fc = 0

ROCZNTKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria 1: PRACE MATEMATYCZNE XXI (1979)

J. Morchalo (Poznan)

A functional differential inequality for linear integro-differential equation with retardation

1. The aim of this paper is a formulation of theorems of Chaplygins type on differential inequalities [2], [4] for the integro-differential equation with retardations

(1) T [x ] = x (n){t)+ £ [ A o ( 0 * (fe)( 0 + J K-koit, т)х(к)(т)^т] +

fc = 0

n — 2 m t

+ ï I [Akj (t) x (k) (ffj (t)) + j K kj (t , t) x ik) (aj (t)) dx] = / ( f )

k = O j = l tQ

with the initial conditions

(2) *“»(() = 0 (к = 0 , 1 , . . . . n —2), t e E ,o, x<-i>(t„) = 0, where

Et0 = {t: °j(t) < to, t e l = ( t 0, T } , j = 1, 2 , m], Akj e C ( I , R ) (к = 0, j = 0, 1,

K kj e C (t0 t ^ t ^ T,R) {к = 0, 1 , л - l , j = 0, 1 , m), oj e C ( l , R), a0(t) = t, Oj{t) ^ t (j =

f e C(I, R).

The results presented here generalize the results obtained in [3].

By the solution of problem (l)-(2) we mean any function x e C ? { I , R ) satisfying equation (1) with initial conditions (2).

Lemma 1. Problem ( l ) - ( 2 ) has a unique solution, and this solution has the form of the integral

t

(3) x(t) = J X { t , s ) f { s ) d s , t e l ,

12 — Prace Matematyczne 21.1

where

Х е С ? ( G, R), G = {(f*, T } x ( t 0, T ) } , t* = inf » - oo is in turn the unique solution of the equation

(4) X (n)( ï , s ) + £ [Ak0{t)X(k)(t,s) + $ K k0( t, T) X(k){T,s)dT] +

k = O s

n— 2 m f

+ Z Z [ Akj ( t ) X lk)(<Tj(t),s)+fKkj(t,T)X<k>(<rj(r),s)dT] = 0,

к = O j — 1 s

for ( t , s ) e G1 — {s $5 t ^ T, to ^ s ^ T | , with the initial conditions (5) X\k)(t,s)\t=s = ^k„-i for ( t , s ) e G2 = {t* < t ^ s , t0 ^ s ^ T}, (к = 0, 1, « — 1).

The proof of the lemma is given in [1].

Theorem 1. Suppose that the following conditions hold:

Г x{t) is the solution of problem (l)-(2),

2° z (t ) eC "( I, R) is the solution of the inequality (6) T W - f ( t ) = h ( t ) > 0 ( h ( t ) < 0) with the initial conditions (2),

3° X (t , s) $5 0 for t0 ^ s ^ t ^ Г.

Then

z(t) ^ x(f) (z(t) ^ x(t)) /or t e / . P ro o f. The proof follows from the equality

t

u(t) = J X ( t , s)h(s)ds, where u(t) = z(t )—x(t).

*0

Theorem 2. ITe assume that the following conditions hold: (i) x(t) is file solution of problem (1)—(2),

(ii) z ( t ) e C ? ( I , R) is the solution of the inequality

Т И - Z W ^ o,

with the initial conditions (2),

(iii) X w (f,s) ^ 0 for t0 ^ s ^ t ^ T (к = 0, 1 , n — 1).

Then

z{k)(t) ^ x(k)(t) /or t e l {к = 0, 1, . . .,n — 1).

P roof. From (1), (3) and (6) we have Г М - Г И = / ( t ) - r [ z ] ,

7 [ > - z ] = / ( r ) - r [ z ] ,

x ( t ) - z ( t ) = J X ( f , s ) [ / ( s ) - T [ z ] ] d s ;

*0 hence

= x »)(t)+ j X<k’( t ,s ) [ T [ z ] - / ( s ) ] d s (к = 0, 1,.... л —1).

