# Convergence Theorems for Maximal Monotone Operators, Weak Relatively Nonexpansive Mappings and Equilibrium Problems

Authors: Kamonrat Nammanee; Suthep Suantai; Prasit Cholamjiak
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We introduce hybrid-iterative schemes for solving a system of the zero-finding problems of maximal monotone operators, the equilibrium problem, and the fixed point problem of weak relatively nonexpansive mappings. We then prove, in a uniformly smooth and uniformly convex Banach space, strong convergence theorems by using a shrinking projection method. We finally apply the obtained results to a system of convex minimization problems.


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## Body

## 1. Introduction

LetE be a real Banach space and C a nonempty subset of E. Let E* be the dual space of E. We denote the value of x*∈E* at x∈E by 〈x*,x〉. Let T:C→C be a nonlinear mapping. We denote by F(T) the fixed points set of T, that is, F(T)={x∈C:x=Tx}. Let A:E→2E* be a set-valued mapping. We denote D(A) by the domain of A, that is, D(A)={x∈E:Ax≠∅} and also denote G(A) by the graph of A, that is, G(A)={(x,x*)∈E×E*:x*∈Ax}. A set-valued mapping A is said to be monotone if 〈x*-y*,x-y〉≥0 whenever (x,x*),(y,y*)∈G(A). It is said to be maximal monotone if its graph is not contained in the graph of any other monotone operators on E. It is known that if A is maximal monotone, then the set A-1(0*)={z∈E:0*∈Az} is closed and convex.The problem of finding a zero point of maximal monotone operators plays an important role in optimizations. This is because it can be reformulated to a convex minimization problem and a variational inequality problem. Many authors have studied the convergence of such problems in various spaces (see, e.g., [1–16]). Initiated by Martinet [17], in a real Hilbert space H, Rockafellar [18] introduced the following iterative scheme: x1∈H and
(1.1)xn+1=Jλnxn,∀n≥1,
where {λn}⊂(0,∞), Jλ is the resolvent of A defined by Jλ:=JλA=(I+λA)-1 for all λ>0, and A is a maximal monotone operator on H. Such an algorithm is called the proximal point algorithm. It was proved that the sequence {xn} generated by (1.1) converges weakly to an element in A-1(0) provided that liminfn→∞λn>0. Recently, Kamimura and Takahashi  [19] introduced the following iteration in a real Hilbert space: x1∈H and
(1.2)xn+1=αnxn+(1-αn)Jλnxn,∀n≥1,
where {αn}⊂[0,1] and {λn}⊂(0,∞). The weak convergence theorems are also established in a real Hilbert space under suitable conditions imposed on {αn} and {λn}.In 2004, Kamimura et al. [20] extended the above iteration process to a much more general setting. In fact, they proposed the following algorithm: x1∈E and
(1.3)xn+1=J-1(αnJ(xn)+(1-αn)J(Jλnxn)),∀n≥1,
where {αn}⊂[0,1], {λn}⊂(0,∞), and Jλ:=JλA=(J+λA)-1J for all λ>0. They proved, in a uniformly smooth and uniformly convex Banach space, a weak convergence theorem.LetF:C×C→ℝ, where ℝ is the set of real numbers, be a bifunction. The equilibrium problem is to find x∈C such that
(1.4)F(x,y)≥0,∀y∈C.
