Open Access

Hybrid shrinking iterative solutions to convex feasibility problems for countable families of relatively nonexpansive mappings and a system of generalized mixed equilibrium problems

Fixed Point Theory and Applications20142014:148

https://doi.org/10.1186/1687-1812-2014-148

Received: 22 February 2014

Accepted: 13 June 2014

Published: 22 July 2014

Abstract

We propose a new hybrid shrinking iterative scheme for approximating common elements of the set of solutions to convex feasibility problems for countable families of relatively nonexpansive mappings of a set of solutions to a system of generalized mixed equilibrium problems. A strong convergence theorem is established in the framework of Banach spaces. The results extend those of other authors, in which the involved mappings consist of just finitely many ones.

MSC:47H09, 47H10, 47J25.

Keywords

relatively nonexpansive mappings hybrid iteration scheme convex feasibility problems generalized mixed equilibrium problems

1 Introduction

Throughout this paper we assume that E is a real Banach space with its dual E , C is a nonempty, closed, convex subset of E, and J : E 2 E is the normalized duality mapping defined by
J x = { f E : x , f = x 2 = f 2 } , x E .
(1.1)
In the sequel, we use F ( T ) to denote the set of fixed points of a mapping T. A point p in C is said to be an asymptotic fixed point of T if C contains a sequence { x n } which converges weakly to p such that the lim n ( x n T x n ) = 0 . The set of asymptotic fixed points of T will be denoted by F ˆ ( T ) . A mapping T : C C is said to be nonexpansive if
T x T y x y , x , y C .
(1.2)
A mapping T : C C is said to be relatively nonexpansive if F ( T ) = F ˆ ( T ) and
ϕ ( p , T x ) ϕ ( p , x ) , x C , p F ( T ) ,
(1.3)
where ϕ : E × E R 1 denotes the Lyapunov functional defined by
ϕ ( x , y ) = x 2 2 x , J y + y 2 , x , y E .
(1.4)
It is obvious from the definition of ϕ that
( x y ) 2 ϕ ( x , y ) ( x + y ) 2 ,
(1.5)
ϕ ( x , y ) = ϕ ( x , z ) + ϕ ( z , y ) + 2 x z , J z J y ,
(1.6)
and
ϕ ( x , y ) = x , J x J y + y x , J y x J x J y + y x y .
(1.7)
The asymptotic behavior of a relatively nonexpansive mapping was studied in [14]. In 1953, Mann [5] introduced the iteration as follows: a sequence { x n } is defined by
x n + 1 = α n x n + ( 1 α n ) T x n ,
(1.8)
where the initial element x 0 C is arbitrary and { α n } is a sequence of real numbers in [ 0 , 1 ] . The Mann iteration has been extensively investigated for nonexpansive mappings. One of the fundamental convergence results was proved by Reich [6]. In an infinite-dimensional Hilbert space, a Mann iteration can yield only weak convergence (see [7, 8]). Attempts to modify the Mann iteration method (1.8) so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi [9] proposed the following modification of Mann iteration method (1.8) for a nonexpansive mapping T from C into itself in a Hilbert space: from an arbitrary x 0 C ,
{ y n = α n x n + ( 1 α n ) T x n , C n = { z C : y n z x n z } , Q n = { z C : x n z , x 0 x n 0 } , x n + 1 = P C n Q n x 0 , n N { 0 } ,
(1.9)

where P K denotes the metric projection from a Hilbert space H onto a closed convex subset K of H and proved that the sequence { x n } converges strongly to P F ( T ) x 0 . A projection onto the intersection of two half-spaces is computed by solving a linear system of two equations with two unknowns (see [[10], Section 3]).

Let θ : C × C R 1 be a bifunction, ψ : C R 1 a real-valued function, and B : C E a nonlinear mapping. The so-called generalized mixed equilibrium problem ( G M E P ) is to find an u C such that
θ ( u , y ) + y u , B u + ψ ( y ) ψ ( u ) 0 , y C ,
(1.10)

whose set of solutions is denoted by Ω ( θ , B , ψ ) .

