Statistical Limit Points of Sequences on Intuitionistic Fuzzy Normed Spaces

VOLUME 6,NUMBER 1 JANUARY 2008

ISSN:1548-5390 PRINT,1559-176X ONLINE

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On Anti Fuzzy Structures in BCC-Algebras

Servet Kutukcu

_{and Sushil Sharma}

a

_{Department of Mathematics, Faculty of Science and Arts}

Ondokuz Mayis University, 55139 Kurupelit, Samsun, Turkey

E-mail: [email protected]

b

_{Department of Mathematics, Madhav Vigyan Mahavidhyalaya}

Vikram University, Ujjain-456010, India

E-mail: [email protected]

September 6, 2007

Abstract

In this paper, we define the notions of anti intuitionistic fuzzy BCC-subalgebras and anti intuitionistic fuzzy ideals of the BCC-algebras with respect to arbitrary t-conorms and t-norms, and obtain some related results.

Keywords: t-norm, t-conorm, anti intuitionistic fuzzy subalgebra, anti intuitionistic fuzzy ideal, BCC-algebra.

AMS Subject Classifications: 06F35, 03G25, 94D05

1

Introduction and preliminaries

The notion of fuzzy sets was introduced by Zadeh [24]. Since then, this concept has been

applied to many mathematical branches, such as group, functional analysis, probability

theory, topology and so on. In 1991, Xi [23] applied this concept to BCK-algebras and

Dudek et al.

[8-11] studied fuzzy structures in BCC-algebras.

A BCK-algebra is an

important class of logical algebras introduced by Iseki [14]. Iseki [14] posed the interesting

problem of whether the class of BCK-algebras is a variety. In connection with this problem,

Komori [18] introduced the notion of BCC-algebras and Dudek [6,7] modified the notion of

BCC-algebras by using a dual form of the ordinary definition in the sense of Komori [18].

In the present paper, using the idea of Kutukcu and Yildiz [19], we introduce the notion

of anti intuitionistic fuzzy BCC-subalgebras of the BCC-algebras with the help of arbitrary

t-conorms and t-norms as a generalization of anti fuzzy subalgebras. We also introduce the

notion of anti intuitionistic fuzzy ideals as a generalization of anti fuzzy ideals and prove

that an intuitionistic fuzzy subset of a BCC-algebra is an intuitionistic fuzzy ideal if and

only if the complement of this intuitionistic fuzzy subset is an anti intuitionistic fuzzy ideal.

We prove that if an intuitionistic fuzzy subset is an anti intuitionistic fuzzy ideal then so is

the fuzzifications of its upper and lower level cuts.

Let us recall [18] that a BCC-algebra is a nonempty set X with a constant 0 and

a binary operation ∗ which satisfies the following conditions, for all x, y, z ∈ X: (i)

((x ∗ y) ∗ (z ∗ y)) ∗ (x ∗ z) = 0; (ii) x ∗ x = 0; (iii) 0 ∗ x = 0; (iv) x ∗ 0 = x; (v)

x ∗ y = 0 and y ∗ x = 0 imply x = y. A nonempty subset G of a BCC-algebra X is

called a BCC-subalgebra of X if x ∗ y ∈ G for all x, y ∈ G (see also [15,23]).

By a triangular conorm (shortly t-conorm) S [22], we mean a binary operation on the

unit interval [0, 1] which satisfies the following conditions, for all x, y, z ∈ [0, 1]: (i) S(x, 0) =

x; (ii) S(x, y) ≤ S(x, z) if y ≤ z; (iii) S(x, y) = S(y, x); (iv) S(x, S(y, z)) = S(S(x, y), z).

Some important examples of t-conorms are S

(x, y) = min {x + y, 1}, S

(x, y) = x + y −xy

and S

(x, y) = max {x, y} .

By a triangular norm (shortly t-norm) T [22], we mean a binary operation on the unit

interval [0, 1] which satisfies the following conditions, for all x, y, z ∈ [0, 1]: (i) T (x, 1) = x;

(ii) T (x, y) ≤ T (x, z) if y ≤ z; (iii) T (x, y) = T (y, x); (iv) T (x, T (y, z)) = T (T (x, y), z).

Some important examples of t-norms are T

(x, y) = max {x + y − 1, 0}, T

(x, y) = xy and

T

(x, y) = min {x, y} .

A t-conorm S and a t-norm T are called associated [20], i.e. S(x, y) = 1 −T (1−x, 1−y)

for all x, y ∈ [0, 1]. For example t-conorm S

and t-norm T

are associated [12,17,19,20].

Also it is well known [12,17] that if S is a t-conorm and T is a t-norm, then max {x, y} ≤

S(x, y) and min {x, y} ≥ T (x, y) for all x, y ∈ [0, 1], respectively.

Note that, the concepts of t-conorms and t-norms are known as the axiomatic skeletons

that we use for characterizing fuzzy unions and intersections, respectively. These concepts

were originally introduced by Menger [21] and several properties and examples for these

concepts were proposed by many authors (see [1,5,12,16,17,19-22]).

A fuzzy set A in an arbitrary non-empty set X is a function µ

_{: X → [0, 1]. The}

complement of µ

, denoted by µ

A

, is the fuzzy set in X given by µ

(x) = 1 − µ

(x) for all

x ∈ X.

For any fuzzy set µ

_{in X and any α ∈ [0, 1], Dudek et al. [11] defined two sets}

U (µ

_{; α) = {x ∈ X : µ}

_{(x) ≥ α} and V (µ}

_{; α) = {x ∈ X : µ}

_{(x) ≤ α}}

which are called an upper and lower α-level cut of µ

, respectively, and can be used to the

characterization of µ

.

Definition 1.1 ([8]). A fuzzy set A in a algebra X is called a fuzzy

BCC-subalgebra of X if

µ

_{(x ∗ y) ≥ min {µ}

(x), µ

_(y)}

for all x, y ∈ X.

Definition 1.2 ([8]). A fuzzy set A in a algebra X is called a fuzzy

BCC-subalgebra of X with respect to a t-norm T (or simply, a T -fuzzy BCC-BCC-subalgebra of X)

if

µ

_{(x ∗ y) ≥ T (µ}

(x), µ

(y))

for all x, y ∈ X. Every algebra is a fuzzy algebra and so a T -fuzzy

BCC-subalgebra but the converse is not true (see [6,8-10]).

Definition 1.3 ([9]). A fuzzy set A in a BCC-algebra X is called a fuzzy ideal of X if

µ

_{(0) ≥ µ}

_{(x) ≥ min {µ}

_{(x ∗ y), µ}

_(y)}

for all x, y ∈ X.

Definition 1.4 ([9,13]). A fuzzy set A in a BCK-algebra X is called an anti fuzzy

subalgebra of X if

µ

_{(x ∗ y) ≤ max {µ}

(x), µ

_(y)}

for all x, y ∈ X.

