Research Article | Open Access

# Strong Convergence Theorems for Quasi-Bregman Nonexpansive Mappings in Reflexive Banach Spaces

**Academic Editor:**Giuseppe Marino

#### Abstract

We study a strong convergence for a common fixed point of a finite family of quasi-Bregman nonexpansive mappings in the framework of real reflexive Banach spaces. As a consequence, convergence for a common fixed point of a finite family of Bergman relatively nonexpansive mappings is discussed. Furthermore, we apply our method to prove strong convergence theorems of iterative algorithms for finding a common solution of a finite family equilibrium problem and a common zero of a finite family of maximal monotone mappings. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

#### 1. Introduction

Throughout this paper, is a real reflexive Banach space with the dual space . The norm and the dual pair between and are denoted by and , respectively. We also assume that is a proper, lower semicontinuous, and convex function. Denote the domain of by ; that is, . Let . The subdifferential of at is the convex set defined by
The* Fenchel conjugate* of is the function defined by .

Let and . The right-hand derivative of at in the direction of is defined by . The function is called* GĂ˘teaux differentiable* at if exists for any and hence coincides with , the value of the gradient of at . The function is said to be* GĂ˘teaux differentiable* if it is GĂ˘teaux differentiable for any . Furthermore, is said to be* FrĂ©chet differentiable *at if this limit is attained uniformly in and it is called* uniformly FrĂ©chet differentiable* on a subset of if the limit is attained uniformly for and .

Let be a GĂ˘teaux differentiable function. The function defined by
is called the* Bregman distance *with respect to [1].

When is a smooth Banach space, setting for all , we have , for all , where is the normalized duality mapping from into , and hence reduces to , for all , which is the Lyapunov function introduced by Alber [2]. If , a Hilbert space, is identity mapping and hence becomes , for all , .

Let be a nonempty and convex subset of and let be a mapping. is said to be* nonexpansive* if for all , and is said to be* quasinonexpansive* if and , for all and , where stands for the fixed point set of ; that is, . A point is said to be an asymptotic fixed point of (see [3]) if contains a sequence which converges weakly to such that . The set of asymptotic fixed points of is denoted by .

A mapping with is called(i)*quasi-Bregman nonexpansive* [4] if
(ii)*Bregman relatively nonexpansive* [4] if
(iii)*Bregman firmly nonexpansive* [5] if, for all ,
or, equivalently,
Iterative methods for approximating fixed points of nonexpansive, quasinonexpansive mappings and their generalizations have been studied by various authors (see, e.g., [6â€“16] and the references therein) in Hilbert spaces. But extending this theory to Banach spaces encountered some difficulties because the useful examples of nonexpansive operators in Hilbert spaces are no longer nonexpansive in Banach spaces (e.g., the resolvent , for , of a monotone mapping and the metric projection onto a nonempty, closed, and convex subset of ). To overcome these difficulties, Bregman [1] discovered techniques with the use of Bregman distance function instead of norm in the process of designing and analyzing feasibility and optimization problems. This opened a growing area of research for solving variational inequalities and approximating solutions or fixed points of nonlinear mappings (see, e.g., [1, 17â€“21] and the references therein).

The existence and approximation of fixed points of Bregman firmly nonexpansive mappings were studied in [5]. It is also known that if is Bregman firmly nonexpansive and is* Legendre function* which is bounded, uniformly FrĂŞchet differentiable, and* totally convex* on bounded subsets of , then and is closed and convex (see [5]). It also follows that every Bregman firmly nonexpansive mapping is Bregman relatively nonexpansive and hence quasi-Bregman nonexpansive mapping.

A* Bregman projection* [1] of onto the nonempty closed and convex set dom is the unique vector satisfying

*Remark 1. *If is a smooth Banach space and for all , then the Bregman projection reduces to the generalized projection (see, e.g., [2]) which is defined by
where .

Very recently, by using Bregman projection, Reich and Sabach [4] introduced an algorithm for finding a common zero of many finitely maximal monotone mappings satisfying in a reflexive Banach space as follows:
where . Under suitable conditions, they proved that if, for each , we have that and the sequences of errors satisfy , then the sequence converges strongly to .

