On the iterative method of solving differential equations with retardations

M. Kwapisz (Gdańsk)

On the iterative method of solving differential equations with retardations

1. In this paper we shall consider a certain modification of Picard’s method of successive approximations for solving differential equations with retarded parameters of the form:

V'(t) = f ( t , y ( t ) , y ( t ) , y( t -r (t ) ), y ( t- r{ t) ), y' {t j) .

A similar method was considered in the previous paper [1 ]; it was applied to a special case of differential equations called “ differential- difference equations” . The theorems concerning the convergence of suc­

cessive approximations which will be given here are a generalization of similar theorems included in papers [2], [3], [4], [5]. A modification of Picard’s successive approximations method gives a sequence of suc­

cessive approximations which is often more rapidly convergent than the sequence obtained by Picard’s usual method of successive approxi­

mations.

The method used in this paper is analogous to that given by T. Wa- żewski in a general form in paper [6].

2. We introduce the equation

( 1) X(t) = <p(tj fx(s)ds + y)(t0), Jx( s) ds+y( t0),

+j x{s)dsJr y)(t0), +J x{s)ds-\-y>{t0), : » ( « } ) +

where for te<t0, a) and t—r(t) < ź0

t— r(t)

+f x(s)ds + ip(t0) r(t)j,

and the following assumptions:

58 M. Kwapi s z

Assumptions H0.

1 ° The function cp{t, иг, u2, uz, щ, %) is defined and continuous for ( t , u x, u2, u3, u4, u5) eD where

I): t e <tQ, a), щ ^ 0, г = 1 , 2 , . . . , 5 .

2° The function у is non-negative and non-decreasing in each of щ, i = 1, 2, 3, 4 in D.

3° If the function у depends on u5, then it is increasing in u5.

4° The function r{t) is defined, continuous and non-negative in

«о , a).

5° The function ip(t) is defined, continuous and non-negative in E0, E0 fj = t ~ r ( t ) < tQ, te( t0, a)].

6° For each continuous and non-negative function h(t ) defined in <tQ, a) equation (1 ) has a non-negative, continuous solution in <t0, a ) .

7° For each non-negative real number p the equation

® = <р(*о, Wito), Wito), W (to— г ( к ) ) , w ( t o ~ r ( t 0)), х ) + [л

has a unique non-negative solution x.

Assumptions Hj.

1 ° (p(t, 0, 0, 0, 0, 0) se0 for te(t0, a).

2° If h(t) ~ 0, te(t0, a) and f(t) = 0, teE0, then s(t) ее0, te(t0, a) is the unique solution of equation (1 ).

How we shall prove two lemmas.

Lemma 1. I f the assumptions H 0 are fulfilled and the function u(t) is continuous for te<t0, a) and also satisfies the inequality

t t

(2) u(t) ^ w(t, ju(s)ds + w(to), j и (s)ds f-W (to),

+ ^t-T(t)

J u( s)ds f-y( t0), +J u(s)dsf-w(to),u(t))) Jt-h(t)

for te(t0, a) , then the inequality u ( t ) ^ . w h(t) holds for t€(t0, a), where wh(t) is the maximum solution of equation (1 ) defined in (to, a).

I f the weak inequality (2) is replaced by the strong inequality, then the inequality u ( t ) < wh(t) holds for each solution of equation (1 ).

P r o o f . We shall prove Lemma 1 only in the case where the func­

tion 9? depends on щ . Evidently, in the other case the proof is very simple.

Let wh.(t) denote the solution of equation (1) when h(t) is replaced by hit) = h(t)+l/i.

First we shall show that

u(t0) < wh.(t0).

Indeed, we have

w h i { t 0) = < p ( to> V ( t o ) i V>( t o) t V>( t o— * ( t o ) ) ł М \ { и ) ) л - Ы к ) -

Put

= «^(*0) —$>(<„, V(to), v(to), f(<0 — v(<0- t( O)i Щ( *о));

then the last equation can be written as

*»(*%(<<>)) = ЫЧ)-

This equation has a unique solution with respect to wh.(t0) for each h*(<0) (this is ensured by the assumptions H0, 7°); therefore we have

w^ito) = > 0.

