In the present note we shall approach this problem differently, and solutions will be given in the form of definite integrals.

ANNALES SOCIETATIS MATIIEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X IX (1976) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PEACE MATEMATYCZNE X IX (1976)

Piotr Bapp (Poznan)

O n particular solutions of non-liomogeneous linear differential equations with constant coefficients and constant deviation

of argument

1. Concerning linear differential equations witli constant coefficients and constant deviation of argument, their solutions may be obtained, for instance, in the form of a periodic function [1], or by a Laplace trans­

form, [1], [2]. Problems connected with approximate solutions of dif­

ferential equations with deviating argument are considered e.g. in [3], [4].

In the present note we shall approach this problem differently, and solutions will be given in the form of definite integrals.

2. Let A , B {, i — 1, 2, ..., m, be square matrices of degree n with

complex coefficients, G(B) — a space of bounded continuous functions on В = real line mapping В into Gn, where Gn is a product Cn = C1 x ...

G1 of n complex planes G1. diag[«q] denotes a diagonal matrix with respect

j

to j = 1, ..., n. We shall use the following standard norms:

for a vector œ = (a?1} ..., xn)eGn ||я?|| = тахЦ-1,

j П

for a matrix A = [akj] \\A\\ = max V \akj\, k j t i for a function feC (B) ||/|| = su p ||/p )||.

teR

All eigenvectors considered in this paper will be chosen so th a t for each of them an element with the largest absolute value be equal +1.

We shall deal with the following equation with deviating argument m

(1) x(t) = Aæ( t ) + 2 B M t - T i ) + g ( t ) ,

i = 1

where x: B ->C, В for i — 1, 2, ..., m. We shall assume th a t g is con­

tinuous on B, or if necessary, gtG{B). We shall further consider the properties of functions xs continuous on В and satisfying on В equa­

tion (1) and the condition

(2) ®a(«o)

—

t0€ В .

1 2 8 P . R a p p

Quasi-polynomial characteristic of (1) has the form

m

<3) det [A + J ? B ie~Xxi - A2?) = 0,

i =1

where E is a unit matrix.

Let A2, ..., l n be different roots of (3), and W 1, ..., W n — linearly independent vectors such th at

Ш

(4) ( л + ^ В (е~гП - ^ Е ) ш > = 0 , j = l , . . . , n . г = 1

We shall show, taking for instance real Ay’s and TP’s, th at the above assumptions are reasonable. To this end we shall prove the following

m

Theorem 1 . I f a m atrix А ф ^ В { has n different eigenvalues , . . . , rjn,

г= 1

then there exists real £ > 0 such that for |тг-| < £, i — 1, . . . , m, there is n

■different real numbers

A2

(3)

P ro o f. Let F 1, . . . , Vn be vectors in R n. Let [F 1, ..., Vn] denote a m atrix whose columns are given vectors. Assume th at y1 < % < • • • < yn-

Ш

The eigenvector of A -j- ^ B { corresponding to the eigenvalue rjj we denote г = 1

W3 V. Then det[TF*, ..., IF”] Ф 0. Let <5 > 0 be such th at if for each j = 1, ...,

there holds the inequality \\V} — TF^|| < <5, then det [F 1, ..., Fn]

Ф 0. Such a ô exists, since a value of a determinant is a continuous function of its elements.

Consider the expression

m

d e t(j. + y = 0 .

i = 1

m

For every fixed A the number

fa is j-th eigenvalue of

+

i =1

'The eigenvector corresponding to the eigenvalue у fa we denote as Wfa.

