A family of Halley-Chebyshev iterative schemes for non-Fréchet differentiable operators
Finding zeros of nonlinear equations is a classical mathematical problem, one of the most important techniques to study these equations is the use of iterative processes, starting from an initial approximation \(x_0\). Newton type methods are the most used. Third order methods have been successfully used in the solution of nonlinear integral equations. Such methods are useful to derive results on the existence and uniqueness of a solution. Here, nonlinear equations in Banach spaces are considered. Fréchet differentiability is not required. Let \(B\) be an open convex set of a Banach space \(X\) and \(F:B\to X\). The authors want to find \(x\in B\) such that \(F(x)=0\). The classical Chebyshev \((\beta=0)\), Halley \((\beta=\tfrac 12)\) and super-Halley \(\beta=1)\) methods can be written as \[ x_{n+1}=x_n-(I+\tfrac 12L_F(x_n))[I-\beta L_F(x_n)]^{-1} [F'(x_n)]^{-1}F(x_n), \] where \[ L_F(x_n)=[F'(x_n)]^{-1}F''(x_n) [F'(x_n)]^{-1}F(x_n). \] In the present paper a modification of these classical third order iterative methods is studied. The new methods do not need to evaluate any derivative. A convergence and uniqueness theorem is proved. A numerical comparison of the proposed methods with a Steffensen type scheme is given. Finally, an example is analyzed where the conditions formulated in the paper are fulfilled and the classical ones fail.
scientific article; zbMATH DE number 5559073