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Fixed Point Theorems for Occasionally Weakly Compatible Mappings in Dislocated-Metric Spaces
Kastriot Zoto1,
, Elida Hoxha2, Panda Sumati Kumari3
1Department of Mathematics and Computer Sciences, Faculty of Natural Sciences, University of Gjirokastra Gjirokastra, Albania
2Department of Mathematics, Faculty of Natural Sciences, University of Tirana; Tirana, Albania
3KL University, Green Fields, Vaddeswaram, Guntur District, Andhra Pradesh, India
Abstract
In this paper we prove some fixed point theorems for one and two pairs of selfmaps which are occasionally weakly compatible and satisfy a “max” and 
contractive conditions. Also, some existing results are derived as corollaries from theorems of this paper in the framework of dislocated metric spaces.
Keywords: occasionally weakly compatible, dislocated metric, contraction condition, common fixed point
Turkish Journal of Analysis and Number Theory, 2014 2 (2),
pp 37-41.
DOI: 10.12691/tjant-2-2-2
Received March 01, 2014; Revised April 03, 2014; Accepted April 10, 2014
Copyright © 2013 Science and Education Publishing. All Rights Reserved.Cite this article:
- Zoto, Kastriot, Elida Hoxha, and Panda Sumati Kumari. "Fixed Point Theorems for Occasionally Weakly Compatible Mappings in Dislocated-Metric Spaces." Turkish Journal of Analysis and Number Theory 2.2 (2014): 37-41.
- Zoto, K. , Hoxha, E. , & Kumari, P. S. (2014). Fixed Point Theorems for Occasionally Weakly Compatible Mappings in Dislocated-Metric Spaces. Turkish Journal of Analysis and Number Theory, 2(2), 37-41.
- Zoto, Kastriot, Elida Hoxha, and Panda Sumati Kumari. "Fixed Point Theorems for Occasionally Weakly Compatible Mappings in Dislocated-Metric Spaces." Turkish Journal of Analysis and Number Theory 2, no. 2 (2014): 37-41.
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1. Introduction
Hitzler and Seda in [18] introduced the concept of a dislocated metrics as a generalization of metrics where the self distance for any point need not to be equal to zero. They generalize the celebrated Banach contraction principle in dislocated metric spaces. Since then, many research papers have dealt with fixed point theory for single-valued mappings in dislocated metric spaces as a larger class than that of metric spaces (see, e.g., [11, 13, 14, 15, 19, 20, 21].
Al-Thagafi and Shahzad [2] defined the concept of occasionally weakly compatible mappings which is more general than the concept of weakly compatible maps. Bhatt et al. [3] have given application of occasionally weakly compatible mappings in dynamical system. Motivated by the works of many authors for occasionally weakly compatible maps in metric spaces, in this paper we give some fixed point theorems for occasionally weakly compatible mappings satisfying 
-weakly contractive condition in the setting of dislocated metric spaces. Our theorems unify and generalize various known results from metric spaces to dislocated metric spaces.
2. Preliminaries
Definition 2.1 [18] Let 
be a non-empty and let 
be a function, called a distance function if for all 
, satisfies:
 |
 |
 |
 |
If 
satisfies the condition 
, then 
is called a metric on X. If it satisfies the conditions
, 
and
it is called a quasi-metric. If 
satisfies conditions
,
and 
it is called a dislocated metric (or simply 
-metric). If 
satisfies only 
and 
then 
is called a dislocated quasi-metric (or simply dq-metric) on X. A nonempty set X with dq-metric 
, i. e., 
is called a dislocated quasi-metric space.
Definition 2.2 [18] A sequence 
in 
-metric space 
is called Cauchy if for all 
, 
such that 
, 
.
Definition 2.3 [18] A sequence 
dislocated converges or
-converges to x if 
. In this case x in called a
-limit of 
and we write xn →x.
Definition 2.4 [18] A 
-metric space 
is complete if every Cauchy sequence in it is 
-convergent.
Lemma 2.5 [18] Every subsequence of 
-convergent sequence to a point 
is 
-convergent to 
.
Definition 2.6 [18] Let 
be a
-metric space. A mapping 
is called contraction if there exists 
such that:
 |
Lemma 2.7 [18] Let 
be a 
-metric space. If 
is a contraction function, then 
is a Cauchy sequence for each 
∈ X.
Lemma 2.8 [18] 
-limits in a 
-metric space are unique.
Definition 2.9 [9] Let 
and 
be mappings of a set 
into itself. Then, 
and 
are said to be weakly compatible if they commute at their coincidence point; that is 
for some 
implies 
.
Definition 2.10 [9] Two self-maps
and 
of a set
are occasionally weakly compatible (owc) iff there is a point 
in 
which is a coincidence point of 
and 
at which 
and 
commute.
Example 2.11 Let 
with dislocated metric 
. Define 
by 
. Then
, 
and 
. Thus the pair 
is occasionally weakly compatible but not weakly compatible.
3. Main Results
After recalling some definitions and lemmas in dislocated metric space, we state the following theorems.
Theorem 3.1 Let 
be a dislocated-metric space and 
and 
are occasionally weakly compatible self-mappings of 
, satisfying the condition:
 | (1) |
for all 
and 
. Then 
and 
have a unique common fixed point.
Proof. Since 
and 
are occasionally weakly compatible there exists a point 
in 
such that 
. We claim that 
is the unique common fixed point of 
and 
. Let show that 
is a fixed point of 
. Consider:
 |
From this inequality and since
, we have
. Thus 
and from 
we see that
is a common fixed point of 
and 
.
