Structural Stability, the Theory of Catastrophes and by P. Hilton

By P. Hilton

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1) be s a t i s f i e d . in J. e. e. means "almost everywhere". PROOF. 1. 7) 2 = -Cl (l+r)-2'4(Kv) + ( 7 - Cl (I+r)-26) + l~Ik2cl(l+r)-2~Ivi 2x - ({ci(I+r)-26C0 z -Cl(l+r)-26(Kv ) + J l ( r ) + J2(r). (Bv'v)x + Co}v'V)xl 31 Jl(r) ~ 0 for sufficiently for sufficiently large r . C~ (r)Ir = O(r -2~) (r-~o) too. 4) is v a l i d f o r large r ~ RI. D. 3. 9) satisfy there e x i s t (Nv)(r) PROOF. Set w = edv. 1. 17) + m(k 2 4~r-21JZo9 r ) r l-~J + {mk 2 _ 4rl-31J(Zo9 r ) 2 } r I-~L => k2/2 with s u f f i c i e n t l y large (r__>R4,m>__l) R4>R 3 , R4 can be taken independent of m~l .

4. D. 5, we have to prepare two lemmas. LEMMA 4 . 1 . 25) set in (~+. :_ 1 + Ilfll S, holds f o r any r a d i a t i v e k2 > 0 and Proof. function v for {L,k,2[f]} Proposition all by r c I. k = k I + ik 2 ~ K, f e L2,~(I,X). 7, there e x i s t s a unique r a d i a t i v e such that with v , v ' • L 2 , ~ ( I , X ). 3 and v Moreover, satisfies imaginary part. 26) R to L 2 , ~ ( I , X ). v • D(1) rq H ~ ' B ( I , X ) I o c M u l t i p l y the both sides of (l+r) 2-23, fc - (l+T)2-251m(v'(T),v(T)) T - 2klk 2 ~ ( l + r ) 2 - 2 6 1 v l ~ d r + (l+R)2-2~Im(v'(R),v(R))x T = Im / ( l + r ) 2 - 2 d ( f , v ) x d r .

32) f ~ L2(I,X~o c . Iv'(r)- holds for a l l Then 2 2 + k2v(r)l~ + kllV(r)[x ik v ( r ) [ # = i v ' ( r ) + (~>~Co(I'X)) 2 4k~k211vllo, (O,r) + 2kllm(f'V)o,(O,r) r c I, where k I = Rek and k 2 = Imk PROOF. 33) as n ~ =~ , where {~n} c C~(R N) { '#n ~ v = U-Iv in H2(]RN) loc" I (T-k2)~n ~ f = U - I f in L2(RN)Ioc T = -4 + ~(y). 34) -~ v ' ( r ) (L-k2)# n -~ f in L 2 ( I , X ) I o c 37 as n > ~, from Set 0 to r fn = (L-k2)%n ' integrate and make use of p a r t i a l ((L-k 2) ~i~n, (1)n) x = (fn %n)x integration.

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