数理解析研究所講究録
第 1544 巻 2007 年 144-152

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SOME FIXED POINT THEOREMS FORCONTRACTIVE TYPE MULTI-VALUED MAPPINGSJEONG SHEOK UME
Department of Applied Mathematics, Changwon National University
Changwon 641-773, Korea
E-mail: jsume@changwon.ac.kr
ABSTRACT. In this paper, using more general mapping than Hausdorff metric we obtain
fixed points for a multi-valued $mapp\dot{u}lg$.

1. INTRODUCTION

Fixed point theory has important applications in diverse disciplines of mathematics,
statistics, engineering and economics in dealing with problems arising in approximation
theory, potential theory, game theory, mathematical economics, etc. Many authors [113] have proved some fixed point theorems for various generahzations of contraction
mapping in metric spaces. Extensions of the Banach contraction mapping principle
to multi-valued mapping were initiated independently by Markin [9] and Nadler [10].
Further results on fixed points of contraction type multi-valued mappings were given by\v{C}iri6 [3], Dube and Singh [5], Kubiaczyk [7], Kubiak [8], Ray [11] and others.
In 1995, Chang et al. [2] proved the following theorem: Let (X, ) be a complete
metric space and let $T$ : $Xarrow CB(X)$ be a multi-valued mapping. If there exists$k\in(O, 1)$ such that for all $x,y\in X$$d$$H(Tx,Ty)\leq kd(x,y)+k|d(x,Tx)-d(y,Ty)|$ ,then has a fixed point in $X$ , where $CB(X)$ is the collection of all nonempty boundedclosed subsets of $X$ .
In 1996, Kada-Suzuki-Tahhashi [6] introduced the concept of w-distance and, byusing this concept, proved a nonconvex minimization theorem and some fixed point
theorems in complete metric spaces.
In this paper, usIng more general mapping than Hausdorff metric we obtai fixed
points for a multi-valued mapping.
$T$1991 Mathematics Subject Classification. $47H10$ .
Key words and phrases. Fixed point, w-distance, multi-valued mapping, inwardness set.
Typeset by $A_{\lambda\#Iffl}$

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2. PRELIMINARIESDefinition 2.1 [6]. Let (X, ) be a metric space. Then a function $p:X\cross Xarrow[0, \infty$ )is called a w-distance on $X$ if the following are satisfied:(1) $p(x, z)\leq p(x,y)+p(y, z)$ for all$z\in X$ ;(2) for any $x\in X,$ $p(x, \cdot):Xarrow[0, \infty)$ is lower semicontinuous;(3) for any, there exists $\delta>0$ such that $p(z,x)\leq\delta$ and $p(z,y)\leq\delta$ imply$d$$x,$ $y,$$\epsilon>0$$d(x, y)\leq\epsilon$.

Example 2.2 [6]. Let (X, ) be a metric space. Then clearly
$d$

$d$

is a w-distance on $X$

Example 2.3. Let (X, ) be a metric space with a continuous w-distance and a continuous w-distance . Then $q:X\cross Xarrow[0, \infty$ ) defined by $q(x,y)= \max[p(x,y),r(y,x)]$
for all $x,y\in X$ is a w-distance on $X$ .
$d$

$p$

$r$

The following is an easy consequence ffom the definition of w-distance$p$.

Lemma 2.4. Let (X, ) be a metric space Utth a w-distance . Letandbe sequences in $X$ , letandbe sequences in) converging to , and let$z\in X$ .
$n\in N_{f}$ thenanyandconverges to .
If
for
$d$

$\{x_{n}\}$$p$$\{\alpha_{n}\}$$p(x_{n},y_{n})\leq\alpha_{n}$$\{\beta_{n}\}$$[0, \infty$$\{y_{\mathfrak{n}}\}$$\theta$$p(x_{n}, z)\leq\beta_{n}$$\{y_{n}\}$$z$

Now we recall the following generalization of Caristi’s fixed point theorem in [6],which will be used in the proof of our results.

Lemma 2.5 [6]. Let (X, ) be a complete metric space with a w-distance , let $F$ :be a mapping and let: $Xarrow[0, \infty$ ) be a prvper lower semicontinuous
fimction, bounded from below such that$d$$Xarrow X$$p$$\varphi$$p(x,Fx)\leq\varphi(x)-\varphi(Fx)$for all $x\in X.?$ ?

