Stable odd solutions of some periodic equations modeling satellite motion.
type: Item
instance of: scholarly article
title: Stable odd solutions of some periodic equations modeling satellite motion.
zbMATH Open document ID: 1034.34051
DOI: 10.1016/S0022-247X(03)00057-X
author: Pedro J. Torres
author: Daniel E. Nuñez
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-05-25T00:00:00Z
review text: 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 \textit{W. V. Petryshyn} and \textit{Z. S. Yu} [ Nonlinear Anal., Theory Methods Appl. 9, 969--975 (1985; Zbl 0581.70024)].
reviewed by: Zaihong Wang
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C15
zbMATH DE Number: 1915780
zbMATH Keywords: Stable odd solutions
zbMATH Keywords: satellite motion
zbMATH Keywords: existence
zbMATH Keywords: stability
zbMATH Keywords: periodic boundary value problem
zbMATH Keywords: Lyapunov stability
zbMATH Keywords: satellite equation
MaRDI profile type: Publication
cites work: Q5554207
cites work: Chaos in nonlinear planar oscillation of a satellite in an elliptical orbit under the influence of third body torque
cites work: Q5512842
cites work: The index of Lyapunov stable fixed points in two dimensions
cites work: Q4347835
cites work: Note on a differential equation describing the periodic motion of a satellite in its elliptical orbit
cites work: Multiple solutions for a nonlinear second order differential equation
cites work: On periodic solutions of a certain nonlinear equation
cites work: Q3294369
cites work: The method of lower and upper solutions and the stability of periodic oscillations
cites work: Periodic solutions of twist type of an Earth satellite equation.
cites work: The twist coefficient of periodic solutions of a time-dependent Newton's equation
cites work: Periodic solutions of a Newtonian equation: Stability by the third approximation
cites work: The number of stable periodic solutions of time-dependent Hamiltonian systems with one degree of freedom
cites work: On the solvability of an equation describing the periodic motions of a satellite in its elliptic orbit
cites work: Periodic solutions of nonlinear second-order equations which are not solved for the highest derivative
cites work: Q4070368
cites work: Qualitative analysis of periodic oscillations of an earth satellite with magnetic attitude stabilization.
cites work: Stability of planar oscillations of a satellite in an elliptic orbit
full work available at URL: s0022-247x(03)00057-x
OpenAlex ID: W2096994853
description: scientific article; zbMATH DE number 1915780