Convergence Behavior for Newton-Steffensen’s Method under 𝛾-Condition of Second Derivative

Abstract

The present paper is concerned with the semilocal as well as the local convergence problems of Newton-Steffensen’s method to solve nonlinear operator equations in Banach spaces. Under the assumption that the second derivative of the operator satisfies 𝛾-condition, the convergence criterion and convergence ball for Newton-Steffensen’s method are established.

1. Introduction

Let 𝑋 and 𝑌 be real or complex Banach spaces, let 𝐷⊂𝑋 be an open subset, and let 𝐹∶𝐷⊂𝑋→𝑌be the Fréchet differentiable nonlinear operator. Approximating a solution of a nonlinear equation 𝐹(𝑥)=0 (1) is widely studied in both theoretical and applied areas of mathematics.

One of the most famous methods to solve this problem is Newton’s method defined by 𝑥_𝑛+1=𝑥_𝑛−𝐹^′𝑥_𝑛⁻¹𝐹𝑥_𝑛,𝑛=0,1,2,…, (2) where 𝑥₀∈𝐷 is an initial point. Usually, the study about convergence issue of Newton’s method includes local and semilocal convergence analyses. The local convergence issue is, based on the information around a solution, to seek estimates of the radii of convergence balls, while the semilocal one is, based on the information around an initial point, to give criteria ensuring the convergence. Among the semilocal convergence results on Newton’s method, one of the famous results is Smale’s point estimate theory which gives a convergence criterion of Newton’s method only based on the information at the initial point for analytic functions; see for example, [1–6]. To extend and improve Smale’s theory, Wang and Han proposed in [7, 8] the notion of𝛾-condition, which is weaker than Smale’s assumption in [5] for analytic operators.

There are several kinds of cubic generalizations for Newton’s method. The most important family is the Euler-Halley family and its variations which include Chebyshev’s method and Halley’s method as special cases; see for example, [9–16] and references therein. However, the disadvantage of this family is that the evaluation of the second derivative of the operator𝐹is required at every step, the operation cost of which may be very large in fact. To reduce the operation cost but also retain the cubic convergence, Sharma in [17] proposed the following Newton-Steffensen’s method which avoids the computation of the second Fréchet derivative. Let𝑓∶ℝ→ℝ. The method is defined as follows: 𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓^′𝑥_𝑛,𝑥_𝑛+1=𝑥_𝑛−𝑓𝑥_𝑛𝑔𝑥_𝑛,𝑛=0,1,…, (3) where𝑔(𝑥_𝑛)=((𝑓(𝑦_𝑛)−𝑓(𝑥_𝑛))/(𝑦_𝑛−𝑥_𝑛)). The author obtained cubic convergence for (3) under the assumption that𝑓is sufficiently smooth in the neighborhood of the solution.

Motivated by the work mentioned above, we extend this method to Banach spaces and present its semilocal and local convergence. The extension is described as follows: 𝑦_𝑛=𝑥_𝑛−𝐹^′𝑥_𝑛⁻¹𝐹𝑥_𝑛,𝑥_𝑛+1=𝑥_𝑛−𝑦_𝑛,𝑥_𝑛;𝐹⁻¹𝐹𝑥_𝑛,𝑛=0,1,…, (4) where the divided difference operator is defined by 𝑦_𝑛,𝑥_𝑛;𝐹=¹₀𝐹^′𝑥_𝑛+𝑡𝑦_𝑛−𝑥_𝑛d𝑡. (5)

In Section 2, we introduce some preliminary notions and important majorizing functions with properties. In Sections 3 and 4, we study the semilocal convergence and local convergence results of Newton-Steffensen’s method under𝛾-condition, respectively. We obtain the uniqueness ball and the convergence ball.

2. Notations and Preliminary Results

Throughout this paper, we assume that 𝑋 and 𝑌 are two Banach spaces. Let 𝐷⊂𝑋 be an open subset and let 𝐹∶D⊂𝑋→𝑌 be a nonlinear operator with the continuous twice Fréchet derivative. For𝑥∈𝑋and𝑟>0, we use𝐁(𝑥,𝑟)and𝐁(𝑥,𝑟)to denote the open ball with radius𝑟and center𝑥and its closure, respectively. Let𝑥∈𝐷be such that 𝐹^′(𝑥)⁻¹exists and𝐁(𝑥,𝑟)⊂𝐷.

Let𝛾be some positive constant and0<𝑟⩽(1/𝛾). We say that𝐹satisfies𝛾-condition on𝐁(𝑥,𝑟)if the following relation holds: 𝐹^′(𝑥)⁻¹𝐹^′′(𝑥)⩽2𝛾1−𝛾‖𝑥−𝑥‖³,𝑥∈𝐁(𝑥,𝑟). (6)

For simplicity, we write 𝑟₀=11−√21𝛾. (7)

The lemma below is useful in the next two sections.

Lemma 1. Suppose that𝑟⩽𝑟₀and that𝐹satisfies𝛾-condition (6) on𝐁(𝑥,𝑟). Then for any𝑥∈𝐁(𝑥,𝑟),𝐹^′(𝑥)⁻¹exists and the following inequality holds: 𝐹^′(𝑥)⁻¹𝐹^′(𝑥⩽‖)1−𝛾𝑥−𝑥‖²21−𝛾‖𝑥−𝑥‖².−1 (8)

Proof. We can derive the following relation: 𝐼−𝐹^′(𝑥)⁻¹𝐹^′(𝑥)=−¹₀𝐹^′(𝑥)⁻¹𝐹^′′(𝑥+𝑡(𝑥−𝑥))(𝑥−𝑥)d𝑡. (9) For any𝑥∈𝐵(𝑥,𝑟), it follows from𝛾-condition and𝑟⩽𝑟₀that 𝐼−𝐹^′(𝑥)⁻¹𝐹^′(𝑥)⩽¹₀2𝛾‖𝑥−𝑥‖1−𝛾‖𝑥−𝑥‖𝑡³d𝑡=11−𝛾‖𝑥−𝑥‖²−1<1. (10) Then, by Banach lemma, one has that𝐹^′(𝑥)⁻¹exists and the following inequality holds: 𝐹^′(𝑥)⁻¹𝐹^′(𝑥⩽1)1−1/1−𝛾‖𝑥−𝑥‖²=‖−11−𝛾𝑥−𝑥‖²21−𝛾‖𝑥−𝑥‖².−1 (11)

Let𝛽>0be some positive constant. The following majorizing functionℎintroduced by Wang and Han in [18] will be used to obtain a Smale-type semilocal convergence criterion: ℎ(𝑡)=𝛽−𝑡+𝛾𝑡²11−𝛾𝑡,𝑡∈0,𝛾. (12)

Let{𝑠_𝑛}and{𝑡_𝑛}denote the corresponding sequences generated by Newton-Steffensen’s method for the majorizing functionℎwith the initial point𝑡₀=0; that is, 𝑠_𝑛=𝑡_𝑛−ℎ𝑡_𝑛ℎ^′𝑡_𝑛,𝑡_𝑛+1=𝑡_𝑛−ℎ𝑠_𝑛−ℎ𝑡_𝑛𝑠_𝑛−𝑡_𝑛⁻¹ℎ𝑡_𝑛,𝑛=0,1,…. (13)

The following lemma taken from [19] describes some useful properties aboutℎ.