The equilibrium problem is a unifying model for several problems arising in physics, engineering, science optimization, economics, transportation, network and structural analysis, Nash equilibrium problems in noncooperative games, and others. It has been shown that variational inequalities and mathematical programming problems can be viewed as a special realization of the abstract equilibrium problems. Many authors have proposed some useful methods to solve the EP (equilibrium problem), GEP (generalized equilibrium problem), MEP (mixed equilibrium problem), and GMEP.

In 2007, Plubtieng and Ungchittrakool [11] established strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space by using the following hybrid method in mathematical programming:
{ x 0 = x C , y n = J 1 [ α n J x n + ( 1 α n ) J z n ] , z n = J 1 [ β n ( 1 ) J x n + β n ( 2 ) J T x n + β n ( 3 ) J S x n ] , H n = { z C : ϕ ( z , y n ) ϕ ( z , x n ) } , W n = { z C : x n z , J x J y 0 } , x n + 1 = P H n W n x , n N { 0 } .
(1.11)

Their results extended and improved the corresponding ones announced by Nakajo and Takahashi [9], Martinez-Yanes and Xu [12], and Matsushita and Takahashi [4].

Recently, Su and Qin [13] modified iteration (1.9), the so-called monotone CQ method for nonexpansive mapping, as follows: from an arbitrary x 0 C ,
{ y n = α n x n + ( 1 α n ) T x n , C 0 = { z C : y 0 z x 0 z } , Q 0 = C , C n = { z C n 1 Q n 1 : y n z x n z } , Q n = { z C n 1 Q n 1 : x n z , x 0 x n 0 } , x n + 1 = P C n Q n x 0 , n N { 0 } ,
(1.12)

and proved that the sequence { x n } converges strongly to P F ( T ) x 0 .

Inspired and motivated by the studies mentioned above, in this paper, we use a modified hybrid iteration scheme for approximating common elements of the set of solutions to convex feasibility problem for a countable families of relatively nonexpansive mappings, of set of solutions to a system of generalized mixed equilibrium problems. A strong convergence theorem is established in the framework of Banach spaces. The results extend those of the authors, in which the involved mappings consist of just finitely many ones.

2 Preliminaries

We say that E is strictly convex if the following implication holds for x , y E :
x = y = 1 , x y x + y 2 < 1 .
(2.1)
It is also said to be uniformly convex if for any ϵ > 0 , there exists δ > 0 such that
x = y = 1 , x y ϵ x + y 2 1 δ .
(2.2)
It is well known that if E is a uniformly convex Banach space, then E is reflexive and strictly convex. A Banach space E is said to be smooth if
lim t 0 x + t y x t
(2.3)

exists for each x , y S ( E ) : = { x E : x = 1 } . E is said to be uniformly smooth if the limit (2.3) is attained uniformly for x , y S ( E ) .

Following Alber [14], the generalized projection P C : E C is defined by
P C = arg inf y C ϕ ( y , x ) , x E .
(2.4)

Lemma 2.1 [14]

Let E be a smooth, strictly convex and reflexive Banach space and C be a nonempty, closed, convex subset of E. Then the following conclusions hold:
  1. (1)

    ϕ ( x , P C y ) + ϕ ( P C y , y ) ϕ ( x , y ) for all x C and y E .

     
  2. (2)

    If x E and z C , then z = P C x z y , J x J z 0 , y C .

     
  3. (3)

    For x , y E , ϕ ( x , y ) = 0 if and only if x = y .

     

Lemma 2.2 [15]

Let E be a uniformly convex and smooth Banach space and let r > 0 . Then there exists a continuous, strictly increasing, and convex function h : [ 0 , 2 r ] [ 0 , ) such that h ( 0 ) = 0 and
h ( x y ) ϕ ( x , y )
(2.5)

for all x , y B r : = { z E : z r } .

Lemma 2.3 [16]

Let E be a uniformly convex and smooth Banach space and let { x n } and { y n } be two sequences of E. If ϕ ( x n , y n ) 0 , where ϕ is the function defined by (1.4), and either { x n } or { y n } is bounded, then x n y n 0 .

Remark 2.4 The following basic properties for a Banach space E can be found in Cioranescu [17].
  1. (i)

    If E is uniformly smooth, then J is uniformly continuous on each bounded subset of E.

