A fixed point theorem in gauge spaces and applications to Ulam-Hyers-Rassias-stability of delay differential equations

First, we prove a new alternative fixed point theorem in generalized gauge (or generalized uniformizable) spaces. This is a generalization of a famous result of Diaz-Margoli. Next, using this theorem, we show the stability of a delay differential equation where the unknown mapping takes its values in a locally convex space (in particular into a Banach space). Examples are given to support our results. We also point out some gaps in the literature and fix them.