Fuzzy Sets and Systems 151 (2005) 581 – 599
www.elsevier.com/locate/fss
Generalizations of the differentiability of fuzzy-number-valued
functions with applications to fuzzy differential equations
Barnabás Bede∗ , Sorin G. Gal
Department of Mathematics, University of Oradea, Str. Armatei Romane no. 5, 410087 Oradea, Romania
Received 14 November 2003; received in revised form 25 March 2004; accepted 2 August 2004
Available online 11 September 2004
Abstract
The usual concept of differentiability of fuzzy-number-valued functions, has the following shortcoming: if c is
a fuzzy number and g : [a, b] → R is an usual real-valued function differentiable on x0 ∈ (a, b) with g (x0 ) 0,
then f (x) = c g(x) is not differentiable on x0 . In this paper we introduce and study generalized concepts of
differentiability (of any order n ∈ N), which solves this shortcoming. Newton–Leibnitz-type formula is obtained
and existence of the solutions of fuzzy differential equations involving generalized differentiability is studied. Also,
some concrete applications to partial and ordinary fuzzy differential equations with fuzzy input data of the form
c g(x), are given.
© 2004 Elsevier B.V. All rights reserved.
Keywords: Fuzzy-number-valued functions; Generalized differentiability; Fuzzy differential equations; Fuzzy partial
differential equations
1. Introduction
The H-derivative of a fuzzy-number-valued function was introduced in [13] and it is studied in several
papers (see e.g. [1,17]). In [14] is defined the Hukuhara derivative of a fuzzy-number-valued function and
fuzzy initial value problem is studied. This derivative has its starting point in the Hukuhara derivative of
multivalued functions. Differential equations in fuzzy setting are a natural way to model uncertainty of
dynamical systems. There are different approaches to this very quickly developing area of fuzzy analysis.
Let us mention some of them. First approach uses the above-mentioned H-derivative or its generalization,
∗ Corresponding author.
E-mail addresses: [email protected] (B. Bede), [email protected] (S.G. Gal).
0165-0114/$ - see front matter © 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.fss.2004.08.001
582
B. Bede, S.G. Gal / Fuzzy Sets and Systems 151 (2005) 581 – 599
the Hukuhara derivative. Under this setting are mainly obtained existence and uniqueness theorems
for the solution of a fuzzy differential equation (see e.g. [11,12,14,18,15], etc.). This approach has the
disadvantage that it leads to solutions with increasing support, fact which is solved by interpreting a
fuzzy differential equation as a system of differential inclusions (see e.g. [10,6]). But this last mentioned
approach has at its turn some shortcomings. The main shortcoming is that one cannot talk about the
derivative of a fuzzy-number-valued function, since a fuzzy differential equation is directly interpreted
with the help of differential inclusions without having a derivative. Also solutions are not necessarily
fuzzy-number-valued functions. Another approach can be found in [16], which provides solutions only in
the class of pyramidal fuzzy numbers. In [5] are presented other two methods for solving fuzzy differential
equations. These last two approaches suffer the same disadvantage as the method of differential inclusions,
i.e. the derivative has no meaning.
In this paper we solve this shortcoming by introducing the weakly generalized differential of a fuzzynumber-valued function.
The study of usual known concept of differentiability of fuzzy-number-valued functions (see e.g. [1] or
[17]) shows us that if g : (a, b) → R is differentiable on x0 ∈ (a, b) with g (x0 ) < 0, then f (x) = cg(x),
∀x ∈ (a, b), where c is a fuzzy number, is not in general differentiable on x0 . To solve this shortcoming, in
Section 3 we introduce and study a new generalized concept of differentiability which extends the present
concept and which allows us to have f (x) = c g (x), for all x ∈ (a, b) when g is differentiable. Also,
higher order concepts of differentiability are considered such that to have f (n) (x) = c g (n) (x), for all
x ∈ (a, b), (n 1), when exists g (n) (x) ∈ R.
This generalization allows us to solve in Sections 4 and 5, in a simple way, some higher order partial
and ordinary fuzzy differential equations, whose fuzzy input data (coefficients) are of the form of above
function f , i.e. products of fuzzy numbers with classical real-valued functions.
2. Preliminaries
Given a set X = ∅, a fuzzy subset of X is a mapping u : X → [0, 1] (see [19]).
Let us denote by RF the class of fuzzy subsets of the real axis (i.e. u : R → [0, 1]) satisfying the
following properties:
(i) ∀u ∈ RF , u is normal, i.e. ∃x0 ∈ R with u(x0 ) = 1;
(ii) ∀u ∈ RF , u is convex fuzzy set (i.e. u(tx + (1 − t)y) min{u(x), u(y)}, ∀t ∈ [0, 1], x, y ∈ R);
(iii) ∀u ∈ RF , u is upper semicontinuous on R;
(iv) {x ∈ R; u(x) > 0} is compact, where A denotes the closure of A.
Then RF is called the space of fuzzy numbers (see e.g. [8]). Obviously, R ⊂ RF . Here R ⊂ RF
is understood as R = {{x} ; x is usual real number}. For 0 < r 1, denote [u]r = {x ∈ R; u(x) r}
and [u]0 = {x ∈ R; u(x) > 0}. Then it is well-known that for each r ∈ [0, 1], [u]r is a bounded closed
interval. For u, v ∈ RF , and ∈ R, the sum u⊕v and the product u are defined by [u⊕v]r = [u]r +[v]r ,
[ u]r = [u]r , ∀r ∈ [0, 1], where [u]r + [v]r means the usual addition of two intervals (subsets) of
R and [u]r means the usual product between a scalar and a subset of R (see, e.g. [8,17]).
r |, |ur − v r |}, where [u]r =
Defining D : RF × RF → R+ ∪ {0} by D(u, v) = supr∈[0,1] max{|ur− − v−
+
+
r
r
r
r
r
[u− , u+ ], [v] = [v− , v+ ], the following properties are well-known (see e.g. [9] or [17]):
D(u ⊕ w, v ⊕ w) = D(u, v), ∀u, v, w ∈ RF ,
D(k u, k v) = |k|D(u, v), ∀k ∈ R, u, v ∈ RF ,
B. Bede, S.G. Gal / Fuzzy Sets and Systems 151 (2005) 581 – 599
583
D(u ⊕ v, w ⊕ e) D(u, w) + D(v, e), ∀u, v, w, e ∈ RF
and (RF , D) is a complete metric space.
Also, the following results and concepts are known.
Theorem 1. (See e.g. [2].) (i) If we denote 0 = {0} then 0 ∈ RF is neutral element with respect to ⊕,
i.e. u ⊕ 0 = 0 ⊕ u = u, for all u ∈ RF .
(ii) With respect to 0, none of u ∈ RF \ R, has inverse in RF (with respect to ⊕).
(iii) For any a, b ∈ R with a, b 0 or a, b 0 and any u ∈ RF , we have (a + b) u = a u ⊕ b u;
For general a, b ∈ R, the above property does not hold.
