%!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: lyapunof.dvi %%Pages: 13 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: A4 %%EndComments %DVIPSCommandLine: dvips -t a4 -o temp.ps lyapunof %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.01.04:1704 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies 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Fg(Ly)m(apuno)m(v)34 b(cen)m(ter)g(theorem)f(for)g (nonlinear)h(PDE's)935 b Fl(5)0 299 y Fm(Remark)153 b(2.3.)33 b Fg(By)g(the)g(second)i(of)e(\(2.4\))f Fk(u)1819 314 y Fj(\017)1891 299 y Fg(has)i(norm)e(of)h(order)h Fk(\017)p Fg(.)0 461 y Fm(Pro)s(of.)h Fl(First)f(w)m(e)h(in)m(tro)s(duce)g(a)f (suitable)h(Hilb)s(ert)e(space)j Fi(H)f Fl(of)g(p)s(erio)s(dic)f (functions.)48 b(Let)35 b Fk(q)j Fl(ha)m(v)m(e)d(the)0 580 y(represen)m(tation)1420 722 y Fk(q)t Fl(\()p Fk(t)p Fl(\))27 b(:=)1815 627 y Fc(X)1742 842 y Fj(j)t Ff(\025)p Fh(0)p Fj(;l)p Ff(\025)p Fh(1)2049 722 y Fk(q)2093 737 y Fj(j)t(l)2178 722 y Fl(cos\()p Fk(j)6 b(t)p Fl(\))p Fk(e)2518 737 y Fj(l)3764 722 y Fl(\(2)p Fk(:)p Fl(5\))0 1034 y(De\014ne)35 b Fi(H)28 b(\032)g Fk(H)617 997 y Fh(1)662 1034 y Fl(\([0)p Fk(;)17 b Fl(2)p Fk(\031)t Fl(])p Fk(;)g(`)1050 997 y Fh(2)1050 1058 y Fj(s)1091 1034 y Fl(\))33 b(to)g(b)s(e)g(the)h(space)g(of)g(all)e(the)h(function) i(of)e(the)h(form)e(\(2.5\))g(with)622 1352 y Fi(k)p Fk(q)t Fi(k)770 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5344 y Fc(\024)1420 5484 y Fk(f)1480 5443 y Fh(\(0\))1586 5484 y Fl(\()p Fk(\030)1669 5499 y Fh(1)1735 5484 y Fl(+)g Fk(\017)1875 5443 y Fj(r)r Ff(\000)p Fh(1)2022 5484 y Fk(q)2066 5499 y Ff(?)2133 5484 y Fl(\))f Fi(\000)h Fk(f)2354 5443 y Fh(\(0\))2460 5484 y Fl(\()p Fk(\030)2543 5499 y Fh(1)2587 5484 y Fl(\))f(+)2777 5417 y(1)p 2760 5462 V 2760 5553 a Fk(\017)2800 5524 y Fj(r)2856 5484 y Fk(f)2916 5443 y Fh(\(1\))3023 5484 y Fl(\()p Fk(\017\030)3146 5499 y Fh(1)3212 5484 y Fl(+)h Fk(\017)3352 5443 y Fj(r)3396 5484 y Fk(q)3440 5499 y Ff(?)3508 5484 y Fl(\))3547 5344 y Fc(\025)3812 5484 y Fl(\()p Fk(Q)p Fl(\))p eop %%Page: 6 6 6 5 bop 0 60 a Fl(6)1685 b Fg(D.)33 b(Bam)m(busi)0 299 y Fl(It)48 b(is)f(useful)j(to)d(consider)i(the)f(system)f(formed)h(b)m (y)g(the)g(equations)g(\()p Fk(P)s(;)17 b(Q;)g(!)t Fl(\))45 b(for)j(the)g(unkno)m(wns)0 418 y(\()p Fk(\014)5 b(;)17 b(q)189 433 y Ff(?)256 418 y Fk(;)g(\017)p Fl(\).)43 b(It)30 b(is)g(clear)h(that)f(an)m(y)g(solution)g(of)h(this)f(system)g 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Ff(H)1140 3269 y Fl(the)34 b(scalar)g(pro)s(duct)g(in)g Fi(H)q Fl(.)45 b(Remark)32 b(that,)h(for)h(\014xed)g Fk(q)3302 3284 y Ff(?)3370 3269 y Fl(,)f(the)h(quan)m(tit)m(y)0 3389 y Fk(\014)56 3404 y Fh(1)134 3389 y Fl(tends)g(to)f(zero)h(as)f Fk(\017)28 b Fi(!)g Fl(0.)199 3508 y(Finally)-8 b(,)33 b(w)m(e)i(need)g(to)f(solv)m(e)g(eq.)g(\()p Fk(!)t Fl(\).)46 b(Recall)34 b(that)g Fk(!)k Fl(is)c(\014xed,)g(so)h(eq.