Convergence Theorems for Bregman<i> K</i>-Mappings and Mixed Equilibrium Problems in Reflexive Banach Spaces

Ali, Bashir; Harbau, M. H.

doi:https://doi.org/10.1155/2016/5161682

Journal of Function Spaces

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Research Article | Open Access

Volume 2016 | Article ID 5161682 | https://doi.org/10.1155/2016/5161682

Convergence Theorems for Bregman K-Mappings and Mixed Equilibrium Problems in Reflexive Banach Spaces

Bashir Ali¹and M. H. Harbau^1,2

Academic Editor: Adrian Petrusel

Received19 May 2016

Accepted25 Jul 2016

Published18 Sept 2016

Abstract

We introduce a new mixed equilibrium problem with a relaxed monotone mapping in a reflexive Banach space and prove the existence of solution of the equilibrium problem. Using Bregman distance, we introduce the concept of Bregman -mapping for a finite family of Bregman quasiasymptotically nonexpansive mappings and show the fixed point set of the Bregman -mapping is the set of common fixed points of . Using the Bregman -mapping, we introduce an iterative sequence for finding a common point in the set of a common fixed points of the finite family of Bregman quasiasymptotically nonexpansive mappings and the set of solutions of some mixed equilibrium problems. Strong convergence of the iterative sequence is proved. Our results generalise and improve many recent results in the literature.

1. Introduction

Let be a real Banach space and let be the dual of . Let be a nonempty closed and convex subset of . A mapping is called nonexpansive if . A point is a fixed point of if . We denote by the fixed point set of ; that is, . Let be a bifunction. The equilibrium problem with respect to and in the sense of Blum and Oettli [1] is to find such thatThe set of solutions of equilibrium problem is denoted by ; that is,In order to solve equilibrium problem (1), the bifunction is usually assumed to satisfy the following conditions: (C1) for all ;(C2) is monotone; that is, for all ;(C3)for all , ;(C4)for all is convex and lower semicontinuous.Fang and Huang [2] introduced the concept of relaxed monotone mapping for solving mixed equilibrium problems.

A mapping is said to be relaxed monotone (see also [3]) if there exist a mapping and a function with for all and , where such thatParticularly if and , where and are two constants, then is called monotone; see [4, 5].

Fang and Huang [2] proved that under some suitable assumptions, the following variational inequality is solvable: find such thatwhere is a function from to . They also proved that the following inequality is equivalent to variational inequality (4): find such thatThe mixed equilibrium problem (see [6, 7]) is to find such thatWe denote the set of solutions of mixed equilibrium problem (6) by . It is easily seen that if , then mixed equilibrium problem (6) reduces to variational inequality (4). In the case of and , then, coincides with .

Equilibrium problems and mixed equilibrium problems have been used as tools for solving problems arising from linear and nonlinear programming, optimization problems, variational inequalities, fixed point problems, and also problems in physics, economics, engineering, and so forth (see, e.g., [1, 6, 8–15] and the references therein).

Let be the normalised duality mapping defined by where denotes the generalised duality pairing. It is well known that if is smooth, strictly convex, and reflexive, then is single-valued, one-to-one, and onto.

Let be a convex function. We denote by the domain of ; that is, . The function is said to be coercive if . is said to be strongly coercive if . The Fenchel conjugate of is the function defined by The subdifferential of is a mapping defined by It is well known that (see [16]) if and only if for all . It is also known that if is a proper, convex, and lower semicontinuous function, then is a proper, convex, and lower semicontinuous function; see, for example, [17].

For any convex function , let and The right-hand derivative of at in the direction is defined byThe function is said to be Gâteaux differentiable at if exists for any . In this case coincides with , the value of the gradient of at . is said to be Gâteaux differentiable if it is Gâteaux differentiable at each . If the limit in (10) is attained uniformly in , then is said to be Fréchet differentiable at . is said to be uniformly Fréchet differentiable on a subset of if the limit in (10) is attained uniformly for every and . We know that if is uniformly Fréchet differentiable on bounded subset of , then is uniformly continuous on bounded set of (see, e.g., [18]).

The function is said to be essentially smooth if is both 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 function if it is both essentially smooth and essentially strictly convex. When the subdifferential of is single-valued, it coincides with the gradient; that is, ; see, for example, [19].