(iv) For any ∈ R and any u, v ∈ RF , we have (u ⊕ v) = u ⊕ v;
(v) For any , ∈ R and any u ∈ RF , we have ( u) = ( · ) u;
Definition 2. (See e.g. [13].) Let x, y ∈ RF . If there exists z ∈ RF such that x = y ⊕ z, then z is called
the H-difference of x and y and it is denoted by x − y.
Definition 3. (See e.g. [13].) A function f : (a, b) → RF is called H-differentiable on x0 ∈ (a, b) if for
h > 0 sufficiently small there exist the H-differences f (x0 + h) − f (x0 ), f (x0 ) − f (x0 − h) and an
element f (x0 ) ∈ RF such that
f (x0 + h) − f (x0 ) f (x0 ) − f (x0 − h) , f (x0 ) = lim D
, f (x0 ) .
0 = lim D
h0
h0
h
h
(Here h at denominator means h1 .)
Let C[0, 1] = {F : [0, 1] → R; F bounded on [0, 1], left continuous for any x ∈ (0, 1], right continuous
on 0 and F has right limit for any x ∈ [0, 1)}. Endowed with the norm ||F || = sup{|F (x)|; x ∈ [0, 1]},
C[0, 1] is a Banach space. It is known the following result which embeds RF into C[0, 1] × C[0, 1]
isometrically and isomorphically.
Theorem 4. (See e.g. [18].) If we define j : RF → C[0, 1] × C[0, 1] by j (u) = (u− , u+ ), where
u− , u+ : [0, 1] → R, u− (r) = ur− , u+ (r) = ur+ , then j (RF ) is a closed convex cone with vertex 0 in
C[0, 1] × C[0, 1] (here C[0, 1] × C[0, 1] is a Banach space with the norm ||(f, g)|| = max{||f ||, ||g||})
and j satisfies:
(i) j (s u ⊕ t v) = sj (u) + tj (v), ∀u, v ∈ RF , s, t 0;
(ii) D(u, v) = ||j (u) − j (v)||.
3. Generalized differentiabilities
Let c ∈ RF and g : (a, b) → R+ be differentiable on x0 ∈ (a, b). Define f : (a, b) → RF by f (x) =
0)
cg(x), for all x ∈ (a, b). Firstly, let us suppose that g (x0 ) > 0. Then by g (x0 ) = limh0 g(x0 +h)−g(x
,
h
it follows that for h > 0 sufficiently small we have g(x0 + h) − g(x0 ) = (x0 , h) > 0. Multiplying by c,
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B. Bede, S.G. Gal / Fuzzy Sets and Systems 151 (2005) 581 – 599
it follows c g(x0 + h) = c g(x0 ) ⊕ c (x0 , h), i.e. there exists the H-difference f (x0 + h) − f (x0 ).
0 −h)
Similarly, by g (x0 ) = limh0 g(x0 )−g(x
, reasoning as above we get that there exists the H-difference
h
f (x0 ) − f (x0 − h) too. Also, simple reasoning shows in this case that f (x0 ) = c g (x0 ).
Now, if we suppose g (x0 ) < 0, we easily see that we cannot use the above kind of reasoning to prove
that the H-differences f (x0 +h)−f (x0 ), f (x0 )−f (x0 −h) and the derivative f (x0 ) exist. Consequently,
by Definition 3 we cannot say that exists f (x0 ). This shortcoming can be solved by introducing some
generalized concepts of differentiability as follows.
Firstly, let us recall the following concept proposed at the end of paper [3].
Definition 5. Let f : (a, b) → RF and x0 ∈ (a, b). We say that f is strongly generalized differentiable
on x0 , if there exists an element f (x0 ) ∈ RF , such that
(i) for all h > 0 sufficiently small, ∃f (x0 + h) − f (x0 ), f (x0 ) − f (x0 − h) and the limits (in the
metric D)
lim
h0
f (x0 + h) − f (x0 )
f (x0 ) − f (x0 − h)
= lim
= f (x0 ),
h0
h
h
or
(ii) for all h > 0 sufficiently small, ∃f (x0 ) − f (x0 + h), f (x0 − h) − f (x0 ) and the limits
lim
h0
f (x0 ) − f (x0 + h)
f (x0 − h) − f (x0 )
= lim
= f (x0 ),
h0
(−h)
(−h)
or
(iii) for all h > 0 sufficiently small, ∃f (x0 + h) − f (x0 ), f (x0 − h) − f (x0 ) and the limits
lim
h0
f (x0 + h) − f (x0 )
f (x0 − h) − f (x0 )
= lim
= f (x0 ),
h0
h
(−h)
or
(iv) for all h > 0 sufficiently small, ∃f (x0 ) − f (x0 + h), f (x0 ) − f (x0 − h) and the limits
lim
h0
f (x0 ) − f (x0 + h)
f (x0 ) − f (x0 − h)
= lim
= f (x0 ).
h0
(−h)
h
(h and (−h) at denominators mean h1 and − h1 , respectively).
Remark 6. (1) This definition is not contradictory, i.e. if for f and x0 , at least two from the possibilities
(i)–(iv) simultaneously hold, then we do not obtain a contradiction. Indeed, let us suppose, for example,
that (i) and (iii) hold. Then by f (x0 + h) = f (x0 ) ⊕ A, f (x0 ) = f (x0 − h) ⊕ B, f (x0 − h) = f (x0 ) ⊕ C,
with A, B, C ∈ RF , we get f (x0 ) = f (x0 ) ⊕ [B ⊕ C], i.e. B ⊕ C = 0, which implies B = C = 0,
case when f (x0 ) = 0, or B, C ∈ R, B = −C, case when f (x0 ) ∈ R. Here R ⊂ RF is understood as
R = {{x} ; x is usual real number}. But in all these cases it is easy to see that all the limits in the previous
definition are equal. Similar conclusion follows for any other combination from (i) to (iv).
(2) If f is strongly generalized differentiable on x0 ∈ (a, b) by Definition 5(i), then by [17, Lemma
3.2] it follows that j ◦ f : (a, b) → C[0, 1] × C[0, 1] is Fréchet differentiable at x0 and (j ◦ f ) (x0 ) =
j (f (x0 )). To characterize the other possibilities too in Definition 5, we introduce another embedding by
B. Bede, S.G. Gal / Fuzzy Sets and Systems 151 (2005) 581 – 599
585
j : RF → C[0, 1] × C[0, 1], j (u) = j ((−1) u), u ∈ RF . It is easy to prove the following properties:
j (u) − j (v) = D(u, v), j (RF ) = j (RF ) and j (s u ⊕ t v) = s j (u) ⊕ t j (v), for all
u, v ∈ RF , t, s 0.
We have
Theorem 7. Let f : (a, b) → RF be strongly generalized differentiable on each point x ∈ (a, b) in the
sense of Definition 5(iii) or 5(iv). Then f (x) ∈ R for all x ∈ (a, b).
