Semilocal Convergence Analysis for Inexact Newton Method under Weak Condition

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 𝑥_𝑛+1=𝑥_𝑛−𝐹^𝑥_𝑛⁻¹𝐹𝑥_𝑛,𝑥₀∈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 𝐹^ involved, the other is to solve the exact solution of Newton equations: 𝐹^𝑥_𝑛𝑥_𝑛+1−𝑥_𝑛=−𝐹𝑥_𝑛,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 𝑥_𝑘 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: 𝑥_𝑛+1=𝑥_𝑛+𝑠_𝑛,𝐹^𝑥_𝑛𝑠_𝑛=−𝐹𝑥_𝑛+𝑟_𝑛,𝑛=0,1,2,…,(1.4) where {𝑟_𝑛} is a sequence in 𝑌.

As is well known, the convergence behavior of the inexact Newton method depends on the residual controls of {𝑟_𝑛} 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 ‖𝑟_𝑛‖≤𝜂_𝑛‖𝐹(𝑥_𝑛)‖ were adopted in [10, 12]; in [15] the affine invariant conditions ‖𝐹′(𝑥₀)⁻¹𝑟_𝑛‖≤𝜂_𝑛‖𝐹′(𝑥₀)⁻¹𝐹(𝑥_𝑛)‖ were considered; while in [21] Shen has analyzed the semilocal convergence behavior in some manner such that the relative residuals {𝑟_𝑛} satisfy ‖‖𝐹′𝑥₀⁻¹𝑟_𝑛‖‖≤𝜂_𝑛‖‖𝐹^𝑥₀⁻¹𝐹𝑥_𝑛‖‖^1+𝜅,0≤𝜅≤1,𝑛=0,1,2,….(1.5) Assuming that the residuals satisfy ‖𝑃_𝑛𝑟_𝑛‖≤𝜃_𝑛‖𝑃_𝑛𝐹(𝑥_𝑛)‖^1+𝜅, where {𝑃_𝑛} 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, ‖‖𝐹′𝑥₀⁻¹𝑟_𝑛‖‖≤𝜂_𝑛‖‖𝐹^𝑥₀⁻¹𝐹𝑥_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 𝐵(𝑥₀,𝑟), 𝐹′(𝑥₀)⁻¹𝐹′ exists and 𝐹′(𝑥₀)⁻¹𝐹′ satisfies the weak Lipschitz condition: ‖‖𝐹^𝑥₀⁻¹𝐹^(𝑥)−𝐹^𝑥^‖‖≤^{𝜌(𝑥𝑥′)𝜌(𝑥)}𝐿(𝑢)𝑑𝑢,∀𝑥∈𝐵𝑥₀,𝑟,∀𝑥′∈𝐵(𝑥,𝑟−𝜌(𝑥)),(1.7) 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 𝜂_𝑛=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>0,0≤𝜆<1,𝜔≥1, and 𝜎≥0. Define 𝜑(𝑡)=𝜓𝛽(−𝑡)(=1−𝛽−𝜆)𝑡𝑡++𝜔𝜎𝑡²_𝑡0+𝜔^𝑡0𝐿(𝑢)(𝑡−𝑢)𝑑𝑢,0≤𝑡≤𝑅,𝐿(𝑢)(𝑡−𝑢)𝑑𝑢,0≤𝑡≤𝑅.(2.1) Obviously, 𝜑^(𝑡)=𝜓−(1−𝜆)+2𝜎𝑡+𝜔^𝑡0𝐿(𝑢)𝑑𝑢,0≤𝑡≤𝑅,(2.2)_𝜑(𝑡)=−1+𝜔^𝑡0𝐿(𝑢)𝑑𝑢,0≤𝑡≤𝑅,(2.3)_(𝑡)=2𝜎+𝜔𝐿(𝑡)>0,0≤𝑡≤𝑅.(2.4) Set 𝑟_𝜆∶=𝑏sup𝑟∈(0,𝑅)∶𝜔^𝑟0𝐿(𝑢)𝑑𝑢+2𝜎𝑟≤1−𝜆,(2.5)_𝜆∶=(1−𝜆)𝑟_𝜆−𝜎𝑟^2𝜆−𝜔^𝑟𝜆₀𝐿(𝑢)𝑟_𝜆−𝑢𝑑𝑢.(2.6) Write 𝛿=𝜔∫^𝑅0𝐿(𝑢)𝑑𝑢+2𝜎𝑅. Then 𝑟_𝜆=𝑟𝑅,if𝛿<1−𝜆,_𝜆,if𝛿≥1−𝜆,(2.7) where 𝑟^𝜆∈[0,𝑅] is such that 𝜔∫^𝑟′𝜆₀𝐿(𝑢)𝑑𝑢+2𝜎𝑟^𝜆=1−𝜆. Furthermore, it follows that 𝑏𝑏_𝜆≥𝜔^𝑟𝜆₀𝐿(𝑢)𝑢𝑑𝑢+𝜎𝑟^2𝜆,if𝛿<1−𝜆,_𝜆=𝜔^𝑟𝜆₀𝐿(𝑢)𝑢𝑑𝑢+𝜎𝑟^2𝜆,if𝛿≥1−𝜆.(2.8) Let 𝑡₀=0,𝑡_𝑛+1=𝑡_𝑛−𝜑𝑡^𝑛𝜓^𝑡_𝑛,𝑛=0,1,2,….(2.9)

