Keywords: Fuzzy real numbers; First-order fuzzy differential equations; Periodic boundary value problems; Generalized differentiability
1. Introduction
Differentiability in the sense of Hukuhara is one of the first approaches to define the concept of solution to a fuzzydifferential equation and to study the existence of such solutions. For instance, in Ref. [9], the existence and uniquenessproblem is addressed for first-order fuzzy differential equations. The particularities of this definition of differentiabilityproduce the nondecreasing behavior of the diameter of the level sets of a differentiable function, which weakens thepotential of this approach in the study of periodic boundary value problems for fuzzy differential equations. In orderto model periodic phenomena through fuzzy systems under Hukuhara differentiability, one can try, for instance, tointroduce impulses in the model [18]. The basic concepts on fuzzy sets, fuzzy differentials and fuzzy differentialequations can be found in [6,7,9,10,17]. See [5,16] for the expression of the solution to initial value problems forlinear fuzzy differential equations with constant coefficients under Hukuhara differentiability. Some results on fuzzyfunctional differential equations are included in [6,14], and other approaches in the analysis of uncertain systems areprovided in [8,4,13]. On the other hand, for strongly generalized differentiability, see [1–3,21]. Some applications ofgeneralized differentiability to numerical calculus of solutions and the study of Nth-order fuzzy differential equationsare included in [15,11].
Ref. [21] is focused on interval differential equations, whose solution is calculated by proving a characterization byODEs and using a numerical scheme. Some initial value problems for fuzzy linear differential equations are solved bycombination of two types of derivatives using a switching point. See also [20] for a generalization of the difference ofHukuhara and the division to the context of real intervals and fuzzy numbers and applications to fuzzy equations.
2 R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 –26
In [12], following the ideas in [21], existence of solution to a periodic boundary value problem for a class of first-order fuzzy differential equations is analyzed under strongly generalized differentiability. The problem considered isy′(t) = a(t)y(t) + b(t), t ∈ J , y(0) = y(T ), where J = [0, T ], and a : [0, T ] → R, b : [0, T ] → RF are continuousfunctions, with a changing its sign only once in the interval J. In this paper, we establish sufficient conditions for theexistence of solution to the above-mentioned periodic boundary value problem, where the coefficient a may have anarbitrary number of zeros on the interval J and a solution is built piecewise through a well-defined combination of (i)-and (ii)-solutions, extending the results in [12] and completing [19], where the levelset problems were addressed. Wealso analyze the existence of solutions which are crisp (or real) at the switching points where the piecewise solutionchanges from (ii)- to (i)-differentiability. Finally, we show some examples to illustrate how this type of fuzzy differentialequations can be used, using this approach, to model phenomena which present the same behavior at two differentinstants. We have adopted a notation which allows a direct application of the Hukuhara difference’s properties and ismore consistent with the general theoretical results available on the solvability of fuzzy linear differential equations.
2. Basic concepts
Consider the space RF of fuzzy intervals, that is, the class of elements u : R → [0, 1] satisfying that
(i) u is normal, i.e., there exists s0 ∈ R such that u(s0) = 1,(ii) u is upper semicontinuous on R,
(iii) u is fuzzy-convex, that is, u(ts + (1 − t)r ) � min{u(s), u(r )}, ∀t ∈ [0, 1], and s, r ∈ R,(iv) cl{s ∈ R|u(s) > 0} is compact, where cl denotes the closure of a set.
For u ∈ RF , and each 0 < � � 1, the�-level set of u is defined as the nonempty compact interval [u]� = {s ∈ R|u(s) � �}and [u]0 = cl{s ∈ R|u(s) > 0}.
We represent [u]� = [u�l , u�r ], so that diam([u]�) = u�r − u�l . The functions u and u given by u(�) = u�l andu(�) = u�r represent the lower and upper branches of u, respectively.
For the study of equation y′(t) = a(t)y(t) + b(t) in RF , we need to define the addition and multiplication by a scalarin the space RF , which are defined levelsetwise. If we consider the metric
D(u, v) = sup�∈[0,1]
max{|u�l − v�l |, |u�r − v�r |}, for u, v ∈ RF
then (RF , D) is a complete metric space.Another operation between fuzzy intervals which is used throughout this paper is the difference of Hukuhara, defined
as follows.
Definition 2.1. Given x, y ∈ RF , if there exists z ∈ RF with x = y + z, we say that z is the H-difference of x and y,denoted by x�y.
In the following, we need some properties of the Hukuhara difference, whose proofs are immediate.
Lemma 2.2. Let A, B, C ∈ RF . If A�C exists, then (A + B)�C exists and (A + B)�C = (A�C) + B.
Lemma 2.3. Let A, C, D ∈ RF . The following assertions hold:
• If the differences A�C and (A�C)�D exist, then A�(C + D) exists and A�(C + D) = (A�C)�D.• If A�(C + D) exists, then the differences A�C and (A�C)�D exist and (A�C)�D = A�(C + D).
Lemma 2.4. Let A, B, C, D ∈ RF . If A�C and B�D exist, then there exist (A + B)�C and (A + B)�(C + D)(and, therefore, by Lemma 2.3, ((A + B)�C)�D) exists and coincides with (A + B)�(C + D)).
Lemma 2.5. Let A, B ∈ RF and � > 0 a real number. Then
diam([A + B]�) = diam([A]�) + diam([B]�) for every � ∈ [0, 1],
diam([A�B]�) = diam([A]�) − diam([B]�) for every � ∈ [0, 1],
R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 –26 3
diam([�A]�) = � diam([A]�) for every � ∈ [0, 1].
In the following sections, the concept of differentiability of fuzzy-valued functions is considered in the sense ofgeneralized differentiability (see [1,2]).
3. Solutions to the periodic boundary value problem
Consider the periodic boundary value problem{y′(t) = a(t)y(t) + b(t), t ∈ J,
y(0) = y(T ),(1)
where T > 0, J = [0, T ], a : [0, T ] → R and b : [0, T ] → RF are continuous functions. Problem (1) is related toproblem (6) in Ref. [3] (see also [12]).
Since function a is allowed to change its sign an arbitrary number of times in the interval J , we select the set ofpoints where the change of sign occurs. To this purpose, in the interval J = [0, T ], we consider a finite sequence ofreal numbers �k ∈ (0, T ), k = 1, 2, . . . , m, in such a way that 0 = �0 < �1 < · · · < �m < �m+1 = T .
To define the concept of solution to problem (1), we consider the following space of piecewise-defined functionswhich are differentiable in the sense of generalized differentiability.
Definition 3.1. Let J = [0, T ] be a real interval and consider the sequence of real numbers�k ∈ (0, T ), k = 1, 2, . . . , m,satisfying that 0 = �0 < �1 < · · · < �m < �m+1 = T . We define the space F{�k } = F{�1,. . .,�m }, consisting on thefunctions u ∈ C(J, RF ) which are differentiable in the sense of generalized differentiability on (0, T ), with switchingpoints at �1, . . . , �m , and such that there exist the one-sided limits u′(0+) and u′(T −) in RF .
In this work, we consider particular solutions to problem (1) which belong to the space F{�k }. Note that the followingdefinition also involves continuity of the generalized derivative of the solution in (0, T ) \ {�1, . . . , �m}.
Definition 3.2. By a solution to (1), we understand a function in the space F{�k } = F{�1,. . .,�m } which satisfies theconditions in (1).
In the following results, we use the notation
A j (t) = e∫ t� j
a(u) duand B j (t) =
∫ t
� j
b(s)
A j (s)ds, for t � � j and j = 0, . . . , m.
The following properties are also useful to the procedure.
Lemma 3.3. A0(�k)Ak(t) = A0(t), for t � �k and k = 1, . . . , m.
Lemma 3.4. Let j = 0, . . . , m and � ∈ R, � > 0. The Hukuhara differences A��(−B j (t)) exist, for every t ∈(� j , � j+1], if and only if A��(−B j (� j+1)) exists.
Proof. If A��(−B j (� j+1)) exists, then we check that
diam([A]�) � � diam([B j (t)]�) for every � ∈ [0, 1],
A�l + �B j (t)�r is nondecreasing in �,
A�r + �B j (t)�l is nonincreasing in �
for every t ∈ (� j , � j+1], assuming their validity for t = � j+1. Indeed,
� diam([B j (t)]�) = �
∫ t
� j
diam([b(s)]�)
A j (s)ds � �
∫ � j+1
� j
diam([b(s)]�)
A j (s)ds
= � diam([B j (� j+1)]�) � diam([A]�).
