# Iterative Methods for Equilibrium Problems and Monotone Inclusion Problems in Hilbert Spaces

Authors: Huan-chun Wu; Cao-zong Cheng
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We introduce a new iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of zeros of the sum of maximal monotone operators, and we obtain strong convergence theorems in Hilbert spaces. We also apply our results to the variational inequality and convex minimization problems. Our results extend and improve the recent result of Takahashi et al. (2012).


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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 is to find x¯∈C such that
(1)f(x¯,y)≥0∀y∈C.
The set of solutions of (1) is denoted by EP(f). Numerous problems in physics, optimization, and economics are reduced to find the solution of an equilibrium problem (e.g., see [1]). For solving an equilibrium problem, we assume that the bifunction f 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 everyx,y,z∈C, limsupt↓0f(tz+(1-t)x,y)≤f(x,y);(A4)
f(x,·) is convex and lower semicontinuous for each x∈C. Equilibrium problems have been studied extensively (see [1–6]).LetB be a mapping of H into 2H. The effective domain of B is denoted by dom(B); that is, dom(B)={x∈H:Bx≠∅}. A multivalued mapping B is said to be monotone if
(2)〈x-y,u-v〉≥0∀x,y∈dom(B),u∈Bx,v∈By.
A monotone operator B is said to be maximal if its graph is not properly contained in the graph of any other monotone operator. For a maximal monotone operator B on H and r>0, we may define a single-valued operator Jr=(I+rB)-1:H→dom(B), which is called the resolvent of B for r. It is known that Jr is firmly nonexpansive. A basic problem for maximal monotone operator B is to
(3)findx∈Hsuchthat0∈Bx.
The classical method for solving problem (3) is the proximal point algorithm. Rockafellar [7] established the weak convergence of the proximal point algorithm for maximal monotone operators. The question whether the strong convergence of the proximal point algorithm holds was answered in the negative by Güler [8]. In order to obtain the strong convergence theorem, some researchers have been devoted to modifying the proximal point algorithm (see [9, 10]). Given a positive constant α, a mapping A:C→H is said to be α-inverse strongly monotone if
(4)〈x-y,Ax-Ay〉≥α∥Ax-Ay∥2∀x,y∈C.
Generally, monotone inclusion problem is to
(5)findx∈Csuchthat0∈(A+B)x,
where the mapping A:C→H is inverse strongly monotone and B is maximal monotone. We write (A+B)-10 for the set of solutions of problem (5); that is, (A+B)-10={x∈C:0∈(A+B)x}. It is well known that (A+B)-10=Fix(Jλ(I-λA)) (see [11]). Takahashi et al. [12] constructed the following iterative sequence. Let u∈C, x1=x∈C and let {xn} be a sequence generated by
(6)xn+1=βnxn+(1-βn)(αnu+(1-αn)Jλn(xn-λnAxn)).
Under appropriate conditions, they proved that the sequence {xn} converges strongly to a point z0∈(A+B)-10. Lin and Takahashi [13] introduced an iterative sequence that converges strongly to an element of (A+B)-10∩F-10, where F is another maximal monotone operator.Motivated by the above results, especially by Chuang et al. [2] and Takahashi et al. [12], we introduce a new iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of zeros of the sum of maximal monotone operators, and we obtain strong convergence theorems in Hilbert spaces. We also apply our results to the variational inequality and convex minimization problems. Our results extend and improve the recent result of [12].


