# Convergence Analysis for a System of Equilibrium Problems and a Countable Family of Relatively Quasi-Nonexpansive Mappings in Banach Spaces

Authors: Prasit Cholamjiak; Suthep Suantai
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

We introduce a new hybrid iterative scheme for finding a common element in the solutions set of a system of equilibrium problems and the common fixed points set of an infinitely countable family of relatively quasi-nonexpansive
mappings in the framework of Banach spaces. We prove the strong convergence theorem by the shrinking projection method. In addition, the results obtained in this paper can be applied to a system of variational inequality problems and to a system of convex minimization problems in a Banach space.


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## Body

## 1. Introduction

LetE be a real Banach space, and let E* be the dual of E. Let C be a closed and convex subset of E. Let {fj}j∈Λ be bifunctions from C×C to ℝ, where ℝ is the set of real numbers and Λ is an arbitrary index set. The system of equilibrium problems is to find x̂∈C such that(1.1)fj(x̂,y)≥0,∀y∈C,j∈Λ.
If Λ is a singleton, then problem (1.1) reduces to find x̂∈C such that(1.2)f(x̂,y)≥0,∀y∈C.
The set of solutions of the equilibrium problem (1.2) is denoted by EP(f).Combettes and Hirstoaga [1] introduced an iterative scheme for finding a common element in the solutions set of problem (1.1) in a Hilbert space and obtained a weak convergence theorem.In 2004, Matsushita and Takahashi [2] introduced the following algorithm for a relatively nonexpansive mapping T in a Banach space E: for any initial point x0∈C, define the sequence {xn} by(1.3)xn+1=ΠCJ-1(αnJxn+(1-αn)JTxn),n≥0,
where J is the duality mapping on E, ΠC is the generalized projection from E onto C, and {αn} is a sequence in [0,1]. They proved that the sequence {xn} converges weakly to fixed point of T under some suitable conditions on {αn}.In 2008, Takahashi and Zembayashi [3] introduced the following iterative scheme which is called the shrinking projection method for a relatively nonexpansive mapping T and an equilibrium problem in a Banach space E:(1.4)x0=x∈C,C0=C,yn=J-1(αnJxn+(1-αn)JTxn),un∈Csuchthatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)},xn+1=ΠCn+1x0,n≥0.
They proved that the sequence {xn} converges strongly to ΠF(T)∩EP(f)x0 under some appropriate conditions.


## 2. Preliminaries and Lemmas

LetE be a real Banach space, and let U={x∈E:∥x∥=1} be the unit sphere of E. A Banach space E is said to be strictly convexif, for any x,y∈U,(2.1)x≠yimplies∥x+y2∥<1.
It is also said to be uniformly convex if, for each ε∈(0,2], there exists δ>0 such that, for any x,y∈U,(2.2)∥x-y∥≥εimplies∥x+y2∥<1-δ.
It is known that a uniformly convex Banach space is reflexive and strictly convex. The function δ:[0,2]→[0,1] which is called themodulus of convexity of E is defined as follows:(2.3)δ(ε)=inf{1-∥x+y2∥:x,y∈E,∥x∥=∥y∥=1,∥x-y∥≥ε}.
