ndu.edu.azndu.edu.az/public_html/files/uploader/elmieserler/deqiq 89.pdfndu.edu.az
TRANSCRIPT
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NAXIVAN DVLT UNVERSTET. ELM SRLR, 2017, 8 (89)
NAKHCHIVAN STATE UNIVERSITY. SCIENTIFIC WORKS, 2017, 8 (89)
. , 2017, 8 (89)
RYAZYYAT
MHMMD HACIYEV [email protected]
Naxvan Dvlt Universiteti UOT: 5801.01 ; 517.2; 517.3; 514.02
FUNKSYANIN TRMSNN TP MSLLRN HLLN TTBQ
Aar sz: riyaziyyat, metodika, tip msl, trm, msl hlli.
Key words: mathematics, methods, typical solve, derivative, typical task.
: , , , , .
Mktb riyaziyyat kursunun v elc d ali mktb riyaziyyatnn sas msllrindn biri
trm anlaynn msl hllin ttbiqi mslsidir.
Qeyd edk ki, bu msl eyni zamanda metodiki icra baxmndan mktb mllimlrini d
dndrn msllrdndir [5, s.159]. Mqalnin ilnilmsind sas mqsd qarya qoyulan
mqsdl bal olub, bir ne misallar nmunsind trmnin msl hllin ttbiqi il bal
nmunlr baxmaqdr ki, bu kimi nmunlr mktb riyaziyyat kursunda kifayt qdr zn yer
tapmayb. Qarya qoyulan mqsd mvafiq olaraq misal nmunsin baxaq:
1) yriy toxunann tnliyi il laqli olan msllr.
2) Funksiyann n byk v n kiik qiymti il bal olan msllr.
3) Sahlrin taplmas il bal olan msllr.
Yuxarda qeyd olunan tiplrl bal msllrin hlli metodlarn rh etmk mqsdi il hr
bir tipl bal olan msllr gtrrk, onlarn hlli yollar il trafl olaraq tan olaq.
yriy toxunann tnliyi il laqli olan msllr.
Aydndr ki, absisi 0x olan nqtd diferensiallanan )(xf funksiyasnn qrafikin toxunann
tnliyi ))(()( 0 oo xxxfxfy kimidir. Yeri glmikn qeyd edk ki, mktb riyaziyyat
kursunda sasn el tip msllr baxlr ki, hmin msllrd 0x nqtsi v bu nqtd f
funksiyas verilir ki, bel msllrin hlli prosesind agirdlr mexaniki kild )( oxf v )( oxf
ddlrini taprlar [1, sh.77-79].
Baxlan misal nmunsind is 0x deyil, )(xf funksiyas verilir (ola bilr ki, )(xf
trmsi verilsin). V baxlan misal nmunsind )( oxf v yaxud )( oxf ifadlrinin,
qiymtlrinin verilmsindn istifad edrk vvlc 0x -i taprq, sonra is yriy toxunann
tnliyini taprq. Bel msllrin hlli agirdlr n tinliklr trtmir. Bel ki, bu msllrin
icras prosesind onlar trmnin ttbiqi il bal biliklr yiylnirlr, onlarda bacarq v vrdilr
formalar [4, s.188-189].
Msl 1. sbat edin ki, 354 xy dz xtti 85323 xxxy yrisin toxunur v
toxunma nqtsinin koordinatlarn tapn.
Hlli: Tutaq ki, 354 xy dz xtti 85323 xxxy yrisin toxunur. Onda
toxunma nqtsinin absisi 4)( xf tnliyini dyr ki, burada mslnin rtin gr
853)( 23 xxxxf -dir.
mailto:[email protected] -
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Onda ,563)( 2 xxxf hminin 4)( xf olduundan alrq ki, 45632 xx
olar v buradan da ,0963 2 xx yaxud da 0322 xx olduunu alrq.
0322 xx tnliyini hll edrk alrq ki, tnliyin kklri 1;3 xx olar. Onda tbiidir ki,
7)1(;23)3( ff olar v alrq ki, (-3; 23) nqtsind 85323 xxxy yrisin toxunann
tnliyi ,23)3(4 xy yaxud ,354 xy kimi olar. Bellikl, gstrdik ki, 354 xy
dz xtti verilmi yriy toxunur. Eyni zamanda almdq ki, 1x nqtsi d 0322 xx
tnliyinin kkdr, baqa szl, alrq ki, (1 ; 7) nqtsind toxunan 7)1(4 xy tnliyi
kimi frqli olan bir tnlikl d verilir. Yni 7)1(4 xy yaxud 34 xy tnliyi msld
veriln yriy toxunann tnliyidir. Buradan alnan qnat onu gstrir ki, toxunma nqtsi (-3;
23) nqtsidir.
Funksiyann n byk v n kiik qiymti il bal olan msllr.
Mktb riyaziyyat kursunda sasn parada tyin olunmu, verilmi funksiyalara baxlr. V
baxlan funksiyann parann uc nqtlrind, parann verilmi myyn nqtsind funksiyann
qiymtinin taplmas, bu hallara uyun olaraq n byk, n kiik qiymtinin taplmas, kritik
nqtlrin taplmas kimi msllr baxlr [3, sh.275-277]. Lakin bel ilmlrin myyn
tinliklr yaratdn, lverisiz olduunu qeyd etmliyik. Onu da qeyd edk ki, gr funksiyann
verilm oblast, para olmazsa (btn ddi ox v s. kimi), onda yuxarda qeyd etdiyimiz kimi
yanama daha da mmknszl bilr. Mlumdur ki, bel yanamalar zaman funksiyann n
byk v n kiik qiymtlrinin taplmas funksiyann monoton olduu paralarn taplmasndan
sonra aradrlr.
Bu kimi msllrin hll zaman trmdn istifad hll prosesini asanladrmaqla yana,
msl hllind optimal sullarn ttbiqinin smrliliyini d ortaya xarr. Qeyd edk ki, bel hll
prosesi n fnnin znn tdrisi il bal, n d agirdlr n vaxtla, qavrama il bal lav
tinliklr yaratmr. Bu kimi yanamalar tmayll sinif v mktblrd, fakltativ mllrd
yerin yetirmk d olar.
Msl 2. 13166)( 234 xxxxf funksiyasnn 3;0 parasndak n byk qiymtini tapn.
Hlli: vvlc )(xf funksiyasnn )(xf trmsini tapaq.
);18(664824)( 323 xxxxxxxf
0018400)( 2 xxxyaxudxxf v buradan alrq ki, 2
52 x .
Alrq ki, 3;0 parasnn daxilind veriln funksiya il bal bir kritik nqt var:
2
52 x (nki digr alnan kk 3;0 parasna aid deyil). Yuxarda da qeyd etdiyimiz kimi
funksiyann n byk qiymtini tapmaq n ;3;0 xx2
52 x qiymtlrind
funksiyann mvafiq qiymtlrini hesablamaq lazmdr. Yni; )3();0( ff v )2
52(
f
qiymtlrini hesablamaq lazmdr. Sonra is alnan bu qiymtlri mqayis edrk, n byk olan
qiymti gtrrk ki, bu da funksiyann 3;0 parasndak n byk qiymtidir. Bu mqsdl aadak cdvli trtib edirik:
x
2
52;0
2
52 3;2
52
)(xf 0 +
-
5
)(xf
Cdvldn d grndy kimi, baxdmz 13166)( 234 xxxxf funksiya n byk
qiymtini 3;0 parasnn uc nqtlrind alr. Bel ki, 26)3(;1)0( ff olur. Odur ki, funksiyann 3;0 parasnda ald n byk qiymtinin 26 olduunu alrq.
Qeyd edk ki, 2
52 x nqtsind funksiya n kiik qiymt aldndan funksiyann
bu nqtd ald )2
52(
f qiymtini hesablamaq lazm glmir.
Sahlrin taplmas il bal olan msllr.
Msl 3. 123 xxy yrisi v absisi - 1 olan nqtd bu yriy toxunanla mhdud olan
fiqurun sahsini tapn.
Hlli: lkin olaraq yriy toxunann tnliyini yazaq. gr 12)( 3 xxxf is onda
,1)1(,2)1(,23)( 2 ffxxf onda alrq ki, yriy toxunann tnliyi 2)1(1 xy v yaxud
3 xy kimi olar.
12)( 3 xxxf yrisi il 3 xy toxunannn ortaq nqtlrini tapaq. Bu mqsdl
3123 xxx tnliyini hll edrk, alrq ki, .0233 xx Mslnin rtin gr 1x
toxunma nqtsinin absisi olmaqla bu tnliyin kklrindn biridir. Onda son tnliyin sol trfi
1x - blnr. Onda 0233 xx tnliyind myyn evirmlr aparaq:
.2102
010)2)(1(0)1(2)1(022023
2
2233
xdayaxxx
dayaxudxxxxxxxxxxxx
Y
3 xy
323 xxy
-1 0 2 X
Yuxardak 3123 xxx brabrliyini )3()12()( 3 xxxxg klind olan funksiya
kimi gstrk. Aydndr ki, bu funksiya ksilmyndir (ksilmyn funksiyalarn cmi
olduqlarndan) v onun 2;1 intervalnda kk yoxdur. Alrq ki, bu funksiya z iarsini saxlayr. )(xg funksiyasn 2;1 intervalnda aradrmaq n intervala aid olan nqtlrdn birind )(xg funksiyasnn qiymtini hesablayaq. Msln, .231)0(0 golduqdax Demli,
2;1 intervalna aid olan btn x-lr n 0)0( g . Btn bu aldmz nticlr sas verir ki, msl il bal olan qrafiki kk.
kilmi qrafik imkan verir ki, inteqrallama srhdlrini qrafik uyun olaraq gtrmkl,
mslni hll edk.
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Axtarlan sahni tapmaq n myyn inteqraldan [2,sh.193] istifad etmkl mslni hll
edk:
.75,6.4
27)2
2
3
4
1())12()3((
2
1
243
2
1
SYaxudxxxdxxxxS
DBYYAT
1. zizov O. Q. Ali riyaziyyatdan mhazirlr. Metodiki vsait. Bak. 2007. 245 s.
2. .. . .II.,
1966, 800 .
3. Quliyev . A. Riyaziyyatn tdrisind mumildirm. Bak, 2009, Azrbaycan Milli
Kitabxanas, www. anl.az/el/q/qe_rtu.pdf.
4.Tahirov B.., Namazov F.M., fndi S.N., Qasmov E.A., Abdullayeva Q.Z. Riyaziyyatn tdrisi
sullar. Bak, 2007, Mechmath. Bsu. Edu. Az/.../riyaziyyat v onun tdrisi metodikas kafedras.
5. Hacyev M.. Riyaziyyatn tdrisi metodikas (mumi metodika, bakalavr pillsi-riyaziyyat
v riyaziyyat-informatika ixtisaslar n). Drs vsaiti. Naxvan, 2017, 178 s.
.
,
,
,
. ,
.
ABSTRACT
M.Hajiyev
Applying the derivative function of the solution of a typical task
The method of solving some typical problems related to the equation of the tangent to the
graph of differentiable functions, finding the largest and smallest values of the function, calculating
the areas of plane figures and determining the number of roots of the equation is considered.
Problems of this type are completely absent in the school textbook, although they provide useful
skills in the field of application of the derivative of a differentiable function to the solution of
typical problems and contribute to a deeper and informal learning of the material under study.
