Strong convergence theorems for solving equilibrium problems and fixed point problems of -strict...

Journal of Computational and Applied Mathematics 234 (2010) 722–732

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Strong convergence theorems for solving equilibrium problems andfixed point problems of ξ -strict pseudo-contraction mappings by twohybrid projection methodsI

Chaichana Jaiboon, Poom Kumam ∗Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand

a r t i c l e i n f o

Article history:Received 11 July 2008Received in revised form 7 August 2009

MSC:46C0547D0347H0947H1047H20

Keywords:ξ -strict pseudo-contraction mappingsFixed point problemsEquilibrium problemHybrid projection methods

a b s t r a c t

In this paper, we introduce an iterative scheme by using the hybrid projection methods forfinding a common element of the set of solutions of an equilibrium problem and the setof fixed points of a ξ -strict pseudo-contraction mapping in Hilbert spaces. We obtain twostrong convergence theorems under mild assumptions on parameters for the sequencesgenerated by these processes. The results presented in the paper extend and improvesome recent results of Marino and Xu [G. Marino, H.K. Xu, Weak and strong convergencetheorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007)336–346], Tada and Takahashi [A. Tada, W. Takahashi, Weak and strong convergencetheorems for a nonexpansive mapping and an equilibrium problem, J. Optim. Theory Appl.133 (2007) 359–370] and Ceng et al. [L.C. Ceng, S. Al-Homidan, Q.H. Ansari, J.C. Yao, Aniterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings, J. Comput. Appl. Math. 223 (2009) 967–974] and many others.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Let H be a real Hilbert space with inner product 〈·, ·〉 and norm ‖ · ‖, and let C be a nonempty closed convex subset of H .We denote the weak convergence and the strong convergence of a sequence {xn} to x in H by notations xn ⇀ x and xn → x,respectively. For a given sequence xn ⊂ H , ωw(xn) denotes the weak ω-limit set of {xn}, that is, ωw(xn) := {x ∈ H : xnj ⇀ xfor some subsequence {nj} of {n}}.Let F be a bifunction of C × C into R, where R is the set of real numbers. The equilibrium problem for F : C × C → R is

to find x ∈ C such that

F(x, y) ≥ 0, ∀y ∈ C . (1.1)

The set of solutions of (1.1) is denoted by EP(F), that is,

EP(F) = {x ∈ C : F(x, y) ≥ 0, ∀y ∈ C} .

Given a mapping S : C → H , let F(x, y) = 〈Sx, y − x〉 for all x, y ∈ C . Then, z ∈ EP(F) if and only if 〈Sz, y − z〉 ≥ 0 for ally ∈ C , that is, z is a solution of the variational inequality. The formulation (1.1) covers monotone inclusion problems, saddlepoint problems, minimization problems, optimization problems, variational inequality problems, and Nash equilibria in

I This research was partially supported by a grant from the Commission on Higher Education and the Thailand research fund under Grant MRG5180034.∗ Corresponding author. Tel.: +662 4708822; fax: +662 4284025.E-mail addresses: [email protected] (C. Jaiboon), [email protected] (P. Kumam).

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C. Jaiboon, P. Kumam / Journal of Computational and Applied Mathematics 234 (2010) 722–732 723

noncooperative games. Recently, many authors studied the problem of finding a common element of the set of fixed pointsof a nonexpansivemapping and the set of solutions of an equilibriumproblem in the framework of Hilbert spaces and Banachspaces, respectively; see, for instance, [1–12] and the references therein. In 2005, Combettes and Hirstoaga [13] introducedan iterative scheme of finding the best approximation to the initial data when EP(F) is nonempty and they also proved astrong convergence theorem.Recall that a mapping S : C → C is said to be nonexpansive if

‖Sx− Sy‖ ≤ ‖x− y‖, ∀x, y ∈ C .

In the sequel, we will use F(S) to denote the set of fixed points of S; that is, F(S) = {x ∈ C : Sx = x}. It is well known thatif C ⊂ H is nonempty, bounded, closed and convex and S is a nonexpansive self-mapping on C , then F(S) is nonempty;see, for example, [14,15]. The mapping S : C → C is called a ξ -strict pseudo-contraction mapping if there exists a constant0 ≤ ξ < 1 such that

‖Sx− Sy‖2 ≤ ‖x− y‖2 + ξ‖(I − S)x− (I − S)y‖2, ∀x, y ∈ C . (1.2)

Recall also that S : C → C is said to be a ξ -quasi-strict pseudo-contraction if the set of fixed point of S, F(S), is nonempty andif there exists a constant 0 ≤ ξ ≤ 1 such that

‖Sx− p‖2 ≤ ‖x− p‖2 + ξ‖x− Sx‖2, ∀x ∈ C and p ∈ F(S). (1.3)

