Open Access

Strong convergence theorems for modifying Halpern iterations for a totally quasi-ϕ-asymptotically nonexpansive multi-valued mapping in reflexive Banach spaces

Journal of Inequalities and Applications20132013:126

https://doi.org/10.1186/1029-242X-2013-126

Received: 21 October 2012

Accepted: 26 February 2013

Published: 26 March 2013

Abstract

In this paper, we discuss an iterative sequence for a totally quasi-ϕ-asymptotically nonexpansive multi-valued mapping for modifying Halpern’s iterations and establish some strong convergence theorems under certain conditions. We utilize the theorems to study a modified Halpern iterative algorithm for a system of equilibrium problems. The results improve and extend the corresponding results of Chang et al. (Appl. Math. Comput. 218:6489-6497, 2012).

MSC:47J05, 47H09, 49J25.

Keywords

multi-valued mappingquasi-ϕ-asymptotically nonexpansivetotal quasi-ϕ-asymptotically nonexpansiveHalpern iterative sequence

1 Introduction

Throughout this paper, we denote by N and R the sets of positive integers and real numbers, respectively. Let D be a nonempty closed subset of a real Banach space X. A mapping T : D D is said to be nonexpansive if T x T y x y for all x , y D . Let N ( D ) and CB ( D ) denote the family of nonempty subsets and nonempty bounded closed subsets of D, respectively. The Hausdorff metric on CB ( D ) is defined by
H ( A 1 , A 2 ) = max { sup x A 1 d ( x , A 2 ) , sup y A 2 d ( y , A 1 ) }

for A 1 , A 2 CB ( D ) , where d ( x , A 2 ) = inf { x y , y A 2 } . The multi-valued mapping T : D CB ( D ) is called nonexpansive if H ( T x , T y ) x y for all x , y D . An element p D is called a fixed point of T : D CB ( D ) if p T ( p ) . The set of fixed points of T is represented by F ( T ) .

In the sequel, denote S ( X ) = { x X : x = 1 } . A Banach space X is said to be strictly convex if x + y 2 1 for all x , y S ( X ) and x y . A Banach space is said to be uniformly convex if lim n x n y n = 0 for any two sequences { x n } , { y n } S ( X ) and lim n x n + y n 2 = 0 . The norm of the Banach space X is said to be Gâteaux differentiable if for each x , y S ( X ) , the limit
lim t 0 x + t y x t
(1.1)

exists. In this case, X is said to be smooth. The norm of the Banach space X is said to be Fréchet differentiable if for each x S ( X ) , the limit (1.1) is attained uniformly for y S ( x ) , and the norm is uniformly Fréchet differentiable if the limit (1.1) is attained uniformly for x , y S ( X ) . In this case, X is said to be uniformly smooth.

Let X be a real Banach space with dual X . We denote by J the normalized duality mapping from X to 2 X which is defined by
J ( x ) = { x X : x , x = x 2 = x 2 } , x X ,

where , denotes the generalized duality pairing.

Remark 1.1 The following basic properties for the Banach space X and for the normalized duality mapping J can be found in Cioranescu [1].
  1. (1)

    X ( X , resp.) is uniformly convex if and only if X (X, resp.) is uniformly smooth.

     
  2. (2)

    If X is smooth, then J is single-valued and norm-to-weak continuous.

     
  3. (3)

    If X is reflexive, then J is onto.

     
  4. (4)

    If X is strictly convex, then J x J y Φ for all x , y X .

     
  5. (5)

    If X has a Fréchet differentiable norm, then J is norm-to-norm continuous.

     
  6. (6)

    If X is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of X.

     
  7. (7)

    Each uniformly convex Banach space X has the Kadec-Klee property, i.e., for any sequence { x n } X , if x n x X and x n x , then x n x X .

     
Next we assume that X is a smooth, strictly convex, and reflexive Banach space and D is a nonempty closed convex subset of X. In the sequel, we always use ϕ : X × X R + to denote the Lyapunov bifunction defined by
ϕ ( x , y ) = x 2 2 x , J y + y 2 , x , y X .
(1.2)
It is obvious from the definition of the function ϕ that
(1.3)
(1.4)
and
ϕ ( x , J 1 ( α J y + ( 1 α ) J z ) ) α ϕ ( x , y ) + ( 1 α ) ϕ ( x , z )
(1.5)

for all α [ 0 , 1 ] and x , y , z X .

