2719
Fixed Points and Coupled Fixed Points in Hausdorff
Intuitionistic L-Fuzzy Metric Spaces
S.Yahya Mohamed1, E.Naargees Begum2
1
PG & Research Department of Mathematics, Government Arts College, Trichy-22. Email: yahya_md@yahoo.com
2
Department of Mathematics, Sri Kailash Women’s College, Thalaivasal,Tamilnadu. Email:.mathsnb@gmail.com
Abstract: Fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets are introduced by Lotfi.A.Zadeh [27] as an extension of the classical notion of sets. Intuitionistic fuzzy set can be utilized as a proper tool for representing hesitancy concerning both membership and non-membership of an element to a set. Atanassov [1] introduced and studied the concept of Intuitionistic fuzzy sets.The idea of Hausdorff fuzzy metric space introduced by Rodriguez-Lopez and Romaguera [16]. In this paper, a new concept of intuitionistic fuzzy fixed point theorems in Hausdorff intuitionistic L-fuzzy metric spaces is introduced and some properties and theorems about fixed points in Hausdorff intuitionistic L-fuzzy metric space are discussed.
Keywords: Intuitionistic Fuzzy mapping, Intuitionistic fuzzy point, fuzzy metric spaces, Hausdorff intuitionistic fuzzy metric spaces, Hausdorff intuitionistic L- fuzzy metric spaces.
AMS Classification 2010: 03B47, 03G25, 03G10, 03E72, 06D99.
1. INTRODUCTION
In 1965, Zadeh [27] introduced the concept of fuzzy set as a method of finding uncertainty. In 1986, the idea of intuitionistic fuzzy set was introduced by Atanassov [1] which is a generalization of fuzzy set. In recent years there has been an increasing interest in mathematical aspects of intuitionistic fuzzy sets. To use this concept in topology and analysis, many authors have extensively developed the theory of intuitionistic fuzzy sets and applications. The study of fixed point theorems of maps satisfying contractive type conditions in fuzzy metric spaces has been a very active field of research activity recently. George and Veeramani [11] introduced the concept of fuzzy metric spaces in different ways. Grabiec [13] obtained the fuzzy version of Banach contraction principle. Mishra et al. [22] proved common fixed point theorems for compatible maps on fuzzy metric spaces. In such ways, Park [23], V. Gregori [9] has defined intuitionistic fuzzy metric spaces and obtained several classical theorems on this new structure. In 2008, R. Saadati et.al [24] introduced Modified intuitionistic fuzzy metric space.
In 2004, Rodriguez- Lopez and Romaguera [16] introduced Hausdorff fuzzy metric on the set of the non-empty compact subsets of a given fuzzy metric space. Later, several authors proved some fixed point theorems for in intuitionistic fuzzy metric spaces, it can be found in [2, 3, 4, 17, 18, 19, 20, 21]. In this paper, the notion of Hausdorff intuitionistic L-fuzzy metric space is introduce and also the theorems
on intuitionistic fuzzy fixed points and coupled Fuzzy Fixed Points in Hausdorff Intuitionistic L- Fuzzy Metric space are stated and proved.
2. PRELIMINARIES
Definition: 2.1
An L-fuzzy set ϕ on U is a mapping ϕ:U → L, where L is a „transitive partially ordered
set‟. In this work, we assume that (L,≤) is a preordered set. Notice that it is natural to assume that the relation
≤ is not anti-symmetric; if x, y ∈ L are synonyms, that is, words or expressions that are used with the same meaning, then and , but still x and y are distinct words.
Example 2.1
Suppose that U consists of a group of people. The
L-fuzzy set, whose membership function ϕ , describes how well the persons in U can ski. For instance, there exist people who can ski very well, some ski badly, and some are moderate skiers.
Definition 2.2
If is a topological space, X is said to be metrizable if there exist a metric d on the set X that induces the topology on X.
A metric space is a metrizable space X together with a specific metric d that gives the topology on X.
Definition 2.3
2720 there exist a neighbourhoods of and
respectively, such that .
Definition 2.4
A binary operation [ ] [ ] [ ] is called a t-norm if it satisfies the following condition.
(i) is associative and commutative. (ii) for every ∈ [ ]
(iii) whenever and
, for ∈ [ ].
In addition, is continuous, then is called a continuous t-norm.
Definition 2.5
A binary operation
[ ] [ ] [ ] is a continuous
t-conorm if satisfies the following conditions, 1) is associative and commutative,
2) is continuous,
3) for all ∈ [ ],
4) whenever and , for all
∈ [ ].
