Nonexpansive-type operators from nonlinear operator-valued measures
Year: 2025
Type: article
Abstract: This paper is grounded in the long-standing, purely abstract theory of integration with respect to nonlinear operator-valued measures $\mu :\mathcal{R}\to N(S,F)$ μ : R → N ( S , F ) , which defines the integral operator $Tf\equiv \int f\,\mathrm {d}\mu $ T f ≡ ∫ f d μ in full generality. By leveraging Pettis and Bochner Radon–Nikodým-type theorems and imposing natural assumptions on measurability and semivariation, we establish that this abstract integral can be represented by a classical scalar integral with a Carathéodory kernel $\varphi :S\times X\to F$ φ : S × X → F . This representation, a core contribution of this work, translates the abstract operator integral into the more familiar form $Tf=\int _{X}\varphi (f(y),y)\,dm(y)$ T f = ∫ X φ ( f ( y ) , y ) d m ( y ) . This approach not only provides a powerful analytical framework but also enables the direct use of standard tools for nonlinear integral operators. Moreover, by introducing a Lipschitz condition on the kernel, we derive the nonexpansive inequality $||Tf-Tg||\le ||L||_{L^{q}}||f-g||_{L^{p}}$ | | T f − T g | | ≤ | | L | | L q | | f − g | | L p , thereby classifying T as a nonexpansive operator (or a strict contraction when $||L||_{L^{q}}<1$ | | L | | L q < 1 ). Consequently, this framework enables the direct application of classical fixed-point algorithms, such as Picard iterations for contractions and Krasnoselskii–Mann or Halpern schemes for nonexpansive mappings. This work bridges the gap between the abstract theory of operator-valued measures and the practical application of nonexpansive-type operator theory and algorithms.
Author: Tomasz Szczypiński
Institution: Andrzej Frycz Modrzewski Krakow University
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $1,790
OpenAlex ID: W7115883694