## 2. Preliminaries

Throughout this paper, letH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥ and let C be a nonempty closed convex subset of H. A mapping S:C→H is nonexpansive if ∥Sx-Sy∥≤∥x-y∥ for all x,y∈C. The set of fixed points of S is denoted by Fix(S); that is, Fix(S)={x∈C:Sx=x}. It is well known that Fix(S) is closed and convex. We denote the strong convergence and the weak convergence of {xn} to x by xn→x and xn⇀x, respectively. For any x∈H, there exists a unique point PCx∈C such that
(7)∥x-PCx∥≤∥x-y∥,∀y∈C.PC is called the metric projection of H onto C. Note that PC is a nonexpansive mapping. For x∈H and z∈C, we have
(8)z=PCx⟺〈x-z,y-z〉≤0foreveryy∈C.Now consider inverse strongly monotone operators. Forα>0, let A:C→H be an α-inverse strongly monotone operator. If 0<λ≤2α, then I-λA is a nonexpansive mapping. Indeed, for x,y∈C  and 0<λ≤2α, we get
(9)∥(I-λA)x-(I-λA)y∥2=∥(x-y)-λ(Ax-Ay)∥2=∥x-y∥2-2λ〈x-y,Ax-Ay〉+λ2∥Ax-Ay∥2≤∥x-y∥2-2λα∥Ax-Ay∥2+λ2∥Ax-Ay∥2≤∥x-y∥2+λ(λ-2α)∥Ax-Ay∥2≤∥x-y∥2.
Therefore, the operator I-λA is a nonexpansive mapping of C into H.We collect some important facts and tools.Lemma 1 (see [1]).
LetC be a nonempty closed convex subset of H and let f be a bifunction from C×C to ℝ satisfying (A1)–(A4). If r>0 and x∈H, then there exists z∈C such that
(10)f(z,y)+1r〈y-z,z-x〉≥0∀y∈C.Lemma 2 (see [4]).
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, define a mapping Tr:H→2C as follows:
(11)Tr(x)={z∈C:f(z,y)+1r〈y-z,z-x〉≥0∀y∈C}.
Then, the following hold:  (i)
Tr is single valued;(ii)
Tr is firmly nonexpansive; that is, for any x,y∈H, 〈x-y,Trx-Try〉≥∥Trx-Try∥2;(iii)
Fix(Tr)=EP(f);(iv)
EP(f) is closed and convex.Lemma 3 (see [6]).
Suppose that (A1)–(A4) hold. Ifx,y∈H and r1,r2>0, then
(12)∥Tr2y-Tr1x∥≤∥y-x∥+|r2-r1|r2∥Tr2y-y∥.Lemma 4 (see [14]).
LetH be a real Hilbert space and let B be a maximal monotone operator on H. Then, the following holds:
(13)s-ts〈Jsx-Jtx,Jsx-x〉≥∥Jsx-Jtx∥2
for all s,t>0 and x∈H.Lemma 5 (see [15]).
Let{xn} and {yn} be bounded sequences in a Banach space X and let {βn} be a sequence in [0,1] with 0<liminfn→∞βn≤limsupn→∞βn<1. Suppose xn+1=(1-βn)xn+βnynfor all integers n≥1 and limsupn→∞(∥yn+1-yn∥-∥xn+1-xn∥)≤0. Then, limn→∞∥yn-xn∥=0.The following lemma is an immediate consequence of the inner product onH.Lemma 6.
For allx,y∈H, the inequality ∥x+y∥2≤∥x∥2+2〈y,x+y〉 holds.Lemma 7 (demiclosedness principle [16]).
LetC be a nonempty closed convex subset of H and S:C→H a nonexpansive mapping. Let x be a point in H and let {xn} be a sequence in C. Suppose that xn⇀x and that xn-Sxn→0. Then, x∈Fix(S).Lemma 8 (see [17]).
Let{an} be a sequence of nonnegative real numbers satisfying an+1≤(1-αn)an+αnβn, where(i)
{αn}⊂(0,1), ∑n=1∞‍αn=∞;(ii)
limsupn→∞βn≤0. Then, limn→∞an=0.


## 3. Strong Convergence Theorems

In this section, we introduce a new iterative method for finding a common element of the set of zeros of the sum of maximal monotone operators and the set of solutions of an equilibrium problem, and we prove the strong convergence theorems in Hilbert spaces.Theorem 9.
