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Mathematische Operationsforschung und Statistik.Series Optimization: A Journal of MathematicalProgramming and Operations ResearchPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/gopt19

Some applications of fixed point theorems formultivalued mappings on minimax problems intopological vector spacesOlga Hadžić a & Ljiljana Gaji a

a Department of Mathematics, University of Novi Sad, Faculty of Sciences, Dr Ilije Djuričića,Novi sad, 21000, YugoslaviaPublished online: 27 Jun 2007.

To cite this article: Olga Hadžić & Ljiljana Gaji (1984): Some applications of fixed point theorems for multivalued mappingson minimax problems in topological vector spaces, Mathematische Operationsforschung und Statistik. Series Optimization: AJournal of Mathematical Programming and Operations Research, 15:2, 193-201

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4Iath. Operationsforach. u. Statist., eer. optimization, 15 (1984) 2, 193-2001

Some Applications of Fixed Point Theorems for Mdtivaiued Mappings on Minimax Problems in Topological Vector Spaces

Summary: Using some f i xed point theorems for m.dtivalued mappings in topological vector space, which is not necessarily locally convex, in this paper some generalizations of Theorem 13 [I], Theorem 15 [I] and Theorem 4 [16] are proved.

' 0. Introduction

It is that so::is: rrlirsrr.,sx principles are related to the of nonerLp+y fntersection of sets with convex sections [i], [2j a d i22]. Some theorems on nonempty intersection of sets with convex section are proved in [I], l2] and [22]

x:--, r v A u t theorems far ri'i'dtivdued mappings. v y r l a ~ d -.?*'*

In [5] and [lo] we have obtained some new fixed point theorems for multi- valued mappings in not necessarily locally convex topological vector spaces and here we shall give some applications of this results to the problem of nonempty intersection of sets with convex section and on minimax problem. In this way we shall obtain some generalizations of BROWDRR'S and FAN'S results on nonempty intersection of sets with convex sections and minimax problem.

In the recent investigations' in the fixed point theory there is an increasing interest in the fixed point theory in not necessarily locally convex topological vector spaces. Some usefd results fa this diractian are ~b ta ined by S. XAXN in [Il l . Fixed point theorems for singlevalued mappings in topological vector spaces, which are not necessarily locally convex, are obtained in [4], [7], [8], [14], [l8], [I91 and 1201.

1. Some Theorems on Nonempty Intersection of a Family of Subsets in Topological Vector Spaces

First we shall introduce the definition of a set of Z ( Z ~ T A ' s type [23]) type in topological vector spaces and later in the text we shall give some examples of such sets. Let X be a HAUSDORFF topological vector space and ll be the family of neighbourhoods of zero in X.

1 University of Novi Sad, Faculty of Sciences, Department of Mathemetics, 21000 Wovi Sad, Dr Ilije DjuriEi6tl 4, Yugoslavia

1 3 optimization, Vol. 15, No. 2

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Definition 1 : Let K 2 X. The set K is of Z type if and only if for every VE U there exists UE U so that convex hull of the set Cni (K - K ) is in 7'.

Remark: If S is a locally convex topological vector space then every subset ZEX - is nf Z type since in this case we c m take U-.- V because n.e h2ve t!xt uun-cex hull of t7 is 7'.

Let us denote by 2" the family of all nonempty subsets of R and by R ( K ) the fam~iy of ail nonempty, ciosed and convex subsets of K.

Tn [ 5 ] have p r o ~ e d the follo~ing fjxec! point tlvorem fa- multiralued mappings in topological vector spaces.

Theorem A : Set X be a ; E h ~ v s n o ~ s ~ topological vector space, R 56 (3 ~bnaed an.d

convex su,bset of X , T : K - RjK) be a closed mapping, T ( K ) be compact and T j K ) be of % type. Then there exists x E ir' such that xE Tjxj.

Some appiications of this fixed point theorem are given in [ 5 ] . C'sing Theorem -4 we shall prove a generalization of Theorem 13 from [I].

" - - - - the ne:it text denote by co (LTSXj the coziv ex huli oi t h e set L7.

