# Composite Algorithms for Minimization over the Solutions of Equilibrium Problems and Fixed Point Problems

Authors: Yonghong Yao; Yeong-Cheng Liou
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

The purpose of this paper is to solve the minimization problem of findingx∗ such that x∗=argminx∈Γ∥x∥2, where Γ stands for the intersection set of the solution set of the equilibrium problem and the fixed points set of a nonexpansive mapping. We first present two new composite algorithms (one implicit and one explicit). Further, we prove that the proposed composite algorithms converge strongly to x∗.


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## Body

## 1. Introduction

In the present paper, our main purpose is to solve the minimization problem of findingx* such that(1.1)x*=argminx∈Γ∥x∥2,
where Γ stands for the intersection set of the solution set of the equilibrium problem and the fixed points set of a nonexpansive mapping. This problem is motivated by the following least-squares solution to the constrained linear inverse problem:(1.2)Bx=b,x∈C,
where C is a nonempty closed convex subset of a real Hilbert space H, B is a bounded linear operator from H to another real Hilbert space H1 and b is a given point in H1. The least-squares solution to (1.2) is the least-norm minimizer of the minimization problem(1.3)minx∈C∥Bx-b∥2.
Let Sb denote the solution set of (1.2) (or equivalently (1.3)). It is known that Sb is nonempty if and only if PB(C)¯(b)∈B(C). In this case, Sb has a unique element with minimum norm (equivalently, (1.2) has a unique least-squares solution); that is, there exists a unique point x†∈Sb satisfying(1.4)∥x†∥=min{∥x∥:x∈Sb}.
The so-called C-constrained pseudoinverse of B is then defined as the operator BC† with domain and values given by (1.5)D(BC†)={b∈H:PB(C)¯(b)∈B(C)},BC†(b)=x†,b∈D(BC†),
where x†∈Sb is the unique solution to (1.4).Note that the optimality condition for the minimization (1.3) is the variational inequality (VI)
(1.6)x̂∈C,〈B*(Bx̂-b),x-x̂〉≥0,x∈C,
where B* is the adjoint of B.Ifb∈D(BC†), then (1.3) is consistent and its solution set Sb coincides with the solution set of VI (1.6). On the other hand, VI (1.6) can be rewritten as
(1.7)x̂∈C,〈(x̂-λB*(Bx̂-b))-x̂,x-x̂〉≤0,x∈C,
where λ>0 is any positive scalar. In the terminology of projections, (1.7) is equivalent to the fixed point equation (1.8)x̂=PC(x̂-λB*(Bx̂-b)).
It is not hard to find that for 0<λ<2/∥B∥2, the mapping x↦PC(x-λB*(Bx-b)) is nonexpansive. Therefore, finding the least-squares solution of the constrained linear inverse problem is equivalent to finding the minimum-norm fixed point of the nonexpansive mapping x↦PC(x-λB*(Bx-b)).Based on the above facts, it is an interesting topic of finding the minimum norm fixed point of the nonexpansive mappings. In this paper, we will consider a general problem. We will focus on to solve the minimization problem (1.1). At this point, we first recall some definitions on the fixed point problem and the equilibrium problem as follows.LetC be a nonempty closed convex subset of a real Hilbert space H. Recall that a mapping A:C→H is called α-inverse strongly monotone if there exists a positive real number α such that 〈Ax-Ay,x-y〉≥α∥Ax-Ay∥2, for all x,y∈C. It is clear that any α-inverse strongly monotone mapping is monotone and 1/α-Lipschitz continuous. Let f:C→H be a ρ-contraction; that is, there exists a constant ρ∈[0,1) such that ∥f(x)-f(y)∥≤ρ∥x-y∥ for all x,y∈C. A mapping S:C→C is said to be nonexpansive if ∥Sx-Sy∥≤∥x-y∥, for all x,y∈C. Denote the set of fixed points of S by Fix(S).LetA:C→H be a nonlinear mapping and F:C×C→R be a bifunction. The equilibrium problem is to find z∈C such that
(1.9)F(z,y)+〈Az,y-z〉≥0,∀y∈C.
