On the $\mathcal L$-Characteristic of the Superposition Operator in Lebesgue Spaces with Mixed Norm

Abstract

We consider the superposition operator Fx(t,s) = f(t,s,x(t,s)) of functions of two variables in spaces with mixed norm [L_p \to L_q] . After establishing a necessary and sufficient acting condition, we get some conclusions on the \mathcal L -characteristic of F . We also prove some theorems, which imply that F is uniformly continuous on balls in the interior of its \mathcal L -characteristic.