Existence of solutions to fuzzy differential equations with generalized Hukuhara derivative via contractive-like mapping principles

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Existence of solutions to fuzzy differential equations with generalized Hukuhara derivative via contractive-like mapping principles ✩ Elder J. Villamizar-Roa a,1 , Vladimir Angulo-Castillo a , Yurilev Chalco-Cano b,∗ a Universidad Industrial de Santander, Escuela de Matemáticas, A.A.678, Bucaramanga, Colombia b Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile

Received 28 October 2013; received in revised form 29 April 2014; accepted 20 July 2014

Abstract We study the existence and uniqueness of solution for fuzzy initial value problems in the setting of a generalized Hukuhara derivative and by using some recent results of fixed point of weakly contractive mappings on partially ordered sets. © 2014 Elsevier B.V. All rights reserved. Keywords: Contractive mappings; Fuzzy differentiability; Fuzzy differential equations

1. Introduction Fuzzy differential equations (FDE) are a suitable tool to model dynamic systems in which uncertainties or vagueness pervade. This theory has been developed in several theoretical directions, and a wide number of applications in many different real problems have been considered (see for instance [3–5,9,18,22,28,34,37–39]). Several settings for studying FDE have been considered. The first and the most popular approach is using the Hukuhara differentiability (H-differentiability) for fuzzy value functions (see for instance [21,28,32,34]). However, this approach has the drawback that the solution of a fuzzy differential equation needs to have increasing length of its support, so, is in this case, the qualitative theory very poor compared to ODEs [3,5,8,11,14,15]. In order to overcome this weakness some alternatives have been proposed; in fact, in [4] was introduced the concept of strongly generalized differentiability (GH-differentiability) which allows us to obtain new solutions of fuzzy differential equations as it was shown in [5,11,23,24]. This concept of differentiability is based on four forms (types) of lateral derivatives. The differentiability in the first form (i) coincides with the H-differentiability and then the GH-differentiability is more general ✩

This work has been partially supported by Conicyt-Chile through Projects Fondecyt 1120665 and VIE-UIS, proyecto:C-2013-01.

* Corresponding author. Tel.: +56 58 2230334.

E-mail addresses: [email protected] (E.J. Villamizar-Roa), [email protected] (Y. Chalco-Cano). 1 The first author has been supported by VIE-UIS, proyecto:C-2013-01.

http://dx.doi.org/10.1016/j.fss.2014.07.015 0165-0114/© 2014 Elsevier B.V. All rights reserved.

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than H-differentiability. Contrary to the previous case, if we consider a fuzzy initial value problem (FIVP) with the differentiability in the second form (ii), the solution needs to have decreasing length of its support. On the other hand, since the GH-differentiability in the third form (iii) and fourth form (iv) are linked with the so called switching points, combining the solutions obtained using GH-differentiability in the first form (i) and second form (ii), we can obtain more than two solutions for an FIVP [5,24]. So, the study of the existence and uniqueness of solutions of an FIVP considering the concept of GH-differentiability in the first form (i) and second form (ii) is very important. In this direction, several results of existence and uniqueness of solutions have been obtained, see for instance [4,5,21,25,28, 34,37–39]. In the present article, in order to obtain the existence and uniqueness of solution for an FIVP, in place of using the classical Banach fixed point theorem, we use some fixed point theorems, established in [19], on weakly contractive functions defined on partially ordered sets. It should be noted that the space of fuzzy numbers is not a Banach space but it is a quasilinear space and partially ordered. This paper is organized as follows: In Section 2 we give some preliminaries about the GH-derivative, which will be necessary to study the existence and uniqueness of solution for an FIVP. In Section 3 we present some fixed point results of weakly contractive mappings on partially ordered sets. In Section 4 we begin with a discussion on the existence of many solutions for an FIVP and then we prove the existence and uniqueness of an (i)-solution, as well as, an (ii)-solution, for an FIVP. 2. Preliminaries about generalized Hukuhara derivative Let us denote by Kcn the space of all nonempty, compact and convex subsets of the n-dimensional Euclidean space Rn . If A, B ∈ Kcn and  ·  denotes the Euclidean norm in Rn , the Hausdorff metric d on Kcn is defined by   d(A, B) = max sup inf a − b, sup inf a − b . a∈A b∈B

b∈B a∈A

For A, B ∈ Kcn and λ ∈ R, the following operations are known as Minkowski operations: A + B = {a + b | a ∈ A, b ∈ B} and

λA = {λa | a ∈ A}.

(1)

The couple (Kcn , d) is a complete metric space (cf. [33]); moreover the metric d verifies the following properties for all A, B, C, D ∈ Kcn , λ ∈ R: (i) d(λA, λB) = |λ|d(A, B), (ii) d(A + B, C + D) ≤ d(A, C) + d(B, D), (iii) d(A + C, B + C) = d(A, B). It is known that in Kcn , in general, A + (−A) = {0}, where −A = (−1)A = {−a | a ∈ A}, and thus Kcn is not a linear space. In order to overcome this difficulty some alternatives have been proposed. In fact, initially in [20] the Hukuhara difference (or H -difference for short) was introduced. The H -difference between two sets A and B of Kcn , denoted by A H B, is defined as A H B = C

⇐⇒

A = B + C.

