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Convergence theorems for a system of equilibrium problems and fixed point problems of a strongly nonexpansive sequence

Fixed Point Theory and Applications20132013:193

https://doi.org/10.1186/1687-1812-2013-193

  • Received: 29 August 2012
  • Accepted: 9 July 2013
  • Published:

Abstract

The purpose of this paper is to prove a strong convergence theorem of an iterative scheme associated to a strongly nonexpansive sequence for finding a common element of the set of equilibrium problems and the set of fixed point problems of a pair of sequences of nonexpansive mappings where one of them is a strongly nonexpansive sequence. Moreover, in the last section, by using our main result, we obtain a strong convergence theorem of an iterative scheme associated to a strongly nonexpansive sequence for finding a common element of the set of a finite family of equilibrium problems and the set of fixed point problems of a pair of sequences of nonexpansive mappings where one of them is a strongly nonexpansive sequence in a Hilbert space, and we also give some examples to support our main result.

Keywords

  • nonexpansive mappings
  • strongly nonexpansive sequence
  • equilibrium problem
  • fixed point

1 Introduction

Throughout this paper, we assume that H is a real Hilbert space with the inner product , and the norm . A mapping T of C into itself is called nonexpansive if T x T y x y for all x , y H . The set of fixed points of T is denoted by F ( T ) , i.e., F ( T ) = { x H : T x = x } . It is known that F ( T ) is closed and convex if T is nonexpansive. Let P C be a metric projection of H onto C, i.e., for x H , P C x satisfies the property
x P C x = min y C x y .
We use and to denote weak and strong convergence, respectively. Let { T n } be a sequence of mappings of C into H. The set of common fixed points of { T n } is denoted by F ( { T n } ) = n = 1 F ( T n ) . Recall the main concepts as follows:
  1. (1)

    A sequence { z n } in C is said to be an approximate fixed point sequence of { T n } if z n T n z n 0 . The set of all bounded approximate fixed point sequences of { T n } is denoted by F ˜ ( { T n } ) ; see [1]. It is clear that if { T n } has a common fixed point, then F ˜ ( { T n } ) is nonempty.

     
  2. (2)
    A sequence { T n } is said to be a strongly nonexpansive sequence if each T n is nonexpansive and
    x n y n ( T n x n T n y n ) 0 ,

    whenever { x n } and { y n } are sequences in C such that { x n y n } is bounded and x n y n T n x n T n y n 0 .

     
  3. (3)

    A sequence { T n } having a common fixed point is said to satisfy the condition (Z) if every weak cluster point of { x n } is a common fixed point whenever { x n } F ˜ ( { T n } ) .

     
  4. (4)
    A sequence { T n } of nonexpansive mappings of C into H is said to satisfy the condition (R) if
    lim n sup y D T n + 1 y T n y = 0

    for every nonempty bounded subset D of C; see [2].

     

Example 1.1 Let be a set of real numbers. For every n N , the mapping T n : R R is defined by T n x = 1 n x for all x R .

Then { T n } is a nonexpansive sequence, but it is not a strongly nonexpansive sequence.

Example 1.2 For every n N , the mapping T n : [ 0 , 1 ] [ 0 , 1 ] is defined by T n x = ( 1 1 n ) x for all x [ 0 , 1 ] .

Then { T n } is a strongly nonexpansive sequence.

Solution It is easy to see that T n is a nonexpansive mapping for all n N .

Let { x n } and { y n } be sequences in [ 0 , 1 ] with { x n y n } being bounded and | x n y n | | T n x n T n y n | 0 as n .

Since x n y n ( T n x n T n y n ) = 1 n ( x n y n ) , for all n N , then we have
x n y n ( T n x n T n y n ) 0 as  n .

Then { T n } is a strongly nonexpansive sequence.

Let G : C × C R be a bifunction. The equilibrium problem for G is to determine its equilibrium points, i.e., the set
EP ( G ) = { x G : G ( x , y ) 0 , y C } .

It is a unified model of several problems, namely, variational inequality problem, complementary problem, saddle point problem, optimization problem, fixed point problem, etc.; see [35]. Several iterative methods have been proposed to solve the equilibrium problem; see, for instance, [68]. In 2005, Combettes and Hirstoaga [4] introduced some iterative schemes of finding the best approximation to the initial data when EP ( G ) is nonempty and proved a strong convergence theorem.

Also in [4], Combettes and Hiratoaga, following [3], defined S r : H C by
S r ( x ) = { z C : G ( z , y ) + 1 r y z , z x 0 , y C } .

They proved that under suitable hypotheses S r is single-valued and firmly nonexpansive with F ( S r ) = EP ( G ) .

In 2007, Takahashi and Takahashi [9] proved the following theorem.

Theorem 1.3 Let C be a nonempty closed convex subset of H. Let G be a bifunction from C × C to satisfying
  1. (A1)

    G ( x , x ) = 0 , x C ;

     
  2. (A2)

    G is monotone, i.e., G ( x , y ) + G ( y , x ) 0 , x , y C ;

     
  3. (A3)

    x , y , z C , lim t 0 + G ( t z + ( 1 t ) x , y ) G ( x , y ) ;

     
  4. (A4)

    x C , y G ( x , y ) is convex and lower semicontinuous;

     
and let S be a nonexpansive mapping of C into H such that F ( S ) EP ( G ) . Let f be a contraction of H into itself, and let { x n } and { u n } be sequences generated by x 1 H and
G ( u n , y ) + 1 r n y u n , u n x n 0 , y C , x n + 1 = α n f ( x n ) + ( 1 α n ) S u n
for all n N , where { α n } [ 0 , 1 ] and { r n } ( 0 , 1 ) satisfy
  1. (C1)

    lim n α n = 0 ;

     
  2. (C2)

    n = 1 α n = ;

     
  3. (C3)

    n = 1 | α n + 1 α n | < ;

     

and lim inf n r n > 0 and n = 1 | r n + 1 r n | < .

Then { x n } and { u n } converge strongly to z F ( S ) EP ( G ) , where z = P F ( S ) EP ( G ) f ( z ) .

Very recently, in 2011, Aoyama and Kimura [10] proved a strong convergence theorem of the iterative scheme of { x n } associated to a strongly nonexpansive sequence as follows.

Theorem 1.4 Let H be a Hilbert space, let C be a nonempty closed convex subset of H, and let { S n } and { T n } be sequences of nonexpansive self-mappings of C. Suppose that F = F ( { S n } ) F ( { T n } ) is nonempty, both { S n } and { T n } satisfy the conditions (R) and (Z), and { S n } or { T n } is a strongly nonexpansive sequence. Let { α n } and { β n } be sequences in [0, 1] such that
lim n α n = 0 , n = 1 α n = and 0 < lim inf n β n lim sup n β n < 1 .
Let x , u C and let { x n } be a sequence in C defined by x 1 = x C and
x n + 1 = β n x n + ( 1 β n ) S n ( α n u + ( 1 α n ) T n x n )

for all n N . Then { x n } converges strongly to P F u .

