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Additional info for (2,k)-Factor-Critical Graphs and Toughness

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T) = H(t, u (t) ) , 1 < q < „ . and II... I Then the sequence ( ( 0 , u ) } is weakly lower closed. 1) holds with H = 1 and (j, the modulus of continuity, suitably defined for large values of 33 5, If f is Lipschitz L. D. 1) holds with |a(6) = 6 , p = 1, equal to the Lipschitz constant. 2) a l s o holds in these c a s e s . In the linear plant quadratic criterion problem we have f(t, x, w) = A(t)x + B(t)w and f°(t,x,w)= + < x , P ( t ) w > + < w , R ( t ) w > , where A and B have entries in L [G] and P, Q, R have entries in L [G] .

4) o> x (x)u x (y) = Jx_ w(x,y,5)o^(§)d§, 0 < y < x . F r o m the e s t i m a t e s in s e c t i o n 2 we m a y deduce t h a t t r a n s l a t i o n is a bounded o p e r a t o r on c e r t a i n s p a c e s . is a positive m e a s u r a b l e function on R, If we denote by f ( R ) I the s p a c e of m e a s u r a b l e functions on x e R+ S i m i l a r l y Y (R 00 function. 6) holds and tt(x) = 1 if p>0, < p < 0 . (cf. , t h e o r e m 4). F r o m the e x t e n s i o n of t h e o r e m 4 m e n t i o n e d above we define T h e o r e m 5.

J In Theorem 4 . 3 we use property (Q*) to show that for a. e. t in G, 37 J L. D. BERKOVITZ (X (t), y(t) )' € Q+(t, z(t)). 1 and 4. 2 we can define a sequence co. and a sequence a. such that for w . ( t ) - X . (t) - f . (t) ) case also, function e Q+(t, z(t) ) . (X (t), y(t) ) e v: G •+ R m ° J Q (t, z ( t ) ) . such that o X (t) ^ f (t, z(t), v(t) ) . It then follows that in this Thus there is a y(t) = f(t, z(t), v(t) ) and We then use an extension of Filippov's lemma to show that we can replace v by a measurable function u.