*0

By virtue of assumptions (ii) and (iii) we obtain

z{k)(t) ^ x (fc)(t) for t e l (k = 0 , 1 , n — 1).

Th e o r e m 3. I f we assume that

(i) x(r) is the solution of problem (l)-(2),

(ii) y(t)eC"(I, R) is the solution of the inequality T { y ] - m

s* о

with the initial conditions (2),

(iii) X (k)(t,s) > 0 for t0 ^ s ^ t ^ T (к = 0, 1, ..., n — 1), then

x ik)(t) ^ >,(k)(0 far t e l (к = 0, l , . . . , n —1).

The proof is analogous as in Theorem 2.

2. In this section we shall consider a system of integral equations of Volterra type with retardation

r n — 1 m

(7) <M*) = J £ X Rtkj(t >'c)'l'k{<rj('c))dT + gi(t), t e l ,

<0 k = 0 J = 0 where

ФгЮ = 0 for £ ф1 {i = 0, 1 , . .. , n - 1 ) ,

Rikje C { t0 ^ г ^ t ^ T,R) (i, /с = 0 , 1 , n - 1 , у = 0, 1, ...,m), 0;e C ( I , R) (i = 0, 1 , и — 1),

Oje C{I, R), o0(t) = t, O j { t ) ^ t , ( j = \ , . . . , m ) .

Lemma 2. The system of integral equations (7) has a unique continuous solution Ijjft) for t e l . This solution is the limit of the sequence of the

Picard successive approximations

<А«о(0 = 9i(t),

t n — 1 m

Pipit) = 0i(O+ ( I I Кцс](*,т)Фкр-i((Tj{T))dT, t0 k = 0 j= O

Ф(р^) = о for t $ I {p = 0, 1 , . . . , n , . . . ; i = 0, 1 , 1 ) .

If, moreover, Rikj( t, x) ^ 0 for t0 ^ т < t ^ T and ^ 0 for t e l , then Piit) > gi(t) > 0 for t e l {i = 0, 1, . . . , n - 1).

Proof. It is sufficient to apply a theorem given in [5].

Lemma 3. If the functions of system (7) satisfy the following conditions (i) Rikj{t, t) ^ 0 for t0 ^ t ^ t ^ T,

(ii) Æ(fcj(t, t) = 0 for t < t,

(iü) MO ^ 0 (^ 0) for t e l , g f a it)) = 0 /or а ^ ) ф1, g f a i t ) ) = 9 d p jit)) M a j ( t ) e l ,

then

hi t) ^ ^ ( î) ^ ôf£(t) (&(г) ^ (t) ^ M 0 ), where iM 0 is a solution of the system of integral equations (7).

Proof. Denote by Aa the operator on C ( I ) defined by the formula

t n — 1 m

( At< / 0 ( 0 = j Z 2 ’ T) (о-j (t>) .

t0 к = 0 j' = 0

Acting on both sides of (7) with the operator I + Aa (where 1 denotes the identity operator), we obtain

(8)

t n — 1 m Oj(x) n — 1 m

•m o = ! Z Z ^ W l -o } J Z Z Rirp(^ji't),s)\i/r{cTpis))ds} dT+ hiit),

t Q k = 0 j = 0 t Q r = О p = 0

where

t n — 1 m

( 9 ) M O = 0 « ( O + J . Z Z RikjityT)gi(<Tjix))dT.

tQ k = 0 j = 0

In virtue of the theorem on inverting the order of integration, from (8) we have

(i° )

t n — 1 m t n — 1 m

Pi it) = J Z Z {f Z Z Rikj(t, s )Rirp((Jj (s), x)ds} x!jr(op{'t))dT + hi {t).

tQ r = 0 p = 0 t k = 0 j = 0

The systems of integral equations of Volterra type (7) and (10) are equivalent.

