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De Malafosse, Bruno
- Solvability of Sequence Spaces Equations Using Entire and Analytic Sequences and Applications
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Affiliations
1 Universite du Havre, FR
1 Universite du Havre, FR
Source
The Journal of the Indian Mathematical Society, Vol 81, No 1-2 (2014), Pagination: 97-114Abstract
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 special sequence spaces equations (SSE) with operators, which are determined by an identity each term of which is a sum or a sum of products of sets of the form Xa (T) and Xf(x) (T) where U+ maps to itself, x is either of the symbols s, s0, Γ, or Λ. We solve (SSE) of the form Xa+Xx' = Xb', and systems of the form Xa+Xx'(Δ) = Xb', and Xb' ⊂ Xx', where X, X' are any of the symbols so, s(c), s, Γ, or Λ. For instance the system sa(c) + sx(c) (Δ) = sb(c) and sb(c) ⊂ sx(c) where Δ is the operator of the first difference means that, bn/xn → l1 (n → ∞), for some l1 ∈ C, and for any given y ∈ ω, we have yn/bn → l1 (n → ∞ if and only if there are u, v ∈ s such that y = u + v and un/an → l1 and Δνn/xn → l" (n → ∞) for some scalars l, l' and l".Keywords
A—Entire Sequence, A—Analytic Sequence, Multiplier of Sets of Sequences, Sequence Spaces Inclusion Equations, Sequence Spaces Equations with Operator.
- Solvability of the Sequence Spaces Equations With Operators Ca+Cx = (Cb)B(r,s) And Ca+Cx = (Cb)G(α,α)
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Authors
Affiliations
1 Department of Mathematics, University of Le Havre, 76600 Le Havre, FR
1 Department of Mathematics, University of Le Havre, 76600 Le Havre, FR
Source
The Journal of the Indian Mathematical Society, Vol 89, No 1-2 (2022), Pagination: 84–99Abstract
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 ? E; in particular, ca, (or s(c) a ) denotes the set of all sequences y such that y/a converges. In this paper, we solve sequence spaces equations of the form (Ex)B(r,s) = Ea, where E 2 {c0, c, `1}. Then we apply these results to the solvability of each of the (SSE) with operators ca + cx = (cb)B(r,s) and ca + cx = (cb)G(,), where B (r, s) is a double band matrix, and G(, ) is the factorable matrix with positive sequences and , that is, the triangle whose the nonzero entries are defined by [G(, )]nk = nk.
Keywords
BK space, matrix transformations, multiplier of sequence spaces, sequence spaces equations, sequence spaces equations with operator.References
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