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Abstract and Applied Analysis
VolumeΒ 2012Β (2012), Article IDΒ 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-ChihΒ Yao

Copyright Β© 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 FrΓ©chet differentiable nonlinear operator from a convex subset 𝐷 of Banach space 𝑋 to Banach space π‘Œ. Finding solutions of a nonlinear operator equation: 𝐹(π‘₯)=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 π‘₯𝑛+1=π‘₯π‘›βˆ’πΉξ…žξ€·π‘₯π‘›ξ€Έβˆ’1𝐹π‘₯𝑛,π‘₯0∈𝐷,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 [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 evaluate πΉξ…ž involved, the other is to solve the exact solution of Newton equations: πΉξ…žξ€·π‘₯𝑛π‘₯𝑛+1βˆ’π‘₯𝑛π‘₯=βˆ’πΉπ‘›ξ€Έ,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 π‘₯π‘˜ 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: π‘₯𝑛+1=π‘₯𝑛+𝑠𝑛,πΉξ…žξ€·π‘₯𝑛𝑠𝑛π‘₯=βˆ’πΉπ‘›ξ€Έ+π‘Ÿπ‘›,𝑛=0,1,2,…,(1.4) where {π‘Ÿπ‘›} is a sequence in π‘Œ.

As is well known, the convergence behavior of the inexact Newton method depends on the residual controls of {π‘Ÿπ‘›} under the hypothesis that 𝐹′ satisfies different conditions. Some relative results can be found in [10–24], for example.

Under the Lipschitz continuity assumption on 𝐹′, different residual controls were used. For example, the residual controls β€–π‘Ÿπ‘›β€–β‰€πœ‚π‘›β€–πΉ(π‘₯𝑛)β€– were adopted in [10, 12]; in [15] the affine invariant conditions ‖𝐹′(π‘₯0)βˆ’1π‘Ÿπ‘›β€–β‰€πœ‚π‘›β€–πΉβ€²(π‘₯0)βˆ’1𝐹(π‘₯𝑛)β€– were considered; while in [21] Shen has analyzed the semilocal convergence behavior in some manner such that the relative residuals {π‘Ÿπ‘›} satisfy β€–β€–ξ€·π‘₯𝐹′0ξ€Έβˆ’1π‘Ÿπ‘›β€–β€–β‰€πœ‚π‘›β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1𝐹π‘₯𝑛‖‖1+πœ…,0β‰€πœ…β‰€1,𝑛=0,1,2,….(1.5) Assuming that the residuals satisfy β€–π‘ƒπ‘›π‘Ÿπ‘›β€–β‰€πœƒπ‘›β€–π‘ƒπ‘›πΉ(π‘₯𝑛)β€–1+πœ…, where {𝑃𝑛} is a sequence of invertible operators from π‘Œ to 𝑋, and that 𝐹′(π‘₯0)βˆ’1𝐹′ satisfies the HΓΆlder condition around π‘₯0, 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 πœ…=1, that is, β€–β€–ξ€·π‘₯𝐹′0ξ€Έβˆ’1π‘Ÿπ‘›β€–β€–β‰€πœ‚π‘›β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1𝐹π‘₯𝑛‖‖2,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 𝐡(π‘₯0,π‘Ÿ), 𝐹′(π‘₯0)βˆ’1𝐹′ exists and 𝐹′(π‘₯0)βˆ’1𝐹′ satisfies the weak Lipschitz condition: β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1ξ€·πΉξ…ž(π‘₯)βˆ’πΉξ…žξ€·π‘₯ξ…žβ€–β€–β‰€ξ€œξ€Έξ€ΈπœŒ(π‘₯π‘₯β€²)𝜌(π‘₯)ξ€·π‘₯𝐿(𝑒)𝑑𝑒,βˆ€π‘₯∈𝐡0ξ€Έ,π‘Ÿ,βˆ€π‘₯β€²βˆˆπ΅(π‘₯,π‘Ÿβˆ’πœŒ(π‘₯)),(1.7) where π‘Ÿ is a positive number, 𝜌(π‘₯)=β€–π‘₯βˆ’π‘₯0β€–,𝜌(π‘₯π‘₯ξ…ž)=𝜌(π‘₯)+β€–π‘₯ξ…žβˆ’π‘₯β€–β‰€π‘Ÿ, and 𝐿 is a positive integrable nondecreasing function on [0,π‘Ÿ]. 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 πœ‚π‘›=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 π‘Œ be Banach spaces. Throughout this paper, 𝑅>π‘Ÿ are two positive numbers, 𝐿 is a positive integrable nondecreasing function on any involved intervals, and 𝐡(π‘₯,𝑅) is an open ball in 𝑋 with center π‘₯ and radius 𝑅. Let 𝛽>0,0β‰€πœ†<1,πœ”β‰₯1, and 𝜎β‰₯0. Define πœ‘(𝑑)=π›½βˆ’(1βˆ’πœ†)𝑑+πœŽπ‘‘2ξ€œ+πœ”π‘‘0ξ€œπΏ(𝑒)(π‘‘βˆ’π‘’)𝑑𝑒,0≀𝑑≀𝑅,πœ“(𝑑)=π›½βˆ’π‘‘+πœ”π‘‘0𝐿(𝑒)(π‘‘βˆ’π‘’)𝑑𝑒,0≀𝑑≀𝑅.(2.1) Obviously, πœ‘ξ…ž(ξ€œπ‘‘)=βˆ’(1βˆ’πœ†)+2πœŽπ‘‘+πœ”π‘‘0πœ“πΏ(𝑒)𝑑𝑒,0≀𝑑≀𝑅,(2.2)ξ…žξ€œ(𝑑)=βˆ’1+πœ”π‘‘0πœ‘πΏ(𝑒)𝑑𝑒,0≀𝑑≀𝑅,(2.3)ξ…žξ…ž(𝑑)=2𝜎+πœ”πΏ(𝑑)>0,0≀𝑑≀𝑅.(2.4) Set π‘Ÿπœ†ξ‚»ξ€œβˆΆ=supπ‘Ÿβˆˆ(0,𝑅)βˆΆπœ”π‘Ÿ0𝑏𝐿(𝑒)𝑑𝑒+2πœŽπ‘Ÿβ‰€1βˆ’πœ†,(2.5)πœ†βˆΆ=(1βˆ’πœ†)π‘Ÿπœ†βˆ’πœŽπ‘Ÿ2πœ†ξ€œβˆ’πœ”π‘Ÿπœ†0ξ€·π‘ŸπΏ(𝑒)πœ†ξ€Έβˆ’π‘’π‘‘π‘’.(2.6) Write βˆ«π›Ώ=πœ”π‘…0𝐿(𝑒)𝑑𝑒+2πœŽπ‘…. Then π‘Ÿπœ†=ξ‚»π‘Ÿπ‘…,if𝛿<1βˆ’πœ†,ξ…žπœ†,if𝛿β‰₯1βˆ’πœ†,(2.7) where π‘Ÿξ…žπœ†βˆˆ[0,𝑅] is such that πœ”βˆ«π‘Ÿβ€²πœ†0𝐿(𝑒)𝑑𝑒+2πœŽπ‘Ÿξ…žπœ†=1βˆ’πœ†. Furthermore, it follows that π‘πœ†ξ€œβ‰₯πœ”π‘Ÿπœ†0𝐿(𝑒)𝑒𝑑𝑒+πœŽπ‘Ÿ2πœ†π‘,if𝛿<1βˆ’πœ†,πœ†ξ€œ=πœ”π‘Ÿπœ†0𝐿(𝑒)𝑒𝑑𝑒+πœŽπ‘Ÿ2πœ†,if𝛿β‰₯1βˆ’πœ†.(2.8) Let 𝑑0=0,𝑑𝑛+1=π‘‘π‘›βˆ’πœ‘ξ€·π‘‘π‘›ξ€Έπœ“ξ…žξ€·π‘‘π‘›ξ€Έ,𝑛=0,1,2,….(2.9)