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Due the restrictive nature of the H -difference, in [39,40] the authors proposed the generalized Hukuhara difference of A and B (gH-difference for short), denoted by A gH B, which is defined as follows: for A, B ∈ Kcn , the difference A gH B is the element C ∈ Kcn such that  (i) A = B + C, or A gH B = C ⇐⇒ (3) (ii) B = A + (−1)C. Clearly, the gH-difference A gH B is a generalization of the H -difference. Moreover, it satisfies, among others, the following basic properties for all A, B ∈ Kcn : (i) if C = A gH B exists, it is unique, (ii) A gH A = {0}, (iii) (A + B) gH B = A, (iv) A gH (A − B) = B, (v) A gH (A + B) = −B. Moreover, unlike to the H -difference, in Kc1 the difference A gH B always exists (cf. [37]). Now we recall some preliminaries about the fuzzy sets defined on Rn (cf. [41]). A fuzzy set on Rn is a mapping u: Rn → [0, 1], where the value u(x) denotes the degree of membership of the element x to the fuzzy set u. For 0 < α ≤ 1, the α-level of u is defined by the set [u]α = {x ∈ Rn | u(x) ≥ α}. For α = 0, the support of u is defined as the set [u]0 = supp(u) = {x ∈ Rn | u(x) > 0}. If u, v are two fuzzy sets, then u = v if and only if [u]α = [v]α , for all α ∈ [0, 1]. From now on, we will use the symbol F n in order to denote the collection of fuzzy sets u on Rn satisfying:

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(i) (ii) (iii) (iv)

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u is normal, that is, there exists x0 ∈ Rn such that u(x0 ) = 1. u is fuzzy convex, that is, u(λx + (1 − λ)y) ≥ min{u(x), u(y)} for any x, y ∈ Rn and 0 ≤ λ ≤ 1. u is upper semicontinuous, that is, [u]α is closed for all α ∈ [0, 1]. [u]0 is compact.

According to Zadeh’s Extension Principle [41], operations of addition and scalar multiplication on F n are defined as:   (u + v)(x) = sup min u(y), v(z) ,

 and

y+z=x

(λu)(x) =

u( xλ ) χ{0} (x)

λ = 0, λ = 0,

where χ{0} is the characteristic function of {0}. Moreover, the following relations hold: [u + v]α = [u]α + [v]α ,

and

[λu]α = λ[u]α ,

∀u, v ∈ F n ,

∀α ∈ [0, 1].

Furthermore, the Hausdorff metric d in Kcn can be extended to F n defining the distance   d∞ (u, v) = sup d [u]α , [v]α , ∀u, v ∈ F n .

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(5)

α∈[0,1]

Thus, (F n , d∞ ) is a complete metric space (cf. [33]). In F 1 , in addition to the partial order defined by the inclusion of the α-levels, another partial order can be defined; indeed, if u, v ∈ F 1 with [u]α = [uαl , uαr ] and [v]α = [vlα , vrα ], we can define the partial order  as uv

⇔

uαl ≤ vlα

and uαr ≤ vrα ,

α ∈ [0, 1].

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We denote the converse of the partial order  by . An interesting property of the partial order  is that if u  v, then u + w  v + w, for u, v, w ∈ F 1 . On the other hand, we say that (uk )k∈N ⊂ F 1 is a nondecreasing sequence if uk  uk+1 for all k ∈ N; analogously, (vk )k∈N ⊂ F 1 is a nonincreasing sequence if vk+1  vk for all k ∈ N. Lemma 2.1. (See [28].) On F 1 the following properties hold: (i) If (uk )k∈N ⊂ F 1 is a nondecreasing sequence such that uk → u in F 1 , then uk  u for all k ∈ N. (ii) If (uk )k∈N ⊂ F 1 is a nonincreasing sequence such that uk → u in F 1 , then uk  u for all k ∈ N. Let a J be a closed interval and denote by C(J, F 1 ) the set of all continuous functions on the interval J with values in F 1 . We will consider the following partial order on C(J, F 1 ):   f, g ∈ C J, F 1 , f  g ⇐⇒ f (t)  g(t), ∀t ∈ J . (7) Moreover, considering the metric D on C(J, F 1 ), defined by     D(f, g) = sup d∞ f (t), g(t) , f, g ∈ C J, F n ,

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t∈J

it holds that (C(J, F 1 ), D) is a complete metric space (cf. [13]). Lemma 2.2. (See [28].) On C(J, F 1 ) the following properties hold: (i) If (fk )k∈N ⊂ C(J, F 1 ) is a nondecreasing sequence, with the order , such that fk → f in C(J, F 1 ), then fk  f for all k ∈ N. (ii) If (fk )k∈N ⊂ C(J, F 1 ) is a nonincreasing sequence, with the order , such that fk → f in C(J, F 1 ), then fk  f for all k ∈ N. On the spaces (F 1 , ) and (C(J, F 1 ), ), any pair of elements always has an upper bound (cf. [28]). Definition 2.3. Let u, v, w ∈ F n . An element w is called the Hukuhara difference (H -difference for short) of u and v, if it verifies the equation u = v + w. If the H -difference there exists, it will be denoted by u H v. Clearly, u H u = {0}, and if u H v exists, it is unique.

