12
positive
n
MIECZYSLAW ALTMAN
{r.} and q1
(1~i~n)
with
1
(a)
l::
a.krk~qri(i=1,
... ,n)
k=1
1
conditions for a matrix to satisfy (a) are given by Matkowski [1,2].
THEOREM. If P is closed on D and -PXEfx(P) for all XED with
such that
llh-11· B.(
II
p
II·'·
1 1 - 1 X 1
h={h.(x)}
1
for BiElB;, (i=1, ... ,n), (see (10.3), then Px=O has a solution XED.
15. GENERALIZED DIRECTIONAL CONTRACTIONS. Let X=X
1
x ••• x
Xn and F;:X +X;
be nonlinear. Then F={F;} is a generalized directional contraction if for
arbitrary x,yEX there exist Oe1 such that
n
I!F.(x+ey)-
F.xl!.~e
l::
a.ki!Ykllk'
1 1 1
k= 1
1
for i = 1, ... ,n, where {aik} is a given matrix with property (a) given in
14. If y is replaced by y={Fix-x;}, then F is called a generalized
directional contraction in the narrow sense.
A FIXED POINT THEOREM. A closed generalized directional contraction in the
narrow sensed has a fixed point, i.e., F;x=xi for some x={xi}. The above
fixed point theorem generalizes the fixed point theorem of Matkowski [1,2],
Pavaloiu [1], and Rus [1].
16. THE METHODOLOGY OF PROOFS IN THE THEORY OF CONTRACTOR DIRECTIONS. Four
different methods of proofs are now available in the theory of contractor
directions:
1. The transfinite induction argument.
2. An argument based on the Brezis-Browder principle on ordered sets and its
generalization.
3. The method of interations.
4. The Caristi-Kirk fixed point argument.
REFERENCES
[1] M. Altman, Contractors and Contractor Directions. Theory and
Applications. M. Dekker 1977.
[2] Recent development of the theory of contractors and contractor
directions and its applications, International Conference on Nonlinear Analysis
and Applications, Memorial University of Newfoundland, St. John's, Newfound-
land, Canada, 1981. Ed. S.P. Singh.
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