On the existence of solutions to periodic boundary value problems for fuzzy linear differential equations

Available online at www.sciencedirect.com

Fuzzy Sets and Systems 219 (2013) 1 – 26 www.elsevier.com/locate/fss

On the existence of solutions to periodic boundary value problems for fuzzy linear differential equations Rosana Rodríguez-López∗ Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain Received 11 December 2010; received in revised form 31 October 2012; accepted 3 November 2012 Available online 21 November 2012

Abstract In this work, we provide sufficient conditions which guarantee the existence of solutions to periodic boundary value problems for first-order linear fuzzy differential equations by using generalized differentiability and switching points. In comparison with some previous works, we consider equations whose coefficient may change its sign a finite number of times in the interval of interest. We also study the existence of solutions which are crisp (or real) at the switching points where the diameter of the level sets changes from nonincreasing to nondecreasing character. © 2012 Elsevier B.V. All rights reserved. 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 fuzzy differential equation and to study the existence of such solutions. For instance, in Ref. [9], the existence and uniqueness problem is addressed for first-order fuzzy differential equations. The particularities of this definition of differentiability produce the nondecreasing behavior of the diameter of the level sets of a differentiable function, which weakens the potential of this approach in the study of periodic boundary value problems for fuzzy differential equations. In order to model periodic phenomena through fuzzy systems under Hukuhara differentiability, one can try, for instance, to introduce impulses in the model [18]. The basic concepts on fuzzy sets, fuzzy differentials and fuzzy differential equations can be found in [6,7,9,10,17]. See [5,16] for the expression of the solution to initial value problems for linear fuzzy differential equations with constant coefficients under Hukuhara differentiability. Some results on fuzzy functional differential equations are included in [6,14], and other approaches in the analysis of uncertain systems are provided in [8,4,13]. On the other hand, for strongly generalized differentiability, see [1–3,21]. Some applications of generalized differentiability to numerical calculus of solutions and the study of Nth-order fuzzy differential equations are included in [15,11]. Ref. [21] is focused on interval differential equations, whose solution is calculated by proving a characterization by ODEs and using a numerical scheme. Some initial value problems for fuzzy linear differential equations are solved by combination of two types of derivatives using a switching point. See also [20] for a generalization of the difference of Hukuhara and the division to the context of real intervals and fuzzy numbers and applications to fuzzy equations. ∗ Tel.: +34 881 81 10 00; fax: +34 881 813197.

E-mail address: [email protected]. 0165-0114/$ - see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2012.11.007

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 firstorder fuzzy differential equations is analyzed under strongly generalized differentiability. The problem considered is y  (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 ] → R F are continuous functions, with a changing its sign only once in the interval J. In this paper, we establish sufficient conditions for the existence of solution to the above-mentioned periodic boundary value problem, where the coefficient a may have an arbitrary 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. We also analyze the existence of solutions which are crisp (or real) at the switching points where the piecewise solution changes from (ii)- to (i)-differentiability. Finally, we show some examples to illustrate how this type of fuzzy differential equations can be used, using this approach, to model phenomena which present the same behavior at two different instants. We have adopted a notation which allows a direct application of the Hukuhara difference’s properties and is more consistent with the general theoretical results available on the solvability of fuzzy linear differential equations. 2. Basic concepts Consider the space R F of fuzzy intervals, that is, the class of elements u : R → [0, 1] satisfying that (i) (ii) (iii) (iv)

u is normal, i.e., there exists s0 ∈ R such that u(s0 ) = 1, u is upper semicontinuous on R, u is fuzzy-convex, that is, u(ts + (1 − t)r ) ⱖ min{u(s), u(r )}, ∀t ∈ [0, 1], and s, r ∈ R, cl{s ∈ R|u(s) > 0} is compact, where cl denotes the closure of a set.

For u ∈ R F , and each 0 <  ⱕ 1, the -level set of u is defined as the nonempty compact interval [u] = {s ∈ R|u(s) ⱖ } 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 and u() = 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 R F , we need to define the addition and multiplication by a scalar in the space R F , which are defined levelsetwise. If we consider the metric D(u, v) = sup max{|u l − vl |, |u r − vr |}, for u, v ∈ R F ∈[0,1]

then (R F , 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 ∈ R F , if there exists z ∈ R F with x = y + z, we say that z is the H-difference of x and y, denoted by xdy. In the following, we need some properties of the Hukuhara difference, whose proofs are immediate. Lemma 2.2. Let A, B, C ∈ R F . If AdC exists, then (A + B)dC exists and (A + B)dC = (AdC) + B. Lemma 2.3. Let A, C, D ∈ R F . The following assertions hold: • If the differences AdC and (AdC)dD exist, then Ad(C + D) exists and Ad(C + D) = (AdC)dD. • If Ad(C + D) exists, then the differences AdC and (AdC)dD exist and (AdC)dD = Ad(C + D). Lemma 2.4. Let A, B, C, D ∈ R F . If AdC and BdD exist, then there exist (A + B)dC and (A + B)d(C + D) (and, therefore, by Lemma 2.3, ((A + B)dC)dD) exists and coincides with (A + B)d(C + D)). Lemma 2.5. Let A, B ∈ R F and  > 0 a real number. Then diam([A + B] ) = diam([A] ) + diam([B] ) for every  ∈ [0, 1], diam([AdB] ) = 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 of generalized 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 ] → R F are continuous functions. Problem (1) is related to problem (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 of points where the change of sign occurs. To this purpose, in the interval J = [0, T ], we consider a finite sequence of real 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 functions which 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{ , ..., } , consisting on the m 1 functions u ∈ C(J, R F ) which are differentiable in the sense of generalized differentiability on (0, T ), with switching points at 1 , . . . , m , and such that there exist the one-sided limits u  (0+ ) and u  (T − ) in R F . In this work, we consider particular solutions to problem (1) which belong to the space F{k } . Note that the following definition 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{ , ..., } which satisfies the m 1 conditions in (1). In the following results, we use the notation  t t b(s)  j a(u) du A j (t) = e ds, for t ⱖ  j and j = 0, . . . , m. and B j (t) =  j A j (s) The following properties are also useful to the procedure. Lemma 3.3. A0 (k )Ak (t) = A0 (t), for t ⱖ k and k = 1, . . . , m. Lemma 3.4. Let j = 0, . . . , m and  ∈ R,  > 0. The Hukuhara differences Ad(−B j (t)) exist, for every t ∈ ( j ,  j+1 ], if and only if Ad(−B j ( j+1 )) exists. Proof. If Ad(−B j ( j+1 )) exists, then we check that diam([A] ) ⱖ  diam([B j (t)] ) for every  ∈ [0, 1], Al + B j (t)r is nondecreasing in , Ar + B j (t)l is nonincreasing in  for every t ∈ ( j ,  j+1 ], assuming their validity for t =  j+1 . Indeed,  t   j+1 diam([b(s)] ) diam([b(s)] ) ds ⱕ  ds  diam([B j (t)] ) =  A j (s) A j (s) j j

=  diam([B j ( j+1 )] ) ⱕ diam([A] ).

4

R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 – 26

Besides, for  ⱕ , Al + B j ( j+1 )r ⱕ Al + B j ( j+1 )r and, hence, for every t ∈ ( j ,  j+1 ],  t   j+1 b(s)r − b(s)r b(s)r − b(s)r (B j (t)r − B j (t)r ) =  ds ⱕ  ds A j (s) A j (s) j j = (B j ( j+1 )r − B j ( j+1 )r ) ⱕ Al − Al . The proof of the nonincreasing character of Ar + B j (t)l in the variable , for every t ∈ ( j ,  j+1 ], follows similarly.  First, for any (finite) arbitrary number of terms in the sequence {k }, we suppose that the continuous real function a : [0, T ] → R is such that   a > 0 on (k , k+1 ) for every even number k with k ⱕ m, (2) a < 0 on (k , k+1 ) for every odd number k with k ⱕ m. 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 ] → R F be continuous functions such that (2) holds. Suppose that the Hukuhara differences ⎛ ⎞ j−1 j   1 1 Bk (k+1 )d ⎝ (−Bk (k+1 ))⎠ exist in R F for every j odd with 1 ⱕ j ⱕ m, A0 (k ) A0 (k ) k=0,k even

k=1,k odd

(3) and 

T

a(u) du < 0.

