- Open Access
Strong convergence theorem for quasi-Bregman strictly pseudocontractive mappings and equilibrium problems in Banach spaces
© Ugwunnadi et al.; licensee Springer. 2014
Received: 18 March 2014
Accepted: 16 October 2014
Published: 17 November 2014
In this paper, we introduce a new iterative scheme by a hybrid method and prove a strong convergence theorem of a common element in the set of fixed points of a finite family of closed quasi-Bregman strictly pseudocontractive mappings and common solutions to a system of equilibrium problems in reflexive Banach space. Our results extend important recent results announced by many authors.
Let be a map, a point is called a fixed point of T if , and the set of all fixed points of T is denoted by . The mapping T is called L-Lipschitzian or simply Lipschitz if there exists , such that , and if , then the map T is called nonexpansive.
Numerous problems in physics, optimization and economics reduce to finding a solution of equilibrium problem. Some methods have been proposed to solve the equilibrium problem in Hilbert spaces; see for example Blum and Oettli , Combettes and Hirstoaga . Recently, Tada and Takahashi [3, 4] and Takahashi and Takahashi  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 a nonexpansive mapping in Hilbert space. In particular, Takahashi and Zembayashi  established a strong convergence theorem for finding a common element of the two sets by using the hybrid method introduced in Nakajo and Takahashi . They also proved such a strong convergence theorem in a uniformly convex and uniformly smooth Banach space.
Let E be a real reflexive Banach space with norm and the dual space of E. Throughout this paper, we shall assume is a proper, lower semi-continuous and convex function. We denote by the domain of f.
The function f is said to be Gâteaux differentiable at x if exists for any y. In this case, coincides with , the value of the gradient ∇f of f at x. The function f is said to be Gâteaux differentiable if it is Gâteaux differentiable for any . The function f is said to be Fréchet differentiable at x if this limit is attained uniformly in . Finally, f is said to be uniformly Fréchet differentiable on a subset C of E if the limit is attained uniformly for and . It is well known that if f is Gâteaux differentiable (resp. Fréchet differentiable) on , then f is continuous and its Gâteaux derivative ∇f is norm-to-weak∗ continuous (resp. continuous) on (see also [12, 13]). We will need the following results.
Lemma 1.1 
If is uniformly Fréchet differentiable and bounded on bounded subsets of E, then ∇f is uniformly continuous on bounded subsets of E from the strong topology of E to the strong topology of .
Definition 1.2 
essentially smooth, if ∂f is both locally bounded and single-valued on its domain;
essentially strictly convex, if is locally bounded on its domain and f is strictly convex on every convex subset of ;
Legendre, if it is both essentially smooth and essentially strictly convex.
Examples of Legendre functions were given in [15, 16]. One important and interesting Legendre function is () when E is a smooth and strictly convex Banach space. In this case the gradient ∇f of f is coincident with the generalized duality mapping of E, i.e., (). 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 1.4 
if and only if , ;
, , .
Lemma 1.5 
the function f is totally convex at x;
- (ii)for any sequence ,
Lemma 1.6 
The function f is totally convex on bounded sets if and only if the function f is sequentially consistent.
Lemma 1.7 
Let be a Gâteaux differentiable and totally convex function. If and the sequence is bounded, then the sequence is bounded too.
Lemma 1.8 
Let be a Gâteaux differentiable and totally convex function, and let C be a nonempty, closed, and convex subset of E. Suppose that the sequence is bounded and any weak subsequential limit of belongs to C. If for any , then converges strongly to .
A point is said to be asymptotic fixed point of a map T, if there exists a sequence in C which converges weakly to p such that . We denote by the set of asymptotic fixed points of T. A point is said to be strong asymptotic fixed point of a map T, if there exists a sequence in C which converges strongly to p such that . We denote by the set of strong asymptotic fixed points of T. Let , a mapping is said to be Bregman relatively nonexpansive  if , and for all and . The map is said to be Bregman weak relatively nonexpansive if , and for all and . T is said to be quasi-Bregman relatively nonexpansive if , and for all and . In  quasi-Bregman relatively nonexpansive is called left quasi-Bregman relatively nonexpansive. A map is called right quasi-Bregman relatively nonexpansive  if , and for all and . T is said to be quasi-Bregman strictly pseudocontractive if there exist a constant and such that for all and . In particular, T is said to be quasi-Bregman relatively nonexpansive if and T is said to be quasi-Bregman pseudocontractive if .
where is the generalized projection from E onto . They proved that the sequence converges strongly to .
