«JlllllOO A l\loJhcma ti col .l o ur,1t1l \ 'ol 9, ,\' :J. '5). 1'rrrmbtT 1!007, Co n c r ete a lge br a ic co h omo log for t h e gro u p (IR,+) o r h ow to o lv t h funct io n a l cq uat io n f (.r + y) - f (x) - f (y ) = g(x, y ) Mi l iai P n111 · u ll o 1m'<'kt•r Sof!wm't'l' nt wiC'kluug, F'rl'ibur~. Gnm1rny lt Mlllllt' of ~latllt'lllll! lc's o f t lw n o mn11i nn ¡.\ cndrmy, Dudum .. "Sl, no 111 011ia . ~ lihni . Prn1 1l'~t ll q.111 01 h.uni·frl'ibtirR.dt.• ADST llACT llw funr11uunl l"< 1uulio u f(.1· +y) J (.r) - /(JI) 9(.r.lf) hM n lio lut.iou f th.11 IM"lonlh 10 Cº(R) if 11 11d o ul y if 1 ht' sy nunt'lnr roc\·ch- 9 IX'kmg.-t to Cº(R'). lf tlw ~\muwtlic rory rl(• y iH ll'<·mi;ilvdy n1>prc-)(1111nblt•. thr«" t"XL""t 11 so l111l ou f v.h1~h L .. 11..:-t1,...1H•ly 11p p1u xh11111Jl e 11 lso. Ir 9 bdon1ts lO C 1\RJ) then t l 1('1~ cxl~L~ tu• 111h'1't11tl t·xpu'"S.'lio u in !I for 11 so luti ou f tlmt bdon~ to C 1(R), oud th1: s11 m t• h.1P1wru for tlw cln.,...._~ ('~, ('"", 111111l y t1 c 11nd pO))IU'llllMI RE ' 11EN L\ t"("u.-.c1ón funrio nnl f (.~ 1 y) /(.r) /(JI) 9(.1:. JI) tJt·•H' unn so lu ció n f c¡u• p. rl<"llt'C"\' 11 C'°(R) t1l .V 11610 s i el codrlo Mllltlnco 9 ¡>1:rH'llC('l' n Cº(R2). S1 c-1 COI lo ""lm1•1 1 ito f1 ('!! np rox i11111hlt• rl.'l'Ul"lll\'i\11\l'lllt'. t'Xl!ill.' unn !>0l11ci ó 11 f 111 tll•"'I l.lmhll'n 1 nproxi nmbl l' n ·c urslvn rn l' ntl.'. S1 9 P<'rtl.'nl"n' ft ('1 (R 1 ), l' UI O IL Cl'S t'U"'"" unA 4'"\:Jltt""ión inl('~rn l l' ll g 1mrn 111111 sa luc10n f qut" pcrl<'n (' 11 C' 1(R) y lo> lllL'IUKt •Uc('(lt• ¡mm ln.i dm•l'S : ('4, ('....,, 11nulh1cn, 1><'hnom1al K t')' w rd ,. 11 11rl p h rn 11c11: 11/grl>mtc C'ohomoi091,f\iru:-1tono/ 8quot1011, J\11ol¡ir1c Prophc• \IMh . S ubj . ' l11 1u1.: 39805, 39821 IU i\ lil ml Prunr-<11 l In t r o l u ct i o 11 Tlw 1•xi. .. h.'UC'\' oí,, f1111 c li o 11 J(.1') su t b¡ fyiu~ 1lw ÍUUC'lionul lot1uo1\011: f(.•· 1 y ) J(r ) /(u) g(r. y) is ich..>ntical wi1h ! lu.· 2.1·o bo1 111 d11r y co 11d ition ror lht• funrtion g{r,y), fl.'i dcfin1.:d in t lw 1ll1itehrnic- rohom log,v o f nlw l i o1i g ro 11ps. Thi:; th rory ~i\'rs in gl'11ernl no1H·o ns t 1ucti w ¡>roof" for the exb tt •!lt'C o f 1 lt e so l 11t io 11 , nnd s111tfü.