Lemma 2. Suppose that 𝛼∶=𝛽𝛾⩽3−2√2. (14) Thenℎhas two zeros in[0,1/𝛾)denoted by𝑡^∗and𝑡^∗∗. They satisfy the following relations: 𝛽⩽𝑡^∗⩽𝑟₀⩽𝑡^∗∗⩽1.2𝛾 (15) Moreover,ℎis decreasing monotonically in interval[0,𝑟₀], while it is increasing monotonically in interval[𝑟₀,1/(2𝛾)].

The lemma below describes the convergence property of the sequences{𝑠_𝑛}and{𝑡_𝑛}, which is crucial for the semilocal convergence analysis of Newton-Steffensen’s method (4) under𝛾-condition.

Lemma 3. Suppose that (14) holds. Let{𝑠_𝑛}and{𝑡_𝑛}be the sequences generated by (13). Then 0⩽𝑡_𝑛<𝑠_𝑛<𝑡_𝑛+1<𝑡^∗∀𝑛⩾0. (16) Moreover,{𝑠_𝑛}and{𝑡_𝑛}converge increasingly to the same point𝑡^∗.

Proof. To show that (16) holds for𝑛=0, we note that0=𝑡₀<𝑠₀=𝛽and that𝑡₁=(𝛽(1−𝛾𝛽)/(1−2𝛾𝛽)). By (15), we have 1−𝛾𝛽1−2𝛾𝛽>1. (17) This implies that𝑡₁>𝛽=𝑠₀. It remains to show that𝑡₁<𝑡^∗for the case𝑛=0. To this end, we define a real function as Φ(𝑡)=1−𝛾𝑡[).1−𝛾𝑡,𝑡∈0,+∞ (18) It is clear thatΦ(𝑡)=−(ℎ(𝑡)−𝛽)/𝑡and thatΦ(𝑡)is decreasing monotonically in[0,+∞). It follows from (15) thatΦ(𝛽)>Φ(𝑡^∗). In view of the fact that𝑡^∗is the unique zero ofℎin[0,𝑟₀], we obtain𝛽/𝑡^∗=Φ(𝑡^∗)<Φ(𝛽). This is equivalent to 𝑡^∗>𝛽1−𝛾𝛽/1−𝛾𝛽=𝑡₁. (19) Hence (16) holds for𝑛=0.
Now we assume that 0⩽𝑡_𝑛−1<𝑠_𝑛−1<𝑡_𝑛<𝑡^∗forsome𝑛⩾1. (20) From Lemma 2, we haveℎ(𝑡)⩾0, for each𝑡∈[0,𝑡^∗], andℎ(𝑡_𝑛)/ℎ^′(𝑡_𝑛)<0. The later one implies that𝑠_𝑛>𝑡_𝑛. Define function 𝑁(𝑡)ℎ(𝑡)=𝑡−ℎ^′(𝑡),𝑡∈0,𝑡^∗. (21) Then,𝑁^′(𝑡)=ℎ(𝑡)ℎ^′′(𝑡)/ℎ^′(𝑡)²>0, which implies that𝑁is increasing monotonically in[0,𝑡^∗]. Hence, we have 𝑠_𝑛=𝑡_𝑛−ℎ𝑡_𝑛ℎ^′𝑡_𝑛<𝑡^∗−ℎ𝑡^∗ℎ^′(𝑡^∗)=𝑡^∗. (22) Sinceℎis convex in[0,𝑡^∗], we getℎ^′(𝑡_𝑛)<(ℎ(𝑠_𝑛)−ℎ(𝑡_𝑛))/(𝑠_𝑛−𝑡_𝑛)and so𝑠_𝑛<𝑡_𝑛+1.
Furthermore, we claim that 𝑡^′−ℎ(𝑡)−ℎ𝑡^′𝑡−𝑡^′−1ℎ𝑡^′<𝑡^′′−ℎ𝑡^′′−ℎ(𝑡)𝑡^′′−𝑡⁻¹ℎ𝑡^′′ (23) for all𝑡^′, 𝑡, 𝑡^′′∈[0,𝑡^∗]and𝑡^′<𝑡<𝑡^′′. Indeed, it follows from the convexity ofℎthat −ℎ𝑡^′′ℎ^′(𝑡)ℎ𝑡⩽−^′′−ℎ(𝑡)𝑡^′′−𝑡⁻¹ℎ𝑡^′′ℎ𝑡⩽−^′′ℎ^′𝑡^′′,−ℎ𝑡^′ℎ^′𝑡^′ℎ(𝑡)𝑡⩽−−ℎ^′𝑡−𝑡^′−1ℎ𝑡^′ℎ𝑡⩽−^′ℎ^′(𝑡), (24) from which we have 𝑡^′′−𝑡^′+ℎ𝑡^′−ℎ𝑡^′′ℎ^′(𝑡)⩽𝑡^′′−𝑡^′+𝑇^′′−𝑇^′⩽𝑡^′′−𝑡^′−ℎ𝑡^′′ℎ^′𝑡^′′+ℎ𝑡^′ℎ^′𝑡^′, (25) where 𝑇^′ℎ(𝑡)𝑡=−−ℎ^′𝑡−𝑡^′−1ℎ𝑡^′,𝑇^′′=−ℎ(𝑡^′′)−ℎ(𝑡)𝑡^′′−𝑡⁻¹ℎ𝑡^′′. (26) Noting that−1<ℎ^′(𝑡)<0for all𝑡∈[0,𝑡^∗], we obtain 𝑡^′′−𝑡^′+ℎ𝑡^′−ℎ𝑡^′′ℎ^′(𝑡)>𝑡^′′−𝑡^′+ℎ𝑡^′′−ℎ𝑡^′⩾0. (27) Then (23) follows. By (23), we conclude that 𝑡_𝑛+1=𝑡_𝑛−ℎ𝑠_𝑛−ℎ𝑡_𝑛𝑠_𝑛−𝑡_𝑛⁻¹ℎ𝑡_𝑛<𝑡^∗−ℎ𝑡^∗−ℎ𝑠_𝑛𝑡^∗−𝑠_𝑛⁻¹ℎ𝑡^∗=𝑡^∗. (28) Therefore, (16) holds for all𝑛⩾0. The inequlities in (16) imply that{𝑠_𝑛}and{𝑡_𝑛}converge increasingly to some same points, say𝜏. Clearly𝜏∈[0,𝑡^∗]and𝜏is a zero ofℎin[0,𝑡^∗]. Noting that𝑡^∗is the unique zero ofℎin[0,𝑟₀], one has that𝜏=𝑡^∗. The proof is complete.

