Type-2 Fuzzy Initial Value Problems for Second-order T2FDEs
Norihiro Someyama, Hiroaki Uesu, Kimiaki Shinkai, Shuya Kanagawa
TType-2 Fuzzy Initial Value Problems for Second-orderT2FDEs
N. Someyama ∗ , H. Uesu ∗ , K. Shinkai ∗ & S. Kanagawa ∗ ∗ Shin-yo-ji Buddhist Temple, 5-44-4 Minamisenju, Arakawa-ku, Tokyo 116-0003, Japan [email protected] ∗ Kanazawa Institute of Technology, 7-1 Ohgigaoka, Nonoichi-shi, Ishikawa 921-8501, Japan [email protected] ∗ Tokyo Kasei Gakuin University, 2600 Aihara, Machida-shi, Tokyo 194-0211, Japan [email protected] ∗ Tokyo City University, 1-28-1 Tamazutsumi, Setagaya-ku, Tokyo 158-8557, Japan [email protected]
Abstract
Type-2 fuzzy differential equations (T2FDEs) of order 1 are already known and the so-lution method of type-2 fuzzy initial value problems (T2FIVPs) for them was given by M.Mazandarani and M. Najariyan [16] in 2014. We give the solution method of second-orderT2FIVPs in this paper. Furthermore, we would like to propose new notations for type-2 fuzzytheory where symbols tend to be complicated and misleading. In particular, the Hukuharadifferential symbols introduced experimentally in this paper will give us clearler meanings andexpressions.
Keywords : Type-1 / type-2 fuzzy number, Type-1 / type-2 fuzzy-valued function, Type-1 / type-2fuzzy H-derivative, Type-1 / type-2 fuzzy differential equation, Type-1 / type-2 fuzzy initial value problem. : 03E72, 26E50, 34A07, 35E15, 65L05.
Corresponding Author : Norihiro Someyama < [email protected] > In 1965, fuzzy set theory [19] was introduced as the origin of mathematical theory of ambiguity byL.A. Zadeh (1921-2017). A fuzzy set A on the universal set X is characterized via the membershipfunction µ A : X → [0 ,
1] and the membership function value µ A ( x ) which represents the grade ofambiguity for each x ∈ X . Any membership function however is basically formed from its individualfunction values determined by the subjectivity of the observer. So, the grades { µ A ( x ) : x ∈ X } may also contain ambiguity. Focusing on this, Zadeh [20] introduced type-2 fuzzy set theoryin 1975. Type-2 fuzzy sets are fuzzy sets whose grades are fuzzy. In other words, the type-2fuzzy set includes not only the uncertainty of the data, but also the membership function whichindicates the uncertainty. Then, we consider the membership function of a type-2 fuzzy set A as µ A : X → [0 , [0 , .The concept of fuzzy derivatives was proposed by Chang & Zadeh [4]. In addition, fuzzyderivatives using the extension principle were proposed by Dubois & Prade [6], and some otherconcepts related to fuzzy derivatives were discussed by Puri & Ralescu [17]. Fuzzy initial value1 a r X i v : . [ m a t h . G M ] J a n SUSK problems have been researched since Kaleva [10] and Bede & Gal [1], and several attempts have beenproposed to define the differentiability of fuzzy functions. Among them, Hukuhara differentiabilityand strongly generalized differentiability [1, 17] have attracted particular attention. For the sakeof simplicity, ‘type-1 / type-2 fuzzy differential equations’ and ‘type-1 / type-2 fuzzy initial valueproblems’ are often abbreviated as ‘T1/T2FDEs’ and ‘T1/T2FIVPs’ respectively in this paper.Now, for example, suppose that we have a highly experienced expert and an inexperiencedstudent measure the temperature of a certain substance and ask them to indicate the membershipfunction of temperature (More specifically, see Section 5 of [16]). In this case, there is a possibilitythat different membership functions will be expressed. Therefore, type-2 fuzzy sets can be usefulif the exact form of the membership function is not known, or if the grade of the membershipfunction itself is ambiguous or inaccurate. Since there are so many problems where the exact formof the membership function cannot be determined, type-2 fuzzy sets are suitable for dealing withhigh levels of uncertainty that involve more complicated calculations. Moreover, the parametersand state variables in real problems are very imprecise and can be better modeled by type-2 fuzzyvariables and hence T2FDEs.In 2014, Mazandarani & Najariyan [16] studied first-order T2FDEs and took up some concreteT2FIVPs for them. Related to this, we study, in this paper, second order T2FDEs and T2FIVPsfor them. We also present and prove other theorems that may not be known in the case of type-1and type-2 first order. We attack the case of crisp coefficients in this paper. The equations thatdescribe the physical phenomena of our world are usually second-order differential equations, sowe believe that our study will be necessary and useful in fields such as mathematical physics. Infact, we can think that it is appropriate to set initial values as fuzzy numbers if an expert measuresthe values.Moreover, it may be necessary to seek the best notation because type-2 fuzzy theory tends tobe complicated in notations. Then, we would like to propose some new notations on a trial basisin this paper.
We arrange, in this section, definitions of terms etc., notations and previous results necessary forthis paper.From beginning to end, we equate fuzzy sets and the membership functions in notations. Wenamely denote ‘the fuzzy set A : X → [0 , µ A ( x ) ofa fuzzy set A , with the membership function µ A : X → [0 , x ∈ X is represented by A ( x ).The opposite of ‘fuzzy’ is called ‘crisp.’ Definition 2.1.
A fuzzy set u : R → [0 , is the type-1 fuzzy number if and only ifi) u is normal, that is, there exists an x ∈ R such that u ( x ) = 1 ;ii) u is fuzzy convex, that is, u ( tx + (1 − t ) y ) ≥ min { u ( x ) , u ( y ) } for any x, y ∈ R and t ∈ [0 , ;iii) u is upper semi-continuous;iv) the support supp( u ) := cl( { x ∈ R : u ( x ) > } ) of u is compact, where cl( S ) stands for theclosure of the crisp set S . It is well known in the study of fuzzy theory so far that the argument of fuzzy sets can bereduced to that of the level cut sets. The α -cut set [ A ] α of a type-1 fuzzy set A on X is defined by[ A ] α := { x ∈ X : A ( x ) ≥ α } , α ∈ (0 , A ] := supp( u ) , α = 0 , ype-2 Fuzzy Initial Value Problems for Second-order T2FDEs A : A = (cid:91) α ∈ [0 , α [ A ] α where α [ A ] α : X → { , α } is a type-1 fuzzy set. Thus, it is sufficient to consider and argue α -cutsets for most problems. In particular, the type-1 fuzzy number is the fuzzy set whose arbitrary α -cut set is a bounded and closed interval, so we put the following notation. Definition 2.2.
We denote the α -cut set of a type-1 fuzzy number u : R → [0 , by [ u ] α = [ u − ,α , u + ,α ] (2.1) for any α ∈ [0 , . (2.1) is called the parametric form of u . Remark 2.1.
We write 0 for crisp zero in this paper and its α -cut set is represented as the closedinterval [0 , r as the closed interval [ r, r ] andbring crisp numbers into the group of fuzzy numbers. Remark 2.2.
Each end of [ u ] α = [ u − ,α , u + ,α ] should satisfy the followings:1) u − ,α is bounded, monotone increasing, left-continuous with respect to α ∈ (0 ,
1] and right-continuous on α = 0;2) u + ,α is bounded, monotone decreasing, left-continuous with respect to α ∈ (0 ,
1] and right-continuous on α = 0;3) u − ,α ≤ u + ,α for any α ∈ [0 , u ] α that does not satisfy any of the above property cannot be called the fuzzy number.We hereafter omit the description ‘ α ∈ [0 , α -cut sets or parametric forms. Definition 2.3.
Let u, v be type-1 fuzzy numbers on R . We denote the α -cut sets of u, v by [ u ] α = [ u − ,α , u + ,α ] , [ v ] α = [ v − ,α , v + ,α ] (2.2) respectively. Then, u = v if and only if [ u ] α = [ v ] α , i.e. u − ,α = v − ,α and u + ,α = v + ,α for any α ∈ [0 , . Definition 2.4.
Let u, v be type-1 fuzzy numbers on R and k ∈ R . We denote the α -cut sets ofthem by (2.2). The sum u + v of u and v is defined by [ u + v ] α := [ u ] α + [ v ] α , i.e. [ u − ,α , u + ,α ] + [ v − ,α , v + ,α ] = [ u − ,α + v − ,α , u + ,α + v + ,α ] . Moreover, the scalar multiple ku of u is defined by [ ku ] α := k [ u ] α = (cid:40) [ ku − ,α , ku + ,α ] , k ≥ ku + ,α , ku − ,α ] , k < . SUSK
Definition 2.5.
