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Title: On generalized solutions of linear differential equations of order "n"

Author: Jan Ligęza

Citation style: Ligęza Jan. (1973). On generalized solutions of linear differential equations of order "n". "Prace Naukowe Uniwersytetu Śląskiego w Katowicach.

Prace Matematyczne" (Nr 3 (1973), s. 101-107).

brought to you by CORE View metadata, citation and similar papers at core.ac.uk

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P R A C E N A U K O W E U N IW E R S Y T E T U Ś L Ą S K IE G O W K A T O W IC A C H N R 30 P R A C E M A T E M A T Y C Z N E II I , 1973

J A N L IG Ę Z A

O N G E N E R A L IZ E D S O L U T IO N S OF L IN E A R D IF F E R E N T IA L E Q U A T IO N S O F ORDER n

§ 1. IN T R O D U C T IO N . This paper deals w ith the follow in g differential equation:

x<">(t) + P 1 ( t ) x < n _ 1 > ( t ) + . . . + p „(t)x (t) + pn+1(t) = 0, (*) in which Pi(t) fo r i = 1,. . . . , n + 1 are know measures in the spaces

R 1, unknown is the distribution x (t). The d erivative is understood in the distribution sense.

In this paper w e shall prove the theorem on the existence o f a unique solution o f Cauchy problem fo r the equation (*) in the class all distri­

butions, which (n — 1) derivatives in the distribution sense can be identified with functions of finite variations in the space R 1. The follow in g theorem generalized some result o f A . Lasota and F.H. Szafraniec (see [8]). Our theorems can be aplied to some equations for which the theorems of J. K u rzw eil (see [7]) cannot be used.

Principal result o f this paper is based on the sequential theory o f the distribution (see [9]).

§ 2. N O T A T IO N S .

Definition 1. A sequence o f smooth and non-negative functions <5„(t) satisfying:

OO

f dn(t) dt = 1

— OO

2° There is a sequence of positive numbers an convergent to zero such that

àn{t) = 0 fo r |t| > an 3° There are numbers M 0, M l t .. . such that

(3)

holds fo r n = 1,2,. . . and every order k.

àn(t) ôn( t)

w ill be called the delta sequence.

The condition 4° was suggested by P. Antosik (see [4]).

Definition 2. B y a regular sequence fo r a distribution / w e understand any sequence the n — therm o f which is (see [10]):

f n - f * à n -

D efinition 3. I f for every regular sequence f n(x ) fo r distribution f ( x) the sequence f n(x 0) is convergent, then the lim it lim /„(x0) w ill be called

71—> o o

the mean value of distribution f ( x) in the point x Q and denoted b y f ( x Q).

This definition o f the value of distribution in the point may be found in [4].

D efinition 4. W e say that a distribution j is a measure, if there exists fo r f a fundamental sequence f n such that, to each finite interval I , the sequence o f numbers J \fn\ is bounded (see [2]).

i

D efinition 5. B y a solution of equation (*) w e understand any distribu­

tion x (t), which satisfies this equation in the space R 1.

Definition 6. W e shall denote by V n_1 a class of all distributions, which (n — 1) derivatives in the distribution sense can be identified with functions o f fin ite variations in the space R 1.

§ 3. LE M M A S . W e shall prove tw o lemmas.

L E M M A 1. I f distribution P ( i ) is a fu nction of fin ite variation in R 1, then there exists a value o f P ( i ) in every point t ^ R 1 in the sense of d efin i­

tion 3 and

P (to ) = ^ [ P ( t i ) + P ( t - ) ] ,

where P (t+ ), P (t~ ) denote respectively rig h t and le ft lim its o f a fnuction P (t ) in t 0 (see [4]).

P r o o f . From the definition 3 and condition 4° fo r delta sequence w e have:

o an an

P (t 0) = lim [ J P ( t —r)ô n(r)d r + j' P ( t —r) ô„(r) d i]t=to = lim [ f(P (t + s) +

«-><» _ an o «->“ = o

an

+ P ( t - S ) ) ô„(s) dsjt^to - lim [(P (tj + P (t + )) f ôn(s) ds]t=t0,

CO Ó

which ends the proof o f the lemma.

