FOR QUADRATIC FUZZY EQUATIONS

FOR QUADRATIC FUZZY EQUATIONS

JUAN J. NIETO AND ROSANA RODR´IGUEZ-L ´OPEZ Received 8 November 2004 and in revised form 8 March 2005

Some results on the existence of solution for certain fuzzy equations are revised and extended. In this paper, we establish the existence of a solution for the fuzzy equation Ex²+Fx+G=x, whereE,F,G, andxare positive fuzzy numbers satisfying certain con- ditions. To this purpose, we use fixed point theory, applying results such as the well- known fixed point theorem of Tarski, presenting some results regarding the existence of extremal solutions to the above equation.

1. Preliminaries

In [1], it is studied the existence of extremal fixed points for a map defined in a subset of the setE¹of fuzzy real numbers, that is, the family of elementsx:R→[0, 1] with the properties:

(i)xis normal: there existst0∈Rwithx(t0)=1.

(ii)xis upper semicontinuous.

(iii)xis fuzzy convex, xλt1+ (1−λ)t2

≥minxt1

,xt2

, ∀t1,t2∈R,λ∈[0, 1]. (1.1) (iv) The support ofx, supp(x)=cl({t∈R:x(t)>0}) is a bounded subset ofR. In the following, for a fuzzy numberx∈E¹, we denote theα-level set

[x]^α=

t∈R:x(t)≥α (1.2)

by the interval [x_αl,x_αr], for eachα∈(0, 1], and [x]⁰=cl∪_α∈(0,1][x]^α=

x0l,x0r

. (1.3)

Note that this notation is possible, since the properties of the fuzzy numberxguarantee that [x]^αis a nonempty compact convex subset ofR, for eachα∈[0, 1].

Copyright©2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:3 (2005) 321–342 DOI:10.1155/FPTA.2005.321

We consider the partial ordering≤inE¹given by

x,y∈E¹, x≤y⇐⇒x_αl≤y_αl, x_αr≤y_αr, ∀α∈(0, 1], (1.4) and the distance that providesE¹the structure of complete metric space is given by

d∞(x,y)= sup

α∈[0,1]

d_H[x]^α, [y]^α, forx,y∈E¹, (1.5) beingdH the Hausdorffdistance between nonempty compact convex subsets ofR(that is, compact intervals).

For each fuzzy numberx∈E¹, we define the functionsxL: [0, 1]→R,xR: [0, 1]→R given byxL(α)=xαlandxR(α)=xαr, for eachα∈[0, 1].

Theorem1.1 [1, Theorem 2.3]. Letu0,v0∈E¹,u0< v0. Let B⊂

u0,v0

=

x∈E¹:u0≤x≤v0

(1.6)

be a closed set ofE¹such thatu0,v0∈B. Suppose thatA:B→Bis an increasing operator such that

u0≤Au0, Av0≤v0, (1.7)

andAis condensing, that is,Ais continuous, bounded andr(A(S))< r(S)for any bounded setS⊂Bwithr(S)>0, wherer(S)denotes the measure of noncompactness ofS. ThenAhas a maximal fixed pointx^∗and a minimal fixed pointx∗inB, moreover

x^∗= lim

n−→+∞vn, x∗= lim

n−→+∞un, (1.8)

wherevn=Avn−1andun=Aun−1,n=1, 2,...and

u0≤u1≤ ··· ≤un≤ ··· ≤vn≤ ··· ≤v1≤v0. (1.9) Corollary1.2 [1, Corollary 2.4]. In the hypotheses ofTheorem 1.1, ifAhas a unique fixed pointx¯inB, then, for anyx0∈B, the successive iterates

xn=Axn−1, n=1, 2,... (1.10)

converge tox, that is,¯ d∞(x_n, ¯x)→0asn→+∞.

Theorem 1.1is used in [1] to solve the fuzzy equation

Ex²+Fx+G=x, (1.11)

whereE,F,Gandxare positive fuzzy numbers satisfying some additional conditions. In this direction, consider the class of fuzzy numbersx∈E¹satisfying

(i)x >0,xL(α),xR(α)≤1/6, for eachα∈[0, 1].

