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Differentiability on semilinear spaces George N. Galanis ∗ Section of Mathematics, Hellenic Naval Academy, Xatzikyriakion, Piraeus 185 39, Greece

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Article history: Received 11 February 2009 Accepted 20 March 2009 MSC: 26E25 34G20 46G05 54C60 Keywords: Semilinear spaces Differentiation on semilinear spaces Hukuhara differentiation

abstract A new differentiability theory for mappings between semilinear spaces is introduced in this work. The main aim is to develop a solid analysis framework that may cover any type of mapping in the semilinear framework and not only curves with real domain as the majority of recent methods in this area do. As a result, a convenient framework is provided within which problems that go beyond the borders of classical analysis and differential geometry for single-valued mappings can be studied. Such issues could include stochastic manifolds, set valued and fuzzy differential equations, etc. On the other hand, the proposed environment keeps all the characteristics of a differentiability theory, being a natural generalization of the most widely used differentiation theories both for singlevalued and set valued functions. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction In recent years there has been increasing research activity in the study of mappings between semilinear spaces due to the rapid development of new techniques for addressing multivalued functions, set and fuzzy differential equations, optimizations theory through set valued mappings, etc. In particular, there is an increased need for analysis tools that can be used in the semilinear framework, like the space of compact, convex subsets of a Banach space, the space of fuzzy sets with basis a Euclidean or a Banach space, etc. (see, e.g, [1–5] and [9]). In order to support such activities, a number of approaches have been developed, proposing different analysis tools for multivalued mappings or, more generally, for mappings on semilinear spaces. The notion of Hukuhara differentiation [6,7] maintains a principal role in this framework, while the embedding of the semilinear space under study into a relevant topological vector space is always the final way out. However, such analysis environments impose significant restrictions, since they are mainly focused on the differentiability of curves – mappings whose domain is a real interval – leaving therefore no space for the study of transition functions, cocycles or even the differential operator itself. This restriction makes impossible any generalization to differential geometry and, therefore, notions like stochastic manifolds cannot be supported. In this paper, a new unified approach to the differentiability of mappings between semilinear spaces is proposed. The techniques adopted give a natural way out of the above mentioned problems since they: - cover any type of mapping or operator between semilinear spaces, - preserve all the basic characteristics of a classical differentiability theory, and - include as special cases the differentiability of single-valued mappings between locally convex spaces as well as the Hukuhara differentiability of multivalued mappings.

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0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.03.045

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In this way, the proposed notions and techniques provide a convenient framework for the development of several relevant issues of pure and applied mathematics, like set and fuzzy differential equations, stochastic manifolds, optimization theory, etc. 2. Preliminaries The necessary background for developing a novel analysis framework for semilinear spaces is presented in this section. Moreover, some basic notions and results for the Hukuhara differentiability, that has been utilized successfully for certain classes of semilinear spaces, are discussed. We begin with the basic definitions for semilinear spaces (see also [10]): Definition 2.1. A semilinear space is a set S endowed with two operations: addition (+ : S × S → S) and scalar multiplication by nonnegative reals (· : R+ × S → S) which satisfy the following properties for every x, y ∈ S and λ, µ ∈ R+ : (i) (S , +) is a commutative cancellative semigroup with 0, (ii) λ · (x + y) = λ · x + λ · y, (iii) (λ + µ) · x = λ · x + µ · x, (iv) (λµ) · x = λ · (µ · x), (iv) 1 · x = x and 0 · x = 0 Definition 2.2. A topological semilinear space is a semilinear space S endowed with a Hausdorff topology with respect to which the two operations of S are continuous. Definition 2.3. A metric semilinear space is a topological semilinear space S whose topology is introduced by a metric d : S × S → R+ that satisfies the additional condition d(λ · x + z , λ · y + z ) = λd(x, y), for every x, y, z ∈ S and λ ∈ R+ . Definition 2.4. A mapping f : S1 → S2 between two semilinear spaces is called semilinear if it is compatible with the operations of S1 , S2 : f (x + y) = f (x) + f (y), f (λ · x) = λ · f (x) for every x, y ∈ S1 and λ ∈ R+ . Examples. 1. Every vector space is a semilinear space. 2. Every topological vector space is a topological semilinear space. 3. Every normed vector space is a metric semilinear space. 4. Every Banach space is a complete metric semilinear space (and so is every Fréchet space). 5. A convex cone in a topological vector space is a topological semilinear space. 6. The space Kc (E) is that of all nonempty compact, convex subsets of a Banach (or Fréchet) space E is a complete metric semilinear space. Its topology is defined by the Hausdorff metric:

