Difference methods for impulsive differential-functional equations

APPLIED NUMERICAL MATHEMATICS ELSEVIER

Applied NumericalMathematics16 (1995) 401-416

Difference methods for impulsive differential-functional equations D r u m i D. B a i n o v a,., Z d z i s l a w K a m o n t b, E m i l M i n c h e v c a Southwestern University, Blagoevgrad, Bulgaria b University of Gdansk, Gdansk, Poland c Sofia University, Sofia, Bulgaria

Abstract

We consider a class of difference methods for initial value problems for first-order impulsive partial differentialfunctional equations. We give sufficient conditions for the convergence of a sequence of approximate solutions under the assumptions that the right-hand sides satisfy the nonlinear estimates of Perron type with respect to the functional argument. The proof of the stability of difference methods is based on a general theorem on the error estimate of approximate solutions for difference-functional equations of Volterra type with an unknown function of several variables.

1. I n t r o d u c t i o n

Many real processes and p h e n o m e n a studied in biology, mechanics and population dynamics are characterized by the fact that at certain moments of their development the system parameters undergo rapid changes. A natural tool for mathematical simulation of such processes and p h e n o m e n a is the theory of impulsive differential equations. The mathematical investigations of this theory mark their beginning with the work of V. Mil'man and A. Myshkis [14]. At first the theory of impulsive differential equations developed slowly, due to some difficulties of technical and theoretical character related to the presence of some characteristic peculiarities such as " b e a t i n g " , "dying", " m e r g i n g " , noncontinuability of the solutions, loss of the property of autonomy, etc. Recently, however, a considerable increase in the number of publications is observed in various branches of this theory [2-4,20]. The first works on impulsive partial differential equations were published after 1991 [1,6]. These are devoted to the qualitative theory of impulsive partial differential equations. It is

* Corresponding author. 0168-9274/95/$09.50 © 1995 ElsevierScience B.V. All rightsreserved SSDI 0168-9274(95)00006-2

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D.D. Bainov et al. /Applied Numerical Mathematics 16 (1995) 401-416

necessary to investigate the numerical methods for this new theory in order to use these works for mathematical simulation. In recent years a n u m b e r of papers concerned with difference methods for first-order partial differential equations [7,8,10,11,16,21] and for differential-functional equations [5,9,17,18,22] were published. A m e t h o d of difference inequalities and simple theorems on recurrent inequalities are used in the investigation of stability. Brandi et al. [5], Kamont and Przadka [8], Kowalski [10,11], Plis [16] and Przadka [17,18] have assumed that given functions have partial derivatives with respect to all arguments with the exception of (x, y). In [9] Kamont and Przadka initiated investigations of difference methods for partial equations with nonlinear estimates of Perron type with respect to the functional argument. In this paper we extend the results of [8,10,17,18] on the case of differential-functional equations with impulses. We consider a general class of difference schemes. The basic tool in the investigation of stability is a t h e o r e m on the error estimate for difference-functional equations of Volterra type. In this paper we use general ideas for finite difference approximations which were introduced in [12,13,15,19].

2. Preliminary notes For any two metric spaces X and Y we denote by C(X, Y) the class of all continuous functions from X into Y. We use the symbol E to denote the H a a r pyramid

i= 1 , . . . , n } , where a > 0, M/>1 0 and b i - M i a > 0 for i = 1 , . . . , n . Write b = ( b l , . . . , b , , ) , M = ( M 1 , . . . , M n) E = {(x, y) = (x, Yl , ' ' ' , Y n ) ~ ~l+n: X ~ [0, a], [Yi[ ~ b i - M i x ,

and E 0 = [ - r 0 , 0] × I - b , b] where r 0 ~ ~+, ~ + = [0, +oo). Suppose that 0 < a I < a 2 < a k < a are given numbers and a 0 --- 0, ak+ 1 = a. We define

--" <

E(i)={(x,y)~E:aiR such that (i) the functions z le,~, i = O, 1,..., k, and z lE° are continuous, (ii) for each i, 0 <~i <<.k, (ai, y) ~ E , there exists lim z(t, ~ ) = z ( a ~ , y), (t, ~) ---~(ai,Y) t>a i

(iii) for each i, 1 <~i <~k + 1, lim (t, ~)--*(ai,Y) t
(ai,

y) ~ E , there exists

z(t, ~ ) = z ( a ; ,

y),

(iv) z(ai, y ) = z ( a +, y) for i = 0 , 1 , . . . , k , y E [ - b + M a i , for y ~ [ - b + Ma, b - Ma]. For

z ~ Cimp[E 0 U E , ~] w e

define

Az(ai,

y ) = z ( a i , y ) -- z ( a T ,

[ - b + Mai, b - Mai]. Let Eimp = {(x, Y): x ~ { a l .... ,ak}, y ~ [ - b + M x , • = (E\Eim,)XCimp[EoUE,

~] X ~ n,

b - M a i ] , and z(a, y ) = z ( a - , y) y),

i = 1 , . . . , k,

b-Mx]},

Zimp=EimpXfimp[EoUE,

~].

