NOCHIZIQLI INTEGRO-DIFFERENSIAL TENGLAMALAR YECHIMLARI HAQIDA

Аннотация

Impulsiv, maksimallarga ega oddiy integro-differensial tenglamalar sistemasi uchun nolokal chegaraviy masala o‘rganildi. Chegaraviy masala integral shart bilan berilgan. Chegaraviy masala yechimining mavjudligi isbotlangan. Kalit so'zlar: impulsiv integro-differensial tenglamalar, nolokal chegaraviy shart, ketma-ket yaqinlashishlar, yechimning mavjudligi va yagonaligi, yechimning uzluksiz bog‘liqligi. Quyidagi integro-differensial tenglamalar sistemasini ko‘rib chiqamiz t[0,T], t t , i =1,2,...,plar uchun i T (  ) x(t) =H(t,s)x(s)ds+ f t,x(t),max x()|h ;h  (1)  1 2 0 tenglama nolokal chegaraviy shart bilan T Ax(0)+ K(t)x(t)dt = B (2) 0 va impuls x ( t+) − x ( t−) = I ( x(t )) , i =1,2,...,p, (3) i i i i bilan berilgan bo‘lsin, bu yerda 0=t t ...t t =T, ARnn,K(t)Rnn 0 1 p p+1 T berilgan matritsa va detQ  0, Q = A+ K(t)dt, f :[0,T]RnRn →Rn, 0 I :Rn →Rn berilgan funksiyalar; i 0 h  h t, h = h (t,x(t)), j =1,2,  noldan farqli haqiqiy parametr, 1 2 j j x ( t+) = lim x(t +h), x ( t−) = lim x(t −h) x(t) funksiyaning mos ravishda t =t i i i i i h→0+ h→0− nuqtadagi o'ng va chap limitlari. 48 International scientific journal “Interpretation and researches” Volume 1 issue 3 (49) | ISSN: 2181-4163 | Impact Factor: 8.2 C ( [0,T],Rn) fazoni aniqlanish sohasi [0,T], qiymatlar sohasi Rn bo‘lgan x(t) uzluksiz vektor funksiyalardan tashkil topgan Banax fazosi sifatida kiritamiz. Bu fazoda norma quyidagicha kiritilgan: n x = max x (t) . j t[0,T] j=1 PC ( [0,T],Rn) chiziqli fazoni quyidagicha belgilaymiz: PC (0,T,Rn) =  x:[0,T]→Rn; x(t)C ((t ,t ,Rn) ,i =1,...,p  , i i+1 bu yerda x ( t+) va x ( t−) (i =0,1,..., p) mavjud va chegaralangan; x ( t−) = x ( t ) . i i i i Malumki, PC ( [0,T],Rn) chiziqli fazo Banax fazosi bo‘lib, bu fazoda norma quyidagicha kiritiladi: x = max x , i =1,2,..., p . PC C((t ,t ]) i i+1 Izlanayotgan x(t)PC ( [0,T],Rn) funksiya, barcha t[0,T],t t , i =1,2,...,p i lar uchun (1) integro-differensial tenglamani qanoatlantiradi, t =t i =1,2,..., p, 0t t ...t T lar uchun (2) nolokal integral shartni va (3) i 1 2 p chegaraviy shartni qanoatlantiradi. Aytaylik, x(t)PC ( [0,T],Rn) funksiya (1)-(3) masalaning yechimi bo‘lsin. T m (  ) x(t) =a (t)b (s)x(s)ds + f t,x(t),max x()|h ;h  . k k  1 2 0k=1 yuqoridagi tenglamada quyidagicha belgilash kiritamiz: T c = b (s)x(s)ds k k 0 so‘ng integro-differensial tenglama quyidagi ko‘rinishga keladi: m (  ) x(t)=a (t)c + f t,x(t),max x()|h ;h  . k k  1 2 k=1 U holda oxirgi tenglamani t(0,t , intervalda integrallash orqali quyidagi i+1 tenglikka ega bo‘lamiz: t  m   a (s)c + f (s,x(s),) ds = k k 0 k=1 t = x(s)ds = x(t )− x ( 0+)+ x(t )− x ( t+)+...