The solution set of (1.9) is denoted by EP. If A=0, then (1.9) reduces to the following equilibrium problem of finding z∈C such that
(1.10)F(z,y)≥0,∀y∈C.
If F=0, then (1.9) reduces to the variational inequality problem of finding z∈C such that(1.11)〈Az,y-z〉≥0,∀y∈C.
We note that the problem (1.9) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games and others, see, for example, [1–4].We next briefly review some historic approaches which relate to the fixed point problems and the equilibrium problems.In 2005, Combettes and Hirstoaga [5] introduced an iterative algorithm of finding the best approximation to the initial data and proved a strong convergence theorem. In 2007, by using the viscosity approximation method, S. Takahashi and W. Takahashi [6] introduced another iterative scheme for finding a common element of the set of solutions of the equilibrium problem and the set of fixed point points of a nonexpansive mapping. Subsequently, algorithms constructed for solving the equilibrium problems and fixed point problems have further developed by some authors. In particular, Ceng and Yao [7] introduced an iterative scheme for finding a common element of the set of solutions of the mixed equilibrium problem (1.9) and the set of common fixed points of finitely many nonexpansive mappings. Maingé and Moudafi [8] introduced an iterative algorithm for equilibrium problems and fixed point problems. Yao et al. [9] considered an iterative scheme for finding a common element of the set of solutions of the equilibrium problem and the set of common fixed points of an infinite nonexpansive mappings. Noor et al. [10] introduced an iterative method for solving fixed point problems and variational inequality problems. Their results extend and improve many results in the literature. Some works related to the equilibrium problem, fixed point problems, and the variational inequality problem in[1–45] and the references therein.However, we note that all constructed algorithms in [2, 4, 6–10, 14, 15, 21, 23–40] do not work to find the minimum-norm solution of the corresponding fixed point problems and the equilibrium problems. It is our main purpose in this paper that we devote to construct some algorithms for finding the minimum-norm solution of the fixed point problems and the equilibrium problems. We first suggest two new composite algorithms (one implicit and one explicit) for solving the above minimization problem. Further, we prove that the proposed composite algorithms converge strongly to the minimum norm element x*.


## 2. Preliminaries

LetC be a nonempty closed convex subset of a real Hilbert space H. Throughout this paper, we assume that a bifunction F:C×C→R satisfies the following conditions: (H1)
F(x,x)=0, for all x∈C;(H2)
F is monotone, that is, F(x,y)+F(y,x)≤0 for all x,y∈C;(H3)
for eachx,y,z∈C, limt↓0F(tz+(1-t)x,y)≤F(x,y);(H4)
for eachx∈C, y↦F(x,y) is convex and lower semicontinuous.The metric (or nearest point) projection fromH onto C is the mapping PC:H→C which assigns to each point x∈C the unique point PCx∈C satisfying the property (2.1)∥x-PCx∥=infy∈C∥x-y∥=:d(x,C).
It is well known that PC is a nonexpansive mapping and satisfies (2.2)〈x-y,PCx-PCy〉≥∥PCx-PCy∥2,∀x,y∈H.
We need the following lemmas for proving our main results.Lemma 2.1 (see [5]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let F:C×C→R be a bifunction which satisfies conditions (H1)–(H4). Let r>0 and x∈C. Then, there exists z∈C such that
(2.3)F(z,y)+1r〈y-z,z-x〉≥0,∀y∈C.
Further, if Tr(x)={z∈C:F(z,y)+(1/r)〈y-z,z-x〉≥0,∀y∈C}, then the following hold: (i)
Tr is single-valued and Tr is firmly nonexpansive, that is, for any x,y∈H, ∥Trx-Try∥2≤〈Trx-Try,x-y〉;(ii)
EP is closed and convex and EP=Fix(Tr).Lemma 2.2 (see [17]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let the mapping A:C→H be α-inverse strongly monotone and r>0 be a constant. Then, one has
(2.4)∥(I-rA)x-(I-rA)y∥2≤∥x-y∥2+r(r-2α)∥Ax-Ay∥2,∀x,y∈C.