The space E is uniformly convex if and only if δ(ε)>0 for all ε∈(0,2]. A Banach space E is said to besmooth if the limit(2.4)limt→0∥x+ty∥-∥x∥t
exists for all x,y∈U. It is also said to be uniformly smooth if the limit (2.4) is attained uniformly for x,y∈U. The duality mappingJ:E→2E* is defined by(2.5)J(x)={x*∈E*:〈x,x*〉=∥x∥2=∥x*∥2}
for all x∈E. If E is a Hilbert space, then J=I, where I is the identity operator. It is also known that, if E is uniformly smooth, then J is uniformly norm-to-norm continuous on bounded subset of E (see [4] for more details).LetE be a smooth Banach space. The function ϕ:E×E→ℝ is defined by(2.6)ϕ(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2
for all x,y∈E. In a Hilbert space H, we have ϕ(x,y)=∥x-y∥2 for all x,y∈H.LetC be a closed and convex subset of E, and let T be a mapping from C into itself. A point p in C is said to be an asymptotic fixed point of T [5] if C contains a sequence {xn} which converges weakly to p such that limn→∞∥xn-Txn∥=0. The set of asymptotic fixed points of T will be denoted by F̂(T). A mapping T is said to be relatively nonexpansive [6–8] if F̂(T)=F(T) and ϕ(p,Tx)≤ϕ(p,x) for all p∈F(T) and x∈C. The asymptotic behavior of a relatively nonexpansive mapping was studied in [6, 7]. T is said to be relatively quasi-nonexpansive if F(T)≠∅ and ϕ(p,Tx)≤ϕ(p,x) for all p∈F(T) and x∈C. It is obvious that the class of relatively quasi-nonexpansive mappings is more general than the class of relatively nonexpansive mappings. The class of relatively quasi-nonexpansive mappings was studied by many authors (see, for example, [9–12]). Recall that T is closed if(2.7)xn→x,Txn→yimplyTx=y.The aim of this paper is to introduce a new hybrid projection algorithm for finding a common element in the solutions set of a system of equilibrium problems and the common fixed points set of an infinitely countable family of closed and relatively quasi-nonexpansive mappings in the frameworks of Banach spaces.We will need the following lemmas.Lemma 2.1 (Kamimura and Takahashi [8]).
LetE be a uniformly convex and smooth Banach space, and let {xn},{yn}be two sequences of E. If ϕ(xn,yn)→0 and either {xn} or {yn} is bounded, then ∥xn-yn∥→0 as n→∞.LetE be a reflexive, strictly convex, and smooth Banach space, and let C be a nonempty, closed, and convex subset of E. The generalized projection mapping, introduced by Alber [13], is a mapping ΠC:E→C that assigns to an arbitrary point x∈E the minimum point of the function ϕ(y,x); that is, ΠCx=x¯, where x¯ is the solution of the minimization problem(2.8)ϕ(x¯,x)=min{ϕ(y,x):y∈C}.
The existence and uniqueness of the operator ΠC follows from the properties of the functional ϕ and strict monotonicity of the duality mapping J (see, for instance, [4, 8, 13–15]). In a Hilbert space, ΠC is coincident with the metric projection.Lemma 2.2 (Alber [13], Kamimura and Takahashi [8]).
LetC be a nonempty, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space E, let x∈E, and let z∈C. Then z=ΠCx if and only if
(2.9)〈y-z,Jx-Jz〉≤0,∀y∈C.Lemma 2.3 (Alber [13], Kamimura and Takahashi [8]).
LetC be a nonempty, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space E. Then
(2.10)ϕ(x,ΠCy)+ϕ(ΠCy,y)≤ϕ(x,y)∀x∈C,y∈E.Lemma 2.4 (Qin et al.  [16]).
LetE be a uniformly convex, smooth Banach space, and let C be a closed and convex subset of E. Let T be a closed and relatively quasi-nonexpansive mapping from C into itself. Then F(T) is closed and convex.For solving the equilibrium problem, let us assume that a bifunctionf satisfies the following conditions:(A1)
f(x,x)=0 for all x∈C;(A2)
f is monotone; that is, f(x,y)+f(y,x)≤0 for all x,y∈C;(A3)
for allx,y,z∈C, lim supt↓0f(tz+(1-t)x,y)≤f(x,y);(A4)
for allx∈C,f(x,·) is convex and lower semicontinuous.Lemma 2.5 (Blum and Oettli [17]).
LetC be a closed and convex subset of a smooth, strictly convex, and reflexive Banach space E, let f be a bifunction from C×C to ℝ which satisfies conditions (A1)–(A4), and let r>0 and x∈E. Then there exists z∈C such that
(2.11)f(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C.Lemma 2.6 (Takahashi and Zembayashi [18]).
