On $U$-startpoints and $U$-fixed points in quasi-uniform spaces: a revisit and extensions

Abstract

We extend the $\U$-fixed-point and $\U$-startpoint framework for quasi-uniform spaces. We reformulate the original existence framework through the filter axioms and supply a two-sided contraction condition on a generating family of quasi-pseudometrics. For this condition we prove a Banach-style fixed-point theorem on $T_0$ bicomplete quasi-uniform spaces, with a Picard-type error bound; a Boyd--Wong companion and a stability estimate follow. The conjugate quasi-uniformity yields an endpoint version, a selector argument extends the result to multivalued maps and to common startpoints of commuting families, a Knaster--Tarski variant covers monotone selectors on order-complete spaces, and a direct quasi-Hausdorff route handles weakly contractive multivalued maps. Moreover, the $T_0$-quotient removes the $T_0$ assumption from the corresponding involution-based fixed-point theorem.