on convergence of certain nonlinear durrmeyer operators at lebesgue points

the aim of this paper is to study the behaviour of certain sequence of nonlinear durrmeyer operators $nd_{n}f$ of the form $$(nd_{n}f)(x)=intlimits_{0}^{1}k_{n}left( x,t,fleft( tright) right) dt,,,0leq xleq 1,,,,,,nin mathbb{n}, $$ 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.