The solutions set of (1.4) is denoted by EP(F).For solving the equilibrium problem, we assume that(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, limsupt↓0F(tz+(1-t)x,y)≤F(x,y),(A4)
for allx∈C,F(x,·) is convex and lower semi-continuous.Recently, Takahashi and Zembayashi [21] introduced the following iterative scheme for a relatively nonexpansive mapping T:C→C in a uniformly smooth and uniformly convex Banach space: x1∈C and
(1.5)C1=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+1(x1),∀n≥1,
where {αn}⊂[0,1] and {rn}⊂(0,∞). Such an algorithm is called the shrinking projection method which was introduced by Takahashi et al. [22]. They proved that the sequence {xn} converges strongly to an element in F(T)∩EP(F) under appropriate conditions. The equilibrium problem has been intensively studied by many authors (see, e.g., [23–31]).Motivated by the previous results, we introduce a hybrid-iterative scheme for finding a zero point of maximal monotone operatorsAi:E→2E* (i=1,2,…,N) which is also a common element in the solutions set of an equilibrium problem for F and in the fixed points set of weak relatively nonexpansive mappings Ti:C→C  (i=1,2,…). Using the projection technique, we also prove that the sequence generated by a constructed algorithm converges strongly to an element in [⋂i=1N‍Ai-1(0*)]∩[⋂i=1∞‍F(Ti)]∩EP(F) in a uniformly smooth and uniformly convex Banach space. Finally, we apply our results to a system of convex minimization problems.


## 2. Preliminaries and Lemmas

In this section, we give some useful preliminaries and lemmas which will be used in the sequel.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+y‖<2.
A Banach space E is said to be uniformly convex if, for each ɛ∈(0,2], there exists δ>0 such that for any x,y∈U,
(2.2)‖x-y‖≥ɛimplies‖x+y‖<2(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 the modulus of convexity of E is defined as follows:
(2.3)δ(ɛ)=inf⁡{1-‖x+y2‖:x,y∈E,‖x‖=‖y‖=1,‖x-y‖≥ɛ}.
Then E is uniformly convex if and only if δ(ɛ)>0 for all ɛ∈(0,2]. A Banach space E is said to be smooth if the limit
(2.4)lim⁡t→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 mapping  J:E→2E* is defined by
(2.5)J(x)={x*∈E*:〈x,x*〉=‖x‖2=‖x*‖2}
for all x∈E. It is also known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on bounded subsets of E (see [32] 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. From the definition of ϕ, we see that
(2.7)(∥x∥-∥y∥)2≤ϕ(x,y)≤(∥x∥+∥y∥)2,ϕ(x,y)=ϕ(x,z)+ϕ(z,y)+2〈x-z,Jz-Jy〉
for all x,y,z∈E.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 [33] 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 F^(T). A mapping T is said to be relatively nonexpansive [33, 34] if F^(T)=F(T) and ϕ(p,Tx)≤ϕ(p,x) for all p∈F(T) and x∈C. A point p in C is said to be a strong asymptotic fixed point of T if C contains a sequence {xn} which converges strongly to p such that limn→∞∥xn-Txn∥=0. The set of strong asymptotic fixed points of T will be denoted by F~(T). A mapping T is said to be weak relatively nonexpansive [35] if F~(T)=F(T) and ϕ(p,Tx)≤ϕ(p,x) for all p∈F(T) and x∈C. It is obvious by definition that the class of weak relatively nonexpansive mappings contains the class of relatively nonexpansive mappings. Indeed, for any mapping T:C→C, we see that F(T)⊂F~(T)⊂F^(T). Therefore, if T is a relatively nonexpansive mapping, then F(T)=F~(T)=F^(T).Nontrivial examples of weak relatively nonexpansive mappings which are not relatively nonexpansive can be found in [36].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 [37], 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, ΠC(x)=x-, where x- is the solution to the minimization problem
(2.8)ϕ(x-,x)=min⁡{ϕ(y,x):y∈C}.
In a Hilbert space, ΠC is coincident with the metric projection denoted by PC.Lemma 2.1 (see [38]).
LetE be a uniformly convex and smooth Banach space and let {xn},{yn} be two sequences in E. If limn→∞ϕ(xn,yn)=0 and either {xn} or {yn} is bounded, then limn→∞∥xn-yn∥=0.Lemma 2.2 (see [37, 38]).
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=ΠC(x) if and only if 〈y-z,Jx-Jz〉≤0 for all y∈C.Lemma 2.3 (see [37, 38]).
LetC be a nonempty, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space E. Then
(2.9)ϕ(x,ΠCy)+ϕ(ΠCy,y)≤ϕ(x,y)∀x∈C,y∈E.Lemma 2.4 (see [39]).
