On the Convergence Region of Multi-step Chebyshev-Halley-type Schemes for Solving Equations

The aim of this article is to extend the convergence region of certain multi-step Chebyshev-Halley-type schemes for solving Banach space valued nonlinear equations. In particular, we find an at least as small region as the region of the operator involved containing the iterates. This way the majorant functions are tighter than the ones related to the original region, leading to a finer local as well as a semi-local convergence analysis under the same computational effort. Numerical examples complete this article.

In particular, we consider the Chebyshev-Halley-type scheme defined for each ..., , 2 , ] ( ) ( ), (in the local convergence case) and [ ], (in the semi-local convergence case). Iterative schemes-type (1.2) have been considered in [19]. However, in this article, we study the local as well as the semi-local convergence of scheme (1.2) under generalized ω-conditions. Moreover, by introducing the center ω-condition, we locate a subset of D containing the iterates. This subset helps us define tighter majorant functions and parameters than before leading to larger radius of convergence (i.e., we obtain a wider choice of initial guesses); tighter error bounds on the distances * + − − x x x x n n n , 1 (i.e., fewer iterates are needed to obtain a desired error tolerance ) 0 > ε and an at least as precise information on the location of the solution. Scheme (1.2) is especially useful, when F ′ ′ is a constant. Other favorable cases can be found in [19].
The design of the article is as follows: Section 2 and Section 3 contain the local and semi-local convergence of scheme (1.2), respectively. The numerical examples appear in the concluding Section 4.

Local Convergence Analysis
We rely on some parameters and scalar functions to show the local convergence Suppose that the equation has at least one positive solution. We denote by 0 ρ the smallest such solution. Let The We denote by 1 ρ the smallest positive Suppose that equation has at least one positive solution. We denote by 2 ρ the smallest such solution. Define the functions 6 5 , µ µ  , continuous and increasing such that for each given in ( The aforementioned hypotheses (H) and notation lead to the local convergence result for method (1.2). On are given previously and r is defined in The definition of the convergence radius r guarantees that for each The proof is based on the estimates (2.9)-(2.17) and mathematical induction. Let Using (h 1 ), (h 2 ), (2.5) and (2.9), we have: which together with the Banach Perturbation Lemma [3][4][5], imply that We can write by (h 1 ), Then, by (h 3 ) and (2.20) In particular, for , By the definition of r, 0 v and (2.22), Then, by the last condition in (h 3 ), we get that We also have that 1 x is well defined by the in the preceding estimates. Then, from the estimate ).
, r x U * The uniqueness of the solution part, is shown by letting Then, by (h 1 ), (h 2 ) and (h 5 ) we obtain that Finally, in view of the identity The radius L L + = ρ Notice that the convergence radius for Newton's method given independently by Rheinboldt [17] and Traub [18] is given by   [19]. We can use the computational order of convergence (COC) [3][4][5] , ln condition (2.9) can be replaced by

Semi-local convergence analysis
We study the semi-local convergence analysis of scheme (1.2) in an analogous way to the local convergence analysis appearing in Section 2. That is why we omit the proofs for which you can also see [19]. The hypotheses on which we base our analysis are (A): is twice continuously differentiable operator in the sense of Fréchet and there exists 0 , 0 , continuous and non-decreasing , 0 continuous and non-decreasing such that for each In the literature the following conditions are used instead of (a 2 ), (a 3 ) and (a 4 ). and Hence, w, K can replace , where ( ) has at least a root in .
is increasing function of u. We have the following estimates [19].

Numerical Examples
The numerical examples are presented in this section.