LEM M A 4.1. Defining A r = {(*, y ) E R 2n \ \ x ~ y \ < r}_then there exist 0 < r0 < 1 and for each e > 0 a continuous function w€: [0;7] x Aro -> [0; + °°) which is Lipschitz continuous and differentiable at each point o f [0;7] x Aro and satisfies

w«(t, x , y ) + H(x, w^(t , X , y)) - H (y , - w ey(t, x, y)) > 0 for (t, x, y) E [0 ;n x A,#,

ivs(i, x, x) < e for x E R n ; wf(x, y, r) > ^ for (t , x, y) E [0;7] X 2Aro

and

lim inf Iw^O, x, y); I x - y I a: r} = + for 0 < r < r0.