(4)

L E M M A 2. I f p n(t ) is an arbitrary regular sequence fo r measure p (t) and x n(t ) an arbitrary sequence o f smoth functions almost u n iform ly convergent to a fu n ction x (t), then

lim (d ) p n(t )x n(t) = p {t)x (t),

n—> co

where the convergence is understood in the distribution sense (what is denoted by lim (d )).

P r o o f . L e t x n(t) be an arbitrary regular sequence fo r the continuous function x (t). Then for an arbitrary number e > 0 there exists such a num­

ber nQ that fo r every number n > n0 w e have:

t i

I J (a?n(s)-x„(s)) Pn(s) ds [ < e IJ |p„(s) |ds|.

a a

This last ineqûality finished the proof.

§ 4. P R IN C IP A L R E S U LT.

THEOREM . L e t p i(t) fo r i = 1 , . . . , n + 1 be measures and P i(t) a lo ­ cally integrable fu n ction in the space R 1. Then the problem

I x<">(t) + P i ( t ) + .. . + p n(t) x (t) + pn+1(t) = 0

\ x<V(a) = xu 1 = 0 , . . . , n — 1 1 ’

has exactly one solution in the class V™-1 (x (l)(a) is the mean value of distribution x<l)(t) in a).

A t first w e shall prove tw o lemmas.

L E M M A 3. L e t P u c( t ) be arbitrary regular sequence fo r the measures p i(t) fo r i = 1, . . , n + 1 and xikf or 1 = 0 , . . . , n — 1 be arbitrary conver­

gent sequences respectively to xt as k-+oo. Then sequences x k<l>(t) defined by

t n

X k(l)( t ) = - [ J ( t - s ) » - l ( J T pik(s) X k< »-«(s ) + Pn+lîc(s)) ds ] W +

are locally equibounded in R 1.

P r o o f . Replacing equation

n

x (n>(t) = - pik(t) x<"-*>(t) + pn+ïfc(t)] (2) i= 1

by a system of equations w e have the follow in g estimation (see [6] p. 72):

\\Yk(t)\\ < {HYfcfaJjj + j J ÜB^sjlldsDexpl J |!AkCs;||ds|, (3 )

a a

(5)

Xk(t) « O k “ 0

Xfc'(t) « l k 0

Y k(t ) = , Y k(o) = , B k(t ) = *

« ) _ « n — l k ~ P n + l k f f )

~ 0 1 0 0 . 0 ”

0 0 1 0 . 0

A k (t ) = 0 0 0 0 . .

1 _ - p . lk(t) . . . - P l k ( t ) _

Xk(n)(t ) = - [J T Pifc(t) X fc( " - « ) ( t ) + P „ + I!c (t ) ] , (=1

the symbol |[A|| denotes the sum o f the absolute values o f all elements o f the m atrix A . From inequality 3 it is easy to prove our lemma.

L E M M A 4. I f the assumptions of lem m a 3 are satisfied, then there exist subsequences o f { x kV ( t ) } convergent to a functions x('-'(t) o f finite variations in R 1.

P r o o f . From lemma 3 and the definition 4 fo r each finite interval I w e have:

£ I (tm) — Xk<l)(tm -l) I < OO, m=1

where p is an arbitrary natural number and t Çf, y — 0 p. From H e lly ’s theorem (see [12] p. 372) it follow s that some subsequences x [l'kit) o f x'kl> (t ) are convergent to x<l>(t) as t Ç I and 1 = 0 , . . . , n — 1. L e t a x be an arbitrary decreasing sequence, bx strongly increasing sequence such that

I = [a, b] = [a 1; bj], lim ax — — oo, lim bx = oo, X->oo X —> oc

and x ph.a)(t ) subsequences o f x p_ lk(t ) for p = 2,3, . . . and I = 0, . . . n — 1 and that their lim its are functions of finite variations in [ap, bp]. Let Us consider the m atarix

~xu M ft) x 12( » ( t ) . . . x 1K<»(t) . . . "

Xn (l)(t ) Xr2(l)(t ) . . . XTk<l>(t) . . .

The diagonal sequences x kk(U(t) are convergent to some1 functions x<l>(t) of finite variations in R 1, which completes the proof of lemma 4.

W ithout loss o f generality w e can assume that

(6)

X k k (l' ( t ) := X k (l)( t ) .

