On the global existence of solutions of fuzzy differential equations

ISSN 0012-2661, Differential Equations, 2006, Vol. 42, No. 10, pp. 1391–1403. c© Pleiades Publishing, Inc., 2006.Original Russian Text c© A.A. Martynyuk, V.I. Slyn’ko, 2006, published in Differentsial’nye Uravneniya, 2006, Vol. 42, No. 10, pp. 1324–1336.

ORDINARY DIFFERENTIAL EQUATIONS

On the Global Existence of Solutionsof Fuzzy Differential Equations

A. A. Martynyuk and V. I. Slyn’koInstitute of Mechanics, National Academy of Sciences, Kiev, Ukraine

Received November 3, 2004

DOI: 10.1134/S0012266106100041

1. INTRODUCTION

The theory of fuzzy differential equations permits one to refine the description of a real processoccurring in the presence of inaccuracy by analyzing an equation or a system of differential equationsdefined on a space of fuzzy sets. This theory is based on Zadeh’s idea [1] on the existence of fuzzysets.

Zadeh’s idea, together with the Hukuhara derivative [2, p. 42], was used in the monograph[2, p. 49] to construct a theory of fuzzy differential equations. The resulting differential equations,as well as contingent and paratingent differential equations, are the object of study in the theoryof differential equations with multivalued right-hand side. There are numerous papers (of whichwe note only [3–5]) dealing with the construction of mathematical analysis and theory of suchequations.

Elements of the theory of fuzzy differential equations are widely used in the construction ofcontrol systems for fuzzy systems and Takagi–Sugeno systems (see [6–9] and the bibliographytherein). Note also that the relationship between the theory of fuzzy sets and probability theorywas discussed in [10] and other papers.

Let us describe a fuzzy differential equation following [2].Let u(t) be a function mapping the space R+ into the space E

n to be defined below. We usethe derivative of the mapping u(t), which generalizes the corresponding definition of the Hukuharaderivative for multimappings. Let [u]α be the α-level of the fuzzy set u ∈ E

n; then [u]α is a convexcompact set, and the Hukuhara derivative is well defined for it.

Suppose that there exists a fuzzy set w ∈ En such that

[w(t)]α = D[u]α,

where D is the Hukuhara derivative; then the fuzzy mapping u(t) is said to be differentiable, andw is referred to as the derivative of the mapping u.

A differential equation of the form

du

dt= f(t, u), u (t0) = u0,

where the right-hand side is the Hukuhara derivative of fuzzy sets in the sense of the above-represented definition, t ∈ R+, and

f ∈ (R+ × En, En) ,

is a fuzzy differential equation.In the present paper, we use a new approach to the analysis of dynamic properties of solutions of

a fuzzy differential equation. The key idea of our approach is to split the phase space En in which

the fuzzy system of differential equations is defined and project the solutions onto the productR

n × En� , where E

n� is the quotient space of the space E

n/Rn. This idea permits one to consider

numerous problems of qualitative theory of fuzzy systems of differential equations on the basis ofapproaches that are to a wider extent within reach for practical use.

1391

1392 MARTYNYUK, SLYN’KO

This approach is used in the present paper to derive conditions, less conservative than thosein [2, p. 63], for the global existence of solutions.

2. ON THE DECOMPOSITION OF THE SPACE En

Taking into account the results of [2], we consider the following spaces En, SE

n, and En1 of fuzzy

sets.

Definition 2.1. The space En is defined if and only if the following axioms hold.

(1) u is a mapping Rn → [0, 1].

(2) The sets [u]α = {x ∈ Rn | u(x) ≥ α} are compact subsets of the space R

n for all α ∈ (0, 1].(3) The set [u]0 = cl

⋃α∈(0,1][u]α is bounded.

(4) The mapping u is fuzzy convex; i.e.,

u(λx + (1 − λ)y) ≥ min[u(x), u(y)]

for all x, y ∈ Rn and λ ∈ [0, 1].

Note that, throughout the following, the operations of addition of elements of the space En and

their multiplication by a nonnegative scalar number are treated in the sense of fuzzy set theory [3].Note that condition (4) ensures that all α-levels [u]α, α ∈ [0, 1], of a fuzzy system are convex.

Definition 2.2. A mapping u ∈ SEn is defined if and only if assumptions (1)–(4) of Defini-

tion 2.1 are valid and all α-levels [u]α, α ∈ I = [0, 1], are centrally symmetric sets with commonsymmetry center.

Definition 2.3. A mapping u ∈ En1 is well defined if and only if assumptions (1)–(4) of

Definition 2.1 are valid and the 1-level [u]1 is a singleton.

Let K nC be the space of nonempty compact convex sets in the space R

n, and let dH(· , ·) be theHausdorff metric in K n

C given by the formula

dH(A,B) = max{

supx∈A

R(x,B), supy∈B

R(y,A)}

,

where R(x,A) = infy∈A ‖x − y‖ and ‖ · ‖ is the Euclidean norm on Rn. Then in the space E

n,one can introduce the Hausdorff metric

DH(u, v) = supα∈[0,1]

dH ([u]α, [v]α) . (2.1)

The space (En,DH) is a complete metric space.

