On Stancu type generalization of (p,q)-Baskakov-Kantorovich operators

Abstract

In the current paper, we introduce Stancu type generalization of Baskakov-Kantorovich operators based on (p, q)-integers and estimate the moments. We show the convergence of the new operators via the weighted Korovkin theorem. Then we investigate direct results by using Peetre’s Kfunctional and modulus of continuity. In addition, we give pointwise estimation by the help of functions belonging to Lipschitz class. Moreover, we demonstrate the Voronovskaya-type theorem for our operators. In the last section, we represent some illustrative graphics to show the convergence of the constructed operators to the selected function by using MATLAB.

Keywords

• weighted Korovkin theorem,q integers,voronovskaya type theorem

References

1. Acar, T., Mursaleen, M., Mohiuddine, S. A., Stancu Type (p,q)-Szasz-Mirakyan-Baskakov operators, Commun. Fac. Sci. Univ. Ank. Series A1, 67 (1), (2018), 116-128.
2. Acar, T., Aral, A., Mohiuddine, S. A., Mursaleen, M., Approximation by (p,q)-Baskakov-Durrmeyer-Stancu operators, Complex Anal. Oper. Theory, 12, (2018), 1453-1468.
3. Acar, T., Mohiuddine, S. A., On Kantorovich modification of (p,q)-Baskakov operators, J. Inequal. Appl., (2016), 2016:98.
4. Aral, A., Gupta, V., (p,q)-type beta functions of second kind, Adv. Oper. Theory, 1 (1), 2016, 134-146.
5. Aral, A., Gupta, V., Generalized q-Baskakov operators, Mathematica Slovaca, 61 (4), (2011), 619-634.
6. Başcanbaz-Tunca, G., Erencin, A., Taşdelen F., On a sequence of Kantorovich type operators via Riemann type q-integral, Bull. Korean Math. Soc., 51 (2), (2014), 303-315.
7. DeVore, R. A., Lorentz, G. G., Constructive Approximation. Springer, Berlin, 1993.
8. Erençin, A., Olgun, A., Taşdelen, F., Generalized Baskakov type operators, Mathematica Slovaca, 67 (5), (2017), 1269-1277.

9. Gadzhiev, A. D., Theorems of the type of P. P. Korovkin type theorems, Math. Zametki, 20 (5), (1976), 781-786. English Translation, Math. Notes 20 (5-6) (1976), 996-998.
10. Gupta, V., (p,q)-Baskakov-Kantorovich operators, Appl. Math. Inf. Sci., 10 (4), (2016), 1551-1556.
11. Gupta, V., Rassias, Th. M., (p,q)-Genuine Baskakov-Durrmeyer operators, Int. J. Nonlinear Anal. Appl., 7 (2), (2016), 69-76.
12. Hounkonnou, M. N., Désiré J., Kyemba, B., R(p,q)-calculus: differentiation and integration, SUT J. Math., 49 (2), (2013), 145-167.
13. Lenze, B., On Lipschitz type maximal functions and their smoothness spaces, Ned. Akad. Indag. Math., 50, (1988), 53-63.
14. Mishra, V. N., Pandey, S., On (p,q) Baskakov-Durrmeyer-Stancu operators, Adv. Appl. Clifford Algebras, 27, (2017), 1633-1646.
15. Mursaleen, M., Ansari, K. J., Khan, A., On (p,q)-analogue of Bernstein operators, Applied Mathematics and Computation, 266, 2015, 874-882, (Erratum: Appl. Math. Comput. 266 (2015) 874-882.
16. Mursaleen, M., Ansari, K. J. and Khan, A., Some approximation results by (p,q)-analogue of Bernstein-Stancu operators, Applied Mathematics and Computation, 264, (2015), 392-402. [Corrigendum: Appl. Math. Comput, 269 (2015) 744-746]
17. Sadjang, P. N., On the fundamental theorem of (p,q)-calculus and some (p,q)-Taylor formulas, arXiv:1309.3934v1 [math.QA], 2013.
18. Sharma, H., Gupta, C., On (p,q)-generalization of Szasz-Mirakyan Kantorovich Operators, Boll Unione Mat Ital, 8, (2015), 213-222.
19. Yüksel, İ., Dinlemez Kantar, Ü., Altın, B., On the (p,q)-Stancu generalization of a Genuine Baskakov-Durrmeyer type operators, Int. J. of Analysis and App., 15 (2), (2017), 138-145.