Let u, v be type-1 fuzzy numbers on R . We denote the α -cut sets of them by (2.2).The product uv of u and v is defined by [ uv ] α = [ u ] α [ v ] α = (cid:20) min i,j ∈{− , + } u i,α v j,α , max i,j ∈{− , + } u i,α v j,α (cid:21) for any α ∈ [0 , . Definition 2.6.
Let u, v be type-1 fuzzy numbers on R . We denote the α -cut sets of them by (2.2).Then, the order relationship between them, u ≤ v , is defined as [ u ] α ≤ [ v ] α , i . e . u − ,α ≤ v − ,α and u + ,α ≤ v + ,α for any α ∈ [0 , . In particular, the non-negativity (resp. positivity) of a type-1 fuzzy number u , u ≥ (resp. u > ), is defined by u − ,α ≥ . u − ,α > for any α ∈ [0 , . Definition 2.7.
Let u, v be type-1 fuzzy numbers on R . We denote the α -cut sets of them by (2.2).The fuzzy Hausdorff distance d H of u and v is defined by d H ( u, v ) := sup α ∈ [0 , max {| u − ,α − v − ,α | , | u + ,α − v + ,α |} . We write T ( R ) for the type-1 fuzzy number space equipped with the d H -topology. Let I be an interval which is a proper subset of R or let I = R . We can consider the type-1 fuzzyfunction F : T ( I ) → T ( R ) by using Zadeh’s extension principle for a crisp function f : I → R .We set T ( I ) = I in this paper, that is, we consider crisp-variable type-1 fuzzy number-valuedfunctions exclusively. The α -cut set of F : I → T ( R ) is represented by[ F ( x )] α := [ F − ,α ( x ) , F + ,α ( x )]for all x ∈ I and any α ∈ [0 , F . Definition 2.8.
Let u, v ∈ T ( R ) . If there exists some w ∈ T ( R ) such that u = v + w , we write w = u − v and call it the T1-Hukuhara difference of u and v . Remark 2.3.
For any u ∈ T ( R ), − u stands for 0 − u , i.e.[ − u ] α = [ − u − ,α , − u + ,α ] , whereas [( − u ] α = [ − u + ,α , − u − ,α ] . We should remark that, in general, u + ( − v (cid:54) = u − v for any u, v ∈ T ( R ). ype-2 Fuzzy Initial Value Problems for Second-order T2FDEs † and double dagger ‡ to denote fuzzy derivatives in the senseof Hukuhara. We use prime (cid:48) as the crisp derivative notation. Definition 2.9.
Let F : I → T ( R ) and h > be a crisp number. F is T1-differentiable in thefirst form at some x ∈ I if and only if there exist F ( x + h ) − F ( x ) and F ( x ) − F ( x − h ) satisfying that the fuzzy limit F † ( x ) := lim h ↓ F ( x + h ) − F ( x ) h = lim h ↓ F ( x ) − F ( x − h ) h exists. Moreover, F is T1-differentiable in the second form at some x ∈ I if and only if thereexist F ( x ) − F ( x + h ) and F ( x − h ) − F ( x ) satisfying that the fuzzy limit F ‡ ( x ) := lim h ↓ F ( x ) − F ( x + h ) − h = lim h ↓ F ( x − h ) − F ( x ) − h exists. Here the above differences (resp. limits) are due to the meaning of T1-Hukuhara (resp. d H ).If F is T1-differentiable in both senses at any x ∈ I , F † and F ‡ is called the (1)-T1-derivative and(2)-T1-derivative of F , respectively. Remark 2.4.
In addition to the above two forms, it is possible to consider the following twoforms, that is,i) the third form: lim h ↓ F ( x + h ) − F ( x ) h = lim h ↓ F ( x − h ) − F ( x ) − h , (2.3)ii) the fourth form: lim h ↓ F ( x ) − F ( x + h ) − h = lim h ↓ F ( x ) − F ( x − h ) h . (2.4)However, it is known ([1], Theorem 7) that (2.3) becomes the crisp number if there exist F ( x + h ) − F ( x ) and F ( x − h ) − F ( x ). (2.4) is also so if F ( x ) − F ( x + h ) and F ( x ) − F ( x − h ).We shall thus ignore the third and fourth forms. Remark 2.5.
We mention the validity of our dagger symbols † , ‡ meaning as fuzzy derivatives.First of all, we can avoid confusion between the crisp derivative and the fuzzy derivative by using † and ‡ (see e.g. Theorems 2.1, 2.2, 3.6 and 3.7 later). Secondly, if we want to the number ofthe differential order to 2 or less, we can use † , ‡ in the same sense as prime (cid:48) to clarify whatkind of fuzzy derivative it is (in particular, see Theorems 2.4, 3.1, 3.4 and 3.5 later). Moreover,the acronym for a letter is often used to represent a mathematical concept or quantity, such as ∆( D elta) for differences or D for derivatives. From this point of view, ‘dagger’ has an appropriateacronym D to represent derivatives.By using this definition repeatedly, we find that the n th-order T1-derivative of F has 2 n forms.For example, when applied to fuzzy differential equations, we need to choose the most appropriatesolution from these 2 n solutions. Theorem 2.1.
Let F : I → T ( R ) be T1-differentiable on I . Then, the parametric forms of itsT1-derivatives are given by1) the first parametric form: [ F † ( x )] α = [ F (cid:48)− ,α ( x ) , F (cid:48) + ,α ( x )] ,
2) the second parametric form: [ F ‡ ( x )] α = [ F (cid:48) + ,α ( x ) , F (cid:48)− ,α ( x )] . SUSK
Theorem 2.2 ([10], Theorem 5.2; [3], Theorem 5) . Let F : I → T ( R ) be second-order T1-differentiable on I . Then, the parametric forms of the second-order T1-derivatives are given by1) the first and first parametric form: [ F †† ( x )] α = [ F (cid:48)(cid:48)− ,α ( x ) , F (cid:48)(cid:48) + ,α ( x )] ,
2) the first and second parametric form: [ F †‡ ( x )] α = [ F (cid:48)(cid:48) + ,α ( x ) , F (cid:48)(cid:48)− ,α ( x )] ,
3) the second and first parametric form: [ F ‡† ( x )] α = [ F (cid:48)(cid:48) + ,α ( x ) , F (cid:48)(cid:48)− ,α ( x )] ,
4) the second and second parametric form: [ F ‡‡ ( x )] α = [ F (cid:48)(cid:48)− ,α ( x ) , F (cid:48)(cid:48) + ,α ( x )] . We first introduce type-2 fuzzy sets. A type-2 fuzzy set is defined by its membeship function witha fixed input order (See Remark 2.6, 1), later for details).
Definition 2.10 ([15]) . A type-2 fuzzy set A on X is characterized by A := { ( x, u ; ν A ( x, u )) : x ∈ X, u ∈ R ( µ A ( x )) ⊂ [0 , } (2.5) where • ν A is the membership function of A from the ordered pair ( x, u ) to [0 , , • R ( µ A ( x )) , for each x ∈ X , is the range of µ A : X → [0 , [0 , called the primary membershipfunction of A , • x is called the primary variable of A , • u is called the secondary variable of A . Remark 2.6. ν A is not just a two-variable function X × R ( µ A ) → [0 , ν A , x isinput and then u must be input.2) (2.5) is often called the point-valued representation of A .3) The above µ A ( x ) (resp. u ) should be recognized as the fuzzy grade (resp. the grade of thefuzzy grade) of A at x ∈ X . Definition 2.11 ([21]) . Let A be a type-2 fuzzy set on X and x ∈ X a fixed point. We define thetype-1 fuzzy set κ A ( x ) := (cid:90) u ∈ R ( µ A ( x )) ν x A ( u ) /u where ν x A := ν A ( x , · ) : R ( µ A ( x )) → [0 , is called the secondary membership function of A at x . Moreover, ν A ( x , u ) is called the secondary grade of x . Here, similar to the well-knownnotation for type-1 fuzzy sets, the above integral symbol does not mean the conventional continuoussum, but just a continuous union. ype-2 Fuzzy Initial Value Problems for Second-order T2FDEs Definition 2.12 ([8], Definition 2.8.1) . Let A be a type-2 fuzzy number on X . The β -cut set of A is defined by [ A ] β := (cid:10) A β , A β (cid:11) := (cid:90) x ∈ X (cid:90) u ∈ R ( µ A ) { ( x, u ) : ν x A ( u ) ≥ β } for any β ∈ [0 , . Then, A β and A β are called the lower membership function and the uppermembership function of A respectively. Moreover, the α -cut set of [ A ] β is defined by [ A ] αβ := (cid:10) [ A β ] α , [ A β ] α (cid:11) (2.6) for any α ∈ [0 , , where A β and A β are two type-1 fuzzy numbers on X that appear when A is cutby β . Remark 2.7.
