Ostrowski type inequalities for interval-valued functions using generalized Hukuhara derivative

Ostrowski type inequalities for interval-valued functions using generalized Hukuhara derivative^*

Y. Chalco-Cano; A. Flores-Franulič; H. Román-Flores

Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica-Chile. E-mails: [e-mail] / [e-mail] / [e-mail]

ABSTRACT

The present paper is devoted to obtaining some Ostrowski type inequalities for interval-valued functions. In this context we use the generalized Hukuhara derivative for interval-valued functions. Also some examples and consequences are presented.

Mathematical subject classification: Primary: 26E25; Secondary: 35A23.

Key words: Ostrowski type inequalities, interval-valued functions, gH-differentiability and integrability of interval-valued functions.

1 Introduction

The importance of the study of set-valued analysis from a theoretical point of view as well as from their application is well known [5,7] . Many advances in set-valued analysis have been motivated by control theory and dynamical games [6] . Optimal control theory and mathematical programming were a motivating force behind set-valued analysis since the sixties [6] . Interval Analysis is a particular case and it was introduced as an attempt to handle interval uncertainty that appears in many mathematical or computer models of some deterministic real-world phenomena. The first monograph dealing with interval analysis was given by Moore [14] . Moore is recognized to be the first to use intervals in computational mathematics, now called numerical analysis. He also extended and implemented the arithmetic of intervals to computers. One of his major achievements was to show that Taylor series methods for solving differential equations not only are more tractable, but also more accurate [15] .

The following inequality is known in the literature as Ostrowski's inequality

where f ∈ C¹([a,b] ), x ∈ [a, b] . Inequality (1) is sharp, see [3] . Since 1938 when A. Ostrowski (see [16] ) presented his famous inequality many researchers have been working about and around it, in many different directions and with a lot of applications. In the book edited by Dragomir and Rassias [11] and recently in the book of Anastassiou [1] are given a brief review of state of art about Ostrowski type inequalities and its applications.

Continuing that tradition, in [2] the Ostrowski type inequality has been extended to context of fuzzy-valued functions. In this context has been used the concept of Hukuhara-derivative for fuzzy-valued functions. Note that interval-valued functions are fuzzy-valued functions. Thus, the fuzzy Ostrowski type inequalities obtained in [2] is valid for interval-valued functions. However, the concept of H-derivative for interval-valued functions is very restrictive, see [8,9] . Generalized Hukuhara differentibility it is the most general differentiability concept for interval-valued functions, see [8,9,19] .

Motivated by [1,2,3,11] and [8,9,13,19] we extend Ostrowski type inequality (1) for gH-differentiable interval-valued functions.

2 Basic concepts

Let be the one-dimensional Euclidean space. Following [10] , let _C denote the family of all non-empty compact convex subsets of , that is,

The Hausdorff metric H on

_C is defined by

where

It is well known that (

_C, H) is a complete metric space (see [5, 10] ). The Minkowski sum and scalar multiplication are defined by

The space

_C is not a linear space since it does not possess an additive inverse and therefore subtraction is not well defined (see [5,9,10,19] ). Actually, _C is a quasilinear space [4,17] .

A crucial concept in obtaining a useful working definition of derivative for interval-valued functions is considering a suitable difference between two intervals. Toward this end, one way is to use (2) by requiring

However, this definition of difference has the drawback that

in general (the exception is when we have a zero width interval, A = [a,a] , that is, a real number). One of the first attempts to overcome (3) was due to Hukuhara [12] who defined what has become to be known as the Hukuhara difference (H-difference). If A = B + C, then the H-difference of A and B, denoted by A -_H B, is equal to C. The H-difference of two intervals does not always exists for arbitrary pairs of intervals. It only exists for intervals A and B for which the widths are such that

where for A = [a,] , µ(A) = - a is the lenght of the interval A.

