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Convergence theorems of modified Mann iterations
Fixed Point Theory and Applications volume 2013, Article number: 282 (2013)
Abstract
In this paper, we introduce the modified iterations of Mann’s type for nonexpansive mapping and asymptotically nonexpansive mapping to have the strong and weak convergence in a uniformly convex Banach space. We also proved strong convergence theorems of our modified Mann’s iteration processes for nonexpansive semigroups and asymptotically nonexpansive semigroups. The results presented in the paper give a partially affirmative answer to the open question raised by Kim and Xu (Nonlinear Anal. 64:1140-1152, 2006). Applications to the accretive operators are also included.
MSC:47H09, 47H10, 47J25.
1 Introduction
Let E be a real Banach space, C a nonempty closed convex subset of E, and a mapping. Recall that T is a nonexpansive mapping [1] if for all , and T is asymptotically nonexpansive [2] if there exists a sequence with for all n and and such that for all integers and . The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [2] as an important generalization of the class of nonexpansive mappings, who proved that if C is a nonempty closed convex subset of a real uniformly convex Banach space, and T is an asymptotically nonexpansive mapping from C into itself, then T has a fixed point. A point is a fixed point of T provided . Denote by the set of fixed points of T; that is, .
A family is said to be an asymptotically nonexpansive semigroup [3] on C with Lipschitzian constants if
-
(1)
is a bounded, measurable, continuous mapping from ;
-
(2)
;
-
(3)
for each , is a mapping from C into itself, and for each ;
-
(4)
for each and ;
-
(5)
for each ;
-
(6)
for each , the mapping is continuous.
is said to be nonexpansive semigroup on C if for all . We use to denote the common fixed point set of the semigroup ; that is, . Note that for an asymptotically nonexpansive semigroup Γ, we can always assume that the Lipschitzian constants are such that for all . L is nonincreasing in t, and ; otherwise, we replace for each , with .
As is well known, the construction of fixed point of nonexpansive mappings and asymptotically nonexpansive mappings (and of common fixed points of nonexpansive semigroups and asymptotically nonexpansive semigroups) is an important subject in the theory of nonexpansive mappings, nonlinear operator theory and their applications: in particular, in image recovery, convex feasibility problem, convex minimization problem and signal processing problem [4–9].
Iterative approximation of a fixed point for nonexpansive mappings, asymptotically nonexpansive mappings, nonexpansive semigroups and asymptotically nonexpansive semigroups in Hilbert or Banach spaces including Mann [10], Ishikawa [11] and Halpern and Mann-type iteration algorithm [12] have been studied extensively by many authors to solve nonlinear operator equations as well as variational inequalities. However, the Mann iteration for nonexpansive mappings has in general only weak convergence even in a Hilbert space. More precisely, a Mann’s iteration procedure is a sequence , which is generated by
where the initial guess is chosen arbitrarily. For example, Reich [13] proved that if E is a uniformly convex Banach space with a Fréchet differentiable norm, and if is chosen such that , then the sequence defined by (1.1) converges weakly to a fixed point of T.
Some attempts to modify the Mann iteration method (1.1) so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi [14] proposed the following modification of the Mann iteration method (1.1) for a nonexpansive mapping T in a Hilbert space H:
where denotes the metric projection from H onto a closed convex subset K of H. They proved that if the is bounded above from one, the sequence generated by (1.2) converges strongly to . Moreover, they introduced and studied an iteration process of a nonexpansive semigroup in a Hilbert space H:
Under the same condition of the sequence , and is positive real divergent sequence, the sequence generated by (1.3) converges strongly to .
Kim and Xu [15], in 2006, adapted iteration (1.2) and (1.3) to asymptotically nonexpansive mapping and asymptotically nonexpansive semigroup. More precisely, they introduced the following iteration processes for asymptotically nonexpansive mapping T and asymptotically nonexpansive semigroup , respectively, with C a closed convex bounded subset of a Hilbert space H:
where as and
where as .
They proved that both iteration processes (1.4) and (1.5) converge strongly to a fixed point of T and a common fixed point of , respectively, provided for all integers n, and is a positive real divergent sequence, using the boundedness of the closed convex subset of C and Lipschitzian constant of the mapping .
