Open Access

Uniformly closed replaced AKTT or AKTT condition to get strong convergence theorems for a countable family of relatively quasi-nonexpansive mappings and systems of equilibrium problems

Fixed Point Theory and Applications20142014:103

https://doi.org/10.1186/1687-1812-2014-103

Received: 9 February 2014

Accepted: 31 March 2014

Published: 6 May 2014

Abstract

The purpose of this paper is to construct a new iterative scheme and to get a strong convergence theorem for a countable family of relatively quasi-nonexpansive mappings and a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space using the properties of generalized f-projection operator. The notion of uniformly closed mappings is presented and an example will be given which is a countable family of uniformly closed relatively quasi-nonexpansive mappings but not a countable family of relatively nonexpansive mappings. Another example shall be given which is uniformly closed but does not satisfy condition AKTT and AKTT. Our results can be applied to solve a convex minimization problem. In addition, this paper clarifies an ambiguity in a useful lemma. The results of this paper modify and improve many other important recent results.

MSC:47H05, 47H09, 47H10.

Keywords

relatively quasi-nonexpansive mappingequilibrium problemsgeneralized f-projection operatorhybrid algorithmuniformly closed mappings

1 Introduction and preliminaries

Let E be a real Banach space and C be a nonempty closed convex subset of E. A mapping T : C C is called nonexpansive if
T x T y x y , x , y C .
Let E be a real Banach space and C be a nonempty closed convex subset of E. A point p C is said to be an asymptotic fixed point of T if there exists a sequence { x n } n = 0 C such that x n p and lim n x n T x n = 0 . The set of asymptotic fixed point is denoted by F ˆ ( T ) . We say that a mapping T is relatively nonexpansive (see [14]) if the following conditions are satisfied:
  1. (I)

    F ( T ) ;

     
  2. (II)

    ϕ ( p , T x ) ϕ ( p , x ) , x C , p F ( T ) ;

     
  3. (III)

    F ( T ) = F ˆ ( T ) .

     

If T satisfies (I) and (II), then T is said to be relatively quasi-nonexpansive. It is easy to see that the class of relatively quasi-nonexpansive mappings contains the class of relatively nonexpansive mappings.

Let E be a real Banach space. The modulus of smoothness of E is the function ρ E : [ 0 , ) [ 0 , ) defined by
ρ E ( τ ) = sup { 1 2 ( x + y + x y ) 1 : x 1 , y τ } .
E is uniformly smooth if and only if
lim τ 0 ρ E τ τ = 0 .

Let dim E 2 . The modulus of convexity of E is the function δ E ( ϵ ) : = inf { 1 x + y 2 : x = y = 1 ; ϵ = x y } . E is uniformly convex if for any ϵ ( 0 , 2 ] , there exists δ = δ ( ϵ ) > 0 such that if x , y E with x 1 , y 1 and x y ϵ , then 1 2 ( x + y ) 1 δ . Equivalently, E is uniformly convex if and only if δ E ( ϵ ) > 0 for all ϵ ( 0 , 2 ] . A normed space E is called strictly convex if for all x , y E , x y , x = y = 1 , we have λ x + ( 1 λ ) y < 1 , λ ( 0 , 1 ) .

Let E be the dual space of E. We denote by J the normalized duality mapping from E to 2 E defined by
J ( x ) = { f E : x , f = x 2 = f 2 } .
The following properties of J are well known (see [57] for more details):
  1. (1)

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

     
  2. (2)

    If E is reflexive, then J is a mapping from E onto E .

     
  3. (3)

    If E is smooth, then J is single valued.

     
Throughout this paper, we denote by ϕ the functional on E × E defined by
ϕ ( x , y ) = x 2 2 x , J ( y ) + y 2 , x , y E .
(1.1)
Let E be a smooth, strictly convex, and reflexive real Banach space and let C be a nonempty closed convex subset of E. Following Alber [8], the generalized projection Π C from E onto C is defined by
Π C ( x ) = arg min y C ϕ ( y , x ) , x E .
The existence and uniqueness of Π C follows from the property of the functional ϕ ( x , y ) and strict monotonicity of the mapping J. It is obvious that
( x y ) 2 ϕ ( x , y ) ( x + y ) 2 , x , y E .
(1.2)
Next, we recall the notion of generalized f-projection operator and its properties. Let G : C × E R { + } be a functional defined as follows:
G ( ξ , φ ) = ξ 2 2 ξ , φ + φ 2 + 2 ρ f ( ξ ) ,
(1.3)
where ξ C , φ E , ρ is a positive number and f : C R { + } is proper, convex, and lower semi-continuous. From the definitions of G and f, it is easy to see the following properties:
  1. (i)

    G ( ξ , φ ) is convex and continuous with respect to φ when ξ is fixed;

     
  2. (ii)

    G ( ξ , φ ) is convex and lower semi-continuous with respect to ξ when φ is fixed.

