A stability condition for nth order difference equations

Abstract

Consider the system of difference equationsin which the unknown x(t) is a complex m-vector, t is a real variable and a1, …, an are complex m × m matrices whose elements are functions of t, x(t), x(t+1), …, x(t+n – 1). A positive definite hermitian form V(x1x2, …, xn), with constant coefficients, is called a strong autonomous quadratic Lyapunov function (written strong AQLF) of (1) if there exists a constant K > 1 such that K2v(t+1) < v(t) for all non-zero solutions x(t)of (1), where v(t) = V(x(t), x(t+ 1), …, x(t+n —1)). The existence of a strong AQLF is a sufficient condition for the trivial solution x =0 of (1) to be globally asymptotically stable. It is a necessary condition only in the special case of an equation

Relevance of osteoporosis in women with fracture of the femoral neck.

“…"3 The lower limit of normal for the hand score is 440%. *4 16 The mean hand scores were plotted on a nomogram relating hand score to age, derived from 317 healthy Glaswegian women without biochemical evidence of metabolic bone disease, who had not sustained fractures of the femoral neck. '6 17 The normal range for the six metacarpal hand score was established from measurements on 64 endocrinologically and biochemically normal premenopausal women aged 19-39 who were attending hospital with menstrual problems.…”