NDU-nun Elmi urasnn 29 dekabr 2017-ci il tarixli qrar il
apa tvsiyy olunmudur (Protokol 04).
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NAXIVAN DVLT UNVERSTET. ELM SRLR, 2017, 8 (89)
NAKHCHIVAN STATE UNIVERSITY. SCIENTIFIC WORKS, 2017, 8 (89)
. , 2017, 8 (89)
ELAD AAYEV
SAHB LYEV
Naxvan Dvlt Universiteti
SFA LYEV Naxvan Universiteti
UOT: 517.9
BR KNC TRTB KVAZXTT ELLPTK TP TNLK N
QEYR MHDUD OBLASTDA DRXLE MSLS
Aar szlr : diferensial tnliklr, elliptik, xeyri xtti, klassik hll
Key words: differential equations, elliptic type, non linear, classic solution
: , , , , -
1.sas anlaylar.Triflr.
Kvazixtti elliptik tip tnliklr nzriyysinin bzi sas anlaylarn verk.
Rn il n ll hqiqi evklid fzan iar edk.Tutaq ki, nR , n 2 hr hans qeyri
mhdud oblastdr v tutaq ki, el R 0 v 0 < 0 < 1 rtini dyn el 0 ddlri mvcuddur ki, nRx olduqda
nR RBmes 00 \ brabrsizliyi dnir.
Burada xRB mrkzi x nqtsind, radiusu is R - brabr olan aq krdir.
Bu rtlri dyn oblastn R v 0 parametrli silindr tipli oblast adlandraq.
Tutaq ki, oblastnda
xcx
uxbxx
uuxALi
n
ji
i
ji
ij
,,,1,
2
klind kvazixtti elliptik operatoru tyin olunmudur.
Burada x = (x1,...,xn) , nxxx
uuuu ,...,,21
Ai.j =Aji L operatorun Aij (x,z,p)
msallar (x,z,p)- nin btn qiymtlri n R Rn oxluunda tyin olunmudur.
Tutaq ki, u R Rn oxluunun hr hans alt oxluudur.
gr pzxAij ,, matrisi hr bir (x,z,p) U qiymtlri n msbt tyin olunarsa,L operatoruna U oxluunda elliptik operator deyirlr.
Bu o demkdir ki, (x,z,p), pzx ,, uyun olaraq pzxAij ,, matrisinin minimal v maksimal mxsusi qiymtlridirs, onda aadak brabrsizlik dorudur:
Burada, upzxRnn ,,,0\,...,, 21
gr bundan baqa,
nisbti U oxluunda mhdud olursa, L operatorunda U oxluunda
mntzm elliptik operator deyilir.
gr L operatoru btn xRxRn oxluunda elliptikdirse (mntzm elliptikdirs), onda
deyirlr ki, L operatoru U oblastnda elliptikdir. (mntzm elliptikdir)
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Tutaq ki, u C/ ().L operatoruna u funksiyasna nzrn elliptik deyirlr. O zaman ki, Aij
(x, u, u) matrisi x qiymti n msbt tyin olunsun. silindir tipli oblastnda
Lu = (x,u)
Diferensial tnliyin 0u rtini dyn hllinin aradracaq. Bu tnliyin hlli dedikd ikinci trtibdn ksilmz diferensiallanan v oblastnn hr bir x
nqtsind baxlan tnliyi dyn u (x) funksiyas baa dcyik.
gr biz bel funksiya mlum olarsa, onda bu funksiyan L operatorunda yerin yazmaqla
aadak kild xtti operatoru alarq:
n
ji
n
ji ji
jiijxx
xaxqraduxuxAL1, 1,
2
,,,
Bu id biz hllin varl mslsin baxmayacaq.Frz edcyik ki, bz tnliyin hlli
verilir v bu hllin xasslrini tdqiq etmk lazmdr.Ona gr d, biz L kvazixtti operatoru
vzin
n
ji
jiij
ji
ij aaxx
xaL1,
2
,
xtti operatoruna baxacaq.
Tutaq ki, bu operator mntzm elliptikdir.
Baqa szl, xRn v 0 sabiti n
21.1,
2
jijin
ji
xa
brabrsizliyi btn oblastnda dorudur.
2. Mslnin qoyuluu v hlli
Rn il n ll evklid fzasn iar edk.Tutaq ki, nR hr hans qeyri mhdud
oblastdr v tutaq ki, el R 0 v 0 < 0 < 1 rtini dyn el 0 ddlri mvcuddur ki,
nnRn RBmesRx 0\ brabrsizliyi dnir.Burada n
RB mrkzi x nqtsind radiusu
is R - brabr olan aq krdir.
Bu rtlri dyn oblastn R 0 v 0 parametrli silindr tipli oblast adlandraq. Tutaq ki, oblastnda
n
ji
x
n
i
ixxiju uxuxcuxbuxaL iji1, 1
0, (1)
Tnliyinin msbt hlli tyin olunmudur.
Burada
nljiMxbMxaaaxcx
xbyx
xaL i
n
ji
ijjiij
i
n
i
iij ,,,,,1,
,
12
2
Mntzm elliptik operatordur. funksiyas is
sUuxsignusign
2,1min1,,, 1
(2)
rtini dyir.
c(x) zrin is c (x) 0 rtini qoyaq.
jiijji
n
i
ij
Gx GGxa
xa
e
1
1,
1
1
sup
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Mnasibti il tyin olunan e ddini L operatorunun elliptik sabiti adlandraq.
Tutaq ki, s msbt ddi s e 2 brabrsizliyini dyir.
Tnliyin hlli dedikd klassik hll baa dcyik.
oblastnda (1) tnliyin (2) rtini dyn hllini aradrmaq n maksimum
prinsipi v bym haqqnda lemma nn aadak formasn verk.
Maksimum prinsipi. Tutaq ki, u (x) (1) tnliyinin G aq oblastnda tyin olunmu
G oxluunda ksilmz olan msbt hllidir. (x,u) funksiyas is sgn =sgnu rtini
dyir.
Onda uuG
maxsup
brabrsizliyi dorudur.
sbat: ksini frz edk.Tutaq ki, Gxxuxu
G
00 ,max maksimum nqt olduundan
nlix
uuxa
n
jixx
i
xxji ji,,0,0)(
1,
0
, 0
mnasibti dorudur.
sgn(x,u)=sgnu v u(x) 0 rtlrindn (x,u)|x=x 0 alnr.
Digr trfdn, (1) tnliyind c(x) 0 rtini d nzr alsaq (1) brabrliyi dnmz.Alnm
zidiyyt ks frziymizin dzgn olmadn gstrir.
Bym haqqnda lemma: Tutaq ki,4
10,04 RBG R aq oxluqdur v
00 ,\ RR BGGBH qbul edk.Tutaq ki, u(x) (1) tnliyinin G oblastnda tyin
olunmu, G oxluunda ksilmz v ,
srhddind 0| u rtini dyn msbt hllidir.
funksiyas is (2) rtini dyir.Onda
0
)(max
61)(max
RBG
xu
G R
Hmesxu
brabrsizliyi dorudur.
Burada 0 s ddindn asl olan sabitdir. sbat: sbatn sas anlarn verk.
S
xx 0
1
funksiyasna baxaq. Burada x G, x
0 R
n is hr hans qeyd olunmu nqtdir.
Onda s e 2 olduqda kifayt qdr kiik r n r < r0 rtind Sxx 0
1
funksiyas
0\0 xBG xR oblastnda (1) tnliyi n subelliptik olar.Burada r0 s- dn v operatordan asldr. Bu halda zrin dn rti tapaq:
1
00
11,
SS
xxxxx
|x-x0| =r vzlmsinin aparaq.
s(1+) < s+2
s +s < s+2
s
-
10
s
2 v ya
)2
,1min(s
Bellikl, E.M.Landinsin bym haqqnda lemmasnn btn rtlri dnir.
Baqa szl bym haqqnda lemmann hkm dorudur.
Tutaq ki, nR (n 2) R v 0 parametrli silindr tipli oblast olub 0
RB krsinin
mrkzini z daxilin alr.
Aadak teorem dorudur.
Teorem: Tutaq ki, nR silindr tipli oblastdr v bu oblastda (1) tnliyinin
coxluunda ksilmyn bu oblastn srhddind sfra brabr qiymt alan msbt hlli tyin
olunmudur.
Tutaq ki, (x,u) funksiyas (2) rtini dyir. )()( max xurMrx
qbul edk.Onda s- dn
elliptik sabiti e- dn, fzann ls olan n dn asl olan el c 0 sabiti mvcuddur ki,
R
rC
erM )(
Brabrsizliyi dorudur.
sbat: Tutaq ki, koordinat balanc 0 , mrkzi koordinat balancnda, radiusu is R -
brabr olan
0
RR krsin baxaq v 0
RR oblastnda bym haqqnda lemman ttbiq edk.
Maksimum prinsipin gr u (x) funksiyas 0
RR oblastnin qapayannda znn n
byk qiymtini bu krnin sthind hr hans x1 nqtsind alr.
Onda alrq:
)0(
\1
)(max)(
0
01u
R
Bmes
B
xuxu
n
R
R
rt gr, oblast silindr tipli oblast olduundan
0
0
0 )\(
n
n
n
R
R
R
R
Bmes
mnasibti dnr.Ona gr d u (x1) / u (0) olar. Burada / = 1+ 0
ndi is 1x
RB krsin baxaq v 1x
RR oblast n bym haqqnda lemman ttbiq edk.
Onda alarq:
)0()()()(max
)( 2101 1uxu
B
xuxu
x
R
nixi ,1, ardclln aadak quraq.:
xi nqtsi u (x) funksiyasnn 11 xR oblastnin qapaqyannda ala bilcyi maksimum
nqtdir. Analoji qaydada bym haqqnda lemman ttbiq etsk, alarq:
)0()()(max
)(1
uB
xuxu
nx
R
n
-
11
rn =| xn | qbul edk.Akardr ki, rn =| xn | n R (Bu bucaq brabrsizliyindn
xr)Axrnc
brabrsizlikdn alrq ki R
xnn
Ona gr d )0()()( uxu Rrn
n
Demli, ),0()(ln
max uexu Rr
x
v ya ,)( conserMR
rc
Burada c s dn, n- dn v 0 dan asl olan msbt dddir.
Qeyd: hr hans rA < qiymtindn balayaraq M (r) = ola bilr.
DBYYAT
1. E.. . .1971, .287
2. .. . , 1, . . 1991, 4, 15-19.
3. . -. . 32, , 1988.
4. Aayev E.V. Bir qeyri xtti 2-citrtib elliptic tip tnlik cn Dirixle-mslsi. NDU, Elmi srlr. Fizika-Riyaziyyat v Texnika Elmlri seriyas, 2014, 3(59), s.11-14
ABSTRACT
Elshad Agayev, Sahib Aliyev, Safa Aliyev
On Dirikchlet problem for one nonlinear elliptical equation of the second
order in unbounded domain.
In this parer the behavior infinity of the of positive solition u (x) of nonlinear elliptic
equation of the second order in a narrow area theth parameter turninq into zero on the
boundary of the area is considered.
The incuasing speed of the solution is determined depending on the eqution and
parametrs of the area.Note that in Therom the ellipticity is used only to ensure to that
maximum principle holds.Howerer, as is well known, the maximum principle is true for a
whole series of equtions with nonegative characterictic form which are not elliptic.
, , Sefa Ae
u(x)
- a
R u 0, .
.
,
.
