# A Halpern-Mann Type Iteration for Fixed Point Problems of a Relatively Nonexpansive Mapping and a System of Equilibrium Problems

Authors: Utith Inprasit; Weerayuth Nilsrakoo
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

A new modified Halpern-Mann type iterative method is constructed. Strong convergence of the scheme to a common element of the set of fixed points of a relatively nonexpansive mapping and the set of common solutions to a system of equilibrium problems in a uniformly convex real Banach space which is also uniformly smooth is proved. The results presented in this work improve on the corresponding ones announced by many others.


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## Body

## 1. Introduction

Throughout this paper, we denote byℕ and ℝ the sets of positive integers and real numbers, respectively. Let E be a Banach space, E* the dual space of E, and C a nonempty closed convex subset of E. Let F:C×C→ℝ be a bifunction. The equilibrium problem is to find x∈C such that(1.1)F(x,y)≥0∀y∈C.
The set of solutions of (1.1) is denoted by EP(F). The equilibrium problems include fixed point problems, optimization problems, variational inequality problems, and Nash equilibrium problems as special cases. Some methods have been proposed to solve the equilibrium problems (see, e.g., [1, 2]). In 2005, Combettes and Hirstoaga [3] introduced an iterative scheme of finding the best approximation to the initial data when EP(F) is nonempty, and they also proved a strong convergence theorem.LetE be a smooth Banach space and J the normalized duality mapping from E to E*. Alber [4] considered the following functional φ:E×E→[0,∞) defined by (1.2)φ(x,y)=‖x‖2-2〈x,Jy〉+‖y‖2(x,y∈E).
Using this functional, Matsushita and Takahashi [5, 6] studied and investigated the following mappings in Banach spaces. A mapping S:C→E is relatively nonexpansive if the following properties are satisfied:  (R1)
F(S)≠∅,(R2)
φ(p,Sx)≤φ(p,x) for all p∈F(S) and x∈C,(R3)
F(S)=F̂(S),whereF(S) and F̂(S) denote the set of fixed points of S and the set of asymptotic fixed points of S, respectively. It is known that S satisfies condition (R3) if and only if I-S is demiclosed at zero, where I is the identity mapping; that is, whenever a sequence {xn} in C converges weakly to p and {xn-Sxn} converges strongly to 0, it follows that p∈F(S). In a Hilbert space H, the duality mapping J is an identity mapping and φ(x,y)=∥x-y∥2 for all x,y∈H. Hence, if S:C→H is nonexpansive (i.e., ∥Sx-Sy∥≤∥x-y∥ for all x,y∈C), then it is relatively nonexpansive. Several articles have appeared providing methods for approximating fixed points of relatively nonexpansive mappings (see, e.g., [5–19] and the references therein). Matsushita and Takahashi [5] introduced the following iteration: a sequence {xn} defined by(1.3)xn+1=ΠCJ-1(αnJxn+(1-αn)JSxn)n=1,2,…,
where x1∈C is arbitrary, {αn} is an appropriate sequence in [0,1], S is a relatively nonexpansive mapping, and ΠC denotes the generalized projection from E onto a closed convex subset C of E. They proved that the sequence {xn} converges weakly to a fixed point of T. Moreover, Matsushita and Takahashi [6] proposed the following modification of iteration (1.3):(1.4)x1∈Cisarbitrary,yn=J-1(αnJxn+(1-αn)JSxn),Cn={z∈C:φ(z,yn)≤φ(z,xn)},Qn={z∈C:〈xn-z,Jx1-Jxn〉≥0},xn+1=ΠCn∩Qnx1,n=1,2,…,
and proved that the sequence {xn} converges strongly to ΠF(S)x1. The iteration (1.4) is called the hybrid method. To generate the iterative sequence, we use the generalized metric projection onto Cn∩Qn for n∈ℕ. It always exists, because each Cn∩Qn is nonempty, closed, and convex. However, in a practical case, it is not easy to be calculated. In particular, as n becomes larger, the shape of Cn∩Qn becomes more complicate, and the projection will take much more time to be calculated.In order to overcome this difficulty, Nilsrakoo and Saejung [15] modified Halpern and Mann's iterations for finding a fixed point of a relatively nonexpansive mapping in a Banach space as follows: x∈E,x1∈C and(1.5)xn+1=ΠCJ-1(αnJx+βnJxn+γnJSxn),n=1,2,…,
where {αn}, {βn}, and {γn} are appropriate sequences in [0,1] with αn+βn+γn≡1, and they proved that {xn} converges strongly to ΠF(S)x.Many authors studied the problems of finding a common element of the set of fixed points for a mapping and the set of common solutions to a system of equilibrium problems in the setting of Hilbert space and uniformly smooth and uniformly convex Banach space, respectively (see, e.g., [20–33] and the references therein). In a Hilbert space H, S. Takahashi and W. Takahashi [34] introduced the iteration as follows: sequence {xn} generated by x,x1∈C, (1.6)un∈Csuch thatF(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnx+(1-αn)Sun,n=1,2,…,
