- Research Article
- Open access
- Published:
On Mappings with Contractive Iterate at a Point in Generalized Metric Spaces
Fixed Point Theory and Applications volume 2010, Article number: 458086 (2011)
Abstract
Using the setting of generalized metric space, the so-called G-metric space, fixed point theorems for mappings with a contractive and a generalized contractive iterate at a point are proved. These results generalize some comparable results in the literature. A common fixed point result is also proved.
1. Introduction
Sehgal in [1] proved fixed point theorem for mappings with a contractive iterate at a point and therefore generalized a well-known Banach theorem.
Theorem 1.1.
Let be a complete metric space and let
be a continuous mapping with property that for every
there exists
so that for every
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ1_HTML.gif)
Then has a unique fixed point
in
and
, for each
.
Guseman [2] extended Sehgal's result by removing the condition of continuity of and weakening (1.1) to hold on some subset
of
such that
, where, for some
,
contains the closure of the iterates of
. Further extensions appear in [3, 4]. Our aim in this study is to show that these results are valid in more general class of spaces.
In 1963, S. Gähler introduced the notion of 2-metric spaces but different authors proved that there is no relation between these two function and there is no easy relationship between results obtained in the two settings. Because of that, Dhage [5] introduced a new concept of the measure of nearness between three or more object. But topological structure of so called -metric spaces was incorrect. Finally, Mustafa and Sims [6] introduced correct definition of generalized metric space as follows.
Definition 1.2 (see [6]).
Let be a nonempty set, and let
be a function satisfying the following properties:
if
;
; for all
, with
;
, for all
, with
;
, (symmetry in all three variables);
, for all
.
Then the function is called a generalized metric, or, more specifically, a
-metric on
, and the pair
is called a
-metric space.
Clearly these properties are satisfied when is the perimeter of triangle with vertices at
,
, and
, moreover taking
in the interior of the triangle shows that
is the best possible.
Example 1.3.
Let be an ordinary metric apace, then
can define
-metrics on
by
,
.
Example 1.4 (see [6]).
Let . Define
on
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ2_HTML.gif)
and extend to
by using the symmetry in the variables. Then it is clear the
is a
-metric space.
Definition 1.5 (see [6]).
Let be a
-metric space, and let
be sequence of points of
, a point
is said to be the limit of the sequence
, if
, and one says that the sequence
is
-convergent to
Thus, if in a
-metric space
, then for any
, there exists
such that
, for all
.
Definition 1.6 (see [6]).
Let be a
-metric space, a sequence
is called
-Cauchy if for every
, there is
such that
, for all
; that is, if
as
.
A -metric space
is said to be
-complete (or complete
-metric) if every
-Cauchy sequence in
is
-convergent in
.
Proposition 1.7 (see [6]).
Let be a
-metric space, then the function
is jointly continuous in all three of its variables.
Definition 1.8 (see [6]).
A -metric space
is called symmetric
-metric space if
, for all
.
Proposition 1.9 (see [6]).
Every -metric space
will define a metric space
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ3_HTML.gif)
Note that if is a symmetric
-metric space, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ4_HTML.gif)
However, if is nonsymmetric, then by the
-metric properties it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ5_HTML.gif)
and that in general these inequalities cannot be improved.
Proposition 1.10 (see [6]).
A -metric space
is
-complete if and only if
is a complete metric space.
In recent years a lot of interesting papers were published with fixed point results in -metric spaces, see [7–18]. This paper is our contribution to the fixed point theory in
-metric spaces.
2. Fixed Point Results
Let be a
-metric space,
a mapping,
such that for some
and each for
there exists a positive integer
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ6_HTML.gif)
for all . Then we write
. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ7_HTML.gif)
for all , we write
.
Theorem 2.1.
Let or
. Let
, with
. If there exists
such that for
,
, then
is the unique fixed point of
in
. Moreover,
,
, for any
for
, and for
if
.
Proof.