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

Lemma 2.1. Suppose that 𝛽≤𝑏_𝜆 and 𝜑 is defined by (2.1). Then the function 𝜑 is strictly decreasing and has exact one zero 𝑡^∗ on [0,𝑟_𝜆] satisfying 𝛽<𝑡^∗.

Proof. By (2.4) and (2.5), we know 𝜑′ is strictly increasing on [0,𝑟_𝜆] and has the values 𝜑′(0)<0 and 𝜑′(𝑟_𝜆)≤0. This implies that 𝜑 is strictly decreasing on [0,𝑟_𝜆]. Note that 𝜑(0)=𝛽>0 and 𝜑(𝑟_𝜆)≤0 by the definition of 𝑏_𝜆. Thus, 𝜑(𝑡)=0 has exact one solution 𝑡^∗ on [0,𝑟_𝜆]. Since 𝜑(𝛽)=𝜆𝛽+𝜎𝛽²+𝜔^𝛽0𝐿(𝑢)(𝛽−𝑢)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,𝑟_𝜆]. Suppose that 𝛽≤𝑏_𝜆 and the sequence {𝑡_𝑛} is defined by (2.9). Then 𝑡_𝑛<𝑡_𝑛+1<𝑡^∗,𝑛=0,1,2,….(2.11) Consequently, {𝑡_𝑛} 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 𝑡_𝑛−1<𝑡_𝑛<𝑡^∗.(2.12) Since 𝜓^(𝑡)=𝜔𝐿(𝑡)>0,−𝜓^ is strictly decreasing on [0,𝑟_𝜆]. Hence, −𝜓^𝑡_𝑛>−𝜓^𝑡^∗≥−𝜓^𝑟_𝜆=−𝜑^𝑟_𝜆+𝜆+2𝜎𝑟_𝜆≥0.(2.13) Moreover, 𝜑(𝑡_𝑛)>0 by of Lemma 2.1. It follows that 𝑡_𝑛+1=𝑡_𝑛−𝜑𝑡^𝑛𝜓^𝑡_𝑛>𝑡^𝑛.(2.14) Define a function 𝑁(𝑡) on [0,𝑡^∗] by 𝑁(𝑡)∶=𝑡−𝜑(𝑡)𝜓^(𝑡),𝑡∈0,𝑡^∗.(2.15) Note that 𝜓′(𝑡)<0,𝑡∈[0,𝑡^∗], unless 𝜆=0, 𝜎=0 and 𝑡=𝑡^∗=𝑟_𝜆, for which we adopt the convention that lim_{𝑡→𝑡∗}(𝜑(𝑡)/𝜓^(𝑡))=0 and 𝑁(𝑡^∗)=𝑡^∗−lim_{𝑡→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 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^∗.(2.16) Hence, 𝑁(𝑡) is monotonically increasing on [0,𝑡^∗). This together with (2.9) and (2.14) implies that 𝑡_𝑛<𝑡_𝑛+1=𝑁𝑡_𝑛<𝑁𝑡^∗=𝑡^∗.(2.17) Therefore, by mathematical induction, (2.11) holds. Consequently, {𝑡_𝑛} is increasing, bounded, and converges to a point 𝑡^∗𝜆, which satisfies 𝜑(𝑡^∗𝜆)=0. Hence, 𝑡^∗=𝑡^∗𝜆. 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)𝑑𝑢≤1. Then 𝐹′(𝑥) is invertible in the ball 𝐵(𝑥₀,𝑟) and ‖‖𝐹^(𝑥)⁻¹𝐹^𝑥₀‖‖≤1−^0𝜌(𝑥)𝐿(𝑢)𝑑𝑢⁻¹.(2.18)

