Ann. Acad. Rom. Sci.

Ser. Math. Appl.

ISSN 2066-6594

Vol. 18, No. 2/2026

EXTENDED FOUR PARAMETER

CHEBYSHEV-HALLEY-TYPE METHODS OF

ORDER SIX^∗

Ioannis K. Argyros^†

Stepan Shakhno^‡

Halyna Yarmola^§

Communicated by G. Moro¸sanu

DOI

10.56082/annalsarscimath.2026.2.111

Abstract

A study of the local and the semilocal convergence is carried out

for the Chebyshev-Halley-type iterative methods under ω-type condi-

tions. The conditions are imposed only on the ﬁrst-order derivatives.

In both cases, the convergence region and the region of uniqueness of

the solution is established. The new technique is a usefull alternative

to expensive Taylor series used to study the convergence of iterative

methods requiring high order derivatives not on the methods. The re-

sults of a numerical experiment are presented to check the convergence

conditions.

Keywords: complete normed space, Chebyshev-Halley-type methods,

local and semi-local convergence, order six.

MSC: 65J15, 65H10, 65G99, 47H30.

^∗Accepted for publication on December 02, 2025

^†iargyros@cameron.edu, Department of Computing and Mathematical Sciences,

Cameron University, Lawton, OK 73505, USA

^‡stepa.shakhno@lnu.edu.ua, Department of Theory of Optimal Processes, Ivan

Franko National University of Lviv, Lviv, Ukraine

^§halyna.yarmola@lnu.edu.ua, Department of Computational Mathematics, Ivan

Franko National University of Lviv, Lviv, Ukraine

111

Extended Chebyshev-Halley-type methods

112

1 Introduction

Mathematical models of the complex physical or technological processes de-

scribed often by nonlinear problems, in particular by systems of nonlinear

algebraic or transcendental equations, nonlinear integral equations, nonlin-

ear boundary value problems and other. It is extremely rare to ﬁnd an exact

solution to such problems. Therefore, research and development of methods

for numerically solving nonlinear problems is an important task. In general,

these problems are written in the form of an operator equation [1,8,11,12,14]

F(x) = 0.

(1)

Here the operator F : Ω ⊂ B₁→ B₂is Frech´et-diﬀerentiable, Ω is an open

and convex set, whereas B₁and B₂are Banach spaces.

There is a large number of iterative methods for the numerical solving

of the equation (1), in particular Newton method or methods with divided

diﬀerences. One of the characteristics of iterative methods is the amount of

calculations of the values of F and its derivatives (or divided diﬀerences). In

most cases, at each step of single-step methods, it is necessary to calculate

one value of F and ﬁnd one inverse operator for F⁰or its approximation.

To increase the convergence order, it is necessary to increase the number

of calculations of F and F⁰which leads to increasing computational costs

and a decreasing of the eﬃciency of the method. Such methods had been

investigated in [2–7,9,10,13].

In this paper we consider the three step method with four real parameters

b, c, p and q (b = 0), which is deﬁned for x₀∈ Ω and each n = 0, 1, 2, . . .

by [9 ]

2

y_n= x_n− F⁰(x_n)⁻¹F(x_n),

A_n= I − F⁰(x_n)⁻¹F⁰(y_n), M_n= I − A_n,

3

c

b

1 3b − 2

1 9b²− 3b − 4c + 2

, (2)

T_n= I + a₁A_n+ a₂A², a₁=

, a₂=

n

4

b

8

b²

ꢀ

ꢁ

1

z_n= x_n− I +

M_n⁻¹A_nT_nF⁰(x_n)⁻¹F(x_n),

2b

x_n+1= z_n− (pI + qA_n)F⁰(x_n)⁻¹F(z_n).

The local convergence order six is shown in [9] for B₁= B₂= IR^m,

3

p = 1, q =

using the Taylor series technique. There are constraints with

2

this technique.

I.K. Argyros, S. Shakhno, H. Yarmola

113

Motivation for writing this paper.

(E₁) The local analysis (see [9 ]) is restricted only on IR^mand the assumption

is made that F⁽⁵⁾(the ﬁfth derivative exists and is bounded on Ω. But

this high order derivative is not on the method (2). Hence, the results

in [9] are applicable to solve equation (1)) provided that the derivatives

reaching up to order ﬁve exist, which may not be true. As an example,

let m = 1 and say Ω = (−1.3, 1.3). Deﬁne the function F : Ω → IR by

(

d₁t⁵log t²+ d₂t⁶+ d₃t⁷, t = 0

F(t) =

0,

t = 0,

where d₁= 0 and d₂+ d₃= 0. It follows by this deﬁnition that

t^∗= 1 ∈ Ω solves the equation F(t) = 0. But the function F⁽⁵⁾is

unbounded at t = 0 ∈ Ω. Thus, the results in [9] cannot assure the

convergence of the method (2) to t^∗. But the method (2 ) converges to

3

t^∗= 1 if say x₀= 0.95, b = c = p = 1 and q = . This observation

2

indicates that if an alternative to the Taylor series technique is used

the suﬃcient convergence conditions in [9] may be weakened.

(E₂) No radius of convergence is given in [9]. So, the initial points x₀are

given in not known.

