Open Access

Halpern-Mann’s iterations for Bregman strongly nonexpansive mappings in reflexive Banach spaces with applications

Journal of Inequalities and Applications20132013:146

https://doi.org/10.1186/1029-242X-2013-146

Received: 4 November 2012

Accepted: 5 March 2013

Published: 2 April 2013

Abstract

We investigate strong convergence for Bregman strongly nonexpansive mappings by modifying Halpern and Mann’s iterations in the framework of a reflexive Banach space. As applications, we apply our main result to problems of finding zeros of maximal monotone operators and equilibrium problems in reflexive Banach spaces.

MSC:47H05, 47H09, 47J25.

Keywords

Bregman strongly nonexpansive mapping Bregman projection Legendre function totally convex function Halpern’s iteration Mann’s iteration

1 Introduction

Throughout this paper, we denote by and the sets of positive integers and real numbers, respectively. Let E be a real reflexive Banach space, and let C be a nonempty, closed and convex subset of E. Let T : C C be a nonlinear mapping. The fixed point set of T is denoted by F ( T ) , that is, F ( T ) = { x C : x = T x } . A mapping T is said to be nonexpansive if
T x T y x y

for all x , y C .

Many problems in nonlinear analysis can be reformulated as a problem of finding a fixed point of a nonexpansive mapping. In 1953, Mann [1] introduced the following iterative sequence { x n } which is defined by
x n + 1 = α n x n + ( 1 α n ) T x n ,

where the initial guess x 1 C 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, Mann’s iteration may fail to converge strongly; for example, see [2].

Some attempts to construct an iteration method guaranteeing the strong convergence have been made. For example, Halpern [3] proposed the following so-called Halpern iteration:
x n + 1 = α n u + ( 1 α n ) T x n ,

where u , x 1 C are arbitrary and { α n } is a real sequence in [ 0 , 1 ] .

Because of a simple construction, Halpern’s iteration is widely used to approximate a solution of fixed points for nonexpansive mappings and other classes of nonlinear mappings by mathematicians in different styles [412].

The purpose of this work is to consider strong convergence results for Bregman strongly nonexpansive mappings in reflexive Banach spaces by modifying Halpern and Mann’s iterations. We note that there are many examples which are Bregman strongly nonexpansive such as the Bregman projection, the resolvents of maximal monotone operators, the resolvents of equilibrium problems, the resolvents of variational inequality problems and others (see, for example, [1316]). Finally, we give some applications concerning the problems of finding zeros of maximal monotone operators and equilibrium problems.

2 Preliminaries and lemmas

In the sequel, we begin by recalling some preliminaries and lemmas which will be used in the proof.

Let E be a real reflexive Banach space with the norm , and let E be the dual space of E. Throughout this paper, f : E ( , + ] is a proper, lower semi-continuous and convex function. We denote by domf the domain of f, that is, the set { x E : f ( x ) < + } .

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

Furthermore, the equality holds if x f ( x ) (see also [[17], Theorem 23.5]).

The set lev f ( r ) = { x E : f ( x ) r } for some r R is called a sublevel of f.

A function f on E is coercive [18] if the sublevel set of f is bounded; equivalently,
lim x + f ( x ) = + .
A function f on E is said to be strongly coercive [19] if
lim x + f ( x ) x = + .
For any x int dom f and y E , 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 of f at x. The function f is said to be Gâteaux differentiable if it is Gâteaux differentiable for any x int dom 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 C of E if the limit is attained uniformly for x C and y = 1 . It is known that if f is Gâteaux differentiable (resp. Fréchet differentiable) on int dom f , then f is continuous and its Gâteaux derivative f is norm-to-weak continuous (resp. continuous) on int dom f (see also [20, 21]). We will need the following result.

Lemma 2.1 [22]

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

Definition 2.2 [23]

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 2.3 Let E be a reflexive Banach space. Then we have
  1. (i)

    f is essentially smooth if and only if f is essentially strictly convex (see [[23], Theorem 5.4]).

     
  2. (ii)
    ( f ) 1 = f
    (see [21]).
     
  3. (iii)

    f is Legendre if and only if f is Legendre (see [[23], Corollary 5.5]).

     
  4. (iv)

    If f is Legendre, then f is a bijection satisfying f = ( f ) 1 , ran f = dom f = int dom f and ran f = dom f = int dom f (see [[23], Theorem 5.10]).

