CHTJOliG 5
D,FT!~'l':; "l ,;?Tr"'i !,T"??,)D -::>D"..-~~"- -,--,-'1. '"p',",-.,,"
.1". '..~ -' " ' J. ' "~"Xe t ffi ~ t oan Olen t uan hoan eno pr:' ong "r~nn L~enard
x"(t) + f(x(t»)x'(t) + g(t,x(t» =Bet), tEl (5.1)
(5.2)x (c) - x (2Ti) =x' (0) - x' (2TI) =0
tror:g db I =L 0,21\J va g: IxiR~IR th~a
I
(A.1) ;;:C.,x) do Gueetren I vch r:1oi XE-IR, ;:Ct,.) lien tue trenl?~ , . ~ ,
0'." - T . ~'. . >0 -~' t ' r TI (l) ,v l n.r.. tE-, va Vvl m.Ol r , ton Et~ ELI sac enor
Idt ,x)l ~ j1
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(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.
._.