Fuzzy Sets and Systems 152 (2005) 173 – 190
www.elsevier.com/locate/fss
Fuzzy quasilinear spaces and applications
M.A. Rojas-Medara,∗,1 , M.D. Jiménez-Gamerob,2 , Y. Chalco-Canoa,3 ,
A.J. Viera-Brandãoc
a IMECC-UNICAMP, CP 6065, 13081-970, Campinas-SP, Brazil
b Departamento de Estadística e Investigación Operativa, Universidad de Sevilla, Facultad de Matemáticas,
41012 Sevilla, Spain
c DEMAT-ICEB-UFOP, Ouro Preto-MG, Brazil
Received 8 July 2003; received in revised form 14 September 2004; accepted 27 September 2004
Available online 19 October 2004
Abstract
We introduce the concept of fuzzy quasilinear space and fuzzy quasilinear operator. Moreover we state some
properties and give results which extend to the fuzzy context some results of linear functional analysis.
© 2004 Elsevier B.V. All rights reserved.
Keywords: Fuzzy sets; Random fuzzy variables; Fuzzy-valued mapping; Analysis
1. Introduction
In [1] Assev presented an approach to the study of spaces of subsets and function spaces of multivalued
mappings, by weakening the requirement of linearity in the construction of linear functional analysis.
Assev introduced the concepts of quasilinear spaces and quasilinear operators which enable us to
consider both linear and nonlinear spaces of subsets and multivalued mappings from a single point of
∗ Corresponding author.
E-mail addresses: [email protected] (M.A. Rojas-Medar), [email protected] (M.D. Jiménez-Gamero),
[email protected] (Y. Chalco-Cano), [email protected] (A.J. Viera-Brandão).
1 M.A. Rojas-Medar is partially supported by CNPq-Brazil, grant No 301354/03-0, FAPESP-Brasil, grant 01/07557-3 and
MCYT-Spain, grant BFM 2003-06579.
2 M.D. Jiménez-Gamero is partially supported by MEC-Spain, grant MTM2004-01433.
3 Y. Chalco-Cano is supported by Fapesp-Brazil, grant No 00/00055-0.
0165-0114/$ - see front matter © 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.fss.2004.09.011
174
M.A. Rojas-Medar et al. / Fuzzy Sets and Systems 152 (2005) 173 – 190
view. He stated properties and theorems which are “quasilinear” counterparts of some results in linear
functional analysis and differential calculus in Banach spaces. His work has motivated us to extend the
notion of quasilinear spaces to the fuzzy context.
The purpose of this paper is to present the concept of fuzzy quasilinear spaces and to introduce a theory
of fuzzy quasilinear operators. We give some properties about fuzzy quasilinear spaces and we state some
propositions and theorems concerning a fuzzy “quasilinear” operator theory. For instance, we state a
fuzzy version of the Banach–Steinhauss theorem and we introduce a duality theory for fuzzy quasilinear
operators. We also present some results related to fuzzy differential inclusions. Others applications of
this new theory in the direction of a fuzzy differential calculus have been developed in [4].
The idea of fuzzy linear space is not new and other proposals have been published. Examples can be
found in [25] and [26]. The approach followed in these papers is quite different from the one presented
here.
The paper is organized as follows. Section 2 contains the concepts of normed quasilinear space and
quasilinear operator, which were introduced by Assev [1]. In Section 3 we study the quasilinear space
F (X), the space of all fuzzy compact sets defined on X, with some norms, the associated metrics and give
some convergence results. We also consider quasilinear operators defined between two fuzzy quasilinear
spaces, obtaining results which are analogous to some well-known ones in functional analysis: the relation
between quasilinear operators and the continuity and the Banach–Steinhaus theorem. Section 4 is devoted
to the study of fuzzy quasilinear operators, which are operators defined on a normed linear space taking
values in a quasilinear fuzzy space. By using the representation theorem of Negoita and Ralescu, we
define the adjoint of a fuzzy quasilinear operator. Finally, in Section 5 we obtain some results on fuzzy
differential inclusions.
2. Preliminaries
Definition 2.1. A set X is called a quasilinear space if a partial order relation , an algebraic sum
operation + and an operation of multiplication by real numbers ·, are defined on it and satisfy the
following properties for any elements x, y, z, v ∈ X and any real numbers , ∈ R:
(q.1) x x,
(q.2) x y, y z ⇒ x z,
(q.3) x y, y x ⇒ x = y,
(q.4) x + y = y + x,
(q.5) x + (y + z) = (x + y) + z,
(q.6) there exists an element ∈ X , called neutral element, such that x + = x,
(q.7) · (· x) = (· )· x,
(q.8) · (x + y) = · x + · y,
(q.9) 1· x = x,
(q.10) 0· x = ,
(q.11) ( + )· x · x + · x,
(q.12) x y and z v ⇒ x + z y + v,
(q.13) x y ⇒ · x · y.
An element x ∈ X is called an inverse of x ∈ X , if x + x = .
M.A. Rojas-Medar et al. / Fuzzy Sets and Systems 152 (2005) 173 – 190
175
Obviously, if an element x has an inverse x , then it is unique. If any element x in the quasilinear space
X has an inverse element x ∈ X , then the partial order on X is determined by equality and consequently
X is a linear space with scalars in R.
Definition 2.2. Let X be a quasilinear space. A real function · X : X → R is called a norm if the
following conditions hold for any x, y ∈ X and any ∈ R:
(n.1) x = ⇒ xX > 0,
(n.2) x + yX xX + yX ,
(n.3) · xX = || xX ,
(n.4) x y ⇒ xX yX ,
(n.5) if for any > 0, there exists an element x ∈ X such that
x y + x and x X ⇒ x y.