In particular, if 0≤r≤2α, then I-rA is nonexpansive.Lemma 2.3 (see [28]).
LetC be a closed convex subset of a real Hilbert space H and let S:C→C be a nonexpansive mapping. Then, the mapping I-S is demiclosed, that is, if {xn} is a sequence in C such that xn→x* weakly and (I-S)xn→y strongly, then (I-S)x*=y.Lemma 2.4 (see [22]).
Assume{an} is a sequence of nonnegative real numbers such that
(2.5)an+1≤(1-γn)an+δnγn,
where {γn} is a sequence in (0,1) and {δn} is a sequence such that (1)
∑n=1∞γn=∞;(2)
lim supn→∞δn≤0 or ∑n=1∞|δnγn|<∞.Then limn→∞an=0.


## 3. Main Results

In this section we will introduce two algorithms for finding the minimum norm elementx* of Γ:=EP∩Fix(S). Namely, we want to find the unique point x* which solves the following minimization problem:(3.1)x*=argminx∈Γ∥x∥2.LetS:C→C be a nonexpansive mapping and A:C→H be an α-inverse strongly monotone mapping. Let F:C×C→R be a bifunction which satisfies conditions (H1)–(H4). Let r and μ be two constants such that r∈(0,2α) and μ∈(0,1). In order to find a solution of the minimization problem (3.1), we construct the following implicit algorithm
(3.2)xt=μPC[(1-t)Sxt]+(1-μ)Tr(xt-rAxt),∀t∈(0,1),
where Tr is defined as Lemma 2.1. We will show that the net {xt} defined by (3.2) converges to a solution of the minimization problem (3.1). As matter of fact, in this paper, we will study the following general algorithm.Letf:C→H be a ρ-contraction. For each t∈(0,1), we consider the following mapping Wt given by
(3.3)Wtx=μPC[tf(x)+(1-t)Sxt]+(1-μ)Tr(I-rA)xt,∀x∈C.
Since the mappings S, PC, Tr and I-rA are nonexpansive, then we can check easily that ∥Wtx-Wty∥≤[1-(1-ρ)μt]∥x-y∥ which implies that Wt is a contraction. Using the Banach contraction principle, there exists a unique fixed point xt of Wt in C, that is,
(3.4)xt=μPC[tf(xt)+(1-t)Sxt]+(1-μ)Tr(I-rA)xt,t∈(0,1).In this point, we would like to point out that algorithm (3.4) includes algorithm (3.2) as a special case due to the contraction f is a possible nonself-mapping.In the sequel, we assume(1)
C is a nonempty closed convex subset of a real Hilbert space H;(2)
S:C→C is a nonexpansive mapping, A:C→H is an α-inverse strongly monotone mapping and f:C→H is a ρ-contraction;(3)
F:C×C→R is a bifunction which satisfies conditions (H1)–(H4);(4)
Γ≠∅.In order to prove our first main result, we need the following lemmas.Lemma 3.1.
The net{xt} generated by the implicit method (3.4) is bounded.Proof.
Setut=Tr(xt-rAxt) and yt=tf(xt)+(1-t)Sxt for all t∈(0,1). Take z∈Γ. It is clear that Sz=z=Tr(z-rAz). Since Tr is nonexpansive and A is α-inverse strongly monotone, we have from Lemma 2.2 that
(3.5)∥ut-z∥2≤∥xt-rAxt-(z-rAz)∥2≤∥xt-z∥2+r(r-2α)∥Axt-Az∥2≤∥xt-z∥2.
So, we have that
(3.6)∥ut-z∥≤∥xt-z∥.
It follows from (3.4) that
(3.7)∥xt-z∥=∥μPC[yt]+(1-μ)ut-z∥≤μ∥PC[yt]-z∥+(1-μ)∥ut-z∥≤μ(t∥f(xt)-z∥+(1-t)∥Sxt-z∥)+(1-μ)∥xt-z∥≤μ(tρ+1-t)∥xt-z∥+μt∥f(z)-z∥+(1-μ)∥xt-z∥=[1-(1-ρ)μt]∥xt-z∥+tμ∥f(z)-z∥,
that is,
(3.8)∥xt-z∥≤∥f(z)-z∥1-ρ.