Without knowing the rate of convergence of (1.2), Kim and Xu [16] in 2005, proposed a simpler modification of Mann’s iteration method (1.1) for a nonexpansive mapping T in a uniformly smooth Banach space E,
where is an arbitrary fixed point element in C. They proved that and are two sequences in , satisfying certain assumptions, then defined by (1.6) converges to a fixed point of T.
In [15], Kim and Xu adapted iteration (1.2) and (1.3) to asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups. At the same time, they also raised the following open question.
Open question [15]
Apparently, the iteration method (1.6) is simpler than (1.2). However, we do not know if we can adapt the method (1.6) to asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups.
It is the purpose of this paper to develop iteration (1.6) to the processes for nonexpansive mappings, asymptotically nonexpansive mappings, nonexpansive semigroups and asymptotically nonexpansive semigroups in the frame of uniformly convex Banach space in Section 3 and Section 4. More precisely, we introduce the following modified Mann iteration processes for nonexpansive mappings, asymptotically nonexpansive mappings T and nonexpansive semigroups, asymptotically nonexpansive semigroups , respectively, with C a closed convex subset of a Banach space E:
and
The strong and weak convergence of the sequence to a fixed point of nonexpansive mappings, asymptotically nonexpansive mappings T are established. Strong convergence theorems for nonexpansive semigroups and asymptotically nonexpansive semigroups are also obtained. Therefore, results presented in the paper give a partially affirmative answer to the open question raised by Kim and Xu [15].
Our second modification of Mann’s iteration method (1.1) is adaption to (1.6) for finding a zero of an m-accretive operator A, for which we assume that the zero set . Our iterations process is given by
and another sequence as follows:
where for each , is the resolvent of A. We prove that only in a uniformly convex Banach space and under certain appropriate assumptions on the sequences , and which will be made precise in Section 5 that defined by (1.9) and (1.10) converge strongly to a zero of A.
We write to indicate that the sequence converges weakly to x. Similarly, will symbolize strong convergence.
2 Preliminaries
This section collects some lemmas, which will be used in the proofs for the main results in the next section.
Lemma 2.1 [17]
Let , and be sequences of nonnegative real numbers satisfying the inequality
If and , then
-
(1)
exists;
-
(2)
whenever .
Lemma 2.2 [18]
Suppose that E is a uniformly convex Banach space, and for all . Let and be two sequences of E such that , and hold for some , then .
Lemma 2.3 [19]
Let C be a nonempty closed convex subset of a uniformly convex Banach space E, and be an asymptotically nonexpansive mapping. Then is demiclosed at zero, i.e., if and , then .
Lemma 2.4 [20]
A real Banach space E is said to satisfy Opial’s condition if the condition implies
for all , .
Lemma 2.5 [21]
A mapping with a nonempty fixed point set F in C will be said to satisfy Condition (I):
If there is a nondecreasing function with , for all such that for all , where .
Lemma 2.6 [22]
Let C be a nonempty closed convex subset of a uniformly convex Banach space E, D a bounded closed convex subset of C and a nonexpansive semigroup (asymptotically nonexpansive semigroup) on C, such that . For each , then
Lemma 2.7 [23]
For and and , the following identity holds
3 Convergence to a fixed point of nonexpansive mapping and asymptotically nonexpansive mapping
In this section, we prove weak and strong convergence theorems for asymptotically nonexpansive mappings and strong convergence theorem for nonexpansive mappings.
Theorem 3.1 Let C be a nonempty closed convex subset of a uniformly convex Banach space E, and let be a nonexpansive mapping satisfying Condition (I) and . Given a point , and given that and are two sequences in such that .
Define a sequence in C by algorithm (1.6), then strongly converges to a fixed point of T.
Proof First, we observe that is bounded, if we take an arbitrary fixed point q of , noting that
we have
By Lemma 2.1 and , thus, exists. Denote
Hence, is bounded, so is . Now
By , we obtain
Since , which implies that
so that (3.2) and (3.3) give
Moreover, implies that
Thus,
given by Lemma 2.2 that
By (3.1) and , then we have
That is,
where for all , for all and for and .
