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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].