ndu.edu.azndu.edu.az/public_html/files/uploader/elmieserler/deqiq 89.pdfndu.edu.az

3

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]

4

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.

6

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).

7



. , 2017, 8 (89)

ELAD AAYEV

[email protected]

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)

8

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

9

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).

12



. , 2017, 8 (89)

BLFZ MMMDOV


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



. , 2017, 8 (89)

YAQUB MMMDOV

Naxvan Mllimlr nstitutu

[email protected]

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



. , 2017, 8 (89)

DASHQIN SEYIDOV Nakhchivan State University

[email protected]

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



. , 2017, 8 (89)

KNL MMMDOVA

[email protected]


GNEL HSNOVA

[email protected]

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



. , 2017, 8 (89)

[email protected]


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



. , 2017, 8 (89)

MMMD RCBOV


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



. , 2017, 8 (89)

ZMRD SFROVA


[email protected]

HAMLET QULYEV

Bak Dvlt Universiteti

[email protected]

UD

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