On periodic solutions, in a given set, for differential systems. (English) Zbl 0725.34039

Fernandez, M. L. C.; Zanolin, F.

Riv. Mat. Pura Appl. 8, 107-130 (1991).

The existence of periodic solutions of the differential system \(x'=f(t,x)\), which remain in a given set M is proved. The set M is a piece of a convex set lying between two level surfaces of a Lyapunov-like function. In one example considered the convex cone is \({\mathcal R}^ m\) and M is the part of the cone which is situated between two spheres. The theory exploits the fixed point index combined with suitable geometric conditions concerning the behaviour of f on the boundary of M.

Reviewer: P.Smith (Keele)

MSC:

34C25	Periodic solutions to ordinary differential equations
34C11	Growth and boundedness of solutions to ordinary differential equations
34D40	Ultimate boundedness (MSC2000)

Keywords:

periodic solutions; Lyapunov-like function; fixed point index