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WP-EMS Working Papers Series in Economics, Mathematics and Statistics

“SOME NOTES ON GENERALIZED HUKUHARA DIFFERENTIABILITY OF INTERVAL-VALUED FUNCTIONS AND INTERVAL DIFFERENTIAL EQUATIONS”

• Luciano Stefanini, (U. Urbino) • Barnabás Bede, (DigiPen Inst. of Technology, Redmond, Washington, USA)

WP-EMS # 2012/8

Some notes on generalized Hukuhara di¤erentiability of interval-valued functions and interval di¤erential equations Luciano Stefanini, Barnabás Bede Faculty of Economics, University of Urbino Carlo Bo, Italy Department of Mathematics, DigiPen Institute of Technology, Redmond, Washington, USA Email adresses: [email protected], [email protected]

Abstract Our paper "Generalized Hukuhara Di¤erentiability of Interval-valued Functions and Interval Di¤erential Equations", appeared in Nonlinear Analysis 71 (2009), contains some imprecisions. In this note we correct them and give few new results on the topic.

1

Introduction

In this note we give a brief account of the new concepts of interval (and fuzzy) generalized di¤erence (gH-di¤erence) and generalized di¤eretiability (gH-di¤erentiability and gH-derivative) for interval-valued functions, introduced in [32] and [33] and in [34]. Our paper [34], "Generalized Hukuhara Di¤erentiability of Interval-valued Functions and Interval Di¤erential Equations", appeared in Nonlinear Analysis 71 (2009), contains some imprecisions. In this note we correct them and give few new results on the topic. These new generalized derivatives are motivated by their usefulness in a very quickly developing area at the intersection of set-valued analysis and fuzzy sets, namely, the area of fuzzy analysis and fuzzy di¤erential equations. It is well-known that the usual Hukuhara di¤erence between two fuzzy numbers exists only under very restrictive conditions [9], [13]. The gH-di¤erence of two fuzzy numbers exists under much less restrictive conditions, however it Preprint submitted to Elsevier Science

12th June 2012

does not always exist [31], [32]. The g-di¤erence proposed in [33] overcomes these shortcomings of the above discussed concepts and the g-di¤erence of two fuzzy numbers always exists. The same remark is valid if we regard di¤erentiability concepts in fuzzy setting. These concepts are connected to the ideas presented in [17], [18], [21]. The note is organized as follows; section 2 introduces the de…nitions and key results on generalized di¤erence and generalized Hukuhara derivative of an interval (fuzzy) valued function. In section 3 we correct some imprecisions published in [34].

2

gH-di¤erence and gH-derivative

One of the …rst de…nitions of di¤erence and derivative for set-valued functions was given by Hukuhara [12] (H-di¤erence and H-derivative); it has been extended to the fuzzy case in [27] and applied to fuzzy di¤erential equations (FDE) by many authors in several papers. But the H-derivative in FDE su¤ers certain disadvantages (see [2], [6], [7], [9]) related to the properties of the space Kn of all nonempty compact sets of Rn and in particular to the fact that Minkowski addition does not possess an inverse subtraction. On the other hand, a more general de…nition of subtraction for compact convex sets, and in particular for compact intervals, has been introduced by several authors. Markov [17], [18], [21] de…ned a non-standard di¤erence, also called inner-di¤erence, and extended its use to interval arithmetic and to interval calculus, including interval di¤erential equations (see [19], [20]). In the setting of Hukuhara di¤erence, the interval and fuzzy generalized Hukuhara di¤erences have been recently examined in [32], [33]. Let KCn be the space of nonempty compact and convex sets of Rn . The set of real intervals will be denoted by I (i.e., I = KC1 ). The Hukuhara H-di¤erence has been introduced as a set C for which A•H B = C () A = B + C and an important property of •H is that A •H A = f0g 8A 2 KCn and (A+B)•H B = A, 8A; B 2 KCn . The H-di¤erence is unique, but it does not always exist (a necessary condition for A •H B to exist is that A contains a translate fcg+B of B). A generalization of the Hukuhara di¤erence aims to overcome this situation. The generalized Hukuhara di¤erence of two sets A; B 2 KCn (gH-di¤erence for short) is de…ned as follows A •gH B = C ()

8 >
: or (b) B = A + ( 1)C

2

(1)

The inner-di¤erence in [21], denoted with the symbol " introducing the inner-sum of A and B by A+ B = and then