Theorem 1 : Let Ki!iE 1) be a nonempty, convex and compact subset o f HAUSDORFP topological vector space E,(i € I ) , K =n Ki and for every i E I , Ki =n Ki.

iiI i%=i Pu&ei., let foi. every i i i , Ei=sd be iii k7 aiiiid for eagy z i K iiizd eccyy i ~1 tXC jef Si(x) is a nonempty and convex subset of the set K where:

I f for every i i I the set Ki is of Z type then n Si+ W . i E l

Proof : The proof is similar to the proof of Theorem 13 from [I]. Let us define the mapping T : K - R ( K ) in the following way : y € T ( x ) (x E K ) o y = (yi) , where yiESi(x), for every iE1. This means that T(x) =u Si(x), for every xEK. We

iEI

shall prove that the mapping T and the set K satisfy all the conditions of Theorem A. ,

First, let us prove that T(x)E R ( K ) , for every X E K . Let nl : K,xK:-K, and 7d2 : K i ~ l i i - - ~ ; are the projection operators. As in [I] it follows that

Si(x) =nl (n,,'(gi) nSi) , for every x E K

and that Si(x) is compact, for every i € I and every xEK. Since T ( x ) =n Si(x), i E I

we conclude that T ( x ) is compact for every x E K . Since for every x E h' and every C E I the set Si(x) is convex it follows that T ( x ) is convex for every X E K and so T ( x ) E R ( K ) , for every X E K . Now, we shall prove that the set I< is of Z type. Let us denote by 23 the fundamental system of neighbourhoods of zero of E in the TIHOXOV product topology and by 'Bi the fundamental system of neighbour- hoods of zero in Ei, where E =n Ei. This means that for every V E 23 there

iEI

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Some -1pplications of Fixed Point Theorems for Multivalued Mappings 195

exists l i e 23 such that

Suppose that VE $3. Then there exists a finite set (i,, i), ... , in) I such that : V ==g I:. where 3: is defined in the folloririg way :

1 r z

where ViE ai ( i ~ {i,, i2, ..., in)) . Since Ki2Ei (2~1) and Ki is of Z type there exist U,c Bi such that

Now, let us prove that for the set U the relation ( 1 ) holds, where: ( 7 7 ' ,-TIC.' - I I lui, S t l \I", ..., ,CnJ

u = ~ E ; and E:'=\& iil i c { i l , i2 , ..., in} .

Suppose that z co ( U f7 ( K -K) j. This impiies that there exist r, (k= 1, 2, ... , m,)

and uk€ U n ( K - - K) ( k = l , 2, ...; m ) SO that ... ...

r ,nOjk=1,2 ,..., m j , r r k = i , z = C r k u k . ?.+ l i - l ,-i

Let us prove that zE V. We shall show that proj zE Vi, for every i€{i,, i z , ..., in). From (2) it follows that Ei

projz= Z r k p r o j u k , forevery i E I . Ei k - i Ei

Suppose now that i~ {ii, i,, ..., in}. Since proj u k € EI'n (Ki-Ki) = U i n (Kc-Ki) S Ei

& Vi i t follows that proj z E co (Ui fl (Ki - Ki) ) 5 Vi. As in [I] i t follows that Ei

the mapping T is closed. For the cornpletness of the proof we shall give the proof that Gr (T) is closed. Suppose that (x,.y)BGr ( T ) . From the definition of the mapping T it follows that there exists i E I so that [ y , Zi] 4Si. The set Si is compact and so there exists a neighbourhood Ni(yi) & IIi and a neighbourhood N2(Zi) &I<: so that ( N i X N 2 ) flSi= 0. Then for the sets N; = N i x K; and N; =KiXN2 we have that ( N ; x N ; ) n Gr ( T ) = 0 which means that the set Gr ( T ) is closed. So all the conditions of Theorem A are satisfied and there exists U E K such that uE T ( u ) . Frurn this it follows that Pui, dt] E Si, for every i i I which implies that U E n si+O.

i ' ~ I Using the princip of duality [ 2 1 ] , in [ l o ] we have proved the following fixed

point theorem.