Hence we get the equality

m i

(a + у B t e - h t - у ( ц Е ) W { t) = 0

г'= 1

for j

1, 2,

Eigenvectors and eigenvalues of a matrix are contin­

uous functions of its elements, so we find a real

0, th at if

1| < e, then

— y f <

< |miin \rjk — rjt\ and \\Wfa — TFJIK <5. Let £ be such

кф1

a real number th a t for Ae —

, yn A c ] and |тг-| < £ holds the inequality

N on-homogeneous linear differential equations 1 2 9

\е~*ч —1\ ^ £. Solving these mec^nojlities we infer tlmt ^ ^ In (1 -f- f ) / /(c + max (1^1, \Vn\))' Let \ri\ ^ £• Then a mapping As->у(ц for each j = 1, ..., n is a continuons function mapping the interval [rjj — c, rjj + e]

into itself. From Brouwer’s fixed point theorem it follows

th a t there exists a point Aj€ [rjj — c, rjj + c] such th a t Aj = y(i.). The numbers Aj satisfy (3) and are different, since the intervals [rjj — c, rjj + c] are panwise disjoint. Vectors W3 = WfXl) satisfy (4) and are linearly independent, because IIT ^ -W jH < <5, hence det [T F ^, ..., TFfo] = det[TFx, .. . , TF”]

Ф 0. The theorem is now proved.

Let W = [TF1, ..., IF™]. We determine a matrix H, of whose eigen­

values are , ...,

and corresponding eigenvectors

..., TFn. From the definition of a canonic form BL — TFdiag[Ay] TF-1.

3

This together with (4) gives

m

H W = W diag [Aj’] = ^TF + £ B i TFdiag[e~Vi],

i i=\ j

hence

Ш

(5) H = A + £ TF diag [e“ Vi] TF-1.

i t i j 3.

2. Operator L defined by:

m t~ri

(6) Lf(t) = f 1Fdiag[eV*-Ti-«>] W~lf{u)du

i—l t i

maps G(B) into C{B).

P ro o f. ЪеЬ feC(B). I t suffices to show th a t уз = AfeG{B). On account of formula (6) we get

Ш

(7) №№11 < W f W - m - IITF-'H J T ||B(||-|ri|-max||cliag[e,<(‘-'< -“)]||,

i=l u i

where maximum is taken with respect to

(8) Ue[t — r{,t] , if r ^ > 0 , or ue [t, t — rf}, if тг- < 0 . Let a{j denote the number

a^ = max |eV<-Ti-M)| = т ^ хе а^1~т~ и\

U U

\vhere Aj = a} + i • fa, and и is defined in (8).

Then

1 foi* OLj > 0 ,

e~a3ri for ajTi < 0.

9 — R oczniki PTM — P race M a tem a ty czn e X IX

130 P. R a p p

We hence get

(9) max||diag[e^(<-Ti-M)]|| = ||diag[a^]|| = m a x a y .

U j j j

Substitution of (9) into (7) gives

(Ю) ы Ь \\ < M -\\f\\,

where

га

(1 1 ) M = ||T F |N |W - 1|| У р у - | т г- | т а х ^ < oo.

i=i i

Passing in (10) to the lower upper bound we get IM! = s u p | H * ) i K ^ J / l l < oo-

UR

Therefore y>eC{B), and the proof is accomplished.

Theorem 3.

I f

are different roots of a quasi-polynomial

and W 1, ..., W n are linearly independent vectors such that formulae (4) hold, and i f functions g and r are continuous on B, then the function æs defined as

t

(12) ws{t) = f T P d i a g W ~xr{u)du

<o i

is a solution of equation (1) i f and only if, on В is satisfied the equality

(13) r = Lr + g,

where L is an operator defined in (6).

P ro o f. Let r be such a function th a t <ca of (12) is a solution of equation (1) on В and satisfies condition (2). Then we have the equality:

Ш

(14) x8(t) = A x a{ t ) + ^ В & у - т д + д у ) , t e B .

i = 1

Substituting to (14) xs defined by (12) we obtain, for each te B, an equality :

m

r(t) = g(t) + (^4 + ^ B f w diag [е~л?'г*] TF-1 — H^æs(t) + L r{t).

i = l

i

Taking into account (5) we obtain (13), which is, in view Theorem 1, reasonable. Equation (13) was obtained from (12) by equivalent tran s­

formations, and therefore, the theorem is proved.

4. Let {Hv} be an infinite sequence of complex square matrices of degree n, and let Xvx, .. . , Xvn be eigenvalues and W\, ..., TF” be correspond­

ing eigenvectors of a m atrix H v.

