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

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.

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{xy,y A 1 }. The multi-valued mapping T:DCB(D) is called nonexpansive if H(Tx,Ty)xy for all x,yD. An element pD is called a fixed point of T:DN(D) if pT(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 ,uD,

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 xintdomf. 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 ) ,xX, x X .

Furthermore, equality holds if x f(x) (see [4]). The set lev f (r):={xX:f(x)r} for some rR 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 xintdomf and yX, 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 xintdomf. 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 xD 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 domf;

  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:XR 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 :domf×intdomf[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 xintdomf onto a nonempty, closed and convex set Ddomf 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 xintdomf 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 xintdomf, 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 :intdomf×[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 pD 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:DN(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),xD,p F ˆ (T),zT(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,yD, x Tx, y Ty.

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:CN(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),zTx,

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 2x,Jy+ y 2 .

Let D be a nonempty, closed and convex subset of X. Let f:XR be a Gâteaux differentiable and totally convex function and xX. It is known from [14] that z= P D f (x) if and only if

f ( x ) f ( z ) , y z 0,yD.

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,yD.

Let f:XR 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 ,xX, x X .

Then V f is nonnegative and V f (x, x )= D f (x, f ( x )) for all xX 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 ) ,xX, 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 iN. 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 kN:

α m k α m k + 1 and α k α m k + 1 .

In fact, m k =max{jk: 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:XR be a Gâteaux differentiable and totally convex function, and let T:DN(D) be a multi-valued mapping such that F(T)= F ˆ (T) is nonempty, closed and convex. Suppose that uD 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 vD 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:DN(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:DN(D) be a Bregman strongly nonexpansive multi-valued mapping on X such that F(T)= F ˆ (T). Suppose that uX 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 ,n1,
(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 (Lf(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 )), nN, 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 jN. By Lemma 2.3, there exists a strictly increasing sequence { m k } of positive integers such that the following properties hold for all kN:

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:DN(D) be a strongly relatively nonexpansive multi-valued mapping on X such that F(T)= F ˆ (T). Suppose that uD 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 ,n1,

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×DR be a bifunction satisfying the conditions:

  1. (A1)

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

  2. (A2)

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

  3. (A3)

    for each x,y,zD, lim t 0 G(tz+(1t)x,y)G(x,y);

  4. (A4)

    for each given xD, the function yf(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 yD. 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 } ,xX.
(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 xX and for all pF( 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×DR be a bifunction which satisfies conditions (A1)-(A4) such that EP(G). Suppose that uX 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 ),n1,
(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.

References

  1. Halpern B: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0

    Article  MATH  Google Scholar 

  2. Reich S: Approximating fixed points of nonexpansive mappings. Panam. Math. J. 1994, 4: 23–28.

    MATH  Google Scholar 

  3. Suthep S, Yeol JC, Prasit C: Halpern’s iteration for Bregman strongly nonexpansive mappings in reflexive Banach spaces. Comput. Math. Appl. 2012, 64: 489–499. 10.1016/j.camwa.2011.12.026

    Article  MATH  MathSciNet  Google Scholar 

  4. Rockafellar RT: Level sets and continuity of conjugate convex functions. Trans. Am. Math. Soc. 1966, 123: 46–63. 10.1090/S0002-9947-1966-0192318-X

    Article  MATH  MathSciNet  Google Scholar 

  5. Hiriart-Urruty JB, Lemaréchal C Grundlehren der mathematischen Wissenschaften 306. In Convex Analysis and Minimization Algorithms II: Advanced Theory and Bundle Methods. Springer, Berlin; 1993.

    Chapter  Google Scholar 

  6. Zǎlinescu C: Convex Analysis in General Vector Spaces. World Scientific, River Edge; 2002.

    Book  Google Scholar 

  7. Asplund E, Rockafellar RT: Gradients of convex functions. Trans. Am. Math. Soc. 1969, 139: 443–467.

    Article  MATH  MathSciNet  Google Scholar 

  8. Bonnans JF, Shapiro A: Perturbation Analysis of Optimization Problems. Springer, New York; 2000.

    Book  MATH  Google Scholar 

  9. 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/S0219199701000524

    Article  MATH  MathSciNet  Google Scholar 

  10. Bauschke HH, Borwein JM, Combettes PL: Bregman monotone optimization algorithms. SIAM J. Control Optim. 2003, 42: 596–636. 10.1137/S0363012902407120

    Article  MATH  MathSciNet  Google Scholar 

  11. Bauschke HH, Borwein JM: Legendre functions and the method of random Bregman projections. J. Convex Anal. 1997, 4: 27–67.

    MATH  MathSciNet  Google Scholar 

  12. 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.

    MATH  MathSciNet  Google Scholar 

  13. 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.

    Article  Google Scholar 

  14. 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 84919

    Google Scholar 

  15. Butnariu D, Iusem AN: Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization. Kluwer Academic, Dordrecht; 2000.

    Book  MATH  Google 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.

    Google Scholar 

  17. 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.

    Google Scholar 

  18. Reich S, Sabach S: Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces. Nonlinear Anal. 2010, 73: 122–135. 10.1016/j.na.2010.03.005

    Article  MATH  MathSciNet  Google Scholar 

  19. Borwein JM, Reich S, Sabach S: A characterization of Bregman firmly nonexpansive operators using a new monotonicity concept. J. Nonlinear Convex Anal. 2011, 12: 161–184.

    MATH  MathSciNet  Google Scholar 

  20. Martín-Márquez V, Reich S, Sabach S: Right Bregman nonexpansive operators in Banach spaces. Nonlinear Anal. 2012, 75: 5448–5465. 10.1016/j.na.2012.04.048

    Article  MATH  MathSciNet  Google Scholar 

  21. Martín-Márquez V, Reich S, Sabach S: Bregman strongly nonexpansive operators in reflexive Banach spaces. J. Math. Anal. Appl. 2013, 400: 597–614. 10.1016/j.jmaa.2012.11.059

    Article  MATH  MathSciNet  Google Scholar 

  22. Martín-Márquez V, Reich S, Sabach S: Iterative methods for approximating fixed points of Bregman nonexpansive operators. Discrete Contin. Dyn. Syst. 2013, 6: 1043–1063.

    Article  MATH  Google Scholar 

  23. 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.

    Google Scholar 

  24. Censor Y, Lent A: An iterative row-action method for interval convex programming. J. Optim. Theory Appl. 1981, 34: 321–353. 10.1007/BF00934676

    Article  MATH  MathSciNet  Google Scholar 

  25. Phelps RP Lecture Notes in Mathematics. In Convex Functions, Monotone Operators, and Differentiability. 2nd edition. Springer, Berlin; 1993.

    Google Scholar 

  26. Kohsaka F, Takahashi W: Proximal point algorithms with Bregman functions in Banach spaces. J. Nonlinear Convex Anal. 2005, 6: 505–523.

    MATH  MathSciNet  Google Scholar 

  27. Maingé PE: The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces. Comput. Math. Appl. 2010, 59: 74–79. 10.1016/j.camwa.2009.09.003

    Article  MATH  MathSciNet  Google Scholar 

  28. Ambrosetti A, Prodi G: A Primer of Nonlinear Analysis. Cambridge University Press, Cambridge; 1993.

    Google Scholar 

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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).

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All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

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Li, Y., Liu, H. & Zheng, K. Halpern’s iteration for Bregman strongly nonexpansive multi-valued mappings in reflexive Banach spaces with application. Fixed Point Theory Appl 2013, 197 (2013). https://doi.org/10.1186/1687-1812-2013-197

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