A continuation method for weakly contractive mappings under the interior condition
Let \(X\) be a complete metric space and \(U\) open in \(X\). In this paper, it is shown that the property of having a fixed point is invariant by a certain class of homotopies for weakly contractive maps \(H : \bar{U}\times [0,1] \to X\) satisfying the interior condition: \[ \begin{aligned}\text{(I-C)}\quad &\text{there exists \(r > 0\) such that \(x \neq H(x,t)\) for every \(t \in [0,1]\), \(f(x) \not\in \bar{U}\),}\\ &\text{and \(x \in U_r = \{y \in U : d(y,\partial{U}) < r\}\).} \end{aligned} \] The notion of weakly contractive maps was introduced by \textit{J.\,Dugundji} and \textit{A.\,Granas} [Bull.\ Greek Math.\ Soc.\ 19, 141--151 (1978; Zbl 0417.54010)]. An analogous result is also obtained for a class of homotopies satisfying a generalized contractive condition which was introduced by \textit{G.\,E. Hardy} and \textit{T.\,D. Rogers} [Can.\ Math.\ Bull.\ 16, 201--206 (1973; Zbl 0266.54015)].
scientific article; zbMATH DE number 5627868