LetE be a smooth and strictly convex Banach space, and let C be a nonempty, closed, and convex subset of E. Let T be a mapping from C into itself such that F(T) is nonempty and ϕ(u,Tx)≤ϕ(u,x) for all (u,x)∈F(T)×C. Then F(T) is closed and convex.LetE be a reflexive, strictly convex, and smooth Banach space. It is known that A:E→2E* is maximal monotone if and only if R(J+λA)=E* for all λ>0, where R(B) stands for the range of B.Define theresolvent of A by JλA=(J+λA)-1J for all λ>0. It is known that JλA is a single-valued mapping from E to D(A) and A-1(0*)=F(JλA) for all λ>0. For each λ>0, the Yosida approximation of A is defined by
(2.10)Aλ(x)=1λ(J(x)-JJλA(x))
for all x∈E. We know that Aλ(x)∈A(JλA(x)) for all λ>0 and x∈E.Lemma 2.5 (see [5]).
LetE be a smooth, strictly convex, and reflexive Banach space, let A⊂E×E* be a maximal monotone operator with A-1(0*)≠∅, and let JλA=(J+λA)-1J for each λ>0. Then
(2.11)ϕ(p,JλA(x))+ϕ(JλA(x),x)≤ϕ(p,x)
for all λ>0, p∈A-1(0*), and x∈E.Lemma 2.6 (see[40]).
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 satisfying (A1)–(A4), and let r>0 and x∈E. Then, there exists z∈C such that
(2.12)F(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C.Lemma 2.7 (see [41]).
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 satisfying (A1)–(A4). For all r>0 and x∈E, define the mapping Tr:E→C as follows:
(2.13)Tr(x)={z∈C:F(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C}.
Then, the following holds:(1)
Tr is single-valued;(2)
Tr is a firmly nonexpansive-type mapping [42], that is, for all x,y∈E,
(2.14)〈Trx-Try,JTrx-JTry〉≤〈Trx-Try,Jx-Jy〉;(3)
F(Tr)=EP(F);(4)
EP(F) is closed and convex.Lemma 2.8 (see [41]).
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 satisfying (A1)–(A4), let r>0. Then
(2.15)ϕ(p,Trx)+ϕ(Trx,x)≤ϕ(p,x),
for all x∈E and p∈F(Tr).


## 3. Strong Convergence Theorems

In this section, we are now ready to prove our main theorem.Theorem 3.1.
LetE be a uniformly smooth and uniformly convex Banach space, and let C be a nonempty, closed and convex subset of E. Let Ai:E→2E* (i=1,2,…,N) be maximal monotone operators, let F:C×C→ℝ be a bifunction, and let Ti:C→C  (i=1,2,…) be weak relatively nonexpansive mappings such that ℱ:=[⋂i=1N‍Ai-1(0*)]∩[⋂i=1∞‍F(Ti)]∩EP(F)≠∅. Let {en}n=1∞⊂E be the sequence such that limn→∞en=0. Define the sequence {xn}n=1∞ in C as follows:
(3.1)x1∈C1=C,yn=JλnNAN∘JλnN-1AN-1∘⋯∘Jλn1A1(xn+en),un=Trnyn,Cn+1={z∈Cn:supi≥1ϕ(z,Tiun)≤ϕ(z,xn+en)},xn+1=ΠCn+1(x1),∀n≥1.
If liminfn→∞λni>0 for each i=1,2,…,N and liminfn→∞rn>0, then the sequence {xn} converges strongly to q=Πℱ(x1).Proof.
We split the proof into several steps as follows.
Step  1. ℱ⊂Cn for all n≥1.
From Lemma2.4, we know that ⋂i=1∞‍F(Ti) is closed and convex. From Lemma 2.7(4), we also know that EP(F) is closed and convex. On the other hand, since Ai (i=1,2,…,N) are maximal monotone, Ai-1(0*) are closed and convex for each i=1,2,…,N; consequently, ⋂i=1NAi-1(0*) is closed and convex. Hence ℱ is a nonempty, closed, and convex subset of C.
