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

Authors: Jinsu He; Jinhua Wang; Jen-Chih Yao
Journal: Abstract and Applied Analysis
 (2013)

---

## 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 generalizedL-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.


---

## Body

## 1. Introduction

In a vector space framework, whenf 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,
(1)xn+1=xn-f′(xn)-1f(xn),n=0,1,….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 generalizedL-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
(2)ϱ(x*,x0)<3-222γ,
then the sequence {xn} 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
(3)x0=x*expv,∥v∥≤a0γ,
with a0=0.081256… being the smallest positive root of the equation a0/(1-4a0+2a02)2=3-22. Clearly,
(4)a0γ<3-222γ.
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)<(3-22)/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 Section2, 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 TeG 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 groupG. We define for each y∈G the left translation Ly:G→G by
(5)Ly(z)=y·zforeachz∈G.
The differential of Ly at z is denoted by (Ly′)z which clearly determines a linear isomorphism from TzG to the tangent space T(y·z)G. In particular, the differential (Ly′)e of Ly at e determines a linear isomorphism form 𝔤 to the tangent space TyG. 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 σu:ℝ→G be the one-parameter subgroup of G determined by the left invariant vector field Xu:y↦(Ly′)e(u); that is, σu satisfies that
(6)σu(0)=e,σu′(t)=Xu(σu(t))=(Lσu(t)′)e(u)foreacht∈ℝ.
The value of the exponential map exp at u is then defined by
(7)exp(u)=σu(1).
Moreover, we have that
(8)exp(tu)=σtu(1)=σu(t)foreacht∈ℝ,u∈𝔤,exp(t+s)u=exp(tu)·exp(su)foranyt,s∈ℝ,u∈𝔤.
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,
(9)exp(u+v)=exp(u)·exp(v),∀u,v∈𝔤.
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
(10)exp(w)=exp(u)·exp(v),
for all u,v in a neighborhood of 0∈𝔤, where w is defined by
(11)w:=u+v+12[u,v]+112([u,[u,v]]+[v,[v,u]])+⋯.Letf:G→𝔤 be a C1-map and let x∈G. We use fx′ 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 Δx∈TxG and any nontrivial smooth curve c:(-ε,ε)→G with c(0)=x and c′(0)=Δx, one has
(12)fx′Δx=(ddt(f∘c)(t))t=0.
In particular,
(13)fx′Δx=(ddtf(x·exp(t(Lx-1′)xΔx)))t=0((Lx-1′)xΔx)foreachΔx∈TxG.
Define the linear map dfx:𝔤→𝔤 by
(14)dfxu=(ddtf(x·exp(tu)))t=0foreachu∈𝔤.
Then, by (13),
(15)dfx=fx′∘(Lx′)e.
Also, in view of definition, we have that, for all t≥0,
(16)ddtf(x·exp(tu))=dfx·exp(tu)uforeachu∈𝔤,(17)f(x·exp(tu))-f(x)=∫0t‍dfx·exp(su)udsforeachu∈𝔤.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
(18)ϱ(x,y)≔inf{∑i=1k‍∥ui∥∣thereexistk≥1andu1,…,uk∈𝔤suchthaty=x·expu1⋯expuk∑i=1k‍∥ui∥∣thereexistk≥1andu1},
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.Letx∈G and r>0. We denoted the corresponding ball of radius r around x of G by Cr(x); that is,
(19)Cr(x):={y∈G∣ϱ(x,y)<r}.
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 ∫0R‍(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.
Letr>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 Cr(x0) if
(20)∥T(x·expu)-T(x)∥≤∫ϱ(x0,x)ϱ(x0,x)+∥u∥‍L(s)ds
holds for any x∈Cr(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. Letr>0. We use B(0,r) to denote the open ball at 0 with radius r on 𝔤; that is,
(21)B(0,r):={v∈𝔤∣∥v∥<r}.
Write N(x*,r):=x*exp(B(0,r)). Clearly, N(x*,r)⊆Cr(x*). Let r^>0 be such that
(22)1r^∫0r^‍L(u)(r^-u)du=1.Theorem 2.