Following Alber [2], the generalized projection Π D : X D is defined by
Π D ( x ) = arg inf y D ϕ ( y , x ) , x X .

Many problems in nonlinear analysis can be reformulated as a problem of finding a fixed point of a nonexpansive mapping.

Remark 1.2 (see [3])

Let Π D be the generalized projection from a smooth, reflexive and strictly convex Banach space X onto a nonempty closed convex subset D of X, then Π D is a closed and quasi-ϕ-nonexpansive from X onto D.

In 1953, Mann [4] introduced the following iterative sequence { x n } :
x n + 1 = α n x n + ( 1 α n ) T x n ,
where the initial guess x 1 D is arbitrary and { α n } is a real sequence in [ 0 , 1 ] . It is known that under appropriate settings the sequence { x n } converges weakly to a fixed point of T. However, even in a Hilbert space, the Mann iteration may fail to converge strongly [5]. Some attempts to construct an iteration method guaranteeing the strong convergence have been made. For example, Halpern [6] proposed the following so-called Halpern iteration:
x n + 1 = α n u + ( 1 α n ) T x n ,
where u , x 1 D are arbitrarily given and { α n } is a real sequence in [ 0 , 1 ] . Another approach was proposed by Nakajo and Takahashi [7]. They generated a sequence as follows:
{ x 1 X is arbitrary ; y n = α n u + ( 1 α n ) T x n , C n = { z D : y n z x n z } , Q n = { z D : x n z , x 1 x n 0 } , x n + 1 = P C n Q n x 1 ( n = 1 , 2 , ) ,
(1.6)

where { α n } is a real sequence in [ 0 , 1 ] and P K denotes the metric projection from a Hilbert space H onto a closed convex subset K of H. It should be noted here that the iteration above works only in the Hilbert space setting. To extend this iteration to a Banach space, the concept of relatively nonexpansive mappings and quasi-ϕ-nonexpansive mappings have been introduced by Aoyama et al. [8], Chang et al. [9, 10], Chidume et al. [11], Matsushita et al. [1214], Qin et al. [15], Song et al. [16], Wang et al. [17] and others.

Inspired by the work of Matsushita and Takahashi, in this paper, we introduce modifying Halpern-Mann iterations sequence for finding a fixed point of a multi-valued mapping T : D CB ( D ) and prove some strong convergence theorems. The results presented in the paper improve and extend the corresponding results in [9].

2 Preliminaries

In the sequel, we denote the strong convergence and weak convergence of the sequence { x n } by x n x and x n x , respectively.

Lemma 2.1 (see [2])

Let X be a smooth, strictly convex and reflexive Banach space, and let D be a nonempty closed convex subset of X. Then the following conclusions hold:
  1. (a)
    ϕ ( x , y ) = 0
    if and only if x = y .
     
  2. (b)
    ϕ ( x , Π D y ) + ϕ ( Π D y , y ) ϕ ( x , y )
    , x , y D .
     
  3. (c)

    If x X and z D , then z = Π D x if and only if z y , J x J z 0 , y D .

     

Lemma 2.2 (see [9])

Let X be a real uniformly smooth and strictly convex Banach space with the Kadec-Klee property, and let D be a nonempty closed convex subset of X. Let { x n } and { y n } be two sequences in D such that x n p and ϕ ( x n , y n ) 0 , where ϕ is the function defined by (1.2), then y n p .

Definition 2.1 A point p D is said to be an asymptotic fixed point of a multi-valued mapping T : D CB ( D ) if there exists a sequence { x n } D such that x n x X and d ( x n , T ( x n ) ) 0 . Denote the set of all asymptotic fixed points of T by F ˆ ( T ) .

Definition 2.2
  1. (1)

    A multi-valued mapping T : D CB ( D ) is said to be relatively nonexpansive if F ( T ) Φ , F ˆ ( T ) = F ( T ) and ϕ ( p , z ) ϕ ( p , x ) , x D , p F ( T ) , z T ( x ) .

     
  2. (2)

    A multi-valued mapping T : D CB ( D ) is said to be closed if for any sequence { x n } D with x n x X and d ( y , T ( x n ) ) 0 , then d ( y , T ( x ) ) = 0 .

     

Remark 2.1 If H is a real Hilbert space, then ϕ ( x , y ) = x y 2 and Π D is the metric projection P D of H onto D.

Next, we present an example of a relatively nonexpansive multi-valued mapping.

