R E S E A R C H
Open Access
Simple projection algorithm for a countable
family of weak relatively nonexpansive
mappings and applications
Jingling Zhang, Yongfu Su
*and Qingqing Cheng
*Correspondence:
[email protected]; [email protected]
Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, China
Abstract
LetEbe a uniformly convex and uniformly smooth Banach space, letCbe a nonempty closed convex subset ofE, let{Tn}:C→Cbe a countable family of weak relatively nonexpansive mappings such thatF=∞n=1F(Tn)=∅. For any given gauss x0∈C, define a sequence{xn}inCby the following algorithm:
⎧ ⎪ ⎨ ⎪ ⎩
C0=C,
Cn+1={z∈Cn:
φ(
z,Tnxn) =φ
(z,xn)}, n= 0, 1, 2, 3,. . .,xn+1=
Cn+1x0.
Then{xn}converges strongly toq=
Fx0.
MSC: 47H05; 47H09; 47H10
Keywords: relatively nonexpansive mapping; weak relatively nonexpansive mapping; generalized projection; projection algorithm; monotone hybrid algorithm
1 Introduction
LetEbe a real Banach space with the dual E*. We denote byJ the normalized duality mapping fromEto E* defined by
Jx=f∈E*:x,f= x = f ,
where·,·denotes the generalized duality pairing. The duality mappingJhas the follow-ing properties: () ifEis smooth, thenJis single-valued; () ifEis strictly convex, thenJis one-to-one; () ifEis reflexive, thenJis surjective; () ifEis uniformly smooth, thenJis uniformly norm-to-norm continuous on each bounded subset ofE; () ifE*is uniformly convex, thenJis uniformly continuous on bounded subsets ofEandJis singe-valued and also one-to-one (see [–]).
LetEbe a smooth Banach space with the dualE*. The functionalφ:E×E→Ris defined by
φ(x,y) = x – x,Jy+ y ,
for allx,y∈E.
LetCbe a closed convex subset ofE, and letT be a mapping fromCinto itself. We denote byF(T) the set of fixed points ofT. A pointpinCis said to be an asymptotic fixed point ofT [] ifCcontains a sequence{xn}which converges weakly topsuch that the stronglimn→∞(xn–Txn) = . The set of asymptotic fixed points ofT will be denoted byF(T). A mappingT fromCinto itself is called nonexpansive if Tx–Ty ≤ x–y for allx,y∈Cand relatively nonexpansive ifF(T) =F(T) andφ(p,Tx)≤φ(p,x) for allx∈C
andp∈F(T). The asymptotic behavior of a relatively nonexpansive mapping was studied in [, –].
Three classical iteration processes are often used to approximate a fixed point of a non-expansive mapping. The first one was introduced in by Mann [] and is well known as Mann’s iteration process defined as follows:
⎧ ⎨ ⎩
x chosen arbitrarily,
xn+=αnxn+ ( –αn)Txn, n≥,
(.)
where the sequence{αn}is chosen in [, ]. Fourteen years later, Halpern [] proposed the new innovation iteration process which resembled Mann’s iteration (.). It is defined by
⎧ ⎨ ⎩
x chosen arbitrarily,
xn+=αnu+ ( –αn)Txn, n≥,
(.)
where the elementu∈Cis fixed. Seven years later, Ishikawa [] enlarged and improved Mann’s iteration (.) to the new iteration method, which is often cited as Ishikawa’s iter-ation process and defined recursively by
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
x chosen arbitrarily,
yn=βnxn+ ( –βn)Txn,
xn+=αnxn+ ( –αn)Tyn, n≥,
(.)
where{αn}and{βn}are sequences in the interval [, ].
In both Hilbert space [–] and uniformly smooth Banach space [–] the iteration process (.) has been proved to be strongly convergent if the sequence{αn}satisfies the following conditions:
(i) αn→; (ii) ∞n=αn=∞;
(iii) ∞n=|αn+–αn|<∞orlimn→∞ααnn+= .
