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Approximating Fixed Points of Non-Lipschitzian Mappings by Metric Projections
Fixed Point Theory and Applications volume 2011, Article number: 976192 (2011)
Abstract
We define and study a new iterative algorithm inspired by Matsushita and Takahashi (2008). We establish a strong convergence theorem of the proposed algorithm for asymptotically nonexpansive in the intermediate sense mappings in uniformly convex and smooth Banach spaces by using metric projections. This theorem generalizes and refines Matsushita and Takahashi's strong convergence theorem which was established for nonexpansive mappings.
1. Introduction and Preliminaries
Let be a real Banach space, and let
be a nonempty subset of
. A mapping
is said to be asymptotically nonexpansive if there exists a sequence
in
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ1_HTML.gif)
for all and each
. If
, then
is known as nonexpansive mapping.
is said to be asymptotically nonexpansive in the intermediate sense [1] provided
is uniformly continuous and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ2_HTML.gif)
is said to be Lipschitzian if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ3_HTML.gif)
for all .
It follows from the above definitions that every asymptotically nonexpansive mapping is asymptotically nonexpansive in the intermediate sense and Lipschitzian mapping but the converse does not hold such as in the following example.
Example 1.1.
Let ,
and
. We define
and for each
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ4_HTML.gif)
We see that is continuous on the compact interval
and so it is uniformly continuous. Consider the function
defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ5_HTML.gif)
Then, for all
and
uniformly. On the other hand, compactness of
gives that for each
there exist
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ6_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ7_HTML.gif)
Thus, is asymptotically nonexpansive in the intermediate sense.
It is easy to see that is differentiable on
and
for all
. Let there exist
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ8_HTML.gif)
for all . Now, choose
such that
. Then, for each
with
, it follows from (1.8) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ9_HTML.gif)
This contradiction shows that is not Lipschitzian mapping and so it is not asymptotically nonexpansive mapping. Another example of an asymptotically nonexpansive in the intermediate sense mapping which is not asymptotically nonexpansive can be found in [2].
It is known [3] that if is a uniformly convex Banach space and
is asymptotically nonexpansive in the intermediate sense self-mapping of a bounded closed convex subset
of
, then
, where
denotes the set of all fixed points of
. Let
be the dual of
. We denote the value of
at
by
. When
is a sequence in
, we denote strong convergence of
to
by
and weak convergence by
. A Banach space
is said to be strictly convex if
for all
with
and
. A Banach space
is also said to be uniformly convex if
for any two sequences
and
in
such that
and
. A Banach space
is said to have Kadec-Klee property if for every sequence
in
,
and
imply that
. Every uniformly convex Banach space has the Kadec-Klee property [4]. Let
be the unit sphere of
. Then the Banach space
is said to be smooth if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ10_HTML.gif)
exists for each . The normalized duality mapping
from
to
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ11_HTML.gif)
for all . It is known that a Banach space
is smooth if and only if the normalized duality mapping
is single-valued. Some properties of duality mapping have been given in [4–6]. Let
be a closed convex subset of a reflexive, strictly convex and smooth Banach space
. Then for any
there exists a unique point
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ12_HTML.gif)
The mapping defined by
is called the metric projection from
onto
. Let
and
. Then, it is known that
if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ13_HTML.gif)
Fixed points of nonlinear mappings play an important role in solving systems of equations and inequalities that often arise in applied sciences. Approximating fixed points of asymptotically nonexpansive and nonexpansive mappings with implicit and explicit iterative schemes has been studied by many authors (see, e.g., [8–14]).
On the other hand, using the metric projection, Nakajo and Takahashi [15] introduced an iterative algorithm in the framework of Hilbert spaces and gave strong convergence theorem for nonexpansive mappings. Xu [16] extended Nakajo and Takahashi's theorem to Banach spaces by using the generalized projection. Recently, Matsushita and Takahashi [17] introduced an iterative algorithm for nonexpansive mappings in Banach spaces as follows.
Let be a nonempty convex bounded subset of a uniformly convex and smooth Banach space
, and let
be a nonexpansive self-mapping of
. For a given
, compute the sequence
by the iterative algorithm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ14_HTML.gif)
where denotes the convex closure of the set
and
is a sequence in
with
. They proved that
generated by (1.14) converges strongly to a fixed point of
.
Inspired and motivated by these facts, we introduce a new iterative algorithm to find fixed points of asymptotically nonexpansive in the intermediate sense mappings in a uniformly convex and smooth Banach space. Let , and compute the sequence
by the iterative algorithm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ15_HTML.gif)
where is a sequence in
with
and
is the metric projection from
onto
.