The function m(z) is monotonie because it is continuous and invertible.

Hence, and from the inequality

m(0) = — (p(t0, y(t0),ip(t0), f ( t 0—r(t0)), y)(t0—r(t0)), O)

< 0 < hiito) = m(wh.(t0))

it follows that the function m(z) is increasing.

Further we get

m(ii {t0)) < hi{t0) and u(t0) < = «^($0).

If there exists a te ( tQ,a) for which the inequality u(t) > wh (t) holds, let у = inf A , where A = [t: u(t) > wh (t0) f te(t0, a)].

For te( t0, y) we have the inequality u ( t ) < wh.(t) but for t = у we get

У У y - r ( y )

u ( y) <<p( y, j u(8)ds-\-ip(tQ), f u(s)ds + y(t0), + f u(s)ds + y(t0), 4

y - r ( y )

+J u(8)d8 + y){t0) ,u(y)} + hi(y) 4

У V Y — r ( y )

f w h.{s)dsĄ-y)(t0), j w hi{s)ds + y{t0) , + j wh.(s)ds + ip{t0),

<0 t0 <Q

У-Цу)

7 Wh.(s)ds + tp(t0), wh.(y)\ + hi(y) = w h.{y).

ł0

60 M. Kwapi s z

The last inequality contradicts the definition of number у and there­

fore we can state that the inequality u ( t ) < w h.(t) holds in the whole interval </0, «)• Notice that from such considerations we get the second part of Lemma 1.

Using a similar argument we infer that if h( t ) < Jc(t) for te<f0, a) and k(t) is continuous, then the inequality wh(t) < wk(t) is satisfied in the interval <tQ, a).

The sequence of the non-negative functions whi(t), i = 1 , 2 , . . . is decreasing, and therefore it has a limit w{t)\ w(t) satisfies equation (1 ) and also the inequality w{t) < wh(t) but, on the other hand, the inequali­

ty w(t) > wh{t) follows from the inequality wh ( t \ > w h(t).

Finally we conclude that w(t) = wh(t) and u(t) < wh(t) for te <<0, «).

This completes the proof of Lemma 1.

Lemma 2. I f the assumptions H0 and Hx are satisfied and z0(t) — whl(t) and zn+1(t), n — 0, 1 , . . . , is the maximum solution of the equation

t t t—r(t)

(3) s(t) = (p{t, fs(v)dv, f z n(v)dv+ipn{t0) , +f s(v)dv,

fo Ч 4

f (r) dv-f- ipn(to) , zn(/)|,

*0

where tp0{t) — y>(t), ipn(t) = 0, n = 1 , 2 , ..., for te£]0, then (4) 0 < zntl{t) < zn{t) for n = 0 , 1 , . .. , te(t0, a),

and the sequence {zn(t)}, n = 0, 1 , . . . , is uniformly convergent in every closed interval contained in <(t0, a) to the function z(t) = 0 for te(t0,a).

In short, we say that it is almost uniformly convergent in the interval (t0, a) . P r o o f . From Lemma 1 and the inequality

t t

zx{t)<<p[ti J fzo(v)dv + y>0(t0),

f0 10

+f z1(v)dv-hf0(t0) , +J z0(v)dv + y>0(t0), Zoiflj + hiit)

10 (0

we conclude that the inequality zx(t) < z0(t) is satisfied in <t0, a).

We assume that for a certain integer n the inequality 0 < zn+1(t)

< zn(t) is satisfied. Then we have

t i t-r( t) t-x(t)

ф J’znj.x(v)dv, j z n+1(v)dv, J' zn_x_x(v)dv, J znJrX{y)dv, znJrX{t)^

tQ

<Q <Q <Q

and

t t t—z{t)

~ {t i Jzn+i{v)dv , J%n+i(r)dv, J znJt.x{y)dv ,

*0 *0 *0

t-r(t)

J' %n+l (^) dv J ^M+l (^) j H- Оио_1 (t) , where an+x(t) > 0 for t€<t0, a). 10

Using the definition of the function zn+2(t) and Lemma 1 , we obtain the inequality 0 < zn+2(t) < zn+x(t) for te(t0, a).

