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Strong convergence theorems for equilibrium problems involving a family of nonexpansive mappings

Abstract

We give new hybrid variants of extragradient methods for finding a common solution of an equilibrium problem and a family of nonexpansive mappings. We present a scheme that combines the idea of an extragradient method and a successive iteration method as a hybrid variant. Then, this scheme is modified by projecting on a suitable convex set to get a better convergence property under certain assumptions in a real Hilbert space.

MSC:65K10, 65K15, 90C25, 90C33.

1 Introduction

In this paper, we always assume that ℋ is a real Hilbert space with the inner product 〈⋅,⋅〉 and the induced norm ∥⋅∥. Let C be a nonempty closed convex subset of ℋ and the bifunction f:C×C→R. Then f is called strongly monotone on C with β>0 iff

f(x,y)+f(y,x)≤−β ∥ x − y ∥ 2 ∀x,y∈C;

monotone on C iff

f(x,y)+f(y,x)≤0∀x,y∈C;

pseudomonotone on C iff

f(x,y)≥0impliesf(y,x)≤0∀x,y∈C;

Lipschitz-type continuous on C in the sense of Mastroeni [1] iff there exist positive constants c 1 >0, c 2 >0 such that

f(x,y)+f(y,z)≥f(x,z)− c 1 ∥ x − y ∥ 2 − c 2 ∥ y − z ∥ 2 ∀x,y,z∈C.

An equilibrium problem, shortly EP(f,C), is to find a point in

Sol(f,C)= { x ∗ ∈ C : f ( x ∗ , y ) ≥ 0 ∀ y ∈ C } .

Let a mapping T of C into itself. Then T is called contractive with constant δ∈(0,1) iff

∥ T ( x ) − T ( y ) ∥ ≤δ∥x−y∥∀x,y∈C.

The mapping T is called strictly pseudocontractive iff there exists a constant k∈[0,1) such that

∥ T ( x ) − T ( y ) ∥ 2 ≤ ∥ x − y ∥ 2 +k ∥ ( I − T ) ( x ) − ( I − T ) ( y ) ∥ 2 .

In the case k=0, the mapping T is called nonexpansive on C. We denote by Fix(T) the set of fixed points of T.

Let T i :C→C, i∈Γ, be a family of nonexpansive mappings where Γ stands for an index set. In this paper, we are interested in the problem of finding a common element of the solution set of problem EP(f,C) and the set of fixed points F= ⋂ i ∈ Γ Fix( T i ), namely:

Find  x ∗ ∈F∩Sol(f,C),
(1.1)

where the function f and the mappings T i , i∈Γ, satisfy the following conditions:

(A1) f(x,x)=0 for all x∈C and f is pseudomonotone on C,

(A2) f is Lipschitz-type continuous on C with constants c 1 >0 and c 2 >0,

(A3) f is upper semicontinuous on C,

(A4) For each x∈C, f(x,⋅) is convex and subdifferentiable on C,

(A5) F∩Sol(f,C)≠∅.

Under these assumptions, for each r>0 and x∈C, there exists a unique element z∈C such that

f(z,y)+ 1 r 〈y−z,z−x〉≥0∀y∈C.
(1.2)

Problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, equilibrium equilibriums, fixed point problems (see, e.g., [2–7]). Recently, it has become an attractive field for many researchers in both theory and its solution methods (see, e.g., [3, 4, 8–12] and the references therein). Most of these algorithms are based on inequality (1.2) for solving the underlying equilibrium problem when F∩Sol(f,C)≠∅. Motivated by this idea for finding a common point of Sol(f,C) and the fixed point set Fix(T) of a nonexpansive mapping T, Takahashi and Takahashi [13] first introduced an iterative scheme by the viscosity approximation method. The sequence { x n } is defined by