~ thC' oru.111cle~ fo r 1 lw funNionnl t'<¡muion to bt• :-0111\ik· (ii1 t..liL' so·cidl rd cohomolof!,i<' no n· trivinl C'n .. -:cs). i\ ly gon l h(·r~· i~ to study .rn11 lytk p r o pe r1Í t'S o f lilL' :so lu1i o n rrnd its L'Xf)n.._,_..._._¡bility i11 !ht~ r ~·ril rn .. ~· Thl' word ro11<:rtlC' tt sl'd i 11 Lli e lil,ll' C:fln he nlso undel'!i tood ns t. he co 111hi nntion of cont111uou .. • md d1.scmle. In ¡v'nC'ral, IC:'l /\" 11 11cl l ... \w 1.wo fl lw li nn group:s nnd le1 g: l .. x L - ¡,· ben fun rlion tr thNt' i-;. ,, fonctio 11 f : /., _, ¡,· ver ifyi 11g lhe íunc1 ionnl ec:¡u11tio11 for 1\!l .r, !J E /_, t ht1 n g(.r.y) tull:>l \"er i(v 1hc fo ll owi 11 µ, co ndit ions: g(.r. y) must bC' S.)'lt lll l('1,r ic, t,lrnt. is: g(:t ,y) =g(y,:r), g(.r.y) m11 s t bl· n 2-cocycll: nrcordiu g to the trivinl nct ion of L 1~ /\" , llmt is: 'J(.•·, y) 1 .'!("· + 11· :) = g(.•·, !I ... :) + 9(y, :). \\"l• ob:-.l"' rvc t lrnt. if fu : IJ - f ¡,· i:i n purtic11h1 r solulion o f the fun c1 iono l ~1quntion, then the se t of nll so l11t.io 11:; is {!0 +-ó1 ó E l lo m (/_,, /\' )} = /o+ 11 0 111 (/_,, /\' ). Thc> cocycle co mlit. io11 for y = O givc.s g(.r,O) = g(O . .:) = g(O,O). 11 c c11 11 11lw11ys sup1~ tlmt g(O,O) = O. l11 deed, if f is 11 soluti on o f tlw funct io na l r.>qm1tio n, th t•n /(O) - -g(O, O). 1 f y(O, O) i- O t,l1~n we re plncc g(.r, y) b~· g(:r. y) - !J(O, O). Tht• 1ww t''<(UOtion hn."' C'Xnct ly tlw so lu1.i ons f (.r) + g (O.O ), whcre / (.r) Me tlw M> lu tions for g(r.y). Tlw íollowi ng fn ct s 1n c provvd iu !=JI. pg. 2:n - 2J9_ Th res ult s go bnck 10 Eill~nhcrg ,uui ~ll1c l.Amt', . ..;vv [2]. lfg lx / .. -· h' isnsy 111 111 0l l'i l'cocyck·wi1hg(O.O) O th1111tht•!>t'IG: /\" x /_, with tlw o¡wrnt ion ( 11 .. r) o (v,y) :-=( u + ti+- g(.r,u).r + 11) i!\ an nb>Hn n g ro up ~uch dtllt tlw nbl.'lio n g rou ps /( 1 G' 1111d l.. forni n ~hort t'Xllct -..'quenn•: 1m:ordini;t to thl~ t'mht· ddi11 g 1: u /\' ..... (u.O) E G 1\ml to .Ju• projl'<"ti 11 ¡1: (11,.r) e-.. .r l. In 1h is s it u t1 ti o 11 OlW :mys tlmt ' i!; nn f"Xl' IL .. ÍOll of "' by l ... Two t'Xll'n'ion ..... G 1111d G' 1if f\' by /_, nrc called ~"(1ui,'í\l1•m 1( th• 11• is 011 iso morphis 111 of 1\bdi.m ~TOUl)S T.¡! : e . (,'' sur h tl \l\t 1' ~·1 nnrl ,,. ,,_ Lt:t llS dt'llOh' i; impl y lw /, l ilw trh•ill l l'X k11 s io11 of ¡,· h~· / .. , rom~pamlmit 10 1lw s.v um1l'lric coc,•dC' !J(I,JI) O. Th1· t'X h •ns io 11 Gis 1-'<1ui \o¡\l1•nt with K lf l 1f .u1d u nl y if tlwn• j¡.¡ nn l..'