Convergence of one-step difference methods for nonlinear parabolic differential-functional systems with initial boundary conditions of Dirichlet type

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXX (1991)

H

L

(Gdansk)

Convergence of one-step difference methods for nonlinear parabolic differential-functional systems with initial boundary conditions of Dirichlet type

Abstract. This paper deals with the differential-functional problem

Dxzt(x, y) = fi(x , y, z(x, y), z, D z,(x, y), D zt(x, y)), ( x , y ) e E, i = l , . . . , k , (i)

z(x, y) = cp(x, y) for (x, y ) e E (0),

where z = (zl s zk), £ = ( 0 , a ] x G c R 1+n, and E(0) = [ —t , a] x G k\ E is an initial set.

The corresponding one-step difference method is of the form

d (0m)z<m) = Ф(л°(х<то), Bz(m), z, A{m)z[m), d (2'm,z<m)) on Eh, i = 1 , Ic, (ii)

z(m) = ф(х(то), y<m)) + (x<mo), Ут,)) on E[°\

where d (0m), A<m), A(2,m) are difference operators. We give sufficient conditions for the convergence of the sequence {uA}, where uh is a solution of (ii), to a solution of (i).

We prove that if the method (ii) is stable and satisfies a consistency condition with respect to (i) then it is convergent.

Introduction. The problems of finite difference approximation to initial

boundary problems for nonlinear parabolic equations were considered by

many authors and under various assumptions: Kamont, Kwapisz ([1], [2]),

Kowalski ([3], [4]), Malec ([5]—[14]), Plis ([15]) and others. The main problem

in these investigations is to find a difference equation that satisfies some

consistency conditions with respect to the differential problem and is stable. In

[5]—[14] the stability of difference schemes is established by means of simple

recurrent inequalities. Difference methods for nonlinear parabolic differential

equations with initial boundary conditions of Dirichlet type were considered in

[1]> [4]—[14]. In [9], [10], [13], [14] Malec investigated the convergence of

difference methods for differential-functional equations by means of difference

inequalities and recurrent inequalities. In [1] parabolic differential-functional

systems with initial conditions of Dirichlet type and corresponding one-step

difference methods with some general difference operators introduced by

Przqdka ([16]) are considered. The methods and results of our paper may be

viewed as some generalization and modification of [1]. We generalize the difference operators of [1] and we consider the differential-functional problem on the cartesian product of an interval and an open set (in [1] there were only products of intervals). Notice that some uniqueness results for such sets are known (see e.g. [17], [18]). In [1] it was assumed that the right-hand side of the difference scheme was quasimonotone and nondecreasing with respect to the functional argument. Instead of this we introduce the assumption that the difference operator satisfies the Lipschitz condition.

In the first part of the paper we establish some estimates of the difference between solutions of recurrent equations. These estimates are basic tools in the investigation of the stability of difference methods.

I. Let N = { — n0, —1,0, 1, . . . , n*}, 1V(0) = { —и0, ..., 0}, N* = (0, 1 ,..., n* — 1}, where n0, n* are natural numbers. For each m0e N let Qmo, Г mo c Yo be fixed finite sets which satisfy the following conditions:

1) Гто c Qmo for m0e N , 2) Qmo Ф 0 for m0e N ,

3) Гто Ф 0 iff m0 = 1, 2, ..., n*.

Let Q = {(m0, t): m0e N , t e Q mo} and Г = {(m0, t): m0e N , t e r mo} and Г = {(m0, t): (m0 + 1, t)eT}. Suppose that f cz Q.

Denote by ^ { X , Y) the class of all functions defined on X taking values in Y (X, Y being arbitrary sets). Let F : Г x Rfc)-»Rfc. The function F is said to satisfy the Volterra condition if for all (m0, t ) e f , z, z e ^ ( Q , Rfc) such that z(m0, T) = z(m0, T) for —n0 ^ m 0 ^ m 0 and t e Q mo, we have F{m0, t, z) = F(m0, t, z).

Consider the problem

z(m0 + 1, t) = F(m0, t, z) for (m0, t ) e f (m0 = 0, ..., n — 1), z(m0, t) = co(m0, t) for (m0, t)EÜ\r,

where со: (Q\r)-+Rk is a given function. If we assume that F satisfies the Volterra condition then (1) has exactly one sqlution. Let V: ^ { Q , Rk)->- ^ { N , R+) be the operator defined by

(Vz){m0) = ((Fiz)(m„),..., (Vkz)(m0)) = ( sup I z ^/W q , t)|, ..., sup |zk(m0, 01),

teQm0 teQmo

where z = (zt , ..., zk) e ^ ( Q , Rfc), —n0 ^ m 0 ^ n * , (m0, t)eQ.

Let Q: N* x ^ ( N , R+)->R+. Q is said to satisfy the Volterra condition if for a, âe£F(N, R+) such that a(m0) = a(m0) for — n0 ^ m0 ^ m0, we have Q(m0, a) = Q(m0, a). If p = (p^ pk) e R k then we define \p\ = f l p j ,..., \pk\).

Let F1: f x J^(0, Rk) ^ R k. Consider the second problem z(m0 + l, 0 = ^ i K , t, z) for (m0, t ) e f (m = 0, ..., n-1),

z(m0, 0 = a>i(m0, t) for (m0, t) e Q \r ,

(2)

where œt : (£2\Г)->Rk is a given function. We are now able to formulate a lemma on the estimate of the difference between solutions of (1) and (2) in the form convenient for our purposes.

L emma 1. Suppose that

1) F, f x 3F(Q, Rfe)->Rfc, F satisfies the Volterra condition, со, (OVO-R* are given functions,

2) Q : N* x , R+) -> R+ is nondecreasing with respect to the functional argument and satisfies the Volterra condition.

3) IF(m0, t, z) — F(m0, t, z)\ ^ Q(m0, V{z-z))for (m0, t ) e f , z, z e ^ ( Q , Rfc), 4) u,ve3F(Q, Rfc) are solutions of (1) and (2), respectively,

5) there is a function ft: N-+ Rk such that

(3) M m 0, 01 < P{m0) far {m0, t) e Q \r ,

(4) /i(m0 + 1) ^ Q(m0, Р) + У(щ) far m0eN*, where y: N*-+Rk+, (5) \F(m0, t, v ) - F 1{m0, t, i>)| ^ y{m0) for (m0, t ) e f .

