Quasiperiodic solutions of semilinear Liénard equations (Q1770128)

This paper deals with the equation \[ x''+F_x (x,t)x'+a^2 x+\phi (x)+e(x,t)=0, \] where \(F\) and \(e\) are \(2\pi\)-periodic in time and \(a\) is a positive constant. Among the nonlinear terms \(\phi\) is the dominant one and the prototype is \(\phi (x)=| x| ^{\alpha -2}x,\; 0<\alpha <1\). Some symmetry conditions are imposed so that if \(x(t)\) is a solution then \(-x(-t)\) is also a solution. The author uses the theory of reversible maps to prove the existence of large amplitude quasi-periodic solutions. The quasi-periodicity is understood in the classical sense or as in the Aubry-Mather theory. In a previous paper with Kunze and Küpper, the author had obtained related results when \(F\) and \(e\) were bounded, now these functions can be unbounded and dominated by \(\phi\). The paper also contains a result on equations of pendulum type.