Abstract
The purpose of this paper is to introduce a new class of Bregman asymptotically quasi-nonexpansive mappings in the intermediate sense. A strong convergence theorem of the shrinking projection method with the modified Mann iteration is established to find fixed points of the mappings in reflexive Banach spaces. This theorem generalizes some known results in the current literature.
MSC:47H09, 47J25.
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1 Introduction
Fixed point theory is an important branch of nonlinear analysis and has been applied in numerous studies of nonlinear phenomena. Many problems in nonlinear functional analysis is related to finding fixed points of nonlinear mappings of nonexpansive type. From the standpoint of real world applications, we want to construct an iterative process to approximate fixed points of mappings of nonexpansive type. Many authors have considered the problem of iterative algorithms for mappings of nonexpansive type which converge to some fixed points.
Let C be a nonempty subset of a real Banach space and T a nonlinear mapping from C into itself. We denote by the set of fixed points of T. Recall that T is said to be nonexpansive if
More generally, T is said to be asymptotically nonexpansive (cf. [1]) if there exists a sequence with such that
In the framework of Hilbert spaces, Takahashi, Takeuchi and Kubota [2] have introduced a new hybrid iterative scheme called a shrinking projection method for nonexpansive mappings. It is an advantage of projection methods that the strong convergence of iterative sequences is guaranteed without any compact assumptions. Moreover, Schu [3] has introduced a modified Mann iteration to approximate fixed points of asymptotically nonexpansive mappings in uniformly convex Banach spaces. Motivated by [2, 3], Inchan [4] has introduced a new hybrid iterative scheme by using the shrinking projection method with the modified Mann iteration for asymptotically nonexpansive mappings. The mapping T is said to be asymptotically nonexpansive in the intermediate sense (cf. [5]) if
If is nonempty and (1.1) holds for all and , then T is said to be asymptotically quasi-nonexpansive in the intermediate sense. It is worth mentioning that the class of asymptotically nonexpansive mappings in the intermediate sense contains properly the class of asymptotically nonexpansive mappings, since the mappings in the intermediate sense are not Lipschitz continuous in general.
Recently, many authors have studied further new hybrid iterative schemes in the framework of real Banach spaces; for instance, see [6–8]. Qin and Wang [9] have introduced a new class of mappings which are asymptotically quasi-nonexpansive with respect to the Lyapunov functional (cf. [10]) in the intermediate sense. By using the shrinking projection method, Hao [11] has proved a strong convergence theorem for an asymptotically quasi-nonexpansive mapping with respect to the Lyapunov functional in the intermediate sense.
In 1967, Bregman [12] has discovered an elegant and effective technique for the using of the so-called Bregman distance function (see Section 2) 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 not only iterative algorithms for solving feasibility and optimization problems, but also algorithms for solving variational inequalities, for approximating equilibria and for computing fixed points of nonlinear mappings.
The purpose of this paper is to prove strong convergence theorems for asymptotically quasi-nonexpansive mappings with respect to Bregman distances in the intermediate sense by using the shrinking projection method. Many authors have studied iterative methods for approximating fixed points of mappings of nonexpansive type with respect to the Bregman distance; see [13–16]. However, nonlinear mappings which are not Lipschitz continuous with respect to the Bregman distance have not been studied yet. Against this background, we introduce a new class of asymptotically quasi-nonexpansive mappings which is an extension of with respect to the Bregman distance in the intermediate sense. Motivated by the results above, we design a new hybrid iterative scheme for finding fixed points of the mapping in reflexive Banach spaces. This iterative method is expected to be applicable to many other problems in nonlinear functional analysis relating to Bregman distances.
In this paper, we introduce a new class of nonlinear mappings which is an extension of asymptotically quasi-nonexpansive mappings with respect to the Bregman distance in the intermediate sense. Motivated by [4, 12], we design a new hybrid iterative scheme for finding a fixed point of mappings in the new class by using the shrinking projection method with respect to Bregman distances in reflexive Banach spaces. We prove a new strong convergence theorem for the mappings, which is an extension of results of [11]. In Section 2, we present several preliminary definitions and results. In Section 3, we introduce the new class of mappings with respect to the Bregman distance and prove closedness and convexity of the set of fixed points of the mappings. In Section 4, we prove a strong convergence theorem for finding a fixed point of mappings in the new class by using the shrinking projection method.
