Local Convergence of Newton’s Method on Lie Groups and Uniqueness Balls

Abstract

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 L-average Lipschitz condition. As applications, we get estimations of radius of convergence ball under the Kantorovich condition and the 𝛾-condition, respectively. In particular, under the 𝛾-condition, our results improve the corresponding results in (Li et al. 2009, Corollary 4.1) as showed in Remark 17. Finally, applications to analytical mappings are also given.

1. Introduction

In a vector space framework, when 𝑓 is a differentiable operator from some domain 𝐷 in a real or complex Banach space 𝑋 to another 𝑌, Newton’s method is one of the most important methods for finding the approximation solution of the equation 𝑓(𝑥)=0, which is formulated as follows: for any initial point 𝑥₀∈𝐷, 𝑥_𝑛+1=𝑥_𝑛−𝑓^′𝑥_𝑛⁻¹𝑓𝑥_𝑛,𝑛=0,1,…. (1)

As is well known, one of the most important results on Newton’s method is Kantorovich’s theorem (cf. [1]), which provides a simple and clear criterion ensuring quadratic convergence of Newton’s method under the mild condition that the second Fréchet derivative of 𝑓 is bounded (or more generally, the first derivative is Lipschitz continuous) and the boundedness of ‖𝑓^′(𝑥)⁻¹‖ on a proper open metric ball of the initial point 𝑥₀. Another important result on Newton’s method is Smale’s point estimate theory (i.e., 𝛼-theory and 𝛾-theory) in [2], where the notions of approximate zeros were introduced and the rules to judge an initial point 𝑥₀ to be an approximate zero were established, depending on the information of the analytic nonlinear operator at this initial point and at a solution 𝑥^∗, respectively. There are a lot of works on the weakness and/or the extension of the Lipschitz continuity made on the mappings; see, for example, [3–7] and references therein. In particular, Wang introduced in [6] the notion of Lipschitz conditions with 𝐿-average to unify both Kantorovich’s and Smale’s criteria.

In a Riemannian manifold framework, an analogue of the well-known Kantorovich theorem was given in [8] for Newton’s method for vector fields on Riemannian manifolds while the extensions of the famous Smale 𝛼-theory and 𝛾-theory in [2] to analytic vector fields and analytic mappings on Riemannian manifolds were done in [9]. In the recent paper [10], the convergence criteria in [9] were improved by using the notion of the 𝛾-condition for the vector fields and mappings on Riemannian manifolds. The radii of uniqueness balls of singular points of vector fields satisfying the 𝛾-conditions were estimated in [11], while the local behavior of Newton’s method on Riemannian manifolds was studied in [12, 13]. Furthermore, in [14], Li and Wang extended the generalized 𝐿-average Lipschitz condition (introduced in [6]) to Riemannian manifolds and established a unified convergence criterion of Newton’s method on Riemannian manifolds. Similarly, inspired by the previous work of Zabrejko and Nguen in [7] on Kantorovich’s majorant method, Alvarez et al. introduced in [15] a Lipschitz-type radial function for the covariant derivative of vector fields and mappings on Riemannian manifolds and gave a unified convergence criterion of Newton’s method on Riemannian manifolds.

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 [16], where the algorithm presented is independent of affine connections on the Lie group. This means that Newton’s method on Lie groups is different from the one defined on Riemannian manifolds. On the other hand, motivated by looking for approaches to solv ordinary differential equations on Lie groups, Owren and Welfert also studied in [17] Newton’s method, independent of affine connections on the Lie group, and showed the local quadratical convergence. Recently, Wang and Li [18] established Kantorovich’s theorem (independent of the connection) for Newton’s method on Lie group. More precisely, under the assumption that the differential of 𝑓 satisfies the Lipschitz condition around the initial point (which is in terms of one-parameter semigroups and independent of the metric), the convergence criterion of Newton’s method is presented. Extensions of Smale’s point estimate theory for Newton’s method on Lie groups were given in [19].

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 𝐿-average Lipschitz condition. As applications, we obtain estimations of radius of convergence ball under the Kantorovich condition and the 𝛾-condition, respectively. In particular, under the 𝛾-condition, we get that (see Theorem 16) if 𝑓(𝑥^∗)=0 and 𝜚𝑥^∗,𝑥₀<3−2√2,2𝛾 (2) then the sequence {𝑥_𝑛} generated by Newton’s method (28) with initial point 𝑥₀ is well defined and converges quadratically to a zero 𝑦^∗ of 𝑓. This improves the corresponding results in [19, Corollary 4.1], where it was proved under the following assumption: there exists 𝑣∈𝔤 such that 𝑥₀=𝑥^∗‖𝑣‖≤𝑎exp𝑣,⁰𝛾, (3) with 𝑎₀=0.081256… being the smallest positive root of the equation 𝑎₀/(1−4𝑎₀+2𝑎²₀)²=3−2√2. Clearly, 𝑎₀𝛾<3−2√2.2𝛾 (4) Note also that in general, there dose not exist 𝑣∈𝔤 satisfying 𝑥₀=𝑥^∗exp𝑣 because the exponential map is not surjective global, even if 𝜚(𝑥^∗,𝑥₀)<(3−2√2)/2𝛾. In view of this, our results somewhat improve the corresponding results in [19, Corollary 4.1].

The remainder of the paper is organized as follows. Some preliminary results and notions are given in Section 2, while the estimation of uniqueness ball is presented in Section 3. In Section 4, the main results about estimations of convergence ball are explored. Theorems under the Kantorovich condition and the 𝛾-condition are provided in Section 5. In the final section, we get the estimations of uniqueness ball and convergence ball under the assumption that 𝑓 is analytic.

2. Notions and Preliminaries

Most of the notions and notations which are used in the present paper are standard; see, for example, [20, 21]. A Lie group (𝐺,⋅) is a Hausdorff topological group with countable bases which also has the structure of an analytic manifold such that the group product and the inversion are analytic operations in the differentiable structure given on the manifold. The dimension of a Lie group is that of the underlying manifold, and we will always assume that it is 𝑚-dimensional. The symbol 𝑒 designates the identity element of 𝐺. Let 𝔤 be the Lie algebra of the Lie group 𝐺 which is the tangent space 𝑇_𝑒𝐺 of 𝐺 at 𝑒, equipped with Lie bracket [⋅,⋅]∶𝔤×𝔤→𝔤.

