Nonlinear contractive and nonlinear compatible type mappings in Menger probabilistic metric spaces
Journal of Inequalities and Applications volume 2014, Article number: 347 (2014)
In this paper some new fixed point theorems for nonlinear contractive type and nonlinear compatible type mapping in complete Menger probabilistic metric spaces are proved.
MSC:54E40, 54E35, 54H25.
1 Introduction and preliminaries
K Menger introduced the notion of a probabilistic metric space in 1942. Since then the theory of probabilistic metric spaces has developed in many directions [1, 2]. The idea of K Menger was to use distribution functions instead of non-negative real numbers as values of the metric. The notion of a probabilistic metric space corresponds to situations when we do not know exactly the distance between two points, but we know probabilities of possible values of this distance. A probabilistic generalization of metric spaces appears to be of interest in the investigation of physical quantities and physiological thresholds. It is also of fundamental importance in probabilistic functional analysis.
The purpose of this paper is to prove some existence theorems of fixed points for nonlinear contractive type and nonlinear compatible type mapping in complete Menger probabilistic metric spaces. In the sequel, we shall adopt the usual terminology, notation and conventions of the theory of probabilistic metric, as in [1–6].
Throughout this paper, let ℝ be the set of all real numbers and be the set of all non-negative real numbers. A mapping is called a distribution function (briefly, d.f.), if it is left-continuous and non-decreasing with
In the sequel, we denote by the set of all distribution functions on ℝ. The space is partially ordered by the usual point-wise ordering of functions, i.e., if and only if for all . The maximal element for in this order is the d.f. given by
A mapping is called a continuous t-norm, if T satisfies the following conditions:
T is commutative and associative;
T is continuous;
for all ;
whenever , , and .
Two typical examples of continuous t-norm are and .
Now t-norms are recursively defined by and
for all and , for all .
Definition 1.2 A Menger Probabilistic Metric space (briefly, Menger PM-space) is a triple , where X is a non-empty set, T is a continuous t-norm, and ℱ is a mapping from into satisfying the following conditions: for all (in the sequel, we use to denote ):
(PM1) , , if and only if ;
(PM3) , and .
Definition 1.3 Let be a Menger PM-space.
A sequence in X is said to be convergent to if, for every and , there exists a positive integer N such that whenever .
A sequence in X is called a Cauchy sequence if, for every and , there exists a positive integer N such that whenever .
A Menger PM-space is said to be complete if and only if every Cauchy sequence in X is convergent to a point in X.
Definition 1.4 Let be a Menger PM space, be a given point.
For any given and the set
is called the strong -neighborhood of p.
The strong neighborhood system for X is the union , where .
Definition 1.6 
Let X be a non-empty set, be a family of mappings from into . The ordered pair is called a generating space of quasi-metrics family, and is called the family of quasi-metrics on X if, it satisfies the following conditions:
(QM-1) for all , if and only if ;
(QM-2) for all and ;
(QM-3) for any given , there exists such that
(QM-4) for any give , the function is nonincreasing and left-continuous.
Lemma 1.7 Let be a Menger PM-space with a t-norm T satisfying the following conditions:
For any given , define a mapping as follows:
is a family of quasi-metrics on X and is a generating space of the quasi-metrics family ;
the topology induced by quasi-metric family on X coincides with the -topology on X.
Proof (1) From Definition 1.6, it is easy to see that the family satisfies the conditions (QM-1) and (QM-2).
Next we prove that is left-continuous in . In fact, for any given and , by the definition of , there exists such that and . Letting and , we have
This implies that
Hence we have
which shows is left-continuous.
Now we prove that, for any given , is nonincreasing in .
In fact, for any with , we have
Condition (QM-4) is proved.
Finally, we prove that also satisfies condition (QM-3).
In fact, for any given , and , by condition (1.1), there exists such that
Letting , from (1.2) for any , we have
and so we have
This implies that
By the arbitrariness of , we have
for any . Especially, if , then condition (QM-3) is proved. The conclusion (1) is proved.
Now we prove the conclusion (2).
For the purpose, it is sufficient to prove that, for any given and ,
In fact, if , then from (1.2) we have . Conversely, if , since is a left-continuous distribution function, there exists a such that , and so .
This completes the proof of Lemma 1.7. □
Remark 1.8 From Lemma 1.7, it is easy to see that a sequence in a Menger PM-space is convergent in the -topology σ, if and only if , . Also a sequence in a Menger PM-space is a Cauchy sequence in the -topology, if and only if (as ).
Definition 1.9 A function is said to satisfy condition (Φ), if it is non-decreasing and for all , where denotes the n th iterate function of .