By the restriction of condition (ii), it is widely believed that Halpern’s iteration process (.) has slow convergence though the rate of convergence has not been determined. Halpern [] proved that conditions (i) and (ii) are necessary in the strong convergence of (.) for a nonexpansive mappingTon a closed convex subsetCof a Hilbert spaceH. Moreover, Wittmann [] showed that (.) converges strongly toPF(T)uwhen{αn}satisfies (i), (ii) and (iii), wherePF(T)(·) is the metric projection ontoF(T).
while the process (.) can still converge for a Lipschitz pseudo-contractive mapping in a Hilbert space []. For example, Reich [] proved that ifEis a uniformly convex Banach space with the Fréchet differentiable norm and if{αn}is chosen such that∞n=αn( –αn) = ∞, then the sequence{xn}defined by (.) converges weakly to a fixed point ofT. However, we note that Mann’s iteration process (.) has only weak convergence even in a Hilbert space [].
Some attempts to modify the Mann iteration method so that strong convergence is guar-anteed have recently been made. Nakajo and Takahashi [] proposed the following mod-ification of the Mann iteration method for a single nonexpansive mappingT in a Hilbert spaceH:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C chosen arbitrarily,
yn=αnxn+ ( –αn)Txn,
Cn={z∈C: yn–z ≤ xn–z },
Qn={z∈C:xn–z,x–xn ≥},
xn+=PCn∩Qn(x),
(.)
whereCis a closed convex subset ofH,PK denotes the metric projection fromH onto a closed convex subsetKofH. They proved that if the sequence{αn}is bounded above from one, then the sequence{xn}generated by (.) converges strongly toPF(T)(x), where
F(T) denotes the fixed point set ofT.
The ideas to generalize the process (.) from a Hilbert space to a Banach space have recently been made. By using available properties on a uniformly convex and uniformly smooth Banach space, Matsushita and Takahashi [] presented their ideas as the following method for a single relatively nonexpansive mappingTin a Banach spaceE:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C chosen arbitrarily,
yn=J–(αnJx+ ( –αn)JTxn),
Cn={z∈C:φ(z,yn)≤φ(z,xn)},
Qn={z∈C:xn–z,Jx–Jxn ≥},
xn+=Cn∩Qn(x).
(.)
They proved the following convergence theorem.
Theorem MT Let E be a uniformly convex and uniformly smooth Banach space,let C be a nonempty closed convex subset of E,let T be a relatively nonexpansive mapping from C into itself,and let{αn}be a sequence of real numbers such that≤αn< andlim supn→∞αn< .
Suppose that{xn}is given by(.),where J is the duality mapping on E.If F(T)is nonempty,
then{xn}converges strongly toF(T)x,whereF(T)(·)is the generalized projection from C
onto F(T).
Theorem SK Let E be a uniformly convex and uniformly smooth real Banach space,let C
be a nonempty closed convex subset of E,let T,S be two relatively nonexpansive mappings from C into itself with F:=F(T)∩F(S)is nonempty.Let a sequence{xn}be defined by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C chosen arbitrarily,
yn=J–(αnJxn+ ( –αn)Jzn),
zn=J–(βn()Jxn+βn()JTxn+βn()JSxn),
Hn={z∈C:φ(z,yn)≤φ(z,xn)},
Wn={z∈C:xn–z,Jx–Jxn ≥},
xn+=Hn∩Wn(x)
(.)
with the following restrictions: (i) ≤αn< ,lim supn→∞αn< ;
(ii) ≤βn(),βn(),βn()≤,limn→∞βn()= ,lim infn→∞βn()βn()> .
Then{xn}converges strongly toFx,whereFis the generalized projection from C onto F.
Theorem SK Let E be a uniformly convex and uniformly smooth Banach space,let C be
a nonempty closed convex subset of E,let T,S be two relatively nonexpansive mappings from C into itself with F:=F(T)∩F(S)is nonempty.Let a sequence{xn}be defined by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C chosen arbitrarily,
yn=J–(αnJx+ ( –αn)Jzn),
zn=J–(βn()Jxn+βn()JTxn+βn()JSxn),
Hn={z∈C:φ(z,yn)≤φ(z,xn) +αn( x + z,Jxn–Jx)},
Wn={z∈C:xn–z,Jx–Jxn ≥},
xn+=Hn∩Wn(x)
(.)
with the following restrictions: (i) <αn< ,lim supn→∞αn< ;
(ii) ≤βn(),βn(),βn()≤,limn→∞βn()= ,lim infn→∞βn()βn()> .
Then{xn}converges strongly toFx,whereFis the generalized projection from C onto F.
In , Su, Xu and Zhang [] proposed the following hybrid algorithms for two count-able families of weak relatively nonexpansive mappings in a Banach space and proved the following convergence theorems.
