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Bregman weak relatively nonexpansive mappings in Banach spaces
Fixed Point Theory and Applications volume 2013, Article number: 141 (2013)
Abstract
In this paper, we introduce a new class of mappings called Bregman weak relatively nonexpansive mappings and propose new hybrid iterative algorithms for finding common fixed points of an infinite family of such mappings in Banach spaces. We prove strong convergence theorems for the sequences produced by the methods. Furthermore, we apply our method to prove strong convergence theorems of iterative algorithms for finding common fixed points of finitely many Bregman weak relatively nonexpansive mappings in reflexive Banach spaces. These algorithms take into account possible computational errors. We also apply our main results to solve equilibrium problems in reflexive Banach spaces. Finally, we study hybrid iterative schemes for finding common solutions of an equilibrium problem, fixed points of an infinite family of Bregman weak relatively nonexpansive mappings and null spaces of a γ-inverse strongly monotone mapping in 2-uniformly convex Banach spaces. Some application of our results to the solution of equations of Hammerstein-type is presented. Our results improve and generalize many known results in the current literature.
MSC:47H10, 37C25.
1 Introduction
The hybrid projection method was first introduced by Hangazeau in [1]. In a series of papers [2–12], authors investigated the hybrid projection method and proved strong and weak convergence theorems for the sequences produced by their method. The shrinking projection method, which is a generalization of the hybrid projection method, was first introduced by Takahashi et al. in [13]. Throughout this paper, we denote the set of real numbers and the set of positive integers by ℝ and ℕ, respectively. Let E be a Banach space with the norm and the dual space . For any , we denote the value of at x by . Let be a sequence in E. We denote the strong convergence of to as by and the weak convergence by . The modulus δ of convexity of E is denoted by
for every ϵ with . A Banach space E is said to be uniformly convex if for every . Let . The norm of E is said to be Gâteaux differentiable if for each , the limit
exists. In this case, E is called smooth. If the limit (1.1) is attained uniformly for all , then E is called uniformly smooth. The Banach space E is said to be strictly convex if whenever and . It is well known that E is uniformly convex if and only if is uniformly smooth. It is also known that if E is reflexive, then E is strictly convex if and only if is smooth; for more details, see [14, 15].
Let C be a nonempty subset of E. Let be a mapping. We denote the set of fixed points of T by , i.e., . A mapping is said to be nonexpansive if for all . A mapping is said to be quasi-nonexpansive if and for all and . The concept of nonexpansivity plays an important role in the study of Mann-type iteration [16] for finding fixed points of a mapping . Recall that the Mann-type iteration is given by the following formula:
Here, is a sequence of real numbers in satisfying some appropriate conditions. The construction of fixed points of nonexpansive mappings via Mann’s algorithm [16] has been extensively investigated recently in the current literature (see, for example, [17] and the references therein). In [17], Reich proved the following interesting result.
Theorem 1.1 Let C be a closed and convex subset of a uniformly convex Banach space E with a Fréchet differentiable norm, let be a nonexpansive mapping with a fixed point, and let be a sequence of real numbers such that and . Then the sequence generated by Mann’s algorithm (1.2) converges weakly to a fixed point of T.
However, the convergence of the sequence generated by Mann’s algorithm (1.2) is in general not strong (see a counterexample in [18]; see also [19]). Some attempts to modify the Mann iteration method (1.2) so that strong convergence is guaranteed have recently been made. Bauschke and Combettes [4] proposed the following modification of the Mann iteration method for a single nonexpansive mapping T in a Hilbert space H:
where C is a closed and convex subset of H, denotes the metric projection from H onto a closed and convex subset Q of H. They proved that if the sequence is bounded above from one, then the sequence generated by (1.3) converges strongly to as .
Let E be a smooth, strictly convex and reflexive Banach space and let J be a normalized duality mapping of E. Let C be a nonempty, closed and convex subset of E. The generalized projection from E onto C [20] is defined and denoted by
where . Let C be a nonempty, closed and convex subset of a smooth Banach space E, let T be a mapping from C into itself. A point is said to be an asymptotic fixed point [21] of T if there exists a sequence in C which converges weakly to p and . We denote the set of all asymptotic fixed points of T by . A point is called a strong asymptotic fixed point of T if there exists a sequence in C which converges strongly to p and . We denote the set of all strong asymptotic fixed points of T by .
