Convergence Theorem for a Family of New Modified Halley’s Method in Banach Space

Abstract

We establish convergence theorems of Newton-Kantorovich type for a family of new modified Halley’s method in Banach space to solve nonlinear operator equations. We present the corresponding error estimate. To show the application of our theorems, two numerical examples are given.

1. Introduction

Solving the nonlinear operator equation is an important issue in the engineering and technology field. Finding solutions of the nonlinear operator equation in Banach spaces also is a very general subject which is widely used in both theoretical and applied areas of mathematics for many years. Many problems may be formulated in terms of finding zeros. These roots cannot be expressed in closed form generally. Thus, in order to solve nonlinear equations, we have to use approximate methods (see [1]). In this study, we consider establishing the Newton-Kantorovich convergence theorems for a family of new modified Halley’s method with third-order Banach space by using majorizing function which is used to solve the nonlinear operator equation. Consider 𝐹(𝑥)=0, (1) where 𝐹 is defined as an open convex Ω of a Banach space 𝑋 with values in a Banach space 𝑌.

There are kinds of methods to find a solution of (1). Iterative methods are often used to solve this problem (see [1]). If we use the famous Newton’s method, we can do as follows: 𝑥_𝑛+1=𝑥_𝑛−𝐹^′(𝑥_𝑛)⁻¹𝐹𝑥_𝑛,(𝑛≥0)𝑥₀∈Ω. (2) Under the reasonable hypothesis, Newton’s method is second-order convergence.

Since Kantorovich presented the famous convergence result (see [2]), many Newton-Kantorovich type convergence theorems were gotten (see [3–11]). To improve the convergence order, many modified methods have been presented (see [12–15]). The famous Halley’s method is third-order convergence which was widely discussed (see [16–23]). The famous Halley’s method is defined as follows: 𝑥_𝑛+1=𝑥_𝑛−1𝐼+2𝐿_𝐹𝑥_𝑛1𝐼−2𝐿_𝐹𝑥_𝑛⁻¹×𝐹^′(𝑥_𝑛)⁻¹𝐹𝑥_𝑛,𝑛=0,1,…, (3) where 𝐿_𝐹(𝑥)=𝐹^′(𝑥)⁻¹𝐹^′′(𝑥)𝐹^′(𝑥)⁻¹𝐹(𝑥),𝑥∈Ω. (4)

In the year of 2001, Gutiérrez and Hernández [24] proposed the Super-Halley method, which is an acceleration of Newton’s method with third order. Consider 𝑥_𝑛+1=𝑥_𝑛−1𝐼+2𝐿_𝐹𝑥_𝑛𝐼−𝐿_𝐹𝑥_𝑛⁻¹𝐹^′(𝑥_𝑛)⁻¹𝐹𝑥_𝑛. (5) Using the majorant principle, the authors also established a semilocal convergence theorem for the Super-Halley method under weaker conditions, which is defined as follows: 𝐹^′𝑥₀⁻¹𝐹^′′(𝑥)−𝐹^′′𝑦≤𝑘𝑥−𝑦. (6)

Extending the family of scalar iterative processes considered by Hernandez and Salanova in [25], Gutiérrez and Hernández [26] presented a one-parameter family of iterative processes 𝑥_𝛼,𝑛+1=𝑥_𝛼,𝑛−1𝐼+2𝐿_𝐹𝑥_𝛼,𝑛𝐼−𝛼𝐿_𝐹𝑥_𝛼,𝑛⁻¹×𝐹^′𝑥_𝛼,𝑛⁻¹𝐹𝑥_𝛼,𝑛. (7) This family includes, as particular cases, Chebyshev’s method (𝛼=0), Halley’s method (𝛼=1/2), and convex acceleration of Newton’s method (𝛼=1). Under Kantorovich-type assumptions, the authors obtained results on existence and uniqueness of solution of (1).

The calculation of the second derivative of the function 𝐹 is needed in the above iterative methods. For avoiding the calculation 𝐹^′′(𝑥_𝑛) and keeping higher order of convergence at the same time, some authors have studied the convergence of the iteration (3) by using difference quotient of the first derivative to replace the second derivative.

Based on Halley’s method in which the second derivative is replaced with a finite difference between first derivatives, that is, 𝐹^′′𝑥_𝑛≃𝐹^′𝑦_𝑛−𝐹^′𝑥_𝑛𝑦_𝑛−𝑥_𝑛, (8) a class of iterative methods with free second derivative is obtained [27]. Consider 𝑥_𝑛+1=𝑥_𝑛−2𝜃𝐹𝑥_𝑛(2𝜃−1)𝐹^′𝑥_𝑛+𝐹^′𝑦_𝑛, (9) where 𝜃∈(0,1] and 𝑦_𝑛=𝑥_𝑛−𝜃𝐹(𝑥_𝑛)/𝐹^′(𝑥_𝑛).

