Approximation properties of a certain nonlinear Durrmeyer operators

Abstract

The present paper is concerned with a certain sequence of the nonlinear Durrmeyer operators NDn, very recently introduced by the author [22] and [23], of the form (NDnf)(x)=?10 Kn (x,t,f(t))dt, 0 ? x ? 1, n ? N, acting on Lebesgue measurable functions defined on [0,1], where Kn (x,t,u) = Fn (x,t)Hn(u) satisfy some suitable assumptions. As a continuation of the very recent papers of the author [22] and [23], we estimate their pointwise convergence to functions f and ??|f| having derivatives are of bounded (Jordan) variation on the interval [0,1]. Here ?o|f| denotes the composition of the functions ? and |f|. The function ? : R0+ ? R0+ is continuous and concave with (0) = 0, ?(u) > 0 for u > 0. This study can be considered as an extension of the related results dealing with the classical Durrmeyer operators.