Phương pháp giải tích hàm trong phương trình phi tuyến

(t) ~ r 'lei Lh. tEl va r:1?i x ~[-r,rJ va ho3:e1a (A.2) (i) ::-3rl t!ii'EL1(I) S3.0 eho , . -1 ( ) ./.Ll!: sup X g t,x ~ r(t) jxl-,> 00 -' '., 1-deu vch n. ",. t,; I (ii) ro~ tai de 83'thue Et,A,r va P. vdi a~ A, r< O<R sao eho vdi h.h. t E I, g(t,x) ~- A khi x ~R v~ g(t,x) ~ a khi x~ r. v hoae (A.3 )" 2ii "-,~n_c1' I ' ~,(v"(C\".,~ - -';'..1.-J, I c.:.w \, I V(;J,. d" ,,;U.t / '-', sac C.10 Y( t) ~ -1 1im.infx ~g(t,x) ~ \x 1->IX) lim sup x-lg(t,x) ~ Ixl~oo r (t) de~'leA h.h. tEL . I ~ 33i to~n (5.1)-(5.2) '101~ thoa (A.I),(A.2) hay (A.l),(A.3) d~ dJde kh;o s~t b;i Lazeri23J, Chang~J,~Etrtel1iBlJ , Mawhini32J .. . - Heissig [39}, Gu,;:t3.[243, ,-':aNr.in-'ilardl35,363 va Gupta-H:J.\vhin[27}, I I-" ~I ,I [ I [Jtrong do cae ket lUEt tot nhat d~t du~c trong 27J. Trang 27 , Gupta -'\,~".'".;-("n'~::-n-;.;r.h~1 d;r. hI ~'~;"+-; h."' ( C I) - ( c,? ) ;)~c-."r~va --::t,,:1~n ~.lU._", ."," "ae -;U- J tou v?-- c..J..J' /8- vu~ o. _..oa ~A.l),p..2) Doae CL1)-(A.3),vdi r =1 +[1+P ~i' E 11(1), r <: Loon). 01001... ,0-0 ' .,r,:;.:) , va [' E:L1( 1) sac cho0 j1 (t) ~ 1 . o,~, 3 ' ~ ~ ~ - , . voi h. to tEl, voi bat dang thuc ng~t tren m?t t~p con cua I co do do QUang va \i'iL0-a + (1i2/3)ii'iL1 I (' t r onz doc.; (r ) > 0 sa0 cho ~ 0 ,2ir . 2 C2Ti)-.L 1 (x'(t) - 0 I 1, v8i I::>i x E E~( I) vci <6Ci' )0 \ " ' " I' Dacbi8~, neu f 1a hang,va r = r = 0 thi ~et qua tren un~ vdi. . 0 0-0 ~ d 'A' ..- c o 1 ) C \ 2) d '" ~' t .l~ ..",', °n l 3/-leu ~ler, :;..~ . .'1.. u.oe cal len ...«ann II _1 < II L 2 roCt)x(t) )dt ;;. 2)\ fxCtJdt =O. 0 2 f ( I') IxIHl ." ' ; ,A. .-'..A A ~'. ,~ ]~o~g ecuo~g nay, chung tOl. eho dleu ~l~n tren so n~ng tleu hao f sac cho (5.1)-(5.2) co nghi~m val g tt;a CA.l),(A.2) hay ( ) ) ,', " .' v- n _l C ) . o~, , , , -, \. A.1 ,(A.3 VOl. bat KY 0 ,I E- L I. Cae Ket qua cua chung tOl lllv , ; -'. ?,' 0;)' ' t 'H'" '0>;1.'" I",'r?::g cae ~e:; qua 'Cuang u.ng cua \.Jup a va ,'lawrun troll§; tru.;,;Lg n9P .!.I " ,'.-. .,' I.-'? .,'" ~ c > 0 va c du ldn. Mot pnan cae Ket qua trans c~uong nay Q~?C. C 0, ~~ .~::; ..» - ~- \1;;>'.'0 ~V ""V.'6 l J. TO' , _° C' ~ .-:'; \ ~" ., I'" ~ Vlnn .1.; ~.~. ~la su CA.1),(h.2) ~uoc thoa. 01.a stl (.. )\- " -'.;) ,!::>l.-;,;'l~UC'!~' !f!?c>O (ii) 11'/_1< (2ir)-1/2c1/2. 'J 'TI""" "e' )C "~ ' .A af, -::. 