




































©2025 Ada Academica https://adac.eeEur. J. Math. Anal. 5 (2025) 18doi: 10.28924/ada/ma.5.18
On Local and Semi-Local Convergence Analysis of A High-Order Iterative Method for Solving

Nonlinear Systems Without High Derivatives

Ioannis K. Argyros1,∗, Stepan Shakhno2, Yurii Shunkin2, Samundra Regmi3,Christopher I. Argyros4
1Department of Computing and Mathematical Sciences, Cameron University, Lawton, OK 73505, USA

iargyros@cameron.edu
2Department of Theory of Optimal Processes, Ivan Franko National University of Lviv, Universytetska Str.

1, 79000 Lviv, Ukraine
stepan.shakhno@lnu.edu.ua, yuriy.shunkin@lnu.edu.ua

3Department of Mathematics, University of Houston, Houston, TX 77014, USA
sregmi5@uh.edu

4Georgia Institute of Technology, 225 North Avenue NW, Atlanta, GA 30313, USA
cargyros3@gatech.edu

∗Correspondence: iargyros@cameron.edu

Abstract. In this paper, we study a general high-order iterative method for solving nonlinear sys-tems in Banach spaces without requiring higher-order derivatives. The proposed method constructseach iteration by combining evaluations of the operator and its derivative, together with an adaptedcorrection scheme. A detailed local convergence analysis under majorant conditions is provided, es-tablishing the convergence to the solution. We also show a semi-local convergence by introducingnew majorizing sequences. The theoretical results are illustrated with examples, and confirm thetheoretical predictions.

1. Introduction
Numerous problems in applied mathematics, scientific computing, and engineering are modeledby nonlinear systems of the form

G : D ⊂ B0 → B,

where D is an open and convex subset of the Banach space B0, and B is another Banach space.The goal is to find x∗ ∈ D satisfying
G(x) = 0. (1)

Received: 1 Jun 2025.
Key words and phrases. high-order methods; nonlinear systems; semi-local convergence; iterative schemes.1

https://adac.ee
https://doi.org/10.28924/ada/ma.5.18


Eur. J. Math. Anal. 10.28924/ada/ma.5.18 2Finding exact analytical solutions to such nonlinear systems is typically difficult or impossible.Consequently, iterative methods are commonly employed. Newton’s method, defined by
xn+1 = xn − G′(xn)−1G(xn),

is among the most classical and efficient iterative schemes, offering quadratic convergence undersuitable conditions. However, many real-world problems demand faster convergence with lowercomputational cost, motivating the development of high-order methods.Following this line, we study a general high-order iterative method, which extends classicalschemes by incorporating additional correction steps while using only evaluations of G and G′(without requiring higher derivatives). Inspired by the high-order framework of Behl et al. [9], whichattains order 3(k−1) for systems in Rm by reusing a frozen inverse Jacobian and relying solely onfirst-order information, the present study generalizes that scheme to Banach spaces, discards theseventh-derivative assumptions underpinning their local Taylor analysis, and furnishes a unifiedlocal and semi-local convergence theory with computable error bounds, larger attraction regions,and sharpened uniqueness criteria—thereby achieving broader applicability.Let k ≥ 3 be a natural number and x0 ∈ D an initial point. Then, the method is defined for each
n = 0, 1, 2, . . . by

y
(1)
n = xn − G′(xn)−1G(xn),

y
(2)
n = xn − 2T−1G(xn),

y
(3)
n = y

(2)
n −MG(y

(2))
n ,

· · ·

xn+1 = y
(k)
n = y

(k−1)
n −MG(y

(k−1)
n ),

(2)

where the operators T , L, and M are given by
T = Tn = G′(xn) + G′(y

(1))
n ,

L = Ln = 3F ′(y
(2))
n − G′(xn),

M = Mn = L−1TG′(xn)−1.

The method (2) is shown in [9] to possess convergence order 3(j − 1) for j = 3, . . . , k usingTaylor expansions when B0 = B = Rm (m natural number), assuming the existence of at least theseventh derivative G(7), although the derivatives G′′, G(3), ..., G(7) are not explicitly required in theiteration steps.Unlike many classical methods that impose strong smoothness assumptions, this method is de-signed to work under weaker differentiability conditions. It builds upon ideas from previous studies,including Parhi and Gupta [14], Wang et al. [18], and Cordero et al. [10], while aiming for a betterbalance between convergence speed, computational cost, and robustness.