As a generalization of the notion of fuzzy sets in X, Atanassov [2] introduced

the concept of intuitionistic fuzzy sets defined on X as objects having the form A =

{(x, µ

(x), λ

(x)) : x ∈ X} where the functions µ

: X → [0, 1] and λ

: X → [0, 1] denote

λ

(x)) of each element x ∈ X to the set A, respectively, and 0 ≤ µ

(x) + λ

(x) ≤ 1 for all

x ∈ X.

In [3], for every two intuitionistic fuzzy sets A and B in X, we have

(i) A ⊆ B iff µ

(x) ≤ µ

(x) and λ

(x) ≥ λ

(x) for all x ∈ X,

(ii)

¤A = {(x, µ

(x), µ

_{(x)) : x ∈ X} ,}

(iii)

♦A = {(x, λ

(x), λ

(x)) : x ∈ X} .

For the sake of simplicity, we shall use the symbol A = (µ

, λ

) for the intuitionistic

fuzzy set A = {(x, µ

(x), λ

(x)) : x ∈ X} as Dudek et al. [11].

2

(S,T)-anti intuitionistic fuzzy BCC-subalgebras

Definition 2.1. A fuzzy set A in a algebra X is said to be an anti fuzzy

BCC-subalgebra of X if

µ

_{(x ∗ y) ≤ max {µ}

(x), µ

_(y)}

for all x, y ∈ X.

Definition 2.2. An intuitionistic fuzzy set A = (µ

, λ

) in a BCC-algebra X is said

to be an anti intuitionistic fuzzy BCC-subalgebra of X if

(i) µ

_{(x ∗ y) ≤ max {µ}

(x), µ

_{(y)} ,}

(ii) λ

(x ∗ y) ≥ min {λ

(x), λ

(y)}

for all x, y ∈ X.

Definition 2.3. An intuitionistic fuzzy set A = (µ

, λ

) in a BCC-algebra X is said

to be an anti intuitionistic fuzzy BCC-subalgebra of X with respect to a t-conorm S and a

t-norm T (or simply, an (S, T )-anti intuitionistic fuzzy BCC-subalgebra of X) if

(i) µ

_{(x ∗ y) ≤ S(µ}

(x), µ

(y)),

(ii) λ

(x ∗ y) ≥ T (λ

(x), λ

(y))

for all x, y ∈ X.

Remark 2.1. Every anti intuitionistic fuzzy BCC-subalgebra of a BCC-algebra is an

(S, T )-anti intuitionistic fuzzy BCC-subalgebra of X, but it is clear that the converse is not

true. If λ

(x) = 1−µ

(x) for all x ∈ X, then every anti intuitionistic fuzzy BCC-subalgebra

of a BCC-algebra X is an anti fuzzy BCC-subalgebra of X. Also, if λ

(x) = 1 − µ

(x) for

all x ∈ X, S = S

and T = T

, then every (S, T )-anti intuitionistic fuzzy BCC-subalgebra

of a BCC-algebra X is an anti fuzzy BCC-subalgebra of X.

Example. Let X = {0, 1, 2, 3} be a BCC-algebra with the Cayley table as follows

∗ | 0 1 2 3

0

1

2

3

¯

¯

¯

¯

¯

¯

¯

¯

0

0

0

0

1

0

0

1

2

1

0

2

3

3

3

0

µ

(x) =







0,

x = 0

1/2,

x = 1 or 2

1,

x = 3

and λ

(x) =







1,

x = 0

1/3,

x = 1 or 2

0,

x = 3

It is easy to check that 0 ≤ µ

(x) + λ

(x) ≤ 1, µ

(x ∗ y) ≤ S

(µ

(x), µ

(y)) and

λ

(x ∗ y) ≥ T

(λ

(x), λ

(y)) for all x, y ∈ X. Hence A = (µ

, λ

) is an (S

, T

)-anti

intuitionistic fuzzy BCC-subalgebra of X. Also note that t-conorm S

and t-norm T

are

not associated.

Example. Let X = {0, a, b, c, d} be a proper BCC-algebra with the Cayley table as

follows

∗ | 0 a b c d

0

a

b

c

d

¯

¯

¯

¯

¯

¯

¯

¯

¯

¯

0

0

0

0

0

a

0

a

0

0

b

b

0

0

0

c

c

a

0

0

d

c

d

c

0

Define an intuitionistic fuzzy set A = (µ

, λ

) in X by

µ

(x) =

½

t

,

x ∈ {0, a, b}

t

,

otherwise

and λ

(x) =

½

t

,

x ∈ {0, a, b}

t

,

otherwise.

where 0 ≤ t

, t

, t

, t

≤ 1 such that t

< t

, t

< t

and t

+ t

+ t

+ t

= 1.

It is easy to check that 0 ≤ µ

(x) + λ

(x) ≤ 1, µ

(x ∗ y) ≤ S

(µ

(x), µ

(y)) and

λ

(x ∗ y) ≥ T

(λ

(x), λ

(y)) for all x, y ∈ X. Hence A = (µ

, λ

) is an (S

, T

)-anti intuitionistic fuzzy BCC-subalgebra of X. Also note that t-conorm S

and t-norm T

are not associated.

Remark 2.2. Note that, the above examples hold even with the t-conorm S

and

t-norm T

, and hence A = (µ

, λ

) is an (S

, T

)-anti intuitionistic fuzzy BCC-subalgebra

of X in such examples. Hence every anti intuitionistic fuzzy BCC-subalgebra of X is an

(S, T )-anti intuitionistic fuzzy BCC-subalgebra, but the converse is not true.

Lemma 2.1. If A = (µ

, λ

) is an (S, T )-anti intuitionistic fuzzy BCC- subalgebra

of a BCC-algebra X, then so is

¤A = (µ

, µ

A

) such that t-conorm S and t-norm T are

associated.

Proof. Since A = (µ

, λ

) is an (S, T )-anti intuitionistic fuzzy BCC-subalgebra of X,

we have

µ

_{(x ∗ y) ≤ S(µ}

(x), µ

(y))

for all x, y ∈ X and so

1 − µ

(x ∗ y) ≤ S(1 − µ

(x), 1 − µ

(y))

which implies

1 − S(1 − µ

(x), 1 − µ

(y)) ≤ µ

(x ∗ y).

Since S and T are associated, we have

T (µ

(x), µ

_{(y)) ≤ µ}

_{(x ∗ y).}

This completes the proof.

Lemma 2.2. If A = (µ

, λ

) is an (S, T )-anti intuitionistic fuzzy BCC- subalgebra

of a BCC-algebra X, then so is

♦A = (λ

, λ

) such that t-conorm S and t-norm T are

Proof. The proof is similar to the proof of Lemma 2.1.

