Nonexpansive-type operators from nonlinear operator-valued measures

Abstract This paper is grounded in the long-standing, purely abstract theory of integration with respect to nonlinear operator-valued measures $\mu :\mathcal{R}\to N(S,F)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> <mml:mo>:</mml:mo> <mml:mi>R</mml:mi> <mml:mo>→</mml:mo> <mml:mi>N</mml:mi> <mml:mo>(</mml:mo> <mml:mi>S</mml:mi> <mml:mo>,</mml:mo> <mml:mi>F</mml:mi> <mml:mo>)</mml:mo> </mml:math> , which defines the integral operator $Tf\equiv \int f\,\mathrm {d}\mu $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> <mml:mi>f</mml:mi> <mml:mo>≡</mml:mo> <mml:mo>∫</mml:mo> <mml:mi>f</mml:mi> <mml:mspace/> <mml:mi>d</mml:mi> <mml:mi>μ</mml:mi> </mml:math> 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 $\varphi :S\times X\to F$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>φ</mml:mi> <mml:mo>:</mml:mo> <mml:mi>S</mml:mi> <mml:mo>×</mml:mo> <mml:mi>X</mml:mi> <mml:mo>→</mml:mo> <mml:mi>F</mml:mi> </mml:math> . This representation, a core contribution of this work, translates the abstract operator integral into the more familiar form $Tf=\int _{X}\varphi (f(y),y)\,dm(y)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> <mml:mi>f</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mo>∫</mml:mo> <mml:mi>X</mml:mi> </mml:msub> <mml:mi>φ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> <mml:mspace/> <mml:mi>d</mml:mi> <mml:mi>m</mml:mi> <mml:mo>(</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:math> . 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 $||Tf-Tg||\le ||L||_{L^{q}}||f-g||_{L^{p}}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>|</mml:mo> <mml:mo>|</mml:mo> <mml:mi>T</mml:mi> <mml:mi>f</mml:mi> <mml:mo>−</mml:mo> <mml:mi>T</mml:mi> <mml:mi>g</mml:mi> <mml:mo>|</mml:mo> <mml:mo>|</mml:mo> <mml:mo>≤</mml:mo> <mml:mo>|</mml:mo> <mml:mo>|</mml:mo> <mml:mi>L</mml:mi> <mml:mo>|</mml:mo> <mml:msub> <mml:mo>|</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msup> </mml:msub> <mml:mo>|</mml:mo> <mml:mo>|</mml:mo> <mml:mi>f</mml:mi> <mml:mo>−</mml:mo> <mml:mi>g</mml:mi> <mml:mo>|</mml:mo> <mml:msub> <mml:mo>|</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:msub> </mml:math> , thereby classifying T as a nonexpansive operator (or a strict contraction when $||L||_{L^{q}}&lt;1$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>|</mml:mo> <mml:mo>|</mml:mo> <mml:mi>L</mml:mi> <mml:mo>|</mml:mo> <mml:msub> <mml:mo>|</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msup> </mml:msub> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> </mml:math> ). 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.