4 R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 –26
Besides, for � � �, A�l + �B j (� j+1)�r � A�l + �B j (� j+1)�r and, hence, for every t ∈ (� j , � j+1],
�(B j (t)�r − B j (t)�r ) = �∫ t
� j
b(s)�r − b(s)�r
A j (s)ds � �
∫ � j+1
� j
b(s)�r − b(s)�r
A j (s)ds
= �(B j (� j+1)�r − B j (� j+1)�r ) � A�l − A�l .
The proof of the nonincreasing character of A�r + �B j (t)�l in the variable �, for every t ∈ (� j , � j+1], followssimilarly. �
First, for any (finite) arbitrary number of terms in the sequence {�k}, we suppose that the continuous real functiona : [0, T ] → R is such that{
a > 0 on (�k, �k+1) for every even number k with k � m,
a < 0 on (�k, �k+1) for every odd number k with k � m.
}(2)
Note that a(T ) can be different from zero but, obviously, a(�k) = 0, for every k = 1, . . . , m.
Theorem 3.5. Let a : [0, T ] → R and b : [0, T ] → RF be continuous functions such that (2) holds. Suppose thatthe Hukuhara differences
j−1∑k=0,k even
1
A0(�k)Bk(�k+1)�
⎛⎝ j∑k=1,k odd
1
A0(�k)(−Bk(�k+1))
⎞⎠ exist in RF for every j odd with 1 � j � m,
(3)
and ∫ T
0a(u) du < 0. (4)
Then there exists a solution u to problem (1) in the space F{�1,. . .,�m }. Furthermore, this solution u is (i)-differentiable
on⋃
k even(�k, �k+1) and (ii)-differentiable on⋃
k odd (�k, �k+1).
Proof. To prove this result, we define an operator in the following manner: we map each fuzzy interval y0 into thefuzzy interval which is obtained as the value at the instant T of a solution (in the sense of Definition 3.2) to the linearequation subject to the initial condition y0. This solution is built piecewise by using switching points between intervalsof (i)-differentiability and (ii)-differentiability depending on the changes of sign of the coefficient a, following theideas in [21] and the expressions of the (i)- and (ii)-solutions given in [3]. Indeed, if we fix y0 ∈ RF , since a > 0 in(0, �1), then
y(t) = e∫ t
0 a(u) du(
y0 +∫ t
0b(s) e− ∫ s
0 a(u) du ds
)= A0(t)(y0 + B0(t)), t ∈ (0, �1]
and hence, for the calculus of the solution on the next interval (�1, �2], we take as initial condition
R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 –26 5
provided that the Hukuhara differences in this expression exist for t ∈ (�1, �2]. In the previous calculus, we have usedLemma 3.3. Moreover, the Hukuhara differences
B0(�1)�1
A0(�1)(−B1(t))
exist in RF , for every t ∈ (�1, �2], since they exist for t = �2 by hypothesis (3) with j = 1 and, hence, it is justifiedthat the expression y is well-defined on [0, �2] independently of the initial condition y0 ∈ RF .
Continuing this process on J = [0, T ], the sought solution y ∈ C(J, RF ) to the linear fuzzy differential equationy′(t) = a(t)y(t) + b(t), t ∈ J , is given recursively as follows:
y(t) ={
A j (t)(y(� j ) + B j (t)) for t ∈ (� j , � j+1], and j even,
A j (t)(y(� j )�(−B j (t))) for t ∈ (� j , � j+1], and j odd(7)
for y(�0) = y(0) = y0 the initial value chosen.Next, in order to check the good-definition of the solution y on the intervals of (ii)-differentiability, we justify the
existence in RF of the Hukuhara differences y(� j )�(−B j (t)), for every t ∈ (� j , � j+1], and j odd. We have alreadychecked this fact for j = 1. To this purpose, we calculate, in terms of y0, the value of y at the switching points � j , forj odd, at the same time we justify the good-definition of y up to � j . In fact, we calculate y(� j ) in terms of y0, for everyj = 1, . . . , m + 1. Indeed, we prove that y is well-defined on [0, T ] and that
y(� j ) = A0(� j )
⎛⎝y0 +j−1∑
k=0,k even
1
A0(�k)Bk(�k+1)�
j−1∑k=1,k odd
1
A0(�k)(−Bk(�k+1))
⎞⎠ (8)
for j = 1, . . . , m + 1.We have proved that y is well-defined on [0, �2] and that (5) holds, which coincides with (8) for j = 1 since
A0(0) = 1. Besides, from (6), we have
y(�2) = A0(�2)
(y0 + B0(�1)�
1
A0(�1)(−B1(�2))
),
which is equal to (8) for j = 2. On the other hand, it is clear that y is well-defined on (�2, �3] and
y(�3) = A2(�3)(y(�2) + B2(�3))
= A2(�3)
(A0(�2)
(y0 + B0(�1)�
1
A0(�1)(−B1(�2))
)+ B2(�3)
)= A0(�3)
(y0 + B0(�1) + 1
A0(�2)B2(�3)�
1
A0(�1)(−B1(�2))
)equal to (8) for j = 3. Next, we justify that y is well-defined on (�3, �4], proving the existence of the differencesA3(t)(y(�3)�(−B3(t))), for t ∈ (�3, �4], which is equivalent to the existence of the difference for t = �4. Indeed,
y(�4) = A3(�4)(y(�3)�(−B3(�4)))
= A3(�4)
(A0(�3)
(y0 + B0(�1) + 1
A0(�2)B2(�3)�
1
A0(�1)(−B1(�2))
)�(−B3(�4))
)= A0(�4)
(y0 + B0(�1) + 1
A0(�2)B2(�3)�
(1
A0(�1)(−B1(�2)) + 1
A0(�3)(−B3(�4))
)).
By hypothesis (3) for j = 3, the previous Hukuhara difference exists. Therefore, y is well-defined on [0, �4] and theprevious expression for y(�4) coincides with (8) for j = 4.
By induction, suppose that y is well-defined on [0, � j−1] for 3 � j � m odd and that (8) holds for j − 1, i.e.,
y(� j−1) = A0(� j−1)
⎛⎝y0 +j−2∑
k=0,k even
1
A0(�k)Bk(�k+1)�
j−2∑k=1,k odd
1
A0(�k)(−Bk(�k+1))
⎞⎠
6 R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 –26
and we prove that y is well-defined on [0, � j+1] and that (8) holds for j and j + 1. Indeed, since y is well-defined on[0, � j−1], with j − 1 � 2 even, then y is well-defined on (� j−1, � j ], and, using the induction hypothesis and (7),
y(� j ) = A j−1(� j )(y(� j−1) + B j−1(� j ))
= A j−1(� j )
⎛⎝A0(� j−1)
⎛⎝y0 +j−2∑
k=0,k even
1
A0(�k)Bk(�k+1)�
j−2∑k=1,k odd
1
A0(�k)(−Bk(�k+1))
⎞⎠ + B j−1(� j )
⎞⎠= A0(� j )
⎛⎝y0 +j−1∑
k=0,k even
1
A0(�k)Bk(�k+1)�
j−1∑k=1,k odd
1
A0(�k)(−Bk(�k+1))
⎞⎠since j − 1 is even, so that (8) is true for j . Now, since y is defined on [0, � j ], for j odd, then the Hukuhara differencesy(t) = A j (t)(y(� j )�(−B j (t))) exist for t ∈ (� j , � j+1], since it exists for t = � j+1 due to (3). Then, y is well-definedon [0, � j+1] and, besides, since j is odd,
y(� j+1) = A j (� j+1)(y(� j )�(−B j (� j+1)))
= A j (� j+1)
⎛⎝A0(� j )
⎛⎝y0 +j−1∑
k=0,k even
1
A0(�k)Bk(�k+1)�
j−1∑k=1,k odd
1
A0(�k)(−Bk(�k+1))
⎞⎠�(−B j (� j+1))
⎞⎠= A0(� j+1)
⎛⎝y0 +j−1∑
k=0,k even
1
A0(�k)Bk(�k+1)�
j∑k=1,k odd
1
A0(�k)(−Bk(�k+1))
⎞⎠expression which coincides with (8) for j + 1. This justifies that y is well-defined on [0, T ] and that (8) is valid forj = 1, . . . , m + 1.
Therefore, we define the operator
G : RF −→ RF
y0 � G(y0) = y(T ),
where y is defined in (7).The operator is well-defined, by virtue of condition (3), and G is a contractive mapping due to the integral condition
(4). Hence, by Banach fixed point theorem, G has a unique fixed point, which is indeed a solution to the periodicboundary value problem related to the linear fuzzy differential equation studied.
