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1

Uniformly bounded set-valued composition operators in the spaces of functions of bounded variation in the sense of Wiener

Guerrero J. A., Matkowski J., Merentes N., Wróbel M.

Journal of Applied Mathematics and Computational Mechanics

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2015

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Vol. 14, nr 4

41--51

EN

We show that the one-sided regularizations of the generator of any uniformly bounded set-valued Nemytskij composition operator mapping the space of bounded variation functions in the sense of Wiener into the space of bounded variation functions with closed bounded convex values (in the sense of Wiener) are affine functions.

2

On K-superquadratic set-valued functions

Troczka-Pawelec K.

Journal of Applied Mathematics and Computational Mechanics

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2015

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Vol. 14, nr 1

101--109

EN

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.

3

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

Troczka-Pawelec K.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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2014

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Vol. 19

227--235

EN

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.

4

K-continuity of K-subquadratic set-valued functions

Troczka-Pawelec K., Tyrala I.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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2014

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Vol. 19

237--244

EN

Let X = (X, +) be an arbitrary topological group. A set-valued function F : X → n(Y) is called K-subquadratic if 2F(s) + 2F(t) ⊂ F(s + t) + F(s - t) + K, for all s, t ϵ X, where Y denotes a topological vector space and where K is a cone in this space. In this paper the K-continuity problem of multifunctions of this kind will be considered with respect to weakly K-boundedness. The case where Y = R N will be considered separately.

5

K-subquadratic set-valued functions

Troczka-Pawelec K.

Commentationes Mathematicae

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2014

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Vol 54, No. 2

147--157

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)⊂F(s+t)+F(s−t)+K,s,t∈X, with values in a topological vector space and where K is a cone in this space.

6

Continuity of subquadratic set-valued functions

Troczka-Pawelec K.

Demonstratio Mathematica

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2012

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Vol. 45, nr 4

939-946

EN

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.

7

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

Ludew J. J.

Journal of Applied Analysis

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2011

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Vol. 17, nr 2

277-290

EN

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.

8

Uniformly continuous set-valued composition operators in the spaces of functions of bounded variation in the sense of Wiener

Azocar A., Guerrero J. A., Matkowski J., Merentes N.

Opuscula Mathematica

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2010

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Vol. 30, no. 1

53-60

EN

We show that the one-sided regularizations of the generator of any uniformly continuous and convex compact valued composition operator, acting in the spaces of functions of bounded variation in the sense of Wiener, is an affine function.

9

Uniformly continuous set-valued composition operators in the spaces of functions of bounded variation in the sense of Riesz

Aziz W., Guerrero J., Merentes N.

Bulletin of the Polish Academy of Sciences. Mathematics

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2010

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Vol. 58, no 1

39-45

EN

We show that any uniformly continuous and convex compact valued Nemytskii composition operator acting in the spaces of functions of bounded φ-variation in the sense of Riesz is generated by an affine function.

10

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

Ludew J.

Demonstratio Mathematica

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2008

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Vol. 41, nr 2

403-414

EN

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.

11

On Lipschitzian operators of substitution generated by set-valued functions

Ludew J. J.

Opuscula Mathematica

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2007

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Vol. 27, no. 1

13-24

EN

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.

12

Set-valued version of Sincov's functional equation

Smajdor W.

Demonstratio Mathematica

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2006

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Vol. 39, nr 1

101-105

EN

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

13

Characterization of globally Lipschitz Nemytskii operators between spaces of set-valued functions of bounded [phi]- variation in the sense of Riesz

Merentes N., Hernandez S.

Bulletin of the Polish Academy of Sciences. Mathematics

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2004

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Vol. 52, nr 4

417-430

EN

Let (X, || . ||) and [Y, || . ||] be two normed spaces and K be a convex cone in X. Let CC(Y) be the family of all non-empty convex compact subsets of Y. We consider the Nemytskii operators, i.e. the composition operators defined by [Nu)(t) = H(t,u[t)), where H is a given set-valued function. It is shown that if the operator N maps the space RV[phi]1 ([a, b]; K) into RW[phi]2([a, b]; CC[Y)) (both are spaces of functions of bounded [phi]- variation in the sense of Riesz), and if it is globally Lipschitz, then it has to be of the form H(t,u[t)) = A(t]u(t)+B(t), where A(t) is a linear continuous set-valued function and B is a set-valued function of bounded [phi]2-variation in the sense of Riesz. This generalizes results of G. Zawadzka [12], A. Smajdor and W. Smajdor [II], N. Merentes and K. Nikodem [5], and N. Merentes and S. Rivas [7].

14

Note on Jensen and Pexider functional equations

Smajdor W.

Demonstratio Mathematica

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1999

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Vol. 32, nr 2

363-376

EN

We determine the general solutions of the Jensen functional equation 2f (x+y):2=f(x) + f(y), x,y zawiera się M and the Pexider functional equation f(x+y)=g(x)+h(y), x,y zawiera się M , for f, g, h : M --+ S , where M is an Abelian semigroup with the division by 2 and S is an abstract convex cone satisfying the cancellation law. Some applications to set-valued versions of these equations are given.

15

On multivalued integral mean

Ślepak W.

Demonstratio Mathematica

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1998

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Vol. 31, nr 2

339-344

EN

In the present note we prove that if a set valued function F : [0, b] -+ n(Y), where n(Y) denote all nonempty subsets of Banach space, is convex or starshaped then the multifunction defined by the formula.