©2025 Ada Academica https://adac.eeEur. J. Math. Anal. 5 (2025) 11doi: 10.28924/ada/ma.5.11
Majorizing Sequences for Newton-Like Method and Their Limit Points

Ioannis K. Argyros1,∗ , Santhosh George2 , Michael Argyros3
1Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA

iargyros@cameron.edu
2Department of Mathematical and Computational Sciences,National Institute of Technology Karnataka,

India-575 025
sgeorge@nitk.edu.in

3Department of Computer Sciences, Franklin University, Ohio, USA
argyro01@email.franklin.edu

∗Correspondence: iargyros@cameron.edu

Abstract. A plethora of problems from diverse disciplines of Mathematics, Mathematical Biology,Chemistry, Medicine, physics and Engineering to mention a few reduce to solving nonlinear equationsor systems of equations usually in the finite dimensional Euclidean or more general spaces. Thesolutions of such equations are numbers or vectors of functions and can be found in closed form onlyin special cases. That is why researchers and practitioners develop mostly iterative methods whichgenerate sequences approximating the solutions. The least number of iterations to be carried outin order to obtain a pre-decided error tolerance on the distances between consecutive iterates aswell as the choice of initial points ensuring the convergence of the methods is very important. Thesetwo objectives can be achieved by introducing real majorizing sequences which control the behaviourof the iterates. Moreover, the closed form of the limits of the real sequences determine the radiusof the ball that contains the initial points. In this paper we contribute by introducing more precisemajorizing sequences and limit points.

1. Introduction
Majorizing sequences have been used extensively to study the semi-local convergence of New-ton’s method defined for x0 ∈ D and each n = 0, 1, 2, ... by

xn+1 = xn − F ′(xn)−1F (xn), (1.1)
where F : D ⊂ B1 → B2 is a Fŕechet- differentiable operator between Banach spaces B1, B2 and
D is an open and convex set [1, 3, 4, 6, 7]. The usually sufficient semi-local convergence conditionsdiffer in general as well as the majorizing sequences and their limit points. We try to relate theseconditions, sequences and limit points in a unified way without additional hypotheses.

Received: 10 Nov 2024.
Key words and phrases. Newton-like method; majorizing sequences; Fréchet derivative; Banach spaces.1

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Eur. J. Math. Anal. 10.28924/ada/ma.5.11 22. Lipschitz conditions
The symbols L(B1, B2), U(x, r) are used to denote the space of bounded linear operators from

B1 into B2 and the open ball centered at x ∈ B1 and of radius r > 0, respectively.We introduce Lipscits conditions used to control F ′. Then, we copare them to each other.
Definition 2.1. Suppose M ∈ L (B1, B2) is an invertible operator and x0 ∈ D. We say that F ′ is
center-Lipschitz continuous if there exists L0 > 0 such that

M−1(F ′(x)−M)‖ ≤ L0‖x − x0‖ for each x ∈ D0. (2.1)
Define the region

D0 = D ∩ U(x0,
1

L0
). (2.2)

Definition 2.2. Suppose M ∈ L (B1, B2) is an invertible operator. We say that F ′ is restricted
Lipschitz continuous if there exists L > 0 such that

‖M−1(F ′(y)− F ′(x))‖ ≤ L‖y − x‖ for each x, y ∈ D0. (2.3)
Definition 2.3. Suppose M ∈ L (B1, B2) is an invertible operator. We say that F ′ is Lipschitz
continuous if there exists L1 > 0 such that

‖M−1(F ′(y)− F ′(x))‖ ≤ L1‖y − x‖ for each x, y ∈ D. (2.4)
REMARK 2.4. It follows by these definitions that since D0 ⊆ D, we have

L0 ≤ L1 (2.5)
and

L ≤ L1. (2.6)
It is worth noting that L0 and L1 depend on x0, F ′ and D. But L depends on x0, F ′ and D0.Moreover, in practice the computation of L1 requires that of L0 and L as special cases. Theseconstants are related to majorizing sequences in Section 3.

