On some integral formulation for differential equations with deviated argument

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXI (1979)

Pi o t r Ra p p

(Poznan)

On some integral formulation for differential equations with deviated argument

In this paper we consider a linear scalar equation with constant coefficients and constant deviation of arguments. For any root of a charac­

teristic quasi-polynomial of a differential equation we construct an integral equation of a second order, which together with some family of free terms is equivalent to an original differential equation. In the case of small devia­

tions we can easily obtain bounded solutions of a differential equation with deviated argument, defined on the whole real line or the real halfline.

1. We shall consider a differential equation with a deviated argument

(1) x ( t ) = a x ( t ) + £ b i X ( t - T i ) + f ( t ) .

i= 1

We assume that a , b i e C , t , T i e R , functions / and x are complex valued.

Here C and R denote respectively complex and real numbers. It is assumed throughout that / is continuous in the interval, where it is defined. Let y be a complex-valued function satisfying the following condition:

Function y is defined and continuous alternatively:

(i) in the interval

— max if, f2 — min

,), if max i f ^ 0, min т,• ^ 0, (a) (ii) in the interval

— max ть

if т£ ^ 0 for i = 1 ,..., n,

(iii) in the interval

— min i,), if x,• ^ 0 for i = 1 where

— oo ^ q < t 2 ^ -b o o .

If / is defined in

and x satisfies (a) and, in

equation (1), then x is continuous in

A characteristic quasi-polynomial F of (1) is of the form

A —a —

bie~Xxi.

i = 1 (

2

Th e o r e m 1.

If a function x satisfies

and is a solution of

in ( t i , t 2), then for every number X such that F(X) = 0 the function x satisfies in ( t i , t 2) the equation

(3) x{t) = y ( t ) - Y biT; j e~Xt*l~e)x { t - eT i )d e ,

i= 1 О

where у satisfies

(4)

in ( t i , t 2), with the initial condition

(5) y(to) = x ( t 0) + Y bi zi i е~Хф~в)x{to- eTi) de i = 1 О

for any t

e ( t i , t2).

P ro o f. Let X and x satisfy the assumptions of the theorem. Then П

X — a — Y bie~kz‘ = 0. Multiplying this equality by x ( t ) , t e ( t

, t 2), and enfo­

ld 1

tracting from (1) we get the identity П

x(t) = Xx(t)+ Y bi l x (t — T;) — e~XT‘x(ty\+f(t)- i= 1

On the other hand, there holds an identity d t-Ti

—— f T* Si)x(si)dsi dt J t

= J biex(t z‘ Si)x(si)dsi + bi l x(t —

— e~kZix(t)].

t

From both the identities we get for t e ( t

, t 2) the identity - 4 " [ x ( 0 - Z I bi eHt~zi - s‘)x{si)dsi' ] -

dt i=1 ,

- X [ x ( t ) ~ £ f bi eMt~z‘~s‘}x(si)dsi] = f ( t ) ,

i = i t

from which it follows that a function у defined in , t

) by the formula (6) y(t) = x ( t ) - Y S bi eA(t_T-~s-)x (si)rfsi

i = 1 t

satisfies in (f

, t

) equation (4) with the initial condition y(to) = x ( t 0) ~ Ÿ I bi eX(to - zi~si)x(si)dsi

i

— 1 tn

(

7

for any г0 е ( ^ , г 2)- Changing variables in the integrals appearing in equations (6) and (7) to 9 according to the formulae st = t-Oxi, we get (3) and (5), what accomplishes the proof of Theorem 1.

Theorem 2.

Let a function x satisfy

Let x be in (ti , t 2) a solution of equation (3), where у denotes any function satisfying (4) in (t

, t 2). Let, moreover, F (A) = 0.

Then x is a solution of equation (1) in ( t i , t 2).