     

Examples of Legendre functions were given in [23, 24]. One important and interesting Legendre function is 1 p p ( 1 < p < ) when E is a smooth and strictly convex Banach space. In this case the gradient f of f is coincident with the generalized duality mapping of E, i.e., f = J p ( 1 < p < ). In particular, f = I the identity mapping in Hilbert spaces. In the rest of this paper, we always assume that f : E ( , + ] is Legendre.

Let f : E ( , + ] be a convex and Gâteaux differentiable function. The function D f : dom f × int dom f [ 0 , + ) defined as follows:
D f ( y , x ) : = f ( y ) f ( x ) f ( x ) , y x

is called the Bregman distance with respect to f [25].

Recall that the Bregman projection [26] of x int dom f onto the nonempty closed and convex set C dom f is the necessarily unique vector P C f ( x ) C satisfying
D f ( P C f ( x ) , x ) = inf { D f ( y , x ) : y C } .

Concerning the Bregman projection, the following are well known.

Lemma 2.4 [27]

Let C be a nonempty, closed and convex subset of a reflexive Banach space E. Let f : E R be a Gâteaux differentiable and totally convex function, and let x E . Then
  1. (a)
    z = P C f ( x )
    if and only if f ( x ) f ( z ) , y z 0 , y C .
     
  2. (b)
    D f ( y , P C f ( x ) ) + D f ( P C f ( x ) , x ) D f ( y , x ) , x E , y C .
    (2.1)
     
Let f : E ( , + ] be a convex and Gâteaux differentiable function. The modulus of total convexity of f at x int dom f is the function ν f ( x , ) : [ 0 , + ) [ 0 , + ] defined by
ν f ( x , t ) : = inf { D f ( y , x ) : y dom f , y x = t } .
The function f is called totally convex at x if ν f ( x , t ) > 0 whenever t > 0 . The function f is called totally convex if it is totally convex at any point x int dom f and is said to be totally convex on bounded sets if ν f ( B , t ) > 0 for any nonempty bounded subset B of E and t > 0 , where the modulus of total convexity of the function f on the set B is the function ν f : int dom f × [ 0 , + ) [ 0 , + ] defined by
ν f ( B , t ) : = inf { ν 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 [[27], Theorem 2.10]).

The next lemma turns out to be very useful in the proof of our main results.

Proposition 2.5 [28]

If x int dom f , then the following statements are equivalent.
  1. (i)

    The function f is totally convex at x.

     
  2. (ii)
    For any sequence { y n } dom f ,
    lim n + D f ( y n , x ) = 0 lim n + y n x = 0 .
     
Recall that the function f is called sequentially consistent [27] if for any two sequences { x n } and { y n } in E such that the first one is bounded,
lim n + D f ( y n , x n ) = 0 lim n + y n x n = 0 .

Lemma 2.6 [29]

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

Let C be a convex subset of int dom f , and let T be a self-mapping of C. A point p C is called an asymptotic fixed point of T (see [30, 31]) if C contains a sequence { x n } which converges weakly to p such that lim n x n T x n = 0 . We denote by F ˆ ( T ) the set of asymptotic fixed points of T.

Definition 2.7 A mapping T with a nonempty asymptotic fixed point set is said to be:
  1. (i)
    Bregman strongly nonexpansive (see [15, 32]) with respect to a nonempty F ˆ ( T ) if
    D f ( p , T x ) D f ( p , x ) , x C , p F ˆ ( T ) ,
    (2.2)
     
and if whenever { x n } C is bounded, p F ˆ ( T ) and
lim n ( D f ( p , x n ) D f ( p , T x n ) ) = 0 ,
it follows that
lim n D f ( x n , T x n ) = 0 .
  1. (ii)
    Bregman firmly nonexpansive [13, 16, 33] if, for all x , y C ,
    f ( T x ) f ( T y ) , T x T y f ( x ) f ( y ) , T x T y
     
or, equivalently,
D f ( T x , T y ) + D f ( T y , T x ) + D f ( T x , x ) + D f ( T y , y ) D f ( T x , y ) + D f ( T y , x ) .

The existence and approximation of Bregman firmly nonexpansive mappings was studied in [16]. It is also known that if T is Bregman firmly nonexpansive and f is a Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of E, 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 ) .

Let f : E R be a convex, Legendre and Gâteaux differentiable function. Following [34] and [25], we make use of the function V f : E × E [ 0 , + ) associated with f, which is defined by
V f ( x , x ) = f ( x ) x , x + f ( x )
(2.3)

for all x E , x E . Then V f is nonnegative and V f ( x , x ) = D f ( x , f ( x ) ) for all x E and x E . We know the following lemma (see [35]).

