Abstract
Quite recently, Alotaibi and Mohiuddine (Adv. Differ. Equ. 2012:39, 2012) studied the idea of a random 2-normed space to determine some stability results concerning the cubic functional equation. In this paper, we define and study the concepts of I-convergence and I * -convergence for double sequences in random 2-normed spaces and establish the relationship between these types of convergence, i.e. ,w e show that I * -convergence implies I-convergence in random 2-normed spaces. Furthermore, we have also demonstrated through an example that, in general, I-convergence does not imply I * -convergence in random 2-normed spaces. MSC: 40A05; 46A70
Highlights
Menger [ ] generalized the metric axioms by associating a distribution function with each pair of points of a set
We study the concept of I-convergence and I*-convergence in a more general setting, i.e., in a random -normed space
3 Main results we study the concept of I-convergence and I*-convergence of double sequences in a random -normed space
Summary
Menger [ ] generalized the metric axioms by associating a distribution function with each pair of points of a set. We say that a double sequence x = (xjk) is said to be I*-convergent in (X, F , ∗) or IF* -convergent to if there exists a subset K = {(jm, km) : j < j < · · · ; k < k < · · · } of N × N such that K ∈ F(I) (i.e., N × N \ K ∈ I) and F - limm xjmkm = In this case we write IF* - lim x = and is called the IF* -limit of the double sequence x = (xjk).
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