A quasilinear space X with a norm defined on it is called a normed quasilinear space.
If any x ∈ X has an inverse element x ∈ X , then the concept of a normed quasilinear space coincides
with the concept of a real normed linear space.
From now on, we will denote (−1) · x by −x.
Let X be a normed quasilinear space. The Hausdorff metric on X is defined by
HX (x, y) = inf{r 0/ x y + a1r , y x + a2r , air ∈ X , air X , r, i = 1, 2}.
Since x y + (x − y) and y x + (y − x), the quantity HX (x, y) is defined for any elements x, y ∈ X
and HX (x, y) x − yX . It is easy to see that the function HX (x, y) satisfies all the axioms of a metric.
The proof of the next lemma is a consequence of the definition of the Hausdorff metric and of the
axioms of quasilinear spaces.
Lemma 2.3. The sum operation +, the multiplication by real numbers and the norm .X are continuous
with respect to the Hausdorff metric HX .
A quite interesting example of normed quasilinear spaces is the following: let X be a real normed linear
space. Denote by K(X) the space of nonempty closed and bounded subsets of X, i.e.,
K(X) = {A ⊂ X/ A is nonempty closed and bounded}
and denote by KC (X) its subspace of convex subsets of X,
KC (X) = {A ∈ K(X)/ A is convex}.
The algebraic sum operation in K(X) and multiplication by a real number ∈ R are defined by the
expressions
A + B = {a + b/ a ∈ A, b ∈ B},
· A = {· a/ a ∈ A}.
The space K(X), with the partial order given by the inclusion, satisfies the conditions (q.1)–(q.13). A
norm in K(X) is given by AK = supa∈A a. Consequently K(X) and KC (X) are normed quasilinear
176
M.A. Rojas-Medar et al. / Fuzzy Sets and Systems 152 (2005) 173 – 190
spaces. In this case the Hausdorff metric is defined, as usual, by
H (A, B) = inf{r 0/ A ⊂ B + rS1 (0), B ⊂ A + rS1 (0)},
where Sa (x) is the closed ball of radius a centered at x ∈ X, Sa (x) = {y ∈ X/ x − y a}.
The considered partial order in K(X) is that given by the inclusion, but one can take many other
partial orders. Nevertheless, one must take care when choosing a partial order since not all of them satisfy
conditions (q.1)–(q.13). For example, when X = R and K(R) = {[a, b] : a, b ∈ R, a b}, a commonly
used partial order , which extends the ordering on the real line, is the following
[a, b][c, d] ⇔ a c and b d.
The problem with this partial order is that it does not satisfy (q.13).
Definition 2.4. Let X and Y be two quasilinear spaces. An application : X → Y is called a quasilinear
operator if it satisfies the following conditions:
(o.1) (x) = (x), ∀ ∈ R,
(o.2) (x + y) (x) + (y), ∀x, y ∈ X ,
(o.3) x y ⇒ (x) (y).
If X and Y are normed quasilinear spaces, then we say that quasilinear operator : X → Y is bounded
if there exists a k > 0 such that
(x)Y kxX ,
∀x ∈ X .
Let X and Y be two normed quasilinear spaces and let L(X , Y ) denote the space of all bounded
quasilinear operators from X to Y . We write 1 2 if 1 (x) 2 (x), ∀x ∈ X . Multiplication by
real numbers is defined on L(X , Y ) by the equality ()(x) = (x). Moreover, the algebraic sum on
L(X , Y ) is defined by the equality (1 + 2 )(x) = 1 (x) + 2 (x). With these operations L(X , Y ) is a
quasilinear space.
A norm on L(X , Y ) is defined by
L = sup (x)Y .
xX =1
Consequently, L(X , Y ) is a normed quasilinear space.
3. Fuzzy quasilinear space
Let X be a Banach space. A fuzzy set on X is a function u : X → [0, 1]. For 0 < 1, we denote by
[u] = {x ∈ X/ u(x) } the -level of u and by [u]0 = supp(u) = {x ∈ X/ u(x) > 0} that it is called
the support of u.
A fuzzy set u : X → [0, 1] is called fuzzy compact set (fuzzy compact convex set, respectively) if [u]
is compact for all ∈ [0, 1] (if [u] is compact convex for all ∈ [0, 1], respectively).
We denote by F (X) (FC (X), respectively) the space of all fuzzy compact sets u : X → [0, 1] (the
space of all fuzzy compact convex sets u : X → [0, 1], respectively). The next proposition contains some
basic results (see [6]).
M.A. Rojas-Medar et al. / Fuzzy Sets and Systems 152 (2005) 173 – 190
177
Proposition 3.1. If u ∈ F (X), then the family {[u] / ∈ [0, 1]} satisfies the following properties:
(a) [u]0 ⊇ [u] ⊇ [u] , ∀ 0 .
n
(b) If n ↑ ⇒ [u] = ∞
n=1 [u] , i.e., the level-application is left-continuous.
(c) u = v ⇔ [u] = [v] , ∀ ∈ [0, 1].
(d) [u] = ∅, ∀ ∈ [0, 1], is equivalent to u(x) = 1 for some x ∈ X. If u satisfies this condition we say
that u is normal.
(e) The relation ⊆, defined by
u ⊆ v ⇔ u(x) v(x), ∀x ∈ X ⇔ [u] ⊆ [v] , ∀ ∈ [0, 1],
(1)
is a partial order on F (X).
The algebraic sum operation and multiplication by a real number ∈ R on F (X) is defined by
x if = 0,
u (u + v)(x) = sup min{u(y), v(x − y)} and (u)(x) =
(x)
if
= 0,
{0}
y∈X
respectively, where for any subset A of X, A denotes the characteristic function of A. With these definitions
we obtain that [u + v] = [u] + [v] and [u] = [u] , for all u, v ∈ F (X), ∈ [0, 1] and ∈ R.
