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1

Operators with memory in Schramm spaces

Wróbel Małgorzata

Journal of Applied Mathematics and Computational Mechanics

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2023

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Vol. 22, nr 3

69-80

EN

We show that every operator with memory acting between Banach spaces CΦBV(I) of continuous functions of bounded variation in the sense of Schramm defined on a compact interval I of a real axis, is a Nemytskij composition operator with the continuous generating function. Moreover, some consequences for uniformly bounded operators with memory will be given. As a by-product, we obtain that a Banach space CΦBV(I) has the uniform Matkowski property.

2

The Nemitskij operator on Lipk-type and BVk-type spaces

Giménez J., López L., Merentes N.

Demonstratio Mathematica

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2013

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Vol. 46, nr 3

543--558

EN

In this paper, we discuss and present various results about acting and boundedness conditions of the autonomous Nemitskij operator on certain function spaces related to the space of all real valued Lipschitz (of bounded variation, absolutely continuous) functions defined on a compact interval of R. We obtain a result concerning the integrability of products of the form (…) and a generalized version of the chain rule for functions a.e differentiable, in the sense of Lebesgue. As an application, we get a generalization of a theorem due to V. I. Burenkov for the case of functions of bounded Riesz-p-variation.

3

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.

4

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

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

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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2011

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Vol. 16

23--32

EN

We show that the one-sided regularizations of the generator of any uniformly continuous set-valued Nemytskij operator, acting between the spaces of functions of bounded variation in the sense of Schramm, is an affine function. Results along these lines extend the study [1].

5

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.

6

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.