- Research Article
- Open access
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Robustness of Mann Type Algorithm with Perturbed Mapping for Nonexpansive Mappings in Banach Spaces
Fixed Point Theory and Applications volume 2010, Article number: 734181 (2010)
Abstract
The purpose of this paper is to study the robustness of Mann type algorithm in the sense that approximately perturbed mapping does not alter the convergence of Mann type algorithm. It is proven that Mann type algorithm with perturbed mapping remains convergent in a Banach space setting where
,
a nonexpansive mapping,
,
, errors and
a strongly accretive and strictly pseudocontractive mapping.
1. Introduction
Let be a nonempty closed convex subset of a real Banach space
, and
a nonexpansive mapping (i.e.,
for all
). We use
to denote the set of fixed points of
; that is,
. Throughout this paper it is assumed that
. Construction of fixed points of nonlinear mappings is an important and active research area. In particular, iterative methods for finding fixed points of nonexpansive mappings have received vast investigation since these methods find applications in a variety of applied areas of variational inequality problems, equilibrium problems, inverse problems, partial differential equations, image recovery, and signal processing (see, e.g., [1–17]).
In 1953, Mann [18] introduced an iterative algorithm which is now referred to as Mann's algorithm. Most of the literature deals with the special case of the general Mann's algorithm; that is, for an arbitrary initial guess , the sequence
is generated by the recursive manner
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ1_HTML.gif)
where is a convex subset of a Banach space
is a mapping and
is a sequence in the interval
. It is well known that Mann's algorithm can be employed to approximate fixed points of nonexpansive mappings and zeros of (strongly) accretive mappings in Hilbert spaces and Banach spaces. Many convergence theorems have been announced and published by a large number of authors. A typical convergence result in connection with the Mann's algorithm is the following theorem proved by Ishikawa [19].
Theorem IS (see [19])
Let be a nonempty subset of a Banach space
and let
be a nonexpansive mapping. Let
be a real sequence satisfying the following control conditions:
(a);
(b).
Let be defined by (1.1) such that
for all
. If
is bounded then
as
.
The interest and importance of Theorem IS lie in the fact that strong or weak convergence of the sequence can be achieved under certain appropriate assumptions imposed on the mapping
, the domain
or the space
. In an infinite-dimensional space
, Mann's algorithm has only weak convergence, in general. In fact, it is known that if the sequence
is such that
, then Mann's algorithm converges weakly to a fixed point of
provided that the underlying space
is a Hilbert space or more general, a uniformly convex Banach space which has a Fréchet differentiable norm or satisfies Opial's property. See, for example, [20, 21].
The study of the robustness of Mann's algorithm is initiated by Combettes [22] where he considered a parallel projection method algorithm in signal synthesis (design and recovery) problems in a real Hilbert space as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ2_HTML.gif)
where for each ,
is the (nearest point) projection of a signal
onto a closed convex subset
of
[23] (
is interpreted as the
th constraint set of the signals),
is a sequence of relaxation parameters in
are strictly positive weights such that
, and
stands for the error made in computing the projection onto
at iteration
. Then he proved the following robustness result of algorithm (1.2).
Theorem 1.1 (see [22]).
Assume . Assume also
(i)
(ii).
Then the sequence generated by (1.2) converges weakly to a point in
.
Define a mapping by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ3_HTML.gif)
and put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ4_HTML.gif)
Since is a projection, the mapping
is nonexpansive. Thus
and algorithm (1.2) can be rewritten as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ5_HTML.gif)
where is given by (1.3). Note that
can be written as
and thus
is nonexpansive. Note also that
. Furthermore, conditions (i) and (ii) in Theorem 1.1 can be stated as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_IEq72_HTML.gif)
.
Very early, some authors had considered Mann iterations in the setting of uniformly convex Banach spaces and established strong and weak convergence results for Mann iterations; see, e.g., [24, 25]. Recently, Kim and Xu [26] studied the robustness of Mann's algorithm for nonexpansive mappings in Banach spaces and extended Combettes' robustness result (Theorem 1.1 above) for projections from Hilbert spaces to the setting of uniformly convex Banach spaces.
Theorem 1.2 (see [26, Theorem 3.3]).
Assume that is a uniformly convex Banach space. Assume, in addition, that either
has the KK- property or
satisfies Opial's property. Let
be a nonexpansive mapping such that
. Given an initial guess
. Let
be generated by the following perturbed Mann's algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ6_HTML.gif)
where and
satisfy the following properties:
(i),
(ii).
Then the sequence converges weakly to a fixed point of
.
Further, Kim and Xu [26] also extended the robustness to nonexpansive mappings which are defined on subsets of a Hilbert space and to accretive operators.
Theorem 1.3 (see [26, Theorem 4.1]).
