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877

The Journal of Fuzzy Mathematics Vol. 20, No. 4, 2012 Los Angeles

Fixed Points and Common Fixed Points in a -fuzzy Pseudo-ordered Sets Abdelkader Stouti * Center for Doctoral Studies: Sciences and Techniques, Laboratory of Mathematics and Applications, Faculty of Sciences and Techniques, University Sultan Moulay Slimane, P. O. Box 523, 23000 Beni-Mellal, MOROCCO. E-mail: [email protected]

Lemnaouar Zedam Laboratory of Pure and Applied Mathematics, M’sila University, P. O. Box 166 Ichbilia, M’sila 28105, ALGERIA. E-mail: [email protected]

Abstract: In this paper, we first introduce the notions of a -fuzzy pseudo-order and a -fuzzy trellis. Secondly, we establish the existence of the greatest and the least fixed points of a -fuzzy monotone maps defined on a nonempty a -fuzzy pseudo-ordered set. Furthermore, we prove that the set of all fixed points of two classes of a -fuzzy monotone maps defined on a nonempty complete a -fuzzy trellis is also a nonempty complete a -fuzzy trellis. As consequences, we obtain an a -fuzzy version of a Skala’s results and we establish that the set of all common fixed points of finite commutative families of ra -fuzzy monotone maps defined on of two classes of nonempty ra -fuzzy compete trellises is also a nonempty ra -fuzzy compete trellis. Keywords and phrases: Fuzzy pseudo-ordered set, fixed point, common fixed point, fuzzy monotone map, fuzzy complete trellis.

1. Introduction and preliminaries Let X be a nonempty set. A fuzzy subset A of X is characterized by its membership function A : X Æ [0,1] and A (x ) is interpreted as the degree of membership of the __________________________ Received May, 2011 *

Corresponding author 1066-8950/12 $8.50 © 2012 International Fuzzy Mathematics Institute Los Angeles

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Abdelkader Stouti and Lemnaouar Zedam

element x in the fuzzy subset A for each x Œ X . In this paper, we introduce the notion of a -fuzzy pseudo-order as follows. Definition 1.1. Let X be a nonempty crisp set and a Œ ]0,1] . An a -fuzzy pseudoorder on X is a fuzzy subset ra of X ¥ X satisfying the following two properties: (i) for all x Œ X , ra (x, x) = a , ( a -fuzzy reflexivity);

(ii) for al x, y ΠX , ra (x, y ) + ra (y, x) > a implies x = y , ( a -fuzzy antisymmetry).

A nonempty set X with an a -fuzzy pseudo-order ra defined on it, is called ra -

fuzzy pseudo-ordered set ( ra -fposet, for shrot) and we denote it by (X , ra ) .

If Y is a subset of a ra -fposet (X , ra ) , then the a -fuzzy pseudo-order ra is an a -

fuzzy pseudo-order on Y and is called the induced a -fuzzy pseudo-order. An a -fuzzy pseudo-order ra is total on X if for every x, y ΠX , we have ra (x, y ) > a 2 or ra (y, x) > a 2 . An a -fuzzy pseudo-ordered set ( X , ra ) in which ra

is total is called a ra -fuzzy chain.

Let ( X , ra ) be a ra -fuzzy pseudo-ordered set and let A be a nonempty subset of X .

(a) An element u ΠX is called a ra -upper bound of A if ra (x, u ) > a 2 for all x ΠA . If u is a ra -upper bound of A and u ΠA , then u is called a ra -greatest

element of A . In this case, we shall write u = max ra (A ) . (b) An element

ΠX is a ra -lower bound of A if ra

is a ra -lower bound of A and this case, we shall write

ΠA , then

= min ra (A ) .