In the sequel, we will make use of the left translation of the Lie group 𝐺. We define for each 𝑦∈𝐺 the left translation 𝐿_𝑦∶𝐺→𝐺 by 𝐿_𝑦(𝑧)=𝑦⋅𝑧foreach𝑧∈𝐺. (5) The differential of 𝐿_𝑦 at 𝑧 is denoted by (𝐿^′_𝑦)_𝑧 which clearly determines a linear isomorphism from 𝑇_𝑧𝐺 to the tangent space 𝑇_{(𝑦⋅𝑧)}𝐺. In particular, the differential (𝐿^′_𝑦)_𝑒 of 𝐿_𝑦 at 𝑒 determines a linear isomorphism form 𝔤 to the tangent space 𝑇_𝑦𝐺. The exponential map exp∶𝔤→𝐺 is certainly the most important construction associated with 𝐺 and 𝔤 and is defined as follows. Given 𝑢∈𝔤, let 𝜎_𝑢∶ℝ→𝐺 be the one-parameter subgroup of 𝐺 determined by the left invariant vector field 𝑋_𝑢∶𝑦↦(𝐿^′_𝑦)_𝑒(𝑢); that is, 𝜎_𝑢 satisfies that 𝜎_𝑢(0)𝜎=𝑒,_′𝑢(𝑡)=𝑋_𝑢𝜎_𝑢(𝑡)=𝐿^′_{𝜎𝑢(𝑡)𝑒}(𝑢)foreach𝑡∈ℝ. (6) The value of the exponential map exp at 𝑢 is then defined by exp(𝑢)=𝜎_𝑢(1). (7) Moreover, we have that exp(𝑡𝑢)=𝜎_𝑡𝑢(1)=𝜎_𝑢(𝑡)()()()foreach𝑡∈ℝ,𝑢∈𝔤,exp𝑡+𝑠𝑢=exp𝑡𝑢⋅exp𝑠𝑢forany𝑡,𝑠∈ℝ,𝑢∈𝔤. (8) Note that the exponential map is not surjective in general. However, the exponential map is a diffeomorphism on an open neighborhood of 0∈𝔤. In the case when 𝐺 is Abelian, exp is also a homomorphism from 𝔤 to 𝐺; that is, exp(𝑢+𝑣)=exp(𝑢)⋅exp(𝑣),∀𝑢,𝑣∈𝔤. (9) In the non-Abelian case, exp is not a homomorphism and, by the Baker-Campbell-Hausdorff (BCH) formula (cf. [21, page 114]), (9) must be replaced by exp(𝑤)=exp(𝑢)⋅exp(𝑣), (10) for all 𝑢,𝑣 in a neighborhood of 0∈𝔤, where 𝑤 is defined by 1𝑤∶=𝑢+𝑣+2[]+1𝑢,𝑣[+[[12([𝑢,𝑢,𝑣]]𝑣,𝑣,𝑢]])+⋯. (11)

Let 𝑓∶𝐺→𝔤 be a 𝐶¹-map and let 𝑥∈𝐺. We use 𝑓^′_𝑥 to denote the differential of 𝑓 at 𝑥. Then, by [22, Page 9] (the proof given there for a smooth mapping still works for a 𝐶¹-map), for each Δ_𝑥∈𝑇_𝑥𝐺 and any nontrivial smooth curve 𝑐∶(−𝜀,𝜀)→𝐺 with 𝑐(0)=𝑥 and 𝑐^′(0)=Δ_𝑥, one has 𝑓^′_𝑥Δ_𝑥=𝑑(𝑡)𝑑𝑡𝑓∘𝑐_𝑡=0. (12) In particular, 𝑓^′_𝑥Δ_𝑥=𝑑𝑓𝑡𝐿𝑑𝑡𝑥⋅exp^′_𝑥−1𝑥Δ_𝑥𝑡=0foreachΔ_𝑥∈𝑇_𝑥𝐺. (13) Define the linear map 𝑑𝑓_𝑥∶𝔤→𝔤 by 𝑑𝑓_𝑥𝑑𝑢=𝑓()𝑑𝑡𝑥⋅exp𝑡𝑢_𝑡=0foreach𝑢∈𝔤. (14) Then, by (13), 𝑑𝑓_𝑥=𝑓^′_𝑥∘𝐿^′_𝑥𝑒. (15) Also, in view of definition, we have that, for all 𝑡≥0, 𝑑𝑓()𝑑𝑡𝑥⋅exp𝑡𝑢=𝑑𝑓_{𝑥⋅exp(𝑡𝑢)}𝑓()(𝑥)=𝑢foreach𝑢∈𝔤,𝑥⋅exp𝑡𝑢−𝑓^𝑡₀𝑑𝑓_{𝑥⋅exp(𝑠𝑢)}𝑢𝑑𝑠foreach𝑢∈𝔤. (16)(17)

For the remainder of the present paper, we always assume that ⟨⋅,⋅⟩ is an inner product on 𝔤 and ‖⋅‖ is the associated norm on 𝔤. We now introduce the following distance on 𝐺 which plays a key role in the study. Let 𝑥,𝑦∈𝐺 and define 𝜚𝑥,𝑦∶=inf^𝑘_𝑖=1𝑢_𝑖∣thereexist𝑘≥1and𝑢₁,…,𝑢_𝑘∈𝔤suchthat𝑦=𝑥⋅exp𝑢₁⋯exp𝑢_𝑘, (18) where we adapt the convention that inf∅=+∞. It is easy to verify that 𝜚(⋅,⋅) is a distance on 𝐺 and that the topology induced by this distance is equivalent to the original one on 𝐺.

Let 𝑥∈𝐺 and 𝑟>0. We denoted the corresponding ball of radius 𝑟 around 𝑥 of 𝐺 by 𝐶_𝑟(𝑥); that is, 𝐶_𝑟(𝑥)∶=𝑦∈𝐺∣𝜚𝑥,𝑦<𝑟. (19) Let ℒ(𝔤) denote the set of all linear operators on 𝔤. Below, we will modify the notion of the Lipschitz condition with 𝐿-average for mappings on Banach spaces to suit sections. Let 𝐿 be a positive nondecreasing integrable function on [0,𝑅], where 𝑅 is a positive number large enough such that ∫^𝑅₀(𝑅−𝑠)𝐿(𝑠)𝑑𝑠≥𝑅. The notion of Lipschitz condition in the inscribed sphere with the 𝐿 average for operators on Banach spaces was first introduced in [23] by Wang for the study of Smale’s point estimate theory.

Definition 1. Let 𝑟>0,??𝑥₀∈𝐺 and let 𝑇 be a mapping from 𝐺 to ℒ(𝔤). Then 𝑇 is said to satisfy the 𝐿-average Lipschitz condition on 𝐶_𝑟(𝑥₀) if 𝑇𝑥⋅exp𝑢−𝑇(𝑥)≤^{𝜚(𝑥0,𝑥)+‖𝑢‖}_{𝜚(𝑥0,𝑥)}𝐿(𝑠)𝑑𝑠 (20) holds for any 𝑥∈𝐶_𝑟(𝑥₀) and 𝑢∈𝔤 such that ‖𝑢‖<𝑟−𝜚(𝑥,𝑥₀).