Next, we prove that is a Cauchy sequence.
Since arbitrarily, and exists, consequently, exists by Lemma 2.5. From Lemma 2.5 and (3.4), we get
Since is a nondecreasing function satisfying , for all , therefore, we have
Let , since and , therefore, there exists a constant such that for all , we have
in particular,
There must exist , such that
From (3.5), it can be obtained that when ,
This implies that is a Cauchy sequence in a closed convex subset C of a Banach space E. Thus, it must converge to a point in C, let .
For all , as , thus, there exists a number such that when ,
In fact, implies that using number above, when , we have . In particular, . Thus, there must exist , such that
From (3.6) and (3.7), we get
As ϵ is an arbitrary positive number, thus, , so converges strongly to a point of T. □
Theorem 3.2 Let C be a nonempty closed convex subset of a uniformly convex Banach space E, and let be an asymptotically nonexpansive mapping satisfying Condition (I) and . Given a point , and given that and are two sequences in , the following conditions are satisfied:
-
(i)
;
-
(ii)
.
Define a sequence in C by algorithm (1.7), then strongly converges to a fixed point of T.
Proof First, we observe that is bounded, if we take an arbitrary fixed point q of , noting that
we have
Put
Thus, sequence is bounded, by Lemma 2.1 and Conditions (i), (ii), thus, exists. Denote
Hence, is bounded, so is . Now
By assumption (i), we obtain . Since , which implies that
so that gives
Moreover, implies that
Thus,
given by Lemma 2.2 that
Now,
Hence, by (3.9),
Also note that
so that Condition (i) and (3.10) give
Next, we show
We have
Hence, by (3.9) and (3.11), we get
By (3.8), we have , where
and then we assume that , so for , now
By Condition (i) and the convergence of , that is,
where , for all , for all and for .
Next, we prove that is a Cauchy sequence.
Since arbitrarily, and exists, consequently, exists by Lemma 2.5. From Lemma 2.5 and (3.12), we get
Since is a nondecreasing function satisfying , for all , therefore, we have
Let , since and , therefore, there exists a constant such that for all , we have
in particular,
There must exist , such that
From (3.13), it can be obtained that when ,
This implies that is a Cauchy sequence in a closed convex subset C of a Banach space E. Thus, it must converge to a point in C, let .
For all , as , thus, there exists a number such that when ,
In fact, implies that using number above, when , we have . In particular, . Thus, there must exist , such that
From (3.14) and (3.15), we get
As ϵ is an arbitrary positive number, thus, , so converges strongly to a point of T. □
Theorem 3.3 Let E be a uniformly convex Banach space, and let T, C and be taken as in Theorem 3.2. Assume that E satisfies Opial’s condition. If , then converges weakly to a fixed point of T.
Proof Since E is uniformly convex, from [23], E is reflexive. Again by Theorem 3.2, is bounded, there exist two arbitrary subsequences and of which are weakly convergent to x and y in C, respectively. By Theorem 3.2, and is demiclosed with respect to zero by Lemma 2.3. It follows that and . Next, we prove the uniqueness. Assuming that , and taking into account the fact that and are weakly convergent to x and y, respectively, it follows from Opial’s condition that
Arriving at a contradiction, so , then given by converges weakly to a fixed point of T. □
4 Strong convergence to a common fixed point of asymptotically nonexpansive semigroups and nonexpansive semigroups
4.1 Strong convergence theorem for nonexpansive semigroups
Theorem 4.1 Let C be a closed convex subset of a uniformly convex Banach space E, and let be a nonexpansive semigroup on C satisfying Condition (I) such that . Given a point , and given sequences and in such that and is a positive real divergent sequence.
Define a sequence in C by (1.8), then strongly converges to a common fixed point of .
Proof We first show that is bounded, if we take a fixed point q of .
we have
Now, an induction yields
Hence, is bounded, so is . We now denote D, the subset of C,
Also
As in the proof of Theorem 3.1, we get
Moreover, implies that
Thus,
given by Lemma 2.2 that
Now,
by Lemma 2.6, we get
for every . From (4.1), we obtain
for every .
