# Strong Convergence Theorems for Equilibrium Problems and Fixed Point Problems in Hilbert Spaces

Authors: Jian-Wen Peng; Yan Wang
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

We introduce an Ishikawa iterative scheme by the viscosity approximate method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in Hilbert space. Then, we prove some strong convergence theorems which extend and generalize S. Takahashi and W. Takahashi's results (2007).


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## Body

## 1. Introduction

LetH be a real Hilbert space and let C be a nonempty closed convex subset of H. Let F be a bifunction from C×C to R, where R is the set of real numbers. The equilibrium problem for F:C×C→R 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). Given a mapping T:C→H, let F(x,y)=〈Tx,y-x〉 for all x,y∈C. Then, z∈EP(F) if and only if 〈Tz,y-z〉≥0 for all y∈C. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.1); for more details, see [1, 2].Recall that a self-mappingS of a closed convex subset C of H is nonexpansive [3] if there holds that
(1.2)∥Sx-Sy∥≤∥x-y∥,∀x,y∈C.
We denote the set of fixed points of S by F(S). There are some methods for approximation of fixed points of a nonexpansive mapping. In 2000, Moudafi [4] introduced the viscosity approximation method for nonexpansive mappings (see [5] for further developments in both Hilbert and Banach spaces). Some methods have been proposed to solve the equilibrium problem; see, for instance, [1, 2, 6, 7]. Recently, Combettes and Hirstoaga [6] introduced an iterative scheme of finding the best approximation to the initial data when EP(F) is nonempty and proved a strong convergence theorem. S. Takahashi and W. Takahashi [7] introduced a Mann iterative scheme by the viscosity approximation method for finding a common element of the set of solution (1.1) and the set of fixed points of a nonexpansive mapping in a Hilbert space and proved a strong convergence theorem.On the other hand, Ishikawa [8] introduced the following iterative process defined recursively by(1.3)xn+1=αnxn+(1-αn)Syn,yn=βnxn+(1-βn)Sxn,∀n∈N,
where the initial guess x0 is taking in C arbitrarily, {αn} and {βn} are sequences in the interval [0,1].In this paper, motivated by the ideas in [4–8], we introduce an Ishikawa iterative scheme by the viscosity approximation method for finding a common element of the set of solution (1.1) and the set of fixed points of a nonexpansive mapping in a Hilbert space.Starting with an arbitraryx1∈H, define sequences {xn}, {yn}, and {un} by(1.4)F(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnf(xn)+(1-αn)Syn,yn=βnxn+(1-βn)Sun,∀n∈N,
where {αn},{βn}⊂[0,1] and {rn}⊂(0,∞).We will prove in Section3 that if the sequences {αn}, {βn}, and {rn} of parameters satisfy appropriate conditions, then the sequences {xn}, {yn}, and {un} generated by (1.4) converge strongly to z∈F(S)∩EP(F). The results in this paper extend and generalize S. Takahashi and W. Takahashi's results [7].


## 2. Preliminaries

LetH be a real Hilbert space with inner product 〈·,·〉, and norm ∥·∥ and let C be a nonempty closed convex subset of H. xn→x implies that {xn} converges strongly to x and xn⇀x means that {xn} converges weakly to x. In a real Hilbert space H, we have(2.1)∥λx+(1-λ)y∥2=λ∥x∥2+(1-λ)∥y∥2-λ(1-λ)∥x-y∥2
for all x,y∈H and λ∈R; see [9].For anyx∈H, there exists a unique nearest point in C, denoted by PC(x), such that ∥x-PC(x)∥≤∥x-y∥ for all y∈C. Such a PC is called the metric projection of H onto C. It is also known that y=PC(x) is equivalent to 〈x-y,y-z〉≥0 for all z∈C.For solving the equilibrium problem, let us assume that the 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 any x,y∈C;(A3)
for eachx,y,z∈C,
(2.2)limt↓0F(tz+(1-t)x,y)≤F(x,y);(A4)
for eachx∈C,y↦F(x,y) is convex and lower semicontinuous.We recall some lemmas needed later.Lemma 2.1 (see [2]).
LetC be a nonempty closed convex subset of H and let F be a bifunction from C×C to R satisfying (A1)–(A4). Let r>0 and x∈H. Then, there exists z∈C such that
(2.3)F(z,y)+1r〈y-z,z-x〉≥0,∀y∈C.Lemma 2.2 (see [5]).
LetC be a nonempty closed convex subset of H, and let F be a bifunction from C×C to R satisfying (A1)–(A4). For r>0 and x∈H, define a mapping Tr:H→C as follows:
(2.4)Tr(x)={z∈C:F(z,y)+1r〈y-z,z-x〉≥0,∀y∈C}
for all x∈H. Then, the following statements hold: (1)
Tr is single-valued;(2)
Tr is firmly nonexpansive, that is, for any x,y∈H,(2.5)∥Tr(x)-Tr(y)∥2≤〈Tr(x)-Tr(y),x-y〉;(3)
F(Tr)=EP(F);(4)
EP(F) is closed and convex.Lemma 2.3 (see [10]).
Let{an} be a sequence of nonnegative real numbers such that
(2.6)an+1≤(1-cn)an+bn,∀n∈N,
where {bn} is a sequence of real numbers and {cn} is a sequence in (0,1) such that(i)
∑n=1∞cn=∞,(ii)
lim supn→∞(bn/cn)≤0 or ∑n=1∞|bn|<∞.Then, limn→∞an=0.