Definition 2.4. A subset E n of En closed in the topology determined by the Hausdorff metric

and satisfying the conditions u + v ∈ E n and λu ∈ E n for all u, v ∈ E n and λ ≥ 0 is called asubspace of the space E

n.

Obviously, the sets SEn and E

n1 are subspaces of E

n. Throughout the following, E n stands forsome subspace of E

n.On the space E n, we introduce a relation � between elements u and v by the following rule.We say that elements u, v ∈ E n satisfy the relation � if and only if there exists a vector x ∈ R

n

such that [u]α = [v]α + x for all α ∈ [0, 1].One can readily see that � is an equivalence. The quotient space of the space E n with respect

to � is referred to as the space E n� .

Note that the space E n� consists of classes of equivalent elements of E n; therefore, throughout

the following, the elements of E n� are denoted by uppercase Latin letters and the corresponding

representatives are denoted by lowercase Latin letters; we write U = {u} if u is a representativeof U .

DIFFERENTIAL EQUATIONS Vol. 42 No. 10 2006

ON THE GLOBAL EXISTENCE OF SOLUTIONS OF FUZZY DIFFERENTIAL . . . 1393

The operations of Minkowski addition and multiplication by a nonnegative scalar number in thespace E n induce the corresponding operations in the space E n

� .Let U, V ∈ E n

� , U = {u}, and V = {v}; then

U + V = {u + v}, (2.2)λU = {λu}. (2.3)

Note that relations (2.2) and (2.3) are well defined, i.e., are independent of the choice of represen-tatives of equivalence classes with respect to �.

In the space E n� , we introduce a metric d : E n

� × E n� → R+ by the relation

d(U, V ) = infx∈Rn

DH(u, v + x), (2.4)

where U = {u}, V = {v}, and DH is the Hausdorff metric in the space En.

Assertion 2.1. Formula (2.4) gives a well-defined metric in the space E n� .

Proof. The fact that the metric (2.4) is well defined is a consequence of the following propertyof the Hausdorff metric:

DH(u + x, v + x) = DH(u, v).Let us show that the metric (2.4) indeed defines a distance in the space E n

� . Obviously, d(U,U) = 0.Let d(U, V ) = 0; then infx∈Rn DH(u, v+x) = 0, and therefore, there exists a sequence {xn}∞n=1 ⊂ R

n

such that DH (u, v + xn) < 1/n. This implies the inequality dH ([u]α, [v]α + xn) < 1/n for eachα ∈ [0, 1]. The sequence {xn}∞n=1 is bounded, since the zero levels of fuzzy sets are bounded; there-fore, there exists a subsequence {xnk

}∞k=1 of the sequence {xn}∞n=1 converging to some x0 ∈ Rn. Since

the metric DH(· , ·) is continuous, we find that dH ([u]α, [v]α + x0) = 0; i.e., U = V . The propertyd(U, V ) = d(V,U) is obvious.

Let us prove the triangle inequality. Let U = {u}, V = {v}, and W = {w}, x, y ∈ Rn; then

DH(x + u, y + w) ≤ DH(x + u, v) + DH(v, y + w),

and consequently,

infx∈Rn

infy∈Rn

DH(x + u, y + w) ≤ infx∈Rn

DH(x + u, v) + infy∈Rn

DH(v, y + w).

This implies thatd(U,W ) ≤ d(U, V ) + d(V,W ).

The proof of the assertion is complete.Next, consider the space R

n × E n� . It is the main space in the analysis of qualitative properties

of solutions of fuzzy differential equations. Let us introduce the notions of sum and product by anonnegative scalar number in the space R

n × E n� . Let (x,U), (y, V ) ∈ R

n × E n� ; then

(x,U) + (y, V ) = (x + y, U + V ), λ(x,U) = (λx, λU), λ ∈ R+. (2.5)

In addition, we introduce a metric on the space Rn × E n

� , by the formula

D ((x,U), (y, V )) = ‖x − y‖ + d(U, V ),

where ‖ · ‖ is the Euclidean norm on Rn.

Let us establish a one-to-one correspondence between elements of the spaces Rn × E n

� and E n.

Definition 2.5. A selection mapping X : E n� → E n is referred to as a decomposition operator

if and only ifX (U + V ) = X U + X V, X (λU) = λX U

for all λ ≥ 0 and U, V ∈ E n� .



Assertion 2.2. Let X be a decomposition operator ; then the mapping B (x,U) = x + X U isan isomorphic one-to-one correspondence between elements of the spaces R

n × E n� and E n.

Proof. Obviously, B (x,U) ∈ E n. For each u ∈ E n, there exist (x,U) ∈ Rn × E n

� such thatB (x,U) = u. Indeed, if U = {u} and x = u − XU , then B (x,U) = u. Let B (x,U) = B (y, V ),then x + X U = y + X V . Hence we find that X U is related to X V by �; therefore, U = V ,X U = X V , and x = y. The relations

B (x + y, U + V ) = x + y + X (U + V ) = x + X U + y + X V = B (x,U) + B (y, V ),B (λx, λU) = λx + X (λU) = λB (x,U)

complete the proof.

Definition 2.6. The space (E n, D ), where E n is a subspace of En with the metric D(u, v) =

D (B −1(u), B −1(v)), is referred to as the space (E n, X ) of fuzzy sets.