For a Legendre function , the following properties are well known:(i) is essentially smooth if and only if is essentially strictly convex; see [16];(ii); see [20];(iii) is Legendre if and only if is Legendre function; see [16];(iv)if is Legendre function, then is bijection satisfying , , and ; see [16].If is smooth and strictly convex, the function , , is Legendre function; see, for example, [21]. In this case , . In particular if is a Hilbert space we have , the identity mapping.

Let be a convex and Gâteaux differentiable function. The function defined byis called Bregman distance corresponding to ; see [22, 23]. It follows from the strict convexity of that and if and only if ; see [24].

Bregman projection with respect to of onto the nonempty closed convex subset of is the unique vector satisfying

Remark 1. If is smooth and strictly convex Banach space and , then we have and hence which is the Lyapunov function introduced by Alber [25] and the Bregman projection reduces to the generalised projection which is defined by If , a Hilbert space, then the Bregman projection reduces to the metric projection of onto .

Observing (11), we havewhich is called the three-point identity.

Let be a convex, Legendre, and Gâteaux differentiable function. Following [23, 25] we make use of the function associated with defined byThen is nonnegative and and Also from definition (15), it is obvious that and is convex in the second variable. Therefore for and , we haveMoreover by subdifferential inequality [26], we haveRecall that a mapping is said to be -quasi nonexpansive if and and . is -quasiasymptotically nonexpansive if and there exists a real sequence such that as and and . is called Bregman quasi nonexpansive if and and . is Bregman quasiasymptotically nonexpansive if and there exists a real sequence such that as and and . is said to be closed if for any sequence with and , .

It is worth mentioning that several iterative schemes have been constructed and proposed for finding points which solve fixed point problems and mixed equilibrium problems with relaxed monotone mappings in various settings. In 2010 Wang et al. [7] introduced the following iterative scheme for finding a common element of the set of solutions of a mixed equilibrium problem with relaxed monotone mapping and the set of fixed points of nonexpansive mappings in Hilbert spaces:where is a relaxed monotone mapping and is a nonexpansive mapping. Under some mild conditions on the three control sequences , , and , they obtained strong convergence of scheme (18) to common solution of mixed equilibrium problems and fixed point of nonexpansive mapping.

Recently, Chen et al. [27] introduced a new mixed equilibrium problem with the relaxed monotone mapping in uniformly convex and uniformly smooth Banach spaces and proved the existence of solutions of the mixed equilibrium problem. They also proposed the following iterative scheme to find the common element of the set of solutions of the mixed equilibrium problem and the set of fixed points of a quasi--nonexpansive mapping:where is a quasi--nonexpansive mapping from into itself and is the normalised duality mapping. Under some assumptions on the parameter sequences and , they obtained strong convergence of scheme (19) to common solution of mixed equilibrium problems and fixed point of nonexpansive mapping.

Motivated and inspired by the above results, in this paper we introduce and prove the existence of solutions of the mixed equilibrium problem with relaxed monotone mapping in reflexive Banach spaces. Using Bregman distance, we introduce the concept of Bregman -mapping of a finite family of Bregman quasiasymptotically nonexpansive mappings and propose an iterative sequence for finding a common element of the set of fixed points of a finite family of Bregman quasiasymptotically nonexpansive mappings and the set of solutions of mixed equilibrium problem.

2. Preliminaries

Let be a convex and Gâteaux differentiable function. The modulus of total convexity of at is the function defined by The function is totally convex at if for all and is totally convex if it is totally convex at each point . Let be a bounded subset of . For , define a functional on , defined by is totally convex on bounded set if for any bounded subset of and , where is the total convexity of the function on the set .

Let , for all and . The function is bounded if is bounded for all and is uniformly convex on bounded subsets of [28] if the function defined by satisfies where is called the gauge of uniform convexity of .

The gauge of uniform smoothness of is the function defined by The function is said to be uniformly smooth on if . We know that from [28, 29] is totally convex on bounded sets if and only if is uniformly convex on bounded sets.

Definition 2 (see [2]). Let be a Banach space with the dual and be a nonempty closed convex subset of . Let and be two mappings. Then is said to be -hemicontinuous if for any fixed , the function defined by is continuous at .

Definition 3 (see [30]). Let be a Banach space with the dual and be a nonempty subset of . A mapping is called a KKM mapping if for any finite set one has .

Remark 4. Observe that from Definition 3, if is a KKM mapping and such that for all , then is a KKM mapping.

In the sequel we will need the following lemmas.