(

, x) > a 2 for all x ΠA . It

is called a ra -least element of A . In

(c) An element s ΠX is the ra -supremum of A if s is a ra -upper bound of A , and

for all ra -upper bound u of A , we have ra (s, u ) > a 2 . When s exists, we shall write s = supra (A ) . Similarly,

ΠX is the ra -infimum of A if

and for all ra -lower bound k of A , we have ra (k, write

)>a

= inf ra (A ) .

is a ra -lower bound of A 2 . When

exists we shall

Next, we shall give an example of a -fuzzy pseudo-order. Let X = {a, b, c , d} . Then the fuzzy subset ra defined on X ¥ X by the following table: a

a b c d

a

0.15a 0.35a 0.60a

b 0.60a

a

0.20a 0.20a

c 0.45a 0.55a

a

0.25a

Is an a -fuzzy pseudo-order on X . In this work, we shall need the following definition.

d 0.40a 0.70a 0.65a

a

Fixed Points and Common Fixed Points in a -fuzzy Pseudo-ordered Sets

879

Definition 1.2. A fpsoset (X , ra ) is said to be a ra -fuzzy trellis if every pair of

elements ( X , ra ) has a greatest ra -lower bound and a least ra -upper bound. A fpsoset

(X , ra )

is said to be a complete ra -fuzzy trellis if every nonempty subset of X has a

ra -infimum and a ra -supremum.

In this paper, we shall use the following definition of a -fuzzy monotonicity. Definition 1.3.

Let ( X , ra ) be a nonempty ra -fuzzy pseudo-ordered set and

f : X Æ X be a map. We shall say that f is ra -fuzzy monotone if for every x, y Œ X

with ra (x, y ) > a 2 , then we have ra (f (x) , f (y )) > a 2 .

Let X be a nonempty set and f : X Æ X be a map. An element x of X is said to be a fixed point of f if f (x ) = x . The set of all fixed points of f is denoted by Fix (f ) .

A family F of maps f of X into X is said to be commutative if for every f , g ΠF , we have f g = g f . An element x of X is said to be a common fixed point for the family F if for every f ΠF , we have f (x) = x . We denote by Fix ( F ) the set

of all common fixed points of F . Let us note that

Fix ( F ) = ∩ Fix (f ) . f ŒF

In this work, we shall need the following notion of inverse fuzzy relation. Definition 1.4. Let X be a nonempty set and let r be a fuzzy relation on its. The inverse fuzzy relation s of r if defined for every x, y ΠX by: s (x, y ) = r (y, x) .

In this paper, we shall we need the two following technical lemmas which their proofs will be given in the appendix. Lemma 1.5. Let ra be an a -fuzzy pseudo-order defined on a nonempty set X and let sa be its inverse fuzzy relation. Then, sa is an a -fuzzy pseudo-order on X . Lemma 1.6. Let ra be an a -fuzzy pseudo-order defined on a nonempty set X , let sa be the inverse fuzzy relation of ra and let A be a nonempty subset of X . Then, we get

(i) if supra (A ) exists, so inf sa (A ) exists too and supra (A ) = infsa (A ) ; (ii) if inf ra (A ) exists, hence supsa (A ) exists also and inf ra (A ) = supsa (A ) ; (iii) if min ra (A ) exists, then max sa (A ) exists too and min ra (A ) = max sa (A ) ;

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Abdelkader Stouti and Lemnaouar Zedam

(iv) if max ra (A ) exists, so min sa (A ) exists also and max ra (A ) = min sa (A ) ; (v) if f : ( X , ra ) Æ ( X , ra ) is a ra -fuzzy monotone map, then f : ( X , sa ) Æ ( X , sa ) is a sa -fuzzy monotone map. In [3], the first author introduced the notion of a -fuzzy order. Definition 1.7 [3]. Let X be a nonempty crisp set and a Œ ]0,1] . An a -fuzzy order

on X is a fuzzy subset ra of X ¥ X satisfying the following three properties: (i) for all x Œ X , ra (x, x ) = a , ( a -fuzzy reflexivity); (ii) for all x, y Œ X , ra (x, y ) + ra (y, x) > a implies x = y , ( a -fuzzy antisymmetry); (iii) for all x, z Œ Xra (x, z ) ≥ supy ŒX ÈÎ min {ra (x, y ) , ra (y, z )}˘˚ , ( a -fuzzy transitivity). A nonempty set X with an a -fuzzy order ra defined on it, is called a -fuzzy ordered set ( a -foset, for short) and we denote it by ( X , ra ) .

