# Some results on fixed points in M-fuzzy metric space.

INTRODUCTION

In [14] Song has mentioned that Grabiec [4] proved a fuzzy Banach contraction theorem and Vasuki [15] generalized the results of Grabiec for common fixed point theorem for a sequence of mappings in the fuzzy metric space. Song [14] has pointed out there are some errors in the papers of Vasuki [15] and Grabiec [4], because definition of Cauchy sequence given by Grabiec [4] is weaker than the one proposed by Song [14] and hence conditions of Vasuki's theorem and its corollary. We present more general conditions.

DEFINITION 1.1:

A mapping T: [0, 1] x [0, 1] [right arrow] [0, 1] is called a triangular norm (shortly t-norm) if it satisfies the following conditions.

(i) T(a, 1) = a for every a [member of] [0, 1]

(ii) T(a, b) = T(b, a) for every a, b [member of] [0, 1]

(iii) T(a, c) [greater than or equal to] T(b, d) for a [greater than or equal to] b; c [greater than or equal to] d

(iv) T(a, T(b, c,)) = T(T (a, b), c) for all a,b,c [member of] [0,1]

EXAMPLE 1.2:

The minimum t- norm, [T.sub.M], is defined by [T.sub.M](x, y) = min (x, y), the product t- norm [T.sub.P], is defined by [T.sub.P](x, y) = xy, the Lukasiewicz t- norm, [T.sub.L] is defined by [T.sub.L](x, y) = max {x +y-1, 0} and finally the weakest t- norm the drastic product [T.sub.D] is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

DEFINITION. 1.3:

The triple (X, M, [DELTA]) is a M-fuzzy metric space if X is an arbitrary set, [DELTA] is a continuous t--norm and M is a fuzzy set in [X.sup.3] x [0,[infinity]) satisfying the following conditions for each x, y, z, a [member of] X and t, s >0

(1) M(x, y, z, t) > 0, for all x, y, z [member of] X

(2) M (x, y, z, t) = 1 iff x = y = z, for all t > 0,

(3) M (x, y, z, t) = M (p{x, y, z}, t), where p is a permutation function,

(4) [DELTA](M (x, y, a, t), M (a, z, z, s)) [less than or equal to] M (x, y, z, t + s),

(5) M (x, y, z, .) : [0, [infinity]) [right arrow] [0,1] is continuous.

LEMMA. 1.1: [8]

Let (X, M, *) be a M-fuzzy metric space. Then M is continuous function on [X.sup.3] x [0,[infinity]).

LEMMA. 1.2:

M (x, y, z, .) is non decreasing for all x, y, z [member of] X.

Proof :

Let M (x, y, z, t + s) [greater than or equal to] [DELTA]{M (x, y, w, t), M (w, z, z, s)} Put w = z on both sides We get M (x, y, z, t + s) [greater than or equal to] [DELTA]{M (x, y, z, t), M (z, z, z, s)}. = M (x, y, z, t) for all t > 0

Hence M (x, y, z, t + s) [greater than or equal to] M (x, y, z, t) for all t > 0, and x, y, z [member of] X, for any s > 0. There fore M--monotonically increasing sequence.

In order to prove a M-fuzzy metric space is a Hausdorff topological space. We needed the following definitions and lemmas.

LEMMA. 1.3:

Let (X, M, *) be a M-fuzzy metric space. Then for every t >0 and for every x, y [member of] X we have M (x, x, y, t) = M (x, y, y, t).

Proof :

For each [member of] > 0 by triangular inequality We have

(i) M (x, x, y, [member of] + t) = M (x, x, x, [member of]) * M (x, y, y, t) = M (x, y, y, t)

(ii) M (y, y, x, [member of] + t) = M (y, y, y, [member of]) * M (y, x, x, t) = M (y, x, x, t).

By taking limits of (i) and (ii) when [member of] [right arrow] 0, we obtain M (x, x, y, t) = M (x, y, y, t).

