Behaviour at infinity of solutions of some linear functional equations in normed spaces

Abstract

Let K ∈ {ℝ, ℂ}, I = (d, ∞), φ: I → I be unbounded continuous and increasing, X be a normed space over K, F: = {f ∈ X I : lim t → ∞ f(t) exists in X}, â ∈ K, A(â): = {α ∈ K I : lim t → ∞ α(t) = â}, and X: = {x ∈ X I : limsup t → ∞ {double pipe}x(t){double pipe} < ∞}. We prove that the limit lim t → ∞ x(t) exists for every f ∈ F, α ∈ A(â) and every solution x ∈ X of the functional equation x(φ(t)) = α(t)x(t) + f(t) if and only if {pipe}â{pipe} ≠ 1. Using this result we study behaviour of bounded at infinity solutions of the functional equation (Formual Presented) under some conditions posed on functions α j (t), j = 0, 1,..., k - 1, φ and f.