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Developments on the Convergence Analysis of Newton-Kantorovich Method for Solving

Nonlinear Equations

Samundra Regmi1, Ioannis K. Argyros2,∗, Santhosh George3, Michael I. Argyros4
1Department of Mathematics, University of Houston, Houston, TX 77204, USA

sregmi5@uh.edu
2Department of Computing and 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 Computer Science, University of Oklahoma, Norman, 73019, OK, USA
michael.i.argyros-1@ou.edu

∗Correspondence: iargyros@cameron.edu

Abstract. Developments are presented for the semi-local convergence of Newton’s method to solveBanach space-valued nonlinear equations. By utilizing a new methodology, we provide a finer con-vergence analysis with no additional conditions than in earlier results. In particular, this is done byintroducing the center-Lipschitz condition by which we construct a stricter domain than the originaldomain of the operator. Then, the Lipschitz constants in the new domain are at least as small asthe original constants leading to weaker sufficient convergence criteria, tighter error bounds on theerror distances involved, and a piece of better information on the location of the solution. Thesebenefits are obtained under the same computational cost since in practice the computation of theoriginal constants requires the computation of the new constants as special cases. The same benefitsare obtained if the Lipschitz conditions are replaced by Hölder conditions or even more general ω−continuity conditions. This methodology can be applied to other methods using such as the Secant,Stirling’s Newton-like, and other methods along the same lines. Numerical examples indicate thatthe new results can be utilized to solve nonlinear equations, but not earlier ones.

1. Introduction
Consider the problem of finding a solution x∗ ∈ Ω of the equation

F (x) = 0, (1.1)
Received: 26 Jan 2023.
Key words and phrases. Newton-Kantorovich method; Convergence; Banach space.1

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Eur. J. Math. Anal. 10.28924/ada/ma.3.15 2
where F : Ω −→ E2 is a continuously differentiable operator in the Fréchet-sense, E1,E2 areBanach spaces and Ω ⊂E1 is an open set.The solution x∗ in closed form is desirable. But this is possible only in special cases. So,most solution methods for (1.1) are iterative methods. The convergence regions for these methodsare small in general, so their applicability is reduced. The error bounds are also pessimistic (ingeneral).Among the iterative methods, the most famous one is Newton’s method (NM) defined for n =
0, 1, 2, . . . by

xn+1 = xn −F ′(xn)−1F (xn) (1.2)Kantorovich provided the semi-local convergence analysis of NM utilizing the contraction map-ping principle attributed to Banach. In particular, he presented two different proofs using majorantfunctions or recurrence relations [15]. His so-called Newton-Kantorovich Theorem is that no as-sumption on the solution is made and at the same time, the existence of the solution x∗ is established.Numerous researchers used this theorem in applications and also as a theoretical tool [1–16]. Butthe convergence criteria may not hold although NM may converge. Motivated by these concernsand optimization considerations we present new results that not only extend the convergence regionbut also provide more precise error estimates and better knowledge of the location of the solution.The novelty of the article is that these benefits require no additional conditions. This is how theusage of NM is extended. The technique used can be applied to extend other iterative methodsalong the same lines.
2. Convergence Analysis

Let α > 0, λ ≥ 0 and x0 ∈ Ω be such that ‖F ′(x0)−1‖ ≤ α, ‖F ′(x0)−1F (x0)‖ ≤ λ and
F ′(x0)

−1 ∈ L(E2,E1), the space of bounded linear operators from E2 to E1. By B(x,b),B[x,b] wedenote the open and closed balls in E1, respectively with center x ∈E1 and of radius b > 0.Some Lipschitz-type conditions are needed.
Definition 2.1. Operator F ′ is center-Lipschitz continuous about x0 on Ω if there exists L0 > 0
such that for all u ∈ Ω

‖F ′(u) −F ′(x0)‖≤ L0‖u −x0‖. (2.1)
Set

Ω0 = B(x0,
1

αL0
) ∩ Ω. (2.2)

Definition 2.2. Operator F ′ is 1−Restricted Lipschitz continuous on Ω0 if there exists L > 0 such
that