The β -cut set of a type-2 fuzzy set is also called the β -plane of it. Strictly speaking,the α -cut set of the β -cut set of a type-2 fuzzy set A should be represented as [[ A ] β ] α , but we willwrite it like (2.6) since that is annoying.The argument of type-2 fuzzy sets can be reduced to that of the level cut sets as with type-1sets. In fact, it is known [8] that the level cut sets make up the original type-2 fuzzy set A : A = (cid:90) x ∈ X (cid:32)(cid:90) u ∈ R ( µ A ( x )) ν x A ( u ) /u (cid:33)(cid:30) x = (cid:91) β ∈ [0 , β [ A ] β = (cid:91) β ∈ [0 , β (cid:91) α ∈ [0 , α [ A ] αβ where α [ A ] αβ : X → { , α } is a type-1 fuzzy set. Thus, it is sufficient to consider and argue β -cutsets or these α -cut sets for most problems. We hereinafter write S A ( x ; β ) for the β -cut set of thesecondary membership function ν x A of A and do the same for B . We hereafter omit the description‘ α ∈ [0 , β ∈ [0 , α, β )-cut sets. Definition 2.13.
Let A and B be type-2 fuzzy numbers on X . We denote the β -cut sets of themby [ A ] β := (cid:10) A β , A β (cid:11) , [ B ] β := (cid:10) B β , B β (cid:11) . (2.7) Then, A = B if and only if S A ( x ; β ) = S B ( x ; β ) for all x ∈ X and any β ∈ [0 , . Definition 2.14 ([8]) . Let A and B be type-2 fuzzy numbers on X and k ∈ R . We denote the β -cut sets of A and B by (2.7). The sum A + B of A and B is defined by [ A + B ] β := (cid:10) [ A β + B β ] α , [ A β + B β ] α (cid:11) = (cid:10) [ A β ] α + [ B β ] α , [ A β ] α + [ B β ] α (cid:11) . Moreover, the scalar multiple k A of A is defined by [ k A ] β := (cid:10) [ kA β ] α , [ kA β ] α (cid:11) = (cid:10) k [ A β ] α , k [ A β ] α (cid:11) . Definition 2.15.
Let A , B be type-2 fuzzy numbers on X . We denote the β -cut sets of them by(2.7). Then, the order relationship between them, A ≤ B , is defined as [ A ] β ≤ [ B ] β SUSK for any β ∈ [0 , . In particular, the non-negativity (resp. positivity) of a type-2 fuzzy number A , A ≥ (resp. A > ), is defined by [ A β ] α ≥ A β ] α ≥ . [ A β ] α > A β ] α > for any α, β ∈ [0 , . The following concepts are important to consider the type-2 version of a type-1 triangular fuzzynumber and it is effective when we actually solve concrete T2FIVPs in Section 4.
Definition 2.16 ([13]) . Let A be a type-2 fuzzy number on X . The union of all secondary domains FP( A ) := (cid:91) x ∈ X R ( µ A ( x )) of A is called the foot-print set (or foot-print of uncertainty) of A . Definition 2.17 ([8], Definition 2.3.9) . Let A be a type-2 fuzzy number on X . Suppose that thereexists at least one u ∈ R ( µ A ( x )) satisfying ν x A ( u ) = ν A ( x, u ) = 1 for any x ∈ X . If we rewrite u x for each such point u ∈ R ( µ A ( x )) , every u x is equal to themembership function value of the type-1 fuzzy set which is uniquely determined. Then, that type-1fuzzy set is called the principle set of A and is denoted by P( A ) . Definition 2.18 ([8], Section 3.4) . Let
A ∈ T ( R ) . A is perfect if and only ifi) the upper and lower membership functions of FP( A ) are equal as type-1 fuzzy numbers, andii) the upper and lower membership functions of P( A ) are equal as type-1 fuzzy numbers.Moreover, if a perfect A also satisfies thatiii) A can be completely determined by using its FP( A ) and P( A ) ,such a A is called the perfect quasi-type-2 fuzzy set on R . We write QT ( R ) for the space of them. Example 2.1.
Consider
A ∈ T ( R ) such that • Primary: µ A ( x ) = max { − | x − | , } , • Secondary: ν x A ( u ) = max { − | u − x | , } = max (cid:110) − (cid:12)(cid:12)(cid:12) u − max { − | x − | , } (cid:12)(cid:12)(cid:12) , (cid:111) (0 ≤ u ≤ . Then, the lower membership function of FP( A ) is given bymax (cid:110) − (cid:12)(cid:12)(cid:12) u − max { − | x − | , } (cid:12)(cid:12)(cid:12) , (cid:111) = 0 , whereas, for e.g. x ∈ [1 , (cid:110) − (cid:12)(cid:12)(cid:12) u − | x − | (cid:12)(cid:12)(cid:12) , (cid:111) = 0implies u = 1 − | x − | ± . The lower membership function of FP( A ) in this case, thus, is given by u = 910 − | x − | < . Hence, this is not normal, so u / ∈ T . This thing implies that A is not perfect. ype-2 Fuzzy Initial Value Problems for Second-order T2FDEs A ∈ QT ( R ) is triangular if and only if [ A ] αβ has[ A β ] α = (cid:104) L αA β , R αA β (cid:105) , (2.8) L αA β = X αA − (1 − β ) (cid:16) X αA − L αA (cid:17) , (2.9) R αA β = Y αA + (1 − β ) (cid:16) R αA − Y αA (cid:17) , (2.10) L αA = C A − (1 − α ) (cid:0) C A − L A (cid:1) , (2.11) R αA = C A + (1 − α ) (cid:0) R A − C A (cid:1) (2.12)and [ A β ] α = (cid:104) L αA β , R αA β (cid:105) , (2.13) L αA β = X αA − (1 − β ) (cid:16) X αA − L αA (cid:17) , (2.14) R αA β = Y αA + (1 − β ) (cid:16) R αA − Y αA (cid:17) , (2.15) L αA = C A − (1 − α ) (cid:0) C A − L A (cid:1) , (2.16) R αA = C A + (1 − α ) (cid:0) R A − C A (cid:1) (2.17)where X αA = C A − (1 − α )( C A − X A ) , (2.18) Y αA = C A + (1 − α )( Y A − C A ) (2.19)are, in this paper, called the left principle number and right principle number of A respectivelyand C A stands for the core of A , that is, the crisp value [ A ] . These meet L αA ≤ X αA ≤ L αA ≤ C A ≤ R αA ≤ Y αA ≤ R αA . (See Figure 1 later.) In particular, the supports of A ,[ A ] α = (cid:104) L αA , R αA (cid:105) and [ A ] α = (cid:104) L αA , R αA (cid:105) , represent the α -cut sets of the lower and upper membership functions of FP( A ) respectively. Also,[ A ] α = (cid:2) X αA , Y αA (cid:3) is the α -cut set of P( A ). The triangular type-1 number u is determined by its left end l , core c and right end r : u = (cid:104)(cid:104) l, c, r (cid:105)(cid:105) , but the triangular perfect quasi-type-2 fuzzy number A is determined by its upper left end L A , leftprinciple number X A , lower left end L A , core C A , lower right end R A , right principle number Y A and upper right end R A : A = (cid:104)(cid:104) L A , X A , L A ; C A ; R A , Y A , R A (cid:105)(cid:105) . We here reconfirm the significance of type-2 fuzzy numbers. For example, let us denote3 ∼ = ‘about 3’0 SUSK
Figure 1: A view of A from directly aboveby a triangular fuzzy number. Then, the core of 3 ∼ is of course 3, but how should we determine theleft and right ends of it? The simple representation of 3 ∼ is the (more precise) isosceles triangularfuzzy number 3 ∼ = (cid:104)(cid:104) − δ, , δ (cid:105)(cid:105) for some δ >
0. So it is important to determine this δ appropriately, but it is generally difficult todetermine δ objectively. By setting 3 − δ (resp. 3 + δ ) as the left (resp. right) principle numberand reconsidering ‘about 3’ as the triangular perfect quasi-type-2 fuzzy number, (2.8)-(2.10) and(2.13)-(2.15) thus determine subjective δ . Herein lies the necessity and usefulness of type-2 fuzzynotion. We introduce the following Hung-Yang distance so as to consider the type-2 fuzzy topology.
