- Research
- Open access
- Published:
Common fixed point theorems for nonlinear contractions in a Menger space
Fixed Point Theory and Applications volume 2013, Article number: 166 (2013)
Abstract
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.
MSC:47H10.
1 Introduction
Probabilistic metric space has been introduced and studied in 1942 by Menger in USA [1], 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 [10] 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.
Jungck’s fixed point theorem [11] has many applications in nonlinear analysis. This theorem is extended by several authors; see [12–16] and the references therein.
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.
2 Preliminaries
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 2.1 A mapping is called a distribution if it is nondecreasing left continuous with and . We shall denote by L the set of all distribution functions. The specific distribution function is defined by
Definition 2.2 ([13])
Probabilistic metric space (PM-space) is an ordered pair , where X is an abstract set of elements and is defined by , where , where the functions satisfy the following:
-
(a)
for all if and only if ;
-
(b)
;
-
(c)
;
-
(d)
and , then .
Definition 2.3 A mapping is called a t-norm if
-
(e)
and for all ;
-
(f)
for all ;
-
(g)
for all with and ;
-
(h)
for all .
Definition 2.4 A Menger space is a triplet , where is a PM-space and t is a t-norm such that for all and all ,
Definition 2.5 ([13])
Let be a Menger space and .
-
(1)
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 .
-
(2)
A sequence in X is said to be Cauchy if for each and , there is a positive integer such that for all with .
-
(3)
A Menger space is said to be complete if every Cauchy sequence in X converges to a point of it.
-
(4)
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 .
-
(5)
f is said to be continuous on X if f is continuous at every point in X.
Definition 2.6 ([4])
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 [5]).
From Definition 2.1-Definition 2.5, we can prove easily the following lemmas.
Lemma 2.7 ([10])
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 [11], 1976)
Let f be a continuous mapping of a complete metric space into itself and let be a map that satisfies the following conditions:
for all and for some . Then f and g have a unique common fixed point.
Definition 3.1 Let be a Menger space with for all and let be two self-mappings of X. We will say that f and g are Jungck-type generalized contraction if
for all and , where is a mapping such that for all , and for all and , is the same as in Definition 2.2.
Remark 3.2
-
(1)
It is clear that (∗1) implies (∗2) if for all , , and for all , where .
-
(2)
In Example 3.10, we shall show that the condition (∗2) is satisfied, but the condition (∗1) is not satisfied.
Definition 3.3 Let be a mapping such that for all . We say that φ is the U-generalized contraction if
for all and .
Lemma 3.4 Let be as in (c) of Theorem A and let be defined by
Then φ is an U-generalized contraction.
Proof It follows from hypotheses that for all ,
Now we shall show that condition (∗3) is satisfied. Since
there are three cases which need to be considered.
Case 1. Let and . Then, since
(∗3) is satisfied.
Case 2. Let and with . Then, since
(∗3) is satisfied.
Case 3. Let and with . Then, since
(∗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).
Example 3.5 Let and be as in Lemma 3.4. Let be defined by and . Define by
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 [11].
Theorem 3.7 Let be a complete Menger space with continuous t-norm and for all , let f be a continuous self-mapping on X and let be a mapping that satisfies the following conditions:
-
(i)
;
-
(ii)
g commutes with f;
-
(iii)
f, g and φ satisfy (∗2) and (∗3);
-
(iv)
φ is a strictly increasing and bijective;
-
(v)
for each , where is n-times repeated composition of with itself.
Then f and g have a unique common fixed point.
Proof
Let . By (i), there exists a sequence in X such that
From (iii) and (3.2), we have
By virtue of (iv), (3.2) and (3.3), we obtain
for all and . In view of (3.4), we have
for all and . By repeated application of (3.5), we have
for all and . From (iii), we have
On account of (3.7), we obtain that
In terms of (3.8), we get that
Now we shall show that is a Cauchy sequence.
From (iii), (iv), (3.5)-(3.10) and Definition 2.4, we deduce that
for all and with . In terms of (iii), (v) and Definition 2.2, we have
By (3.11), (3.12) and Definition 2.2, is a Cauchy sequence in X. Since X is complete and is a Cauchy sequence in X, there exists such that
On account of (3.2) and (3.13), we have
By (ii), (3.1), (3.13), (3.14) and hypotheses,
and
From (3.15), we get that
In view of (ii), (3.16) and (∗2), we have
for all .
By (iv) and (3.17),
for all and . Due to (v), (3.18), Definition 2.1 and Definition 2.2, we get that
From (ii), (3.16) and (3.19), we have
By (3.20), is a common fixed point of f and g. To prove the uniqueness of a common fixed point of f and g, let u and w be common fixed points of f and g. Then and . Putting and in (∗2), we get
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.