(ii)|xL(α)−xL(β)|<(M/6)|α−β| and |xR(α)−xR(β)|<(M/6)|α−β|, for every α,β∈[0, 1].

Denote this class byᏲ.

Theorem1.3 [1, Theorem 2.9]. LetM >0 be a real number. Suppose thatE,F,G∈Ᏺ.

Then (1.11) has a solution in BM=

x∈E¹: 0≤x≤1,|xL(α)−xL(β)| ≤M|α−β|,

|xR(α)−xR(β)| ≤M|α−β|,∀α,β∈[0, 1]. (1.12) Here, 0,1 referred to fuzzy numbers represent, respectively, the characteristic functions of 0 and 1, that is,χ{0_}andχ{1_}.

In the proof ofTheorem 1.3, in addition toTheorem 1.1, the following results are used.

Theorem1.4 [1, Theorem 2.6]. For each fuzzy numberx, functions

x_L: [0, 1]−→R, x_R: [0, 1]−→R (1.13) are continuous.

Theorem1.5 [1, Theorem 2.7]. Suppose thatxandyare fuzzy numbers, then d∞(x,y)=maxx_L−y_L∞,x_R−y_R∞

. (1.14)

Theorem1.6 [1, Theorem 2.8]. BMis a closed subset ofE¹. Lemma1.7 [1, Lemma 2.10]. Suppose thatB⊂E¹. If

BL=

xL:x∈B, BR=

xR:x∈B (1.15)

are compact in(C[0, 1], · ∞), thenBis a compact set inE¹.

InSection 2, we point out some considerations about the previous results and justify the validity of the proof ofTheorem 1.3given in [1], presenting a more general existence result. Then, inSection 3, we study the existence of solution to (1.11) by using some fixed point theorems such as Tarski’s fixed point theorem, proving the existence of extremal solutions to (1.11) under less restrictive hypotheses.

2. Revision and extension of results in[1]

First of all,Theorem 1.4[1, Theorem 2.6] is not valid. Indeed, take for example,x:R→ [0, 1] defined as

t∈R−→x(t)=











 1

2, t∈[−1, 0)∪(0, 1], 1, t=0,

0, otherwise,

(2.1)

which represents [2, Proposition 6.1.7] and [3, Theorem 1.5.1] a fuzzy real number since the level sets ofxare the nonempty compact convex sets

[x]^α=









[−1, 1], if 0≤α≤1 2, {0}, if1

2< α≤1.

(2.2)

Then,x_L: [0, 1]→Ris given by

x_L(α)=









−1, if 0≤α≤1 2, 0, if1

2< α≤1,

(2.3)

andxR: [0, 1]→Ris

xR(α)=









1, if 0≤α≤1 2, 0, if1

2< α≤1,

(2.4)

which are clearly discontinuous. Note thatx_Landx_R are left-continuous see [3, Theo- rem 1.5.1] and [2, Propositions 6.1.6 and 6.1.7]. In the proof ofTheorem 1.4[1, The- orem 2.6], it is considered a sequenceαn> αwithαn→αasn→+∞. ThenxL(αn) is a nonincreasing and bounded sequence, hence,x_L(α_n) converges to a numberL. At this point, one cannot affirm thatx(L)≤αn. For example, in the previous case, takingα=1/2 andαn=1/2 + 1/n, with n >2, thenxL(αn)=0. HencexL(αn) converges toL=0, but x(L)=x(0)=1> α_n=1/2 + 1/nfor alln >2.

A fuzzy number is not necessarily a continuous function, just upper semicontinuous, thusTheorem 1.4[1, Theorem 2.6] is not valid in the general context of fuzzy real num- bers. However, it is valid for continuous fuzzy numbers, that is, fuzzy numbers continu- ous in its membership grade, as we state below. Here᏷¹_Cdenotes the space of nonempty compact convex subsets ofRfurnished with the HausdorffmetricdH.

Definition 2.1. We say that a fuzzy numberx:R→[0, 1] is continuous if the function

[x]^·: [0, 1]−→᏷¹_C (2.5)

given byα→[x]^αis continuous on (0, 1], that is, for everyα∈(0, 1], and>0, there exists a numberδ(,α)>0 such thatdH

[x]^α, [x]^β<, for everyβ∈(α−δ,α+δ)∩[0, 1].