1() ',.1::1. D3.l. ~O3.r,).1 - .J.c:.) cong;h1.emv l mOl e EOL I ~ 2i\ 3. ~. .e::J\ ;' "' (t 'A~ <j~ )...,,-- 2 TI A () Chu'n::: :::i,.h. , ' Gia su 0 <a< L Xet phuong trinh X"(tJ + ax(t) =e(t) - fCx(t)x'(t) - g(t,x(t»)+ax(tJ == ?';x(t) tUOEe: 11ong vdi (5.1). ,f,:-', 1, ) -'. ~'-.,.- -. ..2,1 ( ) ? VOl ::::)1.u E I, "I , 1:.on tal duy nhat mot ngnle:n x =J:\.uG::it I cua , phuong trinh xl! + ax = u, x(O) - x(2IT) =x'(O) - x'(2IT) =0 ':"'.,~', .. 1 () 1 () ' " ..~"'~' ~De tnay ant x,a Kli: C I.~ C I la compact va cac c.~embat a;;mg ") ,. ~.., ~ I ,.. I eua f~ la cae nghi~meua (5.1)-(5.2). ~e ap d~ng d~nh ly diem bat d?ng Leray-~ctauder, ta tim mot bing so' K) 0 sac eho ix 1,,1v < K d" , " ' I .,"..., , ~:l ~ ,V l mo~ ngnl~:n co the co eua ho pn~ungtr~nn x = ;\'Kilx, \ E:(O,l) (5.3) ., I ~ay gia su x ls. mot nghi~m eJa (5.3) vdi\E:(O,l)., ,-n,.ln~ x"(t) + >.f(x(t»x'(t)+(l-}..)ax(t) +/\g(t,x(t» =,\e(t) xCc) - x (27i) =x' (0) - x' (2IT) =0 (5.4) (5.5) ",'... "':T1 (5 !' ) "~~"' I 'r-~c," P.l"'_. .' ". e.l e..o 2JT ZIT (1->.)& J x(t)dt + \' J (g(t,x(t))- e(t»dt =0 0 0 (5.6 ) I . Neu x(t)? 2 \j t 6 I t:J.i ?IT J glt,x(t)dt ~. ZITA 0 ~ ZTr ~e(t)dt 0 , '. ~ '" ( ~ .- )mail tnuan VOl ).0. l'LIong "<,;,, '} I. ~, ~ su ;,,:(t) ~ r' V LEI c.dn tdi mo;;, n:au thu'3.n. V~yto~tai c: E I sac ehe IxC-d I ~ max(R,-r) (5.7) I ~han (5.4) vOl x'Ct) va tieD phan cho, do (i) c:: I '., I C A I -2 .u :::; 2Ti \Cie(t)! +!g(t,x(t)!)jx'(t)idt 0 (5.8) Do x' lien tue tuy~t dei, Ix'i eung lien t~e tuyet dei. vay ix' I kh~ vi h§.u he't va ix' I' = x" sgnx' . Nhan (5.4) vdi sgnx'(t) va tich phan , ta duoc 88 21' 2IT" 2« >-flfCx(tJ)X'CtJ!dt~)' f(ie(t)i+!g(t,x(t»[)dt+ (l-)..)aJlxtt)!dt 0 0 0 (5.9) rJ (5.4) v~ (5.9J, ta suy ra Ix"i_1 ~ 2 (lel_1 + ig(t,x)'_l) + 2(1-A)aixl_1 ~ L ~ L 2IT Do x'(O) =x'(2IT) =0 va ~ x'(s)ds = 0, ta co 0 (5.10) Ix'i C ~ (1/2Jix"l 1 L~ tJ do suy ra , do (5.10) \x'i <C"' ig(t,x)!Tl +~ . I 1alx L- + I e ILl (5.1l) '"'~. ~I. ) I ) A + a .r '~ ~at g\t,x =g\t,x - 2 '. h~ .~ ~ - '" g(t,xJ? - 2 ~~ 0 k~i x ~ R - a - ~ . g( t ,xJ ~ 2 -- ~ 0 khi x ~ r , va _1 -' <Em sup x ~g(t,x) jl (t) lxl-7OV (5.12) delI theo t £:1. Dat r(tJ =\ pet) +t trong db 0 < E. < C2ITJ-1[(2ITJ-1/2c1/2_Jrll].- L , .- 1/ " 1 /211" !I"'I 1 < (2 - ) - c. .'