A multipoint method of third order

LetF be a mapping of the Banach spaceX into itself. A convergence theorem for the iterative solution ofF(x)=0 is proved for the multipoint algorithmx _n+1=x _n ?ø(x _n ), where $$\phi (x) = F\prime_x^{ - 1} \left {F(x) + F\lgroup {x - F\prime_x^{ - 1} F(x)} \rgroup} \right]$$ andF′x is the Frechet derivative ofF. The theorem guarantees that, under appropriate conditions onF, the multipoint sequence {x _n } generated by ø converges cubically to a zero ofF. The algorithm is applied to the nonlinear Chandrasekhar integral equation $$\frac{1}{2}\omega _0 x(t)\int_0^1 {\frac{{tx(s)}}{{s + t}}ds - x(t) + 1 = 0}$$ where ω₀>0. A discretization of the equations of iteration is discussed, and some numerical results are given.