10

MIECZYSLAW ALTMAN

13. ITERATIVE METHODS OF CONTRACTOR DIRECTIONS. Since the general theory of

contractor directions deals mainly with existence of solutions of general

operator equations under rather weak conditions, many theorems obtained in this

way do not indicate how to compute the solutions whose existence has been

proved. Thus, there is a need in a constructive version of the theory of

contractor direction in order to find existing solutions in a practical way.

The first attempt has already been made in A[6,7J. Among other things,

the Newton-Kantorovich method with small steps is investigated there, and

convergence is proved under less restrictive hypotheses. As a consequence,

the initial approximate solution is not required to be as good as in the case

with step-size equal to one. However, the method automatically turns into

the classical Newton-Kantorovich method, i.e., with stepsize equal to one, as

soon as the approximate solution improves sufficiently. A general theory of

iterative methods based upon the concept of contractor directions is developed

in AC7J. In case of P=I-F, the iterative process converges to a fixed point

of F. The crucial part of the iterative method is based on the following

fundamental lemma (see A[7J).

Let P:D{P)cX

+

Y and

u

0=D(P)nS, where S is an open ball with center

x0ED(P) and radius r. We assume that -PXEfx(P)=rx(P,q} for all XEU0,

e.g., relation (10.1) holds with y=-Px, and strategic directions h=h(x)

satisfying II h

II~

B( II Px II) for all XEU0, where BEB (see 10.3), and

r.:_(l-qr 1 J~s- 1 B(s}

ds with

a~IPx 0 11

exp(1-q}.

LEMMA. (AC139J) Let xn+1=xn+enhn, (n=0,1, ... ), where

Oen~1

and hn are

and

Then {xn} lies in

u

0

and both {xn} and {Pxn} are Cauchy sequences.

If

E

E

=oo, then II Pxn II

-+

0.

n=O n

There is also a practical way of choosing {en}: put

~(e,x,h)

= IIP(x+eh) - (1-E)Pxll/c

Then put en=1 if

~(l.xn,hn}~qll

Pxn II, otherwise, under very general

conditions, one can choose Oen1 so that

Bqll Pxn

II~P(en,xn,hn}~qll

Pxn II,

MIECZYSLAW ALTMAN

13. ITERATIVE METHODS OF CONTRACTOR DIRECTIONS. Since the general theory of

contractor directions deals mainly with existence of solutions of general

operator equations under rather weak conditions, many theorems obtained in this

way do not indicate how to compute the solutions whose existence has been

proved. Thus, there is a need in a constructive version of the theory of

contractor direction in order to find existing solutions in a practical way.

The first attempt has already been made in A[6,7J. Among other things,

the Newton-Kantorovich method with small steps is investigated there, and

convergence is proved under less restrictive hypotheses. As a consequence,

the initial approximate solution is not required to be as good as in the case

with step-size equal to one. However, the method automatically turns into

the classical Newton-Kantorovich method, i.e., with stepsize equal to one, as

soon as the approximate solution improves sufficiently. A general theory of

iterative methods based upon the concept of contractor directions is developed

in AC7J. In case of P=I-F, the iterative process converges to a fixed point

of F. The crucial part of the iterative method is based on the following

fundamental lemma (see A[7J).

Let P:D{P)cX

+

Y and

u

0=D(P)nS, where S is an open ball with center

x0ED(P) and radius r. We assume that -PXEfx(P)=rx(P,q} for all XEU0,

e.g., relation (10.1) holds with y=-Px, and strategic directions h=h(x)

satisfying II h

II~

B( II Px II) for all XEU0, where BEB (see 10.3), and

r.:_(l-qr 1 J~s- 1 B(s}

ds with

a~IPx 0 11

exp(1-q}.

LEMMA. (AC139J) Let xn+1=xn+enhn, (n=0,1, ... ), where

Oen~1

and hn are

and

Then {xn} lies in

u

0

and both {xn} and {Pxn} are Cauchy sequences.

If

E

E

=oo, then II Pxn II

-+

0.

n=O n

There is also a practical way of choosing {en}: put

~(e,x,h)

= IIP(x+eh) - (1-E)Pxll/c

Then put en=1 if

~(l.xn,hn}~qll

Pxn II, otherwise, under very general

conditions, one can choose Oen1 so that

Bqll Pxn

II~P(en,xn,hn}~qll

Pxn II,