Following Matsushita and Takahashi [22], a mapping is said to be relatively nonexpansive if the following conditions are satisfied:
-
(1)
is nonempty;
-
(2)
, , ;
-
(3)
.
In 2005, Matsushita and Takahashi [22] proved the following strong convergence theorem for relatively nonexpansive mappings in a Banach space.
Theorem 1.2 Let E be a uniformly smooth and uniformly convex Banach space, let C be a nonempty, closed and convex subset of E, let T be a relatively nonexpansive mapping from C into itself, and let be a sequence of real numbers such that and . Suppose that is given by
If is nonempty, then converges strongly to .
1.1 Some facts about gradient
For any convex function we denote the domain of g by . For any and any , we denote by the right-hand derivative of g at x in the direction y, that is,
The function g is said to be Gâteaux differentiable at x if exists for any y. In this case, coincides with , the value of the gradient ∇g of g at x. The function g is said to be Gâteaux differentiable if it is Gâteaux differentiable everywhere. The function g is said to be Fréchet differentiable at x if this limit is attained uniformly in . The function g is Fréchet differentiable at (see, for example, [[23], p.13] or [[24], p.508]) if for all , there exists such that implies that
The function g is said to be Fréchet differentiable if it is Fréchet differentiable everywhere. It is well known that if a continuous convex function is Gâteaux differentiable, then ∇g is norm-to-weak∗ continuous (see, for example, [[23], Proposition 1.1.10]). Also, it is known that if g is Fréchet differentiable, then ∇g is norm-to-norm continuous (see [[24], p.508]). The mapping ∇g is said to be weakly sequentially continuous if as implies that as (for more details, see [[23], Theorem 3.2.4] or [[24], p.508]). The function g is said to be strongly coercive if
It is also said to be bounded on bounded subsets of E if is bounded for each bounded subset U of E. Finally, g is said to be uniformly Fréchet differentiable on a subset X of E if the limit (1.5) is attained uniformly for all and .
Let be a set-valued mapping. We define the domain and range of A by and , respectively. The graph of A is denoted by . The mapping is said to be monotone [25] if whenever . It is also said to be maximal monotone [26] if its graph is not contained in the graph of any other monotone operator on E. If is maximal monotone, then we can show that the set is closed and convex. A mapping is called γ-inverse strongly monotone if there exists a positive real number γ such that for all , .
1.2 Some facts about Legendre functions
Let E be a reflexive Banach space. For any proper, lower semicontinuous and convex function , the conjugate function of g is defined by
for all . It is well known that for all . It is also known that is equivalent to
Here, ∂g is the subdifferential of g [27, 28]. We also know that if is a proper, lower semicontinuous and convex function, then is a proper, weak∗ lower semicontinuous and convex function; see [15] for more details on convex analysis.
Let be a mapping. The function g is said to be:
-
(i)
essentially smooth, if ∂g is both locally bounded and single-valued on its domain;
-
(ii)
essentially strictly convex, if is locally bounded on its domain and g is strictly convex on every convex subset of ;
-
(iii)
Legendre, if it is both essentially smooth and essentially strictly convex (for more details, we refer to [[29], Definition 5.2]).
If E is a reflexive Banach space and is a Legendre function, then in view of [[30], p.83],
Examples of Legendre functions are given in [29, 31]. One important and interesting Legendre function is (), where the Banach space E is smooth and strictly convex and, in particular, a Hilbert space.
1.3 Some facts about Bregman distance
Let E be a Banach space and let be the dual space of E. Let be a convex and Gâteaux differentiable function. Then the Bregman distance [32, 33] corresponding to g is the function defined by
It is clear that for all . In that case when E is a smooth Banach space, setting for all , we obtain that for all and hence for all .
Let E be a Banach space and let C be a nonempty and convex subset of E. Let be a convex and Gâteaux differentiable function. Then we know from [34] that for and , if and only if
Furthermore, if C is a nonempty, closed and convex subset of a reflexive Banach space E and is a strongly coercive Bregman function, then for each , there exists a unique such that
The Bregman projection from E onto C is defined by for all . It is also well known that has the following property:
for all and (see [23] for more details).
1.4 Some facts about uniformly convex and totally convex functions
Let E be a Banach space and let for all . Then a function is said to be uniformly convex on bounded subsets of E [[35], pp.203, 221] if for all , where is defined by
for all . The function is called the gauge of uniform convexity of g. The function g is also said to be uniformly smooth on bounded subsets of E [[35], pp.207, 221] if for all , where is defined by
for all . The function g is said to be uniformly convex if the function , defined by
satisfies that .
