On the continuous dependences ofC r solutions of a functional equation on the given functions

On the Continuous Dependences of c' Solutions of a Functional

Equation on the Given Functions

J. MATKOWSKI (Katowice, Poland)

215

We consider the problem of the continuous dependence on the given functions for the solutions of the equation

q~ (x) = H (x, ~b I f (x)]),

where q~ (x) is an unknown function, f and H are given real valued functions of real variables defined in an interval L resp. a domain f2.

We denote by C r [A] the class of functions which have continuous derivatives up to order r in A, 0 < r < ~ .

In this paper we shall apply the results on the existence and uniqueness of C r solutions of this equation which are due to Choczewski [1] (see also [3], Chapter IV) and the author [5]. Under the assumptions which guarantee the existence and uniqueness of a solution qS~C r [ I ] for every equation

q~ (x) = H. (x, q5 [ f . (x)]), n = O, 1, 2 . . . . . (1)

we shall prove that i f f . tends to fo in I and/4, tends to H 0 in ~, with all derivatives up to order r, uniformly on every compact, then q~n tends to 4~o in I, with derivatives up to order r, uniformly on every compact set contained in L

§ 1. Let I be an interval and let 0~L By St[I] we denote the class of functions f ~ C r [ I ] which fulfil the condition

f (x) 0 < < 1 for x~I , x # 0 .

x

This definition together with the following Lemma may be found in [3], p. 20-21.

LEMMA 1. I f f e S ' [ I ] , then f ( 0 ) = 0 and for every x~l , x#O, the sequence f " (x) of the iterates o f f is strictly monotonic and l im,_., f " (x) = 0.

We assume the following hypotheses.

(I) f , ~ S ' [ I ] f o r n= 1, 2, ...; if fo¢Sr[I] , then f o ( x ) = x or fo(x)---O for xeI . (II) H,~C'[O] for n=0, 1, 2 .... ; f2 is a domain fulfilling the following conditions: (i) for every x ~ l the set f2 x = {y: (x, y)eg2} is a nonempty open interval; (ii) for every x ~ I we have

n . ( f (x), f2S(x) ) = f~x;

Received May 25, 1970 and in revised form October 26, 1970.

216 J.Matkowski AEQ. MATH.

(iii) (0, 0)~0 and H.(13, O)=Ofor n=O, 1, 2, ...; (III) f , tends to fo in 1, H n tends to H o in f2, together with derivatives up to order r,

in both cases uniformly on every compact. Let us define the functions H,, k by the recurrent relations

~an"0x c3y~'yon" i

H,, ~(x, y, Yl) = ~ (x, y) + f~(x)-z~, (x , Y)Ya %_ ) (2)

H . . k + , ( x , y , y I . . . . . yk+l)= +f . ' (X) \ Yt + ' " + ~ ; 2 - Y g + t •

The following two lemmas have been proved in [1] (also [3], p. 85).

LEMMA 2. I f hypotheses (I), (II) are fulfilled, then the functions H,, k (k = 1,..., r) are defined and of cIass C ' -~ in O x R k. Moreover, we have

oH. n, ,k (x, y, Y t , . . . , Yk) = G,,k (x, y, Yl ..... Yk-~) + If" (X)] k ffyy (x, y)Yl (3)

where G~,kEC~-k[f2 X Rk-1]. By A [I] we denote the class of functions q~ such that for xe I we have 4' I f (x)] e O~,

and q~ (13) =0.

LEMMA 3. Let hypotheses (I) and (II) be fulfilled. I f ~b~eA[I] is a C" solution of equation (1) in 1, then its derivatives $(k) satisfy the equations

~k) (x) = H.,k (X, dp. I f . (X)], qS" [f. (X)] ..... qS~ k) If. (X)]) (4)

for x ~ l and k = 1,...r. From Lemma 2 by induction we obtain

LEMMA 4, Under the assumptions (I)-(III) the sequence H., k tends to Ho,k in I2 x R ~, G,,~ tends to Go, k in f2 x R k- ~ (k= l . . . . , r ), uniformly on every compact.