     

Definition 1.1 [9]

Let E be a real Banach space with its dual E . Let C be a nonempty, closed, and convex subset of E. We say that Π C f : E 2 C is a generalized f-projection operator if
Π C f φ = { u C : G ( u , φ ) = inf ξ C G ( ξ , φ ) } , φ E .

For the generalized f-projection operator, Wu and Huang [9] proved in the following theorem some basic properties.

Lemma 1.2 [9]

Let E be a real reflexive Banach space with its dual E . Let C be a nonempty, closed, and convex subset of E. Then the following statements hold:
  1. (i)

    Π C f is a nonempty closed convex subset of C for all φ E .

     
  2. (ii)
    If E is smooth, then for all φ E , x Π C f φ if and only if
    x y , φ J x + ρ f ( y ) ρ f ( x ) 0 , y C .
     
  3. (iii)

    If E is strictly convex and f : C R { + } is positive homogeneous (i.e., f ( t x ) = t f ( x ) for all t > 0 such that t x C where x C ), then Π C f is a single-valued mapping.

     

Fan et al. [10] showed that the condition f is positive homogeneous which appeared in Lemma 1.2 can be removed.

Lemma 1.3 [10]

Let E be a real reflexive Banach space with its dual E and C a nonempty, closed, and convex subset of E. Then if E is strictly convex, then Π C f is a single-valued mapping.

Recall that J is a single-valued mapping when E is a smooth Banach space. There exists a unique element φ E such that φ = J x for each x E . This substitution in (1.3) gives
G ( ξ , J x ) = ξ 2 2 ξ , J x + x 2 + 2 ρ f ( ξ ) .
(1.4)

Now, we consider the second generalized f-projection operator in a Banach space.

Definition 1.4 [11]

Let E be a real Banach space and C a nonempty, closed, and convex subset of E. We say that Π C f : E 2 C is a generalized f-projection operator if
Π C f x = { u C : G ( u , J x ) = inf ξ C G ( ξ , J x ) } , x E .
Obviously, the definition of relatively quasi-nonexpansive mapping T is equivalent to
  1. (1)

    F ( T ) ;

     
  2. (2)

    G ( p , J T x ) G ( p , J x ) , x C , p F ( T ) .

     

Lemma 1.5 [12]

Let E be a Banach space and f : E R { + } be a lower semi-continuous convex functional. Then there exist x E and α R such that
f ( x ) x , x + α , x E .

We know that the following lemmas hold for operator Π C f .

Lemma 1.6 [13]

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

    Π C f is a nonempty, closed, and convex subset of C for all x E ;

     
  2. (ii)
    for all x E , x ˆ Π C f x if and only if
    x ˆ y , J x J x ˆ + ρ f ( y ) ρ f ( x ) 0 , y C ;
     
  3. (iii)

    if E is strictly convex, then Π C f x is a single-valued mapping.

     

Lemma 1.7 [13]

Let C be a nonempty, closed, and convex subset of a smooth and reflexive Banach space E. Let x E and x ˆ Π C f x . Then
ϕ ( y , x ˆ ) + G ( x ˆ , J x ) G ( y , J x ) , y C .

The fixed points set F ( T ) of a relatively quasi-nonexpansive mapping is closed convex as given in the following lemma.

Lemma 1.8 [14, 15]

Let C be a nonempty closed convex subset of a smooth, uniformly convex Banach space E. Let T be a closed relatively quasi-nonexpansive mapping of C into itself. Then F ( T ) is closed and convex.

Also, this following lemma will be used in the sequel.

Lemma 1.9 [16]

Let C be a nonempty closed convex subset of a smooth, uniformly convex Banach space E. Let { x n } n = 0 and { y n } n = 0 be sequences in E such that either { x n } n = 0 or { y n } n = 0 is bounded. If lim n ϕ ( x n , y n ) = 0 , then lim n x n y n = 0 .

Lemma 1.10 [17]

Let p > 1 and r > 0 be two fixed real numbers. Then a Banach space X is uniformly convex if and only if there is a continuous, strictly increasing and convex function g : R + R + , g ( 0 ) = 0 , such that
λ x + ( 1 λ ) y p λ x p + ( 1 λ ) y p W p ( λ ) g ( x y )

for all x , y B r and 0 λ 1 , where W p ( λ ) = λ ( 1 λ ) p + λ p ( 1 λ ) .

Remark We can see from the Lemma 1.10 that the function g has no relation with the selection of x, y and λ. However, the key point above, in the process of generalization and application about this lemma, has been ambiguous gradually. For instance, the following lemma states that the function g has something to do with λ, which always leads to failure in the proof.