Remark 1.1 Let E be a Banach space, let be a constant and let be a convex function which is uniformly convex on bounded subsets. Then
for all and , where is the gauge of uniform convexity of g.
Let be a convex and Gâteaux differentiable function. Recall that, in view of [[23], Section 1.2, p.17] (see also [36]), the function g is called totally convex at a point if its modulus of total convexity at x, that is, the function , defined by
is positive whenever . The function g is called totally convex when it is totally convex at every point . Moreover, the function f is called totally convex on bounded subsets of E if for any bounded subset X of E and for any , where the modulus of total convexity of the function g on the set X is the function defined by
It is well known that any uniformly convex function is totally convex, but the converse is not true in general (see [[23], Section 1.3, p.30]).
It is also well known that g is totally convex on bounded subsets if and only if g is uniformly convex on bounded subsets (see [[37], Theorem 2.10, p.9]).
Examples of totally convex functions can be found, for instance, in [23, 37].
1.5 Some facts about resolvent
Let E be a reflexive Banach space with the dual space and let be a proper, lower semicontinuous and convex function. Let A be a maximal monotone operator from E to . For any , let the mapping be defined by
The mapping is called the g-resolvent of A (see [38]). It is well known that for each (for more details, see, for example, [14]).
Examples and some important properties of such operators are discussed in [39].
1.6 Some facts about Bregman quasi-nonexpansive mappings
Let C be a nonempty, closed and convex subset of a reflexive Banach space E. Let be a proper, lower semicontinuous and convex function. Recall that a mapping is said to be Bregman quasi-nonexpansive [40] if and
A mapping is said to be Bregman relatively nonexpansive [40] if the following conditions are satisfied:
-
(1)
is nonempty;
-
(2)
, , ;
-
(3)
.
Now, we are in a position to introduce the following new class of Bregman quasi-nonexpansive type mappings. A mapping is said to be Bregman weak relatively nonexpansive if the following conditions are satisfied:
-
(1)
is nonempty;
-
(2)
, , ;
-
(3)
.
It is clear that any Bregman relatively nonexpansive mapping is a Bregman quasi-nonexpansive mapping. It is also obvious that every Bregman relatively nonexpansive mapping is a Bregman weak relatively nonexpansive mapping, but the converse in not true in general. Indeed, for any mapping , we have . If T is Bregman relatively nonexpansive, then . Below we show that there exists a Bregman weak relatively nonexpansive mapping which is not a Bregman relatively nonexpansive mapping.
Example 1.1 Let , where
Let be a sequence defined by
where
for all . It is clear that the sequence converges weakly to . Indeed, for any , we have
as . It is also obvious that for any with n, m sufficiently large. Thus, is not a Cauchy sequence. Let k be an even number in ℕ and let be defined by
It is easy to show that for all , where
It is also obvious that
Now, we define a mapping by
It is clear that and for any ,
If , then we have
Therefore, T is a Bregman quasi-nonexpansive mapping. Next, we claim that T is a Bregman weak relatively nonexpansive mapping. Indeed, for any sequence such that and as , since is not a Cauchy sequence, there exists a sufficiently large number such that for any . If we suppose that there exists such that for infinitely many , then a subsequence would satisfy , so and , which is impossible. This implies that for all . It follows from that and hence . Since , we conclude that T is a Bregman weak relatively nonexpansive mapping.
Finally, we show that T is not Bregman relatively nonexpansive. In fact, though and
as , but . Thus we have .
Let us give an example of a Bregman quasi-nonexpansive mapping which is neither a Bregman relatively nonexpansive mapping nor a Bregman weak relatively nonexpansive mapping (see also [41]).
Example 1.2 Let E be a smooth Banach space, let k be an even number in ℕ and let be defined by
Let be any element of E. We define a mapping by
for all . It could easily be seen that T is neither a Bregman weak relatively nonexpansive mapping nor a Bregman relatively nonexpansive mapping. To this end, we set
Though () as and
as , but . Therefore, and .
In [42], Bauschke and Combettes introduced an iterative method to construct the Bregman projection of a point onto a countable intersection of closed and convex sets in reflexive Banach spaces. They proved a strong convergence theorem of the sequence produced by their method; for more detail, see [[42], Theorem 4.7].
