Periodic solutions of non-autonomous second order systems with \(\gamma\)- quasisubadditive potential. (English) Zbl 0824.34043

Tang, Chunlei

J. Math. Anal. Appl. 189, No. 3, 671-675 (1995).

The author considers the vector differential equation \(u''(t)= \nabla F(t, u(t))\) in \(0\leq t\leq T\) with periodic boundary conditions \(u(0)= u(T)\), \(u'(0)= u'(T)\). The function \(F(t, x)= F(t, x_ 1,\dots, x_ m)\) is measurable in \(t\) for \(x\) fixed, continuously differentiable for almost every \(t\) fixed and satisfies certain growth conditions. The problem is existence of a solution that minimizes the functional \[ \phi(u)= \int^ T_ 0 (\textstyle{{1\over 2}} | u'(t)|^ 2+ F(t, u(t))) dt. \] Existence is proved under growth conditions on the potential \(F(t, x)\); roughly speaking,
\(F(t, x)= F_ 1(t, x)+ F_ 2(t, x)\), where \(F\) and \(F_ 2\) satisfy summability conditions on \(t\) and \(F(t, x)\) is \(\gamma\)-quasisubadditive in \(x\) (this means \(F(t, x+ y)\leq \gamma(F(t,x )+ F(t, y)))\) for almost all \(t\). The results have applications to Hamiltonian systems.

Reviewer: H.O.Fattorini (Los Angeles)

MSC:

34C25	Periodic solutions to ordinary differential equations
49J15	Existence theories for optimal control problems involving ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
37J99	Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems

Keywords:

vector differential equation; periodic boundary conditions; functional; Hamiltonian systems