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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 982925, 13 pages
http://dx.doi.org/10.1155/2012/982925
Research Article

Semilocal Convergence Analysis for Inexact Newton Method under Weak Condition

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

Received 29 May 2012; Accepted 5 August 2012

Academic Editor: Jen-Chih Yao

Copyright © 2012 Xiubin Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 and 𝛾-Conditions are provided and some well-known convergence theorems for Newton's method are obtained as corollaries.

1. Introduction

Let 𝐹 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: 𝐹(𝑥)=0(1.1) 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 𝑥𝑛+1=𝑥𝑛𝐹𝑥𝑛1𝐹𝑥𝑛,𝑥0𝐷,𝑛=0,1,2,.(1.2)

There is a huge literature on local as well as semilocal convergence for Newton's method under various assumptions (see [19]). Besides, there are a lot of works on the weakness of the hypotheses made on the underlying operators, see for example [2, 3, 59] 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 evaluate 𝐹 involved, the other is to solve the exact solution of Newton equations: 𝐹𝑥𝑛𝑥𝑛+1𝑥𝑛𝑥=𝐹𝑛,𝑛=0,1,2,.(1.3) 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: 𝑥𝑛+1=𝑥𝑛+𝑠𝑛,𝐹𝑥𝑛𝑠𝑛𝑥=𝐹𝑛+𝑟𝑛,𝑛=0,1,2,,(1.4) 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 [1024], for example.

Under the Lipschitz continuity assumption on 𝐹, different residual controls were used. For example, the residual controls 𝑟𝑛𝜂𝑛𝐹(𝑥𝑛) were adopted in [10, 12]; in [15] the affine invariant conditions 𝐹(𝑥0)1𝑟𝑛𝜂𝑛𝐹(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 𝑥𝐹01𝑟𝑛𝜂𝑛𝐹𝑥01𝐹𝑥𝑛1+𝜅,0𝜅1,𝑛=0,1,2,.(1.5) Assuming that the residuals satisfy 𝑃𝑛𝑟𝑛𝜃𝑛𝑃𝑛𝐹(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, 𝑥𝐹01𝑟𝑛𝜂𝑛𝐹𝑥01𝐹𝑥𝑛2,𝑛=0,1,2,;(1.6) 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: 𝐹𝑥01𝐹(𝑥)𝐹𝑥𝜌(𝑥𝑥)𝜌(𝑥)𝑥𝐿(𝑢)𝑑𝑢,𝑥𝐵0,𝑟,𝑥𝐵(𝑥,𝑟𝜌(𝑥)),(1.7) where 𝑟 is a positive number, 𝜌(𝑥)=𝑥𝑥0,𝜌(𝑥𝑥)=𝜌(𝑥)+𝑥𝑥𝑟, 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. Section 2 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

Let 𝑋 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>0,0𝜆<1,𝜔1, and 𝜎0. Define 𝜑(𝑡)=𝛽(1𝜆)𝑡+𝜎𝑡2+𝜔𝑡0𝐿(𝑢)(𝑡𝑢)𝑑𝑢,0𝑡𝑅,𝜓(𝑡)=𝛽𝑡+𝜔𝑡0𝐿(𝑢)(𝑡𝑢)𝑑𝑢,0𝑡𝑅.(2.1) Obviously, 𝜑(𝑡)=(1𝜆)+2𝜎𝑡+𝜔𝑡0𝜓𝐿(𝑢)𝑑𝑢,0𝑡𝑅,(2.2)(𝑡)=1+𝜔𝑡0𝜑𝐿(𝑢)𝑑𝑢,0𝑡𝑅,(2.3)(𝑡)=2𝜎+𝜔𝐿(𝑡)>0,0𝑡𝑅.(2.4) Set 𝑟𝜆=sup𝑟(0,𝑅)𝜔𝑟0𝑏𝐿(𝑢)𝑑𝑢+2𝜎𝑟1𝜆,(2.5)𝜆=(1𝜆)𝑟𝜆𝜎𝑟2𝜆𝜔𝑟𝜆0𝑟𝐿(𝑢)𝜆𝑢𝑑𝑢.(2.6) Write 𝛿=𝜔𝑅0𝐿(𝑢)𝑑𝑢+2𝜎𝑅. Then 𝑟𝜆=𝑟𝑅,if𝛿<1𝜆,𝜆,if𝛿1𝜆,(2.7) where 𝑟𝜆[0,𝑅] is such that 𝜔𝑟𝜆0𝐿(𝑢)𝑑𝑢+2𝜎𝑟𝜆=1𝜆. Furthermore, it follows that 𝑏𝜆𝜔𝑟𝜆0𝐿(𝑢)𝑢𝑑𝑢+𝜎𝑟2𝜆𝑏,if𝛿<1𝜆,𝜆=𝜔𝑟𝜆0𝐿(𝑢)𝑢𝑑𝑢+𝜎𝑟2𝜆,if𝛿1𝜆.(2.8) Let 𝑡0=0,𝑡𝑛+1=𝑡𝑛𝜑𝑡𝑛𝜓𝑡𝑛,𝑛=0,1,2,.(2.9)

