1 ( d α n ( x , y ) ) .

 □

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:XX is a J-contraction whose A(α) is finite for any αA. If T satisfies the following conditions:

  1. (i)

    for each αA, there exists c α Φ such that

    ϕ α i (t) c α (t),

    for any ( α k )A(α), iN, t0, and

  2. (ii)

    there exists x 0 X such that for each αA, ( α k )A(α), iN and n,mN, we have

    d α i ( T n x 0 , T m x 0 ) M α ( x 0 ),

    for some M α ( x 0 )R,

then T has a fixed point. Moreover, if for each αA and x,yX, there exists F α (x,y) R 0 + such that

d α i (x,y) F α (x,y),

for all ( α k )A(α) and iN, then the fixed point of T is unique.

Proof For each αA and n,m,NN, since ϕ α is non-decreasing, we have

d α ( T n x 0 , T m x 0 ) α 1 J ( α ) ϕ α ( d α 1 ( T n 1 x 0 , T m 1 x 0 ) ) α 1 J ( α ) ϕ α ( sup { d α 1 ( T n 1 x 0 , T m 1 x 0 ) : n , m N } ) ,

and by letting h N α :=sup{ d α ( T n x 0 , T m x 0 ):n,mN}, it follows that

h N α α 1 J ( α ) ϕ α ( sup { d α 1 ( T n 1 x 0 , T m 1 x 0 ) : n , m N } ) = α 1 J ( α ) ϕ α ( h N 1 α 1 ) α 1 J ( α ) α 2 J ( α 1 ) ϕ α ( ϕ α 1 ( h N 2 α 2 ) ) ( α k ) A N 1 ( α ) ϕ α ϕ ( α k ) N 1 ( h 1 α N 1 ) ( α k ) A N 1 ( α ) c α N ( M α ( x 0 ) ) | A ( α ) | c α N ( M α ( x 0 ) ) .
(1)

Also, for a given t0, since 0 c α N (t)= c α ( c α N 1 (t))< c α N 1 (t), we have lim N c α N (t)= r α for some r α 0. Since c α is right continuous, we have lim N c α ( c α N 1 (t))= c α ( r α ), and hence c α ( r α )= r α . Therefore, r α =0. By (1), it follows that lim N h N α =0. Since α is arbitrary, ( T k x 0 ) is a Cauchy sequence and, by sequential completeness, converges to some zX. Notice also that z must be a fixed point of T by continuity.

Now suppose that for each x,yX and αA, there exists F α (x,y) R 0 + such that d α i (x,y) F α (x,y) for all ( α k )A(α) and iN. If x, y are fixed points of T, then by Lemma 2.8, we have for each αA and nN,

d α ( x , y ) = d α ( T n x , T n y ) ( α k ) A n ( α ) ϕ α ϕ ( α k ) n 1 ( d α n ( x , y ) ) ( α k ) A n ( α ) c α n ( d α n ( x , y ) ) | A ( α ) | c α n ( F α ( x , y ) ) .

Since lim n c α n ( F α (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:XX is Φ-contraction. If

  1. (i)

    for each αA, there exists c α Φ such that ϕ j n ( α ) (t) c α (t) for all nN and t0,

  2. (ii)

    for each nN, sup{ d j n ( α ) (x,y):x,yX}p(α):=sup{ d α (x,y):x,yX},

then there exists a unique fixed point xX of T.

Proof For each x 0 ,x,yX, αA, ( α k )A(α) and i,m,nN, by letting J(α)={j(α)} and M α ( x 0 )=p(α)= F α (x,y), we have A(α)={(α,j(α), j 2 (α),, j k (α),)}, d α i ( T m x 0 , T n x 0 )= d j i ( α ) ( T m x 0 , T n x 0 ) M α ( x 0 ) and d α i (x,y) F α (x,y). Hence, by Theorem 2.9, T has a unique fixed point. □

Theorem 2.11 Suppose X is sequentially complete and T:XX is a self-map satisfying: for each αA and kN, there exist ϕ α , k Φ, a finite set D α , k and a map P α , k : D α , k A such that

d α ( T k x , T k y ) γ D α , k ϕ α , k ( d P α , k ( γ ) ( x , y ) ) ,

for any x,yX.

