Ann.
Acad.
Rom.
Sci.
Ser.
Math.
Appl.
ISSN
2066-6594
Vol.
18,
No.
4/2026
EXTENDED
KANTOROVICH-TYPE
RESULTS
ON
NEWTON’S
METHOD
FOR
SOLVING
GENERALIZED
EQUATIONS
∗
Ioannis
K.
Argyros
†
Santhosh
George
‡
Dedicated
to
Professor
Constantin
Fetecau
on
the
occasion
of
his
80th
anniversary
DOI 10.56082/annalsarscimath.2026.4.129
Abstract
A
semi-local
convergence
analysis
is
developed
for
Newton’s
method
in
order
to
approximate
a
solution
of
a
generalized
equations
involving
Banach
space
valued
operators.
The
convergence
conditions
are
based
on
generalized
continuity
conditions
on
the
Fr´
echet
derivative
of
the
operator
involved
and
the
Aubin
property.
The
specialized
cases
of
our
results
extend
earlier
ones
using
the
same
information.
Suitable
examples
complement
the
theory.
Keywords:
Banach
space,
semi-local
convergence,
Newton’s
method,
generalized
equation.
MSC:
34A34,
65B99,
65P30,
65H05.
1
Introduction
A
plethora
of
applications
from
diverse
disciplines
can
be
reduced
using
Mathematical
Modelling
[
1
–
10
,
12
–
14
,
18
–
27
]
to
solving
the
generalized
equa-
∗
Accepted
for
publication
on
July
17,
2026
†
iargyros@cameron.edu
,
Department
of
Mathematical
Sciences,
Cameron
University,
Lawton,
OK
73505
‡
sgeorge@nitk.edu.in
,
Department
of
Mathematical
and
Computational
Sciences,
National
Institute
of
Technology
Karnataka,
India
129
Extended
Kantorovich-type
results
on
Newton’s
method
130
tion
of
the
form
g
(
x
)
+
G
(
x
)
0
.
(1)
Here,
S
1
,
S
2
are
denoting
complete
normed
spaces,
g
:
S
1
−
→
S
2
is
a
con-
tinuously
differentiable
operator,
and
G
:
S
1
⇒
S
2
is
a
set-valued
operator
with
a
closed
nonempty
graph
[
12
,
13
].
The
local
convergence
analysis
of
the
Newton’s
method
g
(
x
n
)
+
g
(
x
n
)(
x
n
+1
−
x
n
)
+
G
(
x
n
+1
)
0
,
n
=
0
,
1
,
2
,
.
.
.
(2)
for
approximating
a
solution
x
∗
∈
S
1
of
the
generalized
equation
(
1
)
when
G
=
0
or
not
is
given
in
[
1
–
10
,
12
–
15
]
under
various
conditions
such
as
Lipschitz,
H¨
older
continuity
on
the
Fr´
echet
derivative
of
the
operator
g.
Such
conditions
are
important
in
the
convergence
of
Newton’s
method,
since
they
control
the
derivative
[
1
–
10
,
12
–
15
].
The
following
concerns
exist
with
the
aforementioned
works
in
general:
(1)
The
convergence
domain
is
small.
(2)
Error
estimates
on
the
distances
x
n
−
x
∗
are
pessimistic,
where
x
∗
stands
for
a
solution
of
the
equation
(
1
).
(3)
The
uniqueness
of
the
convergence
ball
is
small
in
general.
Items
(1)-(3)
restrict
the
applicability
of
the
method.
That
is
why,
by
developing
more
precise
majorizing
sequences
and
the
concept
of
generalized
continuity,
we
positively
address
these
concerns.
In
particular,
the
semi-local
convergence
analysis
of
Newton’s
method
(
2
)
is
developed
in
[
12
]
using
the
Aubin
property
which
is
related
to
metric
regularity
as
it
is
shown
by
Dontchev
and
Rockafellar
in
[
12
,
13
].
It
turns
out
that
the
following
advantages
are
obtained:
(1)
Weaker
convergence
criteria.
(2)
Tighter
error
distances
x
n
−
x
∗
and
(3)
An
at
least
as
precise
information
on
the
location
of
the
solution.
It
is
also
worth
noting
that
these
advantages
are
realized
with
no
addi-
tional
conditions,
since
the
new
majorant
functions
specialize
to
the
ones
used
in
earlier
studies
[
8
,
9
,
15
].
Moreover,
we
make
a
special
comment
on
the
notable
studies
by
Cibulka
et
al.
[
8
,
9
]
and
Ferreira
et
al.
[
14
,
15
],
where
the
semi-local
convergence
analysis
of
the
method
(
2
)
is
developed
under
Lip-
schitz
continuouty
conditions
and
Kantorovich-type
assumptions.
A
direct
and
favorable
to
us
comparison
is
given
at
the
end
of
the
article.
More-
over,
we
use
generalized
Lipschitz-type
conditions
in
our
analysis
in
order
to
include
a
larger
class
of
problems.
I.K.
Argyros,
S.
George
131
The
rest
of
the
paper
includes:
Preliminaries
in
Section
2;
the
semi-local
convergence
in
Section
3
and
the
special
cases
and
numerical
examples
in
Section
4.
2
Preliminaries
We
assume
familiarity
with
the
concepts
of
graph
(gph
G),
the
domain
(dom),
the
range
(rge)
and
the
inverse
G
−
1
of
a
set-valued
operator
G
[
8
–
10
,
12
–
15
].
Moreover,
we
use
d,
e
which
are
the
standard
notations
for
the
distance
and
excess,
respectively
between
two
subsets
of
S
1
.
Let
R
>
0
.
Define
the
linearization
error
for
some
continuously
differentiable
func-
tion
f
:
[0
,
R
)
−
→
(
−∞
)
,
+
∞
)
e
f
(
v
1
,
v
2
)
=
f
(
v
2
)
−
f
(
v
1
)
−
f
(
v
1
)(
v
2
−
v
1
)
for
each
v
1
,
v
2
∈
[0
,
R
)
.