3. Convergence of Majorizing sequences.
Let Ω ≥ 0, L1 > 0 and λ ≥ 1 be parameters. Define µ and β by

µ = λΩ and β =
λ

1 + (λ− 1)L1µ
. (3.1)

Moreover, define the quadratic majorizing function f1 by
f1(t) =

βL1t
2

2
− t + µ. (3.2)

Further more, define the scalar sequence {vn} for v0 = 0 and each n = 0, 1, 2, ... by
vn+1 = vn −

f1(vn)

f ′1(vn)
. (3.3)

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Eur. J. Math. Anal. 10.28924/ada/ma.5.11 3An auxiliary result is needed for the convergence of the sequence {vn}.
LEMMA 3.1. Suppose

h1 = 2βL1µ ≤ 1. (3.4)
Then, the following assertions hold(i) The zeros of the function f1 are real and given by

v∗ =
1−
√

1− 2βL1µ

βL1
and v∗∗ =

1 +
√

1− 2βL1µ

βL1
. (3.5)

(ii)
vn+1 − vn =

βL1(vn − vn−1)2

2(1− βL1vn)
= −

f1(vn)

f ′1(vn)
. (3.6)

(iii) The sequence {vn} is increasingly convergent to v∗ and can also be written in closed for
as

vn =

∑2n−2
j=0 qj1∑2n−1
j=0 qj1

v∗, n = 1, 2, ..., (3.7)
where,

q1 =
v∗

v∗∗
=

1−
√

1− 2βL1µ

1 +
√

1− 2βL1µ
. (3.8)

Proof. (i) The zeros of the function f are real by (3.4).By setting f (t) = 0 and using the quadratic formula we obtain v∗ and v∗∗.(ii) Let
vn+1 = g1(vn), v0 = 0, n = 0, 1, ... (3.9)where,

g1(t) =
1
2βL1t

2 − µ
βL1t − 1

. (3.10)
Multiply (3.9) by (1− βL1vn) and simplify to get

(1− βL1vn)vn+1 = µ−
1

2
βL1v

2
n ,or (3.11)

1

2
βL1v

2
n − βL1vnvn+1 +

1

2
βL1v

2
n+1 =

1

2
βL1v

2
n+1 − vn+1 + µ,

so
1

2
βL1(vn+1 − vn)2 =

1

2
βL1v

2
n+1 − vn+1 + µ.Thus, we can write

vn+1 − vn =
1
2βL1(vn − vn−1)2

1− βL1vn
= −

f1(vn)

f ′1(vn)
. (3.12)

(iii) The proof can be found in [5].
�

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Eur. J. Math. Anal. 10.28924/ada/ma.5.11 4

REMARK 3.2. (i) In view of (3.1) the results of the Lemma 3.1 can be given without β. For
example (3.4) becomes

µL1(λ+ 1) ≤ 1. (3.13)
(ii) Let L > 0. Define the quadratic majorizing function f by

f (t) =
βLt2

2
− t + µ, (3.14)

and the scalar sequence {un}f oru0 = 0 and each n = 0, 1, 2, ... by

un+1 = un −
f (un)

f ′(un)
. (3.15)

Denote the corresponding zeros of f (t) = 0 by v∗ and v∗∗, respectvely provided that

h2 = 2βLµ ≤ 1 (3.16)
Clearly, the results of the Lemma 3.1 hold, if L replaces L1 and

un+1 − un =
βL(un − un−1)2

2(1− βLun)
. (3.17)

Let L > 0. Define the sequence {sn} for 0 = 0,

s1 = µ, s2 = s1 +
βL0(s1 − s0)2

2(1− L0βs1)
and (3.18)

sn+1 = sn +
βL(sn − sn−1)2

2(1− L0βsn)
.

Next, we compare the sequences {vn}, {un}, and {sn}.

LEMMA 3.3. Suppose (2.5),(2.6) and (3.4) hold.
Then, the following assertions hold

0 ≤ sn ≤ sn+1,

0 ≤ un ≤ un+1

0 ≤ vn ≤ vn+1,

0 ≤ sn ≤ un ≤ vn

and

0 ≤ s∗ = lim
n→+∞

≤ u∗ = lim
n→+∞

=
1−
√

1− 2βLµ

βL
≤ v∗.

Proof. It follows by simple induction (2.5),(2.6) and the definition of these sequences. �

In the next section, we relate sequences {vn}, {un} and {sn} to {xn}.

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Eur. J. Math. Anal. 10.28924/ada/ma.5.11 54. Convergence of Newton’s method
The celebrated Newton-Kantorovich Theorem for solving nonlinear equations using Newton’smethod is stated next. The proof can be found in [3, 6] for M = F ′(x0). Moreover, the proof forgeneral M follows by simply using M instead of F ′(x0) in the Newton-Kantorovich Theorem.