Recently, the authors studies in [4] different properties of a -fuzzy orders. In 1971, H. Skala introduced the crisp notions of pseudo-ordered sets and trellises and gave some fixed point theorems in this setting (see Theorems 36 and 37 in [5]). In [3], the first author established some fixed points theorems in a -fuzzy ordered sets. In this paper, we first introduce the notion of a -fuzzy pseudo-order. Secondly, we establish the existence of the greatest and the least fixed points of a -fuzzy monotone maps defined on nonempty a -fuzzy pseudo-ordered sets (see Theorem 2.1 and 2.3). Thirdly, we prove that the set of all fixed point of two classes of a -fuzzy monotone maps defined on nonempty complete a -fuzzy trellises is also a nonempty complete a -fuzzy trellis (see Theorems 3.1 and 3.2). As consequences, we obtain an a -fuzzy version of the Skala’s result [Theorem 37, 5]. Furthermore, we establish that the set of all common fixed points of finite commutative family of ra -fuzzy monotone maps defined on of two classes of nonempty ra -fuzzy compete trellises is also a nonempty ra -fuzzy compete trellis. 2. Least and greatest fixed points of a -fuzzy monotone maps in a -fuzzy pseudoordered sets

In this section, we shall establish the existence of the greatest and the least fixed points of a -fuzzy monotone maps defined on nonempty a -fuzzy pseudo-ordered sets. First, we shall give our key result in this paper. Theorem 2.1. Let (X , ra ) be a nonempty ra -fuzzy pseudo-ordered set with a ra least element

. Assume that every nonempty subset of X has a ra -supremum. Then,

the set of all fixed point of every ra -fuzzy monotone map f : ( X , ra ) Æ ( X , ra ) is nonempty and has a ra -least element.

Fixed Points and Common Fixed Points in a -fuzzy Pseudo-ordered Sets

881

Let ( X , ra ) be a nonempty ra -fuzzy pseudo-ordered set with a ra -least

Proof.

and let f : ( X , ra ) Æ ( X , ra ) be a ra -fuzzy monotone map.

element

First step. We have: Fix (f ) π 0/ . Indeed, let A be the family of all subsets A of

X satisfying the following three conditions:

(i)

ŒA ;

(ii) f (A ) à A ; (iii) for every nonempty subset B of A , we have supra (B ) Œ A . Let us note that A π 0/ because X ŒA . Now, let C = ∩ AŒA A . Claim 1. The subset C is the nonempty least element of A for the crisp inclusion relation. Indeed, as Œ A for every A ŒA , so Œ C . Since C = ∩ AŒA A , then we Ê ˆ have f (C ) = f Á ∩ A˜ à ∩ f (A ) à ∩ A . Thus, we get f (C ) à C . Now, let Y be a Ë AŒA ¯ AŒA A ŒA

nonempty subset of C . Then, Y à A for every A Œ A . So, supra (Y ) Œ A for every A Œ A . Hence, we obtain sup ra (Y ) Œ C . Therefore, C is the least nonempty element

of A for the crisp inclusion order relation. Now, we set s = sup ra (C ) .

Claim 2. We have: s Œ Fix (f ) . Indeed, since s Œ C and f (C ) à C , then f (s) Œ C .

As s = sup ra (C ) , so we get

ra (f (s) , s) > a 2 .

(2.1)

Now, we consider the following subset D of C defined by D = {x ΠC : ra (x, f (s)) >

a 2} . As

Œ D , so D π 0/ . We claim that D Œ A . As D à C , then we get ra (x, s)

> a 2 for every x ΠD . From the monotonicity of f we obtain ra (f (x) , f (s)) > a 2 ,

for every x Œ D . Then, we have f (D ) à D . Let E be a nonempty subset of D and

m = sup ra (E ) . As E à D , so E à C and m Œ C . By our definition of D we deduce

that f (s) is a ra -supper bound of E . Then, we get ra (m, f (s)) > a 2 . Thus, we obtain m Œ D . Therefore, we get D Œ A . As D à C and C is the least nonempty element of A for the crisp inclusion order relation, then we get C = D . On the other hand, we know that s = sup ra (C ) Œ C . So, s Œ D . Hence, we obtain ra (s, f (s)) > a 2 .