where {αn} is an appropriate sequence in [0,1], S is nonexpansive, and {rn} is an appropriate positive real sequence. They proved that {xn} converges strongly to an element in F(S)∩EP(F). In 2009, Takahashi and Zembayashi [30] proposed the iteration in a uniformly smooth and uniformly convex Banach space as follows: a sequence {xn} generated by u1∈E, (1.7)xn∈Csuch thatF(xn,y)+1rn〈y-xn,Jxn-Jun〉≥0,∀y∈C,un+1=J-1(αnJxn+(1-αn)JSxn),n=1,2,…,
where S is relatively nonexpansive, {αn} is an appropriate sequence in [0,1], and {rn} is an appropriate positive real sequence. They proved that if J is weakly sequentially continuous, then {xn} converges weakly to an element in F(S)∩EP(F). Consequently, there are many results presented strong convergence theorems for finding a common element of the set of fixed points for a mapping and the set of common solutions to a system of equilibrium problems by using the hybrid method. However, Nilsrakoo [35] introduced the Halpern-Mann iteration guaranteeing the strong convergence as follows: x∈C,u1∈E and(1.8)xn∈Csuch thatF(xn,y)+1rn〈y-xn,Jxn-Jun〉≥0,∀y∈C,yn=ΠCJ-1(αnJx+(1-αn)Jxn),un+1=J-1(βnJxn+(1-βn)JSyn),n=1,2,…,
and proved that {un} and {xn} converge strongly to ΠF(S)∩EP(F)x.Motivated by Nilsrakoo and Saejung [15] and Nilsrakoo [35], we present a strong convergence theorem of a new modified Halpern-Mann iterative scheme to find a common element of the set of fixed points of a relatively nonexpansive mapping and the set of common solutions to a system of equilibrium problems in a uniformly convex real Banach space which is also uniformly smooth. The results in this work improve on the corresponding ones announced by many others.


## 2. Preliminaries

We collect together some definitions and preliminaries which are needed in this paper. We say that a Banach spaceE is strictly convex if the following implication holds for x,y∈E: (2.1)‖x‖=‖y‖=1,x≠yimply‖x+y2‖<1.
It is also said to be uniformly convex if for any ɛ>0, there exists δ>0 such that (2.2)‖x‖=‖y‖=1,‖x-y‖≥ɛimply‖x+y2‖≤1-δ.
It is known that if E is a uniformly convex Banach space, then E is reflexive and strictly convex. We say that E is uniformly smooth if the dual space E* of E is uniformly convex. A Banach space E is smooth if the limit limt→0((∥x+ty∥-∥x∥)/t) exists for all norm one elements x and y in E. It is not hard to show that if E is reflexive, then E is smooth if and only if E* is strictly convex.LetE be a smooth Banach space. The function φ:E×E→ℝ (see [4]) is defined by (2.3)φ(x,y)=‖x‖2-2〈x,Jy〉+‖y‖2(x,y∈E),
where the duality mappingJ:E→E* is given by (2.4)〈x,Jx〉=‖x‖2=‖Jx‖2(x∈E).
It is obvious from the definition of the function φ that (2.5)(‖x‖-‖y‖)2≤φ(x,y)≤(‖x‖+‖y‖)2,(2.6)φ(x,y)=φ(x,z)+φ(z,y)+2〈x-z,Jz-Jy〉,
for all x,y,z∈E. Moreover,(2.7)φ(x,J-1(∑i=1nλiJyi))≤∑i=1nλiφ(x,yi),
for all λi∈[0,1] with ∑i=1nλi=1 and x,yi∈E.The following lemma is an analogue of Xu's inequality [36, Theorem  2] with respect to φ.Lemma 2.1 (see [15, Lemma  2.2]).
LetE be a uniformly smooth Banach space and r>0. Then, there exists a continuous, strictly increasing, and convex function g:[0,2r]→[0,∞) such that g(0)=0 and
(2.8)φ(x,J-1(λJy+(1-λ)Jz))≤λφ(x,y)+(1-λ)φ(x,z)-λ(1-λ)g(‖Jy-Jz‖),
for all λ∈[0,1], x∈E and y,z∈Br:={z∈E:∥z∥≤r}.It is also easy to see that if{xn} and {yn} are bounded sequences of a smooth Banach space E, then xn-yn→0 implies that φ(xn,yn)→0.Lemma 2.2 (see [37, Proposition  2]).
LetE be a uniformly convex and smooth Banach space, and let {xn} and {yn} be two sequences of E such that {xn} or {yn} is bounded. If φ(xn,yn)→0, then xn-yn→0.Remark 2.3.
For any bounded sequences{xn} and {yn} in a uniformly convex and uniformly smooth Banach space E, we have
(2.9)φ(xn,yn)⟶0⟺xn-yn⟶0⟺Jxn-Jyn⟶0.LetC be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space E. It is known that [4, 37] for any x∈E, there exists a unique point x̂∈C such that (2.10)φ(x̂,x)=min⁡y∈Cφ(y,x).
Following Alber [4], we denote such an element x̂ by ΠCx. The mapping ΠC is called the generalized projection from E onto C. It is easy to see that in a Hilbert space, the mapping ΠC coincides with the metric projection PC. Concerning the generalized projection, the followings are well known.Lemma 2.4 (see [37, Propositions  4 and  5]).
LetC be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space E, x∈E and x̂∈C. Then,  (a)
x̂=ΠCx if and only if 〈y-x̂,Jx-Jx̂〉≤0 for all y∈C,(b)
φ(y,ΠCx)+φ(ΠCx,x)≤φ(y,x) for all y∈C.Remark 2.5.