N on-homogeneous linear differential equations 131

Let W v denote a matrix, whose columns are vectors F P j,...,T F ”.

We assume th a t numbers ..., Xvn are different for every v — 1,2, ...

The sequence {.H„} is defined by recurrence formula

m _ A* - i T

(15) H, = A + y B 1W._1diag[e 1

<=i

i

where

Ш

(16) H 1 = A Jr У\ B iP i

i= 1

and P f s are some fixed matrices defining the first element of {Hv}

In this paper we shall not be concerned with the investigation of the convergence of the above defined sequence. Examples of converning sequences {Hv} can easily be obtained by assuming th at matrices A, in (15) and (16) are diagonal and taking appropriate elements of P { and appropriately small elements of B { and sufficiently small deviations ri .

Let the sequence {Hv} converge to some matrix _H*. Then {A}} con­

verges to some number A*, and {Wl} to some vector TFÎ for j = 1, ..., n.

Let numbers A* be different. Then they are eigenvalues of Л*, and vectors W£, ..., FT* — corresponding eigenvectors of fi*. We shall th a t A*, j are roots of a quasi-polynomial characteristic (3), and vectors IF*, j = 1, ..., n, satisfy equations (4). Passing to the limit in (15) and taking into account th a t И* = Ж* diag [A*] W z 1 one obtains

3

m _ л*

IF* diag [A*] w z 1 = A + У B {W* diag [e J H] IFF1-

i

i

Multiplying this equality on both sides by IF* and comparing columns of in such a way obtained matrices, we verify th a t

m *

(17) ( i + ^ V sT , - x;e ) w i = o, j =

1 , 2 ,

i = l

Because th e vectors IFÎ are non-zero, equations (17) can bold if and only if

m - Л * т -

d e t(A + ^ B {e j 4 — A* = 0, j = 1 ,2 , ..., w.

i = l

This equality, together with (17), accomplish the proof of the theorem.

5.

In this section we shall present some considerations concerning

convergence of recurrence sequences, and their applications to solving

equations in Banach spaces. The considerations will be applied in the

sequel for constructing a particular solution by successive approximations

method.

132 P. E a p p

Let B be a Banach space, B {X ) a space of bounded linear operators mapping X into itself. In monograph [5] (p. 278-279) the following theo­

rem is jiroved

Th e o r e m 4 .

I f X is a Banach space and i f the operator L e B ( X ) is a norm limit of a sequence of commonly bounded operators {Ln} c B(X),

IlL n\\< q < 1, then the equation

(18) x — Lx + x^

has a unique solution being a limit of a sequence {xn} of the approximating equations solutions:

Xn Ln xn -f X

, n 1 , 2 , . . .

Basing on this theorem we shall prove the following

Th e o r e m 5 .

I f the hypotheses of Theorem

are satisfied, then the se­

quence {yn} defined by the recurrence formula

(19) yn = L nyn_ i+ ^ o , y0 = æ Q1 n = 1 , 2 , . . .

is norm convergent for each x {) e X, and Us limit is the unique solution of equation (18).

P ro o f. We shall demonstrate th at

1° the sequence {yn} is bounded for each x0, 2° \\Уп ~~Уп-г\\ as n -+ o

3° from Theorem 4 and 2° it follows Theorem 5.

1° Applying (19) и-times лее infer th at yn in the sequence {yn} is equal to

Уп — A* • • • L 1 x0 + ... + L n. . . L {x0 + ... + L nx0 + x0.

This yields the estimation

\\yn\\ ^ (IIAJI- • • IIAII+• • • + I A ll • • • IIAII+- • •+IIAII + 1) M

1 — qn+1 1

< (gn+ . . . + î + l ) I W = —;--- I M O - .--- IKII-

i ^{- q} i

This inequality holds for every n, and hence the sequence {yn} is bounded.

2° It may happen th at in the sequence {yn} appear groups containing a finite number of equal, successively following elements. From each of such groups, if any, we remove all elements but one.

