Wyniki wyszukiwania

1

Some results on stability and on characterization of K-convexity of set-valued functions

100%

Cardinali T., Papalini F.

Annales Polonici Mathematici

|

1993

|

tom 58

|

nr 2

185-192

EN

We present a stability theorem of Ulam-Hyers type for K-convex set-valued functions, and prove that a set-valued function is K-convex if and only if it is K-midconvex and K-quasiconvex.

2

Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik

Approximation of set-valued functions by single-valued one

100%

Ginchev I., Hoffmann A.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

|

2002

|

tom 22

|

nr 1

33-66

EN

Let $Σ: M → 2^Y\{∅}$ be a set-valued function defined on a Hausdorff compact topological space M and taking values in the normed space (Y,||·||). We deal with the problem of finding the best Chebyshev type approximation of the set-valued function Σ by a single-valued function g from a given closed convex set V ⊂ C(M,Y). In an abstract setting this problem is posed as the extremal problem $sup_{t ∈ M} ρ(g(t), (t)) → inf$, g ∈ V. Here ρ is a functional whose values ρ(q,S) can be interpreted as some distance from the point q to the set S ⊂ Y. In the paper, we are confined to two natural distance functionals ρ = H and ρ = D. H(q,S) is the Hausdorff distance (the excess) from the point q to the set cl S, and D(q,S) is referred to as the oriented distance from the point q to set cl conv S. We prove that both these problems are convex optimization problems. While distinguishing between the so called regular and irregular case problems, in particular the case V = C(M,Y) is studied to show that the solutions in the irregular case are obtained as continuous selections of certain set-valued maps. In the general case, optimality conditions in terms of directional derivatives are obtained of both primal and dual type.

3

K-subquadratic set-valued functions

100%

Troczka-Pawelec K.

Commentationes Mathematicae

|

2014

|

tom 54

|

nr 2

EN

Let $X=(X,+)$ be an arbitrary topological group. The aim of the paper is to prove a regularity theorem for K-subquadratic set-valued functions, that is, solutions of the inclusion $$ 2F(s)+2F(t)\subset F(s+t)+F(s-t)+K, \quad s,t\in X, $$ with values in a topological vector space and where $K$ is a cone in this space.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Commentationes Mathematicae

1 Discussiones Mathematicae, Differential Inclusions, Control and Optimization

1 Cardinali T.

1 Ginchev I.

1 Hoffmann A.

1 Papalini F.

1 Troczka-Pawelec K.

1 2014

1 2002

1 1993

Some results on stability and on characterization of K-convexity of set-valued functions

Approximation of set-valued functions by single-valued one

K-subquadratic set-valued functions