Some results on common fixed points of compatible mappings

Fixed Points in Menger Space for Compatible Mappings of Type ()

“…[6] A sequence {x n } in a Menger space (X, F, t) is said to be convergent and converges to a point x in X if and only if for each  > 0 and  > 0, there is an integer M(, ) such that…”

“…[6] Self mappings A and S of a Menger space (X, F, t) are said to be compatible if F ASx n , SAx n (x)  1 for all x > 0, whenever {x n } is a sequence in X such that Ax n , Sx n  u for some u in X, as n . Definition 2.8.…”

“…Jungck [3] soon enlarged this concept to compatible maps. The notion of compatible mapping in a Menger space has been introduced by Mishra [6].…”

Fixed Points of Converse Commuting Mappings Using an Implicit Relation

“…Lemma 2.8. ( [17]) Let (X, F, ) be a Menger space. If there exists a constant k ∈ (0, 1) such that F x,y (kt) ≥ F x,y (t) for all t > 0 with fixed x, y ∈ X then x = y.…”

Common Fixed Point Theorem for Compatible Maps of Type (β) and Type (α) using Integral Type Mapping

“…In 1988, Grabiec [3] defined contraction and contractive mappings on a fuzzy metric space and extended fixed point theorems of Banach and Edelstein in such spaces. Following Grabiec's approach, Mishra et al [4] obtained common fixed point theorems for asymptotically commuting mappings on fuzzy metric spaces. In 1998, Vasuki [5] established a generalization of Grabiec's fuzzy contraction theorem wherein he proved a common fixed point theorem for a sequence of mappings in a fuzzy metric space.…”