This is a multi-part message in MIME format. ---------------0406041533864 Content-Type: text/plain; name="04-178.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-178.keywords" Superconductivity, Ginzburg-Landau system, subsolution. ---------------0406041533864 Content-Type: application/postscript; name="root.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="root.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86f Copyright 2001 Radical Eye Software %%Title: root.dvi %%Pages: 27 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips root.dvi -o root.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2004.04.21:2255 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}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 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b(d)p Fn([)p Fm(:)1985 5871 y Fn(4)p eop %%Page: 5 5 5 4 bop 0 162 a Fw(3)135 b(Construction)45 b(of)g(subsolutions)h(and)e (sup)t(ersolutions)i(of)f Fd(\()p Fc(GL)p Fd(\))3491 119 y Fb(s)3491 196 y(d)0 369 y Fv(3.1)112 b(Construction)37 b(of)g(subsolutions)0 540 y Fn(Let)31 b(us)f(recall)f(the)i (de\014nition)d(of)i(a)h(subsolution)d(of)i(\()p Fm(GL)p Fn(\))2049 507 y Fl(s)2049 568 y(d)2091 540 y Fn(.)0 721 y Fe(De\014nition)35 b(3.1)46 b Ff(A)32 b(triplet)i Fn(\()p Fm(f)5 b(;)15 b(A)p Fn(;)g Fm(h)p Fn(\))34 b Ff(is)e(a)h(subsolution)h(of)f Fn(\()p Fm(GL)p Fn(\))2350 688 y Fl(s)2350 749 y(d)2424 721 y Ff(if)f(it)h(satis\014es)g(the)g(c) -5 b(onditions)1023 841 y Fk(8)1023 923 y(<)1023 1086 y(:)1146 957 y Fh(\000)p Fm(\024)1269 924 y Fi(\000)p Fg(2)1363 957 y Fm(f)1418 924 y Fi(00)1480 957 y Fn(+)20 b Fm(f)10 b Fn(\()p Fh(\000)p Fn(1)20 b(+)g Fm(f)1943 924 y Fg(2)2002 957 y Fn(+)g Fm(A)2161 924 y Fg(2)2200 957 y Fn(\))26 b Fh(\024)f Fn(0)p Fm(;)66 b Fh(8)25 b Fm(x)g Fh(2)g Fn([0)p 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5335 y Fg(0)2580 5313 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\))p Fm(;)1009 b Fn(\(3.5\))0 5509 y Ff(wher)-5 b(e)1639 5622 y Fm(g)s Fn(\()p Fm(y)s Fn(\))27 b(=)e(exp)o(\()p Fh(\000)p Fm(\013)2228 5636 y Fg(0)2268 5622 y Fm(y)s Fn(\))p Fm(;)1985 5871 y Fn(5)p eop %%Page: 6 6 6 5 bop 0 162 a Ff(then,)28 b(the)f(pr)-5 b(oblem)28 b Fn(\()p Fm(GL)p Fn(\))907 129 y Fl(s)907 190 y(d)975 162 y Ff(admits)g(a)f(p)-5 b(ositive)28 b(subsolution.)41 b(This)27 b(subsolution)g(is)g(given,)g(for)g(suitable)g(c)-5 b(onstants)0 275 y Fm(C)65 289 y Fg(1)137 275 y Fm(C)209 242 y Fi(0)202 300 y Fg(1)241 275 y Ff(,)33 b(by)975 419 y Fm(f)10 b Fn(\()p Fm(x)p Fn(\))25 b(=)g(tanh)1470 318 y Fk(\020)1561 383 y Fl(\024)p 1534 398 95 4 v 1534 407 a Fi(p)p 1593 407 36 3 v 55 x Fg(2)1638 419 y Fn(\(1)c(+)f Fm(C)1895 433 y Fg(1)1934 419 y Fm(\024)1986 386 y Fg(1)p Fi(\000)p Fl(\017)2130 419 y Fn(+)g Fm(C)2293 386 y Fi(0)2286 443 y Fg(1)2325 419 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))p Fm(x)j Fn(+)d Fm(x)2793 434 y Fl(\024;h)2898 318 y Fk(\021)975 615 y Fn(+)1056 553 y Fm(\024)1108 520 y Fg(2)1147 553 y Fm(h)1199 520 y Fg(2)p 1056 594 184 4 v 1077 679 a Fn(4)p Fm(f)1177 648 y Fg(3)1167 705 y(0)1264 615 y Fn(exp\()p Fh(\000)p Fn(2)p Fm(f)1599 629 y Fg(0)1638 615 y Fm(x)p Fn(\))h(+)f Fm(\025)1890 630 y Fl(d;\024;h)2066 615 y Fn(exp\()p Fh(\000)2311 535 y(p)p 2387 535 46 4 v 80 x Fn(2)p Fm(\024)p Fn(\()p Fm(d)h Fh(\000)f Fm(x)p Fn(\)\))p Fm(E)5 b Fn(\()2917 553 y Fm(x)p 2917 594 52 4 v 2919 677 a(d)2981 615 y Fn(\))p Fm(;)3829 529 y Fn(\(3.6\))0 824 y Ff(wher)-5 b(e)34 b(the)f(c)-5 b(onstants)34 b Fm(x)859 839 y Fl(\024;h)997 824 y Ff(and)f Fm(\025)1226 839 y Fl(d;\024;h)1420 824 y Ff(ar)-5 b(e)33 b(determine)-5 b(d)35 b(by)d(the)h(c)-5 b(onditions)35 b(:)1588 1015 y Fm(f)1643 977 y Fi(0)1666 1015 y Fn(\(0\))26 b(=)f(0)p Fm(;)67 b(f)2095 977 y Fi(0)2117 1015 y Fn(\()p Fm(d)p Fn(\))27 b(=)e(0)p Fm(:)1402 b Fn(\(3.7\))0 1205 y Fe(Pro)s(of)36 b(of)f(Prop)s(osition)h(3.2)31 b Fn(Let)g(us)e(in)m(tro)s(duce,)1800 1396 y Fm(\027)i Fn(:=)25 b(1)c Fh(\000)f Fm(\017;)1613 b Fn(\(3.8\))1052 1582 y(~)1032 1606 y Fm(f)1077 1620 y Fg(1)1116 1606 y Fn(\()p Fm(x)p Fn(\))26 b(:=)f(tanh\()1647 1545 y Fm(\024)p 1612 1585 122 4 v 1612 1603 a Fh(p)p 1688 1603 46 4 v 75 x Fn(2)1743 1606 y(\(1)c(+)f Fm(C)2000 1620 y Fg(1)2040 1606 y Fm(\024)2092 1568 y Fl(\027)2155 1606 y Fn(+)g Fm(C)2318 1568 y Fi(0)2311 1628 y Fg(1)2351 1606 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))p Fm(x)i Fn(+)e Fm(x)2818 1621 y Fl(\024;h)2923 1606 y Fn(\))p Fm(;)846 b Fn(\(3.9\))1775 1807 y(~)1755 1831 y Fm(f)1800 1845 y Fg(2)1865 1831 y Fn(:=)25 b Fm(f)k Fh(\000)2171 1807 y Fn(~)2151 1831 y Fm(f)2196 1845 y Fg(1)2235 1831 y Fm(;)1523 b Fn(\(3.10\))1607 1991 y Fm(z)1649 2006 y Fl(\024;h)1780 1991 y Fn(:=)25 b(exp\()p Fh(\000)p Fn(2)p Fm(x)2243 2006 y Fl(\024;h)2348 1991 y Fn(\))p Fm(;)1375 b Fn(\(3.11\))613 2212 y Fm(f)658 2226 y Fg(1)696 2212 y Fn(\()p Fm(x)p Fn(\))26 b(:=)g(tanh)o(\()1227 2150 y Fm(\024)p 1192 2191 122 4 v 1192 2209 a Fh(p)p 1268 2209 46 4 v 75 x Fn(2)1324 2212 y(\(1)21 b(+)f Fm(C)1581 2226 y Fg(1)1620 2212 y Fm(\024)1672 2174 y Fl(\027)1736 2212 y Fn(+)g Fm(C)1899 2174 y Fi(0)1892 2234 y Fg(1)1931 2212 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))p Fm(x)j Fn(+)d Fm(x)2399 2227 y Fl(\024;h)2504 2212 y Fn(\))g(+)g Fm(\024)2702 2174 y Fg(2)2776 2150 y Fm(h)2828 2117 y Fg(2)p 2752 2191 140 4 v 2752 2276 a Fn(4)p Fm(f)2852 2244 y Fg(3)2842 2302 y(0)2916 2212 y Fn(exp\()p Fh(\000)p Fn(2)p Fm(f)3251 2226 y Fg(0)3291 2212 y Fm(x)p Fn(\))p Fm(;)380 b Fn(\(3.12\))0 2420 y(and)1229 2533 y Fm(f)1274 2547 y Fg(2)1313 2533 y Fn(\()p Fm(x)p Fn(\))26 b(:=)f Fm(\025)1635 2548 y Fl(d;\024;h)1811 2533 y Fn(exp\()p Fh(\000)2056 2454 y(p)p 2132 2454 46 4 v 79 x Fn(2)q Fm(\024)p Fn(\()p Fm(d)c Fh(\000)f Fm(x)p Fn(\)\))p Fm(E)5 b Fn(\()2663 2472 y Fm(x)p 2663 2512 52 4 v 2665 2596 a(d)2726 2533 y Fn(\))p Fm(:)997 b Fn(\(3.13\))0 2721 y Fe(Step)34 b(1:)47 b(Estimates)33 b(for)i Fm(x)1063 2736 y Fl(\024;h)1168 2721 y Fe(,)g Fm(z)1274 2736 y Fl(\024;h)1414 2721 y Fe(and)g Fm(\025)1669 2736 y Fl(d;\024;h)1860 2721 y Fn(The)29 b(constan)m(t)j Fm(x)2465 2736 y Fl(\024;h)2600 2721 y Fn(is)d(suc)m(h)h(that)h Fm(f)3148 2688 y Fi(0)3171 2721 y Fn(\(0\))26 b(=)f(0,)31 b(so)f(it)g(satis\014es)946 2979 y(\(1)21 b Fh(\000)f Fn(tanh)1319 2942 y Fg(2)1359 2979 y Fn(\()p Fm(x)1446 2994 y Fl(\024;h)1551 2979 y Fn(\)\)\(1)i(+)e Fm(C)1879 2993 y Fg(1)1918 2979 y Fm(\024)1970 2942 y Fl(\027)2034 2979 y Fn(+)g Fm(C)2197 2942 y Fi(0)2190 3002 y Fg(1)2229 2979 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))j Fh(\000)2603 2843 y(p)p 2679 2843 46 4 v 75 x Fn(2)p Fm(\024h)2828 2885 y Fg(2)p 2603 2958 266 4 v 2666 3043 a Fn(2)p Fm(f)2766 3012 y Fg(2)2756 3069 y(0)2903 2979 y Fn(=)i(0)p Fm(:)0 3224 y Fn(As)30 b Fm(h)h Fn(satis\014es)f(\(3.5\),) i(w)m(e)f(get)642 3464 y(\(1)21 b Fh(\000)f Fn(tanh)1016 3427 y Fg(2)1055 3464 y Fn(\()p Fm(x)1142 3479 y Fl(\024;h)1248 3464 y Fn(\)\)\(1)i(+)d Fm(C)1575 3478 y Fg(1)1615 3464 y Fm(\024)1667 3427 y Fl(\027)1731 3464 y Fn(+)h Fm(C)1894 3427 y Fi(0)1887 3487 y Fg(1)1926 3464 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))28 b(=)d(1)20 b Fh(\000)g Fm(f)2511 3427 y Fg(2)2501 3487 y(0)2570 3464 y Fn(+)2678 3403 y Fm(C)2743 3417 y Fg(0)2783 3403 y Fm(\024)2835 3370 y Fl(\027)p 2671 3443 215 4 v 2671 3461 a Fh(p)p 2747 3461 46 4 v 76 x Fn(2)p Fm(f)2847 3505 y Fg(2)2837 3563 y(0)2916 3464 y Fn(+)3017 3403 y Fm(C)3089 3370 y Fi(0)3082 3427 y Fg(0)3121 3403 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\))p 3017 3443 322 4 v 3070 3461 a Fh(p)p 3146 3461 46 4 v 76 x Fn(2)p Fm(f)3246 3505 y Fg(2)3236 3563 y(0)3348 3464 y Fm(;)0 3712 y Fn(then)297 3909 y(tanh)479 3872 y Fg(2)519 3909 y Fn(\()p Fm(x)606 3924 y Fl(\024;h)711 3909 y Fn(\))297 4067 y(=)25 b Fm(f)448 4034 y Fg(2)438 4091 y(0)507 4067 y Fn(+)598 3966 y Fk(\020)652 4067 y Fn(\(1)c Fh(\000)f Fm(f)899 4034 y Fg(2)889 4091 y(0)938 4067 y Fn(\))p Fm(C)1038 4081 y Fg(1)1098 4067 y Fh(\000)1259 4030 y Fl(C)1309 4039 y Fp(0)p 1199 4046 206 4 v 1199 4111 a Fg(2)1234 4056 y Fi(p)p 1293 4056 36 3 v 55 x Fg(2)p Fl(f)1369 4088 y Fp(2)1362 4132 y(0)1414 3966 y Fk(\021)1483 4067 y Fm(\024)1535 4034 y Fl(\027)1599 4067 y Fn(+)1690 3966 y Fk(\020)1744 4067 y Fn(\(1)h Fh(\000)f Fm(f)1991 4034 y Fg(2)1981 4091 y(0)2030 4067 y Fn(\))p Fm(C)2137 4034 y Fi(0)2130 4091 y Fg(1)2190 4067 y Fh(\000)2351 4018 y Fl(C)2406 3995 y Fo(0)2401 4039 y Fp(0)p 2291 4046 206 4 v 2291 4111 a Fg(2)2326 4056 y Fi(p)p 2385 4056 36 3 v 55 x Fg(2)p Fl(f)2461 4088 y Fp(2)2454 4132 y(0)2506 3966 y Fk(\021)2575 4067 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\))j(+)d Fh(O)s Fn(\()p Fm(\024)3066 4034 y Fg(2)p Fl(\027)3145 4067 y Fn(\))g(+)g Fh(O)s Fn(\()p Fm(g)s Fn(\()p Fm(\024d)p Fn(\))3616 4034 y Fg(2)3658 4067 y Fn(\))p Fm(:)0 4281 y Fn(Th)m(us,)30 b(w)m(e)g(get)321 4496 y(tanh\()p Fm(x)590 4511 y Fl(\024;h)695 4496 y Fn(\))321 4654 y(=)25 b Fm(f)462 4668 y Fg(0)521 4654 y Fn(+)612 4553 y Fk(\020)676 4605 y Fg(1)p Fi(\000)p Fl(f)807 4582 y Fp(2)800 4626 y(0)p 676 4633 166 4 v 707 4685 a Fg(2)p Fl(f)776 4694 y Fp(0)852 4654 y Fm(C)917 4668 y Fg(1)977 4654 y Fh(\000)1138 4617 y Fl(C)1188 4626 y Fp(0)p 1078 4633 206 4 v 1078 4698 a Fg(2)1113 4643 y Fi(p)p 1172 4643 36 3 v 55 x Fg(2)p Fl(f)1248 4675 y Fp(3)1241 4719 y(0)1293 4553 y Fk(\021)1362 4654 y Fm(\024)1414 4621 y Fl(\027)1478 4654 y Fn(+)1569 4553 y Fk(\020)1633 4605 y Fg(1)p Fi(\000)p Fl(f)1764 4582 y Fp(2)1757 4626 y(0)p 1633 4633 166 4 v 1664 4685 a Fg(2)p Fl(f)1733 4694 y Fp(0)1809 4654 y Fm(C)1881 4621 y Fi(0)1874 4678 y Fg(1)1934 4654 y Fh(\000)2095 4605 y Fl(C)2150 4582 y Fo(0)2145 4626 y Fp(0)p 2035 4633 206 4 v 2035 4698 a Fg(2)2070 4643 y Fi(p)p 2129 4643 36 3 v 55 x Fg(2)p Fl(f)2205 4675 y Fp(3)2198 4719 y(0)2250 4553 y Fk(\021)2319 4654 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\))e(+)c Fh(O)s Fn(\()p Fm(\024)2809 4621 y Fg(2)p Fl(\027)2889 4654 y Fn(\))h(+)g Fh(O)s Fn(\()p Fm(g)s Fn(\()p Fm(\024d)p Fn(\))3360 4621 y Fg(2)3402 4654 y Fn(\))p Fm(:)3783 4575 y Fn(\(3.14\))0 4873 y(Therefore,)30 b(for)h Fm(\024)f Fn(and)g Fm(g)s Fn(\()p Fm(\024d)p Fn(\))i(small,)d(w)m(e)i(get)964 5114 y Fm(z)1006 5129 y Fl(\024;h)1137 5114 y Fn(=)1243 5050 y(1)20 b Fh(\000)g Fn(tanh\()p Fm(x)1668 5065 y Fl(\024;h)1773 5050 y Fn(\))p 1243 5093 567 4 v 1243 5176 a(1)g(+)g(tanh\()p Fm(x)1668 5191 y Fl(\024;h)1773 5176 y Fn(\))1844 5114 y(=)1950 5052 y(1)h Fh(\000)e Fm(f)2151 5066 y Fg(0)p 1950 5093 241 4 v 1950 5176 a Fn(1)i(+)e Fm(f)2151 5190 y Fg(0)2221 5114 y Fn(+)g Fh(O)s Fn(\()p Fm(\024)2473 5076 y Fl(\027)2517 5114 y Fn(\))i(+)f Fh(O)s Fn(\()p Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))p Fm(:)3783 5116 y Fn(\(3.15\))0 5358 y(Let)31 b(us)f(remark)g(that,)h(from)f(\(3.15\),)j(there)d(exists)g Fm(\024)1851 5372 y Fg(0)1921 5358 y Fn(suc)m(h)g(that,)i(for)e Fm(\024)25 b Fh(\024)g Fm(\024)2713 5372 y Fg(0)2753 5358 y Fn(,)31 b(w)m(e)g(ha)m(v)m(e,)g(for)g(all)e Fm(f)3488 5372 y Fg(0)3552 5358 y Fh(2)c Fn([)p Fm(\016)3703 5372 y Fg(0)3743 5358 y Fm(;)15 b(\016)3823 5372 y Fg(1)3863 5358 y Fn(],)1246 5530 y(1)21 b Fh(\000)f Fm(\016)1443 5544 y Fg(1)p 1246 5571 237 4 v 1246 5654 a Fn(1)h(+)f Fm(\016)1443 5668 y Fg(1)1513 5592 y Fn(+)g Fh(O)s Fn(\()p Fm(\024)1766 5554 y Fl(\027)1810 5592 y Fn(\))g(+)g Fh(O)s Fn(\()p Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))28 b Fh(\024)d Fm(z)2482 5607 y Fl(\024;h)2613 5592 y Fh(\024)g Fn(1)p Fm(:)1004 b Fn(\(3.16\))1985 5871 y(6)p eop %%Page: 7 7 7 6 bop 0 171 a Fn(The)30 b(constan)m(t)i Fm(\025)607 186 y Fl(d;\024;h)798 171 y Fn(is)d(suc)m(h)h(that)h Fm(f)1346 138 y Fi(0)1369 171 y Fn(\()p Fm(d)p Fn(\))26 b(=)f(0)p Fm(;)31 b Fn(so)g(that)g Fm(\025)2071 186 y Fl(d;\024;h)2257 171 y Fn(=)25 b Fh(\000)2434 122 y Fl(f)2475 99 y Fo(0)2468 143 y Fp(1)2502 122 y Fg(\()p Fl(d)p Fg(\))p 2434 150 160 4 v 2446 160 a Fi(p)p 2505 160 36 3 v 55 x Fg(2)p Fl(\024)2603 171 y Fm(:)31 b Fn(Therefore,)f(w)m(e)h(get)620 460 y Fm(\025)673 475 y Fl(d;\024;h)859 460 y Fn(=)1023 398 y Fm(\024h)1127 365 y Fg(2)p 965 439 261 4 v 965 532 a Fn(2)1010 457 y Fh(p)p 1086 457 46 4 v 75 x Fn(2)q Fm(f)1187 501 y Fg(2)1177 558 y(0)1251 460 y Fn(exp)o(\()p Fh(\000)p Fn(2)p Fm(f)1585 474 y Fg(0)1625 460 y Fm(d)p Fn(\))620 704 y Fh(\000)p Fn(2\(1)21 b(+)f Fm(C)993 718 y Fg(1)1032 704 y Fm(\024)1084 671 y Fl(\027)1148 704 y Fn(+)g Fm(C)1311 671 y Fi(0)1304 728 y Fg(1)1343 704 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))1721 641 y Fm(z)1763 656 y Fl(\024;h)1884 641 y Fn(exp\()p Fh(\000)2129 566 y(p)p 2205 566 V 75 x Fn(2)p Fm(\024d)p Fn(\(1)i(+)e Fm(C)2607 655 y Fg(1)2646 641 y Fm(\024)2698 608 y Fl(\027)2762 641 y Fn(+)g Fm(C)2925 608 y Fi(0)2918 665 y Fg(1)2957 641 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\)\))p 1603 683 1755 4 v 1603 777 a(\(1)k(+)c Fm(z)1840 792 y Fl(\024;h)1960 777 y Fn(exp\()p Fh(\000)2205 701 y(p)p 2281 701 46 4 v 76 x Fn(2)p Fm(\024d)p Fn(\(1)i(+)e Fm(C)2683 791 y Fg(1)2722 777 y Fm(\024)2774 750 y Fl(\027)2838 777 y Fn(+)g Fm(C)3001 745 y Fi(0)2994 803 y Fg(1)3033 777 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\)\))3318 750 y Fg(2)3371 704 y Fm(:)3783 586 y Fn(\(3.17\))0 972 y(F)-8 b(rom)28 b(\(3.16\))i(and)d(\(3.17\),)k(w)m(e)d(deduce)f(the)h(existence)g(of)f Fm(\024)2077 986 y Fg(0)2145 972 y Fn(and)g Fm(d)2366 986 y Fg(0)2433 972 y Fn(suc)m(h)g(that,)i(for)f(all)e Fm(f)3160 986 y Fg(0)3224 972 y Fh(2)f Fn([)p Fm(\016)3375 986 y Fg(0)3415 972 y Fm(;)15 b(\016)3495 986 y Fg(1)3535 972 y Fn(],)29 b(for)e Fm(\024)f Fh(\024)f Fm(\024)3976 986 y Fg(0)0 1085 y Fn(and)30 b Fm(\024d)c Fh(\025)f Fm(d)445 1099 y Fg(0)484 1085 y Fn(,)31 b(w)m(e)g(get)578 1313 y Fm(\025)631 1328 y Fl(d;\024;h)817 1313 y Fn(=)25 b Fh(\000)p Fn(2\(1)c(+)f Fm(C)1286 1327 y Fg(1)1326 1313 y Fm(\024)1378 1280 y Fl(\027)1441 1313 y Fn(+)g Fm(C)1604 1280 y Fi(0)1597 1337 y Fg(1)1637 1313 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))p Fm(z)1929 1328 y Fl(\024;h)2036 1313 y Fm(:)15 b Fn(exp)q(\()p Fh(\000)2322 1238 y(p)p 2398 1238 V 75 x Fn(2\(1)21 b(+)f Fm(C)2700 1327 y Fg(1)2739 1313 y Fm(\024)2791 1280 y Fl(\027)2855 1313 y Fn(+)g Fm(C)3018 1280 y Fi(0)3011 1337 y Fg(1)3050 1313 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))p Fm(\024d)p Fn(\))578 1471 y Fh(\002)664 1397 y Fk(\000)705 1471 y Fn(1)h(+)f Fh(O)s Fn(\(exp\()p Fh(\000)1217 1396 y(p)p 1293 1396 V 75 x Fn(2)p Fm(\024d)p Fn(\(1)i(+)e Fm(C)1695 1485 y Fg(1)1734 1471 y Fm(\024)1786 