https://doi.org/10.28924/ada/ma.5.18


Eur. J. Math. Anal. 10.28924/ada/ma.5.18 3To illustrate the need for such an approach, consider the scalar function g : D → R defined by
g(t) =

b1t7 log t + b1t
8 + b2t

9, t 6= 0,

0, t = 0,

where D = [−2, 2), and b1, b2 are real constants with b1 6= 0 and b1 + b2 = 0. Although g hasa zero at t∗ = 1, the seventh derivative g(7)(t) does not exist at t = 0, showing that classicalconvergence assumptions based on higher derivatives are not satisfied.This observation motivates the use of generalized majorant conditions rather than strict smooth-ness hypotheses. Moreover, method not only achieves high-order local convergence but also pro-vides a semi-local convergence analysis by constructing suitable majorizing sequences.The main contributions of this paper are as follows:
• A local convergence analysis is established that depends only on G, G′, and suitablyconstructed auxiliary operators, thereby eliminating any need for higher-order derivatives.
• Semi-local convergence results are provided through the use of majorizing sequences, guar-anteeing convergence even when the initial guess is relatively far from the solution.
• Computable radii of convergence and explicit error bounds are derived, permitting a prioriestimates of the number of iterations required to achieve a prescribed accuracy.
• Conditions are specified that ensure the uniqueness of the solution within a neighborhoodof the limit point.The remainder of the paper is organized as follows. In Section 2, we introduce the assumptionsand establish the local convergence theorems, including uniqueness and error estimates. Sec-tion 3 presents the semi-local convergence analysis via majorizing sequences. Section 4 discussesexamples and practical aspects. Finally, conclusions are drawn in Section 5.

2. Convergence Analysis
2.1. Local. Some real functions which are defined on the interval A = [0,+∞) play a crucial rolein the local convergence analysis of the method (2).Suppose

(H1) There exists a nondecreasing and continuous function φ0 : A → A such that the function
1 − φ0(t) has a smallest positive zero in A, which is denoted by s0. Define the interval
A0 = [0, s0).

(H2) There exists a nondecreasing and continuous function φ : A0 → A such that for h1 : A0 → Adefined by
h1(t) =

1∫
0

φ((t − η)t)dη (3)

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Eur. J. Math. Anal. 10.28924/ada/ma.5.18 4the function 1− h1(t) has a smallest positive zero in the interval A0, which is denoted by
r1.

(H3) For p : A0 → A defined by
p(t) =

1

2
(φ0(t) + φ0(h1(t)t)) (4)

the function 1 − p(t) has a smallest positive zero in the interval A0, which is denoted by
s1. Define the interval A1 = [0, s1).

(H4) For φ : A1 → A, h2 : A1 → A, p : A1 → A, defined by
φ(t) =


φ((1 + h1(t))t)

or

φ0(t) + φ0(h1(t)t),

p(t) =
1

2
(φ0(t) + φ0(h1(t)t)) ,

and

h2(t) =

1∫
0

φ0((1− η)t)dη

1− φ0(t)
+

φ(t)

(
1 +

1∫
0

φ0(ηt)dη

)
2(1− φ0(t))(1− p(t))

,

the function 1− h2(t) has a smallest positive zero in the interval A1, which is denoted by
r2.

(H5) For q : A1 → A defined by
q(t) =

1

2
(φ0(t) + 3φ0(h1(t)t))

the function 1 − q(t) has a smallest positive zero, which is denoted by s2. Define theinterval A2 = [0, s2).
(H6) For j = 3, . . . , k , Aj−1 = [0, sj−1), hj : Aj−1 → Athe functions 1− φ0(hj−1(t)t) and 1− hj(t) have smallest positive zeros in the interval

Aj−1, which are denoted by sj−1 and rj , respectively, where
hj(t) =


1∫
0

φ0((1− η)hj−1(t)t)dη

1− φ0(hj−1(t)t)
+

(1 + φ0(h1(t)t))

(
1 +

1∫
0

φ0(ηhj−1(t)t)dη

)
2(1− φ0(t))(1− q(t))

 hj−1(t)
Define

r∗ = min{rj}, m = 1, 2, . . . , k and A∗ = [0, r∗) (5)
It follows by these definitions and conditions (H1)− (H6) that for each t ∈ A∗

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Eur. J. Math. Anal. 10.28924/ada/ma.5.18 5

0 ≤ φ0(t) < 1, (6)
0 ≤ p(t) < 1, (7)
0 ≤ q(t) < 1, (8)
0 ≤ φ0(hj−1(t)t) < 1, (9)and (10)
0 ≤ hj(t) < 1. (11)

Notice also that the parameter r is shown to be a radius of convergence for the method(2) (see Theorem 1).Next, we relate functions φ0 and φ to the operators in the method (2).
(H7) There exists a solution x∗ ∈ D and a linear operator E ∈ L(B0, B) which is invertible suchthat for each z ∈ D

‖E−1(G(z)− E)‖ ≤ φ0(‖z − x∗‖).

