Abstract

We introduce an iterative process for finding common fixed point of finite family of quasi-Bregman nonexpansive mappings which is a unique solution of some equilibrium problem.

1. Introduction

Let be a real reflexive Banach space and a nonempty subset of . Let be a map, a point is called a fixed point of if , and the set of all fixed points of is denoted by . The mapping is called -Lipschitzian or simply Lipschitz if there exists , such that , and if , then the map is called nonexpansive.

Let be a bifunction. The equilibrium problem with respect to is to find The set of solutions of equilibrium problem is denoted by . Thus Numerous problems in physics, optimization, and economics reduce to finding a solution of the equilibrium problem. Some methods have been proposed to solve equilibrium problem in Hilbert spaces; see, for example, Blum and Oettli [1], Combettes and Hirstoaga [2]. Recently, Tada and Takahashi [3, 4] and S. Takahashi and W. Takahashi [5] obtain weak and strong convergence theorems for finding a common element of the set of solutions of an equilibrium problem and set of fixed points of nonexpansive mapping in Hilbert space. In particular, Takahashi and Zembayashi [4] establish a strong convergence theorem for finding a common element of the two sets by using the hybrid method introduced in Nakajo and Takahashi [6]. They also proved such a strong convergence theorem in a uniformly convex and uniformly smooth Banach space.

In 1967, Bregman [7] discovered an elegant and effective technique for using so-called Bregman distance function ; see (3) in the process of designing and analyzing feasibility and optimization algorithms. This opened a growing area of research in which Bregman’s technique has been applied in various ways in order to design and analyze iterative algorithms for solving feasibility and optimization problems.

Let be a convex and Gâteaux differentiable function. The function defined as is called the Bregman distance with respect to (see [8]). It is obvious from the definition of that We observed from (4) that, for any , the following holds:

Recall that the Bregman projection [7] of onto the nonempty closed and convex set is the necessarily unique vector satisfying A mapping is said to be Bregman firmly nonexpansive [9], if, for all ,or, equivalently, A point is said to be asymptotic fixed point of a map , if, for any sequence in which converges weakly to , . We denote by the set of asymptotic fixed points of . Let ; a mapping is said to be Bregman relatively nonexpansive [10] if , , and for all and . is said to be quasi-Bregman relatively nonexpansive if , and for all and .

Recently, by using the Bregman projection, in 2011 Reich and Sabach [9] proposed algorithms for finding common fixed points of finitely many Bregman firmly nonexpansive operators in a reflexive Banach space: Under some suitable conditions, they proved that the sequence generated by (9) converges strongly to and applied the result for the solution of convex feasibility and equilibrium problems.

In 2011, Chen et al. [11] introduced the concept of weak Bregman relatively nonexpansive mappings in a reflexive Banach space and gave an example to illustrate the existence of a weak Bregman relatively nonexpansive mapping and the difference between a weak Bregman relatively nonexpansive mapping and a Bregman relatively nonexpansive mapping. They also proved strong convergence of the sequences generated by the constructed algorithms with errors for finding a fixed point of weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings under some suitable conditions.

Recently in 2014, Alghamdi et al. [12] proved a strong convergence theorem for the common fixed point of finite family of quasi-Bregman nonexpansive mappings. Pang et al. [13] proved weak convergence theorems for Bregman relatively nonexpansive mappings, while Zegeye and Shahzad in [14, 15] proved a strong convergence theorem for the common fixed point of finite family of right Bregman strongly nonexpansive mappings and Bregman weak relatively nonexpansive mappings in reflexive Banach space, respectively.

In 2015 Kumam et al. [16] introduced the following algorithm: where , is a Bregman strongly nonexpansive mapping. They proved that the sequence which is generated by algorithm (10) converges strongly to the point , where

Motivated and inspired by the above works, in this paper, we prove a new strong convergence theorem for finite family of quasi-Bregman nonexpansive mapping and system of equilibrium problem in a real Banach space.

2. Preliminaries

Let be a real reflexive Banach space with the norm and the dual space of . Throughout this paper, we will assume is a proper, lower semicontinuous, and convex function. We denote by the domain of .

Let ; the subdifferential of at is the convex set defined by where the Fenchel conjugate of is the function defined by We know that the Young-Fenchel inequality holds: A function on is coercive [17] if the sublevel set of is bounded; equivalently, A function on is said be strongly coercive [18] if For any 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 . The function is said to be Gâteaux differentiable if it is Gâteaux differentiable for any . The function is said to be Fréchet differentiable at if this limit is attained uniformly in Finally, is said to be uniformly Fréchet differentiable on a subset of if the limit is attained uniformly for and . It is known that if is Gâteaux differentiable (resp., Fréchet differentiable) on , then is continuous and its Gâteaux derivative is norm-to- continuous (resp., continuous) on int (see also [19, 20]). We will need the following results.