Moreover,
define
analogously
E
g
(
u
1
,
u
2
)
=
g
(
u
2
)
−
g
(
u
2
)
−
g
(
u
1
)(
u
2
−
u
1
)
for
each
u
1
,
u
2
∈
S
1
and
E
g
+
G,x
∗
(
v
)
=
g
(
x
)
+
g
(
x
)(
v
−
x
∗
)
+
G
(
v
)
for
each
x,
v
∈
S
1
.
The
definition
of
the
Aubin
property
and
the
following
version
of
the
contraction
mapping
principle
are
given
in
order
to
make
the
article
as
self
contained
as
possible.
More
information
can
be
found
in
[
1
–
3
,
8
–
10
,
12
,
13
].
The
notation
T
(
x,
α
)
is
used
to
denote
an
open
ball
centered
at
x
∈
S
1
and
of
radius
α
>
0
.
Moreover,
T
[
x,
α
]
stands
for
the
closure
of
T
(
x,
α
)
.
Definition
1.
[
13
]
Let
˜
y
∈
S
2
for
˜
x
∈
S
1
.
Then,
the
inverse
operator
G
−
1
of
G
is
said
to
have
the
Aubin
property
at
(˜
y,
˜
x
)
with
modulus
c,
provided
that
˜
x
∈
G
−
1
(˜
y
)
,
if
gphG
−
1
is
locally
closed
at
(˜
y,
˜
x
)
,
for
˜
x
∈
G
−
1
(˜
y
)
,
and
there
exist
constants
a,
b
>
0
so
that
e
−
1
(
G
−
1
(
y
)
∩
T
[˜
x,
a
]
,
G
−
1
(
y
1
))
≤
c
y
−
y
1
for
each
y,
y
1
∈
T
[˜
y,
b
]
.
If
in
addition
the
operator
S
1
w
−
→
G
−
1
(
w
)
∩
(˜
x,
a
)
is
not
multivalued
on
S
1
,
then
G
is
said
to
be
strongly
regular.
Then,
the
operator
G
is
Lipschitz
continuous
on
S
1
.
Extended
Kantorovich-type
results
on
Newton’s
method
132
Theorem
1.
[
10
]
Let
Ψ
:
S
1
⇒
S
1
be
a
set-valued
operator
and
z
∈
S
1
.
Suppose:
There
exist
scalars
ξ
>
0
and
ξ
1
∈
(0
,
1)
so
that
gph
Ψ
∩
(
T
[
z,
ξ
]
×
T
[
z,
ξ
])
is
closed
and
(i)
d
(
z,
Ψ(
z
))
≤
ξ
(1
−
ξ
1
)
(ii)
e
(Ψ(
x
)
∩
T
[
z,
ξ
]
,
Ψ(
z
1
))
≤
ξ
1
x
−
z
1
for
each
ξ
1
,
z
1
∈
T
[
z,
ξ
]
.
Then,
there
exists
z
∗
∈
T
[
z,
β
]
so
that
z
∗
∈
Ψ(
z
∗
)
,
i.e.
Ψ
admits
a
fixed
point
in
T
[
z,
ξ
]
.
3
Local
convergence
We
introduce
and
compare
certain
majorant
conditions.
Let
α,
q
1
be
posi-
tive
constants
and
x
0
∈
S
1
.
Definition
2.
Let
g
be
a
differentiable
operator
on
the
ball
T
(
x
0
,
α
)
.
We
say
that
g
satisfies
the
center
majorant
condition
on
the
ball
T
(
x
0
,
α
)
if
there
exists
a
continuous
and
nondecreasing
(CN)
function
v
0
:
[0
,
α
)
−
→
R
so
that
for
each
x
∈
T
(
x
0
,
α
)
g
(
x
)
−
g
(
x
0
)
≤
v
0
(
x
−
x
0
)
.
(c1)
Assume
that
there
exists
a
minimal
solution
of
the
equation
qv
0
(
t
)
−
1
=
0
for
some
q
>
0
satisfying
ρ
∈
(0
,
α
]
.
Define
the
set
S
0
=
S
1
∩
T
(
x
0
,
ρ
)
.
Definition
3.
Let
g
be
a
differentiable
operator
on
the
ball
T
(
x
0
,
ρ
)
.
We
say
that
g
satisfies
the
restricted
majorant
condition
on
the
set
S
0
if
there
exists
a
CN
function
v
:
[0
,
ρ
)
−
→
R
so
that
for
each
x,
˜
x
∈
T
(
x
0
,
ρ
)
g
(˜
x
)
−
g
(
x
)
≤
v
(
˜
x
−
x
)
.
Definition
4.
Let
g
be
a
differentiable
operator
on
the
ball
T
(
x
0
,
α
)
.
We
say
that
g
satisfies
the
majorant
condition
on
the
ball
T
(
x
0
,
α
)
if
there
exists
a
CN
v
1
:
[0
,
α
)
→
R
so
that
for
each
x,
˜
x
∈
T
(
x
0
,
ρ
)
g
(˜
x
)
−
g
(
x
)
≤
v
1
(
˜
x
−
x
)
.
Remark
1.
These
definitions
imply
that
for
each
s
∈
[0
,
ρ
)
v
0
(
s
)
≤
v
1
(
s
)
(3)
I.K.
Argyros,
S.
George
133
and
v
(
s
)
≤
v
1
(
s
)
.
(4)
Define
the
sequence
{
s
n
}
for
s
0
=
0
,
some
s
1
≥
0
,
by
s
2
=
s
1
+
qv
0
(
s
1
)
1
−
qv
0
(
s
1
)
and
s
n
+2
=
s
n
+1
+
qv
(
s
n
+1
)
1
−
qv
0
(
s
n
+1
)
for
each
n
=
1
,
2
,
.
.
.
.
(5)
The
sequence
{
s
n
}
plays
the
role
of
a
majorizing
sequence
in
the
semi-local
analysis
of
convergence
for
the
Newton
method
(
2
)
(see
Theorem
2
).
However,
an
auxiliary
result
is
useful
for
the
convergence
of
it.
(c2)
There
exists
ρ
0
∈
[0
,
ρ
)
so
that
for
each
n
=
0
,
1
,
2
,
.