     
  2. (ii)

    If E is reflexive and strictly convex, then J 1 is norm-weak-continuous.

     
  3. (iii)

    If E is a smooth, strictly convex and reflexive Banach space, then the normalized duality mapping J : E 2 E is single valued, one-to-one, and onto.

     
  4. (iv)

    A Banach space E is uniformly smooth if and only if E is uniformly convex.

     
  5. (v)

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

     

Lemma 2.5 [18]

Let E be a real uniformly convex Banach space and let B r ( 0 ) be the closed ball of E with center at the origin and radius r > 0 . Then there exists a continuous strictly increasing convex function g : [ 0 , ) [ 0 , ) with g ( 0 ) = 0 such that
λ x + μ y + γ z 2 λ x 2 + μ y 2 + γ z 2 λ μ g ( x y )
(2.6)

for all x , y , z B r ( 0 ) and λ , μ , γ [ 0 , 1 ] with λ + μ + γ = 1 .

Lemma 2.6 [19]

The unique solutions to the positive integer equation
n = i n + ( m n 1 ) m n 2 , m n i n , n = 1 , 2 , 3 ,
(2.7)
are
i n = n ( m n 1 ) m n 2 , m n = [ 1 2 2 n + 1 4 ] , n = 1 , 2 , 3 , ,
(2.8)

where [ x ] denotes the maximal integer that is not larger than x.

3 Main results

Theorem 3.1 Let E be a real uniformly smooth and strictly convex Banach space, and C be a nonempty, closed, convex subset of E. Let { T i } : C C and { S i } : C C be two sequences of relatively nonexpansive mappings with F : = i = 1 ( F ( T i ) F ( S i ) ) . Let { x n } be the sequence generated by
{ x 0 = x C , H 1 = W 1 = C , y n = J 1 [ λ n J x n + ( 1 λ n ) J z n ] , z n = J 1 [ α n J x n + β n J T i n x n + γ n J S i n x n ] , H n = { z H n 1 W n 1 : ϕ ( z , y n ) ϕ ( z , x n ) } , W n = { z H n 1 W n 1 : x n z , J x J y 0 } , x n + 1 = P H n W n x , n N { 0 } ,
(3.1)
where { λ n } , { α n } , { β n } , and { γ n } are sequences in [ 0 , 1 ] satisfying
  1. (1)

    0 λ n < 1 , n N { 0 } ; lim sup n λ n < 1 ;

     
  2. (2)

    α n + β n + γ n = 1 ; lim n α n = 0 and lim inf n β n γ n > 0 ;

     
and i n is the solution to the positive integer equation n = i n + ( m n 1 ) m n 2 ( m n i n , n = 1 , 2 , ), that is, for each n 1 , there exists a unique i n such that
i 1 = 1 , i 2 = 1 , i 3 = 2 , i 4 = 1 , i 5 = 2 , i 6 = 3 , i 7 = 1 , i 8 = 2 , i 9 = 3 , i 10 = 4 , i 11 = 1 , .

Then { x n } converges strongly to P F x , where P F x is the generalized projection from C onto F.

Proof We divide the proof into several steps.
  1. (I)

    H n and W n ( n N { 0 } ) both are closed and convex subsets in C.

     
This follows from the fact that ϕ ( z , y n ) ϕ ( z , x n ) is equivalent to
2 z , J x n J y n x n 2 y n 2 .
(3.2)

(II) F is a subset of n = 0 ( H n W n ) .