LetC be a closed and convex subset of a uniformly smooth, strictly convex, and reflexive Banach space E, and let f be a bifunction from C×C to ℝ which satisfies conditions (A1)–(A4). For all r>0 and x∈E, define the mapping Trf:E→C as follows:
(2.12)Trf(x)={z∈C:f(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C}.
Then, the following statements hold:(1)
Trf is single valued;(2)
Trf is of firmly nonexpansive type [19]; that is, for all x,y∈E,
(2.13)〈Trfx-Trfy,JTrfx-JTrfy〉≤〈Trfx-Trfy,Jx-Jy〉;(3)
F(Trf)=EP(f);(4)
EP(f) is closed and convex.Lemma 2.7 (Takahashi and Zembayashi [18]).
LetC be a closed and convex subset of a smooth, strictly, and reflexive Banach space E, let f be a bifunction from C×C to ℝ which satisfies conditions (A1)–(A4), and let r>0. Then, for all x∈E and q∈F(Trf),
(2.14)ϕ(q,Trfx)+ϕ(Trfx,x)≤ϕ(q,x).


## 3. Strong Convergence Theorems

Theorem 3.1.
LetE be a uniformly convex and uniformly smooth Banach space, and let C be a nonempty, closed, and convex subset of E. Let {fj}j=1M be bifunctions from C×C to ℝ which satisfies conditions (A1)–(A4), and let {Ti}i=1∞ be an infinitely countable family of closed and relatively quasi-nonexpansive mappings from C into itself. Assume that F:=(⋂i=1∞F(Ti))∩(⋂j=1MEP(fj))≠∅. For any initial point x0∈E with x1=ΠC1x0 and C1=C, define the sequence {xn} as follows:
(3.1)yn,i=J-1(αnJxn+(1-αn)JTixn),un,i=TrM,nfMTrM-1,nfM-1⋯Tr1,nf1yn,i,Cn+1={z∈Cn:supi≥1ϕ(z,un,i)≤ϕ(z,xn)},xn+1=ΠCn+1x0,n≥1.
Assume that {αn} and {rj,n} for j=1,2,…,M are sequences which satisfy the following conditions:(B1)
limsupn→∞αn<1;(B2)
liminfn→∞rj,n>0.
Then the sequence{xn} converges strongly to ΠFx0.Proof.
We divide our proof into six steps as follows.
Step 1.
F⊂Cn for all n≥1.
From Lemma2.4 we know that F(Ti) is closed, and convex for all i≥1. From Lemma 2.6(4), we also know that EP(fj) is closed and convex for each j=1,2,…,M. Hence F:=(⋂i=1∞F(Ti))∩(⋂j=1MEP(fj)) is a nonempty, closed and convex subset of C. Clearly C1=C is closed and convex. Suppose that Ck is closed and convex for some k∈ℕ. For each z∈Ck and i≥1, we see that ϕ(z,uk,i)≤ϕ(z,xk) is equivalent to
(3.2)2〈z,Jxk〉-2〈z,Juk,i〉≤∥xk∥2-∥uk,i∥2.
By the construction of the set Ck+1, we see that
(3.3)Ck+1={z∈Ck:supi≥1ϕ(z,uk,i)≤ϕ(z,xk)}=⋂i=1∞{z∈Ck:ϕ(z,uk,i)≤ϕ(z,xk)}‍.
Hence Ck+1 is also closed and convex.
It is obvious thatF⊂C1=C. Now, suppose that F⊂Ck for some k∈ℕ, and let p∈F:=(⋂i=1∞F(Ti)‍)∩(⋂j=1MEP(fj)‍). Then
(3.4)ϕ(p,uk,i)=ϕ(p,TrM,nfMTrM-1,nfM-1⋯Tr1,nf1yk,i)≤ϕ(p,TrM-1,nfM-1TrM-2,nfM-2⋯Tr1,nf1yk,i)⋮≤ϕ(p,Tr1,nf1yk,i)≤ϕ(p,yk,i)=ϕ(p,J-1(αkJxk+(1-αk)JTixk))=∥p∥2-2〈p,αkJxk+(1-αk)JTixk〉+∥αkJxk+(1-αk)JTixk∥2≤∥p∥2-2αk〈p,Jxk〉-2(1-αk)〈p,JTixk〉+αk∥xk∥2+(1-αk)∥Tixk∥2=αkϕ(p,xk)+(1-αk)ϕ(p,Tixk)≤ϕ(p,xk).