", is de…ned by …rst

8 > < X if X solves ( A) + X = B > : Y if Y solves ( B) + Y = A

(2)

A B = A + ( B). (3) It is not di¢ cult to see that A gH B = A B; in fact, A + ( B) = C means ( A) + C = ( B) i.e. case (b) of (1), or ( ( B)) + C = A i.e. case (a) of (1). In case (a) of (1) the gH-di¤erence is coincident with the H-di¤erence. Thus the gH-di¤erence, or the inner-di¤erence, is a generalization of the H-di¤erence. The gH-di¤erence (1) or, equivalently, the inner-di¤erence (3) for intervals or for compact convex sets is the basis for the de…nition of a new di¤erence in the fuzzy context. Real intervals will be denoted A = [a ; a+ ]; B = [b ; b+ ]; = [c ; c+ ] with a a+ ; b b+ ; c c+ , etc. The following properties were obtained in [33]. Proposition 1. ([33]) Let A; B be two intervals; then i) if the gH-di¤erence exists, it is unique; ii) A •gH B = A •H B or A •gH B = (B •H A) whenever the expressions on the right exist; in particular, A •gH A = A •H A = 0; iii) if A •gH B exists in the sense (i), then B •gH A exists in the sense (ii) and vice versa, iv) (A + B) •gH B = A, v) 0 •gH (A •gH B) = B •gH A; vi) A •gH B = B •gH A = C if and only if C = C; furthermore, C = 0 if and only if A = B. Proposition 2. The gH-di¤erence A gH B is the smallest interval C in the sense of inclusion such that A B + C and B A C. PROOF. Let A gH B = C; the inclusions A B + C and B A C follow from the de…nition of gH . The minimality of C follows from the cancellation for addition of intervals: A B () A + X B + X. The following property is interesting, related to multiple application of gHdi¤erences. 3

+ Proposition 3. Let Ak = [ak ; a+ k ] and Bk = [bk ; bk ] be given intervals for k = 1; 2; :::; n; if the n gH-di¤erences Ck = Ak gH Bk are all of the same type (i) (i.e., Ak = Bk +Ck for all k) or all of the same type (ii) (i.e., Bk = Ak Ck for all k), then we have n X

Ak

k=1

!

gH

n X

!

Bk =

k=1

n X

(Ak

gH

Bk )

k=1

PROOF. Assume …rst that Ak = Bk + Ck for all k (case (i)); then n P

k=1 n P

k=1 n P

Bk

(Ak

k=1

n P

(Bk + Ck ) = =

n P

k=1

n P

Ck so that, by de…nition of

k=1

Ck . If Bk = Ak

k=1

Ck ) =

Bk +

n P

k=1

Ak

n P

gH ,

n P

k=1

Ck for all k (case (ii)), then

Ck so that,

k=1

n P

k=1

Ak

gH

n P

k=1

Bk =

n P

Ak =

k=1

Ak n P

gH

Bk =

k=1 n P

Ck .

k=1

Generalized di¤erentiability concepts were …rst considered for interval-valued functions in the works of Markov ([19], [20]). This line of research is continued by several papers, e.g. [2], [8], [26], [34], dealing with interval and fuzzy-valued functions. Based on the gH-di¤erence we obtain the following de…nition (for intervalvalued functions, the same de…nition was suggested in [20] using inner-di¤erence): De…nition 4. Let t0 2]a; b[ and h be such that t0 + h 2]a; b[, then the gHderivative of an interval-valued function f :]a; b[! I at t0 is de…ned as 1 0 fgH (t0 ) = lim [f (t0 + h) •gH f (t0 )]: h!0 h

(4)

0 If fgH (t0 ) 2 I satisfying (4) exists, we say that f is generalized Hukuhara di¤erentiable (gH-di¤erentiable for short) at t0 .