Define the region D0 = D∩U(x∗, s0), where U(x, s) stands for an open ball in B0 centeredat x and of some radius s > 0. The set U[x, s] denotes the closure of U(x, s), which is aclosed set.
(H8) ‖E−1(G′(z2)− G′(z1))‖ ≤ φ(‖z2 − z1‖) for each z1, z2 ∈ D0.
(H9) U[x∗, r∗] ⊂ D.

Remark 1. Some possible selections for the linear operator E can be E = I , the identity operator,or E = G′(z̄) for some z̄ ∈ D with z̄ 6= x∗ or E = G′(x∗). The last choice of E implies x∗ is asimple solution of the equation G(x) = 0. It is worth noting, though, that such an assumption isnot made or implied by the conditions (H1)–(H9).
The local convergence analysis of the method (2) is provided in the next result. Let U0 =

U(x∗, r∗)− {x∗}.
Theorem 1. Suppose that the conditions (H1)–(H9) hold. Then, the sequence {xn} generated for
the starting point x0 ∈ U0 is convergent to the solution x∗ of the equation G(x) = 0.

Proof. The following assertions shall be established using induction on n = 0, 1, 2, . . .

‖y (1)n − x∗‖ ≤ g1(‖xn − x∗‖)‖xn − x∗‖ ≤ ‖xn − x∗‖ < r∗, (12)
‖y (2)n − x∗‖ ≤ g2(‖xn − x∗‖)‖xn − x∗‖ ≤ ‖xn − x∗‖, (13)
‖y (j)n − x∗‖ ≤ gj(‖xn − x∗‖)‖xn − x∗‖ ≤ ‖xn − x∗‖, (14)

· · ·

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Eur. J. Math. Anal. 10.28924/ada/ma.5.18 6

‖xn+1 − x∗‖ = ‖y (k)n − x∗‖ ≤ gk(‖xn − x∗‖)‖xn − x∗‖ ≤ ‖xn − x∗‖. (15)
Pick u ∈ U0. It follows by conditions (H1), (H7), and (5), (6)

‖E−1(G′(u)− E)‖ ≤ φ0(‖u − x∗‖) ≤ φ0(r∗) < 1. (16)
So, the linear operator G′(u) is invertible by the lemma due to Banach [13] and

‖G′(u)−1E‖ ≤
1

1− φ0(‖u − x∗‖)
. (17)

In particular, if u = x0, the iterate y (1)0 exists by the first substep of the method (2) for n = 0,and we can write
y
(1)
0 − x

∗ = x0 − x∗ − G′(x0)−1G(x0)

=
[
G′(x0)

−1E
] 1∫

0

E−1(G′(x0 + η(x∗ − x0))− G′(x0))dη(x0 − x∗)

 (18)
Using the condition (H8), (5), (11) (for i = 1), (17), and (18), we get from (18)
‖y (1)0 −x

∗‖ ≤

1∫
0

φ((1− η)‖x0 − x∗‖)dη‖x0 − x∗‖

1− φ0(‖x0 − x∗‖)
≤ q1(‖x0−x∗‖)‖x0−x∗‖ ≤ ‖x0−x∗‖ < r∗. (19)

Thus, the assertion (12) holds if n = 0 and the iterate y (1)0 ∈ U0. Next, we show T0 is alsoinvertible. In view of the conditions (H7), (5), (7), and (19), we can have
‖(2E)−1(T0 − 2E)‖ ≤

1

2

(
φ0(‖x0 − x∗‖) + φ0(‖y (1)0 − x

∗‖)
)

≤
1

2

(
φ0(‖x0 − x∗‖) + φ0

(
g1(‖x0 − x∗‖)‖x0 − x∗‖

))
= p0 < 1.

Thus, the linear operator T0 is invertible and
‖T−10 E‖ ≤

1

2(1− p0)
. (20)

Moreover, the iterate y (2)0 is well defined by the second substep of method (2), from which wecan also write
y
(2)
0 − x

∗ = x0 − x∗ − G′(x0)−1G(x0) + (G′(x0)
−1 − 2T−10 )G(x0)

= x0 − x∗ − G′(x0)−1G(x0)− (2T−10 − G
′(x0)

−1)G(x0)

= x0 − x∗ − G′(x0)−1G(x0)− T−10 (G′(x0)− G′(y (1)0 )G′(x0))−1G(x0)

= x0 − x∗ − G′(x0)−1G(x0) + [T−10 E][E−1(G′(y
(1)
0 )− G′(x0))]G′(x0)

−1G(x0)

(21)

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Eur. J. Math. Anal. 10.28924/ada/ma.5.18 7By the conditions (H1), (H7), (H8), (17) (for u = x0), (19), (11) (for i = 2), (20) and (21), weobtain
‖y (2)0 − x

∗‖ ≤


1∫
0

φ((1− η)‖x0 − x∗‖)dη

1− φ0(‖x0 − x∗‖)
+

φ0(1 +
1∫
0

φ0(η‖x0 − x∗‖)dη)

2(1− φ0(‖x0 − x∗‖))(1− p0)

 ‖x0 − x∗‖
≤ g2(‖x0 − x∗‖)‖x0 − x∗‖ ≤ ‖x0 − x∗‖.