Combining the above two lemmas, it is easy to see that the following theorem is valid.

Theorem 2.1. A = (µ

, λ

) is an (S, T )-anti intuitionistic fuzzy BCC- subalgebra

of a BCC-algebra X if and only if

¤A and ♦A are (S, T)-anti intuitionistic fuzzy

BCC-subalgebra of X such that t-conorm S and t-norm T are associated.

Corollary 2.1. A = (µ

, λ

) is an (S, T )-anti intuitionistic fuzzy BCC- subalgebra

of a BCC-algebra X if and only if µ

and λ

are anti fuzzy BCC-subalgebra of X such that

t-conorm S and t-norm T are associated.

If A = (µ

, λ

) is an intuitionistic fuzzy set in a BCC-algebra X and f is a self mapping

of X, we define mappings

µ

_{[f ] : X → [0, 1] by µ}

[f ](x) = µ

(f (x))

and

λ

[f ] : X → [0, 1] by λ

[f ](x) = λ

(f (x))

for all x ∈ X, respectively.

Proposition 2.1. If A = (µ

, λ

) is an (S, T )-anti intuitionistic fuzzy BCC- subalgebra

of a BCC-algebra X and f is an endomorphism of X, then (µ

[f ], λ

[f ]) is an (S, T )-anti

intuitionistic fuzzy BCC-subalgebra of X.

Proof. For any given x, y ∈ X, we have

µ

_{[f ](x ∗ y) = µ}

_{(f (x ∗ y)) = µ}

_{(f (x) ∗ f(y)) ≤ S(µ}

(f (x)), µ

(f (y)))

= S(µ

[f ](x), µ

[f ](y)),

λ

[f ](x ∗ y) = λ

(f (x ∗ y)) = λ

(f (x) ∗ f(y)) ≥ T (λ

(f (x)), λ

(f (y)))

= T (λ

[f ](x), λ

[f ](y)).

This completes the proof.

If f is a self mapping of a BCC-algebra X and B = (µ

, λ

) is an intuitionistic fuzzy

set in f (X), then the intuitionistic fuzzy set A = (µ

, λ

) in X defined by µ

= µ

◦ f

and λ

= λ

◦ f (i.e., µ

(x) = µ

(f (x)) and λ

(x) = λ

(f (x)) for all x ∈ X) is called the

preimage of B under f .

Theorem 2.2. An onto homomorphic preimage of an (S, T )-anti intuitionistic fuzzy

BCC-subalgebra is an (S, T )-anti intuitionistic fuzzy BCC-subalgebra.

Proof. Let f : X → Y be an onto homomorphism of BCC-algebras, B = (µ

, λ

) be

an (S, T )-anti intuitionistic fuzzy BCC-subalgebra of Y , and A = (µ

, λ

) be preimage of

B under f . Then, we have

µ

_{(x ∗ y) = µ}

_{(f (x ∗ y)) = µ}

_{(f (x) ∗ f(y)) ≤ S(µ}

(f (x)), µ

(f (y)))

= S(µ

(x), µ

(y)),

λ

(x ∗ y) = λ

(f (x ∗ y)) = λ

(f (x) ∗ f(y)) ≥ T (λ

(f (x)), λ

(f (y)))

= T (λ

(x), λ

(y))

for all x, y ∈ X. Hence, A = (µ

, λ

) is an (S, T )-anti intuitionistic fuzzy BCC-subalgebra

If f is a self mapping of a BCC-algebra X and A = (µ

, λ

) is an intuitionistic fuzzy

set in X, then the intuitionistic fuzzy set A

= (µ

, λ

) in f (X) defined by

µ

(y) =

inf

µ

(x) and λ

(y) =

sup

λ

(x)

for all y ∈ f(x), is called image of A = (µ

, λ

) under f .

An intuitionistic fuzzy set A = (µ

, λ

) in X is said to be have (inf-sup) property if

there exists a t

∈ T such that µ

(t

) = inf

µ

(t) and λ

(t

) = sup

λ

(t) for every

subset T ⊆ X.

Proposition 2.2. An onto homomorphic image of an anti intuitionistic fuzzy

BCC-subalgebra with (inf-sup) property is an anti intuitionistic fuzzy BCC-BCC-subalgebra.

Proof. Let f : X → Y be an onto homomorphism of BCC-algebras and A = (µ

, λ

)

be an anti intuitionistic fuzzy BCC-subalgebra of X with (inf-sup) property. For given

x

, y

_{∈ Y , let x}

∈ f

(x

) and y

∈ f

(y

) such that µ

(x

) = inf

µ

(t),

µ

(y

) = inf

µ

(t), λ

(x

) = sup

λ

(t) and λ

(y

) = sup

λ

(t),

respectively. Then

µ

(x

_{∗ y}

) =

inf

µ

(z) ≤ max {µ

(x

), µ

(y

)}

= max

½

inf

µ

(t),

inf

µ

(t)

¾

= max

n

µ

(x

), µ

(y

)

o

,

λ

(x

_{∗ y}

) =

sup

λ

(z) ≥ min {λ

(x

), λ

(y

)}

= min

(

sup

λ

(t),

sup

λ

(t)

)

= min

n

λ

(x

), λ

(y

)

o

.

Hence, A

= (µ

, λ

) is an anti intuitionistic fuzzy BCC-subalgebra of Y .

Remark 2.3. It is well known [12,17] that max {x, y} ≤ S(x, y) and min {x, y}

≥ T (x, y) for all x, y ∈ [0, 1]. Therefore, it is easy to see that the above proposition is

also true in the case of (S, T )-anti intuitionistic fuzzy BCC-subalgebras.

Lemma 2.3 ([12]). Let S and T be a t-conorm and a t-norm, respectively. Then

S(S(x, y), S(z, t)) = S(S(x, z), S(y, t)),

T (T (x, y), T (z, t)) = T (T (x, z), T (y, t))

for all x, y, z, t ∈ [0, 1].

Theorem 2.3. Let S be a t-conorm, T be a t-norm and X = X

× X

be the

direct product BCC-algebra of BCC-algebras X

and X

.

If A

= (µ

, λ

) (resp.

