Abstract
The aim of this paper is to investigate some fixed point results in dislocated quasi metric (dq-metric) spaces. Fixed point results for different types of contractive conditions are established, which generalize, modify and unify some existing fixed point theorems in the literature. Appropriate examples for the usability of the established results are also given. We notice that by using our results some fixed point results in the context of dislocated quasi metric spaces can be deduced.
MSC:47H10, 54H25.
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1 Introduction
Fixed point theory is one of the most dynamic research subjects in nonlinear analysis. In this area, the first important and significant result was proved by Banach in 1922 for a contraction mapping in a complete metric space. The well-known Banach contraction theorem may be stated as follows: ‘Every contraction mapping of a complete metric space X into itself has a unique fixed point’ (Bonsall 1962).
Dass and Gupta [1] generalized the Banach contraction principle in a metric space for some rational type contractive conditions.
The role of topology in logic programming has come to be recognized (see [2–6] and the references cited therein). Particularly, topological methods are applied to obtain fixed point semantics for logic programs. Such considerations motivated the concept of dislocated metric spaces. This idea was not new and it had been studied in the context of domain theory [4] where the dislocated metrics were known as metric domains.
Hitzler and Seda [3] investigated the useful applications of dislocated topology in the context of logic programming semantics. In order to obtain a unique supported model for these programs, they introduced the notation of dislocated metric space and generalized the Banach contraction principle in such spaces.
Furthermore, Zeyada et al. [7] generalized the results of Hitzler and Seda [3] and introduced the concept of complete dislocated quasi metric space. Aage and Salunke [8, 9] derived some fixed point theorems in dislocated quasi metric spaces. Similarly, Isufati [10] proved some fixed point results for continuous contractive condition with rational type expression in the context of a dislocated quasi metric space. Kohli et al. [11] investigated a fixed point theorem which generalized the result of Isufati. In [12] Zoto gave some new results in dislocated and dislocated quasi metric spaces. For a continuous self-mapping, a fixed point theorem in dislocated quasi metric spaces was investigated by Madhu Shrivastava et al. [13]. In 2013, Patel and Patel [14] constructed some new fixed point results in a dislocated quasi metric space.
In the current manuscript, we establish some fixed point results for single and a pair of continuous self-mappings in the context of dislocated quasi metric spaces which generalize, modify and unify the results of Aage and Salunke [8, 9], Manvi Kohli [11], Patel and Patel [14], Madhu Shrivastava et al. [13] and Zeyada et al. [7]. Throughout the paper represents the set of non-negative real numbers.
2 Preliminaries
Definition 2.1 ([7])
Let X be a non-empty set, and let be a function satisfying the following conditions:
() ;
() implies that ;
() for all ;
() for all .
If d satisfies the conditions from to , then it is called a metric on X, if d satisfies conditions to , then it is called a dislocated metric (d-metric ) on X, and if d satisfies conditions and , only then it is called a dislocated quasi metric (dq-metric) on X.
It is evident that every metric on X is a dislocated metric on X, but the converse is not necessarily true as is clear from the following example.
Example 2.1 Let define the distance function by
Furthermore, from the following example one can say that a dislocated quasi metric on X needs not be a dislocated metric on X.
Example 2.2 Let , we define the function as
In our main work we will use the following definitions which can be found in [7].
Definition 2.2 A sequence in a dq-metric space is called a Cauchy sequence if for there exists a positive integer N such that for , we have .
Definition 2.3 A sequence is called dq-convergent in X if for , we have , where x is called the dq-limit of the sequence .
Definition 2.4 A dq-metric space is said to be complete if every Cauchy sequence in X converges to a point of X.
Definition 2.5 Let be a dq-metric space, a mapping is called a contraction if there exists such that
The following statement is well known (see [7]).
Lemma 1 Limit in a dq-metric space is unique.
In [15] Kannan defined a contraction of the following type.
Definition 2.6 Let be a metric space, and let be a self-mapping. Then T is called a Kannan mapping if
Kannan [15] established a unique fixed point theorem for a mapping which satisfies condition (1) in metric spaces.
Definition 2.7 ([16])
Let be a metric space, a self-mapping is called a generalized contraction if and only for all , there exist , , , such that and
Ciric [16] investigated a unique fixed point theorem for a mapping which satisfies condition (2) in the context of metric spaces.
In the following theorem, Zeyada et al. [7] generalized the Banach contraction principle in dislocated quasi metric spaces.
Theorem 2.1 Let be a complete dq-metric space, be a continuous contraction, then T has a unique fixed point in X.
Aage and Salunke [8] established the following results for single and a pair of continuous mappings in dislocated quasi metric spaces.
Theorem 2.2 Let be a complete dq-metric space and be a continuous self-mapping satisfying the following condition:
where with and for all . Then T has a unique fixed point.
Theorem 2.3 Let be a complete dq-metric space and be continuous self-mappings satisfying the following condition:
where with and for all . Then S and T have a unique common fixed point.
Furthermore, Aage and Salunke [9] derived the following fixed point theorems with a Kannan-type contraction and a generalized contraction in the setting of dislocated quasi metric spaces, respectively.
Theorem 2.4 Let be a complete dq-metric space and be a continuous self-mapping satisfying the following condition:
where with and for all . Then T has a unique fixed point.
