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Fuzzy Ostrowski type inequalities

Abstract

We present optimal upper bounds for the deviation of a fuzzy continuous function from its fuzzy average over [a,b] <FONT FACE=Symbol>Ì</FONT> R, error is measured in the D-fuzzy metric. The established fuzzy Ostrowski type inequalities are sharp, in fact attained by simple fuzzy real number valued functions. These inequalities are given for fuzzy Hölder and fuzzy differentiable functions and these facts are reflected in their right-hand sides.

Fuzzy inequalities; Ostrowski inequality; Fuzzy real analysis


Fuzzy Ostrowski type inequalities

George A. Anastassiou

Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152 U.S.A., E-mail: ganastss@memphis.edu

ABSTRACT

We present optimal upper bounds for the deviation of a fuzzy continuous function from its fuzzy average over [a,b] Ì R, error is measured in the D-fuzzy metric. The established fuzzy Ostrowski type inequalities are sharp, in fact attained by simple fuzzy real number valued functions. These inequalities are given for fuzzy Hölder and fuzzy differentiable functions and these facts are reflected in their right-hand sides.

Mathematical subject classification: 26D07, 26D15, 26E50.

Key words: Fuzzy inequalities, Ostrowski inequality, Fuzzy real analysis.

0 Introduction

Ostrowski inequality (see [8]) has as follows

where f Î C1 ([a,b]), x Î [a,b]. Inequality (*) is sharp, see [1].

Since 1938 when A. Ostrowski proved his famous inequality, see [8], many people have been working about and around it, in many different directions and with a lot of applications in Numerical Analysis and Probability, etc.

One of the most notable works extending Ostrowski's inequality is the work of A.M. Fink, see [6]. The author in [1] continued that tradition.

This current article is mainly motivated by [1], [6], [8], [11] and extends Ostrowski type inequalities into the fuzzy setting, as fuzzyness is a natural reality genuine feature different than randomness and determinism. To the best of our knowledge this is the first attempt of such extension into the fuzzy environment, hoping to find wide continuations and lots of applications.

1 Background

We start with

Definition 1 (see [10]). Let m: ® [0,1] with the following properties:

(i) is normal, i.e., $x0Î ; m(x0) = 1.

(ii) m(lx + (1 – l)y) > min{m(x), m(y)}, "x,y Î , "l Î [0,1] (m is called a convex fuzzy subset).

(iii) m is upper semicontinuous on

, i.e., "x0Î and " e > 0, $ neighborhood V(x0): m(x) < m(x0) + e, "x Î V(x0).

(iv) The set

is compact in
(where supp(m) := { x Î ; m(x) > 0}).

We call m a fuzzy real number. Denote the set of all m with .

E.g.,

{x0}Î , for any x0Î , where {x0} is the characteristic function at x0.

For 0 < r < 1 and m Î define [m]r := { x Î : m(x) > r} and

Then it is well known that for each r Î [0,1], [m]r is a closed and bounded interval of . For u,v Î and l Î , we define uniquely the sum u Å v and the product l u by

where [u]r + [v]r means the usual addition of two intervals (as subsets of ) and l[u]r means the usual product between a scalar and a subset of (see, e.g., [10]). Notice 1 u = u and it holds u Å v = v Å u, l u = u l. If 0 < r1< r2< 1 then [u]r2Í [u]r1. Actually [u]r = , where Î , "r Î [0,1].

Define

by

where [v]r = ; u,v Î . We have that D is a metric on . Then (, D) is a complete metric space, see [10], with the properties

We need

Lemma 1 (Lemma 2.2 of [5]). For any a,b Î : a,b > 0 and any u Î we have

where õ Î is defined by õ := {0}.

We also need

Definition 2 (see [10]). Let x,y Î . If there exists a z Î such that x = y + z, then we call z the H-difference of x and y, denoted by z := x – y.

Definition 3 (Definition 3.3 of [10]). Let T := [x0, x0 + b] Ì , with b > 0. A function f : T ® is H-differentiable at x Î T if there exists a f'(x) Î such that the limits (with respect to metric D)

exist and are equal to f'(x). We call f' the derivative or H-derivative of f at x. If f is H-differentiable at any x Î T, we call f differentiable or H-differentiable and it has H-derivative over T the function f'.

The last definition was given first by M. Puri and D. Ralescu [9].

We use a particular case of the Fuzzy Henstock integral (d(x) = ) introduced in [10], Definition 2.1.

That is,

Definition 4 (Definition 13.14 of [7], p. 644). Let f : [a,b] ® . We say that f is Fuzzy-Riemann integrable to I Î if for any e > 0, there exists d > 0 such that for any division P = {[u,v];x} of [a,b] with the norms D(P) < d, we have

where å* denotes the fuzzy summation. We choose to write

We also call an f as above (FR)-integrable.

Corollary 1 (Corollary 13.2 of [7]). If f Î C([a,b],) then f is (FR) integrable on [a,b].

