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Abdusaidov S. U. Assistant Jizzax Politexnika instituti Abriyev N. T. Assistant Jizzax Politexnika institute
MAKSIMUM BELGISI OSTIDA FUNKSIONAL PARAMETRNI O'Z ICHIGA OLGAN TENGLAMALAR SISTEMASI UCHUN BOSHLANG'ICH MASALA
Annotatsiya : Ushbu maqolada turlarning chiziqli bo 'Imagan integro-defferensial tenglamalarsistemasi ko'rib chiqiladi
Kalit so'zlar boshlang'ich shart, integro-defferensial tenglamalar tizimi, funksional parametr, Inter jarayonini, cheklangan yopiq ko'phadlar, segment
Abdusaidov S.U. Assistant
Jizzakh Polytechnic Institute Abriyev N.T. Assistant
Jizzakh Polytechnic Institute
AN INITIAL PROBLEM FOR A SYSTEM OF INTEGRO-DIFFERENTIAL EQUATIONS CONTAINING A FUNCTIONAL PARAMETER UNDER THE SIGN OF THE MAXIMUM
Abstract: The system of non-linear integro-differential equations of species is considered in this article.
Keywords: initial condition, system of integro-differential equations, functional parameter, Inter process, limited closed polynomials, segment
Ushbu maqolada turlarning chiziqli bo'lmagan integro-defferensial tenglamalar tizimi ko'rib chiqiladi.
i t \
x'(t) = F
t, x(t), J K (t, 0) max {x(r) | r e[0- h;0]} dd, u(t)
t > 0 (1)
Quydagicha boshlang'ich shart berilgan
x(t) = cp(t), t e E0 = [-a, 0] (2)
Bu yerda x e X c Rn vektor holati , u eU œ Rm funksional parameter X, U cheklangan yopiq ko'phadlar , h = h(t,x(t), u(t ) ) kechikish T vaqtiga bog'liq ,
kerakli x(t) funksiyadan va u (t) funksional parametrdan, K (t, 6) {0 < 6 < t < T} shu oraliqda uzluksiz n x n matrissali-funksiya, t - h(t, x, u) > -a0 = const,
maxsimum. Quydagi ifodani soddalashtirish uchun biz quydagi belgilashni qabul
qildik.
t
p = JK(t,6)max{x(r) | r e [6- h;6]}16,hm = h(6,xm (0),u(0)),
0
t t pm = J K (t, 6) max {xm (r) | r e [6 - hm; 6]} d6, J ||K (t, 6)\\l6 < a < « 0 0 Keling, nima sodir bo'lishini isbotlaylik Teorema. Quyidagilar o'rinli bo'lsin:
1 F(t,x,p) e C([0;T] x X x Rn x U) n Bnd(M) n Lip(LyXp); (3)
2 t - h (t, x, u )>-a0= const (4) Va h (t, x, u ) e C ([0; T ] x X x U) n Lip (L2/x ) (5) 3.^(t) e Lip (L3) (6) Bu yerda X = {x e Rn ||x - <p(0< r}, r = max {||p(t) - <p(0t e ^0}.
Keyin
0; t
, t < T segmentda (2) boshlang'ich shart bilan x(t) e Xfunksiyaning
yagona yechimi mavjud.
Isbot. Biz ketma-ket yondashuvlar yordamida (1) tenglamani (2) ga aylantiramiz.
x0(t) = p(t),t e E0,x0(t) = p(t),t > 0 xm+i(t) = p(t),t e E0,m = 0,1,2,...
x
m+1
t
(t) = <(t) + J F (6, xm (6), Pm (6), u(6)) d6, t > 0
(7)
Barcha yondashuvlar X t
0; t
da qolishiga ishonch hosil qilish qiyin emas.
Inter jarayonining farqini baholaymiz (7) da . x1(t) - x0(t) farqi uchun (3) hisobga olinsa , biz quydagicha baholaymiz.
(t) - x0 (t)|| < Mt, t
e
0; t
Bundan tashqari farq uchun x2 (t) - x1 (t) bizda bor
t
||x1(t) - x0(t < L1J (I x1(6) - x0(6)|| + ||p(6) - P0(6)||)d6.