(2.1)

Then there nists $x_{0}\in X$ such that $Fx_{0}=x_{0}$ and $p(x_{0},x_{0})=0$ .

Deflnition 2.6. Let (X, ) be a metric space with a w-distance$d$(i) For any $x\in X$ and $A\subseteq X,$ $p(x, A)$ $:= \inf\{p(x,y) :$\inf\{p(y,x) :y\in A\}$

.

$p$

.

y\in A\}$and $p(A,x):=$(ii) $CB_{p}(X)=$ { $A|A$ :nonempty closed subset of $X$ and $\sup_{x,y\in A}p(x,y)<\infty$ }.
(iii) For $A,$ $B\in CB_{p}(X)$ ,$G(A, B)$ $:= mas\mathfrak{c}\{\sup_{u\in A}p(u, B),\sup_{u\in}p(B,u),\sup_{v\in}p(v, A),\sup_{v\in}p(A,v)\}$.

146) be a normed vector space, $D$ a nonempty subset ofDefinition 2.7 [2]. Let (X,$x\inD$, the set
X. For any given$\Vert\cdot\Vert$$I_{D}(x)=\{x+a(y-x) :y\in D, a\geq 0\}$is called the inwardness set of $D$ at .
For given $x\in D$ and $a\geq 0$ we denote$x$$I_{D,a}(x)=\{x+a(y-x) :y\in D\}$.

3. MAIN RESULTS
Theorem 3.1. Let (X, ) be a complete metric space with a continuous w-distanceand $T:Xarrow CB_{p}(X)$ be a multi-valued mapping such that
$d$

$p$

$G(Tx,Ty) \leq k\max[p(x,y),p(y,x)]$$+k| \max[p(x,Tx),p(Tx,x)]$(3.1)$- \max[p(y,Ty),p(Ty,y)]|$for all $x,y\in X$ and some $k\in(O, 1)$ ,(3.2)$\inf_{u\in A}\{\max[p(x,u),p(u,x)]\}\leq\max[p(x,A),p(A,x)]$

for all $A\in CB_{p}(X)$ and each$x$

ト$x\in X_{f}$and$arrow\max[p(x,Tx),p(Tx,x)]$

for all $x\in X$ ,(3.3)

is lower semicontinuous.
Then $T$ has a fixed point in $X$ .

Prvof. If $\max[p(x,Tx),p(Tx,x)]=0$ for some $x\in X$ , then, by Lemma 2.4, has afixed point in . Next we may assume that $\max[p(x,Tx),p(Tx,x)]>0$ for all $x\in X$ .
$T$

$X$

Take $\alpha>1$ such that$\alpha\cdot k<1$

. By (3.2), for each $x\in X$ there exists$z_{x}\in Tx$$0< \max[p(x,z_{x}),p(z_{x},x)]<\alpha\cdot\max[p(x,Tx),p(Tx,x)]$

.

Since
[ (

), $p(Tz_{x},z_{x})$ ] $\leq G(Tx,Tz_{x})$$\max$ $p$ $z_{x},T$ ち$\leq k\cdot\max[p(x,z_{x}),p(z_{x},x)]$$+k| \max[p(x,Tx),p(Tx,x)]$$- \max[p(z_{x},Tz_{x}),p(Tz_{x},z_{x})]|$,

such that

147
we have$\max[p(z_{x},Tz_{x}),p(Tz_{x}, z_{x})]<\alpha\cdot k\cdot\max[p(x, Tx),p(Tx,x)]$$+k$

I $\max[p(x,Tx),p(Tx,x)]$$-mm[p(z_{x},Tz_{x}),p(Tz_{x}, z_{x})]|$

(3.4)

.

Suppose that$\max[p(x,Tx),p(Tx,x)]<\max[p(z_{x},Tz_{x}),p(Tz_{x},z_{x})].$.

Then, by (3.4), we obtain $1<\alpha\cdot k$ , which is a contradiction. Thus we have$\max[p(z_{x},Tz_{x}),p(Tz_{x}, z_{x})]\leq maxb(x,Tx),p(Tx, x)]$and$\max[p(x, z_{x}),p(z_{x},x)]<(1+k)(\frac{1}{\alpha}-k)^{-1}\{\max[p(x,Tx),p(Tx,x)]$$- \max[p(z_{x},Tz_{x}),p(Tz_{x}, z_{x})]\}$.