Theorem B : Let K be a nonempty, compact subset of HAUSDORPF topological vector space E such that K is of Z type, T : II--2E be an uppersemicontinuous 13.

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mapping such that K s T ( R ) , T ( K ) =z T j R ) , for every X E K , T ( x ) be closed and for ewry x E T ( X ) i% T- i (x ) = T-i(x j . Then there exists x E K such that x E T ( x ) .

Simiiariy as in i 2 i j we can prove the foiioming theorem.

Theorem 2 : .Let ITi ( i c I ) he 0. n.onem.ptyj cnwpact and con?:e.r sub.set g,f !f type of HAGSDCI~FF iopoiiiy icai sectw spa';eiCi ji tr lj, R = n .Ki a d /or every ,i c 1, S, =Si X. For every x = (xi) E K let : iEI

and S,(rvi) = (i$, ei E K:, [xi, gi] E Sij , I<: = n Ki(i E I ) .

j+i

I f the conditions I., 2 . and 3 . are satisfied, where:

2. For every x C K , the set Si(x) is nonempty, for every i € I . 3. For every zE K and ever!/ i E I , the set Sijxi) is convex, then I! Si =I= 0.

iEI

P r o o f : As ill [2i] let us define tile mapping F : K - 2 " by:

Tjx) = A(x) , for every x E K . From 2. it follnws that T ( z ) + @, fnr every XCK E L E ~ as Theorem 1 it fg!!ows that T ( x ) is compact for every X E K . So the set T ( x ) is closed for every X E K . Further T- l ( y ) = n Si(yi) X Ki and Si(yi) =n2 (nri(y i ) nXi) where ni : K i x Iii+Ki, n2 :

iEI K,xK:+K:, i c I . From this i t follows that T- l (y ) = T- l (y ) , for every yE T IK) . Since, as in Theorem 1, i t follows that K is of Z type, K= T ( K ) and T is uppersemicontinuous from Theorem B i t foliows that there exists ztEK such that uE T ( u ) which implies that uE n S i + 0.

i € I Now, we shall give two corollaries of Theorem 1. Let E be a linear space over

the rea l er complex number field. The function j j I\*: E-10, ==) will be called paranorm if and only if: 1. /]XI]* = Oox = 0 . 2. / I -XI/* =ljxlj*, for every X E E . 3. Ijx+ yjl* sI~xII* + / I @ , for every x, y E E. 4. If jlxfl-xo/l* +O and r,-ro then jjr,xfl-roxojl*+O. Then ( E , 1 1 (I*) is a paranormed space. The paranormed space ( E , I / I/*) is a topologi- cal vector space if the fundamental system of neighbourhoods of zero in E is given by the family {VTjT,,, where V,.= {x , x E E , Ilxjj* 4 r ) .

In [23 ] Z N A has proved a generalization of SCHAUDER fixed point theorem in paranormed space for the mapping f : K - K , where K is a convex, closed and bounded subset of paranormed space E, f is completely continuous mapping and there exists C>O such that /ltxll* s Ct/]x/I*, for every t €[O, I] and every x E f ( K ) - f ( K ) .

I n [ 3 ] the following lemma is proved.

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Some Appliccttions of Fixed Point Theorems for Multivalued Mappings . 197

Lemma: Let ( E , / I I\*) be a paranormed space and K is a subset of E such that there exists C ( K ) > 0 SO that :

ijtxji* ~ tC(K) / l x l i* , for every t 6 LO, 11

nnd euery L C R -K. Then for every r > 0, co ( V , /I ( K - K ) ) & V,, whzch means that H is of Z type.