.
.
qv
0
(
s
n
)
<
1
and
s
n
≤
ρ
0
.
This
condition,
the
definition
(
2
)
and
a
simple
inductive
argument
that
for
each
n
=
0
,
1
,
2
,
.
.
.
0
≤
s
n
≤
s
n
+1
≤
ρ
0
.
Thus,
the
sequence
{
s
n
}
is
nondecreasing
and
bounded
from
above
by
ρ
0
and
as
such
it
is
convergent
to
some
s
∗
∈
[0
,
ρ
0
]
.
The
limit
point
s
∗
is
the
least
upper
bound
of
the
sequence
{
s
n
}
which
is
the
only
one.
(c3)
Let
β
and
q
be
positive
numbers.
The
operator
x
→
H
(
x
)
=
g
(
x
0
)
+
g
(
x
0
)(
x
−
x
0
)
+
G
(
x
)
(6)
is
said
to
be
metrically
regular
at
x
0
for
y
0
with
constant
q
related
to
the
balls
T
(
x
0
,
α
)
and
T
(
x
0
,
β
)
.
(c4)
Let
q
1
>
q
and
take
s
1
=
q
1
y
0
.
Assume:
s
∗
≤
α
and
y
0
+
q
1
0
v
0
((1
−
θ
)
s
∗
)
+
1
0
v
((1
−
θ
)
s
∗
)
dθs
∗
≤
β.
The
main
semi-local
analysis
of
convergence
for
the
Newton
method
(
2
)
follows
using
the
conditions
(c1)-(c4)
and
the
preceeding
notation.
Theorem
2.
Assume
that
the
conditions
(c1)-(c4)
are
valid,
where
the
func-
tions
v
0
and
v
are
as
in
the
Definitions
2
and
3
,
respectively.
Then,
there
exists
a
sequence
{
x
n
}
starting
at
x
0
,
generated
by
the
Newton
method
(
2
)
which
remains
in
T
(
x
0
,
α
)
and
is
convergent
to
a
solution
x
∗
∈
T
[
x
0
,
α
]
of
the
generalized
equation
(
1
).
Extended
Kantorovich-type
results
on
Newton’s
method
134
Moreover,
the
error
estimate
is
valid
for
each
n
=
0
,
1
,
2
,
.
.
.
x
n
−
x
∗
≤
s
∗
−
s
n
.
(7)
In
particular,
since
the
operator
H
is
strongly
metrically
regular,
{
x
n
}
is
the
only
sequence
generated
by
the
Newton
method
(
2
)
which
stays
in
the
ball
T
(
x
0
,
α
)
.
Proof.
Induction
on
the
natural
number
m
is
employed
to
establish
the
assertion
x
m
+1
−
x
m
≤
s
m
+1
−
s
m
,
(8)
for
some
sequence
{
x
m
}
⊂
T
(
x
0
,
α
)
.
If
y
0
=
0
and
m
=
0
,
set
x
1
=
x
0
.
Let
y
0
=
0
.
Then,
since
0
∈
T
(
y
0
,
β
)
and
y
0
∈
H
(
x
0
)
q
g
(
x
0
)
+
g
(
x
0
)(
w
−
x
0
)
−
g
(
x
m
)
−
g
(
x
m
)(
w
−
x
m
)
≤
q
1
0
v
0
((1
−
θ
)
w
−
x
0
)
dθ
w
−
x
0
+
1
0
v
((1
−
θ
)
w
−
x
m
)
dθ
w
−
x
m
≤
q
1
0
[
v
0
((1
−
θ
)
s
∗
)
+
v
((1
−
θ
)
s
∗
)]
dθs
∗
≤
β
−
y
0
,
so
g
(
x
0
)
+
g
(
x
0
)(
w
−
x
0
)
−
g
(
x
m
)
−
g
(
x
m
)(
w
−
x
m
)
∈
T
(
y
0
,
β
)
.
Define
the
numbers
p
0
:=
q
1
0
v
0
((1
−
θ
)
x
1
−
x
0
)
dθ
x
1
−
x
0
1
−
q
(
x
1
−
x
0
)
and
p
:=
q
1
0
v
((1
−
θ
)
x
m
−
x
m
−
1
)
dθ
x
m
−
x
m
−
1
1
−
qv
0
(
x
m
−
x
0
)
for
each
n
=
0
,
1
,
2
,
.
.
.
By
the
induction
hypothesis
and
(
5
),
we
get
p
0
≤
q
1
0
v
0
((1
−
θ
)(
s
1
−
s
0
))
dθ
(
s
1
−
s
0
)
1
−
qv
0
(
s
0
)
=
s
2
−
s
1
,
p
≤
q
1
0
v
((1
−
θ
)(
s
m
−
s
m
−
1
))
dθ
(
s
m
−
s
m
−
1
)
1
−
qv
0
(
s
m
)
=
s
m
+1
−
s
m
.
I.K.
Argyros,
S.
George
135
Then,
we
have,
since
x
m
∈
T
(
x
0
,
α
)
is
produced
by
the
Newton
method
(
2
)
given
x
m
−
1
and
for
each
n
=
0
,
1
,
2
,
.
.
.
g
(
x
0
)
+
g
(
x
0
)(
x
m
−
x
0
)
−
g
(
x
m
−
1
)
−
g
(
x
m
−
1
)(
x
m
−
x
m
−
1
)
∈
H
(
x
m
)
.
(9)
Next,
define
the
set-valued
operator
S
1
w
→
Ψ
m
(
x
)
=
H
−
1
(
g
(
x
0
)
+
g
(
x
0
)(
w
−
x
0
)
−
g
(
x
m
)
(10)
−
g
(
x
m
)(
w
−
x
m
))
⊂
S
1
.
If
x
m
=
x
m
−
1
,
then
set
x
m
+1
=
x
m
.
Assume
that
x
m
=
x
m
−
1
.