Then

(6) |w(m0, t)-v (m Q, 01 < P(m0) for (m0, t)eQ.

P ro o f. The proof is by induction on m0e N . For — n0 < m0 0 condition (6) is equivalent to (3). Let m0eN*. For (m0, t ) e f we have

Iu(m0 + 1, 0 ~ v(m0 + 1, 01 < \F(m0, t, u) - F(m0, t, u)|

+ \F{m0, t, v) — F 1(m0, t, t>)|.

We conclude from assumption 3) and conditions (4), (5) that

|w(m0 + l, t)-v{m 0 + l, 01 < Q(m0, V{u-v)) + y{m0)

< 0K>> P) + y(m0) ^ P(m0 + 1).

This and (3) complete the proof.

R em ark 1. Suppose that assumptions 1), 4) of Lemma 1 hold and IF(m0, t, z) — F(m0, t, z)| < Q(m0, \ z - z \ mo),

where

\z~ z\mo = ( max (Fi(z-z))(m 0), ..., max (Vk{z-z))(rh0)),

-n o ^ m o ^ m o -n o ^ m o ^ m o

Q : N* x Rfe+ -» R + is nondecreasing with respect to the second argument. There are functions /?: N -> R +, y: N* ->R + such that conditions (3) and (5) are satisfied and

P(m0 + l) ^ Q(m0, max (3(m0)) + y(m0) for m0eN*.

- n

0

0

Under these assumptions we have (6).

IL We denote by C(X, У) the class of all continuous functions defined on X taking values in Y (X, Y arbitrary metric spaces). Let Z denote the closure of the set Z. Let G, Gt <= R" and G <= G1. We assume that G and Gl are open bounded domains. Let E = (0, a] x G, D = [ — t , a] x Gl5 where a > 0, т ^ 0 are fixed real numbers. Let £ (0) = D\E. If w: £-> R is a function of the variables (x, у) = (x, y1} ..., y„), and the derivatives Dy.w, Dy.yiw, j, l = 1, ...

..., n, exist then we write Dyw = (Dyiw, . . . , Dynw), Dyyw = [DyjyiwYj,i = i' For a function z = (z1? ..., zk): Rk of the variables (x, y) = (x, y1?..., y„) we write Dxz = , ..., L>*zfc), Dyjz = (.Dy.zx, ..., Dy.zk), Dyjyiz = {Dy.yiz x, . . . ..., Dyjyizk), j, l = 1 ,..., n. In a similar way we define Dy.yyytz if z has the corresponding derivatives of the third order.

Let Z = E x Rk x C{D, Rk) xR "x R"2 and assume that / = ( / ! , . . . , f k)- Z-+Rk, (p = (<jol5 ..., <pfe): jE(0)-*Rk are given functions. For z = (z1?..., zk)\

D-+ Rk we write

f( x , y, z(x, y), z, Dyz(x, y), Dyyz(x, y)) = (/^ x , y, z(x, y), z, DyZ^x, y), DyyZ^x, y)),..., / k(x, y, z(x, y), z, Dyzk{x, y), D„zk(x, y»), (x, y)e £.

We consider the differential-functional problem

Dxz(x, y) = f( x , y, z(x, y), z, Dyz(x, y), Dyyz(x, y)), (x, y) e E, z(x, y) = <p(x, y) for (x, y )e £ (0).

We define a mesh in D in the following way. Let y(0) = {y(i \ ..., yj,0)) be a fixed point of G. are fixed subsets of R + = (0, сю) such that 0 is a point of density of 11, Let / = {h0 e R + : there exists a natural number n* such that h0n* = a}.

Let I 0 <z {h = (h0, h'): h0e l , h' = (hl9 ..., h ^ E l x х . . . х / J . For h0e l let n0 be a natural number such that n0h0 ^ т < (n0 + l)/i0. Let m0 be an integer, m' = (m1, ..., m„)eR". We define x(mo) = h0m0 and y(m) = ( у ^ - Ь ^ т ^ ..., y!0) + /i„mn). For / j e / 0 we define Th = {m = (m0, m'):

— n0 < m0 < и*, m '= (m1? ..., m„), are integers, and (x(Wo), y(m'>)eZ)}, Moreover, set T% = {m = (m0, m')e 7),: (x(mo), y(w ))eF} and T* = {m = (m0, m ') e fh: (x(mo + 1), y(m))e£ }.

Denote by where 1 ^ 1 is a fixed natural number, the set of all (Si, ..., sn) such that sx s„ are integers and — A ^ s;- < Я for j — 1 ,..., n.

If s e S 1, m = (m0, m')eTh, let am>s = max {a: 0 ^ a ^ 1, (х(и,о), y(m+as))eD}.

Let Th = {m = (m0, m'): m e T h or there exist s e S t and m = (m0, m')e % such that m '= m'+ am,ss} • Let Tj,0) = Th\T*. For m = (m0, m')e T* let

;(m) = (m0, m15..., т 7_15 rn^.+cx^, mJ+1, ..., mj, where s = (<5Д, ..., (Sjr is the Kronecker symbol), and let —j(m) = (m0, m1, ..., m^-i, rrij—am>s, mj+1, ...

where s = ( - ^ l5 ..., - ô j ^ e S ^

For h e l 0 let Eh = {(x(mo), y(m<)): m = (m0, m')eT%}, Eh = {(x(mo), y(m)):

m = (m0, m') e f * } , Dh = {(x™, / тУ. m = (m0, m')e Th}, Ei°> = /"'>):

m = (m0, m')e Tj,0)}. For a function w: Dh-+R we write w(m) = w(x(mo), у(ж')) for me Th. If —т ^ х ^ а then we set Hx = {(£, q) = (<!;, ql t rjn) e D: < x}. If p = ( p j , p k) e Rfc we put ||p|| = For zeC(D, Rk) we introduce a norm

||z||x = max ||z(£, rç)||, —т ^ x < a.

