ON REGULAR MULTIVALUED COSINE FAMILIES Let X, Y, Z be ...

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274Andrzej SmajdorWe see that every x 2- belongs to A, Fi(xi) = {i} and ||ar,-|| = 1 for everyi £ N. Therefore the assertion of Lemma 4 does not hold.REMARK 2. A convex cone K in Lemma 4 is of the second category inK if one of the three following cases holds true:a) X is a Banach space and intA ^ 0,b) X is a Banach space and K is closed,c) X is a normed space and dim A — dim (K - K) < +oo.Cases a) and b) are obvious. In case c), let n = dim (A' - A). Thenthere exist a basis {ci - d\,c„ - d n] of lin A = K - A, such that theset {ci, ...c n, di, ...d n} is a subset of A'. This subset is a spanning set ofK - A', therefore it contains a basis {ei,e,J C K of A' - A'. The formula\\x\\ = Y^i=z\ If iii f° r x — fi e i + ••• + fn^m defines a norm in A - A. It iseasy to check that the ball B(x 0,r 0) centered at x 0= + ... + ^e nwiththe radius r 0= ^ is a subset of A. So the interior of K is nonempty.Let T and 5 be two metric spaces and let c(S) denote the set of allcompact elements of n(S). The Hausdorff distance derived from the metricin S is a metric in c(5). A set-valued function F : T —> c(S) is said to becontinuous iff it is continuous as a single-valued function from T into themetric space c(5).Let Y be a normed space. We denote by cc(Y) the family of all convexmembers of c(Y). Observe that each linear set-valued function with closedvalues has to have convex ones.LEMMA 5. Let X and Y be two real normed spaces and let d be theHausdorff distance derived from the norm in Y. Suppose that K is a convexcone in X with nonempty interior. Then there exists a positive constant Mosuch that for every linear continuous set-valued function F : A" —> c(Y) theinequalityd(F(x),F(y))
On regular multivalued cosine families275for A > 0 andfor A < 0.The functionalX[A, B] = [-XB, -XA]\\[A,B]\\:= d(A, £),is a norm in Z (see [5]).Now, let F : K —> c(Y) be a linear continuous set-valued function. Thenthe function / : K —• Z given byf(x) = [F(x),{0}},is linear. Moreover, let x 0G K and (x n) be a sequence of elements of K suchthat XQ — lim n->oo s-n- Then F(a;o) =li m n-+ooF(x„) andlim ||/(a:n)-/(so)||= Hm d(F(x n), F(x 0)) = 0,n—>con-+ooso / is continuous. The function / can be extended to a linear function/ : X —• Z. This function'is also continuous. Thereforelim f(x) = lim f(x) = /(0) = 0andlim d(F(x), {0}) = lim \\[F(x), {0}]|| = lim ||/(z)|| - 0.By Lemmas 1 and 2, F and / are bounded. Fix a z e intft'. There existsan € > 0 such that \z + S C where 5 is the closed unit ball in X. Ifv e S and w = \z + v, then u € K and ||«|| < ||\z\\ + 1, therefore w e n a v e- /Mil = Ik - 2/llll/(pffji)H < ll/IMI* - y||,where M 0:= 1 + 2||^||. This implies thatd(F(x), F(y)) = ||[F(*), F(y)}\\ = \\f(x) - /(y)|| < M 0||F||||* - y\\.18 *
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ON REGULAR MULTIVALUED COSINE FAMILIES Let X, Y, Z be ...

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