Convergence of a parameter based iterative method for solving nonlinear equations in Banach spaces

Abstract

In this paper, we introduce a new class of a family of third order iterative methods which depend on a real parameter for solving nonlinear equations in Banach spaces. The semilocal convergence of iterative method under the assumption that the second order Fŕechet derivative of the operator satisfies the Holder continuity condition and the R-order \(2+p\) of the iteration family is established. An existence-uniqueness theorem and error estimate in terms of a parameter \(\alpha \in [0,1]\) are provided for these iterations using a technique based on a new system of recurrence relations. Numerical application of nonlinear Hammerstein integral equations of the mixed type is provided.