# Approximating Common Fixed Points of Nonspreading-Type Mappings and Nonexpansive Mappings in a Hilbert Space

Authors: Kyung Soo Kim
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We obtain some fundamental properties fork-strictly pseudo-nonspreading mappings in a Hilbert space. We study approximation of common fixed points of k-strictly pseudo-nonspreading mappings and nonexpansive mappings in a Hilbert space by using a new iterative scheme. Furthermore, we suggest some open problems.


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## Body

## 1. Introduction

LetH  be a real Hilbert space and  let  C  be a nonempty closed convex subset of  H. Then a mapping  T:C→C  is said to be nonexpansive if
(1.1)∥Tx-Ty∥≤∥x-y∥
for all x,y∈C.Recently, Kohsaka and Takahashi [1] introduced a class of mappings called nonspreading mappings: Let E be a real smooth, strictly convex, and reflexive Banach space, and let  J  denote the duality mapping  of E.  Let  C  be a nonempty closed convex subset of  E. They called a maping  T:C→C is said to be nonspreading if
(1.2)ϕ(Tx,Ty)+ϕ(Ty,Tx)≤ϕ(Tx,y)+ϕ(Ty,x)
for all x,y∈C,  where  ϕ(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2  for all  x,y∈E. They considered the class of nonspreading mappings to study the resolvents of a maximal monotone operator in the Banach space. Observe that, if  E  is a real Hilbert space, then  J  is the identity and
(1.3)ϕ(x,y)=∥x∥2-2〈x,y〉+∥y∥2=∥x-y∥2
for all  x,y∈E.  Thus, if  C  is a nonempty closed convex subset of a Hilbert space, then T:C→C  is nonspreading if
(1.4)2∥Tx-Ty∥2≤∥Tx-y∥2+∥Tx-y∥2
for all  x,y∈C. It is shown in  [2] that  (1.4)  is equivalent to
(1.5)∥Tx-Ty∥2≤∥x-y∥2+2〈x-Tx,y-Ty〉
for all x,y∈C.Following the terminology of Browder-Petryshyn [3, (page 198)], a mapping T:C→H  is k-strictly pseudo-nonspreading if there exists  k∈[0,1)  such that (1.6)∥Tx-Ty∥2≤∥x-y∥2+2〈x-Tx,y-Ty〉+k∥x-Tx-(y-Ty)∥2,
for all  x,y∈C.  Clearly, every nonspreading mapping is  k-strictly pseudo-nonspreading.The following is an example of nonspreading mapping which is not nonexpansive mapping.Example 1.1 (see [2]).
LetH  be a Hilbert space. Set  E={x∈H:∥x∥≤1},  D={x∈H:∥x∥≤2}  and C={x∈H:∥x∥≤3}.  Define a mapping   T:C→C  as follows:
(1.7)Tx={0ifx∈D,PExifx∈C∖D,
where PE is the metric projection of H onto E. Then, T is not nonexpansive but nonspreading mapping.The following example shows that the class ofk-strictly pseudo-nonspreading mapping is more general than the class of nonspreading mappings.Example 1.2 (see [4]).
Letℝ denotes the real numbers with the usual norm. Let T:ℝ→ℝ  be defined for each  x∈ℝ  by
(1.8)Tx={xifx∈(-∞,0),-2xifx∈[0,∞).
Then, T  is  k-strictly pseudo-nonspreading but not nonspreading mapping.Remark 1.3 (see [4]).
LetC be a nonempty closed convex subset of a real Hilbert space H, and let S:C→C be a k-strictly pseudo-nonspreading mapping.  If  F(S)≠∅,  then it is closed and convex.Iemoto and Takahashi [2] introduced the following Moudafi iterative procedure [5]:(1.9)x1∈C,xn+1=(1-αn)xn+αn{βnSxn+(1-βn)Txn}
for finding the approximation of common fixed points of nonspreading mapping S and nonexpansive mapping T in a Hilbert space.In this paper, we obtain some fundamental properties fork-strictly pseudo-nonspreading mappings in a Hilbert space. We study approximation of common fixed points of k-strictly pseudo-nonspreading mappings and nonexpansive mappings in a Hilbert space by using a new iterative scheme. Furthermore, we suggest some open problems.


