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1

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

Ereú T., Merentes N., Sanchez J. L., Wróbel M.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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2012

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Vol. 17

37-47

EN

We prove that, under some general assumptions, the one-sided regularizations of the generator of any uniformly bounded set-valued composition operator, acting in the spaces of functions of bounded variation in the sense of Schramm with nonempty bounded closed and convex values is an aﬃne function. As a special case, we obtain an earlier result ([15]).

2

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.

3

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].

4

On a certain generalization of the Krasnosel`skii theorem

Galewski M.

Journal of Applied Analysis

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2003

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Vol. 9, nr 1

139-147

EN

We provide a generalization of a well known Krasnosel'skii theorem on continuity of the Nemytskii operator for functions taking values in separable Banach spaces. We follow the results obtained in [6] for the finite dimensional case.

5

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.