On $\left( p,q\right) $-analog of Stancu operators of rough $\lambda$- statistically $\rho$-Cauchy convergence of triple sequence spaces

Year 2021, Volume: 3 Issue: 1, 20 - 29, 29.04.2021

Arulmani Indumathı , Ayhan Esi , Nagarajan Subramanian , Mustafa Kemal Özdemir

https://doi.org/10.47087/mjm.900156

https://izlik.org/JA74BS58GY

Abstract

In this work, using the concept of natural density, we introduce the $\left(
p,q\right) $-analogue of the Stancu-beta operators of rough $\lambda$%
-statistically $\rho$-Cauchy convergence on triple sequence spaces. We define
the set of Bernstein Stancu beta opeators of rough statistical limit points of
a triple sequence spaces and obtain to $\lambda-$statistical convergence
criteria associated with this set. Also, we examine the relations between the
set of Bernstein-Stancu beta operators of rough $\lambda$-statistically
$\rho$-Cauchy convergence of triple sequences.

Keywords

$\left( p,q\right) -$ calculus , $\rho$-Cauchy , $r\lambda $-statistical convergence , Bernstein Stancu beta operators , rough statistical convergence , natural density , triple sequences

References

• S. Aytar, Rough statistical convergence, Numerical Functional Analysis Optimization \textbf{29(3)} (2008) 291-303.
• A. Esi, On some triple almost lacunary sequence spaces defined by Orlicz functions, Research and Reviews: Discrete Mathematical Structures \textbf{1(2)} (2014) 16-25.
• A. Esi and M. Necdet Catalbas, Almost convergence of triple sequences, Global Journal of Mathematical Analysis \textbf{2(1)} (2014) 6-10.
• A. Esi and E. Savas, On lacunary statistically convergent triple sequences in probabilistic normed space, Appl. Math. and Inf. Sci. \textbf{9(5)} (2015) 2529-2534.
• A. Esi, S. Araci and M. Acikgoz, Statistical convergence of Bernstein operators, Appl. Math. and Inf. Sci. \textbf{10(6)} (2016) 2083-2086.
• A. J. Dutta, A. Esi and B. C. Tripathy, Statistically convergent triple sequence spaces defined by Orlicz function, Journal of Mathematical Analysis \textbf{4(2)} (2013) 16-22.
• A. Esi, S. Araci and Ayten Esi, $\lambda$-statistical convergence of Bernstein polynomial sequences, Advances and Applications in Mathematical Sciences \textbf{16(3)} (2017) 113-119.
• A. Esi, N. Subramanian and Ayten Esi, On triple sequence space of Bernstein operator of rough $I_{\lambda}$-convergence pre-Cauchy sequences, Proyecciones Journal of Mathematics \textbf{36(4)} (2017) 567-587.
• S. Debnath, B. Sarma and B. C. Das, Some generalized triple sequence spaces of real numbers, Journal of Nonlinear Analysis and Optimization \textbf{6(1)} (2015) 71-79.
• S. K. Pal, D. Chandra and S. Dutta, Rough ideal convergence, Hacettepe Journal Mathematics and Statistics \textbf{42(6)} (2013) 633-640.
• H. X. Phu, Rough convergence in normed linear spaces, Numerical Functional Analysis Optimization \textbf{22} (2001) 201-224.
• A. Sahiner, M. Gurdal and F. K. Duden, Triple sequences and their statistical convergence, Selcuk J. Appl. Math. \textbf{8(2)} (2007) 49-55.
• A. Sahiner, B. C. Tripathy, Some $I$-related properties of triple sequences, Selcuk J. Appl. Math. \textbf{9(2)} (2008) 9-18.
• N. Subramanian and A. Esi, The generalized tripled difference of $\chi^{3}$ sequence spaces, Global Journal of Mathematical Analysis \textbf{3(2)} (2015) 54-60.