R O C Z N IK I P O L S E IE G O T O W A R Z Y S T W A M A T EM A T Y C Z N EG O Sé ria I : P R A C E M A T EM AT Y C ZN E X X I I I (1983)
Je r z y Tr o ja n (G-liwice)
On a majorant principle îor operator equations
Let X, Y be the Banach spaces, and let an operator T : X -> Y be given. We will consider the equation
(1) T(a>) = 0 .
Let ns define a sequence of iterations
(2 ) x0 e X, xn + l= x n + D l
where Dk is computed from the system of equations 0
(3)
о =
Tn+Tii>l>i
.. . + т*я2#*..
]/l,l
0 =
Tn + Tll?l’k + T llîi’kjil’k
+ ...
+ T k
«i>ï*-
• • JJnT)k,kik
where, by definition, Tn = T(ocn), T3n = — T (j){xn).The sequence is defined for & and fulfils the con
ditions:
if г — l for s < r{l), where 1 < r(l) < l, (4)
ifj < l in the other case.
Regarding the symmetry of the multilinear operators T jn, we can assume without loss of generality that the elements {ij,i}j -are non-decreasing.
The case к — 1 is known as the N ewton-Kantorovich method [1]. If к — 2 , then we must consider two possibilities.
(a)
r(
2) = 1.
The system (3) has the form
144 J . T r o j a n
This is the generalization of the Chebyshev method in the case of operator equations. One can find it in [3].
This is the so-called method o f tangent hyperbolas studied b*y M. Altman in [4].
If r(l) = 1 (l = 1, 2, .. ., k) and isj}l — l — l for j ^ 2 , then the condition assuring the convergence of sequence (2 ) can be derived from the theorem of L. Collatz ([2], § 19).
Further on we shall apply the method of real majorant investigated in papers [3]-[5] in the proof of the convergence of the sequence (2) in the general case.
Let it be given a real functional equation
and assume that there exists a real positive solution z*. The function / has 7v + 1 continuous derivatives (in the interval <0, z*)). Analogously to formulas (2) and (3), one can define the following sequence:
(b) r(2 ) = 2 .. We have here: \
(5) /(*) = o
(6) z0 — «я+1 = « я + ^jk'n>
where dkn are derived from the equations
Let us introduce the functions
Lemma. I f the equation gk(z) = 0 has a positive root, conditions (4)
are fulfilled, and
(8) / » > 0 , Л > 0 , f ' < 0 ,
(9) **,!< *', 1+1 f ° r } < * < l = 1, 2,
then
■ , 0 < d l < d l < ... < dk.
P ro o f. The Fourier theorem implies that the equation gk(z) = 0 has two positive roots. Let z*k be the smallest one. Using (8), we can stato that also the equation gl(z) = 0 , l — 1, 2 , ..., Je — 1, has positive roots, and the sequence of inequalities 0 < z* < z* < ... < z\ is valid. Kow, let us see that
л
[fir'(s)]” = 2f l + 6fnz + ... + l ( l - l ) f j - 2 > 0 for z > 0.
For gl(0) = /* > 0 then gl(z) is decreasing in the interval <0, z*). Further, mathematical induction will be useful.
(i) d\ = - f j f l > 0, g1{dln) = 0, < = z*.
(ii) We assume .that
(10) 0 < dl < ... < dln 1 and d £ < s!, p = 1, 2, ..., I — 1 By (4) and (7), for l > 2 , dl = i r+ 1 d r+l’l -i-f u’n * Jn
First of all we shall prove that
(11) fn + & 'n ‘l + ••• + r J i ’1 ■■■ <£■' < K(d), where d = max
I t is hlr (0) = f l < 0. Assume that there exists z < zf_l such that hlr(z) = 0. We obtain
<12) g\z)
*=/»+Л+1*г+1 + •••
+ f J > 9 (0)
and this is a contradiction, because gl (z) is decreasing. This implies inequal ity hlr (d) < 0, and further (11).
146 J . T r o j a n
Kow, we show that dln < z*.
Л Л dl = < £ # . . . # + - + / r < é r ... 4 i l l + f n  {à l~l)l + ... + / Г 1(<~1Г 1+/, f n + fn < ~ 1+ ... + / ; « - 1Г 1  ( Z*Ÿ + ..." + fn +1(ZÎ)r+ ljrfn * — Zi . / J+ / & .+ ... Ш * ) * - 1
Рог the completeness of the proof we must show that dl~* < d\
a 4 ’z. . . 4 z+ d l - d l - 1 = -■r+\ .r + 1 + Г + Х - 1 • •• < +w+/„ d:l - l Л + / “4 2’Ч . . . ii iji.i „■r+1 , r+ 1 f nd i 1- 1 l-l л 1,1-1 > f n * . dУ + . •. +/;+1^ y ... +1>* + r j - 4 y . • • л + / £ $ * + ... + M ’l - - 4 ’1 .1—1 -r+ 1 .r+ 1 ,r .r . .«y-1'1- 1+ ... + r n+id ï i- \ . +/x m_1. • . < ’г“х+ ... +/„ л + / Х 2- ч . . . + д а > г... <£•* 0. Additionally, we have proved that dk < z*k. ■
Колу, we can prove a theorem concerning convergence of the iteration
sequence (6).
