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A new mapping for finding a common element of the sets of fixed points of two finite families of nonexpansive and strictly pseudo-contractive mappings and two sets of variational inequalities in uniformly convex and 2-smooth Banach spaces

Fixed Point Theory and Applications20132013:157

https://doi.org/10.1186/1687-1812-2013-157

Received: 26 January 2013

Accepted: 28 May 2013

Published: 18 June 2013

Abstract

In this paper we introduce a new mapping in a uniformly convex and 2-smooth Banach space to prove a strong convergence theorem for finding a common element of the set of fixed points of a finite family of nonexpansive mappings and the set of fixed points of a finite family of strictly pseudo-contractive mappings and two sets of solutions of variational inequality problems. Moreover, we also obtain a strong convergence theorem for a finite family of the set of solutions of variational inequality problems and the set of fixed points of a finite family of strictly pseudo-contractive mappings by using our main result.

Keywords

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

1 Introduction

Throughout this paper, we use E and E to denote a real Banach space and a dual space of E, respectively. For any pair x E and f E , x , f instead of f ( x ) . The duality mapping J : E 2 E is defined by J ( x ) = { x E : x , x = x 2 , x = x } for all x E . It is well known that if E is a Hilbert space, then J = I , where I is the identity mapping. Recall the following definitions.

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 A Banach space E is said to be smooth if for each x S E = { x E : x = 1 } , there exists a unique functional j x E such that x , j x = x and j x = 1 .

It is obvious that if E is smooth, then J is single-valued which is denoted by j.

Definition 1.3 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 .

It is well known that every uniformly smooth Banach space is smooth.

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.

A mapping T : C C 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 ) . We give some examples for a strictly pseudo-contractive mapping as follows.

Example 1.1 Let be a real line endowed with the Euclidean norm and let C = ( 0 , ) . Define the mapping T : C C by
T x = 2 x 2 3 + 2 x , x C .

Then T is a 1 9 -strictly pseudo-contractive mapping.

Example 1.2 (See [1])

Let be a real line endowed with the Euclidean norm. Let C = [ 1 , 1 ] and let T : C C be defined by
T x = { x if  x [ 1 , 0 ] ; x x 2 if  x ( 0 , 1 ] .

Then T is a λ-strictly pseudo-contractive mapping where λ min { λ 1 , λ 2 } and λ 1 1 2 , λ 2 < 1 .

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 [2] provided P ( x + t ( x P ( x ) ) ) = P ( x ) for all x C and t 0 , whenever x + t ( x P ( x ) ) C . A 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.

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 exist 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 1.3 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. [3]. 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)

Several problems in pure and applied science, numerous problems in physics and economics reduce to finding an element in (1.4); see, for instance, [46].

Recall that normal Mann’s iterative process was introduced by Mann [7] 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 1967, Halpern has introduced the iteration method guaranteeing the strong convergence as follows:
{ x 1 C , x n + 1 = ( 1 α n ) x 1 + α n T x n , n 1 ,
(1.6)

where { α n } ( 0 , 1 ) . Such an iteration is called Halpern iteration if T is a nonexpansive mapping with a fixed point. He also pointed out that the conditions lim n α n = 0 and n = 1 α n = are necessary for the strong convergence of { x n } to a fixed point of T.

Many authors have modified the iteration (1.6) for a strong convergence theorem; see, for instance, [810].

In 2008, Zhou [11] 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 1.4 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 sequences { α n } , { β n } , { γ n } and { δ n } in ( 0 , 1 ) , the following control conditions are satisfied:
(i) a α n λ K 2 for some a > 0 and for all  n 0 , (ii) β n + γ n + δ n = 1 for all n 0 , (iii)  lim n β n = 0 and n = 1 β n = , (iv) α n + 1 α n 0 , as n , (v)  0 < lim inf n γ n lim sup n γ n < 1 .
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 .

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 2006, Aoyama et al. introduced a Halpern-type iterative sequence and proved that such a sequence converges strongly to a common fixed point of nonexpansive mappings as follows.

Theorem 1.5 Let E be a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable and let C be a nonempty closed convex subset of E. Let { T n } be a sequence of nonexpansive mappings of C into itself such that n = 1 N F ( T i ) is nonempty and let { α n } be a sequence of [ 0 , 1 ] such that lim n α n = 0 and n = 1 α n = . Let { x n } be a sequence of C defined as follows: x 1 = x C and
x n + 1 = α n x + ( 1 α n ) T n x n
for every n N . Suppose that n = 1 sup { T n + 1 z T n z : z B } < for any bounded subset B of C. Let T be a mapping of C into itself defined by T z = lim n T n z for all z C and suppose that F ( T ) = n = 1 F ( T n ) . If either
(i) n = 1 | α n + 1 α n | < or (ii) α n ( 0 , 1 ] for every n N and lim n α n α n + 1 ,

then { x n } converges strongly to Qx, where Q is the sunny nonexpansive retraction of E onto F ( T ) = i = 1 F ( T n ) .

In 2005, Aoyama et al. [3] proved a weak convergence theorem for finding a solution of problem (1.3) as follows.

Theorem 1.6 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 that 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 2009, Kangtunykarn and Suantai [12] introduced the S-mapping generated by a finite family of mappings and real numbers as follows.

Definition 1.4 Let C be a nonempty convex subset of a real Banach space. Let { T i } i = 1 N be a finite family of mappings of C into itself. For each j = 1 , 2 , , N , let α j = ( α 1 j , α 2 j , α 3 j ) I × I × I , where I [ 0 , 1 ] and α 1 j + α 2 j + α 3 j = 1 . Define the mapping S : C C as follows:
U 0 = I , U 1 = α 1 1 T 1 U 0 + α 2 1 U 0 + α 3 1 I , U 2 = α 1 2 T 2 U 1 + α 2 2 U 1 + α 3 2 I , U 3 = α 1 3 T 3 U 2 + α 2 3 U 2 + α 3 3 I , U N 1 = α 1 N 1 T N 1 U N 2 + α 2 N 1 U N 2 + α 3 N 1 I , S = U N = α 1 N T N U N 1 + α 2 N U N 1 + α 3 N I .
(1.7)

This mapping is called the S-mapping generated by T 1 , T 2 , , T N and α 1 , α 2 , , α N .