We know that the class of strict pseudo-contraction mappings strictly includes the class of nonexpansive mappings. That is,S is nonexpansive if and only if S is 0-strict pseudo-contraction.In 1953, Mann [16] introduced the following iterative scheme:

xn+1 = αnxn + (1− αn)Sxn, (1.4)

where the initial guess element x0 ∈ C is arbitrary and {αn} is a real sequence in [0, 1]. The Mann’s iteration has beenextensively investigated for nonexpansive mappings. One of the fundamental convergence results is proved in [17]. In aninfinite-dimensional Hilbert space, the Mann’s iteration can conclude only weak convergence [18]. In 1967, Browder andPetryshyn [19] established the first convergence result for a ξ -strict pseudo-contraction in a real Hilbert space. They proved aweak and strong convergence theorem by using iteration (1.4) with a constant control sequence {αn} = α for all n. However,this scheme has only weak convergence even in a Hilbert space. Therefore, many authors try to modify the normal Mann’siteration process to have strong convergence; see, for example, [20–24] and the references therein.Some attempts tomodify theMann iterationmethod so that strong convergence is guaranteed have recently beenmade.

In 2003, Nakajo and Takahashi [23] proposed the following modification of the Mann iteration method (1.4) by using thehybrid projection method for a nonexpansive mapping S in a Hilbert space H:

x0 ∈ C is arbitrary,yn = αnxn + (1− αn)Sxn,Cn = {z ∈ C : ‖yn − z‖ ≤ ‖xn − z‖} ,Qn = {z ∈ C : 〈xn − z, x0 − xn〉 ≥ 0} ,xn+1 = PCn∩Qnx0,

(1.5)

for every n ∈ N ∪ {0}, where PC denotes the metric projection from H onto a closed convex subset C of H . They proved thatif the sequence {αn} bounded above from one, then {xn} defined by (1.5) converges strongly to PF(S)x0.In 2007, Takahashi et al. [25] proved the following strong convergence theorem for a nonexpansive mapping by using

the shrinking projection method in mathematical programming. For C1 = C and x1 = PC1x0, they defined a sequence asfollows:{yn = αnxn + (1− αn)Sxn,

Cn+1 = {z ∈ Cn : ‖yn − z‖ ≤ ‖xn − z‖} ,xn+1 = PCn+1x0,

(1.6)

for every n ∈ N, where 0 ≤ αn < a < 1. They proved that the sequence {xn} generated by (1.6) converges weakly toz ∈ F(S), where z = PF(S)x0.In 2007, Marino and Xu [26] proved the following strong convergence theorem by using the hybrid projection method

(CQ method) for a strict pseudo-contraction. For x0 ∈ C , they defined a sequence as follows:yn = αnxn + (1− αn)Sxn,Cn =

{z ∈ C : ‖yn − z‖2 ≤ ‖xn − z‖2 + (1− αn)(ξ − αn)‖xn − Sxn‖2

},

Qn = {z ∈ C : 〈xn − z, x0 − xn〉 ≥ 0} ,xn+1 = PCn+1x0,

(1.7)

for every n ∈ N ∪ {0}, where 0 < αn < 1. They obtained a strong convergence theorem for a ξ -strict pseudo-contractionmapping in a real Hilbert space.

724 C. Jaiboon, P. Kumam / Journal of Computational and Applied Mathematics 234 (2010) 722–732

On the other hand, for finding an element of EP(F) ∩ F(S), Tada and Takahashi [27] introduced the following iterativescheme by using the hybrid projection method for a nonexpansive mapping in a Hilbert space: x0 = x ∈ H and let

un ∈ C such that F(un, y)+1rn〈y− un, un − xn〉 ≥ 0, ∀y ∈ C,

wn = (1− αn)xn + αnSun,Cn = {z ∈ H : ‖wn − z‖ ≤ ‖xn − z‖} ,Qn = {z ∈ C : 〈xn − z, x0 − xn〉 ≥ 0} ,xn+1 = PCn∩Qnx0,

(1.8)

for every n ∈ N ∪ {0}, where {αn} ⊂ [a, b] for some a, b ∈ (0, 1) and {rn} ⊂ (0,∞) satisfies lim infn→∞ rn > 0. Then, theyproved {xn} and {un} converge strongly to z ∈ F(S) ∩ EP(F), where z = PF(S)∩EP(F)x0.In 2009, Ceng et al. [3] introduced the following iterative scheme for a ξ -strict pseudo-contraction mapping in a Hilbert

space: x1 = x ∈ H{un ∈ C such that F(un, y)+

1rn〈y− un, un − xn〉 ≥ 0, ∀y ∈ C,

yn = αnxn + (1− αn)Sun,(1.9)

for every n ∈ N, where αn ⊂ [a, b] for some a, b ∈ (ξ , 1) and {rn} ⊂ (0,∞) satisfies lim infn→∞ rn > 0. Further, theyproved {xn} and {un} generated by (1.9) converge weakly to z ∈ F(S) ∩ EP(F), where z = PF(S)∩EP(F)x.

