# Common Fixed Points for Suzuki-Generalized Nonexpansive Maps

Authors: Maryam A. Alghamdi; Sompong Dhompongsa; Naseer Shahzad
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

A common fixed point theorem for a pair of maps satisfying condition (C) is proved under certain conditions. We extend the well-knownDeMarr's fixed point theorem to the case of noncommuting family of maps satisfying condition (C). As for an application, an invariant approximation theorem is also derived.


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## Body

## 1. Introduction

Jungck [1] initiated the systematic study of finding a common fixed point of a pair of commuting maps. This problem of finding a common fixed point has been of significant interest in the area of fixed point theory and has been studied by many authors such as in [2–6]. At the first time, the commutativity for two maps was always assumed to find a common fixed point. Later, it was found that the two maps were not necessarily commutative at each point, and then weaker forms of commutativity were defined to obtain a common fixed point for maps on a metric space. For example, the notions of weakly commutative maps [2], compatible maps (weakly compatible maps) [7], biased maps [8], R-subweakly commuting maps [4], and occasionally weakly compatible [9] have been introduced and used to find common fixed points of maps.Recently, Chen and Li [5] introduced the class of Banach operator pairs and, in [10], they investigated the common fixed point problem for nonexpansive maps where (I,T) is a Banach operator pair. Also, they extended the well-known De Marr's fixed point theorem to the noncommuting case.More recently, Suzuki [11] introduced a condition on maps, called condition (C) (maps satisfying condition (C) are also known as Suzuki-generalized nonexpansive maps), and obtained some fixed point theorems and convergence theorems for such maps. Dhompongsa et al. [12] and Dhompongsa and Kaewcharoen [13] made significant contribution to fixed point theory for maps satisfying condition (C). For more results see [14].In this paper, we discuss a common fixed point problem for a Banach operator pair satisfying condition (C). A family of maps satisfying condition (C) is also investigated. As for an application, an invariant approximation theorem is obtained.