## 2. Preliminaries

Throughout this paper, we denote byℕ the set of positive integers and by ℝ  the set of real numbers.  Let  H  be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥.Definition 2.1.
LetE  be a real Banach space. A mapping  T with domain  D(T) and range  R(T) in  E is said to be demiclosed at a point  p∈D(T)  if whenever  {xn} is a sequence in D(T) which converges weakly to a point  x∈D(T) and  {Txn}  converges strongly to p, then  Tx=p.Lemma 2.2 (see [2]).
LetH be a real Hilbert space. Then, the following well known results hold:  (1)
∥tx+(1-t)y∥2=t∥x∥2+(1-t)∥y∥2-t(1-t)∥x-y∥2  for all  x,y∈H and for all  t∈[0,1],(2)
2〈x-y,z-w〉=∥x-w∥2+∥y-z∥2-∥x-z∥2-∥y-w∥2  for all  x,y,z,w∈H.Lemma 2.3 (see [6]).
LetC be a nonempty closed convex subset of  H. If T:C→C  is nonexpansive, then  I-T:C→H  is  1/2-inverse strongly monotone, that is,
(2.1)12∥(I-T)x-(I-T)y∥2≤〈x-y,(I-T)x-(I-T)y〉
for all x,y∈C.The following lemma is one of the characterizations of ak-strictly pseudo-nonspreading mapping.Lemma 2.4.
LetC be a nonempty closed convex subset of H. Then, a mapping S:C→C is k-strictly pseudo-nonspreading if and only if
(2.2)2∥Sx-Sy∥2≤∥Sx-y∥2+∥x-Sy∥3+k∥(I-S)x-(I-S)y∥2
for all x,y∈C.Proof.
We have that for allx,y∈C,
(2.3)∥Sx-Sy∥2≤∥x-y∥2+2〈x-Sx,y-Sy〉+k∥x-Sx-(y-Sy)∥2⟺2∥Sx-Sy∥2≤∥Sx-Sy∥2+∥x-y∥2+2〈Sx-x,x-Sx-(y-Sy)〉+2∥Sx-x∥2+k∥x-Sx-(y-Sy)∥2=∥Sx-Sy∥2+2〈x-Sx,Sx-Sy〉+∥x-Sx∥2+∥x-y∥2+2〈Sx-x,x-y〉+∥Sx-x∥2+k∥x-Sx-(y-Sy)∥2=∥x-Sy∥2+∥Sx-y∥2+k∥(I-S)x-(I-S)y∥2.
This completes the proof of Lemma2.4.Lemma 2.5.
LetC be a nonempty closed convex subset of H. Let S be a k-strictly pseudo-nonspreading mapping of C into itself and let  A=I-S.
Then,(2.4)(2-k)∥Ax-Ay∥2≤2〈x-y,Ax-Ay〉+∥Ax∥2+∥Ay∥2
for all x,y∈C.Proof.
LetA=I-S.  We have
(2.5)∥Ax-Ay∥2=〈Ax-Ay,Ax-Ay〉=〈x-y-(Sx-Sy),Ax-Ay〉=〈x-y,Ax-Ay〉-〈Sx-Sy,Ax-Ay〉
for all x,y∈C. From Lemma 2.2-(2) and Lemma 2.4, we get
(2.6)2〈Sx-Sy,Ax-Ay〉=2〈Sx-Sy,x-y-(Sx-Sy)〉=2〈Sx-Sy,x-y〉-2∥Sx-Sy∥2≥∥Sx-y∥2+∥Sy-x∥2-∥Sx-x∥2-∥Sy-y∥2-(∥Sx-y∥2+∥x-Sy∥2+k∥Ax-Ay∥2)=-∥x-Sx∥2-∥y-Sy∥2-k∥Ax-Ay∥2=-∥Ax∥2-∥Ay∥2-k∥Ax-Ay∥2.
So, from (2.5) and (2.6), we have
(2.7)∥Ax-Ay∥2≤〈x-y,Ax-Ay〉+12(∥Ax∥2+∥Ay∥2)+k2∥Ax-Ay∥2.
Therefore, we get
(2.8)(2-k)∥Ax-Ay∥2≤2〈x-y,Ax-Ay〉+∥Ax∥2+∥Ay∥2
for all x,y∈C.Lemma 2.6 (see [4]).