(8)
(9)
0
Baholash uchun p (t) - p0 (t) biz bu faгqni quydagicha yozamiz
t
Pi(t ) - Po(t ) = J K ( t ,e)(max { Xi(r ) j r e[e - ^;в]]-
- max {X (r ) j r e[e - h0 ;в]]^в = J K ( t ,e)(max {x (r ) j r e[e - h ;в]]-
0
- max {x0 (r ) j r e[e - h ;в]]) - (max {x0 (r ) j r e[e - h ;в]] -
- max { x0 (r )j r e[e- ho;e]])de. (10)
(S) tenglikning biгinchi fa^i uchun (10) hisobga olinsa, quydagicha olish mumkin
max {X (r) j r e [t - h ; t ]]-max {x0 (r)j r e [t - h ; t ]] <
< I j max {( X (г) - х0 (r )) j r e[t - \ ; t ]] < Mt
max {X (г) j г e[t - h ; t]] = max {(x (г) - x0 (г) + x0 (г)) j г e[t - h0 ; t ]] <
< max {(x (r) - x0 (r))j r e[t - h ; t]] + max {x0 (r)j r e[t - h ; t ]] max {X (r) j r e [t - h ; t ]]-max {x0 (r) j r e [t - h ; t ]]<
< max {(X (r) - X0 (r)) j r e[t - h ; t]]<
< max {(X (r) - X0 (r)) j r e[t - h ; t]]
max {X0 (r) j r e [t - h ; t ]]-max {x (r) j r e [t - h ; t ]]<
< max {(Xq (r) - x (r)) j r e[t - \ ; t]], t > 0 Xuddi shunday, biz quyidagi natijani olamiz
max {x (r) j r e[t - h ; t]]- max {x (r) j r e[t - h ; t ]]<
< max {(Xq (r) - x (r)) j r e[t - \ ; t]], t > 0
Ushbu tengsizlikni chap va o'ng qismidan maksimum t ni olib , biz isbotlashni, istalgan tengsizlikni qo'lga kiritamiz. (4)-(6) ni hisobga olgan holda , o'ng tengsizlikdagi ikkinchi faгq uchun quyidagi fa^ni olamiz.
0
max Ix0 (r) I r g[í - \ ; t]}- max jx0 (r) | r g [t - h0 ; t]}||< < L \\h(t, X (t), u(t)) - h(t, x0 (t), u(t))|| < ML2 J x (t ) - x0 (t )|| < M2 L2t, M = max IM, L3} pl(t)p)|| < Ma(l+ML2)t
Shuning uchun
||Pl(t ) -PG« )|| < Ma(l+ML2)t
Keyin (9) quyidagicha yoziladi:
t
X (t ) — x (t )|| < L { M [l + a(l + ML2 )]вdв = LXM [l + a(l + ML2 )]
G t
X3(t) — x2(t)|| < A {(Ix2(в) — хв)! + p2 (в) — Pl(в)ув <
t_ 2!
< Ll
l + a(l +
Xl' (в)
L
t
{ LM [l + a(l + ML2 )
в dв = 2!
2 t З!
= L2 M [l + a(l + ML2 )
X3
(t) - X2(t ) farqni quydagicha olamiz
t
11X3 (t) — X2 (t)|| < Ll { (||X2 (в) — Xl (в)|| + P2 (в) — Pl (в)<
t в < Ll j(| X2(в) — Xl^l + {|| K (9¿)\
G G
[||maxIx2(r) I r g [^ — h2,— max (Xl(r)I r g [^ — h2,£]}|| +
max Ix (r)I r g [^ — h2, — max (x (r)I r g [^ — \, £]}||^^в <
+
= L2 M
l + a(l + ML2 )
21_
З!
Demak, ushbu baholashlardan m da ||z(t) - xm (t)|| ^ G manashu G; t segmentdagi t da teng ravishda ko'rinadi. Shundan kelib chiqadiki, Ixm (t)}
ketma-ketligini talab qiladigan (l) tenglamaning yechimi yagona.
G; t
segmentida
G
G
Teorema isbotlandi.
Adabiyotlar
1 . Efendiev M., Vougalter V. Solvability of some integro-differential equations with drift. Osaka J. Math. 2020. 57, 247-265.
2 El-Sayeda A.M.A., Aahmedb R.G., Solvability of the functional integro-differential equationwith self-reference and state-dependence. Journal of Nonlinear Sciences and Applications. 2020. 13. p. 1-8.