If is a symmetric space than
and (2.1) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ8_HTML.gif)
and (2.2) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ9_HTML.gif)
thus the result follows from Theorem 12 in [3] and it is valid for any . Suppose now that
is nonsymmetric space. Then by inequality (1.5) we have that (2.1) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ10_HTML.gif)
and (2.2) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ11_HTML.gif)
Since need not be less then 1 we can use metric fixed point results only for
. On the other side, using the concept of
-metric space, we are going to prove the result, if the first case for any
, and in the second one for
. This means that our results are real generalization in the case of nonsymmetric
-metric spaces.
Let . Uniqueness follows from (2.1), since for
, it follows that
. Now
implies that
.
Let , and assume
for each
. For
sufficiently large write
,
, and
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ12_HTML.gif)
so ,
.
If , uniqueness follows from (2.2) since for
, it follows that
and further
. Now for any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ13_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ14_HTML.gif)
For we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ15_HTML.gif)
which is a contradiction, and therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ16_HTML.gif)
If then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ17_HTML.gif)
So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ18_HTML.gif)
Therefore, , where
. For
,
,
.
For the set
is called the orbit for
.
Theorem 2.2.
Let be a complete
-metric space and let
be a mapping. Suppose that for some
the orbit
is complete, and that: for some
and each
there is an integer
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ19_HTML.gif)
for all .
Then ,
, converges to some
and for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ20_HTML.gif)
If inequality in (2.14) holds for all , then
and
,
.
Moreover, if , then
is the fixed point of
.
Proof.
If is a symmetric
-metric space the statement easily follows from Guseman fixed point result [2]. Let
be nonsymmetric
-metric space. Then by inequality (1.5)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ21_HTML.gif)
Thus, one can use the fixed point result in metric space only for . But here, using the concept of
-metric, we prove the result for any
. At first let us show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ22_HTML.gif)
For any , sufficiently large, there exist
,
such that
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ23_HTML.gif)
Now, for each
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ24_HTML.gif)
For all ,
, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ25_HTML.gif)
so is Cauchy sequence and there exists
, for some
, and inequality (2.15) is proved.
If we suppose that inequality in (2.14) is satisfied for all , then, for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ26_HTML.gif)
so .
On the other hand,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ27_HTML.gif)
implies that .
Since is continuous it means that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ28_HTML.gif)
Hence .
Now, let us suppose that . Since
by Theorem 2.1
is the fixed point of
in
and
.
For , in inequality (2.14) independently on
, we are going to simplify the proof and to relax the condition in (2.14).
Corollary 2.3.
Let be a complete
-metric apace and let
. Suppose that there exist a point
and
with
complete and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ29_HTML.gif)
for each . Then
converges to some point
and for all
,
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ30_HTML.gif)
If (2.24) holds, for all or
is orbitally continuous at
, then
is a fixed point of
.
Proof.
If is a symmetric space than
so (2.24) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ31_HTML.gif)
and result follows from Theorem 2 in [19].
Now, let be a nonsymmetric
-metric space. Then since
,
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ32_HTML.gif)
so for all ,
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ33_HTML.gif)
and there exists . If (2.24) holds for all
, then by Theorem 2.2, since
, it follows that
.
The fact that is orbitally continuous at
, and that
, implies that
, and therefore
.
Remark 2.4.
Let us note that this result is very close to Theorem 2.1 in [8].
Remark 2.5.
In the statements above does not have to be continuous.
The next theorems are generalizations of Ćirić fixed point results in [4].
Theorem 2.6.
Let be a complete metric space and
a mapping. Suppose that for each
there exists a positive integer
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ34_HTML.gif)
holds for some and all
. Then
has a unique fixed point
. Moreover, for every
,
.
Proof.
If is a symmetric space then
and inequality (2.29) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ35_HTML.gif)
for all . Then the result follows from Theorem 2.1 in [4] and it is true for all
.
Now suppose that is nonsymmetric space. Then, by definition of the metric
and inequality (1.5) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ36_HTML.gif)
But need not to be less than 1, so we will prove the statement by using
-metric.
First, let us prove prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ37_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ38_HTML.gif)
Clearly (2.32) is true for . Suppose that
, and that (2.32) is true for
and let us prove it for
. Let
. Now
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ39_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ40_HTML.gif)
If , then (2.34) imply
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ41_HTML.gif)
If , then (2.34) imply
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ42_HTML.gif)
Thus by induction we obtain (2.32).
