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Common Fixed Point Theorems for a Finite Family of Discontinuous and Noncommutative Maps
Fixed Point Theory and Applications volume 2011, Article number: 847170 (2011)
Abstract
We study common fixed point theorems for a finite family of discontinuous and noncommutative single-valued functions defined in complete metric spaces. We also study a common fixed point theorem for two multivalued self-mappings and a stationary point theorem in complete metric spaces. Throughout this paper, we establish common fixed point theorems without commuting and continuity assumptions. In contrast, commuting or continuity assumptions are often assumed in common fixed point theorems. We also give examples to show our results. Results in this paper except those that generalized Banach contraction principle and those improve and generalize recent results in fixed point theorem are original and different from any existence result in the literature. The results in this paper will have some applications in nonlinear analysis and fixed point theory.
1. Introduction and Preliminaries
Let be a metric space and
be a multivalued map. We say that
is a stationary point of
if
. The existence theorem of stationary point was first considered by Dancs et al. [1]. If
is a self-mapping (multivalued or single valued) defined on
, we denote
the collection of all the fixed points of
. In this paper,we need the following definitions.
Definition 1.1.
A function is called
(i)contraction if there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ1_HTML.gif)
(ii)kannan if there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ2_HTML.gif)
(iii)quasicontractive if there is a constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ3_HTML.gif)
where .
(iv)weakly contractive if there exists a lower semicontinuous and nondecreasing function with
if and only if
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ4_HTML.gif)
It is known that every contraction and every Kannan mapping has a unique fixed point in complete metric spaces Banach [2], Kannan [3] and every quasicontractive mapping has a unique fixed point in Banach spaces Ćirić [4], Rhoades [5]. In 2001, Rhoades [6] proved that every weakly contractive mapping has a unique fixed point in a complete metric space. Let and
be self-maps defined on
; the following inequality was considered in the study of common fixed points theorems Rhoades [5], Chang [7]:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ5_HTML.gif)
for some constant and function
.
If and
satisfy the inequality (1.5) with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ6_HTML.gif)
then and
are said to be a couple of quasicontractive mappings which is studied by Rhoades [5]. Chang [7] prove that every couple of quasicontractive mappings has a unique common fixed point in Banach spaces. Recently, Zhang and Song [8] proved a common fixed point theorem in complete metric spaces under the following assumption:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ7_HTML.gif)
where .
The result of Zhang and Song [8] generalized the results in [2, 3, 5, 6]. Motivated by Chang [7], Zhang and Song [8], it is natural to ask whether there is a common fixed point of and
in
satisfy inequality (1.5) with
. In this paper, we give a positive answer to this question in complete metric spaces.
Let be a finite family of self-mappings on
. If there is a nondecreasing, lower semicontinuous function
with
if and only if
such that for every
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ8_HTML.gif)
where , for all
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ9_HTML.gif)
Here, is the identity map defined on
and
. We show that
have a unique common fixed point if
is complete. As a special case of this result, we give a common fixed point theorem in complete metric spaces under the assumption that inequality (1.5) holds with
. One of our results generalized Banach contraction principle, an example is given (Example 2.12) to show that the maps
above need not to be continuous. The assumption of continuity is often used in the existence theorems of fixed points [6, 9–14]. We also give an example to show that the family
above is not necessary to be commuting, and in contrast that the commutativity assumption is often used in the existence theorems of common fixed points [9, 10, 13, 15, 16]. Finally, we generalize some of our results to the case of multivalued maps.
Let be multivalued maps satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ10_HTML.gif)
for some (where
denotes the Hausdorff metric). In fact, under the hypothesis that inequality (1.10) holds, we can show that
and
for all
if
and
have nonempty closed bounded values. Further we give a new stationary point theorem in complete metric spaces and illustrate with examples (Examples 3.4 and 3.8).
2. Fixed Point Theorems
Throughout this paper, let be a complete metric space and let
be the set of all positive integers. In this section, all the self-maps on
are single valued. The following theorem is the main result in this section.
Theorem 2.1.