Th e o r e m 1. Assume that z* is the smallest one positive root of equation
(5), /(0) > 0, /"(0) > 0, ...,/<*>«» > 0,/'(0) < 0,
(13) /(fc+1)(s) > 0 for 0 < z < z*
and conditions (4) and (9) are fulfilled. Then the sequence {zn} defined by
formula (6 ) is increasing and converges to z*.
Proof. By (13) there is f U)(z )> 0 for 0 < z < z*, j = 2 ,3 , . . . , Jcr and f ( z ) < 0 in the same interval.
To show monotonical convergence of the sequence {znj it is sufficient, using the lemma, ascertain that the equation gk (z) = 0 has a positive root (for every n).
I t is obvious that z0 = 0 < z*‘, suppose zn < z*. Define d: — z* — zn and put down the Taylor series of the function f in the neighbourhood of гя
о =/(«*) = f n+ f nd + ... +/£(d)* + ■ ■■ A - . , f ik+14 * ) W t+'.
Using (13) we obtain that the equation
9k (z) — f n
+ • • •
— 0
has a real root z*. < d. Condition (12) implies zn+l < z*. We have also proved that the ' sequence {zn} is bounded. Suppose now z — Итяп'.
* п-нэо
Hence lim dln = 0 , l = 1 ,2 , ...,1c, and W—>00 ik k . ] l , k 0 = lim dt = — lim lim oo m f n . 7 ’ Jn ik d k>ka + •r+l + Л М Л ■r+1 * ! +1* + f n f l + & l7 + ... +/;<C2’fc... d Y 0 = l i m f{z n) = f(\ im zn) = f(z ). П->00 П~> 00 This means that z = z*. ш
Condition (9) assures monotonieal convergence {zH} to the root z*, but does not assure a high order of convergence. To obtain this one must add one more assumption. The following theorem formulates this result. For brief notation we define here d: —z — zn,d k : = d k, f : —f n.
Th e o r e m 2. Let all conditions of Theorem 1 remain valid. Moreover,
let (11) i f o ^ l — 8 + 1 fo r s< l = 1, 2, ..., Jc. Then dk = d + 0({d)k+1)-Proof. (i) f + f W = 0, f + p d + 1/" (z) { d f = 0, 1 f"(z) = i f (a) (d f, (P = d + - J - ± L (d)‘ = d + 0 ((d)»). (ii) Assume that dp — <?+0((d)3,+1) for p ^ l — 1.
148 J . T r o j a n
From this assumption we have dl = d + 0 ((d f ) (because dl 1 < dl < d) and by dl~l — d + 0((d)lj there is dl = d1' 1 A О ((d)1) and dl I F 1 = 1 + 0 ((d)*-x). 0 = f + f 4 l + [l-bO ((d)l- 1) ] f4 l- 1c F l + ... + [ 1 + 0({d)l- 1)]frdl~1dh F . . •r+l r+l ... d%r’l + f r+ldh’1 ... d!r+1’l + ... + f d h’1... d i,i о = / + / 1<î‘ + [l + 0 ((Æ),- ,)l/2[d + 0 ((<J)i)j2+ ... + [ l + 0 ((d)!- 1)[x x r [ d + 0((d)‘- r+2)Y+ ••• +/[«î + 0 ((d)»)]‘, I
о
= f + f 1# + [ 1 + O ( ( d f- ')]/* [(d)>+ О ((d),+1)J + ... ... + [ l + 0 ((d)’- 1)|/'-[W + 0 (W'+1) ] + . . . +/[(«!)' + 0 ((d),+1)], 0 = / + / 1dl +/aW + - . . . +/'(d)! + 0 ((<i)'+1). IComparing just obtained formula with Taylor series 0 = f + f ' d + f 4 d ) 2+ ... + / W + 0 ((d)‘+‘) we have got the desired result. ■
Now, we are prepared to formulate the essential result of this paper, a theorem concerning convergence of the sequence defined by (2) and (3).