To check the contractive character of G, we prove that, for every y0, y0 ∈ RF ,
where y and y are the solutions corresponding, respectively, to the initial values y0, y0 ∈ RF , as defined in (7).Indeed, for j = 0, D(y(�0), y(�0)) = D(y0, y0) = A0(�0)D(y0, y0). For j = 1, . . . , m + 1, using (8) and the
properties of the Hukuhara differences and the distance D, we get
D(y(� j ), y(� j )) = D
⎛⎝A0(� j )
⎧⎨⎩y0 +j−1∑
k=0,k even
1
A0(�k)Bk(�k+1)�
j−1∑k=1,k odd
1
A0(�k)(−Bk(�k+1))
⎫⎬⎭A0(� j )
⎧⎨⎩y0 +j−1∑
k=0,k even
1
A0(�k)Bk(�k+1)�
j−1∑k=1,k odd
1
A0(�k)(−Bk(�k+1))
⎫⎬⎭⎞⎠
= A0(� j )D(y0, y0).
R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 –26 7
Therefore, for j = m + 1, T = �m+1, we get, independently of the case m even or odd, that
0 a(u) du is a contractivity constant and the proof is finished. �
Remark 3.6. The function u which existence is given by Theorem 3.5 is not necessarily the unique solution to (1) inF{�1,. . .,�m }, but it is the unique solution to (1) which is (i)-differentiable on
⋃k even(�k, �k+1) and (ii)-differentiable on⋃
k odd(�k, �k+1). Under the hypotheses of Theorem 3.5, the initial condition y0 ∈ RF leading to this solution u canbe obtained as
y0 = A0(T )
1 − A0(T )
⎛⎝ m∑k=0,k even
1
A0(�k)Bk(�k+1)�
m∑k=1,k odd
1
A0(�k)(−Bk(�k+1))
⎞⎠ . (10)
Remark 3.7. Under the hypotheses of Theorem 3.5, it is clear from the expression (8) that, for j = 1, . . . , m + 1 and� ∈ [0, 1],
diam([y(� j )]�) = A0(� j )
⎛⎝diam([y0]�) +j−1∑k=0
(−1)k
A0(�k)diam([Bk(�k+1)]�)
⎞⎠= A0(� j )
⎛⎝diam([y0]�) +j−1∑k=0
(−1)k∫ �k+1
�k
1
A0(s)diam([b(s)]�) ds
⎞⎠ .
Remark 3.8. To check the validity of condition (3), it is crucial to control the length of the diameter of the level setsof both terms in the difference, that is, it is necessary to impose that
j∑k=0
(−1)k 1
A0(�k)diam([Bk(�k+1)]�) � 0
for every � ∈ [0, 1] and every odd number j with 1 � j � m, which is equivalent to
j∑k=0
(−1)k∫ �k+1
�k
1
A0(s)diam([b(s)]�) ds � 0
for every � ∈ [0, 1] and every odd number j with 1 � j � m. Besides, to get (3), we have to check, for every oddnumber j with 1 � j � m, the following conditions:
j−1∑k=0,k even
∫ �k+1
�k
b(s)�l
A0(s)ds +
j∑k=1,k odd
∫ �k+1
�k
b(s)�r
A0(s)ds is nondecreasing in �
and
j−1∑k=0,k even
∫ �k+1
�k
b(s)�r
A0(s)ds +
j∑k=1, k odd
∫ �k+1
�k
b(s)�l
A0(s)ds is nonincreasing in �.
Corollary 3.9. Consider 0 = �0 < �1 < �2 < �3 < �4 = T , and a : [0, T ] → R, b : [0, T ] → RF continuousfunctions satisfying that a > 0 on (0, �1) ∪ (�2, �3) and a < 0 on (�1, �2) ∪ (�3, T ). Suppose that:∫ �1
0
1
A0(s)diam([b(s)]�) ds �
∫ �2
�1
1
A0(s)diam([b(s)]�) ds, for every � ∈ [0, 1], (11)
8 R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 –26
∑k=1,3
∫ �k+1
�k
1
A0(s)diam([b(s)]�) ds �
∑k=0,2
∫ �k+1
�k
1
A0(s)diam([b(s)]�) ds, for every � ∈ [0, 1], (12)
∫ �1
0
b(s)�l
A0(s)ds +
∫ �2
�1
b(s)�r
A0(s)ds is nondecreasing in �, (13)
∑k=0,2
∫ �k+1
�k
b(s)�l
A0(s)ds +
∑k=1,3
∫ �k+1
�k
b(s)�r
A0(s)ds is nondecreasing in �, (14)
∫ �1
0
b(s)�r
A0(s)ds +
∫ �2
�1
b(s)�l
A0(s)ds is nonincreasing in �, (15)
∑k=0,2
∫ �k+1
�k
b(s)�r
A0(s)ds +
∑k=1,3
∫ �k+1
�k
b(s)�l
A0(s)ds is nonincreasing in � (16)
and, moreover, that condition (4) holds. Then there exists a solution to problem (1) which is (i)-differentiable on(0, �1) ∪ (�2, �3) and (ii)-differentiable on (�1, �2) ∪ (�3, T ).
Proof. It follows from Theorem 3.5, taking m = 3, since (3) (or conditions in Remark 3.8) must be checked for j = 1and j = 3, that is, there must exist
B0(�1)�1
A0(�1)(−B1�2))
and
B0(�1) + 1
A0(�2)B2(�3)�
(1
A0(�1)(−B1(�2)) + 1
A0(�3)(−B3(�4))
)in RF
while (4) provides the contractive character of the operator G. �
Corollary 3.10. Suppose that 0 = �0 < �1 < �2 < �3 = T , a : [0, T ] → R is a continuous real function satisfyingthat a > 0 on (0, �1) ∪ (�2, T ), a < 0 on (�1, �2), b : [0, T ] → RF is continuous, and assume that the conditions(4), (11), (13), (15) hold. Then there exists a solution to problem (1) which is (i)-differentiable on (0, �1) ∪ (�2, T ) and(ii)-differentiable on (�1, �2).
Proof. The proof derives from Theorem 3.5, taking m = 2, since conditions (11), (13), (15) are related to the existenceof the difference
B0(�1)�1
A0(�1)(−B1(�2)). �
Remark 3.11. If we consider m = 1, a > 0 on (0, �) and a < 0 on (�, T ), then Theorem 3.5 produces the resultsin [12].
On the other hand, consider that the continuous function a : [0, T ] → R satisfies{a < 0 on (�k, �k+1) for every even number k with k � m
a > 0 on (�k, �k+1) for every odd number k with k � m
}. (17)
Again a(�k) = 0, for every k = 1, . . . , m, due to continuity.
R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 –26 9
Theorem 3.12. Let a : [0, T ] → R and b : [0, T ] → RF be continuous functions such that (17) holds. Suppose thatthe following differences
A0(T )m∑
k= j+1,k odd
1
A0(�k)Bk(�k+1) +
j−1∑k=0,k odd
1
A0(�k)Bk(�k+1)
�
⎛⎝ j∑k=0,k even
1
A0(�k)(−Bk(�k+1)) + A0(T )
m∑k= j+1,k even
1
A0(�k)(−Bk(�k+1))
⎞⎠exist in RF , for every j even with 0 � j � m (18)
and that (4) holds, that is,∫ T
0 a(u) du < 0. Then there exists a solution u to problem (1) in the space F{�1,. . .,�m }.Furthermore, this solution u is (i)-differentiable on ∪k odd (�k, �k+1) and (ii)-differentiable on ∪k even(�k, �k+1).
Proof. Consider the following nonempty closed subset of RF
C∗ =⎧⎨⎩y0 ∈ RF : y0 +
j−1∑k=0,k odd
1
A0(�k)Bk(�k+1)�
j∑k=0,k even
1
A0(�k)(−Bk(�k+1))
exist in RF , for every j even with 0 � j � m} .
