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2 Some Jensen's Inequalities for Fuzzy-Interval-Valued Function

# 2 Some Jensen's Inequalities for Fuzzy-Interval-Valued Function

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462

T. M. da Costa et al.

We now provide our fuzzy versions of Jensen’s integral inequality based on the

LRF ´convexity, CWF ´convexity, and on the CWF˚ ´convexity.

Theorem 15 (Fuzzy Jensen’s Inequality with the LRF ´convexity). Let

g : r0, 1s Ñ pa, bq be a Lesbesgue-integral function. Given a LRF ´convex fuzzy˜

interval-valued function

” F : ra, bs Ñ FC

ı pRq whose α´levels Fα : ra, bs Ñ KC

are given by Fα ptq “ f α pgptqq, f α pgptqq , where f α , f α : ra, bs Ñ R and pf α ˝ gq

and pf α ˝ gq are Lebesgue-integrable over r0, 1s for all α P r0, 1s, then

ˆż 1

˙

ż1

F˜ pgptqqdt.

gptqdt ĺLRF pF Aq

0

(37)

0

Proof. From definition of ĺLRF and definition of pF Aq´integral, and from

Remark 2, it follows that (37) holds if and only if

ˆż 1

˙

ż1

˜

F˜α pgptqqdt for all α P r0, 1s.

gptqdt ĺLR pIAq

(38)

0

0

Since F˜ is LRF ´convex, then from item (ii) in Proposition 2, it follows that Fα

is LR´convex for all α P r0, 1s. Then, applying Theorem 5 for each α P r0, 1s, it

follows that (38) holds. Therefore, (37) also holds.

\

[

Using similar argumentation to that used in the proof of Theorem 15, one

can obtain easily a proof of the following two results.

Theorem 16 (Fuzzy Jensen’s Inequality with the CWF ´ convexity). Let

g : r0, 1s Ñ pa, bq be a Lesbesgue-integral function. Given a CWF ´convex fuzzy˜

interval-valued function

” F : ra, bs Ñ FC

ı pRq whose α´levels Fα : ra, bs Ñ KC

are given by Fα ptq “ f α pgptqq, f α pgptqq , where f α , f α : ra, bs Ñ R and pf α ˝ gq

and pf α ˝ gq are Lebesgue-integrable over r0, 1s for all α P r0, 1s, then

ˆż 1

˙

ż1

F˜ pgptqqdt.

gptqdt ĺCW F pF Aq

0

(39)

0

Theorem 17 (Fuzzy Jensen’s Inequality with the CW ˚F ´convexity). Let

g : r0, 1s Ñ pa, bq be a Lesbesgue-integral function. Given a CW ˚F ´convex fuzzy˜

interval-valued function

” F : ra, bs Ñ FC

ı pRq whose α´levels Fα : ra, bs Ñ KC

are given by Fα ptq “ f α pgptqq, f α pgptqq , where f α , f α : ra, bs Ñ R and pf α ˝ gq

and pf α ˝ gq are Lebesgue-integrable over r0, 1s for all α P r0, 1s, then

ˆż 1

˙

ż1

˜

˚

F˜ pgptqqdt.

F

gptqdt ĺCW F pF Aq

0

7

(40)

0

Conclusion

In this presentation we introduce Theorems 5, 6, 7, 15, 16 and 17 that provide

new interval and fuzzy versions of Jensen’s integral inequality. Diﬀerent from the

Order Relations, Convexities, and Jensen’s Integral Inequalities

463

inequalities given in , these Jensen’s inequalities are extensions of the classical

Jensen’s integral inequality for real-valued functions, and these inequalities are

interpreted by means of order relations that allow us to compare non-nested

intervals as well as to compare non-nested fuzzy intervals. From analytic viewpoint, these inequalities allow one to obtain interval numeric estimations for a

class of integrals of interval-valued function and for a class of integrals of fuzzyinterval-valued functions (also interpreted as fuzzy interval expected values as

is done by de Barros et al. in  and by Puri and Ralescu in ) through the

values that the interval integrands and fuzzy integrands assume at real numbers, respectively. Our next step is to try to extend these inequalities for fuzzy

functions, that is, for applications of type F˜˜ : FC pRq Ñ FC pRq.