Definition 2.19 ([9]) . Let A , B be type-2 fuzzy sets on X . Then, we set a distance between A and B as d HY ( A , B ) := (cid:90) ba H f ( κ A ( x ) , κ B ( x )) dx (2.20) where H f ( κ A ( x ) , κ B ( x )) := (cid:82) βd H ( S A ( x ; β ) , S B ( x ; β )) dβ (cid:82) β dβ = 2 (cid:90) βd H ( S A ( x ; β ) , S B ( x ; β )) dβ and the above integrals are defined in the sense of Riemann. We denote the space of type-2 fuzzynumbers on X equipped with d HY -topology by T ( X ) . Remark 2.8.
See Theorem 2.3 of [16] to make sure T ( X ) is a crisp metric space, that is, d HY satisfies the metric axiom. Definition 2.20 ([16], Definition 4.1) . Let A , B ∈ T ( X ) . If there exists some C ∈ T ( X ) suchthat A = B + C , we call C the T2-Hukuhara difference of A and B . Then, we write C as A − B as with type-1. ype-2 Fuzzy Initial Value Problems for Second-order T2FDEs Theorem 2.3 ([16], Theorem 4.1) . Let A , B ∈ T ( X ) . We denote the β -cut sets of them by (2.7).Then, the β -cut set of the T2-Hukuhara difference of A and B is the T1-Hukuhara difference ofthe upper and lower membership functions of A , B : [ A − B ] β = (cid:68) ( A − B ) β , ( A − B ) β (cid:69) = (cid:10) A β − B β , A β − B β (cid:11) . Zadeh’s extension principle derives the type-2 fuzzy number-valued function F : T ( I ) → T ( R ) via a crisp function f : I → R in the same way as type-1. We consider, in this paper,the case of T ( I ) = I . Also, type-2 fuzzy number-valued functions are simply called type-2 fuzzyfunctions. If the β -cut set of F : I → T ( R ) is represented by[ F ( x )] β := (cid:10) [ F β ( x )] α , [ F β ( x )] α (cid:11) , we write [ F β ( x )] α := [ F β, − ,α ( x ) , F β, + ,α ( x )] , [ F β ( x )] α := [ F β, − ,α ( x ) , F β, + ,α ( x )]for all x ∈ I and any α, β ∈ [0 , Definition 2.21 ([16], Definition 4.4) . Let F : I → T ( R ) and h > be a crisp number. F isT2-differentiable in the first form at some x ∈ I if and only if there exist F ( x + h ) − F ( x ) and F ( x ) − F ( x − h ) satisfying that the fuzzy limit F † ( x ) := lim h ↓ F ( x + h ) − F ( x ) h = lim h ↓ F ( x ) − F ( x − h ) h (2.21) exists. Moreover, F is T2-differentiable in the second form at some x ∈ I if and only if thereexist F ( x ) − F ( x + h ) and F ( x − h ) − F ( x ) satisfying that the fuzzy limit F ‡ ( x ) := lim h ↑ F ( x ) − F ( x + h ) − h = lim h ↑ F ( x − h ) − F ( x ) − h (2.22) exists. Here the above differences (resp. limits) are due to the meaning of T2-Hukuhara (resp. d HY ). If F is T2-differentiable in both senses at any x ∈ I , F † and F ‡ is called the (1)-T2-derivative and (2)-T2-derivative of F , respectively. Remark 2.9.
1) Like Remark 2.4, we shall ignore T2-derivatives in the third and fourth forms. The limits ofboth the forms become crisp numbers as with type-1. This thing has already been mentionedin Note 4.1 of [16].2) As with type-1, second-order T2-derivatives are obtained by applying first-order T2-derivativesto (2.21) and (2.22).In what follows, F †† is called the (1,1)-T2-derivative of F , and the other cases are similar. Theorem 2.4 ([16], Theorem 4.2) . Let F : I → T ( R ) be T2-differentiable on I . Then, theparametric forms of its T2-derivatives are given by1) the (1)-parametric form: [ F † ( x )] αβ = (cid:68) [ F † β ( x )] α , [ F † β ( x )] α (cid:69) = (cid:68) [ F (cid:48) β, − ,α ( x ) , F (cid:48) β, + ,α ( x )] , [ F (cid:48) β, − ,α ( x ) , F (cid:48) β, + ,α ( x )] (cid:69) , SUSK
2) the (2)-parametric form: [ F ‡ ( x )] αβ = (cid:68) [ F ‡ β ( x )] α , [ F ‡ β ( x )] α (cid:69) = (cid:68) [ F (cid:48) β, + ,α ( x ) , F (cid:48) β, − ,α ( x )] , [ F (cid:48) β, + ,α ( x ) , F (cid:48) β, − ,α ( x )] (cid:69) . Theorem 2.5 ([16], Corollary 4.1) . Let F : I → QT ( R ) be triangular, that is, F ( x ) = (cid:104)(cid:104) L F ( x ) , X F ( x ) , L F ( x ) ; C F ( x ) ; R F ( x ) , Y F ( x ) , R F ( x ) (cid:105)(cid:105) .
1) If F is (1)-T2-differentiable on I , then F † ( x ) = (cid:104)(cid:104) L F † ( x ) , X F † ( x ) , L F † ( x ) ; C F † ( x ) ; R F † ( x ) , Y F † ( x ) , R F † ( x ) (cid:105)(cid:105) .
2) If F is (2)-T2-differentiable on I , then F ‡ ( x ) = (cid:104)(cid:104) R F ‡ ( x ) , Y F ‡ ( x ) , R F ‡ ( x ) ; C F ‡ ( x ) ; L F ‡ ( x ) , X F ‡ ( x ) , L F ‡ ( x ) (cid:105)(cid:105) . Theorem 3.1.
Let F : I → T ( R ) be second-order T2-differentiable on I . Then, the parametricforms of its second-order T2-derivatives are given by1) the (1,1)-parametric form: [ F †† ( x )] αβ = (cid:68) [ F †† β ( x )] α , [ F †† β ( x )] α (cid:69) = (cid:68) [ F (cid:48)(cid:48) β, − ,α ( x ) , F (cid:48)(cid:48) β, + ,α ( x )] , [ F (cid:48)(cid:48) β, − ,α ( x ) , F (cid:48)(cid:48) β, + ,α ( x )] (cid:69) ,
2) the (1,2)-parametric form: [ F †‡ ( x )] αβ = (cid:68) [ F †‡ β ( x )] α , [ F †‡ β ( x )] α (cid:69) = (cid:68) [ F (cid:48)(cid:48) β, + ,α ( x ) , F (cid:48)(cid:48) β, − ,α ( x )] , [ F (cid:48)(cid:48) β, + ,α ( x ) , F (cid:48)(cid:48) β, − ,α ( x )] (cid:69) ,
3) the (2,1)-parametric form: [ F ‡† ( x )] αβ = (cid:68) [ F ‡† β ( x )] α , [ F ‡† β ( x )] α (cid:69) = (cid:68) [ F (cid:48)(cid:48) β, + ,α ( x ) , F (cid:48)(cid:48) β, − ,α ( x )] , [ F (cid:48)(cid:48) β, + ,α ( x ) , F (cid:48)(cid:48) β, − ,α ( x )] (cid:69) ,
4) the (2,2)-parametric form: [ F ‡‡ ( x )] αβ = (cid:68) [ F ‡‡ β ( x )] α , [ F ‡‡ β ( x )] α (cid:69) = (cid:68) [ F (cid:48)(cid:48) β, − ,α ( x ) , F (cid:48)(cid:48) β, + ,α ( x )] , [ F (cid:48)(cid:48) β, − ,α ( x ) , F (cid:48)(cid:48) β, + ,α ( x )] (cid:69) . Proof.
The proof can be obtained by substituting F † or F ‡ for F in Theorem 2.4. Theorem 3.2.
Let F : I → QT ( R ) be triangular, that is, F ( x ) = (cid:104)(cid:104) L F ( x ) , X F ( x ) , L F ( x ) ; C F ( x ) ; R F ( x ) , Y F ( x ) , R F ( x ) (cid:105)(cid:105) . ype-2 Fuzzy Initial Value Problems for Second-order T2FDEs
1) If F is (1,1)-T2-differentiable on I , then F †† ( x ) = (cid:104)(cid:104) L F †† ( x ) , X F †† ( x ) , L F †† ( x ) ; C F †† ( x ) ; R F †† ( x ) , Y F †† ( x ) , R F †† ( x ) (cid:105)(cid:105) .