LetC be a nonempty closed convex subset of a real Hilbert space H and A an α-inverse strongly monotone operator of C into H with α>0. Suppose that B is a maximal monotone operator on H and f is a bifunction from C×C to ℝ with (A1)–(A4). Assume that Ω:=EP(f)∩(A+B)-10≠∅. Let {αn} and {βn} be sequences in (0,1) and let {un} be a sequence in H. Let {xn} be a sequence generated by
(14)x1∈Cchosenarbitrarily,yn=αnun+(1-αn)Jλn(I-λnA)xn,zn∈C,suchthatf(zn,y)+1rn〈y-zn,zn-yn〉≥0kkkkkkkkkkkkkkkk∀y∈C,xn+1=βnxn+(1-βn)zn.
Assume the following conditions are satisfied:(c1)
limn→∞un=u for some u∈H;(c2)
limn→∞αn=0 and ∑n=1∞‍αn=∞;(c3)
0<liminfn→∞βn≤limsupn→∞βn<1;(c4)
0<d≤rn<∞ and limn→∞(rn+1-rn)=0;(c5)
0<a≤λn≤2α and limn→∞(λn+1-λn)=0. Then, the sequence {xn} converges strongly to PΩu.Proof.
Note that the setΩ is closed and convex since (A+B)-10 and EP(f) are closed and convex.
With the help of Lemmas1 and 2, we have zn=Trnyn, and, for any ω∈Ω,
(15)∥zn-ω∥=∥Trnyn-Trnω∥≤∥yn-ω∥.
Notice that
(16)∥yn-ω∥≤αn∥un-ω∥+(1-αn)∥Jλn(I-λnA)xn-ω∥≤αn∥un-ω∥+(1-αn)∥xn-ω∥.
It follows that
(17)∥xn+1-ω∥=∥βn(xn-ω)+(1-βn)(zn-ω)∥≤βn∥xn-ω∥+(1-βn)×[αn∥un-ω∥+(1-αn)∥xn-ω∥]≤[1-αn(1-βn)]∥xn-ω∥+αn(1-βn)∥un-ω∥≤max{∥xn-ω∥,∥un-ω∥}.
By condition (c1), the sequence {un} is bounded. Hence, there exists a positive number M such that supn{∥un-ω∥}≤M. From a simple inductive process, we have
(18)∥xn+1-ω∥≤max{∥x1-ω∥,M},
which yields that {xn} is bounded. So are {yn} and {zn}.
We have from Lemmas3 and 4 that
(19)∥zn+1-zn∥=∥Trn+1yn+1-Trnyn∥≤∥yn+1-yn∥+|rn+1-rn|rn+1∥Trn+1yn+1-yn+1∥≤∥αn+1un+1+(1-αn+1)Jλn+1(I-λn+1A)xn+1kkkk-αnun-(1-αn)Jλn(I-λnA)xn∥+|rn+1-rn|rn+1∥Trn+1yn+1-yn+1∥≤∥Jλn+1(I-λn+1A)xn+1-Jλn(I-λnA)xn∥+αn+1[∥un+1∥+∥Jλn+1(I-λn+1A)xn+1∥]+αn[∥un∥+∥Jλn(I-λnA)xn∥]+|rn+1-rn|rn+1∥Trn+1yn+1-yn+1∥≤∥Jλn+1(I-λn+1A)xn+1-Jλn+1(I-λn+1A)xnkkkk+Jλn+1(I-λn+1A)xn-Jλn+1(I-λnA)xnkkkk+Jλn+1(I-λnA)xn-Jλn(I-λnA)xn∥+αn+1[∥un+1∥+∥Jλn+1(I-λn+1A)xn+1∥]+αn[∥un∥+∥Jλn(I-λnA)xn∥]+|rn+1-rn|rn+1∥Trn+1yn+1-yn+1∥≤∥xn+1-xn∥+|λn+1-λn|∥Axn∥+|λn+1-λn|a×∥Jλn+1(I-λnA)xn-(I-λnA)xn∥+αn+1[∥un+1∥+∥Jλn+1(I-λn+1A)xn+1∥]+αn[∥un∥+∥Jλn(I-λnA)xn∥]+|rn+1-rn|d∥Trn+1yn+1-yn+1∥.