For x 1 , u , v C , let { u n } , { v n } , { y n } and { x n } be the sequences defined by
{ F 1 ( u n , u ) + 1 r n u u n , u n x n 0 , F 2 ( v n , v ) + 1 s n v v n , v n x n 0 , y n = δ n u n + ( 1 δ n ) v n , x n + 1 = β n x n + ( 1 β n ) S n ( α n f ( T n y n ) + ( 1 α n ) T n y n ) , n 1 ,
(1.1)

where f : C C is a contractive mapping with α ( 0 , 1 2 ) and { S n } , { T n } are sequences of nonexpansive mappings, one of them is a strongly nonexpansive sequence.

In this paper, inspired and motivated by [10] and [9], we prove that a strong convergence theorem of the iterative scheme { x n } defined by (1.1) converges strongly to z = P F f ( z ) , where F = EP ( F 1 ) EP ( F 2 ) F ( { S n } ) F ( { T n } ) , under the conditions (R) and (Z) and suitable conditions of { r n } , { s n } , { α n } , { β n } and { δ n } .

2 Preliminaries

In this section, we need the following lemmas to prove our main result in the next section.

Lemma 2.1 (See [11])

Given x H and y C . Then P C x = y if and only if the following inequality holds:
x y , y z 0 , z C .

Lemma 2.2 (See [12])

Let { s n } be a sequence of nonnegative real numbers satisfying
s n + 1 = ( 1 α n ) s n + α n β n , n 0 ,
where { α n } , { β n } satisfy the conditions
  1. (1)

    { α n } [ 0 , 1 ] , n = 1 α n = ;

     
  2. (2)

    lim sup n β n 0 or n = 1 | α n β n | < .

     

Then lim n s n = 0 .

Lemma 2.3 (See [13])

Let { x n } and { z n } be bounded sequences in a Banach space X, and let { β n } be a sequence in [ 0 , 1 ] with 0 < lim inf n β n lim sup n β n < 1 . Suppose that
x n + 1 = β n x n + ( 1 β n ) z n
for all integers n 0 and
lim sup n ( z n + 1 z n x n + 1 x n ) 0 .

Then lim n x n z n = 0 .

Lemma 2.4 (See [14])

Let C be a closed convex subset of a strictly convex Banach space E. Let { T n : n N } be a sequence of nonexpansive mappings on C. Suppose that n = 1 F ( T n ) is nonempty. Let { λ n } be a sequence of positive numbers with n = 1 λ n = 1 . Then a mapping S on C defined by
S ( x ) = n = 1 λ n T n x

for all x C is well defined, nonexpansive and F ( S ) = n = 1 F ( T n ) holds.

Lemma 2.5 (See [4])

Let C be a nonempty closed convex subset of a Hilbert space H, and let G : C × C R satisfy
  1. (A1)

    G ( x , x ) = 0 , x C ;

     
  2. (A2)

    G is monotone, i.e., G ( x , y ) + G ( y , x ) 0 , x , y C ;

     
  3. (A3)

    x , y , z C , lim t 0 + G ( t z + ( 1 t ) x , y ) G ( x , y ) ;

     
  4. (A4)

    x C , y G ( x , y ) is convex and lower semicontinuous.

     
For x H and r > 0 , define a mapping S r : H C as follows:
S r ( x ) = { z C : G ( z , y ) + 1 r y z , z x 0 , y C } .
Then S r is well defined and the following hold:
  1. (1)

    S r is single-valued;

     
  2. (2)

    S r is firmly nonexpansive, i.e., S r ( x ) S r ( y ) 2 S r ( x ) S r ( y ) , x y , x , y H ;

     
  3. (3)

    F ( S r ) = EP ( G ) ;

     
  4. (4)

    EP ( G ) is closed and convex.

     

Lemma 2.6 (See [11]) (Demiclosedness principle)

Assume that T is a nonexpansive self-mapping of a closed convex subset C of a Hilbert space H. If T has a fixed point, then I T is demiclosed. That is, whenever { x n } is a sequence in C weakly converging to some x C and the sequence { ( I T ) x n } converges strongly to some y, it follows that ( I T ) x = y . Here, I is the identity mapping of H.

Lemma 2.7 Let H be a real Hilbert space. Then, for all x , y H ,
x + y 2 x 2 + 2 y , x + y .

Lemma 2.8 (See [10])

Let H be a Hilbert space, let C be a nonempty subset of H, and let { S n } and { T n } be the sequences of nonexpansive self-mappings of C. Suppose that { S n } and { T n } satisfy the condition (R) and that { T n y : n N , y D } is bounded for any bounded subset D of C. Then { S n T n } satisfies the condition (R).

Lemma 2.9 (See [1])

Let H be a Hilbert space, let C be a nonempty subset of H, and let { S n } and { T n } be the sequences of nonexpansive self-mappings of C. Suppose that { S n } or { T n } is a strongly nonexpansive sequence and that F ˜ ( { S n } ) F ˜ ( { T n } ) is nonempty. Then F ˜ ( { S n } ) F ˜ ( { T n } ) = F ˜ ( { S n T n } ) .

3 Main result

Theorem 3.1 Let H be a Hilbert space, let C be a nonempty closed convex subset of H. Let F 1 and F 2 be two bifunctions from C × C into satisfying (A1)-(A4), respectively, and let { S n } and { T n } be sequences of nonexpansive self-mappings of C with F = EP ( F 1 ) EP ( F 2 ) F ( { S n } ) F ( { T n } ) . Let { T n } or { S n } be a sequence of strongly nonexpansive mappings, and let f : C C be a contractive mapping with α ( 0 , 1 2 ) . Let { x n } , { u n } , { v n } be sequences generated by x 1 , u , v C and
{ F 1 ( u n , u ) + 1 r n u u n , u n x n 0 , F 2 ( v n , v ) + 1 s n v v n , v n x n 0 , y n = δ n u n + ( 1 δ n ) v n , x n + 1 = β n x n + ( 1 β n ) S n ( α n f ( T n y n ) + ( 1 α n ) T n y n ) , n 1 ,
(3.1)
where { α n } , { β n } [ 0 , 1 ] , { r n } , { s n } ( a , b ) [ 0 , 1 ] . Assume that the following conditions hold:
  1. (i)

    lim n α n = 0 and n = 1 α n = ;

     
  2. (ii)

    0 < lim inf n β n lim sup n β n < 1 ;

     
  3. (iii)

    n = 0 | r n + 1 r n | , n = 0 | s n + 1 s n | < ;

     
  4. (iv)

    lim n δ n = δ ( 0 , 1 ) ;

     
  5. (v)

    { S n } and { T n } satisfy the conditions R and Z.