3. Uniqueness Ball of Zero Points of Mappings

This section is devoted to the study of uniqueness ball of zero points of mappings. Let 𝑟>0. We use 𝐁(0,𝑟) to denote the open ball at 0 with radius 𝑟 on 𝔤; that is, 𝐁(0,𝑟)∶={𝑣∈𝔤∣‖𝑣‖<𝑟}. (21) Write 𝑁(𝑥^∗,𝑟)∶=𝑥^∗exp(𝐁(0,𝑟)). Clearly, 𝑁(𝑥^∗,𝑟)⊆𝐶_𝑟(𝑥^∗). Let 𝑟̂>0 be such that 1𝑟̂₀𝑟̂𝐿(𝑢)(𝑟̂−𝑢)𝑑𝑢=1. (22)

Theorem 2. Let 0<𝑟≤𝑟̂. Suppose that 𝑓(𝑥^∗)=0 and 𝑑𝑓_𝑥⁻¹_∗𝑑𝑓 satisfies the 𝐿-average Lipschitz condition in 𝑁(𝑥^∗,𝑟). Then 𝑥^∗ is the unique zero point of 𝑓 in 𝑁(𝑥^∗,𝑟).

Proof. Let 𝑦^∗∈𝑁(𝑥^∗,𝑟) be another zero point of 𝑓 in 𝑁(𝑥^∗,𝑟). Then, there exists 𝑣∈𝔤 such that 𝑦^∗=𝑥^∗exp𝑣 and ‖𝑣‖<𝑟. As 𝐿(⋅) is a positive function, it follows from [6] that the function 𝜓 defined by 𝜓(𝑡)=1𝑡^𝑡₀𝐿(𝑠)(𝑡−𝑠)𝑑𝑠,∀𝑡∈(0,𝑟]̂ (23) is strictly monotonically increasing. set 1𝜆∶=‖𝑣‖₀^‖𝑣‖𝐿(𝑠)(‖𝑣‖−𝑠)𝑑𝑠. (24) Then, by (22), we get 1𝜆<𝑟̂₀𝑟̂𝐿(𝑠)(𝑟̂−𝑠)𝑑𝑠=1. (25) To complete the proof, it suffices to show that ‖𝑣‖≤𝜆‖𝑣‖. (26) Granting this, one has that 𝑥^∗=𝑦^∗. Now, ‖𝑣‖=−𝑑𝑓_𝑥⁻¹_∗𝑓𝑦^∗−𝑓𝑥^∗≤+𝑣−𝑑𝑓_𝑥⁻¹_∗¹₀𝑑𝑓_{𝑥∗exp(𝑠𝑣)}≤𝑣𝑑𝑠+𝑣₁₀𝑑𝑓_𝑥⁻¹_∗𝑑𝑓_{𝑥∗exp(𝑠𝑣)}−𝑑𝑓𝑥^∗𝑣‖≤‖𝑑𝑠_{100𝑠‖𝑣‖}𝐿(𝑡)𝑑𝑡‖𝑣‖𝑑𝑠=₀^‖𝑣‖𝐿(𝑡)(‖𝑣‖−𝑡)𝑑𝑡=𝜆‖𝑣‖, (27) where the third inequality holds because of (20) by selecting 𝑥=𝑥₀=𝑥^∗. Therefore, (26) is seen to hold and the proof is completed

4. Convergence Ball of Newton’s Method

Following [17], we define Newton’s method with initial point 𝑥₀ for 𝑓 on a Lie group as follows: 𝑥_𝑛+1=𝑥_𝑛⋅exp−𝑑𝑓_𝑥⁻¹_𝑛𝑓𝑥_𝑛foreach𝑛=0,1,…. (28)

Let 𝑟₀>0 and 𝑏>0 be such that ^𝑟0₀𝐿(𝑠)𝑑𝑠=1,𝑏=^𝑟0₀𝐿(𝑠)𝑠𝑑𝑠. (29)

Remark 3. (i) Since 𝐿(⋅) is a positive function, we always have 𝑏≤𝑟₀. Indeed, 𝑏−𝑟₀=^𝑟0₀𝐿(𝑠)𝑠𝑑𝑠−𝑟₀^𝑟0₀𝐿(𝑠)=𝑑𝑠_𝑟00𝐿(𝑠)𝑠−𝑟₀𝑑𝑠≤0. (30)
(ii) Consider 𝑟₀≤𝑟̂. Indeed, recall from [6] that the function 𝜓 defined by 𝜓(𝑡)=1𝑡^𝑡₀𝐿(𝑠)(𝑡−𝑠)𝑑𝑠,∀𝑡∈(0,𝑟]̂ (31) is strictly monotonically increasing. Sine 𝜓(𝑟₀)≤1=𝜓(𝑟)̂, we get 𝑟₀≤𝑟̂.

The following proposition plays a key role in this section, which is taken from [24].

Proposition 4. Suppose that 𝑥₀∈𝐺 is such that 𝑑𝑓_𝑥⁻¹₀ exists and 𝑑𝑓_𝑥⁻¹₀𝑑𝑓 satisfies the 𝐿-average Lipschitz condition on 𝐶_𝑟0(𝑥₀) and that 𝛽∶=d𝑓_𝑥⁻¹₀𝑓𝑥₀≤𝑏. (32) Then the sequence {𝑥_𝑛} generated by Newton’s method (28) with initial point 𝑥₀ is well defined and converges to a zero point 𝑥^∗ of 𝑓 and 𝜚(𝑥^∗,𝑥₀)<𝑟₀.

The remainder of this section is devoted to an estimate of the convergence domain of Newton’s method on 𝐺 around a zero 𝑥^∗ of 𝑓. Below we will always assume that 𝑥^∗∈𝐺 is such that 𝑑𝑓_𝑥⁻¹_∗ exists.

