In the present paper some Newton-like iteration methods are developed to enclose solutions of nonlinear operator equations
of the kindF(x)=0. HereF maps a certain subset of a partially ordered vector space into another partially ordered vector space. The obtained results
are proved without any special properties of the orderings by taking use of a new kind of a generalized divided difference
operator, so that they even hold for nonconvex operators. Furthermore a method for constructing including starting points
is presented and two examples are given.