Scientific articles
One method for investigating the solvability of boundary value problems for an implicit differential equation
W. Merchelaab a St. Petersburg University
b Derzhavin Tambov State University
Abstract:
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.
Keywords:
implicit differential equation, linear boundary conditions, existence of solutions to a boundary value problem, covering mapping of metric spaces.
UDC:
517.988.6,
517.922
MSC: 34A09,
34B15,
47J05,
47N20 Received: 29.09.2021
DOI:
10.20310/2686-9667-2021-26-136-404-413
Fulltext:
PDF file (602 kB)