Given any sequence z = (zn) n>1 of positive real numbers and any set E of complex sequences, we write Ez for the set of all sequences y = (yn) n\geq 1 such that y/z = (y_n/z_n) n\geq 1 \in E; in particular, c_z denotes the set of all sequences y such that y/z converges. We denote by w_\infty the set of all sequences y such that sup_{n\geq 1} (n^{-1}\sum_{k=1}^{n}|y_k|) < \infty by \Delta we denote the operator of the rst difference defined by \Delta_ y= y_n -y_{n-1} for all sequences y and all n \geq 1, with the convention y_0 = 0. In this paper we state some results on the (SSE) (E_a)_{\Delta} + F_x = F_b, where c_0\subset E \subset \ell_\infty1, and F \subset \ell_infty. Then for r, u > 0 we deal with the solvability of the (SSE) (Er)_{\Delta} + F_x = F_u, where E, F \in {c_0; c; \ell_\infty} and on the (SSE) (Wr) + cx = cu. For instance the solvability of the (SSE) (Wr)_{\Delta} + c_x = c_u consists in determining the set of all positive sequences x, for which the next statement holds. The condition y_n/u^n \to \ell_1 holds if and only if there are two sequences and with y = + for which sup_{n\geq 1} (n^{-1}\sum_{k=1}^{n}|\Delta_k\alpha|)r^_-k} < \infty and \beta_n/x_n \to \ell_2 (n\to \infty) for all sequences y and for some scalars l\ell_1 and \ell_2.