Example 2.1 (see [18])

Let X be a smooth, strictly convex and reflexive Banach space, let D be a nonempty closed and convex subset of X, and let f : D × D R be a bifunction satisfying the conditions: (A1) f ( x , x ) = 0 , x D ; (A2) f ( x , y ) + f ( y , x ) 0 , x , y D ; (A3) for each x , y , z D , lim t 0 f ( t z + ( 1 t ) x , y ) f ( x , y ) ; (A4) for each given x D , the function y f ( x , y ) is convex and lower semicontinuous. The so-called equilibrium problem for f is to find an x D such that f ( x , y ) 0 , y D . The set of its solutions is denoted by  EP ( f ) .

Let r > 0 , x D and define a multi-valued mapping T r : D N ( D ) as follows:
T r ( x ) = { z D , f ( z , y ) + 1 r y z , J z J x 0 , y D } , x D ,
(2.1)

then (1) T r is single-valued, and so { z } = T r ( x ) ; (2) T r is a relatively nonexpansive mapping, therefore, it is a closed quasi-ϕ-nonexpansive mapping; (3) F ( T r ) = EP ( f ) .

Definition 2.3
  1. (1)

    A multi-valued mapping T : D CB ( D ) is said to be quasi-ϕ-nonexpansive if F ( T ) Φ and ϕ ( p , z ) ϕ ( p , x ) , x D , p F ( T ) , z T x .

     
  2. (2)
    A multi-valued mapping T : D CB ( D ) is said to be quasi-ϕ-asymptotically nonexpansive if F ( T ) Φ and there exists a real sequence k n [ 1 , + ) , k n 1 , such that
    ϕ ( p , z n ) k n ϕ ( p , x ) , x D , p F ( T ) , z n T n x .
    (2.2)
     
  3. (3)
    A multi-valued mapping T : D CB ( D ) is said to be totally quasi-ϕ-asymptotically nonexpansive if F ( T ) Φ and there exist nonnegative real sequences { v n } , { μ n } with v n , μ n 0 (as n ) and a strictly increasing continuous function ζ : R + R + with ζ ( 0 ) = 0 such that
    (2.3)
     

Remark 2.2 From the definitions, it is obvious that a relatively nonexpansive multi-valued mapping is a quasi-ϕ-nonexpansive multi-valued mapping, and a quasi-ϕ-nonexpansive multi-valued mapping is a quasi-ϕ-asymptotically nonexpansive multi-valued mapping, and a quasi-ϕ-asymptotically nonexpansive multi-valued mapping is a total quasi-ϕ-asymptotically nonexpansive multi-valued mapping, but the converse is not true.

Lemma 2.3 Let X and D be as in Lemma 2.2. Let T : D CB ( D ) be a closed and totally quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences { v n } , { μ n } and a strictly increasing continuous function ζ : R + R + with ζ ( 0 ) = 0 . If v n , μ n 0 (as n ) and μ 1 = 0 , then F ( T ) is a closed and convex subset of D.

Proof Let { x n } be a sequence in F ( T ) such that x n x . Since T is a totally quasi-ϕ-asymptotically nonexpansive multi-valued mapping, we have
ϕ ( x n , z ) ϕ ( x n , x ) + v 1 ζ [ ϕ ( x n , x ) ]
for all z T x and for all n N . Therefore,
ϕ ( x , z ) = lim n ϕ ( x n , z ) lim n { ϕ ( x n , x ) + v 1 ζ [ ϕ ( x n , x ) ] } = ϕ ( x , x ) = 0 .

By Lemma 2.1(a), we obtain z = x . Hence, T x = { x } . So, we have x F ( T ) . This implies F ( T ) is closed.

Let p , q F ( T ) and t ( 0 , 1 ) , and put w = t p + ( 1 t ) q . Next we prove that w F ( T ) . Indeed, in view of the definition of ϕ, letting z n T n w , we have
ϕ ( w , z n ) = w 2 2 w , J z n + z n 2 = w 2 2 t p + ( 1 t ) q , J z n + z n 2 = w 2 + t ϕ ( p , z n ) + ( 1 t ) ϕ ( q , z n ) t p 2 ( 1 t ) q 2 .
(2.4)
Since
(2.5)
Substituting (2.4) into (2.5) and simplifying it, we have
ϕ ( w , z n ) t v n ζ [ ϕ ( p , w ) ] + ( 1 t ) v n ζ [ ϕ ( q , w ) ] + μ n 0 ( as  n ) .

