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Kyungpook Mathematical Journal 2019; 59(1): 83-99

Published online March 31, 2019

Weak and Strong Convergence of Hybrid Subgradient Method for Pseudomonotone Equilibrium Problems and Nonspreading-Type Mappings in Hilbert Spaces

Department of Mathematics and Computer Scicence, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi, Pathum Thani 12110, Thailand

Received: October 16, 2016; Revised: January 22, 2019; Accepted: January 28, 2019

In this paper, we introduce a hybrid subgradient method for finding an element common to both the solution set of a class of pseudomonotone equilibrium problems, and the set of fixed points of a finite family of κ-strictly presudononspreading mappings in a real Hilbert space. We establish some weak and strong convergence theorems of the sequences generated by our iterative method under some suitable conditions. These convergence theorems are investigated without the Lipschitz condition for bifunctions. Our results complement many known recent results in the literature.

Let H be a real Hilbert space in which the inner product and norm are denoted by 〈·, ·〉 and || · ||, respectively. Let C be a nonempty closed convex subset of H. Let T : CC be a mapping. A point xC is called a fixed point of T if Tx = x and we denote the set of fixed points of T by F(T). Recall that a mapping T : CC is said to be nonexpansive if

Tx-Tyx-y,for all x,yC,

and it is said to be quasi-nonexpansive if F(T) ≠ ∅︀ and

Tx-Tyx-y,for all xC,and yF(T).

A mapping T : CC is said to be a strict pseudocontraction if there exists a constant k ∈ [0, 1) such that

Tx-Ty2x-y2+k(I-T)x-(I-T)y2,x,yC,

where I is the identity mapping on H. If k = 0, then T is nonexpansive on C.

In 2008, Kohsaka and Takahashi [15] defined a mapping T in a in Hilbert spaces H to be nonspreading if

2Tx-Ty2Tx-y2+Ty-x2,for all x,yC.

Following the terminology of Browder-Petryshyn [10], Osilike and Isiogugu [17] called a mapping T of C into itself κ-strictly pseudononspreading if there exists κ ∈ [0, 1) such that

Tx-Ty2x-y2+2x-Tx,y-Ty+κx-Tx-(y-Ty)2,for all x,yC.

Clearly, every nonspreading mapping is κ-strictly pseudononspreading but the converse is not true; see [17]. We note that the class of strict pseudocontraction mappings and the class of κ-strictly pseudononspreading mappings are independent.

In 2010, Kurokawa and Takahashi [16] obtained a weak mean ergodic theorem of Baillon’s type [7] for nonspreading mappings in Hilbert spaces. Furthermore, using the idea of mean convergence in Hilbert spaces, they also proved a strong convergence theorem of Halpern’s type [12] for this class of mappings. After that, in 2011, Osilike and Isiogugu [17] introduced the concept of κ-strictly pseudononspreading mappings and they proved a weak mean convergence theorem of Baillon’s type similar to [16]. They further proved a strong convergence theorem using the idea of mean convergence. This theorem extended and improved the main theorems of [16] and gave an affirmative answer to an open problem posed by Kurokawa and Takahashi [16] for the case when the mapping T is averaged. In 2013 Kangtunyakarn [14] proposed a new technique, using the projection method, for κ-strictly pseudononspreading mappings. He obtained a strong convergence theorem for finding the common element of the set of solutions of a variational inequality, and the set of fixed points of κ-strictly pseudononspreading mappings in a real Hilbert space.

On the other hand, 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 xC such that

F(x,y)0for all yC.

The set of solutions of (1.1) is denoted by EP(F, C). It is well known that there are several problems, such as complementarity problems, minimax problems, the Nash equilibrium problem in noncooperative games, fixed point problems, optimization problems, that can be written in the form of an EP. In other words, the EP is a unifying model for several problems arising in physics, engineering, science, optimization, economics, etc.; see [6, 8, 11] and the references therein.