2 Preliminaries
Throughout this paper, we denote by N and R the sets of all nonnegative integers and real numbers, respectively, and we assume that E is a real reflexive Banach space with the norm , the dual space of E and the pairing between E and . When is a sequence in E, we denote the strong convergence of to x by and the weak convergence by .
Let be a function. The effective domain of f is defined by
When we say that f is proper. We denote by int domf the interior of the effective domain of f. We denote by ranf the range of f.
The function f is said to be strongly coercive if . Given a proper and convex function , the subdifferential of f is a mapping defined by
The Fenchel conjugate function of f is the convex function defined by
We know that if and only if for ; see [17].
Proposition 2.1 ([18], Proposition 2.47)
Let be a proper, convex and lower semicontinuous function. Then the following conditions are equivalent:
-
(i)
and is bounded on bounded subsets of ;
-
(ii)
f is strongly coercive.
Let be a convex function and . For any , we define the right-hand derivative of f at x in the direction y by
The function f is said to be Gâteaux differentiable at x if the limit (2.1) exists for any y. In this case, the gradient of f at x is the function defined by for all . The function f is said to be Gâteaux differentiable if it is Gâteaux differentiable at each . If the limit (2.1) is attained uniformly in , then the function f is said to be Fréchet differentiable at x. The function f is said to be uniformly Fréchet differentiable on a subset C of E if the limit (2.1) is attained uniformly for and . We know that if f is uniformly Fréchet differentiable on bounded subsets of E, then f is uniformly continuous on bounded subsets of E (cf. [19]). We will need the following results.
Proposition 2.2 ([20], Proposition 2.1)
If a function is convex, 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 .
Proposition 2.3 ([21], Proposition 3.6.4)
Let be a convex function which is bounded on bounded subsets of E. Then the following assertions are equivalent:
-
(i)
f is strongly coercive and uniformly convex on bounded subsets of E;
-
(ii)
is Fréchet differentiable and is uniformly norm-to-norm continuous on bounded subsets of .
A function is said to be admissible if it is proper, convex, and lower semicontinuous on E and Gâteaux differentiable on int domf. Under these conditions we know that f is continuous in int domf, ∂f is single-valued and ; see [17, 22]. An admissible function is called Legendre (cf. [17]) if it satisfies the following two conditions:
(L1) the interior of the domain of f, int domf, is nonempty, f is Gâteaux differentiable and ;
(L2) the interior of the domain of , , is nonempty, is Gâteaux differentiable and .
Let f be a Legendre function on E. Since E is reflexive, we always have . This fact, when combined with conditions (L1) and (L2), implies the following equalities:
Conditions (L1) and (L2) imply that the functions f and are strictly convex on the interior of their respective domains.
Example 2.4 The following functions are Legendre on : Let .
-
(i)
Halved energy: .
-
(ii)
Boltzmann-Shannon entropy:
-
(iii)
Burg entropy:
Note that in (i), whereas in (ii) and (iii).
Let be a convex function on E which is Gâteaux differentiable on int domf. The bifunction given by
is called the Bregman distance with respect to f (cf. [23]). In general, the Bregman distance is not a metric, since it is not symmetric and does not satisfy the triangle inequality. However, it has the following important property, which is called the three point identity (cf. [24]): for any and ,
Example 2.5 The Bregman distances corresponding to the Legendre functions of Example 2.4 are as follows ():
-
(i)
Euclidean distance: .
-
(ii)
Kullback-Leibler divergence: .
-
(iii)
Itakura-Saito divergence: .
With a Legendre function , we associate the bifunction defined by
Proposition 2.6 ([13], Proposition 10)
Let be a Legendre function such that is bounded on bounded subsets of . Let . If the sequence is bounded, then the sequence is also bounded.
Proposition 2.7 ([13], Proposition 1)
Let be a Legendre function. Then the following statements hold:
-
(i)
The function is convex for all ;
-
(ii)
for all and .
