On differential inequality for non-linear integro-differential equation with retardation

R O CZN IKI POLSK IEG O TOWARZYSTWA MATEM ATYCZNEGO Séria I: PRACE MATEMATYCZNE XXII (1980)

J. Mo r c h a l o (Poznan)

On differential inequality for non-linear integro-differential equation with retardation

Abstract. In this paper we consider the Cauchy problem for integro-differential equation of the form

L [x ] = x{n)(t)+ X [A k0(t)xik)( t ) + $ K k0( t , s ) x {k)(s)ds] +

+ Z Z lA /( O * (k)(0;(r))+ 1 K G (r,s)x<lo(0j(s)Ms] = h(t)

k = 0 j = 1 a

with initial conditions

x m (t) = 0 (k = 0, 1, n — 2), t e E a, x(n~ 1}(a) = 0 , where

f i t ) ,

t

F ( t , x(t), x ig ^ tj ), . . . , x ( g m(t)), j К (t, s, x ( s ) , x (0m(s)))dsV

^ a '

t

F ( t , x ( t ) , x (0! (f)),..., X(gm(Г))) + J K (t, s, x(s), x (gt (s)), ...,x ( g m(s)))ds,

a

g ( t , x ( t ) )

F ( t , x ( t ) , J K ( t , s , x ( s ) ) d s } , h(t) =

Ea — 9 j (0 ^ a, t e l , j = 1 , 2 , m}.

In this article we prove the theorem on differential inequalities for the non-linear integro-differential equation with retardations by using function of Cauchy type.

I. In first part of this paper we formulate lemmas concerning integral inequalities the proofs of theorems on which are based.

We consider the integral equation

(¹) x(t) = ^ X( t , s) {F(s, x(s), x( gl ( s ) ) , x ( g m(s))) +

a

+ 1 K (S, 2 , x(z), x ( g ï (z)), . . . , x ( g m(z)))dz} ds + h0(t),

a

under the following assumptions:

As s u m p t io n H0. Suppose that Г I = {t: t e fa, b}},

2° I, = (s: a < s ^ t\,

3° x = (x0, x l s ..., xw), x ,e ( —oo, oo), i = 0, 1 , m, 4° F ( t , x ) e CUx(I x ( — oo, со); (-a o ,o o )),

5° K( t , s , x) e Ct SX(I x / f x ( — со, oo); ( - с о , go)), 6° F (t, x)e C * (/x ( —oo, oo)),

7° X (t, s, x)e Cl (/ x It x ( — oo, oo)),

8° X (t , s)e Cts (/ x Л), (f, s) ^ 0 for any a ^ s ^ t ^ b, 9° h0(t)e Ct ( /,( — oo, со)),

iœ g{(t)e Ct (I, /), 0,(0 t for i = 1,2,

a x a x , ч a x a x

11° —— ^ 0 , ^ 0 in ( / x ( - o o , oo)), —— ^ 0 , ^ 0 in

a x

a x m a x

a x m

(/ x I t x ( — oo, oo)),

a F

a x

12° max —— = F 0, max - — = F x, ..., max —— = Fm, max - — = X 0,

i a x 0 i a x j j a x TO i*it vx0

д к „ dK ^

m ax —— = X , , ma x --- = Xm,

Зх, /X/,

t m m

13° L = m a x fX (t,s ){ X F, + ( s - a ) X K jd s < 1-

I a i = 0 i = 0

Le m m a 1. / / :

Г assumption H0 is satisfied,

2° t/iere exist functions u(t), v(t)e Ct (I), such that the inequalities t

u(t)~ J X( t , s) {f (s, u(s), u(gl (s)),..., u(0w(s))) +

(2)

S

+ j X (s, z, и (z), w(0i(z)),..., M(ôfm(z)))jz}rfs-Ao(0 ^ 0,

a t

v(t)~ j X( t , s ) {f (s, t?(s), u(0 i(s)), •••, v(gm(s))) +

a s

+ J K ( s , z , v ( z ) , v ( g1(z)),...,v(gm(z)))dz}ds-h0(t) ^ 0

are satisfied for any t e l , then the integral equation (1) has the unique solution x (t) e Ct (7) satisfying the inequalities

(3) u(t) < x(t ) ^ v(t) for t e l .

