R E S E A R C H
Open Access
Modified block iterative procedure for solving
the common solution of fixed point problems
for two countable families of total
quasi-
φ
-asymptotically nonexpansive
mappings with applications
Pongrus Phuangphoo
1,2and Poom Kumam
1**Correspondence:
1Department of Mathematics,
Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bang Mod, Thung Kru, Bangkok, 10140, Thailand Full list of author information is available at the end of the article
Abstract
In this paper, we introduce a new iterative procedure which is constructed by the modified block hybrid projection method for solving a common solution of fixed point problems for two countable families of uniformly total quasi-φ-asymptotically nonexpansive and uniformly Lipschitz continuous mappings. Under suitable conditions, some strong convergence theorems are established in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. Finally, we apply the problem of a strong convergence theorem concerning maximal monotone operators in Banach spaces.
MSC: 47H09; 47H10; 47H20; 47J20
Keywords: strong convergence theorem; total quasi-φ-asymptotically nonexpansive mapping; fixed point problems
1 Introduction
Throughout this paper, we assume thatEis a real Banach space,E*is the dual space of E. LetCbe a nonempty, closed, and convex subset ofE and·,·be the pairing between
EandE*. We denote the strong convergence and weak convergence of a sequence{x
n}
byxn→xandxnx, respectively, andJ:E→E
*
is thenormalized duality mapping
defined by
Jx=f∈E*:x,f=x=f, ∀x∈E. (.)
In the sequel, we useF(T) to denote the set of fixed points of a mappingT and useR andR+to denote the set of all real numbers and the set of all nonnegative real numbers, respectively.
Definition . Let E be a Banach space.
() Eis said to bestrictly convexifx+y< for allx,y∈UE={z∈E:z= }withx=y.
() Eis said to beuniformly convexif for eachε∈(, ], there existsδ> such that
x+y
≤ –δfor allx,y∈UEwithx–y>ε.
() Eis said to besmoothif the limit(.)
lim
t–→
x+ty–x
t (.)
exists for eachx,y∈UE.
() Eis said to beuniformly smoothif the limit(.)is attained uniformly for all
x,y∈UE.
Remark . The basic properties below hold (see [, ]).
() IfEis a real uniformly smooth Banach space, thenJis uniformly continuous on each bounded subset ofE.
() IfEis a strictly convex reflexive Banach space, thenJ–is hemicontinuous, that is,
J–is norm-to-weak*-continuous.
() IfEis a smooth and strictly convex reflexive Banach space, thenJis single-valued, one-to-one, and onto.
() A Banach spaceEis uniformly smooth if and only ifE*is uniformly convex. () Each uniformly convex Banach spaceEhas theKadec-Klee property; that is, for any
sequence{xn} ⊂E, if{xn}x∈Eandxn → x, thenxn→x.
() A Banach spaceEis strictly convex if and only ifJis strictly monotone; that is,
x–y,x*–y*> , wheneverx,y∈E,x=y, andx*∈Jx,y*∈Jy.
() Both uniformly smooth Banach spaces and uniformly convex Banach spaces are reflexive.
() E*is uniformly convex, thenJis uniformly norm-to-norm continuous on each bounded subset ofE.
LetEbe a smooth and strictly convex reflexive Banach space, and letCbe a nonempty, closed, and convex subset ofE. We assume that theLyapunov functionalφ:E×E→R+ is defined by [, ]
φ(x,y) =x– x,Jy+y, ∀x,y∈E.
From the definition ofφ, it is easy to see that
x–y≤φ(x,y)≤x+y. (.)
LetCbe a nonempty, closed, and convex subset ofE. For eachx∈E, thegeneralized projection[]C:E→Cis defined by
C(x) =arg min y∈Cφ(x,y).
Lemma .[, ] If C is a nonempty,closed,and convex subset of a smooth and strictly convex reflexive real Banach space E,then
() forx∈Eandu∈C,one has
() φ(x,C(y)) +φ(C(y),y)≤φ(x,y),∀x∈C,y∈E.
() φ(x,y) = if and only ifx=y,∀x,y∈C.
Remark . IfEis a real Hilbert spaceH, thenφ(x,y) =x–yand
C=PC(the metric
projection ofHontoC).
Definition . LetEbe a smooth, strictly convex, and reflexive real Banach space, C be a nonempty, closed, and convex subset ofE,T:C→Cbe a mapping, andFix(T) be the set of fixed points ofT.
() A pointp∈Cis said to be anasymptotic fixed pointofT if there exists a sequence
{xn} ⊂Csuch thatxnpandxn–Txn →. We denote the set of all asymptotic
fixed points ofTbyF(T).
() A pointp∈Cis said to be astrong asymptotic fixed pointofTif there exists a sequence{xn} ⊂Csuch thatxn→pandxn–Txn →. We denote the set of all
strong asymptotic fixed points ofTbyF(T).
Definition . LetEbe a smooth, strictly convex, and reflexive real Banach space, and let
Cbe a nonempty, closed, and convex subset ofE.
() A mappingT:C→Cis said to beclosedif for each{xn} ⊂Cwithxn→xand
Txn→y, thenTx=y.
() A mappingT:C→Cis said to berelatively nonexpansive[, ] ifFix(T)=∅, Fix(T) =F(T), and
φ(p,Tx)≤φ(p,x), ∀x∈C,p∈Fix(T).
() A mappingT:C→Cis said to beweak relatively nonexpansive[] ifFix(T)=∅, Fix(T) =F(T), and
φ(p,Tx)≤φ(p,x), ∀x∈C,p∈Fix(T).
() A mappingT:C→Cis said to bequasi-φ-nonexpansive(relatively quasi-nonexpansive) ifFix(T)=∅and
φ(p,Tx)≤φ(p,x), ∀x∈C,p∈Fix(T).
() A mappingT:C→Cis said to bequasi-φ-asymptotically nonexpansive (asymptotically relatively nonexpansive) ifFix(T)=∅and there exists a sequence
{kn} ⊂[,∞)withkn→such that
φp,Tnx≤knφ(p,x), ∀x∈C,p∈Fix(T), and∀n≥.