(Z,Tj)eHx

We say that the differential-functional problem (7) satisfies the Volterra condition if for xe(0, a] and z, zeC(D, Rk) such that z\Hx = z\Hx we have f(x , y, p, z, q, r) = /(x , y, p, z, q, r) for (x, y, p, z, q, r), (x, y, p, z, q, r)eZ.

For —п0 ^ т 0 ^ п * we define

Ehtmo = {{x{m° \ y (m,))e D h: m0 ^ m 0}, ||z||^ = max |z(*°’m,)|,

(x<mo>,y<m'>)e£h,m0

where z e ^ ( D h, R k). If z = (zl5 zk) e ^ { D h, Rk) then we write \z\hymo = ( b i l h . mo , I zk\hymo), where

\ Zi \ h, mo= . max \zf°’mX i ^{= 1,} k.

(x(mo),y(m'))e£h1m0

For a natural number A ^ 1 and m e f £ , let S%x = {s = (sl 5 s„)eR":

— A ^ S j ^ A , j = l , . . . , n , (m0, m' + s ) e Th}. Now we define operators

A™ = w r . . . . 4 ” ’)> BS"1 = ( B if . . . S ir ’), Ci"> = (CS">. • • •, D f> =

= 1...k - If w e ^№ ),i R) ‘hen

= £ aS swl"°-”' +s>,

seSh.A

( 8 )

B ^ w (m)= X b%!?w{mo'm'+s\

seS^A

C |fw (m) = X i c K )w(W0’m'+s),

seS h.A b j

D\ ÿwim)=

where i, i ' = 1 , fc, j, l = 1, . . n, m e t ? , and for seSfo, a g s, Ьй£\

cm,s> d(mi’!) are real numbers (h = (h0, hn) e l 0).

If z = (z!, ..., zk) g ^ { D h, Rk) then we write 4 (m)z(m) = ( A ^ z ^ , ...

A[m)zim)), B\m)z(m) = (B ^]z(r \ - Я £ Ч т))- If w G ^ (D h, R) then C<M)w(m> = (Cji,)w(m), C ^ w (m)), Dim)w(,n) = [D $ w lmy] l l = l , i = 1, k.

Let I h = Ëh x R* x ^ { D h, Rk) x R" x R"2. Suppose that for h e l 0 we have

afunction Фй = (Ф^,1), 4>hk)): I h^ R k. If z — (zx, z k) e#' ( Dh, Rk) then for

(m0, m ')e f* we write

Фк(х{то), у(п,,), B(m)z(m), z, C(m)z(m), D(m)z(m))

= (Ф[1){х(то\ у(т'\ B{? }z(m), z, C[m)z[m), D ^ z ^ ) , ..., Ф[к)(х(то), у(т'\ B{m)z{m), z, C{m)z{m), D{m}z(km))).

In a similar way we define /( x (mo), у{т'\ B(m)z(m), z, C(m)z(m), D(m)z(m)) for m = (m0, m')eTh and ze C(D, Rk). It will cause no confusion if we use the same letter z to designate a member of C(D, Rfc) and its restriction to Dh.

Consider the one-step difference method for the problem (7) z(mo+l,m') = A(m)z(m) + ho0 h{x(mo), y(ml, B(m)z(w), z, C(m)z(m\ D(m)z(m)),

(9) m0 = 0, ..., n* — 1, ( X < 4 e Eh,

z(m) = pim) + flm) fQr m £ fjO) ^ where S: Eh0)->Rk.

Suppose that uh is a solution of (9) and v is a solution of (7). We give sufficient conditions for the convergence lim^^o ||wj,m) — t>(m)|| = 0 on Eh. The function Фк = (Ф^\ ..., Ф^)' Zh-+ Rfc is said to satisfy the Volterra condition if for m0 = 0 , n* — 1 and for z, z e ^ ( D h, Rk) such that z\Eh<mo = z\Eh>mo we have Фк(х(то), y(m’\ p, z, q, r) = Фк(х(то), y{m'\ p, z, q, r) for (x(mo), y(m'\ p, z, q, r), (x{m°\ Ут ), p, z, q, r)eZh. We introduce the following

A

H

.

1° The function Фй: I h^ R k, h e l 0, of the variables (x, y, p, z, q, r), satisfies the Volterra condition.

2° For h e I 0, {x, y, z)eËhx ^ ( D h, Rk) the function Фк{х, у, -, z, -, -):

Rk x R" x R"2 -*■ Rk is continuous.

3° The derivatives Dpf f i \ Drjl Ф{к exist on Zh and for each h e l 0,{x, y, pl 5 Pi-U pi+u ..., pki z)eËhx R k~1 x 3?{Dh, Rk), i = l , . . . , k , j\ l = 1, ..., n, the functions

£>р.Ф(л°(х, y , p 1, . . . , p i- l , - , p i+1, . . . , p k, z , -, •), DqjtfP(X, У, Pi, . . . . Pi-1, % Pi+1, • Pfc, 2, *, •), Drj^ { x , y, pj, Pi_i, -, Pi+1, ..., pk, z, -, •),

of the variables (pi5 q,r), i = 1 , k, are continuous on R x R "x R "2.

4° Drj}Ф]р = Dri^ j P on Zh, for h e l 0, i = 1, k, j, l = 1, n.

5° There exists a matrix A = [Aif<]kr==1 with ^ 0, i, ï = 1, . . . , k, such that for h e l 0 we have

(10) |Фл(х(то), y{m'\ p, z, q, г )- Ф к(х(то), y(m'}, p, z, q, r)|r

^ AUp-p\ + \z -z \htmJ T,

where (x(mo), y(m'\ p, z, q, r), (x(mo), y(m'\ p, z, q, r)eZh.

A

H2. For each Q eE h and seS™x, m = (m0, m')e T\f, i = 1 , k, we have

j = 1 n j

+ t j,I=l П]П1 ^0.

