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

ON REGULAR MULTIVALUED COSINE FAMILIESANDRZEJ SMAJDORTo the memory of Professor Győrgy TargonskiAbstract. Let K be a convex cone in a real normed space X. A one-parameterfamily {F t: t > o} of set-valued functions F t: K —> n(K), where n(K) :={D : D C A', D jl 0}, is called cosine iff F i+S+ F,_ s= 2F to F s, whenever0 < s < t and F 0is the identity map. A cosine family {F t: t > o} is regular ifflim f-».o+ F t(x) = {1} for every x.The growth and the continuity of regular cosine families are investigated.Let X, Y, Z be nonempty sets and let n(Y) denote the set of all nonemptysubsets of Y. We recall that the superposition G o F of set-valuedfunctions F : X —> n(Y) and G : Y —¥ n(Z) is denned by the formula(GoF)(x) :=(J{G(y):y£F(x)} for x £ X.A subset K of a real vector space X is called a cone if tK C K for allt € (0, +oo). A cone is said to be convex if it is a convex set.A set-valued function F : K —>• n(Y), where is a convex cone in X,is said to be superadditive iff F(x) + F(y) C F(x + y) for x,y £ K.Let A' be a convex cone in X and let Q +denote the set of all positiverational numbers. A set-valued function F : K -> n(Y) is said to beQ +-homogeneous if F(Xx) — \F{x) for A € Q +, a; G A'.Now, we assume that X and F are arbitrary real normed spaces. Aset-valued function F : K —> n(Y) is called /ou>er semicontinuous at XQ 6Received: January 5, 1999.AMS (1991) subject classification: Primary 39B52, 47D09, 26E25.

272 Andrzej SmajdorK iff for every open set V in Y such that F(x 0) C\V ^ 0 there exists aneighbourhood U of zero in X such that F(x) D V ^ 0 for x 6 (x 0+ U) D/

On regular multivalued cosine families273The set of all nonempty bounded subsets of a normed space Y will bedenote by 5(F).LEMMA 3 (Theorem 3 in [7]). Let X arid Y be two real normed spacesand let K be a convex cone in X. Suppose that (F,- : i £ /) is a family ofsuperadditive lower semicontinuous in K and Q+-homogeneous set-valuedfunctions Fj : K —> n(Y). If F(x) = {J ieIF,(x) and the set B = {x £ K :F(x) £ B(Y)} is of the second category in A, then F is bounded and B = A'.Lemma 3 and the same considerations as in the proof of Theorem 4 in[7] allow to derive the following lemma.LEMMA 4. Let X and Y be two real normed spaces and let K be aconvex cone in X. Suppose that (F; : i £ I) is a family of superadditivelower semicontinuous in A' and Q +-homogeneous set-valued functions F; :A —>• n(Y). If K is of the second category in K and Uie/ Fi{ x ) 6 B(Y) forx £ A', then there exists a constant M £ (0, +oo) such thatsup \\Fi(x)\\ < M\\x\\ for x £ A'.ieiREMARK 1. The assumption that the cone A' is a set of the secondcategory in A' is essential and it can not be replaced by the completnessof X.In order to prove it we use an example from Chapter III, §3.7 ofN. Bourbaki's book [ll. Let X — {x £ C(R,R) : lim x(t) = lim x{t) =0}, ||x|| = sup{|x(i)| : t £ R} for x £ X and let A = {x £ X : supp x £c(R)}, where c(R) is the set of all nonempty compact subsets of the set R ofall real numbers. We can check that in this case (X, \\ ||) is a Banach space,A' is a convex cone and set-valued functions F; : A' —> n(R), i — 1,2,...defined by formulasFi{x) = {ix(t)}are additive, continuous and are Q +-homogeneous in A'. Moreover, sets|jF i(a;) = {ża:(t):t€N}are finite. So almost all assumptions of Lemma 4 hold except the "category"assumption. Let functions XJ, i £ N be defined as follows0, if -oo

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 *

276Andrzej SmajdorThis is a stronger version of Lemma 16 in [3](see also Lemma 7 in [4]).The application of the Radstrom's equivalence relation allows to omit theassumption that X is a separable Banach space. This is an idea of dr JoannaSzczawińska.LEMMA 6 (Lemma 1.9 in [6]). Let X be a metric space with a metric pand let F be a set-valued function from X into X. If for a positive numberM the inequalityholds for every x, y € X, thend(F(x),F(y)) 0} ofset-valued functions F t: K —> n{K) is said to be a cosine family iffF 0= /,where / denotes the identity map and(2) F t+s+ F t_ s= 2F toF s,whenever 0 < s < t.EXAMPLES:1. K = (—oo, +oo), F t(x) — x[cost,cosht],2. K = (—co, +oo), Ft{x) — x[cosi, 1].3. K = [0,-r-co), F t(x) = x[l,cosht].Let X be a real normed linear space. A cosine family {F t: t > 0} isregular iff\\m +d(F t(x),{x}) = 0,where d is the Hausdorff distance derived from the norm in X.THEOREM 1. Let X be a real normed space, and let K be a convex conein X of the second category in K. If {F t: t > 0} is a regular cosine familyof continuous superadditive Q^-homogeneous set-valued functions F t: K —>c(A'), then there exist two constants M > 0 and u > 0 such that||F^l < Me ul for t > 0.

