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Strong convergence theorem for quasi-Bregman strictly pseudocontractive mappings and equilibrium problems in Banach spaces
Fixed Point Theory and Applicationsvolume 2014, Article number: 231 (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 E be a real Banach space and C a nonempty closed convex subset of E. The normalized duality map from E to ( is the dual space of E) denoted by J is defined by
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.
Let be a bifunction. The equilibrium problem with respect to g is to find
The set of solution of equilibrium problem is denoted by . Thus
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 denote the Lyapunov functional defined by
A mapping is said to be quasi-ϕ strictly pseudocontractive, see , if and there exists a constant such that
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.
Let ; the subdifferential of f at x is the convex set defined by
where the Fenchel conjugate of f is the function defined by
We know that the Young-Fenchel inequality holds:
A function f on E is coercive  if the sublevel set of f is bounded; equivalently,
A function f on E is said be strongly coercive  if
For any and , the right-hand derivative of f at x in the direction y is defined by
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 
The function f is said to be:
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.
Remark 1.3 Let E be a reflexive Banach space. Then we have:
f is essentially smooth if and only if is essentially strictly convex (see , Theorem 5.4);
f is Legendre if and only if is Legendre (see , Corollary 5.5);
if f is Legendre, then ∇f is a bijection satisfying , and (see , Theorem 5.10).
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.
Let be a convex and Gâteaux differentiable function. The function , defined as follows:
is called the Bregman distance with respect to f (see ). It is obvious from the definition of that
Recall that the Bregman projection  of onto the nonempty, closed, and convex set is the necessarily unique vector satisfying
Concerning the Bregman projection, the following are well known.
Lemma 1.4 
Let C be a nonempty, closed, and convex subset of a reflexive Banach space E. Let be a Gâteaux differentiable and totally convex function and let . Then:
if and only if , ;
, , .
Let be a convex and Gâteaux differentiable function. The modulus of the total convexity of f at is the function defined by
The function f is called totally convex at x if whenever . The function f 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 B of E and , where the modulus of the total convexity of the function f on the set B is the function defined by
Lemma 1.5 
If , then the following statements are equivalent:
the function f is totally convex at x;
for any sequence ,
Recall that the function f called sequentially consistent  if for any two sequences and in E such that the first one is bounded
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 mapping T is said to be Bregman firmly nonexpansive , if for all ,
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 .
Very recently, Zhou and Gao  introduced this definition of a quasi-strict pseudocontraction related to the function ϕ and proved the convergence of a hybrid projection algorithm to a fixed point of a closed and quasi-strict pseudocontraction in a smooth and uniformly convex Banach space. They studied the strong convergence of the following scheme:
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.
Lemma 2.1 Let be a Legendre function which is uniformly Fréchet differentiable and bounded on 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, for any , and the following hold:
Proof Let , and , by definition of T, we have
and, from (1.2), we obtain
This completes the proof. □
Lemma 2.2 
Let E be a real reflexive Banach space, be a proper lower semi-continuous function, then is a proper weak ∗ lower semi-continuous and convex function. Thus, for all , we have
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.
The resolvent of a bifunction g  is the operator defined by
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 
Let E be a real reflexive Banach space and C be a nonempty closed convex subset of E. Let be a Legendre function. If the bifunction satisfies the conditions (A1)-(A4), then the following hold:
is a Bregman firmly nonexpansive operator;
is closed and convex subset of C;
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.
Proof Let be nonempty set. First we show that is closed. Let be a sequence in such that as , we need to show that . From Lemma 2.1, we obtain
From (3.1), we have , and from , Lemma 7.3, it follows that . Therefore is closed.
Next, we show that is convex. Let , for any ; putting , we need to show that . From Lemma 2.1, we obtain, respectively,
Multiplying (3.2) by t and (3.3) by and adding the results, we obtain
which implies , and from , Lemma 7.3, it follows that . Therefore is also convex. This completes the proof. □
We now prove the following theorem.
Theorem 3.2 Let C be a nonempty, closed, and convex subset of a real reflexive Banach space E and a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subset of E. For each , let be a bifunction from to ℝ satisfying (A1)-(A4) and let be a finite family of -Lipschitzian, , closed and quasi-Bregman strictly pseudocontractive self mappings of C such that . Let be a sequence generated by , and
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.
Step II. Show that is closed and convex for all . For , is closed and convex. Assume that is closed and convex for some . For , one obtains
using the fact that and are continuous and linear in E, for , is closed and convex.
Step III. Show that for every . Note that . Suppose , for , then for all , since for each , from (2.2) and Lemma 2.3, we have
also from (2.2) and (2.1), we obtain
But, from (1.2),
From (3.6), (3.7), and (3.8), we obtain
This shows that , which implies for every .
Step IV. Show that exists. From (3.5), , which from (a) of Lemma 1.4 implies
Since , we have
From (b) of Lemma 1.4 we have
This implies that is bounded, from Lemma 1.7, is bounded. By the construction of , we have , and , for any positive integer . Then we obtain
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 .
Since , as , we have in particular
and this further implies that
Step V. Next we show that .
Since , we have from (3.5)
which implies that . Since f is totally convex on bounded subsets of E, f is sequentially consistent (see ). It follows that
From (3.14) and (3.17), we have
Since f is uniformly Fréchet differentiable, it follows from Lemma 1.1 that ∇f is uniformly continuous and f is uniformly continuous on bounded subsets of E (see , Theorem 1.8). Hence
Since , it follows from (3.6), (3.7) that
which implies from (3.20), (3.18), (3.13), and (3.14) that
From the sequential consistency of f, we have
from (3.14) and (3.21), we obtain
which implies that , since . From the uniform continuity of ∇f, we have
From (3.5), we have
which implies from (3.23) that
Since f is strongly coercive and uniformly convex on bounded subsets of E, is uniformly Fréchet differentiable on bounded sets. Moreover, is bounded on bounded sets, and from (3.24) we obtain
On the other hand, we see that
for all , where . It follows from (3.14) and (3.25) that
for all . Thus
for all . Since as , by the closedness of for each , we obtain .
Also, since as , we have from Lemma 2.3, for each ,
Then we have from Lemma 1.5 that , for each . Consequently, we have
From the uniform continuity of ∇f, for each we have
From (2.3), we have, for ,
Furthermore, using (A2) in the last inequality, we obtain
By (A4), (3.28), and as , we have
Let for and . This implies that . This yields . It follows from (A1) and (A4) that
From condition (A3), we obtain
This implies that , for each . Thus, . Hence we have .
Step VI. Finally, we show that . Setting in (3.10), we obtain
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.
Proof From the definition it is clear that f is proper, lower semi-continuous, and convex, and also . By the definition of quasi-Bregman strict pseudocontractivity, we find such that for all and . Now,
From (3.30), (3.31), and (3.32), we obtain
Hence, T is a quasi-Bregman strictly pseudocontractive map.
Since , for all and for all ,
cannot hold for any . Hence, T is not a quasi-ϕ strictly pseudocontractive map. □
4 Numerical example
In this section we discuss the direct application of Theorem 3.2 on a typical example on a real line. Consider the following:
then the scheme can be simplified as
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The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.