Remarks in connection with a paper of S. Czerwik “On a differential equation with deviating argument”

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXI (1979)

Bo g d a n Rz e p e c k i (Poznan)

Remarks in connection with a paper of S. Czerwik “On a differential equation with

deviating argument”

1. In [4] is proved under suitable assumptions that the equation y'(t) = f ( t , y(t), y ( g i ( t ) ) , y ( g m(t)), 2) (cf. [1], [5] and [2], [3]) has exactly one solution defined in the interval [0, oo) and fulfilling the initial condition y(0) = x , and this solution depends continuously on x and real parameter 2.

In this note we shall treat the case m = 1, since for m > 1 the proof is similar and the reader can repeat it himself.

Our results are proved by the theorem [11], p. 355, of the type of Banach fixed-point principle. Note that this theorem is true if В that appears there is a i?*-space.

2. Suppose that / = [0, oo), E is a Banach space with the norm || ||, and A is a metric space with the metric d.

Let p > 0 and let L: I -*■ [0 , oo) be a locally integrable function. We use the following notations:

A t — the space of continuous functions у: I -* E such that ||y(f)||

t

= О (exp (p ■ j L (s) ds)) for t e l , with the norm M defined by

о t

Iy l = sup [exp ( —p ■ § L(s)ds) • ||y(t)||: t e l } ; о

— the set of all continuous functions / : I x E x E x A - * E satisfying the following conditions:

Il f { t , U, V, и, v, Я)|| ^ L(t) • (||м —m|| + ||f - ÿ ||) for every (t, и, v, 2), {t, U, v, 2 )g/ x E x E x A ;

t t

|| J /( s , 0, 0, 2)ds|| = О (exp (p ■ J L (s)ds)) for t e l

о о

and for every fixed 2gz1 (0 denotes the zero of the space E).

16 - Prace Matematyczne 21.1

242 B. Rzepecki

We shall deal with the set x as an if*-space [8] endowed with convergence : lim /„ = f0 meaning that

Л QC

11

A)II

t

L(t) ■ exp (p ■ J L(s)ds) о

for every A e A . Moreover, . f j x f i x y l will be the i?*-product ([8], p. 86) of the if*-spaces $FX, E , A .

The following theorem holds:

Th e o r e m 1. Let h: I -* I be a continuous function. Suppose that there exists a constant q < 1 such that

t h(t) t

exp (p •

J

)

J

^

f

0 0 0

for every t e l . Then, for an arbitrary f e . W i , x e E and А е Л , there exists a unique function У(/,х>я)еЛ 1 such that У(/,х,х) (0) == * and

y'(f,x,X) (0 — / ( f » У(/,х,А) {t), У(/,х,Х) (0)?

for every t e l .

Assume, moreover, that there exist a locally integrable function Q : I -> I and a function œ: I -> I such that

t t

j()(s)d s = 0 (ex p (p-jL (s)ds)) for t e l , œ{t) - ^ 0 as t -* 0+ ,

о v 0

and

( t , u , v ) e ! x E x E } - + 0 as и oo

!|/(t, и, V, A ) - f ( t , и, V,A)J) ^ Q(t) co(d(A, I)) for all f e & x and ( t , u , v , A ) , (t, u, v, A)e I x E x E x A.

Then the function ( /, x , А) ь> У(/,Х,У) maps continuously JT*-space x E x A into the Banach space A x.

As the proof of the above theorem is similar to the proof of Theo­

rem 2 given in Section 3 it will be omitted.

3. The hyperbolic differential equation zxy = f ( x , y , z ) is a two-dimen­

sional analog of the ordinary differential equation z' ~ f ( t , z ) . Therefore, the result presented above can also be obtained for the equation

( + ) ~ f o g — ' = f ( x , y , z ( x , y ) , z ( h l {x),h2{y)),A),

where f , hx and h2 are given real functions and A is a parameter.

We shall give a theorem of this kind.

Suppose that / = [0, c » ) , P = l x I, Rn is an «-dimensional Euclidean

space, (A, d) is a metric space, and the functions ht: I -* I (i = 1,2) are continuous.

Let p > 0, 0 ^ q < 1, let œ: I -* I be a function such that m (f)->0 as t-> 0 + , let Ф: P -> / and W: P -*• / be locally integrable functions such that

* y

exp (p ■ j j Ф(м, u)dMdi;) + exp (p •

о 0

h f x ) h 2 ( y )

0 0 Ф(и, v)dudv)

* у

p- q- exp (p • J j Ф(и, v)dudv),

о 0

x у X у

J j \j/(u, v)dudv = 0 (exp (p • J J Ф(м, v)dudv))

o o v о о 7

for every (x , y ) e P .

