## Fixed-Point Theory, Variational Inequalities, and Its Approximation Algorithms

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# Approximation of Fixed Points of Weak Bregman Relatively Nonexpansive Mappings in Banach Spaces

**Academic Editor:**Yonghong Yao

#### Abstract

We introduce a concept of weak Bregman relatively nonexpansive mapping which is distinct from Bregman relatively nonexpansive mapping. By using projection techniques, we construct several modification of Mann type iterative algorithms with errors and Halpern-type iterative algorithms with errors to find fixed points of weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings in Banach spaces. The strong convergence theorems for weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings are derived under some suitable assumptions. The main results in this paper develop, extend, and improve the corresponding results of Matsushita and Takahashi (2005) and Qin and Su (2007).

#### 1. Introduction

Throughout this paper, without other specifications, we denote by the set of real numbers. Let be a real reflexive Banach space with the dual space . The norm and the dual pair between and are denoted by and , respectively. Let be proper convex and lower semicontinuous. The Fenchel conjugate of is the function defined by
We denote by the domain of , that is, . Let be a nonempty closed and convex subset of a nonlinear mapping. Denote by , the set of fixed points of . is said to be * nonexpansive* if for all .

In 1967, Brègman [1] discovered an elegant and effective technique for the using of the so-called Bregman distance function (see, Section 2, Definition 2.1) in the process of designing and analyzing feasibility and optimization algorithms. This opened a growing area of research in which Bregman's technique is applied in various ways in order to design and analyze iterative algorithms for solving not only feasibility and optimization problems, but also algorithms for solving variational inequalities, for approximating equilibria, for computing fixed points of nonlinear mappings, and so on (see, e.g., [1–25], and the references therein).

Nakajo and Takahashi [26] introduced the following modification of the Mann iteration method for a nonexpansive mapping in a Hilbert space as follows: where and is the metric projection from onto a closed and convex subset of . They proved that generated by (1.2) converges strongly to a fixed point of under some suitable assumptions. Motivated by Nakajo and Takahashi [26], Matsushita and Takahashi [27] introduced the following modification of the Mann iteration method for a relatively nonexpansive mapping in a Banach space as follows: where , for all , is the duality mapping of and is the generalized projection (see, e.g., [2, 3, 28]) from onto a closed and convex subset of . They also proved that generated by (1.3) converges strongly to a fixed point of under some suitable assumptions. Martinez-Yanes and Xu [29] gave a Halpern-type iterative algorithm for a nonexpansive mapping as follows: where . They derived that generated by (1.3) converges strongly to a fixed point of under some suitable assumptions. Qin and Su [30] generalized the results of Martinez-Yanes and Xu [29] to a uniformly convex and uniformly smooth Banach space for a relatively nonexpansive mapping and proposed the following iterative algorithm: where , is the generalized projection (see, e.g., [2, 3, 28]) from onto a closed and convex subset of . They also obtained that generated by (1.5) converges strongly to a fixed point of under some suitable assumptions. In 2003, Butnariu et al. [13] studied several notions of convex analysis: uniformly convexity at a point, total convexity at a point, uniformly convexity on bounded sets, and sequential consistency, which are useful in establishing convergence properties for fixed point and optimization algorithms in infinite dimensional Banach spaces. They established connections between these concepts and used these relations in order to obtain improved convergence results concerning the outer Bregman projection algorithm for solving convex feasibility problems and the generalized proximal point algorithm for optimization in Banach spaces. In 2006, Butnariu and Resmerita [14] presented a Bregman-type iterative algorithms and studied the convergence of the Bregman-type iterative method of solving operator equations. Resmerita [19] investigated the existence of totally convex functions in Banach spaces and, further, established continuity and stability properties of Bregman projections. Very recently, by using Bregman projection, Reich and Sabach [21] presented the following algorithms for finding common zeroes of maximal monotone operators in reflexive Banach space as follows: Further, under some suitable conditions, they obtained two strong convergence theorems of maximal monotone operators in reflexive Banach spaces. Reich and Sabach [22] studied the convergence of two iterative algorithms for finitely many Bregman strongly nonexpansive operators in Banach spaces and obtained two strong convergence theorems for finitely many Bregman strongly nonexpansive operators under some assumptions. In [24], Reich and Sabach proposed the following algorithms for finding common fixed points of finitely many Bregman firmly nonexpansive operators in reflexive Banach space as follows: if Under some suitable conditions, they proved that the sequence generated by (1.8) converges strongly to and applied it to the solution of convex feasibility and equilibrium problems.

