On covering mappings in generalized metric spaces in studying implicit differential equations

Abstract

Let on a set $X\neq \emptyset$ a metric $\rho :X\times X \to [0,\infty]$ be defined, while on $Y\neq\emptyset$ a distance $d :Y\times Y \to [0,\infty],$ be given, which satisfies only the identity axiom. We define the notion of covering and of Lipschitz property for the mappings $X\to Y$. We formulate conditions ensuring the existence of solutions $x\in X$ to equations of form $F(x,x)=y,$ $y \in Y,$ with a mapping $F:X\times X \to Y,$ being covering in one variable and Lipschitz in the other. These conditions are employed for studying the solvability of a functional equation with a deviation variable and of a Cauchy problem for an implicit differential equation. In order to do this, on the space $S$ of Lebesgue measurable functions $z:[0,1]\to \mathbb{R}$ we define the distance \begin{equation*} d (z_1,z_2)=\mathrm{vrai}\sup_{t\in[0,1]}\theta(z_1(t),z_2(t)),\qquad z_1,z_2\in S, \end{equation*} where each continuous function $\theta:\mathbb{R}\times \mathbb{R} \to [0,\infty) $ satisfies $\theta(z_1,z_2)=0$ if and only if $z_1=z_2.$