Departamento de Andlisis Matemdtico, Facultad de Matemdticas,
Universidad de Santiago de Compostela, Santiago de Compostela, 15782, Spain
amrosana@usc. es
DANIEL FRANCO
Departamento de Matemdtica Aplicada I, Escuela Tecnica Superior de Ingenieros Industrials,
Universidad Nacional de Educacion a Distancia, Apartado de Correos 60149, Madrid, 28080, Spain.
dfranco @ind. uned. es
Received 14 December 2004 Revised 30 October 2006
We give the expression for the solution to some particular initial value problems in the space E1 of fuzzy subsets of K. We deduce some interesting properties of the diameter and the midpoint of the solution and compare the solutions with the corresponding ones in the crisp case.
Keywords: Fuzzy sets, Fuzzy systems, Linear systems, Fuzzy differential equations, Fuzzy initial value problems.
1. In t roduct ion
Expression for the unique solution to the initial value problem for linear first-order ordinary differential equations is well known. For I a real interval, M G i a constant, a : I —> R, and u : I —> R, the two following equations are equivalent:
independently of the sign of the constant M. The solution to those equations for an initial condition u(0) = no, UQ £ R, (0 G I) , can be obtained as:
u(t) = uoe~Mt + / <r(s) eM(*"*) ds,tel,t> 0. Jo
For the fuzzy case, the corresponding equations are not equivalent, even in the particular case where a(t) = X{o}? f° r t £ I, and X{o} the characteristic function of {0}. Recall that, if x, y £ E1 are such that x-\-y = X{o}> then x, y are real numbers and y = — x, but x + (—x) = X{o} is n ° t necessarily true for x £ E1 (take, for instance, x = X[o,i])- Equation
u'(t)+Mu(t) = X{o}
implies that the solution is crisp and
u'{t) = -Mu(t),
but the converse is not true in general. Besides, the multiplication of x G E1 by a negative constant implies the inversion of the endpoints of the level sets, that is,
-l[x]a = -l[XahXar\ = \-Xar,-Xal], Va G [0 ,1] ,
and this fact produces that the expressions of the solutions for
ur(t)+u(t) = X{0},
and
u'{t) -u(t) = X{0},
are completely different. In this paper, and for M G M, M > 0, we analyze the existence of solution for
the initial value problems associated to the fuzzy equations
u'(t)+Mu(t) =cr(t), te I,
Uf(t) = -Mu(t) + cr(t), t G / ,
uf(t) =Mu{t) + a{t), t G J,
u\t) -Mu(t) =cr(t), te I,
and compare their solutions by calculating the midpoint and the diameter of their respective level sets. We point out that the study of midpoints for fuzzy sets has many applications in Artificial Intelligence (Ref. 1). In Refs. 2-5, the expression of the solution to analogous problems for a(t) = X{o>5 t £ I, and a fixed M is studied. Other references dealing with fuzzy equations or their integral equivalent formulation are Refs. 6-9, and, recently, Refs. 10-15, while the basic theory about fuzzy metric spaces is included in Ref. 4.
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Linear First-Order Fuzzy Differential Equations 689
For M > 0, we calculate the expression of the solution to the initial value problems corresponding to equations
Uf(t) = -Mu(t) + cr(t), t G I,
u'{t) =Mu(t)+a(t), t e J,
whose existence and uniqueness can be deduced in the case of I bounded from the application of the fuzzy Picard-Lipschitz Theorem (Theorem 3.2.1 in Ref. 5, Theorem 13.2.1 in Ref. 4 and Ref. 8), since functions f,g : I x E1 —> E1 given, respectively, by
f(t,x) = -Mx + a(t),
and
g(t,x) = Mx + a(t),
are Lipschitzian in x and continuous in (£, x) for a continuous. Indeed, let x,y G E1
and analogously for g. As we show, existence of the solution for initial value problems relative to
u'(t)+Mu(t) = cr(t), tel,
and
u\t) -Mu(t) = cr(t), tel,
with M > 0, is subject to the verification of some conditions of compatibility involving the constant M, the function a and the initial data UQ. In fact, if we try to determine the existence and uniqueness of solution for
u'(t)+Mu(t) = cr(t), tel,
by using the fuzzy Picard-Lipschitz Theorem, we can write the equation in its equivalent expression
u'(t) = a(t) -H Mu(t), t G I,
whose right-hand side function /*(£, x) = a(t)—nMx would represent a Lipschitzian function in case it made sense, but the Hukuhara differences a(t) —H Mu(t) in the previous equation are not well-defined unless we can guarantee that
diam[a(t)]a > Mdiam[u(t)]a, Vt,Va,
and additional hypotheses are required in order to obtain that the level set Hukuhara differences define a fuzzy real number. This shows that it is not a trivial question and that further analysis should be made.