The following two lemmas describe some properties about the majorizing function 𝜑 and the convergence property of {𝑡𝑛}.

Lemma 2.1. Suppose that 𝛽𝑏𝜆 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 𝜑(𝛽)=𝜆𝛽+𝜎𝛽2+𝜔𝛽0𝐿(𝑢)(𝛽𝑢)𝑑𝑢>0,(2.10) we have 𝛽<𝑡. The proof is complete.

Lemma 2.2. Let 𝑡 be the positive solution of equation 𝜑(𝑡)=0 on [0,𝑟𝜆]. Suppose that 𝛽𝑏𝜆 and the sequence {𝑡𝑛} is defined by (2.9). Then 𝑡𝑛<𝑡𝑛+1<𝑡,𝑛=0,1,2,.(2.11) Consequently, {𝑡𝑛} is strictly increasing and converges to 𝑡.

Proof. We prove the lemma by mathematical induction. Note that 0=𝑡0<𝑡1=𝛽<𝑡. For 𝑛>1, assume that 𝑡𝑛1<𝑡𝑛<𝑡.(2.12) Since 𝜓(𝑡)=𝜔𝐿(𝑡)>0,𝜓 is strictly decreasing on [0,𝑟𝜆]. Hence, 𝜓𝑡𝑛>𝜓𝑡𝜓𝑟𝜆=𝜑𝑟𝜆+𝜆+2𝜎𝑟𝜆0.(2.13) Moreover, 𝜑(𝑡𝑛)>0 by of Lemma 2.1. It follows that 𝑡𝑛+1=𝑡𝑛𝜑𝑡𝑛𝜓𝑡𝑛>𝑡𝑛.(2.14) Define a function 𝑁(𝑡) on [0,𝑡] by 𝑁(𝑡)=𝑡𝜑(𝑡)𝜓(𝑡),𝑡0,𝑡.(2.15) Note that 𝜓(𝑡)<0,𝑡[0,𝑡], unless 𝜆=0, 𝜎=0 and 𝑡=𝑡=𝑟𝜆, for which we adopt the convention that lim𝑡𝑡(𝜑(𝑡)/𝜓(𝑡))=0 and 𝑁(𝑡)=𝑡lim𝑡𝑡(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 𝑁𝜑(𝑡)=1(𝑡)𝜓(𝑡)𝜑(𝑡)𝜓(𝑡)(𝜓(𝑡))2=𝜓(𝑡)(𝜆+2𝜎𝑡)+𝜑(𝑡)𝜓(𝑡)(𝜓(𝑡))2>0,𝑡0,𝑡.(2.16) Hence, 𝑁(𝑡) is monotonically increasing on [0,𝑡). This together with (2.9) and (2.14) implies that 𝑡𝑛<𝑡𝑛+1𝑡=𝑁𝑛𝑡<𝑁=𝑡.(2.17) 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 that 𝐹 has a continuous derivative satisfying the weak Lipschitz condition (1.7). Let 𝑟 satisfy 𝑟0𝐿(𝑢)𝑑𝑢1. Then 𝐹(𝑥) is invertible in the ball 𝐵(𝑥0,𝑟) and 𝐹(𝑥)1𝐹𝑥010𝜌(𝑥)𝐿(𝑢)𝑑𝑢1.(2.18)

