CUBO, A Mathematical Journal
Vol.22, No¯ 01, (55–70). April 2020

http: // doi. org/ 10. 4067/ S0719-06462020000100055

Super-Halley method under majorant conditions in Banach
spaces

Shwet Nisha and P. K. Parida

Department of Applied Mathematics,

Central University of Jharkhand,

Ranchi-835205, India.

shwetnisha1988@gmail.com, pkparida@cuj.ac.in

ABSTRACT

In this paper, we have studied local convergence of Super-Halley method in Banach

spaces under the assumption of second order majorant conditions. This approach allows

us to obtain generalization of earlier convergence analysis under majorizing sequences.

Two important special cases of the convergence analysis based on the premises of Kan-

torovich and Smale type conditions have also been concluded. To show efficacy of our

approach we have given three numerical examples.

RESUMEN

En este art́ıculo, hemos estudiado la convergencia local del método Super-Halley en

espacios de Banach, asumiendo condiciones mayorantes de segundo orden. Este punto

de vista nos permite obtener generalizaciones de análisis de convergencia bajo sucesiones

mayorantes obtenidos anteriormente. También se han concluido dos casos especiales

del análisis de convergencia basados en las premisas de condiciones tipo Kantorovich y

Smale. Para mostrar la eficacia de nuestro enfoque, damos tres ejemplos numéricos.

Keywords and Phrases: Nonlinear equations; Super-Halley method; Majorant conditions; Local

Convergence; Semilocal Convergnce; Smale-type conditions; Kantorovich-type conditions.

2010 AMS Mathematics Subject Classification: 65D10, 65G99, 65K10, 47H17, 49M15,

47H99.

http://doi.org/10.4067/S0719-06462020000100055


56 Shwet Nisha & P. K. Parida CUBO
22, 1 (2020)

1 Introduction

Let f be a given operator that maps from some nonempty open convex subset Ω of a Banach space

X to another Banach space Y. Approximating a locally unique solution x̄ of a nonlinear equation

f(x) = 0 (1.1)

is widely studied in both theoretical and applied areas of mathematics. Generally, this is done by

using some iterative processes. An iterative process is a mathematical procedure that, from one or

several initial approximations of a solution of (1.1), a sequence of iterates {xn}n∈N is constructed

so that each subsequent iterate of the sequence is a better approximation to the previous approxi-

mation to the solution of (1.1); that is, the sequence {‖xn − x̄‖}n∈N is convergent to zero. Usually,
in order to study convergence analysis of the method, we could consider the study of local and

semilocal convergence analysis. If the convergence analysis seeks assumptions around a solution

x̄, then it is called local convergence and this type of analysis estimates the radii of convergence

balls, where as we could also consider assumptions around an initial point x0 to study convergence

analysis of the method. In that case the convergence analysis is called semilocal one and it gives

criteria ensuring the convergence. It is also very important to give convergence ball of an iterative

method, because that shows the extent to which we can choose an initial guesses for that method.

One of the most important iterative methods to solve this problem is Newton’s method given

by

xn+1 = xn − f
′(xn)

−1f(xn), k = 0, 1, 2, . . . (1.2)

where x0 ∈ Ω is an initial point. As anybody can recall that one of the most famous results to
study convergence of Newton’s method (1.2) is the well known Kantorovich method[16], which

guarantees convergence of the method to a solution, using semilocal conditions. It does not require

a priori existence of a solution, proving instead the existence of the solution and its uniqueness

on some region. Many researches have also done works related to Kantorovich-like method (for

details see [4, 8, 10, 26, 27, 29] and references there in). Also, Smale’s point theory [21] assumes

that the nonlinear operator is analytic at the initial point, which is an important result concerning

Newton’s method. Wan and Han[25, 22] has discussed the generalization and the particular cases

of Smale’s point estimate theory.

