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CUBO A Mathematical Journal
Vol.13, No¯ 03, (1–15). October 2011

On the semilocal convergence of Newton–type methods,
when the derivative is not continuously invertible

Ioannis K. Argyros

Department of Mathematics Sciences,

Cameron University,

Lawton, OK 73505, USA.

email: iargyros@cameron.edu

and

Säıd Hilout

Laboratoire de Mathématiques et Applications,

Poitiers University,

Bd. Pierre et Marie Curie, Téléport 2, B.P. 30179

86962 Futuroscope Chasseneuil Cedex, France

email: said.hilout@math.univ--poitiers.fr

ABSTRACT

We provide a semilocal convergence analysis for Newton–type methods to approximate a

locally unique solution of a nonlinear equation in a Banach space setting. The Fréchet–

derivative of the operator involved is not necessarily continuous invertible. This way

we extend the applicability of Newton–type methods [1]–[12].

We also provide weaker sufficient convergence conditions, and finer error bound on the

distances involved (under the same computational cost) than [1]–[12], in some intersting

cases. Numerical examples are also provided in this study.



2 Ioannis K. Argyros and Säıd Hilout CUBO
13, 3 (2011)

RESUMEN

Ofrecemos un análisis de convergencia semilocal de los métodos de Newton type para

aproximar una solución local única de una ecuación no lineal en un entorno de un

espacio de Banach. L derivada de Fréchet del operador en cuestión no es necesariamente

invertible continua. De esta manera ampliamos la aplicabilidad de los métodos del tipo

Newton [1]–[12].

También proporcionamos condiciones suficientes mas dbiles de convergencia, y una cota

de error más fina de las distancias involucradas que [1]–[12] (en el mismo coste com-

putacional), en algunos casos interesantes. también presentamos ejemplos numéricos.

Keywords: Newton–type methods, Banach space, small divisors, non–invertible operators, semilo-

cal convergence, Newton–Kantorovich–type hypothesis.

Mathematics Subject Classification: 65H10, 65G99, 65J15, 47H17, 49M15.

1 Introduction

In this study we are concerned with the problem of approximating a locally unique solution x⋆ of

equation

F(x) = 0, (1.1)

where, F is a Fréchet–differentiable operator defined on a convex subset D of a Banach space X
with values in a Banach space Y.

A large number of problems in applied mathematics and also in engineering are solved by find-

ing the solutions of certain equations. For example, dynamic systems are mathematically modeled

by difference or differential equations, and their solutions usually represent the states of the sys-

tems. For the sake of simplicity, assume that a time–invariant system is driven by the equation

ẋ = T (x), for some suitable operator T , where x is the state. Then the equilibrium states are deter-

mined by solving equation (1.1). Similar equations are used in the case of discrete systems. The

unknowns of engineering equations can be functions (difference, differential, and integral equa-

tions), vectors (systems of linear or nonlinear algebraic equations), or real or complex numbers

(single algebraic equations with single unknowns). Except in special cases, the most commonly

used solution methods are iterative–when starting from one or several initial approximations a

sequence is constructed that converges to a solution of the equation. Iteration methods are also

applied for solving optimization problems. In such cases, the iteration sequences converge to an

optimal solution of the problem at hand. Since all of these methods have the same recursive struc-

ture, they can be introduced and discussed in a general framework.



CUBO
13, 3 (2011)

On the semilocal convergence of Newton–type · · · 3

The most popular method for generating a sequence {xn} (n ≥ 0), approximating x⋆ is un-
doubtedly Newton’s method:

xn+1 = xn − F
′(xn)

−1
F(xn) (n ≥ 0), (x0 ∈ D). (1.2)

There is an extensive literature on local as well as semilocal convergence results for Newton’s

method [1]–[12].

However, there many problems for which Newton’s method is not applicable in its original

form. A case of interest occurs when the derivative is not continuously invertible, as for instance,

dealing with problems involving small divisors, or other important examples [4], [6]–[10]. Several

Newton–type methods have addressed this problem [1]–[12]. Moret in [10] unified a large class of

such Newton–type methods, where, at each step, the inverse of the derivative, is replaced by a

linear operator, which is obtained recursively from the previous one.

Two iterative schemes were provided in [10]:

Scheme 1. Let the following be given: a Banach space Z, an operator valued mapping B : D −→
L(X , Y), x0 ∈ D, S−1 ∈ L(Z, Y), R−1 ∈ L(Z, X ).