Lemma 2.4. Let 0≤𝑐<𝑅 and define 𝜒(𝑡)=1𝑡²^𝑡0𝐿(𝑐+𝑢)(𝑡−𝑢)𝑑𝑢,0≤𝑡<𝑅−𝑐.(2.19) Then, 𝜒 is increasing on [0,𝑅−𝑐).

3. Semilocal Convergence Analysis

Recall that 𝐹∶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 {𝑟_𝑛} satisfying (1.6) and assume that 𝜂=sup_𝑛≥0𝜂_𝑛<1. Thus, if 𝑛≥0 and {𝑥_𝑛} is well defined, then ‖‖𝐹′𝑥₀⁻¹𝑟_𝑛‖‖≤𝜂_𝑛‖‖𝐹^𝑥₀⁻¹𝐹𝑥_𝑛‖‖²≤𝜂‖‖𝐹^𝑥₀⁻¹𝐹𝑥_𝑛‖‖².(3.1) Let 𝛼=‖‖𝐹^𝑥₀⁻¹𝐹𝑥₀‖‖,𝛽=1+√𝜂𝛼.(3.2) Write 𝜔=1+√𝜂,𝜎=𝜂1+√𝜂1+∫^𝑅0𝐿(𝑢)𝑑𝑢²1−√𝜂².(3.3) Recall that 𝑟_𝜆 is determined by (2.5), 𝜑(𝑡^∗)=0, and {𝑡_𝑛} is generated by (2.9) with 𝜔 and 𝜎 given in (3.3).

Lemma 3.1. Let {𝑥_𝑛} 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≤𝑏_𝜆. For an integer 𝑚≥1, if √𝜂‖‖𝐹^𝑥₀⁻¹𝐹𝑥_𝑛−1‖‖≤1,‖‖𝑥_𝑛−𝑥_𝑛−1‖‖≤𝑡_𝑛−𝑡_𝑛−1(3.4) hold for each 1≤𝑛≤𝑚, then the following assertions hold: 1+√√𝜂‖‖𝐹^𝑥₀⁻¹𝐹𝑥_𝑚‖‖≤𝜑𝑡_𝑚;𝜂‖‖𝐹^𝑥₀⁻¹𝐹𝑥_𝑚‖‖≤1.(3.5)

