@article { author = {Karsli, H.}, title = {On convergence of certain nonlinear Durrmeyer operators at Lebesgue points}, journal = {Bulletin of the Iranian Mathematical Society}, volume = {41}, number = {3}, pages = {699-711}, year = {2015}, publisher = {Iranian Mathematical Society (IMS)}, issn = {1017-060X}, eissn = {1735-8515}, doi = {}, 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.}, keywords = {nonlinear Durrmeyer operators,bounded variation,Lipschitz condition,pointwise convergence}, url = {http://bims.iranjournals.ir/article_643.html}, eprint = {http://bims.iranjournals.ir/article_643_a61793a4bf19240e5ae4ac83d5dad504.pdf} }