Therefore, in spite of any disadvantages which it may possess, this definition is better adapted than others to the study of many problems in double sequences and series.
In this paper, we define an analogoue of Sargent's sequence in the double sequence space Ω.
In this paper, in a natural way, we first define statistical convergence for double sequences in 2-normed spaces using a two-valued measure and also prove some interesting theorems.
In this presentation, our goal is to extend a few results known in the literature from ordinary (single) sequences to double sequences in topological groups and to give some important inclusion theorems.
In the recent past, Mursaleen and Mohiuddine [4, 5] defined these notions for double sequences in locally solid Riesz spaces as well as in intuitionistic fuzzy normed spaces and proved some interesting results.
In this paper we introduce and study lacunary statistical convergence for double sequences in topological groups and we shall also present some inclusion theorems.
In 1926, Robison presented a four-dimensional analogue of regularity for double sequences in which he added an additional assumption of boundedness.
If x = ( x j k ) is a double sequence in X such that I F - lim x = ℓ, then I F ∗ - lim x = ℓ.
Observe that for F in this example, the double sequence { ( x n, y n ) } n ≥ 0, defined by the Picard-type iteration x n + 1 = F ( x n, y n ), y n + 1 = F ( y n, x n ), n ≥ 0, (2.4).
By a double sequence in E we mean a function f : N × N → E defined by f ( n, m ) = x n, m ∈ E. The double sequence { x n, m } is said to converge strongly to x ∗ if for a given ϵ > 0, there exist integers N, M > 0 such that ∀ n ≥ N, m ≥ M, we have that ∥ x n, m − x ∗ ∥ < ϵ.
By a double sequence in E is meant a function f : N × N → E defined by f ( n, m ) = x n, m ∈ E. A double sequence { x n, m } is said to converge strongly to x ∗ if given any ϵ > 0, there exist N, M > 0 such that ∥ x n, m − x ∗ ∥ < ϵ for all n ≥ N, m ≥ M. If ∀ n, r ≥ N, ∀ m, t ≥ M, we have ∥ x n, r − x m, t ∥ < ϵ, then the double sequence is said to be Cauchy.