2) If F is (1,2)-T2-differentiable on I , then F †‡ ( x ) = (cid:104)(cid:104) R F †‡ ( x ) , Y F †‡ ( x ) , R F †‡ ( x ) ; C F †‡ ( x ) ; L F †‡ ( x ) , X F †‡ ( x ) , L F †‡ ( x ) (cid:105)(cid:105) .
3) If F is (2,1)-T2-differentiable on I , then F ‡† ( x ) = (cid:104)(cid:104) R F ‡† ( x ) , Y F ‡† ( x ) , R F ‡† ( x ) ; C F ‡† ( x ) ; L F ‡† ( x ) , X F ‡† ( x ) , L F ‡† ( x ) (cid:105)(cid:105) .
4) If F is (2,2)-T2-differentiable on I , then F ‡‡ ( x ) = (cid:104)(cid:104) L F ‡‡ ( x ) , X F ‡‡ ( x ) , L F ‡‡ ( x ) ; C F ‡‡ ( x ) ; R F ‡‡ ( x ) , Y F ‡‡ ( x ) , R F ‡‡ ( x ) (cid:105)(cid:105) . Proof.
The proof can be obtained by substituting F † or F ‡ for F in Theorem 2.5. Definition 3.1.
Let F : I → T ( R ) and h > be a crisp number. F is T2-continuous on I ifand only if there exists the limit in d HY : lim h ↓ {F ( x + h ) − F ( x ) } = lim h ↓ {F ( x ) − F ( x − h ) } = 0 (3.1) for any x ∈ I . We write C ( I ; T ( R )) for the space of T2-continuous fuzzy functions. Theorem 3.3. If F : I → T ( R ) is T2-differentiable on I , then F is T2-continuous on I .Proof. F is T2-differentiable on I by the assumption, so the limitlim h ↓ F ( x + h ) − F ( x ) h = lim h ↓ F ( x ) − F ( x − h ) h exists. Also, we can denote F ( x + h ) − F ( x ) = F ( x + h ) − F ( x ) h h (3.2) F ( x ) − F ( x − h ) = F ( x ) − F ( x − h ) h h (3.3)for any x ∈ I . Hence, we have (3.1) by letting h ↓ F : I → T ( R ) is second-order T2-differentiable on I in the samecase of differentiability, if F † and F ‡ are (1)-T2-differentiable and (2)-T2-differentiable on I , re-spectively. Theorem 3.4.
Let F , G : I → T ( R ) be second-order T2-differentiable on I in the same case ofdifferentiability. Then, F + G : I → T ( R ) is second-order T2-differentiable on I and ( F + G ) † ( x ) = F † ( x ) + G † ( x ) , (3.4)( F + G ) ‡ ( x ) = F ‡ ( x ) + G ‡ ( x ) , (3.5)( F + G ) †† ( x ) = F †† ( x ) + G †† ( x ) , (3.6)( F + G ) ‡‡ ( x ) = F ‡‡ ( x ) + G ‡‡ ( x ) (3.7) for x ∈ I . Moreover, if there exists the T2-Hukuhara difference F − G , then all of (3.4)-(3.7) holdeven if + is replaced with the T2-Hukuhara difference − . SUSK
Proof.
It is obvious for sums from Definition 2.21. We prove only(
F − G ) † ( x ) = F † ( x ) − G † ( x ) , (3.8)( F − G ) †† ( x ) = F †† ( x ) − G †† ( x ) (3.9)for x ∈ I . The other two cases can be shown in the same way. Since F − G exists, there is sometype-2 fuzzy function W : I → T ( R ) such that F ( x ) = G ( x ) + W ( x ) , F ( x ± h ) = G ( x ± h ) + W ( x ± h )where h > F ( x + h ) − F ( x ) = ( G ( x + h ) + W ( x + h )) − ( G ( x ) + W ( x ))= ( G ( x + h ) − G ( x )) + ( W ( x + h ) − W ( x )) (3.10)by virtue of Lemma 3.1 mentioned later. Similarly, we also have F ( x ) − F ( x − h ) = ( G ( x ) − G ( x − h )) + ( W ( x ) − W ( x − h )) . (3.11) F and G are T2-differentiable on I , so F ( x + h ) − F ( x ), F ( x ) − F ( x − h ), G ( x + h ) − G ( x ) and G ( x ) − G ( x − h ) exist for any x ∈ I . Thus, the following derivative in d HY exists:( F − G ) † ( x ) = lim h ↓ W ( x + h ) − W ( x ) h = lim h ↓ W ( x ) − W ( x − h ) h , x ∈ I. This limit is F † ( x ) − G † ( x ), x ∈ I , from (3.10) and (3.11). Hence we have gained (3.8). (3.9) canbe obtained by repeating the above discussion for ( F − G ) † . This completes the proof. Lemma 3.1.
Let u j , v j ∈ T ( R ) ( j = 1 , . Suppose that ( u + v ) − ( u + v ) , u − u and v − v exist. Then, the distributive law for T1-Hukuhara differences in the following sense hold: ( u + v ) − ( u + v ) = ( u − u ) + ( v − v ) . Moreover, the same result holds for type-2 fuzzy numbers.Proof.
We prove only for type-1 fuzzy numbers, since the definition of T2-Hukuhara differencesfor type-2 fuzzy numbers is essentially equal to that for type-1 fuzzy numbers. The T1-Hukuharadifference is the difference between the left ends and the right ends of two intervals. Let the α -cutsof u j , v j ( j = 1 ,
2) be represented as[ u j ] α := [ u j, − ( α ) , u j, + ( α )] , [ v j ] α := [ v j, − ( α ) , v j, + ( α )] . Then, we have[( u + v ) − ( u + v )] α = [ u + v ] α − [ u + v ] α = ([ u ] α + [ v ] α ) − ([ u ] α + [ v ] α )= ([ u , − ( α ) , u , + ( α )] + [ v , − ( α ) , v , + ( α )]) − ([ u , − ( α ) , u , + ( α )] + [ v , − ( α ) , v , + ( α )])= [ u , − ( α ) + v , − ( α ) , u , + ( α ) + v , + ( α )] − [ u , − ( α ) + v , − ( α ) , u , + ( α ) + v , + ( α )]= [( u , − ( α ) − u , − ( α )) + ( v , − ( α ) − v , − ( α )) , ( u , + ( α ) − u , + ( α )) + ( v , + ( α ) − v , + ( α ))]= [ u , − ( α ) − u , − ( α ) , u , + ( α ) − u , + ( α )] + [ v , − ( α ) − v , − ( α ) , v , + ( α ) − v , + ( α )]= [ u − u ] α + [ v − v ] α = [( u − u ) + ( v − v )] α , so the claim of this lemma has been obtained. ype-2 Fuzzy Initial Value Problems for Second-order T2FDEs Remark 3.1.
We can generally have the similar results to Theorem 3.4 for any order N ∈ N ∪ { } . Theorem 3.5.