Lemma (stated in [[11], Lemma 2.10])

Let E be a uniformly convex real Banach space. For arbitrary r > 0 , let B r ( 0 ) : = { x E : x r } and λ [ 0 , 1 ] . Then there exists a continuous strictly increasing convex function
g : [ 0 , 2 r ] R , g ( 0 ) = 0
such that for every x , y B r ( 0 ) , the following inequality holds:
λ x + ( 1 λ ) y 2 λ x 2 + ( 1 λ ) y 2 λ ( 1 λ ) g ( x y ) .

Let F be a bifunction of C × C into R. The equilibrium problem is to find x C such that F ( x , y ) 0 , for all y C . We shall denote the solutions set of the equilibrium problem by E P ( F ) . Numerous problems in physics, optimization, and economics reduce to find a solution of equilibrium problem. The equilibrium problems include fixed point problems, optimization problems, and variational inequality problems as special cases.

For solving the equilibrium problem for a bifunction F : C × C R , let us assume that F satisfies the following conditions:

(A1) F ( x , x ) = 0 for all x C ;

(A2) F is monotone, i.e., F ( x , y ) + F ( y , x ) 0 for all x , y C ;

(A3) for each x , y C , lim t 0 F ( t z + ( 1 t ) x , y ) F ( x , y ) ;

(A4) for each x C , y F ( x , y ) is convex and lower semi-continuous.

Lemma 1.11 [18]

Let C be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space E and let F be a bifunction of C × C into R satisfying (A1)-(A4). Let r > 0 and x E . Then there exists z C such that
F ( z , y ) + 1 r y z , J z J x 0 , y K .

Lemma 1.12 [19]

Let C be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space E. Assume that F : C × C R satisfies (A1)-(A4). For r > 0 and x E , define a mapping T r F : E C as follows:
T r F ( x ) = { z C : F ( z , y ) + 1 r y z , J z J x 0 , y C }
for all z E . Then the following hold:
  1. (1)

    T r F is single valued;

     
  2. (2)
    T r F is a firmly nonexpansive-type mapping, i.e., for any x , y E ,
    T r F x T r F y , J T r F x J T r F y T r F x T r F y , J x J y ;
     
  3. (3)

    F ( T r F ) = E P ( F ) ;

     
  4. (4)

    E P ( F ) is closed and convex.

     

Lemma 1.13 [19]

Let C be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space E. Assume that F : C × C R satisfies (A1)-(A4) and let r > 0 . Then for each x E and q F ( T r F ) ,
ϕ ( q , T r F x ) + ϕ ( T r F x , x ) ϕ ( q , x ) .
Let { T n } be a sequence of mappings from C into E, where C is a nonempty closed convex subset of a real Banach space E. For a subset B of C, we say that
  1. (i)
    ( { T n } , B ) satisfies condition AKTT (see [15]) if
    n = 1 sup { T n + 1 x T n x : x B } < ;
     
  2. (ii)
    ( { T n } , B ) satisfies condition AKTT (see [15]) if
    n = 1 sup { J T n + 1 x J T n x : x B } < .
     

Recently, Shehu [11] proved strong convergence theorems for approximation of common element of set of common fixed points of countably infinite family of relatively quasi-nonexpansive mappings and set of common solutions to a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space using the properties of generalized f-projection operator. The author obtained the following theorem.

Theorem 1.14 [11]

Let E be a uniformly convex real Banach space which is also uniformly smooth. Let C be a nonempty closed convex subset of E. For each k = 1 , 2 , , m , let F k be a bifunction from C × C satisfying (A1)-(A4) and let { T n } n = 1 be an infinite family of relatively quasi-nonexpansive mappings of C into itself such that F : = ( n = 1 F ( T n ) ) ( k = 1 m E P ( F k ) ) . Let f : E R be a convex and lower semi-continuous mapping with C int ( D ( f ) ) and suppose { x n } n = 0 is iteratively generated by x 0 C , C 1 = C , x 1 = Π C 1 f x 0 ,
{ y n = J 1 ( α n J x n + ( 1 α n ) J T n x n ) , u n = T r m , n F m T r m 1 , n F m 1 T r 2 , n F 2 T r 1 , n F 1 y n , C n + 1 = { w C n : G ( w , J u n ) G ( w , J x n ) } , x n + 1 = Π C n + 1 f x 0 , n 1 ,
(1.5)

where J is the duality mapping on E. Suppose { α n } n = 1 is a sequence in ( 0 , 1 ) such that lim inf n α n ( 1 α n ) > 0 { r k , n } n = 1 ( 0 , ) ( k = 1 , 2 , , m ) satisfying lim inf n r k , n > 0 ( k = 1 , 2 , , m ). Suppose that for each bounded subset B of C, the ordered pair ( { T n } , B ) satisfies either condition AKTT or condition AKTT. Let T be the mapping from C into E defined by T x = lim n T n x for all x C and suppose that T is closed and F ( T ) = n = 1 F ( T n ) . Then { x n } n = 0 converges strongly to Π F f x 0 .