The closed character of C∗ is clear. To prove that C∗ is nonempty, we have to find an element y0 ∈ RF such that theHukuhara differences in C∗ exist. To this purpose, we check three conditions. The first one concerns the length of thediameter of the level sets, that is,
diam([y0]�) �
j∑k=0
(−1)k∫ �k+1
�k
diam([b(s)]�)
A0(s)ds
for every � ∈ [0, 1] and every j even with 0 � j � m. For its validity, it suffices to select y0 ∈ RF such thatdiam([y0]�) �R, for every � ∈ [0, 1], where
maxj even,0 � j � m
⎧⎨⎩j∑
k=0
(−1)k∫ �k+1
�k
diam([b(s)]�)
A0(s)ds
⎫⎬⎭ � maxj even,0 � j � m
⎧⎨⎩j∑
k=0
∫ �k+1
�k
diam([b(s)]�)
A0(s)ds
⎫⎬⎭� max
j even,0 � j � m
∫ � j+1
0
diam([b(s)]0)
A0(s)ds � K
∫ T
0
1
A0(s)ds =: R
and K � diam([b(t)]0), for every t ∈ [0, T ]. Additionally, we have to check that
(y0)�l +j−1∑
k=0,k odd
Bk(�k+1)�l
A0(�k)+
j∑k=0,k even
Bk(�k+1)�r
A0(�k)
is nondecreasing in �, for every j even with 0 � j � m. Indeed, for � � �, we prove that
(y0)�l − (y0)�l �
j−1∑k=0,k odd
∫ �k+1
�k
b(s)�l − b(s)�l
A0(s)ds +
j∑k=0,k even
∫ �k+1
�k
b(s)�r − b(s)�r
A0(s)ds,
which is trivially satisfied if, for � � �,
(y0)�l − (y0)�l � maxj even,0 � j � m
j∑k=0,k even
∫ �k+1
�k
b(s)�r − b(s)�r
A0(s)ds
=m∑
k=0,k even
∫ �k+1
�k
b(s)�r − b(s)�r
A0(s)ds =
m∑k=0,k even
∫ �k+1
�k
b(s)�r
A0(s)ds −
m∑k=0,k even
∫ �k+1
�k
b(s)�r
A0(s)ds
10 R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 –26
then it suffices to select y0 in such a way that (y0)�l increases in � faster (or at the same speed) than
(∑m
k=0,k even
∫ �k+1�k
b(s)/A0(s) ds)�r decreases. On the other hand, to obtain that
(y0)�r +j−1∑
k=0,k odd
Bk(�k+1)�r
A0(�k)+
j∑k=0,k even
Bk(�k+1)�l
A0(�k)
is nonincreasing in �, for every j even with 0 � j � m, it suffices to select y0 in such a way that
(y0)�r − (y0)�r �
m∑k=0,k even
∫ �k+1
�k
b(s)�l
A0(s)ds −
m∑k=0,k even
∫ �k+1
�k
b(s)�l
A0(s)ds
for � � �, that is, with (y0)�r decreasing in � faster (or at the same speed) than (∑m
k=0,k even
∫ �k+1�k
b(s)/A0(s) ds)�l
increases. Therefore, C∗ is nonempty.We consider, for each fixed initial condition y0 ∈ C∗, the solution y ∈ C(J, RF ) to the linear fuzzy differential
equation y′(t) = a(t)y(t) + b(t), t ∈ J = [0, T ], defined recursively by
y(t) ={
A j (t)(y(� j )�(−B j (t))) for t ∈ (� j , � j+1] and j even,
A j (t)(y(� j ) + B j (t)) for t ∈ (� j , � j+1] and j odd,(19)
where the initial condition is y(�0) = y(0) = y0. According to the notation in [3], this function corresponds to takingthe expression y1 on the intervals (� j , � j+1] for j such that a > 0 on (� j , � j+1) and y2 on the intervals (� j , � j+1] forj such that a < 0 on (� j , � j+1). Note that the existence in RF of the Hukuhara differences y(� j )�(−B j (t)), for everyt ∈ (� j , � j+1] and j even, 0 � j � m, is equivalent to their existence at � j+1.
Due to the choice of C∗, for an initial condition y0 in C∗, the Hukuhara differences included in the expression ofy exist for t ∈ (� j , � j+1] and j even, 0 � j � m. In consequence, y is well defined as long as the initial condition y0belongs to C∗.
We calculate y(� j ) in terms of y0 ∈ C∗, at the same time that we justify the good-definition of y up to � j . In fact,for y0 ∈ C∗, we prove that y is well-defined on [0, T ] and
y(� j ) = A0(� j )
⎛⎝y0 +j−1∑
k=0,k odd
1
A0(�k)Bk(�k+1)�
j−1∑k=0,k even
1
A0(�k)(−Bk(�k+1))
⎞⎠ (20)
for j = 1, . . . , m + 1.Indeed, for j = 0, y(�0) = y0 ∈ C∗. Besides, y(t) = A0(t)(y0�(−B0(t))) exists for t ∈ (0, �1], since it exists at �1
(y0 ∈ C∗, see the expressions in C∗ for j = 0). The expression y(�1) = A0(�1)(y0�(−B0(�1))) coincides with (20)for j = 1. Thus, it is clear that y is well-defined on (�1, �2] and
which coincides with (20) for j = 2. To justify that y(t) = A2(t)(y(�2)�(−B2(t))) is well-defined on (�2, �3], wecheck that the expression makes sense at �3 (see the condition in C∗ for j = 2) and
y(�3) = A2(�3)(y(�2)�(−B2(�3)))
= A2(�3)
(A0(�2)
(y0 + 1
A0(�1)B1(�2)�(−B0(�1))
)�(−B2(�3))
)= A0(�3)
(y0 + 1
A0(�1)B1(�2)�
(−B0(�1) − 1
A0(�2)B2(�3)
)),
which coincides with (20) for j = 3.By induction, suppose that y is well-defined on [0, � j ] with j � 1 odd and that (20) holds for j , and check that y is
well-defined on [0, � j+2] and that (20) holds for j + 1 and j + 2. Indeed, since j is odd and y is defined on [0, � j ], itis clear that y is well-defined on [0, � j+1] and, by the induction hypothesis and the fact that j is odd, we get
y(� j+1) = A j (� j+1)(y(� j ) + B j (� j+1))
R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 –26 11
= A0(� j+1)
⎛⎝y0 +j∑
k=0,k odd
1
A0(�k)Bk(�k+1)�
j−1∑k=0,k even
1
A0(�k)(−Bk(�k+1))
⎞⎠ ,
which implies the validity of (20) for j + 1. Now, j + 1 is even and y(t) = A j+1(t)(y(� j+1)�(−B j+1(t))) exists fort ∈ (� j+1, � j+2] since it exists at t = � j+2,
y(� j+2) = A j+1(� j+2)(y(� j+1)�(−B j+1(� j+2)))
= A j+1(� j+2)A0(� j+1)
×⎛⎝y0 +
j∑k=0,k odd
1
A0(�k)Bk(�k+1)�
j−1∑k=0,k even
1
A0(�k)(−Bk(�k+1))�
1
A0(� j+1)(−B j+1(� j+2))
⎞⎠= A0(� j+2)
⎛⎝y0 +j∑
k=0,k odd
1
A0(�k)Bk(�k+1)�
j+1∑k=0,k even
1
A0(�k)(−Bk(�k+1))
⎞⎠ ,
which coincides with (20) for j + 2, since j + 1 is even. Hence, for y0 ∈ C∗, y is well-defined on [0, T ] and (20) holdsfor every j = 0, . . . , m + 1.
Next, we define the operator G : C∗ −→ RF , by G(y0) = y(T ), for y0 ∈ C∗. Condition (18) guarantees that G mapsC∗ into itself. To prove this fact, we take an arbitrary y0 ∈ C∗, for which the difference
y0 +j−1∑
k=0,k odd
Bk(�k+1)
A0(�k)�
j∑k=0,k even
(−Bk(�k+1))
A0(�k)
exists in RF for every j even with 0 � j � m, and prove that Gy0 belongs to C∗, showing that the differences
A0(T )
⎛⎝y0 +m∑
k=0,k odd
1
A0(�k)Bk(�k+1)�
m∑k=0,k even
1
A0(�k)(−Bk(�k+1))
⎞⎠+
j−1∑k=0,k odd
1
A0(�k)Bk(�k+1)�
j∑k=0,k even
1
A0(�k)(−Bk(�k+1))
exist in RF , for every j even with 0 � j � m. Indeed, these differences exist due to (18) and y0 ∈ C∗, which provesthat G(y0) ∈ C∗, and G is well-defined.
Finally, condition (4) provides the contractive character of the mapping G. We prove that, for y0, y0 ∈ C∗,
where functions y and y come from (19) taking, respectively, the initial values y0 and y0. Indeed, for j = 0,D(y(�0), y(�0)) = D(y0, y0) = A0(�0)D(y0, y0). For j = 1, . . . , m + 1, by (20), we get D(y(� j ), y(� j )) =A0(� j )D(y0, y0). In particular,
D(G(y0), G(y0)) = D(y(T ), y(T ))
= A0(�m+1)D(y0, y0) = A0(T )D(y0, y0) = e∫ T
0 a(u) du D(y0, y0),
where A0(T ) = e∫ T
0 a(u) du < 1 and G is a contractive mapping.Hence, there exists a unique fixed point of G, which is one solution to problem (1). �
Remark 3.13. From (20), for each � ∈ [0, 1] and j = 1, . . . , m + 1, we have
diam([y(� j )]�) = A0(� j )
⎛⎝diam([y0]�) +j−1∑k=0
(−1)k+1∫ �k+1
�k
diam([b(s)]�)
A0(s)ds
⎞⎠ . (22)
12 R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 –26
Remark 3.14. Condition (18) is reduced to check that, for every j even with 0 � j � m, the following conditions hold:
A0(T )m∑
k= j+1
(−1)k+1 diam([Bk(�k+1)]�)
A0(�k)�
j∑k=0
(−1)k diam([Bk(�k+1)]�)
A0(�k), ∀� ∈ [0, 1],
A0(T )
⎧⎨⎩m∑
k= j+1,k odd
Bk(�k+1)�l
A0(�k)+
m∑k= j+1,k even
Bk(�k+1)�r
A0(�k)
⎫⎬⎭+
j−1∑k=0,k odd
Bk(�k+1)�l
A0(�k)+
j∑k=0,k even
Bk(�k+1)�r
A0(�k)is nondecreasing in �
and
A0(T )
⎧⎨⎩m∑
k= j+1,k odd
Bk(�k+1)�r
A0(�k)+
m∑k= j+1,k even
Bk(�k+1)�l
A0(�k)
⎫⎬⎭+
j−1∑k=0,k odd
Bk(�k+1)�r
A0(�k)+
j∑k=0,k even
Bk(�k+1)�l
A0(�k)is nonincreasing in �.
