A new class of nonlinear mappings is introduced which contains, in the linear case, the strictly and irreducibly diagonally dominant matrices as well as other classes of matrices introduced by Duffin and Walter. We then extend some of the properties of the above mentioned matrices to these weakly $\Omega$-diagonally dominant functions, and point out their connection to the M- and P- functions studied by Rheinboldt, and More and Rheinboldt, respectively. Finally, new convergence theorems for the nonlinear Jacobi and Gauss-Seidel iterations are presented.