Let F , G : I → T ( R ) be second-order T2-differentiable on I such that • if F is (1,1)-T2-differentiable then G is (2,1)-T2-differentiable, or • if F is (1,2)-T2-differentiable then G is (2,2)-T2-differentiable, or • if F is (2,1)-T2-differentiable then G is (1,1)-T2-differentiable, or • if F is (2,2)-T2-differentiable then G is (1,2)-T2-differentiable.Suppose that there exists the T2-Hukuhara difference F ( x ) − G ( x ) for any x ∈ I . Then, F − G : I → T ( R ) is second-order T2-differentiable on I and ( F − G ) † ( x ) = F † ( x ) + ( − G ‡ ( x ) , (3.12)( F − G ) ‡ ( x ) = F ‡ ( x ) + ( − G † ( x ) , (3.13)( F − G ) †† ( x ) = F †† ( x ) + ( − G ‡† ( x ) , (3.14)( F − G ) †‡ ( x ) = F †‡ ( x ) + ( − G ‡‡ ( x ) , (3.15)( F − G ) ‡† ( x ) = F ‡† ( x ) + ( − G †† ( x ) , (3.16)( F − G ) ‡‡ ( x ) = F ‡‡ ( x ) + ( − G †‡ ( x ) (3.17) for x ∈ I .Proof. We prove only (3.12) and (3.14) in the first case. The other cases and equations can besimilarly proved.Let us first prove (3.12) in the first case. Since, on I , F is (1)-T2-differentiable and G is (2)-T2-differentiable, F ( x + h ) − F ( x ), F ( x ) − F ( x − h ), G ( x ) − G ( x + h ) and G ( x − h ) − G ( x ) existfor any x ∈ I . Moreover, there exists a type-2 fuzzy number-valued function W such that F ( x + h ) = F ( x ) + W ( x, h ) , F ( x ) = F ( x − h ) + W ( x, h ) , for h >
0, with lim h ↓ W ( x, h ) h = lim h ↓ W ( x, h ) h = F † ( x )for x ∈ I and G ( x ) = G ( x + h ) + W ( x, h ) , G ( x − h ) = G ( x ) + W ( x, h ) , for h >
0, with lim h ↓ W ( x, h ) h = lim h ↓ W ( x, h ) h = ( − G ‡ ( x )for x ∈ I . Thus, we have F ( x + h ) + G ( x ) = F ( x ) + G ( x + h ) + W ( x, h ) + W ( x, h ) , but we obtain F ( x + h ) − G ( x + h ) = F ( x ) − G ( x ) + W ( x, h ) + W ( x, h )6 SUSK since F ( x ) − G ( x ) exists for any x ∈ I by the assumption. This implies that {F ( x + h ) − G ( x + h ) } − {F ( x ) − G ( x ) } exists and {F ( x + h ) − G ( x + h ) } − {F ( x ) − G ( x ) } h = W ( x, h ) h + W ( x, h ) h (3.18)for any x ∈ I . Similarly, we can find the existence of {F ( x ) − G ( x ) } − {F ( x − h ) − G ( x − h ) } andobtain {F ( x ) − G ( x ) } − {F ( x − h ) − G ( x − h ) } h = W ( x, h ) h + W ( x, h ) h (3.19)for any x ∈ I . Both the left-hand sides of (3.18) and (3.19) have the common limit ( F − G ) † as h ↓
0. Similarly, both the right-hand sides of (3.18) and (3.19) have the common limit F † + ( − G ‡ as h ↓
0. Hence, we have obtained (3.12).Let us next prove (3.14) in the first case. We can however derive it from (3.12) and Theorem3.4 immediately.This completes the proof.
Theorem 3.6.
Let f : I → R be second-order differentiable and G : I → T ( R ) be second-orderT2-differentiable on I . Then,1) if f ( x ) f (cid:48) ( x ) > and G is (1)-T2-differentiable, then f G is (1)-T2-differentiable and ( f G ) † ( x ) = f (cid:48) ( x ) G ( x ) + f ( x ) G † ( x ) (3.20) for x ∈ I ;2) if f ( x ) f (cid:48) ( x ) < and G is (2)-T2-differentiable, then f G is (2)-T2-differentiable and ( f G ) ‡ ( x ) = f (cid:48) ( x ) G ( x ) + f ( x ) G ‡ ( x ) (3.21) for x ∈ I ;3) if f ( x ) f (cid:48) ( x ) > , f (cid:48) ( x ) f (cid:48)(cid:48) ( x ) > and G is (1,1)-T2-differentiable, then f G is (1,1)-T2-differentiable and ( f G ) †† ( x ) = f (cid:48)(cid:48) ( x ) G ( x ) + 2 f (cid:48) ( x ) G † ( x ) + f ( x ) G †† ( x ) (3.22) for x ∈ I ;4) if f ( x ) f (cid:48) ( x ) < , f (cid:48) ( x ) f (cid:48)(cid:48) ( x ) < and G is (2,2)-T2-differentiable, then f G is (2,2)-T2-differentiable and ( f G ) ‡‡ ( x ) = f (cid:48)(cid:48) ( x ) G ( x ) + 2 f (cid:48) ( x ) G ‡ ( x ) + f ( x ) G ‡‡ ( x ) (3.23) for x ∈ I .Proof. We prove only 2) and 4) because the other two cases are proved in the same way.2) We suppose that f ( x ) > f (cid:48) ( x ) <
0. Let h > f isdifferentiable, there exist ε j ( x, h ) ( j = 1 ,
2) such that f ( x ) = f ( x + h ) + ε ( x, h ) , (3.24) f ( x − h ) = f ( x ) + ε ( x, h ) . (3.25)(Recall the first-order approximation of the differentiable function.) Remark that ε ( x, h ) > ε ( x, h ) = f ( x ) − f ( x + h ) and f is monotone decreasing. The same applies to ε ( x, h ). ype-2 Fuzzy Initial Value Problems for Second-order T2FDEs G is (2)-T2-differentiable on I , G ( x ) − G ( x + h ) and G ( x − h ) − G ( x ) exist for any x ∈ I ,that is, there exist type-2 fuzzy functions W j ( j = 1 ,
2) such that G ( x ) = G ( x + h ) + W ( x, h ) , G ( x − h ) = G ( x ) + W ( x, h ) . We thus have G ( x ) = G ( x + h ) + W ( x, h ) , (3.26) G ( x − h ) = G ( x ) + W ( x, h ) . (3.27)One has f ( x ) G ( x ) = { f ( x + h ) + ε ( x, h ) }{G ( x + h ) + W ( x, h ) } = f ( x + h ) G ( x + h ) + f ( x + h ) W ( x, h ) + ε ( x, h ) G ( x + h ) + ε ( x, h ) W ( x, h )by virtue of (3.24) and (3.26), so f ( x ) G ( x ) − f ( x + h ) G ( x + h ) exists and f ( x ) G ( x ) − f ( x + h ) G ( x + h ) − h = f ( x + h ) W ( x, h ) − h + ε ( x, h ) − h G ( x + h ) + ε ( x, h ) − h W ( x, h ) (3.28)for any x ∈ I . Similarly, from (3.25) and (3.27), f ( x − h ) G ( x − h ) − f ( x ) G ( x ) exists and wealso have f ( x − h ) G ( x − h ) − f ( x ) G ( x ) − h = f ( x − h ) W ( x, h ) − h + ε ( x, h ) − h G ( x − h ) + ε ( x, h ) − h W ( x, h ) (3.29)for any x ∈ I . Theorem 3.3 implies G ∈ C ( I ; T ( R )), so W j ( x, h ) → j = 1 ,
2) as h ↓ h ↓ f ( x ) < f (cid:48) ( x ) >
0. Hence, (3.21) has been derived.4) (3.23) is immediately proved by repeating (3.21).This completes the proof.At the end of this section, we deal with the composition of crisp and type-1 / type-2 fuzzyfunctions and its derivative.
Theorem 3.7.
Let f : I → R be differentiable and G : R → T ( R ) T1-differentiable. We considerthe type-1 fuzzy composite function G ◦ f : I → T ( R ) . Suppose that T1-Hukuhara differences ( G ◦ f )( x + h ) − ( G ◦ f )( x ) and ( G ◦ f )( x ) − ( G ◦ f )( x − h ) exist for any x ∈ I and h > sufficiently small. Then, G ◦ f is T1-differentiable on I and ( G ◦ f ) † = G † ( f ( x )) f (cid:48) ( x ) , (3.30)( G ◦ f ) ‡ = G ‡ ( f ( x )) f (cid:48) ( x ) . (3.31) Moreover, the same result holds for G : R → T ( R ) . SUSK
Proof.
We prove only for type-1 G because we can prove for type-2 G in the same way. Let h > α -cut set of the right difference quotient of ( G ◦ f ) is (cid:20) ( G ◦ f )( x + h ) − ( G ◦ f )( x ) h (cid:21) α = (cid:20) ( G ◦ f ) − ,α ( x + h ) − ( G ◦ f ) − ,α ( x ) h , ( G ◦ f ) + ,α ( x + h ) − ( G ◦ f ) + ,α ( x ) h (cid:21) = (cid:20) G − ,α ( f ( x + h )) − G − ,α ( f ( x )) h , G + ,α ( f ( x + h )) − G + ,α ( f ( x )) h (cid:21) = (cid:20) G − ,α ( f ( x + h )) − G − ,α ( f ( x )) f ( x + h ) − f ( x ) f ( x + h ) − f ( x ) h , G + ,α ( f ( x + h )) − G + ,α ( f ( x )) f ( x + h ) − f ( x ) f ( x + h ) − f ( x ) h (cid:21) . Similarly, the α -cut set of the left difference quotient of ( G ◦ f ) is (cid:20) ( G ◦ f )( x ) − ( G ◦ f )( x − h ) h (cid:21) α = (cid:20) G − ,α ( f ( x )) − G − ,α ( f ( x − h )) f ( x ) − f ( x − h ) f ( x ) − f ( x − h ) h , G + ,α ( f ( x )) − G + ,α ( f ( x − h )) f ( x ) − f ( x − h ) f ( x ) − f ( x − h ) h (cid:21) . The rightmost-hand sides of the above two equations converge as h ↓
0, since f ( x ± h ) → f ( x ) as h ↓ f . Thus (3.30) can be obtained. (3.31) is similar. Hence, thiscompletes the proof. We actually solve, in this section, some concrete type-2 fuzzy initial value problems for second-order T2FDEs. We write all first-order Hukuhara derivatives † and ‡ together as D. That is, D denotes all second-order Hukuhara derivatives †† , †‡ , ‡† and ‡‡ together. Italic numbers , , , , , , , , , stand for concrete type-2 fuzzy numbers. We abbreviate the type-2 fuzzy initial value problem(resp. condition) as T2FIVP (resp. T2FIVC) in this subsection.We consider T2FIVPs on I = [0 , r ] for some r > I = [0 , + ∞ ): D Y ( x ) + a D Y ( x ) + b Y ( x ) = 0 , Y (0) = U ∈ T ( R ) , D Y (0) = V ∈ T ( R ) . (4.1) Definition 4.1.
