# Semilocal Convergence Analysis for Inexact Newton Method under Weak Condition

Authors: Xiubin Xu; Yuan Xiao; Tao Liu
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

Under the hypothesis that the first derivative satisfies some kind of weak Lipschitz conditions, a new semilocal convergence theorem for inexact Newton method is presented. Unified convergence criteria ensuring the convergence of inexact Newton method are also established. Applications to some special cases such as the Kantorovich type conditions andγ-Conditions are provided and some well-known convergence theorems for Newton's method are obtained as corollaries.


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## Body

## 1. Introduction

LetF be a continuously Fréchet differentiable nonlinear operator from a convex subset D of Banach space X to Banach space Y. Finding solutions of a nonlinear operator equation:
(1.1)F(x)=0
in Banach space is a basic and important problem in applied and computational mathematics. A classical method for finding an approximation of a solution of (1.1) is Newton's method which is defined by
(1.2)xn+1=xn-F′(xn)-1F(xn),x0∈D,n=0,1,2,….There is a huge literature on local as well as semilocal convergence for Newton's method under various assumptions (see [1–9]). Besides, there are a lot of works on the weakness of the hypotheses made on the underlying operators, see for example [2, 3, 5–9] and references therein. In particular, Wang in [7, 8] introduced the notions of Lipschitz conditions with L average, under which Kantorovich like convergence criteria and Smale's point estimate theory can be put together to be investigated.However, Newton's method has two disadvantages. One is to evaluateF′ involved, the other is to solve the exact solution of Newton equations:
(1.3)F′(xn)(xn+1-xn)=-F(xn),n=0,1,2,….
In many applications, for example, those in Euclidean spaces, computing the exact solutions using a direct method such as Gaussian elimination can be expensive if the number of unknowns is large and may not be justified when xk is far from the searched solution. While using linear iterative methods to approximate the solutions of (1.3) instead of solving it exactly can reduce some of the costs of Newton's method. One of the methods is inexact Newton method which can be found in [10] and takes the following form:
(1.4)xn+1=xn+sn,F′(xn)sn=-F(xn)+rn,n=0,1,2,…,
where {rn} is a sequence in Y.As is well known, the convergence behavior of the inexact Newton method depends on the residual controls of{rn} under the hypothesis that F′ satisfies different conditions. Some relative results can be found in [10–24], for example.Under the Lipschitz continuity assumption onF′, different residual controls were used. For example, the residual controls ∥rn∥≤ηn∥F(xn)∥ were adopted in [10, 12]; in [15] the affine invariant conditions ∥F′(x0)-1rn∥≤ηn∥F′(x0)-1F(xn)∥ were considered; while in [21] Shen has analyzed the semilocal convergence behavior in some manner such that the relative residuals {rn} satisfy
(1.5)‖F′(x0)-1rn‖≤ηn‖F′(x0)-1F(xn)‖1+κ,0≤κ≤1,n=0,1,2,….
Assuming that the residuals satisfy ∥Pnrn∥≤θn∥PnF(xn)∥1+κ, where {Pn} is a sequence of invertible operators from Y to X, and that F′(x0)-1F′ satisfies the Hölder condition around x0, Li and Shen established the local and semilocal convergence in [16, 20], respectively. Besides, the γ-condition was also introduced into inexact Newton method in [22] by considering residual controls (1.5) with κ=1, that is,
(1.6)‖F′(x0)-1rn‖≤ηn‖F′(x0)-1F(xn)‖2,n=0,1,2,…;
Smale's α-theory for the inexact Newton method was established there.In the present paper, by considering the residual controls (1.6), we will study the convergence of inexact Newton method under the assumption that F has a continuous derivative in a closed ball B(x0,r)¯, F′(x0)-1F′ exists and F′(x0)-1F′ satisfies the weak Lipschitz condition:
(1.7)‖F′(x0)-1(F′(x)-F′(x′))‖≤∫ρ(x)ρ(xx′¯)L(u)du,∀x∈B(x0,r),∀x′∈B(x,r-ρ(x))¯,
where r is a positive number, ρ(x)=∥x-x0∥,ρ(xx′¯)=ρ(x)+∥x′-x∥≤r, and L is a positive integrable nondecreasing function on [0,r]. We also establish the unified convergence criteria, which include Kantorovich type and Smale type convergence criteria as special cases. In particular, in the special case when ηn=0(n=0,1,2,…), (1.4) reduces to Newton's method and our result extends the corresponding one in [7].The paper is organized as follows. Section2 gives some lemmas which are used in the proof of our main theorem. In Section 3, the semilocal convergence of inexact Newton method is studied under the weak Lipschitz condition (1.7). Its applications to some special cases are provided in Section 4.