Lemma 2.8 Let E be a reflexive Banach space, let f : E R be a convex, Legendre and Gâteaux differentiable function, and let V f be as in (2.3). Then
V f ( x , x ) + y , f ( x ) x V f ( x , x + y )
(2.4)

for all x E and x , y E .

Let E be a real reflexive Banach space, let f : E ( , + ] be a proper lower semi-continuous function, then f : E ( , + ] is a proper weak lower semi-continuous and convex function (see [36]). Hence V f is convex in the second variable. Thus, for all z E , we have
D f ( z , f ( i = 1 N t i f ( x i ) ) ) i = 1 N t i D f ( z , x i ) ,
(2.5)

where { x i } i = 1 N E and { t i } i = 1 N ( 0 , 1 ) with i = 1 N t i = 1 .

The following results are of fundamental importance for the techniques of analysis used in this paper.

Lemma 2.9 [37]

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 , n 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 .

Lemma 2.10 [38]

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

3 Main results

In this section, we modify Halpern and Mann’s iterations for finding a fixed point of a Bregman strongly nonexpansive mapping in a real reflexive Banach space.

Lemma 3.1 [39]

Let C be a nonempty, closed and convex subset of a real reflexive Banach space E. Let f : E R be a Gâteaux differentiable and totally convex function, and let T : C C be a mapping such that F ˆ ( T ) = F ( T ) is nonempty, closed and convex. Suppose that u C and { x n } is a bounded sequence in C such that lim n x n T x n = 0 . Then
lim sup n f ( u ) f ( p ) , x n p 0 ,

where p = P F ( T ) f ( u ) and P F ( T ) f is the Bregman projection of C onto F ( T ) .

Theorem 3.2 Let E be a real reflexive Banach space E, and let f : E R be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of E. Let T be a Bregman strongly nonexpansive mapping on E such that F ( T ) = F ˆ ( T ) . Suppose that u E and define the sequence { x n } as follows: x 1 E and
x n + 1 = f ( α n f ( u ) + ( 1 α n ) ( β n f ( x n ) + ( 1 β n ) f ( T x n ) ) ) , n 1 ,
(3.1)

where { α n } and { β n } are sequences in ( 0 , 1 ) satisfying

(C1) lim n α n = 0 ;

(C2) n = 1 α n = ;

(C3) 0 < lim inf n β n lim sup n β n < 1 .

Then { x n } converges strongly to P F ( T ) f ( u ) , where P F ( T ) f is the Bregman projection of E onto F ( T ) .

Proof We note, by Reich and Sabach [16], that F ( T ) is closed and convex. Let p = P F ( T ) f ( u ) F ( T ) = F ˆ ( T ) and y n = f ( β n f ( x n ) + ( 1 β n ) f ( T x n ) ) for all n N . Then
x n + 1 = f ( α n f ( u ) + ( 1 α n ) f ( y n ) )
(3.2)
for all n N . By using (2.5) and (2.2), we have
D f ( p , y n ) = D f ( p , f ( β n f ( x n ) + ( 1 β n ) f ( T x n ) ) ) β n D f ( p , x n ) + ( 1 β n ) D f ( p , T x n ) β n D f ( p , x n ) + ( 1 β n ) D f ( p , x n ) = D f ( p , x n )
(3.3)
and
D f ( p , x n + 1 ) = D f ( p , f ( α n f ( u ) + ( 1 α n ) f ( y n ) ) ) α n D f ( p , u ) + ( 1 α n ) D f ( p , y 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 induction, we have
D f ( p , x n + 1 ) max { D f ( p , u ) , D f ( p , x 1 ) }

for all n N . This implies that { D f ( p , x n ) } is bounded, and hence { D f ( p , y n ) } is bounded.

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

Hence { f ( x n ) } is contained in the sublevel set lev ψ ( M f ( p ) ) , where ψ = f , p . Since f is lower semicontinuous, f is weak lower semicontinuous. Hence the function ψ is coercive by Moreau-Rockafellar theorem (see [[40], Theorem 7A]). This shows that { f ( x n ) } is bounded. Since f is strongly coercive, f is bounded on bounded sets (see [[23], Theorem 3.3]). Hence f is also bounded on bounded subsets of E (see [[29], Proposition 1.1.11]). Since f is a Legendre function, it follows that x n = f ( f ( x n ) ) is bounded for all n N . Therefore { x n } is bounded. So are { y n } , { T x n } , { f ( y n ) } and { f ( T x n ) } . Indeed, since f is a bounded function defined on bounded subsets of E, f is also bounded on bounded subsets of E (see [[29], Proposition 1.1.11]). Therefore { f ( T x n ) } is bounded.