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690 J. J. NietOf R. Rodriguez-Lopez & D. Franco
2. Expression of Solutions
2.1. uf{t) + Mu(t) = o-(t), t E I, M > 0.
Let M > 0, J a real interval i" = [0,T] with T > 0 or J = [0,+oo), cr G C^.E1), uo <E E1 and consider problem
V(£)+Mw(£) = cr(t), £ G J,
(1) w(0) = 1̂ 0-
Theorem 1. Problem (1) has a unique solution in I, given by
U{t) = U0X{e-Mt} + / C r(^)X{eM(s-*)} ds> t e ^ ( 2 ) JO
if, for each t £ I, there exists f3 > 0 such that the Hukuhara differences
u(t + h) —H u(t) and u(t) —H u(t — h)
exist, for all 0 < h < (3.
Proof. Taking
[u(t)}a = [u(t)ahu(t)ar}J
and
Ual(t) = u(t)aU Uar(t) = u(t)ar,
(1) is written levelwise as
u'cd(t) + Mual(t)=<ral(t), tel,
< r ( t ) + Muar(t) = <rar(t), t e I,
Ual(0) = (U0)ah ^a r (O) = (u0)ar.
Using an integrating factor, we get
uai(t) = (u0)aie-Mt + f aal(s)eM^-^ ds, Jo
and, analogously for uar(t), producing (2). Now we study the differentiability of u in the sense of Hukuhara. Let t G I, and h > 0, then, for every a G [0,1],
u(t + h)-Hu(t)\ -Mtfe~Mh-l ~~ [uojaie h
rt / „-Mh _ i \ p-Mh rt+h + J a(s)ale
M^ ds (° h 'J + ^ 1 <s)aleM^ ds,
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Linear First-Order Fuzzy Differential Equations 691
and
u(t + ft) —H u(t) ft
(uo)are~ -Mt -Mh
ft / 0-Mh _ i \ p-Mh rt+ti
+ J a(s)areM^ ds {* h
Lj + t-j- J a(s)areM^ ds.
The limits of these functions as ft —• 0 + , uniformly in a are, respectively, z(t)ai = <r(t)ai - M(u0)aie~ -Mt •M [ a{s)ale
M^s-tUs1 Jo
and
z(t)ar = (J(t)ar ~ M{u0)are-Mt - M f a ( s ) a r e M ^ ds, Jo
since (txo)a/> (uo)ary a r e bounded uniformly in a G [0,1], cr^ai, cr(s)arj are bounded on [0,£] uniformly in a (a is bounded in the compact [0,£] by continuity),
»-Mh _ 1 + Mh
lim ft
0,
and i pt+h
lim - / a(s)aleM^s-tUs = a(t)ah
h^o+ ft Jt
lim - / a(s)areM^s-tUs = a(t)ar,
h^o+ ft Jt
uniformly in a by continuity of a at t. The same behavior can be checked for the left-sided Hukuhara quotients
u(t) —H u(t — ft) ft
and
for ft > 0. This proves that
dH u(t + ft) —H u(t)
ft
U>(t) ~H u(t — ft) ft
[z(t)ahz(t)Q
as ft —>• 0+ , uniformly in a, so that
where, for £ G J, z(£) given levelwise by
[*(*)]°H*(*)aZ,*(*)ar],
is a fuzzy number since E1 is complete. We have proved that u'(t) = z(t) G .E1, for all t G J. It is easy to check that
u'(t) + Mu(t) = z(t) + Mu(t) = cr(t), t G i",
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and u(0) = uo, so that we obtain the solution to (1). •
Jo is a nondecreasing function in t. But this is a real differentiate function and, by
(3),
jdiam{[u{t))a) = diam([a(t)]a) - Me~Mt
diam([u0}a)+ f diam([a(s))a)eMs ds) > 0.