In view of Lemma 2 and (10) we obtain Фi(t) ^ hi(t) ^ 0,

and, in view of (7),

«MO < MO-

This completes the proof.

3. Write

l^jto (01 +^4fco (0

A k0 (0 ^{Akç> (0} l^fcO (01 — ^fcO (0

K k+0 (t,s) = \Kko (t ,

s)|

s)

^-fcO (L 0 —|Xfe0( L O | - ^ k o ( L O

А Ш \Akj(t)\ + Akj(t)

Akj (0 \Akj(t)\ — Akj(t)

K kj(t,s) = ^{\K-kj (t, S)\} + K kj(t, S)

K kj{ t , s ) \Kkj( t , s ) \ - K kj(t,s)

Then we can write (1) as follows:

(1*) x<->(0+ £ [Лк+О( 0 * (к)(0 + } K U t , T)xw (x)di] + k = 0

n —2 m t

+ I I [

+ J

,

) x(fc) (<Tj (

))

‘ = oi =i

=

Z

к = 0

n — 2 m t

+ I I [4fcj(0 *(k)(ffj(0) + I Kjÿ(?’ 0x(k)(a J(T))dxl + / ( 0 » t e l ,

k = 0 j = l t 0

x ik)(t) = 0 (fc = 0, l , . . . , n - 2 ) , t e E to, x(n_1)(t0) = 0, or

n — 1 m t

(il) x(n)(o+Z Z [45(o*(fc)fo(o)+1^k,(L0^(fe)(^(0)^]

n — 1 m t

= Z Z [4y (0*(k)(<L(0) + J ^k}

0

(s))

,

k = Q j = Q t Q

xm (t) = О (к = 0, 1, —2), te£ ,„ , ₌ 0,

where

a0(t) = t, 4 +- u ( 0 = 4 .-1 j(t) = t) = t) = 0

( j = 1, 2 , m) .

Let 2f+(t,s) be the Cauchy function [6] associated with the integro- differential equation

n — 1 m t

X{n)(t)+

Z Z

k = 0 j= 0 t0

with initial conditions

x'*>(f) = 0 (k = 0 , 1 , . . . , « - 2 ) , t e £ , 0, x<"-‘ 4 fo ) = 0 .

Then we get an equation equivalent to (1*) with the initial conditions (2)

t n — 1 m

(12) x ( t ) = f X + ( M ) { I £ [^(s)x<*»(<Tj(s)) +

t0 k = 0 j=0

S t

+ j K kj(s, t) x(k)(gj(t))dr] } ds + § X + ( t , s ) f ( s ) d s , t e l ,

l 0 l 0

x«'(t) = 0 (к = 0, 1. . . И- 2) , t e E tQ, x * - l)(t0) = 0.

We reduce problem (12) by substitution x(k)(t) = yk(t) (к — 0, l , . . . , n —1) to the following system of equations

t n — 1 m

(13) y t (t) = f X ^ ( t , s ) [

Z Z

t0 k = 0 j=0

+ J K kj{s, т)ук(<х,(т))^т]} d s + J X {$(t, s)f(s)ds, i ^ n - 1,

*0 fo

or

t n — 1 m

(14) >’«•(*) =

J Z Z

t0 к = 0 j= 0

where

(15) £ * ,( !. s) = AT'i*((, s) ^ ( s )-i- j А-ф(r, x) АГад (т, s)dr,

s

9i(t) =

J

*0

Theorem 4. I f X (l2 { t , s ) ^ 0 /or t0 ^ s ^ t ^ T" (i == 0 , 1 , . . . , и — 1), then X {i)(t, s) ^ Х (;Чс s) ^ 0 for t0 ^ s ^ t ^ T.