Then considering that p i( t ) is locally integrable function by lemma 2 and [3] (p. 642) w e obtain

lim (d) [x k(n>(t ) + plk(t ) x k(n~ v (t) H- . . + p „ k (t) x k(t) + P n + i M ] = k~*oo

= [x<n)(t) + P i(t) X<n~ x>(t) + . . . + P n (t) x (t ) + Pn +i(t)].

Obviously x® (a) = x\ fo r 1 = 0 , . . . , n — 2. W e shall prove that x(n_1>(a) =

= «n—i- In fact, let f lk, . . . , f„k be regular sequences formeasures P i(t) . x<n~"l>(t) , . . . , p n( t ) ■ x ( t ) respectively, The sequence

dl

1 k ( t ) f i k ( s) ~ . . . ~t- f n k (s) "b P n + lk f^ jj d s T Xn—l a

is convergent to a function Y (t) o f finite variation. B y conditions (* *) w e obtain Y (t ) = x n- x(t). Then from definition 3 and lemma 1 w e have:

x< n -»(a ) = —lim [lim f ( f lflk(s) + ■ • • + Pn+ik(s)) ds) <5, (r)dx](=n + xn- t =

v) -*■ o o k —> o o — oo a

= —lim J t) — F j(a) + .. . + P n+1(t — T) — P n+1(a)]ói](x) J | t=a +

k - > o o — OO

”b xn—1 Xn—i, w h ere it w as assumed, that

F ' i ( t ) = f i ( t) = P i ( t ) x (n~D(t), P 'n+1(t) = p „ +1( t ) , i = 1 , . . . , n.

N o w w e shall prove the uniqueness o f the solution of problem (* *) in the class V n_1. Suppose that x t(t ) e V n~x, x (t ) e V ” - 1 are solutions of problem (* *) and x 1( t ) ^ x 1(t). L e t us denote b y x lk(t), x k(t ) regular sequences fo r x t(t ) and x (t ) respectively. W e consider { Y lk<l>(t)}, { Y ^ f t ) } fo r 1 = 0 , . . . , n — 1.

d f t J1 (n i)

Y Ik(l)(t ) = - [ / ( t - s ) n - * ( ^ P l k ( s ) x lk(s) + pn+lk(s )d s ] (l> +

+ g * w

r 1 ^ (n _i) \ 1 n\ _1_

Y k<»(t) = - [ / ( 2 1 Pik(s) Xk( S) + Pn+lk( s ) ) d s ] a> +

a i = l

/V-J (t — a Y \

+ - f n

i —i

( t - a y \ o > . (5)

The sequences Y lk<l>, Y k<l> are locally equibound in the space R 1 and there exist subsequences o f these sequences convergent to some function Y/b(t) and Y<l>(t) of finite variations. W ithout loss of generality' w e can assume

(7)

that the sequences Y lk<-l>(t) and Y kW(t) are convergent to functions Y ^ y t), Y<l>(t) (for 1 < n — 1 it is an almost uniform convergence). B y lemma 2, (* *), [3] (p. 642), [9] (p. 21) w e have:

Y j v ( t ) = xp >(t), Y «> (t) = x<V(t).

L e t’s put

siH(t) = ||x lk« > ( t ) - : c k«>(t) I - \ZkV(t)\l (6) where

\Ylk<l> ( t ) - Y k(0(t)\ = \Zko>(t)\, 1 = 0 , . . . , n — 1.

A sequence £ik(t) fo r I < n — 1 and k-+oo is almost uniform ly convergent to zero in R 1. The sequence sn- lk(t ) for k —>oo tends to zero in the distri­

bution sense (it is locally equibounded and almst everyw here convergent to zero). From (4), (5) w e have:

\Zk<n)(t)\ < £ M l » B-*W| + B k(t), (7)

i — 1

w here

df

B k(t ) £ok( t) |Pnk(t)| + . . . + £ «—ifc(t)| Ptk(t)|-

Replacing inequality (7) by a system of inequalities w e obtain by [6]

(p. 39— 40, 73):

\\Uk(t)\\ < {||t/kfa;|| + \ jB k(s)ds\}exp\j\\Ak(s)\\ ds\, (8)

a a

where

~ z k(t ) - - 0 - ~ 0 1 0 0 . . 0

Zk'(t) 0 0 0 1 0 . . 0

u k(t) = ,U k(a) =

, A k(t ) =

0 0 0 0 . . 1

_ Z k(n -D (t)_ _ 0 _ |Pnk| • -l plkl _

From inequality (8) and [3] (p. 642) it follow s that xtf't) = x (t), what com­

pletes the proof of the theorem.

