About extension of upper semicontinuous multi-valued maps and applications

Ab out extension of upp er semicon tin uous m ulti-v alued maps and applications Y. Ask oura Octob er 29, 2018 Gretia, INRETS, 2 Av enue du G ´ en ´ eral Malleret-Jo inville, 94114 Arcueil, F ra nce email : ask our a @inrets.fr, Abstract W e form ulate a m ulti-v alued ve rsion of the Tietze-Urysohn extension theorem. Pr e- cisely , we pro v e that an y upp er semicon tin uous multi-v alued map with nonempt y closed con v ex v alues d efined on a closed su b set (resp. closed p erfectly n ormal subset) of a com- pletely normal (resp. of a n ormal) space X in to the u nit interv al [0 , 1] can b e extended to the whole space X . Th e extension is upp er semicon tin uous with n onempt y closed con v ex v alues. W e apply this result for the extension of real semiconti nuous functions, the c haracterizatio n of completely norm al spaces, the existence of Gale-Mas-Col ell and Shafer-Sonnen s c hein t yp e fixed p oint theorems and the existence of equilibrium for qualitativ e games. Mathematics Sub ject Classification: 54C60, 54C20 Keyw ords: Extension of m ulti-v alued maps, Tietze-Urysohn extension theorem, Fixed p oin ts, Maximal elemen ts, Qualitativ e equilibrium 1 In tr o duction The aim of this pap er is to extend upp er semicon tin uous ( usc in short) m ulti-v alued maps o v er spaces as large as p ossible. In particular, w e w an t to av oid metrizabilit y in definition domains. W e are in terested by extending usc m ulti-v alued maps defined o n a closed subset of a giv en top ological space t o another. Similar results are already obtained b y Cellina [6 ], Bro dskii [4, 5], T an and W u [21] and Ma [15], using t he metrizabilit y of the domain. A more general result, in this direction, is that of Borges [2], whic h established extensions for usc maps defined on closed subsets of stratifiable spaces to any top ological space. C ite also results of Drozdo vskii and Filipp o v [8] and Shishk ov [20] whic h extended usc maps defined on closed subsets of paracompact completely normal spaces t o completely metrizable ones. Another t yp e of extensions, whic h is not concerned here, is to extend maps defined on dense subsets [11, 14]. 1 In the sequel, when s p eaking ab out extension o f maps, we signify extensions of the same t yp e (single v alued if the map is single v alued and m ulti-v alued if the map is m ulti-v alued) and preserving the giv en con tin uity concept (contin uit y if the original map is single v alued and con tin uous, up p er semicon tin uit y if t he or ig inal map is multi-v alued and upp er semi- con tin uo us). If we try to compare t he extension of usc m ult i- v a lued maps with contin uous single-v alued maps, t w o things app ear. First, for usc maps, the extension need only b e usc, then w e are tempted to sa y that the first exte nsion is easier. But, within the definition domains, for a map, to b e con tin uous and single-v alued is very constraining comparatively with the fact to b e usc and m ulti-v alued, then prov ides us additio na l prop erties . So the tw o problems are, a prio ri, quite differen t, and without eviden t comparison b et w een them. The results of this pap er (a nd some of the cited o nes ) prov e, in fact, that the extension of m ulti- v alued maps is more constraining. W e consider an usc m ulti-v alued map T : A ⊂ E → I , where A is a closed subset of a top ological space E and I the unit real in terv al. W e o bta in an