Let be a finite family of self-mappings on
. If there is a nondecreasing, lower semicontinuous function
with
if and only if
such that for every
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ11_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ12_HTML.gif)
for all , and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ13_HTML.gif)
is the identity map defined on
and
.
Then, have a unique common fixed point.
Proof.
For any fixed , take
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ14_HTML.gif)
Continuing in this way, we obtain by induction a sequence in
such that
, whenever
with
and
. Then, if
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ15_HTML.gif)
If for some
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ16_HTML.gif)
Therefore is a decreasing and bounded below sequence,and there exists
such that
. Since
is lower semicontinuous,
. Taking upper limits as
on two sides of the following inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ17_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ18_HTML.gif)
Then, and, hence,
.
is a Cauchy sequence in
. Indeed, let
,
. Then
is a decreasing sequence. If
, we are done. Suppose that
, choose
small enough and select
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ19_HTML.gif)
By the definition of , there exist
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ20_HTML.gif)
Since , for all
. Replace
and
if necessary, we may assume that
,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ21_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ22_HTML.gif)
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ23_HTML.gif)
We consider the following two cases:
(i)
(ii).
If , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ24_HTML.gif)
Then, . Since
is arbitrary small positive number, if we take
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ25_HTML.gif)
This yields a contradiction.
If , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ26_HTML.gif)
Then . This also yields a contradiction.
Therefore, and
is a Cauchy sequence in
. Since
is complete,
converges to a point in
, say
.
In order to show that is the unique common fixed point of
. We first claim that
, for all
.
Indeed, for each ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ27_HTML.gif)
We consider the following three cases:
(i),
(ii),
(iii).
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ28_HTML.gif)
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ29_HTML.gif)
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ30_HTML.gif)
Continuing in this process, we show that . By the same argument as in the case above, we see that
.
Then, we see that , for all
. Next, we claim that
is the unique fixed point of
. Indeed, for any
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ31_HTML.gif)
Then, and
. Therefore,
is the unique fixed point of
and we complete the proof.
Remark 2.2.
-
(a)
The sequence
approaching to the unique common fixed point in Theorem 2.1 is different from those in [8, 11, 12, 16–19].
-
(b)
The finite family
of self-mappings in Theorem 2.1 is neither commuting nor continuous, which are often assumed in common fixed point theorems, see [6, 9–16]. In fact, the commuting and continuity assumptions are not needed throughout this paper and we will give examples (Examples 2.12–2.15) to show this fact.
As special cases of Theorem 2.1, we have the following theorems and corollaries.
Theorem 2.3.
Let , be self-mappings on
. If there is a nondecreasing, lower semicontinuous function
with
if and only if
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ32_HTML.gif)
for all . Then
and
have a unique common fixed point.
Proof.
Take ,
and
in Theorem 2.1, then Theorem 2.3 follows from Theorem 2.1.
Corollary 2.4.
Let be self-mappings on
. If there is a nondecreasing, lower semicontinuous function
with
if and only if
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ33_HTML.gif)
Then and
have a unique common fixed point.
Corollary 2.5.
Let be a self-mapping on
. If there is a nondecreasing, lower semicontinuous function
with
if and only if
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ34_HTML.gif)
Then has a unique fixed point.
Proof.
Take in Theorem 2.3, then Corollary 2.4 follows from Corollary 2.4.
Remark 2.6.
-
(a)
Since
for all
implies
(2.25)
Corollary 2.5 generalizes Theorem 1 in Rhoades [6].
-
(b)
Corollary 2.4 is equivalent to Corollary 2.5.
Proof.
It suffices to show that in Corollary 2.4. Indeed, for each
, there exists
such that
. By the hypothesis in Corollary 2.4,
and we complete the proof.
-
(c)
In Theorem 2.3, the map
is not necessary equal
, see Example 2.14. In fact, the maps
and
in Theorem 2.3 are not necessary to be commuting, see Example 2.15.
Theorem 2.7.
Let be a finite family of self-mappings on
. If there exists
such that for every
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ36_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ37_HTML.gif)
and is the identity map defined on
and
.