Lemma 5 (see [31]). Let be a nonempty subset of a Hausdorff topological vector space and let be a mapping. If is closed in for all and compact for some , then .

Lemma 6 (see [32]). If , then the following statements are equivalent: (i)the function is totally convex at ;(ii)for any sequence

Recall that a function is called sequentially consistent [29] if for any two sequences and in such that is bounded,

Lemma 7 (see [24]). The function is totally convex on bounded sets if and only if is sequentially consistent.

Lemma 8 (see [26]). Let be a Legendre function such that is bounded on bounded subsets of . Let . If the sequence is bounded, then the sequence is bounded.

Lemma 9 (see [33]). Let be a constant and let be a uniformly convex function on bounded subsets of . Then for any and ,where is the gauge of the uniform convexity of .

Lemma 10 (see [34]). Let be a uniformly Fréchet differentiable function and bounded on bounded subsets of . Then is uniformly continuous on bounded subsets of from the strong topology of to strong topology of .

Lemma 11 (see [28]). Let be a strongly coercive function. If is uniformly continuous on bounded subsets of , then is uniformly convex on bounded subsets of .

Lemma 12 (see [28]). Let be a convex function which is bounded on bounded subsets of . Then the following assertions are equivalent: (i) is strongly coercive and uniformly convex on bounded subsets of ;(ii) is Fréchet differentiable and is uniformly norm-to-norm continuous on bounded subsets of .

Lemma 13 (see [29]). Let be a nonempty closed and convex subset of . Let be a Gâteaux differentiable and totally convex function. Let . Then (i) if and only if for all ;(ii) for all .

Lemma 14 (see [35]). Let be a reflexive Banach space and be a nonempty closed and convex subset of . Let be a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex which is bounded on bounded subsets of . Let be a closed and Bregman quasiasymptotically nonexpansive mapping. If , then it is closed and convex.

Lemma 15. Let be a reflexive Banach space and be a nonempty closed and convex subset of . Let be a Legendre, uniformly Fréchet differentiable, strongly coercive, and totally convex function on bounded subsets of . Then the Bregman projection is continuous.

Proof. Let be a sequence in such that as . Let and . By Lemma 13(ii), we haveFrom inequality (28), we have Since converges, it is bounded and using the above inequality it follows that is bounded. The function is strongly coercive and totally convex which is bounded on bounded subsets of ; therefore in view of Lemma 12 is uniformly norm-to-norm continuous on bounded subsets of and consequently is bounded. Hence by Lemma 8 we obtain that is bounded.
Since we have . Therefore Since is uniformly Fréchet differentiable on bounded subsets of , it follows that is uniformly continuous on bounded subsets of (see, e.g., [18]). Thus, taking liminf as of both sides of the above inequality, we obtain Now let . Using (28), we have where is some natural number. As and , we obtain By total convexity of , we get as . This completes the proof.

3. Main Results

Lemma 16. Let be a nonempty, closed, and convex subset of a reflexive Banach space with the dual . Let be a convex and Gâteaux differentiable function. Let be -hemicontinuous and relaxed monotone mapping and be a bifunction satisfying and . Let be proper, convex, and lower semicontinuous. For and , suppose the following conditions hold: (i);(ii) is convex for fixed .Then problems (34) and (35) are equivalent.
Find such thatFind such that

Proof. Suppose (34) holds. Let be a solution of (34); then Since is relaxed monotone, we obtain This shows that is a solution of (35).
Conversely, suppose (35) holds; that is, is a solution of (35). Let such that ; then . Let , . Since is a solution of (35), we have Hence by (C1), (C4), (i), and (ii), we obtain Thus, Since is -hemicontinuous and , by allowing we obtain This shows that is a solution of (34)

Lemma 17. Let be a reflexive Banach space with the dual and let be a nonempty closed, convex, and bounded subset of . Let be a Legendre function and uniformly Fréchet differentiable on bounded subsets of . Let be -hemicontinuous and relaxed monotone mapping and be a bifunction satisfying and . Let be proper, convex lower semicontinuous. For and , suppose (i) for all ,(ii),(iii) is convex and lower semicontinuous for fixed ,(iv) is weakly lower semicontinuous.Then there exists such that

Proof. Define two set-valued mappings as follows: We claim is a KKM mapping. By contradiction suppose then there do not exist and such that and . This implies It follows that which is a contradiction. Thus is a KKM mapping.
Next we show that .
Let . Then Since is relaxed monotone, we have Showing that for all . By Remark 4 it follows that is a KKM mapping.
We claim also that is closed in the weak topology of . Let and be the weak closure point of . Since is reflexive, there exists a sequence such that as Observe that is equivalent toBy (iii) and (iv) and taking liminf as of both sides of (49) we obtain That is, . This implies is weakly closed for all . Since is weakly compact, then is weakly compact in for all .
It is clear that the solution sets of problem (34) and (35) are and . Using Lemmas 16 and 5 we obtain Hence there exists such that This completes the proof.

