Dynamic Equations and Almost Periodic Fuzzy Functions on Time Scales 3031112350, 9783031112355

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Synthesis Lectures on Mathematics & Statistics

Chao Wang · Ravi P. Agarwal

Dynamic Equations and Almost Periodic Fuzzy Functions on Time Scales

Synthesis Lectures on Mathematics & Statistics Series Editor Steven G. Krantz, Department of Mathematics, Washington University, Saint Louis, MO, USA

This series includes titles in applied mathematics and statistics for cross-disciplinary STEM professionals, educators, researchers, and students. The series focuses on new and traditional techniques to develop mathematical knowledge and skills, an understanding of core mathematical reasoning, and the ability to utilize data in specific applications.

Chao Wang · Ravi P. Agarwal

Dynamic Equations and Almost Periodic Fuzzy Functions on Time Scales

Chao Wang Department of Mathematics Yunnan University Yunnan, China

Ravi P. Agarwal Department of Mathematics Texas A&M University-Kingsville Kingsville, TX, USA

ISSN 1938-1743 ISSN 1938-1751 (electronic) Synthesis Lectures on Mathematics & Statistics ISBN 978-3-031-11235-5 ISBN 978-3-031-11236-2 (eBook) https://doi.org/10.1007/978-3-031-11236-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

We dedicate this book to our wives Wei Du and Sadhna Agarwal

Preface

Stefan Hilger introduced the concept of time scales in 1988 which can be applied to unify the study of differential equations for continuous case and difference equations for discrete case (see [1]). Since then, time scale theory has developed rapidly (see [2]) and it was used to develop function calculus in various mathematical fields such as the calculus for real-valued functions (see [2, 4]), quaternion dynamic equations (see [5, 6]), measure theory (see [7]), set-valued functions (see [8, 9]) and fuzzy-valued functions (see [10, 11, 12]), etc. When studying the dynamical behavior of an object on time scales it is significant to unify a larger range of time scales to obtain more comprehensive results (see [13, 14, 15]). For a more accurate description of the real world phenomena, it is necessary to consider a number of uncertain factors and this leads naturally to fuzzy dynamical models (see [16, 17, 18, 19, 20, 21, 22, 23, 24]) and set-valued functions, fuzzy-valued functions and their related applications to dynamic equations on time scales (see [10, 11, 12]). Almost periodic theory was initiated by Bohr during the period 1923–1925 (see [25, 26]) which is an important theory to precisely describe almost periodic approximation phenomena in engineering, life sciences, information sciences and control theory (see [27, 28, 29, 30, 31, 32, 33]) and in particular it arises in celestial mechanics, bioengineering and electronic circuits (see [28, 30]). The study of almost periodicity of solutions on time scales was considered in [34, 35, 36]. In 2001, Park, Jung and Lee considered the existence and stability of almost periodic solutions for fuzzy functional differential equations (see [37]), then in 2004, Bede and Gal developed a theory of almost periodic fuzzy number-valued functions and studied a class of almost periodic fuzzy dynamical systems (see [38]). In 2016, Hong and Peng extended the concept of almost periodic functions to set-valued functions on periodic time scales and established the existence of almost periodic solutions for set-valued dynamic equations (see [39]). Unfortunately, the validity of all results involved above must be restricted to traditional periodic time scales (i.e., periodic time scale under a translation) which have a nice closedness under translations. It is worth noting that these results cannot be applied to some important irregular time scales like (−q)Z = {(−q)n : q > 1, n ∈ Z} ∪ {0} (which has important applications in 1 quantum theory), ±N 2 and T = Tn , the space of harmonic numbers, because the concept vii