The space F (X) with the sum, multiplication by real numbers and the partial order defined in (1) is a
quasilinear space with neutral element {0} .
We can consider several norms on F (X). Two examples are
u1 = sup [u] K
01
and
1
u2 =
0
[u] K d.
For the norm · 1 , the Hausdorff metric is defined by
D1 (u, v) = inf{r 0/ u ⊆ v + w1r , v ⊆ u + w2r , wi ∈ F (X), wir 1 r, i = 1, 2},
or equivalently
D1 (u, v) = inf{r 0/ u ⊆ v + r , v ⊆ u + r },
where = S1 (0) . We can also write D1 as follows,
D1 (u, v) = sup H ([u] , [v] ).
∈[0,1]
For the norm · 2 , the associated Hausdorff metric is defined by
D2 (u, v) = inf{r 0/ u ⊆ v + w1r , v ⊆ u + w2r , wi ∈ F (X), wir 2 r, i = 1, 2}.
By using the -levels of u and v, D2 can be equivalently expressed as
1
D2 (u, v) =
H ([u] , [v] ) d.
0
178
M.A. Rojas-Medar et al. / Fuzzy Sets and Systems 152 (2005) 173 – 190
The space F (X) extends K(X) in the sense that, for each A ∈ K(X), its characteristic function A
belongs to F (X). Clearly if A, B ∈ K(X), then
D1 (A , B ) = D2 (A , B ) = H (A, B).
It is well known that (F (X), D1 ) is a complete metric space, but is not separable and (F (X), D2 ) is a
separable metric space, but is not complete (see [6,14]).
Hereafter, the space F (X) with norm .F (X) (where .F (X) is either one of the previously considered
norms or any other norm), will be called fuzzy normed quasilinear space. The Hausdorff metric derived
from .F (X) will be denoted by D(·, ·).
Lemma 3.2. Let F (X) be a fuzzy normed quasilinear space and let u0 , v0 , un , vn , zn ∈ F (X), ∀n ∈ N.
(a) Suppose that un → u0 and vn → v0 . If un ⊆ vn , ∀n n0 , for some n0 ∈ N, then u0 ⊆ v0 .
(b) Suppose that un → u0 and zn → u0 . If un ⊆ vn ⊆ zn , ∀n n0 , for some n0 ∈ N, then vn → u0 .
(c) Suppose that un + vn → u0 and vn → {0} , then un → u0 .
Proof. (a) If un → u0 and vn → v0 , then for any > 0 there exists an N such that for any n N there
exist elements an , bn ∈ F (X), with an F (X) and bn F (X) , such that
u0 ⊆ un + an and vn ⊆ v0 + bn .
Thus,
u0 ⊆ v0 + an + bn
for n N. Since an + bn F (X) an F (X) + bn F (X) 2, it follows from (n.5) that u0 ⊆ v0 . The
proofs of (b) and (c) are analogous. The proof of the next lemma is quite easy and so we omit it.
Lemma 3.3. Let F (X) and F (Y ) be two fuzzy quasilinear spaces. If : F (X) → F (Y ) is a quasilinear
operator, then ({0} ) = {0} .
Lemma 3.4. Let F (X) and F (Y ) be two fuzzy normed quasilinear spaces. A quasilinear operator :
F (X) → F (Y ) is bounded if and only if, it is continuous at {0} ∈ F (X).
Proof. Suppose first that the operator is bounded. Then there exists a k > 0 such that
(u)F (Y ) kuF (X) ,
∀u ∈ F (X).
So, given > 0, if
D(u, {0} ) = uF (X) < = /k,
then
D((u), ({0} )) = (u)F (Y ) kuF (X) < k = ,
M.A. Rojas-Medar et al. / Fuzzy Sets and Systems 152 (2005) 173 – 190
179
and hence is continuous at {0} . Now, suppose that is a quasilinear operator continuous at {0} ∈ F (X).
Then, for any > 0, there exists a > 0 such that D(u, {0} ) = uF (X) < implies
D((u), ({0} )) = D((u), {0} ) = (u)F (Y ) < .
So, for any u ∈ F (X),
u
2
=
(u)F (Y ) < ⇒ (u)F (Y ) < uF (X) ,
2uF (X) F (Y ) 2uF (X)
and therefore is bounded. Lemma 3.5. Let F (X) and F (Y ) be two fuzzy normed quasilinear spaces. Let : F (X) → F (Y ) be a
quasilinear operator. If is continuous at {0} ∈ F (X), then is uniformly continuous on F (X).
Proof. Suppose that is continuous at {0} . Then, for any > 0, there exists a > 0 such that
if uF (X) < , then (u)F (Y ) < . Let u0 ∈ F (X) be given. If D(u, u0 ) < , then there exist
w1 , w2 ∈ F (X) such that
wi F (X) , i = 1, 2, and u ⊆ u0 + w1 , u0 ⊆ u + w2 .
As is a quasilinear operator, it follows that
(u) ⊆ (u0 ) + (w1 ) and (u0 ) ⊆ (u) + (w2 ).
Since wi F (X) < , then (wi )F (Y ) < , i = 1, 2, and thus D((u), (u0 )) < . Example 3.6. If f : X → Y is a function, its Zadeh extension (see [6]), f¯ : F (X) → F (Y ), is defined
by

 sup u(y)
if f −1 (x) = ∅,
−1
¯
f (u)(x) = y∈f (x)

0
if f −1 (x) = ∅.
If f is continuous, then f¯ is a well-defined function (see [6]) and
f¯(u) = f ([u] ), ∀ ∈ [0, 1], ∀u ∈ F (X).
