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Some fixed point results in dislocated quasi metric (dqmetric) spaces
Journal of Inequalities and Applications volume 2014, Article number: 278 (2014)
Abstract
The aim of this paper is to investigate some fixed point results in dislocated quasi metric (dqmetric) 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.
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 wellknown 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 selfmapping, 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 selfmappings 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 {\mathbb{R}}^{+} represents the set of nonnegative real numbers.
2 Preliminaries
Definition 2.1 ([7])
Let X be a nonempty set, and let d:X\times X\to {\mathbb{R}}^{+} be a function satisfying the following conditions:
({d}_{1}) d(x,x)=0;
({d}_{2}) d(x,y)=d(y,x)=0 implies that x=y;
({d}_{3}) d(x,y)=d(y,x) for all x,y,z\in X;
({d}_{4}) d(x,y)\le d(x,z)+d(z,y) for all x,y,z\in X.
If d satisfies the conditions from {d}_{1} to {d}_{4}, then it is called a metric on X, if d satisfies conditions {d}_{2} to {d}_{4}, then it is called a dislocated metric (dmetric ) on X, and if d satisfies conditions {d}_{2} and {d}_{4}, only then it is called a dislocated quasi metric (dqmetric) 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 X={\mathbb{R}}^{+} define the distance function d:X\times X\to {\mathbb{R}}^{+} 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 X=[0,1], we define the function d:X\times X\to {\mathbb{R}}^{+} as
In our main work we will use the following definitions which can be found in [7].
Definition 2.2 A sequence \{{x}_{n}\} in a dqmetric space is called a Cauchy sequence if for \u03f5>0 there exists a positive integer N such that for m,n\ge N, we have d({x}_{m},{x}_{n})<\u03f5.
Definition 2.3 A sequence \{{x}_{n}\} is called dqconvergent in X if for n\ge N, we have d({x}_{n},x)<\u03f5, where x is called the dqlimit of the sequence \{{x}_{n}\}.
Definition 2.4 A dqmetric space (X,d) is said to be complete if every Cauchy sequence in X converges to a point of X.
Definition 2.5 Let (X,d) be a dqmetric space, a mapping T:X\to X is called a contraction if there exists 0\le \alpha <1 such that
The following statement is well known (see [7]).
Lemma 1 Limit in a dqmetric space is unique.
In [15] Kannan defined a contraction of the following type.
Definition 2.6 Let (X,d) be a metric space, and let T:X\to X be a selfmapping. 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 (X,d) be a metric space, a selfmapping T:X\to X is called a generalized contraction if and only for all x,y\in X, there exist {c}_{1}, {c}_{2}, {c}_{3}, {c}_{4} such that sup\{{c}_{1}+{c}_{2}+{c}_{3}+2{c}_{4}\}<1 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 (X,d) be a complete dqmetric space, T:X\to X 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 (X,d) be a complete dqmetric space and T:X\to X be a continuous selfmapping satisfying the following condition:
where a,b,c\ge 0 with a+b+c<1 and for all x,y\in X. Then T has a unique fixed point.
Theorem 2.3 Let (X,d) be a complete dqmetric space and S,T:X\to X be continuous selfmappings satisfying the following condition:
where a,b,c\ge 0 with a+b+c<1 and for all x,y\in X. Then S and T have a unique common fixed point.
Furthermore, Aage and Salunke [9] derived the following fixed point theorems with a Kannantype contraction and a generalized contraction in the setting of dislocated quasi metric spaces, respectively.
Theorem 2.4 Let (X,d) be a complete dqmetric space and T:X\to X be a continuous selfmapping satisfying the following condition:
where a\ge 0 with a<\frac{1}{2} and for all x,y\in X. Then T has a unique fixed point.
Theorem 2.5 Let (X,d) be a complete dqmetric space and T:X\to X be a continuous selfmapping satisfying the following condition:
where a,b,c,e\ge 0 with a+b+c+2e<1 and for all x,y\in X. 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 (X,d) be a complete dqmetric space and T:X\to X be a continuous selfmapping satisfying the following condition:
where a,b>0 with a+b<1 and for all x,y\in X. Then T has a unique fixed point.
