Common Fixed Point Theorems of Multi-Valued Maps in Ultra Metric Space ()
1. Introduction
Roovij in [1] introduced the concept of ultra metric space. Later, C. Petalas, F. Vidalis [2] and Ljiljana Gajic [3] studied fixed point theorems of contractive type maps on a spherically complete ultra metric spaces which are generalizations of the Banach fixed point theorems. In [4] K. P. R. Rao, G. N. V. Kishore and T. Ranga Rao obtained two coincidence point theorems for three or four self maps in ultra metric space.
J. Kubiaczyk and A. N. Mostafa [5] extend the fixed point theorems from the single-valued maps to the setvalued contractive maps. Then Gajic [6] gave some generalizations of the result of [3]. Again, Rao [7] proved some common fixed point theorems for a pair of maps of Jungck type on a spherically complete ultra metric space.
In this article, we are going to establish some results on coincidence and common fixed point for two pair of multi-valued and single-valued maps in ultra metric spaces.
2. Basic Concept
First we introducing a notation.
Let
denote the class of all non empty compact subsets of
. For
, the Hausdorff metric is defined as
![](https://www.scirp.org/html/22-7401296\0ff8fecf-1974-428f-84b4-49d0b235867c.jpg)
where
.
The following definitions will be used later.
Definition 2.1 ([1]) Let
be a metric space. If the metric
satisfies strong triangle inequality
![](https://www.scirp.org/html/22-7401296\9cc17c4c-f778-4ffc-b93e-5abe82b69d48.jpg)
Then
is called an ultra metric on
and
is called an ultra metric space.
Example. Let
,
, then
is a ultra metric space.
Definition 2.2 ([1]) An ultra metric space is said to be spherically complete if every shrinking collection of balls in
has a non empty intersection.
Definition 2.3 An element
is said to be a coincidence point of
and
if
. We denote
![](https://www.scirp.org/html/22-7401296\af317dad-cc38-4f89-8e50-a42b70485769.jpg)
the set of coincidence points of
and
.
Definition 2.4 ([7]) Let
be an ultra metric space,
and
.
and
are said to be coincidentally commuting at
if
implies
.
Definition 2.5 ([8]) An element
is a common fixed point of
and
if
.
3. Main Results
The following results are the main result of this paper.
Theorem 3.1 Let
be an ultra metric space. Let
be a pair of multi-valued maps and
a pair of single-valued maps satisfying
(a)
is spherically complete;
(b) ![](https://www.scirp.org/html/22-7401296\de17d3ce-56d5-4e80-8a29-67f686e85e04.jpg)
for all
, with
;
(c)
;
(d)
.
Then there exist point
and
in
, such that
.
Proof. Let
![](https://www.scirp.org/html/22-7401296\b6fa3ffc-28c8-42f5-b7e6-bf3b1a490d82.jpg)
denote the closed sphere with centered
and radius
.
Let
be the collection of all the spheres for all
.
Then the relation
if ![](https://www.scirp.org/html/22-7401296\79e8b624-7d45-4f64-b355-cc30b0b24942.jpg)
is a partial order on
.
Consider a totally ordered sub family
of
.
Since
is spherically complete, we have
![](https://www.scirp.org/html/22-7401296\cf73958b-df65-4d3e-86c8-93f6ed6b3d1e.jpg)
Let
where
and
.
Then
. Hence
(1)
If
then
. Assume that
.
Let
, then
![](https://www.scirp.org/html/22-7401296\8ec7ead1-68ae-422e-b166-a1408078e899.jpg)
Since
is nonempty compact set, then
such that
;
is a nonempty compact set, then
such that
.
![](https://www.scirp.org/html/22-7401296\7b40002f-5ec2-42d3-88d2-ba54f9bf010b.jpg)
from (a) (b) and Equation (1)
Now
![](https://www.scirp.org/html/22-7401296\f0754f2a-87e2-4fb2-814c-6d3a6b0bc95c.jpg)
So
, we have just proved that
for every
. Thus
is an upper bound in
for the family
and hence by Zorn’s Lemma, there is a maximal element in
, say
. There exists
such that
.
Suppose
.
Since
are nonempty compact sets, then
such that
(2)
(3)
From (b), (c) and Equation (2), we have
(4)
(5)
From (b), (c) and Equations (2)-(5)
(6)
(7)
From Equation (4) and Equation (6) we have
(8)
From Equation (5) and Equation (7) we have
(9)
If
![](https://www.scirp.org/html/22-7401296\feb03451-38b9-48ae-9d76-125559a72e95.jpg)
Then from Equation (8),
. Hence
. It is a contradiction to the maximality of
in
, since ![](https://www.scirp.org/html/22-7401296\30117ca6-8e5b-4c5d-9270-7ffb2acdb166.jpg)
If
![](https://www.scirp.org/html/22-7401296\8454ccd4-5f4c-442e-9e68-6d259d840b2c.jpg)
Then from Equation (9),
. Hence
.It is a contradiction to the maximality of
in
, since
.
So
(10)
In addition,
.
Using (b), (c) and Equation (10), we obtain
![](https://www.scirp.org/html/22-7401296\11d43b51-5fa7-4e43-8be2-90ea8954de5f.jpg)
Hence
.
Then the proof is completed.
Theorem 3.2 Let
be an ultra metric space. Let
be a pair of multi-valued maps and
be a single-valued maps satisfying
(a)
is spherically complete;
(b) ![](https://www.scirp.org/html/22-7401296\a083b7f7-cf49-4338-9e09-7beadc2a2ab8.jpg)
for all
,with
;
(c)
;
(d)
.
Then
and
have a coincidence point in
.
Moreover, if
and
,
and
are coincidentally commuting at
and
, then
and
have a common fixed point in
.
Proof. If
in Theorem 2.1, we obtain that there exist points
and
in
such that
.
As
,
and
ipipare coincidentally commuting at
and
.
Write
, then
.
Then we have
![](https://www.scirp.org/html/22-7401296\4fbaa4d9-8a74-4598-8381-a15cf9af5b82.jpg)
and
.
Now, since also
,
and
are coincidentally commuting at
and
, so we obtain
.
Thus, we have proved that
, that is,
is a common fixed point of
and
.
Corollary 3.3 Let
be a spherically complete ultra metric space. Let
be a pair of multi-valued maps satisfying
(a)
for all
,with
;
(b)
.
Then, there exists a point
in
such that
and
.
Remark 1 If
in Corollary 3.3, then we obtain the Theorem of Ljiljana Gajic [6].
Remark 2 If in Theorem 3.1,
, we obtain Theorem 9 of K. P. R. Rao at [7].
Remark 3 If
and
in Theorem 3.1 are single-valued maps, then: 1) we obtain the results of K. P. R. Rao [4]; 2)
, we obtain the result of Ljiljana Gajic [3]; 3)
, then, we obtain Theorem 4 of K. P. R. Rao at [7].
4. Conclusion
In this paper, we get coincidence point theorems and common fixed point theorems for two pair of multi-valued and single-valued maps satisfying different contractive conditions on spherically complete ultra metric space, which is generalized results of [3-7].
5. Acknowledgements
Foundation item: Science and Technology Foundation of Educational Committee of Tianjin (11026177).
NOTES