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


Year of publication


Published in

Fasciculi Mathematici

Journal year: 2015 | Journal number: nr 55

Article type

scientific article

Publication language


  • BK space
  • spaces of strongly bounded sequences
  • sequence spaces equations
  • sequence spaces
  • equations with operator

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≥1E; 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/xM (F,F). The question is: what happens when we consider the perturbed equation E + Fx = Fb where E is a special linear space of sequences? In this paper we deal with the perturbed sequence spaces equations (SSE), defined by (Ea)∆ + sx(c)= sb(c) where E=c0, or lp, (p >1) and ∆ is the operator of the first difference defined by ∆ny= ynyn−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 = (xn)n that satisfy the next statement. The condition yn/bnL1 holds if and only if there are two sequences u, v with y = u + v such that ∆nu/an → 0 and vn/xnL2 (n→∞)for all y and for some scalars L1 and L2. Then we deal with the resolution of the equation (Ea)∆ + sx0 = sb0 for E = c, or s1, and give applications to particular classes of (SSE).

Pages (from - to)

109 - 131



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

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