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An iterative algorithm to approximate a common element of the set of common fixed points for a finite family of strict pseudo-contractions and of the set of solutions for a modified system of variational inequalities

Fixed Point Theory and Applications20132013:143

https://doi.org/10.1186/1687-1812-2013-143

Received: 14 February 2013

Accepted: 17 May 2013

Published: 4 June 2013

Abstract

In this paper, we introduce a new iterative algorithm for finding a common element of the set of fixed points of a finite family of κ i -strictly pseudo-contractive mappings and the set of solutions of new variational inequalities problems in Hilbert space. By using our main results, we obtain an interesting theorem involving a finite family of κ-strictly pseudo-contractive mappings and two sets of solutions of the variational inequalities problem.

Keywords

pseudo-contractive mappingmodification of a general system of variational inequalitiesS-mapping

1 Introduction

Let H be a real Hilbert space whose inner product and norm are denoted by and , , respectively. Let C be a nonempty closed convex subset of H. A mapping S : C C is called nonexpansive if
S x S y x y ,

for all x , y C .

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

for all x , y C .

It is easy to see that every noexpansive mapping is a κ-strictly pseudo-contractive mapping.

Let A : C H . The variational inequality problem is to find a point u C such that
A u , v u 0
(1.1)

for all v C . The set of solutions of (1.1) is denoted by VI ( C , A ) .

Variational inequalities were initially studied by Kinderlehrer and Stampacchia [1] and Lions and Stampacchia [2]. Such a problem has been studied by many researchers, and it is connected with a wide range of applications in industry, finance, economics, social sciences, ecology, regional, pure and applied sciences; see, e.g., [39].

A mapping A of C into H is called α-inverse-strongly monotone, see [10], if there exists a positive real number α such that
x y , A x A y α A x A y 2

for all x , y C .

Let D 1 , D 2 : C H be two mappings. In 2008, Ceng et al. [11] introduced a problem for finding ( x , z ) C × C such that
{ λ 1 D 1 z + x z , x x 0 , x C , λ 2 D 2 x + z x , x z 0 , x C ,
(1.2)
which is called a system of variational inequalities where λ 1 , λ 2 > 0 . By a modification of (1.2), we consider the problem for finding ( x , z ) C × C such that
{ x ( I λ 1 D 1 ) ( a x + ( 1 a ) z ) , x x 0 , x C , z ( I λ 2 D 2 ) x , x z 0 , x C ,
(1.3)

which is called a modification of system of variational inequalities, for every λ 1 , λ 2 > 0 and a [ 0 , 1 ] . If a = 0 , (1.3) reduce to (1.2).

In 2008, Ceng et al. [11] introduce and studied a relaxed extragradient method for finding solutions of a general system of variational inequalities with inverse-strongly monotone mappings in a real Hilbert space as follows.

Theorem 1.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let the mappings A , B : C H be α-inverse-strongly monotone and β-inverse-strongly monotone, respectively. Let S : C C be a nonexpansive mapping such that F ( S ) Ω , where Ω is the set of fixed points of the mapping G : C C , defined by G ( x ) = P C ( P C ( x μ B x ) λ A P C ( x μ B x ) ) , for all x C . Suppose that x 1 = u C and { x n } is generated by
{ y n = P C ( x n μ B x n ) , x n + 1 = α n u + β n x n + γ n P C ( x n λ A x n ) ,
(1.4)
where λ ( 0 , 2 α ) , μ ( 0 , 2 β ) and { α n } , { β n } , { γ n } are three sequences in [ 0 , 1 ] such that
(i) α n + β n + γ n = 1 , n 1 , (ii) lim n α n = 0 and n = 1 α n = , (iii) 0 < lim inf n β n lim sup n β n < 1 .

Then { x n } converges strongly to x ˜ = P F ( S ) Ω u and ( x ˜ , y ˜ ) is a solution of problem (1.2), where y ˜ = P C ( x ˜ μ B x ˜ ) .

In the last decade, many author studied the problem for finding an element of the set of fixed points of a nonlinear mapping; see, for instance, [1214].

From the motivation of [11] and the research in the same direction, we prove a strong convergence theorem for finding a common element of the set of fixed points of a finite family of κ i -strictly pseudo-contractive mappings and the set of solutions of a modified general system of variational inequalities problems. Moreover, in the last section, we prove an interesting theorem involving the set of a finite family of κ i -strictly pseudo-contractive mappings and two sets of solutions of variational inequalities problems by using our main results.

2 Preliminaries

In this section, we collect and give some useful lemmas that will be used for our main result in the next section.

Let C be a closed convex subset of a real Hilbert space H, let P C be the metric projection of H onto C, i.e., for x H , P C x satisfies the property
x P C x = min y C x y .
It is well known that P C is a nonexpansive mapping and satisfies
x y , P C x P C y P C x P C y 2 , x , y H .
Obviously, this immediately implies that
( x y ) ( P C x P C y ) 2 x y 2 P C x P C y 2 , x , y H .

