323
internat. J. Math. & Math. Sci.
Vol. 9 No. 2 (1986) 323-329
ON COMMON FIXED POINTS OF WEAKLY COMMUTING MAPPINGS
AND SET-VALUED MAPPINGS
S. SESSA
Istituto di Hatematica
Facolta di Architettura
Universita di Napoli
Via Monteoliveto 3
80134 Napoli, Italy
and
B. FISHER
Department of Mathematics
The University
Leicester, ENGLAND
(Received September 7, 1984)
Our main theorem establishes the uniqueness of the common fixed point of tw
ABS]RACT.
set-valued mappings and of two single-valued mappings defined on a complete metric space,
under a contractive condition and a weak commutativity concept.
This improves a theorem
of the second author.
K’Y WORDS AND PHRASES.
Common
fixed point, set-valued mappin., weak eomutativity.
1080 AMS SUBJECT CLASSIFICATION CODES.
Let
(X,d)
be a complete metric space and let
bounded subsets of
As in [I], let
X
sup {d(a,b)
(A,B)
A, B
for all
A
If
B
(A,B)
A, b
a
B(X)
be the set of all nonempty,
be the function defined by
B}
B(X).
in
consists of a single point
6(A,B)
and if
54H25, 47H10.
we write
a
(a,B)
also consists of a single point
(A,B)
b
we write
d(a,b)
It follows immediately from the definition that
(A,B)
6(A,B)
for all
A
B
C
in
O, (A,A)
(B,A)
6(A,C) + 6(C,B)
B(X).
We say that a subset
sets of
a
N
n
is in
X
if each point
A
such that
n
for
A !
n
n
diam A,
of
X
is the limit of a sequence
a
in
A
is the limit of a convergent sequence
1,2
for
{A
A
and if for arbitrary
n
N
where
A
n
of nonempty sub-
{a }, where
n
O, there exists an integer
is the union of all open spheres with
324
S. SESSA and B. FISHER
A
centres in
LEMM_A
I.
and radius
{A
If
{B
and
n
A
verge to the bounded sets
(X,d)
are sequences of bounded subsets of
n
B
and
which con-
{6(An,Bn)}
respectively, then the sequence
(A,B).
converges to
This lemma was proved in [2].
Now let
F
be a mapping of
in
X
if whenever
point
{Fx
the sequence
each point
x
z
point
F
and
for all
x
in
is continuous at the
is a sequence of points in
X
converging to
F
is continuous at
Fx
converges to
F
B(X).
in
If
X
is a continuous mapping of
is said to be a fixed point of
F
z
if
is
in
Two commuting mappings
F
commuting mappings
A
B(X).
into
Fz.
I to be weakly commuting on X if
(Flx,IFx) ! max{(Ix,Fx), diam IFx}
X.
x
(X,d), the authors of [3], extending the results of [2] and [4],
For a selfmap I of
defined
F
B(X)
into
n
X, we say that
in
X
in
B(X)
in
n
{x
We say that
X
I
and
F
(I.I)
I
and
clearly commute, but two weakly
do not necessarily commute as is shown in the following
example.
EXAMPLE I.
[0,I], let
X
Let
6
be the function induced by the euclidean metric
d
and define
x
X
in
x
h
where
X
in
2.
a
F
Then for any non-zero
[0, x/(ax +
ah+l)]--
x
X
in
we have
IFx
we have
x/(x + a h+l)
6(Flx, IFx)
Note that if
x/a
Ix
and
h+l
[0, x/(x + a
)]
FIx
but for any
x/(x+ah)],
[0,
Fx
for all
x/a
(Ix, Fx)
is a single-valued mapping, then the set
single point and therefore diam
{IFx}
0
for all x
in
X.
{IFx} consists of a
Condition (I.I) therefore
reduces to the condition given in [5], i.e.
for all
x
d(Flx, IFx) ! d(Ix, Fx)
X.
in
(1.2)
An extensive literature exists about (common) fixed points of set-valued mappings
satisfying contractive conditions controlled from non-negative real functions
[0, )
into
[0,).