DEFINITION. 1.4:

Let (X, M, *) be a M-fuzzy metric space, For t >0, the open ball [B.sub.M] (x, r, t) with center x [member of] X and radius o < r <1 is defined by [B.sub.M] (x, r, t) = {y [member of] X : M (x, y, y, t) > 1 - r}. The family {BM (x, r, t): x [member of] X, 0 < r < 1, t > 0} is a neighborhood's system for a Hausdorff topology on X, which is called the topology induced by the generalized fuzzy metric M which is denoted by [J.sub.M] topology.

THEOREM. 1.4:

Every M-fuzzy metric space is Hasusdorff.

Proof:

Let (X, M, [DELTA]) be the given M-fuzzy metric space. Let x, y be two distinct points of X.

Then 0 < M (x, y, y, t) < 1. Put M (x, y, y, t) = r for some r [member of] (0, 1). For each r with r < [r.sub.0] < 1, there exists [r.sub.1] such that [DELTA]([r.sub.1], [r.sub.1]) = [r.sub.0]. Now consider the open balls [B.sub.M] (x, 1 - [r.sbu.1], t/2) and [B.sub.M] (y, 1 - [rs.ub.1], t/2). Clearly, [B.sub.M] (x, 1 - [r.sub.1], t/2) [intersection] [B.sub.M] (y, 1 - [r.sub.1], t/2) = [empty set]. For if there exists z [member of] [B.sub.M] (x, 1 - [r.sub.1], t/2) [intersection] [B.sub.M] (y, 1 - [r.sub.1], t/2), then r = M (x, y, y, t) = M (x, x, y, t) [greater than or equal to] [DELTA](M (x, x, z, t/2), M (z, y, y, t/2)) = [DELTA](M (x, z, z, t/2), M (y, z, z, t/2)) = [DELTA]([r.sub.1], [r.sub.1]) = [r.sub.0] > r.

which is contradiction. Hence (X, M, [DELTA]) is Hausdorff.

THEOREM. 1.5:

Let (X, M, [DELTA]) be a generalized fuzzy metric space [J.sub.M] be the topology induced by fuzzy metric M. Then for a sequence {[x.sub.n]} in X, [x.sub.n] [??] x iff M ([x.sub.n], x, x, t) [right arrow] 1 as n [right arrow] for all t > 0.

DEFINITION. 1.5:

A sequence {[x.sub.n]} in a generalized fuzzy metric space (X, M, [DELTA]) is [J.sub.M]--Cauchy iff for each t [member of] (0, 1), t > 0 there exists [n.sub.0] [member of] N, such that M ([x.sub.n], [x.sub.n], [x.sub.m], t) > 1-[member of] for all n > [n.sub.0] and p, q > 0.

A generalized fuzzy metric space in which every [J.sub.M]--Cauchy sequence is [J.sbu.M]--convergent and it is called a [J.sbu.M]--complete fuzzy metric space.

DEFINITION. 1.6:

A mapping F: R [right arrow] [R.sup.+] is called a distribution function if it is nondecreasing and left continuous and it has the following properties.

i) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let [D.sup.+] be the set of all distribution function F such that F(0) = 0 (F is a nondecreasing, left continuous mapping from R into [0, 1] such that F(x) = 1.