LetC be a nonempty closed convex subset of a real Hilbert space H, and let  S:C→C  be a k-strictly pseudo-nonspreading mapping. Then  I-S is demiclosed at  0.Tan and Xu [7] proved the following; see also [6, 8].Lemma 2.7.
Let{an} and {bn} are sequences of nonnegative real numbers such that  an+1≤an+bn  for all  n∈ℕ.  If ∑n=1∞bn<∞,  then  limn→∞an  exists.Lemma 2.8.
Let{αn}, {βn} be sequences of nonnegative real numbers such that ∑n=1∞αn=∞. If ∑n=1∞αnβn<∞,  then liminfn→∞βn=0.


## 3.   Main Theorem

In this section, we prove our main theorem for finding common fixed points ofk-strictly pseudo-nonspreading mapping and nonexpansive mapping in a Hilbert space.Theorem 3.1.
LetC be a nonempty closed convex subset of a real Hilbert space H and let S:C→C be a k-strictly pseudo-nonspreading mapping and let  T:C→C  be a nonexpansive mapping such that  F(S)∩F(T)≠∅.  Let  {αn}, {βn}, {γn} be sequences in  [0,1]  such that  βn∈(k,1].  Define a sequence {xn} as  follows:(3.1)x1∈C,xn+1=(1-αn)(βnxn+(1-βn)Sxn)+αn(γnxn+(1-γn)Txn)
for alln∈ℕ.  Then, the followings hold:  (1)
ifliminfn→∞αn(βn-γn)>0, ∑n=1∞‍αn(1-γn)<∞, and 1+k<(2-αn)βn+αnγn, then  {xn}  converges weakly to  q∈F(S),(2)
ifβn>γn, ∑n=1∞(1-βn)<∞, 2βn-1-αn(βn-γn)>0, and liminfn→∞αn(βn-γn)(2βn-1-αn(βn-γn))>0,  then {xn}  converges weakly to  q∈F(T),(3)
ifliminfn→∞αn>0, liminfn→∞(1-αn)>0, liminfn→∞(1-βn)>0, and liminfn→∞γn(1-γn)>0, then {xn} converges weakly to q∈F(S)∩F(T).Proof.
PuttingUn=βnI+(1-βn)S  and Vn=γnI+(1-γn)T. We first show that the sequence {xn} is bounded. Indeed, from Lemma 2.2-(1) and S as a k-strictly pseudo-nonspreading mapping, we have
(3.2)∥Unx-Uny∥2=∥βn(x-y)+(1-βn)(Sx-Sy)∥2=βn∥x-y∥2+(1-βn)∥Sx-Sy∥2-βn(1-βn)∥x-Sx-(y-Sy)∥2≤βn∥x-y∥2+(1-βn)×(∥x-y∥2+2〈x-Sx,y-Sy〉+k∥x-Sx-(y-Sy)∥2)-βn(1-βn)∥x-Sx-(y-Sy)∥2=∥x-y∥2+2(1-βn)〈x-Sx,y-Sy〉-(1-βn)(βn-k)∥x-Sx-(y-Sy)∥2≤∥x-y∥2+2(1-βn)〈x-Sx,y-Sy〉=∥x-y∥2+21-βn〈x-Unx,y-Uny〉,
for allx,y∈C.  Let p∈F(S)∩F(T), then
(3.3)Unp=βnp+(1-βn)Sp=p.
From (3.2) and (3.3), we have
(3.4)∥Unxn-p∥=∥Unxn-Unp∥≤∥xn-p∥.
Since T is a nonexpansive mapping and  F(T)≠∅,  we get
(3.5)∥Vnxn-p∥=∥γnxn+(1-γn)Txn-p∥≤γn∥xn-p∥+(1-γn)∥Txn-p∥≤∥xn-p∥
for all  p∈F(S)∩F(T).  From (3.4) and (3.5), we get
(3.6)∥xn+1-p∥2=∥(1-αn)Unxn+αnVnxn-p∥2≤(1-αn)∥Unxn-p∥2+αn∥Vnxn-p∥2≤∥xn-p∥2
for all n∈ℕ. Therefore, there exists limn→∞∥xn-p∥ and hence {xn} is bounded.
Let(3.7)limn→∞∥xn-p∥=c.
