$(L^p,L^q)$ Hyers-Ulam stability

We introduce a new concept of Hyers-Ulam stability, in which in the size of a pseudosolution of a given ordinary differential equation and its deviation from an exact solution are measured with respect to different norms. These norms are associated to $L^p$-spaces for $p\in [1,\mathrm{\infty }]$. Our main objective is to formulate sufficient conditions under which semilinear ordinary differential equations exhibit such property. In addition, in certain special cases we obtain explicit formulas for the best Hyers-Ulam constant.