We next show thatCn is closed and convex for all n≥1. Obviously, C1=C is closed and convex. Now suppose that Ck is closed and convex for some k∈ℕ. Then, for each z∈Ck and i≥1, we see that ϕ(z,Tiuk)≤ϕ(z,xk) is equivalent to
(3.2)2〈z,Jxk〉-2〈z,JTiuk〉≤‖xk‖2-‖Tiuk‖2.
By the construction of the set Ck+1, we see that
(3.3)Ck+1={z∈Ck:sup⁡i≥1⁡ϕ(z,Tiuk)≤ϕ(z,xk)}=⋂i=1∞‍{z∈Ck:ϕ(z,Tiuk)≤ϕ(z,xk)}.
Hence, Ck+1 is closed and convex. This shows, by induction, that Cn is closed and convex for all n≥1. It is obvious that ℱ⊂C1=C. Now, suppose that ℱ⊂Ck for some k∈ℕ. For any p∈ℱ, by Lemmas 2.5 and 2.8, we have
(3.4)ϕ(p,Tiuk)≤ϕ(p,uk)=ϕ(p,Trkyk)≤ϕ(p,yk)=ϕ(p,JλkNAN∘JλkN-1AN-1∘⋯∘Jλk1A1(xk+ek))≤ϕ(p,JλkN-1AN-1∘JλkN-2AN-2∘⋯∘Jλk1A1(xk+ek))⋮≤ϕ(p,Jλk2A2∘Jλk1A1(xk+ek))≤ϕ(p,Jλk1A1(xk+ek))≤ϕ(p,xk+ek).
This shows that ℱ⊂Ck+1. By induction, we can conclude that ℱ⊂Cn for all n≥1.
Step  2. limn→∞ϕ(xn,x1) exists.
Fromxn=ΠCn(x1) and xn+1=ΠCn+1(x1)∈Cn+1⊂Cn, we have
(3.5)ϕ(xn,x1)≤ϕ(xn+1,x1),∀n≥1.
From Lemma 2.3, for any p∈ℱ⊂Cn, we have
(3.6)ϕ(xn,x1)=ϕ(ΠCn(x1),x1)≤ϕ(p,x1)-ϕ(p,xn)≤ϕ(p,x1).
Combining (3.5) and (3.6), we conclude that limn→∞ϕ(xn,x1) exists.
Step  3. limn→∞∥J(Tiyn)-J(xn+en)∥=0.
Sincexm=ΠCm(x1)∈Cm⊂Cn for m>n≥1, by Lemma 2.3, it follows that
(3.7)ϕ(xm,xn)=ϕ(xm,ΠCn(x1))≤ϕ(xm,x1)-ϕ(ΠCn(x1),x1)=ϕ(xm,x1)-ϕ(xn,x1).
Letting m,n→∞, we have ϕ(xm,xn)→0. By Lemma 2.1, it follows that ∥xm-xn∥→0 as m,n→∞. Therefore, {xn} is a Cauchy sequence. By the completeness of the space E and the closedness of C, we can assume that xn→q∈C as n→∞. In particular, we obtain that
(3.8)lim⁡n→∞‖xn+1-xn‖=0.
Since en→0, we have
(3.9)lim⁡n→∞‖xn+1-(xn+en)‖=0.
Since xn+1=ΠCn+1(x1)∈Cn+1, for each i≥1,
(3.10)ϕ(xn+1,Tiun)≤ϕ(xn+1,xn+en)=〈xn+1,J(xn+1)-J(xn+en)〉+〈xn+1-(xn+en),J(xn+1)〉.