In [22], Reich and Sabach proposed an algorithm for finding a common fixed point of many finitely Bregman firmly nonexpansive mappings satisfying in a reflexive Banach space as follows:
They proved that, under suitable conditions, the sequence generated by (10), converges strongly to and applied it to the solution of convex feasibility and equilibrium problems.

*Remark 2. *But it is worth mentioning that the iteration processes (9) and (10) seem difficult in the sense that, at each stage of iteration, the set(s) and/or are/is computed and the next iterate is taken as the Bregman projection of onto the intersection of and/or . This seems difficult to do in applications.

In this paper, we investigate an iterative scheme for finding a common fixed point of a finite family of quasi-Bregman nonexpansive mappings in reflexive Banach spaces. We prove strong convergence theorems for the sequences produced by the method. Furthermore, we apply our method to prove strong convergence theorems for finding a solution of a finite family of equilibrium problems and for finding a common zero of a finite family of maximal monotone mappings. Our results improve and generalize many known results in the current literature (see, e.g., [4, 23])

#### 2. Preliminaries

Let be a GĂ˘teaux differentiable function. The function is said to be* essentially smooth* if is both locally bounded and single-valued on its domain. It is called* essentially strictly convex*, if is locally bounded on its domain and is strictly convex on every convex subset of . is said to be a* Legendre*, if it is both essentially smooth and essentially strictly convex. When the subdifferential of is single-valued, it coincides with the gradient (see [24]).

We note that if is a reflexive Banach space, then we have the following.(i) is essentially smooth if and only if is essentially strictly convex (see [25, Theorem 5.4]).(ii) (see [26]).(iii) is Legendre if and only if is Legendre (see [25, Corollary 5.5]).(iv)If is Legendre, then is a bijection satisfying , , and (see [25, Theorem 5.10]).When is a smooth and strictly convex Banach space, one important and interesting example of Legendre function is . In this case the gradient of coincides with the generalized duality mapping of ; that is, . In particular, , the identity mapping in Hilbert spaces.

Let be a Banach space and let , for all and . Then a function is said to be* uniformly convex* on bounded subsets of [27, page 203] if , for all , where is defined by
for all . The function is called the* gauge of uniform convexity of *. The function is also said to be* uniformly smooth* on bounded subsets of [27, page 207] if , for all , where is defined by
for all .

In the sequel, we will need the following lemmas.

Lemma 3 (see [28]). *Let be a Banach space, let be a constant, and let be a uniformly convex on bounded subsets of . Then
**
for all ,â€‰â€‰, and with , where is the gauge of uniform convexity of .*

Lemma 4 (see [24]). *Let be a proper, lower semicontinuous, and convex function; then is a proper, lower semicontinuous, and convex function. Thus, for all , we have
*

Lemma 5 (see [29]). *Let be a GĂ˘teaux differentiable on such that is bounded on bounded subsets of . Let and . If is bounded, so is the sequence .*

A function on is* coercive* [30] if the sublevel set of is bounded; equivalently, . A function on is said to be* strongly coercive* [27] if .

Lemma 6 (see [27]). *Let be a reflexive Banach space and let be a continuous convex function which is strongly coercive. Then the following assertions are equivalent.*(1)* is bounded on bounded subsets and uniformly smooth on bounded subsets of ;*(2)* is FrĂ©chet differentiable and is uniformly norm-to-norm continuous on bounded subsets of ;*(3)*, is strongly coercive and uniformly convex on bounded subsets of .*

Lemma 7 (see [5]). *Let be a Legendre function. Let be a nonempty closed convex subset of and let be a quasi-Bregman nonexpansive mapping. Then is closed and convex.*

Let be a GĂ˘teaux differentiable function. The modulus of total convexity of at is the function defined by
The function is called* totally convex* at if , whenever . The function is called totally convex if it is totally convex at any point and is said to be totally convex on bounded sets if for any nonempty bounded subset of and , where the modulus of total convexity of the function on the set is the function defined by
We know that is totally convex on bounded sets if and only if is uniformly convex on bounded sets (see [18, Theorem 2.10]). The following lemmas will be useful in the proof of our main results.