Question 1. Can we extend the Mann’s iteration in (1.8) to a more general class of strict pseudo-contraction mappings?

Question 2. Can we extend and modify the Mann’s iterative algorithm (1.9) for finding a common element of the set ofsolutions of (1.1) and the set of fixed points to strong convergence theorems?

Motivated and inspired by the above-mentioned results, it is the purpose of this paper to introduce the Mann iterationprocess for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of aξ -strict pseudo-contractionmapping in Hilbert spaces by using the hybrid projectionmethods. Furthermore, we obtain twonew strong convergence theorems under mild assumptions on parameters. The results obtained in this paper extend andimprove several recent results in this area.

2. Preliminaries

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H . For every point x ∈ H , there exists aunique nearest point in C , denoted by PCx such that

‖x− PCx‖ ≤ ‖x− y‖, ∀y ∈ C .PC is called themetric projection of H onto C . It is well known that PC is a nonexpansive mapping of H onto C and satisfies

〈x− y, PCx− PCy〉 ≥ ‖PCx− PCy‖2,for every x, y ∈ H . Moreover, PCx is characterized by the following properties: PCx ∈ C and

〈x− PCx, y− PCx〉 ≤ 0,‖x− y‖2 ≥ ‖x− PCx‖2 + ‖y− PCx‖2,

for all x ∈ H, y ∈ C .

Lemma 2.1 ([26]). Let C be a nonempty closed convex subset of a real Hilbert space H. Given x ∈ H and z ∈ C. Then,

z = PCx⇐⇒ 〈x− z, z − y〉 ≥ 0, ∀y ∈ C .

Lemma 2.2 ([26]). Let H be a real Hilbert space. There hold the following identities (which will be used in the various places inthe proof of the result)(1) ‖x− y‖2 = ‖x‖2 − ‖y‖2 − 2〈x− y, y〉, ∀x, y ∈ H;(2) ‖tx+ (1− t)y‖2 = t‖x‖2 + (1− t)‖y‖2 − t(1− t)‖x− y‖2, ∀t ∈ [0, 1], ∀x, y ∈ H;(3) If {xn} is a sequence in H weakly convergent to z, then

lim supn→∞

‖xn − y‖2 = lim supn→∞

‖xn − z‖2 + ‖z − y‖2, ∀y ∈ H.

Lemma 2.3 ([22]). Let H be a real Hilbert space. Given a closed convex subset C ⊂ H and point x, y, z ∈ H. Given also a realnumber a ∈ R, the set{

v ∈ C : ‖y− v‖2 ≤ ‖x− v‖2 + 〈z, v〉 + a},

is convex and closed.


Lemma 2.4 ([28]). Let C be a closed convex subset of H. Let {xn} be a bounded sequence in H. Assume that

(1) the weak ω-limit set ωw(xn) ⊂ C,(2) for each z ∈ C, limn→∞ ‖xn − z‖ exists.

Then {xn} is weakly convergent to a point in C.

Lemma 2.5 ([22]). Let C be a closed convex subset of H. Let {xn} be a sequence in H and u ∈ H. Let q = PCu. If {xn} isωw(xn) ⊂ Cand satisfies the condition ‖xn − u‖ ≤ ‖u− q‖ for all n. Then xn → q.

Lemma 2.6 ([26]). Assume that C is a closed convex subset of Hilbert space H, and let S : C → C be a self-mapping of C.

(i) If S is a ξ -strict pseudo-contraction, then S satisfies the Lipschitz condition

‖Sx− Sy‖ ≤1+ ξ1− ξ

‖x− y‖ ∀x, y ∈ C .

(ii) If S is a ξ -strict pseudo-contraction, then the mapping I − S is demiclosed (at 0). That is, if {xn} is a sequence in C such thatxn ⇀ z and (I − S)xn → 0, then (I − S)z = 0.

(iii) If S is a ξ -quasi-strict pseudo-contraction, then the fixed point set F(S) of S is closed and convex so that the projection PF(S)is well defined.

For solving the equilibriumproblem for a bifunction F : C×C → R, let us assume that F satisfies the following conditions:

(A1) F(x, x) = 0 for all x ∈ C;(A2) F is monotone, i.e., F(x, y)+ F(y, x) ≤ 0 for all x, y ∈ C;(A3) for each x, y, z ∈ C , limt↓0 F(tz + (1− t)x, y) ≤ F(x, y);(A4) for each x ∈ C, y 7→ F(x, y) is convex and lower semicontinuous.

The following lemma appears implicitly in [1].

Lemma 2.7 ([1]). Let C be a nonempty closed convex subset of H and let F be a bifunction of C × C into R satisfying (A1)–(A4).Let r > 0 and x ∈ H. Then, there exists z ∈ C such that

F(z, y)+1r〈y− z, z − x〉 ≥ 0, ∀y ∈ C .