A

= (µ

, λ

)) is an (S, T )-anti intuitionistic fuzzy BCC-subalgebra of X

(resp. X

),

then A = (µ

, λ

) is an (S, T )-anti intuitionistic fuzzy BCC-subalgebra of X defined by

µ

= µ

_{× µ}

and λ

= λ

× λ

such that

µ

(x

, x

) = (µ

× µ

)(x

, x

) = S(µ

(x

), µ

(x

)),

λ

(x

, x

) = (λ

× λ

)(x

, x

) = T (λ

(x

), λ

(x

))

Proof. Let x = (x

, x

) and y = (y

, y

) be any elements of X. Since X is a

BCC-algebra, we have

µ

_{(x ∗ y) = µ}

((x

, x

) ∗ (y

, y

)) = µ

(x

∗ y

, x

∗ y

)

= (µ

_{× µ}

)(x

∗ y

, x

∗ y

)

= S(µ

(x

∗ y

), µ

(x

∗ y

))

≤ S(S(µ

(x

), µ

(y

)), S(µ

(x

), µ

(y

)))

= S(S(µ

(x

), µ

(x

)), S(µ

(y

), µ

(y

)))

= S((µ

_{× µ}

)(x

, x

), (µ

× µ

)(y

, y

))

= S(µ

(x), µ

(y)),

λ

(x ∗ y) = λ

((x

, x

) ∗ (y

, y

)) = λ

(x

∗ y

, x

∗ y

)

= (λ

× λ

)(x

∗ y

, x

∗ y

)

= T (λ

(x

∗ y

), λ

(x

∗ y

))

≥ T (T (λ

(x

), λ

(y

)), T (λ

(x

), λ

(y

)))

= T (T (λ

(x

), λ

(x

)), T (λ

(y

), λ

(y

)))

= T ((λ

× λ

)(x

, x

), (λ

× λ

)(y

, y

))

= T (λ

(x), λ

(y)).

This completes the proof.

3

(S,T)-anti intuitionistic fuzzy ideals

In this section, we shall define the notion (S, T )-anti intuitionistic fuzzy ideal of a

BCC-algebra with the help of arbitrary t-conorms and t-norms. We investigate some relations

between (S, T )-anti intuitionistic fuzzy ideals and (S, T )-anti intuitionistic fuzzy

BCC-subalgebras and prove some results on them.

Definition 3.1. A fuzzy set A in a BCC-algebra X is said to be an anti fuzzy ideal of

X if

(i) µ

_{(0) ≤ µ}

(x),

(ii) µ

_{(x) ≤ max{µ}

_{(x ∗ y), µ}

_(y)}

for all x, y ∈ X.

Definition 3.2. An intuitionistic fuzzy set A = (µ

, λ

) in a BCC-algebra X is said

to be an anti intuitionistic fuzzy ideal of X if

(i) µ

_{(0) ≤ µ}

(x) and λ

(0) ≥ λ

(x),

(ii) µ

_{(x) ≤ max {µ}

_{(x ∗ y), µ}

_{(y)} ,}

(iii) λ

(x) ≥ min {λ

(x ∗ y), λ

(y)}

for all x, y ∈ X.

Definition 3.3. An intuitionistic fuzzy set A = (µ

, λ

) in a BCC-algebra X is said

to be an anti intuitionistic fuzzy ideal of X with respect to a t-conorm S and a t-norm T

(or simply, an (S, T )-anti intuitionistic fuzzy ideal of X) if

(i) µ

_{(0) ≤ µ}

(x) and λ

(0) ≥ λ

(x),

(ii) µ

_{(x) ≤ S(µ}

_{(x ∗ y), µ}

(y)),

(iii) λ

(x) ≥ T (λ

(x ∗ y), λ

(y))

for all x, y ∈ X.

Remark 3.1. Every anti intuitionistic fuzzy ideal of a BCC-algebra is an (S, T

)-anti intuitionistic fuzzy ideal of X, but it is clear that the converse is not true.

If

λ

(x) = 1 − µ

(x) for all x ∈ X, then every anti intuitionistic fuzzy ideal of a

BCC-algebra X is an anti fuzzy ideal of X. Also, if λ

(x) = 1 − µ

(x) for all x ∈ X, S = S

and T = T

, then every (S, T )-anti intuitionistic fuzzy ideal of a BCC-algebra X is an anti

fuzzy ideal of X.

Example. In Example 1, it is easy to show that A = (µ

, λ

) is also an (S, T )-anti

intuitionistic fuzzy ideal of X.

Lemma 3.1. Let A = (µ

, λ

) be an (S, T )-anti intuitionistic fuzzy ideal of a

BCC-algebra X. If ≤ is a partial ordering on X then µ

(x) ≤ µ

(y) and λ

(y) ≤ λ

(x) for all

x, y ∈ X.

Proof. Let X be a BCC-algebra. It is known [13] that ≤ is a partial ordering on X

defined by x ≤ y if and only if x∗y = 0 for all x, y ∈ X. Let A be a (S, T )-anti intuitionistic

fuzzy ideal of X. Then

µ

_{(x) ≤ S(µ}

_{(x ∗ y), µ}

(y)) = S(µ

(0), µ

(y)) = µ

(y)

and

λ

(x) ≥ T (λ

(x ∗ y), λ

(y) = T (λ

(0), λ

(y)) = λ

(y).

These complete the proof.

Theorem 3.1. Let A = (µ

, λ

) be an (S, T )-anti intuitionistic fuzzy ideal of a

BCC-algebra X. If x ∗ y ≤ x holds in X, A is an (S, T )-anti intuitionistic fuzzy BCC-subBCC-algebra

of X.

Proof. Let A = (µ

, λ

) be an (S, T )-anti intuitionistic fuzzy ideal of X. Since x∗y ≤ x

for all x, y ∈ X, it follows from Lemma 4 that µ

(x ∗ y) ≤ µ

(x) and λ

(x) ≤ λ

(x ∗ y).

Then

µ

_{(x ∗ y) ≤ µ}

_{(x) ≤ S(µ}

_{(x ∗ y), µ}

_{(y)) ≤ S(µ}

(x), µ

(y))

and

λ

(x ∗ y) ≥ λ

(x) ≥ T (λ

(x ∗ y), λ

(y)) ≥ T (λ

(x), λ

(y))

and so A is an (S, T )-anti intuitionistic fuzzy BCC-subalgebra of X.

Remark 3.2. The converse of the above theorem does not hold in general. In fact,

suppose that X be the BCC-algebra in Example 1. It is clear that x ∗ y ≤ x for all x, y ∈ X.

Define an intuitionistic fuzzy set A = (µ

, λ

) in X by

µ

(x) =







0,

x = 0

1/2,

x = 1

1,

x = 2 or 3

and λ

(x) =







1,

x = 0

1/3,

x = 1

0,

x = 2 or 3

By routine calculations, we know that A = (µ

, λ

) is an (S, T )-anti intuitionistic

fuzzy BCC-subalgebra of X but not an (S, T )-anti intuitionistic fuzzy ideal of X because

µ

(2) = 1 > S(µ

_{(2 ∗ 1), µ}

(1)) and λ

(2) = 0 < T (λ

(2 ∗ 1), λ

(1)).

