- Open Access
Common fixed point theorems for nonlinear contractions in a Menger space
© Ume; licensee Springer. 2013
Received: 6 September 2012
Accepted: 4 June 2013
Published: 26 June 2013
The main purpose of this paper is to introduce a new class of Jungck-type contraction and to present some common fixed point theorems for this mapping. Several examples are given to show that our result is a proper extension of many known results.
Probabilistic metric space has been introduced and studied in 1942 by Menger in USA , and since then the theory of probabilistic metric spaces has developed in many directions [2–6]. 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 situation 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 connection with both string and E-infinity which were introduced and studied by a well-known scientific hero El Naschie [7–9].
It is observed by many authors that the contraction condition in a metric space may be exactly translated into a probabilistic metric space endowed with min norms. Sehgal and Bharucha-Reid  obtained a generalization of the Banach contraction principle on a complete Menger space, which is a milestone in developing fixed point theorems in a Menger space.
In this paper, we introduce a new class of Jungck-type contraction and present some common fixed point theorems for this mapping. Several examples are given to show that our result is a proper extension of many known results.
Throughout this paper we denote by N the set of all positive integers, by Q the set of all rational numbers, by the set of all nonnegative integers, by R the set of all real numbers and by the set of all nonnegative real numbers. We shall recall some definitions and lemmas related to a Menger space.
Definition 2.2 ()
for all if and only if ;
and , then .
and for all ;
for all ;
for all with and ;
for all .
Definition 2.5 ()
A sequence in X is said to converge to a point p in X (written as ) if for every and , there exists a positive integer such that for all .
A sequence in X is said to be Cauchy if for each and , there is a positive integer such that for all with .
A Menger space is said to be complete if every Cauchy sequence in X converges to a point of it.
f is said to be continuous at a point p in X if for every sequence in X, which converges to p, the sequence in X converges to .
f is said to be continuous on X if f is continuous at every point in X.
Definition 2.6 ()
A t-norm t is said to be of H-type if a family of functions is equicontinuous at , that is, for any , there exists such that and imply . The t-norm is a trivial example of a t-norm of H-type, but there are t-norms of H-type with (see, e.g., Hadzic ).
From Definition 2.1-Definition 2.5, we can prove easily the following lemmas.
Lemma 2.7 ()
If is a metric, then the metric induces a mapping , defined by , and . Further, if the t-norm is defined by for all , then is a Menger space. It is complete if is complete.
Lemma 2.8 In a Menger space , if for all , then for all .
3 Jungck-type fixed point theorems
In 1976, Jungck proved the following theorem.
Theorem A (Jungck , 1976)
for all and for some . Then f and g have a unique common fixed point.
for all and , where is a mapping such that for all , and for all and , is the same as in Definition 2.2.
It is clear that (∗1) implies (∗2) if for all , , and for all , where .
In Example 3.10, we shall show that the condition (∗2) is satisfied, but the condition (∗1) is not satisfied.
for all and .
Then φ is an U-generalized contraction.
there are three cases which need to be considered.
(∗3) is satisfied.
(∗3) is satisfied.
(∗3) is satisfied. From (∗4), Case 1, Case 2 and Case 3, φ is U-generalized contraction. □
The following example shows that f and g do not have a common fixed point even though f, g and φ satisfy (∗2) and (∗3).
where and H are the same as in Definition 2.1 and Definition 2.2. Let be defined by for all . Then, by Lemma 3.4 and simple calculations, (∗2) and (∗3) are satisfied. But f and g do not have a common fixed point.
Remark 3.6 It follows from Example 3.5 that f and g must satisfy (∗2) and (∗3), and other conditions additionally in order to have a common fixed point of f and g.
The following is Jungck-type common fixed point theorem which is a generalization of Jungck’s common fixed point theorem .
g commutes with f;
f, g and φ satisfy (∗2) and (∗3);
φ is a strictly increasing and bijective;
for each , where is n-times repeated composition of with itself.
Then f and g have a unique common fixed point.
for all .
for all , which gives . Thus is a unique common fixed point of f and g. □
Now we give an example to support Theorem 3.7.
Let the mappings , H and t be as in Example 3.5. Then from Lemma 2.7, is a complete Menger space. By the same method as in Lemma 3.4 and simple calculations, the conditions of Theorem 3.7 are satisfied. Thus f and g have a unique common fixed point 0.
From Theorem 3.7, we have the following corollary.
f is continuous;
g commutes with f;
for all , and for some . Then f and g have a unique common fixed point.
From (b) and (d), we deduce that g is continuous. Thus, by (3.23), the same method as in Lemma 3.4 and simple calculations, the conditions of Theorem 3.7 are satisfied. Therefore f and g have a unique common fixed point.
In the next example, we shall show that all the conditions of Theorem 3.7 are satisfied, but condition (d) in Corollary 3.9 and condition (∗1) in Theorem A are not satisfied. □
which implies that condition (∗1) in Theorem A is not satisfied. Therefore Theorem 3.7 is a proper extension of Theorem A and Corollary 3.9.
A natural question arises from Example 3.5.
Question Would Theorem 3.7 remain true if (i)-(v) in Theorem 3.7 were substituted by some suitable conditions?
The author would like to thank referees for careful reading and useful comments. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2002165).
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