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In an analogous way to the case of Kcn , a generalization of the H -difference for fuzzy context was introduced in [40] (see also [39]). Definition 2.4. (See [39].) For u, v ∈ F n , the generalized Hukuhara difference of u and v (gH-difference for short), denoted by u gH v, is defined as the element z ∈ F n such that  (i) u = v + z, or u gH v = z ⇐⇒ (9) (ii) v = u + (−1)z. Notice that if u gH v and u H v exist, u gH v = u H v; if (i) and (ii) in (9) are satisfied simultaneously, then z is a crisp number2 ; also, u gH u = u H u = {0}, and if u gH v exists, it is unique. Moreover, if u, v ∈ F 1 , where v = {c}, c ∈ R, then u gH v = u − v and v gH u = v − u. For the unidimensional case F 1 , in [39] it was showed that u gH v does not always exist, but there is a characterization which guarantees the existence of u gH v. In fact, the following proposition holds: Proposition 2.5. (See [39].) Let u, v ∈ F 1 be two fuzzy numbers with α-levels given by [u]α = [uαl , uαr ] and [v]α = [vlα , vrα ], respectively. The gH-difference u gH v ∈ F 1 exists if and only if one of the following two conditions is satisfied: (i) len([u]α ) ≥ len([v]α ) for all α ∈ [0, 1], uαl − vlα is increasing with respect to α, and uαr − vrα is decreasing with respect to α (here len(A) denotes the length of the interval A ∈ Kc1 ), or (ii) len([u]α ) ≤ len([v]α ) for all α ∈ [0, 1], uαr − vrα is increasing with respect to α, and uαl − vlα is decreasing with respect to α. Theorem 2.6. (See [17].) Let u ∈ F 1 be a fuzzy number with α-levels given by [u]α = [uαl , uαr ]. Then u is completely determined by any pair u = (ul , ur ) of functions ul , ur : [0, 1] → R, defining the endpoints of the α-level sets, satisfying the following conditions: (i) ul : α → uαl ∈ R is a bounded nondecreasing left-continuous function in (0, 1] and it is right-continuous at 0. (ii) ur : α → uαr ∈ R is a bounded nonincreasing left-continuous function in (0, 1] and it is right-continuous at 0. (iii) uαl ≤ uαr , ∀α ∈ [0, 1]. 2.1. Fuzzy differentiability In recent years, several authors have established different concepts of fuzzy differentiability due to the necessity to enlarge the class of differentiable fuzzy functions and treat, in a best way, FDE as a natural way to model the uncertainty in dynamical systems (cf. [2,4,5,9,18,21,26,33,37,38]). A first generalization of the concept of fuzzy H -differentiability [34], is given by the following definition. Definition 2.7. (See [4].) Let f : (a, b) → F n and t0 ∈ (a, b). f is said to be strongly generalized differentiable (or GH-differentiable for short) at t0 , if there exists an element fG (t0 ) ∈ F n such that (i) there exist the differences f (t0 + h) H f (t0 ), f (t0 ) H f (t0 − h) and lim

h→0+

f (t0 + h) H f (t0 ) f (t0 ) H f (t0 − h) = lim = fG (t0 ), h h h→0+

(10)

or (ii) there exist the differences f (t0 ) H f (t0 + h), f (t0 − h) H f (t0 ) and lim

h→0+

f (t0 ) H f (t0 + h) f (t0 − h) H f (t0 ) = lim = fG (t0 ), + (−h) (−h) h→0

2 In the fuzzy context a crisp number is an element of Rn .

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or (iii) there exist the differences f (t0 + h) H f (t0 ), f (t0 − h) H f (t0 ) and lim

h→0+

f (t0 + h) H f (t0 ) f (t0 − h) H f (t0 ) = lim = fG (t0 ), h (−h) h→0+

(12)

or (iv) there exist the differences f (t0 ) H f (t0 + h), f (t0 ) H f (t0 − h), and lim

h→0+

f (t0 ) H f (t0 + h) f (t0 ) H f (t0 − h) = lim = fG (t0 ). + (−h) h h→0

(13)

Here, the limit is taken in the metric space (F n , d∞ ). We say that f is (i)-GH-differentiable (respectively, (ii)-GH-differentiable, (iii)-GH-differentiable and (iv)GH-differentiable) at t0 if f is GH-differentiable in the first form (i) (respectively, second form (ii), third form (iii) and fourth form (iv)) of Definition 2.7. Notice that the (i)-GH-differentiability coincides with the H -differentiability. Thus, the GH-differentiability is a concept of differentiability for fuzzy function more general than the H -differentiability. Given a fuzzy function f : (a, b) −→ F 1 , the α-levels of f (t) are compact intervals, that is, [f (t)]α = α [fl (t), frα (t)], where flα and frα are real functions on (a, b) which are the endpoint functions of f . Example 2.8. We consider the fuzzy mapping f : R → F 1 defined by f (t) = C · t , where C is a fuzzy number defined via its α-levels by [C]α = [1 + α, 2(3 − α)]. Then 

α [(1 + α)t, 2(3 − α)t] if t ≥ 0, f (t) = [2(3 − α)t, (1 + α)t] if t < 0. We can see that the endpoint functions flα and frα are not differentiable at t = 0. However f is GH-differentiable on R and fG (t) = C. In this case, f is (iii)-GH-differentiable at t = 0. In general we have the following result on the connection between the GH-differentiability of f and its endpoint functions flα and frα . Theorem 2.9. (See [12].) Let f : (a, b) → F 1 be a fuzzy function. If F is GH-differentiable at t0 ∈ (a, b), then we have the following cases: (a) If f is (i)-GH-differentiable at t0 ∈ (a, b) then, for each α ∈ [0, 1], flα and frα are differentiable functions at t0 and 

α    

fG (t0 ) = flα (t0 ), frα (t0 ) . (b) If f is (ii)-GH-differentiable at t0 ∈ (a, b) then, for each α ∈ [0, 1], flα and frα are differentiable functions at t0 and 

α    

fG (t0 ) = frα (t0 ), flα (t0 ) . (c) If f is (iii)-GH-differentiable at t0 ∈ (a, b) then, for each α ∈ [0, 1], there are the lateral derivatives (flα )+/− and (frα )+/− at t0 and 

α    

   

fG (t0 ) = flα + (t0 ), frα + (t0 ) = frα − (t0 ), flα − (t0 ) . (d) If f is (iv)-GH-differentiable at t0 ∈ (a, b) then, for each α ∈ [0, 1], there are the lateral derivatives (flα )+/− and (frα )+/− at t0 and 

α    

   

fG (t0 ) = frα + (t0 ), flα + (t0 ) = flα − (t0 ), frα − (t0 ) .