In [40], Reich and Sabach introduced a proximal method for finding common zeros of finitely many maximal monotone operators in a reflexive Banach space. More precisely, they proved the following strong convergence theorem.
Theorem 1.3 Let E be a reflexive Banach space and let , , be N maximal monotone operators such that . Let be a Legendre function that is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of E. Let be a sequence defined by the following iterative algorithm:
If, for each , and the sequences of errors satisfy , then each such sequence converges strongly to as .
Let C be a nonempty, closed and convex subset of a reflexive Banach space E. Let be a proper, lower semicontinuous and convex function. Recall that a mapping is said to be Bregman firmly nonexpansive (for short, BFNE) if
for all . The mapping T is called quasi-Bregman firmly nonexpansive (for short, QBFNE) [43], if and
for all and . It is clear that any quasi-Bregman firmly nonexpansive mapping is Bregman quasi-nonexpansive. For more information on Bregman firmly nonexpansive mappings, we refer the readers to [38, 44]. In [44], Reich and Sabach proved that for any BFNE operator T, .
In [43], Reich and Sabach introduced a Mann-type process to approximate fixed points of quasi-Bregman firmly nonexpansive mappings defined on a nonempty, closed and convex subset C of a reflexive Banach space E. More precisely, they proved the following theorem.
Theorem 1.4 Let E be a reflexive Banach space and let , , be N QBFNE operators which satisfy for each and . Let be a Legendre function that is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of E. Let be a sequence defined by the following iterative algorithm:
If, for each , the sequences of errors satisfy , then each such sequence converges strongly to as .
Let E be a reflexive Banach space and let be a convex and Gâteaux differentiable function. Let C be a nonempty, closed and convex subset of E. Recall that a mapping is said to be (quasi-)Bregman strongly firmly nonexpansive (for short, BSNE) with respect to a nonempty if and
for all and , and if whenever is bounded and , then we have
The class of (quasi-)Bregman strongly nonexpansive mappings was first introduced in [21, 45] (for more details, see also [46]). We know that the notion of a strongly nonexpansive operator (with respect to the norm) was first introduced and studied in [47, 48].
In [46], Reich and Sabach introduced iterative algorithms for finding common fixed points of finitely many Bregman strongly nonexpansive operators in a reflexive Banach space. They established the following strong convergence theorem in a reflexive Banach space.
Theorem 1.5 Let E be a reflexive Banach space and let , , be N BSNE operators which satisfy for each and . Let be a Legendre function that is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of E. Let be a sequence defined by the following iterative algorithm:
If, for each , the sequences of errors satisfy , then each such sequence converges strongly to as .
But it is worth mentioning that, in all the above results for Bregman nonexpansive-type mappings, the assumption is imposed on the map T.
Remark 1.2 Though the iteration processes (1.10) and (1.12), as introduced by the authors mentioned above, worked, it is easy to see that these processes seem cumbersome and complicated in the sense that at each stage of iteration, two different sets and are computed and the next iterate taken as the Bregman projection of on the intersection of and . This seems difficult to do in application. It is important to state clearly that the iteration process (1.11) involves computation of only one set at each stage of iteration. In [49], Sabach proposed an excellent modification of algorithm (1.10) for finding common zeros of finitely many maximal monotone operators in reflexive Banach spaces.
Our concern now is the following:
Is it possible to obtain strong convergence of modified Mann-type schemes (1.10)-(1.12) to a fixed point of a Bregman quasi-nonexpansive type mapping T without imposing the assumption on T?
In this paper, using Bregman functions, we introduce new hybrid iterative algorithms for finding common fixed points of an infinite family of Bregman weak relatively nonexpansive mappings in Banach spaces. We prove strong convergence theorems for the sequences produced by the methods. Furthermore, we apply our method to prove strong convergence theorems of iterative algorithms for finding common fixed points of finitely many Bregman weak relatively nonexpansive mappings in reflexive Banach spaces. These algorithms take into account possible computational errors. We also apply our main results to solve equilibrium problems in reflexive Banach spaces. Finally, we study hybrid iterative schemes for finding common solutions of an equilibrium problem, fixed points of an infinite family of Bregman weak relatively nonexpansive mappings and null spaces of a γ-inverse strongly monotone mapping in 2-uniformly convex Banach spaces. Some application of our results to the solution of equations of Hammerstein type is presented. No assumption is imposed on the mapping T. Consequently, the above concern is answered in the affirmative in reflexive Banach space setting. Our results improve and generalize many known results in the current literature; see, for example, [4, 7, 8, 11, 22, 40, 42–44, 46, 50–52].
