Fixed Point Results of Contractive Mappings by Altering Distances and C-Class Functions in b-Dislocated Metric Spaces ()
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1. Introduction
The study of metric fixed point theory in dislocated metric spaces was considered by P. Hitzler and A. K. Seda in [1] who introduced this metric as a generalization of usual metric, and generalized the Banach contraction principle on this space. Since then a lot of papers have been written on this topic treating the problem of existence and uniqueness of fixed points for mappings satisfying different contractive conditions, see [2] - [14] . N. Hussain et al. in [15] introduced the b-dislocated metric spaces associated with some topological aspects and properties. These spaces can be seen as generalizations of dislocated metric spaces and also as generalization of b-metric space introduced by Bakhtin in [16] and extensively used by Czerwik in [8] . Recently, there are many papers on existence and uniqueness of fixed point and common fixed point for one, two or more mappings under different types of contractive conditions in the setting of dislocated spaces and b-dislocated metric spaces.
Since altering distance functions were introduced by Khan et al. in [17] , the study of the existence of fixed points of contractive maps in metric spaces and generalized metric spaces has a lot of interest for many authors which are based on this category of functions (see [17] - [23] ). In September 2014, a class of functions called as C-class is presented by A. H. Ansari, see in [24] [25] and is important, see example 2.15.
The present paper is organized in two sections. Using concepts mentioned above, in the first section, we develop some coincidence and common fixed point theorems (existence and uniqueness) for two pairs of weakly compatible mappings in the framework of
-dislocated metric space, using weak generalized
contractive conditions. In the second section, we prove common fixed point theorems for a pair of mappings using generalized
contractive condition and the concept of T-contractions. The related results generalize and improve various theorems in recent literature.
2. Preliminaries
Consistent with [1] and [15] , the following definitions, notations, basic lemma and remarks will be needed in the sequel.
Definition 2.1 [1] Let X be a nonempty set and a mapping
is called a dislocated metric (or simply
-metric) if the following conditions hold for any
:
1) If
, then
2)
3)
The pair
is called a dislocated metric space (or d-metric space for short). Note that for
,
may not be 0.
Definition 2.2 [15] Let X be a nonempty set and a mapping
is called a b-dislocated metric (or simply
-dislocated metric) if the following conditions hold for any
and
:
1) If
, then
2)
3)
The pair
is called a b-dislocated metric space. And the class of b-dislocated metric space is larger than that of dislocated metric spaces, since a b-dislocated metric is a dislocated metric when
.
Example 2.3 If
, then
defines a dislocated metric on X.
Definition 2.4 [1] A sequence
in
-metric space
is called:
1) a Cauchy sequence if, for given
, there exists
such that for all
, we have or
;
2) convergent with respect to
if there exists
such that
as
. In this case, x is called the limit of
and we write
.
A
-metric space X is called complete if every Cauchy sequence in X converges to a point in X.
In [15] , it was shown that each
-metric on X generates a topology
whose base is the family of open
-balls
.
Also in [15] , there are presented some topological properties of
-metric spaces.
Definition 2.5 [15] Let
be a
-metric space, and
be a sequence of points in X. A point
is said to be the limit of the sequence
if
and we say that the sequence
is
-convergent to x and denote it by
as
.
The limit of a
-convergent sequence in a
-metric space is unique ( [15] , Proposition 1.27).
Definition 2.6 [15] A sequence
in a
-metric space
is called a
-Cauchy sequence if, given
, there exists
such that for all
, we have
or
. Every
-convergent sequence in a
-metric space is a
-Cauchy sequence.
Remark 2.7 The sequence
in a
-metric space
is called a
-Cauchy sequence if
for all
.
Definition 2.8 [15] A
-metric space
is called complete if every
-Cauchy sequence in X is
-convergent.
Example 2.9 If
, then
defines a b-dislocated metric on X with parameter
.
Example 2.10 Let
and any constant
. Define function
by
. Then, the pair
is a dislocated metric space.
If
for some
, then x is called the coincidence point of F and S. Furthermore, if the mappings commute at each coincidence point, then such mappings are called weakly compatible [4] .
Definition 2.11 [17] The altering distances functions
and
are defined as
The following lemmas are used to prove our results.
Lemma 2.12 Let
be a b-dislocated metric space with parameter
. Then
1) If
then
;
2) If
is a sequence such that
, then we have
;
3) If
, then
;
Proof. It is clear.
Lemma 2.13 [15] Let
be a b-dislocated metric space with parameter
. Suppose that
and
are
-convergent to
, respectively. Then we have
In particular, if
, then we have
. Moreover, for each
, we have
In particular, if
, then we have
.
Definition 2.14. [24] [25] We say that a function
is called a C-class function if it is continuous and satisfies the following properties.
We denote C-class functions as C.
