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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.

MSC:

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
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Full Text: Euclid