Note that the decomposition operator specifies the metric of the space E n. The choice of suchan operator should be determined by the properties of the problem to be solved and the spaces offuzzy sets to be considered.

Let us indicate some methods for introducing a decomposition operator for the above-describedspaces SE

n, En1 , and E

n of fuzzy sets.

2.1. The Space SEn

In this space, for each u ∈ SEn, we introduce the set W (u) ⊂ SE

n, W (u) = {x + u | x ∈ Rn}.

Let Q u be the element of best approximation to the zero element of the space SEn by the set W (u)

in the metric DH(· , ·).

Assertion 2.3. For each u ∈ SEn, there exists a unique element Q u.

Proof. Let us prove the existence of an element X u for each u ∈ SEn. Consider the function

f(x) = DH(u + x, 0), x ∈ Rn. It follows from the general properties of the metric DH that

|f(x) − f(y)| ≤ ‖x − y‖;

i.e., the function f(x) satisfies the Lipschitz condition and is bounded above; therefore, it followsfrom general theorems of approximation theory [11] that there exists an element Q u of best approx-imation. Let us prove its uniqueness. Suppose that there exist two elements of best approximation,which, without loss of generality, can be assumed to have the form u + x and u − x, where x ∈ R

n

and x �= 0; then dH(u+x, 0) = dH(u−x, 0) = d. It follows from general theorems of approximationtheory that the element u is also an element of best approximation; i.e., dH(u, 0) = d.

The parallelogram identity

‖x + y‖2 + ‖y − x‖2 = 2(‖x‖2 + ‖y‖2

)

implies the inequality

2(‖x‖2 + ‖y‖2

)≤ sup

α∈[0,1]

supy∈[u]α

‖y + x‖2 + supα∈[0,1]

supy∈[u]α

‖y − x‖2

= D2H(u + x, 0) + D2

H(u − x, 0).

Hence we obtain the inequality

2(‖x‖2 + D2

H(u, 0))≤ D2

H(u + x, 0) + D2H(u − x, 0),

which implies that x = 0. The proof of the assertion is complete.Let us prove some properties of the operator Q .



Assertion 2.4. Let U, V ∈ SEn� , U = {u}, V = {v}, and u, v ∈ SE

n. Then

DH(Q u, Q v) = d(U, V ).

Proof. By definition,

DH(Q u, Q v) = supα∈[0,1]

dH ([Q u]α, [Q v]α)

= supα∈[0,1]

inf {δ > 0 | [Q v]α ⊂ [X u]α + δK, [Q u]α ⊂ [Q v]α + δK} ;

therefore, by virtue of the relation1 Q (u + δK) = Q u + δK, where K is the unit ball with centerthe origin and δ > 0, we obtain

inf {δ > 0 | [Q v]α ⊂ [Q u]α + δK, [Q u]α ⊂ [Q v]α + δK}≤ inf {δ > 0 | [v]α ⊂ [u]α + δK, [u]α ⊂ [v]α + δK} .

Consequently, DH(Q u, Q v) ≤ DH(u, v). Note also that, from this, we have DH(Q u, Q v) = d(U, V ).Indeed, for each positive ε > 0, there exists an xε ∈ R

n such that DH (u + xε, v) ≤ d(U, V ) + ε;therefore,

DH(Q u, Q v) ≤ DH (Q (u + xε) , Q v) ≤ DH (u + xε, v) ≤ d(U, V ) + ε.

By passing to the limit as ε → 0 in the inequality d(U, V ) ≤ DH(Q u, Q v) ≤ d(U, V ) + ε, onecompletes the proof of the assertion.

Let us show that Q is a linear positively homogeneous operator.

Assertion 2.5. The relations

Q (u + v) = Q u + Q v, Q (λu) = λQ u

are valid for arbitrary u, v ∈ SEn and λ ≥ 0.

Proof. Let u, v ∈ SEn; then W (Q u + Q v) = W (u + v), which, together with the above-proved

uniqueness of the element of best approximation, implies that

Q (u + v) = Q (Q u + Q v) = Q u + Q v + x. (2.6)

Then, by virtue of well-known properties of the Hausdorff metric [2, p. 5] and the uniqueness ofthe element of best approximation, we have

DH(Q v + x, 0) = DH(Q u + Q v + x, Q u) = DH(Q (Q u + Q v), Q u)≤ DH(Q u + Q v, Q u) = DH(Q v, 0).

But, by definition, DH(Q v + x, 0) ≥ DH(Q v, 0); therefore, DH(Q v + x, 0) = DH(Q v, 0). This,together with the uniqueness of the element of best approximation, implies that x = 0. The propertyQ (λu) = λQ u is obvious. The proof of the assertion is complete.

Let us construct a mapping X : SEn� → SE

n by the rule X U = Q u, where {u} = U . By virtueof Assertions 2.2–2.5, X is a decomposition operator. Let us establish a relationship between themetrics DH and D in the space SE

n.

Assertion 2.6. The inequalityDH(u, v) ≤ D(u, v)

is valid for arbitrary elements u, v ∈ SEn.

