On some properties of rough convergence in 2-normed spaces

Abstract

The concepts of convergence on 2-normed space and rough convergence in normed spaces are important in summability theory. The concept of 2-normed spaces was initially introduced by Gähler in the 1960's. Since then, this concept has been studied by many authors. Gürdal and Pehlivan (2009) studied statistical convergence and statistical Cauchy sequence and investigated some properties of statistical convergence in 2-normed spaces. Gürdal and Açık (2008) investigated I-Cauchy and I^*-Cauchy sequences in 2-normed spaces. Sarabadan and Talebi (2011) studied statistical convergence and ideal convergence of sequences of functions in 2-normed spaces. Yegül and Dündar (2017) investigated statistical convergence of sequences of functions in 2-normed spaces. Arslan and Dündar (2018a, 2018b) investigated the concepts of I-convergence, I^*-convergence, I-Cauchy and I^*-Cauchy sequences of functions in 2-normed spaces. The idea of rough convergence was first introduced by Phu (2001) in finite-dimensional normed spaces. In [19], he showed that the set LIM^r x_i is bounded, closed, and convex; and he introduced the notion of rough Cauchy sequence and investigated some properties. He also investigated the relations between rough convergence and other convergence types and the dependence of LIM^r x_i on the roughness degree r. Arslan and Dündar (2018c) defined r-convergence and r-Cauchy sequence in 2-normed space and also investigated some properties of r-convergence. In this study, we investigated relationships between rough convergence and classical convergence and studied some properties about the notion of rough convergence, the set of rough limit points and rough cluster points of a sequence in 2-normed space. Also, we examined the dependence of r-limit LIM_2^r x_n of a fixed sequence (x_n) on varying parameter r in 2-normed space.