2 Preliminaries
In this section, we begin by recalling some preliminaries and lemmas which will be used in the sequel.
The following definition is slightly different from that in Butnariu and Iusem [23].
Definition 2.1 [24]
Let E be a Banach space. The function is said to be a Bregman function if the following conditions are satisfied:
-
(1)
g is continuous, strictly convex and Gâteaux differentiable;
-
(2)
the set is bounded for all and .
The following lemma follows from Butnariu and Iusem [23] and Zălinscu [35].
Lemma 2.1 Let E be a reflexive Banach space and let be a strongly coercive Bregman function. Then
-
(1)
is one-to-one, onto and norm-to-weak∗ continuous;
-
(2)
if and only if ;
-
(3)
is bounded for all and ;
-
(4)
, is Gâteaux differentiable and .
Now, we are ready to prove the following key lemma.
Lemma 2.2 Let E be a Banach space, let be a constant and let be a convex function which is uniformly convex on bounded subsets of E. Then
for all , , and with , where is the gauge of uniform convexity of g.
Proof Without loss of generality, we may assume that and . By induction on n, for , in view of Remark 1.1 we get the desired result. Now suppose that it is true for , i.e.,
Now, we prove that the conclusion holds for . Put and observe that . Since g is convex, given assumption, we conclude that
This completes the proof. □
Lemma 2.3 Let E be a Banach space, let be a constant and let be a continuous and convex function which is uniformly convex on bounded subsets of E. Then
for all , , and with , where is the gauge of uniform convexity of g.
Proof Let and . Put and observe that for all . In view of Lemma 2.2, we obtain that
Since g is continuous and as , we have
Letting in (2.1), we conclude that
which completes the proof. □
We know the following two results; see [[35], Proposition 3.6.4].
Theorem 2.1 Let E be a reflexive Banach space and let be a convex function which is bounded on bounded subsets of E. Then the following assertions are equivalent:
-
(1)
g is strongly coercive and uniformly convex on bounded subsets of E;
-
(2)
, is bounded on bounded subsets and uniformly smooth on bounded subsets of ;
-
(3)
, is Fréchet differentiable and is uniformly norm-to-norm continuous on bounded subsets of .
Theorem 2.2 Let E be a reflexive Banach space and let be a continuous convex function which is strongly coercive. Then the following assertions are equivalent:
-
(1)
g is bounded on bounded subsets and uniformly smooth on bounded subsets of E;
-
(2)
is Fréchet differentiable and is uniformly norm-to-norm continuous on bounded subsets of ;
-
(3)
, is strongly coercive and uniformly convex on bounded subsets of .
Let E be a Banach space and let be a convex and Gâteaux differentiable function. Then the Bregman distance [32, 33] satisfies the three point identity that is
In particular, it can be easily seen that
Indeed, by letting in (2.2) and taking into account that , we get the desired result.
Lemma 2.4 Let E be a Banach space and let be a Gâteaux differentiable function which is uniformly convex on bounded subsets of E. Let and be bounded sequences in E. Then the following assertions are equivalent:
-
(1)
;
-
(2)
.
Proof The implication (1) ⟹ (2) was proved in [23] (see also [24]). For the converse implication, we assume that . Then, in view of (2.3), we have
The function g is bounded on bounded subsets of E and therefore ∇g is also bounded on bounded subsets of (see, for example, [[23], Proposition 1.1.11] for more details). This, together with (2.3)-(2.4), implies that , which completes the proof. □
The following result was first proved in [37] (see also [24]).
Lemma 2.5 Let E be a reflexive Banach space, let be a strongly coercive Bregman function and let V be the function defined by
Then the following assertions hold:
-
(1)
for all and .
-
(2)
for all and .
Corollary 2.1 [35]
Let E be a Banach space, let be a proper, lower semicontinuous and convex function and let with and . Then the following statements are equivalent.
-
(1)
There exists such that g is ρ-convex with for all .
-
(2)
There exists such that for all ; .
3 Strong convergence theorems without computational errors
In this section, we prove strong convergence theorems without computational errors in a reflexive Banach space. We start with the following simple lemma whose proof will be omitted since it can be proved by a similar argument as that in [[44], Lemma 15.5].
Lemma 3.1 Let E be a reflexive Banach space and let be a convex, continuous, strongly coercive and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, closed and convex subset of E. Let be a Bregman weak relatively nonexpansive mapping. Then is closed and convex.
