Semicompatibility and fixed point theorems in fuzzy metric space using implicit relation

Cho et al. [2] introduced the notion of semicompatible maps in a d-topological space. They define a pair of self-maps (S,T) to be semicompatible if conditions (i) Sy = Ty implies that STy = TSy; (ii) for sequence {xn} in X and x ∈ X , whenever {Sxn} → x,{Txn} → x, then STxn → Tx, as n→∞, hold. However, in Fuzzy metric space (ii) implies (i), taking xn = y for all n and x = Ty = Sy. So, we define a semicompatible pair of self-maps in fuzzy metric space by condition (ii) only. Saliga [9] and Sharma et. al [10] proved some interesting fixed point results using implicit real functions and semicompatibility in d-complete topological spaces. Recently, Popa in [8] used the family F4 of implicit real functions to find the fixed points of two pairs of semicompatible maps in a d-complete topological space. Here, F4 denotes the family of all real continuous functions F : (R+)4 →R satisfying the following properties. (Fh) There exists h ≥ 1 such that for every u ≥ 0, v ≥ 0 with F(u,v,u,v) ≥ 0 or F(u,v,v,u) ≥ 0, we have u≥ hv. (Fu) F(u,u,0,0) < 0, for all u > 0. Jungck and Rhoades [6] (also Dhage [3]) termed a pair of self-maps to be coincidentally commuting or equivalently weak compatible if they commute at their coincidence points. This concept is most general among all the commutativity concepts in this field as every pair of commuting self-maps is R-weakly commuting, each pair of R-weakly commuting self-maps is compatible and each pair of compatible self-maps is weak compatible but the reverse is not always true. Similarly, every semicompatible pair of self-maps is weak compatible but the reverse is not true always. The main object of this paper is to obtain some fixed point theorems in the setting of fuzzy metric space using weak compatibility,