Positive periodic solutions for a class of nonlinear delay equations (Q1765202)

This paper deals with the existence of periodic solutions for a class of neutral functional-differential equations (NFDEs) with multiple discrete delays. First, sufficient criteria are established for the existence of positive periodic solutions for NFDEs of the form \[ N'(t)=N(t)F(t,N(t), N(t-\tau_1(t)),\cdots,N(t-\tau_n(t)), N'(t-\gamma_1(t)),\cdots,N'(t-\gamma_n(t))), \] where \(F\in C(\mathbb{R}^{2n+2},\mathbb{R}), F(t+T)=F(t),\tau_i,\gamma_i\in C(\mathbb{R},[0,+\infty)), \tau_i(t+T)=\tau_i, \gamma_i(t+T)=\gamma_i(t)\) and \(T\) is a positive constant. Then, the author applies the established sufficient criteria to the single species model \[ N'(t)=N(t)[a(t)- \beta(t)N(t)-\sum\limits_{i=1}^nb_i(t)N(t-\tau_i(t)) -\sum\limits_{i=1}^nc_i(t)N'(t-\gamma_i(t))], \] where \(a(t),\beta(t),b_i(t),c_i(t),\tau_i(t),\gamma_i(t)\), \(i=1,\cdots,n\), are nonnegative continuous \(T\)-periodic functions. The criteria established in this paper extend and improve some known results for the above systems. The approach is based on the theory of topological degree, which has been widely applied to explore the existence of periodic solutions of differential equations, impulsive differential equations and difference equations.