To prove (1), let
(3.8)zn+1=(1-αn)Unxn+αn(γnxn+(1-γn)Sxn)
and A=I-S.  Then, we have
(3.9)∥xn+1-zn+1∥=∥(1-αn)Unxn+αnVnxn-(1-αn)Unxn-αn(γnxn+(1-γn)Sxn)∥=αn∥γnxn+(1-γn)Txn-γnxn-(1-γn)Sxn∥=αn(1-γn)∥Txn-Sxn∥.
Since ∑n=1∞αn(1-γn)<∞, we have limn→∞∥xn-zn∥=0. From the continuity of ∥·∥, we get
(3.10)limn→∞∥zn-p∥=limn→∞∥xn-p∥=c.
Since  Un=βnI+(1-βn)S,  we get
(3.11)Unxn-Sxn=βnxn+(1-βn)Sxn-Sxn=βn(xn-Sxn)=βn(I-S)xn=βnAxn,Unxn-xn=βnxn+(1-βn)Sxn-xn=-(1-βn)(xn-Sxn)=-(1-βn)Axn.
So, we have from Lemma 2.5, Lemma 2.6 (;Ap=0), as well as (3.4), (3.11) that
(3.12)∥zn+1-p∥=∥(1-αn)Unxn+αn(γnxn+(1-γn)Sxn)-p∥2=∥Unxn-p-αn(Unxn-Sxn)+αnγn(xn-Sxn)∥2=∥Unxn-p-αnβnAxn+αnγnAxn∥2=∥Unxn-p-αn(βn-γn)Axn∥2≤∥Unxn-p∥2-2αn(βn-γn)〈Unxn-p,Axn〉+αn2(βn-γn)2∥Axn∥2≤∥xn-p∥2-2αn(βn-γn)〈Unxn-p,Axn-Ap〉+αn2(βn-γn)2∥Axn∥2=∥xn-p∥2-2αn(βn-γn)〈Unxn-xn,Axn-Ap〉-2αn(βn-γn)〈xn-p,Axn-Ap〉+αn2(βn-γn)2∥Axn∥2≤∥xn-p∥2+2αn(1-βn)(βn-γn)〈Axn,Axn-Ap〉+αn(βn-γn)(∥Axn∥2+∥Ap∥2-(2-k)∥Axn-Ap∥2)+αn2(βn-γn)2∥Axn∥2=∥xn-p∥2+2αn(1-βn)(βn-γn)∥Axn∥2-(1-k)αn(βn-γn)∥Axn∥2+αn2(βn-γn)2∥Axn∥2=∥xn-p∥2-αn(βn-γn)(1-k-2(1-βn)-αn(βn-γn))∥Axn∥2.
Hence
(3.13)αn(βn-γn)(1-k-2(1-βn)-αn(βn-γn))∥Axn∥2≤∥xn-p∥2-∥zn+1-p∥2.
Since  liminfn→∞αn(βn-γn)>0, we get
(3.14)limn→∞∥xn-Sxn∥=limn→∞∥Axn∥=0.
Since {xn} is a bounded sequence, there exists a subsequence {xni}⊂{xn} such that {xni} converges weakly to q. From Lemma 2.6, we obtain q∈F(S). To show our conclusion, it is sufficient to show that for another subsequence {xnj}⊂{xn} such that {xnj} converges weakly to v∈F(S), q=v. Before proving this, we show that for any z∈F(S), limn→∞∥xn-z∥ exists. Since
(3.15)Unz=βnz+(1-βn)Sz=z
and, from (3.2), we get
(3.16)∥Unxn-z∥2=∥Unxn-Unz∥2≤∥xn-z∥2
for all z∈F(S). Hence, we have
(3.17)∥zn+1-z∥=∥(1-αn)Unxn+αn(γnxn+(1-γn)Sxn)-z∥=∥(1-αn)(Unxn-z)+αn(γn(xn-z)+(1-γn)(Sxn-z))∥≤(1-αn)∥Unxn-z∥+αnγn∥xn-z∥+αn(1-γn)∥Sxn-z∥≤(1-αn)∥xn-z∥+αn∥xn-z∥+αn(1-γn)∥Sxn-z∥=∥xn-z∥+αn(1-γn)∥Sxn-z∥≤∥zn-z∥+∥xn-zn∥+αn(1-γn)∥Sxn-z∥
for all z∈F(S). From Lemma 2.7 and (3.9), limn→∞∥zn-z∥ exists. So, there exists limn→∞∥xn-z∥ for all z∈F(S) because limn→∞(xn-zn)=0. Suppose that q≠v. We have from Opial's theorem [9] that
(3.18)limn→∞∥xn-q∥=limj→∞∥xni-q∥<limj→∞∥xni-v∥=limn→∞∥xn-v∥=limj→∞∥xnj-v∥<limj→∞∥xnj-q∥=limn→∞∥xn-q∥.