Suitable properties of
f
f
from
guarantee the convergence to the
(common) fixed point of the sequence of successive approximations: see for example the
papers of Barcz [6], Chen and Shih [7], Guay, Singh, and Whitfield [8], Miczko and
Palezewski [9], Nhan [I0], Papageorgiou [II], Popa [12], Sharma [13] and Wegrzyk [14].
In this paper
we consider the family
F
of functions
f
such that
(a)
is non-decreasing,
(aa)
is continuous from the right,
(aaa)
LEMMA 2.
f(t) < t
For any
for all
t > 0
t
0.
lim
fn(t)
O.
The proof of this lemma is obvious but see also [15].
from [0, )
into
[0, )
325
FIXED POINTS OF WEAKLY COMMUTING AND SET-VALUED MAPPINGS
Further details about the usage of functions with properties similar to (a), (a),
can be found in the papers of Benedykt and Matkowski [16], Browder [17],
and (aaa)
Conserva and Fedele [18], HegedHs and Szilgyi [19], Hikida [20], Park and Rhoades [21],
Jhoades [22], and Singh and Kasahara [23].
RESULTS IN COMPLETE METRIC SPACES.
2.
F,
Let
G
Xo
yo
(Resp.
{yn })
(resp. yn
resp.
:n
B(X)
into
I
and let
J
be two
such that
F(X) ! l(X),
l.et
X
be two set-valued mappings of
X
elfmaps of
(2.1)
G(X) i J(X)
X
be an arbitrary point in
and define inductively a sequence
Xn_l (resp.
such that, having defined the point
IXn
with
JYn)
(resp.
’I’his can be done since the range of
(resp.
I
(resp.
FXn_l
in
J)
yn_l),
GYn_ 1)
for
{i
choose a point
n
|,2,
(resp. G).
F
contains the range of
Further, assume that
sup{6(FXn,GYo), 6(GYn,FX O)
i<EMARK I.
IF
n
<
1,2
(2.2)
is bounded then (2.2) will always be satisfied for all
X
x, y
in
X.
We consider the following conditions:
continuous,
(y|)
(y2)
F
continuous and
([)
(2)
J
continuous,
G
continuous and
IFx ! Fix
for all
x
in
X
JGx ! GJx
for all
x
in
X
of [I] we are now able to prove the following:
Modifying the proof of theorem
F, G
Let
THEOREM I.
X
be two selfmaps of
X
be two set-valued mappings of
B(X)
into
I, J
and let
satisfying (2.1) and
6(Fx,Gy) ! f(max{d(Ix,Jy), 6(Ix,Gy), 6(Jy,Fx) })
for all
x, y
with
and
X
in
J
where
f
respectively.
(2.2) and if the conditions
and
J
(yi)
and
(%j)
F
F
Further let
have a unique common fixed point
is the unique common fixed point of
PROOF.
F
is in
If there exist points
with
z
and
x
and of
YO
1,2,
i,
Fz
Further,
I
G
and
and
0
(2.3)
G
X
satisfying
hold, then
{z}
Gz
and
weakly commute
in
and
F
z
G
is the
J
Since
6(FXr,GYs)
_<
+
6(FXr,GYo)
6(GYo’Fxo)
+
6(Fxo’GYs)’
it follows from (2.2) that
M
sup
{6(FXr,GYs)
r, s
0,1,2
is finite.
If
fP(M)
O, then for arbitrary
M
e
by lemma 2.
If
M
As in the proof of theorem
and property
e > O, we can choose an integer
O, then
fP(M)
0 < e
for any integer
fP(max{6(FXr’Gyq)
<6
such that
p.
of [24], we have on using inequality (2.3)
(a):
6(FXm,GYn)
p
m
p _< r _< m
n-
p!q!n})
p times
S. SESSA and B. FISHER
326
for
n > p.