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Given a fuzzy function f , for each α ∈ [0, 1], we define the length function len(f )α given by len(f )α (t) = frα − flα . Then, from Theorem 2.9 we can distinguish two kinds of GH-differentiable fuzzy functions in relation to the differentiability of its endpoint functions. The (i) and (ii)-GH-differentiability are linked with the monotonicity of the length function while (iii) and (iv)-GH-differentiability are linked with the change of monotonicity of the length function. More precisely, we have the following result. Proposition 2.10. Let f : (a, b) → F 1 be a fuzzy function. (i) If f is (i)-GH-differentiable then, for each α ∈ [0, 1], the function len(f )α is nondecreasing. (ii) If f is (ii)-GH-differentiable then, for each α ∈ [0, 1], the function len(f )α is nonincreasing. Example 2.11. Consider the fuzzy function f : R → F 1 defined by f (t) = C · t , where C is a fuzzy number such that [C]α = [α − 1, 1 − α] for all α ∈ [0, 1]. Then

α

f (t) = (α − 1)|t|, (1 − α)|t| and len(f )α = 2(1 − α)|t|, for all α ∈ [0, 1]. In this case f is GH-differentiable and fG (t) = C for all t ∈ R. Now, on the interval (−∞, 0) f is (ii)-GH-differentiable and len(f )α is nonincreasing while on the interval (0, +∞) f is (i)-GH-differentiable and len(f )α is nondecreasing. On the other hand, f is (iii)-GH-differentiable at t = 0 and in this point there is a change of monotonicity of the function len(f )α . It is interesting to see how the switch between the two cases (i) and (ii) can occur. Definition 2.12. (See [6].) Let f : (a, b) −→ F 1 be a GH-differentiable fuzzy function. An element t0 ∈ (a, b) is said a switching point for the differentiability of f , if for every neighborhood of t0 there exist points t1 < t0 < t2 such that (i) (type I) f is (i)-GH-differentiable at t1 while f is not (ii)-GH-differentiable at t1 , and f is (ii)-GH-differentiable at t2 while f is not (i)-GH-differentiable at t2 , or (ii) (type II) f is (ii)-GH-differentiable at t1 while f is not (i)-GH-differentiable at t1 , and f is (i)-GH-differentiable at t2 while f is not (ii)-GH-differentiable at t2 . Proposition 2.13. Let f : (a, b) −→ F 1 be a GH-differentiable fuzzy function and t0 ∈ (a, b). (i) If t0 is a switching point for the differentiability of f of type I, then f is (iv)-GH-differentiable at t0 . (ii) If t0 is a switching point for the differentiability of f of type II, then f is (iii)-GH-differentiable at t0 . Proof. The proof is a consequence of Proposition 3 in [9]. 2 Recently, based on the gH-difference on F 1 (see [39]), the following definition of differentiability was analyzed. Definition 2.14. (See [6].) A fuzzy function f : (a, b) −→ F 1 is said gH-differentiable at t0 ∈ (a, b), if there exists  (t ) ∈ F 1 such that fgH 0

1 f (t0 + h) gH f (t0 ) . h→0 h

 fgH (t0 ) = lim

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Here, the limit is taken in the metric space (F 1 , d∞ ). The gH-differentiability is more general than the GH-differentiability. However, both are equivalent when the set of switching points is finite. Theorem 2.15. (See [6].) Let f : (a, b) −→ F 1 be a fuzzy function. The following statements are equivalent: (i) f is GH-differentiable. (ii) f is gH-differentiable and the set of switching points is finite.

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2.2. Fuzzy integrability Now we present some definitions and properties of the fuzzy integral which will be used in Section 4. Given f : [a, b] −→ F 1 , we say that f is integrable bounded, if there exists an integrable function h: [a, b] −→ R such that x ≤ h(t) for all x ∈ [f (t)]0 . Moreover, f is said strongly measurable if the set-valued mapping fα : [a, b] −→ Kc1 defined as fα (t) = [f (t)]α , is measurable for all α ∈ [0, 1] (see [21]). Definition 2.16. (See [21].) A fuzzy function f : [a, b] −→ F 1 is integrable if f is integrable bounded and strongly measurable. Definition 2.17. (See [21].) Let f : [a, b] −→ F 1 . The integral of f on [a, b], denoted by levelwise by the equation α b b

α f (t) dt, α ∈ [0, 1]. f (t) dt = a

b a

f (t) dt, is defined

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a

The following theorem summarizes some basic properties of the integral of fuzzy functions (see [21]). Theorem 2.18. Let f, g: [a, b] −→ F 1 be two fuzzy functions and c ∈ R. Then, (i) (ii) (iii) (iv) (v)

b b b (f + g)(t)dt = a f (t)dt + a g(t)dt . a b b a cf (t)dt = c a f (t)dt. d∞ (f (t), g(t)) is integrable. b b b d∞ ( a f (t)dt, a g(t)dt) ≤ a d∞ (f (t), g(t))dt. b b If f  g and f, g are continuous, then a f (t) dt  a g(t)dt .

Theorem 2.19. (See [5].) Let f : [a, b] −→ F 1 be continuous. Then x (i) The fuzzy function F (x) = a f (t) dt is (i)-GH-differentiable and FG (x) = f (x). b (ii) The fuzzy function G(x) = x f (t) dt is (ii)-GH-differentiable and GG (x) = −f (x). Lemma 2.20. Let x ∈ F 1 be such that the functions xl and xr defined as in Theorem 2.6 are differentiable, with xl increasing and xr decreasing on [0, 1], such that there exist constants c1 > 0, c2 < 0 satisfying (xlα ) ≥ c1 and (xrα ) ≤ c2 for all α ∈ [0, 1]. Let f : [a, b] −→ F 1 be continuous with respect to t and M, M1 , M2 constants such that len([f (t)]1 ) ≤ M for all t ∈ [a, b], | b≤

|c2 | M2

and b ≤

len([x]1 ) M

∂flα (t) ∂α |

≤ M1 and |

∂frα (t) ∂α |

≤ M2 . Moreover, suppose that b is such that b ≤

c1 M1 ,

. If

(a) xl (1) < xr (1) or if (b) xl (1) = xr (1) and the set [f (s)]1 consists of exactly one element for any s ∈ [a, b], then the H -difference

t x H

f (s) ds a

exists for any t ∈ [a, b]. Proof. The proof follows from Lemma 2.2 in [5]. 2 Remark  t 2.21. If x and f verify the hypothesis of Lemma 2.20, then the condition (b) of Proposition 2.5 holds for x and a f (s) ds.

