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1

Solvability of sequence spaces equations of the from (Ea)Δ + Fx = Fb

Malafosse B. de

Fasciculi Mathematici

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2015

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Nr 55

109--131

EN

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 Є E; in particular, sa(c) denotes the set of all sequences y such that y/a converges. For any linear space F of sequences, we have Fx = Fb if and only if x/b and b/x Є M (F, F). The question is: what happens when we consider the perturbed equation Ɛ + Fx = Fb where Ɛ is a special linear space of sequences? In this paper we deal with the perturbed sequence spaces equations (SSE), defined by (Ea)Δ + sx(c) = sb(c) where E = c0, or lp, (p > 1) and Δ is the operator of the first difference defined by Δny = yn - yn-1 for all n ≥ 1 with the convention y>sub>0 = 0. For E = c0 the previous perturbed equation consists in determining the set of all positive sequences x = (xn)n that satisfy the next statement. The condition yn/bn → L1 holds if and only if there are two sequences u, v with y = u + v such that Δnu/an → 0 and vn/xn → L2 (n → ∞) for all y and for some scalars L1 and L2. Then we deal with the resolution of the equation (Ea)Δ + sx0 = sb>0 for E = c, or s1, and give applications to particular classes of (SSE).

2

Solvability of certain sequence spaces inclusion equations with operators

Malafosse B. de

Demonstratio Mathematica

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2013

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Vol. 46, nr 2

299--314

EN

In this paper, we deal with sequence spaces inclusion equations (SSIE), which are determined by an inclusion where each term is a sum or a sum of products of sets of the form a(T) and f(x)(T) where f maps U+ to itself, and (...), the sequence x is the unknown and T is a given triangle. Here, we determine the set of all sequences x with positive entries such that (…) and (…) where (...). We are led to study, among other things, the inclusion equations (…) and (…) where (…) is the operator of first differences defined by (…) for (…) with (…). The first (SSIE) leads to determine the set of all sequences x such that (…) and (…) implies (…). These results generalize some of the results given in [1].

3

Sets of sequences that are strongly r-bounded and matrix transformations between these sets

Malafosse B. de

Demonstratio Mathematica

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2003

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Vol. 36, nr 1

155--171

EN

In this paper we are giving some new properties of the operator of first-difference mapping a space sr into itself and we are dealing with the spaces sr(A h) and sr((A+) ). Next are given some properties of the spaces wr^(A,//),