Deformed super-Halley’s iteration in Banach spaces and its local and semilocal convergence

Abstract

Deformed super-Halley’s iteration for nonlinear equations is studied in Banach spaces with its local and semilocal convergence. The local convergence is established under Holder continuous first Frechet derivative. A theorem for the existence and uniqueness of solution is provided and the radii of convergence balls are obtained. For semilocal convergence, the second order Frechet derivative is Holder continuous. The Holder continuous first Frechet derivative is not used as it leads to lower R-order of convergence. Recurrence relations depending on two parameters are obtained. A theorem for the existence and uniqueness along with the estimation of bounds on errors is also established. The R-order convergence comes out to be $$(2+p), p \in (0,1]$$ . Nonlinear integral equations and a variety of numerical examples are solved to demonstrate our work.