A

H 3. 77ie operators A(m\ B(m), C(m), D(m) satisfy

1° Z m fl«.» = Z « Ь£',#) = 1 and b&P ^ 0 for m = (m0, m')e Tjf,

seS h,A seSh.A

Î, Г = \ , к,

2° Z m = z « = 0 /or m = (m0, m') e T%, i = 1, ..., к,

seS h,A seSh.A

j , l = \ , . . . , n .

Now we give sufficient conditions for the stability of the difference scheme (9).

T

1. Suppose that

1) Assumptions H 1- H 3 are satisfied,

2) veC(D, Rk), v\E(0) — q> and there exists у = (yl5 yk): J0->R+ such that

(11) \v^m° + 1,m) — A(m)v(m) — h0Ф/,(х(т°’\ y(m,), B(m)v(m), v, C(m)v(m\ D{m)v(m))|

^ h 0y{h) f ° r m0 = 0 , n * - l , m = (m0, 3) u,, is a solution of (9),

4) у = (yl9 yk): 70->R+ is a function such that |t)(m)| ^ y(h)for me Tj,0).

Then (12)

mo - 1

|i4m) — v(m)\T ^ (I + 2h0A)moyT{h) + h0 £ (/ + 2h0A)vyT{h)

v = O

/or m0 = 0, 1, и*, m e T h.

P ro o f. We apply Lemma 1. Let Y0 = {m'eR": there exists m0 =

~ n 0, . . . , n* such that (m0, m ')e T j, Qmo= Y 0 for m0e N , Гто = {т 'еR":

(m0, m')e T*} for m0 = 1, n* {Гто = Гх for m0 = 1, ..., n*), Г =T% = (N \ N i0)) x r i , f = N * x r i ,Q = N x Y 0, where N, N (0), N* are sets defined in I. To each function z: Dh-^R k there corresponds a function z: N xQ ->Rk given by z(m) = z(x(mo), y(m )) for m eT h = Q. We will use the same notation for elements of the sets Dh, Rk) and & (Q, Rk). Let F = (F(1), F(k)): F x ^ ( Q , Rk\ ^ R k be defined by

F(i)(m0, m', z) = Т И + гИ + / 1 0Ф № (то), /»'>, B\m)z(m), z, C\m)z\m), D\m)z\m)),

ï = 1, k, m e f , ze!F{Q, Rk). Then we have u% no + 1,m') = F(m0, m', uh) for

me Г and uj,m) = (p{m) + ô(m) for т е Т [ 0). It follows from Assumption that F satisfies the Volterra condition, thus problem (1), with co(m0, m') = (p(m) + d(m\

has only one solution.

Now we prove that F satisfies assumption 3) from Lemma 1. Suppose that m0eN*, (m0, m')ef% and z, ze3F(Q, Rk). Then we have

(13) \F(i)(m0, m \ z ) - F w(m0, m', z)\

^ |y4jm)z ^ - A ^ z<.m) + h0(<Pf(x(ma), y(m'\ 5 (.m)z(m), z, С|т)г‘ш), D|m)z(im)))

_ ф « ( х(т°»! ym'))

^

^ , B(.f)z (.m), В-Г+ ! Bjj^zjf0, z, C|m)z |m), Djm)z |m)»|

+ /i0| ^ i,(x(mo), y(m,), B[™]z (™\ B{J")z{m), B ^ z ^ , z , C|m)z |m), D[m)z[m)) - Ф^(х(то), y (m'\ J3<m)z (m), z, Cj.m)z<m), Djm,z1m))|, i = 1, fc.

From Assumption H A and from the mean value theorem it follows that

|Fli>(m0. m'. z ) - F (0(m0, m', z)| < |zf!"l(z<m|- z i ml)

+ й0В1Г|(2!”)- г ! ”>)0,„Ф1‘»(е,)+Ао Ê C W (zi">-zT ’) C ,,4 f (Q ,)

+ л 0 i f a )(zi", -z !"')B rjl<pp(a)i+/!0 i i„.iB!?,(z ri- z r >)i

Л«= 1 ^{i' = l} _{Г # i}

+ ^0 Z i'=l

for an intermediate point QtE l h, i = 1, k.

By the definition of A(m), B{m), C(m), D(m) and from Assumption H 2 it follows that

(14) |F(i)(m0, m', z ) \ - F (i)(m0, m', z)\

^ £ \ z ^ ' + * - z ^ ’+%<№a + hob ™ D „ m Q Ù

seSÎT.A

+

i _=i £ < 4 г в .,« т +

i rtw*0ч,*р<од) _=i

+ h0 £ l u. B W № - z F \ + h0 £ Я„.|2г - 2 , 4 , то

f' = l

for е ге Г А, i = 1 , k. From Assumption H 3 it follows that

|F(i)(m0, m', z ) - F (i)(m0, m', z)\

к к

^ |Zj Zj|A>mo( l + + /ig ^ l^ i '^ i 'I h . m o T - X! ^i'lh.mo

i' = 1 i' = 1

к

^ |Z f ^ ilh .m o “b 2 / l 0 ^ ^ i i 'I Z i ' ^ i 'l h ,m 0

i' = l

for Qte Z h, i = 1 ,..., k. Thus for m0 = 0 , . . . , n* - 1 , (x(mo), Уш )) g Ëh we have (15) \F(m'0, m’, z)-F {m 0, m', z)\T < (I+ 2h 0 A ) \ z - z \ l mo = Q(m0, V {z -z ))T. def

The function Q satisfies 2), 3) from Lemma 1. Let ^ ( т 0, m', z) = l;<mo + 1’m') for meT% and z e ^ { Q , Rfc). Assumptions 2) from Theorem 1 imply IF{m0, m!, v) — F 1 {m0, m', p)| ^ h 0 y{h) for m e f£ . Let fi be a solution of the problem

(w(mo + 1) = (I+ 2h 0 A)w(mo) + h 0 yT(h) for m 0 = 0, — 1,

| w(m0) _ for — n0 ^ m0 ^ 0.