Let be a convex function on E which is Gâteaux differentiable on int domf. The function f is said to be totally convex at a point if its modulus of total convexity at , defined by
is positive whenever . The function f is said to be totally convex when it is totally convex at every point of int domf. The function f is said to be totally convex on bounded sets if, for any nonempty bounded set , the modulus of total convexity of f on B, is positive for any , where is defined by
We remark in passing that f is totally convex on bounded sets if and only if f is uniformly convex on bounded sets; see [25, 26].
Proposition 2.8 ([25], Proposition 4.2)
Let be a convex function whose domain contains at least two points. If f is lower semicontinuous, then f is totally convex on bounded sets if and only if f is uniformly convex on bounded sets.
Proposition 2.9 ([27], Lemma 3.1)
Let be a totally convex function. If and the sequence is bounded, then the sequence is also bounded.
Let be a convex function on E which is Gâteaux differentiable on int domf. The function f is said to be sequentially consistent (cf. [26]) if for any two sequences and in int domf and domf, respectively, such that the first one is bounded,
Proposition 2.10 ([22], Proposition 2.1.2)
A function is totally convex on bounded subsets of E if and only if it is sequentially consistent.
Let C be a nonempty, closed, and convex subset of E. Let be a convex function on E which is Gâteaux differentiable on int domf. The Bregman projection with respect to f (cf. [23]) of onto C is the minimizer over C of the functional , that is,
Proposition 2.11 ([28], Corollary 2.1)
Let be an admissible, strongly coercive, and strictly convex function. Let C be a nonempty, closed, and convex subset of domf. Then exists uniquely for all .
Remark Let for .
-
(i)
If E is a Hilbert space, then the Bregman projection is reduced to the metric projection onto C.
-
(ii)
If E is a smooth Banach space, then the Bregman projection is reduced to the generalized projection which is defined by
where is the Lyapunov functional (cf. [10]) defined by for all .
Proposition 2.12 ([26], Corollary 4.4)
Let be a totally convex function. Let C be a nonempty, closed, and convex subset of int domf and . If , then the following statements are equivalent:
-
(i)
The vector is the Bregman projection of x onto C.
-
(ii)
The vector is the unique solution z of the variational inequality
-
(iii)
The vector is the unique solution z of the inequality
3 Bregman asymptotically quasi-nonexpansive in the intermediate sense
Let C be a nonempty, closed, and convex subset of E and T a mapping from C into itself. Let be an admissible function. Recall that the mapping T is said to be Bregman quasi-nonexpansive (cf. [14]) if and
The mapping T is said to be Bregman asymptotically quasi-nonexpansive (cf. [29]) if and there exists a sequence with such that
Every Bregman quasi-nonexpansive mapping is Bregman asymptotically quasi-nonexpansive with .
We introduce a new class of mappings; the mapping T is said to be Bregman asymptotically quasi-nonexpansive in the intermediate sense if and
Put
This implies . Then (3.1) is reduced to the following:
Bregman asymptotically quasi-nonexpansive mappings in the intermediate sense are not Lipschitz continuous in general.
Example 3.1 Assume that , and defined by
Note that and for all and . If is a Legendre function, then T is Bregman asymptotically quasi-nonexpansive in the intermediate sense since
However, T above is not Lipschitzian with respect to Bregman distances in Example 2.5. Indeed, suppose that there exists such that for all . By Taylor’s theorem, there exists such that
-
(i)
Let on and for all . Put and . Since , we have
This implies , which is a contradiction.
(ii) Let on and for all and . Note that . Put . By (3.2), we have
and
If , we have
This implies , which is a contradiction.
-
(iii)
Let on and for all . Note that . Put . By (3.2), we have
and
If , we have
This implies , which is a contradiction.
Theorem 3.2 Let be a Legendre function which is totally convex on bounded subsets of E. Suppose that is bounded on bounded subsets of . Let C be a nonempty, closed, and convex subset of int domf. Let be a closed and Bregman asymptotically quasi-nonexpansive mapping in the intermediate sense. Then is closed and convex.