In fact, we note by [[4], Proposition 2.4] that for each i 1 , F ( S i ) and F ( T i ) are closed convex sets and so is F. It is clear that F C = H 1 W 1 . Suppose that F C n 1 Q n 1 for some n N . For any u F , by the convexity of 2 , we have
ϕ ( u , z n ) = ϕ ( u , J 1 [ α n J x n + β n J T i n x n + γ n J S i n x n ] ) = u 2 2 u , α n J x n + β n J T i n x n + γ n J S i n x n + α n J x n + β n J T i n x n + γ n J S i n x n 2 u 2 2 α n u , J x n 2 β n u , J T i n x n 2 γ n u , J S i n x n + α n x n 2 + β n T i n x n 2 + γ n S i n x n 2 = α n ϕ ( u , x n ) + β n ϕ ( u , T i n x n ) + γ n ϕ ( u , S i n x n ) α n ϕ ( u , x n ) + β n ϕ ( u , x n ) + γ n ϕ ( u , x n ) = ϕ ( u , x n ) ,
(3.3)
and then
ϕ ( u , y n ) = ϕ ( u , J 1 [ λ n J x n + ( 1 λ n ) J z n ] ) = u 2 2 u , λ n J x n + ( 1 λ n ) J z n + λ n J x n + ( 1 λ n ) J z n 2 u 2 2 λ n u , J x n 2 ( 1 λ n ) u , J z n + λ n x n 2 + ( 1 λ n ) z n 2 = λ n ( u 2 2 u , J x n + x n 2 ) + ( 1 λ n ) ( u 2 2 u , J z n + z n 2 ) = λ n ϕ ( u , x n ) + ( 1 λ n ) ϕ ( u , z n ) λ n ϕ ( u , x n ) + ( 1 λ n ) ϕ ( u , x n ) = ϕ ( u , x n ) .
(3.4)
This implies that F H n . It follows from x n = P H n 1 W n 1 x and Lemma 2.1(2) that
x n z , J x J x n 0 , z H n 1 W n 1 .
(3.5)
Particularly,
x n z , J x J x n 0 , u F ,
(3.6)
and hence F W n , which yields F H n W n . By induction, F n = 0 ( H n W n ) .
  1. (III)

    lim n x n T i n x n = lim n x n S i n x n = 0 .

     
In view of x n + 1 = P H n W n x H n and the definition of H n , we also have
ϕ ( x n + 1 , y n ) ϕ ( x n + 1 , x n ) , n N .
(3.7)
This implies that
lim n ϕ ( x n + 1 , y n ) = lim n ϕ ( x n + 1 , x n ) = 0 .
(3.8)
It follows from Lemma 2.2 that
lim n x n + 1 y n = lim n x n + 1 x n = 0 .
(3.9)
Since J is uniformly norm-to-norm continuous on bounded sets, we have
lim n J x n + 1 J y n = lim n J x n + 1 J x n = 0
(3.10)
and
J x n + 1 J y n ( 1 λ n ) J x n + 1 J z n λ n J x n + 1 J x n , n N { 0 } .
(3.11)
This implies that
J x n + 1 J z n 1 1 λ n ( J x n + 1 J y n + λ n J x n + 1 J x n ) 1 1 λ n ( J x n + 1 J y n + J x n + 1 J x n ) .
(3.12)
From (3.10) and lim sup n λ n < 1 , we have lim n J x n + 1 J z n = 0 . Since J 1 is also uniformly norm-to-norm continuous on bounded sets, we obtain
lim n x n + 1 z n = lim n J 1 ( J x n + 1 ) J 1 ( J z n ) = 0 .
(3.13)
From x n z n x n x n + 1 + x n + 1 z n we have lim n x n z n = 0 . Since { x n } is bounded, ϕ ( p , T i n x n ) ϕ ( p , x n ) and ϕ ( p , S i n x n ) ϕ ( p , x n ) for any p F . We also find that { J x n } , { J T i n x n } and { J S i n x n } are bounded, and then there exists an r > 0 such that { J x n } , { J T i n x n } , { J S i n x n } B r ( 0 ) . Therefore Lemma 2.5 is applicable and we observe that
ϕ ( p , z n ) = p 2 2 p , α n J x n + β n J T i n x n + γ n J S i n x n + α n J x n + β n J T i n x n + γ n J S i n x n 2 p 2 2 α n p , J x n 2 β n p , J T i n x n 2 γ n p , J S i n x n + α n x n 2 + β n T i n x n 2 + γ n S i n x n 2 β n γ n g ( J T i n x n J S i n x n ) = α n ϕ ( p , x n ) + β n ϕ ( p , T i n x n ) + γ n ϕ ( p , S i n x n ) β n γ n g ( J T i n x n J S i n x n ) ϕ ( p , x n ) β n γ n g ( J T i n x n J S i n x n ) .
(3.14)
That is,
β n γ n g ( J T i n x n J S i n x n ) ϕ ( p , x n ) ϕ ( p , z n ) ,
(3.15)

where g : [ 0 , ) [ 0 , ) is a continuous strictly convex function with g ( 0 ) = 0 .

