Nonexpansive-type operators from nonlinear operator-valued measures
| dc.contributor.author | Szczypiński, Tomasz | |
| dc.date.accessioned | 2026-02-12T14:24:16Z | |
| dc.date.available | 2026-02-12T14:24:16Z | |
| dc.date.issued | 2025-12-18 | |
| dc.description.abstract | Abstract: "This paper is grounded in the long-standing, purely abstract theory of integration with respect to nonlinear operator-valued measures μ :ℛ→N(S, F), which defines the integral operator Tf ≡ ∫︁ f dμ 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 φ : S×X →F. This representation, a core contribution of this work, translates the abstract operator integral into the more familiar form Tf = ∫︁X φ(f (y), y)dm(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|| ≤ ||L||Lq ||f – g||Lp , thereby classifying T as a nonexpansive operator (or a strict contraction when ||L||Lq < 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."(...) | |
| dc.identifier.citation | Fixed Point Theory and Algorithms for Sciences and Engineering, 2026, issue 1 | |
| dc.identifier.doi | https://doi.org/10.1186/s13663-025-00818-0 | |
| dc.identifier.issn | 2730-5422 | |
| dc.identifier.uri | https://hdl.handle.net/11315/31553 | |
| dc.language.iso | en | |
| dc.publisher | Springer Nature | |
| dc.rights | Creative Commons Attribution 4.0 | |
| dc.subject | Operator-valued measure | |
| dc.subject | Integration with respect to nonlinear operator-valued measures | |
| dc.subject | Pettis integral | |
| dc.subject | Bochner integral | |
| dc.subject | Radon-Nikodym representation | |
| dc.subject | Carathéodory kernel | |
| dc.subject | Nonexpansive operator | |
| dc.subject | Krasnoselskii-Mann iteration | |
| dc.subject | Halpern iteration | |
| dc.subject | Fixpoint theory | |
| dc.subject.other | Matematyka | |
| dc.title | Nonexpansive-type operators from nonlinear operator-valued measures | |
| dc.type | Artykuł |