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1

Characterization of inclusion among Riesz−Medvedev variation spaces

Aziz W., Ereú T.

Commentationes Mathematicae

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2016

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Vol 56, No. 2

225--237

EN

We present a characterization of inclusion among Riesz−Medvedev bounded variation spaces, i.e., we shall present necessary and sufficient conditions for the Young functions φ1 and φ2 so that RVφ1[a,b]⊂RVφ2[a,b] or RV∗φ1[a,b]⊂RV∗φ2[a,b] .

2

On the Nemytskii operator in the space of functions of bounded (p, 2, α)-variation with respect to the weight function

Aziz W.

Demonstratio Mathematica

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2014

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Vol. 47, nr 4

933--948

EN

In this paper, we consider the Nemytskii operator (H f) (t) = h(t, f(t)), generated by a given function h. It is shown that if H is globally Lipschitzian and maps the space of functions of bounded (p, 2, α)-variation (with respect to a weight function α) into the space of functions of bounded (q, 2, α)-variation (with respect to α) 1 < q < p, then H is of the form (H f) (t) = A(t)f(t) + B(t). On the other hand, if 1 < p < q then H is constant. It generalize several earlier results of this type due to Matkowski–Merentes and Merentes. Also, we will prove that if a uniformly continuous Nemytskii operator maps a space of bounded variation with weight function in the sense of Merentes into another space of the same type, its generator function is an affine function.

3

On Nemytskii operator in the space of set-valued functions of bounded p-variation in the sense of Riesz with respect to the weight function

Aziz W., Guerrero J. A., Merentes N.

Fasciculi Mathematici

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2013

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Nr 50

17--32

EN

In this paper we consider the Nemytskii operator (Hf) (t) = h(t, f (t)), generated by a given set-valued function h is considered. It is shown that if H is globally Lipschitzian and maps the space of functions of bounded p-variation (with respect to a weight function α) into the space of set-valued functions of bounded q-variation (with respect to α) ) 1 < q < p, then H is of the form (Hϕ)(t) = A(t)ϕ(t) + B(t). On the other hand, if 1 < p < q, then H is constant. It generalizes many earlier results of this type due to Chistyakov, Matkowski, Merentes-Nikodem, Merentes-Rivas, Smajdor-Smajdor and Zawadzka.

4

Uniformly continuous composition operator in the space of functions of two variables of Bounded ɸ-variation in the sense of Schramm

Aziz W., 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

7-16

EN

We prove in this paper that if the composition operator H, generated by a function h : I b a x C(Iba) Y , maps ɸBV (Iba ,C) into ɸ2 BV (Iba , Y ) and is uniformly continuous, then the left-left regularization h* of h is an affine function with respect to the third variable.

5

Uniformly continuous set-valued composition operators in the space of total phi-bidimensional variation in the sense of Riesz

Aziz W., Gimenez J., Merentes N., Sanchez J. L.

Opuscula Mathematica

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2010

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Vol. 30, no. 3

241-248

EN

In this paper we prove that if a Nemytskij composition operator, generated by a function of three variables in which the third variable is a function one, maps a suitable large subset of the space of functions of bounded total φ-bidimensional variation in the sense of Riesz, into another such space, and is uniformly continuous, then its generator is an affine function in the function variable. This extends some previous results in the one-dimensional setting.

6

Functions of two variables with bounded φ-variation in the sense of Riesz

Aziz W., Leiva H., Merentes N., Sánchez J. L.

Journal of Mathematics and Applications

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2010

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Vol. 32

5-23

EN

In this paper we introduce the concept of bounded φ- variation function, in the sense of Riesz, dened in a rectangle [wzór]. We prove that the linear space [wzór] generated by the class [wzór] of all φ-bounded variation functions is a Banach algebra. Moreover, we give necessary and sucient conditions for the Nemytskii operator acting in the space [wzór] to be globally Lipschitz.

7

Uniformly continuous composition operator in the space of φ-variation functions in the sense of Riesz

Acosta A., Aziz W., Matkowski J., Merentes N.

Fasciculi Mathematici

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2010

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Nr 43

5-11

EN

Assuming that a Nemytskii operator maps a subset of the space of bounded variation functions in the sense of Riesz into another space of the same type, and is uniformly continuous, we prove that the generator of the operator is an affine function.

8

A Representation Theorem for '-Bounded Variation of Functions in the Sense of Riesz

Aziz W., Leiva H., Merentes N., Rzepka B.

Commentationes Mathematicae

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2010

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Vol. 50, [Z] 2

109-120

EN

In this paper we extend the well known Riesz lemma to the class of bounded φ-variation functions in the sense of Riesz defined on a rectangle [...].This concept was introduced in [2], where the authors proved that the space [...] of such functions is a Banach Algebra. Moreover, they characterized also the Nemytskii operator acting in this space. Thus our result creates a continuation of the paper [2].

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.