Semilocal convergence for Halley’s method under weak Lipschitz condition

Abstract

The semilocal convergence properties of Halley’s method for nonlinear operator equations are studied under the hypothesis that the second derivative satisfies some weak Lipschitz condition. The method employed in the present paper is based on a family of recurrence relations which will be satisfied by the involved operator. An application to a nonlinear Hammerstein integral equation of the second kind is provided.

Introduction

Let X and Y be Euclidean spaces or more generally Banach spaces, and D be a nonempty open convex subset of X. We assume that the nonlinear operator $F:D\subset X\to Y$ is continuously twice Fréchet differentiable. Consider a nonlinear operator equation denoted by

$$F(x)=0\text{.}$$

Finding solutions of such an operator equation is a very general subject which is widely used in both theoretical and applied areas of mathematics.

Newton’s method with initial point $x_0$ is defined by

$$x_{k+1}=x_k-F^{\prime }(x_k{)}^{-1}F(x_k)\text{,}k=0\text{,}1\text{,}2\text{,}\dots \text{,}$$

which is the most efficient method known for solving such an equation. One of the most important results on Newton’s method is the well-known Newton–Kantorovich theorem [8]. Recent progress in the Newton method is referred to [10], [11], [13], [14]. The classical Newton–Kantorovich theorem assumes that the derivative of the operator involved satisfies Lipschitz condition:

$$\Vert F^{\prime }(x)-F^{\prime }(y)\Vert ⩽L\Vert x-y\Vert \text{,}x\text{,}y\in D\text{,}$$

and then provides a simple and clear convergence criterion for Newton’s iteration based on the data around the initial point. Real majorizing sequences are usually used to prove the convergence of Newton’s method. Furthermore, Kantorovich-type results provide the error estimates and regions where the solutions are located. Instead of assuming that the derivative $F^{\prime }$ satisfies a Lipschitz-type condition, Gutiérrez and Hernández [7] investigated the following condition:

$$\Vert F^{\prime }(x_0{)}^{-1}[F^{\prime }(x)-F^{\prime }(x_0)]\Vert ⩽L\Vert x-x_0\Vert \text{,}\forall x\in D\text{,}$$

and obtained the semilocal convergence.

Another well-known property of Newton’s method is that the iterates $\{x_k\}$ are invariant under any affine transformation

$$F\to G≔\mathit{AF}\text{,}$$

where A denotes a nonsingular bounded linear operator. Some important affine invariant convergence results applying on discretized nonlinear operator equations are obtained by Deuflhard et al. [2], [4], [12].

The Halley method (refer to [9], [15] for the details) denoted by

$$x_{k+1}=x_k-[I-L_F(x_k){]}^{-1}{F}^{\prime }(x_k{)}^{-1}F(x_k)\text{,}k=0\text{,}1\text{,}2\text{,}\dots \text{,}$$

where operator $L_F(x)=\frac{1}{2}{F}^{\prime }(x{)}^{-1}{F}^{″}(x)F^{\prime }(x{)}^{-1}F(x)$, is another famous iteration for solving nonlinear equation (1). The results concerning convergence of this method with its modification have recently been studied under assumptions of Newton–Kantorovich-type, see for example [1], [16], [5], [6]. In particular, Candela [1] investigated the Halley iteration under the condition

$$\Vert F^{″}(x)-F^{″}(y)\Vert ⩽L\Vert x-y\Vert$$

based on the use of recurrence relations instead of the classical majorizing sequences, and gave the semilocal convergence.

Similar to Newton’s method, we can show that the iterates $\{x_k\}$ generated by (5) are also invariant under the affine transformation (4). In fact, consider

$$x_{k+1}=x_k-[I-L_G(x_k){]}^{-1}{G}^{\prime }(x_k{)}^{-1}G(x_k)\text{,}k=0\text{,}1\text{,}2\text{,}\dots \text{,}$$

where $G=\mathit{AF}$ and $L_G(x)=\frac{1}{2}{G}^{\prime }(x{)}^{-1}{G}^{″}(x)G^{\prime }(x{)}^{-1}G(x)$. Since

$$G^{\prime }(x_k{)}^{-1}G(x_k)=[{\mathit{AF}}^{\prime }(x_k){]}^{-1}[\mathit{AF}(x_k)]=F^{\prime }(x_k{)}^{-1}F(x_k)\text{,}$$

and

$$L_G(x)=\frac{1}{2}[{\mathit{AF}}^{\prime }(x){]}^{-1}[{\mathit{AF}}^{″}(x)][{\mathit{AF}}^{\prime }(x){]}^{-1}[\mathit{AF}(x)]=\frac{1}{2}{F}^{\prime }(x{)}^{-1}{F}^{″}(x)F^{\prime }(x{)}^{-1}F(x)=L_F(x)\text{,}$$

we have

$$[I-L_F(x_k){]}^{-1}{F}^{\prime }(x_k{)}^{-1}F(x_k)=[I-L_G(x_k){]}^{-1}{G}^{\prime }(x_k{)}^{-1}G(x_k)\text{.}$$

Therefore, any transformation of type (4) won’t affect the convergence or divergence of the Halley iteration sequence $\{x_k\}$ generated by (5).

Motivated by the idea of Gutiérrez and Hernández in [7], together with the affine invariant theory developed by Deuflhard et al. (see [3] for the details), in the rest of this paper, we assume that the operator $F^{″}$ satisfies

$$\Vert F^{\prime }(x_0{)}^{-1}[F^{″}(x)-F^{″}(x_0)]\Vert ⩽L\Vert x-x_0\Vert \text{,}\forall x\in D\text{.}$$

We notice that (6) obviously implies (7). However, the reverse is false, that is, there are functions satisfying (7) but not (6). For instance, let $f(x)=x^{\frac{5}{2}}$ for $x\in D=[0\text{,}\infty )$. It is obvious that $f^{″}$ does not satisfy the Lipschitz condition (6). But, if we take $x_0>0$, we obtain

$$|f^{″}(x)-f^{″}(x_0)|=\frac{15}{4}|\sqrt{x}-\sqrt{x_0}|=\frac{15}{4}\frac{|x-x_0|}{\sqrt{x}+\sqrt{x_0}}⩽\frac{15}{4\sqrt{x_0}}|x-x_0|\text{,}\forall x\in D\text{,}$$

that is, $f^{″}$ satisfies (7) for any given $x_0>0$.

In this paper, we focus on the semilocal convergence of Halley method (5). Under the assumption that the second derivative of F satisfies the preceding weak Lipschhitz condition (7) on the ball $B(x_0\text{,}R)\subseteq D$ for some proper $R>0$, we establish a semilocal convergence for Halley method (5), which is based on the values of the operator and its second derivative at the initial point as well as the Lipschitz constant. In addition, we give the error estimate and the domain where the solution is located and unique. We end this section by briefly describing the organization of this paper. The recurrence relations are developed in Section 2. The main results about the semilocal convergence and error estimate are stated and proved in Section 3. And finally in Section 4, an application to a nonlinear Hammerstein integral equation is given.