On existence of positive periodic solutions (Q1386707)

Let \(\Omega\) be an open convex subset of \(\mathbb{R}^n\). A differential equation \[ \dot x=f(t,x) \tag{1} \] on \(\mathbb{R}\times \text{cl} \Omega\) is considered. It is assumed that the right-hand side of (1) is \(T\)-periodic in \(t\). The main theorem asserts the existence of a fixed point of the Poincaré map associated to (1) in \(\Omega\) provided there exist two so-called periodic isolated segments in \([0,T]\times \text{cl} \Omega\) satisfying some topological conditions and properly located with respect to the equation. The theorem is applied to results on planar \(T\)-periodic nonautonomous Lotka-Volterra-type systems \[ \dot x=x(a(t)-b(t)y),\qquad \dot y=y(c(t)-d(t)x) \tag{2} \] on \([0,\infty)^2\) and \[ \dot x=a(t)x+b(t,y),\qquad \dot y=yc(t,y) \tag{3} \] on \(\mathbb{R}\times [0,\infty)\). The system (2) has a \(T\)-periodic solution in \((0,\infty)^2\) provided the maps \(a\), \(b\), \(c\), and \(d\) have positive values. If \(c(t,y)\) is positive for each \(y\geq 0\) small enough and every \(t\), and negative for some positive \(y\) and every \(t\) then (3) has a \(T\)-periodic solution in \(\mathbb{R}\times (0,\infty)\) provided, for every \((t,y)\), \(a(t)>0\) and \(b(t,y)\geq 0\) (or \(b(t,y)\leq 0\)).