Define a imction $\varphi:Xarrow[0, \infty$ ) by$\varphi(x)=(1+k)(\frac{1}{\alpha}-k)^{-1}${m&sp(x, $Tx),p(Tx,x)]$ }.

for all $x\in X$ . Define a mapping $F:Xarrow X$ by $Fx=z_{x}$ for all $x\in X$ . Then, we have$\max[p(x,Fx),p(Fx,x)]\leq\varphi(x)-\varphi(Fx)$for all $x\in X$ . Since all conditions of Lemma 2.5 are satisfied, there exists $u\in X$ suchthat $u=Fu=z_{u}\in Tu$ . Therefore has a fixed point in X.
$T$$\square$Corollary 3.2 [2]. Let (X, ) be a complete metric space and let $T:Xarrow CB(X)$ be
a multi-valued mapping. If there exists $k\in(O, 1)$ such that for all $x,y\in X$$d$$H(Tx,Ty)\leq kd(x,y)+k|d(x, Tx)-d(y,Ty)|$ ,

then

$T$

has a fixed point in $X$ .

Proof Since the metric

Corollary 3.2$f_{0}n_{oWS}$

is a w-distance and all conditions of Theorem 3.1 are satisfied,from Theorem 3.1.
$d$$\square$

The following simple example shows that Theorem 3.1 is more general than Theorem
2 of Chang et al. [2].

148
Example. Let $X=[0,1]$ be the closed bounded interval with the usual metric and let$p:X\cross Xarrow[0, \infty)$ be a mapping defined by $p(x, y)=y$ for all $x,y\in X$ . Suppose that
$T:Xarrow CB_{p}(X)$ is a multi-valued mapping such that $Tx= \{\frac{k}{2}x\}$ for all $x\in X$ , whereis a fixed element of $(0,1)$ . Then all conditions of Theorem 3.1 are satisfied but notsatisfied all conditions of Corollary 3.2, since is a w-distance but not a metric.
$k$

$p$

be a metric on $X$ induced by thenorm
as $d(x,y)=||x-y\Vert$ with a continuous w-distance , and $D$ be a nonempty
closed subset of X. Assume that the w-distance satisfies(i) for any $xEX,$ $p(x,y)=p(x-y,O)=p(O,y-x)$ ,(ii) for any $x\in X$ and for any $\alpha>0,$ $p(\alpha x, \alpha y)=\alpha p(x,y)$ .
(iii) for each $s\in X$ ,Theorem 3.3. Let (X,$\Vert\cdot\Vert$) be a Banach space,

$d$

$p$

$\Vert\cdot\Vert$$p$$\inf_{v\in D}\{\max[p(s,v),p(v,s)]\}\leq\max[p(s,D),p(D, s.)]$.

Let : $Darrow CB_{p}(X)$ be a multi-valued mapping satishing (S.1), $(S.Z),$ $(S.S)$ and) such thatthe following condition: theoe exists a constant$T$$\delta\in[0, \frac{1-k}{1+k}$$\inf_{h\in(0,1]}\{\sup_{z\in Tx}\frac{1}{h}\max[p((1-h)x+hz,D),p(D, (1-h)x+hz)]\}$$\leq\delta\cdot\max[p(x,Tx),p(Tx,x)]$(3.5)

in $D$ .
Then $T$ has a fixed point in $X$ .

for each$x$

Proof. Assume that maxb$(x,Tx),p(Tx,x)$] $>0$ for $aUx\in D$ . Let$suchthatq<\frac{1-k}{1+k,q)}-\delta,$$\eta=qfor\bm{t}y\epsilon\in(0,andx\in D,$ $+ \delta andk<\alpha<\frac{1-\eta}{1+\eta,1}.Thenwethereexistsh\in(0,$obtainbe. By (3.5),$q,\eta,$ $\alpha\in(0,1)$$k< \frac{1-\eta}{1+\eta}$] $suchthat$$\sup_{z\in Tx}\{\max[p((1-h)x+hz,D),p(D, (1-h)x+hz)]\}$$<h \{\delta\cdot\max[p(x,Tx),p(Tx,x)]$$+(q- \epsilon)\cdot\max[p(x,Tx),p(Tx,x)]\}$$=h( \eta-\epsilon)\cdot\max[p(x,Tx),p(Tx,x)]$

.