Let0<r≤r^. Suppose that f(x*)=0 and dfx*-1df satisfies the L-average Lipschitz condition in N(x*,r). Then x* is the unique zero point of f in N(x*,r).Proof.
Lety*∈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
(23)ψ(t)=1t∫0t‍L(s)(t-s)ds,∀t∈(0,r^]
is strictly monotonically increasing. set
(24)λ:=1∥v∥∫0∥v∥‍L(s)(∥v∥-s)ds.
Then, by (22), we get
(25)λ<1r^∫0r^‍L(s)(r^-s)ds=1.
To complete the proof, it suffices to show that
(26)∥v∥≤λ∥v∥.
Granting this, one has that x*=y*. Now,
(27)∥v∥=∥-dfx*-1(f(y*)-f(x*))+v∥≤∥-dfx*-1∫01‍dfx*exp(sv)vds+v∥≤∫01‍∥dfx*-1(dfx*exp(sv)-df(x*))∥∥v∥ds≤∫01‍∫0s∥v∥‍L(t)dt∥v∥ds=∫0∥v∥‍L(t)(∥v∥-t)dt=λ∥v∥,
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:
(28)xn+1=xn·exp(-dfxn-1f(xn))foreachn=0,1,….Letr0>0 and b>0 be such that
(29)∫0r0‍L(s)ds=1,b=∫0r0‍L(s)sds.Remark 3.
(i) SinceL(·) is a positive function, we always have b≤r0. Indeed,
(30)b-r0=∫0r0‍L(s)sds-r0∫0r0‍L(s)ds=∫0r0‍L(s)(s-r0)ds≤0.
(ii) Considerr0≤r^. Indeed, recall from [6] that the function ψ defined by
(31)ψ(t)=1t∫0t‍L(s)(t-s)ds,∀t∈(0,r^]
is strictly monotonically increasing. Sine ψ(r0)≤1=ψ(r^), we get r0≤r^.The following proposition plays a key role in this section, which is taken from [24].Proposition 4.
Suppose thatx0∈G is such that dfx0-1 exists and dfx0-1df satisfies the L-average Lipschitz condition on Cr0(x0) and that
(32)β:=∥dfx0-1f(x0)∥≤b.
Then the sequence {xn} 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 onG around a zero x* of f. Below we will always assume that x*∈G is such that dfx*-1 exists.Lemma 5.
Let0<r≤r0 and let x0∈Cr(x*) be such that there exist j≥1 and w1,…,wj∈𝔤 satisfying
(33)x0=x*·expw1⋯expwj,
and ρ(x*,x0):=∑i=1j‍∥wi∥<r. Suppose that dfx*-1df satisfies the L-average Lipschitz condition on Cr(x*). Then dfx0-1 exists,
(34)∥dfx0-1dfx*∥≤11-∫0ρ(x*,x0)‍L(s)ds,(35)∥dfx0-1f(x0)∥≤∫0ρ(x*,x0)‍L(s)(ρ(x*,x0)-s)ds+ρ(x*,x0)1-∫0ρ(x*,x0)‍L(s)ds.Proof.
It follows from [24, Lemma 2.1] that dfx0-1 exists and (34) holds. Write y0=x*,  yi=yi-1·expwi and ρi:=∑l=1i‍∥wl∥ for each i=1,…,j. Thus, by (33), we have yj=x0 and so ρj=ρ(x*,x0). Fix i, one has from (17) that
(36)f(yi)-f(yi-1)=∫01‍dfyi-1·exp(τwi)widτ,
which implies that
(37)dfx*-1(f(yi)-f(yi-1))=∫01‍dfx*-1(dfyi-1·exp(τwi)-dfx*)widτ+wi.
Since dfx*-1df satisfies the L-average Lipschitz condition on Cr(x*), it follows that
(38)∥dfx*-1(dfyi-1·exp(τwi)-dfx*)∥≤∥dfx*-1(dfyi-1·exp(τwi)-dfyi-1)∥+∑l=1i-1‍∥dfx*-1(dfyl-dfyl-1)∥≤∫ρi-1ρi-1+τ∥wi∥‍L(s)ds+∑l=1i-1‍∫ρl-1ρl‍L(s)ds≤∫0ρi-1+τ∥wi∥‍L(s)ds,
where ρ0=0. Noting that f(x0)=∑i=1j‍(f(yi)-f(yi-1)), we have from (37) and (38) that
(39)∥dfx*-1f(x0)∥≤∑i=1j‍(∫01‍∥dfx*-1(dfyi-1·exp(τwi)-dfx*)∥×∥wi∥dτ+∥wi∥∫01‍∥dfx*-1(dfyi-1·exp(τwi)-dfx*)∥)≤∑i=1j‍(∫01‍∫0ρi-1+τ∥wi∥‍L(s)ds∥wi∥dτ+∥wi∥).
Write F(t):=∫0t‍L(s)ds. Then,
(40)∫01‍∫0ρi-1+τ∥wi∥‍L(s)ds∥wi∥dτ=∫01‍F(ρi-1+τ∥wi∥)∥wi∥dτ=∫ρi-1ρi‍F(t)dt,
and so
(41)∑i=1j‍∫01‍∥dfx*-1(dfyi-1·exp(τwi)-dfx*)∥∥wi∥dτ≤∑i=1j‍∫ρi-1ρi‍F(t)dt=∫0ρ(x*,x0)‍F(t)dt=∫0ρ(x*,x0)‍L(s)(ρ(x*,x0)-s)ds.
This, together with (39), yields that
(42)∥dfx*-1f(x0)∥≤∫0ρ(x*,x0)‍L(s)(ρ(x*,x0)-s)ds+ρ(x*,x0).
Combining this with (34) implies that
(43)∥dfx0-1f(x0)∥≤∥dfx0-1dfx*∥∥dfx*-1f(x0)∥≤∫0ρ(x*,x0)‍L(s)(ρ(x*,x0)-s)ds+ρ(x*,x0)1-∫0ρ(x*,x0)‍L(s)ds,
which completes the proof of the lemma.We make the following assumption throughout the remainder of the paper:(44)x*∈Gsuchthatf(x*)=0,dfx*-1exists.
Theorem 6 below gives an estimation of convergence ball of Newton’s method.Theorem 6.