R e m a r k 1. The theorem above generalizes some result? of A.

Lasota and F. H. Szafraniec (see [8]). Our theorem can be applied to some equations for which the theorems of J. K u rzw eil [7] cannot be used. One of such equation is

x (”>(t) + x<n~*)(t) + b(t) + .. . + ô (t) x (t ) + ô ( t ) = 0, (where ô (t) denotes Dirac’s delta).

(8)

R e m a r k 2. L e t’s consider the equation

x '(t ) = - x (t)

w ith the initial condition x ( l ) = 0. It is known that the last problem has tw o solutions X j(t) = 0 and x 2(t) = ô(t) (see [5]). This exam ple shows that without additional conditions the problem o f Cauchy can have more than one solution.

R E FE R E N C E S

[1] P . A n t o s i k, O rd e r w ith respect to m easure and its a p p lica tion in in v e s ti­

ga tion o f prod u ct o f generalized fu n ction s (in Russian), Studia Math., 26 (1966), 247— 261.

[2] P. A n t o s i k, On the modulus o f distribution, Bull. Acad. Polon. Sci.Ser.

math. astr. et phys., 15 (1967), 717— 722.

[3] P. A n t o s i k, Som e cond itions fo r mean convergence, Bull. Acad. Polon. Sci.

Ser. math. astr. et phys., 8 (1968), 641— 646.

[4] P. A n t o s i k, A m ean valu e o f d istrib u tion (in p reparation).

[5] P. A n t o s i k, J. M i k u s i ń s k i, R. S i k o r s k i , T h e elem entary th eory o f distributions o f single rea l variable (to appear).

[6] P. H a r t m a n , O rd in a ry d iffe re n tia l equations (in Russian), M oskw a 1970.

[7] J. K u r z w e i 1, L in e a r d iffe re n tia l equations w ith distribu tion s as c o e ffi­

cients, Bull. Acad. Polon. Sei. Ser. math. astr. et phys., 7 (1959), 557— 560.

[8] A . L a s o t a, F. H. S z a f r a n i e c, A p p lica tio n o f the d iffe re n tia l equations w ith d istrib u tion a l coefficie n ts to the o p tim a l c o n tro l theory, Z eszyty Naukow e UJ, P ra ce M atem atyczne 12 (1968), 31— 37.

[9] J. M i k u s i ń s k i, R. S i k o r s k i , Elem entarna teoria d ystrybucji, P W N , W arszawa, 1964.

[10] J. M i k u s i ń s k i , Irre g u la r operations on distributions, Studia Math., 20 (1961), 163— 169.

[11] J. M i k u s i ń s k i , S equ en tia l th eory o f the co n v o lu tio n o f distributions, Studia Math., 29 (1968) 151— 160.

[12] R. S i k o r s M , F u n k cje rzeczyw iste T. I, P W N ., W arszaw a 1958.

J A N L IG Ę Z A

O 'R O Z W IĄ Z A N IA C H U O G Ó L N IO N Y C H R Ó W N A Ń L IN IO W Y C H n -T E G O R ZĘ D U

S t r e s z c z e n i e

W pracy tej udowodniono tw ierd zen ie o istnieniu i jednoznaczności ro zw ią ­ zania problem u

| a*»>(t) + p t(t) a S »-«(t) + . . . + pn(t) x (t) + pB+1(t) = 0 j xW(a) — «j, l = 0 n — 1

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w klasie dystrybucji, których (n — 1) pochodne w sensie dystrybucyjnym można utożsamić z funkcjami o wahaniu skończonym w przestrzeni Ri, przy czym pt(t) dla i = 1...■ n + 1 oznaczają dane miary (por. [2]), zaś x (i)(a ) oznacza średnią wartość dystrybucji x (i)(t) w punkcie a (por. [4]). Powyższe twierdzenie uogólnia

pewien wynik A. Lasoty i F. H. Szafrańca (por. [8]). Ponadto twierdzenie to może być zastosowane do pewnych typów równań, dla których twierdzenia J. K urzweila

(por. [7]) nie mogą być stosowane

Zasadniczy wynik tej pracy jest oparty na ciągowej teorii dystrybucji (por. [9]).

Oddano do Redakcji 25,11.71.

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