NDU-nun Elmi urasnn 29 dekabr 2017-ci il tarixli qrar il apa
tvsiyy olunmudur (Protokol 04).
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12
NAXIVAN DVLT UNVERSTET. ELM SRLR, 2017, 8 (89)
NAKHCHIVAN STATE UNIVERSITY. SCIENTIFIC WORKS, 2017, 8 (89)
. , 2017, 8 (89)
BLFZ MMMDOV
Naxvan Dvlt Universiteti
UOT 517.957
NC TRTB KSLN MSALLI BR SAD OPERATOR- DFERENSAL
TNLK N QOYULMU BALANIC- SRHD MSLSNN REQULYAR
HLL OLUNANLII HAQDA Aar szlr: normal operator, hilbert fzas, operator-diferensial tnlik, requlyar
hll, requlyar hll olunanlq
Key words: normal operator, Hilbert space, operator-differential equation, reqular solution,
reqular solvability
: , , -
, ,
Separabel H hilbert fzasnda
),,0(),()()()( 3
3
3
RttftuAtdt
tud (1)
0)0()0( " uu (2)
kimi bir balanc-srhd mslsin baxaq, burada )(),( tutf R -da sanki hr yerd tyin
olunmu, qiymtlri H hilbert fzasndan olan vektor-funksiyalardr, trmlr mumilmi
mnada baa dlr [1] v A operatoru il )(t msal zrin aadak rtlr qoyulur:
1) A tamam ksilmz 1A trsin malik v spektri
6
0,|arg:| S
bucaq sektorunda yerln normal operatordur;
2)
),1[,
),1,0(,)(
3
3
t
tt
v 0, olmaqla .
Hilbert fzasnda normal operatorlarn spektral nzriyysindn mlumdur ki, 1) rtini
dyn A operatorunu UCA klind gstrmk olar, bel ki, C z-zn qoma msbt-
myyn, U is unitar operatordur.
)0( H il A operatorunun dourduu hilbert fzalarnn kalasn iar edk, yni
)()( CDADH v H -da skalyar hasil ),(),(),( yCxCyAxAyx
kimi tyin
olunub. Hesab edcyik ki, HH 0 v Hyxyx ),(),( 0 .
Aadak hilbert fzalarn tyin edk [1]:
21
0
2
);(2)(:);(
2
dttfffHRLHHRL
,
-
13
21
2
);(
3
32
);(
3
);(2
3
3
33
2
2
2
32
),;(,:);(
HRLHRLHRW dt
uduAuHRLuA
dt
uduHRW ,
0)0()0(),;(:)2;0;;( "3232 uuHRWuuHRW . Mlumdur ki, )2;0;;(32 HRW hilbert fzas );(
3
2 HRW hilbert fzasnn tam alt fzasdr [1].
Trif-1. gr );(32 HRWu vektor-funksiyas R -da sanki hr yerd (1) tnliyini v (2)
balanc-srhd rtlrini
0)(lim,0)(lim2
102
50
tututt
mnada dyirs, onda ona (1)-(2) balanc srhd mslsinin requlyar hlli deyilir.
Trif-2. gr istniln );(2 HRLf n (1)-(2) balanc-srhd mslsinin requlyar
hlli varsa v bu hll
);();( 2
32 HRLHRW
fconstu
brabrsizliyini dyirs, onda (1)-(2) balanc-srhd mslsi requlyar hll olunan msl
adlanr.
Tqdim olunan mqald (1)-(2) balanc-srhd mslsinin requlyar hll olunanl isbat
edilckdir. Qeyd edk ki, 0)0()0( ' uu v 0)0()0( ''' uu balanc-srhd rtlri daxilind
(1) tnliyinin requlyar hll olunanl mllif trfindn ([4,5]) isbat edilmidir.
Teorem. gr A operatoru 1) rtini v )(t ddi funksiyas is 2) rtini dyirs, onda
(1)-(2) balanc-srhd mslsi requlyar hll olunandr.
sbat. Funksiyann Furye evirmsini ttbiq etsk, asnlqla yoxlamaq olar ki, istniln
);()( 2 HRLtf n
ddsesfAEitu sti
0
)(1333
1 )(2
1)(
v
ddsesfAEitu sti
0
)(1333
2 )()(
funksiyalar R -da sanki hr yerd uyun olaraq
)(333
3
tfuAdt
ud v )(33
3
3
tfuAdt
ud
tnliklrini dyir. Gstrk ki, );()(),( 3221 HRWtutu .
Akardr ki, )(),( 21 tutu vektor funksiyalarnn Furye evirmlri uyun olaraq
)()( ^1333^1 fAEiu
(3)
v
)()( ^1333^2 fAEiu
(4)
klinddir, burada )(^ f )(tf vektor-funksiyasnn Furye evirmsidir.
Planerel teoremin gr
2
);(
^
1
32
);(
^
1
32
);(1
3
2
);(
3
1
32
);(122;2
2
32
)()(HRLHRLHRL
HRL
HRWuAuuA
dt
udu
. (5)
(5) brabrliyi gstrir ki, );()( 321 HRWtu olduunu gstrmk n kifaytdir ki,
-
14
);()( 2^
1
3 HRLu v );()( 2^
1
3 HRLuA olduunu gstrk.
A operatorunun spektral ayrlna gr, istniln R n aadak qiymtlndirm
dorudur: )||( ie
3cos
13sinsup
3sin2sup3sin3cossup
supsup
32
1266616663
||0
2
13336663
||0
13333
||0
133333
||0
13333
)(
13333
ii
eiiAEiA i
A
(3)- v sonuncu brabrsizliyi nzr alsaq
);(3);(
^13333
);(
^13333
);(
^
1
3
22
22
)(3cos
1)(.
)()(
HRLHRL
HRLHRL
tffAEiA
fAEiAuA
alarq ki, bu da );()( 2^
1
3 HRLuA olduunu gstrir.
ndi gstrk ki, );()( 2^
1
3 HRLu .
);(
13333
);(
^13333
);(
^13333
);(
^
1
3
22
22
)(.sup)(.sup
)()(
HRLHRL
HRLHRL
tfAEifAEi
fAEiu
(6)
brabrsizliyind 13333 AEi normasn qiymtlndirk. Onda, A operatorunun
spektral ayrlndan, istniln R n alarq:
21
2
1
2
16663
0
2
16633363
0
2
16633363
||0
2
123332663
||0
1333333
)(
13333
)(
13333
)(
13333
3sin13sin1sup
3sin2sup3sin2sup
3sin3cossup3cos3sinsup
)3sin3(cossupsup
i
iiiAEi
A
AA
Bu sonuncu brabrsizliyi (6)-da nzr alsaq );()( 2^
1
3 HRLu alarq. Onda (5)-
gr );()( 321 HRWtu olar.
Anoloji qayda il isbat edilir ki, );()( 322 HRWtu .
)(1 tu vektor-funksiyasnn ]1,0( yarmintervalna, )(2 tu vektor-funksiyasnn is ),1[
yarmintervalna sxlmasn uyun olaraq )(),( 21 tt il iar etsk, akardr ki,
)];1,0(()( 321 HWt v ));,1([)(3
22 HWt olar. Onda, izlr haqda teorem gr ]1[
2,0;2,1,)0(2
13
)(
jiHj
j
i olar.
-
15
),1(,)()(
],01(,)()()(
5
)1(
4
)1(
22
3
)1(
2111
21
321
teett
teeetttu
AtAt
AttAtA
vektor-funksiyasn quraq, burada )5,1(,1,2
3
2
1,
2
3
2
125321 jHii j v
tAtAtAeee 321 ,,
is uyun olaraq AAA 321 ,, operatorlarnn dourduu operator
yarmqruplardr. )5.1( jj vektorlar )2;0;;(3
2 HRWu rtindn tyin edilir. Bunun n
)2,0)(1()1(,0)0()0( )(2)(
1
"
11 jjj brabrliklrindn istifad edrk )5,1( jj
mchullarna nzrn aadak tnliklr sistemini alm oluruq:
)).1()1((
))1()1((
)1()1(
0
0
"
1
"
2
2
5
2
2
2
4
2
1
2
3
2
3
2
2
2
2
2
1
2
1
2
'
1
'
2
1
5241332211
1254321
3
2
3
2
2
2
2
2
1
2
1
2
321
21
21
21
3
3
Aee
Aee
ee
e
e
AA
AA
AA
A
A
(7)
,
00
00
)(
2
2
22
1
22
3
22
2
22
1
2
21321
2
3
22
2
22
1
2
21
21
21
3
3
EEee
EEEee
EEEee
eEE
eEE
A
AA
AA
AA
A
)1()1((
)1()1((
)1()1(
0
0
~,~
"
1
"
2
2
'
1
'
2
1
22
5
4
3
2
1
A
A
iar etsk (7) tnliklr sistemini
~~)( A (8)
matris tnliyi klind yazarq, burada 5~,~ H . Gstrsk ki, )(A operator-matrisi
trslnndir, onda alarq ki, (8)-in 5H hilbert fzasnda 0~ hlli var. Bunun n )(A
operator-matrisind A operatorunun yerin kompleks dyinini yazb )( matrisin baxaq.
Onda akardr ki, S olmaqla olsa
0)(
111
.11
)(
00
00
11100
000
00011
)(det
2
2
22
1
22
3
2
2132
2
22
1
2
2
2
22
1
22
3
2
213
2
2
22
1
2
O
O
-
16
olar, burada 0)(
O . Bu is o demkdir ki, el N nmrsi var ki, N olan istniln
S n .0)(det oc
ndi gstrk ki, N olan istniln S n 0)(det . Bunun n ksini frz
edk, yni frz edk ki, N olan el S var ki, 0)(det olur. Bu is o demkdir ki, el
sfrdan frqli 554321 ),,,,(
~ C vektoru var ki, ~)( , burada 5C sfr vektordur.
Onda akardr
0)0()0(
0)()()(
"
3
3
3
qq
tqtdt
tqd
(9)
balanc-srhd mslsinin )(32 RW fzasndan olan hlli var v bu hll
),1(,
],01(,)(
5
)1(
4
)1(
3
)1(
21
21
321
tee
teeetq
tt
ttt
klind axtarlmaldr. Gstrk ki, (9) balanc-srhd mslsinin R -da sanki hr yerd yalnz
0)( tq hlli var, yni )5,1(0 ii . Dorudanda, (9)-dan
)(
'3
)(
''''
22))(),()(())(),((
RLRL tqtqttqtq
alarq ki, bu brabrliyi
0 0
'3''' )()()()()( dttqtqtdttqtq
sklind yazb 2) rtini v 0)0()0( '' qq srhd rtini nzr alb hiss-hiss inteqrallama
dsturunu ttbiq etsk
0 1
'3
1
0
'332
'' )()()()(()( dttqtqdttqtqdttq
alarq. Sonuncu brabrlikdn alarq ki,
.)()(Re)()(Re)(1
Re0
1
0 1
'3'3''
3
dttqtqdttqtqdttq
(10)
0)0()0( '' qq rtlrini nzr alb hesablama aparsaq
1
0 1
2
'
2
'
2
)1()()(Re,
2
)1()()(Re
qdttqtq
qdttqtq
alarq. Onda (10) brabrliyind sonuncular nzr alsaq alarq:
0
2
332
''
3.