Thus, the level set Hukuhara differences exist. The intervals
[u(t + h)al- U(t)al, U(t ^h)^- U(t)ar],
[u(t)al ~ U>(t ~ h)ai, U(t)ar -U(t- h)ar],
define the level sets of a fuzzy set, for each t and 0 < h < j3:
• Using (3), we obtain that
U(t + h)al ~ U(t)al < U(t + h)ar ~ u(t)ar, Va,
u(t)ai - u(t - h)ai < u(t)ar - u(t - ft)or, Va,
• By (4), u(t + h)ai — u(t)ai, u(t)ai — u(t — h)ai, are nondecreasing functions in a and u(t + h)ar — u(t)ar, u{t)ar — u(t — h)ar are nonincreasing in a.
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• Since u(t), u(t + /i), u(t — h) (h < t) are elements in E1, then
U(t + h)al ~ U(t)al, U(t + h)ar - U(t)ar,
U(t)al ~ U(t ~ h)au U{t)ar - u(t ~ ft)or,
are left-continuous in a.
R e m a r k 2. Condition (4) can be written as: for t G I, there exists f3 > 0 such that, for 0 < h < /?, and 0 < a < b < 1,
e M h _ ! / t ( ^ ( s ) o r - C r ( s )6 r ) e M s d s
> (^o)ar - (^o)6r + JQ (a(s)ar ~ Cf(s)hr)eMs ds
!_e-Mh Jl_h{(j(s)u - (j(s)ai)eMs ds
> (uo)bl - (uo)al + Jo (a(S)bl ~ &(s)al)eMs ds
!_e-Mfe ft-h(a(s)ar ~ a(s)br)eMs ds
> Mar ~ Mbr + JQ (a(s)ar ~ Cf(s)br)eMs ds
R e m a r k 3. For x G E1, and a G [0,1], mp([x]a) denotes the midpoint of [x]a, that is, \{xai + xar). Note that n, the solution to (1), verifies that
diam([u(t)]a) = diam([u0]a)e-Mt + / ^am([cr(5)]a)eM ( s- t ) ds, t G I,
Jo and
(5)
mp(Kt)] a ) = mp( [^ 0 ] a )e - M t + / m p ( [ ( j ( s ) ] a ) e M ^ ^ s Jo
J((t*o)ai + Mar)e~Mt + \ j (a(s)al + a ( s ) a r ) e M ^ ) ds, £ G J. 2
R e m a r k 4. If <r = X{c}, then diam([cr(t)]a) = 0, for every t, a, and (3) is reduced to
Me~Mtdiam([u0]a) < 0, for all a, and all t,
that is,
^o = X{c /}-
In such a case, (4) is trivially true and the solution is crisp
u(t) = X{cfe-Mt + ^{l-e-Mt)}-
In the ordinary case, u$ G R, cr G C(I, R), conditions (3), (4) are valid, the Hukuhara differences always exist as ordinary differences and the unique solution is given by
U(t) = U0 e~Mt+ [ta(s)eM^s-tUs1teL Jo
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2.2. u'(t) = -Mu(t) + cr(t), t E I, M > 0.
Let M > 0, I = [0,T] with T > 0, or J = [0,+oo), a G ^ ( I , ^ 1 ) , u0 G £ \ and consider problem
u'(t) = -Mu(t) + cr(t), t G J, (6)
w(0) = u0.
