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Iterative scheme for a nonexpansive mapping, an η-strictly pseudo-contractive mapping and variational inequality problems in a uniformly convex and 2-uniformly smooth Banach space

Fixed Point Theory and Applications20132013:23

https://doi.org/10.1186/1687-1812-2013-23

  • Received: 21 July 2012
  • Accepted: 13 January 2013
  • Published:

Abstract

In this paper, we introduce an iterative scheme by the modification of Mann’s iteration process for finding a common element of the set of solutions of a finite family of variational inequality problems and the set of fixed points of an η-strictly pseudo-contractive mapping and a nonexpansive mapping. Moreover, we prove a strong convergence theorem for finding a common element of the set of fixed points of a finite family of η i -strictly pseudo-contractive mappings for every i = 1 , 2 , , N in uniformly convex and 2-uniformly smooth Banach spaces.

Keywords

  • nonexpansive mapping
  • strictly pseudo-contractive mapping
  • variational inequality problem

1 Introduction

Let E be a Banach space with its dual space E and let C be a nonempty closed convex subset of E. Throughout this paper, we denote the norm of E and E by the same symbol . We use the symbol → to denote the strong convergence. Recall the following definition.

Definition 1.1 A Banach space E is said to be uniformly convex iff for any ϵ, 0 < ϵ 2 , the inequalities x 1 , y 1 and x y ϵ imply there exists a δ > 0 such that x + y 2 1 δ .

Definition 1.2 Let E be a Banach space. Then a function ρ E : R + R + is said to be the modulus of smoothness of E if
ρ E ( t ) = sup { x + y + x y 2 1 : x = 1 , y = t } .
A Banach space E is said to be uniformly smooth if
lim t 0 ρ E ( t ) t = 0 .

Let q > 1 . A Banach space E is said to be q-uniformly smooth if there exists a fixed constant c > 0 such that ρ E ( t ) c t q . It is easy to see that if E is q-uniformly smooth, then q 2 and E is uniformly smooth.

Definition 1.3 A mapping J from E onto E satisfying the condition
J ( x ) = { f E : x , f = x 2  and  f = x }

is called the normalized duality mapping of E. The duality pair x , f represents f ( x ) for f E and x E .

Definition 1.4 Let C be a nonempty subset of a Banach space E and T : C C be a self-mapping. T is called a nonexpansive mapping if
T x T y x y

for all x , y C .

T is called an η-strictly pseudo-contractive mapping if there exists a constant η ( 0 , 1 ) such that
T x T y , j ( x y ) x y 2 η ( I T ) x ( I T ) y 2
(1.1)
for every x , y C and for some j ( x y ) J ( x y ) . It is clear that (1.1) is equivalent to the following:
( I T ) x ( I T ) y , j ( x y ) η ( I T ) x ( I T ) y 2
(1.2)

for every x , y C and for some j ( x y ) J ( x y ) .

Let C and D be nonempty subsets of a Banach space E such that C is nonempty closed convex and D C , then a mapping P : C D is sunny [1] provided P ( x + t ( x P ( x ) ) ) = P ( x ) for all x C and t 0 , whenever x + t ( x P ( x ) ) C . The mapping P : C D is called a retraction if P x = x for all x D . Furthermore, P is a sunny nonexpansive retraction from C onto D if P is a retraction from C onto D which is also sunny and nonexpansive. The subset D of C is called a sunny nonexpansive retraction of C if there exists a sunny nonexpansive retraction from C onto D.

An operator A of C into E is said to be accretive if there exists j ( x y ) J ( x y ) such that
A x A y , j ( x y ) 0 , x , y C .
A mapping A : C E is said to be α-inverse strongly accretive if there exists j ( x y ) J ( x y ) and α > 0 such that
A x A y , j ( x y ) α A x A y 2 , x , y C .

Remark 1.1 From (1.1) and (1.2), if T is an η-strictly pseudo-contractive mapping, then I T is η-inverse strongly accretive.

The variational inequality problem in a Banach space is to find a point x C such that for some j ( x x ) J ( x x ) ,
A x , j ( x x ) 0 , x C .
(1.3)
This problem was considered by Aoyama et al. [2]. The set of solutions of the variational inequality in a Banach space is denoted by S ( C , A ) , that is,
S ( C , A ) = { u C : A u , J ( v u ) 0 , v C } .
(1.4)

Numerous problems in physics, optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games reduce to find an element of (1.4); see [3, 4].

Recall that the normal Mann’s iterative process was introduced by Mann [5] in 1953. The normal Mann’s iterative process generates a sequence { x n } in the following manner:
{ x 1 C , x n + 1 = ( 1 α n ) x n + α n T x n , n 1 ,
(1.5)

where the sequence { α n } ( 0 , 1 ) . If T is a nonexpansive mapping with a fixed point and the control sequence { α n } is chosen so that n = 1 α n ( 1 α n ) = , then the sequence { x n } generated by normal Mann’s iterative process (1.5) converges weakly to a fixed point of T.

In 2008, Cho et al. [6] modified the normal Mann’s iterative process and proved strong convergence for a finite family of nonexpansive mappings in the framework of Banach spaces without any commutative assumption as follows.

Theorem 1.2 Let C be a closed convex subset of a uniformly smooth and strictly convex Banach space E. Let { T i } be a nonexpansive mapping from C into itself for i = 1 , 2 , , N . Assume that F = i = 1 N F ( T i ) . Given a point u C and given sequences { α n } , { β n } ( 0 , 1 ) , the following conditions are satisfied:
Let { x n } be a sequence generated by u , x 0 = x C and
{ y n = β n x n + ( 1 β n ) W n x n , x n + 1 = α n u + ( 1 α n ) y n , n 0 ,
(1.6)

where W n is the W-mapping generated by T 1 , T 2 , , T N and γ n 1 , γ n 2 , , γ n N . Then { x n } converges strongly to x F , where x = Q ( u ) and Q : C F is the unique sunny nonexpansive retraction from C onto F.