If ~b, eA[1] c~ C']1] is a solution of equation (1), then it follows from (4) that the values

~1 = # . ( 0 ) . . . . . ~ , = C ) ( 0 ) (5)

satisfy the system of equations

1~ k = Hn, k(O, O, n l . . . . . nk) , k = 1 . . . . . r . (6)

As has been proved in [5] (cf. also [1] and [3] p. 95), i f f , eS ' [1] , H~ satisfies assumption (II) and

If. (x)l ~< 1, Ixl ~< Co, ~o > 0, (7)

Vol. 6, 1971 On the Continuous Dependence of C r Solutions 217

[fg(0)]' ell, 0) , d-y-(0, < I, (8)

then for any system of~/i,.. . , r/, fulfilling equations (6) there exists exactly one solution q ~ A [ I ] ~ C" [I] of equation (I) fulfilling conditions (5).

On account of (5), (6) and (3) the values ~b'~ (0),..., ~b(~ O (0) fulfil the system of equations

- (o, o) 0~')(o) = G.,,(O, O, ~.(0) ..... ~ ' - ' ) (0) ) ,

Hence we get (9)

LEMM A 5. Let fo~S ~ [1], and let hypotheses (I), (II) be fulfilled. Suppose, further, that inequalities (7) and (8) hoM. Then

(i) equation (1) has a unique solution dpn¢ A [IJ n C r [ I ] / f and only i f

[f~ (0)] k t3~ (0, 0) :~ 1, k = 1 .. . . . r - 1 ; (10)

(ii) i f for some k = k (n) we have

aHn (0, O) = 1 (11) [f~ (0)]k(:) a y

then equation (1) has a solution qb~eA[1] w C'[1] i f and only i f

Gn, k(O, O, qS"(O) .. . . . qS(~k- I)(O)) = O, (12)

where the values ~b'n(O),..., q)(~k-1) (O) are uniquely determined by relation (9). More- over, i f (11) and (12) hold, then a solution (aned[I ] c~ C'[I] contains the parameter t = ~ ) ( o ) .

Now we prove

LEM MA 6. Let hypotheses (I)-(III) and inequalities (10) be fulfilled for n= O, 1, 2, .... I f ~n~A[1] c7 Cr[I] is a solution of equation (1) then

lim q~k)(0) = qs~0k)(0), k = 1 . . . . . r . n. .~ oo

Proof. We have by (9) and Lemma 4

OH, (o, o))-'- an,(o, o)

dHo -1 = ( 1 - f~(0)~yy (0, 0)) Go,, (0, 0 ) = qS;(0).

218 J. Matkowski AEQ. MATH.

Suppose that lim,_.oo~b~')(0)=~bcj)(0) for s<k . Now by (9) and Lemma 4 we have

lira ~k)(O) = lim 1 -- [f. '(O)] k OH. (0, 0) G..k(O, O, ~,~(0), q~k-O(O)) t l - v ~ n - ~ \ ~ " ' ' ~

= 1 -- [ f ; (O)] k (0, O) Go,,(O, O, qS~)(O) . . . . . qS~ok-1) (0))

=

and induction completes the proof. L E M M A 7. Let hypotheses (I)-(III) and condition (8) be fulfilled. Suppose that

relations (11) and (12) hold with k ( n ) = k ( O ) = k for n = 0 , 1, 2 . . . . . Then

0 < f , ' (0) < 1, n = 0 , 1 , 2 . . . . . I f we assume that

$~k)(0) = t,, n = 0, 1, 2 . . . . . lim t, = to (13) II--~ oO

then

Proof. f.' (0) ~ 0 and f.' (0) ~ 1.

Similarly to Lemma 6 one can prove the second assertion of the Lemma.

lim ~b~ +) (0) = ~b~oS)(0), s = 1 . . . . . r . I1.--~ CO

It follows f rom (I) that 0~<f ' (0)~<l . Now, by (11) and (8) we obtain

L E M M A 8. Let (I)-(III) and (8) be fulfilled. Suppose that for every n, n = 0, 1, 2 . . . . there exists a positive integer k (n ) such that relation (11) holds. Then k ( n ) < r and there is a pos[tive integer n o such that k (n ) = k ( O ) for n >>- no.