Proof. Assume that (3.4) holds for each 1≤𝑛≤𝑚. Write 𝑥^𝜏𝑚−1=𝑥_𝑚−1+𝜏(𝑥_𝑚−𝑥_𝑚−1),𝜏∈[0,1]. Applying (1.4), we have 𝐹𝑥_𝑚=𝐹𝑥_𝑚−𝐹𝑥_𝑚−1−𝐹^𝑥_𝑚−1𝑥_𝑚−𝑥_𝑚−1+𝑟_𝑚−1=¹⁰𝐹^𝑥^𝜏𝑚−1−𝐹^𝑥_𝑚−1𝑑𝜏𝑥_𝑚−𝑥_𝑚−1+𝑟_𝑚−1.(3.6) Hence, ‖‖𝐹^𝑥₀⁻¹𝐹𝑥_𝑚‖‖≤‖‖‖+𝐹‖‖′𝐹𝑥′₀𝑥⁻¹₀¹⁰𝐹^𝑥^𝜏𝑚−1−𝐹^𝑥_𝑚−1𝑑𝜏𝑥_𝑚−𝑥_𝑚−1‖‖‖⁻¹𝑟_𝑚−1=𝐼₁‖‖+𝐼₂.(3.7) To estimate 𝐼₁, by (3.4), we notice that ‖‖𝑥^𝜏𝑚−1−𝑥₀‖‖=≤‖‖𝑥_{𝑚−1𝑚−1}+𝜏𝑥_𝑚−𝑥_𝑚−1−𝑥₀‖‖_𝑛=1≤𝑡_𝑚−1‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+𝜏‖‖𝑥_𝑚−𝑥_𝑚−1‖‖=(1−+𝜏)𝜏𝑡𝑡_𝑚𝑚−1−𝑡_𝑚−1+𝜏𝑡_𝑚<𝑡^∗.(3.8) In particular, ‖‖𝑥_𝑚−1−𝑥₀‖‖≤𝑡_𝑚−1<𝑡^∗,‖‖𝑥_𝑚−𝑥₀‖‖≤𝑡_𝑚<𝑡^∗.(3.9) Thus, by the weak Lipschitz condition (1.7), we obtain 𝐼₁≤^{‖𝑥𝑚−𝑥𝑚−1‖0}‖‖𝑥_𝑚−𝑥_𝑚−1‖‖−𝑢𝐿‖‖𝑥_𝑚−1−𝑥₀‖‖+𝑢𝑑𝑢.(3.10) Below we estimate 𝐼₂. We firstly notice that (3.1) and (3.4) yield ‖‖𝐹^𝑥₀⁻¹𝐹^𝑥_𝑚−1𝑥_𝑚−𝑥_𝑚−1‖‖≥‖‖𝐹′𝑥₀⁻¹𝐹𝑥_𝑚−1‖‖−‖‖𝐹′𝑥₀⁻¹𝑟_𝑚−1≥‖‖𝐹_‖‖𝑥₀⁻¹𝐹𝑥_𝑚−1‖‖−𝜂‖‖𝐹′𝑥₀⁻¹𝐹𝑥_𝑚−1‖‖²≥1−√𝜂‖‖𝐹^𝑥₀⁻¹𝐹𝑥_𝑚−1‖‖.(3.11) Since ‖‖𝐹^𝑥₀⁻¹𝐹^𝑥_𝑚−1‖‖≤=‖‖1𝐼++_{𝜌(𝑥𝑚−1)0}𝐹^𝑥₀⁻¹𝐹^𝑥_𝑚−1−𝐹^𝑥₀‖‖≤1+_𝑅0𝐿(𝑢)𝑑𝑢𝐿(𝑢)𝑑𝑢,(3.12) we have ‖‖𝐹^𝑥₀⁻¹𝐹𝑥_𝑚−1‖‖≤‖‖𝐹^𝑥₀⁻¹𝐹^𝑥_𝑚−1‖‖‖‖𝑥_𝑚−𝑥_𝑚−1‖‖1−√𝜂≤1+∫^𝑅0𝐿(𝑢)𝑑𝑢1−√𝜂‖‖𝑥^𝑚−𝑥_𝑚−1‖‖.