INTRODUCTION
The concept of fuzzy sets was introduced initially by Zadeh
(1965). To use this concept in topology and analysis many researchers have
developed a theory of fuzzy sets and applications. George
and Veeramani (1994) introduced the concept of fuzzy metric spaces. In this
study we state some of the basic facts about fuzzy metric and normed spaces.
Definition 1: A binary operation *[0.1]x[0,1]→[0,1] is ontinuous
tnorm if * is satisfying the following conditions:
• 
* is commutative and associative 
• 
* is continuous 
• 
a*1 = a for all aε[0,1] 
• 
a*b≤c*d whenever a≤c and b≤d and a,b,c,d ε[0,1] 
Two typical examples of continuous tnorm are a*b = ab and a*b = min (a,b).
Definition 2: A triple (X, M, *) is called a fuzzy metric space if X
is an arbitrary (nonempty) set, * is a continuous tnorm, and M is a fuzzy
set on X^{2}x(0,∞), satisfying the following conditions for each
x, y zεX and t.s>0:
• 
M (x, y, t)>0 
• 
M (x, y, t) = 1 if and only if x = y 
• 
M (x, y, t) = M (y, x, t) 
• 
M (x, y, t) * M (y, z, t+s)≤M(y, z, t+s) 
• 
M (x, y):]0,∞)→]0,1] is continuous 
Example 1: Let (X.d) be a metric space, define a*b = ab or a*b = min
(a,b) and
which is called the standard fuzzy metric induced by metric d.
Let (X, M, *) be a fuzzy metric space . For t>0, the open ball B(x, r, t)
with center x ε X and radius 0<r<1 is defined by:
Let (X, M, *) be a fuzzy metric space. Let τ be the set of all A⊂X with x ε A if and only if there exist t>0 and 0<r<1 such that B(x, r, t) ⊂ A.
Then τ is a topology on x (induced by the fuzzy metric M). This topology is Hausdorff and first countable. A sequence {x_{n}} in X converges to x if and only M(x_{n}, x, t)→1 as n→∞, for each t>0. It is called a Cauchy sequence if for each 0<ε<1 and t>0, there exits n_{0} ε N such that M(x_{n}, x_{m}, t)>1ε for each n, m≥n_{0}. The fuzzy metric space (x, M, *) is said to be complete if every Cauchy sequence is convergent.
Lemma 1: Let (X, M, *) be fuzzy metric space. Then, M (x, y, t) is nondecreasing with respect to t, for all x, y in X.
Definition 3: The triple (X, N, *) is said to be a fuzzy normed space
if X is a vector space, * is a continuous tnorm and N is a fuzzy set on Xx(0,∞)
satisfying the following conditions for every x, y ε X and t, s>0:
• 
N (x, t)>0 
• 
N (x, t) = 1 if and only if x = 0 
• 

• 
N (x, t)*N(y, s)≤N(x+y, t+s) 
• 
N (x) : (0, ∞)→[0, 1] is continuous 
where, in (c), α is in the scalar field of X. In this case N is called
a fuzzy norm.
Lemma 2: Let (X, N,*) be a fuzzy normed space. If define:
then, M is a fuzzy metric on X, which is said to be induced by the fuzzy norm
N.
The fuzzy normed space (X, N, *) is said to be a fuzzy Banach space whenever X is complete with respect to the fuzzy metric induced by fuzzy norm.
Lemma 3: Let (X, M, *) be fuzzy metric space and define
X_{λ,M}:X^{2}→R^{+} ∪ {0} by
E_{λ,M} (x, y) = inf{t>0:M(x, y, t)>1λ} 
for each λε]0,1[ and x, y ε X. Then
(i) For any με]0,1[ there exists λε]0,1[ such that:
E_{μ,M} (x_{1}, x_{n})≤E_{λ,
M}(x_{1}, x_{2})+...+E_{μ,M} (x_{n1},
x_{n}) 
for any x_{1},...,x_{n} εX
(ii) The sequence {x_{n}}_{nεN} is convergent with respect to fuzzy metric M if and only if E_{λ,M}(x_{n}, x)→0. Also the sequence {x_{n}} is a Cauchy sequence with respect to fuzzy metric M if and only if it is a Cauchy sequence with E_{λ,M}.
Proof: For (i), for every μ ε]0,1[, there is a λε]0,1[ such that (1λ)*...*(1λ)>(1μ). By the triangular inequality:
for every δ>0, which implies that:
Definition 4: The fuzzy metric space (X, M, *) said that has the property (C), if it satisfies the following condition:
M (x, y, t) = C, for all t>0 implies C = 1
Lemma 4: Let the function φ(t) satisfies the following condition:
(φ) φ(t):[0,∞] is nondecreasing and
for all t>0, when φ^{n}(t) denotes the nth iterative function
of φ (t), then φ(t)<t for all t>0.