y

D.D. Bainov et al. /Applied Numerical Mathematics 16 (1995) 401-416

403

Suppose that f : Z - - > ~,

q~: E0-~ ~,

g " ~imp ---->~ ,

are given functions. We take into consideration the Cauchy problem with impulses

Dxz(X, y ) = f ( x ,

y, z ( ' ) , DyZ(X, y))

z ( x , y ) = q ~ ( x , y) Az(x, y)=g(x,

y, z ( ' ) )

for (x, y) ~ E\Eimp, for (x, y) s E0, for (x, y) ~ Eimp,

(1)

where D y Z = (Dy Z , . . . , Dyz). We consider classical solutions of (1). A function 2, : E 0 U E --> is a solution of (it) if 2" ~ Cirnp[E 0 U E, ~], there exist derivatives Dx2(X, y) a n d D y Z ( X , y) for (x, y) ~ E \ E i m p and 2 satisfies (1). For each x ~ [0, a] we define

E[x]={(t,~)~EoUE:t<~x

},

E(-)[x]={(t,~)~EoUE:t
T h e function f : ~ ~ [~ is called to satisfy the Volterra condition if for each (x, y, z, q) E Z, . we have f ( x , y, z, q ) = f ( x , y, ~, q). T h e function I E[x~ g " "~imp ---> []~ is said to satisfy the condition V (-) if for each (x, y, z) E Zimp, 2 ~ Cimp[E0 U E, [~] such that ztE~-~x I ='~lE'-'t.xj we have g(x, y, z ) = g ( x , y, 2). For z ~ Cimp[E 0 g E, E] we define

2 ~ C i m p [ E o U E , ~] such that z I E[x I = 2

I [ z l l ~ = s u p { I z ( t , ~:)l: (t, ~ : ) ~ E [ x ] } . We will consider problem (1) with f and g satisfying the Volterra condition and the condition V (-) respectively.

3. Main results

3.1. Approximate solutions of difference-functional equations We will d e n o t e by b~(X, Y) the class of all functions defined on X and taking values in Y, where X and Y are arbitrary sets. We define a mesh in E o U E . Let d = ( d 0, dl . . . . , d , ) ~ l+n and d i > O for i = 0 , 1 , . . . , n . Suppose that for h = (ho, h i , . . . , h n) ~ (0, d] there exist M 0, N 0, N = ( N 1. . . . , Am), where M0, N/, i = 0, 1 , . . . , n, are natural n u m b e r s and Moh o = ~o, Noho = a, Nt.h i = b i, i = 1,..., n. We denote by I a the set of all h ~ (0, d] having the above property. In the r e m a i n d e r of this p a p e r we adopt additional assumptions concerning h and I d. Let Z be the set of integers. For m = ( m 0, m l , . . . , m n) ~ 2~1+" we d e n o t e m' = ( m l , . . . , m n ) and

X~omO)= moho '

y~m,) = mihi '

y(m') = ( y]m,) .... , y~m,)).

We write

Eo.h = {(x(mo), y(m')): _ M 0 ~< mo ~< 0, - N ~
{(x,m0) y,m')): O <~mo <~Uo, l y[m,)l <~b i _

Mix(mo), i = 1 . . . . . n},

404

D.D. Bainov et al. /Applied Numerical Mathematics 16 (1995) 401-416 Xo. h = {x(i): i = - M o , - M 0 + 1 . . . . . 0},

X h = {xU): i = 0, 1 , . . . , No} E [ h , i] = {(x (m°), y(m')) ~Eo. h U Eh: mo <~i}, X[h,i]

= { x O ) ~ X o . h UXh: j<~i},

where i = 0, 1 , . . . , N o. For z " Eo. h U E h --+ • we write z(m) = z ( x(mo) y(m')).

Let Zh =

{(x(m0) y(m'))

~ E h .(x(mo+l) y(m')) ~ E h } '

X 'h = {x u)" i = O, 1 , . . . , N o - 1}. Suppose that

Fh " Ah X g-( Eo.h U Eh, R) ~ ~, are given functions. Fh(X(,,,o), y(m'), Z). The (x(mo), y(m')) ~ A h , z,2 Fh[m, ~']. We take into z(m¢'+l'm')=Fh[m,

¢h" E0.h--+~,

h ~ Id,

For (x (mo), y(m'), z ) ~ A h × 9 - ( E o . h U E h ' ~) we write Fh[m, z ] = function F h is said to satisfy the Volterra condition if for each ~ Je-(E o• h U E h, R) such that z I E [ h , m o ] = ~"[ E [ h , m o j . we have Fh[m, z] = consideration the difference-functional problem Z],

Z(m)=~(hm)

on Eo. h.

