By J. M. Boardman, R. M. Vogt

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D~ o R --n Here ~"~ = ~i~ I + Let ... + ~n~n . f] ~ ~ ~ f(~) ~ etc. - v. 0

By Lemma 3a L(t-llxl)L(t-l) -I -~ i as t -+ ~ . Therefore t lira gm~) -- gin(x) 47 Theorem 3d. for all i. x E R Given and all ~n ¢ > 0 there exists given ¢ > 0 such that t > I ~ ( x ) <__A(e)Itl ~ + ~ , ii. , p . Note: Proof. We have ~) for _x ~ Rtm " = t~L(t-l)-If(t-lx+~_m)_ is bounded we clearly have i. , _x E Rtm " Since f By assumption eL-~mI)l~_'~-ml ~) f (~--) = ~m (['~-m)L (I~_-~__mI )l ~-~_mI~ + o (Ll as ~ ~ ~_m " It follows from this that we can find a constant a neighborhood, Nm, of ~-m ' Nm = {~:]~'~--mI < ~} 0 _< f(~) _< ~<1~-~t)1~-~t ~' for ~ E N Thus if _x E a(tjm)N m c Rtm we have 0 ~ f(='l~+~m) ~ ML(t-IIml)t-~Iml~ Therefore t 0 _< gin(x) < ML(t'l)'IL(t-llx l)Ixl ~ Applying Lemma 3b we get such that M and 48 t 0 _< gin(x) <_ M'M(,)[J~[ e for x E ~ ( t , m ) Nm .

P} > 0 We are given a positive constant function L(t)t -~ ~) defined L(t) We have p and a positive con- 0 < t < ~ w h i c h is slowly is slowly oscillating at ~(e) > 0 is decreasing for for w such that 0 < t < ~(e) L(t)t ¢ 0 if for every is increasing and • non-negative m e a s u r a b l e rO if some trans- 6 > 0 . iii. > 0 its m i n i m u m [~-~m ] ~8 inf {f^(~): tinuous the closure of is star-shaped. _ for each R is star--11 is star-shaped (the closure of We have a real function ~m in is star-shaped with respect to the origin.