() A mappingT:C→Cis said to betotal quasi-φ-asymptotically nonexpansiveif Fix(T)=φand there exists nonnegative real sequences{νn}and{μn}withνn→,
μn→(asn→ ∞), and a strictly increasing continuous functionζ:R+→R+with
ζ() = such that
φp,Tnx≤φ(p,x) +νnζ
Remark . From Definition ., it is easy to know that () every relatively nonexpansive mapping is closed;
() every quasi-φ-asymptotically nonexpansive mapping is a total quasi-φ -asymptotically nonexpansive mapping, but the converse is not true;
() every quasi-φ-nonexpansive mapping is a quasi-φ-asymptotically nonexpansive mapping with{kn= }, but the converse is not true;
() every weak relatively nonexpansive mapping is a quasi-φ-nonexpansive mapping because it does not require the conditionFix(T) =F(T), but the converse is not true; () every relatively nonexpansive mapping is a weak relatively nonexpansive mapping,
but the converse is not true.
Regarding the iterative methods of nonlinear operator equations for relatively nonex-pansive mappings, in Matsushita and Takahashi [] and in Plubtieng and Ung-chittrakool [] proved the following result, respectively.
Theorem MT Let E be a uniformly convex and uniformly smooth Banach space,let C
be a nonempty closed and convex subset of E.Let T:C→C be a relatively nonexpansive mapping,and let{αn}be a real sequence in[, ) withlim supn→∞αn< .Let{xn}be a
sequence defined by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
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), ∀n≥,
(.)
where J is the duality mapping on E.IfFix(T)=∅,then the sequence{xn}converges strongly
toFix(T)(x),whereFix(T)(·)is the generalized projection from C ontoFix(T).
Theorem PU Let E be a uniformly convex and uniformly smooth Banach space,let C be a nonempty,closed,and convex subset of E.Let T,S:C→C be two relatively nonexpansive mappings with:=Fix(T)∩Fix(S)=∅.Let{xn}be a sequence defined by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C chosen arbitrarily,
yn=J–(αnJx+ ( –αn)JTzn),
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), ∀n≥,
(.)
with the following restrictions:
() <αn< andlim supn→∞αn< ;
() ≤βn(),βn(),βn()≤,limn→∞βn()= ,andlim infn→∞βn()βn()> .
In , Suet al.[] introduced the concept of a countable family of weak relatively nonexpansive mappings and proved the following theorem which extends and improves Theorem MT and Theorem PU.
Theorem SXZ Let E be a uniformly convex and uniformly smooth real Banach space,let C
be a nonempty,closed,and convex subset of E.Let{Tn}∞n=,{Sn}∞n=:C→C be two
count-able families of weak relatively nonexpansive mappings such that:= (∞n=Fix(Tn))∩
(∞n=Fix(Sn))=∅.
Let{xn}be a sequence defined by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C chosen arbitrarily,
yn=J–(αnJxn+ ( –αn)Jzn),
zn=J–(βn()Jxn+βn()JTnxn+βn()JSnxn),
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), ∀n≥,
(.)
with the conditions:
() ≤αn≤α< for someα∈(, );
() βn(),βn(),βn()∈(, )such thatβn()+βn()+βn()= for eachn≥;
() lim infn→∞βn()βn()> andlim infn→∞βn()βn()> .
Then the sequence{xn}converges strongly to(x),where(·)is the generalized projec-tion from C onto.
In , Changet al.[] proved some approximation theorems for common fixed points of countable families of total quasi-φ-asymptotically nonexpansive mappings which con-tain several kinds of mappings as their special cases in Banach spaces. Next, Changet al.
[] modified the Halpern-type iteration algorithm for a total quasi-φ-asymptotically non-expansive mapping to have the strong convergence under a limit condition only in the framework of Banach spaces. Recently, Chang, Lee, and Chan [] introduced a block hybrid projection algorithm for solving the convex feasibility problem and the general-ized equilibrium problems for an infinite family of total quasi-φ-asymptotically nonex-pansive mappings and they proved strong convergence theorems in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property.
Theorem CLCY Let E be a uniformly smooth and strictly convex Banach space with
the Kadec-Klee property, let C be a nonempty, closed, and convex subset of E, and let {Ti}∞i= :C→C be a countable family of closed,uniformly Li-Lipschitz continuous,and
μ= ,νn→,μn→ (as n→ ∞),andζ() = .Let{xn}be a sequence generated by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C chosen arbitrarily,C=C,
zn=J–(αnJxn+ ( –αn)Jzn),
yn=J–(βn,Jxn+
∞
i=βn,iJTinxn),
Cn+={v∈Cn:φ(v,yn)≤φ(v,xn) +ξn},
xn+=Cn+(x), ∀n≥,
(.)
whereξn=νnsupp∈ζ(φ(p,xn)) +μn,Cn+is the generalized projection of E onto Cn+.Let {βn,},{βn,i},and{αn}be sequences in[, ]satisfying the following conditions:
() for eachn≥,βn,+ ∞
i=βn,i= ;
() lim infn→∞βn,βn,i> for alli≥;
() ≤αn≤α< for someα∈(, ).
If:=∞i=Fix(Ti)is a nonempty and bounded subset of C,then the sequence{xn}converges
strongly to(x).
Theorem CLCZ Let E be a uniformly smooth and strictly convex Banach space with
the Kadec-Klee property, let C be a nonempty, closed, and convex subset of E, and let {Tm}∞m=:C→C be a countable family of closed,uniformly Lm-Lipschitz continuous,and
uniformly total quasi-φ-asymptotically nonexpansive mappings with nonnegative real se-quences {νn},{μn},and a strictly increasing continuous function ζ :R+→R+ such that
μ= ,νn→,μn→ (as n→ ∞),andζ() = .Let{xn}be a sequence generated by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈E chosen arbitrarily,C=C,
yn,m=J–(αnJx+ ( –αn)JTmnxn), m≥,
Cn+={z∈Cn:supm≥φ(z,yn,m)≤αnφ(z,x) + ( –α)φ(z,xn) +ξn},
xn+=Cn+(x), ∀n≥,
(.)
whereξn=νnsupp∈ζ(φ(p,xn)) +μn,andCn+is the generalized projection of E onto Cn+. If:=∞m=Fix(Tm)is a nonempty and bounded subset of C,then the sequence{xn}
con-verges strongly to(x).