(2)

If F h satisfies the Volterra condition, then there exists exactly one solution v h • Eo. h O E h --+ of (2). We prove a t h e o r e m on the estimate of the difference between the exact and approximate solution of (2). We will denote by V h " ~ ( E o . h U Eh, ~ ) --+ 9 - ( g o . h U Xh, [~+) the operator given by ( V h Z )( X `i,) =

sup{ I z(i' m') I : I y(mj)

t ~ bj - M i x `i,, j = 1 , . . . , n} ,

i = - M o, - M o + 1 . . . . , N O. Suppose that o h • X~, × Y(X0. h U X h, ~+) --+ R+. If (x (i), rl) ~X'h × J ( X o . h U X h, R+), then we write oh[i, "O] = oh(x {°, rl). The function oh is said to satisfy the Volterra condition if for each i, 0 ~
oh[i, ~7] =oh[i, ~]. For z ~ ,9-(Eo. h U Eh, ~), - M o ~< i ~
II z II h,/= sup{I Z(J'm')l: (X (j), y(m')) ~ E[h, i]}. We denote by I1" II the n o r m in ~n and write 17(i) = ~(x(i)), - - M 0 ~ i <~N o.

Ihl = h 0 + h l +

" ' " + h n. For ~ 7 " X o . h U X h ~

we

Theorem 1. Suppose that (1) the function F h • A h × 3 - ( E o . h U Eh, ~ ) --+ ~, h ~ Id, satisfies the Volterra condition; (2) there exists oh : X~ x 3-(X0. h U X h, ~+) --+ ~+ such that (i) oh is non-decreasing with respect to the functional argument and satisfies the Volterra condition,

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D.D. Bainov et al. /Applied Numerical Mathematics 16 (1995) 401-416

(ii) for z,~ ~ ~-(Eo. h U E h, ~) we have on A h I Fh[m, z ] - - F h [ m ,

~]l~
(3) ~Oh ~ 3-( Eo.h, ~) and v h ~ 5~-(Eo. h to Eh, ~) is a solution o f (2); (4) u h ~ 9-(Eo. h tO Eh, ~) and there are 13h • Xo. h tOX h --+ ~+, Yh" X~ ~ ~+, h ~ Id, such that the initial estimate lu~"°-v~m) l <~B~m°),

(x 'm°), y t m " ) ~ E o , h ,

(3)

is satisfied and lUthm°+l'm')--Fh[m, Uh]l <~3'~m°~ for (x ¢mo), y ( m ' ) ) ~ A h ,

(4)

13(m°+l)~°'h[mo, J~h] +T(hm°), m o = 0 , 1 , . . . , N o -

(5)

1.

Under these assumptions we have ]v~hm) - U~h')[ ~
(6)

where (x tmo), y~m')) ~ Eh" Proof. It follows from (3) that (6) is satisfied for m o = O, (x m), y ~ " ) ) ~ E h. Assume that (6) holds true for 0 <~m o <~i, ( x ~m°), ytm')) ~ Eh" It follows from assumption (2) and (3)-(5) that

I v i+ l'm" - u2i+ "m"l

lu + l'm" - Fh[ i, uh] ] + I Fh[ i, Uh] -- Fh[ i, Vhl l ~(i, + Crh[i, Vh(U h _ Vh)] ~y(hi)+6rh[i, 13hi

Hence the proof is complete by induction.

~
R e m a r k 1. If the assumptions of Theorem 1 are satisfied and /3h is non-decreasing on X o . h t O X h, then IlUh--Vhllh,i<...~{hi) for i = 0 , 1 , . . . , N o. 3.2. Difference methods for nonlinear equations We denote by M[~+] the class of all function a:~+---, ~+ such that limt_+o+a(t)=0. Suppose that natural numbers No{1) < No~2) < . . - < No~k) are defined by ho g(i) < a i ~ ho(g(o i) + 1), i = 1. . . . , k. Let

Eimp= {(X(mo), y,m')), h

mo ~ {N(o,,, ' ' ' ,N~ok)}, y ( m ' ) ~ [ - b + M x `m°), b - M x

(x(m0+l)

`m°)] ,

y(m'))~Eh} '

S~--{S=(S1 . . . . ,Sn)" S i ~ { - 1 , 0, 1}, i = 1 , . . . , n } . Let z" Eo. h tO E h ~ ffL We define difference operators a0, 6 = (61,..., 6 n) by aoz~m~=hffl[z(m°+l'm')-Az~m~],

A z ~m~= Y'. a , z (m°'"'+=),

(7)

s~S 6i Z(m) = h71 Y'~ bs.i Z(m°'m'+s), s~S

i = 1 . . . . , n,

(8)

D.D. Bainovet al./Appfied NumericalMathematics16 (1995)401-416

406

where a s, bs. i E [~ and 6 z (m) = (61z(m),..., (~nZ(m)). Let

~h=(gh\gihmP) X g - ( g o . h U E h , ~ ) x ~ n ,

~ihmP=E~mPXSr(Eo.hUEh,~ )

and suppose that for each h e I,~ we have fh : 2h ---' R, gh" £]mp ___,R, q~h" E0. h ~ R" We will approximate solutions of (1) by means of solutions of problem

~OZ(m)=fh(X(mo), y(m'), Z('), ~Z (m))

for

(x (m°), y(m'))~Ah\E~mp ,

Z (m) = ~O(h m) Z(mo+l,m')=z(m) +gh(X(mo), y(m'), Z ( ' ) )

for

(x (m°), y(m'))

for

(X (m°), y(m')) ''- L, K'imp h .

(9)

~ Eo.h '

If~h and gh satisfy the Volterra condition, then there exists a unique solution u h : Eo. h U E h ~ of (9).

