Fiberwise contraction mappings principle
A metric on a mapping \(f\) of a nonempty set \(X\) to a topological space \(Z\) is a pseudometric \(p\) on \(X\) such that it is a metric on every fibre \(f^{-1}z\) of \(f, z \in Z\). A metric mapping \((f, p)\) consisting of a mapping \(f\) of a set to a space and of a metric mapping \((f, p)\) is called a fibrewise complete if \(p\) is a complete metric on every \(f\). For details refer to \textit{B. A. Pasynkov} [Mosc. Univ. Math. Bull. 54, No. 3, 29--32 (1999); translation from Vestn. Mosk. Univ., Ser. I 1999, No. 3, 29--32 (1999; Zbl 0949.54038)]. The main purpose of this paper is to extend the well-known classical Banach contraction theorem to metric mappings.
scientific article; zbMATH DE number 6361150