Nonlinear contractive and nonlinear compatible type mappings in Menger probabilistic metric spaces
© Tang et al.; licensee Springer. 2014
Received: 24 February 2014
Accepted: 7 August 2014
Published: 4 September 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].
T is commutative and associative;
T is continuous;
for all ;
whenever , , and .
Two typical examples of continuous t-norm are 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 .
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.
- (1)For any given and the set
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-4) for any give , the function is nonincreasing and left-continuous.
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).
which shows is left-continuous.
Now we prove that, for any given , is nonincreasing in .
Condition (QM-4) is proved.
Finally, we prove that also satisfies condition (QM-3).
for any . Especially, if , then condition (QM-3) is proved. The conclusion (1) is proved.
Now we prove the conclusion (2).
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 
for any , and .
then is a Cauchy sequence in X.
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
converges to in the -topology of X.
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 .
From Remark 1.10, it follows that , i.e., , .
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. .
where , and , then S, G, f, g have a unique common fixed point z in X.
By Remark 1.10, , i.e., .
and so , i.e., .
This implies that , i.e., , and so is a common fixed point of f, S, g, G in X.
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.
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.
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