For n ≥ 0, let Nn ∈ L(Z, Z), and set:

Sn = Sn−1 Nn + B(xn) Rn−1,
Rn = Rn−1 + Rn−1 Nn,
xn+1 = xn + Rn ∆n,

∆n being a possible approximate solution of

Sn ∆n = −F(xn).

That is ∆n satisfies an equation of type

Sn ∆n = −(F(xn) + rn),

for a suitable null sequence {rn} ⊂ Y.

Moret [10] provided a semilocal convergence analysis for Newton’s method under general

conditions on the mapping B, the starting guesses x0, S−1, R−1, the operators Nn, and the
sequence {rn}.

Similar results were given for scheme 2.



4 Ioannis K. Argyros and Säıd Hilout CUBO
13, 3 (2011)

There are two type of problems common to both schemes: the convergence may be too slow,

and the schemes may not even be applicable. For example in both schemes a condition of the form:

‖ IY − T R−1 ‖< 1 (1.3)

is required, where,

T = F′(x0) (Scheme 1) (1.4)

or

T = B(x0) (Scheme 2). (1.5)

There are simple numerical examples to show that condition (1.3) is violated.

Let scalar function F given by:

F(x) =
1

128
x

2 −
1

64
cos x, (1.6)

and choose x0 = 0.

Then F′(x0) = 0, which shows that (1.3) is violated for T = B(x0) = F′(x0). Hence, Schemes
1 and 2 cannot be used to approximate x⋆. In particular, the classical Newton–Moser method [4],

[6]–[10], obtained from Scheme 1, by setting Z = Y, B = F′, S−1 = IY , Nn = IY − F′(xn) Rn−1,
and rn = 0 (n ≥ 0) cannot be used.

To address these problems, we consider the Newton–type methods of the form (NTM):

xn+1 = xn − (F
′(xn) − A)−1 F(xn) (1.7)

or more generally (NLM):

xn+1 = xn − (C(xn) − A)−1 F(xn), (1.8)

where, C(x) ∈ L(X , Y) is an approximation of F′(x), (x ∈ D), and A a given linear operator.

In the case of function F given in (1.6) (NTM) can be used to approximate x⋆, if A is an
invertible operator.

Methods (1.7), and (1.8) can be combined into one, even more general (GNTM):

xn+1 = xn − A(xn)−1 F(xn), (1.9)

where, A(x) ∈ L(X , Y), (x ∈ D).



CUBO
13, 3 (2011)

On the semilocal convergence of Newton–type · · · 5

Note that if:

A(x) = F′(x) − A (1.10)

or

A(x) = C(x) − A, (1.11)

we obtain (1.7), and (1.8), respectively.

Sufficient conditions for semilocal convergence of (GNTM), and estimates on the distances

‖ xn+1 − xn ‖, ‖ xn − x⋆ ‖ have been given by several authors [1]–[6], [11], [12].

However, in the special case of (1.10) (or (1.11)), we can do better by a direct approach. That

is, we can provide (under the same computational cost) weaker sufficient convergence conditions,

and finer error estimates on ‖ xn+1 − xn ‖, ‖ xn − x⋆ ‖ (n ≥ 0).

2 Semilocal convergence analysis of (GNTM)

Let L0 > 0, L > 0, η > 0, a > 0, c > 1, be given constants.

Set:

b =
c

2
(L η + 2 a), r =

c − 1

c L0
.

We need the following result on majorizing sequences for (GNTM):

Lemma 2.1. Assume:

c ∈ (1,
1

a
), (2.1)

and

η ≤
(c − 1) (1 − c a)

c

(

L0 +
c − 1

2
L

) . (2.2)

Then, scalar sequence {tn} (n ≥ 0) given by

t0 = 0, t1 = η, tn+2 = tn+1 +
L (tn+1 − tn) + 2 a

2 (1 − L0 tn+1)
(tn+1 − tn) (2.3)

is non–decreasing, bounded by

t
⋆⋆ =

c − 1

c L0
, (2.4)



6 Ioannis K. Argyros and Säıd Hilout CUBO
13, 3 (2011)

and converges to its unique least upper bound t⋆, satisfying:

0 ≤ t⋆ ≤
η

1 − b
≤ t⋆⋆. (2.5)

Moreover the following estimates hold for all n ≥ 0:

tn+2 − tn+1 ≤ b (tn+1 − tn) ≤ bn+1 η. (2.6)