We next show that if there exists a subsequence { x n k } of { 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 , T x n k ) D f ( p , x n k ) ) = 0 .
In fact, since { f ( y n k ) } is bounded and α n k 0 , from (3.2) we have
lim k f ( x n k + 1 ) f ( y n k ) = lim k α n k f ( u ) f ( y n k ) = 0 .
(3.4)
Since f is strongly coercive and uniformly convex on bounded subsets of E, f is uniformly Fréchet differentiable on bounded subsets of E (see [[19], Proposition 3.6.2]). Since f is Legendre, by Lemma 2.1, we have
lim k x n k + 1 y n k = lim k f ( f ( x n k + 1 ) ) f ( f ( y n k ) ) = 0 .
(3.5)
On the other hand, since f is uniformly Fréchet differentiable on bounded subsets of E, f is uniformly continuous on bounded subsets of E (see [[41], Theorem 1.8]). It follows that
lim k | f ( x n k + 1 ) f ( y n k ) | = 0 .
(3.6)
We now consider the following equality.
D f ( p , y n k ) D f ( p , x n k ) = f ( p ) f ( y n k ) f ( y n k ) , p y n k D f ( p , x n k ) = f ( p ) f ( x n k + 1 ) + f ( x n k + 1 ) f ( y n k ) f ( x n k + 1 ) , p x n k + 1 + f ( x n k + 1 ) , p x n k + 1 f ( y n k ) , p y n k D f ( p , x n k ) = D f ( p , x n k + 1 ) + ( f ( x n k + 1 ) f ( y n k ) ) + f ( x n k + 1 ) , p x n k + 1 f ( y n k ) , p y 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 ( y n k ) ) + f ( x n k + 1 ) f ( y n k ) , p x n k + 1 f ( y n k ) , x n k + 1 y n k .
(3.7)
It follows from (3.4)-(3.6) that
lim k ( D f ( p , y n k ) D f ( p , x n k ) ) = 0
(3.8)
and
D f ( p , y n k ) D f ( p , x n k ) β n k D f ( p , x n k ) + ( 1 β n k ) D f ( p , T x n k ) D f ( p , x n k ) = ( 1 β n k ) ( D f ( p , T x n k ) D f ( p , x n k ) ) .
By virtue of condition (C3), (2.2) and (3.8), we have
lim k ( D f ( p , T x n k ) D f ( p , x n k ) ) = 0 .
(3.9)

We next consider the following two cases.

Case 1. D f ( p , x n + 1 ) D f ( p , x n ) for all sufficiently large n. Hence the sequence { D f ( p , x n ) } is bounded and nonincreasing. So, the limit lim n D f ( p , x n ) exists. This shows that lim n ( D f ( p , x n + 1 ) D f ( p , x n ) ) = 0 , and hence
lim n ( D f ( p , T x n ) D f ( p , x n ) ) = 0 .
Since T is Bregman strongly nonexpansive, we have
lim n D f ( x n , T x n ) = 0 .
Since f is totally convex on bounded subsets of E, by Lemma 2.6, we have
lim n x n T x n = 0 .
(3.10)
From (2.5) we have
D f ( T x n , y n ) = D f ( T x n , f ( β n f ( x n ) + ( 1 β n ) f ( T x n ) ) ) β n D f ( T x n , x n ) + ( 1 β n ) D f ( T x n , T x n ) = β n D f ( T x n , x n ) 0
(3.11)
and
D f ( y n , x n + 1 ) α n D f ( y n , u ) + ( 1 α n ) D f ( y n , y n ) = α n D f ( y n , u ) 0 .
(3.12)
From (3.11), (3.12) and Lemma 2.6, we get
lim n T x n y n = 0 and lim n y n x n + 1 = 0 .
(3.13)
From (3.10), (3.13) and invoking Lemma 3.1, we have
lim n sup f ( u ) f ( p ) , x n + 1 p = lim n sup f ( u ) f ( p ) , x n p 0 .
(3.14)
Finally, we show that x n p . In fact, by using (2.4), we obtain that
D f ( p , x n + 1 ) = D f ( p , f ( α n f ( u ) + ( 1 α n ) f ( y n ) ) ) = V f ( p , α n f ( u ) + ( 1 α n ) f ( y n ) ) V f ( p , α n f ( u ) + ( 1 α n ) f ( y n ) α n ( f ( u ) f ( p ) ) ) + α n ( f ( u ) f ( p ) ) , f ( α n f ( u ) + ( 1 α n ) f ( y n ) ) p = V f ( p , α n f ( p ) + ( 1 α n ) f ( y n ) ) + α n f ( u ) f ( p ) , x n + 1 p = D f ( p , f ( α n f ( p ) + ( 1 α n ) f ( y n ) ) ) + α n f ( u ) f ( p ) , x n + 1 p α n D f ( p , p ) + ( 1 α n ) D f ( p , y 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 .
(3.15)