## 2. Preliminaries

LetE be a Banach space. E is said to be(i)
strictly convex if∥x+y∥<2 for all x,y∈E with ∥x∥=∥y∥=1 and x≠y,(ii)
uniformly convex in every direction (UCED) if, forε∈(0,2] and z∈E with ∥z∥=1, there exists δ(ε,z)>0 such that
(1)∥x+y∥≤2(1-δ(ε,z))
for all x,y∈E with ∥x∥≤1,∥y∥≤1 and x-y∈{tz:t∈[-2,-ε]∪[+ε,+2]}.It is obvious that being UCED implies strict convexity.LetK be a nonempty subset of E and let T be a self-map of K. We denote by F(T) the set of fixed points of T; that is, F(T)={x∈K:Tx=x}. Also, if I and T are self-maps of K, we denote by F(I,T) the set of common fixed points of I and T; that is, F(I,T)={x∈K:Ix=Tx=x}. If H is a nonempty family of self-maps of E, a point x∈E is called common fixed point of H if it is the fixed point of each member of H.The mapT is called(i)
nonexpansive if(2)∥Tx-Ty∥≤∥x-y∥,∀x,y∈K,(ii)
quasi-nonexpansive ifF(T)≠ϕ and
(3)∥Tx-p∥≤∥x-p∥,forx∈K,p∈F(T).Suzuki [11] introduced a condition on maps, called condition (C), which is weaker than nonexpansiveness and stronger than quasi-nonexpansiveness.Definition 1 (see [11]).
A  self-mapT of K is said to satisfy condition (C) if
(4)12∥x-Tx∥≤∥x-y∥implies∥Tx-Ty∥≤∥x-y∥
for all x,y∈K.Example 2 (see [13]).
Define a mapT on [0,3(1/2)] by
(5)Tx={0ifx∈[0,3]4x-12ifx∈[3,314]-4x+14ifx∈[314,312].
Then T is a continuous map satisfying condition (C) and T is not nonexpansive.Proposition 3 (see [11]).
LetK be a nonempty subset of a Banach space E. Assume that T:K→K is a nonexpansive map. Then T satisfies condition (C).Proposition 4 (see [11]).
LetK be a nonempty subset of a Banach space E. Assume that a map T:K→K satisfies condition (C) and has a fixed point. Then T is a quasi-nonexpansive map.Chen and Li [5] introduced the class of Banach operator pairs.Definition 5 (see [5]).
Let(X,d) be a metric space; the pair (I,T) of two self-maps I and T of X is called a Banach operator pair if the set F(T) of fixed points of T is I-invariant; that is, I(F(T))⊆F(T).A Banach operator pair(I,T) depends on the order of I and T; that is, if (I,T) is a  Banach operator pair, (T,I) need not be such a pair. It is well known that for two self-maps I and T of a metric space X, the pair (I,T) is a Banach pair if and only if I and T commute on the set F(T) [5].Example 6 (see [5]).
Letf and g be two self-maps of X=ℝ2 defined by
(6)f(s,t)=(s2+t2+s-1,s2+t2+t-1),g(s,t)=((s-t)2+2s-t,(s-t)2+s)
for (s,t)∈ℝ2. Directly, we have
(7)F(f)={(s,t)∈ℝ2:s2+t2-1=0},F(g)={(s,t)∈ℝ2:s-t=0ors-t+1=0}.
The following assertions can be verified: (i)
f(F(g))⊆F(g), and hence (f,g) is a Banach operator pair on ℝ2; equivalently, f and g commute on the set F(g).(ii)
(g,f) is not a Banach operator pair, since for (1,0)∈F(f),g(1,0)=(3,2) is not in F(f).The following proposition for Banach operator pairs can be found in [10].Proposition 7.
IfF(T) is a q-star shaped set (i.e., tx+(1-t)q∈F(T) for any x∈F(T) and 0≤t≤1) with q∈F(T), then (I,T) is a Banach operator pair if and only if the pairs (Ik,T)′s are Banach operator pairs for all k∈[0,1], where Ikx=(1-k)Ix+kq.Definition 8 (see [10]).
LetT and I be two self-maps of a metric space X. The pair (I,T) is called symmetric Banach operator pair if both (T,I) and (I,T) are Banach operator pairs; that is, T(F(I))⊂F(I) and I(F(T))⊂F(T).The pair(I,T) is a symmetric Banach operator pair if and only if T and I are commuting on F(T)∪F(I). It is easy to see that a Banach operator pair may not be a symmetric Banach operator pair; see [10].Definition 9 (see [10]).
LetH be a nonempty family of self-maps of a metric space X.  H is called a Banach operator family if for all I,T∈H, (I,T) is a symmetrical Banach operator pair.In 1963, DeMarr [15] stated the following well-known fixed point theorem for a family of commuting nonexpansive maps.Theorem 10 (DeMarr [15]).
IfK is a nonempty compact convex subset of a Banach space X and H is a nonempty family of commuting nonexpansive maps of K into itself, then the family H has a common fixed point in K.Recently Chen and Li [10] extended DeMarr's theorem to the noncommuting case.Theorem 11 (see [10]).
LetK be a nonempty closed convex subset of a normed space E and let H be a nonempty family of nonexpansive maps of K into itself. If H is a Banach operator family and there exists a T∈H such that T(K)¯ is compact, then H has a common fixed point in K.We now collect some results about condition (C) which will be used in the sequel.Lemma 12 (see [11]).
LetK be a nonempty closed subset of a Banach space E. Assume that T:K→K satisfies condition (C). Then F(T) is closed. Moreover, if E is strictly convex and K is convex, then F(T) is also convex.Theorem 13 (see [14]).
LetK be a closed bounded convex subset of a Banach space E. Assume that T:K→K is a map satisfying condition (C) and that T(K)¯ is compact. Then T has a fixed point.Lemma 14 (see [11]).
LetK be a nonempty subset of a Banach space E. Assume that T:K→K is a map satisfying condition (C). Then for x,y∈K, the following hold: (i)
∥Tx-T2x∥≤∥x-Tx∥,(ii)
either12∥x-Tx∥≤∥x-y∥ or 12∥Tx-T2x∥≤∥Tx-y∥ holds,(iii)
either∥Tx-Ty∥≤∥x-y∥ or ∥T2x-Ty∥≤∥Tx-y∥ holds.


