Two generalizations of linear (matrix) operator are considered: discrete sublinear and discrete superlinear operators. It is shown that a number of operators considered in literature can be reduced to them. We investigate contractive properties of these operators and the asymptotic behaviour of the sequence $$ x^{t+1}=H(x^t),\qquad t=0,1,\ldots, $$ where $x^0$ is an arbitrary non-negative initial vector and $H$ is an operator. We introduce the notion of left eigen-element of an operator which is applied to solve one problem of mathematical economics, namely, the problem to find the effective functional in the Neumann–Leontiev model.