In other words, the validity of (18) is equivalent to the validity, for every j even with 0 � j � m, of the following:
A0(T )m∑
k= j+1
(−1)k+1∫ �k+1
�k
diam([b(s)]�)
A0(s)ds �
j∑k=0
(−1)k∫ �k+1
�k
diam([b(s)]�)
A0(s)ds, ∀� ∈ [0, 1],
A0(T )
⎧⎨⎩m∑
k= j+1,k odd
∫ �k+1
�k
b(s)�l
A0(s)ds +
m∑k= j+1,k even
∫ �k+1
�k
b(s)�r
A0(s)ds
⎫⎬⎭+
j−1∑k=0,k odd
∫ �k+1
�k
b(s)�l
A0(s)ds +
j∑k=0,k even
∫ �k+1
�k
b(s)�r
A0(s)ds is nondecreasing in �
and
A0(T )
⎧⎨⎩m∑
k= j+1,k odd
∫ �k+1
�k
b(s)�r
A0(s)ds +
m∑k= j+1,k even
∫ �k+1
�k
b(s)�l
A0(s)ds
⎫⎬⎭+
j−1∑k=0,k odd
∫ �k+1
�k
b(s)�r
A0(s)ds +
j∑k=0,k even
∫ �k+1
�k
b(s)�l
A0(s)ds is nonincreasing in �.
Theorem 3.12 extends some results given in [12] for m = 1 and a < 0 on (0, �), a > 0 on (�, T ), as we show in thefollowing remark.
Remark 3.15. For m = 1, condition (18) is reduced to
A0(T )∫ T
�1
diam([b(s)]�)
A0(s)ds �
∫ �1
0
diam([b(s)]�)
A0(s)ds for every � ∈ [0, 1],
A0(T )∫ T
�1
b(s)�l
A0(s)ds +
∫ �1
0
b(s)�r
A0(s)ds is nondecreasing in �
and
A0(T )∫ T
�1
b(s)�r
A0(s)ds +
∫ �1
0
b(s)�l
A0(s)ds is nonincreasing in �.
R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 –26 13
Theorem 3.16. Let a : [0, T ] → R and b : [0, T ] → RF be continuous such that (17) is satisfied. Suppose that (4)and also the following conditions hold:
j−1∑k=1,k odd
Bk(�k+1)
A0(�k)�
j∑k=1, k even
(−Bk(�k+1))
A0(�k)exists, for every j even with 2 � j � m (23)
and
m∑k=1, k odd
Bk(�k+1)
A0(�k)�
⎛⎝ m∑k=1,k even
(−Bk(�k+1))
A0(�k)+ (−B0(�1))
A0(T )
⎞⎠ exists in RF . (24)
Then there exists a solution u to the periodic boundary value problem (1) in the space F{�1,. . .,�m } which is (i)-differentiable on ∪k odd (�k, �k+1) and (ii)-differentiable on ∪k even(�k, �k+1).
Proof. The mapping G defined in the proof of Theorem 3.12 can also be defined in the closed set
C = {y0 ∈ RF : y0�(−B0(�1)) exists} ={
y0 ∈ RF : y0�∫ �1
0
(−b(s))
A0(s)ds exists
}.
The set C is nonempty, since there exists y0 ∈ RF such that
diam([y0]�) �
∫ �1
0
diam([b(s)]�)
A0(s)ds, for every � ∈ [0, 1],
(y0)�l +∫ �1
0
b(s)�r
A0(s)ds is nondecreasing in �,
(y0)�r +∫ �1
0
b(s)�l
A0(s)ds is nonincreasing in �.
To prove this fact, similar to the proof of Theorem 3.12, it suffices to select y0 ∈ RF in such a way that the diameter of
its level sets are large enough, (y0)�l increases in � faster (or at the same speed) than (∫ �1
0 b(s)/A0(s) ds)�r decreases,and the corresponding condition for the upper branch of y0.
On the other hand, condition (23) guarantees the existence of the Hukuhara differences in the expression of y forevery y0 ∈ C (so that the operator G is well-defined), and (24) provides that G maps C into itself.
We check that, for y0 ∈ C , the Hukuhara differences y(� j )�(−B j (t)) exist for every t ∈ (� j , � j+1] ( j even,0 � j � m). Indeed, take y0 ∈ C , then y0�(−B0(�1)) exists and, therefore, y0�(−B0(t)) exist on (0, �1]. Hence y iswell-defined on [0, �2] and
y(�2) = A0(�2)
(y0 + 1
A0(�1)B1(�2)�(−B0(�1))
).
For the existence of the differences y(�2)�(−B2(t)) for t ∈ (�2, �3], it suffices to prove the existence of
y(�2)�(−B2(�3)) = A0(�2)
(y0 + 1
A0(�1)B1(�2)�(−B0(�1))
)�(−B2(�3))
= A0(�2)
(y0�(−B0(�1)) + 1
A0(�1)B1(�2)�
1
A0(�2)(−B2(�3))
),
which is deduced from the choice of y0 ∈ C , (23) for j = 2, and Lemma 2.4.By induction, suppose that y is well-defined on [0, � j ] with j � 1 odd, and check that y is well-defined on [0, � j+2].
Besides, the expression for y(� j ) is given by (20). If y is defined on [0, � j ] with j odd, then y is well-defined on(� j , � j+1] and
y(� j+1) = A0(� j+1)
⎛⎝y0 +j∑
k=0,k odd
Bk(�k+1)
A0(�k)�
j−1∑k=0,k even
(−Bk(�k+1))
A0(�k)
⎞⎠ .
14 R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 –26
To check the good definition of y on (� j+1, � j+2] for j +1 even, we prove that y(� j+1)�(−B j+1(� j+2)) exists. Indeed,
y(� j+1)�(−B j+1(� j+2)) = A0(� j+1)
⎛⎝y0 +j∑
k=0,k odd
Bk(�k+1)
A0(�k)�
j∑k=0,k even
(−Bk(�k+1))
A0(�k)�
(−B j+1(� j+2))
A0(� j+1)
⎞⎠= A0(� j+1)
⎛⎝y0�(−B0(�1)) +j∑
k=0,k odd
Bk(�k+1)
A0(�k)�
j+1∑k=1,k even
(−Bk(�k+1))
A0(�k)
⎞⎠ ,
which exists from the choice of y0, the properties of the Hukuhara differences and condition (23), so that y is well-definedon [0, � j+2].
Finally, we check that (24) implies that G maps C into itself. Indeed, take y0 ∈ C , and prove that
G(y0) = A0(T )
⎛⎝y0 +m∑
k=0,k odd
Bk(�k+1)
A0(�k)�
m∑k=0,k even
(−Bk(�k+1))
A0(�k)
⎞⎠belongs to C . To this purpose, we justify that the following difference exists, using that y0 ∈ C , the properties of theHukuhara differences and (24), as follows
G(y0)�(−B0(�1)) = A0(T )
⎛⎝y0 +m∑
k=0,k odd
Bk(�k+1)
A0(�k)�
⎛⎝ m∑k=0,k even
(−Bk(�k+1))
A0(�k)+ (−B0(�1))
A0(T )
⎞⎠⎞⎠= A0(T )
⎛⎝y0�(−B0(�1))+m∑
k=0,k odd
Bk(�k+1)
A0(�k)�
⎛⎝ m∑k=1,k even
(−Bk(�k+1))
A0(�k)+ (−B0(�1))
A0(T )
⎞⎠⎞⎠ .
