Application of the infinite matrix theory to the solvability of sequence spaces inclusion equations with operators
UDC 517.9 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/a=(y_n/a_n)_n≥ 1E. 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 FE_a+F_x’ with eF and explicitly find the solutions of these SSIE when a=(r^n)_n≥ 1, F is either c or s_1, and E, F’ are any sets c_0, c, s_1, _p, w_0, and w_∞ . Then we determine the sets of all positive sequences satisfying each of the SSIE cD_r* (c_0)_Δ +c_x and cD_r* (s_1)_Δ+c_x, where Δ is the operator of the first difference defined by Δ _ny=y_n-y_n-1 for all n≥ 1 with y_0=0. Then we solve the SSIE cD_r* E_C_1 +s_x^(c) with E{ c,s_1} and s_1D_r*(s_1) _C_1 +s_x, where C_1, is the Cesaro operator defined by (C_1) _ny=n^-1 olimits_k=1^ny_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}., number=8, journal=Ukrains’kyi Matematychnyi Zhurnal, author=de, Malafosse B., year=2019, month=Aug., pages=1040–1052