For 𝜃=1, a third-order method is obtained [28]: 𝑥_𝑛+1=𝑥_𝑛−2𝐹𝑥_𝑛𝐹^′𝑥_𝑛+𝐹^′𝑥_𝑛−𝐹𝑥_𝑛/𝐹^′𝑥_𝑛. (10)

For 𝜃=1/2, another third-order method is obtained [29, 30]: 𝑥_𝑛+1=𝑥_𝑛−𝐹𝑥_𝑛𝐹^′𝑥_𝑛−𝐹𝑥_𝑛/2𝐹^′𝑥_𝑛. (11) These methods do not require the second derivative although they can converge cubically. Thus, the approach to remove the second derivative is important and interesting for deriving more new high-order iterative methods free from second derivative from third-order iterative methods with second derivative.

By directly replacing 𝐿_𝐹(𝑥_𝑛) with a new approximation 𝐿_𝐹𝑥_𝑛=12𝐹^′′𝑥_𝑛𝐹𝑥_𝑛𝐹^′𝑥_𝑛²≃𝐹𝑦_𝑛+(𝜃−1)𝐹𝑥_𝑛𝜃²𝐹𝑥_𝑛, (12) a class of modifications of Halley’s method free from second derivative [31] is obtained; that is, 𝑥_𝑛+1=𝑥_𝑛−𝜃²𝐹𝑥_𝑛𝜃²−𝜃+1𝐹𝑥_𝑛−𝐹𝑦_𝑛𝐹𝑥_𝑛𝐹^′𝑥_𝑛, (13) where 𝜃∈𝑅,  𝜃≠0, and 𝑦_𝑛=𝑥_𝑛−𝜃𝐹(𝑥_𝑛)/𝐹^′(𝑥_𝑛). This modified Halley’s method is cubically convergent for any nonzero real number 𝜃.

Now, we consider a new finite difference approximation of 𝐹^′′(𝑥): 𝑝𝐹^′′𝑥_𝑛𝑦_𝑛−𝑥_𝑛≃𝐹^′𝑥_𝑛+𝑝𝑦_𝑛−𝑥_𝑛−𝐹^′𝑥_𝑛, (14) where 𝑦_𝑛=𝑥_𝑛−𝐹^′𝑥_𝑛⁻¹𝐹𝑥_𝑛,𝜆>0, (15) and from Halley’s method (3), we obtain a family of new modified Halley’s method with parameters 𝑝 and 𝛼: 𝑦_𝑛=𝑥_𝑛−𝐹^′𝑥_𝑛⁻¹𝐹𝑥_𝑛,𝐻𝑥_𝑛,𝑦_𝑛=1𝑝𝐹^′𝑥_𝑛⁻¹𝐹^′𝑥_𝑛+𝑝𝑦_𝑛−𝑥_𝑛−𝐹^′𝑥_𝑛,[](],𝑥𝛼∈0,1,𝑝∈0,1_𝑛+1=𝑦_𝑛−12𝐻𝑥_𝑛,𝑦_𝑛𝐼−𝛼𝐻𝑥_𝑛,𝑦_𝑛𝑦_𝑛−𝑥_𝑛. (16)

This includes the modified Chebyshev iteration (see [8, 9]) for 𝑝=1/2,  𝛼=0 and the Jarratt iteration (see [32, 33]) for 𝑝=2/3,  𝛼=1. In this paper, we discuss the convergence of modified Halley’s methods for solving nonlinear operator equations in Banach spaces and establish convergence theorems of Newton-Kantorovich’s type. The corresponding error estimate is also given. Finally, two examples are provided to show the application of our theorem.

2. Convergence Theorem

In the section, we establish a Newton-Kantorovich type convergence theorem and present the error estimate. Denote 𝑔(𝑡)=((1/2)𝐾𝑡²)−(𝑡/𝛽)+(𝜂/𝛽), where 𝐾, 𝛽, 𝜂 are positive real numbers. Write ℎ=𝐾𝛽𝜂, 𝑡^∗=((1−√1−2ℎ)/ℎ)𝜂, and 𝑡^∗∗=((1+√1−2ℎ)/ℎ)𝜂, where 𝑡^∗, 𝑡^∗∗ are the roots of the equation 𝑔(𝑡)=0. Let 𝑠_𝑛=𝑡_𝑛−𝑔^′𝑡_𝑛⁻¹𝑔𝑡_𝑛,𝑡₀𝐻=0,_𝑔𝑡_𝑛,𝑠_𝑛=1𝑝𝑔^′𝑡_𝑛⁻¹𝑔^′𝑡_𝑛+𝑝𝑠_𝑛−𝑡_𝑛−𝑔^′𝑡_𝑛,[](],𝑡𝛼∈0,1,𝑝∈0,1_𝑛+1=𝑠_𝑛−12𝐻_𝑔𝑡_𝑛,𝑠_𝑛1−𝛼𝐻_𝑔𝑡_𝑛,𝑠_𝑛𝑠_𝑛−𝑡_𝑛. (17)

Firstly, we get some lemmas.

