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Developments of Newton’s Method under Hölder Conditions

Samundra Regmi1, Ioannis K. Argyros2,∗, Santhosh George3, Christopher I. Argyros4
1Learning Commons, University of North Texas at Dallas, Dallas, TX, USA

samundra.regmi@untdallas.edu
2Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA

iargyros@cameron.edu
3Department of Mathematical and Computational Sciences,National Institute of Technology Karnataka,

India-575 025
sgeorge@nitk.edu.in

4Department of Computing and Technology, Cameron University, Lawton, OK 73505, USA
christopher.argyros@cameron.edu

∗Correspondence: iargyros@cameron.edu

Abstract. The semi-local convergence criteria for Newton’s method are weakened without new con-ditions. Moreover, tighter error distances are provided as well as a more precise information on thelocation of the solution.

1. Introduction
The computation of a solution x∗ of nonlinear equation

F (x) = 0 (1.1)is important in computational sciences, since many applications can be written as (1.1). Here
F : Ω ⊆ X −→ Y is Fréchet-differentiable operator, X, Y are Banach spaces and Ω 6= ∅ is aconvex and open set. But this can be attained only in special cases. That explains why mostsolution methods for (1.1) are iterative. There is a plethora of methods for solving (1.1) [1–14].Among them Newton’s method (NM) defined by

x0 ∈ Ω, xn+1 = xn − F ′(xn)−1F (xn) (1.2)
seems to be the most popular [2,4]. But the convergence domain is small, limiting the applicability ofNM. That is why we have developed a technique that determines a subset Ω0 of Ω also containingthe iterates {xn}. Hence, the Hölder constants are at least as tight as the ones in Ω. This crucial

Received: 20 Mar 2022.
Key words and phrases. Banach space; Hölder condition; semi-local convergence; convergence criteria.1

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https://doi.org/10.28924/ada/ma.2.18


Eur. J. Math. Anal. 10.28924/ada/ma.2.18 2modification leads to: weaker sufficient convergence criteria, the extension of the convergencedomain, tighter error estimates on ‖x∗ − xn‖, ‖xn+1 − xn‖ and a more precise information on x∗.It is worth noticing that these advantages are obtained without additional conditions, since inpractice the evolution of the old Hölderian constants require that of the new conditions as specialcases.
2. Convergence

We introduce certain Hölder conditions crucial for the semi-local convergence. Let p ∈ (0, 1].Suppose there exists x0 ∈ Ω such that F ′(x0)−1 ∈ L(Y,X).

Definition 2.1. Operator F ′ is center Hölderian on Ω if there exists H0 > 0 such that

‖F ′(x0)−1F ′(w)− F ′(x0)‖ ≤ H0‖w − x0‖p (2.1)
for all w ∈ Ω.

Set
Ω0 = U(x0,

1

H
1
p

0

) ∩Ω. (2.2)
Definition 2.2. Operator F ′ is center Hölderian on Ω0 if there exists H > 0 such that

‖F ′(x0)−1F ′(w)− F ′(u)‖ ≤ H̃‖w − u‖p, (2.3)
where H̃ =

{
H, w = u − F ′(u)−1F (u), u ∈ D0
K, w, u ∈ Ω0.

.

We present the results with H although K can be used too. But notice H ≤ K.
Definition 2.3. Operator F ′ is center Hölderian on Ω if there exists H1 > 0 such that

‖F ′(x0)−1F ′(w)− F ′(u)‖ ≤ H1‖w − u‖p (2.4)
for all w, u ∈ Ω.

REMARK 2.4. It follows from (2.2), that

Ω0 ⊆ Ω. (2.5)
Then, by (2.1)-(2.5) the following items hold

H0 ≤ H1 (2.6)
and

H ≤ H1. (2.7)
We shall assume that

H0 ≤ H. (2.8)

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Eur. J. Math. Anal. 10.28924/ada/ma.2.18 3

Otherwise the results that follow hold with H0 replacing H. Notice that H0 = H0(x0,Ω), H1 =

H1(x0,Ω), H = H(x0,Ω0) and H0
H1

can be small (arbitrarily) [2–4]. In earlier studies [1, 5–14] the
estimate

‖F ′(z)−1F ′(x0)‖ ≤
1

1−H1‖z − x0‖
1
p

(2.9)
for all z ∈ U(x0,

1

H
1
p
1

) was found using (2.4). But, if we use (2.1) to obtain the weaker and more

precise estimate

‖F ′(z)−1F ′(x0)‖ ≤
1

1−H0‖z − x0‖
1
p

(2.10)
for all z ∈ U(x0,

1

H
1
p
0

). This modification in the proofs and exchanging H1 by H leads to the

advantages as already mentioned in the introduction. That is why we omit the proofs in our results
that follow. Notice also that in practice the computation of H1 require that of H0 and H as special
cases. Hence, the applicability of NM is extended without additional conditions.