Inspired and motivated by the works, we introduce the concept of weak Bregman relatively nonexpansive mappings in reflexive Banach space and give an example to illustrate the existence of weak Bregman relatively nonexpansive mapping and the difference between weak Bregman relatively nonexpansive mapping and Bregman relatively nonexpansive mapping. Secondly, by using the conception of the Bregman projection (see, e.g., [1, 13, 14]), we construct several modification of Mann type iterative algorithms with errors and Halpern-type iterative algorithms with errors to find fixed points of weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings in Banach spaces. The strong convergence theorems for weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings are derived under some suitable assumptions. Moreover, the convergence rate of our algorithms is faster than that of Matsushita and Takahashi [27] and Qin and Su [30]. The main results in this paper develop, extend, and improve the corresponding results in the literature.

#### 2. Preliminaries

Let be a nonempty closed convex subset of a real reflexive Banach space , and let be a nonlinear mapping. A point is called an * asymptotic fixed point* of (see, e.g., [2, 3]) if contains a sequence which converges weakly to such that . A point is called an *strong asymptotic fixed point* of (see, e.g., [2, 3]) if contains a sequence which converges strongly to such that . We denote the sets of asymptotic fixed points and strong asymptotic fixed points of by and , respectively. When is a sequence in , we denote strong convergence of to by . For any and , the *right-hand derivative* of at in the direction defined by is called *Gâteaux differentiable* at if, for all exists. In this case, coincides with , the value of the gradient of at . is called *Gâteaux differentiable* if it is Gâteaux differentiable for any . is called *Fréchet differentiable* at if this limit is attained uniformly for . We say is * uniformly Fréchet differentiable* on a subset of if the limit is attained uniformly for and .

Legendre function is defined in [7]. From [7], if is a reflexive Banach space, then is Legendre if and only if it satisfies the following conditions (*L*1) and (*L*2):(*L*1) the interior of the domain of , , is nonempty, is Gâteaux differentiable on , and ,(*L*2) the interior of the domain of , , is nonempty, is Gâteaux differentiable on , and .

Since is reflexive, we know that (see, e.g., [31]). This, by (*L*1) and (*L*2), implies
By Theorem 5.4 [7], conditions (*L*1) and (*L*2) also yield that the functions and are strictly convex on the interior of their respective domains. From now on, we assume that the convex function is Legendre.

We first recall some definitions and lemmas which are needed in our main results.

*Definition 2.1 (see [1, 13]). * Let be a Gâteaux differentiable and convex function. The function , defined by
is called *the Bregman distance* with respect to .

*Remark 2.2 (see [24]). * The Bregman distance has the following properties:(i)the three point identity, for any and ,(ii) the four point identity, for any and ,

*Definition 2.3 (see [1]). * Let be a Gâteaux differentiable and convex function. The Bregman projection of onto the nonempty closed and convex set is the necessarily unique vector satisfying

*Remark 2.4. *(i) If is a Hilbert space and for all , then the Bregman projection is reduced to the metric projection of onto .

(ii) If is a smooth Banach space and for all , then the Bregman projection is reduced to the generalized projection (see, e.g. [3]) which defined by
where is the normalized duality mapping from to .

*Definition 2.5 (see[12, 21]). * Let be a nonempty closed and convex set of . The operator with is called: (i)*quasi-Bregman nonexpansive * if (ii)* Bregman relatively nonexpansive* if
and ,(iii)* Bregman firmly nonexpansive* if
or equivalently

*Definition 2.6. *Let be a nonempty closed and convex set of . The operator with is called *weak Bregman relatively nonexpansive* if and

*Remark 2.7. *It is easy to see that each nonexpansive mapping is quasi-Bregman nonexpansive mapping with respect to for all . Moreover, every relatively nonexpansive mapping also is Bregman relatively nonexpansive mapping, where is called relatively nonexpansive mapping (see, e.g., [32]) if the following conditions are satisfied:

Now, we give an example which is weak Bregman relatively nonexpansive mapping but not Bregman relatively nonexpansive mapping.

*Example 2.8. *Let for all , where
and for any . It is well known that is a Hilbert space. Let be a sequence defined by , , where
for all .

