Granural differentiability of fuzzy valued functions and applications

Thedifferential equations in which parameters and/or conditions are uncertain, and thisuncertainty is expressed by a class of fuzzy sets - usually fuzzy numbers - are calledfuzzy differe

HANOI PEDAGOGICAL UNIVERSITY 2 DEPARTMENT OF MATHEMATICS

DUONG THU HOAN

GRANULAR DIFFERENTIABLITY OF FUZZY-VALUED FUNCTIONS AND APPLICATIONS

BACHELOR THESIS Speciality: Analysis

Hanoi, 2019

HANOI PEDAGOGICAL UNIVERSITY 2 DEPARTMENT OF MATHEMATICS

DUONG THU HOAN

GRANULAR DIFFERENTIABLITY OF FUZZY-VALUED FUNCTIONS AND APPLICATIONS

BACHELOR THESIS Speciality: Analysis

Hanoi, 2019

The thesis was written on the basis of my study under the guidance of soc Prof Khuat Van Ninh ,Nguyen Phuong Dong and my effort I have studied andpresented the results from bibliographies The thesis does not coincide others

As-The author

Duong Thu Hoan

On this occation, firstly, I would like to thank all people who helped me in

my study and preparation of this thesis I emphasize to thank Hanoi PedagogicalUniversity 2 where I finished this thesis with the teaching of lectures

Especially, I would like to express my profound gratitude to my supervisor,Assoc Prof Khuat Van Ninh and Nguyen Phuong Dong who helped me carefully inthe processing of researching and writing of this thesis, for their valuable instructivecomments and their illuminating advices as well My sincere thanks are also sent to myteachers in the Department of Mathematics who educated me over around four years

I take this opportunity to thank all my friends who always help and encourage me Ialso give special thanks and deep gratitude towards my family for their vital supportand encouragement

Finally, this thesis may have some errors, anyway I am very pleased and wouldlike to receive constructive comments and suggestions to improve the quality of thisthesis

Ha Noi, May 2019Duong Thu Hoan

Introduction 1

1 Preliminaries 4

1.1 The space of fuzzy numbers 4

1.2 Characterization of fuzzy numbers 5

1.2.1 Some types of fuzzy numbers 6

1.2.2 Zadeh’s Extension Principle 9

1.2.3 The Sum and Scalar Multiplication 9

1.2.4 The product of fuzzy numbers 12

1.2.5 The difference of fuzzy numbers 13

1.3 Fuzzy derivative and integral 14

2 Granular differentiability of fuzzy-valued functions 15

2.1 Granular representation 15

2.2 Granular operations 16

2.3 The granular metric space (E, D gr) 16

2.4 Granular derivative and intergral 21

3 Applications to fuzzy differential equations 27

3.1 Theoretical results 27

3.2 Numerical examples 28

4 Conclusions 32

References 32

1 Rationale

In many real world problems, there is often a need to interpret and solve theproblems operating in the environment inherent uncertainties and vagueness Whenengineers want to handle these disadvantages, they may use either stochastic and sta-tistical models or fuzzy models, but stochastic and statistical uncertainty occur due

to the natural randomness in the process It is generally expressed by a probabilitydensity or frequency distribution function For the estimation of the distribution, itrequires sufficient information about the variables and parameters involved in it Onthe other hand, fuzzy set theory refers to the uncertainty when we may have lack

of knowledge or incomplete information about the variables and parameters In eral, science and engineering systems are governed by ordinary and partial differentialequations, but the type of differential equation (DEs) depends upon the applications,domains, complicated environments, the effect of coupling, and so on As such, thecomplicacy needs to be handled by recently developed differential equations containeduncertainty or fuzziness Since the first time introduced in 1965 by Zadeh, many ex-tensive research have been studied on the applications of the fuzzy sets in various fields

gen-of sciences, e.g in control theory, in medicine and so forth In the recent years, theapplication of the fuzzy sets in differential equations has captured much attention Thedifferential equations in which parameters and/or conditions are uncertain, and thisuncertainty is expressed by a class of fuzzy sets - usually fuzzy numbers - are calledfuzzy differential equations (FDEs) Since without any definition of the derivative,differential equations make no sense, in the recent years, several typical definitions of

a derivative of fuzzy-valued functions have been proposed such as Hukuhara derivative(H-derivative), generalized Hukuhara derivative (gH-derivative), generalized derivative(g-derivative) and Granular derivative (gr-derivative) Among the mentioned defini-tions the gr-derivative is more effective and practical than the others in the viewpoint

of computation and engineering The point about the horizontal membership functionapproach (or the use of granular derivative) is that it not only circumvents short-comings associated to the previously mentioned approaches, but also inherits some

of their benefits The most essentially important merits of gr-derivative and dimensional fuzzy arithmetic based on relative-distance-measure in fuzzy dirrerentialequations studies are outlined below:

multi-• Obtaining fuzzy function derivative and/or solving FDEs is simple;