Suppose thatdfx*-1df satisfies the L-average Lipschitz condition on Cr0(x*). Suppose that ϱ(x*,x0)<(b/2). Then the sequence {xn} 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.
Sinceϱ(x*,x0)<(b/2), there exist j≥1 and w1,…,wj∈𝔤 satisfying
(45)x0=x*·expw1⋯expwj,
and ρ(x*,x0):=∑i=1j‍∥wi∥<(b/2)≤r0, where the last inequality holds because of Remark 3(i). By Lemma 5, dfx0-1 exists and
(46)β:=∥dfx0-1f(x0)∥≤∫0ρ(x*,x0)‍L(s)(ρ(x*,x0)-s)ds+ρ(x*,x0)1-∫0ρ(x*,x0)‍L(s)ds.
Write
(47)L-(s):=L(s+ρ(x*,x0))1-∫0ρ(x*,x0)‍L(s)ds.
Let r-0,b- be such that
(48)∫0r-0‍L-(s)ds=1,b-=∫0r-0‍L-(s)sds.
This gives that
(49)∫0r-0‍L(s+ρ(x*,x0))ds=1-∫0ρ(x*,x0)‍L(s)ds=∫0r0‍L(s)ds-∫0ρ(x*,x0)‍L(s)ds=∫ρ(x*,x0)r0‍L(s)ds.
Hence,
(50)∫ρ(x*,x0)r-0+ρ(x*,x0)‍L(s)ds=∫ρ(x*,x0)r0‍L(s)ds.
As L(·) is a nondecreasing and positive integrable function, one has
(51)r-0+ρ(x*,x0)=r0.
Therefore,
(52)b-=∫0r-0‍L-(s)sds=∫0r0-ρ(x*,x0)‍L-(s)sds=∫ρ(x*,x0)r0‍L(s)(s-ρ(x*,x0))ds1-∫0ρ(x*,x0)‍L(s)ds.
Below, we will show that
(53)β=∥dfx0-1f(x0)∥≤b-.
To do this, by (46), it remains to show that
(54)∫0ρ(x*,x0)‍L(s)(ρ(x*,x0)-s)ds+ρ(x*,x0)≤∫ρ(x*,x0)r0‍L(s)(s-ρ(x*,x0))ds;
that is,
(55)ρ(x*,x0)≤∫0r0‍L(s)(s-ρ(x*,x0))ds=∫0r0‍L(s)sds-ρ(x*,x0)∫0r0‍L(s)ds=b-ρ(x*,x0),
which always holds because ρ(x*,x0)≤(b/2) by assumption. Hence, (53) is seen to hold.
Then in order to ensure that Proposition4 is applicable, we have to show the following assertion: dfx0-1df satisfies the L--average Lipschitz condition in Cr-0(x0). To do this, let x∈Cr-0(x0) be such that there exist v,v1,…,vl∈𝔤 satisfying x=x0expv1⋯expvl,ρ(x0,x):=∑i=1l‍∥vi∥ and ∥v∥+ρ(x0,x)<r-0. Since dfx*-1df satisfies the L-average Lipschitz condition in Cr0(x0) and
(56)ρ(x*,x0)+∥v∥+ρ(x0,x)<ρ(x*,x0)+r-0=r0
thanks to (51), we obtain that
(57)∥dfx*-1(dfxexpv-df(x))∥≤∫ρ(x*,x0)+ρ(x0,x)ρ(x*,x0)+ρ(x0,x)+∥v∥‍L(s)ds.
Combining this with (34) yields that
(58)∥dfx0-1(dfxexpv-df(x))∥≤∥dfx0-1dfx*∥∥dfx*-1(dfxexpu-df(x))∥≤11-∫0ρ(x*,x0)‍L(s)ds∫ρ(x*,x0)+ρ(x0,x)ρ(x*,x0)+ρ(x0,x)+∥v∥‍L(s)ds=∫ρ(x0,x)ρ(x0,x)+∥v∥‍L(s+ρ(x*,x0))1-∫0ρ(x*,x0)‍L(s)dsds=∫ρ(x0,x)ρ(x0,x)+∥v∥‍L-(s)ds.
Hence, dfx0-1df satisfies the L--average Lipschitz condition in Cr-0(x0). Thus, we apply Proposition 4 to conclude that the sequence {xn} 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)<r-0. And
(59)ϱ(y*,x*)≤ϱ(y*,x0)+ϱ(x*,x0)≤ρ(x*,x0)+r-0=r0.
The proof of the theorem is completed.Theorem6 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 thatdfx*-1df satisfies the L-average Lipschitz condition on Cr0(x*). Suppose that Cr0(e)⊆exp(B(0,r0)) and ϱ(x*,x0)<(b/2). Then, the sequence {xn} generated by Newton’s method (28) with initial point x0 is well defined and converges quadratically to x*.Proof.
Sinceϱ(x*,x0)<(b/2), we apply Theorem 6 to conclude that the sequence {xn} 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 Cr0(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≤ru by Remark 3(ii), Theorem 2 is applicable, and so y*=x*.Recall that in the special case whenG 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.
LetG be a compact connected Lie group and endowed with a bi-invariant Riemannian metric. Suppose that dfx*-1df satisfies the L-average Lipschitz condition on Cr0(x*). Suppose that ϱ(x*,x0)<(b/2). Then, the sequence {xn} generated by Newton’s method (28) with initial point x0 is well defined and converges quadratically to x*.Proof.
By Theorem6, the sequence {xn} 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≤r^ by Remark 3(ii), Theorem 2 is applicable, and so y*=x*.


