Working Paper Article Version 1 This version is not peer-reviewed

# New Results on the (SSIE) with Operator of the Form F E + F0x Involving the Spaces of Strongly Summable and Convergent Sequences by the Cesàro Method

Version 1 : Received: 18 June 2021 / Approved: 24 June 2021 / Online: 24 June 2021 (16:42:44 CEST)

A peer-reviewed article of this Preprint also exists.

de Malafosse, B. New Results on the SSIE with an Operator of the form FΔEFx′ Involving the Spaces of Strongly Summable and Convergent Sequences using the Cesàro Method+. Axioms 2021, 10, 157. de Malafosse, B. New Results on the SSIE with an Operator of the form FΔ⊂EFx′ Involving the Spaces of Strongly Summable and Convergent Sequences using the Cesàro Method+. Axioms 2021, 10, 157.

## Abstract

Given any sequence a = (an)n1 of positive real numbers and any set E of complex sequences, we write Ea for the set of all sequences y = (yn)n1 such that y=a = (yn=an)n1 2 E. In this paper, we use the spaces w1, w0 and w of strongly bounded, summable to zero and summable sequences, that are the sets of all sequences y such that ? n?1Pn k=1 jykj  n is bounded, tends to zero and such that y ? le 2 w0, for some scalar l, respectively, (cf. [24, 22]). These sets where used in the statistical convergence, (cf. [17, Chapter 4]). Then we deal with the solvability of each of the (SSIE) F  E + F0x where E is a linear space of sequences, F = c0, c, 1, w0, w, or w1 and F0 = c0, c, or 1. For instance, the solvability of the (SSIE) w  w0+s(c) x consists in determining the set of all sequences x = (xn)n1 2 U+ that satisfy the following statement. For every sequence y that satisfy the condition limn!1 n?1Pn k=1 jyk ? yk?1 ? lj = 0, there are two sequences u and v, with y = u+v such that limn!1 n?1Pn k=1 jukj = 0 and limn!1 (vn=xn) = L for some scalars l and L. These results extend those stated in [11, 12, 10].

## Keywords

BK space, matrix transformations, multiplier of sequence spaces, sequence spaces inclusion equations.

## Subject

Computer Science and Mathematics, Algebra and Number Theory