     

Then the sequences { x n } , { u n } , { v n } , { y n } converge strongly to z = P F f ( z ) .

Proof Let v F . From the definition of x n , we have
x n + 1 v = β n ( x n v ) + ( 1 β n ) ( S n ( α n f ( T n y n ) + ( 1 α n ) T n y n ) v ) β n x n v + ( 1 β n ) α n f ( T n y n ) + ( 1 α n ) T n y n v β n x n v + ( 1 β n ) ( α n f ( T n y n ) v + ( 1 α n ) T n y n v ) β n x n v + ( 1 β n ) ( α n f ( T n y n ) f ( v ) + α n f ( v ) v + ( 1 α n ) T n y n v ) β n x n v + ( 1 β n ) ( α n α y n v + α n f ( v ) v + ( 1 α n ) y n v ) = β n x n v + ( 1 β n ) ( α n f ( v ) v + ( 1 α n ( 1 α ) ) y n v ) .
(3.2)
From Lemma 2.5 and (3.1), we have EP ( F 1 ) = F ( S r n ) , EP ( F 2 ) = F ( S s n ) , S r n x n = u n and S s n x n = v n . By v F and the nonexpansiveness of S r n and S s n , we have
y n v = δ n ( u n v ) + ( 1 δ n ) ( v n v ) δ n u n v + ( 1 δ n ) v n v = δ n S r n x n v + ( 1 δ n ) S s n x n v x n v .
(3.3)
Substituting (3.3) into (3.2), we have
x n + 1 v β n x n v + ( 1 β n ) ( α n f ( v ) v + ( 1 α n ( 1 α ) ) y n v ) β n x n v + ( 1 β n ) ( α n f ( v ) v + ( 1 α n ( 1 α ) ) x n v ) = β n x n v + ( 1 β n ) α n f ( v ) v + ( 1 β n ) ( 1 α n ( 1 α ) ) x n v = ( 1 β n ) α n f ( v ) v + ( 1 α n ( 1 β n ) ( 1 α ) ) x n v max { x n v , f ( v ) v 1 α } .

By induction we can conclude that { x n } is bounded and so are { u n } , { v n } , { y n } . Next, we show that F ˜ ( { S n A n } ) = F ˜ ( { S n } ) and F ˜ ( { A n T n } ) = F ˜ ( { T n } ) , where A n = α n f + ( 1 α n ) I .

Let { z n } be a bounded sequence in C. From the nonexpansiveness of S n , we have
S n A n z n S n z n A n z n z n = α n f ( z n ) z n .
(3.4)
From (3.4) and α n 0 as n , we have
lim n S n A n z n S n z n = 0 .
(3.5)
Let { z n } F ˜ ( { S n A n } ) , then we have
z n S n z n z n S n A n z n + S n A n z n S n z n .
From (3.5), we have
lim n z n S n z n = 0 ,
which implies that { z n } F ˜ ( { S n } ) . It follows that
F ˜ ( { S n A n } ) F ˜ ( { S n } ) .
(3.6)
Let { z n } F ˜ ( { S n } ) , then we have
z n S n A n z n z n S n z n + S n z n S n A n z n .
From (3.5), we have
lim n z n S n A n z n = 0 ,
which implies that { z n } F ˜ ( { S n A n } ) . It follows that
F ˜ ( { S n } ) F ˜ ( { S n A n } ) .
(3.7)
From (3.6) and (3.7), we have
F ˜ ( { S n } ) = F ˜ ( { S n A n } ) .
(3.8)
Let { z n } be a bounded sequence in C, then we have { T n z n } is bounded and so is { f ( T n z n ) } . Since
A n T n z n T n z n = α n f ( T n z n ) T n z n
and α n 0 as n , we have
lim n A n T n z n T n z n = 0 .
(3.9)
Let { z n } F ˜ ( { A n T n } ) , then we have
z n T n z n z n A n T n z n + A n T n z n T n z n .
From (3.9), we have
lim n z n T n z n = 0 ,
which implies that
{ z n } F ˜ ( { T n } ) .
It follows that
F ˜ ( { A n T n } ) F ˜ ( { T n } ) .
(3.10)
Let { z n } F ˜ ( { T n } ) , then we have
z n A n T n z n z n T n z n + T n z n A n T n z n .
From (3.9), we have
lim n z n A n T n z n = 0 ,
which implies that
{ z n } F ˜ ( { A n T n } ) .
It follows that
F ˜ ( { T n } ) F ˜ ( { A n T n } ) .
(3.11)
From (3.10) and (3.11), we have
F ˜ ( { T n } ) = F ˜ ( { A n T n } ) .
(3.12)
Next, we show that
F ˜ ( { S n A n T n } ) = F ˜ ( { S n } ) F ˜ ( { T n } ) .
Since is nonempty, from (3.8), (3.12), we have
F ˜ ( { S n A n } ) F ˜ ( { T n } ) = F ˜ ( { S n } ) F ˜ ( { T n } )
(3.13)
and
F ˜ ( { S n } ) F ˜ ( { A n T n } ) = F ˜ ( { S n } ) F ˜ ( { T n } ) .
(3.14)
Suppose that { S n } is a strongly nonexpansive sequence. From (3.14) and Lemma 2.9, we have
F ˜ ( { S n A n T n } ) = F ˜ ( { S n } ) F ˜ ( { A n T n } ) = F ˜ ( { S n } ) F ˜ ( { T n } ) .
(3.15)
On the other hand, suppose that { T n } is a strongly nonexpansive sequence. From (3.13) and Lemma 2.9, we have
F ˜ ( { S n A n T n } ) = F ˜ ( { S n A n } ) F ˜ ( { T n } ) = F ˜ ( { S n } ) F ˜ ( { T n } ) .
(3.16)
From (3.16) and (3.15), we have F ˜ ( { S n A n T n } ) = F ˜ ( { S n } ) F ˜ ( { T n } ) . Next, we show that { A n } and { S n A n T n } satisfy the condition (R). It is easy to see that A n is a nonexpansive mapping for every n N and that { A n y : n N , y D } is bounded, where D is a bounded subset of C. Let y D , then we have
A n + 1 y A n y = α n + 1 f ( y ) + ( 1 α n + 1 ) y α n f ( y ) ( 1 α n ) y | α n + 1 α n | f ( y ) + | α n + 1 α n | y .
From the condition (i), we have
lim n sup y D A n + 1 y A n y = 0 .

It follows that { A n } satisfies the condition (R). From Lemma 2.8, we have that { S n A n } satisfies the condition (R). From the nonexpansiveness of T n and F , we have { T n y : n N , y D } is bounded for any bounded subset D of C. From Lemma 2.8, we have that { S n A n T n } satisfies the condition (R).