()llUUJ)hi-.m ' : C • /\' X /., Of tlll' forlll 1,•(u,.r) /(r}, r) if lilld o nl y lf WllO 11. ll~l>\Hl ( 2 1) •¡(l. 1)) "c¡11.lhtw,/u{1+ l) /n(1) fo( ! ) f/ (1, \ ) íor1 lton l o1w gds thc t'X p rrssio11 fnr n O. On tlw o tlwr htnid /(O) O nnd /( l ) /(l) g( 1, ! ) 9( 1, 1). ·-· Thls \'•lhl• L .. !h. -.ion. 1tnd 0 11 the ot her h1111d \\'I' kuow !h.,1 1lwr1· 1·xbt¡.; 11 soJ111 k.1 11. 1 E. nm1>ll' l...t ~I Z >< Z • Z bl' giwn by y{r.11) ry_ Tlds funcl ion is 11 !()'111111Nrir ('Ol:\'fh'. 1\ sol111io11 f: Z · Z b ~iwn hy /{n) " 1"l' IJ. Tht•:.t• lll't' lllt' l111ml(ulu nmnhn~. t'X lt•ml1•d owr tht• whok• Z. ~ow lt•t q R '< R • R Kivt•n 11¡.;11i 11 hr 11(r 1,1 ~- .\11 íu nrtio11s f: R • R ~in.•n by /(r) ; +-11• , ,.,,,, / R · R 1/ 111ul 011/y 1/ 1/1(' -'Jl"1r11ctnr COt"ydr g : R'l · R 18 n 12 ~ liluu Pr m ... u rv11lmuo J 1"• lrnrJ / ri 1111 .~ rww / or 11Jl Jo 1- O /U • rl tJu:: Jollo1rmq l .\ t n u ' /"t 11tl. 11 'R tllf~ r rnt~ 1J·1" ·t111 1w1' 1·1111 f 111r wu\ \Ulul1m1 J R • R , ,u-h tlwt /( .ru) Proor H f ¡..;ton; i nuo us, t ht· 11 nis() y . 11p1 1h, 1t g L" 1' co111in110 11 -. s y 11111 wtri1 l()( n J,,_ "•lhout tt'sl rk ti11 µ; ! lit• µ;1 · 1 u •mllt~· w1lh g(O.U) U. Cm1 .. 1rm1 tlw .. 110rl t•.x, 1t t ""1111'111 •'uf lopolo¡.i, kn l µ; 1oup.'i: · R • -R ·ll ll 1·n• ¡_.. (; R x R with tht· l'lwlldi 11 11 topo lQKy .u to ló~i C';\I gro111 >: tl 1t' i11 \'t·1" ' (u,.r) 1 • ( " !J(.r , .r ) , .r ) i:; n l ~o ll ro nt111 111> u-. •'Plllit,Hió n . Tl w (• m ht"t ldin R 1 ". ¡, . (u.O) E e 111111 tlu· pro jl't'li Oll fl: (11 . .r) t G - .r E l .. lllt' ho mnmor ph bm .. u f 1 opul~1l,tl p,ro u ps. ;\('1'0l"di11 ¡.i; 10 u fundum t• nt .11 th t'O rt• m o f ,\ hn·ko ff (:.t'I.' 1·1Jl 11 1opologir.1l jtruup i:-; iso 111 o rp l1 k wi1h son tt' t•ur lid in n group {R". +. O) if nml on lr if 11 i:- ,,bdi.\n. 1-1 ,m~do rfl", locn ll.Y rO IH]J flC' t , C'O ll llt'<"l t') .rZ. w hich i~ unlxm nd c'CI. lf .r O tlt t·11 7r 1(< (u,0) >) uZ w h k h is al-.o 1mho 11 11 d.-'IOJ!;ir.,1 ~roup~ i.p: (.' • IR.2 . lni m .;1{R) '' 1111('l'lo1 ·. /i11r . l ndt,'(I. ,;1(Qr) QJ,t11(.r ). [f t ht• dn~>cl sub~roup ,:1{R) C'Oll lllins R· lh w11r h• indt • 11':'mlrnl t•l1·1111·111i. 11 1 .111d !J'l , 1 ht• 11 i1 w~m ld ro nl a in 1 lw .,.., o f llll rn l io nnl rnmb iuo t ion !I !JI • Qy1 .lnd it i. t'l os u n., so i1 \\'\"l til d ht• llw w holt• R1. whic h is 11 r o ut rndit 1iu 11 . So .;1(R) L .. 1h1• to1w lo,11.k nl clos 11 n • of Q.,_"1( 1). whi r h is,\ n · \I \'i'<" l <'l r· lhw . 