By Stephen R. Bernfeld

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We s h a l l f i r s t show t h a t at the end points of J the desired inequality holds. and hence that > B(a) which implies from ( i v ) t h a t a2 # 0 a' (a) > p ' ( a ) . Then t h e r e e x i s t s a 6 > 0 such > p ( t ) on [a,6). (a) Let a(t) monotony of and condition ( i i i ) , we obtain f D - 0' - D-p' (t) (t) - _> L [ a ' ( t ) B ' ( t ) ] , which, by t h e theory of d i f f e r e n t i a l inequalities, y i e l d s a' ( t ) - B ' ( t ) 2 [a'( a ) - B ' ( a ) le L(t-a) This, together with a ' ( t ) U'(t) p'(t) on [a,6].

Show condition. 2. 1) such t h a t _< x ( t ) 5 p ( t ) on J . Then, by (i)and ( i i i ) it follows that ' ~ ( a , x ( a )5 ) X I(a) Assume now t h a t there exists a 5 Jr(a,x(a)). to E (a,b) such t h a t < ' p ( t o J x ( t o ) ) . 4. NACUMO'S CONDITION and f u r t h e r t h a t This i s a c o n t r a d i c t i o n because of assumption ( i i ) . f o r e conclude t h a t merit shows t h a t x'(t) x'(t) -> ' p ( t , x ( t ) ) -< J r ( t , x ( t ) ) on on J We there- A s i m i l a r argu- J. and t h e proof i s complete.

Lim sup 1 [V(t + h , x + h x ' , x ' + h f ( t , x , x ' ) ) h+O+ h We s h a l l often use s e v e r a l f'unctions ViJ i = 1,2,. ,n, which we c a l l I(fapunov-like f'unctions and f o r n o t a t i o n a l consistency we write Vi of + D V. 1. 1; t h e r e e x i s t two Wapunov functions wi x [x: x ,< a ( t ) ] xR, D2 = J i = 1 J 2 J . where B(t)]xR, f(t,X,X'). D1 = J E C[D~,R], x [x: x > W ( t , x , x ' ) = 0 i f x = a ( t ) , Wl(t,x,xl)>O 1 x < a ( t ) , W2(t,x,xt) = 0 i f x = B(t), W2(t,x,x1) > O i f if such t h a t x > B(t); (iii) for i = 1,2, g locally Lipschitzian i n + D WP(t,x,xt) + D W2F(t,x,xt) i E + C[JxR , R ] , Wi(t,x,xl) is ( x , x t ) and _> gl(t,Wl(t,x,xl)) g2(t,W2(t,x,x1)) 39 i n t h e i n t e r i o r of D1j i n t h e i n t e r i o r of D2; 1.

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