AbstractNonlinear systems may be solved by functional iterations whose convergence properties depend on the properties of contraction mapping defining the iterative process. The path of contraction is defined by a sequence of matrices, known as decaying matrices of D-matrices. When so defined, contraction-mapping, which is a local property of the iteration mapping, becomes a D-mapping which decays and contracts globally. D-mappings reveal convergence or divergence of nonlinear functional iterations. They also give an apriori rate of convergence.