Abstract :
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. We denote by w∞ and w0 the sets of all sequences y such that supn( n^-1∑n k=1 |yk|) ∞ and limn→ ∞ ( n-1∑n k=1 |yk|) = 0. We also use the sets of analytic and entire sequences denoted by Λ and Γ and defined by supn|yn|1/n ∞ and limn→ ∞ |yn|1/n = 0, respectively. In this paper we explicitly calculate the solutions of (SSE) of the form ET+Fx = Fb in each of the cases E = c0, c,(ell)∞ ,(ell)p, (p≥1), w0, Γ, or Λ, F = c, or(ell)∞, and T is either of the triangles∆, or ∑, where∆ is the operator of the first difference, and ∑ is the operator defined by ∑ny = ∑n k=1 yk. For instance the solvability of the (SSE) Γ∑+Λx = Λb consists in determining the set of all positive sequences x = (xn)n that satisfy the statement: supn n{(|yn|/bn)1/n} ∞ if and only if there are u, v Єw with y = u+v such that limn→ ∞│∑n k=1 uk│1/n = 0 and sup n {( |vn| /xn)1/n } ∞ for all y.