‖F ′(u) −F ′(v)‖≤ L‖u −v‖ (2.3)
for all u ∈ Ω0, v = u −F ′(u)−1F (u) ∈ Ω0.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.15 3
Definition 2.3. Operator F ′ is 2−Restricted Lipschitz continuous on Ω0 if there exists L1 > 0 such
that for all u,v ∈ Ω0

‖F ′(u) −F ′(v)‖≤ L1‖u −v‖. (2.4)
Definition 2.4. Operator F ′ is Lipschitz continuous on Ω if there exists L2 > 0 such that for all
u,v ∈ Ω

‖F ′(u) −F ′(v)‖≤ L2‖u −v‖. (2.5)
Definition 2.5. Assume:

λαL0 < 1 (2.6)
and

Ω1 = B(x1,
1

αL0
−‖x1 −x0‖) ⊂ Ω (2.7)

Then, operator is 3− restricted Lipschitz continuous on Ω1 if there exists a constant K > 0 such
that for all u ∈ Ω1

‖F ′(u) −F ′(v)‖≤ K‖u −v‖ (2.8)
for v = u −F ′(u)−1F (u) ∈ Ω1.

REMARK 2.6. By the definition of sets Ω0 and Ω1, we get

Ω0 ⊆ Ω, (2.9)
and

Ω1 ⊆ Ω0. (2.10)
Indeed, if y ∈ Ω1, then we obtain

‖y −x1‖ ≤
1

αL0
−‖x1 −x0‖⇒‖y −x1‖ + ‖x1 −x0‖≤

1

αL0

⇒ ‖y −x0‖≤
1

αL0
⇒ y ∈ Ω0 ⇒ Ω1 ⊆ Ω0.

It follows by these definitions, (2.9) and (2.10) that if the best constants are chosen in the Definitions
2.1-2.5, then

L ≤ L1 ≤ L2, (2.11)
L0 ≤ L2, (2.12)

and

K ≤ L. (2.13)
Hence, parameter K can replace results on Newton’s using the constants L,L1 and L2. Notice also
that L0 = L0(F ′, Ω), L = L(F ′, Ω0),L1 = L1(F ′, Ω0), L2 = L2(F ′, Ω) and K = K(F, Ω0, Ω1).
Examples, where (2.9)-(2.13) are strict can be found in the Numerical Section.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.15 4
Notice that the computation of the constant L2 requires the computation of the other constants

as special cases. Hence, no additional effort is needed to compute them. Moreover, they all depend
on the initial data (x0,F, Ω).

It is also worth noticing that under (2.1) we obtain

‖F ′(u)−1‖≤
α

1 −αL0‖u −x0‖
. (2.14)

This is a tighter estimate than using the stronger (2.5) to get

‖F ′(u)−1‖≤
α

1 −αL2‖u −x0‖
. (2.15)

We assume from now on that
L0 ≤ K. (2.16)

But if K < L0 then, the following results hold with L0 replacing K.
Based on the above we present two extended theorems on Newton’s method.

An important role is played in the convergence of NM by the majorizing sequence {sn} definedby
s0 = 0,sn+1 − sn = −

p(sn)

p′0(sn)

=
αK(sn − sn−1)2

1 −L0αsn
,

p(s) =
K

2
s2 −

s

α
+
λ

α
,

p0(s) =
L0
2
s2 −

s

α λ
α

.

THEOREM 2.7. (Extended Newton-Kantorovich Theorem [1, 2, 10, 12, 13, 15, 16]) Under conditions
(2.1), (2.6)-(2.8) further suppose B(x0,s∗) ⊂ Ω,

H = Kαλ ≤
1

2
. (2.17)

Then, Newton’s method (1.2) initiated at x0 ∈ Ω generates a sequence {xn} such that:{xn} ⊆
B(x0,s∗), limn−→∞xn = x∗ ∈ B[x0,s∗].

‖xn+1 −xn‖≤ sn+1 − sn (2.18)
‖x∗ −xn‖≤ s∗ − sn, (2.19)

where, limn−→∞ sn = s∗ = 1−
√
1−2H
Kα

and s∗∗ = 1+
√
1−2H
Kα

. Moreover, the following items hold for
τ = s∗

s∗∗

s∗ − sn =

{
(s∗∗−s∗)τ2

n

1−τ2n , if s∗ < s∗∗
1
2n
s∗, if s∗ = s∗∗.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.15 5
Furthermore, the element x∗ is the unique solution of equation F (x) = 0 in B[x0, s̄], where s̄ =
2
L0α
− s∗ if L0αs∗ < 2.

