Generalized Bernstein–Durrmeyer Operators and the Associated Limit Semigroup Original Research Article

The aim of this paper is the study of a new sequence of positive linear approximation operators Mn, λ on C([0, 1]) which generalize the classical Bernstein–Durrmeyer operators. After proving a Voronovskaja-type result, we show that there exists a strongly continuous positive contraction semigroup on C([0, 1]) which may be expressed in terms of powers of these operators. As a direct consequence, we are able to represent explicitly the solutions of the Cauchy problems associated with a particular class of second order differential operators.