Semilocal and Local Convergence of a Three Step Fifth Order Iterative Methods under General Continuity Condition in Banach Spaces

Abstract

In this paper, First of all, we study the semilocal convergence ofthe fifth order iterative method using recurrence relation under theassumption that first order Fr\'echet derivative satisfies the moregeneral $\omega$-continuity condition. We calculate also the R-orderof convergence and provide some a priori error bounds. Based onthis, we give existence and uniqueness region of the solution for anonlinear Hammerstein integral equation of the second kind. Next, wediscuss the local convergence of iterative method under theassumptions that the first order Fr\'echet derivative satisfies thesame $\omega-$continuity condition. Also, Numerical Example isworked out to demonstrate the efficacy of our approach.