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

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