It is evident that the assumptions of Lemma 1 are satisfied, and we have

|м!т) — v(m)\T < /?(m0) for m0 = 0, ..., n*, (x(Mo), y(m))eD h. Since m0- 1

j3(m0) = (I + 2h 0 A r ° y T(h) + h 0 £ (/ + 2/i0yl)vyr (/i), the proof is complete.

R em ark 2. Suppose that the assumptions of Theorem 1 hold. If L — ||Л|| > 0 then

ç 2 aL — ^

||ujrl- f ("lll < e2‘LIlfWH h№)ll for m0 = 0 ...n * ,m e T h.

If

= 0 then ||î4M)- i ; (m)|| ^

(A)

-I-a

y (A)

for mQ = 0 , . . . , n*, m e Th, where |jЛ |j = m a x ! = 1 |Я[Т|.

A

H4.

1° The operators A(m) defined by (8) satisfy the conditions

E am,sSj = Z a^\ssjsl = 0, i = 1, ..., k, j, l = 1 ,..., n, me T%.

seSÎT.A se sît jA

2° T/ie difference operators C\m), D\m) satisfy

Z Cm'$Sl = àjh

seSîT.A

V diW ,s -Sv = 1о г ) ф 1 ’ Y é ‘M s~ = о 1 ”"s J I 4 ; « V 7 W - / . , 4 ”“ ' ’ me T*, Ï =

, ..., /c, j , f , l, Ï =

, ..., П.

3° TTiere exists c0 > 0 such that hjh f 1 ^: c 0 and / i / îj / iô 1 < c0 (h e l 0, Л / = 1, ..., n).

4° Г/iere exists c0 > 0 such that \a$J ^ c0, ^ <4 |c ^ | ^ c0, |d^iJ)! ^ c0 for h e l 0, seS^>x, m ef% , i, Г = 1, fe, j, l = 1, ..., n.

10 — Comment. Math. 30.2

Now we prove that if the method (9) is stable and it is consistent with (7) then it is convergent.

T heorem 2. Suppose that

1) Assumptions H x-H 4 are satisfied,

2) / e C ( I , Rfc) and veC{D, R*) is a solution of (7) such that v\E is of class C3,

3) there exists fi: I 0 -> R + such that (16) IФ„(х(то), y(m'\ v(m), v, C(m)v(m\ D(MV m))

- f ( x {mo), y(m'\ v(m), v, C(m)v(m\ D(m)v(m))\ ^ p(h), m0 = 0 , . n* — 1, (x(mo), y(m,))e Ë h, and lrni|ftH0 \\Р(Щ\ = 0,

4) there exists y: / 0->R + such that |<5(m)| ^ y(h) for meTj,0) and lim|fc|_o \\y(h)\\ = 0,

5) uh is a solution of (9).

Then lim^i-o Цwim) — i/m)|| —0 on Eh.

P ro o f. Let c ^ O be a constant such that ||i>(x, y)||, \\Dy.v(x, y)\\,

\\Dyjytv(x, y)||, ||Dyjyjiyiv{x, y)|| < c, (x, y)eE, j , f , l = 1 ,..., n. Using for vt the Taylor expansion of the third order with respect to у we get for some 0fe(O, 1), i = 1 ,..., k,

itf"o+ i + + me f t ,

seSn.A

and \R{i$(h)\ < P${h) for (x(mo), y(m,)) e Ë h, where fâ ](h) = {\h\/6)cQc0cX3n3S^x (S™x denotes the cardinality of the set S™x). Let f30(h) = (f^ih ), ffiih)). We define

R$(h) = Ф^(х(то), y(m’\ B\m)vim), v, C(im)v$m), Djm)v(im)) Т Ф^(х(то), y(m'}, v(m), v, C\m)v\m), D\m)v\m)),_

R\™)(h) = Ф^(х(то), y(m'\ v(m), v, C[m)v[m), D\m)vimy) - f i ( x (mo), y{m'\ v(m\ v, C\m)v\m\ D\m)v\m)), (17) Rf$(h) = /Дх(то), y(m'\ v(m), v, C\m)v\m), D\m)v\m))

- f i ( x (mo\ y(m’\ v(m), v, Dyv\m), Dyyv\m)), R\m 2(h, VÙ = y(m'\ v(m), v, Dyv<T\ Dyyvt*)

~ fi(x(mo) + V iK ’ yim )’ v{x{mo) + Vih0,

v,

D y V f x ^ + ViK, y ^ ^ D y y V f x ^ + ViK’ У(т)))-

Let py(h) = (Ffih), ..., № Щ , v = 1, 2, 3, 4, and

(18)

Л°№)= max JRR'W I, v = 1, 2, 3,

(

Pf{h) = max max |RjJ(fc, |.

( т о , т ' ) е Т н г ц е [ 0 , 1 ]

From Assumptions H 3 and H4 it follows that

(19)

||D 5 «)0(-)_fljjwB<-)|| ^ (|ft|/6)ccgc023n2s^.

From (10), (17), (18), (19) we have Нтщ^о |l(?i№)ll = 0, From (16)—(18) we have Итц|_0 ||/S2(A)|| = 0. Since/is continuous we see that Нтщ^о ||$3(Л)|| = 0 and lim|fc|_>o \\PM\\ = 0. Since

|i)-m°+1,m ) — A\m^y|m) — /ig Ф),*)(x(m°), y(w'\ Г, D|mMm))|

4

v = 0

meT%, i = 1, k, taking y(/i) = £*=о/М^) we have (11) from Theorem 1, where lim^i^o \\y(h)\\ = 0. This completes the proof.

III.