Proof. Simply replace L2 by K and use (2.14) instead of (2.15) in the proof of the version ofNewton-Kantorovich Theorem given in [10] (see also [3–9, 14–16].
�

REMARK 2.8. (i)If K = L2, the result of Theorem 2.7 reduces to one in the Newton-Kantorovich
Theorem where

HK = L2αλ ≤
1

2
, (2.20)

t0 = 0,tn+1 − tn = −
p̄(tn)

p̄′(tn)

=
αL2(tn − tn−1)2

1 −L2αtn
,

p̄(s) =
L2
2
s2 −

s

α
+
λ

α
,

and limn−→∞ tn = t∗ = 1−
√
1−2KK
L2α

and t∗∗ = 1+
√
1−2KK
L2α

, ¯̄s = 2
L2α
− t∗, µ = t∗t∗∗ ,

t∗ − tn =


(t∗∗−t∗)µ2

n

1−µ2n , if t∗ < t∗∗
1
2n
t∗, if t∗ = t∗∗.

Then, in view of estimates (2.11)-(2.13) we have

HK ≤
1

2
⇒ H ≤

1

2
, (2.21)

s∗ ≤ t∗, ¯̄s ≤ s̄, (2.22)
0 ≤ sn+1 − sn ≤ tn+1 − tn (2.23)

and

0 ≤ s∗ − sn ≤ t∗ − tn. (2.24)
Estimates (2.21)-(2.24) justify the advantages (A) as stated in the introduction.
(ii)A more careful look at the proof shows that tighter sequence {rn} defined by

r0 = 0, r1 = λ, r2 = r1 +
αL0(r1 − r0)2

2(1 −L0αr1)
,

rn+2 = rn+1 +
Kα(rn+1 − rn)2

2(1 −L0αrn+1)
,

also majorizes sequence {xn}. The sufficient convergence criterion for this sequence is given by

HA = K̄αλ ≤
1

2
, (2.25)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.15 6
where K̄ = 1

8
(4L0 +

√
KL0 + 8L

2
0 +
√
L0K). This criterion was given by us in [4] for K = L− 2.

Notice that

H ≤
1

2
⇒ HA ≤

1

2
. (2.26)

Hence, if (2.25) and {rn} replace (2.17) and {sn} the conclusions of Theorem 2.7 hold with these
changes too.
(iii)Suppose that there exist a > 0,b > 0 such that

‖F ′(x0 + θ(x1 −x0)) −F ′(x0)‖≤ τa‖x1 −x0‖ (2.27)
and

‖F ′(x1) −F ′(x0)‖≤ b‖x1 −x0‖ (2.28)
for all τ ∈ [0, 1]. Then, it was shown in [5] that sequence {qn} defined by

q0 = 0,q1 = λ, q2 = q1 +
αa(q1 −q0)2

2(1 −bαq1)
,

qn+2 = qn+1 +
Kα(qn+1 −qn)2

2(1 −L0αqn+1)

is also majorizing for sequence {xn}. The convergence criterion for sequence {qn} is given by

HAA =
λ

2c
≤

1

2
, (2.29)

where

p1(s) = (Ka + 2dL0(a− 2b))s2 + 4p(L0 + b)s − 4d,

d =
2K

K +
√
K2 + 8L0K

,

and

c =


1

L0+b
, Ka + 2dL0(a− 2b) = 0positive root of p1, Ka + 2dL0(a− 2b) > 0smaller positive root of p1, Ka + 2dL0(a− 2b) < 0.

Notice that b ≤ a ≤ L0. Hence, {qn} is a tighter majorizing sequence than {rn}.
Criterion (2.29) was given by us in [4] for K = L2. Therefore (2.29) and {qn} can also replace

(2.17) and {sn} in Theorem 2.7.
(iv) It follows from the definition of sequence {sn} that if

L0αsn < 1. (2.30)
Then, sequence {sn} is such that 0 ≤ sn ≤ sn+1 and limn−→∞ sn = s∗ ≤ 1L0α. Hence, weaker than
all conditions (2.30) can be used in Theorem 2.7.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.15 7
3. Examples

We test the convergence criteria.
EXAMPLE 3.1. Defined the real function f on Ω = B[x0, 1 −δ], x0 = 1, δ ∈ (0, 12) by

f (s) = s3 −δ.