Lemma 8 (see [31]). *The function is totally convex on bounded subsets of if and only if, for any two sequences and in and , respectively, such that the first one is bounded,
*

Lemma 9 (see [27, 32]). *Let be a strongly coercive and uniformly convex on bounded subsets of ; then is bounded on bounded sets and uniformly FrĂ©chet differentiable on bounded subsets of .*

Lemma 10 (see [33]). *Let be uniformly FrĂ©chet differentiable and bounded on bounded sets of ; then is uniformly continuous on bounded subsets of from the strong topology of to the strong topology of .*

Lemma 11 (see [18]). *Let be a nonempty, closed, and convex subset of . Let be a GĂ˘teaux differentiable and totally convex function and let ; then *(i)* if and only if .*(ii)*. *

Let be a Legendre and GĂ˘teaux differentiable function. Following [2, 34], we make use of the function associated with , which is defined by Then is nonnegative and Moreover, by the subdifferential inequality, where and (see [35]).

Lemma 12 (see [36]). *Let be a sequence of nonnegative real numbers satisfying the following relation:
**
where and satisfying the following conditions: , and . Then, .*

Lemma 13 (see [37]). *Let be sequences of real numbers such that there exists a subsequence of such that for all . Then there exists an increasing sequence such that and the following properties are satisfied by all (sufficiently large) numbers :
**
In fact, is the largest number in the set such that the condition holds.*

#### 3. Main Results

We now prove the following theorem.

Theorem 14. *Let be a strongly coercive Legendre function which is bounded, uniformly FrĂ©chet differentiable, and totally convex on bounded subsets of . Let be a nonempty, closed, and convex subset of and let , for , be a finite family of quasi-Bregman nonexpansive mappings such that , for . Assume that is nonempty. For let be a sequence generated by
**
where and satisfying , , and . Then, converges strongly to .*

*Proof. *Lemma 7 ensures that each , for , and are closed and convex. Thus, is well defined. Let . Then, from (23), Lemmas 11 and 4, and property of we get that
Moreover, from (23), (18), and (19) we get that
Since is uniformly FrĂ©chet differentiable function we have that is uniformly smooth and hence by Theorem of [27] we get that is uniformly convex. This, with Lemmas 3 and (24), gives that
for each . Thus, by induction,
which implies that is bounded. Now, let . Then we note that . Since is strongly coercive, uniformly convex, uniformly FrĂ©chet differentiable, and bounded on bounded subsets of by Lemmas 10, 9, and 6 we have that and are bounded on bounded sets and hence and are bounded. In addition, using (19), (20), and property of we obtain that
Furthermore, from (26) and (29) we have that
Now, we consider two cases.*Case **1.* Suppose that there exists such that is nonincreasing for all . In this situation, is convergent. Then, from (30) we have that
which implies, by the property of , that
Now, since is a strongly coercive and uniformly convex on bounded subsets of , is uniformly FrĂ©chet differentiable on bounded subsets of (see [27, Proposition ]) and is Legendre; we have by Lemma 10 that
for each .

Furthermore, Lemma 11, property of , and the fact that , as , imply that
and hence by Lemma 8 we get that
Since is bounded and is reflexive, we choose a subsequence of such that and . Then, from (36) we get that
Thus, from (34) and the fact that we obtain that , for each and hence .

Therefore, by Lemma 11, we immediately obtain that . It follows from Lemma 12 and (31) that , as . Consequently, by Lemma 8 we obtain that .*Case **2.* Suppose that there exists a subsequence of such that
for all . Then, by Lemma 13, there exists a nondecreasing sequence such that , and , for all . Then from (30) and the fact that we obtain that
for each . Thus, following the method of proof of Case 1, we obtain that , , , as , and hence we obtain that
Then from (31) we have that
Now, since , inequality (41) implies that
In particular, since , we get
Then, from (40) we obtain , as . This together with (41) gives , as . But , for all ; thus we obtain that . Therefore, from the above two cases, we can conclude that converges strongly to and the proof is complete.

