Approximation Properties of a Certain Nonlinear Durrmeyer Operators

Abstract

The present paper is concerned with a certain sequence of the nonlinear Durrmeyer operators ND_{n}, very recently introduced by the author <cite>karslipam</cite> and <cite>karslibimj</cite>, of the form

(ND_{n}f)(x)=∫₀¹K_{n}(x,t,f(t))dt,0≤x≤1,n∈N,

acting on Lebesgue measurable functions defined on [0,1], where

K_{n}(x,t,u)=F_{n}(x,t)H_{n}(u)

satisfy some suitable assumptions. As a continuation of the very recent papers of the author <cite>karslipam</cite> and <cite>karslibimj</cite>, we estimate their pointwise convergence to functions f and ψ∘|f| having derivatives are of bounded (Jordan) variation on the interval [0,1].Here ψo|f| denotes the composition of the functions ψ and |f|. The function ψ:R₀⁺→R₀⁺ is continuous and concave with ψ(0)=0, ψ(u)>0 for u>0.This study can be considered as an extension of the related results dealing with the classical Durrmeyer operators.