The following two lemmas describe some properties about the majorizing function πœ‘ and the convergence property of {𝑑𝑛}.

Lemma 2.1. Suppose that π›½β‰€π‘πœ† and πœ‘ is defined by (2.1). Then the function πœ‘ is strictly decreasing and has exact one zero π‘‘βˆ— on [0,π‘Ÿπœ†] satisfying 𝛽<π‘‘βˆ—.

Proof. By (2.4) and (2.5), we know πœ‘β€² is strictly increasing on [0,π‘Ÿπœ†] and has the values πœ‘β€²(0)<0 and πœ‘β€²(π‘Ÿπœ†)≀0. This implies that πœ‘ is strictly decreasing on [0,π‘Ÿπœ†]. Note that πœ‘(0)=𝛽>0 and πœ‘(π‘Ÿπœ†)≀0 by the definition of π‘πœ†. Thus, πœ‘(𝑑)=0 has exact one solution π‘‘βˆ— on [0,π‘Ÿπœ†]. Since πœ‘(𝛽)=πœ†π›½+πœŽπ›½2ξ€œ+πœ”π›½0𝐿(𝑒)(π›½βˆ’π‘’)𝑑𝑒>0,(2.10) we have 𝛽<π‘‘βˆ—. The proof is complete.

Lemma 2.2. Let π‘‘βˆ— be the positive solution of equation πœ‘(𝑑)=0 on [0,π‘Ÿπœ†]. Suppose that π›½β‰€π‘πœ† and the sequence {𝑑𝑛} is defined by (2.9). Then 𝑑𝑛<𝑑𝑛+1<π‘‘βˆ—,𝑛=0,1,2,….(2.11) Consequently, {𝑑𝑛} is strictly increasing and converges to π‘‘βˆ—.

