Approximation properties of the fractional q-integral of Riemann-Liouville integral type Szasz-Mirakyan-Kantorovich operators

Abstract

In the present paper, we introduce the fractional q-integral of Riemann-Liouville integral type Szász-Mirakyan-Kantorovich operators. Korovkin-type approximation theorem is given and the order of convergence of these operators are obtained by using Lipschitz-type maximal functions, second order modulus of smoothness and Peetre's K-functional. Weighted approximation properties of these operators in terms of modulus of continuity have been investigated. Then, for these operators, we give a Voronovskaya-type theorem. Moreover, bivariate fractional q- integral Riemann-Liouville fractional integral type Szász-Mirakyan-Kantorovich operators are constructed. The last section is devoted to detailed graphical representation and error estimation results for these operators.

Keywords

• Szasz-Mirakyan-Kantorovich operators
• q-integral of Riemann-Liouville
• Voronovskaya-type

References

1. Ditzian, Z, Totik, V., Moduli of Smoothness, Springer Series in Computational Mathematics, New-York Springer, 1987.
2. Aral, A., Limmam, L.M, Özsaraç, F., Approximation properties of Szasz-Mirakjan-Mirakyan Kantorovich type operators, Math. Methods Appl. Sci., 42(16) (2019), 5233-5240. https://doi:10.1002/mma.5280
3. Duman, O., Özarslan, M.A., Vecchia, B.D., Modified Sz´asz-Mirakjan-Kantorovich operators preserving linear functions, Turk J Math., 33 (2009), 151-158. https://doi:10.3906/mat-0801-2
4. Aral, A., Inoan, D., Rasa, I., On the generalized Sz´asz-Mirakyan operators, Results Math., 65 (2014), 441-452. https://doi:10.1007/s00025-013-0356-0
5. Acar, T., Aral, A., Cardenas-Morales, D., Garrancho, P., Szasz-Mirakyan type operators which fix exponentials, Results in Math., 72 (2017), 1393-1404. https://doi:10.1007/s00025-017-0665-9
6. Acar, T., Aral, A., Gonska, H., On Szasz-Mirakyan operators preserving $e^2ax$, $a > 0$, Mediterr. J. Math., 14(6) (2017). https://doi.org/10.1007/s00009-016-0804-7
7. Gupta, V., Approximation with Positive Linear Operators and Linear Combinations, Springer International Publishing, 2017.
8. Gupta, V., Aral, A., A note on Szasz-Mirakyan-Kantorovich type operators preserving $e^-x$, Positivity, 22 (2018), 415-423. https://doi.org/10.1007/s11117-017-0518-5