(22)

Hence, the assertion (13) holds if n = 0 and the iterate y (2)0 ∈ U0. Similarly, from (8) and (H7)we can have
‖(2E−1)(L− 2E)‖ =

1

2
‖E−1

(
3(G′(y

(1))
0 − E) + (G′(x0)− E)

)
‖

≤
1

2

(
3φ0(‖y (1)0 − x

∗‖) + φ0(‖x0 − x∗‖)
)
≤ q0 < 1,so

‖L−1E‖ ≤
1

2(1− q0)
. (23)

Thus, the iterates y (3)0 , . . . , y
(k)
0 = x1 exist, since L is invertible and we can write for j =

3, 4, . . . , k

y
(j)
0 − x

∗ = y
(j−1)
0 − x∗ − G′(y (j−1)0 )−1G(y

(j−1)
0 ) + (I −M0)G(y

(j−1)
0 )

= y
(j−1)
0 − x∗ − G′(y (j−1)0 )−1G(y

(j−1)
0 )− L−1G′(y (j−1)0 )G′(x0)

−1G(y
(2)
0 )

(24)
which can be implied by (5), (11), (22), and (23)

‖y (j)0 − x
∗‖ ≤


1∫
0

φ0((1− η)‖y (j−1)0 − x∗‖)dη

1− φ0(‖y (j−1)0 − x∗‖)

+

(1 + φ0(‖y
(j−1)
0 − x∗‖))(1 +

1∫
0

φ0(η‖y (j−1)0 − x∗‖)dη)

2(1− φ0(‖x0 − x∗‖))(1− q0)

 ‖y (j−1)0 − x∗‖

≤ gj(‖y (j−1)0 − x∗‖)‖y (j−1)0 − x∗‖ ≤ ‖x0 − x∗‖

(25)

where we have also used the estimates
‖E−1(G′(y (j)0 )− G′(x0))‖ ≤ φ(‖y (1)0 − x0‖) ≤ φ(‖y (1)0 − x

∗‖+ ‖x0 − x∗‖) ≤ φ0or
‖E−1(G′(y (1)0 )− G′(x0))‖ ≤ ‖E−1(G′(y (1)0 )− G′(x∗))‖+ ‖E−1(G′(x0)− G′(x∗))‖

≤ φ0(‖y (1)0 − x
∗‖) + φ0(‖x0 − x∗‖) ≤ φ0,

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Eur. J. Math. Anal. 10.28924/ada/ma.5.18 8

‖E−1G(y
(1)
0 )‖ = ‖E−1(G(y

(1)
0 )− E + E)‖ ≤ 1 + ‖E−1(G′(y (1)0 )− E)‖ ≤ 1 + φ0(‖y (1)0 − x

∗‖),

and
‖E−1G(y

(1)
0 )‖ =

∥∥∥∥∥∥
1∫
0

E−1(G′(x∗ + η(y
(1)
0 − x

∗)))dη(y
(1)
0 − x

∗)

∥∥∥∥∥∥
=

∥∥∥∥∥∥
1∫
0

E−1(G′(x∗ + η(y
(1)
0 − x

∗))− E + E)dη(y
(1)
0 − x

∗)

∥∥∥∥∥∥
≤
(

1 +

∫ 1
0

φ0(η‖y (1)0 − x
∗‖)dη

)
‖y (1)0 − x

∗‖.

Therefore, the assertions (14) and (15) hold if n = 0 and the iterate y (j,)0 x1 ∈ U0.The induction for assertions (12)–(15) is completed if x0, y
(1,)
0 y

(2,)
0 y

(j)
0 are replaced by

xj , y
(1,)
j y

(2,)
j y

(j)
j respectively.Furthermore, by estimate (15) and dk = gk(‖x0 − x∗‖) ∈ [0, 1), we can get

‖xn+1 − x∗‖ = ‖y (k)n − x∗‖

≤ dk‖y (k−1)n − x∗‖

≤ dkdk−1‖y (k−2)n − x∗‖

≤ · · · ≤ dkdk−1 · · · d3‖y (2)n − x∗‖

≤ dkdk−1 · · · d2‖xn − x∗‖.