Theorem 2.7 Let (X,d) be a complete dqmetric space and T:X\to X be a continuous selfmapping satisfying the following condition:
where a,b,c>0 with sup\{a+2b+2c\}<1 and for all x,y\in X. 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 (X,d) be a complete dqmetric space and T:X\to X be a continuous selfmapping satisfying the following condition:
where a,b,c>0 with a+b+c<1 and for all x,y\in X. 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 (X,d) be a complete dqmetric space and T:X\to X be a continuous selfmapping satisfying the following condition:
where a,b,c>0 with a+b+c<1 and for all x,y\in X. 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 (X,d) be a complete dqmetric space, and let T:X\to X be a continuous selfmapping satisfying the following condition:
where {c}_{1},{c}_{2},{c}_{3},{c}_{4},{c}_{5}\ge 0 with {c}_{1}+{c}_{2}+{c}_{3}+2({c}_{4}+{c}_{5})<1 and for all x,y\in X. 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 selfmappings in the context of dislocated quasi metric spaces.
Theorem 3.1 Let (X,d) be a complete dqmetric space, and let T:X\to X be a continuous selfmapping satisfying the following condition:
where {a}_{1},{a}_{2},{a}_{3},{a}_{4},{a}_{5},{a}_{6},{a}_{7}\ge 0 with {a}_{1}+2({a}_{2}+{a}_{3})+{a}_{4}+{a}_{5}+3{a}_{6}+{a}_{7}<1 and for all x,y\in X. Then T has a unique fixed point in X.
Proof Let {x}_{0} be arbitrary in X, we define a sequence \{{x}_{n}\} by the rule
Now we show that \{{x}_{n}\} is a Cauchy sequence in X. Suppose
By using condition (3) we have
Let
Clearly, h<1 because {a}_{1}+2{a}_{2}+2{a}_{3}+{a}_{4}+{a}_{5}+3{a}_{6}+{a}_{7}<1.
So,
Similarly,
Thus
Continuing the same procedure, we have
But 0\le h<1 so {h}^{n}\to 0 as n\to \mathrm{\infty}, which shows that \{{x}_{n}\} is a Cauchy sequence in a complete dqmetric space. So there exists z\in X such that {x}_{n}\to z as n\to \mathrm{\infty}.
Now we show that z is a fixed point of T. Since {x}_{n}\to z as n\to \mathrm{\infty}, using the continuity of T, we have
which implies that
Thus Tz=z. Hence z is a fixed point of T.
Uniqueness. Suppose that T has two fixed points z and w for z\ne w. Consider
Since z and w are fixed points of T, therefore condition (3) implies that d(z,z)=0 and 0=d(w,w). Finally, from (4) we get
Similarly, we have
Subtracting (6) from (5) we have
Since ({a}_{1}+{a}_{2})({a}_{3}+{a}_{6})<1, so the above inequality (7) is possible if
Taking equations (5), (6) and (8) into account, we have d(z,w)=0 and d(w,z)=0. Thus by ({d}_{2}) z=w. Hence T has a unique fixed point in X. □
Example 3.1 Let X=[0,1] with a complete dqmetric defined by
and define the continuous selfmapping T by Tx=\frac{x}{2} with {a}_{1}=1/8, {a}_{2}=1/10, {a}_{3}=1/12, {a}_{4}=1/15, {a}_{5}=1/20, {a}_{6}=1/24, {a}_{7}=1/30. Then T satisfies all the conditions of Theorem 3.1, and x=0 is the unique fixed point of T in X.
Remarks In the above Theorem 3.1:

If {a}_{4}={a}_{5}={a}_{6}={a}_{7}=0, then we get the result of Isufati [10].

If {a}_{2}={a}_{3}={a}_{4}={a}_{7}=0, then we get the result of Madhu Shrivastava et al.[13].

If {a}_{2}={a}_{3}={a}_{4}={a}_{6}={a}_{7}=0, then we get the result of Isufati [10].

