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

Abstract

Given any sequence z = (zn)n≥1 of positive real numbers and any set E of complex sequences, we write Ez for the set of all sequences y = (yn)n≥1 such that y/z = (yn/zn)n≥1 ∈ E; in particular, sz(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) sx(c) (B(r, s)) sx(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 = ryn+syn−1 for all n ≥ 2 and (B(r, s)y)1 = ry1. We also determine the set of all positive sequences x for which ryn + syn−1 /xn → l implies r'yn + s'yn−1 /xn → 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 sx(c) (B(r, s)) ⊂ sa(c)