.ehl 1- He. L Do (5.12), t5~ tai 3 ) max(R,-r) sao eho vdi Ixl> B, -1 ~ Ix -g(t,x») ~ r (t) I vdi 11.r" t t: I. V~y Ig(t,x)1 ~ IrCt)]ixl+\p(t)\ , ' v di motpto L..L(1) Td (5.11) va (5.13), ta suy ra h.h. tEI,VxGIR (5.13) 39 \x'ic ~ (\rILl + 2TIa) IxlC + 1~ILl + Ie ILl (5.1L,) , I , Do (5.7) va Gong thuG trung tint Ix I,., ~ max(R,-r) + ix'l.lv L (5.15) fJ (5.13)-(5.15) SHY ra ZiT oS(iett>l +jg(t,x(t»)1 )/x'(t)\dt'::; \x'IC(irILllxIC +j~ILl +leIL2.) ~[(lrILl + 2Tia)lxlC +1~ILl + leiLl] .[ir\llxlc +1~iLl + ielLl J ,-J 2 2 ( CI I'lL 1 + 211a) I x Ie + c ~ (l rJ..l + 2iTa)2 Ix' I 21 + c21x' j 1+ c31 L L L 2 < c' Ix'! 2-..; . L + c4 (5.1'~ , voi 0 <c' < c ne~ a ducc chon kh~ nhb sao cho (\Ptl+ 2ITa)2< c/2TI. Td (5.8) v~ (5.16) SHY ra I x' I 2 « c5L tu'db suy ra, do (5.14) va (5.15), Ix! 1 < eh C; Dinh l~ 5.1 du~e ehJ~g mint. i ') ~ 7 ") 7 Vinh ly 5.2. Gia su (A.l),tA.3) duoe thoa. Gia ill (i) f: IR.~IR lien tUG va If I~,c >o. (ii) \pl.l<: 2: l.J troni; db net) ==max(IY(t)], [P(t)!) va S'> 0 saa-~-- . eho 3f (1 + 16&2) < 2ITa2c 1 21\, a ==min(411 ~o( t) dt ,1) 0 Th:L bai to~n (5.1)-(5.2) cb nghiem vai e6i eE Ll(I). 0n .Iv , , 1 ZIT ChllL;!' minh. Gis. s110 <.b ..(min(4r. ~o(t)dt,l) sac eha 3 ,..2 (1 -,.. ,,2) <. ' ~-' 2 0 C +.L::>(; ':::1\:) e , ~- J -" ~ - A " I fa chi e~n chung min~ s~ ton mi eua mot eh~n tien nghiem eha cae '",A 1.,-' 17, ,',.J b ,-ng~lem co tne co eua no oal taan len x!l(t) +>--f(x(t»)x'Ct) +.\g(t,xCt» + (l->-)bx(t) =Ae(t) (5.17) (5.18):<:(0) - xl2TI) =x'(O) - x'(2IT) = O,AE(O,l), tEl . ,? ", - - ,,' ., -" .3 -:, "r,ay gB su x la ::ot ngra~::1eua ().1()-(5.1 ) val .\Eev,l). 21\ Chon 0 .( ~ <.(1/4Ti) ~Y(t)dt sac eho pet) =p(t)+ E thoa lP\L1 <:S 0 , , .' Do gi3.'Chiet, to:-, tal r > 0 sao eto -1 . yet) - E ~ x g(t,x)::; r(t) +E (5.13') vch , ' I\x I ~':' va h.h. 'C '" I, t 11do suy ra -( r ( t) - Y( t ) t..) \ (..) i 2.( (7( ~ ) - r( t )+3'(t ) I (t )' 12<:(pct )- ¥(t) t) 2I+-)2 + X,v ...,XC)."'x 2 x - 2 + x \" I vc3i Ix!~. r va n.r:. tE 1. Vav , " \g(t,x)i ~ ~(t)ixi-;. 'let) ai, A, T1 'T),. "o~.,. n Coi. ,Y ~ -.,:, v '1 ~ ~ \ - . (5.19) y',,~'- ( - '~ ) .:?~ ,.1"- ) ,.::: t .'~'" h;:; ..,.la., )...1.( \V.l. A \..., ya. lVH p.:>n eno c Ix 1 \~2 ~ c:'i, Sl\eldi + \ g(t,x»i )\x'(t)\dt 0 (5.20) 7\C.,) ~-" ~,,'~- .-" ,?~ "" 1- .L. " ~,dle.. v.ong v!lU.,o ,"",-f... c.