Anti-periodic Solution forutt−(σ(ux))x−uxxt=f(x, t)

Abstract Existence of a smooth anti-periodic solution for the quasilinear equationu_{tt}-(\sigma(u_x))_x-u_{xxt}=f(x,t)\qquad\hbox{in $[0,\pi]\times{\bfR}$}with the boundary conditionu(0, t)=u(π, t)=0 is proved for a class of σ(v) including \sigma(v)=v/\sqrt{1+v^2} , wheref(x, t) is a given anti-periodic function int.