In this paper we will construct a new iterative scheme and will get strong convergence theorem for a countable family of relatively quasi-nonexpansive mappings and a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space using the properties of generalized f-projection operator. The notion of uniformly closed mappings is presented and an example will be given which is a countable family of uniformly closed relatively quasi-nonexpansive mappings but not a countable family of relatively nonexpansive mappings. Another example shall be given which is uniformly closed but not satisfy condition AKTT and AKTT.

2 Main results

Now, we shall first introduce the notion of uniformly closed mappings and give an example which is a countable family of uniformly closed relatively quasi-nonexpansive mappings but not a countable family of relatively nonexpansive mappings in the sense of G. Another example shall be given which is uniformly closed but not satisfy condition AKTT and AKTT.

Definition 2.1 Let E be a Banach space, C be a nonempty closed convex subset of E. Let { T n } n = 1 : C E be a sequence of mappings of C into E such that n = 1 F ( T n ) is nonempty. { T n } n = 1 is said to be uniformly closed, if p n = 1 F ( T n ) , whenever { x n } C converges strongly to p and x n T n x n 0 as n .

Example 1 Let E = l 2 , where
l 2 = { ξ = ( ξ 1 , ξ 2 , ξ 3 , , ξ n , ) : n = 1 | ξ n | 2 < } , ξ = ( n = 1 | ξ n | 2 ) 1 2 , ξ l 2 , ξ , η = n = 1 ξ n η n , ξ = ( ξ 1 , ξ 2 , ξ 3 , , ξ n , ) , η = ( η 1 , η 2 , η 3 , , η n , ) l 2 .
It is well known that l 2 is a Hilbert space, so that ( l 2 ) = l 2 . Let { x n } E be a sequence defined by
x 0 = ( 1 , 0 , 0 , 0 , ) , x 1 = ( 1 , 1 , 0 , 0 , ) , x 2 = ( 1 , 0 , 1 , 0 , 0 , ) , x 3 = ( 1 , 0 , 0 , 1 , 0 , 0 , ) , x n = ( ξ n , 1 , ξ n , 2 , ξ n , 3 , , ξ n , k , ) ,
where
ξ n , k = { 1 , if  k = 1 , n + 1 , 0 , if  k 1 , k n + 1 ,

for all n 1 .

Define a countable family of mappings T n : E E as follows:
T n ( x ) = { n n + 1 x n , if  x = x n , x , if  x x n ,

for all n 0 .

Conclusion 2.2 { T n } n = 0 has a unique fixed point 0, that is, F ( T n ) = { 0 } , n 0 .

Proof The conclusion is obvious. □

Let { T n } n = 1 be a countable family of quasi-relatively quasi-nonexpansive mappings, if
n = 0 F ( T n ) = F ˆ ( { T n } n = 0 ) ,
the { T n } n = 1 is said to be a countable family of relatively nonexpansive mappings in the sense of functional G, where
F ˆ ( { T n } n = 0 ) = { p C : x n p , x n T n x n 0 , x n C }

is said to be the asymptotic fixed point set of { T n } n = 1 .

Conclusion 2.3 { T n } n = 0 is a countable family of relatively quasi-nonexpansive mappings but not a countable family of relatively nonexpansive mappings in the sense of functional G.

Proof By Conclusion 2.2, we only need to show that G ( 0 , J T n x ) G ( 0 , J x ) , x E . Note that E = l 2 is a Hilbert space, for any n 0 we can derive
G ( 0 , J T n x ) G ( 0 , J x ) x E ϕ ( 0 , T n x ) ϕ ( 0 , x ) 0 T n x 2 0 x 2 T n x 2 x 2 .
It is obvious that { x n } converges weakly to x 0 = ( 1 , 0 , 0 , ) , and
x n T n x n = n n + 1 x n x n = 1 n + 1 x n 0 ,

as n , so x 0 is an asymptotic fixed point of { T n } n = 0 . Joining with Conclusion 2.2, we can obtain n = 0 F ( T n ) F ˆ ( { T n } n = 0 ) .

Thus, { T n } n = 0 is a countable family of relatively quasi-nonexpansive mappings but not a countable family of relatively nonexpansive mappings in the sense of G. □

Conclusion 2.4 { T n } n = 0 is a countable family of uniformly closed relatively quasi-nonexpansive mappings in the sense of functional G.

Proof In fact, for any strong convergent sequence { z n } E such that z n z 0 and z n T n z n 0 as n , there exists a sufficiently large nature number N, such that z n x m for any n , m > N (since x n is not a Cauchy sequence it cannot converge to any element in E). Then T n z n = z n for n > N , it follows from z n T n z n 0 that 2 z n 0 and hence z n z 0 = 0 .