By Lemma 2.9, we can conclude that lim n D f ( p , x n ) = 0 . Therefore, by Lemma 2.6, x n p since f is totally convex on bounded subsets of E.

Case 2. Suppose that there exists a subsequence { D f ( p , n i ) } of { D f ( p , x n ) } such that
D f ( p , x n i ) < D f ( p , x n i + 1 )
for all i N . Then, by Lemma 2.10, there exists a nondecreasing sequence { m k } N such that m k
D f ( p , x m k ) D f ( p , x m k + 1 ) and D f ( p , x k ) D f ( p , x m k + 1 )
for all k N . So, 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 , y n ) D f ( p , x n ) ) lim sup n ( α n D f ( p , u ) + ( 1 α n ) ( β n D f ( p , x n ) + ( 1 β n ) D f ( p , T x n ) ) D f ( p , x n ) ) = lim sup n ( α n D f ( p , u ) + ( 1 α n ) ( 1 β n ) ( D f ( p , T x n ) ) D f ( p , x n ) ) α n ( D f ( p , x n ) ) lim sup n α n ( D f ( p , u ) D f ( p , x n ) ) = 0 .
(3.16)
This implies
lim k ( D f ( p , x m k + 1 ) D f ( p , x m k ) ) = 0 .
(3.17)
Following the proof line in Case 1, we can verify
lim k f ( u ) f ( p ) , x m k + 1 p 0
(3.18)
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 .
Since D f ( p , x m k ) D f ( p , x m k + 1 ) , we have
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 .
In particular, since α m k > 0 , we get
D f ( p , x m k ) f ( u ) f ( p ) , x m k + 1 p .
Hence it follows from (3.18) that lim k D f ( p , x m k ) = 0 . Using this and (3.17) together, we conclude that
lim k sup D f ( p , x k ) lim k D f ( p , x m k + 1 ) = 0 .

This completes the proof. □

Letting β n β gives the following result.

Corollary 3.3 Let E be a real reflexive Banach space E, and let f : E R be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of E. Let T be a Bregman strongly nonexpansive mapping on E such that F ( T ) = F ˆ ( T ) . Suppose that u E and define the sequence { x n } as follows: x 1 E and
x n + 1 = f ( α n f ( u ) + ( 1 α n ) ( β f ( x n ) + ( 1 β ) f ( T x n ) ) )
(3.19)

for all n N , where { α n } is a sequence in ( 0 , 1 ) satisfying conditions (C1) and (C2), and β ( 0 , 1 ) . Then { x n } converges strongly to P F ( T ) f ( u ) .

4 Application to a zero point problem of maximal monotone mappings and equilibrium problems

Let E be a real reflexive Banach space. Let A : E 2 E be a set-valued mapping. The domain of A is denoted by dom A = { x E : A x } , and also the graph of A is denoted by G ( A ) = { ( x , x ) E × E : x A x } . A is said to be monotone if x y , x y 0 for each ( x , x ) , ( y , y ) G ( A ) . It is said to be maximal monotone if its graph is not contained in the graph of any other monotone operator on E. It is known that if A is maximal monotone, then the set A 1 ( 0 ) = { z E : 0 A z } is closed and convex. Now, we apply Theorem 3.2 to the problem of finding x E such that 0 A x , which strongly relates to the convex minimization problems in optimization, economics and applied sciences.

