Existence of periodic solutions to nonlinear difference equations at full resonance. (English) Zbl 1321.39020

Summary: The purpose of this paper is to search for periodic solutions to a system of nonlinear difference equations of the form \[ \Delta x(t) = f(\epsilon,t,x(t)). \] The corresponding linear homogeneous system has an \(n\)-dimensional kernel, i.e. the system is at full resonance. We provide sufficient conditions for the existence of periodic solutions based on asymptotic properties of the nonlinearity \(f\) when \(\epsilon=0\). To this end, we employ a projection method using the Lyapunov-Schmidt procedure together with Brouwer’s fixed point theorem.


39A23 Periodic solutions of difference equations
39A10 Additive difference equations
39A12 Discrete version of topics in analysis
34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text: Euclid