Convergence theorem for common fixed points of a finite family of multi-valued Bregman relatively nonexpansive mappings
© Shahzad and Zegeye; licensee Springer. 2014
Received: 24 February 2014
Accepted: 27 June 2014
Published: 22 July 2014
In this paper, it is our purpose to introduce an iterative process for the approximation of a common fixed point of a finite family of multi-valued Bregman relatively nonexpansive mappings. We prove that the sequence of iterates generated converges strongly to a common fixed point of a finite family of multi-valued Bregman nonexpansive mappings in reflexive real Banach spaces.
MSC:47H05, 47H09, 47H10, 47J25, 49J40, 90C25.
The Fenchel conjugate of f is the function defined by . It is not difficult to check that when f is proper and lower semicontinuous, so is .
The function f is said to be essentially smooth if ∂f is both locally bounded and single-valued on its domain. It is called essentially strictly convex, if is locally bounded on its domain and f is strictly convex on every convex subset of . f is said to be Legendre, if it is both essentially smooth and essentially strictly convex.
If the limit in (1.2) exists then f is called Gâteaux differentiable at x. In this case, coincides with , the value of the gradient ∇f of f at x. The function f is called Gâteaux differentiable if it is Gâteaux differentiable for any . The function f called Fréchet differentiable at x if the limit in (1.2) is attained uniformly for all such that and f is said to be uniformly Fréchet differentiable on a subset C of E if the limit is attained uniformly for and . When the subdifferential of f is single-valued, it coincides with the gradient (see ).
If , a Hilbert space, then J is the identity mapping and hence , where I is the identity mapping in H.
In this paper, E is a reflexive real Banach space, is a proper, lower semicontinuous, and convex function, and is the Fenchel conjugate of f.
If , a Hilbert space, then J is the identity mapping and hence the Bregman distance becomes , for , and the Bregman projection reduces to the metric projection of H on to C.
Let C be a nonempty closed and convex subset of . Let be a mapping. An element is called a fixed point of T if . The set of fixed points of T is denoted by . A point p in C is said to be an asymptotic fixed point of T (see ) if C contains a sequence which converges weakly to p such that . The set of asymptotic fixed points of T will be denoted by . T is said to be nonexpansive if for each , and is called quasi-nonexpansive if and for all and . The mapping T is called relatively nonexpansive if (A1) ; (A2) for and , and (A3) and is said to be Bregman relatively nonexpansive with respect to f if (B1) ; (B2) for , and (B3) . We remark that the class of relatively nonexpansive mappings is contained in a class of Bregman relatively nonexpansive mappings with respect to .
for all , where is the distance from the point a to the subset B.
Let be a mapping. T is said to be nonexpansive if , for all . An element is called a fixed point of T, if , where . A point is called an asymptotic fixed point of T, if there exists a sequence in C which converges weakly to p such that . T is called relatively nonexpansive if (A1)′ ; (A2)′ for all , , and (A3)′ . A mapping T is called quasi-Bregman nonexpansive with respect to f if and for all , , and is called Bregman relatively nonexpansive with respect to f if (B1)′ ; (B2)′ for , , , and (B3)′ .
We note that the class of multi-valued relatively nonexpansive mappings is contained in a class of multi-valued Bregman relatively nonexpansive mappings which includes the class of single-valued Bregman relatively nonexpansive mappings. Hence, the class of multi-valued Bregman relatively nonexpansive mappings is a more general class of mappings. An example of a multi-valued Bregman relatively nonexpansive mapping is given now.
Since , we have and hence . Therefore, T is a multi-valued Bregman relatively nonexpansive mapping.
The approximations of fixed points of nonexpansive, quasi-nonexpansive, relatively nonexpansive, and relatively quasi-nonexpansive mappings when they exist have been intensively studied for almost 40 years or so by various authors (see, e.g., [8–18] and the references therein) in Banach spaces.
In 1967, Bregman  discovered an effective technique using the so-called Bregman distance function in the process of designing and analyzing feasibility and optimization algorithms. This opened a growing area of research in which Bregman’s technique is applied in various ways in order to design and analyze iterative algorithms for solving not only feasibility and optimization problems, but also algorithms for solving variational inequality problems, equilibrium problems, fixed point problems for nonlinear mappings, and so on (see, e.g., [7, 19, 20], and the references therein).