Lemma 1 (see [21]). If is uniformly Fréchet differentiable and bounded on bounded subsets of , then is uniformly continuous on bounded subsets of from the strong topology of to the strong topology of .

Definition 2 (see [22]). The function is said to be (i)essentially smooth, if is both locally bounded and single-valued on its domain,(ii)essentially strictly convex, if is locally bounded on its domain and is strictly convex on every convex subset of ,(iii)Legendre, if it is both essentially smooth and essentially strictly convex.

Remark 3. Let be a reflexive Banach space. Then we have the following:(i) is essentially smooth if and only if is essentially strictly convex (see [22], Theorem ).(ii) (see [20]).(iii) is Legendre if and only if is Legendre (see [22], Corollary ).(iv)If is Legendre, then is a bijection satisfying , , and (see [22], Theorem ).The following result was proved in [23] (see also [24]).

Lemma 4. Let be a Banach space, let be a constant, let be the gauge of uniform convexity of , and let be a convex function which is uniformly convex on bounded subsets of . Then, (i)for any and , (ii)for any , (iii)if, in addition, is bounded on bounded subsets and uniformly convex on bounded subsets of then, for any , , and ,

Lemma 5 (see [25]). Let be a Banach space, let be a constant, and let be a continuous and convex function which is uniformly convex on bounded subsets of . Then for all , and with , where is the gauge of uniform convexity of

We know the following two results; see [18].

Theorem 6. Let be a reflexive Banach space and let be a convex function which is bounded on bounded subsets of . Then the following assertions are equivalent: (1) is strongly coercive and uniformly convex on bounded subsets of .(2), is bounded on bounded subsets and uniformly smooth on bounded subsets of .(3), is Fréchet differentiable and is uniformly norm-to-norm continuous on bounded subsets of .

Theorem 7. 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 Fréchet differentiable and is uniformly norm-to-norm continuous on bounded subsets of .(3), is strongly coercive and uniformly convex on bounded subsets of .

The following result was first proved in [26] (see also [27]).

Lemma 8. Let be a reflexive Banach space, let be a strongly coercive Bregman function, and let be the function defined by Then the following assertions hold: (1) for all and .(2) for all and .

Examples of Legendre functions were given in [22, 28]. One important and interesting Legendre function is when is a smooth and strictly convex Banach space. In this case the gradient of is coincident with the generalized duality mapping of ; that is, . In particular, , the identity mapping in Hilbert spaces. In the rest of this paper, we always assume that is Legendre.

Concerning the Bregman projection, the following are well known.

Lemma 9 (see [26]). Let be a nonempty, closed, and convex subset of a reflexive Banach space . Let be a Gâteaux differentiable and totally convex function and let Then (a) if and only if (b)

Let be a convex and Gâteaux 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

Lemma 10 (see [29]). If , then the following statements are equivalent: (i)The function is totally convex at .(ii)For any sequence ,

Recall that the function is called sequentially consistent [26] if for any two sequences and in such that the first one is bounded

Lemma 11 (see [30]). The function is totally convex on bounded sets if and only if the function is sequentially consistent.

Lemma 12 (see [31]). Let be a Gâteaux differentiable and totally convex function. If and the sequence is bounded, then the sequence is bounded too.

Lemma 13 (see [31]). Let be a Gâteaux differentiable and totally convex function, , and let be a nonempty, closed, and convex subset of . Suppose that the sequence is bounded and any weak subsequential limit of belongs to . If for any , then converges strongly to .

Lemma 14 (see [32]). Let be a real reflexive Banach space, let be a proper lower semicontinuous function, and then is a proper lower semicontinuous and convex function. Thus, for all , one has

In order to solve the equilibrium problem, let us assume that a bifunction satisfies the following conditions [1]:(A1), (A2) is monotone; that is, (A3)(A4)The function is convex and lower semicontinuous.

The resolvent of a bifunction [2] is the operator defined by From Lemma  , in [33], if is a strongly coercive and Gâteaux differentiable function and satisfies conditions (A1)–(A4), then . The following lemma gives some characterization of the resolvent .

Lemma 15 (see [33]). Let be a real reflexive Banach space and let be a nonempty closed convex subset of . Let be a Legendre function. If the bifunction satisfies the conditions (A1)–(A4), then, the following hold: (i) is single-valued.(ii) is a Bregman firmly nonexpansive operator.(iii).(iv) is closed and convex subset of .(v)For all and for all , one has

Lemma 16 (see [34]). Let be a sequence of nonnegative real numbers satisfying the following relation: where and is a real sequence satisfying the following conditions: Then, .