{ x 0 ∈ C , f ( u n , y ) + 1 r n 〈 y − u n , u n − x n 〉 ≥ 0 ∀ y ∈ C , x n + 1 = α n g ( x n ) + ( 1 − α n ) T ( u n ) ∀ n ≥ 0 ,

where g:C→C is contractive. Under certain conditions over the parameters { α n } and { r n }, they showed that the sequences { x n } and { u n } strongly converge to z= Pr Fix ( T ) ∩ Sol ( f , C ) g(z), where Pr C denotes the projection on C. At each iteration n in all of these algorithms, it requires to solve approximation auxiliary equilibrium problems for finding a common solution of an equilibrium problem and a fixed point problem. In order to avoid this requirement, Anh [14] recently proposed a hybrid extragradient algorithm for finding a common point of the set Fix(T)∩Sol(f,C). Starting with an arbitrary initial point x 0 ∈C, iteration sequences are defined by

{ y k = argmin { λ k f ( x k , y ) + 1 2 ∥ y − x k ∥ 2 : y ∈ C } , t k = argmin { λ k f ( y k , t ) + 1 2 ∥ t − x k ∥ 2 : t ∈ C } , x k + 1 = α k x 0 + ( 1 − α k ) T ( x k ) .
(1.3)

Under certain conditions onto parameters { λ k } and { α k }, he showed that the sequences { x k }, { y k } and { t k } weakly converge to the point x∈Fix(T)∩Sol(f,C) in a real Hilbert space. At each main iteration n of the scheme, he only solved strongly convex problems on C, but the proof of convergence was still done under the assumptions that x n + 1 − x n →0.

For finding a common point of a family of nonexpansive mappings T i (i∈Γ), as a corollary of Theorem 2.1 in [15], Zhou proposed the following iteration scheme:

{ x 0 ∈ H  chosen arbitrarily, C 1 , i = C , C 1 = â‹‚ i ∈ Γ C 1 , i , x 1 = Pr C 1 ( x 0 ) , y n , i = ( 1 − α n , i ) x n + α n , i T i ( x n ) , C n + 1 , i = { z ∈ C n , i : α n , i ( 1 − 2 α n , i ) ∥ x n − T i ( x n ) ∥ 2 ≤ 〈 x n − z , y n , i − T i ( y n , i ) 〉 } , C n + 1 = â‹‚ i ∈ Γ C n + 1 , i , x n + 1 = Pr C n + 1 ( x 0 ) .
(1.4)

Under the restrictions of the control sequences 0< lim inf n → ∞ α n , i ≤ lim sup n → ∞ α n , i ≤ a i < 1 2 , he showed that the sequence { x n } defined by (1.4) strongly converges to x ∗ = Pr F ( x 0 ) in a real Hilbert space ℋ, where F= ⋂ i ∈ Γ Fix( T i ).

In this paper, motivated by Ceng et al. [16, 17], Wang and Guo [18], Zhou [15], Nadezhkina and Takahashi [10], Cho et al. [19], Takahashi and Takahashi [13], Anh [6, 12] and Anh et al. [20, 21], we introduce several modified hybrid extragradient schemes to modify the iteration schemes (1.3) and (1.4) to obtain new strong convergence theorems for a family of nonexpansive mappings and the equilibrium problem EP(f,C) in the framework of a real Hilbert space â„‹.

To investigate the convergence of this scheme, we recall the following technical lemmas which will be used in the sequel.

Lemma 1.1 ([14], Lemma 3.1)

Let C be a nonempty closed convex subset of a real Hilbert space ℋ. Let f:C×C→R be a pseudomonotone and Lipschitz-type continuous bifunction. For each x∈C, let f(x,⋅) be convex and subdifferentiable on C. Suppose that the sequences { x n }, { y n }, { t n } are generated by scheme (1.3) and x ∗ ∈Sol(f,C). Then

∥ t n − x ∗ ∥ 2 ≤ ∥ x n − x ∗ ∥ 2 −(1−2 λ n c 1 ) ∥ x n − y n ∥ 2 −(1−2 λ n c 2 ) ∥ y n − t n ∥ 2 ∀n≥0.

Lemma 1.2 Let C be a closed convex subset of a real Hilbert space ℋ, and let Pr C be the metric projection from ℋ on to C (i.e., for x∈H, Pr C is the only point in C such that ∥x− Pr C x∥=inf{∥x−z∥:z∈C}). Given x∈H and z∈C. Then z= Pr C x if only if there holds the relation 〈x−z,y−z〉≤0 for all y∈C.

