Widely applicable periodicity results for higher order difference equations

Abstract

In this paper we study the periodicity of higher order nonlinear equations. They are defined by a recursion which is generated by a mapping , where X is a state set. Our main objective is to prove sharp conditions for the global periodicity of our equations assuming the weakest possible assumptions on the state set X. As an application of our general algebraic-like conditions we prove a new linearized global periodicity theorem assuming that X is a normed space. We needed a new proof-technique since in the infinite dimensional case the Jacobian does not exist. We give new necessary and/or sufficient conditions as well as new examples for global periodicity, for instance whenever the state set X is a group.

Keywords::

• difference equation
• periodic solutions
• global periodicity
• linearized periodicity
• generalized Lozi equation
• periodicity over a group

MSC (2010) Classification::

• 39A10
• 39A23

Acknowledgement

Supported by Hungarian National Foundations for Scientific Research Grant No. K101217.

Notes

^1. Email: [email protected]