Some convergence results for nonlinear Baskakov-Durrmeyer operators

Abstract

This paper is an introduction to a sequence of nonlinear Baskakov-Durrmeyer operators $(NBD_{n})$ of the form \[ (NBD_{n})(f;x) =\int_{0}^\infty K_{n}(x,t,f(t))\,dt \] with $x\in [0,\infty)$ and $n\in\mathbb{N}$. While $K_{n}(x,t,u)$ provide convenient assumptions, these operators work on bounded functions, which are defined on all finite subintervals of $[0,\infty)$. 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.

Keywords

• bounded variation
• nonlinear operator
• $(l-\psi)$ lipschitz condition
• pointwise convergence