Let d ≥ 0 be such that
‖F ′(x0)−1F (x0)‖ ≤ d. (2.11)

We assume that (2.1)-(2.3) hold from now on unless otherwise stated. First we extend the resultsby Keller [11] for NM. Similarly the results for the chord method can also be extended. We leavethe details to the motivated reader. For brevity we skip the extensions on the radii of convergenceballs, and only mention convergence criteria and error estimates.
THEOREM 2.5. Assume:

Hrλ <
1 + λ

2 + λ
,

d ≤
[

1−
2 + λ

1 + λ
Hrλ

]
λ

and

Ū(x0, r) ⊂ Ω.

Then, limn−→∞ xn = x∗ ∈ U(x0, r0) and F (x∗) = 0. Furthermore,

‖x∗ − xn‖ ≤

(
µ
1
λ

2 + λ

)(1+λ)p
r

µ
1
λ

,

where µ = Hrλ

1−H0rλ
1
1+λ < 1.

Proof. See Theorem 2 in [11].
�

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Eur. J. Math. Anal. 10.28924/ada/ma.2.18 4

THEOREM 2.6. Assume:

Hdλ <
1

2 + λ

(
λ

1 + λ

)λ
and

Ū(x0, r) ⊂ Ω.

Then, limn−→∞ xn = x∗ ∈ U(x0, r0), F (x∗) = 0 and

‖x∗ − xn‖ ≤

(
λ
1
p

1− λ

)(1+p)n
d

µ
1
p

,

where λ =
Hrp0
1−H0rp0

(
d
r0

)p
1
1+p < 1 and r0 is the minimal positive root of scalar equation

(2 + p)Ht1+p − (1 + p)(t − d) = 0

provided that r ≥ r0.

Proof. See Theorem 4 in [11].
�

THEOREM 2.7. Assume:
Hdp ≤ 1−

(
p

1 + p

)p
,

R ≥
1 + p

2 + p − (1 + p)p
d

and
Ū(x0, r) ⊂ Ω.

Then, limn−→∞ xn = x∗ ∈ U(x0, r), F (x∗) = 0 and

‖x∗ − xn‖ ≤
(

1

1 + p

)n
[(1 + p)H

1
p d ](1+p)

n

H
1
p .

Proof. See Theorem 5 in [11].
�Next, we extend a result given in [6] which in turn extended earlier ones [1,7–14]. It is convenientto define function on the interval [0,∞) by

g(t) =
H

1 + p
t1+p − t + d

gβ(t) =
βH

1 + p
t1+p − t + d (β ≥ 0)

h(t) =
t1+p + (1 + p)t

(1 + p)1+p − 1
,

v(p) = max
t≥0

h(t),

δ(p) = min{β ≥ 1 : max h(t) ≤ β, 0 ≤ t ≤ t(β)}

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Eur. J. Math. Anal. 10.28924/ada/ma.2.18 5and scalar sequence {sn} by
s0 = 0, sn = sn−1 −

gd(sn−1)

g′(sn−1)
.

Then, we can show:
THEOREM 2.8. Assume:

d ≤
1

v(p)

(
p

1 + p

)p
and

U(x0, r̄) ⊆ Ω,

where r̄ is the minimal solution of equation gv (p) = 0,

gv (t) =
v(p)H

1 + p
t1+p − t + d.

Proof. See Theorem 2.2 in [6].
�Next, we present the extensions of the work by Rokne in [13] but for the Newton-like method(NLM)

xn+1 = xn − L−1n F (xn),

where Ln is a linear operator approximating F ′(xn).

