Approximation of convex set-valued functions

Approximation of set-valued functions is introduced and discussed under a convexity assumption. In particular, a theorem on positive linear operators is given.     

Upper semicontinuity of set-valued functions

Various types of upper semcontinuity properties for set-valued functions have been used in the past to obtain closure and lower closure theorems in optimal control theory as well as selection theorems and fixed-point theorems in topology. This paper unifies these various concepts by using semiclosure operators, extended topologies, and lattice theoretic operations and obtains general closure theorems. In addition, analytic criteria are given for this generalized upper semicontinuity. In particular, set-valued functions which are maximal in terms of certain properties (e.g., maximal monotone multifunctions) are shown to be necessarily upper semicontinuous.     

Some representations of midconvex set-valued functions

Summary In this note we establish conditions under which every midconvex set-valued function can be represented as sum of an additive function and a convex set-valued function. These results improve some theorems obtained in [8], [10] and [3]. Some results on local Jensen selections of midconvex set-valued functions are also given.     

Additive selections of superadditive set-valued functions

Summary A set-valued functionF from a coneC with a cone-basis of a topological vector spaceX into the family of all non-empty compact convex subsets of a locally convex spaceY is called superadditive provided thatF(x) + F(y) F(x + y), for allx, y C. We show that every superadditive set-valued function admits an additive selection.Dedicated to Professor Otto Haupt on his 100th birthday     

Affine selections of convex set-valued functions

Summary It is shown that every convex set-valued function defined on a cone with a cone-basis in a real linear space with compact values in a real locally convex space has an affine selection. Similar results can be obtained for midconvex set-valued functions.     

On some equation for set-valued functions

In his paper Smajdor (Aequat. Math. 75 149–162, 2008) showed that the equation H + tH ² = (I + tH) ○ H, t ≥ 0 is a necessary and sufficient condition under which the family {F ^t , t ≥ 0} of set-valued functions ${F^t(x):=\sum_{n=0}^{\infty} \frac{t^n}{n!}H^n(x), x \in K}$ is an iteration semigroup. We present a simple proof of a generalization of this result, independent of the coefficients of the series.     

Nonlinear Chebyshev approximation to set-valued functions

In this paper, we give a characterization of best Chebyshev approximation to set-valued functions from a family of continuous functions with the weak betweeness property. As a consequence, we obtain a characterization of Kolmogorov type for best simultaneous approximation to an infinity set of functions. We introduce the concept of a set-sun and give a characterization of it. In addition, we prove a property of Amir–Ziegler type for a family of real functions and we get a characterization of best simultaneous approximation to two functions     

Scalar representation and conjugation of set-valued functions

We consider functions with values in the power set of a pre-ordered, separated locally convex space with closed convex images. To each such function, a family of scalarizations is given which completely characterizes the original function. A concept of a Legendre–Fenchel conjugate for set-valued functions is introduced and identified with the conjugates of the scalarizations. To the set-valued conjugate, a full calculus is provided, including a biconjugation theorem, a chain rule and weak and strong duality results of the Fenchel–Rockafellar type.     

Optimality conditions for maximizations of set-valued functions

The maximization with respect to a cone of a set-valued function into possibly infinite dimensions is defined, and necessary and sufficient optimality conditions are established. In particular, an analogue of the Fritz John necessary optimality conditions is proved using a notion of derivative defined in terms of tangent cones.