1 Indeed, DH(u + δK, 0) = supα∈[0,1] supp∈Sn−1 |hQ u(α, p) + δ| = δ + DH(u, 0). By replacing u by u + x and by

evaluating the greatest lower bound with respect to x ∈ �n , we obtain DH(Q (u + δK), 0) = DH(Q u, 0) + δ. On theother hand, by the rectangle inequality, DH(Q u + δ, 0) ≤ DH(Q u, 0) + δ ≤ infx∈�n DH(Q (u + �K) + x, 0). Therefore,DH(Q u + �K, 0) = DH(Q (u + �K), 0), which, together with the assertion on the uniqueness of the element of bestapproximation, implies that Q (u + �K) = Q u + �K.



Proof. Let u = x + Q u and v = y + Q v; then

DH(u, v) ≤ ‖x − y‖ + DH(Q u, Q v) = ‖x − y‖ + d(U, V ) = D(u, v).

2.2. The Space En1

We introduce a mapping X : (En1 )� → E

n1 by the relation X U = u − [u]1, where U = {u} and

u ∈ En1 .

Assertion 2.7. The mapping X is a decomposition operator.

Proof. The fact that X is a selector is obvious. Let U = {u} = {u∗}; then there exists a vectorx ∈ R

n such that u∗ = u + x; therefore, u∗ − [u∗]1 = u + x − [u + x]1 = u − [u]1, and the selectorX is well defined. It is also obvious that X is a linear positively homogeneous operator.

Assertion 2.8. The metrics D and DH of the space En1 are related by the inequality

D(u, v) ≤ 2DH(u, v).

Proof. Let u = x + X u and v = y + X v; then ‖x − y‖ = DH ([u]1, [v]1) ≤ DH(u, v) andD(u, v) = ‖x − y‖ + d(U, V ) ≤ DH(u, v) + d(U, V ) ≤ 2DH(u, v).

2.3. The Space En

Let u ∈ En, and let hu(α, p) be the support function of u. By H we denote the cone of support

functions in the space Y of mappings f : [0, 1] × Sn−1 → R satisfying the following conditions:(1) f(·, p) is measurable; (2) f(α, ·) is continuous, where Sn−1 is the unit sphere of the space R

n.By [2, p. 8], the cone H with the uniform metric is isomorphic and isometric to the space (En,DH)of fuzzy sets. On the space H , we introduce the metric of L2 (I × Sn−1) by the formula

DL2(I×Sn−1)(u, v) =

⎡

⎣

1∫

0

∫

Sn−1

|hu(α, p) − hv(α, p)|2 dα dω

⎤

⎦

1/2

,

where dω is the area element of the sphere Sn−1. Suppose that u ∈ En and consider the set

W (u) = {u + x | x ∈ Rn} ⊂ E

n. Let Q u be an element of best approximation to the zero elementof the space E

n by the set W (u) in the metric DL2(I×Sn−1).

Assertion 2.9. For each u ∈ En, there exists a unique element Q u.

This assertion is a straightforward consequence of general theorems of approximation theoryand the fact that DL2(I×Sn−1) is a Euclidean metric.

The following assertion implies that Q is a linear positively homogeneous operator.

Assertion 2.10. The relations

Q (u + v) = Q u + Q v, Q (λu) = λQ u

are valid for arbitrary u, v ∈ En and λ ≥ 0.

Proof. Let x = (x1, . . . , xn)T ∈ Rn be a solution of the problem

infz∈Rn

DL2(I×Sn−1)(u, z) = DL2(I×Sn−1)(u, x);



then x is found as the minimum of the bilinear function

ϕ(z) =

⎡

⎣

1∫

0

∫

Sn−1

|hu(α, p) − z1p1 − · · · − znpn|2 dα dω

⎤

⎦

1/2

.

The relations ∂ϕ/∂zi|z=x = 0 providing a necessary condition for a minimum, result in the systemof linear equations

∫

Sn−1

p1pidω x1 + · · · +∫

Sn−1

pnpidω xn =

1∫

0

∫

Sn−1

pihu(α, p)dα dω, i = 1, . . . , n. (2.7)

Let us show that the determinant Δ =∣∣∫

Sn−1 pipjdω∣∣ni,j=1

of this system is nonzero. To this end,we introduce the spherical coordinates

p1 = cos ϑ1, p2 = sinϑ1 cos ϑ2, . . . ,

pn−1 = sin ϑ1 sin ϑ2 . . . sin ϑn−2 cos ϑn−1, pn = sin ϑ1 sin ϑ2 . . . sinϑn−2 sin ϑn−1,

dω = sinn−2 ϑ1 sinn−3 ϑ2 . . . sin ϑn−2dϑ1dϑ2 . . . dϑn−1,

where the angular coordinates ϑ1, . . . , ϑn−2 range in the interval [0, π] and the coordinate ϑn−1

ranges in the interval [0, 2π].Let us show that

∫Sn−1 pipjdω = 0 for i �= j. To be definite, we assume that j > i and j �= i;

then the product pipjdω contains a factor of the form cos ϑj sinn−1−j ϑjdϑj; therefore, the integral∫Sn−1 pipjdω contains a factor of the form

∫ π

0cos ϑj sinn−1−j ϑjdϑj, which is zero. If j �= i and j = n,

then the integral∫

Sn−1 pipjdω contains the vanishing factor∫ 2π

0sin ϑn−1dϑn−1.