• This approach does not compel that solution support closure length of FDE benecessarily monotonic;

• Solving each FDE is equivalent to solving just one individual differential equationcalled granular differential equation That is, it avoids the doubling propertydisadvantage;

• An FDE has only one solution on condition that its equivalent granular differentialequation has a solution That is, it avoids the multiplicity of solutions drawback;

• This approach does not result in unnatural behavior in modeling phenomenon.Motivated by aforesaid, in this graduation thesis, we pay more attention instudying this novel concept of differentiability of fuzzy-valued functions and extending

a new concept of integrability, named granular integrability (gr-integrability), thatplay vital roles to investigate the class of fuzzy differential equations under granulardifferentiability

2 Aim of the study

Study the granular differentiability and integrability of fuzzy-valued function

3 Task of the study

• Study horizontal membership function representation of fuzzy numbers and valued functions;

fuzzy-• Study the granular differentiability - integrability of fuzzy-valued function andcompare with some other previous concepts;

• Apply to solve some classes of fuzzy differential equations and then simulate theirsolutions

4 The object and scope of the study

4.1 The object of the study: fuzzy-valued functions and fuzzy differential equations.4.2 The scope of the study: The granular differentiability and integrability

5 Research method

In the thesis, we use some techniques of set-valued functional analysis combinedwith fuzzy analysis to study the granular differentiability and integrability of fuzzy-valued functions

6 Overview of the study

The graduation thesis consists of 4 chapters

Chapter 1: Preliminaries,

Chapter 2: Granular differentiability of fuzzy-valued functions,

Chapter 3: Applications to fuzzy differential equations,

Chapter 4: Conclusions.

The thesis is written on the basis of the paper: “Granular Differentiability

of Fuzzy-Numbers Valued Functions”, IEEE Tran Fuzzy Syst., 26(1)(2018), 310-323 of M Mazandarani, N Pariz and A.V Kamyad and the manuscript “The solvability of Cauchy problem to fuzzy fractional evolution equations under Caputo generalized Hukuhara-differentiability, J Science, HPU2, (2019)”

of N.P Dong and D.T Hoan

Chapter 1

Preliminaries

1.1 The space of fuzzy numbers

A fuzzy number is a generalization of regular real number in the sense that itdoes not refer to one single value but rather to a contected set of possible values, whereeach possible value has its own weight between 0 and 1 The fuzzy number concept isfundamental for fuzzy analysis and fuzzy differential equations and a very useful tool

in several applications of fuzzy sets and fuzzy logic

Definition 1.1.1 [1] Consider a fuzzy set u: R → [0, 1] of the real line u: R → [0, 1].

Then u is said to be a fuzzy number if it satisfies following properties:

(i) u is normal, i.e., ∃x0 ∈ R such that u(x0) = 1;

(ii) u is fuzzy convex, i.e., u(tx + (1 − t)y) ≥ min{u(x), (y)}, for all t ∈ [0, 1], x, y ∈ R;

(iii) u is upper semicontinuous on R, i.e., for all ϵ > 0, ∃δ > 0 such that |x − x0| < δ u(x) − u(x0) < ϵ;

(iv) u is compactly supported, i.e., cl {x ∈ R; u(x) > 0} is compact, where cl(A)

denotes the closure of the set A

Let us denote by E the space of fuzzy numbers on the real line.

Example 1.1.1 The fuzzy set u: R → [0, 1], given by

is a fuzzy number (see Figure 1.1)

Example 1.1.2 The fuzzy set represented in Figure 1.2 is not a fuzzy number since

it is not fuzzy convex

Remark 1.1.1 Any real number is also a fuzzy number,

R ={χ {x} |x is a real number}

that means, R ⊂ E

Here, χ {x} is said to be a singleton fuzzy number for any given real number x ∈ R and

it can be identified with x ∈ R (see Figure 1.3).

Also, fuzzy numbers generalize closed intervals Indeed, if I denotes the set of all real

Figure 1.1: Example of a fuzzy number and its level sets.

interval, then I ∈ E, where

I ={χ [a,b] | [a,b] is real interval};Example of a closed interval represented in Figure 1.4 is a fuzzy number

1.2 Characterization of fuzzy numbers

Definition 1.2.1 [1, 3] For 0 < α ≤ 1, we denote

[u] α ={x ∈ R | u(x) ≥ α} ,

[u]0 = cl{x ∈ R | u(x) > 0}

Then, [u] α is called the α-level sets of the fuzzy number u The 1-level set is

called the core of the fuzzy number, while the 0-level set is called the support of thefuzzy number

Theorem 1.2.1 (Stacking theorem, [1]) Let u ∈ E be a fuzzy number whose α - level sets is given by [u] α for all α ∈ [0, 1] Then

(i) [u] α is a closed interval of the form [u] α = [u − α , u+

Figure 1.2: Example of a fuzzy set that is not a fuzzy number.