  1. 1.

    If there exists x 0 X such that for each αA there exists M α ( x 0 ) R 0 + so that k N | D α , k | ϕ α , k ( M α ( x 0 ))< and

    d P α , k ( γ ) ( x 0 ,T x 0 ) M α ( x 0 ),

for all kN and γ D α , k , then T has a fixed point in X.

  1. 2.

    If for each αA and x,yX, there exists F α (x,y) R 0 + such that k N | D α , k | ϕ α , k ( F α (x,y))< and

    d P α , k ( γ ) (x,y) F α (x,y),

for all kN and γ D α , k , then T has a unique fixed point in X and, for any xX, the sequence ( T n x) converges to the fixed point of T.

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

  1. 1.

    For any αA and m>nN, we have

    d α ( T n x 0 , T m x 0 ) n i < m d α ( T i x 0 , T i + 1 x 0 ) n i < m γ D α , i ϕ α , i ( d P α , i ( γ ) ( x 0 , T x 0 ) ) n i < m | D α , i | ϕ α , i ( M α ( x 0 ) ) .

Also, since k N | D α , k | ϕ α , k ( M α ( x 0 ))<, ( 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.

  1. 2.

    For any xX, αA and m>nN, we have

    d α ( T n x , T m x ) n i < m d α ( T i x , T i + 1 x ) n i < m γ D α , i ϕ α , i ( d P α , i ( γ ) ( x , T x ) ) n i < m | D α , i | ϕ α , i ( F α ( x , T x ) ) .

Also, since k N | D α , k | ϕ α , k ( F α (x,Tx))<, ( 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 αA, kN and x,yF(T),

d α ( x , y ) = d α ( T k x , T k y ) γ D α , k ϕ α , k ( d P α , k ( γ ) ( x , y ) ) γ D α , k ϕ α , k ( F α ( x , y ) ) = | D α , k | ϕ α , k ( F α ( x , y ) ) ,

and lim k | D α , k | ϕ α , k ( F α (x,y))=0, we have the uniqueness. □

Corollary 2.12 (Theorem 5 in [1])

Let us suppose

  1. (i)

    for each αA and n>0, there exist ϕ α , n Φ and j(α,n)A such that

    d α ( T n x , T n y ) ϕ α , n ( d j ( α , n ) ( x , y ) ) ,

    for any x,yX,

  2. (ii)

    there exists x 0 X such that d j ( α , n ) ( x 0 ,T x 0 )p(α)< (n=1,2,), n ϕ α , n (p(α))< and j:A×NA.

Then T has at least one fixed point in X.

Proof By letting D α , k ={j(α,k)} for any αA and kN and P α , k = π k | D α , k . Then for each iN, we have | D α , i |=1 and M α ( x 0 )=p(α). By Theorem 2.11(2), T has a fixed point. □

Theorem 2.13 Suppose X is sequentially complete and T:XX is a J-contraction whose A(α) is finite for each αA. If, for each αA, there exists c α Φ satisfying:

  1. (i)

    c α (t)/t is non-decreasing in t,

  2. (ii)

    ϕ α n (t) c α (t) for any ( α k )A(α), nN and t[0,), and

  3. (iii)

    there exist x 0 X and M α ( x 0 ) R + such that d α n ( x 0 ,T x 0 ) M α ( x 0 ) for any ( α k )A(α) and nN,

then T has a fixed point in X.