## 2. Preliminaries

LetX and Y be Banach spaces. Throughout this paper, R>r are two positive numbers, L is a positive integrable nondecreasing function on any involved intervals, and B(x,R) is an open ball in X with center x and radius R. Let β>0,0≤λ<1,ω≥1, and σ≥0. Define
(2.1)φ(t)=β-(1-λ)t+σt2+ω∫0tL(u)(t-u)du,0≤t≤R,ψ(t)=β-t+ω∫0tL(u)(t-u)du,0≤t≤R.
Obviously,
(2.2)φ′(t)=-(1-λ)+2σt+ω∫0tL(u)du,0≤t≤R,(2.3)ψ′(t)=-1+ω∫0tL(u)du,0≤t≤R,(2.4)φ′′(t)=2σ+ωL(t)>0,0≤t≤R.
Set
(2.5)rλ:=sup⁡{r∈(0,R):ω∫0rL(u)du+2σr≤1-λ},(2.6)bλ:=(1-λ)rλ-σrλ2-ω∫0rλL(u)(rλ-u)du.
Write δ=ω∫0R‍L(u)du+2σR. Then
(2.7)rλ={R,ifδ<1-λ,rλ′,ifδ≥1-λ,
where rλ′∈[0,R] is such that ω∫0rλ′‍L(u)du+2σrλ′=1-λ. Furthermore, it follows that
(2.8)bλ≥ω∫0rλ‍L(u)udu+σrλ2,ifδ<1-λ,bλ=ω∫0rλ‍L(u)udu+σrλ2,ifδ≥1-λ.
Let
(2.9)t0=0,tn+1=tn-φ(tn)ψ′(tn),n=0,1,2,….The following two lemmas describe some properties about the majorizing functionφ and the convergence property of {tn}.Lemma 2.1.
Suppose thatβ≤bλ and φ is defined by (2.1). Then the function φ is strictly decreasing and has exact one zero t* on [0,rλ] satisfying β<t*.Proof.
By (2.4) and (2.5), we know φ′ is strictly increasing on [0,rλ] and has the values φ′(0)<0 and φ′(rλ)≤0. This implies that φ is strictly decreasing on [0,rλ]. Note that φ(0)=β>0 and φ(rλ)≤0 by the definition of bλ. Thus, φ(t)=0 has exact one solution t* on [0,rλ]. Since
(2.10)φ(β)=λβ+σβ2+ω∫0βL(u)(β-u)du>0,
we have β<t*. The proof is complete.Lemma 2.2.
Lett* be the positive solution of equation φ(t)=0 on [0,rλ]. Suppose that β≤bλ and the sequence {tn} is defined by (2.9). Then
(2.11)tn<tn+1<t*,n=0,1,2,….
Consequently, {tn} is strictly increasing and converges to t*.Proof.
We prove the lemma by mathematical induction. Note that0=t0<t1=β<t*. For n>1, assume that
(2.12)tn-1<tn<t*.
Since ψ′′(t)=ωL(t)>0,-ψ′ is strictly decreasing on [0,rλ]. Hence,
(2.13)-ψ′(tn)>-ψ′(t*)≥-ψ′(rλ)=-φ′(rλ)+λ+2σrλ≥0.
Moreover, φ(tn)>0 by of Lemma 2.1. It follows that
(2.14)tn+1=tn-φ(tn)ψ′(tn)>tn.
Define a function N(t) on [0,t*] by
(2.15)N(t):=t-φ(t)ψ′(t),t∈[0,t*].
Note that ψ′(t)<0,t∈[0,t*], unless λ=0, σ=0 and t=t*=rλ, for which we adopt the convention that limt→t*(φ(t)/ψ′(t))=0 and N(t*)=t*-limt→t*(φ(t)/ψ′(t))=t*. Hence, the function N(t) is well defined and continuous on [0,t*].
Moreover, by (2.2) and (2.3), we have
(2.16)N′(t)=1-φ′(t)ψ′(t)-φ(t)ψ′′(t)(ψ′(t))2=-ψ′(t)(λ+2σt)+φ(t)ψ′′(t)(ψ′(t))2>0,t∈[0,t*).
Hence, N(t) is monotonically increasing on [0,t*). This together with (2.9) and (2.14) implies that
(2.17)tn<tn+1=N(tn)<N(t*)=t*.
Therefore, by mathematical induction, (2.11) holds. Consequently, {tn} is increasing, bounded, and converges to a point tλ*, which satisfies φ(tλ*)=0. Hence, t*=tλ*. The proof is complete.To prove our main result, we need two more lemmas. The first can be found in [23] and the second in [7].Lemma 2.3.
Suppose thatF has a continuous derivative satisfying the weak Lipschitz condition (1.7). Let r satisfy ∫0r‍L(u)du≤1. Then F′(x) is invertible in the ball B(x0,r) and
(2.18)‖F′(x)-1F′(x0)‖≤(1-∫0ρ(x)L(u)du)-1.Lemma 2.4.
Let0≤c<R and define
(2.19)χ(t)=1t2∫0tL(c+u)(t-u)du,0≤t<R-c.
Then, χ is increasing on [0,R-c).


