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.
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