One method for investigating the solvability of boundary value problems for an implicit differential equation

The article concernes a boundary value problem with linear boundary conditions of general form for the scalar differential equation
$$\begin{equation*} f \big(t, x (t), \dot{x} (t) \big)= \widehat{y}(t), \end{equation*}$$
not resolved with respect to the derivative $\dot{x}$ of the required function. It is assumed that the function $f$ satisfies the Caratheodory conditions, and the function $\widehat{y}$ is measurable. The method proposed for studying such a boundary value problem is based on the results about operator equation with a mapping acting from a metric space to a set with distance (this distance satisfies only one axiom of a metric: it is equal to zero if and only if the elements coincide). In terms of the covering set of the function $f(t, x_1, \cdot): \mathbb{R} \to \mathbb{R}$ and the Lipschitz set of the function $f (t,\cdot,x_2): \mathbb{R} \to \mathbb{R}$, conditions for the existence of solutions and their stability to perturbations of the function $f$ generating the differential equation, as well as to perturbations of the right-hand sides of the boundary value problem: the function $\widehat{y}$ and the value of the boundary condition, are obtained.