A Common Fixed Point Theorem in Fuzzy Metric Space Using E.A. Property and Implicit Function

Abstract

In this paper, we prove common fixed point theorems in fuzzy metric spaces for weakly compatible mappings along with property (E. A.) satisfying implicit relation. Property (E. A.) buys containment of ranges without any continuity requirement besides minimizing the commutativity conditions of the maps to commutativity at their point of coincidence. Moreover, property (E. A.) allows replacing the completeness requirement of the space with a more natural condition of closeness of the range.

A Note on: A Common Fixed Point Theorem in Fuzzy Metric Space Using E. A. Property and Implicit Function

“…Unfortunately, Theorem 4.1 and hence in Corollary 4.1 are not true in its present form. On critical examination of the results given in paper [1], we notice one crucial error.…”

“…Proof of Theorem 4.1 and hence in Corollary 4.1 given in paper [1] are wrong as the existence of the limit lim n→∞ Ay n = z is not proved. It is proved only that, if this limit exists, then it must be equal to z.…”

“…In order to avoid repetition, we assume the terminology and the notations utilized in [1] rather than presenting the same again. Utilizing the concept of property (E.A), M. Verma et al [1] proved Theorem 4.1 and hence in Corollary 4.1 in fuzzy metric spaces.…”