Proof. According to Lemma 7 we have 0 < f , ' (0) < 1. Now by (8) we get k (n) < r. It follows f rom (11) that

an. k(n) = - l n ~ - y (0, O) [ In f , ( 0 ) ] -1 .

Hypothesis (III) implies that lim~..,ook(n)=k(O). Since k(n) is a positive integer, it follows that k (n )=k (O) for n sufficiently large.

Finally, let us quote the following simple result (cf. [4])

L E M M A 9. I f d is a compact metric space and the sequence of transformations T, fulfils the conditions:

1 °. T . ( . ~ ) ~ d , n = 0 , 1,.. . , 2 °. T, is continuous in d for n = 0 , 1 . . . . . 3 °. T,(cp)=q~ i fandonly ifqg=q~,, qg,~d, n = 0 , 1,.. . , 4 °. T, tends to To uniformly in d ,

then q~n is convergent and lim,.+~o ~p,= q~o-


§ 2. Let us note that if for n =0 relation (10) holds, then it follows by (III) that (10) holds for n sufficiently large. Thus we can assume (10) to hold for every n = 0, 1 .....

If relation (10) is not fulfilled for n = 0 then there exists a positive integer k(0) such that (11) holds for n=0 . The following three cases may occur:

for every sufficiently large n relation (1 1) holds, for every sufficiently large n relation (10) holds, neither of the above two cases occurs.

In the last case we can form the two subsequences of sequence (1) defined by relations (11) and (10), accordingly. Now we can consider separately each of the two subsequences.

Taking into account Lemma 5 we see that it is sufficient to consider the following three cases.

(A). For every n=0 , 1,... relation (113) holds. (B). For every n=0 , 1, ... relations (11) and (12) hold. (C). For n = 0 relations (11) and (12) hold and for n = l , 2,... (10) holds.

§ 3. In this section we shall deal with the continuous dependence of C" solutions of the functional equation on the given functions in a neighbourhood of x = 0.

We shall prove the following

THEOREM 1. Let hypotheses (I)-(III) be fulfilled. Suppose that for n=0 , 1, 2 .... inequalities (7), (8), (10) hold. Then equation (1) has for n=0 , 1, 2, ... exactly one C r solution Cn in a common interval containing O. Moreover, 49~ tends to ¢o together with derivatives up to order r, uniformly in this interval

Proof. If fo~Sr[I] then by Lemma 5 equation (1) has exactly one solution ¢ , ~ A [ I ] c~ C~[I'], for n=0 , 1, 2,. . . . I f fo(x)=-0, then evidently, ~bo(x)=Ho(x , 0) is the unique solution of equation (1) of class C' [I] . I f f o ( x ) - x then by (8) and implicit function theorem there exists exactly one solution tko of class C ~ in an interval ( - e l , 81) n 1, l>0.

We may assume that 0 is the left endpoint of L If 0 is the right endpoint, the argument is similar. I f 0 is an inner point o f / , we consider separately each of the two parts into which I is divided by 0 (with 0 included in both).

Let us write ¢, in the form

where

k=l

(14)

(14a)

220 J.Matkowski AEQ. MATIt.

Now we define the functions

H*(x, y) = Hn(x, P. I f . (x)] + y) - P,,(x). (15)

Let fax= (a(x), b(x)). We see that H* are defined and of class C" in the domain

O* = {(x, y ) :xe I , a(x) - P. [f . (x)] < y < b(x) - P. [f~(x)]}

and we have (0, 0)E f2*, n * ( o , o ) = 0, n = 0 , 1 , 2 . . . . .

