Error bounds for the midpoint method in Banach spaces (Q2782933)

The paper deals with the nonlinear operator equation \(F(x)=0\), where \(F\) is defined on an open convex subset of a Banach space \(E_1\) containing the closed ball with center \(x_0\), with values in a Banach space \(E_2\). Using the majorant theory, the author shows that under certain Newton-Kantorovich assumptions on the part \((F, \;x_0)\) the midpoint method converges to a locally unique zero of the operator equation. Error bounds are derived. Using these results the author suggests new approaches to the solution of quadratic integral equations.