Karsli, H. (2015). On convergence of certain nonlinear Durrmeyer operators at Lebesgue points. Bulletin of the Iranian Mathematical Society, 41(3), 699-711.

H. Karsli. "On convergence of certain nonlinear Durrmeyer operators at Lebesgue points". Bulletin of the Iranian Mathematical Society, 41, 3, 2015, 699-711.

Karsli, H. (2015). 'On convergence of certain nonlinear Durrmeyer operators at Lebesgue points', Bulletin of the Iranian Mathematical Society, 41(3), pp. 699-711.

Karsli, H. On convergence of certain nonlinear Durrmeyer operators at Lebesgue points. Bulletin of the Iranian Mathematical Society, 2015; 41(3): 699-711.

On convergence of certain nonlinear Durrmeyer operators at Lebesgue points

^{}Department of Mathematics, Abant Izzet Baysal University, Faculty of Science and Arts, P.O. Box 14280, Bolu, Turkey

Receive Date: 07 December 2013,
Revise Date: 06 March 2014,
Accept Date: 08 April 2014

Abstract

The aim of this paper is to study the behaviour of certain sequence of nonlinear Durrmeyer operators $ND_{n}f$ of the form $$(ND_{n}f)(x)=\int\limits_{0}^{1}K_{n}\left( x,t,f\left( t\right) \right)
dt\,\,\,0\leq x\leq 1,\,\,\,\,\,n\in \mathbb{N},
$$
acting on bounded functions on an interval $\left[ 0,1\right] ,$ where $%
K_{n}\left( x,t,u\right) $ satisfies some suitable assumptions. Here we
estimate the rate of convergence at a point $x$, which is a Lebesgue point
of $f\in L_{1}\left( [0,1]\right) $ be such that $\psi o\left\vert
f\right\vert \in BV\left( [0,1]\right) $, where $\psi o\left\vert
f\right\vert $ denotes the composition of the functions $\psi $ and $%
\left\vert f\right\vert $. 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.