CONTRACTORS AND FIXED POINTS 11

where 081. Such a choice implies that IIPxnll .... 0 as n .... oo.

A GENERALIZATION OF THE BANACH CONTRACTION PRINCIPLE. Let F be a contraction

and let

with 0£ 1

n-

Moreover,

and arbitrary x0. Then x .... x=Fx whenever

n

11xn-x112. (1-qr 1

ll

x0-Fx011 exp((1-q)(l-tn)),

n-1

E

£ =oo,

n=O n

where t 0=o, t = r £., and Oq1 is the Lipschitz constant for F. This

n i=O

1

is another extension of Banach's contraction principle, and follows from the

above theory without using Banach's theorem.

The iterative method of contractor directions mentioned above can also

be applied to the Newton-Kantorovich method. Then we obtain the following

procedure

where 0£ 1 are properly chosen.

n-

Notice that by taking larger 0£n2.1 we get a faster convergence. But

if £n=1 is admissible, then it will appear in (13.1). Hence, the process

(13.1) turns into the Newton-Kantorovich method automatically if the

Kantorovich hypotheses are satisfied for some xn.

The method (13.1) is convergent if

'f

£ =oo and the Kantorovich

n=O n

hypotheses are satisfied. Kantorovich has proved the convergence in case of

£n=Q1 for all n=0,1, ...

GENERALIZED CONTRACTOR DIRECTIONS. Let Xi,Yi(i=1, ... ,n) be Banach spaces and

Pi:DcX1

x ••• x

Xn .... Vi be nonlinear operators. Then y={yi} E

v

1 x ••• x

Yn is

a generalized contractor direction for P={P.} at

1

i.e., yErx(P), if there exist 0£1 and h={hi}

such that

x={xi}EX1

x ••• x

Xn'

(called strategic direction)

IIP;(x1+£h 1, ... ,xn+£hn)- P;(x1, ... ,xn)- E.Y;IIi

n

2.£

l:

a.kjj ykjjk'

k=1

1

where {aik} is a fixed nonnegative n

x

n real matrix such that there exist

where 081. Such a choice implies that IIPxnll .... 0 as n .... oo.

A GENERALIZATION OF THE BANACH CONTRACTION PRINCIPLE. Let F be a contraction

and let

with 0£ 1

n-

Moreover,

and arbitrary x0. Then x .... x=Fx whenever

n

11xn-x112. (1-qr 1

ll

x0-Fx011 exp((1-q)(l-tn)),

n-1

E

£ =oo,

n=O n

where t 0=o, t = r £., and Oq1 is the Lipschitz constant for F. This

n i=O

1

is another extension of Banach's contraction principle, and follows from the

above theory without using Banach's theorem.

The iterative method of contractor directions mentioned above can also

be applied to the Newton-Kantorovich method. Then we obtain the following

procedure

where 0£ 1 are properly chosen.

n-

Notice that by taking larger 0£n2.1 we get a faster convergence. But

if £n=1 is admissible, then it will appear in (13.1). Hence, the process

(13.1) turns into the Newton-Kantorovich method automatically if the

Kantorovich hypotheses are satisfied for some xn.

The method (13.1) is convergent if

'f

£ =oo and the Kantorovich

n=O n

hypotheses are satisfied. Kantorovich has proved the convergence in case of

£n=Q1 for all n=0,1, ...

GENERALIZED CONTRACTOR DIRECTIONS. Let Xi,Yi(i=1, ... ,n) be Banach spaces and

Pi:DcX1

x ••• x

Xn .... Vi be nonlinear operators. Then y={yi} E

v

1 x ••• x

Yn is

a generalized contractor direction for P={P.} at

1

i.e., yErx(P), if there exist 0£1 and h={hi}

such that

x={xi}EX1

x ••• x

Xn'

(called strategic direction)

IIP;(x1+£h 1, ... ,xn+£hn)- P;(x1, ... ,xn)- E.Y;IIi

n

2.£

l:

a.kjj ykjjk'

k=1

1

where {aik} is a fixed nonnegative n

x

n real matrix such that there exist