Proof. Suppose that f is differentiable on x as in Definition 5(iii). Then the H-differences f (x+h)−f (x)
and f (x − h) − f (x) exist for h > 0 sufficiently small. Then we have
f (x + h) = f (x) ⊕ u(x, h)
(1)
f (x − h) = f (x) ⊕ v(x, h)
(2)
and
for h > 0 sufficiently small. If we take in (2) x = x + h we obtain :
f (x) = f (x + h) ⊕ v(x + h, h)
(3)
for h > 0 sufficiently small.
By adding (1) and (3) it follows that u(x, h) ⊕ v(x + h, h) = 0. By Theorem 1(ii) we obtain
∈ R.
u(x, h), v(x+h, h) ∈ R for h > 0 sufficiently small. Then it is easy to see that f (x) = limh0 u(x,h)
h
If f is differentiable on x as in Definition 5(iv), the reasonings are similar. Theorem 8. Let f : (a, b) → RF and x0 ∈ (a, b).
(i) If f is strongly generalized differentiable on x0 according to Definition 5(ii), then j ◦ f is Fréchet
differentiable on x0 and (j ◦ f ) (x0 ) = −
j (f (x0 )); (Here j is defined by Remark 6(2) after Definition
5).
(ii) If f is strongly generalized differentiable on x0 according to Definition 5(iii), then j ◦ f is Fréchet
differentiable at right and at left on x0 and (j ◦ f )r (x0 ) = j (f (x0 )), (j ◦ f )l (x0 ) = −
j (f (x0 )).
(iii) If f is strongly generalized differentiable on x0 according to Definition 5(iv), then j ◦ f is Fréchet
differentiable at left and at right on x0 and (j ◦ f )l (x0 ) = j (f (x0 )), (j ◦ f )r (x0 ) = −
j (f (x0 )).
Proof. Firstly, let us recall that a mapping F : (a, b) → X, where (X, ·) is a real normed space, is called
Fréchet differentiable at right on x0 if there exists a linear continuous mapping, denoted F (x0 ) : R → X,
such that
F (x0 + h) − F (x0 ) − F (x0 )(h)
lim
= 0.
h0
|h|
If the above limit is taken for h0 then we get the concept of Fréchet differentiability at left on x0 . The
Fréchet differentiability on x0 coincides with the simultaneous right and left Fréchet differentiability.
Since the proofs of (i), (ii) and (iii) are similar, to have a sample, here we will prove, for example, (ii).
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B. Bede, S.G. Gal / Fuzzy Sets and Systems 151 (2005) 581 – 599
Let h0. We have
(j ◦ f )(x0 + h) − (j ◦ f )(x0 )
−
j
(f
(x
))
0
h
(j ◦ f )(x0 + h) − (j ◦ f )(x0 )
f (x0 + h) − f (x0 ) −j
h
h
f (x0 + h) − f (x0 )
+
− j (f (x0 ))
j
.
h
By f (x0 + h) = f (x0 ) ⊕ u we get j (f (x0 + h)) = j (f (x0 )) + j (u), i.e. j (u) = j (f (x0 + h) − f (x0 )) =
j (f (x0 + h)) − j (f (x0 )), which immediately implies that
f (x0 + h) − f (x0 )
(j ◦ f )(x0 + h) − (j ◦ f )(x0 )
j
=
.
h
h
Also,
f (x0 + h) − f (x0 )
+
h)
−
f
(x
)
f
(x
h0
0
0
j
− j (f (x0 ))
, f (x0 ) −→ 0.
=D
h
h
Therefore, passing to h0, it follows
(j ◦ f )(x0 + h) − (j ◦ f )(x0 ) − j (f (x0 )) · h = 0,
lim
h0 |h|
i.e. j ◦ f is Fréchet differentiable at right on x0 , and (j ◦ f )r (x0 ) = j (f (x0 )).
On the other hand, for the same h0, we have
(j ◦ f )(x0 − h) − (j ◦ f )(x0 )
−
[−
j
(f
(x
))]
0
(−h)
(j ◦ f )(x0 − h) − (j ◦ f )(x0 )
= + j (f (x0 ))
(−h)
(j ◦ f )(x0 − h) − (j ◦ f )(x0 )
f
(x
−
h)
−
f
(x
)
0
0
+
j
(−h)
−h
f (x0 − h) − f (x0 )
+
−
j (f (x0 ))
j
.
−h
(We used a + b = (a + c) + (b − c), a + b a + c + b − c = a + c + c − b.)
But
f (x0 − h) − f (x0 )
− j (f (x0 ))
j
=
−h
f (x0 − h) − f (x0 ) h0
D
, f (x0 ) −→ 0.
−h
Also,
f (x0 − h) − f (x0 )
j (f (x0 − h)) − j (f (x0 ))
f (x0 − h) − f (x0 )
j
=j
=
,
−h
h
h
B. Bede, S.G. Gal / Fuzzy Sets and Systems 151 (2005) 581 – 599
587
because by f (x0 − h) − f (x0 ) = u, we get f (x0 − h) = f (x0 ) ⊕ u, j (f (x0 − h)) − j (f (x0 )) = j (u)
and j (u) = j (u) for 0.
As a conclusion,
(j ◦ f )(x0 − h) − (j ◦ f )(x0 )
f (x0 − h) − f (x0 ) = 0,
+j
(−h)
−h
and therefore denoting k = −h,
(j ◦ f )(x0 + k) − (j ◦ f )(x0 )
lim − [−
j (f (x0 )]
= 0,
k0
k
i.e. (j ◦ f )l (x0 ) = −
j (f (x0 )). Remark 9. It is easy to show that j (f (x0 )) = −
j (f (x0 )) if and only if f (x0 ) ∈ R.
Another important result is
Theorem 10. If g : (a, b) → R is differentiable on (a, b) such that g has at most a finite number of
roots in (a, b) and c ∈ RF , then f (x) = c g(x) is strongly generalized differentiable on (a, b) and
f (x) = c g (x), ∀x ∈ (a, b).
Proof. For x0 ∈ (a, b) we have the possibilities:
(i) g(x0 ) > 0, g (x0 ) > 0; (ii) g(x0 ) > 0, g (x0 ) < 0; (iii) g(x0 ) > 0, g (x0 ) = 0; (iv) g(x0 ) < 0,
g (x0 ) > 0; (v) g(x0 ) < 0, g (x0 ) < 0; (vi) g(x0 ) < 0, g (x0 ) = 0; (vii) g(x0 ) = 0, g (x0 ) > 0; (viii)
g(x0 ) = 0, g (x0 ) < 0; (ix) g(x0 ) = 0, g (x0 ) = 0.
