Almost Automorphic Solutions to Nonlinear Difference Equations

Abstract

In the present work, we concentrate on a certain class of nonlinear difference equations and propose sufficient conditions for the existence of their almost automorphic solutions. In our analysis, we invert an appropriate mapping and obtain the main existence outcomes by utilizing the contraction mapping principle. As the second objective of the manuscript, we reconsider one of the landmark results, namely the Bohr–Neugebauer theorem, in the qualitative theory of dynamical equations, and we investigate the relationship between the existence of almost automorphic solutions and the existence of solutions with a relatively compact range for the proposed difference equation type. Thus, we provide a discrete counterpart of the Bohr–Neugebauer theorem due to the almost automorphy notion under some technical conditions.

Keywords

• discrete almost automorphic
• discrete bi-almost automorphic
• fixed point
• contraction
• Bohr–Neugebauer