\()p Fk(!)t Fl(\))e(is)h(an)g(equation)g(for)g Fk(\017)p Fl(.)0 3628 y(Up)s(on)f(insertion)h(of)f(\(2.8\),)f(it)g(b)s(ecomes)1226 3785 y Fk(!)1292 3744 y Fh(2)1364 3785 y Fl(=)c Fk(!)1535 3744 y Fh(2)1531 3810 y(1)1602 3785 y Fi(\000)23 b Fk(\017)1742 3744 y Fj(r)r Ff(\000)p Fh(1)1888 3785 y Fl(\()p Fk(\014)1983 3800 y Fh(0)2050 3785 y Fl(+)g Fk(\014)2206 3800 y Fh(1)2251 3785 y Fl(\()p Fk(\017;)17 b(q)2423 3744 y Fj(!)2419 3810 y Ff(?)2486 3785 y Fl(\()p Fk(\017)p Fl(\)\)\))32 b Fk(:)1022 b Fl(\(2)p Fk(:)p Fl(9\))0 3942 y(If)34 b(the)g(term)f Fk(\014)567 3957 y Fh(1)645 3942 y 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b(therefore)h(suc)m(h)g(a)e Fk(\034)52 b Fl(is)40 b(outside)h(of)1624 3129 y Fc(S)1719 3207 y Fb(l)p Fd(\024)p Fb(l)1821 3217 y Fd(\003)1863 3207 y Fe(\()p Fb(j)s Fe(\))1751 3267 y Fb(j)s Fd(\024)p Fb(J)1876 3277 y Fd(\003)1982 3204 y Fi(R)2066 3219 y Fj(j)t(l)2134 3204 y Fl(\()p Fk(\015)6 b Fl(\),)41 b(and)g(therefore)h (outside)f(of)g(the)g(l.h.s.)65 b(of)0 3366 y(\(3.6\).)43 b(So,)33 b(the)g(thesis)h(follo)m(ws.)p 3891 3292 78 4 v 3891 3362 4 70 v 3965 3362 V 3891 3366 78 4 v 0 3526 a Fm(Lemma)155 b(3.6.)34 b Fg(Fix)f Fl(\012)986 3541 y Fj(c)1055 3526 y Fi(\021)c Fl(\()p Fk(!)1262 3541 y Fh(2)1306 3526 y Fk(;)17 b(!)1413 3541 y Fh(3)1457 3526 y Fk(;)g(!)1564 3541 y Fh(4)1608 3526 y Fk(;)g(:::)p Fl(\))p Fg(,)32 b(and)i(let)g Fk(!)2237 3541 y Fh(1)2310 3526 y Fi(2)29 b Fl([)p Fk(!)2495 3541 y Fj(m)2570 3526 y Fk(;)17 b(!)2677 3541 y Fj(M)2765 3526 y Fl(])33 b Fg(b)s(e)h Fk(\015)39 b Fg(strongly)33 b(nonresonan)m(t)0 3646 y(with)g Fl(\012)299 3661 y Fj(c)339 3646 y Fg(.)44 b(Let)e 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b(of)g(theorem)e(3.2.)46 b Fl(W)-8 b(e)35 b(extract)e(no)m(w)h(the)h(subset)g Fi(A)2406 314 y Fj(\015)2491 299 y Fl(of)2639 274 y(~)2605 299 y Fi(A)2685 314 y Fj(\015)2769 299 y Fl(formed)f(b)m(y)g(the)h(p)s(oin) m(ts)f(whic)m(h)0 418 y(are)f(accum)m(ulation)g(p)s(oin)m(ts)g(b)s(oth) h(from)e(the)i(righ)m(t)f(and)g(from)g(the)h(left.)199 538 y(De\014ne)860 663 y Fi(B)925 678 y Fh(1)997 663 y Fl(:=)1130 553 y Fc(n)1197 663 y Fk(\034)k Fi(2)1408 638 y Fl(~)1373 663 y Fi(A)1453 678 y Fj(\015)1565 663 y Fl(:)60 b Fi(9)p Fk(\017)1748 678 y Fj(\034)1827 663 y Fk(>)28 b Fl(0)33 b Fk(s:t:)f Fl(\()p Fk(\034)6 b(;)17 b(\034)31 b Fl(+)23 b Fk(\017)2535 678 y Fj(\034)2585 663 y Fl(\))f Fi(\\)2769 638 y Fl(~)2735 663 y Fi(A)2815 678 y Fj(\015)2893 663 y Fl(=)28 b Fi(;)3048 553 y Fc(o)854 882 y Fi(B)919 897 y Fh(2)992 882 y Fl(:=)1125 772 y Fc(n)1191 882 y Fk(\034)39 b Fi(2)1402 857 y Fl(~)1368 882 y Fi(A)1448 897 y Fj(\015)1559 882 y Fl(:)61 b Fi(9)p Fk(\017)1743 897 y Fj(\034)1821 882 y Fk(>)28 b Fl(0)33 b Fk(s:t:)g Fl(\()p Fk(\034)f Fi(\000)23 b Fk(\017)2436 897 y Fj(\034)2486 882 y Fk(;)17 b(\034)11 b Fl(\))21 b Fi(\\)2769 857 y Fl(~)2735 882 y Fi(A)2815 897 y Fj(\015)2893 882 y Fl(=)28 b Fi(;)3048 772 y Fc(o)0 1119 y Fl(and)40 b Fi(B)i Fl(:=)c Fi(B)515 1134 y Fh(1)586 1119 y Fi([)27 b(B)744 1134 y Fh(2)789 1119 y Fl(.)63 b(Then)1181 1094 y(~)1147 1119 y Fi(A)1227 1134 y Fj(\015)1317 1119 y Fl(is)39 b(the)h(disjoin)m(t)g(union)g(of)g