(4)

0

Then there exists a solution u to problem (1) in the space F{ , ..., } . Furthermore, this solution u is (i)-differentiable m 1   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 the fuzzy interval which is obtained as the value at the instant T of a solution (in the sense of Definition 3.2) to the linear equation subject to the initial condition y0 . This solution is built piecewise by using switching points between intervals of (i)-differentiability and (ii)-differentiability depending on the changes of sign of the coefficient a, following the ideas in [21] and the expressions of the (i)- and (ii)-solutions given in [3]. Indeed, if we fix y0 ∈ R F , since a > 0 in (0, 1 ), then

 t t s a(u) du − a(u) du b(s) e 0 ds = A0 (t)(y0 + B0 (t)), t ∈ (0, 1 ] y0 + y(t) = e 0 0

and hence, for the calculus of the solution on the next interval (1 , 2 ], we take as initial condition y(1 ) = A0 (1 )(y0 + B0 (1 )) so that y(t) = e

t

1

a(u) du



 y(1 )d

(5)

t

1

(−b(s)) e

−

s

1

a(u) du

ds

= A1 (t){y(1 )d(−B1 (t))} = A1 (t){A0 (1 )(y0 + B0 (1 ))d(−B1 (t))}

1 = A1 (t)A0 (1 ) y0 + B0 (1 )d (−B1 (t)) A0 (1 )

1 = A0 (t) y0 + B0 (1 )d (−B1 (t)) , t ∈ (1 , 2 ] A0 (1 )

(6)

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 used Lemma 3.3. Moreover, the Hukuhara differences B0 (1 )d

1 (−B1 (t)) A0 (1 )

exist in R F , for every t ∈ (1 , 2 ], since they exist for t = 2 by hypothesis (3) with j = 1 and, hence, it is justified that the expression y is well-defined on [0, 2 ] independently of the initial condition y0 ∈ R F . Continuing this process on J = [0, T ], the sought solution y ∈ C(J, R F ) to the linear fuzzy differential equation y  (t) = a(t)y(t) + b(t), t ∈ J , is given recursively as follows:  for t ∈ ( j ,  j+1 ], and j even, A j (t)(y( j ) + B j (t)) (7) y(t) = A j (t)(y( j )d(−B j (t))) for t ∈ ( j ,  j+1 ], and j odd 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 R F of the Hukuhara differences y( j )d(−B j (t)), for every t ∈ ( j ,  j+1 ], and j odd. We have already checked this fact for j = 1. To this purpose, we calculate, in terms of y0 , the value of y at the switching points  j , for j 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 every j = 1, . . . , m + 1. Indeed, we prove that y is well-defined on [0, T ] and that ⎛ ⎞ j−1 j−1   1 1 (8) Bk (k+1 )d (−Bk (k+1 ))⎠ y( j ) = A0 ( j ) ⎝ y0 + A0 (k ) A0 (k ) k=0,k even

k=1,k odd

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

1 y(2 ) = A0 (2 ) y0 + B0 (1 )d (−B1 (2 )) , A0 (1 ) 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 ))

1 = A2 (3 ) A0 (2 ) y0 + B0 (1 )d (−B1 (2 )) + B2 (3 ) A0 (1 )

1 1 = A0 (3 ) y0 + B0 (1 ) + B2 (3 )d (−B1 (2 )) A0 (2 ) A0 (1 ) equal to (8) for j = 3. Next, we justify that y is well-defined on (3 , 4 ], proving the existence of the differences A3 (t)(y(3 )d(−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 )d(−B3 (4 )))

1 1 = A3 (4 ) A0 (3 ) y0 + B0 (1 ) + B2 (3 )d (−B1 (2 )) d(−B3 (4 )) A0 (2 ) A0 (1 )

1 1 1 = A0 (4 ) y0 + B0 (1 ) + B2 (3 )d (−B1 (2 )) + (−B3 (4 )) . A0 (2 ) A0 (1 ) A0 (3 ) By hypothesis (3) for j = 3, the previous Hukuhara difference exists. Therefore, y is well-defined on [0, 4 ] and the previous expression for y(4 ) coincides with (8) for j = 4. By induction, suppose that y is well-defined on [0,  j−1 ] for 3 ⱕ j ⱕ m odd and that (8) holds for j − 1, i.e., ⎛ ⎞ j−2 j−2   1 1 y( j−1 ) = A0 ( j−1 ) ⎝ y0 + Bk (k+1 )d (−Bk (k+1 ))⎠ A0 (k ) A0 (k ) k=0,k even

k=1,k odd

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 ⱖ 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

⎛ = A0 ( j ) ⎝ y0 +

j−1  k=0,k even

1 Bk (k+1 )d A0 (k )

1 Bk (k+1 )d A0 (k )

j−1  k=1,k odd

j−2  k=1,k odd

⎞ ⎞ 1 (−Bk (k+1 ))⎠ + B j−1 ( j )⎠ A0 (k )

⎞ 1 (−Bk (k+1 ))⎠ A0 (k )

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 differences y(t) = A j (t)(y( j )d(−B j (t))) exist for t ∈ ( j ,  j+1 ], since it exists for t =  j+1 due to (3). Then, y is well-defined on [0,  j+1 ] and, besides, since j is odd, y( j+1 ) = A j ( j+1 )(y( j )d(−B j ( j+1 ))) ⎛ ⎛ j−1  = A j ( j+1 ) ⎝ A0 ( j ) ⎝ y0 + k=0,k even

⎞

1 Bk (k+1 )d A0 (k )

j−1  k=1,k odd

⎞ 1 (−Bk (k+1 ))⎠ A0 (k )

d(−B j ( j+1 ))⎠ ⎛ = A0 ( j+1 ) ⎝ y0 +

j−1  k=0,k even

1 Bk (k+1 )d A0 (k )

⎞

j  k=1,k odd

1 (−Bk (k+1 ))⎠ A0 (k )

expression which coincides with (8) for j + 1. This justifies that y is well-defined on [0, T ] and that (8) is valid for j = 1, . . . , m + 1. Therefore, we define the operator G : R F −→ R F 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 periodic boundary value problem related to the linear fuzzy differential equation studied. y0 ∈ R F , To check the contractive character of G, we prove that, for every y0 ,  D(y( j ),  y( j )) = A0 ( j )D(y0 ,  y0 ), ∀ j = 0, 1, . . . , m,

(9)

where y and  y are the solutions corresponding, respectively, to the initial values y0 ,  y0 ∈ R F , 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 ⎧ ⎫ ⎛ j−1 j−1 ⎨ ⎬   1 1 Bk (k+1 )d (−Bk (k+1 )) D(y( j ),  y( j )) = D ⎝ A0 ( j ) y0 + ⎩ ⎭ A0 (k ) A0 (k ) k=0,k even

⎧ ⎨ A0 ( j )  y0 + ⎩

j−1  k=0,k even

= A0 ( j )D(y0 ,  y0 ).

1 Bk (k+1 )d A0 (k )

k=1,k odd

j−1  k=1,k odd

⎫⎞ ⎬ 1 (−Bk (k+1 )) ⎠ ⎭ A0 (k )

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 y0 )) = D(y(T ),  y(T )) = D(y(m+1 ),  y(m+1 )) D(G(y0 ), G( y0 ) = A0 (T )D(y0 ,  y0 ) = e = A0 (m+1 )D(y0 ,  where the Lipschitz constant A0 (T ) = e

T

a(u) du

0

T 0

a(u) du

D(y0 ,  y0 ),

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) in F{ , ..., } , but it is the unique solution to (1) which is (i)-differentiable on k even (k , k+1 ) and (ii)-differentiable on m  1 k odd (k , k+1 ). Under the hypotheses of Theorem 3.5, the initial condition y0 ∈ R F leading to this solution u can be obtained as ⎞ ⎛ m m  1 1 A0 (T ) ⎝  (10) Bk (k+1 )d (−Bk (k+1 ))⎠ . y0 = 1 − A0 (T ) A0 (k ) A0 (k ) k=0,k even

k=1,k odd

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], ⎛ ⎞ j−1 k  (−1) diam([y( j )] ) = A0 ( j ) ⎝diam([y0 ] ) + diam([Bk (k+1 )] )⎠ A0 (k ) k=0 ⎛ ⎞  k+1 j−1  1 = A0 ( j ) ⎝diam([y0 ] ) + (−1)k diam([b(s)] ) ds ⎠ . A (s) 0 k k=0

Remark 3.8. To check the validity of condition (3), it is crucial to control the length of the diameter of the level sets of both terms in the difference, that is, it is necessary to impose that j 

(−1)k

k=0

1 diam([Bk (k+1 )] ) ⱖ 0 A0 (k )

for every  ∈ [0, 1] and every odd number j with 1 ⱕ j ⱕ m, which is equivalent to j 

(−1)

k

k=0

 k+1 k

1 diam([b(s)] ) ds ⱖ 0 A0 (s)

for every  ∈ [0, 1] and every odd number j with 1 ⱕ j ⱕ m. Besides, to get (3), we have to check, for every odd number j with 1 ⱕ j ⱕ m, the following conditions: j−1  k=0,k even

 k+1 b(s)l ds + A0 (s) k

j  k=1,k odd

 k+1 b(s)r ds is nondecreasing in  A0 (s) k

and j−1  k=0,k even

 k+1 b(s)r ds + A0 (s) k

j  k=1, k odd

 k+1 b(s)l ds is nonincreasing in . A0 (s) k

Corollary 3.9. Consider 0 = 0 < 1 < 2 < 3 < 4 = T , and a : [0, T ] → R, b : [0, T ] → R F continuous functions satisfying that a > 0 on (0, 1 ) ∪ (2 , 3 ) and a < 0 on (1 , 2 ) ∪ (3 , T ). Suppose that:  1  2 1 1  (11) diam([b(s)] ) ds ⱖ diam([b(s)] ) ds, for every  ∈ [0, 1], A (s) A 0 0 (s) 0 1