(2.2)

ra (s, f (s)) + ra (f (s) , s) > a .

(2.3)

Combining (2.1) and (2.2), we get

Then, by using (2.3) and the antisymmetry of the a -fuzzy pseudo-order ra , we deduce that we have f (s) = s . Thus, s Œ Fix (f ) . Therefore, Fix (f ) π 0/ .

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Abdelkader Stouti and Lemnaouar Zedam

Second step. Fix (f ) has a least element in (X , ra ) . Indeed, from the first step

above, we know that Fix (f ) π 0/ . Next, we shall show that Fix (f ) has a ra -least element. Now, we consider the following subset S of X defined by S = {x Œ X : ra (x, z ) > a 2 for every z Œ Fix (f )} .

As

= min ra ( X ) , then

ΠS . So,

= min ra (S ) . Let L be a nonempty subset of

S . Hence, by our hypothesis the subset L has a ra -supremum, t , say. As L Ã S , so

every element z of Fix (f ) is a ra -upper bund of L . Then, we have ra ( t, z ) > a 2 for

every z ΠFix (f ) . So, t ΠS . Thus, we have sup ra (L ) ΠS . Hence, sup ra (L) is the ra -supremum of L in the a -fuzzy pseudo-ordered set (S , ra ) .

Now, let x ΠS . Then by our definition of S , we get ra (x, z ) > a 2 for every

z ΠFix (f ) .

Hence, by the monotonicity of f we get ra (f (x) , z ) > a 2 for every

z Œ Fix (f ) . Thus we have, f (x ) Œ S for every x Œ S . Then, f (S ) à S .

Therefore, all hypothesis of the first step above are satisfied for the map f/ S : S Æ S

defined by f/ S = f (x) for every x Œ S . Then, we get Fix (f/ S ) π 0/ . So, there exists at

least an element a ΠS such that f (a ) = a and ra (a, z ) > a 2 for every z ΠFix (f ) .

Thus, a ΠFix (f ) and a is a ra -lower bound of Fix (f ) . So, a is the ra -minimum of Fix (f ) .

As a corollary of Theorem 2.1, we get the following result. Corollary 2.2. Let ( X , ra ) be a nonempty ra -fuzzy ordered set with a least element . Assume that every nonempty subset of X has a ra -supremum. Then, the set of all

fixed points of every ra -fuzzy monotone map f : ( X , ra ) Æ ( X , ra ) is nonempty and has

a ra -least element. Next, by using Lemmas 1.5 and 1.6 and Theorem 2.1, we obtain the following dual result. Theorem 2.3. Let (X , ra ) be a nonempty ra -fuzzy pseudo-ordered set with a ra greatest element g . Assume that every nonempty subset of X has a ra -infimum. Then,

the set of all fixed points of every ra -fuzzy monotone map f : ( X , ra ) Æ ( X , ra ) is

nonempty and has a ra -greatest element. Proof. Let ( X , ra ) be a nonempty ra -fuzzy pseudo-ordered set with a ra -greatest

element g , let f : ( X , ra ) Æ ( X , ra ) be a ra -fuzzy monotone map and let sa be its inverse fuzzy relation. From Lemma 1.5, we know that sa is an a -fuzzy pseudo-order

relation on X . On the other hand, by Lemma 1.6, min sa ( X ) exists and we have

Fixed Points and Common Fixed Points in a -fuzzy Pseudo-ordered Sets

883

min sa (X ) = g . As by our hypothesis f : ( X , ra ) Æ ( X , ra ) is ra -fuzzy monotone, so

from Lemma 1.5 the map f : ( X , sa ) Æ ( X , sa ) is sa -fuzzy monotone.