Hence F⊂Ck+1. By induction, we can conclude that F⊂Cn for all n≥1.Step 2.
limn→∞ϕ(xn,x0) exists.
Fromxn=ΠCnx0 and xn+1=ΠCn+1x0∈Cn+1⊂Cn, we have
(3.5)ϕ(xn,x0)≤ϕ(xn+1,x0),n≥1.
From Lemma 2.3 we get that
(3.6)ϕ(xn,x0)=ϕ(ΠCnx0,x0)≤ϕ(p,x0)-ϕ(p,xn)≤ϕ(p,x0).
Combining (3.5) and (3.6), we get that limn→∞ϕ(xn,x0) exists.Step 3.
{xn} is a Cauchy sequence in C.
Sincexm=ΠCmx0∈Cm⊂Cn for m>n, we obtain from Lemma 2.3 that
(3.7)ϕ(xm,xn)=ϕ(xm,ΠCnx0)≤ϕ(xm,x0)-ϕ(ΠCnx0,x0)=ϕ(xm,x0)-ϕ(xn,x0).
We see that ϕ(xm,xn)→0 as m,n→∞, which implies with Lemma 2.1 that ∥xm-xn∥→0 as m,n→∞. Therefore {xn} is a Cauchy sequence. By the completeness of the space E and the closedness of the set C, we can assume that xn→q∈C as n→∞. Moreover, we get that
(3.8)limn→∞ϕ(xn+1,xn)=0.
Since xn+1=ΠCn+1x0∈Cn+1, we have for all i≥1 that
(3.9)ϕ(xn+1,un,i)≤ϕ(xn+1,xn)→0.
Applying Lemma 2.1 to (3.8) and (3.9), we derive
(3.10)limn→∞∥un,i-xn∥=0,∀i≥1.
This shows that un,i→q as n→∞ for all i≥1. Since J is uniformly norm-to-norm continuous on bounded subsets of E, we obtain that
(3.11)limn→∞∥Jun,i-Jxn∥=0,∀i≥1.Step 4.
q∈⋂i=1∞F(Ti).
DenoteΘnj=Trj,nfjTrj-1,nfj-1⋯Tr1,nf1 for any j∈{1,2,…,M} and Θn0=I for all n≥1. We note that un,i=ΘnMyn,i for all i≥1. From (3.4) we observe that
(3.12)ϕ(p,ΘnM-1yn,i)≤ϕ(p,ΘnM-2yn,i)≤⋯≤ϕ(p,yn,i)≤ϕ(p,xn),∀i≥1.
Since p∈EP(fM)=F(TrM,nfM) for all n≥1, it follows from (3.12) and Lemma 2.7 that
(3.13)ϕ(un,i,ΘnM-1yn,i)≤ϕ(p,ΘnM-1yn,i)-ϕ(p,un,i)≤ϕ(p,xn)-ϕ(p,un,i).
From (3.10) and (3.11), we get that limn→∞ϕ(un,i,ΘnM-1yn,i)=0 for all i≥1. From Lemma 2.1, we have
(3.14)limn→∞∥un,i-ΘnM-1yn,i∥=0,∀i≥1.
From (3.10) and (3.14), we have
(3.15)limn→∞∥xn-ΘnM-1yn,i∥=0,∀i≥1,
and hence,
(3.16)limn→∞∥Jxn-JΘnM-1yn,i∥=0,∀i≥1.