Lemma 18. Let be a reflexive Banach space with the dual and let be a nonempty closed, convex, and bounded subset of . Let be Legendre and Gâteaux differentiable function. Let be -hemicontinuous and relaxed monotone mapping and be a bifunction satisfying , , and . Let be proper, convex, and lower semicontinuous. For and , define a map by Assume that (i);(ii) is convex and lower semicontinuous for fixed ;(iii) is weakly lower semicontinuous;(iv).Then (1) is single-valued;(2) is a Bregman firmly nonexpansive type mapping; that is,(3);(4) is a Bregman quasi nonexpansive satisfying (5) is closed and convex.

Proof. First we show that is single-valued. Let ; thenBy using (C2), adding (56) yields By (i) we have Since is relaxed monotone, we obtain ThusInterchanging and in (60), we haveAdding (60) and (61), we have Hence By (iv), we have Thus,Since is convex and Gâteaux differentiable we haveBy (65) and (66) we obtain Since is Legendre function, then .
Next we show that is Bregman firmly nonexpansive type. Let ; thenAdding (68), using (i) and (C2), we obtain so that Since is relaxed monotone and , we haveAlso interchanging the roles of and in (71) and applying (iv), we have Hence,showing that is Bregman firmly nonexpansive type.
We now show that . Indeed Next we prove that is Bregman quasi nonexpansive mapping.
Sincewe haveLet ; then from (76) we obtain This shows that is Bregman quasi nonexpansive mapping which is Bregman quasiasymptotically nonexpansive. Also from (75), we haveAs , we obtain Lastly, using (3), (4), and Lemma 14 we obtain that is closed and convex.

Definition 19. Let be a nonempty, closed, and convex subset of a real Banach space . Let be a finite family of Bregman quasiasymptotically nonexpansive mappings. For any , define a mapping as follows:Such a mapping is called the Bregman -mapping generated by and ,

Using the above definition, we have the following Lemma.

Lemma 20. Let be a reflexive Banach space with the dual and let be a nonempty, closed, convex, and bounded subset of . Let be strongly coercive, Legendre, uniformly Fréchet differentiable, and totally convex function which is bounded on bounded subsets of . Let be a finite family of continuous Bregman quasiasymptotically nonexpansive mappings of into itself such that . Let be a real sequence in such that . Let be Bregman -mapping generated by in (80). Then (i) for all and , where as ;(ii);(iii) is a closed mapping.

Proof. Let and . Then by Lemma 13(ii) and inequality (16) we have Hence, Observe that as for all . Let ; then as and Next we show that . It is obvious that . Now let and be any point in .
Let for . Then Since is uniformly Fréchet differentiable function which is bounded on bounded subsets of , then by Lemma 10 is uniformly continuous on bounded subsets and consequently from Lemma 11 is uniformly convex. Therefore in view of Lemma 9 we have Continuing in this fashion we obtain From (86), we have Since , we have By the property of , we obtainIn similar fashion and assuming we haveSince by Lemma 12 is uniformly continuous, we obtain from (89) that as . By our assumption, is continuous and so , that is, . Also from (90), we obtainOn the other hand Since we getFrom (91) and (93), we have and . Applying the same argument we can conclude that for . Thus, it follows that .
Next we show that is closed.
From (80)Let be a sequence in such that and as Since is continuous for each , are uniformly continuous and applying Lemma 15 we haveBy uniqueness of limit, we get showing that is closed. Using (95), we have that is closed. Continuing in this way we obtain that is closed.

Now we prove the strong convergence theorems.

Theorem 21. Let be a reflexive Banach space with the dual and let be a nonempty, closed, convex, and bounded subset of . Let be a strongly coercive, Legendre, uniformly Fréchet differentiable, and totally convex function which is bounded on bounded subsets of . For each , let be -hemicontinuous and relaxed monotone mappings and be bifunctions satisfying . Let be proper, convex, and lower semicontinuous mappings. Let be a finite family of continuous Bregman quasiasymptotically nonexpansive mappings of into itself such that . Let be the Bregman -mapping generated by in (80). Assume that the conditions of Lemma 18 and the following condition are satisfied:(v)For all one has Let be iteratively defined as follows:where is a real sequence in satisfying , for some and . Then converges strongly to .