So, {xt} is bounded. Hence {ut} and {f(xt)} are also bounded. This completes the proof.According to Lemma3.1, we can choose some appropriate constant M>0 such that M satisfies the following request.Lemma 3.2.
The net{xt} generated by the implicit method (3.4) is relatively norm compact as t→0.Proof.
From (3.4) and (3.5), we have
(3.9)∥xt-z∥2=∥μ(PC[yt]-z)+(1-μ)(ut-z)∥2≤μ∥PC[yt]-z∥2+(1-μ)∥ut-z∥2≤μ∥t(f(xt)-z)+(1-t)(Sxt-z)∥2+(1-μ)∥ut-z∥2≤μ[t∥f(xt)-z∥2+(1-t)∥Sxt-z∥2]+(1-μ)∥ut-z∥2≤μ[tM+(1-t)∥xt-z∥2]+(1-μ)∥ut-z∥2.
It follows that
(3.10)∥xt-z∥2≤1-μ1-μ+μt∥ut-z∥2+μtM1-μ+μt≤∥ut-z∥2+tM≤∥xt-z∥2+r(r-2α)∥Axt-Az∥2+tM,
that is,
(3.11)r(2α-r)∥Axt-Az∥2≤tM→0.
Since r(2α-r)>0, we derive
(3.12)limt→0∥Axt-Az∥=0.
From Lemmas2.1 and 2.2, we obtain (3.13)∥ut-z∥2=∥Tr(xt-rAxt)-Tr(z-rAz)∥2≤〈(xt-rAxt)-(z-rAz),ut-z〉=12(∥(xt-rAxt)-(z-rAz)∥2+∥ut-z∥2-∥(xt-z)-r(Axt-Az)-(ut-z)∥2)≤12(∥xt-z∥2+∥ut-z∥2-∥(xt-ut)-r(Axt-Az)∥2)=12(∥xt-z∥2+∥ut-z∥2-∥xt-ut∥2+2r〈xt-ut,Axt-Az〉-r2∥Axt-Az∥2),
which implies that
(3.14)∥ut-z∥2≤∥xt-z∥2-∥xt-ut∥2+2r〈xt-ut,Axt-Az〉-r2∥Axt-Az∥2≤∥xt-z∥2-∥xt-ut∥2+2r∥xt-ut∥∥Axt-Az∥≤∥xt-z∥2-∥xt-ut∥2+M∥Axt-Az∥.
By (3.10), and (3.14), we have
(3.15)∥xt-z∥2≤∥ut-z∥2+tM≤∥xt-z∥2-∥xt-ut∥2+(∥Axt-Az∥+t)M.
It follows that
(3.16)∥xt-ut∥2≤(∥Axt-Az∥+t)M.
This together with (3.12) imply that
(3.17)limt→0∥xt-ut∥=0.
It follows that
(3.18)limt→0∥xt-PC[yt]∥=limt→01-μμ∥xt-ut∥=0.
Hence,
(3.19)∥xt-Sxt∥≤∥xt-PC[yt]∥+∥PC[yt]-Sxt∥≤∥xt-PC[yt]∥+∥yt-Sxt∥≤∥xt-PC[yt]∥+t∥f(xt)∥→0.
Next, we show that {xt} is relatively norm compact as t→0. Let {tn}⊂(0,1) be a sequence such that tn→0 as n→∞. Put xn:=xtn and un:=utn. From (3.19), we get
(3.20)∥xn-Sxn∥→0.
By (3.4), we deduce
(3.21)∥xt-z∥2=∥μ(PC[yt]-z)+(1-μ)(ut-z)∥2≤μ∥PC[yt]-z∥2+(1-μ)∥ut-z∥2≤μ∥yt-z∥2+(1-μ)∥xt-z∥2,
that is,
(3.22)∥xt-z∥2≤∥t(f(xt)-z)+(1-t)(Sxt-z)∥2=(1-t)2∥Sxt-z∥2+2t(1-t)〈f(xt)-f(z),Sxt-z〉+2t(1-t)〈f(z)-z,Sxt-z〉+t2∥f(xt)-z∥2≤(1-t)2∥xt-z∥2+2t(1-t)ρ∥xt-z∥2+2t(1-t)〈f(z)-z,Sxt-z〉+t2∥f(xt)-z∥2≤[1-2(1-ρ)t]∥xt-z∥2+2t〈f(z)-z,Sxt-z〉+t2M.