P ro o f. We shall consider instead of (3) its equivalent form (6) which, after differentiation, takes the form

x (t)-A Y J bi eHt~xi~s‘)x(si)dsi - Y bt [ х ( £ - т г) - е ~ Ят‘х(£)] = ÿ(t).

i= 1 t

= 1

Substituting here a sum containing the integrals from equation (6), we get

П П

x (f) = (A— Y, bie~Xzi) x ( t ) + Y biX(t — Ti)+ÿ(t) — Ày{t).

i= 1 i— 1

Taking into account that F (A) = 0, it follows from the above and (4) that П

x(t) = ax(£)+ Y bi X( t- Ti )+ f (t ), i= 1

hence the theorem holds.

From Theorems 1 and 2 it follows that among solutions satisfying (a), equation (1) is, for every A which satisfies F (A) = 0, equivalent to the integral equation (3) together with the family of its free terms {y} consisting of all solutions of (4).

Example 1.

Let

be an equation with a retarded argument, i.e.

^ 0 for i = 1 ,..., n. Denote by q> a function defined and continuous in (tx— max г t j . An initial problem for (1) with an initial function (p consists in finding a continuous solution x^ for t > t x such that x^it) = q>(t) for re(t! — max i i5 ti] and <p(£i) = .х^^ + О). A function x defined as

(p(t) for ££(?! — max тг, tx], xv (t) for t > £i, ( p i t j = хф (£i + 0),

satisfies (a) for —c o < t

< t 2 = + oo. For an initial problem formulated in such manner Theorems 1 and 2 hold true.

■ 2. If equation (1) is homogeneous, i.e. if it has form П

(8) x(£) = ax(t)+ Y biX{t — Ti),

i= 1

then a corresponding equation (3) is, in general, non-homogeneous and has

the form

n 1

x(t) = CxeAt — Y Tf { e ' /Xiil~

)x ( t —Qx^dB,

i —^{1 О}

where Cx is a constant determined by (5) for any t

e ( t

, t 2)-

Let Cx = 0; then the above equation becomes homogeneous and can be written in the form

n 1

(9) (Lxx)(t) = x ( t ) + Y x(t — 0T^d6 = 0.

i = l о

Homogeneous equations (8) and (9) are not equivalent, therefore besides common properties they have some different ones.

Ex a m p l e

2. Let

be a /с-multiple root of a quasi-polynomial F. For any constant M a function M x pX(t) = M t p eAt satisfies (8) for 0 < p < к - 1, which can be verified by substitution. Function M x PtX satisfies (9) for 0 < p

^ к — 2 only, and does not satisfy (9) for p = к — 1. Indeed, the computation shows that

(10) ( L , M x pJ ( f ) = — - X ( " t 1) P + 1

From the fact that F is an entire function and A is its /с-multiple zero it follows that

F(X) = F'{X) = ... = = 0 and F (k)(A) Ф 0.

From this and (10) we get LxM x p<x = 0 for

^ p ^ k

and L; M xk_ 1>;

Ф 0. The set of all solutions of (9) is a proper subset of the set of all solutions of (8) as M t k~l eXt satisfies (8) and does not satisfy (9).

Ex a m p l e

3. Let ft be a fc-multiple zero of a quasi-polynomial F and let ft Ф X. We shall show that in this case M x P

p(t) = M t Pept is, for 0 ^ p ^ к — 1, a solution of (8) and also of (9). Indeed, we have

F(jff) = F'(P) = ... = F(k_1)(/i) = 0 and F(k)( $ # 0, and

H j l P

(Li M xw,)(t) = — T [ ( ' f ( i ) - I ( 4 ) t ’-

F№m ] +

A — p j

+ j ~ p { Lx M x P-i,li)(t)-

Therefore, since F (A) = 0, we have Lx M x p<p = 0 for 0 ^ P ** к - 1.

3. Let C ( R) denote a space of complex continuous functions bounded

on the whole real line R and with the sup norm. By Ax denote the operator

defined by the formula

n 1

(TAx)(f) = — Y, biti J е~Хт‘(1~в) x (t — вх^сШ.

i= 1

0

Ax maps C(R) into itself and there holds an estimation

( 12 )

П ш ^ i

i= 1

sup \e AT,V|.

O^y^ 1

In some simple cases it is possible to determine the norm \\AX\\.

Ex a m p l e 4.

