Note on a differential equation describing the periodic motion of a satellite in its elliptical orbit.

*(English)*Zbl 0669.70028Consider 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.

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.

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\textit{Dang Dinh Hai}, Nonlinear Anal., Theory Methods Appl. 12, No. 12, 1337--1338 (1988; Zbl 0669.70028)

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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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