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Halpern’s iteration for Bregman strongly nonexpansive multi-valued mappings in reflexive Banach spaces with application

Fixed Point Theory and Applications20132013:197

https://doi.org/10.1186/1687-1812-2013-197

  • Received: 20 December 2012
  • Accepted: 5 July 2013
  • Published:

Abstract

Bregman strongly nonexpansive multi-valued mapping in reflexive Banach spaces is established. Under suitable limit conditions, some strong convergence theorems for modifying Halpern’s iterations are proved. As an application, we utilize the main results to solve equilibrium problems in the framework of reflexive Banach spaces. The main results presented in the paper improve and extend the corresponding results in the work by Suthep et al. (Comput. Math. Appl. 64:489-499, 2012).

MSC:47J05, 47H09, 49J25.

Keywords

  • Bregman strongly nonexpansive multi-valued mapping
  • Legendre functions
  • Bregman projection
  • fixed point
  • Halpern’s iteration sequence

1 Introduction

Let D be a nonempty and closed subset of a real Banach space X. Let N ( D ) and CB ( D ) denote the family of nonempty subsets and nonempty, closed and bounded 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 all A 1 , A 2 CB ( D ) , where d ( x , A 1 ) = inf { x y , y A 1 } . 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 N ( D ) if p T ( p ) . The set of fixed points of T is denoted by F ( T ) .

In recent years, several types of iterative schemes have been constructed and proposed in order to get strong convergence results for finding fixed points of nonexpansive mappings in various settings. One classical and effective iteration process is defined by
x n + 1 = α n u + ( 1 α n ) T x n , x 1 , u D ,

where α n ( 0 , 1 ) . Such a method was introduced in 1967 by Halpern [1] and is often called Halpern’s iteration. In fact, he proved, in a real Hilbert space, strong convergence of { x n } to a fixed point of the nonexpansive mapping T, where α n = n a , a ( 0 , 1 ) .

Now, because of a simple construction, Halpern’s iteration is widely used to approximate fixed points of nonexpansive mappings and other classes of nonlinear mappings. Reich [2] also extended the result of Halpern from Hilbert spaces to uniformly smooth Banach spaces. In 2012, Halpern’s iteration for Bregman strongly nonexpansive mappings in reflexive Banach spaces was introduced and a strong convergence theorem for Bregman strongly nonexpansive mappings by Halpern’s iteration in the framework of reflexive Banach spaces was proved.

In this paper, Bregman strongly nonexpansive multi-valued mappings in reflexive Banach spaces are introduced. Under suitable limit conditions, strong convergence theorems for the proposed modified Halpern’s iterations are proved. As an application, we use our results to solve equilibrium problems in the framework of reflexive Banach spaces. The results presented in the paper improve and extend the corresponding results in [3].

2 Preliminaries

In the sequel, we begin by recalling some preliminaries and lemmas which will be used in our proofs. Let X be a real reflexive Banach space with a norm and let X be the dual space of X. Let f : X ( , + ] be a proper, lower semi-continuous and convex function. We denote by domf the domain of f.

Let x intdom f . The subdifferential of f at x is the convex set defined by
f ( x ) = { x X : f ( x ) + x , y x f ( y ) , y X } .
(2.1)
The Fenchel conjugate of f is the function f : X ( , + ] defined by
f ( x ) = sup { x , x f ( x ) : x X } .
We know that the Young-Fenchel inequality holds, that is,
x , x f ( x ) + f ( x ) , x X , x X .

Furthermore, equality holds if x f ( x ) (see [4]). The set lev f ( r ) : = { x X : f ( x ) r } for some r R is called a sublevel of f.

A function f on X is called coercive [5] if the sublevel sets of f are bounded, equivalently,
lim x + f ( x ) = + .
A function f on X is said to be strongly coercive [6] if
lim x + f ( x ) x = + .
For any x intdom f and y X , the right-hand derivative of f at x in the direction y is defined by
f ( x , y ) = lim t 0 + f ( x + t y ) f ( x ) t .

The function f is said to be Gâteaux differentiable at x if lim t 0 + f ( x + t y ) f ( x ) t exists for any y. In this case, f ( x , y ) coincides with f ( x ) , the value of the gradient f ( x ) of f at x. The function f is said to be Gâteaux differentiable if it is Gâteaux differentiable for any x intdom f . The function f is said to be Fréchet differentiable at x if this limit is attained uniformly in y = 1 . Finally, f is said to be uniformly Fréchet differentiable on a subset D of X if the limit is attained uniformly for x D and y = 1 . It is known that if f is Gâteaux differentiable (resp. Frêchet differentiable) on intdomf, then f is continuous and its Gâteaux derivative f is norm-to-weak continuous (resp. continuous) on intdomf (see [7] and [8]).