Therefore, { T n } n = 0 is a countable family of uniformly closed relatively quasi-nonexpansive mappings but not a countable family of relatively nonexpansive mappings in the sense of functional G. □

Now, we give an example which is a countable family of uniformly closed quasi-nonexpansive mappings but not satisfied condition AKTT and AKTT.

Example 2 Let X = 2 . For any complex number x = r e i θ X , define a countable family of quasi-nonexpansive mappings as follows:
T n : r e i θ r e i ( θ + n π 2 ) , n = 1 , 2 , 3 , .
Proof It is easy to see that n = 1 F ( T n ) = { 0 } . We first prove that { T n } is uniformly closed. In fact, for any strong convergent sequence { x n } X such that x n x 0 and x n T n x n 0 as n , there must be x 0 = 0 n = 1 F ( T n ) . Otherwise, if x n x 0 0 , and
x 4 n + 1 T 4 n + 1 x 4 n + 1 0 ,
since T 1 is continuous, we have
x 4 n + 1 T 4 n + 1 x 4 n + 1 = x 4 n + 1 T 1 x 4 n + 1 x 0 T 1 x 0 0 .

This is a contradiction. Therefore, { T n } is uniformly closed.

Besides, take any x = r e i θ 0 . For any n by the definition of T n , we have
T n x T n + 1 x = r e π i 2 = r > 0
and
J T n x J T n + 1 x = r e π i 2 = r > 0 .

That is to say, { T n } does not satisfied condition AKTT and AKTT. □

Now we are in a position to present our main theorems.

Theorem 2.5 Let { T n } n = 1 be a countable family of uniformly closed relatively quasi-nonexpansive mappings of C into itself and other conditions are the same as Theorem 1.14 except for condition AKTT, AKTT and condition ‘Let T be the mapping from C into E defined by T x = lim n T n x for all x C and suppose that T is closed and F ( T ) = n = 1 F ( T n ) . Then the sequence { x n } n = 0 generated by (1.5) converges strongly to Π F f x 0 .

Proof We first show that C n , n 1 , is closed and convex. It is obvious that C 1 = C is closed and convex. Suppose that C n is closed convex for some n > 1 . From the definition of C n + 1 , we have z C n + 1 implies G ( z , J u n ) G ( z , J x n ) . This is equivalent to
2 ( z , J x n z , J u n ) x n 2 u n 2 .

This implies that C n + 1 is closed convex for the same n > 1 . Hence, C n is closed and convex for all n 1 . This shows that Π C n + 1 f x 0 is well defined for all n 0 .

By taking θ n k = T r k , n F k T r k 1 , n F k 1 T r 2 , n F 2 T r 1 , n F 1 , k = 1 , 2 , , m and θ n 0 = I for all n 1 , we obtain u n = θ n m y n .

We next show that F C n , n 1 . From Lemma 1.12, one sees that T r k , n F k , k = 1 , 2 , , m , is relatively nonexpansive mapping. For n = 1 , we have F C = C 1 . Now, assume that F C n for some n 2 . Then for each x F , we obtain
G ( x , J u n ) = G ( x , J θ n m y n ) G ( x , J y n ) = G ( x , ( α n J x n + ( 1 α n ) J T n x n ) ) = x 2 2 α n x , J x n 2 ( 1 α n ) x , J T n x n + α n J x n + ( 1 α n ) J T n x n 2 + 2 ρ f ( x ) x 2 2 α n x , J x n 2 ( 1 α n ) x , J T n x n + α n J x n 2 + ( 1 α n ) J T n x n 2 + 2 ρ f ( x ) = α n G ( x , J x n ) + ( 1 α n ) G ( x , J T n x n ) G ( x , J x n ) .
(2.1)

So, x C n . This implies that F C n , n 1 and the sequence { x n } n = 0 generated by (1.5) is well defined.

We now show that lim n G ( x n , J x 0 ) exists. Since f : E R is a convex and lower semi-continuous, applying Lemma 1.5, we see that there exist u E and α R such that
f ( y ) y , u + α , y E .
It follows that
G ( x n , J x 0 ) = x n 2 2 x n , J x 0 + x 0 2 + 2 ρ f ( x n ) x n 2 2 x n , J x 0 + x 0 2 + 2 ρ x n , u + 2 ρ α = x n 2 2 x n , J x 0 ρ u + x 0 2 + 2 ρ α x n 2 2 x n J x 0 ρ u + x 0 2 + 2 ρ α = ( x n J x 0 ρ u ) 2 + x 0 2 J x 0 ρ u 2 + 2 ρ α .
(2.2)
Since x n = Π C n f x 0 , it follows from (2.2) that
G ( x , J x 0 ) G ( x n , J x 0 ) ( x n J x 0 ρ u ) 2 + x 0 2 J x 0 ρ u 2 + 2 ρ α
for each x F ( T ) . This implies that { x n } n = 1 is bounded and so is { G ( x n , J x 0 ) } n = 0 . By the construction of C n , we have C m C n and x m = Π C m f x 0 C n for any positive integer m n . It then follows from Lemma 1.7 that
ϕ ( x m , x n ) + G ( x n , J x 0 ) G ( x m , J x 0 ) .
(2.3)
It is obvious that
ϕ ( x m , x n ) ( x m x n ) 2 0 .
In particular,
ϕ ( x n + 1 , x n ) + G ( x n , J x 0 ) G ( x n + 1 , J x 0 )
and
ϕ ( x n + 1 , x n ) ( x n + 1 x n ) 2 0 ,

