On convergence of certain nonlinear Bernstein operators

The present paper concerns with the very recently introduced nonlinear Bernstein operators NB_{n}f of the form (NB_{n}f)(x)=∑_{k=0}ⁿP_{n,k}(x,f((k/n))),0≤x≤1,n∈N, acting on bounded functions on an interval [0,1], where P_{n,k} satisfy some suitable assumptions. As a continuation of the very recent paper of the authors kta , we estimate their pointwise convergence to a function f having derivatives of bounded (Jordan) variation on the interval [0,1]. We note that our results are strict extensions of the classical ones, namely, the results dealing with the linear Bernstein polynomials.