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Note on Limit-Periodic Solutions of the Difference Equation x_{t + 1} − [h(x_{t}) + λ]x_{t} = r_{t}, λ > 1

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Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic

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Centre PMF, Laboratory SAMM, Université Paris I Panthéon—Sorbonne, 90, Rue de Tolbiac, 75 634 Paris CEDEX 13, France

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Author to whom correspondence should be addressed.

Received: 7 January 2019 / Revised: 31 January 2019 / Accepted: 1 February 2019 / Published: 5 February 2019

(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)

As a nontrivial application of the abstract theorem developed in our recent paper titled “Limit-periodic solutions of difference and differential systems without global Lipschitzianity restricitions”, the existence of limit-periodic solutions of the difference equation from the title is proved, both in the scalar as well as vector cases. The nonlinearity h is not necessarily globally Lipschitzian. Several simple illustrative examples are supplied.