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Theory and Modern Applications

Newton-Kantorovich convergence theorem of a new modified Halley’s method family in a Banach space

Abstract

A Newton-Kantorovich convergence theorem of a new modified Halley’s method family is established in a Banach space to solve nonlinear operator equations. We also present the main results to reveal the competence of our method. Finally, two numerical examples arising in the theory of the radiative transfer, neutron transport and in the kinetic theory of gasses are provided to show the application of our theorem.

Introduction

In the last two centuries, remarkable contributions have been made to both the theory and application of nonlinear equations. Suppose that we have to find a solution of the nonlinear equation

F(x)=0,
(1)

where F is defined on an open convex subset Ω of a Banach space X with values in a Banach space Y.

These equations are increasingly used to model problems in engineering applications, such as material science, electrical engineering, civil engineering, chemical engineering, mechanics and numerical optimization. There are several iterative methods [16] used to find a solution of nonlinear equations. One of those iterative methods is the famous Newton’s method

x n + 1 = x n F ( x n ) 1 F( x n )(n0)( x 0 Ω)
(2)

often used to solve the nonlinear operator equation under the reasonable hypotheses. However, Newton’s method is only the second-order convergence. Kantorovich presented the famous convergence result [7], and afterward, many Newton-Kantorovich-type convergence theorems have been attained [817]. Furthermore, many deformed methods [1822] have been presented to improve the convergence order. The famous Halley’s method, which has been widely discussed [2330], is the third-order convergence. The famous Halley’s method is defined as

x n + 1 = x n [ I + 1 2 L F ( x n ) ( I 1 2 L F ( x n ) ) 1 ] F ( x n ) 1 F( x n ),n=0,1,,

where

L F (x)= F ( x ) 1 F (x) F ( x ) 1 F(x),xΩ.

Now, we consider Halley’s method with a parameter λ in the form

x λ , n + 1 = x λ , n [ I + 1 2 L F ( x λ , n ) ( I λ L F ( x λ , n ) ) 1 ] F ( x λ , n ) 1 F( x λ , n ),n=0,1,.

One can see that Halley’s method and super-Halley’s method are the special cases for λ= 1 2 and λ=1. In this method, in every step, one needs to compute the second order derivatives of the function F. The computing cost will be the high. To avoid the computation of F ( x n ), and to maintain the high order convergence, many researchers have replaced the second order derivative with the first order divided differences. They presented the modified Halley’s methods with the parameters p, λ. Their modified Halley’s method is as follows [12]:

{ y n = x n F ( x n ) 1 F ( x n ) , H ( x n , y n ) = 1 p F ( x n ) 1 [ F ( x n + p ( y n x n ) ) F ( x n ) ] , λ [ 0 , 1 ] , p ( 0 , 1 ] , x n + 1 = y n 1 2 H ( x n , y n ) [ I λ H ( x n , y n ) ] ( y n x n ) .
(3)

For p= 1 2 , λ=0, the method becomes Chebyshev’s iterative method (see [13]). For p= 2 3 , λ=1, the method becomes inverse-free Jarratt iterative method (see [14, 15]). In this paper, we establish a Kantorovich-type third-order convergence theorem for this kind of method by using majorizing function to improve the result [12].

1 Main results

In this section, we establish a Newton-Kantorovich convergence theorem via majorizing function. Let g(t)= 1 6 K t 3 + 1 2 γ t 2 t+η, where K, γ, η are positive real numbers. Denote

α= 2 γ + γ 2 + 2 K ,β=α 1 6 K α 3 1 2 γ α 2 = 2 ( γ + 2 γ 2 + 2 K ) 3 ( γ + γ 2 + 2 K ) 2 .

