On an iterative method without inverses of derivatives for solving equations

We present the semi-local convergence analysis of a Potra-type method to solve equations involving Banach space valued operators. The analysis is based on our ideas of recurrent functions and restricted convergence region. The study is completed using numerical examples.


Introduction
Many problems in applied sciences can be reduced to the mathematical equation, where F : D ⊆ B 1 −→ B 2 is a Fréchet-differentiable operator, B 1 and B 2 are Banach spaces and D is a nonempty open convex subset of B 1 . Iterative methods are useful in solving many equations of the form (1). For example the following method is considered in [16,17] x The convergence of iterative algorithms is analyzed in two categories: semi-local convergence analysis (i.e., based on the information around an initial point, to obtain conditions ensuring the convergence of these algorithms) and local convergence analysis (i.e., based on the information around a solution to find estimates of the computed radii of the convergence balls).
In this study, we introduce the following iterative method defined for each n = 0, 1, 2, . . . by where x 0 , y 0 are an initial points, A n = [x n , y n ; F ]+[x n+1 , x n ; F ]−[x n+1 , y n ; F ], [., .; F ] : D×D −→ L(B 1 , B 2 ) is the finite difference of order one. We do not choose x 0 = y 0 in practice to avoid injecting derivatives in the method or study variants of Newton's method involving both divided differences and derivatives. But clearly our results can specialize to such methods if derivatives are allowed. Using Lipschitz-type conditions we find computable radii of convergence as well as error bounds on the distances involved. The order of convergence is found using computable order of convergence (COC) or approximate computational order of convergence (ACOC) [24] (see Remark 2.4) that do not require usage of higher order derivatives. This way we expand the applicability of three step method (3) under weak conditions.
The rest of the study is organized as follows: Section 2 contains the local convergence of method (3), whereas in the concluding Section 3 applications and numerical examples can be found.

Semi-local convergence
The semi-local convergence of method (3) is given in this Section. We need two auxiliary results on majorizing sequences for method (3).
In case (3) is not satisfied we have the alternative result.
Next, we present the semi-local convergence analysis of method (3) using {t n } as a majorizing sequence.
where {t n } and t * are defined in the preceding Lemmas. Furthermore, if there exists R > t * such that and then, the point x * is the only solution of equation Proof. We shall show using induction that the following assertions hold.
To complete the proof we show the uniqueness of the solution inŪ (x 0 , R). Let w ∈Ū (x 0 , R) be such that F (w) = 0. By (27), we have in turn that It follows that [x * , w; F ] −1 exists. Then from the identity we conclude that x * = w.
Remark 2.4. (a) The limit point t * can be replaced by t * * given by (5) in Theorem 2.3.
(c) In view of (4) it follows that L 0 and not L is needed in the computation of [x 0 , y 0 ; F ] −1 F (x 1 ) .

Numerical Examples
We shall use the divided difference given by [x, y; F ] = 1 2 (F (x) + F (y)) in both examples. Example 3.1. Let D =Ū (x 0 , 1 − γ), x 0 = 1, y 0 = x 0 + 10 −3 , γ ∈ [0, 1). Define function F on D by We have that Next, we verify that all conditions of Lemma 2.1 hold. In fact, by the definition of polynomial p, we get that α ≈ 0.6543. We also have We see by now that all conditions of Theorem 2.3 are satisfied. Hence, Theorem 2.3 applies.