It follows from (14) and (15) that y. is the unique C r solution of the equation

y(x) = H* (x, y [ / . (x)] ) , (16)

such that y. (0)=0. Moreover, by (14) and (14a) we get

. . . . . = o, n = o , 1, 2 . . . . . ( 17 )

By hypothesis (III) and Lemma 6 it is seen that (2" tends to (2~ as n -+ oo. Let us choose an a, O<a<<.e t, and a b, b>0, in such a manner that the rectangle

A = { ( x , y ) : 0 ~ < x ~ < a , - b ~ < y ~ < b }

is contained in ~*. There exists a number N such that for n 1> N, we have A = ~2". Without loss of generality we may assume that N = 0. Now, it follows from Lemma 6 and (III) that H* tends to H* together with derivatives up to order r uniformly in A. By Whitney's [6] theorem there exist functions hn such that

h " e C r [ P ] ' P = ( ( x ' y ) : O < ~ x < ~ a ' - ~ < Y < + ~ } } h,,(x, y) = H*(x, y), (x, y )~A. (18)

(It is easy to verify that the functions h, satisfy hypothesis (II) with I2=P). Moreover, (cf. [2], p. 591, proof of Lemma 1) we may require that

h. tends to ho together with derivatives up to order r, (19)

uniformly on every compact K ~ P.

According to the definition of h. we have

Oh. (0, O) = ~H, (o , o ) ( 20 )

and consequently (cf. the remarks in the beginning of this proof) the equation

~o (x) = h. (x, ~o I f . (x)]) (21)

has for every n, n=0 , I, 2 .... exactly one solution ~o.eCr [(0, a)] such that 9 . ( 0 ) = 0 and, moreover,

q~. (x) = y. (x) (22)

Vol. 6, 1971 On the Continuous Dependence of C" Solutions 221

in an interval <0, a,> c <0, a>, a, > 0. If, however, we can prove that tp, tends to tp o in <0, c) , 0 < c < a uniformly together with derivatives up to order r, then relation (22) must be valid in an interval <0, d), 0 < d < c (independent o f n) for every nl). Thus, in this interval V, tends to 7o uniformly together with all derivatives up to order r. Finally, by (14) and Lemma 6 we get the last assertion of Theorem 1. Hence, it is sufficient to prove that q~, tends to ~0 o together with derivatives up to order r uniformly in an interval <0, c) , where 0 < c < a.

For h. we may define the functions hn, k and gn,k analogously a s Hn, k and Gn, k (with f 2 = P ) and we may prove analogues of Lemmas 2, 3 and 4. Now, by (17), (18) and Lemma 3 it is seen that

h..k(0, 0, 0 . . . . . 0) = 0, k = 1 . . . . . r . (23)

By (7), (8), (19) and (20) there exist numbers 0 < 0 < l, O < c ' < a and d ' > O such that for n = 0, 1, 2 .. . . we have

I f ' ( x ) l ~ < l , 0 ~ < x ~ < c ' |

~ h , ( x , y ) ~<0 in <0, d> x < - d ' , d ' > . / (24) [ f" (x)]'

Let us fix a K > 0 . F rom (17). (22) and (9) (with the change of G,., into g,.,) we get

g . , , ( o . . . . . o ) = o .

Since g, , , tends to go,, uniformly on every compact set contained in P, in view of (III) and (19) we may assume that c' and d ' have been choosen in saach a manner that for n =0 , 1, 2, ... the following inequalities hold

I g . . . ( x , y , Yl . . . . . y . _ l ) l ~ < ( 1 - 0 ) K in <O,c') x < - d ' , d ' ) " ]

Oh. [f,~ (0)]" / Oh"(x, y_ [f / , (x)]" (0 ,0) ~ < ( 1 - 0 ) K in (25) ay

<o, c'> x < - a', d'>. Now we choose a c so that

0 < c ~< m i n ( c ' , 1, ~ ) (26)

and we define the set D c P x R "-1 as follows

D = {(x, y, Yl . . . . . y r - a ) : 0 ~< x ~< c, lYl ~< g x , lYkl <<- K x , k = 1 . . . . . r - 1}. (27)

To a given e > 0 we assign (i -

I+K

i) Indeed, it follows from the continuity of ~0o and from the condition 90(0) = 0 that there exists an a, 0 < a < c such that for 0 ~< x ~< a we have -- b < (0o (x) < b, i.e. the graph of ~oo (x) is contained in the rectangle zl. Thus, ~00 (x) = y0 (x) for x e <0, a>. According to the uniform convergence of (on to ~oo in <0, ~> there exists an N such that for n/> N and 0 ~< x ~< a we have -- b <<. tpn(x) <~ b (i.e. the graphs of ~on are contained in A). This implies that (22) holds for n ~> N. By taking a smaller d we can obtain relation (22) for every n, n = 0, 1, 2 .....