Then,
it
follows
by
the
definition
of
the
distance,
(
9
)
and
(
11
)
in
turn
that
dist
(
x
m
,
Ψ
m
(
x
m
))
=
dist
(
x
m
,
H
−
1
(
g
(
x
0
)
+
g
(
x
0
)(
x
n
−
x
0
)
−
g
(
x
m
)))
≤
qdist
(
g
(
x
0
)
+
g
(
x
0
)(
x
m
−
x
0
)
−
g
(
x
m
)
,
H
(
x
m
))
≤
q
g
(
x
m
)
−
g
(
x
m
−
1
)
−
g
(
x
m
−
1
)(
x
m
−
x
m
−
1
)
≤
q
1
0
v
((1
−
θ
)
x
m
−
x
m
−
1
)
dθ
x
m
−
x
m
−
1
<
q
1
0
v
((1
−
θ
)
x
m
−
x
m
−
1
)
dθ
x
m
−
x
m
−
1
1
−
qv
0
(
x
m
−
x
0
)
×
(1
−
qv
0
(
x
m
−
x
0
))
=
¯
p
(1
−
qv
0
(
x
m
−
x
0
))
,
where
¯
p
=
p
0
,
if
n
=
0
p,
if
n
=
1
,
2
,
.
.
.
Take
w
1
,
w
2
∈
T
(
x
m
,
s
∗
−
s
m
)
and
w
3
∈
Ψ
m
(
w
1
)
∩
T
(
x
m
,
s
∗
−
s
m
)
.
It
follows
g
(
x
0
)
+
g
(
x
0
)(
w
1
−
x
0
)
−
g
(
x
m
)
−
g
(
x
m
)(
w
1
−
x
m
)
∈
H
(
w
3
)
.
Consequently,
we
get
in
turn
that
dist
(
w
3
,
ψ
m
(
w
2
))
=
dist
(
w
3
,
H
−
1
(
g
(
x
0
)
+
g
(
x
0
)(
w
2
−
x
0
)
−
g
(
x
m
)
−
g
(
x
m
)(
w
2
−
x
m
)))
≤
qdist
(
g
(
x
0
)
+
g
(
x
0
)(
w
2
−
x
0
)
−
g
(
x
m
)
−
g
(
x
m
)(
w
2
−
x
m
)
,
H
(
w
3
))
≤
q
g
(
x
0
)
+
g
(
x
0
)(
w
2
−
x
0
)
−
g
(
x
0
)
−
g
(
x
0
)(
w
2
−
x
m
)
−
(
g
(
x
0
)
+
g
(
x
0
)(
w
1
−
x
0
)
−
g
(
x
m
)
−
g
(
x
m
)(
w
1
−
x
m
))
≤
q
g
(
x
0
)
−
g
(
x
m
)

u
1
−
u
2
≤
qv
0
(
x
m
−
x
0
)
u
1
−
u
2
.
Extended
Kantorovich-type
results
on
Newton’s
method
136
By
applying
Theorem
1
and
since
T
(
x
m
,
¯
p
)
⊂
T
(
x
m
,
s
∗
−
s
m
)
there
exists
a
fixed
point
x
m
+1
∈
T
(
x
n
,
¯
p
)
of
the
operator
ψ
m
.
Therefore,
we
have
the
met-
ric
regularity
of
the
operator
(H)
in
(
6
)
implies
dist
(
x
0
,
H
−
1
(0))
≤
q
y
0
≤
q
1
y
0
.
Thus,
there
exists
x
1
∈
H
−
1
(0)
so
that
x
1
−
x
0
<
q
1
y
0
=
s
1
=
s
1
−
s
0
.
Assume
that
for
some
natural
number
m,
we
have
determined
the
iterates
x
0
,
x
1
,
.
.
.
x
m
∈
T
(
x
0
,
α
)
so
that
x
i
−
x
i
−
1
≤
s
i
−
s
i
−
1
for
i
=
0
,
1
,
2
,
.
.
.
,
m.
We
can
assume
without
losing
any
generality
that
x
m
=
x
0
.
Otherwise,
there
is
nothing
to
show.
Then,
by
the
induction
hypothesis
x
m
−
x
0
≤
m
i
=1
x
i
−
x
i
−
1
≤
m
i
=1
(
s
i
−
s
i
−
1
)
=
s
n
−
s
0
=
s
n
−
0
=
s
n
<
s
∗
<
α.
Moreover,
for
each
w
∈
T
(
x
m
,
s
∗
−
s
m
)
⊂
T
(
x
0
,
s
∗
)
it
follows
by
the
Defini-
tion
2
,
the
Definition
3
and
(
c
4)
that
q
g
(
x
0
)
+
g
(
x
0
)(
w
−
x
0
)
−
g
(
x
m
)
−
g
(
x
m
)
−
g
(
x
m
)(
w
−
x
m
)
≤
q
g
(
x
)
−
g
(
x
0
)
−
g
(
x
m
)(
w
−
x
0
)
+
q
g
(
x
)
−
g
(
x
m
)
−
g
(
x
m
)(
w
−
x
m
)
so
x
m
+1
∈
H
−
1
(
g
(
x
0
)
+
g
(
x
0
)(
x
m
+1
−
x
0
)
−
g
(
x
m
)
−
g
(
x
m
)(
x
m
+1
−
x
m
))
.
Thus,
x
m
+1
is
obtained
from
the
Newton
method
(
2
)
given
x
m
.
Further-
more,
we
have
x
m
+1
−
x
m
≤
¯
p
≤
s
m
+1
−
s
m
.
and
x
m
+1
−
x
0
≤
m
+1
i
=1
x
i
−
x
i
−
1
≤
m
+1
i
=1
(
s
i
−
s
i
−
1
)
=
s
m
+1
−
s
0
=
s
m
+1
<
s
∗
≤
α.
The
induction
for
the
assertion
(
8
)
is
terminated.
It
follows
by
the
last
two
inequalities
that
the
sequence
{
x
m
}
is
Cauchy
in
a
complete
normed
space
and
as
such
it
is
convergent
to
some
x
∗
∈
T
(
x
0
,
s
∗
)
.
By
letting
n
→
+
∞
in
the
Newton
method
(
2
),
we
conclude
that
x
∗
solves
the
problem
(
1
).