(iv) For any sequence {α n } which converges from above to 0, we have

cl (∪∞

n=1 u α n ) = u0.

Theorem 1.2.2 (Negoita-Ralescu characterization theorem, Negoita-Ralescu, [1]) Let

{M α | α ∈ [0, 1]} be a family of subsets that satisfies following conditions:

(i) M α is a non-empty closed interval for each α ∈ [0, 1];

(ii) If 0 ≤ α1 ≤ α2 ≤ 1 then M α2 ⊆ M α1;

(iii) For any sequence {α n } which converges from below to α ∈ [0, 1], we have

∩∞

n=1 M α n = M α ; (iv) For any sequence {α n } which converges from above to 0, we have

cl (∪∞

n=1 M α n ) = M0 Then, there exists a unique u ∈ E such that [u] α = M α for any α ∈ [0, 1].

1.2.1 Some types of fuzzy numbers

The so-called L-R fuzzy numbers are considered as an important fuzzy number

in the theory of fuzzy sets L-R fuzzy numbers, and their particular cases, as e.g.,triangular and trapezoidal fuzzy numbers, are very useful in applications

Figure 1.3: A singleton fuzzy number.

Definition 1.2.2 Let L, R : [0, 1] → [0, 1] be two continuous, increasing functions

A trapezoidal fuzzy number u can be represented by the quadruple (a, b, c, d)

where a ≤ b ≤ c ≤ d, the functions L and R are linear (see Figure 1.5).

Figure 1.4: Example of a closed interval interpreted as a fuzzy number.

If we have b = c then the fuzzy number u is called a triangular fuzzy number Then

a triplet (a, b, c) ∈ R3, a ≤ b ≤ c is sufficient to represent the triangular fuzzy number

α, 3 − 2 √ α] and the shape depicted in Figure 1.7.

A Gaussian fuzzy number has the membership degree given by

Figure 1.5: Example of a trapezoidal fuzzy number

of it respectively, and a represents a tolerance value Its graph is shown in Figure 1.8

1.2.2 Zadeh’s Extension Principle

Definition 1.2.3 [1] Let a function f : X → Y , where X and Y are crisp sets Then,

it can be extended to a function F : F(X) → F(Y ) (a fuzzy-valued function) such

that v = F (u), where

v(y) =

{sup{u(x) : x ∈ X, f(x) = y}, when f −1 (y) ̸= ∅

Then, F is called Zadeh’s extension of f

Theorem 1.2.3 [1] Let f : X → R be a continuous real-valued function Then, it can be extended to a fuzzy-valued function

F : E → E

u 7→ F (u) = v, where v is a fuzzy number whose level sets [v] α = [v − α , v α+] is defined as follows

v − α = inf{f(x) | x ∈ [u] α } , α ∈ [0, 1],

v+α = sup{f(x) | x ∈ [u] α } 1.2.3 The Sum and Scalar Multiplication

For u, v ∈ E and λ ∈ R, based on the Zadeh’s extension principle, one can

define the sum of two fuzzy numbers u + v and the scalar multiplication λu via their

Figure 1.6: Example of a triangular fuzzy number.

level set wises

[u + v] α ={x + y | x ∈ [u] α , y ∈ [v] α } = [u] α + [v] α ,

(λ · u) α

={λx | x ∈ [u] α } = λ[u] α

, ∀α ∈ [0, 1],

where [u] α +[v] αis the sum of two intervals (as subsets ofR) and λ[u] αis the product of

a number and a subset ofR So, fuzzy arithmetic is extended from interval arithmetic

Example 1.2.2 Let u = (1, 2, 3) and v = (2, 3, 5) be triangular fuzzy numbers with

respective level sets

[u] α = [1 + α, 3 − α],

[v] α = [2 + α, 5 − 2α],

for all α ∈ [0, 1] Then, sum of u and v is an element w which can be represented in

following parametric form

[w] α = [u] α + [v] α = [3 + 2α, 8 − 3α].

As a consequence of Theorem 1.2.2, we immediately obtain w = (3, 5, 8).

By similar arguments, we can see that the fuzzy number z = (2, 4, 6) is the result obtained by multiplying the fuzzy number u by 2.

The following theorem deals with the algebraic properties of E

Theorem 1.2.4 (i) The addition in the space of fuzzy numbers is associative and

commutative i.e.,

u + v = v + u,

Figure 1.7: Example of an L-R fuzzy number.

and

u + (v + w) = (u + v) + w, for all u, v, w ∈ E.

