Dependence on roughness degree in 2-normed spaces

Abstract

Recently, in summability theory, some mathematicians have studied the concepts of convergence on 2-normed space and rough convergence in normed space. 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. Statistical convergence of sequences of functions in 2-normed spaces was studied by Yegül and Dündar (2017). Arslan and Dündar (2018a) investigated the concepts of I-convergence and I^*-convergence of functions in 2-normed spaces. Also, Arslan and Dündar (2018b) introduced the concepts of I-Cauchy and I^*-Cauchy sequences of functions in 2-normed spaces. Phu (2001) first introduced the idea of rough convergence 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 the concepts of -convergence and -Cauchy sequence in 2-normed space and also investigated some properties such as convexity, boundedness and closeness of -convergence. In this paper, 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.