(E₃) There are no previous knowledge of the integer N such that

kx^∗− x_nk < ε ∀n ≥ N and some ε ≥ 0.

(E₄) There are no isolation of the solution results.

(E₅) The more diﬃcult and important semi-local analysis of method (2) is

not previously studied.

The technique of this paper positively addresses all issues (E₁)-(E₅) and se-

ries as an alternative to Taylor series for studying the convergence of method

(2) as well as other similar methods analogously [4–7,10,13]. In particular,

we achieve:

(E₁)⁰The local analysis of method (2) is shown by using only the operators

on it, i.e. F and F⁰.

(E₂)⁰A computable radius of convergence is provided. So, the initial points

ate selected from a certain ball about x^∗or x₀.

(E₃)⁰The number of iterations to be carried out, i.e. N is known a priori.

Extended Chebyshev-Halley-type methods

114

(E₄)⁰Domains are determined which contain only one solution.

(E₅)⁰Majorizing scalar sequences [1 ] are employed to determine the conver-

gence of {x_n} generated by the method (2). The analysis is presented

in the more general setting of a Banach space. Moreover, the derivative

F⁰is controlled by generalized continuity conditions used to control it

and also sharpen the error distances kx^∗− x_nk [1–3].

The new local convergence analysis is shown using only the operators on

the method, i.e. F and F⁰. Moreover, the semi-local analysis not previously

studied is presented using majorizing sequences. Both analyses provided in

the more general setting of a Banach space and also depend on generalized

continuity which controls the derivative F⁰. The same technique can be used

to extend the applicability of other methods along the same lines [4–7,10,13].

2 Local convergence

First we introduce the following notation and conditions used in the study

of local convergence of the method (2).

Denote by U(x, r) and U[x, r] open and closed balls, respectively, with

center at the point x and of radius r. Let us set S = [0, ∞).

Suppose

(LC₁) There exists a function ω₀: S → S, which is continuous and strictly

increasing on S such that equation ω₀(t) − 1 = 0 has at least one posi-

tive root.

We denote by %₀the smallest such root and set

S₀= [0, %₀).

(LC₂) There exists a function ω : S₀→ S, which is continuous and strictly

increasing on S₀such that for function g₁: S₀→ S given by

ꢂ

ꢃ

R

¹w((1 − θ)t)dθ +

1 + ¹w₀(θt)dθ

1

3

0

g₁(t) =

,

(3)

1 − w₀(t)

equation g₁(t) − 1 = 0 has at least one root in the interval (0, %₀). We

denote by r₁the smallest such root.

(LC₃) The equation µ(t)−1 = 0 has at least one root in the interval (0, %₀).

We denote by % the smallest such root. Set S₁= [0, %).

Deﬁne the function g₂: S₁→ S by

ꢂ

ꢃ

R

λ(t) 1 + ¹w₀(θt)dθ

R

¹w((1 − θ)t)dθ

0

g₂(t) =

+

1 − w₀(t)

I.K. Argyros, S. Shakhno, H. Yarmola

115

ꢀ

ꢁ

Z

1

(1 + λ(t))w(t)

+

1 +

w₀(θt)dθ ,

2|b|(1 − w₀(t))²(1 − µ(t))

0

where



w((1 + g₁(t))t)



ꢄ ꢄ

c

w(t)

ꢄ ꢄ

w(t) =

or

µ(t) =

,

ꢄ ꢄ



b 1 − w₀(t)

w₀(t) + w₀(g₁(t)t),

ꢀ

ꢁ

w(t)

λ(t) =

|a₁| + |a₂|

.

1 − w₀(t)

The equation

g₂(t) − 1 = 0

has at least one root in the interval (0, r₁). We denote by r₂the

smallest such root.

Deﬁne the function g₃: S₁→ S by

ꢂ

ꢃ



R

w(t) 1 + ¹w₀(θg₂(t)t)dθ

R

¹w((1 − θ)g₂(t)t)dθ

0



g₃(t) =

+

1 − w₀(g₂(t)t)

(1 − w₀(t))(1 − w₀(g₂(t)t))

ꢂ

ꢃ



R

1 + ¹w₀(θt)dθ

ꢀ

ꢁ

|q|w(t)

0



+ |p − 1| +

g₂(t),

1 − w₀(t)

where



w((1 + g₂(t))t)

or



w(t) =



w₀(t) + w₀(g₂(t)t),

The equation

g₃(t) − 1 = 0

has at least one root in the interval (0, r₂). We denote by r₃the

smallest such root.

Deﬁne the parameter

r^∗= min{r_j},

j = 1, 2, 3.

(4)

This parameter is shown to be a radius of convergence in Theorem 1

for the method (2). Set S₂= [0, r^∗).

Extended Chebyshev-Halley-type methods

116

It follows by these deﬁnitions that

0 ≤ ω₀(t) < 1 ∀t ∈ S₂and 0 ≤ ω₀(g₂(t)t) < 1 ∀t ∈ S₂

(5)

(6)

and ∀t ∈ S₂

0 ≤ g_i(t) < 1, i = 1, 2, 3.

(LC₄) There exists a solution x^∗∈ Ω of the equation F(x) = 0, L ∈

L(B₁, B₂) such that L⁻¹∈ L(B₂, B₁) and

kL⁻¹(F⁰(x) − L)k ≤ ω₀(kx − x^∗k) ∀x ∈ Ω.

