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

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.