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International Centre for Theoretical Physics zrrl^^^'C^z ic/89/30i INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS ON THE APPROXIMATIVE NORMAL VALUES OF MULTIVALUES OPERATORS IN TOPOLOGICAL VECTOR SPACE Nguyen Minh Chuong and INTERNATIONAL ATOMIC ENERGY Khuat van Ninh AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION IC/89/301 SUNTO International Atomic Energy Agency and In questo articolo si considera il problemadell'approssimaztone di valori normali di oper- United Nations Educational Scientific and Cultural Organization atori chiusi lineari multivoci dallo spazio vettoriale topologico de Mackey, a valori in un E-spazio. INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS Si dimostrano l'esistenza di un valore normale e la convergenza dei valori approssimanti il valore normale. ON THE APPROXIMATIVE NORMAL VALUES OF MULTIVALUED OPERATORS IN TOPOLOGICAL VECTOR SPACE * Nguyen Minn Chuong * International Centre for Theoretical Physics, Trieste, Italy. and Khuat van Ninh Department of Numerical Analysis, Institute of Mathematics, P.O. Box 631, Bo Ho, 10 000 Hanoi, Vietnam. ABSTRACT In this paper the problem of approximation of normal values of multivalued linear closed operators from topological vector Mackey space into E-space is considered. Existence of normal value and covergence of approximative values to normal value are proved. MIRAMARE - TRIESTE September 1989 To be submitted for publication. Permanent address: Department of Numerical Analysis, Institute of Mathematics, P.O. Box 631, Bo Ho 10 000 Hanoi, Vietnam. 1. Introduction Let X , Y be closed Eubspaces of X,Y respectively, where X is normed In recent years problems on normal value and on approximation of space and Y is Hackey space. Let T be a linear multivalued operator from normal value of multivalued linear operators have been investigated Y to X . Assume that the operator T satisfies the following conditions by many authors ( see [l],[2],[}]) .In these works for these problems 1) Vy e D(T) : 3x e X : Ty = x + XQ the used spaces in studying are normed , metric or topological locally 2) <l"Jx = y + Y convex . The last ones were used <Mly in a Bide of studied equation. o 3) TY = X In this paper we shall solve these problems with vector topolog-ical Mackey space in the other side of considered equation. where T is the multivalued inverse operator of T in the sense that its graph coincides with the graph of T 2. Notations and definitions. Letting X = x/x , Y = Y/Y we define the following operator : Definition I ([?]) . A normed space X is called E-space if it is T a reflexive strictly convex space and satisfying the following con- for every (x,y) satisfying conditions l),2) we set dition : for every sequence (x ) weakly converging- to x and II* then (x ) converges to i in X where [y] = y + YQ , [x] - x + XQ Definition 2 ([4]) • A vector locally convex separated space X is Note that T is linear single-valued inversible operator from Y into called Mickey space if ev£Ty linear bounded single-valued operator from X into vector topological locally convex space Y is continuous. X and T is closed map if and onlji if TL is closed map ([i]). Definition 3 ([2]). A multivalued operator T is called a closed operator If Y is Mackey space then by properties I and 2 there exists a set of if its graph G(T) = |(f,u) : ft D(T) , ut-Tf } is a closed set. subspaces iY ,V<:V] ofY and closed subspace HQ of £__ Y^ cuch Let Y be a Hackey space , we state some needed properties of this space. that Y is isomorphism to )/M , where 5Z. T is the direct topol Property l([4])- A vector topolog-ical locally convex separated space gical sum of Y , v c V is Mackey space if and only if it is the induce limit of sequence of normed spaces. In addition Kackey complete and separated space is the Let k be a isomorphism from Y onto ( induce limit of Banach spaces. j is a inclusion map of Yv into Property 2 ([4]) If X is the induce limit of the sequence of vector t is a canonical map from 51Y onto ( •topological locally convex spaces /Y t vtV t by mappings u . Then Y is isomorphism to a quotient space of direct topological sum Let (EL Y )/H Kc— v '' 0 The proofs of these properties are given in [4] . -3- ^ 9 — Pri.oi' T is a linear