The following characterizes the projection P C .

Lemma 2.1 (See [15])

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

Lemma 2.2 (See [16])

Let { s n } be a sequence of nonnegative real numbers satisfying
s n + 1 = ( 1 α n ) s n + α n β n , n 0 ,
where { α n } , { β n } satisfy the conditions
( 1 ) { α n } [ 0 , 1 ] , n = 1 α n = , ( 2 ) lim sup n β n 0 or n = 1 | α n β n | < .

Then lim n s n = 0 .

Lemma 2.3 (See [17])

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

Then lim n x n z n = 0 .

Definition 2.1 (See [18])

Let C be a nonempty convex subset of a real Hilbert space. Let { T i } i = 1 N be a finite family of κ i -strict pseudo-contractions 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 .
(2.1)

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

Lemma 2.4 (See [18])

Let C be a nonempty closed convex subset of a real Hilbert space. Let { T i } i = 1 N be a finite family of κ-strict pseudo-contractive mappings of C into C with i = 1 N F ( T i ) and κ = max { κ i : i = 1 , 2 , , N } and let α j = ( α 1 j , α 2 j , α 3 j ) I × I × I , j = 1 , 2 , 3 , , N , where I = [ 0 , 1 ] , α 1 j + α 2 j + α 3 j = 1 , α 1 j , α 3 j ( κ , 1 ) for all j = 1 , 2 , , N 1 and α 1 N ( κ , 1 ] , α 3 N [ κ , 1 ) , α 2 j [ κ , 1 ) for all j = 1 , 2 , , N . Let S be a mapping generated by T 1 , T 2 , , T N and α 1 , α 2 , , α N . Then F ( S ) = i = 1 N F ( T i ) and S is a nonexpansive mapping.

Lemma 2.5 (See [19])

Let E be a uniformly convex Banach space, C be a nonempty closed convex subset of E and let S : C C be a nonexpansive mapping. Then I S is demi-closed at zero.

Lemma 2.6 In a real Hilbert space H, the following inequality holds:
x + y 2 x 2 + 2 y , x + y

for all x , y H .

Lemma 2.7 Let C be a nonempty closed convex subset of a Hilbert space H and let D 1 , D 2 : C H be mappings. For every λ 1 , λ 2 > 0 and a [ 0 , 1 ] , the following statements are equivalent:
  1. (a)

    ( x , z ) C × C is a solution of problem (1.3),

     
  2. (b)
    x is a fixed point of the mapping G : C C , i.e., x F ( G ) , defined by
    G ( x ) = P C ( I λ 1 D 1 ) ( a x + ( 1 a ) P C ( I λ 2 D 2 ) x ) ,
     

where z = P C ( I λ 2 D 2 ) x .

Proof (a) (b) Let ( x , z ) C × C be a solution of problem (1.3). For every λ 1 , λ 2 > 0 and a [ 0 , 1 ] , we have
{ x ( I λ 1 D 1 ) ( a x + ( 1 a ) z ) , x x 0 , x C , z ( I λ 2 D 2 ) x , x z 0 , x C .
From the properties of P C , we have
{ x = P C ( I λ 1 D 1 ) ( a x + ( 1 a ) z ) , z = P C ( I λ 2 D 2 ) x .
It implies that
x = P C ( I λ 1 D 1 ) ( a x + ( 1 a ) P C ( I λ 2 D 2 ) x ) = G ( x ) .

Hence, we have x F ( G ) , where z = P C ( I λ 2 D 2 ) x .

(b) (a) Let x F ( G ) and z = P C ( I λ 2 D 2 ) x . Then, we have
x = G ( x ) = P C ( I λ 1 D 1 ) ( a x + ( 1 a ) P C ( I λ 2 D 2 ) x ) = P C ( I λ 1 D 1 ) ( a x + ( 1 a ) z ) .
From the properties of P C , we have
{ x ( I λ 1 D 1 ) ( a x + ( 1 a ) z ) , x x 0 , x C , z ( I λ 2 D 2 ) x , x z 0 , x C .

