Internat. J. Math. & Math. Sci.
VOL. 16 NO.
(1993) 87-94
87
COINCIDENCE POINT THEOREMS FOR MULTIVALUED MAPPINGS
SHIH-SEN CHANG
Department of Mathematics
Sichuan University
Chengdu, Sichuan 610064
People’s Republic of China
YONG-CHENG PENG
Department of Mathematics
Shaoguan University
Shaoguan, Guangdong 512005
People’s Republic of China
(Received December 17, 1990 and in revised form March 2, 1991)
ABSTRACT. Some new coincidence point and fixed point theorems for multivalued mappings in complete
metric space are proved. The results presented in this paper enrich and extend the corresponding results
in [5-16, 20-25, 29].
KEY WORDS AND PHRASES: Multivalued mapping, coincidence point, and fixed point.
1991 AMS SUBJECT CLASSIFICATION CODES. 47H10, 54H25.
1.
INTRODUUHON AND PRELIMINARIES.
In recent years, the existence and uniqueness of coincidence points and fixed points for commuting
mappings, weakly commuting mappings and compatible mappings have been considered by several authors
(see [2, 3, 6, 8, 17-28]). The purpose of this paper is to study the existence of coincidence points and fixed
point for multivalued mappings in complete metric space from different aspects. The results presented in
this paper enrich and extend the corresponding results in [5-16, 20-25, 29].
Throughout this paper, letR [0, +oo) and (X,d) a complete metric space. For any nonempty subsets
A and B of X, we denote
d(x,A )- inf{d(x,a): a .A }(x _X),
d(A,B
inf{d(a,b ):
a _A, b _B
,
H(A,B)-max[suPocA d(a,b), su d(b,A)},
CC(X)-{A: A
CB(X)-{A: A
is a nonempty compact subset of
X},
is a nonempty closed and bounded subset of X}.
and H(., .) is called the Hausdorff metric on CB(X).
S.S. CHANG AND Y.C. PENG
88
LEMMA 1 [4, Lemma 2.2]. Let (X,d) be a metric space, A CX a nonempty compact subset and
B CX a closed subset. Ifd(A,B)-O, thenA CIB
REMARK 1. Even ira and B are both bounded closed subsets, the conclusion of Lemma 1 need not
hold. This can be seen from the following
EXAMPLE 1. Let X R and d the Euclidean metric on R 2. Letting
p(., .) min { 1, d(., .)},
z. Therefore (R2,p) is also a metric space, and it is bounded.
Now we consider the following subsets of (R z, p):
it is easy to verify that p(.,.) is a metric onR
A
(x,y).R’:
y
--, x
Then A and B are both bounded and closed and d(A,B) 0, but A t3B
LEMMA 2 [5, Theorem 1]. Let : R
O.
R be an increasing function such that
tI)(t +)<t for all t>0
(I.I)
ytl,’(t) is finite for all t>0.
(1.2)
and
Then there exists a strictly increasing function
:R
R such that
tIt) < t) for all
>0
(1.3)
> 0.
(1.4)
and
"(t)
LEMMA 3 [5]. (i) If : R
(ii) Let : R
Then (1.2) holds.
(iii) Let : R
-
is finite for
R is strictly increasing and satisfies (1.2), then
satisfies (1.1).
R be increasing and satisfies (1.1). If ZtlY’(tl) is convergent for some tl > 0.
R be increasing and satisfies (1.1). If
O(t) then
0.
MAIN RESULTS
Recently, Kaneko and Sessa [6] extended the definition of compatibility to include multivalued
mappings and proved the following theorem:
THEOREM 1. Let f: X X and T: X CB(X) be compatible continuous mapping such that
T(X) C f(X) and
2.
H(Tx, Ty) max d(fx,fy), d,Tx), d(fy, Ty),
-(d(fx, Ty) + d(fy,Tx))
for all x, y in X, where 0 h < 1. Then there exists a point x. (E X such that fx. E Txo.
