Asymptotically \(\jmath\)-Lacunary statistical equivalent of order $\alpha$ for sequences of sets
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Authors
Ekrem Savaş
- Istanbul Commerce University, Department of Mathematics, Sutluce-Istanbul, Turkey.
Abstract
This paper presents the following definition which is a natural combination of the definition for asymptotically equivalent of
order \(\alpha\), where \(0 < \alpha \leq 1\), \(\jmath\)-statistically limit, and \(\jmath\)-lacunary statistical convergence for sequences of sets. Let \((X, \rho)\) be a metric
space and \(\theta\) be a lacunary sequence. For any non-empty closed subsets \(A_k, B_k \subseteq X\) such that \(d(x,A_k) > 0\) and \(d(x, B
_k) > 0\) for
each \(x \in X\), we say that the sequences \(\{A_k\}\) and \(\{B_k\}\)are Wijsman asymptotically \(\jmath\)-lacunary statistical equivalent of order \(\alpha\) to
multiple L, where \(0 < \alpha \leq 1\), provided that for each \(\varepsilon > 0\) and each \(x \in X\),
\[\{r\in \mathbb{N}: \frac{1}{h^\alpha_r}|\{k\in I_r: |d(x;A_k,B_k)-L|\geq\varepsilon\}|\geq\delta\}\in \jmath,\]
(denoted by \(\{A_k\}^{s\frac{1}{\theta}(\jmath_W)^\alpha}\{B_k\}\) ) and simply asymptotically \(\jmath\)-lacunary statistical equivalent of order \(\alpha\) if \(L = 1\). In addition, we
shall also present some inclusion theorems. The study leaves some interesting open problems.
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ISRP Style
Ekrem Savaş, Asymptotically \(\jmath\)-Lacunary statistical equivalent of order $\alpha$ for sequences of sets, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 6, 2860--2867
AMA Style
Savaş Ekrem, Asymptotically \(\jmath\)-Lacunary statistical equivalent of order $\alpha$ for sequences of sets. J. Nonlinear Sci. Appl. (2017); 10(6):2860--2867
Chicago/Turabian Style
Savaş, Ekrem. "Asymptotically \(\jmath\)-Lacunary statistical equivalent of order $\alpha$ for sequences of sets." Journal of Nonlinear Sciences and Applications, 10, no. 6 (2017): 2860--2867
Keywords
- Asymptotical equivalent
- sequences of sets
- ideal convergence
- Wijsman convergence
- \(\jmath\)-statistical convergence
- \(\jmath\)-lacunary statistical convergence
- statistical convergence of order \(\alpha\).
MSC
References
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