Fixed point theorems for contraction mappings in vector norm (Q1056895)

By means of an extension of a fixed point theorem of Edelstein [\textit{E. Hille}, Methods in Classical and Functional Analysis (1972; Zbl 0223.46001) p. 173], the following theorem is obtained generalising a theorem of \textit{F. Robert} [Matrices nonnégatives et normes vectorielles. Grenoble, 1973-74]: Theorem: Let \(\psi\) be a mapping defined in the non-negative orthant \({\mathbb{R}}^ k_+=\{t\in {\mathbb{R}}^ k| t_ 1\geq 0,...,t_ k\geq 0\}\) with values in \({\mathbb{R}}^ k_+\) which verifies (a) \(\psi (0)=0\), (b) Given \(t\geq 0:\psi_ i(t)<t_ i\) if \(t_ i>0\), \(\psi_ i(t)=0\) if \(t_ i=0\) are satisfied, (c) \(\psi\) is continuous. Let X be a complete normed space with a vector norm p with a size of \(k\geq 1\). Let \(D\subseteq X\) be a closed set and F be a mapping of D in itself, which satisfies \(p(F(x)-f(y))\leq \psi (p(x-y))\) for every pair of points x,y\(\in D\). Then F has a unique fixed point. This theorem is applicable to cases which are not covered by the theorem of Robert.