If, in Theorem 14, we consider a single quasi-Bregman nonexpansive mapping, we get the following corollary.

Corollary 15. *Let be a coercive Legendre function which is bounded, uniformly FrĂ©chet differentiable, and totally convex on bounded subsets of . Let be a nonempty, closed, and convex subset of and let be a quasi-Bregman nonexpansive mapping such that . Let be a sequence generated by
**
where and satisfying and . Then converges strongly to .*

If, in Theorem 14, we assume that each , for , is Bregman relatively nonexpansive mapping, then we have that for each . Thus, we get the following corollary.

Corollary 16. *Let be a strongly coercive Legendre function which is bounded, uniformly FrĂ©chet differentiable, and totally convex on bounded subsets of . Let be a nonempty, closed, and convex subset of and let , for , be a finite family of Bregman relatively nonexpansive mappings. Assume that is nonempty. For let be a sequence generated by
**
where and satisfying , , and . Then, converges strongly to .*

If, in Theorem 14, we assume that each is Bregman firmly nonexpansive, then we get the following corollary.

Corollary 17. *Let be a strongly coercive Legendre function which is bounded, uniformly FrĂ©chet differentiable, and totally convex on bounded subsets of . Let be a nonempty, closed, and convex subset of and let , for , be a finite family of Bregman firmly nonexpansive mappings with . Let be a sequence generated by
**
where and satisfying , , and . Then converges strongly to .*

#### 4. Applications

##### 4.1. Equilibrium Problems

Let be a coercive Legendre function. Let be a nonempty, closed, and convex subset of . Let be a bifunction with the following conditions (see example [38]).(B1), for all ;(B2) is monotone; that is, , for all ;(B3)for all , ;(B4)for each , is convex and lower semicontinuous.The equilibrium problem corresponding to is to find such that The set of solutions of (47) is denoted by EP. Equilibrium problem is a unified model of several problems, namely, variational inequality problem, complementary problem, saddle point problem, optimization problem, fixed point problem, and so forth; see [20, 30, 38â€“43].

The resolvent of a bifunction [39] is the operator , defined by We know the following lemma in [23].

Lemma 18. *Let be a coercive Legendre function. Let be a nonempty, closed, and convex subset of . If the bifunction satisfies conditions (B1)â€“(B4), then*(1)* is single-valued and ;*(2)* is a Bregman firmly nonexpansive mapping;*(3)* is a closed and convex subset of and ;*(4)*For all and for all , we have
*

In addition, if is uniformly FrĂ©chet differentiable and bounded on bounded subsets of , then by Reich and Sabach [5] we have that . Thus, considering , for , in Corollary 17, we get the following result.

Theorem 19. *Let be a strongly coercive Legendre function which is bounded, uniformly FrĂ©chet differentiable, and totally convex on bounded subsets of . Let be a nonempty, closed, and convex subset of and let , for , be bifunctions which satisfy the conditions (B1)â€“(B4) and is nonempty. For let be a sequence generated by
**
where , , and satisfy , , and . Then, converges strongly to .*

##### 4.2. Zeroes of Maximal Monotone Operators

In this section we present an algorithm for finding a common zero of a finite family of maximal monotone mappings.

Let be a mapping with range and domain . Then, is said to be* monotone* if, for any , we have
It is said to be* maximal monotone* if is monotone and the graph of is not a proper subset of the graph of any other monotone mappings, where graph of is given by . It is known that if is maximal monotone, then the set is closed and convex.

Let be maximal monotone mapping and is Legendre function which is bounded, uniformly FrĂ©chet differentiable on bounded subsets of . The resolvent of with respect to is the mapping defined by It is known that is single-valued, Bregman firmly nonexpansive and (see [17]). Furthermore, the result by Reich and Sabach [5] implies that . If, in Theorem 14, we assume that , then we obtain an algorithm for finding a common zero of a finite family of maximal monotone mappings.

Theorem 20. *Let be a Legendre function which is bounded, uniformly FrĂ©chet differentiable, and totally convex on bounded subsets of . Let be a nonempty, closed, and convex subset of and let , be maximal monotone mappings with . Then, for each , the sequence is defined by
**
where , , and satisfy , , and . Then, converges strongly to .*

For a common point of a solution of equilibrium problem and a zero of maximal monotone mapping we have the following corollary.