## 3. Strong Convergence Theorem

In this section, we show a strong convergence theorem which solves the problem of finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space.Theorem 3.1.
LetC be a nonempty closed convex subset of H. Let F be a bifunction from C×C to R satisfying (A1)–(A4) and let S be a nonexpansive mapping of C into H such that F(S)∩EP(F)≠∅. Let f be a contraction of H into itself and let {xn}, {un}, and {yn} be sequences generated by x1∈H and (1.4). If {αn},{βn}⊂[0,1] and {rn}⊂(0,∞) satisfy the following conditions:
(3.1)limn→∞αn=0,∑n=1∞αn=∞,∑n=1∞|αn+1-αn|<∞,0<lim infn→∞βn≤lim supn→∞βn<1,∑n=1∞|βn+1-βn|<∞,lim infn→∞rn>0,∑n=1∞|rn+1-rn|<∞,
then, {xn}, {yn}, and {un} converge strongly to z∈F(S)∩EP(F), where z=PF(S)∩EP(F)f(z).Proof.
LetQ=PF(S)∩EP(F). Then Qf is a contraction of H into itself. In fact, there exists a∈[0,1) such that ∥f(x)-f(y)∥≤a∥x-y∥ for all x,y∈H. So, we have that
(3.2)∥Qf(x)-Qf(y)∥≤∥f(x)-f(y)∥≤a∥x-y∥
for all x,y∈H. Since H is complete, there exists a unique element z∈H such that z=Qf(z). Such a z∈H is an element of C.
Letv∈F(S)∩EP(F). Then from un=Trnxn, we have
(3.3)∥un-v∥=∥Trnxn-Trnv∥≤∥xn-v∥
for all n∈N. Put M=max{∥x1-v∥,(1/(1-a))∥f(v)-v∥}. It is obvious that ∥x1-v∥≤M. Suppose ∥xn-v∥≤M. Then, we have
(3.4)∥xn+1-v∥≤αn∥f(xn)-v∥+(1-αn)∥Syn-v∥≤αn∥f(xn)-f(v)∥+αn∥f(v)-v∥+(1-αn)∥Syn-v∥≤aαn∥xn-v∥+αn∥f(v)-v∥+(1-αn)∥yn-v∥.
On the other hand
(3.5)∥yn-v∥≤βn∥xn-v∥+(1-βn)∥Sun-v∥≤βn∥xn-v∥+(1-βn)∥un-v∥≤βn∥xn-v∥+(1-βn)∥xn-v∥=∥xn-v∥.
Putting (3.5) into (3.4), we have
(3.6)∥xn+1-v∥≤aαn∥xn-v∥+αn∥f(v)-v∥+(1-αn)∥xn-v∥=[1-αn(1-a)]∥xn-v∥+αn(1-a)∥f(v)-v∥1-a≤[1-αn(1-a)]M+αn(1-a)M=M.
So, we have that ∥xn+1-v∥≤M for any n∈N. And hence {xn} is bounded. We also obtain that {un},{yn},{Sun},{Syn}, and {f(xn)} are bounded. Next, we show that limn→∞∥xn+1-xn∥=0. In fact,
(3.7)∥yn-yn-1∥=∥βnxn+(1-βn)Sun-[βn-1xn-1+(1-βn-1)Sun-1]∥=∥βn(xn-xn-1)+(βn-βn-1)xn-1+(1-βn)(Sun-Sun-1)+(βn-1-βn)Sun-1∥≤|βn-βn-1|∥xn-1∥+βn∥xn-xn-1∥+(1-βn)∥un-un-1∥+|βn-βn-1|∥Sun-1∥,
and hence
