Stable odd solutions of some periodic equations modeling satellite motion. (English) Zbl 1034.34051

This paper is concerned with the existence and stability of solutions of the periodic boundary value problem \[ (m(t)x')'+f(t,x)=0,\quad x(0)=x(T), x'(0)=x'(T) , \tag{1} \] where \(f\in C^{0, 4}(\mathbb{R}/TZ\times \mathbb{R}, \mathbb{R})\), \(m\in C(\mathbb{R}/TZ, \mathbb{R}^+)\) satisfy the symmetry condition \[ f(-t, x)=-f(t,x), \qquad m(t)=m(-t). \] By using a result due to the first author [Nonlinear Anal., Theory Methods Appl. 51, 1207–1222 (2002; Zbl 1043.34044)], the authors establish a new stability theorem for equations of the form \[ (m(t)x')'+a(t)x+b(t)x^2+c(t)x^3+R(t,x)=0. \tag{2} \] On the basis of this new theorem, they further prove that equation (1) has an odd solution which is of twist type by using the method of lower and upper solutions. Finally, the parameters’ region of Lyapunov stability for the satellite equation \[ (1+e\cos (t))x''-2e\sin (t)x'+\lambda \sin x=4e\sin(t) \] is studied. For related works, one can see W. V. Petryshyn and Z. S. Yu [ Nonlinear Anal., Theory Methods Appl. 9, 969–975 (1985; Zbl 0581.70024)].


34C25 Periodic solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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