1438 y Fl(\027)1830 1471 y Fn(\)\)\))1935 1397 y Fk(\001)1993 1471 y Fm(:)3783 1383 y Fn(\(3.18\))0 1681 y(It)30 b(results)g(that,)h(for)f Fm(\024)c Fh(\024)f Fm(\024)975 1695 y Fg(0)1045 1681 y Fn(and)k Fm(\024d)d Fh(\025)f Fm(d)1489 1695 y Fg(0)886 1903 y Fm(\025)939 1918 y Fl(d;\024;h)1126 1903 y Fn(=)g Fh(O)1312 1802 y Fk(\020)1366 1903 y Fn(exp\()p Fh(\000)1611 1823 y(p)p 1687 1823 V 80 x Fn(2\(1)c Fh(\000)f(O)s Fn(\()p Fm(\024)2086 1866 y Fl(\027)2130 1903 y Fn(\)\))p Fm(\024d)p Fn(\))2334 1802 y Fk(\021)2405 1903 y Fm(;)66 b Fn(with)49 b Fm(\025)2776 1918 y Fl(d;\024;h)2963 1903 y Fm(<)25 b Fn(0)p Fm(:)654 b Fn(\(3.19\))0 2132 y Fe(Step)35 b(2:)46 b(Estimates)34 b(for)h Fm(f)1067 2099 y Fi(0)1124 2132 y Fe(and)g Fm(f)40 b Fn(F)-8 b(rom)30 b(\(3.6\),)j(w)m(e)e(get)g(:)922 2385 y Fm(f)977 2352 y Fi(0)999 2385 y Fn(\()p Fm(x)p Fn(\))26 b(=)1713 2324 y Fm(\024)p Fn(\(1)21 b(+)f Fm(C)2022 2338 y Fg(1)2061 2324 y Fm(\024)2113 2291 y Fl(\027)2177 2324 y Fn(+)g Fm(C)2340 2291 y Fi(0)2333 2348 y Fg(1)2372 2324 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))p 1253 2365 1832 4 v 1253 2383 a Fh(p)p 1329 2383 46 4 v 75 x Fn(2\(cosh)q(\(2)1662 2431 y Fi(\000)p Fg(1)p Fl(=)p Fg(2)1828 2458 y Fm(\024)p Fn(\(+)p Fm(C)2051 2472 y Fg(1)2090 2458 y Fm(\024)2142 2432 y Fl(\027)2206 2458 y Fn(+)g Fm(C)2369 2427 y Fi(0)2362 2484 y Fg(1)2401 2458 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))p Fm(x)j Fn(+)d Fm(x)2869 2473 y Fl(\024;h)2974 2458 y Fn(\)\))3044 2432 y Fg(2)922 2623 y Fh(\000)1003 2561 y Fm(\024)1055 2528 y Fg(2)1094 2561 y Fm(h)1146 2528 y Fg(2)p 1003 2602 184 4 v 1025 2687 a Fn(2)p Fm(f)1125 2656 y Fg(2)1115 2713 y(0)1211 2623 y Fn(exp\()p Fh(\000)p Fn(2)p Fm(f)1546 2637 y Fg(0)1585 2623 y Fm(x)p Fn(\))922 2845 y(+)p Fm(\024\025)1098 2860 y Fl(d;\024;h)1274 2845 y Fn(exp\()p Fh(\000)1519 2770 y(p)p 1595 2770 46 4 v 75 x Fn(2)p Fm(\024)p Fn(\()p Fm(d)h Fh(\000)f Fm(x)p Fn(\)\))2023 2717 y Fk(\022)2091 2770 y Fh(p)p 2167 2770 V 75 x Fn(2)p Fm(E)5 b Fn(\()2329 2783 y Fm(x)p 2329 2824 52 4 v 2331 2907 a(d)2392 2845 y Fn(\))20 b(+)2575 2783 y(1)p 2548 2824 100 4 v 2548 2907 a Fm(\024d)2658 2845 y(E)2730 2807 y Fi(0)2753 2845 y Fn(\()2798 2783 y Fm(x)p 2798 2824 52 4 v 2800 2907 a(d)2861 2845 y Fn(\))2896 2717 y Fk(\023)2978 2845 y Fm(:)0 3094 y Fn(W)-8 b(e)32 b(assume)e(that)1666 3207 y Fm(C)1731 3221 y Fg(1)1796 3207 y Fm(<)25 b Fn(0)p Fm(;)91 b(C)2125 3169 y Fi(0)2118 3229 y Fg(1)2183 3207 y Fm(<)25 b Fn(0)p Fm(:)1434 b Fn(\(3.20\))0 3374 y(F)-8 b(rom)31 b(\(3.20\),)i(and)d(as)g Fm(\025)868 3389 y Fl(d;\024;h)1054 3374 y Fm(<)25 b Fn(0)31 b(\(see)g(\(3.19\)\),)j Fm(E)c Fh(\025)25 b Fn(0)31 b(and)f Fm(E)2250 3341 y Fi(0)2299 3374 y Fh(\025)25 b Fn(0,)31 b(it)f(results)f(that)909 3633 y Fm(f)964 3595 y Fi(0)987 3633 y Fn(\()p Fm(x)p Fn(\))d Fh(\024)1700 3571 y Fm(\024)p Fn(\(1)21 b(+)f Fm(C)2009 3585 y Fg(1)2049 3571 y Fm(\024)2101 3538 y Fl(\027)2164 3571 y Fn(+)g Fm(C)2327 3538 y Fi(0)2320 3595 y Fg(1)2360 3571 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))p 1240 3612 1832 4 v 1240 3630 a Fh(p)p 1316 3630 46 4 v 75 x Fn(2)q(\(cosh\(2)1649 3679 y Fi(\000)p Fg(1)p Fl(=)p Fg(2)1815 3705 y Fm(\024)p Fn(\(+)p Fm(C)2038 3719 y Fg(1)2078 3705 y Fm(\024)2130 3679 y Fl(\027)2194 3705 y Fn(+)f Fm(C)2356 3674 y Fi(0)2349 3731 y Fg(1)2389 3705 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))p Fm(x)j Fn(+)e Fm(x)2856 3720 y Fl(\024;h)2961 3705 y Fn(\)\))3031 3679 y Fg(2)3081 3633 y Fm(:)677 b Fn(\(3.21\))0 3901 y(According)30 b(to)h(\(3.14\))i(and)d(\(3.15\),)i(for)f(all)e Fm(f)1581 3915 y Fg(0)1645 3901 y Fh(2)c Fn([)p Fm(\016)1796 3915 y Fg(0)1836 3901 y Fm(;)15 b(\016)1916 3915 y Fg(1)1956 3901 y Fn(],)31 b(for)f(all)f Fm(x)d Fh(2)50 b Fn([0)p Fm(;)15 b(d)p Fn(],)32 b(w)m(e)f(get,)h(for)e(some)h(p)s(ositiv)m(e)e Fm(C)1573 4120 y Fh(\000)p Fm(C)7 b(\024)25 b Fh(\024)g Fm(f)1944 4082 y Fi(0)1966 4120 y Fn(\()p Fm(x)p Fn(\))h Fh(\024)f Fn(2)2255 4082 y Fi(\000)2320 4055 y Fp(1)p 2321 4067 31 3 v 2321 4108 a(2)2365 4120 y Fm(\024:)1341 b Fn(\(3.22\))0 4324 y Fe(Upp)s(er)34 b(b)s(ound)g(and)f(lo)m(w)m(er)h (b)s(ound)f(for)h Fm(f)k Fn(F)-8 b(rom)30 b(Lemma)f(8.5)i(p.)39 b(75)30 b(in)e([8)q(],)i(the)f(function)f Fm(f)3438 4338 y Fg(1)3506 4324 y Fn(is)g(increasing.)0 4437 y(As)i Fm(\025)187 4452 y Fl(d;\024;h)374 4437 y Fm(<)25 b Fn(0,)31 b(w)m(e)f(ha)m(v)m(e)i Fh(8)25 b Fm(x)g Fh(2)g Fn([0)p Fm(;)15 b(d)p Fn(])p Fm(;)67 b(f)10 b Fn(\()p Fm(x)p Fn(\))26 b Fh(\024)f Fm(f)1771 4451 y Fg(1)1810 4437 y Fn(\()p Fm(x)p Fn(\))g Fh(\024)g Fm(f)2098 4451 y Fg(1)2137 4437 y Fn(\()p Fm(d)p Fn(\))p Fm(:)32 b Fn(Moreo)m(v)m(er,)h(using)1174 4641 y(tanh\()p Fm(y)s Fn(\))26 b(=)e(1)d Fh(\000)f Fn(2)15 b(exp)q(\()p Fh(\000)p Fn(2)p Fm(y)s Fn(\))20 b(+)g Fh(O)s Fn(\(exp)q(\()p Fh(\000)p Fn(4)p Fm(y)s Fn(\)\))p Fm(;)942 b Fn(\(3.23\))0 4845 y(when)29 b Fm(y)f Fh(!)e Fn(+)p Fh(1)p Fn(,)k(w)m(e)h(get,)g(for)g Fm(\024d)g Fn(large)f(and)g Fm(\024)g Fn(small)f(and)h(for)g Fm(x)25 b Fh(2)g Fn([0)p Fm(;)15 b(d)p Fn(])32 b(:)217 5050 y Fm(f)10 b Fn(\()p Fm(x)p Fn(\))25 b Fh(\024)g Fn(1)20 b Fh(\000)g Fn(2)p Fm(z)758 5065 y Fl(\024;h)879 5050 y Fn(exp\()p Fh(\000)1124 4970 y(p)p 1200 4970 46 4 v 80 x Fn(2)p Fm(\024d)p Fn(\(1)i(+)e Fm(C)1602 5064 y Fg(1)1641 5050 y Fm(\024)1693 5012 y Fl(\027)1757 5050 y Fn(+)g Fm(C)1920 5012 y Fi(0)1913 5072 y Fg(1)1953 5050 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\)\)\(1)j(+)d Fh(O)s Fn(\(exp\()p Fh(\000)2787 4970 y(p)p 2863 4970 V 80 x Fn(2)q Fm(\024d)p Fn(\(1)h(+)f Fm(C)3265 5064 y Fg(1)3305 5050 y Fm(\024)3357 5012 y Fl(\027)3400 5050 y Fn(\)\)\)\))p Fm(:)218 b Fn(\(3.24\))0 5302 y(F)-8 b(or)31 b Fm(x)25 b Fh(2)g Fn([0)p Fm(;)447 5240 y(d)p 447 5281 48 4 v 448 5364 a Fn(2)505 5302 y(],)30 b(as)h Fm(f)742 5316 y Fg(2)781 5302 y Fn(\()p Fm(x)p Fn(\))26 b(=)f(0)31 b(and)e Fm(f)1322 5316 y Fg(1)1392 5302 y Fn(is)g(increasing,)g(w)m(e)i(ha)m(v)m(e)1746 5532 y Fm(f)10 b Fn(\()p Fm(x)p Fn(\))25 b Fh(\025)g Fm(f)2089 5546 y Fg(1)2128 5532 y Fn(\(0\))p Fm(:)1515 b Fn(\(3.25\))1985 5871 y(7)p eop %%Page: 8 8 8 7 bop 0 162 a Fn(F)-8 b(rom)31 b(\(3.5\))h(and)e(\(3.14\),)j(w)m(e)e (get)900 382 y Fm(f)945 396 y Fg(1)984 382 y Fn(\(0\))c(=)e(tanh)o(\()p Fm(x)1490 397 y Fl(\024;h)1596 382 y Fn(\))20 b(+)g Fm(\024)1794 349 y Fg(2)1862 346 y Fl(h)1903 322 y Fp(2)p 1844 361 112 4 v 1844 419 a Fg(4)p Fl(f)1920 397 y Fp(3)1913 441 y(0)900 551 y Fn(=)25 b Fm(f)1041 565 y Fg(0)1100 551 y Fn(+)1191 450 y Fk(\020)1255 502 y Fg(1)p Fi(\000)p Fl(f)1386 479 y Fp(2)1379 523 y(0)p 1255 530 166 4 v 1287 583 a Fg(2)p Fl(f)1356 592 y Fp(0)1431 551 y Fm(C)1496 565 y Fg(1)1556 551 y Fh(\000)1717 514 y Fl(C)1767 523 y Fp(0)p 1657 530 206 4 v 1657 595 a Fg(2)1692 540 y Fi(p)p 1751 540 36 3 v 55 x Fg(2)p Fl(f)1827 572 y Fp(3)1820 616 y(0)1872 450 y Fk(\021)1942 551 y Fm(\024)1994 518 y Fl(\027)2057 551 y Fn(+)2148 450 y Fk(\020)2212 502 y Fg(1)p Fi(\000)p Fl(f)2343 479 y Fp(2)2336 523 y(0)p 2212 530 166 4 v 2244 583 a Fg(2)p Fl(f)2313 592 y Fp(0)2388 551 y Fm(C)2460 518 y Fi(0)2453 576 y Fg(1)2513 551 y Fh(\000)2674 502 y Fl(C)2729 479 y Fo(0)2724 523 y Fp(0)p 2614 530 206 4 v 2614 595 a Fg(2)2649 540 y Fi(p)p 2708 540 36 3 v 55 x Fg(2)p Fl(f)2784 572 y Fp(3)2777 616 y(0)2829 450 y Fk(\021)2898 551 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\))900 727 y(+)p Fh(O)s Fn(\()p Fm(\024)1133 694 y Fg(2)p Fl(\027)1212 727 y Fn(\))c(+)f Fh(O)s Fn(\()p Fm(g)s Fn(\()p Fm(\024d)p Fn(\))1684 694 y Fg(2)1725 727 y Fn(\))h(+)f Fh(O)s Fn(\()p Fm(\024)p Fn(\))p Fm(:)3783 545 y Fn(\(3.26\))0 985 y(On)30 b(the)g(in)m(terv)-5 b(al)30 b([)672 923 y Fm(d)p 672 964 48 4 v 673 1047 a Fn(2)729 985 y Fm(;)15 b(d)p Fn(],)32 b(w)m(e)e(remark,)h(using)e (the)h(fact)i(that)f Fm(f)2184 999 y Fg(1)2253 985 y Fn(is)e(increasing)g(and)h Fm(f)2991 999 y Fg(2)3060 985 y Fn(is)g(decreasing)g(that)625 1244 y Fm(f)10 b Fn(\()p Fm(x)p Fn(\))26 b Fh(\025)f Fm(f)969 1258 y Fg(1)1008 1244 y Fn(\()p Fm(d=)p Fn(2\))d(+)e Fm(f)1373 1258 y Fg(2)1412 1244 y Fn(\()p Fm(d)p Fn(\))625 1429 y(=)25 b Fm(\025)774 1444 y Fl(d;\024;h)956 1429 y Fn(+)20 b(tanh)o(\()1300 1393 y Fl(\024d)p 1273 1408 130 4 v 1273 1472 a Fg(2)1308 1417 y Fi(p)p 1368 1417 36 3 v 1368 1472 a Fg(2)1413 1429 y Fn(\(1)h(+)f Fm(C)1670 1443 y Fg(1)1709 1429 y Fm(\024)1761 1396 y Fl(\027)1825 1429 y Fn(+)g Fm(C)1988 1396 y Fi(0)1981 1453 y Fg(1)2020 1429 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))j(+)d Fm(x)2436 1444 y Fl(\024;h)2541 1429 y Fn(\))h(+)e Fm(\024)2739 1396 y Fg(2)2813 1367 y Fm(h)2865 1334 y Fg(2)p 2789 1408 140 4 v 2789 1493 a Fn(4)p Fm(f)2889 1461 y Fg(3)2879 1519 y(0)2953 1429 y Fn(exp\()p Fh(\000)p Fm(f)3243 1443 y Fg(0)3282 1429 y Fm(d)p Fn(\))p Fm(:)3783 1349 y Fn(\(3.27\))0 1673 y(Using)30 b(\(3.23\),)i(w)m(e)f(get)h(the)e(estimate)232 1892 y(tanh)o(\()485 1856 y Fl(\024d)p 458 1871 130 4 v 458 1936 a Fg(2)493 1881 y Fi(p)p 553 1881 36 3 v 553 1936 a Fg(2)598 1892 y Fn(\(1)21 b(+)f Fm(C)855 1906 y Fg(1)894 1892 y Fm(\024)946 1859 y Fl(\027)1010 1892 y Fn(+)g Fm(C)1173 1859 y Fi(0)1166 1917 y Fg(1)1206 1892 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))i(+)e Fm(x)1621 1907 y Fl(\024;h)1726 1892 y Fn(\))232 2055 y(=)25 b(1)20 b Fh(\000)g Fn(2)p Fm(z)571 2070 y Fl(\024;h)692 2055 y Fn(exp\()p Fh(\000)p Fn(2)982 2022 y Fi(\000)p Fg(1)p Fl(=)p Fg(2)1147 2055 y Fm(\024d)p Fn(\(1)i(+)e Fm(C)1504 2069 y Fg(1)1543 2055 y Fm(\024)1595 2022 y Fl(\027)1659 2055 y Fn(+)g Fm(C)1822 2022 y Fi(0)1815 2079 y Fg(1)1855 2055 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\)\))i(+)e Fh(O)s Fn(\(exp)q(\()p Fh(\000)p Fn(2)2654 2022 y Fg(1)p Fl(=)p Fg(2)2764 2055 y Fm(\024d)p Fn(\(1)i(+)e Fm(C)3121 2069 y Fg(1)3160 2055 y Fm(\024)3212 2022 y Fl(\027)3276 2055 y Fn(+)g Fm(C)3439 2022 y Fi(0)3432 2079 y Fg(1)3471 2055 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\)\))p Fm(:)0 2313 y Fn(According)30 b(to)h(\(3.18\))i(and)d(\(3.27\),)i(w)m(e)f(deduce)f (the)h(estimate,)g(for)f Fm(\024d)h Fn(large)g(and)f(for)g Fm(\024)g Fn(small,)g(for)g Fm(x)25 b Fh(2)g Fn([)3744 2252 y Fm(d)p 3744 2292 48 4 v 3745 2375 a Fn(2)3801 2313 y Fm(;)15 b(d)p Fn(])148 2566 y Fm(f)10 b Fn(\()p Fm(x)p Fn(\))25 b Fh(\025)g Fn(1)c Fh(\000)f Fn(2)p Fm(z)690 2581 y Fl(\024;h)811 2566 y Fn(exp)965 2465 y Fk(\020)1019 2566 y Fh(\000)p Fn(2)1135 2529 y Fi(\000)p Fg(1)p Fl(=)p Fg(2)1300 2566 y Fm(\024d)p Fn(\(1)i(+)e Fm(C)1657 2580 y Fg(1)1696 2566 y Fm(\024)1748 2529 y Fl(\027)1812 2566 y Fn(+)g Fm(C)1975 2529 y Fi(0)1968 2589 y Fg(1)2007 2566 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))2257 2465 y Fk(\021)2329 2566 y Fm(:)15 b Fn(\()q(1)20 b(+)g Fh(O)s Fn(\(exp\()p Fh(\000)p Fn(2)p Fm(\024d)p Fn(\(1)j(+)d Fm(C)3319 2580 y Fg(1)3358 2566 y Fm(\024)3410 2529 y Fl(\027)3454 2566 y Fn(\)\)\)\))c Fm(:)148 b Fn(\(3.28\))0 2779 y(F)-8 b(rom)33 b(\(3.5\),)i(\(3.25\),)g(\(3.26\))f(and)e (\(3.28\),)j(it)d(results)f(that)i(there)g(exists)e Fm(\024)2631 2793 y Fg(0)2700 2779 y Fm(>)d Fn(0)k(suc)m(h)g(that,)i(for)e(all)f Fm(\024)e Fh(\024)f Fm(\024)3809 2793 y Fg(0)3849 2779 y Fn(,)33 b(for)0 2892 y Fm(\024d)e Fn(large,)g(for)f(all)f Fm(f)688 2906 y Fg(0)752 2892 y Fh(2)c Fn([)p Fm(\016)903 2906 y Fg(0)943 2892 y Fm(;)15 b(\016)1023 2906 y Fg(1)1064 2892 y Fn(])30 b(and)g(for)g Fm(x)25 b Fh(2)g Fn([0)p Fm(;)15 b(d)p Fn(],)32 b(w)m(e)f(get)900 3112 y Fm(f)10 b Fn(\()p Fm(x)p Fn(\))25 b Fh(\025)g Fn(tanh\()p Fm(x)1467 3127 y Fl(\024;h)1573 3112 y Fn(\))20 b(+)g Fm(\024)1771 3079 y Fg(2)1839 3076 y Fl(h)1880 3052 y Fp(2)p 1821 3091 112 4 v 1821 3149 a Fg(4)p Fl(f)1897 3126 y Fp(3)1890 3171 y(0)900 3281 y Fn(=)25 b Fm(f)1041 3295 y Fg(0)1100 3281 y Fn(+)1191 3180 y Fk(\020)1255 3232 y Fg(1)p Fi(\000)p Fl(f)1386 3209 y Fp(2)1379 3253 y(0)p 1255 3260 166 4 v 1287 3313 a Fg(2)p Fl(f)1356 3322 y Fp(0)1431 3281 y Fm(C)1496 3295 y Fg(1)1556 3281 y Fh(\000)1717 3244 y Fl(C)1767 3253 y Fp(0)p 1657 3260 206 4 v 1657 3325 a Fg(2)1692 3270 y Fi(p)p 1751 3270 36 3 v 55 x Fg(2)p Fl(f)1827 3302 y Fp(3)1820 3346 y(0)1872 3180 y Fk(\021)1942 3281 y Fm(\024)1994 3248 y Fl(\027)2057 3281 y Fn(+)2148 3180 y Fk(\020)2212 3232 y Fg(1)p Fi(\000)p Fl(f)2343 3209 y Fp(2)2336 3253 y(0)p 2212 3260 166 4 v 2244 3313 a Fg(2)p Fl(f)2313 3322 y Fp(0)2388 3281 y Fm(C)2460 3248 y Fi(0)2453 3306 y Fg(1)2513 3281 y Fh(\000)2674 3232 y Fl(C)2729 3209 y Fo(0)2724 3253 y Fp(0)p 2614 3260 206 4 v 2614 3325 a Fg(2)2649 3270 y Fi(p)p 2708 3270 36 3 v 55 x Fg(2)p Fl(f)2784 3302 y Fp(3)2777 3346 y(0)2829 3180 y Fk(\021)2898 3281 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\))900 3457 y(+)p Fh(O)s Fn(\()p Fm(\024)1133 3424 y Fg(2)p Fl(\027)1212 3457 y Fn(\))c(+)f Fh(O)s Fn(\()p Fm(g)s Fn(\()p Fm(\024d)p Fn(\))1684 3424 y Fg(2)1725 3457 y Fn(\))h(+)f Fh(O)s Fn(\()p Fm(\024)p Fn(\))p Fm(:)3783 3275 y Fn(\(3.29\))0 3666 y Fe(Step)38 b(3:)53 b(Upp)s(er)38 b(b)s(ound)h(for)f Fm(A)1264 3633 y Fg(2)1304 3666 y Fm(f)k Fn(W)-8 b(e)34 b(shall)d(need)i(estimates)h(for)f Fm(A)2593 3633 y Fg(2)2633 3666 y Fm(f)10 b Fn(.)48 b(F)-8 b(or)33 b(this)f(purp)s(ose,)h(as)g(in)f([13)q(,)h(14)q(],)0 3779 y(w)m(e)h(split)e(the)i(in)m(terv)-5 b(al)33 b([0)p Fm(;)15 b(d)p Fn(])36 b(in)m(to)d(t)m(w)m(o)j(in)m(terv)-5 b(als)32 b([0)p Fm(;)15 b(\024)1945 3746 y Fi(\000)p Fl(\017)2034 3779 y Fn(])34 b(and)g([)p Fm(\024)2351 3746 y Fi(\000)p Fl(\017)2439 3779 y Fm(;)15 b(d)p Fn(].)52 b(Let)34 b(us)f(recall)g(that)i(\(see)f([13)r(])g(Lemma)0 3892 y(A.1.)42 b(p.)e(185\))977 4060 y Fm(A)1045 4027 y Fg(2)1084 4060 y Fn(\()p Fm(x)p Fn(\))p Fm(f)10 b Fn(\()p Fm(x)p Fn(\))26 b Fh(\024)1574 3999 y Fm(h)1626 3966 y Fg(2)p 1515 4039 210 4 v 1515 4122 a Fm(f)10 b Fn(\(0\))1685 4096 y Fg(2)1750 4060 y Fn(exp\()p Fh(\000)p Fn(2)p Fm(f)g Fn(\(0\))p Fm(x)p Fn(\))p Fm(f)g Fn(\()p Fm(x)p Fn(\))p Fm(;)92 b Fh(8)25 b Fm(x)g Fh(2)g Fn([0)p Fm(;)15 b(d)p Fn(])p Fm(:)3783 4057 y Fn(\(3.30\))0 4283 y(F)-8 b(or)28 b(\()p Fm(\024;)15 b(d)p Fn(\))29 b(suc)m(h)d(that)i Fm(\024d)e Fh(\025)f Fn(2)i(and)g Fm(d)e Fh(\025)g Fn(2,)j(w)m(e)g(ha)m (v)m(e)g(\()p Fm(\024d)p Fn(\))2032 4250 y Fl(\017)2066 4283 y Fm(d)2113 4250 y Fg(1)p Fi(\000)p Fl(\017)2262 4283 y Fh(\025)d Fn(2)p Fm(;)i Fn(so)h(that)f Fm(k)2807 4250 y Fi(\000)p Fl(\017)2920 4283 y Fh(\024)3026 4247 y Fl(d)p 3026 4262 37 4 v 3027 4314 a Fg(2)3072 4283 y Fm(:)h Fn(Moreo)m(v)m(er,)i(from)c(\(3.22\),)0 4405 y(w)m(e)31 b(get)g(0)26 b Fh(\024)f Fm(f)10 b Fn(\()p Fm(x)p Fn(\))25 b Fh(\024)g Fm(f)10 b