As an improvement and generalization of Theorem 1, we have the following
THEOREM 2. Let F: X CC(X), $, T: X CB(X) be three multivalued mappings such that
S(X) U T(X) C F(X), F(X) is closed and
H(Sx, Ty
@(max l
d(Fx,Fy ), d(Fx, Sx ), d(Fy, Ty ),
1/2(d(Fx, Ty
+ d(Fy,Sx
))[ )
COINCIDENCE POINT THEOREMS FOR MULTIVALUED MAPPINGS
for all x, y in X, where O: R
there exists a point z
89
R is an increasing function satisfying conditions (1.1) and (1.2). Then
EX such that
Fz N Sz CI Tz #
.
PROOF. By Lemma 2, there exists a strictly increasing function
(1.3) and (’1.4). For any x, y in X let us denote
(, y) ma (F,Fy l, d(F,Sx), a(Fy, ry),
:R
g(d(F, ry)
R satisfying conditions
/
a(Fy,s)).
Then (2.1) can be reduced as follows:
(s, ry) (,y)).
For any xo EX, sinceS(X) C F(X), there exists anx EX such that Tx CIS , 0. Let y
Fx f3Sxo,
then we have
d(Yl, TX,l) : H(Sxo, Tx) (since y, Sxo)
(a) IfA(xo, x)-O, thend(Fxo, Sxo)- 0. By Lemma 1,Fxof’lSxo , O. Takingz EFxo f3Sx then we
have
H(a, Txo) H(Sxo, Txu)
(xo, Xo)
(max {O,
O, d(z, Txo),
"
1/2d(z, rxo)})
d(z, rxo)).
mma 3 (iii) d(z, T)- O. Sin T clo z T.
eorem 2 proved.
) A(x) > 0, en, by (1.3) we ve
By
nquently, we n find an y
erefore
is
e nclusion of
Tx such Sat
d@,, y
(2.2)
(x,)).
Since T(X) C F(X), for y Tx C F(X), Sere exism a int X such Sat y F. is implies Sat
we n find an y F Tx such at (2.2) holds.
On e other han by e aption we have
d( Y9 H( rx,)
(x,)).
ff A(x)- O, by e me way
e f of (a) we n prove at e nclusion of
stated
eorem 2 e. ffA()> 0, reating e me way mentioned above, we n find x X and
y
F
such at
duaively, we
d@ Y9 ffi (x,)).
define o quenee {x. }, {y.} C X such at
F.
n -0,,
(2.3)
S.S. CHANG AND Y.C. PENG
90
and
/1, y2. /2)-: A(x,
n-0,1,2
(2.4)
d(y2/3 y/2).:(A(x/2, x/1))
Now we prove that {y. } is a Cauchy sequence in X. In fact, for any positive integer n we have
A(x2. x2,,/l)-max {d(Fx2., Fx/l), d(Fx, Sx2.), d(Fx2./l Tx/l),
(d(Fx2., Tx2. t) + d Fx2.
max {d(Y2.,
Y2./1), d(y,, Y2./1), d(Y,/I,
1
max {d(y2., Y2,/1,
d(y2./1, Y2,/2)}.
By the same way we can prove that
A(x2./z, x2./1)’:max{d(Y,/1, Y2./2), d(Y2./2, Y2./3)}Consequently, in general, we have
n- 1,2,...
(max {d(y., Y./t), d(Y./t, Y./2)}),
If d(y. i, Y. +2) > d(y., y. i) 0, then, by (2.5) and Lemma 3 we have
d(Y, l, Y, +) d(Y, t, Y,+))<d(Y,+t,
a contradiction. Therefore we have d(y, 1, Y, +_) d(y,, y, 1)- Hence we have
a(y., y./).(a(y._, y.))....
_
. -afy,
.
yg).
n
,z
(2.5)
(z.6)
Hence
If d(yl, Y2) "0, i.e. Yl Y2, denoting z Yt Y, then z Yt Fxt fqS z Y2 Fx2 CI Tx.
z Fxt tq Tx. Similarly using the proof in (a) we can prove z Sxl. Hence the conclusion of Theorem 2
is proved.