DH (A, B) = max sup d(x, A), sup d(y, B) x∈B

y∈A

where d(x, A) = inf[d(x, y) : y ∈ A], A, B ∈ Kc (E). 7. The space E n of fuzzy sets with base Rn consisting of all mappings u : Rn → [0, 1] such that: (i) u(x0 ) = 1, for some x0 ∈ Rn , (ii) u(λ · x + (1 − λ) · y) ≥ min {u(x), u(y)} for any x, y ∈ Rn and λ ∈ [0, 1], (iii) u is upper semi-continuous, (iv) [u]0 = {u ∈ E n : u(x) > 0} is compact, is a metric semilinear space under D0 (u, v) = sup

DH ([u]a , [v]a ) ,

where [u]a = {x ∈ Rn : u(x) ≥ a}.

0≤a≤1

The semilinear spaces of the last two examples are the frame sets in which the theories of set and fuzzy, respectively, differential equations have been developed. The lack of subtraction here, as in all semilinear spaces, imposed the development of new analysis tools. Two were the main directions so far – either to embed the semilinear space of interest in a topological vector space and translate there any differentiability question (see e.g. [4]) or to adopt generalizations of the classical differentiation that fit the semilinear environment.

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The most popular and successful approach of the latter type is the use of Hukuhara differentiation. This is based on the fact that, although there is no multiplication with negative elements, a type of difference can be defined, namely the Hukuhara difference. To be more precise, if for two fixed elements x and y in a semilinear space S there exists a third one z ∈ S such that x = y + z, then z is called the Hukuhara difference of x and y and is denoted by x − y. It is worthy of note that the Hukuhara difference of two elements does not always exist, but if it does it is unique. On the basis of this, the notion of Hukuhara differentiable mappings can be introduced: Definition 2.5. Let S be a metric semilinear space. A mapping f : I ⊂ R → S, where I is a real interval, is called Hukuhara differentiable at t0 ∈ I if there exists an element DH f (t0 ) ∈ S such that lim

h→0+

f (t0 + h) − f (t0 ) h

= lim

f (t0 ) − f (t0 − h)

h→0+

h

= DH f (t0 ) ∈ S .

The differences in the numerators are the Hukuhara differences of the elements involved. In this case, the element DH f (t0 ) is called the Hukuhara derivative of f at t0 . It is important to note that if the semilinear space S is replaced by a topological vector space, e.g. instead of multivalued mappings in Kc (E) one uses single-valued functions, then the previous notion reduces to the classical Gâteaux differentiability. Although the above mentioned approaches support, up to a point, the study of differential equations for multivalued and fuzzy mappings (see, e.g., [1–3,7,4,5]), several restrictions are imposed that make impossible the complete study of functions with values in semilinear spaces. Indeed, the definition of Hukuhara differentiability can be applied only for mappings whose domain is an interval of real numbers; the differential operator DH cannot be approached as a potential differentiable mapping, etc. As a result, any attempt to obtain a generalized analysis framework that could support the study of transition functions, cocycles and other tools of analysis and differential geometry seems to be doomed to failure. Trying to contribute to this subject, we propose in the following section a new differentiability notion on semilinear spaces that generalizes the classical differentiation of single-valued mappings between topological vector spaces as well as the Hukuhara differentiation of set valued curves. This new platform proves to be compatible with the classical techniques of analysis and differential geometry and, therefore, can be used for different applications within the framework of pure or applied mathematics. 3. Differentiability on semilinear spaces Having established in the previous section all the necessary background, we proceed here to the main target of this paper, namely the development of a new analysis framework for semilinear spaces. There are two main difficulties that one has to deal with, namely the absence of subtraction and the way in which the differential operator should be treated. Let S1 , S2 be two topological semilinear spaces. Definition 3.1. A continuous mapping f : A ⊂ S1 → S2 , where A ⊂ S1 is open, is called differentiable at x0 ∈ A if there exist: (i) a continuous and semilinear with respect to the second-variable mapping df : U0 × S1 → S2 , where U0 is an open neighborhood of x0 , (ii) a continuous mapping φ : I × U0 × S1 → S2 , where I is an open interval of 0 ∈ R, with limt →0+ ϕ(t , x, y) = 0, for every x ∈ U0 and y ∈ S1 , such that f (x + t · y) = f (x) + t · df (x, y) + t · ϕ(t , x, y), for any t ∈ I, x ∈ U0 and y ∈ S1 . In this case, the continuous and semilinear mapping df (x0 ) : S1 → S2 : y 7→ df (x0 , y) is called the differential of f at x0 . It is obvious that this approach is not restricted to just curves with domain in the real line, like the Hukuhara differentiability, but can be potentially applied to any mapping between semilinear spaces. In the next proposition some first basic properties of this new differentiation are discussed: Proposition 3.2. Under the assumptions of the previous definition the following results hold true: 1. 2. 3. 4.