Theorem 1 Suppose that X and Y are the Banach spaces, and Ω is an open convex subset of X, F:ΩXY has the second order Fréchet derivative, F ( x 0 ) 1 exists for x 0 Ω, and the following conditions hold:

F ( x 0 ) 1 F ( x 0 ) η , F ( x 0 ) 1 F ( x 0 ) γ , F ( x 0 ) 1 ( F ( x ) F ( y ) ) N x y , x , y Ω , 2 + 3 p 2 3 p N K , η < β , S ( x 0 , r 1 ) ¯ Ω ,
(4)

where r 1 r 2 are two positive real roots of the function g(t). Then, for 0<p< 2 3 , the sequence { x n } n 0 generated by (3) is well defined, x n S ( x 0 , r 1 ) ¯ and converges to the unique solution x of equation (1) in S( x 0 ,α).

Theorem 2 Suppose that X and Y are the Banach spaces, and Ω is an open convex subset of X, F:ΩXY has the third-order Fréchet derivative, F ( x 0 ) 1 exists for x 0 Ω, and the following conditions hold:

F ( x 0 ) 1 F ( x 0 ) η , F ( x 0 ) 1 F ( x 0 ) γ , F ( x 0 ) 1 F ( x ) N , F ( x 0 ) 1 ( F ( x ) F ( y ) ) L x y , x , y Ω , ( 1 + 2 p 2 ) L 6 γ ( 1 p ) + N K , η < β , S ( x 0 , r 1 ) ¯ Ω .
(5)

Then for 0<p 2 3 , the sequence { x n } n 0 generated by (3) is well defined, x n S ( x 0 , r 1 ) ¯ and converges to the unique solution x of equation (1) in S( x 0 ,α).

To prove Theorems 1 and 2, we first give some lemmas.

Lemma 1 If ηβ, the polynomial g(t) has two positive real roots r 1 , r 2 (let 0< r 1 < r 2 <+), and a negative real root r 0 ( r 0 >0).

Proof From definition of the function g(t), there follows that g(0)=η>0, lim t g(t)=, hence g(t) has a negative root. Denote it r 0 . We get that g (t)= 1 2 K t 2 +γt1 has the unique positive root α= 2 γ + γ 2 + 2 K , and for t0, g (t)=Kt+γ>0. So, the necessary and sufficient condition that g(t) has two positive roots for t0 is that the minimum of g(t) satisfies g(α)0, that is also ηβ. This completes the proof of Lemma 1. □

Lemma 2 (see [12])

Suppose that the sequences { t n } n 0 and { s n } n 0 are generated by the following iteration t 0 =0,

{ s n = t n g ( t n ) 1 g ( t n ) , H g ( t n , s n ) = 1 p g ( t n ) 1 [ g ( t n + p ( s n t n ) ) g ( t n ) ] , t n + 1 = s n 1 2 H g ( t n , s n ) [ I λ H g ( t n , s n ) ] ( s n t n ) .
(6)

Then for ηβ, { t n }, { s n } are increasing and converge to r 1 .

Lemma 3 Suppose that F(x) satisfies conditions (4) of Theorem  1, xB( x 0 , r 1 ), F ( x ) 1 exists and satisfies the following inequalities:

( I ) F ( x ) 1 F ( x 0 ) g ( x x 0 ) 1 , ( II ) F ( x 0 ) 1 F ( x ) g ( x x 0 ) .

Proof

F ( x 0 ) 1 F ( x ) = F ( x 0 ) 1 F ( x 0 ) + F ( x 0 ) 1 [ F ( x ) F ( x 0 ) ] γ + N x x 0 γ + K x x 0 = g ( x x 0 ) .

By the proof process of Lemma 1, we get g (t)<0, t[0, r 1 ). Hence, for xB( x 0 , r 1 ),

F ( x 0 ) 1 F ( x ) I = F ( x 0 ) 1 [ F ( x ) F ( x 0 ) F ( x 0 ) ( x x 0 ) + F ( x 0 ) ( x x 0 ) ] 0 1 F ( x 0 ) 1 [ F ( x 0 + t ( x x 0 ) ) F ( x 0 ) ] d t ( x x 0 ) + γ x x 0 0 1 N t d t x x 0 2 + γ x x 0 1 2 K x x 0 2 + γ x x 0 = 1 + g ( x x 0 ) < 1 .