Now, suppose that f is linear. Then, f¯ is a quasilinear operator and from the continuity of f, it follows that
f¯ is bounded. Consequently, f¯ is uniformly continuous.
From now to the end of this section we will assume that F (X) and F (Y ) are two fuzzy normed
quasilinear spaces.
The space L(F (X), F (Y )) of all the bounded quasilinear operators from F (X) to F (Y ), with the norm
defined by
L =
sup
uF (X) =1
(u)F (Y )
is a normed quasilinear space.
180
M.A. Rojas-Medar et al. / Fuzzy Sets and Systems 152 (2005) 173 – 190
The proof of the next lemma is very easy and so we omit it.
Lemma 3.7. (a) Any element ∈ L(F (X), F (Y )) satisfies the Lipschitz condition with constant L .
(b) If 1 ∈ L(F (X), F (Y )) and 2 ∈ L(F (Y ), F (Z)), then the operator = 2 ◦ 1 is in the space
L(F (X), F (Z)).
From the properties of the limit we have the following immediate result.
Lemma 3.8. Suppose that the sequence {n } ∈ L(F (X), F (Y )) converges at each point u ∈ F (X).
Then
(u) = lim n (u)
n→+∞
is a quasilinear operator.
We define the function : [0, +∞) → F (Y ) by (t) = t , where = S1 (0) , the characteristic
function of the closed ball of radius 1 centered at 0 ∈ Y . It satisfies the following properties:
u ⊆ (uF (Y ) ),
t s ⇒ (t) ⊆ (s),
(t + s) = (t) + (s).
Lemma 3.9. The operator : F (X) → F (Y ), defined by
(u) = (uF (X) )
belongs to L(F (X), F (Y )).
Proof. Let u, v ∈ F (X) be such that u ⊆ v, then uF (X) vF (X) . By (2) we have that
(uF (X) ) ⊆ (vF (X) )
and
(u + vF (X) ) ⊆ (uF (X) ) + (vF (X) ).
We also have that (uF (X) ) = || (uF (X) ). Since, = − = (−1), it follows that
(uF (X) ) = (uF (X) ).
Consequently, (u) is a quasilinear operator from F (X) to F (Y ). Since,
(u)F (Y ) = (uF (X) )F (Y ) = uF F = uF (X) F (Y ) ,
we conclude that ∈ L(F (X), F (Y )). (2)
M.A. Rojas-Medar et al. / Fuzzy Sets and Systems 152 (2005) 173 – 190
181
Remark 3.10. (a) L = 1.
(b) If L L , then ⊆ .
(c) L · .
The Hausdorff metric on L(F (X), F (Y )) is given by
HL (1 , 2 ) = inf{ r > 0/ 1 2 + r1 , 2 1 + r2 ,
ri ∈ L(F (X), F (Y )), ri L r, i = 1, 2},
or equivalently,
HL (1 , 2 ) = inf{r > 0 : 1 2 + r , 2 1 + r }.
Theorem 3.11. If (F (Y ), D) is a complete metric space, then the normed quasilinear space (L(F (X),
F (Y )), HL ) is a complete metric space.
Proof. Let {n }n∈N be a sequence of Cauchy in L(F (X), F (Y )). Then, for any > 0, there exists an N
m,n satisfying
such that for all n, m N there exist n,m
, m,n
n,m
m,n
n m + n,m
, m n + , L , L .
Consequently, D(n (u), m (u)) uF (X) , for any u ∈ F (X). So the sequence {n (u)} is Cauchy on
F (Y ). Since F (Y ) is complete, there exists an element (u) ∈ F (Y ) such that
(u) = lim n (u).
n→∞
From Lemma 3.8, we have that is a quasilinear operator from F (X) to F (Y ). Furthermore,
n (u)F (Y ) m (u)F (Y ) + n,m
(u)F (Y ) (m L + )uF (X) ,
for any n, m N. Fixing m N and taking the limit as n → ∞ we get that
(u)F (Y ) (m L + )uF (X) .
Therefore, ∈ L(F (X), F (Y )). We now prove that {n } converges to in the quasilinear space
L(F (X), F (Y )). Since
D(n (u), (u)) D(n (u), m (u)) + D(m (u), (u))
uF (X) + D(m (u), (u))
for n, m N, taking the limit as m → ∞ we obtain that
D(n (u), (u)) uF (X) ∀n N.
So, it follows that
n + , n + ∀n N
and hence HL (, n ) , ∀n N. The next result is an analogous to the Banach–Steinhaus theorem.
182
M.A. Rojas-Medar et al. / Fuzzy Sets and Systems 152 (2005) 173 – 190
Theorem 3.12. Suppose that (F (X), D) is a complete metric space. Let {i }i∈I be a family of elements
in L(F (X), F (Y )) such that
sup i (u)F (Y ) < ∞,
i∈I
∀u ∈ F (X).
Then,
sup i L < ∞,
i∈I
or equivalently, there exists a c > 0 such that
i (u)F (Y ) cuF (X) ,
∀u ∈ F (X), ∀i ∈ I.
Proof. For each n 1, we defined the set
Zn = {u ∈ F (X)/ i (u)F (Y ) n, ∀i ∈ I }.
Since i is uniformly continuous ∀i ∈ I , then Zn is closed in F (X) for each n ∈ N, and ∞
n=1 Zn =
F (X). Consequently, by using Baire’s Lemma we get that intZn0 = ∅, for some n0 1, and so there
exist u0 ∈ F (X) and r > 0 such that Sr (u0 ) ⊂ Zn0 . Hence,
i (v)F (Y ) n0 ,
∀v ∈ Sr (u0 ),
∀i ∈ I.