For a positive number α and x ∈ X, we consider B(x, α) to stand for the open ball with radius
α and center x and B̄(x, α) is the corresponding close ball. In [7, 6], Ferreira and Svaiter had

studied the local convergence of Newton’s method (1.2) under the following majorant conditions:

‖f′(x̄)−1[f′(y) − f′(x)]‖ ≤ h′(‖y − x‖ + ‖x − x̄‖) − h′(‖x − x̄‖), (1.3)

for x, y ∈ B(x̄, R), R > 0, where ‖y−x‖ +‖x− x̄‖ < R and h : (0, R) → R is a continuously differen-
tiable, convex and strictly increasing function that satisfies h(0) > 0, h′(0) = −1 and has a zero in



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Super-Halley method under majorant conditions in Banach spaces 57

(0, R). Note that by this study the assumptions for guaranteeing Q-quadratic convergence of the

respective iterative methods has been relaxed and a new estimate of the Q-quadratic convergence

has been obtained.

Recently, inspired by these ideas, Ling and Xu[17] have presented a new convergence analysis of

Halley’s method which makes a relationship of the majorizing function h and the nonlinear operator

f under majorant conditions similar to given in (1.3). Argyros and Ren[1] also presented a local

convergence of Halley’s method which gives a ball convergence of the method under assumptions

similar to (1.3).

On the other hand, one of the famous third order iterative method to solve nonlinear equation

(1.1) in Banach space is the Super-Halley method denoted by

xn+1 = xn −
[

I +
1

2
Lf(xn)[I − Lf(xn)]

−1
]

f′(xn)
−1f(xn), (1.4)

where for x ∈ X, Lf(x) is the linear operator defined as

Lf(x) = f
′(x)−1f′′(x)f′(x)−1f(x).

The results concerning the convergence of this method have been studied in [2, 9, 20] under

different types of assumptions by using recurrence relations. On the other hand Ezquerro and

Hernández[5] and Gutiérrez and Hernández[11] have studied semilocal convergence of this method

by using majorizing sequences. Now, if the nonlinear operator f is analytic at the initial point

then motivated by the ideas of Argyros and Ren[1] and Ling and Xu[17], we have studied local

convergence of Super-Halley method (1.4) using second order majorant condition. This majorant

condition generalizes the earlier results on Super-Halley method[5, 11] using majorizing sequences.

Two particular cases namely results based of affine invariant Lipschitz-type condition and Smale-

type condition have also been derived. Numerical efficacy of the method has also been derived by

way of a number of numerical examples.

Rest of the paper is organized as follows. Some preliminaries results are contained in section

2. In section 2.1, we studied local convergence analysis of Super-Halley method. Two special cases

of main result are presented in section 3. In section 4, we have shown a number of numerical

examples to show efficacy of our study. Section 5 forms the conclusion part of the paper.

2 Preliminaries

In this section we provide some basic results which is required for our convergence analysis of the

method.

Assume a > 0 and φ : (0, a) → R be a twice continuously differentiable function. Let

x, y ∈ B(x̄, a) ⊂ Ω, with ‖y − x‖ + ‖x − x̄‖ < a. We say that the operator f satisfy a second order



58 Shwet Nisha & P. K. Parida CUBO
22, 1 (2020)

majorizing function φ at x̄ if the following conditions hold on f:

‖f′(x̄)−1[f′′(y) − f′′(x)]‖ ≤ φ′′(‖y − x‖ + ‖x − x̄‖) − φ′′(‖x − x̄‖), (2.1)

with the assumptions:

(M1) φ(0) > 0, φ′′(0) > 0, φ′(0) = −1,

(M2) φ′′ is convex and strictly increasing in (0, a),

(M3) φ has atleast one zero in (0, a) with t∗ as the smallest zero and φ′(t∗) < 0.





(2.2)

and

‖f′(x̄)−1f(x̄)‖ ≤ φ(0), ‖f′(x̄)−1f′′(x̄)‖ ≤ φ′′(0). (2.3)

In this paper we assume that φ is the majorizing function of f. Note that if we define

Θf(x) := x −
[

I +
1

2
Lf(x)[I − Lf(x)]

−1
]

f′(x)−1f(x) (2.4)

where Lf(x) = f
′(x)−1f′′(x)f′(x)−1f(x), then Θf(x) can be taken as the iterative function of Super-

Halley method as it can be written as xn+1 = Θf(xn). Also the scalar sequence {tn} can be

generated by applying the method to φ(t). In this case we can write tn+1 = Θφ(tn) with

Θφ(t) := t −
[

1 +
Lφ(t)

2(1 − Lφ(t))

]

φ(t)

φ′(t)
, Lφ(t) =

φ(t)φ′′(t)

φ′(t)2
, t ∈ (0, a). (2.5)

Now we can easily establish some basic properties of the majorizing function φ, the iterative

functions Θf(x) and Θφ(t) which are described in the following lemmas.