In 2008, Zhou [7] proved a strong convergence theorem for the modification of normal Mann’s iteration algorithm generated by a strict pseudo-contraction in a real 2-uniformly smooth Banach space as follows.

Theorem 1.3 Let C be a closed convex subset of a real 2-uniformly smooth Banach space E and let T : C C be a λ-strict pseudo-contraction such that F ( T ) . Given u , x 0 C and the sequences { α n } , { β n } , { γ n } and { δ n } in ( 0 , 1 ) , the following control conditions are satisfied:
Let a sequence { x n } be generated by
{ y n = α n T x n + ( 1 α n ) x n , x n + 1 = β n u + γ n x n + δ n y n , n 0 .
(1.7)

Then { x n } converges strongly to x F ( T ) , where x = Q F ( T ) ( u ) and Q F ( T ) : C F ( T ) is the unique sunny nonexpansive retraction from C onto F ( T ) .

In 2005, Aoyama et al. [2] proved a weak convergence theorem for finding a solution of problem (1.3) as follows.

Theorem 1.4 Let E be a uniformly convex and 2-uniformly smooth Banach space and let C be a nonempty closed convex subset of E. Let Q C be a sunny nonexpansive retraction from E onto C, let α > 0 and let A be an α-inverse strongly accretive operator of C into E with S ( C , A ) . Suppose x 1 = x C and { x n } is given by
x n + 1 = α n x n + ( 1 α n ) Q C ( x n λ n A x n )

for every n = 1 , 2 ,  , where { λ n } is a sequence of positive real numbers and { α n } is a sequence in [ 0 , 1 ] . If { λ n } and { α n } are chosen so that λ n [ a , α K 2 ] for some a > 0 and α n [ b , c ] for some b, c with 0 < b < c < 1 , then { x n } converges weakly to some element z of S ( C , A ) , where K is the 2-uniformly smoothness constant of E.

In this paper, motivated by Theorems 1.2, 1.3 and 1.4, we prove a strong convergence theorem for finding a common element of the set of solutions of a finite family of variational inequality problems and the set of fixed points of a nonexpansive mapping and an η-strictly pseudo-contractive mapping in uniformly convex and 2-uniformly smooth spaces. Moreover, by using our main result, we prove a strong convergence theorem for finding a common element of the set of fixed points of a finite family of η i -strictly pseudo-contractive mappings for every i = 1 , 2 , , N in uniformly convex and 2-uniformly smooth Banach spaces.

2 Preliminaries

In this section, we collect and prove the following lemmas to use in our main result.

Lemma 2.1 (See [8])

Let E be a real 2-uniformly smooth Banach space with the best smooth constant K. Then the following inequality holds:
x + y 2 x 2 + 2 y , J ( x ) + 2 K y 2

for any x , y E .

Definition 2.1 (See [9])

Let C be a nonempty convex subset of a real Banach space. Let { T i } i = 1 N be a finite family of nonexpanxive mappings of C into itself and let λ 1 , , λ N be real numbers such that 0 λ i 1 for every i = 1 , , N . Define a mapping K : C C as follows:
U 1 = λ 1 T 1 + ( 1 λ 1 ) I , U 2 = λ 2 T 2 U 1 + ( 1 λ 2 ) U 1 , U 3 = λ 3 T 3 U 2 + ( 1 λ 3 ) U 2 , U N 1 = λ N 1 T N 1 U N 2 + ( 1 λ N 1 ) U N 2 , K = U N = λ N T N U N 1 + ( 1 λ N ) U N 1 .
(2.1)

Such a mapping K is called the K-mapping generated by T 1 , , T N and λ 1 , , λ N .

Lemma 2.2 (See [9])

Let C be a nonempty closed convex subset of a strictly convex Banach space. Let { T i } i = 1 N be a finite family of nonexpanxive mappings of C into itself with i = 1 N F ( T i ) and let λ 1 , , λ N be real numbers such that 0 < λ i < 1 for every i = 1 , , N 1 and 0 < λ N 1 . Let K be the K-mapping generated by T 1 , , T N and λ 1 , , λ N . Then F ( K ) = i = 1 N F ( T i ) .

Remark 2.3 From Lemma 2.2, it is easy to see that the K mapping is a nonexpansive mapping.

Lemma 2.4 (See [10])

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
x n + 1 = β n x n + ( 1 β n ) z n
for all integer 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 2.5 (See [11])

Let X be a uniformly convex Banach space and B r = { x X : x r } , r > 0 . Then there exists a continuous, strictly increasing and convex function g : [ 0 , ] [ 0 , ] , g ( 0 ) = 0 such that
α x + β y + γ z 2 α x 2 + β y 2 + γ z 2 α β g ( x y )

for all x , y , z B r and all α , β , γ [ 0 , 1 ] with α + β + γ = 1 .

Lemma 2.6 (See [2])

Let C be a nonempty closed convex subset of a smooth Banach space E. Let Q C be a sunny nonexpansive retraction from E onto C and let A be an accretive operator of C into E. Then for all λ > 0 ,
S ( C , A ) = F ( Q C ( I λ A ) ) .

