J. Comp. & Math. Sci. Vol.2 (1), 103-110 (2011)
Fixed Point Theorems in Intuitionistic Fuzzy metric space M. S. CHAUHANa , BIJENDRA SINGHb and BHARAT SINGH* a
Nehru govt. P. G. Collage, Agar (Malwa) Distt., Shajapur, M. P. India b F-24, Vikram University Campus, Ujjain, M. P. India * SOC. and E. IPS. Academy Indore, M. P., India email:bharat_singhips@yahoo.com ABSTRACT In this paper we prove common fixed point theorem in IFMspace using using the notion of reciprocal continuity .moreover we have to show that in the context of reciprocal continuity the notion of compatibility and semi compatibility of maps becomes equivalent. Our result improves recent results of urmila Sharma et .al. In the sense that all the maps involved in the theorems are discontinuous even at common fixed point.
1. INTRODUCTION The concept of fuzzy sets was given by Zadeh11 becomes the foundation of fuzzy metric space. To use this concept in topology and analysis several researchers defined fuzzy metric space in various ways. Singh and Chauhan8 introduced more general concept namely compatibility than that of weak –commutivity of maps in the setting of fuzzy metric space. Recently Singh and Jain10 have been introduced semicompatibility of maps in fuzzy metric space. George and Veeramani4 had showed that every fuzzy metric space is an intuitionistic fuzzy metric space and found a necessary and sufficient condition for an intuitionistic fuzzy metric space to be complete. Chaudhary13 introduced mutually contractive sequence of self maps and proved a fixed point theorem. Kramosil and Michalek6 introduced the notion of Cauchy sequences in an intuitionistic fuzzy metric space and proved the well known fixed point theorem. Turkoglu et al12 gave the generalization of
Jungek’s common fixed point theorem to intuitionistic fuzzy metric space. They first formulate the definition of weakly commuting and R-weakly commuting mapping in intuitionistic fuzzy metric spaces and proved the intuitionistic fuzzy version of Pant’s theorem. Our result improves recent results of urmila Sharma et al.25 and 27, in the sense that all the maps involved in the theorems are discontinuous even at common fixed point 2. PRELIMINARIES Definition 2.1 [7] A binary operation is a continuous t norm if it satisfies the following condition 2.1.1 is commutative and associative 2.1.2 is continuous Address of corresponding author 2.1.3
for all
Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)
M. S. Chauhan, et al., J. Comp. & Math. Sci. Vol.2 (1), 103-110 (2010)
104 2.1.4
whenever
and
for all Definition 2.2 [7] A binary operation is continuous tconorm if ◊ is satisfying the following condition 2.2.1 is commutative and associate 2.2.2 ◊ is continuous 2.2.3 for all whenever and 2.2.4 for all Definition 2.3 [4] the 3-tuple (X, M, ) is called a fuzzy metric space (FM-space) if X is an arbitrary set is a continuous t-norm and M is a fuzzy set in satisfying the following conditions for all and . 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5 is continuous. Remark 2.1 since is continuous, it follows from (2.3.4) that the limit of a sequence in FM-space is uniquely determined Definition 2.4 [16] A five –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 tconorm and M, N are fuzzy sets on satisfying the following conditions for all 2.4.1 2.4.2
2.4.3 2.4.4 2.4.5 is continuous 2.4.6 2.4.7 2.4.8 is 2.4.9 continuous Then (M, N) is called an intuitionistic fuzzy metric On X , the function and adenote the degree of nearness and the degree of non- nearness between x and y with respect to t respectively. Remark 2.2 Every fuzzy metric space (X, is an intuitionistic fuzzy metric space M, form such that t-norm and t-conorm ◊ are associated ie for any but the converse is not true. Definition 2.5 two maps F and G from intuitionistic fuzzy metric space into itself are said to be Rweakly commuting if there exist a positive real number R such that for each and for all . Definition 2.6 two self mappings F and G of an intuitionistic fuzzy metric space are said to be compatible if
Definition 2.7 two self maps F and G of intuitionistic fuzzy metric space are said to be weak
Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)
M. S. Chauhan, et al., J. Comp. & Math. Sci. Vol.2 (1), 103-110 (2010)