- c(m

Now, we shall glve an example of a pranoriiied space (X, /; ,I) and of % closed, convex subset K c E so that .

jltxll l C ( K ) tllxll, for every t E [0, 11 and every x E K -K . Let S(9, 1) be the spacc of finite mcasurablo functions (classes) or, the fnter\ra! [O, I ] mith the metric d(2, ij), 2 , $ E S (0 , 1 ) defined by :

- - - . .. is hi.,=wn that S(3, 1) is an a ~ m l s s i ~ ~ e mgtriz topoiogicai ~ e c t o r spare ~ , ~ d the convergence in X(6, 1) is the c o ~ ~ v q e ~ ~ c e in 1;ileasizi.e p,. The space S(C, 1) is complete and- nonlocally coilvex. Let 17.i';-0 and RM be t h e subset of E(O, 1) defined by :

K,= {z, , f ~ S ( ( j , i j , for an {x ( i j ) c . f , i , j f j l sM, "L'-', LJ] "' . The space S(0 , I ) is also a paranormed space with the paranorme

and we shall prove that for every f! $ c K , and every t € [ O , 11 we have: Ilt2115 s t (1 + 21M) 11211, e =2 -$.

Let {x( t )} € 2 ~ K, and { y ( t ) ) € g € K , so that :

j~(tjlS?kf, l t~ ( t ) / sf, for every ti [ G , I] . Then for z ( t ) = x( t ) - y(t) , t E [0, 11 we have :

This implies that :

and so

I t is easy to see that KM is a closed and convex subset of the space S(0, 1). Namely

if 2, -2 ( f n € K,) in the topology induced by the paranorme then & -f: 2 and so

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there exists Zn, so that (xn,(t))) converges almost everywhere to ( x ( t ) ] from

which i t follows easy that ~ E K , . Il'ow. we can formulate the following corollary.

Corollary I : Let be a family of nonempty, compact and con.zex subsets cf parancrned sFaces (E,, jl il;)f i 5 1 !X,SE, , for ewry i E I ) , K -n Kt crnd K:,

? E i

S , and S , (x ) ( x E K , i E 1 ) are us i n Theorem 1 . If for every i GI there exists C, =- 0 svch fhaf :

ijtziig zCitllzjj$. for every t 6 [ O , 11 and every XEK,:-& then n 5',+ 0. . icI

Let E be a vector space over R (real or complex number field), R, be the set of all mappings from d into R with the TIHOTOV p r ~ d u c t toPolbgy a d the opera- tions + and scalar multiplication as usual. I f f , gE R, we say that f s g if and only i f f(t) s g ( t ) , for every t E d . S y P, we shal! denote the cone of nonnegative elements ir: R,. I:: [I?] is intrnduccd the ::ntinr, of psraxormed space (we shs!! san Y G-para- normed space).

. . ." " - "

DefinitiGn 2 : The triplet (3, j ; Ij, @) is a $-paranmi-& spGcc ;; or;!jr of 1 1 1 1 : E-P,, @ is a linear, continuous and positive mapping from iR, into R, such that the fo^ollowing conditions are satisfieti: 1. ljxlj = 0-z = 0. 2. jjqj = pi \ \ x i ] , fGi eversT sEE and et-ery tE 9.

3. jjx+yl\ s@(ijxll) +@(ljyil), for every x , y €8. The topology in (E, 1 1 , @) is introduced in the following way. By U we shall denote the family of neighbourhoods of zero in R, and for every UEU we shall denote the set (z, xEE, /jx/[E U) by V,. Then E is a topological vector space in which {Trv)v,', is the fundamental family of neighbourhoods of zero in 3. Every HAUSDORFF topological vector space E is a @-paranormed space (E, I j 1 1 , @) over a topological semifield R, [17].

Definition 3 : [3] Let (E, 1 1 11, @) be a @-paranormed space over a topological semifield R, and I< Z E . If for every n E N , every zbiEI< - K (i E { I , 2 , ..., n ) ) and

1z

(si , s2, ..., s,) E [0, 1In si= 1 the inequality: a-1

is satisfied, we say that the set K is of @-type. I n [4] is proved the following lemma.

Lemma 2: Let ( E , jl 1 1 , @) be a @-paranormed space and K S E be a subset of @-type. Then the set K i s of Z type.

