Nontrivial Periodic Solutions for Nonlinear Second-Order Difference Equations

This paper is concerned with the existence of nontrivial periodic solutions and positive periodic solutions to a nonlinear second-order difference equation. Under some conditions concerning the first positive eigenvalue of the linear equation corresponding to the nonlinear second-order equation, we establish the existence results by using the topological degree and fixed point index theories

On a class of difference equations involving a linear map with two dimensional kernel

We establish necessary and sufficient conditions for the existence of periodic solutions to second-order nonlinear difference equations of the form $\Delta^2x_i+\lambda x_i+\Delta f(x_i)=e_i$, $i\in{\mathbb N}$, and for a simpler equation with difference-free nonlinearity. The linear part of the equation has two-dimensional kernel

On a class of difference equations involving a linear map with two dimensional kernel

We establish necessary and sufficient conditions for the existence of periodic solutions to second-order nonlinear difference equations of the form ∆ 2xi + λxi + ∆f(xi) = ei , i ∈ N, and for a simpler equation with difference-free nonlinearity. The linear part of the equation has two-dimensional kernel

Periodic solutions of nonlinear second-order difference equations

We establish conditions for the existence of periodic solutions of nonlinear, second-order difference equations of the form y(t + 2) + by (t + 1) + cy(t) = f (y(t)), where f: ℝ → ℝ and β > 0 is continuous. In our main result we assume that f exhibits sublinear growth and that there is a constant uf (u) > 0 such that |u| ≥ β whenever c = 1. For such an equation we prove that if N is an odd integer larger than one, then there exists at least one N-periodic solution unless all of the following conditions are simultaneously satisfied: |b| < 2, N across-1(-b/2), and π is an even multiple of c ≠ 0

Periodic Travelling Waves in Dimer Granular Chains

We study bifurcations of periodic travelling waves in granular dimer chains from the anti-continuum limit, when the mass ratio between the light and heavy beads is zero. We show that every limiting periodic wave is uniquely continued with respect to the mass ratio parameter and the periodic waves with the wavelength larger than a certain critical value are spectrally stable. Numerical computations are developed to study how this solution family is continued to the limit of equal mass ratio between the beads, where periodic travelling waves of granular monomer chains exist