Example 5. Let f (t) = p(t)A where p is a crisp di¤erentiable function and A 2 I, then it follows relatively easily that the gH-derivative exists and it is 0 fgH (t) = p0 (t)A. The next result gives the expression of the gH-derivative in terms of the derivatives of the endpoints of the intervals and it is a characterization of the gH-di¤erentiability for an important class of interval-valued functions. Theorem 6. Let f :]a; b[! I be such that f (t) = [f (t); f + (t)]. Suppose that the functions f (t) and f + (t) are real-valued functions, di¤erentiable w.r.t. t. 4

Then the function f (t) is gH-di¤erentiable at a …xed t 2]a; b[ and we have 0

0 fgH (x) = [minf f

0

0

(t); f + (t)g; maxf f

0

(t); f + (t)g]

(5)

According to Theorem 6, for the de…nition of gH-di¤erentiability when f (x) and f + (x) are both di¤erentiable, we distinguish two cases.. De…nition 7. Let f : [a; b] ! I and t0 2]a; b[ with f (t) and f + (t) both di¤erentiable at t0 . We say that f is (i)-gH-di¤erentiable at t0 if 0 (t0 ) = [ f fgH

(i.) -

0

0

(t0 ); f + (t0 )]

(6)

f is (ii)-gH-di¤erentiable at t0 if 0

0

0 fgH (t0 ) = [ f + (t0 ); f

(ii.)

(7)

(t0 )]:

It is possible that f : [a; b] ! I is gH-di¤erentiable at t0 and not (i)-gHdi¤erentiable nor (ii)-gH-di¤erentiable, as illustrated by the following example, taken from [29]. Example 8. Consider f :]

1; 1[ ! I de…ned by

f (t) =

"

1 1 ; 1 + jtj 1 + jtj

#

(8)

1 1 and f + (x) = (1+jxj) : For all t 6= 0, both f and f + are i.e. f (x) = (1+jxj) di¤erentiable and satisfy conditions of Theorem 6; at the origin t = 0 the two functions f and f + are not di¤erentiable; they are, respectively, left and right di¤erentiable but left derivative and right derivative are di¤erent, in fact

(f )0 (t) =

8 > > > > >
> > > > :

1 (1+t)2

8 > > > > >
0

Now, for the gH-di¤erence and h 6= 0 we have f (h)

"

> > > > :

#

t0

1 1 1 = ; gH [ 1; 1] h h (1 + jhj) (1 + jhj) " ( ) # 1 jhj jhj = min ; ; maxfidemg h (1 + jhj) (1 + jhj) " # " # 1 jhj jhj 1 1 = ; = ; h (1 + jhj) (1 + jhj) (1 + jhj) (1 + jhj) gH

f (0)

1 (1 t)2 (1+ )

5

It follows that the limit exists 0 fgH (0) = lim

f (h)

gH

h

h !0

f (0)

= [ 1; 1]

and f is gH-di¤erentiable at t = 0 but f and f + are not di¤erentiable at t = 0; observe that f is (i)-gH-di¤erentiable if t < 0 and is (ii)-gH-di¤erentiable if t > 0. The following properties are obtained from Theorem 6. Proposition 9. If f : [a; b] ! I is gH-di¤erentiable (or right or left gHdi¤erentiable) at t0 2 [a; b] then it is continuous (or right or left continuous) at t0 . Proposition 10. The (i)-gH-derivative and (ii)-gH-derivative are additive operators, i.e., if f and g are both (i)-gH-di¤erentiable or both (ii)-gH-di¤erentiable then 0 0 (i) (f + g)0(i) gH = f(i) gH + g(i) gH , 0 0 (ii) (f + g)0(ii) gH = f(ii) gH + g(ii) gH .

PROOF. Consider (i) and suppose that f and g are both (i)-gH-di¤erentiable; then we have, f 0 = [(f )0 ; (f + )0 ] and g 0 = [(g )0 ; (g + )0 ] with (f )0 (f + )0 and (g )0 (g + )0 ; it follows that (f + g)0 = [(f + g ); (f + + g + )]0 = [(f + g )0 ; (f + + g + )0 ] = [(f )0 + (g )0 ; (f + )0 + (g + )0 ] = [(f )0 ; (f + )0 ] + [(g )0 ; (g + )0 ] = f 0 + g0; the case of f and g both (ii)-gH-di¤erentiable is similar. Remark 11. From Proposition 10, it follows that (i)-gH-derivative and (ii)gH-derivative are semi-linear operators (i.e. additive and positive homogeneous). They are not linear in general since we have (kfgH )0(i) gH = k(fgH )0(ii) gH , if k < 0.

3

Corrections to published results

We remark the following errors and imprecisions in our paper [34] (the notation for gH-di¤erence and gH-derivative in [34] was g-di¤erence and g-derivative; 6

after that paper, we have changed notation due to subsequent generalizations of the concepts, in particular in the setting of fuzzy-valued functions). 1. The only-if part of Theorem 17 is incorrect, as is illustrated in section 2. The correct statement of the theorem is Theorem 17. Let f : [a; b] ! I be such that f (x) = [f (x); f + (x)] with f (x) and f + (x) di¤erentiable real-valued functions; then function f (x) is gH-di¤erentiable and f 0 (x) = [minf f

0

0

0

(x); f + (x)g; maxf f

0

(x); f + (x)g].

