Middle East Journal of Applied Science & Technology (MEJAST)
Volume 6, Issue 1, Pages 01-12, January-March 2023
Non-unique Fixed Points of Self Mappings in Bi-metric Spaces
Khan Ihsanullah1*, Muhammad Umair2 & Hassan Asad3
1,3
Jiangsu Normal University, China. 2International Islamic University ISB, Pakistan.
Corresponding Author (Khan Ihsanullah) - 3106337078@qq.com*
DOI: https://doi.org/10.46431/MEJAST.2023.6101
Copyright © 2023 Khan Ihsanullah et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Article Received: 03 December 2022
Article Accepted: 23 January 2023
Article Published: 04 February 2023
ABSTRACT
In this paper, we prove a few non-unique fixed-point results of mapping on a set with bi-metrics using θ – contraction. We also give an example that
justifies our results. In the literature, our result generalized many results.
Keywords: Fixed point, θ – contraction, Bi-metric, Non-unique fixed point, Bi-metric space.
░ Introduction
This paper involves an extension of some non-unique fixed point theorems of mappings in bi-metric space using
– contraction. First, we discuss a brief history of fixed points and applications of the fixed points. Fixed point
theory plays an essential role in various applied and theoretical fields, such as differential and integral equations,
theory of dynamic systems, game theory, approximation theory, and nonlinear analysis. In the research aspect, it is
considered one of the most active and favorable subjects, thus numerous mathematicians such as Cauchy,
Liouville, Peano, Lipschitz and Picard did work in its related field. The significant result in fixed-point theory is the
famous result by Brouwer in 1910. He also proved the result for n–dimensional Euclidean space. Birkhoff and
Kellogg investigated the first infinite dimensional fixed point theorem. Later, Banach [2] gave the idea of a fixed
point for contraction mappings which is known as the “Banach Contraction Principle” in 1922. This principle is
useful for the uniqueness and existence of fixed points. There have been many generalizations of this principle in
distinct directions [1, 3, 4, 6, 7]. Marr gave the first important result for non-expensive mappings. In 1941,
Kakutani studied the fixed point problem of multifunction and generalizations of this type of mapping discussed in
[5, 8, 9, 10].
Main Body
Firstly, consider a theorem about non-unique fixed points of self mappings in bi-metric spaces.
Theorem-1: Let
with bi-metrics
and
on
and
be
satisfies
(i)
( (
(ii) (
(iii)
))
) is
,
(iv) , (
[ ( (
))] ,
𝑓𝑜𝑟 𝑎𝑙𝑙
(
𝑎𝑛𝑑
)
– orbitally complete metric space
* (
) (
) (
,
* (
) (
)+-
)-
ISSN: 2582-0974
( , (
)+-
)-) ,
( , (
𝑓𝑜𝑟 𝑎𝑙𝑙
)-)
𝑓𝑜𝑟
𝑎𝑛𝑑
[1]
(
)
(
)
– orbitally continuous mapping which
Middle East Journal of Applied Science & Technology (MEJAST)
Volume 6, Issue 1, Pages 01-12, January-March 2023
Then
has fixed point.
Proof: Suppose
𝑎𝑡
be any arbitrary point. We wish to prove that the sequence of iterates
is Cauchy sequence. If
for some
sequence, we may suppose that
,
,
,
𝑛* (
,
for each
) (
𝑛* (
) (
) (
𝑛* (
, 𝑛* (
) (
𝑛* (
, 𝑛* (
) (
let
)+-
) (
)+-
( , (
)+)+-
) +-
and
( , (
)+-
) (
is a positive integer then *
where
( , (
)-)
( , (
)-)
we have
)-)
)-)
which implies
,
𝑛* (
By (iv)
,
, (
)+, (
)-
𝑛* (
) (
(
Let
,
) (
𝑛* (
)( , (
)+-
( , (
)+-
, (
)-
( , (
, (
)-
, (
)
)-)
𝑛* (
) and suppose
(
𝑓𝑜𝑟
) (
)+
(
)
)-)
)-)
, (
)-
)-
which is contradiction. So,
, (
)-
( , (
)-)
, (
)-
( , (
)-)
, (
Letting
,(
ISSN: 2582-0974
)-
( , (
)-)
( , (
)-)
𝑓𝑜𝑟 𝑎𝑙𝑙
we get
)-
, (
)[2]
( , (
+ is a Cauchy
)-)
Middle East Journal of Applied Science & Technology (MEJAST)
Volume 6, Issue 1, Pages 01-12, January-March 2023
which implies
(
)
Now proceeding in similar way, such that there exists
(
)
we get
𝑓𝑜𝑟 𝑎𝑙𝑙
Now for
(
)
(
)
(
)
).