+ x(t)− x ( t+) =  1   2 1   i  0 49 International scientific journal “Interpretation and researches” Volume 1 issue 3 (49) | ISSN: 2181-4163 | Impact Factor: 8.2 =−x(0)−x ( t+) − x(t )−x ( t+) − x(t )−...−x ( t+) − x(t )+ x(t).  1 1   2 2   i i  (2) integral shartni oxirgi tenglikka qo‘llasak, quyidagi tenglikka ega bo‘lamiz: t  m  x(t)= x(0)+ a (s)c + f (s,x(s),) ds+  I ( x(t )) . (4) k k i i 0 k=1 0t i t (4) dagi x(t)PC ( [0,T],Rn) funksiya (2) chegaraviy shartni qanoatlantiradi:  T  A+ K(t)dtx(0)= 0 T t  m  T = B−K(t) a (s)c + f (s,x(s),) dsdt −K(t)  I ( x(t )) dt. (5) k k i i 0 0 k=1 0 0t i t detQ 0dan va (5) tenglikdan biz quyidagi tenglamaga ega bo‘lamiz:  T t  m  T  x(0)=Q−1 B−K(t) k (s)c k + f (s,x(s),) dsdt −K(t)  I i ( x(t i )) dt. 0 0 k=1 0 0t i t (6) (6) tenglikni (4) tenglikka qo‘yib, quyidagi  T t  m  T  x(t)=Q−1 B−K(t) k (s)c k + f (s,x(s),) dsdt −K(t)  dt+ 0 0 k=1 0 0t i t t + f (s,x(s),)ds+  I ( x(t )) . (7) i i 0 0t i t tenglamaga ega bo‘lamiz. Quyidagi T t  m  TT  m  K(t) a (s)c + f (s,x(s),) dsdt = K(s)ds a (t)c + f (t,x(t),) dt, k k k k 0 0 k=1 0 t k=1 T T K(t)  I ( x(t )) dt =  K(t)dtI ( x(t )) , i i i i 0 0t i t 0t i Tt i tengliklarni (7) tenglikka qo‘yib TT  m  x(t) =Q−1B−Q−1K(s)ds a (t)c + f (t,x(t),) dt − k k 0 t k=1 T t  m  −Q−1  K(t)dtI ( x(t )) +  a (s)c + f (s,x(s),) ds+  I ( x(t )) . i i k k i i 0t i tt i 0 k=1 0t i t (8) tenglamaga ega bo‘lamiz. 50 International scientific journal “Interpretation and researches” Volume 1 issue 3 (49) | ISSN: 2181-4163 | Impact Factor: 8.2 Berilgan (8) ifodani biroz soddalashtiraylik. U holda quyidagi tenglik hosil bo‘ladi: t  m  TT  m   a (s)c + f (s,x(s),) ds−Q−1K(s)ds a (t)c + f (t,x(t),) dt = k k k k 0 k=1 0 t k=1 t    m  =Q−1A+ K(s)ds a ()c + f (,x(),) k k 0  0  k=1  TT  m  −Q−1K(s)ds a ()c + f (,x(),) d; (9) k k 0 t k=1 T  I ( x(t )) −Q−1  K(t)dtI ( x(t )) = i i i i 0t i t 0t i Tt i  t i  T =Q−1  A+ K(t)dtI ( x(t )) −  Q−1K(t)dtI ( x(t )) . (10)   i i i i 0t i t 0  tt i+1 T t i (9) va (10) tengliklarni hisobga olib, (8) tenglikni quyidagi integral tenglama ko‘rinishida yozamiz: t(t ,t , i =0