3. Convergence Criterion

Throughout this subsection, let𝑥₀∈𝐷be the initial point such that the inverse𝐹^′(𝑥₀)⁻¹exists and let𝐁(𝑥₀,𝑟₀)⊂𝐷, where𝑟₀is defined by (7). Moreover, we assume that𝐹satisfies𝛾-condition on𝐁(𝑥₀,𝑟₀); that is, the following relation holds:

𝐹^′𝑥₀⁻¹𝐹^′′(𝑥)⩽2𝛾1−𝛾𝑥−𝑥₀³,𝑥∈𝐁𝑥₀,𝑟₀. (29) Then, for any𝑥∈𝐁(𝑥₀,𝑟₀), it follows from Lemma 1 that𝐹^′(𝑥)⁻¹exists and the following inequality holds: 𝐹^′(𝑥)⁻¹𝐹^′𝑥₀⩽1−𝛾𝑥−𝑥₀²21−𝛾𝑥−𝑥₀².−1 (30)

Below we list two useful lemmas.

Recall that the divided difference operator[𝑦,𝑥;𝐹]is defined by (5). The following lemma gives the expressions of some desired estimates in the proof of Lemma 5.

Lemma 4. Let𝑥∈𝐁(𝑥₀,𝑟₀). Define 𝑦∶=𝑥−𝐹^′(𝑥)⁻¹𝐹(𝑥),𝑥∶=𝑥−𝑦,𝑥;𝐹⁻¹𝐹(𝑥). (31) Then the following formulas hold: (i)[𝑦,𝑥;𝐹]−𝐹^′(𝑥₀)=∫¹₀∫¹₀𝐹^′′(𝑥₀+𝑠[(𝑥−𝑥₀)+𝑡(𝑦−𝑥)])[(𝑥−𝑥₀)+𝑡(𝑦−𝑥)]d𝑠d𝑡. (ii)𝑥−𝑦=−𝐹^′(𝑥)⁻¹∫¹₀∫¹₀𝐹^′′(𝑥+𝑠𝑡(𝑦−𝑥))(𝑦−𝑥)(𝑥−𝑥)𝑡d𝑠d𝑡.(iii)𝐹(𝑥)=∫¹₀∫¹₀𝐹^′′(𝑥+𝑡(𝑦−𝑥)+𝑠𝑡(𝑥−𝑦))(𝑥−𝑥)(𝑥−𝑦)𝑡d𝑠d𝑡.

Proof. For (i), we notice that 𝑦,𝑥;𝐹−𝐹^′𝑥₀=¹₀𝐹^′𝑥+𝑡𝑦−𝑥−𝐹^′𝑥₀d𝑡=¹₀¹₀𝐹^′′𝑥₀+𝑠𝑥−𝑥₀×+𝑡𝑦−𝑥𝑥−𝑥₀+𝑡𝑦−𝑥d𝑠d𝑡. (32) As for (ii), one has 𝑥−𝑦=𝐹^′(𝑥)⁻¹𝐹(𝑥)−𝑦,𝑥;𝐹⁻¹𝐹(𝑥)=𝐹^′(𝑥)⁻¹𝐹^′(𝑥)−¹₀𝐹^′𝑥+𝑡𝑦−𝑥d𝑡×(𝑥−𝑥)=−𝐹^′(𝑥)⁻¹×¹₀¹₀𝐹^′′𝑥+𝑠𝑡𝑦−𝑥𝑦−𝑥(𝑥−𝑥)𝑡d𝑠d𝑡. (33) Similarly, we obtain 𝐹(𝑥)=𝐹(𝑥)−𝐹(𝑥)−𝑦,𝑥;𝐹()=𝑥−𝑥₁₀𝐹^′(𝑥+𝑡(𝑥−𝑥))(𝑥−𝑥)d𝑡−¹₀𝐹^′𝑥+𝑡𝑦−𝑥(𝑥−𝑥)d𝑡=¹₀¹₀𝐹^′′𝑥+𝑡y−𝑥+𝑠𝑡×(𝑥−𝑦𝑥−𝑥)𝑥−𝑦𝑡d𝑠d𝑡. (34) The proof is complete.

Lemma 5. Suppose that (14) holds. Then the sequence{𝑥_𝑛}generated by (4) with the initial point𝑥₀is well defined and the following estimates hold for any natural number𝑛⩾1: (i)‖𝑦_𝑛−1−𝑥_𝑛−1‖⩽𝑠_𝑛−1−𝑡_𝑛−1,‖𝑥_𝑛−𝑥_𝑛−1‖⩽𝑡_𝑛−𝑡_𝑛−1,‖𝑥_𝑛−𝑦_𝑛−1‖⩽𝑡_𝑛−𝑠_𝑛−1. (ii)‖[𝑦_𝑛−1,𝑥_𝑛−1;𝐹]⁻¹𝐹^′(𝑥₀)‖⩽−((𝑠_𝑛−1−𝑡_𝑛−1)/(ℎ(𝑠_𝑛−1)−ℎ(𝑡_𝑛−1))). (iii)‖𝐹^′(𝑥₀)⁻¹𝐹(𝑥_𝑛)‖⩽ℎ(𝑡_𝑛)(‖𝑥_𝑛−𝑥_𝑛−1‖/(𝑡_𝑛−𝑡_𝑛−1))²(‖𝑦_𝑛−1−𝑥_𝑛−1‖/(𝑠_𝑛−1−𝑡_𝑛−1)).