The functions g,,, are uniformly continuous in D, f~' (x) are uniformly continuous in (0, c), and gh,/dy are uniformly continuous in the set D' = {(x, y): 0 ~< x ~< c, [y[ ~< Kx}. Hence, and by the uniform convergence of these functions in D, (0, c) and D', respectively, there exist positive numbers ~51 (5) and ~5 z (e) such that for n = 0, I, 2,... we have

Ig,,,(£g,P, fil . . . . . fi,-1) - gn.,(~, Y, Yt . . . . . .if,-l)] ~<e' /

~hy. ()2. 27)[f. ()2)],- ~y. (.~..~)if. (~)] , 4 , j (28)

for I)2-~1~<6~(8), I)7-y1~<62 (e), I)Tk--ykl..<62 (8), k = l , . . . , r - l , ()2,)7, )71 ..... )7,_~), (~, y, y~, . . . , y,_~)~D.

Now we put

and define .~r as the set of those functions e (x) which are defined in (0, c) and fulfil the condition

[e()2) - e(~)l < e whenever I)2 - ~[ < ~.

Let o~- be the set of functions ueC' [ (O, c)]. For u e 5 we define the norm

I[ull = max(sup [u(x)b sup [u'(x)l ..... sup lu(')(x)l). <0, c) <o, c) (0, c)

Consequently ~" is a normed vector space over the field of real numbers and the convergence of a sequence u~e~" means the uniform convergence together with derivatives up to order r in (0, c). Since ~- is complete, it is a Banach space.

Now we define ~¢ as the set of those functions rpe~" which fulfil the following conditions

~o (0) = ~o' (0) . . . . . rp ~') (0) = 0, (29)

I~o(°(x)l ~< K, 0 ~< x ~< c, (30)

~ (o~ ,~ . (31)

Let us note that if ~ot, ~2~,.~, then

IIq~a - q~:ll = sup Icp~°(x)- qg~')(x)l. (32) (O,c>

In fact, by the mean-value theorem we have

I q , [ ~ - ' ( x ) - ~ 0 ~ - ' ( x ) l = xl~o?>()2) - ~ ) ( g ) l , 0 < )2 < x ,

Vol. 6, 1971 On the Cont inuous Dependence of C r Solutions 223

k = 1,,.., r. Hence and from (26) we get

s u p l Ik-1)(x) - c s u p - <0, c> <0, c>

~< sup ]~o~k)(x)- q~(2k)(x)]. <O,c>

Further, for 9 E d by (26), (29), (30) and by the mean-value theorem we obtain

19(2)1 ~< K x , I~0(k)(/)l ~< K x , 0 ~< x ~< c, k = 1 . . . . . r - 1. (33)

Next we define the transformation q~ = T. [rp] by the formula

~0 (x) = h. (x, ~o [ f . (x)]), ~0 ~ d . (34)

We shall prove that d and 7". fulfill the hypotheses of Lemma 9. It follows from (30), (31), (32) and from the theorem of Arzela that d is compact . Let q~e d and let • be given by (34). Differentiating (34) k times we obtain

~/(k)(X) = hn, k(X , qo[ f . ( x ) ] , qo' [f~(x)] . . . . . 9 ( ~ ) [ f . ( x ) ] ) , k = 1 . . . . . r . (35)

According to Lemma 2 (for h..,) the function q~(r) is continuous in <0, c> and consequently ~oeo~. Putting x = 0 in (34) and (35) we obtain by (29), (18) and (23)

0 (0) = h. (0, 0) = H* (0, 0) = 0

O ( k ) ( 0 ) = h . , k ( 0 . . . . . 0 ) = 0 , k = 1 . . . . . r .

