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

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.

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
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