Periodic constant-sign solutions for systems of Hill's equations

Abstract

We consider the system of Hill's equations

u″_i(t)+a_i(t)u_i(t)=F_i(t, u₁(t), u₂(t), …, u_n(t)), 1≤i≤n,

where a_i and F_i are periodic in t, and the non-linearities F_i(t, x₁, x₂, …, x_n) can be singular at x_j=0 where j∈{1, 2, …, n}. Criteria are offered for the existence of periodic constant-sign solutions, i.e., θ_iu_i(t)≥0 for each 1≤i≤n, where θ_i∈{1, −1} is fixed. The main tool used is Schauder's fixed point theorem. We also include examples to illustrate the usefulness of the results obtained.