extension of T when E is completely normal or E is norma l and A is p erfectly nor ma l. As it is kno wn, this type of results provides a c haracterization of completely nor ma l spaces. An exte nsion result (for m ulti-v alued maps) in an infinite uncoun table pro duct of spaces giv es directly some existence results of maximal elemen ts and fixed po ints ([10], [19] and [12]) nee ded in game theory . The problem is to a void in the pro of, prop erties which a re not satisfied in an uncountable pro ducts of usual (or a simply important class of ) spaces (lik e metrizabilit y). Unfortunately , the complete no r malit y is n ot, in general, a prop ert y of an infinite uncoun table pro duct of spaces. So, the results c hara cterizing the complete normalit y by extensions of usc m ulti-v alued maps, suggest us to searc h maximal elemen ts (resp. equilibrium for qualitativ e games) for an uncountable usc m ulti-v alued maps (resp. with a n uncoun ta ble set of play ers and usc preference corresp ondence s) with additional requiremen ts. Suc h applications are giv en in the second par t of this pap er. Our result can b e seen as a m ulti-v alued v ersion of the Tietze-Urysohn extension theorem. As it is know n, the original pro of of this fundamen tal result uses the uniform con v ergence of a sequence of functions. Ho wev er, some researc hers ([1 , 16, 17, 18, 22]) aske d for a p ossibilit y to pro v e this result without t he use of un iform con v ergence. The proo f of our ex tension theorem is inspired pa r t icularly by the pap er of Ossa [17], and prov es that his tec hnics are success ful for usc m ulti-v alued maps. It is direct and elemen tary . In this pap er, usc means upp er semicon t inuous, lsc means lo w er semicon tinuous, co ( A ) means the con v ex hull of A . If Y is a top ological space and X a subset of Y , then int Y ( X ) refer to the interior of X in Y and X is the adherence (or the closure) of X in Y . F or a multi-v a lued map T : E 1 → E 2 , w e denote D om ( T ) = { x ∈ E 1 , T ( x ) 6 = ∅} . In the whole of this do cumen t, the subsets are endo wed with the induced top ology . 2 Multi-v alued v ers i on of the Tietze-U r ys ohn exten- sion theo rem Recall t w o definitions : 2 A separated top ological space X is said to b e completely normal if it is hereditarily normal, that is : ev ery subspace of X is normal. This definition is equiv alen t to the f o llo wing : X is completely normal if and o nly if all subsets A and B of X satisfying A ∩ B = A ∩ B = ∅ can b e separated by o pen sets, i.e. there exists t w o op en subsets of X, U and V , A ⊂ U, B ⊂ V suc h that U ∩ V = ∅ . A separated top ological space X is said to b e p erfectly normal if each closed subset of X is a G δ -set, i.e. interse ction of a countable op en sets. Or equiv alen tly , X is p erfectly normal if eac h op en subset of X is an F σ -set, i.e. a coun table union o f closed sets. The p erfect normalit y imply complete normality and the conv erse is false. See [9] for more details ab out these notio ns . The follow ing theorem is the main result in this w ork. Theorem 1. L et X b e a sep ar ate d top olo gic al sp ac e, A a close d subset of X and T : A → [0 , 1] , a usc multi-value d map with close d c onvex va lues. Supp ose that one of the two c o ndi- tions holds : C 1) X is c om pletely normal, C 2) X is normal and A is p erfe ctly normal, Then, ther e exi s ts a usc e x tension of T with close d c onv e x values d efine d on X i n to [0 , 1 ] . i.e. ∃ e T : X → [0 , 1] usc w i th clos e d c on vex va lues such that e T | A ≡ T . Pr o of. Let B 0 = { 0 , 1 } , ..., B n = { i/ 2 n , i ∈ { 0 , ..., 2 n }} . D efin e the set B = ∪ n ∈ I N B n the set of all dy adic n um b ers of [0 , 1]. W e ha v e, B n +1 = B n ∪ { ( r i + r i +1 ) / 2 , i ∈ { 0 , ..., 2 n − 1 }} , r i = i/ 2 n ∈ B n } . It is well kno wn that B is dense in [0 , 1]. Let for ev ery r ∈ B , A r = T − 1 ([0 , r ]) = { x ∈ A, ∃ y ∈ T ( x ) , y ≤ r } . Since T is usc on A , f or all r ∈ [0 , 1], A r is closed. After this, w e construct closed subsets X r , r ∈ B , o f X satisfying the follo wing three condi- tions : 1) X r ∩ A = A r , 2) int A ( A r ) ⊂ int X ( X r ) , 3) X r ⊂ X s, if s, r ∈ B and r < s. Put X 1 = X . In the followin g step, the condition C 1) or C 2) of the theorem is needed. W e illustrate the use of eac h of them. Begin by the condition C 1) : the space X is completely normal. W e ha v e, int A ( A 0 ) ∩ ∁ A A 0 = int A ( A 0 ) ∩ ∁ A A 0 = ∅ . Then, w e can separate int A ( A 0 ) and ∁ A A 0 b y op en sets (in X ) O 0 and O c resp ec tiv ely . This giv es O 0 ∩ A ⊂ ∁ A ( O c ∩ A ) ⊂ A 0 . W e conclude the t w o relations O 0 ∩ A = int A ( A 0 ) and O 0 ∩ A ⊂ A 0 , put X 0 = A 0 ∪ O 0 . T hen, the sets X r , r ∈ B 0 , satisfying 1) − 3) are defined. F or the s ame result, let us use the condition C 2) : X is normal and A is perfectly nor- mal. In this case, int A ( A 0 ) and ∁ A A 0 are F σ -sets, then, it can b e written us a coun t- able unions of closed sets, let int A ( A 0 ) = ∪ i ∈ I N F i and ∁ A A 0 = ∪ i ∈ I N G i . F urthermore, w e ha v e int A ( A 0 ) ∩ ∁ A A 0 = int A ( A 0 ) ∩ ∁ A A 0 = ∅ . W e apply a Bona n’s lemma [1] to separate int A ( A 0 ) 3 and ∁ A A 0 b y op en sets, and w e define X 0 b y the same w ay . Let the sets X r , r ∈ ∪ k ≤ n B k satisfying 1) − 3) b e giv en. Then w e obtain the sets X r , r ∈ B n +1 lik e this : Let r ∈ B n +1 \ B n and i ∈ { 0 , ..., 2 n − 1 } such tha t r = ( r i + r i +1 ) / 2 , with r i = i/ 2 n , r i +1 = ( i + 1) / 2 n are elemen ts of B n . W e pro ceed as previously by the use o f condition C 1) or C 2), when constructing X 0 , with the set A r in the place of A 0 . W e obtain an op en set O ′ r of X suc h that O ′ r ∩ A ⊂ A r and O ′ r ∩ A = int A ( A r ) . Sinc e int A ( A r ) ⊂ int A ( A r i +1 ) ⊂ int X ( X r i +1 ) , the set O r = O ′ r ∩ int X ( X r i +1 ) is op en in X and the t w o relations O r ∩ A ⊂ A r and O r ∩ A = int A ( A r ) are obta ined. Put X r = A r ∪ O r ∪ X r i . W e easily v erify that conditions 1) − 3) are satisfied (the v erification is down for the sets X r i , X r and X r i +1 ). A t this moment, the recursiv e pro cess for the con- struction of all the sets X r , r ∈ B , satisfying conditions 1)-3) is giv en. Thereafter, w e define the map F : X → [0 , 1] , as follow s : F ( x ) =  T ( x ) if x ∈ A, inf { r , x ∈ X r } otherwise. V erify that F , as a m ap defined on X is usc at e ac h p oint of A ( i.e. with res p ect to the top ology of X ). Let x 0 ∈ A and a, b ∈ [0 , 1] suc h that T ( x 0 ) = [ t 1 , t 2 ] ⊂ ] a, b [. The other cases are analogous and explained next. Note also that the conv exit y of the v alues of T is considered here. Let r 1 , r 2 ∈ B suc h that t 2 < r 1 < b and a < r 2 < t 1 . In one hand, x 0 ∈ int A ( A r 1 ) ( b ecause T is usc on A and T ( x 0 ) ⊂ [0 , r 1 [) a nd since int A ( A r 1 ) ⊂ int X ( X r 1 ), there exists an op en neigh b orho od O 1 of x 0 (in X ) suc h that ∀ x ∈ O 1 \ A, F ( x ) ≤ r 1 . In other