The following lemma was also given in [13].

Lemma 2.8 ([13]). Assume that F : C × C → R satisfies (A1)–(A4). For r > 0 and x ∈ H, define a mapping Tr : H → C asfollows:

Tr(x) ={z ∈ C : F(z, y)+

1r〈y− z, z − x〉 ≥ 0, ∀y ∈ C

}for all z ∈ H. Then, the following hold:

(1) Tr is single-valued;(2) Tr is firmly nonexpansive; i.e., for any x, y ∈ H,

‖Trx− Try‖2 ≤ 〈Trx− Try, x− y〉;

(3) F(Tr) = EP(F);(4) EP(F) is closed and convex.

3. Strong convergence theorems

3.1. The shrinking projection method

In this section, wewill use the shrinking projectionmethod to prove a strong convergence theorem for finding a commonelement of the set of solutions of the equilibrium problem (1.1) and the set of fixed points of a ξ -strict pseudo-contractionmapping in Hilbert spaces.

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C × C into Rsatisfying (A1)–(A4) and let S be a ξ -strict pseudo-contraction mapping for some 0 ≤ ξ < 1 such that F(S) ∩ EP(F) 6= ∅. Let


{xn} and {un} be sequences generated by x0 ∈ H, C1 = C, x1 = PC1x0 and letun ∈ C such that F(un, y)+

1rn〈y− un, un − xn〉 ≥ 0, ∀y ∈ C,

yn = αnun + (1− αn)Sun,Cn+1 =

{z ∈ Cn : ‖yn − z‖2 ≤ ‖xn − z‖2

},

xn+1 = PCn+1x0,

(3.1)

for every n ∈ N, where {αn} ⊂ [α, β] for some α, β ∈ [ ξ, 1), and {rn} ⊂ (0,∞) satisfies lim infn→∞ rn > 0. Then, the sequence{xn} converges strongly to PF(S)∩EP(F)x0.

Proof. For any p ∈ F(S) ∩ EP(F). From the definition of Tr , we note that un = Trnxn. It follows that

‖un − p‖ = ‖Trnxn − Trnp‖ ≤ ‖xn − p‖.

Next, we divide the proof of Theorem 3.1 into four steps.Step 1. We show that F(S) ∩ EP(F) ⊂ Cn for every n ∈ N.This can be proved by induction on n ∈ N. In fact, for n = 1, we have C = C1. It is obvious that F(S) ∩ EP(F) ⊂ C = C1.

Suppose that F(S) ∩ EP(F) ⊂ Ck for some k ∈ N. Hence, for p ∈ F(S) ∩ EP(F) ⊂ Ck and Lemma 2.2, we have

‖yk − p‖2 = ‖αk(uk − p)+ (1− αk)(Suk − p)‖2

= αk‖uk − p‖2 + (1− αk)‖Suk − p‖2 − αk(1− αk)‖uk − Suk‖2

≤ αk‖uk − p‖2 + (1− αk){‖uk − p‖2 + ξ‖uk − Suk‖2

}− αk(1− αk)‖uk − Suk‖2

= ‖uk − p‖2 + (1− αk)(ξ − αk)‖uk − Suk‖2

= ‖uk − p‖2 − (1− αk)(αk − ξ)‖uk − Suk‖2 (3.2)≤ ‖uk − p‖2 ≤ ‖xk − p‖2

and hence p ∈ Ck+1 as required. This implies that

F(S) ∩ EP(F) ⊂ Cn, for each n ∈ N.

Step 2. We show that Cn is closed and convex for every n ∈ N.By the assumptions, we see that C1 = C is closed and convex. Assume that Ck is closed and convex for some k ∈ N. Next,

we show that Ck+1 is closed and convex. For any p ∈ Ck, we obtain

‖yk − p‖2 ≤ ‖xk − p‖2

is equivalent to

‖yk − xk‖2 + 2〈yk − xk, xk − p〉 ≤ 0. (3.3)

Thus Ck+1 is closed and convex. Then, Cn is closed and convex for every n ∈ N. By Lemma 2.7, this implies that {xn} iswell-defined.Step 3. We show that limn→∞ ‖xn+1 − xn‖ = 0 and limn→∞ ‖un − Sun‖ = 0.Indeed, from xn = PCnx0, one has

〈x0 − xn, xn − y〉 ≥ 0

for each y ∈ Cn. Since F(S) ∩ EP(F) ⊂ Cn, we also have

〈x0 − xn, xn − p〉 ≥ 0, ∀p ∈ F(S) ∩ EP(F) and n ∈ N.

