Study of the boundary value problem for a differential inclusion

The boundary value problem with respect to an absolutely continuous function $x:[a,b]\to \mathbb{R}^n$ for the differential inclusion
$$\begin{equation*}\label{kr} F(t,x,\dot{x},\dot{x}) \ni 0, \quad t \in [a,b], \end{equation*}$$
with the condition $\alpha x(a) +\beta x(b)=\widetilde{\gamma}$ and additional restriction on the derivative of the desired function $(\mathcal{L}x)(t)\doteq \dot{x }(t) - \lambda x(t) \in B(t),$ $t \in [a,b]$ is under discussion. It is assumed that the boundary value problem with the same conditions for the linear differential equation $\mathcal{L}x =y$ is uniquely solvable for any summable function $y.$ Using Green's function of this «auxiliary» linear boundary value problem, the original problem is reduced to an equivalent integral inclusion with respect to the summable function $\dot{x}.$ To the inclusion obtained, the results on operator inclusion with an orderly covering multivalued mapping are applied.
\noindent In the first section of the work, the information about multivalued mappings of partially ordered spaces used in this study is given.
\noindent In the main section of the work, conditions for the existence and estimates of solutions to the boundary value problem under investigation are obtained in the form of a statement similar to Chaplygin’s theorem on differential inequality. These results are illustrated by an example of studying a periodic boundary value problem for a differential equation which is not resolved with respect to the derivative.