Theorem2.2. Letxbe a fuzzy number, thenxis continuous if and only if functions x_L: [0, 1]−→R, x_R: [0, 1]−→R (2.6) are continuous.

Proof. Suppose thatx∈E¹is continuous and letα∈(0, 1] and>0. Sincexis continu- ous atα, then there existsδ(,α)>0 such that for everyβ∈(α−δ,α+δ)∩[0, 1],

d_H[x]^α, [x]^β=max|x_αl−x_βl|,|x_αr−x_βr|

=max|xL(α)−xL(β)|,|xR(α)−xR(β)|

<, (2.7) which implies that

x_L(α)−x_L(β)<, x_R(α)−x_R(β)<, (2.8)

for everyβ∈(α−δ,α+δ)∩[0, 1], proving the continuity ofx_Landx_Ratα. Reciprocally, continuity ofx_Landx_Rtrivially implies the continuity ofx.

Remark 2.3. For a givenx∈E¹,x, [x]^·,x_Landx_Rare trivially continuous atα=0. Indeed, let>0. The 0-level set ofx(support ofx) is the closure of the union of all of the level sets, that is,

[x]⁰=cl∪β∈(0,1]

xL(β),xR(β). (2.9)

SincexL(β) is nondecreasing inβandxR(β) is nonincreasing inβand those values are bounded, then

x_L(0)= inf

β∈(0,1]x_L(β), x_R(0)= sup

β∈(0,1]

x_R(β). (2.10)

For>0, there existβ1,,β2,∈(0, 1], such that xL(0)≤xL

β1,

< xL(0) +, x_R(0)−< x_Rβ2,

≤x_R(0). (2.11)

By monotonicity,

xL(0)≤xL(β)≤xL β1,

< xL(0) +, for 0< β≤β1,, x_R(0)−< x_Rβ2,

≤x_R(β)≤x_R(0), for 0< β≤β2,. (2.12) Hence, takingδ=min{β1,,β2,}>0, we obtain

x_L(0)≤x_L(β)< x_L(0) +, x_R(0)−< x_R(β)≤x_R(0), (2.13) for every 0< β≤δ, and

dH

[x]⁰, [x]^β=maxxL(0)−xL(β),xR(0)−xR(β)<, ∀β∈[0,δ]. (2.14)

As a particular case of continuous fuzzy numbers, we present Lipschitzian fuzzy num- bers.

Definition 2.4. We say thatx∈E¹is a Lipschitzian fuzzy number if it is a Lipschitz func- tion of its membership grade, in the sense that

dH

[x]^α, [x]^β≤K|α−β|, (2.15) for everyα,β∈[0, 1] and some fixed, finite constantK≥0.

This property of fuzzy numbers is equivalent (see [2, page 43]) to the Lipschitzian character of the support functions_x(·,p) uniformly inp∈S⁰, where

sx(α,p)=sp, [x]^α=supp,a:a∈[x]^α, (α,p)∈[0, 1]×S⁰, (2.16) andS⁰is the unit sphere inR, that is, the set{−1, +1}.

If we consider a Lipschitzian fuzzy numberx, thenx is continuous and, in conse- quence, x_L and x_R are continuous functions. Moreover, we prove that these are Lips- chitzian functions.

Theorem2.5. Letx∈E¹. Thenxis a Lipschitzian fuzzy number, with Lipschitz constant K≥0, if and only ifxL: [0, 1]→RandxR: [0, 1]→RareK-Lipschitzian functions.

Proof. It is deduced from the identity dH

[x]^α, [x]^β=max|xαl−xβl|,|xαr−xβr|

=max|x_L(α)−x_L(β)|,|x_R(α)−x_R(β)|

, for everyα,β∈[0, 1].

(2.17) Note thatTheorem 1.5 [1, Theorem 2.7] is valid for · ∞ considered in the space L^∞[0, 1], but not in C[0, 1], since for an arbitrary fuzzy numberx,x_L andx_R are not necessarily continuous. Nevertheless, fromTheorem 2.2, we deduce that the distanced∞

can be characterized for continuous fuzzy numbers in terms of the sup norm inC[0, 1], and also for Lipschitzian fuzzy numbers.