Proof. We prove the lemma by mathematical induction. Note that 0=𝑑0<𝑑1=𝛽<π‘‘βˆ—. For 𝑛>1, assume that π‘‘π‘›βˆ’1<𝑑𝑛<π‘‘βˆ—.(2.12) Since πœ“ξ…žξ…ž(𝑑)=πœ”πΏ(𝑑)>0,βˆ’πœ“ξ…ž is strictly decreasing on [0,π‘Ÿπœ†]. Hence, βˆ’πœ“ξ…žξ€·π‘‘π‘›ξ€Έ>βˆ’πœ“ξ…žξ€·π‘‘βˆ—ξ€Έβ‰₯βˆ’πœ“ξ…žξ€·π‘Ÿπœ†ξ€Έ=βˆ’πœ‘ξ…žξ€·π‘Ÿπœ†ξ€Έ+πœ†+2πœŽπ‘Ÿπœ†β‰₯0.(2.13) Moreover, πœ‘(𝑑𝑛)>0 by of Lemma 2.1. It follows that 𝑑𝑛+1=π‘‘π‘›βˆ’πœ‘ξ€·π‘‘π‘›ξ€Έπœ“ξ…žξ€·π‘‘π‘›ξ€Έ>𝑑𝑛.(2.14) Define a function 𝑁(𝑑) on [0,π‘‘βˆ—] by 𝑁(𝑑)∢=π‘‘βˆ’πœ‘(𝑑)πœ“ξ…žξ€Ί(𝑑),π‘‘βˆˆ0,π‘‘βˆ—ξ€».(2.15) Note that πœ“β€²(𝑑)<0,π‘‘βˆˆ[0,π‘‘βˆ—], unless πœ†=0, 𝜎=0 and 𝑑=π‘‘βˆ—=π‘Ÿπœ†, for which we adopt the convention that limπ‘‘β†’π‘‘βˆ—(πœ‘(𝑑)/πœ“ξ…ž(𝑑))=0 and 𝑁(π‘‘βˆ—)=π‘‘βˆ—βˆ’limπ‘‘β†’π‘‘βˆ—(πœ‘(𝑑)/πœ“β€²(𝑑))=π‘‘βˆ—. Hence, the function 𝑁(𝑑) is well defined and continuous on [0,π‘‘βˆ—].
Moreover, by (2.2) and (2.3), we have π‘ξ…žπœ‘(𝑑)=1βˆ’ξ…ž(𝑑)πœ“ξ…ž(𝑑)βˆ’πœ‘(𝑑)πœ“ξ…žξ…ž(𝑑)(πœ“ξ…ž(𝑑))2=βˆ’πœ“ξ…ž(𝑑)(πœ†+2πœŽπ‘‘)+πœ‘(𝑑)πœ“ξ…žξ…ž(𝑑)(πœ“ξ…ž(𝑑))2ξ€Ί>0,π‘‘βˆˆ0,π‘‘βˆ—ξ€Έ.(2.16) Hence, 𝑁(𝑑) is monotonically increasing on [0,π‘‘βˆ—). This together with (2.9) and (2.14) implies that 𝑑𝑛<𝑑𝑛+1𝑑=𝑁𝑛𝑑<π‘βˆ—ξ€Έ=π‘‘βˆ—.(2.17) Therefore, by mathematical induction, (2.11) holds. Consequently, {𝑑𝑛} is increasing, bounded, and converges to a point π‘‘βˆ—πœ†, which satisfies πœ‘(π‘‘βˆ—πœ†)=0. Hence, π‘‘βˆ—=π‘‘βˆ—πœ†. 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 π‘Ÿ satisfy βˆ«π‘Ÿ0𝐿(𝑒)𝑑𝑒≀1. Then 𝐹′(π‘₯) is invertible in the ball 𝐡(π‘₯0,π‘Ÿ) and β€–β€–πΉξ…ž(π‘₯)βˆ’1πΉξ…žξ€·π‘₯0ξ€Έβ€–β€–β‰€ξ‚΅ξ€œ1βˆ’0𝜌(π‘₯)𝐿(𝑒)π‘‘π‘’βˆ’1.(2.18)

Lemma 2.4. Let 0≀𝑐<𝑅 and define 1πœ’(𝑑)=𝑑2ξ€œπ‘‘0𝐿(𝑐+𝑒)(π‘‘βˆ’π‘’)𝑑𝑒,0≀𝑑<π‘…βˆ’π‘.(2.19) Then, πœ’ is increasing on [0,π‘…βˆ’π‘).

3. Semilocal Convergence Analysis

Recall that πΉβˆΆπ·βŠ†π‘‹β†’π‘Œ is a nonlinear operator with continuous FrΓ©chet derivative. Let 𝐡(π‘₯0,𝑅)βŠ†π· and π‘₯0∈𝐷 be such that 𝐹′(π‘₯0)βˆ’1 exists. In the present paper, we adopt the residuals {π‘Ÿπ‘›} satisfying (1.6) and assume that πœ‚=sup𝑛β‰₯0πœ‚π‘›<1. Thus, if 𝑛β‰₯0 and {π‘₯𝑛} is well defined, then β€–β€–ξ€·π‘₯𝐹′0ξ€Έβˆ’1π‘Ÿπ‘›β€–β€–β‰€πœ‚π‘›β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1𝐹π‘₯𝑛‖‖2β€–β€–πΉβ‰€πœ‚ξ…žξ€·π‘₯0ξ€Έβˆ’1𝐹π‘₯𝑛‖‖2.(3.1) Let ‖‖𝐹𝛼=ξ…žξ€·π‘₯0ξ€Έβˆ’1𝐹π‘₯0ξ€Έβ€–β€–ξ‚€βˆš,𝛽=1+πœ‚ξ‚π›Ό.(3.2) Write βˆšπœ”=1+πœ‚ξ€·βˆšπœ‚,𝜎=1+πœ‚ξ€Έξ‚€βˆ«1+𝑅0𝐿(𝑒)𝑑𝑒2ξ€·βˆš1βˆ’πœ‚ξ€Έ2.(3.3) Recall that π‘Ÿπœ† is determined by (2.5), πœ‘(π‘‘βˆ—)=0, and {𝑑𝑛} is generated by (2.9) with πœ” and 𝜎 given in (3.3).