0)
0 −h)
Case (i): Let g (x0 ) = limh0 g(x0 +h)−g(x
= limh0 g(x0 )−g(x
. For h > 0, sufficiently small,
h
h
g(x0 + h) > 0, g(x0 − h) > 0, g(x0 + h) − g(x0 ) = 1 (x0 , h) > 0, g(x0 ) − g(x0 − h) = 2 (x0 , h) > 0,
i.e. g(x0 + h) = g(x0 ) + 1 (x0 , h), g(x0 ) = g(x0 − h) + 2 (x0 , h). Multiplying by c ∈ RF , we get
(x0 )
that there exist f (x0 + h) − f (x0 ), f (x0 ) − f (x0 − h) and that f (x0 ) = limh0 f (x0 +h)−f
=
h
f (x0 )−f (x0 −h)
limh0
= c g (x0 ), i.e. f is strongly generalized differentiable by Definition 5(i).
h
0 +h)
0)
Case (ii): Let g (x0 ) = limh0 g(x0 )−g(x
= limh0 g(x0 −h)−g(x
. For h > 0, sufficiently small, as
−h
−h
above g(x0 + h) > 0, g(x0 − h) > 0, and then g(x0 ) − g(x0 + h) = 1 (x0 , h) > 0, g(x0 − h) − g(x0 ) =
2 (x0 , h) > 0. Multiplying by c ∈ RF , we get that there exist f (x0 ) − f (x0 + h), f (x0 − h) − f (x0 )
and that f (x0 ) = c g (x0 ), according to Definition 5(ii).
Case (iii): Firstly, let us suppose that x0 is an extremum point. If x0 is a minimum point then for h > 0
sufficiently small we have g(x0 + h) g(x0 ) > 0, g(x0 − h) g(x0 ) > 0, i.e. g(x0 + h) − g(x0 ) =
1 (x0 , h) 0, g(x0 − h) − g(x0 ) = 2 (x0 , h) 0. Multiplying as above by c ∈ RF we easily get that f
is differentiable on x0 according to Definition 5(iii), and that f (x0 ) = c g (x0 ) = c 0 = 0. If x0 is
a maximum point, then the reasonings are similar.
Now, if g (x0 ) = 0 but x0 is not an extremum point, then there exists an interval (x0 − , x0 + ) ⊂ (a, b),
such that g (x) = 0, ∀x ∈ (x0 − , x0 + ) \ {x0 } and since g has the Darboux’s property, it follows that
g (x) > 0, ∀x ∈ (x0 − , x0 + ) \ {x0 } or g (x) < 0, ∀x ∈ (x0 − , x0 + ) \ {x0 }.
If g (x) > 0, ∀x ∈ (x0 − , x0 + ) \ {x0 }, it follows g(x0 + h) − g(x0 ) = 1 (x0 , h) 0, g(x0 ) −
g(x0 − h) = 2 (x0 , h) 0 and f is differentiable according to Definition 5(i).
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B. Bede, S.G. Gal / Fuzzy Sets and Systems 151 (2005) 581 – 599
If g (x) < 0, ∀x ∈ (x0 − , x0 + ) \ {x0 }, it follows g(x0 − h) − g(x0 ) = 1 (x0 , h) 0, g(x0 ) −
g(x0 + h) = 2 (x0 , h) 0 for all 0 < h < , and f is differentiable according to Definition 5(ii).
The proofs of cases (iv)–(viii) are similar to the proofs of the above cases (i), (ii), (iii). The proof of
the last case (ix) is similar to that of case (iii), which proves the theorem. The concept in Definition 5 can be generalized by the following.
Definition 11. Let f : (a, b) → RF and x0 ∈ (a, b). For a sequence hn 0 and n0 ∈ N, let us denote
(1)
An(1)
=
n
n
;
∃E
:
=
f
(x
+
h
)
−
f
(x
)
0
0
n
0 ,
n
0
(2)
An(2)
=
n
n
;
∃E
:
=
f
(x
)
−
f
(x
+
h
)
0
0
0
n ,
n
0
(3)
=
n
n
;
∃E
:
=
f
(x
)
−
f
(x
−
h
)
An(3)
0
0
0
n ,
n
0
(4)
An(4)
=
n
n
;
∃E
:
=
f
(x
−
h
)
−
f
(x
)
0
0
n
0 .
n
0
We say that f is weakly generalized differentiable on x0 , if for any sequence hn 0, there exists n0 ∈ N,
(1)
(2)
(3)
(4)
such that An0 ∪ An0 ∪ An0 ∪ An0 = {n ∈ N; n n0 } and moreover, there exists an element in RF
(j )
denoted by f (x0 ) such that if for some j ∈ {1, 2, 3, 4} we have card(An0 ) = +∞, then
(j )
En
lim D
, f (x0 ) = 0.
n→∞
(−1)j +1 hn
(j )
n∈An
0
(1)
(2)
Remark 12. (1) Definition 11 is not contradictory. Indeed, if there exists n1 ∈ An0 ∩ An0 , then we get
f (x0 +hn1 ) = f (x0 )⊕c1 , f (x0 ) = f (x0 +hn1 )⊕c2 , which implies f (x0 ) = f (x0 )⊕c1 ⊕c2 . By Theorem
(1)
(2)
1(ii), we easily get c1 = −c2 ∈ R and therefore f (x0 ) = f (x0 + hn1 ). Now if card[An0 ∩ An0 ] = +∞,
(3)
(4)
then it is immediate that f (x0 ) = 0̃. The same conclusion is obtained if there exists n1 ∈ An0 ∩ An0 .
(2) Let f : (a, b) → RF and let us suppose that there exists f (x) for all x ∈ (a, b), then we will say
that f is weakly generalized differentiable of order two on x0 ∈ (a, b), if the conditions in Definition 11
are satisfied for f replaced by f . By iterations, we can define the differentiability of order n ∈ N, n > 2.
Similarly we can introduce strongly generalized differentiability of higher order based on Definition 5.
The concept in Definition 11 allows to prove the following.
Theorem 13. Let c ∈ RF and g : (a, b) → R. If g is differentiable on x0 (in usual sense), then the function
f : (a, b) → RF defined by f (x) = c g(x), ∀x ∈ (a, b), is weakly generalized differentiable on x0 and
we have f (x0 ) = c g (x0 ).
Proof. Concerning the function g we have the following three possibilities: g(x0 ) > 0, g(x0 ) < 0, and
g(x0 ) = 0. Also to the above three cases correspond the following possibilities: g (x0 ) > 0, g (x0 ) < 0,
B. Bede, S.G. Gal / Fuzzy Sets and Systems 151 (2005) 581 – 599
589
and g (x0 ) = 0. Then we have the following nine possibilities:
(i) g(x0 ) > 0, g (x0 ) > 0; (ii) g(x0 ) > 0, g (x0 ) < 0; (iii) g(x0 ) > 0, g (x0 ) = 0;
(iv) g(x0 ) < 0, g (x0 ) > 0; (v) g(x0 ) < 0, g (x0 ) < 0; (vi) g(x0 ) < 0, g (x0 ) = 0;
(vii) g(x0 ) = 0, g (x0 ) > 0; (viii) g(x0 ) = 0, g (x0 ) < 0; (ix) g(x0 ) = 0, g (x0 ) = 0.