E xample 1. Let G = {yeR": |<y1- y i 0)| < r l 5 b/„-y<,0)| < r„} and Gx = {ye R": b h - У Л < Qi, •••, b „ - ^ 0)l < Qn}> where j = and

£ = (0, a] x G, D = [ — t , a] x Gl5 E{0) = D \£. Let / 0<= [h = (h0, hlf hn)e

g R++": there exist integers n*, nf, j = \ , . . . , n , such that h 0 n* = a and hjnf = ôp ; = Let x imo) = h 0 m0, / m,) = y{0) + (h 1 m1, . . . , h nmn), where m 1 , m n are integers and — nf < m;- < nf, j — 1, n. Let rij be a natural number such that rijhj < r} ^ (rij+ l)hj. Let 1 ^ Л <

Hj)’ We define operators A, В, C, D by (see [1]) - л у т ) = £ a(9w(»o.*'+»)ï BiVw(m) = X fc(M')H,(m°’m'+s),

( 20 ) ss S a sg S a

Ci7w(m) = X i c^ V mo’m' +s), DyIw(m)= X - i - d ^ ' ’l)w(mo’m'+s),

s eSa h j s sSa “ A

where w e ^ ( D h, R), (x(mo), y{m,)) e Ë h, i, Г = 1 , к, j, l = 1, n. Now we replace Assumptions H 2, H 3, H4 by the following

A ssumption H'2. For Q e l h we have

aÿ>+h 0 bf»D plm Q ) + i ^ 4 ,J4 J,H')(e)

j = i n j

+ Î r t d l ‘M Dr„ m Q ) S , 0 ,

j , l = 1 П] П1

i = 1, k.

A ssumption H3. For i , i' — 1 » J ’ j > /, /' = 1, . . . , П 1 1° a(‘> = 1, b(‘:’n > 0, = 1,

2° ^S€S a «?"" = X » s, d(U,l) =

X j sg S a sr d f j’l) = o, 3° X/ se S a SjSta f ■ - = 0, X. ss S a c -(»J)

‘V Ls = 4° X sg S a sr svd(f j,l) — | ' àjyàv

2ôjy5u>

/or /o r

j ^ l, j = h

5° there exists с0 > 0 such that hjhr 1 ^ C0’ hjhjiô1 j, I = 1, n).

Then we obtain the following

( h e l 0,

T heorem 3. Suppose that

1) Assumption is satisfied with the above given sets I 0, E, D, 2) Assumptions H 2, H 3 hold true,

3) assumptions 2)-5) from Theorem 2 are satisfied with A(m), B(m), C{m), D(m) replaced by А, В, C, D.

Then lim i^ o !|uj,m) — E(m)|| = 0 on Eh,

Jn [1] a similar theorem is proved without the Lipschitz condition (10) and with the assumption to quasimonotonicity of Фн with respect to p and monotonicity of Фь with respect to the functional argument. We omit the quasimonotonicity and monotonicity assumptions and we introduce the Lipschitz condition. Our Theorem 3 is a simple corollary from Theorem 2. The operators (20) were introduced in [16] by Przqdka and used in [1], [2] and other papers. Our operators (8) are some modification of the operators given in [16].

E xample 2. Let £ be a Haar pyramid, E — {(x, y)eR 1 + ": 0 < x ^ a and IУ]~У?}\ < bj — xlj,_ j = 1, - n}, where y(0) = (y(10), ..., >i0))eR", 6-, > 0,

; = 1 ,..., n. £> = E u [ - t , 0 ] x [ y (O)- h , y(0) + 6], where [y(0)- 6 , y(d) + fi] =

= [yi0)~~bx, yi0) + bf\ x ... x[y<0>-fi„,y<°> + f ij, and £<°> = D\E. Let I0 a {heR++n: there exist natural numbers n*, kj, j — 1, ..., n, such that hj = h0lj/kj}. Let x(mo) = h0m0, y(m<) = + ..., hnm„), and let A, B, C, D be defined by (20), where 1 ^ A ^ mini ^ j ^ nkj. Let

Eh = {(x(mo), y(m'>): (x(Wo), y(mr))eE}, Eh = {(x(,Ho), y(m'>): (x(mo + 1), / n'))eEh}, Dh = {(x(,no), y(m,)): (x(Wo), y(M'>)e D}, E{0) = Dh\Eh.

Consider the differential-functional problem (7) and the corresponding differ­

ence problem (9) with the operators A, B, C, D оn the above sets E, E{0) and

Eh, £[0), respectively. Suppose that Assumption H x is satisfied with E, D, E(0),

I 0, Eh, E[0), and that Assumptions H'2, H 3 and 2)-5) from Theorem 2 are

satisfied. Then using Lemma 1 we can prove that l im ^ o |)wj,m) — v{m)\\ = 0

on Eh.

Ex a m p l e

3. Let À

1 and let D, E, E(0), I 0, Dh, Eh, Ëh, Fj,0) be the sets defined in Example 1 with y(0) = (0, . . 0 ) . Consider the problem (7)

with П

f ( x ,

y, p, z, q, r) = /(x , y, p, z,q)+ £ ам(х, y)rjh J,i =

where an — (a\}\ aft): D-* Rfca n d / = (J \ , . . . , f k): E xR* x C(D, Rk) x R"-> Rfc (an almost linear problem, see [1]).

We define difference operators A = (A l t . . . , A n), A (i,m) = [d{i,m)]"/ = 1 by А У т) = (2hj)~

[w0(m)) — w(

7 =

n,

A ftm) w(ra) = (2 hjhJ ~ 1 [ - w0(m)) - w(,(m)) - w( _Лм)) - w( “ '(m)) + 2w(m) wUU(m))) w(~j( ~ i(»«)))-j

(21) if (/, = {0, /): j, I = 1, ..., n, j ф l, aft{x{mo), y(w,)) ^ 0}, Aft m)w(m) = (2/ îj .Z î J “ 1 [w°‘(m)) + w(!(m)) + w( ~j(m)) + w( ~l(m)) - 2w<m)

■w( - j(Hm))) _ w(j'( - l(m)))

if j = I or 0, = {O', /): j, l = 1, П, j ф l, aft{x{mo\ y(m,)) < 0}, where w: Dh->R, (x(mo), y(m )) e Ê h. Consider the difference method

z | m o + l,m ') = yl . z j« ) + /j0 ^ ) ( x (mo)> >, ( « ') > ^ - Z (m), z , J Z H )

( 22 ) К

+ У - ~ а # ( х (то), y(m')) d ^ " ,)z)M), (х(то),У м,)) е £ h’

