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Note on a differential equation describing the periodic motion of a satellite in its elliptical orbit. (English) Zbl 0669.70028
Consider the following periodic boundary value problem (1) $$x''(t)+e \cos t\cdot x''-2e \sin t\cdot x'+\alpha \sin x=4e \sin t;$$ (2) $$x(0)- x(2\pi)=0=x'(0)-x'(2\pi)$$, where e and $$\alpha$$ are real numbers. For $$0<e<1$$ and $$| \alpha | \leq 3$$, (1)-(2) describes the periodic motion of a satellite in the plane of its elliptic orbit. The problem was recently considered by (*) W. V. Petryshyn and Z. S. Yu [ibid. 9, 969-975 (1985; Zbl 0581.70024)] where the aim was to determine the domains of e and $$\alpha$$ for which (1)-(2) has a solution (cf. also the references in (*)). We found that in (*), the conditions imposed on e and $$\alpha$$ are too restrictive. The purpose of this note is to relax these restrictions.
In fact, we have the following Theorem: Let $$| e| <1$$ and let $$\alpha$$ be any real number. Then (1)-(2) has at least one solution. Before proving the theorem, we observe that in (*), the authors proved, using the theory of degree for A-proper mappings, that (1)-(2) has a solution under the conditions $$0\leq e<2/\pi | \alpha |$$ and (8$$\sqrt{2}+3)e+2| \alpha | <1.$$
For our proof, we use variational methods. In fact, we prove that (for $$| e| <1)$$ problem (1)-(2) admits a solution that is a minimum of the functional on a closed ball with radius depending on e and $$\alpha$$, in a suitable function space.

##### MSC:
 70M20 Orbital mechanics 37-XX Dynamical systems and ergodic theory
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##### References:
 [1] Beletskii, V.V., On the oscillation of a satellite, Iskusst sputn. zemli, 3, 1-3, (1959), (In Russian.) [2] Petryshyn, W.V.; Yu, Z.S., On the solvability of an equation describing the periodic motions of a satellite in its elliptic orbit, Nonlinear analysis, 9, 969-975, (1985) · Zbl 0581.70024
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