(26)

It follows by the definition of dk that there exists d ∈ [0, 1) such that
d2, d3, . . . , dk ≤ d (27)

So, by (26) and (27), we get
‖xn+1 − x∗‖ ≤ dk−1‖xn − x∗‖ ≤ d (k−1)(n+1)‖x0 − x∗‖ < r∗. (28)

Finally, if we let n → +∞ in (28), we conclude that lim
n→+∞

xn = x∗, and all the iterates
{xn} ⊆ U0. �

The uniqueness of the solution x∗ is established in a neighborhood of it next.
Proposition 1. Suppose that the condition (H7) holds in the ball U(x∗, R1), for some R1 > 0 and
there exists R2 ≥ R1 such that

1∫
0

φ0(ηR2)dη < 1. (29)
Define the region D1 = D ∩ U[x∗, R2].

Then, x∗ is the only solution of the equation G(x) = 0 in the region D1.

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Eur. J. Math. Anal. 10.28924/ada/ma.5.18 9

Proof. Suppose that there exists a solution z∗ ∈ D1 of the equation G(x) = 0 such that z∗ 6= x∗.Then, define the linear operator
E1 =

1∫
0

G′(x∗ + η(z∗ − x∗))dη.

By this definition, the condition (H7) and (29) we get in turn
‖E−11 (E1 − E)‖ ≤

1∫
0

φ0(η‖z∗ − x∗‖)dη ≤
1∫
0

φ0(ηR2)dη < 1.

Thus, the linear operator E1 is invertible. It follows by the identity
z∗ − x∗ = E−11 (G(z∗)− G(x∗)) = E−11 (0) = 0,

and we conclude z∗ = x∗. �

Remark 2. Under all the conditions (H4)–(H9), one can set R1 = r∗ in Proposition 1.
2.2. Semi-local. The calculations and formulae are as in Section 2.1, but x∗, φ0, φ are exchangedby x0, ψ0, and ψ, respectively.Suppose

(C1) There exists a nondecreasing and continuous function ψ0 : A → A such that the function
1−ψ0(t) has a smallest positive solution in the interval A, which is denoted by t0. Definethe interval S = [0, t0).

(C2) There exists a nondecreasing and continuous function ψ : S → A.Define the sequences {αin} for α00 = 0, some α10 ≥ 0, i = 0, . . . , k , and each n =

0, 1, 2, . . . by
ψn =


ψ(α1n − α0n),or
ψ0(α

0
n) + ψ0(α

1
n),

pn =
1

2

(
ψ0(α

0
n) + ψ0(α

1
n)
)
,

α2n = α1n +
ψn(α1n − α0n)

2(1− pn)
,

(30)

λj−1n =

1∫
0

ψ((1− η)(αj−1n − α0n))dη · (αj−1n − α0n) + (1 + ψ0(α
0
n))(αj−1n − α1n),

qn =
1

2

(
3ψ0(α

1
n) + ψ0(α

0
n)
)
,

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Eur. J. Math. Anal. 10.28924/ada/ma.5.18 10

αjn = αj−1n +
(ψ0(α

0
n) + ψ0(α

1
n) + 2)λj−1n

2(1− ψ0(α0n))(1− qn)
,

µn+1 =

1∫
0

ψ((1− η)(α0n+1 − α0n))dη(α0n+1 − α0n) + (1 + ψ0(α
0
n))(α0n − α0n),

and
α1n+1 = α0n+1 +

µn+1

1− ψ0(α0n+1)
,

where again α0n+1 = αkn .The sequence {αin}n is shown to be majorizing for {y (i)n }n in Theorem 2.But let us first provide a convergence condition for it.
(C3) There exists t ∈ [0, t0) such that for each i = 0, 1, 2, . . . , k and each n = 0, 1, 2, . . .

ψ0(α
0
n) < 1, pn < 1, qn < 1, and αin ≤ t.

It follows by this condition and (30) that the sequence {αin} is nondecreasing andbounded from above by t and as such it converges to some α∗ ∈ [0, t]. The limit point α∗is the unique least upper bound of the sequence {αin}.As in the local analysis, the operators on the method (2) connect to the functions ψ0 and
ψ.

(C4) There exist x0 ∈ D and a linear operator E such that for each u ∈ D
‖E−1(G′(u)− E)‖ ≤ ψ0(‖u − x0‖).

It follows by the conditions (C1), (C4), and (30) that if u = x0, we get
‖E−1(G′(x0)− E)‖ ≤ ψ0(0) < 1.

So, the linear operator G′(x0) is invertible, in which case we can take
α10 ≥ ‖G′(x0)−1G(x0)‖.

Define the region
D2 = U[x0, α

∗] ∩D.

(C5)

‖E−1(G′(u2)− G′(u1))‖ ≤ ψ(‖u2 − u1‖), for each u2, u1 ∈ D2.

(C6)

U[x0, α
∗] ⊂ D.