Theorem 2. Problem (6) has a unique solution in I, given by the following expression for all t G Ij a G [0,1],
eMt e-Mt
<*)ai = 2"tfi(*,a) + —2-U*(t>a)> (?)
eMt e-Mt <t)ar = —C/ i ( t , a ) + - ^ - E / 2 ( t , a ) , (8)
where
U1(t1a) = diam([u0]a) + [ diam([a(s)}a)e-
Ms ds, Jo
2 ( t , a ) = (w0)oZ + Mar + / (cr(5)aZ + S"0)ar ) Jo
[72(t, a) = (uo)nj + (Mn)ftr + / (o-(s)nj + cr(s)ftr) e s ds.
Proof. Take
[w(t)]° = [w(t)oZ,w(t)ar].
We obtain (Refs. 2, 3, 5) that the function w given, for t G I, a G [0,1], by
w(t)ai = - ^ i a m ( [ t / 0 ] a ) e M t + ± (K) a z + Mar) e~M\ (9)
thus K; +1; is the solution to (6). We seek a solution to (11) of the type
At)ar)~\-eMte-^)\car(t))' (U)
for t £ I, a £ [0,1], such that
[v(0)}a = [v(0)ahv(0)ar} = {0}, Va,
that is, for a G [0,1],
/ 0 \ = / V ( 0 ) a A = / 1 l \ fcal(0)\ \0J \v{°)arj \~1 1 J \car(0) J '
which is a nonsingular homogeneous linear system, hence the unique solution is
cai(0)=0, c a r ( 0 ) = 0 , V a e [ 0 , l ] .
Now, for the expression
fv(t)al \ ( cal(t)eMt + car(t)e-Mt \
\v{t)ar) \-cal(t)eMt+car(t)e-Mt)> [L6)
to define an element in E1, it is necessary that cai(t) is a nonpositive function. Besides, given v by
[v(t)]a = [v(t)ahv(t)ar}1
where v(t)ai, v(t)ar are expressed in (13), for the existence of the Hukuhara differences v(t + h) —H v(t) and v(t) —H v(t — /i), for h > 0 small enough, it is necessary that
diam([v(t)]a) = -cal(t)eMt + car(t)e-Mt - cal(t)e
Mt - c a r( t )e~M t = -2cal(t)eMt
is nondecreasing in t, (for instance, if cai(t) is a nonincreasing function in t). Note that, for t G I, a G [0,1],
Ht)]a = [cal(t)eMt + car(t) e~M\ -cal(t)e
Mt + car(t) e-M% (14)
Now, we calculate cai(t) and car(t) in order to obtain the solution of (11). Passing to the level sets, we obtain, for all a G [0,1], t G / ,
cai(t) < 0, for all t, a, nonincreasing function in t.
Taking into account (14), we get for t € I, and a G [0,1],
v(t)ai = - \ ( e-Msdiam([a(s)]a)ds eMt
* Jo + \ J eMS {aal{s) + aar{s)) ds e~Mt> (15) Mt l rt
v(t)ar = 7: e-Msdiam([a(s)}a)dse 2 Jo
+ l- J eMs (aal(s) + aar(s)) ds e~Mt. (16)
For checking that v(t) defines a fuzzy number, note that a is continuous, diam([(T(s)}a) and a(s)ar decrease in a and a(s)ai increases in a.
