ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria I: PRACE MATEMATYCZNE X (1966)
ANNALES SOCIETATIS MATHEMATICAE POLONAE Serio I: COMMENTATIONES MATHEMATICAE X (1966)
G. Majcher (Kraków)
O n a linear functional equation of order q
§ 1. Introduction. The purpose of the paper is to present some properties of some sequences of functions and to ^construct the solution of the following linear functional equation
Q
(1.1) < p ( x ) ~ ^ A i (a;) f ( f ( x ) ] = F( x) ,
1 = 1
where Афх), i — 1, 2, ..., q, f ( x) and F(x) are given and <p(x) is the un
known function. f k{x) denotes here the fcth iteration of the function f (oo') i 0
f ( x ) = x, f ( x ) = f [ f ~ 1{ x )l fc = 1, 2, ...
We assume that А а(х) Ф 0 in the interval <a, by or the equation (1.1) is of order q (see [1], p. 199).
The complete bibliography concerning the equation (1.1) and a more general functional non-linear equation is given by M, Kuczma (see [1], p. 199, [2], p. 41) and it is not necessary to quote it here. The solution of the equation (1.1) is obtained in this paper by using weaker assumptions. The above mentioned sequences of functions have not been dealt with yet. In the previous paper [4] (p. 40) we cited Theorem 1 that is being analyzed in this paper, but without details (the paper [4]
concerns a system of non-linear functional equations).
The results presented here and in [4] have found an application to the solution of some problems for the hyperbolic equations (see [5], [«]>•
§ 2. Properties of some functional sequences. We assume that:
(Zx) The function f {x) is defined and continuous in the interval (a, by.
(Z2) a < f(x) < b for xe <a, by.
(Z3) The functions Aj(x), j = 1, ...,q , and F{x) are defined and continuous in <a, by and A q(x) Ф 0 for a?e<u, by.
(Z4) There exists a number L such that 0 < L < 1 and Q
JT*\Ai(x)\ < L < 1 for x e ( a , b y .
Let us consider the functional sequence {An{x)}, n = 0,1, ..., x e (a, by, which is defined in the following way:
(2.1)
Л)(я) — A^{x),
К (ж) =
2 1 [ / " +1- # W ] Л „ _ , И + 4 „ +1 И ,
7 = 1
<7
j = l
for 1 < тг < q,
for n > q.
From this definition there come out the following properties of the sequence {Xn{x)}:
1° For n < q , Xn{x) is sum of n-\-1 components. For n > q, Xn{x) is sum of q components.
2° All the functions An(x) are defined and continuous in the interval
<a, by. It comes out from the assumptions (Z4)-(Z 3) and from the defi
nition of Xn(x).
After elementary modifications we can easily get for n > 1 the fol
lowing equivalent form for the functions Яп(х):
(2.2) ^fl{x) --
n
JT Aj{x)Xn_j [ f { x ) ] + A n+i{x) /-i
a
] ? A ; ( x ) A n„ j [ f ( x ) ]
3= 1
for 1 < n < q,
for n < q.
Le m m a 1. I f the assumptions (Z4)-(Z 4) are failfilled then the inequa
lities
(2.3) \K{®)\ < l f n+l),ą for n — 0 , 1 , . . . , x c ( a , b y are true, the sequence {Xn(x)} is in <a, by uniformly convergent, and (2.4) \imXn(x) = 0, x e ( a , b y .
n —>oo
P r o o f. We utilize the formulas (2.2), the identity Л0(х) — A ^ x ) (see(2.1)) and the assumption (Z4). Hence we have for x e (a , by (in order to simplify the formulae we omit x)
U i(® )K №>[Л114а|+|4,| < 14,1+^ U
ia
>(®)
i< lAimiWii+KOTiWaH-Hsi < 14,1+1^.1+14,1
<7 - 1 g
I V i M I < A | 4 , ' I I V w [ / ] l + | 4 , l < < £ ,
/=1
7 = 1
On a linear functional equation 23
however,
4. 4.
u „ w i « 2 1 i ^ i i V i [ / ''] i < £2>
?=i ?=i
etc. So we get the inequalities
\h{% )\ <
L for n = 0, 1, . . . ;
L2 for 11= Ъ ff+1, •••,22-1,
Lk for 11= (fc - l)g , ( f c - l ) 2 + l ,
from which, owing to 0< L < 1 and к > (nĄ-l)/q for n — {k —l)g, (k—l ) g - f l , ..., kq—1, we get the inequalities (2.3).
