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Seventh Order Derivative-Free Methods for Non-differentiable Operator Equations

Sunil Kumar1, Janak Raj Sharma2, Ioannis K. Argyros3,∗, Samundra Regmi4
1Department of Mathematics, University Centre for Research and Development, Chandigarh University,

Mohali-140413, India
sfageria1988@gmail.com

2Department of Mathematics, Sant Longowal Institute of Engineering & Technology, Longowal, Punjab
148106, India

jrshira@yahoo.co.in
3Department of Computing and Mathematical Sciences, Cameron University, Lawton, OK 73505, USA

iargyros@cameron.edu
4Department of Mathematics, University of Houston, Houston, TX, 77024, USA

sregmi5@uh.edu
∗Correspondence: iargyros@cameron.edu

Abstract. In nonlinear problems where function’s derivatives are difficult or expensive to compute,derivative-free iterative methods are good options to find the numerical solution. One of the importantparts in the development of such methods is to study their convergence properties. In this paper, wereview the concepts of local and semi-local convergence for a derivative-free method for nonlinearequations. In the earlier study of the considered method, the convergence analysis was carried outassuming the existence of higher order derivatives while no derivative is used in the method. Suchassumptions certainly restrict its applicability. The present study further provides the estimate ofconvergence radius and bounds on the error for the given method. Thus, the applicability of themethod clearly seems to be extended over the wider class of problems. We also review some of therecent developments in this area. The results presented in this paper can be useful for practitionersand researchers in developing and analyzing derivative-free numerical algorithms.

1. Introduction
There are several numerical methods such as Newton’s method, Broyden’s method, secant methodand Steffensen’s method [3–11,13,14,17] that can be used to approximate x∗ of the equation

F (x) = 0, (1.1)
for F : Ω ⊂ Z → Z, F is a continuous operator, acting between Banach space Z and itself.Newton’s method is a popular iterative method used to find the roots of a nonlinear equation.

Received: 30 Apr 2023.
Key words and phrases. divided difference; Banach space; convergence; order of convergence.1

https://adac.ee
https://doi.org/10.28924/ada/ma.3.24


Eur. J. Math. Anal. 10.28924/ada/ma.3.24 2Iterative solution methods are commonly used when it is not possible to obtain the solution x∗ inclosed or analytical form. Instead, these methods generate a sequence of approximate solutionsthat converge towards the true solution x∗.Steffensen’s method [5, 9] defined for each n = 0, 1, 2, . . . by
xn+1 = xn − B−1F (xn), (1.2)

where B = Bn = [un, xn;F ] and un = xn + F (xn), has been used extensively to generate such asequence converging quadratically to x∗.Many iterative approaches have been developed to improve efficiency and order convergence(see [1, 2, 15,16]). An approach established in [16] that is defined for x0 ∈ Ω by
un = xn + F (xn), vn = xn − F (xn), D = Dn = [un, vn;F ],

yn = xn −D−1F (xn),

zn = yn − (3I − 2D−1[yn, xn;F ])D−1F (yn),

xn+1 = zn −
(13

4
I −D−1[zn, yn;F ]

(7

2
I −

5

4
D−1[zn, yn;F ]

))
D−1F (zn), (1.3)

has received significant attention in this paper. The convergence order seven is shown in [16],when Z = Rm etc. using assumption on F i , i = 1, 2, . . . , 8 not present in the method, significantlyreducing its applicability although it may converge.Consider the function
F (t) =

{
7t3 log(t) + 5t5 − 5t4, t 6= 0

0, t = 0
(1.4)

Then, in any neighborhood of 0 and 1, say F ′′′ is unbounded. Hence, the results in [16] cannotassure convergence to t∗ = 1. But the method converges.In this article we study convergence of the method (1.3) that includes mainly the local andsemi-local convergence (not provided in [16]).Local convergence analysis uses information about the actual solution to determine the rateand radius of convergence of the method. This typically involves estimating the size of the regionaround the true solution where the method is guaranteed to converge. This type of analysis alsousually involves deriving upper bounds on the error norms, which provide an estimate of how closethe iterates of the method are to the true solution.In contrast, in semi-local convergence analysis, the convergence behavior of the method is studiedusing information from the initial point, typically by deriving sufficient conditions that guaranteeconvergence of the method. This analysis is usually carried out without any knowledge of theactual solution of the problem.Generalized Lipschitz-type conditions are often used in both semi-local and local convergenceanalysis. These conditions involve bounding the difference between the iterates of the method

https://doi.org/10.28924/ada/ma.3.24


Eur. J. Math. Anal. 10.28924/ada/ma.3.24 3and the true solution using a Lipschitz constant or a related quantity. These conditions can beused to derive sufficient conditions for convergence, as well as to estimate the rate and radius ofconvergence of the method.It is crucial to examine how technique (1.3) converges in both the local (Section 2) and thesemi-local (Section 3) cases. Moreover, our approach gives a prior error estimates and isolationof the solution results not provided before and in Banach space. This approach also enables acomparison of the convergence criteria of method. If the approach is examined separately, the newconvergence criteria may be weaker than those that have been provided. The numerical examplesare included in Section 4, and the conclusions are discussed in Section 5.
2. Local convergence

