New Results on the Sequence Spaces Equations Using the Operator of the First Difference

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≥1 such that y/z = (yn/zn)n≥1 ∈ E; in particular, cz denotes the set of all sequences y such that y/z converges. By w∞, we denote the set of all sequences y such that supn≥1(n −1 ∑n k=1 |yk|) < ∞. By ∆ we denote the operator of the first difference defined by ∆ny = yn − yn−1 for all sequences y and all n ≥ 1, with the convention y0 = 0. In this paper, we state some results on the (SSE) (Ea)∆ + Fx = Fb, where c0 ⊂ E ⊂ `∞ and F ⊂ `∞. Then for r, u > 0, we deal with the solvability of the (SSE) (Er)∆ + Fx = Fu, where E, F ∈ {c0, c, `∞} and on the (SSE), (Wr)∆+cx = cu. For instance, the solvability of the (SSE) (Wr)∆+cx = cu consists in determining the set of all positive sequences x, for which the next statement holds. The condition yn/u → l1 holds if and only if there are two sequences α and β with y = α+ β, for which supn≥1(n −1 ∑n k=1 |∆kα|r) <∞ and βn/xn → l2 (n→∞) for all sequences y and for some scalars l1 and l2.