Case (i) : Let g(x0 ) > 0 and g (x0 ) > 0. Because g is continuous on x0 , for h > 0 sufficiently small
we have g(x0 + h) > 0. Also, from g (x0 ) > 0 we get g(x0 + h) − g(x0 ) = (x0 , h) > 0, for h > 0,
sufficiently small. It follows c g(x0 + h) = c g(x0 ) ⊕ c (x0 , h) (see Theorem 1(iii)) and
f (x0 + h) − f (x0 )
c (x0 , h)
g(x0 + h) − g(x0 )
=
=c
,
h
h
h
(1)
for h > 0 sufficiently small. Let hn 0. It follows that there exists n0 ∈ N such that An0 = {n ∈
(2)
(3)
(4)
g(x0 +hn )−g(x0 )
N; n n0 }, An0 = An0 = An0 = ∅ and because limhn 0
= g (x0 ), we easily get that
hn
there exists f (x0 ) and f (x0 ) = c g (x0 ) (according to Definition 11).
Case (ii) : Let g(x0 ) > 0 and g (x0 ) < 0. Then, as above g(x0 + h) > 0 for h > 0 sufficiently small
0 +h)
and by g (x0 ) = limh0 g(x0 )−g(x
, we get g(x0 ) − g(x0 + h) = (x0 , h) > 0, for h > 0, sufficiently
−h
small. Multiplying by c ∈ RF , we get
f (x0 ) − f (x0 + h)
c (x0 , h)
g(x0 ) − g(x0 + h)
=
=c
.
−h
−h
−h
(2)
(1)
(3)
(4)
Then for hn 0, there exists n0 ∈ N such that An0 = {n ∈ N; n n0 } and An0 = An0 = An0 = ∅,
which easily implies that there exists f (x0 ) (according to Definition 11) and f (x0 ) = c g (x0 ).
Case (iii) : As above, g(x0 + h) > 0 for h > 0 sufficiently small. Let hn 0. We have
0 = limhn 0 g(x0 +hhnn)−g(x0 ) . Let us denote B+ = {n ∈ N; g(x0 + hn ) − g(x0 ) 0} and B− = {n ∈
N; g(x0 + hn ) − g(x0 ) < 0}. Obviously N = B+ ∪ B− . But if n ∈ B+ , then reasoning as in the above
(1)
(2)
case (i) we get that n ∈ A1 and if n ∈ B− , then reasoning as in the above case (ii) we get that n ∈ A1 .
In both cases, according to Definition 11, there exists f (x0 ) and f (x0 ) = c g (x0 ) = 0.
Case (iv) : Let g(x0 ) < 0 and g (x0 ) > 0. For h > 0 sufficiently small it follows g(x0 +h) < 0. Also by
0 +h)
g (x0 ) = limh0 g(x0 )−g(x
> 0, it follows that for h > 0, sufficiently small that g(x0 ) − g(x0 + h) =
−h
(x0 , h) < 0, i.e. g(x0 ) = g(x0 + h) + (x0 , h). Multiplying by c ∈ RF , we get that there exists
(2)
f (x0 ) − f (x0 + h), for h > 0 sufficiently small. Let hn 0. There exists n0 ∈ N such that An0 = {n ∈
(1)
(3)
(4)
N; n n0 } and An0 = An0 = An0 = ∅, and by
f (x0 ) − f (x0 + hn )
g(x0 ) − g(x0 + hn )
=c
−hn
−hn
we easily get that there exists f (x0 ) = c g (x0 ).
Case (v): Let g(x0 ) < 0, g (x0 ) < 0. For h > 0 sufficiently small, g(x0 + h) < 0. Also by
0)
limh0 g(x0 +h)−g(x
= g (x0 ) < 0, we get g(x0 + h) − g(x0 ) = (x0 , h) < 0, for h > 0, suffih
ciently small. Then c g(x0 + h) = c g(x0 ) ⊕ c (x0 , h), i.e. there exists f (x0 + h) − f (x0 ), and
reasoning as in the above case (i), we get f (x0 ) = c g (x0 ).
Case (vi): Let g(x0 ) < 0 and g (x0 ) = 0. Firstly we have g(x0 + h) < 0 for all h > 0, sufficiently
0 +hn )
small. Let hn 0. We have 0 = limhn 0 g(x0 )−g(x
. The rest of the proof is similar to the proof
−hn
of (iii).
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B. Bede, S.G. Gal / Fuzzy Sets and Systems 151 (2005) 581 – 599
0)
Case (vii) : Let g(x0 ) = 0 and g (x0 ) > 0. By limh0 g(x0 +h)−g(x
= g (x0 ) > 0, it follows that
h
g(x0 + h) > 0 for all h > 0, sufficiently small. By reasoning similar to those in (i) we get f (x0 ) =
c g (x0 ).
0 +h)
Case (viii) : Let g(x0 ) = 0 and g (x0 ) < 0. By limh0 g(x0 )−g(x
= g (x0 ) < 0, it follows that
−h
g(x0 + h) < 0 for all h > 0, sufficiently small. By reasoning similar to those in (v) we get f (x0 ) =
c g (x0 ).
Case (ix) : Let g(x0 ) = 0 and g (x0 ) = 0. We get limh0 g(x0h+h) = 0. Let hn 0. Reasoning as in the
case of (iii) we get f (x0 ) = c g (x0 ) = c 0 = 0.
The theorem is proved. Remark 14. From the proof of Theorem 13, we see that for the differentiability of f (x) = c g(x),
(1)
(2)
with c ∈ RF and g : (a, b) → R, we used only the sets An0 and An0 in Definition 11, the other two sets
(3)
(4)
An0 and An0 being empty. However, in general there exist f : (a, b) → RF which are not of the form in
(1)
(2)
(4)
Theorem 13 such that for some hn 0 and x0 ∈ (a, b) we have, for example, An0 = An0 = An0 = ∅
(3)
and card(An0 ) = +∞.
Corollary 15. Let g : (a, b) → R, c ∈ RF and define f : (a, b) → RF by f (x) = c g(x), for all
x ∈ (a, b). If g is differentiable on (a, b) and g is differentiable on x0 ∈ (a, b), then f is differentiable
on (a, b) and twice differentiable on x0 , with f (x0 ) = c g (x0 ).
Proof. It is an immediate consequence of Theorem 13.
Remark 16. In general, if the above g is n − 1 times differentiable on (a, b) and g (n−1) is differentiable
on x0 , then f = c g is differentiable of order n on x0 and f (n) (x0 ) = c g (n) (x0 ).
4. Applications to fuzzy differential equations
In this section we discuss the existence and uniqueness of the solutions of a fuzzy differential equation.
The following theorem is the Newton–Leibniz formula for integrals of fuzzy-number-valued functions
corresponding to strongly generalized differentiability. In the case of differentiability in the sense of
Definition 5(i) it can be found in [17, Theorem 3.6].