2
)1()()(
3cos qdttq
Axrnc brabrlik gstrir ki, olduqda R -da sanki hr yerd 0)('' tq , yni
battq )( olur. )()( 2 RLtq olduundan R -da sanki hr yerd 0 ba , yni 0)( tq olar.
ndi frz edk ki, . Onda (9)-dan alarq:
.)(),()(),()(
1)(
''3
)(
'''''
2
2
RL
RL
tqtqtqtqt
halnda olduu kimi 2) rtini v 0)0()0( '' qq srhd rtini nzr alb hiss-
hiss inteqrallama dsturunu ttbiq etsk
-
17
0
2'3
1
0 1
'''''
3
'''''
3)(Re)()(
1)()(
1Re dttqdttqtqdttqtq
alarq ki, burdan da
0
'32
''
33)(3cos)1(
11dttqq
olar. 011
33
v 03cos olduundan R -da sanki hr yerd 0)(
' tq , yni consttq )(
alarq. Yen )()( 2 RLtq olduundan R -da sanki hr yerd 0)( tq olar.
Btn bunlar nzr aldqda )5,1(0 ii alarq ki, bu da ~)( matris tnliyinin
~ hllinin olmasna ziddir. Bu ziddiyyt sbb ksini frz etmyimizdr. Demli, frziymiz
doru deyil, yni istniln S n .0)(det Bu is o demkdir ki, )(A operator-matrisi 5H hilbert fzasnda trslnndir. Onda (8)-dn birqiymtli olaraq ~)(~ 1 A alarq. .
),,,,(~ 54321 vektorunun koordinatlarn )(tu -nin ifadsind nzr alndqda (1)-(2)
balanc-srhd mslsinin hllini tapm olarq. )(A operator-matrisi trslnn olduundan
0)0()0(
0)(
''
3
3
3
uu
uAtdt
ud
bircins balanc-srhd mslsi yalnz 0u trivial hllin malik olar. Bu sbbdn
33
3
0 )( Atdt
dP operatoru )2;0;;(32 HRW tam hilbert fzasn );(2 HRL hilbert fzas zrin
izomorf inikas etdirir. Hminin istniln );(32 HRWu n
2
);(
2
);(
3
2
);(
3
32
);(
366
2
);(
3
3
2
);(
3
2
);(
3
3
);(
3
3
3
);(0
32
2
2
2
2
2
22
2
).,max(2
)(2)(
HRW
HRLHRL
HRLHRL
HRLHRLHRL
HRL
uconst
uAdt
udconstuA
dt
ud
uAtdt
uduAt
dt
uduP
olduundan );()2;0;;(: 23
20 HRLHRWP operatoru mhduddur. Onda trs operator haqda
Banax teoremin gr
)2;0;;();(: 3221
0 HRWHRLP
trs operatoru var v );(2 HRL zrind mhduddur, yni
);();(
1
0);( 23
2
32 HRLHRWHRW
fconstfPu
olur. Bu is, trif gr, (1)-(2) balanc- srhd mslsinin requlyar hll olunan olduunu
gstrir. Teorem isbat olundu.
DBYYAT
1. .-., .. . .
M, , 1971, 361 .
2. .. -
.
3. .. -
. // .,
-
18
.7(15), 1997, . 18-25.
4. Abulfaz M. Mamedov. On a boundary value problem for third order operator-differential
equations./ Riyaziyyat v Mexanika Institutunun SRLR, XXV cild, BAKU-2006, Elm
5. blfz Mmmdov. Ksiln msall trtibli sad operator tnlik n bir srhd
mslsinin hll olunmas haqda., NDU Elmi srlr, Fizika-Riyaziyyat v Texnika elmlri
seriyas 3 (28), Naxvan, NDU, Qeyrt-2009
-
-
- , -
),0( R .
ABSTRACT
On reqular solvability of unital-boundary problem for one ordinary operator-differential
equation of third order with discontinuouns coeficient
In this work the definition of reqular solution and reqular solvability of unital-boundary
problem for one ordinary operator-differential equation of third order with discontinuouns
coefficient in ),0( R has feen given and the reqular solvability of that problem has been
proved.
NDU-nun Elmi urasnn 29 dekabr 2017-ci il tarixli qrar il
apa tvsiyy olunmudur (Protokol 04).
-
19
NAXIVAN DVLT UNVERSTET. ELM SRLR, 2017, 8 (89)
NAKHCHIVAN STATE UNIVERSITY. SCIENTIFIC WORKS, 2017, 8 (89)
. , 2017, 8 (89)
YAQUB MMMDOV
Naxvan Mllimlr nstitutu
UOT: 517.51
HEYZENBERQ GRUPUNDA TYN OLUNMU MUMLM BESOV -
MORR FZASINDA SOBOLEV STEYN DAXLOLMALARI V TTBQLR
Key words: Heisenberg groups, SobolevStein embedding, generalized Morrey spaces,
apriori estimate, finite norm
: , -,
, ,
Heyzenberq qrupu kvant fizikasnda v riyaziyyatn mxtlif sahlrind, o cmldn Furye
analizind, kompleks dyinli funksiyalar nzriyysind, hnds v topologiyada ttbiq olunur.
Son vaxtlar Heyzenberq qrupunda funksional fzalar nzriyysi tdqiqatlarn byk diqqtini
clb etmidir. Bu diqqt dyin msall diferensial tnliklr n mslnin oxobrazlda
hllolunma problemlri il baldr. Bizim yanamada nH Heyzenberq qrupu balancda
12 nR evklid fzas il st-st dr v bellikl heyzenberq v uyun evkld anizotropiyasnn
bilavasit mqayissi imkan yaranr.
Trif1.
,2
1=),,,(=
,
10000
1000
0010
1
=
100
10
1
=][
00
2
1
21
xxxxxxx
x
x
xxx
x
x
x
n
n
n
tn
t
(1)
klind yuxar bucaq matrislr ym n trtibli Heyzenberq qrupu adlanr, qrup mliyyat
olaraq matrislrin vurulmas gtrlr.
Burada n(1) yazl n ll vahid matris, ),,(= 1 nxxx str vektorunu, xt - stun
vektorunu ( ),,(= 21 nn xxx strin transponir olunur), 0 n sfrdan ibart stri, 0 t
n sfrdan
ibart stunu gstrir. Qeyd edk ki, (1) matrisind ba diaqonaldan aadak elementlrin hams
sfra brabrdir.
Trifdn grnr ki, qrupun ls 12 n - brabrdir. oxobrazl kimi onu yalnz bir
xritni saxlayan atlasn kmyi il tsvir etmk olar. Bel olaraq 12 nR evklid fzas gtrlr,
koordinatlar nnn xxxxx 2110 ,,,,,, olur (burada ,2
1=0 xxx ini
n
ixxxx 1== ).
12 nR -in
nqtlri qsaca olaraq ),,(= 0 xxxx kimi iar edilir, burada ),,,(= 1 nxxx
).,,(= 21 nn xxx Gstriln nqtvi realizd matrislrin vurulmasnn qrup mliyyat
mliyyatna keir v ),,,(= 0 xxxx ),,(= 0 yyyy nqtlrin yxzzzz ),,(= 0
-
20
nqtsini qar qoyur: ).,),(2
1(= 00 yxyxyxyxyxz
yx nqtsini qsaca olaraq xy il iar edcyik.
Bellikl model olaraq (bunu nH il iar edirik) Heyzenberq qrupu qrup mliyyat
klind tyin olunan 12,, nRyx nqtlr ym kimi x edir. (1)-in ],,[=][ 0 xxxx
matrislri v ),,(= 0 xxxx nqtlri arasnda matrislrin vurulmas mliyyatn qrup
mliyyatna gtirn (v trsin) qarlql birqiymtli uyunluq yaradlr (qrup izomorfizmi). Bu
trifl veriln Heyzenberq qrupunu nHeis simvolu il iar edk. Daxil edilnlr sasn dey
bilrik ki, znn Riman v afin strukturuna gr xtti olan v mli il tmin olunan 12 nR fzas
nn HeisH il st-st dr.
Glckd biz zaman gldikc nHeis qrupunun veriln trifin mracit etsk bel, onun nH klind nqtvi realizasiyasndan istifad edirik. Bu realiz xsusi halda ona gr rahatdr ki,
nHeis qrupunun ][e vahid elementi ( 22 n trtibli vahid matris) nn
He
12
,00,0,=
nqtsi il
tsvir olunur, nHxxxx ),,(= 0 nqtsinin trs elementi is
nHxxxx ),,(= 0 olur.
Qeyd edk ki, nHx elementin trs olan element 1x (baqa szl xx =1 ) simvolu il
iar edilir. Msln, yx )( vzin yx1
yazacaq. nH Heyzenberq qrupunda G
n Li cbrin baxaq. Mlumdur ki, o nH -d sol invariant
vektorlar meydannn bazis sistemindn dour
.,,1,2= ,2
1=)(
,2
1= ,=
0
0
1
0
0
nix
xx
xX
xx
xxX
xX
i
in
in
n
i
i
K&
(2)
nH -d ZY , vektorlar meydan verilrs onlarn komutatorunu ZY , il iar edirik ZYYZZY =, .
(2) vektorlar meydan sistemi n
[ ,,1,2,=,,=], 0 njiXXX ijjni (3)
komutasiya mnasibti dorudur (ij - Kronekker simvoludur). Digr komutatorlarn hams
eynilikl sfra yaxnlar
.,1,2,=, 0,=],[=],[=],[=],[ 00 njiXXXXXXXX jnijninji
,jX ,jnX nj ,1,2,= meydann baza adlandrrq.
Hr bir 0>r n nH qrupunda r avtomorfizmi mvcuddur (nH -dn nH - qrup
mliyyatn saxlayan inikas) v
).,,(= 02 xrxrxrxr (4)
mnasibti il tyin olunur.
r avtomorfizmi dilatasiya adlanr v zn evklid fzas hndssinin analoqu kimi tqdim edir. (4) avtomorfizmi v (3) komutasiya mnasibti nH -d ml gln anizotroplarn
miqdarn gstrir. Bu anizotropiya hminin elementin H bircinslilik normas anlaynn daxil
edilmsini ortaya qoyur. )~,(= 0 xxx ),,,(=),(=
~221 nxxxxxx gtrsk
-
21
.=~,~=||1/2
22
1=
1/442
0
i
n
i
H xxxxx
H - bircinslilik onunla baldr ki, HHr
xrx = . Bellikl H
x funksiyas birinci trtibdn
bircinsdir. H - bircinslilik normasnn kmyil H - metrika anlay daxil edilir
.~~)(2
1==),(
1/4
42
00
1
yxyxyxyxxyyx
H (5)
nH - d evklid laplasiannn daha yaxn analoqu sublaplasian adlanan ikinci trtib
fXfL j
n
j
22
1=
0 = operatoru olur. Qeyd edk ki, 0X operatoru 0L - a akar daxil olmur. Onda
hlledici rolu njX j 21, operatorlar oynayr.
jX baza operatorlarnn v sublaplasiann varl nH =d Sobolev fzasnn, hminin
Riss v Bessel potensiallar fzalarnn analoqlarn tyin etmy imkan verir. ),(n
p HL
-
22
Teorem 1[1]. Tutaq ki,
-
23
Teorem3. Tutaq ki,
-
24
1. Guliyev V.S., Eroglu A., Mammadov Y.Y. Riesz potential in generalized Morrey spaces on the
Heisenberg group //Journal of Mathematical Sciences. Vol. 189, No. 3, March, 2013, 365-382
2. Jerison D.S. The Dirichlet problem for the Kohn Laplacian on the Heisenberg group // I., J.
Funct. Anal., 43 (1981), pp. 97142
3. Jerison D.S. The Dirichlet problem for the Kohn Laplacian on the Heisenberg group // II., J.
Funct. Anal., 43 (1981), pp. 224257.