The
assertion
(
8
)
follows
from
the
estimate
x
m
+
j
−
x
m
≤
s
m
+
j
−
s
m
for
j
→
∞
.
We
still
need
to
show
the
uniqueness
of
the
sequence
x
m
in
T
(
x
0
,
s
∗
)
.
I.K.
Argyros,
S.
George
137
Assume
that
there
is
another
sequence
{
u
m
}
in
T
(0
,
β
)
with
u
m
+1
=
x
m
+1
.
Then,
it
follows
that
x
m
+1
=
ψ
m
(
x
m
+1
)
and
u
m
+1
=
ψ
m
(
u
m
+1
)
.
By
applying
the
definition
of
the
excess
u
m
+1
−
x
m
+1
=
e
(
H
m
(
u
m
+1
)
∩
[
x
∗
,
s
m
]
,
H
m
(
x
m
+1
))
≤
(1
−
qv
0
(
x
m
−
x
0
)
u
m
+1
−
x
m
+1
<
u
m
+1
−
x
m
+1
.
4
Special
cases
and
comparisons
to
earlier
works
A
direct
comparison
cannot
be
given
with
similar
results
such
as
in
[
12
,
15
]
for
G
=
0
or
in
[
14
]
for
G
=
0
.
However,
such
a
comparison
can
be
given
if
we
specialize
the
majorant
functions
v
0
,
v
and
v
1
(see
also
(
3
)
and
(
4
)).
Case
G
=
0
.
We
compare
our
results
to
the
ones
in
[
12
],
(see
also
[
14
],
[
15
])
by
spe-
cializing
the
functions
v
0
and
v
to
v
0
(
t
)
=
q
l
0
2
t
2
−
t
+
η,
s
1
=
η
=
q
y
0
,
v
(
t
)
=
q
l
2
t
2
−
t
+
η
and
v
1
(
t
)
=
q
l
1
2
t
2
−
t
+
η.
Clearly,
we
have
l
0
≤
l
and
l
≤
l
1
.
These
inequalities
can
be
strict
(see
the
numerical
example).
Then,
the
majorizing
sequence
{
t
n
}
in
[
12
],(
see
also
[
14
],
[
15
])
is
t
0
=
0
,
t
1
=
s
1
,
t
n
+2
=
t
n
+1
−
q
1
l
1
(
t
n
+1
−
t
n
)
2
2(1
−
ql
1
(
t
n
+1
))
,
n
=
0
,
1
,
2
,
.
.
.
(11)
Moreover,
the
error
bounds
are
available
x
n
−
x
∗
≤
n
,
(12)
where
for
λ
=
1
−
√
1
−
2
h
1
+
√
1
−
2
h
∈
[0
,
1)
n
=
2
√
1
−
2
hλ
2
n
ql
1
(1
−
λ
2
n
)
,
n
=
1
,
2
,
.
.
.
(13)
Extended
Kantorovich-type
results
on
Newton’s
method
138
and
provided
that
the
famous
for
its
simplicity
and
clarity
Newton-
Kam-
torovich
semi-local
condition
[
1
–
7
,
16
–
18
,
22
,
23
]
for
the
convergence
of
New-
ton’s
method
is
fulfilled,
which
is
given
by
h
1
=
L
1
η
<
1
2
,
L
1
=
ql
1
..
(14)
In
our
case
the
majorizing
sequence
{
s
n
}
reduces
to
s
0
=
0
,
s
1
=
η,
s
2
=
s
1
+
ql
0
(
s
1
−
s
0
)
2
2(1
−
l
0
q
)
(15)
and
s
n
+2
=
s
n
+1
+
ql
(
s
n
+1
−
s
n
)
2
2(1
−
l
0
qs
n
+1
)
,
n
=
1
,
2
,
.
.
.
Set
for
simplicity
L
0
=
ql
0
and
L
=
ql.
The
convergence
of
the
sequence
{
s
n
}
has
been
studied
extensively
by
us
in
[
4
–
7
].
In
particular,
the
corresponding
to
(
14
)
semi-local
convergence
criteria
in
[
[
5
],
see
Lemma
2.1]
are
h
=
¯
Lη
≤
1
2
,
(16)
where
¯
L
=
1
8
(4
L
0
+
L
2
L
0
+
8
L
2
0
+
L
0
L
2
)
,
L
2
=
L
1
or
L
2
=
L.
Then,
under
(
14
)
and
(
16
)
a
simple
inductive
argument
shows
0
≤
s
n
≤
t
n
,
(17)
0
≤
s
n
+1
−
s
n
≤
t
n
+1
−
t
n
,
(18)
so
lim
n
→
+
∞
s
n
=
s
∗
≤
t
∗
=
lim
n
→
+
∞
t
n
=
1
L
1
(1
−
1
−
2
h
1
)
(19)
and
h
1
<
1
2
=
⇒
h
≤
1
2
,
(20)
I.K.
Argyros,
S.
George
139
but
not
necessarily
vice
versa
unless
l
0
=
l
=
l
+
1
.
(see
also
the
numerical
example
where
(
14
)
is
violated
but
(
16
)
is
valid).
It
is
worth
noting
also
that
as
l
1
l
1
→
0
,
then
h
h
1
→
0
(21)
Therefore,
we
justify
the
items
(1)’-(3)’
as
stated
in
the
introduction
of
this
paper.
It
is
also
worth
stating
(
21
)
indicates
by
how
many
times
(infinitely)
our
approach
extends
the
applicability
of
Newton
method
(
2
).
Case
G
=
0
.
We
compare
our
results
to
the
ones
in
[
14
,
15
]
by
specializing
the
functions
v
0
,
v
and
v
1
to
be
v
0
(
t
)
=
ϕ
0
(
t
)
−
ϕ
0
(0)
,
v
((1
−
θ
)
t
)
=
ϕ
(
t
)
−
ϕ
(
θt
)
and
v
1
((1
−
θ
)
t
)
=
ϕ
1
(
t
)
−
ϕ
1
(
θt
)
,
where
the
functions
ϕ
0
,
ϕ
and
ϕ
1
are
twice
continuously
differentiable,
con-
vex,
strictly
increasing
and
satisfy
ϕ
0
(0)
=
ϕ
(0)
=
ϕ
1
(0)
=
0
and
ϕ
0
(0)
=
ϕ
(0)
=
ϕ
1
(0)
=
−
1
.
