V. F(x) -< F(u%, d(u, x) -< l} C (u, l) and. fS(l) C S(I), where. B denotes the closed ball. Throughout this paper,. (V, d) denotes a metric space and. B(V) denotes the.
663
Internat. J. Math. & Math. Sci. Vol. 8 No. 4 (1985) 663-667
ON FIXED POINTS OF SET-VALUED DIRECTIONAL CONTRACTIONS SEHIE PARK Department of Mathematics Seou| National University Seoul 151, KOREA (Received June 4, 1984 and in revised form April 25, 1985)
Using equivalent formulations of
ABSTRACT.
Ekeland’s theorem,
we improve fixed point
thcorems of Clarke, Sehgal, Sehgal-Smithson, and Kirk-Ray on directional contractions
by giving geometric estimations of fixed points.
Ki,’Y
.: aL ;onary
a.
AMS SWBJECT CLASSIFICATION CODES.
I.8,v
1.
.
function, (weak) directional contraction, fixed point, f,oint, Ha.doff p,uedom,:tric.
I.
WORDS AND PHRASES.
47H10, 54H25.
INTRODUCTION AND PRELIMINARIES
In [1 I, [2 l, Sehgal and Smithson proved fixed point theorems for set-valued weak drectional contractions which extend earlier rosults of Clarke [3], Kirk and Ray [4], In the present paper, results in [1], [2] are substantialIy
ad Assad and Kirk [51.
strengthened by giving geometric estimations of locations of fixed points. The foliowing equivalent formulations [61 of the weil-known central resuit of Ekeland [7
I, [81
n the variational principle for approximate solutions of aintmization
problems is used in the proofs of the main resuIts.
[ +o
1.s.c. function, point
V
u
Let
bounded from beiow.
> 0
{x
S(X)
F(x) -< F(u)
V
X-ld(u,x)}.
e
and
0
Then the folIowing equivaient condi-
ttons hold:
(i)
There exists a point
(ii)
If
S(%)
T
F(y) T
v
S(X)
If
x e S(X), then
In Theorem 1,
V
F(x)
F(fx)
2
V
w)
satisfying for
v.
w
is a set-valued map satisfying the condition
F(x)
_
