Differential properties of mappings with $s$-average characteristic

In this paper, we develop the geometric method of moduli of curve families and consider the problem of differential properties of nonhomeomorphic mappings with $s$-averaged characteristic. These properties can be applied in the theory of multidimensional quasiconformal mappings and their generalizations. We prove that if $f$ is an extremal mapping with $s$-averaged characteristic, then it belongs to the class $W^2_2$.