Applied Mathematics and Mechanics (English Edition, Vol. 10, No. 3, Mar. 1989)

Published by SUT, Shanghai, China

SOME COINCIDENCE T H E O R E M S

OF S E T - V A L U E I ~ M A P P I N G S *

Ding Xie-ping (Tl~-x~)

( Sichuan Normal University, Chengdu) (Received Dec. 7, 1987)

Abstract BrowderIll obtained the sharpenedforms of the Schauder fixed point theorem. Many authors generalized Browder's results in several aspects. Recently, H.M. Ko and K.K. Tanl~.31 generalized Browder's theorems to the coincidence theorems of set-valued mappings. In this paper, we also show some coincidence theorems of set-valued mappings. They improve and generalize the important results in [1,2,3]. I.

P~'eliminaries We shall denote by R the real line and, for any nonempty set X, by 2x the family of all nonempty

subsets of X. Let X and Y be topological spaces. A mapping f : X -> 2u is said to be upper

x o E X if for each open set G c Y with f ( x o ) c G , there is a neighborhood U o f x 0 s u c h t h a t f ( y ) c G forall y E U . f is said to be u.s.c, o n X i f f i s u . s . c , at

semicontinuous (u.s.c.) at

each point of X. Also, if ~ c 2 r continuous at ( x , A ) E X

, then a mapping

x s"2 if for each e ~ 0

p: X•

is said to be ultimately

, there exist a neighborhood U o f x and an open

s e t G i n Ywith A ~ G such that for all ( y , B ) E X x D with y E U and B ~ G , [ P ( t l , B ) - p ( x , A ) I d e . p is said to be ultimately continuous on X x ~Q ifp is ultimately continuous at each point of X •

9 We note that if D = { { y } :

continuity coincide. Now let E be a vector space. K c E

yEY} , the notions o f ultimate continuity and

is a nonempty set and

xEK

. Define the sets Ix(x) and

Ox(x) as follows: Ix(x)={ glEE : there exist uE.K and r ~ 0 s u c h t h a t y = x + r ( u - x ) } , O h ( x ) = I : t E E : there exist u E K and r > 0 such t h a t l / = x + r ( u - x ) } . The sets lA.(x) and Ox(x) are called the inward set and outward set of K at x, respectively. A subset of 2 Eis said to be convex if for each A , B E g ? and for tlfi[0,1] , t A + ( 1 - - t ) B E g 2 . lfEisa topological vector space, we shall denote by ,.ge'(E) the family of all compact convex sets in 2 E. We shall need the following lemmas. L e m m a 1.1131 Let E be a topological vector space, let / s K ~ and

be nonempty and let fg."

2 E be u.s.c, such that for each xf:t,5 ,fix) and,q(x) are both compact and aE/? 9T h e n f + , q

a f are both u.s.c.. L e m m a 1.~.t-'l Let E be a Hausdorff locally convex topological vector space, K ~ E

nonempty para-compact convex, Lc_2Is

be nonempty compact convex. If S: K ~

*Projects Supported by the Science Fund of the Chinese Academy of Sciences. 205

be

2 ~ satisfies

206

Ding Xie-ping (i) S(x) is convex for each x E K , ( i i ) S - ' ( y ) is open for each yqEK ,

(iit) S ( x ) ~ L for each x E K \ L then S has a fixed point in K. Lemma

,

1.3141 Let K be a closed convex subset o f the H a u s d o r f f topological vector space E.

T: K ~ 2 K is a mapping such that for each x E K , T(x) is convex and for each y E K open. If there exist a compact set L c K

and

y,EK" such that y , E T ( x )

, T ~(y) is

for each x f . K \ L

,

then T has a fixed point in K.

II.

