This is an announcement for the paper “Using an implicit function to prove common fixed point theorems” by Mohammad Imdad, Rqeeb Gubran and Md Ahmadullah.
Abstract: We observe that, Theorem 2 due to Berinde and Vetro (Fixed Point Theory Appl. 2012:105) is not correct in its present form. In this paper, by correcting and enriching aforementioned theorem, we prove common fixed point results for a self-mappings satisfying an implicit function which is general enough to cover a multitude of known as well as unknown contractions. Our results unify, extend and generalize many relevant results of the existing literature. Interestingly, unlike many other cases, our main results deduce a nonlinear order-theoretic version of a well-known fixed point theorem (proved for quasi-contraction) due to $'{C}iri'{c}$ (Proc. Amer. Math. Soc. (54) 267-273, 1974). Finally, in the setting of metric spaces, we drive a sharpened version of Theorem 1 due to Berinde and Vetro (Fixed Point Theory Appl. 2012:105).
The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1605.05743