Lemma 2.4. Let 0𝑐<𝑅 and define 1𝜒(𝑡)=𝑡2𝑡0𝐿(𝑐+𝑢)(𝑡𝑢)𝑑𝑢,0𝑡<𝑅𝑐.(2.19) Then, 𝜒 is increasing on [0,𝑅𝑐).

3. Semilocal Convergence Analysis

Recall that 𝐹𝐷𝑋𝑌 is a nonlinear operator with continuous Fréchet derivative. Let 𝐵(𝑥0,𝑅)𝐷 and 𝑥0𝐷 be such that 𝐹(𝑥0)1 exists. In the present paper, we adopt the residuals {𝑟𝑛} satisfying (1.6) and assume that 𝜂=sup𝑛0𝜂𝑛<1. Thus, if 𝑛0 and {𝑥𝑛} is well defined, then 𝑥𝐹01𝑟𝑛𝜂𝑛𝐹𝑥01𝐹𝑥𝑛2𝐹𝜂𝑥01𝐹𝑥𝑛2.(3.1) Let 𝐹𝛼=𝑥01𝐹𝑥0,𝛽=1+𝜂𝛼.(3.2) Write 𝜔=1+𝜂𝜂,𝜎=1+𝜂1+𝑅0𝐿(𝑢)𝑑𝑢21𝜂2.(3.3) 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 𝛽𝑏𝜆. For an integer 𝑚1, if 𝜂𝐹𝑥01𝐹𝑥𝑛1𝑥1,𝑛𝑥𝑛1𝑡𝑛𝑡𝑛1(3.4) hold for each 1𝑛𝑚, then the following assertions hold: 1+𝜂𝐹𝑥01𝐹𝑥𝑚𝑡𝜑𝑚;𝜂𝐹𝑥01𝐹𝑥𝑚1.(3.5)