Set Ω₀= Ω ∩ U(x^∗, %₀).

(LC₅) kL⁻¹(F⁰(x) − F⁰(y))k ≤ ω(kx − yk) ∀x, y ∈ Ω₀.

(LC₆) U(x^∗, r^∗) ⊂ Ω.

Theorem 1. Suppose that conditions (LC₁)-(LC₆) hold and x₀∈ U(x^∗, r^∗).

Then, the sequence {x_n} generated by method (2) is well deﬁned in U(x^∗, r^∗)

for each n = 0, 1, . . . and converges to the solution x^∗∈ Ω of the equation

(1). Moreover, the following error estimates hold for each n = 0, 1, . . .

ky_n− x^∗k ≤ g₁(r^∗)kx_n− x^∗k ≤ kx_n− x^∗k < r^∗,

(7)

(8)

kz_n− x^∗k ≤ g₂(r^∗)kx_n− x^∗k ≤ kx_n− x^∗k

and

kx_n+1− x^∗k ≤ g₃(r^∗)kx_n− x^∗k ≤ kx_n− x^∗k.

(9)

Proof. The proof is carried out by mathematical induction. Using conditions

(LC₄) and (5), we obtain in turn that

kL⁻¹(F⁰(x_n) − L)k ≤ ω₀(kx_n− x^∗k) ≤ ω₀(r^∗) < 1.

(10)

The Banach Lemma on invertible linear operators [1,11] and (10) imply that

F⁰(x_n)⁻¹∈ L(B₂, B₁) and

1

kF⁰(x_n)⁻¹Lk ≤

.

(11)

1 − ω₀(kx_n− x^∗k)

I.K. Argyros, S. Shakhno, H. Yarmola

117

Then, by the ﬁrst substep of (2), we can write in turn

1

y_n− x^∗= x_n− x^∗− F⁰(x_n)⁻¹F(x_n) + F⁰(x_n)⁻¹F(x_n)

3

ꢀ

ꢁ

Z

1

= F⁰(x_n)⁻¹F⁰(x_n) −

F⁰(x^∗+ θ(x_n− x^∗))dθ (x_n− x^∗)

0

ꢀ

ꢁ

Z

1

+ F⁰(x_n)⁻¹

F⁰(x^∗+ θ(x_n− x^∗))dθ − F⁰(x^∗) + F⁰(x^∗)

3

0

×(x_n− x^∗).

Taking into account conditions (LC₄), (LC₅), (6), (11) and the last equality,

we have

ꢂ

ꢃ

R

¹w((1 − θ)kx_n− x^∗k)dθ +

1 + ¹w₀(θkx_n− x^∗k)dθ

1

3

0

ky_n− x^∗k ≤

1 − w₀(kx_n− x^∗k)

×kx_n− x^∗k ≤ g₁(kx_n− x^∗k)kx_n− x^∗k ≤ kx_n− x^∗k < r^∗.

Using conditions (LC₄), (LC₅) and (11), we have

kA_nk = kI − F⁰(x_n)⁻¹F⁰(y_n)k = kF⁰(x_n)⁻¹(F⁰(x_n) − F⁰(x^∗)

ꢂ

+F⁰(x^∗) − F⁰(y_n))k ≤ kF⁰(x_n)⁻¹Lk kL⁻¹(F⁰(x_n) − F⁰(x^∗))k

ꢃ

ω₀(kx_n− x^∗k) + ω₀(ky_n− x^∗k)

+kL⁻¹(F⁰(x^∗) − F⁰(y_n))k ≤

1 − ω₀(kx_n− x^∗k)

ω₀(kx_n− x^∗k) + ω₀(g₁(kx_n− x^∗k)kx_n− x^∗k)

=

1 − ω₀(kx_n− x^∗k)

or

kA_nk = kI − F⁰(x_n)⁻¹F⁰(y_n)k = kF⁰(x_n)⁻¹(F⁰(x_n) − F⁰(y_n))k

ꢅ

ꢆ

≤ kF⁰(x_n)⁻¹Lk kL⁻¹(F⁰(x_n) − F⁰(y_n))k

ω(kx_n− y_nk)

ω((1 + g₁(kx_n− x^∗k))kx_n− x^∗k)

≤

1 − ω₀(kx_n− x^∗k)

and

ω(kx_n− x^∗k)

kA_nk ≤

.

(12)

1 − ω₀(kx_n− x^∗k)

In view of the following equality holds

I − T_n= −a₁A_n− a₂A²,

n

Extended Chebyshev-Halley-type methods

118

we obtain using the estimates (10) and (12)

ꢀ

ꢁ

2

w(kx_n− x^∗k)

kI − T_nk ≤ |a₁|

+ |a₂|

1 − w₀(kx_n− x^∗k)

= λ(kx_n− x^∗k) = λ_n,

(13)

(14)

kT_nk ≤ 1 + λ_n,

ꢇ

ꢄ ꢄ

c

b

c

w(kx_n− x^∗k)

ꢄ ꢄ

A_n

≤

= µ(kx_n− x^∗k) = µ_n< 1

ꢄ ꢄ

b 1 − w₀(kx_n− x^∗k)

and

ꢇ

ꢂ

ꢃ_−1ꢇ

c

I − A_n

b

1

kM_n⁻¹k = k(I − (I − M_n))⁻¹k =

≤

.