By Lemma 2.2, we have z n w . This implies that z n + 1 ( T T n w ) w . Since T is closed, we have T w = { w } , i.e., w F ( T ) . This completes the proof of Lemma 2.3. □

Definition 2.4 A mapping T : D CB ( D ) is said to be uniformly L-Lipschitz continuous if there exists a constant L > 0 such that x n y n L x y , where x , y D , x n T n x , y n T n y .

3 Main results

Theorem 3.1 Let X be a real uniformly smooth and strictly convex Banach space with the Kadec-Klee property, let D be a nonempty closed convex subset of X, and let T : D CB ( D ) be a closed and uniformly L-Lipschitz continuous totally quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences { v n } , { μ n } , v n , μ n 0 (as  n ) and a strictly increasing continuous function ζ : R + R + with ζ ( 0 ) = 0 satisfying condition (2.3). Let { α n } be a sequence in [ 0 , 1 ] such that α n 0 . If { x n } is the sequence generated by
{ x 1 X is arbitrary ; D 1 = D , y n = J 1 [ α n J x 1 + ( 1 α n ) J z n ] , z n T n x n , D n + 1 = { z D n : ϕ ( z , y n ) α n ϕ ( z , x 1 ) + ( 1 α n ) ϕ ( z , x n ) + ξ n } , x n + 1 = Π D n + 1 x 1 ( n = 1 , 2 , ) ,
(3.1)

where ξ n = v n sup p F ( T ) ζ [ ϕ ( p , x n ) ] + μ n , F ( T ) is the fixed point set of T, and Π D n + 1 is the generalized projection of X onto D n + 1 . If F ( T ) is nonempty and μ 1 = 0 , then lim n x n = Π F ( T ) x 1 .

Proof (I) First, we prove that D n is a closed and convex subset in D.

By the assumption, D 1 = D is closed and convex. Suppose that D n is closed and convex for some n 1 . In view of the definition of ϕ, we have
D n + 1 = { z D n : ϕ ( z , y n ) α n ϕ ( z , x 1 ) + ( 1 α n ) ϕ ( z , x n ) + ξ n } = { z D : ϕ ( z , y n ) α n ϕ ( z , x 1 ) + ( 1 α n ) ϕ ( z , x n ) + ξ n } D n = { z D : 2 α n z , J x 1 + 2 ( 1 α n ) z , J x n 2 z , J y n α n x 1 2 + ( 1 α n ) x n 2 y n 2 } D n .
This shows that D n + 1 is closed and convex. The conclusions are proved.
  1. (II)

    Next, we prove that F ( T ) D n for all n 1 .

     
In fact, it is obvious that F ( T ) D 1 . Suppose that F ( T ) D n . Hence, for any u F ( T ) D n , by (1.5), we have
ϕ ( u , y n ) = ϕ ( u , J 1 ( α n J x 1 + ( 1 α n ) J z n ) ) α n ϕ ( u , x 1 ) + ( 1 α n ) ϕ ( u , z n ) α n ϕ ( u , x 1 ) + ( 1 α n ) { ϕ ( u , x n ) + v n ζ [ ϕ ( u , x n ) ] + μ n } α n ϕ ( u , x 1 ) + ( 1 α n ) { ϕ ( u , x n ) + v n sup p F ( T ) ζ [ ϕ ( p , x n ) ] + μ n } = α n ϕ ( u , x 1 ) + ( 1 α n ) ϕ ( u , x n ) + ξ n .
(3.2)
This shows that u F ( T ) D n + 1 , and so F ( T ) D n .
  1. (III)

    Now we prove that { x n } converges strongly to some point p .