Hence, we have ( x , z ) C × C is a solution of (1.3). □

3 Main results

Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H and let D 1 , D 2 : C H be d 1 , d 2 -inverse strongly monotone mappings, respectively. Define the mapping G : C C by G ( x ) = P C ( I λ 1 D 1 ) ( a x + ( 1 a ) P C ( I λ 2 D 2 ) x ) for all x C , λ 1 , λ 2 > 0 and a [ 0 , 1 ) . Let { T i } i = 1 N be a finite family of κ-strict pseudo-contractive mappings of C into C with F = i = 1 N F ( T i ) F ( G ) and κ = max { κ i : i = 1 , 2 , , N } and let α j = ( α 1 j , α 2 j , α 3 j ) I × I × I , j = 1 , 2 , 3 , , N , where I = [ 0 , 1 ] , α 1 j + α 2 j + α 3 j = 1 , α 1 j , α 3 j ( κ , 1 ) for all j = 1 , 2 , , N 1 and α 1 N ( κ , 1 ] , α 3 N [ κ , 1 ) , α 2 j [ κ , 1 ) for all j = 1 , 2 , , N . Let S be a mapping generated by T 1 , T 2 , , T N and α 1 , α 2 , , α N . Suppose that x 1 , u C and let { x n } be the sequence generated by
{ y n = P C ( I λ 2 D 2 ) x n , x n + 1 = α n u + β n x n + γ n S P C ( a x n + ( 1 a ) y n λ 1 D 1 ( a x n + ( 1 a ) y n ) ) , x n + 1 = n 1 ,
(3.1)
where λ 1 ( 0 , 2 d 1 ) , λ 2 ( 0 , 2 d 2 ) and { α n } , { β n } , { γ n } are sequences in [ 0 , 1 ] . Assume that the following conditions hold:
(i) α n + β n + γ n = 1 , (ii) lim n α n = 0 and n = 1 α n = , (iii) 0 < lim inf n β n lim sup n β n < 1 .

Then { x n } converges strongly to x 0 = P F u and ( x 0 , y 0 ) is a solution of (1.3), where y 0 = P C ( I λ 2 D 2 ) x 0 .

Proof First, we show that P C ( I λ 1 D 1 ) and P C ( I λ 2 D 2 ) are nonexpansive mappings for every λ 1 ( 0 , 2 d 1 ) , λ 2 ( 0 , 2 d 2 ) . Let x , y C . Since D 1 is d 1 -inverse strongly monotone and λ 1 < 2 d 1 , we have
( I λ 1 D 1 ) x ( I λ 1 D 1 ) y 2 = x y λ 1 ( D 1 x D 1 y ) 2 = x y 2 2 λ 1 x y , D 1 x D 1 y + λ 1 2 D 1 x D 1 y 2 x y 2 2 d 1 λ 1 D 1 x D 1 y 2 + λ 1 2 D 1 x D 1 y 2 = x y 2 + λ 1 ( λ 1 2 d 1 ) D 1 x D 1 y 2 x y 2 .
(3.2)
Thus ( I λ 1 D 1 ) is a nonexpansive mapping. By using the same method as (3.2), we have ( I λ 2 D 2 ) is a nonexpansive mapping. Hence, P C ( I λ 1 D 1 ) , P C ( I λ 2 D 2 ) are nonexpansive mappings. It is easy to see that the mapping G is a nonexpansive mapping. Let x F . Then we have x = S x and
x = G ( x ) = P C ( I λ 1 D 1 ) ( a x + ( 1 a ) P C ( I λ 2 D 2 ) x ) .
Put w n = P C ( I λ 1 D 1 ) ( a x n + ( 1 a ) y n ) and y = P C ( I λ 2 D 2 ) x , we can rewrite (3.1) by
x n + 1 = α n u + β n x n + γ n S w n , n 1 ,

and x = P C ( I λ 1 D 1 ) ( a x + ( 1 a ) y ) .

From the definition of x n , we have
x n + 1 x = α n ( u x ) + β n ( x n x ) + γ n ( S w n x ) α n u x + β n x n x + γ n w n x = α n u x + β n x n x + γ n P C ( I λ 1 D 1 ) ( a x n + ( 1 a ) y n ) P C ( I λ 1 D 1 ) ( a x + ( 1 a ) P C ( I λ 2 D 2 ) x ) α n u x + β n x n x + γ n a ( x n x ) + ( 1 a ) ( P C ( I λ 2 D 2 ) x n P C ( I λ 2 D 2 ) x ) α n u x + β n x n x + γ n ( a x n x + ( 1 a ) x n x ) = α n u x + ( 1 α n ) x n x max { u x , x 1 x } .

By induction we can conclude that x n x max { u x , x 1 x } for all n N . It implies that { x n } is bounded and so are { y n } and { w n } .

Next, we show that lim n x n + 1 x n = 0 .

Let
x n + 1 = ( 1 β n ) z n + β n x n ,
(3.3)

where z n = x n + 1 β n x n 1 β n .