Lemma 5. Let 0<𝑟≤𝑟₀ and let 𝑥₀∈𝐶_𝑟(𝑥^∗) be such that there exist 𝑗≥1 and 𝑤₁,…,𝑤_𝑗∈𝔤 satisfying 𝑥₀=𝑥^∗⋅exp𝑤₁⋯exp𝑤_𝑗, (33) and 𝜌(𝑥^∗,𝑥₀)∶=∑^𝑗_𝑖=1‖𝑤_𝑖‖<𝑟. Suppose that d𝑓_𝑥⁻¹_∗d𝑓 satisfies the 𝐿-average Lipschitz condition on 𝐶_𝑟(𝑥^∗). Then d𝑓_𝑥⁻¹₀ exists, d𝑓_𝑥⁻¹₀d𝑓_𝑥∗≤11−∫^{𝜌(𝑥∗,𝑥0)}₀𝐿(𝑠),d𝑠d𝑓_𝑥⁻¹₀𝑓𝑥₀≤∫^{𝜌(𝑥∗,𝑥0)}₀𝐿(𝑠)𝜌𝑥^∗,𝑥₀−𝑠d𝑠+𝜌𝑥^∗,𝑥₀1−∫^{𝜌(𝑥∗,𝑥0)}₀𝐿(𝑠).d𝑠 (34)(35)

Proof. It follows from [24, Lemma 2.1] that 𝑑𝑓_𝑥⁻¹₀ exists and (34) holds. Write 𝑦₀=𝑥^∗,? 𝑦_𝑖=𝑦_𝑖−1⋅exp𝑤_𝑖 and 𝜌_𝑖∶=∑^𝑖_𝑙=1‖𝑤_𝑙‖ for each 𝑖=1,…,𝑗. Thus, by (33), we have 𝑦_𝑗=𝑥₀ and so 𝜌_𝑗=𝜌(𝑥^∗,𝑥₀). Fix 𝑖, one has from (17) that 𝑓𝑦_𝑖−𝑓𝑦_𝑖−1=¹₀𝑑𝑓_{𝑦𝑖−1⋅exp(𝜏𝑤𝑖)}𝑤_𝑖𝑑𝜏, (36) which implies that 𝑑𝑓_𝑥⁻¹_∗𝑓𝑦_𝑖−𝑓𝑦_𝑖−1=¹₀𝑑𝑓_𝑥⁻¹_∗𝑑𝑓_{𝑦𝑖−1⋅exp(𝜏𝑤𝑖)}−𝑑𝑓_𝑥∗𝑤_𝑖𝑑𝜏+𝑤_𝑖. (37) Since 𝑑𝑓_𝑥⁻¹_∗𝑑𝑓 satisfies the 𝐿-average Lipschitz condition on 𝐶_𝑟(𝑥^∗), it follows that 𝑑𝑓_𝑥⁻¹_∗𝑑𝑓_{𝑦𝑖−1⋅exp(𝜏𝑤𝑖)}−𝑑𝑓_𝑥∗≤𝑑𝑓_𝑥⁻¹_∗𝑑𝑓_{𝑦𝑖−1⋅exp(𝜏𝑤𝑖)}−𝑑𝑓_𝑦𝑖−1+^𝑖−1_𝑙=1𝑑𝑓_𝑥⁻¹_∗𝑑𝑓_𝑦𝑙−𝑑𝑓_𝑦𝑙−1≤^{𝜌𝑖−1+𝜏‖𝑤𝑖‖}_𝜌𝑖−1𝐿(𝑠)𝑑𝑠+^𝑖−1_𝑙=1^𝜌𝑙_𝜌𝑙−1𝐿(𝑠)≤𝑑𝑠_{𝜌𝑖−1+𝜏‖𝑤𝑖‖0}𝐿(𝑠)𝑑𝑠, (38) where 𝜌₀=0. Noting that 𝑓(𝑥₀)=∑^𝑗_𝑖=1(𝑓(𝑦_𝑖)−𝑓(𝑦_𝑖−1)), we have from (37) and (38) that 𝑑𝑓_𝑥⁻¹_∗𝑓𝑥₀≤^𝑗_𝑖=1¹₀𝑑𝑓_𝑥⁻¹_∗𝑑𝑓_{𝑦𝑖−1⋅exp(𝜏𝑤𝑖)}−𝑑𝑓_𝑥∗×𝑤_𝑖𝑑𝜏+𝑤_𝑖≤^𝑗_𝑖=1¹₀^{𝜌𝑖−1+𝜏‖𝑤𝑖‖}₀𝐿(𝑠)𝑑𝑠𝑤_𝑖𝑑𝜏+𝑤_𝑖. (39) Write 𝐹(𝑡)∶=∫^𝑡₀𝐿(𝑠)𝑑𝑠. Then, ¹₀^{𝜌𝑖−1+𝜏‖𝑤𝑖‖}₀𝐿(𝑠)𝑑𝑠𝑤_𝑖=𝑑𝜏₁₀𝐹𝜌_𝑖−1+𝜏𝑤_𝑖𝑤_𝑖=𝑑𝜏_{𝜌𝑖𝜌𝑖−1}𝐹(𝑡)𝑑𝑡, (40) and so ^𝑗_𝑖=1¹₀𝑑𝑓_𝑥⁻¹_∗𝑑𝑓_{𝑦𝑖−1⋅exp(𝜏𝑤𝑖)}−𝑑𝑓_𝑥∗𝑤_𝑖≤𝑑𝜏_{𝑗𝑖=1𝜌𝑖𝜌𝑖−1}𝐹(𝑡)=𝑑𝑡_{𝜌(𝑥∗,𝑥0)0}𝐹(𝑡)=𝑑𝑡_{𝜌(𝑥∗,𝑥0)0}𝐿(𝑠)𝜌𝑥^∗,𝑥₀−𝑠𝑑𝑠. (41) This, together with (39), yields that 𝑑𝑓_𝑥⁻¹_∗𝑓𝑥₀≤^{𝜌(𝑥∗,𝑥0)}₀𝐿(𝑠)𝜌𝑥^∗,𝑥₀−𝑠𝑑𝑠+𝜌𝑥^∗,𝑥₀. (42) Combining this with (34) implies that 𝑑𝑓_𝑥⁻¹₀𝑓𝑥₀≤𝑑𝑓_𝑥⁻¹₀𝑑𝑓_𝑥∗𝑑𝑓_𝑥⁻¹_∗𝑓𝑥₀≤∫^{𝜌(𝑥∗,𝑥0)}₀𝐿(𝑠)𝜌𝑥^∗,𝑥₀−𝑠𝑑𝑠+𝜌𝑥^∗,𝑥₀1−∫^{𝜌(𝑥∗,𝑥0)}₀𝐿(𝑠),𝑑𝑠 (43) which completes the proof of the lemma.

We make the following assumption throughout the remainder of the paper: 𝑥^∗∈𝐺suchthat𝑓𝑥^∗=0,𝑑𝑓_𝑥⁻¹_∗exists. (44) Theorem 6 below gives an estimation of convergence ball of Newton’s method.

Theorem 6. Suppose that 𝑑𝑓_𝑥⁻¹_∗𝑑𝑓 satisfies the 𝐿-average Lipschitz condition on 𝐶_𝑟0(𝑥^∗). Suppose that 𝜚(𝑥^∗,𝑥₀)<(𝑏/2). Then the sequence {𝑥_𝑛} generated by Newton’s method (28) with initial point 𝑥₀ is well defined and converges quadratically to a zero point 𝑦^∗ of 𝑓 and 𝜚(𝑦^∗,𝑥^∗)<𝑟₀.