Some real functions assist in the local analysis of the method. Set T = [0,+∞). Assume:
(H1) There exist continuous as well as nondecreasing functions (CN) f1 : T → T, f2 : T → T,and w0 : T × T → R so that the equation

w0(f1(t), f2(t))− 1 = 0

admits a smallest solution (SS) denoted by δ ∈ T − {0}. Let T0 = [0, δ).
(H2) There exist (CN) functions w : T0 → T, w1 : T0×T0×T0 → T, and w2 : T0×T0×T0 → Tsuch that the equations

hi(t)− 1 = 0, i = 1, 2, 3have (SS) solutions denoted by δi ∈ T0 − {0}, provided that
h1(t) =

w1(f1(t), f2(t), t)

1− w0(f1(t), f2(t))
,

h2(t) =
[w1(h1(t)t, f1(t), f2(t))

1− w0(f1(t), f2(t))
+

2w2(t, h1(t)t, f1(t), f2(t))(1 + w(t))

(1− w0(f1(t), f2(t)))2

]
h1(t),

h0(t) =
1

4

[
5
(w2(h1(t)t, h2(t)t, f1(t), f2(t))

1− w0(f1(t), f2(t))

)2
+ 4

w2(h1(t)t, h2(t)t, f1(t), f2(t))

1− w0(f1(t), f2(t))

]
,

h3(t) =
[w1(f1(t), f2(t), h2(t)t)

1− w0(f1(t), f2(t))
+
h0(t)(1 + w(h2(t)t))

1− w0(f1(t), f2(t))

]
h2(t).

Consider, the parameter δ∗ given as
δ∗ = min{δi}. (2.5)

Let T1 = [0, δ∗). These definitions imply
0 ≤ w0(f1(t), f2(t)) < 1 (2.6)

and
0 ≤ hi(t) < 1, (2.7)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.24 4for all t ∈ T1.Let B(x̄ , r), B̄[x̄ , r ] abbreviate open and closed balls in S1, respectively so that thecenter is x̄ and the radius is some r > 0. The preceding real functions are associated tothe divided difference [., .;F ] as:
(H3) There exists an invertible operator L ∈ L(Z) so that for each x ∈ Ω, u = x + F (x),

v = x − F (x)

||L−1([u, v ;F ]− L)|| ≤ w0(||u − x∗||, ||v − x∗||),

||u − x∗|| ≤ f1(||x − x∗||),

||v − x∗|| ≤ f2(||x − x∗||).

Let B0 = B(x∗, δ).

(H4)

||L−1([x, x∗;F ]− L)|| ≤ w(||x − x∗||),

||L−1([x, y ;F ]− [z, x∗;F ])|| ≤ w1(||x − x∗||, ||y − x∗||, ||z − x∗||)

and
||L−1([u, v ;F ]− [y , x ;F ])|| ≤ w2(||x − x∗||, ||y − x∗||, ||u − x∗||, ||v − x∗||),

for each x, y , z, u, v ∈ B0.
(H5) B[x∗, δ∗] ⊂ Ω.The local analysis is based on the conditions (H1)− (H5) under the preceding notations.

Theorem 2.1 Assume the conditions (H1)− (H5) are validated. If x0 ∈ B(x∗, δ∗)− {x∗}, then the
following items are valid

{xn} ⊂ B(x∗, δ∗), (2.8)
‖yn − x∗‖ ≤ h1(‖xn − x∗‖)‖xn − x∗‖ ≤ ‖‖xn − x∗‖ < δ∗, (2.9)
‖zn − x∗‖ ≤ h2(‖xn − x∗‖)‖xn − x∗‖ ≤ ‖‖xn − x∗‖, (2.10)

‖xn+1 − x∗‖ ≤ h3(‖xn − x∗‖)‖xn − x∗‖ ≤ ‖‖xn − x∗‖ (2.11)
and the sequence {xn} is convergent to x∗.
Proof. These items are shown by mathematical induction. By the condition (H3), estimate (2.7)for u0 = x0 + F (x0), v0 = x0 − F (x0) it follows

‖L−1([u0, v0;F ]− L)‖ ≤ w0(‖u0 − x∗‖, ‖v0 − x∗‖) ≤ w0(f1(‖x0 − x∗‖), f2(‖x0 − x∗‖))

≤ w0(f1(u), f2(v)) < 1. (2.12)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.24 5Then, the existence of [u0, v0;F ]−1 is assured by the estimate (2.12) and the perturbation Lemmaon linear operators with inverses attributed to Banach [6]. We also have
‖[u0, v0;F ]−1L‖ ≤

1

1− w0(f1(‖u0 − x∗‖), f2(‖u0 − x∗‖)
. (2.13)

Thus, the iterate y0 is well defined and
y0 − x∗ = x0 − x∗ − [u0, v0;F ]−1F (x0).