Lemma 2.1. Let φ satisfies assumptions (M1) − (M3). Then

(i) φ′ is strictly convex and strictly increasing on (0, a).

(ii) φ is strictly convex on (0, a), φ(t) > 0 for t ∈ (0, t∗) and equation φ(t) = 0 has at most one
root in (t∗, a).

(iii) −1 < φ′(t) < 0 for t ∈ (0, t∗).

(iv) 0 ≤ Lφ(t) ≤
1

2
for t ∈ [0, t∗].

Proof. The proof is similar to one given in [17], so omitted.

Lemma 2.2. Let φ satisfies assumptions (M1)−(M3). Then for all t ∈ (0, t∗), t < Θφ(t) < t∗.
Moreover, φ′(t∗) < 0 if and only if there exist t ∈ (t∗, a) such that φ(t) < 0.



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Super-Halley method under majorant conditions in Banach spaces 59

Proof. It is not to be mentioned that by using Lemma 2.1, one can have φ(t) > 0, −1 <

φ′(t) < 0 and 0 ≤ Lφ(t) ≤ 12 for t ∈ (0, t
∗). This gives

[

1 +
Lφ(t)

2(1 − Lφ(t))

]

φ(t)

φ′(t)
< 0 and hence

t < Θφ(t).

Also, for any t ∈ (0, t∗), from the definition of directional derivative and assumption (M2)
it follows that since φ′′(t) is increasing in (0, a) and t < t∗ < a, we have φ′′(t) < φ′′(t∗) and

φ′′(t) > 0 which implies that left directional derivative D−φ′′(t) > 0.

As φ′′(t)φ(t)−2φ′(t)2 ≤ −4φ′′(t)φ(t), we obtain D−Θφ(t) = 1+
2φ′(t)φ(t)D−φ′′(t)

(φ′′(t)φ(t) − 2φ′(t)2)φ′′(t)
>

0 for t ∈ (0, t∗).
And this implies that Θφ(t) < Θφ(t

∗) = t∗ for any t ∈ (0, t∗). So the first part of this lemma
is complete. Now, if φ′(t∗) < 0, then it is obvious that there exists t ∈ (t∗, a) such that φ(t) < 0.
Conversely, noting that φ′(t∗) = 0, then we have φ(t) > φ(t∗)+φ′(t)(t∗ −t) for t ∈ (t∗, a), which
implies that φ′(t∗) < 0. This completes the proof.

Remark Following properties are implied by the condition φ′(t∗) < 0 in (M3):

• φ(t∗∗) = 0 for some t∗∗ ∈ (t∗, a).

• φ(t) < 0 for some t ∈ (t∗, a).

Lemma 2.3. Let φ satisfies assumptions (M1) − (M3). Then

t∗ − Θφ(t) ≤
[

1

2

φ′′(t∗)2

φ′(t∗)2
+

1

3

D−φ′′(t∗)

−φ′(t∗)

]

(t∗ − t)3, t ∈ (0, t∗). (2.6)

Proof. We can derive the following relation, by using the definition of Θφ in (2.5)

t∗ − Θφ(t) =
1

1 − Lφ(t)

[

(1 − Lφ(t))(t
∗ − t) +

φ(t)

2φ′(t)
(1 − Lφ(t)) +

φ(t)

2φ′(t)

]

= −
1

φ′(t)(1 − Lφ(t)

∫1

0

[φ′′(t + σ(t∗ − t)) − φ′′(t)](t∗ − t)2(1 − σ)dσ

+
(t∗ − t)φ′′(t)

2(1 − Lφ(t))φ
′(t)2

∫1

0

φ′′(t + σ(t∗ − t))(t∗ − t)2(1 − σ)dσ

−
(t∗ − t)φ′′(t)

2(1 − Lφ(t))φ
′(t)

(

φ(t) +
φ(t)2

φ′(t)(t∗ − t)

)

Since φ′′ is convex and t < t∗, it follows from Lemma 2.1 that

φ′′(t + σ(t∗ − t)) − φ′′(t) ≤ [φ′′(t∗) − φ′′(t)]σ(t
∗ − t)