P roof. We prove that u(t) — x ( t ) ^ 0 for t e l . Suppose, that it is not true. Then we have

t m

0 < u( t ) - x( t ) ^ J X( t , s ) {f0[_u(s) -x(s)]+ £ Fi [u(gi(s))-x(gi(s))] +

a i = 1

s m

+ f [K o(w (z)-x(z))+ £ K i(u(gi{z))-x(gi (z)))]dz}ds

a i = 1

^ L max [u (t) — x (r)] ^ ... ^ C max [u (f) — x (r)].

Hence and by 13° we have L" max [u(£) —x(f)] -> 0 as n oo, which contra­

dicts our assumptions. Thus we show that u(t) ^ x(t). Inequality x(t) ^ v(t) we can prove analogously.

As s u m p t io n Hj. Suppose that

Г F(t, x, y)e Ct x y (I x ( — oo, oo) x ( — oo, oo); ( — 00, 00)), 2° K ( t , s , x ) e C t'S'X( I x I t x ( - 00, 00); ( - 00, 00)),

3° F ( t , x , y ) e U p x<y(F0(t), . . . , Fm(t),Fm + 1(t); 7 x ( - o o , o o )x (-o o , 00)), 4° K(t, s , x ) e U p x (K0( t , s ) , ..., K m(t,s); 7 x It x ( - 00, go)),

5° if x ^ x, y ^ ÿ, then F ( t , x , y) ^ F ( t , x , y ) in I x ( — 00, oo)x ( — 00, 00), t

where y (t) = J K(t , s, x0, x x, ..., xm)ds,

a

6° X (t , s )e C M(/ x 7f), ЛТ(f, s) ^ 0 for any a ^ s ^ t < b, 7° gi(t)e Ct (I, I), 0,(t) ^ t for i = l , .. ., m ,

8° h0( t ) eCt( I — со, 00)),

г m s m

9“ K, = max J X (f, s) { £ F*(s) + T» + 1H £ K*(s, z)dz}rfs '< 1.

1 a fc = 0 a 0

Le m m a 2. 7/:

Г assumption H x is satisfied,

2° there exist functions u(t), v(t)e C,(I), such that the inequalities

t s

(4) u(t)~ J X( t , s ) F( s , и (s), u(gx(sj) , ..., и (gm(s)), J K(s, ^z, u(z) ,

a a

u(gAz)), ... ,u(gm(z]))dz)ds~h0(t) 0 ,

t s

(5) v ( t ) - J X( t , s ) F ( s , v { s ) , v ( g i (s)),...,v(gm(s)),$ K( s , z , v ( z ) ,

a a

v(g1(z)),...,v(gm{z)))dz)ds-h0(t) ^ 0 , are satisfied for any t e l , then the integral equation

8 — Prace Matematyczne 22.1

(6) x(t) = j X ( t , s ) F ( s , x ( s ) , x ( g1(s)), . . . , x(gm(s)),^ K( s , z , x { z ) ,

a a

X(g! (z)), . . . , x ( g m(z)))dz) ds + h0(t) has the unique solution x (t) 6 C, (/) satisfying the inequalities

(7) u(t) ^ x(t) ^ v(t) for t e l .

P ro o f. We prove that u(t) ^ x(t) for t e l . From (4) we have

t s

и (t) ~ f X (t, s) F (5, и (s), u(gt (s)),..., u(gm(s)), J K (s, z, u(z),

a a

u(g1{z)),...,u(gm(z)))dz)ds + h0(t)-r(t); r ( t ) ^0, t e l . Let us define two sequences (t)}

x0(t) = u0(t), t e l ,

t , s

x n + 1(t) = f I ( f , s)F (s, x„(s), (5)), х„(зга(«)), j x ( s , 2, x„(z),

a l a

x n(g1{z)),:..,xn(gm{z)))dz)ds + hQ(t), t e l , n = 0, 1,

t s

un + 1(t) = j X ( t , s ) F ( s , un{s), w„(gf1(s)),...,u„(ôfm(s))J K( s , z , u ( z ) ,

a a

ип(дЛ2)), •••> un(9m(z)))dz)ds + h0(t)-r{t), t e l , n = 0, 1 ,...