Proof. Since 𝜚(𝑥^∗,𝑥₀)<(𝑏/2), there exist 𝑗≥1 and 𝑤₁,…,𝑤_𝑗∈𝔤 satisfying 𝑥₀=𝑥^∗⋅exp𝑤₁⋯exp𝑤_𝑗, (45) and 𝜌(𝑥^∗,𝑥₀)∶=∑^𝑗_𝑖=1‖𝑤_𝑖‖<(𝑏/2)≤𝑟₀, where the last inequality holds because of Remark 3(i). By Lemma 5, 𝑑𝑓_𝑥⁻¹₀ exists and 𝛽∶=𝑑𝑓_𝑥⁻¹₀𝑓𝑥₀≤∫^{𝜌(𝑥∗,𝑥0)}₀𝐿(𝑠)𝜌𝑥^∗,𝑥₀−𝑠𝑑𝑠+𝜌𝑥^∗,𝑥₀1−∫^{𝜌(𝑥∗,𝑥0)}₀𝐿(𝑠).𝑑𝑠 (46) Write 𝐿(𝑠)𝐿𝑥∶=𝑠+𝜌^∗,𝑥₀1−∫^{𝜌(𝑥∗,𝑥0)}₀𝐿(𝑠).𝑑𝑠 (47) Let 𝑟₀,𝑏 be such that ^𝑟0₀𝐿(𝑠)𝑑𝑠=1,𝑏=^𝑟0₀𝐿(𝑠)𝑠𝑑𝑠. (48) This gives that ^𝑟0₀𝐿𝑠+𝜌𝑥^∗,𝑥₀𝑑𝑠=1−^{𝜌(𝑥∗,𝑥0)}₀𝐿(𝑠)=𝑑𝑠_𝑟00𝐿(𝑠)𝑑𝑠−^{𝜌(𝑥∗,𝑥0)}₀𝐿(𝑠)=𝑑𝑠_{𝑟0𝜌𝑥∗,𝑥0}𝐿(𝑠)𝑑𝑠. (49) Hence, ^{𝑟0+𝜌(𝑥∗,𝑥0)}_{𝜌𝑥∗,𝑥0}𝐿(𝑠)𝑑𝑠=^𝑟0_{𝜌𝑥∗,𝑥0}𝐿(𝑠)𝑑𝑠. (50) As 𝐿(⋅) is a nondecreasing and positive integrable function, one has 𝑟₀+𝜌𝑥^∗,𝑥₀=𝑟₀. (51) Therefore, 𝑏=^𝑟0₀𝐿(𝑠)𝑠𝑑𝑠=^{𝑟0−𝜌(𝑥∗,𝑥0)}₀𝐿(𝑠)=∫𝑠𝑑𝑠_{𝑟0𝜌𝑥∗,𝑥0}𝐿(𝑠)𝑠−𝜌𝑥^∗,𝑥₀𝑑𝑠1−∫^{𝜌(𝑥∗,𝑥0)}₀𝐿(𝑠).𝑑𝑠 (52) Below, we will show that 𝛽=𝑑𝑓_𝑥⁻¹₀𝑓𝑥₀≤𝑏. (53) To do this, by (46), it remains to show that ^{𝜌(𝑥∗,𝑥0)}₀𝐿(𝑠)𝜌𝑥^∗,𝑥₀−𝑠𝑑𝑠+𝜌𝑥^∗,𝑥₀≤^𝑟0_{𝜌𝑥∗,𝑥0}𝐿(𝑠)𝑠−𝜌𝑥^∗,𝑥₀𝑑𝑠; (54) that is, 𝜌𝑥^∗,𝑥₀≤^𝑟0₀𝐿(𝑠)𝑠−𝜌𝑥^∗,𝑥₀=𝑑𝑠_𝑟00𝐿(𝑠)𝑠𝑑𝑠−𝜌𝑥^∗,𝑥₀^𝑟0₀𝐿(𝑠)𝑥𝑑𝑠=𝑏−𝜌^∗,𝑥₀, (55) which always holds because 𝜌(𝑥^∗,𝑥₀)≤(𝑏/2) by assumption. Hence, (53) is seen to hold.
Then in order to ensure that Proposition 4 is applicable, we have to show the following assertion: 𝑑𝑓_𝑥⁻¹₀𝑑𝑓 satisfies the 𝐿-average Lipschitz condition in 𝐶_𝑟0(𝑥₀). To do this, let 𝑥∈𝐶_𝑟0(𝑥₀) be such that there exist 𝑣,𝑣₁,…,𝑣_𝑙∈𝔤 satisfying 𝑥=𝑥₀exp𝑣₁⋯exp𝑣_𝑙,𝜌(𝑥₀,𝑥)∶=∑^𝑙_𝑖=1‖𝑣_𝑖‖ and ‖𝑣‖+𝜌(𝑥₀,𝑥)<𝑟₀. Since 𝑑𝑓_𝑥⁻¹_∗𝑑𝑓 satisfies the 𝐿-average Lipschitz condition in 𝐶_𝑟0(𝑥₀) and 𝜌𝑥^∗,𝑥₀+‖𝑣‖+𝜌𝑥₀,𝑥<𝜌𝑥^∗,𝑥₀+𝑟₀=𝑟₀ (56) thanks to (51), we obtain that 𝑑𝑓_𝑥⁻¹_∗𝑑𝑓_𝑥exp𝑣−𝑑𝑓(𝑥)≤^{𝜌(𝑥∗,𝑥0)+𝜌(𝑥0,𝑥)+‖𝑣‖}_{𝜌(𝑥∗,𝑥0)+𝜌(𝑥0,𝑥)}𝐿(𝑠)𝑑𝑠. (57) Combining this with (34) yields that 𝑑𝑓_𝑥⁻¹₀𝑑𝑓_𝑥exp𝑣(𝑥≤−𝑑𝑓)𝑑𝑓_𝑥⁻¹₀𝑑𝑓_𝑥∗𝑑𝑓_𝑥⁻¹_∗𝑑𝑓_𝑥exp𝑢(𝑥≤1−𝑑𝑓)1−∫^{𝜌(𝑥∗,𝑥0)}₀𝐿(𝑠)𝑑𝑠^{𝜌(𝑥∗,𝑥0)+𝜌(𝑥0,𝑥)+‖𝑣‖𝜌(𝑥∗,𝑥0)+𝜌(𝑥0,𝑥)}𝐿(𝑠)=𝑑𝑠_{𝜌(𝑥0,𝑥)+‖𝑣‖𝜌(𝑥0,𝑥)}𝐿𝑠+𝜌𝑥^∗,𝑥₀1−∫^{𝜌(𝑥∗,𝑥0)}₀𝐿(𝑠)=𝑑𝑠𝑑𝑠_{𝜌(𝑥0𝜌𝑥,𝑥)+‖𝑣‖0,𝑥}𝐿(𝑠)𝑑𝑠. (58) Hence, 𝑑𝑓_𝑥⁻¹₀𝑑𝑓 satisfies the 𝐿-average Lipschitz condition in 𝐶_𝑟0(𝑥₀). Thus, we apply Proposition 4 to conclude that the sequence {𝑥_𝑛} generated by Newton’s method (28) with initial point 𝑥₀ is well defined and converges to a zero 𝑦^∗ of 𝑓 and 𝜚(𝑦^∗,𝑥₀)<𝑟₀. And 𝜚𝑦^∗,𝑥^∗≤𝜚𝑦^∗,𝑥₀+𝜚𝑥^∗,𝑥₀≤𝜌𝑥^∗,𝑥₀+𝑟₀=𝑟₀. (59) The proof of the theorem is completed.