De f i n i t i o n. Equation (1) has a real majorant (5) if the following conditions are fulfilled
(1 ) Р У К Я 0 );
(2)
(!„= [Г,]]-1
exists and||G0|1<
-l//'(0); '
(3) 11П1КЯ i = 1 , 2 ,
(4) ||Г("+1>И|| 1)(0) j or \\x — Xq\\ ^ z — z0 < z*, where z* is the smal
lest one positive root of the equation (5).
Th e o r e m 3. Assume that equation (1) has the real majorant (5) and
conditions (4) and (9) are valid. Then the process defined by formulas (2 ) and (3) exists and converges to x*, where x* is one o f the roots o f equation (1 ).
The error can be estimated by the formula
(15), ■хЛ < z — zK
‘ Pr oof. Let us see that condition {3) can be replaced by the following
one:
The inequality
i
|UT“>(a>)||< ||1«(®о)11 + / ||2’|i+1>(*„ + t(x - » 0))||dt■ Has—os.ll 0
1
< f {i)(z0) + f f (i+1)(z0+ 4 s - z 0))dt-(z~z0) = / №(я) 0
motivates it.
Now, we try to use mathematical induction
№511 = I K W I < I I G j
i i t.
h< - 4 = d j.
JoSuppose that ЦЮ^Н < d% for p < 1 — 1 and put down Zth line of the system (3):
ir ir ir+1 ir+1
[2’5 + . . . +Г„1>„гл ■■■ П о'Ч Ч = -[т „+ т '+1п >1-‘ . . .d; +i j + ...
. . . + T ‘ ^ J . . . j 4 J b The left-hand side can be transformated as follows:
• TJ + . . . +
. . . D p= [ I +
T l D $ le„+ . . . +
T I D p . .. I ) p G >„]П. Using the inductive assumption and formula (11), we get the estimation+ •
•
•
+
< - 4
(&£U
+ •
•
•
+ f/P ■ ■ ■ 4 ‘)
Jo
= 1 /o+/o<?o2>4 ... + /o 4 V •••<#’*
/о < 1 .
So, we can compute I)\
1 5 0 J . T r o j a n
and this implies the formula
(18) \\xx- 0O!I<
Now, we check whether conditions (l)-(4) remain valid when replacing a?0 and z0 by xx and zx. The formula
ll® -® o ll< ll® -®1l l + K - ® o l l < « - « i + » i - « o = Z - Z 0
proves that conditions (3) and (4) are valid.
One can verify condition (2 ) in the following way i ||7-e0r!|| < 1КЗД ||Т;-ТЦ <
J
||2,"(«!b+<(®1-*»))||<« 0 1 h f i - ( f"(20+43i-Z o))dt = 1 Jo J a fo < 1 .Since G0T{ = I — (I — G0T{), this inequality implies existence of the oper ator Gx. We have and hence
IIGill <
№0\\ 1-\ \ I-GqT\\ 1 7 iCondition (1) is much harder to verify. First, let us estimate the ex
pression — Dj}||. There is Щ - Щ j-i I f ,-r+ l . . .d; +i-'+. ... +T\I?}-1 I < !|[TJ+ ••• + 2î^à3-1- - ^ o ,‘r 1x X [T„ + T ]D!,-1 + ... + T iD l- ' n f1... D<r;- ‘ + ... + T'0r>}J ... d; 4 ii f j y . . . 4 ‘ + ... +/0ч - ‘ +/. = 4 - 4 “‘ . From the triangle inequality we have
Now, let us substract from the Taylor series
T i = г 0+ а д + а д л { + . . . + а д ... +
’ a - t f
+ J — Ï'<l+I> {a, + t (ж, -x„))dt • x>î Dk-*^0 1
7rth equation of the system (2). We obtain
Tx = т \ в\ в\ -т \ т }}^ к 1
/
0 and analogously, f . . . + T * D j . . . D J ■ - Z ? ! # - * . . . 2 # -* + + ! - 1 T{k+1) (x0 +/(®i -a?0)) dt Dj. л = / o « ~ f J o ' ki 2,k + . . . + / w ... a* - f № k ... # • *+ + ji
: ( i z «)fc fc! /(fc+1) («0 +< («1 -»o)) dt • ( 4 ) fc + 1* By condition (4) it is sufficient to prove thatl '
«or / = 2,3, . . . , * . This we can do, using formulas (17) and (19)
н а д . . . а д а д у . . .
d
S = п т а д - ^ Ч ^ ' * ••• t > o ' * +
+
î î
(
d î
- 2 ) ^ )
b
{ ^ . . . j ) ^ - * + . . . + т а д - в * ' а д . . .
d$\
л л
< f M - d j * ) d ^ k . . . d ’-k+ ... +fl(dk
- o 4 ...
d
= f>dk . . . < % - f j } - k . . . 4k-Now, we can repeat this argumentation, replacing x0 by xx and so on for every n. This yields the formula
152 J . T r o j a n
The sequence {zn} tends by virtue of Theorem 1 to z* so it must fulfil the Cauchy condition. By (20), the sequence {xn} also fulfils it. Since X is a Banach space, there exists an element æ* such that xn~>x*.