Thus, conditions (c2), (c4), and (c5) yield that
(20)limsupn→∞(∥zn+1-zn∥-∥xn+1-xn∥)≤0.
Recall that xn+1=βnxn+(1-βn)zn. It follows from Lemma 5 that
(21)limn→∞∥zn-xn∥=0.
Consequently,
(22)limn→∞∥xn+1-xn∥=limn→∞(1-βn)∥xn-zn∥=0.
For anyω∈Ω, according to Lemma 2, we have
(23)∥zn-ω∥2=∥Trnyn-Trnω∥2≤〈yn-ω,zn-ω〉=12[∥yn-ω∥2+∥zn-ω∥2-∥yn-zn∥2],
which yields that
(24)∥zn-ω∥2≤∥yn-ω∥2-∥zn-yn∥2.
By (14), one has
(25)∥xn+1-ω∥2=∥βnxn+(1-βn)zn-ω∥2≤βn∥xn-ω∥2+(1-βn)∥zn-ω∥2≤βn∥xn-ω∥2+(1-βn)×[∥yn-ω∥2-∥zn-yn∥2]≤βn∥xn-ω∥2+(1-βn)×[αn∥un-ω∥2+(1-αn)∥xn-ω∥2kkkkk-∥zn-yn∥2]≤[1-αn(1-βn)]∥xn-ω∥2+αn(1-βn)∥un-ω∥2-(1-βn)∥zn-yn∥2,
which implies that
(26)(1-βn)∥zn-yn∥2≤∥xn-ω∥2-∥xn+1-ω∥2+αn(1-βn)∥un-ω∥2≤∥xn-xn+1∥(∥xn-ω∥+∥xn+1-ω∥)+αn(1-βn)∥un-ω∥2.
This together with (22) deduces that
(27)limn→∞∥zn-yn∥=0.
Next, we prove that(28)limsupn→∞〈u-x¯,xn-x¯〉≤0,
where x¯=PΩu. In order to show this inequality, we can choose a subsequence {xni} of {xn} such that
(29)limsupn→∞〈u-x¯,xn-x¯〉=limi→∞〈u-x¯,xni-x¯〉.
Owing to the boundedness of {xni}, without loss of generality, we assume that xni⇀p. Now we show that p∈Ω. As {xn} is contained in C and C is a closed convex set, one has p∈C.
We firstly prove thatp∈EP(f). By (14), we have
(30)f(zn,y)+1rn〈y-zn,zn-yn〉≥0,∀y∈C.
The monotonicity of f yields that
(31)1rn〈y-zn,zn-yn〉≥f(y,zn),∀y∈C.
Replacing n by ni, we obtain
(32)〈y-zni,zni-ynirni〉≥f(y,zni),∀y∈C.
It follows from (21), (27), and (A4) that
(33)f(y,p)≤0,∀y∈C.
For 0<t≤1, y∈C, set yt=ty+(1-t)p. Then, yt∈C and f(yt,p)≤0. Thus,
(34)0=f(yt,yt)≤tf(yt,y)+(1-t)f(yt,p)≤tf(yt,y).
Dividing by t, we see that
(35)f(yt,y)≥0.
Letting t↓0, we get
(36)f(p,y)≥0,∀y∈C;
namely, p∈EP(f).