Lemma 1. Assume {𝑡_𝑛} and {𝑠_𝑛} are the sequences generated by (17). If ℎ≤1/2, then the sequences {𝑡_𝑛} and {𝑠_𝑛} are monotonically increasing and converge to 𝑡^∗. Moreover, one has 0≤𝑡_𝑛<𝑠_𝑛<𝑡_𝑛+1<𝑠_𝑛+1<⋯<𝑡^∗,𝑡^∗−𝑡_𝑛+1=𝑡^∗−𝑡_𝑛³𝑡^∗−𝑡_𝑛³+4𝑡^∗−𝑡_𝑛²𝑡^∗∗−𝑡_𝑛+5𝑡^∗−𝑡_𝑛𝑡^∗∗−𝑡_𝑛²+(2−2𝛼)𝑡^∗∗−𝑡_𝑛³×𝑡^∗−𝑡_𝑛+𝑡^∗∗−𝑡_𝑛⁵⁻¹,𝑡^∗∗−𝑡_𝑛+1=𝑡^∗∗−𝑡_𝑛³𝑡^∗∗−𝑡_𝑛³+4𝑡^∗∗−𝑡_𝑛²𝑡^∗−𝑡_𝑛+5𝑡^∗∗−𝑡_𝑛𝑡^∗−𝑡_𝑛²+(2−2𝛼)𝑡^∗−𝑡_𝑛³×𝑡^∗−𝑡_𝑛+𝑡^∗∗−𝑡_𝑛⁵⁻¹. (18)

Proof. By (17) we can get 𝑔(𝑡)=𝐾2𝑡^∗−𝑡𝑡^∗∗,𝑔−𝑡_′(𝑡)𝐾=−2𝑡^∗−𝑡+𝑡^∗∗,𝑠−𝑡_𝑛−𝑡_𝑛=−𝑔^′𝑡_𝑛⁻¹𝑔𝑡_𝑛=𝑡^∗−𝑡_𝑛𝑡^∗∗−𝑡_𝑛𝑡^∗−𝑡_𝑛+𝑡^∗∗−𝑡_𝑛,𝐻_𝑔𝑡_𝑛,𝑠_𝑛=𝑔^′𝑡_𝑛⁻¹𝐾𝑠_𝑛−𝑡_𝑛=−2𝑡^∗−𝑡_𝑛𝑡^∗∗−𝑡_𝑛𝑡^∗−𝑡_𝑛+𝑡^∗∗−𝑡_𝑛²,𝑡_𝑛+1−𝑠_𝑛1=−2𝐻_𝑔𝑡_𝑛,𝑠_𝑛1−𝛼𝐻_𝑔𝑡_𝑛,𝑠_𝑛𝑠_𝑛−𝑡_𝑛. (19) By a simple calculation and mathematical induction, it is easy to prove Lemma 1.

Lemma 2. Let 𝐹(𝑥) be a nonlinear operator defined on a convex domain Ω of a Banach space 𝑋 with values in a Banach space 𝑌. Assume that 𝐹 has second-order continuous Frechet derivatives on Ω. If {𝑥_𝑛}, {𝑦_𝑛} are the sequences generated by (16), then 𝐹𝑥_𝑛+1=¹₀𝐹^′′𝑦_𝑛+𝑡𝑥_𝑛+1−𝑦_𝑛(1−𝑡)𝑑𝑡𝑥_𝑛+1−𝑦_𝑛²+¹₀𝐹^′′𝑥_𝑛+𝑡𝑦_𝑛−𝑥_𝑛()−11−𝑡2𝐹^′′𝑥_𝑛+𝑝𝑡𝑦_𝑛−𝑥_𝑛𝑑𝑡𝑦_𝑛−𝑥_𝑛²−1−𝛼2¹₀𝐹^′′𝑥_𝑛+𝑝𝑡𝑦_𝑛−𝑥_𝑛×𝑦𝑑𝑡_𝑛−𝑥_𝑛𝐻𝑥_𝑛,𝑦_𝑛𝑦_𝑛−𝑥_𝑛−𝛼2¹₀𝐹^′′𝑥_𝑛+𝑡𝑦_𝑛−𝑥_𝑛−𝐹^′′𝑥_𝑛+𝑝𝑡𝑦_𝑛−𝑥_𝑛×𝑦𝑑𝑡_𝑛−𝑥_𝑛𝐻𝑥_𝑛,𝑦_𝑛𝑦_𝑛−𝑥_𝑛+𝛼2¹₀𝐹^′′𝑥_𝑛+𝑡𝑦_𝑛−𝑥_𝑛×𝑦𝑑𝑡_𝑛−𝑥_𝑛𝐻𝑥_𝑛,𝑦_𝑛𝐻𝑥_𝑛,𝑦_𝑛𝑦_𝑛−𝑥_𝑛. (20)