Now, if uF (X) r, then u0 + u ∈ Sr (u0 ). Since u ⊆ (u0 + u) − u0 , it follows that
i (u) ⊆ i (u0 + u) − i (u0 ),
∀i ∈ I,
and thus
i (u)F (Y ) i (u0 + u)F (Y ) + i (u0 )F (Y ) 2n0 ,
∀i ∈ I.
Now, for any u ∈ F (X),
ru = r,
u
F (X) F (X)
and therefore,
i (u)F (Y ) 2n0
uF (X) ,
r
∀i ∈ I.
Lemma 3.13. Suppose that (F (X), D) is a complete metric space. Let {i }i∈I be a family of bounded
quasilinear operators, i : F (X) → K(Y ), ∀i ∈ I , such that the family {i (u)}i∈I is bounded in Y for
each u ∈ F (X). Then, the application R : F (X) → K(Y ) defined by
R(u) = ∪i∈I i (u)
is a bounded quasilinear operator.
M.A. Rojas-Medar et al. / Fuzzy Sets and Systems 152 (2005) 173 – 190
183
Proof. Let u, v ∈ F (X) be such that u ⊆ v. Then,
R(u) = ∪i∈I i (u) ⊆ ∪i∈I i (v) = R(v).
Further, given u, v ∈ F (X),
R(u + v) =
⊆
⊆
=
∪i∈I i (u + v)
∪i∈I (i (u) + i (v))
∪i∈I i (u) + ∪i∈I i (v)
R(u) + R(v).
Moreover,
R(u) = ∪i∈I i (u) = ∪i∈I i (u) = ∪i∈I i (u) = R(u),
for any u ∈ F (X) and any ∈ R. Consequently, R : F (X) → K(Y ) is a quasilinear operator. Now, we
will prove that R is bounded. We define i : F (X) → F (Y ) by i (u) = {i (u)} . Since i is a bounded
quasilinear operator for each i ∈ I , it follows that i is a quasilinear operator for each i ∈ I . Now,
sup i (u)F (Y ) = sup i (u) F (Y ) = sup i (u)K < ∞.
i∈I
i∈I
i∈I
Then, by Theorem 3.12, there exists a c > 0 such that
i (u)K = i (u)F (Y ) cuF (X) ,
∀u ∈ F (X),
∀i ∈ I,
and therefore,
R(u)K = ∪i∈I i (u)K cuF (X) ,
∀u ∈ F (X).
Consequently, R is bounded. Denote by F (X) the space L(F (X), K(R)) and by coF (X) the space L(F (X), KC (R)). A quasilinear operator from F (X) to K(R) will be called quasilinear functional.
Let ∈ L(F (X), F (Y )). Then, to each ∈ F (Y) we can
associate
an
element
∈
F
(X)
through
(u) = ( ◦ )(u). Consequently, the operator : F (Y ) → F (X) given by () = ◦ is
well defined.
Proposition 3.14. Let ∈ L(F (X), F (Y )). Then,
(a) ∈ L(F (Y ) , F (X) ).
(b) L = L .
Proof. (a) Let 1 , 2 ∈ F (Y )
and ∈ R. Then,
(1 + 2 )(u) = (1 + 2 )((u))
= 1 (
(u)) + 2
((u))
= ( (1 ) + (2 ))(u).
Consequently,
(1 + 2 ) (1 ) + (2 ).
184
M.A. Rojas-Medar et al. / Fuzzy Sets and Systems 152 (2005) 173 – 190
We also have that,
()(u) = ((u)) = ()(u),
and thus () = (). Now, suppose that 1 2 . Then,
(1 )(u) = 1 ((u)) 2 ((u)) = (2 )(u),
and hence (1 ) (2 ). Consequently, the operator Since
()L =
sup
uF (X) =1
: F (X)
→ F (Y )
is quasilinear.
((u))K L L ,
the quasilinear operator is bounded and L L .
(b) We will prove that L L . The element 0 (u) = [−uF , uF ] is in the space F (Y )
and 0 L = 1. Then,
L = sup ()L (0 )L =
L =1
=
and therefore
sup
sup
uF (X) =1
0 ((u))K
(u)F (Y ) = L ,
uF (X) =1
L L .
4. Fuzzy quasilinear operator
Let X and Y be two normed linear spaces. The space L(X, Y ) consisting of all bounded linear operators
from X to Y is also a normed linear space. Consequently, the normed quasilinear space K(L(X, Y )) is
well defined. Each element T ∈ K(L(X, Y )) defines a bounded set-valued quasilinear mapping (x) =
T x = {Ax : A ∈ T } from X to K(Y ).
Definition 4.1. We say that an element ∈ L(X, K(Y )) has a linear representation if there exists an
element T ∈ K(L(X, Y )) such that
(x) = T x = {Ax ; A ∈ T }.
The next result, proved in [1], is important and necessary to define the adjoint of a quasilinear
operator.
Theorem 4.2. Suppose that ∈ L(X∗ , KC (R)). Then there exists a unique closed bounded convex subset
F ⊂ X such that for any x ∗ ∈ X∗
(x ∗ ) = F, x ∗ = {f, x ∗ ; f ∈ F }.
Corollary 4.3. If X is a complete reflexive normed linear space, then any bounded quasilinear operator
: X → KC (R) has a linear representation.
M.A. Rojas-Medar et al. / Fuzzy Sets and Systems 152 (2005) 173 – 190
185
Let X and Y be two normed linear spaces. An application : X → F (Y ) is called a fuzzy quasilinear
operator if satisfies (o.1) and (o.2), since (o.3) is automatically satisfied.
Let : X → F (Y ) be a bounded fuzzy quasilinear operator. Then, for any ∈ [0, 1], the application
of level : X → K(Y ), defined by
(x) = [(x)]
is a bounded quasilinear operator.