The differential operator df is unique if it exists. Any constant mapping is differentiable with trivial differential. Any continuous semilinear mapping f : S1 → S2 is differentiable with differential coinciding with f . Any continuous bi-semilinear mapping g : S1 × S2 → S3 is differentiable at any pair (x0 , y0 ) ∈ S1 × S2 with dg (x0 , y0 , x, y) = g (x0 , y) + g (x, y0 );

(x, y) ∈ S1 × S2 .

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5. The differential operator is semilinear: d(λf + µg ) = λdf + µdg , for every differentiable mappings f , g and any nonnegative real λ. Proof. 1. This is mainly a consequence of the fact that the Hukuhara differences in a semilinear space are unique if they exist. Therefore, the part df (x, y) + ϕ(t , x, y) is uniquely determined. Taking also into account that ϕ(0, x, y) = 0, we obtain the result. 2. In this case the requirements of the definition are obviously satisfied by df = 0 and ϕ = 0, for every x ∈ S1 . 3. The semilinearity of f leads to f (x + t · y) = f (x) + t · f (y) + t · 0, for any x, y ∈ S1 and t ∈ R+ . Therefore, f is differentiable at any x ∈ S1 with df (x0 ) = f and ϕ = 0. 4. The assumption of bilinearity leads to g ((x1 , y1 ) + t · (x2 , y2 )) = g (x1 , y1 ) + t · g (x1 , y2 ) + t · g (x2 , y1 ) + t 2 · g (x2 , y2 ). Therefore, on setting G : S1 × S2 × S1 × S2 → S3 : (x1 , y1 , x2 , y2 ) 7→ g (x1 , y2 ) + g (x2 , y1 ) and φ : R × S1 × S2 × S1 × S2 → S3 : (t , x1 , y1 , x2 , y2 ) 7→ t · g (x2 , y2 ), it is not difficult to check that the first mapping is continuous and semilinear with respect to the second factor, while the second is continuous and vanishes as the first variable approaches 0 ∈ R+ . Moreover, g ((x1 , y1 ) + t · (x2 , y2 )) = g (x1 , y1 ) + t · G(x1 , y1 , x2 , y2 ) + t · ϕ(t , x1 , y1 , x2 , y2 ). As a result, g is differentiable with dg (x0 , y0 ) = G(x0 , y0 , ·, ·). 5. The linearity of the differential operator is a direct result of (1) and the fact that for any f , g differentiable and λ ∈ R+ the following condition holds true:

(f + g )(x + t · y) = f (x) + g (x) + t · (df (x, y) + dg (x, y)) + t · ϕf (t , x, y) + ϕg (t , x, y) , where ϕf , ϕg are the mappings satisfying the definition of differentiability for f , g respectively.