Proof. Consider 𝐹𝑥_𝑛+1=𝐹𝑥_𝑛+1−𝐹𝑦_𝑛−𝐹^′𝑦_𝑛𝑥_𝑛+1−𝑦_𝑛+𝐹𝑦_𝑛+𝐹^′𝑦_𝑛𝑥_𝑛+1−𝑦_𝑛=¹₀𝐹^′′𝑦_𝑛+𝑡𝑥_𝑛+1−𝑦_𝑛(1−𝑡)𝑑𝑡𝑥_𝑛+1−𝑦_𝑛²+𝐹𝑦_𝑛+𝐹^′𝑦_𝑛𝑥_𝑛+1−𝑦_𝑛,𝐹^′𝑥_𝑛𝐻𝑥_𝑛,𝑦_𝑛=1𝑝𝐹^′𝑥_𝑛+𝑝𝑦_𝑛−𝑥_𝑛−𝐹^′𝑥_𝑛=¹₀𝐹^′′𝑥_𝑛+𝑝𝑡𝑦_𝑛−𝑥_𝑛𝑑𝑡𝑦_𝑛−𝑥_𝑛,𝐹𝑦_𝑛+𝐹^′𝑦_𝑛𝑥_𝑛+1−𝑦_𝑛=𝐹𝑦_𝑛−𝐹𝑥_𝑛−𝐹^′𝑥_𝑛𝑦_𝑛−𝑥_𝑛−12𝐹^′𝑦_𝑛𝐻𝑥_𝑛,𝑦_𝑛𝐼−𝛼𝐻𝑥_𝑛,𝑦_𝑛𝑦_𝑛−𝑥_𝑛=¹₀𝐹^′′𝑥_𝑛+𝑡𝑦_𝑛−𝑥_𝑛(1−𝑡)𝑑𝑡𝑦_𝑛−𝑥_𝑛²−12𝐹^′𝑦_𝑛−𝐹^′𝑥_𝑛𝐻𝑥_𝑛,𝑦_𝑛𝐼−𝛼𝐻𝑥_𝑛,𝑦_𝑛×𝑦_𝑛−𝑥_𝑛−12𝐹^′𝑥_𝑛𝐻𝑥_𝑛,𝑦_𝑛𝐼−𝛼𝐻𝑥_𝑛,𝑦_𝑛𝑦_𝑛−𝑥_𝑛=¹₀𝐹^′′𝑥_𝑛+𝑡𝑦_𝑛−𝑥_𝑛(1−𝑡)𝑑𝑡(𝑦_𝑛−𝑥_𝑛)²−12¹₀𝐹^′′𝑥_𝑛+𝑝𝑡𝑦_𝑛−𝑥_𝑛𝑑𝑡(𝑦_𝑛−𝑥_𝑛)²+𝛼2¹₀𝐹^′′𝑥_𝑛+𝑝𝑡𝑦_𝑛−𝑥_𝑛𝑑𝑡𝑦_𝑛−𝑥_𝑛×𝐻𝑥_𝑛,𝑦_𝑛𝑦_𝑛−𝑥_𝑛−12¹₀𝐹^′′𝑥_𝑛+𝑡𝑦_𝑛−𝑥_𝑛𝑑𝑡𝑦_𝑛−𝑥_𝑛×𝐻𝑥_𝑛,𝑦_𝑛𝑦_𝑛−𝑥_𝑛+𝛼2¹₀𝐹^′′𝑥_𝑛+𝑡𝑦_𝑛−𝑥_𝑛𝑑𝑡𝑦_𝑛−𝑥_𝑛𝐻𝑥_𝑛,𝑦_𝑛×𝐻𝑥_𝑛,𝑦_𝑛𝑦_𝑛−𝑥_𝑛. (21) Hence, 𝐹𝑥_𝑛+1=¹₀𝐹^′′𝑦_𝑛+𝑡𝑥_𝑛+1−𝑦_𝑛(1−𝑡)𝑑𝑡𝑥_𝑛+1−𝑦_𝑛²+¹₀𝐹^′′𝑥_𝑛+𝑡𝑦_𝑛−𝑥_𝑛()−11−𝑡2𝐹^′′𝑥_𝑛+𝑝𝑡𝑦_𝑛−𝑥_𝑛𝑑𝑡𝑦_𝑛−𝑥_𝑛²−1−𝛼2¹₀𝐹^′′𝑥_𝑛+𝑡𝑦_𝑛−𝑥_𝑛×𝑦𝑑𝑡_𝑛−𝑥_𝑛𝐻𝑥_𝑛,𝑦_𝑛𝑦_𝑛−𝑥_𝑛−𝛼2¹₀𝐹^′′𝑥_𝑛+𝑡𝑦_𝑛−𝑥_𝑛−𝐹^′′𝑥_𝑛+𝑝𝑡𝑦_𝑛−𝑥_𝑛×𝑦𝑑𝑡_𝑛−𝑥_𝑛𝐻𝑥_𝑛,𝑦_𝑛𝑦_𝑛−𝑥_𝑛+𝛼2¹₀𝐹^′′𝑥_𝑛+𝑡𝑦_𝑛−𝑥_𝑛×𝑦𝑑𝑡_𝑛−𝑥_𝑛𝐻𝑥_𝑛,𝑦_𝑛𝐻𝑥_𝑛,𝑦_𝑛𝑦_𝑛−𝑥_𝑛. (22)
This completes the proof.

