2
MIECZYSLAW ALTMAN
(1.1) xn+1=xn-r(xn)Pxn' n=0,1, ... , which converges to a fixed point of F,
where Fx=x-r(x)Px. Hence it follows that Px=O. This process is the basis for
a unified theory of a very large class of iterative methods, including the most
important ones: a) the method of successive approximations, b) the Newton-
Kantorovich method for nonlinear operator equations, c) the Newton-Altman
method for finding roots of a nonlinear functional F defined on a Banach
space X
( 1.2)
of F
F(x )
xn+1=xn- '( n) Yn• n=0,1, ... , where F' is the Frechet derivative
F xn Yn
and F'(xn)Yn with I!Ynll = 1 is approximately equal to IIF'(xn)ll.
d) the method of steepest descent and other gradient type methods. Both
existence and convergence theorems can be obtained (see A[1J) and the con-
tractor method also reveals the character of the convergence which depends on
the corresponding majorant function. The method also yields error estimates.
In the particular case where Px=x-Fx, the solution of the equation
Px=O yields a fixed point of F. Thus the theory of contractors also pro-
vides a general method of obtaining fixed point theorems by means of iterative
procedures.
2. A GENERALIZATION OF THE BANACH CONTRACTION PRINCIPLE. Let
F:X~X
be a
nonlinear operator such that P is closed on its domain D(P), where
Px=x-Fx. Suppose that P has a bounded contractor r, i.e., II r (x) II ::_ B for
all XED(P) and some BO, and the contractor relation IIFx-F(x+r(x)y) -
{I-r(x))yll ::_ qll Yll holds for all XED(P) and some q with Oql. Then F
has a fixed point x=Fx.
It is clear that if r(x)=I, then the above contractor relation means that
F is a contraction.
3. THE NEWTON-KANTOROVICH METHOD. let
P:X~Y
be a nonlinear operator which
is Frechet differentiable and its Frechet derivative P'(x) has a bounded
inverse p'(x)-1 which can be considered as a contractor r(x) under some
additional conditions. Then the Newton-Kantorovich iterative process
(3.1) xn+1=xn-P'(xn)-
1Pxn'
n=0,1, ... , is actually a contractor type method
(1.1) where r(xn)=P'(xn)-
1.
As Kantorovich has proved, the iterative process
(3.1) converges to a solution x of the equation Px=O. On the other hand,
this solution x can be viewed as a fixed point of the operator F, where
F(x)=x-P'(x)-lPx.
4. CONTRACTOR TYPE FIXED POINTS AND THEIR APPLICATIONS. Suppose that
P:X~Y
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