(15)

ꢇ

1 − µ_n

Then, by the second substep of (2), we can write in turn

z_n− x^∗= x_n− x^∗− F⁰(x_n)⁻¹F(x_n) + (I − T_n) F⁰(x_n)⁻¹F(x_n)

1

−

M_n⁻¹A_nT_nF⁰(x_n)⁻¹F(x_n)

2b

ꢀ

ꢁ

Z

1

= F⁰(x_n)⁻¹F⁰(x_n) −

F⁰(x^∗+ θ(x_n− x^∗))dθ (x_n− x^∗)

0

ꢀ

Z

1

+ (I − T_n) F⁰(x_n)⁻¹

−F⁰(x^∗) + F⁰(x^∗) (x_n− x^∗)

F⁰(x^∗+ θ(x_n− x^∗))dθ

0

ꢆ

ꢀ

Z

1

−

M_n⁻¹A_nT_nF⁰(x_n)⁻¹

F⁰(x^∗+ θ(x_n− x^∗))dθ

2b

0

ꢆ

−F⁰(x^∗) + F⁰(x^∗) (x_n− x^∗).

By using (LC₄), (LC₅), estimates (6), (11), (12 )–(15 ), we get

"

R

¹w((1 − θ)kx_n− x^∗k)dθ

kz_n− x^∗k ≤

0

1 − w₀(kx_n− x^∗k)

ꢂ

ꢃ

R

λ_n1 + ¹w₀(θkx_n− x^∗k)dθ

1 −⁰w₀(kx_n− x^∗k)

+

(1 + λ_n)w(kx_n− x^∗k)

2|b|(1 − w₀(kx_n− x^∗k))²(1 − µ_n)

ꢀ

ꢁꢈ

Z

1

× 1 +

w₀(θkx_n− x^∗k)dθ

kx_n− x^∗k

= g₂(kx_n− ⁰x^∗k)kx_n− x^∗k ≤ kx_n− x^∗k.

I.K. Argyros, S. Shakhno, H. Yarmola

119

Then, by the third substep of (2), we can write in turn

ꢅ

ꢆ

x_n+1− x^∗= z_n− x^∗− F⁰(z_n)⁻¹F(z_n) + F⁰(z_n)⁻¹− F⁰(x_n)⁻¹F(z_n)

−(p − 1)F⁰(x_n)⁻¹F(z_n) − qA_nF⁰(x_n)⁻¹F(z_n),

ꢀ

ꢁ

Z

1

= F⁰(z_n)⁻¹F⁰(z_n) −

F⁰(x^∗+ θ(z_n− x^∗))dθ (z_n− x^∗)

0

ꢀ

Z

1

ꢅ

ꢀ

ꢆ

+ F⁰(z_n)⁻¹− F⁰(x_n)⁻¹

F⁰(x^∗+ θ(z_n− x^∗))dθ

0

ꢆ

−F⁰(x^∗) + F⁰(x^∗) (z_n− x^∗) − ((p − 1)I + qA_n) F⁰(x_n)⁻¹

ꢁ

Z

1

×

F⁰(x^∗+ θ(z_n− x^∗))dθ − F⁰(x^∗) + F⁰(x^∗)

0

×(z_n− x^∗).

Using condition (LC₃), (5 ) and (8), we obtain in turn that

kL⁻¹(F⁰(z_n) − L)k ≤ ω₀(kz_n− x^∗k) ≤ ω₀(g₂(r^∗)r^∗) < 1.

(16)

The Banach Lemma on invertible linear operators [1,11] and (16) imply that

F⁰(z_n)⁻¹∈ L(B₂, B₁) and

1

kF⁰(z_n)⁻¹Lk ≤

.

(17)

1 − ω₀(kz_n− x^∗k)

By the following equality

ꢅ

ꢆ

F⁰(z_n)⁻¹− F⁰(x_n)⁻¹= F⁰(x_n)⁻¹F⁰(x_n) − F⁰(z_n) F⁰(x_n)⁻¹

,

and taking into account the conditions (LC₄), (LC₅), estimates (6 ), (8),

(11), (12) and (17), we have

ꢂ

ꢃ



R

w(kx_n− x^∗k) 1 + ¹w₀(θkz_n− x^∗k)dθ

0

kx_n+1− x^∗k ≤



(1 − w₀(kx_n− x^∗k))(1 − w₀(kz_n− x^∗k))

R

¹w((1 − θ)kz_n− x^∗k)dθ

0

+

1 − w₀(kz_n− x^∗k)

ꢀ

ꢁ

|q|w(kx_n− x^∗k)

+ |p − 1| +

1 − w₀(kx_n− x^∗k)

ꢂ

R

ꢃ



1 + ¹w₀(θkz_n− x^∗k)dθ

0

∗



×

kz_n− x k

1 − w₀(kx_n− x^∗k)

≤ g₃(kx_n− x^∗k)kx_n− x^∗k ≤ kx_n− x^∗k.