For every i = 1 , 2 , , N , put α 3 j = 0 in (1.7), then the S-mapping generated by T 1 , T 2 , , T N and α 1 , α 2 , , α N reduces to the K-mapping generated by T 1 , T 2 , , T N and α 1 1 , α 1 2 , , α 1 N , which is defined by Kangtunyakarn and Suantai [13].

Recently, Kangtunyakarn [14] introduced 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 as follows.

Theorem 1.7 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 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 ,
(1.8)
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:
(i) lim n α n = 0 and n = 1 α n = , (ii) { γ n } , { δ n } [ c , d ] ( 0 , 1 ) for some c , d > 0 and n 1 , (iii) n = 1 | β n + 1 β n | , n = 1 | γ n + 1 γ n | , n = 1 | δ n + 1 δ n | < , (iv) 0 < lim inf n β n lim sup n β n < 1 .
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 .

Question How can we prove a strong convergence theorem for the set of fixed points of a finite family of nonexpansive mappings and the set of fixed points of a finite family of strictly pseudo-contractive mappings and the set of solutions of variational inequality problems in a uniformly convex and 2-uniformly smooth Banach space?

Motivated by the S-mapping, we define a new mapping in the next section to answer the above question, and from Theorems 1.4, 1.5, 1.6 and 1.7 we modify the Halpern iteration for finding a common element of two sets of solutions of (1.3) and the set of fixed points of a finite family of nonexpansive mappings and the set of fixed points of a finite family of strictly pseudo-contractive mappings in a uniformly convex and 2-uniformly smooth Banach space. Moreover, by using our main result, we also obtain a strong convergence theorem for a finite family of the set of solutions of (1.3) and the set of fixed points of a finite family of strictly pseudo-contractive mappings.

2 Preliminaries

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

Lemma 2.1 (See [15])

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 .

Lemma 2.2 (See [16])

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 2.3 (See [3])

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 2.4 (See [15])

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 2.5 (See [17])

Let C be a closed and convex subset of a real uniformly smooth Banach space E and let T : C C be a nonexpansive mapping with a nonempty fixed point F ( T ) . If { x n } C is a bounded sequence such that lim n x n T x n = 0 . Then there exists a unique sunny nonexpansive retraction Q F ( T ) : C F ( T ) such that
lim sup n u Q F ( T ) u , J ( x n Q F ( T ) u ) 0

for any given u C .

Lemma 2.6 (See [18])

Let { s n } be a sequence of nonnegative real numbers satisfying
s n + 1 = ( 1 α n ) s n + δ n , n 0 ,
where { α n } is a sequence in ( 0 , 1 ) and { δ n } is a sequence such that
( 1 ) n = 1 α n = , ( 2 ) lim sup n δ n α n 0 or n = 1 | δ n | < .

Then lim n s n = 0 .

From the S-mapping, we define the mapping generated by two sets of finite families of the mappings and real numbers as follows.

Definition 2.1 Let C be a nonempty convex subset of a Banach space. Let { S i } i = 1 N and { T i } i = 1 N be two finite families of mappings of C into itself. For each j = 1 , 2 , , N , let α j = ( α 1 j , α 2 j , α 3 j ) I × I × I , where I [ 0 , 1 ] and α 1 j + α 2 j + α 3 j = 1 . We define the mapping S A : C C as follows:
U 0 = T 1 = I , U 1 = T 1 ( α 1 1 S 1 U 0 + α 2 1 U 0 + α 3 1 I ) , U 2 = T 2 ( α 1 2 S 2 U 1 + α 2 2 U 1 + α 3 2 I ) , U 3 = T 3 ( α 1 3 S 3 U 2 + α 2 3 U 2 + α 3 3 I ) , U N 1 = T N 1 ( α 1 N 1 S N 1 U N 2 + α 2 N 1 U N 2 + α 3 N 1 I ) , S A = U N = T N ( α 1 N S N U N 1 + α 2 N U N 1 + α 3 N I ) .
(2.1)

This mapping is called the S A -mapping generated by S 1 , S 2 , , S N , T 1 , T 2 , , T N and α 1 , α 2 , , α N .

Lemma 2.7 Let C be a nonempty closed convex subset of a 2-uniformly smooth and uniformly convex Banach space. Let { S i } i = 1 N be a finite family of κ i -strict pseudo-contractions of C into itself and let { T i } i = 1 N be a finite family of nonexpansive mappings of C into itself with i = 1 N F ( S i ) i = 1 N F ( T i ) and κ = min { κ i : i = 1 , 2 , , N } with K 2 κ , where K is the 2-uniformly smooth constant of E. Let α j = ( α 1 j , α 2 j , α 3 j ) I × I × I , where I = [ 0 , 1 ] , α 1 j + α 2 j + α 3 j = 1 , α 1 j ( 0 , 1 ] , α 2 j [ 0 , 1 ] and α 3 j ( 0 , 1 ) for all j = 1 , 2 , , N . Let S A be the S A -mapping generated by S 1 , S 2 , , S N , T 1 , T 2 , , T N and α 1 , α 2 , , α N . Then F ( S A ) = i = 1 N F ( S i ) i = 1 N F ( T i ) and S A is a nonexpansive mapping.