In , Reich and Sabach proposed the following algorithms for finding common fixed points of finitely many Bregman firmly nonexpansive operators defined on a nonempty, closed and convex subset C of a reflexive Banach space E (see also [22, 23]). The construction of fixed points for Bregman-type single-valued mappings via iterative processes has been investigated in, for example, [21, 24–27]. This now leads to the following important question.
Question Is it possible to obtain the results of Reich and Sabach  for the class of multi-valued Bregman relatively nonexpansive mappings?
The study of fixed points for multi-valued nonexpansive mappings using the Hausdorff metric was introduced by Markin  (see also ). Later, an interesting and rich fixed point theory for such mappings was developed which has applications in control theory, convex optimization, differential inclusion, and economics (see, for example,  and references therein). Moreover, the existence of fixed points for multi-valued nonexpansive mappings in uniformly convex Banach spaces was proved by Lim  (see also ).
where for all and . They proved that if J is weakly sequentially continuous then the sequence converges weakly to a fixed point of T. Furthermore, it is shown that the scheme converges strongly to a fixed point of T if the interior of is nonempty.
where and , for , satisfy certain conditions, converges strongly to an element of ℱ.
In this paper, it is our purpose to introduce an iterative scheme which converges strongly to a common fixed point of a finite family of multi-valued Bregman relatively nonexpansive mappings. We prove strong convergence theorems for the sequences produced by the method. Our results improve and generalize many known results in the current literature (see, for example, [33, 34]).
for all . The function is called the gauge of the uniform convexity of f. We know that f is totally convex on bounded sets if and only if f is uniformly convex on bounded sets (see , Theorem 2.10).
If f is uniformly convex, then the following lemma is known.
Lemma 2.1 
for all , , , and with , where is the gauge of uniform convexity of f.
In the sequel, we shall need the following lemmas.
Lemma 2.2 
Lemma 2.3 
Let be a strongly coercive and uniformly convex on bounded subsets of E, then is bounded and uniformly Fréchet differentiable on bounded subsets of .
Lemma 2.4 
Let be a uniformly Fréchet differentiable and bounded on bounded sets of E, then ∇f is uniformly continuous on bounded subsets of E from the strong topology of E to the strong topology of .
Lemma 2.5 
Lemma 2.6 
Let be a Gâteaux differentiable on such that is bounded on bounded subsets of . Let and . If is bounded, so is the sequence .
Lemma 2.7 
if and only if , .
and (see ).
Lemma 2.8 
where and satisfy the following conditions: , , and . Then .
Lemma 2.9 
In fact, is the largest number n in the set such that the condition holds.
3 Main result
In the sequel we shall use the following proposition.
Proposition 3.1 Let be a uniformly Fréchet differentiable and totally convex on bounded subsets of E. Let C be a nonempty, closed, and convex subset of and be a Bregman relatively nonexpansive mapping. Then is closed and convex.
Thus, by Lemma 2.2 we get . Hence, and is convex. Therefore, is closed and convex. □
where and satisfy , and . Then converges strongly to .
Now, we consider two cases.
for each . Since is bounded and E is reflexive, we choose a subsequence of such that and . Thus, from (3.12) and the fact that each is Bregman relatively nonexpansive mapping we obtain , for each and hence .
It follows from Lemma 2.8 and (3.8) that as . Consequently, by Lemma 2.2 we obtain .
Then, from (3.15) and (3.13), we obtain as . This, together with (3.14), gives as . But for all , and hence we obtain . Therefore, from the above two cases, we can conclude that converges strongly to and the proof is complete. □
If in Theorem 3.2, we assume that , then we get the following corollary.
where and satisfy , . Then converges strongly to .
If, in Theorem 3.2, we assume that each , is a single-valued Bregman relatively nonexpansive mapping, we get the following corollary.
where and satisfy , and . Then converges strongly to .
If, in Theorem 3.2, we assume that each , , is a multi-valued quasi-Bregman relatively nonexpansive mapping, we get the following corollary.
where and satisfy , and . Then converges strongly to .
Remark 3.6 (i) Theorem 3.2 improves and extends the corresponding results of Homaeipour and Razani  and Zegeye and Shahzad  to the class of multi-valued Bregman relatively nonexpansive mappings in a reflexive real Banach spaces. (ii) The requirement that the interior of F is nonempty is dispensed with.
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The first author acknowledges with thanks DSR for financial support. The authors are grateful to the anonymous reviewers for useful comments.
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