Then have a unique common fixed point.
Proof.
Take for all
, then Theorem 2.7 follows from Theorem 2.1.
Theorem 2.8.
Let be self-mappings on
. If there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ38_HTML.gif)
for all . Then
and
have unique common fixed point.
Proof.
Take, ,
and
in Theorem 2.7, then Theorem 2.8 follows from Theorem 2.7.
Corollary 2.9.
Let be self-mappings on
and if there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ39_HTML.gif)
Then, and
have a unique common fixed point.
Corollary 2.10.
Let be a self-map on
and if there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ40_HTML.gif)
Then has a unique fixed point.
Proof.
Take in Corollary 2.9, then Corollary 2.10 follows from Corollary 2.9.
Remark 2.11.
-
(i)
If
is contractive, then there exists
such that
for all
, but the converse is not true. It is obvious that Corollary 2.9 is a special case of Corollaries 2.4 and 2.10 is a generalization of Banach contraction principle. Further we see that Corollary 2.9 is equivalent to Corollary 2.10 by the same argument as in Remark 2.6.
-
(ii)
A map
satisfies
for all
and for some
is neither continuous nor nonexpansive. We give an example (Example 2.12) to show this fact.
Example 2.12.
Let be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ41_HTML.gif)
Then, for all
and
is not continuous.
Proof.
We consider the following three cases:
(i),
(ii),
(iii).
If ,
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ42_HTML.gif)
If ,
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ43_HTML.gif)
If ,
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ44_HTML.gif)
and, hence . It is obvious that
is not continuous at
but
for all
.
Example 2.13.
Let be the same as in Example 2.12. and take
for all
. Since
for all
. By Example 2.12 and Corollary 2.10, we see that
has a unique common fixed point. But for each
,
is not continuous, the results in [13, 16] do not work in this example. Further it is obvious that the family
have a unique common fixed point 0.
Example 2.14.
Let and define maps
by
and
. Then
and we see that
and
have a unique common fixed point.
Proof.
It suffices to show that there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ45_HTML.gif)
for all .
We have to consider the following two cases:
(i),
(ii).
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ46_HTML.gif)
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ47_HTML.gif)
If we take , then by Theorem 2.8, we see that
and
have a unique common fixed point. In fact, 0 is the unique common fixed point of
and
.
By the same argument as in Example 2.14, we give the following example to show that the maps and
in Theorem 2.8 are not necessary to be commuting.
Example 2.15.
Let and define maps
by
and
. Then
and
are not commuting and we see that
and
have a unique common fixed point.
Proof.
It suffices to show that there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ48_HTML.gif)
for all .
We have to consider the following two cases:
(i),
(ii).
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ49_HTML.gif)
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ50_HTML.gif)
If we take , then by Theorem 2.8, we see
and
have a unique common fixed point. In fact, 0 is the unique common fixed point of
and
.
3. A Common Fixed Point Theorem of Set-Valued Maps and a Stationary Point Theorem
In this section, we study a fixed point theorem and a stationary point theorem which generalize a fixed point theorem in Section 2.
In this section, let be the class of all nonempty bounded closed subsets of
and for
, let
be the Hausdorff metric of
and
and let
for all
.
Lemma 3.1 (see [20]).
For all ,
and
, there exists
such that
.
Theorem 3.2.
Let be multivalued maps. If there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ51_HTML.gif)
Then and
for all
.
Proof.
For any fixed and
. Take
, and let
. By Lemma 3.1, we may choose
such that
,
such that
,
such that
. Continuing in this process, we obtain by induction a sequence
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ52_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ53_HTML.gif)
Therefore, for all
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ54_HTML.gif)
This shows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ55_HTML.gif)
and is a Cauchy sequence. Since
is complete, there exists
such that
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ56_HTML.gif)
and ,
. Therefore
and
.
To complete the proof, it suffices to show the following four cases:
(i) and
for all
,
(ii),
(iii) for all
,
(iv).
For any ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ57_HTML.gif)
This shows that and
. Further
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ58_HTML.gif)
and .