Assumption HI.

Suppose that (1) the operators A and 6 satisfy the conditions: ~] as=

~_, sja s = O,

s~S

Ebs.i--O,

1,

sES

i=1 .... ,n,

s~S Y'~ sjbs.i = ~ij' s~S

(10)

i, j = 1 . . . . , n ,

where ~ij is the Kronecker symbol; (2) for h ~ I d we have h i <~Miho, i - - 1 , . . . , n , i,j=l,...,n.

and there is c o > 0 such that h i h 7 a ~
R e m a r k 2. Suppose that Assumption H 1 is satisfied and (i) u ~ Cimp[E 0 U E, R] and uleu~, i = O, 1 , . . . , k, are of class C 2, (ii) the second partial derivatives of u are b o u n d e d on E (°) U • • • U E (~'). T h e n there exists C > 0 such that

16ou(m)-OxU("Ol<~flh[

,

116u(m)-Oyu(m)ll<~flh[,

for ( x ~mo), y(m')) E E (0) U • • • U E (k). Assumption H z.

Suppose that the functions

O'h :(Xh\X~mp) X3U(go. h U g h , ~+)"~ ~+,

O'h :g~mPX "~(gO.h U g h , ~+)-") ~+

satisfy the conditions: (1) o-h and 6 h are non-decreasing with respect to the functional argument and fulfill the Volterra conditions; (2) Oh[i, Oh] = 0 for i ~ {0, 1 , . . . , N O- 1} \ { N o(1). . . . , N(o k)} and O'h[i, Oh] = 0 for i E {N(ol),..., N(o k)} where O
~7(i+l)=~7(i)+hoO'h[i, "q]

i~{0,

-0(°=0

for i ~ { - M 0 , . . . , 0

T](i+l) ~- Tl(i) W O'h[i, "17]

for i ~ { N(ol),. . . , N(o~)},

1,...,N

0-

1}\{N(ol),...,N(ok)}, },

(11)

D.D. Bainov et al. /Applied NumericalMathematics 16 (1995) 401-416

407

is stable in the following sense: if r/h is a solution of the problem

9~(i+1) = g~(i) + hoO'h[ i, rl] + hot ( I h

n (i)=ao(Ih I)

I)

for i ~ { 0 , 1 .... , N O-

1}\{N(ol),...,N(ok)},

for i ~ { - M o . . . . ,0},

gl(i+l)='rl(i)+O'h[i, g/]

+9(Ihl)

for i ~ {No(i),..., N(o~'}, (12)

where y,~/,ao~M[~+], 1 , . . . , N o.

then there exists /3~M[I~+] such that ~7~°<~[3(Ihl), i = 0 ,

~' imp ~ ~ satisfy the Volterra Assumption H 3. Suppose that functions f h ' ~ h ~ ~ and gh:'~h condition and (1) for each P = (x, y, z, q) ~ 2h there exist derivatives ( D < f h ( P ) , . . . , Dq, fh(P)) = Dqfh(P) and Dqfh(x, y, z, • ) ~ C(~ n, [~"); (2) for each s ~ S we have n

as +ho Y'.h[lb~.iDqJh(x, y, z, q) >~O, (x, y, z, q ) ~ . S h ; (13) i=1 (3) there are % : ( X h \ X ~ m:') × 3-(Xo.h tOXh, E+) ~ E+ and ~h :X~ mp × 3-(Xo.h t'OXh, ~+) ~+ such that

I fh(x(i), y(m'), Z, q ) - - f h ( X (i), y(m'), ~, q)l ~O'h[i, Vh(Z--~.)]~, (X
Igh(x
(X ~i), y(m'))~E~mp,

where z,Y. ~ 9-( Eo. h t.-JEh, ~); (4) the functions o-h and ~7h satisfy Assumption H 2. Now we prove a t h e o r e m on the convergence of the m e t h o d (9). Theorem 2. Suppose that Assumptions H 1 - H 3 are satisfied and (1) f ~ C ( 2 , ~), g ~ C(-Yimp, ~) fulfill the Volterra condition and the condition V ~-) respec-

tively, ~o ~ C( Eo, ~), (2) u~Cirnp[EoUE, ~] is a solution of (1), u is of class C 2 on E \ E i m p and the partial

derivatives of the first and second order of u are bounded on E \Eimp; Vh " Eo.h tO E h ~ ~ is a solution of (9) and there exists a o ~ M[~+] such that I q~
[fh(X(mo) y,m'),

Uh(. ), ~Su~m)) __f ( x,mo), y,m'), U(" ), 6u
( x(m°)' Y(m')) ~-A~''h\~h \ ~Timp,

(14)

gh(x(mo), yO"), Uh(.)) _g(amo ' yfm'), U(')) ~
(X
(15)

D.D. Bainov et aL /Applied Numerical Mathematics 16 (1995) 401-416

408

Then there exists ~/ : I a ~ ~+ such that

luLm)-v(hm)[<. ~/(h) on Eh, Proof.

lim3;(h)=0. h-*O We apply Theorem 1 to prove (16). Let /~h be defined by

~oh(hm)=fh(X(mo), y(m'), Uh(.), ~U(hm)) + ~(hm)

on

(16)

