Aspects regarding the existence of fixed points of the iterates of Stancu operators

Abstract

In the papers Iterates of Stancu Operators, via Contraction Principle (2002), respectively Iterates of Bernstein Operators, via Contraction Principle (2004), author I. A. Rus studied the existence of fixed points for Stancu operators Pn,α,β and Bernstein operators Bn. The aim of this paper is to find conditions for which the Stancu operators Pn,α,β are contractions on the graph, in order to demonstrate that the contraction principle can be applied for the study of the existence of fixed points for iterates of Stancu operators. The method used for this paper is the spectral method, which was also used in the paper Over-iterates of Bernstein-Stancu operators (2007), authors Gonska, Piţul and Raşa. The study began with finding constant C∈[0,1[ that would satisfy the inequality ||Pn,α,β2 (f)-Pn,α,β (f)|| ≤ C ||Pn,α,β (f)-f||, for any f∈ C[0,1]. The conclusion is that there are conditions for which the Stancu operators are contractions on the graph, and the methods used for the study of the existence of fixed points of their iterates can also be extended to the study of the existence of fixed points of other linear operators.

Keywords

• iterate operators
• fixed point
• stancu operators