Proof. We shall show using induction on the integer k:

0 ≤ tk+2 − tk+1 =
L (tk+1 − tk) + 2 a

2 (1 − L0 tk+1)
(tk+1 − tk) ≤ b (tk+1 − tk), (2.7)

and

tk+1 < r0. (2.8)

Estimates (2.7), and (2.8) hold for k = 0, by the initial conditions. Assume (2.7), and (2.8)

hold for all m ≤ k. Then, we have:

0 ≤ tk+2 − tk+1 ≤ b (tk+1 − tk)
≤ b (b (tk − tk−1)) = b2 (tk − tk−1) ≤ · · ·
≤ bk+1 η

and
tk+1 ≤ tk + bk η

≤ tk−1 + bk−1 η + bk η
≤ t1 + b1 η + · · · + bk η

=
1 − bk+1

1 − b
η <

η

1 − b
≤ r by (2.1), and (2.2),

which complete the induction for (2.7), and (2.8).

Finally, sequence {tn} is non–decreasing, and bounded above by t
⋆⋆, and as such it converges

to its unique least upper bound t⋆.

That completes the proof of Lemma 2.1. ♦

We shall show the following semilocal convergence theorem for (GNTM) in the special case,

when A is given by (1.10).

Theorem 2.1. Let F : D ⊆ X −→ Y be a Fréchet–differentiable operator, and let A(x) ∈ L(X , Y)
be given by (1.10). Assume that there exist an open convex subset D of X , x0 ∈ D, a bounded



CUBO
13, 3 (2011)

On the semilocal convergence of Newton–type · · · 7

inverse A(x0)−1 of A(x0), and constants L0 > 0, L > 0, a ≥ 0, and η > 0, such that for all
x, y ∈ D:

‖ A(x0)−1 F(x0) ‖≤ η, (2.9)
‖ A(x0)−1 [F′(x) − F′(y)] ‖≤ L ‖ x − y ‖, (2.10)
‖ A(x0)−1 [F′(x) − A(x)] ‖≤ a, (2.11)
‖ A(x0)−1 [A(x) − A(x0)] ‖≤ L0 ‖ x − x0 ‖, (2.12)
U(x0, t

⋆) = {x ∈ X , ‖ x − x0 ‖≤ t⋆} ⊆ D, (2.13)

and

the hypotheses of Lemma 2.1 hold.

Then, sequence {xn} (n ≥ 0) generated by (GNTM) is well defined, remains in U(x0, t⋆) for
all n ≥ 0, and converges to a solution x⋆ of equation F(x) = 0 in U(x0, t⋆).

Moreover, the following estimates hold for all n ≥ 0:

‖ xn+1 − xn ‖≤ tn+1 − tn, (2.14)

and

‖ xn − x⋆ ‖≤ t⋆ − tn, (2.15)

where, sequence {tn} (n ≥ 0), and t⋆ are given in Lemma 2.1.

Furthermore, the solution x⋆ of equation (1.1) is unique in U(x0, t
⋆) provided that:

(

L

2
+ a + L0

)

t
⋆ + a < 1. (2.16)

Proof. We shall show using induction on m ≥ 0:

‖ xm+1 − xm ‖≤ tm+1 − tm, (2.17)

and

U(xm+1, t
⋆ − tm+1) ⊆ U(xm, t⋆ − tm). (2.18)

For every z ∈ U(x1, t⋆ − t1),

‖ z − x0 ‖ ≤ ‖ z − x1 ‖ + ‖ x1 − x0 ‖
≤ t⋆ − t1 + t1 = t⋆ − t0,



8 Ioannis K. Argyros and Säıd Hilout CUBO
13, 3 (2011)

implies z ∈ U(x0, t⋆ − t0). We also have

‖ x1 − x0 ‖=‖ A(x0)−1 F(x0) ‖≤ η = t1 − t0.

That is (2.17), and (2.18) hold for m = 0.

Given they hold for n ≤ m, then:

‖ xm+1 − x0 ‖ ≤
m+1∑

i=1

‖ xi − xi−1 ‖

≤
m+1∑

i=1

(ti − ti−1) = tm+1 − t0 = tm+1,

and

‖ xm + θ (xm+1 − xm) − x0 ‖ ≤ tm + θ (tm+1 − tm)
≤ t⋆,

for all θ ∈ (0, 1).