Z h \ E ~ mp

and U (m°+l'm') = u(m)+gh(x(m°) ,

y(m'),Uh('))+l~(rn)

g7 imp on --h

It follows from Assumption H 1 and from the consistency conditions (14) and (15) that there exists y ~M[N+] such that [ iOh(m)[ ~<,/(Ihl) on A h. We define

Fh[m , z] =Az(m)+h OJhl, ¢ [x (mO), y(m'), Z('), ~Z (m)) f o r (x (m°), y ( m ' ) ) E A h \ E himp , Fh[m, z] =Z(m)+gh(x(m°), y(m'), Z(')) for (x (m°), y(m'))~E~mP. Then v h satisfies (2) and

lu(hm°+l'm') - Fh[m, uh]l <~hoT(Ihl ) lU(hm°+l'm')--Fh[m, Uh][<<.3'(Ihl)

f o r (x (m°), y(m')) ~ Z h \ E h i m p ,

for (x (m°), y(m'))~Effn°.

We have also the initial estimate

]U(hm)--v(hm)l
where Q = ( x (m°), y(m'), Z("), ~(rn) + ~(~z(m) __ ~ ( m ) ) ) , ~ ~ (0, 1). It follows from (10) and from \ E himp Assumption n I that we have on 2qth\

I Fh[m, z] -- Fh[m, z.]l <~Vh(Z - ~r)(m°) + hoO'h[mo, Vh(Z -- Z')]. If

(X (m°), y ( " ) )

it?imp, ~ "~h

then we have for the above z and ~,

I Fh[m, z] --Fh[m, ~] I <'~Vh( Z -- 7. ) (m°) + ~rh[m o, Vh(Z -- 2)]

(x(mo) y(m')) = lTimp

Consider the problem

~7(i+~)=n(°+ho~rh[i, "O] + h 0 T ( i h [ )

for i ~ { 0 , 1 .... , N o - 1}\{No(1). . . . . No(k)},

77(i) = O~0( I h I)

for i ~ { - M o , . . . , 0 } ,

~(i+ l) = Tl(i) + O'h[ i , 'rl] + r ( l h l )

for i ~ {3/o(1),..., N(ok)}.

(17)

D.D. Bainov et al./Applied NumericalMathematics 16 (1995) 401-416

409

Denote by ~lh" Xo.h UXh --) ~+ the solution of (17). It follows from T h e o r e m 1 that

lU(hm) -- U(hm)l ~ ~(hm°) on E h. Now we obtain (16) for ~(h) Example 1.

=

~-Ih(a).

[]

Suppose that there are Lo,L ~ ~+ such that for z,~ ~ J ( Eo. h U Eh, ~) we have

fh(x(i), y(m'), Z, q ) - - f h ( X (i), y(m'), ~, q) ~Zllz--~llh,i, where (x (i), y(m')) ~ A h \E~rnp and

gh(x(i), y(m'), Z ) - - g h ( X (i), y(m'), ~.) <~toIIZ--ellh,i,

(X(i), y(m'))~ ELmP.

The problem (17) is equivalent to v(i+I)=v(~)(1 + L h o ) + h o r ( I h l ) ,

i ~ { 0 , 1 . . . . , N o - 1}\{No(1),...,NO(k)},

rl(/+a) = r/(i)(1 + Lo) ,

i ~ {N(X),...,NO(k)},

(18)

n (°) = So( I h I).

Write No(°) = 0, No(k+~) = N o. Then the solution r/h of (18) has the form

i-1 n(h0=ao(Ihl)(l+Lho) ~+hor(Ihl)

E(l+Lho)

~,

i = 0 , 1 . . . . ,No(1),

"r=0

i-2 T/(N°(/)+/) = ( 1 + to)~7(N°~J))(1 +Lho) i-1 + h o r ( I h I) E (1 +Lho) ~, "r=0

i = 1, 2,

• .o~V

~r(J+~)_ ~,f(g) j = 1,. 0 ~t'r 0 ~

k,

..~

where E~=o -1 = 0. For L > 0 we define

Yo(h) = a o ( I h l ) e La~ + y ( I h l ) Z - l [ e

L ~ ' - 1],

Y/(h)=(l+Lo)Yi_leL(~+'-~')+y(lhl)L-l[eL(a'+'-~)-l],

Yk( h ) = (1 q- L o ) Y k_ 1eL(a-aD q- 3'( I h I)t-l[eL(a-aD -- 1]. If L = 0, then we put

Yo(h) = a o ( I h I) + ~/(I h I)al, Y~(h) = (1 + Lo)Y~_ 1 + y ( I h l ) ( a i + 1 - ai),

i = 1 .... , k - 1,

Y k ( h ) = ( 1 +Lo)Y~_ 1 + r ( I h l ) ( a - a ~ ) . We have the estimates

I]u(i,'n')-v(i,m')llh,i <~Y~(h), and limh _~oYk(h) = O.