Let { T i n k x n k S i n k x n k } be any subsequence of { T i n x n S i n x n } . Since { x n k } is bounded, there exists a subsequence { x n j } of { x n k } such that for any p F ,
lim j ϕ ( p , x n j ) = lim sup k ϕ ( p , x n k ) : = a .
(3.16)
From (1.6) we have
ϕ ( p , x n j ) = ϕ ( p , z n j ) + ϕ ( z n j , x n j ) + 2 p z n j , J z n j J x n j ϕ ( p , z n j ) + ϕ ( z n j , x n j ) + M J z n j J x n j
(3.17)
for some appropriate constant M > 0 . Since
lim j ϕ ( z n j , x n j ) = 0 = lim j J z n j J x n j ,
(3.18)
it follows that
a = lim inf j ϕ ( p , x n j ) lim inf j ϕ ( p , z n j ) .
(3.19)
From (3.3), we have
lim sup j ϕ ( p , z n j ) lim sup j ϕ ( p , x n j ) = a
(3.20)
and hence lim j ϕ ( p , x n j ) = a = lim j ϕ ( p , z n j ) . By (3.15), we observe that, as j ,
β n j γ n j g ( J T i n j x n j J S i n j x n j ) ϕ ( p , x n j ) ϕ ( p , z n j ) 0 .
(3.21)
Since lim inf n β n γ n > 0 , it follows that lim j g ( J T i n j x n j J S i n j x n j ) = 0 . By the properties of the mapping g, we have lim j J T i n j x n j J S i n j x n j = 0 . Since J 1 is also uniformly norm-to-norm continuous on bounded sets, we obtain
lim j T i n j x n j S i n j x n j = lim j J 1 ( J T i n j x n j ) J 1 ( J S i n j x n j ) = 0 ,
(3.22)
and then lim n T i n x n S i n x n = 0 . Next, we note by the convexity of 2 and (1.7) that, as n ,
ϕ ( T i n x n , z n ) = T i n x n 2 2 T i n x n , α n J x n + β n J T i n x n + γ n J S i n x n + α n J x n + β n J T i n x n + γ n J S i n x n 2 T i n x n 2 2 α n T i n x n , J x n 2 β n T i n x n , J T i n x n 2 γ n T i n x n , J S i n x n + α n x n 2 + β n T i n x n 2 + γ n S i n x n 2 = α n ϕ ( T i n x n , x n ) + β n ϕ ( T i n x n , S i n x n ) 0 ,
(3.23)
since α n 0 . By Lemma 2.3, we have lim n T i n x n z n = 0 and hence
T i n x n x n T i n x n z n + z n x n 0
(3.24)
as n . Moreover, we observe that
S i n x n x n S i n x n T i n x n + T i n x n x n 0
(3.25)
as n .
  1. (IV)

    x n P F x as n .

     
It follows from the definition of W n and Lemma 2.1(2) that x n = P W n x . Since x n + 1 = P H n W n x W n , we have
ϕ ( x n , x ) ϕ ( x n + 1 , x ) , n 1 .
(3.26)
Therefore, { ϕ ( x n , x ) } is nondecreasing. Using x n = P W n x and Lemma 2.1(1), we have
ϕ ( x n , x ) = ϕ ( P W n x , x ) ϕ ( p , x ) ϕ ( p , x n ) ϕ ( p , x )
(3.27)
for all p F and for all n N , that is, { ϕ ( x n , x ) } is bounded. Then
lim n ϕ ( x n , x )  exists .
(3.28)
In particular, by (1.5), the sequence { ( x n x ) 2 } is bounded. This implies that { x n } is bounded. Note again that x n = P W n x and for any positive integer k, x n + k W n + k 1 W n . By Lemma 2.1(1),
ϕ ( x n + k , x n ) = ϕ ( x n + k , P W n x ) ϕ ( x n + k , x ) ϕ ( P W n x , x ) = ϕ ( x n + k , x ) ϕ ( x n , x ) .
(3.29)
By Lemma 2.2, we have, for m , n N with m > n ,
h ( x m x n ) ϕ ( x m , x n ) ϕ ( x m , x ) ϕ ( x n , x ) ,
(3.30)
where h : [ 0 , ) [ 0 , ) is a continuous, strictly increasing, and convex function with h ( 0 ) = 0 . Then the properties of the function g show that { x n } is a Cauchy sequence in C, so there exists x C such that
x n x ( n ) .
(3.31)
Now, set N i = { k N : k = i + ( m 1 ) m 2 , m i , m N } for each i N . Note that T i k = T i and S i k = S i whenever k N i . By Lemma 2.6 and the definition of N i , we have N 1 = { 1 , 2 , 4 , 7 , 11 , 16 , } and i 1 = i 2 = i 4 = i 7 = i 11 = i 16 = = 1 . Then it follows from (3.15) and (3.24) that
lim N i k T i x k x k = lim N i k S i x k x k = 0 , i N .
(3.32)