Since x n + 1 β n x n = α n u + γ n S w n and (3.3), we have
z n + 1 z n = x n + 2 β n + 1 x n + 1 1 β n + 1 x n + 1 β n x n 1 β n = α n + 1 u + γ n + 1 S w n + 1 1 β n + 1 α n u + γ n S w n 1 β n γ n + 1 S w n 1 β n + 1 + γ n + 1 S w n 1 β n + 1 = ( α n + 1 1 β n + 1 α n 1 β n ) u + γ n + 1 1 β n + 1 ( S w n + 1 S w n ) + ( γ n + 1 1 β n + 1 γ n 1 β n ) S w n = ( α n + 1 1 β n + 1 α n 1 β n ) u + γ n + 1 1 β n + 1 ( S w n + 1 S w n ) + ( α n 1 β n α n + 1 1 β n + 1 ) S w n .
It follows that
z n + 1 z n | α n + 1 1 β n + 1 α n 1 β n | u + γ n + 1 1 β n + 1 S w n + 1 S w n + | α n + 1 1 β n + 1 α n 1 β n | S w n = | α n + 1 1 β n + 1 α n 1 β n | ( u + S w n ) + γ n + 1 1 β n + 1 w n + 1 w n = | α n + 1 1 β n + 1 α n 1 β n | ( u + S w n ) + γ n + 1 1 β n + 1 P C ( I λ 1 D 1 ) ( a x n + 1 + ( 1 a ) y n + 1 ) P C ( I λ 1 D 1 ) ( a x n + ( 1 a ) y n ) | α n + 1 1 β n + 1 α n 1 β n | ( u + S w n ) + γ n + 1 1 β n + 1 a ( x n + 1 x n ) + ( 1 a ) ( y n + 1 y n ) | α n + 1 1 β n + 1 α n 1 β n | ( u + S w n ) + γ n + 1 1 β n + 1 ( a x n + 1 x n + ( 1 a ) P C ( I λ 2 D 2 ) x n + 1 P C ( I λ 2 D 2 ) x n ) | α n + 1 1 β n + 1 α n 1 β n | ( u + S w n ) + x n + 1 x n .
From conditions (ii) and (iii), we have
lim sup n ( z n + 1 z n x n + 1 x n ) 0 .
From Lemma 2.3 and (3.3) we have lim n z n x n = 0 . Since x n + 1 x n = ( 1 β n ) ( z n x n ) , then we have
lim n x n + 1 x n = 0 .
(3.4)
From the definition of w n , we have
w n + 1 w n P C ( I λ 1 D 1 ) ( a x n + 1 + ( 1 a ) y n + 1 ) P C ( I λ 1 D 1 ) ( a x n + ( 1 a ) y n ) a x n + 1 x n + ( 1 a ) y n + 1 y n = a x n + 1 x n + ( 1 a ) P C ( I λ 2 D 2 ) x n + 1 P C ( I λ 2 D 2 ) x n a x n + 1 x n + ( 1 a ) x n + 1 x n = x n + 1 x n .
From (3.4), we obtain
lim n w n + 1 w n = 0 .
(3.5)
From the definition of x n , we have
x n + 1 x n = α n ( u x n ) + γ n ( S w n x n ) .
From (3.4), conditions (ii) and (iii), we have
lim n S w n x n = 0 .
(3.6)
From the definition of y n , we have
y n + 1 y n = P C ( I λ 2 D 2 ) x n + 1 P C ( I λ 2 D 2 ) x n x n + 1 x n .
(3.7)
From (3.4) and (3.7), we derive
lim n y n + 1 y n = 0 .
(3.8)
From the nonexpansiveness of P C ( I λ 1 D 1 ) and P C ( I λ 2 D 2 ) , we have
x n + 1 x 2 α n u x 2 + β n x n x 2 + γ n S w n x 2 α n u x 2 + β n x n x 2 + γ n w n x 2 = α n u x 2 + β n x n x 2 + γ n P C ( I λ 1 D 1 ) ( a x n + ( 1 a ) y n ) P C ( I λ 1 D 1 ) ( a x + ( 1 a ) y ) 2 α n u x 2 + β n x n x 2 + γ n ( a x n x 2 + ( 1 a ) y n y 2 ) = α n u x 2 + β n x n x 2 + γ n ( a x n x 2 + ( 1 a ) P C ( I λ 2 D 2 ) x n P C ( I λ 2 D 2 ) x 2 ) α n u x 2 + β n x n x 2 + γ n ( a x n x 2 + ( 1 a ) ( I λ 2 D 2 ) x n ( I λ 2 D 2 ) x 2 ) = α n u x 2 + β n x n x 2 + γ n ( a x n x 2 + ( 1 a ) ( x n x ) λ 2 ( D 2 x n D 2 x ) 2 ) = α n u x 2 + β n x n x 2 + γ n ( a x n x 2 + ( 1 a ) ( x n x 2 2 λ 2 x n x , D 2 x n D 2 x + λ 2 2 D x n D x 2 ) ) α n u x 2 + β n x n x 2 + γ n ( a x n x 2 + ( 1 a ) ( x n x 2 2 λ 2 d 2 D 2 x n D 2 x 2 + λ 2 2 D x n D x 2 ) ) = α n u x 2 + β n x n x 2 + γ n ( a x n x 2 + ( 1 a ) ( x n x 2 λ 2 ( 2 d 2 λ 2 ) D 2 x n D 2 x 2 ) ) = α n u x 2 + β n x n x 2 + γ n ( x n x 2 λ 2 ( 1 a ) ( 2 d 2 λ 2 ) D 2 x n D 2 x 2 ) α n u x 2 + x n x 2 λ 2 γ n ( 1 a ) ( 2 d 2 λ 2 ) D 2 x n D 2 x 2 .