Define a mapping by
for all . It is easy to see that , and so, converges weakly to . Indeed, for any , we have
From , it shows that . Moreover,
Next, for any , one has ; that is, is not a Cauchy sequence. Owing to , we obtain
Then, is an asymptotic fixed point of , but . So, is not Bregman relatively nonexpansive mapping.

For any strong convergent sequence such that and as . Then, there exists a sufficiently large nature number such that for any . Thus, for , which implies that and as . That is, is a strong asymptotic fixed point of , and so, . Since
Therefore, is a weak Bregman relatively nonexpansive mapping.

*Definition 2.9 (see [12]). * Let be a convex and Gâteaux differentiable function. is called: (i)*totally convex at * if its modulus of total convexity at ; that is, the function defined by
is positive whenever ,(ii)*totally convex* if, it is totally convex at every point ,(iii)*totally convex on bounded sets* if is positive for any nonempty bounded subset of and , where the modulus of total convexity of the function on the set is the function defined by

*Definition 2.10 (see [12, 21]). * The function is called:(i)*cofinite *if ,(ii)*sequentially consistent* if, for any two sequences and in such that the first is bounded, and

Lemma 2.11 (see [21, Proposition 2.3]). * If is Fréchet differentiable and totally convex, then is cofinite. *

Lemma 2.12 (see [14, Theorem 2.10]). * Let be a convex function whose domain contains at least two points. Then, the following statements hold: *(i)* is sequentially consistent if and only if it is totally convex on bounded sets, *(ii)*if is lower semicontinuous, then is sequentially consistent if and only if it is uniformly convex on bounded sets, *(iii)*if is uniformly strictly convex on bounded sets, then it is sequentially consistent and the converse implication holds when is lower semicontinuous, Fréchet differentiable on its domain, and the Fréchet derivative is uniformly continuous on bounded sets.*

Lemma 2.13 (see [20, Proposition 2.1]). * Let be a uniformly Fréchet differentiable and bounded on bounded subsets of . Then, is uniformly continuous on bounded subsets of from the strong topology of to the strong topology of .*

Lemma 2.14 (see [21, Lemma 3.1]). * Let be a Gâteaux differentiable and totally convex function. If and the sequence is bounded, then the sequence is also bounded. *

Lemma 2.15 (see [21, Proposition 2.2]). * Let be a Gâteaux differentiable and totally convex function, , and let be a nonempty closed convex subset of . Suppose that the sequence is bounded and any weak subsequential limit of belongs to . If for any , then converges strongly to .*

In [23], Reich and Sabach proved the following result.

Lemma 2.16 (see[23, Lemma 15.5]). *Let be a Legendre function. Let be a nonempty closed convex subset of and a Bregman firmly nonexpansive mapping with respect to . Then, is closed and convex.*

Motivated by Lemma 2.16, we get the similar result for quasi-Bregman nonexpansive mapping.

Proposition 2.17. *Let be a Legendre function. Let be a nonempty closed convex subset of and a quasi-Bregman nonexpansive mapping with respect to . Then, is closed and convex.*

*Proof. *Without loss of generality, set is nonempty. Firstly, we show that is closed. Let be a sequence in such that . By the definition of quasi-Bregman nonexpansive mapping, we have
Since is a Legendre function, is continuous at . Then, from the definition of Bregman distance,
From (2.24) and (2.25), it follows that , and so, from [7, Lemma 7.3(vi), page 642], . Therefore, , and so, is closed.

We now show that is convex. For any and , it yields that . From the definition of quasi-Bregman nonexpansive mapping, it follows that
Again, from [7, Lemma 7.3(vi), page 642], we get . Therefore, is convex. This completes the proof.

From the definitions of Bregman distance and the Fenchel conjugate of , we have the following result.

Lemma 2.18. *Let be a Gâteaux differentiable and proper convex lower semicontinuous. Then, for all ,
**
where and with .*

Lemma 2.19 (see [14, Corollary 4.4]). *Let be a Gâteaux differentiable and totally convex on . Let and a nonempty closed convex set. If , then the following statements are equivalent:*(i)*the vector is the Bregman projection of onto with respect to ,*(ii)* the vector is the unique solution of the variational inequality *(iii)* the vector is the unique solution of the inequality *

#### 3. Main Results

In this section, we introduce several modification of Mann-type iterative algorithms with errors and Halpern-type iterative algorithms with errors to find fixed points of weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings in Banach spaces. The strong convergence theorems for weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings are proved under some suitable conditions.