Coincidence T h e o r e m s

T h e o r e m 2.1 Let K be a nonempty para-compact convex subset o f a H a u s d o r f f locally convex topological vector space E. L c K is a nonempty c o m p a c t convex set. Let f , g : K-~..~g" ( E ) be u.s.c. ~z:z:2 s

beconvex such that forall x, y E K ,

E-Q 9 Suppose that the m a p p i n g p: K • for each x E K , p ( x , .) is a convex function on (i)

p(x,x--f(x))~p(x,a(x)--f(x))

(ii)

p(x,x--f(x))<-~p(x,y--x+g(x) --f(x))

(iii) for each x E K

- f ( x ) ) <_.9(x, x - / ( x )

(VxEL)

(Vx,yEKkL)

with f ( x ) ["ly(x)=~b . there exists b ' E K such that p ( x , y - - x + g ( x )

) 9

then there exists an xoE/'( Proof

x--f(x)E.Oandy--x+g(x)--f(x)

is ultimately continuous on .Q such that .Q . If the following conditions are satisfied

such that f ( x , ) [']g(x,)g:ck.

For each .x~K , define

m(x) = x + / ( x )

--9(x),

n(z) =x--f(x)

Then m(.v) and n(x) are both compact and. by Lemma I.I, the mappings re.n: K - ~ 2 j are u.s.c.. Suppose that foreach x E K , f ( x ) I ' ] . q ( x ) = ~ , so that t h c s e t S(x)=-,[yEK:

9( p ( x , n ( x ) ) } is nonempty by (iii). Thus S:K.-*2 E. Let Ya, y t E S ( x ) p ( x , y ~ - r e ( x ) ) < p ( x , n ( x ) ) . i= 1,2. Since p(x,-) is convex, we have

and

p(x,y--m(x)) tE[0,1]

, then

p ( x , ty t + ( i - - t ) y t - r e ( x ) ) = p ( X , t(ll t --re(X) ) + ( 1 - - t ) (lit ~ m ( x ) ) )

~.~tp( x,ll~ --re( x ) ) + ( 1 --t ) p( x, Yz--m( x) ) ~ p ( x , n( x) ) and so ty t + ( 1 - - t ) y a E S ( x )

. Thus for each x E K , S(x) is convex.

Now we shall prove that for each

yEK,

S-t(y)

is open. Indeed. if x E S - ~ ( y )

, then

liES(x) , so that p ( x , y - - m ( x ) ) ( p ( x , n ( x ) ) . Let e = [ p ( x , n ( x ) ) - - p ( x , y - - m ( x ) ) ]/9.. Since p is ultimately continuous at ( x , n ( x ) ) E K • . there exist an open n e i g h b o r h o o d UtofxinKandanopensetGinEwith n ( x ) ~ G such that I p ( z , A ) - - p ( x , n ( x ) ) l < e forall ( z , A ) ~ . K x O with zlfiU~ and A ~ G 9 Also since n is u.s.c, at x, there exist an open neighborhood U, of x in K such that for each zEU2 , n( z ) ~ G .Let V ~ = U t n U: , t h e n for each zEI/'~ , we have

I P ( z , n(z) ) - - p ( x , n(x) ) I < e

(2.1)

Next, for fixed / t E N , p is ultimately continuous at ( x , l l - - m ( x ) ) i E K x O . Thus there exist an open neighborhood U 3 o f x in K and an open set G a in E with l i - - m ( x ) ~ G ~ such that

[p(z,A)-- p(x,y--m(x))l
Some Coincidence Theorems of,~"t-Valued Mappings

m(z)Ctt-G, for each zEV,

foreach zEU, and so

.Let Va=UaNU ,,thenwehave

207

zEU3 and y - - r n ( z ) c G j

Ip ( z , y - - m ( z ) ) - - p ( x , u - - m ( x ) ) ] ,
(2.2)

Let V ' = Y t N V : . T h e n V i s a n e i g h b o r h o o d o f x . From (2.1), (2.2) and the definition of follows that for al! z E V

e

it

P(z,lJ--m(z) ) < p ( x , l l - - m ( x ) ) +~ =p(x,n(x))--e
p( x,, y ( x , ) -- f (x,) ) = p ( x , , x 0 - - m ( x , ) =p(x~

) < p ( x,,n( x,) ) ).

which contradicts condition (i). Thus there exists x~ such that f(x~ [7y ( X o ) ~ . C o r o l l a r y 9-.1 Let E be a Hausdorfflocally convex topological vector space and K ~ E be a nonempty para-compact convex set. L ~ K is a nonempty compact convex set and f,9: K ~ E be continuous. Suppose that p: K • E ~ R is continuous such that for each . ~ K , p(x..) is a convex function on E. If the following condition hold: (i) (ii)

p(x,x--f(x))•p(x,y(x)--f(x)) (VxEL) p(x,x--f(x))~p(x,tl--x+g(x)--f(x)) (Vx,~IEK\L)

