- Research
- Open access
- Published:
Halpern’s iteration for Bregman strongly nonexpansive multi-valued mappings in reflexive Banach spaces with application
Fixed Point Theory and Applications volume 2013, Article number: 197 (2013)
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 and denote the family of nonempty subsets and nonempty, closed and bounded subsets of D, respectively. The Hausdorff metric on is defined by
for all , where . The multi-valued mapping is called nonexpansive if for all . An element is called a fixed point of if . The set of fixed points of T is denoted by .
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
where . 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 to a fixed point of the nonexpansive mapping T, where , .
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 be the dual space of X. Let be a proper, lower semi-continuous and convex function. We denote by domf the domain of f.
Let . The subdifferential of f at x is the convex set defined by
The Fenchel conjugate of f is the function defined by
We know that the Young-Fenchel inequality holds, that is,
Furthermore, equality holds if (see [4]). The set for some is called a sublevel of f.
A function f on X is called coercive [5] if the sublevel sets of f are bounded, equivalently,
A function f on X is said to be strongly coercive [6] if
For any and , the right-hand derivative of f at x in the direction y is defined by
The function f is said to be Gâteaux differentiable at x if exists for any y. In this case, coincides with , the value of the gradient of f at x. The function f is said to be Gâteaux differentiable if it is Gâteaux differentiable for any . The function f is said to be Fréchet differentiable at x if this limit is attained uniformly in . Finally, f is said to be uniformly Fréchet differentiable on a subset D of X if the limit is attained uniformly for and . 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
-
(i)
essentially smooth if ∂f is both locally bounded and single-valued on its domain;
-
(ii)
essentially strictly convex if is locally bounded on its domain and f is strictly convex on every convex subset of ;
-
(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
-
(a)
f is essentially smooth if and only if is essentially strictly convex;
-
(b)
;
-
(c)
f is Legendre if and only if is Legendre;
-
(d)
If f is Legendre, then ∂f is a bijection which satisfies
Examples of Legendre functions can be found in [11]. One important and interesting Legendre function is () 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., . In particular, 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 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 .
Let be a convex and Gâteaux differentiable function. The function , defined by
is called the Bregman distance with respect to f.
Recall that the Bregman projection [13] of onto a nonempty, closed and convex set is the necessarily unique vector (for convenience, here we use for ) satisfying
The modulus of total convexity of f at is the function defined by
The function f is called totally convex at x if whenever . The function f is called totally convex if it is totally convex at any point , and it is said to be totally convex on bounded sets if for any nonempty bounded subset B and , where the modulus of total convexity of the function f on the set B is the function defined by
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 and in X, such that the first sequence is bounded, the following implication holds:
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 is called an asymptotic fixed point of T if D contains a sequence , which converges weakly to p, such that (as ).
We denote by the set of asymptotic fixed points of T.
Definition 2.3 A multi-valued mapping with a nonempty fixed point set is said to be:
-
(i)
Bregman strongly nonexpansive with respect to a nonempty if
and if, whenever is bounded, and , then , where .
-
(ii)
Bregman firmly nonexpansive if
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 and is closed and convex (see [16]). It also follows that every Bregman firmly nonexpansive mapping is Bregman strongly nonexpansive with respect to . 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 [18–22] 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 is called a strongly relatively nonexpansive multi-valued mapping on X if and
and if, whenever is bounded, and , then , where and .
Let D be a nonempty, closed and convex subset of X. Let be a Gâteaux differentiable and totally convex function and . It is known from [14] that if and only if
We also know the following characterization:
Let be a convex, Legendre and Gâteaux differentiable function. Following [23] and [24], we make use of the function associated with f, which is defined by
Then is nonnegative and for all and . Moreover, by the subdifferential inequality (see [22], Proposition 1(iii), p.1047),
In addition, if is a proper and lower semi-continuous function, then is a proper, weak∗ lower semi-continuous and convex function (see [25]). Hence is convex in the second variable (see [22], Proposition 1(i), p.1047). Thus,
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 be a sequence of real numbers such that there exists a nondecreasing subsequence of , that is, for all . Then there exists a nondecreasing subsequence such that , and the following properties are satisfied for all sufficiently large numbers sequence :
In fact, .
Lemma 2.4 (see [3], Lemma 2.5, p.493)
Assume that is a sequence of nonnegative real numbers such that
where is a sequence in and is a sequence such that
-
(a)
, ;
-
(b)
.
Then .
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 be a Gâteaux differentiable and totally convex function, and let be a multi-valued mapping such that is nonempty, closed and convex. Suppose that and is a bounded sequence in D such that . Then
Proof Since X is reflexive and is bounded, there exists a subsequence such that as and
On the other hand, since , then . By the definition of Bregman projection, we have
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 be a Legendre function and let D be a nonempty, closed and convex subset of intdomf. Let be a Bregman firmly nonexpansive multi-valued mapping with respect to f. Then is closed and convex.