In this case, the solution is found in the set C , but it is the same solution given by Theorem 3.12. �
Remark 3.17. Under the hypotheses of Theorem 3.16, the initial condition y0 ∈ C leading to the solution to problem(1) which is (i)-differentiable on
⋃k odd(�k, �k+1) and (ii)-differentiable on
⋃k even(�k, �k+1) can be obtained as
y0 = A0(T )
1 − A0(T )
⎛⎝ m∑k=0,k odd
Bk(�k+1)
A0(�k)�
m∑k=0,k even
(−Bk(�k+1))
A0(�k)
⎞⎠ . (25)
The same expression of y0 is the initial condition derived in Theorem 3.12. In particular, under the assumptions ofTheorem 3.12, it is deduced that y0 ∈ RF belongs to the set C∗.
In the proof of Theorems 3.5, 3.12 and 3.16, we could have followed a constructive approach, avoiding the applicationof a fixed point result and checking, directly, the good definition of the solution starting at the initial condition y0 whichis appropriate to the periodic boundary condition. However, in our approach, the emphasis is made on the hypotheseswhich allow to guarantee the good definition of the solution y to the linear differential equation for initial conditionsy0 in a certain subset of RF , and not only for the solution to the boundary value problem. In any case, the expressionfor the suitable initial condition is also provided.
Finally, we study sufficient conditions for the existence of a solution which is crisp (respectively, real) at the switchingpoints where differentiability changes from (ii)-differentiability to (i)-differentiability. Note that we do not impose thisrestriction at �m+1, not even in the case where the solution is (ii)-differentiable on (�m, T ), since if the solution wasreal (or crisp) at t = T , then the same property would affect the initial condition, due to the type of boundary valueproblem considered. Thus, according to our construction, if (2) holds, then we refer to points � j where j is an evennumber with 2 � j � m; on the other hand, if (17) is satisfied, then we refer to points � j where j is an odd numberwith 1 � j � m.
We proceed with the study of the first case.
R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 –26 15
Theorem 3.18. Assume that the hypotheses of Theorem 3.5 hold, and suppose that the expression
j−1∑k=0
(−1)k diam([Bk(�k+1)]�)
A0(�k)+ A0(T )
m∑k= j
(−1)k diam([Bk(�k+1)]�)
A0(�k)
is constant in �, for each j even with 2 � j � m. Then the solution u to problem (1) given by Theorem 3.5 is crisp at thepoints � j where j is an even number with 2 � j � m.
Proof. The solution u provided by Theorem 3.5 satisfies that the value at � j is given by (8), where the initial conditionis specified in (10). To prove that the solution u is crisp at the points � j where j is an even number with 2 � j � m, wecheck that diam([u(� j )]�) is constant in � for each of those values of j . Indeed, for each j even with 2 � j � m,
diam([y(� j )]�) = A0(� j )
⎛⎝diam([y0]�) +j−1∑k=0
(−1)k diam([Bk(�k+1)]�)
A0(�k)
⎞⎠= A0(� j )
⎛⎝ A0(T )
1 − A0(T )
m∑k=0
(−1)k diam([Bk(�k+1)]�)
A0(�k)+
j−1∑k=0
(−1)k diam([Bk(�k+1)]�)
A0(�k)
⎞⎠= A0(� j )
1 − A0(T )
⎛⎝ j−1∑k=0
(−1)k diam([Bk(�k+1)]�)
A0(�k)+ A0(T )
m∑k= j
(−1)k diam([Bk(�k+1)]�)
A0(�k)
⎞⎠is constant in �, by hypotheses. �
Similarly, we obtain the following result.
Theorem 3.19. Assume that the hypotheses of Theorem 3.5 hold, and suppose that
j−1∑k=0
(−1)k diam([Bk(�k+1)]0)
A0(�k)+ A0(T )
m∑k= j
(−1)k diam([Bk(�k+1)]0)
A0(�k)= 0
for each j even with 2 � j � m. Then the solution u to problem (1) given by Theorem 3.5 is real at the points � j wherej is an even number with 2 � j � m.
The additional hypothesis in Theorem 3.19 is equivalent to
j−1∑k=0
(−1)k∫ �k+1
�k
diam([b(s)]0)
A0(s)ds + A0(T )
m∑k= j
(−1)k∫ �k+1
�k
diam([b(s)]0)
A0(s)ds = 0
for each j even with 2 � j � m.Next, we study the second case.
Theorem 3.20. Assume that the hypotheses of Theorem 3.12 (or Theorem 3.16) hold, and suppose that the expression
j−1∑k=0
(−1)k+1 diam([Bk(�k+1)]�)
A0(�k)+ A0(T )
m∑k= j
(−1)k+1 diam([Bk(�k+1)]�)
A0(�k)
is constant in �, for each j odd with 1 � j � m. Then the solution u to problem (1) given by Theorem 3.12 (or Theorem3.16) is crisp at the points � j where j is an odd number with 1 � j � m.
Proof. In this case, the solution u provided by Theorem 3.12 (or Theorem 3.16) satisfies that its value at � j is givenby (20), and the initial condition comes from (25). Again, to prove that u is crisp at the points � j where j is odd with
16 R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 –26
1 � j � m, we check that diam([u(� j )]�) is constant in � for each of those values of j . Indeed, if we fix j odd with1 � j � m, then
diam([y(� j )]�) = A0(� j )
⎛⎝diam([y0]�) +j−1∑k=0
(−1)k+1 diam([Bk(�k+1)]�)
A0(�k)
⎞⎠= A0(� j )
⎛⎝ A0(T )
1 − A0(T )
m∑k=0
(−1)k+1 diam([Bk(�k+1)]�)
A0(�k)+
j−1∑k=0
(−1)k+1 diam([Bk(�k+1)]�)
A0(�k)
⎞⎠= A0(� j )
1 − A0(T )
⎛⎝ j−1∑k=0
(−1)k+1 diam([Bk(�k+1)]�)
A0(�k)+ A0(T )
m∑k= j
(−1)k+1 diam([Bk(�k+1)]�)
A0(�k)
⎞⎠is constant in �, by hypothesis. �
Theorem 3.21. Assume that the hypotheses of Theorem 3.12 (or Theorem 3.16) hold, and suppose that
j−1∑k=0
(−1)k+1 diam([Bk(�k+1)]0)
A0(�k)+ A0(T )
m∑k= j
(−1)k+1 diam([Bk(�k+1)]0)
A0(�k)= 0 (26)
for each j odd with 1 � j � m. Then the solution u to problem (1) given by Theorem 3.12 (or Theorem 3.16) is real atthe points � j where j is an odd number with 1 � j � m.
Remark 3.22. For m = 1 and �1 = �, condition (26) is written ( j = 1) as
−∫ �
0
diam([b(s)]0)
A0(s)ds + A0(T )
∫ T
�
diam([b(s)]0)
A0(s)ds = 0.
We state the previous existence results in other terms.
Theorem 3.23. Assume that the hypotheses of Theorem 3.16 hold and suppose, moreover, that the differences incondition (23) are crisp (resp. real) for every j even with 2 � j < m, and that the difference in (24) is crisp (resp. real).Then the solution u to problem (1) given by Theorem 3.16 is crisp (resp. real) at the points � j where j is an odd numberwith 1 � j � m.
Proof. It is straightforward, since the initial condition y0 given by (25) belongs to RF and satisfies that the correspondingsolution (19) is crisp (resp. real) at the points � j where j is an odd number with 1 � j � m. Indeed, we check thaty0�(−B0(�1)) exists and it is crisp (resp. real) and that y0 ∈ RF , which follow from the substitution of the correspondingexpressions
A0(T )
1 − A0(T )
⎧⎨⎩m∑
k=1, k odd
Bk(�k+1)
A0(�k)�
⎛⎝ m∑k=1,k even
(−Bk(�k+1))
A0(�k)+ (−B0(�1))
A0(T )
⎞⎠⎫⎬⎭and using the properties:
• If A�(C + D) exists and it is crisp, then A�C exists (non necessarily crisp) and (A�C)�D exists and it is crisp(it coincides with A�(C + D)).
• If A�(C + D) exists and it is real, then A�C exists (non necessarily real) and (A�C)�D exists and it is real(it coincides with A�(C + D)).
By property (23) and the restriction imposed for j even with 2 � j < m, the solution y given by (19) is well-definedon [0, T ] and it is crisp (resp. real) at the remaining required points. �
One qualitative difference between the crisp and the real case is that, in the real case, the Hukuhara differences inthe hypotheses can be split in differences of two fuzzy numbers, taking as basis the following property.
R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 –26 17
Remark 3.24. Let A, B, C, D ∈ RF , and suppose that A�C and (A + B)�(C + D) exist and are crisp. If B�Dexists in RF , then it is crisp. However, B�D does not necessarily exist.
On the other hand, if A�C and (A + B)�(C + D) are real, then B�D is real. Obviously, if A�C and B�D existand are real, then (A + B)�(C + D) is real.