The resolvent of A, denoted by Res A f : E 2 E , is defined as follows [13]:
Res A f ( x ) = ( f + A ) 1 f ( x ) .
It is known that F ( Res A f ) = A 1 ( 0 ) , and Res A f is single-valued and Bregman firmly nonexpansive (see [13]). If f is a Legendre function which is bounded, uniformly Fréchet differentiable on bounded subsets of E, then F ˆ ( Res A f ) = F ( Res A f ) (see [16]). The Yosida approximation A λ : E E , λ > 0 , is also defined by
A λ ( x ) = 1 λ ( f ( x ) f ( Res λ A f ( x ) ) )

for all x E . From Proposition 2.7 in [42], we know that ( Res λ A f ( x ) , A λ ( x ) ) G ( A ) and 0 A x if and only if 0 A λ x for all x E and λ > 0 . Using these facts, we obtain the following result by replacing T = Res λ A f , λ > 0 in Theorem 3.2.

Theorem 4.1 Let C be a nonempty, closed and convex subset of a real reflexive Banach space E, and let f : E R be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of E. Let A : E 2 E be a maximal monotone operator such that A 1 ( 0 ) . Suppose that u E and define the sequence { x n } as follows: x 1 E and
x n + 1 = f ( α n f ( u ) + ( 1 α n ) ( β n f ( x n ) + ( 1 β n ) f ( Res λ A f x n ) ) ) , n 1 ,

where λ > 0 and { α n } and { β n } are sequences in ( 0 , 1 ) satisfying

(C1) lim n α n = 0 ;

(C2) n = 1 α n = ;

(C3) 0 < lim inf n β n lim sup n β n < 1 .

Then { x n } converges strongly to P A 1 ( 0 ) f ( u ) , where P A 1 ( 0 ) f is the Bregman projection of E onto A 1 ( 0 ) .

Let C be a nonempty, closed and convex subset of E, and let Θ : C × C R be a bifunction. Now, we apply Theorem 3.2 to the problem of finding x C such that Θ ( x , y ) 0 for all y C . Such a problem is called an equilibrium problem and the solutions set is denoted by EP ( Θ ) . Numerous problems in economics, physics and applied sciences can be reduced to finding solutions of equilibrium problems.

In order to solve the equilibrium problem, let us assume that a bifunction Θ : C × C R satisfies the following conditions [43]:

( A 1 ) Θ ( x , x ) = 0 , x C .

( A 2 ) Θ is monotone, i.e., Θ ( x , y ) + Θ ( y , x ) 0 , x , y C .

( A 3 ) lim sup t 0 Θ ( x + t ( z x ) , y ) Θ ( x , y ) , x , z , y C .

( A 4 ) The function y Θ ( x , y ) is convex and lower semi-continuous.

The resolvent of a bifunction Θ [44] is the operator Res Θ f : E 2 C defined by
Res Θ f ( x ) = { z C : Θ ( z , y ) + f ( z ) f ( x ) , y z 0 , y C } .

From Lemma 1 in [15], if f : E ( , + ] is a strongly coercive and Gâteaux differentiable function, and Θ satisfies conditions ( A 1 )-( A 4 ), then dom ( Res Θ f ) = E .

The following lemma gives us some characterizations of the resolvent Res Θ f .

Lemma 4.2 [15]

Let E be a real reflexive Banach space, and let C be a nonempty closed convex subset of E. Let f : E ( , + ] be a Legendre function. If the bifunction Θ : C × C R satisfies the conditions ( A 1 )-( A 4 ). Then the following hold:
  1. (i)
    Res Θ f
    is single-valued;
     
  2. (ii)
    Res Θ f
    is a Bregman firmly nonexpansive operator;
     
  3. (iii)
    F ( Res Θ f ) = EP ( Θ )
    ;
     
  4. (iv)
    EP ( Θ )
    is a closed and convex subset of C;
     
  5. (v)
    for all x E and for all p F ( Res Θ f ) , we have
    D f ( p , Res Θ f ( x ) ) + D f ( Res Θ f ( x ) , x ) D f ( p , x ) .
     

In addition, by Reich and Sabach [16], if f is uniformly Fréchet differentiable and bounded on bounded subsets of E, then we have from Lemma 4.2 that F ( Res Θ f ) = F ˆ ( Res Θ f ) = EP ( Θ ) is closed and convex. Also, by replacing T = Res Θ f in Theorem 3.2, we obtain the following result.

Theorem 4.3 Let C be a nonempty, closed and convex subset of a real reflexive Banach space E, and let f : E R be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of E. Let Θ : C × C R be a bifunction which satisfies the conditions ( A 1 )-( A 4 ) such that EP ( Θ ) . Suppose that u E and define the sequence { x n } as follows: x 1 E and
x n + 1 = f ( α n f ( u ) + ( 1 α n ) ( β n f ( x n ) + ( 1 β n ) f ( Res Θ f x n ) ) ) , n 1 ,

where { α n } and { β n } are sequences in ( 0 , 1 ) satisfying

(C1) lim n α n = 0 ;

(C2) n = 1 α n = ;

(C3) 0 < lim inf n β n lim sup n β n < 1 .