In recent years the problem of finding an element common to the set of solutions of a equilibrium problems, and the set of fixed points of nonlinear mappings, has become a fascinating subject, and various methods have been developed by many authors for solving this problem (see [1, 4, 5, 20]). Most of all the existing algorithms for this problem are based on applying the proximal point method to the equilibrium problem EP(F, C), and using a Mann’s iteration to the fixed point problems of nonexpansive mappings. The convergence analysis has been considered when the bifunction F is monotone. This is because the proximal point method is not valid when the underlying operator F is pseudomonotone.

Recently, Anh [2] introduced a new hybrid extragradient iteration method for finding a element common to the set of fixed points of a nonexpansive mapping and the set of solutions of equilibrium problems for a pseudomonotone bifunctions. In this algorithm the equilibrium bifunction is not required to satisfy any monotonicity property, but it must satisfy a Lipschitz-type continuous bifunction i.e. there are two Lipschitz constants c1 > 0 and c2 > 0 such that

f(x,y)+f(y,z)f(x,z)-c1x-y2-c2y-z2,x,y,zC.

They obtained strongly convergent theorems for the sequences generated by these processes in a real Hilbert space.

Anh and Muu [3] reiterated that the Lipschitz-type condition (1.2) is not in general satisfied, and if it is, that finding the constants c1 and c2 is not easy. They further observed that solving strongly convex programs is also difficult except in special cases when C has a simple structure. They introduced and studied a new algorithm, which is called a hybrid subgradient algorithm for finding a common point in the set of fixed points of nonexpansive mappings and the solution set of a class of pseudomonotone equilibrium problems in a real Hilbert space. The proposed algorithm is a combination of the well-known Mann’s iterative scheme for fixed point and the projection method for equilibrium problems. Furthermore, the proposed algorithm uses only one projection and does not require any Lipschitz condition for the bifunctions. To be more precise, they proposed the following iterative method:

{x0C,wnɛnF(xn,·)xn,un=PC(xn-γnwn),γn=βnmax{σn,wn},xn+1=αnxn+(1-αn)Tun,for each n=1,2,3,,

where εF(x, ·)(x) stands for ε-subdifferential of the convex function F(x, ·) at x and {εn}, {γn}, {βn}, {σn}, and {αn} were chosen appropriately. Under certain conditions, they prove that {xn} converges strongly to a common point in the set of a class of pseudomonotone equilibrium problems and the set of fixed points of nonexpansive mapping. Using the idea of Anh and Muu [3], Thailert et al. [21] proposed a new algorithm for finding a common point in the solution set of a class of pseudomonotone equilibrium problems and the set of common fixed points of a family of strict pseudocontraction mappings in a real Hilbert space. Then Thailert et al. [22] introduced new general iterative methods for finding a common element in the solution set of pseudomonotone equilibrium problems and the set of fixed points of nonexpansive mappings which is a solution of a certain optimization problem related to a strongly positive linear operator. Under suitable control conditions, They proved the strong convergence theorems of such iterative schemes in a real Hilbert space.

In this paper, motivated by Anh and Muu [3], Kangtunyakarn [14], and other research going on in this direction, we proposed a hybrid subgradient method for the pseudomonotone equilibrium problem and the finite family of κ-strictly pseudononspreading mapping in a real Hilbert space. The weak and strong convergence of the proposed methods is investigated under certain assumptions. Our results improve and extend many recent results in the literature.

Let H be a real Hilbert space with inner product 〈·, ·〉 and norm || · ||, respectively. It is well-known that for all x, y, zH and α, β, γ ∈ [0, 1], with α+β+γ = 1 there holds

x-y2=x2-y2-2x-y,y,

and

αx+βy+γz2=αx2+βy2+γz2-αβx-y2-βγy-z2.

Let C be a nonempty closed convex subset of H. Then, for any xH, there exists a unique nearest point of C, denoted by PCx, such that || xPCx ||≤|| xy || for all yC. Such a PC is called the metric projection from H into C. We know that PC is nonexpansive. It is also known that, PCxC and

x-PCx,PCx-z0,         for all xHand zC.