Lemma 17 (see [35]). Let be a sequence of real numbers such that there exists a subsequence of such that for all . Then there exists a nondecreasing sequence such that and the following properties are satisfied by all (sufficiently large) numbers : In fact, .

3. Main Results

We now prove the following theorem.

Theorem 18. Let be a nonempty, closed, and convex subset of a real reflexive Banach space and a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subset of . For each , let be a bifunction from to satisfying (A1)–(A4) and let be a finite family of quasi-Bregman nonexpansive self-mapping of such that , where and Let be a sequence generated by , and where and and satisfying and . Then converges strongly to , where is the Bregman projection of onto

Proof. Let from Lemma 15; we obtain Now from (32), we obtainAlso from (32), (26), and (34), we have Thus, by induction we obtain which implies that is bounded and hence , , , and are all bounded for each . Now from (32) let . Furthermore since as , we obtain Since is strongly coercive and uniformly convex on bounded subsets of , is uniformly Fréchet differentiable on bounded sets. Moreover, is bounded on bounded sets; from (37), we obtainOn the other hand, in view of (3) in Theorem 6, we know that and is strongly coercive and uniformly convex on bounded subsets. Let and be the gauge of uniform convexity of the conjugate function . Now from (32) and Lemmas 4 and 8, we obtainNow, we consider two cases.
Case  1. Suppose that there exists such that is nonincreasing. In this situation is convergent. Then from (40) we obtainwhich implies, by the property of and since , Since is strongly coercive and uniformly convex on bounded subsets of , is uniformly Fréchet differentiable on bounded sets. Moreover, is bounded on bounded sets; from (43), we obtain Now from (4), we obtain and therefore Also, from (28) in Lemma 15, we haveThen, we have from Lemma 10 that Also, from (b) of Lemma 9, we have Then, we have from Lemma 10 that From (38) and (48), we obtain From (50) and (51), we obtain Since is strongly coercive and uniformly convex on bounded subsets of , is uniformly Fréchet differentiable on bounded sets. Moreover, is bounded on bounded sets; from (52), we obtain Also from (44) and (52) Now from (4) and (34), we obtain therefore, from (53), we obtainAlso thus Also, from (56) Then, we have from Lemma 10 that Then from (32) and (44), we have This impliesfrom (44), (52), and (60), we obtain This implies that Also from (52) and (62), we obtain Butas Hence From the uniformly continuous , we have from (66) that From (4), (35), and (69), we obtain which implies Also from quasi-Bregman nonexpansivity of , we have which implies and from the uniform continuous , we obtain Also from (4) and (64), we obtain From (64), (66), and (73), we obtain which from uniform continuous implies and from (4) and (77), we obtain From (4), (71), (77), and (78) Also from (4), (71), and (79) Using the quasi-Bregman nonexpansivity of for each , we obtain the following finite table: Then, applying Lemma 10 on each line above, we obtain and adding up this table, we obtain Using this and (68), we obtain Also from quasi-Bregman nonexpansivity of , for each , we have Then, we have from Lemma 10 that Sincethen, from (52), (84), and (86), we obtain Following the argument from (85), (86), and (88) by replacing with and using (51), we obtainLet be a subsequence of . Since is bounded and is reflexive, without loss of generality, we may assume that for some and since as , then . Since the pool of mappings of is finite, passing to a further subsequence if necessary, we may further assume that, for some , from (89), we get and also Noticing that for each , we obtainHence From (A2), we note that, for each , Taking the limit as in above inequality and from (A4) and , we have for each . For and , define . Noticing that , we obtain , which yield that . It follows from (A1) that That is, for each , we have
Let ; from (A3), we obtain for any , for each . This implies that Hence It follows from the definition of the Bregman projection that It follows from Lemma 16 and (41) that as . Consequently, from Lemma 10, we obtain as
Case  2. Suppose is not monotone decreasing sequences; then set and let be a mapping defined for all for some sufficiently large by Then by Lemma 17 is a nondecreasing sequence such that as and , for . Then from (40) and the fact that , we obtain that Following the same argument as in Case  1, we obtain and also we obtain Then from (41), we obtain that It follows from (101) and , that as . Thus Furthermore, for , if (i.e., ), because for it then follows that for all we have This implies that , and hence as . Consequently, from Lemma 10, we obtain as Therefore from the above two cases, we conclude that converges strongly to and this completes the proof.