For any ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ59_HTML.gif)
This shows that . Till now, we see that
and
for all
.
For any ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ60_HTML.gif)
Hence .
It remains to show that .
Indeed, for any ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ61_HTML.gif)
and . Then
and
.
Remark 3.3.
-
(a)
If one of
and
in Theorem 3.2 is single valued, then the set
is singleton and the maps
and
have a unique common fixed point in
. Therefore, Theorem 3.2 is a generalization of Corollary 2.9, but Theorem 3.2 is not a generalization of Theorem 5 Nadler [20].
-
(b)
The sequence
approaches to the common fixed point
of
and
in Theorem 3.2 is different from those in [20–25].
-
(c)
By Example 2.12, we see that both
and
in Theorem 3.2 are neither to be upper semicontinuous nor to be lower semicontinuous (multivalued maps). Further the maps
and
are not necessary to be commuting. We give an example below.
Example 3.4.
Let and let maps
be defined by
and
. Then we see that
and
have a unique common fixed point.
Proof.
It suffices to show that there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ62_HTML.gif)
We have to consider the following two cases:
(i)
(ii)
If , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ63_HTML.gif)
If , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ64_HTML.gif)
If we take , then by Theorem 3.2, we see
and
have a unique common fixed point. In fact, 0 is the unique common fixed point of
and
.
Corollary 3.5.
Let be a multivalued map with nonempty compact values and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ65_HTML.gif)
Then and
for all
.
Similarly, we have the following existence theorem of stationary points.
Theorem 3.6.
Let be a multivalued map,
be a single valued function. If
is a nondecreasing, lower semicontinuous function with
for all
and
if and only if
. Suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ66_HTML.gif)
Then has a unique stationary point, say
. In fact,
and
.
Proof.
For any fixed , let
,
,
,
. Continuing in this process, we obtain by induction a sequence
such that
and
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ67_HTML.gif)
Then, is a decreasing and bounded below sequence, and hence there exist
such that
. Since
is lower semicontinuous,
. Taking upper limits as
on two sides of the following inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ68_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ69_HTML.gif)
Then, and hence
.
is a Cauchy sequence in
. Indeed, let
. Then
is a decreasing sequence. If
, we are done. Suppose that
, choose
small enough and select
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ70_HTML.gif)
By the definition of , there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ71_HTML.gif)
Since , for all
. Replace
and
if necessary, we may assume that
,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ72_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ73_HTML.gif)
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ74_HTML.gif)
and . Since
is arbitrary small positive number, if we take
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ75_HTML.gif)
This yields a contradiction. Therefore and
is a Cauchy sequence in
. By the completeness of
,
converges to a point in
, say
.
Since,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ76_HTML.gif)
It follows that .
Further for all ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ77_HTML.gif)
Then, and
.
For all ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ78_HTML.gif)
Then, and hence
.
Remark 3.7.
-
(a)
The single valued map
in Theorem 3.6 is not necessary to be continuous (see Example 2.12), but the continuity assumption is used in Theorem 3.2 [21, 22, 25] and Theorem 2.1 Ćirić and Ume [23]. We give an example to show that
and
in Theorem 3.6 are not necessary to be commuting.
-
(b)
Theorems 3.2 and 3.6 are different and Theorem 3.6 is also a generalization of Corollary 2.9.
Example 3.8.
Let and let maps
and
be defined by
and
. Then we see that
has a unique stationary point.
Proof.
It suffices to show that there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ79_HTML.gif)
We have to consider the following two cases:
(i),
(ii).
If , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ80_HTML.gif)
If , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F847170/MediaObjects/13663_2010_Article_1433_Equ81_HTML.gif)
If we take . Then by Theorem 3.6, we see
has a unique stationary point. In fact, 0 is the unique stationary point of
.
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Lin, LJ., Wang, SY. Common Fixed Point Theorems for a Finite Family of Discontinuous and Noncommutative Maps. Fixed Point Theory Appl 2011, 847170 (2011). https://doi.org/10.1155/2011/847170
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DOI: https://doi.org/10.1155/2011/847170