Lemma 2.7 (See [12])

Let C be a closed convex subset of a strictly convex Banach space X. Let { T n : n N } be a sequence of nonexpansive mappings on C. Suppose 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 x C is well defined, non-expansive and F ( S ) = n = 1 F ( T n ) holds.

Lemma 2.8 (See [8])

Let r > 0 . If E is uniformly convex, then there exists a continuous, strictly increasing and convex function g : [ 0 , ) [ 0 , ) , g ( 0 ) = 0 such that for all x , y B r ( 0 ) = { x E : x r } and for any α [ 0 , 1 ] , we have α x + ( 1 α ) y 2 α x 2 + ( 1 α ) y 2 α ( 1 α ) g ( x y ) .

Lemma 2.9 (See [13])

Let X be a uniformly smooth Banach space, C be a closed convex subset of X, T : C C be a nonexpansive mapping with F ( T ) and let f C where C is to denote the collection of all contractions on C. Then the sequence { x t } defined by x t = t f ( x t ) + ( 1 t ) T x t converses strongly to a point in F ( T ) . If we define a mapping Q : C F ( T ) by Q ( f ) = lim t 0 x t for all f C , then Q ( f ) solves the following variational inequality:
( I f ) Q ( f ) , j ( Q ( f ) p ) 0

for all f C , p F ( T ) .

Lemma 2.10 (See [14])

In a Banach space E, the following inequality holds:
x + y 2 x 2 + 2 y , j ( x + y ) , x , y E ,

where j ( x + y ) J ( x + y ) .

Lemma 2.11 (See [15])

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

Then lim n s n = 0 .

Lemma 2.12 Let C be a nonempty closed convex subset of a 2-uniformly smooth Banach space E and let T : C C be a nonexpansive mapping and S : C C be an η-strictly pseudocontractive mapping with F ( S ) F ( T ) . Define a mapping B A : C C by B A x = T ( ( 1 α ) I + α S ) x for all x C and α ( 0 , η K 2 ) , where K is the 2-uniformly smooth constant of E. Then F ( B A ) = F ( S ) F ( T ) .

Proof It is easy to see that F ( T ) F ( S ) F ( B A ) . Let x 0 F ( B A ) and x F ( T ) F ( S ) , we have
x 0 x 2 = T ( ( 1 α ) x 0 + α S x 0 ) x 2 ( 1 α ) x 0 + α S x 0 x 2 = x 0 x + α ( S x 0 x 0 ) 2 x 0 x 2 + 2 α S x 0 x 0 , j ( x 0 x ) + 2 K 2 α 2 S x 0 x 0 2 = x 0 x 2 + 2 α S x 0 x , j ( x 0 x ) + 2 α x x 0 , j ( x 0 x ) + 2 K 2 α 2 S x 0 x 0 2 = x 0 x 2 + 2 α S x 0 x , j ( x 0 x ) 2 α x 0 x 2 + 2 K 2 α 2 S x 0 x 0 2 x 0 x 2 + 2 α ( x 0 x 2 η ( I S ) x 0 2 ) 2 α x 0 x 2 + 2 K 2 α 2 S x 0 x 0 2 = x 0 x 2 2 α η x 0 S x 0 2 + 2 K 2 α 2 S x 0 x 0 2 = x 0 x 2 2 α ( η K 2 α ) x 0 S x 0 2 .
(2.2)
(2.2) implies that
2 α ( η K 2 α ) x 0 S x 0 2 x 0 x 2 x 0 x 2 = 0 .

Then we have S x 0 = x 0 , that is, x 0 F ( S ) .

Since x 0 F ( B A ) , from the definition of B A , we have
x 0 = B A x 0 = T ( ( 1 α ) x 0 + α S x 0 ) = T x 0 .

Then we have x 0 F ( T ) . Therefore, x 0 F ( T ) F ( S ) . It follows that F ( B A ) F ( T ) F ( S ) . Hence, F ( B A ) = F ( T ) F ( S ) . □

Remark 2.13 Applying (2.2), we have that the mapping B A is nonexpansive.

3 Main results

Theorem 3.1 Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E. Let Q C be the sunny nonexpansive retraction from E onto C. For every i = 1 , 2 , , N , let A i : C E be an α i -inverse strongly accretive mapping. Define a mapping G i : C C by Q C ( I λ i A i ) x = G i x for all x C and i = 1 , 2 , , N , where λ i ( 0 , α i K 2 ) , K is the 2-uniformly smooth constant of E. Let B : C C be the K-mapping generated by G 1 , G 2 , , G N and ρ 1 , ρ 2 , , ρ N , where ρ i ( 0 , 1 ) , i = 1 , 2 , , N 1 and ρ N ( 0 , 1 ] . Let T : C C be a nonexpansive mapping and S : C C be an η-strictly pseudo-contractive mapping with F = F ( S ) F ( T ) i = 1 N S ( C , A i ) . Define a mapping B A : C C by T ( ( 1 α ) I + α S ) x = B A x , x C and α ( 0 , η K 2 ) . Let { x n } be the sequence generated by x 1 C and
x n + 1 = α n f ( x n ) + β n x n + γ n B x n + δ n B A x n , n 1 ,
(3.1)
where f : C C is a contractive mapping and { α n } , { β n } , { γ n } , { δ n } [ 0 , 1 ] , α n + β n + γ n + δ n = 1 and satisfy the following conditions:
Then the sequence { x n } converses strongly to q F , which solves the following variational inequality:
q f ( q ) , j ( q p ) 0 , p F .

Proof First, we will show that G i is a nonexpansive mapping for every i = 1 , 2 , , N .