THE MAIN RESULTS
Theorem 1: Let {A_{n}} be a sequence of mappings A_{i}
of a complete fuzzy metric space (X, M, *), which this space has the property
(C), into itself such that, for any two mappings A_{i}, A_{j}:
for some m, x, yεX and for all t>0.
Here, φ_{i,j}:[0, ∞)→[0, ∞) is a function such that φ_{i,j} (t)<φ(t) for i, j = 1,2,... and the function φ(t): [0, ∞)→on to [0,∞) is strictly increasing and satisfies condition φ. Then the sequence {A_{n}} has a unique common fixed point in X.
Proof: Let x_{0} be an arbitrary point in X and define a sequence
{x_{n}} in X by:
Then
and:
and so on. By induction,
which implies
for every λε]0,1[.
Now, showed that {x_{n}} is a Cauchy sequence. For every με]0,1[,
there exists γε]0,1[such that:
as m, n > 1. Since, X is complete, there is x ε X such that
Now is proved that x is a periodic point of A_{i} for any i = 1, 2,
.... Notice:
as n→∞. Thus and
is got
To show uniqueness, assume that y ≠ x is another periodic point of A_{i}.
Then:
On the other hand, by Lemma 1 implies that:
Hence, M(x, y, t) = C for all t>0. Since, M has the property (C), it follows:
That C = 1, i.e., X = Y. Also:
i.e., A_{i} (x) is also a periodic point of A_{i}. Therefore, x = A_{i} (x), i.e., x is a unique common fixed periodic point of the mappings A_{n} for n = 1,2... . This completes the proof.
Theorem 2: Let (X, M, *) be a complete fuzzy metric space, let (X, M,
*) has the property (C) and let f, g: X→X be maps that satisfy the following
conditions:
• 
g (x) ⊆ f(X) 
• 
f is continuous 
• 
M (g(x), g(y), φ(t))≥M(f(x), f(y),t) for all x, y ε X where,
the function is
strictly increasing and satisfies condition φ 
Then f and g have a unique common fixed point provided f and g commute.
Proof: Let x_{0} εX. By (a) there is x_{1} such that f(x_{l}) = g (x_{0}). By induction, we can define a sequence {x_{n}}_{n} such that f(x_{n}) = g(x_{n1}). By induction again:
for n = 12,..., which implies that:
for every λε]0,1[.
Now, have been showed that {f(x_{n})} is a Cauchy sequence. For every με]0,1[, there exists γε]0,1[ such that, for m≥n,
as m,n→∞. Since, X is complete, there exists y ε X such that lim_{n→∞}f(x_{n}) = y. So, g(x_{n1}) = f(x_{n}) tends to y. It can be seen from (c) that the continuity of f implies that to g.
Thus {g(f(x_{n}))}_{n} converges to g(y). However, g(f(x_{n}))
= f(g(x_{n})) by the commutativity of f and g. Thus f(g (x_{n}))
converges to f (y). Because the limits are unique, f(y) = g(y). So, f(f(y))
= f(g(y)) by commutativity and:
On the other hand, Lemma 1 implies that:
Hence, M (g(y), g(g(y), t) = C for all t>0. Since, M has the property (C), it follows that C = 1, i.e., g(y) = g(g(y)). Thus, g (y) = g(g(y)) = f (g(y)). So, g(y) is a common fixed point of f and g.
If y and z are two fixed points common to f and g, then:
On the other hand, by Lemma 1
Hence, M(y, z, t) = C for all t>0. Since, M has the property (C), it follows that C = 1, i.e., y = z.
Theorem 3: Let W be a closed and convex subset of a fuzzy Banach space
(V, N, *) and f: W→W a mapping which satisfies the condition:
for all x, yεW and for all t>0. The function is
strictly increasing and satisfies condition φ. Then f has at least a fixed
point.
Proof: Let x_{0} in W be arbitrary and let a sequence {x_{n}} be defined by:
x_{n+1} = [x_{n} + f(x_{n})]/2 (n = 0,1,2,...)
For this sequence
and hence,
Therefore, for x = x_{n1} and y = x_{n} the condition (Eq.
1) states:
By condition (Eq. 2):
Hence,
By Lemma 2 as proof of Theorem 1 is concluded that {x_{n}} is Cauchy
sequence in W and converges to some uεW. Since:
Hence,
Now, let us put in Eq. 1 x = u and y = x_{n} and
use Eq. 2. Then:
If now n tend to infinity one has
which implies that fu = u and this theorem is established.