Example 2.15 [24] [25] The following functions
are elements of C, for all
:
1)
2)
3)
4)
5)
For
, we have
6)
7)
8)
9)
, here
is continuous and such that
and
for
.
3. Main Results
Before we give the main result we denote with letter
the following set
(3.1.1)
for all
.
Motivated by the works of [15] [21] - [29] we extend the concept of
-weakly contractive maps to four maps in a b-dislocated metric space, giving the following definition.
Definition 3.1 Let
be four self maps of a b-dislocated metric space
with parameter
. If there exists
,
and
such that
(A)
for all
, where
is defined as in (3.1.1) then
and T are said to satisfy a generalized
weakly contractive condition.
Theorem 3.2 Let
be a b-dislocated metric space with parameter
and
are self-mappings such that (a)
,
and satisfy generalized
weakly contractive condition. If one of
or
is a complete subspace of X, then
and
have a point of coincidence in X. Moreover if suppose that
and
are weakly compatible pairs, then
have a unique common fixed point.
Proof. Let
be an arbitrary point in X. Since
we can choose
such that
. And since
corresponding to
we can choose
such that
. Continuing the same process we obtain sequences
and
in X such that:
for all ![]()
We consider following steps:
Step 1. If
(that means
) for some n, then
. Hence
is a coincidence point of G and T. Using definition of
and lemma 2.12 we have,
![]()
![]()
Thus
(3.2.1)
Using condition (A) and property of C-class, we have that
![]()
By property of
we have,
(3.2.2)
As a result we get,
.
Again from contractive condition of theorem have,
![]()
The inequality above implies,
![]()
By property of function f of C-class we obtain
or
.
And also by property of
we get
so that
and then
. Also
is a coincidence point of F and S.
Step 2. Suppose
that means
for all n by condition (3.1.1) we have:
![]()
![]()
If
then
(3.2.3)
Also from condition of theorem we have:
![]()
By property of function
we have
(3.2.4)
From (3.2.3) and (3.2.4) we get
(3.2.5)
Also from condition of theorem and (3.2.5) we have,
![]()
The above inequality implies:
![]()
which means
.
From property of C-class we obtain
.
So we have
that is a contradiction since we suppose
.
So we have
.
In a similar way as above we have
. As a result
the sequence
is non increasing and bounded below. And so there exists
such that,
.
Suppose that
. Since
is continuous and
is lower semi continuous we have:
(3.2.6)
If we consider condition (A) we have,
(3.2.7)
taking the upper limit as
in (3.2.7) and using (3.2.6) we have that,
(3.2.8)
From (3.2.8) and property of
we get
that is a contradiction. Hence
(3.2.9)
Now we prove that
is a
-Cauchy sequence. Assume the contrary that
is not a
-Cauchy sequence. Then there exists
for which we can
find subsequences
and
of
so that
is the smallest index for which
, that
(3.1.10)
and
(3.2.11)
From property c) of definition 2.2 we have:
(3.2.12)
Taking the upper limit as
in (3.2.12) and using result (3.129) and (3.2.11), we get
(3.2.13)
Also we have
(3.2.14)
Hence taking the upper limit in above inequality, we obtain
(3.2.15)
Again from property c) of definition 2.2, we have
(3.2.16)
Thus from 3.2.9; 3.2.15 we have
(3.2.17)
As a result,
(3.2.18)
Similarly,
(3.2.19)
Taking the upper limit in (3.2.19) and using 3.2.9, we get
(3.2.20)
Similarly,
![]()
Taking the upper limit in above inequality and using (3.2.9), we have
(3.2.21)
Also,
![]()
Taking the upper limit and using 3.2.9; 3.2.18 we get
(3.2.22)
So, by (3.2.21) and (3.2.22) we have
(3.2.23)
According to the set (3.1.1) we have:
(3.2.24)
Taking the upper limit in (3.2.24) and using results 3.2.9; 3.2.18; 3.2.13; 3.2.23 we get
(3.2.25)
Similarly, we can show,
(3.2.26)
From contractive condition of theorem, we have
(3.2.27)
Taking the upper limit as
in (3.2.27) and using 3.2.25; 3.2.26, we obtain
![]()
From this inequality and since
is non decreasing follows that
.
That is a contradiction since we supposed
. Thus
is a
-Cauchy sequence in b-dislocated metric space
. Also the subsequences
,
,
,
are
-Cauchy. Let we suppose that
is a complete subspace of X, since the subsequence
is
-Cauchy then there exists
such that
. Then we have,
. (3.2.28)
Since
, then there exists
such that
. According to (3.1.1) consider
(3.2.29)
Taking the upper limit and using lemma 2.13, result (3.2.9) and (3.2.28) we obtain
(3.2.30)
Using contractive condition (A) of theorem we have,
(3.2.31)
Taking the upper limit in (3.2.31) and using (3.2.30) we get
![]()
This implies
and so
. Thus
, so y is a point of coincidence of the pair
.