The generalized projection mappingΠC above is relatively nonexpansive and F(ΠC)=C.LetE be a reflexive, strictly convex, and smooth Banach space. The duality mapping J* from E* onto E**=E coincides with the inverse of the duality mapping J from E onto E*; that is, J*=J-1. We make use of the following mapping V:E×E*→ℝ studied in Alber [4]:(2.11)V(x,x*)=‖x‖2-2〈x,x*〉+‖x*‖2,
for all x∈E and x*∈E*. Obviously, V(x,x*)=φ(x,J-1(x*)) for all x∈E and x*∈E*. We know the following lemma (see [4] and [38, Lemma  3.2]).Lemma 2.6.
LetE be a reflexive, strictly convex, and smooth Banach space, and let V be as in (2.11). Then
(2.12)V(x,x*)+2〈J-1(x*)-x,y*〉≤V(x,x*+y*),
for all x∈E and x*,y*∈E*.Lemma 2.7 (see [39, Lemma  2.1]).
Let{an} be a sequence of nonnegative real numbers. Suppose that
(2.13)an+1≤(1-γn)an+γnδn
for all n∈ℕ, where the sequences {γn} in (0,1) and {δn} in ℝ satisfy conditions: limn→∞γn=0, ∑n=1∞γn=∞, and limsupn→∞δn≤0. Then limn→∞an=0.Lemma 2.8 (see [40, Lemma  3.1]).
Let{an} be a sequence of real numbers such that there exists a subsequence {ni} of {n} such that ani<ani+1 for all i∈ℕ. Then, there exists a nondecreasing sequence {mk}⊂ℕ such that mk→∞ and the following properties are satisfied by all (sufficiently large) numbers k∈ℕ:
(2.14)amk≤amk+1,ak≤amk+1.
In fact, mk=max{j≤k:aj<aj+1}.For solving the equilibrium problem, we usually assume that a bifunctionF:C×C→ℝ satisfies the following conditions (see, e.g., [1, 3, 30]):  (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 semicontinuous.The following lemma is a result which appeared in Blum and Oettli [1, Corollary  1].Lemma 2.9 (see [1, Corollary  1]).
LetC be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E. Let F:C×C→ℝ be a bifunction satisfying conditions (A1)–(A4), and let r>0 and x∈E. Then, there exists z∈C such that
(2.15)F(z,y)+1r〈y-z,Jz-Jx〉≥0∀y∈C.The following lemma gives a characterization of a solution of an equilibrium problem.Lemma 2.10 (see [30, Lemma  2.8]).
LetC be a nonempty closed convex subset of a reflexive, strictly convex, and uniformly smooth Banach space E. Let F:C×C→ℝ be a bifunction satisfying conditions (A1)–(A4). For r>0, define a mapping TrF:E→C so-called the resolvent  of F as follows:
(2.16)TrF(x)={z∈C:F(z,y)+1r〈y-z,Jz-Jx〉≥0∀y∈C},
for all x∈E. Then, the followings hold:  (i)
Tr is single-valued,(ii)
Tr is a firmly nonexpansive-type mapping [11], that is, for all x,y∈E(2.17)〈TrFx-TrFy,JTrFx-JTrFy〉≤〈TrFx-TrFy,Jx-Jy〉,(iii)
for allx∈E and p∈EP(F),(2.18)φ(p,TrFx)≤φ(z,TrFx)+φ(TrFx,x)≤φ(p,x),(iv)
F(TrF)=EP(F),(v)
EP(F) is closed and convex.Remark 2.11.
Some well-known examples of resolvents of bifunctions satisfying conditions  (A1)–(A4) are presented in [3, Lemma  2.15].Lemma 2.12  2.12 (see [8, Lemma  2.3]).
LetC be a nonempty closed convex subset of a Banach space E, F a bifunction from C×C→ℝ satisfying conditions  (A1)–(A4), and z∈C. Then, z∈EP(F) if and only if F(y,z)≤0 for all y∈C.Lemma 2.13  2.13 (see [6], Proposition  2.4).
LetC be a nonempty closed convex subset of a strictly convex and smooth Banach space E and S:C→E a relatively nonexpansive mapping. Then F(S) is closed and convex.


## 3. Main Results

In this section, we introduce a modified Halpern-Mann type iteration withoutusing the generalized metric projection and prove a strong convergence theorem for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions to a system of equilibrium problems in a uniformly convex and uniformly smooth Banach space.Theorem 3.1.
LetE a uniformly convex and uniformly smooth Banach space, C a nonempty closed convex subset of E, {Fi}i=1m be a finite family of a bifunction of C×C into ℝ satisfying conditions (A1)–(A4), and S:C→E a relatively nonexpansive mapping such that Ω:=F(S)∩(∩i=1mEP(Fi))≠∅. Let {Tri,nFi}i=1m be a finite family of the resolvents of Fi with positive real sequences {ri,n} such that liminfn→∞ri,n>0 for all i=1,2,…,m. Let {xn} be a sequence generated by x,x1∈E and
(3.1)xn+1=J-1(αnJx+βnJxn+γnJSTrm,nFmTrm-1,nFm-1,⋯Tr1,nF1xn)(n≥1),
where {αn}, {βn}, and {γn} are sequences in [0,1] satisfying the following conditions:  (i)
αn+βn+γn≡1,(ii)
limn→∞αn=0,(iii)
∑n=1∞αn=∞,(iv)
liminfn→∞βn(1-βn)>0. Then, {xn} converges strongly to ΠΩx.Proof.
