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

Let \({\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}, I = (d, \infty), \phi : I \to I}\) be unbounded continuous and increasing, X be a normed space over \({\mathbb{K}, \mathcal{F} : = \{f \in X^I : {\rm lim}_{t \to \infty} f(t) {\rm exists} \, {\rm in} X\},\hat{a} \in \mathbb{K}, \mathcal{A}(\hat{a}) : = \{\alpha \in \mathbb{K}^I : {\rm lim}_{t \to \infty} \alpha(t) = \hat{a}\},}\) and \({\mathcal{X} : = \{x \in X^I : {\rm lim} \, {\rm sup}_{t \to \infty} \|x(t)\| < \infty\}}\) . We prove that the limit lim _{t → ∞} x(t) exists for every \({f \in \mathcal{F}, \alpha \in \mathcal{A}(\hat{a})}\) and every solution \({x \in \mathcal{X}}\) of the functional equation $$x(\phi(t)) = \alpha(t) x(t) + f(t)$$ if and only if \({|\hat{a}| \neq 1}\) . Using this result we study behaviour of bounded at infinity solutions of the functional equation $$x(\phi^{k]}(t)) = \sum_{j=0}^{k-1} \alpha_j(t) x (\phi^{j]}(t)) + f(t),$$ under some conditions posed on functions \({\alpha_j(t), j = 0, 1,\ldots, k - 1,\phi}\) and f.