Theorem SKZ Let E be a uniformly convex and uniformly smooth real Banach space,
(∞n=F(Sn))=∅.Define a sequence{xn}in C by the following algorithm:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C chosen arbitrarily,
zn=J–(βn()Jxn+βn()JTnxn+βn()JSnxn),
yn=J–(αnJxn+ ( –αn)Jzn),
Cn={z∈Cn–∩Qn–:φ(z,yn)≤φ(z,xn)},
C={z∈C:φ(z,y)≤φ(z,x)},
Qn={z∈Cn–∩Qn–:xn–z,Jx–Jxn ≥},
Q=C,
xn+=Cn∩Qn(x)
(.)
with the conditions
(i) lim infn→∞βn()βn()> ; (ii) lim infn→∞βn()βn()> ;
(iii) ≤αn≤α< for someα∈(, ).
Then{xn}converges strongly toFx,whereFis the generalized projection from C onto F.
Unfortunately, in recent years, many hybrid algorithms have been very complex, so these complex algorithms are not applicable or are very difficult in applications. Naturally, we hope to obtain some simple and practical algorithms. The purpose of this article is to present a simple projection algorithm for a countable family of weak relatively nonexpan-sive mappings and to prove strong convergence theorems in Banach spaces.
In addition, we shall give an example which is a countable family of weak relatively non-expansive mappings, but not a countable family of relatively nonnon-expansive mappings.
2 Preliminaries
LetEbe a smooth Banach space with the dualE*. The functionalφ:E×E→Ris defined by
φ(x,y) = x – x,Jy+ y , (.)
for allx,y∈E. Observe that in a Hilbert spaceH, (.) reduces toφ(x,y) = x–y ,x,y∈H. Recall that ifCis a nonempty, closed and convex subset of a Hilbert spaceHandPC:
H→Cis the metric projection ofHontoC, thenPCis nonexpansive. This is true only whenHis a real Hilbert space. In this connection, Alber [] has recently introduced a generalized projection operatorCin a Banach spaceEwhich is an analogue of the metric projection in Hilbert spaces. The generalized projectionC:E→Cis a map that assigns to an arbitrary pointx∈Ethe minimum point of the functionalφ(y,x), that is,Cx=x¯, wherex¯is the solution to the minimization problem
φ(¯x,x) =min
y∈Cφ(y,x). (.)
obvious from the definition of the functionalφthat
x – y ≤φ(y,x)≤ y + x (.)
and
φ(x,y) =φ(x,z) +φ(z,y) – x–z,Jz–Jy (.)
for allx,y∈E. See [] for more details.
This section collects some definitions and lemmas which will be used in the proofs for the main results in the next section. Some of them are known; others are not hard to derive.
Remark . IfEis a reflexive strictly convex and smooth Banach space, then forx,y∈E,
φ(x,y) = if and only ifx=y. It is sufficient to show that ifφ(x,y) = thenx=y. From (.), we have x = y . This impliesx,Jy= x = Jy . From the definition ofJ, we haveJx=Jy. SinceJis one-to-one, then we havex=y; see [, , ] for more details.
In this paper, we give the definitions of a countable family of relatively nonexpansive mappings and a countable family of weak relatively nonexpansive mappings which are generalizations of a relatively nonexpansive mapping and a weak relatively nonexpansive mapping respectively. We also give an example which is a countable family of weak rela-tively nonexpansive mappings, but not a countable family of relarela-tively nonexpansive map-pings.
LetCbe a closed convex subset ofE, and let{Tn}∞n=be a countable family of mappings from Cinto itself. We denote byF the set of common fixed points of{Tn}∞n=. That is
F=∞n=F(Tn), whereF(Tn) denotes the set of fixed points ofTnfor alln≥. A pointp inCis said to be anasymptotic fixed pointof{Tn}n∞=ifCcontains a sequence{xn}which converges weakly topsuch thatlimn→∞ Tnxn–xn = . The set of asymptotic fixed points of{Tn}∞n=will be denoted byF({Tn}∞n=). A pointpinCis said to be astrong asymptotic fixed pointof{Tn}∞n=ifCcontains a sequence{xn}which converges strongly topsuch thatlimn→∞ Tnxn–xn = . The set of strong asymptotic fixed points of{Tn}∞n=will be denoted byF({Tn}∞n=).