References

1. Aubin, J.P., Cellina, A.: Diﬀerential Inclusions: Set-Valued Maps and Viability

Theory. Grundlehren der mathematischen Wissenschaften. Springer, Heidelberg

(1984). https://doi.org/10.1007/978-3-642-69512-4

2. Aumann, R.J.: Integrals of set-valued functions. J. Math. Anal. Appl. 12(1), 1–12

(1965)

3. Bede, B.: Mathematics of Fuzzy Sets and Fuzzy Logic. Studies in Fuzziness and

Soft Computing, vol. 295. Springer, Heidelberg (2013). https://doi.org/10.1007/

978-3-642-35221-8

4. Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics, vol. 580. Springer, Berlin (1977). https://doi.org/10.

1007/BFb0087685

5. Chalco-Cano, Y., Lodwick, W.A., Ruﬁan-Lizana, A.: Optimality conditions of type

KKT for optimization problem with interval-valued objective function via generalized derivative. Fuzzy Optim. Decis. Mak. 12, 305–322 (2013)

6. Costa, T.M.: Jensen’s inequality type integral for fuzzy-interval-valued functions.

Fuzzy Sets Syst. 327, 31–47 (2017)

7. de Barros, L.C., Bassanezi, R.C., Lodwick, W.A.: A First Course in Fuzzy Logic,

Fuzzy Dynamical Systems, and Biomathematics: Theory and Applications. Studies

in Fuzziness and Soft Computing, vol. 347. Springer, Heidelberg (2017). https://

doi.org/10.1007/978-3-662-53324-6

8. Diamond, P., Kloeden, P.E.: Metric Spaces of Fuzzy Sets: Theory and Applications.

World Scientiﬁc, Singapore (1994)

9. Goetschel, R., Voxman, W.: Elementary fuzzy calculus. Fuzzy Sets Syst. 18(1),

31–43 (1986)

10. Ishibuchi, H., Tanaka, H.: Multiobjective programming in optimization of the interval objective function. Eur. J. Oper. Res. 48, 219–225 (1990)

11. Kaleva, O.: Fuzzy numbers fuzzy diﬀerential equations. Fuzzy Sets Syst. 24(3),

301–317 (1987)

12. Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliﬀs (1966)

13. Negoita, C.V., Ralescu, D.A.: Applications of Fuzzy Sets to Systems Analysis.

Wiley, New York (1975)

14. Puri, M.L., Ralescu, D.A.: Fuzzy random variables. J. Math. Anal. Appl. 114,

409–422 (1986)

Fuzzy Initial Value Problem:

A Short Survey

Marina Tuyako Mizukoshi(B)

Instituto de Matem´

atica e Estat´ıstica, Universidade Federal de Goi´

as,

Campus II - Samambaia, Goiˆ

ania, GO 74690-900, Brazil

tuyako@ufg.br

http://www.ime.ufg.br

Abstract. This article provides a survey of the available literature on

Fuzzy Initial Value Problem (FIVP) and various diﬀerent interpretations. The fuzzy diﬀerential equations can be studied using the derivative concept or without it. The Malthusian population model with fuzzy

initial condition is used to illustrate the diﬀerent approaches, namely,

Hukuhara derivative, gh-diﬀerentiability, π−derivative and Zadeh’s

extension applied to derivative operator using the diﬀerentiability and

diﬀerential inclusion theory, Zadeh’s extension principle applied in deterministic solution without derivative concept.

Keywords: FIVP · Diﬀerential inclusion

Stability · Hukuhara derivative

1

· Fuzzy diﬀerentiability

Introduction

The modeling of various phenomenon is frequently made by using deterministic

diﬀerential systems

(1)

x (t) = f (t, x(t)); x(0) = x0 ,

where x(t), x0 ∈ Rn and f : R × Rn → Rn is a function that satisﬁes some

existence conditions.

