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A parameter based set of third order iterative method and the semilocal convergence analysis of this methods using majorizing sequence approach for solving nonlinear equations in Banach
spaces is investigated by Ezquerro and Hernandez [4]. This method is a weighted mean between the Chebyshev and the Halley methods, the weight being and 1−, where 2 R. A convergence theorem and corresponding error bounds provided. We have recurrence relation approach to discuss the semilocal convergence of iterative methods. This is motivated us to discuss the semilocal convergence. In this paper, mainly we focus on to discuss the semilocal convergence of parameter based iterative method developed by [4] using recurrence relations approach under the assumption that F00 is bounded and a punctual condition. Also, we established the R-order of convergence and provided some a priori error bounds. Finally, we discuss some numerical examples that where the Smale-like theorem fails but our bounded condition satisfy. We calculate the existence and uniqueness region for the Numerical examples. Also, we calculate the error bounds for parameter alpha= 0, 1, 2. We observed that the existence region obtained by our approach is superior than Ezquerro and Hernandez [4] for each value of parameter alpha = 0, 1, 2.

The Halley’s method, The Convex acceleration of Newton’s method, A Continuation method, Banach space, Lipschitz condition, Fr´echet derivative.