Next, we show that
lim n x n + 1 x n = 0 .
(3.17)
Put
x n + 1 = β n x n + ( 1 β n ) w n ,
(3.18)
where w n = S n ( α n f ( T n y n ) + ( 1 α n ) T n y n ) . From the definition of w n , we have
w n + 1 w n = S n + 1 A n + 1 T n + 1 y n + 1 S n A n T n y n S n + 1 A n + 1 T n + 1 y n + 1 S n A n T n y n + 1 + S n A n T n y n + 1 S n A n T n y n sup y D S n + 1 A n + 1 T n + 1 y S n A n T n y + y n + 1 y n ,
(3.19)
where D is a bounded subset of C. Besides, we have
y n + 1 y n = δ n + 1 u n + 1 + ( 1 δ n + 1 ) v n + 1 δ n u n ( 1 δ n ) v n = δ n + 1 u n + 1 δ n + 1 u n + δ n + 1 u n ( 1 δ n + 1 ) v n + ( 1 δ n + 1 ) v n + ( 1 δ n + 1 ) v n + 1 δ n u n ( 1 δ n ) v n = δ n + 1 ( u n + 1 u n ) + ( δ n + 1 δ n ) u n + ( 1 δ n + 1 ) ( v n + 1 v n ) + ( δ n δ n + 1 ) v n δ n + 1 u n + 1 u n + | δ n + 1 δ n | u n + ( 1 δ n + 1 ) v n + 1 v n + | δ n δ n + 1 | v n .
(3.20)
From (3.1) and Lemma 2.5, we have u n = S r n x n . This implies that
F 1 ( u n , u ) + 1 r n u u n , u n x n 0 for all  u C
(3.21)
and
F 1 ( u n + 1 , u ) + 1 r n + 1 u u n + 1 , u n + 1 x n + 1 0 for all  u C .
(3.22)
Putting u = u n + 1 in (3.21) and u = u n in (3.22), we have
F 1 ( u n , u n + 1 ) + 1 r n u n + 1 u n , u n x n 0
(3.23)
and
F 1 ( u n + 1 , u n ) + 1 r n + 1 u n u n + 1 , u n + 1 x n + 1 0 .
(3.24)
Summing up the last two inequalities and using (A2), we obtain
u n + 1 u n , u n x n r n u n + 1 x n + 1 r n + 1 0 .
This implies that
u n + 1 u n , u n u n + 1 + u n + 1 x n r n r n + 1 ( u n + 1 x n + 1 ) 0 .
Hence,
u n + 1 u n 2 u n + 1 u n , u n + 1 x n r n r n + 1 ( u n + 1 x n + 1 ) = u n + 1 u n , u n + 1 x n + 1 + x n + 1 x n r n r n + 1 ( u n + 1 x n + 1 ) = u n + 1 u n , x n + 1 x n + ( 1 r n r n + 1 ) ( u n + 1 x n + 1 ) u n + 1 u n ( x n + 1 x n + 1 r n + 1 | r n + 1 r n | u n + 1 x n + 1 ) u n + 1 u n ( x n + 1 x n + 1 a | r n + 1 r n | u n + 1 x n + 1 ) .
Then we have
u n + 1 u n x n + 1 x n + 1 a | r n + 1 r n | u n + 1 x n + 1 .
(3.25)
From (3.1) and Lemma 2.5, we have v n = S s n x n . This implies that
F 2 ( v n , v ) + 1 s n v v n , v n x n 0 for all  v C .
By using the same method as (3.25), we have
v n + 1 v n x n + 1 x n + 1 a | s n + 1 s n | v n + 1 x n + 1 .
(3.26)
Substituting (3.25) and (3.26) into (3.20), we have
y n + 1 y n δ n + 1 u n + 1 u n + | δ n + 1 δ n | u n + ( 1 δ n + 1 ) v n + 1 v n + | δ n δ n + 1 | v n δ n + 1 ( x n + 1 x n + 1 a | r n + 1 r n | u n + 1 x n + 1 ) + ( 1 δ n + 1 ) ( x n + 1 x n + 1 a | s n + 1 s n | v n + 1 x n + 1 ) + 2 M | δ n δ n + 1 | x n + 1 x n + 1 a | r n + 1 r n | u n + 1 x n + 1 + 1 a | s n + 1 s n | v n + 1 x n + 1 + 2 M | δ n δ n + 1 | ,
(3.27)
where M = sup n N { u n , v n } . Substituting (3.27) into (3.19), we have
w n + 1 w n sup y D S n + 1 A n + 1 T n + 1 y S n A n T n y + y n + 1 y n sup y D S n + 1 A n + 1 T n + 1 y S n A n T n y + x n + 1 x n + 1 a | r n + 1 r n | u n + 1 x n + 1 + 1 a | s n + 1 s n | v n + 1 x n + 1 + 2 M | δ n δ n + 1 | .
(3.28)
From (3.28), the conditions (iii), (iv) and { S n A n T n } satisfying the condition (R), we have
lim sup n ( w n + 1 w n x n + 1 x n ) 0 .
(3.29)
From Lemma 2.3 and the definition of x n , we have
lim n x n w n = 0 .
(3.30)
From the definition of x n , we have
x n + 1 x n = ( 1 β n ) ( w n x n ) .
(3.31)
From (3.30), (3.31) and the condition (ii), we have
lim n x n + 1 x n = 0 .
Next, we show that
lim n y n x n = 0 .
From the definition of y n , we have
y n x n δ n u n x n + ( 1 δ n ) v n x n .
(3.32)
Next, we show that
lim n u n x n = lim n v n x n = 0 .
Let v F . From the definition of x n , we have
x n + 1 v 2 β n x n v 2 + ( 1 β n ) S n ( α n f ( T n y n ) + ( 1 α n ) T n y n ) v 2 β n x n v 2 + ( 1 β n ) α n ( f ( T n y n ) v ) + ( 1 α n ) ( T n y n v ) 2 β n x n v 2 + ( 1 β n ) ( α n f ( T n y n ) v 2 + ( 1 α n ) T n y n v 2 ) β n x n v 2 + ( 1 β n ) ( α n f ( T n y n ) v 2 + ( 1 α n ) y n v 2 ) β n x n v 2 + ( 1 β n ) ( α n f ( T n y n ) v 2 + ( 1 α n ) ( δ n u n v 2 + ( 1 δ n ) v n v 2 ) ) .
(3.33)
From the firm nonexpansiveness of S r n and u n = S r n x n , we have
u n v 2 = S r n x n S r n v 2 u n v , x n v = 1 2 ( u n v 2 + x n v 2 u n x n 2 ) .
It implies that
u n v 2 x n v 2 u n x n 2 .
(3.34)
Since S s n is a firmly nonexpansive mapping and v n = S s n x n , by using the same method as (3.34), we have
v n v 2 x n v 2 v n x n 2 .
(3.35)
Substituting (3.34), (3.35) into (3.33), we have
x n + 1 v 2 β n x n v 2 + ( 1 β n ) ( α n f ( T n y n ) v 2 + ( 1 α n ) ( δ n u n v 2 + ( 1 δ n ) v n v 2 ) ) β n x n v 2 + ( 1 β n ) ( α n f ( T n y n ) v 2 + ( 1 α n ) ( δ n ( x n v 2 u n x n 2 ) + ( 1 δ n ) ( x n v 2 v n x n 2 ) ) ) = β n x n v 2 + ( 1 β n ) ( α n f ( T n y n ) v 2 + ( 1 α n ) ( δ n x n v 2 δ n u n x n 2 + ( 1 δ n ) x n v 2 ( 1 δ n ) v n x n 2 ) ) = β n x n v 2 + ( 1 β n ) ( α n f ( T n y n ) v 2 + ( 1 α n ) ( x n v 2 δ n u n x n 2 ( 1 δ n ) v n x n 2 ) ) = β n x n v 2 + ( 1 β n ) α n f ( T n y n ) v 2 + ( 1 α n ) ( 1 β n ) ( x n v 2 δ n u n x n 2 ( 1 δ n ) v n x n 2 ) = β n x n v 2 + ( 1 β n ) α n f ( T n y n ) v 2 + ( 1 α n ) ( 1 β n ) x n v 2 δ n ( 1 α n ) ( 1 β n ) u n x n 2 ( 1 δ n ) ( 1 α n ) ( 1 β n ) v n x n 2 x n v 2 + α n f ( T n y n ) v 2 δ n ( 1 α n ) ( 1 β n ) u n x n 2 ( 1 δ n ) ( 1 α n ) ( 1 β n ) v n x n 2 .
(3.36)
From (3.36), we have
δ n ( 1 α n ) ( 1 β n ) u n x n 2 x n v 2 x n + 1 v 2 + α n f ( T n y n ) v 2 ( 1 δ n ) ( 1 α n ) ( 1 β n ) v n x n 2 ( x n v + x n + 1 v ) x n + 1 x n + α n f ( T n y n ) v 2 ( 1 δ n ) ( 1 α n ) ( 1 β n ) v n x n 2 ( x n v + x n + 1 v ) x n + 1 x n + α n f ( T n y n ) v 2 .
From the conditions (i), (ii), (iv) and (3.17), we have
lim n u n x n = 0 .
(3.37)
By using the method as (3.37), we have
lim n v n x n = 0 .
(3.38)
From (3.32), (3.37) and (3.38), we have
lim n y n x n = 0 .
(3.39)
Next, we show that
{ y n } F ˜ ( { S n } ) F ˜ ( { T n } ) .
(3.40)
Since
S n A n T n y n y n S n A n T n y n x n + x n y n = w n x n + x n y n ,
from (3.30) and (3.39), we have
lim n S n A n T n y n y n = 0 .
Since { y n } is bounded, we have
{ y n } F ˜ ( { S n A n T n } ) .
(3.41)