In practice, exact knowledge of the initial condition or parameters of (1)

may be unavailable, or diﬃcult to obtain. Generally, their values are imprecise

because they are either approximately known, or result from observations prone

to error. As Rouvray , “all scientiﬁc pronouncements have some inherent

uncertainty about them and cannot be assumed to be strictly valid”. A way

to address imprecise initial conditions is to rewrite (1) as Fuzzy Initial Value

Problem (FIVP)

x (t) = f (t, x(t)); X(0) = X0 ,

where X0 ∈ F(R) and f : [0, T ] × F(Rn ) → F(Rn ) with x(t) ∈ F(Rn ) or

f : [0, T ] × Rn → F(Rn ) for x(t) ∈ Rn .

c Springer International Publishing AG, part of Springer Nature 2018

G. A. Barreto and R. Coelho (Eds.): NAFIPS 2018, CCIS 831, pp. 464–476, 2018.

https://doi.org/10.1007/978-3-319-95312-0_40

Fuzzy Initial Value Problem: A Short Survey

465

Uncertainty was formally admitted into sciences about four centuries ago

and since then the modeling has been dominated by stochastic methods. In the

1930’s, diﬀerential inclusions theory was introduced by the Polish and French

mathematicians Zaremba  and Marchaud  as a generalization of diﬀerential equations by considering the uncertainty in direction the vector velocity.

They studied the so-called paratingent and contingent equations, respectively.

In 1962, Wazewski  proved that the solutions can be understood in the

Caratheodory sense of absolute continuity satisfying the diﬀerential inclusion

almost everywhere for all time.

Fuzzy set and possibility theory are notions that have introduced recently.

Interval analysis and fuzzy set theory emerged in 1959 and 1965, respectively

with Moore  and Zadeh . Subsequently in  was proposed the study of

the interval theory using Constraint Interval Arithmetic (CIA). According with

Lodwick and Dubois  interval analysis is not only useful but necessary to

the understanding of fuzzy interval analysis especially in the context of linear

systems.

The initial value problem is discussed in  and a comprehensive overview

of the computational aspects are given in . Diﬀerential inclusion in the framework of fuzzy set theory was ﬁrst discussed by Baidosov  as follows:

x (t) ∈ F (t, x(t)),

where the right side is a fuzzy multivalued function. Alternatively, Aubin 

assumes taht the right side of diﬀerential inclusion is a fuzzy set. Hullermeier

 has suggests to solve the FIVP looking at it as a family of diﬀerential

inclusions.

The term fuzzy diﬀerential equation was introduced in 1978 by Kandel and

Byatt  and an extended version of this short note was published two years

later . The concept of diﬀerentiability and integrability for fuzzy multivalued

were introduced by Puri and Ralescu . The Cauchy problem for ﬁrst-order

fuzzy diﬀerential equation was investigated by Kaleva , Seikkala , Ouyang

and Wu  using a extension of Hukuhara derivative.

Fuzzy diﬀerential equations can be studied from a point of view discrete

[2,27,28,42,43] or continuous [7,15,21,22,45]. In the continuous case we have

two diﬀerent approaches: in the ﬁrst one without the derivative concept by diﬀerential inclusion theory [13,14] or Zadeh’s extension principle [8,33]; in the second

case, the diﬀerentiability is considered, Hukuhara derivative , π−derivative

, gh-diﬀerentiability , extension in derivative operator . The spaces of

all closed and bounded interval of R are not linear spaces and therefore the

subtraction is not well deﬁned. As a consequence, alternative formulations for

subtraction have been suggested and so on there are diﬀerent deﬁnitions for the

diﬀerentiability in a fuzzy diﬀerential equation. The concept of fuzzy derivative,

leads the connection between interval and fuzzy theories. By considering the

generalized derivative some authors obtained both the solution to linear interval

systems and to fuzzy diﬀerential equations. Another point of view is the dynamic

systems obtained by means of a Mamdani type fuzzy rule-based system [19,38].

466

M. T. Mizukoshi

Oberguggenberger and Pittshmann [20,36] studied diﬀerential equations system

with fuzzy parameter by applying the Zadeh’s extension in equations and solution operators. Finally, fuzzy periodic solutions were studied in [10,11,14,35].