THEOREM 2.9. Assume:

‖L(x)− L(x0)‖ ≤ M0‖x − x0‖p

for all x ∈ Ω. Set Ω0 = U(x0,
1

(γ2M0)
1
p

).

‖F ′(x)− F ′(y)‖ ≤ M̄‖x − y‖p

for all x, y ∈ Ω0,

‖F ′(x)− L(x)‖ ≤ γ0 + γ1‖x − x0‖p

for all x ∈ Ω0, and some γ0 ≥ 0, γ1 ≥ 0. L(x0)
−1 ∈ L(Y,X) with ‖L(x0)

−1‖ ≤ γ2 and
‖L(x0)

−1F (x0)‖ ≤ γ3, function q defined by

q(t) = t1+p(γ2γ0 + γ2M0) + t(
γ2M̄d

p

1 + p
+ γ2γ0 − 1)− γ2M0γ3tp + γ3

has a smallest positive zero R > γ3,

γ2M̄R
p < 1,

ρ =
p

1− γ2M̄Rp

[
γ2M̄d

p

1 + p
+ γ2γ0 + γ2γ1R

p

]
< 1,

Ū(x0, R) ⊂ Ω. Then limn−→∞ xn = x∗ and F (x∗) = 0.

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Eur. J. Math. Anal. 10.28924/ada/ma.2.18 6

Proof. See Theorem 1 in [13].
�Many results on Newton’s method were also reported in the elegant book in [9]. Next, we showhow to extend one of them. The details of how to extend the result of them are left to the motivatedreader.

THEOREM 2.10. Suppose: conditions (2.1), (2.3), (2.8), and
(C) h0 = Hdp ∈ (0, ρ) where ρ is the only solution of equation

(1 + p)p(1− t)1+p − tp = 0, p ∈ (0, 1]

in (0, 12 ] and U(x0, s) ⊂ Ω, where s = (1+p)(1−h0)
(1+p)−(2+p)h0 hold. Then, sequence {xn} converges to a

solution x∗ of equation F (x) = 0. Moreover, {xn}, x∗ ∈ U[x0, s] and x∗ is the only solution in
Ω ∩ U(x0,

d

h
1/p
0

). Moreover, the following error estimates hold

‖xn − x∗‖ ≤ en,

where en = δ
(1+p)n−1

p2 An

1−δ
(1+p)n
p A

d, with δ = h1
h0
, A = 1 − h0, h1 = h0f1(h0)

1+pf2(h0)
p, f1(t) = 1

1−t

and f2(t) = t
1+p .

Finally, we extend the results by F. Cianciaruso and E. De Pascale in [6] who in turn extendedearlier ones [1, 5, 7, 11,12,14]. Define scalar sequence {vn} for h = dpH by
v0 = 0, v1 = h

1
p ,

vn+1 = vn +
(vn − vn−1)1+p

(1 + p)(1− vpn )
. (2.12)

Next, we extend Theorem 2.1 and Theorem 2.3 in [6], respectively.
THEOREM 2.11. Let function f : [1,∞) −→ [0,∞), R : [0,∞) −→ [0,∞) be defined by

f (t) = (1−
1

t
)

1 + p

((1 + p)
1
1−p + (t(t − 1)p)

1
1−p )1−p

and

R(t) =
(1 + p)

1
p

((1 + p)
1
1−p + (t(t − 1)p)

1
1−p )1−p

.

Suppose that
h ≤ f (M), (2.13)

where M is a global maximum for function f , given explicitly by M =
1+
√
1+4(1+p)pp1−p

2 . Then, the
following assertion hold

vn ≤ R(M)(1−
1

Mn
), (2.14)

vn+1
vn
≤

1− 1
Mn+1

1− 1
Mn

, (2.15)

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Eur. J. Math. Anal. 10.28924/ada/ma.2.18 7

vn ≤ vn+1 ≤ R(M) < 1

and limn−→∞ vn = v∗ ∈ [0, R(M)].

Simply use H for H1 in [6].
�

THEOREM 2.12. Under condition (2.13) further suppose that r∗ = H−
1
p v∗ ≤ ρ and U(x0, ρ) ⊆ Ω.

Then, sequence {xn} generated by NM is well defined in U(x0, v
∗), stays in U(x0, v

∗) and converges
to the unique solution x∗ ∈ U[x0, v

∗] of equation F (x) = 0, so that

‖xn+1 − xn‖ ≤ vn+1 − vn

and

‖x∗ − xn‖ ≤ v∗ − vn.

