On continuity and compactness of some nonlinear operators in the spaces of functions of bounded variation

Bugajewski, Dariusz; Gulgowski, Jacek; Kasprzak, Piotr

doi:10.1007/s10231-015-0526-7

Cited by 14 publications

(16 citation statements)

References 28 publications

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“…For that we will use the well-known concept of Bernstein polynomials. For the definition and properties (especially those required in the proof below) of Bernstein polynomials we would like to refer the reader to [5] and references therein. Proof.…”

Section: Continuity Of the Autonomous Superposition Operatormentioning

confidence: 99%

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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Section: Continuity Of the Autonomous Superposition Operatormentioning

confidence: 99%

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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“…To prove that F 2 maps bounded subsets of BV [0, 1] into relatively compact subsets of CBV [0, 1], we will use similar techniques to those used in [7]. Fix R 2 > 0 and let (x n ) n∈N be an arbitrary sequence of BV -functions such that x n BV ≤ R 2 for n ∈ N. By Helly's selection theorem there exists a subsequence (x n k ) k∈N of (x n ) n∈N and a function x ∈ BV [0 , 1] such that x n k → x pointwise on [0, 1] and x BV ≤ R 2 (see [2,Theorem 1.11] or [30, 1.4.5]).…”

Section: Existence Results Via Krasnosel ′ Slkiȋ's Theoremmentioning

confidence: 99%

“…In the second part of the paper (see Section 4), by means of some version of Krasnosel ′ slkiȋ's theorem for the sum of two operators, we study the existence of solutions of the following ing the compactness of nonlinear integral operators in the space of functions of bounded variation developed in [7]. (It may seem surprising, but sufficient conditions guaranteeing the compactness of Hammerstein integral operators in the space BV [0, 1] of functions of bounded variation, were not described until very recently -for more details see [7]. )…”

Section: Introductionmentioning

confidence: 99%

Solvability of Hammerstein Integral Equations with Applications to Boundary Value Problems

Bugajewska¹,

Infante²,

Kasprzak³

2017

Z. Anal. Anwend.

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In this paper we present some new results regarding the solvability of nonlinear Hammerstein integral equations in a special cone of continuous functions. The proofs are based on a certain fixed point theorem of Leggett and Williams type. We give an application of the abstract result to prove the existence of nontrivial solutions of a periodic boundary value problem. We also investigate, via a version of Krasnosel ′ slkiȋ's theorem for the sum of two operators, the solvability of perturbed Hammerstein integral equations in the space of continuous functions of bounded variation in the sense of Jordan. As an application of these results, we study the solvability of a boundary value problem subject to integral boundary conditions of Riemann-Stieltjes type. Some examples are presented in order to illustrate the obtained results.2010 Mathematics Subject Classification. Primary 45G99; Secondary 34B10, 47H10, 47H30.

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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%

“…The problem of acting conditions and related problems have been studied extensively for different concepts of variation (see the classical papers [17,20], more recent results [6,[9][10][11]21] and also [1] for an overview). Therefore, it seems to be important to answer this question also in this case.…”

Section: Superposition Operatormentioning

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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On continuity and compactness of some nonlinear operators in the spaces of functions of bounded variation

Cited by 14 publications

References 28 publications

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

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

Solvability of Hammerstein Integral Equations with Applications to Boundary Value Problems

On integral bounded variation

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