Remark 3. As in the local analysis, possible selections for E can be E = I or E = G′(z̄) for someauxiliary point x̄ ∈ D such that x̄ 6= x0, or E = G′(x0), or some other selection.

https://doi.org/10.28924/ada/ma.5.18


Eur. J. Math. Anal. 10.28924/ada/ma.5.18 11The semi-local analysis of the method (2) follows in the next result.
Theorem 2. Suppose the conditions (C1) − (C6) hold. Then, the sequence {xn} generated by
the method (2) is well-defined in U(x0, α

∗), remains in U(x0, α
∗), and is convergent to a solution

x∗ ∈ U[x0, α
∗] of the equation G(x) = 0 such that for each n = 0, 1, 2, . . .

‖x∗ − xn‖ ≤ α∗αn.

Proof. As in the local analysis, induction is used to first establish the assertions
‖y (1)n − xn‖ ≤ α1n − α0n, (31)
‖y (2)n − y (1)n ‖ ≤ α2n − α1n, (32)
‖y (j)n − y (j−1)n ‖ ≤ αjn − αj−1n . (33)

By switching the conditions (H1)− (H9) by (C1)− (C5) but using the same formulas, we get inturn
y
(2)
n − y (1)n = T

(
G′(y

(1)
n )− G′(xn)

)
G′(xn)−1G(xn)

= −
[
TE−1

] [
E−1(G′(y

(1)
n )− G′(xn))

]
(y
(1)
n − xn),

‖y (2)n − y (1)n ‖ ≤
ψn(α1n − α0n)

2(1− pn)
≤ α2n − α1n,

‖y (2)n − x0‖ ≤ ‖y (2)n − y (1)n ‖+ ‖y (1)n − x0‖

≤ α2n − α1n + α1n − α0n = α2n < α∗.

So, the estimate (32) holds and the iterate y (2)n ∈ U[x0, α
∗].Then, by the identity

G(y
(j−1)
n ) = G(y

(j−1)
n )− G(xn)− G′(xn)(y

(1)
n − xn),

= G(y
(j−1)
n )− G(xn)− G′(xn)(y

(j−1)
n − xn) + G′(xn)(y

(j−1)
n − y (1)n ),

which can imply
‖E−1G(y

(j−1)
n )‖ ≤

1∫
0

ψ((1−η)(αj−1n −α0n))dη(αj−1n −α0n)+(1+ψ0(α
0
n))(αj−1n −α1n) = λj−1n (34)

‖y (j)n − y (j−1)n ‖ ≤
(ψ0(α

0
n) + ψ0(α

1
n) + 2)λj−1n

2(1− ψ0(α0n))(1− qn)
= αjn − αj−1n .

Thus,
‖y (j)n − x0‖ ≤ ‖y (j)n − y (j−1)n ‖+ ‖y (j−1)n − x0‖ ≤ αjn − αj−1n + αj−1n − α00 = αjn < α∗.

Thus, the assertions (33) hold and all the iterates {y (j)n } ⊂ U(x0, α
∗).It is left to show that assertion (31) holds if n + 1 replaces n.

https://doi.org/10.28924/ada/ma.5.18


Eur. J. Math. Anal. 10.28924/ada/ma.5.18 12But we can write in turn
G(xn+1) = G(xn+1)− G(xn)− G′(xn)(y1 − xn)

= G(xn+1)− G(xn)− G′(xn)(xn+1 − xn) + G′(xn)(xn+1 − y (1)n ),which can give, as in (34),
‖E−1G(xn+1)‖ ≤

1∫
0

ψ((1−η)(α0n+1−α0n))dη(α0n+1−α0n)+(1+ψ0(α
0
n))(α0n+1−α1n) = µn+1. (35)

Consequently, we obtain
‖y (1)n+1 − xn+1‖ ≤ ‖G

′(xn+1)
−1E‖‖E−1G(xn+1)‖,

≤
µn+1

1− ψ0(α0n+1)
= α1n+1 − α0n+1

and
‖y (1)n+1 − x0‖ ≤ ‖y

(1)
n+1 − xn+1‖+ ‖xn+1 − x0‖

≤ (α1n+1 − α0n+1) + (α0n+1 − α00) = α1n+1 < α∗.

Thus, the induction for assertions (31)–(33) is completed, and all the iterates {y (i)n } ∈ U(x0, α
∗).It also follows that the sequence {x jn} is complete in Banach space B0, since {αin} is alsocomplete as convergent by the condition (C4). Therefore, there exists x∗ ∈ U[x0, α

∗] such that
lim
n→+∞

y
(k)
n = x∗ or lim

n→+∞
xn = x∗.