This is a contradiction. So, {xn}  converges weakly to  q∈F(S).
To prove (2), let(3.19)zn+1=(1-αn)(βnxn+(1-βn)Txn)+αnVnxn
and  B=I-T.  It follows that
(3.20)∥xn+1-zn+1∥=∥(1-αn)Unxn+αnVnxn-(1-αn)(βnxn+(1-βn)Txn)-αnVnxn∥=(1-αn)∥βnxn+(1-βn)Sxn-βnxn-(1-βn)Txn∥≤(1-βn)∥Sxn-Txn∥.
So, from the boundedness of {xn}, {zn} is also bounded. Since T is a nonexpansive, by Lemma 2.3,  B is 1/2-inverse strongly monotone and  Bp=0,  we have
(3.21)∥zn+1-p∥2=∥(1-αn)(βnxn+(1-βn)Txn)+αnVnxn-p∥2=∥βn(xn-p)+(1-βn)(Txn-p)-αn(βn-γn)Bxn∥2=∥βn(xn-p)+(1-βn)(Txn-p)∥2-2αn(βn-γn)〈βn(xn-p)+(1-βn)(Txn-p),Bxn〉+αn2(βn-γn)2∥Bxn∥2≤(βn∥xn-p∥+(1-βn)∥Txn-p∥)2-2αnβn(βn-γn)〈xn-p,Bxn-Bp〉-2αn(1-βn)(βn-γn)〈Txn-p,Bxn-Bp〉+αn2(βn-γn)2∥Bxn∥2≤∥xn-p∥2-2αnβn(βn-γn)〈xn-p,Bxn-Bp〉-2αn(1-βn)(βn-γn)〈Txn-xn,Bxn-Bp〉-2αn(1-βn)(βn-γn)〈xn-p,Bxn-Bp〉+αn2(βn-γn)2∥Bxn∥2=∥xn-p∥2+2αn(1-βn)(βn-γn)〈Bxn,Bxn〉-2αn(βn-γn)〈xn-p,Bxn-Bp〉+αn2(βn-γn)2∥Bxn∥2≤∥xn-p∥2+2αn(1-βn)(βn-γn)∥Bxn∥2-αn(βn-γn)∥Bxn-Bp∥2+αn2(βn-γn)2∥Bxn∥2=∥xn-p∥2-αn(βn-γn)(1-2(1-βn)-αn(βn-γn))∥Bxn∥2
and hence
(3.22)αn(βn-γn)(1-2(1-βn)-αn(βn-γn))∥Bxn∥2≤∥xn-p∥2-∥zn+1-p∥2
for  p∈F(S)∩F(T). Summing from  n=1 to N, from (3.20), we have
(3.23)∑n=1Nαn(βn-γn)(2βn-1-αn(βn-γn))∥Bxn∥2≤∥x1-p∥2+∑n=1N-1(∥xn+1-p∥2-∥zn+1-p∥2)-∥zN+1-p∥2≤∥x1-p∥2+∑n=1N-1(∥xn+1-p∥+∥zn+1-p∥)∥xn+1-zn+1∥≤∥x1-p∥2+∑n=1N-1(1-βn)(∥xn+1-p∥+∥zn+1-p∥)∥Sxn-Txn∥≤∥x1-p∥2+M∑n=1N-1(1-βn),
where M=supn∈ℕ{(∥xn+1-p∥+∥zn+1-p∥)∥Sxn-Txn∥}. Letting N→∞, from ∑n=1∞(1-βn)<∞, we have
(3.24)∑n=1∞αn(βn-γn)(2βn-1-αn(βn-γn))∥Bxn∥2≤∥x1-p∥2+M∑n=1∞(1-βn)<∞.
Since  ∑n=1∞αn(βn-γn)(2βn-1-αn(βn-γn))=∞,  from  Lemma 2.8, we get
(3.25)liminfn→∞∥xn-Txn∥=liminfn→∞∥Bxn∥=0.