## 5. Theorems under the Kantorovich Condition and theγ-Condition

This section is devoted to the study of some applications of the results obtained in the preceding sections. At first, ifL(·) is a constant, then the L-average Lipschitz condition is reduced to the classical Lipschitz condition.Letr>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 Cr(x0) if
(60)∥T(x·expu)-T(x)∥≤L∥u∥
holds for any u,u0,…,uk∈𝔤 and x∈Cr(x0) such that x=x0expu0expu1⋯expuk and ∥u∥+ρ(x,x0)<r, where ρ(x,x0)=∑i=0k‍∥ui∥.Hence, in the case whenL(·)≡L, we obtain from (22) and (29) that
(61)r0=1L,b=12L,r^=2L.
Thus, by Theorems 2 and 6, we have the following results, where Theorem 10 has been given in [18].Theorem 9.
Let0<r≤(2/L), Suppose that dfx*-1df 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 thatdfx*-1df satisfies the L Lipschitz condition on C1/L(x*). Suppose that ϱ(x*,x0)<(1/4L). Then the sequence {xn} 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 Corollaries7 and 8, one has the following results.Corollary 11.
Suppose thatdfx*-1df satisfies the L Lipschitz condition on C1/L(x*). Suppose that C1/L(e)⊆exp(B(0,(1/L))) and ϱ(x*,x0)<(1/4L). Then, the sequence {xn} generated by Newton’s method (28) with initial point x0 is well defined and converges quadratically to x*.Corollary 12.
LetG be a compact connected Lie group and endowed with a bi-invariant Riemannian metric. Suppose that dfx*-1df satisfies the L Lipschitz condition on C1/L(x*). Suppose that ϱ(x*,x0)<(1/4L). Then, the sequence {xn} generated by Newton’s method (28) with initial point x0 is well defined and converges quadratically to x*.Letk be a positive integer, and assume further that f:G→𝔤 is a Ck-map. Define the map dkfx:𝔤k→𝔤 by
(62)dkfxu1⋯uk=(∂k∂tk⋯∂t1f(x·exptkuk⋯expt1u1))tk=⋯=t1=0,
for each (u1,…,uk)∈𝔤k. In particular,
(63)dkfxuk=(dkdtkf(x·exptu))t=0foreachu∈𝔤.
Let 1≤i≤k. Then, in view of the definition, one has
(64)dkfxu1⋯uk=dk-i(dif·(u1⋯ui))xui+1⋯ukd11k-iforeach(u1,…,uk)∈𝔤k.
In particular, for fixed u1,…,ui-1, ui+1,…,uk∈𝔤,
(65)difxu1⋯ui-1(u)=d(di-1f(u1⋯ui-1))(u)=d(di-1f(u1⋯ui-1))(u)(u)foreachu∈𝔤.
This implies that difxu1⋯ui-1u is linear with respect to u∈𝔤 and so is dkfxu1⋯ui-1uui+1⋯uk. Consequently, dkfx is a multilinear map from 𝔤k to 𝔤 because 1≤i≤k is arbitrary. Thus, we can define the norm of dkfx by
(66)∥dkfx∥:=sup{∥uj∥∥dkfxu1u2⋯uk∥∣(u1,…,uk)∈𝔤kwitheach∥uj∥=1∥dkfxu1u2⋯uk∥∥uj∥}.For the remainder of the paper, we always assume thatf is a C2-map from G to 𝔤. Then taking i=2, we have
(67)d2fzvu=d(df·v)zuforanyu,v∈𝔤andeachz∈G.
Thus, (17) is applied (with df·v in place of f(·) for each v∈𝔤) to conclude the following formula:
(68)dfx·exp(tu)-dfx=∫0t‍d2fx·exp(su)udsdfx·exp(tu)-dfxforeachu∈𝔤,t∈ℝ.Theγ-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.
Letx0∈G be such that dfx0-1 exists. f is said to satisfy the γ-condition at x0 on Cr(x0) if, for any x∈Cr(x0) with x=x0expu0expu1⋯expuk such that ρ(x,x0):=∑i=0k‍∥ui∥<r,