(3.6)

By (3.2), choosing $z\in Tx$ such that$\max[p(x,z),p(z,x)]<(1+h\epsilon)\max[p(x,Tx),p(Tx,x)]$(3.7)

and for this , taking $y\in D$ in (3.6) from (iii), we have$z$maxb $((1-h)x+hz-y,0),p(O, (1-h)x+hz-y)$ ]$<h( \eta-\epsilon)\max[p(x,Tx),p(Tx,x)]$. (3.8)

149
Ftom (3.8) and $z\in Tx$ , we get $y\neq x$ . Letting $u=(1-h)x+hz$ , we obtain$\max[p(u,y),p(y,u)]<h\cdot\eta\cdot\max[p(x,Tx),p(Tx,x)]$$-h \cdot\epsilon\cdot\max[p(x,Tx),p(Tx,x)]$$\leq h\cdot\eta\cdot\max[p(x, z),p(z,x)]$$-h \cdot\epsilon\cdot\max[p(x,Tx),p(Tx,x)]$$= \eta\cdot\max[p(u,x),p(x,u)]$$-h \cdot\epsilon\cdot\max[p(x,Tx),p(Tx,x)]$

.

(3.9)

Thus we have$\max[p(x,y),p(y,x)]\leq\max[p(x,u),p(u,x)]$$+maxb(u,y),p(y,u)]$$\leq(1+\eta)\max[p(x,u),p(u,x)]$.

(3.10)Rom (3.1) and $k<\alpha$ , we get$l= \alpha\cdot\max[p(x,y),p(y,x)]+k|\max[p(x, Tx),p(Tx,x)]$$- \max[p(y,Ty),p(Ty,y)]|-G(Tx,Ty)>0$.

By (3.2), there exists $b\in Ty$ such that$\max[p(z,b),p(b,z)]<G(Tx,Ty)+l$.

(3.11)

Thus we have$maac[p(y,Ty),p(Ty,y)]\leq\max[p(y,b),p(b,y)]$$\leq ma\partial c[p(y,u),p(u,y)]$$+mascb(u, z),p(z,u)]$$+ \max[p(z,b),p(b,z)]$.

(3.12)

Using (3.7), (3.9), (3.11) and (3.12), we obtain$\max[p(y,Ty),p(Ty,y)]<(\eta-1)\cdot\max[p(x,u),p(u,x)]$$+ \max[p(x,Tx),p(Tx,x)]$$+\alpha\cdot maxb(x,y),p(y,x)]$$+k|m\alpha[p(x,Tx),p(Tx,x)]$$- \max[p(y,Ty),p(Ty,y)]|$.

150
Rom (3.10) and $\eta<1$ , we get

maxb $(y,Ty),p(Ty,y)$ ]$<( \alpha+\frac{\eta-1}{\eta+1})\max[p(x,y),p(y, x)]$$+ \max[p(x,Tx),p(Tx,x)]$$+k| \max[p(x,Tx),p(Tx,x)]$$- \max[p(y,Ty),p(Ty,y)]|$Suppose that$\max[p(x,Tx),p(Tx,x)]\leq\max[p(y,Ty),p(Ty,y)]$.

.

Then we have$\max[p(y,.Ty),p(Ty,y)]<\frac{1}{1+k}(\alpha+\frac{\eta-1}{\eta+1})\max[p(x,y),p(y,x)]$$+ \max[p(x,Tx),p(Tx,x)]$$< \max[p(x,Tx),p(Tx,x)]$.

This is a contradiction. Thus we get$\max[p(y,Ty),p(Ty,y)]<\max[p(x,Tx),p(Tx,x)]$and$\frac{1}{1+k}(\frac{1-\eta}{\eta+1}-\alpha)\cdot maxb(x,y),p(y,x)]$$\leq\max[p(x,Tx),p(Tx,x)]-\max[p(y,Ty),p(Ty,y)]$.

(3.13)On the other hand, from (3.7), (3.8) and (3.13), there exists a function(3.14)$F:Darrow D$such that for any $x\in D,$ $Fx:=y,$ $y\neq x$ and$\max[p((x,Fx),p(Fx,x)]<\varphi(x)-\varphi(Fx)$

,

where
$\varphi(x)=(1+k)(\frac{1-\eta}{1+\eta}-\alpha)^{-1}\max[p(x,Tx),p(Tx,x)]$.

