A new existence theory for positive periodic solutions to functional differential equations
The aim of the paper is to obtain the existence of positive periodic solutions for the nonlinear functional-differential equation \[ y'(t)=-a(t)y(t)+g(t,y(t-\tau(t))), \tag{1} \] where \(a\in C(\mathbb{R},(0,\infty))\), \(\tau\in C(\mathbb{R},\mathbb{R})\), \(g\in C(\mathbb{R}\times[0,\infty),[0,\infty))\) and \(a,\tau, g(t,\cdot)\) are all \(\omega\)-periodic functions, \(\omega>0\) is a constant. By using a fixed-point theorem in a cone, the authors establish the following result: equation (1) has at least one positive \(\omega\)-periodic solution provided one of the following conditions holds: \[ \liminf_{u\downarrow 0}\min_{0\leq t\leq\omega}\frac{g(t,u)}{a(t)u}>1\quad \text{and} \quad \limsup_{u\uparrow\infty}\max_{0\leq t\leq\omega}\frac{g(t,u)}{a(t)u}<1;\tag{i} \] \[ \limsup_{u\downarrow 0}\max_{0\leq t\leq\omega}\frac{g(t,u)}{a(t)u}<1\quad \text{and} \quad \liminf_{u\uparrow\infty}\min_{0\leq t\leq\omega}\frac{g(t,u)}{a(t)u}=\infty.\tag{ii} \]
scientific article; zbMATH DE number 2142385