Now, we prove that v is differentiate in the sense of Hukuhara and that the derivative of v in the sense of Hukuhara at t is —Mv(t) + a(t). Note that it is trivially true that the levelset Hukuhara differences exist since
diam([v(t)]a) = -2cal(t)eMt = [ e-
Msdiam([a(s)]a) eMtds Jo
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is nondecreasing in t. Moreover, the Hukuhara differences
v(t + h) -H v(t), v(t) -H v(t - h)
exist for h small enough. To prove this fact, let t £ I. Using (15) and (16), we get, for t £ I and a £ [0,1],
(v(t + h) -H v(t))al = \ f e-Msdiam([a(s)}a) ds eMt(l - eMh) ^ Jo
since the continuity of a implies that J0 <r(s)x{eRs} ds, Jt o-(s)x{eRs} ds are el
ements in E1, for R = ± M , t £ I, h > 0. Finally, we have to prove that (v(t + h) —H v(t))ai is nondecreasing in a and (v(t + h) —H v(t))ar is nonincreasing in a. Take a, b £ [0,1], a < 6, then
The same procedure is valid for the case of the left-sided Hukuhara derivative of v at t. This completes the proof, since v is differentiate in the sense of Hukuhara at every t and v'(t) = —Mv(t) +<r(£). Note that adding w to v, we obtain (7) and (8), which provide the solution to (6). •
Remark 5. Expressions (7) and (8) are valid for / bounded or / = [0, +oo), with aeCil.E1).
Remark 6. In particular, if a = X{o}, then (6) is
u'(t) = -Mu(t), u(0) = u0,
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and the solution is given by (9)-(10). If UQ = X{o}> then the solution is v in the previous Theorem.
R e m a r k 7. If we denote by mp([x\a) the midpoint of [x]a, that is, ^(xai + xa
and by r([x]c
expressed as and by r([x]a) the radius of [x]a, that is, r([x]°) = hdiam{[x\a), then (7) can be
u{t)al = - (r([u0]a) + j r([a(s)]a)e-Ms ds ) eMt
mp([u0]a)+ f mp([a(s)}a)eMs ds
\ Jo
and (8) as
<t)ar = (r([u0]a) + f r([a(s)]a)e-Ms ds\ eMt
„Ms An \ -Mt + lmp({u0}a) + / mp([a(s)}a)eMsds) e
This solution is calculated taking into account the midpoint and the diameter of the level sets (of the initial data as well as function a), which characterize the level sets.
R e m a r k 8. If uo and a are crisp, then (3) and (4) are valid, and the solutions given by (2) and (7)-(8) are the same crisp function:
u(t) = u0e~Mt + / a(s)eMis~t) ds, t e I . Jo
Note that, in this case,
r([uo]a) = 0, mp([uo]a) = UQ, for all a,
and
r(W(s)]a) = °> mp([°(s)}a) = <Ks), for all s, and all a.
In fact, in the crisp case, (1) and (6) are equivalent problems.
R e m a r k 9. In the general case, if u is the solution to (6), then
diam([u(t)]a) = diam([u0]a)eMt + f diam{[a{s)]a)eM{t-s) ds, t e I,
= \((uo)ai + {uo)ar)e-Mt + \ J (a(s)al + v(s)ar)eM(*-» ds, t e I.
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(18)
2.3. u'(t) = Mu(t) + cr(t), t G I , M > 0
Let M > 0, I a real interval, J = [0,T] with T > 0 or I = [0,+oo), a G C{I,EX), uo £ E1 and consider problem
u'(t) = Mu(t)+a(t)J t el, (17)
u(0) = w0.
Theorem 3. Problem (17) has a unique solution in I, given by
u(t) = u0X{eMt} + / o-(s)x{eM(t-sn ds, t G I. Jo
Proof. Take [t^(t)]a = [u(t)ai,u(t)ar], and (17) is written levelwise as
u'al(t) = Mual(t)+aal(t), tel,
u'ar{t) = Muar(t) + <Tar(£), t G I.