By the Banach theorem, we know ( F ( x 0 ) 1 F ( x ) ) 1 = F ( x ) 1 F ( x 0 ) exists, and

F ( x ) 1 F ( x 0 ) 1 1 I F ( x 0 ) 1 F ( x ) g ( x x 0 ) 1 .

This completes the proof of Lemma 3. □

Lemma 4 Suppose that the nonlinear operator F:ΩXY is defined on an open convex subset Ω of a Banach space X with values in a Banach space Y, F has the second-order Frechét derivative, and the sequences { x n }, { y n } are generated by (3). Then the following formula holds for all natural numbers n:

F ( x n + 1 ) = 0 1 F ( y n + t ( x n + 1 y n ) ) ( 1 t ) d t ( x n + 1 y n ) 2 + 0 1 [ F ( x n + t ( y n x n ) ) ( 1 t ) 1 2 F ( x n + p t ( y n x n ) ) ] d t ( y n x n ) 2 1 λ 2 0 1 F ( x n + p t ( y n x n ) ) d t ( y n x n ) H ( x n , y n ) ( y n x n ) 1 2 0 1 [ F ( x n + t ( y n x n ) ) F ( x n + p t ( y n x n ) ) ] d t ( y n x n ) × H ( x n , y n ) ( y n x n ) + λ 2 0 1 F ( x n + t ( y n x n ) ) d t ( y n x n ) H ( x n , y n ) H ( x n , y n ) ( y n x n ) .

Proof

F ( x n + 1 ) = F ( x n + 1 ) F ( y n ) F ( y n ) ( x n + 1 y n ) + F ( y n ) + F ( y n ) ( x n + 1 y n ) F ( x n + 1 ) = 0 1 F ( y n + t ( x n + 1 y n ) ) ( 1 t ) d t ( x n + 1 y n ) 2 + F ( y n ) + F ( y n ) ( x n + 1 y n ) , F ( x n ) H ( x n , y n ) = 1 p [ F ( x n + p ( y n x n ) ) F ( x n ) ] = 0 1 F ( x n + p t ( y n x n ) ) d t ( y n x n ) , F ( y n ) + F ( y n ) ( x n + 1 y n ) = F ( y n ) F ( x n ) F ( x n ) ( y n x n ) 2 1 2 F ( y n ) H ( x n , y n ) [ I λ H ( x n , y n ) ] ( y n x n ) = 0 1 F ( x n + t ( y n x n ) ) ( 1 t ) d t ( y n x n ) 2 1 2 [ F ( y n ) F ( x n ) ] H ( x n , y n ) [ I λ H ( x n , y n ) ] ( y n x n ) 1 2 F ( x n ) H ( x n , y n ) [ I λ H ( x n , y n ) ] ( y n x n ) = 0 1 F ( x n + t ( y n x n ) ) ( 1 t ) d t ( y n x n ) 2 1 2 0 1 F ( x n + p t ( y n x n ) ) d t ( y n x n ) 2 + λ 2 0 1 F ( x n + p t ( y n x n ) ) d t ( y n x n ) H ( x n , y n ) ( y n x n ) 1 2 0 1 F ( x n + t ( y n x n ) ) d t ( y n x n ) H ( x n , y n ) ( y n x n ) + λ 2 0 1 F ( x n + t ( y n x n ) ) d t ( y n x n ) H ( x n , y n ) H ( x n , y n ) ( y n x n ) .

Hence,

F ( x n + 1 ) = 0 1 F ( y n + t ( x n + 1 y n ) ) ( 1 t ) d t ( x n + 1 y n ) 2 + 0 1 [ F ( x n + t ( y n x n ) ) ( 1 t ) 1 2 F ( x n + p t ( y n x n ) ) ] d t ( y n x n ) 2 1 λ 2 0 1 F ( x n + p t ( y n x n ) ) d t ( y n x n ) H ( x n , y n ) ( y n x n ) 1 2 0 1 [ F ( x n + t ( y n x n ) ) F ( x n + p t ( y n x n ) ) ] d t ( y n x n ) × H ( x n , y n ) ( y n x n ) + λ 2 0 1 F ( x n + t ( y n x n ) ) d t ( y n x n ) H ( x n , y n ) H ( x n , y n ) ( y n x n ) .