The remaining elements will from a subsequence {ynJ of {yn}. In this subsequence no two neighbouring elements are equal. I t is easy to notice th a t:

(2°) (\\Упр-У п

^ 9 ? as

О (ЦУп-Уп-Л ->0, as n-*oo),

N on-homogeneous linear differential equations 133

and th at {yn^ may be formed by the recurrence relation

( 2 4 ) УпР -^ПуУnv_i œ0 , Уп0 *^'о г v 4 > ^ > • • • ?

where {.Lnv} is a subsequence of {Ln}.

To continue the proof of 2° let us assume that

(2 2)

as v -> с».

Making use of (21) we verify th a t for each v there holds the equality

• V 3 4 У п р — l ) У п р) У п

„_1

Then, taking into account th a t \\уп„ —

0 for each v, we get

(23)

H^»v + J У п Р II

11Ч -Я -и 11

У п р

II

From assumption (22) it follows th at the sequence {\\уПр~Уп,,_1\\}

may have more than one accumulation point.

There are two cases, i.e.

1. none of the accumulation points is zero, 2. one of the accumulation points is zero.

We shall consider both these cases.

Assume then th a t none of the accumulation points of the sequence

{ \ \ У п Р ~ 2 / n v_ 1ll}

is zero. By (22), taking into consideration th a t {AnJ is

norm convergent, the term on the right-hand side of (23) tends to zero as v — > oo. We have therefore

II - ^ П р ( У п р 2 /n „ _ 1)ll l i m --- = 1 .

* - > 0 0 11У й, + 1- У« „ 1 1

We deduce from this there is a sequence {gP} of real numbers such th a t \\ynv+1- y nJ\ = gA Lnv{ynp- y nv_^\\ and g , -+1 as v oo. Then

( 2 4 ) \ \ УпР + 1 - У п г \\ < g J L n J l \ \ Уп р - У п Р_ 1 \ \ < 9 г * \ \ У п р - У п Р_ ^

Since q < 1, gv ->1 as v -> oo, then there is a natural number v0 such tth a t for v > v0 it is q0 = supgvq < 1. From this and (24) it follows th a

’’>”0

\ \ УпР + 1 - У п „ \ \ < % \ \ У п р - У п р _ г \ \ , V > V o -

For any s > 0 there exists therefore a natural s, such th at

%+8+]

%+ i Уп”« < e,

and this means th a t \\ynv — || ->0 as v -> oo, hence contradiction

Consider now the second case when the sequence {\\уПу — ||} has

a zero accumulation point. On account of (22) and of boundedness of

134 P. R a p p

th e sequence {s/w}, the sequence {\\yn — ||} has besides at least one non-zero accumulation point. Let e > 0 be sufficiently small fixed positive number such th a t in the interval ( — s, e) there is no non-zero accumulation point. Denote Ayv — уп„ — Уп

and F e will denote a set those elements of Ш п .-У п ^ И } = {IИуЛ} for Which \\Ayv\\< e. Removing from {\\Ayv\\}

all elements of F e we obtain a subsequence {\\AyV]^\} without non-zero accumulation points, i.e. for each к holds the inequality

(25) \\AyVk\\> e.

On account of (23) we have for each к

(26) \\AyVl

< --- *--- ||у I I\AyvJ

Passing to the limit as к ->

by (25) the right-hand side of (26) tends to zero, hence

(27) lim

k-+oo

114,-1 ЛУ’г

I = l .

Let k0 be a sufficiently large natural number. Reasoning similary as in the previous case, from (27) we deduce th a t for each к > k0 there holds the inequality

(28) U y Vj}\< qoWAy^W.

If \\Ayv г\\е then from (28) and the definition of F e we get I\AyVk\\< e , k > k0.

If, on the other hand, \\Ayv x\\4 F e, then for k > kQ there is a natural s0 such th at

U yV]} \< qso\\^yVks 0\\

and \\AyVk_So\\€ F e. Therefore there holds the inequality 2o-£< e.

I t follows then th a t {\\Ayv\\} tends to zero, contrary to our assumption.

Therefore p a rt 2° of the proof is shown.

