(1)α-FUZZY FIXED POINTS FOR α-FUZZY MONOTONE MULTIFUNCTIONS A

α

α

A. STOUTI

Abstract. In this note, we prove the existence of maximal, minimal, greatest and leastα-fuzzy fixed points forα-fuzzy monotone multifunctions.

1. Introduction

LetX be a nonempty set. A fuzzy subsetAofX is a function ofX into [0,1] (see [14]). A fuzzy multifunction is a mapT :X →[0,1]^X such that for everyx∈X, T(x) is a nonempty fuzzy set. Let α∈]0,1] and let T :X →[0,1]^X be a fuzzy multifunction. We say that an element x ofX is an α-fuzzy fixed point of T if T(x)(x) =α.When α= 1,the elementx is called a fixed point ofT.

During the last few decades several authors established fixed points theorems in fuzzy setting, see for example [1] – [12]. Recently, in [9], we introduced the notion ofα-fuzzy ordered sets in which we established some fixed points theorems for fuzzy monotone multifunctions.

The aim of this note is to study the existence ofα-fuzzy fixed points forα-fuzzy monotone multifunctions. First, we prove the existence of maximal and minimal α-fuzzy fixed points (see Theorems 3.1 and 3.3). Second, we establish the existence of greatest and leastα-fuzzy fixed points (see Theorems 4.1 and 4.2).

2. Preliminaries First, we recall the definition ofα-fuzzy order.

Definition 2.1. [9] LetX be a nonempty set andα∈]0,1].Anα-fuzzy order onX is a fuzzy subsetrα ofX×X satisfying the following three properties:

(i) for allx∈X, rα(x, x) =α,(α-fuzzy reflexivity);

(ii) for allx, y∈X, rα(x, y) +rα(y, x)> αimpliesx=y.(α-fuzzy antisymme- try);

(iii) for allx, z∈X, rα(x, z)≥supy∈X[min{rα(x, y), rα(y, z)}] (α-fuzzy transi- tivity).

Received February 4, 2004.

2000Mathematics Subject Classification. Primary 04A72, 03E72, 06A06, 47H10.

Key words and phrases. Fuzzy set, α-fuzzy order relation, monotone multifunction, fixed point.

The pair (X, rα),whererαis aα-fuzzy order onX is called arα-fuzzy ordered set. An α-fuzzy order rα is said to be total if for all x 6= y we have either rα(x, y)> ^α₂ or rα(y, x)> ^α₂. Arα-fuzzy ordered setX on which the orderrα is total is calledrα-fuzzy chain.

Let (X, rα) be a nonemptyrα-fuzzy ordered set and Abe a subset ofX.

An elementuofX is said to be arα-upper bound ofAifrα(x, u)> ^α₂ for all x∈A.

Ifx is a rα-upper bound of Aand x ∈A, then it is called a greatest element ofA.

An elementmofAis called a maximal element ofAif there isx∈Asuch that rα(m, x)> ^α₂,thenx=m.

An element l ofX is said to be a rα-lower bound of A if rα(l, x)> ^α₂ for all x∈A.

Ifl is arα-lower bound ofAandl∈A,then it is called the least element ofA.

An elementnofAis called a minimal element ofAif there isx∈Asuch that rα(x, n)>^α₂,thenx=n.As usual,

sup_rα(A) := the least element ofrα-upper bounds ofA(if it exists), infrα(A) := the greatest element ofrα-lower bounds ofA(if it exists), maxrα(A) := the greatest element ofA(if it exists),

minrα(A) := the least element ofA(if it exists).

Next, we shall give four examples ofα-fuzzy orders.

Examples.

1. LetX ={0,1,2}and rα be theα-fuzzy order relation defined on X by:

rα(0,0) =rα(1,1) =rα(2,2) =α, rα(0,2) = 0.55α

rα(2,0) = 0.1α

rα(2,1) = 0.2α rα(1,2) = 0.6α

rα(1,0) = 0.7α rα(0,1) = 0.15α.

As properties ofrα,we have infrα(X) = 0 and sup_rα(X) = 2.