     
In fact, since x n = Π D n x 1 , from Lemma 2.1(c), we have
x n y , J x 1 J x n 0 , y D n .
Again since F ( T ) D n , we have
x n u , J x 1 J x n 0 , u F ( T ) .
It follows from Lemma 2.1(b) that for each u F ( T ) and for each n 1 ,
ϕ ( x n , x 1 ) = ϕ ( Π D n x 1 , x 1 ) ϕ ( u , x 1 ) ϕ ( u , x n ) ϕ ( u , x 1 ) .
(3.3)
Therefore, { ϕ ( x n , x 1 ) } is bounded and so is { x n } . Since x n = Π D n x 1 and x n + 1 = Π D n + 1 x 1 D n + 1 D n , we have ϕ ( x n , x 1 ) ϕ ( x n + 1 , x 1 ) . This implies that { ϕ ( x n , x 1 ) } is nondecreasing. Hence lim n ϕ ( x n , x 1 ) exists. Since X is reflexive, there exists a subsequence { x n i } { x n } such that x n i p (some point in D = D 1 ). Since D n is closed and convex and D n + 1 D n . This implies that D n is weakly closed and p D n for each n 1 . In view of x n i = Π D n i x 1 , we have
ϕ ( x n i , x 1 ) ϕ ( p , x 1 ) , n i 1 .
Since the norm is weakly lower semi-continuous, we have
lim n i inf ϕ ( x n , x 1 ) = lim n i inf ( x n i 2 2 x n i , J x 1 + x 1 2 ) p 2 2 p , J x 1 + x 1 2 = ϕ ( p , x 1 ) ,
and so
ϕ ( p , x 1 ) lim n i inf ϕ ( x n , x 1 ) lim n i sup ϕ ( x n , x 1 ) = ϕ ( p , x 1 ) .
This shows that lim n i ϕ ( x n i , x 1 ) = ϕ ( p , x 1 ) , and we have x n i p . Since x n i p , by virtue of the Kadec-Klee property of X, we obtain that x n i p . Since { ϕ ( x n , x 1 ) } is convergent, this together with lim n i ϕ ( x n i , x 1 ) = ϕ ( p , x 1 ) shows that lim n i ϕ ( x n , x 1 ) = ϕ ( p , x 1 ) . If there exists some subsequence { x n j } { x n } such that x n j q , then from Lemma 2.1 we have
ϕ ( p , q ) = lim n i , n j ϕ ( x n i , x n j ) = lim n i , n j ϕ ( x n i , Π D n j x 1 ) lim n i , n j [ ϕ ( x n i , x 1 ) ϕ ( Π D n j x 1 , x 1 ) ] = lim n i , n j [ ϕ ( x n i , x 1 ) ϕ ( x n j , x 1 ) ] = ϕ ( p , x 1 ) ϕ ( p , x 1 ) = 0 ,
i.e., p = q , and hence
x n p .
(3.4)
By the way, from (3.4), it is easy to see that
ξ n = v n sup p F ( T ) ζ [ ϕ ( p , x n ) ] + μ n 0 .
(3.5)
  1. (IV)

    Now we prove that p F ( T ) .

     
In fact, since x n + 1 D n + 1 , from (3.1), (3.4) and (3.5), we have
ϕ ( x n + 1 , y n ) α n ϕ ( x n + 1 , x 1 ) + ( 1 α n ) ϕ ( x n + 1 , x n ) + ξ n 0 .
(3.6)
Since x n p , it follows from (3.6) and Lemma 2.2 that
y n p .
(3.7)
Since { x n } is bounded and T is a totally quasi-ϕ-asymptotically nonexpansive multi-valued mapping, T n x n is bounded. In view of α n 0 , from (3.1), we have
lim n J y n J z n = lim n α n J x 1 J z n = 0 .
(3.8)
Since J y n J p , this implies J z n J p . From Remark 1.1, it yields that
z n p .
(3.9)
Again, since
z n p = J z n J p J z n J p 0 ,
(3.10)
this together with (3.9) and the Kadec-Klee-property of X shows that
z n p .
(3.11)
On the other hand, by the assumption that T is L-Lipschitz continuous, we have
d ( T z n , z n ) d ( T z n , z n + 1 ) + z n + 1 x n + 1 + x n + 1 x n + x n z n ( L + 1 ) x n + 1 x n + z n + 1 x n + 1 + x n z n .
(3.12)
From (3.11) and x n p , we have that d ( T z n , z n ) 0 . In view of the closedness of T, it yields that T ( p ) = { p } , which implies that p F ( T ) .
  1. (V)

    Finally, we prove that p = Π F ( T ) x 1 and so x n Π F ( T ) x 1 .

     
Let w = Π F ( T ) x 1 . Since w F ( T ) D n , we have ϕ ( p , x 1 ) ϕ ( w , x 1 ) . This implies that
ϕ ( p , x 1 ) = lim n ϕ ( x n , x 1 ) ϕ ( w , x 1 ) ,
(3.13)

which yields that p = w = Π F ( T ) x 1 . Therefore, x n Π F ( T ) x 1 . The proof of Theorem 3.1 is completed. □

By Remark 2.2, the following corollaries are obtained.