We shall deal with the set C1 (/) of continuously differentiable functions on 1 as an if*-space endowed with convergence: lim tp„ = <p0 meaning

t n -+00

sup 1<М*)-фо(*)1

x у

exp (p • J J Ф(и, v)dudv)

о 0

(x, у)еР 0 as n -► oo.

We use the following notations:

C(P) — the Banach space of bounded continuous functions on P, with the usual supremum norm Ц|-|||;

A2 — the space of continuous functions z: P -> R1 such that z (x ,y ) x у

= 0 (ex p (p - j J Ф(и, v)dudv)\ for (x , y ) e P , with the norm 0 0

x у

IIzII = sup [|z(x, y)| • exp ( —p • J j Ф(и, v)dudv): ( x , y ) e P } ;

о о

Ж 2 — the set of all continuous functions / : P x R2 x A ^ > R1 satisfying

x у X у

the following conditions: j j / ( u , v, 0 ,0 , A) dudv = О (exp (p- j j Ф(и, v)dudv)\

0 0 ' 00 7

for every ( x , y ) e P and for every fixed л еЛ ; |/ ( x , y, u, v, A)—/( x , y , П, v, A)|

^ Ф(х, у) [|u — Щ + jr> — û|] for every { x , y , u , v,À), (x, y , U , v , X)e P x R2 x A;

\ f { x , y , u , v , X ) - f ( x , y , u , v , l ) \ ^ ф(х, y)- co(d(X,l)) for every (x, y, u, v, Я), (x, y, u, v, A ) e P x R2 x A;

Ж — the set of points (cr, т)е C1 (/) x C1 (7) such that <x (0) = i(0) and

X у

сг(х)+т(у) = 0 (exp (p J J Ф(и, v)dudv)} for every ( x ,y )e P .

244 B. Rzepecki

A sequence (/„) of elements of ^ 2 will be called convergent to f0 if sup \fn(x, y , u , v , A) - f 0 {x, y, u, v, A)|

X у

Ф(х, у) • exp (р • j J Ф(и, v)dudv) о о

(х, у, и, v ) e P x R2 -+ 0 as п —* со

for every А е А . The set & 2 supplied with this convergence is an i?*-space.

Moreover, ^ 2 х Ж x A is considered as a j£?*-product of the j£?*-spaces J*2, С1*/), СЧ1) and A.

We have the following

Theorem 2. For an arbitrary / е ^ 2, (cr, т)e3C and А е Л there exists a unique function z(/>CT>I.jA) e A2 satisfying equation ( + ) on P and such that

0) = <x(x), z(/iffft>A)(0, у) = т(у) for x ^ 0 and у ^ 0. Moreover, the function

( / , < 7 , T , Я ) Н » Z(f,a,x,k) maps continuously 2 x хЛ into Л2.

P roof. Let us put В = F 2 x f х Л , M = C(P), and X у

T ( z , ( f , о, т,А ))(х,у) = z(x , y) • exp ( —p • J J Ф(и, v)dudv), о 0

S ( z ,( /, <т, т, A))(x,y) = (<т(х)+т(у)-б7(0) +

X у X у

+ | J f [ u , v, z(u, v), z(/ii (u), h2(v)), A] dudi;) • exp ( — p- j j Ф(и, v)dudv)

0 0 ' 0 0

for z e Л 2, (/, и , t, A) e B and (x, y )e P. Obviously, T: Л2 х В - > М . Modyfying the reasoning form [4] and [11], we obtain: S maps the set A2 x B into M, {S{z,r])\ z eA 2} cz (T(z, q): z e A 2j = M for all q e B and |||S(zb rç)-S(z2,rç)|||

^ ^ •|||T (z1, r j ) - T ( z 2, rj)\\\ for every rjeB and z lKz2e A 2.

Let us fix ц in B. Then the equation S(z,rj) = T(z,rj) has at most one solution in A2 since T(-,rj) is one-to-one in Л2.