Secondly, we prove thatp∈(A+B)-10. Thanks to (c5), there is a subsequence {λnij} of {λni} such that λnij→λ0∈[a,2α]. Without loss of generality, we assume that λni→λ0. Observe that
(37)∥xni-Jλ0(I-λ0A)xni∥≤∥xni-yni∥+∥yni-[αniuni+(1-αni)Jλ0(I-λ0A)xni]∥+∥αniuni+(1-αni)Jλ0(I-λ0A)xnikkkkkkkk-Jλ0(I-λ0A)xni∥≤∥xni-yni∥+(1-αni)×∥Jλni(I-λniA)xni-Jλ0(I-λ0A)xni∥+αni∥uni-Jλ0(I-λ0A)xni∥≤∥xni-yni∥+(1-αni)×[∥Jλni(I-λniA)xni-Jλn0(I-λniA)xni∥kkkkkkkk+∥Jλn0(I-λniA)xni-Jλn0(I-λn0A)xni∥]+αni∥uni-Jλ0(I-λ0A)xni∥≤∥xni-yni∥+(1-αni)×[|λ0-λni|λ0∥Jλn0(I-λniA)xni-(I-λniA)xni∥kkkkkkkk+|λni-λn0|∥Axni∥|λ0-λni|λ0]+αni∥uni-Jλ0(I-λ0A)xni∥.
Note that expressions (21) and (27) imply that
(38)limn→∞∥xn-yn∥=0.
Therefore,
(39)limi→∞∥xni-Jλ0(I-λ0A)xni∥=0.
Since Jλ0(I-λ0A) is nonexpansive and xni⇀p, Lemma 7 gives p∈Fix(Jλ0(I-λ0A)); that is, p∈(A+B)-10. Thus, we have p∈Ω.
By (29) and the property of metric projection, we have
(40)limsupn→∞〈u-x¯,xn-x¯〉=limi→∞〈u-x¯,xni-x¯〉=〈u-x¯,p-x¯〉≤0.
Inequality (28) arrives.
Finally, we prove thatxn→x¯. Using (14) again, we obtain
(41)∥xn+1-x¯∥2≤βn∥xn-x¯∥2+(1-βn)∥yn-x¯∥2≤βn∥xn-x¯∥2+(1-βn)×[(1-αn)2∥xn-x¯∥2+2αn〈un-x¯,yn-x¯〉]≤[1-αn(1-βn)]∥xn-x¯∥2+2αn(1-βn)×(〈un-u,yn-x¯〉kkkkkk+〈u-x¯,yn-xn〉+〈u-x¯,xn-x¯〉).
Note that
(42)limsupn→∞〈u-x¯,xn-x¯〉≤0.
It follows from Lemma 8 that {xn} converges strongly to x¯.The following results are direct consequences of Theorem9.Corollary 10.
LetC be a nonempty closed convex subset of a real Hilbert space H and A an α-inverse strongly monotone operator of C into H with α>0. Suppose that B is a maximal monotone operator on H with (A+B)-10≠∅. Let {αn} and {βn} be sequences in (0,1) and let {un} be a sequence in H. Let {xn} be a sequence generated by
(43)x1∈Cchosenarbitrarily,yn=αnun+(1-αn)Jλn(I-λnA)xn,xn+1=βnxn+(1-βn)PCyn.
If conditions (c1)–(c3) and (c5) are satisfied, then the sequence {xn} converges strongly to P(A+B)-10u.Proof.
Lettingf(x,y)≡0 for all x,y∈C and rn=1 in Theorem 9, the desired result follows.Remark 11.
Whenun≡u∈C, the algorithm in Corollary 10 corresponds to the one in Theorem 10 of Takahashi et al. [12]. Thus, Corollary 10 includes, as a special case, Theorem 10 of [12].Corollary 12.
LetC be a nonempty closed convex subset of a real Hilbert space H and let f be a bifunction from C×C to ℝ with (A1)–(A4). Assume that the set EP(f) is nonempty. Let {αn} and {βn} be sequences in (0,1) and let {un} be a sequence in H. Let {xn} be a sequence generated by
(44)x1∈Cchosenarbitrarily,yn=αnun+(1-αn)xn,zn∈C,suchthatf(zn,y)+1rn〈y-zn,zn-yn〉≥0,kkkkkkkkkkkk∀y∈C,xn+1=βnxn+(1-βn)zn.
If conditions (c1)–(c4) are satisfied, then the sequence {xn} converges strongly to PEP(f)u.Proof.