Definition: 2.6
A 5-tuple(X, M, N, *, ◊) is said to be an intuitionistic fuzzy metric space if X is an arbitrary set, * is a continuous t-norm, ◊ is a continuous t- conorm and, M, N are fuzzy sets on X2 × [0, ∞) satisfying the
conditions:
1. 2.
3.
4.
5. 6. [
[ ]
7. for all, x,y ϵ X and t
˃ 0;
8.
9.
10.
11. 12. [
[ ]
13. for all x,y ϵ X.
The functions and denote the degree of nearness and the degree of non-nearness between x and y w.r.t. t respectively.
Definition 2.7
Let be a fuzzy metric space. The Hausdorff intuitionistic fuzzy metric
is defined by
{ ∈ (
∈ ) ∈ (
∈ ) }
{ ∈ (
∈ ) ∈ (
∈ ) }
for all ∈ ∈ and
Where denotes the set of its nonempty compact subsets.
Definition 2.8
A fuzzy point in X is called a fixed point of a fuzzy mapping F,if ,
i.e., or ∈ [ ] , i.e., the fixed degree of in F is at least .
If ∈ then is called a fixed point of the fuzzy mapping F.
Definition 2.9
Let X be an arbitrary nonempty set, T be fuzzy mapping from X into and z∈ If there exists [ ] such that z∈ [ ] , then a point z is called an - fuzzy fixed point of T.
Definition 2.10
An element ∈ is called a coupled fixed fuzzy point of the mapping
, if
,
That is, ∈ [ ] ∈ [ ] . .
3. FIXED POINTS AND COUPLED FIXED
POINTS IN HAUSDORFF INTUITIONISTIC L-FUZZY METRIC SPACES
Definition 3.1
Let be an intuitionistic fuzzy metric space. The Hausdorff intuitionistic L- fuzzy metric and is defined by
{ ∈ (
∈ ) ∈ (
∈ ) }
{ ∈ ( ∈ ) ∈ ( ∈ ) }
for all ∈ and
Where denotes the set of its nonempty compact subsets.
Lemma 3.1
2721
where
. Suppose that
……… (1)
… .… (2)
For all t > 0 and h > 1. Then { } is a cauchy sequence.
PROOF:
Given, For each ∈ We have M , N
M ( )
( ) ( )
N ( )
( ) ( )
Thus for each ∈ , We get
M ( )
N ( )
Pick the constant and ∈ such that hk < 1
and ∑
Hence, for , We get
M ( ( ) )
( ) ( )
( )
( ) ( )
( )
(
) ( )
( )
(
)
N ( ( ) )
( ) ( )
( )
( ) ( )
( )
(
) ( )
(
)
∞ ( ) Then, from the above, we have
( ) ,
(
) For each t > 0. Therefore, we get
For each t > 0 and so { } is a Cauchy sequence.
Hence the proof.
Theorem 3.1
Let be a complete intuitionistic -fuzzy metric space and be a mapping such that [ ] is a nonempty compact subset of X for
all ∈ . Suppose that is an intuitionistic fuzzy mapping such that
([ ] [ ] ) ……… (3) ([ ] [ ] ) ……… (4)
For all t > 0, where ∈ . If there exist ∈ ∈ [ ] such that
……… (5)
……… (6)
For all t > 0 and h > 1, then T has an -intuitionistic fuzzy fixed point.
Proof :
We start from ∈ and ∈ [ ]
and under the hypothesis. From the assumption,
We have [ ] is a nonempty compact subset of X.
If [ ] [ ] then ∈ [ ] and so is an -fuzzy fixed point of T and the proof is finished.
Therefore , We may assume that
[ ] [ ] . Since ∈ [ ] satisfying
M ∈ [ ]
([ ] [ ] )
2722
N ∈ [ ]
([ ] [ ] )
If [ ] [ ] , then ∈ [ ]
This implies that -intuitionistic fuzzy fixed point of T and then the proof is finished.
Therefore, we may assume that
[ ] [ ]
Since ∈ [ ] and [ ] is a nonempty compact subset of X,
We know that from equation (3) and (4), there exists
∈ [ ] satisfying
M ∈ [ ]
M ([ ] [ ] )
M
N ∈ [ ]
N ([ ] [ ] )
N
By induction, We can construct the sequence { } in X such that ∈ [ ] and
M , for all
∈ .
N , for all ∈ .
From lemma 3.1, we get { } is a Cauchy sequence. Since is a complete -intuitionistic fuzzy metric space, there exists ∈ such that
,
Which means
for each
t > 0.
Now we claim that ∈ [ ] .
Since ([ ] [ ] )
([ ] [ ] )
and , then for each t > 0.