(3.13) Combining this with (3.1) implies that 𝐼₂≤𝜂‖‖𝐹^𝑥₀⁻¹𝐹𝑥_𝑚−1‖‖²≤𝜂1+∫^𝑅0𝐿(𝑢)𝑑𝑢²1−√𝜂²‖‖𝑥_𝑚−𝑥_𝑚−1‖‖².(3.14) Consequently, by (3.7), (3.10), (3.14) and Lemma 2.4, we get 1+√𝜂‖‖𝐹^𝑥₀⁻¹𝐹𝑥_𝑚‖‖≤≤11++√√𝜂𝐼₁+𝐼₂+𝜂1+𝜂√^{‖𝑥𝑚−𝑥𝑚−1‖0}‖‖𝑥_𝑚−𝑥_𝑚−1‖‖−𝑢𝐿‖‖𝑥_𝑚−1−𝑥₀‖‖+𝑢𝑑𝑢𝜂1+∫^𝑅0𝐿(𝑢)𝑑𝑢²1−√𝜂²‖‖𝑥_𝑚−𝑥_𝑚−1‖‖²=+𝜔𝜎‖‖^{‖𝑥𝑚−𝑥𝑚−1‖0}𝑥_𝑚‖‖𝑥_𝑚−𝑥_𝑚−1‖‖−𝑢𝐿‖‖𝑥_𝑚−1−𝑥₀‖‖+𝑢𝑑𝑢−𝑥_𝑚−1‖‖²=𝜔‖‖𝑥_𝑚−𝑥_𝑚−1‖‖²^{‖𝑥𝑚−𝑥𝑚−1‖0}×𝐿‖‖𝑥_𝑚−1‖‖𝑥_𝑚−𝑥_𝑚−1‖‖−𝑢−𝑥₀‖‖+𝑢𝑑𝑢+𝜎‖‖𝑥_𝑚−𝑥_𝑚−1‖‖²≤𝜔𝑡_𝑚−𝑡_𝑚−1²^{𝑡𝑚−𝑡𝑚−1}₀𝑡_𝑚×𝑡−𝑡_𝑚−1−𝑢𝐿𝑡_𝑚−1+𝑢𝑑𝑢+𝜎_𝑚−𝑡_𝑚−1²=+𝜔𝜎^{𝑡𝑚−𝑡𝑚−1}₀𝑡_2𝑚𝑡_𝑚−𝑡_𝑚−1−𝑢𝐿𝑡_𝑚−1+𝑢𝑑𝑢=𝜑𝑡_𝑚−𝑡^2𝑚−1−2𝑡_𝑚−1𝑡_𝑚−𝑡_𝑚−1−𝜑𝑡_𝑚−1−𝜑^𝑡_𝑚−1𝑡_𝑚−𝑡_𝑚−1.(3.15) Noting that 𝜑′(𝑡)=𝜓′(𝑡)+𝜆+2𝜎𝑡 and −𝜑(𝑡_𝑚−1)−𝜓′(𝑡_𝑚−1)(𝑡_𝑚−𝑡_𝑚−1)=0, we have 1+√𝜂‖‖𝐹^𝑥₀⁻¹𝐹𝑥_𝑚‖‖=≤𝜑𝜑𝑡𝑡_𝑚𝑚−𝜑𝑡_𝑚−1−𝜑^𝑡_𝑚−1𝑡_𝑚−𝑡_𝑚−1≤𝜑𝑡_𝑚−𝜆+2𝜎𝑡_𝑚−1𝑡_𝑚−𝑡_𝑚−1.(3.16) Moreover, since 𝜑 is decreasing on [0, 𝑡^∗], one has 1+√𝜂‖‖𝐹^𝑥₀⁻¹𝐹𝑥_𝑚‖‖≤𝜑𝑡_𝑚≤𝜑𝑡₀=𝛽.(3.17) And therefore √𝜂‖‖𝐹^𝑥₀⁻¹𝐹𝑥_𝑚‖‖≤√𝜂1+√𝜂𝛽=√𝜂‖‖𝐹^𝑥₀⁻¹𝐹𝑥₀‖‖≤1.(3.18) That is, (3.5) holds, and the proof is complete.