Lemma 3.1. Let {π‘₯𝑛} be a sequence generated by (1.4). Suppose that F satisfies the weak Lipschitz condition (1.7) on 𝐡(π‘₯0,π‘‘βˆ—)βŠ†π΅(π‘₯0,𝑅) and that π›½β‰€π‘πœ†. For an integer π‘šβ‰₯1, if βˆšπœ‚β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1𝐹π‘₯π‘›βˆ’1ξ€Έβ€–β€–β€–β€–π‘₯≀1,π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–β‰€π‘‘π‘›βˆ’π‘‘π‘›βˆ’1(3.4) hold for each 1β‰€π‘›β‰€π‘š, then the following assertions hold: ξ‚€βˆš1+πœ‚ξ‚β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1𝐹π‘₯π‘šξ€Έβ€–β€–ξ€·π‘‘β‰€πœ‘π‘šξ€Έ;βˆšπœ‚β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1𝐹π‘₯π‘šξ€Έβ€–β€–β‰€1.(3.5)

Proof. Assume that (3.4) holds for each 1β‰€π‘›β‰€π‘š. Write π‘₯πœπ‘šβˆ’1=π‘₯π‘šβˆ’1+𝜏(π‘₯π‘šβˆ’π‘₯π‘šβˆ’1),𝜏∈[0,1]. Applying (1.4), we have 𝐹π‘₯π‘šξ€Έξ€·π‘₯=πΉπ‘šξ€Έξ€·π‘₯βˆ’πΉπ‘šβˆ’1ξ€Έβˆ’πΉξ…žξ€·π‘₯π‘šβˆ’1π‘₯ξ€Έξ€·π‘šβˆ’π‘₯π‘šβˆ’1ξ€Έ+π‘Ÿπ‘šβˆ’1=ξ€œ10ξ€ΊπΉξ…žξ€·π‘₯πœπ‘šβˆ’1ξ€Έβˆ’πΉξ…žξ€·π‘₯π‘šβˆ’1ξ€·π‘₯ξ€Έξ€»π‘‘πœπ‘šβˆ’π‘₯π‘šβˆ’1ξ€Έ+π‘Ÿπ‘šβˆ’1.(3.6) Hence, β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1𝐹π‘₯π‘šξ€Έβ€–β€–β‰€β€–β€–β€–ξ€·π‘₯𝐹′0ξ€Έβˆ’1ξ€œ10ξ€ΊπΉξ…žξ€·π‘₯πœπ‘šβˆ’1ξ€Έβˆ’πΉξ…žξ€·π‘₯π‘šβˆ’1ξ€·π‘₯ξ€Έξ€»π‘‘πœπ‘šβˆ’π‘₯π‘šβˆ’1ξ€Έβ€–β€–β€–+β€–β€–ξ€·π‘₯𝐹′0ξ€Έβˆ’1π‘Ÿπ‘šβˆ’1β€–β€–=𝐼1+𝐼2.(3.7) To estimate 𝐼1, by (3.4), we notice that β€–β€–π‘₯πœπ‘šβˆ’1βˆ’π‘₯0β€–β€–=β€–β€–π‘₯π‘šβˆ’1ξ€·π‘₯+πœπ‘šβˆ’π‘₯π‘šβˆ’1ξ€Έβˆ’π‘₯0β€–β€–β‰€π‘šβˆ’1𝑛=1β€–β€–π‘₯π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–β€–β€–π‘₯+πœπ‘šβˆ’π‘₯π‘šβˆ’1β€–β€–β‰€π‘‘π‘šβˆ’1𝑑+πœπ‘šβˆ’π‘‘π‘šβˆ’1ξ€Έ=(1βˆ’πœ)π‘‘π‘šβˆ’1+πœπ‘‘π‘š<π‘‘βˆ—.(3.8) In particular, β€–β€–π‘₯π‘šβˆ’1βˆ’π‘₯0β€–β€–β‰€π‘‘π‘šβˆ’1<π‘‘βˆ—,β€–β€–π‘₯π‘šβˆ’π‘₯0β€–β€–β‰€π‘‘π‘š<π‘‘βˆ—.(3.9) Thus, by the weak Lipschitz condition (1.7), we obtain 𝐼1β‰€ξ€œβ€–π‘₯π‘šβˆ’π‘₯π‘šβˆ’1β€–0ξ€·β€–β€–π‘₯π‘šβˆ’π‘₯π‘šβˆ’1‖‖𝐿‖‖π‘₯βˆ’π‘’π‘šβˆ’1βˆ’π‘₯0β€–β€–ξ€Έ+𝑒𝑑𝑒.(3.10) Below we estimate 𝐼2. We firstly notice that (3.1) and (3.4) yield β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1πΉξ…žξ€·π‘₯π‘šβˆ’1π‘₯ξ€Έξ€·π‘šβˆ’π‘₯π‘šβˆ’1ξ€Έβ€–β€–β‰₯β€–β€–ξ€·π‘₯𝐹′0ξ€Έβˆ’1𝐹π‘₯π‘šβˆ’1ξ€Έβ€–β€–βˆ’β€–β€–ξ€·π‘₯𝐹′0ξ€Έβˆ’1π‘Ÿπ‘šβˆ’1β€–β€–β‰₯β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1𝐹π‘₯π‘šβˆ’1ξ€Έβ€–β€–β€–β€–ξ€·π‘₯βˆ’πœ‚πΉβ€²0ξ€Έβˆ’1𝐹π‘₯π‘šβˆ’1ξ€Έβ€–β€–2β‰₯ξ‚€βˆš1βˆ’πœ‚ξ‚β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1𝐹π‘₯π‘šβˆ’1ξ€Έβ€–β€–.