We get,
([ ] [ ] ) ……… (7) ([ ] [ ] ) ……… (8) This implies that
∈ [ ]
= 1
∈ [ ]
= 0
And thus there exists a sequence { } [ ] such
that
,
for each t > 0 . For each ∈ , We have
M ( ) ( )
Since
( ) ( )
We get,
N ( ) ( )
Since
( ) ( )
We get, ………(9)
That is, .
It follows from [ ] being a compact subset of X
and ∈ [ ] that ∈ [ ] . Therefore, x is an -intuitionistic fuzzy fixed point of T.
Corollary 3.2
Let (X,d) be a complete metric space and
be a mapping such that [ ] is a nonempty compact subset of X for all ∈ .
Suppose that be an intuitionistic fuzzy mapping such that
([ ] [ ] )
([ ] [ ] )
For all t > 0, where ∈ . Then T has an -intuitionistic fuzzy fixed point.
Proof:
Let be standard Intuitionistic -fuzzy metric space induced by the metric d with
Now we show that the condition of Theorem 3.6 is satisfied.
Since (X,d) is a complete metric space then
is complete. It is easy to see that
satisfies equation (7) in Theorem 3.6.
For each nonempty compact subset of X, We have
…………. (10)
…………. (11)
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([ ] [ ] )
([ ] [ ] )
([ ] [ ] ) ( )
[ ] [ ]
([ ] [ ] ) for
each t > 0 and each ∈
([ ] [ ] )
([ ] [ ] )
([ ] [ ] ) ( )
([ ] [ ] ) ([ ] [ ] ) for
each t > 0 and each ∈
Therefore, T has an -intuitionistic fuzzy fixed point.
Theorem 3.2
Let be a complete intuitionistic -fuzzy metric space and an intuitionistic fuzzy mapping. Suppose that, for all ∈ and , the
following condition holds:
( ( ) ) , ……… (12)
( ( ) ) , ……… (13) Where ∈ . Then F has a fixed fuzzy point.
PROOF :
Let be a given point in X and ∈ . Since is compact, We can choose ∈
such that
M ( ( ) )
∈ ( ( ) )
( ( ) )
( ( ) )
∈ ( ( ) )
( ( ) )
Thus, We have
M
N
Continuing this way, We can obtain a sequence
{ } in X such that
∈ for all , and
M
N
Thus the sequence
{ } { } is non-decreasing.
If follows from an assumption on that
( ) ( )
And hence there exist and ∈ such that
( ) , ( ) for all . ……… (14)
As non-decreasing, then from equation (6) We have
( ( ) )
( ( ) )
for
all ∈ with .
( ( ) )
( ( ) )
for all ∈ with .
Hence we have
,
for all ∈ with
Continuing this way, for all , We can obtain
( )
( )
( )
( )
( )
( )
Now, we note that the sequence
is an s-increasing sequence,
Since , for all .
Next fix ∈ , then there exists ∈ with
such that
∏ ( )
,
∏ ( )
for all .
Then, for all and
2724 Since ∑
, We write
M ( ∑ )
( )
(
)
M( )
∏ ( )
N ( ∑ )
( )
(
)
N( )
∏ ( )
Hence
for all and the sequence { } is a Cauchy sequence.
Since is a complete intuitionistic -fuzzy metric space, we have
̅ for each , ……… (15) ̅ for each ,
……… (16) For some ̅ in x.
By given assumption, We have
( ̅ )
( ̅ )
Thus there exists with such that
( ̅ ) ………… (17) ( ̅ ) ……… (18)
Now, by (1) and (4), We have
̅ ( ̅ ( ̅ ) )
̅ ̅
̅ ( ̅ ( ̅ ) ) ̅ ̅
On taking limit as , We obtain
̅
̅
Since ∈ , it follows that
∈
∈
And hence there exists a sequence { } with ∈ ̅ such that
…… (19) …… (20)
For each Now, for each ∈ ,
̅ ( ) ( ̅ ) …… (21)
̅ ( ) ( ̅ ) ……… (22) On taking limit as in equation (10) and by equation (7) and (9),We have
̅ ̅
That is, ̅
Since ̅ is compact, ̅ ∈ ̅ that is, ̅ ̅.
Hence ̅ is a fixed intuitionistic fuzzy point of F.
4. CONCLUSION
Last three decades were very productive for fuzzy mathematics and the recent literature has observed the fuzzy application in almost every direction of mathematics. In this paper a general analysis has been done to reveal the link between the fixed point properties and the Hausdorff Intuitionistic L- fuzzy metric spaces. The analysis can be used to form some new fuzzy fixed point theorems by using different types. In future from this concept can able to extend fuzzy concept for find out various solution in various spaces.
(A.1)
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