We consider the Fuzzy Initial Value Problem, in particular one dimensional

Malthusian model to concretize our review of the diﬀerent approaches to solution

and stability of fuzzy diﬀerential equations.

2

Basic Concepts

First of all, we provide some notation and recall known results.

We denote by Kn the family of all the nonempty compact subsets of Rn . For

A, B ∈ Kn and λ ∈ R the operations of addition and scalar multiplication are

deﬁned by

rA + B = {a + b| a ∈ A, b ∈ B}

λA = {λa | a ∈ A} .

Let X be metric space. A fuzzy subset U of X is given by a mapping μU :

X → [0, 1] such that the set of ordered pairs (x, μU (x)), x ∈ X indicates the

degree of each x in U. The degrees 0 and 1 represent, respectively, the nonbelonging and the maximum belonging of x to fuzzy subset U . To simplify the

notation we indicate the membership function μU by U.

Let U be a fuzzy set in Rn , the n-dimensional Euclidian space, we deﬁne

α

[U ] = {x ∈ Rn / U (x) ≥ α} the α-level of U , with 0 < α ≤ 1. For α = 0 we

have [U ]0 = supp(U ) = {x ∈ Rn | U (x) > 0}, the support of U .

A fuzzy set U is called compact if [U ]α ∈ Kn , ∀α ∈ [0, 1]. We will denote by

F(Rn ) the space of all the compact fuzzy sets whose α−level are compact and

connected set in Rn .

The operations of addition and scalar multiplication on F(Rn ) for all

α−levels are deﬁned by

and [λU ]α = λ[U ]α , ∀α ∈ [0, 1].

[U + V ]α = [U ]α + [V ]α ;

(2)

The metric on F(U ) is given by

d∞ (U, V ) = sup dH ([U ]α , [V ]α ),

0≤α≤1

where dH is the usual Pompieu-Hausdorﬀ metric deﬁned for compact subsets of

Rn . This metric turns the space (F(R), d∞ ) into a complete metric space .

Zadeh  proposed the so called extension principle , which became an

important tool in fuzzy set theory. The idea is that each function, f : X → Y,

induces a corresponding function f : F(X) → F(Y ) (i.e., f is a function mapping

fuzzy sets in X to fuzzy sets in Y ) deﬁned for each fuzzy set U in X by

sup

f (U )(y) =

u∈f −1 (y)

U (u), if f −1 (y) = ∅

0, if f −1 (y) = ∅.

(3)

Fuzzy Initial Value Problem: A Short Survey

467

The function f is said to be obtained from f by the extension principle.

An important result of extension principle is the characterization of the levels

of the image of a fuzzy set through f , where f is a continuous function.

Theorem 1 . If f : Rn −→ Rn is continuous, then the Zadeh’s extension

f : F(Rn ) −→ F(Rn ) is well-defined and

f (U )

α

= f ([U ]α ), ∀α ∈ [0, 1].

(4)

Relation (4) continues to be valid if f : W → Rn , and W is an open subset

in Rn . Moreover, according to Rom´an-Flores et al.  it was shown that f is

a continuous function with respect to Pompieu-Hausdorﬀ metric extended to

F(Rn ).

Let us consider the following diﬀerential inclusion,

x (t) ∈ F (t, x(t))

x(t0 ) = x0 ∈ X0

(5)

where F : [t0 , T ] × Rn → Kn is a set-valued function and X0 ∈ Kn .

A function x(t, x0 ) with the initial condition x0 ∈ X0 is a solution of (5) in

interval [t0 , T ] if it is absolutely continuous and satisﬁes (5) for all t ∈ [t0 , T ],

(for more details, see ). The attainable set in time t ∈ [t0 , T ], associated with

problem (5), is the subset of Rn given by

At (X0 ) = {x(t, x0 ) / x(·, x0 ) is solution of (5) with x0 ∈ X0 }.

The set-valued function F allows modeling of certain types of uncertainty 

and because for each pair (t, x) ∈ [t0 , T ] × Rn , the derivative may not be known

precisely, but known to be an element of the set F (t, x).

The following presents a survey of Fuzzy Initial Value Problems (FIVP) considering diﬀerential inclusion, extension principle and fuzzy diﬀerentiability.