8

R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 – 26

  k+1 k=1,3 k

  k+1 1 1  diam([b(s)] ) ds ⱕ diam([b(s)] ) ds, for every  ∈ [0, 1], A0 (s) A0 (s)  k k=0,2

(12)

 1  2 b(s)l b(s)r ds + ds is nondecreasing in , A (s) 0 0 1 A0 (s)

(13)

  k+1 b(s)r   k+1 b(s)l ds + ds is nondecreasing in , A0 (s) A0 (s) k=0,2 k k=1,3 k

(14)

 1  2 b(s)r b(s)l ds + ds is nonincr easing in , A0 (s) 0 1 A0 (s)

(15)

  k+1 b(s)r   k+1 b(s)l ds + ds is nonincreasing in  A0 (s) A0 (s)   k k k=0,2 k=1,3

(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 = 1 and j = 3, that is, there must exist B0 (1 )d

1 (−B1 2 )) A0 (1 )

B0 (1 ) +

1 B2 (3 )d A0 (2 )

and

1 1 (−B1 (2 )) + (−B3 (4 )) A0 (1 ) A0 (3 )

in R F

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 satisfying that a > 0 on (0, 1 ) ∪ (2 , T ), a < 0 on (1 , 2 ), b : [0, T ] → R F 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 existence of the difference B0 (1 )d

1 (−B1 (2 )). A0 (1 )



Remark 3.11. If we consider m = 1, a > 0 on (0, ) and a < 0 on (, T ), then Theorem 3.5 produces the results in [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 ⱕ m . a > 0 on (k , k+1 ) for every odd number k with k ⱕ m Again a(k ) = 0, for every k = 1, . . . , m, due to continuity.

(17)

R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 – 26

9

Theorem 3.12. Let a : [0, T ] → R and b : [0, T ] → R F be continuous functions such that (17) holds. Suppose that the following differences m 

A0 (T )

k= j+1,k odd

⎛

j 

d⎝

1 Bk (k+1 ) + A0 (k )

j−1  k=0,k odd

1 (−Bk (k+1 )) + A0 (T ) A0 (k )

k=0,k even

1 Bk (k+1 ) A0 (k ) m  k= j+1,k even

⎞

1 (−Bk (k+1 ))⎠ A0 (k )

(18) exist in R F , for every j even with 0 ⱕ j ⱕ m T and that (4) holds, that is, 0 a(u) du < 0. Then there exists a solution u to problem (1) in the space F{ , ..., } . m 1 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 R F ⎧ j−1 j ⎨   1 Bk (k+1 )d C ∗ = y0 ∈ R F : y0 + ⎩ A0 (k ) k=0,k odd

k=0,k even

1 (−Bk (k+1 )) A0 (k )

exist in R F , for every j even with 0 ⱕ j ⱕ m} . The closed character of C ∗ is clear. To prove that C ∗ is nonempty, we have to find an element y0 ∈ R F such that the Hukuhara differences in C ∗ exist. To this purpose, we check three conditions. The first one concerns the length of the diameter of the level sets, that is, diam([y0 ] ) ⱖ

j 

(−1)k

k=0

 k+1 diam([b(s)] ) ds A0 (s) k

for every  ∈ [0, 1] and every j even with 0 ⱕ j ⱕ m. For its validity, it suffices to select y0 ∈ R F such that diam([y0 ] ) ⱖ R, for every  ∈ [0, 1], where ⎧ ⎧ ⎫ ⎫  k+1 j j  k+1 ⎨ ⎨ ⎬    diam([b(s)] ) diam([b(s)] ) ⎬ (−1)k ds ⱕ max ds max ⎭ ⎭ j even,0 ⱕ j ⱕ m ⎩ j even,0 ⱕ j ⱕ m ⎩ A0 (s) A0 (s) k k=0 k=0 k   j+1  T diam([b(s)]0 ) 1 ⱕ max ds ⱕ K ds =: R j even,0 ⱕ j ⱕ m 0 A0 (s) 0 A0 (s) and K ⱖ diam([b(t)]0 ), for every t ∈ [0, T ]. Additionally, we have to check that j−1 

(y0 )l +

Bk (k+1 )l + A0 (k )

k=0,k odd

j  k=0,k even

Bk (k+1 )r A0 (k )

is nondecreasing in , for every j even with 0 ⱕ j ⱕ m. Indeed, for  ⱕ , we prove that j−1 

(y0 )l − (y0 )l ⱖ

k=0,k odd

 k+1 b(s)l − b(s)l ds + A0 (s) k

j  k=0,k even

 k+1 b(s)r − b(s)r ds, A0 (s) k

which is trivially satisfied if, for  ⱕ ,  k+1 b(s)r − b(s)r ds j even,0 ⱕ j ⱕ m A0 (s) k=0,k even k  k+1  k+1 m  b(s)r − b(s)r b(s)r ds = ds − A0 (s) A0 (s) k k

(y0 )l − (y0 )l ⱖ =

m  k=0,k even

max

j 

k=0,k even

m  k=0,k even

 k+1 b(s)r ds A0 (s) k

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  k+1  b(s)/A0 (s) ds)r decreases. On the other hand, to obtain that ( m k=0,k even  k

(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 ⱕ j ⱕ m, it suffices to select y0 in such a way that  k+1  k+1 m m   b(s)l b(s)l ds − ds (y0 )r − (y0 )r ⱖ A (s) A0 (s) 0 k k k=0,k even

k=0,k even

 k+1  for  ⱕ , that is, with (y0 )r decreasing in  faster (or at the same speed) than ( m b(s)/A0 (s) ds)l k=0,k even k increases. Therefore, C ∗ is nonempty. We consider, for each fixed initial condition y0 ∈ C ∗ , the solution y ∈ C(J, R F ) to the linear fuzzy differential equation y  (t) = a(t)y(t) + b(t), t ∈ J = [0, T ], defined recursively by  A j (t)(y( j )d(−B j (t))) for t ∈ ( j ,  j+1 ] and j even, y(t) = (19) for t ∈ ( j ,  j+1 ] and j odd, A j (t)(y( j ) + B j (t)) where the initial condition is y(0 ) = y(0) = y0 . According to the notation in [3], this function corresponds to taking the 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 ] for j such that a < 0 on ( j ,  j+1 ). Note that the existence in R F of the Hukuhara differences y( j )d(−B j (t)), for every t ∈ ( j ,  j+1 ] and j even, 0 ⱕ j ⱕ 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 of y exist for t ∈ ( j ,  j+1 ] and j even, 0 ⱕ j ⱕ m. In consequence, y is well defined as long as the initial condition y0 belongs 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 ⎛ ⎞ j−1 j−1   1 1 (20) y( j ) = A0 ( j ) ⎝ y0 + Bk (k+1 )d (−Bk (k+1 ))⎠ A0 (k ) A0 (k ) k=0,k odd

k=0,k even

for j = 1, . . . , m + 1. Indeed, for j = 0, y(0 ) = y0 ∈ C ∗ . Besides, y(t) = A0 (t)(y0 d(−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 d(−B0 (1 ))) coincides with (20) for j = 1. Thus, it is clear that y is well-defined on (1 , 2 ] and

1 B1 (2 )d(−B0 (1 )) , y(2 ) = A1 (2 )(A0 (1 )(y0 d(−B0 (1 ))) + B1 (2 )) = A0 (2 ) y0 + A0 (1 ) which coincides with (20) for j = 2. To justify that y(t) = A2 (t)(y(2 )d(−B2 (t))) is well-defined on (2 , 3 ], we check that the expression makes sense at 3 (see the condition in C ∗ for j = 2) and y(3 ) = A2 (3 )(y(2 )d(−B2 (3 )))

1 = A2 (3 ) A0 (2 ) y0 + B1 (2 )d(−B0 (1 )) d(−B2 (3 )) A0 (1 )

1 1 = A0 (3 ) y0 + B1 (2 )d −B0 (1 ) − B2 (3 ) , A0 (1 ) A0 (2 ) which coincides with (20) for j = 3. By induction, suppose that y is well-defined on [0,  j ] with j ⱖ 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 ], it is 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