Theorem 6 gives an estimate of the convergence domain for Newton’s method. However, we do not know whether the limit 𝑦^∗ of the sequence generated by Newton’s method with initial point 𝑥₀ from this domain is equal to the zero 𝑥^∗. The following corollary provides the convergence domain from which the sequence generated by Newton’s method with initial point 𝑥₀ converges to the zero 𝑥^∗. Recall that 𝑒 designates the identity element of 𝐺.

Corollary 7. Suppose that 𝑑𝑓_𝑥⁻¹_∗𝑑𝑓 satisfies the 𝐿-average Lipschitz condition on 𝐶_𝑟0(𝑥^∗). Suppose that 𝐶_𝑟0(𝑒)⊆exp(𝐁(0,𝑟₀)) and 𝜚(𝑥^∗,𝑥₀)<(𝑏/2). Then, the sequence {𝑥_𝑛} generated by Newton’s method (28) with initial point 𝑥₀ is well defined and converges quadratically to 𝑥^∗.

Proof. Since 𝜚(𝑥^∗,𝑥₀)<(𝑏/2), we apply Theorem 6 to conclude that the sequence {𝑥_𝑛} generated by Newton’s method (28) with initial point 𝑥₀ is well defined and converges quadratically to a zero point 𝑦^∗ of 𝑓 and 𝜚(𝑦^∗,𝑥^∗)<𝑟₀; that is, 𝜚((𝑥^∗)⁻¹𝑦^∗,𝑒)<𝑟₀. Since 𝐶_𝑟0(𝑒)⊆exp(𝐁(0,𝑟₀)), there exists 𝑢∈𝔤 such that ‖𝑢‖≤𝑟₀ and (𝑥^∗)⁻¹𝑦^∗=exp𝑢; that is, 𝑦^∗=𝑥^∗exp𝑢. Hence, 𝑦^∗∈𝑁(𝑥^∗,𝑟₀)∶=𝑥^∗exp(𝐁(0,𝑟₀)). As 𝑟₀≤𝑟_𝑢 by Remark 3(ii), Theorem 2 is applicable, and so 𝑦^∗=𝑥^∗.

Recall that in the special case when 𝐺 is a compact connected Lie group, 𝐺 has a bi-invariant Riemannian metric (cf. [22, page 46]). Below, we assume that 𝐺 is a compact connected Lie group and endowed with a bi-invariant Riemannian metric. Therefore, an estimate of the convergence domain with the same property as in Corollary 7 is described in the following corollary.

Corollary 8. Let 𝐺 be a compact connected Lie group and endowed with a bi-invariant Riemannian metric. Suppose that 𝑑𝑓_𝑥⁻¹_∗𝑑𝑓 satisfies the 𝐿-average Lipschitz condition on 𝐶_𝑟0(𝑥^∗). Suppose that 𝜚(𝑥^∗,𝑥₀)<(𝑏/2). Then, the sequence {𝑥_𝑛} generated by Newton’s method (28) with initial point 𝑥₀ is well defined and converges quadratically to 𝑥^∗.

Proof. By Theorem 6, the sequence {𝑥_𝑛} generated by Newton’s method (28) with initial point 𝑥₀ is well defined and converges to a zero, say 𝑦^∗, of 𝑓 with 𝜚(𝑥^∗,𝑦^∗)<𝑟₀. Clearly, there is a minimizing geodesic 𝑐 connecting 𝑥^∗−1⋅𝑦^∗ and 𝑒. Since 𝐺 is a compact connected Lie group and endowed with a bi-invariant Riemannian metric, it follows from [20, page 224] that 𝑐 is also a one-parameter subgroup of 𝐺. Consequently, there exists 𝑢∈𝔤 such that 𝑦^∗=𝑥^∗⋅exp𝑢 and ‖𝑢‖=𝜚(𝑥^∗,𝑦^∗)<𝑟₀. Hence, 𝑦^∗∈𝑁(𝑥^∗,𝑟₀)∶=𝑥^∗exp(𝐁(0,𝑟₀)). As 𝑟₀≤𝑟̂ by Remark 3(ii), Theorem 2 is applicable, and so 𝑦^∗=𝑥^∗.

5. Theorems under the Kantorovich Condition and the 𝛾-Condition

This section is devoted to the study of some applications of the results obtained in the preceding sections. At first, if 𝐿(⋅) is a constant, then the 𝐿-average Lipschitz condition is reduced to the classical Lipschitz condition.

Let 𝑟>0,??𝑥₀∈𝐺 and let 𝑇 be a mapping from 𝐺 to ℒ(𝔤). Then 𝑇 is said to satisfy the 𝐿 Lipschitz condition on 𝐶_𝑟(𝑥₀) if 𝑇𝑥⋅exp𝑢−𝑇(𝑥)≤𝐿‖𝑢‖ (60) holds for any 𝑢,𝑢₀,…,𝑢_𝑘∈𝔤 and 𝑥∈𝐶_𝑟(𝑥₀) such that 𝑥=𝑥₀exp𝑢₀exp𝑢₁⋯exp𝑢_𝑘 and ‖𝑢‖+𝜌(𝑥,𝑥₀)<𝑟, where 𝜌(𝑥,𝑥₀)=∑^𝑘_𝑖=0‖𝑢_𝑖‖.

Hence, in the case when 𝐿(⋅)≡𝐿, we obtain from (22) and (29) that 𝑟₀=1𝐿1,𝑏=,̂22𝐿𝑟=𝐿. (61) Thus, by Theorems 2 and 6, we have the following results, where Theorem 10 has been given in [18].