Theorem 3.1. *Let be a nonempty closed convex subset of a real reflexive Banach space and a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subset of , and let be a weak Bregman relatively nonexpansive mapping such that . Define a sequence in by the following algorithm:
**
where such that , and is an error sequence in with as . Then, the sequences and converge strongly to the point , where is the Bregman projection of onto .*

*Proof. *By Proposition 2.17, it follows that is a nonempty closed and convex subset of . It is easy to verify that , and are closed and convex. Suppose that and are closed and convex. Then, is closed and convex. For any ,
which implies that and are closed and convex. As a consequence, and are closed and convex for all . Taking arbitrarily,
that is, , and so, for all . We now show that for all . Clearly, . Assume that for all . Note that , and we have
Therefore,
which yields that . Then, for all . Consequently, and is nonempty closed and convex for all . Moreover, is well defined.

Secondly, we show that is a Cauchy sequence and bounded. Since
it follows that . Therefore, by ,
Taking arbitrarily. From Lemma 2.19, it yields that
Moreover, one has
Hence, is bounded and so , , and are also bounded. From (3.7), it shows that exists. In the light of for any , by Lemma 2.19,
that is,
Consequently, one has
Since is totally convex on bounded subsets of , by Lemma 2.12 and (3.12), we have
Thus, is a Cauchy sequence, and so,
Since as , one has
Let . Then, .

Thirdly, we show that converges strongly to a point of . Since is uniformly Fréchet differentiable on bounded subsets of , from Lemma 2.12, is norm-to-norm uniformly continuous on bounded subsets of . So, by (3.15),
It follows from that
By the uniformly Fréchet differentiable of on bounded subsets of , is also uniformly continuous on bounded subsets of . Hence, from (3.12) and ,
As a consequence, and so, . Moreover, one has
Since and as . Noticing that
Therefore,
In view of and from both (3.16) and (3.19), one has
Furthermore, we have
and so, by (3.14),
Since and , we get .

Finally, we show . Since , it follows from that . By Lemma 2.15, as . Therefore, and converge strongly to . This completes the proof.

Theorem 3.2. *Let be a nonempty closed convex subset of a real reflexive Banach space and a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subset of , and let be a Bregman relatively nonexpansive mapping such that . Assume that such that , and is an error sequence in with as . Then, the sequences and generated by (3.1) converge strongly to the point , where is the Bregman projection of onto .*

*Proof. *As in the proof of Theorem 3.1, we know that the sequences and converge strongly to , and so,
Then, for any subsequence of converges weakly to ,
Therefore, . By the similar proof of Theorem 3.2, the sequences and converge strongly to . This completes the proof.

If , , and for all , , then from Remark 2.4 and Theorem 3.1, we have the following result.

Corollary 3.3. *Let be a nonempty closed convex subset of a real reflexive, smooth, and strictly convex Banach space , and let be a relatively nonexpansive mapping such that . Define a sequence in by the following algorithm:
**
where is the duality mapping on , such that . Then, the sequences and converge strongly to the point , where is the generalized projection (see, e.g., [2, 3, 28]) of onto . **In [27], Matsushita and Takahashi proved the following result.*

Theorem MT (see [27, Theorem 3.1]). *Let be a nonempty closed convex subset of a real uniformly convex and uniformly smooth Banach space , and let be a relatively nonexpansive mapping such that . Assume that is a sequence of real numbers such that and . Then, the sequence generated by (1.3) converges strongly to the point , where is the generalized projection (see, e.g., [2, 3, 28]) of onto .*

*Remark 3.4. *Corollary 3.3 extends Theorem MT [27] from uniformly convex and uniformly smooth Banach spaces to reflexive, smooth, and strictly convex Banach space.

Now, we investigate convergence theorems for Halpern-type iterative algorithms with errors.

Theorem 3.5. *Let be a nonempty closed convex subset of a real reflexive Banach space and a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subset of , and let be a weak Bregman relatively nonexpansive mapping such that . Define a sequence in by the following algorithm:
**
where such that and , and is an error sequence in with as . Then, the sequences and converge strongly to the point , where is the Bregman projection of onto .*

*Proof. *By Proposition 2.17, it follows that is a nonempty closed and convex subset of . It is easy to see that is closed and is closed and convex for all . For any , ,