The second part of Lemma 1 follows from the first one. This com
pletes the proof.
Having given the sequence (Яи(ж)} let us consider for every of its terms, beginning from n — 1, the system of q functions
Ям 2 {x), . .., hiq (®))>
namely (see formulas (2.1)):
(2.5) Xni(x) =
п+г- 1
£ A j [ f l+i / (ж)]Я,1+^_у_1(ж)+Нп+Дж),
for 1 < n < q, г = 1, 2, ..., q—n,
a
/ = г
for 1 < w < г = w-f-1, ..., q and for n > q, i — 1, 2, ..., q.
We get from this definition the following properties of Xni{x):
1° The functions Xni{x) are defined and continuous in <«,&>. This comes out from the assumptions (Z^^Zg), from the property 2° of the sequence {Xn{x)} and from (2.5).
2° Xnl == Xn(x) for n = 1, 2, ...
3° If g > 1, so
Я(м+1)г(^) (ж)] Яп1 (ж) ~ЬЯм(г+1) (ж)
for п — 1, 2, . .., г — 1 , 2 , . . . , q—1.
The functional sequence (ги(ж)} is now so defined:
a
Гп(х) — |Ямг|) . ^ == 1) •••>
i = l
(2.6) Же <6t, b).
The properties of {rn{x)} are:
1° All the functions rn{x) are defined, non-negative and continuous in <a, by. It comes out from the assumptions (ZJ^Zg) and from the definitions of the functions Xn(x), l ni{%) and rn(x).
2° Every term of the sequence (2.6) fulfils (when we have summed up the moduli of all particular terms with regard to their columns) the inequality
(2.7) rn(a?).<
n q q
2 , ( 2 W / n+1'* (* )]i) i V i W i + A i a h i
i= 1 j = i j —n+1
for 1 < n < q,
q a
У ( У | Л - [ Г +1“ ((* )]| )Я ,-4(*)| for n > q .
i= 1 j = i
Le m m a 2 . Under the assumptions (Z j)-(Z 4) the sequence {:rn(x)} is in the interval <a, b > uniformly convergent and
(2.8) lim r j® ) = 0 for x e ( a , b y . П-+0О
P r o o f. Let n ^ q. Then from the second part of the inequality (2.7) and from the assumption (Z4) we have for n = q, g + 1 , ...
Q
г=1
The statement of Lemma 2 comes out from Lemma 1 because every sequence out of q sequences (q is a constant number) being on the right side of the above inequality is uniformly convergent to 0 in (a, by or, because of (2.3), we have
\rn(x)\ < where В = •
§ 3. Solution of the equation (1.1) in the clasg of continuous functions Theorem 1 (see [4], p. 40). Under assumptions (Z J -(Z 4) the equa
tion (1 .1 ) has exactly one solution in <a, by which is continuous in this interval. It has the form
CO
(3.1) ?>(*) = F ( X ) + ^ K ( x ) F [ r + l (x)1,
n=0
where the functions Xn(x) are of the form (2.1 ), or (2.2).
P r o o f. The existence and the uniqueness of the solution of the equation (1 .1 ) follows easily from the known theorem of Banach-Cac- ciopoli (see[3], p. 37). We consider the space of all functions <p(x) con-
On a linear functional equation 25
tinuous in <a, 6> and with the snpremnm metric Q{<P, 4>) = sup 1^»(я?) — ^ (я?)I = \\<p-y\\.
(a,by
The operation
a
(3.2) (Bep){x) = F{%)+ ^ A 3-(x)(p[f(x)], j=i
transforms ^ into itself. Moreover for every two arbitrary elements cp (x), q){x)e^ we have, on account of (Z4),
9
q(B<P, Bq>) = SUp|B<p-£<£| < s u p { Y | ^ /И М Л ® ) ] - ? [ /( ® ) ] | }
<a,by {a,by \
9
< sup \ ( Y |Ay(a?)|) sup|ę>(®)-£(®)l} < Lo{<p, Й.