Theorem 3. Let 𝑋 and 𝑌 be Banach space and let Ω⊂𝑋 be an open convex domain. Assume 𝐹∶Ω⊂𝑋→𝑌 has second-order continuous Frechet derivatives. For an initial value 𝑥₀∈Ω and fixed parameters 𝛼∈[0,1),  𝑝∈(0,1], if 𝐹^′(𝑥₀)⁻¹ exists and the conditions𝐹^′𝑥₀⁻¹≤𝛽,𝑦₀−𝑥₀=𝐹^′𝑥₀⁻¹𝐹𝑥₀𝐹≤𝜂,_′′(𝑥)≤𝑀,𝐹^′′(𝑥)−𝐹^′′𝑦,𝑀≤𝑁𝑥−𝑦𝑥,𝑦∈Ω,max𝑁1+2+3𝑝6𝛽(1−𝛼)𝑀²,𝑀³𝑁1+1−𝑝2𝛽𝑀²≤𝐾,𝑆𝑥₀,𝑡^∗⊂Ω, (23) are satisfied and ℎ=𝐾𝛽𝜂≤1/2, then the sequence {𝑥_𝑛}_𝑛≥0 generated by (16) is well defined and converges to a unique solution 𝑥^∗ of (1) in 𝑆(𝑥₀,𝑡^∗∗).

Theorem 4. Assume 𝐹 satisfies conditions in Theorem 3. Denoting 𝜃=(𝑡^∗/𝑡^∗∗)=(1−√1−2ℎ)/(1+√1−2ℎ), one has the following:(i)when ℎ=𝐾𝛽𝜂<6√2−8=0.4852⋯, ‖𝑥_𝑛−𝑥^∗‖≤𝑡^∗−𝑡_𝑛≤1−𝜃²𝜂1−1/√2√2𝜃^3𝑛√2𝜃^3𝑛−1, (24)(ii)when 𝛼≥3/8,ℎ≤4(9√5−20)=0.4984⋯, 𝑥_𝑛−𝑥^∗≤𝑡^∗−𝑡_𝑛≤1−𝜃²𝜂1−2/√5√5/2𝜃^3𝑛√52𝜃^3𝑛−1. (25)