(t∗ − t)



60 Shwet Nisha & P. K. Parida CUBO
22, 1 (2020)

So by noting that φ′′ is strictly increasing, we have

t∗ − Θφ(t) ≤ −
φ′′(t∗) − φ′′(t)

6φ′(t)(1 − Lφ(t))
(t∗ − t)2 +

φ′′(t∗)φ′′(t)

4φ′(t)2(1 − Lφ(t))
(t∗ − t)3

−
φ(t)φ′′(t)

2(1 − Lφ(t))φ
′(t)

(t∗ − t) −
φ(t)2φ′′(t)

2(1 − Lφ(t))φ
′(t)2

Since φ′(t) < 0, φ′′(0) > 0 and φ′, φ′′ are strictly increasing on (0, t∗) and 0 ≤ Lφ(t) ≤ 12 for
t ∈ [0, t∗] by Lemma 2.1, so we have

t∗ − Θφ(t) ≤
[

1

2

φ′′(t∗)2

φ′(t∗)2
+

1

3

D−φ′′(t∗)

−φ′(t∗)

]

(t∗ − t)3. (2.7)

As φ′ is increasing, φ′(t∗) < 0 and φ′(t) < 0 t in (0, t∗), we have

φ′′(t∗)) − φ′′(t)

−φ′(t)
≤ φ

′′(t∗) − φ′′(t)

−φ′(t∗)
=

1

−φ′(t∗)

φ′′(t∗)) − φ′′(t)

t∗ − t
(t∗ − t) ≤ D

−φ′′(t∗)

−φ′(t∗)
(t∗ − t)

where the last inequality follows from definitions of directional derivative. Combining the above

inequality with (2.7), we conclude that (2.6) holds. This completes the proof. �

Let {tk} denote the majorizing sequence generated by

t0 = 0, tk+1 = Θφ(tk) = tk −
[

I +
Lφ(tk)

2(1 − Lφ(tk))

]

φ(tk)

φ′(tk)
, k = 0, 1, 2, . . . (2.8)

We arrive at the following theorem using Lemma 2.3 that

Theorem 2.4. Let the sequence {tk} be defined by (2.8). Then {tk} is well defined, strictly
increasing and is contained in (0, t∗). Moreover, {tk} satisfies (2.6) and converges to t∗ with
Q − cubic, i.e.:

t∗ − tk+1 ≤
[

1

2

φ′′(t∗)2

φ′(t∗)2
+

1

3

D−φ′′(t∗)

−φ′(t∗)

]

(t∗ − tk)
3, tk ∈ (0, t∗).

2.1 Local convergence results for Super-Halley method

This section is devoted to giving the local convergence analysis of (1.4). For that the following

lemmas will play important role.

Lemma 2.5. Assume ‖x − x̄‖ ≤ t < t∗. If φ : (0, t∗) → R is a twice continuously differentiable
function which majorizes f at x̄, then

(i) f′(x) is nonsingular and

‖f′(x)−1f′(x̄)‖ ≤ − 1
φ′(‖x − x̄‖)

≤ − 1
φ′(t)

. (2.9)



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Super-Halley method under majorant conditions in Banach spaces 61

(ii) ‖f′(x̄)−1f′′(x)‖ ≤ φ′′(‖x − x̄‖) ≤ φ′′(t).

Proof. Let us take x ∈ B(x̄, t), 0 ≤ t < t∗. Since

f′(x) = f′(x̄) +

∫1

0

[f′′(x̄ + σ(x − x̄)) − f′′(x̄)](x − x̄)dσ + f′′(x̄)(x − x̄),

we get

‖I − f′(x̄)−1f′(x)‖ ≤
∫1

0

‖f′(x̄)−1[f′′(x̄ + σ(x − x̄)) − f′′(x̄)]‖‖x − x̄‖dσ

+‖f′(x̄)−1f′′(x̄)‖‖x − x̄‖

≤
∫1

0

[φ′′(σ(‖(x − x̄)‖)) − φ′′(0)]‖x − x̄‖dσ + φ′′(0)‖x − x̄‖

= φ′(‖(x − x̄)‖) − φ′(0).