Recently, Stefanini and Bede [19] introduced the concept of generalizedHukuhara difference of two sets A, B ∈ _C (gH-difference for short) and it is defined as follows

In case (a), the gH-difference is coincident with the H-difference. Thus, the gH-difference is a generalization of the H-difference. On the other hand, gH-difference exists for any two compact intervals A = [a, b] , B = [c, d] ∈ _C and

For more details and properties of gH-difference see [19,20] .

3 Calculus for interval-valued functions

Henceforth T = [a, b] denotes a closed interval. Let F : T → _C be an interval-valued function. We will denote F(t) = [f(t),(t)] , where f(t) < (t), ∀t ∈ T. The functions f and are called the lower and the upper (endpoint) functions of F, respectively.

For interval-valued functions it is clear that F : T → _C is continuous at t₀∈ T if

where the limit is taken in the metric space (

_C, H). Consequently, F is continuous at t₀∈ T if and only if its endpoint functions f and are continuous functions at t₀∈ T.

We denote by C([a, b] ,_C) the family of all continuous interval-valuedfunctions. Then, C([a, b] ,_C) is a quasilinear spaces, see [4,17] . On the quasilinear space C([a, b] ,_C) we can define a quasinorm ||·||_∞ given by

For more details and properties of quasilinear spaces and quasinorms see [4,17] .

Definition 3.1. ([5] ) Let F : T → _C be an interval-valued function. The integral (Aumann integral) of F over T is defined as

where S(F) is the set of all integrable selectors of F, i.e.:

If S(F) ≠ ∅, then the integral exists and F is said to be integrable (Aumann integrable).

Note that if F is measurable then has a measurable selector (see [5,7,10] ) which is integrable and, consequently, S(F) ≠ ∅. More precisely.

Theorem 3.2. ([5] ) Let F : T → _C be a measurable and integrably bounded interval-valued function. Then it is integrable and F(t)dt ∈ _C.

Corollary 3.3. ([5,10] ) A continuous interval-valued function F : T → _C is integrable.

The Aumann integral satisfies the following properties.

Proposition 3.4. ([5,10] ) Let F, G : T → _C be two measurable and integrably bounded interval-valued functions. Then

Theorem 3.5. ([8] ) Let F : T → _C be a measurable and integrably bounded interval-valued function such that F(t) = [f(t),(t)] . Then f and are integrable functions and

The H-derivative (differentiability in the sense of Hukuhara) for interval-valued functions was initially introduced in [12] and it is based on the H-difference of intervals.

Definition 3.6. ([12] ) Let F : T → _C be interval-valued function. We saythat F is differentiable at t₀∈ T if there exists an element F'(t₀) ∈ _C such that the limits

exist and are equal to F'(t₀).

Here the limits are taken in the metric space (

_C,H). Note that the H-derivative is very restrictive. For example, if we consider the interval-valued function F(t) = (1 - t³)[-2, 1] , since F(0 + h) -_HF(0) = (1 - h³)[-2, 1] -_H[-2, 1] , the H-difference F(0 + h) -_HF(0) does not exist as h → 0⁺. Therefore, the H-derivative of F does not exist at t = 0. In general, if F(t) = C · g(t) where C is an interval and g : [a, b] → ⁺ is a function with g'(t₀) < 0, then F is not differentiable at t₀ ([8,9] ). To avoid this difficulty, in [19] the authorshave introduced a more general definition of derivative for interval-valuedfunctions. For more details see [9,19] .

Definition 3.7. ([19] ) The gH-derivative of an interval-valued function F : T → _C at t₀∈ T is defined as

If F'(t₀) ∈ _C satisfying (6) exists, we say that F is generalized Hukuhara differentiable (gH-differentiable) at t₀.

In connection with the endpoint functions of F we have the following result.

