Existence and uniqueness of positive periodic solutions of functional differential equations (Q1827065)

This paper deals with the existence and uniqueness of a positive periodic solution for the first-order functional-differential equation with a parameter \[ y'(t)= -a(t)y(t)+ \lambda h(t)f(y(t-\tau(t))), \tag{1} \] where \(a(t)\) and \(\tau(t)\) are continuous \(T\)-periodic functions and \(h(t)\) is a positive continuous \(T\)-periodic function, \(\lambda\) is a positive parameter. Additionally it is assumed \(\int_0^T a(u)\, du>0\). The authors establish several sufficient conditions for the existence of a positive \(T\)-periodic solution of (1) by using the eigenvalue theory of operators. Using the theory of \(\alpha\)-concave operators, they prove the uniqueness of the solution. Dependence on the parameter \(\lambda\) is discussed and finally, as an application, the authors present sufficient conditions for the existence of a positive periodic solution of the Nicholson blow flies model.