1 0 1w r.111 i.up pnH• t lt nt ip1(1R.) / {O} x R; if not , 'n' :-; ul~lltuh'.; wit h To{), wh t• n • T i!I ,\ '>llMll tot,1• ion C'o ns ick r 1lu• 11 p pl itatio n 15: R · R giwn hy J(.r ): p..; 1(0 , .r ). J ¡... '' honmmot)l his 111 o f !o po lo g k rd Al'Oups, so i s nddiliw .u u l ro nl inuous. T his 1111 •1111 s th.n ,¡ ¡., ,\ ro n1in11 ous :-;o l111i o 11 lo r 1hl' fun r tio n11 l l'<¡muio11 o f C'a ur hy 6(r +y) d(rl · J(y) owr R. ll1 •11< 't' 11t t' l'l' is 1m 11 R !'1 11 h thl\l d(r) a.r, ,111 d a ¡ O h1't' í\11 -.1• T" 1 {O} x R) "/. kl•111 p . \\',, ro1i..tn11 1 11 11 11 1ipli rn ti o 11 () : R 2 · Rl s at is Í\·ing tlll' foll owin J:; nindl l io n:; : O(;<(R)) R x {O} ; o.,,.¡ 1) (l.o) ; o ' 101 a (r- 1 r). u ThL"' '-" 1lo1w lw 1lw liiw111 nppl icn ti o n O s11d1 th nl 0(~1(1)) tJ O} .111d fJ( O, 1) (0 , 11 ). 11 L. ... 11\ 1somorphis 111 o f 1opoloj.1 ir11 I p; rnups. \,JI" fl.;. t' : 1/11 nud ¡/ : fJ\.'. 1 Th11 11 t'(u) (fl. 0 ) und 1l{ u, r ) z, m JMtHrul111 ,,,,, O. lt fo llows 111111 \.' ii. 11 11 1 morphi.."111 lw tw ('(' ll tl w r xn r t .. bnrt 11!11("(111• nct-s of l 11pulog if'n l p; rot tps (R,1, G.p. R) .111<1 R. 1'. Rl , p ', R) , so t h1•n • i!'I ll runtmuou .. fun1 1lo 11 f : R • R !l UC" h 1h 1l for 1l ll (u . ..r) E R 2 11 hold s \.1( 11 ,r) {u / .r) \ flO lrll11 ¡i. 10 lh l' tl's ult s c111oh'fl in 1 h~· hi11odu1 11011 , tlw <' 1'h l'O l'l' 111 2.2 Jf tl1(' ro 11tn111n11.'i s1111w1ftr'lc rocr¡clt' g R~ - R u rrc11r . .,11rcly o¡i- 1irnrir11t1hlr tJu11 1111 rr nn· ra11ltr11w11.~ MJlullori. / • R • R uluch ort nl'lo rY'('tir.vwcly fll'/lnliltllllbh Proor Ld / R • R ht• auy so\ 111 io11 of tlw f1111 ct1C"11MI c-qu1uion. As Wt' kuow, / fo + tS. wht•T\' / o Is 1\ c-o1H inuous iK>lutio u nnd 6 nu .ulchtin· hom()lnorphism of R. lt fo\lo,,-s tlMt f lo / o lo + o.r lo fo t wmt' fl E R Th b nw.ul.! t lutt / lo : Q · R is 11lw11~·14 c-ontmuous Ou 1 lu• o t ht'I' l111ml, l'Onlinuo us soluuon .... dt·fuwd ow•r Q or ow1· R llrt' unh11ll'ly dt1h •rmhll'd hy fl vn llll' in SOllll' Jo--/: o. ror t·:i...'lmplt• h y f( l ). l.A·I n Z lw 11 ndK .. ul>an.l UJl ()f R. onsidl'rlng tlh' :olmllnr íunc;tion,\I t-qmu ion co11t'spo 11d i11 ¡:. to ,,¡..,7 n1 tuul lh<' form for t lw solution gi\·L'll in tfl(' Rf'nMrk 1.2 writtt: n in inll'¡.tL'r 11111lu11h-.. oí o "'' ""' tlmt t l li!Sl' dist·11·ll' 1'0lu1ions M<' ill."O uniqu<'IY clt•tt·rmin('d by /(1) L1•111111n 2.3 ú t q R x R · R. /J1• a S!111w1(n) n m l / 't hy thc vnlm' / 1 (~l But / 1 1 .. z is n solu tiou for tlw ... \m• lll'Ohlt-•m 11s fn· l h.