Thus ~h fulfils condition (29). By (35) and (3) we have

10(')(x)l < [g. . , (x , ~o [ f . (x)] . . . . . 9 ( ' -O [ f . (x)] ) l

Oh. (x, rp [f~(x)]) [ f" (x)] 'I I~o (') [ f . (x) ] [ • + Oy

In view of (33), (26) and Lemma 1 we have

[(p [f~(x)][ ~< K c <<. d', 19 (k) [ f . (x) ] [ ~ K c <. d', k = 1 . . . . . r - 1.

Using (24), (25) and (30) we obtain

10(r)(x)l ~< (1 - 8) K + 8 K = K , 0 <. x <~ c ,

i.e., 0 (r) fulfils condition (30). Let us take an ~ > 0. Let 2, ~ e <0, c> and let [ 2 - ~l < 5 (5). We write ~ = cp [ f . (2)],

)~k = (p(k) i f . 07)] ' ~ = ~o [ f . (~)], Yk = qCk) i f . (~)], k = 1 ..... r - 1. Thus we have

10(°(2) - 0(')(N)I <~ Ig., r(2, ~ . . . . . ~ , - , ) -- g., .(~, Y . . . . . Y,- t)l

Oh" (x ' Oy Oh. - - i f - (~)]r + ko(O [f . (~)][ fi) [ f" (2)]" -- ~ (N,N)

(2, ~) i f ; (2) ] , l (p(r)[ . f . (2)] _ t#(,) [ f . ( ~ ) ] I • + v y


By the first of the inequalities (24) we get

[fn(2) - fn(~)l ~< [2 - 21. (36)

F rom (33) and (36) we have

l Y - Yl = 19 [ fn(2)] -~0 [f , (N)]l ~< I~'(t)[ I f , ( 2 ) - f , (~) f ~< K [ 2 - NI,

IPk - Ykl = I ~ ( ~ [ L ( ~ ) ] - 9 (~ [ f . (.~)]1 ~< 19(~+" (t~)l If. (2) - f . ( ~ ) l <. K c 12 - 21 <~ K 12 - ~1

for k = 1,... , r - 2 , and by (30) we have

I L - 1 - L - 1 1 = 19 ( r - 1) I f . (2)] - 9 (r- 1 ) I f . (~)]l

~< I~0(')(t,)l If, (~?) - f . ( 2 ) l ~< K I~ - ~1

where t, tk are points between f~(2) and f , (~ ) , k = l . . . . , r - 1 . Hence we get I P - Y l < ~ g 6 ( e ) < ~ S z , l yk - -yk l< .g6(e )<<.Sz , k = l . . . . . r - 1 . Since 12- .~[<fi (e)~<51, we may use the inequalities (28). By (31) and (36) we get

19(') [ f . ( 2 ) ] - 9 (~) [ f . ( 2 ) ] l ~< ~.

Now by (28), (30) and (24) we obtain

I~b(')(2) - ~,(° (~)1 ~< , ' + K d + ,9e = e,

i.e. $ fulfils condit ion (31). This completes the p roo f of the inclusion T, (~¢) ~ ~¢. Let 9~e..~¢ and let 9~ tend to 90 (in the sense of the convergence in ~- ; this

convergence means that 9 , (x) --* 90 (x), 9~ k) (x) ~ 9(o k) (x), k = 1 . . . . , r, uniformly in (0 , c}). Let us write

$,(x) = h.(x, 9, [ f . ( x ) ] ) , So(X) = h.(x, 9o[fn(x)']).

Evidently ~ (x) tends to ~o (x) uniformly in (0, c) . It follows f rom (33) and (30) that the point (x, 9 , l - f . (x ) ] . . . . , 9 ~ ° [ f ~ ( x ) ] ) belongs to D x ( - K , K} for x e ( 0 , c) . Now by the uni form continuity of h,, , in D x ( - K , K ) we obtain that

~9~ "1 (x) = h.,~ (x, 9s [-fn (x)] . . . . . 9~ r) I f . (x)]) tends to

¢o'~(x) = h . , . (x , 90 [ f . (O] ..... 9 2 [f.(x)])

uniformly in <0, c> (as s - , oo). In view of (32) this completes the p roo f of 2 °. Evidently, every fixed point of t ransformation (34) is a solution of equat ion (21).