Proof Let x 0 F ( S A ) and x i = 1 N F ( S i ) i = 1 N F ( T i ) , we have
x 0 x 2 = T N ( α 1 N S N U N 1 + α 2 N U N 1 + α 3 N I ) x 0 x 2 α 1 N ( S N U N 1 x 0 x ) + α 2 N ( U N 1 x 0 x ) + α 3 N ( x 0 x ) 2 = ( 1 α 3 N ) ( α 1 N 1 α 3 N ( S N U N 1 x 0 x ) + α 2 N 1 α 3 N ( U N 1 x 0 x ) ) + α 3 N ( x 0 x ) 2 ( 1 α 3 N ) α 1 N 1 α 3 N ( S N U N 1 x 0 x ) + α 2 N 1 α 3 N ( U N 1 x 0 x ) 2 + α 3 N x 0 x 2 = ( 1 α 3 N ) α 1 N 1 α 3 N ( S N U N 1 x 0 x ) + ( 1 α 1 N 1 α 3 N ) ( U N 1 x 0 x ) 2 + α 3 N x 0 x 2 = ( 1 α 3 N ) α 1 N 1 α 3 N ( S N U N 1 x 0 U N 1 x 0 ) + U N 1 x 0 x 2 + α 3 N x 0 x 2 ( 1 α 3 N ) ( U N 1 x 0 x 2 + 2 α 1 N 1 α 3 N S N U N 1 x 0 U N 1 x 0 , j ( U N 1 x 0 x ) + 2 K 2 ( α 1 N 1 α 3 N ) 2 S N U N 1 x 0 U N 1 x 0 2 ) + α 3 N x 0 x 2 = ( 1 α 3 N ) ( U N 1 x 0 x 2 + 2 α 1 N 1 α 3 N S N U N 1 x 0 x , j ( U N 1 x 0 x ) + 2 α 1 N 1 α 3 N x U N 1 x 0 , j ( U N 1 x 0 x ) + 2 K 2 ( α 1 N 1 α 3 N ) 2 S N U N 1 x 0 U N 1 x 0 2 ) + α 3 N x 0 x 2 ( 1 α 3 N ) ( U N 1 x 0 x 2 + 2 α 1 N 1 α 3 N ( U N 1 x 0 x 2 κ ( I S N ) U N 1 x 0 2 ) 2 α 1 N 1 α 3 N x U N 1 x 0 2 + 2 K 2 ( α 1 N 1 α 3 N ) 2 S N U N 1 x 0 U N 1 x 0 2 ) + α 3 N x 0 x 2 = ( 1 α 3 N ) ( U N 1 x 0 x 2 2 α 1 N 1 α 3 N κ ( I S N ) U N 1 x 0 2 + 2 K 2 ( α 1 N 1 α 3 N ) 2 S N U N 1 x 0 U N 1 x 0 2 ) + α 3 N x 0 x 2 = ( 1 α 3 N ) ( U N 1 x 0 x 2 2 α 1 N 1 α 3 N ( κ K 2 ( α 1 N 1 α 3 N ) ) ( I S N ) U N 1 x 0 2 ) + α 3 N x 0 x 2 ( 1 α 3 N ) U N 1 x 0 x 2 + α 3 N x 0 x 2 ( 1 α 3 N ) ( ( 1 α 3 N 1 ) U N 2 x 0 x 2 + α 3 N 1 x 0 x 2 ) + α 3 N x 0 x 2 = j = N 1 N ( 1 α 3 j ) U N 2 x 0 x 2 + ( 1 j = N 1 N ( 1 α 3 j ) ) x 0 x 2 j = 3 N ( 1 α 3 j ) U 2 x 0 x 2 + ( 1 j = 3 N ( 1 α 3 j ) ) x 0 x 2 = j = 3 N ( 1 α 3 j ) T 2 ( α 1 2 S 2 U 1 + α 2 2 U 1 + α 3 2 I ) x 0 x 2 + ( 1 j = 3 N ( 1 α 3 j ) ) x 0 x 2 j = 3 N ( 1 α 3 j ) α 1 2 ( S 2 U 1 x 0 x ) + α 2 2 ( U 1 x 0 x ) + α 3 2 ( x 0 x ) 2 + ( 1 j = 3 N ( 1 α 3 j ) ) x 0 x 2 = j = 3 N ( 1 α 3 j ) ( 1 α 3 2 ) ( α 1 2 1 α 3 2 ( S 2 U 1 x 0 x ) + α 2 2 1 α 3 2 ( U 1 x 0 x ) ) + α 3 2 ( x 0 x ) 2 + ( 1 j = 3 N ( 1 α 3 j ) ) x 0 x 2 j = 3 N ( 1 α 3 j ) ( ( 1 α 3 2 ) α 1 2 1 α 3 2 ( S 2 U 1 x 0 x ) + α 2 2 1 α 3 2 ( U 1 x 0 x ) 2 + α 3 2 x 0 x 2 ) + ( 1 j = 3 N ( 1 α 3 j ) ) x 0 x 2 = j = 3 N ( 1 α 3 j ) ( ( 1 α 3 2 ) α 1 2 1 α 3 2 ( S 2 U 1 x 0 x ) + ( 1 α 1 2 1 α 3 2 ) ( U 1 x 0 x ) 2 + α 3 2 x 0 x 2 ) + ( 1 j = 3 N ( 1 α 3 j ) ) x 0 x 2 = j = 3 N ( 1 α 3 j ) ( ( 1 α 3 2 ) α 1 2 1 α 3 2 ( S 2 U 1 x 0 U 1 x 0 ) + U 1 x 0 x 2 + α 3 2 x 0 x 2 ) + ( 1 j = 3 N ( 1 α 3 j ) ) x 0 x 2 j = 3 N ( 1 α 3 j ) ( ( 1 α 3 2 ) ( U 1 x 0 x 2 + 2 α 1 2 1 α 3 2 S 2 U 1 x 0 U 1 x 0 , j ( U 1 x 0 x ) + 2 K 2 ( α 1 2 1 α 3 2 ) S 2 U 1 x 0 U 1 x 0 2 ) + α 3 2 x 0 x 2 ) + ( 1 j = 3 N ( 1 α 3 j ) ) x 0 x 2 j = 3 N ( 1 α 3 j ) ( ( 1 α 3 2 ) ( U 1 x 0 x 2 2 α 1 2 1 α 3 2 ( κ K 2 ( α 1 2 1 α 3 2 ) ) ( I S 2 ) U 1 x 0 2 ) + α 3 2 x 0 x 2 ) + ( 1 j = 3 N ( 1 α 3 j ) ) x 0 x 2 j = 3 N ( 1 α 3 j ) ( 1 α 3 2 ) ( U 1 x 0 x 2 + α 3 2 x 0 x 2 ) + ( 1 j = 3 N ( 1 α 3 j ) ) x 0 x 2 = j = 2 N ( 1 α 3 j ) U 1 x 0 x 2 + ( 1 j = 2 N ( 1 α 3 j ) ) x 0 x 2 = j = 2 N ( 1 α 3 j ) α 1 1 ( S 1 U 0 x 0 x ) + α 2 1 ( U 0 x 0 x ) + α 3 1 ( x 0 x ) 2 + ( 1 