Let be a nonempty closed convex subset of a Hilbert space
and
a nonexpansive mapping with
. Let
be generated from an arbitrary
via one of the following algorithms (1.7) and (1.7):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ7_HTML.gif)
where the sequences and
are such that
(i),
(ii).
Then converges weakly to a fixed point of
.
Theorem 1.4 (see [26, Theorem 5.1]).
Let be a uniformly convex Banach space. Assume in addition that either
has the KK- property or
satisfies Opial's property. Let
be an
-accretive operator in
such that
. Moreover, assume that
and
satisfy the following properties:
(i);
(ii);
(iii), where
and
are two constants;
(iv).
Then the sequence generated from an arbitrary
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ8_HTML.gif)
converges weakly to a point of .
Let be a real reflexive Banach space. Let
be a nonexpansive mapping with
. Assume that
is
-strongly accretive and
-strictly pseudocontractive with
where
. In this paper, inspired by Combettes' robustness result (Theorem 1.1 above) and Kim and Xu's robustness result (Theorem 1.2 above) we will consider the robustness of Mann type algorithm with perturbed mapping, which generates, from an arbitrary initial guess
, a sequence
by the recursive manner
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ9_HTML.gif)
where and
are sequences in
and in
, respectively, such that
(i);
(ii);
(iii).
More precisely, we will prove under conditions (i)–(iii) the weak convergence of the algorithm (1.9) in a uniformly convex Banach space which either has the KK- property or satisfies Opial's property. This theorem extends Kim and Xu's robustness result (Theorem 1.2 above) from Mann's algorithm (1.6) with errors to Mann type algorithm (1.9) with perturbed mapping
. On the other hand, we also extend Kim and Xu's robustness results (Theorems 1.3 and 1.4 above) for nonexpansive mappings which are defined on subsets of a Hilbert space and accretive operators in a uniformly convex Banach space from Mann's algorithm with errors to Mann type algorithm with perturbed mapping.
Throughout this paper, we use the following notations:
(i)stands for weak convergence and
for strong convergence,
(ii) denotes the weak
-limit set of
.
2. Preliminaries
Let be a real Banach space. Recall that the norm of
is said to be Fréchet differentiable if, for each
, the unit sphere of
, the limit
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ10_HTML.gif)
exists and is attained uniformly in . In this case, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ11_HTML.gif)
for all , where
is the normalized duality map from
to
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ12_HTML.gif)
is the duality pairing between
and
, and
is a function defined on
such that
. Examples of Banach spaces which have a Fréchet differentiable norm include
and
for
(these spaces are actually uniformly smooth).
It is known that a Banach space is Fréchet differentiable if and only if the duality map
is single-valued and norm-to-norm continuous.
We need the concept of the KK-property. A Banach space is said to have the KK-property (the Kadec-Klee property) if, for any sequence
in
, the conditions
and
imply that
. It is known [27, Remark 3.2] that the dual space of a reflexive Banach space with a Fréchet differentiable norm has the KK-property.
Recall now that satisfies Opial's property [28] provided that, for each sequence
in
, the condition
implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ13_HTML.gif)
It is known [28] that each enjoys this property, while
does not unless
. It is known [29] that any separable Banach space can be equivalently renormed so that it satisfies Opial's property.
Recall that a Banach space is said to be uniformly convex if, for each
, the modulus of convexity
of
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ14_HTML.gif)
is positive.
We need an inequality characterization of uniform convexity.
Lemma 2.1 (see [30]).
Given a number . A real Banach space
is uniformly convex if and only if there exists a continuous strictly increasing function
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ15_HTML.gif)
for all and
such that
and
.
A mapping with domain
and range
in
is called
-strongly accretive if for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ16_HTML.gif)
for some .
is called
-strictly pseudocontractive if for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ17_HTML.gif)
for some . It is easy to see that (2.8) can be rewritten as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ18_HTML.gif)
The following proposition will be used frequently throughout this paper. For the sake of completeness, we include its proof.
Proposition 2.2.
Let be a real Banach space and
a mapping.
(i)If is a
-strictly pseudocontractive, then
is Lipschitz continuous with constant
(ii)If is
-strongly accretive and
-strictly pseudocontractive with
, then for each fixed
, the mapping
has the following property:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ19_HTML.gif)
Proof.
-
(i)
From (2.9), we derive
(2.11)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ21_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ22_HTML.gif)
and so is Lipschitz continuous with constant
.
-
(ii)
From (2.8) and (2.9), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ23_HTML.gif)
Since , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ24_HTML.gif)
Consequently, for each fixed , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ25_HTML.gif)
This shows that inequality (2.10) holds.
Proposition 2.3.
Let be a uniformly convex Banach space and
a nonempty closed convex subset of
.