Some further examples of multivalued functions in the space Kc (R) that are differentiable in the proposed sense follow: Examples. 1. The mapping F : [0, 1] → Kc (R) : x 7→ [0, x] is differentiable with df (x, h) = [0, h]. 2. The mapping G : [0, 1] → Kc (R) : x 7→ [0, g (x)], where g is any smooth real function, is differentiable with dg (x, h) = [0, g 0 (x) · h]. Apart from these, easily proved but essential, properties of the proposed differentiation, the following, main for the applications, results can be also stated. Theorem 3.3. Let f : A ⊂ S1 → S2 , g : B ⊂ S2 → S3 be continuous mappings between topological semilinear spaces such that f is differentiable at x0 ∈ A and g at f (x0 ) ∈ B. Then, their composition g ◦ f is differentiable at x0 ∈ A and the corresponding differential satisfies the chain rule: d(g ◦ f )(x0 ) = dg (f (x0 )) ◦ df (x0 ). Proof. Let φ1 : I1 × U0 × S1 → S2 , φ2 : I2 × V0 × S2 → S3 be the continuous mappings that fulfill the requirements of the differentiability definition for f , g respectively, where I1 , I2 are real intervals containing 0, U0 an open neighborhood of x0 ∈ A and V0 an open neighborhood of f (x0 ) ∈ B appropriately restricted so that f (U0 ) ⊂ V0 . We set G : U0 × S1 → S3 : (x, y) 7→ dg (f (x), df (x, y)),

φ3 : (I1 ∩ I2 ) × U0 × S1 → S3 : (t , x, y) 7→ dg (f (x), φ1 (t , x, y)) + φ2 (t , f (x), df (x, y) + φ1 (t , x, y)) . G is continuous as a composition of continuous mappings. It is also semilinear with respect to the second factor since for any x, y ∈ S1 and t ∈ R+ , the corresponding semilinearity of df , dg gives G(x, y1 + t · y2 ) = dg (f (x), df (x, y1 + t · y2 ))

= dg (f (x), df (x, y1 ) + t · df (x, y2 )) = dg (f (x), df (x, y1 )) + t · dg (f (x), df (x, y2 )) = G(x, y1 ) + t · G(x, y2 ).

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On the other hand, φ3 is also continuous as constructed by continuous mappings and lim ϕ3 (t , x, y) = lim (dg (f (x), φ1 (t , x, y)) + (φ2 (t , f (x), df (x, y) + φ1 (t , x, y))))

t →0+

t →0+

= dg (f (x), 0) + φ2 0, f (x), df (x, y) + lim φ1 (t , x, y) t →0+

= dg (f (x), 0) + φ2 (0, f (x), df (x, y)) = 0. Moreover, the composition g ◦ f calculated on any x, y ∈ S1 and t ∈ I1 ∩ I2 gives

(g ◦ f )(x + t · y) = g (f (x) + t · (df (x, y) + ϕ1 (t , x, y))) = g (f (x)) + t · dg (f (x), df (x, y) + ϕ1 (t , x, y)) + t · ϕ2 (t , f (x), df (x, y) + ϕ1 (t , x, y)) = (g ◦ f )(x) + t · dg (f (x), df (x, y)) + t · (dg (f (x), ϕ1 (t , x, y)) + ϕ2 (t , f (x), df (x, y) + ϕ1 (t , x, y))) = (g ◦ f )(x) + t · G(x, y) + t · φ3 (t , x, y) which completes the proof.

Another important issue that needs to be addressed is the relation between the proposed differentiability and relevant notions of differentiation for single-valued and set valued mappings: Proposition 3.4. If the semilinear spaces used in Definition 3.1 are topological vector spaces, then the differentiability proposed coincides with the classical Gâteux differentiation. The connection between the two differentiability notions is obvious and no proof is needed. Proposition 3.5. Let f : I ⊂ R → Kc (E) be a set valued function which is differentiable at t0 ∈ I. Then, f is also Hukuhara differentiable and the relevant differential operators are connected through DH f (t0 ) = df (t0 , 1). Proof. The differentiability of f at t0 ensures that f (t0 + h) = f (t0 ) + h · df (t0 , 1) + h · ϕ(h, t0 , 1), f (t0 ) = f (t0 − h) + h · df (t0 − h, 1) + h · ϕ(h, t0 − h, 1). As a result, the limits required in the Hukuhara differentiability take the form lim

h→0+

lim

f (t0 + h) − f (t0 ) h f (t0 ) − f (t0 − h)

h→0+

h

= lim (df (t0 , 1) + ϕ(h, t0 , 1)) = df (t0 , 1), h→0+

= lim (df (t0 − h, 1) + ϕ(h, t0 − h, 1)) h→0+

= df (t0 , 1)

which yields the result.