and so { G ( x n , J x 0 ) } n = 0 is nondecreasing. It follows that the limit of { G ( x n , J x 0 ) } n = 0 exists.

By the fact that C m C n and x m = Π C m f x 0 C n for any positive integer m n , we obtain
ϕ ( x m , u n ) ϕ ( x m , x n ) .
Now, (2.3) implies that
ϕ ( x m , u n ) ϕ ( x m , x n ) G ( x m , J x 0 ) G ( x n , J x 0 ) .
(2.4)
Taking the limit as m , n in (2.4), we obtain
lim n ϕ ( x m , x n ) = 0 .

It then follows from Lemma 1.9 that x m x n 0 as m , n . Hence, { x n } n = 0 is a Cauchy sequence. Since E is a Banach space and C is closed and convex, there exists p C such that x n p as n .

Now since ϕ ( x m , x n ) 0 as m , n we have in particular that ϕ ( x n + 1 , x n ) 0 as n and this further implies that lim n x n + 1 x n = 0 . Since x n + 1 = Π C n = 1 f x 0 C n + 1 we have
ϕ ( x n + 1 , u n ) ϕ ( x n + 1 , x n ) , n 0 .
Then we obtain
lim n ϕ ( x n + 1 , u n ) = 0 .
Since E is uniformly convex and smooth, we have from Lemma 1.9
lim n x n + 1 x n = 0 = lim n x n + 1 u n .
So,
x n u n x n + 1 x n + x n + 1 u n .
Hence,
lim n x n u n = 0 .
(2.5)
Since J is uniformly norm-to-norm continuous on bounded sets and lim n x n u n = 0 , we obtain
lim n J x n J u n = 0 .
(2.6)
Let r = sup n 1 { x n , T n x n } . Since E is uniformly smooth, we know that E is uniformly convex. Then from Lemma 1.10, we have
G ( x , J u n ) = G ( x , J θ n m y n ) G ( x , J y n ) = G ( x , ( α n J x n + ( 1 α n ) J T n x n ) ) = x 2 2 α n x , J x n 2 ( 1 α n ) x , J T n x n + α n J x n + ( 1 α n ) J T n x n 2 + 2 ρ f ( x ) x 2 2 α n x , J x n 2 ( 1 α n ) x , J T n x n + α n J x n 2 + ( 1 α n ) J T n x n 2 α n ( 1 α n ) g ( J x n J T n x n ) + 2 ρ f ( x ) = α n G ( x , J x n ) + ( 1 α n ) G ( x , J T n x n ) α n ( 1 α n ) g ( J x n J T n x n ) G ( x , J x n ) α n ( 1 α n ) g ( J x n J T n x n ) .
It then follows that
α n ( 1 α n ) g ( J x n J T n x n ) G ( x , J x n ) G ( x , J u n ) .
But
G ( x , J x n ) G ( x , J u n ) = x n 2 u n 2 2 x , J x n J u n x n 2 u n 2 + 2 | x , J x n J u n | | x n u n | ( x n + u n ) + 2 x J x n J u n x n u n ( x n + u n ) + 2 x J x n J u n .
From (2.5) and (2.6), we obtain
G ( x , J x n ) G ( x , J u n ) 0 , n .
Using the condition lim inf n α n ( 1 α n ) > 0 , we have
lim n g ( J x n J T n x n ) = 0 .
By the properties of g, we have lim n J x n J T n x n = 0 . Since J 1 is also uniformly norm-to-norm continuous on bounded sets, we have
lim n x n T n x n = 0 .

Since { T n } n = 1 are uniformly closed, and { x n } n = 1 is a Cauchy sequence. Then p F ( T ) = n = 1 F ( T n ) .