Then { x n } converges strongly to P EP ( Θ ) f ( u ) , where P EP ( Θ ) f is the Bregman projection of E onto EP ( Θ ) .

Declarations

Acknowledgements

The authors would like to express their thanks to the referees for their helpful comments and suggestions. This work was supported by the Scientific Research Fund of Sichuan Provincial Education Department (12ZA295), the Scientific Research Fund of Science Technology Department of Sichuan Province (2011JYZ011) and the Natural Science foundation of Yibin university (2011Z08).

Authors’ Affiliations

(1)
Institute of Mathematics, Yibin University
(2)
College of Statistics and Mathematics, Yunnan University of Finance and Economics

References

  1. Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3View ArticleGoogle Scholar
  2. Genel A, Lindenstrauss J: An example concerning fixed points. Isr. J. Math. 1975, 22: 957–962.MathSciNetView ArticleGoogle Scholar
  3. Halpern B: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0View ArticleGoogle Scholar
  4. Lions PL: Approximation de points fixes de contractions. C. R. Math. Acad. Sci. Paris, Sér. A-B 1977, 284: A1357-A1359.Google Scholar
  5. Reich S: Approximating fixed points of nonexpansive mappings. Panam. Math. J. 1994, 4: 23–28.Google Scholar
  6. Wittmann R: Approximation of fixed points of nonexpansive mappings. Arch. Math. 1992, 58: 486–491. 10.1007/BF01190119MathSciNetView ArticleGoogle Scholar
  7. Shioji N, Takahashi W: Strong convergence of approximated sequences for nonexpansive mapping in Banach spaces. Proc. Am. Math. Soc. 1997, 125: 3641–3645. 10.1090/S0002-9939-97-04033-1MathSciNetView ArticleGoogle Scholar
  8. Xu HK: Another control condition in an iterative method for nonexpansive mappings. Bull. Aust. Math. Soc. 2002, 65: 109–113. 10.1017/S0004972700020116View ArticleGoogle Scholar
  9. Cho YJ, Kang SM, Zhou H: Some control conditions on iterative methods. Commun. Appl. Nonlinear Anal. 2005, 12: 27–34.MathSciNetGoogle Scholar
  10. Chidume CE, Chidume CO: Iterative approximation of fixed points of nonexpansive mappings. J. Math. Anal. Appl. 2006, 318: 288–295. 10.1016/j.jmaa.2005.05.023MathSciNetView ArticleGoogle Scholar
  11. Suzuki T: A sufficient and necessary condition for Halpern-type strong convergence to fixed points of nonexpansive mappings. Proc. Am. Math. Soc. 2007, 135: 99–106.View ArticleGoogle Scholar
  12. Saejung S: Halpern’s iteration in Banach spaces. Nonlinear Anal. TMA 2010, 73: 3431–3439. 10.1016/j.na.2010.07.031MathSciNetView ArticleGoogle Scholar
  13. Bauschke HH, Borwein JM, Combettes PL: Bregman monotone optimization algorithms. SIAM J. Control Optim. 2003, 42: 596–636. 10.1137/S0363012902407120MathSciNetView ArticleGoogle Scholar
  14. Butnariu D, Kassay G: A proximal-projection method for finding zeroes of set-valued operators. SIAM J. Control Optim. 2008, 47: 2096–2136. 10.1137/070682071MathSciNetView ArticleGoogle Scholar
  15. Reich S, Sabach S: Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces. Nonlinear Anal. TMA 2010, 73: 122–135. 10.1016/j.na.2010.03.005MathSciNetView ArticleGoogle Scholar
  16. Reich S, Sabach S: Existence and approximation of fixed points of Bregman firmly nonexpansive mappings in reflexive Banach spaces. In Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer, New York; 2011:301–316.View ArticleGoogle Scholar
  17. Rockafellar RT: Convex Analysis. Princeton University Press, Princeton; 1970.View ArticleGoogle Scholar
  18. Hiriart-Urruty J-B, Lemaréchal C Grundlehren der mathematischen Wissenschaften 306. In Convex Analysis and Minimization Algorithms II. Springer, Berlin; 1993.View ArticleGoogle Scholar
  19. Zǎlinescu C: Convex Analysis in General Vector Spaces. World Scientific, River Edge; 2002.View ArticleGoogle Scholar
  20. Asplund E, Rockafellar RT: Gradients of convex functions. Trans. Am. Math. Soc. 1969, 139: 443–467.MathSciNetView ArticleGoogle Scholar
  21. Bonnans JF, Shapiro A: Perturbation Analysis of Optimization Problems. Springer, New York; 2000.View ArticleGoogle Scholar