m
Thus
(Fxm,Fxn) _<
for
n > p
m
z
6(GYs,FXr)
2
Z
X
in
n
{Fx
converges to the set
n
{Jyn
Similarly, it can be proved that the sequence
{Gy n}
the sequence of sets
is independent of the particu-
z
where
{Ix
It follows in particular that the sequence
and the sequence of sets
z
+
is therefore a Cauchy sequence in the complete
and so has a limit
lar choice of each
to
{z
The sequence
X
metric space
6(FXm,GYs)
converges
n
{z}
converges to a point
and
w
{w}
converges to the set
Using (2.3) we have
6(Fx n,Gy n) _<
tend to infinity and using lemma
n
Letting
(yl)
Now suppose that
Iz
n
1,2,
{Iz}
and
I
Let
w
< 6(Fix
Iz
and
(aaa)
and
it is
Iz
{Fix
for
n
we have on using (I.I)
F
2(12Xn+ I’ IFXn)}
+ 6(IFx n’ Iz)
n
{w
we see that the sequence
is independent of the particular choice of
means that the sequence of sets
Fix
converge
+ (IFx n Iz)
tend to infinity and using lemma
But
{IFXn
be an arbitrary point in
n
max{6(Ix n’ Fx n)
to
{12Xn
weakly commutes with
6(FlXn,lZ)
n’ IFXn)
_<
d(Wn,IZ)
n
(aa)
Then the sequence
holds.
respectively.
Then since
Letting
and properties
z.
w
seen that
to
f(max{d(IXn,JYn), 6 (Ixn’Gy n), 6(JYn,FXn
converges to the set
w
n
in
Fix
converges
n
and this
n
{Iz}
Using inequality (2.3) we have
(FlXn’GYn) -< f(max{d(12Xn’JYn ), 6(12Xn,GY n),
tend to infinity and using lemma
n
Letting
6(Jy n,Flxn)})
and property (aa), we have
d(Iz,z) _< f(d(Iz,z))
Iz
which implies
z
(aaa).
by
Since
(Fz,GYn)
we have on letting
which gives
Fz
n
_<
f(max{d(Iz,Jyn),
and
z
Gz
is
G
and
J
and condition
(i)
implies
{z}
using inequality (2.3)
f
and property
(Fz,z) _< f((z,Fz))
{z} by ().
Now assume that
since
GYn),6(JYn,FZ)})
tend to infinity and using lemma
Similarly, the weak commutativity of
Jz
(Iz,
(y2)
holds.
Then the sequence
{FIXn
converges to
Fz
and
we have
6(FlXn’GYn) -< f(max{d(12Xn’JYn ), 6(12Xn,GY n), 6(JYn,FlXn)})
f(max{6(Flx Jyn
6(Flx n GYn) 6(Jy n FlXn)})
n’
and so 12x
is in IFXn_
is in Fx
non-decreasing and Ix
n
n-I
n
FlXn_
FIXED POINTS OF WEAKLY COMMUTING AND SET-VALUED MAPPINGS
tend to infinity and using lemma
n
Letting
6(Fz,z) ! "f(6(Fz,z))
{z} by (aaa)
Fz
which implies
such that
lu
and property ()
327
we have
Thus by (2.1) there must exist a point
u
X
in
z.
Using inequality (2.3) we have
f(max{d(lu,JYn), (lu,GYn), (JYn,FU)}).
(Fu,Gy n)
tend to infinity and using lemma
n
Letting
(a), we obtain the ine-
and property
quality
(Fu,z) ! f(max{d(lu,z), 6(z,Fu)})
f((z,Fu))
{z} by (aaa) and since F and I weakly commute, we have
Fu
Thus
Fz
z
Iz
It follows that
Iz}.
IFu
Flu
z.
(%2)
Similarly property
{z} and Jz
assures that Gz
We have therefore shown that if the conditions
Iz
hold then
Jz
z
That
F, G
Let
be two selfmaps of
x, y
X
If
F
I
or
c <
I.
and
G
have a unique common fixed point
F
common fixed point of
IFx
Fix
I, 2,
i,
I
and
G
and of
J
and
Since
GJx
and
X
B(X) and let
into
(2.1) and
I
and
F
and
JGx
if we assume that f(t)
G
G
and of
for all
x
and
G
commute with
Fz
and
{z
Gz
and
I
and
F, G, I and J
z
is the unique
J.
of [I], it is proved that (2.2) holds for any
commute with
ct for all t
F
J are continuous, then
Further,
z.
(2.4)
Further, let
or
As in the proof of theorem
X.
in
x0’ Y0
F
be two set-valued mappings of
satisfying
X, where 0
in
respectively.
PROOF.
with
c.max{d(Ix,Jy),(Ix,Gy),(Jy,Fx)}
(Fx,Fy)
for all
(j),
This completes the proof of the theorem.