.
- -
,
.
, , ,
.
.
-
- .
ABSTRACT
Y.Mammadov
SobolevStein embedding on a generalized BesovMorrey spaces on the
Heisenberg group and application
Actuality of theory Sobolev space in Heisenberg group based on exploration of solution
features of subelliptic differential equation, teaching of quasiconform analysis and varions mixed
issues. The Heisenberg group appears in quantum physics and many fields of mathematics,
including Fourier analysis, functions of several complex variables, geometry, and topology.
Recently theory functional space in the Heisenberg group had attracted the investigators
attention.
In the article analogues of Sobolev-Stein embedding theorem on a generalized Besov-Morrey
space in Heisenberg group had been received.
NDU-nun Elmi urasnn 29 dekabr 2017-ci il tarixli
qrar il apa tvsiyy olunmudur (Protokol 04).
-
25
NAXIVAN DVLT UNVERSTET. ELM SRLR, 2017, 8 (89)
NAKHCHIVAN STATE UNIVERSITY. SCIENTIFIC WORKS, 2017, 8 (89)
. , 2017, 8 (89)
DASHQIN SEYIDOV Nakhchivan State University
UOT:577
EIGENSUBSPACES OF WEIGHTED ENDOMORPHISMS OF
UNIFORM ALGEBRAS
Aar szlr: Rezonansl mxsusi dd, rezonansl monom, mxsusi altfza, endomorfizm,
mntzm cbr
Key words: resonancing eigenvalue, resonancing monom, eigensubspaces, endomorphism,
uniform algebra
: o , o ,
,,
In this work we investigate the relation between eigenvalues and eigensubspaces of weighted
endomorphisms induced by selmappings (of compact, where our uniform algebra defined on this
compact) with Denjoy-Wolff type fixed points on the uniform algebras with analytical structure
and eigenvalues and eigensubspaces of endomorphisms of algebras of convergent power series
of n variables = (1, , ) . In [2] Kamowitz was considered the weighted composition operator on the disc-algebra (i.e.
the algebra of continuous functions on the closed unit disc and analytic in the interior of its) and
was determined its spectrum in the case when is compact. In [3] we have more generally results inclusing multidimensional cases. In [3] was considered the weighted composition operators
actings on uniform spaces of analytic functions, which induced by the compressly mappings on the
bounded domains ( 1) and was determined its spectrum. Another words, if is a bounded domain and : is holomorphic mapping (where denote closure of ), then in [3] was considered the operators of the form : , , for every , where is fixed function and is Banach-A(D) module, which is uniform subspace of space of holomorphic functions on equipped with uniform topology. It is well known the mapping has a unique fixed point in . In [3] was shown the spectrum of operator is equal to semigroup induced by eigenvalues of linear part of at the fixed point. Since these operators are compacts, then every eigensubspace corresponding to nonzero eigenvalue has finite dimensions. But from method of [3]
we know about dimensions of eigensubspaces, if only case when differential of mapping at the fixed point has differently, nonzero and multiplicativly independent eigenvalues.
In this work avoid the results of [3], we will calculate directly the eigenvalues of the weighted
endomorphisms : () (), of uniform algebras (with analytical structure) (), defined on the compact , where the selfmap : has a Denjoy-Wolff type fixed point 0 (the operator maybe non-compact operator, no so as [4]) . We may assume, as so as [4], weighted function is identity function, and in finite dimensional cases, domain of , which induced the weighted endomorphism contains the origin of coordinate and it is fixed point for mapping . We will show that in this case between eigenvalues and eigensubspaces of operator and eigenvalues and eigensubspaces of endomorphism of algebra of formal (or convergent) series
there are bijective mapping.
Investigation of spectral properties (for example, spectrum, eigenvalues, eigensubspaces and
so) of endomorphisms, also weighted endomorphisms on different algebras (for example, on the
uniform algebras, especially on the function algebras with analytic structure, etc), usually leads to
investigation these problems on the algebras formally convergent power series (instance, in the case
-
26
algebra of analytic functions, we have the algebra of germs of functions at the fixed points, etc).
Moreover, in many cases studying some algebraic and spectral properties of endomorphisms, or
weighted endomorphisms induced by compression mappings (for example, see [3]), or more
generally, by the mappings which have fixed points, in some sense(for example, in the Denjoy-
Wolff sense fixed point, and so) on the function algebras with analytic structure, again leads to
studying endomorphisms of above mentioned algebras. Especially, on the uniform algebras
spectrum of the compact, or quazi-compact weighted endomorphisms described by the
eigennumbers of linear part of endomorphism at the origin, which modules less than 1 (see [3]). So,
in this work we will assume that modules of eigennumbers of the linear part of mapping (which
induced the given endomorphism) on initial point of coordinate system less than 1.
Definition 2.1. A point 0 is called the Denjoy-Wolff fixed point of : , if the
sequence convergent to 0 uniformly on the compact (in generally, if is any domain, then the sequence convergent to 0 uniformly on the compact subsets of ), where
denote the iterate of , . . , 0() = and () = (1()) for and 1 . In this section we will investigate the relation between eigennumbers and corresponding
eigenfunctions of the endomorphisms induced by the selfmap : , which has a Denjoy-
Wolff type fixed point 0 (on the algebra () which has analytical structure) and []0 on the
algebras of convergent power series of n variables = (1, , ), i.e., on the algebra 0() the -algebra of germs of the function of () at the point zero (for simplity we assume the point zero is a Denjoy-Wolff fixed point and we consider the case 1). We represent the maping in the form () = + (), where is a linear mapping while |()| ||2 for all . It is clear that every eigennumber (1 ) of A satisfies the condition || < 1 . Let () be eigensubspaces of operator T corresponding to eigennumber .
Theorem 2.1. If a matrix of linear part of at the 0 = 0 is diagonalizable, then
eigennumbers of compact operator and []0 are coinsides and for every nonzero
eigennumber 0, there is a biholmorphic isomorphism between eigensubspaces () and
([]0) .
Proof. Since the matrix of is diagonalizable, then by Poincare-Dulacs theorem in a small neighborhood of the fixed point by biholomophic changing coordinate system we can reduce the
mapping to polynomial normal form consisting of resonancing monoms (see [1]). We recall that an eigenvalue (1 ) of is said a resonancing eigenvalue, if there exist nonnegative
integers 1, , , such that 2 and = 11
; in this case = 11
22
is called resonancing monom corresponding to , where is a basis vector. In other words there exist the neighborhoods , of 0 = 0 and the biholomorphism such that, the mapping is reducing to form as =
1 ( + ) = 1 0 on the neighborhoods (i.e., the mapping have a normal form as 0 = + on the algebras of convergent power series of n variables = (1, , )) :
= 1 ( + ) = 1 0 , 0 = + ,
where, is a polynomial, which consisting only of resonancing monoms. Since || < 1 for all 1 , so we can choose the neighborhoods , such that 0() .
Let () be an eigenfunction of the operator () ()
which corresponding to the eigennumber , . ., (()) = () for all .
Since {}=1 uniformly convergence to zero as , so there exists a natural number
such that () ; assume that, this number is fixed. Put 0() = (1()),
-
27
. Then we have
[]00() = 0(0) = (1(0)) = (
101()) == ((1()))
= (1()) = 0() and this show that the germ 0 is an eigenfunction of the operator []0
corresponding to eigennumber . Conversely, let 0 be a germ of holomorphic function on , which is an eigenfunction of the
operator []0 corresponding to eigennumber
0 < || < 1, i.e., 0(0()) = 0(), .
Put () =1
0(()), . Since, for all we have () , so ()
, i.e., the function () is well defined and it is clear that (). Moreover, for all we have:
()() = (()) =1
0 ((())) =
1
0 ((())) ==
1
0 (
1(()))
=1
0 (0 ((()))) ==
0 ((()))
= ()
Theorem 2.1 is proved.
In this section we will consider the generalization of endomorphisms on the uniform algebras
with analytical structure (also, generalization of composition operators), namely, the operators of
weighted compositions, i.e., the operators
() () of the forms () = ()()() = ()(()) ( ()),
where () is a fixed function and is a fixed continuous self- mapping of, holomorphic on the (and certainly, we assume that the mapping has a Denjoy-Wolff type fixed point). Analogously, by above mentioned agreement we will assume that zero is
a Denjoy-Wolff fixed point of . Theorem3.1. On the above mentioned conditions, between eigennumbers (and also,
corresponding eigenfunctions, i.e., eigensubspaces) of operators and 0 there is a bijective relation.
Proof. Let be an eigennumber of operator () () and () is a
corresponding eigenfunction, i.e., we have = , or for any we have ()(()) =
(); Now, we consider the iteration: 2 = () = = 2, or
2() = ()(())(2()) = 2() for any . By the same way using the iteration,
we have that
(()) (()) = ()() = ()
1
=0
Such that, the mapping has a Denjoy-Wolff type fixed point in the compact , so, by using this iteration we can show that, the sequence of functions
(()) 1=0
convergent to some function (when the index n tends to infinity), which belong to the uniform algebra (). So, we can constructive a weighted type endomorphism (in generally, weighted type composition operator) of algebras of convergent power series of n variables =(1, , ):
0 with the weighted germ function of . Further, analogously as Theorem 2.1 we can constructive a bijective relation between eigennumbers (and also, corresponding eigenfunctions, i.e.,
eigensubspaces) of operators and 0 . Theorem 3.1 is proved.
-
28
References
1. Arnold V.I., Complementary chapters of theory of ordinary differential equations, Nauka, 1978
2. Kamowitz H., Compact operator of the form , Pacific Jour. Of Math., 1979, vol.80, No1, pp.205-211.
3. Shahbazov A.I., Spectrum of compact operator of weighted composition in certain Banach
spaces of holomorphic functions, Jour. Sov. Math., 1990, vol.48, No 6, pp.696-701.
4. Shahbazov A.I., Imamquliyev R.A., Compact weighted composition operators on the space of
holomorphic functions, Trans. Nas of Azerb., 2007, vol.27, No 1, pp.123-128.
XLAS
Daqn Seyidov Bu mqald analitik struktural mntzm cbrlrd Dencoy-Volf tipli trpnmz nqty
malik inikaslarn yaratd kili endomorfizmlrin mxsusi ddlri v onlara uyun mxsusi
altfzalar il, = (1, , ) n dyinlrinin ylan qvvt sralar cbrlrinin uyun endomorfizmlrinin mxsusi ddlri v onlara uygun mxsusi altfzalar arasnda laq tdqiq
olunmudur .
-
-
n = (1, , ) .
NDU-nun Elmi urasnn 29 dekabr 2017-ci il tarixli qrar il apa
tvsiyy olunmudur. (Protokol 04).