P ro o f. In view of Lemma 2, system (14) has a unique solution _у*(г) ^ 0 (i = 0, 1 , n — 1) for arbitrary f ( t ) ^ 0. Since x {k)(t) = yk(t), we conclude

that x (k)(t) ^ 0 (к = О, 1, ..., n — 1). And from (3) we obtain x(t) = J X { t , s ) f ( s ) d s .

l 0

Differentiating i-times both sides of the above equality we obtain t

x (l)(t) = J s)f(s)ds (i = 0, 1, n — 1)

or

(16) y i ( t ) = j X {i)(t, s)f(s)ds.

Comparing (14) and (16), we see that

t t n — 1 m t

J X {i)(t, s) f( s)ds = $ X I Rikj( t , s ) y k((jj{s))ds+ J X ^ { t , s)f{s)ds.

t k = о j= о Hence

$l XM( t , s) - Xy( t , s) - ] f ( s) ds^ 0

for arbitrary f ( s ) ^ 0. From this it follows that X (l)(t, s) ^ .Y + ^ s ) ^ 0 for t0 ^ s ^ t < T. This completes the proof of Theorem 4.

Write

Dj = { t e ( t 0, Г ) : (Tj{t)e(t0, T>}, Ej = { t e ( t 0, T>: Gj{t)^<tо, T>},

1 if s e D j ,

“ ' ( s ) = ^0 ; = о л ... m-

Theorem 5. / /

% i к м * : » * ] . , ( . ) * «

then X {$ ( t ,s ) ^ 0 for t0 ^ s ^ t ^ T (i = 0, l , . . . , n —1).

P roof. Put in (4) instead of Akj{t) and K kj(t, t) respectively, 4 5 (t) and Kfj(t,T) and integrating «-times; then

a?)

4 z£^ - i 4 r ! ^ - { " i ‘ £ 45(

) * ¥ ^

.*)+

( « - 1 ) ! (n—1)! k = 0 j = 0

+ j ^{K î j}(т, rç) (ffJ- fa), s) d r j ] d x,

where Л„+_ и (г) = Х„+_^-(г, т) = 0, 7 = 1,2, By differentiating (17) i-times we have

X y ( t , s )

(n — 1 — i) ! - 1

( f - T ) " - 1 - '

(и — 1 — j) !

n - 1

! Z

m

Z [ 4 5 (t)X (^ )(o-j(t) ,s) + j =0

+ S K kj (t, * (+ (ffj 0?), s) <fy} (1 = 0 , 1 , . . . , n 1), using the substitution X {+](t,s) = Yk+ (t,s).

In virtue of the theorem on inverting the order of integration we obtain

/ у __„yi — i — 1 t n — 1 m

(18) Yt+ (t, s) = ———— - — S £ £ Rikj (t , x) Yk+ (tTj ⁽t) , s) dx,

V 1 1 0 - s k = 0 7 = 0

where

Rikj(t,*) ( t - T ) n~ 1~ i

(w — 1 — i) ! 43 (t)+ j T

( t - x f ~ 1-f

Kkj (ту rj)drj.

The system of equations (18), for each value of parameter s, is equivalent to system (7) for which Lemma 3 is valid.

Conditions (i), (ii) of Lemma 3 are true for system (18). We show, moreover, that (iii) of Lemma 3 holds. Indeed, the inequality X {$(tyS) ^ 0 for t0 ^ s ^ t ^ T holds when

(19)

Let

(п — 1 — i)! j t

+ 1^-

t

n — 1 m

1 i

( г - x f - ' - '

(n — 1 — i) ! 4 у (т ) + (t- т ) " - 1-*

(n — 1 — i)\ Kl} (T, n)dr\ (<T,.(t) -s)" 1

(n — 1 — i) ! dx ^ 0.

K (t) -s)+ (Jj(x)-s 0

if Oj(x) > s,

if Uj(x) ^ s, ^{, m .} From the inequality

(t — s)”~ 1 -l

(20) - 1

П - 1 m (t — r\n~ 1

<” — 1 — 0 ! ; L » . o j=o ( И - 1 - 1 ) !

