On the determination of the number of periodic (or closed) solutions of a scalar differential equation with convexity
The scalar differential equation \(\dot{x} = f(t, x)\), where \(f(t, x)\) is continuous, \(T\)-periodic in \(t\) and weekly convex or concave in \(x\) has been investigated with respect to the number of \(T\)-periodic solutions by many authors. The general result is that there can be no, one or two \(T\)-periodic solutions or a connected band of \(T\)-periodic solutions. The last mentioned possibility can be excluded if \(f(t, x)\) is strictly convex or concave for some \(t\) in the period interval. The problem whether it is possible to determine the actual number of \(T\)-periodic solutions of a given equation of this type is treated in the present paper (assuming in addition a continuous derivative \(f_x'(t, x)\)) using the idea to view a periodic solution as a fixpoint of the successor function (the Poincaré map) \(S(\xi),\) i.e., as a zero of the deviation function \(P(\xi) = S(\xi) - \xi.\) It turns out that there are three cases. In each of these cases, using some monotonicity properties, the actual number of periodic solutions can be determined.
scientific article; zbMATH DE number 5149557
scientific article