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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

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.

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<ϵ2, the inequalities x1, y1 and xyϵ imply there exists a δ>0 such that x + y 2 1δ.

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 q2 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 xE.

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

TxTyxy

for all x,yC.

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,yC and for some j(xy)J(xy). 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,yC and for some j(xy)J(xy).

Let C and D be nonempty subsets of a Banach space E such that C is nonempty closed convex and DC, then a mapping P:CD is sunny [1] provided P(x+t(xP(x)))=P(x) for all xC and t0, whenever x+t(xP(x))C. The mapping P:CD is called a retraction if Px=x for all xD. 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(xy)J(xy) such that

A x A y , j ( x y ) 0,x,yC.

A mapping A:CE is said to be α-inverse strongly accretive if there exists j(xy)J(xy) and α>0 such that

A x A y , j ( x y ) α A x A y 2 ,x,yC.

Remark 1.1 From (1.1) and (1.2), if T is an η-strictly pseudo-contractive mapping, then IT 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,xC.
(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 uC 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 =xC 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:CF 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:CC 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 ) :CF(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 =xC 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,yE.

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:CC 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,,N1 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 n0 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 ={xX:xr}, 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 αβg ( 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 :nN} 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 Sx= n = 1 λ n T n x for xC 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)={xE:xr} and for any α[0,1], we have α x + ( 1 α ) y 2 α x 2 +(1α) y 2 α(1α)g(xy).

Lemma 2.9 (See [13])

Let X be a uniformly smooth Banach space, C be a closed convex subset of X, T:CC 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 =tf( x t )+(1t)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 , pF(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,yE,

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 ,n0,

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:CC be a nonexpansive mapping and S:CC be an η-strictly pseudocontractive mapping with F(S)F(T). Define a mapping B A :CC by B A x=T((1α)I+αS)x for all xC 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

2α ( 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 :CE be an α i -inverse strongly accretive mapping. Define a mapping G i :CC by Q C (I λ i A i )x= G i x for all xC and i=1,2,,N, where λ i (0, α i K 2 ), K is the 2-uniformly smooth constant of E. Let B:CC 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,,N1 and ρ N (0,1]. Let T:CC be a nonexpansive mapping and S:CC 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 :CC by T((1α)I+αS)x= B A x, xC 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 ,n1,
(3.1)

where f:CC 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 qF, which solves the following variational inequality:

q f ( q ) , j ( q p ) 0,pF.

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

Let x,yC. 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:CC 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 zF; 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 =0and 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:CC by Lx=(1ϵ)Bx+ϵ B A x for all xC 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 =qF and x t begins the fixed point of the contraction

xtf(x)+(1t)Lx.

Then x t solves the fixed point equation x t =tf( x t )+(1t)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 n1. 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 t0, 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 qF. 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 qF. 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:CE be a ν-inverse strongly accretive mapping. Let T:CC be a nonexpansive mapping and S:CC be an η-strictly pseudo-contractive mapping with F=F(S)F(T)S(C,A). Define a mapping B A :CC by T((1α)I+αS)x= B A x, xC 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 (IλA) x n + δ n B A x n ,n1,

where f:CC 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 qF, which solves the following variational inequality:

q f ( q ) , j ( q p ) 0,pF.

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 :CE be an α i -inverse strongly accretive mapping. Define a mapping G i :CC by Q C (I λ i A i )x= G i x for all xC and i=1,2,,N, where λ i (0, α i K 2 ), K is the 2-uniformly smooth constant of E. Let B:CC 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,,N1 and ρ N (0,1]. Let T:CC 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 ,n1,

where f:CC 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 qF, which solves the following variational inequality:

q f ( q ) , j ( q p ) 0,pF.

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 :CE be an α i -inverse strongly accretive mapping. Define a mapping G i :CC by Q C (I λ i A i )x= G i x for all xC and i=1,2,,N, where λ i (0, α i K 2 ), K is the 2-uniformly smooth constant of E. Let B:CC 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,,N1 and ρ N (0,1]. Let S:CC be an η-strictly pseudo-contractive mapping with F=F(S) i = 1 N S(C, A i ). Define a mapping B A :CC by (1α)x+αSx= B A x, xC 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 ,n1,

where f:CC 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 qF, which solves the following variational inequality:

q f ( q ) , j ( q p ) 0,pF.

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:CC be an η-strictly pseudo-contractive mapping with F(P). Then F(P)=S(C,IP).

Proof It is easy to see that F(P)S(C,IP). Put A=IP and z F(P). Let z 0 S(C,IP), then there exists j(x z 0 )J(x z 0 ) such that

( I P ) z 0 , j ( x z 0 ) 0,xC.
(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,IP)F(P). Hence, we have S(C,IP)=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,IP)=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 :CE be an η i -strictly pseudo-contractive mapping. Define a mapping G i :CC by Q C (I λ i (I S i ))x= G i x for all xC and i=1,2,,N, where λ i (0, η i K 2 ), K is the 2-uniformly smooth constant of E. Let B:CC 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,,N1 and ρ N (0,1]. Let T:CC be a nonexpansive mapping and S:CC be an η-strictly pseudo-contractive mapping with F=F(S)F(T) i = 1 N F( S i ). Define a mapping B A :CC by T((1α)I+αS)x= B A x, xC 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 ,n1,

where f:CC 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 qF, which solves the following variational inequality:

q f ( q ) , j ( q p ) 0,pF.