We now give the main result.

Theorem 3.2. Suppose that 𝛽≤min{1/√𝜂,𝑏_𝜆} 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 {𝑥_𝑛} generated by the inexact Newton method (1.4) converges to a solution 𝑥^∗ of (1.1). Moreover, ‖‖𝑥_𝑛−𝑥^∗‖‖≤𝑡^∗−𝑡_𝑛,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: ‖‖𝑥₁−𝑥₀‖‖≤≤𝛼‖‖𝐹+^𝜂𝛼𝑥₀₂⁻¹𝐹𝑥₀‖‖+‖‖𝐹′𝑥₀⁻¹𝑟₀‖‖≤𝛼+√𝜂𝛼=1+√𝜂𝛼=𝛽=𝑡₁−𝑡₀.(3.20) Assume that (3.4) holds for all 𝑛≤𝑚. Then, Lemma 3.1 is applicable to concluding that 1+√√𝜂‖‖𝐹^𝑥₀⁻¹𝐹𝑥_𝑚‖‖≤𝜑𝑡_𝑚;𝜂‖‖𝐹^𝑥₀⁻¹𝐹𝑥_𝑚‖‖≤1.(3.21) Hence, by (3.5), together with the weak Lipschitz condition (1.7) and Lemma 2.3, one has ‖‖𝑥_𝑚+1−𝑥_𝑚‖‖≤≤‖‖𝐹^𝑥_𝑚1⁻¹𝐹^𝑥₀‖‖‖‖𝐹^𝑥₀⁻¹𝐹𝑥_𝑚‖‖+‖‖𝐹′𝑥₀⁻¹𝑟_𝑚‖‖1−∫^{𝜌(𝑥𝑚)0}𝐿(𝑢)𝑑𝑢‖‖𝐹^≤1+√𝑥₀⁻¹𝐹𝑥_𝑚‖‖+𝜂‖‖𝐹′(𝑥_𝑚)⁻¹𝐹(𝑥_𝑚)‖‖²𝜂1−𝜔∫^{𝜌(𝑥𝑚)0}𝐿(𝑢)𝑑𝑢‖‖𝐹^≤−𝜑𝑡_𝑚𝑥₀⁻¹𝐹𝑥_𝑚‖‖𝜓^𝑡_𝑚=𝑡^𝑚+1−𝑡_𝑚.(3.22) Therefore, (3.4) holds for 𝑛=𝑚+1 and so for each 𝑛≥1. Consequently, for 𝑛≥0 and 𝑘≥0, ‖‖𝑥_𝑘+𝑛−𝑥_𝑛‖‖≤^𝑘_𝑖=1‖‖𝑥_𝑖+𝑛−𝑥_{𝑖+𝑛−1}‖‖≤^𝑘_𝑖=1𝑡_𝑖+𝑛−𝑡_{𝑖+𝑛−1}=𝑡_𝑘+𝑛−𝑡_𝑛.(3.23) This together with Lemma 2.2 means that {𝑥_𝑛} is a Cauchy sequence and so converges to some 𝑥^∗. While taking 𝑘→∞ in (3.23), we obtain ‖‖𝑥_𝑛−𝑥^∗‖‖≤𝑡^∗−𝑡_𝑛,𝑛=0,1,2,….(3.24) The proof is complete.

