On the existence and uniqueness of a positive solution to a boundary value problem for a nonlinear ordinary differential equation of $4n$ order

The paper considers a two-point boundary value problem with homogeneous boundary conditions for a single nonlinear ordinary differential equation of order $4n$. Using the well-known Krasnoselsky theorem on the expansion (compression) of a cone, sufficient conditions for the existence of a positive solution to the problem under consideration are obtained. To prove the uniqueness of a positive solution, the principle of compressed operators was invoked. In conclusion, an example is given that illustrates the fulfillment of the obtained sufficient conditions for the unique solvability of the problem under study.