COROLLARY I.
J
(yi)
{z}.
Gz
is the unique common fixed point of
follows easily.
I, J
Fz
z and
z.
and
X.
in
I
and
J
respectively, we have
The thesis then follows from theorem
O.
The result of this corollary was given in [I].
We now give an example in which theorem
EXAMPLE 2.
for all
x
X.
in
By example i, F
for all
x
and
G
weakly commute with
F(X)
[0,I/5]
[0,3
I(X),
G(X)
[0,1/9]
[0,1/2]
J(X),
IFx
[0,x/(2x + 8)]
JGx
[0,x/(2x + 16)]
in
is not applicable.
x
[0,x/(x + 4)],
Fx
holds but corollary
[o,I] with 6 induced by the euclidean metric
Gx
[0,x/(x + 8)],
Jx
Ix
Let X
[0,x/(x + 8)]
[O,x/(x + 16]
X.
Since
6(Fx,Gy)
I.
max{x/(x+4), y/(y+8)}
! max{x/(x+4), y/(y+4)}
Further, we have
Fix
GJx
d
and let
S. SESSA and B. FISHER
328
! max{x, y}
(Ix,Gy),
(Jy,Fx),
t
f(t)
for all
F, G
point of
x
if
x < y
are satisfied if we assume
is bounded all the hypotheses of theorem
X
and since
y,
if
f
Clearly
0.
t
F
is in
0
and
is the’unique common fixed
I
and
is a stronger result than corollary I, even if the mappings under consid-
Theorem
eration are commutative, as is shown in the following example.
EXAMPLE 3.
X
Let
6
be the reals with
{0},
Ix
for all
x
0 < x
[0, i/4],
if
x >
{0},
if
x
if
0 < x !
if
x > I,
O, if
x_< 0
x, if
0 < x <_
I, if
x
X
in
x
if
[0, x/(l + 2x)],
{1/3},
Gx
and let
f
let
0
if
[0, x/(1 + 3x)],
Fx
d
induced by the euclidean metric
0
_< O,
O, if
x
x, if
x > O,
Jx
in
F
be given by
t/(l + 2t),
if
0 < t <
t/3,
if
t >
f(t)
We have
0
(Fx,Gy)
y/(l+2y)
1/3 < y/3
(Fx,Gy)
(Fx,Gy)
x,y <_ 0
f(d(Ix,Jy)), if
x < 0
and
0 < y _< I,
f(y)= f(d(Ix,Jy)), if
x _< 0
and
y >
f(y)
x/(l+3x)<x/(l+2x)=f(x)=f(d(Ix,Jy)),
(Fx,Gy)
<
(Fx,Gy)
(Fx,Gy)
(Fx,Gy)
(Fx,Gy)
_
f(d(Ix,Jy)), if
(Fx,Gy)
max{x/(l + 3x), y/(1 + 2y)}
max{x/(l + 2x), y/(l + 2y)}
f(y)
f((Fx,Jy)),
if
and
Gx
G(X)
F
[0,1/3]
for any
cable.
and
and
0
x
0
G
and
Y0
_< O,
f(x)
f((Ix,Gy)),
if x > y, and if 0 < x, y _< I,
I/3 < y/3
f(y)
f((Jy,Fx)),
if 0 < x _<
and y > I,
I/4 < I/3
f(1)
f(d(Ix,Jy)),
if x >
and y _< O,
max{I/4, y/(l + 2y)} _< 1/3
f(1)
f((Ix,Gy)), if x
and
0 < y_<
I/3 < y/3
f(y)
f(6(Jy,Fx)), if x,y >
[0,1.4]
[0,I/3]
commute with
for all
and y
x < y
Condition (2.3) therefore holds in every case since
F(X)
if 0 < x <
[0,I]
[0,]
I
and
f
is non-decreasing.
Further
I(X)
J(X)
J
respectively. Since Fx _c [0,1/4] and
X, it is easily seen that M _< I/3 and so (2.2) holds
chosen in X. As I and J are continuous, theorem
is applix
in
However, the conditions of the corollary are
y < I, condition (2.4) should imply
not satisfied.