Lemma 1.3 Let â„‹ be a real Hilbert space. Then the following equations hold:

  1. (i)

    ∥ x − y ∥ 2 = ∥ x ∥ 2 − ∥ y ∥ 2 −2〈x−y,y〉 for all x,y∈H.

  2. (ii)

    ∥ t x + ( 1 − t ) y ∥ 2 =t ∥ x ∥ 2 +(1−t) ∥ y ∥ 2 −t(1−t) ∥ x − y ∥ 2 for all t∈[0,1] and x,y∈H.

2 Convergence theorems

Now, we prove the main convergence theorem.

Theorem 2.1 Let C be a nonempty closed convex subset of a real Hilbert space ℋ. Suppose that assumptions (A1)-(A5) are satisfied and { T i } i ∈ Γ is a family of nonexpansive mappings from C into itself and a nonempty common fixed points set F. Let { x n } be a sequence generated by the following scheme:

{ x 0 ∈ H  chosen arbitrarily , C 1 , i = D 1 , i = C , C 1 = â‹‚ i ∈ Γ C 1 , i , D 1 = â‹‚ i ∈ Γ D 1 , i , x 1 = Pr C 1 ∩ D 1 x 0 , y n = argmin { λ n f ( x n , y ) + 1 2 ∥ y − x n ∥ 2 : y ∈ C } , z n = argmin { λ n f ( y n , y ) + 1 2 ∥ z − x n ∥ 2 : z ∈ C } , y n , i = ( 1 − α n , i ) z n + α n , i T i z n , C n + 1 , i = { z ∈ C n , i : α n , i ( 1 − 2 α n , i ) ∥ z n − T i z n ∥ 2 ≤ 〈 z n − z , y n , i − T i y n , i 〉 } , C n + 1 = â‹‚ i ∈ Γ C n + 1 , i , D n + 1 , i = { z ∈ D n , i : ∥ y n , i − z ∥ ≤ ∥ x n − z ∥ } , D n + 1 = â‹‚ i ∈ Γ D n + 1 , i , x n + 1 = Pr C n + 1 ∩ D n + 1 x 0 , 0 < lim inf α n , i ≤ lim sup α n , i < 1 , { λ n } ⊂ [ a , b ]  for some  a , b ∈ ( 0 , 1 L ) ,  where  L = max { 2 c 1 , 2 c 2 } .

Then the sequences { x n }, { y n } and { z n } strongly converge to the same point Pr F ∩ Sol ( f , C ) x 0 .

Proof The proof of this theorem is divided into several steps.

Step 1. Claim that C n and D n are closed and convex for all n≥0.

We have to show that for any fixed point but arbitrary i∈Γ, C n , i is closed and convex for every n≥0. This can be proved by induction on n. It is obvious that C 1 , i =C is closed and convex. Assume that C n , i is closed and convex for some n∈ N ∗ ={1,2,…}. We have that the set

A= { z ∈ C : α n , i ( 1 − 2 α n , i ) ∥ z n − T i z n ∥ 2 ≤ 〈 z n − z , y n , i − T i y n , i 〉 }

is closed and convex, and C n + 1 , i = C n , i ∩A, hence C n + 1 , i is closed and convex. Then C n is closed and convex for all n≥0. We can write D n + 1 , i under the form

D n + 1 , i = { z ∈ D n , i : ∥ y n , i − x n ∥ 2 + 2 〈 y n , i − x n , x n − z 〉 ≤ 0 } .

Then D n + 1 , i is closed and convex. Thus, D n is closed and convex.

Step 2. Claim that F∩Sol(f,C)⊆ C n ∩ D n for all n∈ N ∗ .

First, we show that F⊆ C n by induction on n. It suffices to show that F⊆ C n , i .

