B. P. Liu and C. V. Pao [Appl. Anal. 14, 293-301 (1983; Zbl 0515.35077)] showed that the problem to find the periodic travelling wave solutions \(u(x,t)=v(x-bt),v(y)=v(y+2T)\) of the equation \(u_ t+f(u)_ x+u_{xxx}=0\) can be reduced to one of solving the equation \[ F(v)\equiv v(y)-\int^{2T}_{0}K(y,z)f(v(z))dz=0, \] where \(K(y,z)=(2\lambda sh \lambda T)^{-1}ch \lambda (T-| y-z|)-(2T \lambda^ 2)^{-1}.\)
The author proposes a Newton’s iteration procedure to solve the equation \(F(v)=0\) and establishes the convergence of the iteration under some technical suppositions.