Next, we show that p k = 1 m E P ( F k ) . From (2.1), we obtain
ϕ ( x , u n ) = ϕ ( x , θ n m y n ) = ϕ ( x , T r m , n F m θ n m 1 y n ) ϕ ( x , θ n m 1 y n ) ϕ ( x , x n ) .
(2.7)
Since x E P ( F m ) = F ( T r m , n F m ) for all n 1 , it follows from (2.7) and Lemma 1.13 that
ϕ ( u n , θ n m 1 y n ) = ϕ ( T r m , n F m θ n m 1 y n , θ n m 1 y n ) ϕ ( x , θ n m 1 y n ) ϕ ( x , u n ) ϕ ( x , x n ) ϕ ( x , u n ) .
From (2.5) and (2.6), we obtain lim n ϕ ( θ n m y n , θ n m 1 y n ) = lim n ϕ ( u n , θ n m 1 y n ) = 0 . From Lemma 1.9, we have
lim n θ n m y n θ n m 1 y n = lim n u n θ n m 1 y n = 0 .
(2.8)
Hence, we have from (2.8) that
lim n J θ n m y n J θ n m 1 y n = 0 .
(2.9)
Again, since x E P ( F m 1 ) = F ( T r m 1 , n F m 1 ) for all n 1 , it follows from (2.7) and Lemma 1.13 that
ϕ ( θ n m 1 y n , θ n m 2 y n ) = ϕ ( T r m 1 , n F m 1 θ n m 2 y n , θ n m 2 y n ) ϕ ( x , θ n m 2 y n ) ϕ ( x , θ n m 1 y n ) ϕ ( x , x n ) ϕ ( x , u n ) .
Again, from (2.5) and (2.6), we obtain lim n ϕ ( θ n m 1 y n , θ n m 2 y n ) = 0 . From Lemma 1.9, we have
lim n θ n m 1 y n θ n m 2 y n = 0
(2.10)
and hence,
lim n J θ n m 1 y n J θ n m 2 y n = 0 .
(2.11)
In a similar way, we can verify that
lim n θ n m 2 y n θ n m 3 y n = = lim n θ n 1 y n y n = 0 .
(2.12)
From (2.8), (2.10), and (2.12), we can conclude that
lim n θ n k y n θ n k 1 y n = 0 , k = 1 , 2 , , m .
(2.13)
Since x n p , n , we obtain from (2.5) that u n p , n . Again, from (2.8), (2.10), (2.12), and u n p , n , we have that θ n k y n p , n for each k = 1 , 2 , , m . Also, using (2.13), we obtain
lim n J θ n k y n J θ n k 1 y n = 0 , k = 1 , 2 , , m .
Since lim inf n r k , n > 0 , k = 1 , 2 , , m ,
lim n J θ n k y n J θ n k 1 y n r k , n = 0 .
(2.14)
By Lemma 1.12, we have for each k = 1 , 2 , , m
F k ( θ n k y n , y ) + 1 r k , n y θ n k y n , J θ n k y n J θ n k 1 y n 0 , y C .
Furthermore, using (A2) we obtain
1 r k , n y θ n k y n , J θ n k y n J θ n k 1 y n F k ( y , θ n k y n ) .
(2.15)
By (A4), (2.14), and θ n k y n p , we have for each k = 1 , 2 , , m
F k ( y , p ) 0 , y C .
For fixed y C , let z t = t y + ( 1 t ) p for all t ( 0 , 1 ] . This implies that z t C . This yields F k ( z t , p ) 0 . It follows from (A1) and (A4) that
0 = F k ( z t , z t ) t F k ( z t , y ) + ( 1 t ) F k ( z t , p ) t F k ( z t , y )
and hence
0 F k ( z t , y ) .
From condition (A3), we obtain
F k ( p , y ) 0 , y C .

This implies that p E P ( F k ) , k = 1 , 2 , , m . Thus, p k = 1 m E P ( F k ) . Hence, we have p F = k = 1 m E P ( F k ) ( n = 1 F ( T n ) ) .

Finally, we show that p = Π F f x 0 . Since F = k = 1 m E P ( F k ) ( n = 1 F ( T n ) ) is a closed and convex set, from Lemma 1.6, we know that Π F f x 0 is single valued and denote w = Π F f x 0 . Since x n = Π c n f x 0 and w F C n , we have
G ( x n , J x 0 ) G ( w , J x 0 ) , n 0 .
We know that G ( ξ , J φ ) is convex and lower semi-continuous with respect to ξ when φ is fixed. This implies that
G ( p , J x 0 ) lim inf n G ( x n , J x 0 ) lim sup n G ( x n , J x 0 ) G ( w , J x 0 ) .