j = 2 N ( 1 α 3 j ) ) x 0 x 2 = j = 2 N ( 1 α 3 j ) α 1 1 ( S 1 x 0 x ) + ( 1 α 1 1 ) ( x 0 x ) 2 + ( 1 j = 2 N ( 1 α 3 j ) ) x 0 x 2 = j = 2 N ( 1 α 3 j ) α 1 1 ( S 1 x 0 x 0 ) + x 0 x 2 + ( 1 j = 2 N ( 1 α 3 j ) ) x 0 x 2 j = 2 N ( 1 α 3 j ) ( x 0 x 2 + 2 α 1 1 S 1 x 0 x 0 , j ( x 0 x ) + 2 K 2 ( α 1 1 ) 2 S 1 x 0 x 0 2 ) + ( 1 j = 2 N ( 1 α 3 j ) ) x 0 x 2 = j = 2 N ( 1 α 3 j ) ( x 0 x 2 + 2 α 1 1 S 1 x 0 x , j ( x 0 x ) + 2 α 1 1 x x 0 , j ( x 0 x ) + 2 K 2 ( α 1 1 ) 2 S 1 x 0 x 0 2 ) + ( 1 j = 2 N ( 1 α 3 j ) ) x 0 x 2 j = 2 N ( 1 α 3 j ) ( x 0 x 2 + 2 α 1 1 ( x 0 x κ S 1 x 0 x 0 2 ) 2 α 1 1 x x 0 2 + 2 K 2 ( α 1 1 ) 2 S 1 x 0 x 0 2 ) + ( 1 j = 2 N ( 1 α 3 j ) ) x 0 x 2 = j = 2 N ( 1 α 3 j ) ( x 0 x 2 2 α 1 1 ( κ K 2 α 1 1 ) S 1 x 0 x 0 2 ) + ( 1 j = 2 N ( 1 α 3 j ) ) x 0 x 2 = x 0 x 2 j = 2 N ( 1 α 3 j ) 2 α 1 1 ( κ K 2 α 1 1 ) S 1 x 0 x 0 2 x 0 x 2 .
(2.2)
For every j = 1 , 2 , , N and (2.2), we have
U j x 0 x 2 x 0 x 2 .
(2.3)
For every k = 1 , 2 , , N 1 and (2.2) we have
x 0 x 2 j = k + 1 N ( 1 α 3 j ) U k x 0 x 2 + ( 1 j = k + 1 N ( 1 α 3 j ) ) x 0 x 2 = j = k + 1 N ( 1 α 3 j ) T k ( α 1 k S k U k 1 + α 2 k U k 1 + α 3 k I ) x 0 x 2 + ( 1 j = k + 1 N ( 1 α 3 j ) ) x 0 x 2 j = k + 1 N ( 1 α 3 j ) α 1 k ( S k U k 1 x 0 x ) + α 2 k ( U k 1 x 0 x ) + α 3 k ( x 0 x ) 2 + ( 1 j = k + 1 N ( 1 α 3 j ) ) x 0 x 2 = j = k + 1 N ( 1 α 3 j ) ( 1 α 3 k ) ( α 1 k 1 α 3 k ( S k U k 1 x 0 x ) + α 2 k 1 α 3 k ( U k 1 x 0 x ) ) + α 3 k ( x 0 x ) 2 + ( 1 j = k + 1 N ( 1 α 3 j ) ) x 0 x 2 j = k + 1 N ( 1 α 3 j ) ( ( 1 α 3 k ) α 1 k 1 α 3 k ( S k U k 1 x 0 x ) + α 2 k 1 α 3 k ( U k 1 x 0 x ) 2 + α 3 k x 0 x 2 ) + ( 1 j = k + 1 N ( 1 α 3 j ) ) x 0 x 2 = j = k + 1 N ( 1 α 3 j ) ( ( 1 α 3 k ) α 1 k 1 α 3 k ( S k U k 1 x 0 x ) + ( 1 α 1 k 1 α 3 k ) ( U k 1 x 0 x ) 2 + α 3 k x 0 x 2 ) + ( 1 j = k + 1 N ( 1 α 3 j ) ) x 0 x 2 = j = k + 1 N ( 1 α 3 j ) ( ( 1 α 3 k ) α 1 k 1 α 3 k ( S k U k 1 x 0 U k 1 x 0 ) + U k 1 x 0 x 2 + α 3 k x 0 x 2 ) + ( 1 j = k + 1 N ( 1 α 3 j ) ) x 0 x 2 j = k + 1 N ( 1 α 3 j ) ( ( 1 α 3 k ) ( U k 1 x 0 x 2 + 2 α 1 k 1 α 3 k S k U k 1 x 0 U k 1 x 0 , j ( U k 1 x 0 x ) + 2 K 2 ( α 1 k 1 α 3 k ) 2 S k U k 1 x 0 U k 1 x 0 2 ) + α 3 k x 0 x 2 ) + ( 1 j = k + 1 N ( 1 α 3 j ) ) x 0 x 2 = j = k + 1 N ( 1 α 3 j ) ( ( 1 α 3 k ) ( U k 1 x 0 x 2 + 2 α 1 k 1 α 3 k S k U k 1 x 0 x , j ( U k 1 x 0 x ) + 2 α 1 k 1 α 3 k x U k 1 x 0 , j ( U k 1 x 0 x ) + 2 K 2 ( α 1 k 1 α 3 k ) 2 S k U k 1 x 0 U k 1 x 0 2 ) + α 3 k x 0 x 2 ) + ( 1 j = k + 1 N ( 1 α 3 j ) ) x 0 x 2 j = k + 1 N ( 1 α 3 j ) ( ( 1 α 3 k ) ( U k 1 x 0 x 2 + 2 α 1 k 1 α 3 k ( U k 1 x 0 x 2 κ ( I S k ) U k 1 x 0 ) 2 α 1 k 1 α 3 k x U k 1 x 0 2 + 2 K 2 ( α 1 k 1 α 3 k ) 2 S k U k 1 x 0 U k 1 x 0 2 ) + α 3 k x 0 x 2 ) + ( 1 j = k + 1 N ( 1 α 3 j ) ) x 0 x 2 = j = k + 1 N ( 1 α 3 j ) ( ( 1 α 3 k ) ( U k 1 x 0 x 2 2 α 1 k 1 α 3 k ( κ K 2 ( α 1 k 1 α 3 k ) ) ( I S k ) U k 1 x 0 2 ) + α 3 k x 0 x 2 ) + ( 1 j = k + 1 N ( 1 α 3 j ) ) x 0 x 2 j = k + 1 N ( 1 α 3 j ) ( ( 1 α 3 k ) ( x 0 x 2 2 α 1 k 1 α 3 k ( κ K 2 ( α 1 k 1 α 3 k ) ) ( I S k ) U k 1 x 0 2 ) + α 3 k x 0 x 2 ) + ( 1 j = k + 1 N ( 1 α 3 j ) ) x 0 x 2 ,
which implies that
U k 1 x 0 = S k U k 1 x 0
(2.4)