(3.11) Since β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1πΉξ…žξ€·π‘₯π‘šβˆ’1ξ€Έβ€–β€–=‖‖𝐼+πΉξ…žξ€·π‘₯0ξ€Έβˆ’1ξ€ΊπΉξ…žξ€·π‘₯π‘šβˆ’1ξ€Έβˆ’πΉξ…žξ€·π‘₯0β€–β€–ξ€œξ€Έξ€»β‰€1+𝜌(π‘₯π‘šβˆ’1)0πΏξ€œ(𝑒)𝑑𝑒≀1+𝑅0𝐿(𝑒)𝑑𝑒,(3.12) we have β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1𝐹π‘₯π‘šβˆ’1ξ€Έβ€–β€–β‰€β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1πΉξ…žξ€·π‘₯π‘šβˆ’1ξ€Έβ€–β€–β€–β€–π‘₯π‘šβˆ’π‘₯π‘šβˆ’1β€–β€–βˆš1βˆ’πœ‚β‰€βˆ«1+𝑅0𝐿(𝑒)π‘‘π‘’βˆš1βˆ’πœ‚β€–β€–π‘₯π‘šβˆ’π‘₯π‘šβˆ’1β€–β€–.(3.13) Combining this with (3.1) implies that 𝐼2β€–β€–πΉβ‰€πœ‚ξ…žξ€·π‘₯0ξ€Έβˆ’1𝐹π‘₯π‘šβˆ’1ξ€Έβ€–β€–2β‰€πœ‚ξ‚€βˆ«1+𝑅0𝐿(𝑒)𝑑𝑒2ξ€·βˆš1βˆ’πœ‚ξ€Έ2β€–β€–π‘₯π‘šβˆ’π‘₯π‘šβˆ’1β€–β€–2.(3.14) Consequently, by (3.7), (3.10), (3.14) and Lemma 2.4, we get ξ‚€βˆš1+πœ‚ξ‚β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1𝐹π‘₯π‘šξ€Έβ€–β€–β‰€ξ‚€βˆš1+πœ‚ξ‚ξ€·πΌ1+𝐼2ξ€Έβ‰€ξ‚€βˆš1+πœ‚ξ‚ξ€œβ€–π‘₯π‘šβˆ’π‘₯π‘šβˆ’1β€–0ξ€·β€–β€–π‘₯π‘šβˆ’π‘₯π‘šβˆ’1‖‖𝐿‖‖π‘₯βˆ’π‘’π‘šβˆ’1βˆ’π‘₯0β€–β€–ξ€Έ+πœ‚ξ€·βˆš+𝑒𝑑𝑒1+πœ‚ξ€Έξ‚€βˆ«1+𝑅0𝐿(𝑒)𝑑𝑒2ξ€·βˆš1βˆ’πœ‚ξ€Έ2β€–β€–π‘₯π‘šβˆ’π‘₯π‘šβˆ’1β€–β€–2ξ€œ=πœ”β€–π‘₯π‘šβˆ’π‘₯π‘šβˆ’1β€–0ξ€·β€–β€–π‘₯π‘šβˆ’π‘₯π‘šβˆ’1‖‖𝐿‖‖π‘₯βˆ’π‘’π‘šβˆ’1βˆ’π‘₯0β€–β€–ξ€Έβ€–β€–π‘₯+𝑒𝑑𝑒+πœŽπ‘šβˆ’π‘₯π‘šβˆ’1β€–β€–2=ξƒ©πœ”β€–β€–π‘₯π‘šβˆ’π‘₯π‘šβˆ’1β€–β€–2ξ€œβ€–π‘₯π‘šβˆ’π‘₯π‘šβˆ’1β€–0ξ€·β€–β€–π‘₯π‘šβˆ’π‘₯π‘šβˆ’1β€–β€–ξ€Έξ€·β€–β€–π‘₯βˆ’π‘’Γ—πΏπ‘šβˆ’1βˆ’π‘₯0β€–β€–ξ€Έξƒͺβ€–β€–π‘₯+𝑒𝑑𝑒+πœŽπ‘šβˆ’π‘₯π‘šβˆ’1β€–β€–2β‰€ξƒ©πœ”ξ€·π‘‘π‘šβˆ’π‘‘π‘šβˆ’1ξ€Έ2ξ€œπ‘‘π‘šβˆ’π‘‘π‘šβˆ’10ξ€·π‘‘π‘šβˆ’π‘‘π‘šβˆ’1ξ€ΈπΏξ€·π‘‘βˆ’π‘’π‘šβˆ’1ξ€Έξƒͺ×𝑑+𝑒𝑑𝑒+πœŽπ‘šβˆ’π‘‘π‘šβˆ’1ξ€Έ2ξ€œ=πœ”π‘‘π‘šβˆ’π‘‘π‘šβˆ’10ξ€·π‘‘π‘šβˆ’π‘‘π‘šβˆ’1ξ€ΈπΏξ€·π‘‘βˆ’π‘’π‘šβˆ’1𝑑+𝑒𝑑𝑒+𝜎2π‘šβˆ’π‘‘2π‘šβˆ’1βˆ’2π‘‘π‘šβˆ’1ξ€·π‘‘π‘šβˆ’π‘‘π‘šβˆ’1𝑑=πœ‘π‘šξ€Έξ€·π‘‘βˆ’πœ‘π‘šβˆ’1ξ€Έβˆ’πœ‘ξ…žξ€·π‘‘π‘šβˆ’1π‘‘ξ€Έξ€·π‘šβˆ’π‘‘π‘šβˆ’1ξ€Έ.(3.15) Noting that πœ‘β€²(𝑑)=πœ“β€²(𝑑)+πœ†+2πœŽπ‘‘ and βˆ’πœ‘(π‘‘π‘šβˆ’1)βˆ’πœ“β€²(π‘‘π‘šβˆ’1)(π‘‘π‘šβˆ’π‘‘π‘šβˆ’1)=0, we have ξ‚€βˆš1+πœ‚ξ‚β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1𝐹π‘₯π‘šξ€Έβ€–β€–ξ€·π‘‘β‰€πœ‘π‘šξ€Έξ€·π‘‘βˆ’πœ‘π‘šβˆ’1ξ€Έβˆ’πœ‘ξ…žξ€·π‘‘π‘šβˆ’1π‘‘ξ€Έξ€·π‘šβˆ’π‘‘π‘šβˆ’1𝑑=πœ‘π‘šξ€Έβˆ’ξ€·πœ†+2πœŽπ‘‘π‘šβˆ’1π‘‘ξ€Έξ€·π‘šβˆ’π‘‘π‘šβˆ’1ξ€Έξ€·π‘‘β‰€πœ‘π‘šξ€Έ.(3.16) Moreover, since πœ‘ is decreasing on [0, π‘‘βˆ—], one has ξ‚€βˆš1+πœ‚ξ‚β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1𝐹π‘₯π‘šξ€Έβ€–β€–ξ€·π‘‘β‰€πœ‘π‘šξ€Έξ€·π‘‘β‰€πœ‘0ξ€Έ=𝛽.(3.17) And therefore βˆšπœ‚β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1𝐹π‘₯π‘šξ€Έβ€–β€–β‰€βˆšπœ‚βˆš1+πœ‚βˆšπ›½=πœ‚β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1𝐹π‘₯0‖‖≀1.(3.18) That is, (3.5) holds, and the proof is complete.