From the definition of Π F f x 0 and p F , we see that p = w . This completes the proof. □

Corollary 2.6 Let E be a uniformly convex and uniformly smooth real Banach space, and let C be a nonempty closed convex subset of E. For each k = 1 , 2 , , m , let F k be a bifunction from C × C satisfying (A1)-(A4) and let { T n } n = 1 be a countable family of uniformly closed relatively quasi-nonexpansive mappings of C into itself such that F : = ( n = 1 F ( T n ) ) ( k = 1 m E P ( F k ) ) . Suppose { x n } n = 0 is iteratively generated by x 0 C , C 1 = C , x 1 = Π C 1 f x 0 ,
{ y n = J 1 ( α n J x n + ( 1 α n ) J T n x n ) , u n = T r m , n F m T r m 1 , n F m 1 T r 2 , n F 2 T r 1 , n F 1 y n , C n + 1 = { w C n : ϕ ( w , u n ) ϕ ( w , x n ) } , x n + 1 = Π C n + 1 x 0 , n 1 ,

where J is the duality mapping on E. Suppose { α n } n = 1 is a sequence in ( 0 , 1 ) such that lim inf n α n ( 1 α n ) > 0 , and { r k , n } n = 1 ( 0 , ) ( k = 1 , 2 , , m ) satisfying lim inf n r k , n > 0 ( k = 1 , 2 , , m ). Then { x n } n = 0 converges strongly to Π F x 0 .

Proof Take f ( x ) = 0 for all x E in Theorem 2.5, then G ( ξ , J x ) = ϕ ( ξ , x ) and Π C f x 0 = Π C x 0 . Then Corollary 2.6 holds. □

Take F k 0 ( k = 1 , 2 , , m ), it is obvious that the following holds.

Corollary 2.7 Let E be a uniformly convex and uniformly smooth real Banach space, and let C be a nonempty closed convex subset of E. Let { T n } n = 1 be a countable family of uniformly closed relatively quasi-nonexpansive mappings of C into itself such that F = ( n = 1 F ( T n ) ) . Let f : E R be a convex and lower semi-continuous mapping with C int ( D ( f ) ) and suppose { x n } n = 0 is iteratively generated by x 0 C , C 1 = C , x 1 = Π C 1 f x 0 ,
{ y n = J 1 ( α n J x n + ( 1 α n ) J T n x n ) , C n + 1 = { w C n : G ( w , J y n ) G ( w , J x n ) } , x n + 1 = Π C n + 1 f x 0 , n 1 ,

where J is the duality mapping on E. Suppose { α n } n = 1 is a sequence in ( 0 , 1 ) such that lim inf n α n ( 1 α n ) > 0 , and { r k , n } n = 1 ( 0 , ) ( k = 1 , 2 , , m ) satisfying lim inf n r k , n > 0 ( k = 1 , 2 , , m ). Then { x n } n = 0 converges strongly to Π F x 0 .

3 Applications

Let φ : C R be a real-valued function. The convex minimization problem is to find x C such that
φ ( x ) φ ( y ) ,
(3.1)
y C . The set of solutions of (3.1) is denoted by C M P ( φ ) . For each r > 0 and x E , define the mapping
T r φ ( x ) = { z C : φ ( y ) + 1 r y z , J z J x φ ( z ) , y C } .
Theorem 3.1 Let E be a uniformly convex and uniformly smooth real Banach space, and let C be a nonempty closed convex subset of E. For each k = 1 , 2 , , m , let φ k be a bifunction from C × C satisfying (A1)-(A4) and let { T n } n = 1 be a countable family of uniformly closed relatively quasi-nonexpansive mappings of C into itself such that F : = ( n = 1 F ( T n ) ) ( k = 1 m C M P ( φ k ) ) . Let f : E R be a convex and lower semi-continuous mapping with C int ( D ( f ) ) and suppose { x n } n = 0 is iteratively generated by x 0 C , C 1 = C , x 1 = Π C 1 f x 0 ,
{ y n = J 1 ( α n J x n + ( 1 α n ) J T n x n ) , u n = T r m , n φ m T r m 1 , n φ m 1 T r 2 , n φ 2 T r 1 , n φ 1 y n , C n + 1 = { w C n : G ( w , J u n ) G ( w , J x n ) } , x n + 1 = Π C n + 1 f x 0 , n 1 ,

where J is the duality mapping on E. Suppose { α n } n = 1 is a sequence in ( 0 , 1 ) such that lim inf n α n ( 1 α n ) > 0 and { r k , n } n = 1 ( 0 , ) ( k = 1 , 2 , , m ) satisfying lim inf n r k , n > 0 ( k = 1 , 2 , , m ). Then { x n } n = 0 converges strongly to Π F f x 0 .

Proof Define F k ( x , y ) = φ k ( y ) φ k ( x ) , x , y C and k = 1 , 2 , , m . Then F ( T r k F k ) = E P ( F k ) = C M P ( φ k ) = F ( T r k φ k ) for each k = 1 , 2 , , m , and therefore { F k } k = 1 m satisfies conditions (A1) and (A2). Furthermore, one can easily show that { F k } k = 1 m satisfies (A3) and (A4). Therefore, from Theorem 2.5, we obtain Theorem 3.1. □

Declarations

Acknowledgements

This project is supported by the National Natural Science Foundation of China under grant (11071279).

Authors’ Affiliations

(1)
Department of Mathematics, Tianjin Polytechnic University

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© Zhang et al.; licensee Springer. 2014

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