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2 Mainka-Niemczyk E.

2 2012

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Some properties of set-valued sine families

Mainka-Niemczyk E.

Opuscula Mathematica

2012

Vol. 32, no. 1

159-170

Let {Ft : t ≥ 0} be a family of continuous additive set-valued functions defined on a convex cone K in a normed linear space X with nonempty convex compact values in X. It is shown that (under some assumptions) a regular sine family associated with {Ft : t ≥ 0} is continuous {Ft : t ≥ 0} is a continuous cosine family.

Integral representation of set-valued sine families

Mainka-Niemczyk E.

Journal of Applied Analysis

2012

Vol. 18, nr 2

243-258

Let K be a closed convex cone with nonempty interior in a real Banach space. The main objective of this paper is to show that if {Gt : t ≥ 0} is a regular sine family of continuous additive set-valued functions associated with {Ft : t ≥ 0}, then Gt are of the form Gt(x) = ∫ t 0 Fu(x)du for x ∈ K,t ≥ 0.