Definition 2.1 (cf. [9])

The function f is said to be
  1. (i)

    essentially smooth if ∂f is both locally bounded and single-valued on its domain;

     
  2. (ii)

    essentially strictly convex if ( f ) 1 is locally bounded on its domain and f is strictly convex on every convex subset of dom f ;

     
  3. (iii)

    Legendre if it is both essentially smooth and essentially strictly convex.

     

Remark 2.1 (cf. [10])

Let X be a reflexive Banach space. Then we have
  1. (a)

    f is essentially smooth if and only if f is essentially strictly convex;

     
  2. (b)

    ( f ) 1 = f ;

     
  3. (c)

    f is Legendre if and only if f is Legendre;

     
  4. (d)
    If f is Legendre, then ∂f is a bijection which satisfies
    f = ( f ) 1 , ran f = dom f = intdom f and ran f = dom f = intdom f .
     

Examples of Legendre functions can be found in [11]. One important and interesting Legendre function is 1 p p ( 0 < p < + ) when X is a smooth and strictly convex Banach space. In this case, the gradient f of f is coincident with the generalized duality mapping of X, i.e., f = J p . In particular, f = I the identity mapping in Hilbert spaces. In this paper, we always assume that f is Legendre.

The following crucial lemma was proved by Reich and Sabach [12].

Lemma 2.1 (cf. [12])

If f : X R is uniformly Fréchet differentiable and bounded on bounded subsets of X, then f is uniformly continuous on bounded subsets of X from the strong topology of X to the strong topology of X .

Let f : X ( , + ] be a convex and Gâteaux differentiable function. The function D f : dom f × intdom f [ 0 , + ) , defined by
D f ( y , x ) : = f ( y ) f ( x ) f ( x ) , y x ,

is called the Bregman distance with respect to f.

Recall that the Bregman projection [13] of x intdom f onto a nonempty, closed and convex set D dom f is the necessarily unique vector proj D f ( x ) D (for convenience, here we use P D f ( x ) for proj D f ( x ) ) satisfying
D f ( proj D f ( x ) , x ) = inf { D f ( y , x ) : y D } .
The modulus of total convexity of f at x intdom f is the function v f ( x , t ) : [ 0 , + ) [ 0 , + ) defined by
v f ( x , t ) : = inf { D f ( y , x ) : y dom f , y x = t } .
The function f is called totally convex at x if v f ( x , t ) > 0 whenever t > 0 . The function f is called totally convex if it is totally convex at any point x intdom f , and it is said to be totally convex on bounded sets if v f ( B , t ) > 0 for any nonempty bounded subset B and t > 0 , where the modulus of total convexity of the function f on the set B is the function v f : intdom f × [ 0 , + ) [ 0 , + ) defined by
v f ( B , t ) = inf { v f ( x , t ) : x B dom f } .

We know that f is totally convex on bounded sets if and only if f is uniformly convex on bounded sets (see [14]).

Recall that the function f is said to be sequentially consistent [14] if for any two sequences { x n } and { y n } in X, such that the first sequence is bounded, the following implication holds:
lim n + D f ( x n , y n ) = 0 lim n + x n y n = 0 .

The following crucial lemma was proved by Butnariu and Iusem [15].

Lemma 2.2 (cf. [15])

The function f is totally convex on bounded sets if and only if it is sequentially consistent.

Definition 2.2 (cf. [16])

Let D be a convex subset of intdomf and let T be a multi-valued mapping of D. A point p D is called an asymptotic fixed point of T if D contains a sequence { x n } , which converges weakly to p, such that d ( x n , T x n ) 0 (as n ).

We denote by F ˆ ( T ) the set of asymptotic fixed points of T.

Definition 2.3 A multi-valued mapping T : D N ( D ) with a nonempty fixed point set is said to be:
  1. (i)
    Bregman strongly nonexpansive with respect to a nonempty F ˆ ( T ) if
    D f ( p , z ) D f ( p , x ) , x D , p F ˆ ( T ) , z T ( x ) ,

    and if, whenever { x n } D is bounded, p F ˆ ( T ) and lim n [ D f ( p , x n ) D f ( p , z n ) ] = 0 , then lim n D f ( x n , z n ) = 0 , where z n T x n .

