, (1). -, [9], [1]. 1.. T =[ a] R: _(t)=f((t)) _ L(t) () = x f, L(t) T., L(t), L() = L(a ; ) = L(a). (2) - : L n (t) =(L n )(t) = 1=n R supp [ 1], 1R

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Download ", (1). -, [9], [1]. 1.. T =[ a] R: _(t)=f((t)) _ L(t) () = x f, L(t) T., L(t), L() = L(a ; ) = L(a). (2) - : L n (t) =(L n )(t) = 1=n R supp [ 1], 1R"

Nelson Park
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1 25 3(514) ,..,.. -.,.., -, -. : _x(t) =f(t x(t)) _ L(t) (1) L(t) _.. - -, f(t x(t)) L(t). _ ([1],. 1, x 8,. 41),. [2]{[4],., [2]{[4],, [1]. - x(t) =x f( x())dl() t {, {.. [5]{[7]. L(t),. [8] (1), L(t). ([1],. 4,. 143) : x(t) =x t f( x())dl c () i<t S( i x( i ; ) L( i )) S( p x( p ; ) L( p ))(t ; p ) L c (t) L(t), i L(t), L( i ) = L d ( i ); L d ( i ; ), (t), S( i x( i ; ) L( i )).,, - [1],. 23

2 , (1). -, [9], [1]. 1.. T =[ a] R: _(t)=f((t)) _ L(t) () = x f, L(t) T., L(t), L() = L(a ; ) = L(a). (2) - : L n (t) =(L n )(t) = 1=n R supp [ 1], 1R (s)ds =1. n (t h n ) ; n (t) =f n ( n (t))[l n (t h n ) ; L n (t)] n (t)j [ hn) = n (t): L(t s) n (s)ds, f n = f n, n (t) =n(nt), (t) 2 C 1 (R), (t), t T. t t = t m t h n, t 2 [ h n ), m t 2 N., (3) n (t) = n ( t ) f n ( n ( t kh n ))[L n ( t (k 1)h n ) ; L n ( t kh n )]: 1. f 2 CB(R). 1 n! 1, h n!, h n = o(1=n) t 2 T, n (t) (3) Y (L(t)), Y (v) Y (v) =x Z v t 2 T j n ( t ) ; x j!. (2) (3) f(y (u))du (4) 1., Y (L(t)) - x(t) =x f(x())dl c () S(x( i ; ) L( i )) t it R S(x u)='(1 x u);'( x u), '(t x u) '(t x u)=xu t f('(s x u))ds., 1, n!1, h n!, h n = o(1=n), - (3) [1] (2). 2. f 2 CB(R). 1 n!1, h n!, 1=n = o(h n ), t 2 T n (t) (3) (t) =x t 2 T j n ( t ) ; x j!. f((s ; ))dl(s) (5) 24

3 2., (3) - (2), [5]{[7] n Z n1 A n A k n B k Z k (6) A, A k, B k Z k > k = n. - n n Z n1 A A k exp B k :. (6), Z n1 A n B k Z k n A k B k Z k B n Z n A B k Z k = A n n A k B n A A k A k B n A A k (B n 1) B k Z k A A k B n A A k B (B n 1) B 1 (1 B 2 (1 (1B (1 B n )))) A n ln Z n1 ln A A k. n A n ln(1 B n ) ln A S 2 L n (t) = mt [L n ( t kh n ) ; L n ( t (k ; 1)h n )] 2. n;2 A k ny n A k A k (1 B n ): 2. L(t). S 2 L n (t)! t 2 T, n!1, h n! h n = o(1=n).. L(t) n B n : L(t)=L c (t)l d (t) (7) L c () L d () L(). L(), S 2 L n (t) max jl n ( t kh n ) ; L n ( t (k ; 1)h n )j 1km t m t L(t) max jl n ( t kh n ) ; L n ( t (k ; 1)h n )j: 1km t 25 jl n ( t kh n ) ; L n ( t (k ; 1)h n )j

