Fixed Point Theorems for Multivalued Contractive Mappings in Fuzzy Metric Spaces
Basit Ali^{1, *}, Mujahid Abbas^{2}
^{1}Department of Mathematics, University of Management and Technology, Johar Town Lahore, Pakistan
^{2}Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa
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To cite this article:
Basit Ali, Mujahid Abbas. Fixed Point Theorems for Multivalued Contractive Mappings in Fuzzy Metric Spaces. American Journal of Applied Mathematics. Special Issue: Proceedings of the 1st UMT National Conference on Pure and Applied Mathematics (1st UNCPAM 2015). Vol. 3, No. 3-1, 2015, pp. 41-45. doi: 10.11648/j.ajam.s.2015030301.17
Keywords: Multivalued Mapping, Upper and Lower Semicontinuous, t-norm, Fuzzy Metric Space
1. Introduction
Due to its applications in mathematics and other related disciplines, Banach contraction principle has been generalized in many directions (for details one can see [1, 3, 4, 5, 8, 14, 19, 24, 25].
The concept of fuzzy sets was initiated by Zadeh [26] in 1965. Fuzzy metric spaces were introduced by Kramosil and Michalek [15]. Romaguera [21] introduced Hausdorff fuzzy metric on a set of nonempty closed and bounded subsets of a given fuzzy metric space. Fixed point theory in fuzzy metric spaces has been studied by a number of authors. For a wide survey we refer to ([6, 7, 11, 12, 16, 17, 18, 20, 23]) and the references therein.
Kiany et al. [14] proved fixed point theorems for multivalued fuzzy contraction maps in fuzzy metric spaces and obtained generalization of Banach contraction theorem in fuzzy metric spaces.
The aim of this paper is to obtain fixed point theorems for multivalued mapping in fuzzy metric spaces. As a results we extended the results given in ([8, 14] and reference therein) in fuzzy metric spaces.
Definition 1.1 [22] A binary operation is called a if for all
I. is associative and commutative;
II. is continuous;
III.
IV. whenever and
Definition 1.2 (compare [15]) A fuzzy metric space is a triple such that is a continuous t-norm and is a fuzzy set in such that for all
a.
b. if and only if for all
c.
d. for all
e. is left continuous.
The pair (or simply, if no confusion arises) is said to be a fuzzy metric on It is well known and easy to see that for each is a non-decreasing function on Each fuzzy metric on a set induces a topology on which has a base the family of open balls
where
Observe that a sequence converges to (with respect to if and only if for all
It is also well known [13] that every fuzzy metric space is metrizable, i.e., there exists a metric on whose induced topology agrees with Conversely, if is a metric space and we defin by
and
for all then (where is minimum norm) is a fuzzy metric space and is called the standard fuzzy metric of [9]. Moreover, the topology agrees with the topology induced by
A sequence in a fuzzy metric space is said to be a Cauchy sequence if for each there exists such that for all A fuzzy metric space is said to be complete ([10]) if every Cauchy sequence converges. A subset is said to be closed if for each convergent sequence and implies A subset is said to be compact if each sequence in has a convergent subsequence. The set of all compact subsets of will be denoted by .
Lemma 1.3 [11] For all is nondecreasing.
Definition 1.4 Let be a fuzzy metric space, is said to be continuous on if
whenever is a sequence in which converges to a point that is,
Lemma 1.5 [11] is a continuous function on
Kiany et al. [14] introduced the following Lemma in fuzzy metric spaces.
Lemma 1.6 ([14]) Let be a fuzzy metric space satisfying
(1)
for every and . Suppose is sequences in satisfying
for all and Then is a Cauchy sequence.
Lemma 1.7 [21] Let be a fuzzy metric space. Then, for each and there is such that
Consistent with [21], we recall the notion of Hausdorff fuzzy metric induced by a fuzzy metric as follows: For , and define:
for all where Then is called the Hausdorff fuzzy metric induced by the fuzzy metric The triplet is called Hausdorff fuzzy metric space.
Definition 1.8 A function is called lower semicontinuous, if for any and implies A function is called upper semicontinuous, if for any and implies A multivalued mapping (collection of all nonempty subsets of ) is called upper semicontinuous, if for any and a neighborhood of there is a neighborhood of such that for any we have A multivalued mapping (collection of all nonempty subsets of ) is called lower semicontinuous, if for any and a neighborhood , there is a neighborhood of such that for any we have
2. Fixed Point Theorems for Multivalued Contractive Mappings in Fuzzy Metric Spaces
In the following theorem we obtain fixed point for multivalued mapping satisfying a contractive condition.
Let be a multivalued mapping. Define for For a positive constant we define a set
(2)
Theorem 2.1 Let be a complete fuzzy metric space and be a multivalued mapping. If there exist a constant such that for any there is satisfying
(3)
for Assume that satisfies (1) for some in then has a fixed point provided and is upper semicontinuous.
Proof: Since by Lemma 1.7 is nonempty for any and . Let be arbitrary, there exists satisfying
and for there exists satisfying
Continuing this process, we obtain a sequence in such that satisfying
(4)
On the other hand gives
(5)
From (4) and (5) we obtain
That is
(6)
Let then from (6)
(7)
for all and Pick the constant , such that then and that is we get
(8)
for all and Also, we have
(9)
Then by Lemma 6, we have
This shows that is a Cauchy sequence in Since is complete, there exist such that From (4) and (5), it is clear that is increasing and hence converges to Since is upper semicontinuous, so we have
This implies that so Hence by Lemma 7, we have
Kiany et al. [14] gave the following corollary.
Corollary 2.2 [14] Let be a complete fuzzy metric. Suppose be a multivalued mapping such that
(10)
for each and Furthermore, assume that satisfies (1) for some and Then has a fixed point.
Remark 2.3 Theorem 2.1 is a generalization of above corollary. Let satisfies the conditions of above corollary and if is upper semicontinuous, then from (10) for any we get
Hence satisfies all the conditions of Theorem 9, the existence of fixed point has been proved. Following example shows that Theorem 2.1 is an extension of Corollary 2.2.
Example 2.4 Let
and
for . Then is a complete metric space where defined by Let be defined as given in [8]
Since
which implies that satisfies (1) Moreover
and
There does not exist any such that (10) is satisfied. If it exists then
implies a contradiction. On the other hand
is continuous. There exist for any such that
,
that is
So there exist such that
Then the existence of fixed point follows from Theorem 2.1.
Acknowledgements
The authors are thankful to the anonymous reviewers for their useful suggestions that helped us to improve the presentation of the paper significantly.
References