A Hybrid Extragradient Method for Solving
Ky Fan Inequalities, Variational Inequalities and
Fixed Point Problems
Supak Phiangsungnoen and Poom Kumam,
Member, IAENG.
Abstract—The purpose of this paper is to introduce a new hybrid extragradient iterative method for finding a common element of the set of points satisfying the Ky Fan inequalities, variational inequality and the set of fixed points of a strict pseudocontraction mapping in Hilbert spaces. Consequently, we obtain the strong convergence of an iterative algorithm generated by the hybrid extragradient projection method, under some suitable assumptions the function associated with Ky Fan inequality is pseudomonotone and weakly continuous.
Index Terms—Extragradient Method, Fixed point problems, Hybrid projection method,Variational inequality problems, ξ−strict pseudocontraction, Lipschitz continuity.
I. INTRODUCTION
T
HROUGHOUT this paper, we always assume that H be a real Hilbert space, whose inner product and norm denoted by h·,·i and k · k, respectively. Let C be a closed convex subset ofH andF :C×C→R, whereRdenotesthe set of real number, a bifunction. We concider the Ky Fan inequality which was first introduced by Ky Fan [25] as follows:
find x∈C such that F(x, y)≥0, ∀y∈C. (1) The set of such thatx∈C is denoted byEP(F), i.e.,
EP(F) ={x∈C:F(x, y)≥0, ∀y∈C}.
Numerous problems in physics, optimization, and economics reduce to find a solution of (1). Some methods have been proposed to solve the Ky Fan inequality (or some time called equilibrium problem, see [1,3,8,14]).
In 2005, Combettes and Hirstoaga [2] introduced an iter-ative scheme of finding the best approximation to the initial data whenEP(F)is nonempty and they also proved a strong convergence theorem. A mapping S :C →C is said to be
nonexpansiveif
kSx−Syk ≤ kx−yk,
for all x, y ∈ C. We denote by F ix(S) the set of fixed points of, S i.e., F ix(S) = {x ∈ C : Sx = x}. If C is bounded closed convex and S is a nonexpansive mapping of C into itself, thenF ix(S)is nonempty (see [6]).
This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission.
Department of Mathematics, Faculty of Science, King Mongkut’s Uni-versity of Technology Thonburi, Bangkok 10140, Thailand.
Emails: supuk [email protected] (S. Phiangsungnoen) and [email protected] (P. Kumam)
We write xn → x (xn * x, resp.) if {xn} converges
(weakly, resp.) to x. A mapping A of C into H is called
monotoneif
hAu−Av, u−vi ≥0.
The classicalvariational inequality problemis to findu∈C such that hv−u, Aui ≥ 0 for all v ∈ C. We denoted by V I(A, C) the set of solutions of this variational inequality problem. The variational inequality has been extensively studied in the literature. See, e.g. [19,20] and the references therein. A mapping A of C into H is called α- inverse-strongly-monotone if there exists a positive real number α such that
hAu−Av, u−vi ≥αkAu−Avk2, (2)
for all u, v ∈C.It is obvious that any α -inverse-strongly-monotone mappings A is monotone and Lipschitz continu-ous.
In 1953, Mann [7] introduced the iteration as follows: a sequence{xn} defined by
xn+1=αnxn+ (1−αn)Sxn, (3)
where the initial guess elementx0∈Cis arbitrary and{αn}
is a real sequence in [0,1]. The Mann iteration has been extensively investigated for nonexpansive mappings. One of the fundamental convergence results is proved by Reich [11].
In an infinite-dimensional Hilbert space, the Mann iter-ation can conclude only weak convergence [4]. Attempts to modify the Mann iteration method (3) so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi [9] proposed the following modification of the Mann iteration method (3):
x0∈C is arbitrary,
yn=αnxn+ (1−αn)Sxn,
Cn={z∈C:kyn−zk ≤ kxn−zk},
Qn ={z∈C:hxn−z, x0−xni ≥0},
xn+1=PCn∩Qnx0, n= 0,1,2. . . ,
(4)
wherePC is metric projection on C.
