Recurrence relations for semilocal convergence of a fifth‐order method in Banach spaces

In this paper, we study the semilocal convergence for a fifth‐order method for solving nonlinear equations in Banach spaces. The semilocal convergence of this method is established by using recurrence relations. We prove an existence‐uniqueness theorem and give a priori error bounds which demonstrates the R‐order of the method. As compared with the Jarratt method in Hernández and Salanova (1999) and the Multi‐super‐Halley method in Wang et al. (2011), the differentiability conditions of the convergence of the method in this paper are mild and the R‐order is improved. Finally, we give some numerical applications to demonstrate our approach.