Aubry–Mather sets for relativistic oscillators with anharmonic potentials

Abstract

In this paper, by the Aubry–Mather theory, it is proved that there are many periodic solutions and usual or generalized quasiperiodic solutions for relativistic oscillator with anharmonic potentials models $$\frac{d}{{dt}}\left( {\frac{{\dot x}}{{\sqrt {1 - {{\left| {\dot x} \right|}^2}} }}} \right) + {\left| x \right|^{\alpha - 1}}x = \;p\left( t \right),$$ where p(t) ∈ C0(R1) is 1-periodic and α > 0.