## 3. Main Results

Lemma 15 (see [16] or [15]).
LetM be a nonempty compact subset of a Banach space E. Let δ be the diameter of M. If δ>0, then there exists an element u∈co¯M such that
(8)sup{∥x-u∥:x∈M}<δ,
where co¯M is the smallest closed convex set containing M.Following [15], we are able to prove the following lemma.Lemma 16.
LetK be a nonempty closed convex subset of a Banach space E. Suppose that T:K→K satisfies condition (C) such that there exists a compact set M⊂F(T) not reduced to a point.  Then there exists a nonempty closed convex set K1 such that (1)
K1⊂K and T(K1)⊂K1,(2)
M∩(K1)c≠∅.Proof.
Letδ be the diameter of M. Since M  is not reduced to a point, we have δ>0. According to Lemma 15, there is u∈co¯M such that
(9)δ1=sup{∥x-u∥:x∈M}<δ.
For each x∈M, define
(10)U(x)={y:∥y-x∥≤δ1}.
Since u∈U(x) for each x∈M, it follows that K0=⋂x∈MU(x)≠∅. It is easy to check that K0 is closed and convex. Let K1=K0∩K. Then K1 is not empty since u∈K1. For any x∈K1 and any z∈M, we have x∈U(z); that is, ∥x-z∥≤δ1. Since
(11)12∥z-Tz∥=0≤∥z-x∥,
we obtain that
(12)∥z-T(x)∥=∥T(z)-T(x)∥≤∥z-x∥≤δ1.
That is, T(x)∈U(z). This is true for any z∈M; thus T(x)∈K1. This shows that T(x)∈K1 for all x∈K1. Recalling that M is compact, therefore, there exist x0,x1∈M such that ∥x0-x1∥=δ>δ1. Thus, x1∉U(x0)⊃K1; that is, x1∈M∩(K1)c≠∅.Theorem 17.
LetK be a nonempty closed bounded convex subset of a Banach space E. Suppose that T and I are two self-maps on K satisfying condition (C). If (I,T) is a Banach operator pair, I is nonexpansive, and T(K)¯ is compact, then F(I,T)≠∅.Proof.
LetΓ be the set of all nonempty closed bounded convex subsets A of K such that T(A)⊂A and I(A)⊂A and T(A)¯ is compact. Since K∈Γ, then Γ is nonempty. Define a partial order “≤” by set inclusion on the set Γ; that is, Ai≤Aj whenever Ai⊆Aj.
LetΓ0 be any total ordering subset of Γ and A∈Γ0. Since A  is closed, we have T(A)¯⊂A, and since T(A)¯ is compact, it follows that
(13)∅≠⋂A∈Γ0T(A)¯⊂⋂A∈Γ0A=A0.
It is clear that A0∈Γ. By Zorn’s lemma, Γ has a minimal set K0.
SinceT satisfies condition (C) and T(K0)¯ is compact, then, by Theorem 13, T has a nonempty fixed point set F(T)⊂K0. It follows that F(T) is a closed subset of T(K0)¯ and thus is compact. On the other hand, we have T(F(T))=F(T), and since (I,T) is a Banach operator pair, it implies that I(F(T))⊂F(T). Using Zorn's lemma again, there exists a minimal nonempty compact subset M of F(T) which satisfies T(M)=M and I(M)⊆M (M is not necessarily convex).
Next, we showI(M)=M. If I(M)≠M, then we have I(I(M))⊂I(M), and I(M) is compact because I is continuous. Also, we have T(I(M))=I(M) since I(M)⊂M⊂F(T). This contradicts the minimality of M.
IfM has only one point, the proof is finished. Suppose that M has at least two points. By Lemma 16 there exists a setK1 satisfying T(K1)⊂K1 and M∩(K1)c≠∅. Since I is nonexpansive and I(M)=M, it follows that K1∈Γ which implies that K1 is a proper subset of K0 and this contradicts the minimality of K0. This completes the proof.Theorem 18.
LetK be a nonempty closed bounded convex subset of a strictly convex space E. Suppose that T and I are two self-maps on K satisfying condition (C). If (I,T) is a Banach operator pair and T(K)¯ is compact, then F(I,T)≠∅.Proof.