Proposition 3.1. Let A = (µ

, λ

) be an (S, T )-anti intuitionistic fuzzy ideal of a

BCC-algebra X. If x ∗ y ≤ z holds in X, then µ

(x) ≤ S(µ

(y), µ

(z)) and λ

(x) ≥

T (λ

(y), λ

(z)) for all x, y, z ∈ X.

Proof. Since x ∗ y ≤ z holds for all x, y, z ∈ X, we have

µ

_{(x ∗ y) ≤ S(µ}

_{((x ∗ y) ∗ z), µ}

(z))

≤ S(µ

(z ∗ z), µ

(z))

= S(µ

(0), µ

(z))

= µ

(z)

it follows that

µ

_{(x) ≤ S(µ}

_{(x ∗ y), µ}

_{(y)) ≤ S(µ}

(z), µ

(y))

and

λ

(x ∗ y) ≥ T (λ

((x ∗ y) ∗ z), λ

(z))

≥ T (λ

(z ∗ z), λ

(z))

= T (λ

(0), λ

(z))

= λ

(z)

it follows that

λ

(x) ≥ T (λ

(x ∗ y), λ

(y)) ≥ T (λ

(z), λ

(y)).

They complete the proof.

Proposition 3.2. An intuitionistic fuzzy subset A of a BCC-algebra X is an (S, T

)-intuitionistic fuzzy ideal of X if and only if its complement A

is an (S, T )-anti intuitionistic

fuzzy ideal of X such that t-conorm S and t-norm T are associated.

Proof. Let A be an (S, T )-intuitionistic fuzzy ideal of X such that S and T are associated.

Then

µ

_{(0) = 1 − µ}

_{(0) ≤ 1 − µ}

(x) = µ

(x)

and

λ

_{(0) = 1 − λ}

(0) ≥ 1 − λ

(x) = λ

(x).

We also have

µ

_{(x) = 1 − µ}

_{(x) ≤ 1 − T (µ}

_{(x ∗ y), µ}

(y))

= 1 − T (1 − µ

(x ∗ y), 1 − µ

(y))

= S(µ

_{(x ∗ y), µ}

(y))

and

λ

_{(x) = 1 − λ}

(x) ≥ 1 − S(λ

(x ∗ y), λ

(y))

= 1 − S(1 − λ

(x ∗ y), 1 − λ

(y))

= T (λ

_{(x ∗ y), λ}

(y)).

for all x, y ∈ X. Thus A

is an (S, T )-anti intuitionistic fuzzy ideal of X. The converse also

can be proved similarly.

Theorem 3.2. Let A be an anti intuitionistic fuzzy ideal of a BCC-algebra X. Then

the set

X

:= {x ∈ X : µ

(x) = µ

(0), λ

(x) = λ

(0)}

is an ideal of X.

Proof. Since A is an anti intuitionistic fuzzy ideal of X, we have µ

_{(0) ≤ µ}

(x) and

λ

(0) ≥ λ

(x). Now, suppose that x, y ∈ X such that x ∗ y ∈ X

and y ∈ X

. Then

µ

_{(x ∗ y) = µ}

(0) = µ

(y) and λ

(x ∗ y) = λ

(0) = λ

(y), so we have

µ

_{(x) ≤ max{µ}

_{(x ∗ y), µ}

_{(y)} = max{µ}

(0), µ

_{(0)} = µ}

(0)

and

λ

(x) ≥ min{λ

(x ∗ y), λ

(y)} = min{λ

(0), λ

(0)} = λ

(0)

respectively. Thus, we have µ

(x) = µ

(0) and λ

(x) = λ

(0), and therefore x ∈ X

.

Also, it is easy to see that 0 ∈ X

. This completes the proof.

Theorem 3.3. Let A be an intuitionistic fuzzy subset of a BCC-algebra X. Then A is

an (S, T )-anti intuitionistic fuzzy ideal of X if and only if for each α

, α

∈ [0, 1] such that

α

≤ λ

(0) and α

≥ µ

(0), the upper α

-level cut U (λ

; α

) and the lower α

-level cut

V (µ

; α

) are ideals of X.

Proof. Let A be an (S, T )-anti intuitionistic fuzzy ideal of X and let α

∈ [0, 1] such

that α

≤ λ

(0). Clearly 0 ∈ U(λ

; α

). Let x, y ∈ X such that x ∗ y ∈ U(λ

; α

) and

y ∈ U(λ

; α

). Then

λ

(x) ≥ S(λ

(x ∗ y), λ

(y)) ≥ α

and x ∈ U(λ

; α

). Hence U (λ

; α

) is an ideal of X. Similarly, V (µ

; α

) is also an ideal

of X.

Conversely, we first show that λ

(0) ≥ λ

(x) for all x ∈ X. If not, then there

exists a x

∈ X such that λ

(0) < λ

(x

).

Taking α

=

(λ

(x

) + λ

(0)) then

0 ≤ λ

(0) < α

< λ

(x

) ≤ 1. It follows that x

∈ U(λ

; α

), so U (λ

; α

) 6= ∅.

Since U (λ

; α

) is an ideal of X we have 0 ∈ U(λ

; α

) or λ

(0) ≥ α

which is a

contradiction.

Hence λ

(0) ≥ λ

(x) for all x ∈ X. Next, we prove that λ

(x) ≥

T (λ

(x ∗ y), λ

(y)) for all x, y ∈ X. If not, then there exist x

, y

∈ X such that

λ

(x

) ≥ T (λ

(x

∗ y

), λ

(y

)). Taking α

=

(λ

(x

) + T (λ

(x

∗ y

), λ

(y

))) then

α

< λ

(x

) and 0 ≤ T (λ

(x

∗ y

), λ

(y

)) < α

≤ 1. Thus, we have α

> λ

(x

∗ y

)

and α

> λ

(y

) which imply that x

∗ y

∈ U(λ

; α

) and y

∈ U(λ

; α

). As U (λ

; α

)

is an ideal of X, it follows that x

∈ U(λ

; α

) or λ

(x

) ≥ α

, which is a contradiction.

Similarly, µ

_{(x) ≥ λ}

(0) and µ

(x) ≤ S(µ

(x ∗ y), µ

(y)) for all x, y ∈ X. This completes

the proof.

Theorem 3.4. Let A be an (S, T )-anti intuitionistic fuzzy ideal of a BCC-algebra X.

Two upper level cuts U (λ

; α

) and U (λ

; α

) with α

> α

(resp. two lower level cuts

V (µ

; β

) and V (µ

; β

) with β

> β

_{) are equal if and only if there exists no x ∈ X such}

that α

≥ λ

(x) > α

(resp. β

≥ µ

(x) > β

).