It then immediately follows from (3.31) and (3.32) that x F ( T i ) F ( S i ) for each i N and hence x F .

Put u = P F x . Since u F H n W n and x n + 1 = P H n W n x , we have ϕ ( x n + 1 , x ) ϕ ( u , x ) , n N . Then
ϕ ( x , x ) = lim n ϕ ( x n + 1 , x ) ϕ ( u , x ) ,
(3.33)

which implies that x = u since u = P F x , and hence x n x = P F x as n . This completes the proof. □

Remark 3.2 Note that the algorithm (3.1) is based on the projection onto an intersection of two closed and convex sets. We first give an example [20] of how to compute such a projection onto the intersection of two half-spaces.

Let H be a Hilbert space and suppose that ( x , y , z ) H 3 satisfies
{ w H : w y , x y 0 } { w H : w z , y z 0 } .
(3.34)
Set
π = x y , y z , μ = x y 2 , ν = y z 2 , ρ = μ ν π 2 ,
(3.35)
and
Q ( x , y , z ) = { z , if  ρ = 0  and  π 0 ; x + ( 1 + π / ν ) ( z y ) , if  ρ > 0  and  π ν ρ ; y + ( ν / ρ ) ( π ( x y ) + μ ( z y ) ) , if  ρ > 0  and  π ν < ρ .
(3.36)
In [21], Haugazeau introduced the operator Q as an explicit description of the projector onto the intersection of the two half-spaces defined in (3.34). He proved in [21] that the sequence { y n } defined by y 0 = x and
( n N ) y n + 1 = Q ( x , Q ( x , y n , P B y n ) , P A Q ( x , y n , P B y n ) )
(3.37)

converges strongly to P C x .

Since the algorithm (3.1) involves the projection onto the intersection of two convex sets not necessarily half-spaces, we next give an example [22] to explain and illustrate how the projection is calculated in the general convex case.

Dykstra’s algorithm Let Ω 1 , Ω 2 , , Ω p be closed and convex subsets of R n . For any i = 1 , 2 , , p and x 0 R n , the sequences { x i k } are defined by the following recursive formulas:
{ x 0 k = x p k 1 , x i k = P Ω i ( x i 1 k y i k 1 ) , i = 1 , 2 , , p , y i k = x i k ( x i 1 k y i k 1 ) , i = 1 , 2 , , p ,
(3.38)

for k = 1 , 2 , with initial values x p 0 = x 0 and y i 0 = 0 for i = 1 , 2 , , p . If Ω : = i = 1 p Ω i , then { x i k } converges to x = P Ω ( x 0 ) , where P Ω ( x ) : = arg inf y Ω y x 2 , x R n .

Note Another iterative method termed HAAR (Haugazeau-like Averaged Alternating Reflections) for finding the projection onto intersection of finitely many closed convex sets in a Hilbert space can be found in [[20], Remark 3.4(iii)].

4 Applications

The so-called convex feasibility problem for a family of mappings { T i } i = 1 is to find a point in the nonempty intersection i = 1 F ( T i ) .

Note Although the problem mentioned above is indeed a convex feasibility problem, it is mainly referred to the finite case.

