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

Malafosse B. de

Fasciculi Mathematici

2015

Nr 55

109--131

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).