(69)∥dfx0-1d2fx∥≤2γ(1-γρ(x,x0))3.As shown in Proposition20, if f is analytic at x0, then f satisfies the γ-condition at x0.Letγ>0 and let L be the function defined by
(70)L(s)=2γ(1-γs)3foreach0<s<1γ.
The following proposition shows that the γ-condition implies the L-average Lipschitz condition, which is taken from [24].Proposition 14.
Suppose thatf satisfies the γ-condition at x0 on Cr(x0). Then dfx0-1df satisfies the L-average Lipschitz condition on Cr(x0) with L being defined by (70).In the case whenL is given by (70), we have from (22) and (29) that
(71)r0=2-22γ,b=3-22γ,r^=12γ.
Thus, by Theorems 2 and 6, we have the following results.Theorem 15.
Let0<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 thatf satisfies the γ-condition on C(2-2)/2γ(x*). Suppose that ϱ(x*,x0)<(3-22)/2γ. Then the sequence {xn} 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.
Theorem16 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+2a02)2=3-22. Clearly, (a0/γ)<(3-22)/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)<(3-22)/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 Corollaries7 and 8.Corollary 18.
Suppose thatf 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)<(3-22)/2γ. Then, the sequence {xn} generated by Newton’s method (28) with initial point x0 is well defined and converges quadratically to x*.Corollary 19.
LetG 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)<(3-22)/2γ. Then, the sequence {xn} 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 thatf is analytic on G. For x∈G such that dfx-1 exists, we define
(72)γx:=γ(f,x)=supi≥2∥dfx-1difxi!∥1/(i-1).
Also we adopt the convention that γ(f,x)=∞ if dfx is not invertible. Note that this definition is justified, and, in the case when dfx is invertible, γ(f,x) is finite by analyticity.The following proposition is taken from [19].Proposition 20.
Letγx*:=γ(f,x*) and let r=(2-2)/2γx*. Then f satisfies the γx*-condition at x* on Cr(x*).Thus, by Theorems15 and 16 and Proposition 20, we have the following results.Theorem 21.
Let0<r≤(1/2γx*). Then x* is the unique zero point of f in N(x*,r).Theorem 22.
Suppose thatϱ(x*,x0)<(3-22)/2γx*. Then the sequence {xn} 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)/2γx*.Moreover, we get the following two corollaries from Corollaries7 and 8 and Proposition 20.Corollary 23.
Suppose thatC(2-2)/2γx*(e)⊆exp(B(0,(2-2)/2γx*)) and ϱ(x*,x0)<((3-22)/2γx*). Then, the sequence {xn} generated by Newton’s method (28) with initial point x0 is well defined and converges quadratically to x*.Corollary 24.
LetG be a compact connected Lie group and endowed with a bi-invariant Riemannian metric. Suppose that ϱ(x*,x0)<(3-22)/2γx*. Then, the sequence {xn} generated by Newton’s method (28) with initial point x0 is well defined and converges quadratically to x*.



---

*Source: 367161-2013-11-13.xml*