if {a}_{2}={a}_{3}={a}_{6}={a}_{7}=0, then we get the result of Manvi Kohli [11].
Theorem 3.2 Let (X,d) be a complete dqmetric space, and let S,T:X\to X be two continuous selfmappings satisfying the following condition:
where {c}_{1},{c}_{2},{c}_{3},{c}_{4},{c}_{5}\ge 0 with {c}_{1}+{c}_{2}+{c}_{3}+2{c}_{4}+4{c}_{5}<1 and for all x,y\in X. Then S and T have a unique common fixed point in X.
Proof Let {x}_{0} be arbitrary in X, we define a sequence \{{x}_{n}\} by the rule {x}_{0},{x}_{1}=S{x}_{0},\dots ,{x}_{2n+1}=S{x}_{2n} and {x}_{2}=T{x}_{1},\dots ,{x}_{2n}=T{x}_{2n1} for all n\in \mathrm{N}. We claim that \{{x}_{n}\} is a Cauchy sequence in X. For this consider
By using condition (9) we have
Therefore, finally we have
Let
Then h<1 as {c}_{1}+{c}_{2}+{c}_{3}+2{c}_{4}+4{c}_{5}<1. Thus
and
So
Similarly, we proceed to get
Since 0\le h<1 and n\to \mathrm{\infty} implies that {h}^{n}\to 0, which proved that \{{x}_{n}\} is a Cauchy sequence in a complete dqmetric space. Therefore there exists z in X such that {x}_{n}\to z as n\to \mathrm{\infty}. Also the subsequences \{{x}_{2n}\} and \{{x}_{2n+1}\} converge to z. Since T is a continuous mapping, therefore
Hence
Similarly, taking the continuity of S, we can show that Sz=z.
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 z\ne w. Consider
Since z and w are common fixed points of T and S, condition (9) implies that d(z,z)=0 and d(w,w). Thus equation (10) becomes
Similarly,
Subtracting (12) from (11) we get
Since {c}_{1}<1, so the above inequality is possible if
By combining equations (11), (12) and (13), one can get d(z,w)=0 and d(w,z)=0. Using ({d}_{2}) we have z=w. Hence S and T have a unique common fixed point in X. □
Example 3.2 Let X=[0,1] and complete dqmetric is defined by
where the continuous selfmappings S and T are defined by Sx=0 and Tx=x/5 for all x\in X. Suppose {c}_{1}=1/5, {c}_{2}=1/6, {c}_{3}=1/8, {c}_{4}=1/10, {c}_{5}=1/12.
Then S and T satisfy all the conditions of Theorem 3.2, so x=0 is the unique common fixed point of S and T in X.
Theorem 3.2 yields the following corollaries.
Corollary 3.1 If S=T and all other conditions of Theorem 3.2 are satisfied, then T has a unique fixed point.
Corollary 3.2 Let {c}_{4}={c}_{5}=0, and let S,T:X\to X be two selfcontinuous 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 {c}_{4}={c}_{5}=0, and S=T and all other conditions of Theorem 3.2 be satisfied, then again T has a unique fixed point.
Corollary 3.4 Suppose {c}_{1}={c}_{2}={c}_{3}={c}_{4}=0. Let S,T:X\to X be two selfcontinuous mappings satisfying all other conditions of Theorem 3.2. Then S and T have a unique fixed point in X.
Corollary 3.5 Suppose {c}_{1}={c}_{2}={c}_{3}={c}_{5}=0 and S=T, and all other conditions of Theorem 3.2 are satisfied. Then T has a unique fixed point in X.
Remarks
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Sarwar, M., Rahman, M.U. & Ali, G. Some fixed point results in dislocated quasi metric (dqmetric) spaces. J Inequal Appl 2014, 278 (2014). https://doi.org/10.1186/1029242X2014278
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DOI: https://doi.org/10.1186/1029242X2014278
Keywords
 complete dqmetric space
 contraction mapping
 selfmapping
 Cauchy sequence
 fixed point