This completes the proof of Lemma 4. □

Proof of Theorem 1 By induction, we can prove, for n0, that the following formulae hold:

( I n ) : x n S ( x 0 , t n ) ¯ , ( II n ) : F ( x n ) 1 F ( x 0 ) g ( t n ) 1 , F ( x 0 ) 1 F ( x n ) g ( t n ) , ( III n ) : y n x n s n t n , ( IV n ) : y n S ( x 0 , s n ) ¯ , ( V n ) : x n + 1 y n t n + 1 s n .

In fact, by Lemma 2, we know that { t n } is increasing and converges to the minimum positive root of the function g(t). Hence, t n < r 1 for all natural numbers n. It is easy to verify it for the case n=0. By using mathematical induction, we now suppose the formulae above also hold for n0. Then

( I n + 1 ): x n + 1 x 0 x n + 1 y n + y n x 0 t n + 1 s n + s n = t n + 1 .

By Lemma 3, and the fact that g ( t ) 1 , g (t) are increasing on [0, r 1 ], we get ( II n + 1 ).

( III n + 1 ) : 0 1 F ( x 0 ) 1 [ F ( y n + t ( x n + 1 y n ) ) ( 1 t ) d t ( x n + 1 y n ) 2 0 1 g ( y n x 0 + t ( x n + 1 y n ) ) ( 1 t ) d t ( t n + 1 s n ) 2 0 1 g ( s n + t ( t n + 1 s n ) ) ( 1 t ) d t ( t n + 1 s n ) 2 , 0 1 F ( x 0 ) 1 [ F ( x n + t ( y n x n ) ) ( 1 t ) 1 2 F ( x n + p t ( y n x n ) ) ] d t 0 1 F ( x 0 ) 1 [ F ( x n + t ( y n x n ) ) F ( x n ) ] ( 1 t ) d t + 1 2 0 1 F ( x 0 ) 1 [ F ( x n + p t ( y n x n ) ) F ( x n ) ] d t ( 2 + 3 p ) N 12 y n x n , 0 1 F ( x 0 ) 1 [ F ( x n + t ( y n x n ) ) F ( x n + p t ( y n x n ) ) ] d t 0 1 N ( 1 p ) t y n x n d t ( 1 p ) N 2 ( s n t n ) , H ( x n , y n ) = F ( x n ) 1 0 1 F ( x n + p t ( y n x n ) ) ( y n x n ) d t H ( x n , y n ) = F ( x n ) 1 F ( x 0 ) 0 1 F ( x 0 ) 1 F ( x n + p t ( y n x n ) ) ( y n x n ) d t H ( x n , y n ) g ( t n ) 1 0 1 g ( t n + p t ( s n t n ) ) d t ( s n t n ) = H g ( t n , s n ) .