Theorem CLC Let E be a uniformly smooth and strictly convex Banach space with
the Kadec-Klee property, let C be a nonempty, closed, and convex subset of E, and let {Si}∞i= :C→C be a countable family of closed, uniformly Li-Lipschitz continuous,and
uniformly total quasi-φ-asymptotically nonexpansive mappings with nonnegative real se-quences {νn},{μn},and a strictly increasing continuous function ζ :R+→R+ such that
μ= ,νn→,μn→ (as n→ ∞),andζ() = .Let Bi:C→E* (i= , , , . . . ,N)be
a finite family of continuous and monotone mappings.Letψi:C→R(i= , , , . . . ,N)
be a finite family of lower semi-continuous and convex functions,and let Fi:C×C→R
(i= , , , . . . ,N)be a finite family of bifunctions satisfying the conditions(A)-(A). Sup-pose that:= (Ni=Fi)∩(
∞
i=Fix(Si))is a nonempty and bounded subset of C,whereFi
(i= , , , . . . ,N)is the set of the following generalized mixed quasi-equilibrium problems:
Let{xn}be a sequence generated by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C chosen arbitrarily,C=C,
yn=J–(βnJxn+ ( –βn)Jzn),
zn=J–(αn,Jxn+
∞
i=αn,iJSinxn),
u(ni)=Kfi,riKfi–,ri–· · ·Kf,rKf,ryn, i= , , , . . . ,N, Cn+={v∈Cn:maxi=,,,...,Nφ(v,u(ni))≤φ(v,xn) +ξn},
xn+=Cn+(x), ∀n≥,
(.)
whereξn=νnsupp∈ζ(φ(u,xn)) +μn,∀n≥,Cn+is the generalized projection of E onto Cn+.Let{αn,i},{βn}be sequences in[, ]satisfying the following conditions:
() for eachn≥,∞i=αn,i= ;
() lim infn→∞( –βn)αn,αn,i> for alli≥.
Then the sequence{xn}converges strongly to(x).
In this paper, motivated and inspired by the previously mentioned results, we introduce a new iterative procedure by the modified block hybrid projection method for solving a common solution of fixed point problems for two countable families of uniformly total quasi-φ-asymptotically nonexpansive and uniformly Lipschitz continuous mappings in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. Then, we prove a strong convergence theorem of the iterative procedure generated by these con-ditions. The results obtained in this paper extend and improve several recent results in this area.
2 Preliminaries
Definition . LetCbe a nonempty, closed, and convex subset of a real Banach spaceE. () A mappingT:C→Cis said to benonexpansiveif
Tx–Ty ≤ x–y, ∀x,y∈C.
() A mappingT:C→Cis said to beuniformlyL-Lipschitz continuousif there exists a constantL> such that
Tnx–Tny≤Lx–y, ∀x,y∈C,∀n≥.
Definition . [] LetC be a nonempty, closed, and convex subset of a real Banach
spaceE.
() A countable family of mappings{Ti}∞i=is said to be auniformly quasi-φ
-asymptotically nonexpansive mappingif∞i=Fix(Ti)=∅and there exists a sequence
{kn} ⊂[,∞)withkn→such that for eachi≥,
φp,Tinx≤knφ(p,x), ∀x∈C,p∈
∞
i=
Fix(Ti), and∀n≥.
nonnegative real sequences{νn}and{μn}withνn→,μn→(asn→ ∞), and a
strictly increasing continuous functionζ:R+→R+withζ() = such that for each
i≥,
φp,Tinx≤φ(p,x) +νnζ
φ(p,x)+μn, ∀x∈C,p∈
∞
i=
Fix(Ti), and∀n≥.
Lemma .[] Let E be a uniformly smooth and strictly convex real Banach space,and let{xn}and{yn}be two sequences of E.Ifφ(xn,yn)→and either{xn}or{yn}is bounded,
thenxn–yn →.
Lemma .[] Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property,and let C be a nonempty,closed,and convex subset of E.Let{xn}
and{yn}be two sequences in C and p∈E.If xn→p andφ(xn,yn)→,then yn→p.
Lemma .[] Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property,and let C be a nonempty,closed,and convex subset of E.Let T:C→C be a closed and total quasi-φ-asymptotically nonexpansive mapping with nonnegative real sequences{νn},{μn},and a strictly increasing continuous functionζ:R+→R+such that
νn→,μn→ (as n→ ∞),andζ() = .Ifμ= ,then the fixed point setFix(T)is a
closed and convex subset of C.
Lemma .[] Let E be a uniformly convex Banach space,r> be a positive number,
and Br()be a closed ball of E.Then for any given sequence{xn}∞n=⊂Br()and for any
given{λn}∞n=⊂(, )with ∞
n=λn= ,there exists a continuous,strictly increasing,and
convex function g: [, r)→[,∞)with g() = such that for any positive integers i,j with i<j,
∞
n=
λnxn
≤
∞
n=
λnxn–λiλjg
xi–xj
.
3 Main results
In this section, we shall use the modified block hybrid projection method to study a common solution of fixed point problems for two countable families of closed andLi,i
-Lipschitz continuous and uniformly total quasi-φ-asymptotically nonexpansive mappings in Banach spaces. For the purpose, we assume the following hypotheses.
