On $\Delta$-Uniform and $\Delta$-Pointwise Convergence on Time Scale

Year 2020, , 218 - 225, 25.06.2020

Mustafa Seyyit Seyyidoğlu Ayşe Karadaş

https://doi.org/10.37094/adyujsci.546724

Abstract

In this article, we define the concept of $\Delta$-Cauchy$, \Delta$-uniform convergence and $\Delta$-pointwise convergence of a family of functions $\{f_{j}\}_{j\in \mathbb{J}}$, where $\mathbb{J}$ is a time scale. We study the relationships between these notions. Moreover, we introduced sufficient conditions for interchangeability of $\Delta$-limitation with Riemann $\Delta$-integration or $\Delta$-differentiation. Also, we obtain the analogue of the well-known Dini's Theorem.

Keywords

$ \Delta $-Convergence, $ \Delta $-Cauchy, Statistical Convergence

References

• [1] Hilger, S., Ein maßkettenkalkül mit anwendung auf zentrumsmannigfaltigkeiten, PhD Thesis, 1989.
• [2] Hilger, S., Analysis on measure chains—a unified approach to continuous and discrete calculus, Results in Mathematics 18, 1-2, 18-56, 1990.
• [3] Fast, H., Sur la convergence statistique, Colloquium Mathematicae, 2(3-4), 1951.
• [4] Schoenberg, I.J,.The integrability of certain functions and related summability methods, The American Mathematical Monthly, 66(5), 361-775, 1959.
• [5] Fridy, J.A., On statistical convergence, Analysis,5(4), 301-314, 1985.
• [6] Seyyidoğlu, M.S., Tan, N.Ö,. A note on statistical convergence on time scale, Journal of Inequalities and Applications, 2012(1), 219, 2012.
• [7] Altın, Y., Koyunbakan, H., Yılmaz, E., Uniform statistical convergence on time scales, Journal of Applied Mathematics, vol. 2014, 6 pages, 2014.
• [8] Yılmaz, E., Altın, A., Koyunbakan, H., λ-Statistical convergence on time scales, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 23, 69-78, 2016.
• [9] Ceylan, T., Duman, O., Fundamental Properties of Statistical Convergence and Lacunary Statistical Convergence on Time Scales, Filomat, 31(14), 2017.