hand, x 0 / ∈ A r 2 . Since x 0 is an elemen t of A , x 0 / ∈ X r 2 . Let O 2 b e an op en neigh b orho o d of x 0 in X such that O ∩ X r 2 = ∅ . W e hav e, ∀ x ∈ O 2 \ A , F ( x ) ≥ r 2 . W e take in the last time an op en set O 3 (in X ) suc h that ∀ x ∈ O 3 ∩ A, T ( x ) ⊂ ] a, b [ and define O = O 1 ∩ O 2 ∩ O 3 . W e o bt a in ∀ x ∈ O , F ( x ) ⊂ ] a, b [ , which giv es the fact that F is usc on A . The case of t 2 = b = 1 and T ( x 0 ) ⊂ ] a, b ] (resp. t 1 = a = 0 and T ( x 0 ) ⊂ [ a, b [) is a simple particular case where it suffices to consider only O 2 (resp. O 1 ). In the other case, w e put O = X . Denote b y H the gr a ph of F in X × [0 , 1]. The desired map is e T giv en as follows : e T ( x ) = co { y , ( x, y ) ∈ H } . This map is usc, b ecause its graph is closed and the image space ([0 , 1]) is compact. W e end by applying the upp er semicon tin uity of F on A to prov e that its v alues are not affected on A when passing throw the closure of its graph. Let x 0 ∈ A. W e can separate F ( x 0 ) = T ( x 0 ) and an y p oin t y / ∈ T ( x 0 ) of [0 , 1] by op en sets V 0 , V y resp ec tiv ely . Then, there exists an op en neighborho o d O x 0 of x 0 suc h that F ( O x 0 ) ⊂ V 0 . W e ha v e obtained, ( x 0 , y ) ∈ O x 0 × V y , F ( O x 0 ) ⊂ V 0 and V y ∩ V 0 = ∅ , whic h means that O x 0 × V y ∩ H = ∅ . That is ( x 0 , y ) is no t a p oin t of t he adherence of H . W e can finally affirm that : e T | A ≡ F | A ≡ T Note that the previous theorem, stated only with condition C 1), is pro ve d differen tly b y Shishk ov [20]. 4 3 Applicatio n s No w we giv e applications of Theorem 1 in differen t domains. In the first time, ana lo gously to the c haracterization b y Tie tze-Urysohn extension theorem of nor ma l spaces , w e giv e a c ha racterization of completely normal spaces (after Gutev [13] and Shishk ov [20], using Theorem 1). This prov es that the last theorem, stated only with condition C 1), can not b e impro v ed b y the relaxation of the complete normality imp osed to X . Corollary 1. L et X b e a sep ar ate d top olo gic al sp ac e. Then, X is c ompletely normal if and only if every usc multi-value d map T define d on a close d subse t of X into [0 , 1] with nonemp ty close d c onvex values has a multi-value d usc extensi o n defin e d on the whole of X into [0 , 1] with nonempty clo s e d c onvex values. Pr o of. The necess it y is a straightforw ard consequence of Theorem 1. The sufficiency is pro v ed easily as follows : Le t A a nd B b e subsets o f X suc h that A ∩ B = B ∩ A = ∅ . Define the m ulti-v alued map T : A ∪ B → [0 , 1] , b y : T ( x ) =    [0 , 1] if x ∈ A ∩ B , 0 if x ∈ A \ A ∩ B , 1 if x ∈ B \ A ∩ B . Let us v erify that T is usc on A ∪ B . Let x 0 ∈ A ∪ B and V an op en subset of [0 , 1] con taining T ( x 0 ) . If x 0 ∈ A ∩ B , then V = [0 , 1], we can c ho ose, in this case, O = A ∪ B as a neigh b orhoo d of x 0 suc h that T ( O ) ⊂ V . If x 0 ∈ A \ A ∩ B = ∁ A ∪ B B , choose O = ∁ A ∪ B B as a neigh b orho od of x 0 suc h that T ( O ) ⊂ V . The ot he r case is similar. Then, T is usc on A ∪ B . F rom the h yp othesis , there exists a usc extension e T of T defined o n the whole of X , with closed con v ex v alues. The sets e T − 1 (]1 / 2 , 1]) = { x ∈ X , e T ( x ) ⊂ ]1 / 2 , 1 ] } and e T − 1 ([0 , 1 / 2[) are op en a nd separate A and B . Then X is completely normal Theorem 1 can b e applied to extend real semicon t in uous functions. In what follo ws, w e use the same no tations usc and l s c fo r corresp o nding