If d(y, Y2) > 0, in view of condition (1.4), we know that :E-td(yl, Y2)) is convergent. It follows
from (2.6) that Y_,d(y,, y, t) is convergent too. This implies that {y, } is a Cauchy sequence in X. Let it
converge to some point y. in X. Since y, Fx, C F(X) and F(X) is closed, this shows that y. F(X).
Hence there exists z X such that y. _Fz. By (2.1) and (2.3) we have
aty.,S) afy., y./)+d(y./, Sz)
d(y.,
Y2,
2) + H(Tx2, t, Sz)
<_ a(u., u2. + 2) + (a(z,
-: d(y., Y2,
d(Fx
l,
Tx
l),
Fx
t), d(Fz,Sz),
l) + d(Fx, t, Sz))}
"(d(Fz, Tx2,
d(y,,y2, +2)+ tl(max {d(d(yo, Y2,/l),d(y.,Sz),
d(y2,
Letting n
oo, we have
l,
Y2,
2),
1
Tx
x2,, +
2) + tD(max {d(Fz,
1
(since Y2,
(d(y., Y2,
2) + d(y2, t, Sz))} ).
t)
91
COINCIDENCE POINT THEOREMS FOR MULTIVALUED MAPPINGS
d(y.,Sz) qnax O,d(y.,Sz),
O,-d(y.,Sz)
q)d(y.,Sz)).
By Lemma 3 (iii) we have d(yo, SZ) O. Since Sz is closed, so that y. Sz.
Tz. Therefore we have y. Fz fSz f Tz.
Similarly, we can prove that y.
This completes the proof.
REMARK 2. (i) Theorem 1 is a special case of Theorem 2 with F being a single-valued mapping,
S=T and O(t) h t, where 0 h < 1 and R /.
(ii) Even if the mapping F in Theorem 2 is assumed to satisfy the condition "F(X) is closed",
Theorem 2 still weakens the continuity and compatibility conditions on T in Theorem I. This can be seen
from the following Example:
EXAMPLE 2. Let X R and f and g be two functions from R" into R defined by
if x < I,
g(x) x(x + 1)-1.
f(x)
1,
if x
1,
It is easy to see that f(X) is closed, f and g are continuous, but they are not compatible (see [8, Example
2.5]).
Theorem 2 extends and improves also the corresponding results of [7, 8, 20-25].
(iii)
a
As consequence of Theorem 2 we have the following result:
COROLLARY 1. Let T: X CB(X)(i 1,2, ...) and
H(Trr,
Ty),:tl)(max[d(x,y), d(x, Trr), d(y,TY), 1/2(d(x, Tiy)+d(y,Tx))),
i,j
(2.7)
for all x, y in X, where : R
R is an increasing function satisfying conditions (1.1) and (1.2). Then
T.,x}, 1, 2, are nonempty and equal to each other. Moreover, ff at least
the fixed point set {x: x
one of {T} is continuous, then they are all closed.
PROOF. For the sake of convenience we prove the conclusions of Corollary only for the case of i=
and j=2.
By Theorem 2, there exists an z X such that z Tz T z.
How we prove that the fixed point sets of T1 and T are equal to each other. In fact, if u is a fixed
point of T, i.e. u Tzu, then we have
d(u,T_u)’H(Tlu,T_u)
m.
d(.,I, d(.,rI, (.,r.I,
-(max{0,
0, d(u,T2u ),
g(d(.,rl/(d(.,r,.llt
1/2d(u,T2u)})
q)(d(u, T u )).
By Lemma 3 (iii), we have d(u, T u) O. Since Tz u is closed, u
.
.
T u.
By the same way we can prove that if w is a fixed point of T2 then w is also a fixed point of TI. Hence
we have {x .X: x Tx} {x .X: x Tx}.
S.S. CHANG AND Y.C. PENG
92
Next, we prove that if T (or T2) is continuous, then the set of fixed points {x X: x
set.
Tx } is a closed
oo. Sincex, Tx, and Tx, Tx ash
In fact, let {x,} (7_ {x .X: x _Tx} andx, --,,x asn
we have
d (x
i.e. d(x, Tx)
O. Therefore x
.