Let x , y C . From nonexpansiveness of Q C , we have
G i x G i y 2 = Q C ( I λ i A i ) x Q C ( I λ i A i ) y 2 ( I λ i A i ) x ( I λ i A i ) y 2 = x y λ i ( A i x A i y ) 2 x y 2 2 λ i A i x A i y , j ( x y ) + 2 K 2 λ i 2 A i x A i y 2 x y 2 2 λ i α i A i x A i y 2 + 2 K 2 λ i 2 A i x A i y 2 = x y 2 2 λ i ( α i K 2 λ i ) A i x A i y 2 x y 2 .
Then we have G i is a nonexpansive mapping for every i = 1 , 2 , , N . Since B : C C is the K-mapping generated by G 1 , G 2 , , G N and ρ 1 , ρ 2 , , ρ N and Lemma 2.2, we can conclude that F ( B ) = i = 1 N F ( G i ) . From Lemma 2.6 and the definition of G i , we have F ( G i ) = S ( C , A i ) for every i = 1 , 2 , , N . Hence, we have
F ( B ) = i = 1 N F ( G i ) = i = 1 N S ( C , A i ) .
(3.2)

Next, we will show that the sequence { x n } is bounded.

Let z F ; from the definition of x n , we have
x n + 1 z α n f ( x n ) z + β n x n z + γ n B x n z + δ n B A x n z α n f ( x n ) z + ( 1 α n ) x n z α n f ( x n ) f ( z ) + α n f ( z ) z + ( 1 α n ) x n z α n a x n z + α n f ( z ) z + ( 1 α n ) x n z = ( 1 α n ( 1 a ) ) x n z + α n f ( z ) z max { x 1 z , f ( z ) z 1 a } .

By induction, we can conclude that the sequence { x n } is bounded and so are { f ( x n ) } , { B x n } , { B A x n } .

Next, we will show that
lim n x n + 1 x n = 0 .
(3.3)
From the definition of x n , we can rewrite x n by
x n + 1 = β n x n + ( 1 β n ) z n ,
(3.4)

where z n = α n f ( x n ) + γ n B x n + δ n B A x n 1 β n .

Since
z n + 1 z n = α n + 1 f ( x n + 1 ) + γ n + 1 B x n + 1 + δ n + 1 B A x n + 1 1 β n + 1 ( α n f ( x n ) + γ n B x n + δ n B A x n 1 β n ) = x n + 2 β n + 1 x n + 1 1 β n + 1 x n + 1 β n x n 1 β n = x n + 2 β n + 1 x n + 1 1 β n + 1 x n + 1 β n x n 1 β n + 1 + x n + 1 β n x n 1 β n + 1 x n + 1 β n x n 1 β n x n + 2 β n + 1 x n + 1 1 β n + 1 x n + 1 β n x n 1 β n + 1 + x n + 1 β n x n 1 β n + 1 x n + 1 β n x n 1 β n = 1 1 β n + 1 x n + 2 β n + 1 x n + 1 ( x n + 1 β n x n ) + | 1 1 β n + 1 1 1 β n | x n + 1 β n x n = 1 1 β n + 1 x n + 2 β n + 1 x n + 1 ( x n + 1 β n x n ) + | β n + 1 β n | ( 1 β n ) ( 1 β n + 1 ) x n + 1 β n x n = 1 1 β n + 1 α n + 1 f ( x n + 1 ) + γ n + 1 B x n + 1 + δ n + 1 B A x n + 1 ( α n f ( x n ) + γ n B x n + δ n B A x n ) + | β n + 1 β n | ( 1 β n ) ( 1 β n + 1 ) x n + 1 β n x n = 1 1 β n + 1 ( α n + 1 f ( x n + 1 ) α n f ( x n ) + γ n + 1 B x n + 1 B x n + δ n + 1 B A x n + 1 B A x n + | γ n + 1 γ n | B x n + | δ n + 1 δ n | B A x n ) + | β n + 1 β n | ( 1 β n ) ( 1 β n + 1 ) x n + 1 β n x n 1 1 β n + 1 ( α n + 1 f ( x n + 1 ) + α n f ( x n ) + ( γ n + 1 + δ n + 1 ) x n + 1 x n + | γ n + 1 γ n | B x n + | δ n + 1 δ n | B A x n ) + | β n + 1 β n | ( 1 β n ) ( 1 β n + 1 ) x n + 1 β n x n = α n + 1 1 β n + 1 f ( x n + 1 ) + α n 1 β n + 1 f ( x n ) + γ n + 1 + δ n + 1 1 β n + 1 x n + 1 x n + | γ n + 1 γ n | 1 β n + 1 B x n + | δ n + 1 δ n | 1 β n + 1 B A x n + | β n + 1 β n | ( 1 β n ) ( 1 β n + 1 ) x n + 1 β n x n α n + 1 1 β n + 1 f ( x n + 1 ) + α n 1 β n + 1 f ( x n ) + x n + 1 x n + | γ n + 1 γ n | 1 β n + 1 B x n + | δ n + 1 δ n | 1 β n + 1 B A x n + | β n + 1 β n | ( 1 β n ) ( 1 β n + 1 ) x n + 1 β n x n .
(3.5)
From (3.5) and the conditions (i)-(iv), we have
lim sup n ( z n + 1 z n x n + 1 x n ) 0 .
(3.6)
From Lemma 2.4 and (3.4), we have
lim n z n x n = 0 .
(3.7)
From (3.4), we have
x n + 1 x n = ( 1 β n ) z n x n ,
and from the condition (iv) and (3.7), we have
lim n x n + 1 x n = 0 .
Next, we will show that
lim n B x n x n = 0 and lim n B A x n x n = 0 .
From the definition of x n , we can rewrite x n + 1 by
x n + 1 = α n f ( x n ) + β n x n + γ n B x n + δ n B A x n = α n f ( x n ) + β n x n + ( γ n + δ n ) ( γ n B x n + δ n B A x n ) γ n + δ n = α n f ( x n ) + β n x n + e n z n ,
(3.8)

where e n = γ n + δ n and z n = ( γ n B x n + δ n B A x n ) γ n + δ n .

