Abstract
We prove the existence of common fuzzy fixed points for a sequence of locally contractive fuzzy mappings satisfying generalized Banach type contraction conditions in a complete metric space by using iterations. Our main result generalizes and unifies several well-known fixed-point theorems for multivalued maps. Illustrative examples are also given.
MSC:46S40, 47H10, 54H25.
Similar content being viewed by others
1 Introduction
The Banach contraction theorem and its subsequent generalizations play a fundamental role in the field of fixed point theory. In particular, Heilpern introduced in [1] the notion of a fuzzy mapping in a metric linear space and proved a Banach type contraction theorem in this framework. Subsequently several other authors [2–10] have studied and established the existence of fixed points of fuzzy mappings. The aim of this paper is to prove a common fixed-point theorem for a sequence of fuzzy mappings in the context of metric spaces without the assumption of linearity. Our results generalize and unify several typical theorems of the literature.
2 Preliminaries
Given a metric space , denote by the family of all nonempty closed bounded subsets of . As usual, for and , we define
Then the Hausdorff metric H on induced by d is defined as
for all .
A fuzzy set in is a function with domain X and values in . denotes the collection of all fuzzy sets in X. If A is a fuzzy set and , then the function value is called the grade of membership of ζ in A. The α-level set of a fuzzy set A is denoted by , and it is defined as follows:
According to Heilpern [1], a fuzzy set A in a metric linear space is said to be an approximate quantity if is compact and convex in X, for each , and . The family of all approximate quantities of the metric linear space is denoted by .
Now, for and , define
and
It is well known that is a metric on .
In case that is a (non-necessarily linear) metric space, we also define
whenever and , .
In the sequel the letter ℕ will denote the set of positive integer numbers.
The following well-known properties on the Hausdorff metric (see e.g. [11]) will be useful in the next section.
Lemma 2.1 Let be a metric space and let with , . If , then there exists such that .
Lemma 2.2 Let be a metric space and let be a sequence in such that , for some . If , for all , and , then .
Now, let X be an arbitrary set and let Y be a metric space. A mapping T is called fuzzy mapping if T is a mapping from X into . In fact, a fuzzy mapping T is a fuzzy subset on with membership function . The value is the grade of membership of ξ in .
If is a metric space and T is a (fuzzy) mapping from X into , we say that is a fixed point of T if .
We conclude this section with the notion of contractiveness that will be used in our main result.
Definition 2.3 (compare [12])
Let . A function is said to be a MT-function if it satisfies Mizoguchi-Takahashi’s condition (i.e., , for all ).
Clearly, if is a nondecreasing function or a nonincreasing function, then it is a MT-function. So the set of MT-functions is a rich class.
3 Fixed points of fuzzy mappings
Fixed-point theorems for locally contractive mappings were studied, among others, by Edelstein [13], Beg and Azam [14], Holmes [15], Hu [11], Hu and Rosen [16], Ko and Tasi [17], Kuhfitting [18] and Nadler [19].
Heilpern [1] established a fixed-point theorem for fuzzy contraction mappings in metric linear spaces, which is a fuzzy extension of Banach’s contraction principle. Afterwards Azam et al. [4, 5], and Lee and Cho [10] further extended Banach’s contraction principle to fuzzy contractive mappings in Heilpern’s sense. In our main result (Theorem 3.1 below) we establish a common fixed-point theorem for a sequence of generalized fuzzy uniformly locally contraction mappings on a complete metric space without the requirement of linearity. This is a generalization of many conventional results of the literature.
Let , and . A metric space is said to be ε-chainable if given , there exists an ε-chain from ζ to ξ (i.e., a finite set of points , such that , for all ). A mapping is called an uniformly locally contractive mapping if and , implies . A mapping is called an uniformly locally contractive fuzzy mapping if and , imply . We remark that a globally contractive mapping can be regarded as an uniformly locally contractive mapping and for some special spaces every locally contractive mapping is globally contractive.
Theorem 3.1 Let , a complete ε-chainable metric space and a sequence of fuzzy mappings from X into such that, for each and , . If
for all , where is a MT-function, then the sequence has a common fixed point, i.e., there is such that , for all .