For eachn≥1, setting
(3.2)znk=Trk,nFkTrk1,nFk-1⋯Tr1,nF1xn,(k=1,2,…,m),yn=J-1(βn1-αnJxn+γn1-αnJSznm).
We can see that znk=Tk,nFkznk-1. Since Ω is nonempty, closed, and convex, we put x̂=ΠΩx. By Lemma 2.10(iii), we get
(3.3)φ(x̂,znm)≤φ(x̂,znm-1)-φ(znm,znm-1)≤φ(x̂,znm-2)-φ(znm-1,znm-2)-φ(znm,znm-1)⋮≤φ(x̂,xn)-∑k=1mφ(znk,znk-1),
where zn0=xn. This together with (2.7) gives
(3.4)φ(x̂,yn)≤βn1-αnφ(x̂,xn)+γn1-αnφ(x̂,Sznm)≤βn1-αnφ(x̂,xn)+γn1-αnφ(x̂,znm)≤φ(x̂,xn).
By Lemma 2.6, we obtain
(3.5)φ(x̂,xn+1)=V(x̂,Jxn+1)≤V(x̂,Jxn+1-αn(Jx-Jx̂))-2〈xn+1-x̂,-αn(Jx-Jx̂)〉=φ(x̂,J-1(αnJx̂+(1-αn)Jyn))+2αn〈xn+1-x̂,Jx-Jx̂〉≤αnφ(x̂,x̂)+(1-αn)φ(x̂,yn)+2αn〈xn+1-x̂,Jx-Jx̂〉≤(1-αn)φ(x̂,xn)+2αn〈xn+1-x̂,Jx-Jx̂〉.
Next, we show that {xn} is bounded. We consider
(3.6)φ(x̂,xn+1)≤φ(x̂,J-1(αnJx+βnJxn+γnJSznm))=φ(x̂,J-1(αnJx+(1-αn)Jyn)≤αnφ(x̂,x)+(1-αn)φ(x̂,yn)≤αnφ(x̂,x)+(1-αn)φ(x̂,xn)≤max⁡{φ(x̂,x),φ(x̂,xn)}.
By induction, we have
(3.7)φ(x̂,xn+1)≤max⁡{φ(x̂,x),φ(x̂,x1)},
for all n≥1. This implies that {xn} is bounded, and so are {xn}, {un}, {yn}, {znm}, and {Sznm}. Let g:[0,2r]→[0,∞) be a function satisfying the properties of Lemma 2.1, where r=sup{∥xn∥,∥Sznm∥:n≥1}. It follows from (3.3) that
(3.8)φ(x̂,yn)≤βn1-αnφ(x̂,xn)+γn1-αnφ(x̂,Sznm)-βnγn(1-αn)2g(‖Jxn-JSznm‖)≤βn1-αnφ(x̂,xn)+γn1-αnφ(x̂,znm)-βnγn(1-αn)2g(‖Jxn-JSznm‖)≤φ(x̂,xn)-γn1-αn∑k=1mφ(znk,znk-1)-βnγn(1-αn)2g(‖Jxn-JSznm‖).
The rest of the proof will be divided into two cases.Case 1.
Suppose that there existsn0∈ℕ such that {φ(x̂,xn)}n=n0∞ is nonincreasing. In this situation, {φ(x̂,xn)} is then convergent. Then,
(3.9)φ(x̂,xn)-φ(x̂,xn+1)⟶0.
Notice that
(3.10)φ(x̂,xn+1)≤αnφ(x̂,x)+(1-αn)φ(x̂,yn).
From condition (ii),
(3.11)φ(x̂,xn)-φ(x̂,yn)=φ(x̂,xn)-φ(x̂,xn+1)+φ(x̂,xn+1)-φ(x̂,yn)≤φ(x̂,xn)-φ(x̂,xn+1)+αn(φ(x̂,x)-φ(x̂,yn))⟶0.
It follows from (3.8) that
(3.12)γn1-αn∑k=1mφ(znk,znk-1)+βnγn(1-αn)2g(‖Jxn-JSznm‖)⟶0.
By the assumptions (i), (ii), and (iv),
(3.13)φ(znk,znk-1)⟶0(k=1,2,…,m),g(‖Jxn-JSznm‖)⟶0.
By Remark 2.3, we get
(3.14)znk-znk-1⟶0(k=1,2,…,m).
From g is continuous strictly increasing with g(0)=0, we have
(3.15)znm-Sznm⟶0,φ(xn,Sznm)⟶0.
Consequently,
(3.16)φ(xn,yn)≤βn1-αnφ(xn,xn)+γn1-αnφ(xn,Sznm)=γn1-αnφ(xn,Sznm)⟶0,φ(yn,xn+1)≤αnφ(yn,x)+(1-αn)φ(yn,yn)=αnφ(yn,x)⟶0.
This implies that
(3.17)xn+1-xn⟶0.
Since {xn} is bounded and E is reflexive, we choose a subsequence {xnj} of {xn} such that xnj⇀w and
(3.18)limsup⁡n→∞〈xn-x̂,Jx-Jx̂〉=lim⁡j→∞〈xnj-x̂,Jx-Jx̂〉=〈w-x̂,Jx-Jx̂〉.