4 L n. L n n, (7), = jl n ( t kh n ) ; L n ( t (k ; 1)h n )j = [L( t kh n s) ; L( t (k ; 1)h n s)] n (s)ds max s1=n jlc ( t kh n s) ; ;L c ( t (k ; 1)h n s)j max jt 1;t 2j1=n jl c (t 1 ) ; L c (t 2 )j [L d ( t kh n s) ; L d ( t (k ; 1)h n s)] n (s)ds [L d ( t kh n s) ; L d ( t (k ; 1)h n s)] n (s)ds : n!1 L c () T.. L(), 1 jl( i )j < 1. " > n 2 N, 1 jl( i )j <": i=n L d () L d (t) =L d n (t)l d <n (t) (8) L d n () L d <n () i, n,..i n, n,..i<n,.,, h n < 1=n n, n h [L d ( t kh n s) ; L d ( t (k ; 1)h n s)] n (s)ds ;L d n ( t (k ; 1)h n s)] n (s)ds s)] n (s)ds 1 i=n jl d ( i )j 1 Z 1=n i=n jl d ( i )j n (s)ds L(t) [L d n ( t kh n s) ; [L d <n ( t kh n s) ; L d <n ( t (k ; 1)h n Z i; t;(i;1)h n i; t;ih n nh n " L(t) L d ( i ) n (s)ds nh n :, ". 1. (2) n (t) = n ( t ) f n ( n ( t kh n ))[L n ( t (k 1)h n ) ; L n ( t kh n )]:, Y (u) (4), j n (t) ; Y (L n (t))j = n( t ) ;L n ( t kh n )] ; x ; m Z Ln(t) m f n ( n ( t kh n ))[L n ( t (k 1)h n ) ; f(y (s))ds j n( t ) ; x j Z Ln( t) ; fn ( n ( t kh n )) ; f( n ( t kh n )) [L n ( t (k 1)h n ) ; 26 f(y (s))ds

5 ;L n ( t kh n )] ;L n ( t kh n )] m ; f(n ( t kh n )) ; f(y (L n ( t kh n )) [L n ( t (k 1)h n ) ; m Z Ln( t(k1)h n) [f(y (L n ( t kh n ))) ; f(y (s))] ds = L n( tkh n) = I 1 (t)i 2 (t)i 3 (t)i 4 (t)i 5 (t): M = max x2r jf(x)j, M 1 =max x2r jf (x)j. f 2 C 1 B(R), MM 1 < 1. I 2 (t) MjL n ( t )j:, L, L n f n, : I 3 (t) m ; fn ( n ( t kh n ));f( n ( t kh n )) [L n ( t (k 1)h n );L n ( t kh n )] I 4 (t) = ;L n ( t kh n )] M 1 m ; f(n ( t kh n )) ; f(y (L n ( t kh n ))) [L n ( t (k 1)h n ) ; M 1 L(t)=n: j n ( t kh n ) ; Y (L n ( t kh n ))jjl n ( t (k 1)h n ) ; L n ( t kh n )j: (4) m MM 1 I 5 (t) = Z Ln( t(k1)h n) L n( tkh n) Z Ln( t(k1)h n) L n( tkh n) jl n ( t kh n ) ; sjds = MM 1 2, [f(y (L n ( t kh n ))) ; f(y (s))]ds [L n ( t (k 1)h n ) ; L n ( t kh n )] 2 : j n (t) ; Y (L n (t))j j n ( t ) ; x j MjL n ( t )j M 1 L(t)=n M 1 j n ( t kh n ) ; Y (L n ( t kh n ))jjl n ( t (k 1)h n ) ; L n ( t kh n )j MM 1 2 [L n ( t (k 1)h n ) ; L n ( t kh n )] 2 : 1, j n (t) ; Y (L n (t))j j n ( t ) ; x j MjL n ( t )j M 1 L(t)=n MM 1 2 [L n ( t (k 1)h n ) ; L n ( t kh n )] 2 exp ; M 1 L(t) : n! 1, h n!, h n = o(1=n), (5) (t) =x f((s)) dl c (s) it f(( i ; ))L d ( i ),, i, L d ( i ) L(t). 27