Proof Let be an arbitrary, but fixed element of X. Find such that . Let
be an arbitrary ε-chain from to . (We suppose, without loss of generality, that , for each with .)
Since , we deduce that
Rename as . Since , using Lemma 2.1 we find such that
Similarly we may choose an element such that
Thus we obtain a set of points of X such that and , for , with
for .
Let . Thus the set of points is an ε-chain from to . Rename as . Then by the same procedure we obtain an ε-chain
from to . Inductively, we obtain
with
for .
Consequently, we construct a sequence of points of X with
for all .
For each , we deduce from (2) that is a decreasing sequence of non-negative real numbers and therefore there exists such that
By assumption, , so there exists such that , for all where .
Now put
Then, for every , we obtain
Putting , we have
for all . Hence
whenever .
Since , for all , it follows that is a Cauchy sequence. Since is complete, there is such that . So for each there is such that implies . This in view of inequality (1) implies , for all . Consequently, . Since with , we deduce from Lemma 2.2 that , for all . This completes the proof. □
Corollary 3.2 Let , a complete ε-chainable metric space and a sequence of fuzzy mappings from X into such that, for each and , . If
for all , where , then the sequence has a common fixed point.
Proof Apply Theorem 3.1 when ψ is the MT-function defined as , for all . □
Corollary 3.3 Let , a complete ε-chainable metric linear space and a sequence of fuzzy mappings from X into satisfying the following condition:
for all , where is a MT-function. Then the sequence has a common fixed point.
Proof Since and , for all , the result follows immediately from Theorem 3.1. □
Corollary 3.4 Let , a complete ε-chainable metric linear space and a sequence of fuzzy mappings from X into satisfying the following condition:
for all , where . Then the sequence has a common fixed point.
Corollary 3.5 [4]
Let , a complete ε-chainable metric linear space and , , two fuzzy mappings from X into satisfying the following condition:
for , where is a MT-function. Then and have a common fixed point.
Let , a complete ε-chainable metric linear space and T: an uniformly locally contractive fuzzy mapping. Then T has a fixed point.
Corollary 3.7 Let , a complete ε-chainable metric space and S be a multivalued mapping from X into satisfying the following condition:
where is a MT-function. Then S has a fixed point.
Proof Define a fuzzy mapping T from X into as if and , otherwise. Then , for all , so , for all . Since
for all , we deduce that condition (1) of Theorem 3.1 is satisfied for T. Hence T has a fixed point , i.e., . We conclude that . The proof is complete. □
Corollary 3.8 [13]
Let , a complete ε-chainable metric space and S be a multivalued mapping from X into satisfying the following condition:
where . Then S has a fixed point.
Corollary 3.9 ([20, 21], see also [9, 13])
Let be a complete metric space, S a multivalued mapping from X into and a MT-function such that
for all . Then S has a fixed point in X.
Proof Apply Corollary 3.8 with . □
We conclude the paper with two examples to support Theorem 3.1 and Corollary 3.2.
Example 3.10 Let be the compact, and thus complete, metric space such that , and , for all . Let λ be a constant such that and let be the sequence of fuzzy mappings defined from X into as follows:
For each with , and we have
Hence, for , the conditions of Corollary 3.2, and hence of Theorem 3.1, are satisfied for any , whereas X is not linear. Therefore all previous relevant fixed point results Corollaries 3.3-3.6 on metric linear spaces are not applicable.
Example 3.11 Let be the complete metric space such that , , for all , and whenever (in the sequel we shall write instead of ).
Note that a sequence is a Cauchy sequence in if and only if . Moreover, is the only non-isolated point of X for the topology induced by d.
Let be the MT-function defined as
and let be the sequence of fuzzy mappings defined from X into as follows:
Observe that, for ,
and, for ,
Therefore , for all and (recall that each is an isolated point for the induced topology, so every bounded interval belongs to ).
We show that condition (1) of Theorem 3.1 is satisfied for and ψ as defined above. Indeed, let with and . Assume without loss of generality that .
If , for each , we obtain
Similarly, for each , we obtain
Consequently
If and , we deduce, in a similar way, that
Finally, if , we deduce
We have shown that all conditions of Theorem 3.1 are satisfied (in fact is the only fixed point of T).
