On the Lipschitz perturbation of monotonic functions

Z. Makó, Zs Páles

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2 Citations (Scopus)

Abstract

A real valued function f defined on a real interval I is called d-Lipschitz if it satisfies \ℓ(x) - ℓ(y)\ ≦ d(x, y) for x, y ∈ I. In this paper, we investigate when a function p : I → R can be decomposed in the form p = q + ℓ, where q is increasing and ℓ is d-Lipschitz. In the general case when d : I2 → R is an arbitrary semimetric, a function p : I → R can be written in the form p = q + ℓ if and only if (Equation Presented) is fulfilled for all real numbers t1 < s 1, . . . , tn < sn and u1 < ν1, . . , νm < νm, in I satisfying the condition (Equation Presented) where 1[a,b] denotes the characteristic function of the interval ]a, b]. In the particular case when d : I2 → R is a so-called concave semimetric, a function p : I → R is of the form p = q + ℓ if and only if (Equation Presented) holds for all x0 ≦ x1 < ⋯< x2n ≦ x2n+1 in I.

Original languageEnglish
Pages (from-to)1-18
Number of pages18
JournalActa Mathematica Hungarica
Volume113
Issue number1-2
DOIs
Publication statusPublished - Oct 1 2006

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Keywords

  • Lipschitz function
  • Lipschitz perturbation
  • Monotonicity

ASJC Scopus subject areas

  • Mathematics(all)

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