On the basis of the principle of induction we can state that inequali­

ty (4) is true for n = 0 , 1 , ... The limit of the sequence {«„(<)} satisfies equation (1) for h(t) = 0, tef tQ,a) and ip(t) = 0, teE0; therefore, be­

cause of the assumptions H x, it must equal zero for te ( t0, a). Further, it follows from Dim’s theorem that the seguenee {«„(<)} is almost uni­

formly convergent in the interval </0, a) . Thus, the proof of Lemma 2 is finished.

3. In this part of the paper we are going to consider the differential equation with retardation in the Banach space B.

Let us note further assumptions, which will also be used in our con­

siderations.

Assumptions H 2.

1° Assume that the function f { t , u x, u 2, u 3, u 4, u 5) is defined and continuous for (t, uxi u2, u3, u4, where

2?i: t0 ^ t < а, щ е В , * = 1 , 2 , ..., 5.

2° The values of f(t, ux, u2, u3, u4, u5) are in the Banach space B, i.e. f ( t , 'W’21 у ^4 j ^5 *) € В for {f у , ^4j ^5) f H i.

3° The condition

(5) %? %) %? %)ll

q>(t, \\ux uxК, \\u2 Mgll j !i% %ll > IN4 %ll) 11% %ll) holds for (t, ux, . .. , u5), {t, % , . . . , u5)eJDx, where y> is a function satis­

fying the assumptions H0 and Н г.

We shall deal with a differential equation with retardation of the form

(6) y'(t) = /(< , y( t) , y( t) , у ( t - r( t )) , у ( t- r( t )) , y'(t)j for t e (t0, a) , with the initial condition

(6') y{t) = i(t) for teE0,

where £(t) is a given continuous function; r(t) and E 0 have the same meaning as in the assumptions H0.

62 M. Kwap i s z

Using the substitution y'(t) = u{t) we can write equation (6) as follows

t t

(7) U(t) ^ f ( t , ju(s)ds + f(t0), J « ( e ) d e + f ( $ 0),

^0 ^0

t-T(t) t-x(t)

+f u(s)ds + £{t0), +J u(s)ds+ £(t0),u{t)\.

<0 <0

In this way, the problem of finding a solution of equation (6) with a given initial condition y(t) = tj(t) for te E0 is reduced to the problem of finding a continuous solution of equation (7).

The solution of equation (7) is constructed by the following succes­

sive approximations method: we put u0(t) = 0 for te(t0, a)~\~E0, and if un(t) is defined (n = 0, 1 , ...) then un+1{t) is the solution of the equation

t t

(8) u(t) = = f \t ,J u( s )d s+ £ {t 0) , j u n(s) ds+£( t0),

<0 t-T(ł) ł° t-T(t)

+J u( s) ds + £(t0), +J un(s)ds-\- S(t0), un(t)j,

10 *0

where it is assumed that

t t— X

(t)

f u 0( s) ds +£( t0) = +f u0( s) ds+$( t0) = 0, te<j0ła).

r0 *0

4. At first we shall prove a theorem connected with the existence and location of the successive approximations defined above:

Theorem 1. I f the assumptions H 2 are satisfied, then relation (8) determines uniquely a sequence of successive approximations for t e ( t 0, a) and the inequality

(9) I|m»(*)II < *%(*) for t€<t0, a )

is satisfied, where whl(t) is the solution of equation (1 ) and h(t) = sup Hf(t, 0, 0, 0, 0, 0)H for te<t0, a) while ^{tn < s}

<

v>(t) = snp[||f (e)||: s e E0, s < t].

P r o o f . Let us consider the equation for v(t) (10 v(t) = f [ t , Jv{ s) ds + £(t0), f u ( s ) d s + £(t0),

+ ^{t-x (t)}

f v {s )ds + £{t0) , + j u{s)ds-\r£{t0),u{t)}

where u(t)sB is a given function, continuous in the interval <tQ, a) and 1И t)\\<whl(t) for te( t0,a).