((9))

Proof. In the case of interval-valued functions, the gH-di¤erence always exists. Analyzing all the possible cases of existence of the gH-di¤erences to the left and right we obtain the statement, i.e., if f (x) and f + (x) are di¤er0 0 entiable then f is gH-di¤erentiable and f 0 (x) = [minf(f ) (x); (f + ) (x)g; 0 0 maxf(f ) (x); (f + ) (x)g].

The error in Theorem 17 has been propagated in remark 22: the assumption 0 that f (x) and f + (x) are di¤erentiable (or that fb0 (x) and f (x) exist) is necessary for the validity of equations (12) and (13). The same error is also propagated into the proof of Proposition 24, where 0 the di¤erentiability of fb0 (x) and f (x) is assumed. In [1], a counterexample shows that the superadditivity property is not valid; indeed, the superadditivity property is to be reversed into subadditivity. The correct proposition is Proposition 24. Under the assumptions of Theorem 17, the gH-derivative is a homogeneous and sub-additive operator, i.e., for gH-di¤erentiable functions f; g : [a; b] ! I with di¤erentiable f , g , f + and g + ( f )0 = f 0 for (f + g)0 f 0 + g 0 .

2R

Proof. Using the midpoint representation (we omit reference to D x for simpliE 0 b city of notation), f = ( f ; j jf ) and, by equation (3), ( f ) = ( fb)0 ; j( f )0 j = D

0

E

fb0 ; j jjf j =

D

E

D

0 fb0 ; jf j = f 0 . For the inclusion, we have f +g = fb + gb; f + g

D

E

D

0

E

0

so that (f + g)0 = (fb + gb)0 ; j(f + g)0 j = fb0 + gb0 ; jf + g 0 j ; but jf + g 0 j 0 jf j + jg 0 j so that fb0 + gb0

0

jf + g 0 j

fb0 + gb0 7

0

jf j

0 jg 0 j and fb0 + gb0 + jf + g 0 j

E

0

0

0

fb0 + gb0 +jf j+jg 0 j. Recalling (13) (f +g)0 = [fb0 + gb0 jf +g 0 j; fb0 + gb0 +jf +g 0 j] 0 0 [fb0 + gb0 jf j jg 0 j; fb0 + gb0 + jf j + jg 0 j] = f 0 + g 0 . 2. An error appears in the last part of Theorem 30, where we say that "... Rb 0 f (x)dx = f (b) •g f (a) if and only if f (ci ) is crisp for i = 1; 2; :::; n..."; the a correct result is the following (and the only-if part is not valid): Theorem 30. Let us suppose that function f is GH-di¤erentiable with switching points at a < c1 < c2 < ::: < cn < b and exactly at these points. Then we have ( c0 = a and cn+1 = b for simplicity of notation) f (b) •g f (a) =

n X

"

Rci

i=1 ci

1

f (x)dx •g ( 1) 0

ci+1 R

0

f (x)dx

ci

#

((14))

Also, Rb

f 0 (x)dx =

a

n+1 X

[f (ci ) •g f (ci 1 )]

i=1

((15))

R

and if f (ci ) is crisp for i = 1; 2; :::; n then ab f 0 (x)dx = f (b) f (a). The last di¤erence and summations in (14)-(15) are to be intended in the interval arithmetic (Minkowski) sense. Proof. The proof of (14) and (15) is the same as in [34]. To prove that Rb 0 f (a) if f (ci ) is crisp for i = 1; 2; :::; n , consider that a f (x)dx = f (b) Z

b

0

f (x)dx =

a

Z

c

0

f (x)dx +

Z

c

a

b

f 0 (x)dx = (f (c) •g f (a)) + (f (b) •g f (c)) D

and, if f (c) = hf (c); 0i is a singleton, then f (c) •g f (a) = f (c) D

and f (b) •g f (c) = fb(b) D

f (c)) = fb(b) addition.

E

E

fb(a); f (a)

f (c); f (b) ; it follows that (f (c) •g f (a))+ (f (b) •g E

fb(a); f (a) + f (b) = f (b)

f (a), the standard Minkowski

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