So, we conclude that

‖I − f′(x̄)−1f′(x)‖ ≤ φ′(t) − φ′(0) < 1

as φ′(0) = −1 and −1 < φ′(t) < 0 for (0, t∗) using Lemma 2.1. Therefore, it follows from Banach

lemma that f′(x̄)−1f′(x) is nonsingular and (2.9) holds as

‖f′(x)−1f′(x̄)‖ ≤ 1
1 − φ′(‖x − x̄‖) + φ′(0)

= −
1

φ′(‖x − x̄‖)
≤ − 1

φ′(t)
.

Thus we conclude that, f′ is nonsingular in B(x̄, t∗). By using majorant conditions, we have

‖f′(x̄)−1f′′(x)‖ ≤ ‖f′(x̄)−1[f′′(x) − f′′(x̄)]‖ + ‖f′(x̄)−1f′′(x̄)‖
≤ φ′′(‖x − x̄‖) − φ′′(0) + φ′′(0) = φ′′(‖x − x̄‖) ≤ φ′′(t).

The last inequality holds true because of φ′′ is strictly increasing. This completes the proof. ✷

Now the main local convergence result for the Super-Halley method (1.4) is presented as

follows.

Theorem 2.6. Let f satisfies the second order majorant conditions (2.1)-(2.3). Then, the

sequence of iterates {xn} generated by Super-Halley method (1.4) is well defined, contained in

B(x̄, t∗) and converges to the unique solution x̄ of (1.1). Moreover, the following error estimate

hold

‖x̄ − xk+1‖ ≤ (t∗ − tk+1)
(

‖x̄ − xk‖
t∗ − tk

)3

, k = 0, 1, 2, . . . (2.10)

Thus the sequence {xn} generated by Super-Halley method (1.4) converges Q−cubic as follows

‖x̄ − xk+1‖ ≤
[

1

2

h′′(t∗)2

h′(t∗)2
+

1

3

D−h′′(t∗)

−h′(t∗)

]

‖x̄ − xk‖3, k = 0, 1, 2, . . . (2.11)



62 Shwet Nisha & P. K. Parida CUBO
22, 1 (2020)

Proof. By using f(x̄) = 0 and some standard analytic techniques, one can have

x̄ − xk+1 = −Γf(xk)f
′
(xk)

−1
[−f′(xk)(x̄ − xk) − f(xk)] − Γf(xk)Lf(xk)(x̄ − xk)

+
[

1

2
f′(xk)

−1f(xk) −
1

2
Γf(xk)f

′(xk)
−1f(xk)

]

= −Γf(xk)f
′(xk)

−1

∫1

0

(1 − σ)[f′′(xk + σ(x̄ − xk)) − f
′′(xk)](x̄ − xk)

2dσ

+f′′(xk)Γf(xk)f
′(xk)

−1
[

f′(xk)
−1

∫1

0

(1 − σ)f′′(xk + σ(x̄ − xk))

×(x̄ − xk)2dσ
]

(x̄ − xk)

−
1

2
f′′(xk)Γf(xk)f

′
(xk)

−1
[

f′(xk)
−1

∫1

0

(1 − σ)f′′(xk + σ(x̄ − xk))

×(x̄ − xk)2dσ
]2

,

where Γf(x) = (I − Lf(x))
−1. Using majorant condition, we can get

∫1

0

‖f′(x̄)−1[f′′(xk + σ(x̄ − xk)) − f′′(xk)]‖(1 − σ)dσ ≤
∫1

0

‖[φ′′(σ‖x̄ − xk‖ + ‖xk − x̄‖)

−φ′′(‖xk − x̄‖)]‖(1 − σ)dσ.

Also, we know that, if u, v, w ∈ (0, a) and u < v < w, then because of convexity[7] of φ(x) in
(0, a), we have

φ(v) − φ(u) ≤ [φ(w) − φ(u)] v − u
w − u

.

Therefore,

φ′′(σ‖x̄ − xk‖ + ‖xk − x̄‖) − φ′′(‖xk − x̄‖) ≤ φ′′(σ‖x̄ − xk‖ + tk) − φ′′(tk)

≤ [φ′′(σ(t∗ − tk) + tk) − φ′′(tk)]
‖x̄ − xk‖
t∗ − tk

.

