New sufficient convergence conditions of the secant method for nondifferentiable operators (Q858786)

The problem of approximating a locally unique solution of the equation \(F(x)=0\), where \(F\) is a continuous but not differentiable operator on an open convex domain in a Banach space \(X\), with values in a Banach space \(Y\) is considered. Because the Newton method for a nondifferentiable operator cannot be applied the secant method, which uses divided differences instead of derivative is used. The divided difference \([x,y;F]\) of order one for the operator \(F\) on the points \(x\neq y\) is a linear operator from \(X\) to \(Y\), for which the equation \([x,y:F](x-y)= F(x)-F(y)\) holds. The sufficient condition for convergence of such method is prove based on idea that if \(x_0\) and \(x_{-1}\) are two initial points and \(x_1\) is the first approximation generated by the secant method the condition \(\|x_1-x_i\| \leq\eta\) for \(i=-1\), 0 is used instead of the classical conditions \(\|x_0-x_{-1} \|\leq\alpha\) and \(\|x_1-x_0\|\leq\eta\). An example when the classical method fails and the new method can be applied is given.