It follows that
(3.23)∥xt-z∥2≤11-ρ〈z-f(z),z-Sxt〉+t2(1-ρ)M.
In particular,
(3.24)∥xn-z∥2≤11-ρ〈z-f(z),z-Sxn〉+tn2(1-ρ)M,z∈Γ.
Since{xn} is bounded, without loss of generality, we may assume that {xn} converges weakly to a point x*∈C. Also Sxn→x* weakly. Noticing (3.20) we can use Lemma 2.3 to get x*∈Fix(S).
Now, we showx*∈EP. Since un=Tr(xn-rAxn), for any y∈C, we have (3.25)F(un,y)+1r〈y-un,un-(xn-rAxn)〉≥0.
From the monotonicity of F, we have
(3.26)1r〈y-un,un-(xn-rAxn)〉≥F(y,un),∀y∈C.
Hence,
(3.27)〈y-uni,uni-xnir+Axni〉≥F(y,uni),∀y∈C.
Put zt=ty+(1-t)x* for all t∈(0,1] and y∈C. Then, we have zt∈C. So, from (3.27), we have
(3.28)〈zt-uni,Azt〉≥〈zt-uni,Azt〉-〈zt-uni,uni-xnir+Axni〉+F(zt,uni)=〈zt-uni,Azt-Auni〉+〈zt-uni,Auni-Axni〉-〈zt-uni,uni-xnir〉+F(zt,uni).
Note that ∥Auni-Axni∥≤(1/α)∥uni-xni∥→0. Further, from monotonicity of A, we have 〈zt-uni,Azt-Auni〉≥0. Letting i→∞ in (3.28), we have
(3.29)〈zt-x*,Azt〉≥F(zt,x*).
From (H1), (H4), and (3.29), we also have
(3.30)0=F(zt,zt)≤tF(zt,y)+(1-t)F(zt,x*)≤tF(zt,y)+(1-t)〈zt-x*,Azt〉=tF(zt,y)+(1-t)t〈y-x*,Azt〉
and hence
(3.31)0≤F(zt,y)+(1-t)〈Azt,y-x*〉.
Letting t→0 in (3.31), we have, for each y∈C,
(3.32)0≤F(x*,y)+〈y-x*,Ax*〉.
This implies that x*∈EP. Therefore, x*∈Γ.
We substitutex* for z in (3.24) to get (3.33)∥xn-x*∥2≤11-ρ〈x*-f(x*),x*-Sxn〉+tn2(1-ρ)M.
Hence, the weak convergence of {Sxn} to x* implies that xn→x* strongly. This has proved the relative norm compactness of the net {xt} as t→0. This completes the proof.Now, we show our first main result.Theorem 3.3.
The net{xt} generated by the implicit method (3.4) converges in norm, as t→0, to the unique solution x* of the following variational inequality:
(3.34)x*∈Γ,〈(I-f)x*,x-x*〉≥0,x∈Γ.
In particular, if we take f=0, then the net {xt} defined by (3.2) converges in norm, as t→0, to a solution of the minimization problem (3.1).Proof.
Now we return to (3.24) in Lemma 3.2 and take the limit as n→∞ to get
(3.35)∥x*-z∥2≤11-ρ〈z-f(z),z-x*〉,z∈Γ.
In particular, x* solves the following variational inequality
(3.36)x*∈Γ,〈(I-f)z,z-x*〉≥0,z∈Γ
or the equivalent dual variational inequality:
(3.37)x*∈Γ,〈(I-f)x*,z-x*〉≥0,z∈Γ.
Therefore, x*=(PΓf)x*. That is, x* is the unique fixed point in Γ of the contraction PΓf. Clearly this is sufficient to conclude that the entire net {xt} converges in norm to x* as t→0.