For finding an element ofF ix(S)∩V I(A, C), Takahashi and Toyoda [18] introduced the following iterative scheme:
xn+1=αnxn+ (1−αn)SPC(xn−λnAxn) (5)
for everyn= 0,1,2, ..., where PC is the metric projection
on the set C, x0 = x ∈ C, {αn} is a sequence in (0,1)
and {λn} is a sequence in (0,2α). They proved a weak
On the other hand, for finding an element of F ix(S), Takahashi et al., [17] introduced the following iteration procedure which is usually called the shrinking projection method. Let C be nonempty closed convex subset of a real Hilbert space H. Let {αn} be a sequence in (0,1). Let
x0 ∈H. For C1 =C and x1 =PC1x0, define a sequence
{xn} of C as follows:
yn= (1−αn)xn+αnSnxn,
Cn+1={z∈Cn:kyn−zk ≤ kxn−zk},
xn+1=PCn+1x0, n≥1,
(6)
wherePCn is the metric projection ofH ontoCn and{Sn}
is a family of nonexpansive mappings. They proved that the sequence {xn} generated by (6) converges strongly to z =
PF ix(S)x0, whereF ix(S) =T ∞
n=1F ix(Sn).
Moreover, in 2012, Phan Tu Vuong, et. al., [26], they con-sidered the sequences{xn},{yn},{zn},and{tn}generated
byx0∈C and
yn= arg miny∈C{λnF(xn, y) +12ky−xnk2},
zn= arg miny∈C{λnF(yn, y) +12ky−xnk2},
tn=αnxn+ (1−αn)[βnzn+ (1−βn)Szn],
xn+1=tn,
(7)
for every n ∈ N, where {αn} ⊂ [0,1[, {βn} ⊂]0,1[, and
{λn} ⊂]0,1].
In this paper, we consider the sequences {xn},{yn},
{zn},{wn} and {tn} generated by x0 ∈ H, C1 = C,
x1=PC1x0 and
yn = arg miny∈C{λnF(xn, y) +12ky−xnk2},
zn= arg miny∈C{λnF(yn, y) +12ky−xnk2},
wn=PC(zn−λnAzn),
tn =αnxn+ (1−αn)[(1−µ)Swn+µPC(1−βn)wn],
Cn+1={z∈Cn:ktn−zk ≤ kxn−zk},
xn+1=PCn+1x0,
(8)
for every n∈ N, where µ be a constant in(0,1),{αn} ⊂ [0,1),{βn} ⊂ (0,1),{λn} ⊂ (0,1]. We combine the
equations (5), (6) and (7) above for solving the EP(F)
with a fixed point and variational inequality problems. Consequencely, we obtained the strong convergent theorem for solving fixed point problems, Ky Fan inequality and variational inequality problems.
II. PRELIMINARIES
Let C be a closed convex subset of a Hilbert space H. Then the following hold:y
kx−yk2=kxk2− kyk2−2hx−y, yi (9)
and
kλx+(1−λ)yk2=λkxk2+(1−λ)kyk2−λ(1−λ)kx−yk2
(10)
for all x, y ∈ H and λ ∈ [0,1]. It is also known that H satisfies theOpial’s condition [10], that is, for any sequence
{xn} withxn * x, the inequality lim inf
n→∞ kxn−xk<lim infn→∞ kxn−yk (11)
holds for everyy ∈ H withy 6=x. Also Hilbert space H satisfies the Kadec-Klee property [5, 15], that is, for any sequence{xn}withxn* x and kxnk → kxktogether
implykxn−xk →0.
Let C be a closed convex subset of H. For every point x∈H, there exists a unique nearest point in C, denoted by PCxsatisfying the property
kx−PCxk ≤ kx−yk. (12)
For a givenx∈H andz∈C,
z=PCx⇔ hx−z, z−yi ≥0, ∀y∈C. (13)
It is well known thatPC isfirmly nonexpansiveof H onto
C and satisfies
hx−y, PCx−PCyi ≥ kPCx−PCyk2. (14)
In the context of the variational inequality problem, this implies that
u∈V I(A, C)⇔u=PC(u−λAu), ∀λ >0. (15)
We also have that, for allu, v∈C andλ >0,
k(I−λA)u−(I−λA)vk2 = k(u−v)−λ(Au−Av)k2
≤ ku−vk2+λ(λ−2α)kAu−Avk2. (16)
So, if λ ≤ 2α, then I−λA is a nonexpansive mapping fromC toH.
Let us recall thatSis aξ-strict pseudocontraction mapping if there exists a scalarξ∈[0,1)such that
kSx−Syk2≤ kx−yk2+ξk(x−Sx)−(y−Sy)k2 (17)
for every x, y ∈ C. It is easy to see that a nonexpansive mapping onC is also a0-strict pseudocontraction mapping. Furthermore [24], if S is a ξ-strict pseudocontraction mapping, then the fixed point set F ix(S) is closed and convex and the mapping I−S is demiclosed at zero, i.e., satisfies the property
xn* x and Sxn−xn→0⇒Sx=x.