From Lemmas 3 and 4, we get

F ( x 0 ) 1 F ( x n + 1 ) 0 1 g ( s n + t ( t n + 1 s n ) ) ( 1 t ) d t ( t n + 1 s n ) 2 + ( 2 + 3 p ) N 12 ( s n t n ) 3 + 1 λ 2 0 1 g ( t n + p t ( s n t n ) ) d t ( H g ( t n , s n ) ) ( s n t n ) 2 + 1 2 ( 1 p ) N 2 ( H g ( t n , s n ) ) ( s n t n ) 3 + λ 2 0 1 g ( t n + t ( s n t n ) ) d t ( s n t n ) 2 H g 2 ( t n , s n ) 0 1 g ( s n + t ( t n + 1 s n ) ) ( 1 t ) d t ( t n + 1 s n ) 2 + ( 2 3 p ) 12 2 + 3 p 2 3 p N ( s n t n ) 3 1 λ 2 0 1 g ( t n + p t ( s n t n ) ) d t H g ( t n , s n ) ( s n t n ) 2 1 2 ( 1 p ) K 2 H g ( t n , s n ) ( s n t n ) 3 + λ 2 0 1 g ( t n + t ( s n t n ) ) d t ( s n t n ) 2 H g 2 ( t n , s n ) 0 1 g ( s n + t ( t n + 1 s n ) ) ( 1 t ) d t ( t n + 1 s n ) 2 + ( 2 3 p ) K 12 ( s n t n ) 3 1 λ 2 0 1 g ( t n + p t ( s n t n ) ) d t H g ( t n , s n ) ( s n t n ) 2 1 2 ( 1 p ) K 2 H g ( t n , s n ) ( s n t n ) 3 + λ 2 0 1 g ( t n + t ( s n t n ) ) d t ( s n t n ) 2 H g 2 ( t n , s n ) = g ( t n + 1 ) .

Hence, we get

y n + 1 x n + 1 F ( x n + 1 ) 1 F ( x 0 ) F ( x 0 ) 1 F ( x n + 1 ) y n + 1 x n + 1 g ( t n + 1 ) 1 g ( t n + 1 ) = s n + 1 t n + 1 , ( IV n + 1 ) : y n + 1 x 0 y n + 1 x n + 1 + x n + 1 x 0 ( s n + 1 t n + 1 ) + t n + 1 = s n + 1 , ( V n + 1 ) : x n + 2 y n + 1 = 1 2 H ( x n + 1 , y n + 1 ) [ I λ H ( x n + 1 , y n + 1 ) ] ( y n + 1 x n + 1 ) x n + 2 y n + 1 1 2 H ( x n + 1 , y n + 1 ) [ 1 + λ H ( x n + 1 , y n + 1 ) ] ( y n + 1 x n + 1 ) x n + 2 y n + 1 1 2 H g ( t n + 1 , s n + 1 ) [ 1 λ H g ( t n + 1 , s n + 1 ) ] ( s n + 1 t n + 1 ) x n + 2 y n + 1 = t n + 2 s n + 1 .

So, the sequence { x n } n 0 generated by (3) is well defined, x n S ( x 0 , r 1 ) ¯ and { x n } converges to the solution x of equation (1) on S ( x 0 , r 1 ) ¯ . Now, we prove the uniqueness. If y is also the solution of equation (1) in S( x 0 ,α), then, by the proof of Lemma 1, we know that g (t)<0, t[0,α).

Thus,

F ( x 0 ) 1 0 1 F ( x + t ( y x ) ) d t I = 0 1 F ( x 0 ) 1 { F [ x + t ( y x ) ] F ( x 0 ) } d t 0 1 0 1 F ( x 0 ) 1 F { x 0 + s [ x + t ( y x ) x 0 ] } x x 0 + t ( y x ) d s d t 0 1 0 1 g [ s x x 0 + t ( y x ) ] x x 0 + t ( y x ) d s d t = 0 1 { g [ x x 0 + t ( y x ) ] g ( 0 ) } d t = 0 1 { g [ ( 1 t ) ( x x 0 ) + t ( y x 0 ) ] } d t + 1 < 1 .

By the Banach theorem, we know the inverse of

0 1 F [ x + t ( y x ) ] dt

exists. Since

0=F ( y ) F ( x ) = 0 1 F [ x + t ( y x ) ] dt ( y x ) ,

we have y = x . This completes the proof of uniqueness. Thus, the proof of Theorem 1 is complete. □

Proof of Theorem 2 We know that F:ΩXY has the three-order Fréchet derivative. Then

