8
MIECZYSLAW ALTMAN
one (see AElJ).
10. CONTRACTOR DIRECTIONS. Contractors and contractor directions are the
basic concepts employed in the general theory of contractors and contractor
directions.
Given a nonlinear mapping P:X
~
Y where X
and a positive number q1, then rx(P)=rx(P,q)
and Y are Banach spaces,
is a set of contractor
directions for P at x if there exist
Os=s(x,y)~1
and h=h(x,y)EX
such that
(10.1) liP(x+sh) - Px-
sYII~qsiiYii, YEf/~).
The element h is called the strategic direction at x. In general the
strategic direction is supposed to satisfy the following growth condition
(10.2)
llhii~B(g{x)
· IIYII),
where g is an arbitrary positive functional on X which is bounded on
closed bounded sets, and B is a continuous increasing function such that
B(O)=O, B(s)O for sO, and
(10.3)
j~s-
1
B(s)dsoo,
for aO. The class of such functions B is denoted by ID. The method of
contractor directions yields a unified approach to solvability problems of
nonlinear operator equations. In particular if Y=X and P=I-F where I is
the identity mapping of X, then one can also obtain various fixed points
theorems. In case of an operator with closed range restriction (10.2) is not
required. The function B plays an important part in the theory of
contractor directions, and does not seem to be introduced artificially. In
fact, it is shown in AC127J that in the case of the Newton-Kantorovich
k k
iterative method, the function B is given by the formula
B(s)=(~
K)
2S 2
,
where K is the Lipschitz constant for the Frechet derivative p'.
11. A GENERAL EXISTENCE PRINCIPLE.
THEOREM (AC6J). Suppose that the following hypotheses are satisfied.
Let S=S(x0,r) be an open ball with center x0ED(P) and radius rO,
and put
U=D(P)n~.
where
S
is the closure of S. Suppose that
(a) P:U
+
Y is closed on U
(b) for each XEU0=D(P)nS, we have that -PxErx(P), more precisely, y=-Px
satisfies relations (10.1) and (10.2) with g{x)=C (some positive
constant).
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