From Lemma 2.5 and (3.8), we have
x n + 1 z 2 = α n ( f ( x n ) z ) + β n ( x n z ) + e n ( z n z ) 2 α n f ( x n ) z 2 + β n x n z 2 + e n z n z 2 β n e n g 1 ( z n x n ) = α n f ( x n ) z 2 + β n x n z 2 β n e n g 1 ( z n x n ) + e n ( γ n B x n + δ n B A x n ) γ n + δ n z 2 = α n f ( x n ) z 2 + β n x n z 2 β n e n g 1 ( z n x n ) + e n ( 1 δ n γ n + δ n ) ( B x n z ) + δ n γ n + δ n ( B A x n z ) 2 α n f ( x n ) z 2 + β n x n z 2 β n e n g 1 ( z n x n ) + e n ( ( 1 δ n γ n + δ n ) B x n z + δ n γ n + δ n B A x n z ) 2 α n f ( x n ) z 2 + β n x n z 2 β n e n g 1 ( z n x n ) + e n x n z 2 α n f ( x n ) z 2 + x n z 2 β n e n g 1 ( z n x n ) ,
which implies that
β n e n g 1 ( z n x n ) α n f ( x n ) z 2 + x n z 2 x n + 1 z 2 α n f ( x n ) z 2 + ( x n z + x n + 1 z ) x n + 1 x n .
(3.9)
From the conditions (i), (ii), (iv) and (3.3), we have
lim n g 1 ( z n x n ) = 0 .
From the properties of g 1 , we have
lim n z n x n = 0 .
(3.10)
From Lemma 2.8 and the definition of z n , we have
z n z 2 = ( γ n B x n + δ n B A x n ) γ n + δ n z 2 = ( 1 δ n δ n + γ n ) ( B x n z ) + δ n δ n + γ n ( B A x n z ) 2 ( 1 δ n δ n + γ n ) B x n z 2 + δ n δ n + γ n B A x n z 2 ( 1 δ n δ n + γ n ) δ n δ n + γ n g 2 ( B x n B A x n ) x n z 2 ( 1 δ n δ n + γ n ) δ n δ n + γ n g 2 ( B x n B A x n ) ,
which implies that
( 1 δ n δ n + γ n ) δ n δ n + γ n g 2 ( B x n B A x n ) x n z 2 z n z 2 ( x n z + z n z ) z n x n .
From the condition (iii) and (3.10), we have
lim n g 2 ( B x n B A x n ) = 0 .
From the properties of g 2 , we have
lim n B x n B A x n = 0 .
(3.11)
From the definition of x n , we can rewrite x n + 1 by
x n + 1 = α n f ( x n ) + β n x n + γ n B x n + δ n B A x n = β n x n + γ n B x n + ( α n + δ n ) α n f ( x n ) + δ n B A x n α n + δ n = β n x n + γ n B x n + d n z n ,
(3.12)

where d n = α n + δ n and z n = α n f ( x n ) + δ n B A x n α n + δ n .

From Lemma 2.5 and the convexity of 2 , we have
x n + 1 z 2 = β n ( x n z ) + γ n ( B x n z ) + d n ( z n z ) 2 β n x n z 2 + γ n B x n z 2 + d n z n z 2 β n γ n g 3 ( x n B x n ) = β n x n z 2 + γ n B x n z 2 + d n α n f ( x n ) + δ n B A x n α n + δ n z 2 β n γ n g 3 ( x n B x n ) = β n x n z 2 + γ n B x n z 2 + d n α n α n + δ n ( f ( x n ) z ) + ( 1 α n α n + δ n ) ( B A x n z ) 2 β n γ n g 3 ( x n B x n ) β n x n z 2 + γ n B x n z 2 + d n ( α n α n + δ n f ( x n ) z 2 + ( 1 α n α n + δ n ) B A x n z 2 ) β n γ n g 3 ( x n B x n ) = β n x n z 2 + γ n B x n z 2 + d n α n α n + δ n f ( x n ) z 2 + d n ( 1 α n α n + δ n ) B A x n z 2 β n γ n g 3 ( x n B x n ) β n x n z 2 + γ n x n z 2 + d n α n α n + δ n f ( x n ) z 2 + d n x n z 2 β n γ n g 3 ( x n B x n ) x n z 2 + d n α n α n + δ n f ( x n ) z 2 β n γ n g 3 ( x n B x n ) ,
(3.13)
which implies that
β n γ n g 3 ( x n B x n ) x n z 2 x n + 1 z 2 + d n α n α n + δ n f ( x n ) z 2 ( x n z + x n + 1 z ) x n + 1 x n + d n α n α n + δ n f ( x n ) z 2 .
(3.14)
From the conditions (i), (ii), (iv) (3.14) and (3.3), we have
lim n g 3 ( x n B x n ) = 0 .
From the properties of g 3 , we have
lim n x n B x n = 0 .
(3.15)
From (3.11), (3.15) and
x n B A x n x n B x n + B x n B A x n ,
we have
lim n x n B A x n = 0 .
(3.16)

Define a mapping L : C C by L x = ( 1 ϵ ) B x + ϵ B A x for all x C and ϵ ( 0 , 1 ) . From Lemma 2.7, 2.12 and (3.2), we have F ( L ) = F ( B ) F ( B A ) = i = 1 N S ( C , A i ) F ( S ) F ( T ) = F .