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 :CE be an η i -strictly pseudo contractive mapping. Define a mapping G i :CC by Q C (I λ i (I S i ))x= G i x for all xC and i=1,2,,N, where λ i (0, η i K 2 ), K is the 2-uniformly smooth constant of E. Let B:CC 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,,N1 and ρ N (0,1]. Let T:CC 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 ,n1,

where f:CC 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 qF, which solves the following variational inequality:

q f ( q ) , j ( q p ) 0,pF.

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 :CE be an η i -strictly pseudo contractive mapping. Define a mapping G i :CC by Q C (I λ i (I S i ))x= G i x for all xC and i=1,2,,N, where λ i (0, η i K 2 ), K is the 2-uniformly smooth constant of E. Let B:CC 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,,N1 and ρ N (0,1]. S:CC be an η-strictly pseudo contractive mapping with F=F(S) i = 1 N F( S i ). Define a mapping B A :CC by (1α)x+αSx= B A x, xC 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 ,n1,

where f:CC 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 qF, which solves the following variational inequality:

q f ( q ) , j ( q p ) 0,pF.

References

  1. Reich S: Asymptotic behavior of contractions in Banach spaces. J. Math. Anal. Appl. 1973, 44: 57–70. 10.1016/0022-247X(73)90024-3

    Article  MathSciNet  Google Scholar 

  2. Aoyama K, Iiduka H, Takahashi W: Weak convergence of an iterative sequence for accretive operators in Banach spaces. Fixed Point Theory Appl. 2006., 2006: Article ID 35390. doi:10.1155/FPTA/2006/35390

    Google Scholar 

  3. Chang SS, Lee HWJ, Chan CK: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal. 2009, 70: 3307–3319. 10.1016/j.na.2008.04.035

    Article  MathSciNet  Google Scholar 

  4. Kangtunyakarn A: A new iterative algorithm for the set of fixed-point problems of nonexpansive mappings and the set of equilibrium problem and variational inequality problem. Abstr. Appl. Anal. 2011., 2011: Article ID 562689. doi:10.1155/2011/562689

    Google Scholar 

  5. Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4(3):506–510. 10.1090/S0002-9939-1953-0054846-3

    Article  Google Scholar 

  6. Cho YJ, Kang SM, Qin X: Convergence theorems of fixed points for a finite family of nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2008., 2008: Article ID 856145. doi:10.1155/2008/856145

    Google Scholar 

  7. Zhou H: Convergence theorems for λ -strict pseudo-contractions in 2-uniformly smooth Banach spaces. Nonlinear Anal. 2008, 69: 3160–3173. 10.1016/j.na.2007.09.009

    Article  MathSciNet  Google Scholar 

  8. Xu HK: Inequalities in Banach spaces with applications. Nonlinear Anal. 1991, 16: 1127–1138. 10.1016/0362-546X(91)90200-K

    Article  MathSciNet  Google Scholar 

  9. Kangtunyakarn A, Suantai S: A new mapping for finding common solutions of equilibrium problems and fixed point problems of finite family of nonexpansive mappings. Nonlinear Anal. 2009, 71: 4448–4460. 10.1016/j.na.2009.03.003

    Article  MathSciNet  Google Scholar 

  10. Suzuki T: Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl. 2005, 305(1):227–239. 10.1016/j.jmaa.2004.11.017

    Article  MathSciNet  Google Scholar 

  11. Cho YJ, Zhou HY, Guo G: Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings. Comput. Math. Appl. 2004, 47: 707–717. 10.1016/S0898-1221(04)90058-2

    Article  MathSciNet  Google Scholar 

  12. Bruck RE: Properties of fixed point sets of nonexpansive mappings in Banach spaces. Trans. Am. Math. Soc. 1973, 179: 251–262.

    Article  MathSciNet  Google Scholar 

  13. Xu HK: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 2004, 298: 279–291. 10.1016/j.jmaa.2004.04.059

    Article  MathSciNet  Google Scholar 

  14. Chang SS: On Chidumes open questions and approximate solutions of multivalued strongly accretive mapping equations in Banach spaces. J. Math. Anal. Appl. 1997, 216: 94–111. 10.1006/jmaa.1997.5661

    Article  MathSciNet  Google Scholar 

  15. Xu HK: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 2003, 116(3):659–678. 10.1023/A:1023073621589

    Article  MathSciNet  Google Scholar 

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Acknowledgements

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

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Kangtunyakarn, A. 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 Appl 2013, 23 (2013). https://doi.org/10.1186/1687-1812-2013-23

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