Since  T  is a nonexpansive mapping, from (3.1), we get
(3.26)∥Txn+1-xn+1∥=∥Txn+1-(1-αn)Unxn-αnVnxn∥≤∥Txn+1-Txn∥+(1-αn)∥Txn-Unxn∥+αn∥Txn-Vnxn∥≤∥xn+1-xn∥+(1-αn)∥Txn-Unxn∥+αn∥Txn-Vnxn∥≤(1-αn)∥Unxn-xn∥+αn∥Vnxn-xn∥+(1-αn)∥Txn-βnxn-(1-βn)Sxn∥+αn∥Txn-γnxn-(1-γn)Txn∥=(1-αn)∥βnxn+(1-βn)Sxn-xn∥+αn∥γnxn+(1-γn)Txn-xn∥+(1-αn)∥βn(Txn-xn)+(1-βn)(Txn-Sxn)∥+αn∥γn(Txn-xn)∥≤(1-αn)(1-βn)∥Sxn-xn∥+αn(1-γn)∥Txn-xn∥+(1-αn)∥Txn-xn∥+(1-αn)(1-βn)∥Txn-Sxn∥+αnγn∥Txn-xn∥≤∥Txn-xn∥+(1-βn)(∥Sxn-xn∥+∥Txn-Sxn∥).
Since ∑n=1∞‍(1-βn)<∞, from Lemma 2.7, there exists the limit of {∥Txn-xn∥}. Therefore, from (3.25), we get
(3.27)limn→∞∥Txn-xn∥=0.
Since {xn} is a bounded sequence, there exists a subsequence {xni}⊂{xn} such that {xni} converges weakly to q. Since a nonexpansive mapping T is demiclosed, we have q∈F(T). As in the proof of (1), {xn} converges weakly to q∈F(T).
(3) From (3.6) and (3.7), we have that, for any  p∈F(S)∩F(T),
(3.28)0≤∥xn-p∥2-∥xn+1-p∥2→c2-c2=0
as n→∞. We first show that {xn} converges weakly to some point in F(S). Actually, from (3.4) and (3.5), we have
(3.29)∥xn+1-p∥2≤(1-αn)∥Unxn-p∥2+αn∥Vnxn-p∥2≤(1-αn)∥Unxn-p∥2+αn∥xn-p∥2≤∥xn-p∥2
for p∈F(S)∩F(T). Hence, we get
(3.30)0≤∥xn-p∥2-(1-αn)∥Unxn-p∥2-αn∥xn-p∥2=(1-αn)(∥xn-p∥2-∥βnxn+(1-βn)Sxn-p∥2)≤∥xn-p∥2-∥xn+1-p∥2.
Since liminfn→∞(1-αn)>0, it follows from (3.28) and (3.30) that
(3.31)limn→∞(∥xn-p∥2-∥βnxn+(1-βn)Sxn-p∥2)=0.
From Lemma 2.2-(1), we have
(3.32)∥βnxn+(1-βn)Sxn-p∥2=∥βn(xn-p)+(1-βn)(Sxn-p)∥2=βn∥xn-p∥2+(1-βn)∥Sxn-p∥2-βn(1-βn)∥xn-Sxn∥2.
Since S is a k-strictly pseudo-nonspreading mapping, from (1.6), we get
(3.33)βn(1-βn)∥xn-Sxn∥2=βn∥xn-p∥2+(1-βn)∥Sxn-p∥2-∥βnxn+(1-βn)Sxn-p∥2≤βn∥xn-p∥2+(1-βn)(∥xn-p∥2+k∥xn-Sxn∥2)-∥βnxn+(1-βn)Sxn-p∥2=∥xn-p∥2+k(1-βn)∥xn-Sxn∥2-∥βnxn+(1-βn)Sxn-p∥2
and hence
(3.34)(1-βn)(βn-k)∥xn-p∥2≤∥xn-p∥2-∥βnxn+(1-βn)Sxn-p∥2
for p∈F(S)∩F(T). Since liminfn→∞(1-βn)>0, from (3.31), we get
(3.35)limn→∞∥xn-Sxn∥=0.