viii

Preface

of almost periodic functions in these works become unsuitable and inapplicable for these irregular time scales which are without closedness under translations. To overcome this difficulty, in 2017, inspired by Advar’s work [40], Wang and Agarwal introduced the concept of relatively dense set under shift operators and proposed almost periodic stochastic processes. This development makes it possible to study almost periodic problems on many irregular time scales involving q-difference equations and more general types of dynamic equations (see [15, 41]). On the other hand, fuzzy arithmetic is also a complex and intractable problem including the multiplication and division of fuzzy numbers and fuzzy vectors. Fortunately, Stefanini et al., opened a feasible avenue to fuzzy arithmetic and introduced a generalized Hukuhara difference and division for interval and fuzzy arithmetic (see [42, 24]). Moreover, they also initiated the unidimensional and multidimensional boxes which motivated us to introduce fuzzy (box) vectors and establish the calculus of fuzzy vector-valued functions on time scales (see [43]). Further, in [43], we introduced a new multiplication determined by a determinant algorithm, based on which the classical fundamental formulas of calculus can be derived under fuzzy background, and this provides a powerful calculus tool to solve classical problems of multidimensional fuzzy dynamic systems in various research fields. We organize this book into six chapters. In Chap. 1, some necessary knowledge of interval and fuzzy arithmetic is presented. A generalization of the Hukuhara difference is introduced. First, the case of compact convex sets is investigated which are applied to generalize the Hukuhara difference of fuzzy numbers by using their compact and convex level-cuts. Moreover, a similar approach is presented to propose a generalization of division for real intervals and fuzzy numbers. Some applications are provided to solve interval and fuzzy linear equations and fuzzy differential equations. In Chap. 2, an embedding theorem for fuzzy multidimensional space is established and two new types of multiplication of fuzzy vectors are introduced and studied. In Chap. 3, we introduce the basic notions of gH--derivatives of fuzzy vector-valued functions on time scales and obtain their fundamental properties. Moreover, the -integral of fuzzy vector-valued functions is introduced and studied. Some basic results related to calculus of fuzzy vector-valued functions are established on time scales. In Chap. 4, some necessary knowledge of shift operators and a generalized periodic time scales is presented. A notion of shift almost periodic fuzzy vector-valued functions is addressed and studied on complete-closed time scales under non-translational shifts, some fundamental results of shift almost periodic fuzzy vector-valued functions are established. In Chap. 5, some basic results of fuzzy multidimensional spaces are demonstrated and a new division of multidimensional intervals and fuzzy vectors induced by a determinant algorithm is introduced and studied. In Chap. 6, we develop a theory of almost periodic fuzzy multidimensional dynamic systems on time scales and several applications are provided. In particular, a new type

Preface

ix

of fuzzy dynamic systems called fuzzy q-dynamic systems (i.e., fuzzy quantum dynamic systems) is proposed and studied. This book will establish an almost periodic theory of multidimensional fuzzy dynamic equations and fuzzy vector-valued functions on complete-closed time scales under nontranslational shifts including some commonly irregular time scales, and it involves an almost periodic theory of fuzzy functions on quantum-like time scales. Our results are not only effective on periodic time scales (i.e., T = Z, R or hZ, etc.) but also are valid 1 1 for irregular time scales q Z , −q Z , ±N 2 , N 3 , (−q)Z , etc. The book is written at a graduate level and is intended for university libraries. Graduate students and researchers working in the field of fuzzy dynamic equations on time scales will be able to stimulate further research. The book is also a good reference material for those undergraduates who are interested in fuzzy dynamic equations and functions on time scales and familiar with fuzzy sets and systems and ordinary differential equations. We acknowledge with gratitude the support of National Natural Science Foundation of China (11961077), CAS “Light of West China” Program of Chinese Academy of Sciences and Educational Reform Research Project of Yunnan University (No. 2021Y10). Kunming, China Kingsville, USA

Chao Wang Ravi P. Agarwal

Contents

1 Generalized Hukuhara Difference and Division for Interval and Fuzzy Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Generalized Hukuhara Difference and Properties . . . . . . . . . . . . . . . . . . . . . 1.2 The Case of Compact Intervals in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 g H -Difference of Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Support Functions and Fuzzy g H -Difference . . . . . . . . . . . . . . . . . . 1.3.2 A Decomposition of Fuzzy Numbers and g H -Difference . . . . . . . 1.3.3 Approximated Fuzzy g H -Difference . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Generalized Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 The Fuzzy Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Approximated Fuzzy g-Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Applications of g H -Difference and g-Division . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Interval and Fuzzy Algebraic Equations . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Interval and Fuzzy Differential Equations . . . . . . . . . . . . . . . . . . . . .

1 1 7 10 11 15 19 20 22 23 24 25 28

2 An Embedding Theorem and Multiplication of Fuzzy Vectors . . . . . . . . . . . . 2.1 An Embedding Theorem for Fuzzy Multidimensional Space . . . . . . . . . . . 2.2 Multiplication of Fuzzy Vectors in Fuzzy Multidimensional Space . . . . . .