Definition . The countable family of mappings{Tn}∞n=is said to bea countable family of relatively nonexpansive mappingsif the following conditions are satisfied:
() F({Tn}∞n=)is nonempty;
() φ(u,Tnx)≤φ(u,x),∀u∈F(Tn),x∈C,n≥; () F({Tn}∞n=) =∞n=F(Tn).
Definition . The countable family of mappings{Tn}∞n=is said to bea countable family of weak relatively nonexpansive mappingsif the following conditions are satisfied:
() F({Tn}∞n=)is nonempty;
() φ(u,Tnx)≤φ(u,x),∀u∈F(Tn),x∈C,n≥; () F({Tn}∞n=) =∞n=F(Tn).
() F(T)is nonempty;
() φ(u,Tx)≤φ(u,x),∀u∈F(T),x∈C; () F(T) =F(T).
Definition . The mappingT is said to bea weak relatively nonexpansive mappingif the following conditions are satisfied:
() F(T)is nonempty;
() φ(u,Tx)≤φ(u,x),∀u∈F(T),x∈C; () F(T) =F(T).
Definition . (Definition .) is a special form of Definition . (Definition .) asTn≡
T for alln≥.
The hybrid algorithms for a fixed point of relatively nonexpansive mappings and appli-cations have been studied by many authors; see, for example, [, , , , , ]. In recent years, the definition of a weak relatively nonexpansive mapping has been presented and studied by many authors [, , , ], but they have not given an example of a mapping which is weak relatively nonexpansive, but not relatively nonexpansive.
In the next section, we shall give an example which is a countable family of weak rela-tively nonexpansive mappings, but not a countable family of relarela-tively nonexpansive map-pings.
We need the following lemmas for the proof of our main results.
Lemma .[] Let E be a uniformly convex and smooth real Banach space and let{xn},
{yn}be two sequences of E.Ifφ(xn,yn)→and either{xn}or{yn}is bounded,then xn–
yn →.
Lemma .[, , ] Let C be a nonempty closed convex subset of a smooth real Banach space E and x∈E.Then,x=Cx if and only if
x–y,Jx–Jx ≥ for all y∈C.
Lemma .[, , ] Let E be a reflexive,strictly convex and smooth real Banach space,
let C be a nonempty closed convex subset of E and let x∈E.Then
φ(y,cx) +φ(cx,x)≤φ(y,x) for all y∈C.
Lemma .[] Let E be a uniformly convex Banach space and Br() ={x∈E: x ≤r}
be a closed ball of E. Then there exists a continuous strictly increasing convex function g: [,∞)→[,∞)with g() = such that
λx+μy+γz ≤λ x +μ y +γ z –λμg x–y (.)
for all x,y,z∈Br()andλ,μ,γ ∈[, ]withλ+μ+γ = .
Lemma . Let E be a strictly convex and smooth real Banach space,let C be a closed convex subset of E,and let T be a weak relatively nonexpansive mapping from C into itself.
Then F(T)is closed and convex.
3 Main results
Firstly, we give an example which is a countable family of weak relatively nonexpansive mappings, but not a countable family of relatively nonexpansive mappings in the Banach spacel.
Example LetE=l, where
l=
ξ= (ξ,ξ,ξ, . . . ,ξn, . . .) :
∞
n=
|xn|<∞
,
ξ =
∞
n= |ξn|
, ∀ξ∈l,
ξ,η=
∞
n=
ξnηn, ∀ξ= (ξ,ξ,ξ, . . . ,ξn, . . .),η= (η,η,η, . . . ,ηn. . .)∈l.
It is well known thatlis a Hilbert space, so (l)*=l. Let{xn} ⊂Ebe a sequence defined by
x= (, , , , . . .),
x= (, , , , . . .),
x= (, , , , , . . .),
x= (, , , , , , . . .),
· · ·
xn= (ξn,,ξn,,ξn,, . . . ,ξn,k, . . .),
· · ·,
where
ξn,k=
⎧ ⎨ ⎩
ifk= ,n+ ,
ifk= ,k=n+ ,
for alln≥.
Define a countable family of mappingsTn:E→Eas follows:
Tn(x) =
⎧ ⎨ ⎩
n
n+xn ifx=xn, –x ifx=xn,
Conclusion . {xn}converges weakly to x.
Proof For anyf = (ζ,ζ,ζ, . . . ,ζk, . . .)∈l= (l)*, we have
f(xn–x) =f,xn–x=
∞
k=
ζkξn,k=ζn+→,
asn→ ∞. That is,{xn}converges weakly tox.
