Asymptotically $\jmath$-Lacunary statistical equivalent of order $\alpha$ for sequences of sets

Savaş, Ekrem¹

¹ Istanbul Commerce University, Department of Mathematics, Sutluce-Istanbul, Turkey

Journal of nonlinear sciences and its applications, Tome 10 (2017) no. 6, p. 2860-2867.

Résumé

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.

Détail

DOI : 10.22436/jnsa.010.06.01

Classification : 40A35, 46A45
Keywords: Asymptotical equivalent, sequences of sets, ideal convergence, Wijsman convergence, \(\jmath\)-statistical convergence, \(\jmath\)-lacunary statistical convergence, statistical convergence of order \(\alpha\).

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