Ball Convergence of Efficient Multistep Methods Under Ω ‐Continuity for Banach Space Valued Equations

The convergence order of multistep methods is usually determined using Taylor series, which in turn require the existence of derivatives that are not present in the methods. Other drawbacks of this approach include a lack of a priori error estimates and information on the radius of convergence or the solution′s uniqueness. Therefore, in order to extend the utilization of such methods and in more general space settings than the Euclidean space, the local convergence analysis relies on conditions involving only the operators of the methods. Moreover, the more challenging semilocal convergence analysis, which is absent in these studies, is also developed in this paper using majorizing sequences. To obtain higher error bounds, the derivative is controlled by generalized continuity conditions. Examples are used to show that the sufficient convergence conditions are satisfied.