(ii) The fuzzy set ˆ0 = χ {0} ∈ E is the neutral element w.r.t the addition i.e.,

u + ˆ0 = ˆ0 + u = u,

for any u ∈ E.

(iii) None of u ∈ E \ {R} has an opposite in E (w.r.t the addition).

(iv) For any a, b ∈ R+ with ab ≥ 0 and any u ∈ E, we have

(a + b) · u = (a · u) + (b · u).

In general, if a, b ∈ R then this property does not hold.

(v) For any λ ∈ R and u, v ∈ E , we have

λ · (u + v) = λ · u + λ · v.

(vi) For any λ, µ ∈ R and any u ∈ E, we have

(λ · µ) · u = λ · (µ · u).

Figure 1.8: Example of an exponential fuzzy number.

1.2.4 The product of fuzzy numbers

The product w = u · v of fuzzy numbers u and v, is defined based on Zadeh’s

extension principle Denote

α = (3− α)(5 − 2α),

where (u · v)1 = [6, 6] , (u · v)0 = [2, 15].

Theorem 1.2.5 (i) The singleton fuzzy set ˆ1 = χ {1} ∈ E is the neutral element w.r.t the multiplication i.e., u · ˆ1 = ˆ1 · u = u, for any u ∈ E.

(ii) None of u ∈ E \ R has an inverse in E(w.r.t the multiplication).

(iii) For any u, v, w ∈ E we have

((u + v) · w)λ ⊆ (u · w)λ + (v · w)λ, for all λ ∈ R.

In general, the distribution property does not hold.

(iv) For any u, v, w ∈ E such that none of their supports contains 0, we have

u · (v · w) = (u · v) · w.

Proposition 1.2.6 If u and v are positive fuzzy numbers then the fuzzy number w,

defined by w = u · v, has the level sets [w] α = [w − α , w+

α ], where

w − α = u − α v −1 + u −1v − α − u −

1v −1, and

w+α = u+α v+1 + u+1v+α − u+

1v1+, for every α ∈ [0, 1] Moreover, w is a positive fuzzy number.

1.2.5 The difference of fuzzy numbers

Definition 1.2.4 [3, 4] The Hukuhara difference (H-difference ) of two fuzzy number

u and v is defined by

u ⊖ H v = w ⇔ u = v + w,

where + represents for the standard fuzzy addition

If u ⊖ H v exists, its α − cuts are

Definition 1.2.5 [3, 4] Let u, v ∈ E Then, the generalized Hukuhara difference

(gH-difference for short) is the fuzzy number w, if it exists, such that

Example 1.2.5 Let u = (1, 2, 3) and v = (2, 3, 5) be positive triangular fuzzy numbers.

Then, the α − cuts of u and v are given by

[u] α = [1 + α, 3 − α],

[v] α = [2 + α, 5 − 2α],

respectively Then, according to Proposition 1.2.7, we can see that

[u ⊖ gH v] α = [min{−1, −2 + α} , max {−1, −2 + α}] = [−2, −1].

1.3 Fuzzy derivative and integral

Theorem 1.3.1 [1] Let f : (a, b) ⊆ R → E be a fuzzy function and x0 ∈ (a, b) Then,

f (x) is strongly generalized Hukuhara differentiable (SGH-differentiable) at the point

x0, in the first form, if there exists an element f SGH (x0) ∈ E such that for h > 0 sufficiently near zero, f (x0+ h) ⊖ H f (x0),f (x0)⊖ H f (x0− h) exist and the limits

Definition 1.3.1 [2, 3] Let f : (a, b) ⊆ R → E be a fuzzy function and x0 ∈ (a, b).

Then, f (x) is generalized Hukuhara differentiable (gH-differentiable) at the point x0, if

there exists an element f gH ′ (x) ∈ E such that for h sufficiently near zero the following

differentiable for all α ∈ [0, 1] Then, the fuzzy-valued function f(x) is gH-differentiable

at a fix x ∈ (a, b) if and only if one of the following two cases holds:

1) f α − (x) and f α+(x) are increasing and decreasing functions of α, f α=1 − (x) ≤ f+

The definition of Lebesgue integral for fuzzy-valued functions [1] is in the sense

of Aumann integral for set-valued functions [5]

Definition 1.3.2 [2, 3] Assume that f : [a, b] → E is a fuzzy-valued function with

parametric form [f (t)] α = [f α − (t), f α+(t)] for all t ∈ [a, b], α ∈ [0, 1] and f −

α (t), f α+(t) are measurable and Lebesgue integrable on [a, b] Then the Lebesgue integral of f is

denoted by ∫b

a f (t)dt, which can be written in following parametric form

[∫ b a

f (t)dt

]α

=

[∫ b a

f α − (t)dt,

∫ b a