Corollary 21. *Let be a strongly coercive Legendre function which is bounded, uniformly FrĂ©chet differentiable, and totally convex on bounded subsets of . Let be a nonempty, closed, and convex subset of and let be bifunction which satisfy the conditions (B1)â€“(B4) and let be a maximal monotone mapping. Assume that is nonempty. For let be a sequence generated by
**
where and satisfying , , and . Then, converges strongly to .*

*Remark 22. *Corollary 17 and Corollary 4.3 improve Theorem 1 of Reich and Sabach [23] and Theorem 4.2 of Reich and Sabach [4], respectively, in the sense that at each stage the computation of or is not required.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. 170/130/1434. The authors, therefore, acknowledge with thanks DSR technical and financial support. Also, they would like to thank the anonymous reviewers for their valuable comments and suggestions.

#### References

- L. M. Brègman, â€śA relaxation method of finding a common point of convex sets and its application to the solution of problems in convex programming,â€ť
*USSR Computational Mathematics and Mathematical Physics*, vol. 7, pp. 620â€“631, 1967. View at: Google Scholar | MathSciNet - Y. I. Alber, â€śMetric and generalized projection operators in Banach spaces: properties and applications,â€ť in
*Theory and Applications of Nonlinear Operators of Accretive and Monotone Type*, Lecture Notes in Pure and Applied Mathematics, pp. 15â€“50, 1996. View at: Google Scholar | MathSciNet - S. Reich, â€śA weak convergence theorem for the alternating method with Bregman distances,â€ť in
*Theory and Applications of Nonlinear Operators of Accretive and Monotone Type*, pp. 313â€“318, Marcel Dekker, New York, NY, USA, 1996. View at: Google Scholar - S. Reich and S. Sabach, â€śTwo strong convergence theorems for a proximal method in reflexive Banach spaces,â€ť
*Numerical Functional Analysis and Optimization*, vol. 31, no. 1–3, pp. 22â€“44, 2010. View at: Publisher Site | Google Scholar | MathSciNet - S. Reich and S. Sabach, â€śExistence and approximation of fixed points of Bregman firmly nonexpansive mappings in reflexive Banach spaces,â€ť in
*Fixed-Point Algorithms for Inverse Problems in Science and Engineering*, Springer Optimization and Its Applications, pp. 301â€“316, Springer, New York, NY, USA, 2011. View at: Publisher Site | Google Scholar | MathSciNet - F. E. Browder, â€śConvergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces,â€ť
*Archive for Rational Mechanics and Analysis*, vol. 24, pp. 82â€“90, 1967. View at: Publisher Site | Google Scholar | MathSciNet - B. Halpern, â€śFixed points of nonexpanding maps,â€ť
*Bulletin of the American Mathematical Society*, vol. 73, pp. 957â€“961, 1967. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Reich, â€śStrong convergence theorems for resolvents of accretive operators in Banach spaces,â€ť
*Journal of Mathematical Analysis and Applications*, vol. 75, no. 1, pp. 287â€“292, 1980. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. Zegeye and N. Shahzad, â€śStrong convergence theorems for a finite family of nonexpansive mappings and semigroups via the hybrid method,â€ť
*Nonlinear Analysis. Theory, Methods – Applications*, vol. 72, no. 1, pp. 325â€“329, 2010. View at: Publisher Site | Google Scholar | MathSciNet - H. Zegeye, N. Shahzad, and M. A. Alghamdi, â€śConvergence of Ishikawa's iteration method for pseudocontractive mappings,â€ť
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 74, no. 18, pp. 7304â€“7311, 2011. View at: Publisher Site | Google Scholar | MathSciNet - H. Zegeye, N. Shahzad, and M. A. Alghamdi, â€śMinimum-norm fixed point of pseudocontractive mappings,â€ť
*Abstract and Applied Analysis*, vol. 2012, Article ID 926017, 15 pages, 2012. View at: Publisher Site | Google Scholar | MathSciNet - H. Zegeye and N. Shahzad, â€śStrong convergence theorems for monotone mappings and relatively weak nonexpansive mappings,â€ť