(A) Let{Ti}∞i=:C→Cbe a countable family of closed, uniformlyLi-Lipschitz
contin-uous, and uniformly total quasi-φ-asymptotically nonexpansive mappings with nonneg-ative real sequences{νn},{μn}and a strictly increasing functionζ :R+→R+ such that
νn→,μn→ (asn→ ∞),μ= , andζ() = , and for eachi≥,
φp,Tinx≤φ(p,x) +νnζ
φ(p,x)+μn, ∀x∈C,p∈
∞
i=
Fix(Ti), and∀n≥.
(A) Let{Si}∞i=:C→Cbe a countable family of closed, uniformlyi-Lipschitz
κn→,ωn→ (asn→ ∞),ω= , andρ() = , and for eachi≥,
φp,Snix≤φ(p,x) +κnρ
φ(p,x)+ωn, ∀x∈C,p∈
∞
i=
Fix(Si), and∀n≥.
Theorem . Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property,let C be a nonempty,closed,and convex subset of E,and let{Ti}∞i=
and{Si}∞i=satisfy the above conditions(A)-(A),respectively.
Suppose that := (∞i=Fix(Ti))∩(
∞
i=Fix(Si))is nonempty and bounded in C. Let
{xn}∞n=,{yn}∞n=,and{zn}∞n=be sequences generated by ⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C chosen arbitrarily,C=C,
zn=J–(αn,Jxn+∞i=αn,iJSinyn),
yn=J–(βn,Jxn+
∞
i=βn,iJTinxn),
Cn+={v∈Cn:φ(v,yn)≤φ(v,xn) +ξnandφ(v,zn)≤φ(v,xn) +θn},
xn+=Cn+(x), ∀n≥,
(.)
whereξn=νnsupp∈ζ(φ(p,xn)) +μnandθn=κnsupp∈ρ(φ(p,xn)) +ωn.
Let{αn,i}and{βn,i}be coefficient sequences in[, ]satisfying the following conditions:
() ∞i=αn,i= and
∞
i=βn,i= ;
() lim infn–→∞αn,αn,i> ,andlim infn–→∞βn,βn,i> ,∀i≥.
Then the sequence{xn}∞n=converges strongly to some point p*,where p*=(x). Proof We shall complete this proof of Theorem . in seven steps.
Step . We will show thatandCn+are closed and convex for eachn≥.
In fact, it follows from Lemma . thatFix(Ti) andFix(Si), for anyi≥, are closed and
convex subsets ofC. Therefore,is closed and convex inC.
Clearly,C=Cis closed and convex. Suppose thatCnis closed and convex for some
n≥.
By the assumption ofCn+,
φ(v,yn)≤φ(v,xn) +ξn
is equivalent to
v– v,Jy
n+yn≤ v– v,Jxn+xn+ξn.
So that v,Jxn– v,Jyn= v,Jxn–Jyn ≤ xn–yn+ξn.
Again, by the assumption ofCn+,
φ(v,zn)≤φ(v,xn) +θn
is equivalent to
v– v,Jz
n+zn≤ v– v,Jxn+xn+θn.
Hence,Cn+={v∈Cn: v,Jxn–Jyn ≤ xn–yn+ξnand v,Jxn–Jzn ≤ xn–
zn+θn}is closed and convex.
Step . We will show that{xn}is bounded and{φ(xn,x)}is a convergent sequence for
alln≥.
Indeed, it follows from (.) and Lemma .() that
φ(xn,x) =φ
Cn(x),x
≤φ(p,x) –φp,Cn(x)
≤φ(p,x), ∀n≥,p∈.
This implies that {φ(xn,x)} is bounded. By virtue of (.), the sequence {xn} is also
bounded.
By the assumption ofCn, we haveCn+⊂Cn,xn=Cn(x) andxn+=Cn+(x).
This implies thatxn+∈Cn+⊂Cnand
φ(xn,x)≤φ(xn+,x), ∀n≥.
Therefore,{φ(xn,x)}is a convergent sequence. Without loss of generality, we can assume that
lim
n→∞φ(xn,x) =d≥. (.)
Step . We will show that⊂Cnfor alln≥.
It is obvious that⊂C=C. Suppose that⊂Cnfor somen≥. For any givenp∈,
from (.) and Lemma ., we compute
φ(p,yn) =φ
p,J–
βn,Jxn+
∞
i=
βn,iJTinxn
=p–
p,βn,Jxn+
∞
i=
βn,iJTinxn
+
βn,Jxn+
∞
i=
βn,iJTinxn
≤ p– β
n,p,Jxn–
∞
i=
βn,i
p,JTinxn
+βn,xn+
∞
i=
βn,iTinxn
–βn,βn,igJxn–JTinxn
=βn,φ(p,xn) + ( –βn,)p
–
∞
i=
βn,i
p,JTinxn
+
∞
i=
βn,iTinxn
–βn,βn,igJxn–JTinxn
=βn,φ(p,xn) +
∞
i=
βn,iφ
p,Tinxn
≤βn,φ(p,xn) +
∞
i=
βn,i
φ(p,xn) +νnζ
φ(p,xn)
+μn
–βn,βn,igJxn–JTinxn
=
βn,+
∞
i=
βn,i
φ(p,xn) +
∞
i=
βn,i
νnζ
φ(p,xn)
+μn
–βn,βn,igJxn–JTinxn
≤φ(p,xn) +νnζ
φ(p,xn)
+μn–βn,βn,igJxn–JTinxn
≤φ(p,xn) +νnsup p∈
ζφ(p,xn)
+μn–βn,βn,igJxn–JTinxn
=φ(p,xn) +ξn–βn,βn,igJxn–JTinxn. (.)
It follows that
φ(p,yn)≤φ(p,xn) +ξn, whereξn=νnsup p∈
ζφ(p,xn)
+μn. (.)