They tend uniformly to the functions x(t) and u(t), respectively. From conditions 5° and 6° of assumption H x it follows that for any n e N

un + i (t) ^ x n+l(t), t e l .

Passing to the limit as n -> 00 we obtain x(t) ^ u ( t ) , t e l . Inequality x(t)

^ v(t) we can prove analogously.

Le m m a 3. If:

1° conditions 1°, 2°, 5°, 6°, 7°, 8° of assumption H x are satisfied, 2° there exist functions u(t), v(t)e Ct (I), such that the inequalities

t s

(8) и (t) < [ X (t, s)F (s, u(s), « (jj (s)), .... « (jm(s)), ( K (s, z, «(z),

a a

u(g1{z)), ..., u(gm{z)))dz)ds + h0(t),

t s

(9) v{t) > $ X ( t , s ) F ( s , v ( s ) , v ( g l (s))i ...,v(gm(s)),$ K( s , z , v { z ) ,

a a

v(gi ( z ) ) , v(gm{z]))dz)ds + h0(t) are satisfied for any t e l , then

(10) u{t) < xhQ(t) < v(t) for t e l , where xhQ(t)e Ct (I) is the solution of equation (6).

R em ark . For fixed h0(t) equation (6) might not be uniquely solved. The lemma states that inequality (10) is true for every solution.

P roof. We prove the inequality u(t) < xho(t). Inequality xhQ(t) < v(t) can be shown analogously. From (6) and (8) for t = a we have и (a) < x hQ(a).

In view of continuity function xho(t) and u(t) inequality (10) holds in some neighbourhood of the point a. Let the inequality u(t) < x hQ(t) be not satisfied in an interval 7. Then there exists a point t0 > a, such that u(t) < xhQ(t) for t e ( a , t0) and u(t0) = xho(t0). In view of (6), (8) and the monotonicity of F ( t , x , y ) for t = t0 we have

Xh0 (to) - и (to) > $ X ( t0,s) {f (s, x hQ (s), xhQ (g x (s)),..., xhQ (gm (s)),

a

s

j К ( s , z , x hQ(z), x hQ {gt (z)),..., x„0 (gm (z))) dz) -

a

s

— F (s, u(s), u(gx (s)),..., w(gfm(s)), J K (s, z, u(z), u(gx (z) ) ,...

a

u(gm(z)))dz)} ds ^ 0 which contradicts assumptions that xho(t0) = u(t0) in the point t = t0. Finally we conclude that u(t) < xhQ(t) for t e l . Hence the proof is complete.

We shall now consider the integral equation

* g(t,y(t))

(1) y(x) = J X (x, t)F(t, y(t), j K(t, s, y(s))ds)dt + h0(t), t e l .

a a

As s u m p t io n H 2. Suppose that

Г F (x, y, z)e Cxy z (l x ( — oo, oo )x ( — oo, oo); ( — oo, oo)), 2° K (x, t , y) gCXtUy (7 x Ix x ( - oo, oo); ( - oo, oo)),

3° g(x, y)e Cx y(l x ( — oo,'oo)), a ^ #(x,y(x)) ^ x for any x e l , 4° F(x, y, z ) e L i p ytZ(Ll (x), L2(x); 7x( -co, co)x( -go, go)), 5° K ( x , t , y ) e L i p y(L3(x,t); I x Ix x ( - со, oo)),

6° g ( x , y ) e L i p y(LA(x); 7 x ( - 00, 00)),

* 9(t<-)

T R2 = Il J X ( x ,t) [ L 1( 0 + L 2(t) J L3(t,s)ds + L2(t)L4.it) \\K(t, s, -)ll] dt\\

a a

< 1, where ||w|| = max \u(t)\,

8° X (x, t )e CXft(I x Ix), X (x, t) > 0 for a ^ t ^ x ^ b, 9° if z x > z2, then

9( t , z i (0) g(t , z2 (t))

F (t, Zj(t), f K ( t , s , z1( s ) ) d s ) - F( t , z2(t), J K(t, s, z2(s))ds) ^ 0

a a

for any t, s e 7.

