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

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http://dx.doi.org/10.1090/conm/021/729502