29 36 41

3 Calculus of Fuzzy Vector-Valued Functions on Time Scales . . . . . . . . . . . . . . . 3.1 g H -Δ-Derivative of Fuzzy Vector-Valued Functions on Time Scales . . . . 3.2 Δ-Integral of Fuzzy Vector-Valued Functions on Time Scales . . . . . . . . . .

57 58 79

4 Shift Almost Periodic Fuzzy Vector-Valued Functions . . . . . . . . . . . . . . . . . . . . 4.1 Shift Operators on Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Shift Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Periodicity of Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Complete-Closed Time Scales Under Non-translational Shifts . . . . . . . . . . 4.3 Shift Almost Periodic Fuzzy Vector-Valued Functions . . . . . . . . . . . . . . . . .

85 85 85 89 94 97

xi

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Contents

5 Division of Fuzzy Vector-Valued Functions Depending on Determinant Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Basic Results of Fuzzy Multidimensional Spaces . . . . . . . . . . . . . . . . . . . . . 5.2 A New Division of Multidimensional Intervals and Fuzzy Vectors . . . . . . 5.3 Basic Results of Calculus of Fuzzy Vector-Valued Functions on Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Almost Periodic Generalized Fuzzy Multidimensional Dynamic Equations and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Almost Periodic Generalized Fuzzy Multidimensional Dynamic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Applications on Fuzzy Dynamic Equations and Models on Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115 115 116 122 127 127 140

Appendix: Almost Anti-periodic Discrete Oscillation . . . . . . . . . . . . . . . . . . . . . . .

151

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

183

1

Generalized Hukuhara Difference and Division for Interval and Fuzzy Arithmetic

1.1

Generalized Hukuhara Difference and Properties

In this Section, we will present some basic knowledge of generalized Hukuhara difference and properties which was established in the literature [75] and it will be used in our later chapters. Consider a metric vector space X with the induced topology and in particular the space X = Rn , n ≥ 1, of real vectors equipped with standard addition and scalar multiplication operations. Following Diamond and Kloeden (see [28]), denote by K (X) and KC (X) the spaces of nonempty compact and compact convex sets of X. Given two subsets A, B ⊆ X and k ∈ R, Minkowski addition and scalar multiplication are defined by A + B = {a + b|a ∈ A, b ∈ B} and k A = {ka|a ∈ A} and it is well known that addition is associative and commutative and with neutral element {0}. If k = −1, scalar multiplication gives the opposite −A = (−1)A = {−a|a ∈ A} but, in general, A + (−A)  = {0}, i.e., the opposite of A is not the inverse of A in Minkowski addition (unless A = {a} is a singleton). Minkowski difference is A − B = A + (−1)B = {a − b|a ∈ A, b ∈ B}. A first implication of this fact is that, in general, even if it is true that (A + C = B + C) ⇐⇒ A = B, addition/subtraction simplification is not valid, i.e., (A + B) − B  = A. To partially overcome this situation, Hukuhara [44] introduced the following H -difference: A  B = C ⇐⇒ A = B + C (1.1) and an important property of  is that A  A = {0}, ∀A ∈ K (X) and (A + B)  B = A, ∀A, B ∈ K (X); H -difference is unique, but a necessary condition for A  B to exist is that A contains a translate {c} + B of B. In general, A − B  = A  B.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 C. Wang and R. P. Agarwal, Dynamic Equations and Almost Periodic Fuzzy Functions on Time Scales, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-11236-2_1

1

2

1 Generalized Hukuhara Difference and Division for Interval and Fuzzy Arithmetic

From an algebraic point of view, the difference of two sets A and B may be interpreted both in terms of addition as in (1.1) or in terms of negative addition, i.e., A  B = C ⇐⇒ B = A + (−1)C,