Let k=1,2,…,m be fixed. Then, znjk⇀w as j→∞. From liminfn→∞rk,n>0 and (3.14), we have
(3.19)lim⁡n→∞1rk,n‖Jznk-Jznk-1‖=0.
Then,
(3.20)Fk(znk,y)+1rk,n〈y-znk,Jznk-Jznk-1〉≥0,∀y∈C.
Replacing n by nj, we have from (A2) that
(3.21)1rk,nj〈y-znjk,Jznjk-Jznjk-1〉≥-Fk(znjk,y)≥Fk(y,znjk),∀y∈C.
Letting j→∞, we have from (3.19) and (A4) that
(3.22)Fk(y,w)≤0,∀y∈C.From Lemma 2.12, we have w∈EP(Fk). Since S satisfies condition (R3) and znm-Sznm→0, we have w∈F(S). It follows that w∈Ω. By Lemma 2.4(a), we immediately obtain that
(3.23)limsup⁡n→∞〈xn+1-x̂,Jx-Jx̂〉=limsup⁡n→∞〈xn-x̂,Jx-Jx̂〉=〈w-x̂,Jx-Jx̂〉≤0.
It follows from Lemma 2.7 and (3.5) that φ(x̂,xn)→0. Then, xn→x̂.Case 2.
Suppose that there exists a subsequence{ni} of {n} such that
(3.24)φ(x̂,xni)<φ(x̂,xni+1),
for all i∈ℕ. Then, by Lemma 2.8, there exists a nondecreasing sequence of positive integer numbers {ℓj} such that ℓj→∞,
(3.25)φ(x̂,xlj)≤φ(x̂,xlj+1),φ(x̂,xj)≤φ(x̂,xlj+1),
for all sufficiently large numbers j. We may assume without loss of generality that αℓj>0 for all sufficiently large numbers j. Since
(3.26)φ(x̂,xlj+1)≤αljφ(x̂,x)+(1-αlj)φ(x̂,ylj),
we obtain
(3.27)φ(x̂,xlj)-φ(x̂,ylj)=φ(x̂,xlj)-φ(x̂,xlj+1)+φ(x̂,xlj+1)-φ(x̂,ylj)≤αlj(φ(x̂,x)-φ(x̂,ylj))⟶0.
It follows from (3.8) that
(3.28)γlj1-αlj∑k=1mφ(zljk,zljk-1)+βljγlj(1-αlj)2g(‖Jxlj-JSzljm‖)⟶0.
Using the same proof of Case1, we also obtain
(3.29)limsup⁡j→∞〈xlj+1-x̂,Jx-Jx̂〉≤0.
From (3.5), we have
(3.30)φ(x̂,xlj+1)≤(1-αlj)φ(x̂,xlj)+2αlj〈xlj+1-x̂,Jx-Jx̂〉.
Since φ(x̂,xℓj)≤φ(x̂,xℓj+1), we have
(3.31)αljφ(x̂,xα)≤φ(x̂,xlj)-φ(x̂,xlj+1)+2αlj〈xlj+1-x̂,Jx-Jx̂〉≤2αlj〈xlj+1-x̂,Jx-Jx̂〉.
In particular, since αℓj>0, we get
(3.32)φ(x̂,xmk)≤2〈xlj+1-x̂,Jx-Jx̂〉.
It follows from (3.29) that φ(x̂,xℓj)→0. This together with (3.30) gives
(3.33)φ(x̂,xlj+1)⟶0.
But φ(x̂,xj)≤φ(x̂,xℓj+1) for all sufficiently large numbers j, we conclude that xj→x̂.
From the two cases, we can conclude that{xn} converges strongly to x̂ and the proof is finished.Settingm=1, F1=F≡0, and r1,n≡rn in Theorem 3.1, we have the following.Corollary 3.2.
LetE be a uniformly convex and uniformly smooth Banach space, C a nonempty closed convex subset of E, F a bifunction of C×C into ℝ satisfying conditions (A1)–(A4), and S:C→E be a relatively nonexpansive mapping such that F(S)∩EP(F)≠∅. Let TrnF be the resolvent of F with a positive real sequence {rn} such that liminfn→∞rn>0. Let {xn} be a sequence generated by x,x1∈E and
(3.34)xn+1=J-1(αnJx+βnJxn+γnJSTrnFxn)(n≥1),
where {αn}, {βn}, and {γn} are sequences in [0,1] satisfying the following conditions:  (i)
αn+βn+γn≡1,(ii)
limn→∞αn=0,(iii)
∑n=1∞αn=∞,(iv)
liminfn→∞βn(1-βn)>0. Then, {xn} converges strongly to ΠF(S)∩EP(F)x.SettingF1≡0 and r1,n≡1 in Corollary 3.2, we have the following result.Corollary 3.3.