It is easy to see that (2.3) equivalent to

x-z2x-PCx2+z-PCx2,         for all xHand zC.

Lemma 2.1.([19])

Let H be a real Hilbert space, let C be a nonempty closed convex subset of H and let A be a mapping of C into H. Let uC. Then for λ > 0,

uVI(C,A)u=PC(I-λA)u,

where PC is the metric projection of H onto C.

Recall that a bifunction F : C × C → ℝ is said to be

• η-strongly monotone if there exists a number η > 0 such that F(x,y)+F(y,x)-ηx-y2,   for all x,yC,

• monotone on C if F(x,y)+F(y,x)0,   for all x,yC,

• pseudomonotone on C with respect to xC if F(x,y)0implies F(y,x)0,   for all yC.

It is clear that (i) ⇒ (ii) ⇒ (iii), for every xC. Moreover, F is said to be pseudomonotone on C with respect to AC, if it is pseudomonotone on C with respect to every xA. When AC, F is called pseudomonotone on C.

The following example, taken from [18], shows that a bifunction may not be pseudomonotone on C, but yet is pseudomonotone on C with respect to the solution set of the equilibrium problem defined by F and C:

F(x,y):=2yx(y-x)+xyy-x,for all x,y,C:=[-1,1].

Clearly, EP(F) = {0}. Since F(y, 0) = 0 for every yC, this bifunction is pseudomonotone on C with respect to the solution x* = 0, However, F is not pseudomonotone on C. In fact, both F(−0.5, 0.5) = 0.25 > 0 and F(0.5, −0.5) = 0.25 > 0.

For solving the equilibrium problem (1.1), let us assume that Δ is an open convex set containing C and the bifunction F : Δ × Δ → ℝ satisfies the following assumptions:

• (A1) F(x, x) = 0 for all xC and F(x, ·) is convex and lower semicontinuous on C;

• (A2) for each yC, F(·, y) is weakly upper semicontinuous on the open set Δ;

• (A3) F is pseudomonotone on C with respect to EP(F, C) and satisfies the strict paramonotonicity property, i.e., F(y, x) = 0 for xEP(F, C) and yC implies yEP(F, C);

• (A4) if {xn} ⊆ C is bounded and εn → 0 as n → ∞, then the sequence {wn} with wnnF(xn, ·)xn is bounded, where εF(x, ·)x stands for the ε-subdifferential of the convex function F(x, ·) at x.

The following idea of the ε-subdierential of convex functions can be found in the work of Bronsted and Rockafellar [9] but the theory of ε-subdierential calculus was given by Hiriart-Urruty [13].

Definition 2.2

Consider a proper convex function φ : C → ℝ̄. For a given ε > 0, the ε-subdierential of φ at x0Domφ is given by

ɛφ(x0)={xC:φ(y)-φ(x0)x,y-x0-ɛ,yC}.

Remark 2.3

It is known that if the function φ is proper lower semicontinuous convex, then for every xDomφ, the ε-subdierential εφ(x) is a nonempty closed convex set (see [13]).

Next, throughout this paper, weak and strong convergence of a sequence {xn} in H to x are denoted by xnx and xnx, respectively. In order to prove our main results, we need the following lemmas.

Lemma 2.4.([17])

Let C be a nonempty closed convex subset of a real Hilbert space H, and let T : CC be a κ-strictly pseudonospreading mapping. If F(T) ≠ ∅︀, then it is closed and convex.

Remark 2.5

If T : CC is a κ-strictly pseudononspreading mapping with F(T) ≠ ∅︀, then from Lemma 2.8 in [14] and Lemma 2.1, we have F(T) = VI(C, (IT)) = F(PC(Iλ(IT))), for all λ > 0.