We know that this equat ion has exactly one solution 9 .mC ' [<0 , c>]. Thus for the p roo f of 3 ° it is sufficient to prove that 9 , e , ~ ¢. Let us note that ~¢ is convex. Since ~ ' is compact and conditions 1 ° and 2 ° are fulfilled, on account of Schauder's fixed point

Vol. 6, 1971 On the Con t inuous Dependence of C ~ Solutions 225

principle there exists at least one fixed point ~ o . ~ d of the t ransformat ion T.. This

completes the proof of 3 ° . In order to prove 4 ° it is sufficient to show that for a given e > 0 there exists an

N t> 0 such that for n >I N and for every qo e d we have II T. [~o] - To [~o] I[ ~< ~, or, in view of (32), that for every ~ o e d and x~<0, c>

IT. [~o]( ')(x) -- To [ (p]( ' )(x) l ~< ~. For ~ p e d we have

T. [~o3 (') (x) = h. , . (x, ~o I f . (x ) ] , ~o' [ f . ( x ) ] . . . . . ~o ('} I f . ( x ) ] ) . Thus

IT. [ ¢ ] ( ' ) ( x ) - To [~o]( ')(x)l

~< Ih., , (x , q~ I f . (x)] . . . . . ~o (') I f . (x)])

- h o . . ( x , ~o [ f . ( x ) ] . . . . . q¢') [ f . ( x ) ] ) l

+ Iho,,(x, ~o [ L (x)] . . . . . ~o (') [ f~(x)] )

- h o , . ( x , q~ [ fo (x)] . . . . . ~o (') [ fn (x ) ] ) l .

According to Lemma 4 (we replace H~,, by h.,. and f2 by P ) h.,, tends to ho,, uniformly in D x ( - K , K>. Thus there exists an no such that for n>~n o and xe<0 , c> we

have

lb . , . (x . . . . , ~o ~') [ f ~ ( x ) ] ) - h o , . ( x . . . . . ~o (') [ f ~ ( x ) J ) l ~< ~ . (37)

I t follows from the uniform continuity of ho,. in D × < - K , K> that there is a ~ > 0 such that for IP-Y[ <~ and lY,-#kl <5, k = 1 . . . . , r we have

8 Iho, . (x , 37, 37a . . . . . 37,) - ho, . (x, #, Yl . . . . . #.)[ ~< 2. (38)

Moreover , there exists an n~ such that for n>~n~, for every ~0es¢ and for xe<0, c>

we have I~ ('~ [ f~(x)] - q¢') [fo (x)]l < 6. (39)

In fact, suppose that (39) is false. Then there exists sequences x,, 0~< x. ~< c, % e . ~ '

and e. such that for every n we have

lop (') [ f~. (x . ) ] - ~o~ ') [ fo (x . ) ] l >i 6. (40)

Since ~¢ is compact we can choose from the {~o(~ ")} a subsequence which is uniformly convergent in <0, c) . For the simplicity we may assume that {~o~ °} is uniformly

convergent. Hence q~') [ f~ . (x ) ] -- q)~') [)Co (x) ]

tends uniformly to zero in <0, c}. This contradicts (40) and thus (39) is proved.


Now by (37), (38) and (39) we get

IT. [~o]~O(x) - To [q~]~O(x)[ ~< e, n ~> max(no, nl), q ~ d , 0 ~< x ~< c.

Thus 4 ° is proved and this completes the proof of Theorem 1.

§ 4. In this section we consider the three possibilities (A), (B) and (C). In case (A) we have the following

THEOREM 2. Let fo (x) ~ x. I f hypotheses (I)-(III) and inequalities (7), (8), (10) are fulfilled for n=0, 1, 2, ... then for every n there exists exactly one solution ~,~A[I] n C'[1] of equation (1). Moreover, cb, tends to 4o together with derivatives up to order r uniformly on every compact set contained in L

Proof. We may assume that 0 is the left endpoint of L It follows by Theorem 1 that there exists an interval (0, c) c~ 1 in which ~b. tends to ~b o uniformly together with derivatives up to order r. Let x o be the supremum of all such c. For an indirect proof suppose that I ~ ( 0 , Xo) 50. This implies that xo~L Hence we getfo (Xo) <Xo. This is obvious when fo ( x ) - 0, and for fo ~S" [_1] it follows by Lemma 1, It follows from the continuity offo that there exists an xl6I, xx >Xo, such that

To(x)<Xo for x ~ ( x o ,x l ) .