Proof. Simply use H for H1 used in [6].
REMARK 2.13. (1) If K = H1 the last two results coincide with the corresponding ones in [6].
But if K < H1 then the new results constitute an improvement with benefits already stated in the
introduction. Notice that the majorizing sequence {wn} in [6] was defined for h1 = dpH1 by

w0 = 0, w1 = h
1
p

1 ,

wn+1 = wn +
(wn − wn−1)1+p

(1 + p)(1− wpn )
, (2.16)

and the convergence criterion is

h1 ≤ f (M). (2.17)
It then follows by (2.7), (2.12), (2.13), (2.16) and (2.17) that

h1 ≤ f (M)⇒ h ≤ f (M) (2.18)
but not necessarily vice versa, unless if H = H1,

vn ≤ wn,

0 ≤ vn+1 − vn ≤ wn+1 − wn

and

0 ≤ v∗ ≤ w∗ = lim
n−→∞

wn.

(2) In view of (2.9) and (2.10) sequence {un} defined for each n = 0, 1, 2, . . . by

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Eur. J. Math. Anal. 10.28924/ada/ma.2.18 8

u0 = 0, u1 = h
1
p

1 ,

u2 = u1 +
H0(u1 − u0)1+p

(1 + p)(1−H0up1)
,

un+1 = un +
H(un − un−1)1+p

(1 + p)(1−H0upn)

is a tighter majorizing sequence than {vn} and can replace it in Theorem 2.11 and Theorem 2.12.
Concerning the uniqueness of the solution x∗ we provide a result based only on (2.1).

PROPOSITION 2.14. Suppose:
(1) The point x∗ ∈ U(x0, a) ⊂ Ω is a simple solution of equation F (x) = 0 for some a > 0.

(2) Condition (2.1) holds.
(3) There exist b ≥ a such that

H0

∫ 1
0

((1− τ)a + τb)pdτ < 1. (2.19)
Let G = U[x0, b] ∩Ω. Then, the point x∗ is the only solution of equation F (x) = 0 in the set G.

Proof. Let z∗ ∈ G with F (z∗) = 0. By (2.1) and (2.19), we obtain in turn for Q =
∫ 1
0 F
′(x∗ +

τ(z∗ − x∗))dτ

‖F ′(x0)−1(Q− F ′(x0))‖ ≤ H0

∫ 1
0

‖x∗ + τ(z∗ − x∗)− x0‖pdτ

≤ H0

∫ 1
0

[(1− τ)‖x∗ − x0‖+ τ‖z∗ − x0‖]pdτ

≤ H0

∫ 1
0

((1− τ)a + τb)pdτ < 1,

showing z∗ = x∗ by the invertibility of Q and the approximation Q(x∗− z∗) = F (x∗)−F (z∗) = 0.

�Notice that if K = H1 the results coincide to the ones of Theorem 3.4 in [9]. But, if K < H1then they constitute an extension.
REMARK 2.15. (a) We gave the results in affine invariant form.
(b)The results in this study can be extended more if we consider the set S = U(x1,

1
H1/p
− d)

provided that H1/pd < 1. Moreover, suppose S ⊂ Ω. Then, S ⊂ Ω0, so the Hölderian constant
corresponding to S is at least as small as K, and can replace it in all previous results.

References
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Eur. J. Math. Anal. 10.28924/ada/ma.2.18 9

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https://doi.org/10.28924/ada/ma.2.18
https://doi.org/10.1016/j.jco.2010.08.006
https://doi.org/10.1016/j.jco.2010.08.006
https://doi.org/10.1007/978-0-387-72743-1
https://doi.org/10.1007/978-0-387-72743-1
https://doi.org/10.1081/nfa-120026367
https://doi.org/10.1016/j.jmaa.2005.09.008
https://doi.org/10.4171/zaa/821
https://doi.org/10.1007/978-3-030-48702-7
https://doi.org/10.1016/s0022-0000(70)80009-5
https://doi.org/10.1016/s0022-0000(70)80009-5
https://doi.org/10.1007/bf01406677

	1. Introduction
	2.  Convergence
	References