Fi(B)j Fl(and)d(a)f(set)h Fi(A)3004 1134 y Fj(\015)3094 1119 y Fl(comp)s(osed)g(b)m(y)f(p)s(oin)m (ts)0 1239 y(whic)m(h)32 b(are)f(accum)m(ulation)f(p)s(oin)m(ts)h(b)s (oth)g(from)f(the)h(righ)m(t)g(and)g(from)f(the)h(left.)44 b(W)-8 b(e)31 b(c)m(ho)s(ose)g(the)h(set)f Fi(A)3918 1254 y Fj(\015)0 1358 y Fl(of)39 b(the)f(statemen)m(t)g(of)g(the)h (theorem)f(to)g(b)s(e)g(the)h(one)g(just)g(de\014ned.)61 b(It)38 b(is)g(easy)g(to)g(see)h(that)f Fi(B)3703 1373 y Fj(i)3775 1358 y Fl(is)g(at)0 1502 y(most)31 b(n)m(umerable,)h(so)g (that)f Fi(jA)1197 1517 y Fj(\015)1248 1502 y Fi(j)c Fl(=)1408 1388 y Fc(\014)1408 1447 y(\014)1408 1507 y(\014)1476 1477 y Fl(~)1441 1502 y Fi(A)1521 1517 y Fj(\015)1572 1388 y Fc(\014)1572 1447 y(\014)1572 1507 y(\014)1605 1502 y Fl(.)44 b(Indeed,)33 b(consider)g Fi(B)2479 1517 y Fh(1)2556 1502 y Fl(and)f(\014x)g Fk(\034)38 b Fi(2)28 b(B)3129 1517 y Fh(1)3175 1502 y Fl(,)j(c)m(ho)s(ose)i(a)f(rational)0 1652 y(n)m(um)m(b)s(er)e(in)f(\()p Fk(\034)6 b(;)17 b(\034)24 b Fl(+)15 b Fk(\017)805 1667 y Fj(\034)855 1652 y Fl(\);)30 b(this)g(giv)m(es)f(a)g(biuniv)m(o)s(cal)g(corresp)s(ondence)j(b)s(et)m (w)m(een)f Fi(B)3075 1667 y Fh(1)3150 1652 y Fl(and)f(a)f(subset)h(of)g (the)0 1771 y(rational)i(n)m(um)m(b)s(ers.)199 1891 y(It)g(remains)g (to)g(pro)m(v)m(e)h(that)f(for)h Fk(\034)39 b Fi(2)28 b(A)1690 1906 y Fj(\015)1773 1891 y Fl(the)33 b(frequency)g Fk(!)2456 1906 y Fh(1)2501 1891 y Fl(\()p Fk(\034)11 b Fl(\))32 b(is)g(the)h(accum)m(ulation)f(p)s(oin)m(t)g(of)h(a)0 2010 y(sequence)26 b(of)f(frequencies)h(whic)m(h)f(are)k(~)-54 b Fk(\015)30 b Fl(nonresonan)m(t)25 b(with)g(\012)2370 2025 y Fj(c)2410 2010 y Fl(\()p Fk(\034)11 b Fl(\).)40 b(But)25 b(this)f(is)g(a)h(trivial)d(consequence)0 2130 y(of)31 b(lemma)e(3.6.)43 b(Indeed,)33 b(let)d Fk(\034)1164 2145 y Fj(k)1241 2130 y Fi(2)e(A)1415 2145 y Fj(\015)1496 2130 y Fl(b)s(e)k(a)f(sequence)i(con)m(v)m(erging)e(to)g Fk(\034)11 b Fl(.)43 b(It)31 b(follo)m(ws)g(that)f Fk(!)3567 2145 y Fh(1)3612 2130 y Fl(\()p Fk(\034)3695 2145 y Fj(k)3743 2130 y Fl(\))h(is)g Fk(\015)0 2249 y Fl(strongly)k(nonresonan)m(t)i (with)e(\012)1254 2264 y Fj(c)1294 2249 y Fl(\()p Fk(\034)1377 2264 y Fj(k)1426 2249 y Fl(\))g(and)h(that)f Fi(j)p Fk(!)2004 2264 y Fj(l)2035 2249 y Fl(\()p Fk(\034)2118 2264 y Fj(k)2166 2249 y Fl(\))23 b Fi(\000)h Fk(!)2391 2264 y Fj(l)2422 2249 y Fl(\()p Fk(\034)11 b Fl(\))p Fi(j)30 b(\024)i Fk(c)2765 2264 y Fh(2)2827 2249 y Fi(j)p Fk(\034)2899 2264 y Fj(k)2969 2249 y Fi(\000)23 b Fk(\034)11 b Fi(j)16 b Fk(=l)3250 2213 y Fj(d)3296 2249 y Fl(,)36 b(so,)g(for)f Fk(k)k Fl(large)0 2369 y(enough)h Fk(!)411 2384 y Fh(1)456 2369 y Fl(\()p Fk(\034)539 2384 y Fj(k)587 2369 y Fl(\))f(is)g(also)k (~)-54 b Fk(\015)44 b Fl(strongly)39 b(nonresonan)m(t)i(with)d(\012) 2338 2384 y Fj(c)2379 2369 y Fl(\()p Fk(\034)11 b Fl(\))38 b(\(with)h(an)m(y)k(~)-54 b Fk(\015)43 b(<)38 b(\015)6 b Fl(\).)61 b(So,)41 b Fk(!)3648 2384 y Fh(1)3692 2369 y Fl(\()p Fk(\034)3775 2384 y Fj(k)3824 2369 y Fl(\))e(is)0 2489 y(the)34 