Theorem 9. Let 0<𝑟≤(2/𝐿), Suppose that 𝑑𝑓_𝑥⁻¹_∗𝑑𝑓 satisfies the 𝐿 Lipschitz condition in 𝑁(𝑥^∗,𝑟). Then 𝑥^∗ is the unique zero point of 𝑓 in 𝑁(𝑥^∗,𝑟).

Theorem 10. Suppose that 𝑑𝑓_𝑥⁻¹_∗𝑑𝑓 satisfies the 𝐿 Lipschitz condition on 𝐶_1/𝐿(𝑥^∗). Suppose that 𝜚(𝑥^∗,𝑥₀)<(1/4𝐿). Then the sequence {𝑥_𝑛} generated by Newton’s method (28) with initial point 𝑥₀ is well defined and converges quadratically to a zero 𝑦^∗ of 𝑓 and 𝜚(𝑦^∗,𝑥^∗)<(1/𝐿).

Furthermore, by Corollaries 7 and 8, one has the following results.

Corollary 11. Suppose that 𝑑𝑓_𝑥⁻¹_∗𝑑𝑓 satisfies the 𝐿 Lipschitz condition on 𝐶_1/𝐿(𝑥^∗). Suppose that 𝐶_1/𝐿(𝑒)⊆exp(𝐁(0,(1/𝐿))) and 𝜚(𝑥^∗,𝑥₀)<(1/4𝐿). Then, the sequence {𝑥_𝑛} generated by Newton’s method (28) with initial point 𝑥₀ is well defined and converges quadratically to 𝑥^∗.

Corollary 12. Let 𝐺 be a compact connected Lie group and endowed with a bi-invariant Riemannian metric. Suppose that 𝑑𝑓_𝑥⁻¹_∗𝑑𝑓 satisfies the 𝐿 Lipschitz condition on 𝐶_1/𝐿(𝑥^∗). Suppose that 𝜚(𝑥^∗,𝑥₀)<(1/4𝐿). Then, the sequence {𝑥_𝑛} generated by Newton’s method (28) with initial point 𝑥₀ is well defined and converges quadratically to 𝑥^∗.

Let 𝑘 be a positive integer, and assume further that 𝑓∶𝐺→𝔤 is a 𝐶^𝑘-map. Define the map 𝑑^𝑘𝑓_𝑥∶𝔤^𝑘→𝔤 by 𝑑^𝑘𝑓_𝑥𝑢₁⋯𝑢_𝑘=𝜕^𝑘𝜕𝑡_𝑘⋯𝜕𝑡₁𝑓𝑥⋅exp𝑡_𝑘𝑢_𝑘⋯exp𝑡₁𝑢_{1𝑡𝑘=⋯=𝑡1=0}, (62) for each (𝑢₁,…,𝑢_𝑘)∈𝔤^𝑘. In particular, 𝑑^𝑘𝑓_𝑥𝑢^𝑘=𝑑^𝑘𝑑𝑡^𝑘𝑓𝑥⋅exp𝑡𝑢_𝑡=0foreach𝑢∈𝔤. (63) Let 1≤𝑖≤𝑘. Then, in view of the definition, one has 𝑑^𝑘𝑓_𝑥𝑢₁⋯𝑢_𝑘=𝑑^𝑘−𝑖𝑑^𝑖𝑓⋅(𝑢₁⋯𝑢_𝑖)_𝑥𝑢_𝑖+1⋯𝑢_𝑘foreach𝑢₁,…,𝑢_𝑘∈𝔤^𝑘. (64) In particular, for fixed 𝑢₁,…,𝑢_𝑖−1, 𝑢_𝑖+1,…,𝑢_𝑘∈𝔤, 𝑑^𝑖𝑓_𝑥𝑢₁⋯𝑢_𝑖−1(𝑢)=𝑑𝑑^𝑖−1𝑓𝑢₁⋯𝑢_𝑖−1(𝑢)foreach𝑢∈𝔤. (65) This implies that 𝑑^𝑖𝑓_𝑥𝑢₁⋯𝑢_𝑖−1𝑢 is linear with respect to 𝑢∈𝔤 and so is 𝑑^𝑘𝑓_𝑥𝑢₁⋯𝑢_𝑖−1𝑢𝑢_𝑖+1⋯𝑢_𝑘. Consequently, 𝑑^𝑘𝑓_𝑥 is a multilinear map from 𝔤^𝑘 to 𝔤 because 1≤𝑖≤𝑘 is arbitrary. Thus, we can define the norm of 𝑑^𝑘𝑓_𝑥 by 𝑑^𝑘𝑓_𝑥∶=sup𝑑^𝑘𝑓_𝑥𝑢₁𝑢₂⋯𝑢_𝑘∣𝑢₁,…,𝑢_𝑘∈𝔤^𝑘witheach𝑢_𝑗=1. (66)

For the remainder of the paper, we always assume that 𝑓 is a 𝐶²-map from 𝐺 to 𝔤. Then taking 𝑖=2, we have 𝑑²𝑓_𝑧𝑣𝑢=𝑑𝑑𝑓⋅𝑣_𝑧𝑢forany𝑢,𝑣∈𝔤andeach𝑧∈𝐺. (67) Thus, (17) is applied (with 𝑑𝑓⋅𝑣 in place of 𝑓(⋅) for each 𝑣∈𝔤) to conclude the following formula: 𝑑𝑓_{𝑥⋅exp(𝑡𝑢)}−𝑑𝑓_𝑥=^𝑡₀𝑑²𝑓_{𝑥⋅exp(𝑠𝑢)}𝑢𝑑𝑠foreach𝑢∈𝔤,𝑡∈ℝ. (68)

The 𝛾-condition for nonlinear operators in Banach spaces was first introduced by Wang and Han [25], and explored further by Wang [26] to study Smale’s point estimate theory, which has been extended in [19] for a map 𝑓 from a Lie group to its Lie algebra in view of the map 𝑑²𝑓 as given in Definition 13 below. Let 𝑟>0 and 𝛾>0 be such that 𝛾𝑟≤1.

Definition 13. Let 𝑥₀∈𝐺 be such that 𝑑𝑓_𝑥⁻¹₀ exists. 𝑓 is said to satisfy the 𝛾-condition at 𝑥₀ on 𝐶_𝑟(𝑥₀) if, for any 𝑥∈𝐶_𝑟(𝑥₀) with 𝑥=𝑥₀exp𝑢₀exp𝑢₁⋯exp𝑢_𝑘 such that 𝜌(𝑥,𝑥₀)∶=∑^𝑘_𝑖=0‖𝑢_𝑖‖<𝑟, 𝑑𝑓_𝑥⁻¹₀𝑑²𝑓_𝑥≤2𝛾1−𝛾𝜌𝑥,𝑥₀³. (69)

As shown in Proposition 20, if 𝑓 is analytic at 𝑥₀, then 𝑓 satisfies the 𝛾-condition at 𝑥₀.