Finally, if we takef=0, then (3.35) is reduced to(3.38)∥x*-z∥2≤〈z,z-x*〉,z∈Γ.
Equivalently,
(3.39)∥x*∥2≤〈x*,z〉,z∈Γ.
This clearly implies that
(3.40)∥x*∥≤∥z∥,z∈Γ.
Therefore, x* is a solution of minimization problem (3.1). This completes the proof.Next we introduce an explicit algorithm for finding a solution of minimization problem (3.1). This scheme is obtained by discretizing the implicit scheme (3.4).Algorithm 3.4.
Given thatx0∈C arbitrarily, let the sequence {xn} be generated iteratively by
(3.41)xn+1=μnPC[αnf(xn)+(1-αn)Sxn]+(1-μn)Tr(xn-rAxn),n≥0,
where {αn} and {μn} are two sequences in [0,1].Next, we give several lemmas in order to prove our second main result.Lemma 3.5.
The sequence{xn} generated by (3.41) is bounded.Proof.
Pickz∈Γ. Let un=Tr(xn-rAxn) and yn=tf(xn)+(1-t)Sxn for all n≥0. From (3.41), we get
(3.42)∥un-z∥=∥Tr(xn-rAxn)-Tr(z-rAz)∥≤∥xn-z∥,∥xn+1-z∥=∥μn(PC[yn]-z)+(1-μn)(un-z)∥≤μn∥PC[yn]-z∥+(1-μn)∥un-z∥≤μn∥yn-z∥+(1-μn)∥xn-z∥≤μnαn∥f(xn)-z∥+μn(1-αn)∥Sxn-z∥+(1-μn)∥xn-z∥≤[1-(1-ρ)μnαn]∥xn-z∥+μnαn∥f(z)-z∥.
By induction, we obtain, for all n≥0,
(3.43)∥xn-z∥≤max{∥x0-z∥,∥f(z)-z∥1-ρ}.
Hence, {xn} is bounded. Consequently, we deduce that {un}, {f(xn)} and {Axn} are all bounded. This completes the proof.Lemma 3.6.
Assume the sequences{αn} and {μn} satisfy the following conditions: (i)
limn→∞αn=0 and limn→∞(αn+1/αn)=1;(ii)
0<liminfn→∞μn≤lim supn→∞μn<1 and limn→∞((μn+1-μn)/αn+1)=1.Then limn→∞∥xn-Sxn∥=0.Proof.
From (3.41), we have
(3.44)∥xn+2-xn+1∥=∥μn+1PC[yn+1]+(1-μn+1)un+1-μnPC[yn]-(1-μn)un∥=∥μn+1(PC[yn+1]-PC[yn])+(μn+1-μn)PC[yn]+(1-μn+1)(un+1-un)+(μn-μn+1)un∥≤μn+1∥yn+1-yn∥+(1-μn+1)∥un+1-un∥+|μn+1-μn|(∥PC[yn]∥+∥un∥).
Next we estimate ∥yn+1-yn∥ and ∥un+1-un∥. We have
(3.45)∥yn+1-yn∥=∥αn+1f(xn+1)+(1-αn+1)Sxn+1-αnf(xn)+(1-αn)Sxn∥=∥αn+1(f(xn+1)-f(xn))+(αn+1-αn)f(xn)+(1-αn+1)(Sxn+1-Sxn)+(αn-αn+1)Sxn∥≤[1-(1-ρ)αn+1]∥xn+1-xn∥+|αn+1-αn|(∥f(xn)∥+∥Sxn∥),∥un+1-un∥=∥Tr(xn+1-rAxn+1)-Tr(xn-rAxn)∥≤∥(xn+1-rAxn+1)-(xn-rAxn)∥≤∥xn+1-xn∥.
Then, we obtain
(3.46)∥xn+2-xn+1∥≤[1-(1-ρ)αn+1μn+1]∥xn+1-xn∥+(|αn+1-αn|+|μn+1-μn|)M,
where M>0 is a constant satisfying
(3.47)supn{∥f(xn)∥+∥Sxn∥,∥PC[yn]∥+∥un∥,∥f(xn)-z∥2,2∥xn-z∥,2r∥xn-un∥}≤M.