[u0, v0;F ]−1([u0, v0;F ]− [x0, x
∗;F ])(x0 − x∗). (2.14)

Then, the conditions (H3), (H4), (2.5), (2.7) (for i = 1), (2.13) and (2.14) give in turn
‖y0 − x∗‖ ≤

w1(‖u0 − x∗‖, ‖v0 − x∗‖, ‖x0 − x∗‖)‖x0 − x∗‖
1− w0(f1(‖x0 − x∗‖), f2(‖x0 − x∗‖))

≤h1(‖x0 − x∗‖)‖x0 − x∗‖ ≤ ‖x0 − x∗‖ < δ∗. (2.15)
Hence, the iterate y0 ∈ B(x∗, δ∗) and the item (2.9) is validated for n = 0.Notice that the iterates z0 and x1 are also well defined by the second and the third substep ofthe method (1.3). In particular, we get

z0 − x∗ = y0 − x∗ −D−1F (y0) + 2D−1(D − [yn, xn;F ])D−1F (yn)

=D−1(D − [yn, x
∗;F ])(yn − x∗) + 2D−1([u0, v0;F ]− [y0, x0;F ])D−1F (y0). (2.16)

But we can write by the first substep that
F (y0) = F (y0)− F (x∗) = [y0, x

∗;F ](y0 − x∗),

so by (H4)

‖L−1F (y0)‖ = ‖L−1([y0, x
∗;F ]− L+ L)(y0 − x∗)‖

≤ (1 + w(‖y0 − x∗‖))‖y0 − x∗‖. (2.17)
Consequently, (2.5), (2.7) (for i = 2), (H4), (2.13), (2.16) and (2.17) imply
‖z0 − x∗‖ ≤

[ w1(‖y0 − x∗‖, ‖u0 − x∗‖, ‖v0 − x∗‖)
1− w0(f1(‖x0 − x∗‖), f2(‖x0 − x∗‖))

+
2w2(‖x0 − x∗‖, ‖y0 − x∗‖, ‖u0 − x∗‖, ‖v0 − x∗‖)(1 + w(‖y0 − x∗‖))

(1− w0(f1(‖x0 − x∗‖), f2(‖x0 − x∗‖)))2

]
‖y0 − x∗‖,

≤ h2(‖x0 − x∗‖)‖x0 − x∗‖ ≤ ‖x0 − x∗‖. (2.18)
Thus, the iterate z0 ∈ B(x∗, δ∗) and the item (2.10) is validated for n = 0. Moreover, the thirdsubstep gives

x1 − x∗ = z0 − x∗ −D−1F (z0)− AD−1F (z0)

= D−1(D − [z0, x
∗;F ])(z0 − x∗)− AD−1F (z0), (2.19)

https://doi.org/10.28924/ada/ma.3.24


Eur. J. Math. Anal. 10.28924/ada/ma.3.24 6where
A = −

[9

4
I −

7

2
D−1[zn, yn;F ] +

5

4
(D−1[zn, yn;F ])2

]
= −

1

4

(
5(D−1([zn, yn;F ]− [un, vn;F ]))2 − 4D−1([zn, yn;F ]− [un, vn;F ])

)
.

Therefore,
‖A‖ ≤

1

4

[
5
(w2(‖y0 − x∗‖, ‖z0 − x∗‖, ‖u0 − x∗‖, ‖v0 − x∗‖)

1− w0(f1(‖x0 − x∗‖), f2(‖x0 − x∗‖))

)2
+ 4

w2(‖y0 − x∗‖, ‖z0 − x∗‖, ‖u0 − x∗‖, ‖v0 − x∗‖)
1− w0(f1(‖x0 − x∗‖), f2(‖x0 − x∗‖))

]
= h0. (2.20)

Then, by (2.5), (2.7) (for i = 3), (H4), (2.13) and (2.18)-(2.20)
‖x1 − x∗‖ ≤

[ w1(‖u0 − x∗‖, ‖v0 − x∗‖, ‖z0 − x∗‖)
1− w0(f1(‖x0 − x∗‖), f2(‖x0 − x∗‖))

+
h0(1 + w(‖z0 − x∗‖))

1− w0(f1(‖x0 − x∗‖), f2(‖x0 − x∗‖))

]
‖z0 − x∗‖

≤ h3(‖x0 − x∗‖)‖x0 − x∗‖ ≤ ‖x0 − x∗‖. (2.21)
Hence, the items (2.8) and (2.11) are validated for n = 0 and the iterate x1 ∈ B(x∗, δ∗). If thepreceding calculations are repeated with xm, ym, xm+1, replacing x0, y0, x1, respectively, theinduction for the items (2.8)-(2.11) is terminated. Furthermore, from estimation