Moreover, by letting n → +∞ in (35), we obtain G(x∗) = 0, where the continuity of the operator
G has also been used. Furthermore, by noticing that αkn = αn+1 and αkn = α0n+1, estimate (33)can be rewritten for j = k as

‖xn+1 − xn‖ ≤ αn+1 − αn,so
‖xn+h − xn‖ ≤ αn+h − αn, h = 0, 1, 2, . . . (36)Finally, by letting h → +∞ in (36), we show the assertion (2).

�

Next, we study the uniqueness of a solution in a certain region.
Proposition 2. Suppose there exists a solution y∗ ∈ U(x0, R3) of the equation G(x) = 0 for some
R3 > 0; the condition (C4) holds in the ball U(x0, R3), and there exists R4 ≥ R3 such that

1∫
0

ψ0((1− η)R3 + ηR4)dη < 1. (37)
Define the region D3 = D ∩ U[x0, R4].

Then, y∗ is the only solution of the equation G(x) = 0 in the region D3.

https://doi.org/10.28924/ada/ma.5.18


Eur. J. Math. Anal. 10.28924/ada/ma.5.18 13

Proof. Suppose there exists y∗∗ ∈ D3 solving the equation G(x) = 0 such that y∗∗ 6= y∗.Then define the linear operator
E2 =

1∫
0

G′(y∗ + η(y∗∗ − y∗))dη.

By applying the condition (C4) and (37), we obtain in turn
‖E−1(E2 − E)‖ ≤

1∫
0

ψ0((1− η)‖y∗∗ − x0‖+ η‖y∗ − x0‖)dη ≤
1∫
0

ψ0((1− η)R3 + ηR4)dη < 1.

It follows that the linear operator E2 is invertible. Hence, again we conclude y∗∗ = y∗. �

Remark 4.

• The limit point a∗ given in the condition (C1) can be replaced by t0 in (C6).
• If all conditions (C1)− (C6) hold, then we can set R3 = α∗ and y∗ = x∗ in Proposition 2.

3. Numerical Results
To comprehensively evaluate the performance and robustness of the proposed high-order iterativemethods, we present five numerical examples of different complexity and dimensionality. Thesetest problems have been selected from the literature and include systems with diverse nonlinearcharacteristics, such as trigonometric, exponential, and polynomial structures. The examples aredesigned to assess the methods’ accuracy, convergence speed, and stability.In all numerical experiments, the stopping criterion was based on achieving a residual norm belowcertain ε, ensuring a high level of numerical precision. A maximum of 50 iterations was imposedto prevent excessive computational effort. This limit is justified by empirical evidence indicatingthat well-designed, high-order methods typically achieve convergence within this range. To ensurereliable performance metrics, CPU execution times were averaged over 50 independent runs, therebymitigating the influence of background system noise and transient operational conditionsAll simulations were conducted within a Google Colaboratory runtime environment. This envi-ronment was configured with an Intel Xeon CPU operating at 2.20 GHz, 13 GB of system RAM,and an NVIDIA Tesla K80 GPU equipped with 12 GB of VRAM. Numerical computations wereperformed using the Python library mpmath, with the arithmetic precision set to 100 decimal dig-its. This standardized setup was maintained across all test cases to ensure fair and reproduciblecomparisons.We compare method (2) with several established iterative methods, specifically the sixth-ordermethod (29) of Wang et al. [18], the method (14) by Hueso et al. [12], the scheme (6) of Cordero etal. [10] and the method (14) proposed by Abbasbandy et al. [1]. These benchmark techniques arewell known in the literature for their high-order convergence properties and are frequently used fortesting nonlinear solvers. Our method is evaluated against these in terms of number of iterations,

https://doi.org/10.28924/ada/ma.5.18


Eur. J. Math. Anal. 10.28924/ada/ma.5.18 14residual norm ‖G(x)‖, step difference ‖xn+1 − xn‖, and CPU time. The results are demonstratedin Tables 1–5.
Example 1. Consider the nonlinear system defined as:

Gi(x) = arctan(xi) + 1− 2

20∑
j=1
j 6=i

x2j = 0, i = 1, 2, . . . , 20.

The methods converge to the zero x∗ = (0.1757683, 0.1757683, . . . , 0.1757683)T , starting fromthe initial approximation x0 = (0.15, 0.15, . . . , 0.15)T .

Table 1. Results for Example 1
Method Iterations ‖G(x)‖ ‖xn+1 − xn‖ CPU Time (s)

Method (2) 2 6.7121× 10−31 8.1782× 10−30 0.342917Wang et al. 3 1.8677× 10−52 1.4708× 10−26 0.656792Hueso et al. 4 6.4643× 10−44 8.6687× 10−46 1.112088Cordero et al. 2 1.8677× 10−52 1.0138× 10−10 0.333056Abbasbandy et al. 3 1.2046× 10−23 2.0006× 10−23 0.307392
Example 2. Consider the nonlinear system:

Gi(x) = xi − cos

2πxi −
50∑
j=1

xj

 = 0, i = 1, 2, . . . , 50.

