Dependence on roughness degree in 2-normed spaces

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

On some properties of rough convergence in 2-normed spaces

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

2-Normlu uzaylarda çift fonksiyon dizilerinin I_2-yakınsaklığı ve bazı özellikleri

Çalışmamız boyunca, N tüm doğal sayılar kümesinin ve R tüm gerçek sayıların kümesini belirtecektir. Reel sayı dizilerinin bir genelleştirmesi olan istatistiksel yakınsaklık Fast (1951) tarafından tanımlandı. Daha sonra Schoenberg (1959) ve Fridy (1985) gibi matematikçiler tarafından istatistiksel yakınsaklığın bazı özellikleri incelendi. İstatistiksel yakınsaklığın bir genelleştirmesi olan I-yakınsaklık Kostyrko vd. (2000) tarafından tanılmış olup, bu kavram N doğal sayılar kümesinin alt kümelerinin sınıfı olan I idealinin yapısına bağlıdır. Ayrıca bu çalışmada, I^*-yakınsaklık kavramı tanımlanarak (AP) şartı yardımıyla I-yakınsaklık ile aralarındaki ilişkiler araştırılmıştır. Gezer ve Karakuş (2005) fonksiyon dizilerinin I-noktasal ve düzgün yakınsaklığı ve I^*-noktasal ve düzgün yakınsaklığını araştırıp ve aralarındaki ilişkiyi incelediler. Das vd. (2008) metrik uzaylarda çift dizilerinin I-yakınsaklık kavramını tanıtıp bu bu yakınsaklığın bazı özellilerini incelemişlerdir. Dündar and Altay (2015, 2016) çift fonksiyon dizilerinin noktasal ve düzgün I-yakınsaklık ve I^*-yakınsaklık kavramını ve bununla ilgili özellikleri incelemişlerdir. Dahası Dündar (2015) çift fonksiyon dizilerinin I_2-yakınsaklığının hakkında birçok araştırma yapmıştır. 2-normlu uzay kavramı 1960’lı yıllarda Ga ̈hler tarafından tanıtılmıştır. Gürdal ve Pehlivan (2009) 2-normlu uzaylarda istatistiksel yakınsaklık kavramını tanımlayarak bu kavram ile ilgili özellikleri incelemişlerdir. Gürdal (2006) 2-normlu uzaylarda ideal yakınsaklığı çalışmıştır. 2-normlu uzayda fonksiyon dizilerinin istatistiksel yakınsaklık ve istatistiksel Cauchy dizileri Yegül ve Dündar (2017) tarafından incelenmiştir. Ayrıca. Yegül ve Dündar (2018) 2-normlu uzaylardaki çift fonksiyon dizilerinin istatistiksel yakınsaklık ve istatistiksel Cauchy dizilerinin noktasal ve düzgün yakınsaklık kavramlarını tanımladı. Son zamanlarda, Arslan and Dündar (2018) 2-normlu uzaylardaki fonksiyon dizilerinin I-yakınsaklık ve I-Cauchy dizisi kavramlarını tanımladı. Bu çalışmada, 2-normlu uzaylarda çift fonksiyon dizilerinin I_2-yakınsaklık kavramını tanımlayacağız. Ayrıca bu kavram ile ilgili bazı önemli özellikleri inceleyeceğiz.Throughout the paper, N denotes the set of all positive integers and R the set of all real numbers. Statistical convergence, which is a generalization of the real number sequences, was defined by Fast (1951). Some features of statistical convergence were studied by mathematicians such as Schoenberg (1959) and Fridy (1985). The idea of I-convergence was introduced by Kostyrko et al. (2000) as a generalization of statistical convergence which is based on the structure of the ideal I of subset of N. Also, In this study, I^*-convergence concept was defined and the relations between I-convergence and its relations with (AP) condition were investigated. Gezer and Karakuş (2005) investigated I-pointwise and uniform convergence and I^*-pointwise and uniform convergence of function sequences and they examined the relations between them. Das et al. (2008) introduced the concept of I convergence of double sequences in a metric space and studied some properties of this convergence. Dündar and Altay (2015, 2016) studied the concepts of pointwise and uniformly I convergence and I_2^*-convergence of double sequences of functions and investigated some properties about them. Furthermore, Dündar (2015) investigated some results of I_2-convergence of double sequences of functions. The concept of 2-normed spaces was initially introduced by Ga ̈hler in the 1960's. Gürdal and Pehlivan (2009) describe the concept of statistical convergence in 2-normed spaces and examined the properties related to this concept. Gürdal (2006) studied ideal convergence in 2-normed spaces. Statistical convergence and statistical Cauchy sequence of functions in 2-normed space were studied by Yegül and Dündar (2017). Also, Yegül and Dündar (2018) introduced concepts of pointwise and uniform convergence, statistical convergence and statistical Cauchy double sequences of functions in 2-normed space. Recently, Arslan and Dündar (2018) inroduced I-convergence and I-Cauchy sequences of functions in 2-normed spaces. In this study, we will describe the concept of I_2-convergence of double function sequences in 2-normed spaces. We will also examine some important aspects of this concept