Using (2.8), (2.12), the induction hypotheses, we get:

‖ A(x0)−1 [A(xm+1) − A(x0)] ‖ ≤ L0 ‖ xm+1 − x0 ‖
≤ L0 (tm+1 − t0)
≤ L0 tm+1 < 1.

(2.19)

It follows from (2.19), and the Banach lemma on invertible operators [4], that A(xm+1)−1
exists, and

‖ A(xm+1)−1 A(x0) ‖≤ (1 − L tm+1)−1. (2.20)

Using (1.9), we obtain the approximation:

xm+2 − xm+1 = −A(xm+1)−1 F(xm+1)

= −A(xm+1)−1 A(x0) A(x0)−1
( ∫

1

0

[F′(xm+1 + θ (xm − xm+1)) − F
′(xm)] (xm+1 − xm) dθ+

(F′(xm) − A(xm)) (xm+1 − xm)
)

(2.21)

Using (2.10)–(2.12), (2.17) (2.20), and (2.21), and the induction hypotheses, we obtain in turn:



CUBO
13, 3 (2011)

On the semilocal convergence of Newton–type · · · 9

‖ xm+2 − xm+1 ‖ ≤ (1 − L0 tm+1)−1
(

L

2
‖ xm+1 − xm ‖2 +a ‖ xm+1 − xm ‖

)

≤ (1 − L0 tm+1)−1
(

L

2
(tm+1 − tm) + a

)

(tm+1 − tm)

= tm+2 − tm+1,

(2.22)

which shows (2.17) for all m ≥ 0.

Thus, for every z ∈ U(xm+2, t⋆ − tm+2), we have:

‖ z − xm+1 ‖ ≤ ‖ z − xm+2 ‖ + ‖ xm+2 − xm+1 ‖
≤ t⋆ − tm+2 + tm+2 − tm+1 = t⋆ − tm+1,

which shows (2.18) for all m ≥ 0.

Lemma 2.1 implies that sequence {tn} is Cauchy. Moreover, it follows from (2.17) and (2.18)

that {xn} (n ≥ 0) is also a Cauchy sequence in a Banach space X , and as such it converges to some
x⋆ ∈ U(x0, t⋆) (since U(x0, t⋆) is a closed set).

By letting m −→ ∞ in (2.22), we obtain F(x⋆) = 0. Furthermore estimate (2.15) is obtained
from (2.14) by using standard majorization techniques [1], [4]. Finally to show that x⋆ is the unique

solution of equation (1.1) in U(x0, t
⋆), as in (2.21) and (2.22), we get in turn for y⋆ ∈ U(x0, t⋆),

with F(y⋆) = 0, the estimation:

‖ y⋆ − xm+1 ‖ ≤ ‖ A(xm)−1 A(x0) ‖
( ∫

1

0

‖ A(x0)−1 (F′(xm + θ (y⋆ − xm)) − F′(xm)) ‖ dθ

+ ‖ A(x0)−1 [F′(xm) − A(xm)] ‖
)

‖ y⋆ − xm ‖

≤ (1 − L0 tm+1)−1
(

L

2
‖ y⋆ − xm ‖2 +a ‖ y⋆ − xm ‖

)

≤ (1 − L0 tm+1)−1
(

L

2
(t⋆ − tm) + a

)

‖ y⋆ − xm ‖

≤ (1 − L0 t⋆)−1
(

L

2
(t⋆ − t0) + a

)

‖ x⋆ − xm ‖

< ‖ y⋆ − xm ‖

(2.23)

by (2.20).

It follows by (2.23) that lim
m−→ ∞

xm = y
⋆. But we showed lim

m−→ ∞
xm = x

⋆. Hence, we deduce

x⋆ = y⋆.



10 Ioannis K. Argyros and Säıd Hilout CUBO
13, 3 (2011)

That completes the proof of Theorem 2.1. ♦

Note that t⋆ can be replaced by t⋆⋆ given by (2.4) in the uniqueness hypothesis provided that

U(x0, t
⋆⋆) ⊆ D, or in all hypotheses of the theorem.

3 Applications

Remark 3.1. According to [4], [6], [12], the sufficient convergence condition becomes

η ≤
(1 − a)2

2 L
, (3.1)

and the majorizing iteration {vn} is given by:

v0 = 0, v1 = η, vn+2 = vn+1 +

L

2
v

2

n+1
− (1 − a) vn+1 + η

1 − L0 vn+1
(n ≥ 0). (3.2)

In view of (2.2), and (3.1), our condition is weaker, if:

(1 − a)2

2 L
<

(c − 1) (1 − c a)

c (L0 +
c − 1

2
L)

. (3.3)

Let L0 = p L, for p ∈ [0, 1].

