Open Access

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 ) .

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

Proof By using the same method as (3.2), we can conclude that { G i } i = 1 N is a nonexpansive mapping. From Lemma 2.3, we have F ( G i ) = S ( C , A i ) for all i = 1 , 2 , , N . From Theorem 3.1 we can conclude the desired conclusion. □

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

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© Kangtunyakarn; licensee Springer. 2013