As in the proof of (1), from Lemma 2.6, we obtain that if {xni} converges weakly to v, then v∈F(S). We also show that such v is in F(T). In fact, from (3.4) and (3.5), we get
(3.36)∥xn+1-p∥2≤(1-αn)∥Unxn-p∥2+αn∥Vnxn-p∥2≤(1-αn)∥xn-p∥2+αn∥γnxn+(1-γn)Txn-p∥2≤∥xn-p∥2
for  p∈F(S)∩F(T). Hence, we have
(3.37)0≤∥xn-p∥2-(1-αn)∥xn-p∥2-αn∥γnxn+(1-γn)Txn-p∥2=αn(∥xn-p∥2-∥γnxn+(1-γn)Txn-p∥2)≤∥xn-p∥2-∥xn+1-p∥2.
Since liminfn→∞αn>0, it follows from (3.28) that
(3.38)limn→∞(∥xn-p∥2-∥γnxn+(1-γn)Txn-p∥2)=0.
Since T is a nonexpansive mapping, we have
(3.39)∥γnxn+(1-γn)Txn-p∥2=γn∥xn-p∥2+(1-γn)∥Txn-p∥2-γn(1-γn)∥xn-Txn∥2≤∥xn-p∥2-γn(1-γn)∥xn-Txn∥2,
and hence
(3.40)γn(1-γn)∥xn-Txn∥2≤∥xn-p∥2-∥γnxn+(1-γn)Txn-p∥2.
Since liminfn→∞γn(1-γn)>0, from (3.38), we get
(3.41)limn→∞∥xn-Txn∥=0.
Since {xni} converges weakly to q, we have q∈F(T). Let {xnj} be another subsequence of {xn} such that {xnj} converges weakly to v. Then, we have q=v. In fact, if q≠v, then we obtain
(3.42)limn→∞∥xn-q∥=limi→∞∥xni-q∥<limi→∞∥xni-v∥=limn→∞∥xn-v∥=limj→∞∥xnj-v∥<limj→∞∥xnj-q∥=limn→∞∥xn-q∥.
This is a contradiction. So, we have q=v. Therefore, we conclude that {xn} converges weakly to q∈F(S)∩F(T).For nonspreading mappingS, we have k=0 reduces the followings.Corollary 3.2 (cf. [2]).
LetC be a nonempty closed convex subset of a Hilbert space H and let S:C→C be a nonspreading mapping and let T:C→C be a nonexpansive mapping such that F(S)∩F(T)≠∅. Define a sequence {xn} as follows:
(3.43)x1∈C,xn+1=(1-αn)(βnxn+(1-βn)Sxn)+αn(γnxn+(1-γn)Txn)
for all n∈ℕ, where {αn}, {βn}, {γn}⊂[0,1]. Then, the followings hold:  (1)
ifliminfn→∞αn(βn-γn)>0, ∑n=1∞αn(1-γn)<∞ and 1<(2-αn)βn+αnγn, then {xn}  converges weakly  to  q∈F(S),(2)
ifβn>γn,  ∑n=1∞(1-βn)<∞,  2βn-1-αn(βn-γn)>0 and liminfn→∞αn(βn-γn)(2βn-1-αn(βn-γn))>0, then {xn} converges weakly to q∈F(T),(3)
ifliminfn→∞αn>0,  liminfn→∞(1-αn)>0,  liminfn→∞(1-βn)>0  and  liminfn→∞γn(1-γn)>0, then {xn}  converges weakly  to  q∈F(S)∩F(T).Corollary 3.3 (cf. [2, 10]).
LetC be a nonempty closed convex subset of a Hilbert space H and let  S:C→C be a nonspreading mapping such that F(S)≠∅. Define a sequence  {xn} as follows:
(3.44)x1∈C,xn+1=αnxn+(1-αn)Sxn
for all n∈ℕ, where {αn}⊂[0,1]. If liminfn→∞αn>0, then {xn} converges weakly to q∈F(S).Proof.
Puttingβn=0,γn=1  for  n∈ℕ  in Theorem 3.1, we get the conclusion.Corollary 3.4.
LetC be a nonempty closed convex subset of a Hilbert space H and let T:C→C be a nonexpansive mapping such that  F(T)≠∅. Define a sequence {xn} as follows:
(3.45)x1∈C,xn+1=(1-αn)xn+αnTxn
for alln∈ℕ, where {αn}⊂[0,1]. If ∑n=1∞αn=∞, then {xn} converges weakly to q∈F(T).Proof.
Puttingβn=1,γn=0 for n∈ℕ in Theorem 3.1, we get the conclusion.