     
  2. (ii)
    Bregman firmly nonexpansive if
    f ( x ) f ( y ) , x y f ( x ) f ( y ) , x y , x , y D , x T x , y T y .
     

In particular, the existence and approximation of Bregman firmly nonexpansive single-valued mappings was studied in [16]. It is also known that if T is Bregman firmly nonexpansive and f is the Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of X, then F ( T ) = F ˆ ( T ) and F ( T ) is closed and convex (see [16]). It also follows that every Bregman firmly nonexpansive mapping is Bregman strongly nonexpansive with respect to F ( T ) = F ˆ ( T ) . The class of single-valued Bregman strongly nonexpansive mappings was introduced first in [17]. For a wealth of results concerning this class of mappings, see [1822] and the references therein.

Remark 2.2 Let X be a uniformly smooth and uniformly convex Banach space, and let D be a nonempty, closed and convex subset. An operator T : C N ( D ) is called a strongly relatively nonexpansive multi-valued mapping on X if F ˆ ( T ) Φ and
ϕ ( p , z ) ϕ ( p , x ) , p F ˆ ( T ) , z T x ,

and if, whenever { x n } D is bounded, p F ˆ ( T ) and lim n [ φ ( p , x n ) φ ( p , z n ) ] = 0 , then lim n φ ( x n , z n ) = 0 , where z n T x n and ϕ ( x , y ) = x 2 2 x , J y + y 2 .

Let D be a nonempty, closed and convex subset of X. Let f : X R be a Gâteaux differentiable and totally convex function and x X . It is known from [14] that z = P D f ( x ) if and only if
f ( x ) f ( z ) , y z 0 , y D .
We also know the following characterization:
D f ( y , P D f ( x ) ) + D f ( P D f ( x ) , x ) D f ( y , x ) , x , y D .
Let f : X R be a convex, Legendre and Gâteaux differentiable function. Following [23] and [24], we make use of the function V f : X × X [ 0 , + ) associated with f, which is defined by
V f ( x , x ) = f ( x ) + f ( x ) x , x , x X , x X .
Then V f is nonnegative and V f ( x , x ) = D f ( x , f ( x ) ) for all x X and x X . Moreover, by the subdifferential inequality (see [22], Proposition 1(iii), p.1047),
V f ( x , x ) + y , f ( x ) x V f ( x , x + y ) , x X , x , y X .
In addition, if f : X ( , + ] is a proper and lower semi-continuous function, then f : X ( , + ] is a proper, weak lower semi-continuous and convex function (see [25]). Hence V f is convex in the second variable (see [22], Proposition 1(i), p.1047). Thus,
D f ( z , f ( t f ( x ) + ( 1 t ) f ( y ) ) ) t D f ( z , x ) + ( 1 t ) D f ( z , y ) , t ( 0 , 1 ) , x , y X .
(2.2)

The properties of the Bregman projection and the relative projection operators were studied in [14] and [26].

The following lemmas give us some nice properties of sequences of real numbers which will be useful for the forthcoming analysis.

Lemma 2.3 (cf. [27], Lemma 2.1, p.76)

Let { α n } be a sequence of real numbers such that there exists a nondecreasing subsequence α n i of α n , that is, α n i α n i + 1 for all i N . Then there exists a nondecreasing subsequence { m k } N such that m k , and the following properties are satisfied for all sufficiently large numbers sequence k N :
α m k α m k + 1 and α k α m k + 1 .

In fact, m k = max { j k : α j α j + 1 } .

Lemma 2.4 (see [3], Lemma 2.5, p.493)

Assume that { α n } is a sequence of nonnegative real numbers such that
α n + 1 ( 1 γ n ) α n + γ n δ n ,
where { γ n } is a sequence in ( 0 , 1 ) and { δ n } is a sequence such that
  1. (a)

    lim n γ n = 0 , n = 1 γ n = ;

     
  2. (b)

    lim sup n δ n 0 .

     

Then lim n α n = 0 .

3 Main results

To prove our main result, we first prove the following two propositions.

