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

INTRODUCTION

In [14] Song has mentioned that Grabiec [4] proved a fuzzy fuzz·y
1. Covered with fuzz.

2. Of or resembling fuzz.

3. Not clear; indistinct: a fuzzy recollection of past events.

4.
Banach contraction contraction, in physics
contraction, in physics: see expansion.
contraction, in grammar
contraction, in writing: see abbreviation.

contraction - reduction
theorem theorem, in mathematics and logic, statement in words or symbols that can be established by means of deductive logic; it differs from an axiom in that a proof is required for its acceptance.  and Vasuki [15] generalized gen·er·al·ized
1. Involving an entire organ, as when an epileptic seizure involves all parts of the brain.

2. Not specifically adapted to a particular environment or function; not specialized.

3.
the results of Grabiec for common fixed point theorem for a sequence of mappings in the fuzzy metric space metric space

In mathematics, a set of objects equipped with a concept of distance. The objects can be thought of as points in space, with the distance between points given by a distance formula, such that: (1) the distance from point A to point B is zero if and only if A and
. Song [14] has pointed out there are some errors in the papers of Vasuki [15] and Grabiec [4], because definition of Cauchy sequence (mathematics) Cauchy sequence - A sequence of elements from some vector space that converge and stay arbitrarily close to each other (using the norm definied for the space).  given by Grabiec [4] is weaker than the one proposed by Song [14] and hence conditions of Vasuki's theorem and its corollary corollary: see theorem. . 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 re·pro·duce
v. re·pro·duced, re·pro·duc·ing, re·pro·duc·es

v.tr.
1. To produce a counterpart, image, or copy of.

2. Biology To generate (offspring) by sexual or asexual means.
IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ]

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 Fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets have been introduced by Lotfi A. Zadeh (1965) as an extension of the classical notion of set. In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent  in [X.sup.3] x [0,[infinity infinity, in mathematics, that which is not finite. A sequence of numbers, a1, a2, a3, … , is said to "approach infinity" if the numbers eventually become arbitrarily large, i.e. ]) 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 One possible combination of items out of a larger set of items. For example, with the set of numbers 1, 2 and 3, there are six possible permutations: 12, 21, 13, 31, 23 and 32.

(mathematics) permutation - 1.
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 lemma (lĕm`ə): see theorem.

(logic) lemma - A result already proved, which is needed in the proof of some further result.
. 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 fore

front, e.g. forelimb.

fore cannon
the third metacarpal bone of the horse.
M--monotonically increasing sequence.

In order to prove a M-fuzzy metric space is a Hausdorff topological space Noun 1. topological space - (mathematics) any set of points that satisfy a set of postulates of some kind; "assume that the topological space is finite dimensional"
mathematical space
. We needed the following definitions and lemmas This following is a list of lemmas (or, "lemmata", i.e. minor theorems, or sometimes intermediate technical results factored out of proofs). See also list of axioms, list of theorems and list of conjectures. .

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 inequality, in mathematics, statement that a mathematical expression is less than or greater than some other expression; an inequality is not as specific as an equation, but it does contain information about the expressions involved.  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 topology, branch of mathematics, formerly known as analysis situs, that studies patterns of geometric figures involving position and relative position without regard to size.  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 integer: see number; number theory  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 summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument)  over (n=1)] [[phi].sup.n](t) < [infinity] for all t > 0, when [[phi].sup.n](t) denotes the nth iterative it·er·a·tive
1. Characterized by or involving repetition, recurrence, reiteration, or repetitiousness.

2. Grammar Frequentative.

Noun 1.
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 A group of commands or functions that do not include all the capabilities of the original specification. Software or hardware components designed for the subset will also work with the original. ] 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 con·verge
v. con·verged, con·verg·ing, con·verg·es

v.intr.
1.
a. To tend toward or approach an intersecting point: lines that converge.

b.
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 raj also Raj
n.
Dominion or rule, especially the British rule over India (1757-1947).

[Hindi r
Josph **

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

AMS AMS - Andrew Message System  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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