Then (3.3) holds, if:
(1 − a)2

2
<

(c − 1) (1 − c a)

c (p +
c − 1

2
)

. (3.4)

If p is close enough to zero, and e.g. c = 2, we have (3.4) holds provided:

(1 − a)2 < 2 (1 − 2 a)

or

a ∈ (0,
√

2 − 1).

As example, set p = .1, c = 2, a = .3, then condition (3.4) becomes:

.245 < .333 (3.5)

Hence, for η ∈ (.245, .333), our results can apply, where as the ones in [4], [6], [12] cannot
guarantee convergence, since (3.1) is violated.



CUBO
13, 3 (2011)

On the semilocal convergence of Newton–type · · · 11

Concerning the error estimates, if:

(L − 2 L0) η + 2 (a + 1)

1 − L0 η
(1 − η) + L η < 1 + 2 a, (3.6)

then, we have

v0 = t0, v1 = t1, v2 = t2, t3 − t2 < v3 − v2.

An inductive argument shows:

tn < vn (n ≥ 3),

tn+1 − tn < vn+1 − vn (n ≥ 2),
t
⋆ − tn < v

⋆ − vn (n ≥ 2),
and

t
⋆

< v
⋆
.

Estimates (3.6) holds, for example, let

L0 = .00005, L = .0001, η = .8, a = .0001

to obtain

.400496017 < 1.0002.

Note also that (2.2), and (3.1) also hold, since:

η ≤ 4999,

and

η ≤ 4999.00005,
respectively.

Hence, the claims made in the introduction of this study are now justified.

In practice, on will test (2.2), (3.1), (3.6), and then use the combination of the best results.

Let return back to the numerical examples in the introduction of this study.

Example 3.1. Let D = [−4, 4], and choose c = 2, and a = .2. Then, we have:

L0 = L =
5

32
, η =

5

64
, (3.7)

and (2.2) becomes:

.15625 < 1.28.

Hence, the conclusions of Theorem 2.1 apply to solve equation F(x) = 0.



12 Ioannis K. Argyros and Säıd Hilout CUBO
13, 3 (2011)

Example 3.2. Define the scalar function F by F(x) = c0 x + c1 + c2 sin e
c3 x, x0 = 0, where ci,

i = 1, 2, 3 are given parameters. Then it can easily be seen that for c3 large and c2 sufficiently

small,
L

L0
can be arbitrarily large. That is (2.2) may be satisfied but not (3.1).

We provide two examples, where L0 < L.

Example 3.3. Let X = Y = C[0, 1] be the space of real–valued continuous functions defined on the
interval [0, 1] with norm

‖ x ‖= max
0≤s≤1

|x(s)|.

Let θ ∈ [0, 1] be a given parameter. Consider the ”Cubic” integral equation:

u
3(s) + λ u(s)

∫
1

0

q(s, t) u(t) dt + y(s) − θ = 0. (3.8)

Here the kernel q(s, t) is a continuous function of two variables defined on [0, 1] × [0, 1]; the pa-
rameter λ is a real number called the ”albedo” for scattering; y(s) is a given continuous function

defined on [0, 1] and x(s) is the unknown function sought in C[0, 1]. Equations of the form (3.8)
arise in the kinetic theory of gasses [4], [5]. For simplicity, we choose u0(s) = 0, y(s) = 1, and

q(s, t) =
s

s + t
, for all s ∈ [0, 1], and t ∈ [0, 1], with s + t 6= 0. If we let D = U(u0, 1 − θ), and

define the operator F on D by:

F(x(s)) =
1

3000
x

3(s) + λ x(s)

∫
1

0

q(s, t) x(t) dt + y(s) − θ, (3.9)

for all s ∈ [0, 1], then every zero of F satisfies equation (3.8).

We have the estimates:

max
0≤s≤1

|

∫
s

s + t
dt| = ln 2.

Therefore, if we set a−1 =‖ A(u0)−1 ‖, then it follows from hypotheses of Theorem 2.1:

F
′(u0(s)) = 0,

η = a−1 (|λ| ln 2 + 1 − θ),

L = 2 a−1 (|λ| ln 2 +
2 − θ

1000
) and L0 = a

−1 (2 |λ| ln 2 +
3 − θ

1000
).

