Third-order iterative methods with applications to Hammerstein equations: a unified approach
This article deals with the family of iterations \[ x_{n+1} = x_n - \left(I + \tfrac12L_F(x_n)(I + b(x_n)F'(x_n)^{-1}f(x_n))^{-1}\right)F'(x_n)^{-1}F(x_n), \quad n = 0,1,2,\dots \] (\(I\) is the identity operator on \(X\), \(L_F(x) = F'(x)^{-1}F''(x)F'(x)^{-1}F(x)\)) for approximately solving a nonlinear operator equation \(F(x) = 0\) with a nonlinear operator \(F\) between two Banach spaces \(X\) and \(Y\). This family covers a lot of concrete iterations, in particular, Halley's and Chebyshev's methods. The authors present two theorems about the convergence to a solutions \(x_*\) of \(F(x) = 0\) and the uniqueness of this solution. The assumptions in these theorems are formulated in terms of some estimates for \(F'(x_1)^{-1}\), \(\|F'(x_1)^{-1}f(x_1)\|\), \(\|F''(x_1)\|\), \(\|B(x_1)\|\) and moduli of smoothness for \(F''(x)\) and \(B(x)\), where \(B(x)\) is an auxiliary function with values in the space of bilinear operators of type \(X \times X \to Y\). As an application, the Hammerstein integral equation \[ u(s) = \psi(s) + \int_0^1 H(s,t)f(t,u(t)) \, dt \] is considered. There are some vague places in this paper; in particular, it is not clear why the assumptions in both theorems are given at the point \(x_1\) instead of \(x_0\); on examination of the Hammerstein integral equation, the authors do not fix the spaces \(X\) and \(Y\), and, as a result, their arguments are not correct and so on.
scientific article; zbMATH DE number 5865680