(
(
)
∑
(
)
∑
(
)
∑
(
/
)
).
(
(
/
)
).
(
(
)
+ is C.S. As (
) is
/
∑
). Therefore, *
It is infinite geometric series and is convergent (as
complete, so there exists
– orbitally
such that
By orbital continuity of
Now change some conditions in the above statement, and take a new theorem.
Theorem-2: Suppose
complete and
(
(iii) , (
Then
on . Suppose that (
) is
– orbitally
such that (
there exists
)
for
satisfies
, (
(ii)
and
– orbitally continuous. For
be
and if
(i)
be non-empty set with bi-metrics
)-
[ ( (
)
implies that ,
)-
( , (
))]
)-)
𝑓𝑜𝑟 𝑎𝑙𝑙
and
𝑛* (
,
) (
(
and
) (
(
)
)+-
( , (
)-) for
(
)
)
has periodic point.
Proof: Let we define a subset
{ (
)
𝑓𝑜𝑟 𝑜
and suppose that
ISSN: 2582-0974
[3]
} of positive integer is non-empty. Let
Middle East Journal of Applied Science & Technology (MEJAST)
Volume 6, Issue 1, Pages 01-12, January-March 2023
(
)
𝑓𝑜𝑟
Case-1: If
,
𝑛* (
,
𝑛* (
(
then
) (
) (
)
)+-
for
and
[ ( (
) (
)+-
[ ( (
))]
) (
)+-
[ ( (
))]
we have
))]
By (iii)
,
𝑛* (
(
Let
) and if
𝑛* (
, (
)-
[ ( (
))]
, (
)-
[ ( (
))]
, (
)-
, (
) (
, (
)+
(
)
)-
)-
which is a contradiction. So,
, (
)-
[ ( (
))]
Similarly,
, (
)-
, (
[ ( (
))]
, (
), (
)-
Applying limit
[ ( (
[ ( (
))]
)-
[ ( (
))]
))]
we get
, (
)-
Implies
(
)
Now by same argument of Theorem 1 we get
Case-2: If
(
)
and suppose
𝑓𝑜𝑟 𝑎𝑙𝑙
Then by our assumption
,
𝑛* (
,
𝑛* (
ISSN: 2582-0974
) (
) (
( )
(
) (
) (
)
)+)+-
[ ( (
[ ( (
[4]
))]
))]
Middle East Journal of Applied Science & Technology (MEJAST)
Volume 6, Issue 1, Pages 01-12, January-March 2023
By (iii)
,
𝑛* (
) (
(
Let
) (
) and if
)+-
𝑛* (
, (
)-
[ ( (
))]
, (
)-
[ ( (
))]
, (
)-
( (
))
[ ( (
) (
( (
))]
) (
)+
(
)
))
( )
By using (1)
( )
, (
( )
( )
)-
( )
which is not possible.
If
𝑛* (
) (
) (
)+
(
)
Then by (1) it is again contradiction. So,
, (
)-
[ ( (
))]
, (
)-
[ ( (
))]
( ( ))
Similarly,
, (
)-
[ ( (
, (
)-
[ ( (
))]
))]
[ ( (
))]
[ ( (
))]
[ ( (
))]
In general
, (
)-
[ ( (
, (
)-
, (
)-
[ ( (
))]
))]
[ ( (
( ( ))
))]
𝑓𝑜𝑟 𝑎𝑙𝑙
implies
(
)
Now proceeding in similar way such that there exists
(
ISSN: 2582-0974
)
we get
𝑓𝑜𝑟 𝑎𝑙𝑙
[5]
[ ( (
))]
Middle East Journal of Applied Science & Technology (MEJAST)
Volume 6, Issue 1, Pages 01-12, January-March 2023
Now for
(
)
(
(
)
(
)
∑
(
)
∑
(
)
∑
)
(
)
(
)
(
)
(
(
(
)
)
∑
). Therefore, *
It is infinite geometric series and is convergent (as
is
)
– orbitally complete, so there exists
+ is a Cauchy sequence. As (
)
such that
Now by orbital continuity of
Theorem-3: Suppose
If
satisfies
(i)
( (
(ii)
, (
(iii)
,
))
)-
be non-empty set with bi-metrics
[ ( (
( , (
))]
)-)
* (
) (
) (
,
* (
) (
)+-
Then the sequence *
Proof: Let
𝑎𝑛𝑑
)+-
( (
(
))
on
and
(
)
be
– orbitally continuous.