6 : j n (t) ; (t)j = n( t ) ; x ;L n ( t kh n )] ; j n ( t ) ; x j m f n ( n ( t kh n ))[L n ( t (k 1)h n ) ; f((s))dl c (s) ; f(( i ; ))L d ( i ) it f((s)) dl c (s) kh n )))[L n ( t (k 1)h n ) ; L n ( t kh n )] kh n )))[L n ( t (k 1)h n ) ; L n ( t kh n )] m m m (f n ( n ( t kh n )) ; f( n ( t (f( n ( t kh n )) ; f(( t f(( t kh n ))[L c n( t (k 1)h n ) ; L c n( t kh n )] ; f(( t kh n ))[L c ( t (k 1)h n ) ; L c ( t kh n )] m m f(( t kh n ))[L c ( t (k 1)h n ) ; L c ( t kh n )] ; f((s)) dl c (s) t f(( t kh n ))[L d n( t (k 1)h n ) ; L d n( t kh n )] ; f(( i ; ))L d ( i ) = it = I 1 (t)i 2 (t)i 3 (t)i 4 (t)i 5 (t)i 6 (t)i 7 (t): f 2 C 1 B(R) L(t), I 2 (t) M f n L n, I 3 (t) (M 1 =n) jl n ( t kh n ) ; L n ( t (k 1)h n )jm 1 L(t)=n:, f 2 C 1 B(R), m I 4 (t) M 1 j n ( t kh n ) ; ( t kh n )jjl n ( t (k 1)h n ) ; L n ( t kh n )j: L c (t). t2[ h n) I 5 (t),, (t): = m t I 5 (t) = m ; f(( t kh n ))[L c n( t (k 1)h n ) ; L c ( t (k 1)h n )] ; f(( t kh n ))[L c n( t kh n ) ; L c ( t kh n )] = f(( t (k ; 1)h n ))[L c n( t kh n ) ; L c ( t kh n )] ; ;L c ( t kh n )] m = f(( t kh n ))[L c n( t kh n ) ; [f(( t (k ; 1)h n )) ; f(( t kh n ))][L c n( t kh n ) ; L c ( t kh n )] f(( t (m t ; 1)h n ))(L c n( t m t h n ) ; L c ( t m t h n )) ; 28

7 ;f(( t ))(L c n( t ) ; L c ( t )) M 1 ;( t kh n )j M M 1 (t) M 1 M L(t) m t2[ tkh n t(k1)h n] max t2[ tkh n tkh n1=n] L c (t)m t2[t 1 t 2] L c (t)2m max t2[t 1 t 2] L c (t)2m M 1 m M 1 I 6 (t)= ; max t f ; (bs(s)) ; f ; (s) dlc (s) t2[ tkh n t(k1)h n] (t) L c (t)j( t (k ; 1)h n ) ; L c (t) t2[ tm th n tm th n1=n] max L c (t) t2[t 1 t 2] max L c (t): t2[t 1 t 2] t2[ tkh n t(k1)h n] L c (t) (t) MM 1 L(t) t2[t 1 t 2] L c (t) max t2[t 1 t 2] L c (t) bs(s) = t kh n, s 2 [ t kh n t (k 1)h n ). I 7 (t). (8), I 7 (t) m f(( t kh n ))[L d <n n ( t kh n ) ; L d <n n ( t (k 1)h n )] ; ; it f(( i ; ))L d <n ( i ) ;L d n m f(( t kh n ))[L d n n ( t kh n ) ; n ( t (k 1)h n )] ; f(( i ; ))L d n ( i ) = I 7(t)I 1 7(t): 2 it L d <n (t) n ; 1 T, k i, i 2 [ t k i h n t (k i 1)h n ],, h n < 1=2 min j i1 ; i j, R 1in ;1 k i 6= k j i 6= j.,, 1=n = o(h n ), i;(ki;1)hn;t n (s)ds =1.O i;(k i1)h n; t, L n : Z i;(k i;1)h n; t I7(t)= 1 f(( t (k i ; 1)h n )) n (s)ds i;k ih n; t it f(( t k i h n )) max jl d ( i )j 1in ;1 Z i;k ih n; t i;(k i1)h n; t n (s)ds ; f(( i ; )) n ;1 Z i;(k i;1)h n; t L d <n ( i ) i;k ih n; t n (s)ds(f(( t (k i ; 1)h n )) ; ;f(( t k i h n ))) f(( t k i h n )) ; f(( i ; )) L(t) jf(( t (k i ; 1)h n ) ; n ;1 ;f(( t k i h n ))j jf(( t k i h n )) ; f(( i ; ))j M 1 L(t) n ;1 j( t (k i ; 1)h n ) ; ( t k i h n )j 29 j( t k i h n ) ; ( i ; )j