Proof. For the case𝑛=1in (i), it is clear that‖𝑦₀−𝑥₀‖⩽𝛽=𝑠₀−𝑡₀. By Lemma 4 and (29), we have 𝐹^′𝑥₀⁻¹𝑦₀,𝑥₀;𝐹−𝐹^′𝑥₀=𝐹^′𝑥₀⁻¹×¹₀¹₀𝐹^′′𝑥₀+𝑠𝑡𝑦₀−𝑥₀𝑦₀−𝑥₀𝑡d𝑠d𝑡⩽¹₀¹₀2𝛾𝑠₀−𝑡₀𝑡1−𝛾𝑠₀−𝑡₀𝑠𝑡³d𝑠d𝑡=ℎ𝑠₀−ℎ𝑡₀𝑠₀−𝑡₀+1. (35) In view of the monotonicity ofℎ, one has that(ℎ(𝑠₀)−ℎ(𝑡₀))/(𝑠₀−𝑡₀)<0. Hence, we get from Banach lemma that[𝑦₀,𝑥₀;𝐹]⁻¹exists and satisfies 𝑦₀,𝑥₀;𝐹⁻¹𝐹^′𝑥₀⩽11−ℎ𝑠₀−ℎ𝑡₀/𝑠₀−𝑡₀𝑠+1=−⁰−𝑡₀ℎ𝑠₀−ℎ𝑡₀. (36) Combining (36) inequality with the definitions of{𝑠_𝑛}and{𝑡_𝑛}given in (13), one has 𝑥₁−𝑥₀⩽𝑦₀,𝑥₀;𝐹⁻¹𝐹^′𝑥₀𝐹^′𝑥₀⁻¹𝐹𝑥₀𝑠⩽−⁰−𝑡₀ℎ𝑠₀−ℎ𝑡₀ℎ𝑡₀=𝑡₁−𝑡₀. (37) As for the estimate‖𝑥₁−𝑦₀‖, by Lemma 4, we have 𝑥₁−𝑦₀=−𝐹^′𝑥₀⁻¹×¹₀¹₀𝐹^′′𝑥₀+𝑠𝑡𝑦₀−𝑥₀×𝑦₀−𝑥₀𝑥₁−𝑥₀𝑡d𝑠d𝑡. (38) This together with the obtained bounds‖𝑦₀−𝑥₀‖,‖𝑥₁−𝑥₀‖and (29) yields that 𝑥₁−𝑦₀⩽¹₀¹₀2𝛾𝑠₀−𝑡₀𝑡₁−𝑡₀1−𝛾𝑠₀−𝑡₀𝑠𝑡³𝑡d𝑠d𝑡×𝑦₀−𝑥₀𝑠₀−𝑡₀𝑥₁−𝑥₀𝑡₁−𝑡₀=𝑡₁−𝑠₀𝑦₀−𝑥₀𝑠₀−𝑡₀𝑥₁−𝑥₀𝑡₁−𝑡₀. (39) This implies that statement (i) holds for𝑛=1.
Statement (ii) for the case𝑛=1is justified by (36). Below, we consider the case𝑛=1for (iii). First we have the following expression of𝐹(𝑥₁)due to Lemma 4: 𝐹𝑥₁=¹₀¹₀𝐹^′′𝑥₀+𝑡𝑦₀−𝑥₀+𝑠𝑡𝑥₁−𝑦₀×𝑥₁−𝑥₀𝑥₁−𝑦₀𝑡d𝑠d𝑡, (40) from which we obtain that 𝐹^′𝑥₀⁻¹𝐹𝑥₁⩽¹₀¹₀2𝛾𝑡₁−𝑡₀𝑡₁−𝑠₀𝑡1−𝛾𝑡𝑠₀−𝑡₀+𝑠𝑡𝑡₁−𝑠₀³d𝑠d𝑡×𝑥₁−𝑥₀𝑡₁−𝑡₀²𝑦₀−𝑥₀𝑠₀−𝑡₀=ℎ𝑡₁𝑥₁−𝑥₀𝑡₁−𝑡₀²𝑦₀−𝑥₀𝑠₀−𝑡₀. (41) Therefore statement (iii) holds for𝑛=1.
Assume that statements (i)–(iii) are true for𝑛=𝑘(⩾1). Below, we will show that they also hold for𝑛=𝑘+1. First, by statement (i), we have 𝑥_𝑘−𝑥₀⩽^𝑘−1_𝑖=0𝑥_𝑖+1−𝑥_𝑖⩽^𝑘−1_𝑖=0𝑡_𝑖+1−𝑡_𝑖=𝑡_𝑘<𝑡^∗<𝑟₀. (42) Hence,𝐹^′(𝑥_𝑘)⁻¹exists by Lemma 1.
