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+. Axioms2021, 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.
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+. Axioms2021, 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)n1 of positive real numbers and any set E of complex sequences, we write Ea for the set of all sequences y = (yn)n1 such that y=a = (yn=an)n1 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)n1 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
Copyright:
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