On regular multivalued cosine families 277PROOF. The proof will be divided into three steps.1° There exists an n, 0 < n < 1 such that the function t t-» ||Ff|| isbounded for 0 < t < n.Suppose that it is false. Then there is a sequence (t n) satisfying conditions:t n> 0, lim n-too t n= 0 and \\F tn\\ > n for n = 1, 2,... . From Lemma4 it follows that for some x G K the sequence (||F TN(x)\\) is unbounded contraryto the regularity of the family {F t: t > 0}. Thus there exist an r/,0 < t] < 1 and L > 0 such thatIIFII < L for i € [0,r,].Since ||F 0|| = ||/|| = 1 we have L > 1.2° Let m > 1 be an arbitrary constant. If s > 0 and \\F S\\ < m, then forevery n = 1,2,... we have \\F ns\\ < (3m) n .The proof is by induction on n. The case n = 1 is trivial. If n = 2 weobtainll*2.|| < + 2||F S|| 2 < 2m 2 + 1 < (3m) 2 .Now, we suppose that n > 3 and\\F ks\\ < (3m) fcfor\

278 Andrzej SmajdorA cosine family {F t: i > 0} is continuous iff the function t H-> F t(x) iscontinuous for every x 6 K.THEOREM 2. Let X be a real Banach space and let K be a convex cone inX such thatmtK ^ 0. If{F t: t > 0} is a regular cosine family of continuousadditive set-valued functions F t: K —¥ cc{K), then it is continuous.PROOF. The proof will be divided into eight steps.1° We assume that there exist XQ € K and t 0€ [0, +oo) such that thefunction t H-> F t(xo) is discontinuous at the point to- Since the consideredcosine family is regular, to is positive.2° Let us defineL n:= sup{d(F t(xo),F s(x 0)) : |t-t 0| < ^-,|s-*o| < A * > 0, s > 0}.for every positive integer n.3° There exists L > 0 such that L n> L for every n.Obviously (L„) is a non-negative and non-increasing sequence. Hencethere exists aL = lim L n€ K.We see that L n> L for any n. Suppose that L — 0. For every e > 0 thereexists a positive integer n such thatd(F t(a:o), ^fao)) < e,whenever \t - t 0| < \s - t 0\ < |£, t > 0 and s > 0. This implies thatthe function t i-» F t(xo) is continuous at t 0contrary to our assumption 1°.Therefore L > 0 and we take L = L.4° For every positive integer n there exist two positive numbers s andt such that \s - t 01 < |t - to I < & and d(F t(x 0), F s{x 0)) > L n- £ > 0.Hence t ^ s. Therefore there exist two sequences (t n) and (s„) such thats n> t n> 0, \t n-t 0\ < g£, |sn-*o| < & and d(F tn(xo), F Sn(x 0)) > - £for every n.5° 2t„ — s n€ (0, +oo) for every n.It suffices to show that t n> s n- t nBy 4° clearly«n - = (-Sn ~ to) + (to ~ *n) < ~r~and*n — £o — (^0 — ^n) >8n- 18n

On regular multivalued cosine families 2796°for every n — 1,2,....From 4° we have \s in- t 0\ < < j£ and |(2i 4n - *4n) - *o| =|2(*4n - *0) + (*0 - *4n)| < 2|t 4n - *o| + 1*0 ~ *4n| < |&

280 Andrzej SmajdorREMARK 3. In the proof of Theorems 1 and 2 we have essentialy usedideas of M. Sova [8] for cosine operator functions.REFERENCES[1] N. Bourbaki, Elements de Mathematique, Livre V, Espaces Vectoriels Topologiques, Paris1953-1955.[2] E. Hille, R. S. Phillips, Functional Analysis and Semigroups, Providence, Rhode Island1957.[3] J. Olko, Rodziny wielowartościowych funkcji liniowych, doctoral dissertation.[4] J. Olko, Semigroups of set-valued functions, Publ. Math. 51 (1997), 81-96.[5] H. Radstrom, An embedding theorem for space of convex sets, Proc. Amer. Math. Soc. 3(1952), 165-169.[6] A. Smajdor, Iteration of multivalued functions, Prace Naukowe Uniwersytetu Śląskiego wKatowicach nr 759, Uniwersytet Śląski, Katowice 1985.[7] W. Smajdor, Superadditive set-valued functions an Banach-Steinhaus theorem, RadoviMat. 3 (1987), 203-214.[8] M. Sova, Cosine operator functions, Rozprawy Mat. 49 (1966), 1-47.PEDAGOGICAL UNIVERSITYPODCHORĄŻYCH 2PL-30-084 KRAKÓWPOLAND