The article deals with Newton-like approximations \[ x_{n+1} = x_n - A(x_n)^{-1}(F(x_n) + G(x_n)), \quad n = 0,1,2,\ldots,\tag{1} \] to a nonlinear operator equation \[ F(x) + G(x) = 0 \] with a Fréchet differentiable operator \(F\) and a continuous operator \(G\); here \(A(x)\) are linear operators with the invertible \(A(x_0)\). In particular, in special cases, these approximations are reduced to usual and modified Newton-Kantorovich ones, some modifications of two-step approximations, Halley and Chebyshev-like approximations, and so on. Under different assumptions on \(F\), \(G\) and \(A\), the authors construct scalar majorants for approximations (1) and study their convergence. As a result they obtain some conditions for the convergence of approximations (1). In the end of the article the authors consider special cases and some illustrative numerical examples.