Let E be a smooth, strictly convex, and reflexive Banach space, and C be a nonempty, closed, convex subset of E. Let { B i } i = 1 : C E be a sequence of β i -inverse strongly monotone mappings, { ψ } i = 1 : C R 1 a sequence of lower semi-continuous and convex functions, and { θ i } i = 1 : C × C R 1 a sequence of bifunctions satisfying the conditions:

(A1) θ ( x , x ) = 0 ;

(A2) θ is monotone, i.e., θ ( x , y ) + θ ( y , x ) 0 ;

(A3) lim sup t 0 θ ( x + t ( z x ) , y ) θ ( x , y ) ;

(A4) the mapping y θ ( x , y ) is convex and lower semicontinuous.

A system of generalized mixed equilibrium problems ( G M E P ) for { θ i } i = 1 , { B i } i = 1 and { ψ i } i = 1 is to find an x C such that
θ i ( x , y ) + y x , B i x + ψ i ( y ) ψ i ( x ) 0 , y C , i N ,
(4.1)

whose set of common solutions is denoted by Ω : = i = 1 Ω i , where Ω i denotes the set of solutions to generalized mixed equilibrium problem for θ i , B i , and ψ i .

Define a countable family of mappings { S r , i } i = 1 : E C with r > 0 as follows:
S r , i ( x ) = { z C : τ i ( z , y ) + 1 r y z , J z J x 0 , y C } , i N ,
(4.2)
where τ i ( x , y ) = θ i ( x , y ) + y x , B i x + ψ i ( y ) ψ i ( x ) , x , y C , i N . It has been shown by Zhang [23] that
  1. (1)

    { S r , i } i = 1 is a sequence of single-valued mappings;

     
  2. (2)

    { S r , i } i = 1 is a sequence of closed relatively nonexpansive mappings;

     
  3. (3)

    i = 1 F ( S r , i ) = Ω .

     
Theorem 4.1 Let E be a smooth, strictly convex, and reflexive Banach space, and C be a nonempty, closed, convex subset of E. Let { T i } i = 1 : C C be a sequence of relatively nonexpansive mappings and { S r , i } i = 1 : C C be a sequence of mappings defined by (4.2) with F : = i = 1 ( F ( T i ) F ( S r , i ) ) . Let { x n } be the sequence generated by
{ x 0 = x C , H 1 = W 1 = C , y n = J 1 [ λ n J x n + ( 1 λ n ) J z n ] , z n = J 1 [ α n J x n + β n J T i n x n + γ n J S r , i n x n ] , H n = { z H n 1 W n 1 : ϕ ( z , y n ) ϕ ( z , x n ) } , W n = { z H n 1 W n 1 : x n z , J x J y 0 } , x n + 1 = P H n W n x , n N { 0 } ,
(4.3)
where { λ n } , { α n } , { β n } and { γ n } are sequences in [ 0 , 1 ] satisfying
  1. (1)

    0 λ n < 1 , n N { 0 } ; lim sup n λ n < 1 ;

     
  2. (2)

    α n + β n + γ n = 1 ; lim n α n = 0 and lim inf n β n γ n > 0 ;

     

and i n satisfies the equation n = i n + ( m n 1 ) m n 2 ( m n i n , n = 1 , 2 , ). Then { x n } converges strongly to P F x , which is some common solution to the convex feasibility problem for { T i } i = 1 and a system of generalized mixed equilibrium problems for { S r , i } i = 1 .

Declarations

Acknowledgements

The authors are very grateful to the referees for their useful suggestions, by which the contents of this article has been improved. This study is supported by the National Natural Science Foundation of China (Grant No. 11061037).

Authors’ Affiliations

(1)
School of Statistics and Mathematics, Yunnan University of Finance and Economics
(2)
Architectural Engineering Faculty, Kunming Metallurgy College