b(w)m(on)m(ted)h(sequence)h(of)e(frequencies)i(accum)m (ulating)d(at)g Fk(!)2351 2504 y Fh(1)2396 2489 y Fl(\()p Fk(\034)11 b Fl(\).)45 b(Remark)33 b(also)h(that,)f(since)i Fk(c)3711 2504 y Fh(1)3785 2489 y Fi(6)p Fl(=)29 b(0,)0 2608 y(if)k Fk(\034)135 2623 y Fj(k)212 2608 y Fk(>)28 b(\034)44 b Fl(then)34 b(also)f Fk(!)894 2623 y Fh(1)938 2608 y Fl(\()p Fk(\034)1021 2623 y Fj(k)1070 2608 y Fl(\))27 b Fk(>)h(!)1303 2623 y Fh(1)1348 2608 y Fl(\()p Fk(\034)11 b Fl(\))32 b(and)i(vice-v)m(ersa.)p 3891 2534 78 4 v 3891 2604 4 70 v 3965 2604 V 3891 2608 78 4 v 0 2778 a Fm(Corollary)225 b(3.7.)48 b Fg(Let)h Fk(!)1178 2793 y Fj(l)1209 2778 y Fl(\()p Fk(\013;)17 b(\014)5 b Fl(\))53 b(:=)1669 2691 y Fc(p)p 1768 2691 522 4 v 1768 2778 a Fk(l)1800 2750 y Fh(4)1867 2778 y Fl(+)22 b Fk(\013l)2062 2750 y Fh(2)2129 2778 y Fl(+)h Fk(\014)54 b Fg(b)s(e)49 b(the)g(square)g(ro)s(ots)f(of)h(the)g(Diric)m(hlet)0 2898 y(eigen)m(v)-6 b(alues)38 b(of)g Fk(@)690 2913 y Fj(xxxx)901 2898 y Fi(\000)25 b Fk(\013@)1120 2913 y Fj(xx)1240 2898 y Fl(+)h Fk(\014)5 b Fg(;)40 b(\014x)d Fk(\013)e Fi(\025)g Fl(0)i Fg(then)h Fl(\012\()p Fk(\013;)17 b(\014)5 b Fl(\))34 b Fi(\021)i Fl(\()p Fk(!)2714 2913 y Fh(1)2758 2898 y Fl(\()p Fk(\013;)17 b(\014)5 b Fl(\))p Fk(;)17 b(!)3113 2913 y Fh(2)3157 2898 y Fl(\()p Fk(\013;)g(\014)5 b Fl(\))p Fk(;)17 b(!)3512 2913 y Fh(3)3555 2898 y Fl(\()p Fk(\013;)g(\014)5 b Fl(\))p Fk(;)17 b(:::)p Fl(\))0 3018 y Fg(has)34 b(the)f(prop)s(ert)m(y)h Fk(\015)6 b Fg({NR)32 b(with)h(some)f Fk(\015)6 b Fg(,)32 b(for)i Fk(\014)39 b Fg(in)33 b(a)g(subset)h(of)g Fl([0)p Fk(;)17 b Fl(+)p Fi(1)p Fl(])32 b Fg(ha)m(ving)h(full)g(measure.)0 3386 y(3.2)g(The)g(case)h Fk(d)28 b Fl(=)g(1)199 3656 y(W)-8 b(e)43 b(will)e(use)i(the)g(con)m(tin)m(ued)h(fraction)e(expansion)h (of)g Fk(!)t Fl(.)71 b(Corresp)s(ondingly)42 b(w)m(e)h(will)e(use)i (the)0 3775 y(standard)34 b(notation)e Fk(!)g Fl(=)c([)p Fk(a)1094 3790 y Fh(0)1138 3775 y Fk(;)17 b(a)1236 3790 y Fh(1)1280 3775 y Fk(;)g(a)1378 3790 y Fh(2)1421 3775 y Fk(;)g(:::)p Fl(])31 b(with)i Fk(a)1889 3790 y Fj(i)1955 3775 y Fl(non)h(negativ)m(e)f(in)m(tegers)h(to)f(mean)1460 4062 y Fk(!)f Fl(=)c Fk(a)1712 4077 y Fh(0)1778 4062 y Fl(+)2138 3994 y(1)p 1890 4039 546 4 v 1890 4182 a Fk(a)1943 4197 y Fh(1)2009 4182 y Fl(+)2247 4115 y(1)p 2121 4159 303 4 v 2121 4251 a Fk(a)2174 4266 y Fh(2)2241 4251 y Fl(+)22 b Fk(:::)2480 4062 y(:)0 4510 y Fm(Prop)s(osition)168 b(3.8.)90 b Fg(Let)36 b Fk(!)1260 4525 y Fh(1)1338 4510 y Fl(=)e Fk(a)1502 4525 y Fh(0)1570 4510 y Fl(+)25 b Fk(\013)33 b Fl(=)g([)p Fk(a)1960 4525 y Fh(0)2004 4510 y Fk(;)17 b(a)2102 4525 y Fh(1)2146 4510 y Fk(;)g(a)2244 4525 y Fh(2)2288 4510 y Fk(;)g(:::)p Fl(])34 b Fg(with)i Fl(0)d Fk(<)g(\013)g(<)h Fl(1)i Fg(and)h Fk(!)3455 4525 y Fh(1)3532 4510 y Fi(\025)d Fl(1)p Fk(=)p Fl(2)p Fg(;)j(let)0 4630 y Fi(f)p Fk(!)112 4645 y Fj(l)142 4630 y Fi(g)192 4660 y Fj(l)p Ff(\025)p Fh(2)358 4630 y Fg(b)s(e)d(a)f(sequence)i(of)e (p)s(ositiv)m(e)g(n)m(um)m(b)s(ers)h(satisfying)1552 4913 y Fi(j)p Fk(!)1642 4928 y Fj(l)1694 4913 