From (3.15) and (3.16) and
x n L x n = ( 1 ϵ ) ( x n B x n ) + ϵ ( x n B A x n ) ( 1 ϵ ) x n B x n + ϵ x n B A x n ,
we have
lim n x n L x n = 0 .
(3.17)
Next, we will show that
lim sup n f ( q ) q , j ( x n q ) 0 ,
(3.18)
where lim t 0 x t = q F and x t begins the fixed point of the contraction
x t f ( x ) + ( 1 t ) L x .

Then x t solves the fixed point equation x t = t f ( x t ) + ( 1 t ) L x t .

From the definition of x t , we have
x t x n 2 = t ( f ( x t ) x n ) + ( 1 t ) ( L x t x n ) 2 ( 1 t ) 2 L x t x n 2 + 2 t f ( x t ) x n , j ( x t x n ) ( 1 t ) 2 ( L x t L x n + L x n x n ) 2 + 2 t f ( x t ) x n , j ( x t x n ) ( 1 t ) 2 ( x t x n + L x n x n ) 2 + 2 t f ( x t ) x n , j ( x t x n ) = ( 1 t ) 2 ( x t x n 2 + 2 x t x n L x n x n + L x n x n 2 ) + 2 t f ( x t ) x n , j ( x t x n ) = ( 1 t ) 2 ( x t x n 2 + 2 x t x n L x n x n + L x n x n 2 ) + 2 t f ( x t ) x t , j ( x t x n ) + 2 t x t x n , j ( x t x n ) = ( 1 2 t + t 2 ) x t x n 2 + ( 1 t ) 2 ( 2 x t x n L x n x n + L x n x n 2 ) + 2 t f ( x t ) x t , j ( x t x n ) + 2 t x t x n 2 = ( 1 + t 2 ) x t x n 2 + f n ( t ) + 2 t f ( x t ) x t , j ( x t x n ) ,
(3.19)
where f n ( t ) = ( 1 t ) 2 ( 2 x t x n L x n x n + L x n x n 2 ) . From (3.17), we have
lim n f n ( t ) = 0 .
(3.20)
(3.19) implies that
x t f ( x t ) , j ( x t x n ) t 2 x t x n 2 + 1 2 t f n ( t ) t 2 D + 1 2 t f n ( t ) ,
(3.21)
where D > 0 such that x t x n 2 D for all t ( 0 , 1 ) and n 1 . From (3.20) and (3.21), we have
lim sup n x t f ( x t ) , j ( x t x n ) t 2 D .
(3.22)
From (3.22) taking t 0 , we have
lim sup t 0 lim sup n x t f ( x t ) , j ( x t x n ) 0 .
(3.23)
Since
f ( q ) q , j ( x n q ) = f ( q ) q , j ( x n q ) f ( q ) q , j ( x n x t ) + f ( q ) q , j ( x n x t ) f ( q ) x t , j ( x n x t ) + f ( q ) x t , j ( x n x t ) f ( x t ) x t , j ( x n x t ) + f ( x t ) x t , j ( x n x t ) = f ( q ) q , j ( x n q ) j ( x n x t ) + x t q , j ( x n x t ) + f ( q ) f ( x t ) , j ( x n x t ) + f ( x t ) x t , j ( x n x t ) f ( q ) q , j ( x n q ) j ( x n x t ) + x t q x n x t + a q x t x n x t + f ( x t ) x t , j ( x n x t ) ,
it follows that
lim sup n f ( q ) q , j ( x n q ) lim sup n f ( q ) q , j ( x n q ) j ( x n x t ) + x t q lim sup n x n x t + a q x t lim sup n x n x t + lim sup n f ( x t ) x t , j ( x n x t ) .
(3.24)
Since j is norm-to-norm uniformly continuous on a bounded subset of C and (3.24), then we have
lim sup n f ( q ) q , j ( x n q ) = lim sup t 0 lim sup n f ( q ) q , j ( x n q ) 0 .
Finally, we will show the sequence { x n } converses strongly to q F . From the definition of x n , we have
x n + 1 q 2 = α n ( f ( x n ) q ) + β n ( x n q ) + γ n ( B x n q ) + δ n ( B A x n q ) 2 β n ( x n q ) + γ n ( B x n q ) + δ n ( B A x n q ) 2 + 2 α n f ( x n ) q , j ( x n + 1 q ) ( β n x n q + γ n B x n q + δ n B A x n q ) 2 + 2 α n f ( x n ) f ( q ) , j ( x n + 1 q ) + 2 α n f ( q ) q , j ( x n + 1 q ) ( 1 α n ) 2 x n q 2 + 2 α n f ( x n ) f ( q ) , j ( x n + 1 q ) + 2 α n f ( q ) q , j ( x n + 1 q ) ( 1 α n ) 2 x n q 2 + 2 a α n x n q x n + 1 q + 2 α n f ( q ) q , j ( x n + 1 q ) ( 1 α n ) 2 x n q 2 + a α n x n q 2 + a α n x n + 1 q 2 + 2 α n f ( q ) q , j ( x n + 1 q ) = ( 1 2 α n + α n 2 ) x n q 2 + a α n x n q 2 + a α n x n + 1 q 2 + 2 α n f ( q ) q , j ( x n + 1 q ) = ( 1 2 α n + a α n ) x n q 2 + α n 2 x n q 2 + a α n x n + 1 q 2 + 2 α n f ( q ) q , j ( x n + 1 q ) = ( 1 a α n 2 α n + 2 a α n ) x n q 2 + α n 2 x n q 2 + a α n x n + 1 q 2 + 2 α n f ( q ) q , j ( x n + 1 q ) = ( 1 a α n 2 α n ( 1 a ) ) x n q 2 + α n 2 x n q 2 + a α n x n + 1 q 2 + 2 α n f ( q ) q , j ( x n + 1 q ) ,
which implies that
x n + 1 q 2 ( 1 2 α n ( 1 a ) 1 a α n ) x n q 2 + α n 1 a α n ( α n x n q 2 + 2 f ( q ) q , j ( x n + 1 q ) ) ( 1 2 α n ( 1 a ) 1 a α n ) x n q 2 + 2 α n ( 1 a ) 1 a α n 1 2 ( 1 a ) ( α n x n q 2 + 2 f ( q ) q , j ( x n + 1 q ) ) .