We have F⊆C= C 1 , i is obvious. Suppose F⊆ C n , i for some n∈N. We have to show that F⊆ C n + 1 , i . Indeed, let w∈F, by inductive hypothesis, we have w∈ C n , i and

∥ z n − T i z n ∥ 2 = 〈 z n − T i z n , z n − T i z n 〉 = 1 α n , i 〈 z n − y n , i , z n − T i z n 〉 = 1 α n , i 〈 z n − y n , i , z n − T i z n − ( y n , i − T i y n , i ) 〉 + 1 α n , i 〈 z n − y n , i , y n , i − T i y n , i 〉 = 1 α n , i 〈 z n − y n , i , z n − T i z n − ( y n , i − T i y n , i ) 〉 + 1 α n , i 〈 z n − w + w − y n , i , y n , i − T i y n , i 〉 = 1 α n , i 〈 z n − y n , i , z n − y n , i 〉 + 1 α n , i 〈 z n − y n , i , T i y n , i − T i z n 〉 + 1 α n , i 〈 z n − w , y n , i − T i y n , i 〉 + 1 α n , i 〈 w − y n , i , y n , i − T i y n , i 〉 ≤ 2 α n , i ∥ z n − y n , i ∥ 2 + 1 α n , i 〈 z n − w , y n , i − T i y n , i 〉 + 1 α n , i 〈 w − y n , i , y n , i − T i y n , i 〉 .
(2.1)

On the other hand, for all w∈F and y n , i ∈C, we have

∥ w − y n , i ∥ 2 ≥ 〈 T i w − T i y n , i , w − y n , i 〉 = 〈 w − T i y n , i , w − y n , i 〉 = 〈 w − y n , i + y n , i − T i y n , i , w − y n , i 〉 = ∥ w − y n , i ∥ 2 + 〈 y n , i − T i y n , i , w − y n , i 〉 ,

and hence

〈 w − y n , i , y n , i − T i y n , i 〉 ≤0.

Combining this with (2.1), we obtain

∥ z n − T i z n ∥ 2 ≤ 2 α n , i ∥ z n − y n , i ∥ 2 + 1 α n , i 〈 z n − w , y n , i − T i y n , i 〉 ≤ 2 α n , i ∥ z n − T i z n ∥ 2 + 1 α n , i 〈 z n − w , y n , i − T i y n , i 〉 .

This follows that

α n , i (1−2 α n , i ) ∥ z n − T i z n ∥ 2 ≤ 〈 z n − w , y n , i − T i y n , i 〉 .

By the definition of C n + 1 , i , we have w∈ C n + 1 , i , and so F⊆ C n + 1 , i for all i∈Γ, which deduces that F⊆ C n . This shows that F∩Sol(f,C)⊆ C n for all n∈ N ∗ .

Next, we will prove F∩Sol(f,C)⊆ D n by induction on n∈ N ∗ . It suffices to show that F∩Sol(f,C)⊆ D n , i . Indeed, F⊆C= D 1 , i so F∩Sol(f,C)⊆ D 1 , i . Suppose that F∩Sol(f,C)⊆ D n , i . Let x ∗ ∈F∩Sol(f,C), then x ∗ ∈ D n , i . Using Lemma 1.1, we get

∥ y n , i − x ∗ ∥ 2 = ∥ ( 1 − α n , i ) z n + α n , i T i z n − x ∗ ∥ 2 ≤ ( 1 − α n , i ) ∥ z n − x ∗ ∥ 2 + α n , i ∥ T i z n − T i x ∗ ∥ 2 ≤ ∥ z n − x ∗ ∥ 2 ≤ ∥ x n − x ∗ ∥ 2 − ( 1 − 2 λ n c 1 ) ∥ x n − y n ∥ 2 − ( 1 − 2 λ n c 2 ) ∥ y n − z n ∥ 2 ≤ ∥ x n − x ∗ ∥ 2 .
(2.2)

Then we have x ∗ ∈ D n + 1 , i and hence F∩Sol(f,C)⊆ D n + 1 , i . This shows that F∩Sol(f,C)⊆ D n , which yields that F∩Sol(f,C)⊆ C n ∩ D n for all n∈ N ∗ .