Thus, all

hypothesis of Theorem 2.1 are satisfied. Therefore, the set Fix (f ) of all fixed points of

the map f is nonempty and has a sa -least element in (X , sa ) , m , say. Then, from Lemma 1.6, we get m = max (Fix (f )) = max (Fix (f )) . sa

ra

As a consequence of Theorem 2.3, we get the following. Corollary 2.4. Let (X , ra ) be a nonempty ra -fuzzy ordered set with a ra -greatest element g . Assume that every nonempty subset of X has a ra -infimum. Then, the set of all fixed points of every ra -fuzzy monotone map f : ( X , ra ) Æ ( X , ra ) is nonempty

and has a ra -greatest element. Combining Theorem 2.1 and 2.3, we obtain the following results. Corollary 2.5. Let ( X , ra ) be a nonempty complete ra -fuzzy trellis. Then, the set of

all fixed points of every ra -fuzzy monotone map f : ( X , ra ) Æ ( X , ra ) is nonempty and has a ra -greatest and a ra -least element.

Corollary 2.6. Let ( X , ra ) be a nonempty complete ra -fuzzy lattice. Then, the set of

all fixed points of every ra -fuzzy monotone map f : ( X , ra ) Æ ( X , ra ) is nonempty and has a ra -least and a ra -greatest element.

3. A fuzzy version of Skala’s fixed point theorem for complete trellises

In this section, we shall prove that the set of all fixed points of two classes of ra -fuzzy monotone maps defined on a nonempty ra -fuzzy compete trellis is also a nonempty ra fuzzy compete trellis. First, we shall show the following. Theorem 3.1. Let (X , ra ) be a nonempty complete ra -fuzzy trellis. Then, the set of

all fixed points of every ra -fuzzy monotone map f : ( X , ra ) Æ ( X , ra ) such that for every element x of X , we have ra (x, f (x)) > a 2 is also a nonempty complete ra -fuzzy trellis. Proof. Let ( X , ra ) be a nonempty complete ra -fuzzy trellis and f : ( X , ra ) Æ ( X , ra )

be a ra -fuzzy monotone map such that for every element x of X , we have

ra (x, f (x)) > a 2 . Then, by Corollary 2.5 we know that Fix (f ) is nonempty and has a ra -least and a ra -greatest element. Let A be a nonempty subset of Fix (f ) .

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Abdelkader Stouti and Lemnaouar Zedam

Claim 1. The ra -infimum of A in Fix (f ) belongs to Fix (f ) . Indeed, consider the

following subset D of Fix (f ) defined by

a Ï ¸ D = Ìx Œ Fix (f ) : ra (x, z ) > for every z Œ A ˝ . 2 Ó ˛ From Corollary 2.5, we know that the ra -fuzzy monotone map f has a ra -least fixed

point. So, D π 0/ . Let d = supra (D ) . Then, since every element z of A is a ra -upper

bound of D , so we get ra (d, z ) > a 2 for every z ΠA . Thus, d ΠD . Hence, by using the monotonicity of f , we obtain ra (f (d ) , z ) > a 2 for every z ΠA . So, we get f (d )

ΠD . Then, since d = supra (D ) we obtain ra (f (d ) , d ) > a 2 . On the other hand from

our hypothesis we know that ra (d, f (d )) > a 2 . So, we obtain ra (f(d ), d ) + ra (d,f (d ))

> a . Therefore by using the a -fuzzy antisymmetry of ra , we get f (d ) = d . Thus, d =

supra (D ) = inf ra (A ) ΠFix (f ) . Indeed, let y ΠFix (f ) such that y is a ra -lower bound

of A . Then, y ΠD . As d = supra (D ) , so ra (y, d ) > a 2 . Hence, d is a ra -upper

bound of the set of all ra -lower bounds of A in Fix (f ) . As d ΠFix (f ) , therefore the element d is the ra -infimum of A in Fix (f ) .

Claim 2. The ra -supremum of A in Fix (f ) belongs to Fix (f ) . Indeed, let E be

following subset of Fix (f ) defined by E ={x ΠFix (f ) : ra (z , x) > a 2 for every z ΠA} .