7.1=1 " А

z(m) = ф(м) + <5(m) for (x(Wo) , y(m'>) e £j,0),

where Л, J5 are defined by (20) with л = 1. We replace Assumption H 2 by the following

A ssumption H'2'. For each Q = (х(,ио), y(m ), p, z, q, r)e Z h we have 4 ' + 1 г 0 Щ*ОГ 1 Щ Ч О )-2 £

j=i

+ Î 7nHaS 'V mo1, » 0, W-l hi hl

j*l

“%> + К b U D „ Ф Ш + Dqi# ( Q ) + jjf / ”•>)

- I ~ hjht Д О " ' > °- I = 1 n,

af»m + h0b ^ e)DPM ^ - ~ - D tJ^>(Q) + ^ i S ( ^ < ‘\ y ^ )

E Г Т I4 ? ^ '”01. У”°)1 > 0 , j = l , . . . , n , 1=1 hJhl

t* j

(23)

« 5 ?ад)+ЛоМ('й))вр,^',(е)+ 2 щ 14 ?(^"°|. >'(m'l)i » ° /<”• (л Об Ai»,

а®л . 1т| + й0№ 5(.1((1|)1)и ^ 1 ,> ( е ) + 2 ^ И ? ( х ("»»> У"'»)| » 0 /or O', OeAi»,

a«i»ww)l + fcob(i f (W)lOp,#i,‘>(0 + ^ | aÿ(x'«>', У"'»)! > 0 for (j, Ое Ai»,

a f - m ) + h0 b № m D p& \ Q ) + ^ \ a $ ( x < ™ \ У"’>)| ÿ 0 for (j, 0 б A i ’.

a^ + hob^D pffiiQ ) ^ 0 for [s = j(l(0)) or s = -j(-l(0)), and {j, l)eJÜ~\

or [s = - ;(/(0)) or s =j(-l(0)), and (j , l)e J \ff\ or [s] > 2, w/iere [s] = X"= î N (s = (sx, . . . , s„)).

Now we have

T heorem 4. Suppose that

1) Assumptions H l5 Щ , H'3 1°, 3°, 5° are satisfied,

2) assumptions 2)-4) from Theorem 2 are satisfied with operators А, В, Л,

3) uh is a solution of (22).

Then Шп|й| - 0 || m J,w) — t/m)|| = 0 on Eh.

R em ark 3. Malec ([6]—[13]) introduces the assumption that the Drjl<Pil) are nonnegative or nonpositive on I h. Kamont and Kwapisz ([1]) omit this assumption by definition of the operators d (,,m) which depend on m. They prove a result similar to our Theorem 4 independently of the main theorem of [1]. Theorem 4 and the results of the above-mentioned authors with the assumption that the Drj^ ] are nonnegative (nonpositive) follow easily from our main result (Theorem 2).

E xample 4. Let Я = 3. Suppose that the function Фй does not depend on the variables rjt for j, l = 1 , . n, j ф l. Let

Pj = {m: (x{mo\ y {m,)) e Ë h and (x(mo + 1), y j(m'] e E[0)}, (24) P - j = {m: (x(Mo), yW)) e E h and (x(mo+1), y - ^ e E ^ } ,

Rj = {m: (x(mo), у(т )) е £ й and тфР] и P-j], j = 1 ,...» и.

For s e S t let

Ps = {m: m ePj if Sj = 1, m e P - j if Sj = — 1, m eR j if Sj = 0, j = 1, ..., n}.

Consider the difference scheme

z(mo+l,m') = A ^ z W + h ^ i x ^ , y(m'\ B\m)z(m), z, A\m)z\m), A\ 2 'm)z\m))

(25) on Eh, i = 1 , k,

z(m) = ф(т) + (5(т) 0П E[0),

where Ëh, Ej,0) are defined in Example 1 with y(0) = (0, 0), A {m), B\m) are defined by (8), and d j m> = ( 4 " \ .... A % \ Aj2’m> = [d g ’m)]",, = i, i = 1, .... k,

' h f 1 1 - W ~ jim))~ 2 w(m) + wU(m))- iw (j(j(m)))] for m e P -j, h j 1 [±W(- J(- J(m))) - w(- j(m)) + W m) + s wü(m))] for m e Pp

^ 1 l ^ w i- j(- J{m)))- W ~ j(m))+ W j(m))- - h w iMm)))'\ for meRj, A \fw {m) = <

A\ 2 j m)w(m) = <

(26)

h i 2 [Й ■ w(~j(m))—f w(m)+ \ w°’(m)) + ^ —12 w0W(m))))]

for m e P -j,

h i 2 [ —12 w( ~ * " j( “ J'(m))))+%w(~j<~j(m))) + jw (~j(m))—I w(m) + 12 w0(m))]

for mePj,

h i 2 1 ~ i2 w ( “ J’( ~ j{m))) + f w ( " m ) — I w (m)+ f w°'(m))— ^ 2 w W(m)))]

for meRj, i = 1 ,..., к, j = 1 ,..., n.

Suppose that

(27) a^ + h^D^mi+i i i ^гЧ,Ф№)

2=1

2=1 j

m e P , - m e P - ,•

and that for m e P - j we have

- s i j«i nj

( 28 )

ай-А«+л0г>^-да)£>р,Ф1,о( е ) - ^ ;в ,^ г ,( е ) + ^ | а лфр te» > o,

<m +Kb'Jâo,DPim Q ) + Y D b m Q ) + ^ D rjjm Q ) » 0,

^ M tm + b „ b s ^ mDn m Q ) ~ D ^ m Q H ^ D r Mm Q ) > 0 ,

<УО««»)+'>оЬ£1ш»),>Ор,Ф11,( е ) - 12 Ь А Л4 Ш » 0,

for m e Pj we have

(29)

< - n - m + МЙ’Д n- j mDP,Ф Ш -D,,Ф № + ^В,„Ф _{?>(0 5} = ₀ ,

щ

h 1 \h

a Z m + K b ^ D ^ W + ^ D ^ f W + j ^ D ^ i Q ) Ï 0, and for m e R j we have

< ® . - M - m + h b r - K - r n D » W m + ^ l > « m Q ) - - ^ I > r um Q ) > o,

Л I 1

ай)._д»)+ й 0ь<;^л»)в и Ф1,> ( е ) - ^ в щФг,( е ) + з Л ^ иФг,( е ) > о ,

( 3 0 ) 2

h h

^ т + Ь о Ь ^ О г М Ш + ^ О ^ Ф Ш + з д О ^ Ф Ш ) > 0,

«i?jw» + K b Z u m O PM 4 Q ) - ^ D q^ ( Q ) - ^ D rj^ n Q ) » 0, (x(mo), y(m'}) e Ë h, Q = (x(mo), y (m'), p i z , q , r ) e E h, ; = l , . . . , n , 0 = (O, . . . , 0 ) e S .

