Periodic orbits of nonlinear first-order general periodic boundary value problem

Abstract

In this paper, the existence and multiplicity of periodic orbits are obtained for first-order general periodic boundary value problem x'(t) + a(t)x(t)=f(t,x), t ? [0,T], x(0)=?x(T), where a : [0,T] ? [0,+?) and f : [0,T]x R+ ? R are continuous functions, ? > 0 and T > 0 with ?e??T,0 a(s)ds=1. The proofs are carried out by the use of topological degree theory. We also prove some nonexistence theorems. Our results extend and improve some recent work in the literature.