Consider the integrals∫

Sn−1 p2i dω. The product p2

i dω contains factors of two types, sink dϑ andcos2 ϑ sink ϑ dϑ, where k is an integer. The integral of factors of the first type can be expressed viathe gamma function,

π∫

0

sink ϑ dϑ =√

π Γ(

k + 12

)/

Γ(

k + 22

)

,

and is obviously zero. By using this formula and the well-known property Γ(s + 1) = sΓ(s), s > 0,of the gamma function, we obtain

π∫

0

cos2 ϑ sink ϑ dϑ =√

π Γ(

k + 12

)/

Γ(

k + 22

)

−√

π Γ(

k + 32

)/

Γ(

k + 42

)

=√

π Γ(

k

2

)

Γ(

k + 12

)

k

/(

4Γ(

k + 22

)

Γ(

k + 42

))

�= 0

for k �= 0. If k = 0, then, obviously,∫ π

0cos2 ϑ sink ϑ dϑ �= 0. This implies the inequality Δ �= 0.

By solving the system of linear equations (2.7) by the Cramer formulas, we obtain the followingexpression for xi :

xi =

1∫

0

∫

Sn−1

Ki(p)hu(α, p)dα dω,

where Ki(p) ∈ C (Sn−1, R). By introducing the vector function K(p) = (K1(p), . . . ,Kn(p))T,we have

x =

1∫

0

∫

Sn−1

K(p)hu(α, p)dα dω.



We have thereby obtained the expression

Q u = u −1∫

0

∫

Sn−1

K(p)hu(α, p)dα dω,

which implies the desired assertion and completes the proof.Let us introduce the operator X : E

n� → E

n, X U = Q u. It follows from Assertions 2.9 and 2.10that X is a decomposition operator.

The following assertion can be useful in the construction of decomposition operators in somecases.

Assertion 2.11. Let U = {u} and U ∈ En� . Then the operator X : E

n� → E

n given by theformula

X U = u −1∫

0

∫

Sn−1

K(p)hu(α, p)dα dω, (2.8)

where K ∈ C (Sn−1, Rn) , is a decomposition operator if and only if the vector function K(p) satisfiesthe matrix integral equation ∫

Sn−1

K(p)pTdω = I,

where I is the identity matrix.

Proof. Obviously, the operator (2.8) is linear and positively homogeneous. The fact that it iswell defined can be verified by straightforward computation.

Let us establish a relationship between the metrics of the spaces (En, D) and (En,DH) of fuzzysets.

Assertion 2.12. If the decomposition operator has the form (2.8), then metrics of the spaces(En, D) and (En,DH) are equivalent.

Proof. Let u = x + X U and v = y + X V . By definition, we have

D(u, v) = ‖x − y‖ + d(U, V )

≤∥∥∥∥∥

1∫

0

∫

Sn−1

K(p)hu(α, p)dα dω −1∫

0

∫

Sn−1

K(p)hv(α, p)dα dω

∥∥∥∥∥

+ DH(u, v)

≤ (1 + c)DH(u, v),

where c =∫

Sn−1 ‖K(p)‖dω. On the other hand, we have

DH(X U, X V ) = supα∈[0,1]

supp∈Sn−1

|hQ u(α, p) − hQ v(α, p)| ≤ (1 + c)DH(u, v). (2.9)

For any positive number ε, there exists a vector xε ∈ Rn such that DH (u + xε, v) ≤ d(U, V ) + ε.

Since the operator X U is well defined, the substitution of the element u+xε for u into (2.9) resultsin the inequality

DH(X U, X V ) ≤ (1 + c)DH (u + xε, v) ≤ (1 + c)(d(U, V ) + ε).

By passing to the limit as ε → 0 in the last inequality, we obtain

DH(X U, X V ) ≤ (1 + c)d(U, V ).



Hence it follows that

DH(u, v) ≤ ‖x − y‖ + DH(X U, X V ) ≤ ‖x − y‖ + (1 + c)d(U, V )≤ (1 + c)(‖x − y‖ + d(U, V )) = (1 + c)D(u, v).

The proof of the assertion is complete.

Corollary. The space(E

n� , d

)is a complete metric space.

As an application of the suggested approach, we consider the problem on the global existenceof solutions of fuzzy ODE.

3. CONDITIONS FOR THE GLOBAL EXISTENCEOF SOLUTIONS OF A FUZZY DIFFERENTIAL EQUATION

On the product(R

n × E n� , D

), we consider the fuzzy ordinary differential equations

dx

dt= F (t, x, U), x (t0) = x0 ∈ R

n,dU

dt= G (t, x, U), U (t0) = U0 ∈ E n

� . (3.1)

Here (x,U) ∈ Rn × E n

� , F ∈ C(

T0 × Rn × E n

� , Rn), and G ∈ C

(T0 × R

n × E n� , E n

�

).