We now give the main result.

Theorem 3.2. Suppose that βˆšπ›½β‰€min{1/πœ‚,π‘πœ†} and 𝐡(π‘₯0,π‘‘βˆ—)βŠ†π΅(π‘₯0,𝑅), and that 𝐹′(π‘₯0)βˆ’1𝐹′ satisfies the weak Lipschitz condition (1.7) on 𝐡(π‘₯0,π‘‘βˆ—). Then the sequence {π‘₯𝑛} generated by the inexact Newton method (1.4) converges to a solution π‘₯βˆ— of (1.1). Moreover, β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–β‰€π‘‘βˆ—βˆ’π‘‘π‘›,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: β€–β€–π‘₯1βˆ’π‘₯0β€–β€–β‰€β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1𝐹π‘₯0ξ€Έβ€–β€–+β€–β€–ξ€·π‘₯𝐹′0ξ€Έβˆ’1π‘Ÿ0‖‖≀𝛼+πœ‚π›Ό2βˆšβ‰€π›Ό+ξ‚€βˆšπœ‚π›Ό=1+πœ‚ξ‚π›Ό=𝛽=𝑑1βˆ’π‘‘0.(3.20) Assume that (3.4) holds for all π‘›β‰€π‘š. Then, Lemma 3.1 is applicable to concluding that ξ‚€βˆš1+πœ‚ξ‚β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1𝐹π‘₯π‘šξ€Έβ€–β€–ξ€·π‘‘β‰€πœ‘π‘šξ€Έ;βˆšπœ‚β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1𝐹π‘₯π‘šξ€Έβ€–β€–β‰€1.(3.21) Hence, by (3.5), together with the weak Lipschitz condition (1.7) and Lemma 2.3, one has β€–β€–π‘₯π‘š+1βˆ’π‘₯π‘šβ€–β€–β‰€β€–β€–πΉξ…žξ€·π‘₯π‘šξ€Έβˆ’1πΉξ…žξ€·π‘₯0ξ€Έβ€–β€–ξ‚€β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1𝐹π‘₯π‘šξ€Έβ€–β€–+β€–β€–ξ€·π‘₯𝐹′0ξ€Έβˆ’1π‘Ÿπ‘šβ€–β€–ξ‚β‰€1∫1βˆ’πœŒ(π‘₯π‘š)0‖‖𝐹𝐿(𝑒)π‘‘π‘’ξ…žξ€·π‘₯0ξ€Έβˆ’1𝐹π‘₯π‘šξ€Έβ€–β€–β€–β€–+πœ‚πΉβ€²(π‘₯π‘š)βˆ’1𝐹(π‘₯π‘š)β€–β€–2ξ‚β‰€βˆš1+πœ‚βˆ«1βˆ’πœ”πœŒ(π‘₯π‘š)0‖‖𝐹𝐿(𝑒)π‘‘π‘’ξ…žξ€·π‘₯0ξ€Έβˆ’1𝐹π‘₯π‘šξ€Έβ€–β€–πœ‘ξ€·π‘‘β‰€βˆ’π‘šξ€Έπœ“ξ…žξ€·π‘‘π‘šξ€Έ=π‘‘π‘š+1βˆ’π‘‘π‘š.(3.22) Therefore, (3.4) holds for 𝑛=π‘š+1 and so for each 𝑛β‰₯1. Consequently, for 𝑛β‰₯0 and π‘˜β‰₯0, β€–β€–π‘₯π‘˜+π‘›βˆ’π‘₯π‘›β€–β€–β‰€π‘˜ξ“π‘–=1β€–β€–π‘₯𝑖+π‘›βˆ’π‘₯𝑖+π‘›βˆ’1β€–β€–β‰€π‘˜ξ“π‘–=1𝑑𝑖+π‘›βˆ’π‘‘π‘–+π‘›βˆ’1ξ€Έ=π‘‘π‘˜+π‘›βˆ’π‘‘π‘›.(3.23) This together with Lemma 2.2 means that {π‘₯𝑛} is a Cauchy sequence and so converges to some π‘₯βˆ—. While taking π‘˜β†’βˆž in (3.23), we obtain β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–β‰€π‘‘βˆ—βˆ’π‘‘π‘›,𝑛=0,1,2,….(3.24) The proof is complete.