Conclusion . {xn} is not a Cauchy sequence,so, it does not converge strongly to any element of l.
Proof In fact, we have xn–xm = √
for anyn=m. Then{xn}is not a Cauchy sequence.
Conclusion . Tnhas a unique fixed point,that is,F(Tn) ={}for all n≥.
Proof The conclusion is obvious.
Conclusion . xis an asymptotic fixed point of{Tn}∞n=.
Proof Since{xn}converges weakly toxand
Tnxn–xn =
nn+ xn–xn
=
n+ xn →
asn→ ∞, so,xis an asymptotic fixed point of{Tn}∞n=. Conclusion . {Tn}∞n=has a unique strong asymptotic fixed point,so,
∞
n=F(Tn) =
F({Tn}∞n=).
Proof In fact, for any strong convergent sequence{zn} ⊂Esuch thatzn→zand zn–
Tnzn → asn→ ∞, from Conclusion ., there exists a sufficiently large natural number
N such that zn=xm for anyn,m>N. ThenTzn= –znforn>N, it follows from zn–
Tnzn → that zn→ and hencezn→z= .
Conclusion . {Tn}∞n=is a countable family of weak relatively nonexpansive mappings.
Proof SinceE=Lis a Hilbert space, for anyn≥, we have
φ(,Tnx) = –Tnx = Tnx
≤ x = x– =φ(,x), ∀x∈E.
From Conclusion ., we have∞n=F(Tn) =F({Tn}∞n=), then{Tn}∞n=is a countable family
of weak relatively nonexpansive mappings.
Proof From Conclusions . and ., we have∞n=F(Tn)=F({Tn}∞n=), so,{Tn}∞n=is not
a countable family of relatively nonexpansive mappings.
Secondly, we give another example which is a weak relatively nonexpansive mapping, but not a relatively nonexpansive mapping in the Banach spacel.
Example LetE=l, where
l=
ξ= (ξ,ξ,ξ, . . . ,ξn, . . .) :
∞
n=
|xn|<∞
,
ξ =
∞
n= |ξn|
, ∀ξ∈l,
ξ,η=
∞
n=
ξnηn, ∀ξ= (ξ,ξ,ξ, . . . ,ξn, . . .),η= (η,η,η, . . . ,ηn. . .)∈l.
It is well known thatlis a Hilbert space, so (l)*=l. Let{xn} ⊂Ebe a sequence defined by
x= (, , , , . . .),
x= (, , , , . . .),
x= (, , , , , . . .),
x= (, , , , , , . . .),
· · ·
xn= (ξn,,ξn,,ξn,, . . . ,ξn,k, . . .),
· · ·,
where
ξn,k=
⎧ ⎨ ⎩
ifk= ,n+ ,
ifk= ,k=n+ ,
for alln≥. Define the mappingT:E→Eas follows
T(x) =
⎧ ⎨ ⎩
n
n+xn ifx=xn(∃n≥), –x ifx=xn(∀n≥).
Conclusion . {xn}converges weakly to x.
Proof For anyf = (ζ,ζ,ζ, . . . ,ζk, . . .)∈l= (l)*, we have
f(xn–x) =f,xn–x=
∞
k=
ζkξn,k=ζn+→,
Conclusion . {xn}is not a Cauchy sequence,so,it does not converge strongly to any element of l.
Proof In fact, we have xn–xm = √
for anyn=m. Then{xn}is not a Cauchy sequence.
Conclusion . T has a unique fixed point,that is,F(T) ={}.
Proof The conclusion is obvious.
Conclusion . xis an asymptotic fixed point of T.
Proof Since{xn}converges weakly toxand
Txn–xn =
nn+ xn–xn
=
n+ xn →
asn→ ∞, thenxis an asymptotic fixed point ofT.
Conclusion . T has a unique strong asymptotic fixed point,so,F(T) =F(T).
Proof In fact, for any strong convergent sequence{zn} ⊂Esuch thatzn→zand zn–
Tzn → asn→ ∞, from Conclusion ., there exists a sufficiently large natural number N such thatzn=xm, for anyn,m>N. ThenTzn= –znforn>N, it follows from zn–
Tzn → that zn→ and hencezn→z= .
Conclusion . T is a weak relatively nonexpansive mapping.
Proof SinceE=Lis a Hilbert space, we have
φ(,Tx) = –Tx = Tx
≤ x = x– =φ(,x), ∀x∈E.
