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

摘要

We consider the system of Hill's equationsrnU"_i(t) + a_i(t)u_i(t) = F_i(t,u_1(t),u_2(t),...,u_n(t)), 1≤i≤nrnwhere a_i and F_i are periodic in t, and the non-linearities F_i(t,x_1,X_2,... ,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.