An existence theorem for periodic solutions in convex sets with applications
Let \(C\subset \mathbb{R}^m\) be a closed convex set with nonempty interior and let \(G\subset C\) be bounded and open relatively to \(C\). We prove a continuation theorem which ensures the existence of a solution \(x(\cdot)\) of the boundary value problem \(\dot x=F(t,x)\), \(x(0)=x(T)\), such that \(x(t)\) remains in the closure of \(G\) relatively to \(C\), for all \(t\). To do this, a flow-invariance condition is combined with the functional-analytic framework of coincidence degree; moreover, the usual assumptions on the ``averaged'' vector field \(\bar f_0(x):=(1/T) \int^T_0 f(s,x;0)\, ds\) are replaced by a condition on the fixed point index of the map \(r(I+\bar f_0)\), where \(r: \mathbb R^m\to C\) is the canonical projection. Applications to a periodically perturbed differential system and to a Lotka-Volterra type equation are provided, too.
periodically perturbed differential system
scientific article; zbMATH DE number 4065460
scientific article