Journal of Function Spaces
Volume 2016 (2016), Article ID 5161682, 18 pages
http://dx.doi.org/10.1155/2016/5161682

Research Article

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

Bashir Ali¹ and M. H. Harbau1,2

¹Department of Mathematical Sciences, Bayero University, Kano, Nigeria
²Department of Science and Technology Education, Bayero University, Kano, Nigeria

Received 19 May 2016; Accepted 25 July 2016

Academic Editor: Adrian Petrusel

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.

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 that

The 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 that

Particularly 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 that

where is a function from to . They also proved that the following inequality is equivalent to variational inequality (4): find such that

The mixed equilibrium problem (see [6, 7]) is to find such that

We 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 by

The 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 by

is 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 have

which 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 by

Then is nonnegative and and Also from definition (15), it is obvious that and is convex in the second variable. Therefore for and , we have

Moreover by subdifferential inequality [26], we have

Recall 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 have

From 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 that

Find 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 to

By (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 ; then

By using (C2), adding (56) yields

By (i) we have

Since is relaxed monotone, we obtain

Thus

Interchanging and in (60), we have

Adding (60) and (61), we have

Hence

By (iv), we have

Thus,

Since is convex and Gâteaux differentiable we have

By (65) and (66) we obtain

Since is Legendre function, then .
Next we show that is Bregman firmly nonexpansive type. Let ; then

Adding (68), using (i) and (C2), we obtain

so that

Since is relaxed monotone and , we have

Also 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.
Since

we have

Let ; then from (76) we obtain

This shows that is Bregman quasi nonexpansive mapping which is Bregman quasiasymptotically nonexpansive. Also from (75), we have

As , 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 obtain

In similar fashion and assuming we have

Since 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 obtain

On the other hand

Since we get

From (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 have

By 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 have

But

Thus,

This shows that which implies , . Therefore we obtain .
Next we show that exists.
Since , we have . Therefore

showing that is nondecreasing sequence of real numbers. On the other hand we have from Lemma 13(ii) that

Using (102) and (103) we have that exists. Now

Since exists, we obtain

Since is totally convex on bounded set, by Lemma 7   is sequentially consistent and so we have

Let where . Using (104) we have

Hence

It follows that

This shows that is Cauchy sequence in . Since is reflexive and is weakly closed, then there exists such that

Observe that , . Hence we obtain

By (105) and the fact that as , we get

and so

Now

Therefore using (106) and (113) we obtain

Also by (110) and (115), it follows that

Let . In view of Lemma 9, we have

From (99), we have

Therefore by Lemma 20(i), we have

Observe that

Therefore

From (115), we obtain

By (122) and , it follows from (119) that

From the property of , we deduce By uniform continuity of on bounded subsets of , we obtain

On the other hand

From (110) and (124), we have

Now 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 obtain

From (115) and (129), we have

Again since is uniformly continuous and , we get

From 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 so

Therefore 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 that

Letting 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.

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