Recently, Zegeye and Shahzad  proved a strong convergence theorem for the common fixed point of a finite family of right Bregman strongly nonexpansive mappings in a reflexive Banach space. Alghamdi et al.  proved a strong convergence theorem for the common fixed point of a finite family of quasi-Bregman nonexpansive mappings. Pang et al.  proved weak convergence theorems for Bregman relatively nonexpansive mappings. Shahzad and Zegeye  proved a strong convergence theorem for multivalued Bregman relatively nonexpansive mappings, while Zegeye and Shahzad  proved a strong convergence theorem for a finite family of Bregman weak relatively nonexpansive mappings.
Motivated and inspired by the above works, in this paper, we prove a new strong convergence theorem for a finite family of closed quasi-Bregman strictly pseudocontractive mapping and a system of equilibrium problems in a real reflexive Banach space. These results generalize and improve several recent results. We showed by an example that the class of quasi-Bregman strictly pseudocontractive mappings is a proper generalization of the class of quasi-ϕ-Bregman strictly pseudocontractive mappings.
The next lemma will be useful in the proof of our main results.
This completes the proof. □
Lemma 2.2 
In order to solve the equilibrium problem, let us assume that a bifunction satisfies the following conditions :
(A1) , ;
(A2) g is monotone, i.e., , ;
(A3) , ;
(A4) the function is convex and lower semi-continuous.
From Lemma 1, in , if is a strongly coercive and Gâteaux differentiable function, and g satisfies conditions (A1)-(A4), then . The following lemma gives some characterization of the resolvent .
Lemma 2.3 
is a Bregman firmly nonexpansive operator;
is closed and convex subset of C;
- (v)for all and for all , we have(2.4)
3 Main result
Lemma 3.1 Let be a Legendre function which is uniformly Fréchet differentiable on bounded subsets of E, let C be a nonempty, closed, and convex subset of E and let be a quasi-Bregman strictly pseudocontractive mapping with respect to f. Then is closed and convex.
From (3.1), we have , and from , Lemma 7.3, it follows that . Therefore is closed.
which implies , and from , Lemma 7.3, it follows that . Therefore is also convex. This completes the proof. □
We now prove the following theorem.
where , and , for each , is uniformly continuous; suppose and , are sequences in such that (i) , (ii) , . Then converges strongly to , where is the Bregman projection of E onto F.
Proof The proof is divided into six steps.
Step I. Show that is closed and convex. From Lemma 3.1, is closed and convex and from (iv) of Lemma 2.3, is closed and convex. So, is closed and convex.
using the fact that and are continuous and linear in E, for , is closed and convex.
This shows that , which implies for every .
Since and , we obtain , . This shows that is nondecreasing and hence the limit exists. Thus from (3.12), taking the limit as , we obtain . Since f is totally convex on bounded subsets of E, f is sequentially consistent (see ). It follows that as . Hence is Cauchy sequence in C. As is Cauchy in a complete space E, there exists such that as . Clearly .
Step V. Next we show that .
for all . Since as , by the closedness of for each , we obtain .
This implies that , for each . Thus, . Hence we have .
By (a) of Lemma 1.4, we have . □
Here we give an example of a quasi-Bregman strictly peudocontractive mapping which is not quasi-ϕ strictly pseudocontractive mapping; this shows that the former class is a generalization of the latter.
Example 3.3 Let , and define by and , for all . We want to show that T is a quasi-Bregman strictly pseudocontractive but not quasi-ϕ strictly pseudocontractive.
Hence, T is a quasi-Bregman strictly pseudocontractive map.
cannot hold for any . Hence, T is not a quasi-ϕ strictly pseudocontractive map. □
4 Numerical example
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