Lemma 2.6

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. For every i = 1, 2, …, N, let Ti : CC be a finite family of κi-strictly pseudononspreading mappings withi=1NF(Ti). Let {a1, a2, …, an} ⊂ (0, 1) withΣi=1Nai=1, let κ̄ = max{κ1, κ2, …, κN} and let λ ∈ (0, 1 − κ̄). Then

• i=1NF(Ti)=F(Σi=1NaiPC(I-λ(I-Ti))).

• Σi=1NaiPC(I-λ(I-Ti))x-y2x-y2, for all xC andyi=1NF(Ti), i.e.Σi=1NaiPC(I-λ(I-Ti))is quasi-nonexpansive.

Proof

(i) It easy to see that i=1NF(Ti)F(Σi=1NaiPC(I-λ(I-Ti))). Let xF(Σi=1NaiPC(I-λ(I-Ti))) and let x*i=1NF(Ti)F(Σi=1NaiPC(I-λ(I-Ti))). Note that for every i = 1, 2, 3, …, N we have

PC(I-λ(I-Ti))x-x*2x-x*-λ(I-Ti)2=x-x*2-2λx-x*,(I-Ti)x+λ2(I-Ti)x2.

Put Ai = ITi, for all i = 1, 2, …, N, we have Ti = IAi and

Tix-Tix*2=(I-Ai)x-(I-Ai)x*2=(x-x*)-Aix2=x-x*2-2x-x*,Aix+Aix2x-x*2+κi(I-Ti)x-(I-Ti)x*2+2x-Tix,x*-Tix*=x-x*2+κi(I-Ti)x2,

which implies that

(1-κi)(I-Ti)x22x-x*,Aix,         for all i=1,2,3,,N

From (2.5) and (2.6), we have

PC(I-λ(I-Ti))x-x*2x-x*2-2λx-x*,(I-Ti)x+λ2(I-Ti)x2x-x*2-λ(1-κi)(I-Ti)x2+λ2(I-Ti)x2=x-x*2-λ[(1-κi)-λ](I-Ti)x2x-x*2,

for all i = 1, 2, 3, …, N.

From the definition of x and (2.7), we have

x-x*2=Σi=1NaiPC(I-λ(I-Ti))x-x*2=a1PC(I-λ(I-T1))x-x*2+a2PC(I-λ(I-T2))x-x*2++aNPC(I-λ(I-TN))x-x*2-a1a2PC(I-λ(I-T1))x-PC(I-λ(I-T2))x2-a2a3PC(I-λ(I-T2))x-PC(I-λ(I-T3))x2--aN-1aNPC(I-λ(I-TN-1))x-PC(I-λ(I-TN))x2x-x*2-a1a2PC(I-λ(I-T1))x-PC(I-λ(I-T2))x2-a2a3PC(I-λ(I-T2))x-PC(I-λ(I-T3))x2--aN-1aNPC(I-λ(I-TN-1))x-PC(I-λ(I-TN))x2.

This implies that

PC(I-λ(I-T1))x=PC(I-λ(I-T2))x==PC(I-λ(I-TN))x

Since xF(Σi=1NaiPC(I-λ(I-Ti))), we get that x = PC(Iλ(ITi))x, for all i = 1, 2, 3, …, N From Remark 2.5, we have xF(Ti), for all i = 1, 2, 3, …, N. That is xi=1NF(Ti). Hence F(Σi=1NaiPC(I-λ(I-Ti)))i=1NF(Ti).

(ii) Let xC and yi=1NF(Ti)=F(Σi=1NaiPC(I-λ(I-Ti)))

As the same argument as in (i), we can show that

PC(I-λ(I-Ti))x-y2x-y2,

for all i = 1, 2, 3, …, N. Thus

Σi=1NaiPC(I-λ(I-Ti))x-y2a1PC(I-λ(I-T1))x-y2+a2PC(I-λ(I-T2))x-y2++aNPC(I-λ(I-TN))x-y2Σi=1Naix-y2=x-y2.

Lemma 2.7.([23])

Let {an} and {bn} be two sequences of nonnegative real numbers such that

an+1an+bn,n1,

wheren=0bn<. Then the sequence {an} is convergent.