Theorem 3.8. ([9] ) Let F : T → _C be an interval-valued function such that F(t) = [f(t),(t)] . Then, F is gH-differentiable at t₀∈ T if and only if one of the following cases holds

(a) f and are differentiable at t₀ and

(b) (f)'_-(t₀), (f)'₊(t₀), ()'_-(t₀) and ()'₊(t₀) exist and satisfy (f)'_-(t₀) = ()'₊(t₀) and (f)'₊(t₀) = ()'_-(t₀). Moreover

Example 3.9. Let the interval-valued function F : → _Cdefined by F(t) = [-|t|,|t|] . Then F is gH-differentiable in but the endpoint functionsfand are not differentiable at 0. Also, from Theorem 3 part (a) we have F'(t) = [()'(t), (f)'(t)] = [-1,1] for all t ∈ (- ∞, 0) and F'(t) = [(f)'(t), ()'(t)] = [-1,1] for all t ∈ (0, ∞). From part (b) we have F'(0) = [-1, 1] .

From Example 3.9 we can see that on the interval (- ∞, 0) the lenght of the interval F(t) (for short, len(F(t))) is decreasing while on the interval (0, ∞) the len(F(t)) is increasing and t = 0 is a switching point for the monotonicity of len(F(t)), that is to say, in t = 0, len(F(t)) change its monotonicity. Thus, we establish that (see [19] ):

(I) F is differentiable at t₀∈ T in the first form if f and are differentiable at t₀ and

(II) F is differentiable at t₀∈ T in the second form if f and are differentiable at t₀ and

Even more, a point t₀∈ T is said to be a switching point for the differentiability of F, if in any neighborhood V of t₀ there exist points t₁ < t₀ < t₂ such that

(type I) F is differentiable at t₁ in the first form while it is not differentiable in the second form, and F is differentiable at t₂ in the second form while it is not differentiable in the first form, or

(type II) F is differentiable at t₁ in the second form while it is not differentiable in the first form, and F is differentiable at t₂ in the first form while it is not differentiable in the second form.

Next we give an interval version of the second fundamental theorem of calculus which will be important to obtaining our main results.

Theorem 3.10. ([18] ) Let F : [a, b] → _C be an interval-valued function. If F is gH-differentiable in the first form (or second form) in [a, b] then

Theorem 3.11. Let the interval-valued function F : [a, b] → _C gH-differentiable on [a, b] with a finite number of switching points at a = c₀ < c₁ < c₂ < ... < c_n < c_{n + 1} = b and exactly at these points. Then we have

Proof. For simplicity we consider only one switching point, the case of a finite number of switching points follow similarly. Let us suppose that F is differentiable on [a, c] in the first form and F is differentiable on [c, b] in the second form. Then from Proposition 3 and Theorem 3 we have

This completes the proof.

Remark 3.12. In [19] was presented a similar result to Theorem 3.11, but with different arguments used in the proof. Moreover if c ∈ [a, b] is the only switching point for differentiability of F and F(c) is a singleton not necessarely F'(x)dx = F(b)_gHF(a). For instance, if F is considered as in the Example 3.9, we have F(0) = 0 and F'(x)dx ≠ F(1)_gHF(-1). It corrects the Theorem 30 in [19] .

Next we present a version of mean value theorem for gH-differentiableinterval-valued functions. This result will be also important in the next section.

Theorem 3.13Let F : [a, b] → _C be an gH-differentiability interval-value function on [a, b] with a finite number of switching points at a = c₀ < c₁ < c₂ < ... < c_n < c_{n + 1} = b and exactly at these points. Assume that F' is continuous. Then

Proof. Firstly we suppose that F is gH-differentiable with no switching point in the interval [a, b] then, taking on account the Theorem 3.10, we have

Now, we consider only one switching point, the case of a finite number of switching points follow similarly. Let us suppose that F is differentiable on [a, c] in the first form and F is differentiable on [c, b] in the second form. Then

So the Theorem is established.

