Rough $\Delta \mathcal{I}-$Convergence

Abstract

In this paper, we study the concept of rough $\mathcal{I}-$convergence for difference sequences in $\left( \mathbb{R}^{n},\left\Vert .\right\Vert \right) $ where $ \mathbb{R}^{n}$ denotes the real $n-$dimensional space with the norm $\left\Vert .\right\Vert $. At the same time, we examine some basic properties of the set $\mathcal{I}-\lim_{\Delta x_{I}}^{r}=\lbrace x_{\ast}\in\mathbb{R}^{n}:\Delta x_{i}\overset{r}{\rightarrow}x_{\ast}\rbrace $ which is called as $r$-$\mathcal{I-}$ limit set of the difference sequence $\left( \Delta x_{i}\right) $ and we give some properties of $\mathcal{I}-\lim \inf \Delta x_{i},$ $\mathcal{I}-\lim \sup \Delta x_{i}$ and $\mathcal{I}-$core$\left\{ \Delta x_{i}\right\} .$

Keywords

• Statistical convergence
• I-convergence
• rough convergence
• difference sequences
• I-limit point set
• I-core

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