8 MM 1 L(t) t2[ t(k i;1)h n tk ih n] MM 1 L(t) n ;1 L(t) t2[ i;2h n i] L c (t)" t2[ tk ih n i) : L(t), [ i ; 2h n i ) [ t (k i ; 1)h n i ) L(t) L c (t) ", L(t) L c (t)"., [ i;2h n i) I 2 7(t) = m [ i;2h n i] j n (t) ; (t)j j n ( t ) ; x j M f(( t kh n ))[L d n ( t kh n ) ; L d n ( t (k 1)h n )] ; ; f(( i ; ))L d n ( i ) 2M": it ;( t kh n )jjl n ( t kh n ) ; L n ( t (k 1)h n )j M 1 2M max L c (t)mm 1 t2[t 1 t 2] MM 1 L(t) L c (t)m 1 L(t)=n M 1 j n ( t kh n ) ; t2[ h n) t2[ i;2h n i] max max L c (t)m L(t) t2[t 1 t 2] L c (t) L(t) t2[t 1 t 2] L c (t)" 2M": 1, j n (t) ; (t)j j n ( t ) ; x j M L c (t)m 1 L(t)=n t2[ h n) MM 1 M 1 max L c (t)m L(t)2M t2[t 1 t 2] max max L c (t) L(t)MM 1 L(t) t2[t 1 t 2] 2M" exp ; M 1 L(t) : t2[t 1 t 2] L c (t) t2[ i;2h n i] L c (t)" n!1, h n!, 1=n = o(h n ), "!, L c (t) T,,, j n (t) ; (t)j! t 2 T. 3. (5) : x(t)=x t f(x()) dl c () it S(x( i ; ) L( i )) S(x u) ='(1 x u) ; '( x u), '(t x u) '(t x u) =x u Z [ t) 3 f('(s x u)) d(s)

9 ( 1 s> (s) = s : 1 3, (2) x(t)=x f(x()) dl c () S(x( i ; ) L( i )) (9) t it S(x u) = '(1 x u) ; '( x u) '(t x u) -. \ " n, - (3),, (9), - '(t x u)., n - (2), [2]{[4].,, L(t), (3), -, " " n (9). 1...,... M. {.:, { Antosik., Ligeza J. roducts of measures and functions of nite iations // Generalized functions and operational calculus: roc. Conf. { Varna, { Soa, {. 2{ Ligeza J. On generalized solutions of some dierential non-linear equations of order n // Ann. ol. math. { {V.31.{ 2.{. 115{ Ligeza J. The existence and uniqueness of some systems of non-linear dierential equations // Cas. pestov. math. { { V. 12. { 1. {. 218{ //. { :.. -, {. 63{ Das.C., Sharma R.R. Existence and stability of measure dierential equations // Czech. Math. J. { { V. 22. { 1. {. 145{ andit S.G., Deo S.G. Dierential systems involving impulses // Lect. Notes Math. { { V { 12 p. 8. Kurzweil J. Generalized ordinary dierential equations // Czech. Math. J. { { V. 8. { 1. {. 36{ ,..,.. //... { {. 43. { 2. {. 272{ ,.. //... { 21. {. 42. { 1. {. 87{

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