ON THE SOLVABILITY OF CERTAIN (SSIE) WITH OPERATORS OF THE FORM B(r, s)

Given any sequence z = (z_n)_n≥1 of positive real numbers and any set E of complex sequences, we write Ez for the set of all sequences y = (y_n)_n≥1 such that y/z = (y_n/z_n)_n≥1 ∈ E; in particular, s_z^(c) denotes the set of all sequences y such that y/z converges. In this paper we deal with sequence spaces inclusion equations (SSIE), which are determined by an inclusion each term of which is a sum or a sum of products of sets of sequences of the form Xa(T) and Xx(T) where a is a given sequence, the sequence x is the unknown, T is a given triangle, and Xa(T) and Xx(T) are the matrix domains of T in the set X . Here we determine the set of all positive sequences x for which the (SSIE) s_x^(c) (B(r, s)) s_x^(c)⊂ (B(r', s')) holds, where r, r', s' and s are real numbers, and B(r, s) is the generalized operator of the first difference defined by (B(r, s)y)_n = ry_n+sy_n−1 for all n ≥ 2 and (B(r, s)y)₁ = ry₁. We also determine the set of all positive sequences x for which ry_n + sy_n−1 /x_n → l implies r'y_n + s'y_n−1 /x_n → l (n → ∞) for all y and for some scalar l. Finally, for a given sequence a, we consider the a–Tauberian problem which consists of determining the set of all x such that s_x^(c) (B(r, s)) ⊂ s_a^(c) .