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3. Some fixed points theorems The Banach fixed point theorem is a fundamental and useful tool in mathematics due its vast applicability for solving nonlinear, differential and integral equations, among others. This theorem has been generalized and extended by several authors in various ways, considering, for instance, non-monotone mappings in ordered partially sets (see for instance [1,16,19,27,29–31,35,36] and references therein), and a considerable number of applications in the existence of solutions for first-order ordinary differential equations, quadratic fuzzy equations, Fredholm and Volterra type integral equations, among others, have been explored. In particular, in [16,19,36] some fixed point results of weakly contractive applications on complete metric spaces were obtained; moreover, the existence of a unique solution for first order ordinary differential equations with periodic boundary conditions was studied. Briefly we present some fixed point results obtained in [19], which generalize the other ones of [16,19,28,36]. This results will be used in next section in order to analyze the existence of solutions for an FIVP in the setting of the GH-derivative. Definition 3.1. (See [19].) An altering distance function is a function ψ: [0, ∞) → [0, ∞) such that (i) ψ is continuous and nondecreasing; (ii) ψ(t) = 0 if and only if t = 0. Definition 3.2. (See [19].) Let (X, d) be a metric space and f : X → X a function. It is said that f is weakly contractive if        ψ d f (x), f (y) ≤ ψ d(x, y) − φ d(x, y) , ∀x, y ∈ X, (16) where ψ and φ are altering distance functions. Let (X, ≤) be a partially ordered set and f : X −→ X a function. We say that f is monotone nondecreasing, if every all x, y ∈ X with x ≤ y implies that f (x) ≤ f (y). The function f is monotone nonincreasing, if for all x, y ∈ X with x ≤ y implies that f (x) ≥ f (y). We show some fixed point results obtained in [19] where the function f is not necessarily continuous. Theorem 3.3. (See [19].) Let (X, ≤) be a partially ordered set and suppose that there exists a metric d in X such that (X, d) is a complete metric space. Let f : X −→ X be a monotone nondecreasing function such that        ψ d f (x), f (y) ≤ ψ d(x, y) − φ d(x, y) , for x ≥ y, (17) for some altering distance functions ψ and φ. Suppose that X verifies either that if a nondecreasing sequence (xk )k∈N is convergent to x ∈ X, then xk ≤ x, for all k ∈ N, or that f is continuous. If there exists x0 ∈ X such that x0 ≤ f (x0 ), then f has a fixed point. We obtain a similar result to Theorem 3.3, whose proof is analogous to the proof of Theorem 2.1 in [19]. Theorem 3.4. Let (X, ≤) be a partially ordered set and suppose that there exists a metric d in X such that (X, d) is a complete metric space. Let f : X −→ X be a monotone nondecreasing function such that (17) holds. Suppose that X verifies either that if a nonincreasing sequence (xk )k∈N is convergent to x ∈ X, then x ≤ xk for all k ∈ N, or that f is continuous. If there exists x0 ∈ X such that x0 ≥ f (x0 ), then f has a fixed point. Next theorem guarantees the existence and uniqueness of a fixed point and the global convergence of the method of successive approximations, that is, if (X, ≤) is a partially ordered set and f : X −→ X is a function, the sequence (f k (x))k∈N converges to the fixed point of f for all x ∈ X. Theorem 3.5. (See [19].) Under the assumption of Theorem 3.3 (respectively of Theorem 3.4), if every pair of elements of X has an upper bound or a lower bound, f has a unique fixed point. Moreover, if x is the fixed point of f , then for all x ∈ X it holds límk→∞ f k (x) = x.

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4. Fuzzy initial value problems We consider the following FIVP   x (t) = f (t, x(t)), t ∈ J = [0, T ], x(0) = x0 ,

(18)

where the derivative x  is considered in the sense of GH-differentiable, where at the end points of J we consider only the one-sided derivatives, and the fuzzy function f : J × F 1 −→ F 1 is continuous. The initial data x0 is assumed in F 1 . Let us denote by C 1 (J, F 1 ) the set of all continuous fuzzy functions f : J −→ F 1 with continuous derivative. Definition 4.1. (i) A solution for Problem (18) is a fuzzy function x ∈ C 1 (J, F 1 ) satisfying (18). (ii) A solution for Problem (18) is called (i)-solution if it is (i)-GH-differentiable. (iii) A solution for Problem (18) is called (ii)-solution if it is (ii)-GH-differentiable. There are works articles which study the procedure for obtaining an (i)-solution and an (ii)-solution for Problem (18). Using the concept of switching points we can construct other solutions for Problem (18) as we are going to show in the following example. Example 4.2. Consider the fuzzy initial value problem X  = −X(t),

X(0) = X0 ,

(19)

where the initial condition X0 = (1, 2, 3) is a triangular fuzzy number. If we consider the (i)-GH-derivative of X then we obtain an (i)-solution X 1 (t) for (19) given by X 1 (t) = 2 · χ{e−t } + W · et , where W is the triangular fuzzy number W = (−1, 0, 1). If we consider the (ii)-GH-derivative of X then we obtain an (ii)-solution X 2 (t) for (19) given by X 2 (t) = X0 · e−t . Now, we can combine the derivatives (i) and (ii) in order to obtain other solutions. For example for solving the problem (19), first we consider the (ii)-solution X 2 on the interval [0, 1] and then we consider the following problem: X  = −X(t),

X(1) = X 2 (1),

˜ from which we obtain the (i)-solution X(t) = 2 · χ{e−t } + W problem (19) which is given by  X0 · e−t , 0≤t ≤1 3 X (t) = 2 · χ{e−t } + W · et−2 , t ≥ 1.