Since F ˜ ( { S n A n T n } ) = F ˜ ( { S n } ) F ˜ ( { T n } ) and (3.41), we have (3.40).

Next, we show that
lim n S n m n m n = 0 ,
where m n = α n f ( T n y n ) + ( 1 α n ) T n y n . From the definition of m n , we have
S n m n m n S n m n x n + m n x n = S n m n x n + α n ( f ( T n y n ) x n ) + ( 1 α n ) ( T n y n x n ) w n x n + α n f ( T n y n ) x n + ( 1 α n ) T n y n x n w n x n + α n f ( T n y n ) x n + T n y n y n + y n x n .
From (3.39), (3.40), (3.30) and the condition (i), we have
lim n S n m n m n = 0 .
Next, we show that
lim sup n f ( z ) z , m n z 0 ,

where z = P F f ( z ) . Since { y n } is bounded, there exists a subsequence { y n i } of { y n } converging weakly to v, that is, y n i v as i . From (3.40), { S n } and { T n } satisfying the condition (Z), we have v F ( { S n } ) F ( { T n } ) .

Define the mapping Q : C C by
Q ( x ) = δ S r n x + ( 1 δ ) S s n x for all  x C ,
where lim n δ n = δ ( 0 , 1 ) . From the nonexpansiveness of S r n , S s n and Lemma 2.4, we have
F ( Q ) = F ( S r n ) F ( S s n ) = EP ( F 1 ) EP ( F 2 ) .
From the definitions of y n and Q, we have
x n Q x n x n y n + y n Q x n x n y n + δ n u n + ( 1 δ n ) v n δ S r n x n ( 1 δ ) S s n x n x n y n + | δ n δ | u n + | δ n δ | v n .
(3.42)
From (3.39), (3.42) and the condition (iv), we have
lim n x n Q x n = 0 .
(3.43)
From (3.39) and y n i v as i , we have x n i v as i . By (3.43), x n i v as i and Lemma 2.6, we have
v F ( Q ) = EP ( F 1 ) EP ( F 2 ) .
Hence,
v EP ( F 1 ) EP ( F 2 ) F ( { S n } ) F ( { T n } ) = F .
(3.44)
By (3.40), (3.44) and the condition (i), we have
lim sup n f ( z ) z , m n z = lim sup n ( α n f ( z ) z , f ( T n y n ) T n y n + f ( z ) z , T n y n z ) = lim i ( α n i f ( z ) z , f ( T n i y n i ) T n i y n i + f ( z ) z , T n i y n i z ) = lim i ( α n i f ( z ) z , f ( T n i y n i ) T n i y n i + f ( z ) z , T n i y n i y n i + f ( z ) z , y n i z ) = f ( z ) z , v z 0 .
Finally, we show that the sequence { x n } converges strongly to z = P F f ( z ) . From the definition of { x n } , we have
x n + 1 z 2 = β n ( x n z ) + ( 1 β n ) ( S n m n z ) 2 β n x n z 2 + ( 1 β n ) S n m n z 2 β n x n z 2 + ( 1 β n ) m n z 2 .
(3.45)
Since m n = α n f ( T n y n ) + ( 1 α n ) T n y n , we have
m n z 2 = α n ( f ( T n y n ) z ) + ( 1 α n ) ( T n y n z ) 2 ( 1 α n ) 2 T n y n z 2 + 2 α n f ( T n y n ) z , m n z ( 1 α n ) x n z 2 + 2 α n f ( T n y n ) f ( z ) , m n z + 2 α n f ( z ) z , m n z ( 1 α n ) x n z 2 + 2 α n α x n z m n z + 2 α n f ( z ) z , m n z ( 1 α n ) x n z 2 + α n α ( x n z 2 + m n z 2 ) + 2 α n f ( z ) z , m n z = ( 1 α n ) x n z 2 + α n α x n z 2 + α n α m n z 2 + 2 α n f ( z ) z , m n z = ( 1 α n ( 1 α ) ) x n z 2 + α n α m n z 2 + 2 α n f ( z ) z , m n z .
This implies that
m n z 2 1 α n ( 1 α ) 1 α n α x n z 2 + 2 α n 1 α n α f ( z ) z , m n z = 1 α n α + α n α α n ( 1 α ) 1 α n α x n z 2 + 2 α n 1 α n α f ( z ) z , m n z = ( 1 α n ( 1 2 α ) 1 α n α ) x n z 2 + 2 α n 1 α n α f ( z ) z , m n z .
(3.46)
Substituting (3.46) into (3.45), we have
x n + 1 z 2 β n x n z 2 + ( 1 β n ) m n z 2 β n x n z 2 + ( 1 β n ) ( ( 1 α n ( 1 2 α ) 1 α n α ) x n z 2 + 2 α n 1 α n α f ( z ) z , m n z ) β n x n z 2 + ( 1 β n ) ( 1 α n ( 1 2 α ) 1 α n α ) x n z 2 + 2 α n ( 1 β n ) 1 α n α f ( z ) z , m n z = β n x n z 2 + ( ( 1 β n ) α n ( 1 2 α ) ( 1 β n ) 1 α n α ) x n z 2 + 2 α n ( 1 β n ) 1 α n α f ( z ) z , m n z = ( 1 α n ( 1 2 α ) ( 1 β n ) 1 α n α ) x n z 2 + α n ( 1 β n ) ( 1 2 α ) 1 α n α 2 f ( z ) z , m n z ( 1 2 α ) .
(3.47)