THEOREM 4.1. Suppose that (2.4) and (3.4) hold for λ = 1, µ = Ω and Ω ≥ ‖F ′(x0)−1F (x0)‖.
Then, the sequence {xn} generated by Newton’s method (1.1) is well defined in U(x0, v

∗), remains
in U(x0, v

∗) for each n = 0, 1, 2, ... and converges to a unique solution x∗ ∈ U(x0, r
∗) of the

equation F (x) = 0. Moreover, the sequence {vn} majorizes {xn},

‖xn+1 − xn‖ ≤ vn+1 − vn (4.1)
and

‖x∗ − xn‖ ≤ v∗ − vn. (4.2)
Furthermore, if there exists v̄ ≥ v∗ such that

L1
2

(v∗ + v̄) < 1, (4.3)
then the solution x∗ is more unique in U[x0,

2
L1
− v∗], where U[x0, r ] is the closure of U(x0, r).

REMARK 4.2. In view of (2.6) Theorem (4.1) holds provided that L, {un} replace L1, {vn}, respec-
tively. By Lemma 3.1 and 3.3 the sequence {un} is tighter than {vn} and the limit point u∗ is
atleast as small as v∗. Moreover, they are given in closed form. This is not however the case for
s∗. The convergence condition for {sn} given in [2] for

M = F ′(x0), λ = 1 h3 = 2L̄µ ≤ 1, (4.4)
where

L̄ =
1

8
(4L0 +

√
L0L+ 8L20 +

√
L0L).

Notice that

h1 ≤ 1 =⇒ h2 ≤ 1 and h3 ≤ 1 (4.5)
but not necessarily vice versa unless if L0 = L = L1. Moreover,

h3
h1
→ 0 as L0

L1
→ 0. (4.6)

h3
h1
→ 0 as L0

L
→ 0. (4.7)

In view of (4.5)-(4.7) and the Lemma 3.3 the results using (4.4) improve the ones by Theorem 4.1
infinitely many times. However, s∗ is not given in closed form. But we have

s∗ ≤ s̄ , (4.8)

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Eur. J. Math. Anal. 10.28924/ada/ma.5.11 6

where

s̄ = µ+
L0µ

2

2(1− α)(1− L0µ)
, (4.9)

where

α =
2L

L+
√
L2 + 8L0L

. (4.10)
Next, we shall find an upper bound on s∗ which is given in closed form and may be tighter than

s̄ . Let
a =

1

8L
(4L0 +

√
L0L+ 8L20 +

√
L0L) (4.11)

Then, the condition (4.4) is equivalent to
h = 2aLµ ≤ 1 (4.12)

Then, the corresponding Theorem in [2] can be written as
THEOREM 4.3. Suppose for µ ≥ ‖F ′(x0)−1F (x0)‖ conditions (2.1), (2.2), (4.12) and Ū[x0, s

∗] ⊂
D. Then, the sequences {xn} generated by Newton’s method (1.1) is well defined in U(x0, s

∗),

remains in U(x0, s
∗) for each n = 0, 1, 2, ... and is convergent to a solution x∗ ∈ U[x0, s

∗] of the
equation F (x) = 0. Moreover, the following error estimates hold

‖xn+1 − xn‖ ≤ sn+1 − sn (4.13)
and

‖x∗ − xn‖ ≤ s∗ − sn. (4.14)
Additionally, if for some b ≥ s∗

L0(s
∗ + b) < 1 (4.15)

then, the solution x∗ is unique in the region D ∩ U[x0, b].

Proof. Simply notice that (4.12) is equivalent to (4.4) used in [2]. �

REMARK 4.4. By the definition of a
It follows that

0 < a ≤ 1 if L0 ≤ L (4.16)
and

a ≥ 1 if L ≤ L0. (4.17)
Define the function f by

f (t) =
aLt2

2
− t + µ (4.18)

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Eur. J. Math. Anal. 10.28924/ada/ma.5.11 7

and the sequence {s̄n} for s̄0 = 0, s̄1 = µ,

s̄2 = s̄1 +
L0(s̄1 − s̄0)2

2(1− L0s̄1)
, (4.19)

s̄n+1 = s̄n+1 −
f (s̄n+1)

f ′(s̄n+1)
, n = 1, 2, ... (4.20)

PROPOSITION 4.5. Suppose that the conditions of Theorem 4.3 hold. Then, we have that smallest
solution denoted by s̄∗ of the equation f (t) = 0, i.e.

s̄∗ =
1−
√

1− 2aLµ

aL
(4.21)

is an upper bound in closed form of the sequence {s̄n} and

0 ≤ sn ≤ s̄n, (4.22)
0 ≤ sn+1 − sn ≤ s̄n+1 − sn (4.23)

and
s∗ ≤ s̄∗. (4.24)

Proof. Indeed, this is clear under (4.16), whereas if (4.17) holds the, we have from
‖M−1(F (xn+1 − F (xn)− F ′(xn)(xn+1 − xn)‖

≤
L̃‖xn+1 − xn‖2

2

L̃

2
(s̄n+1 − s̄n)2 =

f (s̄n+1
a

,

where
L̃ =

{
L0, n = 0

L, n = 1, 2, ...
,

‖F ′(xn+1)−1M‖ ≤
1

1− L0‖xn+1 − x0‖
≤

1

1− L0s̄n+1so
‖xn+1 − xn+1‖ ≤ ‖F ′(x−1n+1M‖‖M

−1F (xn+1)‖

≤
f (s̄n+1)

a(1− L0s̄n+1)
≤ −

f (s̄n+1)

f ′(s̄n+1
,

since 1

a(1− L0s̄n+1)
≤

1

1− aLs̄n+1
= −

1

f ′(s̄n+1)
.

