New discontinuity results at fixed point on metric spaces (Q2031287)

Let \((X,d)\) be a complete metric space and \(T\colon X\to X\). For \(u,v\in X\), denote \[ M(u,v)=\max\left\{d(u,v),d(u,Tu),d(v,Tv),\left[\frac{d(u,Tv)+d(v,Tu)}{1+d(u,Tu)+d(v,Tv)}\right]d(u,v)\right\}. \] The following result is proved: If (1) there exists a function \(\phi: \mathbb R^+\to \mathbb R\) such that \(\phi(t)<t\) for each \(t>0\) and \(d(Tu,Tv)\leq\phi(M(u,v))\), for all \(u,v\in X\); (2) for a given \(\varepsilon>0\), there exists a \(\delta>0\) such that \(\varepsilon<M(u,v)<\varepsilon+\delta\) implies \(d(Tu,Tv)\leq\varepsilon\), then \(T\) has a fixed point \(z\in X\) and \(T^nu\to z\) as \(n\to\infty\), for each \(u\in X\). Moreover, \(T\) is discontinuous at \(z\) if and only if \(\lim_{u\to z}M(u,z)\ne0\). Variants of this result for common fixed point problems of two self-mappings and for fixed circle problems are also obtained. Some appropriate examples are provided for each of these results.