Unified ball convergence of third and fourth convergence order algorithms under continuity conditions

There is a plethora of third and fourth convergence order algorithms for solving banach space valued equations. these orders are shown under conditions on higher than one derivatives not appearing on these algorithms. moreover, error estimations on the distances involved or uniqueness of the solution results if given at all are also based on the existence of high order derivatives. but these problems limit the applicability of the algorithms. that is why we address all these problems under conditions only on the first derivative that appear in these algorithms. our analysis includes computable error estimations as well as uniqueness results based on continuity conditions on the fr'echet derivative of the operator involved.