Theorem2.6. Suppose thatxandyare continuous fuzzy numbers (in the sense ofDefinition 2.1), then

d_∞(x,y)=maxx_L−y_L∞,x_R−y_R∞

. (2.18)

Proof. Indeed,

d∞(x,y)= sup

α∈[0,1]

d_H[x]^α, [y]^α

= sup

α∈[0,1]

max|xL(α)−yL(α)|,|xR(α)−yR(α)|

=max

sup

α∈[0,1]

|xL(α)−yL(α)|, sup

α∈[0,1]

|xR(α)−yR(α)|

=maxxL−yL∞,xR−yR∞ .

(2.19)

ForM >0 fixed, consider the set

BM=

x∈E¹:χ{0_}≤x≤χ{1_},xisM-Lipschitzian. (2.20)

Note thatB_Mcoincides with the set with the same name defined inTheorem 1.3[1, Theo- rem 2.9] and thatB_M is a closed set inE¹. For the sake of completeness, we give here another proof. Letxna sequence inBM such that lim_n→+_∞xn=x∈E¹inE¹. We prove thatx∈BM. Given>0, there existsn0∈Nsuch that

d∞

x_n,x= sup

α∈[0,1]

d_H[x_n]^α, [x]^α<, forn≥n0. (2.21)

Then, forn≥n0, dH

[x]^α, [x]^β≤dH

[x]^α, [xn]^α+dH

[xn]^α, [xn]^β+dH

[xn]^β, [x]^β

<2+Mα−β, for everyα,β∈[0, 1]. (2.22) Since>0 is arbitrary, this means that

dH

[x]^α, [x]^β≤M|α−β|, for everyα,β∈[0, 1], (2.23)

andxisM-Lipschitzian. We can easily prove thatχ{0}≤x_n≤χ{1}, for allnimplies that χ{0_}≤x≤χ{1_}.Therefore,x∈BM.

ConcerningLemma 1.7[1, Lemma 2.10] we have to restrict our attention to relatively compact sets, since we are not considering closed sets. On the other hand, ifBcontains noncontinuous fuzzy numbers,BLandBRare not subsets ofC[0, 1]. We prove the corre- sponding result.

Lemma2.7. Suppose thatB⊂E¹consists of continuous fuzzy numbers, hence B_L=

x_L:x∈B, B_R=

x_R:x∈B (2.24)

are subsets ofC[0, 1]. IfBLandBR are relatively compact in(C[0, 1], · ∞), thenBis a relatively compact set inE¹.

Proof. Let{xn}n⊆B a sequence inB and I=[0, 1]. SinceBL is relatively compact in (C(I), · ∞), then{(x_n)_L}_nhas a subsequence{(x_nk)_L}_kconverging inC(I) tof1∈C(I).

Using thatBR is relatively compact in (C(I), · ∞), then{(xnk)_R}k has a subsequence {(xnl)_R}l converging inC(I) to f2∈C(I). We have to prove that {[f1(α),f2(α)] :α∈ [0, 1]}is the family of level sets of some fuzzy numberx∈E¹and, hence,x_L=f1,x_R=f2. Indeed, intervals [f1(α),f2(α)] are nonempty compact convex subsets ofR, since

xnl

L(α)≤ xnl

R(α), ∀α∈[0, 1],l∈N, (2.25) and, thus, passing to the limit asl→+∞,

f1(α)≤ f2(α), ∀α∈[0, 1]. (2.26)

Moreover, if 0≤α1≤α2≤1, x_nlLα1

≤ x_nlLα2

, x_nlRα1

≥ x_nlRα2

, ∀l, (2.27)

so that

f1

α1

≤ f1

α2

, f2

α1

≥f2

α2

, (2.28)

then

f1(α2),f2(α2)⊆

f1(α1),f2(α1). (2.29) Finally, letα >0 and{α_i} ↑α, then{[f1(α_i),f2(α_i)]}is a contractive sequence of compact intervals, and, by continuity of f1and f2,

∩_i≥1

f1(α_i),f2(α_i)=

αilim−→α⁻f1(α_i), lim

αi−→α⁻f2(α_i)=

f1(α),f2(α). (2.30) Applying [2, Proposition 6.1.7] or also [3, Theorem 1.5.1], there existsx∈E¹such that