Since E is uniformly smooth, J is uniformly norm-to-norm continuous on bounded sets. It follows from (3.9) and by the boundedness of {xn} that
(3.11)lim⁡n→∞ϕ(xn+1,Tiun)=0
for all i=1,2,…. So from Lemma 2.1, we have
(3.12)limn→∞∥xn+1-Tiun∥=0,lim⁡n→∞‖Tiun-xn‖=0,
and, since en→0, therefore
(3.13)lim⁡n→∞‖Tiun-(xn+en)‖=0,
for all i=1,2,…. Since J is uniformly norm-to-norm continuous on bounded subsets of E,
(3.14)lim⁡n→∞‖J(Tiun)-J(xn+en)‖=0
for all i=1,2,….
Step  4. limn→∞∥Tiun-un∥=0 for all i=1,2,….
Denote thatΘni=JλniAi∘Jλni-1Ai-1∘…∘Jλn1A1 for each i∈{1,2,…,N} and Θn0=I for each n≥1. We note that yn=ΘnN(xn+en) for each n≥1.
To this end, we will show that(3.15)lim⁡n→∞‖J(Θni(xn+en))-J(Θni-1(xn+en))‖=0
for all i=1,2,…,N.
For anyp∈ℱ, by (3.4), we see that
(3.16)ϕ(p,ΘnN-1(xn+en))≤ϕ(p,ΘnN-2(xn+en))≤ϕ(p,ΘnN-3(xn+en))⋮≤ϕ(p,(xn+en)).
Since p∈ℱ, by Lemma 2.5 and (3.16), it follows that
(3.17)ϕ(yn,ΘnN-1(xn+en))≤ϕ(p,ΘnN-1(xn+en))-ϕ(p,yn)≤ϕ(p,(xn+en))-ϕ(p,yn)≤ϕ(p,(xn+en))-ϕ(p,un)≤ϕ(p,(xn+en))-ϕ(p,Tiun)=‖xn+en‖2-‖Tiun‖2-2〈p,J(xn+en)-J(Tiun)〉.
From (3.13) and (3.14), we get that limn→∞ϕ(yn,ΘnN-1(xn+en))=0. So we obtain that
(3.18)lim⁡n→∞‖yn-ΘnN-1(xn+en)‖=0.
Again, since p∈ℱ,
(3.19)ϕ(ΘnN-1(xn+en),ΘnN-2(xn+en))≤ϕ(p,ΘnN-2(xn+en))-ϕ(p,ΘnN-1(xn+en))≤ϕ(p,(xn+en))-ϕ(p,ΘnN-1(xn+en))≤ϕ(p,(xn+en))-ϕ(p,Tiun).
From (3.13) and (3.14), we get that
(3.20)lim⁡n→∞ϕ(ΘnN-1(xn+en),ΘnN-2(xn+en))=0.
It also follows that
(3.21)lim⁡n→∞‖ΘnN-1(xn+en)-ΘnN-2(xn+en)‖=0.
Continuing in this process, we can show that
(3.22)lim⁡n→∞‖ΘnN-2(xn+en)-ΘnN-3(xn+en)‖=⋯=lim⁡n→∞‖Θn1(xn+en)-(xn+en)‖=0.
So, we now conclude that
(3.23)lim⁡n→∞‖Θni(xn+en)-Θni-1(xn+en)‖=0
for each i=1,2,…,N. By the uniform norm-to-norm continuity of J, we also have
(3.24)lim⁡n→∞‖J(Θni(xn+en))-J(Θni-1(xn+en))‖=0
for each i=1,2,…,N. Using (3.23), it is easily seen that
(3.25)lim⁡n→∞‖yn-(xn+en)‖=0.
From un=Trnyn, by Lemma 2.8, it follows that
(3.26)ϕ(un,yn)=ϕ(Trnyn,yn)≤ϕ(p,yn)-ϕ(p,Trnyn)≤ϕ(p,xn+en)-ϕ(p,un)≤ϕ(p,xn+en)-ϕ(p,Tiun).
This implies that limn→∞ϕ(un,yn)=0 and hence
(3.27)lim⁡n→∞‖un-yn‖=0.
Combining (3.13), (3.25), and (3.27), we obtain that
(3.28)lim⁡n→∞‖Tiun-un‖=0
for all i≥1.