2. Consider theα-fuzzy order relationrα defined onX ={0,1,2}by:

rα(0,0) =rα(1,1) =rα(2,2) =α, rα(0,2) = 0.6α

rα(2,0) = 0.2α

rα(2,1) = 0.2α rα(1,2) = 0.3α

rα(1,0) = 0.3α rα(0,1) = 0.55α.

In this case, we have infrα(X) = 0 and suprα(X) do not exist inX.Note that 1 and 2 are two maximal elements in (X, rα).

3. Letrα be theα-fuzzy order defined on X={0,1,2}by:

rα(0,0) =rα(1,1) =rα(2,2) =α, rα(0,2) = 0.65α

rα(2,0) = 0.15α

rα(2,1) = 0.1α rα(1,2) = 0.7α

rα(1,0) = 0.15α rα(0,1) = 0.10α.

Then, sup_rα(X) = 2 and infrα(X) do not exist inX.In addition, 1 and 0 are two minimal elements in (X, rα).

4. Letrα be theα-fuzzy order defined on X={0,1,2}by:

rα(0,0) =rα(1,1) =rα(2,2) =α, rα(0,2) = 0.8α

rα(2,0) = 0.15α

rα(2,1) = 0.20α rα(1,2) = 0.30α

rα(1,0) = 0.30α rα(0,1) = 0.20α.

In this case, infrα(X) and sup_rα(X) do not exist inX.Also, 1 is a maximal and minimal element of (X, rα).

Next, we recall some definitions and results for subsequent use.

Definition 2.2. [9] Let (X, rα) be a nonempty rα-fuzzy ordered set. The inverseα-fuzzy relationsαofrαis defined bysα(x, y) =rα(y, x),for allx, y∈X.

Let us not that by [9, Proposition 3.5], ifrαis anα-fuzzy order, thensαis also anα-fuzzy order.

In [10], we proved the following lemma.

Lemma 2.3. Let(X, rα) be a rα-fuzzy order set and sα be the inverse fuzzy order relation ofrα.Then,

(i) If a nonempty subsetAofX has arα-supremum, thenAhas asα-infimum andinfsα(A) = suprα(A).

(ii) If a nonempty subsetAofX has arα-infimum, thenAhas asα-supremum andinfrα(A) = supsα(A).

The followingα-fuzzy Zorn’s Lemma is given in [9].

Lemma 2.4. Let(X, rα)be a nonemptyα-fuzzy ordered sets. If every nonemty rα-fuzzy chain inX has a rα-upper bound, then X has a maximal element.

LetT :X →[0,1]^X be a fuzzy multifunction. Then, for everyx∈X,we define the following subset ofX by setting:

T_x^α={y∈X:T(x)(y) =α}.

In this note, we shall use the following definition ofα-fuzzy monotonicity.

Definition 2.5. Let (X, rα) be a nonempty rα-fuzzy ordered set. A fuzzy multifunctionT :X →[0,1]^X is said to berα-fuzzy monotone if the two following properties are satisfied:

(i) for allx∈X, T_x^α6=∅;

(ii) ifrα(x, y)> ^α₂ andx6=y,forx, y∈X,then for all a∈T_x^αandb∈T_y^α, we haverα(a, b)>^α₂.

We denote byF_T^α the set of allα-fuzzy fixed points ofT. 3. Maximal and minimalα-fuzzy fixed points

In this section, we investigate the existence of maximal and minimalα-fuzzy fixed points ofα-fuzzy monotone multifunctions. First, we shall show the following:

Theorem 3.1.Let(X, rα)be anα-fuzzy ordered set with the property that every nonempty rα-fuzzy chain in (X, rα) has arα-supremum. LetT :X →[0,1]^X be arα-fuzzy monotone multifunction. If there exist a, b∈X such that T(a)(b) =α andrα(a, b)> ^α₂,then the setF_T^α of allα-fuzzy fixed points ofT is nonempty and has a maximal element.

Proof. LetHαbe the fuzzy ordered subset ofX defined by Hα=n

x∈X : there existsy∈X, T(x)(y) =αandrα(x, y)>α 2 o

. Sincea∈Hα,then the subsetHαis nonempty.

Claim 1. The subset Hα has a maximal element. Indeed, if C is a nonempty rα-fuzzy chain in Hα and s = suprα(C), then we distinguish the following two cases.