* (f —xY, - 1 - i

K5 (T' (n — 1 — i) ! dx ^ 0

we have (19)

Let (сг(т)—s)+ = max (сгДт)—s)+ ; then inequality (20) is satisfied when

t e l

(21) ( t - s f 1 -i ^t {n— 1 — i)\ 1

t ( n - 1 - 0 ! Inequality (21) is satisfied when

(22)

n - 1 m (t _ yi - l -i

+ J {t- х Г 1 - 1 ^ +K kJ{x, rj)drj ((t(t) -s)+

( и - 1 - 0 ! dx ^ 0.

t (а(т) — ч)п~1 - i (t — тУ~l ~‘ "~1 m f

holds for s ^ x ^ t ^ T.

Form the inequality (23)

(ff(x)-s)n+ i ~l ( t - x ) n~ i ~l ".Z1

1 I - 1 - i

Z Z

+

(T »

( ^t )

^ 1

s k = oj=o

we get (22). Since the function

( a

( t )— s)+“1 ~ ‘

—

1 _ ‘

F(t, x,s)

(n—l — i)l(t — s f— 1 — i

is increasing in t and is decreasing in s, then inequality (22) is satisfy, when for t0 ^ s ^ x ^ T we have

(24) f ( Т - т У - ' И т Ь О Г ' - ( T —toT~l ~'(n— 1 — /)!

n — 1 m f

x Z Z M (T)+ f xi5 (T>

^ 1.

This completes the proof of Theorem 5.

Re ma rk 1. Since (<t(t) — t0)+ ^ x — t0, inequality (24) is satisfied if

T n — 1 m T

(25) J ( Г - т У ‘''(т -Г о Г ’" I I K W + J

to ^{к =} 0 j= 0 t

$ ( T - t or 1_' ( n - l - 0 ! , or

T n - 1

J Z Z

+ J

(T »

(

)

^

г0 k = 0 j= 0 t (T -to )2'" -1-'1

Re ma rk 2. Let Akj(z) + j K kj(z, t])drj ^ М, where М is arbitrarily Г

fixed constant; then inequality (24) is satisfied, when M ( T —t0) ( m + \ ) n

( T - t 0)2(n- 1 - i) Re ma rk 3. Let

n — 1 m

max X Z [Ли(т)+ I *чНт, ^JcOjCr) = M,

те/ к = 0 j = 0 T

where M is arbitrarily fixed constant; then inequality (24) is satisfied if M ( T - t0r '■ ^

(2n — 2i — 1)!

(n — 1 — i) ! c1).

Re ma rk 4. Let in the integro-differential equation (1)

Ako (t) = 0, K k0(t, i) = 0, K kj(t, t) = 0, Akj(t) = 0 for k = 1, 2 , n — 2; then we have the results obtained in [3].

References

[1] Ju. A. V ed\ Z. P a h y ro v , On the boundedness and stability of the solutions of integro-differential equations with deviating argument, Differencialnye Uravnenija 5. 11 (1969), p. 2050-2061 (Russian).

[2] B. N. B a b k in , On S. A. Caplygin's theorem on differential inequalities, Mat. Sborn.

46. 88 (1958), p. 389-398 (Russian).

[3] Ju. I. Z u b k o , On differential inequalities for linear differential equations with deviating argument, Differencialnye Uravnenija §. 3 (1972), p. 534-538 (Russian).

[4] N. A. KaSCeev, The exact limit for applying the theorem of S. A. Caplygin for .linear equation, Dokl. Akad. Nauk SSSR 111. 5 (1956), p. 937-940 (Russian).

[5] N. V. A zbelev, L. F. R a h m a tu llin a , On the linear differential equations with deviating argument, Differencialnye Uravnenija 6.4 (1970), p. 616-628 (Russian).

[6] N. M. M at wee v, Integration methods for ordinary differential equations, Warszawa 1970 (Polish; translation from Russian).

(Ь From (25) and j(b — t)m(t — a)kdt k! ml

(k+m+ 1)! (b-af