Applying (3.47), the conditions (i), (ii) and Lemma 2.2, we have { x n } converges strongly to z = P F f ( z ) . From (3.39), (3.37) and (3.38), it is easy to see that { y n } , { u n } , { v n } converge strongly to z = P F f ( z ) . This completes the proof. □

4 Applications

In this section, we give three examples for a strongly nonexpansive sequence and prove a strong convergence theorem associated to the variational inequality problem.

Before we give three examples, we need the following definition and lemmas.

Definition 4.1 Let C be a nonempty closed convex subset of a real Hilbert space H. A mapping A : C H is called an α-inverse strongly monotone mapping if there exists an α > 0 such that
x y , a x A y α A x A y 2

for all x , y C .

A mapping A : C H is called α-strongly monotone if there exists α > 0 such that
x y , a x A y α x y 2

for all x , y C .

A mapping T : C C is called a κ-strictly pseudo-contractive mapping if there is κ [ 0 , 1 ) such that
T x T y 2 x y 2 + κ ( I T ) x ( I T ) y 2
(4.1)

for all x , y C .

Then (4.1) is equivalent to
x y , ( I T ) x ( I T ) y 1 κ 2 ( I T ) x ( I T ) y 2

for all x , y C .

The set of solutions of the variational inequality problem of the mapping A : C H is denoted by VI ( C , A ) , that is,
VI ( C , A ) = { x C : y x , A x 0 , y C } .
Let A , B : C H be two mappings. In 2013, Kangtunyakarn [15] modified VI ( C , A ) as follows:
VI ( C , a A + ( 1 a ) B ) = { x C : y x , ( a A + ( 1 a ) B ) x 0 , y C  and  a ( 0 , 1 ) } .

Remark 4.1 If T : C C is a κ-strictly pseudo-contractive mapping with F ( T ) , then ( I T ) is a 1 κ 2 -inverse strongly monotone mapping and F ( T ) = VI ( C , I T ) .

Lemma 4.2 (See [16])

Let H be a Hilbert space, let C be a nonempty closed convex subset of H, and let A be a mapping of C into H. Let u C . Then, for λ > 0 ,
u = P C ( I λ A ) u u VI ( C , A ) ,

where P C is the metric projection of H onto C.

Lemma 4.3 (See [15])

Let C be a nonempty closed convex subset of a real Hilbert space H, and let A , B : C H be α and β-inverse strongly monotone mappings, respectively, with α , β > 0 and VI ( C , A ) VI ( C , B ) . Then
VI ( C , a A + ( 1 a ) B ) = VI ( C , A ) VI ( C , B ) , a ( 0 , 1 ) .
(4.2)

Furthermore, if 0 < γ < 2 η , where η = min { α , β } , we have I γ ( a A + ( 1 a ) B ) is a nonexpansive mapping.

Example 4.4 Let T : C C be a κ-strictly pseudo-contractive mapping with F ( T ) . Let { λ n } be a sequence of positive real numbers such that
0 < inf n N λ n sup n N λ n < 1 κ and lim n ( λ n + 1 λ n ) = 0 ,

and let { T n } be a sequence of mappings defined by T n = P C ( I λ n ( I T ) ) . Then { T n } is a strongly nonexpansive sequence satisfying the conditions (R) and (Z).

Proof Since T is a κ-strictly pseudo-contractive mapping, then I T is 1 κ 2 -inverse strongly monotone. From Example 4.3 in [10], we have { T n } is a strongly nonexpansive sequence satisfying the conditions (R) and (Z). □

Example 4.5 Let A , B : C H be α , β -inverse strongly monotone mappings, respectively, with γ ¯ = min { α , β } and VI ( C , A ) VI ( C , B ) . Let { λ n } be a sequence of positive real numbers such that
0 < inf n N λ n sup n N λ n < 2 γ ¯ and lim n ( λ n + 1 λ n ) = 0 ,

and let { T n } be a sequence of mappings defined by T n = P C ( I λ n D ) , where D = a A + ( 1 a ) B for all a ( 0 , 1 ) . Then { T n } is a strongly nonexpansive sequence satisfying the conditions (R) and (Z).

Proof Let x , y C , then we have
x y , D x D y = x y , ( a A + ( 1 a ) B ) x ( a A + ( 1 a ) B ) y a x y , A x A y + ( 1 a ) x y , B x B y a α A x A y 2 + ( 1 a ) β B x B y 2 γ ¯ ( a A x + ( 1 a ) B x a A y ( 1 a ) B y 2 ) γ ¯ D x D y 2 .

Then D is a γ ¯ -inverse strongly monotone mapping. From Example 4.3 in [10], we have that { T n } is a strongly nonexpansive sequence satisfying the conditions (R) and (Z). □

Example 4.6 Let A : C H be an α-strongly monotone and L-Lipschitzian mapping with VI ( C , A ) . Let { λ n } be a sequence of positive real numbers such that
0 < inf n N λ n sup n N λ n < 2 α L 2 and lim n ( λ n + 1 λ n ) = 0 ,

and let { T n } be a sequence of mappings defined by T n = P C ( I λ n A ) . Then { T n } is a strongly nonexpansive sequence satisfying the conditions (R) and (Z).