for every k = 1 , 2 , , N 1 .

From (2.2), it implies that x 0 = S 1 x 0 , that is, x 0 F ( S ) . From the definition of S A , we have
U 1 x 0 = T 1 ( α 1 1 S 1 U 0 x 0 + α 2 1 U 0 x 0 + α 3 1 x 0 ) = T 1 x 0 = x 0 .
(2.5)
From (2.2) and U 1 x 0 = x 0 , we have
x 0 x 2 j = 3 N ( 1 α 3 j ) ( ( 1 α 3 2 ) ( U 1 x 0 x 2 2 α 1 2 1 α 3 2 ( κ K 2 ( α 1 2 1 α 3 2 ) ) ( I S 2 ) U 1 x 0 2 ) + α 3 2 x 0 x 2 ) + ( 1 j = 3 N ( 1 α 3 j ) ) x 0 x 2 = j = 3 N ( 1 α 3 j ) ( ( 1 α 3 2 ) ( x 0 x 2 2 α 1 2 1 α 3 2 ( κ K 2 ( α 1 2 1 α 3 2 ) ) ( I S 2 ) x 0 2 ) + α 3 2 x 0 x 2 ) + ( 1 j = 3 N ( 1 α 3 j ) ) x 0 x 2 = j = 3 N ( 1 α 3 j ) ( 1 α 3 2 ) ( x 0 x 2 2 α 1 2 1 α 3 2 ( κ K 2 ( α 1 2 1 α 3 2 ) ) ( I S 2 ) x 0 2 ) + j = 3 N ( 1 α 3 j ) α 3 2 x 0 x 2 + ( 1 j = 3 N ( 1 α 3 j ) ) x 0 x 2 = j = 2 N ( 1 α 3 j ) ( x 0 x 2 2 α 1 2 1 α 3 2 ( κ K 2 ( α 1 2 1 α 3 2 ) ) ( I S 2 ) x 0 2 ) + ( 1 j = 2 N ( 1 α 3 j ) ) x 0 x 2 .

It implies that x 0 = S 2 x 0 .

From the definition of S A and x 0 = S 2 x 0 , we have
U 2 x 0 = T 2 ( α 1 2 S 2 U 1 + α 2 2 U 1 + α 3 2 I ) x 0 = T 2 x 0 .
(2.6)
From the definition of U 3 and (2.4), we have
U 3 x 0 = T 3 ( α 1 3 S 3 U 2 + α 2 3 U 2 + α 3 3 I ) x 0 = T 3 ( ( 1 α 3 3 ) U 2 x 0 + α 3 3 x 0 ) .
(2.7)
From (2.2), (2.6), (2.7) and E is uniformly convex, we have
x 0 x 2 j = 4 N ( 1 α 3 j ) U 3 x 0 x 2 + ( 1 j = 4 N ( 1 α 3 j ) ) x 0 x 2 = j = 4 N ( 1 α 3 j ) T 3 ( ( 1 α 3 3 ) U 2 x 0 + α 3 3 x 0 ) x 2 + ( 1 j = 4 N ( 1 α 3 j ) ) x 0 x 2 j = 4 N ( 1 α 3 j ) ( 1 α 3 3 ) ( U 2 x 0 x ) + α 3 3 ( x 0 x ) 2 + ( 1 j = 4 N ( 1 α 3 j ) ) x 0 x 2 = j = 4 N ( 1 α 3 j ) ( 1 α 3 3 ) ( T 2 x 0 x ) + α 3 3 ( x 0 x ) 2 + ( 1 j = 4 N ( 1 α 3 j ) ) x 0 x 2 j = 4 N ( 1 α 3 j ) ( ( 1 α 3 3 ) T 2 x 0 x 2 + α 3 3 x 0 x 2 α 3 3 ( 1 α 3 3 ) g 2 ( T 2 x 0 x 0 ) ) + ( 1 j = 4 N ( 1 α 3 j ) ) x 0 x 2 j = 4 N ( 1 α 3 j ) ( x 0 x 2 α 3 3 ( 1 α 3 3 ) g 2 ( T 2 x 0 x 0 ) ) + ( 1 j = 4 N ( 1 α 3 j ) ) x 0 x 2 .
It implies that
g 2 ( T 2 x 0 x 0 ) = 0 .
(2.8)
Assume that T 2 x 0 x 0 , then we have T 2 x 0 x 0 > 0 . From the properties of g 2 , we have
0 = g ( 0 ) < g ( T 2 x 0 x 0 ) = 0 .
(2.9)

This is a contradiction. Then we have T 2 x 0 = x 0 . From (2.6), we have x 0 = T 2 x 0 = U 2 x 0 .