Fn(\(0\))21 b(+)f(2)1077 4372 y Fi(\000)p Fg(1)p Fl(=)p Fg(2)1242 4405 y Fm(\024;)92 b Fh(8)15 b Fm(x)25 b Fh(2)f Fn([0)p Fm(;)15 b(\024)1801 4372 y Fi(\000)p Fl(\017)1890 4405 y Fn(])p Fm(:)31 b Fn(Using)f(\(3.14\),)j(then)d(\(3.30\),)j(w)m(e)d(get)i(for)e Fm(x)25 b Fh(2)g Fn([0)p Fm(;)15 b(\024)3759 4372 y Fi(\000)p Fl(\017)3848 4405 y Fn(],)697 4651 y Fm(A)765 4618 y Fg(2)805 4651 y Fm(f)34 b Fh(\024)990 4590 y Fm(h)1042 4557 y Fg(2)p 990 4630 92 4 v 994 4713 a Fm(f)1039 4727 y Fg(0)1107 4651 y Fn(exp\()p Fh(\000)p Fn(2)p Fm(f)10 b Fn(\(0\))p Fm(x)p Fn(\))697 4842 y Fh(\002)p Fn(\(1)21 b(+)960 4741 y Fk(\020)1014 4842 y Fh(\000)1095 4793 y Fg(1)p Fi(\000)p Fl(f)1226 4769 y Fp(2)1219 4814 y(0)p 1094 4821 166 4 v 1122 4879 a Fg(2)p Fl(f)1198 4857 y Fp(2)1191 4901 y(0)1270 4842 y Fm(C)1335 4856 y Fg(1)1395 4842 y Fn(+)1556 4804 y Fl(C)1606 4813 y Fp(0)p 1496 4821 206 4 v 1496 4885 a Fg(2)1531 4830 y Fi(p)p 1590 4830 36 3 v 55 x Fg(2)p Fl(f)1666 4863 y Fp(4)1659 4907 y(0)1731 4842 y Fn(+)1896 4806 y Fg(1)p 1832 4821 163 4 v 1832 4830 a Fi(p)p 1891 4830 36 3 v 55 x Fg(2)p Fl(f)1960 4894 y Fp(0)2005 4741 y Fk(\021)2074 4842 y Fm(\024)2126 4809 y Fl(\027)2190 4842 y Fn(+)2281 4741 y Fk(\020)2335 4842 y Fh(\000)2416 4793 y Fg(1)p Fi(\000)p Fl(f)2547 4769 y Fp(2)2540 4814 y(0)p 2416 4821 166 4 v 2443 4879 a Fg(2)p Fl(f)2519 4857 y Fp(2)2512 4901 y(0)2592 4842 y Fm(C)2664 4809 y Fi(0)2657 4866 y Fg(1)2716 4842 y Fn(+)2877 4793 y Fl(C)2932 4769 y Fo(0)2927 4814 y Fp(0)p 2817 4821 206 4 v 2817 4885 a Fg(2)2852 4830 y Fi(p)p 2911 4830 36 3 v 55 x Fg(2)q Fl(f)2988 4863 y Fp(4)2981 4907 y(0)3032 4741 y Fk(\021)3102 4842 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\))697 5017 y(+)p Fh(O)s Fn(\()p Fm(\024)930 4984 y Fg(2)p Fl(\027)1009 5017 y Fn(\))f(+)g Fh(O)s Fn(\()p Fm(g)s Fn(\()p Fm(\024d)p Fn(\))1480 4984 y Fg(2)1522 5017 y Fn(\))g(+)g Fh(O)s Fn(\()p Fm(\024)p Fn(\)\))p Fm(:)3783 4812 y Fn(\(3.31\))0 5224 y(F)-8 b(or)30 b Fm(x)c Fh(2)e Fn([)p Fm(\024)402 5191 y Fi(\000)p Fl(\017)491 5224 y Fm(;)15 b(d)p Fn(],)31 b(according)e(to)i(\(3.14\),)h(\(3.24\))g (and)d(\(3.30\),)j(w)m(e)e(get,)i(for)d Fm(\024)h Fn(and)f Fm(g)s Fn(\()p Fm(\024d)p Fn(\))j(small)c(and)h(for)g Fm(\024d)i Fn(large)0 5337 y(enough)f(the)h(other)f(upp)s(er)f(b)s (ound)179 5578 y Fm(A)247 5545 y Fg(2)286 5578 y Fn(\()p Fm(x)p Fn(\))p Fm(f)10 b Fn(\()p Fm(x)p Fn(\))26 b Fh(\024)718 5516 y Fm(h)770 5483 y Fg(2)p 717 5557 94 4 v 717 5642 a Fm(f)772 5610 y Fg(2)762 5668 y(0)836 5578 y Fn(exp\()p Fh(\000)p Fn(2)p Fm(f)10 b Fn(\(0\))p Fm(x)p Fn(\))1398 5477 y Fk(\020)1453 5578 y Fn(1)21 b(+)f Fh(O)s Fn(\()p Fm(\024)1772 5540 y Fl(\027)1816 5578 y Fn(\))g(+)g Fh(O)s Fn(\()p Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))j(+)d Fh(O)s Fn(\(exp\()p Fh(\000)2791 5498 y(p)p 2867 5498 46 4 v 80 x Fn(2)p Fm(\024d)p Fn(\(1)i(+)e Fm(C)3269 5592 y Fg(1)3309 5578 y Fm(\024)3361 5540 y Fl(\027)3404 5578 y Fn(\)\)\))3509 5477 y Fk(\021)3580 5578 y Fm(:)3783 5577 y Fn(\(3.32\))1985 5871 y(8)p eop %%Page: 9 9 9 8 bop 0 162 a Fe(Step)38 b(4:)54 b(The)38 b(function)h Fm(f)47 b Fe(is)39 b(a)f(subsolution)d Fn(Using)d(the)i(splitting)d(of) i Fm(f)10 b Fn(,)34 b Fm(f)40 b Fn(=)30 b Fm(f)3081 176 y Fg(1)3142 162 y Fn(+)22 b Fm(f)3280 176 y Fg(2)3352 162 y Fn(\(see)34 b(\(3.6\),)i(\(3.12\))0 275 y(and)30 b(\(3.35\)\),)j(w)m(e)e(get)726 508 y Fh(\000)p Fm(\024)849 475 y Fi(\000)p Fg(2)943 508 y Fm(f)998 475 y Fi(00)1060 508 y Fn(+)20 b(\()p Fh(\000)p Fn(1)h(+)f Fm(f)1469 475 y Fg(2)1528 508 y Fn(+)g Fm(A)1687 475 y Fg(2)1726 508 y Fn(\))p Fm(f)726 666 y Fn(=)25 b(\(2)p Fm(C)967 680 y Fg(1)1007 666 y Fm(\024)1059 633 y Fl(\027)1123 666 y Fn(+)20 b(2)p Fm(C)1331 633 y Fi(0)1324 691 y Fg(1)1364 666 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\))i(+)e Fh(O)s Fn(\()p Fm(\024)1854 633 y Fg(2)p Fl(\027)1953 666 y Fn(+)g Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))2294 633 y Fg(2)2336 666 y Fn(\))2391 642 y(~)2371 666 y Fm(f)2416 680 y Fg(1)2455 666 y Fn(\(1)h Fh(\000)2667 642 y Fn(~)2647 666 y Fm(f)2702 633 y Fg(2)2692 691 y(1)2741 666 y Fn(\))726 824 y Fh(\000)807 789 y Fl(h)848 765 y Fp(2)p 807 804 76 4 v 810 856 a Fl(f)844 865 y Fp(0)907 824 y Fn(exp\()p Fh(\000)p Fn(2)p Fm(f)1242 838 y Fg(0)1281 824 y Fm(x)p Fn(\))726 1031 y Fh(\000)p Fm(\025)850 1046 y Fl(d;h;\024)1026 1031 y Fn(exp\()p Fh(\000)1271 956 y(p)p 1347 956 46 4 v 75 x Fn(2)p Fm(\024)p Fn(\()p Fm(d)g Fh(\000)f Fm(x)p Fn(\)\))p Fm(:)1800 875 y Fk(")1854 1031 y Fn(2)p Fm(E)5 b Fn(\()2016 995 y Fl(x)p 2016 1010 40 4 v 2018 1062 a(d)2067 1031 y Fn(\))21 b(+)2224 969 y(2)2269 894 y Fh(p)p 2345 894 46 4 v 75 x Fn(2)p 2224 1010 167 4 v 2257 1093 a Fm(\024d)2400 1031 y(E)2472 993 y Fi(0)2496 1031 y Fn(\()2541 969 y Fm(x)p 2541 1010 52 4 v 2543 1093 a(d)2603 1031 y Fn(\))g(+)2842 969 y(1)p 2760 1010 210 4 v 2760 1093 a(\()p Fm(\024d)p Fn(\))2929 1067 y Fg(2)2979 1031 y Fm(E)3051 993 y Fi(00)3094 1031 y Fn(\()3139 969 y Fm(x)p 3139 1010 52 4 v 3141 1093 a(d)3201 1031 y Fn(\))3236 875 y Fk(#)726 1255 y Fn(+)816 1231 y(~)797 1255 y Fm(f)842 1269 y Fg(2)881 1255 y Fm(:)p Fn(\()p Fh(\000)p Fn(1)g(+)f Fm(f)1224 1222 y Fg(2)1282 1255 y Fn(+)g Fm(f)1462 1231 y Fn(~)1443 1255 y Fm(f)1488 1269 y Fg(1)1547 1255 y Fn(+)1657 1231 y(~)1638 1255 y Fm(f)1693 1222 y Fg(2)1683 1280 y(1)1732 1255 y Fn(\))g(+)g Fm(A)1946 1222 y Fg(2)1986 1255 y Fm(f)5 b(:)3783 873 y Fn(\(3.33\))0 1470 y(W)-8 b(e)39 b(ha)m(v)m(e)h(to)f(pro)m(v)m(e)g(that)g(the)f(righ)m(t-hand-side)f(of) h(\(3.33\))j(is)c(negativ)m(e)i(when)f Fm(\024)g Fn(and)g Fm(g)s Fn(\()p Fm(\024d)p Fn(\))i(are)f(small,)g Fm(\024d)g Fn(is)0 1583 y(large,)31 b Fm(C)313 1597 y Fg(1)377 1583 y Fm(<)25 b Fn(0)31 b(and)f Fm(C)798 1550 y Fi(0)791 1608 y Fg(1)856 1583 y Fm(<)24 b Fn(0.)42 b(W)-8 b(e)31 b(again)g(distinguish)26 b(the)31 b(t)m(w)m(o)g(in)m(terv)-5 b(als)30 b([0)p Fm(;)15 b(\024)2777 1550 y Fi(\000)p Fl(\017)2866 1583 y Fn(])30 b(and)g([)p Fm(\024)3175 1550 y Fi(\000)p Fl(\017)3263 1583 y Fm(;)15 b(d)p Fn(].)0 1696 y Fe(Step)35 b(4.1:)47 b(Study)35 b(on)g(the)f(in)m(terv)-6 b(al)35 b Fn([0)p Fm(;)15 b(\024)1622 1663 y Fi(\000)p Fl(\017)1711 1696 y Fn(])66 b(F)-8 b(rom)31 b(\(3.4\),)h(w)m(e)f (remark)f(that)1568 1919 y Fm(E)5 b Fn(\()1685 1858 y Fm(x)p 1685 1898 V 1687 1982 a(d)1747 1919 y Fn(\))26 b(=)f(0)51 b(on)f([0)p Fm(;)15 b(\024)2308 1882 y Fi(\000)p Fl(\017)2397 1919 y Fn(])p Fm(:)1336 b Fn(\(3.34\))0 2149 y(According)30 b(to)h(\(3.6\),)i(\(3.9\))f(and)e(\(3.10\),)i(it)e (results)f(that)878 2330 y(~)858 2354 y Fm(f)903 2368 y Fg(2)942 2354 y Fm(:)p Fn(\()p Fh(\000)p Fn(1)21 b(+)f Fm(f)1285 2316 y Fg(2)1344 2354 y Fn(+)g Fm(f)1524 2330 y Fn(~)1505 2354 y Fm(f)1550 2368 y Fg(1)1609 2354 y Fn(+)1719 2330 y(~)1700 2354 y Fm(f)1755 2316 y Fg(2)1745 2376 y(1)1793 2354 y Fn(\))26 b(=)f Fh(O)s Fn(\()2080 2330 y(~)2060 2354 y Fm(f)2105 2368 y Fg(2)2144 2354 y Fn(\))h(=)f Fh(O)s Fn(\()p Fm(\024)2463 2316 y Fg(2)2503 2354 y Fm(h)2555 2316 y Fg(2)2595 2354 y Fn(\))p Fm(:)15 b Fn(exp\()p Fh(\000)p Fn(2)p Fm(f)3005 2368 y Fg(0)3045 2354 y Fm(x)p Fn(\))p Fm(:)626 b Fn(\(3.35\))0 2558 y(Moreo)m(v)m(er,) 33 b(from\(3.34\),)f(w)m(e)f(get)659 2824 y Fh(\000)p Fm(\025)783 2839 y Fl(d;h;\024)959 2824 y Fn(exp\()p Fh(\000)1204 2745 y(p)p 1280 2745 46 4 v 79 x Fn(2)p Fm(\024)p Fn(\()p Fm(d)22 b Fh(\000)e Fm(x)p Fn(\)\))p Fm(:)p Fn([2)p Fm(E)5 b Fn(\()1906 2763 y Fm(x)p 1906 2803 52 4 v 1908 2887 a(d)1970 2824 y Fn(\))20 b(+)2126 2763 y(2)2171 2687 y Fh(p)p 2247 2687 46 4 v 76 x Fn(2)p 2126 2803 167 4 v 2160 2887 a Fm(\024d)2303 2824 y(E)2375 2787 y Fi(0)2398 2824 y Fn(\()2443 2763 y Fm(x)p 2443 2803 52 4 v 2445 2887 a(d)2506 2824 y Fn(\))g(+)2744 2763 y(1)p 2662 2803 210 4 v 2662 2887 a(\()p Fm(\024d)p Fn(\))2831 2860 y Fg(2)2882 2824 y Fm(E)2954 2787 y Fi(00)2997 2824 y Fn(\()3042 2763 y Fm(x)p 3042 2803 52 4 v 3044 2887 a(d)3104 2824 y Fn(\)])26 b(=)f(0)p Fm(:)427 b Fn(\(3.36\))0 3094 y(As)26 b(0)g Fm(<)317 3070 y Fn(~)297 3094 y Fm(f)342 3108 y Fg(1)406 3094 y Fm(<)f Fn(1,)j Fm(C)665 3108 y Fg(1)730 3094 y Fm(<)d Fn(0)i(and)e Fm(C)1142 3061 y Fi(0)1135 3118 y Fg(1)1200 3094 y Fm(<)g Fn(0,)j(the)e(\014rst)g(term)h (in)e(the)h(righ)m(t-hand-side)f(of)i(\(3.33\))h(is)e(negativ)m(e.)40 b(It)27 b(results)0 3218 y(that)i(the)f(sign)f(of)h(the)g(righ)m (t-hand-side)f(of)h(\(3.33\))i(dep)s(ends)c(on)i(the)h(sign)e(of)h (\003)d(:=)g Fm(A)2969 3185 y Fg(2)3009 3218 y Fm(f)g Fh(\000)3175 3182 y Fl(h)3216 3158 y Fp(2)p 3175 3197 76 4 v 3178 3249 a Fl(f)3212 3258 y Fp(0)3276 3218 y Fn(exp)o(\()p Fh(\000)p Fn(2)p Fm(f)3610 3232 y Fg(0)3650 3218 y Fm(x)p Fn(\))p Fm(:)j Fn(Using)0 3330 y(\(3.14\),)33 b(\(3.31\))g(and)c(expanding)g(the)i(term)f(exp\()p Fh(\000)p Fn(2)p Fm(f)10 b Fn(\(0\))p Fm(x)p Fn(\),)32 b(w)m(e)f(get)909 3590 y(\003)25 b Fh(\024)1103 3529 y Fm(h)1155 3496 y Fg(2)p 1103 3569 92 4 v 1107 3652 a Fm(f)1152 3666 y Fg(0)1220 3590 y Fn(exp\()p Fh(\000)p Fn(2)p Fm(f)1555 3604 y Fg(0)1595 3590 y Fm(x)p Fn(\)\(exp\(\003)1954 3604 y Fg(1)1994 3590 y Fm(x)p Fn(\)\(1)c(+)2297 3529 y(\003)2360 3543 y Fg(1)p 2283 3569 130 4 v 2283 3652 a Fn(2)p Fm(f)2373 3666 y Fg(0)2443 3590 y Fn(+)2623 3529 y(1)p 2544 3569 206 4 v 2544 3587 a Fh(p)p 2619 3587 46 4 v 2619 3663 a Fn(2)q Fm(f)2710 3677 y Fg(0)2759 3590 y Fm(\024)2811 3553 y Fl(\027)2854 3590 y Fn(\))g Fh(\000)f Fn(1\))p Fm(;)0 3843 y Fn(where)185 4053 y(\003)248 4067 y Fg(1)312 4053 y Fn(:=)185 4211 y Fh(\000)p Fn(2)316 4110 y Fk(\020\020)434 4163 y Fg(1)p Fi(\000)p Fl(f)565 4139 y Fp(2)558 4183 y(0)p 434 4190 166 4 v 465 4243 a Fg(2)p Fl(f)534 4252 y Fp(0)610 4211 y Fm(C)675 4225 y Fg(1)735 4211 y Fh(\000)896 4174 y Fl(C)946 4183 y Fp(0)p 836 4190 206 4 v 836 4255 a Fg(2)871 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3554 y Fm(x)p 2294 3594 52 4 v 2296 3678 a(d)2357 3615 y Fn(\))2392 3460 y Fk(!)2479 3615 y Fm(:)0 3886 y Fn(F)-8 b(rom)31 b(\(3.5\))h(and)e(\(3.18\),)j(w)m(e)e(ha)m(v)m(e)485 4107 y(\003)548 4074 y Fi(00)616 4107 y Fn(=)25 b(2)p Fm(g)s Fn(\()p Fm(\024d)p Fn(\))p Fm(z)1014 4122 y Fl(\024;h)1137 4107 y Fn(exp\()p Fh(\000)1382 4032 y(p)p 1457 4032 46 4 v 1457 4107 a Fn(2)q Fm(\024)p Fn(\003)1618 4121 y Fg(2)1658 4107 y Fm(x)p Fn(\)[)p Fm(C)1842 4074 y Fi(0)1835 4132 y Fg(1)1875 4107 y Fm(z)1917 4122 y Fl(\024;h)2042 4107 y Fn(+)20 b(12\()p Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))2508 4074 y Fi(\000)p Fg(1)2606 4107 y Fm(z)2652 4074 y Fg(2)2648 4135 y Fl(\024;h)2754 4107 y Fn(\003)2817 4121 y Fg(2)2856 4107 y Fn(\(exp\()p Fh(\000)3136 4032 y(p)p 3212 4032 V 75 x Fn(2)q Fm(\024)p Fn(\003)3373 4121 y Fg(2)3413 4107 y Fm(d)p Fn(\)\))485 4266 y Fh(\002)15 b Fn(exp\()p Fh(\000)816 4190 y(p)p 892 4190 V 76 x Fn(2)p Fm(\024)p Fn(\003)1052 4280 y Fg(2)1092 4266 y Fn(\()p Fm(d)21 b Fh(\000)f Fm(x)p 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Fm(:)p Fn(\()p Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))2189 5339 y Fi(\000)p Fg(1)2286 5376 y Fm(z)2328 5391 y Fl(\024;h)2449 5376 y Fn(exp)o(\()p Fh(\000)2703 5240 y(p)p 2779 5240 46 4 v 75 x Fn(2)p 2703 5355 122 4 v 2741 5439 a(4)2835 5376 y Fm(\024d)p Fn(\(1)h(+)f(\003)3189 5390 y Fg(2)3229 5376 y Fn(\)\))370 5509 y Fk(\000)412 5583 y Fn(1)g(+)g Fh(O)s Fn(\(exp\()p Fh(\000)923 5508 y(p)p 999 5508 46 4 v 75 x Fn(2)q Fm(\024d)p Fn(\003)1207 5550 y Fi(0)1207 5607 y Fg(2)1247 5583 y Fn(\)\))1317 5509 y Fk(\001)1374 5583 y Fn(\()1419 5521 y(2)1464 5446 y Fh(p)p 1541 5446 V 1541 5521 a Fn(2)p 1419 5562 167 4 v 1453 5645 a Fm(\024d)1596 5583 y Fh(k)p Fm(E)1713 5545 y Fi(0)1737 5583 y Fh(k)1782 5597 y Fi(1)1878 5583 y Fn(+)2061 5521 y(1)p 1979 5562 210 4 v 1979 5645 a(\()p Fm(\024d)p Fn(\))2148 5619 y Fg(2)2198 5583 y Fh(k)p Fm(E)2315 5545 y Fi(00)2359 5583 y Fh(k)2404 5597 y Fi(1)2479 5583 y Fn(\)])p Fm(:)1962 5871 y Fn(10)p eop %%Page: 11 11 11 10 bop 0 162 a Fn(If)36 b Fm(C)169 129 y Fi(0)162 187 y Fg(1)236 162 y Fm(<)f Fn(0,)j(the)e(term)g(b)s(et)m(w)m(een)h (the)g(last)f(square)g(brac)m(k)m(ets,)j(and)c(then)h(quan)m(tit)m(y)h (\003)3042 129 y Fi(00)3120 162 y Fn(will)d(b)s(e)h(negativ)m(e,)k(for) d Fm(\024)0 275 y Fn(small)29 b(enough)h(and)g Fm(\024d)h Fn(large)f(enough)g(if)1188 538 y Fm(g)s Fn(\()p Fm(y)s Fn(\))1352 501 y Fi(\000)p Fg(1)1462 538 y Fn(exp)1616 383 y Fk( )1688 538 y Fh(\000)1769 402 y(p)p 1845 402 46 4 v 75 x Fn(2)p 1769 517 122 4 v 1807 601 a(4)1900 538 y Fm(y)s Fn(\(2)21 b(+)f Fm(C)2205 552 y Fg(1)2244 538 y Fm(\024)2296 501 y Fl(\027)2360 538 y Fn(+)g Fm(C)2523 501 y Fi(0)2516 561 y Fg(1)2555 538 y Fm(g)s Fn(\()p Fm(y)s Fn(\)\))2754 383 y Fk(!)0 801 y Fn(tends)30 b(to)h(0)g(when)e Fm(y)k Fn(tends)d(to)h(+)p Fh(1)p Fn(.)41 b(W)-8 b(e)31 b(deduce)f(the)h(result)e(when)g Fm(g)34 b Fn(satis\014es)1237 1049 y Fm(g)s Fn(\()p Fm(y)s Fn(\))26 b(=)f(exp\()p Fh(\000)p Fm(\013)1826 1063 y Fg(0)1866 1049 y Fm(y)s Fn(\))51 b(with)e(0)26 b Fm(<)f(\013)2452 1063 y Fg(0)2516 1049 y Fm(<)2622 912 y Fh(p)p 2698 912 46 4 v 76 x Fn(2)p 2622 1028 122 4 v 2660 1112 a(2)2753 1049 y Fm(:)0 1261 y Fn(The)30 b(pro)s(of)f(of)i(Prop)s(osition)d(3.2)k(follo)m(ws.)0 1502 y Fv(3.2)112 b(Construction)37 b(of)g(sup)s(ersolutions)g(of)h Fu(\()p Fa(GL)p Fu(\))2142 1466 y Fl(s)2142 1528 y(d)0 1674 y Fn(In)h([13)r(],)k(C.)d(Bolley)g(and)f(B.)i(Hel\013er)f(remark)g (that)h Fm(f)51 b Fh(\021)41 b Fn(1)f(is)g(a)g(sup)s(ersolution)d(of)k (\()p Fm(GL)p Fn(\))3266 1641 y Fl(s)3266 1702 y(d)3307 1674 y Fn(.)70 b(In)40 b(order)g(to)h(get)0 1787 y(appro)m(ximate)g (solutions,)i(w)m(e)e(should)e(construct)j(more)f(accurate)i(sup)s (ersolutions.)70 b(This)39 b(is)h(the)i(goal)f(of)g(the)0 1900 y(follo)m(wing)29 b(prop)s(osition:)0 2087 y Fe(Prop)s(osition)36 b(3.3)46 b Ff(L)-5 b(et)39 b Fm(\017)d Fh(2)p Fn(]0)p Fm(;)15 b Fn(1[)p Ff(,)42 b Fm(\013)1349 2101 y Fg(0)1424 2087 y Fh(2)p Fn(]0)p Fm(;)1606 1995 y Fi(p)p 1665 1995 36 3 v 55 x Fg(2)p 1606 2066 95 4 v 1635 2118 a(2)1710 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Fg(1)2239 3095 y Ff(,)33 b(by)951 3317 y Fm(f)10 b Fn(\()p Fm(x)p Fn(\))25 b(=)g(tanh)1446 3216 y Fk(\020)1537 3281 y Fl(\024)p 1510 3296 95 4 v 1510 3305 a Fi(p)p 1569 3305 36 3 v 55 x Fg(2)1614 3317 y Fn(\(1)c(+)1827 3294 y(\026)1806 3317 y Fm(C)1871 3331 y Fg(1)1911 3317 y Fm(\024)1963 3284 y Fg(1)p Fi(\000)p Fl(\017)2106 3317 y Fn(+)2218 3294 y(\026)2197 3317 y Fm(C)2269 3284 y Fi(0)2262 3341 y Fg(1)2305 3317 y Fn(\026)-49 b Fm(g)t Fn(\()p Fm(\024d)p Fn(\)\))p Fm(x)22 b Fn(+)k(\026)-51 b Fm(x)2769 3332 y Fl(\024;h)2874 