Notice
that
(see
also
(
3
)
and
(
4
))
ϕ
0
(
t
)
≤
ϕ
1
(
t
)
<
0
(22)
and
ϕ
(
t
)
≤
ϕ
1
(
t
)
<
0
(23)
on
some
subinterval
of
[0,
∞
).
These
inequalities
can
be
strict
(see
the
ex-
ample).
The
majorizing
sequence
in
[
14
,
15
]
is
t
n
+1
=
t
n
−
ϕ
1
(
t
n
)
ϕ
1
(
t
n
)
,
(24)
Extended
Kantorovich-type
results
on
Newton’s
method
140
but
ours
is
s
n
+1
=
s
n
ϕ
0
(
s
n
)
−
ψ
(
s
n
)
ϕ
0
(
s
n
)
,
ψ
=
ϕ
or
ψ
=
ϕ
1
(25)
Notice
that
for
ψ
=
ϕ
1
the
sequence
(
25
)
can
be
written
as
s
n
+1
=
s
n
−
ϕ
1
(
s
n
)
ϕ
1
(
s
n
)
ϕ
1
(
s
n
)
ϕ
0
(
s
n
)
.
But
by
(
22
)
and
(
23
)
|
ϕ
1
(
s
n
)
|
≤
|
ϕ
0
(
s
n
)
|
.
Consequently,
the
estimates
(
17
)-(
19
)
are
valid
in
this
more
general
setting.
The
same
is
true
for
the
convergence
criteria
and
uniqueness
of
the
solution
results.
Notice
also
that
in
[
14
]
Newton’s
method
is
shown
to
be
of
order
two
and
the
sequence
{
t
n
}
is
its
majorant
sequence
which
is
of
order
two
(see
also
(
13
)).
Let
us
complete
this
study
with
a
simple
example
that
shows
even
for
scalar
equations
the
condition
(
14
)
is
not
satisfied
but
the
new
condition
(
16
)
is
valid.
Example
1.
Let
S
1
=
S
2
=
(
−∞
,
+
∞
)
,
T
=
T
(1
,
1
−
γ
)
,
γ
∈
[0
,
1
2
)
and
consider
the
real
function
f
:
T
→
(
−∞
,
+
∞
)
defined
by
f
(
t
)
=
t
3
−
γ.
Then,
we
have
η
=
1
2
(1
−
µ
)
,
L
0
=
3
−
µ,
L
=
2(1+
1
1
−
L
0
)
and
L
1
=
2(2
−
µ
)
.
But,
the
Kantorovich
condition
(
14
)
is
not
satisfied,
since
h
1
=
2
3
(2
−
µ
)(1
−
µ
)
>
1
2
for
each
µ
∈
[0
,
1
2
)
.
Thus,
the
approach
in
[
12
,
14
,
15
]
cannot
assure
the
convergence
of
New-
ton
method
to
the
solution
x
∗
=
3
√
γ
although
it
converges.
But
the
new
condition
(
16
)
is
valid
if
we
set
L
2
−
L
1
for
µ
∈
[0
.
463
,
1
2
)
and
even
better
if
L
2
=
L
for
µ
∈
[0
.
447
,
1
2
)
.
Other
examples,
where
v
0
<
v,
v
0
<
v
1
,
v
0
<
v,
l
0
<
l
1
,
l
<
l
1
,
l
0
<
l,
ϕ
0
<
ϕ
1
,
ϕ
<
ϕ
1
and
ϕ
0
<
ϕ
can
be
found
in
[
4
–
9
].
The
applicability
of
the
rest
of
the
methods
for
solving
the
generalized
equation
(
1
)
given
in
[
12
,
14
,
15
]
can
be
extended
along
the
same
lines.
We
leave
the
details
to
the
motivational
reader.
I.K.
Argyros,
S.
George
141
5
Semi-local
convergence
It
is
worth
noticing
that
the
Aubin
property
is
related
to
the
metric
regular-
ity
[
13
].
Consequently,
the
results
are
provided
in
terms
of
metric
regularity.
But
first,
we
need
a
relationship
between
different
types
of
majorant
condi-
tions.
Assume
R
>
0
.
Definition
5.
A
function
h
0
:
[0
,
R
)
−
→
(
−∞
,
+
∞
)
which
is
twice
con-
tinuously
differentiable
and
non-decreasing
is
said
to
be
a
center-majorant
function
for
g
on
T
(
x
0
,
R
)
with
modulus
c
>
0
if
for
each
y
∈
T
[
x
0
,
R
]
(A1)
c
g
(
y
)
−
g
(
x
0
)
≤
h
0
(
y
−
x
0
)
−
h
0
(0)
.
(A2)
The
function
h
0
(
t
)
has
a
smallest
positive
zero
denoted
by
ρ.
Definition
6.
A
function
h
=
h
(
h
0
)
:
[0
,
ρ
)
−
→
(
−∞
,
+
∞
)
which
is
twice
continuously
differentiable
and
non-decreasing
is
said
to
be
a
restricted-
majorant
function
for
g
on
T
(
x
0
,
ρ
)
with
modulus
c
>
0
if
for
each
x,
y
∈
T
(
x
0
,
ρ
)
(A3)
c
g
(
y
)
−
g
(
x
)
≤
h
(
y
−
x
+
x
−
x
0
)
−
h
(
x
−
x
0
)
.
Notice
that
the
function
h
0
depends
on
x
0
and
¯
ρ,
where
as
the
function
h
relies
on
x
0
,
ρ
and
h
0
.
(A4)
Assume:
h
0
(
s
)
≤
h
(
s
)
for
each
s
∈
[0
,
ρ
)
.
(26)
Definition
7.