First case: s∈C,thens∈Hα.

Second case: s 6∈ C. Then, for every c ∈ C, rα(c, s) > ^α₂ and c 6= s. By our definitionT_s^α6=∅.Then, there existsz∈X such thatT(s)(z) =α.Sincec∈Hα, there exists d ∈ X such that T(c)(d) = α and rα(c, d) > ^α₂. As T is rα-fuzzy monotone, we getrα(d, z) > ^α₂. Byα-fuzzy transitivity, we obtainrα(c, z)> ^α₂. As c is a general element of C, then z is a rα-upper bound of C. On the other hand, we know thats= sup_rα(C).Hence,rα(s, z)> ^α₂.From this we deduce that s∈Hα. Therefore every nonemtyrα-fuzzy chain in Hα has a rα-upper bound in Hα.By Lemma 2.4,Hα has a maximal element, saym.

Claim 2. The elementmis a maximalα-fuzzy fixed point ofT.Indeed, byClaim 1, m∈Hα.Hence, there existsy∈X such thatT(m)(y) =αandrα(m, y)> ^α₂.On the other hand, by our hypothesis,T_y^α6=∅.Therefore, there existst∈X such that T(y)(t) =α. Fromrα-fuzzy monotonicity ofT we get rα(y, t)> ^α₂. So, y ∈Hα. ByClaim 1, m is a maximal element of Hα. From this and since T(m)(y) =α, rα(y, m) > ^α₂ and y ∈ Hα, we deduce that we have y = m. So, T(m)(m) = α.

Thus,m∈ F_T^α. Now, letx∈ F_T^α.Then,x∈Hα.So,F_T^α⊆Hα.Asm∈ F_T^α,then

mis a maximal element ofF_T^α.

In order to establish the existence of a minimalα-fuzzy fixed, we shall need the following lemma:

Lemma 3.2. Let(X, rα) be a rα-fuzzy order set and sα be the inverse fuzzy relation ofrα.Then, everyrα-fuzzy monotone multifunction is alsosα-fuzzy mono- tone.

Proof. Let T : X → [0,1]^X be a rα-fuzzy monotone multifunction. Now, let x, y∈X such thatx6=yandsα(x, y)> ^α₂.Then, we haverα(y, x)>^α₂.SinceT is rα-fuzzy monotone, then for alla, b∈X such thatT(x)(a) =αandT(y)(b) =α, we getrα(b, a)>^α₂.Therfore, we obtainsα(a, b)>^α₂. By using Lemmas 2.3 and 3.2 and Theorem 3.1, we obtain the following result.

Theorem 3.3. Let(X, rα)be arα-fuzzy ordered set with the property that every nonempty rα-fuzzy chain has a rα-infimum. Let T : X → [0,1]^X be a rα-fuzzy monotone multifunction. Assume that there exist a, b∈X such that T(a)(b) =α

andrα(b, a)> ^α₂. Then, the setF_T^α of all α-fuzzy fixed points of T is nonempty and has a minimal element.

Proof. Letsαbe the inverse fuzzy order relation ofrα.From Lemma 2.3, every nonemptysα-fuzzy chain has asα-supremum. On the other hand, by Lemma 3.2, we know thatTissα-fuzzy monotone. From this andsα(a, b)>^α₂,by Theorem 3.1, we deduce thatT has a maximalα-fuzzy fixed point,lsay, in (X, sα).Letx∈ F_T^α such that rα(x, l) > ^α₂. Then, sα(l, x) > ^α₂. Since l is a maximal α-fuzzy fixed point ofT in (X, sα),thenl =x.Therefore,l is a minimalα-fuzzy fixed point of

T in (X, rα).

4. Greatest and leastα-fuzzy fixed points

In this section, we shall establish the existence of the greatest and the leastα-fuzzy forα-fuzzy monotone multifunctions. First, we shall prove the following:

Theorem 4.1. Let(X, rα)be arα-fuzzy ordered set with the property that every nonempty fuzzy ordered subset of X has a rα-supremum. Let T :X →[0,1]^X be arα-fuzzy monotone multifunction. If there exist a, b∈X such that T(a)(b) =α andrα(a, b)>^α₂,then T has the greatest α-fuzzy fixed point. Moreover, we have

max(F_T^α) = sup

rα

n

x∈X : there existsy∈X, T(x)(y) =αandrα(x, y)> α 2 o

.