3. Weak Convergence Theorem

In this section, we prove weak convergence theorem for finding a common element in the solution set of a class of pseudomonotone equilibrium problems and the set of fixed points of a finite family of κ-strictly presudononspreading mappings in a real Hilbert space.

Theorem 3.1

Let C be a closed convex subset of a real Hilbert space H and F : C × C → ℝ be a bifunction satisfying (A1)–(A4). Let {κ1, κ2, …, κN} ⊂ [0, 1) and{Ti}i=1Nbe a finite family of κi-strictly pseudononspreading mappings of C into itself such thatΩ:=i=1NF(Ti)EP(F,C). Let x0C and {xn} be a sequence generated by

{x0C,wnɛnF(xn,·)xn,un=PC(xn-ρnwn),ρn=δnmax{σn,wn},xn+1=αnxn+βnΣi=1NaiPC(I-λni(I-Ti))xn+γnun,n,

where a, b, c, d, λ ∈ ℝ, ai ∈ (0, 1), for all i = 1, 2, …, N withΣi=1Nai=1, {αn}, {βn}, {γn} ⊂ [0, 1] with αn + βn + γn = 1 and {δn}, {εn}, {λni}(0,)satisfying the following conditions:

• 0<λλnimin{1-κ1,1-κ2,,1-κN}andΣn=1λni<for all i = 1, 2, …, N;

• 0 < a < αn, βn, γn < b < 1;

• n=0δn=,n=0δn2<, andn=0δnɛn<.

Then the sequence {xn} converges weakly to x̄ ∈ Ω.

Proof

First, we will show that {xn} is bounded. Let p ∈ Ω. Then we have

un-p2=xn-p2-un-xn2+2xn-un,p-unxn-p2+2xn-un,p-un.

Since un = PC(xnρnwn) and pC, we get that

xn-un,p-unρnωn,p-un.

Substuting (3.3) into (3.2), we have

un-p2xn-p2+2ρnwn,p-un=xn-p2+2ρnwn,p-xn+2ρnwn,xn-unxn-p2+2ρnwn,p-xn+2ρnwnxn-unxn-p2+2ρnwn,p-xn+2δnxn-un.

By using un = PC(xnρnwn) and xnC again, we get

xn-un2=xn-un,xn-unρnwn,xn-unρnwnxn-unδnxn-un,

which implies that

xn-unδn.

By condition (iii), we have

limnxn-un=0.

Combining (3.4) and (3.6), we obtain

un-p2xn-p2+2ρnwn,p-xn+2δn2.

Since wnεnF(xn, ·)xn, pC and F(x, x) = 0 for each xC, we obtain that

wn,p-xnF(xn,p)-F(xn,xn)+ɛn=F(xn,p)+ɛn.

Thus, it follows from (3.8) and (3.9) that

un-p2xn-p2+2ρnF(xn,p)+2ρnɛn+2δn2.

Form Lemma 2.6 (ii), we have

Σi=1NaiPC(I-λni(I-Ti))xn-p2xn-p2.

From (3.1), (3.10) and (3.11), we have

xn+1-p2=αnxn+βnΣi=1NaiPC(I-λni(I-Ti))xn+γnun-p2αnxn-p2+βnΣi=1NaiPC(I-λni(I-Ti))xn-p2+γnun-p2-αnβnxn-Σi=1NaiPC(I-λni(I-Ti))xn2αnxn-p2+βnxn-p2+γn(xn-p2+2ρnF(xn,p)+2ρnɛn+2δn2)-αnβnxn-Σi=1NaiPC(I-λni(I-Ti))xn2=xn-p2+2γnρnF(xn,p)+2γnρnɛn+2γnδn2-αnβnxn-Σi=1NaiPC(I-λni(I-Ti))xn2.

Since pEP(F, C) and F is pseudomonotone on F with respect to p, we get that F(xn, p) ≤ 0 for all n ∈ ℕ. Then from (3.12) it follows that

xn+1-p2xn-p2+2γnρnɛn+2γnδn2-αnβnxn-Σi=1NaiPC(I-λni(I-Ti))xn2xn-p2+2γnρnɛn+2γnδn2.