3216 y Fk(\021)951 3513 y Fn(+)p Fm(\024)1074 3476 y Fg(2)1147 3452 y Fm(h)1199 3419 y Fg(2)p 1123 3492 140 4 v 1123 3577 a Fn(4)p Fm(f)1223 3546 y Fg(3)1213 3603 y(0)1288 3513 y Fn(exp\()p Fh(\000)p Fn(2)p Fm(f)1623 3527 y Fg(0)1662 3513 y Fm(x)p Fn(\))21 b(+)1864 3489 y(\026)1860 3513 y Fm(\025)1913 3528 y Fl(d;\024;h)2090 3513 y Fn(exp\()p Fh(\000)2335 3433 y(p)p 2410 3433 46 4 v 2410 3513 a Fn(2)q Fm(\024)p Fn(\()p Fm(d)g Fh(\000)f Fm(x)p Fn(\)\))p Fm(E)5 b Fn(\()2941 3452 y Fm(x)p 2941 3492 52 4 v 2943 3575 a(d)3004 3513 y Fn(\))p Fm(;)3783 3427 y Fn(\(3.43\))0 3765 y Ff(wher)-5 b(e)34 b(the)f(c)-5 b(onstants)40 b Fn(\026)-51 b Fm(x)859 3780 y Fl(\024;h)997 3765 y Ff(and)1177 3741 y Fn(\026)1173 3765 y Fm(\025)1226 3780 y Fl(d;\024;h)1420 3765 y Ff(ar)-5 b(e)33 b(determine)-5 b(d)35 b(by)d(the)h(c)-5 b(onditions:)1588 3956 y Fm(f)1643 3918 y Fi(0)1666 3956 y Fn(\(0\))26 b(=)f(0)p Fm(;)67 b(f)2095 3918 y Fi(0)2117 3956 y Fn(\()p Fm(d)p Fn(\))27 b(=)e(0)p Fm(:)1356 b Fn(\(3.44\))0 4155 y Fe(Remark)34 b(3.4)46 b Ff(L)-5 b(et)38 b(us)g(r)-5 b(emark)39 b(that)45 b Fn(\026)-51 b Fm(x)1419 4170 y Fl(\024;h)1562 4155 y Ff(and)1747 4131 y Fn(\026)1743 4155 y Fm(\025)1796 4170 y Fl(d;\024;h)1995 4155 y Ff(ar)-5 b(e)39 b(determine)-5 b(d)39 b(by)e(\(3.14\))j(and)e(\(3.18\))i(r)-5 b(eplacing)38 b Fm(C)3948 4169 y Fg(0)3988 4155 y Ff(,)0 4268 y Fm(C)65 4282 y Fg(1)104 4268 y Ff(,)33 b Fm(C)237 4235 y Fi(0)230 4292 y Fg(0)302 4268 y Ff(and)g Fm(C)550 4235 y Fi(0)543 4292 y Fg(1)615 4268 y Ff(with)833 4245 y Fn(\026)813 4268 y Fm(C)878 4282 y Fg(0)917 4268 y Ff(,)998 4245 y Fn(\026)978 4268 y Fm(C)1043 4282 y Fg(1)1082 4268 y Ff(,)1163 4245 y Fn(\026)1142 4268 y Fm(C)1214 4235 y Fi(0)1207 4292 y Fg(0)1279 4268 y Ff(and)1476 4245 y Fn(\026)1456 4268 y Fm(C)1528 4235 y Fi(0)1521 4292 y Fg(1)1593 4268 y Ff(in)f(these)h(estimates.)0 4467 y Fn(T)-8 b(o)34 b(construct)h(the)f(sup)s(ersolution,)e(w)m(e)i (should)e(get)k(a)e(lo)m(w)m(er)g(b)s(ound)e(for)i Fm(A)2690 4434 y Fg(2)2729 4467 y Fm(f)44 b Fn(in)32 b(the)i(in)m(terv)-5 b(al)34 b([0)p Fm(;)15 b(\024)3582 4434 y Fi(\000)p Fl(\017)3671 4467 y Fn(].)51 b(In)33 b(the)0 4580 y(in)m(terv)-5 b(al)34 b([)p Fm(\024)410 4547 y Fi(\000)p Fl(\017)498 4580 y Fm(;)15 b(d)p Fn(],)36 b(w)m(e)f(will)d(sho)m(w)i(that)h(w)m(e)g(can)f (neglect)h(this)f(term.)52 b(F)-8 b(ollo)m(wing)34 b(the)h(metho)s(d)f (used)f(in)g([17)q(],)j(w)m(e)0 4693 y(impro)m(v)m(e)30 b(Inequalit)m(y)g(\(2.3\).)0 4870 y Fe(Lemma)j(3.5)46 b Ff(L)-5 b(et)31 b Fn(\()p Fm(\024;)15 b(d)p Fn(\))27 b Fh(2)e Fn(\(]0)p Fm(;)15 b Fn(+)p Fh(1)p Fn([\))1394 4837 y Fg(2)1465 4870 y Ff(and)32 b Fm(\017)25 b Fh(2)p Fn(]0)p Fm(;)15 b Fn(1[)p Ff(.)42 b(L)-5 b(et)31 b Fm(A)g Ff(b)-5 b(e)30 b(the)h(solution)h(of)f(\(3.2\))g(wher)-5 b(e)32 b Fm(f)40 b Ff(is)30 b(de\014ne)-5 b(d)32 b(in)0 4982 y(\(3.43\).)43 b(Ther)-5 b(e)34 b(exist)e Fm(\024)836 4996 y Fg(0)876 4982 y Ff(,)h Fm(d)984 4996 y Fg(0)1056 4982 y Ff(and)g Fm(C)40 b Ff(such)32 b(that)i(for)f(al)5 b(l)33 b Fm(\024)26 b Fh(\024)f Fm(\024)2230 4996 y Fg(0)2270 4982 y Ff(,)32 b(for)h Fm(\024d)26 b Fh(\025)f Fm(d)2743 4996 y Fg(0)2783 4982 y Ff(,)32 b(we)h(have)923 5216 y Fm(A)p Fn(\()p Fm(x)p Fn(\))26 b Fh(\024)f(\000)1383 5155 y Fm(h)p 1316 5195 187 4 v 1316 5278 a Fn(\003\()p Fm(\024)p Fn(\))1512 5216 y(\(1)c Fh(\000)f Fm(C)7 b(\024)1828 5179 y Fg(2)1867 5216 y Fn(\))15 b(exp)q(\()p Fh(\000)p Fn(\003\()p Fm(\024)p Fn(\))p Fm(x)p Fn(\))p Fm(;)92 b Fh(8)25 b Fm(x)g Fh(2)g Fn([0)p Fm(;)15 b(\024)2953 5179 y Fi(\000)p Fl(\017)3042 5216 y Fn(])p Fm(;)691 b Fn(\(3.45\))0 5451 y Ff(wher)-5 b(e)1069 5570 y Fn(\003\()p Fm(\024)p Fn(\))26 b(:=)g Fm(f)10 b Fn(\(0\))20 b(+)1693 5521 y Fg(1)p Fi(\000)p Fl(f)1824 5498 y Fp(2)1817 5542 y(0)p 1693 5549 166 4 v 1729 5559 a Fi(p)p 1788 5559 36 3 v 55 x Fg(2)1869 5570 y Fm(\024)1921 5537 y Fl(\027)1985 5570 y Fn(+)g Fh(O)s Fn(\()p Fm(\024)2238 5537 y Fg(2)p Fl(\027)2317 5570 y Fn(\))g(+)g Fh(O)s Fn(\()p Fm(\024)2625 5537 y Fl(\027)2672 5570 y Fn(\026)-48 b Fm(g)t Fn(\()p Fm(\024d)p Fn(\)\))p Fm(:)3783 5563 y Fn(\(3.46\))1962 5871 y(11)p eop %%Page: 12 12 12 11 bop 0 162 a Fe(Pro)s(of)34 b Fn(F)-8 b(rom)34 b(\(3.14\))i(and)c (\(3.43\),)37 b(as)d Fm(f)42 b Fn(is)33 b(increasing)f(on)h([0)p Fm(;)15 b(\024)2282 129 y Fi(\000)p Fl(\017)2371 162 y Fn(],)34 b(w)m(e)g(remark)f(that,)i(for)e(all)g Fm(x)d Fh(2)g Fn([0)p Fm(;)15 b(\024)3739 129 y Fi(\000)p Fl(\017)3828 162 y Fn(],)34 b(w)m(e)0 275 y(ha)m(v)m(e)e(for)e Fm(\024)g Fn(small)f(and)h Fm(\024d)h Fn(large)565 502 y Fm(f)10 b Fn(\()p Fm(x)p Fn(\))25 b Fh(\024)g Fn(\003\()p Fm(\024)p Fn(\))i(:=)e Fm(f)10 b Fn(\()p Fm(\024)1338 464 y Fi(\000)p Fl(\017)1426 502 y Fn(\))25 b(=)g(tanh\()6 b(\026)-51 b Fm(x)1851 517 y Fl(\024;h)1956 502 y Fn(\))21 b(+)2113 440 y(1)f Fh(\000)g Fm(f)2324 407 y Fg(2)2314 464 y(0)p 2113 481 251 4 v 2177 499 a Fh(p)p 2253 499 46 4 v 75 x Fn(2)2373 502 y Fm(\024)2425 464 y Fl(\027)2489 502 y Fn(+)g Fh(O)s Fn(\()p Fm(\024)2742 464 y Fg(2)p Fl(\027)2821 502 y Fn(\))g(+)g Fh(O)s Fn(\()p Fm(\024)3129 464 y Fl(\027)3176 502 y Fn(\026)-48 b Fm(g)t Fn(\()p Fm(\024d)p Fn(\)\))p Fm(:)334 b Fn(\(3.47\))0 726 y(As)30 b Fm(f)10 b Fn(\(0\))26 b(=)f(tanh\()6 b(\026)-51 b Fm(x)695 741 y Fl(\024;h)800 726 y Fn(\))21 b(+)f Fh(O)s Fn(\()p Fm(\024)p Fn(\),)31 b(w)m(e)g(get)h(the)e(estimate)927 936 y(\003\()p Fm(\024)p Fn(\))d(=)d Fm(f)10 b Fn(\(0\))21 b(+)1526 887 y Fg(1)p Fi(\000)p Fl(f)1657 863 y Fp(2)1650 907 y(0)p 1526 915 166 4 v 1562 924 a Fi(p)p 1621 924 36 3 v 55 x Fg(2)1702 936 y Fm(\024)1754 903 y Fl(\027)1818 936 y Fn(+)f Fh(O)s Fn(\()p Fm(\024)p Fn(\))h(+)f Fh(O)s Fn(\()p Fm(\024)2380 903 y Fg(2)p Fl(\027)2459 936 y Fn(\))g(+)g Fh(O)s Fn(\()p Fm(\024)2767 903 y Fl(\027)2814 936 y Fn(\026)-48 b Fm(g)t Fn(\()p Fm(\024d)p Fn(\)\))p Fm(:)3783 929 y Fn(\(3.48\))0 1135 y(T)-8 b(o)31 b(get)g(an)g(upp)s(er)d(b)s(ound)g(for)i Fm(A)p Fn(,)h(w)m(e)g(compare)g(this)e(function)g(with)g(the)i(unique)d (solution)h(in)g Fm(H)3438 1102 y Fg(2)3478 1135 y Fn(\([0)p Fm(;)15 b(d)p Fn(]\))32 b(of)1412 1229 y Fk(8)1412 1311 y(<)1412 1475 y(:)1535 1345 y Fm(a:)46 b Fh(\000)19 b Fm(y)1792 1312 y Fi(00)1855 1345 y Fn(+)h(\003\()p Fm(\024)p Fn(\))2131 1312 y Fg(2)2172 1345 y Fm(y)28 b Fn(=)d(0)p Fm(;)1535 1503 y(b:)51 b(y)1698 1470 y Fi(0)1721 1503 y Fn(\(0\))26 b(=)f Fm(h;)66 b(y)s Fn(\()p Fm(\024)2236 1470 y Fi(\000)p Fl(\017)2325 1503 y Fn(\))25 b(=)g(0)p Fm(:)3783 1416 y Fn(\(3.49\))0 1704 y(Applying)g(Lemma)j(2.2)g(with)f Fm(u)e Fn(=)g Fm(A)15 b Fh(\000)g Fm(y)29 b Fn(and)e Fm(C)7 b Fn(\()p Fm(x)p Fn(\))25 b(=)g(\003\()p Fm(\024)p Fn(\))2151 1671 y Fg(2)2192 1704 y Fn(,)j(from)f(\(3.47\),)k(it)c (deriv)m(es)g(that)h Fm(A)e Fh(\024)e Fm(y)31 b Fn(on)c([0)p Fm(;)15 b(\024)3876 1671 y Fi(\000)p Fl(\017)3965 1704 y Fn(].)0 1817 y(The)30 b(solution)f(of)h(\(3)p Fm(:)p Fn(49\))864 1831 y Fl(a)939 1817 y Fn(is)f(giv)m(en)h(b)m(y)1068 1993 y Fm(y)s Fn(\()p Fm(x)p Fn(\))c(=)f Fm(C)1425 2007 y Fg(1)1464 1993 y Fn(\()p Fm(\024)p Fn(\))15 b(exp)q(\()p Fh(\000)p Fn(\003\()p Fm(\024)p Fn(\))p Fm(x)p Fn(\))22 b(+)e Fm(C)2297 2007 y Fg(2)2336 1993 y Fn(\()p Fm(\024)p Fn(\))15 b(exp)r(\(\003\()p Fm(\024)p Fn(\))p Fm(x)p Fn(\))p Fm(;)837 b Fn(\(3.50\))0 2169 y(and)30 b(from)g(\(3)p Fm(:)p Fn(49\))622 2184 y Fl(b)658 2169 y Fn(,)h(it)f(results)f(that) 709 2398 y Fm(C)774 2412 y Fg(1)813 2398 y Fn(\()p Fm(\024)p Fn(\))d(=)f Fh(\000)1205 2336 y Fm(h)p 1138 2377 187 4 v 1138 2460 a Fn(\003\()p Fm(\024)p Fn(\))1354 2398 y(+)20 b Fm(C)1510 2412 y Fg(2)1550 2398 y Fn(\()p Fm(\024)p Fn(\))p Fm(;)92 b Fn(where)75 b Fm(C)2162 2412 y Fg(2)2202 2398 y Fn(\()p Fm(\024)p Fn(\))26 b(=)2527 2336 y Fm(h)15 b Fn(exp\()p Fh(\000)p Fn(\003\()p Fm(\024)p Fn(\))p Fm(\024)3076 2303 y Fi(\000)p Fl(\017)3166 2336 y Fn(\))p 2456 2377 816 4 v 2456 2460 a(2\003\()p Fm(\024)p Fn(\))g(cosh)r (\(\003\()p Fm(\024)p Fn(\))p Fm(\024)3147 2434 y Fi(\000)p Fl(\017)3237 2460 y Fn(\))3282 2398 y Fm(:)476 b Fn(\(3.51\))0 2622 y(F)-8 b(rom)28 b(\(3.48\))j(and)c(\(3.51\),)k(w)m(e)d(ha)m(v)m(e) h(for)f Fm(\024)g Fn(small)f(and)g(for)h(all)e Fm(n)f Fh(2)g Fn(I)-15 b(N,)29 b Fm(C)2553 2636 y Fg(2)2592 2622 y Fn(\()p Fm(\024)p Fn(\))d(=)f Fh(O)s Fn(\()p Fm(\024)2998 2589 y Fl(n)3046 2622 y Fn(\).)40 b(F)-8 b(rom)29 b(\(3.50\),)h(it)e (results)0 2744 y(that)39 b(there)f(exist)g Fm(\024)720 2758 y Fg(0)798 2744 y Fn(and)f Fm(d)1029 2758 y Fg(0)1107 2744 y Fn(suc)m(h)h(that)g(for)g Fm(\024)h Fh(\024)e Fm(\024)1922 2758 y Fg(0)2000 2744 y Fn(and)h Fm(\024d)g Fh(\025)g Fm(d)2478 2758 y Fg(0)2518 2744 y Fn(,)i(w)m(e)f(ha)m(v)m(e)g (on)f(the)g(in)m(terv)-5 b(al)37 b([0)p Fm(;)3697 2709 y Fl(\024)3738 2685 y Fo(\000)p Fj(\017)p 3697 2724 119 4 v 3739 2776 a Fg(2)3826 2744 y Fn(])h(the)0 2857 y(estimate)1199 2997 y Fm(y)s Fn(\()p Fm(x)p Fn(\))26 b(=)1491 2869 y Fk(\022)1558 2997 y Fh(\000)1705 2936 y Fm(h)p 1639 2976 187 4 v 1639 3060 a Fn(\003\()p Fm(\024)p Fn(\))1855 2997 y(+)20 b Fh(O)s Fn(\()p Fm(\024)2108 2960 y Fl(n)2155 2997 y Fn(\))2190 2869 y Fk(\023)2273 2997 y Fn(exp\()p Fh(\000)p Fn(\003\()p Fm(\024)p Fn(\))p Fm(x)p Fn(\))p Fm(:)0 3195 y Fn(Finally)-8 b(,)29 b(for)h Fm(n)25 b Fn(=)g(2)31 b(and)e(for)i(some)f Fm(C)7 b Fn(,)30 b(w)m(e)h(get)985 3419 y Fm(y)d Fh(\024)d(\000)1301 3357 y Fm(h)p 1235 3398 V 1235 3481 a Fn(\003\()p Fm(\024)p Fn(\))1430 3419 y(\(1)c Fh(\000)f Fm(C)7 b(\024)1746 3381 y Fg(2)1786 3419 y Fn(\))15 b(exp\()p Fh(\000)p Fn(\003\()p Fm(\024)p Fn(\))p Fm(x)p Fn(\))p Fm(;)93 b Fh(8)25 b Fm(x)g 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Fi(0)1165 5517 y Fg(0)p 1022 5533 261 4 v 1022 5627 a Fn(2)1067 5551 y Fh(p)p 1143 5551 46 4 v 76 x Fn(2)q Fm(f)1244 5595 y Fg(4)1234 5653 y(0)1293 5399 y Fk(!)1383 5554 y Fn(\026)-48 b Fm(g)s Fn(\()p Fm(\024d)p Fn(\))22 b(+)e Fh(O)s Fn(\()p Fm(\024)p Fn(\))h(+)f Fh(O)s Fn(\()p Fm(\024)2179 5521 y Fg(2)p Fl(\027)2258 5554 y Fn(\))h(+)f Fh(O)s Fn(\()p Fm(\024)2567 5521 y Fl(\027)2613 5554 y Fn(\026)-48 b Fm(g)t Fn(\()p Fm(\024d)p Fn(\)\))22 b(+)e Fh(O)s Fn(\()s(\026)-48 b Fm(g)t Fn(\()p Fm(\024d)p Fn(\))3300 5521 y Fg(2)3341 5554 y Fn(\))p Fm(:)3783 5418 y Fn(\(3.54\))1962 5871 y(12)p eop %%Page: 13 13 13 12 bop 0 162 a Fe(Step)35 b(2:)46 b(The)35 b(function)g Fm(f)45 b Fe(is)35 b(a)f(sup)s(ersolution)0 275 y(Step)i(2.1:)51 b(Lo)m(w)m(er)36 b(b)s(ound)h(on)g Fn([0)p Fm(;)15 b(\024)1399 242 y Fi(\000)p Fl(\017)1489 275 y Fn(])32 b(W)-8 b(e)33 b(start)f(from)g(\(3.33\).)47 b(W)-8 b(e)33 b(assume)f(that)3135 252 y(\026)3114 275 y Fm(C)3179 289 y Fg(1)3247 275 y Fm(>)27 b Fn(0)32 b(and)3621 252 y(\026)3601 275 y Fm(C)3673 242 y Fi(0)3666 300 y Fg(1)3733 275 y Fm(>)c Fn(0,)k(so)0 388 y(w)m(e)j(can)h(also)e(eliminate)g(the)h(\014rst)f(term)h(in)f(the) h(righ)m(t-hand-side)e(of)i(\(3.33\).)57 b(As)35 b(in)e(the)i(pro)s(of) g(of)g(Prop)s(osition)0 510 y(3.2,)29 b(the)e(sign)f(of)h(the)g(righ)m (t-hand-side)e(of)i(\(3.33\))i(dep)s(ends)c(on)i(the)g(sign)f(of)h Fm(A)2720 477 y Fg(2)2760 510 y Fm(f)c Fh(\000)2921 475 y Fl(h)2962 451 y Fp(2)p 2921 490 76 4 v 2924 542 a Fl(f)2958 551 y Fp(0)3022 510 y Fn(exp\()p Fh(\000)p Fn(2)p Fm(f)3357 524 y Fg(0)3396 510 y Fm(x)p Fn(\).)40 b(F)-8 b(rom)27 b(\(3.54\))0 623 y(and)j(follo)m(wing)f(the)h(Step)g(3.1.)42 b(in)29 b(the)i(pro)s(of)e(of)i(Prop)s(osition)d(3.2,)k(w)m(e)f(get)351 805 y Fm(A)419 772 y Fg(2)459 805 y Fm(f)e Fh(\000)634 769 y Fl(h)675 746 y Fp(2)p 634 784 V 637 836 a Fl(f)671 845 y Fp(0)734 805 y Fn(exp\()p Fh(\000)p Fn(2)p Fm(f)1069 819 y Fg(0)1109 805 y Fm(x)p Fn(\))c Fh(\025)1327 769 y Fl(h)1368 746 y Fp(2)p 1327 784 V 1330 836 a Fl(f)1364 845 y Fp(0)1427 805 y Fn(exp\()p Fh(\000)p Fn(2)p Fm(f)1762 819 y Fg(0)1802 805 y Fm(x)p Fn(\))1904 704 y Fk(\020)1959 805 y Fn(exp)o(\(\003)2195 819 y Fg(1)2235 805 y Fn(\()p Fm(\024)p Fn(\))p Fm(x)p Fn(\)\(1)e(+)2648 760 y Fg(\003)2697 769 y Fp(1)2731 760 y Fg(\()p Fl(\024)p Fg(\))p 2648 784 180 4 v 2685 836 a(2)p Fl(f)2754 845 y Fp(0)2857 805 y Fh(\000)2958 701 y Fi(p)p 3017 701 36 3 v 55 x Fg(2\(1)p Fi(\000)p Fl(f)3210 733 y Fp(2)3203 777 y(0)3245 756 y Fg(\))p 2958 784 315 4 v 3081 836 a Fl(f)3115 845 y Fp(0)3283 805 y Fm(\024)3335 772 y Fl(\027)3378 805 y Fn(\))e Fh(\000)f Fn(1)3570 704 y Fk(\021)3640 805 y Fm(;)0 986 y Fn(where)34 b(\003)330 1000 y Fg(1)369 986 y Fn(\()p Fm(\024)p Fn(\))i(is)d(giv)m(en)i(b)m(y)f Fh(\000)p Fn(2\(\003\()p Fm(\024)p Fn(\))24 b Fh(\000)f Fm(f)1493 1000 y Fg(0)1532 986 y Fn(\))p Fm(:)35 b Fn(W)-8 b(e)35 b(remark)g(that)g(the)f(term)h(exp)o(\(\003)2922 1000 y Fg(1)2962 986 y Fn(\()p Fm(\024)p Fn(\))p Fm(x)p Fn(\))h(inside)c(the)j(brac)m(k)m(et)h(is)0 1099 y(larger)30 b(than)g(1)h(for)f(all)f Fm(f)856 1113 y Fg(0)921 1099 y Fh(2)24 b Fn([)p Fm(\016)1071 1113 y Fg(0)1112 1099 y Fm(;)15 b(\016)1192 1113 y Fg(1)1232 1099 y Fn(],)31 b(when)e Fm(\024)i Fn(and)e Fm(g)s Fn(\()p Fm(\024d)p Fn(\))k(are)d(small,)g(under)e(the)j(t)m(w)m(o)h(conditions)1006 1251 y(1)21 b Fh(\000)f Fm(\016)1206 1218 y Fg(2)1203 1275 y(0)p 1006 1291 240 4 v 1064 1375 a Fn(2)p Fm(\016)1149 1389 y Fg(0)1277 1289 y Fn(\026)1256 1312 y Fm(C)1321 1326 y Fg(1)1381 1312 y Fn(+)1482 1251 y(1)g Fh(\000)g Fm(\016)1681 1218 y Fg(2)1678 1275 y(0)p 1482 1291 V 1541 1309 a Fh(p)p 1617 1309 46 4 v 76 x Fn(2)1757 1312 y Fm(<)1956 1228 y Fn(\026)1935 1251 y Fm(C)2000 1265 y Fg(0)p 1863 1291 250 4 v 1863 1385 a Fn(2)1908 1309 y Fh(p)p 1984 1309 46 4 v 76 x Fn(2)p Fm(\016)2072 1353 y Fg(3)2069 1411 y(1)2123 1312 y Fm(;)2249 1251 y Fn(1)g Fh(\000)g Fm(\016)2448 1218 y Fg(2)2445 1275 y(0)p 2249 1291 240 4 v 2306 1375 a Fn(2)p Fm(\016)2391 1389 y Fg(0)2519 1289 y Fn(\026)2499 1312 y Fm(C)2571 1275 y Fi(0)2564 1335 y Fg(1)2628 1312 y Fm(<)2827 1228 y Fn(\026)2807 1251 y Fm(C)2879 1218 y Fi(0)2872 1275 y Fg(0)p 2734 1291 250 4 v 2734 1385 a Fn(2)2779 1309 y Fh(p)p 2855 1309 46 4 v 76 x Fn(2)q Fm(\016)2944 1353 y Fg(3)2941 1411 y(1)2994 1312 y Fm(:)764 b Fn(\(3.55\))0 1532 y(The)23 b(sum)f(\003\()p Fm(\024)p Fn(\))i(de\014ned)e(in)f(\(3.46\))26 b(is)c(then)h(p)s(ositiv)m(e)f(for)g(all)g Fm(f)2142 1546 y Fg(0)2204 1532 y Fn(when)g(the)i(t)m(w)m(o)g(follo)m(wing)e (conditions)f(are)j(satis\014ed)1004 1668 y(\026)983 1691 y Fm(C)1048 1705 y Fg(0)p 910 1732 250 4 v 910 1825 a Fn(2)955 1750 y Fh(p)p 1032 1750 46 4 v 1032 1825 a Fn(2)p Fm(\016)1120 1794 y Fg(4)1117 1851 y(1)1196 1752 y Fm(>)1302 1691 y Fn(1)c Fh(\000)g Fm(\016)1501 1658 y Fg(2)1498 1715 y(0)p 1302 1732 240 4 v 1357 1817 a Fn(2)p Fm(\016)1445 1785 y Fg(2)1442 