For any p ∈ F(S) ∩ EP(F), we obtain

0 ≤ 〈x0 − xn, xn − p〉= 〈x0 − xn, xn − x0 + x0 − p〉= −〈x0 − xn, x0 − xn〉 + 〈x0 − xn, x0 − p〉≤ −‖x0 − xn‖2 + ‖x0 − xn‖‖x0 − p‖,

which implies that

‖x0 − xn‖ ≤ ‖x0 − p‖, ∀p ∈ F(S) ∩ EP(F) and n ∈ N. (3.4)

From xn = PCnx0, and xn+1 = PCn+1x0 ∈ Cn+1 ⊂ Cn, we obtain

〈x0 − xn, xn − xn+1〉 ≥ 0. (3.5)


From (3.5), we have

0 ≤ 〈x0 − xn, xn − xn+1〉= 〈x0 − xn, xn − x0 + x0 − xn+1〉= −〈x0 − xn, x0 − xn〉 + 〈x0 − xn, x0 − xn+1〉≤ −‖x0 − xn‖2 + ‖x0 − xn‖‖x0 − xn+1‖.

It follows that

‖x0 − xn‖ ≤ ‖x0 − xn+1‖.

Thus the sequence {‖xn − x0‖} is a bounded and nonincreasing sequence, so limn→∞ ‖xn − x0‖ exists, that is, there exists aconstantm ∈ R such that

m = limn→∞‖xn − x0‖. (3.6)

From (3.5), we get

‖xn − xn+1‖2 = ‖xn − x0 + x0 − xn+1‖2

= ‖xn − x0‖2 + 2〈xn − x0, x0 − xn+1〉 + ‖x0 − xn+1‖2

= ‖xn − x0‖2 + 2〈xn − x0, x0 − xn + xn − xn+1〉 + ‖x0 − xn+1‖2

= ‖xn − x0‖2 − 2〈xn − x0, xn − x0〉 + 2〈xn − x0, xn − xn+1〉 + ‖x0 − xn+1‖2

= −‖xn − x0‖2 + 2〈xn − x0, xn − xn+1〉 + ‖x0 − xn+1‖2

≤ −‖xn − x0‖2 + ‖x0 − xn+1‖2.

By (3.6), we obtain

limn→∞‖xn − xn+1‖ = 0, (3.7)

On the other hand, xn+1 = PCn+1 ∈ Cn+1 ⊂ Cn, which implies that

‖xn+1 − yn‖ ≤ ‖xn+1 − xn‖.

Furthermore, we also obtain

‖yn − xn‖ ≤ ‖yn − xn+1‖ + ‖xn+1 − xn‖≤ ‖xn+1 − xn‖ + ‖xn+1 − xn‖= 2‖xn+1 − xn‖.

In view of (3.7), we obtain

limn→∞‖yn − xn‖ = 0. (3.8)

For p ∈ F(S) ∩ EP(F), note that Tr is firmly nonexpansive (Lemma 2.8), then we see that

‖un − p‖2 = ‖Trnxn − Trnp‖2

≤ 〈Trnxn − Trnp, xn − p〉= 〈un − p, xn − p〉

=12

(‖un − p‖2 + ‖xn − p‖2 − ‖xn − un‖2

),

and hence

‖un − p‖2 ≤ ‖xn − p‖2 − ‖xn − un‖2.

In view of (3.2), we obtain

‖yn − p‖2 ≤ ‖un − p‖2

≤ ‖xn − p‖2 − ‖xn − un‖2.

It follows that

‖xn − un‖2 ≤ ‖xn − p‖2 − ‖yn − p‖2

= (‖xn − p‖ − ‖yn − p‖) (‖xn − p‖ + ‖yn − p‖)≤ ‖xn − yn‖ (‖xn − p‖ + ‖yn − p‖) .


From (3.8), we arrive at

limn→∞‖xn − un‖ = 0. (3.9)

Since lim infn→∞ rn > 0, we obtain

limn→∞

∥∥∥∥xn − unrn

∥∥∥∥ = limn→∞ 1rn ‖xn − un‖ = 0. (3.10)

Using (3.2) again, we have

‖yn − p‖2 ≤ ‖un − p‖2 − (1− αn)(αn − ξ)‖un − Sun‖2

≤ ‖xn − p‖2 − (1− αn)(αn − ξ)‖un − Sun‖2.

It follows that

(1− β)(α − ξ)‖un − Sun‖2 ≤ (1− αn)(αn − ξ)‖un − Sun‖2

≤ ‖xn − p‖2 − ‖yn − p‖2

≤ ‖xn − yn‖ (‖xn − p‖ + ‖yn − p‖) .