From the definition of U 3 , we have
U 3 x 0 = T 3 ( ( 1 α 3 3 ) U 2 x 0 + α 3 3 x 0 ) = T 3 x 0 .
By using the same method as above, we have
x 0 = U 3 x 0 = T 3 x 0 .
Continuing on this way, we can conclude that
x 0 = U i x 0 = T i x 0
(2.10)
for every i = 1 , 2 , , N 1 . From (2.2) and (2.10), we have
x 0 x 2 ( 1 α 3 N ) ( U N 1 x 0 x 2 2 α 1 N 1 α 3 N ( κ K 2 ( α 1 N 1 α 3 N ) ) ( I S N ) U N 1 x 0 2 ) + α 3 N x 0 x 2 = ( 1 α 3 N ) ( x 0 x 2 2 α 1 N 1 α 3 N ( κ K 2 ( α 1 N 1 α 3 N ) ) ( I S N ) x 0 2 ) + α 3 N x 0 x 2 .
It implies that
x 0 = S N x 0 .
(2.11)
From the definition of S A and (2.10), we have
x 0 = S A x 0 = U N x 0 = T N ( α 1 N S N U N 1 + α 2 N U N 1 + α 3 N I ) x 0 = T N x 0 .
Then we have
x 0 i = 1 N F ( T i ) and x 0 i = 1 N F ( U i ) .
(2.12)
Since S k U k 1 x 0 = U k 1 x 0 for every k = 1 , 2 , , N 1 and x 0 i = 1 N F ( U i ) , then we have
S k x 0 = x 0
for every k = 1 , 2 , , N 1 . From (2.11), it implies that
x 0 i = 1 N F ( S i ) .
(2.13)
From (2.12) and (2.13), we have
x 0 i = 1 N F ( T i ) i = 1 N F ( S i ) .
(2.14)

Hence, F ( S A ) i = 1 N F ( T i ) i = 1 N F ( S i ) . It is easy to see that i = 1 N F ( T i ) i = 1 N F ( S i ) F ( S A ) .

Applying (2.2), we have that the mapping S A is nonexpansive. □

Lemma 2.8 [19]

Let C be a closed convex subset of a strictly convex Banach space E. Let T 1 and T 2 be two nonexpansive mappings from C into itself with F ( T 1 ) F ( T 2 ) . Define a mapping S by
S x = λ T 1 x + ( 1 λ ) T 2 x , x C ,

where λ is a constant in ( 0 , 1 ) . Then S is nonexpansive and F ( S ) = F ( T 1 ) F ( T 2 ) .

Applying Lemma 2.8, we have the following lemma.

Lemma 2.9 Let C be a closed convex subset of a strictly convex Banach space E. Let T 1 , T 2 and T 3 be three nonexpansive mappings from C into itself with F ( T 1 ) F ( T 2 ) F ( T 3 ) . Define a mapping S by
S x = α T 1 x + β T 2 x + γ T 3 x , x C ,

where α, β, γ is a constant in ( 0 , 1 ) and α + β + γ = 1 . Then S is nonexpansive and F ( S ) = F ( T 1 ) F ( T 2 ) F ( T 3 ) .

Proof For every x C and the definition of the mapping S, we have
S x = α T 1 x + β T 2 x + γ T 3 x = α T 1 x + ( 1 α ) ( β 1 α T 2 x + γ 1 α T 3 x ) = α T 1 x + ( 1 α ) ( β 1 α T 2 x + ( 1 β 1 α ) T 3 x ) = α T 1 x + ( 1 α ) S 1 x ,
(2.15)

where S 1 = β 1 α T 2 + ( 1 β 1 α ) T 3 . From Lemma 2.8, we have F ( S 1 ) = F ( T 2 ) F ( T 3 ) and S 1 is a nonexpansive mapping. From Lemma 2.8 and (2.15), we have F ( S ) = F ( T 1 ) F ( S 1 ) and S is a nonexpansive mapping. Hence we have F ( S ) = F ( T 1 ) F ( T 2 ) F ( T 3 ) . □

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 a sunny nonexpansive retraction from E onto C and let A, B be α- and β-inverse strongly accretive mappings of C into E, respectively. Let { S i } i = 1 N be a finite family of κ i -strict pseudo-contractions of C into itself and let { T i } i = 1 N be a finite family of nonexpansive mappings of C into itself with F = i = 1 N F ( S i ) i = 1 N F ( T i ) S ( C , A ) S ( C , B ) and κ = min { κ i : i = 1 , 2 , , N } with K 2 κ , where K is the 2-uniformly smooth constant of E. Let α j = ( α 1 j , α 2 j , α 3 j ) I × I × I , where I = [ 0 , 1 ] , α 1 j + α 2 j + α 3 j = 1 , α 1 j ( 0 , 1 ] , α 2 j [ 0 , 1 ] and α 3 j ( 0 , 1 ) for all j = 1 , 2 , , N . Let S A be the S A -mapping generated by S 1 , S 2 , , S N , T 1 , T 2 , , T N and α 1 , α 2 , , α N . Let { x n } be the sequence generated by x 1 , u C and
x n + 1 = α n u + β n x n + γ n Q C ( I a A ) x n + δ n Q C ( I b B ) x n + η n S A x n , n 1 ,
(3.1)
where { α n } , { β n } , { γ n } , { δ n } , { η n } [ 0 , 1 ] and α n + β n + γ n + δ n + η n = 1 and satisfy the following conditions:
(i) lim n α n = 0 , n = 1 α n = , (ii) { γ n } , { δ n } , { η n } [ c , d ] ( 0 , 1 ) , for some c , d > 0 , n 1 , (iii) n = 1 | β n + 1 β n | , n = 1 | γ n + 1 γ n | , n = 1 | δ n + 1 δ n | , (iii) n = 1 | η n + 1 η n | , n = 1 | α n + 1 α n | < , (iv) 0 < lim inf n β n lim sup n β n < 1 , (v) a ( 0 , α K 2 ) and b ( 0 , β K 2 ) .

Then { x n } converges strongly to z 0 = Q F u , where Q F is the sunny nonexpansive retraction of C onto .