Le m m a 4. If:

1° assumption H 2 is satisfied,

2° there exist functions u(x), v(x)e CX(I) such that the inequalities

x g(t,u(t))

(12) u(x) ^ j X( x , t)F(t, u(t), j K(t, s, u(s))ds)dt + h0(x),

a a

x gi t Mt ) )

(13) v(x) ^ j X( x , t)F(t, v(t), J K(t, s, v(s))ds)dt + h0(x)

a a

are satisfied for any x e l , then the integral equation (11) has the unique solution y ( x ) e C x (I) satisfying the inequalities

u(x) ^ j/(x) ^ v(x) for x e l . For a proof of Lemma 4 see Lemma 2.

II. In this section we are going to consider

(14) L [x ] = xw (0 + £ [Ako(t)xm (t)+ f K k0( t , s )x(k)(s)ds] +

k = 0 a

n — 2 m t

+ Z Z

f

k=0 j= 1 a

with initial conditions (15)

where

(16) h(t)

x w {t) = О (к = 0, 1,..., n —2), t e E a, x {n~1}(a) = 0,

Ea = {t: gj(t) < a, t e l , j = 1 , 2 , m}, fit),

F(t, x(t), x( g1(t)),..., x(ôfm(l))) + t

+ J x ( l,s ,^ ( 5 ) ,x ( ^ (s)), ...,x(^w(s)))ds,

a

t

F(t, x(t), x( gj (t)), ^{. . . ,} x(gm(t)), J K( t , s, x(s),x(g1(⁵)),^{. . .}

a

. . . , x( gm(s)))ds), t

F(t, x(t), J K(t, s, x(s))ds),

a g(t,x(t))

F( t , x( t ), j K(t , s, x(s))ds),

a

t

F(t, x(t), x ’(t),..., x("-1)(0, J K(t, s, x(s), x' (s), ..., x("-1)(s))),

Akj(t)e Ct(I, R) (к = О, 1 ,..., n 1, j = 0, 1,

K kj( t , s ) e C t<s(I x /,, R) (к = 0, 1 , n - 1, j = 0, 1 , m), gj ( t ) eCt(I, R), g0(t) = t, gj(t) ^ t for j = 0 , 1 , m,

h ( t ) e Ct(I, R).

This section continues considerations on equation (14) initiated in [3].

Let us consider two equations (17) L [x] = / ( ( ) ,

(18) L [y] = y'”»(t)+ " t [Bt0(t)y<‘>(t)+ J Lk0U, s)y<*4s)ds] +

fc= 0 a

n — 2 m t

+

Z Z

к = 0 j = 1 о

with initial conditions

(19) x {k)(t) = yik)(t) = 0 (к = 0, 1 , —2)

t e E a, x {n~1)(a) = У"_ 1)(а) = 0 . By the solutions of problem (17), (19) and ( 18)—( 19) we mean any function x(t), (y(f))e С"(/, R) satisfying equation (17) or (18) with initial conditions (19).

Denote by X j(t,s ) and X2(t,s) functions of the Cauchy type [1 ,3 ], correspondingly for equations (17) and (18).

Assumption H3. Suppose that

Г the functions Akj(t), Bkj(t)e Ct (I, R) (к = 0, 1 ,..., n — 1 ; j = 0, ...,m ), K kj(t,s), Lkj(t , s)eCttS(I x It, R) (k — 0 , n 1 ; j = 0 ,..., m), f ( t ) e Ct(I, R), gj (t )eCt (I, R), g0(t) = t, g^t) ^ t (or j = 0 , 1 ,...,m ,

2° Bkj(t) ^ Akj(t), L kj(t,s) ^ K kj(t,s), t, s e l {k = 0 ,..., n - 1, / = 0 ,...

