ON THE LOCAL CONVERGENCE OF A NEWTON — TYPE METHOD IN BANACH SPACES UNDER A GAMMA — TYPE CONDITION

We provide a local convergence analysis for a Newton—type method to approximate a locally unique solution of an operator equation in Banach spaces. The local convergence of this method was studied in the elegant work by Werner in [11], using information on the domain of the operator. Here, we use information only at a point and a gamma— type condition [4], [10]. It turns out that our radius of convergence is larger, and more general than the corresponding one in [10]. Moreover the same can hold true when our radius is compared with the ones given in [9] and [11]. A numerical example is also provided. AMS Subject Classification. 65G99, 65K10, 47H17, 49M15.


Introduction
In this paper we are concerned with the problem of approximating a locally unique solution x * of the nonlinear equation where F is a twice-Fréchet-differentiable operator defined on a convex subset D of a Banach space X with values in a Banach space Y .
Let us illustrate how this method is conceived: We start with the identity The linear operator in (1.4) can be approximated in different ways [1], [3], [4], [12].

If for example
then (1.4) suggests the famous Newton's method [1]- [12]: Another choice is given by which leads to the implicit iteration: Unfortunately iterates in (1.8) can only be computed in very restrictive cases, and numerically, the method (1.8) is not a pratical procedure.
That is why we consider y n given in (1.2) as a suitable replacement for x n+1 (n ≥ 0).Hence, we arrive at method (1.2), which requires the computation of two iterates x n and y n .The computation of the additional iterate y n can be seen as a step to calculate the iterate x n+1 using Newton's method (1.8).
This shows that iterate x n+1 , thus defined is corrected by computing the iterate y n+1 using (1.8).Another advantage of method (1.2) is that the particular case x 0 = y 0 corresponds to the classical Newton's method (1.8).Procedure (1.2) has a geometrical interpretation similar to the tangent-Secant method in the scalar case, and was introduced by King [8] (see procedure (I, II), p. 299), and extended into Banach space by Werner in [11], where the R-order 1 + √ 2 local convergence was established.Here, we provide a local convergence analysis of the Newton-type method (1.2) using a γ-type condition (see (2.3) and (2.4)).Our radius of convergence r A (see Theorem 2.2) is larger than the corresponding one denoted by r W (see (2.28)) given in the elegant work by Wang and Zhao [10].Note also that a special choice of γ denoted by γ (see (2.29)) used in [10].As it turns out the radius of convergence can be larger than the radii given in [9], [11] where information on a domain is used (see (2.30) and (2.32)) instead of only information at a point used by us.A numerical example is also provided.

Local convergence analysis of the midpoint method (1.2)
Let us define scalar function f on [0, 1 γ ) by where b ≥ 0, and γ > 0 are given.
It is known [9] that if We use throughout this paper the concept of γ-conditions: (ii) Operator F is thrice-Fréchet-differentiable on D, and for all In view of (2.1), we have and We need the following Lemma: Lemma 2.2.Under the γ-conditions given by Definition 2.1, and for all and Proof.Using the γ-conditions, and the properties of function f , we obtain in turn: Moreover, we have It follows by the Banach Lemma on invertible operators [4], [12] that F 0 (x) −1 ∈ L(Y, X), and That complete the proof of the Lemma.♦ It is convenient for us to define sequences {a n }, {b n }, {c n }, {d n } by ; and functions a, b, c, d on [0, r 0 ) by .
It is simple algebra to see that system of inequalities ). (2.8)We shall also use the identities [4]: and for z = x + y 2 , and all x, y, w ∈ D.
We can show the local convergence theorem for the Newton-type method (1.2): Theorem 2.3.Under the γ-conditions given by Definition 2.1 for x ∈ U (x , r = 5 − √ 13 6 γ ) ⊆ D, sequences {x n }, {y n } generated by the Newtontype method (1.2) are well defined, remain in U (x , r ) for all n ≥ 0, and converge to the unique zero of equation Moreover the following estimates hold for all n ≥ 0: (2.14) Proof.By hypotheses x 0 , y 0 ∈ U (x , r ), and for x = we get F 0 (z 0 ) −1 exists, and We shall show that x k+1 , x k+1 ∈ U (x , r ), and estimates (2.13), (2.14) hold true.
Set λ 1 = g 1 (r ), λ 2 = g 2 (r ), and λ 3 = g 3 (r ).(2.25) In view of (2.7), (2.13), (2.14) and (2.23) we get It then follows from (2.24) and (2.25) that there exists c > 0 such that (2.21) holds true for δ n =k x n − x k, p = 1 and q = 1.Hence, we arrived at: where l is the Lipschitz constant in condition: k F 0 (x ) −1 (F 0 (x) − F 0 (y)) k≤ l k x − y k for all x, y ∈ D. The radius r W W given by Werner in [11] is defined by Moreover note that under (2.2) the existence of x in U (x 0 , )) is guaranteed.However, in practice the existence of x may have been established by another way that avoids condition (2.2).Finally note that enlarging the radius of convergence is very important in computational mathematics since in this case we can obtain a wider range of initial guesses x 0 .
34)wherek F 0 (x ) −1 k≤ Γ (2.35) and k F 0 (x) − F 0 (y) k≤ l 1 k x − y k (2.36)hold true for all x, y ∈ D.Hence, since Γ = l 1 = e, we get by (2.32) r W W = .270670566.(2.37)Hence, we deducer W < r R < r W W < r A .(2.38)By comparing r A and r W we see that it is always true that r W < r A .(2.39)