compatible if they commute at their coincidence points that is implies . Remark 2.1 from the above definitions it is clear that in an intuitionistic fuzzy metric space a pair (F,G) of self maps is R-weakly commuting implies that the pair is compatible, which implies that the pair is weak-compatible. But the converse is not true Example 2.2 let be an intuitionistic fuzzy metric space where t norm is defined and t co-norm is defined by for all and
105
and for all Define self maps F and G on X as follow = 3 1 3
=
Take
=
, then
and
Now
and ) as Hence F and G are not compatible .The set of coincident points of F and G is [1,3]. Now for any and . Thus F and G are weak-compatible but not compatible Lemma 2.7[26]. Let be a sequence in IFM space with the condition and . If there exist a number such that and
,
for all , then sequence in X
is a Cauchy
MAIN RESULT Theorem: Let A, B, S and T be self maps on a complete Intuitionistic fuzzy metric space where is a continuous t-norm and is a continuous t co-norm, defined by and satisfying (i). , (ii). [B,T] is weak compatible (iii). for all and
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106
M. S. Chauhan, et al., J. Comp. & Math. Sci. Vol.2 (1), 103-110 (2010)
and
Where from each
are continuous functions such that for and for each
If [A,S] is semi-compatible pair of reciprocal
continuous maps then A,B, S, and T have a unique common fixed point Proof let be any arbitrary point then for which there exist and thus we can construct a sequence and in X such that and for then by the condition we get
And
Similarly And In general And
Therefore is an increasing and is an decreasing sequence of positive real numbers in [0,1] and tend to limit and we
I.e. And if letting
A contradiction then
claim that
and
if
And on letting
and on
Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)
then
M. S. Chauhan, et al., J. Comp. & Math. Sci. Vol.2 (1), 103-110 (2010)
107
i.e. a contradiction Now for any positive integer p and
Letting
we get
and thus and Thus is a Cauchy sequence in X. since X is complete space converges to a point z(say) in X. Hence the subsequences
Now since A and S are reciprocal continuous and semi-compatible then we have
also converges to z And Therefore we get now we will show for this suppose then by contractive condition we get
A contradiction since such that
and
Putting condition we get
Letting
we get
thus there exists
in contraction
and Letting
we get
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108
M. S. Chauhan, et al., J. Comp. & Math. Sci. Vol.2 (1), 103-110 (2010)
and â&#x;š and the weak-compatible of (B,T) gives again by contractive conditions and assuming we get hence ie finally we get z is a common fixed point of A,B, S and T. the uniqueness follows from contractive condition
This completes the proof Now we prove another common fixed point theorem with different contractive condition Theorem let A, B, S and T be self maps on a complete Intuitionistic fuzzy metric space Satisfying , (i) (ii) (B,T) is weak compatible (iii). for all and
and where
are continuous functions from such that for each and for each if (A,S) is semi-compatible pair of reciprocal continuous maps then A, B, S, and T have a unique common fixed point Proof Let be any arbitrary point then for which there exist and thus we can construct a sequence and in X such that and for then by the condition we get
and
Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)
M. S. Chauhan, et al., J. Comp. & Math. Sci. Vol.2 (1), 103-110 (2010)
109
Similarly we get And In general And
Therefore is an is an increasing and decreasing sequence of positive real numbers in [0,1] and tend to limit and then by the similar technique of above theorem we can easily show that is a Cauchy sequence in X. since X is complete converges to a point z (say) in X.hence the subsequences also converges to z now A, and B are Reciprocal continuous and semi-compatible then we have an d therefore we get now we will show for this suppose then by contractive condition we get a contradiction thus hence by similar techniques of above theorem we can easily show that z is a common fixed point of A,B, S and T. uniqueness of fixed point can be easily verify by contractive condition this completes the proof Example Let be usual metric space and M and N be usual fuzzy metrics on where
be the induced intuitionistic fuzzy metric space with for B, S and T by
we define mappings A,
then A, B, S, and T satisfy all the conditions of the above theorem with and
where
and we have a
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