H g ( x n , y n ) = g ( t n ) 1 0 1 g ( t n + p t ( s n t n ) ) d t ( s n t n ) H g ( x n , y n ) = 1 1 γ t n 1 2 t n 2 0 1 [ K ( t n + p t ( s n t n ) ) + γ ] d t ( s n t n ) γ ( s n t n ) > 0 , 0 1 F ( x 0 ) 1 [ F ( x n + t ( y n x n ) ) ( 1 t ) 1 2 F ( x n + p t ( y n x n ) ) ] d t 0 1 0 1 F ( x 0 ) 1 [ F ( x n + σ t ( y n x n ) ) F ( x n ) ] t ( 1 t ) d σ d t y n x n + p 2 0 1 0 1 F ( x 0 ) 1 [ F ( x n + p σ t ( y n x n ) ) F ( x n ) ] t d σ d t y n x n + 2 3 p 12 F ( x 0 ) 1 F ( x n ) y n x n ( 1 + 2 p 2 ) L 24 y n x n 2 + 2 3 p 12 N y n x n , ( 1 + 2 p 2 ) L 24 y n x n 4 + 1 2 ( 1 p ) N 2 ( H g ( t n , s n ) ) ( s n t n ) 3 [ ( 1 + 2 p 2 ) L 6 ( 1 p ) ( s n t n ) ( H g ( t n , s n ) ) + N ] ( 1 p ) 4 ( H g ( t n , s n ) ) ( s n t n ) 3 [ ( 1 + 2 p 2 ) L 6 γ ( 1 p ) + N ] ( 1 p ) 4 ( H g ( t n , s n ) ) ( s n t n ) 3 ( 1 p ) 4 K H g ( t n , s n ) ( s n t n ) 3 .

Hence,

F ( x 0 ) 1 F ( x n + 1 ) 0 1 g ( s n + t ( t n + 1 s n ) ) ( 1 t ) d t ( t n + 1 s n ) 2 + ( 1 + 2 p 2 ) L 24 y n x n 4 + 2 3 p 12 N y n x n 3 + 1 λ 2 0 1 g ( t n + p t ( s n t n ) ) d t ( H g ( t n , s n ) ) ( s n t n ) 2 + 1 2 ( 1 p ) N 2 ( H g ( t n , s n ) ) ( s n t n ) 3 + λ 2 0 1 g ( t n + t ( s n t n ) ) d t ( s n t n ) 2 H g 2 ( t n , s n ) 0 1 g ( s n + t ( t n + 1 s n ) ) ( 1 t ) d t ( t n + 1 s n ) 2 + ( 2 3 p ) 12 K ( s n t n ) 3 1 λ 2 0 1 g ( t n + p t ( s n t n ) ) d t H g ( t n , s n ) ( s n t n ) 2 ( 1 p ) K 4 H g ( t n , s n ) ( s n t n ) 3 + λ 2 0 1 g ( t n + t ( s n t n ) ) d t ( s n t n ) 2 H g 2 ( t n , s n ) = g ( t n + 1 ) .

Using the same proof method as in Theorem 1, we get assertion of Theorem 2. □

2 Numerical examples

In this section, we apply the convergence ball result and show two numerical examples.

Example 1 Suppose that F(x)= 1 6 x 3 + 1 6 x 2 5 6 x+ 1 3 =0, we consider initial point x 0 =0, Ω=[1,1]. We can choose

η=γ= 2 5 ,N= 6 5 ,L=0.
(7)

Hence,

K=N= 6 5 ,β= 2 ( γ + 2 γ 2 + 2 K ) 3 ( γ + γ 2 + 2 K ) 2 = 3 5 ,η<β.
(8)

Moreover, by Theorem 2, we get that the sequence x n (n0) generated by (3) is well defined and convergent.

