Semilocal convergence of a sixth-order method in Banach spaces
The paper deals with the approximate solution of a nonlinear equation \(F(x)= 0\), where \(F\) is a mapping of a convex set \(\Omega\) of a Banach space \(X\) in a Banach space \(Y\). It is assumed that \(F\) is Fréchet-differentiable of order 3. To solve the equation numerically we know Newton's method, Chebyshev's method, Halley's method, Newton-likes methods, the super-Halley's method. For the solution of nonlinear Hammerstein integral equations some fourth-order methods are used. In this paper, a new iterative method of order six is proposed: Suppose \(x_n\) is given. Then denote \[ \begin{aligned}\Gamma_n &= [F'(x_n)]^{-1},\\ u_n &= x_n-\textstyle{{1\over 3}}\Gamma_n F(x_n),\\ K(x_n) &= \Gamma F''(u_n)\Gamma_n F(x_n),\\ z_n &= x_n -\{I+\textstyle{{1\over 2}} K(x_n)[I- K(x_n)]^{01}\} \Gamma_n F(x_n),\\ x_{n+1} &= z_n- [I+ \Gamma_n F''(u_n)(z)_n- x_n)]^{-1} \Gamma_n F(x_n).\end{aligned} \] The semilocal convergence of the iterations is proved. The existence and the uniqueness of the solution of the nonlinear equation can be proved under certain assumptions. An a priori error estimate is given. Two numerical applications (the integral equation of Chandrasekhar and a nonlinear Hammerstein integral equation of second kind) demonstrate the proposed method. In these examples the numerical results are better than those of other known methods.
nonlinear Hammerstein integral equations
scientific article; zbMATH DE number 6110504
scientific article