Again, since p∈EP(fM-1)=F(TrM-1,nfM-1) for all n≥1, it follows from (3.12) and Lemma 2.7 that
(3.17)ϕ(ΘnM-1yn,i,ΘnM-2yn,i)≤ϕ(p,ΘnM-2yn,i)-ϕ(p,ΘnM-1yn,i)≤ϕ(p,xn)-ϕ(p,ΘnM-1yn,i).
From (3.15) and (3.16), we also have
(3.18)limn→∞∥ΘnM-1yn,i-ΘnM-2yn,i∥=0,∀i≥1.
Hence, from (3.15) and (3.18), we get
(3.19)limn→∞∥xn-ΘnM-2yn,i∥=0,∀i≥1,(3.20)limn→∞∥Jxn-JΘnM-2yn,i∥=0,∀i≥1.
In a similar way, we can verify that
(3.21)limn→∞∥ΘnM-2yn,i-ΘnM-3yn,i∥=⋯=limn→∞∥Θn1yn,i-yn,i∥=0
for all i≥1,
(3.22)limn→∞∥xn-ΘnM-3yn,i∥=⋯=limn→∞∥xn-yn,i∥=0
for all i≥1,
(3.23)limn→∞∥Jxn-JΘnM-3yn,i∥=⋯=limn→∞∥Jxn-Jyn,i∥=0
for all i≥1. Hence, we can conclude that
(3.24)limn→∞∥Θnjyn,i-Θnj-1yn,i∥=0
for each j=1,2,…,M and i≥1. Observe that
(3.25)∥Jyn,i-Jxn∥=∥αnJxn+(1-αn)JTixn-Jxn∥=(1-αn)∥JTixn-Jxn∥,
then we obtain from (B1) and (3.23) that
(3.26)limn→∞∥JTixn-Jxn∥=0,∀i≥1.
Since J-1 is also uniformly norm-to-norm continuous on bounded subsets, we get that
(3.27)limn→∞∥Tixn-xn∥=0,∀i≥1.
Since Ti is closed for all i≥1 and xn→q, we conclude that q∈⋂i=1∞F(Ti).Step 5.
q∈⋂j=1MEP(fj).
From (3.24) and (B2), we have that ∥JΘnjyn,i-JΘnj-1yn,i∥/rj,n→0 as n→∞. Then, for each j=1,2,…,M, we obtain that
(3.28)fj(Θnjyn,i,y)+1rj,n〈y-Θnjyn,i,JΘnjyn,i-JΘnj-1yn,i〉≥0,∀y∈C.
From (A2) we have that
(3.29)∥y-Θnjyn,i∥∥JΘnjyn,i-JΘnj-1yn,i∥rj,n≥1rj,n〈y-Θnjyn,i,JΘnjyn,i-JΘnj-1yn,i〉≥-fj(Θnjyn,i,y)≥fj(y,Θnjyn,i),∀y∈C.
From (A4) and the fact that Θnjyn,i→q for i≥1, we get fj(y,q)≤0 for all y∈C. For each 0<t<1 and y∈C, denote yt=ty+(1-t)q. Then yt∈C, which implies that fj(yt,q)≤0. From (A1) and (A4), we obtain that 0=fj(yt,yt)≤tfj(yt,y)+(1-t)fj(yt,q)≤tfj(yt,y). Thus, fj(yt,y)≥0. From (A3), we have fj(q,y)≥0 for all y∈C and j=1,2,…,M. Hence q∈⋂j=1MEP(fj).Step 6.
q=ΠFx0.
Fromxn=ΠCnx0, we have
(3.30)〈Jx0-Jxn,xn-z〉≥0∀z∈Cn.
Since F⊂Cn, we also have
(3.31)〈Jx0-Jxn,xn-p〉≥0∀p∈F.
Letting n→∞ in (3.31), we obtain that
(3.32)〈Jx0-Jq,q-p〉≥0∀p∈F.
From Lemma 2.2 we conclude that q=ΠFx0. This completes the proof.Remark 3.2.
