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 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 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
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