‖xm+1 − x∗‖ ≤ µ‖xm − x∗‖ < ‖xm − x∗‖, (2.22)
where µ = h3(‖x0 − x∗‖) ∈ [0, 1), we conclude that limm→∞ xm = x∗ and the iterate xm+1 ∈
B(x∗, δ∗). �

Remark 2.2 The second and third hypotheses in (H3) are left as uncluttered as possible. Somepossible choices for the functions f1 and f2 are specified.
un − x∗ = xn − x∗ + F (xn) = (I + [xn, x

∗;F ])(xn − x∗)

= (I + L+ LL−1([xn, x
∗;F ]− L))(xn − x∗),

so
‖un − x∗‖ ≤

(
‖I + L‖+ (‖L‖w(‖(xn − x∗)‖))

)
‖(xn − x∗)‖.Thus, we can choose

f1(t) =
(
‖I + L‖+ ‖L‖w(t)

)
t.Notice also that we can set w(t) = w0(0, t).Similarly, we define

f2(t) =
(
‖I − L‖+ ‖L‖w(t)

)
t.In view of the above the second and third conditions in (H3) can be dropped if (H5) is replaced by

(H5)
′ B[x∗, δ̄] ⊂ Ω, where δ̄ = max{δ∗, f1(δ∗)δ∗, f2(δ∗)δ∗}.

https://doi.org/10.28924/ada/ma.3.24


Eur. J. Math. Anal. 10.28924/ada/ma.3.24 7Possible choices for L are
L = F ′(x∗) (The differentiable case)

or
L = [., .;F ] (The non-differentiable case).

In practice L should be chosen to optimize the results.Next, the point x∗ is shown to be the only solution of the equation F (x) = 0 in a certain set.
Preposition 2.3 Assume: There exists a solution x∗1 ∈ B(x∗, δ4) of the equation F (x) = 0 for some
δ4 > 0; The first assumption in (H3) is validated on the ball B(x∗, δ4) and there exists δ5 ≥ δ4 so
that

w(δ5) < 1.

Let B1 = Ω ∩ B[x∗, δ5]. Then, the only solution of the equation F (x) = 0 in the set B1 is x∗.
Proof. Let Q = [x∗, x∗1 ]. By the assumption it follows

‖L−1(Q− L)‖ ≤ w(‖x∗1 − x∗‖)

≤ w(δ5) < 1,

thus Q−1 ∈ L(Z) and consequently from the approximation
x∗1 − x∗ = Q−1(F (x∗1 )− F (x∗)) = Q−1(0) = 0,

it is concluded that x∗1 = x∗. �Clearly, we can choose δ4 = δ∗.

3. Semi-local Analysis
The role of x∗ is exchanged by x0. But there are some more differences.Assume:

(C1) There exist (CN) functions g1 : T0 → T, g2 : T0 → T and w0 : T0 × T0 → T so that theequation
w0(g1(t), g2(t))− 1 = 0

has a (SS) denoted by r0 ∈ T0 − {0}. Set T3 = [0, r0).Define the scaler sequence {an} for a0 = 0, b0 ∈ [0, r0) and some (CN) functions
g1 : T3 → T, g2 : T3 → T, w2 : T3 × T3 × T3 × T3 → T by
cn = bn +

[w2(an, bn, f1(an), f2(an))

1− w0(g1(an), g2(an))
+

2w2(an, bn, f1(an), f2(an))

(1− w0(g1(an), g2(an)))2

]
(βn − an), (3.23)

βn =
1

4

[
5
(w2(an, bn, f1(an), f2(an))

1− w0(g1(an), g2(an))

)2
+ 4
(w2(an, bn, f1(an), f2(an))

1− w0(g1(an), g2(an))

)2]
,

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Eur. J. Math. Anal. 10.28924/ada/ma.3.24 8

γn = w2(an, bn, g1(an), g2(an))(bn − an),

an+1 = cn +
(1 + w0(bn, cn))(cn − bn)βn + γn

1− w0(g1(an), g2(an))
,

δn+1 = (1 + v0(an, bn))(an+1 − an) + (1 + w0(g1(an), g2(an)))(bn − an)

and
bn+1 = an+1 +

δn+1
1− w0(g1(an+1), g2(an+1))

.

Next, general convergence conditions are developed.
Lemma 3.1 Assume there exists µ1 ∈ [0, r0) such that for each n = 0, 1, 2, . . .,

(C2) w0(g1(an), g2(an)) < 1 and an ≤ µ1.
Then, the following items hold

0 ≤ an ≤ bn ≤ cn ≤ an+1 ≤ µ1

and there exists a∗ ∈ (0, µ1] such that limn→∞ an = a∗.