By Theorem13 and Lemma 12, F(T) is a nonempty closed bounded convex set. It is compact since T(K)¯ is compact. Since I(F(T))⊂F(T), again by Theorem 13, I has a fixed point in F(T); that is, F(T)∩F(I)≠∅.Corollary 19.
LetK be a nonempty closed bounded convex subset of an UCED Banach space E. Suppose that T and I are two self-maps on K satisfying condition (C). If (I,T) is a Banach operator pair and T(K)¯ is compact, then F(I,T)≠∅.Example 20.
Considerℝ with the usual metric and let K=[0,3(1/2)]. Define a map T on K by
(14)Tx={0ifx∈[0,3]4x-12ifx∈[3,314]-4x+14ifx∈[314,312]
and define a map I on K by Ix=x2/7. Then T is a continuous map satisfying condition (C) and T is not nonexpansive (see [13]) and I is nonexpansive and hence satisfies condition (C). Also (I,T) is a Banach operator pair. Therefore, all conditions of Theorem 17 (and Theorem 18) are satisfied and I and T have a common fixed point. Note that Theorem 2.1 in Chen and Li [10] is not applicable here.Next, we show a common fixed point theorem of a countable family of maps satisfying condition (C). We need first the following proposition which shows that for a given mapI there are a lot of maps T such that (I,T) is a symmetric Banach operator pair.Proposition 21 (see [10]).
LetI be a self-map on a convex subset K of a normed space E and let α be a map from K to [0,1] such that the set {x∈X:α(x)=0} is I-invariant; that is, α(Ix)=0,∀x∈{x∈X:α(x)=0}. Define
(15)Tαx=α(x)Ix+(1-α(x))x.
Then (I,Tα) is a symmetric Banach operator pair.Theorem 22.
LetK be a nonempty closed bounded convex subset of a Banach space E. Suppose that H is a nonempty family of self-maps on K satisfying condition (C). If H is a Banach operator family and there exists a T1∈H such that T1(K)¯ is compact and every Tj (except T1) in the family H is nonexpansive, then H has a common fixed point in K.Proof.
LetT1,T2andT3∈H and let Γ be the set of all nonempty closed bounded convex subsets A of K such that T1(A)⊂A, T2(A)⊂A,andT3(A)⊂A and T1(A)¯ is compact. On the set Γ, define a partial order by set inclusion; then we can find a minimal set K0∈Γ.
As in the proof of Theorem17, T1 and T2 have a nonempty compact common fixed point set F=F(T1,T2) in K0 satisfying T1(F)=F and T2(F)=F. Since (T3,T1) and (T3,T2) are Banach operator pairs, we have T3(F)⊂F. Using Zorn's lemma, there exists a minimal nonempty compact subset M of K0 which satisfies T1(M)=M,T2(M)=M, and T3(M)⊂M. Using an argument similar to that in Theorem 17, we can show that T3(M)=M. If M reduces to a point, then F(T1,T2,T3)≠∅. If M has at least two different points, then, by Lemma 16, this contradicts the minimality of K0. Therefore we obtain that K0 is a singleton and F(T1,T2,T3)≠∅.For any finite mapsTj∈H,j=1,2,…,n, we have by induction that F(T1,T2,…,Tn)≠∅. We now let θ={F(T1,T):T∈H}. Thus for any T∈H,F(T1,T) is a nonempty compact set, and for each Tj∈H,j=2,…,n, we have
(16)⋂j=2nF(T1,Tj)=F(T1,T2,…,Tn)≠∅.
This implies that the set family θ has the finite intersect property. Thus,
(17)⋂T∈HF(T1,T)≠∅.
Therefore the family H has a common fixed point in K.