It follows from Theorem 2.1 that if condition (2.2) holds, then problem (3.8) has a unique so-

lution near u0. This assumption can be weaker than the one given before The Newton–Kantorovich

hypothesis (3.1), since L0 < L for all θ ∈ [0, 1].

For

λ = .001, a = .9, θ = .9, c = 1.1,



CUBO
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On the semilocal convergence of Newton–type · · · 13

condition (2.2) becomes:

.111188124 < .223204896.

Hence, the conclusions of Theorem 2.1 apply to solve equation (3.9).

Example 3.4. Consider the following nonlinear boundary value problem [4]

{
u′′ = −u3 − γ u2

u(0) = 0, u(1) = 1.

It is well known that this problem can be formulated as the integral equation

u(s) = s +

∫
1

0

Q(s, t) (u3(t) + γ u2(t)) dt (3.10)

where, Q is the Green function:

Q(s, t) =

{
t (1 − s), t ≤ s
s (1 − t), s < t.

We observe that

max
0≤s≤1

∫
1

0

|Q(s, t)| =
1

8
.

Let X = Y = C[0, 1], with norm
‖ x ‖= max

0≤s≤1
|x(s)|.

Then problem (3.10) is in the form (1.1), where, F : D −→ Y is defined as

[F(x)] (s) = x(s) − s −

∫
1

0

Q(s, t) (x3(t) + γ x2(t)) dt.

It is easy to verify that the Fréchet derivative of F is defined in the form

[F′(x)v] (s) = v(s) −

∫
1

0

Q(s, t) (3 x2(t) + 2 γ x(t)) v(t) dt.

If we set u0(s) = s, and D = U(u0, R), then since ‖ u0 ‖= 1, it is easy to verify that
U(u0, R) ⊂ U(0, R + 1). It follows that 2 γ < 5, then

‖ I − F′(u0) ‖ ≤
3 ‖ u0 ‖2 +2 γ ‖ u0 ‖

8
=

3 + 2 γ

8
,

‖ F′(u0)−1 ‖ ≤
1

1 −
3 + 2 γ

8

=
8

5 − 2 γ
,

‖ F(u0) ‖ ≤
‖ u0 ‖3 +γ ‖ u0 ‖2

8
=

1 + γ

8
,



14 Ioannis K. Argyros and Säıd Hilout CUBO
13, 3 (2011)

‖ F(u0)−1 F(u0) ‖ ≤
1 + γ

5 − 2 γ
.

On the other hand, for x, y ∈ D, we have

[(F′(x) − F′(y))v] (s) = −

∫
1

0

Q(s, t) (3 x2(t) − 3 y2(t) + 2 γ (x(t) − y(t))) v(t) dt.

Consequently,

‖ F′(x) − F′(y) ‖ ≤
‖ x − y ‖ (2 γ + 3 (‖ x ‖ + ‖ y ‖))

8

≤
‖ x − y ‖ (2 γ + 6 R + 6 ‖ u0 ‖)

8

=
γ + 6 R + 3

4
‖ x − y ‖,

‖ F′(x) − F′(u0) ‖ ≤
‖ x − u0 ‖ (2 γ + 3 (‖ x ‖ + ‖ u0 ‖))

8

≤
‖ x − u0 ‖ (2 γ + 3 R + 6 ‖ u0 ‖)

8

=
2 γ + 3 R + 6

8
‖ x − u0 ‖ .

Therefore, conditions of Theorem 2.1 hold with

η =
1 + γ

5 − 2 γ
, L =

γ + 6 R + 3

4
, L0 =

2 γ + 3 R + 6

8
.

Note also that L0 < L.

Conclusion

We provided a semilocal convergence analysis for (GNTM) method in order to approximate a lo-

cally unique solution of an equation in a Banach space, when the derivative of the operator involved

is not continuously invertible.

We provided an analysis with the following advantages over the work in [1]–[12] weaker suf-

ficient convergence conditions, and larger convergence domain. Note that these advantages are

obtained under the same computational cost as in [1]–[12]. Numerical examples further validating

our results are also provided in this study.

Received: September 2009. Revised: October 2009.



CUBO
13, 3 (2011)

On the semilocal convergence of Newton–type · · · 15

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	Introduction
	Semilocal convergence analysis of (GNTM) 
	Applications