4 Ostrowski type inequalities

In this Section we present some Ostrowski type inequalities for gH-differentiable interval-valued functions. We want to remark that the concept of gH-differentiability is the more general concept of differentiability than another concept for interval-valued fuctions. For more details see [9,13,19] .

Theorem 4.1. Let F : [a, b] → _C be a continuously gH-differentiable interval-valued function on [a, b] with a finite number of switching points at a = c₀ < c₁ < c₂ < ... < c_n < c_{n + 1} = b and exactly at these points. Then, for x ∈ [a, b] we have

Proof. Taking in account Theorem 3.13 and properties of Hausdorff metricwe have

And the inequality (7) is proved.

Proposition 4.2. Inequality (7) is sharp at x = a, in fact attained by F(y) = (y - a)(b - a)A, with A ∈ _C being fixed.

Proof. We denote by A = [a, ] , with a < . Since (y - a)(b - a) > 0 then F(y) = (y - a)(b - a)A = [(y - a)(b - a)a,(y - a)(b - a)a] . From Theorem 3.8 F is a continuously gH-differentible interval-valued function and F'(y) = (b - a)A. Thus, we have that

and

So, the equality in (7) is attained.

Example 4.3. We consider the interval-valued function F : [0, π] → _C defined by

or equivalently

Since g(t) = cos(4t) is a continuously differentiable function then F is continuously gH-differentiable and F'(t) = [-16, -8] sin(4t). So, ||F'||_∞ = 16.

On the other hand, F has seven switching points for its gH-differentiability in (0, π) which are {π/8,π/4,3π/8,π/2,5π/8,3π/4,7π/8}.

Figure 1 shows the endpoint functions of F, the solid line curve represent the lower function f and the dashed one represent the upper function .

The left hand of the inequality (7) is given by

while the right hand is

So, the inequality (7) is valid for F.

Note that the inequality (7) is valid for any continuously gH-differentiable interval-valued function on [a, b] with a finite number of switching points. From the example above we can see that F is continuously gH-differentiable and (7) is valid however the endpoint functions are not necessarely differentiables. For this special case, when endpoint functions are differentiables we have the following result, where we omitted that F has a finite number of switching points.

Theorem 4.4.Let F : [a, b] → _C be an interval-valued function such that the endpoint functions f, are continuously differentiables. Then, F is continuously gHdifferentiable and for x ∈ [a, b]

Proof. Taking in account the Ostrowski inequality (1) we have

Thus, the proof is completed.

Next we present another one generalization of the Ostrowski type inequality (1).

Theorem 4.5. Let the interval-valued function F : [a, b] → _C gH-differentiable in (a, b) such that the endpoint functions f,

are continuously differentiables. Let α : [a, b] → [a, b] and β : (a, b] → [a, b] , α(x)< x , β(x)> x. Then, for all x ∈ [a, b] we have

Proof. From Theorem 47 in [11] and properties of Hausdorff metric, we have that

So, the inequality is established.

Remark 4.6. As a consequence of Theorem 4.5 we have the following special inequality: Let the interval-valued function F : [a, b] → _C satisfying the same conditions of Theorem 4.5. Then, if α(x) = and β(x) = we have, for all x ∈ [a ,b] ,

Finally, we stablish that:

a) For this functions α and β we get the best bound for any x ∈ [a, b] because the inequality in Theorem 4.5 contains a sum of squares and the minimun of this expresion occurs when each quadratic terms are zero.

b) If x = (the midpoint of [a, b] ) we obtain an even more accurate formula from Remark 4.6. In fact,

Received: 14/X/11.

Accepted: 06/V/12.

#CAM-430/11.

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^*

The research in this paper has been partially supported by Fondecyt-Chile through projects 1120665 and 1120674, and Ministerio de Ciencia e Innovación, Spain, through grant MTM2008-00018.

Publication Dates

• Publication in this collection
  28 Nov 2012
• Date of issue
  2012

History

• Received
  14 Oct 2011
• Accepted
  06 May 2012