semicon tin uit y concepts for single-v a lued real functions. Corollary 2. L et X b e a sep a r ate d top olo gic al sp ac e and A a c lose d subset of X . Supp ose that one of the c onditions C 1) or C 2) is satisfie d. Given two functions f , g : A → [0 , 1] , such that f is lsc, g is usc and f ( x ) ≤ g ( x ) , for every x ∈ A . Then, ther e exis ts extensions e f and e g of f and g r esp e ctively, s uch that e f is lsc, e g is usc and e f ( x ) ≤ e g ( x ) , for every x ∈ X . Pr o of. Define the m ulti-v alued map T : A → [0 , 1] by T ( x ) = [ f ( x ) , g ( x )], for ev ery x ∈ A. Let us ve rify that T is usc on A. Let x 0 ∈ A a nd a, b ∈ [0 , 1] suc h tha t T ( x 0 ) ⊂ ] a, b [ . The other p ossibilities are part icu lar cases. W e hav e f ( x 0 ) > a and g ( x 0 ) < b. Then, there exists t w o neigh b orho o ds V 1 and V 2 of x 0 (in A ) suc h that f ( x ) > a, for eve ry x ∈ V 1 and g ( x ) < b , for ev ery x ∈ V 2 . Put V = V 1 ∩ V 1 . This giv es T ( V ) ⊂ ] a, b [ . If a = 0, b 6 = 1 and T ( x 0 ) ⊂ [ a, b [ (resp. b = 1, a 6 = 0 and T ( x 0 ) ⊂ ] a, b ]), w e do not need V 1 (resp. V 2 ). If a = 0, b = 1 and T ( x 0 ) ⊂ [ a, b ], w e put V = A. 5 W e infer, b y Theorem 1, an usc extension e T of T with nonempty con v ex c ompact v alues. Define e f and e g as f o llo ws : e f ( x ) = min { y , y ∈ T ( x ) } and e g ( x ) = max { y , y ∈ T ( x ) } , for ev ery x ∈ X . It remains to ve rify the desired prop erties of e f and e g . It is clear that e f | A = f , e g | A = g and e f ( x ) ≤ e g ( x ) , for ev ery x ∈ X . Let λ ∈ [0 , 1]. Then, { x ∈ X, e f ( x ) ≤ λ } = e T − 1 ([0 , λ ]) and { x ∈ X , e g ( x ) ≥ λ } = e T − 1 ([ λ, 1]). Since e T is usc, the last leve l sets are closed. This pro v es that e f is lsc o n X and e g is usc on X W e apply Theorem 1 to pro v e a v ersion of the Ga le-Mas-Colell’s [10], Shafer-Sonnensc hein’s [19] a nd Gourdel’s [12] fixed p oint theorems with an arbitrary n um b er (p ossibly uncoun t a ble) of m ulti- v a lued maps. Theorem 2. L et X α , α ∈ I , I is arbitr ary and X α = [0 , 1] . L et fo r every α in I , T α : Q λ ∈ I X λ → X α an usc multi-value d map with empty or none mpty close d c onvex val ues such that D om ( T α ) is p e rf e ctly norm al. Then, ∃ x ∈ Q λ ∈ I X λ such that : ∀ α ∈ I , either x α ∈ T α ( x ) or T α ( x ) = ∅ . Pr o of. W e apply Theorem 1 to find extensions e T α of T α , α ∈ I . Consider the map F : Q λ ∈ I X λ → Q λ ∈ I X λ , defined b y F ( x ) = Q λ ∈ I e T λ ( x ) . Any fixed p oin t of F (apply an y fixed p oin t theorem for Kakutani maps in lo cally conv ex s paces, for example Glicks b erg’s fixed p oin t t heorem) ensure t he result No w, w e giv e an application t o qualitative games with a n infinitely num b er of pla y ers. A qualitativ e game is a pair ( X i , P i ) i ∈ I , I is the set of play ers, X i is the set of strategies of the pla y er i ∈ I and P i is his preference corresp ondence. F or a literature in qualitativ e games, see [3, 21, 7]. Theorem 3. L et G = ( X i , P i ) i ∈ I b e a qualitat ive game. F or every i ∈ I , X i = [0 , 1] , P i : X = Q j ∈ I X j → X i is usc w i th empty or nonempty close d c on vex values such that D om ( P i ) is p erfe ctly normal and I is an arbitr ary set of indi c es. If ∀ i ∈ I , ∀ x ∈ X , x i / ∈ P i ( x ) , then G has an e quilibrium, that is : ∃ y ∈ X , such that ∀ i ∈ I , P i ( y ) = ∅ . Pr o of. Is a straightforw ard consequence o f Theorem 2 References [1] E. 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