(i)Reference [31] (demiclosedness principle). If is a nonexpansive mapping and if
is a sequence in
such that
and
, then
.
(ii)Reference [32]. If is also bounded, then there exists a continuous, strictly increasing, and convex function
(depending only on the diameter of
) with
and such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ26_HTML.gif)
for all , and nonexpansive mappings
.
We also use the following elementary lemma.
Lemma 2.4 (see [33]).
Let and
be sequences of nonnegative real numbers such that
and
for all
. Then
exists.
3. Robustness of Mann Type Algorithm with Perturbed Mapping
Let be a real reflexive Banach space. Let
be a nonexpansive mapping with
. Assume that
is
-strongly accretive and
-strictly pseudocontractive with
. We now discuss the robustness of Mann type algorithm with perturbed mapping, which generates, from an initial guess
, a sequence
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ27_HTML.gif)
where and
are sequences in
and in
, respectively, such that
(i);
(ii);
(iii).
We remark that Mann type algorithm with perturbed mapping is based on Mann iteration method and steepest-descent method. Indeed, in algorithm (3.1), one iteration step "" is taken from Mann iteration method, and another iteration step "
" is taken from steepest-descent method.
We first discuss some properties of algorithm (3.1).
Lemma 3.1.
Let be generated by algorithm (3.1) and let
Then
exists.
Proof.
We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ28_HTML.gif)
The conclusion of the lemma is a consequence of Lemma 2.4.
Proposition 3.2.
Let be a uniformly convex Banach space.
(i)For all and
,
exists.
(ii)If, in addition, the dual space of
has the
-property, then the weak
-limit set of
,
, is a singleton.
Proof.
-
(i)
For integers
, define the mappings
and
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ29_HTML.gif)
and . It is easy to see that
. First, let us show that
and
are nonexpansive. Indeed, for all
, using Proposition 2.2 no. (ii) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ30_HTML.gif)
Thus is nonexpansive (due to
) and so is
.
Second, let us show that for each ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ31_HTML.gif)
Indeed, whenever , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ32_HTML.gif)
This implies that inequality (3.5) holds for . Assume that inequality (3.5) holds for some
. Consider the case of
. Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ33_HTML.gif)
This shows that inequality (3.5) holds for the case of . Thus, by induction, we know that inequality (3.5) holds for all
.
Now set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ34_HTML.gif)
By Proposition 2.3 no. (ii) and noticing inequality (3.5) we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ35_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ36_HTML.gif)
Since exists and
and
are convergent, we conclude from (3.10) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ37_HTML.gif)
Also, since, for all ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ38_HTML.gif)
it follows from (3.11) and (3.12) that exists.
-
(ii)
This is Lemma 3.2 of [27].
Now we can state and prove the main result of this section.
Theorem 3.3.
Assume that is a uniformly convex Banach space. Assume, in addition, that either
has the
-property or
satisfies Opial's property. Let
be a nonexpansive mapping such that
and
-strongly accretive and
-strictly pseudocontractive with
. Given an initial guess
. Let
be generated by the following Mann type algorithm with perturbed mapping
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ39_HTML.gif)
where and
satisfy the following properties:
(i);
(ii);
(iii).
Then the sequence converges weakly to a fixed point of
.
Proof.
Fix and select a number
large enough so that
for all
. Let
satisfy
for all
. By Lemma 2.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ40_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ41_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ42_HTML.gif)
In particular, . Due to condition (i), we must have that
. Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ43_HTML.gif)
However, since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ44_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ45_HTML.gif)
and, by Lemma 2.4, exists and hence, by (3.17),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ46_HTML.gif)
Notice that, by the demiclosedness principle of , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ47_HTML.gif)
Hence to prove that converges weakly to a fixed point of
, it suffices to show that
is a singleton. We distinguish two cases. First assume that
has the KK-property. Then that
is a singleton is guaranteed by Proposition 3.2 no. (ii).
Next assume that satisfies Opial's property. Take
and let
and
be subsequences of
such that
and
, respectively. If
, we reach the following contradiction:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ48_HTML.gif)
This shows that is a singleton. The proof is therefore complete.
4. The Case Where Mappings Are Defined on Subsets
We observe that if the domain is a proper closed convex subset
of
, then the vectors
and
may not belong to
. In this case the next iterate
may not be well defined by (3.13). In order to consider this situation, we will use the nearest projection
and for the projection to be nonexpansive, we have to restrict our spaces to be Hilbert spaces.
Let be a real Hilbert space with inner product
and norm
. Given a closed convex subset
of
. Recall that the (nearest point) projection
from
onto
assigns each point
with its (unique) nearest point in
which is denoted by
. Namely,
is the unique point in
with the property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ49_HTML.gif)
Note that is nonexpansive.