Proof Let x , y C , then we have
x y , A x A y α x y 2 α L 2 A x A y 2 .

Then A is an α L 2 -inverse strongly monotone mapping. From Example 4.3 in [10], we have that { T n } is a strongly nonexpansive sequence satisfying the conditions (R) and (Z). □

Example 4.7 (See [10])

Let { R n } be a sequence of nonexpansive mappings of C into itself having a common fixed point, and let { μ n } be a sequence in [ 0 , 1 ] . For each n N , a W-mapping [17] T n generated by R n , R n 1 , , R 1 and μ n , μ n 1 , , μ 1 is defined as follows:
U n , n = μ n R n + ( 1 μ n ) I , U n , n 1 = μ n 1 R n 1 U n , n + ( 1 μ n 1 ) I , U n , n 2 = μ n 2 R n 2 U n , n 1 + ( 1 μ n 2 ) I , U n , k = μ k R k U n , k + 1 + ( 1 μ k ) I , U n , 2 = μ 2 R 2 U n , 3 + ( 1 μ 2 ) I , T n = U n , 1 = μ 1 R 1 U n , 2 + ( 1 μ 1 ) I .

If 0 < μ 1 1 and 0 < μ n b , for all n 2 and 0 < b < 1 , then { T n } satisfies the conditions (R) and (Z).

By using our main result and these three examples, we obtain the following results.

Theorem 4.8 Let H be a Hilbert space, let C be a nonempty closed convex subset of H. Let F 1 and F 2 be two bifunctions from C × C into satisfying (A1)-(A4), respectively. Let T : C C be a κ-strictly pseudo-contractive mapping with F ( T ) . Let { λ n } be a sequence of positive real numbers such that
0 < inf n N λ n sup n N λ n < 1 κ and lim n ( λ n + 1 λ n ) = 0 ,
and let { T n } be a sequence of mappings defined by T n = P C ( I λ n ( I T ) ) . Let { R n } be a sequence of nonexpansive mappings of C into itself having a common fixed point, and let { μ n } be a sequence in [ 0 , 1 ] . For each n N , W n is a W-mapping generated by R n , R n 1 , , R 1 and μ n , μ n 1 , , μ 1 . Assume that F = EP ( F 1 ) EP ( F 2 ) F ( { R n } ) F ( T ) . Let f : C C be a contractive mapping with α ( 0 , 1 2 ) . Let { x n } , { u n } , { v n } be sequences generated by x 1 , u , v C and
{ F 1 ( u n , u ) + 1 r n u u n , u n x n 0 , F 2 ( v n , v ) + 1 s n v v n , v n x n 0 , y n = δ n u n + ( 1 δ n ) v n , x n + 1 = β n x n + ( 1 β n ) W n ( α n f ( T n y n ) + ( 1 α n ) T n y n ) , n 1 ,
(4.3)
where { α n } , { β n } [ 0 , 1 ] , { r n } , { s n } ( a , b ) [ 0 , 1 ] . Assume that the following conditions hold:
  1. (i)

    lim n α n = 0 and n = 1 α n = ;

     
  2. (ii)

    0 < lim inf n β n lim sup n β n < 1 ;

     
  3. (iii)

    n = 0 | r n + 1 r n | , n = 0 | s n + 1 s n | < ;

     
  4. (iv)

    lim n δ n = δ ( 0 , 1 ) .

     

Then the sequences { x n } , { u n } , { v n } , { y n } converge strongly to z = P F f ( z ) .

Proof From Example 4.4, we have { T n } is a strongly nonexpansive sequence satisfying the conditions (R) and (Z). From Lemma 4.2, we have F ( T n ) = F ( P C ( I λ n ( I T ) ) ) = VI ( C , I T ) = F ( T ) for all n N . It implies that F ( { T n } ) = F ( T ) . From [18], we have F ( { W n } ) = F ( { R n } ) . It follows that F = EP ( F 1 ) EP ( F 2 ) F ( { W n } ) F ( { T n } ) . From Example 4.7, we have { W n } is a nonexpansive sequence satisfying the conditions (R) and (Z). By Theorem 3.1, we can conclude the desired result. □

Theorem 4.9 Let H be a Hilbert space, let C be a nonempty closed convex subset of H. Let F 1 and F 2 be two bifunctions from C × C into satisfying (A1)-(A4), respectively. Let A , B : C H be α , β -inverse strongly monotone mappings, respectively, with γ ¯ = min { α , β } and VI ( C , A ) VI ( C , B ) . Let { λ n } be a sequence of positive real numbers such that
0 < inf n N λ n sup n N λ n < 2 γ ¯ and lim n ( λ n + 1 λ n ) = 0 ,
and let { T n } be a sequence of mappings defined by T n = P C ( I λ n D ) , where D = a A + ( 1 a ) B for all a ( 0 , 1 ) . Let { R n } be a sequence of nonexpansive mappings of C into itself having a common fixed point, and let { μ n } be a sequence in [ 0 , 1 ] . For each n N , W n is a W-mapping generated by R n , R n 1 , , R 1 and μ n , μ n 1 , , μ 1 . Assume that F = EP ( F 1 ) EP ( F 2 ) F ( { R n } ) VI ( C , A ) VI ( C , B ) . Let f : C C be a contractive mapping with α ( 0 , 1 2 ) . Let { x n } , { u n } , { v n } be sequences generated by x 1 , u , v C and
{ F 1 ( u n , u ) + 1 r n u u n , u n x n 0 , F 2 ( v n , v ) + 1 s n v v n , v n x n 0 , y n = δ n u n + ( 1 δ n ) v n , x n + 1 = β n x n + ( 1 β n ) W n ( α n f ( T n y n ) + ( 1 α n ) T n y n ) , n 1 ,
(4.4)
where { α n } , { β n } [ 0 , 1 ] , { r n } , { s n } ( a , b ) [ 0 , 1 ] . Assume that the following conditions hold:
  1. (i)

    lim n α n = 0 and n = 1 α n = ;

     
  2. (ii)

    0 < lim inf n β n lim sup n β n < 1 ;

     
  3. (iii)

    n = 0 | r n + 1 r n | , n = 0 | s n + 1 s n | < ;

     
  4. (iv)

    lim n δ n = δ ( 0 , 1 ) .

     

Then the sequences { x n } , { u n } , { v n } , { y n } converge strongly to z = P F f ( z ) .