Proof. The conclusions follow immediately by the formula (3.23) and the condition (C2).

�Notice that the limit a∗ is unique, since it is the unique least upper bound of the sequence
{an}.

(C3) There exist an invertible operator L and a point x0 ∈ Ω such that for each x, y ∈ Ω,

u = x + F (x), v = x − F (x)

‖L−1([u, v ;F ]− L)‖ ≤ w0(‖u − x0‖, ‖v − x0‖),

‖u − x0‖ ≤ g1(‖x − x0‖),

‖v − x0‖ ≤ g2(‖x − x0‖)

and w0(g1(‖F (x0)‖), g2(‖F (x0)‖)) < 1.The existence of [u0, v0;F ]−1 is guaranteed, by the Banach lemma and since
L−1‖[u0, v0;F ]− L‖ ≤ w0(‖u0 − x0‖, ‖v0 − x0‖) < 1.

(C4) ‖[u0, v0;F ]−1F (x0)‖ ≤ b0. Let B2 = B(x0, µ0).

(C5) ‖L−1([x, y ;F ]− [u, v ;F ])‖ ≤ w2(‖x−x0‖, ‖y−x0‖, ‖u−x0‖, ‖v−x0‖) for each x, y , u, v ∈
B2.

(C6) B[x0, a
∗] ⊂ Ω.

Next, the semi-local convergence is provided for the method (1.3).

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Eur. J. Math. Anal. 10.28924/ada/ma.3.24 9

Theorem 3.2 Assume the conditions (C1)− (C6) hold. Then, the following items hold

{xn} ⊂ B(x0, a
∗), (3.24)

‖yn − xn‖ ≤ bn − an, (3.25)
‖zn − yn‖ ≤ cn − bn, (3.26)
‖xn+1 − zn‖ ≤ an+1 − cn, (3.27)

and there exists a solution x∗ of the equation F (x) = 0 such that

‖x∗ − xn‖ ≤ a∗ − an. (3.28)
Proof. The items (3.24)-(3.27) are shown by induction. Notice that the iterates y0, z0, x1 existsby the invertibility of [u0, v0;F ] and the method (1.3). The estimate (3.25) is validated for n = 0,since by the condition (C4)

‖y0 − x0‖ = ‖[u0, v0;F ]−1F (x0)‖ ≤ b0 = b0 − a0 ≤ a∗ (3.29)
and the iterate y0 ∈ B(x0, a

∗).Then, as in the local convergence case but using x0, (C) instead of x∗, (H), we obtain from
F (yn) = F (yn)− F (xn)−D(yn − xn) = ([yn, xn;F ]−D)(yn − xn),

so
‖L−1F (yn)‖ ≤ w2(‖xn − x0‖, ‖yn − x0‖, ‖un − x0‖, ‖vn − x0‖). (3.30)

Hence, by the second substep
zn − yn = −D−1F (yn)− 2D−1(D − [yn, xn;F ])−1D−1F (yn),

and
‖zn − yn‖ ≤

[w2(‖xn − x0‖, ‖yn − x0‖, ‖un − x0‖, ‖vn − x0‖)
1− w0(f1(‖xn − x0‖), f2(‖xn − x0‖))

+ 2
(w2(‖xn − x0‖, ‖yn − x0‖, ‖un − x0‖, ‖vn − x0‖)

1− w0(f1(‖xn − x0‖), f2(‖xn − x0‖))

)2]
‖yn − xn‖ ≤ an − bn (3.31)

and
‖zn − x0‖ ≤ ‖zn − yn‖+ ‖yn − x0‖ ≤ an − bn + bn − a0 = cn < a∗,

thus the item (3.26) holds and the iterate z0 ∈ B(x0, a
∗).Moreover, by the third substep

xn+1 − zn = −BD−1F (zn), (3.32)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.24 10where
B = −

1

4

(
13I − 14D−1[zn, yn;F ] + 5(D−1[zn, yn;F ])2

)
= −

1

4

(
5(D−1[zn, yn;F ]− I)2 − 4(D−1[zn, yn;F ]− I) + 4I

)
,

thus
‖B‖ ≤

1

4

(
5
(w2(‖xn − x0‖, ‖yn − x0‖, ‖un − x0‖, ‖vn − x0‖)

1− w0(f1(‖xn − x0‖), f2(‖xn − x0‖))

)2
+ 4
(w2(‖xn − x0‖, ‖yn − x0‖, ‖un − x0‖, ‖vn − x0‖)

1− w0(f1(‖xn − x0‖), f2(‖xn − x0‖))

)
+ 4
)

= β̄ < βn.