9. Otto, S., Generalization of S. Bernstein’s polynomials to the infinite interval, J. Res. Nat. Bur. of Standards, 45(3) (1950), 239-245.
10. Mirakjan, G.M., Approximation of continuous functions with the aid of polynomials, In Dokl. Acad. Nauk SSSR, 31 (1941), 201-205.
11. Devore, R.A., Lorentz, G.G., Constructive Approximation, Springer-Verlang, New York-London, 1993.
12. Gadjieva, A.D., A problem on the convergence of a sequence of positive linear operators on unbounded sets, and theorems that are analogous to P.P. Korovkin’s theorem, Doklady Akademii Nauk SSSR, 218(5) (1974), 1001-1004.
13. Doğru, O., Gadjieva, E., Ağırlıklı uzaylarda Szasz tipinde operatörler dizisinin sürekli fonksiyonlara yaklaşımı, II. Kızılırmak Uluslararası Fen Bilimleri Kongresi Bildiri Kitabı, Kırıkkale, (1998), 29-37.
14. Dhamija, M., Pratap, R., Deo, N., Approximation by Kantorovich form of modified Szasz-Mirakyan operators, Appl. Math. Comput., 317 (2018), 109-120. https://doi.org/10.1016/j.amc.2017.09.004
15. Gupta, V., Acu, A.M., On Baskakov-Szasz-Mirakyan-type operators preserving exponential type functions, 22(3) (2018), 919-929. https://doi.org/10.1007/s11117-018-0553
16. Mursaleen, M., Alotaibi, A., Ansari, K.J., On a Kantorovich variant of Szasz- Mirakjan operators, J. Funct. Spaces, 2016. https://doi.org/10.1155/2016/1035253
17. Acar, T., Gupta, V., Aral, A., Rate of convergence for generalized Szasz operators, Bull. Math. Sci., 1 (2011), 99-113. https://doi.org/10.1007/s13373-011-0005-4
18. Agrawal, P.N., Gupta, V., Kumar, A.S., Kajla, A., Generalized Baskakov-Szasz type operators, Appl. Math. Comput., 236 (2014), 311-324. https://doi.org/10.1016/j.amc.2014.03.084
19. Aral, A., A generalization of Szasz-Mirakyan operators based on q-integers, Math. Comput. Modelling, 47(9-10) (2008), 1052-1062. https://doi.org/10.1016/j.amc.2014.03.084
20. Finta, Z., Govil, N.K., Gupta, V., Some results on modified Szasz-Mirakjan operators, J. Math. Anal. Appl., 327(2) (2007), 1284-1296. https://doi.org/10.3906/mat-0801-2
21. Mazhar, S.M., Totik, V., Approximation by modified Szasz operators, Acta Sci. Math., 49 (1985), 257-269.
22. Totik, V., Approximation by Szasz-Mirakjan-Kantorovich operators in $L_ {p}$ $(p > 1)$, Analysis Mathematica, 9(2) (1983), 147-167. https://doi.org/10.1007/bf01982010
23. Dahmani, Z., Tabharit, L., Taf, S., New generalizations of Gruss inequality using Riemann-Liouville fractional integrals, Bull. Math. Anal. Appl., 2(3) (2010), 93-99.
24. Katugompola, U.N., New approach generalized fractional integral, Applied Math and Comp., 218(3) (2011), 860-865. https://doi.org/10.1016/j.amc.2011.03.062
25. Latif, M.A., Hussain, S., New inequalities of Ostrowski type for co-ordineted convex functions via fractional integrals, Journal of Fractional Calculus and Applications, 2(9) (2012), 1-15.
26. Romero, L.G., Luque, L.L., Dorrego, G.A., Cerutti, R.A., On the k-Riemann Liouville fractional derivative, Int. J. Contemp. Math. Sciences, 8(1) (2013), 41-51. http://dx.doi.org/10.12988/ijcms.2013.13004
27. Tunc, M., On new inequalities for h-convex functions via Riemann-Liouville fractional integration, Filomat, 27(4) (2013), 559–565. https://doi.org/10.2298/FIL1304559T
28. Mahmudov, N.I., On q-Parametric Szasz-Mirakjan operators, Mediterr. J. Math., 7 (2010), 297–311. https://doi.org/10.1007/s00009-010-0037-0
29. Mahmudov, N.I, Approximation properties of complex q-Szasz-Mirakjan operators in compact disks, Computers and Mathematics with Applications, 60(6) (2010), 1784-1791. https://doi.org/10.1016/j.camwa.2010.07.009
30. Aral, A., Gupta, V., The q-derivative and applications to q-Szasz Mirakyan operators, Calcolo, 43(3) (2006), 151-170. https://doi.org/10.1007/s10092-006-0119-3
31. Cai, Q., Zeng, X.M., Cui, Z., Approximation properties of the modification of Kantorovich type q-Szasz operators, J. Computational Analysis and Applications, 15(1) (2013), 176-187.
32. Gal, S., Mahmudov, N.I, Kara, M., Approximation by complex q-Szasz-Kantorovich operators in compact disks, q > 1, Complex Anal. Oper. Theory, 7 (2013), 1853-1867. https://doi.org/10.1007/s11785-012-0257-3
33. Örküvü, M., Doğru, O., q-Szasz Mirakyan Kantorovich type operators preserving some test functions, Appl. Math. Lett., 24(9) (2011), 1588-1593. https://doi.org/10.1016/j.aml.2011.04.001
34. Mahmudov, N.I., Vijay, G., On certain q-analogue of Szasz Kantorovich operators, J. Appl. Math. Comput., 37 (2011), 407-419. https://doi.org/10.1007/s12190-010-0441-4
35. Tariboon, J., Ntouyas, S.K., Agarwal, P., New concepts of fractional quantum calculusand applications to impulsive fractional q-difference equations, Advance in Difference Equations, 18 (2015). https://doi.org/10.1186/s13662-014-0348-8
36. Kac, V., Cheung, P., Quantum Calculus, Universitext, New York, 2002.
37. Becker, M., Global approximation theorems for Szasz -Mirakjan and Baskakov operators in polynomial weight spaces, Indiana Univ. Math. J., 27(1) (1978), 127-142.
38. Lenze, B., On Lipschitz type maximal functions and their smoothness spaces, Nederl. Akad. Indag. Math., 91(1) (1988), 53-63.
39. Lopez-Moreno, A.J., Weighted simultaneous approximation with Baskakov type operators, Acta Mathematica Academiae Scientiarum Hungaricae, 104 (2004), 143-151. https://doi.org/10.1023/B:AMHU.0000034368.81211.23
40. Volkov, V.I., On the convergence of sequences of linear positive operators in the space of continuous functions of two variables, Dokl. Akad. Nauk SSSR, 115(1) (1957), 17-19.
41. Gupta, V., Agarwal, R.P., Convergence Estimates in Approximation Theory, Springer International Publishing, 2014.