creative_2019_28_1_61_68_001

On Wijsman asymptotically lacunary I-statistical equivalence of weight g of sequence of sets

Kişi, Ömer

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creative_2019_28_1_61_68

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This paper presents the following definition which is a natural combination of the definitions of asymptotically equivalence, \mathcal{I}-convergence, statistical limit, lacunary sequence, and Wijsman convergence of weight g; where g:\mathbb{N}\rightarrow \left[ 0,\infty \right) is a function satisfying \lim_{n\rightarrow \infty }g\left( n\right) =\infty and \frac{n}{g\left( n\right) }\nrightarrow 0 as n\rightarrow \infty for sequence of sets. Let (X,\rho ) be a metric space, \theta =\{k_{r}\} be a lacunary sequence and \mathcal{I}\subseteq 2^{ \mathbb{N}} be an admissible ideal. 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 \mathcal{I}-asymptotically lacunary statistical equivalent of multiple L of weight g if for every \varepsilon >0, \delta >0 and for each x\in X,

    \begin{equation*} \left \{ r\in \mathbb{N}:\frac{1}{g\left( h_{r}\right) }\left \vert \left \{ k\in I_{r}:\left \vert \frac{d(x,A_{k})}{d(x,B_{k})}-L\right \vert \geq \varepsilon \right \} \right \vert \geq \delta \right \} \in \mathcal{I} \end{equation*}

(denoted by A_{k}\overset{S_{\theta }^{L}\left( \mathcal{I}_{W}\right) ^{g}} {\sim }B_{k} ). We mainly investigate their relationship and also make some observations about these classes.

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Kişi, Ömer

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