It will be shown that equation (10) has a solution in the interval (t0, a) satisfying the inequality

(1 1 ) 1И0И < whl(t) for te(tQ,a).

Let us construct the sequence of successive approximations for equa­

tion (10) in the following way:

Mt) ~ 0, t€<tb, a) + EQ,

t t

(12) vn+1{t) = f ( t , Jvn{s)ds+£(t0), fu(s)ds-{- £(t0),

1o ł0

t—r(t) t—r(t)

+f vn(s)ds + i(t0) , +J u(s)ds-f £(t0), u(t)J

^0 *0

for te( t0, a), n = 0, 1, ...

We observe that the first approximation v0(t) satisfies the inequality Po(t)\\ < whl{t), te( t0,a).

Further, if we assume that the approximation vn(t) satisfies the inequality

IK(Ż)I1 < te( t0,a), we get

t t t-r (t)

IK +1(t)|| < ||/(«, Jvn(s)ds+£(t0),ju(s)ds4-£(t0), +J vn(s)ds+ £(t0),

tо t0

t-r{t)

+f u(s)ds-t- l(t 0), u ( t ) ) - f ( t , 0, 0, 0, 0, 0)|| + IIf(t, 0, 0, 0, 0, 0)||

ł0 t t

/ IM*)ll<fo+H!(«o)ll> Jl|w(*)li<fc+ll£(<o)ll>

^0 *0

7 IK (s)ll*+ ll£ (« ll, +J ||«(*)||<b+||fMI, ll«W ll)+ll/«,0,0,0 ,0,0)||

*0 *0

t t t—t(<)

< ę » ( < , f w hl(s)ds + v(to)f J w f t 1 ( e ) < f e + v ( t 0 ) » + / w A j ( e ) d s + y > ( t 0 ) ,

<0 <0 <0

t-r(t)

+f wftl (e) ds -F ip (t0) , wAl (t)) + hx (t) = whl (t).

We can prove by induction that the inequality (13) ||«»(<)||< whl(t), te<t0,a ), holds for n — 0, 1 , . . .

Let us now investigate the convergence of the sequence {vn(t)}. We have for any p = 1 , 2 , . . .

IM<) — M 0 l l < wAl( t ) ^ 5 0(t), f€<t0,

a).

64 M. Kwapisz:

If we assume that the inequality \\vn+p(t) — vn(t)\\ < zn(t), te(t0,a), holds for • any p = 1 , 2 , . . . , we get

H^n+i+pW ^«.-4-1 (^) [’

t t-r(t)

^ p /||®n+p(e) ds, 0, j ||^и+р(®) Vn(s)||ds,0,oj

*0 ^0

t t—T(t)

< <p\t, JZn(s)ds, 0, +J zn(s)ds, 0, o) JLSn+1(«).

*0 *0

Hence, again by induction, we have

(14) \\vn+p{ t ) - v n(t)\\ < zn(t), te<t0, a), p = 1 , 2, ...; % = 0 ,1 , ...

The following statement easily follows from the definition of the sequence {zn(t)}, n = 0, 1 , ...:

0 ^ + l ( t ) ^ 3 n { t ) 7 ^ e \^0 7 **) 7 W = 0 , 1 , . . . ,

and that sequence is almost uniformly convergent to zero for £e<£0, a).

Indeed, the monotonicity of the sequence {zn(t)} follows from the inequality

t t-T(t)

z0{t) > < p ( t , f ż 0{s)ds, О , " / z0(s)ds, 0, 0).

*0 * 0

From Lemma 1 and from the inequality z(t) = <p[t, Jz(s)ds, 0 , + J S(s)ds,0,0

t-T(t) t~T{t)

(p{t, Jz(s)ds, Jz(s)ds, 7 z(s)ds,+f z(s)ds,~z(t)|

where z(t) = lim zn(t), te ( t0, a), we infer that z(t) = 0, t e < i 0 , a).

