Generalization of Fixed Point Theorem for φ- Contraction Mapping of a Fuzzy b- Metric Space

Let (X,W,T) be a fuzzy b- metric space, where X is a non-empty set, W is a fuzzy set on X×X×(0,∞) to [0,1], and T is a continuous t-norm, and let a function φ:[0,1]→[0,1] satisfies the following conditions: The function φ is strictly decreasing and continuous, φ(c)=0 If and only if c equals 1 and φ(T(c,a))= T(φ(c),φ(a)), where c and a in X. Which is called φ- function and use it to define φ- Contraction mappings of type Ι and ΙΙ. In this research, we will complete the study of many authors about fixed point theory on fuzzy b-metric spaces as Ashraf (1) and Rakic et al (2) and generalize some results on fixed points theory on fuzzy metric spaces to fuzzy b- metric spaces with simplify different proofs. Shen et,al (3) established many results on compact fuzzy metric spaces and complete fuzzy metric spaces. we generalize results of Shen et al to fuzzy b-metric spaces by using φ – Contraction mappings of type Ι and ΙΙ in both complete and compact fuzzy b- metric spaces to show existence of fixed points for this type of self-mapping.