Using ideas in [22], we can prove the following result.
Theorem 3.1 Let E be a reflexive Banach space and let be a strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets of E. Let C be a nonempty, closed and convex subset of E and let be an infinite family of Bregman weak relatively nonexpansive mappings from C into itself such that . Suppose in addition that for all , where I is the identity mapping on E. Let be a sequence generated by
where ∇g is the right-hand derivative of g. Let and be sequences in satisfying the following control conditions:
-
(1)
, ;
-
(2)
There exists such that , ;
-
(3)
for all and .
Then the sequence defined in (3.1) converges strongly to as .
Proof We divide the proof into several steps.
Step 1. We show that is closed and convex for each .
It is clear that is closed and convex. Let be closed and convex for some . For , we see that
is equivalent to
An easy argument shows that is closed and convex. Hence is closed and convex for each .
Step 2. We claim that for all .
It is obvious that . Assume now that for some . Employing Lemma 2.5, for any , we obtain
This implies that
This proves that . Thus, we have for all .
Step 3. We prove that , , and are bounded sequences in C.
In view of (1.9), we conclude that
This implies that the sequence is bounded and hence there exists such that
In view of Lemma 2.1(3), we conclude that the sequence is bounded. Since is an infinite family of Bregman weak relatively nonexpansive mappings from C into itself, we have for any that
This, together with Definition 2.1 and the boundedness of , implies that the sequence is bounded.
Step 4. We show that for some , where .
By Step 3, we have that is bounded. By the construction of , we conclude that and for any positive integer . This, together with (1.9), implies that
In view of (1.9), we conclude that
It follows from (3.4) that the sequence is bounded and hence there exists such that
In view of (3.3), we conclude that
This proves that is an increasing sequence in ℝ and hence by (3.5) the limit exists. Letting in (3.3), we deduce that . In view of Lemma 2.4, we get that as . This means that is a Cauchy sequence. Since E is a Banach space and C is closed and convex, we conclude that there exists such that
Now, we show that . In view of (3.3), we obtain
Since , we conclude that
This, together with (3.7), implies that
Employing Lemma 2.4 and (3.7)-(3.8), we deduce that
In view of (3.6), we get
From (3.6) and (3.9), it follows that
Since ∇g is uniformly norm-to-norm continuous on any bounded subset of E, we obtain
In view of (3.1), we have
It follows from (3.10)-(3.11) that
Since ∇g is uniformly norm-to-norm continuous on any bounded subset of E, we obtain
Applying Lemma 2.4, we derive that
It follows from the three point identity (see (2.2)) that
as .
The function g is bounded on bounded subsets of E and thus ∇g is also bounded on bounded subsets of (see, for example, [[23], Proposition 1.1.11] for more details). This implies that the sequences , , and are bounded in .
In view of Theorem 2.2(3), we know that and is strongly coercive and uniformly convex on bounded subsets. Let and be the gauge of uniform convexity of the conjugate function . Now, we fix satisfying condition (2). We prove that for any and
Let us show (3.14). For any given and , in view of the definition of the Bregman distance (see (1.7)), (1.6), Lemmas 2.3 and 2.5, we obtain
In view of (3.13), we obtain
In view of (3.14) and (3.15), we conclude that
as . From the assumption , , we have
Therefore, from the property of , we deduce that
Since is uniformly norm-to-norm continuous on bounded subsets of , we arrive at
In particular, for , we have
This, together with (3.16), implies that
Since is an infinite family of Bregman weak relatively nonexpansive mappings, from (3.6) and (3.17), we conclude that , . Thus, we have .
Finally, we show that . From , we conclude that
Since for each , we obtain
Letting in (3.18), we deduce that
In view of (1.8), we have , which completes the proof. □
Remark 3.1 Theorem 3.1 improves Theorem 1.2 in the following aspects.
-
(1)
For the structure of Banach spaces, we extend the duality mapping to a more general case, that is, a convex, continuous and strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets.
-
(2)
For the mappings, we extend the mapping from a relatively nonexpansive mapping to a countable family of Bregman weak relatively nonexpansive mappings. We remove the assumption on the mapping T and extend the result to a countable family of Bregman weak relatively nonexpansive mappings, where is the set of asymptotic fixed points of the mapping T.
-
(3)
For the algorithm, we remove the set in Theorem 1.2.
