CONTRACTORS AND FIXED POINTS

_l

a

II

(c)

r~(1-q)

J0s_l

B(x)ds, q1Px0

exp(1-q)~a.

Then the equation Px=O has a solution x in U.

The above theorem yields a fixed point if Y=X and P=I-F, where

F:X

~

X is a nonlinear mapping.

12. WEAK CONTRACTOR DIRECTIONS AND WEAK DIRECTIONAL CONTRACTIONS FOR AN

OPERATOR WITH CLOSED RANGE. Let X be an abstract set and P:X

~

Y

a mapping of X into a (real or complex) Banach space Y. Given an upper

semicontinuous function q such that Og(s)s for sO, we define sets

* *

rx(P)cY of weak contractor directions for P at XEX as follows: yErx(P)

if there exist a positive

e=e(x,y)~1

and an elem XEX such that

(12.1) IIPx-Px-eyll

~eq(IIYII

).

*

If q(IIYII )=qll Yll with Oq1, then obviously rx(P)=rx(P).

*

THEOREM. If rx(P)=Y for all XEX, then Px=Y.

9

This theorem generalizes an earlier theorem (see A[1J) and recent "nonnal

solvability" theorems of Pohozhayev, Browder, Krasnoselskii-Zabreiko, and

others.

Let X be a Banach space and q an upper semicontinuous function such

that Oq(s)s for sO.

A mapping F:D(P)cX

~

X is called a weak directional contraction if for

each XED(P) and yEX, there exists a positive

e=e(x,y)~1

such that

x+eyED(F) and

(12.2)

IIF(x+ey)-Fxll~eq(ll

Yll ).

THEOREM. Let F:D(F)cX

~

X be a weak directional contraction. If the range

of I-F

is closed in X, where I is the identity mapping, then P=I-F is

a mapping onto X.

A FIXED POINT THEOREM FOR WEAK DIRECTIONAL CONTRACTIONS IN THE NARROW SENSE.

Let WcX be a convex set. A mapping F:W

~

W is called a weak directional

contraction in the narrow sense if relation (12.2) is satisfied for all XEW

and y=FX-x.

THEOREM. Let F:W

~

W be a weak directional contraction in the narrow sense.

If the range (I-F) (W) is closed in X, then F has a fixed poiht in W.

Note that if F:W

~

W is a contraction of a closed convex set W of a Banach

space X, then (I-F) is evidently closed in X. Therefore, the above fixed

point theorem is an extension of the Banach contraction principle, and also

generalizes some known results.

_l

a

II

(c)

r~(1-q)

J0s_l

B(x)ds, q1Px0

exp(1-q)~a.

Then the equation Px=O has a solution x in U.

The above theorem yields a fixed point if Y=X and P=I-F, where

F:X

~

X is a nonlinear mapping.

12. WEAK CONTRACTOR DIRECTIONS AND WEAK DIRECTIONAL CONTRACTIONS FOR AN

OPERATOR WITH CLOSED RANGE. Let X be an abstract set and P:X

~

Y

a mapping of X into a (real or complex) Banach space Y. Given an upper

semicontinuous function q such that Og(s)s for sO, we define sets

* *

rx(P)cY of weak contractor directions for P at XEX as follows: yErx(P)

if there exist a positive

e=e(x,y)~1

and an elem XEX such that

(12.1) IIPx-Px-eyll

~eq(IIYII

).

*

If q(IIYII )=qll Yll with Oq1, then obviously rx(P)=rx(P).

*

THEOREM. If rx(P)=Y for all XEX, then Px=Y.

9

This theorem generalizes an earlier theorem (see A[1J) and recent "nonnal

solvability" theorems of Pohozhayev, Browder, Krasnoselskii-Zabreiko, and

others.

Let X be a Banach space and q an upper semicontinuous function such

that Oq(s)s for sO.

A mapping F:D(P)cX

~

X is called a weak directional contraction if for

each XED(P) and yEX, there exists a positive

e=e(x,y)~1

such that

x+eyED(F) and

(12.2)

IIF(x+ey)-Fxll~eq(ll

Yll ).

THEOREM. Let F:D(F)cX

~

X be a weak directional contraction. If the range

of I-F

is closed in X, where I is the identity mapping, then P=I-F is

a mapping onto X.

A FIXED POINT THEOREM FOR WEAK DIRECTIONAL CONTRACTIONS IN THE NARROW SENSE.

Let WcX be a convex set. A mapping F:W

~

W is called a weak directional

contraction in the narrow sense if relation (12.2) is satisfied for all XEW

and y=FX-x.

THEOREM. Let F:W

~

W be a weak directional contraction in the narrow sense.

If the range (I-F) (W) is closed in X, then F has a fixed poiht in W.

Note that if F:W

~

W is a contraction of a closed convex set W of a Banach

space X, then (I-F) is evidently closed in X. Therefore, the above fixed

point theorem is an extension of the Banach contraction principle, and also

generalizes some known results.