и—*oo

The almost uniform convergence of the sequence is a consequence of Dini’s theorem.

We deduce from inequality (14) that the sequence {vn(t)}, n —

= 0, 1 , . . . , is almost uniformly convergent in the interval <t0, a) to the function v(t) which is the solution of equation (10). We can easily show that the function v{t) is the unique solution of (10). By inequality (13) we infer that inequality (1 1 ) holds. Now by induction we verify that all ap­

proximations un{t) are uniquely determined and fulfill inequality (9).

5. We shall now prove the convergence of successive approximations un(t), n — 0, 1 , . . . , defined by relation (8) and the existence of the unique solution of equation (6).

Theorem 2. I f the assumptions H 2 are satisfied, then the sequence {un(t)}, n — 0, 1 , . . . , defined by means of formula (8) converges almost uniformly on the interval (t0, a) to the unique continuous solution u(t), te (t 0,a), of equation (7), and the error estimate of the form

(15) \\un(t) — u(t)\\ < zn(t), t€<t0, a), n — 0, 1 , ...,

holds, where the functions zn(t) are defined by means of relation (3).

P r o o f. First it will be shown that the sequence {un(t)} of successive approximations satisfies the Cauchy criterion for uniform convergence in the same way as in the proof of Theorem 1.

From Theorem 1 we have

\\up ( t ) - u 0(t)\\ = \\up (t)\\< Whl(t) < z 0(t)

for t e < t 0, a), p = 0 , 1 , . . .

Further if we suppose that the inequality

\\uv + n { t ) — u n {t)\\ < z n {t) , p = 0 , 1 , . . . ,

t e ' ( t 0, a), is satisfied, then we get

t t

\\up + n + i ( t ) — u n+1(t)\\ = || f ( t , j u p+n+1(s)ds-\- £ (t 0) , § u p + n { s ) d s + £ ( t 0) ,

^0 ^0

t-T(t) t-T(t)

J 'd 'p + n + i(&)^{d s}-)- £ { t f ) , J 'd p + n i^ ) ds £ (tQ) , Up^.n (t)^

t0 *0

t t

— f ( t t f u n+ i [ 8 ) d 8 + £ ( t 0) , j u n (s)d s-\ - £ ( t 0) ,

j 'dn+l (s)ds+ £(t0) , + f un(s)ds-j- £(t0), un{t)) ||

'o *0

t . t

(p\t,

^0 *0

J' INp+n+i(s) ^»+l (®)ll d® * J' ll^23+n(^) Un(s)\\ds, || Up^.n{t) ( ^) 11 ^

t *0 t

<p[t, j\\up+n+l{s) — un+1{s)\\ds,jzn{s)ds,

t-T(t)

*0 *0

J ll^p+n+i(^) w » + i ( * ) l l < f o » J" zn(s)ds, zn(t)\.

According to Lemma 1 and the definition of the function zn+1(t) we obtain ll%f+«+i(0 (t)\\ < Sn+i(t)

for te<tQ, a), p = 0 , 1 , ...

Prace Matematyczne IX. 1 8

66 M. K w a p i s z !

By induction we infer that the inequality (16) \\up+n(t) — un(t)\\<zn(t), te<t0,a), holds for any p = 0 , 1 , ... and n = 0, 1 , . . .

Applying Lemma 2 we conclude that the sequence {un{t)}, n =

= 0, 1 , ..., is almost uniformly convergent in the interval <t0, a) to the

* function u(t).

From the uniform convergence of the sequence {un{t)} and the con­

tinuity of the function f(t, ux, иг, uz, щ, uf) it follows that the function и(t) = lim un{t), te(t0,a),

П—ЮО

is a continuous solution of equation (7).

The proof of inequality (15) follows from inequality (16) as p —> &o.

The proof of the uniqueness of the solution of equation (7) is very simple.

We see that the absolute value of the difference of two solutions of (7) should satisfy inequality (2) with h{t) = 0,tp{t) = 0 and therefore, by Lemma 1 and by assumptions H 2, 2°, it must be equal to zero.