Step  5. q∈⋂i=1∞‍F(Ti).
Sincexn→q and en→0, xn+en→q. So from (3.25) and (3.27), we have un→q. Note that Ti (i=1,2,…) are weak relatively nonexpansive. Using (3.28), we can conclude that q∈F~(Ti)=F(Ti) for all i≥1. Hence q∈⋂i=1∞F(Ti).
Step  6. q∈⋂i=1N‍Ai-1(0*).
Noting thatΘni(xn+en)=JλniAiΘni-1(xn+en) for each i=1,2,…,N, we obtain that
(3.29)‖AλniΘni-1(xn+en)‖=1λni‖J(Θni-1(xn+en))-J(Θni(xn+en))‖.
From (3.24) and liminfn→∞λni>0, we have
(3.30)lim⁡n→∞‖AλniΘni-1(xn+en)‖=0.
We note that (Θni(xn+en),AλniΘni-1(xn+en))∈G(Ai) for each i=1,2,…,N. If (w,w*)∈G(Ai) for each i=1,2,…,N, then it follows from the monotonicity of Ai that
(3.31)〈w*-AλniΘni-1(xn+en),w-Θni(xn+en)〉≥0.
We see that Θni(xn+en)→q for each i=1,2,…,N. Thus, from (3.30) and (3.31), we have
(3.32)〈w*,w-q〉≥0.
By the maximality of Ai, it follows that q∈Ai-1(0*) for each i=1,2,…,N. Therefore, q∈⋂i=1N‍Ai-1(0*).
Step  7. q∈EP(F).
Fromun=Trnyn, we have
(3.33)F(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C.
By (A2), we have
(3.34)‖y-un‖‖Jun-Jyn‖rn≥1rn〈y-un,Jun-Jyn〉≥-F(un,y)≥F(y,un),∀y∈C.
Note that ∥Jun-Jyn∥/rn→0 since liminfn→∞rn>0. From (A4) and un→q, we get F(y,q)≤0 for all y∈C. For 0<t<1 and y∈C, define that yt=ty+(1-t)q. Then yt∈C, which implies that F(yt,q)≤0. From (A1), we obtain that 0=F(yt,yt)≤tF(yt,y)+(1-t)F(yt,q)≤tF(yt,y). Thus, F(yt,y)≥0. From (A3), we have F(q,y)≥0 for all y∈C. Hence, q∈EP(F). From Steps 5, 6, and 7, we now can conclude that q∈ℱ.
Step  8. q=Πℱ(x1).
Fromxn=ΠCn(x1), we have
(3.35)〈J(x1)-J(xn),xn-z〉≥0,∀z∈Cn.
Since ℱ⊂Cn, we also have
(3.36)〈J(x1)-J(xn),xn-z〉≥0,∀z∈F.
Letting n→∞ in (3.36), we obtain that
(3.37)〈J(x1)-J(q),q-z〉≥0,∀z∈F.
This shows that q=Πℱ(x1) by Lemma 2.2. We thus complete the proof.As a direct consequence of Theorem3.1, we can also apply to a system of convex minimization problems.Theorem 3.2.
LetE be a uniformly smooth and uniformly convex Banach space, and let C be a nonempty, closed, and convex subset of E. Let fi:E→(-∞,∞]  (i=1,2,…,N) be proper lower semicontinuous convex functions, let F:C×C→ℝ be a bifunction, and let Ti:C→C (i=1,2,…) be weak relatively nonexpansive mappings such that ℱ:=[⋂i=1N‍(∂fi-1)(0*)]∩[⋂i=1∞‍F(Ti)]∩EP(F)≠∅. Let {en}n=1∞⊂E be the sequence such that limn→∞en=0. Define the sequence {xn}n=1∞ in C as follows:
(3.38)x1∈C1=C,zn1=argmin⁡y∈E{f1(y)+12λn1‖y‖2+1λn1〈y,J(xn+en)〉},⋮znN-1=argminy∈E{fN-1(y)+12λnN-1∥y∥2+1λnN-1〈y,J(znN-2)〉},yn=argminy∈E{fN(y)+12λnN∥y∥2+1λnN〈y,J(znN-1)〉},un=Trnyn,Cn+1={z∈Cn:supi≥1ϕ(z,Tiun)≤ϕ(z,xn+en)},xn+1=ΠCn+1(x1),∀n≥1.