Mqalni apa tqdim etdi: Riyaziyyat zr flsf doktoru, dosent
Mhmmd Namazov
-
29
NAXIVAN DVLT UNVERSTET. ELM SRLR, 2017, 8 (89)
NAKHCHIVAN STATE UNIVERSITY. SCIENTIFIC WORKS, 2017, 8 (89)
. , 2017, 8 (89)
KNL MMMDOVA
Naxvan Dvlt Universiteti
GNEL HSNOVA
MEHRBAN KRMOVA
tgadjiev@ mail.az
Azrbaycan Milli Elmlr Akademiyasnn
Riyaziyyat v Mexanika nstitutu
UOT:517
QEYR-XTT ELLPTK-PARABOLK TP TNLKLRN HLLRNN TDQQ
Aar szlr: elliptik-parabolik, qeyri-xtti,srhd mslsi, hellin varl, diffuziya prosesleri
Key words:elliptical-parabolic, nonlinear, boundary problem, exist of solutions, diffusion
reaction
: o-, , , -
,
Biz mxtlif msllr riyazi modellmnin ttbiqind baxacaq. Msln keirici mhitd
elektrikl yklnmi fazann srtnm diffuziya reaksiyasnn keiriciliyi v s. kinci trtib elliptik-
parabolik tnliklrin srhd mslsini tdqiq edcyik.lk df Keldi v Fikera bu tip
msllr baxmlar. -d ikinci trtib elliptik-parabolik tnliklr n srhd mslsi tdqiq
edilmidir.
Tutaq ki, -d mhdud oxluqdur,
. Birinci srhd mslsin nzr yetirk.
(1)
(2)
(3)
oxluunun srhdi n aadak hamarlq rtini qbul edk. Burada ixtiyari
ddlrdir. rtini qbul edirik, v mrkzi x nqtsind,
radiusu R olan ardr.
Tutaq ki, (1)-(3) mslsinin msal aadak srtlri dyir, hqiqi simmetrik
matrisdir v ixtiyari v n aadak dorudur
(4)
Burada funksiyalar
x,t-y gr ll biln funksiyalardr, hr bir . Hminin
(5)
1 2 3
nR),0( TQT
T
n
i i
i
n
i i
i
n
ji i
ij
i
Qxtutxcx
utxb
x
utxb
t
utx
xutxa
xt
u
),(,0),(),(
),(),(),,(
1
11,2
2
),0(),(,),(),( Txtxtfxtu
xxhxu ,)(),0(
0, R
nRRxB ,/),( 00 RR ),( RB
),,( utxaij
TQtx ),( nR
n
ji
jiij utxa1,
212 ),,(
,,1,,),,(,),(,),(,),,(,1,0 njiutxatxbtxcutxa ijiij
TQxt ),(
)(),(,0),( 1 nLtxctxc
-
30
(6)
Frz edk ki, aadak rtlr kili funksiyalar n dorudur
harada ki, Makenhaupt rti
(7)
burada msbt sabitdir. (1)-(3) mslsin nzr yetirk, hans ki,
(8)
(9)
Biz -de finit normal funksiyalar fzas daxil edek
fzas fzasnn alt fzasdr v -dan olan btn
funksiyalarn qapanmasdr v zrind sfra yaxnlar.
Funksiyalar (1)-(3) mslsinin hlli adlanr, gr
(10)
(10) brabrliyi ixtiyari funksiyalar n -da 0-a yaxnlaan funksiyalar
v dnilir.
Teorem1.Tutaq ki, (4)-(9) rtlri dnilir. Onda mlum parametrlrdn asl olan el M1 sabit ddi var ki, (1)-(3) mslsinin hllri
(11)
rtini dyir.
.
Teorem2.Tutaq ki, Teorem 1-in rtlri dnilir. Onda mlum parametrlrdn asl olan el
M2 sabiti var ki, (1)-(3) mslsinin hlli aadak brabrsizliyi dyir.
(12)
Lemma1. Frz edk ki, Teorem1-in rtlri dnilir v aadak brabrsizlik
0),(),(,)(),(2
2 txKctxbLtxb ni
)()()(),( tTtxtx
,,0,0,0,,0)( 11 TCzzzTCt ,)()(,0)0()0( zzz
))(,,0())(,,0()(),( 111
WTLWTLQLxtf T
))(,,0(
LTL
t
f
)()( Lxh
TQ2/1
222
1
22
)(),()(1,1
,2
dxdtutxuuuxu
T
i
Q
ttt
n
i
xQW
)(1,1
,2 QW
)(
1,1
,2 QW )( TQC
)( TQ
0),(
),(),(),,(
0
2
0 1, 1,
dxdttt
utx
dtdxutxcx
utxb
x
u
x
uutxadxdt
t
u
T
T n
ji
n
ji i
i
ij
ij
)( TQC
))(,,0(),(,),0( 1,1,22 TQWLxtfut
1
2
22
21),0(
),(
),()()),(()),((
Mdxdtt
utx
dxdtt
utxdxdt
t
uxdxxtuxtuess
T
TT
Q
QQTt
TT
dsssudsssu00
21 )()(,)()(
TQ
Mdxdtt
utx
t
ux 2
22
),()(
-
31
(13)
dorudur.
.K1 yalnz mlum parametrlrdn asldr.Onda
(14)
(15) brabrliyi il tyin olunan rqmlri znd saxlayr
(15)
DBYYAT
1. Keldysh M.V. On some cases of degeneration of equation of elliptic type the boundary of domain. Dan SSSR, 1951, No 2, pp.181-183. (Russian)
2. Fichera G. On a unified theory of boundary value problem for elliptic-parabolic equations of second order. Boundary problem in differential equation Madison Gadjiev T.S., Kerimova
M.N The solutions degenerate elliptic-parabolic equations. Journal of Advances in
Matematics, 2013, vol 3, No 3, pp.218-235.
ABSTRACT
Konul Mammadova, Gunel Hasanova, Mehriban Karimova
The study of differently problem the reducing to elliptico-parabolic equations. This kind of
problem the firstly studing ty Keldis [1], Fikere [2]. He is finding correctly statements problem. In
paper [3] investigated the boundry-value problem for second order elliptic-parabolic equations. On
based this estimates qualitative property of solutions is studied.We proof that solution of boundary
problem is bounded and investigated Holder property of solutions.For this is apriori estimates for
solutions is obtained .
, ,
- .
(1), (2).
. (3) -
.
-
.
.
.
NDU-nun Elmi urasnn 29 dekabr 2017-ci il tarixli qrar il apa tvsiyy
olunmudur (Protokol 04).
Mqalni apa tqdim etdi: Riyaziyyat zr flsf doktoru, dosent Mhmmd Namazov
1
),0(),( Kdxxtuess q
Tt
2;
2
2 n
n
nq
2
22
2
),0(
),(),(),( Kdx
x
xtuxtudxxtuess
pn
pn
Tt
1)1(
2
q
qp
n
np
-
32
NAXIVAN DVLT UNVERSTET. ELM SRLR, 2017, 8 (89)
NAKHCHIVAN STATE UNIVERSITY. SCIENTIFIC WORKS, 2017, 8 (89)
. , 2017, 8 (89)
Naxvan Dvlt Universiteti
DK 517.9/539.3
Aar szlr: Kirxof tnliyi, birtrfli msl, Qalyorkin metodu, requlyarizasiya sulu,
crim sulu, hiperbolik tnliklr sistemi.
: , ,
, , ,
S. .
),()()1()(2
xtfudxuauuL k
R
kk
ttn
, 0, t ,
nR .
)(a )(f
.,0)( 0 Raa
[1].
.. [2]
.
(. [3,8]).
.
[7] [8]
.
.
S. . nR nR
. )(2 L , ,
. :
1,2 uWuK rrr ,
rr
l
rr
lxuWuW lrr
,2
1,,1,0
,,2
,,1,0,0)(),(22
),0( TQ
2
1
222212
111211
,0)(),,(
,0)(),,(
Rt
Rt
KztuzuuL
KztuzuuL
.
-
33
(1)
,),(),0(),(),0( 1111 xxxuxxu t (2)
xxxuxxu t ),(),0(),(),0( 2222 ,
),,(),()1(),(2
2
2
12121 xtfuuuauuuL ii
lrr
i
l
ittiiiii
21,max iii rrR , i=1,2. (1),(2)
),(21 RRTT
KKHH ,
),(21 RRT
KKH = ),;,0()(:),( 2221 ili WTLuuu ),;,0()( 2ilit WTLu )),(;,0()( 2 LTLuitt ,),( iRit Ktu 2,1.,. i .
,
1. 0),(),()( 021 aaRCa ii
2. 2,1. ilR ii
3. 2,1,))(;,0()(,);,0()( 2222 iLTLfWTLf itl
i
i
4. 2,1,)(,)( 22
2 iWKWi
i
i l
Ri
l
i
:
1. , 1-4. (1), (2)
),(),(2121 RRT
KKHuu .
.
3. .
,, ii lrK :
0,1
,1,:2
2,,
vvWvvK iii
ii
lRl
lR i=1,2.
,,)( ii lRi K , 0
.2,1),(2 iWil
ii (3)
),(21 RRT
KKH :
),( ,,,, 2211 lRlRT KKH = ),;,0()(),;,0()(:),( 22
221
ii l
it
l
i WTLuWTLuuu
2,1,),0(..,),()),(;,0( ,,2 iTtKtuLTLu ii lRititt .
:
.2,1),(),0(),(),0(
..),0(,,0),,( ,,1121
ixxxuxxu
TtKzuzuuL
iitii
lRtii ii
(4)
2. , 1-4. (4)
),(),( ,,,,21 2211 lRlRT KKHuu .
. (4)
.
-
34
),(),( ,,,,21 2211 lRlRT KKHvv ,
.,0))(),(()(~ 12
2
2
121 TCtvtvata ii
rr
ii
:
xxxuxxu
Kvuzxtfutau
iitii
lRitiii
l
i
l
itt ii
i
),(),0(),(),0(
,0),,()(~)1( ,,1
(5)
(.)i
vvvvv iiiiiilllRRR
i
22 11)(
(.)i - il
W2
2 ilW 22 . , 0,0
xxxuxxu
uxtfutau
iitii
itiii
l
i
l
ittii
),(),0(),(),0(
0)(1
),()(~)1( ,
iiu , ),(,, xtuu ii
)(;,0(),;,0(),;,0(: 2222 LTLvWTLvWTLvv ii ltli . ,
0 , ,,222 ,.)(),(;,0(),;,0(),;,0(: iiii lRttt
l
t
l
ii KtvLTLvWTLvWTLvvu (6)
(5).
),,()(,( ,,,,)(2)(1 2211 lRlRTnn KKHuu ,...2,1n , )(,( )(2
)(
1
nn uu
)()(),(
),(),0(),(),0(
,0),,())(),(()1(
)0(
)()(
,,
)()(2)1(
2
2)1()( 21
xtxxtu
xxxuxxu
Kzuzxtfututuau
iii
i
n
iti
n
i
lRi
n
itii
n
i
lnrn
i
r
i
ln
itt ii
iiii
(7)
),( )(2)(1 nn uu
,1),()( tu nitRi (8)
1),()( tu nit
li . (9)
(8),(9) ,
,),()( ctu nit
Ri (10)
-
35
)(),( Ctui
li (11)
0c - 0 n, 0)( C n.
D , ..
)]()([1
)( thththD
(7) iz ),(
)( tuz niti . 0
),,0(,0,.)](,.)([
1),,(
,.)())(),(()1(,.)(
)()(
)(2)1(
2
2)1()( 21
Tttutuxtf
tututuatu
n
it
n
iti
n
i
lnrn
i
r
i
ln
ittiiii
0
).,(,0,.)](,.)([
1),,(
,.)())(),(()1(,.)(
)()(
)(2)1(
2
2)1()( 21
Tttutuxtf
tututuatu
n
it
n
iti
n
i
lnrn
i
r
i
ln
ittiiii
0 ,
),0(,0,.)](),,(,.)())(),(()1(,.)()()(2)1(
2
2)1()( 21 Tttuxtftututuatu nittin
i
lnrn
i
r
i
ln
ittiiii .
