On a differential equation with deviating argument

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R O C Z N IK I P O L SK IE G O TO W A RZY STW A M A TEM A TYCZNEGO Séria I: P R A C E M A TEM A TY CZ N E X I X (1977)

S. Cz e r w ik (Katowice)

On a differential equation with deviating argument

1. In the present paper we are concerned with the differential- functional equation of the form

(1) =

h(x,cp{x)1(p[f1{x)'\7

where <p is an unknown function and h, f, i = 1, ..., n are known func

tions and и is a real parameter.

We shall prove, under suitable assumptions, that equation (1) pos

sesses exactly one solution defined in the interval <0,

oo)

and fulfilling the initial condition

( 2 ) <p{0) = I

and this solution depends continuously on £ and u. It is some generali

zation of results obtained in [1], [3], [5], [6].

For the equation

(3) — = й((я?,9?(я?),9?[/(а?)],м)

the corresponding problem has been investigated by T. Dlotko [4]. For the equation

~ = h {x , (p (x) , <p Ux (^)] ,...,? > [ / » (®)1 ldi (®)], • • •, lgm 00] j* for xe(0,a,y,

a < oo

(under different assumptions) some theorems (Theorems 5 and 7) have been obtained in [7].

2. We assume the hypotheses:

Hy p o t h e sis 1.

(i) Let (В , ||-||) be a Banach space. The function h: I x B n+1 x R->B,

I = <0,

oo),

В — ( —

oo, oo)

is continuous in I — B n+1 x B .

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(ii) There exist real functions L 0, L n, continuous in I, L {(x) > 0 for x I , i = 0,*

1,

..., n, such that for every (r0, ..., rn), (zQ, zn)eBn+1, x e l, ueR we have

П

(4) \\h{x, r0, . . . , r n,u ) - h ( x , z0, ...,z n, u) || < ^ L fx ) ||г<--г<||.

i = 0

(iii) The real functions f iy i = 0,1, n, where f 0{x) — x, are con

tinuous in I and f f l ) c l , i = 1, n.

(iv) There exist constants a > 1, К , 0€<O, 1) and (y0, ..., yn)eBn+1 such that for every fixed ueR, we have

X

(6)

j \\h{s,y^ . . . , y n,u)\\ds^Ko^(aL{x)), xel,

о

where

X П

(6) L(x) = J ^ L i(s)ds,

X e l .

0 i= 0

Moreover,

n n

(7) У в {(х)екр(аВ[Ъ(х)]) < a0exp (ai(a?)) У в {{х), x e l.

i= 0 i = 0

Hy p o t h esis

2. There exist constant M and functions H : I~>I, co: I - + I such that (o(u)->0 as u-+ 0 + and

x •

exp(—

aL{x))

J

N(s)ds

<

M, xel.

о

For every (z0, ..., zn)eBn+1, x e l, иг, u%eR we have

IIh{x, z0, u1) — h(x, z0, ..., zn, tt2)|| < N {

^x

)(

^o

{\

^ux

-

^u

2\).

R em ark . Condition (7) is fulfilled, for example, if f fx ) < x or fi(x)

> x and

L [ f i (x)'] — L{x) < с < а -11па, 4 = 1 , ..., n.

3. Now we shall prove

Theo rem 1.

I f Hypothesis

1

holds, then for every ueR there exists exactly one function у satisfying equation (1) and fulfilling the initial con

dition (2), where £eB, and such that

M®)ll = 0(exp[aL(®)]).

It is the limit of successive approximations.

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P roof. We define G as the space of these functions which are defined and continuous in I and <p(æ)€B for x d and

(8) Il9>(a>)|| = Oj(exp[aL (»)]), œel.

For (peG we define the norm (cf. also [1]) l?>l = sup (||ç>(®)||exp(-aL(®))).

xel

We can easily verify that G is a Banach space. Equation (1) with condition (2) is equivalent with the equation

X

(9) <p{œ) = | + Jü(e,ç>(s),ç>[/i()], ...,ç>[/«(*)L «)<**•*

о We define the transform

(10⁾

Ф(оо) = £ + fh]{s,<p(s), ...,(p [fn(s)], u)ds.

