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

Abstract

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.

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

“…In particular, we provide necessary and sufficient conditions for a linear integral operator to map the space BV[0, 1] into ΦBV[0, 1] and to be continuous/compact, obtaining thus the full characterization of such operators in terms of their generating kernels. Those results extend the results obtained in [3,5,14] and complement the results of [2] (see also the very recent monograph [17,Section 4.3]).…”

“…Remark 11. Theorem 10 extends analogous results established in [3,5] for linear integral operators acting into the spaces BV[0, 1], BV p [0, 1] and ΛBV[0, 1] of functions of bounded Jordan, Wiener and Waterman variation, respectively (cf. also [17,Section 4.3]).…”

On integral bounded variation

“…In particular in [11] there was presented the characterization of kernels k for which the corresponding map K : BV (I ) → BV (I ) is bounded. Quite recently similar necessary and sufficient conditions for the continuity of the integral operator have been also specified for BV spaces (see [9]). We are going to present now two sets of sufficient conditions for the continuity of the integral operator K : L q → I BV q 1 .…”

“…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.…”

On Λ p B V -Solutions of Some Nonlinear Integral Equations on the Plane

“…One of the most important aspects of functions of bounded variation is that they form an algebra of discontinuous functions whose rst derivative exists almost everywhere and are frequently used to de ne generalized solutions to nonlinear problems involving functionals, and partial di erential equations in mathematics, physics, and engineering. In recent decades, the solutions of this type of integral equations have been studied by various authors in various spaces of bounded variation, for example, in the space of the functions of bounded variation in the Jordan sense and in the Waterman sense, see [1,2], in addition to other generalized spaces of bounded variation, some of these have been studied in [3][4][5][6][7][8][9][10][11][12][13][14][15][16].…”