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

100%

Cardinali T. , Papalini F.

nr 2

185-192

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.

On Lipschitzian operators of substitution generated by set-valued functions

100%

Ludew J. J.

tom Vol. 27, no. 1

13-24

We consider the Nemytskii operator, i.e., the operator of substitution, defined by (Nφ)(x) := G(x,φ(x)), where G is a given multifunction. It is shown that if N maps a Hölder space Hα into Hβ and N fulfils the Lipschitz condition then G(x,y) = A(x,y) + B(x), where A(x,·) is linear and A(·,y), B ∈ Hβ. Moreover, some conditions are given under which the Nemytskii operator generated by (1) maps Hα into Hβ and is Lipschitzian.

On K-superquadratic set-valued functions

100%

Troczka-Pawelec K.

tom Vol. 14, nr 1

101--109

In this paper we consider K-superquadratic set-valued functions. We will present here some connections between K-boundedness of K-superquadratic set-valued functions and K-semicontinuity of multifunctions of this kind.

Continuity of subquadratic set-valued functions

100%

Troczka-Pawelec K.

2012

tom Vol. 45, nr 4

939-946

Let X = (X, +) be an arbitrary topological group. The aim of the paper is to prove a regularity theorem for set valued subquadratic functions, that is solutions of the inclusion (…), with values in a topological vector space.

K-continuity problem of K-superquadratic set-valued functions

75%

Troczka-Pawelec K.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2014

tom Vol. 19

227--235

In this paper we study K-superquadratic set-valued functions.We will present here some connections between K-boundedness of K-superquadratic set-valued functions and K-semicontinuity of multifunctions of this kind.

On Nemytskij operator of substitution in the C1 space of set-valued functions

63%

Ludew J.

tom Vol. 41, nr 2

403-414

We consider the Nemytskij operator, i. e., the operator of substitution, defined by (N[...]x) := G(x,<[...](x)), where G is a given multifunction. It is shown that N maps C1 (I, C), the space of all continuously differentiable functions on the interval I with values in a cone C in a Banach space, into C1 (I, cc(Z)), the space of all continuously differentiable set-functions on I with compact and convex values in a Banach space Z and N fulfils the Lipschitz condition if and only if the generator G is of the form G(x,y)=A(x,y) + B(x) where A(x, •) is continuous, linear function, A(.,y) and B are continuously differentiable and the function x— > A(x, •) is Lipschitzian.

On Nemytskij operator in the space of absolutely continuous set-valued functions

63%

Ludew J. J.

tom Vol. 17, nr 2

277-290

We consider the Nemytskij operator, defined by (Nφ)(x) ? G(x, φ(x)), where G is a given set-valued function. It is shown that if N maps AC(I, C), the space of all absolutely continuous functions on the interval I ? [0, 1] with values in a cone C in a reflexive Banach space, into AC(I, K), the space of all absolutely continuous set-valued functions on I with values in the set K, consisting of all compact intervals (including degenerate ones) on the real line R, and N is uniformly continuous, then the generator G is of the form G(x, y) = A(x)(y) + B(x), where the function A(x) is additive and uniformly continuous for every x ∈ I and, moreover, the functions x ? A(x)(y) and B are absolutely continuous. Moreover, a condition, under which the Nemytskij operator maps the space AC(I, C) into AC(I, K) and is Lipschitzian, is given.

Set-valued version of Sincov's functional equation

63%

Smajdor W.

tom Vol. 39, nr 1

101-105

We investigate selections of set-valued solutions of Sincov's functional equation that satisfy the same equation.