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

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.