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ID 52077
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Author
Malafosse, Bruno de
Malkowsky, Eberhard
Abstract
Given any sequence z = (zn)n≥1 of positive real numbers and any set E of complex sequences, we write Ez for the set of all sequences y = (yn)n≥1 such that y/z = (yn/zn)n≥1 ∈ E; in particular, sz(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) sx(c) (B(r, s)) sx(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 = ryn+syn−1 for all n ≥ 2 and (B(r, s)y)1 = ry1. We also determine the set of all positive sequences x for which ryn + syn−1 /xn → l implies r'yn + s'yn−1 /xn → 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 sx(c) (B(r, s)) ⊂ sa(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
2014-01
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
0030-1566
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