3°

Theorem 4 and (19) we get

’^П o?

Уп ^ п У п - i A - ^ o ’

Hence

Уп - ^ n i ^ n У п — l) y n ) A - L n { y n y n —i ) f IК - У п \ \ < 0 . \ \ ® п - У п \ \ + < 1 \ \ У п - У п - 1 Ь

\K - ^{y J} <

Non-homogeneous linear differential equations 135

In view of the fact th a t \\yn — yn-i\\ ->0 as n->oo,

(29) \ K - y J - + ° as №->oo,

Let a? be a solution of equation (18). We have then inequality (30) II

- 2 / J K W œ - æ J + l K - y J .

By Theorem 4, \\x — con\\ ->0 as n-^oo. Taking into account (29) we get from inequality (30) th a t \\x — yn\\ -» 0 as n -> oo. Therefore {yn} is norm convergent, and its limit is the element x, being the unique solution of (18).

This ends the proof of Theorem 5.

6

. Considerations of Sections 4 and 5 will be useful in constructing a particular solution of equation (1) by successive approximation. To this end we define an infinite sequence of differential equations without deviation of an argument such th a t their particular solutions xv8, v — 1 ,2 ,..., under some assumptions, form a sequence almost uniformly convergent to the function xs, being a continuous solution of (1) an R and satisfying condition (2). The solution xs will have form (12). Assume th a t geC(R).

Adopting all notations and conventions of Section 4, we define xvs,

p

= 1,2, ... , in {a?*} as a particular solution of the equation (31) xv(t) = B vxv{t) + rv{t), v = 1 , 2 , . . . , which we obtain from the equation

(32) if(t) ^Aæ"(t)+ *=1,2,...,

г= 1

in which, for i = 1, 2, ..., m it is substituted

( 3 3 ) x 1 ( t Tj) = Р г- я 1 ( * ) ,

* = 2 , 3 , . . . , where we denote, in equation (31)

r M = 9(t),

(34) rv(i) = g{t) + ( A - H v)xvs '(«) + ^ B tx vs 1{ t - r i),

v = 2 , 3 , ..., i=l

and the solution x vs of (31) is represented by

tr, i

x vs(t) = J W vdiag[e*j{t w)]Wv 1 rv(u)du, v = 1 , 2 ,

(35)

136 P. R a p p

From (35) it follows th a t to determine {xvs} it suffices to define the sequence {rv}. Substituting therefore (35) to the second relation of (34), for v = 2 , 3 , . . . , we get

_/ —1

(36) r,(t) = [ A + ' 2 B iW,_ldiag[e 5 '*]>ГгЛ.тя .) * Г , № +

i= 1

}

t — T„-

+ 9(t) + X Bi I diag[eAj (< 4

i= l t 3

Substituting now (15) to (36) shows th a t (34) may be transformed to the form

(37) * 1 = 9 ,

= 0 + A - i V - u v = 2 , 3 , . . . ,

where L v is an operator defined by:

m t~ri v

(38) WA™g\.ef i ~4^ W ; lf(-u)dii.

i= 1 t

Let В [0(B)] be a space of bounded linear operators mapping 0(B ) into itself. From Theorem 2, by the convergence of {Hv}, it follows th a t {.Lv} с: В [0(B)] and th a t {Lv} is commonly bounded, satisfying in B [C (B )]

the Cauchy condition. The completeness of В 10(B)] implies now th a t

*

lim

B[C(B)], hence

is norm convergent in

[0(B)],

V—>OQ

Consider now th at starting from some v0 there holds the inequality (39) \\Lv\ \ < q < l , v > v 0.

We shall show such cases (39) actually exist. One account of Theorem 1

m

and considerations in Section 4 we may assume th a t the matrix A + I B ,

i —l

has different eigenvalues and th a t у > 0 is such a real number th a t for

|гг| < y H* has different eigenvalues.