Uniqueness. Suppose that 
and 
are two common fixed point of 
and 
such that 
and 
and 
.
By condition (1) have:
 |
since 
we have 
. This implies 
. Thus fixed point is unique.
Example 3.2 Let 
with dislocated metric 
. Define 
by 
for 
and 
. Then 
, 
and 
, so the pair 
is occasionally weakly compatible. Also for all 
and 
have,
 |
Thus all conditions of theorem are satisfied and
is the unique common fixed point of 
and 
.
Corollary 3.3 Let 
be a dislocated-metric space and 
and 
are occasionally weakly compatible self-mappings of 
, satisfying the condition:
 |
for all 
and nonnegative constant 
with 
. Then 
and 
have a unique common fixed point.
Proof. This theorem can be obtained as corollary of theorem 3.1 since we have that;
 |
for all 
and constant 
non negative with 
For the following theorem which involve two pairs of self mappings each owc, we use the class of function 
where 
such that 
is a continuous non decreasing with 
, 
such that 
is a continuous function with 
and denote
 |
Fixed point results that we are proving can be considered as continuation or generalization of many results given by [11, 13, 14, 16].
Theorem3.4 Let 
be a dislocated metric space and 
and 
be self-mappings of
. The pairs 
and 
are owc, and satisfy the condition:
 | (2) |
for all
and 
where 
.
Then there exists a unique common fixed point of 
and 
.
Proof. Since the pairs
and
are owc, there are points 
such that 
and 
and 
. We claim that 
. Consider that
 |
By the condition of theorem have:
 |
This inequality is a contradiction unless 
, thus we have 
, i.e. 
. Firstly observe that
 |
Suppose that 
, then inequality (2) gives:
 |
which is a contradiction. Hence 
and so 
.
Thus
is a fixed point of 
and 
.
In the same way, we observe that
 |
If suppose that 
, from condition (2) we have,
 |
which is a contradiction, unless 
. Therefore, we get 
, and 
is a common fixed point of 
and
.
Uniqueness. If we assume that there exists two common fixed points 
and 
of
and
. For 
again from the condition of theorem we get
 |
So
 |
which is a contradiction unless 
and as a result 
. Therefore
is the unique common fixed point of 
and 
.
Corollary 3.5 Let 
be a dislocated metric space and 
and 
be self-mappings of 
. The pairs 
and 
are owc, and satisfy the condition:
 |
for all 
and 
where 
.
Then there exists a unique common fixed point of 
and 
.
Proof. This is clear if in theorem3.4 we put 
.
Corollary 3.6 Let 
be a dislocated-metric space and 
and 
are occasionally weakly compatible self-mappings of 
, satisfying the condition:
 |
for all 
and
and functions 
. Then 
and 
have a unique common fixed point.
Proof. The proof is taken from theorem 3.4 if we take in it 
(identity map)
Example 3.7 Let 
with dislocated metric 
. Define 
by 
. Then 
, 
and 
. Thus the pair 
is occasionally weakly compatible but not weakly compatible, and for functions 
as 
and 
we observe that,
 |
for all 
Thus all conditions of theorem hold and 
is the unique common fixed point of 
and 
.
Corollary 3.8 Let 
be a dislocated metric space and 
and 
be self-mappings of 
. The pairs 
and 
are owc, and satisfy the condition:
 |
for all 
and 
where
.
Then there exists a unique common fixed point of 
and 
.
This corollary is taken from theorem3.4 If we take the function 
as 
.
Corollary 3.9 Let 
be a dislocated-metric space and 
and 
are occasionally weakly compatible self-mappings of 
, satisfying the condition:
 |
for all 
and 
where 
. Then 
and 
have a unique common fixed point.
This corollary is taken from theorem if we put in it 
and 
.
Corollary 3.10 Let
be a dislocated metric space and
and
be self-mappings of
. The pairs
and
are owc, and satisfy the condition:
 |
for all
, 
and 
.
Then there exists a unique common fixed point of
and
.
This corollary is taken from above corollary3.8 if we take in it the function 
for 
.
Let be the class 
of functions 
which are Lebesgue integrable functions and summable nonnegative such that 
for each 
. Now we give the following fixed point theorems for occasionally weakly compatible mappings satisfying contractive conditions of integral type.
Theorem 3.11 Let
be a dislocated metric space and
and
be self-mappings of
. The pairs
and
are owc, and satisfy the condition:
 | (3) |
for all
and 
where 
.
Then
and
have a unique common fixed point.
Proof. If we take 
and 
then we see that the functions
are functions from 
. So on this conditions, we can use theorem 3.4 and the self mappings 
and
have a unique common fixed point.
Corollary 3.12 Let
be a dislocated metric space and
and
be self-mappings of
. The pairs
and
are owc, and satisfy the condition:
 |
for all
, 
and 
where 
.
Then 
and 
have a unique common fixed point.
Proof. If we take 
then from theorem 3.11 we conclude that
and
have a unique common fixed point.
Remark 3.13 These theorems are an extension of many results on fixed point given in dislocated metric spaces by authors [11, 13, 14, 15, 16, 17, 19, 22, 23] for occasionally weakly compatible mappings without imposing conditions on the space or mappings such as completeness, closedness and continuity.
Acknowledgements
The authors are thankful to the editor and referees, for their valuable suggestions for the improvement of the paper.
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