⎛ = A0 ( j+1 ) ⎝ y0 +

j  k=0,k odd

1 Bk (k+1 )d A0 (k )

j−1  k=0,k even

11

⎞ 1 (−Bk (k+1 ))⎠ , A0 (k )

which implies the validity of (20) for j + 1. Now, j + 1 is even and y(t) = A j+1 (t)(y( j+1 )d(−B j+1 (t))) exists for t ∈ ( j+1 ,  j+2 ] since it exists at t =  j+2 , y( j+2 ) = A j+1 ( j+2 )(y( j+1 )d(−B j+1 ( j+2 ))) = A j+1 ( j+2 )A0 ( j+1 ) ⎛ ⎞ j j−1   1 1 1 × ⎝ y0 + Bk (k+1 )d (−Bk (k+1 ))d (−B j+1 ( j+2 ))⎠ A0 (k ) A0 (k ) A0 ( j+1 ) k=0,k odd k=0,k even ⎛ ⎞ j j+1   1 1 = A0 ( j+2 ) ⎝ y0 + Bk (k+1 )d (−Bk (k+1 ))⎠ , A0 (k ) A0 (k ) k=0,k odd

k=0,k even

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) holds for every j = 0, . . . , m + 1.  0 ) = y(T ), for y0 ∈ C ∗ . Condition (18) guarantees that G  maps  : C ∗ −→ R F , by G(y Next, we define the operator G ∗ ∗ C 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 ) d A0 (k )

j  k=0,k even

(−Bk (k+1 )) A0 (k )

y0 belongs to C ∗ , showing that the differences exists in R F for every j even with 0 ⱕ j ⱕ m, and prove that G ⎛ ⎞ m m   1 1 A0 (T ) ⎝ y0 + Bk (k+1 )d (−Bk (k+1 ))⎠ A0 (k ) A0 (k ) k=0,k odd

+

j−1  k=0,k odd

1 Bk (k+1 )d A0 (k )

k=0,k even

j  k=0,k even

1 (−Bk (k+1 )) A0 (k )

exist in R F , for every j even with 0 ⱕ j ⱕ m. Indeed, these differences exist due to (18) and y0 ∈ C ∗ , which proves  0 ) ∈ C ∗ , and G  is well-defined. that G(y  We prove that, for y0 ,  y0 ∈ C ∗ , Finally, condition (4) provides the contractive character of the mapping G. D(y( j ),  y( j )) = A0 ( j )D(y0 ,  y0 ), ∀ j = 0, 1, . . . , m + 1,

(21)

where functions y and  y come from (19) taking, respectively, the initial values y0 and  y0 . Indeed, for j = 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 )) = D(y(0 ),  y0 ). In particular, A0 ( j )D(y0 ,   0 ), G(  y0 )) = D(y(T ),  D(G(y y(T )) = A0 (m+1 )D(y0 ,  y0 ) = A0 (T )D(y0 ,  y0 ) = e

T 0

a(u) du

D(y0 ,  y0 ),

T

 is a contractive mapping. where A0 (T ) = e 0 a(u) du < 1 and G  which is one solution to problem (1).  Hence, there exists a unique fixed point of G, Remark 3.13. From (20), for each  ∈ [0, 1] and j = 1, . . . , m + 1, we have ⎞ ⎛  k+1 j−1 )  diam([b(s)] diam([y( j )] ) = A0 ( j ) ⎝diam([y0 ] ) + ds ⎠ . (−1)k+1 A0 (s) k k=0

(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 ⱕ j ⱕ m, the following conditions hold: m 

A0 (T )

(−1)

k+1 diam([Bk (k+1 )]

A0 (k )

k= j+1

⎧ ⎨

A0 (T )

m 

⎩

k= j+1,k odd

j−1 

+

k=0,k odd

and

⎧ ⎨ A0 (T )

m 

k= j+1,k odd

+

k=0,k odd

Bk (k+1 )l + A0 (k )

Bk (k+1 )l + A0 (k )

⎩

j−1 

)

Bk (k+1 )r + A0 (k )

diam([Bk (k+1 )] ) , ∀ ∈ [0, 1], A0 (k ) k=0 ⎫ m  Bk (k+1 )r ⎬ A0 (k ) ⎭ ⱖ

(−1)k

k= j+1,k even

j 

Bk (k+1 )r A0 (k )

k=0,k even

Bk (k+1 )r + A0 (k )

j 

m  k= j+1,k even

j  k=0,k even

Bk (k+1 )l A0 (k )

is nondecreasing in  ⎫ 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 ⱕ j ⱕ m, of the following: m 

A0 (T )

(−1)

k+1

k= j+1

⎧ ⎨

A0 (T )

m 

⎩

k= j+1,k odd

j−1 

+

k=0,k odd

and

⎧ ⎨ A0 (T )

k= j+1,k odd

+

j−1  k=0,k odd

k= j+1,k even

 k+1 b(s)l ds + A0 (s) k

m 

⎩

 k+1  k+1 j  diam([b(s)] ) diam([b(s)] ) k ds ⱖ ds, ∀ ∈ [0, 1], (−1) A0 (s) A0 (s) k k k=0 ⎫  k+1  k+1 m  b(s)l b(s)r ⎬ ds + ds A0 (s) A0 (s) ⎭ k k j  k=0,k even

 k+1 b(s)r ds + A0 (s) k

 k+1 b(s)r ds + A0 (s) k

 k+1 b(s)r ds is nondecreasing in  A0 (s) k m 

k= j+1,k even

j  k=0,k even

⎫  k+1 b(s)l ⎬ ds A0 (s) ⎭ k

 k+1 b(s)l ds is nonincreasing in . A0 (s) k

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 the following remark. Remark 3.15. For m = 1, condition (18) is reduced to  T  1 diam([b(s)] ) diam([b(s)] ) A0 (T ) ds ⱖ ds for every  ∈ [0, 1], A0 (s) A0 (s) 1 0  1  T b(s)l b(s)r ds + ds is nondecreasing in  A0 (T ) A0 (s) 1 A0 (s) 0 and



T

b(s)r A0 (T ) ds + 1 A0 (s)

 1 b(s)l ds is nonincreasing in . A0 (s) 0

R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 – 26

13

Theorem 3.16. Let a : [0, T ] → R and b : [0, T ] → R F be continuous such that (17) is satisfied. Suppose that (4) and also the following conditions hold: j−1  k=1,k odd

and m  k=1, k odd

Bk (k+1 ) d A0 (k )

j  k=1, k even

⎛ Bk (k+1 ) ⎝ d A0 (k )

(−Bk (k+1 )) exists, for every j even with 2 ⱕ j ⱕ m A0 (k )

m  k=1,k even

⎞ (−Bk (k+1 )) (−B0 (1 )) ⎠ + exists in R F . A0 (k ) A0 (T )

(23)

(24)

Then there exists a solution u to the periodic boundary value problem (1) in the space F{ , ..., } which is (i)m 1 differentiable on ∪k odd (k , k+1 ) and (ii)-differentiable on ∪k even (k , k+1 ).  defined in the proof of Theorem 3.12 can also be defined in the closed set Proof. The mapping G    1 (−b(s))  C = {y0 ∈ R F : y0 d(−B0 (1 )) exists} = y0 ∈ R F : y0 d ds exists . A0 (s) 0  is nonempty, since there exists y0 ∈ R F such that The set C  1 diam([b(s)] )  diam([y0 ] ) ⱖ ds, for every  ∈ [0, 1], A0 (s) 0  1 b(s)r (y0 )l + ds is nondecreasing in , A0 (s) 0  1 b(s)l (y0 )r + ds is nonincreasing in . A0 (s) 0 To prove this fact, similar to the proof of Theorem 3.12, it suffices to select y0 ∈ R F 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 ( 0 1 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 for  is well-defined), and (24) provides that G  maps C  (so that the operator G  into itself. every y0 ∈ C  We check that, for y0 ∈ C, the Hukuhara differences y( j )d(−B j (t)) exist for every t ∈ ( j ,  j+1 ] ( j even,  then y0 d(−B0 (1 )) exists and, therefore, y0 d(−B0 (t)) exist on (0, 1 ]. Hence y is 0 ⱕ j ⱕ m). Indeed, take y0 ∈ C, well-defined on [0, 2 ] and

1 y(2 ) = A0 (2 ) y0 + B1 (2 )d(−B0 (1 )) . A0 (1 ) For the existence of the differences y(2 )d(−B2 (t)) for t ∈ (2 , 3 ], it suffices to prove the existence of

1 y(2 )d(−B2 (3 )) = A0 (2 ) y0 + B1 (2 )d(−B0 (1 )) d(−B2 (3 )) A0 (1 )