[x]^α=

f1(α),f2(α), ∀α∈(0, 1], (2.31) and

[x]⁰=cl∪0<α≤1[f1(α),f2(α)]=

αlim−→0⁺f1(α), lim

α−→0⁺f2(α)=

f1(0),f2(0) (2.32) again by continuity of f1, f2. Note thatxL= f1 andxR= f2are continuous, thusxis a continuous fuzzy number and alsox_nlis, for everyl. Then, byTheorem 2.6,

d∞(x_nl,x)=max(x_nl)_L−f1∞,(x_nl)_R−f2∞ _l→+_∞

−−−−→0, (2.33)

and{x_nl}_l→xinE¹, completing the proof.

Recall equation (1.11)

Ex²+Fx+G=x. (2.34)

Here, the productx·yof two fuzzy numbersxandyis given by the Zadeh’s extension principle:

x·y:R−→[0, 1]

(x·y)(t)=sup

s·s=t

minx(s),y(s). (2.35) Note that [x·y]^α=[x]^α·[y]^α, for everyα∈[0, 1]. See [2, page 4] and [3, page 3].

In the following, we make reference to the canonical partial ordering≤onE¹as well as the orderdefined by

x,y∈E¹,xy⇐⇒[x]^α⊆[y]^α, ∀α∈(0, 1], (2.36) that is,

xαl≥yαl, xαr≤yαr, ∀α∈(0, 1]. (2.37) Remark 2.8. Note that, for a givenx∈E¹, it is not true in general that

x²≥χ{0}, x²χ{0}. (2.38) Indeed, forx=χ[−3,3],

χ[−3,3]

2

=χ[−9,9]≥χ{0}, (2.39) and, fory=χ[1,2], we obtain

χ[1,2]

2

=χ[1,4]χ{0}. (2.40)

The proof ofTheorem 1.3[1, Theorem 2.9] can be completed using the revised results.

In fact, the same proof is valid for a more general situation. Note that, ifG=χ{0_}, then x=χ{0}is a solution to (1.11).

Theorem2.9. LetM >0be a real number, andE,F,Gfuzzy numbers such that (i)E,F,G≥χ{0},d∞(E,χ{0})≤1/6,d∞(F,χ{0})≤1/6,d∞(G,χ{0})≤4/6.

(ii)E,F,Gare(M/6)-Lipschitzian.

Then (1.11) has a solution inB_M. Proof. We define the mapping

A:B_M−→B_M, (2.41)

byAx=Ex²+Fx+G. To check thatAis well-defined, letx∈BMand then (Ax)L(α)−(Ax)L(β)

=E_L(α)x²_L(α) +F_L(α)x_L(α) +G_L(α)−E_L(β)x_L²(β)−F_L(β)x_L(β)−G_L(β)

≤EL(α)−EL(β)x_L²(α) +EL(β)xL(α) +xL(β)·xL(α)−xL(β) +F_L(α)−F_L(β)x_L(α) +F_L(β)x_L(α)−x_L(β)+G_L(α)−G_L(β)

≤M

6 α−β+2M

6 α−β+M

6 α−β+M

6α−β+M

6 α−β

=Mα−β, ∀α,β∈[0, 1],

(2.42)

and, analogously,

(Ax)R(α)−(Ax)R(β)≤M|α−β|, for everyα,β∈[0, 1], (2.43)

therefore, by Theorem 2.5,Ax∈E¹ isM-Lipschitzian and, using the hypotheses and χ{0}≤x≤χ{1}, we obtain

0≤EL(α)x²_L(α) +FL(α)xL(α) +GL(α)=(Ax)L(α)

≤(Ax)_R(α)=E_R(α)x_R²(α) +F_R(α)x_R(α) +G_R(α)