Consequently, we have
‖xn+1 − zn‖ ≤

β̄n(1 + w0(‖yn − x0‖, ‖zn − x0‖))‖zn − yn‖
1− w0(f1(‖xn − x0‖), f2(‖xn − x0‖))

≤
βn(1 + w0(βn, cn))(cn − bn)

1− w0(f1(an), f2(an))
= an+1 − cn < βn (3.33)

and
‖xn+1 − x0‖ ≤ ‖xn+1 − zn‖+ ‖zn − x0‖ ≤ an+1 − cn + cn − a0 = an+1 < a∗.Hence, the item (3.27) is validated and iterate xn+1 ∈ B(x0, a

∗). �

Remark 3.3 As in the local case the functions g1 and g2 can be expressed in terms of the rest ofthe conditions.Assume that there exists a CN function ϕ1 : T → R such that for each x ∈ Ω

‖L−1([x, x0;F ]− L)‖ ≤ ϕ1(‖x − x0‖).

Then, from the estimate
un − x0 = xn − x0 + F (xn)− F (x0) + F (x)

=(I + L+ LL−1([xn, x0;F ]− L))(xn − x0) + F (x0),

so we can choose
g1(t) = (‖1 + L‖+ ‖L‖ϕ1(t))t + ‖F (x0)‖and similarly
g2(t) = (‖1− L‖+ ‖L‖ϕ1(t))t + ‖F (x0)‖.Under these choices of g1 and g2

‖u − x0‖ = g1(‖x − x0‖),

‖v − x0‖ = g2(‖x − x0‖)

and the second and third conditions in (H3) can be dropped.Possible choices for L are
L = F ′(x0) (The differentiable case)

https://doi.org/10.28924/ada/ma.3.24


Eur. J. Math. Anal. 10.28924/ada/ma.3.24 11or
L = [., .;F ] (The non-differentiable case).The condition (C6) can be replaced by

(C6)
′ B[x0, ā] ⊂ Ω, where ā = max{a∗, g1(a∗), g2(a∗)}.A uniqueness of the solution set is specified.

Proposition 3.4 Assume: There exists a solution d ∈ B(x0, δ6) of the equation F (x) = 0 for some
δ6 > 0; the first condition in (H3) holds in the ball B(x0, δ6) and there exists δ7 ≥ δ6 such that

w0(δ6, δ7) < 1.

Let B3 = B[x0, δ7] ∩Ω. Then, the point d is the only solution of the equation F (x) = 0 in the set
B3.
Proof. Let d1 ∈ B3 be such that F (d1) = 0. Define the divided difference [d, d1;F ]. Then,

‖L−1([d, d1;F ]− L)‖ ≤ w0(‖d − x0‖, ‖d1 − x0‖)

≤ w0(δ6, δ7) < 1,

thus d1 = d. �

Remark 3.5 If all the conditions (C1)− (C6) hold, then set d = x∗ and δ6 = a∗.

4. Numerical Tests
In order to validate the theoretical deductions, we take into account the following numericalexamples to estimate the real parameters defined in the preceding sections:

Example 1. Let Z = R× R× R and Ω = B(ξ∗, 1) with ξ∗ = (0, 0, 0)T . Define the mapping F for
ξ = (ξ1, ξ2, ξ3)

T , ξi ∈ R by
F (ξ) =

(
ξ1, e

ξ2 − 1,
(e − 1)

2
ξ23 + ξ3

)T
.

This definition gives that the F ′ of the mapping F is the Jacobian matrix
F ′(ξ) =

1 0 0

0 eξ2 0

0 0 (e − 1)ξ3 + 1

 .
Notice that F (ξ∗) = O and F ′(ξ∗) = I. Then, the conditions (H1)− (H4) are validated if

w0(t1, t2) =
1

2
(e − 1)(t1 + t2),

w(t) =
1

2
(e − 1)t,

w1(t1, t2, t3) =
1

2
(e − 1)(t1 + t2 + t3),

w2(t1, t2, t3, t4) =
1

2
(e − 1)(t1 + t2 + t3 + t4)

https://doi.org/10.28924/ada/ma.3.24


Eur. J. Math. Anal. 10.28924/ada/ma.3.24 12and the functions f1 and f2 are given in Remark 2.2. Then, the radius δ∗ using (2.5) is
δ1 = 0.20415, δ2 = 0.13109, δ3 = 0.11960 and δ∗ = 0.11960.

Figure 1. Graph of radius of convergence of example 1.

∆1

∆2

∆3

-0.2 -0.1 0.0 0.1 0.2

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

t

h
i

t

h
i

h1

h2

h3

Example 2. Let Z = C[0, 1] be the space of continuous functions defined in [0, 1] and Ω = B̄(l∗, 1).Consider the integral equation of the mixed Hammerstein-type [6, 12] by
l(d) =

∫ 1
0

T (d, ω)
(
l(ω)3/2 +

l(ω)2

2

)
dω, (4.34)

T (d, ω) =

{
(1− d)ω, ω ≤ d,
d(1− ω), d ≤ ω.Notice that l∗(d) = 0. Define H : Ω ⊆ [0, 1]→ C[0, 1] as

H(l)(d) = l(d)−
∫ 1
0

T (d, ω)
(
l(ω)3/2 +

l(ω)2

2

)
dω.