3. Proof of Theorem

Proof of Theorem 3. To prove Theorem 3, we first prove that following items are true for all 𝑛≥0:(I_𝑛)𝑥_𝑛∈𝑆(𝑥₀,𝑡_𝑛); (II_𝑛)‖𝐹^′(𝑥_𝑛)⁻¹‖≤−𝑔^′(𝑡_𝑛)⁻¹; (III_𝑛)‖𝑦_𝑛−𝑥_𝑛‖≤𝑠_𝑛−𝑡_𝑛; (IV_𝑛)𝑦_𝑛∈𝑆(𝑥₀,𝑠_𝑛); (V_𝑛)‖𝑥_𝑛+1−𝑦_𝑛‖≤𝑡_𝑛+1−𝑠_𝑛. It is easy to check for the case 𝑛=0 by the initial conditions. By using mathematical induction, assume that the above statements are true for some fixed 𝑛≥0. Then we have (I_𝑛+1): 𝑥_𝑛+1−𝑥₀≤𝑥_𝑛+1−𝑦_𝑛+𝑦_𝑛−𝑥₀≤𝑡_𝑛+1−𝑠_𝑛+𝑠_𝑛=𝑡_𝑛+1, (26) (II_𝑛+1): 𝐹^′𝑥_𝑛+1−𝐹^′𝑥₀≤𝑀𝑥_𝑛+1−𝑥₀≤𝑀𝑡_𝑛+1<𝐾𝑡^∗√=𝐾𝜂1−1−2ℎℎ≤1𝛽=1𝑓^′𝑥₀⁻¹. (27)By Banach Lemma, we get that 𝐹^′(𝑥_𝑛+1)⁻¹ exists and 𝐹^′(𝑥_𝑛+1)⁻¹≤𝐹^′(𝑥₀)⁻¹1−𝐹^′𝑥₀⁻¹⋅𝐹^′𝑥_𝑛+1−𝐹^′𝑥₀≤𝛽1−𝛽𝑀𝑥_𝑛+1−𝑥₀≤11/𝛽−𝐾𝑥_𝑛+1−𝑥₀≤11/𝛽−𝐾𝑡_𝑛+1=−𝑔^′𝑡_𝑛+1⁻¹; (28)(III_𝑛+1): ¹₀𝐹^′′𝑥_𝑛+𝑡𝑦_𝑛−𝑥_𝑛()−11−𝑡2𝐹^′′𝑥_𝑛+𝑝𝑡𝑦_𝑛−𝑥_𝑛≤𝑑𝑡₁₀𝐹^′′𝑥_𝑛+𝑡𝑦_𝑛−𝑥_𝑛−𝐹^′′𝑥_𝑛)+1(1−𝑡𝑑𝑡2¹₀𝐹^′′𝑥_𝑛+𝑝𝑡𝑦_𝑛−𝑥_𝑛−𝐹^′′𝑥_𝑛≤𝑁𝑑𝑡2+3𝑝𝑦12^𝑛−𝑥_𝑛,𝐻𝑥_𝑛+1,𝑦_𝑛+1=1𝑝𝐹^′𝑥_𝑛+1⁻¹¹₀𝐹^′′𝑥_𝑛+1+𝑝𝑡𝑦_𝑛+1−𝑥_𝑛+1×𝑝𝑦_𝑛+1−𝑥_𝑛+1𝑀𝑑𝑡≤−𝑔^′𝑡_𝑛+1𝑦_𝑛+1−𝑥_𝑛+1. (29) By Lemma 2 and 0<−𝑔^′(𝑡_𝑛)≤1/𝛽, 𝐹𝑥_𝑛+1≤𝑀2𝑥_𝑛+1−𝑦_𝑛²+2+3𝑝𝑁×𝑦12_𝑛−𝑥_𝑛³+(1−𝛼)𝑀²−2𝑔^′𝑡_𝑛𝑦_𝑛−𝑥_𝑛³+𝛼𝑀³2𝑔^′𝑡_𝑛²𝑦_𝑛−𝑥_𝑛⁴+𝛼𝑀𝑁1−𝑝−4𝑔^′𝑡_𝑛𝑦_𝑛−𝑥_𝑛⁴≤𝐾2𝑥_𝑛+1−𝑦_𝑛²+𝑁1+1−𝑝2𝛽𝑀²𝛼𝑀³𝑦_𝑛−𝑥_𝑛⁴2𝑔^′𝑡_𝑛²+𝑁1+2+3𝑝6𝛽(1−𝛼)𝑀²(1−𝛼)𝑀²−2𝑔^′𝑡_𝑛𝑦_𝑛−𝑥_𝑛³≤𝐾2𝑡_𝑛+1−𝑠_𝑛²+𝛼𝐾³2𝑠_𝑛−𝑡_𝑛⁴𝑔^′𝑡_𝑛²−(1−𝛼)𝐾²2𝑔^′𝑡_𝑛𝑠_𝑛−𝑡_𝑛³=𝑔𝑡_𝑛+1. (30) Hence, we deduce that 𝑦_𝑛+1−𝑥_𝑛+1≤−𝐹^′𝑥_𝑛+1⁻¹𝐹𝑥_𝑛+1≤−𝑔^′𝑡_𝑛+1⁻¹𝑔𝑡_𝑛+1=𝑠_𝑛+1−𝑡_𝑛+1. (31) Moreover, we have (IV_𝑛+1): 𝑦_𝑛+1−𝑥₀≤𝑦_𝑛+1−𝑥_𝑛+1+𝑥_𝑛+1−𝑥₀≤𝑠_𝑛+1−𝑡_𝑛+1+𝑡_𝑛+1=𝑠_𝑛+1, (32)(V_𝑛+1) ∶𝑥_𝑛+2−𝑦_𝑛+1=−12𝐻𝑥_𝑛+1,𝑦_𝑛+1𝐼−𝛼𝐻𝑥_𝑛+1,𝑦_𝑛+1𝑦_𝑛+1−𝑥_𝑛+1≤12𝑀𝑦_𝑛+1−𝑥_𝑛+1−𝑔^′𝑡_𝑛+1𝑀𝑦1+𝛼^𝑛+1−𝑥_𝑛+1−𝑔^′𝑡_𝑛+1×𝑦_𝑛+1−𝑥_𝑛+11≤−2𝐾𝑠_𝑛+1−𝑡_𝑛+1𝑔^′𝑡_𝑛+1𝐾𝑠1−𝛼^𝑛+1−𝑡_𝑛+1𝑔^′𝑡_𝑛+1𝑠_𝑛+1−𝑡_𝑛+1=𝑡_𝑛+2−𝑠_𝑛+1. (33) By Lemma 1, if ℎ≤1/2, then the sequence {𝑥_𝑛,}_𝑛≥0 generated by (16) is well defined, remains in 𝑆(𝑥₀,𝑡^∗) for all 𝑛≥0, and converges to a solution 𝑥^∗ of (1).
To show uniqueness, let us assume that there exists a second solution 𝑦^∗ of (1) in 𝑆(𝑥₀,𝑡^∗∗). Then 𝐹^′𝑥₀⁻¹¹₀𝐹^′𝑥^∗+𝑡𝑦^∗−𝑥^∗≤𝐹𝑑𝑡−𝐼_′𝑥₀⁻¹¹₀𝐹^′𝑥^∗+𝑡𝑦^∗−𝑥^∗−𝐹^′𝑥₀𝑑𝑡≤𝛽𝑀¹₀𝑥^∗+𝑡𝑦^∗−𝑥^∗−𝑥₀𝑑𝑡≤𝛽𝑀¹₀(1−𝑡)𝑥^∗−𝑥₀+𝑡𝑦^∗−𝑥₀<𝑑𝑡𝛽𝑀2𝑡^∗+𝑡^∗∗≤1. (34)
By Banach Lemma, we can obtain that the inverse of the linear operator ∫¹₀𝐹^′[𝑥^∗+𝑡(𝑦^∗−𝑥^∗)]𝑑𝑡 exists and 𝐹𝑦^∗−𝐹𝑥^∗=¹₀𝐹^′𝑥^∗+𝑡𝑦^∗−𝑥^∗𝑑𝑡𝑦^∗−𝑥^∗. (35)
We conclude that 𝑥^∗=𝑦^∗. The proof of Theorem 3 is completed.

