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).

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).