1 1 = A0 (2 ) y0 d(−B0 (1 )) + B1 (2 )d (−B2 (3 )) , A0 (1 ) A0 (2 )  (23) for j = 2, and Lemma 2.4. which is deduced from the choice of y0 ∈ C, By induction, suppose that y is well-defined on [0,  j ] with j ⱖ 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 ⎞ ⎛ j j−1   B ( ) (−B ( )) k k+1 k k+1 ⎠. d y( j+1 ) = A0 ( j+1 ) ⎝ y0 + A0 (k ) A0 (k ) k=0,k odd

k=0,k even

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 )d(−B j+1 ( j+2 )) exists. Indeed, ⎞ ⎛ j j   (−B j+1 ( j+2 )) B ( ) (−B ( )) k k+1 k k+1 ⎠ y( j+1 )d(−B j+1 ( j+2 )) = A0 ( j+1 ) ⎝ y0 + d d A0 (k ) A0 (k ) A0 ( j+1 ) k=0,k odd k=0,k even ⎞ ⎛ j j+1   B ( ) (−B ( )) k k+1 k k+1 ⎠ , = A0 ( j+1 ) ⎝ y0 d(−B0 (1 )) + d A0 (k ) A0 (k ) k=0,k odd

k=1,k even

which exists from the choice of y0 , the properties of the Hukuhara differences and condition (23), so that y is well-defined on [0,  j+2 ].  maps C  and prove that  into itself. Indeed, take y0 ∈ C, Finally, we check that (24) implies that G ⎞ ⎛ m m   Bk (k+1 ) (−Bk (k+1 )) ⎠  0 ) = A0 (T ) ⎝ y0 + G(y d A0 (k ) A0 (k ) k=0,k odd

k=0,k even

 To this purpose, we justify that the following difference exists, using that y0 ∈ C,  the properties of the belongs to C. Hukuhara differences and (24), as follows ⎛ ⎛ ⎞⎞ m m   B ( ) (−B ( )) ( )) (−B k k+1 k k+1 0 1 ⎠⎠  0 )d(−B0 (1 )) = A0 (T ) ⎝ y0 + G(y d⎝ + A0 (k ) A0 (k ) A0 (T ) k=0,k odd k=0,k even ⎞⎞ ⎛ ⎛ m m   Bk (k+1 ) ⎝ (−Bk (k+1 )) (−B0 (1 )) ⎠⎠ . = A0 (T ) ⎝ y0 d(−B0 (1 ))+ d + A0 (k ) A0 (k ) A0 (T ) k=0,k odd

k=1,k even

 but it is the same solution given by Theorem 3.12. In this case, the solution is found in the set C,



 leading to the solution to problem Remark 3.17. Under the hypotheses y0 ∈ C  of Theorem 3.16, the initial condition  (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 ) d 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 of Theorem 3.12, it is deduced that y0 ∈ R F 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 application of a fixed point result and checking, directly, the good definition of the solution starting at the initial condition y0 which is appropriate to the periodic boundary condition. However, in our approach, the emphasis is made on the hypotheses which allow to guarantee the good definition of the solution y to the linear differential equation for initial conditions y0 in a certain subset of R F , and not only for the solution to the boundary value problem. In any case, the expression for 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 switching points where differentiability changes from (ii)-differentiability to (i)-differentiability. Note that we do not impose this restriction at m+1 , not even in the case where the solution is (ii)-differentiable on (m , T ), since if the solution was real (or crisp) at t = T , then the same property would affect the initial condition, due to the type of boundary value problem considered. Thus, according to our construction, if (2) holds, then we refer to points  j where j is an even number with 2 ⱕ j ⱕ m; on the other hand, if (17) is satisfied, then we refer to points  j where j is an odd number with 1 ⱕ j ⱕ 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 

 diam([Bk (k+1 )] ) diam([Bk (k+1 )] ) (−1)k + A0 (T ) A0 (k ) A0 (k ) m

(−1)k

k=0

k= j

is constant in , for each j even with 2 ⱕ j ⱕ m. Then the solution u to problem (1) given by Theorem 3.5 is crisp at the points  j where j is an even number with 2 ⱕ j ⱕ m. Proof. The solution u provided by Theorem 3.5 satisfies that the value at  j is given by (8), where the initial condition is specified in (10). To prove that the solution u is crisp at the points  j where j is an even number with 2 ⱕ j ⱕ m, we check that diam([u( j )] ) is constant in  for each of those values of j. Indeed, for each j even with 2 ⱕ j ⱕ m, ⎛ ⎞ j−1   diam([Bk (k+1 )] ) ⎠ (−1)k diam([y( j )] ) = A0 ( j ) ⎝diam([y0 ] ) + A0 (k ) k=0 ⎞ ⎛ j−1 m ) )   (T ) diam([B ( )] diam([B ( )] A 0 k k+1 k k+1 ⎠ = A0 ( j ) ⎝ + (−1)k (−1)k 1 − A0 (T ) A0 (k ) A0 (k ) k=0 k=0 ⎛ ⎞ j−1 m ) )   A0 ( j ) diam([B ( )] diam([B ( )] k k+1 k k+1 ⎝ (−1)k ⎠ = (−1)k + A0 (T ) 1 − A0 (T ) A0 (k ) A0 (k ) k=0

k= j

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 

 diam([Bk (k+1 )]0 ) diam([Bk (k+1 )]0 ) (−1)k + A0 (T ) =0 A0 (k ) A0 (k ) m

(−1)k

k=0

k= j

for each j even with 2 ⱕ j ⱕ m. Then the solution u to problem (1) given by Theorem 3.5 is real at the points  j where j is an even number with 2 ⱕ j ⱕ m. The additional hypothesis in Theorem 3.19 is equivalent to j−1  k=0

 k+1  k+1 m  diam([b(s)]0 ) diam([b(s)]0 ) k (−1) (−1) ds + A0 (T ) ds = 0 A0 (s) A0 (s) k k k

k= j

for each j even with 2 ⱕ j ⱕ 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

 diam([Bk (k+1 )] ) diam([Bk (k+1 )] ) (−1)k+1 + A0 (T ) A0 (k ) A0 (k ) m

(−1)k+1

k= j

is constant in , for each j odd with 1 ⱕ j ⱕ m. Then the solution u to problem (1) given by Theorem 3.12 (or Theorem 3.16) is crisp at the points  j where j is an odd number with 1 ⱕ j ⱕ m. Proof. In this case, the solution u provided by Theorem 3.12 (or Theorem 3.16) satisfies that its value at  j is given by (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 ⱕ j ⱕ m, we check that diam([u( j )] ) is constant in  for each of those values of j. Indeed, if we fix j odd with 1 ⱕ j ⱕ m, then ⎛ ⎞ j−1 )  diam([B ( )] k k+1 ⎠ diam([y( j )] ) = A0 ( j ) ⎝diam([y0 ] ) + (−1)k+1 A0 (k ) k=0 ⎛ ⎞ j−1 m     diam([Bk (k+1 )] ) diam([Bk (k+1 )] ) ⎠ A0 (T ) = A0 ( j ) ⎝ + (−1)k+1 (−1)k+1 1 − A0 (T ) A0 (k ) A0 (k ) k=0 k=0 ⎛ ⎞ j−1 m ) )   A0 ( j ) diam([B ( )] diam([B ( )] k k+1 k k+1 ⎝ (−1)k+1 ⎠ (−1)k+1 + A0 (T ) = 1 − A0 (T ) A0 (k ) A0 (k ) k=0

k= j

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

 diam([Bk (k+1 )]0 ) diam([Bk (k+1 )]0 ) + A0 (T ) =0 (−1)k+1 A0 (k ) A0 (k ) m

(−1)k+1

(26)

k= j

for each j odd with 1 ⱕ j ⱕ m. Then the solution u to problem (1) given by Theorem 3.12 (or Theorem 3.16) is real at the points  j where j is an odd number with 1 ⱕ j ⱕ m. Remark 3.22. For m = 1 and 1 = , condition (26) is written ( j = 1) as    T diam([b(s)]0 ) diam([b(s)]0 ) − ds + A0 (T ) ds = 0. A0 (s) A0 (s) 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 in condition (23) are crisp (resp. real) for every j even with 2 ⱕ 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 number with 1 ⱕ j ⱕ m. Proof. It is straightforward, since the initial condition y0 given by (25) belongs to R F and satisfies that the corresponding solution (19) is crisp (resp. real) at the points  j where j is an odd number with 1 ⱕ j ⱕ m. Indeed, we check that y0 d(−B0 (1 )) exists and it is crisp (resp. real) and that y0 ∈ R F , which follow from the substitution of the corresponding expressions ⎧ ⎛ ⎞⎫ m m ⎨   Bk (k+1 ) ⎝ (−Bk (k+1 )) (−B0 (1 )) ⎠⎬ A0 (T ) d + ⎭ 1 − A0 (T ) ⎩ A0 (k ) A0 (k ) A0 (T ) k=1, k odd

k=1,k even

and using the properties: • If Ad(C + D) exists and it is crisp, then AdC exists (non necessarily crisp) and (AdC)dD exists and it is crisp (it coincides with Ad(C + D)). • If Ad(C + D) exists and it is real, then AdC exists (non necessarily real) and ( AdC)dD exists and it is real (it coincides with Ad(C + D)). By property (23) and the restriction imposed for j even with 2 ⱕ j < m, the solution y given by (19) is well-defined on [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 in the 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 ∈ R F , and suppose that AdC and ( A + B)d(C + D) exist and are crisp. If BdD exists in R F , then it is crisp. However, BdD does not necessarily exist. On the other hand, if AdC and (A + B)d(C + D) are real, then BdD is real. Obviously, if AdC and BdD exist and are real, then (A + B)d(C + D) is real. Thus, in the real case, the hypotheses of Theorem 3.23 can be given in simpler differences between fuzzy numbers as follows. Theorem 3.25. Let a : [0, T ] → R and b : [0, T ] → R F be continuous such that (17) is valid. Suppose that (4) and also the following conditions hold: B j−1 ( j ) (−B j ( j+1 )) d exists and it is real for every j even with 2 ⱕ j < m, A0 ( j−1 ) A0 ( j ) if m even, and m  k=m−1,k odd