Otherwise for
x=0
329
FIXED POINTS OF WEAKLY COMMUTING AND SET-VALUED MAPPINGS
Y
<
+ 2y- c.max{y,
6(Fx,Gy)
and so
I/(I + 2y) _<
c
which as
y
Y
+ 2y’ y
tends to zero, gives
cy
c
_> I,
a contradiction.
REFERENCES
.
I.
FISHER, B., "Common fixed point theorems for mappings and set-valued mappings",
Univ. Kuwait Sci., to appear.
2.
FISHER, B., "Common fixed points of mappings and set-valued mappings", Rostock. Math
Kolloq., 18(1981), 69-77.
3.
SESSA, S., KHAN, M.S., and IMDAD, M., "Common fixed point theorems with a weak commutativity condition", submitted for publication.
4.
FISHER,B., "Fixed points of mappings and set-valued mappings", J. Univ. Kuwait Sci.,
9(1982), 175-180.
5.
SESSA, S., "On a weak commutativity condition in fixed point considerations", Publ.
Inst. Math., 32(46)(1982), 149-153.
6.
BARCZ, E. "Some fixed point theorems for multi-valued mappings", Demonstratio Math.,
16(1983), 735-744.
7.
CHEN, M. P. and SHIN, M. H., "Fixed point theorems for point to point and point to
set maps", J. Math. Anal. Appl., 71(1979), 515-524.
8.
GUAY, M.D., SINGH, K. L., and WHITFIELD, J.H.M., "Common fixed points for set-valued
mappings", Bull. Acad. Polon. Sci. Sr. Sci. Math., 30(1982), 545-551.
9.
MICZKO, A. and PALCZEWSKI, B., "On convergence of successive approximations of some
generalized contraction mappings", Ann. Polon. Math., 4(1983), 213-232.
10. DUANG TRONG NHAN, "Pair of nonlinear contraciton mappings common fixed points",
Studia Univ. Babes-Bolyai Math., 26(1981), 34-51.
ii. PAPAGEORGIOU, N. S., "Fixed point theorems for multifunctions in metric and vector
spaces", Nonlinear Anal. TMA 7(1983), 763-770.
12. POPA, V., "Fixed point theorems for multifunctions", Studia Univ. Babes-Bolyai Math.,
27(1982), 21-27.
13. SHARMA, A. K., "Common fixed points of set-valued
Sci. Math., 27(1979), 407-412.
maps", Bull. Acad. Polon.
Sci.
Sr.
14. WEGRZYK, R., "Fixed-point theorems for multi-valued functions and their applications
to funcitonal equations", Dissertations Math., 201, Warszawa (1982), 1-28.
15. SINGH, S.P. and MEADE, B.A.,"On common fixed point theorems", Bull. Austral. Math.
Sco., 16(1977), 49-53.
16. BENEDYKT, Z. and MATKOWSKI, J., "Remarks on fixed point theorems", Demonstratic
Math., 14(1981), 227-232.
i7. BROWDER, F. E., "Remarks on fixed point theorems of contractive
Anal. TMA, 3(1979), 657-661.
type",
Nonlinear
18. CONSERVA, V. and FEDELE, F., "Remark on a Browder’s fixed point theorem", Math.
Japon., 28(1983), 147-157.
19.
SZILGYI,
HEGEDS,
M. and
T., "Equivalent conditions and a new fixed point theorem in
the theory of contractive type mappings", Math. Japon., 25(1980), 147-157.
20. HIKIDA, M.,
137.
"A
note on contractions in ranked
space",
Math.
Japon., __26(1981), 131-
21. PARK, S. and RHOADES, B. E., "Extensions of some fixed point theorems of HegedHs and
Kasahara", Math. Sem. Notes, Kobe Univ., 9(1981), 113-118.
22. RHOADES, B. E., "Contractive definitions revisited", Topological Methods in Nonlinear
Functional Analysis, Contemporary Math. AMS, 21(1983), 189-205.
23
SINGH, S
L
Indian
.
"On some recent results on common fixed points"
and KASAHARA, S
Pure Appl. Math., 13(7)(1982), 757-761.
24. FISHER, B., "Four mappings with a common fixed point", J. Univ. Kuwait Sci.,
131-139.
(1981),