On some properties oî solutions oî a functional equation with parameter

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I : PRACE MATEMATYCZNE X V II (1974)

S

C

(Katowice)

On some properties oî solutions oî a functional equation with parameter

1. In the present paper we are concerned with the functional equation

(1) <p [f ( x ), t] = g {x, t) <p {x, t) + F {x, t) ,

where <p(x, t) is an unknown function and f(x ), g(x, t), F (x , t) are known real functions of a real variables and ^ is a real parameter.

We shall prove that under some assumptions concerning the given functions (in case (C) — cf. [3], and p. 336) the solution cp(x, t) of equation (1) is of class Cr, 1 < r < oo, with respect to the parameter t.

2. We write f°(x) = x, f n+1(x) = f [ f n(x)], n — 0, 1 , . . . We assume the following hypotheses analogous to the hypotheses in [3] :

(I) The functions g(x, t), F (x , t) are continuous in A =

<a , b )x T and g(x, t) Ф 0 in A0 = (а, Ъ) x T , T — interval.

The function f(x ) is continuous and strictly increasing in an interval

<■a , b) and a < f(x ) < x in {a, b).

(П)

(11Г m7 d°g dvF

There exist the derivatives -^-(&, t), -~v- (x, t), v = 1, 2, . tinuous in A.

r, con-

(1УГ) For every v = 1 , . . . , r and for every closed interval <a, /?> с T there exist: interval {a, a + rjf) c: <a, b), r\v > 0, function B v(x, t) bounded in <a , a + rjvy X <a , /?> and a constant 0 < в < 1, such that the inequalities

dvg

di < B v(x, t), BV[ M , Ï] < 0Bv(x, t)

hold in <a , a + rjfy X <a, /?>.

(Vr) For every v = 1, ..., r and for every closed interval (a , /9) a T there exist: internal (a , a

v}

(a , b),

,„ > 0, a function B v(x, t) bounded in (a , a + Qvy X <a, /9) and constant 0 < в < 1, such that the inequalities

d v

~ Ж {ас’ г) < JDv(æ, t), A ,№ ) , t] < 6Dv{x, t) holds in <a, a -f

v}

<a, /?>.

(YI) There exists a function c(t) = c (c — constant) such that for every closed interval (a , /?> с T exist an interval {a, a + d) a (a , 6), d > 0 , a function В (x, t) continuous with respect to the variable t in <a, /9) and bounded in (a , a + x <a, /?) mid a constant 0 < 6 < 1 such that the inequalities

(2) \F{x, $)| < B (x , t), В [f(x ), f] < 6B{x, t)

hold in <a , a-\-d} x (a , /9>, wüere .F(

, t) = F (x , t) + c(t)[g(x, t) — 1].

We put

71 — 1

(3) On(x ,t) = f ] g [ f v(x ),t],

V = 0

(4) G(x, t) = limG„(æ, $).

П— >oo

There are the following three possibilities (i is fixed) :

(A) The limit (4) exists, G(x, t) is continuous and G{x, t) Ф 0 in (a , b).

(B) There exists an interval I <= (a, b) such that limGn(x, t) = 0 n— >oo

uniformly in I.

(C) Neither (A) nor (B) occurs.

3. Now we shall prove the following

T

1. Suppose that hypotheses (I), (II), (III1)-(Y 1), (YI) are fulfilled. If, moreover, g(x, t ) ^ 1 in A and for every te T case (C) occurs, then there exists exactly one solution of equation (1) in A of class C1 with respect to the parameter t in A.

P ro o f. On account of [2] there exists exactly one function <p(x,t) satisfying equation (1) and continuous in A. Let te T be fixed and e > 0 be such that for \h\ < e is t + h t T . The function <p(x,t-\-h) satisfies the equation

(5)

+ = g{x, t-hh)<p{x, t + h) + F (x , t + h).

By (5) and (1) we obtain

<P U iæ)i t + h ] - ( p [f(x ), f\ = g {x, t + b) <p {x, t + h) — g (x, t) cp {x?t) + + F ( x ,t + Ti) — F (x ,t).

According to the Hadamard’s lemma ([

], cf. also [4]) we have

<p[f{x), t + Ti]-<p[f{x), t] --= Ф2{х, Ti)[<p{x, t + Ti)-q>{x, «)] + Фх(ж, h)Ti + -\-уг{х, Ti)Ti, where

1

Ф1 = Ф1(х,Т1) --= f {<р(х, t) + 8[<p{x,t + h ) -<р{х, t)]}gt (x ,t + sh )d s, о

1 1

Ф2 = ф2(х,Ть) — f g (x, t + sh) ds, чрг = ip1(x ,h ) = J F t(x, t-\-sh)ds.

ô о

Let. ns write

(p(x, t-\-Ti) — (p{x, t) = ip(x, h), and then we obtain

У [/(я

> ^) + (Фх + Ух)^?

or? for h Ф

V [f(x ),h ] , у(я, Л)

Ti Ф,

Л ■+ Фх + Ух*

-г, ,,. w (x.h)

Putting y(x, Ti) = —- , we have fb

(

) y [ f ( x ), Tl ] = фг{х, h)y(x, h) + Фг(х, ft) + yi(æ, Tl).