References

  1. Butnariu D, Reich S, Zaslavski AJ: Asymptotic behavior of relatively nonexpansive operators in Banach spaces. J. Appl. Anal. 2001, 7: 151–174.MathSciNetGoogle Scholar
  2. Butnariu D, Reich S, Zaslavski AJ: Weak convergence of orbits of nonlinear operators in reflexive Banach spaces. Numer. Funct. Anal. Optim. 2003, 24: 489–508. 10.1081/NFA-120023869View ArticleMathSciNetGoogle Scholar
  3. Censor Y, Reich S: Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization. Optimization 1996, 37: 323–339. 10.1080/02331939608844225View ArticleMathSciNetGoogle Scholar
  4. Matsushita S, Takahashi W: A strong convergence theorem for relatively nonexpansive mappings in a Banach space. J. Approx. Theory 2005, 134: 257–266. 10.1016/j.jat.2005.02.007View ArticleMathSciNetGoogle Scholar
  5. Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4(3):506–510. 10.1090/S0002-9939-1953-0054846-3View ArticleGoogle Scholar
  6. Reich S: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 1979, 67(2):274–276. 10.1016/0022-247X(79)90024-6View ArticleMathSciNetGoogle Scholar
  7. Bauschke HH, Matoušková E, Reich S: Projection and proximal point methods: convergence results and counterexamples. Nonlinear Anal., Theory Methods Appl. 2004, 56(5):715–738. 10.1016/j.na.2003.10.010View ArticleGoogle Scholar
  8. Genel A, Lindenstrauss J: An example concerning fixed points. Isr. J. Math. 1975, 22(1):81–86. 10.1007/BF02757276View ArticleMathSciNetGoogle Scholar
  9. Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 2003, 279(2):372–379. 10.1016/S0022-247X(02)00458-4View ArticleMathSciNetGoogle Scholar
  10. Solodov MV, Svaiter BF: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Program., Ser. A 2000, 87(1):189–202.MathSciNetGoogle Scholar
  11. Plubtieng S, Ungchittrakool K: Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space. J. Approx. Theory 2007, 149: 103–115. 10.1016/j.jat.2007.04.014View ArticleMathSciNetGoogle Scholar
  12. Martinez-Yanes C, Xu HK: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Anal. 2006, 64: 2400–2411. 10.1016/j.na.2005.08.018View ArticleMathSciNetGoogle Scholar
  13. Su Y, Qin X: Monotone CQ iteration processes for nonexpansive semigroups and maximal monotone operators. Nonlinear Anal., Theory Methods Appl. 2008, 68(12):3657–3664. 10.1016/j.na.2007.04.008View ArticleMathSciNetGoogle Scholar
  14. Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Edited by: Kartosator AG. Dekker, New York; 1996:15–50.Google Scholar
  15. Kohsaka F, Takahashi W: Block iterative methods for a finite family of relatively nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2007., 2007: Article ID 21972Google Scholar
  16. Kamimura S, Takahashi W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 2002, 13: 938–945. 10.1137/S105262340139611XView ArticleMathSciNetGoogle Scholar
  17. Cioranescu I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic, Dordrecht; 1990.View ArticleMATHGoogle Scholar
  18. Zhou HY, Guo GT, Hwang HJ, Cho YJ: On the iterative methods for nonlinear operator equations in Banach spaces. Panam. Math. J. 2004, 14: 61–68.MathSciNetGoogle Scholar
  19. Deng W-Q, Bai P: An implicit iteration process for common fixed points of two infinite families of asymptotically nonexpansive mappings in Banach spaces. J. Appl. Math. 2013., 2013: Article ID 602582Google Scholar
  20. Bauschke HH, Combettes PL, Luke DR: A strongly convergent reflection method for finding the projection onto the intersection of two closed convex sets in a Hilbert space. J. Approx. Theory 2006, 141: 63–69. 10.1016/j.jat.2006.01.003View ArticleMathSciNetGoogle Scholar
  21. Haugazeau, Y: Sur les Inéquations Variationnelles et la Minimisation de Fonctionnelles Convexes, Thèse, Université de Paris, France (1968)Google Scholar
  22. Boyle JP, Dykstra RL: A method for finding projections onto the intersections of convex sets in Hilbert spaces. Lecture Notes in Statistics 37. Advances in Order Restricted Statistical Inference 1986, 28–47.View ArticleGoogle Scholar
  23. Zhang SS: The generalized mixed equilibrium problem in Banach space. Appl. Math. Mech. 2009, 30: 1105–1112. 10.1007/s10483-009-0904-6View ArticleGoogle Scholar

Copyright

© Deng and Qian; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.