(x (i), y(m'))~Eh,

i= l,...,k-1,

D.D. Bainovet aL/AppliedNumericalMathematics16 (1995)401-416

410 R e m a r k 3.

Suppose that 6 o and t~ = ( 6 1 , . . . , 6 n) are defined by

~°z(m)=h°l[z(m°+l'm')-(2n)-l

~" (z(i(m)) + z(-i(m)))]

~iz (m) = ( 2 h i ) - l [ z (i(m)) - Z(-i(m))],

(19)

i = 1,..., n,

(20)

where i ( m ) = ( m o , m l , . . . , m i _ l , mi + l, m i + a , . . . , m , ) , mi+ 1,..., m,). T h e n condition (13) is equivalent to

1-nhoh~llDq, fh(x,y,z,q)[>O

on X h,

-i(m)=(mo,

m 1.... ,mi_ p m i - l ,

i=l,...,n.

Suppose that

Remark 4.

aoz(m) = hol( z(mo+ l, m') _ z(m)), h ? l ( z m) --2 (-i(m))) h3l(z(i(m))-z(m))

6iz(m)=

(21) for i = 1 , . . . , n o, for / = n 0 + l , . . . , n ,

(22)

where 0 ~
1 - h o ]~_,h['lDq, fh(x, y, z, i=1

Dqifh(x, y , z , q ) { ~ < O ' >0,

q)l o,

(23)

i=l,...,n0,

(24)

i----no+ l , . . . , n ,

where (x, y, z, q) ~ Xh. R e m a r k 5. It is easy to prove a t h e o r e m on the convergence of a difference m e t h o d for (1) in the case w h e n E = [0, a] x ~ , E 0 = [ - % , 0] x N". Our results can be extended for weakly coupled differential-functional systems

Dxzi(x, y ) = f i ( x , where z = ( z l , . . . , z k , )

y, z ( ' ) ,

DyZi(X, y)),

i= 1,...,k',

and

f = ( f l , " " ", fk'): (g\gimp) X Cimp[ E 0 t..JE , [~k'] X [~n _.+ uk'. T h e o r e m 2 is a generalization of the results of [8,17,18] where the authors have assumed linear estimates with respect to the functional argument.

3.3. Difference methods for almost linear differential-functional equations with impulses Suppose that

Fo:(E\gimp)>
g : .~imp --->[1~,

q): E o --* ~.

D.D. Bainov et al. /Applied NumericalMathematics 16 (1995) 401-416 In this part of the paper we consider the problem n DxZ(X, y ) = F o ( x , y , z ( ' ) ) + Y'~Fi(x, y)Dyiz(x, y) i=1

z(x, y)=

y)

for (x, y) E E \ E i m p , for (x, y) ~ E 0,

az(x, y) =g(x, y,

411

(25)

for (x, y) ~ Elmp.

Assume that

Fo.h(Eh \E~ rap) ×o~-(Eo. h t-) Eh) + ~, q~h : Eo.h -+ ~,

gh : Ehrnp X 9-(E0. h U E h ) --9 ~. Consider the difference-functional problem n (~OZ(m)=Fo.h(x(rno), y(m'), Z ( ' ) ) + E Fi(x (m°), y(m'))t~iz(m) i=l for (x (m°), y(m')) ~Ah\Ehimp,

Z (m) ~-- q~(hm) on Eo.h, Z(mo+l,m')=z(m) +gh(X(mo), y(m'), Z('))

(26)

for (x (m°), y(m')) ~ E~mp,

where 60 and ~5 are given by (21) and (22). If we apply Theorem 2 to problems (25) and (26), then we need the following assumption on F: the function sign F(x, y) = (sign Fl(x, y ) , . . . , sign Fn(x, y)), (x, y) ~ E, is constant on E (see (24)). Now we prove that this assumption can be omitted if we define

f h71[z(m)-z(-i(m))], ~iz(m)= Ihtl[z(i(m))--z(m)],

if Fi(x (m°), y ( m ' ) ) ~ 0 , if Fi(x era°', y('~") >0.

(27)

Suppose that (1) Assumption H 2 is satisfied and for h ~ I a we have hi <~Miho, i = 1,..., n; (2) Fo. h and gh fulfill the Volterra condition and

Theorem 3.

IFo.h(xti,, y(m'), z)__Fo.h(X(i), y(m'), ~.)[ <~O.h[i, Vh(Z__2) ]

(X (i), y(m')) EEh\E~rno, Igh(x(i), y(,,'), z)_gh(xti), y(m'), for

(28) Vh(z-~)],

for (x ~ 0; i=1

(29)

(30)

D.D. Bainovet al./AppliedNumericalMathematics16 (1995)401-416

412

(4) F o E C((E\Eim p) × Cimp[E o to E, ~], [~) satisfies the Volterra condition, u ~ Cimp[E o tO E, ~] is a solution off (25) and u [ e\E~m, is of class C1; (5) there exists/31, /32 ~ M[[]~+] such that

]Fo.h(x(mo~,y(m'), Uh)--Fo(x(m°) , y(m'), u ) l ~ l ( l h l )

on A h \ E ~ mp, 15-imp

where u h is the restriction of u to the set Eo. h tOEh; (6) vh : Eo. h tOE h ~ E is a solution of (26) and (27) and there exists 0% ~ M [ R + ] such that I~0(m) -- ~(m) l "~<~o(Ih l) on Eo. h. Under these assumptions there exists ~, ~ M [ ~ + ] such that

]U(h'n)--V(hm)l~~,(Ihl ) on Eh. Proof.