Extended Chebyshev-Halley-type methods

120

Moreover, by (9) there exists α ∈ [0, 1) such that

kx_n+1− x^∗k ≤ αkx_n− x^∗k ≤ αⁿ⁺¹kx₀− x^∗k < r^∗.

(18)

Therefore, it follows from (18 ) that the iterate x_n+1∈ U(x^∗, r^∗) and

lim x_n= x^∗.

n→∞

Next, we present a result for uniqueness of the solution of the equa-

tion (1).

Proposition 1. Suppose:

(a) The condition (LC₄) holds in the ball U(x^∗, %₁) for some %₁> 0.

(b) There exists %₂≥ %₁such that

Z

ꢂ

ꢃ

1

ω₀θ%₂dθ < 1.

(19)

0

Set Ω₁= U[x^∗, %₂] ∩ Ω.

Then, the equation F(x) = 0 has the unique solution x^∗in the region Ω₁.

Proof. Suppose that there exists u^∗∈ Ω₁, u^∗= x^∗and F(u^∗) = 0. Let

1

R

T =

F⁰(x^∗+ θ(u^∗− x^∗))dθ. It follows by (a) and (b) that

0

Z

1

kL⁻¹(T − L)k ≤

ω₀(kx^∗+ θ(u^∗− x^∗) − x^∗k)dθ ≤

0

Z

ꢂ

ꢃ

ꢂ

ꢃ

1

≤

ω₀θku^∗− x^∗k dθ

ω₀θ%₂dθ < 1.

0

Hence, the operator T is invertible. Then, by the identity

u^∗− x^∗= T⁻¹(F(u^∗) − F(x^∗)) = T⁻¹(0 − 0) = 0,

we conclude that u^∗= x^∗.

3 Semi-local convergence

Majorizing sequences are used to provide the semi-local analysis.

Suppose:

(SLC₁) There exists a function v₀: S → S which is continuous and nonde-

creasing such that the equation

v₀(t) − 1 = 0

has a smallest solution %₃. Set S₃= [0, %₃).

I.K. Argyros, S. Shakhno, H. Yarmola

121

(SLC₂) There exists a function v : S₃→ S which is continuous and nonde-

creasing.

The majorant sequence {α_n} is deﬁned for α₀= 0,

2

β₀≥ kF⁰(x₀)⁻¹F(x₀)k

3

and ∀n = 0, 1, 2, . . . by



v(β_n− α_n),



v_n=

or



v₀(α_n) + v₀(β_n),

ꢀ

ꢁ

ꢄ ꢄ

v_n

c

v_n

ꢄ ꢄ

λ_n= |a₁| + |a₂|

,

µ =

n

,

ꢄ ꢄ

1 − v₀(α_n)

b 1 − v₀(α_n)

ꢀ

ꢁ

3

2

1

(1 + λ_n)v_n

γ_n= β_n+

+ λ_n+

(β_n− α_n),

3

2|b|(1 − v₀(α_n))(1 − µ_n)

Z

1

ξ_n

=

v((1 − θ)(γ_n− α_n))dθ(γ_n− α_n) + (1 + v₀(α_n))(γ_n− β_n)

0

1

+ (1 + v₀(α_n))(β_n− α_n),

2

ꢀ

ꢁ

v_n

ξ_n

α_n+1= γ_n+ |p| + |q|

,

(20)

1 − v₀(α_n)

Z

1

δ_n+1

=

v((1 − θ)(α_n+1− α_n))dθ(α_n+1− α_n)

0

1

+(1 + v₀(α_n))(α_n+1− β_n) + (1 + v₀(α_n))(β_n− α_n)

2

and

2

δ_n+1

β_n+1= α_n+1

+

.

3 1 − v₀(α_n+1

)

(SLC₃)

Lemma 1. Suppose that ∀n = 0, 1, 2, ...

0 ≤ v₀(α_n) < 1,

µ_n< 1,

0 ≤ α_n< α for some α > 0.

(21)

Then, the sequence {α_n} given by the formula (20) is nondecreasing

and convergent to its unique least upper bound α^∗∈ [0, 1].

Extended Chebyshev-Halley-type methods

122

Proof. The sequence {α_n} is nondecreasing and bounded from above

by α and as such it is convergent to α^∗.

The additional conditions shall be used in the semi-local convergence

analysis of method (2).

(SLC₄) There exist points x₀∈ Ω and parameter β₀such that F⁰(x₀)⁻¹

,

L⁻₀¹∈ £(Y, X) and

2

kF⁰(x₀)⁻¹F(x₀)k ≤ β₀.

3

(SLC₅) kL⁻¹(F⁰(x) − F⁰(x₀))k ≤ v₀(kx − x₀k) ∀ x ∈ Ω.

Set Ω₂= U(x₀, %₃) ∩ Ω.

(SLC₆) kL⁻¹(F⁰(x) − F⁰(y))k ≤ v(kx − yk) ∀ x, y ∈ Ω₂.

(SLC₇) U[x₀, α^∗] ⊂ Ω.

Next, the semi-local convergence of the method (2) is presented based on

the conditions (SLC₁)-(SLC₇) and the preceding terminology.