T,x
s d (x, x,, + d (x,,
Tx
d(x,x,)+H(Tx,,Tx)--O
as n
Tx.
This completes the proof.
REMARK 3. If all the mapping T,
1, 2
in Corollary I are single-valued, then T,
1, 2
have a unique common fixed point in X.
In fact, let u, v X be two common fixed points of T,
d(u, )
1,2
then we have
d(r,u, Tf
I(max {d(u,v), d(u,Tu), d(v, Tf),
l
(d(u, ,) + d(v,
2
))})
-qp(max{d(u,v), O, O, d(u,v)})
qP(d(u, v)), for all i, j, ,j.
Hence we have d(u, v) O, i.e. u v.
REMARK 4. The results of [5, Theorem 9], [9, 10, 11, 12, 13, Theorem 1], [14, Theorem] and [15,
Theorem 1, 3, 4] are all the special cases of Corollary.
DEFINITION. A function (tx, t.z, ts, t4,ts): R /s
R is called to satisfy the condition (q0, if it is
nondecreasing in each variable and there exists an increasing function O(t): R
conditions (1.1) and (1.2) such that
Vt O, a + b 3, a, b 1,2.
(t,t,t, at, bt) l(t),
-
R satisfying the
THEOREM 3. Let F: X CC(X), $, T: X CB(X) be three multivalued mappings such that
S(X) U T(X)(7. F(X), F(X) is closed and satisfies the following conditions:
H(Sx, Ty),,P(d(Fx,Fy), d(Fx, Sx), d(Fy, ry), d(Fx, Ty), d(Fy,Sx))
(2.8)
for allx, y inX, where P(tx, tt,t4,t): R /s
such that Fz CISz CI Tz
#
R satisfies condition (xp).
Ten there exists a point z
J.
PROOF. Let
t"
max
[
d(Fx, Fy), d(Fx, Sx), d(Fy, Ty ),
1/2(d(Fx, Ty + d(Fy,Sx))}.
Without loss of generality we can assume that d(Fx, Ty) d(Fy,Sx) (otherwise, it can be proved similarly).
Then we have
t"
max {d(Fx, Fy),
d(Fx,Sx), d(Fy, Ty)},
l(d(Fx, Ty) + d(Fy,Sx))
2t"
d (Fx, Ty + d (Fy, Sx
d(Fy,Sx)
d (Fx
Ty
93
COINCIDENCE POINT THEOREMS FOR MULTIVALUED MAPPINGS
Using condition () and (2.8) we have
H(Sx, Ty)
tP(t’,t’,t’,2t’,t’) (t’)
Therefore, F, S, T satisfies all conditions of Theorem 2. The conclusion of Theorem 3 follows from
Theorem 2 immediately.
From Theorem 3 we can obtain the following
COROY 2. Let T: X
CB(X),
1,2,
satisfy the following condition
H(Tx, T y)(d(x,y), d(x, rrr), d(y, Tiy), d(x,Ty), d(y,Trx)), Vi,
for all x, y in X, where (t,t.z, ts, ts, ts): R
{x X: x T.,x},
/s
y,
i,j
R satisfies condition Q). Then the: fixed point sets
1, 2,... are nonempty and equal to each other. Moreover, if one of T,
1,2,
is continuous, then they are closed.
REMARK $. Corollary 2 generalizes the corresponding result of [29].
CHANG, S-S., Fixed Point Theorems and Applications, Chongqing Publishing House, Chongqing,
1984.
CHANG, S-S., On Common Fixed Point Theorem for a Family of O-Contraction Mappings, Math
29 (1984), 527-536.
CHANG, S-S., A Common Fixed Point Theorem for Commuting Mappings, Pro. Amer. Math.
Soc.. 83 (1981), 645-652.
CHANG, S-S. and GUO, W-P., Coincidence points and fixed points for 1-Set Contraction
Mappings, J. Math. Research & Exposition (to appear).