Let us fix z in A 2. We prove that S ( z , ) is continuous in B. For

= ifn,(tn,Tn,An), rj0 = (f0, o0,T0,A0) e B such that lim = rj0 and for (x, y ) e P we get

|S(z, rin) { x , y ) - S ( z , r j0) ( x , y ) I

< ( K M - ^<*o M l^{+ К} (у) - ^ч ( y )l+ К (0) - ^<To (0)| +

X у

+ ! ! !|fn[u, V,z(u, v), z f/iju ), h2(v)), ; j - 0 0

- f n\u, v, z(u, v), z (ht (и), h2(v)), A0]\ +

+ \fn[ u , v , z ( u , V), z ( h i (и), h2(и)), Я0] -

X у

—/о [и, V, z(u, v), z(hx (и), h2(vj), Я0] |} dudvj ■ exp ( — p • J j Ф (u, v) dudv) о о

x у

< (M * ) - M *)l + K i y ) - To (у)! + K (0) - cro(0)|) • exp ( - p ■ { f Ф(и, v)dudv) +

о 0

x у X у

+ (o(d(À„, Я0)) • exp ( — p • j J Ф(и, v)dudv) - J Jij/(u, v)dudv +

oo oo

x y

+ exp ( — p • J J Ф(и, v)dudv) x 0 0

s u p m x , y , r , s , X 0) - M x , y , r , s , W .

(w >s)6PxRZ| x

^ ^{( f ) (} Y лЛ . PYn ^{l r \} .1 \ ( b ( u 1 l \ / f a i / f a l \ J

^ Ф(х, y) • exp (p • J J Ф(и, v)dudv) 0 0

JC y X y

x J j Ф(и, и) • exp (p • j J 4>(s, r)dsdr)dudv

0 0 0 0

< sup

( x , y ) e P x y

exp (p • J J Ф(и, v)dudv) 0 0

+ sup

(x ,y )e p

|т„(у)-т0(у)|

x y

exp (p • j J Ф(и, v)dudv) 0 0

■+

+ sup

(x:,y)eP

k „ (0 )-M 0 )l

x y

exp (p • J J Ф(и, v)dudv) 0 0

+ C • w ( d (Я„, Я0)) +

+ P " 1 sup

x , y ^ O r , s e l f 1

I fn(x,У, Г, s, Я0) - / о ( х , у, r, s, Я0)1

X у

Ф(х, у) • exp (p • j J Ф(и, v)dudv) о 0

and therefore |||S(z, rj„)—S(z, rç0)||| ->0 as oo.

Consequently, an application of theorem given in [11] proves our theorem.

4. Let us remark that further results can be obtained if the concept of a metric space with the distance function taking its values in a normal cone in a Banach space ([6], [7], [9]) and the concept of a “generalized metric space” [10] (not every two points have necessarily a finite distance)

will be used. See also [12], [13], [14], [15] and [16].

References

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Phys. 4 (1956), p. 261-264.

246 B. Rzepecki

[2] S. C z erw ik , On the global existence of solutions o f a functional-differential equation.

Period. Math. Hungar. 6 (1975), p. 347-351.

[3] —, Special solutions of a functional equation, Ann. Polon. Math. 31 (1975), p. 141-144.

[4] —, On a differential equation with deviating argument, Comm. Math. 19 (1977), p. 183-187.

[5] T. D lo tk o , О pewnym zastosowaniu twierdzenia Banacha о punkcie stalym, Zesz. Naukowe Wyzszej Szkoly Pedag. w Katowicach 5 (1966), p. 83-88.

[6] M. A. K r a s n o s e l’skiT, G. M. V a tn ik k o , P. P. Z a b re îk o , Ya. B. R u tic k ii and V. Ja. S te c e n k o , Approximate solution of operator equations [in Russian], Moscow 1969.

[7] L. C o ll a t z, Funktionalanalysis und numerische Mathematik, Berlin-GOttingen-Heidelberg 1964.

[8] C. K u ra to w s k i, Topologie, v. I, Warszawa 1958.

[9] M. K w a p isz and J. T u ro , Some integral-functional equations, Funkcialaj Ekvacioj 18 (1975), p. 107-162.

[10] W. A. J. L u x e m b u rg , On the convergence of successive approximations in the theory of ordinary differential equations II, Indag. Math. 20 (1958), p. 540-546.

[11] B. R z ep e ck i, On the Banach principle and its application to the theory o f differential equations, Comm. Math. 19 (1977), p. 355-363.

[12] —, Note on the differential equation p { t, y(t), y(h(t)), ÿ(t)) = 0, Comment. Math. Univ.

Carolinae 19 (1978), p. 627-637.

[13] —, A generalization of Banach's contraction theorem, Bull. Acad. Polon. Sci., Sér. Sci. Math.

Astronom. Phys. 26 (1978), p. 603-609. -

[14] —, On some classes of differential equations, Publ. Inst. Math. 25 (1979). 4 [15] —, Note on hyperbolic partial differential equations, Math. Slovaca 30 (1980).

[16] —, Some remarks on hyperbolic partial differential equations, Banach Center Publications (to appear).