LetE be a uniformly convex and uniformly smooth Banach space, C a nonempty closed convex subset of E, and S:C→E a relatively nonexpansive mapping such that F(S)≠∅. Let {xn} be a sequence generated by x,x1∈E and
(3.35)xn+1=J-1(αnJx+βnJxn+γnJSΠCxn)(n≥1),
where {αn}, {βn}, and {γn} are sequences in [0,1] satisfying the following conditions:  (i)
αn+βn+γn≡1,(ii)
limn→∞αn=0,(iii)
∑n=1∞αn=∞,(iv)
liminfn→∞βn(1-βn)>0. Then {xn} converges strongly to ΠF(S)x.Next, we prove a strong convergence theorem for finding an element of the set of solutions to a system of equilibrium problems in a uniformly convex and uniformly smooth Banach space.Theorem 3.4.
LetE be a uniformly convex and uniformly smooth Banach space, C a nonempty closed convex subset of E, {Fi}i=1m a finite family of a bifunction of C×C into ℝ satisfying conditions (A1)–(A4), and ∩i=1mEP(Fi)≠∅. Let {Tri,nFi}i=1m be a finite family of the resolvents of Fi with positive real sequences {ri,n} such that liminfn→∞ri,n>0 for all i=1,2,…,m. Let {xn} be a sequence generated by x,x1∈E and
(3.36)xn+1=J-1(αnJx+βnJxn+γnJTrm,nFmTrm-1,nFm-1,…,Tr1,nF1xn)(n≥1),
where {αn}, {βn}, and {γn} are sequences in [0,1] satisfying the following conditions:  (i)
αn+βn+γn≡1,(ii)
limn→∞αn=0,(iii)
∑n=1∞αn=∞,(iv)
liminfn→∞βn(1-βn)>0 or liminfn→∞βn=0. Then, {xn} converges strongly to Π∩i=1mEP(Fi)x.Proof.
For eachn≥1, setting
(3.37)znk=Trk,nFkTrk1,nFk-1⋯Tr1,nF1xn,(k=1,2,…,m),yn=J-1(βn1-αnJxn+γn1-αnJznm).
Since ∩i=1mEP(Fi) is nonempty, closed, and convex, we put x̂=Π∩i=1mEP(Fi)x. Using the same proof of Theorem 3.1 when S is the identity operator, we can see that
(3.38)φ(x̂,yn)≤φ(x̂,xn)-γn1-αn∑k=1mφ(znk,znk-1),(3.39)φ(x̂,xn+1)≤(1-αn)φ(x̂,xn)+2αn〈xn+1-x̂,Jx-Jx̂〉.
The rest of the proof will be divided into two cases.Case 1.
Suppose that there existsn0∈ℕ such that {φ(x̂,xn)}n=n0∞ is non-increasing. In this situation, {φ(x̂,xn)} is then convergent. Then,
(3.40)φ(x̂,xn)-φ(x̂,xn+1)⟶0.
Notice that
(3.41)φ(x̂,xn+1)≤αnφ(x̂,x)+(1-αn)φ(x̂,yn).
From condition (ii),
(3.42)φ(x̂,xn)-φ(x̂,yn)=φ(x̂,xn)-φ(x̂,xn+1)+φ(x̂,xn+1)-φ(x̂,yn)≤φ(x̂,xn)-φ(x̂,xn+1)+αn(φ(x̂,x)-φ(x̂,yn))⟶0.
It follows from (3.38) that
(3.43)γn1-αn∑k=1mφ(znk,znk-1)⟶0.
By the assumptions (i), (ii), and (iv),
(3.44)φ(znk,znk-1)⟶0(k=1,2,…,m).
By Remark 2.3, we get
(3.45)znk-znk-1⟶0(k=1,2,…,m).
Consequently,
(3.46)φ(xn,yn)≤βn1-αnφ(xn,xn)+γn1-αnφ(xn,znm)=γn1-αnφ(zn0,znm)⟶0,φ(yn,xn+1)≤αnφ(yn,x)+(1-αn)φ(yn,yn)=αnφ(yn,x)⟶0.
This implies that
(3.47)xn+1-xn⟶0.
Since {xn} is bounded and E is reflexive, we choose a subsequence {xnj} of {xn} such that xnj⇀w and
(3.48)limsup⁡n→∞〈xn-x̂,Jx-Jx̂〉=lim⁡j→∞〈xnj-x̂,Jx-Jx̂〉=〈w-x̂,Jx-Jx̂〉.
Let k=1,2,…,m be fixed. Then, znjk⇀w as j→∞. From liminfn→∞rk,n>0 and (3.14), we have
(3.49)lim⁡n→∞1rk,n‖Jznk-Jznk-1‖=0.
Then,
(3.50)Fk(znk,y)+1rk,n〈y-znk,Jznk-Jznk-1〉≥0,∀y∈C.
Replacing n by nj, we have from (A2) that
(3.51)1rk,nj〈y-znjk,Jznjk-Jznjk-1〉≥-Fk(znjk,y)≥Fk(y,znjk),∀y∈C.
Letting j→∞, we have from (3.49) and (A4) that
(3.52)Fk(y,w)≤0,∀y∈C.
From Lemma 2.12, we have w∈EP(Fk). By Lemma 2.4(a), we immediately obtain that
(3.53)limsup⁡n→∞〈xn+1-x̂,Jx-Jx̂〉=limsup⁡n→∞〈xn-x̂,Jx-Jx̂〉=〈w-x̂,Jx-Jx̂〉≤0.
It follows from Lemma 2.7 and (3.39) that φ(x̂,xn)→0. Then, xn→x̂.Case 2.