For system (3.1), we make the following assumptions.Assumption 3.1. The system of differential equations (3.1) satisfies the following conditions.(a) The condition of the local existence of solutions is valid for arbitrary (x0, U0) ∈ Z ⊂ R

n×E n� ,

where Z is an open subset of Rn × E n

� and t0 ∈ T0 ⊂ R+; i.e., there exists a number a (x0, U0) > 0such that the solution (x (t; t0, x0, U0) , U (t; t0, x0, U0)) exists on the time interval t ∈ [t0, t0 + a).

(b) There exist functions f ∈ C (T0 × R+ × R+, R+) and g ∈ C (T0 × R+ × R+, R+) such that

‖F (t, x, U)‖ ≤ f(t, ‖x‖, d(U, 0)), d(G (t, x, U), 0) ≤ g(t, ‖x‖, d(U, 0))

for (t, x, U) ∈ T0 × Z .Consider the comparison system

dw1

dt= f (t, w1, w2) , w1 (t0) = w10,

dw2

dt= g (t, w1, w2) , w2 (t0) = w20; (3.2)

the properties of its solutions determine the properties of solutions of system (3.1).

Theorem 3.1. Let system (3.1) satisfy Assumption 3.1 and the following conditions.(1) The vector function (f (t, w1, w2) , g (t, w1, w2)) satisfies the Wazewski condition with respect

to the variables (w1, w2) for all t ∈ T0.(2) There exist constants μ1, μ2 > 0 such that the upper solution w+ (t; t0, w0) of the comparison

system (3.2) globally exists and is uniformly [with respect to the initial data wT0 < (μ1, μ2)] bounded

on any finite time interval ; i.e., for arbitrary t0, T ∈ T0, T > t0, there exists a constant Ct0,T > 0such that

w+ (t; t0, w0) < Ct0,T e

for all wT0 < (μ1, μ2) and t ∈ [t0, T ) , where e = (1, 1)T.

Then the solution (x (t; t0, x0, U0) , U (t; t0, x0, U0)) of system (3.1) with initial data satisfying theestimates ‖x0‖ < μ1 and d (U0, 0) ≤ μ2 is globally defined.

Proof. Let x(t) = x (t; t0, x0, U0), and let U (t; t0, x0, U0) be a solution of system (3.1) with initialconditions satisfying the estimates ‖x0‖ ≤ μ1 and d (U0, 0) ≤ μ2; by virtue of Assumption 3.1a, thissolution exists on some interval [t0, β), t0 < β < ∞. Suppose that the value β cannot be increased.We set m1(t) = ‖x(t)‖ and m2(t) = d(U(t), 0); then, following [2, p. 63; 13, p. 65], one can obtainthe estimates

D+m1(t) ≤ f (t,m1(t),m2(t)) , D+m2(t) ≤ g (t,m1(t),m2(t)) ,



which are valid for all t ∈ [t0, β). From the general theorems of the comparison principle [13, p. 28],we obtain the estimates

‖x (t; t0, x0, U0)‖ ≤ w+1 (t; t0, w10, w20) ,

d (U (t; t0, x0, U0) , 0) ≤ w+2 (t; t0, w10, w20) .

(3.3)

It follows from Assumption 2.1a that ‖x (t; t0, x0, U0)‖ < Ct0,β and d (U (t; t0, x0, U0) , 0) < Ct0,β

for all t ∈ [t0, β). Since f and g are continuous functions on the closed compact set t0 ≤ t ≤ β,0 ≤ w ≤ Ct0,βe, it follows that they are bounded by some constant Nt0,β > 0. Let t1, t2 ∈ [t0, β);then we have

‖x (t1; t0, x0, U0) − x (t2; t0, x0, U0)‖ =

∥∥∥∥∥∥

t2∫

t1

F (s, x(s), U(s))ds

∥∥∥∥∥∥

≤t2∫

t1

f(s, ‖x(s)‖, d(U(s), 0))ds ≤ Nt0,β (t2 − t1) ,

d (U (t1) , U (t2)) = d

⎛

⎝

t2∫

t1

G (s, x(s), U(s))ds, 0

⎞

⎠

≤t2∫

t1

g(s, ‖x(s)‖, d(U(s), 0))ds ≤ Nt0,β (t2 − t1) .

(3.4)

Since the limits of the right-hand sides as t1, t2 → β − 0 exist and are finite, we find, by choosingt1, t2 → β − 0 and by using the Cauchy criterion and the estimates (3.4) that there exist limitslimt→β−0 x (t; t0, x0, U0) and limt→β−0 U (t; t0, x0, U0). Since the space R

n × E n� is complete, one can

consider the initial-value problem

dx

dt= F (t, x, U), x(β) = lim

t→β−0x (t; t0, x0, U0) ,

dU

dt= G (t, x, U), U(β) = lim

t→β−0U (t; t0, x0, U0) .

It follows from the local existence of solutions that the solution (x(t), U(t)) can be continuedbeyond β, which contradicts the original assumption.

Theorem 3.2. Let the system of equations (3.1) satisfy Assumption 3.1 and the following con-ditions.

(1) The vector functions (f (t, w1, w2) , g (t, w1, w2)) are monotone with respect to the variables(w1, w2) for all t ∈ T0.