Proposition 3.1 Let D be a nonempty, closed and convex subset of a real reflexive Banach space X. Let f : X R be a Gâteaux differentiable and totally convex function, and let T : D N ( D ) be a multi-valued mapping such that F ( T ) = F ˆ ( T ) is nonempty, closed and convex. Suppose that u D and { x n } is a bounded sequence in D such that lim n d ( x n , T x n ) = 0 . Then
lim sup n f ( u ) f ( p ) , x n p 0 , p = P F ( T ) f ( u ) .
(3.1)
Proof Since X is reflexive and { x n } is bounded, there exists a subsequence { x n k } { x n } such that x n k v D as k and
lim sup n f ( u ) f ( p ) , x n p = f ( u ) f ( p ) , v p .
On the other hand, since lim k d ( x n k , T n k ) = 0 , then v F ˆ ( T ) = F ( T ) . By the definition of Bregman projection, we have
lim sup n f ( u ) f ( p ) , x n p = f ( u ) f ( p ) , v p 0 .

The proof of Proposition 3.1 is now completed. □

The proof of the following result in the case of single-valued Bregman firmly nonexpansive mappings was done in ([16], Lemma 15.5, p.305). In the multi-valued case, the proof is identical and therefore we will omit the exact details. The interested reader will consult [16].

Proposition 3.2 Let f : X ( , + ] be a Legendre function and let D be a nonempty, closed and convex subset of intdomf. Let T : D N ( D ) be a Bregman firmly nonexpansive multi-valued mapping with respect to f. Then F ( T ) is closed and convex.

Theorem 3.1 Let X be a real reflexive Banach space and let f : X ( , + ] be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of X. Let D be a nonempty, closed and convex subset of intdomf and let T : D N ( D ) be a Bregman strongly nonexpansive multi-valued mapping on X such that F ( T ) = F ˆ ( T ) . Suppose that u X and define the sequence { x n } by
x 1 D , x n + 1 = f ( α n f ( u ) + ( 1 α n ) f ( z n ) ) , z n T x n , n 1 ,
(3.2)

where α n ( 0 , 1 ) satisfying lim n α n = 0 and n = 1 α n = . Then { x n } strongly converges to P F ( T ) f ( u ) .

Proof First, by Proposition 3.2, we know that F ( T ) is closed and convex. Letting p = P F ( T ) f ( u ) F ( T ) = F ˆ ( T ) , we have
D f ( p , x n + 1 ) α n D f ( p , u ) + ( 1 α n ) D f ( p , z n ) α n D f ( p , u ) + ( 1 α n ) D f ( p , x n ) max { D f ( p , u ) , D f ( p , x n ) } .

By the induction, the sequence D f ( p , x n ) is bounded.

Next, we show that the sequence { x n } is also bounded. We follow the proof as in [16]. Since D f ( p , x n ) is bounded, there exists L > 0 such that
f ( p ) f ( x n ) , p + f ( f ( x n ) ) = V f ( p , f ( x n ) ) = D f ( p , x n ) L .

Hence { f ( x n ) } is contained in the sublevel set lev ψ ( L f ( p ) ) , where ψ = f , p . Since f is lower semicontinuous, f is weak lower semicontinuous. Hence the function ψ is coercive (see [4]). This shows that { f ( x n ) } is bounded. Since f is strongly coercive, f  is bounded on bounded sets (see [9]). Hence f is also bounded on bounded subsets of E (see [15]). Since f is a Legendre function, it follows that x n = f ( f ( x n ) ) , n N , is bounded. Therefore { x n } is bounded. So are { z n } and { f ( z n ) } .

We next show that if there exists a subsequence { x n k } { x n } such that
lim k [ D f ( p , x n k + 1 ) D f ( p , x n k ) ] = 0 ,
then
lim k [ D f ( p , z n k ) D f ( p , x n k ) ] = 0 ,

where z n k T x n k .

Since { f ( z n ) } is bounded, we have from (3.2)
lim k f ( x n k + 1 ) f ( z n k ) = lim k α n k f ( u ) f ( z n k ) = 0 .
(3.3)
Since f is strongly coercive and uniformly convex on bounded subsets of X, f is uniformly Fréchet differentiable on bounded subsets of X (see [6]). Moreover, f is bounded on bounded sets. Since f is Legendre, by Lemma 2.1, we obtain
lim k x n k + 1 z n k = lim k f ( f ( x n k + 1 ) ) f ( f ( z n k ) ) = 0 .
(3.4)
On the other hand, since f is uniformly Fréchet differentiable on bounded subsets of X, then f is uniformly continuous on bounded subsets of X (see [28]). It follows that
lim k | f ( x n k + 1 ) f ( z n k ) | = 0 .
(3.5)
Since the following equality holds:
D f ( p , z n k ) D f ( p , x n k ) = f ( p ) f ( z n k ) f ( z n k ) , p z n k D f ( p , x n k ) = f ( p ) f ( x n k + 1 ) + f ( x n k + 1 ) f ( z n k ) f ( x n k + 1 ) , p x n k + 1 + f ( x n k + 1 ) , p x n k + 1 f ( z n k ) , p z n k D f ( p , x n k ) = D f ( p , x n k + 1 ) + ( f ( x n k + 1 ) f ( z n k ) ) + f ( x n k + 1 ) , p x n k + 1 f ( z n k ) , p z n k D f ( p , x n k ) = ( D f ( p , x n k + 1 ) D f ( p , x n k ) ) + ( f ( x n k + 1 ) f ( z n k ) ) + f ( x n k + 1 ) f ( z n k ) , p x n k + 1 f ( z n k ) , x n k + 1 z n k ,
it follows from (3.3), (3.4) and (3.5) that
lim k ( D f ( p , z n k ) D f ( p , x n k ) ) = 0 .