DEFINITION. 1.7:

A generalized probabilistic metric space (briefly GPM--space) is an ordered pair (S, F) where S is a non empty set and F: S x S x S [right arrow] [D.sup.+] F(p, q, r) is denoted by [F.sub.p,q,r] for every (p, q, r) [member of] S x S x S) satisfies the following conditions.

i) [F.sub.u,v,w] (x) = 1 for all x > 0 iff u = v = w

ii) [F.sub.u,v,w](x) = [F.sub.v,w,u] (x) = [F.sub.w,u,v] (x) for all u, v, w [member of] S and x[member of]R

iii) If [F.sub.u,v,a](x) = 1 and [F.sub.a,w,w](y) = 1 then [F.sub.u,v,w](x + y) = 1 for u, v, w [member of]S and x, y, z [member of][R.sup.+]

DEFINITION. 1.8:

A generalized probabilistic metric space is a triple (S, F, [DELTA]) where (S, F) is a GPM--space, [DELTA] is a t-norm and the following inequality holds:

[F.sub.u,v,w](x + y) [greater than or equal to] [DELTA] ([F.sub.u,v,a] (x), [F.sub.a,w,w] (y)) for u, v, w, a [member of]S and every x > 0 and y > 0.

then (S, F, [DELTA]) is a Hausdroff topological space in the topology [tau] induced by the family of ([member of], [lambda])--neighborhoods {Up ([member of], [lambda]) : p [member of] S, [member of] > 0, [lambda] >0}, where Up ([member of], [lambda])= {u [member of] S : [F.sub.u,u,p] ([member of]) > 1-[lambda]}.

DEFINITION. 1.9 :

Let (X, F, [DELTA]) be a Probabilistic Metric (PM-) space with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(i) A sequence {[u.sub.n]} in X is said to be [tau]-convergent to u [member of] X (we write un [right arrow] u) if for any given [member of] > 0, [lambda] > 0, there exists a positive integer N = N ([member of], [lambda]) such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] whenever n [greater than or equal to] N.

(ii) A sequence {[u.sub.n]} in X is called [tau]-Cauchy sequence if for any [member of] > 0 and [lambda] > 0, there exists a positive integer N = N([member of], [lambda]) such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([member of]) > 1-[lambda] whenever n, m [greater than or equal to] N.

(iii) A PM-space (S, F, [DELTA]) is said to be [tau]--complete if each [tau]--Cauchy sequence in S is [tau]-convergent to some point in S.

2. A COMMON FIXED POINT THEOREM

In this section, we present a sufficient condition which a sequence to be a Cauchy sequence in the M-fuzzy metric space (X, M, [DELTA]). In addition, we prove a common fixed point theorem. We need definition and the two next lemma.

DEFINITION. 2.1:

[DELTA] is said to be an h-type t-norm, if the family [{[[DELTA].sup.m] (t)}.sub.[infinity].sub.m=1] is equicontinuous at t =1, where [[DELTA].sup.1](t) = [DELTA](t, t,), [[DELTA].sup.2](t) = [DELTA]([[DELTA].sup.1](t), t) and [[DELTA].sup.m](t) = [DELTA]([[DELTA].sup.m-1](t), t) m = 2,3 ...

Obviously, [DELTA] is an h-type t-norm iff for any [lambda] [member of] (0, 1), there exists [delta] ([lambda]) [member of] (0, 1) such that [[DELTA].sup.m](t) > 1-[lambda] for all m[member of] N, when t > [delta] ([lambda]).

EXAMPLE. 2.2 :

t-norm minimum is a h-type t-norm.

LEMMA. 2.1:

Let the function [phi](t) satisfy the following conditions: ([phi])[phi](t): [0,[infinity]) [right arrow] [0, [infinity]) is nondecreasing and [[infinity].summation over (n=1)] [[phi].sup.n](t) < [infinity] for all t > 0, when [[phi].sup.n](t) denotes the nth iterative functions of [phi](t), then [phi](t) < t for all t > 0.

LEMMA. 2.2:

Let (S, F, [DELTA]) be a PM-space with an h-type t-norm [DELTA]. Suppose {[x.sub.n]} [subset] X such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all t > 0, where the function [phi](t) satisfies the condition ([phi]). Then {[x.sub.n]} is a [tau]--Cauchy.

