Contemporary Mathematics
Volume 21, 1983
CONTRACTORS AND FIXED POINTS
Mieczyslaw Altman
0. INTRODUCTION. The theory of contractors and contractor directions as pre-
sented in A[lJ and in a number of subsequent investigations is basically
designed for solving general equations in abstract spaces. The theory is based
on two fundamental concepts: contractors and contractor directions. The
theory of contractors offers a unified approach to a very large class of
iterative methods including the most important ones. From the standpoint of
the fixed point theory the contractor method yields a broad class of fixed
points which can be characterized as contractor type fixed points. For
example, the Newton-Kantorovich method yields a contractor type fixed point.
Another class of fixed points for nonlinear mappings F can be obtained by
applying the contractor method to the operator equation Px = x - Fx = 0. For
example, if F is a contraction, then the contractor for P will be the
identity mapping. The theory of contractor directions appears to be more
general and can be used to obtain existence theorems under rather weak
conditions. It also contains iterative methods with small step size, and can
be utilized to obtain fixed points if applied to equations Px = x - Fx = 0.
1. CONTRACTORS. Let P : X
~
Y be a nonlinear mapping and let r(x) : Y
~
X be
a bounded linear operator associated with .XEX,. where X and Y are Banach
spaces. Then r is a contractor for P with majorant function Q if
II
P(x+r(x}y} - Px - Y
II
~Q(
II
Y
II)
is satisfied for x E X and y E Y as defined by a given problem. If
Q(s) = qs for some Oql, then Q is a linear majorant function.
If Y=X and Px=x-Fx, where F is a contraction of X, then the identity
operator I=r(x) for all XEX, is a contractor for P with linear majorant
function Q(s}=qs, where Oql is the Lipschitz constant of F. This
observation justifies theterm "contractor".
Series of iterates of positive functions constitute an interesting class
of majorant functions (see[A 1,3,4]).
The contractor method is based on the following general iterative
procedure
1
©
1983 American Mathematical Society
0271-4132/83 $1.00
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http://dx.doi.org/10.1090/conm/021/729502
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