It implies that
λ 2 γ n ( 1 a ) ( 2 d 2 λ 2 ) D 2 x n D 2 x 2 α n u x 2 + 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 .
(3.9)
From (3.4), (3.9) conditions (ii) and (iii), we have
lim n D 2 x n D 2 x = 0 .
(3.10)
Put h = a x + ( 1 a ) y and h n = a x n + ( 1 a ) y n . From the definition of x n , we have
x n + 1 x 2 α n u x 2 + β n x n x 2 + γ n w n x 2 = α n u x 2 + β n x n x 2 + γ n P C ( I λ 1 D 1 ) h n P C ( I λ 1 D 1 ) h 2 α n u x 2 + β n x n x 2 + γ n ( I λ 1 D 1 ) h n ( I λ 1 D 1 ) h 2 = α n u x 2 + β n x n x 2 + γ n ( h n h ) λ 1 ( D 1 h n D 1 h ) 2 = α n u x 2 + β n x n x 2 + γ n ( h n h 2 2 λ 1 h n h , D 1 h n D 1 h + λ 1 2 D 1 h n D 1 h 2 ) α n u x 2 + β n x n x 2 + γ n ( h n h 2 2 λ 1 d 1 D 1 h n D 1 h 2 + λ 1 2 D 1 h n D 1 h 2 ) = α n u x 2 + β n x n x 2 + γ n ( h n h 2 λ 1 ( 2 d 1 λ 1 ) D 1 h n D 1 h 2 ) = α n u x 2 + β n x n x 2 + γ n ( a ( x n x ) + ( 1 a ) ( y n y ) 2 λ 1 ( 2 d 1 λ 1 ) D 1 h n D 1 h 2 ) α n u x 2 + β n x n x 2 + γ n ( a x n x 2 + ( 1 a ) P C ( I λ 2 D 2 ) x n P C ( I λ 2 D 2 ) x 2 λ 1 ( 2 d 1 λ 1 ) D 1 h n D 1 h 2 ) α n u x 2 + x n x 2 λ 1 γ n ( 2 d 1 λ 1 ) D 1 h n D 1 h 2 ,
which implies that
λ 1 γ n ( 2 d 1 λ 1 ) D 1 h n D 1 h 2 α n u x 2 + 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 .
(3.11)
From (3.4), (3.11), conditions (ii) and (iii), we can conclude
lim n D 1 h n D 1 h = 0 .
(3.12)
Next, we show that
lim n S w n w n = 0 .
(3.13)
From the definition of y n , we have
y n y 2 = P C ( I λ 2 D 2 ) x n P C ( I λ 2 D 2 ) x 2 x n λ 2 D 2 x n ( x λ 2 D 2 x ) , y n y = 1 2 ( x n λ 2 D 2 x n ( x λ 2 D 2 x ) 2 + y n y 2 x n λ 2 D 2 x n ( x λ 2 D 2 x ) ( y n y ) 2 ) = 1 2 ( x n λ 2 D 2 x n ( x λ 2 D 2 x ) 2 + y n y 2 x n y n ( x y ) λ 2 ( D 2 x n D 2 x ) 2 ) = 1 2 ( x n λ 2 D 2 x n ( x λ 2 D 2 x ) 2 + y n y 2 x n y n ( x y ) 2 + 2 λ 2 x n y n ( x y ) , D 2 x n D 2 x λ 1 2 D 2 x n D 2 x 2 ) .
It implies that
y n y x n λ 2 D 2 x n ( x λ 2 D 2 x ) 2 x n y n ( x y ) 2 + 2 λ 2 x n y n ( x y ) , D 2 x n D 2 x λ 1 2 D 2 x n D 2 x 2 x n x 2 x n y n ( x y ) 2 + 2 λ 2 x n y n ( x y ) , D 2 x n D 2 x λ 1 2 D 2 x n D 2 x 2 .
(3.14)
From the nonexpansiveness of P C ( I λ 1 D 1 ) and (3.14), we have
x n + 1 x 2 α n u x 2 + β n x n x 2 + γ n S w n x 2 α n u x 2 + β n x n x 2 + γ n w n x 2 = α n u x 2 + β n x n x 2 + γ n P C ( I λ 1 D 1 ) ( a x n + ( 1 a ) y n ) P C ( I λ 1 D 1 ) ( a x + ( 1 a ) y ) 2 α n u x 2 + β n x n x 2 + γ n ( a x n x 2 + ( 1 a ) y n y 2 ) α n u x 2 + β n x n x 2 + γ n ( a x n x 2 + ( 1 a ) ( x n x 2 x n y n ( x y ) 2 + 2 λ 2 x n y n ( x y ) , D 2 x n D 2 x λ 1 2 D 2 x n D 2 x 2 ) ) α n u x 2 + β n x n x 2 + γ n ( a x n x 2 + ( 1 a ) x n x 2 ( 1 a ) x n y n ( x y ) 2 + 2 λ 2 x n y n ( x y ) D 2 x n D 2 x ) α n u x 2 + x n x 2 γ n ( 1 a ) x n y n ( x y ) 2 + 2 λ 2 x n y n ( x y ) D 2 x n D 2 x .
It follows that
γ n ( 1 a ) x n y n ( x y ) 2 α n u x 2 + x n x 2 x n + 1 x 2 + 2 λ 2 x n y n ( x y ) D 2 x n D 2 x α n u x 2 + ( x n x + x n + 1 x ) x n + 1 x n + 2 λ 2 x n y n ( x y ) D 2 x n D 2 x .
From condition (ii), (3.4) and (3.10), we have
lim n x n y n ( x y ) = 0 .
(3.15)
From the definition of w n , x , h n , h , we have
w n = P C ( I λ 1 D 1 ) ( a x n + ( 1 a ) y n ) = P C ( I λ 1 D 1 ) h n