REMARK. 2.3:

If (X, M, [DELTA]) be a M-fuzzy metric space, with a continuous t-norm [DELTA] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all x, y, z [member of] X (2.3.1)

Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and therefore [tau]-topology for [F.sub.x,y,z] (t) = M(x, y, z, t) is exactly topology induced by M when [F.sub.x,y,z] (t) = M (x, y, z, t) for all x, y, z [member of] X and t [greater than or equal to] 0. Thus, if(X, M, [DELTA]) is a

M-fuzzy metric space with continuous t-norm [DELTA] which satisfies (2.3.1) then {[x.sub.n]} is a [J.sub.M] -Cauchy if {[x.sub.n]} be a [tau]-Cauchy when ([F.sub.x,y,z], (t) = M (x, y, z (t)) for all x, y, z [member of] X and t [greater than or equal to] 0.

COROLLORY. 2.3:

Let(X, M, [DELTA]) be generalized fuzzy metric space satisfying (2.3.1) with a continuous h-type t-norm [DELTA]. Suppose {[x.sub.n]} [subset] X such that, M ([x.sub.n], [x.sub.n], [x.sub.n+1], [[phi].sup.n](t)) [greater than or equal to] M ([x.sub.0], [x.sub.0], [x.sub.1], t) for all t >0, where the function [phi](t) satisfies the condition [phi]. Then {[x.sub.n]} is a Cauchy ([J.sbu.M]-Cauchy) sequence.

PROOF:

Let [F.sub.x,y,z] (t) = M (x, y, z, t) for all x, y, z [member of] X and t [greater than or equal to] 0. Then (X, F, [DELTA]) is a PM-space with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Now, by applying lemma (2.2) and remark (2.3), the result follows immediately.

THEOREM. 2.4. Let (X, M, [DELTA]) be a complete fuzzy M-metric space. where [DELTA] is a continuous h- type t-norm and T be a selfmap of X such that M (Tx, Ty, Tz, [phi](t)) [greater than or equal to] M (x, y, z, t) for all x, y, z [member of] X and t > 0, where the function [phi](t) satisfies the condition ([phi]). Then T has a fixed point.

PROOF: Let [x.sub.0] [member of]X and [x.sub.n] = [T.sup.n] [x.sub.0], (n [member of] N). Now we have for t > 0 M ([x.sub.n+1], [x.sub.n], [x.sub.n], [[phi].sup.n](t)) = M ([T.sup.n+1] [x.sub.0], [T.sup.n] [x.sub.0], [T.sup.n] [x.sub.0], [[phi].sub.n](t)) [greater than or equal to] M ([T.sup.n] [x.sub.0], [T.sup.n-1] [x.sub.0], [T.sup.n-1] [x.sub.0], [[phi].sup.n-1](t)) [greater than or equal to] M ([T.sup.n-1] [x.sub.0], [T.sup.n-2] [x.sub.0], [T.sup.n-2] [x.sub.0], [[phi].sup.n-2](t)) [greater than or equal to] ... [greater than or equal to] M (T [x.sub.0], [x.sub.0], [x.sub.0], t) = M ([x.sub.1], [x.sub.0], [x.sub.0], t) for all n[member of]N.

Thus corollary (2.3) implies that {[x.sub.n]}is a Cauchy ([J.sub.M]-Cauchy) sequence, hence {[x.sub.n]} is converging to point [x.sub.0] in X.

Since M is continuous and [phi](t) < t we have M (x, x, Tx, t) [greater than or equal to] [DELTA] (M (x, x, [x.sub.n+1], t - [phi](t)), M ([T.sub.x], [x.sub.n+1], [x.sub.n+1] [phi](t))) [greater than or equal to] [DELTA] (M (x, x, [x.sub.n+1], t - [phi](t)), M (x, [x.sub.n], [x.sub.n], t)) [greater than or equal to] [DELTA] (M (x, x, [x.sub.n+1], t - [phi](t)), M (x, x, [x.sub.n], t)) [right arrow] (1,1) = 1 as n [right arrow] [infinity]

There fore [T.sub.x] = x.