Theorem 2. Suppose that the conditions (SLC₁)-(SLC₇) hold. Then, the

sequence {x_n} generated by the method (2) is well deﬁned in U(x₀, α^∗),

remains in U(x₀, α^∗) for all n = 0, 1, 2, ... and converges to a solution

x^∗∈ U[x₀, α^∗] of the equation F(x) = 0. Moreover, the following error

estimates hold

kx^∗− x_nk ≤ α^∗− α_n.

(22)

Proof. By the condition (SLC₄) the iterate y₀is well deﬁned, since from the

ﬁrst substep of the method for n = 0 we have

2

ky₀− x₀k ≤

kF⁰(x₀)⁻¹F(x₀)k ≤ β₀− α₀= β₀< α^∗,

3

and the iterate y₀∈ U(x₀, α^∗).

Suppose that y_i, z_i, x_i+1∈ U(x₀, α^∗) for i = 0, . . . , n − 1 and

ky_i− x_ik ≤ β_i− α_i,

kz_i− y_ik ≤ γ_i− β_i,

kx_i+1− y_ik ≤ α_i+1− β_i,

kx_i+1− z_ik ≤ α_i+1− γ_i.

Then for i = n, we get following estimates.

I.K. Argyros, S. Shakhno, H. Yarmola

123

Using conditions (SLC₃) and (SLC₅), we obtain in turn that

kL⁻¹(F⁰(x_n) − L)k ≤ v₀(kx_n− x₀k) ≤ v₀(α_n) < 1.

(23)

By Banach Lemma [1] and (23) the operator F⁰(x_n)⁻¹∈ L(B₂, B₁) and

1

kF⁰(x_n)⁻¹Lk ≤

.

(24)

1 − v₀(kx_n− x₀k)

By the identity

3

F(x_n) = F(x_n) − F(x_n−1) − F⁰(x_n−1)(y_n−1− x_n−1

)

2

= F(x_n) − F(x_n−1) − F⁰(x_n−1)(x_n− x_n−1) + F⁰(x_n−1)(x_n− y_n−1

)

1

− F⁰(x_n−1)(y_n−1− x_n−1

)

2

1

Z

ꢂ

ꢃ

=

F⁰(x_n−1+ θ(x_n− x_n−1)) − F⁰(x_n−1) dθ(x_n− x_n−1

)

0

1

+F⁰(x_n−1)(x_n− y_n−1) − F⁰(x_n−1)(y_n−1− x_n−1),

2

we get

Z

ꢂ

ꢃ

1

kL⁻¹F(x_n)k ≤

v (1 − θ)(α_n− α_n−1) dθ(α_n− α_n−1

)

0

+(1 + v₀(α_n−1))(α_n− β_n−1

)

1

+ (1 + v₀(α_n−1))(β_n−1− α_n−1) = δ_n.

(25)

2

We have by the ﬁrst substep of the method

2

δ_n

ky_n− x_nk ≤

kF⁰(x_n)⁻¹LkkL⁻¹F(x_n)k ≤

= β_n− α_n

3

3 1 − v₀(α_n)

and

ky_n− x₀k ≤ ky_n− x_nk + kx_n− x₀k ≤ β_n− α_n+ α_n= β_n< α^∗.

Subtracting the third and second substeps of the method gives

1

z_n− y_n= − F⁰(x_n)⁻¹F(x_n) + (I − T_n) F⁰(x_n)⁻¹F(x_n)

3

1

−

M_n⁻¹A_nT_nF⁰(x_n)⁻¹F(x_n).

(26)

2b

Extended Chebyshev-Halley-type methods

124

Using (SLC₅), (SLC₆) and (23), we have

kA_nk = kI − F⁰(x_n)⁻¹F⁰(y_n)k

= kF⁰(x_n)⁻¹(F⁰(x_n) − F⁰(x₀) + F⁰(x₀) − F⁰(y_n))k

ꢂ

≤ kF⁰(x_n)⁻¹Lk kL⁻¹(F⁰(x_n) − F⁰(x₀))k

ꢃ

+kL⁻¹(F⁰(x₀) − F⁰(y_n))k

v₀(kx_n− x₀k) + v₀(ky_n− x₀k)

v₀(α_n) + v₀(β_n)

≤

1 − v₀(kx_n− x₀k)

1 − v₀(α₀)

or

kA_nk = kI − F⁰(x_n)⁻¹F⁰(y_n)k = kF⁰(x_n)⁻¹(F⁰(x_n) − F⁰(y_n))k

v(kx_n− y_nk)

≤ kF⁰(x_n)⁻¹LkkL⁻¹(F⁰(x_n) − F⁰(y_n))k ≤

1 − v₀(kx_n− x₀k)

v₀(α_n− β_n)

≤

1 − v₀(α₀)

and

v(kx_n− y_nk)

v¯_n

kA_nk ≤

≤

.

(27)

1 − v₀(kx_n− x₀k)

1 − v₀(α)

Here v(kx_n− y_nk) = v₀(kx_n− x₀k) + v₀(ky_n− x₀k) or

v(kx_n− y_nk) = v(kx_n− y_nk).

From the following equality

I − T_n= −a₁A_n− a₂A²,

n

and using the estimate (27), we obtain that

ꢀ

ꢁ

2

v(kx_n− x₀k)

kI − T_nk ≤ |a₁|

+ |a₂|

(28)

(29)

1 − v₀(kx_n− x₀k)

≤ λ_n

and

kT_nk ≤ 1 + λ_n.

