Abstract
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.
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1 Introduction
Let E be a reflexive real Banach space E, and its dual. Let be a proper convex and lower semicontinuous function. The subdifferential of f at is the convex set defined by
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.
Let . Then for any and , the right-hand derivative of f at x in the direction of y is defined by
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 [1]).
We remark that if E is a reflexive Banach space. Then we have
-
(1)
f is essentially smooth if and only if is essentially strictly convex (see [2], Theorem 5.4).
-
(2)
(see [3]).
-
(3)
f is Legendre if and only if is Legendre (see [2], Corollary 5.5).
-
(4)
If f is Legendre, then ∇f is a bijection satisfying , and (see [2], Theorem 5.10).
When E is a smooth and strictly convex Banach space, one important and interesting example of Legendre function is (). In this case the gradient (), where is the generalized duality mapping from E into defined by
In particular, is called the normalized duality mapping. It is well known that if is strictly convex, then is single-valued and that
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.
Let be a Gâteaux differentiable function. The function defined by
is called the Bregman distance with respect to f [4]. Since and , it is easy to check that
A Bregman projection [5] of onto the nonempty closed and convex set is the unique vector satisfying
Remark 1.1 If E is a smooth and strictly convex Banach space and for all , then we have , for all , where J the normalized duality mapping and hence the function reduces to which is defined by for all , which is the Lyapunov function introduced by Alber [6], and reduces to the generalized projection (see, e.g., [6]), which is defined by
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 [7]) 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 .
Let and denote the family of nonempty subsets and nonempty closed bounded subsets of C, respectively. Let H be the Hausdorff metric on defined by
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.
Example 1.2 Let , , and . Let be defined by
Let be defined by , , . Clearly, we have for all , and , where satisfies . It is clear that . Let and such that for all . Then, using (1.3), we get
Next, let such that there exists such that , then
Hence, T is a multi-valued quasi-Bregman nonexpansive mapping. Now, we show that . Let be a sequence which converges weakly to h, and . Let , then we have
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 [5] 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 [21], 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 [21] 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 [28] (see also [29]). 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, [30] and references therein). Moreover, the existence of fixed points for multi-valued nonexpansive mappings in uniformly convex Banach spaces was proved by Lim [31] (see also [32]).
Recently, Homaeipour and Razani [33] studied the following iterative scheme for a fixed point of relatively nonexpansive multi-valued mapping in uniformly convex and uniformly smooth Banach space E:
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.
More recently, Zegeye and Shahzad [34], extended the above result to a finite family of multi-valued relatively nonexpansive mappings without the assumption that the interior of is nonempty. In fact, they proved that if C is a nonempty, closed, and convex subset of a uniformly smooth and uniformly convex real Banach space E and , for , are relatively nonexpansive multi-valued mappings with nonempty, then the sequence generated by
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]).
2 Preliminaries
Let E be a reflexive real Banach space and as its dual. Let be a 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 total convexity of the function f on the set B is the function defined by
Let E be a Banach space and let for all and . Then a function is said to be uniformly convex on bounded subsets of E [[35], pp.203] if for all , where is defined by
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 [36], Theorem 2.10).
If f is uniformly convex, then the following lemma is known.
Lemma 2.1 [37]
Let E be a Banach space, let be a constant, and let be a uniformly convex function on bounded subsets of E. Then
for all , , , and with , where is the gauge of uniform convexity of f.
A function f on E is coercive [38] if the sublevel set of f is bounded; equivalently, . A function f on E is said to be strongly coercive [35] if .
In the sequel, we shall need the following lemmas.
Lemma 2.2 [39]
The function is totally convex on bounded subsets of E if and only if for any two sequences and in and domf, respectively, such that the first one is bounded, we have
Lemma 2.3 [35]
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 [26]
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 [1]
Let be a proper, lower semicontinuous and convex function, then is a proper, weak ∗ lower semicontinuous and convex function. Thus, for all , we have
Lemma 2.6 [40]
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 [36]
Let C be a nonempty, closed, and convex subset of E. Let be a Gâteaux differentiable and totally convex function and let . Then
-
(i)
if and only if , .