Lemma 3.2 Let E be a reflexive Banach space and let be a strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets of E. Let A be a maximal monotone operator from E to such that . Let and be the g-resolvent of A. Then is a Bregman weak relatively nonexpansive mapping.
Proof Let be a sequence such that and . Since ∇g is uniformly norm-to-norm continuous on bounded subsets of E, we obtain
It follows from
and the monotonicity of A that
for all and . Letting in the above inequality, we have for all and . Therefore, from the maximality of A, we conclude that , that is, . Hence is Bregman weak relatively nonexpansive, which completes the proof. □
As an application of our main result, we include a concrete example in support of Theorem 3.1. Using Theorem 3.1, we obtain the following strong convergence theorem for maximal monotone operators.
Theorem 3.2 Let E be a reflexive Banach space and let be a strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets of E. Let A be a maximal monotone operator from E to such that . Let such that and be the g-resolvent of A. Let be a sequence generated by
where ∇g is the right-hand derivative of g. Let and be sequences in satisfying the following control conditions:
-
(1)
, ;
-
(2)
There exists such that , ;
-
(3)
for all and .
Then the sequence defined in (3.19) converges strongly to as .
Proof Letting , , in Theorem 3.1, from (3.1) we obtain (3.19). We need only to show that satisfies all the conditions in Theorem 3.1 for all . In view of Lemma 3.2, we conclude that is a Bregman relatively nonexpansive mapping for each . Thus, we obtain
and
where is the set of all strong asymptotic fixed points of . Therefore, in view of Theorem 3.1, we have the conclusions of Theorem 3.2. This completes the proof. □
4 Strong convergence theorems with computational errors
In this section, we study strong convergence of iterative algorithms to find common fixed points of finitely many Bregman weak relatively nonexpansive mappings in a reflexive Banach space. Our algorithms take into account possible computational errors. We prove the following strong convergence theorem concerning Bregman weak relatively nonexpansive mappings.
Theorem 4.1 Let E be a reflexive Banach space and let be a strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets of E. Let and be a finite family of Bregman weak relatively nonexpansive mappings from E into int domg such that is a nonempty subset of E. Suppose in addition that , where I is the identity mapping on E. Let be a sequence generated by
where ∇g is the right-hand derivative of g. Let be a sequence in satisfying the following control conditions:
-
(1)
, ;
-
(2)
There exists such that , .
If, for each , the sequences of errors satisfy , then the sequence defined in (4.1) converges strongly to as .
Proof We divide the proof into several steps.
Step 1. We show that is closed and convex for each .
It is clear that is closed and convex. Let be closed and convex for some . For , we see that
is equivalent to
An easy argument shows that is closed and convex. Hence is closed and convex for all .
Step 2. We claim that for all .
It is obvious that . Assume now that for some . Employing Lemma 2.5, for any , we obtain
This proves that . Consequently, we see that for any .
Step 3. We prove that , and are bounded sequences in E.
In view of (1.9), we conclude that
It follows from (4.3) that the sequence is bounded and hence there exists such that
In view of Lemma 2.1(3), we conclude that the sequence and hence is bounded. Since is a finite family of Bregman weak relatively nonexpansive mappings from E into int domg, for any , we have
This, together with Definition 2.1 and the boundedness of , implies that is bounded.
Step 4. We show that for some , where .
By Step 3, we deduce that is bounded. By the construction of , we conclude that and for any positive integer . This, together with (1.9), implies that
In view of (4.6), we have
This proves that is an increasing sequence in ℝ and hence by (4.4) the limit exists. Letting in (4.6), we deduce that . In view of Lemma 2.4, we obtain that as . Thus we have is a Cauchy sequence. Since E is a Banach space, we conclude that there exists such that
Now, we show that . In view of (4.6), we obtain
Since , for all , in view of Lemma 2.4 and (4.8), we obtain that
The function g is bounded on bounded subsets of E and thus ∇g is also bounded on bounded subsets of (see, for example, [[23], Proposition 1.1.11] for more details). Since , we get
This, together with (4.9), implies that
Employing Lemma 2.4 and (4.9)-(4.10), we deduce that
In view of (4.7) and (4.11), we get
Thus, is a bounded sequence.
From (4.11) and (4.12), it follows that
Since ∇g is uniformly norm-to-norm continuous on any bounded subset of E, we obtain
Applying Lemma 2.4, we deduce that
It follows from the three point identity (see (2.2)) that