## 3. Semilocal Convergence Analysis

Recall thatF:D⊆X→Y is a nonlinear operator with continuous Fréchet derivative. Let B(x0,R)⊆D and x0∈D be such that F′(x0)-1 exists. In the present paper, we adopt the residuals {rn} satisfying (1.6) and assume that η=supn≥0ηn<1. Thus, if n≥0 and {xn} is well defined, then
(3.1)‖F′(x0)-1rn‖≤ηn‖F′(x0)-1F(xn)‖2≤η‖F′(x0)-1F(xn)‖2.
Let
(3.2)α=‖F′(x0)-1F(x0)‖,β=(1+η)α.
Write
(3.3)ω=1+η,σ=η(1+η)(1+∫0RL(u)du)2(1-η)2.
Recall that rλ is determined by (2.5), φ(t*)=0, and {tn} is generated by (2.9) with ω and σ given in (3.3).Lemma 3.1.
Let{xn} be a sequence generated by (1.4). Suppose that F satisfies the weak Lipschitz condition (1.7) on B(x0,t*)⊆B(x0,R) and that β≤bλ. For an integer m≥1, if
(3.4)η‖F′(x0)-1F(xn-1)‖≤1,‖xn-xn-1‖≤tn-tn-1
hold for each 1≤n≤m, then the following assertions hold:
(3.5)(1+η)‖F′(x0)-1F(xm)‖≤φ(tm);η‖F′(x0)-1F(xm)‖≤1.Proof.
Assume that (3.4) holds for each 1≤n≤m. Write xm-1τ=xm-1+τ(xm-xm-1),τ∈[0,1]. Applying (1.4), we have
(3.6)F(xm)=F(xm)-F(xm-1)-F′(xm-1)(xm-xm-1)+rm-1=∫01[F′(xm-1τ)-F′(xm-1)]dτ(xm-xm-1)+rm-1.
Hence,
(3.7)‖F′(x0)-1F(xm)‖≤‖F′(x0)-1∫01[F′(xm-1τ)-F′(xm-1)]dτ(xm-xm-1)‖+‖F′(x0)-1rm-1‖=I1+I2.
To estimate I1, by (3.4), we notice that
(3.8)‖xm-1τ-x0‖=‖xm-1+τ(xm-xm-1)-x0‖≤∑n=1m-1‖xn-xn-1‖+τ‖xm-xm-1‖≤tm-1+τ(tm-tm-1)=(1-τ)tm-1+τtm<t*.
In particular,
(3.9)‖xm-1-x0‖≤tm-1<t*,‖xm-x0‖≤tm<t*.
Thus, by the weak Lipschitz condition (1.7), we obtain
(3.10)I1≤∫0‖xm-xm-1‖(‖xm-xm-1‖-u)L(‖xm-1-x0‖+u)du.
Below we estimate I2. We firstly notice that (3.1) and (3.4) yield
(3.11)‖F′(x0)-1F′(xm-1)(xm-xm-1)‖≥‖F′(x0)-1F(xm-1)‖-‖F′(x0)-1rm-1‖≥‖F′(x0)-1F(xm-1)‖-η‖F′(x0)-1F(xm-1)‖2≥(1-η)‖F′(x0)-1F(xm-1)‖.
Since
(3.12)‖F′(x0)-1F′(xm-1)‖=‖I+F′(x0)-1[F′(xm-1)-F′(x0)]‖≤1+∫0ρ(xm-1)L(u)du≤1+∫0RL(u)du,
we have
(3.13)‖F′(x0)-1F(xm-1)‖≤‖F′(x0)-1F′(xm-1)‖‖xm-xm-1‖1-η≤1+∫0RL(u)du1-η‖xm-xm-1‖.
Combining this with (3.1) implies that
(3.14)I2≤η‖F′(x0)-1F(xm-1)‖2≤η(1+∫0RL(u)du)2(1-η)2‖xm-xm-1‖2.
Consequently, by (3.7), (3.10), (3.14) and Lemma 2.4, we get
(3.15)(1+η)‖F′(x0)-1F(xm)‖≤(1+η)(I1+I2)≤(1+η)∫0‖xm-xm-1‖(‖xm-xm-1‖-u)L(‖xm-1-x0‖+u)du+η(1+η)(1+∫0RL(u)du)2(1-η)2‖xm-xm-1‖2=ω∫0‖xm-xm-1‖‍(‖xm-xm-1‖-u)L(‖xm-1-x0‖+u)du+σ‖xm-xm-1‖2=(ω‖xm-xm-1‖2∫0‖xm-xm-1‖‍(‖xm-xm-1‖-u)×L(‖xm-1-x0‖+u)du+σω‖xm-xm-1‖2)‖xm-xm-1‖2≤(ω(tm-tm-1)2∫0tm-tm-1(tm-tm-1-u)L(tm-1+u)du+σ)×(tm-tm-1)2=ω∫0tm-tm-1(tm-tm-1-u)L(tm-1+u)du+σ[tm2-tm-12-2tm-1(tm-tm-1)]=φ(tm)-φ(tm-1)-φ′(tm-1)(tm-tm-1).
Noting that φ′(t)=ψ′(t)+λ+2σt and -φ(tm-1)-ψ′(tm-1)(tm-tm-1)=0, we have
(3.16)(1+η)‖F′(x0)-1F(xm)‖≤φ(tm)-φ(tm-1)-φ′(tm-1)(tm-tm-1)=φ(tm)-(λ+2σtm-1)(tm-tm-1)≤φ(tm).