Example 2 Consider the following integral equations

x(s)=1+ 1 4 x(s) 0 1 s s + t x(t)dt
(9)

and the space X=C[0,1] with the norm

x= max 0 s 1 | x ( s ) | .
(10)

This equation arises in the theory of the radiative transfer, neutron transport and in the kinetic theory of gasses. Let us define the operator F on X by

F(x)= 1 4 x(s) 0 1 s s + t x(t)dtx(s)+1.
(11)

Then, for x 0 =1, we get the following results:

N = 0 , L = 0 , K = 0 , F ( x 0 ) 1 = 1.5304 , η = F ( x 0 ) 1 F ( x 0 ) = 0.2652 , γ = F ( x 0 ) 1 F ( x 0 ) = 1.5304 × 2 1 4 max 0 s 1 | 0 1 s s + t d t | = 0.5303 , 2 ( γ + 2 γ 2 + 2 K ) 3 ( γ + γ 2 + 2 K ) 2 = 0.9429 > η ,

this means that the hypotheses of Theorem 2 hold.

References

  1. Ortega JM, Rheinbolt WC: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York; 1970.

    Google Scholar 

  2. Cordero A, Hueso JL, Martinez E, Torregrosa JR: Increasing the convergence order of an iterative method for nonlinear systems. Appl. Math. Lett. 2012, 25: 2369-2374. 10.1016/j.aml.2012.07.005

    Article  MathSciNet  Google Scholar 

  3. Babajee DKR, Cordero A, Soleymani F, Torregrosa JR: On a novel fourth-order algorithm for solving systems of nonlinear equations. J. Appl. Math. 2012., 2012: Article ID 165452

    Google Scholar 

  4. Cordero A, Torregrosa JR, Vindel P: Study of the dynamics of third-order iterative methods on quadratic polynomials. Int. J. Comput. Math. 2012, 89: 1826-1836. 10.1080/00207160.2012.687446

    Article  MathSciNet  Google Scholar 

  5. Soleymani F: Some efficient seventh-order derivative-free families in root-finding. Opusc. Math. 2013, 33: 163-173. 10.7494/OpMath.2013.33.1.163

    Article  MathSciNet  Google Scholar 

  6. Soleymani F, Karimi Vanani S: A modified eighth-order derivative-free root solver. Thai J. Math. 2012, 10: 541-549.

    MathSciNet  Google Scholar 

  7. Kantorovich L: On Newton method. Tr. Mat. Inst. Steklova 1949, 28: 104-144.

    Google Scholar 

  8. Wu QB, Zhao YQ: Third-order convergence theorem by using majorizing function for a modified Newton method in Banach space. Appl. Math. Comput. 2006, 175: 1515-1524. 10.1016/j.amc.2005.08.043

    Article  MathSciNet  Google Scholar 

  9. Ezquerro JA, Hemández MA: On the R-order of Halley method. J. Math. Anal. Appl. 2005, 303: 591-601. 10.1016/j.jmaa.2004.08.057

    Article  MathSciNet  Google Scholar 

  10. Gutierrez JM, Hernandez MA: Recurrence relations for the Super-Halley method. Comput. Math. Appl. 1998, 36: 1-8.

    Article  MathSciNet  Google Scholar 

  11. Wu QB, Zhao YQ: Newton-Kantorovich-type convergence theorem for a family of new deformed Chebyshev method. Appl. Math. Comput. 2007, 192: 405-412. 10.1016/j.amc.2007.03.018

    Article  MathSciNet  Google Scholar 

  12. Guo X: The convergence for the second-order-derivative-free iterations. J. Eng. Math. 2001, 18: 29-34. (in Chinese)

    Google Scholar 

  13. Argyros IK, Chen D: Results on the Chebyshev method in Banach spaces. Proyecciones 1993, 12: 119-128.

    MathSciNet  Google Scholar 

  14. Ezquerro JA, et al.: The application of an inverse-free Jarratt-type approximation to nonlinear integral equations of Hammersteintype. Comput. Math. Appl. 1998, 36: 9-20.

