A Fixed Point Theorems in L-Fuzzy Quasi-Metric Spaces

At first we considered the L-fuzzy metric space notation which is useful in modeling some phenomena where it is necessary to study the relationship between two probability functions as well observed in Gregori et al. [A note on intuitionistic fuzzy metric spaces. Chaos, Solitons and Fractals 2006; 28: 902-905]. Then we introduced the concept of fixed point theorem in L-fuzzy metric space and finally, showed that every contractive mapping on an L-fuzzy metric space has a unique fixed point.


INTRODUCTION
The concept of fuzzy sets was introduced initially by Zadeh [1] in 1965. Various concepts of fuzzy metric spaces were considered in George and Veeramani [2] and Mihet [3,4] .
In this research, at first we shall adopt the usual terminology, notation and conventions of L-fuzzy metric spaces introduced by Saadati et al. [5] which are a generalization of fuzzy metric spaces [2] and intuitionistic fuzzy metric spaces [6,7] . Then we consider the fixed point theorem on such spaces and show that every contractive mapping on non-Archimedean Lfuzzy metric space has a unique fixed point. Definitions 1.1: Goguen [8] let L = (L, ≤ L ) be a complete lattice and U a non-empty set called universe. An Lfuzzy set A on U is defined as a mapping. A: U→L. For each u in U, A(u) represents the degree (in L) to which u satisfies A.

Definitions 1.2:
A triangular norm (t-norm) on L is a mapping τ: L 2 →L satisfying the following conditions: x, x ', y, y' L x x 'and y y' x, y x ', y'

Definition 1.4:
The triple (X, M, τ) is said to be an Lfuzzy quasi-metric space if X is an arbitrary (nonempty) set, τ is a continuous t-norm on L and M is an L-fuzzy set on X 2 ×]0,+∞[ satisfying the following conditions for every x, y, z in X and t, s in ]0,+∞[: In this case, M is called an L-fuzzy quasi-metric.
If, in the above definition, the triangular inequality (c) is replaced by Then the triple (X, M, τ) is called a non-Archimedean L-fuzzy quasi-metric space [3,4] . M(x, x, t).
≥ τ ≥ τ Λ Λ ≥ Λ Therefore x 0 ∈B x and B y ⊆B x implies that B x ≤ B y for all B x ∈A 1 . Thus B y is an upper bound in A for family A 1 . Hence by Zorn's Lemma, A has a maximal element, say, B z , for some z∈X. We claim that z = ∆z.
Uniqueness easily follows from contractive condition.

CONCLUSION
In this research we introduce the concept of fixed point theorem in L-fuzzy metric spaces and present some results.