y Fi(\000)23 b Fk(l)r Fi(j)k(\024)1998 4846 y Fk(c)2041 4861 y Fj(\027)p 1998 4890 93 4 v 2029 4981 a Fk(l)2202 4913 y(l)j Fi(\025)e Fl(2)0 5181 y Fg(assume)45 b(that)f(the)h(sequence)h Fk(a)1243 5196 y Fj(i)1321 5181 y Fg(has)f(in\014nitely)f(man)m(y)g (non)m(v)-6 b(anishing)46 b(elemen)m(ts)e(and)i(is)e(b)s(ounded;)0 5300 y(de\014ne)k Fk(\015)353 5315 y Fh(1)448 5300 y Fl(:=)j(inf)725 5315 y Fj(i)p Ff(\025)p Fh(2)877 5300 y Fl(\()p Fk(a)969 5315 y Fj(i)1023 5300 y Fl(+)23 b(2\))1212 5264 y Ff(\000)p Fh(1)1365 5300 y Fg(assume)47 b(also)g Fk(c)1980 5315 y Fj(\027)2080 5300 y Fk(<)k(\015)2260 5315 y Fh(1)2304 5300 y Fk(=)p Fl(4)c Fg(and)g Fl(3)p Fk(\013)j(>)h(c)2993 5315 y Fj(\027)3043 5300 y Fg(,)f(then)d(there)h (exists)f(a)0 5420 y(p)s(ositiv)m(e)39 b Fk(\015)45 b Fg(suc)m(h)d(that)d Fl(\012)h Fg(has)h(the)f(prop)s(ert)m(y)g Fk(\015)6 b Fg(-NR.)39 b(Moreo)m(v)m(er,)j(there)e(is)g(family)e(of)j (frequencies)g Fk(\015)0 5539 y Fg(strongly)32 b(nonresonan)m(t)j(with) d Fl(\012)1246 5554 y Fj(c)1287 5539 y Fg(,)h(accum)m(ulating)f(at)h Fk(!)2143 5554 y Fh(1)2220 5539 y Fg(and)h(ha)m(ving)f(the)g(p)s(o)m(w) m(er)h(of)f(the)g(con)m(tin)m(uum.)p eop %%Page: 11 11 11 10 bop 911 60 a Fg(Ly)m(apuno)m(v)34 b(cen)m(ter)g(theorem)f(for)g (nonlinear)h(PDE's)911 b Fl(11)0 299 y Fm(Pro)s(of.)34 b Fl(W)-8 b(e)34 b(b)s(egin)f(b)m(y)h(pro)m(ving)f(that)g(one)g(has) 1280 502 y Fi(j)p Fk(j)6 b(!)1417 517 y Fh(1)1483 502 y Fi(\000)23 b Fk(l)r Fi(j)k(\025)1787 434 y Fk(\015)1839 449 y Fh(1)p 1787 479 97 4 v 1812 570 a Fk(j)2028 502 y Fl(for)34 b(all)e Fk(l)d Fi(6)p Fl(=)g Fk(a)2537 517 y Fh(0)2581 502 y Fk(j)39 b(:)1075 b Fl(\(3)p Fk(:)p Fl(7\))0 747 y(This)32 b(is)g(a)f(consequence)j(of)e(the)g(follo)m (wing)f(t)m(w)m(o)h(facts)g(\(i\))f(if)h Fi(j)p Fk(j)6 b(!)2453 762 y Fh(1)2516 747 y Fi(\000)20 b Fk(l)r Fi(j)27 b Fk(<)2836 708 y Fh(1)p 2817 724 77 4 v 2817 781 a(2)p Fj(j)2938 747 y Fl(then)32 b Fk(l)r(=j)37 b Fl(is)32 b(a)f(con)m(v)m(ergen)m(t)0 880 y(of)i Fk(!)175 895 y Fh(1)253 880 y Fl(\(see)h([10,)e(Th.5C]\).)g(\(ii\))g(if)i Fk(l)1333 895 y Fj(n)1387 880 y Fk(=j)1478 895 y Fj(n)1565 880 y Fl(is)f(the)h Fk(n)p Fl(-th)f(con)m(v)m(ergen)m(t)h(of)g Fk(!)2731 895 y Fh(1)2809 880 y Fl(then)g(one)f(has)1333 1109 y Fi(j)p Fk(j)1402 1124 y Fj(n)1456 1109 y Fk(!)1518 1124 y Fh(1)1584 1109 y Fi(\000)23 b Fk(l)1714 1124 y Fj(n)1768 1109 y Fi(j)k(\025)1963 1042 y Fl(1)p 1940 1087 96 4 v 1940 1178 a Fk(j)1981 1193 y Fj(n)2124 1109 y Fl(inf)2064 1174 y Fj(i)p Ff(\025)p Fj(n)p Fh(+1)2437 1042 y Fl(1)p 2333 1087 258 4 v 2333 1178 a Fk(a)2386 1193 y Fj(i)2441 1178 y Fl(+)c(2)0 1345 y(\(see)34 b([10,)e(p.23]\).)43 b(Then)34 b(an)g(argumen)m(t)e(equal)h(to)g(that)g(of)g(lemma)e(3.6)i (sho)m(ws)h(that)1111 1575 y Fi(j)p Fk(j)6 b(!)1248 1590 y Fh(1)1314 1575 y Fi(\000)23 b Fk(!)1476 1590 y Fj(l)1507 1575 y Fi(j)k(\025)1667 1464 y Fc(\020)1727 1575 y Fk(\015)1779 1590 y Fh(1)1845 1575 y Fi(\000)1957 1508 y Fk(c)2000 1523 y Fj(\027)p 1957 1552 93 4 v 1978 1643 a Fl(4)2061 1464 y Fc(\021)2149 1508 y