Some convergence results for nonlinear Baskakov-Durrmeyer operators

Keywords:
bounded variation, nonlinear operator, (L−ψ)(L−ψ) Lipschitz condition, pointwise convergenceAbstract
This paper is an introduction to a sequence of nonlinear Baskakov-Durrmeyer operators (NBDn)(NBDn) of the form (NBDn)(f;x)=∫∞0Kn(x,t,f(t))dt(NBDn)(f;x)=∫∞0Kn(x,t,f(t))dt with x∈[0,∞)x∈[0,∞) and n∈N. While Kn(x,t,u) provide convenient assumptions, these operators work on bounded functions, which are defined on all finite subintervals of [0,∞). This paper comprise some pointwise convergence results for these operators in certain functional spaces. As well as this study can be seen as a continuation of studies about nonlinear operators, it is the first study on nonlinear Baskakov-Durrmeyer or modified Baskakov operators, while there were more papers on linear part of the operators.