From (.) and Lemma ., we compute
φ(p,zn) =φ
p,J–
αn,Jxn+
∞
i=
αn,iJSinyn
=p–
p,αn,Jxn+
∞
i=
αn,iJSinyn
+
αn,Jxn+
∞
i=
αn,iJSniyn
≤ p– α
n,p,Jxn–
∞
i=
αn,i
p,JSniyn
+αn,xn+
∞
i=
αn,iSniyn
–αn,αn,igJyn–JSniyn
=αn,φ(p,xn) + ( –αn,)p–
∞
i=
αn,i
p,JSinyn
+
∞
i=
αn,iSniyn
–αn,αn,igJyn–JSniyn
=αn,φ(p,xn) +
∞
i=
αn,iφ
p,Sinyn
–αn,αn,igJyn–JSniyn
≤αn,φ(p,xn) +
∞
i=
αn,i
φ(p,x) +κnρ
φ(p,x)+ωn
–αn,αn,igJyn–JSniyn
=
αn,+
∞
i=
αn,i
φ(p,xn) +
∞
i=
αn,i
κnρ
φ(p,x)+ωn
–αn,αn,igJyn–JSniyn
≤φ(p,xn) +κnρ
φ(p,x)+ωn–αn,αn,igJyn–JSniyn
≤φ(p,xn) +κnsup p∈
ρφ(p,x)+ωn–αn,αn,igJyn–JSinyn
It follows that
φ(p,zn)≤φ(p,xn) +θn, whereθn=κnsup p∈
ρφ(p,x)+ωn. (.)
By the assumptions of{νn},{μn},{κn}, and{ωn}, and from (.) and (.), we obtain
ξn=νnsup p∈
ζφ(p,xn)
+μn→ (asn→ ∞), (.)
and
θn=κnsup p∈
ρφ(p,xn)
+ωn→ (asn→ ∞). (.)
So, we getp∈Cn+. This implies that⊂Cnfor alln≥, and the sequence{xn}is well
defined.
Step . We will show that there exists some pointp*∈Csuch thatx
n→p*.
In fact, since{xn}is bounded andEis reflexive, then there exists a subsequence{xni} ⊂
{xn}such thatxnip
*(some point inC).
SinceCnis closed and convex andCn+ ⊂Cn, it follows thatCnis weakly closed and
p*∈C
nfor eachn≥.
In view ofxni=Cni(x), we have
φ(xni,x)≤φ
p*,x
, ∀ni≥.
Since the norm · is weakly lower semi-continuous, we have
lim inf
ni→∞
φ(xni,x) =lim infn i→∞
xni– xni,Jx+x
≥ p*– p*,Jx
+x=φ
p*,x
,
and so
φp*,x
≤lim inf
ni→∞
φ(xni,x)≤lim sup
ni→∞
φ(xni,x)≤φ
p*,x
.
This implies thatlimni→∞φ(xni,x) =φ(p
*,x), and sox
ni → p*. Sincexnip
*, and by virtue of the Kadec-Klee property ofE, we obtain
lim
ni→∞ xni=p
*, asn
i→ ∞.
The sequence{φ(xn,x)}is convergent, andlimni→∞φ(xni,x) =φ(p
*,x), which implies thatlimn→∞φ(xn,x) =φ(p*,x). If there exists some subsequence{xnj} ⊂ {xn}such that xnjq, then from Lemma .() we have
φp*,q= lim
ni,nj→∞
φ(xni,xnj)
= lim
ni,nj→∞
φxni,Cnj(x)
≤ lim
ni,nj→∞
φ(xni,x) –φ
Cnj(x),x
= lim
ni,nj→∞
φ(xni,x) –φ(xnj,x)
=φp*,x
–φp*,x
= .
This implies thatp*=q, and so
lim
n→∞xn=p
*. (.)
Step . We will show thatlimn→∞Jxn–Jyn= andlimn→∞Jxn–Jzn= asn→ ∞.
Sincexn+∈Cn+⊂Cn, by the definition ofCn+, we have
φ(xn+,xn) =φ
xn+,Cn(x)
≤φ(xn+,x) –φ
Cn(x),x
=φ(xn+,x) –φ(xn,x). (.)
Sincelimn–→∞φ(xn,x) exists, and we are takingn→ ∞in (.), thenφ(xn+,xn)→.
It follows from Lemma . that
lim
n→∞xn+–xn= . (.)
By the definition ofCn+andxn+∈Cn+, we get
φ(xn+,yn)≤φ(xn+,xn) +ξn, and
φ(xn+,zn)≤φ(xn+,xn) +θn.
(.)
Fromφ(xn+,xn)→,ξn→, andθn→, asn→ ∞, we obtain
φ(xn+,yn)→, and φ(xn+,zn)→, asn→ ∞. (.)
Sincelimn–→∞xn=p*, by virtue of Lemma ., we get
lim
n→∞yn=p
*, and lim
n→∞zn=p
*. (.)
It follows from (.) and (.) that
lim
n→∞xn–yn= , and nlim→∞xn–zn= . (.)
SinceJis uniformly continuous on each bounded subset ofE, then
lim
n→∞Jxn–Jyn= , and nlim→∞Jxn–Jzn= . (.)
Step . We will show thatp*∈, where:= (∞i=Fix(Ti))∩(
∞
i=Fix(Si)).
(.) First, we will show thatp*∈∞
For anyi≥ and for anyp∈, it follows from (.), (.), (.), and (.) that
βn,βn,igJxn–JTinxn ≤ φ(p,xn) –φ(p,yn) +ξn
= xn–yn– p,Jxn–Jyn+ξn
= xn+yn
xn–yn
– p,Jxn–Jyn+ξn
≤ xn+yn
xn–yn
– pJxn–Jyn+ξn
→, asn→ ∞.
By the conditionlim infn–→∞βn,βn,i> ,∀i≥, we obtain
gJxn–JTinxn→, asn→ ∞.
It follows from the property ofgthat
Jxn–JTinxn→, asn→ ∞. (.)
Sincexn→p*andJis uniformly continuous on each bounded subset ofE, it yields that
Jxn→Jp*.
Hence, from (.) we get
JTinxn→Jp*, asn→ ∞,∀i≥. (.)
SinceJ–:E*→Eis norm-to-weak*-continuous, we also have
Tinxnp*, asn→ ∞,∀i≥. (.)