and
x = P C ( I λ 1 D 1 ) ( a x + ( 1 a ) y ) = P C ( I λ 1 D 1 ) h .
From the properties of P C , we have
y n w n + ( x y ) 2 = y n y ( w n x ) 2 = y n a x n + a x n a y n + a y n λ 1 D 1 ( a x n + ( 1 a ) y n ) + λ 1 D 1 ( a x n + ( 1 a ) y n y + a x a x + a y a y + λ 1 D 1 ( a x + ( 1 a ) y ) λ 1 D 1 ( a x + ( 1 a ) y ) ( w n x ) 2 = a x n + ( 1 a ) y n λ 1 D 1 ( a x n + ( 1 a ) y n ) ( a x + ( 1 a ) y λ 1 D 1 ( a x + ( 1 a ) y ) ) ( w n x ) + λ 1 ( D 1 ( a x n + ( 1 a ) y n ) D 1 ( a x + ( 1 a ) y ) ) + a ( y n x n y + x ) 2 = ( I λ 1 D 1 ) ( a x n + ( 1 a ) y n ) ( I λ 1 D 1 ) ( a x + ( 1 a ) y ) ( w n x ) + λ 1 ( D 1 ( a x n + ( 1 a ) y n ) D 1 ( a x + ( 1 a ) y ) ) + a ( y n x n y + x ) 2 = ( I λ 1 D 1 ) h n ( I λ 1 D 1 ) h ( P C ( I λ 1 D 1 ) h n P C ( I λ 1 D 1 ) h ) + λ 1 ( D 1 h n D 1 h ) + a ( y n x n y + x ) 2 ( I λ 1 D 1 ) h n ( I λ 1 D 1 ) h ( P C ( I λ 1 D 1 ) h n P C ( I λ 1 D 1 ) h ) 2 + 2 λ 1 ( D 1 h n D 1 h ) + a ( y n x n y + x ) , y n w n + ( x y ) ( I λ 1 D 1 ) h n ( I λ 1 D 1 ) h 2 P C ( I λ 1 D 1 ) h n P C ( I λ 1 D 1 ) h 2 + 2 ( λ 1 D 1 h n D 1 h + a y n x n y + x ) × y n w n + ( x y ) = ( I λ 1 D 1 ) h n ( I λ 1 D 1 ) h 2 w n x 2 + 2 ( λ 1 D 1 h n D 1 h + a y n x n y + x ) × y n w n + ( x y ) ( I λ 1 D 1 ) h n ( I λ 1 D 1 ) h 2 S w n S x 2 + 2 ( λ 1 D 1 h n D 1 h + a y n x n y + x ) × y n w n + ( x y ) ( ( I λ 1 D 1 ) h n ( I λ 1 D 1 ) h + S w n S x ) × ( I λ 1 D 1 ) h n ( I λ 1 D 1 ) h ( S w n x ) + 2 ( λ 1 D 1 h n D 1 h + a y n x n y + x ) × y n w n + ( x y ) = ( ( I λ 1 D 1 ) h n ( I λ 1 D 1 ) h + S w n S x ) × h n h λ 1 ( D 1 h n D 1 h ) ( S w n x ) + 2 ( λ 1 D 1 h n D 1 h + a y n x n y + x ) × y n w n + ( x y ) = ( ( I λ 1 D 1 ) h n ( I λ 1 D 1 ) h + S w n S x ) × x n S w n + ( x h ) ( x n h n ) λ 1 ( D 1 h n D 1 h ) ) + 2 ( λ 1 D 1 h n D 1 h + a y n x n y + x ) × y n w n + ( x y ) ( ( I λ 1 D 1 ) h n ( I λ 1 D 1 ) h + S w n S x ) × ( x n S w n + ( x h ) ( x n h n ) + λ 1 D 1 h n D 1 h ) + 2 ( λ 1 D 1 h n D 1 h + a y n x n y + x ) × y n w n + ( x y ) = ( ( I λ 1 D 1 ) h n ( I λ 1 D 1 ) h + S w n S x ) × ( x n S w n + ( 1 a ) x y x n + y n + λ 1 D 1 h n D 1 h ) + 2 ( λ 1 D 1 h n D 1 h + a y n x n y + x ) × y n w n + ( x y ) .
From (3.6), (3.12) and (3.15), we have
lim n y n w n + ( x y ) = 0 .
(3.16)
Since
x n w n x n y n ( x y ) + y n + ( x y ) w n
and (3.15), (3.16), then we have
lim n x n w n = 0 .
(3.17)
From (3.6) and (3.17), we can conclude that
lim n S w n w n = 0 .
Next we show that
lim sup n u x 0 , x n x 0 0 ,
(3.18)
where x 0 = P F u . To show this inequality, take a subsequence { x n k } of { x n } such that
lim sup n u x 0 , x n x 0 = lim k u x 0 , x n k x 0 .
Without loss of generality, we may assume that x n k ω as k , where ω C . From (3.17), we have w n k ω as k . From Lemma 2.5 and (3.13), we have
ω F ( S ) .
From Lemma 2.4, we have F ( S ) = i = 1 N F ( T i ) . Then we obtain
ω i = 1 N F ( T i ) .
From the nonexpansiveness of the mapping G and the definition of w n , we have
w n G w n = P C ( I λ 1 D 1 ) ( a x n + ( 1 a ) P C ( I λ 2 D 2 ) x n ) G ( w n ) = G x n G w n x n w n .
From (3.17), we have
lim n w n G w n = 0 .
(3.19)
From w n k ω as k , (3.19) and Lemma 2.5, we have
ω F ( G ) .