Definition 4.4. We say that a fuzzy valued mapping : X → F (Y ) has a linear representation if, for
each ∈ [0, 1], the application of level has a linear representation.
The next result follows immediately from Corollary 4.3.
Proposition 4.5. Let X be a complete reflexive normed linear space. Then each bounded fuzzy quasilinear
operator : X → FC (R) has a linear representation.
Let X, Y be two normed linear spaces and let : X → K(Y ) be a bounded quasilinear operator.
Then there exists a unique bounded quasilinear operator ⊗ : Y ∗ → KC (X∗ ) such that (x), y ∗ =
x, ⊗ (y ∗ ), for all x ∈ X and y ∗ ∈ Y ∗ (see [1]). The operator ⊗ ∈ L(Y ∗ , K(X∗ )) is called the adjoint
of the operator ∈ L(X, K(Y )). Hence, given a bounded fuzzy quasilinear operator : X → F (Y ), for
∗
∗
∗
each ∈ [0, 1] there exists an operator ⊗
: Y → K(X ), the adjoint of , such that (x), y =
∗
∗
∗
x, ⊗
(y ) for any x ∈ X and y ∈ Y .
The next result generalizes the concept of adjoint of an operator ∈ L(X, F (Y )).
Theorem 4.6. Let X, Y be two normed linear spaces. Let : X → F (Y ) be a bounded fuzzy quasilinear
operator. Then there exists a unique fuzzy set ⊗ (y ∗ ) : X∗ → [0, 1] such that
∗
[⊗ (y ∗ )] = ⊗
(y ),
for all ∈ [0, 1] and y ∗ ∈ Y ∗ .
∗
Proof. Consider the family {⊗
(y )}∈[0,1] . Such family satisfies the conditions of the theorem of
representation due to Negoita and Ralescu (see [18]). In fact,
∗
∗
(i) By definition, it follows that ⊗
(y ) ∈ K(X ), for all ∈ [0, 1].
(ii) Let , then
∗
∗
∗
⊗ ∗
x, ⊗
(y ) = (x), y ⊆ (x), y = x, (y ),
∀x ∈ X,
∗
⊗ ∗
and hence ⊗
(y ) ⊆ (y ).
(iii) Consider the sequence 1 2 3 · · ·, such that limi→∞ i = , then
∩i 1 i (x) = (x),
186
M.A. Rojas-Medar et al. / Fuzzy Sets and Systems 152 (2005) 173 – 190
and consequently
∗
⊗ ∗
x, ∩i 1 ⊗
i (y ) = x, (∩i 1 i ) (y )
= ∩i 1 i (x), y ∗ = (x), y ∗ ∗
= x, ⊗
(y ).
From the theorem of representation, there exists a unique fuzzy set ⊗ (y ∗ ) : X∗ → [0, 1] such that
∗
[⊗ (y ∗ )] = ⊗
(y ).
Definition 4.7. Let : X → F (Y ) be a bounded fuzzy quasilinear operator. The adjoint of is the
operator ⊗ ∈ L(Y ∗ , F (X∗ )) such that
∗
[⊗ (y ∗ )] = ⊗
(y ),
for all ∈ [0, 1] and y ∗ ∈ Y ∗ .
5. Fuzzy differential inclusions
The following definition of fuzzy differential inclusion was introduced by Zhu and Rao (see [27]),
where they obtained some results concerning existence of solution (see also [2] for the study of classic
differential inclusions). We consider a Banach space X, an interval J of R and we denote by C(J, X) the
set of continuous functions defined on J with values in X.
Definition 5.1. Let : X → F (X) be a fuzzy valued mapping. Let : X → [0, 1] be a function and let
J be an interval of R. We call fuzzy differential inclusion to the following problem: to find x ∈ C(J, X)
such that
ẋ(t) ∈ [(x(t))](x(t)) .
(3)
If : X → F (X) is a fuzzy quasilinear operator, then the problem (3) is called a fuzzy quasilinear
differential inclusion.
Suppose that the bounded fuzzy quasilinear operator : Rn → F (Rn ) has a linear representation, i.e.,
for each ∈ [0, 1], there exists a compact set T in the space of n × n matrices such that (x) = T x =
{Ax ; A ∈ T }. Thus there exists a bounded fuzzy quasilinear operator ⊗ : Rn → F (Rn ), the adjoint
⊗ = {A ; A∗ ∈ T } for each ∈ [0, 1].
of , and ⊗
has linear representation T
n
n
Let : R → F (R ) be a fuzzy quasilinear operator and let ⊗ : Rn → F (Rn ) be the adjoint of .
The fuzzy differential inclusion
ẋ(t) ∈ −[⊗ (x(t))](x(t))
is called the adjoint of the fuzzy differential inclusion (3).
The proof of the next result is easy.
M.A. Rojas-Medar et al. / Fuzzy Sets and Systems 152 (2005) 173 – 190
187
Proposition 5.2. Suppose that the fuzzy quasilinear operator : Rn → F (Rn ) has a linear representation. Then is continuous.
We denote by F C (X) (see [18]) the subspace of F (X) for which the elements u are such that the
mapping → [u] is H-continuous on [0, 1], i.e., given > 0, there exists a > 0 such that | − | < implies H ([u] , [u] ) < . Since [0, 1] is a compact metric space, the application → [u] is, in fact,
uniformly continuous.
Proposition 5.3. Let : Rn → F C (Rn ) be a fuzzy quasilinear operator with a linear representation.
Suppose that : X → [0, 1] is continuous. Then the mapping : X → F C (X), defined by (x) =
[(x)] , is continuous.