According to the uniform convergence of f . tOfo there exists an N such that for n/> N we have

f . ( x )<Xo for x~(xo, x l ) . (41)

Since ~b, satisfies equation (1) for xeI, we have

dp.(x)= H.(x, ~.[f~(x)]) for xE(xo, X~). (42)

Hence and by (41) it follows that ~b, tends to ~b o together with derivatives up to order r uniformly in (0, x~). This contradicts the definition of Xo. This completes the proof.

In case (B) we have the following

THEOREM 3. Let hypotheses (I)-(III) and inequalities (7) and (8) be fulfilled for n =0, I, 2,.... I f case (B) occurs then for every n equation (1) has a one-parameter family of solutions {d?, (x, t ): teR} ~ A [I] n C ' [I]. Moreover, i f for every n, we f ix thepara-

df meter t=t, in such a manner that lim..+~ot,=t0, then q$,(x)=~b.(x, t,) tends to qb o (x)= qb o (x, to) together with derivatives up to order r, uniformly on every compact set contained in L

Proof. It follows by Lemma 7 that fo~Sr[1]. Hence and by Lemma 5 we get the first part of this theorem. We may assume that the number n o defined in Lemma 8


is 0. According to L e m m a 5 we may write q~(x) in fo rm (14) where r

en (x) s! + k ! " / . . d

s = l s C k

By L e m m a 7, P~ tends to Po (with derivatives and uniformly on every compact) . Now, as in the p r o o f of Theo rem 1 we can prove that q~ tends to q~o with derivatives up to order r, uniformly in an interval (0, c). Applying the method used in the p r o o f o f Theorem 2 we obtain Theorem 3.

Similarly we can prove the fol lowing

T H E O R E M 4. Let hypotheses ( I ) - ( I I I ) and inequalities (7) and (8) be fulfilled.

I f case (C) occurs then equation (1) has for every n, n = 1, 2 . . . . . exactly one solution

Sn ~ A [ I ] n C ' [ I ] and f o r n = 0 a one-parameter fami ly o f solutions {4o (x, t): t~ R} c c A l l ] n C ' [ I ] . The sequence ~ is convergent together with derivatives up to order r

uniformly on every compact set contained in I i f and only i f there exists the limit

c = l im (p(~k)(O), k = k(O), (43)

where k(O) is defined by relation (11). Moreover, i f limit (43) exists, then q~. tends to

(x) = 4o (x, c).

REFERENCES

[1] C~ocz~ws~, B., Investigation of the Existence and Uniqueness of Differentiable Solutions of a Functional Equation, Ann. Polon. Math. 15, 117-141 (1964).

[2] FlenTENGOLC, G. M., Kurs differencialnogo i integralnogo isczislenija, I (Moscow 1962). [3] KOCZMA, M., Functional Equations in a Single Variable (Polska Akademia Nauk, Warszawa 1968

(Monografie Matematyczne, Vol. 46]). [4] MATKOWSKI, J., On the Continuous Dependence of Local Analytic Solutions of a Functional

Equation on Given Functions, Ann. Polon. Math. 24, 21-26 (1970). [5] MATKOWSKI, L, On the Uniqueness of D~fferentiable Solutions of a Functional Equation, Bull.

Acad. Polon. Sci. S6r. Sci. Math. Astronom. Phys. Vol. 18, No. 5, 253-255 (1970). [6] Wmrlx~-¢, H., Analytic Extensions of Differentiable Functions Defined in Closed Sets, Trans.

Amer. Math. Soc. 36, 63-89 (1934).

Silesian University

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On the continuous dependences ofC r solutions of a functional equation on the given functions