Chebyshev-secant-type methods for non-differentiable operators (Q1949821)

Let \(F:\Omega\subset X\rightarrow Y\) be a nonlinear operator defined on a non-empty open convex domain of a Banach space \(X\) with values in a Banach space \(Y\). The following Chebyshev-secant-type method was introduced by same authors in [J. Comput. Appl. Math. 235, No. 10, 3195--3206 (2011; Zbl 1215.65102)]: \[ \begin{cases} x_{-1},x_{0} \in\Omega,\\ y_{n} =x_{n}-A_{n}^{-1}F\left( x_{n}\right) ,\;A_{n}=[x_{n-1,}x_{n};F],\\ z_{n} =x_{n}+a\left( y_{n}-x_{n}\right) ,\\ x_{n+1} =x_{n}-A_{n}^{-1}\left( bF\left( x_{n}+cF\left( x_{n}\right) \right) \right),\quad n\geq 0,\end{cases} \] where \(a,b,c\) are nonnegative parameters to be computed such that \(\left\{ x_{n}\right\} \) is convergent to \(x^{\ast},\) \(F\left( x^{\ast}\right) =0.\) The concept of ``family of iterative methods'' is based on these parameters. The connection with other Newton (like) methods is discussed. The main results extends the local convergence of the above iteration to non-differentiable operators. Two examples are provided. The first one deals with nonlinear systems in \(\mathbb R^{2}\) with absolute values and the second one considers nonlinear integral equations of mixed Hammerstein type with kernels given by Green functions.