Remark 17. Everywhere in what follows the integral is the Henstock one in [17, Definition 2.1]. If
b
g(x) is continuous on [a, b], then the Henstock integral a g(x) dx becomes that of Riemann. Here the
b
Riemann integral a g(x) dx means that
n−1
b
g(x) dx,
lim D
(xk+1 − xk ) f (k ) = 0,
(dn ) → 0
a
k=0
for all divisions of (a, b) , dn : x0 = a < x1 < · · · < xk < xk+1 < · · · < b = xn and all k ∈ (xk , xk+1 ) ,
k = 0, n − 1, when (dn ) denotes the norm of division dn .
B. Bede, S.G. Gal / Fuzzy Sets and Systems 151 (2005) 581 – 599
591
Theorem 18. Let f : [a, b] → RF and let a = b0 < b1 < · · · < bn = b be a division of the interval
[a, b] such that f is (i) or (ii)-differentiable in the sense of Definition 5 on each of the intervals [bi−1 , bi ],
i = 1, . . . , n, with the same kind of differentiability on each subinterval. Then
b
f (x) dx =
(f (bi ) − f (bi−1 )) ⊕ (−1) (f (bk−1 ) − f (bk )),
a
i∈I
k∈J
where
I = {i ∈ {1, . . . , n} such that f is (i)-differentiable on (bi−1 , bi )}
and
J = {k ∈ {1, . . . , n} such that f is (ii)-differentiable on (bk−1 , bk )}.
Proof. Suppose that f is (i)-differentiable on (bi−1 , bi ). Then for i ∈ I we have by [17, Theorem 3.6]
bi
f (x) dx = f (bi ) − f (bi−1 ), ∀i ∈ I.
(4)
bi−1
Let k ∈ J . By the Newton–Leibniz formula for functions with values in a Banach space (see [17,
Theorem 3.5]) we have
bk
(j ◦ f ) (x) dx.
(j ◦ f )(bk ) = (j ◦ f )(bk−1 ) +
bk−1
By Theorem 8 there exists (j ◦ f ) (x) and we get
bk
(−
j (f (x))) dx.
(j ◦ f )(bk ) = (j ◦ f )(bk−1 ) +
bk−1
Since the embedding j commutes with the integral (see e.g. [17, Theorem 3.4]) we obtain
bk
f (x) dx .
(j ◦ f )(bk ) = (j ◦ f )(bk−1 ) − j
bk−1
Then it follows that
bk
f (x) dx + (j ◦ f )(bk ) = (j ◦ f )(bk−1 ).
j
bk−1
By the definition of j we obtain
bk
f (x) dx + j (f (bk )) = j (f (bk−1 )).
j (−1) bk−1
By the additivity of the embedding j we have
bk
f (x) dx = f (bk−1 ) − f (bk )
(−1) bk−1
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B. Bede, S.G. Gal / Fuzzy Sets and Systems 151 (2005) 581 – 599
and finally
bk
bk−1
f (x) dx = (−1) (f (bk−1 ) − f (bk )), ∀k ∈ J
(5)
By adding (4) and (5) for all i ∈ I and k ∈ J, the required result is obtained. Remark 19. If on [bi−1 , bi ] and [bi , bi+1 ] f in Theorem 18 is of different differentiability, then it follows
that f is simultaneously (i) and (ii) differentiable at bi which immediately implies that f (bi ) ∈ R. Also,
note that (i)+(ii) differentiability is equivalent to (iii)+(iv) differentiability.
Based on the previous Theorem 18 we can obtain some results on the existence of solutions of fuzzy
differential equations. A fuzzy differential equation is an equation of the form y = f (x, y) where we
search for solutions y : R → RF which are strongly generalized differentiable on any interval (a, b) with
at most a finite set of points in which y is (iii)+(iv)-differentiable. The following lemma transforms the
fuzzy differential equation into integral equations.
Lemma 20. For x0 ∈ R, the fuzzy differential equation y = f (x, y), y(x0 ) = y0 ∈ RF where f :
R × RF → RF is supposed to be continuous, is equivalent to one of the integral equations:
x
f (t, y(t)) dt, ∀x ∈ [x0 , x1 ]
y(x) = y0 ⊕
x0
or
y0 = y(x) ⊕ (−1) x
x0
f (t, y(t)) dt, ∀x ∈ [x0 , x1 ]
on some interval (x0 , x1 ) ⊂ R, depending on the strongly differentiability considered, (i) or (ii), respectively.
Here the equivalence between two equations means that any solution of an equation is a solution too
for the other one.
Proof. It is immediate by Theorem 18. Remark 21. In the case of strongly generalized differentiability, to the fuzzy differential equation y =
f (x, y) we may attach two different integral equations, while in the case of differentiability in the sense
of Definition 3, we may attach
x only one. The second integral equation in Lemma 20 can be written in the
form y(x) = y0 − (−1) x0 f (t, y(t)) dt.
Now, we extend Theorem 4.1 in [18] for fuzzy differential equations with strongly generalized differentiability.
Theorem 22. Let us suppose that the following conditions hold:
(a) Let R0 = [x0 , x0 + p] × B(y0 , q), p, q > 0, y0 ∈ RF , where B(y0 , q) = {y ∈ RF : D(y, y0 ) q}
denote a closed ball in RF and let f : R0 → RF be a continuous function such that D(
0, f (x, y)) =
f (x, y)F M for all (x, y) ∈ R0 .
B. Bede, S.G. Gal / Fuzzy Sets and Systems 151 (2005) 581 – 599
593
(b) Let g : [x0 , x0 + p] × [0, q] → R, such that g(x, 0) ≡ 0 and 0 g(x, u) M1 , ∀x ∈ [x0 , x0 + p],
0 u q, such that g(x, u) is nondecreasing in u and g is such that the initial value problem u (x) =
g(x, u(x)), u(x0 ) = 0 has only the solution u(x) ≡ 0 on [x0 , x0 + p].
(c) We have D(f (x, y), f (x, z)) g(x, D(y, z)), ∀(x, y), (x, z) ∈ R0 and D(y, z) q.
(d) There exists d > 0 such that for x ∈ [x0 , x0 + d] the sequence y n : [x0 , x0 + d] → RF given by
x
y 0 (x) = y0 , y n+1 (x) = y0 − (−1) x0 f (t, y n (t)) dt is defined for any n ∈ N.
Then the fuzzy initial value problem
y = f (x, y),
y(x0 ) = y0
has two solutions (one differentiable as in Definition 5(i)
and the other
one differentiable as in Definition
q
5 (ii)) y, y : [x0 , x0 + r] → B(y0 , q) where r = min p, M
, Mq1 , d and the successive iterations
y0 (x) = y0
yn+1 (x) = y0 ⊕
x
x0
f (t, yn (t)) dt,
(6)
and
y 0 (x) = y0
y n+1 (x) = y0 − (−1) x
x0
f (t, y n (t)) dt
(7)
converge to these two solutions, respectively.