(iii) foreach x E K with [ ( x ) ~ f l ( x ) .

there exists yE.K such that p ( x , y - - x + o ( x ) - -

f(x))
such that/(xo)--.a(x,).

x(K

R e m a r k 2.1 If we letO(x) = x for all x E K 9 then condition (i) of Corollary 2. I is satisfied. Thus Corollary 2. I generalizes Theorem 3.3 of [21 and Proposition 2 of [I ]. Hence Theorem 2. I generalizes the corresponding results of [I.2] to the coincidence theorems of a pair of set-valued mappings. T h e o r e m 2.2 let K be a closed convex subset of a Hausdorff topological vector space E, L ~ K b e a nonempty compact set, f , g z K ~ . j r ' ( E ) be u.s.c, and D e 2 t be a convex set such that for all x,yEK,x--[(x)EI-2 and y - - x + g ( x ) - - f ( x ) E ~ . Suppose ;.hat Pz K x . Q ~ R is ultimately continuous on ,Q such that for each x E K , p ( x , . ) is convex on .Q . If the following conditions hold: (i)

p(x,x--l(x))<-.~P(x,g(x)--/(x))

(VxEK),

(ii) for all x E K \ L with f ( x ) [ ' 1 9 ( x ) = 4 9 there exists / / , E K + 9(x) --/(x) )
such that p ( x , y , - - x

(iii) for each xlEL with y ( x ) 0 f ( x ) = 4 +g(x)--/(x))
such that

then there exists x, EK

, there exists b'tEK

such that f ( x ~ ) I ' l g ( x 0 ) : ~ 4 .

P(x, tt--x

208

Ding Xie-ping

Proof

For each xEK , define

m(x)=x+.f(x) -g(x), Suppose that for each xEK, f ( x ) {]g(x)=~

n(x)=x-.f(x) . Define the mappingS:

K ~ 2 J r a s follows

S(x) = { ~ E K : p ( x , y - - m ( x ) ) < P ( x , n ( x ) ) } From the proof of Theorem 2.1, we see that S(x)--/=q~ for all xEK , S(x) is convex for each xEKandS-l(~l)isopen for each vEK 9Condition (ii) implies that tdoES(x) for all x E K \ L 9 Thus, by Lemma 1.3, there exists xoEK such that xoES(xo) 9Then we have

p(xo, g(xo) --[(Xo) ) =p(xo, xo--ra(xD ) < p (x0, n(xo) ) = P (x0, x0 - - f (x,) which contradicts condition (i). Therefore there exists xoEK

such that f(xo) ~g(xo)=/=~

R e m a r k 2.2 As a special case of Theorem 2.2 with L = K, we obtain Theorem 2.4 of [3]. Futhermore, our proof is simpler. T h e o r e m 2.8 Let K be a nonempty para-compact convex subset of a H a u s d o r f f locally convex topological vector space E, L ~ / - ~ be a nonempty compact convex set, f , g: K--~ ~ ( E ) be u.s.c., oQc::2x be convex such that for all x,9"EK, x--f(x)El'~ and y - - x + g ( x ) - - f ( x ) E D . S'uppose that p: K x t ' 2 ~ R is ultimately continuous on f2 such that for e a c h x E K , 9 ( x , . ) is convex. If the following conditions hold

p(x,x--f(x)) =p(x,g(x) --f(x)) (VxEK) (ii) p(x,x-f(x))<~p(x,v--x+g(x) -f(x)) (Vx,yEK\L) (iii) for each xEK with f(x) ffl.q(x)-----~ , there exists yEIr(x) such t h a t P ( x , y - - x + g(x) --f(x) ) < p (x, x - - f (x)), then there exists xoEK such that f(xo) • O(xD =/=qb . P r o o f For each xEK with f(x) N g(x)-=q~ , by condition (iii), there exists y E I x ( x ) such that P ( x , y - - x + g ( x ) - - f ( x ) ) ~ p ( x , x - - f ( x ) ) . If b ' E K , from Theorem 2.1 it follows that there exists xoEK such that f(xo) n g(xo) 4=q~ . If t,' ~ K , by the definition ofljc(x), there exist uEK and f i E ( 0 , 1 ) such that u = ( 1 - - f l ) x + f l y . By the convexity ofp(x,.)' we have p(x,u--x+g(x)--f(x)))=p(x, (1--fl)x+fly--x+g(x)--f(x)) = p ( x , ( 1 - 8 ) (g(x) - f (x)) + # ' ( y - x + o(x) - f ( x ) ) ) <(1--t~)p(x,g(x) --f(x) ) +fl p ( x , v - - x + g ( x ) - - f ( x ) ). (i)