Example 3.8 Let be the set of reals with the usual metric and let and be mappings defined as follows:
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.
Corollary 3.9 Let be a complete Menger space with continuous t-norm and for all . Let be maps that satisfy the following conditions:
-
(a)
;
-
(b)
f is continuous;
-
(c)
g commutes with f;
-
(d)
for all , and for some . Then f and g have a unique common fixed point.
Proof Let be defined by
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. □
Example 3.10 Let be as in (c) of Theorem A and let be the set of reals with usual metric. Suppose that and are mappings defined as follows:
and
Let the mappings , H and t be the same as in Example 3.5. Then, from Lemma 2.7 and Lemma 3.4, is a complete Menger space and φ satisfies (∗3). Since
for all , we deduce that
for all and , which implies (∗2). By simple calculations, conditions (i), (ii), (iv) and (v) of Theorem 3.7 are satisfied. Thus all the conditions of Theorem 3.7 are satisfied. Hence f and g have a unique common fixed point 0. By hypotheses, there exist , and such that
which implies that condition (d) of Corollary 3.9 is not satisfied. By hypotheses, there exist and such that
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?
References
Menger K: Statistical metrics. Proc. Natl. Acad. Sci. USA 1942, 28: 535–537. 10.1073/pnas.28.12.535
Chang SS, Cho YJ, Kang SM: Nonlinear Operator Theory in Probabilistic Metric Space. Nova Science Publishers, New York; 2001.
Cho YJ, Ha KS, Chang SS: Common fixed point theorems for compatible mappings of type (A) in non-Archimedean Menger PM-spaces. Math. Jpn. 1997, 46: 169–179.
Hadzić O: A fixed point theorem in Menger spaces. Publ. Inst. Math. (Belgr.) 1979, 26: 107–112.
Hadzić O: Fixed point theorems for multivalued mappings in probabilistic metric spaces. Fuzzy Sets Syst. 1997, 88: 219–226. 10.1016/S0165-0114(96)00072-3
Schweizer B, Sklar A North-Holand Series in Probability and Applied Mathematics. In Probabilistic Metric Spaces. North-Holand, New York; 1983.
El Naschie MS: On the uncertainty of Cantorian geometry and two-slit experiment. Chaos Solitons Fractals 1998, 9: 517–529. 10.1016/S0960-0779(97)00150-1
El Naschie MS: A review of E -infinity theory and the mass spectrum of high energy particle physics. Chaos Solitons Fractals 2004, 19: 209–236. 10.1016/S0960-0779(03)00278-9
El Naschie MS: The idealized quantum two-slit Gedanken experiment revisited-criticism and reinterpretation. Chaos Solitons Fractals 2006, 27: 9–13. 10.1016/j.chaos.2005.05.010
Sehgal VM, Bharucha-Reid AT: Fixed points of contraction mappings on probabilistic metric spaces. Math. Syst. Theory 1972, 6: 97–102. 10.1007/BF01706080
Jungck G: Commuting maps and fixed points. Am. Math. Mon. 1976, 83: 261–263. 10.2307/2318216
Jungck G: Common fixed points for commuting and compatible maps on compacta. Proc. Am. Math. Soc. 1988, 103: 977–983. 10.1090/S0002-9939-1988-0947693-2
Mishra SN: Common fixed points of compatible mappings in PM-spaces. Math. Jpn. 1991, 36: 283–289.
Razani A, Fouladgar K: Extension of contractive maps in the Menger probabilistic quasi-metric space. Chaos Solitons Fractals 2007, 34: 1724–1731. 10.1016/j.chaos.2006.05.022
Rezaiyan R, Cho YJ, Saadati R: A common fixed point theorems in Menger probabilistic quasi-metric spaces. Chaos Solitons Fractals 2008, 37: 1153–1157. 10.1016/j.chaos.2006.10.007
Saadati R, Vaezpour SM, Ćirić LB: Generalized distance and some common fixed point theorems. J. Comput. Anal. Appl. 2010, 12: 157–162.
Acknowledgements
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).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that he has no competing interests.
Authors’ contributions
The author completed the paper himself. The author read and approved the final manuscript.
Rights and permissions
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.
About this article
Cite this article
Ume, J.S. Common fixed point theorems for nonlinear contractions in a Menger space. Fixed Point Theory Appl 2013, 166 (2013). https://doi.org/10.1186/1687-1812-2013-166
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2013-166