Proof. Suppose that xn → x in Rn . Then
H ((xn ), (x)) = H [(xn )](xn ) , [(x)](x)
H [(xn )](xn ) , [(x)](xn ) + H [(x)](xn ) , [(x)](x)
D((xn ), (x)) + H [(x)](xn ) , [(x)](x) → 0.
Theorem 5.4. Suppose that the fuzzy quasilinear operator : Rn → F C (Rn ) has a linear representation
and that : Rn → [0, 1] is continuous. Moreover, let x(t) be a solution ofthe fuzzy differential inclusion
(3). Then, there exists a measurable matrix-valued function A : J → t∈J T (x(t)) such that x(t) is
an absolutely continuous solution of the ordinary linear differential equation ẋ = A(t)x. Here J is the
interval on which x(t) is defined.
Proof. We define the set valued mapping ¯ : Rn → K(Rn ) by
¯ (x) = [F (x)](x) .
Denote by G = {(x, y) ∈ Rn × Rn : y ∈ ¯ (x)} the graph of the set valued mapping ¯ . Then G is a
closed set (Proposition 5.3). We define the set valued mapping P : G → K(L(Rn , Rn )) by
P (x, y) = {A ∈ ∪t∈J T (x(t)) / Ax = y}.
We will prove that P has closed graph. Suppose that (xn , yn ) → (x, y) in G, An → A in L(Rn , Rn )
with An xn = yn . Taking the limit as n → ∞ we have that Ax = y. Furthermore, P (x, y) is bounded.
Consequently P is upper semicontinuous. Let H be the set valued mapping given by H (t) = P (x(t), ẋ(t)).
H is measurable, since for any open set U ∈ L(Rn , Rn ), from the upper semicontinuous of P it follows that
{(x, y) : P (x, y) ⊂ U } is an open set. Consequently, {t ∈ J ; H (t) ⊂ U } is measurable. Therefore,
it has a measurable single-valued branch A : J → ∪t∈J T(x(t)) , A(t) ∈ H (t), such that ẋ(t) = A(t)x(t)
for a.e. t ∈ J . Example 5.5. Consider the fuzzy-valued mapping : (0, 1) → F (R) defined, for each x ∈ (0, 1), by
the isosceles triangular fuzzy set with support [−x, x]. Thus, for each ∈ [0, 1] the application of level
188
M.A. Rojas-Medar et al. / Fuzzy Sets and Systems 152 (2005) 173 – 190
is (x) = [(x)] = (1 − )[−1, 1]x. Consequently, is a fuzzy quasilinear operator and has a linear
representation. We consider the function : R → [0, 1] defined by

if x < 0
 1
(x) = 1 − x if x ∈ [0, 1] ,

0
if x > 1
then the fuzzy differential inclusion (3) has the following form
ẋ ∈ [−1, 1]x 2 .
(4)
If we agree the initial condition x(0) = x0 , then, each solution of (4) is a solution of the differential
equation
ẋ = ax 2 ,
x(0) = x0 ,
(5)
with a ∈ [−1, 1]. So, the set of solutions of (4) with J = [0, 1) is given by
x0
−
/a ∈ [−1, 1] .
at − 1
We observe that if is a fuzzy quasilinear operator and has a linear representation, then we can obtain
all the solutions of the fuzzy differential inclusion resolving the differential equation (5).
Example 5.6. Consider : (0, 1) → F (R) defined by
 2t
 x if 0 t x2
(x)(t) = 1 if x2 t 1

0 if t ∈
/ [0, 1].
It is easy to see that [(x)] = 2x , 1 , for each ∈ [0, 1]. Thus, is not a fuzzy quasilinear operator
and has not a linear representation. Consider : R → [0, 1] as in the Example 5.5. Then, the fuzzy
differential inclusion is given by
(1 − x)x
ẋ ∈
,1 .
(6)
2
To obtain a solution of (6), with the initial condition x(0) = x0 , we can use the technique
of selection
x
for set-valued mapping: for example, the function f (x) = 2 is a selection of F (x) = (1−x)x
,
1
for
2
all x ∈ (0, 1). Consequently, a solution of (6) is given by x(t) = x0 et/2 . In this case, we cannot easily
determine all the solutions of (6) as we did in Example 5.5, where the solutions of the fuzzy differential
inclusion were obtained by solving the differential equation (6).
Remark 5.7. Kaleva [12,13] introduced the concept of fuzzy differential equation obtaining some results
of existence and uniqueness on the class of level-Lipschitz fuzzy sets. For more recent results on the
existence of solutions for fuzzy differential equations on fuzzy sets defined on Rn see Song [22] (who
uses the Hukukara derivative for fuzzy mappings), Ding et al. [7] (using elements of topological degree
theory), Wu and Song [24] (using compactness conditions). Other works are [5,10,15,16,21]. The fuzzy
M.A. Rojas-Medar et al. / Fuzzy Sets and Systems 152 (2005) 173 – 190
189
differential inclusion problem with (t) = t, t ∈ [0, 1], was introduced by Hullermeier [11]. This work
was extended by Diamond [5] (see also [3,20]). The main idea used in these papers [3,5,11,20] is to
consider for each ∈ [0, 1] the classical differential inclusion, so a family of associated solutions is
built and, finally, it is proved that this family satisfies the hypotheses of Nagoita–Ralescu Representation
Theorem. This approach is quite useful, since it allows to use the classical theory of differential inclusions.
Other important work is [17], that studies the impulsive functional differential inclusion and gives some
applications to fuzzy population models. The fuzzy differential inclusion problem considered here, that
was introduced by Zhu and Rao [27], is more general than the one considered in the above cited papers
and the approach followed in them is not adequate to solve it. We have seen that the fuzzy quasilinear
spaces and the theory of fuzzy quasilinear operators introduced in this paper can be applied to solve this
more general problem.