We also need

Lemma 2 (Lemma 1 of [2]). If f,g : [a,b] Í ® are fuzzy continuous (with respect to metric D), then the function F : [a,b] ® +È {0} defined by F(x) := D(f(x), g(x)) is continuous on [a,b], and

We mention

Lemma 3 (Lemma 3 of [2]). Let f : [a,b] Í ® be fuzzy continuous. Then

We use the Fuzzy Taylor formula.

Theorem 1 (Theorem 1 of [2]). Let T := [x0 ,x0 + b] Ì , with b > 0. We assume that f(i) : T ® are H-differentiable for all i = 0,1,...,n – 1, for any x Î T. (I.e., there exist in

the H-differences f(i)(x + h) – f(i)(x), f(i)(x) – f(i)(x – h), i = 0,1,...,n – 1 for all small h : 0 < h < b. Furthermore there exist f(i+1)(x) Î such that the limits in D-distance exist and

for all i = 0,1,...,n – 1.) Also we assume that f(n), is fuzzy continuous on T. Then for s > a, s,a Î T we obtain

where

Here Rn(a,s) is fuzzy continuous on T as a function of s.

We use

Proposition 1 (Proposition 1 of [4]). Let F(t) := tn

u, t > 0, n Î , and u Î be fixed. Then (the H-derivative)

In particular when n = 1 then F'(t) = u.

We mention

Proposition 2 (Proposition 6 of [4]). Let I be an open interval of

and let f : I ® be H-fuzzy differentiable, c Î . Then

We use the "Fuzzy Mean Value Theorem".

Theorem 2 (Theorem 1 of [4]). Let f : [a,b] ® be a fuzzy differentiable function on [a,b] with H-fuzzy derivative f' which is assumed to be fuzzy continuous. Then

for any c,d Î [a,b] with d > c.

We finally need the "Univariate Fuzzy Chain Rule".

Theorem 3 (Theorem 2 of [4]). Let I be a closed interval in

. Here g : I ® z := g(I) Í is differentiable, and f : z ®
is H-fuzzy differentiable. Assume that g is strictly increasing. Then (f ° g)'(x) exists and

2 Results

We give the following

Theorem 4. Let f Î C([a,b] ,), the space of fuzzy continuous functions, x Î [a,b] be fixed. We assume that f fulfills the Hölder condition

for some Lf > 0. Then

Proof. We have that

Optimality of (1) comes next.

Proposition 3. Inequality (1) is sharp, in fact, attained by f*(y) := |y – x|a u, 0 < a<1, with u Î fixed. Here x,y Î [a,b].

Proof. Clearly f*Î C([a,b],): for letting yn ® y, yn Î [a,b], then

Furthermore

That is, for Lf* := D(u,õ) we get

So that f* is a Hölder function.

Finally we have

Next comes the basic Ostrowski type fuzzy result in

Theorem 5. let f Î C1([a,b],), the space of one time continuously differentiable functions in the fuzzy sense. Then for x Î [a,b],

Inequality (2) is sharp at x = a, in fact attained by f*(y) := (y – a)(b – a) u, u Î being fixed.

Proof. We observe that

proving (2).

By Propositions 1, 2 and Theorem 3 we get that f*'(y) = (b – a) u. We have that

And

That is equality in (2) is attained.

We conclude with the following Ostrowski type inequality fuzzy generalization in

Theorem 6. Let f Î Cn+1([a,b],), n Î , the space of (n + 1) times continuously differentiable functions on [a,b] in the fuzzy sense. Call

Then

If f(i)(a) = õ, i = 1,...,n. Then

Inequalities (3) and (4) are sharp, in fact attained by

being fixed.

Corollary 2. Let f Î C2([a,b],). Then

When f'(a) = õ, then

Proof of Theorem 6. Let x Î [a,b], then by Theorem 1 we get

where

(here we need x > a). We observe that

We have established inequalities (3) and (4).

Consider g(x) :=

u, x Î [a,b], c > 0, Î +, u Î fixed. We prove that g is fuzzy continuous. Let xn Î [a,b] such that xn ® x as n ® +¥. Then

Hence by the last argument, Propositions 1, 2 and Theorem 3 we obtain that f*Î Cn+1([a,b],).

We see that

That is M = 0. Furthermore it holds

Finally, we notice that

Also we find

Proving (3) and (4) sharp, in fact attained inequalities.

Received: 19/IX/02.

#556/02.

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  • [9] M.L. Puri and D.A. Ralescu, Differentials of fuzzy functions, J. of Math. Analysis and Appl., 91 (1983), 552-558.
  • [10] Congxin Wu and Zengtai Gong, On Henstock integral of fuzzy number valued functions (I), Fuzzy Sets and Systems, 120, No. 3 (2001), 523-532.
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Publication Dates

  • Publication in this collection
    19 July 2004
  • Date of issue
    2003

History

  • Received
    19 Sept 2002
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