<a,by ' {a,by J
Consequently, according to the theorem of Banach - Cacciopoli the ope
ration (3.2) has a unique fixed point in or
9
(3.3) , р(®) = У (*) + 1У ^ ( * ) , > [ / / (®)].
7=1
In order to receive the solution of (1.1) we put in (3.3) subsequently the functions f { x ) , f2{x), ... instead of x. Then we obtain for 1c — 1,2, ...
9
(3.4) ? [ / ( * ) ] = F [ f ( x ) ] + ^ A l [ f ‘ ( x ) M f +t(x)),
7 = 1
and we put it in (3.3) subsequently again. After the first step, that is after substituting in (3.3) <p[f{x)] of the form (3.4) (i = 1), we get (see formulas (2.1), (2.5) and in the case q > 1 the property 2° of hni)
9
<p(x) = 4-л0 (Ж) ^ [/(Ж)] + ^ Лк
i = l
Now we put on the right-hand side of this formula (p[f2{x)] of the form (3.4) (i = 2) and we get
9
<p(x) = F ( x ) + X „ ( x ) F [ f ( x ) ] + ) .1(x)F[J4x)] + ^ X li( x M f +4 x ) ] .
г= 1
Let us suppose that after Tc such steps we have
*-i
<p(x) = F ( x ) + У k { x ) F [ f +,(x)]+yk(x), (3.5)
a
where yk(x) — ^ Ki{x)(p\_f+k (x)j. Putting in (3.5) (p[f1+k(x)~\ of the form
г—1
(3.4) (г = 1+&) we receive
k - l
f ( x ) = F( x) + A , W ? [ / +V ) ] + 4 i W J ,[ /I+‘ (*)] +
г=0 U~ 1
+ v {AiU ' +4 x) lh i (® ) +h (i+i)( x ) M f + ' ‘+i(x)] + ■
J=1
+ A <I[fI+,1(x)]Ui(x)?[f1+k+,‘ (x)l.
Hence, on account of the projierties 2° and 3° (3° if q > 1) of functions l ni, we obtain in (ft + l)-t h step
к q
,f(x) = F ( x ) + jT1 X{ ( * ) F [ f + '(* )]+ ^ A,t+.„-(®)?'[/1+*+i(®)].
г=0 ]=1
We have proved that for every natural n
n~~ 1
<p(%) = F(x) + JT M x ) F [ f l+1( x ) ] + y n(x),
ъ= 0
Q
where yn{x) — ^ l ni{x)(p[f+n(x)]. Moreover, for n = 1, 2, ...
г= 1 Q
ly»(*)l < IMI ^\^ni{x)l < М Ы ® )-
г= 1
Hence, and from Lemma 2, it follows that the sequence {yn{x)} is uniformly convergent to 0 in <a, Ъ>. The same follows from the theorem of Weierstrass, too, because from (2.3) we have
U »(* )-P [r+lW ] K Finally we get the formula (3 .1 ).
References
[1] M. K u c z m a , Bównania funkcyjne i ich znaczenie we współczesnej matema
tyce, Prace Mat. 6(1961), pp. 175-211.
[2] — A survey of the theory of functional equations, Puhlikacije Elektrotech- nickog Fakulteta, Beograd, 130(1964), pp. 1-64.
[3] L . A . L u s t e r n ik i W . L . S o b o le w , Elementy analizy funkcjonalnej, Warszawa 1959.
[4] G. M a jc h e r , On some functional equations, Ann. Polon. Math. 16(1964), pp. 3 5 -4 4 .
[5] — Sur Vapplication de la fonction de Biemann et de Vintegrate le Boux d la solution du probleme generalise de Ooursat, Arch. Mech. Stos. (in press).
[6] — O pewnym zagadnieniu dla równań różniczkowych typu hiperbolicznego, Zeszyty Naukowe Politechniki Krakowskiej (in press).