Again, since for anyi≥,
Tn
ixn–p*=J
Tn
ixn–Jp*≤J
Tn ixn
–Jp*→, asn→ ∞,
from (.) and the Kadec-Klee property ofE, it follows that
lim
n→∞T n
ixn=p*. (.)
On the other hand, by the assumption that for eachi≥,Tiis uniformly Li-Lipschitz
continuous, we get
Tin+xn–Tinxn≤Tin+xn–Tin+xn++Tin+xn+–xn+
+xn+–xn+xn–Tinxn
≤(Li+ )xn+–xn+Tin+xn+–xn++xn–Tinxn.
It follows from (.), (.), and (.) that
lim
n→∞T n+
i xn–Tinxn= , and lim n→∞T
n+
and so
lim
n→∞T n+
i xn= lim n→∞TiT
n
ixn= lim n→∞Tip
*=p*.
In view of (.) and the closeness ofTi, it yields thatTip*=p*for alli≥. This implies
that
p*∈
∞
i=
Fix(Ti). (.)
(.) Next, we will show thatp*∈∞
i=Fix(Si).
For anyi≥ and for anyp∈, it follows from (.), (.), (.), and (.) that
αn,αn,igJyn–JSniyn ≤ φ(p,xn) –φ(p,zn) +θn
= xn–zn– p,Jxn–Jzn+θn
= xn+zn
xn–zn
– p,Jxn–Jzn+θn
≤ xn+zn
xn–zn
– pJxn–Jzn+θn
→, asn→ ∞.
By the conditionlim infn–→∞αn,αn,i> ,∀i≥, we obtain
gJyn–JSniyn→, asn→ ∞.
It follows from the property ofgthat
Jyn–JSniyn→, asn→ ∞. (.)
Sinceyn→p*andJis uniformly continuous on each bounded subset ofE, it yields that
Jyn→Jp*.
Hence, from (.) we have
JSinyn→Jp*, asn→ ∞,∀i≥. (.)
SinceJ–:E*→Eis norm-to-weak*-continuous, we also have
Sniynp* asn→ ∞,∀i≥. (.)
Again, since for anyi≥,
Sniyn–p*=J
Sniyn–Jp*≤J
Sinyn
–Jp*→, asn→ ∞,
from (.) and the Kadec-Klee property ofE, it follows that
lim
n→∞S n
On the other hand, by the assumption that for eachi≥,Si is uniformly i-Lipschitz
continuous, we have
Sin+yn–Sniyn≤Sni+yn–Sin+yn++Sni+yn+–xn++xn+–xn
+xn–yn+yn–Sniyn
≤iyn–yn++Sni+yn+–xn++xn+–xn
+xn–yn+yn–Sniyn.
Fromxn→p*,yn→p*andSinyn→p*, asn→ ∞, we obtain
yn–yn+ →, Sin+yn+–xn+→, and yn–Sinyn→, asn→ ∞.
And it follows from (.) and (.) that
lim
n→∞S
n+
i yn–Sinyn= , and lim n→∞S
n+
i yn=p*,
and so
lim
n→∞S n+
i yn= lim n→∞SiS
n
iyn= lim n→∞Sip
*=p*.
In view of (.) and the closeness ofSi, it yields thatSip*=p*for alli≥. This implies
that
p*∈
∞
i=
Fix(Si). (.)
From (.) and (.), we can conclude thatp*∈:= (∞
i=Fix(Ti))∩(
∞
i=Fix(Si)).
Step . Finally, we will show thatxn→p*=(x).
Letw=(x). Sincew∈⊂Cnandxn=Cn(x), we have φ(xn,x)≤φ(w,x), ∀n≥.
This implies that
φp*,x
= lim
n→∞φ(xn,x)≤φ(w,x). (.)
In view of the definition of(x), from (.), we havep*=w. Therefore,xn→p*=
(x).
This completes the proof of Theorem ..
Theorem . Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property,and let C be a nonempty,closed,and convex subset of E.We assume the following:
(B)Let{Ti}∞i=:C→C be a countable family of closed,uniformly Li-Lipschitz
(B)Let{Si}∞i=:C→C be a countable family of closed,uniformlyi-Lipschitz
continu-ous,and uniformly quasi-φ-asymptotically nonexpansive mappings with a real sequence {ln} ⊂[,∞),ln→.
Suppose that:= (∞i=Fix(Ti))∩(∞i=Fix(Si))is a nonempty and bounded in C.Let
{xn}∞n=,{yn}∞n=,and{zn}∞n=be sequences generated by ⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C chosen arbitrarily,C=C,
zn=J–(αn,Jxn+
∞
i=αn,iJSinyn),
yn=J–(βn,Jxn+
∞
i=βn,iJTinxn),
Cn+={v∈Cn:φ(v,yn)≤φ(v,xn) +ξnandφ(v,zn)≤φ(v,xn) +θn},
xn+=Cn+(x), ∀n≥,
(.)
whereξn=supp∈(kn– )(φ(p,xn))andθn=supp∈(ln– )(φ(p,xn)).
Let{αn,i}and{βn,i}be coefficient sequences in[, ]satisfying the following conditions:
() ∞i=αn,i= and
∞
i=βn,i= ;
() lim infn–→∞αn,αn,i> andlim infn–→∞βn,βn,i> ,∀i≥.
Then the sequence{xn}∞n=converges strongly to some point p*,where p*=(x).
Proof Since {Ti}∞i=, {Si}∞i= are countable families of closed and uniformly quasi-φ -asymptotically nonexpansive mappings, by virtue of Remark .(), {Ti}∞i=, {Si}∞i= are countable families of closed and uniformly total quasi-φ-asymptotically nonexpansive mappings with nonnegative sequences νn=kn– ,μn= , and κn=ln– ,ωn= ,
re-spectively, and a strictly increasing and continuous functionζ(t) =ρ(t) =t,t≥. Hence,
ξn=supp∈(kn– )(φ(p,xn))→ andθn=supp∈(ln– )(φ(p,xn))→ (asn→ ∞).