1843 y(0)1572 1730 y Fn(\026)1551 1752 y Fm(C)1616 1766 y Fg(1)1676 1752 y Fn(+)1777 1616 y Fh(p)p 1853 1616 46 4 v 75 x Fn(2\(1)h Fh(\000)f Fm(\016)2133 1658 y Fg(2)2130 1715 y(0)2173 1691 y Fn(\))p 1777 1732 432 4 v 1953 1815 a Fm(\016)1993 1829 y Fg(0)2219 1752 y Fm(;)2438 1668 y Fn(\026)2417 1691 y Fm(C)2489 1658 y Fi(0)2482 1715 y Fg(0)p 2345 1732 250 4 v 2345 1825 a Fn(2)2390 1750 y Fh(p)p 2466 1750 46 4 v 75 x Fn(2)p Fm(\016)2554 1794 y Fg(4)2551 1851 y(1)2630 1752 y Fm(>)2736 1691 y Fn(1)g Fh(\000)g Fm(\016)2935 1658 y Fg(2)2932 1715 y(0)p 2736 1732 240 4 v 2791 1817 a Fn(2)p Fm(\016)2879 1785 y Fg(2)2876 1843 y(0)3006 1730 y Fn(\026)2986 1752 y Fm(C)3058 1715 y Fi(0)3051 1775 y Fg(1)3090 1752 y Fm(:)668 b Fn(\(3.56\))0 1979 y(W)-8 b(e)32 b(can)e(c)m(ho)s(ose)i(\()668 1956 y(\026)647 1979 y Fm(C)712 1993 y Fg(0)751 1979 y Fm(;)812 1956 y Fn(\026)791 1979 y Fm(C)863 1946 y Fi(0)856 2003 y Fg(0)896 1979 y Fm(;)957 1956 y Fn(\026)936 1979 y Fm(C)1001 1993 y Fg(1)1041 1979 y Fm(;)1102 1956 y Fn(\026)1081 1979 y Fm(C)1153 1946 y Fi(0)1146 2003 y Fg(1)1186 1979 y Fn(\))e(satisfying)g(\(3.55\))i(and)e(\(3.56\).)43 b(W)-8 b(e)31 b(th)m(us)f(get)i(the)f(result)e(in)g([0)p Fm(;)15 b(\024)3580 1946 y Fi(\000)p Fl(\017)3669 1979 y Fn(].)0 2092 y Fe(Step)35 b(2.2:)47 b(Lo)m(w)m(er)35 b(b)s(ound)h(on)f Fn([)p Fm(\024)1305 2059 y Fi(\000)p Fl(\017)1394 2092 y Fm(;)15 b(d)p Fn(])31 b(On)f(the)h(in)m(terv)-5 b(al)29 b([)p Fm(\024)2251 2059 y Fi(\000)p Fl(\017)2340 2092 y Fm(;)15 b(d)p Fn(],)31 b(w)m(e)g(can)g(neglect)h(the)e(p)s (ositiv)m(e)g(term)h Fm(A)3897 2059 y Fg(2)3936 2092 y Fm(f)10 b Fn(.)0 2205 y(F)-8 b(rom)31 b(\(3.33\),)i(w)m(e)e(get)369 2392 y Fh(\000)p Fm(\024)492 2359 y Fi(\000)p Fg(2)587 2392 y Fm(f)642 2359 y Fi(00)703 2392 y Fn(+)20 b(\()p Fh(\000)p Fn(1)h(+)f Fm(f)1112 2359 y Fg(2)1171 2392 y Fn(+)g Fm(A)1330 2359 y Fg(2)1369 2392 y Fn(\))p Fm(f)35 b Fh(\025)369 2550 y Fn(\(2)471 2527 y(\026)449 2550 y Fm(C)514 2564 y Fg(1)554 2550 y Fm(\024)606 2517 y Fl(\027)670 2550 y Fn(+)20 b(2)827 2527 y(\026)806 2550 y Fm(C)878 2517 y Fi(0)871 2574 y Fg(1)914 2550 y Fn(\026)-48 b Fm(g)s Fn(\()p Fm(\024d)p Fn(\))22 b(+)e Fh(O)s Fn(\()p Fm(\024)1401 2517 y Fg(2)p Fl(\027)1500 2550 y Fn(+)j(\026)-48 b Fm(g)t Fn(\()p Fm(\024d)p Fn(\)\))1842 2517 y Fg(2)1883 2550 y Fn(\))1938 2526 y(~)1918 2550 y Fm(f)1963 2564 y Fg(1)2002 2550 y Fn(\(0\))5 b(\026)-50 b Fm(z)2159 2565 y Fl(\024;h)2281 2550 y Fn(exp\()p Fh(\000)2526 2475 y(p)p 2602 2475 46 4 v 75 x Fn(2)p Fm(\024)p Fn(\(1)22 b(+)2912 2527 y(\026)2891 2550 y Fm(C)2956 2564 y Fg(1)2996 2550 y Fm(\024)3048 2517 y Fl(\027)3112 2550 y Fn(+)3223 2527 y(\026)3203 2550 y Fm(C)3275 2517 y Fi(0)3268 2574 y Fg(1)3310 2550 y Fn(\026)-48 b Fm(g)t Fn(\()p Fm(\024d)p Fn(\)\))p Fm(x)p Fn(\))369 2734 y Fh(\000)450 2673 y Fm(h)502 2640 y Fg(2)p 450 2714 92 4 v 454 2797 a Fm(f)499 2811 y Fg(0)552 2734 y Fn(\(1)21 b(+)e Fh(O)s Fn(\()p Fm(\024)905 2697 y Fg(2)946 2734 y Fn(\)\))c(exp\()p Fh(\000)p Fn(2)p Fm(f)1366 2748 y Fg(0)1406 2734 y Fm(x)p Fn(\))369 2974 y Fh(\000)444 2950 y Fn(\026)440 2974 y Fm(\025)493 2989 y Fl(d;h;\024)669 2974 y Fn(exp\()p Fh(\000)914 2898 y(p)p 990 2898 46 4 v 76 x Fn(2)p Fm(\024)p Fn(\()p Fm(d)21 b Fh(\000)f Fm(x)p Fn(\)\))p Fm(:)1443 2818 y Fk(")1497 2974 y Fn(3)p Fm(E)5 b Fn(\()1659 2912 y Fm(x)p 1659 2953 52 4 v 1661 3036 a(d)1722 2974 y Fn(\))21 b(+)1879 2912 y(2)1924 2837 y Fh(p)p 2000 2837 46 4 v 75 x Fn(2)p 1879 2953 167 4 v 1912 3036 a Fm(\024d)2056 2974 y(E)2128 2936 y Fi(0)2151 2974 y Fn(\()2196 2912 y Fm(x)p 2196 2953 52 4 v 2198 3036 a(d)2258 2974 y Fn(\))g(+)2497 2912 y(1)p 2415 2953 210 4 v 2415 3036 a(\()p Fm(\024d)p Fn(\))2584 3010 y Fg(2)2635 2974 y Fm(E)2707 2936 y Fi(00)2749 2974 y Fn(\()2794 2912 y Fm(x)p 2794 2953 52 4 v 2796 3036 a(d)2857 2974 y Fn(\))2892 2818 y Fk(#)369 3198 y Fn(+3)505 3174 y(~)485 3198 y Fm(f)540 3165 y Fg(2)530 3222 y(1)583 3174 y Fn(\026)579 3198 y Fm(\025)632 3213 y Fl(d;h;\024)808 3198 y Fn(exp\()p Fh(\000)1053 3123 y(p)p 1129 3123 46 4 v 75 x Fn(2)p Fm(\024)p Fn(\()p Fm(d)h Fh(\000)e Fm(x)p Fn(\)\))p 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Fn(\)\))c(exp\()p Fh(\000)p Fn(2)p Fm(f)1366 3947 y Fg(0)1406 3933 y Fm(x)p Fn(\))20 b Fh(\000)1608 3909 y Fn(\026)1604 3933 y Fm(\025)1657 3948 y Fl(d;h;\024)1833 3933 y Fn(exp\()p Fh(\000)2078 3853 y(p)p 2154 3853 46 4 v 80 x Fn(2)q Fm(\024)p Fn(\()p Fm(d)h Fh(\000)f Fm(x)p Fn(\)\))p Fm(:)2686 3871 y Fn(1)p 2603 3912 210 4 v 2603 3995 a(\()p Fm(\024d)p Fn(\))2772 3969 y Fg(2)2823 3933 y Fm(E)2895 3895 y Fi(00)2938 3933 y Fn(\()2983 3871 y Fm(x)p 2983 3912 52 4 v 2985 3995 a(d)3045 3933 y Fn(\))369 4129 y(+)444 4105 y(\026)440 4129 y Fm(\025)493 4096 y Fg(3)493 4156 y Fl(d;h;\024)669 4129 y Fn(exp\()p Fh(\000)p Fn(3)959 4054 y Fh(p)p 1035 4054 46 4 v 75 x Fn(2)q Fm(\024)p Fn(\()p Fm(d)h Fh(\000)f Fm(x)p Fn(\)\))p Fm(E)5 b Fn(\()1566 4067 y Fm(x)p 1566 4108 52 4 v 1568 4191 a(d)1629 4129 y Fn(\))1664 4091 y Fg(3)1704 4129 y Fm(:)0 4318 y Fn(F)-8 b(rom)31 b(\(3.41\),)i(for)d (all)f Fm(f)837 4332 y Fg(0)901 4318 y Fh(2)c Fn([)p Fm(\016)1052 4332 y Fg(0)1092 4318 y Fm(;)15 b(\016)1172 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y Fm(x)p Fn(\))1168 4621 y Fk(\020)1222 4722 y Fn(2)1288 4699 y(\026)1267 4722 y Fm(C)1332 4736 y Fg(1)1392 4698 y Fn(~)1372 4722 y Fm(f)1417 4736 y Fg(1)1456 4722 y Fn(\(0\))5 b(\026)-50 b Fm(z)1613 4737 y Fl(\024;h)1740 4722 y Fh(\000)1831 4647 y(p)p 1906 4647 V 1906 4722 a Fn(2)q Fm(\024)2004 4689 y Fi(\000)p Fg(1)p Fi(\000)p Fl(\027)2193 4722 y Fn(\(1)21 b(+)e Fh(O)s Fn(\()p Fm(\024)2546 4689 y Fg(2)2587 4722 y Fn(\)\))c(exp\()p Fh(\000)p Fn(\(2)p Fm(\016)3037 4736 y Fg(0)3098 4722 y Fh(\000)3189 4647 y(p)p 3265 4647 V 75 x Fn(2)p Fm(\024)p Fn(\003)3425 4736 y Fg(2)3465 4722 y Fn(\))p Fm(d)p Fn(\))3582 4621 y Fk(\021)3653 4722 y Fm(:)0 4909 y Fn(F)-8 b(or)184 4886 y(\026)163 4909 y Fm(C)228 4923 y Fg(1)293 4909 y Fm(>)25 b Fn(0,)31 b(\003)553 4876 y Fi(0)606 4909 y Fn(is)f(p)s(ositiv)m(e)f(for)h Fm(\024)h Fn(small)e(and)h Fm(\024d)h Fn(large)f(enough.)41 b(Let)31 b(us)e(no)m(w)i(consider)e (the)h(term)419 5100 y(\003)482 5067 y Fi(00)550 5100 y Fn(:=)25 b(2)737 5077 y(\026)716 5100 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Fn(\)\))p Fm(E)5 b Fn(\()1706 5234 y Fm(x)p 1706 5275 52 4 v 1708 5358 a(d)1769 5296 y Fn(\))1804 5258 y Fg(3)1844 5296 y Fm(:)0 5509 y Fn(If)23 b(w)m(e)h(assume)539 5486 y(\026)519 5509 y Fm(C)591 5476 y Fi(0)584 5534 y Fg(1)648 5509 y Fm(>)h Fn(0,)h(w)m(e)e(can)g(conclude)f(as)h(in)e (Prop)s(osition)g(3.2,)k(taking)h(\026)-48 b Fm(g)s Fn(\()p Fm(y)s Fn(\))26 b(=)f(exp\()p Fh(\000)p Fm(\013)3210 5523 y Fg(0)3250 5509 y Fm(y)s Fn(\))e(with)g Fm(\013)3615 5523 y Fg(0)3680 5509 y Fh(2)p Fn(]0)p Fm(;)3861 5418 y Fi(p)p 3920 5418 36 3 v 55 x Fg(2)p 3861 5488 95 4 v 3891 5541 a(4)3965 5509 y Fn([.)0 5622 y(The)30 b(pro)s(of)f(of)i (Prop)s(osition)d(3.3)k(follo)m(ws.)1962 5871 y(13)p eop %%Page: 14 14 14 13 bop 0 162 a Fw(4)135 b(Existence)46 b(of)f(solutions)h(of)f Fd(\()p Fc(GL)p Fd(\))1938 119 y Fb(s)1938 196 y(d)0 365 y Fn(F)-8 b(ollo)m(wing)33 b(the)h(pro)s(of)e(of)i(Prop)s(osition)d (6.5)k(giv)m(en)e(b)m(y)h(C.)f(Bolley)g(and)g(B.)h(Hel\013er)f(in)g ([8],)i(p.)49 b(66,)36 b(it)d(is)f(su\016cien)m(t)0 478 y(to)k(construct)f(a)h(subsolution)c Fm(f)1131 493 y Fl(sub)1274 478 y Fn(and)i(a)i(sup)s(ersolution)c Fm(f)2145 492 y Fl(sup)2292 478 y Fn(\()k Fm(f)2408 493 y Fl(sub)2548 478 y Fh(\024)d Fm(f)2697 492 y Fl(sup)2809 478 y Fn(\))j(in)e(order)g (to)i(deriv)m(e)f(a)g(solution)0 591 y(\()p Fm(f)5 b(;)15 b(A)p Fn(\))31 b(of)f(\()p Fm(GL)p Fn(\))565 558 y Fl(s)565 619 y(d)607 591 y Fn(.)0 704 y(In)c(the)h(same)h(spirit)c(of)j([10)q(]) h(and)e([16)q(],)i(from)e(Prop)s(ositions)f(3.2)j(and)f(3.3,)h(w)m(e)g (deduce)e(the)h(pro)s(of)g(of)g(Theorem)f(1.1.)0 833 y Fe(Pro)s(of)32 b(of)f(Theorem)e(1.1)f Fn(According)e(Prop)s(ositions) f(3.2)j(and)e(3.3,)j(w)m(e)e(can)g(c)m(ho)s(ose)h Fm(g)h Fn(=)f(\026)-48 b Fm(g)s Fn(,)28 b(taking)f Fm(\013)3615 847 y Fg(0)3680 833 y Fh(2)p Fn(]0)p Fm(;)3861 741 y Fi(p)p 3920 741 36 3 v 55 x Fg(2)p 3861 812 95 4 v 3891 864 a(4)3965 833 y Fn([.)0 945 y Fe(Step)39 b(1:)54 b(Construction)40 b(of)f(subsolutions)h(and)f(sup)s(ersolutions)c Fn(F)-8 b(or)2833 910 y Fg(1)p 2804 925 V 2804 934 a Fi(p)p 2863 934 36 3 v 55 x Fg(2)2939 945 y Fm(<)31 b(\016)3081 959 y Fg(0)3152 945 y Fm(<)f(\016)3293 959 y Fg(1)3364 945 y Fm(<)h Fn(1)j(\014xed,)h(let)e(us)0 1074 y(consider)c Fm(f)399 1088 y Fg(0)463 1074 y Fh(2)c Fn([)p Fm(\016)614 1088 y Fg(0)654 1074 y Fm(;)15 b(\016)734 1088 y Fg(1)775 1074 y Fn(])30 b(and)g(the)h(solutions)d(in)h(]0)p Fm(;)15 b Fn(1[)33 b(of)994 1199 y Fh(p)p 1069 1199 46 4 v 1069 1278 a Fn(2)q Fm(y)1163 1241 y Fg(2)1202 1278 y Fn(\(1)21 b Fh(\000)f Fm(y)1442 1241 y Fg(2)1481 1278 y Fn(\))h(+)f Fm(C)1693 1292 y Fg(0)1732 1278 y Fm(\024)1784 1241 y Fg(1)p Fi(\000)p Fl(\017)1927 1278 y Fn(+)g Fm(C)2090 1241 y Fi(0)2083 1301 y Fg(0)2123 1278 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\))27 b(=)2461 1199 y Fh(p)p 2537 1199 V 79 x Fn(2)p Fm(f)2637 1241 y Fg(2)2627 1301 y(0)2676 1278 y Fn(\(1)21 b Fh(\000)f Fm(f)2923 1241 y Fg(2)2913 1301 y(0)2962 1278 y Fn(\))p Fm(;)807 b Fn(\(4.1\))0 1483 y(where)26 b Fm(C)324 1497 y Fg(0)364 1483 y Fn(,)h Fm(C)488 1450 y Fi(0)481 1507 y Fg(0)548 1483 y Fn(and)f Fm(g)k Fn(are)d(giv)m(en)g(b)m(y)g(Prop)s(osition)e(3.2.)40 b(F)-8 b(or)28 b Fm(\024)f Fn(small)e(enough)i(and)f Fm(\024d)h Fn(large)g(enough,)h(there)f(exist)0 1595 y(t)m(w)m(o)32 b(solutions)d(of)h(\(4.1\))i(denoted)f(b)m(y)f Fm(y)1387 1609 y Fg(1)1426 1595 y Fn(\()p Fm(f)1506 1609 y Fg(0)1545 1595 y Fm(;)15 b(\024)p Fn(\))32 b(and)d Fm(y)1925 1609 y Fg(2)1964 1595 y Fn(\()p Fm(f)2044 1609 y Fg(0)2084 1595 y Fm(;)15 b(\024)p Fn(\).)42 b(W)-8 b(e)31 b(can)g(assume)f Fm(y)2960 1609 y Fg(1)2999 1595 y Fn(\()p Fm(f)3079 1609 y Fg(0)3118 1595 y Fm(;)15 b(\024)p Fn(\))27 b Fm(<)e(\016)3408 1609 y Fg(0)3473 1595 y Fm(<)g(y)3614 1609 y Fg(2)3653 1595 y Fn(\()p Fm(f)3733 1609 y Fg(0)3772 1595 y Fm(;)15 b(\024)p Fn(\).)0 1708 y(F)-8 b(rom)27 b(\(4.1\),)j(as)d(0)f Fm(<)f(f)795 1722 y Fg(0)859 1708 y Fm(<)g Fn(1)i(and)g Fm(y)1246 1722 y Fg(2)1285 1708 y Fn(\()p Fm(f)1365 1722 y Fg(0)1404 1708 y Fm(;)15 b(\024)p Fn(\))26 b Fh(2)p Fn(])1683 1673 y Fg(1)p 1653 1688 95 4 v 1653 1697 a Fi(p)p 1712 1697 36 3 v 55 x Fg(2)1757 1708 y Fn(1[)p Fm(;)i Fn(it)e(results)g(that)i(there)f(exists)2944 1685 y(\026)2923 1708 y Fm(C)33 b Fn(suc)m(h)27 b(that)h(for)e(all)g Fm(y)i Fh(2)p Fn(]0)p Fm(;)15 b Fn(1[,)0 1832 y(for)30 b Fm(\024)h Fn(small)e(enough)h(and)g Fm(\024d)h Fn(large)f(enough,)g (w)m(e)h(ha)m(v)m(e)1302 2036 y Fh(j)p Fm(y)1372 2050 y Fg(2)1411 2036 y Fn(\()p Fm(f)1491 2050 y Fg(0)1530 2036 y Fm(;)15 b(\024)p Fn(\))22 b Fh(\000)e Fm(f)1815 2050 y Fg(0)1854 2036 y Fh(j)25 b(\024)2021 2013 y Fn(\026)2000 2036 y Fm(C)2087 1963 y Fk(\000)2128 2036 y Fm(\024)2180 1999 y Fg(1)p Fi(\000)p Fl(\017)2324 2036 y Fn(+)20 b Fm(g)s Fn(\()p Fm(\024d)p Fn(\))2630 1963 y Fk(\001)2689 2036 y Fm(:)1115 b Fn(\(4.2\))0 2240 y(F)-8 b(rom)24 b(\(3.14\))h(and)e(Prop)s(osition)e(3.2,)26 b(there)e(exist)f(t)m(w)m (o)i(subsolutions)20 b(of)k(the)f(Ginzburg-Landau)f(system,)j(denoted)0 2369 y(b)m(y)30 b Fm(f)181 2321 y Fg(\(1\))171 2399 y Fl(sub)309 2369 y Fn(and)g Fm(f)541 2321 y Fg(\(2\))531 2399 y Fl(sub)668 2369 y Fn(satisfying)488 2596 y Fm(f)543 2548 y Fg(\(1\))533 2625 y Fl(sub)640 2596 y Fn(\(0\))d(=)e(tanh)o(\()p Fm(x)1146 2548 y Fg(\(1\))1146 2625 y Fl(\024;h)1252 2596 y Fn(\))20 b(+)g Fh(O)s Fn(\()p Fm(\024)p Fn(\))p Fm(;)67 b Fn(with)e(tanh)o(\()p Fm(x)2198 2548 y Fg(\(1\))2198 2625 y Fl(\024;h)2304 2596 y Fn(\))25 b(=)g Fm(y)2505 2610 y Fg(1)2544 2596 y Fn(\()p Fm(f)2624 2610 y Fg(0)2664 2596 y Fm(;)15 b(\024)p Fn(\))21 b(+)f Fh(O)e Fn(\()p Fm(\024)3080 2558 y Fl(\027)3144 2596 y Fn(+)i Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))e Fm(;)301 b Fn(\(4.3\))0 2800 y(and)508 2913 y Fm(f)563 2865 y Fg(\(2\))553 2942 y Fl(sub)660 2913 y Fn(\(0\))27 b(=)e(tanh\()p Fm(x)1167 2865 y Fg(\(2\))1167 2942 y Fl(\024;h)1272 2913 y Fn(\))c(+)f Fh(O)s Fn(\()p Fm(\024)p Fn(\))51 b(with)64 b(tanh\()p Fm(x)2178 2865 y Fg(\(2\))2178 2942 y Fl(\024;h)2284 2913 y Fn(\))25 b(=)g Fm(y)2485 2927 y Fg(2)2524 2913 y Fn(\()p Fm(f)2604 2927 y Fg(0)2644 2913 y Fm(;)15 b(\024)p Fn(\))21 b(+)f Fh(O)e Fn(\()p Fm(\024)3060 2875 y Fl(\027)3124 2913 y Fn(+)i Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))d Fm(:)322 b Fn(\(4.4\))0 3080 y(No)m(w,)31 b(for)f Fm(\024)h Fn(small)e(and)h Fm(\024d)h Fn(large,)g(let)f(us)f(consider)36 b(\026)-51 b Fm(y)1894 3094 y Fg(1)1933 3080 y Fn(\()p Fm(f)2013 3094 y Fg(0)2052 3080 y Fm(;)15 b(\024)p Fn(\))32 b(and)k(\026)-52 b Fm(y)2432 3094 y Fg(2)2471 3080 y Fn(\()p Fm(f)2551 3094 y Fg(0)2591 3080 y Fm(;)15 b(\024)p Fn(\))31 b(the)g(solutions)d (in)h(]0)p Fm(;)15 b Fn(1[)33 b(of)994 3205 y Fh(p)p 1069 3205 46 4 v 1069 3284 a Fn(2)q Fm(y)1163 3247 y Fg(2)1202 3284 y Fn(\(1)21 b Fh(\000)f Fm(y)1442 3247 y Fg(2)1481 3284 y Fn(\))h(+)1648 3261 y(\026)1628 3284 y Fm(C)1693 3298 y Fg(0)1732 3284 y Fm(\024)1784 3247 y Fg(1)p Fi(\000)p Fl(\017)1927 3284 y Fn(+)2039 3261 y(\026)2018 3284 y Fm(C)2090 3247 y Fi(0)2083 3307 y Fg(0)2123 3284 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\))27 b(=)2461 3205 y Fh(p)p 2537 3205 V 79 x Fn(2)p Fm(f)2637 3247 y Fg(2)2627 3307 y(0)2676 3284 y Fn(\(1)21 b Fh(\000)f Fm(f)2923 3247 y Fg(2)2913 3307 y(0)2962 3284 y Fn(\))p Fm(;)0 3488 y Fn(where)286 3465 y(\026)265 3488 y Fm(C)330 3502 y Fg(0)402 3488 y Fn(and)601 3465 y(\026)580 3488 y Fm(C)652 3455 y Fi(0)645 3513 y Fg(0)717 3488 y Fn(are)33 b(giv)m(en)f(b)m(y)h(Prop)s(osition)d(3.3.)48 b(W)-8 b(e)34 b(can)e(assume)39 b(\026)-51 b Fm(y)2607 3502 y Fg(1)2646 3488 y Fn(\()p Fm(f)2726 3502 y Fg(0)2765 3488 y Fm(;)15 b(\024)p Fn(\))30 b Fm(<)k Fn(\026)-51 b Fm(y)3066 3502 y Fg(2)3105 3488 y Fn(\()p Fm(f)3185 3502 y Fg(0)3224 3488 y Fm(;)15 b(\024)p Fn(\).)48 b(Similarly)-8 b(,)30 b(from)0 3619 y(Prop)s(osition)36 b(3.3,)k(there)e(exist)f(t)m (w)m(o)i(sup)s(ersolutions)34 b(of)k(the)g(Ginzburg-Landau)d(system,)40 b Fm(f)3322 3571 y Fg(\(1\))3312 3631 y Fl(sup)3462 3619 y Fn(and)d Fm(f)3701 3571 y Fg(\(2\))3691 3631 y Fl(sup)3841 3619 y Fn(suc)m(h)0 3732 y(that:)198 3936 y(tanh)o(\()6 b(\026)-51 b Fm(x)466 3888 y Fg(\(2\))466 3965 y Fl(\024;h)572 3936 y Fn(\))25 b(=)32 b(\026)-52 