Similarly we can show that
, so v is a point of coincidence of the pair
. Therefore we have
. (3.2.32)
Let show that z is a unique point of coincidence of pairs
and
. Suppose that exists another point
such that
.
We consider,
![]()
Using contractive condition of theorem we have,
![]()
The inequality above implies that
so
that means the point of coincidence is unique.
Let prove that z is a common fixed point. By the weak compatibility of the pairs
and
have:
and
.
From condition of theorem we have,
(3.2.33)
This inequality implies
.
And
(3.2.34)
Again from (3.2.33) and (3.2.34) we get,
.
By property of functions
and C-class, we have
(3.2.35)
So we obtained
, that iz
. Therefore
.
Let we prove that z is a fixed point of F.
Again we consider
![]()
By property of
follows
(3.2.36)
where
(3.2.37)
From (3.2.36), (3.2.37) we get
(3.2.38)
In similar way as in (3.2.35) using (3.2.38), property of C-class and functions
we obtain,
and
. Hence z is a common fixed point.
Uniqueness. Let we prove that the fixed point is unique. If suppose that u and z are two common fixed points of F, G, S, T then from condition (b) we have,
![]()
By property of
we get
. Also we have,
![]()
So, ![]()
and
![]()
As a result
and so
.
The following is corollary of theorem 3.2 which is taken for parameter
in a dislocated metric space.
Corollary 3.3 Let
be a dislocated metric space and
are self-mappings such that (a)
,
and exists
,
and
such that satisfy the condition
(A)
for all
, where
is defined as in (3.1.0). If one of
or
is a complete subspace of X, then
and
have a point of coincidence in X. Moreover if suppose that
and
are weakly compatible pairs, then
have a unique common fixed point.
Now we give an example to support our Theorem 3.2.
Example 3.4 Let
and
. Then the pair ![]()
is a b-dislocated metric space with parameter
. We define the functions
and T as follows:
. The pairs ![]()
and
are weakly compatible, functions
and T are continuous and ![]()
We have,
![]()
where
;
and
, for all
.
Thus all conditions of theorem 3.2 are satisfied and
is the unique common fixed point of
and G.
In a similar way as in Theorem 3.2, the following theorem can be proved.
Theorem 3.5 Let
be a complete b-dislocated metric space with parameter
and
are self-mappings such that (a)
and satisfy generalized
weakly contractive condition. If one of
or
is closed, then
and
have a point of coincidence in X. Moreover if suppose that
and
are weakly compatible pairs, then
have a unique common fixed point.
For the different functions f of C-class (refer to example 2.15) we can take the following corollaries.
Corollary 3.6 Let
be a
-dislocated metric space with parameter
and
are self-mappings where (a)
and exists
,
such that satisfies the condition
![]()
for all
, where
is defined as in (3.1.0). If one of
or
is a complete subspace of X, then
and
have a point of coincidence in X. Moreover if suppose that
and
are weakly compatible pairs, then
have a unique common fixed point.
Proof. If we take in Theorem 3.2 the function f as
then we get the corollary.
Corollary 3.7 Let
be a
-dislocated metric space with parameter
and
are self-mappings where (a)
and exists
,
such that satisfies the condition
![]()
for all
, where
is defined as in (3.1.0). If one of
or
is a complete subspace of X, then
and
have a point of coincidence in X. Moreover if suppose that
and
are weakly compatible pairs, then
have a unique common fixed point.
Proof. This corollary is obtained from Theorem 3.2 if we take as f the function
.
Corollary 3.8 Let
be a
-dislocated metric space with parameter
and
are self-mappings where (a)
and exists
,
such that satisfies the condition
![]()
for all
, where
is defined as in (3.1.0). If one of
or
is a complete subspace of X, then
and
have a point of coincidence in X. Moreover if suppose that
and
are weakly compatible pairs, then
have a unique common fixed point.
Proof. If we take in Theorem 3.2 the function f as
then we get the corollary.
Corollary 3.9 Let
be a
-dislocated metric space with parameter
and
are self-mappings where (a)
and exists
,
such that satisfies the condition
![]()
for all
, where
is defined as in (3.1.0). If one of
or
is a complete subspace of X, then
and
have a point of coincidence in X. Moreover if suppose that
and
are weakly compatible pairs, then
have a unique common fixed point.
Proof. If we take in Theorem 3.2 the function f as
then we get the corollary.
Remark 3.10 As a consequence of theorem 3.2 and all corollaries for taking
1) the parameter
.
2) the parameter
and
and
.
3) functions f from the set C and taking
and
.
