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Application of the Infinite Matrix Theory to the Solvability of Sequence Spaces Inclusion Equations with Operators

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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≥1E. In particular, ca denotes the set of all sequences y such that y/a converges. We deal with sequence spaces inclusion equations (SSIE) of the form FEa + \( {F}_x^{\prime } \) with eF and explicitly find the solutions of these SSIE when a = (rn)n≥1, F is either c or s1, and E and F′ are any sets c0, c, s1,p, w0, and w. Then we determine the sets of all positive sequences satisfying each SSIE cDr * (c0) + cx and cDr * (s1) + cx, where Δ is the operator of the first difference defined by Δny = yn− yn−1 for all n ≥ 1 with y0 = 0. Then we solve the SSIE cDr * \( {E}_{C_1}+{s}_x^{(c)} \) with E ∈ {c, s1} and s1Dr * \( {\left({s}_1\right)}_{C_1} \) + sx, where C1 is the Cesàro operator defined by (C1)ny = n1\( {\sum}_{k=1}^n{y}_k \) for all y. We also deal with the solvability of the sequence spaces equations (SSE) associated with the previous SSIE and defined as Dr * \( {E}_{C_1}+{s}_x^{(c)} \) = c with E ∈ {c0, c, s1} and Dr * \( {E}_{C_1} \) + sx = s1 with E ∈ {c, s1}.

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Correspondence to B. de Malafosse.

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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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