Abstract
We consider complete Menger probabilistic quasimetric space and prove common fixed point theorems for weakly compatible maps in this space.
Fixed Point Theory and Applications volume 2009, Article number: 546273 (2009)
We consider complete Menger probabilistic quasimetric space and prove common fixed point theorems for weakly compatible maps in this space.
K. Menger introduced the notion of a probabilistic metric space in 1942 and since then the theory of probabilistic metric spaces has developed in many directions [1]. The idea of Menger was to use distribution functions instead of nonnegative real numbers as values of the metric. The notion of a probabilistic metric space corresponds to the situations when we do not know exactly the distance between two points, we know only probabilities of possible values of this distance. Such a probabilistic generalization of metric spaces appears to be well adapted for the investigation of physiological thresholds and physical quantities particularly in connections with both string and theory; see [2–5]. It is also of fundamental importance in probabilistic functional analysis, nonlinear analysis and applications [6–10].
In the sequel, we will adopt usual terminology, notation, and conventions of the theory of Menger probabilistic metric spaces, as in [7, 8, 10]. Throughout this paper, the space of all probability distribution functions (in short, dfs) is denoted by is left-continuous and nondecreasing on , and and the subset is the set . Here denotes the left limit of the function at the point , . The space is partially ordered by the usual pointwise ordering of functions, that is, if and only if for all in . The maximal element for in this order is the df given by
Definition 1.1 (see [1]).
A mapping is if is satisfying the following conditions:
(a) is commutative and associative;
(b) for all ;
(d) whenever and , and .
The following are the four basic :
Each can be extended [11] (by associativity) in a unique way to an -ary operation taking for the values and
for and , for all .
We also mention the following families of
Definition 1.2.
It is said that the -norm is of Hadžić-type ( for short) and if the family of its iterates defined, for each in , by
is equicontinuous at , that is,
There is a nice characterization of continuous of the class [12].
(i)If there exists a strictly increasing sequence in such that and , then is of Hadžić-type.
(ii)If is continuous and , then there exists a sequence as in (i).The is an trivial example of a of but there are of Hadžić-type with (see, e.g., [13]).
Definition 1.3 (see [13]).
If is a and , then is defined recurrently by 1, if and for all . If is a sequence of numbers from then is defined as (this limit always exists) and as . In fixed point theory in probablistic metric spaces there are of particular interest the -norms and sequences such that and . Some examples of with the above property are given in the following proposition.
Proposition 1.4 (see [13]).
For the following implication holds:
If , then for every sequence in I such that , one has .
Note [14, Remark 13] that if is a for which there exists such that and , then Important class of is given in the following example.
Example 1.5.
The Dombi family of is defined by
The Aczél-Alsina family of is defined by
Sugeno-Weber family of is defined by
In [13] the following results are obtained.
(a)If is the Dombi family of and is a sequence of elements from such that then we have the following equivalence:
(b)Equivalence (1.10) holds also for the family that is,
(c)If is the Sugeno-Weber family of and is a sequence of elements from such that then we have the following equivalence:
Proposition 1.6.
Let be a sequence of numbers from such that and -norm is of . Then
Definition 1.7.
A Menger Probabilistic Quasimetric space (briefly, Menger PQM space) is a triple , where is a nonempty set, is a continuous , and is a mapping from into such that, if denotes the value of at the pair , then the following conditions hold, for all in ,
(PQM1) for all if and only if ;
(PQM2) for all and .
Definition 1.8.
Let be a Menger PQM space.
(1)A sequence in is said to be convergent to in if, for every and , there exists positive integer such that whenever .
(2)A sequence in is called Cauchy sequence [15] if, for every and , there exists positive integer such that whenever ().
(3)A Menger PQM space is said to be complete if and only if every Cauchy sequence in is convergent to a point in .
In 1998, Jungck and Rhoades [16] introduced the following concept of weak compatibility.
Definition 1.9.
Let and be mappings from a Menger PQM space into itself. Then the mappings are said to be weak compatible if they commute at their coincidence point, that is, implies that .
Throughout this section, a binary operation is a continuous -norm and satisfies the condition
where . It is easy to see that this condition implies .
Lemma 2.1.
Let be a Menger PQM space. If the sequence in X is such that for every
for very , where is a monotone increasing functions.Then the sequence is a Cauchy sequence.
Proof.
For every and , we have
for each and . Hence sequence is Cauchy sequence.
Theorem 2.2.
Let be a complete Menger PQM space and let be maps that satisfy the following conditions:
(a);
(b)the pairs and are weak compatible, is closed subset of ;
(c) for all and every , where is a monotone increasing function.
If
then and have a unique common fixed point.
Proof.
Let . By (a), we can find such that and . By induction, we can define a sequence such that and . By induction again,
Similarly, we have
Hence, it follows that
for
Now by Lemma 2.1, is a Cauchy sequence. Since the space is complete, there exists a point such that
It follows that, there exists such that . We prove that . From (c), we get
as , we have
which implies that, . Moreover,
as , we have
which implies that Since, the pairs and are weak compatible, we have hence it follows that Similarly, we get Now, we prove that Since, from (c) we have
as , we have
It follows that . Therefore, . That is is a common fixed point of and .
If and are two fixed points common to and , then
as , which implies that and so the uniqueness of the common fixed point.
Corollary 2.3.
Let be a complete Menger PQM space and let be maps that satisfy the following conditions:
(a);
(b)the pair is weak compatible, is closed subset of ;
(c) for all and where is monotone increasing function.
If
then and have a unique common fixed point.
Proof.
It is enough, set in Theorem 2.2.
Corollary 2.4.
Let be a complete Menger PQM space and let be maps that satisfy the following conditions:
(a)
(b)the pair is weak compatible, is closed subset of ;
(c) for all and , where is monotone increasing function;
If
then have a unique common fixed point.
Proof.
By Corollary 2.3, if set then have a unique common fixed point in . That is, there exists , such that . We prove that , for From (c), we have
By (d), we get
Hence, . Thus , .
Similarly, we have .
Corollary 2.5.
Let be a complete PQM space and let satisfy conditions (a), (b), and (c) of Theorem 2.2. If is a of then there exists a unique common fixed point for the mapping and .
Proof.
By Proposition 1.6 all the conditions of the Theorem 2.2 are satisfied.
Corollary 2.6.
Let for some be a complete PQM space and let satisfy conditions (a), (b), and (c) of Theorem 2.2. If then there exists a unique common fixed point for the mapping and .
Proof.
From equivalence (1.10) we have
Corollary 2.7.
Let for some be a complete PQM space and let satisfy conditions (a), (b), and (c) of Theorem 2.2. If then there exists a unique common fixed point for the mapping and .
Proof.
From equivalence (1.11) we have
Corollary 2.8.
Let for some be a complete PQM space and let satisfy conditions (a), (b), and (c) of Theorem 2.2. If then there exists a unique common fixed point for the mapping and .
Proof.
From equivalence (1.12) we have
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The second author is supported by MNTRRS 144012.
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Sedghi, S., Žikić-Došenović, T. & Shobe, N. Common Fixed Point Theorems in Menger Probabilistic Quasimetric Spaces. Fixed Point Theory Appl 2009, 546273 (2009). https://doi.org/10.1155/2009/546273
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DOI: https://doi.org/10.1155/2009/546273