Extended and unified local convergence of k-step solvers for equations with applications

摘要

There is a plethora of k-step solvers for equations involving operators on Banach spaces. Their convergence is estimated by adopting hypotheses on high-order derivatives which are not even in these iterative solvers. In addition, no computable error bounds or information on the uniqueness of the solution based on Lipschitz-type functions are given. Moreover, the choice of the initial guess is like shooting in the dark. Finally, the criteria of convergence differ from solver to solver, so no comparison can be made between their convergence domains. The novelty of our work is that we address these problems by introducing a generalized k-step solver containing all previous k-step solvers. Moreover, we address all previously stated problems in the earlier papers and utilizing weaker conditions that require only the continuity of the involved operator. Applications are also presented where we test the convergence criteria.