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f-rough cauchy sequences
Abstract
In this article, we narrated the new notion of rough f-statistical convergence and rough f-statistical Cauchy sequences, successively becoming a more generalized version of rough statistical convergence and rough statistical Cauchy sequences. Consecutively, compare the following important theorems with those of List´an-Garc´ıa [Quaest. Math. 37(4) (2014), 525–530], Phu [Numer. Funct. Anal. Optim. 22(1–2) (2001), 199–222 and Numer. Funct. Anal. Optim. 24(3–4) (2003), 285–301], and Aytar [Numer. Funct. Anal. Optim. 29(3–4) (2008), 291–303].
(i) Suppose x = {x n}n∈N is a bounded sequence in some finite dimensional normed space X. Let C denote the cluster point set of this sequence. Then, DX(C) is the minimal Cauchy degree and r X (C) is the minimal convergence degree r̄ of {x n}n∈ N, where
r̄=inf{r ∈ R+ : LIMr x ≠ ∅ }, D X (C) = sup ∥a−b∥, rX(C)= inf sup ∥a−b∥.
a,b∈C a∈X b∈C
That means D X (C) = min{ρ ∈ R+ : {x n} n∈N is a ρ-Cauchy sequence }, and
LIMr x = ∅ for r < r X (C),
≠ ∅, for r ≥ r X (C).
(ii) Suppose ρ ≥ 0 and {x n} n∈N is a ρ-Cauchy sequence in some normed space X.
Then, {x n}n∈N is r-convergent for every r > 2−1 J ( X )ρ. If X is finite dimensional then {x n} n∈N is r-convergent for every r ≥ 2−1 J ( X ) ρ.
(iii) If x = { x n } n∈N is a sequence in a finite dimensional normed space Rn, then
st − LIM r x = ∩ B̅ r (c) = {x∗ ∈ R n : Γx ⊆ B̅r (x∗) }.
c∈Γx
If we reformulate the above three results regarding rough f-statistical convergence and rough f-statistical Cauchy sequences in an infinite dimensional normed space, subsequent assertions will be false. Apart from these, we find the minimal f-statistical convergence degree and the minimal f-statistical Cauchy degree of a sequence in any dimensional normed spaces.
Mathematics Subject Classification (2020): Primary: 40A35, 46B15; Secondary: 46B50.
