ID  52077 
FullText URL  
Author 
Malafosse, Bruno de
Malkowsky, Eberhard

Abstract  Given any sequence z = (z_{n})_{n≥1} of positive real numbers
and any set E of complex sequences, we write Ez for the set of all
sequences y = (y_{n})_{n≥1} such that y/z = (y_{n}/z_{n})_{n≥1} ∈ E; in particular,
s_{z}^{(c)}
denotes the set of all sequences y such that y/z converges. In this
paper we deal with sequence spaces inclusion equations (SSIE), which
are determined by an inclusion each term of which is a sum or a sum
of products of sets of sequences of the form Xa(T) and Xx(T) where
a is a given sequence, the sequence x is the unknown, T is a given
triangle, and Xa(T) and Xx(T) are the matrix domains of T in the set X
. Here we determine the set of all positive sequences x for which the
(SSIE) s_{x}^{(c)}
(B(r, s)) s_{x}^{(c)}⊂
(B(r', s')) holds, where r, r', s' and s are real
numbers, and B(r, s) is the generalized operator of the first difference
defined by (B(r, s)y)_{n} = ry_{n}+sy_{n−1} for all n ≥ 2 and (B(r, s)y)_{1} = ry_{1}.
We also determine the set of all positive sequences x for which
ry_{n} + sy_{n−1} /x_{n}
→ l implies
r'y_{n} + s'y_{n−1}
/x_{n}
→ l (n → ∞) for all y
and for some scalar l. Finally, for a given sequence a, we consider the
a–Tauberian problem which consists of determining the set of all x such
that s_{x}^{(c)} (B(r, s)) ⊂ s_{a}^{(c)} .

Keywords  Matrix transformations
BK space
the spaces s<sub>a</sub>, s<doubleint><sub>a</sub><sup>0</sup></doubleint> and s<sub>a</sub><sup>(c)</sup>
(SSIE)
(SSE) with operator
band matrix B(r, s)
Tauberian result

Published Date  201401

Publication Title 
Mathematical Journal of Okayama University

Volume  volume56

Issue  issue1

Publisher  Department of Mathematics, Faculty of Science, Okayama University

Start Page  179

End Page  198

ISSN  00301566

NCID  AA00723502

Content Type 
Journal Article

language 
英語

Copyright Holders  Copyright©2014 by the Editorial Board of Mathematical Journal of Okayama University

File Version  publisher

Refereed 
True

Submission Path  mjou/vol56/iss1/14

JaLCDOI 