Using an integrating factor, we get
uai(t) = (u0)aieMt + f aal(s)eM^ ds, Jo
and analogously for uar(t), then
[u(t)]a = [u0]aeMt + / [a(s)]aeM^-s) ds,
Jo
for every a G [0,1] and t G / , obtaining (18). It is obvious that u(t) defined by (18) defines a fuzzy number. Now, we check
that u is differentiable in the sense of Hukuhara at every point t with Hukuhara derivative at t equal to Mu(t) + cr(t), which is a fuzzy number for all t,
(Mu(t) + a(t))al = M(u0)aieMt + M [ a(s)al eM^ds + a{t)ah Jo
which taking into account the initial values cai(0) and car(0) give, for t E / , a E [0,1],
cai(t)
Car\t)
-\diam{[u0}a) - \ [ diam([a(s)}a)eMs ds,
1 1 ff
This joint to (24), produces (20) and (21). Hypothesis (22) and continuity of a guarantee that those level sets define a fuzzy number since it is obvious that
U(t)al < U(t) ar
and these functions are left-continuous in a.
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For t G I, and h > 0 small enough, the Hukuhara difference quotients are given
by
'u{t + h) —H u(t)~
-Mh
— J (diam([u0]a) + f diam([a(s)]a)eMs ds
h
\„-Mt l~e
"2 1 / pMh _ 1 \ / /•*
" 2 ^ ( JT~ ) ( {Uo)al + {Uo)ar + J {a{s)al + , J ( s ) - ) e " M S ds
rt+h je-M(t+h)
2h ' — — ~- -
1 /»t+Al
- / diam([a(s)]a)eMsdsi
^ y ( a ( S ) a i + a ( S ) a r ) e - M ^ S e M ^ ) and
i/(£ + h) —H u(t) h
i / p - M h _ i \ / />t
: - e ~ M t ( J ( diam([u0}a) + / diam([a(s)]a)eMs ds
- 2 ^ ( — f t " ) ( M a / + M a r + jf (*(*)a* + ^ a r K ^ dfi
t+/i M{t+h) + — / diam([a(s)]a)eMs ds
+ 2hJ ^s)ai+^)ar)e~Msds
which converge as h —• 0 + uniformly in a, respectively, to g(£)a/ and g(t)a r , where
and
«(*)aJ = *(t)al + ^~MtWi(t,a) + ^-eMtW2(t,a),
q(t)ar = °{t)ar ~ ^e~MtWx(t,a) + ^eMtW2(t,a),
which define a fuzzy number q(t) by [q(t)]a = [q(t)ahq(t)ar], a £ [0,1]. The same element in E1 is the limit in d^ as h —> 0 + of the Hukuhara difference quotients
u(t) —H u{t — h) h *
We have proved that ur(t) = q(t), t £ I and, by the expression of q(t), it is easy to check that, for all a G [0,1], and t G I,
The initial condition u(0) = u$ is trivially verified, in consequence, u is the solution to (19). •
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R e m a r k 12. It is not always true that (20) and (21) define a fuzzy number. For instance, taking a(t) = x^o} crisp, then (20) and (21) are reduced to
e-Mt eMt
u(t)ai = —diam([u0}a) + — ((u0)ai + (^o)or),
e-Mt eMt u(t)ar = -^—diam([u0}
a) + — ((u0)ai + (u0)ar),
for t G I, a G [0,1], which do not define a fuzzy number for an arbitrary initial condition UQ. In fact, taking the following triangular fuzzy number u$ as the initial data
u°(t)= { - ^ ( t - loo), t e [0,100], 0, otherwise,
the level sets are given by
[«o]° =
hence, if we set
a — 1 100 :
-Mt
100(1 - a)
(100(1 -a
aG[0 , l ] ,
a-1
v(t,a)
2 V
QMt , a _ 1
100
100
+ 100(1 - a) ) ,
then
u(t)ai = - / i ( t ,a) + z/(£,a),
u(t)ar = /i(t,a) ^^(t.a),
for t G / , a G [0,1], and it is not valid that
[u(t)u,u(t)br] £ [u(t)al,u(t)ar], &! b > CL.
For this condition to hold, u(t)ai should be a nondecreasing function in a, that is,
—/i(t,a) + u(t,a) < —/i(t, b) + u(t, 6),
for a < 6, but this is equivalent, for 6 > a, to
-100e-M t + 100eMt < ^e~Mt + Mt
100
or
9999eMt < lOOOle" Mt
which is valid for 0 < t small, but it fails if t large, for instance, at t M> since
9999e2 > 9999-2 > 10001.