Proof. For the case of (i)-differentiability one obtains the solution y (see [18, Theorem 4.1])
For the case of (ii)-differentiability we have by (7)
x
D(y n+1 (x), y0 ) D(f (t, y n (t)), 0) dt.
x0
Also, by the properties of the distance D we have
x
D(y 1 (x), y0 (x)) = D y 1 (x), y 1 (x) ⊕ (−1) f (t, y 0 (t)) dt
x0 x
x
f (t, y 0 (t)) dt, 0 D(f (t, y 0 (t)), 0) dt.
=D
x0
x0
(8)
(9)
By the invariance to translation of distance D and (7) we have
D(y n+1 (x), y n (x)) = D(y0 − y n+1 , y0 − y n (x))
x
x
= D (−1) f (t, y n (t)) dt, (−1) f (t, y n−1 (t)) dt
x0
x0
x
D f (t, y n (t)), f (t, y n−1 (t)) dt.
x0
(10)
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B. Bede, S.G. Gal / Fuzzy Sets and Systems 151 (2005) 581 – 599
Let x0 x < x0 + r and h such that x + h x0 + r. We observe that
x+h
y n+1 (x) − y n+1 (x + h) = (−1) f (t, y n (t)) dt.
x
Indeed, we have by direct computation
x+h
y n+1 (x + h) ⊕ (−1) f (t, y n (t)) dt
x
x+h
f (t, y n (t)) dt ⊕ (−1) = y0 − (−1) = y0 − (−1) x
0x+h
x0
x+h
x
(11)
f (t, y n (t)) dt
f (t, y n (t)) dt ⊕
x
f (t, y n (t)) dt − (−1) f (t, y n (t)) dt
x0
x0
x
f (t, y n (t)) dt = y n+1 (x).
= y0 − (−1) ⊕ (−1) x+h
x0
and passing to limit with h0 we have by Definition 5,
x+h
y n+1 (x) − y n+1 (x + h)
1
= lim f (t, y n (t)) dt.
lim
h0
h0 h
−h
x
Multiplying by
1
−h
We observe that
x+h
1
D
f (t, y n (t)) dt, f (x, y n (x))
h
x
x+h
x+h
1
1
f (t, y n (t)) dt, f (x, y n (x)) dt
= D
h
h
x
x
x+h
1
D f (t, y n (t)), f (x, y n (x)) dt
h
x
sup D f (t, y n (t)), f (x, y n (x)) .
|t−x| h
and since f is continuous, for h0 the last term 0 which means that
y n+1 (x) − y n+1 (x + h)
= f (x, y n (x)).
h0
−h
lim
Similar to (11) we can obtain
y n+1 (x − h) − y n+1 (x) = (−1) x
x−h
f (t, y n (t) dt
which by analogous reasonings leads to
y n+1 (x − h) − y n+1 (x)
= f (x, y n (x)).
h0
−h
lim
(12)
B. Bede, S.G. Gal / Fuzzy Sets and Systems 151 (2005) 581 – 599
595
Finally, it follows that y n+1 is (ii)-differentiable and
y n+1 = f (x, y n (x)), ∀x ∈ [x0 , x0 + d].
(13)
Relations (8), (9), (10), (13) allow to prove the existence of the solution y as in the proof of Theorem
4.1 in [18]. The following theorem deals with fuzzy differential equations with input data triangular fuzzy-numervalued functions. We recall that for a < b < c, a, b, c ∈ R, the triangular fuzzy number u = (a, b, c)
determined by a, b, c is given such that ur− = a + (b − a)r and ur+ = c − (c − b)r are the endpoints of
the r-level sets, for all r ∈ [0, 1]. Here u1− = u1+ = b and it is denoted by u1 . The set of triangular fuzzy
numbers will be denoted by RT . The following lemma gives a sufficient condition for the existence of
the H-difference of two triangular fuzzy numbers.
0 − v 0 ) min{u1 −
Lemma 23. Let u, v ∈ RT be such that u1 − u0− > 0, u0+ − u1 > 0 and len(v) = (v+
−
0
0
1
u− , u+ − u }. Then the H-difference u − v exists.
0 len(v) u1 − u0 and v 0 − v 1 len(v) u0 − u1 we obtain
Proof. Since v 1 − v−
−
+
+
0
0
u0− − v−
u1 − v 1 u0+ − v+
.
0 , u1 − v 1 , u0 − v 0 define a triangular fuzzy number w such that w 0 = u0 − v 0 ,
It follows that u0− − v−
+
+
−
−
−
1
1
1
0
0 . It is easy to see that w ⊕ v = u, i.e. the H-difference u − v exists. w = u − v , w+ = u0+ − v+
Corollary 24. Let f : R0T → RT , where R0T = [x0 , x0 + p] × (B(y0 , q) ∩ RT ), and y0 ∈ RT such that
(y0 )1 − (y0 )0− > 0 and (y0 )0+ − (y0 )1 > 0. Let m = min{(y0 )1 − (y0 )0− , (y0 )0+ − (y0 )1 }. Under the
assumptions (a), (b) and (c) of the preceding Theorem 22 the fuzzy initial value problem
y = f (x, y)
y(x0 ) = y0
q
m
has two solutions y, y : [x0 , x0 + r] → B(y0 , q) where r = min p, M
, Mq1 , 2M
and the successive
iterations in relations (6), (7) converge to the two solutions.
Proof. It is easy to check that for any fuzzy number we have len(u) 2 uF . Let h <
denote the functions in (7). Then we have
x0 +h
f (t, y n (t)) dt
len (−1) x0
x0 +h
f (t, y n (t)) dt 2h · M < m.
= len
m
2M
and let y n
x0
By the preceding Lemma 23, for x ∈ [x0 , x0 + r], the H-differences in (7) exist for all n ∈ N, which
completes the proof. 596
B. Bede, S.G. Gal / Fuzzy Sets and Systems 151 (2005) 581 – 599
The following Peano-type theorem provides solutions of fuzzy differential equations with strongly
generalized differentiability, under some relaxed conditions.
Theorem 25. Let R0 = [x0 , x0 +p]×B(y0 , q), p, q > 0, y0 ∈ RF and f : R0 → RF be continuous such
that f (x, y)F M for all (x, y) ∈ R0 and f satisfies the Lipschitz condition D(f (x, y), f (x, z)) L ·
D(y, z), ∀(x, y), (x, z) ∈ R0 and D(y, z) q. If there exists
x d > 0 such that for x ∈ [x0 , x0 + d] the
sequence given by y 0 (x) = y0 , y n+1 (x) = y0 − (−1) x0 f (t, y n (t)) dt is defined for any n ∈ N, then
the fuzzy initial value problem
y = f (x, y),
y(x0 ) = y0
q
has two solutions y, y : [x0 , x0 + r] → B(y0 , q) where r = min p, M
, Mq1 , d and the successive iterations in (6), (7) converge to these two solutions, respectively.