Moreover, since φ is decreasing on [0, t*], one has
(3.17)(1+η)‖F′(x0)-1F(xm)‖≤φ(tm)≤φ(t0)=β.
And therefore
(3.18)η‖F′(x0)-1F(xm)‖≤η1+ηβ=η‖F′(x0)-1F(x0)‖≤1.
That is, (3.5) holds, and the proof is complete.We now give the main result.Theorem 3.2.
Suppose thatβ≤min{1/η,bλ} and B(x0,t*)¯⊆B(x0,R), and that F′(x0)-1F′ satisfies the weak Lipschitz condition (1.7) on B(x0,t*). Then the sequence {xn} generated by the inexact Newton method (1.4) converges to a solution x* of (1.1). Moreover,
(3.19)‖xn-x*‖≤t*-tn,n=0,1,2,….Proof.
We firstly use mathematical induction to prove that (3.4) holds for each n=1,2,…. For n=1, by the above condition and (3.2), the first inequality in (3.4) holds trivially. While the second one can be proved as follows:
(3.20)‖x1-x0‖≤‖F′(x0)-1F(x0)‖+‖F′(x0)-1r0‖≤α+ηα2≤α+ηα=(1+η)α=β=t1-t0.
Assume that (3.4) holds for all n≤m. Then, Lemma 3.1 is applicable to concluding that
(3.21)(1+η)‖F′(x0)-1F(xm)‖≤φ(tm);η‖F′(x0)-1F(xm)‖≤1.
Hence, by (3.5), together with the weak Lipschitz condition (1.7) and Lemma 2.3, one has
(3.22)‖xm+1-xm‖≤‖F′(xm)-1F′(x0)‖(‖F′(x0)-1F(xm)‖+‖F′(x0)-1rm‖)≤11-∫0ρ(xm)L(u)du(‖F′(x0)-1F(xm)‖+η‖F′(xm)-1F(xm)‖2)≤1+η1-ω∫0ρ(xm)L(u)du‖F′(x0)-1F(xm)‖≤-φ(tm)ψ′(tm)=tm+1-tm.
Therefore, (3.4) holds for n=m+1 and so for each n≥1. Consequently, for n≥0 and k≥0,
(3.23)‖xk+n-xn‖≤∑i=1k‖xi+n-xi+n-1‖≤∑i=1k(ti+n-ti+n-1)=tk+n-tn.
This together with Lemma 2.2 means that {xn} is a Cauchy sequence and so converges to some x*. While taking k→∞ in (3.23), we obtain
(3.24)‖xn-x*‖≤t*-tn,n=0,1,2,….
The proof is complete.In the special case whenηn=0(n=0,1,2,…), inexact Newton method (1.4) reduces to Newton's method. Moreover, ω=1,σ=0,β=∥F′(x0)-1F(x0)∥. Thus, Theorem 3.2 reduces to the related theorem of Newton's method.Corollary 3.3.
Assume thatβ≤bλ and B(x0,t*)¯⊆B(x0,R), where bλ=∫0rλ‍L(u)udu and rλ satisfying ∫0rλ‍L(u)du≤1-λ. Suppose that F′(x0)-1F′ satisfies the weak Lipschitz condition (1.7) on B(x0,t*). Then the sequence {xn} generated by Newton's method (1.2) converges to a solution x* of (1.1). Moreover,
(3.25)‖xn-x*‖≤t*-tn,n=0,1,2,…,
where t* and {tn} are defined in Lemma 2.2 for η=0.In more particular, suppose that∫0R‍L(u)du>1 and λ=0. Then Corollary 3.3 reduces to the following result given in (Theorem 3.1, [7]).Corollary 3.4.
Assume thatβ≤bλ0, where bλ0=∫0rλ0‍L(u)udu and ∫0rλ0‍L(u)du=1. Suppose that F′(x0)-1F′ satisfies weak Lipschitz condition (1.7) on B(x0,t*)⊆B(x0,R). Then the sequence {xn} generated by Newton's method (1.2) converges to a solution x* of (1.1). Moreover,
(3.26)‖xn-x*‖≤t*-tn,n=0,1,2,…,
where t* and {tn} are defined in Lemma 2.2 for η=0 and λ=0.