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Linear First-Order Fuzzy Differential Equations 707
Then, if the interval / is large, (20)-(21) do not necessarily define a fuzzy number.
Remark 13. Taking into account that UQ G E1, the continuity of a, and the expression of
(u(t + h) -H U(t))ah (u(t + fy ~H U(t))ar,
(u(t) ~H U(t - h))ah (u(t) ~H U(t - h))arj
for h > 0, we obtain the following sufficient conditions for the existence of the Hukuhara differences u(t + h) — H u(h), u(t) — H u(t — h), for every t and h > 0 small
• Condition (3). • For t £ I, there exists j3 > 0 such that, for 0 < h < /?,
u(t + h)ai — u(t)ai nondecreasing in a u(t + h)ar — u(t)ar nonincreasing in a, . , u(t)ai — u{t — h)ai nondecreasing in a
u(t)ar — u(t — h)ar nonincreasing in a.
Note that the existence of the levelset Hukuhara differences of u comes from (3) and continuity of cr, since
\{{u0)al + (u0)ar)eMt + \ j (a(s)al + a(*)ar)eM<*-> ds, t e I.
3. Relation Among the Solutions to Different Problems
In conclusion, for M > 0, initial value problems for equations
uf(t) = -Mu{t) + cr(t), t e I, (Eq. (6)),
u\t) = Mu(t) + <T(£), t E / , (Eq. (17)),
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708 J. J. NietOf R. Rodriguez-Lopez & D. Franco
have always a unique solution. However, solutions relative to equations
u'{t) + Mu(t) = cr(t), t e I, (Eq. (1)),
u'(t) - Mu(t) = cr(t), t E I, (Eq. (19)),
exist under restrictive conditions, in fact, for a a crisp function, the initial data and the solution are necessarily crisp.
There are two elements which characterize the solution u(t) of a fuzzy problem, the diameter and the midpoint of each level set. If we compare the solution to (1) with the solution to (6), although their expressions seem to be very different, we can appreciate that the midpoint of each level set [u(t)]a is exactly the same. However, the diameter changes, replacing M by — M.
The same happens if we compare the solutions of (17) and (19). That is, the operation of passing the term depending on the constant to the other side of the identity produces a change in the diameter of the level sets of the solutions, but preserves the midpoint invariant.
Now, if we compare (1) and (19), which differ only in the sign of the constant (passing from a positive one to a negative one or vice versa), we find that the diameter of the level sets of the solutions is the same, but in this case, the midpoint of the level sets is different (we can pass from one expression to another, by writing —M instead of M). Similar considerations can be made for solutions of (6) and (17).
Note that the solutions of (1) and (17) have a similar expression, we can obtain one from the other replacing M by — M. The same change applies to the diameter and the midpoint of the level sets of the solution. Analogously for solutions to (6) and (19).
As we have shown, the expression of the solution changes if we change the sign of the constant M or if we change the term Mu(t) to the other side of the equation. Nevertheless, there is a close relation among the different solutions of these problems, that can be appreciated by comparing the diameter and the midpoint of their level sets.
Finally, note that for M = 0 conditions (3), (4), (22), and (25) are trivially fulfilled and the expressions for the solutions to the different problems (1), (6), (17), and (19) are equal to
u(t) = UQ + / a(s) ds, t e J, Jo
the unique solution to problem
( u'(t) = ait), t e / ,
u(0) = u0.
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Linear First-Order Fuzzy Differential Equations 709
A c k n o w l e d g e m e n t s
Research partially supported by Ministerio de Ciencia y Tecnologia / FEDER,
project BFM2001-3884-C02-01; Ministerio de Educacion y Ciencia / F E D E R
project MTM2004-06652-C03-01; and by Xunta de Galicia / FEDER, projects
PGIDIT02PXIC20703PN and PGIDIT05PXIC20702PN.
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