It remains to show that æ* is a solution of (1).
Щд>*)|| = ||T(lima?n)|| = lim||T(ag||< lim/(sj =/(*£) = 0.
oo 00
Estimation (15) we obtain, when in (20) p tends to infinity. ■
Bern ark. Let us introduce an order of convergence of a sequence {яп}, zn->z*, in the following way:
/ 1 1
(21) p({zw}) — exP I lim —
lnln---W o o n zn — z
Then:
(a) Theorem 2 can be reformulated as follows:
The order convergence o f the method defined by (6) and (7) is equal
to &-fl.
(b) If in the assumptions of Theorem 3 we put additionally condition (14), then the method defined by (2) and (3) has the order of convergence equal to fc + l.
Theorem 2 shows that the whole class of iterative methods introduced above has the same order of convergence. However, we can make nlore subtle considerations.
Assume that conditions (8) and (9) are fulfilled and consider two methods 0 = /ra+/X> 0 — fn+fn^hl and o = f n + / X + / Х Ч o = / „ + / X + / X < Ü -Thus d i - d l = - ___L ___ f n + / X f2/?1 J пУ’П + Jn+fn(dhŸ f n m f i + f n < ) [ / « + / X + / n K ) 2] > o.
It is easy to verify that the quantity
is monotonie as a function of d3n for j < l. This implies that the best method (the largest dJff) for fixed r is the following one:
b —
fn+ / X +/X Xt
1
+ • • • + / X X 1)r
1+/:+1A - lr l +.
+ f n « - 1)1, 1 = 1, 2 ,Now, we can show that dln ^ dln for r > r. The first step of induction has been just made.
dl — dl f n + f rn+1« - Y +1+ . . . + / * « ~ 1)* f n + f A - ' - h... + / ; « - 1Г 1 \fn+1(dl- 1Ÿ +
•
Un+ / X
1
+ • •
1 / ,+ / Г Ч < -У +1'+ ••• + A « - 1)* Й + / п < "1+ . . . + / ;« - 1Г 1 .. + y ; ( O r- 1][/ .+ / x ~ 1+ . .. - h f j d 1- 1)1] In virtue of this consideration we state that the “best” method is defined by the formulas0 =/»-h/X>
o — fn + / X + / X X ?
0 = A + / X + / X 4 _1+ ... + / X X ' 1)fc' S and the “worst” one'
o =/„+/X >
^ = / » + / X + / » X ) 2>
о = / * + / Х + Л ( « " 1)2+ ••• + / î( < f
-Theorem 3 does not give the explicit formula of a function / . In prac tice, we can use following procedure. Define
/(*): = \\ТЖ+ ... +\\Г1¥* —-ffr . +l|2’„ll-If the equation f(z) = 0 has a positive root, then define
/л: = sup ||T(A:+1)(æ)||, llæ—x0ll<s*
154 J . T r o j a n
I t is easy to verify that / fulfils conditions (l)-(4). Moreover, if the equation
f(z ) = 0 has a positive root, then it is a majorant equation for equation (1). At the end let us notice two simple facts:
(1 ) I f the sequence xn+1 — xn -f TJ^ is convergent (fulfils assumptions of
Theorem 3), then the sequence xn+1 — xn +D ln for l < Tc is also convergent. (2 ) F or every root x* o f equation (1 ) one can choose such a start point x0
that the assumptions assuring convergence will be fulfilled.
References
[1] L. Y. K a n to r o v ic h , G. P. A k ilo v , Functional analysis in normed spaces, Moscow
1959 (in Russian).
[2] L. C o lla tz , Functional analysis and numerical mathematics, Praha 1970 (in Czech).
[3] M. A ltm a n , Concerning Chebyshev's generalised method of solving non-linear fu n c tional equations, Bull. Acad. Polon. Sci. 9 (1961), 261-265.
[4] —, Concerning the method of tangent hyperbolas for operator equations, ibidem 9
(1961), 633-637.
[5] —, A general majorant principle for functional equations, ibidem 9 (1961), 745-750.
IN S T Y T U T M A T E M A T Y K I P O L IT E C H N IK I ê L A S K I E J I N S T I T U T E O F M A T H E M A T IC S