We can establish many other corollaries in the setting of dislocated and b-dislocated metric spaces.
In this section, we use the notion of T-contractions introduced by Beiranvad et al. in [3] as a new class of contractive mappings, by generalizing the contractive condition in terms of another function. These contractions have been used by many authors. In this direction in order to generalize some other well-known results as in [32] [33] [34] we extend the notion of
generalized weak contractions in the context of T-contractions, giving the following theorem.
Theorem 3.11 Let
be a complete b-dislocated metric space with parameter
and
be an injective, continuous and sequentially convergent mapping. Let
be self-mappings and if exist
,
and
such that
(B)
for all
, where
![]()
then
have a unique common fixed point.
Proof. We divide the proof into two parts as follows.
First part. Each fixed point u of F is a fixed point of G and conversely, and the common fixed point of
is unique.
Let
be a fixed point of F. If
then, follows that
and so u is a fixed point of G. If we suppose that
, we evaluate
as;
![]()
So we have
(3.11.1)
Then by contractive condition (B), we have
![]()
By property of
we have
(3.11.2)
Hence from (3.11.1) and (3.11.2) follows
.
Again
![]()
![]()
By property of
we get
that is
and by injectivity of T follows
.
Thus u is a fixed point of G. Similarly we can prove the other implication.
Second part. We prove that the function F has a fixed point. We define two eterative sequences
as
and
, and
as
for each
.
If for some n, we have
then
and
is a fixed point of F and by the first part
is a fixed point of G and the proof is completed.
Now, we assume that
for all n, and since T is injective we have
; then from condition (B) of theorem, we have
![]()
where
![]()
(3.11.3)
If
then from (3.11.3) we get
(3.11.4)
Using condition (B) and property of C-class, we have
![]()
By property of function
we have
(3.11.5)
From (3.11.4) and (3.11.5) we get
(3.11.6)
Also from condition of theorem and (3.11.6), we have
![]()
Also we have,
![]()
By property of
and
we have
so
which is a contradiction since we supposed
.
Hence, we have
.
Similarly, we have that
![]()
Therefore for all n we have
![]()
and
is a non increasing sequence of nonnegative real numbers
and bounded below. Hence there exists
such that
![]()
By the property of functions
and
, we have
(3.11.7)
If we consider condition (B) we have,
(3.11.8)
Taking the upper limit as
in (3.11.8) and using (3.11.7) we have that,
(3.11.9)
From (3.11.9) and property of
and
follows that
and also
(3.11.10)
In a similar way as in Theorem 3.2 we can show that the sequence
(also
) is a
-Cauchy sequence in b-dislocated metric space
. Since X is complete there exists
such that
. Since T is sequentially convergent, we can deduce that
is convergent to
and the subsequences
converge to u, that means
.
Since T is continuous we have
.
Let we prove that u is a fixed point of F and G (
). If suppose that
then since T is injective follows
(and
)
Consider,
(3.11.11)
Taking the upper limit in (3.11.11) and using lemma (2.13), and result (3.11.10) we get
(3.11.12)
According to contractive condition (B) we have,
(3.11.13)
Taking the upper limit in (3.11.13) and using lemma (2.13), we obtain,
![]()
This implies that
that is
which is a contradiction. As a result
and u is a fixed point of F. By the first part of proof u is a fixed point of G and also a common fixed point.
Easily using the contractive condition (B) of theorem can be proved that the common fixed point is unique.
Example 3.12 Let
be equipped with the b-dislocated metric
for all
, where
. It is clear that
is a complete b-dislocated metric space. Also let be the self-mappings
defined by
. We note, T is continuous and sequentially convergent.
If
,
and
then for each
, we have
![]()
Thus
satisfy all the conditions of Theorem 3.11. Moreover
is the unique common fixed point of
.
If in theorem3.11 we take
we get the following corollary.
Corollary 3.13 Let
be a complete b-dislocated metric space with parameter
and
be two self mappings, where T is injective, continuous and sequentially convergent. If exist
,
and
such that
![]()
for all
, where
![]()
then G has a unique fixed point.
Corollary 3.14 Let
be a complete b-dislocated metric space with parameter
and
be an injective, continuous and sequentially convergent mapping. Let
be self-mappings and if exist
,
such that
![]()
for all
, where
![]()
then
have a unique common fixed point.
Proof. If we take in Theorem 3.11 the function
as
then we get the corollary.
Remark 3.15
1) Theorem 3.11 generalizes, extends and unifies results as Theorem 8 in [32] , Theorem 4 in [33] and many existing results of literature in a set effective larger as b-dislocated metric spaces.
2) The class C of functions has a general character and so according to example 2.15, we can provide many results from theorem 3.11.
3) If we take in theorem 3.11 the parameter
as a consequence, we obtain results in a dislocated metric space.