COMMON FIXED POINT RESULTS FOR HYBRID CONTRACTION IN HAUSDORFF FUZZY METRIC SPACE

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. Hybrid contraction of single and multi-valued fuzzy mappings in Hausdorff fuzzy metric space is discussed in the present article. Here, we introduced the concept of α−η∗−ψ−hybrid contraction for single and multi-valued fuzzy mappings and prove the common fixed point results in Hausdorff fuzzy metric space.

In nonlinear analysis the FPT play a key role. For the existing of FP in FMS, the contractive conditions and implicit function play a key role (see [5,30,37]). Samet et al. [39] first introduced the concept of admissible mapping [AM] for single valued mapping [SVM] and Asl et al. [2] extended the concept of admissible for SVM to multi-valued mappings [MVM]. Latter, Salimi et al. [38], defined an α-AM with respect to η on MS. Afterwards, a number of authors investigated FPT for α * and η * type's AM's in FMS (see [32,43]). Recently, Hong [19] introduced the concept of α * − η * −admissible for set valued mappings in FMS. Motivated by result of Phiangsungnoen [33]

PRELIMINARIES
Recall that a continuous triangular norm (t-norm [40]) with unit 1 is an associative and commutative binary operation : or a b = min(a, b), and a b = ab max{a,b,λ } , for 0 < λ < 1. For Lukasievicz t-norm that is, Definition 2.1.
(c) A fuzzy metric space in which every Cauchy sequence is convergent is said to be complete.
Let X be a non-empty set and α, η : is a fuzzy set. For any fuzzy set and Θ ∈ X , E(Θ) called the membership grade of E in X . The λ −cut of fuzzy set E represented Let us consider W (X ) be a compact sub-collection of all roughly value in X . A fuzzy set E in X is said to be an roughly value iff E λ is compact and convex in X ∀λ ∈ (0, 1] and sup Θ∈X E(Θ) = 1.
is a Hausdorff fuzzy metric space.
Throughout this paper, let (W (X ), H M , ) be a compact HFMS and W (X ) be a compact sub-collection of all roughly values. Then for all ∀E, F ∈ W (X ),t > 0 and λ ∈ (0, 1], we have It is noted that p λ is non-increasing function of λ and thus

It follows immediately from the definition that
Note that M (p) is a non-increasing function for p and H M ( p) is a Hausdorff fuzzy metric induced by fuzzy metric M on W (X ).

MAIN RESULTS
Let T : X → W (X ) be a multi-valued mapping of Hausdorff fuzzy metric space (W (X ), H M , ) If Θ ∈ T Θ then an element Θ ∈ X is called a fixed point of T .

CONCLUSIONS
In this chapter we introduced the concept of α * − η * − ψ−hybrid contraction for single and multi-valued fuzzy mappings and prove the common fixed point results in Hausdorff fuzzy metric space. Our result helps in the applications of integral equation, which is significantly contributed to the existing literature for fixed point theorem.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.