Note that 𝐹^′𝑥₀⁻¹𝐹𝑥_𝑘⩽ℎ𝑡_𝑘 (43) by the inductive hypotheses of (i) and (iii). Then it follows from (30) and (13) that 𝑦_𝑘−𝑥_𝑘⩽𝐹^′𝑥_𝑘⁻¹𝐹^′𝑥₀𝐹^′𝑥₀⁻¹𝐹𝑥_𝑘ℎ𝑡⩽−^𝑘ℎ^′𝑡_𝑘=𝑠_𝑘−𝑡_𝑘. (44) Hence by (29), (44), Lemma 4, and the inductive hypothesis of (i), we have 𝐹^′𝑥₀⁻¹𝑦_𝑘,𝑥_𝑘;𝐹−𝐹^′𝑥₀=𝐹^′𝑥₀⁻¹¹₀¹₀𝐹^′′𝑥₀+𝑠𝑥_𝑘−𝑥₀+𝑡𝑦_𝑘−𝑥_𝑘×𝑥_𝑘−𝑥₀+𝑡𝑦_𝑘−𝑥_𝑘d𝑠d𝑡⩽¹₀¹₀2𝛾1−𝛾𝑠𝑡_𝑘+𝑡𝑠_𝑘−𝑡_𝑘³×𝑡_𝑘+𝑡𝑠_𝑘−𝑡_𝑘d𝑠d𝑡=¹₀11−𝛾𝑡_𝑘+𝑡𝑠_𝑘−𝑡_𝑘²−1d𝑡=ℎ𝑠_𝑘−ℎ𝑡_𝑘𝑠_𝑘−𝑡_𝑘+1<1. (45) It follows from Banach lemma that[𝑦_𝑘,𝑥_𝑘;𝐹]⁻¹𝐹^′(𝑥₀)exists and satisfies
𝑦_𝑘,𝑥_𝑘;𝐹⁻¹𝐹^′𝑥₀⩽11−ℎ𝑠_𝑘−ℎ𝑡_𝑘/𝑠_𝑘−𝑡_𝑘𝑠+1=−^𝑘−𝑡_𝑘ℎ𝑠_𝑘−ℎ𝑡_𝑘. (46) Hence, (ii) holds for𝑛=k+1.
Combining (46) with the inductive hypothesis (iii), one has 𝑥_𝑘+1−𝑥_𝑘⩽𝑦_𝑘,𝑥_𝑘;𝐹⁻¹𝐹^′𝑥₀𝐹^′𝑥₀⁻¹𝐹𝑥_𝑘ℎ𝑠⩽−^𝑘−ℎ𝑡_𝑘𝑠_𝑘−𝑡_𝑘⁻¹ℎ𝑡_𝑘×𝑥_𝑘−𝑥_𝑘−1𝑡_𝑘−𝑡_𝑘−1²𝑦_𝑘−1−𝑥_𝑘−1𝑠_𝑘−1−𝑡_𝑘−1=𝑡_𝑘+1−𝑡_𝑘×𝑥_𝑘−𝑥_𝑘−1𝑡_𝑘−𝑡_𝑘−1²𝑦_𝑘−1−𝑥_𝑘−1𝑠_𝑘−1−𝑡_𝑘−1, (47) which implies that‖𝑥_𝑘+1−𝑥_𝑘‖⩽𝑡_𝑘+1−𝑡_𝑘.
On the other hand, by (29), (30), (44), and Lemma 4, we conclude that 𝑥_𝑘+1−𝑦_𝑘⩽¹₀¹₀𝐹^′𝑥_𝑘⁻¹𝐹^′′𝑥_𝑘+𝑠𝑡𝑦_𝑘−𝑥_𝑘×𝑦_𝑘−𝑥_𝑘𝑥_𝑘+1−𝑥_𝑘𝑡d𝑠d𝑡⩽𝐹^′𝑥_𝑘⁻¹𝐹^′𝑥₀×¹₀¹₀2𝛾𝑦_𝑘−𝑥_𝑘𝑥_𝑘+1−𝑥_𝑘𝑡1−𝛾𝑥_𝑘−𝑥₀+𝑠𝑡𝑦_𝑘−𝑥_𝑘³d𝑠d𝑡⩽1−𝛾𝑡_𝑘²21−𝛾𝑡_𝑘²×−1₁₀₁₀2𝛾𝑠_𝑘−𝑡_𝑘𝑡_𝑘+1−𝑡_𝑘𝑡1−𝛾𝑡_𝑘+𝑠𝑡𝑠_𝑘−𝑡_𝑘³d𝑠d𝑡×𝑦_𝑘−𝑥_𝑘𝑥_𝑘+1−𝑥_𝑘𝑠_𝑘−𝑡_𝑘𝑡_𝑘+1−𝑡_𝑘1=−ℎ^′𝑡_𝑘𝑡_𝑘+1−𝑡_𝑘ℎ𝑠_𝑘−ℎ𝑡_𝑘𝑠_𝑘−𝑡_𝑘−ℎ^′𝑡_𝑘×𝑦_𝑘−𝑥_𝑘𝑥_𝑘+1−𝑥_𝑘𝑠_𝑘−𝑡_𝑘𝑡_𝑘+1−𝑡_𝑘=𝑡_𝑘+1−𝑠_𝑘𝑦_𝑘−𝑥_𝑘𝑥_𝑘+1−𝑥_𝑘𝑠_𝑘−𝑡_𝑘𝑡_𝑘+1−𝑡_𝑘, (48) which leads to‖𝑥_𝑘+1−𝑦_𝑘‖⩽𝑡_𝑘+1−𝑠_𝑘. Thus, (i) holds for𝑛=𝑘+1.
Next, we will show that (iii) also holds for𝑛=𝑘+1. In fact, by using Lemma 4, (29), (44), and (48), we obtain 𝐹^′𝑥₀⁻¹𝐹𝑥_𝑘+1⩽¹₀¹₀2𝛾𝑡_𝑘+1−𝑡_𝑘𝑡_𝑘+1−𝑠_𝑘1−𝛾𝑡_𝑘+𝑡𝑠_𝑘−𝑡_𝑘+𝑠𝑡𝑡_𝑘+1−𝑠_𝑘³𝑡d𝑠d𝑡⋅𝑥_𝑘+1−𝑥_𝑘𝑡_𝑘+1−𝑡_𝑘²𝑦_𝑘−𝑥_𝑘𝑠_𝑘−𝑡_𝑘=ℎ𝑡_𝑘+1−ℎ𝑡_𝑘−𝑡_𝑘+1−𝑡_𝑘𝑠_𝑘−𝑡_𝑘ℎ𝑠_𝑘−ℎ𝑡_𝑘⋅𝑥_𝑘+1−𝑥_𝑘𝑡_𝑘+1−𝑡_𝑘²𝑦_𝑘−𝑥_𝑘𝑠_𝑘−𝑡_𝑘⩽ℎ𝑡_𝑘+1𝑥_𝑘+1−𝑥_𝑘𝑡_𝑘+1−𝑡_𝑘²𝑦_𝑘−𝑥_𝑘𝑠_𝑘−𝑡_𝑘. (49) Therefore statement (iii) holds for𝑛=𝑘+1. Hence (i)–(iii) hold for all𝑛⩾0. Furthermore, by statement (i), one has, for any𝑛⩾0,‖𝑥_𝑛−𝑥₀‖⩽𝑡_𝑛<𝑡^∗<𝑟₀. Thus by Lemma 1 we know that𝐹^′(𝑥_𝑛)⁻¹exists for each𝑛⩾1;thatis,{𝑥_𝑛}is well defined. The proof is complete.