Proof. LetPαbe the fuzzy ordered subset defined by Pα=n

x∈X : there existsy∈X, T(x)(y) =αandrα(x, y)> α 2 o

. Asa∈Pα,then the subsetPα is nonempty. Letg= sup_rα(Pα).

Claim 1. We have: g ∈ Pα. Indeed, assume on the contrary that g 6∈ Pα. Then for all x ∈ Pα, we have x 6= g. As by our definition T_g^α 6= ∅, then there exists z ∈ T_g^α. Let x ∈ Pα. Hence, there exists y ∈ T_x^α such that rα(x, y) > ^α₂. From α-fuzzy monotonicity ofT,we obtainrα(y, z)> ^α₂.Byα-fuzzy transitivity, we get rα(x, z)>^α₂.Asx is a general element ofPα,sozis arα-upper bound ofPα.On the other hand; by our hypothesis; we haveg = sup_rα(Pα). Then, rα(g, z)> ^α₂. Thus,g∈Pα.That is a contradiction, and our claim is proved.

Claim 2. We have:

z∈X:T(g)(z) =αandrα(g, z)>^α₂ = {g}. By absurd, suppose that there exists z ∈ T_g^α such that rα(g, z) > ^α₂ and z 6= g. As T is rα-fuzzy monotone andT_z^α 6=∅,then there exists l∈T_z^α such thatrα(z, l)> ^α₂. Therefore,z ∈ P and rα(z, g)> ^α₂. Hence, we get rα(z, g) +rα(g, z)> α. From this andα-fuzzy antisymmetry, we obtaing=z.That is a contradiction with the fact thatz6=g and our Claim is proved.

Claim 3. The elementgis the greatestα-fuzzy fixed point ofT.Indeed, asg∈Pα, then there existsz∈T_g^αsuch thatrα(g, z)>^α₂.Then byClaim 2, we deduce that z=g andg is aα-fuzzy fixed point ofT. On the other hand, letx be anα-fuzzy fixed point ofT.Sox∈Pα.Thus,F_T^α⊆Pα.Hence,gis arα-upper bound ofF_T^α. Asg∈ F_T^α, therefore,g is the greatest element of F_T^α.

Combining Lemmas 2.3 and 3.2 and Theorem 4.1, we get the following:

Theorem 4.2. Let(X, rα)be arα-fuzzy ordered set with the property that every nonempty fuzzy ordered subset ofXhas arα-infimum. LetT :X →[0,1]^Xbe arα- fuzzy monotone multifunction. Assume that there isa, b∈Xsuch thatT(a)(b) =α andrα(b, a)>^α₂.Then, T has a least α-fuzzy fixed point. Furthermore, we have

min(F_T^α) = inf

rα

n

x∈X : there existsy∈X, T(x)(y) =αandrα(y, x)>α 2 o

. Proof. Let sα be the inverse α-fuzzy order of rα. From Lemma 2.3, every nonempty fuzzy ordered subset ofX has an infimum in (X, sα). By Lemma 3.2, T is sα-fuzzy monotone. Since rα(b, a) > ^α₂, then sα(a, b) > ^α₂. From this and by Theorem 4.1 we deduce that the fuzzy multifunctionT has a greatestα-fuzzy fixed point in (X, sα), m,say. Therefore,mis the leastα-fuzzy fixed point ofT in (X, rα).Sincemis the greatestα-fuzzy fixed ofT in (X, sα),then by Theorem 4.1, we have

m= sup

sα

n

x∈X : there existsy∈X, T(x)(y) =αandsα(x, y)> α 2 o

. Therefore, by Lemma 2.3, we conclude that

m= inf

rα

n

x∈X : there existsy∈X, T(x)(y) =αandrα(y, x)> α 2 o

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A. Stouti, Unit´e de Recherche: Math´ematiques et Applications, Universit´e Cadi Ayyad, Facult´e des Sciences et Techniques de Beni-Mellal, B.P. 523, 23000 Beni-Mellal, Morocco,

e-mail:[email protected]