3

FIVP and Some Interpretations

Consider the FIVP

X (t) = f (t, X(t))

X(0) = X0 .

(6)

To study the stability of solutions of (6) we need to understand what is a solution

of (6).

We consider the deterministic Malthusian problem

x (t) = −ax(t)

,

x(0) = x0 .

(7)

where 0 < a ∈ R and x : [0, T ] × R → R to illustrate the diﬀerent approach to

fuzzy diﬀerential equations.

468

3.1

M. T. Mizukoshi

FIVP with Diﬀerentiability

In this section we consider the problem 6 where is a fuzzy function f that indicates a fuzzy direction and the trajectory for FIVP are or diﬀerent deterministic

solutions with a membership degree to each of them or a function that assigns

to each instant t ∈ [0, T ] a fuzzy subset.

First Approach: The ﬁrst interpretation about (6) appears in 1987 with

Seikkala and Kaleva using the Hukuhara derivative. In this interpretation,

f : [0, T ] × F(Rn ) → F(Rn ), X0 ∈ F(Rn ) and the solution is a fuzzy-setvalued function X : [0, T ] → F(Rn ). Next, we present the concept established

by Seikkala  for F(R), which was used by author to rewrite problem (6) from

the one dimensional case into a bidimensional system of ordinary diﬀerential

equations.

Deﬁnition 1. Let I = [0, T ], T ∈ R+ be a real interval. The application X :

α

I −→ F(R) is called a fuzzy process. We denoted [X(t)]α = [xα

1 (t), x2 (t)], t ∈

I, α ∈ [0, 1]. The derivative X (t) of a fuzzy process X is defined by [X (t)]α =

α

[(xα

1 (t)) , (x2 (t)) ], 0 < α ≤ 1 provided that the equation defines a fuzzy number

X (t) ∈ F(R).

α

Let [X(t)]α = [xα

1 (t), x2 (t)] and f : [0, T ] × F(R) → F(R) is a continuous

mapping. Applying the Zadeh’s extension principle to f , (6) in the one dimensional case can be rewritten as:

α

α

α

α

(xα

1 ) (t) = f1 (x1 , x2 ), x1 (0) = x01

α

α

α

α

α

(x2 ) (t) = f2 (x1 , x2 ), x2 (0) = x02 ,

(8)

for t ∈ [0, T ) e α ∈ [0, 1], where

α

α

f1 (xα

1 , x2 ) = min{f (x)/x ∈ [X] }

α

α

α

f2 (x1 , x2 ) = max{f (x)/x ∈ [X] }.

Note that the fuzzy problem has been reduced to an initial value problem in R2 .

Example 1. : Suppose the size a population occurs in accordance the law of

Malthusian growth. Then, FIVP associated to (7) is given by:

X (t) = −aX(t)

X(0) =

X0 ,

(9)

where X(t), X0 ∈ F(R) , 0 < a ∈ R and the α−levels of X(t) are given by

α

[X]α = [xα

1 , x2 ].

Using the Hukuhara derivative it follows that: [X (t)]α = [(xα

1 ) (t),

α

(x2 ) (t)], α ∈ [0, 1], t ∈ [0, T ]. By Zadeh’s extension principle we have that

f (t, X(t)) = −aX(t) such that the α−levels are:

α

α

α

[f (t, X(t))]α = [min{−axα

1 (t), −ax2 (t)}, max{−ax1 (t), −ax2 (t)}], ∀α ∈

[0, 1], t ∈ [0, T ].

Fuzzy Initial Value Problem: A Short Survey

Thus for a > 0

⎧ α

⎨ (x1 ) (t) = −axα

2 (t)

α

)

(t)

=

−ax

(xα

1 (t)

⎩ α2

α

α

x1 (0) = x01 , x2 (0) = xα

02 , α ∈ [0, 1].

The solutions of (10) are given by

1 α

at

⎨ xα

(x − xα

1 (t) =

02 ) e +

2 01

⎩ xα (t) = + 1 (−xα + xα ) eat +

2

01

02

2

1 α

−at

x + xα

02 e

2 01

1 α

−at

(x + xα

02 ) e

2 01

469

(10)

(11)

Therefore, the solution X(t) of (9) has α−levels given by (11) for a ≥ 0.