A
function
h
1
:
[0
,
R
)
−
→
(
−∞
,
+
∞
)
which
is
twice
continu-
ously
differentiable
and
non-decreasing
is
said
to
be
a
majorant
function
for
g
on
T
(
x
∗
,
ρ
)
with
modulus
a
>
0
if
for
each
x,
y
∈
T
(
x
∗
,
¯
ρ
)
.
(A3)’
a
g
(
y
)
−
g
(
x
)
≤
h
1
(
y
−
x
+
x
−
x
0
)
−
h
1
(
x
−
x
0
)
.
It
follows
by
these
definitions
that
h
0
(
s
)
≤
h
1
(
s
)
and
h
(
s
)
≤
h
1
(
s
)
for
each
s
∈
[0
,
ρ
)
.
(27)
Thus,
the
results
in
the
literature
using
only
h
1
(see
e.g.
[
10
,
14
,
15
]
for
G
=
0)
can
be
replaced
by
the
pair
(
h
0
,
h
)
resulting
to
a
finer
error
distances
with
advantages
as
already
given
in
the
introduction.
These
advantages
are
obtained
under
the
same
computational
cost,
since
the
new
majorant
functions
h
0
,
h
are
specializations
of
the
function
h
1
.
The
additional
conditions
are:
(A4)
h
0
(0)
>
0
,
h
0
(0)
=
−
1;
Extended
Kantorovich-type
results
on
Newton’s
method
142
(A5)
h
0
is
convex
and
strictly
increasing;
(A6)
h
0
(
t
)
=
0
for
some
t
∈
(0
,
R
)
.
Take
t
∗
0
=
min
{
t
∈
[0
,
R
)
:
h
0
(
t
)
=
0
}
;
(A7)
h
(0)
>
0
,
h
(0)
=
−
1;
(A8)
h
is
convex
and
strictly
increasing;
(A9)
h
(
t
)
=
0
for
some
t
∈
[0
,
t
∗
0
)
.
Take
t
∗
=
min
{
t
∈
[0
,
t
∗
0
)
:
h
(
t
)
=
0
}
.
The
conditions
in
[
15
]
are:
(A4)’
h
1
(0)
>
0
,
h
1
(0)
=
−
1;
(A5)’
h
1
is
convex
and
strictly
increasing;
(A6)’
h
1
(
t
)
=
0
for
some
t
∈
(0
,
R
)
.
Take
t
∗
1
=
min
{
t
∈
[0
,
R
)
:
h
1
(
t
)
=
0
}
;
Define
the
majorant
sequences
corresponding
to
these
conditions
to
be
for
¯
h
=
h
0
,
n
=
0
h,
n
=
1
,
2
,
.
.
.
r
0
=
0
,
r
n
+2
=
r
n
+1
+
1
0
¯
h
(
r
n
+
θ
(
r
n
=1
−
r
n
))
dθ
(
r
n
+1
−
r
n
)
1
−
h
0
(
r
n
+1
)
,
(28)
s
0
=
0
,
s
n
+1
=
s
n
−
¯
h
(
s
n
)
h
0
(
s
n
)
,
(29)
and
t
0
=
0
,
t
n
+1
=
t
n
−
h
1
(
t
n
)
h
1
(
t
n
)
.
(30)
The
sequence
{
t
n
}
is
studied
in
[
15
]
under
the
conditions
(A4)’-(A6)’
and
in
[
13
]
if
the
function
h
1
is
constant.
The
sequences
{
r
n
}
and
{
s
n
}
cor-
respond
to
the
conditions
(A1)-(A9).
Clearly,
a
simple
induction
argument
shows
that
0
≤
s
n
≤
r
n
≤
t
n
,
(31)
0
≤
s
n
+1
−
s
n
≤
r
n
+1
−
r
n
≤
t
n
+1
−
t
n
,
(32)
and
lim
n
−
→
+
∞
r
n
≤
lim
n
−
→
+
∞
s
n
=
t
∗
≤
lim
n
−
→
+
∞
t
n
=
t
∗
1
.
(33)
Thus,
the
new
error
analysis
is
tighter.
I.K.
Argyros,
S.
George
143
Let
us
consider
the
conditions:
(A10)
The
set
valued
mapping
u
→
L
−
1
g
+
G,x
0
(
u
)
has
the
Aubin
property
at
0
for
x
1
,
with
modulus
c
>
0
,
and
associated
constants
a
>
0
and
b
>
0
.
Finally,
we
define
the
set-valued
mapping
Ψ
x
:
Ω
⇒
S
1
by
Ψ
x
(
v
)
:=
L
−
1
g
+
G,x
0
(
E
g
(
x,
v
)
−
E
g
(
x
0
,
v
))
.
(34)
Then,
all
the
results
in
[
1
,
3
,
15
]
can
be
rewritten
with
(
h
0
,
h
)
replacing
the
function
h
1
.
Next,
we
only
mention
the
main
ones.
The
details
for
the
rest
are
left
to
the
motivated
reader.
Theorem
3.
Assume
that
the
conditions
(A1)-(A10)
hold.
Moreover,
as-
sume
that
the
iterate
x
1
is
generated
from
the
initial
point
x
0
∈
Ω
by
(
2
)
g
(
x
n
)
+
g
(
x
n
)(
x
n
+1
−
x
n
)
+
G
(
x
n
+1
)
0
.
(35)
Furthermore,
assume
T
[
x
0
,
t
∗
]
⊂
Ω
.
Then,
there
exists
a
sequence
{
x
n
}
generated
by
(
34
)
such
that
the
iterate
x
1
∈
T
(
x
0
,
t
∗
)
∩
T
(
x
1
,
a
)
such
that
x
n
+1
−
x
n
≤
s
n
+1
−
s
n
,
x
n
+1
−
x
n
≤
s
n
+1
−
s
n
(
s
n
−
s
n
−
1
)
2
x
n
−
x
n
−
1
2
,
x
n
+1
−
x
n
≤
1
2
x
n
−
x
n
−
1
.