This together with (i), (ii) and Lemma 2.4 imply that
(3.48)limn→∞∥xn+2-xn+1∥=0.
By the convexity of the norm ∥·∥, we have
(3.49)∥xn+1-z∥2=∥μn(PC[yn]-z)+(1-μn)(un-z)∥2≤μn∥PC[yn]-z∥2+(1-μn)∥un-z∥2≤μn∥αn(f(xn)-z)+(1-αn)(Sxn-z)∥2+(1-μn)∥un-z∥2≤μn[αn∥f(xn)-z∥2+(1-αn)∥Sxn-z∥2]+(1-μn)∥un-z∥2≤μn[αnM+(1-αn)∥xn-z∥2]+(1-μn)∥un-z∥2.
From Lemma 2.2, we get
(3.50)∥un-z∥2=∥Tr(xn-rAxn)-Tr(z-rAz)∥2≤∥(xn-rAxn)-(z-rAz)∥2≤∥xn-z∥2+r(r-2α)∥Axn-Az∥2.
Substituting (3.50) into (3.49), we have
(3.51)∥xn+1-z∥2≤(1-μnαn)∥xn-z∥2+r(r-2α)(1-μn)∥Axn-Az∥2+μnαnM.
Therefore,
(3.52)r(2α-r)(1-μn)∥Axn-Az∥2≤(1-μnαn)∥xn-z∥2-∥xn+1-z∥2+μnαnM≤(∥xn-z∥+∥xn+1-z∥)∥xn-xn+1∥+αnM≤(∥xn-xn+1∥+αn)M.
Since liminfn→∞(1-μn)r(2α-r)>0, ∥xn-xn+1∥→0 and αn→0, we derive
(3.53)limn→∞∥Axn-Az∥=0.
From Lemma2.1 and (3.41), we obtain(3.54)∥un-z∥2=∥Tr(xn-rAxn)-Tr(z-rAz)∥2≤〈(xn-rAxn)-(z-rAz),un-z〉=12(∥(xn-rAxn)-(z-rAz)∥2+∥un-z∥2-∥(xn-z)-r(Axn-Az)-(un-z)∥2)≤12(∥xn-z∥2+∥un-z∥2-∥(xn-un)-r(Axn-Az)∥2)=12(∥xn-z∥2+∥un-z∥2-∥xn-un∥2+2r〈xn-un,Axn-Az〉-r2∥Axn-Az∥2).
Thus, we deduce
(3.55)∥un-z∥2≤∥xn-z∥2-∥xn-un∥2+2r∥xn-un∥∥Axn-Az∥≤∥xn-z∥2-∥xn-un∥2+M∥Axn-Az∥.
By (3.49) and (3.55), we have
(3.56)∥xn+1-z∥2≤μn[αnM+(1-αn)∥xn-z∥2]+(1-μn)∥un-z∥2≤μn[αnM+(1-αn)∥xn-z∥2]+(1-μn)[∥xn-z∥2-∥xn-un∥2+M∥Axn-Az∥]≤∥xn-z∥2-(1-μn)∥xn-un∥2+(∥Axn-Az∥+αn)M.
It follows that
(3.57)(1-μn)∥xn-un∥2≤(∥xn+1-xn∥+∥Axn-Az∥+αn)M.
Since liminfn→∞(1-μn)>0, αn→0, ∥xn+1-xn∥→0 and ∥Axn-Az∥→0, we derive that
(3.58)limn→∞∥xn-un∥=0.
Note that xn+1-xn=μn(PC[yn]-xn)+(1-μn)(un-xn). Hence,
(3.59)∥PC[yn]-xn∥→0.
Therefore,
(3.60)∥Sxn-xn∥≤∥Sxn-PC[yn]∥+∥PC[yn]-xn∥≤∥Sxn-yn∥+∥PC[yn]-xn∥≤αn∥f(xn)-Sxn∥+∥PC[yn]-xn∥→0.
This completes the proof.Now, we show the strong convergence of the sequence{xn} generated by (3.41).Theorem 3.7.