We shall show for each q < 1 there exists such a number 0 < £ < y th a t if |тг-| < Ç, then ||X*|| < q < 1. Indeed, if H v ->Я*, then AJ — a]+ ft]

-> a* -f i • ft* — À*, and denoting a*j = max e j (* 4 u) we can, according U

to Theorem 2, estimate the norm of X* in the following way

m

(40) IIL.II < M * = ||Ж*|М|Ж*-1| | - 2 1 \\В{\\- |т<| -max a*.

i —l 0

Ш

< sup IITW^fl * \ \ W ^ t У 11-8*11*

- m ax

.

K K ? i= i 3

N on-homogeneous linear differential equations 1 3 7

Taking £ = min(#/ sup ||TF*||-||TF* ||FJmaxa*-, у) from (40) we-

|Tjl<y г=1 j

deduce th a t for |т*| < C it is ]|.L*|| < q < 1. Since the norm is continuous then there is a natural v0 such th at (39) holds. In view of above consider­

ations and the remarks concerning of {Hv} it is easy to see th a t there exist examples confirming validity of assumption (39).

Assume then, th a t (39) holds. From the completeness of G(R) and from Theorem 5 it follows th a t the sequence {rv} defined by the recurrence formula (37) converges in norm in the space G(R) to some function r*eG(R) satisfying the relation

(41) r* = L*r*+g.

I t follows then th a t {a?*} defined by (35) is almost uniformly convergent on R to a function x* defined by

t *

(42) x*{t) = J TF*diag[e j(* u)] W*l r*(u)du.

<o j

It is evident th a t x* is a continuous function on R and the condition, a?*(t0) = 0, and so, in view of Theorem 3, is the solution of equation (1)..

7.

Consider the equation

(43) x(t) = x(t)-\-ax(t — T1)-Jrbx{t — T2) Jrg{t), a, beR. Formula (15) takes the form

(44) A” = 1 + ae~xV~lr' + Ъе~к’~1х*

and it is also a characteristic quasi-polynomial of equation (43). Operator L v is defined by

Tj t~~

(45)

= a J e*V(i~Tl~u)f(u)du-]-b J exV(t~t2~u)f(u)du.

t t

^Note th a t L v transforms a constant function into a constant function..

Let

= const. Then

(46) L a Av+1 —(1 + a + b)

I v v = 1 ,2 ,

Take a = b = xx = 2, r2 = —1, g(t) = c. For A1 = 0.5 the sequence- {A”} of (44) converges approximately to A* = limA” A15 = 0.77989.

From (40) we get M яа 0.77265 < 1. Hence we get the estimation for

j

C* = limL v, ||X*||< Ж* ^ 0.77265 < 1. Therefore (39) is satisfied. The

v~>oo

sequence {rv} defined by (37) is convergent.

Assuming th a t g(t) = c, from (46) we get that. limr„ = r* == const.-

V^{—> -00}

In this simple case r* can be easily determined directly from (41), not

138 P . R a p p

necessarily through determining successive terms of {rv}. Equation (41), takes the form, if we consider (46),

By Theorem 3, the solution of (43) such th a t æ*(t0) = 0 is of the form :

A c k n o w le d g m e n t. The author wishes to express his gratitude to Professor Julian Musielak for his valuable conversations which helped to improve the proof of Theorem 5.

[1] Л. Э. Э льсгольц, С. Б. Н оркин, Введение в теорию дифференциальных ур а ­ внений с отклоняющимся аргументом, Наука, Москва 1971.

[2] R. B ellm a n and К. L. Cooke, Differential-difference equations, Acad. Press., New York-London 1963.

[3] Э. M. М аркуш ин, G. H. Ill им ан о в, О сходимости оптимального уравнения сче­

тной системы дифференциальных уравнений, Диф. Уравн. Том 2, 3 (1966), р. 314-323.

£ 4 ] ---Приближенное решение задачи аналитического конструирования регулятора для систем с запаздыванием, Автомат, и Телемех. 29, 3 (1968).

£5] D. P r z e w o r sk a -R o le w ic z and S. R olew icz, Equations in linear spaces, PWN, Warszawa 1968.

A* —(1 -f-a + ô) r* = c + r* ---*--- hence

References