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)
_<
has a fixed point
(iii)
for all
2
V
fx}
S(X) \ T(x) .q y e V
x
tlen
e-Id(v,
F(v)
F(w)
S(X)
v
such that
eX -ld(x, y),
S()).
satisfies
eX-d(x,
fx)
has a fixed point
v e S(X).
denotes the power set of
a
be given, and a
.
such that
F(u) <- inf V F +
Let
V- R U {+}
be a compIete metric space, and
(V, d)
Let
THEOREM 1.
V
Note that
S. PARK
664
S(,,) C {x
F(x) -< F(u%, d(u, x) -< l} C
V
fS(l) C S(I), where
and
B
(u,
l)
denotes the closed ball.
(V, d)
Throughout this paper,
B(V)
denotes a metric space and
V
class of all nonempty bounded subsets of
denotes the
H
with the Hausdorff pseudometric
defined
by
max{SUPx
B)
H(A
C(V)
A|s,,
A
and
E
B)
A d(x
g
SUpy
B
g
A)}
d(y
V
denotes the class of all nonempty compact subsets of
C(V), we
x E V
For an
put
{y e A
[x, A]
For
which is nonempty.
[x, yl
{z e V
(x, y]
Ix, y]
d(x, A)},
d(x, y)
y e V
x
we denote
d(x, z) + d(z, y)
d(x, y)},
and
Let
x e S
S
{x}, (x, y)= (x, y] \ {y}.
\
V
be a nonempty subset of
A e
and
direction of a
and
T
C(V)
S
C(V), the weak directional derivative
is defined by
y e Ix, T(x)]
DT(x, y)
0,
DT(x
Tz)
z)
inf(ll(Tx,
d(x,
y)
(x, y] CS},
z e
’,
A map
k e
T
[0, I)
C(V)
S
For
be a set-valued map.
T
of
at
x
S *
,
in the
y,
if
x
if
(x, y]
if
(x, y] ( S
.
is called a weak directional contraction if there exists a
such that for each
y e [x, T(x)]
x e S, there exists a
DT(x, y)
with
k [2].
A map
for all
2.
S- B(V)
T
k e [0, I)
is called a directional contraction if there exists a
such that for each
z e
Ix, y]
x e S and y e T(x),
H(T(z), T(x)) -< kd(z, x)
S
[7].
S
be a complete subset of
RESULTS.
Let
THEOREH 2.
F(x)
tional contraction for which the function
f()r
any
S
u
s(e) C B(u, e)
PROOF.
r
x
satisfying
F(u) <- (1
and
T
C(1/)
S
a weak direc-
d(x, T(x)), x
S, is 1.s.c. Then
k), T has a fixed point in
S.
Choose a point
x e S(e) \ T(x).
[x, T(x)],
0
and
V
Since
* y, with
T
u
e S
satisfying
F(u) <-
infsF
+ (1
k)e.
Suppose
is a weak directional contraction, there exists a
DT(x, y)
k
Hence, there exists
a
z e (x,
ha
H(T(x), T(z))
k d(x, z).
Since
d(x, z)+ d(z, T(x))
<_
d(x, y)= d(x, T(x)),
we have
d(z, T(z)) -< d(z, T(x))+ H(T(x), T(z))
-< d(x, T(x))
d(x, z) + k d(x, z)
-< d(x, T(x))
(1
k)d(x, z).
yJ
ye
S
such
_
FIXED POINTS OF SET-VALUED DIRECTIONAL CONTRACTIONS
665
T
has a fixed point
ltence, F(z) -< F(x)
v
Theorem 2 is
l’herefore, by Therem l(iii),
/.).
improved version of Theorem (a) of [2] with much simpler proof.
a,
values of
fact, for suitable
],i
k)d(x,
(1
S().
the conclusion gives geometric estimations
k
and
E
However, for Theorem (b) of [2], such estimation seems
of locations of fixed points.
t be hard to get.
of Clarke
Note also that for Theorem
we can apply our Theorem 2.
[]J,
The following improves Corollary of [2] and a result of Kirk and Ray [4].
COROLLARY I.
k e [0, I)
t]ere exists a
[x, T(x)
such that for each
e (0, I)
and a
x
T
and
Suppose
is l.s.c.
y(x)
y
there correspond a
S
x
S
satisfying
k6[[y- x[ [.
x)) -<
H(T(x), T(x + 6(y
follows.
Then the conclusion of Theorem
As in the proof in [2 [,
PROOF.
d(x, T(x)),
F(x)
a map for which the function
C(S)
S
X
be a closed convex subset of a Banach space
S
Let
is a weak directional contraction with the
T
k
cnatant
(a)
for each
(b)
g(x)
S,
x
T
and
satisfies
with T(z) S,
(x, y) \ S
T(x) \ S, there exists a z
y
T
If
a
a directional contraction with the constant
S- B(V)
V
be a closed subset of a complete metric space
S
I.et
THEOREM 3.
lld
then, for any
d(x, T(x))
e S,
u
exists a fixed point
is l.s.c.,
0
e
T
of
v
and
B
a
satisfying
g(u)
(I -B)
S(E) fh S.
in
A
Under the hypothesis of Theorem 3, there exists a map
LEMMA [4 ].
there
B(X)
S
with the following properties
i)
iii)
z
a
if
A(x)
y
A(x)
x e S,
for each
ii)
then
A(x)
and
T(x)
B +a)-]d(x, T(x)),
(1
d(x, y)
S
for some x e S,
A(x)
z(x, y)e (x,y) fh S such that
A(x)
y(x)
y
then there exists a
if
(2.1)
d(x, y) s d(x, T(x)) + (B- a)d(x, z).