Proof. From the definition of upper level cut, it follows that U (λ

; α) = λ

([α, λ

(0)])

for α ∈ [0, 1]. Let α

, α

∈ [0, 1] such that α

> α

. Then

U (λ

; α

)

=

U (λ

; α

) ⇐⇒ λ

([α

, λ

(0)]) = λ

([α

, λ

(0)])

⇐⇒ λ

((α

, α

]) = ∅

⇐⇒ there is no x ∈ X such that α

≥ λ

(x) > α

.

References

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[2] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Systems 20(1986), 87-96.

[3] K. Atanassov, New operations defined over the intuitionistic fuzzy sets, Fuzzy Sets Systems 61(1994), 137-142.

[4] R. Biswas, Fuzzy subgroups and anti fuzzy subgroups, Fuzzy Sets and Systems 35(1990), 121-124.

[5] D. Dubois, H. Prade, New results about properties and semantics of fuzzy set-theoretic operators, In: P.P. Wang, S.K. Chang editors. Fuzzy sets: theory and applications to policy analysis and information systems, New York, Plenum Press, 1980.

[6] W.A. Dudek, The number of subalgebras of finite BCC-algebras, Bull. Inst. Math. Acad. Sinica 20(1992), 129-136.

[7] W.A. Dudek, On proper BCC-algebras, Bull. Inst. Math. Acad. Sinica 20(1992), 137-150. [8] W.A. Dudek, K.H. Kim, Y.B. Jun, Fuzzy BCC-subalgebras of BCC-algebras with respect to

a t-norm, Sci. Math. 3(2000), 99-106.

[9] W.A. Dudek, Y.B. Jun, Fuzzy BCC-ideals in BCC-algebras, Math. Montisnigri 10(1999), 21-30. [10] W.A. Dudek, Y.B. Jun, S.M. Hong, On fuzzy topological BCC-algebras, Discuss Math. Gen.

Algebra Appl. 20(1)(2000), 77-86.

[11] W.A. Dudek, B. Davvaz, Y.B. Jun, On intuitionistic fuzzy sub-hyperquasigroups of hyperqua-sigroups, Inform. Sci. 170(2005), 251-262.

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Math., in press.

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A KANTOROVICH ANALYSIS OF NEWTON

METHODS ON LIE GROUPS

Ioannis K. Argyros

Cameron University, Department of Mathematical Sciences,

Lawton, OK 75505, U.S.A.

E-mail address: [email protected]

Abstract

A local as well as a semilocal Kantorovich-type convergence analysis is provided for Newton methods (Newton’s method and Modified Newton’s method) to solve equations on Lie groups. Motivated by optimization considerations and by using more precise majorizing sequences than before [6], [9], [10], we show that under the same or weaker hypotheses: a larger convergence domain; finer error estimates on the distances involved can be obtained; an at least as precise information on the location of the solution is given semilocal case, and a larger radius of convergence (in the local case).

We also note that our results are obtained under the same compu-tational cost as in [6], [9], [10]. Finally the results are extended to the H¨older case not examined before.

AMS (MOS) subject classification codes: 65J15, 65G99, 65B05, 65L50, 47H17, 49M15.

Key Words: Newton’s method, Modified Newton’s method, Lie groups, Abelian groups, majorizing sequence, local/semilocal convergence, Kan-torovich hypothesis, radius of convergence, Lipschitz conditions.

1

Introduction

In this study we are concerned with the problem of approximating a locally unique zero x∗ of a map f defined on a Lie grup (to be precised in section 1). Numerical algorithms on manifolds are very important in computational mathematics [2], [5], [6], [9], [10], because they appear in connection to eigen-value problems, minimization problems, optimization problems. A convergence analysis of Newton’s method on Riemannian manifolds under various condition similar to the corresponding ones on Banach spaces [1],[4], [7], [8] has been given in [9], [10] and the references there.

Here we are motivated in particular by the elegant work in [9], and optimiza-tion consideraoptimiza-tions with advantages over earlier works [6], [9], [10] as already mentioned in the abstract of the paper.

2

Preliminaries

A Lie group (G,·) is a Hausdorff topological group with countable bases which also has the structure of a smooth manifold such that the group product and the inversion are smooth operations in the differentiable structure given on the manifold. The dimension of a Lie group is that of the underlying manifold, and we shall always assume that it is finite. The symbol e designates the identity element of G. Let g be the Lie algebra of the Lie group G which is the tangent space TeG of G at e, equipped with Lie bracket [·, ·] : g × g → g. In the sequel

we will make use of the left translation of the Lie group G. We define for each y ∈ G

Ly : G → G

z → y · z,

the left multiplication in the group. The differential of Ly at e denoted by

(dLy)e determines an isomorphism of g = TeG with the tangent space TyG via

the relation

(dLy)e(g) = TyG

or, equivalently,

g = (dLy)−1e (TyG) = dLy−1)y(TyG).

The exponential map is a map

exp : g → G u → exp(u),

which is certainly the most important construct associated to G and g. Given u ∈ g, the left invariant vector field Xu : y → (dLy)e(u) determines an

one-parameter subgroup of G σu: R → G such that σu(0) = e and

σ_u0(t) = Xu(σu(t)) = (dLσu(t))e(u).

The exponential map is then defined by the relation exp(u) = σu(1).

Note that the exponential map is not surjective in general. However, the ex-ponential map is a diffeomorphism on an open neighborhood N (0) of 0 ∈ g. Let

N (e) = exp(N (0)).

Then for each y ∈ N (e), there exists υ ∈ N (0) such that y = exp(υ). Further-more, if

exp(u) = exp(υ) ∈ N (e)

for some u, υ ∈ N (0), then u = υ. If G is Abelian, exp is also a homomorphism from g to G, i.e.,

for all u, υ ∈ g = TeG. In the non-abelian case, exp is not a homomorphism and

(1) must be replaced by

exp(ω) = exp(u) · exp(υ),

where ω is given by the Baker-Campbell-Hausdorff (BCH) formula ω = u + υ + 1

2[u, υ] + 1

12([u[u, υ]] + [υ[υ, u]]) + ...,

for all u, υ in an open neighborhood of 0 ∈ g. To analyse convergence, we need a Riemannian metric on the Lie group G.

Following [5] take an inner product h, ieon g and define

hu, υix= h(dLx−1)x(u), (dLx−1)x(υ)ie, for each x ∈ G and u, υ ∈ TxG.

This construction actually produces a Riemannian metric on the Lie group G, see for example [5]. Let k·k_xbe associated norm, where the subscript x is sometimes omitted if there is no confusion. For any two distinct elements x, y ∈ G, let c : [0, 1] → G be a piecewise smooth curve connecting X and y. Then the arc-length of c is defined by l(c) :=R1

0 kc

0_{(t)k dt, and the distance from x to}

y by d(x, y) := infcl(c), where the infimum is taken over all piecewise smooth

curves c : [0, 1 → G connecting x and y. Thus, we assume throughout the whole paper that G is connected and hence (G, d) is a complete metric space. Since we only deal with finite dimensional Lie algebras, every linear mapping ϕ : g → g is bounded and we define its norm by

kϕk = sup

u6=0

kϕ(u)k

kuk = supkuk=1

kϕ(u)k < ∞.