(2) There exist constants μ1, μ2 > 0 such that the upper solution w+ (t; t0, w0) of system (3.2)exists globally for all wT

0 < (μ1, μ2).Then the solution (x (t; t0, x0, U0) , U (t; t0, x0, U0)) of system (3.1) with initial conditions satis-

fying the estimates ‖x0‖ < μ1, d (U0, 0) ≤ μ2, exists globally.

Proof. Just as in the proof of Theorem 3.1, we obtain the estimates (3.3). Let t1, t2 ∈ [t0, β);then, by virtue of the monotonicity of the functions f (t, w1, w2) and g (t, w1, w2), by analogy



with (3.4), we have

‖x (t1; t0, x0, U0) − x (t2; t0, x0, U0)‖ ≤t2∫

t1

f(s, ‖x(s)‖, d(U(s), 0))ds

≤t2∫

t1

f(s,w+

1 (s; t0, w10, w20) , w+2 (s; t0, w10, w20)

)ds

= w+1 (t2; t0, w10, w20) − w+

1 (t1; t0, w10, w20) ,

d (U (t1) , U (t2)) ≤t2∫

t1

g(s, ‖x(s)‖, d(U(s), 0))ds

≤t2∫

t1

g(s,w+

1 (s; t0, w10, w20) , w+2 (s; t0, w10, w20)

)ds

= w+2 (t2; t0, w10, w20) − w+

2 (t1; t0, w10, w20) .

Since the limits limt→β−0 w+1 (t; t0, w10, w20) and limt→β−0 w+

2 (t; t0, w10, w20) exist and are finite,it follows from the argument used in the final part of the proof of Theorem 3.1 that the assertionof Theorem 3.2 is valid.

Theorem 3.3. Let system (3.1) satisfy the following conditions.(1) The function F (t, x, U) is independent of U and is defined in the entire space.(2) The solutions of the Cauchy problem

dx

dt= F (t, x), x (t0) = x0

globally exist for all ‖x0‖ ≤ μ1.(3) The estimate

d(G (t, x, U), 0) ≤ g(t, ‖x‖, d(U, 0))

is valid for all (t, x, U) ∈ T0 × Z , and g(t, · , ·) is a monotone function.(4) The upper solutions w+ (t; t0, w0, ε) of the equations

dw

dt= g(t, ε, w), w (t0) = w0

globally exist for all w0 < μ2 and ε > 0.Then the solution (x (t; t0, x0, U0) , U (t; t0, x0, U0)) globally exists for initial conditions satisfying

the estimates ‖x0‖ < μ1 and d (U0, 0) ≤ μ2.

Proof. By virtue of assumptions (1) and (2) of the theorem, the solution x (t; t0, x0) is boundedon any finite time interval. Suppose that [t0, β), t0 ≤ β < ∞, is the existence interval of the solutionU (t; t0, x0, U0); moreover, β cannot be increased. There exists a constant Ct0,β,x0 > 0 such that‖x (t; t0, x0)‖ ≤ Ct0,β,x0. Let w+ (t; t0, w0, Ct0,β,x0) be an upper solution of the equation

dw

dt= g (t, Ct0,β,x0, w) , w (t0) = w0.

Since the function g(t, · , ·) is monotone, we have the estimate

D+m2(t) ≤ g (t, Ct0,β,x0,m2(t)) ,



which, together with the theorems of the comparison principle [13], implies the inequality

d(U(t), 0) ≤ w+ (t; t0, w0, Ct0,β,x0) .

Let t1, t2 ∈ [t0, β); then, just as above,

d (U (t1) , U (t2)) ≤t2∫

t1

g(s, ‖x(s)‖, d(U(s), 0))ds

≤t2∫

t1

g(s,Ct0,β,x0, w

+ (s; t0, w0, Ct0,β,x0))ds

= w+ (t2; t0, w1, Ct0,β,x0) − w+ (t1; t0, w0, Ct0,β,x0) .

To complete the proof, one should follow the lines of the proof of Theorems 3.1 and 3.2.

4. EXAMPLE AND DISCUSSION OF RESULTS

To compare our results with well-known assertions on the global existence of solutions of fuzzydifferential equation [2, p. 63], in the space E n = SE

n∩En1 of fuzzy sets, we consider the differential

equationdu

dt= F (u), u (t0) = u0, (4.1)

where u ∈ E n, F (u) = [u]1 +∣∣∣‖[u]1‖2 − ‖[u]1‖

∣∣∣ v, v is the fuzzy set defined by the levels [v]α =

(1 − α)K, α ∈ [0, 1], and K is the unit ball in the space Rn with center 0. This system is well

defined in the space E n in the sense that if u0 ∈ E n, then u (t; t0, u0) ∈ E n for all t ≥ t0.Let us analyze the global existence of solutions of system (4.1) on the basis of Theorem 3.7.1

in [2]. To this end, we estimate the Hausdorff distance from F (u) to zero:

DH(F (u), 0) ≤∥∥[u]1

∥∥ +

∣∣∣∥∥[u]1

∥∥2 −

∥∥[u]1

∥∥∣∣∣ .