A set valued mappingT :H →2H is called monotone if
for allx, y∈H, f ∈T xandg∈T yimplyhx−y, f−gi ≥0. A monotone mappingT :H →2H is maximal if the graph
G(T)ofTis not properly contained in the graph of any other monotone mapping. It known that a monotone mappingT is maximal if and only if for(x, f)∈H×H,hx−y, f−hi ≥
0 for every (y, g) ∈ G(T) implies f ∈ T x. Let A be an inverse-strongly monotone mapping ofCintoHand letNCv
be the normal cone toC atv∈C, i.e.,
NCv={w∈H:hv−u, wi ≥0,∀u∈C}
and define
T v=
Av+NCv, v∈C,
∅, v /∈C.
Proposition 1. ([21], Lemma 3.1) For every x∗ ∈EP(F), and everyn∈N, one has
(i) hxn−yn, y−yni ≤λnF(xn, y)−λnF(xn, yn),
∀y∈C;
(ii) kzn−x∗k2≤ kxn−x∗k2−(1−2λnc1)kyn−xnk2− (1−2λnc2)kzn−ynk2.
Lemma 2. [28] LetC be a closed convex subset of H and let{xn}be a bounded sequence in H. Assume that
1) the weakω−limitsetωw(xn)⊂C,
2) for eachz∈C,limn→∞kxn−zk exists. Then{xn} is weakly convergent to a point in C.
Proposition 3. [24] LetKbe a nonempty closed and convex subset ofH. Letu∈H and let{xn}be a sequence inH. If any weak limit point of{xn}belongs toK, andkxn−uk ≤
ku−PKuk for all n∈N, thenxn→PKu.
In order to apply this Proposition, we setK:=EP(F)∩
F ix(S)∩V I(A, C) and u= x0. Furthermore, we impose
that the sequence{xn}generated by our algorithms satisfies,
for alln∈N,kxn−x0k ≤ kxn+1−x0k ≤ kx˜0−x0k
wherex˜0=PKx0. In that case, the sequence{kxn−x0k}
is convergent and the sequence {xn} is bounded. These
properties will be useful to prove that any weak limit point of {xn} belongs toK.
For solving the Ky Fan inequality or equilibrium problem, let us assume that the following conditions (A1) - (A5) are satisfied on the bifunction F :C×C→R.
(A1) F(x, x) = 0for every x∈C;
(A2) F is pseudomonotone on C, i.e.,
F(x, y)≥0⇒F(y, x)≤0, ∀x, y∈C;
(A3) F is jointly weakly continuous onC×C in the sense that, ifx, y∈Cand{xn}and{yn}are two sequences
inC converging weakly toxandy, respectively, then F(xn, yn)→F(x, y);
(A4) F(x,·) is convex, lower semicontinuous, and subdif-ferentiable onC foe veryx∈C;
(A5) F satisfies the Lipschitz-type condition, there exist positive integerc1 andc2, for every x, y, z∈C,
F(x, y)+F(y, z)≥F(x, z)−c1ky−xk2−c2kz−yk2.
IfF satisfies the properties (A1)-(A4), then the setEP(F)
of solutions to the Ky Fan inequality is closed and convex.
Remark 4. A first example of functionF satisfying assump-tion (A5), is given by
F(x, y) =hG(x), y−xi ∀x, y∈C,
where G : C → H is Lipschitz continuous on C (with constantL >0)(see[22]). In that example,c1=c2=L\2.
Another example, related to the Cournot-Nash equilibrium model, is described in [23]. The functionF :C×C→Ris
defined, for every x, y∈C, by
F(x, y) =hG(x) +Q(y) +q, y−xi, with C = {x ∈Rn : Ax ≤b}, F : C →
Rn, Q ∈Rn×n,
a symmetric positive semidefinite matrix, and q ∈R. If F
is Lipschitz continuous on C (with constant L > 0) then F satisfies (A5) with c1 >0, c2 >0 such that 2
√
c1c2 ≥
L+kQk.
III. MAIN RESULT
A. An Algorithm
LetC be a nonempty, closed and convex subset of a real Hilbert space H, let F be a bifunction F : C× C into R satisfying conditions (A1)−(A5), let A be an α-inverse-strongly monotone mapping ofCintoH. Let S be aξ-strict pseudocontraction mapping fromC toC.