Proof Let D α , i = A i (α), P α , i (( α k ))= α i , and ϕ α , i (t)= c α i (t) for any iN, αA, ( α k ) A i (α), and t[0,). Then for any αA and x,yX, we have, by Lemma 2.8,

d α ( T i x , T i y ) ( α k ) A i ( α ) ϕ α ϕ ( α k ) i 1 ( d α i ( x , y ) ) ( α k ) A i ( α ) c α i ( d α i ( x , y ) ) = ( α k ) D α , i ϕ α , i ( d P α , i ( ( α k ) ) ( x , y ) ) .

Since

| D α , i + 1 | ϕ α , i + 1 ( M α ( x 0 ) ) | D α , i | ϕ α , i ( M α ( x 0 ) ) = | A i + 1 ( α ) | c α i + 1 ( M α ( x 0 ) ) | A i ( α ) | c α i ( M α ( x 0 ) ) c α ( c α i ( M α ( x 0 ) ) ) c α i ( M α ( x 0 ) ) c α ( M α ( x 0 ) ) M α ( x 0 ) <1,

for any iN, we have i N | D α , i | ϕ α , i ( M α ( x 0 ))<. Then by Theorem 2.11(1), T has a fixed point. □

Corollary 2.14 (Theorem 2 in [1])

Let us suppose

  1. (i)

    the operator T:XX is a Φ-contraction,

  2. (ii)

    for each αA there exists a Φ-function c α such that ϕ j n ( α ) (t) c α (t) for all nN and c α (t)/t is non-decreasing,

  3. (iii)

    there exists an element x 0 X such that d j n ( α ) ( x 0 ,T x 0 )p(α)< (n=1,2,).

Then T has at least one fixed point in X.

Proof By letting J(α)={j(α)} for any αA and M α ( x 0 )=p(α). Then |A(α)|=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 :XX; i.e., there exist j 1 , j 2 :AA, and for each αA, there exist ϕ 1 , α , ϕ 2 , α Φ such that

d α ( T 1 x, T 1 y) ϕ 1 , α ( d j 1 ( α ) ( x , y ) ) and d α ( T 2 x, T 2 y) ϕ 2 , α ( d j 2 ( α ) ( x , y ) ) ,

for any αA and x,yX. Suppose further that

  1. (i)

    j 1 n + 1 = j 2 n j 1 and j 1 n j 2 = j 2 n + 1 for any nN,

  2. (ii)

    for each αA, ϕ 1 , α (t)= c 1 (α)t and ϕ 2 , α (t)= c 2 (α)t for some c 1 (α)+ c 2 (α)(0,1), and

  3. (iii)

    there exists x 0 X such that d j 1 n ( α ) ( x 0 , T 1 x 0 ) p 1 ( x 0 ,α)< and d j 2 n ( α ) ( x 0 , T 2 x 0 ) p 2 ( x 0 ,α)< for any αA and n=1,2,.

Then H= T 1 + T 2 2 is a J-contraction with J(α)={ j 1 (α), j 2 (α)} and ϕ H , α (t)=( c 1 (α)+ c 2 (α))t. Also, by (i) and (iii), we have |A(α)|=2< and

d α n ( x 0 ,H x 0 ) d α n ( x 0 , T 1 x 0 ) + d α n ( x 0 , T 2 x 0 ) 2 p 1 ( x 0 , α ) + p 2 ( x 0 , α ) 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 ( α ) + d j 2 ( α ) A for some αA, 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 { | | α :αA} and C([0,T];X) the space of all continuous functions from [0,T] into X topologized by the family of seminorms { α :αA}, where x α := sup t [ 0 , T ] | x ( t ) | α for any xC([0,T];X). Let L(X) denote the set of all continuous linear operators on X,

L 0 (X)= { l L ( X ) : α A , M ( α ) > 0 , x X , | l x | α M ( α ) | x | α } ,

and let { S ( t ) } t 0 be a C 0 -semigroup on X such that S:[0,) 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)C([0,T];X) is an operator such that there exists J B :A P f (A) so that for any αA, there is k α , B L loc 1 ([0,T];[0,)) such that

| B x ( t ) B y ( t ) | α k α , B (t) β J B ( α ) | x ( t ) y ( t ) | β ,

for any x,yC([0,T];X).

