Anti-periodic solution for u_tt - (σ(u_x))_x - U_xxt = f(x,t)

Existence of a smooth anti-periodic solution for the quasilinear equation u_tt - (σ(u_x))_x - u_xxt = f(x,t) in [0,π] × R with the boundary condition u(0,t) = u(π,t) = 0 is proved for a class of σ(v) including σ(v) = v/ √1 + v², where f(x, t) is a given anti-periodic function in t.