Proof. Assume that (3.4) holds for each 1𝑛𝑚. Write 𝑥𝜏𝑚1=𝑥𝑚1+𝜏(𝑥𝑚𝑥𝑚1),𝜏[0,1]. Applying (1.4), we have 𝐹𝑥𝑚𝑥=𝐹𝑚𝑥𝐹𝑚1𝐹𝑥𝑚1𝑥𝑚𝑥𝑚1+𝑟𝑚1=10𝐹𝑥𝜏𝑚1𝐹𝑥𝑚1𝑥𝑑𝜏𝑚𝑥𝑚1+𝑟𝑚1.(3.6) Hence, 𝐹𝑥01𝐹𝑥𝑚𝑥𝐹0110𝐹𝑥𝜏𝑚1𝐹𝑥𝑚1𝑥𝑑𝜏𝑚𝑥𝑚1+𝑥𝐹01𝑟𝑚1=𝐼1+𝐼2.(3.7) To estimate 𝐼1, by (3.4), we notice that 𝑥𝜏𝑚1𝑥0=𝑥𝑚1𝑥+𝜏𝑚𝑥𝑚1𝑥0𝑚1𝑛=1𝑥𝑛𝑥𝑛1𝑥+𝜏𝑚𝑥𝑚1𝑡𝑚1𝑡+𝜏𝑚𝑡𝑚1=(1𝜏)𝑡𝑚1+𝜏𝑡𝑚<𝑡.(3.8) In particular, 𝑥𝑚1𝑥0𝑡𝑚1<𝑡,𝑥𝑚𝑥0𝑡𝑚<𝑡.(3.9) Thus, by the weak Lipschitz condition (1.7), we obtain 𝐼1𝑥𝑚𝑥𝑚10𝑥𝑚𝑥𝑚1𝐿𝑥𝑢𝑚1𝑥0+𝑢𝑑𝑢.(3.10) Below we estimate 𝐼2. We firstly notice that (3.1) and (3.4) yield 𝐹𝑥01𝐹𝑥𝑚1𝑥𝑚𝑥𝑚1𝑥𝐹01𝐹𝑥𝑚1𝑥𝐹01𝑟𝑚1𝐹𝑥01𝐹𝑥𝑚1𝑥𝜂𝐹01𝐹𝑥𝑚121𝜂𝐹𝑥01𝐹𝑥𝑚1.(3.11) Since 𝐹𝑥01𝐹𝑥𝑚1=𝐼+𝐹𝑥01𝐹𝑥𝑚1𝐹𝑥01+𝜌(𝑥𝑚1)0𝐿(𝑢)𝑑𝑢1+𝑅0𝐿(𝑢)𝑑𝑢,(3.12) we have 𝐹𝑥01𝐹𝑥𝑚1𝐹𝑥01𝐹𝑥𝑚1𝑥𝑚𝑥𝑚11𝜂1+𝑅0𝐿(𝑢)𝑑𝑢1𝜂𝑥𝑚𝑥𝑚1.(3.13) Combining this with (3.1) implies that 𝐼2𝐹𝜂𝑥01𝐹𝑥𝑚12𝜂1+𝑅0𝐿(𝑢)𝑑𝑢21𝜂2𝑥𝑚𝑥𝑚12.(3.14) Consequently, by (3.7), (3.10), (3.14) and Lemma 2.4, we get 1+𝜂𝐹𝑥01𝐹𝑥𝑚1+𝜂𝐼1+𝐼21+𝜂𝑥𝑚𝑥𝑚10𝑥𝑚𝑥𝑚1𝐿𝑥𝑢𝑚1𝑥0+𝜂+𝑢𝑑𝑢1+𝜂1+𝑅0𝐿(𝑢)𝑑𝑢21𝜂2𝑥𝑚𝑥𝑚12=𝜔𝑥𝑚𝑥𝑚10𝑥𝑚𝑥𝑚1𝐿𝑥𝑢𝑚1𝑥0𝑥+𝑢𝑑𝑢+𝜎𝑚𝑥𝑚12=𝜔𝑥𝑚𝑥𝑚12𝑥𝑚𝑥𝑚10𝑥𝑚𝑥𝑚1𝑥𝑢×𝐿𝑚1𝑥0𝑥+𝑢𝑑𝑢+𝜎𝑚𝑥𝑚12𝜔𝑡𝑚𝑡𝑚12𝑡𝑚𝑡𝑚10𝑡𝑚𝑡𝑚1𝐿𝑡𝑢𝑚1×𝑡+𝑢𝑑𝑢+𝜎𝑚𝑡𝑚12=𝜔𝑡𝑚𝑡𝑚10𝑡𝑚𝑡𝑚1𝐿𝑡𝑢𝑚1𝑡+𝑢𝑑𝑢+𝜎2𝑚𝑡2𝑚12𝑡𝑚1𝑡𝑚𝑡𝑚1𝑡=𝜑𝑚𝑡𝜑𝑚1𝜑𝑡𝑚1𝑡𝑚𝑡𝑚1.(3.15) Noting that 𝜑(𝑡)=𝜓(𝑡)+𝜆+2𝜎𝑡 and 𝜑(𝑡𝑚1)𝜓(𝑡𝑚1)(𝑡𝑚𝑡𝑚1)=0, we have 1+𝜂𝐹𝑥01𝐹𝑥𝑚𝑡𝜑𝑚𝑡𝜑𝑚1𝜑𝑡𝑚1𝑡𝑚𝑡𝑚1𝑡=𝜑𝑚𝜆+2𝜎𝑡𝑚1𝑡𝑚𝑡𝑚1𝑡𝜑𝑚.(3.16) Moreover, since 𝜑 is decreasing on [0, 𝑡], one has 1+𝜂𝐹𝑥01𝐹𝑥𝑚𝑡𝜑𝑚𝑡𝜑0=𝛽.(3.17) And therefore 𝜂𝐹𝑥01𝐹𝑥𝑚𝜂1+𝜂𝛽=𝜂𝐹𝑥01𝐹𝑥01.(3.18) That is, (3.5) holds, and the proof is complete.

