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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 982925, 13 pages
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.
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 -Conditions are provided and some well-known convergence theorems for Newton's method are obtained as corollaries.
Let be a continuously Fréchet differentiable nonlinear operator from a convex subset of Banach space to Banach space . Finding solutions of a nonlinear operator equation: 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
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 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 involved, the other is to solve the exact solution of Newton equations: 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 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  and takes the following form: where is a sequence in .
As is well known, the convergence behavior of the inexact Newton method depends on the residual controls of under the hypothesis that satisfies different conditions. Some relative results can be found in [10–24], for example.
Under the Lipschitz continuity assumption on , different residual controls were used. For example, the residual controls were adopted in [10, 12]; in  the affine invariant conditions were considered; while in  Shen has analyzed the semilocal convergence behavior in some manner such that the relative residuals satisfy Assuming that the residuals satisfy , where is a sequence of invertible operators from to , and that satisfies the Hölder condition around , Li and Shen established the local and semilocal convergence in [16, 20], respectively. Besides, the -condition was also introduced into inexact Newton method in  by considering residual controls (1.5) with , that is, Smale's -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 has a continuous derivative in a closed ball , exists and satisfies the weak Lipschitz condition: where is a positive number, , and is a positive integrable nondecreasing function on . 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 , (1.4) reduces to Newton's method and our result extends the corresponding one in .
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.
Let and be Banach spaces. Throughout this paper, are two positive numbers, is a positive integrable nondecreasing function on any involved intervals, and is an open ball in with center and radius . Let , and . Define Obviously, Set Write . Then where is such that . Furthermore, it follows that Let
The following two lemmas describe some properties about the majorizing function and the convergence property of .
Lemma 2.1. Suppose that and is defined by (2.1). Then the function is strictly decreasing and has exact one zero on satisfying .
Proof. By (2.4) and (2.5), we know is strictly increasing on and has the values and . This implies that is strictly decreasing on . Note that and by the definition of . Thus, has exact one solution on . Since we have . The proof is complete.
Lemma 2.2. Let be the positive solution of equation on . Suppose that and the sequence is defined by (2.9). Then Consequently, is strictly increasing and converges to .
Proof. We prove the lemma by mathematical induction. Note that . For , assume that
Since is strictly decreasing on . Hence,
Moreover, by of Lemma 2.1. It follows that
Define a function on by
Note that , unless , and , for which we adopt the convention that and . Hence, the function is well defined and continuous on .
Moreover, by (2.2) and (2.3), we have Hence, is monotonically increasing on . This together with (2.9) and (2.14) implies that Therefore, by mathematical induction, (2.11) holds. Consequently, is increasing, bounded, and converges to a point , which satisfies . Hence, . The proof is complete.
Lemma 2.3. Suppose that has a continuous derivative satisfying the weak Lipschitz condition (1.7). Let satisfy . Then is invertible in the ball and
Lemma 2.4. Let and define Then, is increasing on .
3. Semilocal Convergence Analysis
Recall that is a nonlinear operator with continuous Fréchet derivative. Let and be such that exists. In the present paper, we adopt the residuals satisfying (1.6) and assume that . Thus, if and is well defined, then Let Write Recall that is determined by (2.5), , and is generated by (2.9) with and given in (3.3).
Proof. Assume that (3.4) holds for each . Write . Applying (1.4), we have Hence, To estimate , by (3.4), we notice that In particular, Thus, by the weak Lipschitz condition (1.7), we obtain Below we estimate . We firstly notice that (3.1) and (3.4) yield Since we have Combining this with (3.1) implies that Consequently, by (3.7), (3.10), (3.14) and Lemma 2.4, we get Noting that and , we have Moreover, since is decreasing on [0, ], one has And therefore That is, (3.5) holds, and the proof is complete.
We now give the main result.
Proof. We firstly use mathematical induction to prove that (3.4) holds for each . For , by the above condition and (3.2), the first inequality in (3.4) holds trivially. While the second one can be proved as follows: Assume that (3.4) holds for all . Then, Lemma 3.1 is applicable to concluding that Hence, by (3.5), together with the weak Lipschitz condition (1.7) and Lemma 2.3, one has Therefore, (3.4) holds for and so for each . Consequently, for and , This together with Lemma 2.2 means that is a Cauchy sequence and so converges to some . While taking in (3.23), we obtain The proof is complete.
Corollary 3.3. Assume that and , where and satisfying . Suppose that satisfies the weak Lipschitz condition (1.7) on . Then the sequence generated by Newton's method (1.2) converges to a solution of (1.1). Moreover, where and are defined in Lemma 2.2 for .
Corollary 3.4. Assume that , where and . Suppose that satisfies weak Lipschitz condition (1.7) on . Then the sequence generated by Newton's method (1.2) converges to a solution of (1.1). Moreover, where and are defined in Lemma 2.2 for and .
This section is divided into two subsections: we consider the applications of our main results specializing, respectively, in Kantorovich type condition and in -condition. In particular, our results reduce some of the corresponding results of Newton's method.
4.1. Kantorovich-Type Condition
Corollary 4.1. Let be a positive constant, and , where and . Assume that satisfies the condition: where . Then the sequence generated by Newton's method (1.2) converges to a solution of (1.1), and satisfies
In the more special case, when and , we obtain the criterion the same with Newton's method in .
Corollary 4.2. Let be a positive constant, and , where and . Assume that F satisfies the condition: where . Then the sequence generated by Newton's method (1.2) converges to a solution of (1.1), and satisfies
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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