Bm−1 (m ) (−Bm (m+1 )) d exists in R F (nonnecessarily r eal), A0 (m−1 ) A0 (m ) ⎛ Bk (k+1 ) ⎝ d A0 (k )

m 

k=m,k even

⎞ (−Bk (k+1 )) (−B0 (1 )) ⎠ + exists and it is real. A0 (k ) A0 (T )

(27)

(28)

(29)

Then there exists a solution u to the periodic boundary value problem (1) in the space F{ , ..., } which is (i)m 1   differentiable on k odd (k , k+1 ), (ii)-differentiable on k even (k , k+1 ) and real at the points  j where j is an odd number with 1 ⱕ j ⱕ m. Proof. It follows from the previous arguments and taking into account that y0 d(−B0 (1 )) can be thought as ⎧ ⎫

m−1 Bk−1 (k ) (−Bk (k+1 )) Bm (m+1 ) (−B0 (1 )) ⎬ A0 (T ) ⎨  d + d 1 − A0 (T ) ⎩ A0 (k−1 ) A0 (k ) A0 (m ) A0 (T ) ⎭ k=2,k even

for m odd, and ⎧ A0 (T ) ⎨ 1 − A0 (T ) ⎩

m−2 



k=2,k even

Bk−1 (k ) (−Bk (k+1 )) d A0 (k−1 ) A0 (k )

⎫

Bm−1 (m ) (−Bm (m+1 )) (−B0 (1 )) ⎬ + d + ⎭ A0 (m−1 ) A0 (m ) 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 conditions on the sign of a (a < 0 on (0, ), a > 0 on (, T )), hypothesis (4) and B1 (2 ) (−B0 (1 )) d exists in R F and it is crisp, A0 (1 ) A0 (T )

(30)

i.e., 

T



b(s) 1 dsd A0 (s) A0 (T )

  (−b(s)) ds exists in R F and it is crisp. A0 (s) 0

(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 consider the 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, for n ∈ Z. Example 4.1. We consider the following periodic boundary value problem: ⎧   7 ⎪ ⎪ , ⎨ y  (t) = cos(t)y(t) + e−t+sin(t) , t ∈ J = 0, 2 7 ⎪ ⎪ , ⎩ y(0) = y 2

(32)

where  is the triangular fuzzy interval defined as [] = [ − 1, 1 − ], for  ∈ [0, 1]. Since a(t) = cos(t), we select the sequence {k } by choosing the endpoints of the interval J , as well as the zeros of a in J , that is, 0 = 0 < 1 =

 3 5 7 < 2 = < 3 = < 4 = 2 2 2 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 7 cos(u) du = sin = −1 < 0 2 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  k+1 1  diam([b(s)] ) (−1)k ds A0 (s) k k=0  3/2  /2 (e−s+sin(s) )(2 − 2)e− sin(s) ds − (e−s+sin(s) )(2 − 2) e− sin(s) ds = /2

0

= (2 − 2)(1 − 2e−/2 + e−3/2 ) ⱖ 0 and 3 

(−1)k

k=0

 k+1 diam([b(s)] ) ds A0 (s) k

= (2 − 2)(1 + e−3/2 − 2e−/2 ) +

 5/2 3/2

e−s (2 − 2) ds −

= (2 − 2)(1 − 2e−/2 + 2e−3/2 − 2e−5/2 + e−7/2 ) ⱖ 0. Besides,  1 0

and

 7/2 5/2

e−s (2 − 2) ds

 2  /2  3/2 b(s)l b(s)r ds + ds = (e−s+sin(s) )( − 1) e− sin(s) ds + (e−s+sin(s) )(1 − ) e− sin(s) ds A0 (s) 1 A0 (s) /2 0 = ( − 1)(e−3/2 + 1 − 2e−/2 )

  k+1 b(s)r   k+1 b(s)l ds + ds A0 (s) A0 (s) k=0,2 k k=1,3 k  /2  3/2 −s+sin(s) − sin(s) = (e )( − 1)e ds + (e−s+sin(s) )(1 − ) e− sin(s) ds 0

/2

R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 – 26

+

 5/2 3/2

(e−s+sin(s) )( − 1)e− sin(s) ds +

 7/2 5/2

19

(e−s+sin(s) )(1 − ) e− sin(s) ds

= ( − 1)(e−7/2 + 1 − 2e−/2 + 2e−3/2 − 2e−5/2 ) are nondecreasing in , and, on the other hand,  1  2 b(s)r b(s)l ds + ds A0 (s) 0 1 A0 (s)  /2  3/2 = (e−s+sin(s) )(1 − )e− sin(s) ds + (e−s+sin(s) )( − 1)e− sin(s) ds 0

= (1 − )(e−3/2 + 1 − 2e−/2 ) and

/2

  k+1 b(s)r   k+1 b(s)l ds + ds A0 (s) A0 (s) k=0,2 k k=1,3 k  3/2  /2 −s+sin(s) − sin(s) (e )(1 − )e ds + (e−s+sin(s) )( − 1) e− sin(s) ds = /2

0

+

 5/2 3/2

(e−s+sin(s) )(1 − )e− sin(s) ds +

 7/2 5/2

(e−s+sin(s) )( − 1) e− sin(s) ds

= (1 − )(e−7/2 + 1 − 2e−/2 + 2e−3/2 − 2e−5/2 ) are nonincreasing in . Therefore, problem (32) has a solution which is (i)-differentiable on (0, /2) ∪ (3/2, 5/2) and (ii)-differentiable on (/2, 3/2) ∪ (5/2, 7/2). Next, we calculate the expression of the mapping G. To this purpose, we fix y0 ∈ R F and consider expression (7), then we get

 /2   /2 s cos(u) du −s+sin(s) − 0 cos(u) du 0 y y0 + =e e e ds 2 0

 /2 = e y0 +  e−s ds = e(y0 + (1 − e−/2 )), 0    3/2

s  3/2  3 − /2 cos(u) du cos(u) du /2 =e d y (−b(s)) e ds y 2 2 /2    3/2  

 3/2   = e−2 y (−e−s+sin(s) )e1−sin(s) ds = e−2 y (−e1−s ) ds d d 2 2 /2 /2 = e−2 (e(y0 + (1 − e−/2 ))d(e1−(3/2) − e1−(/2) )) = e−1 (y0 + (1 − e−/2 )d(e−3/2 − e−/2 )),    s  5/2 5/2 3 5 − cos(u) du cos(u) du y y = e 3/2 + e−s+sin(s) e 3/2 ds 2 2 3/2  

 5/2 3 3 2 −s−1 2 −3/2−1 −5/2−1 e ds = e y −e ) + + (e =e y 2 2 3/2 = e2 (e−1 (y0 + (1 − e−/2 )d(e−3/2 − e−/2 )) + (e−3/2−1 − e−5/2−1 )) = e(y0 + (1 − e−/2 + e−3/2 − e−5/2 )d(e−3/2 − e−/2 )) and

y(T ) = y

7 2

=e

 7/2 5/2

cos(u) du

 7/2 s 5 (−e−s+sin(s) )e− 5/2 cos(u) du ds y d 2 5/2

20

R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 – 26

=e

−2



 7/2 5 5 −s+1 −2 −7/2+1 −5/2+1 y y (−e ) ds = e −e ) d d(e 2 2 5/2

= e−2 (e(y0 + (1 − e−/2 + e−3/2 − e−5/2 )d(e−3/2 − e−/2 ))d(e−7/2+1 − e−5/2+1 )) = e−1 (y0 + (1 − e−/2 + e−3/2 − e−5/2 )d(e−3/2 − e−/2 )d(e−7/2 − e−5/2 )) = e−1 (y0 + (1 − e−/2 + e−3/2 − e−5/2 )d(e−3/2 − e−/2 + e−7/2 − e−5/2 )) and, therefore, the operator G is given by G y0 = e−1 (y0 + (1 − e−/2 + e−3/2 − e−5/2 )d(e−3/2 − e−/2 + e−7/2 − e−5/2 )) for y0 ∈ R F . The level sets of G y0 are given by [G y0 ] = e−1 ([y0 ] + (1 − e−/2 + e−3/2 − e−5/2 )[ − 1, 1 − ] d (e−3/2 − e−/2 + e−7/2 − e−5/2 )[ − 1, 1 − ]) so that, attending to the sign of the coefficients and using the notation u  = u l and u  = u r , we get G y0  = e−1 (y0  + (1 − e−/2 + e−3/2 − e−5/2 )( − 1) − (e−3/2 − e−/2 + e−7/2 − e−5/2 )(1 − )) = e−1 (y0  + (1 − 2e−/2 + 2e−3/2 − 2e−5/2 + e−7/2 )( − 1)) and, similarly, 