  22. Reich S, Sabach S: A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces. J. Nonlinear Convex Anal. 2009, 10: 471–485.MathSciNetGoogle Scholar
  23. Bauschke HH, Borwein JM, Combettes PL: Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces. Commun. Contemp. Math. 2001, 3: 615–647. 10.1142/S0219199701000524MathSciNetView ArticleGoogle Scholar
  24. Bauschke HH, Borwein JM: Legendre functions and the method of random Bregman projections. J. Convex Anal. 1997, 4: 27–67.MathSciNetGoogle Scholar
  25. Censor Y, Lent A: An iterative row-action method for interval convex programming. J. Optim. Theory Appl. 1981, 34: 321–353. 10.1007/BF00934676MathSciNetView ArticleGoogle Scholar
  26. Bregman LM: The relaxation method for finding the common point of convex sets and its application to the solution of problems in convex programming. U.S.S.R. Comput. Math. Math. Phys. 1967, 7: 200–217.View ArticleGoogle Scholar
  27. Butnariu D, Resmerita E: Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces. Abstr. Appl. Anal. 2006., 2006: Article ID 84919Google Scholar
  28. Resmerita E: On total convexity, Bregman projections and stability in Banach spaces. J. Convex Anal. 2004, 11: 1–16.MathSciNetGoogle Scholar
  29. Butnariu D, Iusem AN: Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization. Kluwer Academic, Dordrecht; 2000.View ArticleGoogle Scholar
  30. Censor Y, Reich S: Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization. Optimization 1996, 37: 323–339. 10.1080/02331939608844225MathSciNetView ArticleGoogle Scholar
  31. Reich S: A weak convergence theorem for the alternating method with Bregman distances. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Dekker, New York; 1996:313–318.Google Scholar
  32. Bruck RE, Reich S: Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houst. J. Math. 1977, 3: 459–470.MathSciNetGoogle Scholar
  33. Kohsaka F, Takahashi W: Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces. SIAM J. Control Optim. 2008, 19: 824–835. 10.1137/070688717MathSciNetView ArticleGoogle Scholar
  34. Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operator of Accretive and Monotone Type. Edited by: Kartsatos AG. Dekker, New York; 1996:15–50.Google Scholar
  35. Kohsaka F, Takahashi W: Proximal point algorithms with Bregman functions in Banach spaces. J. Nonlinear Convex Anal. 2005, 6: 505–523.MathSciNetGoogle Scholar
  36. Phelps RP Lecture Notes in Mathematics 1364. In Convex Functions, Monotone Operators, and Differentiability. 2nd edition. Springer, Berlin; 1993.Google Scholar
  37. Xu HK: Another control condition in an iterative method for nonexpansive mappings. Bull. Aust. Math. Soc. 2002, 65: 109–113. 10.1017/S0004972700020116View ArticleGoogle Scholar
  38. Mainge PE: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 2008, 16: 899–912. 10.1007/s11228-008-0102-zMathSciNetView ArticleGoogle Scholar
  39. Suantai S, Cho YJ, Cholamjiak P: Halpern’s iteration for Bregman strongly nonexpansive mappings in reflexive Banach spaces. Comput. Math. Appl. 2012, 64: 489–499. 10.1016/j.camwa.2011.12.026MathSciNetView ArticleGoogle Scholar
  40. Rockafellar RT: Level sets and continuity of conjugate convex functions. Trans. Am. Math. Soc. 1966, 123: 46–63. 10.1090/S0002-9947-1966-0192318-XMathSciNetView ArticleGoogle Scholar
  41. Ambrosetti A, Prodi G: A Primer of Nonlinear Analysis. Cambridge University Press, Cambridge; 1993.Google Scholar
  42. Reich S, Sabach S: Two strong convergence theorems for a proximal method in reflexive Banach spaces. Numer. Funct. Anal. Optim. 2010, 31: 22–44. 10.1080/01630560903499852MathSciNetView ArticleGoogle Scholar
  43. Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.MathSciNetGoogle Scholar
  44. Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 2005, 6: 117–136.MathSciNetGoogle Scholar

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© Zhu and Chang; licensee Springer. 2013

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