Thus by Lemma 2.5, there exists $v\in D$ such that $Fv=v$ . This is a contradiction. This
completes the proof.
$\square$Using Theorem 3.3, we have the following corollary

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Corollary 3.4 [2]. Let (X,) be a Banach space, be a metric induced by the norm
$D$
$d(x,
y)=\Vert
x-y\Vert,$asbe a nonempty closed subset of $X$ and $T:Darrow CB(X)$ be amulti-valued mapping satisfying the following conditions:(i) there
a constant $k\in(O, 1)$ such that for any $x,$ $y\in D$$d$$||\cdot\Vert$$w\dot{w}ts$$H(Tx,Ty)\leq k\Vert x-y\Vert+k|d(x,Tx)-d(y,Ty)|$ ;(ii) there exists a constant$\delta\in[0, \frac{1-k}{1+k}$(3.15)) such that$\inf_{k\in(0,1]}\sup_{z\in Tx}\frac{1}{k}d((1-k)x+kz),$ $D$) $\leq\delta\cdot d(x,Tx)$ .

Then $T$ has a fixed point in $X$ .

Theorem 3.5. Let (X,) be a nomed space with a continuous w-distance connecting with a metric induced by the normas $d(x,y)=\Vert x-y\Vert,$ $D$ be a convex
subset of $X,$ $x\in D$ and $A\in CB_{p}(X)$ . Then$\Vert\cdot||$

$d$

$p$

$\Vert\cdot\Vert$$\inf_{h\in(01]},\sup_{z\in A}\frac{1}{h}\{\max[p((1-h)x+hz,D),p(D, (1-h)x+hz)]\}$$= \inf_{a\geq 0}\sup_{z\in A}\{\max[p(z,I_{D,a}(x)),p(I_{D,a}(x),z)]\}$,(i)$\inf_{a\geq 0}\sup_{z\in A}\{\max[p(z, I_{D,a}(x)),p(I_{D,a}(x),z)]\}$$\geq\sup_{z\in A}\{\max[p(z,I_{D}(x)),p(I_{D}(x), z)]\}$

.

(ii)

Prvof. Since$\inf_{h\in(0.1]}\sup_{z\in A}\frac{1}{h}\{\max[p((1-h)x+hz, D),p(D, (1-h)x+hz)]\}$$= \inf_{a\geq 1}\sup_{z\in A}\{m\mathfrak{W}[p(z, I_{D,a}(x)),p(I_{D,a}(x),z)]\}$

and$\max[p(z,I_{D,1}(x)),p(I_{D,1}(x),z)]\leq\max[p(z,I_{D,a}(x)),p(I_{D,a}(x),z)]$,for all $a\in R$ with $0\leq a<1$ and for all $x\in D,$ $z\in.A$ , we obtain (i). By elementary
calculus, we obtain (ii).
$\square$Using Theorem 3.3 and Theorem 3.5 we have the following Theorem.

152) be a Banach space with a continuous w-distance conTheorem 3.6. Let (X,as $d(x,y)=\Vert x-y\Vert,$ $D$ be a nonempty
necting with a metric induced by the norm
closed convex subset of $X$ satisfying (iii) in Theorem 3. 3 and : $Darrow CB_{p}(X)$ be amulti-valued mapping satisfying (3.1), (3.2), (3.3) and the following condition:$\Vert\cdot\Vert$

$p$

$d$

$||\cdot\Vert$$T$

there exists a constant$\delta\in[0, \frac{1-k}{1+k}$) such that$\inf_{a\geq 0}\sup_{z\in Tx}\{\max[p(z, I_{D,a}(x)),p(I_{D,a}(x), z)]\}$

for all $x\in D$ .

$\leq\delta\cdot\max[p(x,Tx),p(Tx, x)]\}$

point in $D$ .
has a$m_{m}$ Theorem 3.6 we have thefollowin9 corvllary.

Then

$T$

$\ovalbox{\tt\small REJECT} ed$

Corollary 3.7 [2]. Let (X, . Il) be a Banach space, $D$ be a nonempty closed convec
subset of $X$ and $T$ : $Darrow CB(X)$ be a mapping satishing (S.15) and the folloutng
condition:$\Vert$there eaxists a constant$\delta\in[0, \frac{1-k}{1+k}$$\inf_{a\geq 0}\sup_{z\in Tx}d(z,I_{D,a}(x))\leq\delta d(x,Tx)$) such thatfor all $x\in D$ .

Then $T$ has a fixed point in $D$ .

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$p\dot{\alpha}\mathfrak{n}t$$A$$6\ddot{m}ds$$s$