(8)-(11)
),(),( )()( tuCtu niln
itti (12)
C>0 n 0 . (11) (12)
)(),()( Ctu nitt (13)
(8)-(13) ),( )(2)(1 nn uu ), )(2)(1 kk nn uu ,
,)(
i
n
i uuk ),;,0(
2
2il
WTL (14)
,)(
it
n
it uuk ),;,0(
2
2il
WTL (15)
,)(
itt
n
itt uuk - ))(;,0( 2 LTL . (16)
(8) (9) ,
,1)( tuitRi (17)
1)( tuit
li , (18)
)()( Ctuili (19)
(7) iz )1( niti uz .
0)(,),()1()()1()(
2)1(
2
2)1(
1
)( 21 tuuuuuau nitn
it
n
i
lnrnr
i
ln
ittiiii (20)
(7) n n-1 )(niti uz .
0)(,),()1()1()()1(
2)2(
2
2)2(
1
)1( 21 tuuuuuau nitn
it
n
i
lnrnr
i
ln
ittiiii (21)
)1()()( nitn
it
n
i uuw (20), (21) ,
-
36
)()1(
2)2(
2
2)1(
1
2)2(
2
2)2(
1
)()(2
)1(
2
2)1(
1
)(
,.),(),(
,),()1(
2121
21
n
it
ln
i
lnrnrnrnr
n
it
n
i
lnrnr
i
ln
itt
wuuuauua
wwuuaw
iiiiii
iiii
.
t,0 ,
dwuuuauua
dwuuad
d
wuuatw
n
i
lnl
t
nrnr
i
nrnr
i
n
i
lnr
t
nr
i
n
i
lnrnr
i
n
it
iiiiii
iii
iii
)()1(
1
0
2)2(
2
2)2(
1
2)1(
2
2)1(
1
2)(
2)1(
2
0
2)1(
1
2)(
2)1(
2
2)1(
1
2)(
.),(),(
),(2
1
),(2
1)(
2
1
2121
21
21
(22)
(17)-(19)
2)(
2)()1(
2
)1(
2
2)1(
2
2)1(
1
)1(
1
)1(
1
2)1(
2
2)1(
1
2)(
2)1(
2
2)1(
1
.,2),(
,2),(
.,(
2211
1121
21
n
i
ln
i
lnrnrnrnr
i
nrnrnrnr
i
n
i
lnrnr
i
wCwuuuua
uuuua
wuuad
d
ii
t
iiii
t
iiii
iii
(23)
..),(
.),(
),(),(
2)2(
2
2)1(
2
/
,
2)2(
1
2)1(
1,
2)2(
2
2)2(
1
2)1(
2
2)1(
1
22
11
21211
nrnr
Q
nrnr
Q
nrnrnrnr
uuaSup
uuaSup
uuauua
ii
ii
iiii
2)2(
2
2)1(
2,...2,1,0
2)2(
1
2)1(
1,....2,1,0
,:, nn
nTt
nn
nTt
uuSupuuSupQ .
)2(
1
)1(
1
)2(
1
)1(
1
2)2(
1
2)1(
1111111 . nrnrnrnrnrnr uuuuuu iiiiii . (24)
(23) (22) :
dwwwCwwt
n
i
ln
i
ln
i
ln
i
ln
itiiii
0
)1()(2
)(2
)(2
)( . .
tnln
i
ldwCw ii
0
)1(
1
)( ,
!
)( 1)(
n
tcw
nn
i
li .
c, ),( )(2)(1 21 nlnl uu - )()(];,0[ 22 LLTC . )(;,0 2 LTCui i=1,2 kn
i
n
iuu k
)( , ilWTC 2;,0 i=1,2. (25)
, (16), (17)
-
37
,1),( tu tiRi
(26)
(7) n nk, (13)-(15) (26) ,
),( 21 uu (5).
4. 1. (26) ,
,),( CtuiRi (27)
)(2 L iA
.)(1
,)(2
2
xhhA
WAD
ii
i
ll
i
l
i
iA )(iG .
(29) itii uGz )( . ,
)()1()( ill
i GGIii :
0)()()1(),,(),()1(1
2
2
2
121
tuGxtfuuuau tii
ll
i
lrr
i
l
iiiiiii
tt . (28)
0
0)()()1(),,(),()1(2
2
2
121 tuGxtfuuuau tii
ll
i
lrr
i
l
iiiiiii
tt
t0,t :
.0)()1(,.),(
)()1(,),()1(
)()1(,
0
0
2
2
2
1
0
21
duGf
duGuuua
duGu
t
ii
ll
t
ii
ll
i
lrr
i
l
t
ii
ll
i
ii
iiiiii
ii
(29)
0
22
0 000
2
1,.)(
2
1
,)()()1(,
i
l
ti
l
t t
i
l
ii
l
ii
ll
i
ii
iiii
tu
duuGLimduGuLim
. (30)
t
i
l
t
ii
ll
i
lrr
i
lucduGuuua iiiiiii
00
2
2
2
1 )()1(,),()1(21
. (31)
0
dufduGfLimduGfLim
t
i
ll
t
ii
ll
t
ii
ll iiiiii 00
00
0,.),()(,.),()()1(,.),( .
-
38
t
l
t
i
l
t
ii
lldfduduGfLim iiii
0
2
0
2
00
,.)(2
1
2
1)()1(,.),(
. (32)
(29)-(32) (28) ,
.),(2
1),(
2
1
)(2
1),(),(
2
1
0
2
0
2
2
0
22
t
l
t
i
l
i
l
t
i
l
ti
l
dfdu
dtuctu
ii
iii
Ctuti
li ),( , (33)
Cdut
i
li 0
),( , (34)
0C 0 . (15) (29) ,
),(),( tuCtu il
ttii
(35)
(34),(35)
Cdu
t
i 0
2),( . (36)
(30)-(33), (35), (36), k ,
ii uu k *- );,0(
2
22
il
WTL , (37)
itti uu k *- );,0( 2
il
WTL , (38)
ttittiuu
k *- ))(;,0( 22 LTL . (39)
,
,...2,1,,,,, 1 kKK kiikii lRlR
(21) k N Nk
0)(,),(2
2
2
1 tuzuuuautkkkkttk
iiiii ,
..),0(,,, TtKz Nii lRi . (40)
(38)-(40) ,
ii uu k )];,0([ 2
il
WTC (41)
(37)-(39) (41) 0k . ,
..),0(,,0)(,),()1( ,,2
2
2
121 TtKztuzuuuau
Nii
iiii
lRiitii
lrr
i
l
itt , (42)
)(),0(),(),0( xxuxxu iitii . (43)
(43) iRi
Kz , N .
-
39
1. .., , .III, .323-331
2. .., , .,
.96, 1, 1975, .152-166
3. ..,
, , .21, 1, 9985
4. DAncona P., S.Spagnolo. Nonlinear pertubations of the Kirchhoff equation. Comm. Pure
Appl. Math. 47, 1994, 1005-1029
5. Kosuke Ono, Global Existence, Degay and Blowup of Solution for some Mildly Degenerate
Nonlinear Kirchhoff String, J.of. Differential equations, 137, 1997, 273-301
6. Ghisi M., Gobbino M. Global existence and asymptotic behaviour for a mildly degenerate
dissipative hyperbolic equation of Kirchhoff type, asymptotic Analysis, 40, 2004, 25-36
7. ..
. , , .274, 6, 1984, .1341-1344
8. ..
. , , .297, 2, 1987, .271-275
XLAS
Kirxof operatorlar sistemi n variasiya brabrsizliyi
d bir sinif qeyri lokal qeyri xttili Kirxof operatorlar n variasiya brabrsizliklr
sistemi aradrlr. Kompaktlq, requlyarizasiya v crim operatoru metodlarnn kombinasiyasn-
dan istifad edib uyun Koi mslsi hll edilmidir.
ABSTRACT
Variational inequality for systems Kirchhof operators
In this paper we study systems of variation inequalities for a class of Kirchhoff operators with
nonlocal nonlinearities. Using the combined methods of compactness, regularization, and penalty,
the solvability theorem corresponding to the Cauchy problem is proved.
NDU-nun Elmi urasnn 29 dekabr 2017-ci il tarixli qrar il apa tvsiyy
olunmudur (Protokol 04).
Mqalni apa tqdim etdi: Riyaziyyat zr flsf doktoru, dosent Mhmmd
Namazov
-
40
NAXIVAN DVLT UNVERSTET. ELM SRLR, 2017, 8 (89)
NAKHCHIVAN STATE UNIVERSITY. SCIENTIFIC WORKS, 2017, 8 (89)
. , 2017, 8 (89)
MMMD RCBOV
Naxvan Dvlt Universiteti
UOT:519.17
EYLERN QRAFLAR NZRYYSN AD LRNN
YLNCL MSLLRN HLLN TTBQ
Aar szlr: Qraf, til, unikursal fiqur, marrut, mstvi qraf, Eyler dvr, bk
Key words: Graphic, language, unicorn figure, route, plane graph, Eyler cycle, network
: , , , , ,
, .
Qraflar nzriyysi riyaziyyatn mstqil sahsi kimi 1930-cu illrin ortalarndan inkiaf
etmy balamdr. Lakin hl 1679-cu ild grkmli alman riyaziyyats Qotfrid Vilhelm Leybins
(1646-1716) Hollandiya alimi Kristian Hygens (1629-1695) yazd mktublarn birind
gstrirdi. Gman etmk olar ki, riyaziyyatn kmiyytlr mul olan blmsindn baqa yeni bir
blmsi d olmaldr. Leybins gr bu blm vziyyt hndssi adlandrlan fnnin msllrinin
tdqiqi il mul olandr. Vziyyt hndssi adlandrlan fnnin ilk msllrindn biri Leonid
Eyler (1707-1783) trfindn oxzlnn uyun olaraq tplri tillri v zlrinin say arasnda
yaradlan
0 - 1 + 2 = 1 (1)
mnasibt olmudur. Bu mnasibt onunla mhduddur ki,o he bir oxzlnn tillrinin uzunluu,
bucaqlarnn qiymti il laqdar deyil. oxzlnn bu xasssi onlarn metrikasndan asl deyil.
Gman edildiyin gr riyaziyyatn blmlrindn biri olan qraflar nzriyysi L.Eylerin
Kniqsberq (indiki Kaleninqrad) hrind 7 krp haqqnda mhur mhakimsi il baldr.
Vaxtil Peterburq Elmlr Akademiyasnn dvti il Rusiyaya glmi, bu hrd yaam v 7
krp haqqnda mslni hll etmidir.
Bu hrin parkn Preken ay ikisi sahillrd ikisi is ada klind olmaq rtil 4 drd
hissy blr. Adalar v sahillri bir-biril 7 krp birldirir. (kil 1 a)
1. b)
kil 1
hr halisinin n sevimli ylnclrindn biri el marutun taplmasna can atmaq
olmudur ki, o krplrin hamsn, hrsindn bir df kemkl hrktini balancda xlan
sahd qurtarsn.