PuttingA=0, B=∂δC (i.e., the subdifferential of the indicator function of  C) in Theorem 9, we get the result.


## 4. Applications

In this section, we apply the results in the preceding section to variational inequality and optimization problems.Now we consider the variational inequality problem. LetH be a real Hilbert space and let f be a proper lower semicontinuous convex function of H into (-∞,+∞]. Then, the subdifferential ∂f of f is defined as follows:
(45)∂f(x)={z∈H:f(y)-f(x)≥〈z,y-x〉,∀y∈H},
for all x∈H. Rockafellar [18] claimed that ∂f is a maximal monotone operator. Let δC be the indicator function of C; that is,
(46)δC(x)={0,x∈C,+∞,x∉C.
Since δC is a proper lower semicontinuous convex function on H, the subdifferential ∂δC of δC is a maximal monotone operator. The resolvent Jλ of ∂δC for λ>0 is defined by
(47)Jλx=(I+λ∂δC)-1x,∀x∈H.
We have
(48)u=Jλx=(I+λ∂δC)-1x⟺x∈u+λ∂δCu⟺x∈u+λNCu⟺x-u∈λNCu⟺1λ〈x-u,y-u〉≤0∀y∈C⟺〈x-u,y-u〉≤0∀y∈C⟺u=PCx,
where NCu={z∈H:〈z,y-u〉≤0, forally∈C}.LetA:C→H be a nonlinear mapping. The variational inequality problem for A is to find z∈C such that
(49)〈Az,y-z〉≥0,∀y∈C.
The set of its solutions is denoted by VI(A,C). It is easy to see that VI(A,C)=(A+∂δC)-10 (for more details, see [13]). Using Corollary 10, we obtain the strong convergence theorem for the variational inequality problem.Theorem 13.
LetC be a nonempty closed convex subset of a real Hilbert space H and A an α-inverse strongly monotone operator of C into H with α>0. Assume that the set VI(A,C) is nonempty. Let {αn} and {βn} be sequences in (0,1) and let {un} be a sequence in H. Let {xn} be a sequence generated by
(50)x1∈Cchosenarbitrarily,yn=αnun+(1-αn)PC(I-λnA)xn,xn+1=βnxn+(1-βn)PCyn.
If conditions (c1)–(c3) and (c5) are satisfied, then the sequence {xn} converges strongly to PVI(A,C)u.Proof.
Notice thatVI(A,C)=(A+∂δC)-10. Corollary 10 yields the conclusion.Next we study the optimization problem(51)ming(x),x∈C,
where g(x) is a proper lower semicontinuous convex function of H into (-∞,+∞] such that C is contained in domg={x∈H:g(x)<+∞}. We denote by Sol(g,C) the set of solutions of problem (51). Let f:C×C→ℝ be a bifunction defined by
(52)f(x,y)=g(y)-g(x).
It is clear that f(x,y) satisfies (A1)–(A4) and EP(f)=Sol(g,C). Therefore, by Corollary 12, the following result is obtained.Theorem 14.
Letg(x) be a proper lower semicontinuous convex function of H into (-∞,+∞] and let C be a nonempty closed convex subset of H such that C is contained in domg. Suppose that the set Sol(g,C) is nonempty. Let {αn} and {βn} be sequences in (0,1) and let {un} be a sequence in H. Let {xn} be a sequence generated by
(53)x1∈Cchosenarbitrarily,yn=αnun+(1-αn)xn,zn=argminy∈C{g(y)+12rn∥y-yn∥2},xn+1=βnxn+(1-βn)zn.
If conditions (c1)–(c4) are satisfied, then the sequence {xn} converges strongly to PSol(g,C)u.Proof.
Lettingf(x,y)=g(y)-g(x) in Corollary 12, we get the required result.Remark 15.
Letting0=u=limn→∞un in our results, we obtain the algorithms for minimal-norm solutions of corresponding problems.



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