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,
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