Suppose that there exists a subsequence{ni} of {n} such that
(3.54)φ(x̂,xni)<φ(x̂,xni+1),
for all i∈ℕ. Using the same proof of Case2 in Theorem 3.1, we also conclude that xj→x̂.
From the two cases, we can conclude that{xn} converges strongly to x̂.Finally, we give two explicit examples validating the assumptions in Theorem3.1 as follows.Example 3.5 (Optimization).
LetEbe a uniformly convex and uniformly smooth Banach space,Ca nonempty bounded closed convex subset ofE, andf:C→ℝa lower semicontinuous and convex functional. For instance, letE=ℝ,C=[0,1]andf:[0,1]→ℝbe defined dy(3.55)f(x)={0,ifx=0,1;xlog⁡x+(1-x)log⁡(1-x),ifx∈(0,1).
Then f is lower semicontinuous and convex. For each i=1,2,…,m, let Fi:C×C→ℝ be defined by Fi(x,y):=f(y)-f(x) for all x,y∈C. It is known [1, 11] that Fi satisfies conditions (A1)–(A4), and EP(Fi)≠∅. Let S=ΠC. Then, S is relatively nonexpansive of E into C (see [5, 6]) and F(S)=C. Then, Ω:=F(S)∩(∩i=1mEP(Fi))=EP(Fi)≠∅. Applying Theorem 3.1, we conclude that the sequence defined by (3.1) converges strongly to ΠΩx.Example 3.6 (The convex feasibility problem).
LetEbe a real Hilbert space, letC1,C2,…,Cmbe nonempty closed convex subsets ofEsatisfyingC:=∩i=1mCi≠∅(e.g.,C1=C2=⋯=Cm=C≠∅). Let{Fi}i=1mbe a finite family of bifunctions ofE×Eintoℝdefined by(3.56)Fi(x,y)=12〈y-x,x-PCix〉∀x,y∈E,
where PCi is a metric projection from E onto Ci. It is known [3, Lemma  2.15(iv)] that Fi satisfies conditions (A1)–(A4) and EP(Fi)=Ci. Let S=PC. Then, S is relatively nonexpansive of E into C (see [5, 6]) and then Ω:=F(S)∩(∩i=1mEP(Fi))=C≠∅. Applying Theorem 3.1, we conclude that the sequence defined by (3.1) converges strongly to ΠΩx.


## 4. Deduced Theorems in Hilbert Spaces

In Hilbert spaces, ifS is quasi-nonexpansive such that I-S is demiclosed at zero, then S is relatively nonexpansive. We obtain the following result.Theorem 4.1.
LetH be a Hilbert space, C a nonempty closed convex subset of H, {Fi}i=1m a finite family of a bifunction of C×C into ℝ satisfying conditions (A1)–(A4), and S:C→E a quasi-nonexpansive mapping such that I-S is demiclosed at zero and Ω:=F(S)∩(∩i=1mEP(Fi))≠∅. Let {Tri,nFi}i=1m be a finite family of the resolvents of Fi with real sequences {ri,n} such that liminfn→∞ri,n>0 for all i=1,2,…,m. Let {xn} be a sequence generated by x,x1∈H and
(4.1)xn+1=αnx+βnxn+γnSTrm,nFmTrm-1,nFm-1,…,Tr1,nF1xn(n≥1),
where {αn}, {βn}, and {γn} are sequences in [0,1] satisfying the following conditions:  (i)
αn+βn+γn≡1,(ii)
limn→∞αn=0,(iii)
∑n=1∞αn=∞,(iv)
liminfn→∞βn(1-βn)>0. Then {xn} converges strongly to PΩx.Applying Theorem4.1 and using the technique in [41], we have the following result.Theorem 4.2.
LetH be a Hilbert space, C a nonempty closed convex subset of H, f a contraction of H into itself (i.e., there is a∈(0,1) such that ∥f(x)-f(y)∥≤a∥x-y∥ for all x,y∈H), {Fi}i=1m a finite family of a bifunction of C×C into ℝ satisfying conditions (A1)–(A4), and S:C→E be a nonexpansive mapping such that Ω:=F(S)∩(∩i=1mEP(Fi))≠∅. Let {Tri,nFi}i=1m be a finite family of the resolvents of Fi with real sequences {ri,n} such that liminfn→∞ri,n>0 for all i=1,2,…,m. Let {xn} be a sequence generated by x,x1∈H and
(4.2)xn+1=αnf(xn)+βnxn+γnSTrm,nFmTrm-1,nFm-1,…,Tr1,nF1xn,(n≥1),
where {αn}, {βn}, and {γn} are sequences in [0,1] satisfying the following conditions:  (i)
αn+βn+γn≡1,(ii)
limn→∞αn=0,(iii)
∑n=1∞αn=∞,(iv)
liminfn→∞βn(1-βn)>0. Then, {xn} converges strongly to z such that z=PΩf(z).Proof.
We note thatPΩf is contraction. By Banach contraction principle, let z be the fixed point of PΩf and {yn} a sequence generated by y1=x1∈H and
(4.3)yn+1=αnf(z)+βnyn+γnSTrm,nFmTrm-1,nFm-1,…,Tr1,nF1yn,(n≥1).