Proof. It follows from Lemmas 20(ii) and 14 that is closed and convex. On the other hand by Lemma 18(5) is closed and convex for each ; consequently is closed and convex.
Next we prove is closed and convex. The proof is by induction.
For any fixed , is closed and convex which implies is closed and convex.
Now suppose is closed and convex for some . Observe that It follows that is closed and convex and so is closed and convex. Hence is closed and convex for all . This implies that the iterative sequence is well defined.
Next we show that . Obviously . Assume for some and . Let ; then and so . Observe from scheme (97) that . Therefore by Lemma 18(4) we haveBut Thus, This shows that which implies , . Therefore we obtain .
Next we show that exists.
Since , we have . Thereforeshowing that is nondecreasing sequence of real numbers. On the other hand we have from Lemma 13(ii) thatUsing (102) and (103) we have that exists. NowSince exists, we obtainSince is totally convex on bounded set, by Lemma 7 is sequentially consistent and so we haveLet where . Using (104) we have Hence It follows thatThis shows that is Cauchy sequence in . Since is reflexive and is weakly closed, then there exists such thatObserve that , . Hence we obtain By (105) and the fact that as , we get and soNow Therefore using (106) and (113) we obtainAlso by (110) and (115), it follows thatLet . In view of Lemma 9, we have From (99), we have Therefore by Lemma 20(i), we haveObserve that Therefore From (115), we obtainBy (122) and , it follows from (119) that From the property of , we deduce By uniform continuity of on bounded subsets of , we obtainOn the other hand From (110) and (124), we haveNow using (110), (126), and Lemma 20(iii), we obtain . This implies .
Next we show .
From scheme (97) Therefore by (124) and uniform continuity of , we have As is uniformly continuous, we obtainFrom (115) and (129), we haveAgain since is uniformly continuous and , we getFrom scheme (97) Using (C2) and Lemma 17(ii), it follows that Using (116) and (131) and taking liminf as of the above inequality, we get Now for any and , let . Then and soTherefore by (C1), (C4), Lemma 18(i), (ii), and (135), we have That is, Since , we have From (C3), (v), and lower semicontinuity of , we have by allowing This shows that .
Lastly we show .
From and Lemma 13(i) we have Since , this implies thatLetting in (141), we obtain Again by Lemma 13(i) we have . This completes the proof.

If and , in Theorem 21 then we have the following corollary.

Corollary 22. Let be a reflexive Banach space with the dual and let be a nonempty, closed, convex, and bounded subset of . Let be a strongly coercive, Legendre, uniformly Fréchet differentiable, and totally convex function which is bounded on bounded subsets of . Let be -hemicontinuous and relaxed monotone mapping and be bifunctions satisfying . Let be a proper, convex, and lower semicontinuous mapping. Let be a continuous Bregman quasiasymptotically nonexpansive mapping of into itself such that . Assume that the conditions of Lemma 18 and the following condition are satisfied:(v)For all one has Let be iteratively defined as follows:where is a real sequence in satisfying , for some and . Then converges strongly to .

Setting in Theorem 21 then we obtain the following result.

Corollary 23. Let be a reflexive Banach space with the dual and let be a nonempty, closed, convex, and bounded subset of . Let be a strongly coercive, Legendre, uniformly Fréchet differentiable, and totally convex function which is bounded on bounded subsets of . For each , let be bifunctions satisfying and be proper, convex, and lower semicontinuous mappings. Let be a finite family of continuous Bregman quasiasymptotically nonexpansive mapping of into itself such that . Let be the Bregman -mapping generated by in (80). Let be iteratively defined as follows:where is a real sequence in satisfying , for some and . Then converges strongly to .

If and in Theorem 21 then we obtain the following result.

Corollary 24. Let be a reflexive Banach space with the dual and let be a nonempty, closed, convex, and bounded subset of . Let be a strongly coercive, Legendre, uniformly Fréchet differentiable, and totally convex function which is bounded on bounded subsets of . For each , let be bifunctions satisfying . Let be a finite family of continuous Bregman quasiasymptotically nonexpansive mappings of into itself such that . Let be the Bregman -mapping generated by in (80). Let be iteratively defined as follows:where is a real sequence in satisfying , for some and . Then converges strongly to .