SASTRY, IC P. R., NAIDU, S. V. R. and PRASAD, J. IL, Common Fixed Points for Multimaps
in a Metric Space, Nonlinear Anal., 13 (1989), 221-229.
KANEKO, I-L and SESSA S., Fixed Point Theorems for Compatible multi-valued and single-valued
mappings, Intemat. J. Math. & Math. Sci., 12 (1989), 257-262.
KANEKO, H. Single-valued and Multi-valued f-contractions, Boll. Un. Math. Ital., (6) 4-A (1985),
29-33.
JUNGCK, G., Common Fixed Point for Commuting and Compatible Maps on Compact, Proc..
Amer. Math., 103 (1988), 977-983.
KUBIAK, T., Fixed Point Theorems for Contractive Type Mulitvalued Mappings, Math. Japonica.
10.
11.
12.
15.
16.
30 (1985), 89-101.
SINGI-I, IC L and WH]TFIELD, J. H. M., Fixed Points of Contractive Type Multivalued Mappings,
Math. Japonica, 27 (1982), 117-124.
ACHARI, J., Generalized Multi-valued Contractions and Fixed Points, Rev. Roum. Math. Pures
Appl., 24 (1979), 179-182.
ISIKI, IC, Multi-valued Contraction Mappings in Complete Metric Spaces, Rend. Sem. Mat. Univ.
Padova, 53 (1975), 15-19.
KUBIACZYK, L, Some Fixed Point Theorems, Demonstratio Math., 6 (1976), 507-515.
KITA, T., A Common Fixed Point Theorem for Multivalued Mappings, Math. Japonica. 22 (1977),
113-116.
PAPAGEORGION, N. S., Fixed Point Theorems for Multifunctions in Metric and Vector Spaces,
Nonlinear Anal., 7 (1983), 763-770.
KHAN, M. S., Common Fixed Point Theorems for Multivalued Mappings, Pacific J. Math., 96
(1981), 337-347.
94
17.
18.
19.
22.
23.
24.
25.
26.
27.
28.
29.
S.S. CHANG AND Y.C. PENG
KANG, S. M., CHO, C. L. and JUNGCK, G., Common Fixed Point of Compatible Mappings,
Intemat. I, Math. & Math. Sci., 13 (1990), 61-66.
JUNGCK, G., Compatible Mappings and Common Fixed Points, Inlrnat. J. Math & Math. Sci.,
9 (1986), 771-779.
KANG, S. M. and RYU, J. W., A Common Fixed Point Theorem for Compatible Mappings, Math.
Japonica, 35 (1990), 153-157.
JUNGCK, G., Commuting Mappings and Fixed Points, Amer. Math. Monthly, 83 (1976), 251-263.
DAS, K. M. and NAIK, K. V., Common Fixed Point Theorems for Commuting Maps on a Metric
Space, Proc. Amer. Math. Sot., 77 (1979), 369-373.
CHATrERJI, H., A Note on Common Fixed Points and Sequences of Mappings, Bull. Calcutta
Math. Sot., 72 (1980), 139-142.
JAISWAL, A. and SINGH, B., Fixed Point Theorems in Metric Space Over Topological Semifield,
Math. Japonica, 2,1. (1976), 187-196.
M-LIKHERJEE, R. N., Common Fixed Points of Some Nonlinear Mappings, Indian J. Pure Appl.
Math., 12 (1981), 930-933.
CIRIC, LJ. B., A Generalization of Banach’s Contraction Principle, Proc. Amer. Math. Sot., 46
(1974), 267-273.
RHOADES, B. E., PARK, S. and MOON, IC B., On Generalizations of the Meir-Keeler Type
Contraction Maps, I, Math. Anal. Ap_pl., 148 (1990), 482-484.
JUNGCK, G., A Common Fixed Point Theorem for commuting Maps on L-spaces, Math. Japonica,
25 (1980), 81-85.
JUNGCK, G., Periodic and Fixed Points and Commuting Mappings, Proc. Amer. Math. So,, 76
(1979), 333-339.
YEH, C. C., Some Fixed Point Theorems in Complete Metric Spaces, Math. Japonica. 23 (1978),
27-31.