Then, the definitions are satisfied for λ = 1−δ
3
, α = 1

3
, L0 = 3(3 − δ),L2 = 6(2 − δ), L1 =

6(1 + 1
3−δ ),x1 =

2+δ
3
, L =

5(4−δ
3−δ)

3+δ

3(4−δ
3−δ)

2
, a = b = δ + 5,

K = 5h
3+δ
3h2

, and h = δ+2
3

+
3−(1−δ)(3−δ)
3(1−δ) .

Denote by M1,M2,M3,M4 the set of values δ ∈ (0, 12) for which (2.20), (2.17), (2.25) and
(2.29) are satisfied, respectively. Then, by solving these inequalities for δ, we get M1 = ∅, M2 =
(0.0751, 0.5), M3 = (0.1320, 0.5) and M4 = (0.3967, 0.5).

Notice in particular that the Newton-Kantorovich criterion (2.20) [1, 9–15] cannot assure con-
vergence of NM since M1 = ∅.

A second example is provided to show that our conditions can be used to solve equations incases where the ones in [1, 2, 10, 12, 13] cannot.
EXAMPLE 3.2. Consider E1 = E2 = C[0, 1] with the norm-max. Set Ω = B(x0, 3). Define,
Hammerstein-type integral operator M on Ω by

M(z)(w) = z(w) −y(w) −
∫ 1
0

T (w,t)v3(t)dt, (3.1)
w ∈ [0, 1], z ∈ C[0, 1], where y ∈ C[0, 1] is fixed and T is a Green’s Kernel defined by

T (w,u) =

{
(1 −w)u, if u ≤ w
w(1 −u), if w ≤ u.

(3.2)
Then, the derivative M′ according to Fréchet is defined by

[M′(v)(z)](w) = z(w) − 3
∫ 1
0

T (w,u)v2(t)z(t)dt, (3.3)
w ∈ [0, 1], z ∈ C[0, 1]. Let y(w) = x0(w) = 1. Then, using (3.1)-(3.3), we obtain M′(x0)−1 ∈
L(E2,E1), ‖I − M′(x0)‖ < 38, ‖M

′(x0)
−1‖ ≤ 8

5
:= α, λ = 1

5
, L0 =

12
5
, L2 =

18
5
, and Ω0 =

B(1, 3) ∩ B(1, 5
12

) = B(1, 5
12

), so L1 = 32, and L0 < L2, L1 < L2. Set K = L = L1. Then,
the old sufficient convergence criterion is not satisfied, since αλL2 = 15

8
5
18
5

= 144
125

> 1
2

holds.
Therefore, there is no guarantee that Newton’s method (1.2) converges to x∗ under the conditions
of the aforementioned references. But our condition hold, since dba = 1

5
8
5
3
2

= 24
50
< 1
2
. Therefore,

the conclusions of our Theorem 2.7 follow.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.15 8
4. Conclusion

The technique of recurrent functions has been utilized to extend the sufficient conditions forconvergence of NM for solving nonlinear equations. The new results are finer than the earlierones. So, they can replace them. No additional conditions have been used. The technique is verygeneral rendering useful to extend the usage of other iterative methods.
Declarations

The authors declare that there are no competing interests and that all authors contributedequally in conceptualization, methodology, formal analysis, and investigation. The original draftwas prepared by I. K. Argyros and review and editing was done by S. Regmi, S. George, and M.Argyros.
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Eur. J. Math. Anal. 10.28924/ada/ma.3.15 9
[16] R. Verma, New trends in fractional programming, Nova Science Publisher, New York, USA, (2019). https://

novapublishers.com/shop/new-trends-in-fractional-programming/.

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https://novapublishers.com/shop/new-trends-in-fractional-programming/
https://novapublishers.com/shop/new-trends-in-fractional-programming/

	1. Introduction
	2.  Convergence Analysis
	3.  Examples
	4. Conclusion
	Declarations
	References