Let ηn=2γnρnɛn+2γnδn2 for all n ≥ 0. From condition (ii) and (iii), we get that

Σn=0ηn=Σn=0(2γnρnɛn+2γnδn2)2bΣn=0ρnɛn+2bΣn=0δn2<+

Now applying Lemma 2.7 to (3.13), we obtain that the limnxn-p exists, i.e. limnxn-p=a¯ for some āC. Thus {xn} is bounded. Also, it easy to verify that {un} and {Σi=1NaiPC(I-λni(I-Ti))xn} are also bounded.

Next, we will show that lim supnF(xn,p)=0 for any p ∈ Ω. Since F is pseudomonotone on C and F(p, xn) ≥ 0, we have −F(xn, p) ≥ 0. From (3.12) and condition (ii), we have

2γnρn[-F(xn,p)]xn-p2-xn+1-p2+2γnρnɛn+2γnδn2xn-p2-xn+1-p2+2bρnɛn+2bδn2.

Summing up (3.14) for every n, we obtain

02n=0γnρn[-F(xn,p)]x0-p2+2bn=0ρnɛn+2bn=0δn2<+.

By the assumption (A4), we can find a real number w such that ||wn|| ≤ w for every n. Setting Γ := max{σ, w}, where σ is a real number such that 0 < σn < σ for every n, it follows from (ii) that

02aΓn=0δn[-F(xn,p)]2n=0γnρn[-F(xn,p)]<+,

which implies that

0n=0δn[-F(xn,p)]<+.

Combining with −F(xn, p) ≥ 0 and n=0δn=, we can deduced that lim supnF(xn,p)=0 as desired.

Next, we will show that ωω(xn) ⊂ Ω, where ωω(xn) = {xH : xnix for some subsequence {xni} of {xn}}. In deed since {xn} is bounded and H is reflexive, ωω(xn) is nonempty. Let ωω(xn). Then there exists subsequence {xni} of {xn}. converging weakly to , that is xni as i → ∞. By the convexity, C is weakly closed and hence C. Since F(·, p) is weakly upper semicontinuous for p ∈ Ω, we obtain

F(x¯,p)lim supiF(xn,p)=limiF(xni,p)=lim supnF(xn,p)=0.

Since F is pseudomontone with respect to p and F(p, ) ≥ 0, we obtain F(, p) ≤ 0. Thus F(, p) = 0. Furthermore, by assumption (A3), we get that EP(F, C). On the other hand, from (3.13) and conditions (ii)–(iii), we have

αnβnxn-Σi=1NaiPC(I-λni(I-Ti))xn2xn-p2-xn+1-p2+2γnρnɛn+2γnδn2xn-p2-xn+1-p2+2bρnɛn+2bδn2

taking the limit as n → ∞ yields

limnxn-Σi=1NaiPC(I-λni(I-Ti))xn=0.

Now, we will show that x¯i=1NF(Ti). Assume that x¯i=1NF(Ti). By Lemma 2.6, we have x¯F(Σi=1NaiPC(I-λn(I-Ti))). From the Opial’s condition, (3.21) and condition (i), we can write

lim infixni-x¯<lim infixni-Σi=1NaiPC(I-λni(I-Ti))x¯lim infi(xni-Σi=1NaiPC(I-λni(I-Ti))xni+Σi=1NaiPC(I-λni(I-Ti))xni-Σi=1NaiPC(I-λni(I-Ti))x¯)lim infi(xni-x¯+Σi=1Naiλni(I-Ti)xni-(I-Ti)x¯)lim infixni-x¯.

This is a contradiction. Then x¯i=1NF(Ti). Thus EP(F, C) ∩ F(T) = Ω and so ωω(xn) ⊂ Ω.

