Some notes to existence and stability of the positive periodic solutions for a delayed nonlinear differential equations (Q317883)

The authors consider the existence of positive \(\omega\)-periodic solutions for delay differential equations of the form NEWLINE\[NEWLINE\dot{x}(t)=-p(t)x(t)+\sum_{i=1}^nq_i(t)f(x(\tau_i(t))),\quad t\geq t_0,NEWLINE\]NEWLINE whereNEWLINENEWLINE(i) \(p, q_i \in C([t_0,\infty),(0,\infty))\), \(i=1,...,n\), \(f\in C^1(\mathbb{R},\mathbb{R})\), \(f(x)>0\) for \(x>0\),NEWLINENEWLINE(ii) \(\tau_i \in C([t_0,\infty),(0,\infty))\), \(\tau_i(t)<t\) and \(\lim_{t\to\infty}\tau_i(t)=\infty\) for \(i=1,\dots,n\).NEWLINENEWLINEThey provide sufficient conditions to assure the existence and exponential stability of such solutions. It seems that is very difficult to check in practice these conditions. But, in turn, it is not assumed that the functions \(p\), \(q_i\), \(\tau_i\) are periodic.