Affiliations: Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901-6975, USA | KFUPM Chair Professor, Mathematics and Statistics Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia. E-mail: agarwal@fit.edu | Department of Mathematics, National University of Ireland, Galway, Ireland. E-mail: donal.oregan@nuigalway.ie | School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore. E-mail: ejywong@ntu.edu.sg
Abstract: We consider the system of Hill's equations u″i(t)+ai(t)ui(t)=Fi(t, u1(t), u2(t), …, un(t)), 1≤i≤n, where ai and Fi are periodic in t, and the non-linearities Fi(t, x1, x2, …, xn) can be singular at xj=0 where j∈{1, 2, …, n}. Criteria are offered for the existence of periodic constant-sign solutions, i.e., θiui(t)≥0 for each 1≤i≤n, where θi∈{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.
Keywords: constant-sign solutions, system of Hill's equations, singular equations