A(x)
A(x) f S
y(x) e A(x)
y
since there exist
f(x)
satisfying (2.1) by Lemma, let
A(x) ( S
Ix, f(x)]
( S
x
S
and
such that
(x, y)
S
x e S
We claim that for any
z
If
the definition of
know that for any
S
x
z(x, y)
z
such that
(2.2)
T(f(x))) -< a d(x, f(x)).
H(T(x)
This is clear if
and for
and
x e S
for
as follows:
A(x) fl S
be any element of
f(x)
let
S
S
S
Define a map
PROOF OF THEOREM 3.
T
A(x) f S
since
f(x)
T(x)
Set
F(x)
(I
implies (2.2).
and
f(x)g
and
B)-Ig(x)
We
f(x)
y
F(y) < F(x) -d(x, y)
holds as in the proof of [I, Theorem I].
and
S(e) f S
0
satisfying
F(u)
This implies that
the definition of
This contradicts
_< inf
v
A(v) f S
v z f(v)
Therefore, by Theorem
(iii), for any
F + e, there exists a fixed point
S
T(v)
for otherwise
Thus,
Consequently,
v
f(v)
A(v)
v
S
f(v) e (v, y(v)) for some
0
T(v). Since inf S F
f
of
u
S
in
and hence by
y(v)
u
A(v).
can be
666
S. PARK
<_ e
F(u)
chosen so that
d(u, T(u)) -< (I
that is,
B)e.
This completes our proof.
Note that Theorem 3 is a strengthened form of [I, Theorem I].
x, y e X, x
A metric space is said to be convex if for each
a
It is known that if
z e ix, y)
V
space
and
x
S
g
S
y
and
S
there exists
y,
is a closed subset of a complete convex metric
z e [x, y) f 8S
then there exists a
where
is
the boundary.
I, Corollary 13
Now, we obtain the following improved version of
as an immediate
consequence of Theorem 3.
L,t
B(V)
T
T("fi) C S
B,
,,d
in
be a drectional contraction hich the constant
d(x, T(x))
(x)
If
V
be a closed subset of a complete convex metric space
S
Let
COROLLARY 2.
is l.s.c,
g(u)
satisfying
B), there exists
(I
such that
a
then for any
S
on
u c
0
S
a fixed point
of
v
T
S(.) f S.
Also,, the followin improves [1, Corollary 2] and an earlier result of Assad-Kirk
I1.
COROLLARY 3.
T
Suppose
thnt for all
S
Let
B(X)
S
H(T(x), T(y)) -< a d(x, y).
If
T(6S) C S
pint
of
v
then for any
f,)c
T
e F0, 1)
such
S,
x,y
d(u, T(u)) -< (1
V
be a closed subset of a complete convex metric space
is a contraction, that is, there exists an
u e
either
S,
0
and
,
B
a
is a fixed point of
u
T
satisfying
or there exists a fixed
in
S(e) f3 S \ B(u, s)
where
d(u, T(u)) (i + a) -1
s
PROOF.
Since a contraction is a directional contraction and
is continuous, by Corollary
not fixed under
2,
T
has a fixed point
Y e B(u, s)
Then for any
T
S
v e
S(e) f S
g(x)
d(x, T(x))
Suppose
u
is
we have
d(u, T(u)) -< d(u, y)+ d(y, T(u))
s
+ d(y, T(u)),
that is,
cz(l +
a)-Id(u,
d(y, T(u)).
T(u))
ttence,
d(y, T(u))
Suppose
y e T(y)
as
Then we have
HiT(y), T(u))
a contradiction.
ad(y, u).
ad(y, u),
This completes our proof.
ACKNOWLEDGEMENT:
The author expresses his gratitude to the referee for his kind
comments.
This work was partially supported by a grant from the Korea Science and Engineer-
ing Foundation, 1983-84.
667
FIXED POINTS OF SET-VALUED DIRECTIONAL CONTRACTIONS
REFERENCES
I.
SEHGAL, V. M.
2.
SEHGAL, V. M. and SMITHSON, R. E.
S,me fixed point theorems for set valued directional contraction
mappings, Internat. J. Math. & Math. Sci. 3(1980), 455-460.
traction multifunctions, Hath.
A fixed point theorem for weak directional conJaponica 25(1980), 345-348.
3.
CLARKE, F. H. Pointwise contraction criteria for the existence of fixed points,
Canad. Math. Bull. 2(1978), 7-11.
4.
KIRK, W. A.
5.
.
ASSAD, N. A. nd KIRK, ’. A. Fixed point theorems for set valued mappings, Pacific
J. Hth. 4 11972), 553-56|.
7,
EKELAND, I. ), tle
324-353.
8.
I.KELAND, I. N’onvex minimization prblems, Bull. Amer. Math. Soc., 1(1979),
443-474.
,nd RAY, W. O.
A remark on directiona] contractions,
Sc. 66(|977), 279-283.
PARK, S.
Equivalent frmulati, f
Prc. Amer. Hath.
l’.’keland’s variational principle fr approxi-
mate soltis of minimization problems and their applications, The HSRIKj,__r5,_, l’_u_b_l. _1(1985), to alpear.
variati,,,l
principle, J.
M_at_h. Anal..A_pp.1. 4..7(1’)74),