About the Iterates of Some Operators Depending on a Parameter and Preserving the Affine Functions

  • Ana-Maria Acu
  • Voichiţa Adriana RaduEmail author
Research Paper


In this paper, we analyze the behavior of the iterates of some positive linear operators depending on a parameter and which preserve the affine function. We give some quantitative results using first and second moduli of continuity and, also, we show that the iterates converge uniformly toward the linear function that interpolates at the endpoints 0 and 1.


Linear positive operators Iterates Q-integers Degree of approximation First- and second-order moduli of continuity Contraction principles Weakly Picard operators 

2010 MSC

41A36 41A25 47H10 54H25 


  1. Agratini O (2008) On the iterates of a class of summation-type linear positive operators. Comp Math Appl 55:1178–1180MathSciNetCrossRefzbMATHGoogle Scholar
  2. Agratini O, Rus IA (2003) Iterates of a class of discrete linear operators via contraction principle. Comment Math Univ Carol 44(3):555–563MathSciNetzbMATHGoogle Scholar
  3. Altomare F (2013) On some convergence criteria for nets of positive operators on continuous function spaces. J Math Anal Appl 398:542–552MathSciNetCrossRefzbMATHGoogle Scholar
  4. Altomare F, Campiti M (1994) Korovkin type approximation theory and its applications, vol 17. de Gruyter studies in mathematics. Walter de Gruyter, BerlinCrossRefzbMATHGoogle Scholar
  5. Cătinaş T, Otrocol D (2013) Iterates of Bernstein type operators on a square with one curved side via contraction principle. Int J Fixed Point Theory Comput Appl 14(1):97–106MathSciNetzbMATHGoogle Scholar
  6. Gavrea I, Ivan M (2010) On the iterates of positive linear operators preserving the affine functions. J Math Anal Appl 372:366–368MathSciNetCrossRefzbMATHGoogle Scholar
  7. Gavrea I, Ivan M (2011) On the iterates of positive linear operators. J Approx Theory 163:1076–1079MathSciNetCrossRefzbMATHGoogle Scholar
  8. Gonska HH (1979) Quantitative Aussagen zur Approximation durch positive lineare Operatoren, Dissertation, Universitt DuisburgGoogle Scholar
  9. Gonska HH, Kacsó D, Piţul P (2006) The degree of convergence of over-iterated positive linear operators. J Appl Funct Anal 1(4):403–423MathSciNetzbMATHGoogle Scholar
  10. Gonska HH, Raşa I (2006) The limiting semigroup of the Bernstein iterates: degree of convergence. Acta Math Hungar 111:119–130MathSciNetCrossRefzbMATHGoogle Scholar
  11. Karlin S, Ziegler Z (1970) Iterates of positive approximation operators. J Approx Theory 3:310–339CrossRefzbMATHGoogle Scholar
  12. Kelisky RP, Rivlin TJ (1967) Iterates of Bernstein polynomials. Pac J Math 21:511–520MathSciNetCrossRefzbMATHGoogle Scholar
  13. Lupaş A (1987) A q-analogue of the Bernstein operator. Univ Cluj-Napoca Sem Numer Stat Calc Prepr 9:85–92MathSciNetzbMATHGoogle Scholar
  14. Mahmudov NI (2010) Approximation theorems for certain positive linear operators. Appl Math Lett 23:812–817MathSciNetCrossRefzbMATHGoogle Scholar
  15. Mahmudov NI (2010) Approximation properties of complex q-Szasz-Mirakjian operators in compact disks. Comput Math Appl 60(6):1784–1791MathSciNetCrossRefzbMATHGoogle Scholar
  16. Novak G (2009) Approximation properties for generalized q-Bernstein polynomials. J Math Anal Appl 350:50–55MathSciNetCrossRefGoogle Scholar
  17. Oruç H, Tuncer N (2002) On the Convergence and Iterates of q-Bernstein polynomials. J Approx Theory 117:301–313MathSciNetCrossRefzbMATHGoogle Scholar
  18. Ostrovska S (2003) q-Bernstein polynomials and their iterates. J Approx Theory 123:232–255MathSciNetCrossRefzbMATHGoogle Scholar
  19. Phillips GM (1997) Bernstein polynomials based on the q-integers. Ann Numer Math 4:511–518MathSciNetzbMATHGoogle Scholar
  20. Radu VA (2016) Iterates of q and (p, q)-Bernstein operators, via contraction principles. Sci Stud Res Ser Math Inf 26(2):83–94zbMATHGoogle Scholar
  21. Raşa I (2009) Asymptotic behavior of iterates of positive linear operators. Jaen J. Approx. 1:195–204MathSciNetzbMATHGoogle Scholar
  22. Raşa I (2010) \(C_0\) semigroups and iterates of positive linear operators: asymptotic behavior. Rend Circ Mat Palermo 2(Suppl. 82):123–142Google Scholar
  23. Rus IA (1993) Weakly Picard mappings. Comment Math Univ Carol 34(4):769–773MathSciNetzbMATHGoogle Scholar
  24. Rus IA (2004) Iterates of Bernstein operators, via contraction principle. J Math Anal Appl 292:259–261MathSciNetCrossRefzbMATHGoogle Scholar
  25. Rus IA (2010) Iterates of stancu operators (via fixed point principles) revisited. Fixed Point Theory 11:369–374MathSciNetzbMATHGoogle Scholar

Copyright information

© Shiraz University 2017

Authors and Affiliations

  1. 1.Department of Mathematics and InformaticsLucian Blaga University of SibiuSibiuRomania
  2. 2.Department of Statistics-Forecasts-MathematicsBabes-Bolyai University, FSEGACluj-NapocaRomania

Personalised recommendations