(20) · et−2

for t ≥ 1. Thus, we have a third solution

X3

for

In this case, t = 1 is a switching point of type II for differentiability of X 3 . Similarly to process above, we can obtain other solutions for the problem (19). From previous example, we can see that there is not a unique solution for Problem (18) since, under suitable conditions, using the concept of switching points, we can construct other solutions combining the (i)-solution and (ii)-solution. Thus, it is interesting to study the existence of (i)-solution and (ii)-solution for Problem (18). In this section we use the properties of the GH-derivative together the fixed points theorems of Section 3, in order to obtain new results on the existence and uniqueness of (i)-solutions and (ii)-solutions for Problem (18), generalizing some of the results obtained in [5–7,21,28,34,37,38].

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Theorem 4.3. (See [4,10].) (i) A fuzzy function x ∈ C 1 (J, F 1 ) is an (i)-solution of (18) if and only if it satisfies the integral equation

t x(t) = x0 +

  f s, x(s) ds,

t ∈ J = [0, T ].

(21)

0

(ii) A fuzzy function x ∈ C 1 (J, F 1 ) is an (ii)-solution of (18) if and only if it satisfies the integral equation

t x(t) = x0 H (−1)

  f s, x(s) ds,

t ∈ J = [0, T ].

(22)

0

Definition 4.4. A fuzzy function μ ∈ C 1 (J, F 1 ) is said a lower solution for Problem (18) if   μ (t)  f t, μ(t) , t ∈ J, μ(0)  x0 . If μ is (i)-differentiable (respectively, (ii)-differentiable), then μ is said a lower (i)-solution (respectively, a lower (ii)-solution). A fuzzy function μ ∈ C 1 (J, F 1 ) is said an upper solution for Problem (18) if   μ (t)  f t, μ(t) , t ∈ J, μ(0)  x0 . If μ is (i)-differentiable (respectively, (ii)-differentiable), then μ is said an upper (i)-solution (respectively, (ii)-differentiable). In general, under suitable condition on f , it is possible to obtain a lower (upper) solution by inspection. For instance, if there exists a fuzzy interval W such that W  f (t, x), for all (t, x) ∈ J × F 1 , then we can obtain a lower solution μ solving the fuzzy differential equation μ (t) = W . In addition, if we consider the following class of fuzzy differential equations ˜ x), x  (t) = h(t,

x(0) = x0 ,

(23)

where t ∈ J and h˜ is obtained from a continuous function h : J × R → R by applying the Zadeh extension principle in the second argument, then, we can obtain a lower solution μ of (23) by solving the differential equation y  (t) = h(t, y(t)) with initial condition y(0) = x0 0 and considering the fuzzy function μ defined by μ(t) = χ{y(t)} . Also, if we consider the following class of fuzzy differential equations x  (t) = hS (t, x, W ),

x(0) = x0 ,

(24)

F1

where t ∈ J , W ∈ and hS is obtained from a continuous function h : J × R × R → R by applying the fuzzy standard interval arithmetic, then we can obtain a lower solution μ of (24) solving the differential equation y  (t) = min hS (t, y, [W ]0 ) with initial condition y(0) = x0 0 and considering the fuzzy function μ defined by μ(t) = χ{y(t)} . The following example shows the last argument. Example 4.5. Consider the following fuzzy differential equation (logistic model) x  (t) = W · x(1 − x),

x(0) = (0.2, 0.3, 0.4),

(25)

where W = (0.4, 0.5, 0.6). In this case min hS (t, y, [W ]0 ) = 0.4y(1 − y). Now, we take the fuzzy function μ(t) = χ{y(t)} , where y : J → R is a solution of the differential equation y  (t) = 0.4y(1 − y),

y(0) = 0.2.

Since, for each α ∈ [0, 1], [μ(t)]α = [χ{y(t)} ]α = {y(t)} then μ is a lower solution of (25). Our main result on the existence and uniqueness of solutions to an FIVP is given by the following theorem.

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Theorem 4.6. Suppose that there exists a lower (i)-solution μ ∈ C 1 (J, F 1 ) for Problem (18). Let f : J × F 1 −→ F 1 be continuous such that: (i) f is nondecreasing in the second variable, that is, if x  y then f (t, x)  f (t, y), (ii) f is weakly contractive for comparable elements, that is, for some altering distance functions ψ and φ, it holds        ψ d∞ f (t, x), f (t, y) ≤ ψ d∞ (x, y) − φ d∞ (x, y) , if x  y. Then, Problem (18) has a unique (i)-solution defined on J . Proof. We will apply Theorem 3.3. For this, we define the operator A1 : C(J, F 1 ) → C(J, F 1 ) by

t [A1 x](t) = x0 +

  f s, x(s) ds,

t ∈ J.

0

Taking into account Theorem 4.3 part (i), notice that if x ∈ C(J, F 1 ) is a fixed point of A1 , then x ∈ C 1 (J, F 1 ) is a solution of Problem (18) and conversely. −ρT In C(J, F 1 ), for ρ > 0 large enough such that 1−eρ < 1, consider the metric       Dρ (x, y) = sup d∞ x(s), y(s) e−ρs , x, y ∈ C J, F 1 . s∈J

This metric is equivalent to metric D, because Dρ (x, y) ≤ D(x, y) ≤ eρT Dρ (x, y) for all x, y ∈ C(J, F 1 ). Moreover, (C(J, F 1 ), Dρ ) is a complete metric space (cf. [13]). From the assumption (i) and Theorem 2.18 we have

t [A1 x](t) = x0 +



 f s, x(s) ds  x0 +

0

t

  f s, y(s) ds = [A1 y](t),

0

whenever x  y and t ∈ J . Then, A1 y  A1 x whenever y  x, and consequently, the operator A1 is nondecreasing. Now, from (ii), f verifies        ψ d∞ f (t, x), f (t, y) ≤ ψ d∞ (x, y) − φ d∞ (x, y) , if x  y. (26) Then, for all x  y,      ψ d∞ f (t, x), f (t, y) ≤ ψ d∞ (x, y) .