Algorithm 5. Choose the sequences{αn} ⊂[0,1),{βn} ⊂ (0,1),{λn} ⊂(0,1]andµbe a constant in(0,1).
Assume that the following 6 Steps S(1) - S(6) S(1) Letx0∈H,C1=C,x1=PC1x0. Setn= 0.
S(2) Solve successively the strongly convex programs
arg miny∈C{λnF(xn, y) +12ky−xnk2} and arg miny∈C{λnF(yn, y) +12ky−xnk2} to obtain
the unique optimal solution yn andzn, respectively.
S(3) Computewn=PC(zn−λnAzn).
S(4) Compute
tn=αnxn+ (1−αn)[(1−µ)Swn+µPC(1−βn)wn].
Ifyn=xn andtn=xn, then STOP:
xn∈EP(F)∩F ix(S)∩V I(A, C).Otherwise, go to
Step 5.
S(5) Computexn+1=PCn+1x0,
whereCn+1={z∈Cn:ktn−zk ≤ kxn−zk},
S(6) Setn:=n+ 1, and go to Step 2.
In the sequel, we also suppose that the sequences of parameters{αn},{βn},{λn}, ξ andµsatisfy the following
conditions: (1) - (4)
(1) {λn} ⊂ [λmin, λmax], where 0 < λmin ≤ λmax < min 1
2c1, 1 2c2 ;
(2) {αn} ⊂[0, c] for somec <1;
(3) limn→∞βn= 0 andP∞n=1βn =∞;
(4) ξ andµ be constant, where0≤ξ < µ <1.
Now, let {xn},{yn},{zn}, and {wn} be the sequences
generated by combination of the hybrid extragradient method, variational inequality by projection method and the fixed point method described at the beginning of this section.
B. A strong convergence theorem
Here we start our main theorem.
Theorem 6. LetCbe a nonempty, closed and convex subset of a real Hilbert space H, let F be a bifunction F : C×
C into R satisfying conditions (A1)−(A5), let A be an
α-inverse-strongly monotone mapping of C into H. Let S be aξ-strict pseudocontraction mapping from C to C and such thatΩ :=EP(F)∩F ix(S)∩V I(A, C)6=∅. Suppose that the sequences {αn},{βn},{λn} and µ satisfying the conditions(1)−(4). Then the sequence{xn} generated by Algorithm5converges strongly to the projection ofx0on to the setΩ.
Proof. Step 1.We show that{xn}is well defined.
First, we show thatCnis closed and convex for eachn≥1.
Indeed, it is obvious that C1 = C is closed and convex.
Next, we show thatCk+1 is closed and convex for the same
k. Let z1, z2 ∈ Ck+1 and z = tz1+ (1−t)z2, where t ∈ (0,1). Notice that ktn−zk ≤ kxn−zk is equivalent to
ktn−xnk2+ 2htn−xn, xn−zi ≤0
Thus Ck+1 is closed and convex. Then, Cn is closed and
convex for anyn∈N. This implies that{xn}is well defined.
Step 2.Next, we show by induction thatΩ⊂Cnfor each
n≥1. Letx∗∈Ω. Thenx∗=PC(x∗−λnAx∗).
Since wn =PC(zn−λnAzn), we consider
kwn−x∗k2
= k[PC(zn−λnAzn)]−[PC(x∗−λnAx∗)]k2
≤ k(zn−λnAzn)−(x∗−λnAx∗)k2 = kzn−x∗k2−2λnhzn−x∗, Azn−Ax∗i
+λ2nkAzn−Ax∗k2
≤ kzn−x∗k2+λn(λn−2α)kAzn−Ax∗k2 (18)
≤ kzn−x∗k2.
By Proposition1 (ii), we have
kzn−x∗k2 ≤ kxn−x∗k2−(1−2λnc1)kyn−xnk2
−(1−2λnc2)kzn−ynk2 (19)
that is, we obtain kzn−x∗k ≤ kxn−x∗k
andkwn−x∗k ≤ kxn−x∗k.
Setun := (1−µ)Swn+µPC(1−βn)wn, for all n≥0.
Then, we havetn=αnxn+ (1−αn)un, It follows that
kun−x∗k2
= k(1−µ)Swn+µPC(1−βn)wn−x∗k2
≤ k(1−µ)(Swn−x∗) +µ
(1−βn)wn−x∗
k2
≤ kwn−x∗k2−(1−µ)(µ−ξ)kSwn−wnk2
−βnµ2kwnk2 (20)
≤ kwn−x∗k2≤ kxn−x∗k2
that is, kun−x∗k ≤ kxn−x∗k.