It follows from conditions (i) and (iii) that

p(x, u - x + g(x) - f (x) ) < p(x, x - / ( x ) ) . By Theorem 2.1, also there exists xoEK

such that f(xo) f~g(xo)--/=q~

C o r o l l a r y 2.2 Let K be a para-compact convex subset of a Hausdorff locally convex topological vector space E, L ~ K be a nonempty compact convex set. Suppose that f : K o .~-'(E) is u.s.c, and .(2~2 x is convex such that for all x, ~ E K , x--f(x)ES2, y - - f ( x ) E ~ Assume that p: K x .(-2oR is ultimately continuous on s such that for each x E K . p(x,.) is convex on D . If the following conditions hold:

Some Coincidence Theorems of Set-Valued Mappings (i)

p(x,x--f(x))<~p(x,y--f(x))

209

(V x,yEK\L),

(ii) for each xEK with xq~f(x), there existsyEIx(x) such that p ( x , y - - f ( x ) ) < p ( x , x-~C(x)), then there exists xoEK such that xoEf(x,) P r o o f From Theorem 2.3 with g(x)={x} VxEtq we see that this corollary holds. R e m a r k 9..3 Corollary 2.2 is the set-valued generalization of Corollary 3.5 in [2]. Obviously, Theorem 1 of [1] is also a special case of Corollary 2.2. T h e o r e m 9..4 Let K be a nonempty para-compact convex subset of a Hausdorff locally convex topological vector space E, L ~ K be a nonempty compact convex set. Suppose that f,g.. K ~ , ~ r ( E ) a r e u.s,c, and ~c::::2 B is convex such that for all AE~Q , AE,Q and for x,yEK, x - - f ( x ) E ~ , y--x+g(x)--f(x)E~Q . Assume that p. K'x~Q--~R is ultimately continuous on ,Q such that for each xEK ,p(x,.) is convex on ~ . If the following conditions hold :

p ( x , x - f ( x ) ) =p(x,g(x) - f ( x ) ) (VxEK). (ii) p ( x , x - - f (x) )<,~mia{p(x,y--x+ g ( x ) - - f (x) ), P ( x , x - - y + g ( x ) - - f ( x ) )} ( V x, y E K \ L ) , (iii) for each xEK with f(x) nO(x)=r , there exists yEOx(x) such that p ( x , y - - x +g(x) --f(x)) ~ p ( x , x - - f ( x ) ) . then there exists xoEK such that f(xo) f']g(xo)4=r . P r o o f Let h(x)=2x-/(x), r(x)=2x-9(x), VxEK . Then for each xEK 9 f ( x ) NO(x) 4=etch(x) Nr(x)-'/=r N o w f o r a l l ( x , A ) E K x ~ ,define q. K• asfollows: q ( x , A ) = p ( x , --A). Under the hypotheses of this theorem, it is easy to prove that h,r and q satisfy the conditions of Theorem 2.3. From Theorem 2.3 it follow that there exists x,EK such that h (xo) [Tr(xo) 4=r and so f(Xo) ~ g ( x 0 ) ~ r 9 (i)

C o r o l l a r y 2.8 Let K be a nonempty para-compact convex subset of a Hausdorff locally convex topological vector space E and L c K be a nonempty compact convex set. Suppose that f: K--*Eis continuous and p: K x E->R is continuous such that for each xEK , p(x,.) is convex on E. If the" following conditions are satisfied (i)

p(x,x_f(x))
(Vx,yEK\L)

xEK with xq=f(x) , there exists yEOx(x) such that p ( x , y - - f ( x ) )
(ii) for each

then f has fixed point in K. P r o o f Let f a n d g be simple-valued mappings and g ( x ) = x , ' q x E . K in Theorem 2.4, then this corollary holds from Theorem 2.4. C o r o l l a r y 9..4 Let K be a convex nonempty subset of a normed space E and L ~ K be a nonempty compact convex set. If f : K-->Eis continuous and satisfies (i)