From Conclusion ., we haveF(T) =F(T), thenT is a weak relatively nonexpansive
mapping.
Conclusion . T is not a relatively nonexpansive mapping.
Proof From Conclusions . and ., we haveF(T)=F(T), so,Tis not a relatively
non-expansive mapping.
Next, we prove our convergence theorems as follows.
define a sequence{xn}in C by the following algorithm:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
C=C,
Cn+={z∈Cn:φ(z,Tnxn)≤φ(z,xn)}, n= , , , , . . . ,
xn+=Cn+x.
(.)
Then{xn}converges strongly to q=Fx.
Proof Firstly,Cnis closed and convex. SinceT is a closed hemi-relatively nonexpansive mapping, thenF(T)⊆Cn,n= , , , , . . . .
Sincexn=CnxandCn⊂Cn–, then we get
φ(xn,x)≤φ(xn+,x), for alln≥. (.)
Therefore,{φ(xn,x)}is nondecreasing. On the other hand, by Lemma . we have
φ(xn,x) =φ(Cnx,x)
≤φ(p,x) –φ(p,xn)≤φ(p,x),
for allp∈F(T)⊂Cnand for alln≥. Therefore,φ(xn,x) is also bounded. This together with (.) implies that the limit of{φ(xn,x)}exists. Put
lim
n→∞φ(xn,x) =d. (.)
From Lemma ., we have, for any positive integerm, that
φ(xn+m,xn) =φ(xn+m,Cnx)
≤φ(xn+m,x) –φ(Cnx,x)
=φ(xn+m,x) –φ(xn+,x),
for alln≥. This together with (.) implies that
lim
n→∞φ(xn+m,xn) =
holds, uniformly for allm. By using Lemma ., we get that
lim
n→∞ xn+m–xn =
holds, uniformly for allm. Then{xn}is a Cauchy sequence. Therefore, there exists a point
p∈Csuch thatxn→p.
Sincexn+=Cn+x⊂Cn+⊂Cn, then
φ(xn+,Tnxn)≤φ(xn+,xn), n= , , , , . . . .
Finally, we prove thatp=Fx. From Lemma ., we have
φ(p,Fx) +φ(Fx,x)≤φ(p,x).
On the other hand, sincexn+=Cn+xandF⊂Cnfor alln, also from Lemma ., we have
φ(Fx,xn+) +φ(xn+,x)≤φ(Fx,x). (.)
By the definition ofφ(x,y), we know that
lim
n→∞φ(xn+,x) =φ(p,x). (.)
Combining (.) and (.), we know thatφ(p,x) =φ(Fx,x). Therefore, it follows from the uniqueness ofFxthatp=Fx. This completes the proof.
Theorem . Let E be a uniformly convex and uniformly smooth Banach space,let C be a nonempty closed convex subset of E,let{Tn}:C→C be a countable family of weak relatively nonexpansive mappings such that F=∞n=F(Tn)=∅.For any given gauss x∈C,
define a sequence{xn}in C by the following algorithm:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
C=C,
Cn+={z∈Cn:φ(z,Tnxn) =φ(z,xn)}, n= , , , , . . . ,
xn+=Cn+x.
(.)
Then{xn}converges strongly to q=Fx.
Proof Let{xn}be defined by (.). We claim that
xn+∈
z∈Cn:φ(z,Tnxn) =φ(z,xn)
, n= , , , , . . . .
Therefore,
xn+={z∈Cn:φ(z,Tnxn)=φ(z,xn)}x. If not, there existsxn+such that
φ(xn+,Tnxn) <φ(xn+,xn).
We define
z(t) = ( –t)xn++tx∈C, t∈[, ].
Observe thatz() =xn+. Sinceφ(·,Tnxn),φ(·,xn) are continuous, then there existst∈ (, ) such that
φz(t),Tnxn
<φz(t),xn
that is,z(t)∈Cn+. On the other hand, we have
φz(t),x
=φ( –t)xn++tx,x
=( –t)xn++tx
– ( –t)xn++tx,Jx
+ x
≤( –t) xn+ +t x
– ( –t)xn+,Jx– tx,Jx+ x
= ( –t) xn+ +t x
– ( –t)xn+,Jx– t x + x
= ( –t) xn+ + ( –t) x
– ( –t)xn+,Jx
= ( –t)φ(xn+,x) <φ(xn+,x).
This is a contradiction toxn+=Cn+xandz(t)∈Cn+. This completes the proof.