In the special case when πœ‚π‘›=0(𝑛=0,1,2,…), inexact Newton method (1.4) reduces to Newton's method. Moreover, πœ”=1,𝜎=0,𝛽=‖𝐹′(π‘₯0)βˆ’1𝐹(π‘₯0)β€–. Thus, Theorem 3.2 reduces to the related theorem of Newton's method.

Corollary 3.3. Assume that π›½β‰€π‘πœ† and 𝐡(π‘₯0,π‘‘βˆ—)βŠ†π΅(π‘₯0,𝑅), where π‘πœ†=βˆ«π‘Ÿπœ†0𝐿(𝑒)𝑒𝑑𝑒 and π‘Ÿπœ† satisfying βˆ«π‘Ÿπœ†0𝐿(𝑒)𝑑𝑒≀1βˆ’πœ†. Suppose that 𝐹′(π‘₯0)βˆ’1𝐹′ satisfies the weak Lipschitz condition (1.7) on 𝐡(π‘₯0,π‘‘βˆ—). Then the sequence {π‘₯𝑛} generated by Newton's method (1.2) converges to a solution π‘₯βˆ— of (1.1). Moreover, β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–β‰€π‘‘βˆ—βˆ’π‘‘π‘›,𝑛=0,1,2,…,(3.25) where π‘‘βˆ— and {𝑑𝑛} are defined in Lemma 2.2 for πœ‚=0.

In more particular, suppose that βˆ«π‘…0𝐿(𝑒)𝑑𝑒>1 and πœ†=0. Then Corollary 3.3 reduces to the following result given in (Theorem 3.1, [7]).

Corollary 3.4. Assume that π›½β‰€π‘πœ†0, where π‘πœ†0=βˆ«π‘Ÿπœ†00𝐿(𝑒)𝑒𝑑𝑒 and βˆ«π‘Ÿπœ†00𝐿(𝑒)𝑑𝑒=1. Suppose that 𝐹′(π‘₯0)βˆ’1𝐹′ satisfies weak Lipschitz condition (1.7) on 𝐡(π‘₯0,π‘‘βˆ—)βŠ†π΅(π‘₯0,𝑅). Then the sequence {π‘₯𝑛} generated by Newton's method (1.2) converges to a solution π‘₯βˆ— of (1.1). Moreover, β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–β‰€π‘‘βˆ—βˆ’π‘‘π‘›,𝑛=0,1,2,…,(3.26) where π‘‘βˆ— and {𝑑𝑛} are defined in Lemma 2.2 for πœ‚=0 and πœ†=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 ξ‚€1πœ‘(𝑑)=π›½βˆ’(1βˆ’πœ†)𝑑+𝜎+2ξ‚π‘‘πœ”πΏ21,𝑑β‰₯0,πœ“(𝑑)=π›½βˆ’π‘‘+2πœ”πΏπ‘‘2,𝑑β‰₯0.(4.1) By (2.5) and (2.6), we get π‘Ÿπœ†=1βˆ’πœ†πœ”πΏ+𝜎,π‘πœ†=(1βˆ’πœ†)2πœ”πΏ2.(πœ”πΏ+𝜎)(4.2) The convergence criterion becomes β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1𝐹π‘₯0‖‖≀(1βˆ’πœ†)2πœ”πΏ2.(πœ”πΏ+𝜎)(4.3)

Moreover, suppose that πœ‚=0 and πœ†=0. Then criterion (4.3) reduces to the well-known Kantorovich type criterion ‖𝐹′(π‘₯0)βˆ’1𝐹(π‘₯0)‖≀1/2𝐿 of Newton's method in [7].