≤1 6+1

6+4 6⁼1,

(2.44)

forα∈[0, 1], achievingAx∈BM. Moreover,Ais a nondecreasing and continuous map- ping (useTheorem 2.6).Ais bounded, since

d∞

Ax,χ{0}

=d∞

Ex²+Fx+G,χ{0}

≤1, forx∈B_M. (2.45) LetS⊂BM a bounded set (consisting of continuous fuzzy numbers) withr(S)>0, and prove thatA(S) is relatively compact. In that case,

rA(S)=0< r(S) (2.46)

and the proof is complete by application ofTheorem 1.1[1, Theorem 2.3]. LetA(S)⊂E¹ and prove thatA(S)LandA(S)Rare relatively compact inC[0, 1]. Indeed, using that for y∈A(S),χ{0}≤y≤χ{1}, we obtain thatA(S)_Lis a bounded set inC[0, 1],

yL

∞≤d∞ y,χ{0_}

≤1, y∈A(S). (2.47)

Let f ∈A(S)_L, then f isM-Lipschitzian, andA(S)_Lis equicontinuous. This proves that A(S)_Lis relatively compact by Arzel`a-Ascoli theorem, and the same forA(S)_R.Lemma 2.7 guarantees thatA(S) is relatively compact and, therefore,Ais condensing. Besides,χ{0_}

andχ_{1_}are elements inB_M andχ_{0_}≤Aχ_{0_},Aχ_{1_}≤χ_{1_}. This completes the proof. In fact, there exist extremal solutions betweenχ{0}andχ{1}. Remark 2.10. Note that ourTheorem 2.9do not imposeG_R(α)≤1/6 for allα∈[0, 1]

and, therefore, improves the results of [1].

Theorem 2.11. LetE,F,G be Lipschitzian fuzzy numbers withE,F,G≥χ{0_}. Moreover, suppose that there existk >0,S≥0such that

E_R(0)k²+F_R(0)k+G_R(0)≤k, (2.48) MEk²+ER(0)2kS+MFk+FR(0)S+MG≤S, (2.49)

whereM_E,M_F,M_Gare, respectively, the Lipschitz constants ofE,FandG. Then (1.11) has a solution in

B_k,S:=

x∈E¹:χ{0}≤x≤χ{k},xisS-Lipschitzian. (2.50) Proof. Define

A:Bk,S−→E¹, (2.51)

byAx=Ex²+Fx+G. We show thatA(Bk,S)⊆Bk,S. Indeed, forx∈Bk,S, and everyα∈ [0, 1],

0≤EL(α)x²_L(α) +FL(α)xL(α) +GL(α)=(Ax)L(α)

≤(Ax)_R(α)=E_R(α)x²_R(α) +F_R(α)x_R(α) +G_R(α)

≤ER(α)k²+FR(α)k+GR(α)≤ER(0)k²+FR(0)k+GR(0)

≤k,

(2.52)

which proves thatχ{0}≤Ax≤χ{k}.Besides, forx∈Bk,S, andα,β∈[0, 1], (Ax)L(α)−(Ax)L(β)

≤E_L(α)−E_L(β)x²_L(α) +E_L(β)x_L(α) +x_L(β)·x_L(α)−x_L(β) +FL(α)−FL(β)xL(α) +FL(β)xL(α)−xL(β)+GL(α)−GL(β)

≤

M_Ek²+E_L(β)2kS+M_Fk+F_L(β)S+M_G|α−β|

≤

MEk²+ER(0)2kS+MFk+FR(0)S+MG

|α−β| ≤S|α−β|,

(2.53)

and, similarly,

(Ax)R(α)−(Ax)R(β)≤S|α−β|, (2.54) provingAx∈B_k,S. The proof is completed in the same way ofTheorem 2.9.

Remark 2.12. Inequalities (2.48) and (2.49) inTheorem 2.11are equivalent to d∞

E,χ{0}

k²+d∞ F,χ{0}

k+d∞ G,χ{0}

≤k, (2.55)

MEk²+d∞ E,χ{0}

2kS+MFk+d∞ F,χ{0}

S+MG≤S, (2.56) since, forx∈E¹,x≥χ{0},

d∞ x,χ{0}

= sup

α∈[0,1]

maxx_L(α),x_R(α)=x_R(0). (2.57)

Corollary2.13. InTheorem 2.11, takeE_R(0)≤1/6,F_R(0)≤1/6,G_R(0)≤4/6, andM_E= M_F=M_G=M/6, withM >0, to obtainTheorem 2.9.

