Abstract

Under the new Hölder conditions, we consider the convergence analysis of the inverse-free Jarratt method in Banach space which is used to solve the nonlinear operator equation. We establish a new semilocal convergence theorem for the inverse-free Jarratt method and present an error estimate. Finally, three examples are provided to show the application of the theorem.

1. Introduction

We consider the following boundary value problem: Those are equivalent to the following nonlinear integral equation (see [1, 2]): where and is a twice Fréchet-differentiable operator. is the set of all continuous functions in ; is the Green function:

Instead of 2, we can try to solve a nonlinear operator equation , where Solving the nonlinear operator equation is an important issue in the engineering and technology field as these kinds of problems appear in many real-world applications. Economics [3], chemistry [4], and physics [5–8] are some of the examples of the scientific and engineering technology areas applied to solve these type of equations. In this study, we consider to establish a new semilocal convergence theorem of the Jarratt method in Banach space which is used to solve the nonlinear operator equation where is defined on an open convex of a Banach space with values in a Banach space .

There are a lot of methods of finding a solution of equation . Particularly iterative methods are often used to solve this problem (see [1, 2, 9, 10]). If we use the famous Newton method, we can proceed as Under a reasonable hypothesis, Newton’s method is the second-order convergence.

To improve the convergence order, many modified methods have been presented. The famous Halley’s method and the supper-Halley method are the third-order convergence. References [11–22] give the convergence analysis for these methods. Now, we consider the following Jarratt method (see [23–25]):

In this paper, we discuss the convergence of 7 for solving nonlinear operator equations in Banach spaces and establish a new semilocal convergence theorem under the following condition (see [20, 21]): where is a nondecreasing continuous function. Finally, the corresponding error estimate is also given.

2. Main Results

In the section, we establish a new semilocal convergence theorem and present the error estimate. Denote and . Suppose that and are the Banach spaces, is an open convex of the Banach space , and has continuous Fréchet derivative of the third-order. exists, for some , and satisfies (A5) is a nondecreasing continuous real function for such that , and there exists a positive real number such that for and .(A6)Denote ,  . Let , , , ,  , , where First, we get some lemmas.

Lemma 1. Suppose that , are given by 10. Then , is increasing and ; , , is increasing; , , , and  .

Lemma 2. Suppose that , are given by 10. If then(i)the sequences , , are nonnegative and decreasing;(ii), .

Proof. (i) When , Suppose for . By Lemma 1, and are increasing; then (ii) By (i), is decreasing and . So, for all , This completes the proof of Lemma 2.

Lemma 3. Suppose that the conditions of Lemma 2 hold. Denote . Then (i),   ,   ;(ii), .

Proof. First, by induction, we prove that (i) holds. Because and , we have Suppose that (i) holds for . Then we get and from (ii) we get This completes the proof of Lemma 3.

Lemma 4. Suppose that and are Banach spaces, is an open convex of the Banach space , has continuous Fréchet derivative of the second-order, and the sequences , are generated by 7. Then, for all natural numbers , the following formula holds:

Proof. Consider This completes the proof of Lemma 4.
By (A1)–(A6), 10, and 11, if , then where ; hence, . Consider By Banach lemma, exists, and