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By William Hume-Rothery

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F i r s t discovered by Weinstein [9; Such r e s u l t s were 10; 151 i n a c l a s s i c a l s e t t i n g and subsequently g e n e r a l i z e d and improved i n c e r t a i n d i r e c t i o n s by C a r r o l l [l;2; 51 i n a d i s t r i b u t i o n framework; we w i l l f o l l o w 35 SINGULAR AND DEGENERATE CAUCHY PROBLEMS (Other k i n d s o f growth and c o n v e x i t y t h e l a t t e r approach. theorems f o r s i n g u l a r Cauchy problems were developed by C a r r o l l [18; 19; 251 and w i l l be t r e a t e d l a t e r .

13) and a n a l y t i c i n A. The statements f o r Zt(A,t,-r) i n t ( c f . 5). Sim f o l l o w upon d i f f e r e n - We n o t e a l s o t h a t i f QED S e c t i o n 3. 30), a(t)Zt(A,t,o) 1 and represents a continuous f u n c t i o n a n a l y t i c i n A , which s a t i s f i e s (5. 3). (A,t,T), Zt(A,t,-c) E 1 (-l)kJk k=O t i b l x the series as t -+ 0. we must examine t h e behavior o f Thus ( c f . 31)) 49 we s e t SINGULAR AND DEGENERATE CAUCHY PROBLEMS a ( t ) jt(P(<)/P(t))df T now t h a t a [o,sl, and C (o,b] E H(t,T) = 1 ' 2 = a /a E 0 then c l e a r l y ( c f .

QED A A with f E Co(E ) where t A (Ev) w h i l e i n i t i a l c o n d i t i o n s ' @(*,T,T) J A -+ A = T(*) E A E and Q t ( = , ~ , ~ =) 0 are stipulated. 20). 20) legitimate f o r T > 0 and one 0 i n t h e r e s u l t i n g formulas ( c f . 21) with a r e continuous l A To work @(Y,~,T) continuous). 6, A 4 i n E we r e c a l l t h a t (S,T) + ST : x E E i s separately Y Y Y continuous w i t h E a Frechet space and hence t h i s map i s conY tinuous (see Bourbaki [2] and f o r v e c t o r valued i n t e g r a t i o n see Bourbaki [3; 41, C a r r o l l [14]).

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