Hence, we can conclude that ω F .

Since x n k ω as k and ω F , we have
lim sup n u x 0 , x n x 0 = lim k u x 0 , x n k x 0 = u x 0 , ω x 0 0 .
(3.20)
From the definition of x n and x 0 = P F u , we have
x n + 1 x 0 2 = α n ( u x 0 ) + β n ( x n x 0 ) + γ n ( S w n x 0 ) 2 β n ( x n x 0 ) + γ n ( S w n x 0 ) 2 + 2 α n u x 0 , x n + 1 x 0 β n x n x 0 2 + γ n G x n x 0 2 + 2 α n u x 0 , x n + 1 x 0 β n x n x 0 2 + γ n x n x 0 2 + 2 α n u x 0 , x n + 1 x 0 ( 1 α n ) x n x 0 2 + 2 α n u x 0 , x n + 1 x 0 .

From condition (ii), (3.18) and Lemma 2.2, we can conclude that the sequence { x n } converges strongly to x 0 = P F u . This completes the proof. □

Remark 3.2 (1) If we take a = 0 , then the iterative scheme (3.1) reduces to the following scheme:
{ x 1 , u C , y n = P C ( I λ 2 D 2 ) x n , x n + 1 = α n u + β n x n + γ n S P C ( I λ 1 D 1 ) y n , n 1 ,
(3.21)

which is an improvement to (1.4). From Theorem 3.1, we obtain that the sequence { x n } generated by (3.21) converges strongly to x 0 = P i = 1 N F ( T i ) F ( G ) u , where the mapping G : C C defined by G x = P C ( I λ 1 D 1 ) P C ( I λ 2 D 2 ) x for all x C and ( x 0 , y 0 ) is a solution of (1.2) where y 0 = P C ( I λ 2 D 2 ) x 0 .

(2) If we take N = 1 , α 1 1 = 1 and T 1 = T , then the iterative scheme (3.1) reduces to the following scheme:
{ x 1 , u C , y n = P C ( I λ 2 D 2 ) x n , x n + 1 = α n u + β n x n + γ n T P C ( I λ 1 D 1 ) ( a x n + ( 1 a ) y n ) , n 1 ,
(3.22)

From Theorem 3.1, we obtain that the sequence { x n } generated by (3.22) converges strongly to x 0 = P F ( T ) F ( G ) u , where the mapping G : C C defined by G ( x ) = P C ( I λ 1 D 1 ) ( a x + ( 1 a ) P C ( I λ 2 D 2 ) x ) for all x C and ( x 0 , y 0 ) is a solution of (1.3) where y 0 = P C ( I λ 2 D 2 ) x 0 .

4 Applications

In this section we prove a strong convergence theorem involving variational inequalities problems by using our main result. We need the following lemmas to prove the desired results.

Lemma 4.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let T , S : C C be nonexpansive mappings. Define a mapping B A : C C by B A x = T ( α I + ( 1 α ) S ) x for every x C and α ( 0 , 1 ) . Then F ( B A ) = F ( T ) F ( S ) and B A is a nonexpansive mapping.

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 ) . By the definition of B A , we have
x 0 x 2 = B x 0 x 2 = T ( α I + ( 1 α ) S ) x 0 x 2 α x 0 + ( 1 α ) S x 0 x 2 = α x 0 x 2 + ( 1 α ) S x 0 x 2 α ( 1 α ) x 0 S x 0 2 α x 0 x 2 + ( 1 α ) x 0 x 2 α ( 1 α ) x 0 S x 0 2 = x 0 x 2 α ( 1 α ) x 0 S x 0 2 .
(4.1)
From (4.1), it implies that
α ( 1 α ) x 0 S x 0 2 0 .
Then we have x 0 = S x 0 , that is, x 0 F ( S ) . By the definition of B A , we have
x 0 = B A x 0 = T ( α x 0 + ( 1 α ) S x 0 ) = T x 0 .

