Relations between forcing sequences and inexact newton-like iterates in banach space

Abstract

We use inexact Newton-like iterates to approximate a solution of a nonlinear equation in a Banach space. Solving a nonlinear equation using Newton-like iterates at each stage is very expensive in general. That is why we consider inexact Newton-like methods, where the Newton- like equations are solved only approximately and in some unspecified manner. In the elegant paper [6] natural assumptions under which the forcing sequence is uniformly less than one were given based on the first-Fréchet derivative of the operator involved in the special case of inexact Newton iterates. Here, we use assumptions on the first and second Fréchet-derivative. This way, we essentially reproduce all results found earlier but for inexact Newton-like iterates. However, our upper error bounds on the distances involved are smaller

Keywords:

• Inexact Newton-like method
• Banach space
• forcing sequence
• Fréchet-derivative
• superlinear
• strong
• convergence