From the condition (i) and Lemma 2.11, we can imply that { x n } converses strongly to q F . This completes the proof. □

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

Corollary 3.2 Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E. Let Q C be the sunny nonexpansive retraction from E onto C. For every i = 1 , 2 , , N , let A : C E be a ν-inverse strongly accretive mapping. Let T : C C be a nonexpansive mapping and S : C C be an η-strictly pseudo-contractive mapping with F = F ( S ) F ( T ) S ( C , A ) . Define a mapping B A : C C by T ( ( 1 α ) I + α S ) x = B A x , x C and α ( 0 , η K 2 ) , where K is the 2-uniformly smooth constant of E. Let { x n } be the sequence generated by x 1 C and
x n + 1 = α n f ( x n ) + β n x n + γ n Q C ( I λ A ) x n + δ n B A x n , n 1 ,
where f : C C is a contractive mapping and { α n } , { β n } , { γ n } , { δ n } [ 0 , 1 ] , α n + β n + γ n + δ n = 1 , λ ( 0 , ν K 2 ) and satisfy the following conditions:
Then the sequence { x n } converses strongly to q F , which solves the following variational inequality:
q f ( q ) , j ( q p ) 0 , p F .
Corollary 3.3 Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E. Let Q C be the sunny nonexpansive retraction from E onto C. For every i = 1 , 2 , , N , let A i : C E be an α i -inverse strongly accretive mapping. Define a mapping G i : C C by Q C ( I λ i A i ) x = G i x for all x C and i = 1 , 2 , , N , where λ i ( 0 , α i K 2 ) , K is the 2-uniformly smooth constant of E. Let B : C C be the K-mapping generated by G 1 , G 2 , , G N and ρ 1 , ρ 2 , , ρ N , where ρ i ( 0 , 1 ) , i = 1 , 2 , , N 1 and ρ N ( 0 , 1 ] . Let T : C C be a nonexpansive mapping with F = F ( T ) i = 1 N S ( C , A i ) . Let { x n } be the sequence generated by x 1 C and
x n + 1 = α n f ( x n ) + β n x n + γ n B x n + δ n T x n , n 1 ,
where f : C C is a contractive mapping and { α n } , { β n } , { γ n } , { δ n } [ 0 , 1 ] , α n + β n + γ n + δ n = 1 and satisfy the following conditions:
Then the sequence { x n } converses strongly to q F , which solves the following variational inequality:
q f ( q ) , j ( q p ) 0 , p F .
Corollary 3.4 Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E. Let Q C be the sunny nonexpansive retraction from E onto C. For every i = 1 , 2 , , N , let A i : C E be an α i -inverse strongly accretive mapping. Define a mapping G i : C C by Q C ( I λ i A i ) x = G i x for all x C and i = 1 , 2 , , N , where λ i ( 0 , α i K 2 ) , K is the 2-uniformly smooth constant of E. Let B : C C be the K-mapping generated by G 1 , G 2 , , G N and ρ 1 , ρ 2 , , ρ N , where ρ i ( 0 , 1 ) , i = 1 , 2 , , N 1 and ρ N ( 0 , 1 ] . Let S : C C be an η-strictly pseudo-contractive mapping with F = F ( S ) i = 1 N S ( C , A i ) . Define a mapping B A : C C by ( 1 α ) x + α S x = B A x , x C and α ( 0 , η K 2 ) . Let { x n } be the sequence generated by x 1 C and
x n + 1 = α n f ( x n ) + β n x n + γ n B x n + δ n B A x n , n 1 ,
where f : C C is a contractive mapping and { α n } , { β n } , { γ n } , { δ n } [ 0 , 1 ] , α n + β n + γ n + δ n = 1 and satisfy the following conditions:
Then the sequence { x n } converses strongly to q F , which solves the following variational inequality:
q f ( q ) , j ( q p ) 0 , p F .

4 Applications

To prove the next theorem, we needed the following lemma.

Lemma 4.1 Let C be a nonempty closed convex subset of a Banach space E and let P : C C be an η-strictly pseudo-contractive mapping with F ( P ) . Then F ( P ) = S ( C , I P ) .