b Fm(y)773 3950 y Fg(1)812 3936 y Fn(\()p Fm(f)892 3950 y Fg(0)932 3936 y Fm(;)15 b(\024)p Fn(\))21 b(+)f Fh(O)e Fn(\()p Fm(\024)1348 3898 y Fl(\027)1412 3936 y Fn(+)i Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))68 b(and)d(tanh\()6 b(\026)-51 b Fm(x)2302 3888 y Fg(\(2\))2302 3965 y Fl(\024;h)2407 3936 y Fn(\))26 b(=)31 b(\026)-51 b Fm(y)2609 3950 y Fg(2)2648 3936 y Fn(\()p Fm(f)2728 3950 y Fg(0)2767 3936 y Fm(;)15 b(\024)p Fn(\))22 b(+)d Fh(O)g Fn(\()p Fm(\024)3184 3898 y Fl(\027)3248 3936 y Fn(+)h Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))d Fm(:)198 b Fn(\(4.5\))0 4164 y(Making)34 b(similar)d(computations)i(to)h(the)g (previous)e(one,)j(w)m(e)f(can)g(sho)m(w)g(that)g(there)g(exists)3284 4141 y(^)3264 4164 y Fm(C)40 b Fn(suc)m(h)33 b(that)h(for)g(all)0 4277 y Fm(f)45 4291 y Fg(0)109 4277 y Fh(2)p Fn(])234 4241 y Fg(1)p 205 4256 95 4 v 205 4266 a Fi(p)p 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Fn(\026)-51 b Fm(y)2654 4937 y Fg(2)2693 4923 y Fn(\()p Fm(f)2773 4937 y Fg(0)2812 4923 y Fm(;)15 b(\024)p Fn(\))p Fm(:)0 5143 y Fe(Step)39 b(2:)57 b(Comparison)39 b(of)h Fm(f)1148 5095 y Fg(\(2\))1138 5173 y Fl(sub)1285 5143 y Fe(and)g Fm(f)1547 5095 y Fg(\(2\))1537 5155 y Fl(sup)1684 5143 y Fn(F)-8 b(rom)35 b(\(4.2\))h(and)e(\(4.6\),)k(w)m(e) d(deduce)f(the)h(estimate,)i(for)d Fm(\024)h Fn(small)0 5256 y(and)30 b Fm(\024d)h Fn(large)f(enough)968 5460 y(\026)-51 b Fm(y)1007 5474 y Fg(2)1046 5460 y Fn(\()p Fm(f)1126 5474 y Fg(0)1165 5460 y Fm(;)15 b(\024)p Fn(\))22 b Fh(\000)d Fm(y)1449 5474 y Fg(2)1488 5460 y Fn(\()p Fm(f)1568 5474 y Fg(0)1608 5460 y Fm(;)c(\024)p Fn(\))26 b(=)f Fh(O)s Fn(\()p Fm(\024)2019 5423 y Fl(\027)2083 5460 y Fn(+)20 b Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))p Fm(;)68 b Fh(8)15 b Fm(f)2628 5474 y Fg(0)2692 5460 y Fh(2)25 b Fn([)p Fm(\016)2843 5474 y Fg(0)2883 5460 y Fm(;)15 b(\016)2963 5474 y Fg(1)3003 5460 y Fn(])p Fm(:)776 b Fn(\(4.7\))1962 5871 y(14)p eop %%Page: 15 15 15 14 bop 0 162 a Fn(F)-8 b(rom)28 b(the)f(inequalit)m(y)e Fm(y)852 176 y Fg(2)891 162 y Fn(\()p Fm(f)971 176 y Fg(0)1011 162 y Fm(;)15 b(\024)p Fn(\))26 b Fm(<)31 b Fn(\026)-51 b Fm(y)1305 176 y Fg(2)1344 162 y Fn(\()p Fm(f)1424 176 y Fg(0)1463 162 y Fm(;)15 b(\024)p Fn(\),)29 b(and)e(from)f(\(4.5\))j(and)e(\(4.7\),)i(w)m(e)f(deduce)f(that)g (there)h(exists)e Fm(\024)3809 176 y Fg(0)3874 162 y Fm(>)f Fn(0)0 275 y(suc)m(h)30 b(that)h(for)f(all)f Fm(\024)d Fh(\024)f Fm(\024)893 289 y Fg(0)933 275 y Fn(,)30 b(for)h(all)e Fm(f)1299 289 y Fg(0)1363 275 y Fh(2)c Fn([)p Fm(\016)1514 289 y Fg(0)1554 275 y Fm(;)15 b(\016)1634 289 y Fg(1)1674 275 y Fn(],)31 b(w)m(e)g(ha)m(v)m(e)1096 490 y Fm(x)1148 442 y Fg(\(2\))1148 520 y Fl(\024;h)1278 490 y Fm(<)g Fn(\026)-51 b Fm(x)1426 442 y Fg(\(2\))1426 520 y Fl(\024;h)1582 490 y Fn(and)56 b(\026)-51 b Fm(x)1831 442 y Fg(\(2\))1831 520 y Fl(\024;h)1956 490 y Fh(\000)20 b Fm(x)2099 442 y Fg(\(2\))2099 520 y Fl(\024;h)2229 490 y Fn(=)25 b Fh(O)s 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Fg(1)2761 1552 y Fm(\024)2813 1526 y Fl(\027)2877 1552 y Fn(+)g Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))p Fm(x)j Fn(+)d Fm(x)3436 1504 y Fg(\(2\))3436 1581 y Fl(\024;h)3541 1451 y Fk(\021)3605 1457 y Fm(;)153 1722 y(J)203 1736 y Fl(\024)249 1722 y Fn(\()p Fm(x)p Fn(\))26 b(:=)614 1686 y Fl(\024)655 1662 y Fp(2)689 1686 y Fl(h)730 1662 y Fp(2)p 528 1701 323 4 v 528 1755 a Fg(4)5 b(\026)-40 b Fl(y)598 1764 y Fp(2)632 1755 y Fg(\()p Fl(f)693 1764 y Fp(0)728 1755 y Fl(;\024)p Fg(\))816 1736 y Fp(3)876 1722 y Fn(exp\()p Fh(\000)p Fn(2)6 b(\026)-51 b Fm(y)1211 1736 y Fg(2)1250 1722 y Fn(\()p Fm(f)1330 1736 y Fg(0)1370 1722 y Fm(;)15 b(\024)p Fn(\))p Fm(x)p Fn(\))21 b Fh(\000)1792 1686 y Fl(\024)1833 1662 y Fp(2)1868 1686 y Fl(h)1909 1662 y Fp(2)p 1706 1701 V 1706 1755 a Fg(4)p Fl(y)1776 1764 y Fp(2)1811 1755 y Fg(\()p Fl(f)1872 1764 y Fp(0)1907 1755 y Fl(;\024)p Fg(\))1995 1736 y Fp(3)2054 1722 y Fn(exp\()p Fh(\000)p Fn(2)p Fm(y)2389 1736 y Fg(2)2429 1722 y Fn(\()p Fm(f)2509 1736 y Fg(0)2548 1722 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Fn(\(5.1\))1962 5871 y(15)p eop %%Page: 16 16 16 15 bop 0 162 a Fn(and)32 b(where)468 139 y(\026)444 162 y Fm(A)512 176 y Fg(0)584 162 y Fn(is)f(de\014ned)h(in)f(\(1.10\).) 49 b(The)32 b(parameter)39 b(\026)-51 b Fm(x)2083 176 y Fg(0)2155 162 y Fn(is)32 b(determined)f(b)m(y)h(the)h(equalit)m(y) 3380 138 y(\026)3361 162 y Fm(f)3406 176 y Fg(0)3445 162 y Fn(\(0\))d(=)e Fm(f)3734 176 y Fg(0)3773 162 y Fm(;)44 b(f)3887 176 y Fg(0)3955 162 y Fh(2)0 275 y Fn(])65 239 y Fg(1)p 35 254 95 4 v 35 264 a Fi(p)p 94 264 36 3 v 55 x Fg(2)139 275 y Fm(;)15 b Fn(1[.)0 399 y(F)-8 b(or)31 b(tec)m(hnical)f(reasons,)h(w)m(e)g(assume)f(that)1668 603 y Fm(\024d)c Fh(\025)e Fm(\024)1940 565 y Fi(\000)p Fl(\017)2028 603 y Fm(;)66 b(\017)25 b(>)g Fn(0)p Fm(:)1482 b Fn(\(5.2\))0 807 y Fe(Pro)s(of)33 b(of)g(Theorem)e(1.2)e Fn(Let)g(\()p Fm(f)5 b(;)15 b(A)p Fn(\))29 b(b)s(e)e(the)i(solution)e (de\014ned)g(in)g(Theorem)h(1.1.)41 b(By)28 b(construction,)h(w)m(e)f (ha)m(v)m(e)0 920 y(sho)m(wn)i(in)f(the)i(pro)s(of)f(of)h(Theorem)f (1.1)i(that)f Fm(f)40 b Fn(satis\014es)30 b Fm(f)2074 935 y Fl(sub)2207 920 y Fh(\024)c Fm(f)35 b Fh(\024)25 b Fm(f)2525 934 y Fl(sup)2638 920 y Fn(.)41 b(So,)31 b(w)m(e)g(ha)m(v)m(e)h Fm(f)3245 935 y Fl(sub)3373 920 y Fh(\000)3483 896 y Fn(\026)3464 920 y Fm(f)3509 934 y Fg(0)3573 920 y Fh(\024)26 b Fm(f)j Fh(\000)3855 896 y Fn(\026)3835 920 y Fm(f)3880 934 y Fg(0)3945 920 y Fh(\024)0 1033 y Fm(f)45 1047 y Fl(sup)178 1033 y Fh(\000)288 1009 y Fn(\026)269 1033 y Fm(f)314 1047 y Fg(0)353 1033 y Fm(:)0 1146 y Fe(Step)35 b(1:)46 b(Estimate)33 b(for)i Fh(j)p Fm(f)30 b Fh(\000)1181 1122 y Fn(\026)1161 1146 y Fm(f)1206 1160 y Fg(0)1245 1146 y Fh(j)h Fn(W)-8 b(e)31 b(ha)m(v)m(e)576 1379 y Fm(f)621 1393 y Fl(sup)734 1379 y Fn(\()p Fm(x)p Fn(\))21 b Fh(\000)987 1355 y Fn(\026)968 1379 y Fm(f)1013 1393 y Fg(0)1052 1379 y Fn(\()p Fm(x)p Fn(\))576 1537 y(=)k Fm(L)734 1552 y Fl(\024;d)835 1537 y Fn(\()p Fm(x)p Fn(\))c(+)1165 1501 y Fl(\024)1206 1478 y Fp(2)1240 1501 y Fl(h)1281 1478 y Fp(2)p 1079 1516 323 4 v 1079 1571 a Fg(4)5 b(\026)-40 b Fl(y)1149 1580 y Fp(2)1184 1571 y Fg(\()p Fl(f)1245 1580 y Fp(0)1279 1571 y Fl(;\024)p Fg(\))1367 1552 y Fp(3)1427 1537 y Fn(exp\()p Fh(\000)p Fn(2)6 b(\026)-51 b Fm(y)1762 1551 y Fg(2)1801 1537 y Fn(\()p Fm(f)1881 1551 y Fg(0)1921 1537 y Fm(;)15 b(\024)p Fn(\))p Fm(x)p Fn(\))21 b(+)2251 1513 y(\026)2247 1537 y Fm(\025)2300 1552 y Fl(d;\024;h)2476 1537 y Fn(exp\()p Fh(\000)2721 1462 y(p)p 2797 1462 46 4 v 75 x Fn(2)p Fm(\024)p Fn(\()p Fm(d)h Fh(\000)e Fm(x)p Fn(\)\))p Fm(E)5 b Fn(\()3328 1501 y Fl(x)p 3328 1516 40 4 v 3330 1568 a(d)3379 1537 y Fn(\))p Fm(;)3829 1451 y Fn(\(5.3\))0 1750 y(where)714 1958 y Fm(L)776 1973 y Fl(\024;d)876 1958 y Fn(\()p Fm(x)p Fn(\))26 b(:=)1398 1863 y(sinh)1576 1762 y Fk(\020)1666 1827 y Fl(\024)p 1640 1842 95 4 v 1640 1852 a Fi(p)p 1699 1852 36 3 v 55 x Fg(2)1744 1863 y Fn(\()1800 1840 y(\026)1779 1863 y Fm(C)1844 1877 y Fg(1)1884 1863 y Fm(\024)1936 1830 y Fl(\027)1999 1863 y Fn(+)2111 1840 y(\026)2090 1863 y Fm(C)2162 1830 y Fi(0)2155 1887 y Fg(1)2195 1863 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))p Fm(x)c Fn(+)k(\026)-51 b Fm(x)2662 1815 y Fg(\(2\))2662 1892 y Fl(\024;h)2787 1863 y Fh(\000)26 b Fn(\026)-51 b Fm(x)2930 1877 y Fg(0)2970 1762 y Fk(\021)p 1155 1937 2112 4 v 1155 2045 a Fn(cosh)q(\()1400 2009 y Fl(\024)p 1373 2024 95 4 v 1373 2034 a Fi(p)p 1432 2034 36 3 v 55 x Fg(2)1477 2045 y Fn(\(1)21 b(+)1689 2022 y(\026)1669 2045 y Fm(C)1734 2059 y Fg(1)1773 2045 y Fm(\024)1825 2019 y Fl(\027)1889 2045 y Fn(+)2001 2022 y(\026)1980 2045 y Fm(C)2052 2014 y Fi(0)2045 2071 y Fg(1)2084 2045 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))p Fm(x)i Fn(+)j(\026)-51 b Fm(x)2552 1997 y Fg(\(2\))2552 2075 y Fl(\024;h)2657 2045 y Fn(\))15 b(cosh)q(\()2932 2009 y Fl(\024x)p 2925 2024 95 4 v 2925 2034 a Fi(p)p 2984 2034 36 3 v 55 x Fg(2)3049 2045 y Fn(+)26 b(\026)-51 b Fm(x)3192 2059 y Fg(0)3231 2045 y Fn(\))3277 1958 y Fm(:)527 b Fn(\(5.4\))0 2233 y(F)-8 b(rom)31 b(\(4.6\))h(and)e(as)g Fm(\024d)c Fh(\025)f Fm(\024)1017 2200 y Fi(\000)p Fl(\017)1105 2233 y Fn(,)31 b(w)m(e)g(ha)m(v)m(e)1640 2453 y(\026)-51 b Fm(x)1686 2405 y Fg(\(2\))1686 2482 y Fl(\024;h)1812 2453 y Fh(\000)25 b Fn(\026)-51 b Fm(x)1954 2467 y Fg(0)2019 2453 y Fn(=)25 b Fh(O)s Fn(\()p Fm(\024)2277 2415 y Fl(\027)2321 2453 y Fn(\))p Fm(:)1448 b Fn(\(5.5\))0 2671 y(Using)27 b(\(3.8\))i(and)f(\(5.5\),)i(w)m(e)e(deduce)g(that)g(there)g(exists)f Fm(C)2051 2685 y Fg(1)2116 2671 y Fm(>)e Fn(0)j(suc)m(h)f(that)i(on)e (the)h(in)m(terv)-5 b(al)27 b([0)p Fm(;)15 b(\024)3447 2638 y Fi(\000)p Fg(1)p Fi(\000)p Fl(\017)3627 2671 y Fn(],)28 b(w)m(e)g(ha)m(v)m(e)0 2783 y(for)i Fm(\024)h Fn(small)e(enough)938 2941 y(sinh)1115 2813 y Fk(\022)1227 2880 y Fm(\024)p 1192 2920 122 4 v 1192 2939 a Fh(p)p 1268 2939 46 4 v 75 x Fn(2)1323 2941 y(\()1379 2918 y(\026)1358 2941 y Fm(C)1423 2955 y Fg(1)1463 2941 y Fm(\024)1515 2904 y Fl(\027)1579 2941 y Fn(+)1690 2918 y(\026)1670 2941 y Fm(C)1742 2904 y Fi(0)1735 2964 y Fg(1)1774 2941 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))p Fm(x)23 b Fn(+)j(\026)-51 b Fm(x)2242 2893 y Fg(\(2\))2242 2971 y Fl(\024;h)2367 2941 y Fh(\000)26 b Fn(\026)-51 b Fm(x)2510 2955 y Fg(0)2549 2813 y Fk(\023)2641 2941 y Fh(\024)25 b Fm(C)2802 2955 y Fg(1)2842 2941 y Fm(\024)2894 2904 y Fg(1)p Fi(\000)p Fg(2)p Fl(\017)3052 2941 y Fm(:)0 3168 y Fn(As)239 3133 y Fg(1)p 143 3148 229 4 v 143 3201 a(cosh\()p Fl(x)p Fg(\))406 3168 y Fh(\024)g Fn(2)15 b(exp)q(\()p Fh(\000)p Fm(x)p Fn(\))29 b(for)f(all)g Fm(x)d Fh(2)g Fn(I)-23 b(R)1426 3133 y Fg(+)1485 3168 y Fn(,)30 b(and)e(b)s(ecause)2070 3146 y(\026)2050 3168 y Fm(C)2115 3182 y Fg(1)2179 3168 y Fm(>)d Fn(0)30 b(and)2545 3146 y(\026)2525 3168 y Fm(C)2597 3136 y Fi(0)2590 3193 y Fg(1)2654 3168 y Fm(>)25 b Fn(0,)30 b(w)m(e)g(get)g(for)e(all)g Fm(f)3440 3182 y Fg(0)3504 3168 y Fh(2)d Fn([)p Fm(\016)3655 3182 y Fg(0)3695 3168 y Fm(;)15 b(\016)3775 3182 y Fg(1)3815 3168 y Fn(])29 b(and)0 3294 y(all)g Fm(;)15 b(x)26 b Fh(2)f Fn([0)p Fm(;)15 b(d)p Fn(])1283 3475 y(1)p 511 3516 1590 4 v 511 3631 a(cosh)698 3530 y Fk(\020)789 3595 y Fl(\024)p 762 3610 95 4 v 762 3620 a Fi(p)p 821 3620 36 3 v 55 x Fg(2)866 3631 y Fn(\(1)21 b(+)1079 3608 y(\026)1058 3631 y Fm(C)1123 3645 y Fg(1)1163 3631 y Fm(\024)1215 3605 y Fl(\027)1279 3631 y Fn(+)1390 3608 y(\026)1369 3631 y Fm(C)1441 3600 y Fi(0)1434 3657 y Fg(1)1474 3631 y Fm(g)s Fn(\()p Fm(\024d)p Fn(\)\))p Fm(x)h Fn(+)k(\026)-51 b Fm(x)1941 3583 y Fg(\(2\))1941 3660 y Fl(\024;h)2046 3530 y Fk(\021)2404 3475 y Fn(1)p 2121 3516 613 4 v 2121 3631 a(cosh)2308 3530 y Fk(\020)2379 3595 y Fl(\024x)p 2372 3610 95 4 v 2372 3620 a Fi(p)p 2431 3620 36 3 v 55 x Fg(2)2496 3631 y Fn(+)26 b(\026)-51 b Fm(x)2639 3645 y Fg(0)2679 3530 y Fk(\021)2768 3537 y Fh(\024)25 b Fm(C)2929 3551 y Fg(2)2984 3537 y Fn(exp\()p Fh(\000)3229 3457 y(p)p 3305 3457 46 4 v 80 x Fn(2)p Fm(\024x)p Fn(\))p Fm(:)0 3865 y Fn(As)138 3841 y(\026)134 3865 y Fm(\025)187 3880 y Fl(d;\024;h)374 3865 y Fm(<)g Fn(0,)31 b(w)m(e)f(deduce)g(that)h (there)g(exists)f Fm(C)i(>)25 b Fn(0)30 b(suc)m(h)h(that)f(for)h Fm(\024)f Fn(small)f(enough,)i(w)m(e)f(ha)m(v)m(e)1001 4080 y Fm(f)1046 4094 y Fl(sup)1179 4080 y Fh(\000)1289 4056 y Fn(\026)1270 4080 y Fm(f)1315 4094 y Fg(0)1379 4080 y Fh(\024)25 b Fm(C)7 b(\024)1599 4042 y Fg(1)p Fi(\000)p Fg(2)p Fl(\017)1772 4080 y Fn(exp\()p Fh(\000)2017 4000 y(p)p 2093 4000 V 80 x Fn(2)p Fm(\024x)p Fn(\))p Fm(;)92 b Fh(8)15 b Fm(x)25 b Fh(2)g Fn([0)p Fm(;)15 b(\024)2785 4042 y Fi(\000)p Fg(1)p Fi(\000)p Fl(\017)2964 4080 y Fn(])p Fm(:)815 b Fn(\(5.6\))0 4284 y(Moreo)m(v)m(er,)33 b(there)d(exists)g Fm(C)973 4298 y Fg(3)1038 4284 y Fm(>)25 b Fn(0)31 b(suc)m(h)f(that)h(for)f(all)f Fm(f)1922 4298 y Fg(0)1986 4284 y Fh(2)c Fn([)p Fm(\016)2137 4298 y Fg(0)2177 4284 y Fm(;)15 b(\016)2257 4298 y Fg(1)2298 4284 y Fn(],)30 b(for)g(all)g Fm(x)25 b Fh(2)g Fn([0)p Fm(;)15 b(d)p Fn(],)32 b(w)m(e)f(ha)m(v)m(e)1477 4517 y Fm(L)1539 4532 y Fl(\024;d)1640 4517 y Fn(\()p Fm(x)p Fn(\))26 b Fh(\024)f Fm(C)1949 4531 y Fg(3)2003 4517 y Fn(exp\()p Fh(\000)2258 4480 y Fg(2)2293 4425 y Fi(p)p 2352 4425 36 3 v 55 x Fg(2)q Fl(\024x)p 2258 4496 210 4 v 2345 4548 a Fg(3)2478 4517 y Fn(\))p Fm(:)0 4740 y Fn(F)-8 b(rom)31 b(\(5.3\),)h(it)e(deriv)m(es)g(that)h(there)g (exists)f Fm(C)1622 4754 y Fg(4)1686 4740 y Fm(>)25 b Fn(0,)31 b(suc)m(h)f(that,)h(on)g(the)f(in)m(terv)-5 b(al)30 b([)p Fh(\000)p Fm(c\024)3109 4707 y Fi(\000)p Fg(1)3219 4740 y Fn(ln)14 b Fm(\024;)h(d)p Fn([,)32 b(w)m(e)f(get)1409 5006 y Fm(f)1454 5020 y Fl(sup)1587 5006 y Fh(\000)1698 4982 y Fn(\026)1678 5006 y Fm(f)1723 5020 y Fg(0)1787 5006 y Fh(\024)25 b Fm(C)1948 5020 y Fg(4)1988 5006 y Fm(\024)15 b Fn(exp\()p Fh(\000)2310 4869 y(p)p 2386 4869 46 4 v 76 x Fn(2)p 2310 4985 122 4 v 2348 5068 a(2)2441 5006 y Fm(\024x)p Fn(\))p Fm(:)1224 b Fn(\(5.7\))0 5254 y(F)-8 b(or)31 b Fm(\024)g Fn(small)e(enough)h Fm(I)837 5268 y Fg(1)896 5254 y Fh(\\)20 b Fm(I)1017 5268 y Fg(2)1082 5254 y Fh(6)p Fn(=)25 b Fh(;)p Fn(.)41 b(F)-8 b(rom)31 b(\(5.6\))h(and)e(\(5.7\),)i(w)m(e)f(deduce)f(that)h(there)g(exists) 3302 5231 y(^)3282 5254 y Fm(C)g(>)25 b Fn(0)31 b(suc)m(h)f(that)1090 5521 y Fm(f)1135 5535 y Fl(sup)1268 5521 y Fh(\000)1378 5497 y Fn(\026)1359 5521 y Fm(f)1404 5535 y Fg(0)1468 5521 y Fh(\024)1585 5498 y Fn(^)1564 5521 y Fm(C)7 b(\024)1688 5483 y Fg(1)p Fi(\000)p Fg(2)p Fl(\017)1861 5521 y Fn(exp\()p Fh(\000)2116 5384 y(p)p 2192 5384 46 4 v 75 x Fn(2)p 2116 5500 122 4 v 2154 5583 a(2)2247 5521 y Fm(\024x)p Fn(\))p Fm(;)67 b Fh(8)25 b Fm(x)g Fh(2)g Fn([0)p Fm(;)15 b(d)p Fn(])p Fm(:)905 b Fn(\(5.8\))1962 5871 y(16)p eop %%Page: 17 17 17 16 bop 0 162 a Fn(Similarly)-8 b(,)27 b(one)k(can)g(c)m(hec)m(k)h (that)f(there)f(exists)g(a)h(constan)m(t)h Fm(C)2171 176 y Fg(5)2235 162 y Fm(>)25 b Fn(0)31 b(suc)m(h)f(that)h(w)m(e)g(ha)m (v)m(e)928 415 y Fm(f)973 430 y Fl(sub)1081 415 y Fn(\()p Fm(x)p Fn(\))21 b Fh(\000)1334 391 y Fn(\026)1314 415 y Fm(f)1359 429 y Fg(0)1398 415 y Fn(\()p Fm(x)p Fn(\))26 b Fh(\025)f(\000)p Fm(C)1778 429 y Fg(5)1817 415 y Fm(\024)1869 378 y Fg(1)p Fi(\000)p Fl(\017)2008 415 y Fn(exp\()p Fh(\000)2263 278 y(p)p 2338 278 46 4 v 2338 354 a Fn(2)p 2263 394 122 4 v 2301 478 a(2)2394 415 y Fm(\024x)p Fn(\))p Fm(;)92 b Fh(8)15 b Fm(x)25 b Fh(2)f Fn([0)p