..., m), f ( t ) ^ 0 , t e l ,

3° there exist function 1 (20(1,х) ^ 0 (A'*1°(f, .s) ^ 0), a ^ s ^ t ^ b (i = 0 , 1, л - 1),

4° R ikj(t,s) = X ÿ ( t , s ) [ B kj ( s ) - A kJ(s)-]+ j X<‘>(t,z)[Lkj( z , s ) - K kj(z,s)]dz

S

« 0 (RlkJ(t,s) = Ar<1° ( t , s ) [ ^ ( s ) - B „ ( s ) ] + jx y '( f ,z ) [ K w(2, s ) - L w(z,s)]dz

s

« 0), В„_и (г) = Л„_^(Г) = B „_,j(r,s) = s) = 0 0 = 1 ,2 ,..., m),

t n — 1 m

5° hi(t) = Уо(0+ ( I I R ikj(t, z ) yii)(gj (z))dz ^ 0 (/ = 0 , 1 , 1)

a k = 0 j = 0 t n — 1 m

( M 0 = x (,)(0 + J Z Z ^iikj(f,₂)x (i)(^(z))d₂ < 0) (i = 0, 1,..., л —-1 ).

Theorem 1. I f assumption H 3 is satisfied, then

Kit) ^ x (i)(t) ^ У°(0 (x(0(f) ^ y(i)(t) ^ hi(t)),

where x (t), y (t) e C" (I) are solution of equations (17) and (18) with initial conditions (19).

P ro o f. Let us write equation (17) in the form (20) x'">(t)+ Z |A o (t)* <*, W + i Uo(l , s)x<k4s)ds] +

fc = 0 a

n — 2 m t

+

Z Z

к = 0 j = 1 a

= Z

+

+

к = O a

n — 2 m

+ I I {(Bw( t ) - A w(t))x(*»(9j(0) +

к = О у = 1

+ J (L kj (t , s )~ K kj (t, s)) x (k) (gj(s)) ds] + f ( t ) .

a

By assumption H 3 and by Lemma 1 of paper [3] the solution of equation (18) can be presented in the form

y(t) = J X2(t , s)f (s)ds.

a

Hence the function x(t) is the solution of the integral equation (21) x(t) = i x 2(t,s){ " z r(B »o(s)-A o(s))^‘4s) +

a fc = 0

+ (Lk0 (s , z ) - K kо (s, z)) x {k) (z) dz] +

n — 2 m

+ Z I (Bkj( s ) - A kj(s))xw (gj(s))+

k=0 j=l

f

+

J

a

We reduce equation (21) by substitution x {l)(t) = и f t ) (i = 0, . . . , n — 1) to the following system of equations

t n — 1 m

(22)

J

а к = 0 j = 0

Since for system (22) the assumptions of Lemma 3, [3], are satisfied, therefore we have

hi(t) ^ x (l)(t) ^ y{l)(t), t e I (i = 0, 1, n — 1).

Proof of the inequality in brackets process analogously.

As s u m p t io n H4. Suppose that

1° there exist functions x(t), z(t)e C"(7, R) such that the inequality,

t t

(23) L\_x]—F(t, x(t), j K(t, s, x{s))ds) ^ ^ [ z ] - F(t, z(t), j K(t, s, z(s))ds),

a a

is satisfied for any t e l , where

L i [ z ] = z(n)(0 + X [Ako{t)z{k){t)+ j K k0(t, s)z(k){s)ds] +

k = 0 a

n — 2 m t

+ Z Z I e k j(0z ( k ) ( d j(0) + $ R kj { t , s )z(k)( g j(s))d s ] ,

k = 0 j = 1 a

c

x (k)(t) = zw (t) = 0 .(k = 0, 1 ,..., n —2), t e Ea, x in~l)(a) = z(n~ l)(a) = 0, 2° the functions F{t, u, w), K(t , s, u) satisfy assumption H l5

3° there exists function X(£,s) ^ 0,

4° the functions Akj(t), Ckj(t), K kj{t, s), Rkj{t, s) satisfy inequality Ckj(t) ^ Akj(t) for t e l (j = 1 , m, к = 0, 2),

Rkj(t, s) < K kj(t, s) for a ^ s ^ t ^ b {j — 1 , m, к = 0 , n — 2), 5° x {l)(t) ^ 0 or z(i)(£) ^ 0 for t e l (i = 0 ,...,n — 2),

6° the function u(t) = 0 is the unique solution of equation

t t

(24) L [w] = F (t, x + u, j K(t, s, x + u)ds) —F(t, x, § K(t, s, x)ds).

a a

Th e o r e m 2. I f assumption H4 is satisfied, then x(t) ^ z(t) for t e l .

