On convergence of certain nonlinear Durrmeyer operators at Lebesgue points

Abstract

The aim of this paper is to study the behaviour of certain sequence of nonlinear Durrmeyer operators $ND_{n}f$ of the form <span style="font-size: 10px;">$$(ND_{n}f)(x)=intlimits_{0}^{1}K_{n}left( x,t,fleft( tright) right)</span> dt,,,0leq xleq 1,,,,,,nin mathbb{N}, $$ acting on bounded functions on an interval $left[ 0,1right] ,$ where $% K_{n}left( x,t,uright) $ satisfies some suitable assumptions. Here we estimate the rate of convergence at a point $x$, which is a Lebesgue point of $fin L_{1}left( [0,1]right) $ be such that $psi oleftvert frightvert in BVleft( [0,1]right) $, where $psi oleftvert frightvert $ denotes the composition of the functions $psi $ and $% leftvert frightvert $. The function $psi :mathbb{R}_{0}^{+}rightarrow mathbb{R}_{0}^{+}$ is continuous and concave with $psi (0)=0,$ $psi (u)>0$ for $u>0$, which appears from the $left( L-psi right) $ Lipschitz conditions.