^{1}

^{2, 3}

^{4, 5}

^{1}

^{2}

^{3}

^{4}

^{5}

An estimation of uniqueness ball of a zero point of a mapping on
Lie group is established. Furthermore, we obtain a unified estimation of radius of convergence ball of
Newton’s method on Lie groups under a generalized

In a vector space framework, when

As is well known, one of the most important results on Newton’s method is Kantorovich’s theorem (cf. [

In a Riemannian manifold framework, an analogue of the well-known Kantorovich theorem was given in [

Note also that Mahony used one-parameter subgroups of a Lie group to develop a version of Newton’s method on an arbitrary Lie group in [

The purpose of the present paper is to continue the study of Newton’s method on Lie groups. At first, we give an estimation of uniqueness ball of a zero point of a mapping on a Lie group. Second, we establish a unified estimation of radius of convergence ball of Newton’s method on Lie groups under a generalized

The remainder of the paper is organized as follows. Some preliminary results and notions are given in Section

Most of the notions and notations which are used in the present paper are standard; see, for example, [

In the sequel, we will make use of the left translation of the Lie group

Let

For the remainder of the present paper, we always assume that

Let

Let

This section is devoted to the study of uniqueness ball of zero points of mappings. Let

Let

Let

Following [

Let

(i) Since

(ii) Consider

The following proposition plays a key role in this section, which is taken from [

Suppose that

The remainder of this section is devoted to an estimate of the convergence domain of Newton’s method on

Let

It follows from [

We make the following assumption throughout the remainder of the paper:

Suppose that

Since

Then in order to ensure that Proposition

Theorem

Suppose that

Since

Recall that in the special case when

Let

By Theorem

This section is devoted to the study of some applications of the results obtained in the preceding sections. At first, if

Let

Hence, in the case when

Let

Suppose that

Furthermore, by Corollaries

Suppose that

Let

Let

For the remainder of the paper, we always assume that

The

Let

As shown in Proposition

Let

Suppose that

In the case when

Let

Suppose that

Theorem

Moreover, we get the following two corollaries from Corollaries

Suppose that

Let

Throughout this section, we always assume that

The following proposition is taken from [

Let

Thus, by Theorems

Let

Suppose that

Moreover, we get the following two corollaries from Corollaries

Suppose that

Let

The authors declare that there is no conflict of interests regarding the publication of this paper.

The research of the second author was partially supported by the National Natural Science Foundation of China (Grant nos. 11001241 and 11371325) and by Zhejiang Provincial Natural Science Foundation of China (Grant no. LY13A010011). The research of the third author was partially supported by a Grant from NSC of Taiwan (NSC 102-2221-E-037-004-MY3).