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Given any sequence a=(an)n≥1 of positive real numbers and any set E of complex sequences, we can use Ea to represent the set of all sequences y=(yn)n≥1 such that y/a=(yn/an)n≥1∈E. In this paper, we use the spaces w∞, w0 and w of strongly bounded, summable to zero and summable sequences, which are the sets of all sequences y such that n−1∑k=1nykn is bounded and tends to zero, and such that y−le∈w0, for some scalarl. These sets were used in the statistical convergence. Then we deal with the solvability of each of the SSIE FΔ⊂Ɛ+Fx′, where Ɛ is a linear space of sequences, F=c0, c, ℓ∞, w0, w or w∞, and F′=c0, c or ℓ∞. For instance, the solvability of the SSIE wΔ⊂w0+sxc relies on determining the set of all sequences x=xnn≥1∈U+ that satisfy the following statement. For every sequence y that satisfies the condition limn→∞n−1∑k=1nyk−yk−1−l=0, there are two sequences u and v, with y=u+v such that limn→∞n−1∑k=1nuk=0 and limn→∞vn/xn=L for some scalars l and L.

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New Results on the SSIE with an Operator of the form FΔ ⊂ Ɛ + Fx′ Involving the Spaces of Strongly Summable and Convergent Sequences Using the Cesàro Method