Let be a nonexpansive mapping with
and
-strongly monotone and
-strictly pseudocontractive with
. Starting with
and after
in
is defined, we have two ways to define the next iterate
: either applying the projection
to the vectors
and
and defining
as the convex combination of
and
, or projecting a convex combination of
and
onto
to define
. More precisely, we define
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ50_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ51_HTML.gif)
Theorem 4.1.
Let be a nonempty closed convex subset of a Hilbert space
. Let
be a nonexpansive mapping with
and
-strongly monotone and
-strictly pseudocontractive with
. Let
be generated by either (4.2) or (4.3) where the sequences
and
are such that
(i);
(ii);
(iii).
Then converges weakly to a fixed point of
.
Proof.
Given . Assume that
is generated by (4.2). Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ52_HTML.gif)
Hence exists; in particular,
is bounded. Let
be a constant such that
for all
.
We compute
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ53_HTML.gif)
That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ54_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ55_HTML.gif)
In particular (noticing assumption (i)),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ56_HTML.gif)
We also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ57_HTML.gif)
Moreover, noticing
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ58_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ59_HTML.gif)
Similarly, if is generated by algorithm (4.3), then relations (4.4)–(4.11) still hold.
It is now readily seen that (4.11) together with Lemma 2.4 implies that exists, which together with (4.8) further implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ60_HTML.gif)
Equation (4.12) implies that , due to the demiclosedness principle. Finally, repeating the last part of the proof of Theorem 3.3 in the case of Opial's property, we see that
converges weakly to a fixed point of
. The proof is therefore complete.
Finally in this section, we consider the case of accretive operators. Recall that a multivalued operator with domain
and range
in a Banach space
is said to be accretive if, for each
and
, there is
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ61_HTML.gif)
where is the duality map from
to the dual space
. An accretive operator
is
-accretive if
for all
.
Denote by the zero set of
; that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ62_HTML.gif)
Throughout the rest of this paper it is always assumed that is
-accretive and
is nonempty.
Denote by the resolvent of
for
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ63_HTML.gif)
It is known that is a nonexpansive mapping from
to
which will be assumed convex (this is so if
is uniformly convex). It is also known that
for
.
Now consider the problem of finding a zero of an -accretive operator
in a Banach space
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ64_HTML.gif)
We will study the convergence of the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ65_HTML.gif)
where is a perturbed mapping, the initial guess
is arbitrary,
and
are two sequences in
is a sequence of positive numbers, and
is an error sequence in
.
Theorem 4.2.
Let be a uniformly convex Banach space. Assume in addition that either
has the
-property or
satisfies Opial's property. Let
be an
-accretive operator in
such that
and let
be
-strongly accretive and
-strictly pseudocontractive with
. Moreover, assume that
and
satisfy the following properties:
(i);
(ii);
(iii);
(iv), where
and
are two constants;
(v).
Then the sequence generated by algorithm (4.17) converges weakly to a point of
.
Proof.
The proof is a refinement of that of Theorem 3.3 given in Section 3 and [34, Theorem 3.3] together with Proposition 3.2. So we only sketch it.
Let . By (4.17), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ66_HTML.gif)
By Lemma 2.4, we see that exists.
With slight modifications of the proof of Theorem 3.3 (replacing by
), we can obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ67_HTML.gif)
Now noticing
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ68_HTML.gif)
and letting for all
, we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ69_HTML.gif)
By mimicking the proof of Theorem 3.3 in [34], we can show that, in the case of ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ70_HTML.gif)
and in the case of ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ71_HTML.gif)
where is such that
for all
. In either case we conclude from (4.22) and (4.23) that
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ72_HTML.gif)
where fulfills
. By Lemma 2.4, (4.24) implies that
(
) exists. This together with the assumption (iv) and (4.19) implies that
. So, by Lemma 3.3 in [34], we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F734181/MediaObjects/13663_2009_Article_1335_Equ73_HTML.gif)
By the demiclosedness principle, (4.25) ensures that . Repeating the last part of the proof of Theorem 3.3, we conclude that
converges weakly to a point of
.
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Acknowledgments
This research was partially supported by Grant no. NSC 98-2923-E-110-003-MY3 and was also partially supported by the Leading Academic Discipline Project of Shanghai Normal University (DZL707), Innovation Program of Shanghai Municipal Education Commission Grant (09ZZ133), National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), Science and Technology Commission of Shanghai Municipality Grant (075105118), and Shanghai Leading Academic Discipline Project (S30405).
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Ceng, L., Liou, Y. & Yao, J. Robustness of Mann Type Algorithm with Perturbed Mapping for Nonexpansive Mappings in Banach Spaces. Fixed Point Theory Appl 2010, 734181 (2010). https://doi.org/10.1155/2010/734181
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DOI: https://doi.org/10.1155/2010/734181