On ideal convergence Fibonacci difference sequence spaces

Vakeel A Khan,Sameera A A Abdullah,Ayaz Ahmad,Kamal M A S Alshlool,Rami K A Rababah

doi:10.1186/s13662-018-1639-2

Abstract

The Fibonacci sequence was firstly used in the theory of sequence spaces by Kara and Başarir (Casp. J. Math. Sci. 1(1):43–47, 2012). Afterward, Kara (J. Inequal. Appl. 2013(1):38, 2013) defined the Fibonacci difference matrix F̂ by using the Fibonacci sequence (f_{n}) for nin{{0, 1, ldots}} and introduced new sequence spaces related to the matrix domain of F̂. In this paper, by using the Fibonacci difference matrix F̂ defined by the Fibonacci sequence and the notion of ideal convergence, we introduce the Fibonacci difference sequence spaces c^{I}_{0}(hat {F}), c^{I}(hat{F}), and ell^{I}_{infty}(hat{F}). Further, we study some inclusion relations concerning these spaces. In addition, we discuss some properties on these spaces such as monotonicity and solidity.

Highlights

Let N and R denote the sets of natural and real numbers, respectively
Throughout the paper, ∞, c, and c0 are the classes of bounded, convergent, and null sequences, respectively, with norm x ∞ = supk∈N |xk|
We say that A defines a matrix transformation from λ into μ, and we denote it by writing A : λ −→ μ if for every sequence x = ∈ λ, the sequence Ax = {An(x)}, the A-transform of x, is in μ, where

Summary

Introduction

Let N and R denote the sets of natural and real numbers, respectively. By ω we denote the vector space of all real sequences. I-convergence, that is, the notion of I-summability and introduced new sequence spaces cIA and mIA, the I-convergence field and bounded I-convergence field of an infinite matrix A, respectively. By combining the definitions of Fibonacci difference matrix Fand ideal convergence we introduce the sequence spaces cI0(F ), cI(F ), and Definition 1.3 ([24]) A sequence x = (xn) ∈ ω is said to be I-convergent to a number ∈ R if, for every > 0, the set {n ∈ N : |xn – | ≥ } ∈ I, and we write I- lim xn =. Definition 1.9 ([27]) A sequence space E is said to be monotone if it contains the canonical preimages of its step space (i.e., if for all infinite K ⊆ N and (xn) ∈ E, the sequence (αnxn) with αn = 1 for n ∈ K and αn = 0 otherwise belongs to E).

I-Convergence Fibonacci difference sequence spaces

Conclusion