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

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