The rest of the proof will be divided into two parts.

Case 1. Suppose that { D f ( p , x n ) } is eventually decreasing, i.e., there exists a sufficiently large k > 0 such that D f ( p , x n ) > D f ( p , x n + 1 ) for all n > k . In this case, lim n D f ( p , x n ) exists. In this situation, we have that lim n D f ( p , x n ) exists. This shows that lim n ( D f ( p , x n ) D f ( p , x n + 1 ) ) = 0 and hence lim n ( D f ( p , z n ) D f ( p , x n ) ) = 0 .

Since T is a Bregman strongly nonexpansive multi-valued mapping, then
lim n ( D f ( x n , z n ) ) = 0 .
Since f is totally convex on bounded subsets of E, by Lemma 2.2, we have
lim n x n z n = 0 .
By Proposition 3.1, we obtain
lim sup n f ( u ) f ( p ) , x n p 0 .
Finally, we show that x n p as n . Indeed
D f ( p , x n + 1 ) = V f ( p , α n f ( u ) + ( 1 α n ) f ( z n ) ) V f ( p , α n f ( u ) + ( 1 α n ) f ( z n ) α n ( f ( u ) f ( p ) ) ) + α n ( f ( u ) f ( p ) ) , x n + 1 p = V f ( p , α n f ( p ) + ( 1 α n ) f ( z n ) ) + α n f ( u ) f ( p ) , x n + 1 p α n V f ( p , f ( p ) ) + ( 1 α n ) V f ( p , f ( z n ) ) + α n f ( u ) f ( p ) , x n + 1 p = ( 1 α n ) D f ( p , z n ) + α n f ( u ) f ( p ) , x n + 1 p ( 1 α n ) D f ( p , x n ) + α n f ( u ) f ( p ) , x n + 1 p .

By Lemma 2.4, we conclude that lim n D f ( p , x n ) = 0 . Therefore, by Lemma 2.2, since f is totally convex on bounded subsets of X, we obtain that x n p as n .

Case 2. If { D f ( p , x n ) } is not eventually decreasing, there exists a subsequence { D f ( p , x n j ) } { D f ( p , x n ) } such that D f ( p , x n j ) < D f ( p , x n j + 1 ) for all j N . By Lemma 2.3, there exists a strictly increasing sequence { m k } of positive integers such that the following properties hold for all k N :
D f ( p , x m k ) D f ( p , x m k + 1 ) , D f ( p , x k ) D f ( p , x m k + 1 ) .
Since the inequality D f ( p , z n ) D f ( p , x n ) 0 holds by Definition 2.3, hence, by Lemma 2.3, we have
0 lim k ( D f ( p , x m k + 1 ) D f ( p , x m k ) ) lim sup n ( D f ( p , x n + 1 ) D f ( p , x n ) ) lim sup n ( α n D f ( p , u ) + ( 1 α n ) D f ( p , z n ) D f ( p , x n ) ) = lim sup n ( α n ( D f ( p , u ) D f ( p , z n ) ) + D f ( p , z n ) D f ( p , x n ) ) lim sup n α n ( D f ( p , u ) D f ( p , z n ) ) = 0 .
This implies that
lim sup k ( D f ( p , x m k + 1 ) D f ( p , x m k ) ) = 0 .
(3.6)
Following the proof of Case 1, we have
lim sup k f ( u ) f ( p ) , x m k p 0 ,
and
D f ( p , x m k + 1 ) ( 1 α m k ) D f ( p , x m k ) + α m k f ( u ) f ( p ) , x m k + 1 p .
This implies that
α m k D f ( p , x m k ) D f ( p , x m k ) D f ( p , x m k + 1 ) + α m k f ( u ) f ( p ) , x m k + 1 p α m k f ( u ) f ( p ) , x m k + 1 p .
Hence
lim k D f ( p , x m k ) = 0 .
Using this and (3.5) together, we conclude that
lim sup k D f ( p , x k ) lim k D f ( p , x m k + 1 ) = 0 .