Assume that uh is a solution of (25) and assumptions 2)-4) of Theorem 2 are satisfied. Suppose that Assumptions H l5 H 3 and conditions (27)-(30) hold.

Then lim,fc|-*o li^m)- ^ (m)|| = 0 on Eh.

E xample 5. Consider (7) and (9) with f {(x, y, p, q, r) = /.(x , y, p, z) +

°>o Y j ~ i ru>

y«')> p> z> q< r) = фЦ>(х<»«>, p, z) + ft,0 £ r)S, ci ) q > 0.

j = 1

Let E = (0, a] x ( - b , b)n, £ (0) = [ ~ i , a] x [ - b , b]"\£, b > 0. Let h.} = h for

; = l , . . . , n . We define a = co0h0/h2. A particular case of Example 4 is obtained with B p }w(m) = w(m) and

29п<Г) + 31п<иЬ)

A[m)w(m) = U + 6n w ^(ж)

+ a £ [ _ J w(-j(m))_lwü(m))_lwü0'(m))) + l wüüt/(m))))]

J= 1

m eP- j

+ Z [бw(~л' л~ - W ~ j(~jim)))~ W ~ j(m))~ 3w°'(m))]

J=1

+ Z [w<~ j( ~j{m))) — 4w( “ j(m)) — 4w0(m)) + w « ] , j= i

m e P j

where

n n

" I } = Z 1 »

W II tE

J = 1 7=1

m e P - j - m ePj

if we suppose that for i = 1 ,..., k, j = 1, ..., n, Q = (x(mo), y{rn'\ p, z, q, r)eZh we have 1+ /го/)р.Ф|,о( 0 + |и а ^ 0. This condition implies (27)-(30).

R em ark 4. Many examples of particular operators A, В were defined on a product of intervals in [1]. Our examples are modifications of those in [1]. In Example 3 we have difference operators of second order, in Examples 4, 5 — of third order. In the previous examples the set G was a parallelepiped. Below we show an example of another type.

E xample 6. Let G = {(y1 ;..., y je R ": 0 < yp j = 1 , ..., n, p 1 y 1+ ...

+ pnyn < 1}, p 1 , . . . , p n are positive numbers. Let E = (0, a] x G, D = [ - T , a] x G, E{0) = D\E, I 0 cz {h = (h0, h')eR++n: there exist n*, К such that n*h 0 = a and hj = 1 /(pjK)}. Let x(mo) = m 0 h0, y{m>) = (m 1 hl , ..., mnhn), Eh = {(x(mo), y{m): m 0 = 1, ..., n, m1, ..., mn are natural numbers and mx + ... +m„ < K}, Dh = {(x(mo), y(m>): ra0 = — n0, ..., n* and [(m,. ^ 0 are integers, j = 1 , ..., n, and m1+ .. . + m „ ^ X ) or (m' = m! + г(Д0) + ^(0))> where j Ф /, в = (0, . .. , 0) g R" and (ximo), y{m)) e E h and + ... H-m„ = X — 1]},

£j,0) = Dh\E h. We define difference operators d, d (l,m) by (21) if (x(mo), y(m ))e Ë h and mt + ... +mn < К — 1, and by

(31) dJVm)w(m) = (2hj hz) ~1 [ — w0(m)) — w(i(m)) — w( ~ — w( ~?(w)) + | w(»o.«' + (1/2)0(0) + m)) + f(m)))j

if (x(mo), y(m)) e Ë h and mA + ... +m„ = X — 1. Let = w(m) and

1 » 1 »

Л. w(m) = I vHm)---V [ w< “ Лж» + vv°’(m))] H--- У w( ~j(l(m)))

1 2 * j=i 2 n ( n - l ) j ^ 1

j* i Consider the difference scheme

zjM0+1’M') = + й0Ф^)(х(то), y(w,), B*z(m), z, dz)m), d (i-M)zim))

(32) for (x(mo), у(т,))б £ й,

z(M) = <p(m) + <5("° for (x(mo), y(m,)) g £ („0), where d, A(i,m) are defined by (21) and (31).

We introduce the following

A

H'2".

1° / / (x{mo), Ут,)) е £ л then Dr. ^ \ Q ) ^ 0 (or 0) /or each 6 = (x<Wo), p, z, <?, r ) e l h.

2» ! + А0Дп 4>1,|(е )-2 й „ £ p O,

j - i "J №+ k i j,i=i nj ni

if Q e Z h and mx 4 - ... + m„ < К — 1, and

" 1

2 + h0DPi0j})(Q) — 2h0 £ -^Drjjm Q ) > 0

j = l nj

if Q e l h and mi + ... +mn = K — l.

30 > 0

for Q = (x(mo), y{m’\ p, z, q, r)e Z h, i = 1 , k, j = 1, n.

4° 1

2n(n — 1) O I.

for Q = (x (mo), y(m'\ p, z, q, r)eZh, j, 1 = 1 ,.. n.

It is easy to see that under assumptions H l5 H'2', H3 5° and 2)-4) from Theorem 2 we have 1ш1|й|_»0 ||м^т) —i?(m)|| = 0 on Eh, where uh is a solution of (32). Condition 1° from Assumption H'2" is satisfied for example if

П

<Ph(x, y t p, z, q, r) = <!>h{x, y, p, z,q)+ £ an{x, y)rfl

j,i = l

(an almost linear problem).

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