By Corollary 2.5, we know that f has a ra -greatest fixed point. Hence, E π 0/ . Let

e = inf ra (E ) . As e is a ra -lower bound of E , then we get ra (e, x) > a 2 for every x ΠE . Then, by using the a -fuzzy monotonicity of f , we deduce that we have ra (f (e ) , x) > a 2 for every x ΠE . Hence the element f (e ) is a ra -lower bound of

E . So as e = inf ra (E ) , then we get ra (f (e ) , e ) > a 2 . On the other hand as every

element z of A is a ra -lower bound of E , so ra (z , e ) > a 2 .

So, by using the

monotonicity of f , we obtain ra (z , f (e )) > a 2 for every z ΠA .

Hence, we get

f (e ) ΠE .

Thus, we have

Then, since e = inf ra (E ) we get ra (e, f (e )) > a 2 .

ra (e, f (e )) + ra (f (e ) , e ) > a . Therefore, by using the a -fuzzy antisymmetry of ra , we

obtain that f (e ) = e . Thus, we have e = inf ra (D ) = supra (A ) ΠFix (f ) . Indeed, let y ΠFix (f ) such that y is a ra -upper bound of A . Then, y ΠE . As e = inf ra (E ) , so ra (e, y ) > a 2 . Hence, e is a ra -lower bound of the ra -upper bounds of A in Fix (f ) .

As e ΠFix (f ) , therefore the element e is the ra -supremum of A in Fix (f ) .

Using Lemma 1.5 and 1.6 and Theorem 3.1, we get the following dual result.

Fixed Points and Common Fixed Points in a -fuzzy Pseudo-ordered Sets

885

Theorem 3.2. Let (X , ra ) be a nonempty complete ra -fuzzy trellis. Then, the set of

all fixed points of every ra -fuzzy monotone map f : ( X , ra ) Æ ( X , ra ) such that for every element x of X , we have ra (f (x) , x) > a 2 is also a nonempty complete ra -fuzzy trellis. Proof. Let (X , ra ) be a nonempty complete ra -fuzzy trellis, let f : ( X , ra ) Æ ( X , ra )

be a ra -fuzzy monotone map such that for every element x of X , we have

ra (f (x) , x) > a 2 and let sa be the inverse fuzzy relation of ra . From Lemma 1.5, we

know that sa is an a -fuzzy pseudo-order relation on X . On the other hand, by Lemma 1.6, we know that (X , sa ) is a complete sa -fuzzy trellis. Also, the map f is sa -fuzzy

monotone. On the other hand, by our hypothesis we know that for every x ΠX , we have ra (f (x) , x) > a 2 . So, we get sa (x, f (x)) > a 2 for every x ΠX . Thus, all hypothe-sis of Theorem 3.1 are satisfied for the sa -fuzzy monotone map

f : ( X , sa ) Æ ( X , sa ) . Therefore, the set of all fixed points Fix (f ) of the map f is a

nonempty complete sa -fuzzy trellis. Then from Lemma 1.6, we deduce that Fix (f ) is a nonempty complete ra -fuzzy trellis.

4. A tarski type common fixed point theorem for fuzzy compelte trellises

In this section, we shall establish that the set of all common fixed points of a finite commutative family of ra -fuzzy monotone maps defined on two classes of nonempty ra -fuzzy compete trellises is also a nonempty ra -fuzzy compete trellises. First, we shall prove the following result. Theorem 4.1. Let ( X , ra ) be a nonempty complete ra -fuzzy trellis and F be a finite

commutative family of ra -fuzzy monotone maps f : ( X , ra ) Æ ( X , ra ) such that for every

element x of X , we have ra (x, f (x)) > a 2 . Then, the set of all common fixed points Fix ( F ) of F is a nonempty complete ra -fuzzy trellis.