Thus Lemma 2.5 and above inequality implies

‖x̄ − xk+1‖ ≤ −
1

(1 − Lφ(tk))φ
′(tk)

[

∫1

0

[φ′′(σ(t∗ − tk) + tk) − φ
′′(tk)](1 − σ)dσ

]‖x̄ − xk‖3
t∗ − tk

+
φ′′(tk)

(1 − Lφ(tk))φ
′(tk)

2

[

∫1

0

[φ′′(σ(t∗ − tk) + tk)(1 − σ)dσ
]

‖x̄ − xk‖3

+
1

2

φ′′(tk)

(1 − Lφ(tk))φ
′(tk)

2

[

∫1

0

[φ′′(σ(t∗ − tk) + tk)(1 − σ)dσ
]2

‖x̄ − xk‖4

≤ φ(tk)
(1 − Lφ(tk))φ

′(tk)

(‖x̄ − xk‖
t∗ − tk

)3

= (t∗ − tk+1)
(‖x̄ − xk‖

t∗ − tk

)3

.

Finally, we want to show that the solution x̄ of (1.1) is unique in B̄(x̄, t∗). For that assume ȳ be

another solution in B̄(x̄, t∗). Then proceeding similarly as above we get

‖ȳ − xk+1‖ ≤ (t∗ − tk+1)
(‖ȳ − xk‖

t∗ − tk

)3

.



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Super-Halley method under majorant conditions in Banach spaces 63

Since the sequence {xk} converges to x̄ and {tk} converges to t
∗, we conclude that ȳ = x̄. Therefore,

x̄ is the unique zero of (1.1) in B̄(x̄, t∗). ✷

Remark 2.7. It is to be noted that if we replace x̄ with the initial approximation x0 in (2.1),

then after some manipulations we can obtain a semilocal convergence analysis of our iteration

method. This analysis approach enables us to drop out the assumption of existence of a second

root for the majorizing function, still guarantee Q−cubic convergence rate. Thus the semilocal

convergence theorem for the iteration method (1.4) is as follows:

Theorem 2.8. Suppose f : Ω ⊆ X → Y be a twice continuously differentiable nonlinear operator,
and Ω is open and convex. Consider that for a given initial guess x0 ∈ Ω, f′(x0) is nonsingular that
is f′(x0)

−1 exists and that φ is the majorizing function to f at x0 and φ satisfies the assumptions

(M1) − −(M3). Then sequence {xk} generated by the method (1.4) for solving equation (1.1) with
a starting point x0 is well defined, contained in B(x0, t

∗) and converges to a solution x̄ ∈ B̄(x0, t∗)
of the Eq.(1.1). The solution is unique in B(x0, σ), where σ is defined as σ := sup{t ∈ (t∗, R) :
φ(t) ≤ 0}. For k = 0, 1, 2, . . . , a priori error estimate and a posteriori error estimate are given
respectively as

‖x̄ − xk+1‖ ≤ (t∗ − tk+1)
(‖x̄ − xk‖

t∗ − tk

)3

,

and

‖x̄ − xk+1‖ ≤ (t∗ − tk+1)
(‖xk+1 − xk‖

tk+1 − tk

)3

.

Also the method converges Q−cubically as

‖x̄ − xk+1‖ ≤
[

1

2

φ′′(t∗)2

φ′(t∗)2
+

1

3

D−φ′′(t∗)

−φ′(t∗)

]

(‖x̄ − xk‖)3.

3 Special cases and applications

This section consists of two special cases of the local convergence results obtained in previous

section. Namely, convergence results under affine covariant Kantorovich-type condition and the

Smale-type γ-condition.

3.1 Kantorovich-type

Suppose that f satisfies the affine covariant Lipschitz condition (see Han and Wang[12]) as given

by:

‖f′(x̄)−1[f′′(y) − f′′(x)]‖ ≤ λ1‖y − x‖, x, y ∈ Ω. (3.1)

and the following initial conditions

‖f′(x̄)−1f(x̄)‖ ≤ β, (3.2)



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‖f′(x̄)−1f′′(x̄)‖ ≤ λ2. (3.3)

Consider the scalar valued function

φ(t) =
λ1

6
t3 +

λ2

2
t2 − t + β. (3.4)

This function was considered as majorizing function in [28, 5, 11] for establishing convergence of

super-Halley method. If we choose the above cubic polynomial as the majorizing function φ in