Fl(1)p 2149 1552 50 4 v 2151 1643 a Fk(j)2244 1575 y(;)216 b(l)29 b Fi(6)p Fl(=)f Fk(a)2705 1590 y Fh(0)2750 1575 y Fk(j)39 b(:)0 1805 y Fl(Then)e(w)m(e)g(study)g(the)g(case)g Fk(l)e Fl(=)f Fk(a)1309 1820 y Fh(0)1353 1805 y Fk(j)6 b Fl(.)54 b(If)37 b Fk(a)1638 1820 y Fh(0)1715 1805 y Fl(=)d(0)i(then)h Fk(l)e Fl(=)f(0)i(is)g(not)h(allo)m(w)m(ed.)54 b(Assume)36 b Fk(a)3516 1820 y Fh(0)3594 1805 y Fi(\025)d Fl(1,)k(one)0 1925 y(has)1045 2079 y Fi(j)o Fk(!)1134 2094 y Fh(1)1179 2079 y Fk(j)28 b Fi(\000)22 b Fk(!)1409 2094 y Fj(a)1452 2104 y Fe(0)1492 2094 y Fj(j)1534 2079 y Fi(j)27 b(\025)h Fk(\013j)g Fi(\000)1964 2012 y Fk(c)2007 2027 y Fj(\027)p 1939 2056 145 4 v 1939 2148 a Fk(a)1992 2163 y Fh(0)2036 2148 y Fk(j)2122 2079 y Fl(=)2227 1939 y Fc(\022)2301 2079 y Fk(\013j)2412 2038 y Fh(2)2478 2079 y Fi(\000)2592 2012 y Fk(c)2635 2027 y Fj(\027)p 2590 2056 98 4 v 2590 2148 a Fk(a)2643 2163 y Fh(0)2699 1939 y Fc(\023)2801 2012 y Fl(1)p 2801 2056 50 4 v 2803 2148 a Fk(j)2896 2079 y Fl(;)0 2298 y(if)38 b Fk(a)149 2313 y Fh(0)229 2298 y Fi(6)p Fl(=)e(1)i(then)h(it)f(follo)m(ws)f(that)h Fk(!)1381 2313 y Fh(1)1464 2298 y Fl(is)g Fk(\015)43 b Fl(strongly)37 b(nonresonan)m(t)j(with)e(\012)2926 2313 y Fj(c)2966 2298 y Fl(.)59 b(If)38 b Fk(a)3210 2313 y Fh(0)3290 2298 y Fl(=)e(1)i(then)h Fk(j)i Fl(=)c(1)0 2417 y(is)k(not)g(allo)m(w)m(ed,)i(so)e(the)g(same)g(conclusion)h (holds.)68 b(Then)41 b(remark)f(that)h(the)g(frequencies)i Fk(!)i Fl(of)c(the)0 2537 y(form)h(1)p Fk(=)p Fl(2)g Fk(<)i(!)j Fl(=)c Fk(!)849 2552 y Fh(1)922 2537 y Fl(+)29 b Fk(\013)1092 2552 y Fh(1)1180 2537 y Fl(=)44 b([)p Fk(a)1382 2552 y Fh(0)1426 2537 y Fk(;)17 b(b)1514 2552 y Fh(1)1557 2537 y Fk(;)g(b)1645 2552 y Fh(2)1689 2537 y Fk(;)g(:::)p Fl(])41 b(with)h Fi(j)p Fk(\013)2215 2552 y Fh(1)2259 2537 y Fi(j)h Fk(<)g Fl(min)o Fi(f)p Fk(\013=)p Fl(4)p Fk(;)17 b Fl(1)27 b Fi(\000)i Fk(\013)p Fi(g)43 b Fl(and)g(4)p Fk(c)3511 2552 y Fj(\027)3603 2537 y Fk(<)h(\015)3776 2552 y Fh(2)3863 2537 y Fl(:=)0 2656 y(inf)121 2671 y Fj(i)p Ff(\025)p Fh(2)273 2656 y Fl(\()p Fk(b)355 2671 y Fj(i)410 2656 y Fl(+)22 b(2\))598 2620 y Ff(\000)p Fh(1)739 2656 y Fk(<)34 b Fi(1)j Fl(accum)m(ulate)g(at)f Fk(!)1701 2671 y Fh(1)1783 2656 y Fl(and)h(are)k(~)-54 b Fk(\015)43 b Fl(nonresonan)m(t)38 b(with)f(\012)3114 2671 y Fj(c)3191 2656 y Fl(with)g(a)g(suitable)j(~)-53 b Fk(\015)5 b Fl(.)p 0 2702 78 4 v 0 2772 4 70 v 74 2772 V 0 2776 78 4 v 0 2935 a Fm(Corollary)152 b(3.9.)33 b Fg(De\014ne)h Fk(!)1204 2950 y Fj(l)1263 2935 y Fl(:=)1396 2852 y Fi(p)p 1479 2852 286 4 v 83 x Fk(l)1511 2907 y Fh(2)1577 2935 y Fl(+)23 b Fk(m)33 b Fg(to)g(b)s(e)g(the)g(square)h(ro) s(ots)e(of)h(the)g(Diric)m(hlet)g(eigen)m(v)-6 b(alues)0 3055 y(of)30 b Fi(\000)p Fk(@)240 3070 y Fj(xx)352 3055 y Fl(+)16 b Fk(m)31 b Fg(on)f Fl([0)p Fk(;)17 b(\031)t Fl(])28 b Fg(then)i(prop)s(ert)m(y)g Fk(\015)6 b Fg(-NR)30 b(holds)g(for)h Fk(m)f Fg(b)s(elonging)g(to)g(a)f(non{n)m(umerable)i (subset)0 3175 y(of)47 b Fl(I)-17 b(R)p Fg(.)43 b(Moreo)m(v)m(er)30 b(for)f(an)m(y)h