Topological degree and approximation of solutions for nonregular problems of mechanics: Oscillations of satellites on elliptic orbits

We construct the method of approximate solution for the differential equation of oscillations of a satellite on elliptic orbits subject to the gravity torque and the light pressure torque. Parabolic orbits are included as a limiting case. The metric of a weighted Sobolev space is used as a measure of the vicinity. This allows us to construct a uniform approximation of solution with respect to the eccentricity of the orbit.
To prove such an approximation, we use the Leray–Schauder degree theory and the Krasnosel'skij theorem of Galerkin approximations for compact vector fields adapted to the problem under consideration. To establish the uniform estimate of the convergence of approximate solutions to the solution, we also use a modification of an appropriate Krasnosel'skij theorem.