Fm(;)15 b(d)p Fn(])p Fm(:)744 b Fn(\(5.9\))0 643 y(F)-8 b(rom)30 b(\(5.8\))h(and)e(\(5.9\),)i(w)m(e)f(deduce)f(the)h(existence)g(of)f Fm(\024)2002 657 y Fg(0)2071 643 y Fn(and)2268 620 y(\026)2247 643 y Fm(C)j(>)25 b Fn(0)k(suc)m(h)g(that)h(for)f Fm(\024)d Fh(\024)f Fm(\024)3278 657 y Fg(0)3318 643 y Fn(,)30 b(for)f Fm(x)c Fh(2)g Fn([0)p Fm(;)15 b(d)p Fn(],)31 b(w)m(e)0 756 y(ha)m(v)m(e)1373 909 y Fh(j)p Fm(f)f Fh(\000)1583 885 y Fn(\026)1564 909 y Fm(f)1609 923 y Fg(0)1648 909 y Fh(j)25 b(\024)1815 886 y Fn(\026)1794 909 y Fm(C)7 b(\024)1918 871 y Fg(1)p Fi(\000)p Fg(2)p Fl(\017)2091 909 y Fn(exp\()p Fh(\000)2346 772 y(p)p 2422 772 46 4 v 75 x Fn(2)p 2346 888 122 4 v 2384 971 a(2)2478 909 y Fm(\024x)p Fn(\))p Fm(:)0 1097 y Fn(Inequalit)m(y)29 b(\(1.8\))j(follo)m(ws.)0 1210 y Fe(Step)j(2:)46 b(Estimate)33 b(for)i Fh(j)p Fm(A)21 b Fh(\000)1199 1187 y Fn(\026)1175 1210 y Fm(A)1243 1224 y Fg(0)1283 1210 y Fh(j)30 b Fn(W)-8 b(e)32 b(follo)m(w)d(the)i(pro)s(of)f(of)g(Prop)s(osition)e(4.5)k(in)d ([10)q(].)41 b(W)-8 b(e)32 b(set)f Fm(\032)25 b Fn(:=)g(2)p Fm(\017;)31 b Fn(and)1345 1468 y Fm(f)1390 1482 y Fl(max)1559 1468 y Fn(=)1674 1444 y(\026)1655 1468 y Fm(f)1700 1482 y Fg(0)1759 1468 y Fn(+)1870 1445 y(\026)1850 1468 y Fm(C)6 b(\024)1973 1430 y Fg(1)p Fi(\000)p Fl(\032)2119 1468 y Fn(exp\()p Fh(\000)2374 1331 y(p)p 2450 1331 46 4 v 76 x Fn(2)p 2374 1447 122 4 v 2412 1530 a(2)2505 1468 y Fm(\024x)p Fn(\))p Fm(:)1114 b Fn(\(5.10\))0 1697 y(Let)31 b Fm(A)231 1711 y Fl(max)405 1697 y Fn(b)s(e)f(the)g(unique)f (solution)g(in)g Fm(H)1514 1664 y Fg(2)1553 1697 y Fn(\([0)p Fm(;)15 b(d)p Fn(]\))33 b(of)e(the)f(problem)1351 1811 y Fk(8)1351 1893 y(<)1351 2057 y(:)1473 1928 y Fh(\000)p Fm(A)1612 1895 y Fi(00)1612 1950 y Fl(max)1776 1928 y Fn(+)20 b Fm(A)1935 1942 y Fl(max)2079 1928 y Fm(f)2134 1895 y Fg(2)2124 1950 y Fl(max)2293 1928 y Fn(=)25 b(0)p Fm(;)1473 2086 y(A)1541 2053 y Fi(0)1541 2108 y Fl(max)1685 2086 y Fn(\(0\))i(=)e Fm(h;)91 b(A)2159 2100 y Fl(max)2303 2086 y Fn(\()p Fm(d)p Fn(\))27 b(=)d(0)p Fm(:)0 2306 y Fn(F)-8 b(rom)31 b(Lemma)f(2.2,)i(as)f Fm(f)j Fh(\024)25 b Fm(f)1067 2320 y Fl(max)1241 2306 y Fn(and)1437 2282 y(\026)1418 2306 y Fm(f)1463 2320 y Fg(0)1527 2306 y Fh(\024)g Fm(f)1668 2320 y Fl(max)1811 2306 y Fn(,)31 b(it)e(results)h(that,)h(on)f([0)p Fm(;)15 b(d)p Fn(])1259 2501 y Fm(A)25 b Fh(\024)g Fm(A)1516 2515 y Fl(max)1685 2501 y Fm(<)g Fn(0)76 b(and)2149 2478 y(\026)2125 2501 y Fm(A)2193 2515 y Fg(0)2257 2501 y Fh(\024)25 b Fm(A)2421 2515 y Fl(max)2591 2501 y Fm(<)g Fn(0)p Fm(:)1026 b Fn(\(5.11\))0 2697 y(W)-8 b(e)38 b(in)m(tro)s(duce)d Fm(Z)42 b Fn(:=)36 b Fm(A)877 2711 y Fl(max)1045 2697 y Fh(\000)1164 2674 y Fn(\026)1140 2697 y Fm(A)1208 2711 y Fg(0)1248 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b(\(6.4\))i(and)d(\(6.13\),)k Fm(t)2323 958 y Fl(m;n)2467 944 y Fm(g)s Fn(\()p Fm(f)2593 958 y Fl(n)2640 944 y Fm(;)15 b(A)2748 958 y Fl(n)2796 944 y Fn(\))25 b Fh(!)h Fn(0)k(when)f Fm(n)24 b Fh(!)i Fn(+)p Fh(1)p Fn(.)40 b(Coming)0 1057 y(to)32 b(\(6.1\),)j(it)c(results)f(that)i(for)g(all)f Fm(\017)c(>)g Fn(0,)33 b(there)f(exists)f Fm(A)d(>)f Fn(0)32 b(suc)m(h)f(that)h(for)g(all)e(pairs)h(\()p Fm(m;)15 b(n)p Fn(\))28 b Fh(2)f Fn(I)-16 b(N)3572 1021 y Fg(2)3643 1057 y Fn(suc)m(h)31 b(that)0 1169 y Fm(n)25 b Fh(\025)g Fm(A)p Fn(,)31 b(w)m(e)f(get)269 1397 y Fh(h)p Fm(D)s(")424 1412 y Fl(d)465 1397 y Fn(\()p Fm(f)545 1411 y Fl(n)592 1397 y Fm(;)15 b(A)700 1411 y Fl(n)748 1397 y Fn(\)\()p Fm(f)863 1411 y Fl(n)930 1397 y Fh(\000)20 b Fm(f)1066 1411 y Fl(m)1132 1397 y Fm(;)15 b Fn(0\))p Fh(i)27 b Fn(=)e Fm(t)1443 1411 y Fl(m;n)1587 1397 y Fh(h)p Fm(D)s(")1742 1412 y Fl(d)1783 1397 y Fn(\()p Fm(f)1863 1411 y Fl(n)1909 1397 y Fm(;)15 b(A)2017 1411 y Fl(n)2065 1397 y Fn(\)\()p Fm(f)2180 1411 y Fl(n)2247 1397 y Fh(\000)2358 1373 y Fn(\026)2338 1397 y Fm(f)2383 1411 y Fl(m;n)2512 1397 y Fm(;)g Fn(0\))p Fh(i)27 b(\024)e Fm(t)2823 1411 y Fl(m;n)2967 1397 y Fm(g)s Fn(\()p Fm(f)3093 1411 y Fl(n)3140 1397 y Fm(;)15 b(A)3248 1411 y Fl(n)3296 1397 y Fn(\))25 b Fh(\024)g Fm(\017:)3783 1388 y Fn(\(6.14\))0 1608 y(W)-8 b(e)32 b(ma)m(y)f(rev)m(erse)g(the)f(roles)g(of)h Fm(m)f Fn(and)f Fm(n)p Fn(.)41 b(On)29 b(the)i(other)f(hand,)g(w)m(e)h(get)913 1836 y Fh(h)p Fm(D)s(")1068 1851 y Fl(d)1109 1836 y Fn(\()p Fm(f)1189 1850 y Fl(n)1236 1836 y Fm(;)15 b(A)1344 1850 y Fl(n)1392 1836 y Fn(\)\()p Fm(f)1507 1850 y Fl(n)1574 1836 y Fh(\000)20 b Fm(f)1710 1850 y Fl(m)1776 1836 y Fm(;)15 b Fn(0\))p Fh(i)22 b(\000)e(h)p Fm(D)s(")2199 1851 y Fl(d)2240 1836 y Fn(\()p Fm(f)2320 1850 y Fl(m)2386 1836 y Fm(;)15 b(A)2494 1850 y Fl(m)2561 1836 y Fn(\)\()p Fm(f)2676 1850 y Fl(n)2744 1836 y Fh(\000)20 b Fm(f)2880 1850 y Fl(m)2946 1836 y Fm(;)15 b Fn(0\))p Fh(i)913 2027 y Fn(=)1009 1903 y Fk(Z)1100 1929 y Fl(d)1060 2109 y Fg(0)1141 2027 y Fh(f)p Fm(\024)1238 1989 y Fi(\000)p Fg(2)1333 2027 y Fn(\()p Fm(f)1423 1989 y Fi(0)1413 2049 y Fl(n)1480 2027 y Fh(\000)20 b Fm(f)1626 1989 y Fi(0)1616 2049 y Fl(m)1682 2027 y Fn(\))1717 1989 y Fg(2)1777 2027 y Fn(+)g(\()p Fm(f)1948 2041 y Fl(n)2015 2027 y Fh(\000)g Fm(f)2151 2041 y Fl(m)2217 2027 y Fn(\)\()p Fm(A)2355 1989 y Fg(2)2355 2049 y Fl(n)2403 2027 y Fm(f)2448 2041 y Fl(n)2514 2027 y Fh(\000)g Fm(A)2673 1989 y Fg(2)2673 2049 y Fl(m)2740 2027 y Fm(f)2785 2041 y Fl(m)2851 2027 y Fn(\))913 2220 y Fh(\000)p Fn([\(1)h Fh(\000)f Fm(f)1256 2187 y Fg(2)1246 2242 y Fl(n)1295 2220 y Fn(\))p Fm(f)1375 2234 y Fl(n)1442 2220 y Fh(\000)g Fn(\(1)h Fh(\000)f Fm(f)1780 2187 y Fg(2)1770 2242 y Fl(m)1836 2220 y Fn(\))p Fm(f)1916 2234 y Fl(m)1982 2220 y Fn(]\()p Fm(f)2087 2234 y Fl(n)2155 2220 y Fh(\000)g Fm(f)2291 2234 y Fl(m)2357 2220 y Fn(\))p Fh(g)p Fm(dx:)3783 2019 y Fn(\(6.15\))0 2435 y(According)30 b(to)h(\(6.14\))i(and)d(\(6.15\),)i(w)m(e)f(deduce) 799 2668 y Fh(h)p Fm(D)s(")954 2683 y Fl(d)995 2668 y Fn(\()p Fm(f)1075 2682 y Fl(n)1122 2668 y Fm(;)15 b(A)1230 2682 y Fl(n)1277 2668 y Fn(\)\()p Fm(f)1392 2682 y Fl(n)1460 2668 y Fh(\000)20 b Fm(f)1596 2682 y Fl(m)1662 2668 y Fm(;)15 b Fn(0\))p Fh(i)22 b(\000)d(h)p Fm(D)s(")2084 2683 y Fl(d)2125 2668 y Fn(\()p Fm(f)2205 2682 y Fl(m)2272 2668 y Fm(;)c(A)2380 2682 y Fl(m)2447 2668 y Fn(\)\()p Fm(f)2562 2682 y Fl(n)2629 2668 y Fh(\000)20 b Fm(f)2765 2682 y Fl(m)2831 2668 y Fm(;)15 b Fn(0\))p Fh(i)27 b(\024)e Fn(2)p Fm(\017:)3783 2659 y Fn(\(6.16\))0 2891 y(F)-8 b(rom)40 b(\(6.9\))i(and)e(\(6.10\),)45 b(and)39 b(as)h Fm(H)1352 2858 y Fg(1)1392 2891 y Fn(\(]0)p Fm(;)15 b(d)p Fn([\))43 b Fm(,)-15 b Fh(!)41 b Fm(C)1901 2858 y Fg(0)1940 2891 y Fn(\([0)p Fm(;)15 b(d)p Fn(]\),)45 b(w)m(e)c(deduce)e(that)i Fm(f)2970 2905 y Fl(n)3056 2891 y Fn(and)f Fm(A)3311 2905 y Fl(n)3398 2891 y Fn(are)g(b)s(ounded)e(in)0 3004 y Fm(C)72 2971 y Fg(0)111 3004 y Fn(\([0)p Fm(;)15 b(d)p Fn(]\))35 b(and)d Fm(f)622 3018 y Fl(n)697 3004 y Fh(!)c Fm(f)42 b Fn(in)31 b Fm(C)1083 2971 y Fg(0)1122 3004 y Fn(\([0)p Fm(;)15 b(d)p Fn(]\))35 b(when)c Fm(n)e Fh(!)f Fn(+)p Fh(1)p Fn(.)47 b(W)-8 b(e)34 b(deduce)e(that)h(for)f(all)f Fm(\017)e(>)g Fn(0,)k(there)g(exists)3798 2981 y(~)3774 3004 y Fm(A)c(>)f Fn(0)0 3116 y(suc)m(h)i(that)h(for)f(all)f(pairs)g (\()p Fm(m;)15 b(n)p Fn(\))26 b Fh(2)f Fn(I)-15 b(N)1333 3081 y Fg(2)1403 3116 y Fn(suc)m(h)30 b(that)h Fm(m)25 b Fh(\025)2030 3094 y Fn(~)2006 3116 y Fm(A)30 b Fn(et)h Fm(n)25 b Fh(\025)2410 3094 y Fn(~)2386 3116 y Fm(A)p Fn(,)31 b(w)m(e)g(ha)m(v)m(e)454 3382 y Fh(j)494 3259 y Fk(Z)585 3285 y Fl(d)545 3465 y Fg(0)626 3382 y Fh(f)p Fn(\()p Fm(f)751 3396 y Fl(n)818 3382 y Fh(\000)20 b Fm(f)954 3396 y Fl(m)1020 3382 y Fn(\)\()p Fm(A)1158 3345 y Fg(2)1158 3405 y Fl(n)1206 3382 y Fm(f)1251 3396 y Fl(n)1318 3382 y Fh(\000)f Fm(A)1476 3345 y Fg(2)1476 3405 y Fl(m)1543 3382 y Fm(f)1588 3396 y Fl(m)1654 3382 y Fn(\))i Fh(\000)f Fn([\(1)h Fh(\000)f Fm(f)2073 3345 y 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Fm(\017:)1275 b Fn(\(6.17\))0 4269 y(F)-8 b(rom)30 b(\(6.12\))i(and)e(\(6.17\),)i(w)m(e)e(deduce)g (that)g(\()p Fm(f)1676 4283 y Fl(n)1723 4269 y Fn(\))1758 4300 y Fl(n)p Fi(2)p Fn(I)-12 b(N)1972 4269 y(is)28 b(a)j(Cauc)m(h)m (y's)f(sequence)g(in)f Fm(H)3089 4236 y Fg(1)3128 4269 y Fn(\(]0)p Fm(;)15 b(d)p Fn([\).)43 b(It)30 b(deriv)m(es)f(that)0 4382 y Fm(f)35 b Fh(2)24 b Fm(H)248 4349 y Fg(1)288 4382 y Fn(\(]0)p Fm(;)15 b(d)p Fn([\))32 b(and)e Fm(f)794 4396 y Fl(n)866 4382 y Fh(!)25 b Fm(f)5 b(;)65 b Fn(in)50 b Fm(H)1332 4349 y Fg(1)1371 4382 y Fn(\(]0)p Fm(;)15 b(d)p Fn([\))p Fm(:)0 4495 y Fn(Similarly)-8 b(,)38 b(ev)-5 b(aluating)39 b Fh(h)p Fm(D)s(")1016 4510 y Fl(d)1057 4495 y Fn(\()p Fm(f)1137 4509 y Fl(n)1183 4495 y Fm(;)15 b(A)1291 4509 y Fl(n)1339 4495 y Fn(\)\(0)p Fm(;)g(A)1562 4509 y Fl(n)1637 4495 y Fh(\000)25 b Fm(A)1801 4509 y Fl(m)1868 4495 y Fn(\))p Fh(i)i(\000)e(h)p Fm(D)s(")2216 4510 y Fl(d)2257 4495 y Fn(\()p Fm(f)2337 4509 y Fl(m)2404 4495 y Fm(;)15 b(A)2512 4509 y Fl(m)2579 4495 y Fn(\)\(0)p Fm(;)g(A)2802 4509 y Fl(n)2877 4495 y Fh(\000)25 b Fm(A)3041 4509 y Fl(m)3108 4495 y Fn(\))p Fh(i)p Fn(,)42 b(one)d(can)h(c)m(hec)m (k)g(that)0 4608 y(\()p Fm(A)103 4622 y Fl(n)151 4608 y Fn(\))186 4639 y Fl(n)p Fi(2)p Fn(I)-12 b(N)411 4608 y(is)41 b(Cauc)m(h)m(y's)h(sequence)g(in)f Fm(H)1501 4575 y Fg(1)1540 4608 y Fn(\(]0)p Fm(;)15 b(d)p Fn([\).)78 b(Consequen)m(tly)-8 b(,)44 b(there)e(exists)g Fm(A)i Fh(2)g Fm(H)3297 4575 y Fg(1)3336 4608 y Fn(\(]0)p Fm(;)15 b(d)p Fn([\))45 b(suc)m(h)c(that)0 4721 y Fm(A)68 4735 y Fl(n)140 4721 y Fh(!)26 b Fm(A;)66 b Fn(in)49 b Fm(H)625 4688 y Fg(1)664 4721 y Fn(\(]0)p Fm(;)15 b(d)p Fn([\))p Fm(:)33 b Fn(The)d(pro)s(of)g(of)g(Prop)s(osition)f(6.3)i(follo)m(ws.)0 4947 y(No)m(w,)44 b(w)m(e)d(sho)m(w)f(that)h(a)g(critical)e(p)s(oin)m (t)h(of)g Fm(")1633 4962 y Fl(d)1714 4947 y Fn(in)f(the)i(sense)f(of)h (the)g(de\014nition)d(6.1)j(is)f(a)g(solution)f(of)i(\()p Fm(GL)p Fn(\))3949 4914 y Fl(s)3949 4974 y(d)3990 4947 y Fn(.)0 5060 y(W)-8 b(e)32 b(follo)m(w)d(a)i(pro)s(of)f(due)f(to)i(T.) g(Giorgi,)f(S.)g(Alama,)g(L.)h(Bronsard)e(\(see)j(Lemma)e(4.1)i(in)d ([2]\).)0 5247 y Fe(Prop)s(osition)36 b(6.4)46 b Ff(L)-5 b(et)43 b(us)f(assume)h(that)g Fn(\()p Fm(f)5 b(;)15 b(A)p Fn(\))43 b Fh(2)f Fm(C)2001 5261 y Fl(\013)2093 5247 y Ff(with)h Fm(g)s Fn(\()p Fm(f)5 b(;)15 b(A)p Fn(\))44 b(=)e(0)p Ff(.)71 b(Then)43 b Fn(\()p Fm(f)5 b(;)15 b(A)p Fn(\))42 b Ff(is)g(a)h(solution)g(of)0 5360 y Fn(\()p Fm(GL)p Fn(\))203 5327 y Fl(s)203 5388 y(d)245 5360 y Ff(.)1962 5871 y Fn(20)p eop %%Page: 21 21 21 20 bop 0 162 a Fe(Pro)s(of)43 b Fn(Let)f Fm(\036;)15 b( )49 b Fh(2)44 b Fm(C)849 129 y Fi(1)842 187 y Fg(0)923 162 y Fn(\(]0)p Fm(;)15 b(d)p Fn([\))44 b(and)e Fm(\017)i(>)g Fn(0.)76 b(F)-8 b(rom)43 b(\(6.2\))g(and)f(the)g(de\014nition)d(of)j Fm(g)k Fn(\(see)d(\(6.1\)\),)k(w)m(e)42 b(ha)m(v)m(e)0 275 y Fh(h)p Fm(D)s(")155 290 y Fl(d)196 275 y Fn(\()p Fm(f)5 b(;)15 b(A)p Fn(\))p Fm(;)g Fn(\(0)p Fm(;)635 234 y Fl( )p 596 254 127 4 v 596 308 a Fi(jj)p Fl( )r Fi(jj)733 275 y Fn(\))p Fh(i)26 b(\024)f Fm(g)s Fn(\()p Fm(f)5 b(;)15 b(A)p Fn(\))p Fm(:)32 b Fn(Th)m(us)d(as)i Fm(g)s Fn(\()p Fm(f)5 b(;)15 b(A)p Fn(\))26 b(=)f(0,)31 b(w)m(e)g(get)1552 492 y Fh(h)p Fm(D)s(")1707 507 y Fl(d)1748 492 y Fn(\()p Fm(f)5 b(;)15 b(A)p Fn(\))p Fm(;)g Fn(\(0)p Fm(;)g( )s Fn(\))p Fh(i)29 b(\024)c Fn(0)p Fm(:)0 696 y Fn(Replacing)k Fm( )34 b Fn(with)29 b Fh(\000)p Fm( )34 b Fn(in)29 b(this)g(previous)g(inequalit)m(y)-8 b(,)29 b(w)m(e)i(get)1182 901 y Fh(h)p Fm(D)s(")1337 916 y Fl(d)1378 901 y Fn(\()p Fm(f)5 b(;)15 b(A)p Fn(\))p Fm(;)g Fn(\(0)p Fm(;)g( )s Fn(\))p Fh(i)28 b Fn(=)d(0)p Fm(;)67 b Fh(8)24 b Fm( )29 b Fh(2)c Fm(C)2480 863 y Fi(1)2473 923 y Fg(0)2554 901 y Fn(\([0)p Fm(;)15 b(d)p Fn(]\))p Fm(:)952 b Fn(\(6.18\))0 1105 y(W)-8 b(e)32 b(deduce)e(that)1398 1218 y Fh(\000)p Fm(A)1537 1180 y Fi(00)1600 1218 y Fn(+)20 b Fm(Af)1814 1180 y Fg(2)1878 1218 y Fn(=)25 b(0)p Fm(;)66 b Fn(in)50 b Fm(D)2315 1180 y Fi(0)2338 1218 y Fn(\(]0)p Fm(;)15 b(d)p Fn([\))p Fm(:)1168 b Fn(\(6.19\))0 1385 y(W)-8 b(e)36 b(set)f Fm(\036)361 1399 y Fl(\017)427 1385 y Fn(=)d Fh(\000)p Fn(\()p Fm(f)h Fn(+)23 b Fm(\017\036)g Fh(\000)g Fm(\013)p Fn(\))1109 1399 y Fi(\000)1201 1385 y Fh(\025)33 b Fn(0)p Fm(;)114 b(\036)1543 1352 y Fl(\017)1608 1385 y Fn(=)33 b(\()p Fm(f)f Fn(+)23 b Fm(\017\036)h Fh(\000)e Fn(1\))2206 1399 y Fg(+)2299 1385 y Fh(\025)32 b Fn(0)p Fm(;)k Fn(and)e(de\014ne)g Fm(w)3021 1399 y Fl(\017)3089 1385 y Fn(to)h(b)s(e)f(the)h(truncation)g(of)0 1498 y Fm(f)29 b Fn(+)20 b Fm(\017\036)1560 1610 y(w)1625 1624 y Fl(\017)1683 1610 y Fn(=)25 b Fm(f)k Fn(+)20 b Fm(\017\036)g Fn(+)g Fm(\036)2200 1624 y Fl(\017)2253 1610 y Fh(\000)g Fm(\036)2398 1573 y Fl(\017)2431 1610 y Fm(:)0 1777 y Fn(Then)29 b(\()p Fm(w)337 1791 y Fl(\017)370 1777 y Fm(;)15 b(A)p Fn(\))27 b Fh(2)e Fm(C)691 1791 y Fl(\013)740 1777 y Fm(:)31 b Fn(F)-8 b(or)31 b Fm(\017)25 b(>)g Fn(0,)31 b(it)f(results)f(that)1452 1982 y Fh(h)p Fm(D)s(")1607 1997 y Fl(d)1648 1982 y Fn(\()p Fm(f)5 b(;)15 b(A)p Fn(\))p Fm(;)g Fn(\()p Fm(f)31 b Fh(\000)20 b Fm(w)2183 1996 y Fl(\017)2215 1982 y Fm(;)15 b Fn(0\))p Fh(i)27 b(\024)e Fn(0)p Fm(:)1220 b Fn(\(6.20\))0 2186 y(W)-8 b(e)32 b(set)1057 2299 y(\012)1123 2313 y Fl(\017)1180 2299 y Fn(:=)26 b Fh(f)p Fm(x)f Fn(:)h Fm(f)j Fn(+)20 b Fm(\017\036)25 b Fh(\024)g Fm(\013)p Fh(g)p Fm(;)67 b Fn(\012)2113 2261 y Fl(\017)2171 2299 y Fn(:=)25 b Fh(f)p Fm(x)h Fn(:)f Fm(f)30 b Fn(+)20 b Fm(\017\036)25 b Fh(\025)g Fn(1)p Fh(g)p Fm(:)0 2466 y Fn(F)-8 b(rom)31 b(\(6.20\),)i(w)m(e)e(deduce)f(that)683 2670 y(0)c Fh(\024)f Fm(\017)p Fh(h)p Fm(D)s(")1042 2685 y Fl(d)1083 2670 y Fn(\()p Fm(f)5 b(;)15 b(A)p Fn(\))p Fm(;)g Fn(\()p Fm(\036;)g Fn(0\))p Fh(i)23 b(\000)d(h)p Fm(D)s(")1864 2685 y Fl(d)1905 2670 y Fn(\()p Fm(f)5 b(;)15 b(A)p Fn(\))p Fm(;)g Fn(\()p Fm(\036)2262 2632 y Fl(\017)2296 2670 y Fm(;)g Fn(0\))p Fh(i)22 b Fn(+)e Fh(h)p Fm(D)s(")2719 2685 y Fl(d)2760 2670 y Fn(\()p Fm(f)5 b(;)15 b(A)p Fn(\))p Fm(;)g Fn(\()p Fm(\036)3117 2684 y Fl(\017)3151 2670 y Fm(;)g Fn(0\))p Fh(i)p Fm(;)0 2874 y Fn(th)m(us)722 3002 y Fh(h)p Fm(D)s(")877 3017 y Fl(d)918 3002 y Fn(\()p Fm(f)5 b(;)15 b(A)p Fn(\))p Fm(;)g Fn(\()p Fm(\036;)g Fn(0\))p Fh(i)28 b(\025)1564 2941 y Fn(1)p 1564 2982 46 4 v 1568 3065 a Fm(\017)1620 3002 y Fn([)p Fh(h)p Fm(D)s(")1800 3017 y Fl(d)1841 3002 y Fn(\()p Fm(f)5 b(;)15 b(A)p Fn(\))p Fm(;)g Fn(\()p Fm(\036)2198 2965 y Fl(\017)2232 3002 y Fm(;)g Fn(0\))p Fh(i)22 b(\000)e(h)p Fm(D)s(")2655 3017 y Fl(d)2696 3002 y Fn(\()p Fm(f)5 b(;)15 b(A)p Fn(\))p Fm(;)g