(3.8)∥xn+1-xn∥=∥αnf(xn)+(1-αn)Syn-αn-1f(xn-1)-(1-αn-1)Syn-1∥=∥αnf(xn)-αnf(xn-1)+αnf(xn-1)-αn-1f(xn-1)+(1-αn)Syn-(1-αn)Syn-1+(1-αn)Syn-1-(1-αn-1)Syn-1∥≤αna∥xn-xn-1∥+|αn-αn-1|∥f(xn-1)∥+(1-αn)∥yn-yn-1∥+|αn-αn-1|∥Syn-1∥≤αna∥xn-xn-1∥+|αn-αn-1|∥f(xn-1)∥+(1-αn)×[|βn-βn-1|∥xn-1∥+βn∥xn-xn-1∥+(1-βn)∥un-un-1∥+|βn-βn-1|∥Sun-1∥]+|αn-αn-1|∥Syn-1∥=[βn-αn(βn-a)]∥xn-xn-1∥+|αn-αn-1|[∥f(xn-1)∥+∥Syn-1∥]+(1-αn)|βn-βn-1|[∥xn-1∥+∥Sun-1∥]+(1-αn)(1-βn)∥un-un-1∥≤[βn-αn(βn-a)]∥xn-xn-1∥+|αn-αn-1|K1+(1-αn)|βn-βn-1|K2+(1-αn)(1-βn)∥un-un-1∥,
where K1=sup{∥f(xn)∥+∥Syn∥:n∈N} and K2=sup{∥xn∥+∥Sun∥:n∈N}.
On the other hand, fromun=Trnxn and un+1=Trn+1xn+1, we have
(3.9)F(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,(3.10)F(un+1,y)+1rn+1〈y-un+1,un+1-xn+1〉≥0,∀y∈C.
Puttingy=un+1 in (3.9) and y=un in (3.10), we have
(3.11)F(un,un+1)+1rn〈un+1-un,un-xn〉≥0,F(un+1,un)+1rn+1〈un-un+1,un+1-xn+1〉≥0.
So, from the monotonicity of F, we get
(3.12)〈un+1-un,un-xnrn-un+1-xn+1rn+1〉≥0,
and hence
(3.13)〈un+1-un,un-un+1+un+1-xn-rnrn+1(un+1-xn+1)〉≥0.
Without loss of generality, let us assume that there exists a real number b such that rn>b>0 for all n∈N. Then, we have
(3.14)∥un+1-un∥2≤〈un+1-un,xn+1-xn+(1-rnrn+1)(un+1-xn+1)〉≤∥un+1-un∥{∥xn+1-xn∥+|1-rnrn+1|∥un+1-xn+1∥},
and hence
(3.15)∥un+1-un∥≤∥xn+1-xn∥+1rn+1|rn+1-rn|∥un+1-xn+1∥≤∥xn+1-xn∥+1b|rn+1-rn|L,
where L=sup{∥un-xn∥:n∈N}. So from (3.8), we have
(3.16)∥xn+1-xn∥≤[βn-αn(βn-a)]∥xn-xn-1∥+|αn-αn-1|K1+(1-αn)|βn-βn-1|K2+(1-αn)(1-βn)[∥xn-xn-1∥+1b|rn-rn-1|L]=(1-αn(1-a))∥xn-xn-1∥+|αn-αn-1|K1+(1-αn)|βn-βn-1|K2+(1-αn)(1-βn)1b|rn-rn-1|L.
Using Lemma 2.1 in [10], we obtain
(3.17)limn→∞∥xn+1-xn∥=0.
From (3.15) and |rn+1-rn|→0, we have
(3.18)limn→∞∥un+1-un∥=0.
It follows from (3.7) that
(3.19)limn→∞∥yn+1-yn∥=0.
Since xn=αn-1f(xn-1)+(1-αn-1)Syn-1, we have
(3.20)∥xn-Syn∥≤∥xn-Syn-1∥+∥Syn-1-Syn∥≤αn-1∥f(xn-1)-Syn-1∥+∥yn-1-yn∥.
From αn→0, we have ∥xn-Syn∥→0. For v∈F(S)∩EP(F), we have
(3.21)∥un-v∥2=∥Trnxn-Trnv∥2≤〈Trnxn-Trnv,xn-v〉=〈un-v,xn-v〉=12(∥un-v∥2+∥xn-v∥2-∥xn-un∥2),
and hence
(3.22)∥un-v∥2≤∥xn-v∥2-∥xn-un∥2.
Therefore, from the convexity of ∥·∥2, we have
(3.23)∥yn-v∥2≤βn∥xn-v∥2+(1-βn)∥Sun-v∥2≤βn∥xn-v∥2+(1-βn)∥un-v∥2≤βn∥xn-v∥2+(1-βn)[∥xn-v∥2-∥xn-un∥2]=∥xn-v∥2-(1-βn)∥xn-un∥2,