Moreover,
{
x
n
}
⊂
u
(
x
0
,
t
∗
)
∩
T
(
x
1
,
a
)
is
convergent
to
some
point
x
∗
∈
T
[
x
0
,
t
∗
]
∩
T
[
x
1
,
a
]
satisfying
g
(
x
∗
)
+
G
(
x
∗
)
0
such
that
x
∗
−
x
n
≤
s
∗
−
s
n
.
Additionally,
if
the
condition
h
0
(
s
∗
)
<
0
holds,
then
x
n
+1
−
x
n
≤
h

(
s
∗
)
2
|
h
0
(
s
∗
)
|
x
n
−
x
n
−
1
2
.
Hence,
the
sequence
{
x
n
}
is
convergent
Q
−
quadratically
to
x
∗
.
Proof.
Simply
replace
h
1
by
the
pair
(
h
0
,
h
)
as
well
as
the
sequence
{
t
n
}
by
{
s
n
}
in
the
proof
of
Theorem
6
in
[
15
].
Clearly,
these
conclusions
can
be
rewritten
provided
that
tighter
sequence
{
r
n
}
replaces
{
s
n
}
.
Extended
Kantorovich-type
results
on
Newton’s
method
144
Theorem
4.
Assume
that
the
conditions
(A1)-(A10)
hold
and
the
mapping
in
(
34
)
is
single-valued
in
T
(0
,
b
)
.
Then,
there
exists
a
sequence
{
x
n
}
generated
by
(
35
)
belonging
to
T
(
x
0
,
t
∗
)
∩
T
(
x
1
,
a
)
which
converges
to
x
∗
∈
T
[
x
0
,
t
∗
]
∩
T
[
x
1
,
a
]
solving
g
(
x
∗
)+
G
(
x
∗
)
0
.
Moreover,
the
sequence
{
x
n
}
converges
to
x
∗
such
that
x
∗
−
x
n
+1
≤
s
∗
−
s
n
+1
(
s
∗
−
s
n
)
2
x
∗
−
x
n
2
,
and
x
∗
−
x
n
+1
≤
1
2
x
∗
−
x
n
2
.
Additionally,
if
h
0
(
t
∗
)
<
0
,
then
the
sequence
{
x
n
}
is
convergent
Q
−
quadratically
to
x
∗
and
satisfies
x
∗
−
x
n
+1
≤
h

(
s
∗
)
2
|
h
0
(
s
∗
)
|
x
∗
−
x
n
2
.
Moreover,
x
∗
is
the
unique
solution
of
equation
(
2
)
in
T
[
x
0
,
s
∗
]
∩
T
[
x
1
,
a
]
.
Proof.
As
in
the
proof
of
Theorem
3
.
Remark
2.
If
h
0
=
h
=
h
1
,
t
n
=
s
n
,
then
the
results
of
Theorem
3
and
Theorem
4
reduce
to
Theorem
5
and
Theorem
7,
respectively,
in
[
15
].
Oth-
erwise,
they
are
better
with
advantages,
as
already
stated
in
the
introduction.
Clearly,
the
results
in
[
15
]
imply
those
in
this
article
but
not
necessarily
vise
versa.
Let
us
specialize
the
majorant
functions
h
0
,
h,
h
1
by
h
0
(
t
)
=
q
0
t,
h
(
t
)
=
qt,
and
h
1
(
t
)
=
q
1
t.
Clearly,
we
have
0
≤
q
0
≤
q
≤
q
1
.
Then
the
corresponding
sequences
{
r
n
}
,
{
s
n
}
and
{
t
n
}
are
r
0
=
0
,
r
1
=
x
n
−
x
0
,
r
2
=
r
1
+
q
0
((
r
1
−
r
0
)
2
2(1
−
q
0
r
1
)
(36)
r
n
+2
=
r
n
+1
+
q
((
r
n
+1
−
r
n
)
2
2(1
−
q
0
r
n
+1
)
s
0
=
0
,
s
1
=
x
1
−
x
0
,
s
2
=
s
1
+
q
0
((
s
1
−
s
0
)
2
2(1
−
q
0
s
1
)
s
n
+2
=
s
n
+1
+
q
((
s
n
+1
−
s
n
)
2
2(1
−
q
0
s
n
+1
)
(37)
t
0
=
0
,
t
1
=
x
n
−
x
0
,
t
n
+1
=
t
n
+
q
1
((
t
n
−
t
n
−
1
)
2
2(1
−
q
1
t
n
)
,
n
=
1
,
2
,
.
.
.
.
(38)
I.K.
Argyros,
S.
George
145
The
convergence
of
these
sequences
is
studied
in
[
4
–
7
,
16
].
In
particular,
Kantorovich
used
the
sequence
{
t
n
}
to
solve
equation
(
1
)
if
G
=
0
.
The
convergence
condition
is
q
1
x
1
−
x
0
≤
1
2
.
(39)
But
the
convergence
conditions
in
[
4
–
7
,
16
]
are
always
weaker.
Let
us
just
provide
only
one
for
both
sequences
{
r
n
}
and
{
s
n
}
given
by
¯
q
x
1
−
x
0
≤
1
2
,
(40)
where
¯
q
=
q
0
+
q
2
.
Then,
(
39
)
implies
(
40
)
but
not
necessarily
vise
versa
unless
q
0
=
q
=
q
1
.
Examples
where
q
0
≤
q
≤
q
1
are
given
in
[
3
–
7
,
16
].
Clearly,
the
specialized
cases
given
in
[
15
]
are
also
extended
along
the
lines
of
the
above.
6
Conclusion
A
finer
semi-local
convergence
analysis
for
Newton’s
and
related
methods
is
developed
for
solving
the
generalized
equation
(
1
)
with
advantages
as
already
stated
in
the
Introduction
even
if
the
majorant
functions
used
are
specialized
to
the
ones
given
in
earlier
papers
[
11
,
14
,
15
].
The
advantages
are
obtained
by
using
more
precise
majorizing
sequences.
The
new
methodology
can
be
used
on
other
methods
in
order
to
extend
their
applicability
in
a
similar
way
[
1
–
3
,
10
,
11
,
13
–
27
].Clearly,
the
new
approach
can
be
used
to
extend
results
on
the
local
convergence
of
these
methods.
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