Fn(\()p Fm(\036)3053 3016 y Fl(\017)3087 3002 y Fm(;)g Fn(0\))p Fh(i)p Fn(])p Fm(:)0 3203 y Fn(W)-8 b(e)32 b(ha)m(v)m(e)f Fh(h)p Fm(D)s(")521 3218 y Fl(d)562 3203 y Fn(\(1)p Fm(;)15 b(A)p Fn(\))p Fm(;)g Fn(\()p Fm(\036)914 3170 y Fl(\017)950 3203 y Fm(;)g Fn(0\))p Fh(i)26 b Fn(=)1227 3130 y Fk(R)1270 3235 y Fg(\012)1321 3216 y Fj(\017)1371 3203 y Fm(A)1439 3170 y Fg(2)1478 3203 y Fm(\036)1532 3170 y Fl(\017)1565 3203 y Fm(dx:)31 b Fn(It)f(results)g(that)917 3407 y Fh(h)p Fm(D)s(")1072 3422 y Fl(d)1113 3407 y Fn(\()p Fm(f)5 b(;)15 b(A)p Fn(\))p Fm(;)g Fn(\()p Fm(\036)1470 3370 y Fl(\017)1505 3407 y Fm(;)g Fn(0\))p Fh(i)52 b(\025)e(h)p Fm(D)s(")1988 3422 y Fl(d)2029 3407 y Fn(\()p Fm(f)5 b(;)15 b(A)p Fn(\))21 b Fh(\000)e Fm(D)s(")2488 3422 y Fl(d)2529 3407 y Fn(\(1)p Fm(;)c(A)p Fn(\))p Fm(;)g Fn(\()p Fm(\036)2881 3370 y Fl(\017)2916 3407 y Fm(;)g Fn(0\))p Fh(i)p Fm(:)0 3611 y Fn(Consequen)m(tly)-8 b(,)622 3852 y Fh(h)p Fm(D)s(")777 3867 y Fl(d)818 3852 y Fn(\()p Fm(f)5 b(;)15 b(A)p Fn(\))p Fm(;)g Fn(\()p Fm(\036)1175 3819 y Fl(\017)1209 3852 y Fm(;)g Fn(0\))p Fh(i)27 b(\025)1487 3729 y Fk(Z)1537 3935 y Fg(\012)1588 3916 y Fj(\017)1623 3852 y Fn(\()p Fm(\024)1710 3815 y Fi(\000)p Fg(2)1805 3852 y Fm(f)1860 3815 y Fi(0)1882 3852 y Fn(\()p Fm(\036)1971 3815 y Fl(\017)2005 3852 y Fn(\))2040 3810 y Fi(0)2084 3852 y Fn(+)19 b Fm(f)10 b Fn(\()p Fh(\000)p Fn(1)20 b(+)g Fm(f)2546 3815 y Fg(2)2585 3852 y Fn(\))p Fm(\036)2674 3815 y Fl(\017)2727 3852 y Fn(+)g Fm(A)2886 3815 y Fg(2)2926 3852 y Fn(\()p Fm(f)30 b Fh(\000)20 b Fn(1\))p Fm(\036)3261 3815 y Fl(\017)3294 3852 y Fm(dx)622 4059 y Fh(\025)718 3935 y Fk(Z)769 4141 y Fg(\012)820 4122 y Fj(\017)854 4059 y Fn([)p Fm(\024)931 4021 y Fi(\000)p Fg(2)1026 4059 y Fn(\()p Fm(f)30 b Fh(\000)20 b Fn(1\))1307 4021 y Fi(0)1331 4059 y Fn(\()p Fm(f)30 b Fh(\000)19 b Fn(1)i(+)f Fm(\017\036)p Fn(\))1814 4021 y Fi(0)622 4252 y Fn(+)p Fm(A)761 4219 y Fg(2)800 4252 y Fn(\()p Fm(f)30 b Fh(\000)20 b Fn(1\)\()p Fm(f)31 b Fh(\000)20 b Fn(1)g(+)g Fm(\017\036)p Fn(\))h(+)f Fm(f)10 b Fn(\()p Fh(\000)p Fn(1)20 b(+)g Fm(f)2049 4219 y Fg(2)2087 4252 y Fn(\)\()p Fm(f)31 b Fh(\000)20 b Fn(1)g(+)g Fm(\017\036)p Fn(\)])p Fm(dx)622 4423 y Fh(\025)25 b Fm(\017)770 4299 y Fk(Z)821 4505 y Fg(\012)872 4486 y Fj(\017)906 4423 y Fn(\()p Fm(\024)993 4385 y Fi(\000)p Fg(2)1088 4423 y Fm(f)1143 4385 y Fi(0)1166 4423 y Fm(\036)1220 4385 y Fi(0)1263 4423 y Fn(+)20 b Fm(f)10 b Fn(\()p Fh(\000)p Fn(1)20 b(+)g Fm(f)1726 4385 y Fg(2)1765 4423 y Fn(\))p Fm(\036)g Fn(+)g Fm(A)2033 4385 y Fg(2)2073 4423 y Fn(\()p Fm(f)30 b Fh(\000)20 b Fn(1\))p Fm(\036dx:)0 4678 y Fn(As)30 b Fm(mes)p Fn(\(\012)400 4645 y Fl(\017)433 4678 y Fn(\))25 b Fh(!)h Fn(0)k(when)g Fm(\017)25 b Fh(!)g Fn(0)1146 4645 y Fg(+)1206 4678 y Fn(,)30 b(w)m(e)h(get)1487 4882 y Fh(h)p Fm(D)s(")1642 4897 y Fl(d)1683 4882 y Fn(\()p Fm(f)5 b(;)15 b(A)p Fn(\))p Fm(;)g Fn(\()p Fm(\036)2040 4845 y Fl(\017)2074 4882 y Fm(;)g Fn(0\))p Fh(i)27 b(\025)e Fm(o)p Fn(\()p Fm(\017)p Fn(\))p Fm(:)1255 b Fn(\(6.21\))0 5086 y(Similarly)-8 b(,)27 b(w)m(e)k(get)842 5327 y Fh(h)p Fm(D)s(")997 5342 y Fl(d)1038 5327 y Fn(\()p Fm(f)5 b(;)15 b(A)p Fn(\))p Fm(;)g Fn(\()p Fm(\036)1395 5341 y Fl(\017)1429 5327 y Fm(;)g Fn(0\))p Fh(i)27 b Fn(=)1707 5204 y Fk(Z)1757 5410 y Fg(\012)1808 5418 y Fj(\017)1842 5327 y Fn(\()p Fm(\024)1929 5290 y Fi(\000)p Fg(2)2024 5327 y Fm(f)2079 5290 y Fi(0)2102 5327 y Fm(\036)2156 5341 y Fl(\017)2189 5290 y Fi(0)2232 5327 y Fn(+)20 b Fm(f)10 b Fn(\()p Fh(\000)p Fn(1)20 b(+)g Fm(f)2695 5290 y Fg(2)2754 5327 y Fn(+)g Fm(A)2913 5290 y Fg(2)2953 5327 y Fn(\))p Fm(\036)3042 5341 y Fl(\017)3075 5327 y Fm(dx)842 5542 y Fh(\024)25 b(\000)p Fm(\017)1061 5418 y Fk(Z)1111 5624 y Fg(\012)1162 5632 y Fj(\017)1211 5542 y Fm(\024)1263 5504 y Fi(\000)p Fg(2)1358 5542 y Fm(f)1413 5504 y Fi(0)1436 5542 y Fm(\036)1490 5504 y Fi(0)1533 5542 y Fn(+)20 b Fm(A)1692 5504 y Fg(2)1732 5542 y Fm(f)10 b(\036dx)25 b Fn(=)g Fm(o)p Fn(\()p Fm(\017)p Fn(\))p Fm(:)3783 5441 y Fn(\(6.22\))1962 5871 y(21)p eop %%Page: 22 22 22 21 bop 0 162 a Fn(According)30 b(to)h(\(6.21\))i(and)d(\(6.22\),)i (w)m(e)f(get)h Fh(h)p Fm(D)s(")1711 177 y Fl(d)1752 162 y Fn(\()p Fm(f)5 b(;)15 b(A)p Fn(\))p Fm(;)g Fn(\()p Fm(\036;)g Fn(0\))p Fh(i)28 b(\025)2398 126 y Fg(1)p 2398 141 36 4 v 2401 194 a Fl(\017)2443 162 y Fm(o)p Fn(\()p Fm(\017)p Fn(\))p Fm(;)67 b Fh(8)25 b Fm(\017)g(>)g Fn(0)p Fm(;)31 b Fn(th)m(us)1556 330 y Fh(h)p Fm(D)s(")1711 345 y Fl(d)1752 330 y Fn(\()p Fm(f)5 b(;)15 b(A)p Fn(\))p Fm(;)g Fn(\()p Fm(\036;)g Fn(0\))p Fh(i)29 b(\025)c Fn(0)p Fm(:)1324 b Fn(\(6.23\))0 497 y(Since)29 b(w)m(e)i(migh)m(t)f(just)g (as)h(w)m(ell)e(ha)m(v)m(e)j(considered)d Fh(\000)p Fm(\036)h Fn(in)f(the)i(ab)s(o)m(v)m(e)g(calculation,)f(w)m(e)h(obtain)1190 665 y Fh(h)p Fm(D)s(")1345 680 y Fl(d)1386 665 y Fn(\()p Fm(f)5 b(;)15 b(A)p Fn(\))p Fm(;)g Fn(\()p Fm(\036;)g Fn(0\))p Fh(i)28 b Fn(=)d(0)p Fm(;)67 b Fh(8)24 b Fm(\036)i Fh(2)f Fm(C)2472 627 y Fi(1)2465 687 y Fg(0)2546 665 y Fn(\(]0)p Fm(;)15 b(d)p Fn([\))p Fm(;)960 b Fn(\(6.24\))0 833 y(th)m(us)1147 945 y Fh(\000)p Fm(\024)1270 908 y Fi(\000)p Fg(2)1364 945 y Fm(f)1419 908 y Fi(00)1481 945 y Fh(\000)20 b Fm(f)30 b Fn(+)20 b Fm(f)1793 908 y Fg(3)1851 945 y Fn(+)g Fm(A)2010 908 y Fg(2)2050 945 y Fm(f)35 b Fn(=)25 b(0)p Fm(;)66 b Fn(in)49 b Fm(D)2566 908 y Fi(0)2589 945 y Fn(\(]0)p Fm(;)15 b(d)p Fn([\))p Fm(:)917 b Fn(\(6.25\))0 1094 y(F)-8 b(rom)32 b(\(6.19\))h(and)d (\(6.25\),)k(w)m(e)e(deduce)e(that)i Fm(f)k Fh(2)26 b Fm(H)1856 1061 y Fg(2)1895 1094 y Fn(\([0)p Fm(;)15 b(d)p Fn(]\))34 b(and)c Fm(A)d Fh(2)f Fm(H)2623 1061 y Fg(2)2662 1094 y Fn(\([0)p Fm(;)15 b(d)p Fn(]\),)34 b(then)d Fm(f)k Fh(2)26 b(\\)3409 1125 y Fl(m)p Fi(2)p Fn(I)-11 b(N)3612 1094 y Fm(H)3695 1061 y Fl(m)3761 1094 y Fn(\([0)p Fm(;)15 b(d)p Fn(]\))0 1207 y(and)26 b Fm(A)f Fh(2)g(\\)413 1238 y Fl(m)p Fi(2)p Fn(I)-12 b(N)615 1207 y Fm(H)698 1174 y Fl(m)765 1207 y Fn(\([0)p Fm(;)15 b(d)p Fn(]\))p Fm(:)29 b Fn(The)d(functions)e Fm(f)36 b Fn(and)25 b Fm(A)i Fn(are)f Fm(C)2208 1174 y Fi(1)2309 1207 y Fn(and)g(satisfy)f(the)i (Ginzburg-Landau)d(equations)0 1320 y(in)31 b(the)i(usual)e(sense.)47 b(Let)33 b(us)f(sho)m(w)g(that)h(\()p Fm(f)5 b(;)15 b(A)p Fn(\))33 b(satis\014es)f(the)h(b)s(oundary)d(conditions)h(at)i(zero)h (\(see)f(\(1.2\)\).)49 b(Let)12 1409 y(~)0 1433 y Fm(\036)25 b Fh(2)g Fm(C)237 1400 y Fg(1)230 1457 y(0)276 1433 y Fn(\([0)p Fm(;)15 b(d)p Fn([\).)43 b(F)-8 b(rom)31 b(\(6.18\),)i(w)m(e) e(get)1346 1540 y Fk(Z)1437 1566 y Fl(d)1397 1746 y Fg(0)1478 1664 y Fn(\()p Fm(A)1581 1626 y Fi(0)1604 1664 y Fm(f)1659 1626 y Fi(0)1702 1664 y Fn(+)20 b Fm(Af)1916 1626 y Fg(2)1967 1640 y Fn(~)1955 1664 y Fm(\036)p Fn(\))p Fm(dx)h Fn(+)f Fm(h)2319 1640 y Fn(~)2307 1664 y Fm(\036)p Fn(\(0\))27 b(=)e(0)p Fm(:)1114 b Fn(\(6.26\))0 1872 y(On)30 b(the)g(other)h(hand) 1348 1900 y Fk(Z)1439 1927 y Fl(d)1399 2106 y Fg(0)1495 2024 y Fm(A)1563 1987 y Fi(00)1617 2000 y Fn(~)1605 2024 y Fm(\036dx)26 b Fn(=)f([)p Fm(A)1973 1987 y Fi(0)2009 2000 y Fn(~)1997 2024 y Fm(\036)p Fn(])2076 1987 y Fl(d)2076 2047 y Fg(0)2137 2024 y Fh(\000)2228 1900 y Fk(Z)2319 1927 y Fl(d)2278 2106 y Fg(0)2374 2024 y Fm(A)2442 1987 y Fi(0)2478 2000 y Fn(~)2466 2024 y Fm(\036)2520 1987 y Fi(0)2543 2024 y Fm(dx;)0 2226 y Fn(th)m(us,)30 b(taking)h(accoun)m (t)g(\(6.26\),)i(and)d(the)h(fact)g(that)g Fh(\000)p Fm(A)1978 2193 y Fi(00)2040 2226 y Fn(+)20 b Fm(f)2186 2193 y Fg(2)2225 2226 y Fm(A)25 b Fn(=)g(0,)31 b(w)m(e)g(deduce)f(that) 1282 2370 y(~)1270 2394 y Fm(\036)p Fn(\(0\)\()p Fm(A)1542 2356 y Fi(0)1567 2394 y Fn(\(0\))21 b Fh(\000)f Fm(h)p Fn(\))26 b(=)f(0)p Fm(;)67 b Fh(8)2203 2370 y Fn(~)2191 2394 y Fm(\036)24 b Fh(2)h Fm(C)2427 2356 y Fg(1)2420 2416 y(0)2466 2394 y Fn(\([0)p Fm(;)15 b(d)p Fn([\))p Fm(:)0 2561 y Fn(It)30 b(deriv)m(es)g(that)1805 2674 y Fm(A)1873 2637 y Fi(0)1896 2674 y Fn(\(0\))d(=)d Fm(h:)0 2823 y Fn(Similarly)-8 b(,)33 b(using)h(\(6.24\),)39 b(w)m(e)d(can)g(sho)m(w)f(that)h Fm(f)1745 2790 y Fi(0)1768 2823 y Fn(\(0\))e(=)f(0)p Fm(:)j Fn(T)-8 b(o)36 b(sho)m(w)f(that)h Fm(f)2753 2790 y Fi(0)2776 2823 y Fn(\()p Fm(d)p Fn(\))f(=)e(0,)k(w)m (e)f(mak)m(e)g(similar)d(com-)0 2936 y(putations)g(with)g(test)i (functions)1214 2912 y(~)1195 2936 y Fm( )g Fh(2)c Fm(C)1453 2903 y Fg(1)1492 2936 y Fn(\(]0)p Fm(;)15 b(d)p Fn(]\).)54 b(The)33 b(pair)g(\()p Fm(f)5 b(;)15 b(A)p Fn(\))34 b(satis\014es)g (the)g(b)s(oundary)e(conditions.)50 b(The)0 3049 y(pro)s(of)30 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b(and)g(the)g(stable)h(solutions)d(satisfy)i Fm(f)10 b Fn(\(0\))26 b Fm(>)f(\022)s Fn(\()p Fm(\024)p Fn(\).)0 5509 y(Theorems)h(1.3)i(and)e(1.5)i(giv)m(e)f(the)g(existence) h(of)e(non-minimizing)d(solutions,)j(but)g(the)h(approac)m(h)g(used)f (here)h(do)s(es)0 5622 y(not)k(p)s(ermit)d(to)j(determine)f(a)h(lo)s (calization)e(of)i(these)f(solutions.)40 b(This)28 b(question)i(is)f (also)h(an)h(op)s(en)e(problem.)1962 5871 y(24)p eop %%Page: 25 25 25 24 bop 0 162 a Fv(Ac)m(kno)m(wledgemen)m(ts)0 334 y Fn(The)30 b(author)g(w)m(ould)f(lik)m(e)h(to)h(thank)f(C.)g(Bolley)g (and)g(B.)h(Hel\013er)f(for)g(discussions)e(around)h(the)i(sub)5 b(ject.)0 619 y Fw(References)45 822 y Fn([1])47 b(A.)31 b(Aftalion,)187 935 y(On)e(the)i(minimizers)c(of)k(the)f (Ginzburg-Landau)f(energy)h(for)h(high)e(k)-5 b(appa:)40 b(the)31 b(one-dimensional)d(case.)k Fe(8)p Fn(,)187 1048 y Ff(Eur.)g(J.)g(Appl.)h(Math.,)g(\(1997\))j Fn(331-345)45 1197 y([2])47 b(S.)30 b(Alama,)h(L.)f(Bronsard)g(&)g(T.)g(Giorgi,)187 1310 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b(Bolley)g(&)g(B.)h(Hel\013er)f(,) 187 2434 y(Change)g(of)h(stabilit)m(y)e(for)h(symmetric)g(bifurcating)e (solutions)h(in)g(the)h(Ginzburg-Landau)f(equations)187 2547 y Ff(Eur.)j(J.)g(Appl.)h(Math.,)g(Vol.)g(11)g(\(2000\))g Fn(95{119)45 2696 y([6])47 b(C.)30 b(Bolley)g(&)g(B.)h(Hel\013er)f(,) 187 2809 y(Rigorous)39 b(results)e(on)j(the)f(Ginzburg-Landau)e(mo)s (dels)h(in)g(a)i(\014lm)d(submitted)h(to)i(an)f(exterior)g(magnetic)323 2922 y(\014eld.)187 3035 y Ff(Part)33 b(I.)g(Nonline)-5 b(ar)33 b(Studies,)g(No.)g(3,)g(\(1996\))f Fn(1{29)45 3184 y([7])47 b(C.)30 b(Bolley)g(&)g(B.)h(Hel\013er)f(,)187 3297 y(Rigorous)39 b(results)e(on)j(the)f(Ginzburg-Landau)e(mo)s(dels)h (in)g(a)i(\014lm)d(submitted)h(to)i(an)f(exterior)g(magnetic)323 3409 y(\014eld.)187 3522 y Ff(Part)33 b(II.)g(Nonline)-5 b(ar)33 b(Studies,)g(V)-7 b(ol.)33 b(3,)g(No.)f(2,)h(\(1996\))g Fn(1{32)45 3671 y([8])47 b(C.)35 b(Bolley)g(&)g(B.)h(Hel\013er)f(,)i 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b(in)h(a)h(semi-in\014nite)e(\014lm)g(in)h(the)i (w)m(eak)g Fm(\024)f Fn(limit:)66 b(n)m(umerical)43 b(results)g(and)h (appro)m(ximate)323 5134 y(mo)s(dels.)187 5247 y Ff(Math.)33 b(Mo)-5 b(del.)33 b(Num.)f(A)n(nal.,)h(Vol.)f(31,)i(No.)e(1,)h (\(1997\))j Fn(121{165)0 5396 y([12])47 b(C.)30 b(Bolley)-8 b(,)31 b(F.)g(F)-8 b(ouc)m(her)31 b(&)f(B.)h(Hel\013er)f(,)187 5509 y(Sup)s(erheating)e(\014eld)h(for)h(the)h(Ginzburg-Landau)d (equations)i(in)f(the)i(case)g(of)g(a)g(large)f(b)s(ounded)e(in)m(terv) -5 b(al.)187 5622 y Ff(J.)32 b(Math.)h(Phys.)g(V)-7 b(ol.)33 b(41,)g(No.)f(11,)i(\(2000\))e Fn(7263{7289.)1962 5871 y(25)p eop %%Page: 26 26 26 25 bop 0 162 a Fn([13])47 b(C.)30 b(Bolley)-8 b(,)31 b(&)f(B.)h(Hel\013er)f(,)187 275 y(Global)f(sup)s(erheating)g(\014eld)g (for)h(sup)s(erconductors)f(in)g(a)i(large)f(b)s(ounded)e(in)m(terv)-5 b(al.)187 388 y Ff(Physic)g(a)33 b(D)g(172,)g(\(2002\))g Fn(162-189)0 538 y([14])47 b(P)-8 b(.)31 b(Del)f(Castillo)f(,)187 651 y(Tw)m(o)39 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y(Pro)s(of)h(of)g(the)h(P)m(arr)f(form)m (ula)g(for)g(the)h(sup)s(erheating)d(\014eld.)187 2118 y Ff(Pr)-5 b(eprint:)43 b(Math.)33 b(Phys.)g(A)n(r)-5 b(ch.,)32 b(Novemb)-5 b(er)33 b(2003)0 2269 y Fn([18])47 b(P)-8 b(.)31 b(Del)f(Castillo)59 b(and)30 b(S.)g(Me\014re,)187 2381 y(Numerical)22 b(study)g(of)i(the)f(stabilit)m(y)f(of)h(the)h (solutions)d(of)j(the)f(one)h(dimensional)c(Ginzburg-Landau)h(problem) 187 2494 y Ff(In)33 b(pr)-5 b(ep)g(ar)g(ation)0 2645 y Fn([19])47 b(P)-8 b(.)31 b(G.)g(De)g(Gennes,)187 2758 y(Sup)s(erconductivit)m(y)d(selected)j(topics)f(in)f(solid)f(state)k (ph)m(ysics)d(and)h(theoretical)h(Ph)m(ysics.)187 2870 y Ff(Pr)-5 b(o)g(c.)33 b(of)g(8th)h(L)-5 b(atin)33 b(americ)-5 b(an)34 b(scho)-5 b(ol)35 b(of)e(physics,)g(Car)-5 b(ac)g(as.)33 b Fn(\(1966\).)0 3021 y([20])47 b(S.)30 b(J.)g(Di)g(Bartolo,)i(A.)e(T.) h(Dorsey)g(&)f(A.)g(J.)61 b(Dolgert,)187 3134 y(Sup)s(erheating)28 b(\014elds)h(of)i(sup)s(erconductors)d(Asymptotic)j(analysis)e(and)g(n) m(umerical)g(results.)187 3247 y Ff(Physic)-5 b(al)34 b(R)-5 b(eview)33 b(B,)f(Vol.)h(53,)g(No)f(9)p Fn(,)f(1996.)187 3359 y(Erratum.)187 3472 y Ff(Physic)-5 b(al)34 b(R)-5 b(eview)33 b(B,)f(Vol.)h(56,)g(No.)f(5)p Fn(,)f(1997.)0 3623 y([21])47 b(V.)31 b(L.)f(Ginzburg)f(,)187 3736 y(On)g(the)i (theory)g(of)f(sup)s(erconductivit)m(y)-8 b(,)187 3848 y Ff(Nuovo)33 b(Cimento)e Fe(2)f Fn(\(1955\))j(1234.)0 3999 y([22])47 b(V.)31 b(L.)f(Ginzburg)f(,)187 4112 y(On)g(the)i (destruction)e(and)h(the)h(onset)g(of)f(sup)s(erconductivit)m(y)e(in)h (a)i(magnetic)g(\014eld,)187 4225 y Ff(Zh.)h(Eksp)-5 b(erim.)34 b(i)e(T)-7 b(e)i(or.)34 b(Fiz.)c Fe(34)h Fn(\(1958\))i (113{125.)187 4337 y(T)-8 b(ransl.)29 b Ff(Soviet)k(Physics)g(JETP)62 b Fe(7)30 b Fn(\(1958\))j(78{87.)0 4488 y([23])47 b(S.)30 b(Hastings,)187 4601 y(Another)40 b(Lo)s(ok)g(at)h(Some)f(Recen)m(t)i (Results)d(Concerning)g(Bifurcation)g(in)g(One-Dimensional)f(Mo)s(dels) i(of)323 4714 y(Sup)s(erconductivit)m(y)28 b(\(1999\))187 4826 y Ff(Eur.)k(J.)g(Appl.)h(Math.,)g(V)-7 b(ol.)33 b(11)g(\(2000\))g Fn(121{128)0 4977 y([24])47 b(S.)30 b(Hastings,)g(M.)h(K.)g(Kw)m(ong)f(&)g(W.C.)h(T)-8 b(ro)m(y)g(,)187 5090 y(The)30 b(existence)h(of)f(m)m(ultiple)e(solutions)h(for)h(a)h (Ginzburg-Landau)d(t)m(yp)s(e)j(mo)s(del)e(of)i(sup)s(erconductivit)m (y)187 5203 y Ff(Eur.)h(J.)g(Appl.)h(Math.,)g(Vol.)g(7)g(\(2002\))f Fn(559{574)0 5353 y([25])47 b(M.)31 b(K.)f(Kw)m(ong,)187 5466 y(On)f(the)i(one-dimensional)d(Ginzburg-Landau)h(BVP's,)187 5579 y Ff(Di\013er)-5 b(ential)33 b(Inte)-5 b(gr)g(al)34 b(Equations)f(8,)g(\(1995\))g Fn(1395-1405)1962 5871 y(26)p eop %%Page: 27 27 27 26 bop 0 162 a Fn([26])47 b(M.)31 b(Stru)m(w)m(e,)187 275 y(V)-8 b(ariational)30 b(metho)s(ds.)187 388 y(Springer,)e(Berlin,) h(1990.)1962 5871 y(27)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0406041533864--