Continuity of $L_{p}$ Balls and an Application to Input-Output Systems

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 radius $r$ in the space $L_{p}(\Omega,\Sigma,\mu; \mathcal{X})$ centered at the origin, $(\Omega,\Sigma,\mu)$ is a finite and positive measure space, and $\mathcal{X}$ is a separable Banach space. An application to input-output systems described by Urysohn type integral operators is discussed.