On the Solvability of Certain (SSIE) and (SSE), with Operators of the Form B (r, s, t)

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 = (yn/zn)(n >= 1) is an element of E; in particular, s(z)(0) 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 (chi(a))(Lambda) and (chi(x))(Lambda) where chi is any of the symbols s, s(0), or s((c)), a is a given sequence in U+, x is the unknown, and Lambda 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) = ry(2) + sy(1) and (B (r, s, t) y)(n) = ry(n) + syn-1 + tyn-2 for all n >= 3. Then we determine the set of all x that satisfy the (SSIE) (chi x)<(B(r,s,t))over tilde> subset of chi(x), and the (SSE) (chi x)<(B(r,s,t))over tilde> = chi(x), where chi is an element of {s, s(0)} and B(r,]s, t) is the infinite tridiagonal matrix obtained from B(r, s,t) by deleting its first row. For chi= s(0) the solvability of the (SSE) (chi x)<(B(r,s,t))over tilde> = chi(x) consists in determining the set of all x is an element of U+ for which ry(n+1) + sy(n) + ty(n-1/)x -> 0 double left arrow double right arrow y(n) xn -> 0 (n -> infinity) for all y.