The solution is x∗ = (0.5018261, 0.5018261, . . . , 0.5018261)T , with the initial guess x0 =

(0.51, 0.51, . . . , 0.51)T .

Table 2. Results for Example 2
Method Iterations ‖G(x)‖ ‖xn+1 − xn‖ CPU Time (s)

Method (2) 2 2.221× 10−25 4.9027× 10−12 4.127071Wang et al. 7 6.9532× 10−22 3.6049× 10−20 17.881168Hueso et al. 8 5.4021× 10−24 2.8007× 10−22 29.956985Cordero et al. 7 5.5536× 10−23 2.8792× 10−21 15.002068Abbasbandy et al. 9 6.4707× 10−16 7.4808× 10−16 12.348433

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Eur. J. Math. Anal. 10.28924/ada/ma.5.18 15

Example 3. Let us consider the following system of 99 nonlinear equations:
Gi(x) =

xixi+1 − 1 = 0, 1 ≤ i ≤ 98,

x99x1 − 1 = 0, i = 99.

The exact solution is x∗ = (1, 1, . . . , 1)T , with the starting vector x0 = (2, 2, . . . , 2)T .

Table 3. Results for Example 3
Method Iterations ‖G(x)‖ ‖xn+1 − xn‖ CPU Time (s)

Method (2) 2 0.0 9.3704× 10−5 21.838247Wang et al. 4 0.0 4.0358× 10−25 56.250490Hueso et al. 5 0.0 8.9985× 10−39 99.436670Cordero et al. 3 0.0 7.0416× 10−15 33.705407Abbasbandy et al. 4 0.0 1.3554× 10−17 24.382990
Example 4. Consider the following system of nonlinear equations

G(x) =



x1 + x2 − 1 = 0,

2x1 + x2 + 2x3 − 2 = 0,

x1 + x2 + x3 − x4 = 0,

x22 x3
x21 x4
− (0.647)2 = 0.

The solution vector is x∗ = (0.422499, 0.577501, 0.288751, 1.288751)T , obtained from theinitial estimate x0 = (0.8, 0.2, 0.9, 1.8)T .

Table 4. Results for Example 4
Method Iterations ‖G(x)‖ ‖xn+1 − xn‖ CPU Time (s)

Method (2) 4 1.4315× 10−51 4.334× 10−16 0.04373Wang et al. N/A N/A N/A N/AHueso et al. N/A N/A N/A N/ACordero et al. 4 2.3346× 10−53 1.243× 10−21 0.01554Abbasbandy et al. N/A N/A N/A N/A

N/A indicates that the method did not converge to the required solution within theprescribed iteration or tolerance limits.

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Eur. J. Math. Anal. 10.28924/ada/ma.5.18 16

Example 5. Consider the nonlinear system of equations of size 200
Gi(x) = e−xi −

200∑
j=1
j 6=i

xj = 0, i = 1, 2, . . . , 200.

The initial approximation is set as x0 =
(
3
2 ,
3
2 , . . . ,

3
2

)T
, with parameters a = −2.0 and b = 2.0,leading to the solution: x∗ = (0.0050, 0.0050, . . . , 0.0050)T .

Table 5. Results for Example 5
Method Iterations ‖G(x)‖ ‖xn+1 − xn‖ CPU Time (s)

Method (2) 2 3.4388× 10−51 4.7064× 10−37 50.414524Wang et al. 3 6.8725× 10−52 7.9328× 10−26 86.904696Hueso et al. 3 1.1438× 10−51 1.4069× 10−18 129.807516Cordero et al. 3 2.5137× 10−51 3.0846× 10−48 71.584528Abbasbandy et al. 3 6.5891× 10−52 1.7579× 10−20 98.961347
4. Conclusions

This paper presented a general high-order iterative method for solving nonlinear systems withoutrequiring higher-order derivatives. We established both local and semi-local convergence resultsusing majorant conditions and majorizing sequences, providing rigorous guarantees even from dis-tant initial guesses. Numerical experiments on benchmark problems confirmed the method’s ac-curacy, fast convergence, and low residual errors compared to existing high-order methods. Theresults validate the theoretical findings and highlight the method’s applicability to a wide range ofnonlinear problems, suggesting potential for further extensions and refinements.
Author Contributions. Authors contributed equally. All authors have read and agreed to thepublished version of the manuscript.

References
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	1. Introduction
	2. Convergence Analysis
	2.1. Local
	2.2. Semi-local

	3. Numerical Results
	Example 1
	Example 2
	Example 3
	Example 4
	Example 5

	4. Conclusions
	References

