regularly continuous divided differences
On the semilocal convergence of the secant method with regularly continuous divided differences
The article deals with the following iterations NEWLINE\[NEWLINEx_{n+1} = x_n - [x_n,x_{n-1};F]^{-1}F(x_n)\qquad (n = 0,1,\ldots)\tag{1}NEWLINE\]NEWLINE for approximate solution of the nonlinear operator equation \(F(x) = 0\) with a smooth operator \(F\) between Banach spaces \(X\) and \(Y\). The operator (divided difference) \([u,v;F]\) are defined (of course, not uniquely way) by the formula \(F(u) - F(v) = [u,v;F](u - v)\) The authors formulate some new (and unfortunately, cumbersome) conditions under which the sequence \((x_n)\) is well defined and converges to the solution \(x_*\) of the equation \(F(x) = 0\). The main result of the article is obtained under the assumption that \([u,v;F]\) satisfies the following (Galperine-like) condition of \(\omega\)-regular continuity: there exist a function \(\omega\) and constant \(\underline{h} \in [0,\underline{h}([x,y;F])]\) such that for all \(x,y,u,v\) NEWLINE\[NEWLINE\omega^{-1}\big(\min \;\{[x,y;F]\|, \|[u,v;F]\|\} - \underline{h} + \|[x,y;F] - [u,v;F]\|\big) - NEWLINE\]NEWLINE NEWLINE\[NEWLINE- \omega^{-1}\big(\min \;\{[x,y;F]\|, \|[u,v;F]\|\} - \underline{h}\big) \leq \|x - u\| + \|y - v\|.NEWLINE\]
scientific article; zbMATH DE number 6077350