Mslni hll etmk nn lverili variant qlm v kaz gtrb hr iki sahili v adalar
uyun olaraq A,B, C, D, nqtlri il onlar birldirn krplri is bu nqtlri birldirn
xttlrl tsvir etmkdir. Nticd mslnin mstvi zrind tsvirini alrq. (kil 1 b)
gr bu kil diqqtl baxsaq A, B, C, D nqtlri hrlr bu nqtlri birldirn til
adlanan xttlr is hrlraras dmiryollarnn tsviri kimi v yaxud A B C nqtlrin mntqlr
D nqtsin baza, tillr is onlar arasnda yollarn tsviri kimi v s. baxa bilrik.
Qraflar nzriyysin gtiriln qdim bir msly d baxaq: evin hr birindn su
quyusunun hr n biri digrini ksmyn yollar kmk olarm?
gr qlm v kaz l alb bu mslnin hlli il mul olsaq, onda ox kmz ki,
-
41
mvfqiyytsizliy rastlayarq. gr mslnin hllinin mmkn olmadn frz etsk, onda daha
tin problmlr il rastlaarq, nku evdn su quyusuna yolu mstvi zrind tsvir etsk, onda
grrik ki xttlri kifayt qdr oxlu yollarla kmk olar. Dourdan da ola bilsin ki, xttlr bir
ne ilgk czdqdan sonra mqsd atmaq olar. Bu msld artq tfsilat atsaq onun rtini
aadak kimi yaza bilrik:
Hr biri nqtdn ibart olan iki oxluq verilmidir. Bu oxluqlardan birinin hr bir
nqtsini o birinin nqtsi il birldirn el xttlr kmk olar ki, onlar ksimsinlr.(kil 2)
kil 2
Qeyd edk ki, riyaziyyatn bu v ya digr blmsi il mul olan blmsi qraflar
nzriyysi adlanr.
Baxdmz bu iki msldn aydn olur ki, qraflar nzriyysind mstvi zrind
nqtlrl tsvir etdiyimiz iki obyekti birldirn v qrafn tillri adlanan xttlrin dz xtt paras
ksilmz yri xtt qvslri, bu xtlrin uzun v ya qsa olmas he bir hmiyyt malik deyildir.
Burada n mhm cht hmin xtlrin veriln iki nqtni birldirib-birldirilmmsinddir.
Trif: gr qraf el kmk mmkns ki, istniln iki tilin ucunun tpdn baqa he bir ortaq
nqtsi olmasn, bel qraf mstvi qraf adlanr. Balanc v sonu st-st dn yol dvr adlanr.
kil 3 a da tsvir ediln qraf 5 zldr. (A1, A4, A6, A2), (A2, A4, A6, A2), (A3, A1, A6, A3),
(A3, A5, A6, A3), (A2, A5, A6, A2).
Qeyd edk ki, iki qraf o zaman eyni olur ki, 1) tplrin say eyni olsun. 2) millrin say
eyni olsun. 3) uyun tplrin trtibi eyni olsun.
2. b)
kil 3
gr yol qrafn btn tplrindn keirs bel yol Eyler yolu adlanr. Qrafn btn
tplrindn ken dvr Eyler dvr. Bel qraf is Eyler qraf adlanr.
Yuxarda qeyd etdiyimiz kimi qraflarn topoloji xasslrindn biri Eyler dturu il baldr.
Qeyd edk ki. ev v su quyusu mslsinin hllind onu tsvir edn qrafn mstvi qraf
olub-olmamasndan asldr.Qrafn istnikn iki tpsini gtrk. Balanc bu tplrdn birind
sonu is o birind olmaq rti il hr sonra gln tili zndn vvlkinin sonundan balayan tillr
arasnda marrut adlanr.gr qrafn istniln iki tpsini birldirn marrut vardrsa, bel qrafa
rabitli qraf deyilir. kil 3 b-d istniln A v B tplri sek. Grndy kimi onlar arasnda
marrut vardr.
bhsiz rabitli. Sonlu mstvi qraflar mstvini sonlu sayda oblastlara (qrafn zlrin)
blckdir.
kil 4
-
42
kil 4-d tsvir olunmu qrafn 10-tpsi.15-tili.6-z vardr. Qrafn mstvini bldy sonlu
sayda oblastlara qrafn xaricind qalan oblast da qatsaq onda qrafn zlrinin say 6 deyil 7
olacaqdr.
Grkmli riyaziyyat L.Eyler 1752-ci ild hr bir oxzln bu oxzlnn tplrinin say
0, tillrinin say 1 v sonsuz z d daxil olmaqla zlrinin say 2 olduqda geni ttbiq malik
olan 0 1 + 2 = 2 (1) dsturunun doruluunu isbat etmidir.
gr bu dstur kil 4 d tsvir edilmi qrafa ttbiq etsk 0 = 10, 1 = 15, 2 = 7 olur =>
10-15+7=2 alrq. Bu dstur btn rabitli, sonlu v mstvi qraflar n dorudur.
gr qrafn zlrinin sayn onun xaricind qalan z (oblast) yeni bir z kimi lav etsk,
onda Eyler dsturu aadak kimi olar:
0 - 1 + 2 = 1 (2)
Qeyd edk ki, sonlu, rabitli mstvi qraflarn (2) Eyler dsturu il xarakteriz ediln
xasssi qrafn hr hans tilini sildikd d dnilir. nki qrafda hr hans bir tilin silinmsiyl ya
zlrin say, ya da tplrin say bir vahid azalr, odur ki, (2) brabrliyinin sol trfi dyimir.
Dourdan da,
0 (1 -1) + (2 -1) = 0 1 + 1+ 2 + 1 2 -1= 0 1 + 2
yaxud 0 -1-(1 -1)+ 2= 0-1- 1 + 1+ 2 = 0 1 + 2 kil 5 a) da tsvir edilmi qraf n (2) dsturunun doruluu v yuxardak qeydi
asanlqla yoxlamaq olar.
a) b) c) d)
kil 5
Burada 0 = 12, 1 = 19, 2 = 9,
12-20+9 = 1
Tillrdn birini silsk (kil 5 b)
0 = 12, 1 = 19, 2 = 8
12-9+8 = 11
Yenidn bir til silsk (kil 5 c) 0 = 12, 1 = 18, 2 = 7
12-8+7 = 1
Daha bir til silsk tplrin say bir vahid azalr. (kil 5 d)
0 = 11, 1= 17 2 = 7
11-17+7=1
Prosesi sonuncu tp qalanadk davam etdirsk, nhayt
0 = 1, 1 = 0, 2 = 0
0 1 + 2 = 1-0+0 = 1 olur.
Demli Eyler dsturu qraflarn topoloji xasssini xarakteriz edir.
Artq ev v quyu suyu mslsini hll etmk n kifaytdir ki, kil 3-d tsvir edilmi
qrafn mstvi qraf olmadn gstrk.
kil 2-d tsvir edilmi qraf rabitli v sonlu qraflar n Eyler dsturuna gr 0 1 + 2 = 2 mnasibti olmaldr.
Akardr ki, bu qrafda tplri say 0 = 6, tillri say,1 = 9, zlri say 2 = 2-6+9 = 5
olmaldr.
Grldy kimi kil 2-d uzunluu 3 olan sad dvr yoxdur, nki zlrin srhdi 4 tildn
az deyil.
ndi tillrin iki qat qiymtin, yni 21 - baxaq. Digr trfdn hr bir til iki zn srhdi
olduundan biz sonsuz z d nzr alsaq, onda 4 2 tillrin ikiqat sayndan byk ola bilmz.
Baqa szl 42 21 .
-
43
Lakin baxlan halda 21 = 18, 42 =20 olduundan 2018 alrq. Bu ziddiyt gstrir ki,
ev v su quyusu mslsinin kil 2-d tsvir edilmi qraf mstvi qraf deyil. Demli mslnin
hlli yoxdur.
7 krp mslsini hll etmzdn vvl aadak msly baxaq: Qlmi kazdan
ayrmadan v hr bir xttin zri il ikinci df kemdn kil 6-da tsvir edilmi qraflar kmk
olarm?
lk baxda kil 6-da verilmi qraflarn hr hans bir tpsindn balayaraq hr bir tilin zri
il bir dfdn artq getmmk rti il btn tplrdn kerk balanc tpy glmyin
(dvrn) olub olmadni tapmaq ox tin grlr. Lakin baxlan msly qraflar nzriyysinin
ttbiqi mslnin hllini el asanladrr ki, bu maraql v ylncli mslni uaq baasnda v
ibtidai siniflrd uaqlara tklif etmk olar.
kil 6
L.Eyler gstrmidir ki, gr qrafn btn tplri ct trtiblidirs, yni bu tplr yerln
tillrin say ctdrs, onda qrafda istniln bir tpdn balayaraq hr bir tilin zri il bir df
kemk rti il btn tplrdn keib balanc nqty glmk mmkndr. kil 6-d tsvir
edilmi fiqurdan b) v c) bndlrind tsvir edilmi fiqurlar yuxarda gstriln qayda il kmk
mmkndr, nki bu fiqurlarda qrafn btn tplri ct trtiblidir. Qeyd edk ki, bel fiqurlara
unikursal fiqurlar deyilir.
kil 6-da a) v d) bndlrind tsvir edilmi fiqurlarn qrafnda tplrdn bzi ct, bzisi
is tk trtibli olduundan bu fiqurlar qlmi kazdan ayrmadan hr tilin zri il kmk
mmkn deyil.
Artq 7 krp mslsini asanlqla hll ed bilrik. kil 1 b-d tsvir edilmi fiqur unikursal
fiqur olmadndan mslnin hlli yoxdur. nki qrafn tplri ct trtibli deyil. Qeyd edk ki,
Eyler qraflar nzriyysin aid ilrinin qraflar nzriyysinin inkiafnda v qraflarn praktiki
msllrinin hllin ttbiqind mhm hmiyyti olmudur.
DBYYAT
1. O O. . , , 1980
2. .. , , 1979
.
1930-
.
1679
. . ,
.
-
44
,
().
(1707-1783) 1752 , 0-1 + 2 = 2 .
,
.
.
ABSTRACT
M.Rajabov
The application of the theory of graph theory to the
solution of entertainment questions
Application of the work about graph theory by L.Eyler to the solution of entertaining
queries.
The Theory of graphs began to develop from the midst of 1930s as an independent field of
mathematics.
In one of the letters to the Dutch scientist Christian Huygens in 1679, the prominent German
mathematician G.W. Leibniz shows that it is possible to suppose that mathematics should have a
new section except the field dealing with quantities.
As to Leibniz, this field must study the research of the subject queries called positional
geometry.
One of the first queries of positional geometry was formula 0- 1+ 2=2 by Leonid
Eyler(1707-1783) wich was created among the tops edges and the facets of the multifaced.
This formula is famous with that this is not related to the length of the edges of the multi-
faced, the value of the angles.In this study, application of the work about graph theory by L.Eyler to
the solution of entertaining queries was reviewed.
NDU-nun Elmi urasnn 29 dekabr 2017-ci il tarixli qrar il apa tvsiyy olunmudur
(Protokol 04).
Mqalni apa tqdim etdi: Riyaziyyat zr flsf doktoru, dosent Mhmmd Namazov
-
45
NAXIVAN DVLT UNVERSTET. ELM SRLR, 2017, 8 (89)
NAKHCHIVAN STATE UNIVERSITY. SCIENTIFIC WORKS, 2017, 8 (89)
. , 2017, 8 (89)
ZMRD SFROVA
Naxvan Dvlt Universiteti
HAMLET QULYEV
Bak Dvlt Universiteti
UD