THEOREM. 2.5:

(M-Fuzzy Banach Contraction theorem) Let (X, M, [DELTA]) be a [J.sub.M]--complete fuzzy metric space. Where [DELTA] is a continuous h- type t- norm. Suppose T is a self map of X such that M (Tx, Tx, Ty, kt) [greater than or equal to] M (x, y, z, t) for all x, y, z [member of] X and t > 0, where 0 [less than or equal to] k <1, Then T has a unique fixed point.

PROOF:

Let [phi](t) = kt, 0 = k < 1. Theorem (2.4) implies that T has a fixed point, Now we prove that fixed point is unique. Let y be another fixed point of T. Then Ty = y. M (x, y, y, [phi](t)) = M (Tx, Ty, Ty, [phi](t)) [greater than or equal to] M (x, y, y, t) put [phi](t) = kt, 0 [less than or equal to] k < 1. M (x, y, y, kt) [greater than or equal to] M (x, y, y, t) [greater than or equal to] M (x, y, y, t/[k.sup.n]) [right arrow] 1 as n [right arrow] [infinity] M (x, y, y, [phi](t)) = 1. There fore x = y.

THEOREM: 2.6:

Let {[T.sub.n]} be a sequence of mapping of a [Js.ub.M]--complete fuzzy metric space (X, M, [DELTA]) into itself, where [DELTA] is a continuous h--type t-norm such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all x, y, z [member of] X and for any three mappings [T.sub.i], [T.sub.j] and [T.sub.k] and any x, y, z [member of] X. we have M ([T.sub.i]x, [T.sub.j]y, [T.sub.k]z, [[phi].sub.i,j,k](t)) [greater than or equal to] M (x, y, z, t) for all t > 0 where [[phi].sub.i, j, k]: [0, [infinity]) [right arrow] [0, [infinity]) is a function such that [[phi].sub.i, j, k](t) < [phi](t) for all t >0, i, j, k = 1,2....the function [phi](t) :[0, [infinity]) [0, [infinity]) is strictly increasing and satisfies condition ([phi]). Also [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for all t > 0. Then the sequence {[T.sub.n]} has a unique common fixed point.

PROOF:

Suppose [x.sub.0] be an arbitrary point in X.

Define [x.sub.1] = [T.sub.1][x.sub.0] [x.sub.2] = [T.sub.2][x.sub.1] [x.sub.3] = [T.sub.3][x.sub.2]

[x.sub.n] = [T.sub.n][x.sub.n-1]

Then for all t > 0,

M ([x.sub.2], [x.sub.1], [x.sub.1], [phi](t)) [greater than or equal to] M ([x.sub.2], [x.sub.1], [x.sub.1], [[phi].sub.2,1,1](t)) = M ([T.sub.2] [x.sub.1], [T.sub.1] [x.sub.0], [T.sub.1] [x.sub.0], [[phi].sub.2,1,1](t)) [greater than or equal to] M ([x.sub.1], [x.sub.0], [x.sub.1], t)

next we consider

M ([x.sub.3], [x.sub.2], [x.sub.2], [[phi].sup.2](t)) [greater than or equal to] M ([x.sub.3], [x.sub.2], [x.sub.2], [[phi].sub.3,2,2][phi](t)) = M ([T.sub.3] [x.sub.2], [T.sub.2] [x.sub.1], [T.sub.2] [x.sub.1], [[phi].sub.3,2,2][phi](t)) [greater than or equal to] M ([x.sub.2], [x.sub.1], [x.sub.1], [phi] (t)) [greater than or equal to] M ([x.sub.1], [x.sub.0], [x.sub.0], t)

Hence By induction, we obtain

M ([x.sub.n+1], [x.sub.n], [x.sub.n], [[phi].sup.n](t)) [greater than or equal to] M ([x.sub.1], [x.sub.0], [x.sub.0], t) for all n. Thus the sequence {[x.sub.n]} is a [J.sub.M]--Cauchy sequence and hence it convergence to point x [member of]X,