Proof First we show that Q C ( I a A ) and Q C ( I b B ) are nonexpansive mappings. Let x , y C , we have
Q C ( I a A ) x Q C ( I a A ) y 2 x y a ( A x A y ) 2 x y 2 2 a A x A y , j ( x y ) + 2 K 2 a 2 A x A y 2 x y 2 2 a α A x A y 2 + 2 K 2 a 2 A x A y 2 = x y 2 2 a ( α K 2 a ) A x A y 2 x y 2 .
(3.2)

Then we have Q C ( I a A ) is a nonexpansive mapping. By using the same methods as (3.2), we have Q C ( I b B ) is a nonexpansive mapping.

Let x F . From Lemma 2.3, we have x F ( Q C ( I a A ) ) and x F ( Q C ( I b B ) ) . By the definition of x n , we have
x n + 1 x α n u x + β n x n x + γ n Q C ( I a A ) x n x + δ n Q C ( I b B ) x n x + η n S A x n x α n u x + ( 1 α n ) x n x max { u x , x 1 x } .

By induction, we have x n x max { u x , x 1 x } . We can imply that the sequence { x n } is bounded and so are { S A x n } , { Q C ( I a A ) x n } and { Q C ( I b B ) x n } .

Next, we show that lim n x n + 1 x n = 0 . From the definition of x n , we have
x n + 1 x n = α n u + β n x n + γ n Q C ( I a A ) x n + δ n Q C ( I b B ) x n + η n S A x n α n 1 u β n 1 x n 1 γ n 1 Q C ( I a A ) x n 1 δ n 1 Q C ( I b B ) x n 1 η n 1 S A x n 1 | α n α n 1 | u + β n x n x n 1 + | β n β n 1 | x n 1 + γ n Q C ( I a A ) x n Q C ( I a A ) x n 1 + | γ n γ n 1 | Q C ( I a A ) x n 1 + δ n Q C ( I b B ) x n Q C ( I b B ) x n 1 + | δ n δ n 1 | Q C ( I b B ) x n 1 + η n S A x n S A x n 1 + | η n 1 η n | S A x n ( 1 α n ) x n x n 1 + | α n α n 1 | u + | β n β n 1 | x n 1 + | γ n γ n 1 | Q C ( I a A ) x n 1 + | δ n δ n 1 | Q C ( I b B ) x n 1 + | η n 1 η n | S A x n .
Applying Lemma 2.6, we have
lim n x n + 1 x n = 0 .
(3.3)
Next, we show that
lim n Q C ( I a A ) x n x n = lim n Q C ( I b B ) x n x n = lim n S A x n x n = 0 .
(3.4)
From the definition of x n , we have
x n + 1 x 2 = α n ( u x ) + β n ( x n x ) + γ n ( Q C ( I a A ) x n x ) + δ n ( Q C ( I b B ) x n x ) + η n ( S A x n x ) 2 = β n ( x n x ) + γ n ( Q C ( I a A ) x n x ) + ( α n + δ n + η n ) ( α n ( u x ) α n + δ n + η n + δ n ( Q C ( I b B ) x n x ) α n + δ n + η n + η n ( S A x n x ) α n + δ n + η n ) 2 = β n ( x n x ) + γ n ( Q C ( I a A ) x n x ) + c n z n 2 ,

where c n = α n + δ n + η n and z n = α n ( u x ) α n + δ n + η n + δ n ( Q C ( I b B ) x n x ) α n + δ n + η n + η n ( S A x n x ) α n + δ n + η n .

From Lemma 2.2, we have
x n + 1 x 2 β n x n x 2 + γ n Q C ( I a A ) x n x + c n z n 2 β n γ n g 1 ( x n Q C ( I a A ) x n ) ( β n + γ n ) x n x 2 β n γ n g 1 ( x n Q C ( I a A ) x n ) + c n ( α n u x 2 α n + δ n + η n + δ n Q C ( I b B ) x n x 2 α n + δ n + η n + η n S A x n x 2 α n + δ n + η n ) ( β n + γ n ) x n x 2 β n γ n g 1 ( x n Q C ( I a A ) x n ) + α n u x 2 + ( δ n + η n ) x n x 2 x n x 2 β n γ n g 1 ( x n Q C ( I a A ) x n ) + α n u x 2 ,
which implies that
β n γ n g 1 ( x n Q C ( I a A ) x n ) x n x 2 x n + 1 x 2 + α n u x 2 ( x n x + x n + 1 x ) x n + 1 x n + α n u x 2 .
(3.5)
From (3.3) and condition (i), we obtain
lim n g 1 ( x n Q C ( I a A ) x n ) = 0 .
(3.6)
From the property of g 1 , we have
lim n x n Q C ( I a A ) x n = 0 .
(3.7)
By using the same method as (3.7), we can imply that
lim n x n Q C ( I b B ) x n = lim n x n S A x n = 0 .
Define G x = α S A x + β Q C ( I a A ) x + γ Q C ( I b B ) x for all x C and α + β + γ = 1 . From Lemma 2.9, we have F ( G ) = F ( Q C ( I a A ) ) F ( Q C ( I b B ) ) F ( S A ) . From Lemmas 2.3 and 2.7, we have F = F ( G ) = i = 1 N F ( T i ) i = 1 N F ( S i ) S ( C , A ) S ( C , B ) . By the definition of G, we obtain
G x n x n α S A x n x n + β Q C ( I a A ) x n x n + γ Q C ( I b B ) x n x n .
From (3.4), we have
lim n G x n x n = 0 .
(3.8)
From Lemma 2.5 and (3.8), we have
lim sup n u z 0 , j ( x n z 0 ) 0 ,
(3.9)
where z 0 = Q F u . Finally, we prove strong convergence of the sequence { x n } to z 0 = Q F u . From the definition of x n , we have
x n + 1 z 0 2 = α n ( u z 0 ) + β n ( x n z 0 ) + γ n ( Q C ( I a A ) x n z 0 ) + δ n ( Q C ( I b B ) x n z 0 ) + η n ( S A x n z 0 ) 2 = α n ( u z 0 ) + ( 1 α n ) ( β n ( x n z 0 ) 1 α n + γ n ( Q C ( I a A ) x n z 0 ) 1 α n + δ n ( Q C ( I b B ) x n z 0 ) 1 α n + η n ( S A x n z 0 ) 1 α n ) 2 ( 1 α n ) ( β n ( x n z 0 ) 1 α n + γ n ( Q C ( I a A ) x n z 0 ) 1 α n + δ n ( Q C ( I b B ) x n z 0 ) 1 α n + η n ( S A x n z 0 ) 1 α n ) 2 + 2 α n u x 0 , j ( x n + 1 z 0 ) ( 1 α n ) x n z 0 2 + 2 α n u x 0 , j ( x n + 1 z 0 ) .