Assume the sequences{αn} and {μn} satisfy the following conditions: (i)
limn→∞αn=0, ∑n=0∞αn=∞ and limn→∞(αn+1/αn)=1;(ii)
0<liminfn→∞μn≤lim supn→∞μn<1 and limn→∞((μn+1-μn)/αn+1)=1.Then the sequence {xn} generated by (3.41) converges strongly to x* which is the unique solution of variational inequality (3.34). In particular, if f=0, then the sequence {xn} generated by
(3.61)xn+1=μnPC[(1-αn)Sxn]+(1-μn)Tr(xn-rAxn),n≥0,
converges strongly to a solution of the minimization problem (3.1).Proof.
We first prove(3.62)lim supn→∞〈x*-f(x*),x*-Sxn〉≤0
where x*=PΓf(x*).
Indeed, we can choose a subsequence{Sxni} of {Sxn} such that (3.63)lim supn→∞〈x*-f(x*),x*-Sxn〉=limi→∞〈x*-f(x*),x*-Sxni〉.
Without loss of generality, we may further assume that Sxni→x̃ weakly. By the same argument as that of Theorem 3.3, we can deduce that x̃∈Γ. Therefore,
(3.64)lim supn→∞〈x*-f(x*),x*-Sxn〉=〈x*-f(x*),x*-x̃〉≤0.
From (3.41), we have
(3.65)∥xn+1-x*∥2=∥μn(PC[yn]-z)+(1-μn)(un-z)∥2≤μn∥PC[yn]-z∥2+(1-μn)∥un-z∥2≤μn∥yn-x*∥2+(1-μn)∥xn-x*∥2=μn[(1-αn)2∥Sxn-x*∥2+2αn(1-αn)〈f(xn)-f(x*),Sxn-x*〉+2αn(1-αn)〈f(x*)-x*,Sxn-x*〉+αn2∥f(xn)-x*∥2]+(1-μn)∥xn-x*∥2≤μn[(1-αn)2∥xn-x*∥2+2αn(1-αn)ρ∥xn-x*∥2+2αn(1-αn)〈f(x*)-x*,Sxn-x*〉+αn2M]+(1-μn)∥xn-x*∥2≤[1-2(1-ρ)μnαn]∥xn-x*∥2+2αnμn(1-αn)〈f(x*)-x*,Sxn-x*〉+2αn2μnM=(1-γn)∥xn-x*∥2+δnγn,
where γn=2(1-ρ)μnαn and δn=((1-αn)/(1-ρ))〈f(x)*-x*,Sxn-x*〉+αnM/(1-ρ). It is clear that ∑n=0∞γn=∞ and lim supn→∞δ≤0. Hence, all conditions of Lemma 2.4 are satisfied. Therefore, we immediately deduce that xn→x*.
Finally, if we takef=0, by the similar argument as that Theorem 3.3, we deduce immediately that x* is a minimum norm element in Γ. This completes the proof.


## 4. Conclusions

Iterative methods for finding the common element of the equilibrium problem and the fixed point problem have been extensively studied, see, for example, [2, 4, 6, 7, 9, 14, 15, 21, 23–28]. However, iterative methods for finding the minimum norm solution of the equilibrium problem and the fixed point problem are far less developed than those for only finding the common element of the equilibrium problem and the fixed point problem. In the present paper, we suggest two algorithm, one implicit algorithm (3.4) and one explicit algorithm (3.41). We prove the strong convergence of the algorithms (3.4) and (3.41) to the common element of the equilibrium problem and the fixed points set of a nonexpansive mapping. As special cases, we prove that algorithms (3.2) and (3.61) converges to x* which solves the minimization problem (3.1). It should be pointed out that our algorithms and our main results are new even if we assume f is a self-mapping on C.Since in many problems, it is needed to find a solution with minimum norm. Hence, it is a very interesting problem to construct some algorithms for finding the minimum norm solution of some practical problem. The reader can develop iterative algorithms for solving some minimization problems by using our methods and technique contained in the present paper.



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*Source: 763506-2010-10-11.xml*