As a direct consequence of Theorem 21 and Remark 1, we obtain the convergence result for system of the mixed equilibrium problems and finite family of quasi--asymptotically nonexpansive mappings in uniformly convex and uniformly smooth Banach spaces.

Corollary 25. Let be a uniformly convex and uniformly smooth Banach space with the dual and let be a nonempty, closed, convex, and bounded subset of . Let be a normalised duality mapping. For each , let be -hemicontinuous and relaxed monotone mappings and be bifunctions satisfying . Let be proper, convex, and lower semicontinuous mappings. Let be a finite family of continuous Bregman quasiasymptotically nonexpansive mappings of into itself such that . Let be the Bregman -mapping generated by in (80). Assume that the conditions of Lemma 18 and the following condition are satisfied:(v)For all one has Let be iteratively defined as follows:where is a real sequence in satisfying , for some and . Then converges strongly to .

Remark 26. Our theorems and corollaries generalise the main theorem of Chen et al. [27] in the following senses: (1)For the structure of Banach spaces, we extend the duality mapping to more general case: that is, a Legendre, strongly coercive, uniformly Fréchet differentiable, and totally convex function.(2)For the mapping, we consider Bregman quasiasymptotically nonexpansive mappings which contain Bregman quasi nonexpansive mappings as a special case which itself is generalisation of quasi--nonexpansive mappings.(3)In Chen et al. [27] the authors considered mixed equilibrium problems while in this paper a system of equilibrium problems is considered.

Competing Interests

The authors declare that they have no competing interests.

