Affine periodicity is a generalization of the notion of conventional periodicity and it is a symmetry property for classes of functions. This study is concerned with the existence of (Q, T)-affine periodic solutions of discrete dynamical systems. Sufficient conditions for the main results are proposed due to discrete exponential dichotomy and fixed point theory. | Turk J Math (2018) 42: 2260 – 2269 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article On the affine-periodic solutions of discrete dynamical systems 1 Halis Can KOYUNCUOĞLU1,∗,, Murat ADIVAR2 , Department of Engineering Sciences, Faculty of Engineering and Architecture, İzmir Kâtip Çelebi University, İzmir, Turkey 2 Department of Management, Marketing, and Entrepreneurship, Fayetteville State University, Fayetteville, North Carolina, USA Received: • Accepted/Published Online: • Final Version: Abstract: Affine periodicity is a generalization of the notion of conventional periodicity and it is a symmetry property for classes of functions. This study is concerned with the existence of (Q, T ) -affine periodic solutions of discrete dynamical systems. Sufficient conditions for the main results are proposed due to discrete exponential dichotomy and fixed point theory. Obtained results are also implemented for some economical and biological models. In particular cases, our results cover some existing results in the literature for periodic, antiperiodic, or quasiperiodic solutions of difference equations. Key words: Affine periodic, affine symmetric, exponential dichotomy, fixed point 1. Introduction The symmetry property for solutions of dynamical systems has received great interest due to its potential for application in almost all fields of natural and applied sciences, such as physics, engineering sciences, or economics (see [3, 11, 13]). As it is well known, one of the strongest symmetry properties for the solution of a dynamical system is periodicity. By strict periodicity for a solution of a continuous time dynamical system x′ (t) = f (t, x(t)) , t ∈ R, () we mean that solution x satisfies x (t + T ) = x(t) for all t ∈ R , where T is a fixed positive constant. Periodicity is a relaxable and generalizable concept for the classes of functions. As a .