(30)

Moreover,

ꢇ

ꢄ ꢄ

c

v(kx_n− x₀k)

ꢇ

ꢄ ꢄ

A_n

≤

= µ_n< 1,

ꢇ

ꢄ ꢄ

b

b 1 − v₀(kx_n− x₀k)

ꢇ

ꢂ

ꢃ_−1ꢇ

c

I − A_n

b

1

kM_n⁻¹k = k(I − (I − M_n))⁻¹k =

≤

.

(31)

ꢇ

1 − µ_n

I.K. Argyros, S. Shakhno, H. Yarmola

125

Taking into account the equality (26), and estimates (23), (27), (28),

(30), (31), we get in turn

ꢀ

ꢁ

1

3

(1 + λ_n)v_n

3

2

kz_n− y_nk ≤

+ λ_n+

(β_n− α_n) = γ_n− β_n

2|b|(1 − v₀(α_n))(1 − µ_n)

and

kz_n− x₀k ≤ kz_n− y_nk + ky_n− x₀k ≤ γ_n− β_n+ β_n= γ_n< α^∗.

Since,

3

F(z_n) = F(z_n) − F(x_n) − F⁰(x_n)(y_n− x_n)

2

= F(z_n) − F(x_n) − F⁰(x_n)(z_n− x_n) + F⁰(x_n)(z_n− y_n)

1

− F⁰(x_n)(y_n− x_n)

2

ꢀ

ꢁ

Z

1

=

(F⁰(x_n+ θ(z_n− x_n)) − F⁰(x_n))dθ (z_n− x_n)

0

1

+F⁰(x_n)(z_n− y_n) − F⁰(x_n)(y_n− x_n).

2

Then, using (SLC₅) and (SLC₆), we have

Z

1

kL⁻¹F(z_n)k ≤

v((1 − θ)(γ_n− α_n))dθ(γ_n− α_n) + (1 + v₀(α_n))

0

1

×(γ_n− β_n) + (1 + v₀(α_n))(β_n− α_n) = ξ_n.

(32)

2

We get by the last substep of the method, (23), (27) and (32 )

ꢀ

ꢁ

v_n

ξ_n

kx_n+1− z_nk ≤ |p| + |q|

= α_n+1− γ_n

1 − v₀(α_n)

and

kx_n+1− x₀k ≤ kx_n+1− z_nk + kz_n− x₀k ≤ α_n+1− γ_n+ γ_n= α_n+1< α^∗.

Thus, the iterates y_n, z_n, x_n+1∈ U(x₀, α^∗).

It follows from the obtained estimates that the sequence {x_n} is complete

in a Banach space B₁. Hence, it converges to some point x^∗∈ U[x₀, α^∗].

Furthermore, by letting n → ∞ in (25 ) and using the continuity of F we

conclude that F(x^∗) = 0. Finally, by the estimate

kx_n+i− x_nk ≤ kx_n+i− x_n+i−1k + ... + kx_n+1− x_nk

≤ α_n+i− α_n+i−1+ ... + α_n+1− α_n= α_n+i− α_n,

we show (22) if i → ∞.

Extended Chebyshev-Halley-type methods

126

Proposition 2. Suppose:

(1) There exists a solution x_∗∈ U(x₀, %₄) of the equation (1) for some

%₄> 0.

(2) Condition (SLC₅) holds on U(x₀, %₄).

(3) There exist %₅> %₄such that

Z

ꢂ

ꢃ

1

v₀(1 − θ)%₅+ θ%₄dθ < 1.

(33)

0

Set Ω₃= U[x₀, %₅] ∩ Ω.

Then, the equation (1) has unique solution x^∗in the region Ω₃.

Proof. Suppose that there exists y^∗∈ Ω₃, y^∗= x^∗and F(y^∗) = 0. Deﬁne

1

R

the linear operator G by G =

F⁰(y^∗+ θ(x^∗− y^∗))dθ. Using (SLC₂) and

0

(33), we obtain

Z

1

kL⁻¹(G − F⁰(x₀))k ≤

v₀(ky^∗+ θ(x^∗− y^∗) − x₀k)dθ

0

Z

ꢂ

ꢃ

1

≤

v₀(1 − θ)ky^∗− x₀k + θkx^∗− x₀k dθ

Z

ꢂ

ꢃ

v₀(1 − θ)%₅+ θ%₄dθ < 1.

So, the linear operator G is invertible. Therefore, from the identity

y^∗− x^∗= G⁻¹(F(y^∗) − F(x^∗)) = G⁻¹(0 − 0) = 0,

we conclude that y^∗= x^∗.

4 Numerical examples

In this section, we give the results of verifying the convergence conditions of

the Theorems 1 and 2 for the considered method (2). The experiments were

conducted in GNU Octave 7.3.0 software. To stop the iterative process the

condition kx_n+1− x_nk ≤ ε was used. The calculations were performed with

ε = 10⁻⁸and the norms k · k_∞is used.

Example 1. Consider the nonlinear integral equations [1]

Z

1

F(x)(s) = x(s) − 5s

tx(t)³dt, s, t ∈ [0, 1].

0

Here, B₁= B₂= C[0, 1], Ω ⊆ C[0, 1] and the exact solution x^∗(s) = 0.