�

5. A Numerical Example
The convergence conditions, majorizing sequences and limit points are compared with each other.

EXAMPLE 5.1. Let B1 = B2 = R, D = U(x0, 1 − p), p ∈ (0, 1) and x0 = 1. Define the function
Ψ : D → R by

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Eur. J. Math. Anal. 10.28924/ada/ma.5.11 8

Ψ(t) = x3 − p (5.1)
Then, for λ = 1, µ = 1

3(1− p) and β = 1. Moreover, the definitions(2.1)-(2.3) hold if
L0 = 3− p, L = 2(1 + 1

3−p ) and L1 = 2(2− p).
Notice that

L0 < L1, L < L1 for each p ∈ (0, 1).

We also have that

L ≤ L0 if p ∈ (0, 2−
√

3]

and

L0 ≤ L if p ∈ [2−
√

3, 1).

Let us restrict p ∈ (0, 12) Then, the Newton-Kantorovich condition (3.4) [3, 6] does not hold,
since (3.4) is not satisfied for any p ∈ (0, 12). However, our condition (4.12) hold provided that
p ∈ (.46, 12). Thus, the old results [6] cannot guarantee the convergence of Newton’s method for
any p ∈ (0, 12). However, Newton’s method converges to x∗ = 3

√
p if we say p = 0.48. In order to

compare sequences and limit points. Let p = 0.7. Then, both (3.4) and (4.12) hold.
Thus, the old results [3, 5–7] cannot guarantee the convergence of Newton’s method for any

p ∈ (0, 12). However, Newton’s method converges to x∗ = 3
√
p. If say p = 0.48. In order to compare

sequences and limit points. Let p = 0.7. Then, both (3.4) and (4.12) hold.
Then, we have s̄ = 0.3965, v̄ = 0.6511, b = 0.3194. Therefore, the new error bounds and

limit points are tighter than the ones given before [3, 5–7] and under weaker sufficient semi-local
convergence criteria.

Table 1. Comparison Between Majorizing Sequences and their Limit Points
n vn vn+1 − vn sn sn+1 − sn s̄n s̄n+1 − s̄n
0 0 0 0 0 0 0
1 0.1000 0.1000 0.1000 0.1000 0.1000 0.1000
2 0.1176 0.0176 0.1149 0.4350e-03 0.1149 0.7619e-03
3 0.1181 = v∗ 0.0006 0.1154 = s∗ 0.0004e-03 0.1157 = s̄∗ 0.0009e-03

References
[1] I.K. Argyros, The Theory and Applications of Iteration Methods, Second edition, CRC Press, Boca Raton, 2022.[2] I.K. Argyros, S. Hilout, Weaker Conditions for the Convergence of Newton’s Method, J. Complex. 28 (2012), 364–387.

https://doi.org/10.1016/j.jco.2011.12.003.

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Eur. J. Math. Anal. 10.28924/ada/ma.5.11 9

[3] P. Deuflhard, G. Heindl, Affine Invariant Convergence Theorems for Newton’s Method and Extensions to RelatedMethods, SIAM J. Numer. Anal. 16 (1979), 1–10. https://doi.org/10.1137/0716001.[4] P. Deuflhard, Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms, Springer, Berlin,2004. https://doi.org/10.1007/978-3-642-23899-4.[5] W.B. Gragg, R.A. Tapia, Optimal Error Bounds for the Newton–Kantorovich Theorem, SIAM J. Numer. Anal. 11(1974), 10–13. https://doi.org/10.1137/0711002.[6] L.V. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982.[7] A.M. Ostrowski, Solutions of Equations in Euclidean and Banach Spaces, Academic Press, New York, 1973.

https://doi.org/10.28924/ada/ma.5.11
https://doi.org/10.1137/0716001
https://doi.org/10.1007/978-3-642-23899-4
https://doi.org/10.1137/0711002

	1. Introduction
	2. Lipschitz conditions
	3. Convergence of Majorizing sequences.
	4. Convergence of Newton's method
	5. A Numerical Example
	References