6. Concluding remarks
As a way to solve the inexistence of a linear structure on the space of fuzzy sets, in this paper we have
introduced the concepts of fuzzy quasilinear spaces and fuzzy quasilinear operators. We have given fuzzy
analogues of some classical results and we have applied them to solve a quite general fuzzy differential
inclusion problem.
In a future work, we will use the fuzzy quasilinear spaces and the fuzzy quasilinear operators to develop
the differentiability of fuzzy-valued mappings and to establish Lyapunov type stability results. The point
of interest is that the way to obtain these results is completely analogous to the classical theory [4,9,19].
In our opinion, the fuzzy quasilinear spaces and the fuzzy quasilinear operators can become a powerful
tool in the construction of a consistent fuzzy analysis, since we expect that it is possible to go on giving
fuzzy analogues of more classical results.
In addition to this, other extensions of the work in this paper could be done. For example, one might
consider other order relations over fuzzy sets (see for example [8] and [23]).
References
[1] S.M. Assev, Quasilinear operators and their application in the theory of multivalued mappings, Proc. Steklov Inst. Math.
2 (1986) 23–52.
[2] J.P. Aubin, A. Cellina, Differential Inclusions, Springer, Berlin, 1984.
[3] V.A. Boldosov, Differential inclusions with fuzzy right-hand side, Soviet Math. Dokl. 40 (1990) 567–569.
[4] M.D. Jiménez-Gamero, Y. Chalco-Cano, M.A. Rojas-Medar, A.J.V. Brandão, On the differentiability of fuzzy-valued
mappings and the stability of a fuzzy differential inclusion, submitted for publication.
[5] P. Diamond, Time-dependent differential inclusions, cocycle attractors and fuzzy differential equations, IEEE Trans. Fuzzy
Systems 7 (1999) 734–740.
[6] P. Diamond, P. Kloeden, Metric Space of Fuzzy Sets: Theory and Application, World Scientific, Singapore, 1994.
[7] Z. Ding, M. Ma, A. Kandel, Existence of solutions of fuzzy differential equations with parameters, Inform. Sci. 99 (1997)
205–217.
[8] D. Dubois, H. Prade, A unified view of ranking techniques for fuzzy numbers, Proc. 1999 IEEE Int. Conf. Fuzzy Systems,
Seoul, 1328–1333.
[9] E. Fernández-Cara, M.A. Rojas-Medar, A.J.V. Brandão, Inclusiones diferenciales, matemática difusa y aplicaciones, Bol.
Soc. Esp. Mat. Apl. 24 (2003) 31–62.
190
M.A. Rojas-Medar et al. / Fuzzy Sets and Systems 152 (2005) 173 – 190
[10] M. Friedman, M. Ming, A. Kandel, On the validity of the Peano theorem for fuzzy differential equations, Fuzzy Sets and
Systems 86 (1997) 331–334.
[11] E. Hullermeier, An approach to modeling and simulation of uncertain dynamical systems, Int. J. Uncertainty, Fuzziness,
Knowledge-Bases Syst. 5 (1997) 117–137.
[12] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems 24 (1987) 301–317.
[13] O. Kaleva, The Cauchy problem for fuzzy differential equations, Fuzzy Sets and Systems 35 (1990) 389–396.
[14] E.P. Klement, M.L. Puri, D.A. Ralescu, Limit theorems for fuzzy random variables, Proc. Roy. Soc. London A-407 (1986)
171–182.
[15] P.E. Kloeden, Remarks on Peano-like theorems for fuzzy differential equations, Fuzzy Sets and Systems 44 (1991)
161–163.
[16] J.J. Nieto, The Cauchy problem for continuous differential equations, Fuzzy Sets and Systems 102 (1999) 259–262.
[17] M. Guo, X. Xue, R. Li, Impulsive functional differential inclusions and fuzzy population models, Fuzzy Sets and Systems
138 (2003) 601–615.
[18] M. Rojas-Medar, R.C. Bassanezi, H. Román-Flores, A generalization of the Minkowski embedding theorem and
applications, Fuzzy Sets and Systems 102 (1999) 263–269.
[19] M.A. Rojas-Medar, R.C. Bassanezi,Y. Chalco-Cano, M.T. Mizukoshi, Population dynamics by fuzzy differential inclusion,
Kybernetes, to appear.
[20] M.A. Rojas-Medar, H. Román-Flores, A. Flores-Franulic, Inclusiones diferenciales fuzzy, viabilidad y fuzzines, Actas del
5 Congreso Latinoamericano de Biomatemática, Santiago, Chile, 1991.
[21] H. Román-Flores, M.A. Rojas-Medar, Embedding of level-continuous fuzzy sets on Banach spaces, Inform. Sci. 144
(2002) 24–47.
[22] S. Song, L. Guo, C. Feng, Global existence of solutions to fuzzy differential equations, Fuzzy Sets and Systems 115 (2000)
371–376.
[23] X. Wang, E. Kerre, Reasonable properties for the ordering of fuzzy quantities (2 parts), Fuzzy Sets and Systems 118 (2001)
375–405.
[24] C. Wu, S. Song, Existence theorem to the Cauchy problem of fuzzy differential equations under compactness-type
conditions, Inform. Sci. 108 (1998) 123–134.
[25] J. Xiao, X. Zhu, On linearly topological structure and property of fuzzy normed linear space, Fuzzy Sets and Systems 125
(2002) 153–161.
[26] J. Xiao, X. Zhu, Fuzzy normed space of operators and its complements, Fuzzy Sets and Systems 133 (2003) 389–399.
[27] Y. Zhu, L. Rao, Differential Inclusions for Fuzzy Maps, Fuzzy Sets and Systems 112 (2000) 257–261.