•Jll't', lhl· o n\_\' thintt lo do¡_ .. lO chOO~t· /11(;) sud1 llml f i (n) /,... (n ). Oy so lvlng t lw l''t ll íll io n f 0 (0) 2/ '1 ( ~) g(Cl, ~) o m · g t'I s 1 he \'1\l1w /~( ~) / n_(_o) _ _!l_(_o, ~-) So. wh.u '"" IMw 10 d o, Í!'I 'º conslruC't 1lu.• ~'Qllt'lll"t.' or di..<;,('H'll' Í111 1{' tioml /1 , /~ . ft· • J. , wl1 h th(• p ro pc•rty thnt n ll /r}n 1 .z. f.;.,· T lwy iHt' nll n·~1rk1 io1l" of tlw ro1ttin110111! solu1ion /: R · R dt•tt•rmim t b\' /( 1) / 1( 1), w hirh 1111.11 to lw AJ,;.. n ,\ ft'('\lrSin• n·nl. Tlw 1111io n o r nll tht"'4' domoiml f\rt' llll' d ynclic 1111111lwr 111·h1dt .u1• dl'n~ i11 R. 11t1cl tlw u11ion of nll grnph_.. i.-. dt·nM' 111 tl 1t• gropl1 o f / . Sn fe m b· u'CUt!->iwl~· 11pp1oximntt·d. 1 3 las C 1 a n d m o r e Lm u m n 3.1 !Al g Rl • R /ir o .~ymmrtr1r roryc/( o/ ria.~~ C 1 T/1rr1 //ir /oflowm9 11frnM1n he d 11 ~ lll1111 Pnu.- ll • I g{r O) g(U, .) •¡(O, ll) . e. (1J1g)(u.1•} {lJJ.fJ){P,11) . J. (IJ,g)(r.y) (ll,11)(.•· 1 y, O) (i),11)(y.O} (llogl r.y) (IJ,y)(O,.•· 1 11) (81y){O.r). Proor Poi111 1 lms ht'~'11 w ow d i11 l h<' introdun ion Point 2 fol\ow~ by ~yn111w1n· Point 1 follow~ fro 111 2 11 11d :1. To prov1..• :}. con:-id<'r th(' following rcfo1mul111io11s for 1 lu.' l'OCyrh.'-1o:ic'l111 1 fo r .::: / O: y(.•·, /1 1 z ) y(.•·, y) g(r +y.:) g(y. :) '!_(r-J1_+•l _y_é•·, 111 y(.•· i·y.:)-g{r+y,O) g{y,zl Y.(11.0l ~lnki• now ; - O 1111d n.•t·1t!l t,lmt, fJ E (' 1. h follow~ (8,y)(.•·, 11) = (8,y)(r +y. O) {iJ,g)(y.O). Thcor e m 3.2 Tl1r fr111 f'l,irm ol eq1111l.i1.· 11 S)' ltl lllt'lric cO(')'C k· o r c)n.._,, C 1 wit h 9 (0,0) o. Tnk1• f to bt• lhl• func-t ion µ; iw 11 i11 tlw Sll\l ('ltWlll 1\nd ron!-ickr tht• íu nct i 11: ll{r.11) / (I + !J) j(.r) f(y) , ÜÍ ('O lll'S<.', /¡ is(\ :<)'llllllt'lriC COCyclt>, Hlld fl ÍU !lC'l iOll o f dlL'-" C 1 By APJ>lying Lr11w 111 :1. 1 st'\'t'ro l l imcs, onl' rom¡>tth°!' (iloh){r. y) (0211)(...+ 11• O) (8,<¡)(r,O) (il«¡)(O.r ~y) (il1g)(O, r) (il1 •¡)(r, y) (CJ l<)(..-,y) (8,•¡ )(., +y.O) (111•/){y.O) • (8,y)(r,y) L.'l't 110\\ l(r,11): (11 y )(.r,y) C 1 ll1'<'0U~· (éJ1ll(.r.y) O 1md (C-Jil)(r,y) O, llw ÍUU,11011 l(r,y) mus ! b~· co 11:rlngcr Vcrlng, 2000. Cubo a mathematical journal 2007 v9 nª3_0047 Cubo a mathematical journal 2007 v9 nª3_0048 Cubo a mathematical journal 2007 v9 nª3_0049 Cubo a mathematical journal 2007 v9 nª3_0050 Cubo a mathematical journal 2007 v9 nª3_0051 Cubo a mathematical journal 2007 v9 nª3_0052 Cubo a mathematical journal 2007 v9 nª3_0053