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

Next, we show that B A is a nonexpansive mapping. Let x , y C , since
B A x B A y 2 = T ( α I + ( 1 α ) S ) x T ( α I + ( 1 α ) S ) y 2 ( α I + ( 1 α ) S ) x ( α I + ( 1 α ) S ) y 2 = α ( x y ) + ( 1 α ) ( S x S y ) 2 α x y 2 + ( 1 α ) S x S y 2 x y 2 .
(4.2)

Then we have B A is a nonexpansive mapping. □

Lemma 4.2 (See [15])

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

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

Lemma 4.3 Let C be a nonempty closed convex subset of a real Hilbert space H and let D 1 , D 2 : C H be d 1 , d 2 -inverse strongly monotone mappings, respectively, which VI ( C , D 1 ) VI ( C , D 2 ) . Define a mapping G : C C as in Lemma 2.7 for every λ 1 ( 0 , 2 d 1 ) , λ 2 ( 0 , 2 d 2 ) and a ( 0 , 1 ) . Then F ( G ) = VI ( C , D 1 ) VI ( C , D 2 ) .

Proof First, we show that ( I λ 1 D 1 ) , ( I λ 2 D 2 ) are nonexpansive. Let x , y C . Since D 1 is d 1 -inverse strongly monotone and λ 1 < 2 d 1 , we have
( I λ 1 D 1 ) x ( I λ 1 D 1 ) y 2 = x y λ 1 ( D 1 x D 1 y ) 2 = x y 2 2 λ 1 x y , D 1 x D 1 y + λ 1 2 D 1 x D 1 y 2 x y 2 2 d 1 λ 1 D 1 x D 1 y 2 + λ 1 2 D 1 x D 1 y 2 = x y 2 + λ 1 ( λ 1 2 d 1 ) D 1 x D 1 y 2 x y 2 .
(4.3)
Thus ( I λ 1 D 1 ) is nonexpansive. By using the same method as (4.3), we have ( I λ 2 D 2 ) is a nonexpansive mapping. Hence P C ( I λ 1 D 1 ) , P C ( I λ 2 D 2 ) are nonexpansive mappings. From
G ( x ) = P C ( I λ 1 D 1 ) ( a x + ( 1 a ) P C ( I λ 2 D 2 ) x ) ,
for every x C and Lemma 4.1, we have
F ( G ) = F ( P C ( I λ 1 D 1 ) ) F ( P C ( I λ 2 D 2 ) ) .
(4.4)
From Lemma 4.2, we have
F ( G ) = VI ( C , D 1 ) VI ( C , D 2 ) .

 □

Theorem 4.4 Let C be a nonempty closed convex subset of a real Hilbert space H and let D 1 , D 2 : C H be d 1 , d 2 -inverse strongly monotone mappings, respectively. Define the mapping G : C C by G ( x ) = P C ( I λ 1 D 1 ) ( a x + ( 1 a ) P C ( I λ 2 D 2 ) x ) for all x C , λ 1 , λ 2 > 0 and a ( 0 , 1 ) . Let { T i } i = 1 N be a finite family of κ-strict pseudo-contractive mappings of C into C with F = i = 1 N F ( T i ) VI ( C , D 1 ) VI ( C , D 2 ) and κ = max { κ i : i = 1 , 2 , , N } and let α j = ( α 1 j , α 2 j , α 3 j ) I × I × I , j = 1 , 2 , 3 , , N , where I = [ 0 , 1 ] , α 1 j + α 2 j + α 3 j = 1 , α 1 j , α 3 j ( κ , 1 ) for all j = 1 , 2 , , N 1 and α 1 N ( κ , 1 ] , α 3 N [ κ , 1 ) , α 2 j [ κ , 1 ) for all j = 1 , 2 , , N . Let S be a mapping generated by T 1 , T 2 , , T N and α 1 , α 2 , , α N . Suppose that x 1 , u C and let { x n } be a sequence generated by
{ y n = P C ( I λ 2 D 2 ) x n , x n + 1 = α n u + β n x n + γ n S P C ( a x n + ( 1 a ) y n λ 1 D 1 ( a x n + ( 1 a ) y n ) ) , x n + 1 = n 1 ,
(4.5)
where λ 1 ( 0 , 2 d 1 ) , λ 2 ( 0 , 2 d 2 ) and { α n } , { β n } , { γ n } are sequences in [ 0 , 1 ] . Assume that the following conditions hold:
(i) α n + β n + γ n = 1 , (ii) lim n α n = 0 and n = 1 α n = , (iii) 0 < lim inf n β n lim sup n β n < 1 .

Then { x n } converges strongly to x 0 = P F u .

Proof From Lemma 4.3 and 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

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