Proof of Theorem 4. By Lemma 1, we have 𝑡^∗−𝑡_𝑛+1=𝑡^∗−𝑡_𝑛³𝑡^∗−𝑡_𝑛³+4𝑡^∗−𝑡_𝑛²𝑡^∗∗−𝑡_𝑛+5𝑡^∗−𝑡_𝑛(𝑡^∗∗−𝑡_𝑛)²+()×𝑡2−2𝛼_∗∗−𝑡_𝑛³×𝑡^∗−𝑡+𝑡^∗∗−𝑡⁵⁻¹<𝑡^∗−𝑡_𝑛³5𝑡^∗−𝑡_𝑛²𝑡^∗∗−𝑡_𝑛+5𝑡^∗−𝑡_𝑛𝑡^∗∗−𝑡_𝑛²+2𝑡^∗∗−𝑡_𝑛³×𝑡^∗−𝑡+𝑡^∗∗−𝑡⁵⁻¹. (36)
Similarly, 𝑡^∗∗−𝑡_𝑛+1=𝑡^∗∗−𝑡_𝑛³𝑡^∗∗−𝑡_𝑛³+4𝑡^∗∗−𝑡_𝑛²𝑡^∗−𝑡_𝑛+5𝑡^∗∗−𝑡_𝑛𝑡^∗−𝑡_𝑛²+(2−2𝛼)𝑡^∗−𝑡_𝑛³×𝑡^∗−𝑡+𝑡^∗∗−𝑡⁵⁻¹>(𝑡^∗∗−𝑡_𝑛)³𝑡^∗∗−𝑡_𝑛³+4𝑡^∗∗−𝑡_𝑛²𝑡^∗−𝑡_𝑛+5𝑡^∗∗−𝑡_𝑛𝑡^∗−𝑡_𝑛²×𝑡^∗−𝑡+𝑡^∗∗−𝑡⁵⁻¹. (37)
Hence, 𝑡^∗−𝑡_𝑛+1𝑡^∗∗−𝑡_𝑛+1𝑡<2^∗−𝑡_𝑛𝑡^∗∗−𝑡_𝑛³<2⋅2³𝑡^∗−𝑡_𝑛−1𝑡^∗∗−𝑡_𝑛−1³²<⋯<2⋅2³⋯2^3𝑛𝑡^∗−𝑡₀𝑡^∗∗−𝑡₀^3𝑛+1=√2^3𝑛+1−1𝜃^3𝑛+1. (38) Because 𝑡^∗∗−𝑡_𝑛=𝑡^∗−𝑡_𝑛+𝑡^∗∗−𝑡^∗=𝑡^∗−𝑡_𝑛+((1−𝜃²)𝜂)/𝜃, we obtain 𝑡^∗−𝑡_𝑛≤1−𝜃²𝜂1−1/√2√2𝜃^3𝑛√2𝜃^3𝑛−1. (39) When 𝛼≥3/8, 𝑡^∗−𝑡_𝑛+1=𝑡^∗−𝑡_𝑛³𝑡^∗−𝑡_𝑛³+4𝑡^∗−𝑡_𝑛²𝑡^∗∗−𝑡_𝑛+5𝑡^∗−𝑡_𝑛𝑡^∗∗−𝑡_𝑛²+(2−2𝛼)𝑡^∗∗−𝑡_𝑛³×𝑡^∗−𝑡+𝑡^∗∗−𝑡⁵⁻¹<𝑡^∗−𝑡_𝑛³5𝑡^∗−𝑡_𝑛²𝑡^∗∗−𝑡_𝑛+5𝑡^∗−𝑡_𝑛𝑡^∗∗−𝑡_𝑛²+54𝑡^∗∗−𝑡_𝑛³×𝑡^∗−𝑡+𝑡^∗∗−𝑡⁵⁻¹=54𝑡^∗−𝑡_𝑛³4𝑡^∗−𝑡_𝑛²𝑡^∗∗−𝑡_𝑛+4𝑡^∗−𝑡_𝑛𝑡^∗∗−𝑡_𝑛²+𝑡^∗∗−𝑡_𝑛³×𝑡^∗−𝑡+𝑡^∗∗−𝑡⁵⁻¹. (40)
Hence, we get 𝑡^∗−𝑡_𝑛+1𝑡^∗∗−𝑡_𝑛+1<54𝑡^∗−𝑡_𝑛𝑡^∗∗−𝑡_𝑛³<√52^3𝑛+1−1𝜃^3𝑛+1,𝑥_𝑛−𝑥^∗≤𝑡^∗−𝑡_𝑛≤1−𝜃²𝜂1−2/√5√5/2𝜃^3𝑛√52𝜃^3𝑛−1. (41)
The proof of Theorem 4 is completed.

