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
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
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
and an elem XEX such that
If q(IIYII )=qll Yll with Oq1, then obviously rx(P)=rx(P).
THEOREM. If rx(P)=Y for all XEX, then Px=Y.
This theorem generalizes an earlier theorem (see A[1J) and recent "nonnal
solvability" theorems of Pohozhayev, Browder, Krasnoselskii-Zabreiko, and
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
THEOREM. Let F:D(F)cX
X be a weak directional contraction. If the range
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
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.