(N2) f:[0,T]×X×XX is continuous and there exist J f :A P f (A) and K f L loc 1 ([0,T];[0,)) such that for each αA,

| f ( t , u 1 , v 1 ) f ( t , u 2 , v 2 ) | α K f (t) ( β J f ( α ) | u 1 u 2 | β + | v 1 v 2 | α ) ,

for any t[0,T] and u 1 , u 2 , v 1 , v 2 X,

(N3) K f k α , B L loc 1 ([0,T];[0,)).

Consider the integral equation

x(t)=S(t) x 0 + 0 t S(ts)f ( s , x ( s ) , B x ( s ) ) ds;t[0,T]
(2)

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

d x d t =ax+f ( t , x ( t ) , B x ( t ) ) ,

where a denotes the infinitesimal generator of { S ( t ) } t 0 .

We now define an operator G on C x 0 ([0,T];X)={xC([0,T];X):x(0)= x 0 } by

(Gx)(t)=S(t) x 0 + 0 t S(ts)f ( s , x ( s ) , B x ( s ) ) ds,

for any x C x 0 ([0,T];X). Following the proof of Theorem 3 in [8] and for each t>0, S(t) L 0 (X), then we can show that, for any αA, there exists M(α)>0 such that

G x G y α H α ( β J f ( α ) x y β + β J B ( α ) x y β ) ,

where H α =max{M(α) 0 T K f (s)ds,M(α) 0 T K f (s) k α , B (s)ds}. It is easy to see that if for each αA, H α (0,1) and J f (α) J B (α)=, then G is a J-contraction with J G (α)= J f (α) J B (α).

In particular, if we assume further that for each αA, J f (α)={α}, | J B (α)|=1 such that J B J B = J B and H α = H J B ( α ) < 1 2 . Then for any kN and x,y C x 0 ([0,T];X), we have

G k x G k y α H α k x y α + ( i = 1 k ( 2 H J B ( α ) ) k i H α i ) x y J B ( α ) = H α k x y α + ( i = 1 k 2 k i H α k ) x y J B ( α ) 2 k 1 H α k ( x y α + i = 1 k x y J B ( α ) ) .

Now, by letting ϕ α , k (t)= 2 k 1 H α k t, D α , k ={(1,α),(1, J B (α))(2, J B (α)),,(k, J B (α))}, P α , k (γ)= π 2 (γ), and F α (x,y)=max{ x y α , x y J B ( α ) }, we have

  1. (i)

    x y P α , k ( γ ) F α (x,y) for any x,y C x 0 ([0,T];X), kN, αA, and γ D α , k ,

  2. (ii)

    k N | D α , k | ϕ α , k ( F α (x,y))= k N k + 1 2 ( 2 H α ) k F α (x,y)< for any x,y C x 0 ([0,T];X) and αA.

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:XX, whose fixed point set F(T) is nonempty, on a Hausdorff space X is said to be virtually stable [4] if for each xF(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)U for all nN. 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:XX 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 X : the sequence  ( T n x )  converges } .

As in the previous section, let (E,A) be a Hausdorff uniform space whose uniformity is generated by a saturated family of pseudometrics A={ d α :αA} indexed by A and XE. The following theorem gives a general criterion for a self-map on X to be virtually stable.

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

  1. (i)

    for each αA and kN, there exist a finite set D α , k and a map P α , k : D α , k A such that

    d α ( T k x , T k y ) γ D α , k d P α , k ( γ ) (x,y),

    for any x,yX,

  2. (ii)

    there exists NN such that | D α , n || D α , N | and P α , n ( D α , n ) P α , N ( D α , N ) for any nN and αA.