Recall that{𝑠_𝑛}and{𝑡_𝑛}are defined in (13). Based on the preceding useful lemmas, we are now ready to prove a Smale-type semilocal convergence theorem for Newton-Steffensen’s method (4) under𝛾-condition.

Theorem 6. Suppose that (14) holds. Then the sequence{𝑥_𝑛}generated by (4) with the initial point𝑥₀is well defined and converges to a solution𝑥^∗∈𝐁(𝑥₀,𝑡^∗)of (1) with Q-cubic rate, and this solution𝑥^∗is unique in𝐁(𝑥₀,𝑟), where𝑡^∗⩽𝑟<𝑡^∗∗. Moreover, the following error bounds 𝑥^∗−𝑥_𝑛⩽𝑡^∗−𝑡_𝑛𝑥^∗−𝑥_𝑚𝑡^∗−𝑡_𝑚^3𝑛−𝑚∀𝑛⩾𝑚⩾0 (50) are valid, where𝑡^∗and𝑡^∗∗are defined in Lemma 2.

Proof. The uniqueness ball can be obtained by Theorem  5.2 in [19]. It follows from Lemma 1 that{𝑥_𝑛}is well defined. In addition, from Lemmas 3 and 5 (i), we can see that{𝑥_𝑛}is convergent to a limit, say𝑥^∗. Below, we show that𝑥^∗is a solution of (1). It follows from Lemma 5 (iii) that 𝐹^′𝑥₀⁻¹𝐹𝑥_𝑛⩽ℎ𝑡_𝑛∀𝑛⩾0. (51) Letting𝑛→∞in the preceding relation gives that the limit𝑥^∗is a solution of (1). Moreover, we have 𝑥^∗−𝑥_𝑛⩽𝑡^∗−𝑡_𝑛. (52)
Next, we verify that estimate (62) is true. By (29) and Lemma 5, one has 𝑥^∗−𝑦_𝑛=𝐹^′𝑥_𝑛⁻¹×¹₀𝐹^′𝑥_𝑛𝑥^∗−𝑥_𝑛d𝑡+𝐹𝑥_𝑛−𝐹𝑥^∗=𝐹^′𝑥_𝑛⁻¹×¹₀¹₀𝐹^′′𝑥_𝑛+𝑠𝑡𝑥^∗−𝑥_𝑛𝑥^∗−𝑥_𝑛²𝑡d𝑠d𝑡⩽𝐹^′𝑥_𝑛⁻¹𝐹^′𝑥₀×¹₀¹₀𝐹^′𝑥₀⁻¹𝐹^′′𝑥_𝑛+𝑠𝑡𝑥^∗−𝑥_𝑛×𝑥^∗−𝑥_𝑛²𝑡d𝑠d𝑡⩽1−𝛾𝑥_𝑛−𝑥₀²21−𝛾𝑥_𝑛−𝑥₀²×−1₁₀₁₀2𝛾𝑡^∗−𝑡_𝑛²𝑡1−𝛾𝑥_𝑛−𝑥₀+𝑠𝑡𝑥^∗−𝑥_𝑛³⩽𝑡𝑑𝑠𝑑𝑡_∗−𝑠_𝑛𝑥^∗−𝑥_𝑛𝑡^∗−𝑡_𝑛². (53) In order to estimate‖𝑥_𝑛+1−𝑥^∗‖, we first notice that
𝑦_𝑛,𝑥_𝑛;𝐹𝑥_𝑛+1−𝑥^∗=¹₀¹₀𝐹^′′𝑥_𝑛+𝑧_𝑛^𝑠,𝑡𝑥^∗−𝑦_𝑛𝑥^∗−𝑥_𝑛𝑡d𝑠d𝑡, (54) where𝑧_𝑛^𝑠,𝑡=𝑡(𝑦_𝑛−𝑥_𝑛)+𝑠𝑡(𝑥^∗−𝑦_𝑛). This together with Lemma 5(i), (29), (52), and (53) gives that 𝐹^′𝑥₀⁻¹𝑦_𝑛,𝑥_𝑛;𝐹𝑥_𝑛+1−𝑥^∗⩽¹₀¹₀2𝛾𝑥^∗−y_𝑛𝑥^∗−𝑥_𝑛𝑡1−𝛾𝑥_𝑛−𝑥₀+𝑡𝑦_𝑛−𝑥_𝑛+𝑠𝑡𝑥^∗−𝑦_𝑛³d𝑠d𝑡⩽¹₀¹₀2𝛾𝑡^∗−𝑠_𝑛𝑡^∗−𝑡_𝑛𝑡1−𝛾𝑡_𝑛+𝑡𝑠_𝑛−𝑡_𝑛+𝑠𝑡𝑡^∗−𝑠_𝑛³d𝑠d𝑡×𝑥^∗−𝑥_𝑛𝑡^∗−𝑡_𝑛³=−ℎ𝑡_𝑛−ℎ𝑠_𝑛−ℎ𝑡_𝑛𝑠_𝑛−𝑡_𝑛𝑡^∗−𝑡_𝑛𝑥^∗−𝑥_𝑛𝑡^∗−𝑡_𝑛³=𝑡_𝑛+1−𝑡^∗ℎ𝑠_𝑛−ℎ𝑡_𝑛𝑠_𝑛−𝑡_𝑛𝑥^∗−𝑥_𝑛𝑡^∗−𝑡_𝑛³. (55) Combining the above inequality with (46), we have 𝑥_𝑛+1−𝑥^∗⩽𝑦_𝑛,𝑥_𝑛;𝐹⁻¹𝐹^′𝑥₀×𝐹^′𝑥₀⁻¹𝑦_𝑛,𝑥_𝑛;𝐹𝑥_𝑛+1−𝑥^∗ℎ𝑠⩽−^𝑛−ℎ𝑡_𝑛𝑠_𝑛−𝑡_𝑛⁻¹𝑡_𝑛+1−𝑡^∗×ℎ𝑠_𝑛−ℎ𝑡_𝑛𝑠_𝑛−𝑡_𝑛𝑥^∗−𝑥_𝑛𝑡^∗−𝑡_𝑛³⩽𝑡^∗−𝑡_𝑛+1𝑥^∗−𝑥_𝑛𝑡^∗−𝑡_𝑛³. (56) Therefore, the error estimate (62) follows. Also, from the previous inequality, we know that the convergence rate of{𝑥_𝑛}to𝑥^∗is 𝑄-cubic. This completes the proof.

One typical and important class of examples satisfying𝛾-condition is the one of analytic functions. Smale [5] studied the convergence and error estimation of Newton’s method (2) under the hypotheses that𝐹is analytic and satisfies 𝐹^′(𝑥)⁻¹𝐹^(𝑛)(𝑥)⩽𝑛!𝛾^𝑛−1,𝑛⩾2, (57) where𝑥is a fixed point in𝐷and𝛾is defined by 𝛾∶=sup_𝑛>1𝐹^′(𝑥)⁻¹𝐹^(𝑛)(𝑥)𝑛!^1/(𝑛−1). (58)

The following lemma taken from [20] shows that an analytic operator satisfies𝛾-condition.

Lemma 7. Let𝑟₀and𝛾be defined by (7) and (86), respectively. Suppose that𝐹is analytic and satisfies (85). Then𝐹satisfies𝛾-condition 𝐹^′(𝑥)⁻¹𝐹^′′(𝑥)⩽2𝛾1−𝛾‖𝑥−𝑥‖³ (59) on𝐵(𝑥,𝑟₀).

According to this lemma, we can conclude that the semilocal results obtained in Theorem 6 also hold when𝐹is an analytic operator.

Corollary 8. Suppose that (14) holds. Suppose that𝐹is analytic and satisfies 𝐹^′𝑥₀⁻¹𝐹^(𝑛)𝑥₀⩽𝑛!𝛾^𝑛−1,𝑛⩾2, (60) where 𝛾 is defined by 𝛾∶=sup_𝑛>1𝐹^′𝑥₀⁻¹𝐹^(𝑛)𝑥₀𝑛!^1/(𝑛−1). (61) Then the sequence{𝑥_𝑛}generated by (4) with the initial point𝑥₀is well defined and converges to a solution𝑥^∗∈𝐁(𝑥₀,𝑡^∗)of (1) with Q-cubic rate and this solution𝑥^∗is unique in𝐁(𝑥₀,𝑟), where𝑡^∗⩽𝑟<𝑡^∗∗. Moreover, the following error bounds 𝑥^∗−𝑥_𝑛⩽𝑡^∗−𝑡_𝑛𝑥^∗−𝑥_𝑚𝑡^∗−𝑡_𝑚^3𝑛−𝑚∀𝑛⩾𝑚⩾0 (62) are valid, where𝑡^∗and𝑡^∗∗are defined in Lemma 2.

