Rough I-Convergence

Abstract

Abstract. In this work, using the concept of I-convergence and using the concept of rough convergence, we introduced the notion of rough I-convergence and the set of rough I-limit points of a sequence and obtained two rough I-convergence criteria associated with this set. Later, we proved that this set is closed and convex. Finally, we examined the relations between the set of I-cluster points and the set of rough I-limit points of a sequence. Background and introductionThe concept of convergence of a sequence of real numbers has been extended to statistical convergence independently by Fast [4] [14]. In general, statistically convergent sequences satisfy many of the properties of ordinary convergent sequences in metric spaces.The idea of I-convergence was introduced by Kostyrko et al.[6] as a generalization of statistical convergence which is based on the structure of the ideal I of subset of the set of natural numbers. Nuray and Ruckle [9] indepedently introduced the same with another name generalized statistical convergence. Kostyrko et al. [7] studied the idea of I-convergence and extremal I-limit points and Demirci [3] studied the concepts of I-limit superior and limit inferior.Salát, Tripathy and Ziman [13] introduced the notion of c I A and m I A , the I-convergence field and bounded I-convergence field of an infinite matrix A.The idea of rough convergence was first introduced by Phu [10] in finitedimensional normed spaces. In [10], he showed that the set LIM r x is bounded, closed, and convex; and he introduced the notion of rough Cauchy sequence. He also investigated the relations between rough convergence and other 2010 Mathematics Subject Classification: 40A05, 40A35.

Rough I 2 ${I}_{2}$ -lacunary statistical convergence of double sequences

“…In this section, we recall some definitions and notations, which form the base for the present study [6, 10, 11, 23, 32, 33, 35, 38, 40, 42–46]. …”

“…Also, Aytar [39] studied that the rough limit set and the core of a real sequence. Recently, Dündar and Çakan [11, 40], Pal et al [41] introduced the notion of rough -convergence and the set of rough -limit points of a sequence and studied the notion of rough convergence and the set of rough limit points of a double sequence. Further this notion of rough convergence of double sequence has been extended to rough statistical convergence of double sequence by Malik et al [42] using double natural density of in the similar way as the notion of convergence of double sequence in Pringsheim sense was generalized to statistical convergence of double sequence.…”

I - convergence on cone metric spaces

“…Definition 2.9. [9] A sequence x = {x n } in a normed linear space is said to be rough I-convergent of roughness degree r to x * for some r ≥ 0 if for any ε > 0 the set {n ∈ N : x n − x * ≥ r + ε} ∈ I.…”

“…The definition of I-bounded sequence in a normed linear space has been given in [9] as follows: Now we recall the definition of rough convergence in cone metric space from [7].…”

“…Using these ideas Aytar [1], in 2008 gave the concept of rough statistical convergence of a sequence. In 2014, using the concepts of I-convergence and rough convergence Dündar et al [9] introduced the notion of rough I-convergence. Many more works have been done in different direction [2,19,20] by several authors using this idea given by Phu [23].…”