Theorem 3.1 Let X be a real reflexive Banach space and let 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 be a Bregman strongly nonexpansive multi-valued mapping on X such that . Suppose that and define the sequence by
where satisfying and . Then strongly converges to .
Proof First, by Proposition 3.2, we know that is closed and convex. Letting , we have
By the induction, the sequence is bounded.
Next, we show that the sequence is also bounded. We follow the proof as in [16]. Since is bounded, there exists such that
Hence is contained in the sublevel set , where . Since f is lower semicontinuous, is weak∗ lower semicontinuous. Hence the function ψ is coercive (see [4]). This shows that is bounded. Since f is strongly coercive, is bounded on bounded sets (see [9]). Hence is also bounded on bounded subsets of (see [15]). Since f is a Legendre function, it follows that , , is bounded. Therefore is bounded. So are and .
We next show that if there exists a subsequence such that
then
where .
Since is bounded, we have from (3.2)
Since f is strongly coercive and uniformly convex on bounded subsets of X, is uniformly Fréchet differentiable on bounded subsets of (see [6]). Moreover, is bounded on bounded sets. Since f is Legendre, by Lemma 2.1, we obtain
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
Since the following equality holds:
it follows from (3.3), (3.4) and (3.5) that
The rest of the proof will be divided into two parts.
Case 1. Suppose that is eventually decreasing, i.e., there exists a sufficiently large such that for all . In this case, exists. In this situation, we have that exists. This shows that and hence .
Since T is a Bregman strongly nonexpansive multi-valued mapping, then
Since f is totally convex on bounded subsets of E, by Lemma 2.2, we have
By Proposition 3.1, we obtain
Finally, we show that as . Indeed
By Lemma 2.4, we conclude that . Therefore, by Lemma 2.2, since f is totally convex on bounded subsets of X, we obtain that as .
Case 2. If is not eventually decreasing, there exists a subsequence such that for all . By Lemma 2.3, there exists a strictly increasing sequence of positive integers such that the following properties hold for all :
Since the inequality holds by Definition 2.3, hence, by Lemma 2.3, we have
This implies that
Following the proof of Case 1, we have
and
This implies that
Hence
Using this and (3.5) together, we conclude that
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 be a strongly relatively nonexpansive multi-valued mapping on X such that . Suppose that and define the sequence as follows: and
where satisfying and . Then converges strongly to , where is the generalized projection onto .
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 be a bifunction satisfying the conditions:
-
(A1)
for all ;
-
(A2)
for any ;
-
(A3)
for each , ;
-
(A4)
for each given , the function is convex and lower semicontinuous.
The so-called equilibrium problem for G is to find a such that for each . The set of its solutions is denoted by .
The resolvent of a bifunction G [18] is the operator defined by
If is a strongly coercive and Gâteaux differentiable function, and G satisfies conditions (A1)-(A4), then (see [18]). We also know:
-
(1)
is single-valued;
-
(2)
is a Bregman firmly nonexpansive mapping;
-
(3)
;
-
(4)
is a closed and convex subset of D;
-
(5)
for all and for all , we have
(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 is closed and convex. Hence, by replacing 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 be a bifunction which satisfies conditions (A1)-(A4) such that . Suppose that and define the sequence by
where satisfying and . Then converges strongly to .
References
Halpern B: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0
Reich S: Approximating fixed points of nonexpansive mappings. Panam. Math. J. 1994, 4: 23–28.
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
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
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.
Zǎlinescu C: Convex Analysis in General Vector Spaces. World Scientific, River Edge; 2002.
Asplund E, Rockafellar RT: Gradients of convex functions. Trans. Am. Math. Soc. 1969, 139: 443–467.
Bonnans JF, Shapiro A: Perturbation Analysis of Optimization Problems. Springer, New York; 2000.
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
Bauschke HH, Borwein JM, Combettes PL: Bregman monotone optimization algorithms. SIAM J. Control Optim. 2003, 42: 596–636. 10.1137/S0363012902407120
Bauschke HH, Borwein JM: Legendre functions and the method of random Bregman projections. J. Convex Anal. 1997, 4: 27–67.
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.
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.
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
Butnariu D, Iusem AN: Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization. Kluwer Academic, Dordrecht; 2000.
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.
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.
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
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.
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
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
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.
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.
Censor Y, Lent A: An iterative row-action method for interval convex programming. J. Optim. Theory Appl. 1981, 34: 321–353. 10.1007/BF00934676
Phelps RP Lecture Notes in Mathematics. In Convex Functions, Monotone Operators, and Differentiability. 2nd edition. Springer, Berlin; 1993.
Kohsaka F, Takahashi W: Proximal point algorithms with Bregman functions in Banach spaces. J. Nonlinear Convex Anal. 2005, 6: 505–523.
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
Ambrosetti A, Prodi G: A Primer of Nonlinear Analysis. Cambridge University Press, Cambridge; 1993.
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).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
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.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2013-197