4. Convergence Ball

Now we begin to study the local convergence properties for Newton-Steffensen’s method (4) under𝛾-condition. Recall that𝑟₀is defined by (7). Throughout this section, suppose that𝑥^∗∈𝐷such that𝐹(𝑥^∗)=0,𝐁(𝑥^∗,𝑟₀)⊂𝐷, and the inverse𝐹^′(𝑥^∗)⁻¹exists. Moreover, we assume that𝐹satisfies the𝛾-condition on𝐁(𝑥^∗,𝑟₀); that is, the following relation holds: 𝐹^′𝑥^∗−1𝐹^′′(𝑥)⩽2𝛾1−𝛾‖𝑥−𝑥^∗‖³,𝑥∈𝐁𝑥^∗,𝑟₀. (63) Then, for any𝑥∈𝐁(𝑥^∗,𝑟₀), it follows from Lemma 1 that 𝐹^′(𝑥)⁻¹𝐹^′𝑥^∗⩽1−𝛾𝑥−𝑥^∗221−𝛾‖𝑥−𝑥^∗‖².−1 (64)

Let 𝑟₁√∶=5−17.4𝛾 (65) Define function𝐺as follows: 𝐺(𝑡)=𝛾𝑡1−4𝛾𝑡+2𝛾²𝑡²,𝑡∈0,𝑟₁. (66) It is clear that𝑟₁∈(0,𝑟₀)and that𝐺(𝑟₁)=1. Moreover,𝐺increases monotonically in(0,𝑟₁).

Theorem 9. Let𝑟₁be defined in (65). Then, for any𝑥₀∈𝐵(𝑥^∗,𝑟₁), the sequence{𝑥_𝑛}generated by (4) converges to𝑥^∗and satisfies that 𝑥_𝑛−𝑥^∗⩽𝑞^3𝑛−1𝑥₀−𝑥^∗,𝑛=0,1,…, (67) where 𝑞=𝐺𝑡₀<1,𝑡₀=𝑥₀−𝑥^∗. (68)

