2015

Fasciculi Mathematici

Journal year: 2015 | Journal number: nr 55

scientific article

english

EN Given any sequence a = (a_n)_n≥1 of positive real numbers and any set E of complex sequences, we write E_a for the set of all sequences y = (y_n)_n≥1 such that y/a = (y_n/a_n)_n≥1 ∈ E; in particular, s_a^(c) denotes the set of all sequences y such that y/a converges. For any linear space F of sequences, we have F_x = F_b if and only if x/b and b/x ∈ M (F,F). The question is: what happens when we consider the perturbed equation E + F_x = F_b where E is a special linear space of sequences? In this paper we deal with the perturbed sequence spaces equations (SSE), defined by (E_a)∆ + s_x^(c)= s_b^(c) where E=c₀, or l_p, (p >1) and ∆ is the operator of the first difference defined by ∆_ny= y_n − y_n−1 for all n ≥ 1 with the convention y0 = 0. For E=c₀ the previous perturbed equation consists in determining the set of all positive sequences x = (x_n)_n that satisfy the next statement. The condition y_n/b_n→L₁ holds if and only if there are two sequences u, v with y = u + v such that ∆_nu/a_n → 0 and v_n/xn→L₂ (n→∞)for all y and for some scalars L₁ and L₂. Then we deal with the resolution of the equation (E_a)∆ + s_x⁰ = s_b⁰ for E = c, or s₁, and give applications to particular classes of (SSE).

109 - 131

10.1515/fascmath-2015-0018

CC BY-NC-ND (attribution - noncommercial - no derivatives)

Download file

public

10