Proof. Let (X , ra ) be a nonempty complete ra -fuzzy trellis and let F = {f1 ,

, fn }

be a finite commutative family of ra -fuzzy monotone maps f : ( X , ra ) Æ ( X , ra ) such

that for every element x of X , we have ra (x, f (x)) > a 2 . Let g = max ra ( X ) . Then, fi ( g ) = g for every i = 1,

, n . Hence, Fix ( F ) is nonempty and has a ra -greatest

element. From Corollary 2.5, we know that the set Fix (f1 ) is a nonempty complete ra -

fuzzy trellis. Now, we shall show that the set of all common fixed points of the family {f1 , f2 } is also a nonempty complete ra -fuzzy trellis. Indeed, if x ΠFix (f1 ) , then f2 (x) = f2 (f1 (x)) = f1 (f2 (x)) . So, for every x ΠFix (f1 ) we have f2 (x) ΠFix (f1 ) .

886

Abdelkader Stouti and Lemnaouar Zedam

Then, f2 (Fix (f1 )) Ã Fix (f1 ) . Hence as Fix (f1 ) is a nonempty complete ra -fuzzy

trellis and ra (x, f (x)) > a 2 for every x ΠFix (f1 ) , so by Corollary 2.5, the set of all

fixed points of f2 in Fix (f1 ) is a nonempty complete ra -fuzzy trellis. On the other hand, the set of all fixed points of f2 in Fix (f1 ) is equal to Fix ({f1 , f2 }) . Thus, the set of all

common fixed points of the family {f1 , f2 } is a nonempty complete ra -fuzzy trellis. Now,

by induction assume that for every kΠ{1, the family {f1 ,

, n - 1} the set of all common fixed points of

, fk } is a nonempty complete ra -fuzzy trellis. Let x ŒFix({f1, ,fn -1 }) .

Then, fn (x) = fn (fk (x)) = fk (fn (x)) for every k Π{1,

for every k Π{1,

(

, n - 1} . Thus, fn Fix ({f1 ,

hypothesis of induction Fix ({f1 ,

, fn -1 }) Ã Fix ({f1 ,

, fn -1 }) . As by the

, fn -1 }) is a nonempty complete ra -fuzzy trellis and

ra (x, f (x)) > a 2 for every x ΠFix ({f1 ,

that the set Fix ({f1 ,

)

, n - 1} . So, fn (x ) ΠFix (fk )

, fn -1 }) ,then from Corollary 2.5 we deduce

, fn -1 }) is a nonempty complete ra -fuzzy trellis.

Next, by using Theorem 4.1 and Lemma 1.5 and 1.6, we obtain the following dual result. Theorem 4.2. Let ( X , ra ) be a nonempty complete ra -fuzzy trellis and F be a finite

commutative family of ra -fuzzy monotone maps f : ( X , ra ) Æ ( X , ra ) such that for every

element x of X , we have ra (f (x) , x) > a 2 . Then, the set of all common fixed points Fix ( F ) of F is a nonempty complete ra -fuzzy trellis.

Proof.

Let

(X , ra )

be a nonempty complete ra -fuzzy trellis, F be a finite

commutative family of monotone maps f : ( X , ra ) Æ ( X , ra ) such that for every element x of X , we have ra (f (x) , x) > a 2 and let sa be the fuzzy inverse relation of ra .

From Lemma 1.5, we know that sa is an a -fuzzy pseudo-order relation on X . On the other hand, by Lemma 1.6 the a -fuzzy pseudo-ordered relation on X . On the other hand, by Lemma 1.6 the a -fuzzy pseudo-ordered set (X , sa ) is a nonempty complete

ra -fuzzy trellis. Let f Œ F . As by our hypothesis f : ( X , ra ) Æ ( X , ra ) is a ra -fuzzy

monotone map, so from Lemma 1.6 the map f : ( X , sa ) Æ ( X , sa ) is sa -fuzzy Since ra (f (x) , x) > a 2 for every x Œ X and f Œ F , so we get

monotone.

sa (x, f (x)) > a 2 for every x ΠX and f ΠF . Therefore, from Theorem 4.1 we know

(

that the set Fix ({f1 ,

Lemma 1.6 we deduce that trellis.