In the sequel, the following lemmas are needed to prove our main convergence theorem.
Lemma 1.2 (see [18, Lemma  1.5]).
Let be a nonempty bounded closed convex subset of a uniformly convex Banach space
and
be a mapping which is asymptotically nonexpansive in the intermediate sense. For each
, there exist integers
and
such that if
is any integer,
,
and if
for
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ16_HTML.gif)
for any numbers with
.
Lemma 1.3 (see [18, Lemma  1.6]).
Let be a real uniformly convex Banach space, let
be a nonempty closed convex subset of
, and let
be a mapping which is asymptotically nonexpansive in the intermediate sense. If
is a sequence in
converging weakly to
and if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ17_HTML.gif)
then is demiclosed at zero; that is, for each sequence
in
, if
for some
and
, then
.
2. Main Results
In this section, we study the iterative algorithm (1.15) to find fixed points of asymptotically nonexpansive in the intermediate sense mappings in a uniformly convex and smooth Banach space. We first prove that the sequence generated by (1.15) is well-defined. Then, we prove that
converges strongly to
, where
is the metric projection from
onto
.
Lemma 2.1.
Let be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space
, and let
be a mapping which is asymptotically nonexpansive in the intermediate sense. If
, then the sequence
generated by (1.15) is well-defined.
Proof.
It is easy to check that is closed and convex and
for each
. Moreover
and so
. Suppose
. Since
, it follows from (1.13) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ18_HTML.gif)
for all and so for all
, that is
. Thus,
. By mathematical induction, we obtain that
for all
. Therefore,
is well-defined.
In order to prove our main result, the following lemma is needed.
Lemma 2.2.
Let be a nonempty bounded closed convex subset of a uniformly convex and smooth Banach space
, and let
be a mapping which is asymptotically nonexpansive in the intermediate sense. If
is the sequence generated by (1.15), then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ19_HTML.gif)
for all integers .
Proof.
Let be fixed, and let
be arbitrary. We take
for simplicity. Since
, we have
. Since
, there exist elements
in
and numbers
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ20_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ21_HTML.gif)
for all . We put
,
, and
. The inequality (2.4) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ22_HTML.gif)
for all . Now, let
, and choose an integer
and
with
as in Lemma 1.2. Since
and
, we may choose an integer
such that for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ23_HTML.gif)
This together with (2.5) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ24_HTML.gif)
for all and all
. Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ25_HTML.gif)
and so by Lemma 1.2 we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ26_HTML.gif)
where . It follows from (2.3)–(2.9) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ27_HTML.gif)
for all ; that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ28_HTML.gif)
This completes the proof.
Now, we state and prove the strong convergence theorem of the iterative algorithm (1.15).
Theorem 2.3.
Let be a nonempty bounded closed convex subset of a uniformly convex and smooth Banach space
, let
be a mapping which is asymptotically nonexpansive in the intermediate sense and let
be the sequence generated by (1.15). Then
converges strongly to the element
of
.
Proof.
Put . Since
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ29_HTML.gif)
for all . On the other hand, we observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ30_HTML.gif)
and so by uniform continuity of and Lemma 2.2 we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ31_HTML.gif)
Since is bounded, there exists a subsequence
of
such that
. It follows from (2.14) and Lemma 1.3 (demiclosedness of
) that
. From the weakly lower semicontinuity of norm and (2.12), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ32_HTML.gif)
This together with the uniqueness of implies that
, and hence
. This gives that
. By using the same argument as in proof of (2.15), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F976192/MediaObjects/13663_2010_Article_1449_Equ33_HTML.gif)
Since is uniformly convex, by Kadec-Klee property, we obtain that
. It follows that
. This completes the proof.
Since every nonexpansive mapping is asymptotically nonexpansive and every asymptotically nonexpansive mapping is asymptotically nonexpansive in the intermediate sense, we have the following result which generalizes and refines the strong convergence theorem of Matsushita and Takahashi [17, Theorem  3.1].
Corollary 2.4.
Let be a nonempty bounded closed convex subset of a uniformly convex and smooth Banach space
, let
be a nonexpansive self-mapping of
, and let
be the sequence generated by (1.15). Then
converges strongly to the element
of
.
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The authors thank the referees and the editor for their careful reading of the manuscript and their many valuable comments and suggestions for the improvement of this paper.
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Dehghan, H., Gharajelo, A. & Afkhamitaba, D. Approximating Fixed Points of Non-Lipschitzian Mappings by Metric Projections. Fixed Point Theory Appl 2011, 976192 (2011). https://doi.org/10.1155/2011/976192
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DOI: https://doi.org/10.1155/2011/976192