Since tn = αnxn + (1 −αn)un for every x∗ ∈ Ω,
we have
ktn−x∗k2 = kαnxn+ (1−αn)un−x∗k2
= kαn(xn−x∗) + (1−αn)(un−x∗)k2
≤ αnkxn−x∗k2+ (1−αn)kun−x∗k2
≤ αnkxn−x∗k2+ (1−αn)kxn−x∗k2
≤ kxn−x∗k2.
Hence ktn−x∗k ≤ kxn−x∗k. It follows thatx∗∈Cn+1.
This implies that
Ω :=EP(F)∩F ix(S)∩V I(A, C)⊂Cn, ∀n∈N.
Step 3. Next, we show that limn→∞kxn+1−xnk = 0
andlimn→∞ktn−xnk= 0.
From xn =PCnx0, we have
hx0−xn, xn−vi ≥0
for eachv∈Cn. UsingΩ⊂Cn for each we also have
hx0−xn, xn−zi ≥0, ∀z∈Ω, n∈N.
So, forz∈Ω, we have
0 ≤ hx0−xn, xn−zi
= hx0−xn, xn−x0+x0−zi
= −hx0−xn, x0−xni+hx0−xn, x0−zi
≤ −kx0−xnk2+kx0−xnkkx0−zk.
This implies that
kx0−xnk ≤ kx0−zk, ∀z∈Ω, n∈N.
From xn = PCnx0 and xn+1 = PCn+1x0 ∈ Cn+1 ⊂Cn,
we obtain
hx0−xn, xn−xn+1i ≥0. (21)
From (21), we have, forn∈N,
0 ≤ hx0−xn, xn−xn+1i
= hx0−xn, xn−x0+x0−xn+1i
= −hx0−xn, x0−xni+hx0−xn, x0−xn+1i
≤ −kx0−xnk2+kx0−xnkkx0−xn+1k.
It follows that
kx0−xnk ≤ kx0−xn+1k.
Thus the sequence{kxn−x0k}is bounded and nonincreasing
sequence, solimn→∞kxn−x0k exists, that is lim
n→∞kxn−x0k=m. (22)
Indeed, (21), we get
kxn−xn+1k2
= kxn−x0+x0−xn+1k2
= kxn−x0k2+ 2hxn−x0, x0−xn+1i +kx0−xn+1k2
= −kxn−x0k2+ 2hxn−x0, xn−xn+1i +kx0−xn+1k2
≤ −kxn−x0k2+kx0−xn+1k2.
From (22), we obtain
lim
n→∞kxn−xn+1k= 0. (23)
Sincexn+1∈Cn, we have
Cn ={z∈C:ktn−zk ≤ kxn−zk};
ktn−xn+1k ≤ kxn−xn+1k
and
kxn−tnk ≤ kxn−xn+1k+kxn+1−tnk
≤ 2kxn−xn+1k.
By (23), we obtain
lim
n→∞kxn−tnk= 0. (24)
Step 4. We will show that limn→∞kSwn −wnk = 0.
For x∗ ∈ Ω, from (18), (19) and (20), we can choose a constantM >0 such that,
sup n
We observe that
ktn−x∗k2
≤ αnkxn−x∗k2+ (1−αn)kun−x∗k2
≤ αnkxn−x∗k2+ (1−αn)kwn−x∗k2
−(1−µ)(µ−ξ)kSwn−wnk2−βnµ2kwnk2
= kxn−x∗k2−(1−αn)(1−2λnc1)kyn−xnk2
−(1−αn)(1−2λnc2)kzn−ynk2 +(1−αn)λn(λn−2α)kAz−Ax∗k2
−(1−αn)(1−µ)(µ−ξ)kSwn−wnk2
−(1−αn)βnµ2M.
Therefore, we get
(1−αn)(1−2λnc1)kyn−xnk2
≤ kxn−x∗k2− ktn−x∗k2−(1−αn)βnµ2M
=
kxn−x∗k+ktn−x∗k
kxn−tnk
−(1−αn)βnµ2M.
Since limn→∞kxn−tnk= 0,limn→∞βn= 0, 1−αn ≥1−c >0,1−2λnc1>1−2λmaxc1>0
and the sequence {xn},{tn}are bounded, we get lim
n→∞kyn−xnk= 0.