Finally, we prove that {xn} converge weakly to an element of Ω. It’s sufficient to show that ωω(xn) is a single point set. Taking z1, z2ωω(xn) arbitrarily, and let {xnk} and {xnm} be subsequence of {xn} such that xnkz1 and xnmz2 respectively. Since limnxn-p exists for all p ∈ Ω and z1, z2 ∈ Ω, we get that limnxn-z1 and limnxn-z2 exist. Now, assume that z1z2, then by the Opial’s condition,

limnxn-z1=limkxnk-z1<limkxnk-z2=limnxn-z2=limmxnm-z2<limmxnm-z1=limnxn-z1,

which is a contradiction. Thus z1 = z2. This show that ωω(xn) is single point set. i.e. xn. This completes the proof.

If we set κi = 0 for all i = 1, 2, …, N then we get the following Corollary.

Corollary 3.2

Let C be a closed convex subset of a real Hilbert space H and F : C × C → ℝ be a bifunction satisfying (A1)–(A4). Let{Ti}i=1Nbe a finite family of nonspreading mappings of C into itself such thatΩ:i=1NF(Ti)EP(F,C). Let x0C and {xn} be a sequence generated by

{x0C,wnɛnF(xn,·)xn,un=PC(xn-ρnwn),   ρn=δnmax{σn,wn},xn+1=αnxn+βnΣi=1NaiPC(I-λni(I-Ti))xn+γnun,   n,

where a, b, c, d, λ ∈ ℝ, ai ∈ (0, 1), for all i = 1, 2, …, N withΣi=1Nai=1, {αn}, {βn}, {γn} ⊂ [0, 1] with αn + βn + γn = 1 and {δn}, {εn}, {λni}(0,)satisfying the following conditions:

• (i) 0<λλni<1andΣn=1λni<for all i = 1, 2, …, N;

• (ii) 0 < a < αn, βn, γn < b < 1;

• (ii) n=0δn=,n=0δn2<, andn=0δnɛn<.

Then the sequence {xn} converges weakly to x̄ ∈ Ω.

4. Strong Convergence Theorem

In this section, to obtain strong convergence result, we add the control condition limnαn=12, and then we get the strong convergence theorem for finding a common element in the solution set of a class of pseudomonotone equilibrium problems and the set of fixed points of a finite family of κ-strictly presudononspreading mappings in a real Hilbert space.

Theorem 4.1

Let C be a closed convex subset of a real Hilbert space H and F : C × C → ℝ be a bifunction satisfying (A1)–(A4). Let {κ1, κ2, …, κN} ⊂ [0, 1) and{Ti}i=1Nbe a finite family of κi-strictly pseudononspreading mappings of C into itself such thatΩ:=i=1NF(Ti)EP(F,C). Let x0C and {xn} be a sequence generated by

{x0C,wnɛnF(xn,·)xn,un=PC(xn-ρnwn),   ρn=δnmax{σn,wn},xn+1=αnxn+βnΣi=1NaiPC(I-λni(I-Ti))xn+γnun,   n,

where a, b, c, d, λ ∈ ℝ, ai ∈ (0, 1), for all i = 1, 2, …, N withΣi=1Nai=1, {αn}, {βn}, {γn} ⊂ [0, 1] with αn + βn + γn = 1 and {δn}, {εn}, {λni}(0,)satisfying the following conditions:

• 0<λλnimin{1-κ1,1-κ2,,1-κN}andΣn=1λni<for all i = 1, 2, …, N;

• 0 < a < αn, βn, γn < b < 1 andlimnαn=12;

• n=0δn=,n=0δn2< , andn=0δnɛn<.

Then the sequence {xn} converges strongly to x̄ ∈ Ω.

Proof

By a similar argument to the proof of Theorem 3.1 and (2.4), we have

Σi=1NaiPC(I-λni(I-Ti))xn-PΩ(xn)2Σi=1NaiPC(I-λni(I-Ti))xn-xn2-xn-PΩ(xn)2

and

un-PΩ(xn)2un-xn2-xn-PΩ(xn)2.

