In this paper a general asymptotic stability theory for ordinary differential systems is developed. We consider a fairly general system of the form (1) (∇pG(u,u'))' −∇uG(u,u')+qm(t)f(u)=Q(t,u,u′), t∈[T,∞), where u is an unknown vector function, G=G(u,p) is strictly convex and homogeneous of degree m>1 in the variable p∈RN, f=∇uF is a continuous function of gradient type satisfying (f(u),u)>0 for u≠0, and Q is a vector function satisfying a certain growth condition. We present some sufficient conditions for solutions of (1) to be asymptotically stable. Consider the equation (2) u′′+h(t)u′+q2(t)f(u)=0, t∈[T,∞), where f,h are continuous functions with f satisfying the restoring condition (3) uf(u)>0 for u≠0. The main theorems in particular imply the following that under general Lyapunov conditions, every bounded solution of (2) tends to 0 as t→∞. If f+,f−∉L1(R), then the conclusion holds for all solutions of (2). This fact generalizes results of R. J. Ballieu and K. Peiffer [J. Math. Anal. Appl. 65 (1978), 321-332]. Several other applications are also given. We include applications to holonomic mechanical systems with N degrees of freedom and strongly time-dependent restoring potentials, to generalized Matukuma equations, to the canonical equation (2), to the radial Matukuma equation and to related problems for partial differential equations.

Asymptotic stability for ordinary differential systems with time dependent restoring potentials

PUCCI, Patrizia;
1995

Abstract

In this paper a general asymptotic stability theory for ordinary differential systems is developed. We consider a fairly general system of the form (1) (∇pG(u,u'))' −∇uG(u,u')+qm(t)f(u)=Q(t,u,u′), t∈[T,∞), where u is an unknown vector function, G=G(u,p) is strictly convex and homogeneous of degree m>1 in the variable p∈RN, f=∇uF is a continuous function of gradient type satisfying (f(u),u)>0 for u≠0, and Q is a vector function satisfying a certain growth condition. We present some sufficient conditions for solutions of (1) to be asymptotically stable. Consider the equation (2) u′′+h(t)u′+q2(t)f(u)=0, t∈[T,∞), where f,h are continuous functions with f satisfying the restoring condition (3) uf(u)>0 for u≠0. The main theorems in particular imply the following that under general Lyapunov conditions, every bounded solution of (2) tends to 0 as t→∞. If f+,f−∉L1(R), then the conclusion holds for all solutions of (2). This fact generalizes results of R. J. Ballieu and K. Peiffer [J. Math. Anal. Appl. 65 (1978), 321-332]. Several other applications are also given. We include applications to holonomic mechanical systems with N degrees of freedom and strongly time-dependent restoring potentials, to generalized Matukuma equations, to the canonical equation (2), to the radial Matukuma equation and to related problems for partial differential equations.
1995
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11391/117141
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