Since is a nonexpansive semigroup, and is a positive real divergent sequence, then, for all and the bounded closed convex subset D of C containing ,
As in the proof of Theorem 3.1, we have (). □
4.2 Strong convergence theorem for asymptotically nonexpansive semigroups
In this part, assume that is an asymptotically nonexpansive semigroup defined on a nonempty closed convex subset C of a Banach space E. Recall that we use to denote Lipschitzian constant of the mapping , and assume that is bounded and measurable so that the integral exists for all . Recall also that for all , is nonincreasing in t, and . In the rest of this part, we put for each .
Theorem 4.2 Let C be a closed convex subset of a uniformly convex Banach space E, and let be an asymptotically nonexpansive semigroup on C satisfying Condition (I) such that . Given a point , and given sequences and in , is a positive real divergent sequence, the following conditions are satisfied:
-
(i)
;
-
(ii)
.
Define a sequence in C by (1.8), then strongly converges to a common fixed point of .
Proof We first show that is bounded if we take a fixed point q of .
we have
Now, an induction yields
Since , hence, is bounded, so is . We now denote D, the subset of C
Also
Thus, by Condition (i), (ii) and following from Lemma 2.1, there exists .
As in the proof of Theorem 3.2, we get
Moreover, , which implies that
Thus,
given by Lemma 2.2,
Now,
by Lemma 2.6, we get
for every . From (4.2), we obtain
for every .
Since is asymptotically nonexpansive semigroup, and is a positive real divergent sequence, then, for all , and for the bounded closed convex subset D of C containing ,
As in the proof of Theorem 3.2, we have (). □
5 Application
Let E be a real Banach space. Recall that an operator (possibly multivalued) A with domain and range in E is said to be accretive if, for each and (), there exists a such that
where J is the normalized duality map from E to the dual space given by
An accretive operator A is m-accretive if for all . Denote the zero set of A by
For an m-accretive operator A with and convex, the problem of finding a zero of A, i.e.,
has extensively been investigated due to its applications in related problems such as minimization problems, variational inequality problems and nonlinear evolution equations.
It is known that the resolvent of A, defined by
for , is a nonexpansive mapping from E to C, and it is straightforward to see that F coincides with the fixed point set of for any . Therefore, (5.1) is equivalent to the fixed point problem . Then an interesting approach to solving this problem is via iterative methods for nonexpansive mappings. We need the resolvent identity [23].
Theorem 5.1 Let E be a uniformly convex Banach space, and let A be an m-accretive operator in E such that , is nonexpansive for all satisfying Condition (I). Given a point , and given sequences and in , the following conditions are satisfied:
-
(i)
;
-
(ii)
for some and for all .
Define a sequence by (1.9), then strongly converges to a zero of A.
Proof Take any arbitrary , it follows from Lemma 2.1 that exists. From Lemma 2.2, it can be shown that . Since is nonexpansive for all satisfying Condition (I), it follows from Lemma 2.7 that . Therefore, all the conditions in Theorem 3.1 are satisfied. The conclusion of Theorem 5.1 can be obtained from Theorem 3.1 immediately. □
Theorem 5.2 Let E be a uniformly convex Banach space, and let A be an m-accretive operator in E such that , is nonexpansive for all () satisfying Condition (I). Given sequences and in , the following conditions are satisfied:
-
(i)
;
-
(ii)
for some and for all .
Define a sequence by (1.10), then strongly converges to a zero of A.
Proof Only a sketch of the proof is given here. Take any arbitrary , it follows from Lemma 2.1 that exists. From Lemma 2.2, it can be shown that (). Since is nonexpansive for all satisfying Condition (I), it follows from Lemma 2.7 that . Therefore, all the conditions in Theorem 3.1 are satisfied. The conclusion of Theorem 5.2 can be obtained from Theorem 3.1 immediately. □
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant No. 11171046).
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Chen, J., Wu, D. Convergence theorems of modified Mann iterations. Fixed Point Theory Appl 2013, 282 (2013). https://doi.org/10.1186/1687-1812-2013-282
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DOI: https://doi.org/10.1186/1687-1812-2013-282