Proof. For𝑛=0,1,…, we write𝑡_𝑛=‖𝑥_𝑛−𝑥^∗‖. It is sufficient to show that 𝑡_𝑛+1⩽𝑡_𝑛,𝑥_𝑛+1−𝑥^∗⩽𝐺𝑡_𝑛𝑡_𝑛²𝑥_𝑛−𝑥^∗3,𝑛=0,1,…. (69) In fact, by noticing the monotonicity of𝐺/𝑡, we have
𝑥_𝑛+1−𝑥^∗⩽𝐺𝑡_𝑛𝑡_𝑛²𝑥_𝑛−𝑥^∗3⩽𝐺(𝑡₀)𝑡₀²𝑥_𝑛−𝑥^∗3=𝑞𝑡₀²𝑥_𝑛−𝑥^∗3,𝑛=0,1,…. (70) From this we can easily establish (67) by mathematical induction.
We now prove (69). First we get the following expression of𝑥_𝑛+1−𝑥^∗: 𝑥_𝑛+1−𝑥^∗=𝑦_𝑛,𝑥_𝑛;𝐹⁻¹×¹₀𝐹^′𝑥_𝑛+𝑡𝑦_𝑛−𝑥_𝑛𝑥_𝑛−𝑥^∗d𝑡−𝐹𝑥_𝑛−𝐹𝑥^∗=𝑦_𝑛,𝑥_𝑛;𝐹⁻¹×¹₀¹₀𝐹^′′𝑥^∗+(1−𝑡)𝑥_𝑛−𝑥^∗+𝑠𝑡𝑦_𝑛−𝑥^∗×𝑦_𝑛−𝑥^∗𝑥_𝑛−𝑥^∗𝑡d𝑠d𝑡. (71) Similarly, we also have 𝑦_𝑛−𝑥^∗=𝑥_𝑛−𝐹^′𝑥_𝑛⁻¹𝐹^′𝑥_𝑛−𝑥^∗=𝐹^′𝑥_𝑛⁻¹×𝐹𝑥^∗−𝐹𝑥_𝑛+𝐹^′𝑥_𝑛𝑥_𝑛−𝑥^∗=𝐹^′𝑥_𝑛⁻¹×¹₀𝐹^′𝑥_𝑛+𝑡𝑥^∗−𝑥_𝑛−𝐹^′𝑥_𝑛×𝑥^∗−𝑥_𝑛d𝑡=𝐹^′𝑥_𝑛⁻¹×¹₀¹₀𝐹^′′𝑥_𝑛+𝑠𝑡𝑥^∗−𝑥_𝑛𝑥^∗−𝑥_𝑛²𝑡d𝑠d𝑡. (72) This together with (63) and (64) yields 𝑦_𝑛−𝑥^∗⩽𝐹^′𝑥_𝑛⁻¹𝐹^′𝑥^∗×¹₀¹₀𝐹^′𝑥^∗−1𝐹^′′𝑥_𝑛+𝑠𝑡𝑥^∗−𝑥_𝑛𝑡d𝑠d𝑡×𝑥^∗−𝑥_𝑛²⩽11−𝛾𝑥^∗−𝑥_𝑛²×−2₁₀₁₀2𝛾1−𝛾𝑥_𝑛+𝑠𝑡𝑥^∗−𝑥_𝑛−𝑥^∗3d𝑠d𝑡×𝑥^∗−𝑥_𝑛²=11−𝛾𝑡_𝑛²×−2₁₀₁₀2𝛾1−𝛾(1−𝑠𝑡)𝑡_𝑛³d𝑠d𝑡×𝑥^∗−𝑥_𝑛²=𝛾1−4𝛾𝑡_𝑛+2𝛾²𝑡²_𝑛𝑥^∗−𝑥_𝑛²=𝐺𝑡_𝑛𝑡_𝑛𝑥^∗−𝑥_𝑛². (73) On the other hand, we notice that𝐹^′𝑥^∗−1𝑦_𝑛,𝑥_𝑛;𝐹−𝐹^′𝑥^∗=𝐹^′𝑥^∗−1×¹₀¹₀𝐹^′′𝑥^∗+𝑠(1−𝑡)𝑥_𝑛−𝑥^∗+𝑠𝑡𝑦_𝑛−𝑥^∗×(1−𝑡)𝑥_𝑛−𝑥^∗+𝑡𝑦_𝑛−𝑥^∗d𝑠d𝑡. (74) It follows from (63) that 𝐹^′𝑥^∗−1𝑦_𝑛,𝑥_𝑛;𝐹−𝐹^′𝑥^∗⩽¹₀¹₀2𝛾(1−𝑡)𝑥_𝑛−𝑥^∗+𝑡𝑦_𝑛−𝑥^∗1−𝛾𝑠(1−𝑡)𝑥_𝑛−𝑥^∗+𝑠𝑡𝑦_𝑛−𝑥^∗3d𝑠d𝑡. (75) For the case𝑛=0, by (88) and (73), we get 𝑦₀−𝑥^∗⩽𝐺𝑡₀𝑥^∗−𝑥₀⩽𝑥^∗−𝑥₀. (76) Combining Lemma 2 with (75) and (76), we obtain 𝐹^′𝑥^∗−1𝑦₀,𝑥₀;𝐹−𝐹^′𝑥^∗⩽¹₀¹₀2𝛾(1−𝑡)𝑥₀−𝑥^∗+𝑡𝑦₀−𝑥^∗1−𝛾𝑠(1−𝑡)𝑥₀−𝑥^∗+𝑠𝑡𝑦₀−𝑥^∗3d𝑠d𝑡⩽¹₀¹₀2𝛾(1−𝑡)𝑡₀+𝑡𝑡₀1−𝛾𝑠(1−𝑡)𝑡₀+𝑠𝑡𝑡₀³d𝑠d𝑡=11−𝛾𝑡₀²−1<1. (77) It follows from Banach lemma that 𝑦₀,𝑥₀;𝐹⁻¹𝐹^′𝑥^∗⩽11−1/1−𝛾𝑡₀²=−11−𝛾𝑡₀²21−𝛾𝑡₀².−1 (78) This together with (63), (71) and (76) yields 𝑥₁−𝑥^∗⩽𝑦₀,𝑥₀;𝐹⁻¹𝐹^′𝑥^∗×𝐹^′𝑥^∗−1×¹₀¹₀𝐹^′′𝑥^∗+(1−𝑡)𝑥₀−𝑥^∗+𝑠𝑡𝑦₀−𝑥^∗𝑡d𝑠d𝑡×𝑦₀−𝑥^∗𝑥₀−𝑥^∗⩽1−𝛾𝑡₀²21−𝛾𝑡₀²×−1₁₀₁₀2𝛾1−𝛾(1−𝑡)𝑡₀+𝑠𝑡𝑡₀³𝑡d𝑠d𝑡×𝐺𝑡₀𝑥^∗−𝑥₀³=1−𝛾𝑡₀²21−𝛾𝑡₀²1−1𝑡₀11−𝛾𝑡₀²−11−𝛾𝑡₀×𝐺𝑡₀𝑥^∗−𝑥₀³=𝐺𝑡₀𝑡₀²𝑥^∗−𝑥₀³. (79) Hence (69) holds for𝑛=0.
Now assume that the inequalities in (69) hold for up to some𝑛⩾1. Then by (73), one has 𝑦_𝑛+1−𝑥^∗=𝐺𝑡_𝑛+1𝑡_𝑛+1⩽𝐺𝑡₀𝑡_𝑛+1⩽𝑡_𝑛+1. (80) Thus (75) can be further reduced to 𝐹^′𝑥^∗−1𝑦_𝑛+1,𝑥_𝑛+1;𝐹−𝐹^′𝑥^∗⩽¹₀¹₀2𝛾(1−𝑡)𝑡_𝑛+1+𝑡𝑡_𝑛+11−𝛾𝑠(1−𝑡)𝑡_𝑛+1+𝑠𝑡𝑡_𝑛+1³d𝑠d𝑡=11−𝛾𝑡_𝑛+1²−1<1. (81) Using Banach lemma again, one has 𝑦_𝑛+1,𝑥_𝑛+1;𝐹⁻¹𝐹^′𝑥^∗⩽1−𝛾𝑡_𝑛+1²21−𝛾𝑡_𝑛+1².−1 (82) This together with (63), (71), and (73) yields 𝑥_𝑛+2−𝑥^∗⩽𝑦_𝑛+1,𝑥_𝑛+1;𝐹⁻¹𝐹^′𝑥^∗×𝐹^′𝑥^∗−1⋅¹₀¹₀𝐹^′′𝑥^∗+(1−𝑡)𝑥_𝑛+1−𝑥^∗+𝑠𝑡𝑦_𝑛+1−𝑥^∗𝑡d𝑠d𝑡×𝑦_𝑛+1−𝑥^∗𝑥_𝑛+1−𝑥^∗⩽1−𝛾𝑡_𝑛+1²21−𝛾𝑡_𝑛+1²×−1₁₀₁₀2𝛾1−𝛾(1−𝑡)𝑡_𝑛+1+𝑠𝑡𝑡_𝑛+1³𝑡d𝑠d𝑡×𝐺𝑡_𝑛+1𝑥^∗−𝑥₀³=1−𝛾𝑡_𝑛+1²21−𝛾𝑡_𝑛+1²×1−1𝑡_𝑛+111−𝛾𝑡_𝑛+1²−11−𝛾𝑡_𝑛+1×𝐺𝑡_𝑛+1𝑥^∗−𝑥₀³=𝐺𝑡_𝑛+1𝑡_𝑛+1²𝑥^∗−𝑥_𝑛+1³. (83) Thus 𝑡_𝑛+2≤𝐺𝑡_𝑛+1²𝑡_𝑛+1≤𝐺𝑡₀²𝑡_𝑛+1≤𝑡_𝑛+1. (84) This and (83) show that the inequalities in (69) hold for𝑛+1and hence they hold for each𝑛. The proof is complete.

Applying Lemma 7 to the above theorem, we get the following corollary:

Corollary 10. Suppose that𝐹is analytic and satisfies 𝐹^′𝑥^∗−1𝐹^(𝑛)𝑥^∗⩽𝑛!𝛾^𝑛−1,𝑛⩾2, (85) where 𝛾 is defined by 𝛾∶=sup_𝑛>1𝐹^′𝑥^∗−1𝐹^(𝑛)𝑥^∗𝑛!^1/(𝑛−1). (86) Let𝑟₁and𝐺be defined by (65) and (66), respectively. Then, for any𝑥₀∈𝐵(𝑥^∗,𝑟₁), the sequence{𝑥_𝑛}generated by (4) converges to𝑥^∗and satisfies that 𝑥_𝑛−𝑥^∗⩽𝑞^3𝑛−1𝑥₀−𝑥^∗,𝑛=0,1,…, (87) where 𝑞=𝐺𝑡₀<1,𝑡₀=𝑥₀−𝑥^∗. (88)

Conflict of Interests

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

Acknowledgment

Xiubin Xu’s work was supported in part by the National Natural Science Foundation of China (Grant no. 61170109).