INVESTIGATING POSITIVE PERIODIC SOLUTIONS IN SUPERLINEAR FIRST-ORDER DIFFERENTIAL SYSTEMS THROUGH GLOBAL BIFURCATION THEORY

Abstract

This paper investigates the superlinear first-order differential systems of the form u′(t)+a(t)u(t)=λb(t)f(v(t−τ(t))), t R, v′(t)+a(t)v(t)=λb(t)g(u(t−τ(t))), t R, where λ is a parameter and a, b, τ, f, and g satisfy certain assumptions. The aim of this study is to explore the existence of positive periodic solutions for this system and use the bifurcation theory to demonstrate the presence of an unbounded component of positive solutions. The authors use a new technique to directly prove that the component must bifurcate from infinity at λ = 0 without additional conditions of f, g. The research has significant applications in areas such as physics, information, chemistry, engineering, economics, and mathematical biology. The authors also discuss the asymptotic problem in an abstract setting and apply the result to the proof of the main theorem. Overall, the paper contributes to the study of differential equations under superlinear or sublinear conditions and provides useful information for the numerical solutions of such equations