Fk(m)g Fg(in)f(suc)m(h)i(a)e(set,)i(the)e(family)f(of) i Fk(!)t Fg('s)f(whic)m(h)h(are)f Fk(\015)6 b Fg({strongly)28 b(nonresonan)m(t)0 3294 y(with)33 b Fl(\012)299 3309 y Fj(c)373 3294 y Fg(and)g(accum)m(ulate)g(at)g Fk(!)1273 3309 y Fh(1)1318 3294 y Fg(,)g(is)g(not)g(coun)m(table.)199 3513 y Fl(A)g(simple)g(generalization)f(of)i(the)g(ab)s(o)m(v)m(e)f (corollary)f(is)0 3673 y Fm(Corollary)194 b(3.10.)42 b Fg(Let)g Fk(V)1187 3688 y Fh(0)1274 3673 y Fi(2)h Fk(L)1451 3636 y Fh(2)1496 3673 y Fl([0)p Fk(;)17 b(\031)t Fl(])39 b Fg(b)s(e)k(a)e(function)i(with)f(zero)g(a)m(v)m(erage.)71 b(Denote)42 b(b)m(y)g Fk(\025)3815 3688 y Fj(i)3891 3673 y Fl(=)0 3792 y Fk(\025)58 3807 y Fj(i)91 3792 y Fl(\()p Fk(\026;)17 b(m;)g(V)425 3807 y Fh(0)469 3792 y Fl(\))39 b Fg(the)i(Diric)m(hlet)e(eigen)m(v)-6 b(alues)41 b(of)f Fi(\000)p Fk(@)1915 3807 y Fj(xx)2038 3792 y Fl(+)27 b Fk(m)2229 3756 y Fh(2)2300 3792 y Fl(+)g Fk(\026V)2522 3807 y Fh(0)2607 3792 y Fg(on)40 b Fl([0)p Fk(;)17 b(\031)t Fl(])p Fg(.)61 b(Then)41 b(there)g(exists)e(an)0 3912 y(uncoun)m(table)51 b(set)f Fi(A)55 b(\032)73 b Fl(I)-17 b(R)50 b Fg(and)g(a)f(constan)m(t)i Fl(0)k Fi(\024)g Fk(C)7 b Fl(\()p Fk(V)2317 3927 y Fh(0)2362 3912 y Fl(\))55 b Fk(<)h Fi(1)p Fg(,)d(suc)m(h)e(that,)i(if)d Fk(m)55 b Fi(2)h(A)p Fg(,)d(and)0 4031 y Fk(C)7 b Fl(\()p Fk(V)175 4046 y Fh(0)220 4031 y Fl(\))p Fi(j)p Fk(\026)p Fi(j)27 b Fk(<)h(m)594 3995 y Fh(2)639 4031 y Fg(,)33 b(the)g(frequencies)j Fk(!)1444 4046 y Fj(i)1504 4031 y Fl(:=)1637 3954 y Fi(p)p 1720 3954 92 4 v 77 x Fk(\025)1778 4046 y Fj(i)1845 4031 y Fg(ful\014ll)d(prop)s(ert)m(y)g Fk(\015)6 b Fg(-NR)33 b(with)g(some)g Fk(\015)6 b Fg(.)199 4250 y Fl(More)39 b(general)g(examples)f(can)i(b)s(e)f(obtained)g(taking)f(in)m(to)g (accoun)m(t)i(kno)m(wn)f(results)g(of)g(Sturm{)0 4370 y(Liouville)32 b(theory)h([11].)0 4968 y Fm(4.)115 b(Applications)36 b(to)i(nonlinear)e(b)s(eam)i(and)f(w)m(a)m(v)m(e)g(equations)199 5237 y Fl(W)-8 b(e)36 b(are)f(no)m(w)g(ready)g(to)f(pro)m(v)m(e)i (existence)g(of)f(small)e(oscillations)h(in)h(nonlinear)h(b)s(eam)e (and)h(w)m(a)m(v)m(e)0 5356 y(equations.)44 b(W)-8 b(e)33 b(assume)h(that)f(the)g(nonlinearit)m(y)g(has)h(the)f(form)1465 5539 y Fk( )t Fl(\()p Fk(u)p Fl(\))26 b(=)i Fk( )1865 5554 y Fh(0)1910 5539 y Fk(u)1967 5498 y Fj(r)2033 5539 y Fl(+)23 b Fk( )2202 5498 y Fh(\(1\))2308 5539 y Fl(\()p Fk(u)p Fl(\))33 b Fk(;)p eop %%Page: 12 12 12 11 bop 0 60 a Fl(12)1660 b Fg(D.)33 b(Bam)m(busi)0 299 y Fl(with)39 b(some)f Fk(r)i Fi(\025)e Fl(2)h(and)g Fk( )1045 263 y Fh(\(1\))1190 299 y Fl(whic)m(h)h(admits)e(t)m(w)m(o)h (Lipscitz)h(deriv)-6 b(ativ)m(es,)40 b(v)-6 b(anish)39 b(together)g(with)g(its)0 418 y(\014rst)34 b(deriv)-6 b(ativ)m(e)33 b(at)f(zero)i(and)g(ful\014lls)f(the)h(inequalit)m(y)921 682 y(sup)758 770 y Ff(j)p Fj(x)p Ff(j)p Fj(<)p Fh(2)p Fj(\017;)p Ff(j)p Fj(y)s Ff(j)p Fj(<)p Fh(2)p Fj(\017)1251 537 y Fc(\014)1251 597 y(\014)1251 657 y(\014)1251 717 y(\014)1285 571 y(\020)1344 682 y Fk( )1413 641 y Fh(\(1\))1520 571 y Fc(\021)1579 592 y Ff(00)1647 