There-fore, all the conditions in Theorem . are satisfied. The conclusion of Theorem . can
be obtained from Theorem . immediately.
Theorem . Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property,and let C be a nonempty,closed,and convex subset of E.We assume the following:
(C)Let{Ti}∞i=:C→C be a countable family of closed and quasi-φ-nonexpansive
map-pings.
(C)Let{Si}∞i=:C→C be a countable family of closed and quasi-φ-nonexpansive
map-pings.
Suppose that := (∞i=Fix(Ti))∩(
∞
i=Fix(Si))is nonempty and bounded in C. Let
{xn}∞n=,{yn}∞n=,and{zn}∞n=be sequences generated by ⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C chosen arbitrarily,C=C,
zn=J–(αn,Jxn+
∞
i=αn,iJSiyn),
yn=J–(βn,Jxn+
∞
i=βn,iJTixn),
Cn+={v∈Cn:φ(v,yn)≤φ(v,xn)andφ(v,zn)≤φ(v,xn)},
xn+=Cn+(x), ∀n≥,
(.)
and let{αn,i}and{βn,i}be coefficient sequences in[, ]satisfying the following conditions:
() ∞i=αn,i= ,and
∞
() lim infn–→∞αn,αn,i> ,andlim infn–→∞βn,βn,i> ,∀i≥.
Then the sequence{xn}∞n=converges strongly to some point p*,where p*=(x).
Proof Since {Ti}∞i=, {Si}∞i= are countable families of closed and quasi-φ-nonexpansive mappings, by Remark .(), {Ti}∞i=,{Si}∞i= are countable families of closed and
quasi-φ-asymptotically nonexpansive mappings with nonnegative sequenceskn= andln= ,
respectively. Hence,ξn=supp∈(kn– )(φ(p,xn)) = andθn=supp∈(ln– )(φ(p,xn)) = .
Therefore, all the conditions in Theorem . as ‘is bounded inC’ and ‘for eachi≥,
{Ti}∞i=,{Si}∞i= are uniformlyLi,i-Lipschitz continuous’ are of no use here. Thus, all the
conditions in Theorem . are satisfied. The conclusion of Theorem . can be obtained
from Theorem . immediately.
Theorem . Let E be a uniformly smooth and strictly convex real Banach space with the
Kadec-Klee property,and let C be a nonempty,closed,and convex subset of E.We assume the following:
(D)Let{Ti}∞i=:C→C be a countable family of weak relatively nonexpansive mappings. (D)Let{Si}∞i=:C→C be a countable family of weak relatively nonexpansive mappings.
Suppose that := (∞i=Fix(Ti))∩(
∞
i=Fix(Si))is nonempty and bounded in C. Let
{xn}∞n=,{yn}∞n=,and{zn}∞n=be sequences generated by ⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C chosen arbitrarily,C=C,
zn=J–(αn,Jxn+∞i=αn,iJSiyn),
yn=J–(βn,Jxn+
∞
i=βn,iJTixn),
Cn+={v∈Cn:φ(v,yn)≤φ(v,xn)andφ(v,zn)≤φ(v,xn)},
xn+=Cn+(x), ∀n≥,
(.)
and let{αn,i}and{βn,i}be coefficient sequences in[, ]satisfying the following conditions:
() ∞i=αn,i= and
∞
i=βn,i= ;
() lim infn–→∞αn,αn,i> andlim infn–→∞βn,βn,i> ,∀i≥.
Then the sequence{xn}∞n=converges strongly to some point p*,where p*=(x).
Proof Since{Ti}∞i=,{Si}∞i= are countable families of weak relatively nonexpansive map-pings, from Remark .(),{Ti}∞i=,{Si}∞i=are countable families of quasi-φ-nonexpansive mappings. Therefore, all the conditions in Theorem . are satisfied. The conclusion of Theorem . can be obtained from Theorem . immediately.
Theorem . Let E be a uniformly smooth and strictly convex real Banach space with the
Kadec-Klee property,and let C be a nonempty,closed,and convex subset of E.We assume the following:
Suppose that := (∞i=Fix(Ti))∩(
∞
i=Fix(Si))is nonempty and bounded in C. Let
{xn}∞n=,{yn}n∞=,and{zn}∞n=be sequences generated by ⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C chosen arbitrarily,C=C,
zn=J–(αn,Jxn+
∞
i=αn,iJSiyn),
yn=J–(βn,Jxn+
∞
i=βn,iJTixn),
Cn+={v∈Cn:φ(v,yn)≤φ(v,xn)andφ(v,zn)≤φ(v,xn)},
xn+=Cn+(x), ∀n≥,
(.)
and let{αn,i}and{βn,i}be coefficient sequences in[, ]satisfying the following conditions:
() ∞i=αn,i= and
∞
i=βn,i= ;
() lim infn–→∞αn,αn,i> andlim infn–→∞βn,βn,i> ,∀i≥.
Then the sequence{xn}∞n=converges strongly to some point p*,where p*=(x).
Proof Since{Ti}∞i=,{Si}∞i= are countable families of relatively nonexpansive mappings, it follows from Remark .() that {Ti}∞i=,{Si}∞i= are countable families of weak relatively nonexpansive mappings. Therefore, all the conditions in Theorem . are satisfied. The conclusion of Theorem . can be obtained from Theorem . immediately.
Remark . Theorems .-. generalize, improve, and extend the corresponding results in [–, –], and [] in the following aspects:
(a) For the framework of spaces, we extend the space from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property. (Note that each uniformly convex Banach space must have the Kadec-Klee property.)
(b) For the mappings, we extend the mappings from a nonexpansive mapping, a rel-atively nonexpansive mapping, a weakly relrel-atively nonexpansive mapping, a quasi-φ -nonexpansive mapping or a quasi-φ-asymptotically nonexpansive mapping to a total quasi-φ-asymptotically nonexpansive mapping.