## 4. Some Open Problems

LetS be a semigroup. We denote by l∞(S) the Banach space of all bounded real-valued functional on S with supremum norm. For each s∈S, we define the left and right translation operators  l(s)  and  r(s) on l∞(S) by
(4.1)(l(s)f)(t)=f(st),(r(s)f)(t)=f(ts)
for  each t∈S and f∈l∞(S), respectively. Let X be a subspace of l∞(S) containing 1. An element μ in the dual space  X*  of X is said to be a mean on X if ∥μ∥=μ(1)=1. For s∈S, we can define a point evaluation  δs  by  δs(f)=f(s) for each f∈X. It is well known that μ is mean on  X  if and only if
(4.2)infs∈Sf(s)≤μ(f)≤sups∈Sf(s)
for each f∈X.LetX be a translation invariant subspace of l∞(S) (i.e.,  l(s)X⊂X  and r(s)X⊂X for each s∈S) containing 1. Then, a mean μ on X is said to be left invariant (resp. right invariant) if
(4.3)μ(l(s)f)=μ(f)(resp.μ(r(s)f)=μ(f))
for each s∈S and f∈X. A mean μ on X is said to be invariant if μ is both left and right invariant [11–14]. X is said to be left (resp., right) amenable if X has a left (resp. right) invariant mean. X is amenable if X is left and right amenable. Moreover, l∞(S) is amenable when S is a commutative semigroup or a solvable group. However, the free group or semigroup of two generators is not left or right amenable (see [15–17]). A net {μα} of means on X is said to be asymptotically left (resp., right) invariant if
(4.4)limα(μα(l(s)f)-μα(f))=0(resp.limα(μα(r(s)f)-μα(f))=0)
for each f∈X and s∈S, and it is said to be  left (resp.,  right)  strongly asymptotically invariant (or  strong regular) if
(4.5)limα∥l*(s)μα-μα∥=0(resp.limα∥r*(s)μα-μα∥=0)
for each s∈S, where l*(s) and r*(s) are the adjoint operators of l(s) and r(s), respectively. Such nets were first studied by Day in [15] where they were called weak*invariant and norm invariant, respectively.It is easy to see that if a semigroupS is left (resp. right) amenable, then the semigroup S'=S∪{e} where es′=s′e=s′ for all s′∈S is also left (resp. right) amenable and conversely.From now onS  denotes a semigroup with an identity e. S is called left reversible if any two right ideals of S have nonvoid intersection, that is, aS∩bS≠∅ for a,b∈S. In this case, (S,⪯) is a directed system when the binary relation “⪯” on  S is defined by a⪯b if and only if  {a}∪aS⊇{b}∪bS for a,b∈S. It is easy to see that t⪯ts  for all  t,s∈S. Further, if  t⪯s,  then  pt⪯ps for all p∈S. The class of left reversible semigroup includes all groups and commutative semigroups. If a semigroup S is left amenable, then S is left reversible. But the converse is not true ([18–23]).LetS be a semigroup and let C be a closed and convex subset of E. Let F(T) denote the fixed point set of T. Then, ℑ={T(s):s∈S} is called a representation ofSas nonexpansive mappings onC if T(s) is nonexpansive with T(e)=I and T(st)=T(s)T(t) for each s,t∈S. We denote by F(ℑ) the set of common fixed points of {T(s):s∈S}, that is,
(4.6)F(ℑ)=⋂s∈S‍F(T(s))=⋂s∈S‍{x∈C:T(s)x=x}.We know that ifμ is a mean on X and if for each x*∈E* the function s↦〈T(s)x,x*〉 is contained in X and C is weakly compact, then there exists a unique point x0 of E such that μ〈T(·)x,x*〉=〈x0,x*〉 for each x*∈E*. We denote such a point x0 by Tμx. Note that Tμx is contained in the closure of the convex hull of {T(s)x:s∈S} for each x∈C. Note that Tμz=z for each z∈F(ℑ); see [12–14, 17, 20, 22].Open Problems..
(1)
Does Theorem3.1 hold for a Banach space? (cf. [24, 25])(2)
Does Theorem3.1 hold a commutative or amenable or reversible semigroup of nonexpansive mappings in place of T using asymptotically invariant nets?



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*Source: 594218-2012-08-30.xml*