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 70, no. 7, pp. 2707â€“2716, 2009. View at: Publisher Site | Google Scholar | MathSciNet - H. Zegeye and N. Shahzad, â€śA hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems,â€ť
*Nonlinear Analysis: Theory, Methods and Applications*, vol. 74, no. 1, pp. 263â€“272, 2011. View at: Publisher Site | Google Scholar | MathSciNet - D. R. Sahu, V. Colao, and G. Marino, â€śStrong convergence theorems for approximating common fixed points of families of nonexpansive mappings and applications,â€ť
*Journal of Global Optimization*, vol. 56, no. 4, pp. 1631â€“1651, 2013. View at: Publisher Site | Google Scholar | MathSciNet - F. Cianciaruso, G. Marino, and L. Muglia, â€śIterative methods for equilibrium and fixed point problems for nonexpansive semigroups in Hilbert spaces,â€ť
*Journal of Optimization Theory and Applications*, vol. 146, no. 2, pp. 491â€“509, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. Cianciaruso, G. Marino, and L. Muglia, â€śIshikawa iterations for equilibrium and fixed point problems for nonexpansive mappings in Hilbert spaces,â€ť
*Fixed Point Theory*, vol. 9, no. 2, pp. 449â€“464, 2008. View at: Google Scholar | MathSciNet - H. H. Bauschke, J. M. Borwein, and P. L. Combettes, â€śBregman monotone optimization algorithms,â€ť
*SIAM Journal on Control and Optimization*, vol. 42, no. 2, pp. 596â€“636, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - D. Butnariu and E. Resmerita, â€śBregman distances, totally convex functions, and a method for solving operator equations in Banach spaces,â€ť
*Abstract and Applied Analysis*, vol. 2006, Article ID 84919, 39 pages, 2006. View at: Publisher Site | Google Scholar | MathSciNet - N. Shahzad, H. Zegeye, and A. Alotaibi, â€śConvergence results for a common solution of a finite family of variational inequality problems for monotone mappings with Bregman distance function,â€ť
*Fixed Point Theory and Applications*, vol. 2013, article 343, 2013. View at: Publisher Site | Google Scholar | MathSciNet - H. Zegeye and N. Shahzad, â€śA hybrid approximation method for equilibrium, variational inequality and fixed point problems,â€ť
*Nonlinear Analysis. Hybrid Systems*, vol. 4, no. 4, pp. 619â€“630, 2010. View at: Publisher Site | Google Scholar | MathSciNet - H. Zegeye and N. Shahzad, â€śApproximating common solution of variational inequality problems for two monotone mappings in Banach spaces,â€ť
*Optimization Letters*, vol. 5, no. 4, pp. 691â€“704, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Reich and S. Sabach, â€śA projection method for solving nonlinear problems in reflexive Banach spaces,â€ť
*Journal of Fixed Point Theory and Applications*, vol. 9, no. 1, pp. 101â€“116, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Reich and S. Sabach, â€śTwo strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces,â€ť
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 73, no. 1, pp. 122â€“135, 2010. View at: Publisher Site | Google Scholar | MathSciNet - R. P. Phelps,
*Convex Functions, Monotone Operators and Differentiability*, vol. 1364 of*Lecture Notes in Mathematics*, Springer, Berlin, Germnay, 2nd edition, 1993. - H. H. Bauschke, J. M. Borwein, and P. L. Combettes, â€śEssential smoothness, essential strict convexity, and Legendre functions in Banach spaces,â€ť
*Communications in Contemporary Mathematics*, vol. 3, no. 4, pp. 615â€“647, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. F. Bonnans and A. Shapiro,
*Perturbation Analysis of Optimization Problems*, Springer, New York, NY, USA, 2000. View at: Publisher Site | MathSciNet - C. Zalinescu,