Step 3. Claim that the sequence { x n } is bounded and there exists the limit lim n → ∞ ∥ x n − x 0 ∥=c.

From x n = Pr C n ∩ D n x 0 , it follows that

〈 x 0 − x n , x n − y 〉 ≥0∀y∈ C n ∩ D n .
(2.3)

Then, using Step 2, we have F∩Sol(f,C)⊆ C n ∩ D n and

〈 x 0 − x n , x n − w 〉 ≥0∀w∈F∩Sol(f,C).
(2.4)

Combining this and assumption (A5), the projection Pr F ∩ Sol ( f , C ) x 0 is well defined and there exits a unique point p such that p= Pr F ∩ Sol ( f , C ) x 0 . So, we have

0 ≤ 〈 x 0 − x n , x n − p 〉 = 〈 x 0 − x n , x n − x 0 + x 0 − p 〉 ≤ − ∥ x 0 − x n ∥ 2 + ∥ x 0 − x n ∥ ∥ x 0 − p ∥ ,

and hence

∥ x 0 − x n ∥ ≤ ∥ x 0 − p ∥ .

Then the sequence { x n } is bounded. So, the sequences { y n }, { z n }, { y n , i }, { T i y n , i } also are bounded. Since x n + 1 ∈ C n + 1 ∩ D n + 1 ⊂ C n ∩ D n and (2.3), we have

0 ≤ 〈 x 0 − x n , x n − x n + 1 〉 = 〈 x 0 − x n , x n − x 0 + x 0 − x n + 1 〉 ≤ − ∥ x 0 − x n ∥ 2 + ∥ x 0 − x n ∥ ∥ x 0 − x n + 1 ∥ ,

and hence ∥ x 0 − x n ∥≤∥ x 0 − x n + 1 ∥. This together with the boundedness of { x n } implies that the limit lim n → ∞ ∥ x n − x 0 ∥=c exists.

Step 4. We claim that lim n → ∞ x n =q∈C.

Since C m ∩ D m ⊆ C n ∩ D n , x m = Pr C m ∩ D m x 0 ∈ C n ∩ D n for any positive integer m≥n and (2.3), we have

〈 x 0 − x n , x n − x n + m 〉 ≥0.

Then

∥ x n − x n + m ∥ 2 = ∥ x n − x 0 + x 0 − x n + m ∥ 2 = ∥ x n − x 0 ∥ 2 + ∥ x 0 − x n + m ∥ 2 − 2 〈 x 0 − x n , x 0 − x n + m 〉 ≤ ∥ x 0 − x n + m ∥ 2 − ∥ x n − x 0 ∥ 2 − 2 〈 x 0 − x n , x n − x n + m 〉 ≤ ∥ x 0 − x n + m ∥ 2 − ∥ x n − x 0 ∥ 2 .
(2.5)

Passing the limit in (2.5) as n→∞, we get lim n → ∞ ∥ x n − x n + m ∥=0 ∀m∈ N ∗ . Hence, { x n } is a Cauchy sequence in a real Hilbert space ℋ and so lim n → ∞ x n =q∈C.

Step 5. We claim that q= Pr F ∩ Sol ( f , C ) x 0 , where q= lim n → ∞ x n .

First we show that q∈F∩Sol(f,C). Since x n + 1 = Pr C n + 1 ∩ D n + 1 x 0 , we have x n + 1 ∈ D n + 1 . Then x n + 1 ∈ D n + 1 , i and

∥ y n , i − x n + 1 ∥ ≤ ∥ x n − x n + 1 ∥ ,

which yields that

∥ x n − y n , i ∥ ≤ ∥ x n − x n + 1 ∥ + ∥ x n + 1 − y n , i ∥ ≤ 2 ∥ x n − x n + 1 ∥ .