Let Y : I → T ( R ) be second-order T2-differentiable. We denote the β -cut setof Y = Y ( x ) by [ Y ( x )] β = (cid:10) Y β ( x ) , Y β ( x ) (cid:11) . Then, Y is the ( i, j ) -type-2 fuzzy solution of (4.1), ( i, j ) ∈ { , } , if and only if, for each β ∈ [0 , ,i) D i Y β , D i Y β , D i,j Y β , D i,j Y β exist on I , andii) Y β and Y β : I → T ( R ) satisfy [D i,j Y β ( x )] α + [ a D i Y β ( x )] α + [ bY β ( x )] α = 0 , x ∈ I,Y β (0) = u β ∈ T ( R ) , D i Y β (0) = v β ∈ T ( R ) ype-2 Fuzzy Initial Value Problems for Second-order T2FDEs and [D i,j Y β ( x )] α + [ a D i Y β ( x )] α + [ bY β ( x )] α = 0 , x ∈ I,Y β (0) = u β ∈ T ( R ) , D i Y β (0) = v β ∈ T ( R ) for any α ∈ [0 , , respectively. Remark 4.1.
The type-2 fuzzy solution generally becomes the type-1 fuzzy solution if β = 1; itbecomes the crisp solution if α = β = 1.We solve problems in the case of crisp coefficients. We have four candidate solutions for type-1fuzzy differential equations of order 2, but we have eight of them for type-2 ones of order 2.Recall Definition 2.4 and 2.14 for operations. We begin with easy problems, that is, type-2 fuzzy initial value problems of order 2 in the case ofcrisp coefficients.
Problem 4.1.
Let Y : [0 , → T ( R ) be a type-2 fuzzy function. Then, solve T2FIVPs: D Y ( x ) + 3D Y ( x ) = 0 , (4.2) Y (0) = ∈ QT ( R ) , (4.3)D Y (0) = ∈ QT ( R ) . (4.4) Solution.
We consider the ( α, β )-cut set of (4.2): (cid:68) [ Y †† β ( x )] α + 3[ Y † β ( x )] α , [ Y †† β ( x )] α + 3[ Y † β ( x )] α (cid:69) = 0 , α ∈ [0 , . This can be solved by (1,1) or (2,2)-T1-differentiation, so we have the following two T1FIVPs: [ Y †† β ( x )] α + 3[ Y † β ( x )] α = [ Y (cid:48)(cid:48) β, − ,α ( x ) + 3 Y (cid:48) β, − ,α ( x ) , Y (cid:48)(cid:48) β, + ,α ( x ) + 3 Y (cid:48) β, + ,α ( x )] = 0 , (4.5) Y β (0) = 5 β ∈ T ( R ) , (4.6) Y † β (0) = 1 β ∈ T ( R ) , (4.7)and [ Y †† β ( x )] α + 3[ Y † β ( x )] α = [ Y (cid:48)(cid:48) β, − ,α ( x ) + 3 Y (cid:48) β, − ,α ( x ) , Y (cid:48)(cid:48) β, + ,α ( x ) + 3 Y (cid:48) β, + ,α ( x )] = 0 , (4.8) Y β (0) = 5 β ∈ T ( R ) , (4.9) Y † β (0) = 1 β ∈ T ( R ) . (4.10)We can thus solve the given T2FIVP because the solution method of type-1 fuzzy differentialequations is well known.(4.5) can be solved in the form of Y β, ± ,α ( x ) = C , ± e − x + C , ± , (4.11)thus, Y (cid:48) β, ± ,α ( x ) = − C , ± e − x , (4.12)where these equations represent two equations, one for the upper sign and the other for the lowersign. Similarly, (4.8) can be solved in the form of Y β, ± ,α ( x ) = C , ± e − x + C , ± , (4.13)0 SUSK thus, Y (cid:48) β, ± ,α ( x ) = − C , ± e − x , (4.14)where these equations represent two equations, one for the upper sign and the other for the lowersign.Let us determine type-2 fuzzy initial value by setting them as the triangular quasi-type-2fuzzy number = (cid:104)(cid:104) . , , . . , , . (cid:105)(cid:105) . Since (2.16), (2.11), (2.12), (2.17), (2.18) and (2.19) imply that L α = 5 − (1 − α )(5 − .
5) = 32 α + 72 ,L α = 5 − (1 − α )(5 − .
5) = 12 α + 92 ,R α = 5 + (1 − α )(5 . −
5) = − α + 112 ,R α = 5 + (1 − α )(6 . −
5) = − α + 132 ,X α = 5 − (1 − α )(5 −
4) = α + 4 ,Y α = 5 + (1 − α )(6 −
5) = − α + 6 , we have L α β = ( α + 4) − (1 − β ) (cid:26) ( α + 4) − (cid:18) α + 92 (cid:19)(cid:27) = 12 α + 12 αβ − β + 92 ,R α β = ( − α + 6) + (1 − β ) (cid:26)(cid:18) − α + 112 (cid:19) − ( − α + 6) (cid:27) = − α − αβ + 12 β + 112and L α β = ( α + 4) − (1 − β ) (cid:26) ( α + 4) − (cid:18) α + 72 (cid:19)(cid:27) = 32 α − αβ + 12 β + 72 ,R α β = ( − α + 6) + (1 − β ) (cid:26)(cid:18) − α + 132 (cid:19) − ( − α + 6) (cid:27) = − α + 12 αβ − β + 132from (2.9)-(2.10) and (2.14)-(2.15). Therefore, it follows that[ Y β (0)] α = [5 β ] α = (cid:20) α + 12 αβ − β + 92 , − α − αβ + 12 β + 112 (cid:21) , (4.15)[ Y β (0)] α = [5 β ] α = (cid:20) α − αβ + 12 β + 72 , − α + 12 αβ − β + 132 (cid:21) (4.16)from (2.8) and (2.13). Next, let us determine type-2 fuzzy initial value by setting them as thetriangular quasi-type-2 fuzzy number = (cid:104)(cid:104)− . , , . . , , . (cid:105)(cid:105) . Therefore, it follows that[ Y † β (0)] α = [1 β ] α = (cid:20) α + 12 αβ − β + 12 , − α − αβ + 12 β + 32 (cid:21) , (4.17)[ Y † β (0)] α = [1 β ] α = (cid:20) α − αβ + 12 β − , − α + 12 αβ − β + 52 (cid:21) (4.18) ype-2 Fuzzy Initial Value Problems for Second-order T2FDEs Y β, − ,α ( x ) = (cid:18) − α − αβ + 16 β − (cid:19) e − x + (cid:18) α + 23 αβ − β + 143 (cid:19) , (4.19) Y β, + ,α ( x ) = (cid:18) α + 16 αβ − β − (cid:19) e − x + (cid:18) − α − αβ + 23 β + 6 (cid:19) . (4.20)Similarly, substituting (4.16) and (4.18) for (4.13), we have Y β, − ,α ( x ) = (cid:18) − α + 16 αβ − β + 16 (cid:19) e − x + (cid:18) α − αβ + 23 β + 103 (cid:19) , (4.21) Y β, + ,α ( x ) = (cid:18) α − αβ + 16 β − (cid:19) e − x + (cid:18) − α + 23 αβ − β + 223 (cid:19) . (4.22)Hence, the desired solution is composed of the ( α, β )-cut set[ Y ( x )] αβ = (cid:10)(cid:2) Y β, − ,α ( x ) , Y β, + ,α ( x ) (cid:3) , (cid:2) Y β, − ,α ( x ) , Y β, + ,α ( x ) (cid:3)(cid:11) with (4.19)-(4.22).The same result is obtained by solving the given problem in the (1,2)-case. (ANSWER COM-PLETED) Remark 4.2.