P roof. Let z{l)(t) ^ 0 (i = 0, . . . , n — 2). Therefore z{l)(gj(t)) ^ 0 (z{l)(gj(t))

^ 0 by gj (t )el , z{l)(gj(t)) = 0 by gj{t) ^ a). From (23) we obtain

t t

Li |> ] - L [ x ] ^ F (t, z, J K(t , s, z)ds) — F(t, x, j K(t, s, x)ds).

a a

Hence

t t

• L [z] — L [x] ^ F(t, z, J K (t, s, z)ds) — F(t, x, j K (t , s, x)ds) +

a a

n —2 m t

+

'Z Z

k = 0 j = 1 a

^ F(r, z, j K (t, s, z)ds) — F(t, x, j K(t, s, x)ds).

Let z — x = и, therefore from above we get t

(25) L[u] ^ F ( t , u + x, j K(t , s , u + x)ds)~

a

t

— F ( t , x , J K(t , s, x)ds) = G(t, x, u).

a

From (25) we get

(26) L\_u] = G(t, x, u) + m(t), t e l , m(t) ^ 0.

From of Lemma 1 [3] and (26) we have

t s

u(t) = { X( t , s ) [f (s, m(s) + x(s), J K (s, r, u(r) + x(r))dr) —

a a

s t

— F (s,x(s) , j K(s, r, x ( r ) ) d r ) ] ds+ J X (t , s)m(s)ds.

a a

Because X( t , s ) ^ 0, then

t s

u(t) ^ J X( t , s) [f (s, m(s) + x(s), J ^{K (s,} r, u(r)+x(r))dr) —

a a

s

— F(s, x(s) , j K(s, r, x(r))dr) ] ds.

a

In view of Lemma 2 we obtain x(t) ^ z(t) for t e l .

R em ark . If Ckj(t) = Akj(t), R kj(t,s) = K kj(t,s), then Theorem 2 holds without the assumption that z{l)(t) ^ 0 or x(,)(r) ^ 0 .

As s u m p t io n H 5. Suppose that

1° there exist functions x(t), y(t), z(t )e С"(I, R) such that the inequalities

t

L[ x] = F(t, x(t), j K(t, s, x(s))ds),

a

t

L x [y] ^ F(t , y(t ), J K(t, s, y(s))ds),

a t

^ F (f,z (f),J K(t, s,z(s))ds)

a

hold for any t e l , and they satisfy initial conditions

y{k)(t) = z{k)(t) = x {k)(t) = 0 for t e E a (k = 0 , . . . , n - 2), У"*1^ ) = z("_ 1)(a) = x(n_1)(a) = >0 ,

2’ conditions 2 \ 3°, 4°, 6° of Theorem 2 are satisfied, 3° y{i)(t) ^ 0, z(i)(f) ^ 0 for t e l (i = 0, 1 , и — 2).

Th e o r e m 3. I f assumption H 5 is satisfied, then y(t) ^ x(t) ^ z(t) for t e l . The proof is analogous as in Theorem 2.

R em ark. If Ckj(t) = Akj(t), t e I (k = 1 , m, j = 0 , n - 2 ) , Rkj(t,s)

= K kj(t,s), a ^ s ^ t ^ b (к = 1 j = 0, . . . , n — 2), then Theorem 3 holds without assumption 3°.

R em ark. Since one can reduce problem (14)—(15) to the integral form x(t) = j X( t , s)h(s)ds

a

so replacing the function h(t) by one of the functions (16) gives equations of the type 1, 6 or 11. For these equations in turn Lemmas 1-4 hold.