## 4. Application

This section is divided into two subsections: we consider the applications of our main results specializing, respectively, in Kantorovich type condition and inγ-condition. In particular, our results reduce some of the corresponding results of Newton's method.


### 4.1. Kantorovich-Type Condition

Throughout this subsection, letL be a positive constant. By (2.1), we have
(4.1)φ(t)=β-(1-λ)t+(σ+12ωL)t2,t≥0,ψ(t)=β-t+12ωLt2,t≥0.
By (2.5) and (2.6), we get
(4.2)rλ=1-λωL+σ,bλ=(1-λ)2ωL2(ωL+σ).
The convergence criterion becomes
(4.3)‖F′(x0)-1F(x0)‖≤(1-λ)2ωL2(ωL+σ).Moreover, suppose thatη=0 and λ=0. Then criterion (4.3) reduces to the well-known Kantorovich type criterion ∥F′(x0)-1F(x0)∥≤1/2L of Newton's method in [7].Corollary 4.1.
LetL be a positive constant, β=∥F′(x0)-1F(x0)∥ and β≤bλ0, where bλ0=1/2L and rλ0=1/L. Assume that F satisfies the condition:
(4.4)‖F′(x0)-1(F′(x)-F′(x′))‖≤L‖x-x′‖,∀x,x′∈B(x0,r),‖x-x0‖+‖x-x′‖≤r,
where r=(1-1-2Lβ)/L. Then the sequence {xn} generated by Newton's method (1.2) converges to a solution x* of (1.1), and satisfies
(4.5)‖xn-x*‖≤t*-tn,n=0,1,2,….