*Convex Analysis in General Vector Spaces*, World Scientific, River Edge, NJ, USA, 2002. View at: Publisher Site | MathSciNet - E. Naraghirad and J. Yao, â€śBregman weak relatively nonexpansive mappings in Banach spaces,â€ť
*Fixed Point Theory and Applications*, vol. 2013, article 141, 2013. View at: Publisher Site | Google Scholar | MathSciNet - V. Martín-Márquez, S. Reich, and S. Sabach, â€śBregman strongly nonexpansive operators in reflexive Banach spaces,â€ť
*Journal of Mathematical Analysis and Applications*, vol. 400, no. 2, pp. 597â€“614, 2013. View at: Publisher Site | Google Scholar | MathSciNet - J.-B. Hiriart-Urruty and C. Lemarchal,
*Convex Analysis and Minimization Algorithms II*, vol. 306 of*Grundlehren der mathematischen Wissenschaften*, Springer, 1993. - D. Butnariu and A. N. Iusem,
*Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000. View at: MathSciNet - H. H. Bauschke, â€śThe approximation of fixed points of compositions of nonexpansive mappings in Hilbert space,â€ť
*Journal of Mathematical Analysis and Applications*, vol. 202, no. 1, pp. 150â€“159, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Reich and S. Sabach, â€śA strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces,â€ť
*Journal of Nonlinear and Convex Analysis.*, vol. 10, no. 3, pp. 471â€“485, 2009. View at: Google Scholar | MathSciNet - Y. Censor and A. Lent, â€śAn iterative row-action method for interval convex programming,â€ť
*Journal of Optimization Theory and Applications*, vol. 34, no. 3, pp. 321â€“353, 1981. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. Kohsaka and W. Takahashi, â€śProximal point algorithms with Bregman functions in Banach spaces,â€ť
*Journal of Nonlinear and Convex Analysis*, vol. 6, no. 3, pp. 505â€“523, 2005. View at: Google Scholar | MathSciNet - H. Xu, â€śAnother control condition in an iterative method for nonexpansive mappings,â€ť
*Bulletin of the Australian Mathematical Society*, vol. 65, no. 1, pp. 109â€“113, 2002. View at: Publisher Site | Google Scholar | MathSciNet - P. Maingé, â€śStrong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization,â€ť
*Set-Valued Analysis*, vol. 16, no. 7-8, pp. 899â€“912, 2008. View at: Publisher Site | Google Scholar | MathSciNet - E. Blum and W. Oettli, â€śFrom optimization and variational inequalities to equilibrium problems,â€ť
*The Mathematics Student*, vol. 63, no. 1–4, pp. 123â€“145, 1994. View at: Google Scholar | MathSciNet - P. L. Combettes and S. A. Hirstoaga, â€śEquilibrium programming in Hilbert spaces,â€ť
*Journal of Nonlinear and Convex Analysis*, vol. 6, no. 1, pp. 117â€“136, 2005. View at: Google Scholar | MathSciNet - H. Zegeye and N. Shahzad, â€śConvergence theorems for a common point of solutions of equilibrium and fixed point of relatively nonexpansive multivalued mapping problems,â€ť
*Abstract and Applied Analysis*, vol. 2012, Article ID 859598, 16 pages, 2012. View at: Publisher Site | Google Scholar | MathSciNet - H. Zegeye, N. Shahzad, and M. A. Alghamdi, â€śStrong convergence theorems for a common point of solution of variational inequality, solutions of equilibrium and fixed point problems,â€ť
*Fixed Point Theory and Applications*, vol. 2012, article 119, 17 pages, 2012. View at: Publisher Site | Google Scholar | MathSciNet - H. Zegeye and N. Shahzad, â€śStrong convergence theorems for a solution of a finite family of Bregman weak relatively nonexpansive mappings in reflexive Banach spaces,â€ť
*The Scientific World Journal*, vol. 2014, Article ID 493450, 8 pages, 2014. View at: Publisher Site | Google Scholar - H. Zegeye and N. Shahzad, â€śStrong convergence theorems for a solution of finite families of equilibrium and variational inequality problems,â€ť
*Optimization*, vol. 63, no. 2, pp. 207â€“223, 2014. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

Copyright © 2014 Mohammed Ali Alghamdi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.