The proof of Theorem 3.1 is now completed. □

As a direct consequence of Theorem 3.1 and Remark 2.2, we obtain the convergence result concerning strongly relatively nonexpansive multi-valued mappings in a uniformly smooth and uniformly convex Banach space.

Corollary 3.1 Let X be a uniformly smooth and uniformly convex Banach space. Let D be a nonempty, closed and convex subset on X and let T : D N ( D ) be a strongly relatively nonexpansive multi-valued mapping on X such that F ( T ) = F ˆ ( T ) . Suppose that u D and define the sequence { x n } as follows: x 1 D and
x n + 1 = J 1 ( α n J ( u ) + ( 1 α n ) J z n , z n T x n , n 1 ,

where α n ( 0 , 1 ) satisfying α n 0 and n = 1 α n = . Then { x n } converges strongly to Π F ( T ) u , where Π F ( T ) is the generalized projection onto F ( T ) .

4 Application

In this section, we give an application of Theorem 3.1, which is the equilibrium problems in the framework of reflexive Banach spaces.

Let X be a smooth, strictly convex and reflexive Banach space, let D be a nonempty, closed and convex subset of X, and let G : D × D R be a bifunction satisfying the conditions:
  1. (A1)

    G ( x , x ) = 0 for all x D ;

     
  2. (A2)

    G ( x , y ) + G ( y , x ) 0 for any x , y D ;

     
  3. (A3)

    for each x , y , z D , lim t 0 G ( t z + ( 1 t ) x , y ) G ( x , y ) ;

     
  4. (A4)

    for each given x D , the function y f ( x , y ) is convex and lower semicontinuous.

     

The so-called equilibrium problem for G is to find a x D such that G ( x , y ) 0 for each y D . The set of its solutions is denoted by EP ( G ) .

The resolvent of a bifunction G [18] is the operator Res G f : X 2 D defined by
Res G f ( x ) = { z D , G ( z , y ) + f ( z ) f ( x ) z , y z 0 , y D } , x X .
(4.1)
If f : X ( , + ] is a strongly coercive and Gâteaux differentiable function, and G satisfies conditions (A1)-(A4), then dom ( Res G f ) = X (see [18]). We also know:
  1. (1)

    Res G f is single-valued;

     
  2. (2)

    Res G f is a Bregman firmly nonexpansive mapping;

     
  3. (3)

    F ( Res G f ) = EP ( G ) ;

     
  4. (4)

    EP ( G ) is a closed and convex subset of D;

     
  5. (5)
    for all x X and for all p F ( Res G f ) , we have
    D f ( p , Res G f ( x ) ) + D f ( Res G f ( x ) , x ) D f ( p , x ) .
    (4.2)
     

In addition, by Reich and Sabach [16], if f is uniformly Fréchet differentiable and bounded on bounded subsets of X, then we have that F ( Res G f ) = F ˆ ( Res G f ) = EP ( G ) is closed and convex. Hence, by replacing T = Res G f in Theorem 3.1, we obtain the following result.

Theorem 4.1 Let D be a nonempty, closed and convex subset of a real reflexive Banach space X. Let f be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of X. Let G : D × D R be a bifunction which satisfies conditions (A1)-(A4) such that EP ( G ) . Suppose that u X and define the sequence { x n } by
x 1 X , x n + 1 = f ( α n f ( u ) + ( 1 α n ) f ( z n ) , z n Res G f ( x n ) , n 1 ,
(4.3)

where α n ( 0 , 1 ) satisfying α n 0 and n = 1 α n = . Then { x n } converges strongly to P EP ( G ) f u .

Declarations

Acknowledgements

The authors are very grateful to both reviewers for carefully reading this paper and their comments. This work is supported by the Doctoral Program Research Foundation of Southwest University of Science and Technology (No. 11zx7129) and Applied Basic Research Project of Sichuan Province (No. 2013JY0096).

Authors’ Affiliations

(1)
School of Science, Southwest University of Science and Technology, Mianyang Sichuan, 621010, P.R., China

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