Theorem3.1 improves and extends Theorem 3.1 of Takahashi and Zembayashi in [3] in the following senses:(i)
from the case of an equilibrium problem to a finite family of equilibrium problems;(ii)
from a single relatively nonexpansive mapping to an infinitely countable family of relatively quasi-nonexpansive mappings;(iii)
ifM=1 and Ti=T for all i≥1, then our restriction on {αn} is weaker than that of Theorem 3.1 of [3].Remark 3.3.
The iteration (3.1) is a modification of (1.4) in the following ways.(i)
We use the composition of mappings{Trj,nfj}j=1M in the second step.(ii)
We construct the setCn+1 by using the concept of supremum concerning an infinitely countable family of closed and relatively quasi-nonexpansive mappings {Ti}i=1∞. If M=1 and Ti=T for all i≥1, then the iteration (3.1) reduces to that of (1.4).
If we takeαn=0 for all n∈ℕ in Theorem 3.1, then we have the following corollary.Corollary 3.4.
LetE be a uniformly convex and uniformly smooth Banach space, and let C be a nonempty, closed, and convex subset of E. Let {fj}j=1M be bifunctions from C×C to ℝ which satisfies conditions (A1)–(A4), and let {Ti}i=1∞ be an infinitely countable family of closed and relatively quasi-nonexpansive mappings from C into itself. Assume that F:=(⋂i=1∞F(Ti))∩(⋂j=1MEP(fj))≠∅. For any initial point x0∈E with x1=ΠC1x0 and C1=C, define the sequence {xn} as follows:
(3.33)yn,i=Tixn,un,i=TrM,nfMTrM-1,nfM-1⋯Tr1,nf1yn,i,Cn+1={z∈Cn:supi≥1ϕ(z,un,i)≤ϕ(z,xn)},xn+1=ΠCn+1x0,n≥1.
If  liminfn→∞rj,n>0  for each j=1,2,…,M, then {xn} converges strongly to ΠFx0.


## 4. Applications

In this section, we give several applications of Theorem3.1 in the framework of Banach spaces and Hilbert spaces.LetA:C→E* be a nonlinear mapping. The classical variational inequality problem is to find that x̂∈C such that(4.1)〈Ax̂,y-x̂〉≥0∀y∈C.
The solutions set of (4.1) is denoted by VI(C,A). For each r>0 and x∈E, define the mapping TrA:E→C as follows:(4.2)TrA(x)={z∈C:〈Az,y-z〉+1r〈y-z,Jz-Jx〉≥0,∀y∈C}.Theorem 4.1.
LetE be a uniformly convex and uniformly smooth Banach space, and let C be a nonempty, closed, and convex subset of E. Let {Aj}j=1M be continuous and monotone operators from C to E*, and let {Ti}i=1∞ be an infinitely countable family of closed and relatively quasi-nonexpansive mappings from C into itself such that F:=(⋂i=1∞F(Ti))∩(⋂j=1MVI(C,Aj))≠∅. For any initial point x0∈E with x1=ΠC1x0 and C1=C, define the sequence {xn} as follows:
(4.3)yn,i=J-1(αnJxn+(1-αn)JTixn),un,i=TrM,nAMTrM-1,nAM-1⋯Tr1,nA1yn,i,Cn+1={z∈Cn:supi≥1ϕ(z,un,i)≤ϕ(z,xn)},xn+1=ΠCn+1x0,n≥1.
Assume that {αn} and {rj,n} for j=1,2,…,M are sequences which satisfy conditions (B1) and (B2) of Theorem 3.1.
Then the sequence{xn} converges strongly to ΠFx0.Proof.
Definefj(x,y)=〈Ajx,y-x〉 for all x,y∈C and j=1,2,…,M. First, we see that F(Trjfj)=EP(fj)=VI(C,Aj)=F(TrjAj) for each j=1,2,…,M.
Next, we show that{fj}j=1M satisfy conditions (A1)–(A4).(A1)
Considerfj(x,x)=〈Ajx,x-x〉=0 for all x∈C and j=1,2,…,M.(A2)
For eachx,y∈C and j=1,2,…,M, we observe that(4.4)fj(x,y)+fj(y,x)=〈Ajx,y-x〉+〈Ajy,x-y〉=〈Ajx-Ajy,y-x〉.