From the assumption ξ ≤ α ≤ αn ≤ β < 1 and (3.8), we obtain that

limn→∞‖un − Sun‖ = 0. (3.11)

Step 4. We show that ωw(xn) ⊂ F(S) ∩ EP(F) and xn → p = PF(S)∩EP(F)x0 as n→∞.Since {xn} is bounded andH is reflexive,ωw(xn) is nonempty. Let z ∈ ωw(xn) be an arbitrary element. Then, there exists a

subsequence {xni} of {xn}which converges weakly to z. From ‖xn−un‖ → 0, we obtain also that uni ⇀ z. Since uni ⊂ C andC is closed and convex, we obtain z ∈ C . From (3.11) we obtain Suni ⇀ z. First, we show that z ∈ EP(F). Since un = Trnxn,we have

F(un, y)+1rn〈y− un, un − xn〉 ≥ 0, ∀y ∈ C .

It follows from the condition (A2), and we see that

1rn〈y− un, un − xn〉 ≥ −F(un, y) ≥ F(y, un),

and hence⟨y− uni ,

uni − xnirni

⟩≥ F(y, uni).

Sinceuni−xnirni→ 0 and uni ⇀ z, it follows by (A4) that F(y, z) ≤ 0 for all y ∈ C . For t with 0 < t ≤ 1 and y ∈ C , let

yt = ty+ (1− t)z. Since y ∈ C and z ∈ C , we have yt ∈ C and hence F(yt , z) ≤ 0. So, from (A1) and (A4) we have

0 = F(yt , yt) ≤ tF(yt , y)+ (1− t)F(yt , z) ≤ tF(yt , y)

and hence F(yt , y) ≥ 0. From (A3), we have F(z, y) ≥ 0 for all y ∈ C and hence z ∈ EP(F).Next, we show that z ∈ F(S). Assume z 6∈ F(S), since S is a ξ -strict pseudo-contraction mapping, by Lemma 2.6(2) we

know that the mapping I − S is demiclosed at zero. From ‖Sun − un‖ → 0 and uni ⇀ z. Thus, we obtain z ∈ F(S). Theconclusion z ∈ F(S) ∩ EP(F) is proved. Since z was an arbitrary element, we conclude that ωw(xn) ⊂ F(S) ∩ EP(F). Byinequality (3.4) and Lemma 2.5 ensure that the convergence of {xn} to p = PF(S)∩EP(F)x0. This completes the proof. �

Corollary 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H and let F be a bifunction from C × C into Rsatisfying (A1)–(A4) such that EP(F) 6= ∅. Let {xn} and {un} be sequences generated by x0 ∈ H, C1 = C, x1 = PC1x0 and let


Cn+1 ={z ∈ Cn : ‖un − z‖2 ≤ ‖xn − z‖2

},

xn+1 = PCn+1x0,

for every n ∈ N, where {rn} ⊂ (0,∞) satisfies lim infn→∞ rn > 0. Then, the sequence {xn} converges strongly to PEP(F)x0.

Proof. Putting S = I and αn = 0. Then, we get yn = un in Theorem 3.1, we obtain Corollary 3.2. �


Corollary 3.3. Let C be a nonempty closed convex subset of a real Hilbert space H and let S be a ξ -strict pseudo-contractionmapping for some 0 ≤ ξ < 1 such that F(S) 6= ∅. Let {xn} and {un} be sequences generated by x0 ∈ H, C1 = C, x1 = PC1x0 andlet

un ∈ C such that 〈y− un, un − xn〉 ≥ 0, ∀y ∈ C,yn = αnun + (1− αn)Sun,Cn+1 =

{z ∈ Cn : ‖yn − z‖2 ≤ ‖xn − z‖2

},

xn+1 = PCn+1x0,

for every n ∈ N, where {αn} ⊂ [α, β] for some α, β ∈ [ξ, 1). Then, the sequence {xn} converges strongly to PF(S)x0.

Proof. Putting F(x, y) = 0 for all x, y ∈ C and rn = 1 in Theorem 3.1, the conclusion follows. �

Corollary 3.4. Let C be a nonempty closed convex subset of a real Hilbert space H and let S be a ξ -strict pseudo-contractionmapping for some 0 ≤ ξ < 1 such that F(S) 6= ∅. Let {xn} be a sequence generated by x0 ∈ H, C1 = C ∈ H, x1 = PC1x0 and letyn = αnPCxn + (1− αn)SPCxn,Cn+1 =

{z ∈ Cn : ‖yn − z‖2 ≤ ‖xn − z‖2

},

xn+1 = PCn+1x0,

for every n ∈ N, where {αn} ⊂ [α, β] for some α, β ∈ [ξ, 1). Then, the sequence {xn} converges strongly to PF(S)x0.

Proof. Putting F(x, y) = 0 for all x, y ∈ C and rn = 1 in Theorem 3.1, by Lemma 2.1 we have un = PCxn. From Theorem 3.1,the sequence {xn} converges strongly to PF(S)x0. �

Corollary 3.5. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C × C into Rsatisfying (A1)–(A4) and let S be a nonexpansive mapping such that F(S)∩EP(F) 6= ∅. Let {xn} and {un} be sequences generatedby x0 ∈ H, C1 = C, x1 = PC1x0 and let


yn = αnun + (1− αn)Sun,Cn+1 =

{z ∈ Cn : ‖yn − z‖2 ≤ ‖xn − z‖2

},

xn+1 = PCn+1x0,

for every n ∈ N, where {αn} ⊂ [α, β] for some α, β ∈ [0, 1), and {rn} ⊂ (0,∞) satisfies lim infn→∞ rn > 0. Then, the sequence{xn} converges strongly to PF(S)∩EP(F)x0.