Corollary 3.1 Let X and D be as in Theorem 3.1, and let T : D CB ( D ) be a closed and uniformly L-Lipschitz continuous relatively nonexpansive multi-valued mapping. Let { α n } in ( 0 , 1 ) with lim n α n = 0 . Let { x n } be the sequence generated by
{ x 1 X is arbitrary ; D 1 = D , y n = J 1 [ α n J x 1 + ( 1 α n ) J z n ] , z n T x n , D n + 1 = { z D n : ϕ ( z , y n ) α n ϕ ( z , x 1 ) + ( 1 α n ) ϕ ( z , x n ) } , x n + 1 = Π D n + 1 x 1 ( n = 1 , 2 , ) ,
(3.14)

where F ( T ) is the set of fixed points of T, and Π D n + 1 is the generalized projection of X onto D n + 1 , then { x n } converges strongly to Π F ( T ) x 1 .

Corollary 3.2 Let X and D be as in Theorem 3.1, and T : D CB ( D ) be a closed and uniformly L-Lipschitz continuous quasi-ϕ-nonexpansive multi-valued mapping. Let { α n } be a sequence of real numbers such that α n ( 0 , 1 ) for all n N and satisfy lim n α n = 0 . Let { x n } be the sequence generated by (3.14). Then { x n } converges strongly to Π F ( T ) x 1 .

Corollary 3.3 Let X be a real uniformly smooth and strictly convex Banach space with the Kadec-Klee property, let D be a nonempty closed convex subset of X, and let T : D CB ( D ) be a closed and uniformly L-Lipschitz continuous quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences { k n } [ 1 , + ) and k n 1 satisfying condition (2.2). Let { α n } be a sequence in ( 0 , 1 ) and satisfy lim n α n = 0 . If { x n } is the sequence generated by
{ x 1 X is arbitrary ; D 1 = D , y n = J 1 [ α n J x 1 + ( 1 α n ) J z n ] , z n T n x n , D n + 1 = { z D n : ϕ ( z , y n ) α n ϕ ( z , x 1 ) + ( 1 α n ) ϕ ( z , x n ) + ξ n } , x n + 1 = Π D n + 1 x 1 ( n = 1 , 2 , ) ,
(3.15)

where ξ n = ( k n 1 ) sup p F ( T ) ϕ ( p , x n ) , F ( T ) is the fixed point set of T, and Π D n + 1 is the generalized projection of X onto D n + 1 , if F ( T ) is nonempty, then { x n } converges strongly to Π F ( T ) x 1 .

4 Application

We utilize Corollary 3.2 to study a modified Halpern iterative algorithm for a system of equilibrium problems.

Theorem 4.1 Let D, X and { α n } be the same as in Theorem 3.1. Let f : D × D R be a bifunction satisfying conditions (A1)-(A4) as given in Example 2.1. Let T r : X D be a mapping defined by (2.1), i.e.,
T r ( x ) = { x D , f ( z , y ) + 1 r y z , J z J x 0 , y D } , x X .
Let { x n } be the sequence generated by
{ x 1 X is arbitrary ; D 1 = D , f ( u n , y ) + 1 r y u n , J u n J x n 0 , y D , r > 0 , u n T r x n , y n = J 1 [ α n J x 1 + ( 1 α n ) J u n ] , D n + 1 = { z D n : ϕ ( z , y n ) α n ϕ ( z , x 1 ) + ( 1 α n ) ϕ ( z , x n ) } , x n + 1 = D n + 1 x 1 ( n = 1 , 2 , ) .
(4.1)

If F ( T r ) Φ , then { x n } converges strongly to F ( T ) x 1 , which is a common solution of the system of equilibrium problems for f.

Proof In Example 2.1, we have pointed out that u n = T r ( x n ) , F ( T r ) = EP ( f ) and T r is a closed quasi-ϕ-nonexpansive mapping. Hence (4.1) can be rewritten as follows:
{ x 1 X is arbitrary ; D 1 = D , y n = J 1 [ α n J x 1 + ( 1 α n ) J u n ] , u n T r x n , D n + 1 = { z D n : ϕ ( z , y n ) α n ϕ ( z , x 1 ) + ( 1 α n ) ϕ ( z , x n ) } , x n + 1 = D n + 1 x 1 ( n = 1 , 2 , ) .
(4.2)

Therefore the conclusion of Theorem 4.1 can be obtained from Corollary 3.2. □

Declarations

Acknowledgements

The authors are very grateful to both reviewers for carefully reading this paper and for their comments.

Authors’ Affiliations

(1)
School of Science, Southwest University of Science and Technology

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