If satisfies condition (Φ), then , . If , then .
Lemma 1.11 
Let be a Menger PM-space. Suppose that the function is onto and strictly increasing. Then
for any , and .
Lemma 1.12 Let be a Menger PM-space with a t-norm T satisfying condition (1.1) and be a sequence in X such that
where is onto, strictly increasing, and satisfies condition (Φ). If
then is a Cauchy sequence in X.
Proof For any , it follows from Lemma 1.11 and condition (1.4) that
For any given positive integers m, n, , and for given , it follows from (1.5) and Lemma 1.7 that there exists a such that
By Lemma 1.7, is a Cauchy sequence in X.
This completes the proof of Lemma 1.12. □
2 Fixed point theorems of nonlinear contraction type mappings in Menger PM-spaces
Theorem 2.1 Let be a complete Menger PM-space, be a sequence of mappings from X into itself such that, for any two mappings , , and for any and ,
where is onto, strictly increasing, and satisfies condition (Φ). If there exists such that
Then has a unique common fixed point in X, and the sequence defined by
converges to in the -topology of X.
Proof It follows from (2.1) and (2.3) that
If , then from (2.4)
By induction, we can prove that, for any positive integer n,
Let , we have for all . This contradicts that is a distribution function. Hence we have
Similarly, we have
By induction, we can prove that
It follows from condition (2.2) and Lemma 1.12 that is a Cauchy sequence in X. Since X is complete, there exists a point such that .
Next we prove that is the unique common fixed point of . In fact, for any given positive integers i, n, and for any , by Lemma 1.11, we have
By virtue of the continuity of ϕ, we have
From Remark 1.10, it follows that , i.e., , .
Next we prove that is the unique common fixed point of in X. In fact, if is also a common fixed point of , then, for any and any i, j, ,
i.e., by Remark 1.10. And so .
This completes the proof of Theorem 2.1. □
3 Fixed point theorems for nonlinear compatible type mappings in PM-spaces
Definition 3.1 Let be a Menger PM-space, and let f and S be two mappings from X into itself. f and S are called compatible if , , whenever is a sequence in X such that and converge in the -topology to some as .
Remark 3.2 It should be point out that the concept of compatible mappings was introduced by Jungck  in metric space. The concept of compatible mappings introduced here is a generalization of Jungck  and Singh et al. .
Theorem 3.3 Let be a complete Menger PM-space, be mappings satisfying the following conditions:
for all , , where the function is onto strictly increasing and satisfies condition (Φ). If either f or g is continuous and the pairs S, f and G, g both are compatible, and if there exists an such that
where , and , then S, G, f, g have a unique common fixed point z in X.
Proof By condition (i), there exists such that . For , there exists such that . Inductively, we can construct sequences and as follows:
It follows from condition (ii) that, for any ,
If , then from (3.3), , , and so
Since ϕ satisfies condition (Φ), we have
This contradicts that is a distribution function. Therefore
Similarly we can prove that
These show that, for any positive integer , we have
On the other hand, it follows from Lemma 1.11 that, for any ,
By induction, we can prove that
By Lemma 1.7, for any given and for any positive integers m, n, , there exists such that
This implies that is a Cauchy sequence in X. Without loss of generality, we can assume that . Therefore
By the assumption, without loss of generality, we can assume that f is continuous, then and . Since S and f are compatible, we have
and so we have
This shows that
Again for any positive integer and , from Lemma 1.11, we have
Therefore we have
By Remark 1.10, , i.e., .
Similarly, we can prove that
Hence we have
and so , i.e., .
Select such that . Then and for any
This implies that , and so . Since G, g are compatible and , we get
This shows that . Again for any
This implies that , i.e., , and so is a common fixed point of f, S, g, G in X.
If is also a common fixed point of f, S, g, G, then we have
i.e., . This completes the proof of Theorem 3.3. □
Remark 3.4 If the mappings given in Theorem 3.3 are multi-valued, we can also prove that the conclusion of Theorem 3.3 still holds, i.e., . This completes the proof of Theorem 3.3.
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This study was supported by the National Natural Science Foundation of China (Grant No. 11361070). The authors would like to express their thanks to the reviewers for their helpful comments and suggestions.
The authors declare that they have no competing interests.
The authors contributed equally to the writing of the present article. And they also read and approved the final manuscript.
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Tang, Y.K., Chang, Ss. & Wang, L. Nonlinear contractive and nonlinear compatible type mappings in Menger probabilistic metric spaces. J Inequal Appl 2014, 347 (2014). https://doi.org/10.1186/1029-242X-2014-347