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

Proof Let zF(T) and let U be a neighborhood of z. We may assume that U= i = 1 m {wX: d α i (w,z)<ϵ} for some ϵ>0 and α 1 ,, α m A. For each nN, let

V n = i = 1 m γ D α i , n { w X : d P α i , n ( γ ) ( w , z ) < ϵ | D α i , n | } .

By (ii), there exists NN such that | D α i , n || D α i , N | and P α i , n ( D α i , n ) P α i , N ( D α i , N ) for any nN and i=1,,m. Let V= V 1 V 2 V N which is clearly a nonempty open subset of X, yV, lN and i{1,,m}. It follows that

d α i ( T l y , z ) = d α i ( T l y , T l z ) γ D α i , l d P α i , l ( γ ) (y,z).

If l<N, then

d α i ( T l y , z ) < γ D α i , l ϵ | D α i , l | =ϵ.

If lN, since P α i , l (γ) P α i , l ( D α i , l ) P α i , N ( D α i , N ), we have d P α i , l ( γ ) (y,z)< ϵ | D α i , N | for each γ D α i , l , and hence

d α i ( T l y , z ) < γ D α i , l ϵ | D α i , N | = ϵ | D α i , l | | D α i , N | 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). If there exists NN such that | A n (α)|| A N (α)| and π n ( A n (α)) π N ( A N (α)) for any nN and αA, then T is uniformly virtually stable with respect to the sequence of all natural numbers.

Proof By letting D α , n = A n (α) and P α , n = π n | A n ( α ) for any nN and αA, we have

d α ( T l x , T l y ) γ D α , l d P α , l ( γ ) (x,y),

for any x,yX. The result then follows from Theorem 3.2. □

Example 3.4 Let E= 2 equipped with the weak topology and T: 2 2 be defined by

T( x 1 , x 2 ,)= ( | x 1 + x 3 | 3 , | x 2 + x 4 | 3 , x 3 , x 4 , ) ,

for any ( x 1 , x 2 ,) 2 . Then A={|f|:f 2 }, and by Lemma 4.5 and Theorem 4.6 in [7], we have

| f ( T n x T n y ) | 2 f [ 2 9 ( | x 1 y 1 + x 3 y 3 | + | x 2 y 2 + x 4 y 4 | ) + 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 2 ] + f ( 1 3 | x 1 y 1 | + | x 1 y 1 + x 3 y 3 | + 1 3 | x 2 y 2 | + | x 2 y 2 + x 4 y 4 | ) + f | x 1 y 1 | + f | x 2 y 2 | + | f ( x y ) | ,

for each f 2 , nN, x=( x 1 , x 2 ,) and y=( y 1 , y 2 ,) 2 .

By letting J: 2 P( 2 ) be defined by J(f)={|f|,| g 1 |,| g 2 |,| g 3 |,| g 4 |} for each f 2 , where

g 1 ( x ) = f ( 2 2 9 + 2 2 9 6 2 + 1 ) ( x 1 + x 3 ) , g 2 ( x ) = f ( 2 2 9 + 2 2 9 6 2 + 1 ) ( x 2 + x 4 ) , g 3 ( x ) = f ( 2 2 9 6 2 + 4 3 ) x 1 , g 4 ( x ) = f ( 2 2 9 6 2 + 4 3 ) x 2 ,

for each x=( x 1 , x 2 ,) 2 , it follows that T is J-nonexpansive.

Notice that (0,0,) is a fixed point of T, and for each f 2 and n,mN, π n (A(|f|))= π 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,) and y=(1,1,4,4,0,) are fixed points of T, while the convex combination 1 2 x+ 1 2 y=(1,1,1,1,0,) is not.

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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., Songsa-ard, S. Fixed points in uniform spaces. Fixed Point Theory Appl 2014, 134 (2014). https://doi.org/10.1186/1687-1812-2014-134

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Keywords

  • fixed points
  • Φ-contractions
  • uniform spaces