III. (n Sections I and II and in paper [3] is determinate the sign of the function of Cauchy type X( t , s ) of problem ( 14)—(15). In paper [3] were given the conditions for non-negative of the function X( t , s) . In this section we give another conditions which assure the sign of the function X (t,s).

For- this porpous let us present equation (14) in the form (27) L [x ] e LjM + Z J K k0( t , s ) x{k)(s)ds +

k=0 a

n — 2 m t

+

Z Z

(0

W)

j

(c s)-x(k)

(s))

k =0j

=1

with initial conditions

(28) x w (t) = 0 (к = 0 , 1, n — 2), t e E a, x (n~ l){a) = 0 , where

Z

k=0

Let Y(t, s) denote the Cauchy function of the operator L j[ x ]. Then the solution of equation

^ i M = / ( 0 - Z ^ K k0{t , s)xik){s)ds- k —0 a

n — 2 m t

~

Z Z

j

к = 0 j = О a

may be written in the form

•(29) x(f) = J F ( f ,s ) { / ( s ) - Z J K ko(s, z ) xik)( z ) dz -

a k =

0

n— 2 m s

- z ^{Z Г}

^(s)

⁽

^{(s)) +}

x(k)

k = 0

./= 1

Putting in (29) yi(t) = x (l)(t) (i = 0, 1 , n — 1) we get the system of equation ydt) = f Y {i)( t , s ) { f ( s ) - £ I K k0(s, z)yk( z ) dz -

a к = 0 a

n — 1 m s

-

Z Z

k = 0 j = 1 a

or

t n — 1 m

У* (0 = J

Z Z

a к = O j = О

(i = 0 , 1, n - l ) , where

R ikj(t,z) = - { Y (i)( t , z ) Akj(z)+ J Y(i)( t , s ) Kkj(s,z)ds},

9o(z) = z^{> Ao(z)} A n^{- 1}j(z) = x „ _ u (s,z) = 0

(fc = o , и — 1 ; j = 1, m),

h i ( t ) = J

a

The function 7 ( t,z ) of the operator L x [x] is of the form

T (t,z) 1 1 ( t - z f

---—- f N ( s , z ) ( t - s f 4 s + —---- -

(и- l) ! z (n-l

where N ( t , z ) — the solving kernel for the kernel

(30) M ( t , z ) = — ( A n_ l 0 ( t ) + A „ - . 2 o ( t ) ( t:~ z ) + ••• + A 0 0 ( t ) ( t - z r 1 )

(и- l ) ! J

Volterra integral equation [2], and its derivative has the form

Y‘ ‘4 , ’ z) = D !

j

From (30) we easily deduce that Y {l)( t , z ) ^ 0 if and only if the coefficients are non-positive, i.e. A k0(t) ^ 0 (k = 0 , n - l ) , t e l .

As s u m p t io n H6. Suppose that

1° the functions Akj(t), K kj(t, s), gj(t), f (t) satisfy (16'), 2° there is a Cauchy function T (t,s) of the operator L ^ x ] ,

3° the coefficients of equation (27) Akj(t), K kj(t,s) (k = 0 ,..., n - l , j = 0 ,..., m) are non-positive for a ^ s ^ t < b,

4° there exists a function и (t) e С” (I, R) satisfying initial conditions (28) and inequality

Th e o r e m 4. I f assumption H6 is satisfied, then x(,) (t) ^ u{l) (t) for t e l .

For proving one should apply the initial considerations of Section 3 and Lemma 2.

References

[1] Ju. A. V ied, Z. P a h y r o v , Bounded and stability for integrodifferential equations with time lag, Diff. Equations 5. 11 (1969), p. 2050-2061 (in Russian).

[2] N. A. K a sceew , Applications of Chaplygins theorems for linear differential equations, Dokl. Acad. Nauk SSSR, 111. 5 (1956), p. 937-940 (in Russian).

[3] J. M o r c h a lo , A functional differential inequality for linear integro-difjerential equation with retardation, Comment. Math. 21 (1979), p. 177-186.