Combining this and lim n → ∞ ∥ x n − x m ∥=0 for all m∈ N ∗ , we get

lim n → ∞ ∥ x n − y n , i ∥ =0.
(2.6)

For each x ∗ ∈Sol(f,C)∩F, by (2.2) we have

( 1 − 2 b c 1 ) ∥ x n − y n ∥ 2 ≤ ( 1 − 2 λ n c 1 ) ∥ x n − y n ∥ 2 ≤ ∥ x n − x ∗ ∥ 2 − ∥ y n , i − x ∗ ∥ 2 = ( ∥ x n − x ∗ ∥ + ∥ y n , i − x ∗ ∥ ) ( ∥ x n − x ∗ ∥ − ∥ y n , i − x ∗ ∥ ) ≤ ( ∥ x n − x ∗ ∥ + ∥ y n , i − x ∗ ∥ ) ( ∥ x n − y n , i ∥ ) .

Using this, the boundedness of sequences { x n }, { y n , i } and (2.6), we obtain

lim n → ∞ ∥ x n − y n ∥ =0.
(2.7)

By a similar way, we also have lim n → ∞ ∥ z n − y n ∥=0. Then it follows from the inequality

∥ x n − z n ∥ ≤ ∥ x n − y n ∥ + ∥ y n − z n ∥

that

lim n → ∞ ∥ x n − z n ∥ =0.
(2.8)

On the other hand, we have

∥ y n , i − z n ∥ ≤ ∥ y n , i − x n ∥ + ∥ x n − z n ∥ .

Combining this, (2.6) and (2.8), we obtain lim n → ∞ ∥ y n , i − z n ∥=0. By the definition of the sequence { y n , i }, we have

∥ y n , i − z n ∥ = α n , i ∥ T i z n − z n ∥ ,

and hence

lim n → ∞ ∥ T i z n − z n ∥ =0,

which yields that

∥ T i x n − x n ∥ ≤ ∥ T i x n − T i z n ∥ + ∥ T i z n − z n ∥ + ∥ x n − z n ∥ ≤ 2 ∥ x n − z n ∥ + ∥ T i z n − z n ∥ → 0 as  n → ∞

and

lim n → ∞ ∥ T i x n − x n ∥ =0.

It follows from Step 4 that lim n → ∞ T i x n =q. Hence q∈F.

Now we show that q∈Sol(f,C). By Step 5, we have y n →q as n→∞.

Since y n is the unique solution of the strongly convex problem

min { 1 2 ∥ y − x n ∥ 2 + λ n f ( x n , y ) : y ∈ C } ,

we get

0∈ ∂ 2 ( λ n f ( x n , y ) + 1 2 ∥ y − x n ∥ 2 ) ( y n ) + N C ( y n ) .

From this it follows that

0= λ n w+ y n − x n + w ¯ ,

where w∈ ∂ 2 f( x n ,⋅)( y n ) and w ¯ ∈ N C ( y n ). By the definition of the normal cone N C , we have

〈 y n − x n , y − y n 〉 ≥ λ n 〈 w , y n − y 〉 ∀y∈C.
(2.9)

On the other hand, since f( x n ,⋅) is subdifferentiable on C, by the well-known Moreau-Rockafellar theorem, there exists w∈ ∂ 2 f( x n ,⋅)( y n ) such that

f ( x n , y ) −f ( x n , y n ) ≥ 〈 w , y − y n 〉 ∀y∈C.

Combining this with (2.9), we have

λ n ( f ( x n , y ) − f ( x n , y n ) ) ≥ 〈 y n − x n , y n − y 〉 ∀y∈C.

Then, using { λ n }⊂[a,b]⊂(0, 1 L ), (2.7), x n →q, y n →q as n→∞ and the upper semicontinuity of f, we have

f(q,y)≥0∀q∈C.

This means that q∈Sol(f,C). By taking the limit in (2.4), we have

〈 x 0 − q , q − w 〉 ≥0∀w∈F∩Sol(f,C),

which implies that q= Pr F ∩ Sol ( f , C ) x 0 . Thus, the subsequences { x n }, { y n }, { z n } strongly converge to the same point q= Pr F ∩ Sol ( f , C ) x 0 . This completes the proof. □

Now, notice that ∀w∈F

∥ z n − T i z n ∥ 2 = ∥ z n − w + w − T i z n ∥ 2 = ∥ z n − w ∥ 2 + ∥ w − T i z n ∥ 2 + 2 〈 z n − w , w − T i z n 〉 ≤ 2 ∥ z n − w ∥ 2 + 2 〈 z n − w , w − z n + z n − T i z n 〉 = 2 ∥ z n − w ∥ 2 − 2 ∥ z n − w ∥ 2 + 2 〈 z n − w , z n − T i z n 〉 = 2 〈 z n − w , z n − T i z n 〉 .