IIx-/(x)ll<~min{lly-/(x)~, 12x-V--l(x)ll~

(Vx,uEK\L)

210

Ding Xie-ping (ii) for each x E K with .':.:#:f(x). then: c,~st, .v60,. { - ) such that

t ! y - - / ( x ) H--~-llx--f

(x} H. thcn f h a s a fixed point in K Proof Obviom, ly. K is para-compact I)cfinc p: K . t-~R by p ( x , y ) = ~ . q [ { (:c, .tl)EK x E - f h c n the ~.onclu~,ion t,t 'ht~ ~_'t~rollar', follm~, from Corollary 2.3. C o r o l l a r y 2.5

for all

Let K hc a nonempt.', ctm',ex subset of a normed space E and L c : : K

be a

ll~)llclnpty c o n l p ; i c t col1~,cx set. I f / : K -,-/'." {., conlllltlOkls ;And satisfies

;:.-TfE.,:),.
(il

(ill for erich .:-~]( lllcil

with

.....

f(x)

i',2x--!@ -- f (.,:) ~}-

(Vx, yEK\,L),

. f ( - ) E C)~,-C,:)

/ ]l:.i:, :.i lqxetl t'~oini 1il g.

~,,appt)~,c _"E[x ~lth x-~+f(x) . then

Proof

,lECky:f:,:) such that

;,ci: ..,,,,I,

R e m a r k 2.4

Theorem

I!x--/(x) ll>0

. Since

f(x)EOx(x)

I*.y--f(-v) ll<-- I~-'-:--f(x) lt 9 This corollary follows from

Corollaries 2.3,2.4 and 2.5 settle the open problem mentioned by K, and Tan in

2.5

Let K be a closed convex o f a H a u s d o r f f topological vector space E and

J . ~ l ( I~c a noncmpiy c~,mpact .set. Suppose that f , g :

1(-~. 7,"(F.'~ are u.s.c, and

~c_s.2 E

is

y,:sEK, x--.fg..:)E-C2 and ~l--x+g(x)--f(.,.)E_Q . Assume that t': t'(>. ~,.'-,N i~, ulliinatcl.~ continuer: ,,1, .Q such that ['or each ::EK .P(X,') is convex on

c o n \ c \ ,,ut_h that for all Q

II lhc I~ lh'~,,,., ,,,nditions h,,Id p'%

lii~ for all

.,1~)

--t ~ . v ) ) = p t ,., ,..r

xEK\L f(x))

) - -/(x))

with f~x) i ; f f ( x ) = 4 . .

(VxEE), ;here exi`sts

.~,'DEK such that p(x,.tlo--x

p(-,',---f(x)),

:'."; for each xEL with f ( x ) D..q(x)=q~ . there exists .tlESx(.'O such that -> ,.2(x} - - f ( x ) ) < p ( v , x - - f ( x ) ) , il,cn thcrccxist`s x0EK such that ffx0") D g ( x 0 ) 4:~b . Proof

For each xEL

with

f ( x ) D g f x ) = ~ b , by condition (iii), there exists y E l x ( x )

~uchthat p ( x , y - - x + g ( x ) - - . f ( x ) ) < , p(x,,,!--f(x)) c\;sts xoE.K such that f(xo) Ng(.-,.~4:dp . If u ~ K uEI'(and

fiE(0,1)

p(x,!j~x

. I f y E K , then by T h e o r e m 2.2 there . by the definition o f IK(x), there exist

such that u = ( 1 - - f l ) x - ~ - f l y . By the convexity ofp(x,.), we have

p ( x , t~- x ~ 9 ( x ) - f

( x ) ) = t,' ; , ', l - q) (.q ( x ) - [ ( x ) ) + / 5 ' ( , 7 - - x + g ( x ) - - f ( x ) ) )