682 y Fl(\()p Fk(x)p Fl(\))21 b Fi(\000)1903 571 y Fc(\020)1962 682 y Fk( )2031 641 y Fh(\(1\))2138 571 y Fc(\021)2197 592 y Ff(00)2265 682 y Fl(\()p Fk(y)t Fl(\))2396 537 y Fc(\014)2396 597 y(\014)2396 657 y(\014)2396 717 y(\014)2456 682 y Fi(\024)28 b Fk(C)2632 697 y Fh(6)2677 682 y Fk(\017)2717 641 y Fj(r)r Ff(\000)p Fh(2)2863 682 y Fi(j)p Fk(x)22 b Fi(\000)g Fk(y)t Fi(j)33 b 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m(h)g(dep)s(end)h(explicitly)c(on)j(the)f(space)i(v)-6 b(ariable.)0 5355 y Fm(Remark)120 b(4.6.)27 b Fg(The)f(function)h Fk( )t Fl(\()p Fk(u)p Fl(\))e Fg(in)h(\(1.4\))f(and)i(in)f(\(1.5\))f (do)s(es)i(not)f(need)h(to)f(b)s(e)g(o)s(dd.)42 b(Ho)m(w)m(ev)m(er,)27 b(if)0 5474 y(it)22 b(is)h(o)s(dd)g(and)g(has)h Fk(k)t Fl(+)q(1)f Fg(Lipsc)m(hitz)h(deriv)-6 b(ativ)m(es,)25 b(then)e(one)g(can)h(easily)e(pro)m(v)m(e)h(that)f Fk(u)3199 5489 y Fj(\017)3238 5474 y Fl(\()p Fk(t)p Fl(\))27 b Fi(2)h Fk(H)3564 5438 y Fj(k)3613 5474 y Fl(\([0)p Fk(;)17 b(\031)t Fl(]\))p Fg(.)p eop %%Page: 13 13 13 12 bop 911 60 a Fg(Ly)m(apuno)m(v)34 b(cen)m(ter)g(theorem)f(for)g (nonlinear)h(PDE's)911 b Fl(13)1715 299 y Fm(References)44 418 y Fl([1])49 b(S.B.)41 b(Kuksin:)62 b Fa(Hamiltonian)43 b(Perturb)-5 b(ations)43 b(of)g(In\014nite-dimensional)f(Line)-5 b(ar)43 b(Systems)h(with)199 538 y(Imaginary)c(Sp)-5 b(e)g(ctrum.)58 b Fl(F)-8 b(unktsional.)56 b(Anal.)g(i)38 b(Prilozhen.)57 b Fm(21)p Fl(,)39 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b(in)g(higher)199 1780 y(dimension.)k Fl(GAF)-11 b(A)33 b Fm(5)p Fl(,)g(629-639)f (\(1995\))44 1941 y([6])49 b(J.)28 b(Bourgain:)41 b(Nonlinear)27 b(Sc)m(hr\177)-50 b(odinger)30 b(equations.)42 b(In)28 b(\\Hyp)s(erb)s(olic)f(equations)h(and)g(frequency)199 2061 y(in)m(teractions";)46 b(L.)c(Ca\013arelli,)g(E.)f(W)-8 b(einan)42 b(editors.)70 b(IAS/P)m(ark)41 b(Cit)m(y)g(mathematics)e (series)k(5.)199 2180 y(American)33 b(Mathematical)e(So)s(ciet)m(y)i (\(Pro)m(vidence,)h(Rho)s(de)g(Island)f(1999\).)44 2341 y([7])49 b(A.)32 b(Am)m(brosetti,)f(G.)i(Pro)s(di:)43 b(A)32 b(primer)g(of)h(nonlinear)g(analysis.)43 b(Cam)m(bridge)32 b(Univ)m(ersit)m(y)g(Press,)199 2461 y(\(Cam)m(bridge,)g(1993\))44 2622 y([8])49 b(R.)39 b(de)g(la)f(Lla)m(v)m(e:)54 b Fa(V)-8 b(ariational)40 b(metho)-5 b(ds)41 b(for)f(quasi{p)-5 b(erio)g(dic)38 b(solutions)i(of)g(p)-5 b(artial)40 b(di\013er)-5 b(ential)199 2741 y(e)g(quations)p Fl(.)43 b(Preprin)m(t)34 b(1999.)44 2902 y([9])49 b(J.)35 b(Mujica:)48 b(Complex)34 b(Analysis)g(in)h(Banac)m(h)h(Spaces.)50 b(North)35 b(Holland)f (Mathematical)f(Studies)199 3022 y(120.)44 b(Amsterdam)31 b(1986.)-6 3183 y([10])49 b(W.M.)35 b(Sc)m(hmidt:)48 b(Diophan)m(tine)36 b(Appro)m(ximation.)47 b(Lect.)j(Notes)35 b(Math.)49 b(785,)35 b(Springer)g(V)-8 b(erlag)199 3303 y(1980.)-6 3464 y([11])49 b(J.)33 b(P\177)-50 b(osc)m(hel,)34 b(E.)f(T)-8 b(rub)s(o)m(witz:)43 b(In)m(v)m(ersal)34 b(Sp)s(ectral)g(Theory)-8 b(.)43 b(Academic)33 b(Press.)44 b(\(Boston)33 b(1987\))p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF