## Abstract

Using the weakly Picard operators technique, we study the convergence of the iterates of some hivariate and trivariate Cheney-Sharma operators. Also, we generalize the procedure for the multivariate case.

## Authors

T. **Catinas
**(Babes Bolyai Univ.)

D. **Otrocol**

(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

## Keywords

## Cite this paper as:

T. Catinas, D. Otrocol, *Iterates of multivariate Cheney-Sharma operators*, J. Comput. Anal. Appl., Vol. 15 (2013), no. 7, pp. 1240-1246

## About this paper

##### Journal

Journal of Computational Analysis and Applications

##### Publisher Name

Eudoxus Press, Cordova, USA

##### DOI

##### Print ISSN

1521-1398

##### Online ISSN

##### MR

MR3075657

##### ZBL

## Google Scholar

[1] O. Agratini, I.A. Rus, *Iterates of a class of discrete linear operators via contraction principle*, Comment. Math. Univ. Caroline, 44(2003), 555-563.

[2] O. Agratini, I.A. Rus, *Iterates of some bivariate approximation process via weakly Picard operators*, Nonlinear Analysis Forum, 8(2)(2003), 159-168.

[3] F. Altomare, M. Campiti, *Korovkin-type Approximation Theory and its Applications*, de Gruyter Studies in Mathematics, 17, Walter de Gruyter & Co., Berlin, 1994.

[4] A.M. Bica, *On iterates of Cheney-Sharma operator*, J. Comput. Anal. Appl., 11(2009), No. 2, 271-273.

[5] E.W. Cheney, A. Sharma, On a generalization of Bernstein polynomials, Riv. Mat. Univ. Parma, 5(1964), 77-84.

[6] G. Coman, T. Catinas, *Interpolation operators on a triangle with one curved side*, BIT Numerical Mathematics, 50(2010), No. 2, 243-267.

[7] T. Catinas, D. Otrocol, *Iterates of Bernstein type operators on a square with one curved side via contraction principle*, Fixed Point Theory, to appear.

[8] I. Gavrea, M. Ivan, *The iterates of positive linear operators preserving the affine functions*, J. Math. Anal. Appl., 372(2010), 366-368.

[9] I. Gavrea, M. Ivan, *The iterates of positive linear operators preserving the constants*, Appl. Math. Lett., 24(2011), No. 12, 2068-2071.

[10] I. Gavrea, M. Ivan, *On the iterates of positive linear operators*, J. Approximation Theory, 163(2011), No. 9, 1076-1079.

[11] H. Gonska, D. Kacso, P. Pitul, *The degree of convergence of over-iterated positive linear operators*, J. Appl. Funct. Anal., 1(2006), 403-423.

[12] H. Gonska, P. Pitul, I. Rasa, *Over-iterates of Bernstein-Stancu operators*, Calcolo, 44(2007), 117-125.

[13] H. Gonska, I. Rasa, *The limiting semigroup of the Bernstein iterates: degree of convergence*, Acta Math. Hungar., 111(2006), No. 1-2, 119-130.

[14] S. Karlin, Z. Ziegler, *Iteration of positive approximation operators*, J. Approximation Theory 3(1970), 310-339.

[15] R.P. Kelisky, T.J. Rivlin, *Iterates of Bernstein polynomials*, Pacific J. Math., 21(1967), 511-520.

[16] I. Rasa, *Asymptotic behaviour of iterates of positive linear operators*, Jaen J. Approx., 1 (2009), no. 2, 195204.

[17] I.A. Rus, *Generalized contractions and applications*, Cluj Univ. Press, 2001.

[18] I.A. Rus, *Iterates of Stancu operators, via contraction principle*, Stud. Univ. Babes–Bolyai Math., 47(2002), No. 4, 101-104.

[19] I.A. Rus, *Iterates of Bernstein operators, via contraction principle*, J. Math. Anal. Appl., 292(2004), 259-261.

[20] I.A. Rus, A. Petrusel, M.A. Serban, *Weakly Picard operators: Weakly Picard operators, equivalent definitions, applications and open problems*, Fixed Point Theory, 7 (2006), 3-22.

[21] D.D. Stancu, L.A. Cabulea, D. Pop, Approximation of bivariate functions by means of the operator S α,β;a,b m,n , Stud. Univ. Babes–Bolyai Math., 47(2002), No. 4, 105-113.

[22] I. Tascu, *On the approximation of trivariate functions by means of some tensorproduct positive linear operators*, Facta Universitatis (Nis), Ser. Math. Inform., 21 (2006), 23-28.