4. Applications

Example 1. Consider the case as follows: 𝑥(𝑠)1=1+4𝑥(𝑠)¹₀𝑠𝑥(𝑡)𝑠+𝑡𝑑𝑡, (42) where the space is 𝑋=𝐶[0,1] with norm ‖𝑥‖=max_0≤𝑠≤1|𝑥(𝑠)|. (43)
This equation arises in the theory of the radiative transfer, neutron transport, and kinetic theory of gasses. Let us define the operator 𝐹 on 𝑋 by 𝐹(𝑥)=14𝑥(𝑠)¹₀𝑠𝑥(𝑡)(𝑠)𝑠+𝑡𝑑𝑡−𝑥+1. (44) Then, for 𝑥₀=1, we can obtain 1𝑀=2⋅4max_0≤𝑠≤1¹₀𝑠=𝑠+𝑡𝑑𝑡ln22𝐹=0.3465,𝑁=0,𝐾=𝑀=0.3465,𝛽=^′(𝑥₀)⁻¹𝐹=1.5304,𝜂=^′𝑥₀⁻¹𝐹^′𝑥₀=0.2652,ℎ=𝐾𝛽𝜂=0.1406<0.4852,𝑡^∗𝑡=0.2870,_∗∗=3.485,𝜃=0.08240. (45)
That means the hypotheses of the theorem are satisfied and for 𝛼=1/2>3/8 the error bound becomes 𝑥_𝑛−𝑥^∗≤𝑡^∗−𝑡_𝑛≤1−𝜃²𝜂1−2/√5√5/2𝜃^3𝑛√52𝜃^3𝑛−1=0.26341−0.8944⋅(0.07370)^3𝑛(0.07370)^3𝑛−1. (46) For 𝑛=1,2,3, we get 𝑥₁−𝑥^∗≤𝑡^∗−𝑡₁≤2.2371×10⁻³,𝑥₂−𝑥^∗≤𝑡^∗−𝑡₂≤1.3667×10⁻⁹,𝑥₃−𝑥^∗≤𝑡^∗−𝑡₃≤3.1229×10⁻²⁸. (47)
In practical computation, we use a discretization process. By Gauss-Legendre quadrature formula with 8 nodes: ¹₀𝜑(𝑡)𝑑𝑡≃⁸_𝑗=1𝜔_𝑗𝜑𝑡_𝑗, (48) we approximate the integral equation (42), where the nodes 𝑡_𝑗 and the weights 𝜔_𝑗 are known. Denote 𝑥(𝑡_𝑖) by 𝑥_𝑖,𝑖=1,2,…8, so we can transform (42) into the following system of nonlinear equations: 𝑥_𝑖1=1+4𝑥_𝑖⁸_𝑗=1𝑎_𝑖𝑗𝑥_𝑗, (49) where 𝑎_𝑖𝑗=𝑤_𝑗(𝑡_𝑖/(𝑡_𝑖+𝑡_𝑗)). Then, we rewrite the above system in the matrix form. Consider 𝐹(𝑋)1=𝑋−𝑉−4𝐴𝑊, (50) where 𝑋=(𝑥₁,𝑥₂,…,𝑥₈)^𝑇,𝑉=(1,1,…,1)^𝑇,𝐴=(𝑎_𝑖𝑗)⁸_𝑖,𝑗=1, and 𝑊=(𝑥²₁,𝑥²₂,…,𝑥²₈)^𝑇. We also get 𝐹^′(𝑋)1=𝐼−2𝐴diag𝑥₁,𝑥₂,…,𝑥₈. (51)
After 3 iterative steps, we can obtain the numerical solution of (42) by method (16) with 𝑝=2/3, 𝛼=1/2, and the solution 𝑋_∗=(1.2751,1.4418,1.5595,1.6486,1.7189,1.7761,1.8236,1.8638)^𝑇, and we present the error computing result of (42) in Table 1.

Table 1 

Error computing results of (42).

Example 2. Now we employ iterative methods (16) to solve the equation and compare these methods with Newton’s method, Halley’s method, and modified Halley’s methods (16). We define as follows: 𝑓(𝑥)=𝑥^𝑛−𝑅,𝑛>2,𝑅∈(0,+∞). (52) Denote 𝑥^∗=√^𝑛𝑅, by (16) 𝑥_𝑛+1=𝑣(𝑥_𝑛), where 𝑣(𝑥)=𝑥−[1−(1/2)𝐻(𝑥)+(𝛼/2)𝐻²(𝑥)](𝑓(𝑥))/(𝑓^′(𝑥)), 𝐻(𝑥)=(𝑓^′(𝑥−𝑝𝑓(𝑥)/𝑓^′(𝑥))−𝑓^′(𝑥))/(𝑝𝑓^′(𝑥)). We have 𝑣^′(𝑥^∗)=𝑣^′′(𝑥^∗)=𝑣^′′′(𝑥^∗)=0 if 2𝑝=3,𝛼=1. (53) So, we get the convergence of the sequence {𝑥_𝑛} generated by modified Halley’s method (16) with four orders when 𝑝=2/3, 𝛼=1.
Now, we compare some of these methods for the calculus of 𝑥^∗=√⁴16=2. We analyze the errors 𝑥_𝑛−𝑥^∗ for these methods with Newton’s method, Halley’s method, and the family of new modified Halley’s methods (16). In these cases, we have taken 𝑛=4, 𝑥₀=2.2 (see Table 2).

Table 2 

Error computing results (|𝑥_𝑛−𝑥^∗|).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Basic Research 973 Program of China (no. 2011JB105001), National Natural Science Foundation of China (Grant no. 11371320), Zhejiang Natural Science Foundation (Grant no. LZ14A010002), Foundation of Science and Technology Department (Grant no. 2013C31084) of Zhejiang Province, and Scientific Research Fund of Zhejiang Provincial Education Department (nos. Y201431077 and Y201329420).