(2.1), then the majorant condition (2.1) reduced to the Lipschitz condition (3.1) and in this way

the results based on Lipschitz condition have been generalized by our assumptions of majorant

conditions. The assumptions (M1) and (M2) are satisfied for f if the following criterion holds

β ≤ 2(λ2 + 2(λ
2
2 + 2λ1)

1/2)

3(λ2 + 2(λ
2
2
+ 2λ1)

1/2)2
. (3.5)

Therefore, Theorem 2.6 reduces to the following form:

Theorem 3.1. Suppose that f satisfies the conditions (3.1)-(3.3) with the assumptions given

in (3.5). Then, the sequence {xk} generated by Super-Halley method (1.4) for solving equation
(1.1) with a starting point x0 is well defined, contained in B(x̄, t

∗) and converges to a solution

x̄ ∈ B̄(x̄, t∗) of the Eq.(1.1). Note that t∗ is the smallest positive root of φ defined by (3.4) in
[0, r1] where r1 = (−λ2 + (λ

2
2 + 2λ1)

1/2)/λ1 is the positive root of φ
′. The limit x̄ of the sequence

{xk} is the unique zero of Eq.(1.1) in B(x̄, t∗∗), where t∗∗ is the root of φ in the interval (r1, +∞).
Moreover, the following error estimates holds

‖x̄ − xk+1‖ ≤ (t∗ − tk+1)
(‖x̄ − xk‖

t∗ − tk

)3

, k = 0, 1, 2, . . .

and the sequence generated by Super-Halley method (1.4) converges Q-cubic as follows

‖x̄ − xk+1‖ ≤
[

3(λ1 + λ2t
∗)2 + 2λ2(1 − λ1t

∗ − λ2t
∗2/2)

6(1 − λ1t
∗ − λ2t

∗2/2)2

]

(‖x̄ − xk‖)3, k = 0, 1, 2, . . .

3.2 Smale-type

This subsection contains the local convergence results for the Super-Halley method (1.4) under the

γ-Condition.

In [21], Smale has studied the convergence and error estimation of Newton’s method under

the hypotheses that f is analytic and satisfies

‖f′(x̄)−1f(n)(x̄)‖ ≤ n!γn−1, n > 2



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Super-Halley method under majorant conditions in Banach spaces 65

where

γ := sup
k>1

‖f
′(x̄)−1f(n)(x̄)

k!
‖

1

k−1

Smale’s result is completely improved by Wang and Han[24, 25] by introducing a majorizing func-

tion

φ(t) = β − t +
γt2

1 − γt
, t ∈

[

0,
1

γ

)

(3.6)

where

‖f′(x̄)−1f(x̄)‖ ≤ β (3.7)

and

‖f′(x̄)−1f′′(x̄)‖ ≤ 2γ (3.8)

If we choose this function as the majorizing function φ in (2.1), then it reduces to the following

condition:

‖f′(x̄)−1[f′′(y) − f′′(x)]‖ ≤ 2γ
(1 − γ‖y − x‖ − γ‖x − x̄‖)3

−
2γ

(1 − γ‖x − x̄‖)3
, γ > 0 (3.9)

where ‖y − x‖ + ‖x − x̄‖ < 1
γ
, and the assumptions (M1) and (M2) are satisfied for φ. Also, if

α := βγ < 3 − 2
√
2, then assumption (M3) is satisfied for φ. Therefore, the concrete form of

Theorem 2.6 is given as follows.

Theorem 3.2. Suppose f satisfies (3.7)-(3.9). If α := βγ < 3 − 2
√
2, then the sequence {xk}

generated by the super-Halley method (1.4) for solving the equation (1.1) with a starting point x0

is well defined, is contained in B(x̄, t∗) and converges to a solution x̄ ∈ B̄(x̄, t∗) of the Eq.(1.1).
The limit x̄ of the sequence {xk} is unique in B(x̄, t∗∗), where t∗ and t∗∗ are given as

t∗ =
α + 1 −

√

(α + 1)2 − 8α

4γ
and t∗∗ =

α + 1 +
√

(α + 1)2 − 8α

4γ

respectively. Moreover, the following error bound holds: for all k ≥ 0, we have

‖x̄ − xk+1‖ ≤ (t∗ − tk+1)
(‖x̄ − xk‖

t∗ − tk

)3

, k = 0, 1, 2, . . .

and the sequence {xk} converges Q-cubic as follows

‖x̄ − xk+1‖ ≤
2γ2

[2(1 − γt∗)2 − 1]2
(‖x̄ − xk‖)3, k = 0, 1, 2, . . .