In the special case when 𝜂_𝑛=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 𝛽≤𝑏_𝜆 and 𝐵(𝑥₀,𝑡^∗)⊆𝐵(𝑥₀,𝑅), where 𝑏_𝜆=∫^𝑟𝜆₀𝐿(𝑢)𝑢𝑑𝑢 and 𝑟_𝜆 satisfying ∫^𝑟𝜆₀𝐿(𝑢)𝑑𝑢≤1−𝜆. Suppose that 𝐹′(𝑥₀)⁻¹𝐹′ satisfies the weak Lipschitz condition (1.7) on 𝐵(𝑥₀,𝑡^∗). Then the sequence {𝑥_𝑛} generated by Newton's method (1.2) converges to a solution 𝑥^∗ of (1.1). Moreover, ‖‖𝑥_𝑛−𝑥^∗‖‖≤𝑡^∗−𝑡_𝑛,𝑛=0,1,2,…,(3.25) where 𝑡^∗ and {𝑡_𝑛} 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 𝛽≤𝑏_𝜆0, where 𝑏_𝜆0=∫^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 {𝑥_𝑛} generated by Newton's method (1.2) converges to a solution 𝑥^∗ of (1.1). Moreover, ‖‖𝑥_𝑛−𝑥^∗‖‖≤𝑡^∗−𝑡_𝑛,𝑛=0,1,2,…,(3.26) where 𝑡^∗ and {𝑡_𝑛} 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 𝜑(𝑡)=𝛽−(1−𝜆)𝑡+𝜎+12𝜔𝐿𝑡²𝜓(𝑡)=𝛽−𝑡+1,𝑡≥0,2𝜔𝐿𝑡²,𝑡≥0.(4.1) By (2.5) and (2.6), we get 𝑟_𝜆=1−𝜆𝜔𝐿+𝜎,𝑏^𝜆=(1−𝜆)²𝜔𝐿2(𝜔𝐿+𝜎).(4.2) The convergence criterion becomes ‖‖𝐹^𝑥₀⁻¹𝐹𝑥₀‖‖≤(1−𝜆)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 𝛽≤𝑏_𝜆0, where 𝑏_𝜆0=1/2𝐿 and 𝑟_𝜆0=1/𝐿. Assume that 𝐹 satisfies the condition: ‖‖𝐹^𝑥₀⁻¹𝐹^(𝑥)−𝐹^𝑥^‖‖≤𝐿‖‖𝑥−𝑥^‖‖,∀𝑥,𝑥^∈𝐵𝑥₀,𝑟,‖‖𝑥−𝑥₀‖‖+‖𝑥−𝑥′‖≤𝑟,(4.4) where 𝑟=(1−√1−2𝐿𝛽)/𝐿. Then the sequence {𝑥_𝑛} generated by Newton's method (1.2) converges to a solution 𝑥^∗ of (1.1), and satisfies ‖‖𝑥_𝑛−𝑥^∗‖‖≤𝑡^∗−𝑡_𝑛,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 ‖‖𝐹^𝑥₀⁻¹𝐹^(𝑥)‖‖≤2𝛾1−𝛾‖‖𝑥−𝑥₀‖‖³,∀𝑥∈𝐵𝑥₀,1𝛾.(4.6) Let 𝐿(𝑢)=2𝛾(1−𝛾𝑢)³,𝑢∈0,1𝛾.(4.7) Then, by (2.1), we have 𝜑(𝑡)=𝛽−(1−𝜆)𝑡+𝜎𝑡²+𝛾𝑡²𝜓(𝑡)=𝛽−𝑡+𝛾𝑡₂1−𝛾𝑡,0≤𝑡<𝛾1,1−𝛾𝑡,0≤𝑡<𝛾1.(4.8) By (2.5) and (2.6), 𝑟_𝜆 and 𝑏_𝜆 satisfy 𝜔11−𝛾𝑟_𝜆²−1+𝜎𝑟_𝜆=1−𝜆,𝑏_𝜆=𝛾𝑟^2𝜆1−𝛾𝑟_𝜆².(4.9) The convergence criterion becomes ‖‖𝐹^𝑥₀⁻¹𝐹𝑥₀‖‖≤𝛾𝑟^2𝜆1−𝛾𝑟_𝜆².(4.10)

In the more special case, when 𝜂=0 and 𝜆=0, we obtain the criterion ‖𝐹′(𝑥₀)⁻¹𝐹(𝑥₀)‖≤(3−2√2)/𝛾 the same with Newton's method in [7].

Corollary 4.2. Let 𝛾 be a positive constant, 𝛽=‖𝐹′(𝑥₀)⁻¹𝐹(𝑥₀)‖ and 𝛽≤𝑏_𝜆0, where 𝑏_𝜆0=(3−2√2)/𝛾 and 𝑟_𝜆0=(1−(1/√2))(1/𝛾). Assume that F satisfies the condition: ‖‖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,(4.11) where 𝑟=(1+𝛽𝛾−√(1+𝛽𝛾)²−8𝛽𝛾)/4𝛾. Then the sequence {𝑥_𝑛} generated by Newton's method (1.2) converges to a solution 𝑥^∗ of (1.1), and satisfies ‖‖𝑥_𝑛−𝑥^∗‖‖≤𝑡^∗−𝑡_𝑛,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).