Halpern’s iteration for Bregman strongly nonexpansive multi-valued mappings in reflexive Banach spaces with application
© Li et al.; licensee Springer 2013
Received: 20 December 2012
Accepted: 5 July 2013
Published: 22 July 2013
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.
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 .
where . Such a method was introduced in 1967 by Halpern  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  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 .
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.
Furthermore, equality holds if (see ). The set for some is called a sublevel of f.
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  and ).
Definition 2.1 (cf. )
essentially smooth if ∂f is both locally bounded and single-valued on its domain;
essentially strictly convex if is locally bounded on its domain and f is strictly convex on every convex subset of ;
Legendre if it is both essentially smooth and essentially strictly convex.
Remark 2.1 (cf. )
f is essentially smooth if and only if is essentially strictly convex;
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 . 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 .
Lemma 2.1 (cf. )
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 .
is called the Bregman distance with respect to f.
We know that f is totally convex on bounded sets if and only if f is uniformly convex on bounded sets (see ).
The following crucial lemma was proved by Butnariu and Iusem .
Lemma 2.2 (cf. )
The function f is totally convex on bounded sets if and only if it is sequentially consistent.
Definition 2.2 (cf. )
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.
- (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 . 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 ). 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 . For a wealth of results concerning this class of mappings, see [18–22] and the references therein.
and if, whenever is bounded, and , then , where and .
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. , Lemma 2.1, p.76)
In fact, .
Lemma 2.4 (see , Lemma 2.5, p.493)
3 Main results
To prove our main result, we first prove the following two propositions.
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 (, 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 .
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.
where satisfying and . Then strongly converges to .
By the induction, the sequence is bounded.
Hence is contained in the sublevel set , where . Since f is lower semicontinuous, is weak∗ lower semicontinuous. Hence the function ψ is coercive (see ). This shows that is bounded. Since f is strongly coercive, is bounded on bounded sets (see ). Hence is also bounded on bounded subsets of (see ). Since f is a Legendre function, it follows that , , is bounded. Therefore is bounded. So are and .
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 .
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 .
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.
where satisfying and . Then converges strongly to , where is the generalized projection onto .
In this section, we give an application of Theorem 3.1, which is the equilibrium problems in the framework of reflexive Banach spaces.
for all ;
for any ;
for each , ;
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 .
is a Bregman firmly nonexpansive mapping;
is a closed and convex subset of D;
- (5)for all and for all , we have(4.2)
In addition, by Reich and Sabach , 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.
where satisfying and . Then converges strongly to .
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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