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4 Numerical Examples

This section is devoted to illustrate the above theoretical results by a number of numerical examples.

Example 4.1. Let X = Y = R with Ω = B(0, 1) and the function f on Ω is

f(x) = ex − 1 (4.1)

and for x̄ = 0,

f′(x̄) = 1, f′′(x̄) = 1.

Also, we obtain that

λ1 = e, λ2 = 1, β = 0

Therefore, the convergence criterion (3.5) holds and the Theorem 3.1 is applicable to conclude that

the sequence generated by super-Halley method (1.4) with initial point x0 = 0.25 converges to a root

of (4.1). In this case, we have t∗ = 0 and t∗∗ = 1.03304078, that is the existence and uniqueness

ball are B(0.25, 0) and B̄(0.25, 1.03304078) respectively and the error bound is 1.525807581.

Example 4.2. Let X = C[0, 1] the space of continuous functions defined on interval [0, 1]

equipped with max norm and let Ω = U[0, 1] and the function f on Ω is .

f(x)(s) = x(s) − 2λ

∫1

0

G(s, t)x(t)3dt (4.2)

Therefore we have

f′(x)u(s) = u(s) − 6λ

∫1

0

G(s, t)x(t)2u(t)dt, u ∈ Ω,

and

f′′(x)[uv](s) = −λ

∫1

0

G(s, t)x(t)(uv)(t)dt, u, v ∈ Ω,

Now, let M = max
s∈[0,1]

∫1
0
|G(s, t)|dt. Then M = 1

8
.. Also, for any x, y ∈ Ω, we have

‖f′(x̄)−1[f′′(x) − f′′(x̄)]‖ ≤ 3|λ|
2

‖x − x̄‖.

So, we obtain the values of β, λ2 and λ1 in as follows

β = 0, λ2 = 0, λ1 =
3|λ|

2
.

Therefore, the convergence criterion (3.5) holds and the Theorem 3.1 is applicable to conclude that

the sequence generated by Super-Halley method (1.4) with initial point x0 converges to a zero of f

defined by (4.2).



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Super-Halley method under majorant conditions in Banach spaces 67

For the different values of λ i.e. for λ = 0.0625, 0.125, 0.25, 0.5, 1 and the initial point x0 = 0.25

the corresponding domain of existence and uniqueness of solution, are given in Table-4.2.

Table-4.2: Domains of existence and uniqueness of solution for Super-Halley’s method

λ convergence ball in our work

existence uniqueness

0.0625 B̄(0.25, 0) B(0.25, 8)

0.125 B̄(0.25, 0) B(0.25, 5.656854249)

0.25 B̄(0.25, 0) B(0.25, 4)

0.5 B̄(0.25, 0) B(0.25, 2.828427125)

1 B̄(0.25, 0) B(0.25, 2)

Now, we present a numerical example to illustrate the Smale-type conditions.

Example 4.3. Let X = Y = R with Ω = U[0, 1] and the function f on Ω is

f(x) = ex + 2x2 − 1 (4.3)

For x̄ = 0,

f′(x̄) = 1, f′′(x̄) = 5

and we obtain that

β = 0, γ = 2.5

Therefore, the convergence criterion (3.9) holds (which can be seen from the above graph in

case y > x) and the Theorem 3.2 is applicable to conclude that the sequence generated by Super-

Halley method (1.4 ) converges to a zero of f defined by (4.3) with t∗ = 0 and t∗∗ = 0.2.



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5 Conclusions

In this paper, the local convergence of Super-Halley method has been studied under majorant

conditions on second derivative of f. Convergence ball of the method has been included. Two

special cases: one Kantorovich-type conditions and another Smale-type conditions have also been

studied. A number of numerical examples also given to illustrate our study.



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Super-Halley method under majorant conditions in Banach spaces 69

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	Introduction
	Preliminaries
	Local convergence results for Super-Halley method 

	Special cases and applications
	Kantorovich-type 
	Smale-type 

	Numerical Examples
	Conclusions