Proof From Example 4.5, we have { T n } is a strongly nonexpansive sequence satisfying the conditions (R) and (Z). From Lemmas 4.2 and 4.3, we have F ( T n ) = F ( P C ( I λ n D ) ) = VI ( C , D ) = VI ( C , A ) VI ( C , B ) for all n N . It implies that F ( { T n } ) = VI ( C , A ) VI ( C , B ) . From [18], we have F ( { W n } ) = F ( { R n } ) . It follows that F = EP ( F 1 ) EP ( F 2 ) F ( { W n } ) F ( { T n } ) . From Example 4.7, we have { W n } is a nonexpansive sequence satisfying the conditions (R) and (Z). By Theorem 3.1, we can conclude the desired result. □

Theorem 4.10 Let H be a Hilbert space, let C be a nonempty closed convex subset of H. Let F 1 and F 2 be two bifunctions from C × C into satisfying (A1)-(A4), respectively. Let A : C H be an α-strongly monotone and L-Lipschitzian mapping with VI ( C , A ) . Let { λ n } be a sequence of positive real numbers such that
0 < inf n N λ n sup n N λ n < 2 α L 2 and lim n ( λ n + 1 λ n ) = 0 ,
and let { T n } be a sequence of mappings defined by T n = P C ( I λ n A ) . Let { R n } be a sequence of nonexpansive mappings of C into itself having a common fixed point, and let { μ n } be a sequence in [ 0 , 1 ] . For each n N , W n is a W-mapping generated by R n , R n 1 , , R 1 and μ n , μ n 1 , , μ 1 . Assume that F = EP ( F 1 ) EP ( F 2 ) F ( { R n } ) VI ( C , A ) . Let f : C C be a contractive mapping with α ( 0 , 1 2 ) . Let { x n } , { u n } , { v n } be sequences generated by x 1 , u , v C and
{ F 1 ( u n , u ) + 1 r n u u n , u n x n 0 , F 2 ( v n , v ) + 1 s n v v n , v n x n 0 , y n = δ n u n + ( 1 δ n ) v n , x n + 1 = β n x n + ( 1 β n ) W n ( α n f ( T n y n ) + ( 1 α n ) T n y n ) , n 1 ,
(4.5)
where { α n } , { β n } [ 0 , 1 ] , { r n } , { s n } ( a , b ) [ 0 , 1 ] . Assume that the following conditions hold:
  1. (i)

    lim n α n = 0 and n = 1 α n = ;

     
  2. (ii)

    0 < lim inf n β n lim sup n β n < 1 ;

     
  3. (iii)

    n = 0 | r n + 1 r n | , n = 0 | s n + 1 s n | < ;

     
  4. (iv)

    lim n δ n = δ ( 0 , 1 ) .

     

Then the sequences { x n } , { u n } , { v n } , { y n } converge strongly to z = P F f ( z ) .

Proof From Example 4.6, we have { T n } is a strongly nonexpansive sequence satisfying the conditions (R) and (Z). From Lemma 4.2, we have F ( T n ) = F ( P C ( I λ n A ) ) = VI ( C , A ) for all n N . It implies that F ( { T n } ) = VI ( C , A ) . From [18], we have F ( { W n } ) = F ( { R n } ) . It follows that F = EP ( F 1 ) EP ( F 2 ) F ( { W n } ) F ( { T n } ) . From Example 4.7, we have { W n } is a nonexpansive sequence satisfying the conditions (R) and (Z). By Theorem 3.1, we can conclude the desired result. □

Theorem 4.11 Let H be a Hilbert space, let C be a nonempty closed convex subset of H. Let F 1 be a bifunction from C × C into satisfying (A1)-(A4), and let { S n } and { T n } be sequences of nonexpansive self-mappings of C with F = EP ( F 1 ) F ( { S n } ) F ( { T n } ) . Let { T n } or { S n } be a sequence of strongly nonexpansive mappings, and let f : C C be a contractive mapping with α ( 0 , 1 2 ) . Let { x n } , { u n } be sequences generated by x 1 , u C and
{ F 1 ( u n , u ) + 1 r n u u n , u n x n 0 , x n + 1 = β n x n + ( 1 β n ) S n ( α n f ( T n u n ) + ( 1 α n ) T n u n ) , n 1 ,
(4.6)
where { α n } , { β n } [ 0 , 1 ] , { r n } , { s n } ( a , b ) [ 0 , 1 ] . Assume that the following conditions hold:
  1. (i)

    lim n α n = 0 and n = 1 α n = ;

     
  2. (ii)

    0 < lim inf n β n lim sup n β n < 1 ;

     
  3. (iii)

    n = 0 | r n + 1 r n | < ;

     
  4. (iv)

    { S n } and { T n } satisfy the conditions R and Z.

     

Then the sequences { x n } , { u n } converge strongly to z = P F f ( z ) .

Proof Put F 1 F 2 , s n = r n and u n = v n . From Theorem 3.1, we can conclude the desired conclusion. □

The following result can be obtained from Theorem 3.1. We, therefore, omit the proof.

Theorem 4.12 Let H be a Hilbert space, let C be a nonempty closed convex subset of H. Let F i be bifunctions from C × C into , for every i = 1 , 2 , , N , satisfying (A1)-(A4), and let { S n } and { T n } be sequences of nonexpansive self-mappings of C with F = i = 1 N EP ( F i ) F ( { S n } ) F ( { T n } ) . Let { T n } or { S n } be a sequence of strongly nonexpansive mappings, and let f : C C be a contractive mapping with α ( 0 , 1 2 ) . Let { x n } , { u n } , { v n } be sequences generated by x 1 , u i C , for every i 1 , 2 , , N , and
{ F i ( u n i , u i ) + 1 r n i u u n i , u n i x n 0 , y n = i = 1 N δ n i u n i , x n + 1 = β n x n + ( 1 β n ) S n ( α n f ( T n y n ) + ( 1 α n ) T n y n ) , n 1 ,
(4.7)
where { α n } , { β n } [ 0 , 1 ] , { r n } , { s n } ( a , b ) [ 0 , 1 ] . Assume that the following conditions hold:
  1. (i)

    lim n α n = 0 and n = 1 α n = ;

     
  2. (ii)

    0 < lim inf n β n lim sup n β n < 1 ;

     
  3. (iii)

    n = 0 | r n + 1 i r n i | < , i = 1 , 2 , , N ;

     
  4. (iv)

    i = 1 N δ n i = 1 ;

     
  5. (v)

    lim n δ n i = δ i ( 0 , 1 ) , i = 1 , 2 , , N ;

     
  6. (vi)

    { S n } and { T n } satisfy the conditions R and Z.

     

Then the sequences { x n } , { y n } and { u n i } , for every i = 1 , 2 , , N , converge strongly to z = P F f ( z ) .

Declarations

Acknowledgements

This research was supported by the Research Administration Division of King Mongkut’s Institute of Technology Ladkrabang.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok, 10520, Thailand

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