Using Theorem 4.1, we have yn→z=PΩf(z). Since S and Trk,nFk(k=1,2,…,m) are nonexpansive,
(4.4)‖yn+1-xn+1‖≤αn‖f(xn)-f(z)‖+βn‖yn-xn‖+γn‖STrm,nFmTrm-1,nFm-1,…,Tr1,nF1yn-STrm,nFmTrm-1,nFm-1,…,Tr1,nF1xn‖≤αna‖xn-z‖+(βn+γn)‖yn-nn‖≤αna(‖xn-yn‖+‖yn-z‖)+(βn+γn)‖xn-yn‖=(1-αn(1-a))‖yn-xn‖+αn(1-a)(a1-a‖yn-z‖).
Applying Lemma 2.7, yn-xn→0 and so xn→z=PΩf(z).Settingm=1, F1=F≡0, and r1,n≡rn in Theorem 4.1, we have the following.Corollary 4.3.
LetH be a Hilbert space, C a nonempty closed convex subset of H, F a bifunction of C×C into ℝ satisfying conditions (A1)–(A4), and S:C→E a quasi-nonexpansive mapping such that I-S is demiclosed at zero and F(S)∩EP(F)≠∅. Let TrnF be the resolvent of F with a positive real sequence {rn} such that liminfn→∞rn>0. Let {xn} be a sequence generated by x,x1∈H and
(4.5)xn+1=αnx+βnxn+γnSTrnFxn(n≥1),
where {αn}, {βn}, and {γn} are sequences in [0,1] satisfying the following conditions:  (i)
αn+βn+γn≡1,(ii)
limn→∞αn=0,(iii)
∑n=1∞αn=∞,(iv)
liminfn→∞βn(1-βn)>0. Then, {xn} converges strongly to PF(S)∩EP(F)x.Corollary 4.4.
LetH be a Hilbert space, C a nonempty closed convex subset of H, f a contraction of H into itself, F a bifunction of C×C into ℝ satisfying conditions (A1)–(A4), and S:C→E a nonexpansive mapping such that F(S)∩EP(F)≠∅. Let TrnF be the resolvent of F with a positive real sequence {rn} such that liminfn→∞rn>0. Let {xn} be a sequence generated by x,x1∈H and
(4.6)xn+1=αnf(xn)+βnxn+γnSTrnFxn(n≥1),
where {αn}, {βn}, and {γn} are sequences in [0,1] satisfying the following conditions:  (i)
αn+βn+γn≡1,(ii)
limn→∞αn=0,(iii)
∑n=1∞αn=∞,(iv)
liminfn→∞βn(1-βn)>0. Then, {xn} converges strongly to z such that z=PF(S)∩EP(F)f(z).Remark 4.5.
Corollary4.4 improves and extends [42, Theorem  5]. More precisely, the conditions limn→∞(rn+1-rn)=∞ are removed.SettingF≡0 and rn≡1 in Corollary 4.3, we have the following.Corollary 4.6.
LetH be a Hilbert space, C a nonempty closed convex subset of H, and S:C→E a quasi-nonexpansive mapping such that I-S is demiclosed at zero and F(S)≠∅. Let {xn} be a sequence generated by x,x1∈H and
(4.7)xn+1=αnx+βnxn+γnSPCxn(n≥1),
where {αn}, {βn}, and {γn} are sequences in [0,1] satisfying the following conditions:  (i)
αn+βn+γn≡1,(ii)
limn→∞αn=0,(iii)
∑n=1∞αn=∞,(iv)
liminfn→∞βn(1-βn)>0. Then, {xn} converges strongly to PF(S)x.Applying Theorem3.4, we have the following result.Theorem 4.7.
LetH be a Hilbert space, C a nonempty closed convex subset of H, {Fi}i=1m a finite family of a bifunction of C×C into ℝ satisfying conditions (A1)–(A4), and ∩i=1mEP(Fi)≠∅. Let {Tri,nFi}i=1m be a finite family of the resolvents of Fi with positive real sequences {ri,n} such that liminfn→∞ri,n>0 for all i=1,2,…,m. Let {xn} be a sequence generated by x,x1∈H and
(4.8)xn+1=αnx+βnxn+γnTrm,nFmTrm-1,nFm-1,…,Tr1,nF1xn(n≥1),
where {αn}, {βn}, and {γn} are sequences in [0,1] satisfying the following conditions:  (i)
αn+βn+γn≡1,(ii)
limn→∞αn=0,(iii)
∑n=1∞αn=∞,s(iv)
liminfn→∞βn(1-βn)>0 or liminfn→∞βn=0. Then {xn} converges strongly to P∩i=1mEP(Fi)x.Settingm=1, F1=F≡0, r1,n≡rn, and βn≡0 in Theorem  4.4, we have the following result.Corollary 4.8 (see [35, Corollary  4.4]).
LetH be a Hilbert space, C a nonempty closed convex subset of H, F a bifunction of C×C into ℝ satisfying conditions (A1)–(A4), and EP(F)≠∅. Let TrnF the resolvent of F with a positive real sequence {rn} such that liminfn→∞rn>0. Let {xn} be a sequence generated by x,x1∈H and
(4.9)xn+1=αnx+(1-αn)TrnFxn(n≥1),
where {αn} is a sequence in [0,1] satisfying the following conditions:  (i)
limn→∞αn=0,(ii)
∑n=1∞αn=∞, Then, {xn} converges strongly to PEP(F)x.



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*Source: 632857-2011-09-21.xml*