On integral operators and nonlinear integral equations in the spaces of functions of bounded variation

Bugajewski, Dariusz; Gulgowski, Jacek; Kasprzak, Piotr

doi:10.1016/j.jmaa.2016.06.014

Cited by 13 publications

(24 citation statements)

References 22 publications

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“…, I n is an arbitrary finite family of nonoverlapping subintervals of I of the form I i = [a i , b i ]. Then, by [6], Proposition 2 and the condition (ii), for every x ∈ BV (I), we get…”

Section: Proposition 4 Let K : I × I → R Satisfy 2 • (A) -(C) Then mentioning

confidence: 90%

“…We will show that the operator K maps BV (I) into ΛBV (I) and is continuous. Let us note that by [6], Remark 5, the function Kx is well defined for every x ∈ BV (I). Suppose now that I 1 , .…”

Section: Proposition 4 Let K : I × I → R Satisfy 2 • (A) -(C) Then mentioning

confidence: 99%

“…In the paper [6] the authors gave the sufficient and necessary conditions for the integral operator K : BV (I) → BV (I) of the form (12) to be continuous (see [6], Theorem 4). It appears that similar conditions may be stated for such integral operators K : BV (I) → ΛBV (I).…”

Section: Proposition 4 Let K : I × I → R Satisfy 2 • (A) -(C) Then mentioning

confidence: 99%

“…6 we examine embeddings of compact subsets of a given space of functions of bounded Λ-variation into another space of this type. It appears, roughly speaking, that sets of functions which are continuous in Λ-variation possess some nice properties, as far as compactness is concerned (see Theorems 18,19).…”

Section: Introductionmentioning

confidence: 99%

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On some nonlinear operators in $${\varvec{\Lambda }} {\varvec{BV}}$$ Λ B V -spaces

Bugajewski

Czudek

Gulgowski

et al. 2017

J. Fixed Point Theory Appl.

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Abstract. In this paper we investigate autonomous as well as nonautonomous superposition operators acting between spaces of functions of bounded Λ-variation. A particular emphasis is put on acting conditions as well as on continuity problems for such operators. In particular, we give necessary and sufficient conditions for nonautonomous superposition operators to map a space of functions of bounded Λ-variation into itself. Moreover, we prove the continuity of certain autonomous superposition operators acting between various spaces of functions of bounded Λ-variation. We also examine the existence of ΛBV -solutions as well as the topological structure of such solution sets to classical nonlinear integral equations.Mathematics Subject Classification. Primary 47H30, 26A45; Secondary 45G10, 47B34.Keywords. Aronszajn-type theorem, compact embedding, existence theorem, linear integral operator of a Fredholm-type, nonlinear Hammerstein integral equation, superposition operator, variation in the sense of Jordan, Waterman sequence, Λ-variation.

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“…, I n is an arbitrary finite family of nonoverlapping subintervals of I of the form I i = [a i , b i ]. Then, by [6], Proposition 2 and the condition (ii), for every x ∈ BV (I), we get…”

Section: Proposition 4 Let K : I × I → R Satisfy 2 • (A) -(C) Then mentioning

confidence: 90%

Section: Proposition 4 Let K : I × I → R Satisfy 2 • (A) -(C) Then mentioning

confidence: 99%

Section: Proposition 4 Let K : I × I → R Satisfy 2 • (A) -(C) Then mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On some nonlinear operators in $${\varvec{\Lambda }} {\varvec{BV}}$$ Λ B V -spaces

Bugajewski

Czudek

Gulgowski

et al. 2017

J. Fixed Point Theory Appl.

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“…Such operators, considered on spaces of functions of bounded variation in the sense of Jordan and Wiener, have been extensively studied in [10,11]. In particular in [11] there was presented the characterization of kernels k for which the corresponding map K : BV (I ) → BV (I ) is bounded.…”

Section: Integral Operatorsmentioning

confidence: 99%

On integral bounded variation

Gulgowski

2017

RACSAM

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In this paper we will investigate the concept of the q-integral p-variation introduced in 1970's by Terehin. This kind of integral variation has been mainly used, until now, to describe the regularity of functions in L p in terms of the L q -norm (for q > p).In the present paper we show that the class of functions of bounded q-integral p-variation appears a very nice tool to work with integral operators, including these corresponding to Riemann-Liouville fractional integration. We also give the acting conditions for autonomous superposition operators and prove some existence theorems for Hammerstein integral equation in this space.

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