) is a nonempty s -fuzzy complete trellis. Hence, by (Fix ({f , , f }) , r ) is a nonempty complete r -fuzzy

, fn }) , sa

a

1

n

a

a

Fixed Points and Common Fixed Points in a -fuzzy Pseudo-ordered Sets

887

5. Appendix

In this section, we shall give the proofs of Lemma 1.5 and 1.6. Proof of Lemma 1.5. Let ra be an a -fuzzy pseudo-order defined on a nonempty set X and let sa be its inverse fuzzy relation.

(i) a -fuzzy reflexivity. Let x ΠX . Then, sa (x, x) = ra (x, x) = a . So, sa is a -

fuzzy reflexive. (ii) a -fuzzy antisymmetry. Let x, y Œ X such that x π y . Since ra is a -fuzzy antisymmetric, then ra (x, y ) + ra (y, x ) ≥ a .

On the other hand, we know that

sa (x, y ) + sa (y, x) = ra (y, x) + ra (x, y ) . Hence, we get sa (x, y ) + sa (y, x) ≥ a . Thus,

the fuzzy relation sa is a -fuzzy antisymmetric. Proof of Lemma 1.6. Let ra be an a -fuzzy pseudo-order defined on a nonempty set X , let sa be its inverse fuzzy relation and let A be a nonempty subset of X .

(i) Assume that supra (A ) exists. Set s = supra (A ) . Then, for every x ΠA we have

ra (x, s) > a 2 . So, we get sa (s, x) > a 2 for every x ΠA . Thus, s is a sa -lower

bound of A . Let

be an another sa -lower bound of A . So, we have sa

for every x ΠA . Hence, ra (x, s = supra (A ) , so ra (s,

) >a

)>a

2 . Then,

(

, x) > a 2

is a ra -upper bound of A . As

( , s) > a 2 . Thus, s is the greatest sa -lower bound of A . Therefore, we get s = infs (A ) . (ii) Assume that inf r (A ) exists. Let = inf r (A ) and let x be a given element of A . Then, ra ( , x) > a 2 . So, we get sa (x, ) > a 2 for every x ΠA . Thus, is a 2 . Hence, we get sa a

a

a

sa -upper bound of A . Let m be an another sa -upper bound of A . So, we get

sa (x, m ) > a 2 for every x ΠA . Then, ra (m, x) > a 2 . Hence, m is a ra -lower

bound of A . Since Thus,

= inf ra (A ) , then ra (m,

)>a

2 . Hence, we get sa

is the least sa -upper bound of A . Thus, we have

(

, m) > a 2 .

= supsa (A ) .

(iii) Let m = min ra (A ) . Then, m = inf ra (A ) and m ΠA . From (ii) above, we get

m = supsa (A ) . As m ΠA , hence we deduce that m = max sa (A ) .

(iv) Let s = max ra (A ) .

So, s = supra (A ) and s ΠA .

From (ii) above, we get

s = infsa (A ) . As s ΠA , hence we obtain s = min sa (A ) .

(v) Let f : ( X , ra ) Æ ( X , ra ) be a ra -fuzzy monotone map. Let x, y Œ X such that

sa (x, y ) > a 2 . So, we get ra (y, x) > a 2 . As f is ra -fuzzy monotone, hence we

obtain ra (f (y ) , f (x)) > a 2 . So, we deduce that we have sa (f (x) , f (y )) > a 2 . Thus, f is sa -fuzzy monotone.

888

Abdelkader Stouti and Lemnaouar Zedam

References [1] H. L. Skala, Trellis theory, Algebra Universalis, 1 (1971), 218-233. [2] H. L. Skala, Trellis theory, Mem. Amer. Math. Soc., 121 Providence (1972). [3] A. Stouti, Fixed point theory for fuzzy monotone multifunctions, J. Fuzzy Math., 11 (2) (2003), 455466. [4] A. Stouti and L. Zedam, On a -fuzzy orders, J. Fuzzy Math., 18 (1) (2010), 171-192. [5] L. A. Zadeh, Similarity relations and fuzzy orderings, Info. Sci., 3 (1971), 177-200. [6] H. J. Zimmermann, Fuzzy set theory and its applications, Kluwer academic publisher, Dordrecht, (1991).

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