G y0 = e−1 (y0  + (1 − e−/2 + e−3/2 − e−5/2 )(1 − ) − (e−3/2 − e−/2 + e−7/2 − e−5/2 )( − 1)) = e−1 (y0  + (1 − 2e−/2 + 2e−3/2 − 2e−5/2 + e−7/2 )(1 − )) for every  ∈ [0, 1]. From the application of Banach fixed point theorem in the proof of Theorem 3.5, if we choose y0 ∈ R F , then the fixed point of G is attained as limn→∞ G n y0 = y0∗ and this is an appropriate initial condition to obtain a solution to (32). If we denote P = (1 − 2e−/2 + 2e−3/2 − 2e−5/2 + e−7/2 ), we get 

G n y0  = e−1 (G n−1 y0 + P( − 1)) so that G n y0  = e−n y0  + [e−n + · · · + e−1 ]P( − 1) = e−n y0  +

e−1 − e−(n+1) P( − 1) 1 − e−1

and, analogously, 

G n y0 = e−n y0  +

e−1 − e−(n+1) P(1 − ). 1 − e−1

In consequence, the initial condition for the solution to the periodic boundary value problem is limn→∞ G n y0 = y0∗ , where, for every  ∈ [0, 1], y0∗  =

1 − 2e−/2 + 2e−3/2 − 2e−5/2 + e−7/2 e−1 P ( − 1) = ( − 1) P( − 1) = 1 − e−1 e−1 e−1

and 

y0∗ =

1 − 2e−/2 + 2e−3/2 − 2e−5/2 + e−7/2 P (1 − ) = (1 − ). e−1 e−1

The same expressions can be obtained by solving the equations 



G y0∗  = e−1 (y0∗  + P( − 1)) = y0∗  and G y0∗ = e−1 (y0∗ + P(1 − )) = y0∗ for  ∈ [0, 1].



R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 – 26 4

21

left endpoint of supp(y) core of y right endpoint of supp(y)

3 2 1 0 -1 -2 -3 -4 0

2

4

6

8

10

t

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 ⎧ t ! ⎪ t ∈ 0, , ⎪ esin(t) (y0∗ +  0 e−s ds), ⎪ ⎪ 2  ⎪    ⎪ t ⎪  3 ⎪ ⎪ d /2 (−e−s+1 ) ds , , , t∈ ⎪ esin(t)−1 y ⎨ 2 2 2 



y(t) = t 3 5 3 ⎪ ⎪ esin(t)+1 y t∈ +  3/2 e−s−1 ds , , , ⎪ ⎪ ⎪ ⎪ 2

2 2 ⎪ ⎪ t 5 5 7 ⎪ ⎪ d 5/2 (−e−s+1 ) ds , t ∈ , ⎩ esin(t)−1 y 2 2 2 that is,

⎧ ⎪ esin(t) (y0∗ + (1 − e−t )), ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ sin(t)−1     ⎪ −t+1 − e−/2+1 ) , ⎪ e d(e y ⎪ ⎨ 2

y(t) = ⎪ sin(t)+1 y 3 + (e−3/2−1 − e−t−1 ) , ⎪ e ⎪ ⎪ ⎪ ⎪ 2

⎪ ⎪ 5 ⎪ sin(t)−1 −t+1 −5  /2+1 ⎪ y d(e −e ) , ⎩e 2

! , t ∈ 0, 2   3 , , t∈ 2 2  3 5 t∈ , , 2 2 5 7 , . t∈ 2 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 species whose behavior is described by a differential equation similar to that in problem (32), where the number of individuals at each instant is difficult to compute and, thus, imprecise:     t y(t) + e−0.3t+(3/(10)) sin((/3)t) cos t , t ⱖ 0 (33) y  (t) = 0.1 cos 3 3 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, in the seasons with optimal environmental conditions, the population increases rapidly but, in absence of conditions of habitability, it declines severely. We have considered a population which finds optimal conditions of the environment in 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 such as possible diseases affecting the population growth during the hottest months, where its spread can be favored by the abundance of insects, and others which may have a stronger influence during the cold season. Hence, we have adjusted the coefficient a for taking negative values during two seasons a year (understanding that the extreme values are taken in the middle of the season). We set the initial instant in the middle of spring, corresponding to the maximum positive value of function a and, then, assign alternately negative and positive values to this function a, changing its sign after a period of three months (except in the first season, which is not complete). For different populations, the coefficient a should 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 a function 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 of the same species established outside the region of interest. However, it is assumed that these movements are produced independently of the population size but obviously affecting its rate of change. The term   t  b(t) = e−0.3t+(3/(10)) sin((/3)t) cos 3 reflects the migration movements, the immigration movement is produced mainly during spring and autumn and emigration 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 the environment. Once clarified the meaning of the terms of the equation, we ask, in relation with the boundary condition, whether it is possible to obtain, at the end of the next hot dry season, the same number of members as at t = 0, that is, we have to solve the periodic boundary value problem consisting of Eq. (33) and the boundary condition y(0) = y( 33 2 ), and ]. hence, 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 9 15 21 27 33 < 2 = < 3 = < 4 = < 5 = < 6 = 2 2 2 2 2 2

0 = 0,  j =

3(2 j − 1) , j = 1, . . . , 6, and m = 5. 2

or

 It is clear that conditions in Theorem 3.5 hold: a, b are continuous, a > 0 on k even (k , k+1 ) = (0, 23 ) ∪ ( 29 , 15 2 ) ∪ · · ·,  33/2  3 9 15 21 a(u) du = −3/(10) < 0. Besides, condition (3) holds a < 0 on k odd (k , k+1 ) = ( 2 , 2 ) ∪ ( 2 , 2 ) ∪ · · ·, and 0  9/2 for j = 1, since 0 < 0 e−0.3s cos((/3)s) ds 0.024 implies that    9/2   1 B0 (1 )d e−0.3s cos (−B1 (2 )) = s ds [] A0 (1 ) 3 0 and, similarly, for j = 3 and j = 5: ⎛ ⎞  21/2     1 1 ⎝ ⎠ Bk (k+1 )d (−Bk (k+1 )) = s ds[] , e−0.3s cos A0 (k ) A0 (k ) 3 0 k=0,2

k=1,3

⎛

 k=0,2,4

where



0< 0

⎞  33/2    1 1 ⎝ ⎠ Bk (k+1 )d (−Bk (k+1 )) = s ds[] , e−0.3s cos A0 (k ) A0 (k ) 3 0 k=1,3,5

21/2

e−0.3s cos

 33/2     e−0.3s cos s ds 0.215, 0 < s ds 0.247. 3 3 0

R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 – 26

23

1.5

1

0.5

0

-0.5 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:

1 B2 (3 ) B4 (5 ) (−B1 (2 )) (−B3 (4 )) (−B5 (6 )) y0 = 3/(10) B0 (1 ) + + d + + e −1 A0 (2 ) A0 (4 ) A0 (1 ) A0 (3 ) A0 (5 )  33/2    1 e−0.3s cos = 3/(10) s ds [] = G[] , e −1 0 3 where G=

2.7 − 3e−0.3∗33/2 2.46. (e3/(10) − 1)(2 + 0.81)

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, or also (see Fig. 2) ⎧ ⎪ ⎪ ⎨

x − 19 if 1900G ⱕ x ⱕ 2000G, 100G x y0 (x) := − + 21 if 2000G < x ⱕ 2100G, ⎪ ⎪ ⎩ 100G 0 otherwise. Note that the core of the independent term b(t) = e−0.3t+(3/(10)) sin((/3)t) cos

  t  3

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 "     # t , (2100 − 100) cos t , b(t)l = e−0.3t+(3/(10)) sin((/3)t) min (1900 + 100) cos 3 3 "     # t , (2100 − 100) cos t b(t)r = e−0.3t+(3/(10)) sin((/3)t) max (1900 + 100) cos 3 3 for  ∈ [0, 1].

24

R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 – 26

2000

core of b(t)

2000

1500

left endpoint of supp(b(t)) core of b(t) right endpoint of supp(b(t))

1500

1000 b(t)

1000

500

500

0

0

-500

-500

-1000

-1000 0

2

4

6

8

10 t

12

14

16

0

2

4

6

8

10

12

14

16

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 ⱖ 0. (b) Endpoints of the support and core of b(t).

