Bounded solutions of fractional discrete equations of positive non-integer orders

The paper considers a linear fractional discrete equation x^α x(n + 1) = λx(n) + δ(n), n = 0, 1,...
where Δ^α is the fractional α-order difference, α > 0, λ ∈ R and δ: {0, 1,...} → R. A problem is considered of the existence of a solution x: {0, 1,...} → R satisfying |x(n)| < M, n = 0, 1,..., where M is a constant. This problem is also considered for an equation Δα x(n + 1) = λ(n)x(n) + δ(n, x(n), x(n − 1),..., x(0)), n = 0, 1,..., where λ: {0, 1,...} → R, δ: {0, 1,..., n} × R × R × ... × R, (n+1 times)→ R, generalizing the previous one.