It worth noticing here that Definition 3.1 can be inductively generalized in order to cover differentiability of higher order: Definition 3.6. A continuous mapping f : A ⊂ S1 → S2 , where A ⊂ S1 open, is called n-times differentiable at x0 ∈ A if: (i) f is (n − 1)-times differentiable at x0 , (ii) there exists a continuous, symmetric and semilinear with respect to the last (n − 1) variables mapping dn f : U0 ×S1n → S2 , where U0 is an open neighborhood of x0 , and (iii) there exists a continuous mapping φ : I ×U0 ×S1 → S2 , where I is an open interval of 0 ∈ R, with limt →0+ ϕ(t , x, y) = 0, for every x ∈ U0 and y ∈ S1 , such that f (x + t · y) = f (x) + t · df (x, y) +

t2 2

· d2 f (x, y, y) + · · · +

tn n!

· dn f (x, y, y, . . . , y) + t n · ϕ(t , x, y),

for any t ∈ I, x ∈ U0 and y ∈ S1 . A natural question that also arises is that of the possibility of connecting the proposed differentiability with any type of integrability theory. Such theories have already been established for set valued mappings in [8]. To be more precise, for a set valued mapping F : I → Kc (E), where E stands for an arbitrarily chosen Banach space and I for an interval of reals, the integral of F in the sense of Aumann, on a measurable set I0 ⊂ I, is defined as follows:

Z

F (s) ds = I0

Z

f (s) ds : f is a Bochner integrable selector of F I0

.

G.N. Galanis / Nonlinear Analysis 71 (2009) 4732–4738

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For a continuous map F : I → Kc (E) the integral I F (s) ds can be introduced in a natural way in the sense of Bochner. The 0 compatibility of these integrals with Hukuhara differentiation has led to the development of a complete theory for studying and solving set differential equations (see, for example, [2,3,7]). Fortunately, this compatibility holds also for the proposed differentiation. To be more precise, the following result holds true:

R

Proposition 3.7. Let F : I → Kc (E) continuous and integrable. Then, the mapping Φ : I → Kc (E) with

Φ (t ) = U0 +

t

Z

F (s) ds,

U0 ∈ Kc (E), t0 ∈ I ,

t0

is differentiable with dΦ (u, v) = v F (u), (u, v) ∈ I × R. Proof. For any u ∈ I and t , v ∈ R, one obtains

Φ (u + t v) = U0 +

Z

u +t v

F (s) ds = U0 +

u

Z

F (s) ds +

= Φ (u) + t v F (u) + t

u +t v

Z 1 t

F (s) ds

u

t0

t0

u +t v

Z

F (s) ds − v F (u) .

u

By setting G : I × R → Kc (E) with G(u, v) = v FR(u), which is continuous and semilinear with respect to the second factor, u+t v and ϕ : I × I × R → Kc (E) with ϕ(t , u, v) = 1t u F (s) ds − v F (u), it can be easily checked that h:=t v

lim ϕ(t , u, v) = v lim

t →0+

h→0+

u+h

Z 1 h

F (s) ds

− v F (u) = 0.

u

On the other hand,

Φ (u + t v) = Φ (u) + tG(u, v) + t ϕ(t , u, v) which means that Φ is differentiable with dΦ (u, v) = v F (u).

This result gives the opportunity to repeat all the solvability theory obtained so far for set differential equations through Hukuhara differentiation by means of the proposed differentiability. More precisely, let E be an arbitrarily chosen separable Banach space and dU (t , 1) = F (t , U (t ));

U (t0 ) = U0 ∈ Kc (E),

t0 ≥ 0,

(3.1)

an initial value problem (IVP) where F ∈ C [R+ × Kc (E), Kc (E)]. Then, following the lines of [3], one may prove: Proposition 3.8. Let R0 = J × B(U0 , b), where J = [t0 , t0 + a], B(U0 , b) = {U ∈ Kc (E) : DH [U , U0 ] ≤ b, i ∈ N} and DH is the Hausdorff metric of Kc (E). Assume that: (1) The function F is bounded, that is there exists a positive M0 > 0 with DH (F (t , A), 0) ≤ M0 , if t ∈ J = [t0 , t0 + a] and A ∈ Kc (F) with DH (A, U0 ) ≤ b. (2) g ∈ C [J × [0, 2b], R+ ], g (t , w) ≤ M1 on J × [0, 2b], g (t , 0) ≡ 0, g (t , w) is nondecreasing in w , for each t ∈ J, and w(t ) ≡ 0 is the only solution of w 0 = g (t , w), w(t0 ) = 0, on J. (3) DH (F (t , A), F (t , B)) ≤ g (t , DH (A, B)), for every i ∈ N, on R0 . Then, the successive approximations defined by Un+1 (t ) = U0 +