Letting α = β = 1 in (4.19)-(4.22), we have the crisp solution y ( x ) = − e − x + 163to the crisp IVP: y (cid:48)(cid:48) ( x ) + 3 y (cid:48) ( x ) = 0 , y (0) = 5 , y (cid:48) (0) = 1 . We can thus find that type-2 (or type-1) fuzzy differential equation theory is an extension to crispone. Also, the solution if ( α, β ) = (1 / , /
2) with the parametric forms[ Y / ( x )] / = (cid:20) − e − x + 143 , − e − x + 6 (cid:21) , [ Y / ( x )] / = (cid:20) − e − x + 389 , − e − x + 589 (cid:21) is as shown in Figure 2. Problem 4.2.
Let Y : [0 , → T ( R ) be a type-2 fuzzy function. Then, solve T2FIVPs: D Y ( x ) − Y ( x ) = 0 , (4.23) Y (0) = ∈ QT ( R ) , (4.24)D Y (0) = ∈ QT ( R ) . (4.25) Solution.
We consider the ( α, β )-cut set of (4.23): (cid:68) [ Y †† β ( x )] α − [ Y β ( x )] α , [ Y †† β ( x )] α − [ Y β ( x )] α (cid:69) = 0 , α ∈ [0 , . This can be solved by (1,1) or (2,2)-T1-differentiation. So, considering (1,1)-case, we have thefollowing two T1FIVPs: [ Y †† β ( x )] α − [ Y β ( x )] α = [ Y (cid:48)(cid:48) β, − ,α ( x ) − Y β, − ,α ( x ) , Y (cid:48)(cid:48) β, + ,α ( x ) − Y β, + ,α ( x )] = 0 , (4.26) Y β (0) = 5 β ∈ T ( R ) , (4.27) Y † β (0) = 1 β ∈ T ( R ) , (4.28)2 SUSK
Figure 2: The crisp and fuzzy solution of Problem 4.1 if ( α, β ) = (1 / , / [ Y †† β ( x )] α − [ Y β ( x )] α = [ Y (cid:48)(cid:48) β, − ,α ( x ) − Y β, − ,α ( x ) , Y (cid:48)(cid:48) β, + ,α ( x ) − Y β, + ,α ( x )] = 0 , (4.29) Y β (0) = 5 β ∈ T ( R ) , (4.30) Y † β (0) = 1 β ∈ T ( R ) . (4.31)That is, rewriting problems (4.26)-(4.28) and (4.29)-(4.31), we have Y β, − ,α ( x ) = C − e − x + D − e x , (4.32) Y β, − ,α (0) = 12 α + 12 αβ − β + 92 , (4.33) Y † β, − ,α (0) = 12 α + 12 αβ − β + 12 ; (4.34) Y β, + ,α ( x ) = C + e − x + D + e x , (4.35) Y β, + ,α (0) = − α − αβ + 12 β + 112 , (4.36) Y † β, + ,α (0) = − α − αβ + 12 β + 32 ; (4.37) Y β, − ,α ( x ) = C − e − x + D − e x , (4.38) Y β, − ,α (0) = 32 α − αβ + 12 β + 72 , (4.39) Y † β, − ,α (0) = 32 α − αβ + 12 β −
12 ; (4.40) ype-2 Fuzzy Initial Value Problems for Second-order T2FDEs Y β, + ,α ( x ) = C + e − x + D + e x , (4.41) Y β, + ,α (0) = − α + 12 αβ − β + 132 , (4.42) Y † β, + ,α (0) = − α + 12 αβ − β + 52 . (4.43)We can obtain Y β, ± ,α ( x ) and Y β, ± ,α ( x ) by solving these:[ Y β ( x )] α = [ Y β, − ,α ( x ) , Y β, + ,α ( x )]= (cid:20) e − x + 12 ( α + αβ − β + 5) e x , e − x + 12 ( − α − αβ + β + 7) e x (cid:21) , (4.44)[ Y β ( x )] α = [ Y β, − ,α ( x ) , Y β, + ,α ( x )]= (cid:20) e − x + 12 (3 α − αβ + β + 3) e x , e − x + 12 ( − α + αβ − β + 9) e x (cid:21) . (4.45)Hence, the desired solution is composed of the ( α, β )-cut[ Y ( x )] αβ = (cid:10)(cid:2) Y β, − ,α ( x ) , Y β, + ,α ( x ) (cid:3) , (cid:2) Y β, − ,α ( x ) , Y β, + ,α ( x ) (cid:3)(cid:11) . with (4.44) and (4.45).The same result is obtained by solving the given problem in the (2,2)-case. (ANSWER COM-PLETED) Remark 4.3.
Letting α = β = 1 in (4.44) and (4.45), we have the crisp solution y ( x ) = 2 e − x + 3 e x to the crisp IVP: y (cid:48)(cid:48) ( x ) − y (cid:48) ( x ) = 0 , y (0) = 5 , y (cid:48) (0) = 1 . We can thus find that type-2 (or type-1) fuzzy differential equation theory is an extension to crispone. Also, the solution if ( α, β ) = (1 / , /
2) with the parametric forms[ Y / ( x )] / = (cid:20) e − x + 52 e x , e − x + 72 e x (cid:21) , [ Y / ( x )] / = (cid:20) e − x + 136 e x , e − x + 236 e x (cid:21) is as shown in Figure 3. Problem 4.3.
Let Y : [0 , → T ( R ) be a type-2 fuzzy function. Then, solve T2FIVPs: D Y ( x ) + ( − Y ( x ) = 0 , (4.46) Y (0) = ∈ QT ( R ) , (4.47)D Y (0) = ∈ QT ( R ) . (4.48) Solution.
We consider the ( α, β )-cut set of (4.46): (cid:68) [ Y †† β ( x )] α + ( − Y β ( x )] α , [ Y †† β ( x )] α + ( − Y β ( x )] α (cid:69) = 0 , α ∈ [0 , . SUSK
Figure 3: The crisp and fuzzy solution of Problem 4.2 if ( α, β ) = (1 / , / [ Y †† β ( x )] α + ( − Y β ( x )] α = [ Y (cid:48)(cid:48) β, + ,α ( x ) − Y β, + ,α ( x ) , Y (cid:48)(cid:48) β, − ,α ( x ) − Y β, − ,α ( x )] = 0 , (4.49) Y β (0) = 5 β ∈ T ( R ) , (4.50) Y † β (0) = 1 β ∈ T ( R ) , (4.51)and [ Y †† β ( x )] α + ( − Y β ( x )] α = [ Y (cid:48)(cid:48) β, + ,α ( x ) − Y β, + ,α ( x ) , Y (cid:48)(cid:48) β, − ,α ( x ) − Y β, − ,α ( x )] = 0 , (4.52) Y β (0) = 5 β ∈ T ( R ) , (4.53) Y † β (0) = 1 β ∈ T ( R ) . (4.54)We thus gain the same lower and upper equations (4.32)-(4.43) as Problem 4.2. By solving problems(4.49)-(4.51) and (4.52)-(4.54), we can obtain Y β, ± ,α ( x ) and Y β, ± ,α ( x ) respectively. Hence, thedesired solution is the same as 4.2 and is composed of the ( α, β )-cut[ Y ( x )] αβ = (cid:10)(cid:2) Y β, − ,α ( x ) , Y β, + ,α ( x ) (cid:3) , (cid:2) Y β, − ,α ( x ) , Y β, + ,α ( x ) (cid:3)(cid:11) . with (4.44) and (4.45).The same result is obtained by solving the given problem in the (2,1)-case. (ANSWER COM-PLETED) ype-2 Fuzzy Initial Value Problems for Second-order T2FDEs Remark 4.4.
Problem 4.2 and 4.3 imply that Hukuhara differentiation solves the problem of howthe negative coefficients of T1/T2FDEs should be treated. As a result, the negative coefficientmay be unified into the Hukuhara difference.
Acknowledgement
The authors are deeply grateful to Dr. Jiro Inaida for his valuable and heartfelt advice on fuzzy numbertheory.
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The Authors : Norihiro Someyama ;Completed Ph.D. program without a Ph.D. degree, Major in Mathematics and MathematicalPhysics
Hiroaki Uesu ; Ph.D., Major in Mathematics and Information Theory
Kimiaki Shinkai ; Ph.D., Major in Mathematics and Mathematical Education