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Extension of some results on the (SSIE) and the (SSE) of the from F ⊂ Ɛ+F’x and Ɛ+Fx = F

de Malafosse B.

Fasciculi Mathematici

2017

Nr 59

107--123

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 ∈ E. In this paper we deal with the solvability of the (SSIE) of the form l∞ ⊂ Ɛ + F’x where E is a linear space of sequences and F’ is either c0, or l∞ and we solve the (SSIE) c0 ⊂ Ɛ + sx for Ɛ ⊂ (sα)Δ and α ∈ c0. Then we study the (SSIE) c ⊂ Ɛ + s( c) x and the (SSE) Ɛ +s( c ) x = c. Then we apply the previous results to the solvability of the (SSE) of the form (lpr)Δ ) + Fx = F for p ≥ 1 and F is any of the sets c0, c, or l∞. These results extend some of those given in [8] and [9].

Solvability of certain sequence spaces inclusion equations with operators

Malafosse B. de

Demonstratio Mathematica

2013

Vol. 46, nr 2

299--314

In this paper, we deal with sequence spaces inclusion equations (SSIE), which are determined by an inclusion where each term is a sum or a sum of products of sets of the form a(T) and f(x)(T) where f maps U+ to itself, and (...), the sequence x is the unknown and T is a given triangle. Here, we determine the set of all sequences x with positive entries such that (…) and (…) where (...). We are led to study, among other things, the inclusion equations (…) and (…) where (…) is the operator of first differences defined by (…) for (…) with (…). The first (SSIE) leads to determine the set of all sequences x such that (…) and (…) implies (…). These results generalize some of the results given in [1].