This completes the proof of Theorem 2.

6. Conclusion 1 . As a result of the equivalence of equation (6), with the initial condition y(t) = $(t) for teE0, to equation (7), we can state that the function

t

(17) Уп+iit) —

/wn+i(*)(fa+f(<o)» a), i(t), teE0, П = 0, 1, satisfies the differential equation

(18) y'{t) = f [ t , y { t ) , y n{t), y( t—x(t)), yn( t- r( t) ),y' n{tj), te<t0, a) , and the initial condition yn+i(t) = Hi) f or te E0, n = 0 , 1 , . . . , where y 0{t) = 0 for teF 0+ < t0, a).

Sequence (17) is almost uniformly convergent in the interval <t0, a) to the function y(t) which is a solution of equation (6) and satisfies the initial condition у (t) — £(t) for t eE 0. The error estimate is the following:

t

\\yn(t)~y{t)II < Jzn{8)ds.

h

R e m a rk 1. If the function / does not depend on uz, щ and the function cp has the form

(p(t, ux, щ , uz, u±, u5) — М иг-\-NuzĄ-Ещ, M , N , L ^ 0

and 0 < L < 1, then Theorem 2 of this paper follows from Theorem 2 of paper [3].

E em a rk 2. It is evident that the above method of successive approximations can be used for differential equations with more than one retardation, i.e. for equations of the form:

y'(t) =f(t,y(t), y(t), У

,...,y (t—Ti{t))

(t—Tt{t

y'

7. Let us now consider an implicit first order differential equation of the form

where the function g(t, ux, u2, u3, щ, uf) is defined and continuous for (t, щ , u2, u3, щ, u5) e D 1 and its values are in the Banach space B.

Before formulating a theorem on the existence of the unique solu­

tion of equation (19) with initial condition (19'), we state the following * assumptions:

Assumptions H 3. 1° There exists a continuous operator C(t), te ( t0, a) , from В into В such that the equation C(t)x = 0 , x e B , has the unique solution x — 0;

2° The function 0(t, ux, . . . ,ms) = ws + G(t)g(t, ux, ..., uf) satisfies the assumptions H 2.

Now we can formulate the following theorem:

Theorem 3. I f the assumptions H3 are satisfied, then equation (19) with the initial condition (19') has a unique solution in the interval (t0, a), y(t), which is a limit of the sequence {yn(t)}, n — 0, 1 , almost uni­

formly convergent in the interval (t0, a), where y0(t) = 0, and the function Уп+iit), n = 0 ,1 , . .., is the solution of the equation

with the initial condition yn+i{t) = £(t) f or te E0.

P r o o f. As in the case of equation (6), by substituting y'if) = u(t) we get an equation equivalent to equation (19) with the initial condition (19')

(19)

g(h y(t)t y{*)> y(t-*(t)), y(t—r(t

y'(t))

with the initial condition

(19') y(t) — £(t), te E o,

y'if)

( 2 0 )

J u(s)ds+ ę(tQ),u(t)} = 0.

6 8 M. K w a p i s z

Applying the assumptions H3 we can easily see that equation (20) is equivalent to the following one:

t t

u(t) = u(t) + C(t) [.g{t, Jw(s)ds + £{t0), fu(s)ds + £{t0),

t-T(t)

~f U(s)ds-j-i(t0) , +J u(s)ds-\- £(t0), и (0)], i.e.

t

(21) u(t) — O {t, j u{ s) ds + £{t0), fu(s)ds+ £(t0),

in i

+

0 ‘0

u( s )ds +£ (t 0), +j u(s)ds+ £(t0), u(t)j.

f0 10

Further, applying Theorem 2 and Conclusion 1 we conclude the proof of Theorem 3.

Bern ark 4. Theorem 3 is a generalization of Theorems 1 and 2 of [7], where under assumptions of the Lipschitz type a system of differential equations was investigated by Picard’s usual successive approximations method. In this case C(t) is a non-singular constant matrix.

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[3] — Some remarks on a certain method of successive approximations in diffe­

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