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# Continuous dependence of the solution of a system of differential equations with impulses on the initial condition

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ROCZN1KI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria 1: PRACE MATEMATYCZNE XXX (1990)

is h l ie v

a i n o v

(2)

(3)

x a m p l e

h e o r e m

i = 1

(4)

(5)

(6)

t i

t j i

i p ^ i +

(7)

t

t

tp

t p

(8)

t

t i < t 0

4 i < t 0

X i ^ r i i < t

x i

m < t

V i < X i < t 0

### + Z 1МЖ))-*,-(у(ъ))|| + Z I |^i Л (Ti)) ~ h (x (*/;))| I » t eQ.

X i ^ t j i < t V i < X i < t

(9)

M ( r i 1

(10)

& { - e \ =

\ \ *о- у о\\>

### Let £ be an arbitrary positive constant. Then from (35)— (38) for i = p + l and from condition (C) it follows that there exists a positive constant ô p = ô p (e) such that ||x(0" + O) — y(0" + O)|| < ôp implies

Q'p + \ — 0 p + i

### < e>

\\z p + i ( 6 p + i ) ~ z p + i ( 6 p + i ) \ \

(11)

p

h e o r e m

(12)

(13)

V i , r i i < t

### + Z ||7;Д*(^))-^(у(^))|| + К|1*М-у(^)11^

r j i < V i < t 0

## + Z И/л(у(тД-Ыу(ъ))И+ Z ||^(*(тЛ-^(*(^))11’ teQ’

### u r n - m u « + 0 ) 1 ,

(14)

Ы в \ ) - у ( в д \ \ ^

\ \ х ( в "

у { в "

### E4. For x e D and each i = 1, 2 ,... there exists a unit n-vector yf(x) and a function a£: D->R such that

6 i < « ; ( * ) <

Q i ( z ) ,

h e o r e m

(15)

h e o r e m

h e o r e m

t

h s + i ( X S

### and p < oo. Assume that the integral curve (t , y(t)) of the system with impulses

— Commentationes Math. 30.1

(16)

(17)

### [14] —, On one case of continuous dependence of the solutions o f differential equations on a parameter, ibid. 14(3) (1962), 289-298 (in Russian).

HIGHER INSTITUTE OF CHEMICAL TECHNOLOGY, SOFIA, BULGARIA PLOVDIV UNIVERSITY “PAISSII HILENDARSKI”

PLOVDIV, BULGARIA

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