We shall prove that (10) maps G into itself. Ф is evidently continuous in I and Ф(х)еВ for œel. Now, by (4) and (5) we have

X

||Ф(ж)1К f + /|||Ч «> <?>()> •••> <?,[/n(s)L Уо, . . . , y n,u)\\d8 +* 0

x

+

f \\h{s,y0, . . . , y n,u)\\ds

X n

0 Ilf 11+ J ( N -£г(5)1!9[/г(«)]-2/г!|)^« + ^?

0 г= 0

where

J

= / IIM«>

Уо,

•••,

Ут u)\\d8.

о Next

(и )

^цф

(

^й

)||

X П

< ||f|| + m ax(iç>-y0|, ..., \<p-yn\) J ( ^ Х г-(«)ехр(В [/,(*)])) ^ds + ^{J .}

From (7) and (6) we obtain

# ^П j

/ ( j ^ i ( s)exp(a£[/f(s)]))ds < 0 / a ( ^7|X tf(s)) exp(al/(s))ds

“ ■ ■ 0 г = 0

= бехр (ai(s)) |o < бехр (aL(cc)j.

0 i—0

0 i=0

X n

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Consequently, Ъу (11) we have

1|Ф(®)11< Ш + 0 т а х ( |^ - у о1, |

9

>-yJ)exp(aZ(a?))-f-J, whence

||Ф(а?)||ехр(-а£(аО) ^ const for a? > 0.

Thus ФеО. Now we shall prove that (10) is a contraction map. Actually,

(12) t|<P(æ)-!P(e)||

*

< / ||л(*, ç > ( » ) ,

u )- h ( s , y ) { s ) ,

...,?[/„(*)], »)||de

0 x n

< J às

О г==0 x n

< \<P~V\ f ( y x , ( s ) e s p f a L[fd>)])) *

0 г = 0

ж »

< в\<р--у)\ f a ( Хг-($)) exp[a£(s)]ds

О г = 0

= 0 |ç? — y>\ [exp(aZ-(a?)j — l] < 6\<p — y|exp(oZ/(Æ)).

Consequently

(13) \0-W \ < в\<p-y>\.

On account of the Banach theorem for every fixed ueB there exists a unique fixed point of transform (10), i.e., unique function <peG satisfying equation (1) and condition (2) and it is given as the limit of successive approximations. This completes the proof.

Next, we shall prove the following

Th e o r e m 2.

I f Hypotheses

¹

and

²

are fulfilled, then solution of equa

tion (1) depends a continuously on $ and u.

P roof. We assume a(£, u) — 6 < 1 and

X

F = F(a>,<p, £,u ) = Ç + jh {s,(p {s), ...,<p[fn{s)],u)ds.

о From (12) and (13) we have

I F(œ, (p, f, u) — F(æ, y>, f, u)\ < 6 \q> — tp\.

The proof that F is continuous with respect to | and и (on acconut of Hypothesis 2) does not differ from that given for Theorem 3 in [4]

and is therefore omitted. Consequently Theorem 2 follows from [2]

(Theorem 7), which completes the proof.

The author wishes to express his appreciation to the referee for

his valuable suggestions concerning the material in this paper.

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References

[1] A. B ie le c k i, Une remarque sur la méthode de Banach-Oacciopoli-Tikhonov dans la théorie des équations différentielles ordinaires, Bull. Acad. Polon. Sci., Sér.

Sci. Math. Astronom. Phys., Yol. IV, No. 5 (1956), p. 261-265.

[2] — Bownania rôzniczkowe zwyczajne i pewne ich uogôlnienia, Warszawa 1961.

[3] C. C odun ean u, Sur Vexistence et le comportement des solutions d'une classe d'équations différentielles, Bull. Math. Soc. Sci. Math. Phys. B.P.R. 2 (50) Nr. 4 (1958), p. 397-400.

[4] T. D lo t ko, O pewnym zastosowaniu twierdzenia Banacha о punkcie stalym, Zesz. Naukowe Wyzszej Szkoly Pedag. w Katowicach, Nr. 5 (1966), p. 83-88.

[5] — et M. K u czm a, Sur une équation différentielle-fonctionnelle, Colloq. Math.

12 (1964), p. 107-114.

[6] S. D oss and S. K. N asr, On the functional equation dyjdx = f ( x , y(x), y(x + h)), h > 0, Amer. J . Math. 4 (1953), p. 713-716.

[7] M. K w ap isz, On certain differential equations with deviated argument, Comm.

Math. 12 (1968), p. 23-29.

2 — P ra c e M a te m a tyczne 19 z. 2

On a differential equation with deviating argument