Now we prove that x is a unique common fixed point of [T.sub.i]. For t > 0, we have M (x, x, [T.sub.i]x, t) = [DELTA](M (x, x, [x.sub.n+1], t - [phi](t)), M ([T.sub.i] x, [x.sub.n+1], [x.sub.n+1] [phi](t))) [greater than or equal to] [DELTA] (M (x, x, [x.sub.n+1], t - [phi](t)), M ([T.sub.i] x, [T.sub.n+1][x.sub.n], [T.sub.n+1][x.sub.n], [[phi].sub.i,n+1, n+1](t)) [greater than or equal to] [DELTA] (M (x, x, [x.sub.n+1], t - [[phi](t)), M (x, [x.sub.n], [x.sub.n], t)) [right arrow] [DELTA] (1,1) = 1 as n [right arrow] [infinity]

There fore [T.sub.i]x = x. Similarly we can prove that [T.sub.j]x = x and [T.sub.k]x = x.

UNIQUENESS:

Suppose x [not equal to] y such that [T.sub.i]y = [T.sub.j] y = [T.sub.k] y = y. Then M (x, y, y, t) [greater than or equal to] M ([T.sub.i]x, [T.sub.j]y, [T.sub.k]y, t) [greater than or equal to] M (x, y, z, [[phi].sup.-1](t)) [greater than or equal to] M (x, y, z, [[phi].sup.-2](t)) .... [greater than or equal to] M (x, y, z, [[phi].sup.-n](t)) [right arrow] 1 as n [right arrow] [infinity]

Where we have applied [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It means that M (x, y, y, t) = 1 for all t, Thus x = y. Thus x is a unique common fixed point of the sequence {[T.sub.n]}. There fore [T.sub.i]x is also a common fixed point of the sequence {[T.sub.i]}, so [T.sub.i]x = x. Then x is a unique common fixed point of the sequence {[T.sub.n]}.

COROLLARY. 2.7:

Let {[T.sub.n]} be a sequence of mapping of a [J.sub.M]--complete generalized fuzzy metric space (X, M, [DELTA]) into itself, where [DELTA] is a continuous h-type t-norm such that for any three mapping [T.sub.i], [T.sub.j] and [T.sub.k] we have M ([T.sub.i]x, [T.sub.j]y, [T.sub.k]z, [[alpha].sub.i, j, k], t) [greater than or equal to] M (x, y, z, t) for some m, 0 < [[alpha].sub.i, j, k] < [alpha] < 1, i, j, k = 1,2, ....... x, y, z [member of] X. Then the sequence {[T.sub.n]} has a unique common fixed point.

PROOF :

Taking [phi](t) = [alpha]t and [[phi].sub.i, j, k] (t) = [[alpha].sub.i, j, k] (t) when 0< [[alpha].sub.i, j, k]<[alpha] < 1 and i, j, k = 1,2..... The result followed theorem (2.6) immediately.

REMARK. 2.4:

One can obtain theorem (2.4) from theorem (2.6) by taking [[phi].sub.i, j, k] (t) = [phi](t) for all i,j,k = 1,2, ....., t >0 and without condition (I) [phi] is bijection and (II) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all t > 0.

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T. Veera Pandi, M. Jeyaraman * and J. Paul Raj Josph **

Reader in Mathematics, P.M.T. College, Melaneelithanallur--627 953, India.

AMS Mathematics subject classification: 47H10, 54H25.

* Department of Mathematics, Mohamed Sathak Engg. College, Kilakarai--623 806, India. India. E.Mail : vrpnd@yahoo.co.in, E.Mail: snegha20012002@yahoo.co.in

** Department of Mathematics, Manomaniam Sundaranar University, Tirunelveli, India.
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