The aim of this paper is to study the semilocal convergence of the eighth-order iterative method by using the recurrence relations for solving nonlinear equations in Banach spaces. The existence and uniqueness theorem has been proved along with priori error bounds. We have also presented the comparative study of the computational efficiency in case of R m with some existing methods whose semilocal convergence analysis has been already discussed. Finally, numerical application on nonlinear integral equations is given to show our approach.

The use of complex dynamics tools in order to deepen the knowledge of qualitative behaviour of iterative methods for solving non-linear equations is a growing area of research in the last few years with fruitful results. Most of the studies dealt with the analysis of iterative schemes for solving non-linear equations with simple roots; however, the case involving multiple roots remains almost unexplored. The main objective of this paper was to discuss the dynamical analysis of the rational map associated with an existing class of iterative procedures for multiple roots. This study was performed for cases of double and triple multiplicities, giving as a conjecture that the wideness of the convergence regions of the multiple roots increases when the multiplicity is higher and also that this family of parametric methods includes some specially fast and stable elements with global convergence.

The object of the present work is to present the new classes of third-order and fourth-order iterative methods for solving nonlinear equations. Our third-order method includes methods of Weerakoon [10], Homeier [4], Chun [1] e.t.c. as particular cases. After that we make this third-order method to fourth-order (optimal) by using a single weight function rather than using two different weight functions in [2]. Finally some examples are given to illustrate the performance of the our method by comparing with new existing third and fourth-order methods. (2000). 65H05. Mathematics Subject Classification

The prime objective of this paper is to design a new family of eighth-order iterative methods by accelerating the order of convergence and efficiency index of well existing seventh-order iterative method of [1] without using more function evaluations for finding simple roots of nonlinear equations. The presented iterative family requires three function and one derivative evaluations and thus agrees with the conjecture of Kung-Traub for the case n = 4 (i.e. optimal). We have also discussed the derivative free version of the proposed scheme. Numerical comparisons have been carried out to demonstrate the efficiency and the performances of proposed method. (2000). 65H05, 41A25, 65D99. Mathematics Subject Classification

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