multivalued operator from D(T ) into X where It is clear that T is a linear multivalued operator from D(T ) into B(TV) , -I X and k •''•j is a linear continuous operator from Y into Y . T s Definition 4. If at y £ D(T) t 7 i represented only in the for On the other hand I is a closed operator from D(T) into X . Thus we have D ZL T Tv = (TV)-—> X is a closed operator . Since 0 t Y 1 we have YQ (~\ Im(k .t.Jv) / 0 , Vy wht re Q c_ V is a finite subset of V , y £- D(T ) is defined Lot Yo - (k .t.j^)" (YQ) . Since Yo is a closed subspace of Y and by y then the sum t, I V the operators k ,t,j arc continuous , it follows that T is a closed o V~ O B x = tL - x , x inin subepace of Yv . For every yv t (TV) we have (k'^.t.j )(y ) <• O V V V £ Q X is liaid to bo normal value of T at y fc D(T) . 5ettinG (k" .t.j )y = y , by 2) we get Ty - x + X . Then We denote y - pr y on Y Vv = .t.jv)(yv) = x + XQ . By 2} T y + Therefore J. Existence of normal value. Let Y i'i' a complete separated Mackey space , X ~ae a Banach space. (y+YQ) We need the following Lemmas Leminal Let T : D(T)- > X be a linear closed multivalued operator satis- From -this follows the condition 2*), fying the conditions I)—3) mentioned in Section I. Then T is On the other hand from condition 3) a linear closed operator from D(T ) into X and satisfies the following v / —I — I —I -T conditions '. For every v there exists subspace Y of T such that _ rjry _ ^ 0 O I') Vyv e D(TV) , 3 x fc X : x + Xo This implies the condition 3")» Lemma I is proved. 2*) T^x D yv 3') T YV = X ' V O 0 -4- -S- Proof Lemrr.a Therl ihero bo the basic of neighborhoods of 0 in Y. Then | W^ I For tvery yQe B(T) C Y , k(yQ) e ( 21 I finite set of indexee Q C V such that ~ (k(Wd)) is the basic of neighborhoods of 0 in rfd = \ i ' - ^"d Proof Since Y is isomorphism to ( ^> Y )/K by mapping k , it follows k basic of neighborhoods of 0 in ( 2I_ ^ )/M . The vf Q •that (Wd) is the onto ( *^?~~ Y )/M therefore operator t is continuous from T(y0) v f Q t~ (k(W,)) is the basic of neighborhoods of 0 in We note that the set X is defined .Indeed if there is the strongest which On tlifi other hand the topology in JH Y subset of indexes Qj i Q such that ) is a neighborhood in induced topology in Yy . Then Y^f) t'(k(Wd vfc Y , Wo obtain v arld J (y ) belong to tVie same equivaleni Since 2-^ JvCyp V veQ -> class whpre —> o if {wdj- (t.Jv)(yv) vfcQT Lcmma l([2]) Let E be a normeormed reflexivreflexivee strictly convex space , EQ be a closed and subspace of E . Then every equivalent class [e] fe E/E contain! (y ) T fy ) unique element with minimal norm. Theorem I Let Y be a Mackey space such that for every v,Y is a normed reflexive By means of that T satisfies the condition I*) of Lemma I and by Lemma 3 the set T (y ) contains a unique element with minimal space. Let X be a normed reflexive strictly convex Bpace , T be a norm. We denote this element by x . Hence there exists only a linear multivalued operator satisfying the conditions l)-3). normal value of T at y . Theorem I is proved. D(T) "there exists only a normal value of T at y . Then for every Q -7- -6- Proof •'• . Approximation of normal value • It is well known that a closed convex set in normed reflexive stric- AESUHIC that f W . 1 is the "basic of absolute convex closed neigh- ly convex space contains a unique element with minimal norm([2J) , borhoods of 0 in space Y . Then £ yQ + «d ! ie the basic of absolute hence we have only to prove that G. is a closed convex subset in X . d comrci closed neighborhoods of y in space Y . e Indeed , denote by G.Iand S (y i rf) the images of the sets G, Let and s(y ,e^) respectively by canonical maps in Xj. = X/X , Y = = Y/T - Let TT : Y T * XT be a linear multivalued operator ud = TT O IV VI I. defined as in Section I .By Lemma I, T is linear closed map and by notations in Section I, T-. is single-value linear closed map. Since d V\ S( y tej) is a convex closed bounded set in reflexive space Y , it whore Lv is a projection of y^ on follows thBt S (y ,ep is weakly compact set . On the other hand By Lemma 2 operator TT is single-valued linear closed map therefore it is weakly closed map and its image is a weakly closed convex set , It follows v / v where e,> 0 , a 0 when U.
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