By the monotonicity of Aj, we obtain that fj is monotone. Thus {fj}j=1M satisfy condition (A2).(A3)
For eachx,y,z∈C and j=1,2,…,M, we have by the continuity of Aj that(4.5)limsupt↓0fj(tz+(1-t)x,y)=limsupt↓0〈Aj(tz+(1-t)x),y-(tz+(1-t)x)〉=〈Ajx,y-x〉=fj(x,y).
This shows that {fj}j=1M satisfy condition (A3).(A4)
Letu,v∈C and s∈(0,1). Then, for each x∈C and j=1,2,…,M, we have(4.6)fj(x,su+(1-s)v)=〈Ajx,su+(1-s)v-x〉=s〈Ajx,u-x〉+(1-s)〈Ajx,v-x〉=sfj(x,u)+(1-s)fj(x,v).
Thus fj is convex in the second variable. Let un∈C and limn→∞un=u. Then
(4.7)fj(x,u)=〈Ajx,u-x〉=limn→∞〈Ajx,un-x〉=limn→∞fj(x,un).
This shows that fj is lower semicontinuous in the second variable. Hence {fj}j=1M satisfy condition (A4). From Theorem 3.1 we obtain the desired result.If we takeαn=0 for all n∈ℕ in Theorem 4.1, then we have the following corollary.Corollary 4.2.
LetE be a uniformly convex and uniformly smooth Banach space, and let C be a nonempty, closed, and convex subset of E. Let {Aj}j=1M be continuous and monotone operators from C to E*, and let {Ti}i=1∞ be an infinitely countable family of closed and relatively quasi-nonexpansive mappings from C into itself such that F:=(⋂i=1∞F(Ti))∩(⋂j=1MVI(C,Aj))≠∅. For any initial point x0∈E with x1=ΠC1x0 and C1=C, define the sequence {xn} as follows:
(4.8)yn,i=Tixn,un,i=TrM,nAMTrM-1,nAM-1⋯Tr1,nA1yn,i,Cn+1={z∈Cn:supi≥1ϕ(z,un,i)≤ϕ(z,xn)},xn+1=ΠCn+1x0,n≥1.
If  liminfn→∞rj,n>0  for each j=1,2,…,M, then {xn} converges strongly to ΠFx0.
Letφ:C→ℝ be a real-valued function. The convex minimization problem is to find that x̂∈C such that
(4.9)φ(x̂)≤φ(y)∀y∈C.
The solutions set of (4.9) is denoted by CMP(φ). For each r>0 and x∈E, define the mapping Trφ:E→C as follows:
(4.10)Trφ(x)={z∈C:φ(y)+1r〈y-z,Jz-Jx〉≥φ(z),∀y∈C}.Theorem 4.3.
LetE be a uniformly convex and uniformly smooth Banach space, and let C be a nonempty, closed, and convex subset of E. Let {φj}j=1M be lower semicontinuous and convex functions from C to ℝ, and let {Ti}i=1∞ be an infinitely countable family of closed and relatively quasi-nonexpansive mappings from C into itself such that F:=(⋂i=1∞F(Ti))∩(⋂j=1MCMP(φj))≠∅. For any initial point x0∈E with x1=ΠC1x0 and C1=C, define the sequence {xn} as follows:
(4.11)yn,i=J-1(αnJxn+(1-αn)JTixn),un,i=TrM,nφMTrM-1,nφM-1⋯Tr1,nφ1yn,i,Cn+1={z∈Cn:supi≥1ϕ(z,un,i)≤ϕ(z,xn)},xn+1=ΠCn+1x0,n≥1.
Assume that {αn} and {rj,n} for j=1,2,…,M are sequences which satisfy conditions (B1) and (B2) of Theorem 3.1.
Then the sequence{xn} converges strongly to ΠFx0.Proof.