Proof Since T is closed, we can easily conclude that is closed. Now we show the convexity of . Let and , where . We prove that . By the definition of T, we have
for . By the three point identity (2.2), we know that
This implies
for . Combining (3.3) and (3.4) yields
for . Multiplying t and on both sides of (3.5) with and , respectively, yields
This implies that is bounded. By Propositions 2.6 and 2.10, we see that the sequence is bounded and as . By the closedness of T, we have
and hence . Therefore is convex. This completes the proof. □
Theorem 3.2 is reduced to the following results.
Corollary 3.3 ([29], Lemma 1)
Let be a Legendre function which is totally convex on bounded subsets of E. Let C be a nonempty, closed, and convex subset of int domf and a closed and Bergman asymptotically quasi-nonexpansive mapping with the sequence such that as . Then is closed and convex.
4 Main results
In this section, we prove the following strong convergence theorem for finding a fixed point of a Bregman asymptotically quasi-nonexpansive mapping in the intermediate sense. Let C be a nonempty, closed, and convex subset of E and T a mapping from C into itself. The mapping T is said to be asymptotically regular if, for any ,
Theorem 4.1 Let be a Legendre function which is bounded, strongly coercive, uniformly Fréchet differentiable and totally convex on bounded subsets on E. Let C be a nonempty, closed, and convex subset of int domf. Let be a closed and Bregman asymptotically quasi-nonexpansive mapping in the intermediate sense. Suppose that T is asymptotically regular on C and is bounded. Let be a sequence of C generated by
where
and is a sequence satisfying . Then converges strongly to .
Proof We divide the proof into five steps.
Step 1. We show that is closed and convex for all .
It is obvious that is closed and convex. Suppose that is closed and convex for some . We see that, for ,
is equivalent to
Let and , where . By (4.1), we have
and hence . Therefore is closed and convex for all . By Proposition 2.11, is well defined for all .
Step 2. We show that for all .
Let . It is obvious that . Suppose that for some . By Proposition 2.7, we have
This implies . Therefore for all .
Step 3. We show that is bounded.
Let . By Proposition 2.12(iii), we have
for all . This implies that is bounded. By Proposition 2.9, the sequence is bounded.
Step 4. We show that every subsequential limit of belongs to .
Since is bounded and E is reflexive, we may assume that is a weakly convergent subsequence of and denote its weak limit by . Since is closed and convex, we have for all . By the lower semicontinuity of f, we have
This implies
By Proposition 2.12(iii), we have
By Proposition 2.10, we have as . By Proposition 2.2, we have
Since and , we have . This implies that is nondecreasing and the limit of exists as . By Proposition 2.12(iii), we have
for all . This implies
By Proposition 2.10, we have as . By Proposition 2.2, we have
Since , we have for all . By (4.5), we have as . By Proposition 2.10, we have as . By Proposition 2.2, we have
By the definition of , we have
By (4.6) and (4.7), we find from that
We have
By (4.4) and (4.8), we have as . By Propositions 2.3 and 2.8, is uniformly continuous on bounded subsets of and thus as . Since f is asymptotically regular, we have
This implies as . By the closedness of T, we have . Therefore, the limit of belongs to .
Step 5. We show that as .
Since and , we have for all . By (4.3), we have
Thus since . Hence is only strong cluster point of . Therefore as . This completes the proof. □
If for all , then Theorem 4.1 is reduced to the following corollary.
Corollary 4.2 ([11], Theorem 2.1)
Let E be a reflexive, strictly convex and smooth Banach space such that both E and have the Kadec-Klee property. Let C be a nonempty, closed, and convex subset of E. Let be an asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense. Assume that T is asymptotically regular on C and closed, and is bounded. Let be a sequence generated by
where
is the generalized projection from E onto and is a sequence satisfying . Then converges strongly to , where is the generalized projection from C onto .
Proof Using the technique used in the proof of Theorem 4.1 with for all , we find that the sequence generated by (4.9) converges strongly to . □
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Tomizawa, Y. A strong convergence theorem for Bregman asymptotically quasi-nonexpansive mappings in the intermediate sense. Fixed Point Theory Appl 2014, 154 (2014). https://doi.org/10.1186/1687-1812-2014-154
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DOI: https://doi.org/10.1186/1687-1812-2014-154