Given any sequence *a* = (*a*_{n})_{n≥1} of positive real numbers and any set *E* of complex sequences, we write *E*_{a} for the set of all sequences *y* = (*y*_{n})_{n≥1} such that *y/a* = (*y*_{n}*/a*_{n})_{n≥1} ∈ *E.* In particular, *c*_{a} denotes the set of all sequences *y* such that *y/a* converges. We deal with sequence spaces inclusion equations (SSIE) of the form *F* ⊂ *E*_{a} + \( {F}_x^{\prime } \) with *e* ∈ *F* and explicitly find the solutions of these SSIE when *a* = (*r*^{n})_{n≥1}*, F* is either *c* or *s*_{1}*,* and *E* and *F*′ are any sets *c*_{0}*, c, s*_{1}*,* ℓ_{p}*, w*_{0}*,* and *w*_{∞}*.* Then we determine the sets of all positive sequences satisfying each SSIE *c* ⊂ *D*_{r} * (*c*_{0})_{∆} + *c*_{x} and *c* ⊂ *D*_{r} * (*s*_{1})_{∆} + *c*_{x}*,* where Δ is the operator of the first difference defined by Δ_{n}*y* = *y*_{n}*− y*_{n−1} for all *n ≥* 1 with *y*_{0} = 0*.* Then we solve the SSIE *c* ⊂ *D*_{r} * \( {E}_{C_1}+{s}_x^{(c)} \) with *E* ∈ {*c, s*_{1}} and *s*_{1} ⊂ *D*_{r} * \( {\left({s}_1\right)}_{C_1} \) + *s*_{x}*,* where *C*_{1} is the Cesàro operator defined by (*C*_{1})_{n}*y* = *n*^{−1}\( {\sum}_{k=1}^n{y}_k \) for all *y.* We also deal with the solvability of the sequence spaces equations (SSE) associated with the previous SSIE and defined as *D*_{r} * \( {E}_{C_1}+{s}_x^{(c)} \) = *c* with *E* ∈ {*c*_{0}*, c, s*_{1}} and *D*_{r} * \( {E}_{C_1} \) + *s*_{x} = *s*_{1} with *E* ∈ {*c, s*_{1}}*.*

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 8, pp. 1040–1052, August, 2019.

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de Malafosse, B. Application of the Infinite Matrix Theory to the Solvability of Sequence Spaces Inclusion Equations with Operators.
*Ukr Math J* **71, **1186–1201 (2020) doi:10.1007/s11253-020-01717-w

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