I.K. Argyros, S. Shakhno, H. Yarmola

127

The derivative of operator F is deﬁned by the following formula

Z

1

F⁰(x)h(s) = h(s) − 15s

tx(t)²h(t)dt, h ∈ C[0, 1]

0

and we have that F⁰(x^∗) = I (I is an identity operator),

Z

1

F⁰(x)h(s) − F⁰(y)h(s) = 15s

t(y(t) + x(t))(y(t) − x(t))h(t)dt.

0

Let’s choose L = F⁰(x^∗) and Ω = U(x^∗, 1) for a local case. Then, we have

ꢀ

ꢁ

2

ω₀(kx − x^∗k) = 7.5kx − x^∗k, ρ₀

=

, Ω₀

=

x^∗,

and a function

15

ω(kx − yk) = 2kx − yk, x, y ∈ Ω₀. The radii obtained for diﬀerent values of

parameters are given in Table 1.

Table 1: Radii for Example 1.

Parameters

Radius r^∗

5.3551e-02

5.1918e-02

5.1268e-02

b = 3, c = p = q = 1

b = 3, c = p = 1, q = 3/2

b = 1, c = p = 1, q = 3/2

b = −3, c = p = 1, q = 3/2 4.8902e-02

Example 2. Consider the system of m equations

m

X

x_j+ e^x− 1 = 0,

i = 1, . . . , m.

i

j=1

Here, B₁= B₂= IR^m, Ω ⊆ IR^mand the exact solution x^∗= (0, . . . , 0)^T.

For this problem elements of the Jacobian matrix are calculated by for-

mulas

ꢉ

e^x+ 1,

i = j,

2,

i = j,

i

F⁰(x)_ij=

and F⁰(x^∗)_ij=

1,

i = j,

1,

i = j.

Let’s choose L = F⁰(x^∗) and Ω = U(x^∗, 1) for a local case. Then, we

have

⁻¹(F⁰(x) − L) = L⁻¹diag {e^x− 1, . . . , e^x− 1} ,

1

m

L

x_m

1

L

⁻¹(F⁰(x) − F⁰(y)) = L⁻¹diag {e^x− e , . . . , e

y₁

− e^y} .

m

Extended Chebyshev-Halley-type methods

128

Therefore, functions ω₀and ω have the following form

ω₀(kx − x^∗k) = σ(e − 1)kx − x^∗k

and

ω(kx − yk) = σe^{min{1,% }}kx − yk,

.

0

where σ = (F⁰(x^∗))⁻₁₁¹

Local case. Let m = 5, b = 3, c = p = q = 1. Then, Ω₀≈ (x^∗, 0.6984),

r^∗≈ min{0.2658, 0.1393, 0.1267} ≈ 0.1267

and U[x^∗, r^∗] ≈ [−0.1267, 0.1267]^m⊂ Ω₀. The method (2) converges to the

exact solution at two iterations for starting approximation

x₀= (0.12, . . . , 0.12)^T

and error estimates (7)-(9) hold for each n ≥ 0 (see Table 2).

Table 2: Error estimates at iteration for Example 2.

n

0

1

2

kx_n− x^∗k

g₃^∗

ky_n− x^∗k

g₁^∗

kz_n− x^∗k

g₂^∗

1.2000e-01

–

4.0849e-02 6.8860e-02 2.6447e-07 9.7156e-02

1.8766e-09 1.2000e-01 6.2554e-10 1.0769e-09 7.3635e-17 1.5194e-09

7.0097e-17 1.8766e-09

3

Similar results are obtained for b = 3, c = p = 1 and q =

:

2

Ω₀≈ (x^∗, 0.6984), r^∗≈ min{0.2658, 0.1393, 0.1211} ≈ 0.1211 and

U[x^∗, r^∗] ≈ [−0.1211, 0.1211]^m⊂ Ω₀. Error estimates at iteration are given

in Table 3.

Table 3: Error estimates at iteration for Example 2.

n

0

1

2

kx_n− x^∗k

g₃^∗

ky_n− x^∗k

g₁^∗

kz_n− x^∗k

g₂^∗

1.2000e-01

–

4.0849e-02 6.7307e-02 2.6447e-07 1.2775e-01

6.3512e-11 1.2000e-01 2.1171e-11 3.5623e-11 7.6134e-17 4.6702e-11

7.0513e-17 6.3512e-11

0

In Tables 2 and 3 we use following notations: g₁^∗= g₁(r^∗)kx_n− x^∗k,

g₂^∗= g₂(r^∗)kx_n− x^∗k and g₃^∗= g₃(r^∗)kx_n−1− x^∗k.

I.K. Argyros, S. Shakhno, H. Yarmola

129

5 Conclusions

The paper studies the convergence of a three-step iterative method con-

taining inverse linear operators under the weak conditions. Moreover, these

conditions contain only operators that are in the method. A local and a

semilocal convergence analysis of this method under generalized Lipschitz

conditions for only the ﬁrst-order derivatives is presented. The regions of

convergence and uniqueness of the solution are established. The results of

a numerical experiment are also presented. The new technique is an alter-

native to expensive Taylor series usually employed to study the convergence

of iterative methods. The same technique is applicable to extend other

methods [4–14].

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