~:>dln.i 1y ).1, "a suy ra. \x'\(' ~ \,g(t,x)\71 -;. iel,l + blx\ L 1 v ~ L (5.21) , . t~ db sUi ra , rio l5.19), Ix'l(' !; (lpl_1 -;.2i\bJ\xlr + lelrl +v .LJ OJ.u ! a 1_1. lo (5.22 ) , ) ' ,f. Ksy nh~n l5.17 vai xlt) v~ tieh phsn eha C:i\ ?Ti;:> ZIT 2IT \ ~ bl t ,x) x( t) Qt + (1-),) b ~ xl t fdt =~' \x' ( t)\2dt +\O~ ex( tJ dt0 0 0 (5.23) 91 Do (5.18'), to'n t?i PIS L\I) sac eho 2 xg(t,x) > (Y(t)- E.) Ixl - ~(t) I vdi h.h. t G I va moi XGIR. Vf?y 2Ti 2IT 2 211 5 xg(t,x)dt ~ r ('((t)- t )\xl dt - S\~(t)1 dta a a , -, Tieh phan tung phan eho ~ ~ 2 ) (o(t)- E. )x(t)2dt =( f (Y(t)- c. )dt)x(a) - a a 2IT t - 2 r ( f ('((s)- E. )ds)x(t)x'(t)dt a a (5.24) Tli' (5.23) va (5.24) suy ra 2~ 2 2~ 2 2IT (>-/2) ( r y( t ) dt ) !x (0) I + (1 - A) b S x (t) dt ~ S x ' ( t )2dt + 0 a a 2IT 2IT 2IT + 2 (\01.1 + 2ITE) ~!x( t )x, ( t) j dt + i Ie( t) x ( t)j dt + SI~(t) j dt (5.25)Lao 0 Do e3ng th~c trung hint, 2 2 2Ti x (t) ~ x (0) + 2 Slxx'ldt 0 \ J 2112 2iT xL ( t) ~ (1/2IT) r x (s) dt + 2 j' I xxI Ids 0 0 (5.26) (5.27) , rli (5.25)-(5.27) suy ra 2IT ) 2IT (>-/2)( ~O(t)dt)x-(t) + 2TIb(l-)..)x2Ct) ~ (>-/2)( SY(t)dt)x2(O) + 0 0 2~2 ?IT. 2K + (l-\)b}X (t)dt + (>-1¥Ct)dt + 4Iib(l-}..» S lxx'idt a a 0 2IT J 2iT ZIT :£ r x' (t)Ldt + [z(iY1.1 + 2iT£) +\ i nt)dt + 4T1b(l->..)j {Ixx'i dt +0 La 0 '.,Ie. 2IT fl~(t)1 dt 0 2IT Do 2ITb 6 (1/2) S Y(t)dt, 0 2i1 + )Iexldt+ 0 V't6I (5.203) (5.28) k~o thee 21f 2 2iT ZIT 2IT 2ITbx2(t) 6 ~ x'(t) dt + 4\p1L1 J Ixx'idt + flexldt + {IP(t)! dt 0 0 0 0 l5.29), vdi rnoi t E. I. . ~ I Do bat dang thUG Cauchy, 2~ 2IT 2IT 4\p!,.lflxx'ldt ~ (b/4) flx 12dt +(16\p\21 /b) ~'lx'12dt ~ 0 0 L 0 , va ZiT f \ex \dt ~. 0 2 2 (ITb/2 ) I x I" + 0/211 b) lei 1 v L , I tli do suy ra, do (5.29), 1Tblx\; ~ (1 + 16ipl~1/b)lx'I~2 ~ .L 2 + (l/ZTib) I ell L +I~I 1L \j t E I hay Z IxlC ~ (l/TIb) (1 + 16Ip12l/b) Ix' I 22L L .2 2. 2 + (1/2 I~b ) I e I - 1 + L + (l/ITb) i~I. 1 ~ (5.30) Do (5.19),(5.20), (5.22) va (5.3°), . I zc Ix'Z ~ ~ 2rr f (lei + Ig(t,x)\ )Ix'i dt ~ (ie1L1 + Ig(t,x)!LUlx'lC 0 ~ ( \plc1Ixlf' + iel.1 + Iqlr1)« Iplc1 + 2/Tb)ixI C +lel L1 + IqI L1)~ v L u ~ " ') { (3/2) \151,\ ixl~ L~ + c1 IxlC + Cz ~ C3/Z)lpl~1«iTb)-1(l+16b-llpl~U.L ') . Ix I! ,-;:J L- + c3lx'l r2 + c4l.J ~' I \ N I (' t u do suy ra, do p - 1 < 0 ,L ,;);; Ix'i 2 < c5L I ~ " - " "" trong do c5 khong tuy thuoc vaG x va ~ . V~YI do (5.22) va (5.30), Ix I 1 <: c,. C~ 0 trong do c,. khong tuy thuQC vao x va ). . , 0 . i I Dinh 1y 5.2 d~oc chung mint. ._.