Let 𝛾>0 and let 𝐿 be the function defined by 𝐿(𝑠)=2𝛾1−𝛾𝑠³1foreach0<𝑠<𝛾. (70) The following proposition shows that the 𝛾-condition implies the 𝐿-average Lipschitz condition, which is taken from [24].

Proposition 14. Suppose that 𝑓 satisfies the 𝛾-condition at 𝑥₀ on 𝐶_𝑟(𝑥₀). Then 𝑑𝑓_𝑥⁻¹₀𝑑𝑓 satisfies the 𝐿-average Lipschitz condition on 𝐶_𝑟(𝑥₀) with 𝐿 being defined by (70).

In the case when 𝐿 is given by (70), we have from (22) and (29) that 𝑟₀=2−√2√2𝛾,𝑏=3−22𝛾,̂1𝑟=.2𝛾 (71) Thus, by Theorems 2 and 6, we have the following results.

Theorem 15. Let 0<𝑟≤(1/2𝛾). Suppose that 𝑓 satisfies the 𝛾-condition in 𝑁(𝑥^∗,𝑟)∶=𝑥^∗exp(𝐁(0,𝑟)). Then 𝑥^∗ is the unique zero point of 𝑓 in 𝑁(𝑥^∗,𝑟).

Theorem 16. Suppose that 𝑓 satisfies the 𝛾-condition on 𝐶_{√(2−2)/2𝛾}(𝑥^∗). Suppose that 𝜚(𝑥^∗,𝑥₀)<(3−2√2)/2𝛾. Then the sequence {𝑥_𝑛} generated by Newton’s method (28) with initial point 𝑥₀ is well defined and converges quadratically to a zero point 𝑦^∗ of 𝑓 and 𝜚(𝑦^∗,𝑥^∗)<(2−√2)/2𝛾.

Remark 17. Theorem 16 improves the corresponding results in [19, Corollary 4.1], where it was proved under the following assumption: there exists 𝑣∈𝔤 such that 𝑥₀=𝑥^∗exp𝑣 and ‖𝑣‖≤(𝑎₀/𝛾) with 𝑎₀=0.081256… being the smallest positive root of the equation 𝑎₀/(1−4𝑎₀+2𝑎²₀)²=3−2√2. Clearly, (𝑎₀/𝛾)<(3−2√2)/2𝛾. Note also that in general, there dose not exist 𝑣∈𝔤 satisfying 𝑥₀=𝑥^∗exp𝑣 because the exponential map is not surjective global, even if 𝜚(𝑥^∗,𝑥₀)<(3−2√2)/2𝛾. In view of this, our results somewhat improves the corresponding results in [19, Corollary 4.1].

Moreover, we get the following two corollaries from Corollaries 7 and 8.

Corollary 18. Suppose that 𝑓 satisfies the 𝛾-condition on 𝐶_{√(2−2)/2𝛾}(𝑥^∗). Suppose that 𝐶_{√(2−2)/2𝛾}(𝑒)⊆exp(𝐁(0,(2−√2)/2𝛾)) and 𝜚(𝑥^∗,𝑥₀)<(3−2√2)/2𝛾. Then, the sequence {𝑥_𝑛} generated by Newton’s method (28) with initial point 𝑥₀ is well defined and converges quadratically to 𝑥^∗.

Corollary 19. Let 𝐺 be a compact connected Lie group and endowed with a bi-invariant Riemannian metric. Suppose that 𝑓 satisfies the 𝛾-condition on 𝐶_{√(2−2)/2𝛾}(𝑥^∗). Suppose that 𝜚(𝑥^∗,𝑥₀)<(3−2√2)/2𝛾. Then, the sequence {𝑥_𝑛} generated by Newton’s method (28) with initial point 𝑥₀ is well defined and converges quadratically to 𝑥^∗.

6. Applications to Analytic Maps

Throughout this section, we always assume that 𝑓 is analytic on 𝐺. For 𝑥∈𝐺 such that 𝑑𝑓_𝑥⁻¹ exists, we define 𝛾_𝑥∶=𝛾𝑓,𝑥=sup_𝑖≥2𝑑𝑓_𝑥⁻¹𝑑^𝑖𝑓_𝑥𝑖!^1/(𝑖−1). (72) Also we adopt the convention that 𝛾(𝑓,𝑥)=∞ if 𝑑𝑓_𝑥 is not invertible. Note that this definition is justified, and, in the case when 𝑑𝑓_𝑥 is invertible, 𝛾(𝑓,𝑥) is finite by analyticity.

The following proposition is taken from [19].

Proposition 20. Let 𝛾_𝑥∗∶=𝛾(𝑓,𝑥^∗) and let 𝑟=(2−√2)/2𝛾_𝑥∗. Then 𝑓 satisfies the 𝛾_𝑥∗-condition at 𝑥^∗ on 𝐶_𝑟(𝑥^∗).

Thus, by Theorems 15 and 16 and Proposition 20, we have the following results.

Theorem 21. Let 0<𝑟≤(1/2𝛾_𝑥∗). Then 𝑥^∗ is the unique zero point of 𝑓 in 𝑁(𝑥^∗,𝑟).

Theorem 22. Suppose that 𝜚(𝑥^∗,𝑥₀)<(3−2√2)/2𝛾_𝑥∗. Then the sequence {𝑥_𝑛} generated by Newton’s method (28) with initial point 𝑥₀ is well defined and converges quadratically to a zero point 𝑦^∗ of 𝑓 and 𝜚(𝑦^∗,𝑥^∗)<(2−√2)/2𝛾_𝑥∗.

Moreover, we get the following two corollaries from Corollaries 7 and 8 and Proposition 20.

Corollary 23. Suppose that 𝐶_{√(2−2)/2𝛾𝑥∗}(𝑒)⊆exp(𝐁(0,(2−√2)/2𝛾_𝑥∗)) and 𝜚(𝑥^∗,𝑥₀)<((3−2√2)/2𝛾_𝑥∗). Then, the sequence {𝑥_𝑛} generated by Newton’s method (28) with initial point 𝑥₀ is well defined and converges quadratically to 𝑥^∗.

Corollary 24. Let 𝐺 be a compact connected Lie group and endowed with a bi-invariant Riemannian metric. Suppose that 𝜚(𝑥^∗,𝑥₀)<(3−2√2)/2𝛾_𝑥∗. Then, the sequence {𝑥_𝑛} generated by Newton’s method (28) with initial point 𝑥₀ is well defined and converges quadratically to 𝑥^∗.

Conflict of Interests

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

Acknowledgments

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).