Existence theorems for generalized distance on complete metric spaces (Q1958032)

First, the author introduces the concept of \(u\)-distance on a metric space, as a generalization of the \(w\)-distance (concept due to \textit{O. Kada, T. Suzuki} and \textit{W. Takahashi} [Math. Jap. 44, No.~2, 381--391 (1996; Zbl 0897.54029)]). Then he obtains a generalization of Takahashi's minimization theorem based on which several existence theorems are established for problems of the following type:\ Given a complete metric pace \((X,d)\), a \(u\)-distance \(p\) on \(X\) and a self-mapping \(T\) of \(X\), find \(x_0\in X\) such that \(Tx_0=x_0\) and \(p(x_0, x_0)=0\).