For a differential equation of the form

= ax(t)f- + bx(t — x)+f{t), Ax has the form

l

(13) (Лях)(£) = — bx j e~kz(1~e) x(t — 0x)d6.

о This gives, when Re Я # 0,

1ИЯХ||

^

\ \bxe Лт(1 e)\d& =

о

b

Re X (e- Reh- l ) for x e C (R), and, when R e X = 0, we get

Ия *11 < Ы \bx\.

Consider next the function x

(t) = e ,ImAt. Calculate Ax x

from (13) and take supremum of |Аях 0 (£)| with respect to t e R . We get

Ия *oll _{Re Я} ^{b (e}

if Re Я Ф 0 and

H^Xoll = |Ьт| if Re Я = 0.

From the above, taking into account that ||x0 II = 1, we get

and

Ия II _{Re Я} ^{b (e}

if Re Я Ф 0

ИяII ^{= |Ьт|} if Re Я = 0.

4. We shall present in the sequel applications of Theorem 2 to determining solutions of (1) which are defined and bounded on the whole real line R or on the halfline, in the case of small deviations of arguments.

The solutions defined on R for various types of differential equations were considered, for instance, in [5 ]- [7 ] , [8], p. 82-85, [4], p. 58-60.

The solutions defined on the halfline were investigated in [1 ]—[3].

Let t x = —oo, t

= + o o . Applying the method presented in [8], p. 127—

128, which makes use of the Rouché theorem, we can prove the following theorem.

Theorem 3.

For any quasi-polynomial F there are a number

and its zero AT such that if |х*| ^ т < x* and x -* 0, then AT -► a + £ bt and

i

the aboslute values of remaining zeros of this quasi-polynomial F increase to infinity.

From this and (12) it follows that lim || Ax || = 0. For every q < 1 there t->0 '

is a number xq ^ x* such that |xf| ^ xq for i = 1 ,..., n then

(14) \\Akz\ \ ^ q < 1.

In the notation of (11) equation (3) has the form

(15) x = y + Axx.

This, if y e C ( R ) and (14) holds, implies that (15) has a solution which is unique and is of the form

00

(16) * = X A l y .

n= 0

Clearly x eC (R ), since C(R) is complete. This implies that if f e C { R ) and y e C ( R ) , then if (14) holds the function x defined by (16) is a solution of (1) and x e C( R ) .

If in equation (1) there occurs only the delay (x,- ^ 0), then one can also consider its solutions defined on ( — o o ,f2]- Then in place of C(R) one should take C (—o o ,t2] , a space of continuous bounded functions on ( — oo, t2] with a sup norm. Then Ax maps C (—o o ,t2] into itself and the above considerations remain valid.

Similarly, if (1) is an equation with an advanced argument (x, ^ 0), then one can consider the solutions defined on , + oo) in the space C , + oo).

References

[1] St. C z erw ik , On the global existence of solutions of a functional-differential equation, Periodica Math. Hungarica 6 (1975), p. 347-351.

[2] T. D i o t ko, M. K u czm a, Sur une équation différentielle fonctionnelle a argument accéléré, Colloq. Math. 12 (1964), p. 107-114.

[3] S. D oss, S. K. N a sr, On the functional equation dyjdx = / ( x , j;(x), y(x+h)), h > 0, Amer.

J. Math. 75 (1953), p. 713-716.

[4] A. D. M y Ski s, Linear differential equations with delayed argument, Nauka, Moskva 1972 (Russian).

[5] P. R app, On particular solutions of non-homogeneous linear differential equations with constant coefficients and constant deviation of argument, Comm. Math. 19 (1976), p. 127-138.

[6] L. E. ÈFsgol’c, S. B. N o rk in , Introduction to the theory of differential equations with deviated argument, Nauka, Moskva 1971 (Russian).

[7] G. H. R yder, Solutions of a functional differential equation, Amer. Math. Montly 76 (1969), p. 1031-1033.

[8] Ju. A. R jab o v , An application of a small parameter method for control systems with delay, Avtomat. i Telemeh. 21 (1960), p. 729-739 (Russian).