)
, 𝑓𝑜𝑟 𝑎
then
is fixed point of
be a cluster point then
for some
If
for all
𝑛* (
, then
is a fixed point of
then for
𝑛* (
𝑛* (
) (
) (
) (
,
𝑛* (
ISSN: 2582-0974
which is our required result.
we get
) (
,
,
𝑎𝑛𝑑
+ has cluster point
If
,
𝑓𝑜𝑟 𝑎𝑙𝑙
and
)+) (
) (
)+-
)+)+[6]
( (
( (
))
))
Middle East Journal of Applied Science & Technology (MEJAST)
Volume 6, Issue 1, Pages 01-12, January-March 2023
,
𝑛* (
, 𝑛* (
) (
)+) +-
( (
))
( (
))
))
,
𝑛* (
) (
, -
)+-
,
𝑛* (
) (
)+-
( (
) (
)+-
[ ( (
By using (ii)
,
𝑛* (
If 𝑛* (
) (
)+
))]
(
, (
)-
[ ( (
))]
, (
)-
[ ( (
))]
, (
)-
, (
) , then
, (
)-
)-
which is not possible. So,
, (
)-
[ ( (
))]
, (
)-
[ ( (
))]
, (
)-
Applying limit
[ ( (
[ ( (
))]
[ ( (
))]
))]
we have
, (
)-
Implies
(
)
Now proceeding in similar way such that there exists
(
)
we get
𝑓𝑜𝑟 𝑎𝑙𝑙
Now for
(
)
(
)
(
)
ISSN: 2582-0974
(
)
(
)
(
(
)
)
(
∑
[7]
)
(
)
(
)
Middle East Journal of Applied Science & Technology (MEJAST)
Volume 6, Issue 1, Pages 01-12, January-March 2023
(
Hence,
)
. So *
as
+ is Cauchy sequence. So there exists
such that
Now by orbitally continuity of
Now change some conditions in the theorem-3, and take a new theorem.
Theorem-4: Suppose
continuous. If
(i)
( (
(ii)
, (
))
(
[ ( (
( , (
)
))]
)-)
𝑓𝑜𝑟 𝑎𝑙𝑙
,
(
𝑎𝑛𝑑
(
𝑎𝑛𝑑
,
implies that
sequence *
Then
and
on
and
be
𝑛* (
)
)
) (
) (
)+-
( (
)) , for each
+ has cluster point
is a fixed point of .
Proof: Let
be a cluster point i.e.
Then there exists
(
.
such that
)
which gives
(
)
(
)
(
)
Now, we define a set
*
(
𝑓𝑜𝑟 𝑜
)
𝑟
+
is non-empty. Suppose that
*
(
𝑓𝑜𝑟 𝑜
𝑟
then
+
which is our required result. If
then by our assumption
(
Case-1: If
,
)
for some 𝑟
If
for all 𝑟
– orbitally
satisfies
)-
(iii)
be a non-empty set with bi-metrics
𝑛* (
)
then
(
)
) (
) (
and for
)+-
we have
[ ( (
By (ii)
,
𝑛* (
) (
)+-
( , (
If
𝑛* (
) (
)+
(
ISSN: 2582-0974
)-)
) then
[8]
))]
the
Middle East Journal of Applied Science & Technology (MEJAST)
Volume 6, Issue 1, Pages 01-12, January-March 2023
, (
)-
( , (
)-)
, (
)-
( , (
)-)
, (
)-
, (
, (
)-
)-
which is not possible. So,
, (
( , (
)-
)-)
, (
)-
( , (
, (
)-
( , (
Applying limit
( , (
)-)
)-)
)-)
we get
, (
)-
Implies
(
)
Now, we can say that
Case-2: If
suppose that
(
)
for all
,
( )
and by our assumption
𝑛* (
(
)
we have
) (
) (
)+-
, (
( , (
)-
By (ii)
,
𝑛* (
) (
) (
)+-
If
𝑛* (
) (
) (
)+
, (
)-
( , (
)-)
, (
)-
( , (
)-)
, (
)-
( (
))
, (
)-
( (
))
, (
)-
( )
( (
( )
Using (2) we obtain
( )
, (
)-
From above both equations
( )
( )
ISSN: 2582-0974
[9]
))
(
)-)
) then
Middle East Journal of Applied Science & Technology (MEJAST)
Volume 6, Issue 1, Pages 01-12, January-March 2023
which is not possible. If
𝑛* (
) (
) (
)+
, (
)-
( , (
, (
)-
( , (
, (
)-
, (
)-
, (
)-
, (
)-
, (
)-
( )
(
)
)-)
, (
)-)
)-
( )
By (2) we have
( )
( )
Again it is not possible. So,
, (
( , (
), (
)-
)-)
( , (
( , (
)-)
)-)
, (
)-
Applying limit
we get
, (
( , (
)-)
( , (
)-)
)-
implies