Proof. Conditions inTheorem 2.11are valid fork=1 andS=M. Indeed, ER(0)k²+FR(0)k+GR(0)≤1=k,

M_Ek²+E_R(0)2kS+M_Fk+F_R(0)S+M_G

=M

6 +E_R(0)2M+M

6 +F_R(0)M+M 6 ^≤M.

(2.58)

3. Other existence results

Now, we present some results on the existence of extremal solutions to (1.11), based on Tarski’s fixed point Theorem [6]. For the sake of completeness, we present it here, and note that the proof is not constructive.

Theorem3.1. LetXbe a complete lattice and

F:X−→X (3.1)

a nondecreasing function, that is,F(x)≤F(y)wheneverx≤y. Suppose that there exists x0∈Xsuch thatF(x0)≥x0. ThenFhas at least one fixed point inX.

Proof. Consider the setY= {x∈X:F(x)≥x}, which is a nonempty set sincex0∈Y. Letz=supY (x0≤z). Note that, for everyx∈Y,F(x)≥x, so thatF(F(x))≥F(x)≥x andF(x)∈Y. Letx∈Y, thenx≤z, andx≤F(x)≤F(z), which implies thatz≤F(z).

On the other hand,z∈Y, so thatF(z)∈Y, thenF(z)≤zandzis a fixed point forFin X. Note thatzis thus the maximal fixed point inX.

Remark 3.2. In the hypotheses of the previous result, if there exists x1∈X such that F(x1)≤x1, we obtain the minimal fixed point as the infimum of the set Z= {x∈X: F(x)≤x}. If, at the same time, there existx0andx1such thatF(x0)≥x0andF(x1)≤x1, then

z=supY=sup{x∈X:F(x)≥x},

zˆ=infZ=inf{x∈X:F(x)≤x} (3.2) are, respectively, the maximal and minimal fixed points ofF inX. Indeed, since there exists at least one fixed point forF, then ˆz≤z, and any fixed point forF is between ˆz andz.

Lemma3.3. IfE,x,y∈E¹are such thatE≥χ{0}andχ{0}≤x≤y, thenχ{0}≤Ex≤Ey.

Proof. By hypotheses,

0≤xL(a)≤yL(a), 0≤xR(a)≤yR(a), ∀a∈[0, 1],

0≤EL(a), 0≤ER(a), ∀a∈[0, 1], (3.3)

so that, fora∈[0, 1], [Ex]^a=

EL(a)xL(a),ER(a)xR(a), [Ey]^a=

EL(a)yL(a),ER(a)yR(a), (3.4) where

0≤EL(a)xL(a)≤EL(a)yL(a), 0≤ER(a)xR(a)≤ER(a)yR(a), ∀a∈[0, 1], (3.5) hence

χ{0_}≤Ex≤Ey. (3.6)

Theorem3.4. LetE,F,Gbe fuzzy numbers such that

E,F,G≥χ{0_}, (3.7)

and suppose that there existsp >0such that

ER(0)p²+FR(0)p+GR(0)≤p. (3.8) Then (1.11) has extremal solutions in the interval

χ{0_},χ{p} :=

x∈E¹:χ{0_}≤x≤χ{p}

. (3.9)

Proof. Sincep >0,χ{0}< χ{p}. Define

A:χ{0_},χ{p}

−→E¹, (3.10)

byAx=Ex²+Fx+G. We show thatA([χ{0},χ{p}])⊆[χ{0},χ{p}]. Indeed, Aχ{0_}=E(χ{0_})²+Fχ{0_}+G=χ{0_}+χ{0_}+G=G≥χ{0_},

Aχ{p}=E(χ{p})²+Fχ{p}+G, (3.11) so that, using the conditions, for everya∈[0, 1], we have

Aχ{p}_a

=

E_L(a),E_R(a)p²+F_L(a),F_R(a){p}+G_L(a),G_R(a)

=

EL(a)p²+FL(a)p+GL(a),ER(a)p²+FR(a)p+GR(a). (3.12) By hypotheses and using the properties of E_L,E_R,F_L,F_R,G_L,G_R, we obtain, for all a∈ [0, 1],

E_L(a)p²+F_L(a)p+G_L(a)≤E_R(a)p²+F_R(a)p+G_R(a)

≤ER(0)p²+FR(0)p+GR(0)≤p. (3.13)