(27)

Suppose that d∞ (x, y) < d∞ (f (t, x), f (t, y)), for all x  y. Then, given that ψ is nondecreasing it holds ψ(d∞ (x, y)) ≤ ψ(d∞ (f (t, x), f (t, y))). Thus, from inequality (27), ψ(d∞ (x, y)) = ψ(d∞ (f (t, x), f (t, y))), for all x  y. Hence, replacing (26) in (27), it follows that 0 ≤ −φ(d∞ (x, y)), and therefore, φ(d∞ (x, y)) = 0. Then d∞ (x, y) = 0, and thus, ψ(d∞ (f (t, x), f (t, y))) = φ(d∞ (x, y)) = 0. Therefore, since φ is an altering distance function, we get that d∞ (f (t, x), f (t, y)) = 0 reaching a contradiction. Thus,   d∞ f (t, x), f (t, y) ≤ d∞ (x, y), for all x  y. (28) Then, for x  y it holds

    Dρ (A1 x, A1 y) = sup d∞ [A1 x](t), [A1 y](t) e−ρt t∈J





t

= sup d∞ x0 + t∈J



 t

= sup d∞ t∈J

0

  f s, x(s) ds, x0 +

0

  f s, x(s) ds,

t 0

t 0

   −ρt f s, y(s) ds e 

   −ρt f s, y(s) ds e 

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 t ≤ sup t∈J

     d∞ f s, x(s) , f s, y(s) ds e−ρt

0

 t ≤ sup

d∞

t∈J



 x(s), y(s) ds e−ρt

 (by (28))

0

 t = sup

d∞

t∈J





 x(s), y(s) e−ρs eρs ds e−ρt

0

t

≤ sup Dρ (x, y) t∈J

eρs ds e−ρt

1 − e−ρt = sup Dρ (x, y) ρ t∈J Therefore Dρ (A1 x, A1 y) ≤

1−e−ρT ρ





0





 =

1 − e−ρT Dρ (x, y). ρ

Dρ (x, y). It holds that

    1 − e−ρT γ Dρ (A1 x, A1 y) ≤ γ Dρ (x, y) ρ        1 − e−ρT = γ Dρ (x, y) − γ Dρ (x, y) − γ Dρ (x, y) , ρ −ρT

for some increasing altering distance function γ . Then, if Φ(t) = γ (t) − γ ( 1−eρ

t) it follows that for x  y,

      γ Dρ (A1 x, A1 y) ≤ γ Dρ (x, y) − Φ Dρ (x, y) .

(29)

Finally, using the existence of the lower (i)-solution and Theorem 4.3 part (i), it holds

t μ(t) = μ(0) +



t

μ (s) ds  x0 + 0

  f s, μ(s) ds = [A1 μ](t),

t ∈ J.

0

Thus μ  A1 μ. In this way the operator A1 verifies all hypotheses of Theorem 3.3, and therefore, A1 has a fixed point in C(J, F 1 ). Given that C(J, F 1 ) verifies that every pair of elements of C(J, F 1 ) has an upper bound (see Theorem 3.5), it follows that the operator A1 has a unique fixed point. 2 Theorem 4.7. Suppose that there exists a lower (ii)-solution μ ∈ C 1 (J, F 1 ) for Problem (18). Let x0 ∈ F 1 which is not a crisp number and f : J × F 1 −→ F 1 be continuous such that: t t (i) len([x0 ]α ) ≥ len([ 0 f (s, x(s)) ds]α ) for all α ∈ [0, 1], (x0 )αl − 0 flα (t) dt is increasing with respect to α, and t t (x0 )αr − 0 frα (t) dt is decreasing with respect to α, where [x0 ]α = [(x0 )αl , (x0 )αr ] and [ 0 f (s, x(s)) ds]α = t α t α [ 0 fl (t) dt, 0 fr (t) dt]. (ii) f is nondecreasing in the second variable, that is, if x  y then f (t, x)  f (t, y), (iii) f is weakly contractive for comparable elements, that is, for some altering distance functions ψ and φ, it holds        ψ d∞ f (t, x), f (t, y) ≤ ψ d∞ (x, y) − φ d∞ (x, y) , if x  y. Then, Problem (18) has a unique (ii)-solution defined on J . Remark 4.8. The condition (i) of Theorem 4.7 can be obtained if x0 and f verify all the hypotheses of Lemma 2.20.

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Proof. In the same spirit of previous theorem, we define the operator A2: C(J, F 1 ) → C(J, F 1 ) by   t   [A2 x](t) = x0 H − f s, x(s) ds ,

t ∈ J.

0

t Notice that the condition (i) guarantees the existence of x0 H (− 0 f (s, x(s)) ds). Taking into account Theorem 4.3 part (ii), if x ∈ C(J, F 1 ) is a fixed point of A2 , then x ∈ C 1 (J, F 1 ) is a solution for Problem (18) and conversely. Also, it holds that the operator A2 is nondecreasing, since for x  y,    t  t     [A2 x](t) = x0 H − f s, x(s) ds  x0 H − f s, y(s) ds = [A2 y](t), 0

t ∈ J.

