The motive of this paper is to discuss the local convergence of a two-step Newton-type method of convergence rate three for solving nonlinear equations in Banach spaces. It is assumed that the first order derivative of nonlinear operator satisfies the generalized Lipschitz i.e., L-average condition. Also, some results on convergence of the same method in Banach spaces are established under the assumption that the derivative of the operators satisfies the radius or center Lipschitz condition with a weak L-average particularly it is assumed that L is positive integrable function but not necessarily non-decreasing. Our new idea gives a tighter convergence analysis without new conditions. The proposed technique is useful in expanding the applicability of iterative methods. Useful examples justify the theoretical conclusions.
Comparisons between Newton’s and Steffensen-like methods are given for solving systems of equations as well as Banach space valued equations. Our idea of the restricted convergence domain is used to compare the sufficient convergence criteria of these methods under the same conditions as in previous papers. It turns out that the following advantages are shown: enlarged convergence domain; tighter error estimates and a more precise information on the location of the solution. Advantages are obtained under the same or at least as tight Lipschitz constants, which are specializations of earlier ones. Hence, the applicability of these methods is extended. Numerical experiments complete this study.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.