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Common Fixed Point Theorems on Weakly Contractive and Nonexpansive Mappings
Fixed Point Theory and Applications volume 2008, Article number: 469357 (2007)
Abstract
A family of commuting nonexpansive self-mappings, one of which is weakly contractive, are studied. Some convergence theorems are established for the iterations of types Krasnoselski-Mann, Kirk, and Ishikawa to approximate a common fixed point. The error estimates of these iterations are also given.
1. Introduction and Preliminaries
Let be a metric space and
. A mapping
is said to be nonexpansive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ1_HTML.gif)
and it is said to be weakly contractive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ2_HTML.gif)
where is continuous and nondecreasing such that
is positive on
,
and
.
It is evident that is contractive if it is weakly contractive with
, where
, and it is nonexpansive if it is weakly contractive.
As an important extension of the class of contractive mappings, the class of weakly contractive mappings was introduced by Alber and Guerre-Delabriere [1]. In Hilbert and Banach spaces, Alber et al. [1–4] and Rhoades [5] established convergence theorems on iteration of fixed point for weakly contractive single mapping.
Inspired by [2, 5, 6], the purpose of this paper is to study a family of commuting nonexpansive mappings, one of which is weakly contractive, in arbitrary complete metric spaces and Banach spaces.
We will establish some convergence theorems for the iterations of types Krasnoselski-Mann, Kirk, and Ishikawa to approximate a common fixed point and to give their error estimates.
Throughout this paper, we assume that is the set of fixed points of a mapping
, that is,
;
is defined by the antiderivative (indefinite integral) of
on
, that is,
, and
is the inverse function of
.
We define iterations which will be needed in the sequel.
Suppose that is a metric space and
,
is a family of commuting self-mappings of
and
. The iteration
of type Krasnoselski-Mann (see [7, 8]) is cyclically defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ3_HTML.gif)
For convenience, we write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ4_HTML.gif)
where the function takes values in
.
Let be a closed convex subset of the normed space
. Then the iteration
of type Kirk (see [5, 9]) is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ5_HTML.gif)
Again, the iteration of type lshikawa with error (see [10–12]) is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ6_HTML.gif)
where ,
,
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ7_HTML.gif)
We will make use of following result in the proof of Theorem 2.4.
Lemma 1.1 (see [12]).
Suppose that ,
are two sequences of nonnegative numbers such that
, for all
. If
, then
exists.
2. Main Result
Theorem 2.1.
Let be a complete metric space and let
be a family of commuting self-mappings, where
are all nonexpansive and
is weakly contractive, then there is a unique common fixed point
and the iteration
of type Krasnoselski-Mann generated by (1.4) converges in metric to
, with the following error estimate:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ8_HTML.gif)
where is the Gauss integer of
.
Proof.
The uniqueness of fixed point of is clear from (1.2). Hence, the common fixed point of
is unique. Let
be an arbitrary point in
and let
be an iteration of type Krasnoselski-Mann generated by (1.4). Since
is commutative, then we have
. Suppose that
and
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ9_HTML.gif)
Write for fixed
. Then
is a subsequence of
. Since
is nonexpansive and
is weakly contractive, then we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ10_HTML.gif)
which shows , that is,
is a nonincreasing sequence of nonnegative real numbers. Therefore, it tends to a limit
. If
, then, by nondecreasity of
,
Thus, from (2.3) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ11_HTML.gif)
a contradiction for large enough. Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ12_HTML.gif)
By (2.5), for any given , there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ13_HTML.gif)
We claim that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ14_HTML.gif)
In fact, from (2.6) we see that (2.7) holds when . Suppose that
. If
, then from (2.6) we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ15_HTML.gif)
If , then
we also get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ16_HTML.gif)
Therefore, by induction we derive that (2.7) holds. Since is arbitrary,
is a Cauchy sequence. As
is complete, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ17_HTML.gif)
Observe that are all continuous, so is
. From (2.10), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ18_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ19_HTML.gif)
By (1.1), (1.2), and (2.11), we deduce
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ20_HTML.gif)
which shows
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ21_HTML.gif)
From (2.12), it implies that is a common fixed point of
that is,
. Hence,
. By (2.10) and (2.14), we conclude
. Set
From (2.3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ22_HTML.gif)
Since is continuous and nondecreasing, using (2.15), it yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ23_HTML.gif)
Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ24_HTML.gif)
From (2.16) and (2.17), we obtain the error estimate (2.1). This completes the proof.
Remark 2.2.
If in Theorem 2.1, where
is the identity mapping of
, then we conclude that the sequence
converges to the unique common fixed point
of weakly contractive mapping
, with the error estimate
, where
. Thus, our Theorem 2.1 is a generalization of the corresponding theorem of Rhoades [5].
Theorem 2.3.
Let be a Banach space and let
be a nonempty closed convex set. Let
be a family of commuting self-mappings, where
are all nonexpansive and
is weakly contractive. Then, for any
, the iteration
of type Kirk generated by (1.5) converges strongly to a unique common fixed point
, with the following error estimate:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ25_HTML.gif)
Proof.
Applying Theorem 2.1, we can suppose that is a unique common fixed point of
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ26_HTML.gif)
we derive that is a fixed point of
. Since
are all nonexpansive,
is weakly contractive, and
, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ27_HTML.gif)
The inequality (2.20) shows that is weakly contractive. Thus,
is a unique fixed point of
. Set
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ28_HTML.gif)
and converges to
with the following error estimate (see Remark 2.2):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ29_HTML.gif)
Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ30_HTML.gif)
From (2.21)–(2.23), we obtain (2.18). This completes the proof.
Theorem 2.4.
Let be a Banach space and let
be a nonempty closed convex set. Let
be a family of commuting self-mappings, where
are all nonexpansive and
is weakly contractive. For any
, let
be the iteration of type Ishikawa generated by (1.6), where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ31_HTML.gif)
and are all bounded. Then,
converges strongly to a unique common fixed point
with the following estimate:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ32_HTML.gif)
where .
Proof.
Applying Theorem 2.1, we can suppose that is a unique common fixed point of
. Since
are all bounded, we have
. Since
are all nonexpansive and
is weakly contractive, we obtain in proper order that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ33_HTML.gif)
Write Then
and (2.26) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ34_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ35_HTML.gif)
From (2.27) and Lemma 1.1, it implies that exists, and so does
by the continuity of
. From (2.28), it implies that
Since
we conclude that
Therefore,
that is,
converges strongly to
. To establish the error estimate, we set
and
Then, (2.26) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ36_HTML.gif)
Set From (2.29) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ37_HTML.gif)
Since is nondecreasing, from (2.30) we deduce
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ38_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F469357/MediaObjects/13663_2007_Article_1086_Equ39_HTML.gif)
Hence, the estimate (2.25) holds. This completes the proof.
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Acknowledgment
This work is supported by the National Natural Science Foundation of China (10671094).
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Xiao, JZ., Zhu, XH. Common Fixed Point Theorems on Weakly Contractive and Nonexpansive Mappings. Fixed Point Theory Appl 2008, 469357 (2007). https://doi.org/10.1155/2008/469357
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DOI: https://doi.org/10.1155/2008/469357