We now give the main result.

Theorem 3.2. Suppose that 𝛽min{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, 𝑥𝑛𝑥𝑡𝑡𝑛,𝑛=0,1,2,.(3.19)

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: 𝑥1𝑥0𝐹𝑥01𝐹𝑥0+𝑥𝐹01𝑟0𝛼+𝜂𝛼2𝛼+𝜂𝛼=1+𝜂𝛼=𝛽=𝑡1𝑡0.(3.20) Assume that (3.4) holds for all 𝑛𝑚. Then, Lemma 3.1 is applicable to concluding that 1+𝜂𝐹𝑥01𝐹𝑥𝑚𝑡𝜑𝑚;𝜂𝐹𝑥01𝐹𝑥𝑚1.(3.21) Hence, by (3.5), together with the weak Lipschitz condition (1.7) and Lemma 2.3, one has 𝑥𝑚+1𝑥𝑚𝐹𝑥𝑚1𝐹𝑥0𝐹𝑥01𝐹𝑥𝑚+𝑥𝐹01𝑟𝑚11𝜌(𝑥𝑚)0𝐹𝐿(𝑢)𝑑𝑢𝑥01𝐹𝑥𝑚+𝜂𝐹(𝑥𝑚)1𝐹(𝑥𝑚)21+𝜂1𝜔𝜌(𝑥𝑚)0𝐹𝐿(𝑢)𝑑𝑢𝑥01𝐹𝑥𝑚𝜑𝑡𝑚𝜓𝑡𝑚=𝑡𝑚+1𝑡𝑚.(3.22) Therefore, (3.4) holds for 𝑛=𝑚+1 and so for each 𝑛1. Consequently, for 𝑛0 and 𝑘0, 𝑥𝑘+𝑛𝑥𝑛𝑘𝑖=1𝑥𝑖+𝑛𝑥𝑖+𝑛1𝑘𝑖=1𝑡𝑖+𝑛𝑡𝑖+𝑛1=𝑡𝑘+𝑛𝑡𝑛.(3.23) 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 𝑥𝑛𝑥𝑡𝑡𝑛,𝑛=0,1,2,.(3.24) The proof is complete.

In the special case when 𝜂𝑛=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 that 𝛽𝑏𝜆 and 𝐵(𝑥0,𝑡)𝐵(𝑥0,𝑅), where 𝑏𝜆=𝑟𝜆0𝐿(𝑢)𝑢𝑑𝑢 and 𝑟𝜆 satisfying 𝑟𝜆0𝐿(𝑢)𝑑𝑢1𝜆. 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, 𝑥𝑛𝑥𝑡𝑡𝑛,𝑛=0,1,2,,(3.25) where 𝑡 and {𝑡𝑛} are defined in Lemma 2.2 for 𝜂=0.

In more particular, suppose that 𝑅0𝐿(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 that 𝛽𝑏𝜆0, where 𝑏𝜆0=𝑟𝜆00𝐿(𝑢)𝑢𝑑𝑢 and 𝑟𝜆00𝐿(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, 𝑥𝑛𝑥𝑡𝑡𝑛,𝑛=0,1,2,,(3.26) 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 in 𝛾-condition. In particular, our results reduce some of the corresponding results of Newton's method.