The derivative H′ is given by
H′(l)q(d) = q(d)−

∫ 1
0

T (d, ω)
(3

2
l(ω)1/2 + l(ω)

)
dω,

since H′(l∗(d)) = 1, it follows
‖H′(α)−1(H′(l)−H′(q))‖ ≤

5

16
‖l − q‖. (4.35)

In (4.35), switch q by l0
‖H′(α)−1(H′(l)−H′(l0))‖ ≤

5

16
‖l − l0‖.Thus, we take

w0(t1, t2) = t1 + t2,

w(t) = t,

w1(t1, t2, t3) = t1 + t2 + t3,

w2(t1, t2, t3, t4) = 5.

https://doi.org/10.28924/ada/ma.3.24


Eur. J. Math. Anal. 10.28924/ada/ma.3.24 13Hence, we obtain
δ1 = 0.17539, δ2 = 0.02678, δ3 = 0.90819× 10−3 and δ∗ = 0.90819× 10−3.

Figure 2. Graph of radius of convergence of example 2.

∆1
∆2

∆3

-0.6 -0.4 -0.2 0.0 0.2

-20

0

20

40

60

80

t

h
i

t

h
i

h1

h2

h3

Example 3. Let Z = C[0, 1] be the space of continuous functions [12] defined on the interval [0, 1]and Ω = B̄(0, 1). Define the function H on Ω by
H(ϕ)(l) = ϕ(l)− 10

∫ 1
0

lρϕ(ρ)3dρ.

It follows that
H′(ϕ(ξ))(l) = ξ(l)− 30

∫ 1
0

lρϕ(ρ)2ξ(ρ)dρ, for each ξ ∈ Ω.

Since l∗ = 0, so we can set
w0(t1, t2) = 2(t1 + t2),

w(t) =
t

5
,

w1(t1, t2, t3) = 2(t1 + t2 + t3),

w2(t1, t2, t3, t4) = 2(t1 + t2 + t3 + t4).

Hence, we obtain
δ1 = 0.98449× 10−1, δ2 = 0.62003× 10−1, δ3 = 0.55704× 10−1 and δ∗ = 0.55704× 10−1.

Example 4. Lastly, a nondifferentiable nonlinear system on R × R is solved using the method(1.3), where the divided difference is defined by the 2 × 2 matrix given for t̄ = (t1, t2) ∈ R × R,
t̃ = (t3, t4) ∈ R× R, and F = (F1, F2) by

[t̄ , t̃;F ]i ,1 =
Fi(t3, t4)− Fi(s1, s4)

t3 − t1
t3 6= t1

and
[t̄ , t̃;F ]i ,2 =

Fi(s1, s4)− Fi(s1, s2)
s4 − s2

. s4 6= s2.

https://doi.org/10.28924/ada/ma.3.24


Eur. J. Math. Anal. 10.28924/ada/ma.3.24 14

Figure 3. Graph of radius of convergence of example 3.
∆2

∆1

∆3

-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10

-1

0

1

2

t

h
i

t

h
i

h1

h2

h3

Otherwise, we set [., .; .] = O. 7 Let us consider the nonlinear and nondifferentiable system as
3t21 t2 + t22 − 1 + |t1 − 1| = 0,

t41 + t1t
3
2 − 1 + |t2| = 0.

Then, we set F = (F1, F2), where
F1(t1, t2) = F1 = 3t21 t2 + t22 − 1 + |t1 − 1| = 0

and
F2(t1, t2) = F2 = t41 + t1t

3
2 − 1 + |t2| = 0.

Choose initial points (5, 5) and (1, 0). Then, using the aforementioned divided difference andthe method (1.3) we obtain the solution x∗ = (x∗1 , x
∗
2 ) after three iterations with x∗1 =

0.894655074966771 and x∗2 = 0.327826643198819.
5. Conclusion

The focus of this paper is to provide a comprehensive analysis of the local and semilocal con-vergence of a derivative-free seventh-order method in Banach space. It is noteworthy that theconvergence has been investigated in earlier studies by assuming the existence of some higherorder derivatives, which in fact are not used in the iterative method. Contrary to this, our approachonly considers the first-order divided differences that are actually present in the iterative process.This unique feature makes the method applicable to a wider range of functions, thereby expandingits utility. In the analysis, we present an error estimate and convergence ball that bounds the iter-ates, providing further benefits to the analysis of convergence. In addition, the sufficient conditionsare developed to show the uniqueness of solution in the given domain. To verify the theoreticalresults, we have conducted numerical tests on several problems, demonstrating the effectiveness ofthis approach. Moreover, this idea has the potential to be extended to other methods, making it avaluable contribution in the field of the theory of iterative functions [1–17].