For r > 0 we introduced the corresponding ball of radius r around y ∈ G defined by one parameter subgroups of G as

Cr(y) = {z ∈ G : z = y · exp(u), kuk ≤ r}.

We give the following definition on convergence:

Definition 1 Let {xn}n≥0 be a sequence of G and x ∈ G. Then {xn}n≥0 is

said to be

(i) convergent to x if for any ε > 0 there exists a natural number K such that x−1· xn∈ N (e) and

exp−1(x−1· xn)

≤ ε for all n ≥ K;

(ii) quadratically convergent to x if { exp−1(x−1· xn)

} is quadratically con-vergent to 0; that is, {xn}n≥0is convergent to x and there exists a constant

q and an natural number K such that exp−1(x−1) · xn+1) ≤ q exp−1(x−1· xn) 2 for all n ≥ K.

Note that convergence of a sequence {xn}n≥0 in G to x in the sense of

Definition 1 above is equivalent to that limn→+∞d(xn, x) = 0.

In the remainder of this paper, let f : G → g = TeG be a C1 - mapping.

The differential of f at a point x ∈ G is a linear map fx0 : TxG → g defined by

f_x0(4x) =

d

dtf (x · exp(t((dx−1)x)(4x))) |t=0 for any 4x∈ TxG. (2) The differential f0

x can be expressed via a function dfx: g → g given by

dfx= (f ◦ Lx)0e= fx0 ◦ (dLx)e.

Thus, by (8), it follows that dfx(u) = fx0((dLx)e(u)) =

d

dtf (x · exp(tu) |t=0 for any u ∈ g. Therefore the following lemma is clear.

Lemma 2 Let x ∈ G, u ∈ g and t ∈ R. Then d

dtf (x · exp(−tu)) = −dfx·exp(−tu)(u) (3)

and

f (x · exp(tu)) − f (x) = Z t

0

dfx·exp(su)(u)ds. (4)

As in [10] Newton’s method for f with initial point x0 ∈ G is defined as

follows

xn+1= xn· exp(−dfx−1n ◦ f (xn)) (n ≥ 0). (5)

We also define the modified Newton’s method by

xn+1= xnexp(−dfx−10 f (xn) (n ≥ 0). (6)

3

Local convergence analysis of Newton’s method

(5)

We will use the following definition involving Lipschitz conditions:

Definition 3 Let r > 0, and let x0∈ G be such that dfx−10 exists. Then df

−1 x0 df

is said to satisfy: the center Lipschitz condition with constant `0> 0 in C(x0, r)

if

df_x−1₀ (dfx0exp(u)− dfx0)

≤ `0kuk , for each u ∈ g with kuk ≤ r; (7)

the center Lipschitz condition with constant ` in C(x0, r) if

df_x−1₀ (dfx·exp(u)− dfx)

≤ ` kuk (8)

Remark 4 . In general

`0≤ `, (9)

holds, and `

`0 can be arbitrarily large [1],[4].

We can show the following local convergence results for Newton’s method (5).

Theorem 5 Assume that G is un Abelian group. Choose r ∈ (0,_2`2

0+`), and let

x∗_{∈ G such that f (x}∗_{) = 0 and df}−1

x∗ exists. Moreover assume df_x−1∗ df satisfies

condition (8).

Then sequence {xn} generated by Newton’s method (5) is well defined,

re-mains in C(x∗, r) for all n ≥ 0, and converges quadratically to x∗ provided that x0∈ C(x∗, r), with ratio α given by

α = `

2(1 − `0ku0k

, (10)

where u0∈ g with ku0k ≤ r, and x0= x∗exp(u0).

Proof. Set α0 = α ku0k . In view of (10) α0 ∈ [0, 1). We shall show using

induction that for each n ≥ 0, xn is well-defined, remains in C(x∗, r), and there

exists un ∈ g with kunk ≤ r such that

xn = x∗exp(un), and kun+1k ≤ α kunk2≤ α2

n+1

−1

0 kυ0k . (11)

Estimates (11) hold true for n = 0 by the initial conditions. Assume estimates (11) hold true for n ≤ k, xnis well defined and there exist Un ∈ g with kunk ≤ r

such that (11) hold. Using (7) we get

df_x−1∗ (df_x∗_exp(u

k)− dfx∗)

≤ `0kukk < 1. (12)

It follows from the Banach Lemma [8] df_x−1

k exists and df_x−1_kdfx∗ ≤ 1 1 − `0kukk . (13)

That is xk+1is well defined. Set

uk+1= uk− dfx−1k(f (xk)). (14)

In view of (13) and (14) we get in turn kuk+1k = uk− dfx−1k(f (xk) − f (x ∗₎₎ (15) ≤ df_x−1 kdfx∗ Z 1 0 df_x−1∗ (dfxk− dfx∗exp(tuk))uk dt ≤ 1 1 − `0kukk Z 1 0 (1 − t)l kukk2dt = 2 2(1 − `0kukk) kukk 2 ≤ α kukk 2 ≤ α2₀k+1−1ku0k = r,

which establishes the quadratic convergence and kuk+1k ≤ τ. Moreover since G

is an Abelian group

xk+1= x∗exp(uk) exp[−dfx−1kf (xk)] = x

∗_exp(u

k+1). (16)

That completes the induction and the proof of the theorem.

Remark 6 If `0 = `, 5 reduces to Theorem 3.1 in [9]. Otherwise it is an

improvement. Indeed let τW L be the corresponding radius of convergence in [9]

selected in (0,_3l2). Then since (0,_3l2) ⊆ (0,_2l2

0+l) it follows our radius rAis such

that

rW L< rA. (17)

Hence, our approach allows a wider choice of initial guesses x0. Moreover

the ratio is smaller than the corresponding one α in [9] (simply let `0 = ` in

(10) to obtain α),since we have

α < α. (18)

The uniqueness of the solution x∗ is discussed next. Proposition 7 Let r ∈ (0,_l2

0). Assume f (x

∗_{) = 0 and df}−1

x∗ df satisfies (7) in

U (x∗_{, r). Then x}∗ _{is the unique zero of f in U (x}∗_{, r).}

Proof. Let y∗ be a zero of f in U (x∗, τ ). It follows that there exists u ∈ g so that y∗= x∗exp(u) and kuk ≤ τ. We can have in turn

kuk = −df_x−1∗ (f (y∗) − f (x∗)) + u (19) = −df_x−1∗ Z 1 0 dfx∗_exp(tu)(u)dt + u = −dfx−1∗ Z 1 0 (dfx∗_exp(u)− df_x∗)udt ≤ Z 1 0 t`0kuk2dt = `0 2 kuk 2 .