If DH(u, 0) ≤ 1, then ‖[u]1‖ ≤ DH(u, 0) ≤ 1 and DH(F (u)) ≤ 2 ‖[u]1‖. But if DH(u, 0) > 1, thenDH(F (u), 0) ≤ 2 ‖[u]1‖+ ‖[u]1‖2. Therefore, since the function g(t, w) occurring in the comparisonequation

dw

dt= g(t, w), w (t0) = w0, (4.2)

should be continuous, it follows that this function necessarily admits the representation

g(t, w) ={

2w + 1 for w ≤ 12w + w2 for w > 1.

Solutions of the scalar equation (4.2) cannot be continued to the infinite time interval for arbitraryinitial values w0; therefore, it is impossible to prove the global existence of solutions of Eq. (4.1)on the basis of Theorem 3.7.1 [2].

Note that since the decomposition operators described in Sections 2.1 and 2.2. coincide on thespace E n, it follows from Assertions 2.6 and 2.8 that the metric D of the space E n is equivalent tothe Hausdorff metric DH . Therefore, Eq. (4.1) on the space R

n × E n� is equivalent to the system of

equationsdx

dt= x, x (t0) = x0,

dU

dt=

∣∣‖x‖2 − ‖x‖

∣∣ V, U (t0) = U0. (4.3)

One can verify that system (4.3) satisfies all assumptions of Theorem 3.3 with the family of com-parison equations

dw

dt= ε2 + ε, w (t0) = w0.

Therefore, the solutions of system (4.1) exist globally.



CONCLUDING REMARKS

One can treat the space(R

n × En� , D

)as a space of fuzzy sets and, on this basis, construct a

qualitative theory of fuzzy dynamical systems, i.e., dynamical systems in a metric space. In ad-dition, in particular, the general results of the monograph [15, pp. 20–68] can be extended tofuzzy dynamical systems. Note also that the suggested construction permits one to perform thedecomposition of a fuzzy dynamical system defined in the space (En, D) into a finite-dimensionaldynamical system in the space R

n and an infinite-dimensional system in the space En� , which can be

investigated with the use of well-known (but modified) methods of a qualitative theory of equations.The use of the suggested approach in the dynamics of fuzzy systems leads to wider conditions

for the global existence of solutions. This is explained by the fact that, for equations in a finite-dimensional space, conditions for the global existence of solutions are less restrictive than thecorresponding conditions in infinite-dimensional spaces [14, p. 304 of the Russian translation].For this reason, the extraction of a finite-dimensional component from a fuzzy dynamical systemcan weaken existence conditions for solutions and for their boundedness [16, 17] as compared withconditions obtained with the use of other methods.

ACKNOWLEDGMENTS

The authors are grateful to V. Lakshmikantham for the opportunity to get acquainted with themonograph [2].

REFERENCES

1. Zadeh, L.A., Inf. Control , 1965, vol. 8, pp. 338–353.2. Lakshmikantham, V. and Ram Mohapatra, Theory of Fuzzy Differential Equations and Inclusions,

London: Taylor and Francis, 2003.3. Aubin, J.P. and Cellina, A., Differential Inclusions , Berlin: Springer, 1984.4. Filippov, A.F., Differentsial’nye uravneniya s razryvnoi pravoi chast’yu (Differential Equations with

Discontinuous Right-Hand Sides), Moscow: Nauka, 1985.5. Aumann, R.J., J. Math. Anal. Appl., 1965, vol. 12, pp. 1–12.6. Chang-Hua Lien, Nonlinear Dyn. Syst. Theory, 2004, vol. 4, no. 1, pp. 15–30.7. Xinkai Chen and Guisheng Zhai, Nonlinear Dyn. Syst. Theory, 2004, vol. 4, no. 2, pp. 125–138.8. Jing Zhou, Changyun Wen, and Ying Zhang, Nonlinear Dyn. Syst. Theory, 2005, vol. 5, no. 3,

pp. 285–298.9. Benrejeb, M., Gasmi, M., and Borne, P., Nonlinear Dyn. Syst. Theory, 2005, vol. 5, no. 4, pp. 369–380.

10. Puri, M.L. and Ralescu, D.A., J. Math. Anal. Appl., 1986, vol. 114, pp. 409–422.11. Akhiezer, N.I., Lektsii po teorii approksimatsii (Lectures on Approximation Theory), Moscow: Nauka,

1965.12. Korneichuk, N.P., Ekstremal’nye zadachi teorii priblizhenii (Extremal Problems of Approximation The-

ory), Moscow, 1976.13. Lakshmikantham, V., Leela, S., and Martynyuk, A.A., Ustoichivost’ dvizheniya: metod sravneniya

(Stability of Motion: the Comparison Method), Kiev: Naukova Dumka, 1991.14. Dieudonne, J., Foundations of Modern Analysis , New York: Academic, 1960. Translated under the title

Osnovy sovremennogo analiza, Moscow: Mir, 1964.15. Zubov, V.I., Metody A.M. Lyapunova i ikh primenenie (Lyapunov Methods and Their Applications),

Leningrad: Leningr. Univ., 1957.16. Martynyuk, A.A. and Slyn’ko, V.I., Dokl. Akad. Nauk , 2005, vol. 402, no. 3, pp. 303–307.17. Martynyuk, A.A. and Slyn’ko, V.I., Prikl. Mekh., 2005, vol. 42, no. 12, pp. 93–99.


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