Thus, in the real case, the hypotheses of Theorem 3.23 can be given in simpler differences between fuzzy numbersas follows.
Theorem 3.25. Let a : [0, T ] → R and b : [0, T ] → RF be continuous such that (17) is valid. Suppose that (4) andalso the following conditions hold:
B j−1(� j )
A0(� j−1)�
(−B j (� j+1))
A0(� j )exists and it is real for every j even wi th 2 � j < m, (27)
if m even,Bm−1(�m)
A0(�m−1)�
(−Bm(�m+1))
A0(�m)exists in RF (nonnecessarily real), (28)
and
m∑k=m−1,k odd
Bk(�k+1)
A0(�k)�
⎛⎝ m∑k=m,k even
(−Bk(�k+1))
A0(�k)+ (−B0(�1))
A0(T )
⎞⎠ exists and it is real. (29)
Then there exists a solution u to the periodic boundary value problem (1) in the space F{�1,. . .,�m } which is (i)-
differentiable on⋃
k odd (�k, �k+1), (ii)-differentiable on⋃
k even (�k, �k+1) and real at the points � j where j is anodd number with 1 � j � m.
Proof. It follows from the previous arguments and taking into account that y0�(−B0(�1)) can be thought as
A0(T )
1 − A0(T )
⎧⎨⎩m−1∑
k=2,k even
(Bk−1(�k)
A0(�k−1)�
(−Bk(�k+1))
A0(�k)
)+ Bm(�m+1)
A0(�m)�
(−B0(�1))
A0(T )
⎫⎬⎭for m odd, and
A0(T )
1 − A0(T )
⎧⎨⎩m−2∑
k=2,k even
(Bk−1(�k)
A0(�k−1)�
(−Bk(�k+1))
A0(�k)
)+ Bm−1(�m)
A0(�m−1)�
((−Bm(�m+1))
A0(�m)+ (−B0(�1))
A0(T )
)⎫⎬⎭for m even. �
Remark 3.26. If m = 1 and �1 = �, the conditions in Theorem 3.23 in the crisp case are: the corresponding conditionson the sign of a (a < 0 on (0, �), a > 0 on (�, T )), hypothesis (4) and
B1(�2)
A0(�1)�
(−B0(�1))
A0(T )exists in RF and it is crisp, (30)
i.e., ∫ T
�
b(s)
A0(s)ds�
1
A0(T )
∫ �
0
(−b(s))
A0(s)ds exists in RF and it is crisp. (31)
This way, we obtain a solution u to the periodic boundary value problem (1) which is (ii)-differentiable on (0, �),(i)-differentiable on (�, T ) and crisp at �. For a solution with u(�) real (see also Theorem 3.25), it suffices to considerthe difference (30) (equivalently, (31)) to be real, so that this study extends the corresponding results in [12].
18 R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 –26
4. Example
We show an example, where function a is chosen as a(t) = cos(t), which is zero at the points t = (2n − 1)�/2, forn ∈ Z.
Example 4.1. We consider the following periodic boundary value problem:⎧⎪⎪⎨⎪⎪⎩y′(t) = cos(t)y(t) + e−t+sin(t)�, t ∈ J =
[0,
7�
2
],
y(0) = y
(7�
2
),
(32)
where � is the triangular fuzzy interval defined as [�]� = [� − 1, 1 − �], for � ∈ [0, 1]. Since a(t) = cos(t), we selectthe sequence {�k} by choosing the endpoints of the interval J , as well as the zeros of a in J , that is,
�0 = 0 < �1 = �
2< �2 = 3�
2< �3 = 5�
2< �4 = 7�
2
or � j = (2 j − 1)�/2, j = 0, . . . , m + 1, and m = 3.For this example, we check that the hypotheses in Theorem 3.5 are satisfied (see Remark 3.8 and also Corollary
3.9). Indeed, a (real) and b (fuzzy interval-valued) are continuous functions, a > 0 on (�0, �1) ∪ (�2, �3) = (0, �/2) ∪(3�/2, 5�/2), a < 0 on (�1, �2) ∪ (�3, �4) = (�/2, 3�/2) ∪ (5�/2, 7�/2),∫ 7�/2
0cos(u) du = sin
(7�
2
)= −1 < 0
and, for every � ∈ [0, 1] and every odd number j with � j < 7�/2, that is, j = 1 or j = 3, condition (3) is satisfied,since b(t) = e−t+sin(t)�, thus
The same expressions can be obtained by solving the equations
Gy∗0� = e−1(y∗
0� + P(� − 1)) = y∗
0� and Gy∗
0� = e−1(y∗
0� + P(1 − �)) = y∗
0�
for � ∈ [0, 1].
R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 –26 21
-4
-3
-2
-1
0
1
2
3
4
0 2 4 6 8 10t
left endpoint of supp(y)core of y
right endpoint of supp(y)
Fig. 1. Solution to problem (32) which is (i)-differentiable on (0,�/2) ∪ (3�/2, 5�/2), and (ii)-differentiable on (�/2, 3�/2) ∪ (5�/2, 7�/2).
Therefore, a solution to (32) is, taking into account (7), the function given by
y(t) =
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
esin(t)(y∗0 + �
∫ t0 e−s ds), t ∈
[0,
�
2
],
esin(t)−1(
y(�
2
)��
∫ t�/2(−e−s+1) ds
), t ∈
(�
2,
3�
2
],
esin(t)+1(
y
(3�
2
)+ �
∫ t3�/2 e−s−1 ds
), t ∈
(3�
2,
5�
2
],
esin(t)−1(
y
(5�
2
)��
∫ t5�/2(−e−s+1) ds
), t ∈
(5�
2,
7�
2
]that is,
y(t) =
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
esin(t)(y∗0 + �(1 − e−t )), t ∈
[0,
�
2
],
esin(t)−1(
y(�
2
)��(e−t+1 − e−�/2+1)
), t ∈
(�
2,
3�
2
],
esin(t)+1(
y
(3�
2
)+ �(e−3�/2−1 − e−t−1)
), t ∈
(3�
2,
5�
2
],
esin(t)−1(
y
(5�
2
)��(e−t+1 − e−5�/2+1)
), t ∈
(5�
2,
7�
2
].
In Fig. 1, we show the endpoints of the �-level sets of y for � = 0 and � = 1.
The type of problems studied may derive from specific problems which arise in the modeling of real world phenomena.To illustrate this fact, we include an application of the results in this paper to a population model for a single specieswhose behavior is described by a differential equation similar to that in problem (32), where the number of individualsat each instant is difficult to compute and, thus, imprecise:
y′(t) = 0.1 cos(�
3t)
y(t) + e−0.3t+(3/(10�)) sin((�/3)t) cos(�
3t)
�, t � 0 (33)
for � = (1900, 2000, 2100) the triangular fuzzy number given by [�]� = [1900 + 100�, 2100 − 100�], ∀� ∈ [0, 1].The population is measured in number of members and time in months. The coefficient a(t) := 0.1 cos((�/3)t)
represents the population growth rate and it is considered as a function depending on time in such a way that, inthe seasons with optimal environmental conditions, the population increases rapidly but, in absence of conditions ofhabitability, it declines severely. We have considered a population which finds optimal conditions of the environmentin the spring and autumn (temperate climate) but finds difficulties to persist in the habitat during the hot dry season,
22 R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 –26
due to the lack of water and high temperatures, and also during the winter (cold and deficiency of food and sustenance).Apart from resources restrictions, there are also other types of factors which might diminish the population size suchas possible diseases affecting the population growth during the hottest months, where its spread can be favored by theabundance of insects, and others which may have a stronger influence during the cold season. Hence, we have adjustedthe coefficient a for taking negative values during two seasons a year (understanding that the extreme values are takenin the middle of the season). We set the initial instant in the middle of spring, corresponding to the maximum positivevalue of function a and, then, assign alternately negative and positive values to this function a, changing its sign aftera period of three months (except in the first season, which is not complete). For different populations, the coefficient ashould be adapted to the specifics of the habitat and species considered. In summary, if the equation was homogeneous,we are considering that the rate of change in the population is proportional to the total number of members through afunction of t which can take positive and negative values.
We consider that the population is open, so that migration is allowed between the population and other groups ofthe same species established outside the region of interest. However, it is assumed that these movements are producedindependently of the population size but obviously affecting its rate of change. The term
b(t) = e−0.3t+(3/(10�)) sin((�/3)t) cos(�
3t)
�
reflects the migration movements, the immigration movement is produced mainly during spring and autumn andemigration during the hot dry and cold seasons, being also more significant during the middle part of the seasons.The fact that function b in Eq. (33) decreases its amplitude with time can be explained by a certain adaptation to theenvironment.
