Local and semilocal convergences of a fourth-order derivative-free method in Banach spaces and its applications

Purpose The purpose of this paper is to study local and semilocal convergences of a fourth-order derivative-free iterative method in Banach spaces and its applications. Design/methodology/approach The local convergence and semilocal convergence of a fourth-order derivative-free iterative method are analyzed under the assumptions that the first-order divided differences satisfy the weak Lipschitz continuity conditions. Findings In local convergence, the domain of convergence and the error estimation can be obtained. In semilocal convergence, convergence criteria can be obtained, based on any initial point, to ensure that the iterative sequence converges to a unique solution in a given domain. Both of the above convergence analyses can prove the uniqueness of the solution. Originality/value In the proof of convergence order, Taylor expansions require third or higher derivatives. The applicability of the method is restricted. To extend the applicability of the method, local and semilocal convergences are studied under Lipschitz conditions with the first derivative. Finally, to prove the applicability of the method, the method is applied to solve ordinary differential equations, partial differential equations and Beidou satellite positioning. The comparison between the method and the same order methods shows that the method has good behavior of convergence.