-
(ii)
, .
Let be a Legendre and Gâteaux differentiable function. Following [6] and [4], we make use of the function associated with f, which is defined by
Then we observe that is nonnegative and
Moreover, by the subdifferential inequality,
and (see [41]).
Lemma 2.8 [42]
Let be a sequence of nonnegative real numbers satisfying the following relation:
where and satisfy the following conditions: , , and . Then .
Lemma 2.9 [43]
Let be sequences of real numbers such that there exists a subsequence of such that for all . Then there exists an increasing sequence such that and the following properties are satisfied by all (sufficiently large) numbers :
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.
Proof First, we show that is closed. Let be a sequence in such that . Since T is Bregman relatively nonexpansive mapping, we have , for all for all . Therefore,
Thus, by Lemma 2.2 we obtain . Hence, and is closed. Next, we show that is convex. Let and for . We show that . Let , then we have
Thus, by Lemma 2.2 we get . Hence, and is convex. Therefore, is closed and convex. □
Theorem 3.2 Let be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of E. Let C be a nonempty, closed and convex subset of and , for , be a finite family of Bregman relatively nonexpansive mappings such that is nonempty. For let be a sequence generated by
where and satisfy , and . Then converges strongly to .
Proof Proposition 3.1 ensures that each , for , and hence is closed and convex. Thus, is well defined. Let . Then, from (3.2), Lemmas 2.7, 2.5, and the property of , we get
Moreover, from (3.2), (2.3), and (2.2) we get
Since f is uniformly Fréchet differentiable function we find that f is uniformly smooth and hence by Theorem 3.5.5 of [35] we find that is uniformly convex. This, with Lemma 2.1 and (3.3), gives
for each . Thus, by induction,
which implies that is bounded. Furthermore, from (3.2), (2.3), (2.4), and Lemma 2.7 we obtain
Furthermore, from (3.4) and (3.6) we have
Now, we consider two cases.
Case 1. Suppose that there exists such that is non-increasing for all . In this situation, is convergent. Then, from (3.7), we have
which implies, by the property of that
Now, since f is strongly coercive and uniformly convex on bounded subsets of E by Lemma 2.3 we see that is uniformly Fréchet differentiable on bounded subsets of and since f is Legendre by Lemma 2.4 we find that is uniformly continuous on bounded subsets of and hence from (3.10) we get
In addition, since we have
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 .
Therefore, by Lemma 2.7, we immediately obtain
It follows from Lemma 2.8 and (3.8) that as . Consequently, by Lemma 2.2 we obtain .
Case 2. Suppose that there exists a subsequence of such that
for all . Then, by Lemma 2.9, there exists a nondecreasing sequence such that , , and , for all . Then, from (3.7) and the fact that , we obtain
for each . Thus, following the method of proof of Case 1, we obtain as , and hence we obtain
Then, from (3.8), we get
Now, since , inequality (3.14) implies that
Thus, we get
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.
Corollary 3.3 Let be a strongly coercive Legendre function which is bounded, 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 such that is nonempty. For let be a sequence generated by
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.
Corollary 3.4 Let be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of E. Let C be a nonempty, closed and convex subset of and , for , be a finite family of Bregman relatively nonexpansive mappings such that is nonempty. For let be a sequence generated by
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.
Corollary 3.5 Let be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of E. Let C be a nonempty, closed, and convex subset of and , for , be a finite family of quasi-Bregman nonexpansive mappings with , for each . Suppose that is nonempty. For let be a sequence generated by
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 [33] and Zegeye and Shahzad [34] 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.
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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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Shahzad, N., Zegeye, H. Convergence theorem for common fixed points of a finite family of multi-valued Bregman relatively nonexpansive mappings. Fixed Point Theory Appl 2014, 152 (2014). https://doi.org/10.1186/1687-1812-2014-152
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DOI: https://doi.org/10.1186/1687-1812-2014-152