Definefj(x,y)=φj(y)-φj(x) for all x,y∈C and j=1,2,…,M. Then F(Trjfj)=EP(fj)=CMP(φj)=F(Trjφj) for each j=1,2,…,M, and therefore {fj}j=1M satisfy conditions (A1) and (A2).
Next, we show that{fj}j=1M satisfy conditions (A3) and (A4). For each x,y,z∈C, we have by the lower semicontinuity of φj that
(4.12)limsupt↓0fj(tz+(1-t)x,y)=limsupt↓0(φj(y)-φj(tz+(1-t)x))≤φj(y)-φj(x)=fj(x,y).
This implies that {fj}j=1M satisfy condition (A3).
Letu,v∈C and s∈(0,1). For each x∈C, we have by the convexity of φj that
(4.13)fj(x,su+(1-s)v)=φj(su+(1-s)v)-φj(x)≤sφj(u)+(1-s)φj(v)-φj(x)=s(φj(u)-φj(x))+(1-s)(φj(v)-φj(x))=sfj(x,u)+(1-s)fj(x,v).
On the other hand, let un∈C and limn→∞un=u. By the lower semicontinuity of φj we have
(4.14)fj(x,u)=φj(u)-φj(x)≤liminfn→∞(φj(un)-φj(x))=liminfn→∞fj(x,un).
Thus {fj}j=1M satisfy condition (A4). From Theorem 3.1 we also obtain the desired result.If we takeαn=0 for all n∈ℕ in Theorem 4.3, then we have the following corollary.Corollary 4.4.
LetE be a uniformly convex and uniformly smooth Banach space, and let C be a nonempty, closed, and convex subset of E. Let {φj}j=1M be lower semicontinuous and convex functions from C to ℝ, and let {Ti}i=1∞ be an infinitely countable family of closed and relatively quasi-nonexpansive mappings from C into itself such that F:=(⋂i=1∞F(Ti))∩(⋂j=1MCMP(φj))≠∅. For any initial point x0∈E with x1=ΠC1x0 and C1=C, define the sequence {xn} as follows:
(4.15)yn,i=Tixn,un,i=TrM,nφMTrM-1,nφM-1⋯Tr1,nφ1yn,i,Cn+1={z∈Cn:supi≥1ϕ(z,un,i)≤ϕ(z,xn)},xn+1=ΠCn+1x0,n≥1.
If  liminfn→∞rj,n>0  for each j=1,2,…,M, then {xn} converges strongly to ΠFx0.As a direct consequence of Theorem3.1, we obtain the following application in a Hilbert space.Theorem 4.5.
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let {fj}j=1M be bifunctions from C×C to ℝ which satisfies conditions (A1)–(A4), and let {Ti}i=1∞ be an infinitely countable family of closed and quasi-nonexpansive mappings from C into itself such that F:=(⋂i=1∞F(Ti))∩(⋂j=1MEP(fj))≠∅. For any initial point x0∈H with x1=PC1x0 and C1=C, define the sequence {xn} as follows:
(4.16)yn,i=αnxn+(1-αn)Tixn,un,i=TrM,nfMTrM-1,nfM-1⋯Tr1,nf1yn,i,Cn+1={z∈Cn:supi≥1∥z-un,i∥≤∥z-xn∥},xn+1=PCn+1x0,n≥1,
where P is the metric projection. Assume that {αn} and {rj,n} for j=1,2,…,M are sequences which satisfy conditions (B1) and (B2) of Theorem 3.1.
Then the sequence{xn} converges strongly to PFx0.Proof.
TakingE=H in Theorem 3.1, the result is obtained immediately.Remark 4.6.
Theorem4.5 improves and extends the main results of [20–22] in the following senses:(i)
from the case of an equilibrium problem to a finite family of equilibrium problems;(ii)
from the class of nonexpansive mappings to the class of an infinitely countable family of quasi-nonexpansive mappings.



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*Source: 141376-2010-09-06.xml*