Corollary 4.1. Let 𝐿 be a positive constant, 𝛽=‖𝐹′(π‘₯0)βˆ’1𝐹(π‘₯0)β€– and π›½β‰€π‘πœ†0, where π‘πœ†0=1/2𝐿 and π‘Ÿπœ†0=1/𝐿. Assume that 𝐹 satisfies the condition: β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1ξ€·πΉξ…ž(π‘₯)βˆ’πΉξ…žξ€·π‘₯ξ…žβ€–β€–β€–β€–ξ€Έξ€Έβ‰€πΏπ‘₯βˆ’π‘₯ξ…žβ€–β€–,βˆ€π‘₯,π‘₯ξ…žξ€·π‘₯∈𝐡0ξ€Έ,β€–β€–,π‘Ÿπ‘₯βˆ’π‘₯0β€–β€–+β€–π‘₯βˆ’π‘₯β€²β€–β‰€π‘Ÿ,(4.4) where βˆšπ‘Ÿ=(1βˆ’1βˆ’2𝐿𝛽)/𝐿. Then the sequence {π‘₯𝑛} generated by Newton's method (1.2) converges to a solution π‘₯βˆ— of (1.1), and satisfies β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–β‰€π‘‘βˆ—βˆ’π‘‘π‘›,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 β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1πΉξ…žξ…žβ€–β€–β‰€(π‘₯)2𝛾‖‖1βˆ’π›Ύπ‘₯βˆ’π‘₯0β€–β€–ξ€Έ3ξ‚΅π‘₯,βˆ€π‘₯∈𝐡0,1𝛾.(4.6) Let 𝐿(𝑒)=2𝛾(1βˆ’π›Ύπ‘’)3ξ‚Έ1,π‘’βˆˆ0,𝛾.(4.7) Then, by (2.1), we have πœ‘(𝑑)=π›½βˆ’(1βˆ’πœ†)𝑑+πœŽπ‘‘2+𝛾𝑑211βˆ’π›Ύπ‘‘,0≀𝑑<𝛾,πœ“(𝑑)=π›½βˆ’π‘‘+𝛾𝑑211βˆ’π›Ύπ‘‘,0≀𝑑<𝛾.(4.8) By (2.5) and (2.6), π‘Ÿπœ† and π‘πœ† satisfy πœ”ξƒ¬1ξ€·1βˆ’π›Ύπ‘Ÿπœ†ξ€Έ2ξƒ­βˆ’1+πœŽπ‘Ÿπœ†=1βˆ’πœ†,π‘πœ†=π›Ύπ‘Ÿ2πœ†ξ€·1βˆ’π›Ύπ‘Ÿπœ†ξ€Έ2.(4.9) The convergence criterion becomes β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1𝐹π‘₯0ξ€Έβ€–β€–β‰€π›Ύπ‘Ÿ2πœ†ξ€·1βˆ’π›Ύπ‘Ÿπœ†ξ€Έ2.(4.10)

In the more special case, when πœ‚=0 and πœ†=0, we obtain the criterion ‖𝐹′(π‘₯0)βˆ’1𝐹(π‘₯0√)‖≀(3βˆ’22)/𝛾 the same with Newton's method in [7].

Corollary 4.2. Let 𝛾 be a positive constant, 𝛽=‖𝐹′(π‘₯0)βˆ’1𝐹(π‘₯0)β€– and π›½β‰€π‘πœ†0, where π‘πœ†0√=(3βˆ’22)/𝛾 and π‘Ÿπœ†0√=(1βˆ’(1/2))(1/𝛾). Assume that F satisfies the condition: β€–β€–πΉξ…žξ€·π‘₯0ξ€Έβˆ’1ξ€·πΉξ…ž(π‘₯)βˆ’πΉξ…žξ€·π‘₯ξ…žβ€–β€–β‰€1ξ€Έξ€Έξ€·β€–β€–1βˆ’π›Ύπ‘₯βˆ’π‘₯0β€–β€–β€–β€–π‘₯βˆ’π›Ύξ…žβˆ’π‘₯0β€–β€–ξ€Έ2βˆ’1ξ€·1βˆ’π›Ύβ€–π‘₯βˆ’π‘₯0β€–ξ€Έ2,βˆ€π‘₯,π‘₯ξ…žξ€·π‘₯∈𝐡0ξ€Έ,β€–β€–,π‘Ÿπ‘₯βˆ’π‘₯0β€–β€–+β€–π‘₯β€²βˆ’π‘₯β€–β‰€π‘Ÿ,(4.11) where βˆšπ‘Ÿ=(1+π›½π›Ύβˆ’(1+𝛽𝛾)2βˆ’8𝛽𝛾)/4𝛾. Then the sequence {π‘₯𝑛} generated by Newton's method (1.2) converges to a solution π‘₯βˆ— of (1.1), and satisfies β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–β‰€π‘‘βˆ—βˆ’π‘‘π‘›,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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