α

α

α

at

Then, diam(xα

1 (t), x2 (t)) = |x01 − x02 |e , ∀α ∈ [0, 1], t > 0.

This means that, the solution is more fuzzy when t is increasing, implying

that we do not have the stability condition that we had in deterministic theory.

Then, we consider another type of diﬀerentiability in (6) because the

Hukuhara derivative leads to solutions with increasing support. A complete

review about as type of diﬀerentiability of fuzzy multivalued as compared solutions of fuzzy diﬀerential equations can be found in .

Second Approach: Strongly generalized diﬀerentiability was deﬁned by considering the lateral Hukuhara derivative (four cases) and a generalization is given

in , which is called weakly generalized diﬀerentiable. The advantage these definitions is that if g is diﬀerentiable on (a, b), then f : (a, b) → F(R) such that

f (x) = c g(x), ∀x ∈ (a, b) is the strongly generalized diﬀerentiable on (a, b)

and f (x) = c g (x). In this context the fuzzy diﬀerential equation has no

unique solution but we can choose among the solutions to ﬁnd one solution with

increasing or decreasing support. This feature allow us to choose singular points

where the solutions change monotonicity, such points are called switch points.

From Theorem 4.2.4 in , the problem (6) on some interval [t0 , t0 + k]

with X(t), X(t0 ) ∈ F(R) is the union of the following two ordinary diﬀerential

equations:

⎧ α

α

⎨ (x1 (t)) = f1α (t, xα

1 , x2 )

α

α

α

(x (t)) = f2 (t, x1 , xα

(12)

2)

⎩ α2

α

α

,

x

(t

)

=

x

x1 (t0 ) = xα

0

01

2

02

⎧ α

α

α

α

⎨ (x1 (t)) = f2 (t, x1 , x2 )

α

(xα (t)) = f1α (t, xα

(13)

1 , x2 )

⎩ α2

α

α

x1 (t0 ) = x01 , x2 (t0 ) = xα

02 ,

α

α

where were considered the following α−levels [X(t0 )]α = [xα

01 , x02 ], [X(t)] =

α

α

α

α

α

α

α

α

α

[x1 , x2 ], [f (t, X(t))] = [f1 (t, x1 , x2 ), f2 (t, x1 , x2 )] , α ∈ [0, 1].

Third Approach: Chalco-Cano et al.  studied fuzzy diﬀerential equations

using the π−derivative.The spaces of all closed and bounded interval of R

by using the Radstrăứm Embedding Theorem guarantee the existence of a

470

M. T. Mizukoshi

real normed linear space . The π-derivative for fuzzy interval valued functions is a generalization of π−derivative for set-valued mappings. The mapping X : [a, b] → F(R) is called a fuzzy function and [X(t)]α = X α (t) =

[f α (t), g α(t) ], t ∈ [a, b], 0 ≤ α ≤ 1. In this context, we have in (6)

that f : [0, T ] × F(R → F(R is a continuos function and X0 ∈ F(R). A

solution in this case is a fuzzy function x : [0, T ] → F(R which satisﬁes

α

α

=

the FIVP for each t ∈ [0, T ]. Then, if [X(t)]α = [xα

1 (t), x2 (t)], [X(t0 )]

α

α

α

[x01 (t), x02 (t)], [f (t, X(t))] =

α

α

α

α

α

α

α

[f1α (t, xα

1 (t), x2 (t)), f2 (t, x1 (t), x2 (t))] such that π([X(t)] ) = (x1 (t), x2 (t) −

α

α

α

α

α

α

α

x1 (t) = δ (t)), π([X(t0 )] ) = (x01 (t), x02 (t) − x01 (t) = δ0 (t)) and

α

α

α

α

π([f (t, X(t))]α ) = (g1α (t, xα

1 (t), δ (t)), g2 (t, x1 (t), δ (t))). Thus (6) is equivalent

to solving

α

α

α

(xα

1 ) (t) = g1 (t, x1 (t), δ (t))

.