Let us write n

f h [ m , Z]

= Z (m) -4- hoFo.h(X (m°),

y(m'), z) + h o ~., Fi(x(m°), y(m'))SiZ("O i=1

for (x (nO), y(m')) EZh\E~mP, where 8 is given by (27) and

Fh[m, z] = z ( ' ) + 6h~, " /,,(no) .4, , y(-"), z ( ' ) )

for (x (m°),

y(m'))~E~,m°.

It follows from (28) that for (x (m°), y(m')) EAh\E~mo a n d for z,Y. ~ 3r(Eo.h tO Eh, R) we have

'Fh[m, z] --Fh[m, 211 <~ (z(rn)--z(m))[ 1-h°i~J+[m]y'~ hi-lfi(x(m°)' Y(m')) + h°i~J-[m] y'~ hT1fi(x(m°)' Y(m'))}l -4-ho

E h;1z(i(m))Fi(x(m°), y ( m ' ) ) _ E hTlz(-i(m))Fi(X(m°), y(m')) i~J+[m] i~J-[rn] +hooh[m o, Vh(Z--Z.)], (X (m°), y(m'))~Ah\E~mr' , where J + [ m ] = {i:Fi(x (m°), y(m'))> 0} and J-[m] = {1.... , n}\J+[m]. F r o m ( 3 0 ) w e see that

IFh[m, z] --Fh[m, ~]l<~ Vh(Z--2) (m°)+hoOh[mo, Vh(z--z)]

on A h \ E ~ m°.

It follows from (29) that for z,~. ~ o~-(Eo,h tO Eh, [~) we have

IFh[m, z] --Fh[m, ~.]l~Vh(Z--~)(m°) +6"h[mo, Vh(z--Y)] on E~,m°. Analysis similar to that in the proof of T h e o r e m 2 shows that our assertion holds true.

[]

Now we consider examples of nonlinear estimates for F 0. Example 2. We consider problem (25) with E = [0, a] × [ - b , b] (we assume that M i = 0 for i = 1 , . . . , n). Let X 0 = [-~'0, 0], X = [0, a], Xim p -~-{a 1. . . . , a~}. We denote by Cimp[X0 t.)X, ~] the class of all functions w : X 0 u X ~ ~ such that (i) the functions W l(as,ai+O, i = O, 1,..., k, and W lxo are continuous;

D.D. Bainou et al. /Applied NumericalMathematics16 (1995)401-416

413

(ii) for each i, 0 ~< i ~
w'(x)=o-(x,w('))

for XEF-X\Ximp,

Aw(x)=(~(x,w(.))

for x ~ X i m p ,

w(x)=o~(x)

for x ~ X o,

l

I

(31)

where o~ ~ C ( X o, ~+). We denote by V: Cimo[E 0 U E , R] ~ 3 - ( S 0 U S , R+) the operator given

by (Vz)(x)=max{Iz(x,

y)l: y ~ [ - b , b ] } ,

x~SoUS,

where z ~ Cimp[E0 U E, R]. It is easy to see that V." Cimp[ E o U E , ~] ~ Cimp[ S o U X, ~+ ]. We introduce the operator T h : 3-(Eo. h U Eh, ~) ~ 9"(E o U E, R) as follows. Let

S+={e=(eo, Suppose that

e l , . . . , e n ) : egG{O, 1} for i = 0 , 1 . . . . ,n}.

7.~9-(Eo.hYEh,[~) and

(x, y ) ~ E o U E .

There exists m ~ Z l+n such that

(x(m°),y(m')), (X (m°+l), y(m'+D)~Eo, h U E h , m ' + 1 = ( m 1 + 1,...,mn+ 1), and x ( m ° ) ~ X ~ x(mo+l), y(m') ~ y ~y(m'+l). If x ~mo) ~ X ~ rap, then we define ( g - - g(m))e( g - g(m)) l-e (ThZ)(X , y ) = ~_, Z (re+e) 1 e~S+

~

h

h

'

where

h

:

-h-o

h//

i=1

(32)

(1 g--g(m))l--e( x--x(m°)]l--e°( yiIy}mi) = 1

~o

i=lfi

]

1

hi

,

(33)

and we take 0 ° = 1 in (32) and (33). If x ~mo) ~ X~mp, then there exists i, 1 ~< i ~
((ZhZ)(g(oi)ho, y), (Thz)(x, Y)=

(Zhz)((U(oi)4-1)ho,

y)

if x~[g(oi)ho, ai), if x E [ a i , (U(oi)+ l)ho].

T h e n we have T h : 9-(Eo. h U E, ~) ---) Cimo[E o U E, ~]. Suppose that r/:Xo, h U X h ~ ~. We define L[h o, r/]: X 0 UX---) ~ in the following way. Suppose that x ~ X o U X . T h e n there exists i, - M o ~
t[ho, "O] ( x ) = "q(i+ l)[ x - x(i)] ho 1 4- ~(i)[1- ( x - x(i))hol ] .