## 4. Applications

LetK be a subset of the normed space E and x^∈E; then the distance of a point x^ to the subset K is defined by
(18)dist(x^,K)=inf{∥y-x^∥:y∈K}.
The set of best approximants of a point x^ in K is denoted by PK(x^) and defined by
(19)PK(x^)={y∈K:∥y-x^∥=dist(x^,K)}.
It is well known that PK(x^) is always a bounded subset of K and is a closed and convex set if K is so. Also, if K is compact, then PK(x^) is nonempty. For more details, we refer to [17].LetΩ0 denote the class of closed convex subsets of E containing 0. For K∈Ω0 and x^∈E, let
(20)Kx^={x∈K:∥x∥≤2∥x^∥}.
It is clear that PK(x^)⊂Kx^∈Ω0.The following result provides a partial solution of an existence problem of approximation theory in the following result (see also [14]).Theorem 23.
LetE be a Banach space and let T be a self-map of E with x^∈F(T)andK∈Ω0 such that T(Kx^)⊂K. Assume that T satisfies condition (C) on Kx^∪{x^} and T(Kx^)¯ is compact. Then the set of best approximations PK(x^) is nonempty.Proof.
Without loss of generality we may assume thatx^∈E∖K. If x∈K∖Kx^, then
(21)∥x-x^∥≥∥x∥-∥x^∥>2∥x^∥-∥x^∥=∥x^∥≥dist(x^,Kx^)≥dist(x^,K).
As a result
(22)dist(x^,Kx^)=dist(x^,K).
Since T(Kx^)¯ is compact, we can find y∈T(Kx^)¯ such that
(23)dist(x^,T(Kx^)¯)=∥y-x^∥,
and so by Lemma 14,
(24)dist(x^,Kx^)=dist(x^,K)≤dist(x^,T(Kx^)¯)≤∥Tx-Tx^∥≤∥x-x^∥
for all x∈Kx^. Hence
(25)dist(x^,K)=dist(x^,Kx^)=dist(x^,T(Kx^)¯)=∥y-x^∥
and thus y∈PK(x^).The following is an application of Theorem17 to invariant approximations for convex sets.Theorem 24.
LetE be a Banach space, I and T self-maps of E with x^∈F(I,T), and K∈Ω0 with I(Kx^)⊂K and T(Kx^)⊂K. If (I,T) is a Banach operator pair on Kx^, both I and T are maps satisfying condition (C) on Kx^∪{x^},I is nonexpansive, and T(Kx^)¯ is compact, then F(I,T)∩PK(x^)≠∅.Proof.
By Theorem23, PK(x^) is a nonempty. Since K is closed and convex, then PK(x^) is a closed convex set. We now show that PK(x^) is T-invariant. Let y∈PK(x^). Then ∥y-x^∥= dist(x^,K). Since T satisfies condition (C) on Kx^∪{x^}, by Lemma 14, we obtain that
(26)∥T(y)-x^∥=∥T(y)-T(x^)∥≤∥y-x^∥,
and so
(27)dist(x^,K)≤∥T(y)-x^∥≤∥y-x^∥=dist(x^,K).
This implies that T(y)∈PK(x^). Consequently, we have T(PK(x^))⊂PK(x^), and, similarly, we can prove that I(PK(x^))⊂PK(x^). Since T(PK(x^))¯⊂T(Kx^)¯ and T(Kx^)¯ is compact, we have that T(PK(x^))¯ is compact. Now, Theorem 17 guarantees that F(I,T)∩PK(x^)≠∅.



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