Theorem 2.5 Let be a complete dq-metric space and be a continuous self-mapping satisfying the following condition:
where with and for all . Then T has a unique fixed point.
Isufati [10] derived the following two results, where the first one generalized the result of Dass and Gupta [1] in dislocated quasi metric spaces.
Theorem 2.6 Let be a complete dq-metric space and be a continuous self-mapping satisfying the following condition:
where with and for all . Then T has a unique fixed point.
Theorem 2.7 Let be a complete dq-metric space and be a continuous self-mapping satisfying the following condition:
where with and for all . Then T has a unique fixed point.
In [11] Kohli, Shrivastava and Sharma proved the following theorem in the context of dislocated quasi metric spaces which generalized Theorem 2.6.
Theorem 2.8 Let be a complete dq-metric space and be a continuous self-mapping satisfying the following condition:
where with and for all . Then T has a unique fixed point.
For rational type contraction conditions Madhu Shrivastava et al. [13] proved the following theorem in a dislocated quasi metric space.
Theorem 2.9 Let be a complete dq-metric space and be a continuous self-mapping satisfying the following condition:
where with and for all . Then T has a unique fixed point.
In 2013, Patel and Patel [14] derived the following result in dislocated quasi metric spaces.
Theorem 2.10 Let be a complete dq-metric space, and let be a continuous self-mapping satisfying the following condition:
where with and for all . Then T has a unique fixed point.
3 Main results
In this section we derive some fixed point theorems with examples for single and a pair of continuous self-mappings in the context of dislocated quasi metric spaces.
Theorem 3.1 Let be a complete dq-metric space, and let be a continuous self-mapping satisfying the following condition:
where with and for all . Then T has a unique fixed point in X.
Proof Let be arbitrary in X, we define a sequence by the rule
Now we show that is a Cauchy sequence in X. Suppose
By using condition (3) we have
Let
Clearly, because .
So,
Similarly,
Thus
Continuing the same procedure, we have
But so as , which shows that is a Cauchy sequence in a complete dq-metric space. So there exists such that as .
Now we show that z is a fixed point of T. Since as , using the continuity of T, we have
which implies that
Thus . Hence z is a fixed point of T.
Uniqueness. Suppose that T has two fixed points z and w for . Consider
Since z and w are fixed points of T, therefore condition (3) implies that and . Finally, from (4) we get
Similarly, we have
Subtracting (6) from (5) we have
Since , so the above inequality (7) is possible if
Taking equations (5), (6) and (8) into account, we have and . Thus by () . Hence T has a unique fixed point in X. □
Example 3.1 Let with a complete dq-metric defined by
and define the continuous self-mapping T by with , , , , , , . Then T satisfies all the conditions of Theorem 3.1, and is the unique fixed point of T in X.
Remarks In the above Theorem 3.1:
-
If , then we get the result of Isufati [10].
-
If , then we get the result of Madhu Shrivastava et al.[13].
-
If , then we get the result of Isufati [10].
-
if , then we get the result of Manvi Kohli [11].
Theorem 3.2 Let be a complete dq-metric space, and let be two continuous self-mappings satisfying the following condition:
where with and for all . Then S and T have a unique common fixed point in X.
Proof Let be arbitrary in X, we define a sequence by the rule and for all . We claim that is a Cauchy sequence in X. For this consider
By using condition (9) we have
Therefore, finally we have
Let
Then as . Thus
and
So
Similarly, we proceed to get
Since and implies that , which proved that is a Cauchy sequence in a complete dq-metric space. Therefore there exists z in X such that as . Also the sub-sequences and converge to z. Since T is a continuous mapping, therefore
Hence
Similarly, taking the continuity of S, we can show that .
Hence z is the common fixed point of S and T.
Uniqueness. Suppose that S and T have two common fixed points z and w for . Consider
Since z and w are common fixed points of T and S, condition (9) implies that and . Thus equation (10) becomes
Similarly,
Subtracting (12) from (11) we get
Since , so the above inequality is possible if
By combining equations (11), (12) and (13), one can get and . Using () we have . Hence S and T have a unique common fixed point in X. □
Example 3.2 Let and complete dq-metric is defined by
where the continuous self-mappings S and T are defined by and for all . Suppose , , , , .
Then S and T satisfy all the conditions of Theorem 3.2, so is the unique common fixed point of S and T in X.
Theorem 3.2 yields the following corollaries.
Corollary 3.1 If and all other conditions of Theorem 3.2 are satisfied, then T has a unique fixed point.
Corollary 3.2 Let , and let be two self-continuous mappings satisfying all other conditions of Theorem 3.2. Then S and T have a unique common fixed point in X.
Corollary 3.3 Let , and and all other conditions of Theorem 3.2 be satisfied, then again T has a unique fixed point.
Corollary 3.4 Suppose . Let be two self-continuous mappings satisfying all other conditions of Theorem 3.2. Then S and T have a unique fixed point in X.
Corollary 3.5 Suppose and , and all other conditions of Theorem 3.2 are satisfied. Then T has a unique fixed point in X.
Remarks
References
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Sarwar, M., Rahman, M.U. & Ali, G. Some fixed point results in dislocated quasi metric (dq-metric) spaces. J Inequal Appl 2014, 278 (2014). https://doi.org/10.1186/1029-242X-2014-278
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DOI: https://doi.org/10.1186/1029-242X-2014-278