414

D.D. Bainovet al. /Applied NumericalMathematics16 (1995)401-416

If x (i) ~X~ mp, then there exists j, 1 ~
nl(N(o 'ho), L[ho, r/](x)=

if

L[ho, r/](( No(j'+l)ho)'

x e [NoU)h0, aj),

if x ~ [aj, (N(o~' + l)ho].

Consider the difference method (26) for

Fo.h(X , y , z ) = F o ( x , y, Thz),

(x, y , Z ) ~ ( E h \ E ~ m P ) X g - ( E o , hUEh, R),

gh(x(i), y , z ) = g ( a i ,

(x(i), y, Z) E ~K'imp h X J ( E o h U Eh, ~).

y, ZhZ),

(34)

Suppose that functions o- : ( X \ X i m p) x Cimp[X0 U S , ~+] ~ [I~+ and 5 : Ximp X Cimp[X0 U X, E+] -~ R+ satisfy the conditions: (1) or and 5 are continuous and o-(x, O)= 0 for x e X \ X i m p and 5(x, O)= 0 for x ~Xim p where O(x) = 0 for x ~ X 0 UX; (2) if r/,~ ~ Cimp[So U X , ~+], X EX\Ximn~" and ~7,I [ - r 0 , x] = ~,I [ - r 0 , x l., then o-(x, r/)= o-(x, ~); (3) if r/,~ ~ Cimp[Xo UX, ~+], x EXim p and r/I . x)= 71 . x., then (7(x, r/)= ~(x, ~); (4) for (x, y, z), (x, y, ~.)~ ( E \ E i m p) X Cimp[#O b E , ~]Lw°e'have

IFo(x, y, z ) - F o ( x , y, 2)1 ~
(35)

(5) for (x, y, z), (x, y, ~') ~ Eimp X Cimp[Eo U E, ~] we have

Ig(x, y, z ) - g ( x ,

y, ~')1 ~<5(x, V ( z - ~ , ) ) ;

(36)

(6) o- and t~ are non-decreasing with respect to the functional argument. Since ~-, y _ y(m') e' __ y(m')

e'ES+

(

-h"7

)(y 1

h'

= 1,

y(m') <~y ~< y(m'+l),

where S + = {e' = ( e l , . . . , e,): e i ~ {0, 1} for i = 1 . . . . , n } , h' = ( h a , . . . , gh given by (34) satisfy the following estimates: (a) If(x, y, z), (x, y, ~.)~(En\E~mp)xJr(Eo.h UEn, R), then

IFo.h(X, Y,

z)--Fo.h(X, Y,

=lFo(x,Y, T h z ) - F o ( x ,

hn),

we have that F0. h and

Y, ThY.)l

<<.o'(x, V(ThZ - The)) <~o'(x, L[ ho, Vh(z - ~')1). (b) If (x (i), y, z), (x (i), y, z.) ~ "~h r'imp X 9"(E o h U Eh, R), then

[gh(X (i), y, z)--gh(X (i), y,

= [g(a/, y, ThZ ) --g(a i, y,

rh )]

<~6"(ai, V(ThZ - ThY.)) <~&(ai, L[ho, Vh(Z -- 2)1). Thus we see that problem (11) is equivalent with '?~(i+I)=T~(i) + hoor(x (i), L[ho, r/])

for x(i) E----Xh\X~ mp,

'17(i) = 0

for - M o ~
,rl(i+l)='rl(i)-k-~(ai, t [ h o , "r/])

for x ( i ) e x ~ mp.

D.D. Bainov et al. /Applied Numerical Mathematics 16 (1995) 401-416

415

If we assume that problem (31) with oJ(x) = 0 for x ~ [ - z 0, 0] has the right-hand maximum solution ~(x) = 0 for x ~ X 0 u X , then the above problem is stable in the sense of Assumption n 2.

Remark 6.

Suppose that oro "(XkXim p) X ~+''> ~+ and (o " X imp ~ ~+ are given functions and we define o-(x, w ) : o - 0 ( x ,

sup

w(t)),

(X, W) ~ (X\Simp) X Cimp[S 0 U S , l]~+] ,

w(t)),

(X, W) ~Xim p X Cimp[g 0 U S , ~+],

--70<~t<~x

t~(x, w)=t~o(X,

sup --~'o~
then assumptions (35) and (36) have the form

lEo(x, Y, z) - Eo(x, Y, z)l <~tro(X,

Ilz-~llx),

where (x, y, z), (x, y, ~,) ~ (E\Eimp) X Cirnp[ E 0 U E, ~], Ig(x,

y, Z)--g(x, y, ~')1 ~<~0(X, IIz--~" II x),

(X, y, Z), (X, y, ~) ~ E,mo × Cimp[E0 U E, El, and they are typical for theorems on uniqueness of solutions. Remark 7.

It is easy to see that Example 2 can be adopted for the nonlinear differential-functional problem (1). Theorem 3 can be extended for weakly coupled differential-functional systems.

Acknowledgements

The present investigation was partially supported by the Bulgarian Ministry of Science and Education under Grant MM-422.

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