0

Since f verifies (iii), we have that ψ(d∞ (f (t, x), f (t, y))) ≤ ψ(d∞ (x, y)) for all x  y. Then d∞ (f (t, x), f (t, y)) ≤ d∞ (x, y). Thus, whenever x  y it holds that d∞

    t  t      [A2 x](t), [A2 y](t) = d∞ x0 gH − f s, x(s) ds , x0 gH − f s, x(s) ds



 t = d∞

0

  f s, x(s) ds,

0

t



0

  f s, y(s) ds .

0 1−e−ρT

Therefore, it follows that Dρ (A2 x, A2 y) ≤ ρ Dρ (x, y). Then, if γ is some increasing altering distance function, it holds     1 − e−ρT γ Dρ (A2 x, A2 y) ≤ γ Dρ (x, y) ρ        1 − e−ρT . = γ Dρ (x, y) − γ Dρ (x, y) − γ Dρ (x, y) ρ −ρT

Then, if Φ(t) = γ (t) − γ ( 1−eρ

t) and whenever x  y it follows that

      γ Dρ (A2 x, A2 y) ≤ γ Dρ (x, y) − Φ Dρ (x, y) .

(30)

Finally, since μ is a lower (ii)-solution, from Theorem 4.3 part (ii) we get    t  t   μ(t) = μ(0) H − μ (s) ds  x0 H − f s, μ(s) ds = [A2 μ](t), 0

t ∈ J.

0

Thus μ  A2 μ. Then the operator A2 verifies all hypotheses of Theorem 3.3, and therefore, A2 has a fixed point in C(J, F 1 ). Given that C(J, F 1 ) verifies that every pair of elements of C(J, F 1 ) has an upper bound (see Theorem 3.5), the operator A2 has a unique fixed point. 2 Corollary 4.9. Suppose that there exists a lower (i)-solution ((ii)-solution) μ ∈ C 1 (J, F 1 ) of Problem (18). Let x0 ∈ F 1 is not a crisp number and f : J × F 1 −→ F 1 be continuous such that: t t (i) len([x0 ]α ) ≥ len([ 0 f (s, x(s)) ds]α ) for all α ∈ [0, 1], (x0 )αl − 0 flα (t) dt is increasing with respect to α, and  t t (x0 )αr − 0 frα (t) dt is decreasing with respect to α, where [x0 ]α = [(x0 )αl , (x0 )αr ] and [ 0 f (s, x(s)) ds]α = t t [ 0 flα (t) dt, 0 frα (t) dt]. (ii) f is nondecreasing in the second variable, that is, if x  y then f (t, x)  f (t, y),

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(iii) For some altering distance function φ, it holds     d∞ f (t, x), f (t, y) ≤ d∞ (x, y) − φ d∞ (x, y) ,

if x  y.

Then, Problem (18) has a unique (i)-solution ((ii)-solution, respectively) defined on J . Proof. It is enough to consider ψ(t) = t and to use Theorem 4.6 (Theorem 4.7, respectively).

2

Changing the existence of a lower solution by the existence of an upper solution to Theorem 4.6 and Theorem 4.7 we obtain the following result. Theorem 4.10. Replacing the existence of a lower (i)-solution ((ii)-solution) to Problem (18) by an upper (i)-solution ((ii)-solution), the conclusion of Theorem 4.6 and Theorem 4.7 is still valid. Proof. If μ is an upper (i)-solution for Problem (18), we have that

t μ(t) = μ(0) +

t



μ (s) ds  x0 + 0

  f s, μ(s) ds = [A1 μ](t),

t ∈ J.

(31)

0

Also, if μ is an upper (ii)-solution for Problem (18), we have that    t  t    μ(t) = μ(0) H − μ (s) ds  x0 H − f s, μ(s) ds = [A2 μ](t), 0

t ∈ J.

(32)

0

Thus μ ≥ Ai μ, i = 1, 2, which correspond to each case considered in Theorem 4.6 and Theorem 4.7. The existence of a solution for Problem (18) follows from Theorem 3.4, because C(J, F 1 ) verifies that if a nonincreasing sequence (xk )k∈N is convergent to x ∈ C(J, F 1 ), then x  xk for all k ∈ N. Given that every pair of elements of C(J, F 1 ) has an upper bound, the operator Ai μ, i = 1, 2, has a unique fixed point. 2 Corollary 4.11. Replacing the existence of a lower (i)-solution ((ii)-solution) by an upper (i)-solution ((ii)-solution) at to Problem (18), the conclusion of Corollary 4.9 is still valid. Remark 4.12. The solution from Theorem 4.6 and Theorem 4.7 for Problem (18) can be obtained as límk→∞ Aki (x), i = 1, 2, for any x ∈ C(J, F 1 ). In particular, (Aki (μ))k∈N , i = 1, 2, are nondecreasing sequences and convergent in C(J, F 1 ) to each of the two solutions of Problem (18). This property is a consequence of Theorem 3.5 since the space C(J, F 1 ) verifies that every pair of elements of C(J, F 1 ) has an upper bound. Like as the Picard iteration in ordinary differential equations, this iteration scheme suggests a numerical algorithm to construct the solution. Remark 4.13. The results given in [28] on an FIVP use the Hukuhara derivative and require as hypothesis that f be Lipschitz in the second variable for related elements. We obtain a generalization of these results, if we simply change the contractivity hypothesis of f in [28], by the following condition        ψ d∞ f (t, x), f (t, y) ≤ ψ d∞ (x, y) − φ d∞ (x, y) , t ∈ J, (33) where ψ and φ are altering distance functions. 5. Conclusions Using some results of fixed point of weakly contractive mappings on partially ordered sets, in place of using the classical Banach fixed point theorem, we analyzed the existence and uniqueness of solutions for fuzzy initial value problems in the setting of a generalized Hukuhara derivative (GH-derivative). In particular, given that the concept of GH-differentiability is based on four forms (types) of lateral derivatives, we proved the existence and uniqueness of an (i)-solution, as well as, an (ii)-solution, for an FIVP.

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