Proof. Since the class of ξ -strict pseudo-contraction mappings strictly includes the class of nonexpansive mappings withξ = 0, the conclusion follows immediately from Theorem 3.1. This completes the proof. �

3.2. The hybrid projection method

In this section, we derive a strong convergence theorem by using the hybrid projection method (some authors call thisthe CQmethod) which solves a problem of finding a common element of the set of solutions of the equilibrium problem andthe set of fixed points of a ξ -strict pseudo-contraction mapping in a real Hilbert space.

Theorem 3.6. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C × C into Rsatisfying (A1)–(A4) and let S be a ξ -strict pseudo-contraction mapping for some 0 ≤ ξ < 1 such that F(S) ∩ EP(F) 6= ∅. Let{xn} and {un} be sequences generated by x0 = x ∈ H and let


yn = αnun + (1− αn)Sun,Cn =

{z ∈ C : ‖yn − z‖2 ≤ ‖xn − z‖2

},

Qn = {z ∈ C : 〈xn − z, x0 − xn〉 ≥ 0} ,xn+1 = PCn∩Qnx0,

(3.12)

for every n ∈ N ∪ {0}, where {αn} ⊂ [α, β] for some α, β ∈ [ξ, 1) and {rn} ⊂ (0,∞) satisfies lim infn→∞ rn > 0. Then, thesequence {xn} converges strongly to PF(S)∩EP(F)x0.

Proof. From the definition of Cn and Lemma 2.3, we obtain Cn is closed and convex. It is obvious that Qn is closed and convexfor every n ∈ N ∪ {0}. So, Cn ∩ Qn is a closed convex subset of H for every n ∈ N ∪ {0}. Let p ∈ F(S) ∩ EP(F) and let {Trn} bea sequence of mappings defined as in Lemma 2.8. Then un = Trnxn, we obtain

‖un − p‖ = ‖Trnxn − Trnp‖ ≤ ‖xn − p‖, (3.13)


for every n ∈ N. For each p ∈ F(S) ∩ EP(F), by (1.3), (3.13) and Lemma 2.2(2), we have

‖yn − p‖2 = ‖αn(un − p)+ (1− αn)(Sun − p)‖2

= αn‖un − p‖2 + (1− αn)‖Sun − p‖2 − αn(1− αn)‖un − Sun‖2

≤ αn‖un − p‖2 + (1− αn){‖un − p‖2 + ξ‖un − Sun‖2

}− αn(1− αn)‖un − Sun‖2

= ‖un − p‖2 + (1− αn)(ξ − αn)‖un − Sun‖2

= ‖un − p‖2 − (1− αn)(αn − ξ)‖un − Sun‖2

≤ ‖un − p‖2 ≤ ‖xn − p‖2,

for every n ∈ N. This implies that

‖yn − p‖ ≤ ‖un − p‖ ≤ ‖xn − p‖. (3.14)

So, we have p ∈ Cn, and hence

F(S) ∩ EP(F) ⊂ Cn, for all n ∈ N. (3.15)

Next, we show that F(S) ∩ EP(F) ⊂ Cn ∩ Qn for all n ∈ N. We prove this by mathematical induction. For n = 1, we haveF(S) ∩ EP(F) ⊂ C1 and Q1 = H , we get F(S) ∩ EP(F) ⊂ C1 ∩ Q1. Now we assume that F(S) ∩ EP(F) ⊂ Ck ∩ Qk for somek ∈ N. With it we will show that F(S) ∩ EP(F) ⊂ Ck+1 ∩ Qk+1.In fact, if F(S) ∩ EP(F) ⊂ Ck ∩ Qk for some k ∈ N. Then, there exists a unique element xk+1 ∈ Ck ∩ Qk such that

xk+1 = PCk∩Qkx0. Then, by Lemma 2.1 and the convexity of Ck ∩ Qk, we obtain

〈xk+1 − z, x0 − xk+1〉 ≥ 0, for each z ∈ Ck ∩ Qk

and hence z ∈ Qk+1. By definition of Qk+1 implies that F(S) ∩ EP(F) ⊂ Qk+1. This together with (3.15) gives

F(S) ∩ EP(F) ⊂ Ck+1 ∩ Qk+1, for all k ∈ N.

This implies that {xn} is well defined. From Lemma 2.7, the sequence {un} is also well defined.Now, we prove that ‖xn+1 − xn‖ → 0 as n→∞. Since F(S) ∩ EP(F) is a nonempty closed and convex subset of H , then

there exists a unique z∗ ∈ F(S) ∩ EP(F) such that

z∗ = PF(S)∩EP(F)x0.