If liminfn→∞λni>0 for each i=1,2,…,N and liminfn→∞rn>0, then the sequence {xn} converges strongly to q=Πℱ(x1).Proof.
By Rockafellar's theorem [43, 44], ∂fi are maximal monotone operators for each i=1,2,…,N. Let λi>0 for each i=1,2,…,N. Then, zi=Jλi∂fi(x) if and only if
(3.39)0∈∂fi(zi)+1λi(J(zi)-J(x))=∂(fi+1λi(‖⋅‖22-J(x)))(zi),
which is equivalent to
(3.40)zi=argmin⁡y∈E{fi(y)+1λi(‖y‖22-〈y,J(x)〉)}.
Using Theorem 3.1, we thus complete the proof.IfE=H is a real Hilbert space, we then obtain the following results.Corollary 3.3.
LetC be a nonempty, closed and convex subset of a real Hilbert space H. Let Ai:H→2H  (i=1,2,…,N) be maximal monotone operators, let F:C×C→ℝ be a bifunction, and let Ti:C→C  (i=1,2,…) be weak relatively nonexpansive mappings such that ℱ:=[⋂i=1N‍Ai-1(0)]∩[⋂i=1∞‍F(Ti)]∩EP(F)≠∅. Let {en}n=1∞⊂H be the sequence such that limn→∞en=0. Define the sequence {xn}n=1∞ in C as follows:
(3.41)x1∈C1=C,yn=JλnNAN∘JλnN-1AN-1∘⋯∘Jλn1A1(xn+en),un=Trnyn,Cn+1={z∈Cn:supi≥1∥z-Tiun∥≤∥z-(xn+en)∥},xn+1=PCn+1(x1),∀n≥1.
If liminfn→∞λni>0 for each i=1,2,…,N and liminfn→∞rn>0, then the sequence {xn} converges strongly to q=Pℱ(x1).Corollary 3.4.
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let fi:H→(-∞,∞]   (i=1,2,…,N) be proper lower semi-continuous convex functions, let F:C×C→ℝ be a bifunction, and let Ti:C→C (i=1,2,…) be weak relatively nonexpansive mappings such that ℱ:=[⋂i=1N‍(∂fi-1)(0)]∩[⋂i=1∞‍F(Ti)]∩EP(F)≠∅. Let {en}n=1∞⊂H be the sequence such that limn→∞en=0. Define the sequence {xn}n=1∞ in C as follows:
(3.42)x1∈C1=C,zn1=argmin⁡y∈H{f1(y)+12λn1‖y‖2+1λn1〈y,xn+en〉},⋮znN-1=argminy∈H{fN-1(y)+12λnN-1∥y∥2+1λnN-1〈y,znN-2〉},yn=argminy∈H{fN(y)+12λnN∥y∥2+1λnN〈y,znN-1〉},un=Trnyn,Cn+1={z∈Cn:supi≥1∥z-Tiun∥≤∥z-(xn+en)∥},xn+1=PCn+1(x1),∀n≥1.
If liminfn→∞λni>0 for each i=1,2,…,N and liminfn→∞rn>0, then the sequence {xn} converges strongly to q=Pℱ(x1).Remark 3.5.
Using the shrinking projection method, we can construct a hybrid-proximal point algorithm for solving a system of the zero-finding problems, the equilibrium problems, and the fixed point problems of weak relatively nonexpansive mappings.Remark 3.6.
Since every relatively nonexpansive mapping is weak relatively nonexpansive, our results also hold ifTi:C→C (i=1,2,…) are relatively nonexpansive mappings.



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*Source: 804538-2012-06-25.xml*