and hence
(3.24)∥xn+1-v∥2=∥αnf(xn)+(1-αn)Syn-v∥2≤αn∥f(xn)-v∥2+(1-αn)∥Syn-v∥2≤αn∥f(xn)-v∥2+(1-αn)∥yn-v∥2≤αn∥f(xn)-v∥2+(1-αn)[∥xn-v∥2-(1-βn)∥xn-un∥2]≤αn∥f(xn)-v∥2+∥xn-v∥2-(1-αn)(1-βn)∥xn-un∥2.
So, we have
(3.25)(1-αn)(1-βn)∥xn-un∥2≤αn∥f(xn)-v∥2+∥xn-v∥2-∥xn+1-v∥2≤αn∥f(xn)-v∥2+∥xn-xn+1∥(∥xn-v∥+∥xn+1-v∥).
Without loss of generality, let us assume that there exists two real numbers β* and β̅ such that 1>β̅≥βn≥β*>0 for all n∈N. Hence,
(3.26)(1-αn)(1-β̅)∥xn-un∥2≤(1-αn)(1-βn)∥xn-un∥2≤αn∥f(xn)-v∥2+∥xn-xn+1∥(∥xn-v∥+∥xn+1-v∥).
It follows that ∥xn-un∥→0. We also have
(3.27)∥Sun-xn∥≤∥Syn-xn∥+∥Sun-Syn∥≤∥Syn-xn∥+∥un-yn∥≤∥Syn-xn∥+∥un-xn∥+∥xn-yn∥=∥Syn-xn∥+∥un-xn∥+(1-βn)∥xn-Sun∥.
It follows that
(3.28)β*∥Sun-xn∥≤βn∥Sun-xn∥≤∥Syn-xn∥+∥un-xn∥.
Hence, ∥Sun-xn∥→0. Since
(3.29)∥Sun-un∥≤∥Sun-xn∥+∥xn-un∥,
we also have limn→∞∥Sun-un∥=0. Next, we show that
(3.30)lim supn→∞〈f(z)-z,xn-z〉≤0,
where z=PF(S)∩EP(F)f(z). To show this inequality, we choose a subsequence {xni} of {xn} such that
(3.31)limn→∞〈f(z)-z,xni-z〉=lim supn→∞〈f(z)-z,xn-z〉.
Since {uni} is bounded, there exists a subsequence {unij} of {uni} which converges weakly to w. Without loss of generality, we can assume that {uni}⇀w. From ∥Sun-un∥→0, we obtain Suni⇀w. Let us show w∈EP(F). By un=Trnxn, we have
(3.32)F(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C.
From (A2), we also have
(3.33)1rn〈y-un,un-xn〉≥F(y,un),
and hence,
(3.34)〈y-uni,uni-xnirni〉≥F(y,uni).
Since (uni-xni)/rni→0 and uni⇀w, from (A4), we have
(3.35)f(y,w)≤0,∀y∈C.
For t with 0<t≤1 and y∈C, let yt=ty+(1-t)w. Since y∈C and w∈C, we obtain yt∈C and hence F(yt,w)≤0. So we have
(3.36)0=F(yt,yt)≤tF(yt,y)+(1-t)F(yt,w)≤tF(yt,y).
Dividing by t, we get
(3.37)F(yt,y)≥0.
Letting t→0 and from (A3), we get
(3.38)F(w,y)≥0
for all y∈C and hence w∈EP(F). We shall show that w∈F(S). Assume w∉F(S). Since uni⇀w and w≠Sw, from the Opial theorem [11] we have
(3.39)lim infi→∞∥uni-w∥<lim infi→∞∥uni-Sw∥≤lim infi→∞{∥uni-Suni∥+∥Suni-Sw∥}≤lim infi→∞∥uni-w∥.