From xn+1 = PCn∩Qnx0, we have

‖xn+1 − x0‖ ≤ ‖p− x0‖, for all p ∈ F(S) ∩ EP(F) and n ∈ N.

Since z∗ ∈ F(S) ∩ EP(F) ⊂ Cn ∩ Qn, we obtain

‖xn+1 − x0‖ ≤ ‖z∗ − x0‖, for all n ∈ N. (3.16)

Therefore, {xn} is bounded. From (3.14), we know that {un} and {yn} are also bounded. Since xn = PQnx0. Then by xn+1 =PCn∩Qnx0 we know that xn+1 ∈ Qn, and ‖xn− x0‖ ≤ ‖xn+1− x0‖ for all n ∈ N. Since {xn} is bounded, the sequence {‖xn− x0‖}is a bounded and nonincreasing. So limn→∞ ‖xn− x0‖ exists. In addition, by Lemma 2.1, xn = PQnx0 and the convexity of Qn,we get

〈xn+1 − xn, xn − x0〉 ≥ 0 for every n ∈ N. (3.17)

From (3.17) and Lemma 2.2(1), we have

0 ≤ ‖xn+1 − xn‖2 = ‖(xn+1 − x0)− (xn − x0)‖2

= ‖xn+1 − x0‖2 − ‖xn − x0‖2 − 2〈xn+1 − xn, xn − x0〉≤ ‖xn+1 − x0‖2 − ‖xn − x0‖2.

It follows from the previous inequality and limn→∞ ‖xn − x0‖ exists, we have

limn→∞‖xn+1 − xn‖ = 0. (3.18)

From Step 3 in the proof of Theorem 3.1, we obtain

limn→∞‖yn − xn‖ = 0 (3.19)

and

limn→∞‖un − Sun‖ = 0. (3.20)

As the proof that follows Theorem 3.1 and (3.16) to (3.20), we obtain that the sequence {xn} converges strongly to p =PF(S)∩EP(F)x0. This completes the proof of Theorem 3.6. �


Corollary 3.7. Let C be a nonempty closed convex subset of a real Hilbert space H and let F be a bifunction from C × C into Rsatisfying (A1)–(A4) such that EP(F) 6= ∅. Let {xn} and {un} be sequences generated by x0 = x ∈ H and let


Cn ={z ∈ C : ‖yn − z‖2 ≤ ‖xn − z‖2

},


for every n ∈ N ∪ {0}, where {rn} ⊂ (0,∞) satisfies lim infn→∞ rn > 0. Then, the sequence {xn} converges strongly to PEP(F)x0.

Proof. Putting S = I and αn = 0 in Theorem 3.6, the conclusion follows. �

Corollary 3.8. Let C be a nonempty closed convex subset of a real Hilbert space H and let S be a ξ -strict pseudo-contractionmapping for some 0 ≤ ξ < 1 such that F(S) 6= ∅. Let {xn} and {un} be sequences generated by x0 = x ∈ H and let

un ∈ C such that 〈y− un, un − xn〉 ≥ 0, ∀y ∈ C,yn = αnun + (1− αn)Sun,Cn =

{z ∈ C : ‖yn − z‖2 ≤ ‖xn − z‖2

},


for every n ∈ N ∪ {0}, where {αn} ⊂ [α, β] for some α, β ∈ [ξ, 1). Then, the sequence {xn} converges strongly to PF(S)x0.

Proof. Putting F(x, y) = 0 for all x, y ∈ C and rn = 1 in Theorem 3.6. Hence, the conclusion follows. �

Corollary 3.9 (Tada and Takahashi [27]). Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunctionfrom C × C into R satisfying (A1)–(A4) and let S be a nonexpansive mapping such that F(S) ∩ EP(F) 6= ∅. Let {xn} and {un} besequences generated by x0 = x ∈ H and let


yn = αnun + (1− αn)Sun,Cn =

{z ∈ C : ‖yn − z‖2 ≤ ‖xn − z‖2

},


for all n ∈ N ∪ {0}, where {αn} ⊂ [α, β] for some α, β ∈ [0, 1) and {rn} ⊂ (0,∞) satisfies lim infn→∞ rn > 0. Then, thesequence {xn} converges strongly to PF(S)∩EP(F)x0.

Proof. Since S is nonexpansive if and only if S is a 0-strict pseudo-contractionmapping, the conclusion follows immediatelyfrom Theorem 3.6. This completes the proof. �

Acknowledgements

The first authorwas supported by the Faculty of Applied Liberal Arts RMUTR Research Fund and KingMongkut’s Diamondscholarship for fostering special academic skills by KMUTT. Moreover, the authors would like to thank the referees forreading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the originalversion of this paper.

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