Local Linearization - Nonlinear Oligopolies: Stability and Bifurcations - page 281

Consequently for all t 1,

kx .t/ xk q t kx .0/ xk (B.4)

showing that x .t/ !x as t !1 .

A slight modification of the above proof can be used to show the following
sufficient condition for global asymptotic stability of the equilibrium.

Theorem B.2. Assume that D is convex, and g is continuously differentiable on D .

If with some matrix norm, kJ . x / k q<1 for all x 2 D ,where q is a constant,

then x is globally asymptotically stable.

The conditions of this theorem can be relaxed to cases when g is a continuous and
piece-wise differentiable function. Assume now that D is convex and is the union

.1/ , .2/ ;:::, with mutually exclusive interiors. The restriction

of the closed sets D D

.k/

is denoted by g .k/

of g to the region D and we assume that it can be extended to

.k/ and that it is differentiable there. Let J .k/ . x / denote the

an open set containing D

Jacobian of g .k/

.k/ ,that k J .k/ . x / k q<1where
k : k is a matrix norm that is compatible with some vector norm and q is a scaler.
Assume in addition that for the linear segment between

and assume, for all k and x 2 D

x and any x 2 D there are

finitely many values 1 0 D t 0 <t 1 < <t K. x /

D 1 such that for each entire

subsegment,

.k l /

Πx C t l . x x /; x C t lC1 . x x / D

with some k l .

Theorem B.3. Under the above conditions x is globally asymptotically stable

in D .

Proof. Notice that with any x 2 D ,

. g . x C t lC1 . x x // g . x C t l . x x ///

K. x / 1

X

k g . x / x kD k g . x / g . x / kD

l D 0

Z t l C 1

J .k l / . x C t. x x //. x x /dt

K. x / 1

X

t l

lD0

Z t l C 1 K. x / 1

X

k J .k l / .x C t. x x // k : k x x k dt

t l

lD0

K. x / 1

X

q k x x k .t lC1 t l / D q k x x k ;

lD0

and then the proof can follow the lines of the proof of Theorem B.1.

1 K.x/ is the number of subregions that a linear segment goes through between x and the
equilibrium.