By similar way since limn→∞βn= 0,
1−αn ≥1−c >0and1−2λnc2>1−2λmaxc2>0,
we have
lim
n→∞kzn−ynk= 0.
Since limn→∞βn = 0,1−αn≥1−c >0
and−λn(λn−2α)>0,we obtain lim
n→∞kAz−Ax ∗k= 0.
By limn→∞βn= 0, 1−αn≥1−c >0
and(1−µ)(µ−ξ)>0,we have
lim
n→∞kSwn−wnk= 0.
Step 5. We will show that x˜∈Ω.
(5.1). We will show that x˜ ∈ EP(F). Since {xn} is
bounded, there exists a subsequence {xni} of {xn} which
xni *x˜andlimn→∞kxn−ynk= 0, we have thatyni*˜x.
On the other hand, by using Proposition 1(i), we have, for everyy∈C and for everyi∈N, that
hxni−yni, y−ynii ≤λniF(xni, y)−λniF(xni, yni).
Sincekxni−ynik →0 andy−yni* y−x˜ asi→ ∞and
since∀i∈N, 0< λmin≤λni ≤λmax,as i→ ∞, we get
F(˜x, y)≥0,∀y∈C. It means that x˜∈EP(F).
(5.2). We will show that x˜ ∈ F ix(S). Since {wn}
is bounded then there exists a subsequence {wni} of
{wn} which converges weakly to x. Since˜ S is a ξ-strict
pseudocontraction mapping, we know that the mapping I−S is demiclosed at zero. From kSwn−wnk → 0 as
n→ ∞ and{wni}*x. Thus, we obtain that˜ x˜∈F ix(S).
(5.3).Finally, we show thatx˜∈V I(A, C). Define T v=
Av+NCv, v∈C,
∅, v /∈C.
Then, we have thatT is maximal monotone operator. Since w ∈T v =Av+NC(v), we get w−Av ∈ NC(v).
Fromwn∈C, we have
hv−wn, w−Avi ≥0, (n= 1,2,3, ...).
We also havewn =PC(zn−λnAzn) and∀v∈C, we get
v−wn,
wn−zn
λn
+Azn
≥0. Therefore, we have
hv−wni, wi
≥ hv−wni, Avi
≥ hv−wni, Avi −
v−wni,
wni−zni
λni
+Azni
= hv−wni, Av−Awnii+hv−wni, Awni−Aznii
−
v−wni,
wni−zni
λni
.
Usingwni →x˜ andkwni−znik →0 which A is Lipshitz
continuous implies that
hv−x, w˜ i ≥0, as i→ ∞.
Since T is maximal monotone, we have x˜ ∈ T−1(0), and
hencex˜∈V I(A, C). Thus is clear thatx˜∈Ω.
Step 6. Finally, we show thatxn→x, where˜ x˜=PΩx0.
SinceΩis nonempty closed convex subset ofH, there exists a uniquex∗∈Ωsuch thatx∗=P
Ωx0.Sincex∗∈Ω⊂Cn
andxn=PCnx0, we havekxn−x0k ≤ kx˜−x0k.
It follows fromx∗=PΩx0 and the lower semicontinuity of
norm that
kx∗−x0k ≤ kx˜−x0k ≤ lim
i→∞infkxni−x0k
≤ lim
i→∞supkxni−x0k ≤ kx ∗−x
0k.
Thus, we obtain that
lim
i→∞kxni−x0k=kx˜−x0k=kx ∗−x
0k.
Using the Kadec-Klee property ofH, we obtain that
lim
i→∞xni = ˜x=x ∗.
Since {xni} is an arbitrary subsequence of {xn}, we can
conclude that{xn}converges strongly toPΩx0. C. Reduced theorem
Corollary 7. LetCbe a nonempty, closed and convex subset of a real Hilbert space H, let F be a bifunction F : C×
C into R satisfying conditions (A1)−(A5), let A be an
α-inverse-strongly monotone mapping of C into H. Let S be aξ-strict pseudocontraction mapping from C to C and such thatΩ :=EP(F)∩F ix(S)∩V I(A, C)6=∅. Suppose that the sequences {αn},{βn},{λn} and µ satisfying the condition(i)−(iv). Then the sequences{xn} generated by Algorithm5converges strongly to the projection ofx0on to the setΩ.
ACKNOWLEDGMENTS
The first author would like to thank the Faculty of Sci-ence and Department of Mathematics, KMUTT for financial support. Also, the authors were supported by the Higher Education Research Promotion and National Research Uni-versity Project of Thailand, Office of the Higher Education Commission.
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