Now we shall consider the equation

(7) a[f{x), Ti] = Ф2{х, Ti)a{x, Т

) + Фх(х, ft) + yi(® , Л).

Since <

(

, i) ^ l , we have

П

and hence

(

8

03(æ ,*)= / 7

<

, xe (a , b), \Ti\ < £, w- = 1, 2, ...

G»(®, Л) .

Thus case (B) for equation (7) does not occur for any \h\ < e.

Let {hn} be any sequence such that \hn\ < e and hn -> 0. We have (9) in <a, a + rjoy X <2 — s ,t + s}.

In view of (IV1), (V1) and (9), we obtain

i i

(10) |0>x|< j (L + 2Ls)\gt(x, t + shn)\ d s^ 3 L j B x{x, t + shn)d s= B x{x, hn),

о ⁰

xe < a , a + ^x),

l l

/ J ^I ds ^ J

(t^7j $ “j- sJbyi) ds —

(*^j ~^n) ? *1? ^ (л, 6Ь I •

0 0

It is easily seen that

(12) B x[f(x ), hn] < 6B x{x, йя), 2>х[/(я?), hn] < 0£>x(^ hn) in <e, a+£>, where £ = min^x, £x).

According to (10), (11), (

) and [1] for every hn equation (7) has exactly one continuous solution in <я, b) fulfilling condition a (a, hn) =

. It is given by formula

(13) a (x , hn) = —

@v+l (*^J ^n)

If hn = 0, then Ф

(а?, 0) = g(x, t) and for equation (7) case (C) occurs.

On account of (10) and (11) it is easily verify that (VI) hold if we assume e(t) — 0 and F{x, 0) = Фх(х, 0) + у х{х1 0). Consequently in view of [1], Theorem 9, the equation

« № ), 0] = g {x, t)a{x,

) + Фх{x, 0) + tpx(x , 0) has exactly one continuous solution in <a , b) given by the formula

a(x ,

) = - ^

v= o

Ф ЛГ(х),_0] + щ [Г (х ),

]

@v+l{®1 ^)

Let de (a, b) be arbitrarily fixed and let N be chosen so that f N(x)e e (a , a + £> for xe {a, d}. Next, we have

Ф хГ Г И ^ З + ухСГИ, K l

^«+i(^? К ) <\ф1 [ Г ( я ) , К 1\ + \ п 1Г (х ),К ]\

< B x[ f ( x ) , h n] + Dx[ r ( x ) , hn]

< Bv- N(Bx[ f N(x), hn]+ D x[ f N(x), hn])

< 6V~N[ sup B x(x ,h )+ sup Dx(x ,h )].

<a,tZ> x <— е,б> <a, й>х<— e,e>

Thus series (13) uniformly converges in <a, d>x< — s, e) and we have

(14) lima (a?, hn) = a (x ,

).

On the other hand, the function

(15) y(cc, hn) = cp{x, t + hn)-c p (x , t) К

is the continuous solution of equation (7) such that у (a, hn) =

. Since equation (7) has exactly one continuous solution in (a , b) fulfilling the condition a (a, hn) =

, we have

a (x f hn) = y(x , Jin)y 7i =

,

, ...,

and consequently by (14) and (15) for every t < ? T there exists the derivative

— (x , t) and is continuous in <a, b). dcp

Replacing h by hn in equation (7) and passing to the limit as n -> oo, we obtain

(16) dcp dcp

-ft *1 = 9 (®> *) ~ft *) + dg_

dt (x, t)cp(x, t) dF dt (x, t).

In view of (

) and (

) it is easily verify that for equation (16) (YI) holds if we assume c(t) = 0 and F (x , t) = gt(x, t)cp{x, t) + F t{x, t). Hence, according to the condition g{x, t) >

and [

] equation (16) has exactly one solution continuous in A. Therefore the function dcp (x, t) is continuous in A, which was to be proved. 01

T

2. Let the assumptioTis of Theorem 1 be fulfilled (instead of (III'H Y1) we assume (ПГ)-(УГ), 1 < r < oo). Then there exists exactly one solution of equation (1) in A of class Cr with respect to the parameter t in A.

The proof is analogous to the proof of Theorem 1 and Theorem 5 in [4], and is therefore omitted.

References

[1] B. Choc zew ski and M. K uczm a,

Fund. Math. 58 (1966), p. 163-175.

[2] S. C zerw ik,

Prace Naukowe U.&L, Prace Mat. 3 (1973), p. 25-30.

[3] —

this volume, p. 341-345.

[4] —

Ann. Polon. Math. 24 (1971), p. 217-225.

[5] I. P ie tro w s k i,

Warszawa 1953.

3 ~~ Roczniki PTM — Prace M atem atyczne XVII.