(14)

α

α

α

(x2 ) (t) = g2 (t, x1 (t), δ α (t))

α

α

α

α

α

α

whose solution is [xα

1 (t), x1 (t) + δ (t)], if δ (t) > 0 and [x1 (t) + δ (t), x1 (t)], if

α

δ (t) < 0.

Fourth Approach: Actually, Gomes and Barros [16,17] studied (6) by considering that f is a function that indicates the direction of stable variable X

in a time t using extended derivative operator D. This approach is not equivalent to any another considered because as D is a fuzzy derivative, then we can

to consider as Hukuhara derivative as strongly generalized derivative. Besides,

provided some conditions it is possible to obtain the same solution via diﬀerential inclusion theory and Zadeh’s extension of the deterministic solution. In this

context, the FIVP (6) becomes

DX(t) = f (t, X(t))

X(0) = X0 , X0 , DX(t) ∈ F(Rn ).

(15)

Gomes et al.  proved that if f is continuous and has an unique solution

in deterministic autonomous IVP then the solution obtained via extension of

deterministic solution and D−derivative is the same. Besides, in some conditions

for f and X0 the FIVP has at least two solutions.

Example 2. Considering the problem (9) for a > 0 in context of the strong

generalized diﬀerentiability we have that (12) is the same that (10) in according

with Hukuhara diﬀerentiable solution. Now, for (13) we have

⎧ α

⎨ (x1 (t)) = −axα

1 (t)

α

(t))

=

−ax

(xα

(16)

2 (t)

⎩ α2

α

α

,

x1 (t0 ) = x01 , x2 (t0 ) = xα

02

α −at

α −at

, xα

such that

whose solution is xα

1 (t) = x01 e

2 (t) = x02 e

lim xα

2 (t) = 0.

t→+∞

lim xα

1 (t) =

t→+∞

Fuzzy Initial Value Problem: A Short Survey

471

Now, by considering the π−derivative, the solution for FIVP (9) with the

α

initial condition being a fuzzy number whose α−levels are given by [xα

01 , x01 ] is

α

(xα

1 ) (t) = −ax1 (t)

α

α

(δ ) (t) = −aδ (t))

.

α

α

α

α

10 (t0 ) = x01 , δ (t) = x02 − x01

(17)

−at

−at

with solution [X(t)]α = [xα

, xα

].

01 e

01 e

Finally, the solution of (10) via the interpretation of (6) considering the

extended derivative operator is

[DX(.)]α = D[X(.)]α = {Dx(.) : x(t) = x0 e−at , x0 ∈ [X0 ]α } = {ax(.) :

X(t) = x0 e−at , x0 ∈ [X0 ]α } = [aX(.)]α .

Therefore, in all this cases the diameter of solutions are decreasing.

3.2

FIVP Without Diﬀerentiability

In this section we study FIVP without the derivative concept. We consider uncertainties in modelling(coeﬃcient and/or initial condition) via IVP with fuzzy initial condition. The preference less or more about the trajectories are considered

by the value of its membership degrees. For each α− level of fuzzy initial condition we have a family of trajectories with the same membership degree.

First Approach: The interpretation for (6) is considering a family of dierential

inclusions. Hă

ullermeier  proposal that (6), where f : [0, T ] × Rn −→ F(Rn ) is

a fuzzy multivalued and X0 ∈ F(Rn ), can be rewritten as a family of diﬀerential

inclusions

x (t) ∈ [f (t, x(t))]α

(18)

x(0) ∈

[X0 ]α ,

where [f (t, x(t))]α e [X0 ]α are the α−levels of the fuzzy subsets f (t, x(t)) and X0 ,

respectively and x : I −→ Rn . For each α ∈ [0, 1], we say that x : [t0 , T ] −→ Rn ,

is an α-solution of (6) if it is a solution of (5). We will denote by At ([X0 ]α ) := Aα

t,

t0 ≤ t ≤ T, the attainable set of the α-solutions, that is,

α

α

t = At ([X0 ] ) = {x(t, x0 ) / x(., x0 ) is solution of (5) with x0 ∈ [X0 ] }.