Corollary 2 : For every i E I let Ki be a nonempty, convex and compact subset of Ei where (E,, j l / I ,, @,) be a cP,-paranormed space, K =n K i and Xi, I<: and S i ( x )

iEI ( x E I<, i E I ) be as Theorem 1 . If for every i E I the set ICi i s of Qi-type then r) S i i: Q.

i € I

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Some Applications of Fisec! Point Theorems for Muitivalued SIappings

2 . Applications

Theorem 3 : Let topological ?;ector

indexed family cj

R,(i E I ) be a nonempty, compact and convex ~ u b s e t of HACSDORFF spctce Ei(i:cI) R=II X i and let { f i ) i c , be tc correspondi-ngiy

iEI co,ntinnzcozw real-aalued fz~nciions 0n K. Let K i = f l Xj $0 ifiai

j = F i

each point x of K can be written uniquely in the form [xi, 2i] with xi in Ki and Zi i n K:, i f for every i EI and r v w y t F R the set

is conwx mhcrt of K,, f c ~ every EX: and Ri is of Z type there is u point uEK . w ~ h that:

fi(u.) = max f i(yi, tii), i E I . V i € Ki

Si = ju, u EK, fi(uj z m a x fi(yi, dJ), for every i el' Y ~ E E ;

t hex we car, eonc!nde thzt ,rj fii .i. k! and from 21 E ,q A', it follow.: that the relation (3) is satisfied. i c I iEI

Definitio~ 4 [16]: A subset A of topobgical vector space E is & h o s t convex if for every neigbbourhood 'v' or' zero iri 3 &lid e v e v finite set {xi, x,, . . . ,qJ 2 A there exists a subset {z,, z,, ..., 2,)s A such that zi-xi€ V ( i ~ { l , 2, ..., n ) ) and co {zi, zz, ..., z , )SA .

In [ l o ] the following generalization of fixed point theorem from [I61 is proved.

Theorem C : Let K be a nonempty and compact subset of E, T : K -2", E be a HAUSDORFF topological vector space, T' be uppersemicontinuous mapping, T ( x ) be closed for every xE K and convex for every X E A where A is a dense almost convex subset of T ( I ! ) . I f the set T ( K ) is of Z type then there exists X E K such that x E T ( x ) .

Using Theorem C i t is easy to prove the following generalization of Theorem 1 : Let {Ei}iEl be a family of HAUSDORFF topological vector spaces, for every ; € I , K c be a compact subset of Ei and Ai be a dense, almost convex subset of K,. Let A; =n A j , K: =n I$, for every i E I , K =n Ki and {Si)ic, be the fami-

7 C i i+i i € I ly of closed subsets of K so that for every i E I, Kiis of Z type and for every i E I let :

Si(Pi) = {xi, xi E Ki , [xi, Zi] E Sd, ZiE Ki . If Xi(&) i. 0, for every i E I and every Pi€ K ; and X i ( Q is convex for every Zi€ A: then n S i i B .

iEI

The proof is similar to the proof of Theorem 3 from [16]. From the above generalization of Theorem 1, similarly as in [16] we can prove the following minimax theorem, which is a generalization of Theorem 4 from [16].

Theorem 4 : Let K,, K , be compact subset of Z type of the HAUSDORFF topologicaZ vector spaces L,, L,, let A,, A , be dense almost convex subsets of K1 , K2 respectively

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and let f be a continuous real-valued function o n K i X K a . I f for a n y X ~ E - ~ ~ . yoEd2 the sets (x / x €ITi, f(x, yo) =mau f (u , yo)) and { y / y ER?, f (xo, y) = min /(xo. v))

uEEi oEE1 are convex then:

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[6] H A D ~ I ~ , 0.: On multivalued mappings in paranormed spaces, Conirn. Math. Tjniv. Cerolinee, 28 (19Sl) 1, 123-136

[;] if-3EiC, c, : Fixed theGrSEs in r;ecesstzri!jr 130al!jT corii7ex tgpClrjgical yectr;r spaces. Lectures Notes in Mathematics, Conference on Functional Analysis, 945, Springer-Verlag, 1982, 11 8-130

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Some -4ppiications of Fixed Point Theorems for Multivalued Mappings "1

RBumen sowie Anwendungen auf ' F I A ~ ~ ~ ~ ~ R s T E I s S C ~ ~ Gleichungen. Dissertation. ,4achen, 1976.

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Received June 1982, revised March 1983

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