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Abstract: The semi-local convergence of a Newton-type method used to solve nonlinear equations in a Banach space is studied. We also give, as two important applications, convergence analyses of two classes of two-point Newton-type methods including a method mentioned in and the midpoint method studied in . Recently, interest has been shown in such methods . [Copyright &y& Elsevier]

Abstract: Ostrowski provided the sharp sufficient condition for to be an attraction point, for a nonlinear mapping differentiable at a fixed point . This result provides no estimate for the size of the attraction ball. Recently, Cătinaş provided such an estimate in terms of in a Hölder continuity setting. We show that the results by Cătinaş remain valid in a weaker setting by simply replacing the Hölder by the center-Hölder continuity assumption. The radius of convergence of Picard’s iteration is extended, which allows a wider choice of initial guesses. Moreover the estimates of the distances are more precise, which lead to the computation of fewer iterates to achieve a desired accuracy. We also provide examples where our results apply, whereas those by Cătinaş  do not, or where our results are better. [Copyright &y& Elsevier]

Abstract: In this paper, by employing a non-variational version of a max–min principle, new results on existence and uniqueness of generalized solution to the boundary value problem of higher order elliptic partial differential equations at resonance are given. [Copyright &y& Elsevier]

In this study we are concerned with the local convergence of a Newton-type method introduced by us [I.K. Argyros and D. Chen, On the midpoint iterative method for solving nonlinear equations in Banach spaces, Appl. Math. Lett. 5 (1992), pp. 7-9.] for approximating a solution of a nonlinear equation in a Banach space setting. This method has also been studied by Homeier [H.H.H. Homeier, A modified Newton method for rootfinding with cubic convergence, J. Comput. Appl. Math. 157 (2003), pp. 227-230.] and Ozban [A.Y. Ozban, Some new variants of Newton's method, Appl. Math. Lett. 17 (2004), pp. 677-682.] in real or complex space. The benefits of using this method over other methods using the same information have been explained in [I.K. Argyros, Computational theory of iterative methods, in Studies in Computational Mathematics, Vol. 15, C.K. Chui and L. Wuytack, eds., Elsevier Science Inc., New York, USA, 2007.; I.K. Argyros and D. Chen, On the midpoint iterative method for solving nonlinear equations in Banach spaces, Appl. Math. Lett. 5 (1992), pp. 7-9.; H.H.H. Homeier, A modified Newton method for rootfinding with cubic convergence, J. Comput. Appl. Math. 157 (2003), pp. 227-230.; A.Y. Ozban, Some new variants of Newton's method, Appl. Math. Lett. 17 (2004), pp. 677-682.]. Here, we give the convergence radii for this method under a type of weak Lipschitz conditions proven to be fruitful by Wang in the case of Newton's method [X. Wang, Convergence of Newton's method and inverse function in Banach space, Math. Comput. 68 (1999), pp. 169-186 and X. Wang, Convergence of Newton's method and uniqueness of the solution of equations in Banach space, IMA J. Numer. Anal. 20 (2000), pp. 123-134.]. Numerical examples are also provided. [ABSTRACT FROM AUTHOR]

Abstract: An improved regula falsi iterative formula for solving nonlinear equations is proposed in this paper. This method is effective and promising with its third-order convergence of both the sequences of diameters and iterative points sequence. [Copyright &y& Elsevier]

Abstract: In this study, a modification of the classical Secant method for solving nonlinear, univariate and unconstrained optimization problems based on the development of the cubic approximation is presented. The iteration formula including an approximation of the third derivative of f(x) by using the Taylor series expansion is derived. The performance of the new method is analyzed in terms of the number of iterations in comparison with the Secant methods using six test functions. [Copyright &y& Elsevier]

Abstract: Some new iterative formulae having three-order convergence are discussed in this paper. The theoretical analysis and numerical experiments show that new iterative formulae are effective and comparable to well-known method of Newton and Stefensen method. [Copyright &y& Elsevier]