https://doi.org/10.28924/ada/ma.3.24


Eur. J. Math. Anal. 10.28924/ada/ma.3.24 15References
[1] A. Cordero, J.L. Hueso, E. Martinez, J.R. Torregrosa, A modified Newton-Jarratt’s composition, Numer. Algor. 55(2010) 87–99. https://doi.org/10.1007/s11075-009-9359-z.[2] F. Awawdeh, On new iterative method for solving systems of nonlinear equations, Numer. Algor. 54 (2010) 395–409.

https://doi.org/10.1007/s11075-009-9342-8.[3] F. A. Potra, V. Pták, Nondiscrete Induction and Iterative Processes Pitman Publishing, Boston, 1984.[4] I.K. Argyros, Computational Theory of Iterative Methods, Series: Studies in Computational Mathematics, 15, Editors:Chui C.K. and Wuytack L, Elsevier Publ. Co., New York, 2007.[5] I.F. Steffensen, Remarks on iteration, Scand. Actuar. J. 16 (1933) 64–72. https://doi.org/10.1080/03461238.
1933.10419209.[6] I.K. Argyros, Á. A. Magreñán, Iterative Methods and Their Dynamics with Applications, CRC Press: New York, NY,USA, 2017.[7] J.A. Ezquerro, Á. Grau, M. Grau-Sánchez, M. A. Hernández, M. Noguera, Analysing the efficiency of some modifi-cations of the secant method, Comput. Math. Appl. 64 (2012) 2066–2073. https://doi.org/10.1016/j.camwa.
2012.03.105.[8] J. A. Ezquerro, M. A. Hernández, N. Romero, Solving nonlinear integral equations of Fredholm type with high orderiterative methods, J. Comput. Appl. Math. 36 (2011) 1449–1463. https://doi.org/10.1016/j.cam.2011.09.009.[9] J.F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, Englewood Cliffs, New Jersey, 1964.[10] J.M. Ortega, W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, NewYork, 1970.[11] J.R. Sharma, H. Arora, An efficient derivative free iterative method for solving systems of nonlinear equations, Appl.Anal. Discrete Math. 7 (2013) 390–403. https://doi:10.2298/AADM130725016S.[12] J.R. Sharma, S. Kumar, I.K. Argyros, Generalized Kung-Traub method and its multi-step iteration in Banach spaces,Journal of Complexity 54 (2019) 101400. https://doi.org/10.1016/j.jco.2019.02.003.[13] M. Grau-Sánchez, Á, Grau, M. Noguera, Frozen divided difference scheme for solving systems of nonlinear equations,J. Comput. Appl. Math. 235 (2011) 1739–1743. https://doi.org/10.1016/j.cam.2010.09.019.[14] M. Grau-Sánchez, M. Noguera, A technique to choose the most efficient method between secant method and somevariants, Appl. Math. Comput. 218 (2012) 6415–6426. https://doi.org/10.1016/j.amc.2011.12.011.[15] M. Grau-Sanchez, M. Noguera, S. Amat, On the approximation of derivatives using divided difference operatorspreserving the local convergence order of iterative methods, J. Comput. Appl. Math. 237 (2013) 363–372. https:
//doi.org/10.1016/j.cam.2012.06.005.[16] X. Wang, T. Zhang, W. Qian, M. Teng, Seventh-order derivative-free iterative method for solving nonlinear systems,Numer. Algor. 70 (2015) 545–558. https://doi.org/10.1007/s11075-015-9960-2.[17] Z. Liu, Q. Zheng, P. Zhao, A variant of Steffensen’s method of fourth-order convergence and its applications, Appl.Math. Comput. 216 (2010) 1978–1983. https://doi.org/10.1016/j.amc.2010.03.028.

https://doi.org/10.28924/ada/ma.3.24
https://doi.org/10.1007/s11075-009-9359-z
https://doi.org/10.1007/s11075-009-9342-8
https://doi.org/10.1080/03461238.1933.10419209
https://doi.org/10.1080/03461238.1933.10419209
https://doi.org/10.1016/j.camwa.2012.03.105
https://doi.org/10.1016/j.camwa.2012.03.105
https://doi.org/10.1016/j.cam.2011.09.009
https://doi:10.2298/AADM130725016S
https://doi.org/10.1016/j.jco.2019.02.003
https://doi.org/10.1016/j.cam.2010.09.019
https://doi.org/10.1016/j.amc.2011.12.011
https://doi.org/10.1016/j.cam.2012.06.005
https://doi.org/10.1016/j.cam.2012.06.005
https://doi.org/10.1007/s11075-015-9960-2
https://doi.org/10.1016/j.amc.2010.03.028

	1. Introduction
	2. Local convergence
	3. Semi-local Analysis
	4. Numerical Tests
	5. Conclusion
	References

