The midpoint method in Banach spaces and the Pták error estimates (Q1322885)

The authors introduce a midpoint two-step method, which contains a Newton method and a modified Newton method in each iteration, to solve approximately a nonlinear operator equation in a Banach space setting. They give existence and uniqueness theorems as well as an error analysis by the majorant method, assuming the Fréchet derivative of the operator involved satisfies Pták-like estimates. For a special choice of the function involved their method is of order three as other well-known Newton methods of order three.