Abstract
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
if and only if \({|\hat{a}| \neq 1}\). Using this result we study behaviour of bounded at infinity solutions of the functional equation
under some conditions posed on functions \({\alpha_j(t), j = 0, 1,\ldots, k - 1,\phi}\) and f.
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Brzdęk, J., Stević, S. Behaviour at infinity of solutions of some linear functional equations in normed spaces. Aequat. Math. 87, 379–389 (2014). https://doi.org/10.1007/s00010-013-0194-x
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DOI: https://doi.org/10.1007/s00010-013-0194-x