On convergence of certain nonlinear Durrmeyer operators at Lebesgue points

Document Type : Research Paper

Author

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

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.

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  • Receive Date: 07 December 2013
  • Revise Date: 06 March 2014
  • Accept Date: 08 April 2014
  • First Publish Date: 01 June 2015