4.1. Kantorovich-Type Condition

Throughout this subsection, let 𝐿 be a positive constant. By (2.1), we have 1𝜑(𝑡)=𝛽(1𝜆)𝑡+𝜎+2𝑡𝜔𝐿21,𝑡0,𝜓(𝑡)=𝛽𝑡+2𝜔𝐿𝑡2,𝑡0.(4.1) By (2.5) and (2.6), we get 𝑟𝜆=1𝜆𝜔𝐿+𝜎,𝑏𝜆=(1𝜆)2𝜔𝐿2.(𝜔𝐿+𝜎)(4.2) The convergence criterion becomes 𝐹𝑥01𝐹𝑥0(1𝜆)2𝜔𝐿2.(𝜔𝐿+𝜎)(4.3)

Moreover, suppose that 𝜂=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. Let 𝐿 be a positive constant, 𝛽=𝐹(𝑥0)1𝐹(𝑥0) and 𝛽𝑏𝜆0, where 𝑏𝜆0=1/2𝐿 and 𝑟𝜆0=1/𝐿. Assume that 𝐹 satisfies the condition: 𝐹𝑥01𝐹(𝑥)𝐹𝑥𝐿𝑥𝑥,𝑥,𝑥𝑥𝐵0,,𝑟𝑥𝑥0+𝑥𝑥𝑟,(4.4) where 𝑟=(112𝐿𝛽)/𝐿. Then the sequence {𝑥𝑛} generated by Newton's method (1.2) converges to a solution 𝑥 of (1.1), and satisfies 𝑥𝑛𝑥𝑡𝑡𝑛,𝑛=0,1,2,.(4.5)

4.2. 𝛾-Condition

Throughout this subsection, we assume that 𝛾>0 and 𝐹 has continuous second derivative and satisfies 𝐹𝑥01𝐹(𝑥)2𝛾1𝛾𝑥𝑥03𝑥,𝑥𝐵0,1𝛾.(4.6) Let 𝐿(𝑢)=2𝛾(1𝛾𝑢)31,𝑢0,𝛾.(4.7) Then, by (2.1), we have 𝜑(𝑡)=𝛽(1𝜆)𝑡+𝜎𝑡2+𝛾𝑡211𝛾𝑡,0𝑡<𝛾,𝜓(𝑡)=𝛽𝑡+𝛾𝑡211𝛾𝑡,0𝑡<𝛾.(4.8) By (2.5) and (2.6), 𝑟𝜆 and 𝑏𝜆 satisfy 𝜔11𝛾𝑟𝜆21+𝜎𝑟𝜆=1𝜆,𝑏𝜆=𝛾𝑟2𝜆1𝛾𝑟𝜆2.(4.9) The convergence criterion becomes 𝐹𝑥01𝐹𝑥0𝛾𝑟2𝜆1𝛾𝑟𝜆2.(4.10)

In the more special case, when 𝜂=0 and 𝜆=0, we obtain the criterion 𝐹(𝑥0)1𝐹(𝑥0)(322)/𝛾 the same with Newton's method in [7].

Corollary 4.2. Let 𝛾 be a positive constant, 𝛽=𝐹(𝑥0)1𝐹(𝑥0) and 𝛽𝑏𝜆0, where 𝑏𝜆0=(322)/𝛾 and 𝑟𝜆0=(1(1/2))(1/𝛾). Assume that F satisfies the condition: 𝐹𝑥01𝐹(𝑥)𝐹𝑥11𝛾𝑥𝑥0𝑥𝛾𝑥0211𝛾𝑥𝑥02,𝑥,𝑥𝑥𝐵0,,𝑟𝑥𝑥0+𝑥𝑥𝑟,(4.11) where 𝑟=(1+𝛽𝛾(1+𝛽𝛾)28𝛽𝛾)/4𝛾. Then the sequence {𝑥𝑛} generated by Newton's method (1.2) converges to a solution 𝑥 of (1.1), and satisfies 𝑥𝑛𝑥𝑡𝑡𝑛,𝑛=0,1,2,.(4.12)

Acknowledgment

Supported in part by the National Natural Science Foundation of China (Grants no. 61170109 and no. 10971194) and Zhejiang Innovation Project (Grant no. T200905).

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