On the solvability of certain (SSIE) and (SSE), with operators of the form B(r,s,t)

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, s0z denotes the set of all sequences y such that y/z tends to zero. Here, we deal with some extensions of sequence spaces inclusion equations (SSIE) and sequence spaces equations (SSE) with operator. They are determined by an inclusion or identity each term of which is a sum or a sum of products of sets of the form (?a) ? and (?x) ? where ? is any of the symbols s, s0, or s(c), a is a given sequence in U+, x is the unknown, and ? is an infinite matrix. Here, we explicitely calculate the inverse of the triangle B(r,s,t) represented by the operator defined by (B(r,s,t) y)1 = ry1, (B(r,s,t)y)2 = ry2 + sy1 and (B(r,s,t)y) n = ryn + syn-1 + tyn-2 for all n ? 3. Then we determine the set of all x that satisfy the (SSIE) (?x)]B(r,s,t)~ ? ?x, and the (SSE) (?x)B(r,s,t)~ = ?x, where ? ? {s,s0} and B(r,s,t) is the infinite tridiagonal matrix obtained from B(r,s,t) by deleting its first row. For ? = s0 the solvability of the (SSE) (?x) B(r,s,t)~ = ?x consists in determining the set of all x ? U+ for which ryn+1+syn+tyn-1/xn ? 0 ? yn/xn ? 0 (n ? ?) for all y.