Let us prove that is a Cauchy sequence. Let
,
,
, and we define inductively a sequence of integers and a sequence of points
in
as follows:
, and
,
. Evidently,
is a subsequence of the orbit
. Using this sequence we will prove that
is a Cauchy sequence.
Let be any fixed member of
and let
and
be any two members of the orbit which follow after
. Then
and
for some
and
, respectively. Now, using (2.29) we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ43_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ44_HTML.gif)
Similarly, , where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ45_HTML.gif)
Repeating this argument times we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ46_HTML.gif)
Hence . Similarly
, so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ47_HTML.gif)
Since , it follows that
is a Cauchy sequence. Let
. We show that
is a fixed point of
. First, let us prove that
, where
. For
, we now have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ48_HTML.gif)
and on letting tend to infinity it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ49_HTML.gif)
For we have
.
To show that is a fixed point of
, let us suppose that
and let
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ50_HTML.gif)
Since , it follows that
, which implies that
is a fixed point of
.
Let us suppose that for some ,
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ51_HTML.gif)
implies that and thus
is the unique fixed point in
.
If we suppose that is continuous, then we may prove the following theorem.
Theorem 2.7.
Let be a complete
-metric space and let
be a continuous mapping which satisfies the condition: for each
there is a positive integer
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ52_HTML.gif)
for some and all
. Then
has a unique fixed point
and
, for every
.
Proof.
Let be an arbitrary point in
. Then, as in the proof of Theorem 2.6, the orbit
is bounded and is a Cauchy sequence in the complete
-metric space
and so it has a limit
in
. Since by the hypothesis
is continuous,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ53_HTML.gif)
Therefore, is a fixed point of
. By the same argument as in the proof of Theorem 2.6, it follows that
is a unique fixed point of
.
Remark 2.8.
The condition that is a continuous mapping can be relaxed by the following condition:
is continuous at a point
.
3. A Common Fixed Point Result
Now, we are going to prove Hadžić [20] fixed point theorem in 2-metric space, in a manner of -metric spaces.
Theorem 3.1.
Let be a complete
-metric space,
and
one to one continuous mappings,
continuous mapping commutative with
and
. Suppose that there exists a point
such that
is complete and that the following conditions are satisfied:
(i)For every there exists
so that for all
and some
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ54_HTML.gif)
(ii)There exists such that for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ55_HTML.gif)
Then there exists one and only one element such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ56_HTML.gif)
(e.g., there exists a unique common fixed point for , and
)
Proof.
Since starting with
we can define the sequence
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ57_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ58_HTML.gif)
We are going to prove that is Cauchy sequence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ59_HTML.gif)
Similarly one can prove that ,
, for all
,
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ60_HTML.gif)
Thus we proved that is a Cauchy sequence, so there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ61_HTML.gif)
It obvious that .
At first we will prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ62_HTML.gif)
so .
Now, since that and
are continuous we have that
so
.
Further, let us prove that .
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ63_HTML.gif)
implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ64_HTML.gif)
and . Similarly one can see that
, so we prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ65_HTML.gif)
If we suppose that is some other common fixed point for
, and
then we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458086/MediaObjects/13663_2010_Article_1286_Equ66_HTML.gif)
which is contradiction!
So, the common fixed point for , and
is unique, and proof is completed.
Remark 3.2.
For condition (2.14) is satisfied but the Theorem 2.2 is not just a consequence of Theorem 3.1 since in Theorem 2.2 we do not suppose that
is continuous.
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Acknowledgments
The authors are thankful to professor B. E. Rhoades, for his advice which helped in improving the results. This work was supported by grants approved by the Ministry of Science and Technological Development, Republic of Serbia, for the first author by Grant no. 144016, and for the second author by Grant no. 144025.
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Gajić, L., Lozanov-Crvenković, Z. On Mappings with Contractive Iterate at a Point in Generalized Metric Spaces. Fixed Point Theory Appl 2010, 458086 (2011). https://doi.org/10.1155/2010/458086
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DOI: https://doi.org/10.1155/2010/458086