By A.M. Fink

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4 Let Q O(O; be as in the preced,,ing g € d(9,Q,0) i,s lemma and let R") be od,d, and, non-uani,sh,ing on 0e. Then euetl,- be a ball of (sufficiently small) Since 0 e fì, let B(0; ") theoretn, let tlt € C(0; R" ) Tietze's By fl. in contained raclitrs such that Prooft r€o{t ,h@)_ { P(r)'' I r, r€B(0; ") Then ,r/ cloes not vanish for l"l - r and so by Proposition d(P, 0, lemnra) we carì firr

This is useful in getting infornration on the solution set as illustrated by the following example. aCl firnr:tionsur;hthat lg@)l S 1 arrd g'(n) ) 0 for trll rr: € R. Assrunc t,hilt p(1) :0. r1 - 0 Ì ,trtt- 3t:'r12rgQ:) has atleast three solutions in the ball B(0; fo, 1l -+ IR' by ,. (3'5'4) 2). To see this, define H :B(o;2) x H((:r,a),t) - (*t - J*u' + t9@). /3 * Jr2r). We first verify that this does not assurne the value (1,,0) on the circle 12 +U2 :4. Indeed, if H( (*,y),t) : (1,0), then either U : 0 or ll I 0 and U2 : 3r2.

2 Let f ,g € c(a; IR3) . su,ch b). , 1) : glan ' erists h and Then d(f,Q,b) - d(g,Q,b). ,t) : i. Then, by hornotopy [0, 1];R'). ,0) invariance of the degree, dU,A,ù : d(l,Q,b). L I The following two exercises problem to the regular case. 4 Let {oi}irt be a fanrily of pairwise disjoint open sets in IR' whose union is contained in a bounded open set fl. Let f e C(O; R') and b such that /-1(b) c Ujetej. 5 (Product Fornlrla) For i, : 1,2' krt gi € C(q;R',') where Qi C K't is a bounded open set. t, bi f po(Af),).