Local convergence of a one parameter fourth-order Jarratt-type method in Banach spaces

We present a local convergence analysis of a one parameter Jarratt-type method. We use this method to approximate a solution of an equation in a Banach space setting. The semilocal convergence of this method was recently carried out in earlier studies under stronger hypotheses. Numerical examples are given where earlier results such as in [Ezquerro J.A., Hernandez M.A., New iterations of R-order four with reduced computational cost, BIT Numer. Math. 49 (2009), 325-342] cannot be used to solve equations but our results can be applied.