Once clarified the meaning of the terms of the equation, we ask, in relation with the boundary condition, whether itis possible to obtain, at the end of the next hot dry season, the same number of members as at t = 0, that is, we haveto solve the periodic boundary value problem consisting of Eq. (33) and the boundary condition y(0) = y( 33
2 ), andhence, we work in the interval J = [0, 33
2 ].To obtain one solution with these features, we can select the switching points as the points where function a changes
its sign (in this case, zeroes of a), that is,
�0 = 0 < �1 = 3
2< �2 = 9
2< �3 = 15
2< �4 = 21
2< �5 = 27
2< �6 = 33
2or
�0 = 0, � j = 3(2 j − 1)
2, j = 1, . . . , 6, and m = 5.
It is clear that conditions in Theorem 3.5 hold: a, b are continuous, a > 0 on⋃
R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 –26 23
-0.5
0
0.5
1
1.5
4600 4700 4800 4900 5000 5100 5200
Fig. 2. Initial condition y0 for a solution to the periodic boundary value problem.
From Theorem 3.5, we obtain a solution to the periodic boundary value problem if we take the initial condition:
y0 = 1
e3/(10�) − 1
(B0(�1) + B2(�3)
A0(�2)+ B4(�5)
A0(�4)�
((−B1(�2))
A0(�1)+ (−B3(�4))
A0(�3)+ (−B5(�6))
A0(�5)
))= 1
e3/(10�) − 1
∫ 33/2
0e−0.3s cos
(�
3s)
ds [�]� = G[�]�,
where
G = 2.7 − 3�e−0.3∗33/2
(e3/(10�) − 1)(�2 + 0.81) 2.46.
Hence, the initial condition y0 is written as the triangular number y0 = (1900G, 2000G, 2100G) with levels [y0]� =[(1900 + 100�)G, (2100 − 100�)G], for � ∈ [0, 1], where 1900G 4675, 2000G 4921 and 2100G 5168, oralso (see Fig. 2)
y0(x) :=
⎧⎪⎪⎨⎪⎪⎩x
100G− 19 if 1900G � x � 2000G,
− x
100G+ 21 if 2000G < x � 2100G,
0 otherwise.
Note that the core of the independent term
b(t) = e−0.3t+(3/(10�)) sin((�/3)t) cos(�
3t)
�
in (33) is as shown in Fig. 3(a), while the endpoints of some level sets of b(t) are represented in Fig. 3(b), being
b(t)�l = e−0.3t+(3/(10�)) sin((�/3)t) min{
(1900 + 100�) cos(�
3t)
, (2100 − 100�) cos(�
3t)}
,
b(t)�r = e−0.3t+(3/(10�)) sin((�/3)t) max{
(1900 + 100�) cos(�
3t)
, (2100 − 100�) cos(�
3t)}
for � ∈ [0, 1].
24 R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 –26
0
-1000
-500
500
1000
1500
2000
0 2 4 6 8 10 12 14 16t
core of b(t)
-1000
-500
0
500
1000
1500
2000
0 2 4 6 8 10 12 14 16
b(t)
t
left endpoint of supp(b(t))core of b(t)
right endpoint of supp(b(t))
Fig. 3. Aspect of b(t). (a) Core of function b(t), given by 2000 e−0.3t+(3/(10�)) sin((�/3)t) cos ((�/3)t), t � 0. (b) Endpoints of the support and coreof b(t).
On the other hand, the solution to the boundary value problem on the interval [0, 332 ] is defined, according to (7), by
y(t)�l = e3 sin(�t/3)/(10�)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(1900 + 100�)G + (1900 + 100�)∫ t
0 e−0.3s cos(�
3s)
ds if t ∈[
0,3
2
],
e−3/(10�) y
(3
2
)�l
+ (1900 + 100�)∫ t
3/2 e−0.3s cos(�
3s)
ds if t ∈[
3
2,
9
2
],
e3/(10�) y
(9
2
)�l
+ (1900 + 100�)∫ t
9/2 e−0.3s cos(�
3s)
ds if t ∈[
9
2,
15
2
],
e−3/(10�) y
(15
2
)�l
+ (1900 + 100�)∫ t
15/2 e−0.3s cos(�
3s)
ds if t ∈[
15
2,
21
2
],
e3/(10�) y
(21
2
)�l
+ (1900 + 100�)∫ t
21/2 e−0.3s cos(�
3s)
ds, if t ∈[
21
2,
27
2
],
e−3/(10�) y
(27
2
)�l
+ (1900 + 100�)∫ t
27/2 e−0.3s cos(�
3s)
ds if t ∈[
27
2,
33
2
]and
y(t)�r = e3 sin(�t/3)/(10�)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(2100 − 100�)G + (2100 − 100�)∫ t
0 e−0.3s cos(�
3s)
ds if t ∈[
0,3
2
],
e−3/(10�) y
(3
2
)�r
+ (2100 − 100�)∫ t
3/2 e−0.3s cos(�
3s)
ds if t ∈[
3
2,
9
2
],
e3/(10�) y
(9
2
)�r
+ (2100 − 100�)∫ t
9/2 e−0.3s cos(�
3s)
ds if t ∈[
9
2,
15
2
],
e−3/(10�) y
(15
2
)�r
+ (2100 − 100�)∫ t
15/2 e−0.3s cos(�
3s)
ds if t ∈[
15
2,
21
2
],
e3/(10�) y
(21
2
)�r
+ (2100 − 100�)∫ t
21/2 e−0.3s cos(�
3s)
ds if t ∈[
21
2,
27
2
],
e−3/(10�) y
(27
2
)�r
+ (2100 − 100�)∫ t
27/2 e−0.3s cos(�
3s)
ds if t ∈[
27
2,
33
2
],
R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 –26 25
4000
4500
5000
5500
6000
6500
7000
7500
8000
0 2 4 6 8 10 12 14 16
y
t
core of solution y
4000
4500
5000
5500
6000
6500
7000
7500
8000
0 2 4 6 8 10 12 14 16
y
t
left endpoint of supp(y)core of y
right endpoint of supp(y)
Fig. 4. Solution to the boundary value problem. (a) Core of y(t). (b) Support and core of the solution y(t).
for � ∈ [0, 1]. To quantify the estimated size of the population at the switching points for the solution to the boundaryvalue problem, we give approximations of the endpoints of the support at those instants:
y( 32 )0l 6849, y( 9
2 )0l 4291, y( 152 )0l 5867, y( 21
2 )0l 4621, y( 272 )0l 5704
and
y( 32 )0r 7570, y( 9
2 )0r 4743, y( 152 )0r 6484, y( 21
2 )0r 5107, y( 272 )0r 6305.
The core of the solution to the boundary value problem on the interval [0, 332 ] is represented in Fig. 4(a). For the support
and the core of the solution, see Fig. 4(b).This study is useful to give an estimate of the population’s size needed for recovering the initial number of members
after 16.5 months. Another problem of interest in ecology is to study the possibility of making stable a certain population.
5. Conclusions
Note that the solutions provided by the main results in this paper do not constitute all possible solutions, sincethe solutions to the periodic boundary value problem (1) are not necessarily related to the sign of a. There existother possibilities to obtain more general solutions to problem (1) by using the concept of switching point [21] withoutpaying attention to the sign of function a, since there exist two (local) solutions to the problem ((i) and (ii)-differentiablefunctions, respectively) and the switching point is not necessarily located at a zero of the function a. However, we provideone solution to the problem, which is not unique, but it satisfies the periodic boundary condition, and we give sufficientconditions which guarantee that this solution is well-defined on [0, T ] and has the properties of differentiability ofinterest. The particular solutions presented are natural since the switching points are exactly located at the points wherefunction a changes it sign so that, although the solution is not unique, it is useful to show, for this type of equations, thatperiodic boundary value problems and fuzzy differential equations are compatible from the point of view of generalizeddifferentiability. Moreover, the results provided are useful to deduce the existence of periodic solutions to fuzzy lineardifferential equations, just by solving the equation subject to the periodic boundary condition y(0) = y(T ) in caseswhere the coefficients a and b are T -periodic functions.
On the other hand, although the type of equations considered is quite simple, we remark that, in many occasions, wecan approximate the solutions to more general (nonlinear) problems by using the solutions to linear problems with verysimple coefficients (for instance, through the monotone iterative technique, applied in [18] to impulsive differentialequations). Therefore, the study of the existence of solutions to this type of simple problems can be considered anappropriate starting point for the study of the properties of solutions to problems with a more complex fuzzy structure.
26 R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 –26
Acknowledgements
The author is grateful to the editor and the anonymous referees whose interesting and helpful comments and sugges-tions were important to improve the paper. This research is partially supported by Ministerio de Educación y Cienciaand FEDER, Project MTM2010-15314.
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