References
Heilpern S: Fuzzy mappings and fixed point theorems. J. Math. Anal. Appl. 1981, 83: 566–569. 10.1016/0022-247X(81)90141-4
Ali B, Abbas M: Suzuki-type fixed point theorem for fuzzy mappings in ordered metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 9
Arshad M, Azam A: Fixed points solutions of sequence of locally contractive fuzzy mappings via iterative process. International Conference of Mathematical Sciences (ICM) 2012. Al Ain, UAE, 2012, 11-14 March
Azam A, Beg I: Common fixed points of fuzzy maps. Math. Comput. Model. 2009, 49: 1331–1336. 10.1016/j.mcm.2008.11.011
Azam A, Arshad M, Beg I: Fixed points of fuzzy contractive and fuzzy locally contractive maps. Chaos Solitons Fractals 2009, 42: 2836–2841. 10.1016/j.chaos.2009.04.026
Azam A, Arshad M, Vetro P: On a pair of fuzzy- ϕ contractive mappings. Math. Comput. Model. 2010, 52: 207–214. 10.1016/j.mcm.2010.02.010
Azam A, Arshad M: A note on ‘Fixed point theorems for fuzzy mappings’ by P. Vijayaraju and M. Marudai. Fuzzy Sets Syst. 2010, 161: 1145–1149. 10.1016/j.fss.2009.10.016
Azam A, Waseem M, Rashid M: Fixed point theorems for fuzzy contractive mappings in quasi-pseudo-metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 27
Kamran T: Common fixed points theorems for fuzzy mappings. Chaos Solitons Fractals 2008, 38: 1378–1382. 10.1016/j.chaos.2008.04.031
Lee BS, Cho SJ: A fixed point theorem for contractive type fuzzy mappings. Fuzzy Sets Syst. 1994, 61: 309–312. 10.1016/0165-0114(94)90173-2
Hu T: Fixed point theorems for multivalued mappings. Can. Math. Bull. 1980, 23: 193–197. 10.4153/CMB-1980-026-2
Du WS: On coincidence point and fixed point theorems for nonlinear multivalued maps. Topol. Appl. 2012, 159: 49–56. 10.1016/j.topol.2011.07.021
Edelstein M: An extension of Banach’s contraction principle. Proc. Am. Math. Soc. 1961, 12: 7–12.
Beg I, Azam A: Fixed points of multivalued locally contractive mappings. Boll. Unione Mat. Ital., A (7) 1990, 7: 227–233.
Holmes RD: On fixed and periodic points under certain set of mappings. Can. Math. Bull. 1969, 12: 813–822. 10.4153/CMB-1969-106-1
Hu T, Rosen H: Locally contractive and expansive mappings. Proc. Am. Math. Soc. 1982, 86: 656–662.
Ko HM, Tasi YH: Fixed point theorems for localized property. Tamkang J. Math. 1977, 8: 81–85.
Kuhfitting PK: Fixed point of locally contractive and nonexpansive set valued mappings. Pac. J. Math. 1976, 65: 399–403. 10.2140/pjm.1976.65.399
Nadler SB: Multivalued contraction mappings. Pac. J. Math. 1969, 30: 475–488. 10.2140/pjm.1969.30.475
Mizoguchi N, Takahashi W: Fixed point theorems for multi-valued mappings on complete metric spaces. J. Math. Anal. Appl. 1989, 188: 141–177.
Suzuki T: Mizoguchi-Takahashi’s fixed point theorem is a real generalization of Nadler’s. J. Math. Anal. Appl. 2008, 340: 752–755. 10.1016/j.jmaa.2007.08.022
Acknowledgements
The third author thanks the support of the Ministry of Economy and Competitiveness of Spain, Grant MTM2012-37894-C02-01.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The three authors contributed equally in writing this article. They 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
Ahmad, J., Azam, A. & Romaguera, S. On locally contractive fuzzy set-valued mappings. J Inequal Appl 2014, 74 (2014). https://doi.org/10.1186/1029-242X-2014-74
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2014-74