Continuity of the $L_{p}$ Balls and an Application to Input-Output System Described by the Urysohn Type Integral Operator

TL;DR

This paper proves the continuity of $L_p$ balls as a function of p in a Banach space setting and applies this to analyze input-output systems modeled by Urysohn type integral operators.

Contribution

It establishes the continuity of $L_p$ balls with respect to p and applies this result to systems described by Urysohn integral operators.

Findings

Continuity of the set-valued map $p o B_{Ω,𝓍,p}(r)$ is proven.

Application to input-output systems with Urysohn type integral operators is demonstrated.

Provides a mathematical foundation for analyzing such systems in variable $L_p$ spaces.

Abstract

In this paper the continuity of the set valued map $p\rightarrow B_{\Omega,\mathcal{X},p}(r),$ $p\in (1,+\infty),$ is proved where $B_{\Omega,\mathcal{X},p}(r)$ is the closed ball of the space $L_{p}\left(\Omega,\Sigma,\mu; \mathcal{X}\right)$ centered at the origin with radius $r,$ $\left(\Omega,\Sigma,\mu\right)$ is a finite and positive measure space, $\mathcal{X}$ is separable Banach space. An application to input-output system described by Urysohn type integral operator is discussed.