References

E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994.
View at: Google Scholar | MathSciNet
Y. P. Fang and N. J. Huang, “Variational-like inequalities with generalized monotone mappings in Banach spaces,” Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 327–338, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. H. Siddiqi, Q. H. Ansari, and K. R. Kazmi, “On nonlinear variational inequalities,” Indian Journal of Pure and Applied Mathematics, vol. 25, no. 9, pp. 969–973, 1994.
View at: Google Scholar | MathSciNet
D. Goeleven and D. Motreanu, “Eigenvalue and dynamic problems for variational and hemivariational inequalities,” Communications on Applied Nonlinear Analysis, vol. 3, no. 4, pp. 1–21, 1996.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
R. U. Verma, “Nonlinear variational inequalities on convex subsets of Banach spaces,” Applied Mathematics Letters, vol. 10, no. 4, pp. 25–27, 1997.
View at: Publisher Site | Google Scholar | MathSciNet
L.-C. Ceng and J.-C. Yao, “A hybrid iterative scheme for mixed equilibrium problems and fixed point problems,” Journal of Computational and Applied Mathematics, vol. 214, no. 1, pp. 186–201, 2008.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. H. Wang, G. Marino, and F. H. Wang, “Strong convergence theorems for a generalized equilibrium problem with a relaxed monotone mapping and a countable family of nonexpansive mappings in a Hilbert space,” Fixed Point Theory and Applications, vol. 2010, Article ID 230304, 2010.
View at: Publisher Site | Google Scholar
B. Ali, “Fixed points approximation and solutions of some equilibrium and variational inequalities problems,” International Journal of Mathematics and Mathematical Sciences, vol. 2009, Article ID 656534, 17 pages, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
P. Daniele, F. Giannessi, and A. Maugeri, Equilibrium Problems and Variational Models, Kluwer Academic, Dordrecht, The Netherlands, 2003.
View at: Publisher Site | MathSciNet
N. Petrot, K. Wattanawitoon, and P. Kumam, “A hybrid projection method for generalized mixed equilibrium problems and fixed point problems in Banach spaces,” Nonlinear Analysis: Hybrid Systems, vol. 4, no. 4, pp. 631–643, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
W. Takahashi and K. Zembayash, “Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2008, Article ID 528476, 11 pages, 2008.
View at: Publisher Site | Google Scholar
S. Wang and B. Guo, “New iterative scheme with nonexpansive mappings for equilibrium problems and variational inequality problems in Hilbert spaces,” Journal of Computational and Applied Mathematics, vol. 233, no. 10, pp. 2620–2630, 2010.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
K. Wattanawitoon and P. Kumam, “Generalized mixed equilibrium problems for maximal monotone operators and two relatively quasi-nonexpansive mappings,” Thai Journal of Mathematics, vol. 9, no. 1, pp. 165–189, 2011.
View at: Google Scholar | MathSciNet
Q. Yuan and M. J. Shang, “Ky Fan inequalities, ϕ-distances, and generalizesd asymptotically quasi-ϕ-nonexpansive mappings,” Journal Fixed Point Theory, vol. 8, 2013.
View at: Google Scholar
Q. N. Zhang and H. L. Wu, “Hybrid algorithms for equilibrium and common fixed point problems with applications,” Journal of Inequalities and Applications, vol. 2014, article 221, 2014.
View at: Publisher Site | Google Scholar
H. H. Bauschke, J. M. Borwein, and P. L. Combettes, “Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces,” Communications in Contemporary Mathematics, vol. 3, no. 4, pp. 615–647, 2001.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
W. Takahashi, Convex Analysis and Approximation of Fixed Points, Yokahama Publishers, Yokahama, Japan, 2000.
A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, vol. 34 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1993.
View at: MathSciNet
R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, vol. 1364 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2nd edition, 1993.
View at: MathSciNet
J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer, New York, NY, USA, 2000.
View at: Publisher Site | MathSciNet
G. C. Ugwunnadi, B. Ali, I. Idris, and M. S. Minjibir, “Strong convergence theorem for quasi-Bregman strictly pseudocontractive mappings and equilibrium problems in BANach spaces,” Fixed Point Theory and Applications, vol. 2014, article 231, 16 pages, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
L. M. Bregman, “The relation method of finding the common points of convex sets and its application to solution of problems in convex programming, USSR,” Computational Mathematics and Mathematical Physics, vol. 7, pp. 200–217, 1967.
View at: Google Scholar
Y. Censor and A. Lent, “An iterative row-action method for interval convex programming,” Journal of Optimization Theory and Applications, vol. 34, no. 3, pp. 321–353, 1981.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
D. Butnariu and A. N. Lusem, Totally Convex Functions for Fixed Point Computation and Finite Dimensional Optimization, Kluwer Academic, Dordrecht, The Netherlands, 2012.
Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 15–50, 1996.
View at: Google Scholar | MathSciNet
V. Martín-Márquez, S. Reich, and S. Sabach, “Iterative methods for approximating fixed points of Bregman nonexpansive operators,” Discrete and Continuous Dynamical Systems. Series S, vol. 6, no. 4, pp. 1043–1063, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
M. Chen, J. Song, and S. Wang, “New mixed equilibrium problems and iterative algorithms for fixed point problems in Banach spaces,” Journal of Applied Mathematics, vol. 2014, Article ID 193749, 10 pages, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
C. Zălinescu, Convex Analysis in General Vector Spaces, World Scientific Publishing, River Edge, NJ, USA, 2002.
View at: Publisher Site | MathSciNet
D. Butnariu and E. Resmerita, “Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces,” Abstract and Applied Analysis, vol. 2006, Article ID 84919, 39 pages, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
K. K. Marzukiewicz, “Ein beweis des fixpuntsatzen fur n-dimensionale simplexe,” Fundamenta Mathematicae, vol. 14, pp. 132–137, 1929.
View at: Google Scholar
K. Fan, “Some properties of convex sets related to fixed point theorems,” Mathematische Annalen, vol. 266, no. 4, pp. 519–537, 1984.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
E. Resmerita, “On total convexity, Bregman projections and stability in BANach spaces,” Journal of Convex Analysis, vol. 11, no. 1, pp. 1–16, 2004.
View at: Google Scholar | MathSciNet
E. Naraghirad and J.-C. Yao, “Bregman weak relatively nonexpansive mappings in Banach spaces,” Fixed Point Theory and Applications, vol. 2013, article 141, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
S. Reich and S. Sabach, “A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces,” Journal of Nonlinear and Convex Analysis, vol. 10, no. 3, pp. 471–485, 2009.
View at: Google Scholar | MathSciNet
Y. H. Zhao, S. S. Chang, and J. H. Zhu, “Strong convergence theorems for Bregman quasi-asymptotically nonexpansive mappings and equilibrium problem in reflexive Banach spaces,” Mathematical Inequalities & Applications, vol. 16, no. 4, pp. 1171–1183, 2013.
View at: Publisher Site | Google Scholar | MathSciNet

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Copyright © 2016 Bashir Ali and M. H. Harbau. 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.

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