A common fixed point theorem for fuzzy weakly contractive mappings in quasi-metric spaces (Q2906125)

A quasi-metric on a nonempty set \(X\) is a non-negative real-valued function \(d\) on \( X\times X \) such that, for all \(x, y, z\in{X}\), (a) \(d(x, y) = d(y, x) = 0 \Longleftrightarrow x = y\), and (b) \(d(x, y) \leq d(x, z)+d(z, y)\). A pair \((X, d)\) is called a quasi-metric space if \(d\) is a quasi-metric on \(X\). A~sequence \((x_{n})\) in a quasi-metric space \((X, d)\) is called left \(K\)-Cauchy if, for each \(\varepsilon >0\), there is a \(k\in{\mathbb{N}}\) such that \(d(x_{n}, x_{m}) <\varepsilon\) for all \(n,m \in{\mathbb{N}}\) with \(m\geq n\geq k\). A quasi-metric space \((X, d)\) is said to be Smyth-complete if each left \(K\)-Cauchy sequence in \((X, d)\) converges in the metric space \((X, d_{s})\), where \(d_{s}(x, y) = \max\{d(x, y), d(y, x)\}\) for all \(x, y \in{X} \).NEWLINENEWLINEIn the paper under review, the authors establish a common fixed-point theorem in Smyth-complete quasi-metric spaces with a contractive-type condition involving fuzzy mappings, which extends some known theorems.