-~ ( ' --fl) p(x,g(.v) - - f (x) ) + / q p ( x , y - - x + g(x ) - f (x) ) ~ p ( x , x _ f By Thc~)rena 2.2. there also exists yt,E K

(x) )

such that f(xo) I'],q(x0)4=q~ 9

R e m a r k 2.5 As a special case of Theorem 2.5 with K = L, we obtain corollary 2.5 o f [3]. Clearly, Theorem 2.5 also generalizes theorem 1 o f [t]. " I ' h e o r e m 2.6

If c,,nd:~,m ~;~ <,illac,.,r~:ul 2.5 is replaced by the following condition

Some Coincidence Theorems of ~ t - V a l u e d Mappings (iii)" for each x E L

with f ( x ) f ' l g ( x ) = ~ . ,

there exists yEOx(x)

211

such that

p(x,y--x

+ g(x) - f ( x ) ) < p ( x , x-,t" (x) ) , then there also exists x o E K such that f ( x , ) Ng(xo)4:r . P r o o f Using Theorem 2.6 and the methdd o f proving theorem 2.4 it is easy to prove'this theorem. Theorem 2.6 improves and generalizes Corollary 2.6 of [3] and Theorem 2 of

P u a m a r k g.6

Ill. From Theorem 2.5 and 2.6 we easily obtain the following Corollaries. C o r o l l a r y 2.6 Let K be a nonempty closed convex subset of a normed space E and is a nonempty compact set.. If f: K.-*E is continuous and satisfies (i) forall

xEK\L

with x ~ f ( x )

,thereexists

yoEK s u c h t h a t

L~K

"'liyo--f(x) l l ~ x - - f

(x) H, (ii) for each xEL with xq~f(x),thereexists

yEIr(x)

suchthat

Ht/--f(x)]'~[[x--[

(x) X 9 then f has a fixed point in K. P r o o f Let f a n d g be both simple-valued mappings and,q(x) = x for all x E K in Theorm 2.5. Suppose p ( x , y ) = ~//[[ V ( x , y ) E K x E . Then from Theorem 2.5 this corollary holds. C o r o l l a r y 2.7 Let K be a nonempty closed corrvex subset of a normed space E and L ~ K be a nonempty compact set. I f / ' : K ~ E is continuous and satisfies (i) for all x E K \ L with x=/=f(x) , there exists y , E K such that[[yo--f(x) ~
Let K be a nonempty closed convex subset o f a normed space E a n d

be a compact set. If f , K ~ E is continuous and satisfies (i) for all x E K \ L with x ~ f ( x ) . there exists g 0 E K

such that

Ilv0-f(x) ll
f ( x ) fl, (ii) for each

xEL

with x ~ f ( x )

, there exists !/lfiOt(x) such that ~y--f (x) ll< llx--

/ ( x ) II, then f has a fixed point in K. P r o o f Using Theorem 2.6 and the method of proving Corollary 2.6 it is easy to prove that this corollary holds. C o r o l l a r y 2.9

Let K be a nonempty closed convex subset of a normed space E and

L~K"

be a nonempty compact set. If f : K-~E is continuous and satisfies (i) for all x E K \ L

with x ~ f ( x )

, thereexists yoEK such that IIv,--f(x) ~< I x - - f (x) II, with x::/:f(x) , f ( x ) E O x ( x )

(ii) for each xEL then f has a fixed point in K. P r o o f The p r o o f is the same as that in Corollary 2.7. R e m a r k 2,7

Corollaries 2.6,2.7,.2.8 and 2.9 all generalize the Schauder fixed point theorem

in different directions.

212

Ding Xie-ping

References

[ 1 ] Browder, F,E.. On a sharpened form of the Schauder fixed point theorem,Proc. Nat. Acad. Sci. USA, 74 (1~77), 4749-4751. [ 2 ] Ko, H.M. and K,K Tan., A coincidence theorem with applications to minimax inequalities and fixed point tneorems, Tamkang J. Math.. 17 (198b), 37-45. [ 3 ] Tan, K.K., Generalizations of F.E. Browder's sharpened form of the Schauder fixed point theorem, J.Austral. Math. Soc., 49,, (1987). [4] Browder, F.E., Coincidence theorems, minimax theorems, and variational inequalities, Contemp..Math.. 26 ( 1984)~ 67 - 80.