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 $z_1=z_2.$
Keywords:covering mapping, metric space, functional equation with a deviating variable, ordinary differential equation, existence of solution.
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