Proof. The proof is immediate by taking in Theorem 22, g(x, u) = L · u. Remark 26. For fuzzy initial value problem with strongly generalized differentiability, the existence of
two solutions in a neighborhood of a point x0 generates a way of choosing which kind of differentiability
is expected for the solution, as follows. If on an interval we expect solution with increasing support
then we find (i)-differentiable solution. If we expect decreasing support then we find (ii)-differentiable
solution.
Example 27. In [7] it is proved the variation of constants formula for fuzzy differential equations, using
the approach given in [10], which interprets a fuzzy differential equation as differential inclusions. In
[7] it is solved, for example, the fuzzy differential equation y (x) = −2y, y(0) = (0, 1)s , where (c, d)s
denotes the symmetric triangular fuzzy number having the interval [c, d] as support. Since in the crisp
case this equation has a solution that decreases asymptotically to 0, we expect a solution with decreasing
support, i.e. (ii)-differentiable. In [7] it is obtained that the solution
(i.e. a function which
to this problem
−2x
satisfy some differential inclusions) has the level sets [y(x)] = 2 e , 1 − 2 e−2x , ∈ [0, 1], so
y(x) = y(0) e−2x . It is easy to see that y(x) − y(x + h) = y(0) (e−2x − e−2(x+h) ) exists and by
Definition 5(ii) we have y (x) = limh0 y(x)−y(x+h)
= (−2) y(x).
−h
Remark 28. According to the previous example, another possible approach for choosing (i) or (ii)
differentiability can be the following. We solve some differential inclusions for the 0 level sets and
find the nature of the support of a solution (i.e. determine intervals on which it is increasing or decreasing). Then on the intervals on which it is increasing we choose (i)-differentiability and on the intervals
on which it is decreasing we choose (ii)-differentiability.
Remark 29. Lemma 23 suggests us the following question of independent interest: given u, v ∈ RF ,
find simple sufficient conditions for the existence of H-difference u − v.
B. Bede, S.G. Gal / Fuzzy Sets and Systems 151 (2005) 581 – 599
597
5. Concrete examples
In this section we present applications to some concrete examples of fuzzy (partial) differential
equations whose fuzzy input data (fuzzy coefficients) are products of fuzzy numbers with real-valued
functions.
Example 30. Let us consider the initial value problem (for fuzzy partial differential equation)
*u
(t, x)
*t
= (−1) u(t, x), t 0, x ∈ [a, b],
u(0, x) = u0 (x), x ∈ [a, b],
where u : R+ × [a, b] → RF .
If we denote u(t, x) = e−t u0 (x), then u is strongly generalized differentiable with respect to all t,
*u
(t, x) = (−1) e−t u0 (x) (see Theorem 10) and u(t, x) satisfies the above initial value problem.
*t
On the other hand, because e−t > 0 and (e−t ) < 0, ∀t ∈ R+ , we cannot say nothing about the existence
of **ut (t, x) in the H-differentiability sense, since there not exists the H-difference u (t + h, x) − u(t, x),
when u (t, x) ∈ RF \ R.
This shows the advantage of strongly generalized differentiability with respect to the usual differentiability.
Example 31. Let us consider the second order fuzzy differential equation
y (x) = c f (x, y(x), y (x)), x ∈ (a, b)
y(a) = , y(b) = ,
where , , c ∈ RF and f : [a, b] × R × R → R is continuous on [a, b] × R × R (again y (x), y (x) are
considered in the sense of Definition 11).
If = ⊕ c and f (x, y, z) 0, ∀(x, y, z) ∈ [a, b] × R × R or if = (and f arbitrary), then
b
b−x
x−a
⊕
⊕c
G(x, s)[−f (s, y(s), y (s)] ds,
y(x) =
b−a
b−a
a
where G(x, s) is the classical Green function defined by
(s−a)(b−x)
if s x,
b−a
G(x, s) = (x−a)(b−s)
if s x
b−a
satisfies the
above fuzzy differential equation. Indeed
if = ⊕ c and f 0, then we can write y(x) =
b
x−a
⊕ c b−a + a G(x, s)[−f (s, y(s), y (s)] ds and if = (and f arbitrary) then y(x) = ⊕ c b
(s)] ds and by Theorem 13 we easily obtain that y(x) satisfies the mentioned
G(x,
s)[−f
(s,
y(s),
y
a
b
equation in both cases, because it is known that a G(x, s)[−f (s, y(s), y (s)] ds = f (x, y(x), y (x)),
∀x ∈ (a, b).
598
B. Bede, S.G. Gal / Fuzzy Sets and Systems 151 (2005) 581 – 599
Example 32. Let us consider the fuzzy partial differential equation (fuzzy wave equation)
2
2
* u
(x, t)
*x 2
= c12 **t u2 (x, t), ∀t 0, x ∈ R,
u(x, 0) = f (x), **ut (x, 0) = g(x), x ∈ R,
where , ∈ RF , c ∈ R, c > 0, f, g : R → R, f of C2 class, g of C1 class on R. If = k , k ∈ R,
then taking
k
f (x − ct) + f (x + ct)
u(x, t) = +
2
2c
x+ct
x−ct
g() d ,
by Theorem 13 we easily obtain that u(x, t) satisfies the fuzzy wave equation, for all x ∈ R, t 0.
In Example 32 the derivative is considered in sense of Definition 11.
6. Concluding remarks
Strong and weak generalized differentiability are introduced and studied. The main advantage of these
concepts of differentiability is that (c f ) = c f for a crisp differentiable function f and a fuzzy
constant c ∈ RF .
Apparently the disadvantage of strongly generalized differentiability of a function compared to Hdifferentiability and Hukuhara differentiability is that a fuzzy differential equation has no unique solution.
However, this disadvantage can be seen as an advantage since we can choose the singular points where the
support of the solution changes its monotonicity. So we can obtain reversible solutions (in possibilistic
terms), stable and almost periodic solutions (see [3]), and asymptotic behavior of solutions to fuzzy
differential equations.
Generalized differentiability has also advantages with respect to differential inclusions. Firstly, it is
more practical for numerical computation. Secondly, one can use the (partial) derivative of a fuzzynumber-valued function, which is not the case when interpreting a fuzzy differential equation as a system
of differential inclusions, since this last one interprets directly the notion of fuzzy differential equation,
without a derivative. Also, differential inclusions do not always lead to fuzzy-number-valued solutions,
while with generalized differentiability we have to deal only with fuzzy-number-valued functions. As
an application we have obtained explicit solutions to some fuzzy differential and partial differential
equations, e.g. for the fuzzy wave equation, proposed in [4].
Acknowledgements
We thank to an anonymous referee for his remarks which improved the paper. Also, the first author
acknowledges the support of grant 201/03/0455 of the Grant Agency of the Czech Republic.
B. Bede, S.G. Gal / Fuzzy Sets and Systems 151 (2005) 581 – 599
599
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