4 Applications
Now, we apply Theorem . to prove a strong convergence theorem concerning maximal monotone operators in a Banach spaceE.
LetAbe a multi-valued operator fromEtoE*with the domainD(A) ={z∈E:Az=∅} and rangeR(A) ={z∈E:z∈D(A)}. An operator A is said to be monotone if
x–x,y–y ≥
for eachx,x∈D(A) andy∈Ax,y∈Ax. A monotone operator A is said to be maximal if its graph G(A) ={(x,y) :y∈Ax} is not properly contained in the graph of any other monotone operator. We know that ifAis a maximal monotone operator, then A– is closed and convex. The following result is also well known.
Theorem .(Rockafellar []) Let E be a reflexive,strictly convex and smooth Banach space,and let A be a monotone operator from E to E*.Then A is maximal if and only if
R(J+rA) =E*for all r> .
LetEbe a reflexive, strictly convex and smooth Banach space, and letAbe a maximal monotone operator fromEtoE*. Using Theorem . and strict convexity ofE, we obtain that for everyr> andx∈E, there exists a uniquexrsuch that
Jx∈Jxr+rAxr.
Then we can define a single valued mappingJr:E→D(A) byJr= (J+rA)–Jand such aJr is called the resolvent ofA. We know thatA–=F(J
Theorem . Let E be a uniformly convex and uniformly smooth real Banach space,let
A be a maximal monotone operators from E to E*such that A–=∅,let Jrbe the
resol-vent of A,where r> .For any given gauss x∈C=C,define a sequence{xn}in C by the
following algorithm:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
C=C,
Cn+={z∈Cn:φ(z,Jrnxn) =φ(z,xn)}, n= , , , , . . . , xn+=Cn+x,
with the condition,rn> ,lim infn→∞rn> .Then{xn}converges strongly to q=A–x.
Proof We only need to prove that{Jrn}∞n=, is a countable family of weak relatively nonex-pansive mappings.
Firstly, we have∞n=F(Jrn) =A–=∅. Secondly, from the monotonicity ofA, we have
φ(p,Jrnw) = p – p,JJrnw+ Jrnw
= p + p,Jw–JJrnw–Jw+ Jrnw
= p + p,Jw–JJrnw– p,Jw+ Jrnw
= p – Jrnw–p–Jrnw,Jw–JJrnw–Jw
– p,Jw+ Jrnw
= p – Jrnw–p,Jw–JJrnw–Jw
+ Jrnw,Jw–JJrnw– p,Jw+ Jrnw
≤ p + Jrnw,Jw–JJrnw– p,Jw+ Jrnw
= p – p,Jw+ w – Jrnw
+ Jrnw,Jw– w
=φ(p,w) –φ(Jrnw,w)
≤φ(p,w)
for all n≥. Thirdly, we prove the set of strong asymptotic fixed pointsF({Jrn}∞n=) =
∞
n=F(Jrn) =A–.
We first show thatF({Jrn}∞n=)⊂A–. Letp∈F({Jrn}∞n=), then there exists{zn} ⊂Esuch thatzn→pandlimn→∞ zn–Jrnzn = . SinceJis uniformly norm-to-norm continuous
on bounded sets, we obtain
rn
(Jzn–JJrnzn)→.
It follows from
rn
and the monotonicity ofAthat
w–Jrnzn,w*–
rn
(Jzn–JJrnzn)
≥
for allw∈D(A) andw*∈Aw. Lettingn→ ∞, we havew–p,w* ≥ for allw∈D(A) and
w*∈Aw. Therefore, from the maximality ofA, we obtainp∈A–. On the other hand, we know thatF(Jrn) =A–,F(Jrn)⊂F(Jrn) for alln≥; therefore,A– =
∞
n=F(Jrn) = F(∞n=Jrn). From above three conclusions, we have proved{Jrn}∞n=is a countable family of weak relatively nonexpansive mappings. By using Theorem ., we can conclude that {xn}converges strongly toA–x. This completes the proof.
Competing interests
The authors declare that they have no competing interests. Authors’ contributions
All the authors contributed equally to the writing of the present article. All authors read and approved the final manuscript.
Acknowledgements
This project is supported by the National Natural Science Foundation of China under Grant (11071279). Received: 14 April 2012 Accepted: 29 October 2012 Published: 13 November 2012
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doi:10.1186/1687-1812-2012-205