### 4.2.γ-Condition

Throughout this subsection, we assume thatγ>0 and F has continuous second derivative and satisfies
(4.6)‖F′(x0)-1F′′(x)‖≤2γ(1-γ‖x-x0‖)3,∀x∈B(x0,1γ).
Let
(4.7)L(u)=2γ(1-γu)3,u∈[0,1γ).
Then, by (2.1), we have
(4.8)φ(t)=β-(1-λ)t+σt2+γt21-γt,0≤t<1γ,ψ(t)=β-t+γt21-γt,0≤t<1γ.
By (2.5) and (2.6), rλ and bλ satisfy
(4.9)ω[1(1-γrλ)2-1]+σrλ=1-λ,bλ=γrλ2(1-γrλ)2.
The convergence criterion becomes
(4.10)‖F′(x0)-1F(x0)‖≤γrλ2(1-γrλ)2.In the more special case, whenη=0 and λ=0, we obtain the criterion ∥F′(x0)-1F(x0)∥≤(3-22)/γ the same with Newton's method in [7].Corollary 4.2.
Letγ be a positive constant, β=∥F′(x0)-1F(x0)∥ and β≤bλ0, where bλ0=(3-22)/γ and rλ0=(1-(1/2))(1/γ). Assume that F satisfies the condition:
(4.11)‖F′(x0)-1(F′(x)-F′(x′))‖≤1(1-γ‖x-x0‖-γ‖x′-x0‖)2-1(1-γ∥x-x0∥)2,∀x,x′∈B(x0,r),‖x-x0‖+‖x′-x‖≤r,
where r=(1+βγ-(1+βγ)2-8βγ)/4γ. Then the sequence {xn} generated by Newton's method (1.2) converges to a solution x* of (1.1), and satisfies
(4.12)‖xn-x*‖≤t*-tn,n=0,1,2,….


## 4.1. Kantorovich-Type Condition

Throughout this subsection, letL be a positive constant. By (2.1), we have
(4.1)φ(t)=β-(1-λ)t+(σ+12ωL)t2,t≥0,ψ(t)=β-t+12ωLt2,t≥0.
By (2.5) and (2.6), we get
(4.2)rλ=1-λωL+σ,bλ=(1-λ)2ωL2(ωL+σ).
The convergence criterion becomes
(4.3)‖F′(x0)-1F(x0)‖≤(1-λ)2ωL2(ωL+σ).Moreover, suppose thatη=0 and λ=0. Then criterion (4.3) reduces to the well-known Kantorovich type criterion ∥F′(x0)-1F(x0)∥≤1/2L of Newton's method in [7].Corollary 4.1.
LetL be a positive constant, β=∥F′(x0)-1F(x0)∥ and β≤bλ0, where bλ0=1/2L and rλ0=1/L. Assume that F satisfies the condition:
(4.4)‖F′(x0)-1(F′(x)-F′(x′))‖≤L‖x-x′‖,∀x,x′∈B(x0,r),‖x-x0‖+‖x-x′‖≤r,
where r=(1-1-2Lβ)/L. Then the sequence {xn} generated by Newton's method (1.2) converges to a solution x* of (1.1), and satisfies
(4.5)‖xn-x*‖≤t*-tn,n=0,1,2,….


## 4.2.γ-Condition

Throughout this subsection, we assume thatγ>0 and F has continuous second derivative and satisfies
(4.6)‖F′(x0)-1F′′(x)‖≤2γ(1-γ‖x-x0‖)3,∀x∈B(x0,1γ).
Let
(4.7)L(u)=2γ(1-γu)3,u∈[0,1γ).
Then, by (2.1), we have
(4.8)φ(t)=β-(1-λ)t+σt2+γt21-γt,0≤t<1γ,ψ(t)=β-t+γt21-γt,0≤t<1γ.
By (2.5) and (2.6), rλ and bλ satisfy
(4.9)ω[1(1-γrλ)2-1]+σrλ=1-λ,bλ=γrλ2(1-γrλ)2.
The convergence criterion becomes
(4.10)‖F′(x0)-1F(x0)‖≤γrλ2(1-γrλ)2.In the more special case, whenη=0 and λ=0, we obtain the criterion ∥F′(x0)-1F(x0)∥≤(3-22)/γ the same with Newton's method in [7].Corollary 4.2.
Letγ be a positive constant, β=∥F′(x0)-1F(x0)∥ and β≤bλ0, where bλ0=(3-22)/γ and rλ0=(1-(1/2))(1/γ). Assume that F satisfies the condition:
(4.11)‖F′(x0)-1(F′(x)-F′(x′))‖≤1(1-γ‖x-x0‖-γ‖x′-x0‖)2-1(1-γ∥x-x0∥)2,∀x,x′∈B(x0,r),‖x-x0‖+‖x′-x‖≤r,
where r=(1+βγ-(1+βγ)2-8βγ)/4γ. Then the sequence {xn} generated by Newton's method (1.2) converges to a solution x* of (1.1), and satisfies
(4.12)‖xn-x*‖≤t*-tn,n=0,1,2,….



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*Source: 982925-2012-08-30.xml*