Proof It is easy to see that F ( P ) S ( C , I P ) . Put A = I P and z F ( P ) . Let z 0 S ( C , I P ) , then there exists j ( x z 0 ) J ( x z 0 ) such that
( I P ) z 0 , j ( x z 0 ) 0 , x C .
(4.1)
Since P is an η-strictly pseudo-contractive mapping, then there exists j ( z 0 z ) such that
P z 0 P z , j ( z 0 z ) = ( I A ) z 0 ( I A ) z , j ( z 0 z ) = z 0 z ( A z 0 A z ) , j ( z 0 z ) = z 0 z , j ( z 0 z ) A z 0 A z , j ( z 0 z ) = z 0 z 2 A z 0 , j ( z 0 z ) z 0 z 2 η ( I P ) z 0 2 .
(4.2)
From (4.1), (4.2), we have
η z 0 P z 0 2 A z 0 , j ( z 0 z ) = A z 0 , j ( z z 0 ) 0 .

It implies that z 0 = P z 0 , that is, z 0 F ( P ) . Then we have S ( C , I P ) F ( P ) . Hence, we have S ( C , I P ) = F ( P ) . □

Remark 4.2 If C is a closed convex subset of a smooth Banach space E and Q C is a sunny nonexpansive retraction from E onto C, from Remark 1.1, Lemma 2.6 and 4.1, we have
F ( P ) = S ( C , I P ) = F ( Q C ( I λ ( I P ) ) )
(4.3)

for all λ > 0 .

Theorem 4.3 Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E. Let Q C be the sunny nonexpansive retraction from E onto C. For every i = 1 , 2 , , N , let S i : C E be an η i -strictly pseudo-contractive mapping. Define a mapping G i : C C by Q C ( I λ i ( I S i ) ) x = G i x for all x C and i = 1 , 2 , , N , where λ i ( 0 , η i K 2 ) , K is the 2-uniformly smooth constant of E. Let B : C C be the K-mapping generated by G 1 , G 2 , , G N and ρ 1 , ρ 2 , , ρ N , where ρ i ( 0 , 1 ) , i = 1 , 2 , , N 1 and ρ N ( 0 , 1 ] . Let T : C C be a nonexpansive mapping and S : C C be an η-strictly pseudo-contractive mapping with F = F ( S ) F ( T ) i = 1 N F ( S i ) . Define a mapping B A : C C by T ( ( 1 α ) I + α S ) x = B A x , x C and α ( 0 , η K 2 ) . Let { x n } be the sequence generated by x 1 C and
x n + 1 = α n f ( x n ) + β n x n + γ n B x n + δ n B A x n , n 1 ,
where f : C C is a contractive mapping and { α n } , { β n } , { γ n } , { δ n } [ 0 , 1 ] , α n + β n + γ n + δ n = 1 and satisfy the following conditions:
Then the sequence { x n } converses strongly to q F , which solves the following variational inequality:
q f ( q ) , j ( q p ) 0 , p F .

Proof Since S i is an η i -strictly pseudo-contractive mapping, then we have ( I S i ) is an η i -inverse strongly accretive mapping for every i = 1 , 2 , , N . For every i = 1 , 2 , , N , putting A i = I S i in Theorem 3.1, from Remark 4.2 and Theorem 3.1, we can conclude the desired results. □

Next corollaries are derived from Theorem 4.3. We, therefore, omit the proof.

Corollary 4.4 Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E. Let Q C be the sunny nonexpansive retraction from E onto C. For every i = 1 , 2 , , N , let S i : C E be an η i -strictly pseudo contractive mapping. Define a mapping G i : C C by Q C ( I λ i ( I S i ) ) x = G i x for all x C and i = 1 , 2 , , N , where λ i ( 0 , η i K 2 ) , K is the 2-uniformly smooth constant of E. Let B : C C be the K-mapping generated by G 1 , G 2 , , G N and ρ 1 , ρ 2 , , ρ N , where ρ i ( 0 , 1 ) , i = 1 , 2 , , N 1 and ρ N ( 0 , 1 ] . Let T : C C be a nonexpansive mapping with F = F ( T ) i = 1 N F ( S i ) . Let { x n } be the sequence generated by x 1 C and
x n + 1 = α n f ( x n ) + β n x n + γ n B x n + δ n T x n , n 1 ,
where f : C C is a contractive mapping and { α n } , { β n } , { γ n } , { δ n } [ 0 , 1 ] , α n + β n + γ n + δ n = 1 and satisfy the following conditions:
Then the sequence { x n } converses strongly to q F , which solves the following variational inequality:
q f ( q ) , j ( q p ) 0 , p F .
Corollary 4.5 Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E. Let Q C be the sunny nonexpansive retraction from E onto C. For every i = 1 , 2 , , N , let S i : C E be an η i -strictly pseudo contractive mapping. Define a mapping G i : C C by Q C ( I λ i ( I S i ) ) x = G i x for all x C and i = 1 , 2 , , N , where λ i ( 0 , η i K 2 ) , K is the 2-uniformly smooth constant of E. Let B : C C be the K-mapping generated by G 1 , G 2 , , G N and ρ 1 , ρ 2 , , ρ N , where ρ i ( 0 , 1 ) , i = 1 , 2 , , N 1 and ρ N ( 0 , 1 ] . S : C C be an η-strictly pseudo contractive mapping with F = F ( S ) i = 1 N F ( S i ) . Define a mapping B A : C C by ( 1 α ) x + α S x = B A x , x C and α ( 0 , η K 2 ) . Let { x n } be a sequence generated by x 1 C and
x n + 1 = α n f ( x n ) + β n x n + γ n B x n + δ n B A x n , n 1 ,
where f : C C is a contractive mapping and { α n } , { β n } , { γ n } , { δ n } [ 0 , 1 ] , α n + β n + γ n + δ n = 1 and satisfy the following conditions:
Then the sequence { x n } converses strongly to q F , which solves the following variational inequality:
q f ( q ) , j ( q p ) 0 , p F .

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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