Fourth-order iterations for solving Hammerstein integral equations (Q1015905)

The integral equation of Hammerstein type \[ \varphi(s)=x(s)+ \sum _{j=1}^{m}\int_{a}^{b}K_{j}(s,t)H_{j}(x(t))dt \] is considered as a nonlinear operator equation \(F(X)=0\) in a Banach space \(C[a,b]\). To approximate solving this equation the authors use a Newton-Kantorovich method of the fourth order. In application, when the kernel \( K_{j}(s,t)\) is a Green function on \([0,1]\), the integral is replaced using the Gauss-Legendre quadrature formula. Numerical results are obtained for a particulary case of this algorithm.