(c) We extend a countable family of closed and uniformly Li-Lipschitz continuous
and uniformly total quasi-φ-asymptotically nonexpansive mappings to two countable families of closed and uniformlyLi-Lipschitz continuous and uniformly total quasi-φ
-asymptotically nonexpansive mappings.
4 Deduced theorem
If we takei= in Theorem ., then we obtain the following result.
Theorem . Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property,and let C be a nonempty,closed,and convex subset of E.We assume the following:
(F)Let T:C→C be a closed,uniformly L-Lipschitz continuous,and uniformly total quasi-φ-asymptotically nonexpansive mapping with nonnegative real sequences{νn},{μn},
and a strictly increasing functionζ :R+→R+such thatν
n→,μn→ (as n→ ∞),
ζ() = ,and
φp,Tnx≤φ(p,x) +νnζ
(F)Let S:C→C be a closed,uniformly-Lipschitz continuous,and uniformly total quasi-φ-asymptotically nonexpansive mapping with nonnegative real sequences{κn},{ωn},
and a strictly increasing function ρ:R+→R+ such thatκ
n→,ωn→ (as n→ ∞),
ρ() = ,and
φp,Snx≤φ(p,x) +κnρ
φ(p,x)+ωn, ∀x∈C,p∈Fix(S), and∀n≥.
Suppose that:=Fix(T)∩Fix(S)is nonempty and bounded in C.Let{xn}∞n=,{yn}∞n=,
and{zn}∞n=be sequences generated by ⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C chosen arbitrarily,C=C,
zn=J–(αnJxn+ ( –αn)JSnyn),
yn=J–(βnJxn+ ( –βn)JTnxn),
Cn+={v∈Cn:φ(v,yn)≤φ(v,xn) +ξnandφ(v,zn)≤φ(v,xn) +θn},
xn+=Cn+(x), ∀n≥,
(.)
whereξn=νnsupp∈ζ(φ(p,xn)) +μnandθn=κnsupp∈ρ(φ(p,xn)) +ωn.
Let{αn}and{βn}be coefficient sequences in[, ]satisfying the following conditions:
() ≤αn≤α< for someα∈(, );
() ≤βn≤β< for someβ∈(, ).
Then the sequence{xn}∞n=converges strongly to some point p*,where p*=(x). If we setSn
i =I(identity mapping) for alli= , , , . . . in Theorem ., then we obtain the
following result.
Theorem . Let E be a uniformly smooth and strictly convex real Banach space with the
Kadec-Klee property,and let C be a nonempty,closed,and convex subset of E.We assume the following:
(G)Let{Ti}∞i=:C→C be a countable family of closed,uniformly Li-Lipschitz
continu-ous,and uniformly total quasi-φ-asymptotically nonexpansive mappings with nonnegative real sequences{νn},{μn},and a strictly increasing functionζ:R+→R+such thatνn→,
μn→ (as n→ ∞),μ= ,andζ() = ,and for each i≥,
φp,Tinx≤φ(p,x) +νnζ
φ(p,x)+μn, ∀x∈C,p∈
∞
i=
Fix(Ti),and∀n≥.
(G)Suppose that:=∞i=Fix(Ti)is nonempty and bounded in C.Let{xn}∞n=,{yn}∞n=,
and{zn}∞n=be sequences generated by ⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C chosen arbitrarily,C=C,
zn=J–(αnJxn+ ( –αn)Jyn),
yn=J–(βn,Jxn+
∞
i=βn,iJTinxn),
Cn+={v∈Cn:φ(v,yn)≤φ(v,xn) +ξn},
xn+=Cn+(x), ∀n≥,
whereξn=νnsupp∈ζ(φ(p,xn)) +μn.Let{αn}and{βn,i}be coefficient sequences in[, ]
satisfying the following conditions: () ≤αn≤α< for someα∈(, );
() ∞i=βn,i= ;
() lim infn–→∞βn,βn,i> ,∀i≥.
Then the sequence{xn}∞n=converges strongly to some point p*,where p*=(x).
Remark . Theorem . contains the result of Changet al.[].
5 Applications
Now, we apply Theorem . to prove a strong convergence theorem concerning two max-imal monotone operators in Banach spaces.
LetEbe a smooth, strictly convex, and reflexive real Banach space, and letA:E→E* be a maximal monotone operator. For eachr> , we can define a single value mapping
Jr:E→D(A) byJr= (J+rA)–J, and such a mappingJris called theresolventofA. It is easy
to prove thatA–() =F(J
r) for allr> . Using Theorem ., we can obtain the following
strong convergence theorem for maximal monotone operators.
Theorem . Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property,and let C be a nonempty,closed,and convex subset of E.We assume the following:
(H)Let A,B be two maximal monotone operators from E to E*,and let JrA,JrB be the resolvent of A and B,respectively,where r> .
(H)Suppose that:=A–()∩B–()is nonempty,and let{xn}∞n=,{yn}∞n=,and{zn}∞n=
be sequences generated by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C chosen arbitrarily,C=C
zn=J–(αn,Jxn+
∞
i=αn,iJJrBiyn) yn=J–(βn,Jxn+
∞
i=βn,iJJrAixn)
Cn+={v∈Cn:φ(v,yn)≤φ(v,xn)andφ(v,zn)≤φ(v,xn)}
xn+=Cn+(x), ∀n≥,
(.)
where rn> withlim infn→∞rn> ,and let{αn,i}and{βn,i}be coefficient sequences in[, ]
satisfying the following conditions: () ∞i=αn,i= ,and
∞
i=βn,i= ;
() lim infn–→∞αn,αn,i> ,andlim infn–→∞βn,βn,i> ,∀i≥.
Then the sequence{xn}∞n=converges strongly to some point p*,where p*=(x).
Proof It is well known that for eachi≥,JrAiis a relatively nonexpansive mapping (see, for example, [, , ]). Therefore, for eachp∈F(JrAi) andw∈E, we have
φp,JrAiw≤φ(p,w).
Again, by the same method, we can prove that the set of strong asymptotically fixed points
FJrAi∞i==
∞
i=