In view of (19) we deduce kuk ≥_`2

0. Hence, we arrived at a contradiction.

That completes the proof of the Proposition.

4

Semilocal convergence analysis of Newton’s

method (5)

Our semilocal convergence analysis of method (5) depends on the scalar sequence {sn} (n ≥ 0) introduced by us in [1], [4]:

s0= 0, s1= n, sn+2= sn+1+

L(sn+1− sn)2

2(1 − L0sn+1)

for some L0 > 0, L > 0 with L0 ≤ L and η > 0. Sufficient convergence

condi-tions for majorizing sequence {sn} we given in [1],[4]. Here we summarize the

conditions: hδ= (L + δL0)η ≤ δ, δ ∈ [0, 1]. (21) or hδ ≤ δ, δ ∈ [0, 2), (22) 2L0η 2 − δ ≤ 1 (23) and L0δ2 2 − δ ≤ L (24) or hδ ≤ δ, δ ∈ [δ0, 2) (25) where, δ0= −b√b2_{+ 8b} 2 , b = L L0 . (26)

Under any of the above conditions {sn} converges (increasingly) to some s∗ ∈

(0,_2−δ2η ]. Iteration {sn} coincides for L0= L with iteration {tn} used in [9]:

t0= 0, t1= n, tn+2= tn+1+

L(tn+1− tn)

2(1 − Ltn+1)

, (n ≥ 0) (27)

and has been compared favorably with it when L0 < L. Indeed we showed in

[1],[4]: sn< tn (n ≥ 2), (28) sn+1− sn< tn+1− tn, (n ≥ 2), (29) s∗≤ t∗=1 − √ 1 − 2h L , (30) and s∗− sn≤ t∗− tn, (n ≥ 0), (31)

provided that any of (21) or (22)-(24) or (25)-(26) and the famous Newton-Kantorovich condition [8]

h = 2Lη ≤ 1 (32)

hold. Note that,

h ≤ 1 =⇒ h1≤ 1 (33)

but not vice versa unless if L0= L.

We need definitions corresponding to Definition 3 above. Let us first intro-duce the metric closed ball of radius r > 0 about y ∈ G denoted by

U (y, τ ) = {z ∈ G : d(z, y) ≤ r}. (34)

Note that

Definition 8 Let r > 0, and let x0∈ G be such that dfx−10 exists. Then df

−1 x0 df

is said to satisfy: the center Lipschitz condition which constant L0 in U (x0, r)

if

dfx−10 (dfx− dfx0)

≤ L0d(x0, x), for all x ∈ U (x0; r); (36)

the Lipschitz condition in the inscribed sphere with constant L in U (x0, r) if

df_x−1₀ (dfy− dfx)

≤ Ld(x, y) holds for all x, y ∈ U (x0, r) with (37)

d(x0, x) + d(x, y) ≤ r.

We can show the main semilocal convergence result for Newton’s method (5):

Theorem 9 . Let x0∈ G be such that dfx−10 exists and set n =

df−1

x0 (f (x0))

. Assume that either (21) or (22)-(24) or condition (25) hold. Moreover, as-sume df_x−1

0df satisfies (35) and (36). Then sequence {xn} generated by Newton’s

method (5) is well defined, remains in U (x0, τ, s∗) for all n ≥ 0 and converges to

n zero s∗of f in U (x0, s∗).Moreover, the following estimates hold for all n ≥ 0:

d(xn+1, xn) ≤ sn+1− sn, (38)

and

d(xn, x∗) ≤ s∗− sn. (39)

Furthermore, if G is an Abelian group, then there is n zero s∗ of f in C(x0, s∗)

such that for all n ≥ 0, there exists un ∈ g such that xn = x∗exp(un), and for

al n ≥ 1 kunk ≤ L(s∗− tn−1) 2(1 − L0tn−1)  kunk s∗_{− t} n−1 2 . (40)

Proof. We shall show

d(xn+1, xn) ≤ kvnk ≤ tn+1− tn, (41)

where, vn = −dfx−1nf (xn), (n ≥ 0).

Let us define the curve c0(t) = x0exp(tv0), t ∈ [0, 1]. Then c0 is smooth and

connects x0to x1 with leng (c0) = kv0k . That is, d(x1, x0) ≤ leng (c0) = kv0k .

That is, d(x1, x0) ≤ leng(c0) = kvdk . In view of kv0k =

−df−1

x0 f (x0)

≤

η ≤ s1− s0, (40) holds true for n = 0. We assume (40) to hold true for n =

0, 1, ..., k − 1. It follows d(xk, x0) ≤ k−1 X i=0 d(xi+1, xi) ≤ k−1 X i=0 kvik ≤ sk− s0= sk< s∗. (42)

That is xk ∈ U (x0, s∗). As in (13) but using (35) instead of (7) we deduce dfx−1k

exists and df_x−1_kdfx0 ≤ 1 1 − L0sk . (43)

In view of (5), xk+1is well defined. Using (5), (36), and (42) we obtain in turn: df_x−1 0 f (xk) ≤ Z 1 0 df_x−1 0 [dfxk−1exp(tvk−1) − dfxk−1] kvk−1k dt (44) ≤ Z 1 0 Ld(xk−1,xk−1exp(tvk−1)) kvk−1k dt ≤ Z 1 0 L ktvk−1k kvk−1k dt ≤L 2(sk− sk−1) 2_, and kvkk = df_x−1 kdfx0df −1 x0 f (xk) (45) ≤ −dfx−1kdfx0 df_x−1 0 f (xk) ≤ L(sk− sk−1) 2 2(1 − L0− sk−1) = sk+1− sk,

which also shows (37).We define the curve ck(as c0above) by ck(t) = xkexp(tvk)t ∈

[0, 1].As above we have d(xk+1, xk) ≤ leng(ck) = kvkk . That completes the

in-duction for (40). It follows that sequence {xn} is Cauchy and as such it converges

to some x∗ _{∈ U (x}

0, s∗) (since U (x0, s∗) is a closed set). By letting k → ∞ in

(43) we obtain f (x∗_{) = 0. Moreover (38) follows from (37) by using standard}

majorizations techniques. Define

un= − ∞ X k=n vk (n ≥ 0). (46) It follows by (40) that kunk ≤ s∗− sn (n ≥ v). (47)

Let x∗= x0exp(−u0). Then we have x∗∈ C(x0, s∗). Moreover, we get

xk= x0 k−1 Y i=0 exp(vi) = x0exp k−1 X i=0 vi ! .

It follows that clearly xn = x∗exp(un). That is sequence {xn} converges to

x∗ which is a zero of f in C(x0, s∗).To complete the proof we must show (39).