Applying Lemma 2.6 and condition (i), we have lim n x n z 0 = 0 . This completes the proof. □

4 Applications

From our main results, we obtain strong convergence theorems in a Banach space. Before proving these theorem, we need the following lemma which is the result from Lemma 2.7 and Definition 1.4. Therefore, we omit the proof.

Lemma 4.1 Let C be a nonempty closed convex subset of a 2-uniformly smooth and uniformly convex Banach space. Let { S i } i = 1 N be a finite family of κ i -strict pseudo-contractions of C into itself with i = 1 N F ( S i ) and κ = min { κ i : i = 1 , 2 , , N } with K 2 κ , where K is the 2-uniformly smooth constant of E. Let α j = ( α 1 j , α 2 j , α 3 j ) I × I × I , where I = [ 0 , 1 ] , α 1 j + α 2 j + α 3 j = 1 , α 1 j ( 0 , 1 ] , α 2 j [ 0 , 1 ] and α 3 j ( 0 , 1 ) for all j = 1 , 2 , , N . Let S be the S-mapping generated by S 1 , S 2 , , S N and α 1 , α 2 , , α N . Then F ( S ) = i = 1 N F ( S i ) and S is a nonexpansive mapping.

Theorem 4.2 Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E. Let Q C be a sunny nonexpansive retraction from E onto C and let A, B be α- and β-inverse strongly accretive mappings of C into E, respectively. Let { S i } i = 1 N be a finite family of κ i -strict pseudo-contractions of C into itself with F = i = 1 N F ( S i ) S ( C , A ) S ( C , B ) and κ = min { κ i : i = 1 , 2 , , N } with K 2 κ , where K is the 2-uniformly smooth constant of E. Let α j = ( α 1 j , α 2 j , α 3 j ) I × I × I , where I = [ 0 , 1 ] , α 1 j + α 2 j + α 3 j = 1 , α 1 j ( 0 , 1 ] , α 2 j [ 0 , 1 ] and α 3 j ( 0 , 1 ) for all j = 1 , 2 , , N . Let S be the S-mapping generated by S 1 , S 2 , , S N and α 1 , α 2 , , α N . Let { x n } be the sequence generated by x 1 , u C and
x n + 1 = α n u + β n x n + γ n Q C ( I a A ) x n + δ n Q C ( I b B ) x n + η n S x n , n 1 ,
where { α n } , { β n } , { γ n } , { δ n } , { η n } [ 0 , 1 ] and α n + β n + γ n + δ n + η n = 1 and satisfy the following conditions:
(i) lim n α n = 0 , n = 1 α n = , (ii) { γ n } , { δ n } , { η n } [ c , d ] ( 0 , 1 ) for some c , d > 0 , n 1 , (iii) n = 1 | β n + 1 β n | , n = 1 | γ n + 1 γ n | , n = 1 | δ n + 1 δ n | , (iii) n = 1 | η n + 1 η n | , n = 1 | α n + 1 α n | < , (iv) 0 < lim inf n β n lim sup n β n < 1 , (v) a ( 0 , α K 2 ) and b ( 0 , β K 2 ) .

Then { x n } converges strongly to z 0 = Q F u , where Q F is the sunny nonexpansive retraction of C onto .

Proof Put I = T 1 = T 2 = = T N in Theorem 3.1. From Lemma 4.1 and Theorem 3.1 we can conclude the desired result. □

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 a sunny nonexpansive retraction from E onto C. For every i = 1 , 2 , , N , let A i , A, B be α i -, α- and β-inverse strongly accretive mappings of C into E, respectively. Define a mapping G i : C C by Q C ( I λ i A i ) x = G i x , where λ i ( 0 , α i K 2 ) , K is the 2-uniformly smooth constant of E, for all x C and i = 1 , 2 , , N . Let { S i } i = 1 N be a finite family of κ i -strict pseudo-contractions of C into itself and with F = i = 1 N F ( S i ) i = 1 N S ( C , A i ) S ( C , A ) S ( C , B ) and κ = min { κ i : i = 1 , 2 , , N } with K 2 κ . Let α j = ( α 1 j , α 2 j , α 3 j ) I × I × I , where I = [ 0 , 1 ] , α 1 j + α 2 j + α 3 j = 1 , α 1 j ( 0 , 1 ] , α 2 j [ 0 , 1 ] and α 3 j ( 0 , 1 ) for all j = 1 , 2 , , N . Let S A be the S A -mapping generated by S 1 , S 2 , , S N , G 1 , G 2 , , G N and α 1 , α 2 , , α N . Let { x n } be the sequence generated by x 1 , u C and
x n + 1 = α n u + β n x n + γ n Q C ( I a A ) x n + δ n Q C ( I b B ) x n + η n S A x n , n 1 ,
where { α n } , { β n } , { γ n } , { δ n } , { η n } [ 0 , 1 ] and α n + β n + γ n + δ n + η n = 1 and satisfy the following conditions:
(i) lim n α n = 0 , n = 1 α n = , (ii) { γ n } , { δ n } , { η n } [ c , d ] ( 0 , 1 ) for some c , d > 0 , n 1 , (iii) n = 1 | β n + 1 β n | , n = 1 | γ n + 1 γ n | , n = 1 | δ n + 1 δ n | , (iii) n = 1 | η n + 1 η n | , n = 1 | α n + 1 α n | < , (iv) 0 < lim inf n β n lim sup n β n < 1 , (v) a ( 0 , α K 2 ) and b ( 0 , β K 2