This is a contradiction. So, we get w∈F(S). Therefore, w∈F(S)∩EP(F). Since z=PF(S)∩EP(F)f(z), we have
(3.40)lim supn→∞〈f(z)-z,xn-z〉=limi→∞〈f(z)-z,xni-z〉=limi→∞〈f(z)-z,uni-z〉=〈f(z)-z,w-z〉≤0.
From xn+1-z=αn(f(xn)-z)+(1-αn)(Syn-z), we have
(3.41)(1-αn)2∥Syn-z∥2≥∥xn+1-z∥2-2αn〈f(xn)-z,xn+1-z〉,(3.42)∥yn-z∥2=∥βnxn+(1-βn)Sun-z∥2≤βn∥xn-z∥2+(1-βn)∥Sun-z∥2≤βn∥xn-z∥2+(1-βn)∥un-z∥2≤∥xn-z∥2.
It follows that
(3.43)∥xn+1-z∥2≤(1-αn)2∥Syn-z∥2+2αn〈f(xn)-z,xn+1-z〉≤(1-αn)2∥yn-z∥2+2αn〈f(xn)-f(z),xn+1-z〉+2αn〈f(z)-z,xn+1-z〉≤(1-αn)2∥xn-z∥2+2αna∥xn-z∥∥xn+1-z∥+2αn〈f(z)-z,xn+1-z〉≤(1-αn)2∥xn-z∥2+αna{∥xn-z∥2+∥xn+1-z∥2}+2αn〈f(z)-z,xn+1-z〉.
Hence
(3.44)∥xn+1-z∥2≤(1-αn)2+αna1-αna∥xn-z∥2+2αn1-αna〈f(z)-z,xn+1-z〉.
From αn→0, we know that there exists a positive integer n0, such that 1>1-αna>1/2 for all n≥n0. Then
(3.45)(1-αn)2+αna1-αna=1-2αn+αna1-αna+αn21-αna=1-2(1-a)αn1-αna+αn21-αna≤1-2(1-a)αn+2αn2,∀n≥n0.
Putting above inequality into (3.44), we get
(3.46)∥xn+1-z∥2≤(1-2(1-a)αn)∥xn-z∥2+2M̅αn2+2αn1-αnaσn,∀n≥n0,
where M̅=sup{∥xn-z∥2:n∈N}, and σn=〈f(z)-z,xn+1-z〉.
It follows from Lemma2.3 that
(3.47)xn→z∈F(S)∩EP(F).
It follows from ∥xn-un∥→0 and (3.42) that un→z and yn→z.By Theorem3.1, we can obtain the following new result.Corollary 3.2.
LetC be a nonempty closed convex subset of H. Let S be a nonexpansive mapping of C into H such that F(S)≠∅. Let f be a contraction of H into itself and let {xn} and {yn} be sequences generated by x1∈H and
(3.48)xn+1=αnf(xn)+(1-αn)Syn,yn=βnxn+(1-βn)SPCxn,∀n∈N.
If {αn},{βn}⊂[0,1] satisfy the following conditions:
(3.49)limn→∞αn=0,∑n=1∞αn=∞,∑n=1∞|αn+1-αn|<∞,0<lim infn→∞βn≤lim supn→∞βn<1,∑n=1∞|βn+1-βn|<∞,
then, {xn} and {yn} converge strongly to z∈F(S), where z=PF(S)f(z).Proof.
PutF(x,y)=0 for all x,y∈C and rn=1 for all n∈N in Theorem 3.1. Then, we get un=PCxn. So from Theorem 3.1, the sequences {xn} and {yn} converge strongly to z∈F(S), where z=PF(S)f(z).Remark 3.3.
Theorem3.1 and Corollary 3.2, respectively, extend and generalize Theorem 3.2 and Corollary 3.3 in [7] from the Mann iterative form to the Ishikawa iterative form.



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