It follows from (4.2) and condition (ii) that

xn+1-PΩ(xn+1)2αnxn+βnΣi=1NaiPC(I-λni(I-Ti))xn+γnun-PΩ(xn)2αnxn-PΩ(xn)2+βnΣi=1NaiPC(I-λni(I-Ti))xn-PΩ(xn))2+γnun-PΩ(xn)2αnxn-PΩ(xn)2+βn(Σi=1NaiPC(I-λni(I-Ti))xn-xn2-xn-PΩ(xn)2)+γn(un-xn2-xn-PΩ(xn)2)=(αn-(βn+γn))xn-PΩ(xn)2+βnΣi=1NaiPC(I-λni(I-Ti))xn-xn2+γnun-xn2.(2αn-1)xn-PΩ(xn)2+bΣi=1NaiPC(I-λni(I-Ti))xn-xn2+bun-xn2.

Combining (3.7), (3.21), conditions (ii)–(iii), and the boundedness of the sequence {xnPΩ(xn)}, we obtain

limnxn+1-PΩ(xn+1)=0

Since Ω is convex, for all m > n, we have 12(PΩ(xm)+PΩ(xn))Ω, and therefore

PΩ(xm)-PΩ(xn)2=2xm-PΩ(xm)2+2xm-PΩ(xn)2-4xm-12(PΩ(xm)+PΩ(xn))22xm-PΩ(xm)2+2xm-PΩ(xn)2-4xm-PΩ(xm)2=2xm-PΩ(xn)2-2xm-PΩ(xm)2.

Using (3.13) with p = PΩ(xn), we have

xm-PΩ(xn)2xm-1-PΩ(xn)2+ηm-1xm-2-PΩ(xn)2+ηm-1+ηm-2xn-PΩ(xn)2+j=nm-1ηj,

where ηj=2γjρjɛj+2γjδj2. It follows from (4.4) and (4.5) that

PΩ(xm)-PΩ(xn)22xn-PΩ(xn)2+2j=nm-1ηj-2xm-PΩ(xm)2.

Together with (4.3) and j=0ηj<+, this implies that {PΩ(xn)} is a Cauchy sequence, Hence {PΩ(xn)} strongly converges to some point x* ∈ Ω. Moreover, we obtain

x*=limiPΩ(xni)=PΩ(x¯)=x¯,

which implies that PΩ(xi) → x* = ∈ Ω. Then from (4.3) and (4.7), we can conclude that xn. This completes the proof.

If we set κi = 0 for all i = 1, 2, …, N then we get the following Corollary.

Corollary 4.2

Let C be a closed convex subset of a real Hilbert space H and F : C ×C → ℝ be a bifunction satisfying (A1)–(A4). Let{Ti}i=1Nbe a finite family of nonspreading mappings of C into itself such thatΩ:=i=1NF(Ti)EP(F,C). Let x0C and {xn} be a sequence generated by

{x0C,wnɛnF(xn,·)xn,un=PC(xn-ρnwn),   ρn=δnmax{σn,wn},xn+1=αnxn+βnΣi=1NaiPC(I-λni(I-Ti))xn+γnun,   n,

where a, b, c, d, λ ∈ ℝ, ai ∈ (0, 1), for all i = 1, 2, …, N withΣi=1Nai=1, {αn}, {βn}, {γn} ⊂ [0, 1] with αn + βn + γn = 1 and {δn}, {εn}, {λni}(0,)satisfying the following conditions:

• 0<λλni<1andΣn=1λni<for all i = 1, 2, …, N;

• 0 < a < αn, βn, γn < b < 1 andlimnαn=12;

• n=0δn=,n=0δn2< , andn=0δnɛn<.

Then the sequence {xn} converges weakly to x̄ ∈ Ω.

The authors would like to thank the faculty of science and technology, Rajamangala University of Technology Thanyaburi (RMUTT), Thailand for the financial support. Moreover, the authors would like to thank the referees for their valuable suggestions and comments which helped to improve the quality and readability of the paper.

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