On the other hand, the solution to the boundary value problem on the interval [0, 33 2 ] is defined, according to (7), by   ⎧    t −0.3s 3 ⎪ ⎪ (1900 + 100)G + (1900 + 100) 0 e cos s ds if t ∈ 0, , ⎪ ⎪ 3 2 ⎪ ⎪ ⎪

  ⎪   ⎪  t −0.3s 3 9 ⎪ −3/(10) y 3 ⎪ + (1900 + 100) e cos e s ds if t ∈ , , ⎪ ⎪ 3/2 ⎪ 2 l 3 2 2 ⎪ ⎪ ⎪   ⎪   ⎪  t −0.3s 9 15 ⎪ 3/(10) y 9 ⎪ e + (1900 + 100) e cos s ds if t ∈ , , ⎪ 9/2 ⎨ 2 l 3 2 2 3 sin(t/3)/(10) y(t)l = e     t ⎪ −3/(10) 15 ⎪ −0.3s cos  s ds if t ∈ 15 , 21 , ⎪ e y + (1900 + 100) e ⎪ 15/2 ⎪ 2 l 3 2 2 ⎪ ⎪ ⎪

  ⎪   ⎪ t ⎪  21 27 ⎪ e3/(10) y 21 −0.3s ⎪ + (1900 + 100) 21/2 e cos s ds, if t ∈ , , ⎪ ⎪ 2 l 3 2 2 ⎪ ⎪ ⎪   ⎪   ⎪ t 27 27 33 ⎪ ⎪ ⎩ e−3/(10) y + (1900 + 100) 27/2 e−0.3s cos s ds if t ∈ , 2 l 3 2 2 and

⎧    t −0.3s ⎪ ⎪ (2100 − 100)G + (2100 − 100) e cos s ds ⎪ 0 ⎪ 3 ⎪ ⎪ ⎪ ⎪   ⎪ t 3 ⎪ −3/(10  ) ⎪ e s ds y + (2100 − 100) 3/2 e−0.3s cos ⎪ ⎪ ⎪ 2 r 3 ⎪ ⎪ ⎪ ⎪   ⎪  t −0.3s 9 ⎪ 3/(10  ) ⎪ e y + (2100 − 100) e cos s ds ⎪ 9/2 ⎨ 2 r 3 3 sin(t/3)/(10) y(t)r = e   t ⎪ ⎪ −3/(10) y 15 −0.3s cos  s ds ⎪ e + (2100 − 100) e ⎪ 15/2 ⎪ 2 r 3 ⎪ ⎪ ⎪

⎪   ⎪  ⎪ 3/(10) 21 t ⎪ −0.3s cos  s ds ⎪ e y + (2100 − 100) e ⎪ 21/2 ⎪ 2 r 3 ⎪ ⎪ ⎪ ⎪  ⎪ t 27   ⎪ ⎪ ⎩ e−3/(10) y s ds + (2100 − 100) 27/2 e−0.3s cos 2 r 3

  3 if t ∈ 0, , 2   3 9 if t ∈ , , 2 2   9 15 if t ∈ , , 2 2   15 21 if t ∈ , , 2 2   21 27 if t ∈ , , 2 2   27 33 if t ∈ , , 2 2

R. Rodríguez-López / Fuzzy Sets and Systems 219 (2013) 1 – 26

25

8000

8000

left endpoint of supp(y) core of y right endpoint of supp(y)

7500

7500

7000

7000

6500

6500

6000

6000

y

y

core of solution y

5500

5500

5000

5000

4500

4500 4000

4000 0

2

4

6

8

10

12

14

16

0

2

4

6

8

10

12

14

16

t

t

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 boundary value problem, we give approximations of the endpoints of the support at those instants: y( 23 )0l 6849,

y( 29 )0l 4291,

y( 15 2 )0l 5867,

y( 23 )0r 7570,

y( 29 )0r 4743,

y( 15 2 )0r 6484,

y( 21 2 )0l 4621,

y( 27 2 )0l 5704

and y( 21 2 )0r 5107,

y( 27 2 )0r 6305.

The core of the solution to the boundary value problem on the interval [0, 33 2 ] 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, since the solutions to the periodic boundary value problem (1) are not necessarily related to the sign of a. There exist other possibilities to obtain more general solutions to problem (1) by using the concept of switching point [21] without paying attention to the sign of function a, since there exist two (local) solutions to the problem ((i) and (ii)-differentiable functions, respectively) and the switching point is not necessarily located at a zero of the function a. However, we provide one solution to the problem, which is not unique, but it satisfies the periodic boundary condition, and we give sufficient conditions which guarantee that this solution is well-defined on [0, T ] and has the properties of differentiability of interest. The particular solutions presented are natural since the switching points are exactly located at the points where function a changes it sign so that, although the solution is not unique, it is useful to show, for this type of equations, that periodic boundary value problems and fuzzy differential equations are compatible from the point of view of generalized differentiability. Moreover, the results provided are useful to deduce the existence of periodic solutions to fuzzy linear differential equations, just by solving the equation subject to the periodic boundary condition y(0) = y(T ) in cases where 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, we can approximate the solutions to more general (nonlinear) problems by using the solutions to linear problems with very simple coefficients (for instance, through the monotone iterative technique, applied in [18] to impulsive differential equations). Therefore, the study of the existence of solutions to this type of simple problems can be considered an appropriate 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 suggestions were important to improve the paper. This research is partially supported by Ministerio de Educación y Ciencia and FEDER, Project MTM2010-15314. References [1] B. Bede, S.G. Gal, Almost periodic fuzzy-number-valued functions, Fuzzy Sets Syst. 147 (2004) 385–403. [2] B. Bede, S.G. Gal, Generalizations of the differentibility of fuzzy number value functions with applications to fuzzy differential equations, Fuzzy Sets Syst. 151 (2005) 581–599. [3] B. Bede, I.J. Rudas, A.L. Bencsik, First order linear fuzzy differential equations under generalized differentiability, Inf. Sci. 177 (2007) 1648–1662. [4] P. Diamond, Time-dependent differential inclusions, cocycle attractors and fuzzy differential equations, IEEE Trans. Fuzzy Syst. 7 (1999) 734–740. [5] P. Diamond, Brief note on the variation of constant formula for fuzzy differential equations, Fuzzy Sets Syst. 129 (2002) 65–71. [6] P. Diamond, P. Kloeden, Metric Spaces of Fuzzy Sets, World Scientific, Singapore, 1994. [7] D. Dubois, H. Prade, Towards fuzzy differential calculus: part 3, differentiation, Fuzzy Sets Syst. 8 (1982) 225–233. [8] E. Hüllermeier, An approach to modelling and simulation of uncertain dynamical systems, Int. J. Uncertainty Fuzziness Knowl.-Based Syst. 5 (1997) 117–138. [9] O. Kaleva, Fuzzy differential equations, Fuzzy Sets Syst. 24 (1987) 301–317. [10] O. Kaleva, A note on fuzzy differential equations, Nonlinear Anal. 64 (2006) 895–900. [11] A. Khastan, F. Bahrami, K. Ivaz, New Results on Multiple Solutions for N th-order Fuzzy Differential Equations under Generalized Differentiability, Boundary Value Problems, Article ID 395714, 2009, 13 pp. [12] A. Khastan, J.J. Nieto, R. Rodríguez-López, Periodic boundary value problems for first-order differential equations with uncertainty under generalized differentiability, Inf. Sci. 222 (2013) 544–558. [13] P.E. Kloeden, T. Lorenz, Fuzzy differential equations without fuzzy convexity, Fuzzy Sets Syst., in press, http://dx.doi.org/10.1016/j.fss. 2012.01.012. [14] V. Lupulescu, On a class of fuzzy functional differential equation, Fuzzy Sets Syst. 160 (2009) 1547–1562. [15] J.J. Nieto, A. Khastan, K. Ivaz, Numerical solution of fuzzy differential equations under generalized differentiability, Nonlinear Anal.: Hybrid Syst. 3 (2009) 700–707. [16] J.J. Nieto, R. Rodríguez-López, D. Franco, Linear first-order fuzzy differential equations, Int. J. Uncertainty Fuzziness Knowl.-Based Syst. 14 (2006) 687–709. [17] M.L. Puri, D.A. Ralescu, Differentials of fuzzy functions, J. Math. Anal. Appl. 91 (1983) 552–558. [18] R. Rodríguez-López, Periodic boundary value problems for impulsive fuzzy differential equations, Fuzzy Sets Syst. 159 (2008) 1384–1409. [19] R. Rodríguez-López, On boundary value problems for fuzzy differential equations, in: D. Dubois, M.A. Lubiano, H. Prade, M.A. Gil, P. Grzegorzewski, O. Hryniewicz (Eds.), Soft Methods for Handling Variability and Imprecision. Advances in Soft Computing, vol. 48, Springer, Berlin, Heidelberg, New York, 2008. [20] L. Stefanini, A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Sets Syst. 161 (2010) 1564–1584. [21] L. Stefanini, B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Anal. Theory Methods Appl. 71 (2009) 1311–1328.