Total Stability of Sets for Nonautonomous Differential Systems

Abstract

ABSTRACT. The principal purpose of this paper is to present sufficient conditions for total stability, or stability under constantly acting perturbations, of sets of a sufficiently general kind for nonautonomous ordinary differential equations. To do this, two Liapunov-like functions with specific properties are used. The obtained results include and considerably improve the classical results on total stability of isolated equilibrium points. Applications are presented to study the stability of nonautonomous Lurie-type nonlinear equations. The concept of stability is one of those which reaches beyond the general domain in mathematics. In his well-known dissertation Problème général de la stabilité du mouvement, Liapunov [1] has given several criteria for stability and asymptotic stability of solutions of systems of differential equations with the help of certain auxiliary scalar functions which are now commonly called Liapunov functions. The problem of stability and relationships between stabilities and Liapunov functions has been discussed by many authors since Liapunov.The classical Liapunov theorem on asymptotic stability of the zero solution for a differential equation x = f(t, x) uses a positive definite function V (t, x) whose time derivative V(t, x) along solutions of the differential equation has to be negative definite. However, this stronger property is not always possessed by the natural candidate for the Liapunov function, namely, the energy. The damped harmonic oscillator is the standard example.An effective tool to overcome the difficulty was developed by LaSalle, and is

Lyapunov-Type Inequality for Extremal Pucci’s Equations

“…In fact, efforts were made to obtain a Lyapunov-type inequality for PDEs by generalizing the corresponding result for ordinary differential equations in [6]. Cañada et al [6] considered the Neumann problem corresponding to (1)(2), that is,…”

“…a(x) dx ≥ 0 and (1-4) has nontrivial solutions to (1)(2)(3)(4). They defined the following quantity:…”

“…Further, Lee et al [24] considered the equation involving a more general operator than a p-Laplace operator in (1)(2)(3). In fact, they considered the following equation:…”

“…( [1][2][3][4][5] and studied the qualitative properties of β p . They obtained an explicit expression for β p as a function of p, b and c. There are a good number of applications of the Lyapunov inequality; see for instance [1-3, 5, 8, 17, 27] and the references cited therein.…”

“…and studied the qualitative properties of β p and obtained the explicit expression for β p as a function of p, b and c. There are a good number of applications of Lyapunov inequality, see for instance [1,3,4,8,10,17,25]. The result in [9] has been generalised in the context of PDE in [7].…”