According Gomes , the solution is a fuzzy bunch of functions X(t) ∈

F(AC([0, t]; Rn )) whose elements satisfy (18) a.e. in [0, t]. But, here we don’t

have the fuzzy derivative concept in equations of (6).

Diamond, in , uses the Representation Theorem to prove that Aα

t are

the α-levels of a fuzzy set At (X0 ) in Rn for all t0 ≤ t ≤ T . The fuzzy set

At (X0 ) will be said to be the attainable set of problem (6). In  the author

studied Lyapunov stability and periodicity of the fuzzy solution set for both the

time-dependent and autonomous case.

Second Approach: Oberguggenberger and Pittschmann  studied (6) when

the coeﬃcients and initial conditions are fuzzy subsets. The authors deﬁne the

equation, restriction and solution for

x (t) = f (x)

x(0) = x0 , x0 ∈ Rn ,

(19)

472

M. T. Mizukoshi

and apply the Zadeh’s extension principle in operator to obtain a solution for

(6). In addition, they establish in formal way the concepts for fuzzy solution

and fuzzy componentwise solution. Also, in this interpretation we don’t have

the fuzzy derivative concept and f : [0, T ] × Rn → F(Rn ) was obtained from a

continuous function g : [0, T ] × X → Rn by applying Zadeh’s extension principle

and X0 ∈ F(Rn ).

Buckley et al.  in similar way to idea of Oberguggenberger and Pittschmann

also obtained solutions for (6) by fuzzifying the deterministic solution using the

Mizukoshi et al.  was prove that the solution obtained via family of diﬀerential inclusions for (6) and the solution via Zadeh’s extension principle of deterministic solution is equivalent in some conditions. In the context of Mizukoshi

et al., if the coeﬃcients and/or initial condition are fuzzy subsets then X(t, X0 )

is a solution for (6) by applying the Zadeh’s extension principle in deterministic

solution. The point of view in [32,33] was that if the solutions satisfy the concept

of ﬂow, then we can to stablish results equilibrium and stability (for more details

see ).

Let ϕt (x0 ) be the solution (unique) of (19) for each x0 in time t, deﬁned

on its maximal interval of existence I(x0 ). For each t ∈ I(x0 ), the family of

mappings ϕt : X −→ X deﬁned by ϕt (x0 ) = ϕ(t, x0 ) such that ϕ0 = I, where

I is the identity mapping on X and ϕt+s = ϕt ◦ ϕs , t, s ∈ R+ , where “◦” is the

composition operation.

The mapping ϕt : F(X) → F(X) obtained by applying the Zadeh’s extension

principle on the initial condition in ϕt : X → X is a fuzzy ﬂow for (6).

¯ = X,

¯ ∀t ≥

¯ ∈ F(X) is a fuzzy equilibrium for the FIVP if ϕt (X)

Recall that X

¯ α = [X]

¯ α , ∀α ∈ [0, 1].

0 or equivalentely, ϕt (X)]

¯ be an equilibrium point of the FIVP (6). We say that,

Let X

¯ is stable if and only if for all ε > 0, there is a δ > 0, such that, for every

(a) X

¯ < δ, then D(ϕt (X), X)

¯ < ε, ∀t ≥ 0.

X ∈ F(Rn ) with D(X, X)

¯

(b) X is asymptotically stable if it is stable and, in addition, ∃r > 0 such that

¯ = 0, for all X satisfying D(X, X)

¯ < r.

lim D(ϕt (X), X)

t→+∞

Example 3. Consider the fuzzy Malthusian problem

x (t) = ax(t)

x(0) = X0 ,

(20)

where X0 ∈ F(R), a, x(t) ∈ R.

In according Hullermeier’s interpretation, the FIVP (9) is the following family

of diﬀerential inclusions:

X (t) ∈ [−aX(t)]α

(21)

X(0) ∈ [X0 (t)]α

where X : [0, T ] × R → F(R) and X0 is a a fuzzy number. ### Tài liệu bạn tìm kiếm đã sẵn sàng tải về

2 Some Jensen's Inequalities for Fuzzy-Interval-Valued Function

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