On periodic solutions of functional differential equations with infinite delay (Q1319153)

Using Mawhin's continuation theorem in the theory of coincidence degree, the general existence theorem for periodic solutions of functional differential equations with infinite delay \[ {{dx(t)} \over {dt}}= f(t,x_ t), \qquad x(t)\in \mathbb{R}^ n, \] is proved, which is an extension of Mawhin's existence theorem of periodic solutions of functional differential equations with finite delay. As an application an existence theorem of positive periodic solutions of the Lotka-Volterra equations \[ \begin{aligned} {{dx(t)} \over {dt}} &= x(t) \bigl( a-kx(t)- by(t)\bigr),\\ {{dy(t)} \over {dt}} &=- cy(t)+d \int_ 0^{+\infty} x(t-s) y(t-s) d\mu(s)+ p(t) \end{aligned} \] is obtained. Here \(a,b,c,d,k>0\) are constants, \(p(t)\) is continuous and \(\omega\)-periodic and \(\mu(s)\) is monotone increasing with \(\int_ 0^{+\infty} d\mu(s)=1\). Let \(p_ 0= {1\over\omega} \int_ 0^ \omega p(t)dt\), \(p_ 1= \max_{0\leq t\leq\omega} | p(t)- p_ 0|\) and let \(\beta\) be a positive number defined by \(\omega= \beta(2+e^{\beta c})\). The existence of at least one \(\omega\)-periodic solution of the Lotka-Volterra equations is proved if the following three conditions hold: (1) \(p(t)\geq 0\), (2) \(ad-cke^{a(\omega- \beta)}>0\), (3) \(c(ad-ck)- bd(p_ 0+ p_ 1)>0\).