Z

t

F (s, Un (s)) ds,

n = 0, 1, 2, . . . ,

t0

exist on J0 = [t0 , t + η), where η = min{a, Mb }, M = max{M0 , M1 }, as continuous functions and converge uniformly to the unique solution of the IVP on J0 . On the other hand, the connection between the proposed differentiability and Bochner integrals can be also used in order to prove the inverse of Proposition 3.5. Proposition 3.9. Let F : I ⊂ R → Kc (E) be a Hukuhara differentiable set valued function at t0 ∈ I. Then, F is differentiable in the sense of Definition 3.1 with dF (u, v) = v DH F (u).

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Proof. For any u ∈ I and t , v ∈ R, F satisfies F (u + t v) = F (u) +

u +t v

Z

DH F (s) ds

u

= F (u) + t v DH F (u) + t

u+t v

Z 1 t

DH F (s) ds − v DH F (u) .

u

Following the lines of the proof of Proposition 3.7, one may check that the requirements of the definition of the new differentiability are fulfilled by the mappings G : I × R → Kc (E) : (u, v) 7→ v DH F (u),

ϕ : I × I × R → Kc (E) : (t , u, v) 7→ a fact that yields the result.

1 t

u +t v

Z

DH F (s) ds − v DH F (u),

u

Combining Propositions 3.4, 3.5 and 3.9 we may conclude that Theorem 3.10. The notion of the proposed differentiability is a direct generalization of its classical counterparts for single-valued and set valued mappings since it coincides with: - The Gâteux differentiability if one restricts to single-valued mappings between topological vector spaces. - The Hukuhara differentiation for set valued curves. At the same time, the proposed analysis framework provides a smoother and wider environment for the study of mappings between semilinear spaces, since it can be applied to any function and not just to curves, therefore leaving open horizons for use in differential geometry and especially manifolds with set valued mappings as local coordinate systems. References [1] D. Kuroiwa, Existence theorems of set optimization with set-valued maps, J. Inform. Optim. Sci. 24 (1) (2003) 73–84. [2] G. Galanis, T. Gnana Bhaskar, V. Lakshmikantham, P.K. Palamides, Setvalued functions in Fréchet spaces: Continuity, Hukuhara differentiability and applications to set differential equations, Nonlinear Anal. 61 (2005) 559–575. [3] G. Galanis, T.G. Bhaskar, V. Lakshmikantham, Set differential equations in Fréchet spaces, J. Appl. Anal. 14 (1) (2008) 103–113. [4] Y.R. Syau, Differentiability and convexity of fuzzy mappings, Comput. Math. Appl. 41 (2001) 73–81. [5] G. Wand, C. Wu, Directional derivatives and subdifferential of convex fuzzy mappings and application in convex fuzzy programming, Fuzzy Sets Systems 138 (2003) 559–591. [6] M. Hukuhara, Integration of measurable maps with compact, convex set values, Funkc. Ekvac. 10 (1967) 205–223. [7] V. Lakshmikantham, T. Gnana Bhaskar, J. Vasundhara Devi, Theory of Set Differential Equations in a Metric Space, Cambridge Scientific Pub., 2006. [8] A. Tolstonogov, Differential Inclusions in a Banach Space, Kluwer Academic Publishers, Dordrecht, 2000. [9] B. Bede, I.J. Rudas, A.L. Bencsik, First order linear fuzzy differential equations under generalized differentiability, Inform. Sci. 177 (2007) 1648–1662. [10] R.E. Worth, Boundaries of semilinear spaces and semialgebras, Trans. Amer. Math. Soc. 148 (1970) 99–119.

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