    Article  MathSciNet  Google Scholar 

  15. Argyros IK: A new convergence theorem for the Jarratt method in Banach space. Comput. Math. Appl. 1998, 36: 13-18.

    Article  Google Scholar 

  16. Ezquerro JA, Gonzalez D, Hernandez MA: A modification of the classic conditions of Newton-Kantorovich for Newton’s method. Math. Comput. Model. 2013, 57: 584-594. 10.1016/j.mcm.2012.07.015

    Article  MathSciNet  Google Scholar 

  17. Ezquerro JA, Gonzalez D, Hernandez MA: A variant of the Newton-Kantorovich theorem for nonlinear integral equations of mixed Hammerstein type. Appl. Math. Comput. 2012, 218: 9536-9546. 10.1016/j.amc.2012.03.049

    Article  MathSciNet  Google Scholar 

  18. Homeier HHH: A modified Newton method with cubic convergence: the multivariate case. J. Comput. Appl. Math. 2004, 169: 161-169. 10.1016/j.cam.2003.12.041

    Article  MathSciNet  Google Scholar 

  19. Weerakoon S, Fernando TGI: A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 2000, 13: 87-93.

    Article  MathSciNet  Google Scholar 

  20. Frontini M, Sormani E: Some variant of Newton’s method with third-order convergence. Appl. Math. Comput. 2003, 140: 419-426. 10.1016/S0096-3003(02)00238-2

    Article  MathSciNet  Google Scholar 

  21. Kou JS, Li YT, Wang XH: On modified Newton methods with cubic convergence. Appl. Math. Comput. 2006, 176: 123-127. 10.1016/j.amc.2005.09.052

    Article  MathSciNet  Google Scholar 

  22. Chen M, Khan Y, Wu Q, Yildirim A: Newton-Kantorovich convergence theorem of a modified Newton’s method under the gamma-condition in a Banach space. J. Optim. Theory Appl. 2013. 10.1007/s10957-012-0237-9

    Google Scholar 

  23. Chen D, Argyros IK, Qian QS: A note on the Halley method in Banach spaces. Appl. Math. Comput. 2004, 58: 215-224.

    Article  MathSciNet  Google Scholar 

  24. Argyros IK: The super-Halley method using divided differences. Appl. Math. Lett. 1997, 10: 91-95.

    Article  Google Scholar 

  25. Gutiérrez JM, Hernández MA: Recurrence relations for the super-Halley method. Comput. Math. Appl. 1998, 36: 1-8.

    Article  Google Scholar 

  26. Hernández MA, Salanova MA: Indices of convexity and concavity: application to Halley method. Appl. Math. Comput. 1999, 103: 27-49. 10.1016/S0096-3003(98)10047-4

    Article  MathSciNet  Google Scholar 

  27. Ezquerro JA, Hernández MA: A modification of the super-Halley method under mild differentiability conditions. J. Comput. Appl. Math. 2000, 114: 405-409. 10.1016/S0377-0427(99)00348-9

    Article  MathSciNet  Google Scholar 

  28. Ezquerro JA, Hernández MA: On the R-order of the Halley method. J. Comput. Appl. Math. 2005, 303: 591-601.

    Google Scholar 

  29. Kou JS, Li YT, Wang XH: Modified Halley’s method free from second derivative. Appl. Math. Comput. 2006, 183: 704-708. 10.1016/j.amc.2006.05.097

    Article  MathSciNet  Google Scholar 

  30. Gutiérrez JM, Hernández MA: An acceleration of Newton’s method: super-Halley method. J. Appl. Math. Comput. 2001, 117: 223-239. 10.1016/S0096-3003(99)00175-7

    Article  Google Scholar 

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Acknowledgements

This work is supported by the National Basic Research 973 Program of China (No. 2011JB105001), the National Natural Science Foundation of China (Grant No. 11371320), the Foundation of Science and Technology Department (Grant No. 2013C31084) of Zhejiang Province and the Foundation of the Education Department (No. 20120040, Y201329420) of Zhejiang Province of China and by the Grant FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno University of Technology, Czech Republic.

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Lin, R., Zhao, Y., Šmarda, Z. et al. Newton-Kantorovich convergence theorem of a new modified Halley’s method family in a Banach space. Adv Differ Equ 2013, 325 (2013). https://doi.org/10.1186/1687-1847-2013-325

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