(
)
Now proceeding in similar way such that there exists
(
)
we have
𝑓𝑜𝑟 𝑎𝑙𝑙
Now for
(
)
(
)
(
)
As (
(
)
(
)
(
(
)
)
(
)
(
)
)
∑
)
as
. So *
+ is a Cauchy sequence. Since
continuity of
ISSN: 2582-0974
(
[10]
. Now by orbital
Middle East Journal of Applied Science & Technology (MEJAST)
Volume 6, Issue 1, Pages 01-12, January-March 2023
Now, consider the example, in which we apply the new conditions, made in the above theorems. And prove that the
following example satisfies the new conditions, but didn’t satisfy the other conditions.
*
Example-1: Let
|
| and
(
)
and
(
)
(
. Let
and
|
. Let
|
(
)
|
|
|
(
)
| (
|
|
)|
|
|
|
|
|
|
|
Now,
𝑛* (
) (
) (
)+
)
(
𝑛* |
|
+
|
|
𝑛* (
, 𝑛* (
𝑛* |
𝑛* |
, (
)-
,
(
)
) (
|+
|
Now, we consider ( )
,
| |
𝑛* (
|
|
,
𝑛* |
|
( , (
+|+-
)+)+-
( , |
( , (
|-)
)-)
)
which satisfy for all
ISSN: 2582-0974
) (
) (
|
| |
(
.
) (
and
(
and
be a mapping define by
| |
)
)
| for all
|
(
(
|
|
)
+ we define bi-metrics
𝑓𝑜𝑟 𝑎𝑙𝑙
).
[11]
)-)
)+
such that
(
)
for all
Middle East Journal of Applied Science & Technology (MEJAST)
Volume 6, Issue 1, Pages 01-12, January-March 2023
Declarations
Source of Funding
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit
sectors.
Competing Interests Statement
The authors declare no competing financial, professional and personal interests.
Consent for publication
Authors declare that they consented for the publication of this research work.
Availability of data and material
Authors are willing to share data and material according to the relevant needs.
References
[1] Agarwal, R. P., Meehan, M., & O'regan, D. (2001). Fixed point theory and applications (Vol. 141). Cambridge
University Press.
[2] Banach, S. (1922). Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales.
Fund. math, 3(1): 133-181.
[3] Berinde, V. (2008). General constructive fixed point theorems for Ćirić-type almost contractions in metric
spaces. Carpathian Journal of Mathematics, Pages 10-19.
[4] Edelstein, M. (1962). On fixed and periodic points under contractive mappings. Journal of the London
Mathematical Society, 1(1): 74-79.
[5] Eilenberg, S., & Montgomery, D. (1946). Fixed point theorems for multi-valued transformations. American
Journal of mathematics, 68(2): 214-222.
[6] Khamsi, M. A., & Kirk, W. A. (2011). An introduction to metric spaces and fixed point theory. John Wiley &
Sons.
[7] Kirk, W. A. (2015). Metric fixed point theory: a brief retrospective. Fixed Point Theory and Applications,
2015(1): 1-17.
[8] Nadler Jr, S. B. (1969). Multi-valued contraction mappings. Pacific Journal of Mathematics, 30(2): 475-488.
[9] Plunkett, R. L. (1956). A fixed point theorem for continuous multi-valued transformations. Proceedings of the
American Mathematical Society, 7(1): 160-163.
[10] Strother, W. L. (1953). On an open question concerning fixed points. Proceedings of the American
Mathematical Society, 4(6): 988-993.
ISSN: 2582-0974
[12]