Hence

∥ y n , i − w ∥ 2 = ∥ ( 1 − α n , i ) ( z n − w ) + α n , i ( T i z n − w ) ∥ 2 = ( 1 − α n , i ) ∥ z n − w ∥ 2 + α n , i ∥ T i z n − w ∥ 2 − α n , i ( 1 − α n , i ) ∥ T i z n − z n ∥ 2 = ( 1 − α n , i ) ∥ z n − w ∥ 2 + α n , i ∥ T i z n − z n + z n − w ∥ 2 − α n , i ( 1 − α n , i ) ∥ T i z n − z n ∥ 2 = ( 1 − α n , i ) ∥ z n − w ∥ 2 + α n , i ∥ T i z n − z n ∥ 2 + α n , i ∥ z n − w ∥ 2 + 2 α n , i 〈 T i z n − z n , z n − w 〉 − α n , i ( 1 − α n , i ) ∥ T i z n − z n ∥ 2 ≤ ∥ z n − w ∥ 2 + 2 α n , i 〈 z n − w , z n − T i z n 〉 + 2 α n , i 〈 T i z n − z n , z n − w 〉 − α n , i ( 1 − α n , i ) ∥ T i z n − z n ∥ 2 = ∥ z n − w ∥ 2 − α n , i ( 1 − α n , i ) ∥ T i z n − z n ∥ 2 .
(2.10)

From (2.10) and using the methods in Theorem 2.1, we can get the following convergence result.

Theorem 2.2 Let C be a nonempty closed convex subset of a real Hilbert space ℋ. Suppose that assumptions (A1)-(A5) are satisfied and { T i } i ∈ Γ is a family of nonexpansive mappings from C into itself and a nonempty common fixed points set F. Let { x n } be a sequence generated by the following scheme:

{ x 0 ∈ H  chosen arbitrarily , C 1 , i = D 1 , i = C , C 1 = â‹‚ i ∈ Γ C 1 , i , D 1 = â‹‚ i ∈ Γ D 1 , i , x 1 = Pr C 1 ∩ D 1 x 0 , y n = argmin { λ n f ( x n , y ) + 1 2 ∥ y − x n ∥ 2 : y ∈ C } , z n = argmin { λ n f ( y n , y ) + 1 2 ∥ z − x n ∥ 2 : z ∈ C } , y n , i = ( 1 − α n , i ) z n + α n , i T i z n , C n + 1 , i = { z ∈ C n , i : ∥ y n , i − z ∥ 2 ≤ ∥ z n − z ∥ 2 − α n , i ( 1 − α n , i ) ∥ z n − T i z n ∥ 2 } , C n + 1 = â‹‚ i ∈ Γ C n + 1 , i , D n + 1 , i = { z ∈ D n , i : ∥ y n , i − z ∥ ≤ ∥ x n − z ∥ } , D n + 1 = â‹‚ i ∈ Γ D n + 1 , i , x n + 1 = Pr C n + 1 ∩ D n + 1 x 0 , 0 < lim inf α n , i ≤ lim sup α n , i < 1 , { λ n } ⊂ [ a , b ]  for some  a , b ∈ ( 0 , 1 L ) ,  where  L = max { 2 c 1 , 2 c 2 } .

Then the sequences { x n }, { y n } and { z n } converge strongly to the same point Pr F ∩ Sol ( f , C ) x 0 .

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Thanh, D.D. Strong convergence theorems for equilibrium problems involving a family of nonexpansive mappings. Fixed Point Theory Appl 2014, 200 (2014). https://doi.org/10.1186/1687-1812-2014-200

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