(1.2)

where (−1)C is the opposite set of C. Conditions (1.1) and (1.2) are compatible to each other and this suggests a generalization of Hukuhara difference: Definition 1.1 (see [75]) Let A, B ∈ K (X); we define the generalized Hukuhara difference of A and B as the set C ∈ K (X) such that  (i) A = B + C, A g H B = C ⇐⇒ (1.3) or (ii) B = A + (−1)C. Proposition 1.1 (see [75]) (Unicity of A g H B) If C = A g H B exists, it is unique and if also A  B exists then A g H B = A  B. Proof If C = A g H B exists in case (i), we obtain C = A  B which is unique. Suppose that case (ii) is satisfied for C and D, i.e., B = A + (−1)C and B = A + (−1)D; then A + (−1)C = A + (−1)D =⇒ (−1)C = (−1)D =⇒ C = D. If case (i) is satisfied for C and case (ii) is satisfied for D, i.e., A = B + C and B = A + (−1)D, then B = B + C + (−1)D =⇒ {0} = C − D and this is possible only if C = D = {c} is a singleton. This completes the proof.  The generalized Hukuhara difference A g H B will be called the g H -difference of A and B. Remark 1.1 A necessary condition for A g H B to exist is that either A contains a translate of B (as for A  B) or B contains a translate of A. In fact, for any given c ∈ C, we get B + {c} ⊆ A from (i) or A + {−c} ⊆ B from (ii). Remark 1.2 It is possible that A = B + C and B = A + (−)C hold simultaneously; in this case, A and B translate into each other and C is a singleton. In fact, A = B + C implies B + {c} ⊆ A ∀c ∈ C and B = A + (−1)C implies A − {c} ⊆ B ∀c ∈ C i.e., A ⊆ B + {c}; it follows that A = B + {c} and B = A + {−c}. On the other hand, if c , c ∈ C then A = B + {c } = B + {c } and this requires c = c . Remark 1.3 If A g H B exists, then B g H A exists and B g H A = −(A g H B).

1.1

Generalized Hukuhara Difference and Properties

3

Proposition 1.2 (see [75]) The g H -difference g H has the following properties: (1) A g H A = {0}. (2) (a) (A + B) g H B = A; (b) A g H (A − B) = B; (c) A g H (A + B) = −B. (3) A g H B exists if and only if B g H A and (−B) g H (−A) exist and A g H B = (−B) g H (−A) = −(B g H A). (4) In general, B g H A = A g H B does not imply A = B; but A g H B = B g H A = C if and only if C = −C and, in particular, C = {0} if and only if A = B. (5) If B g H A exists then either A + (B g H A) = B or B − (B g H A) = A and both equalities hold if and only if B g H A is a singleton set. (6) If B g H A = C exists, then for all D ∈ K (X) either (B + D) g H A = C + D or B g H (A + D) = C − D. Proof Property 1 is immediate. To prove 2(a) if C = (A + B) g H B then either A + B = C + B or B = (A + B) + (−1)C = B + (A + (−1)C); in the first case it follows that C = A, in the second case A + (−1)C = {0} and A and C are singleton sets so A = C. With a similar argument, 2(b) and 2(c) can be proved. To prove the first part of (3) let C = A g H B according to case (i), i.e., A = B + C, then A = B − (−C) and B g H A = −C according to case (ii); if instead C = A g H B according to case (ii), i.e., B = A − C, then B = A + (−C) and B g H A = −C according to case (i); on the other hand, if A = B + C or B = A − C, then −A = −B + (−C) or −B = −A + C and this means (−B) g H (−A) = C. To see the first part of (4) consider for example the unidimensional case A = [a − , a + ], B = [b− , b+ ]; equality A − B = B − A is valid if a − + a + = b− + b+ and this does not require A = B (unless A and B are singletons). For the second part of (4), from (A g H B) = (B g H A) = C, considering the four combinations derived from (3), one of the following four cases is valid: (A = B + C and B = A + C) or (A = B + C and A = B − C) or (B = A + (−1)C and B = A + C) or (B = A + (−1)C and A = B + (−1)C); in all of them we deduce C = −C and C = {0} if and only if A = B. To see (5), consider that if (B g H A) exists in the sense of (i) the first equality is valid and if it exists in the sense of (ii) the second one is valid. To prove (6), if B = A + C then B + D = A + C + D and (B + D) g H A = C + D according to case (i); if A = B − C then A + D = B − C + D = B − (C − D) and B g H (A + D) = C − D according to  case (ii). Remark 1.4 The equivalence (A  B) = C ⇐⇒ (A  C) = B is valid only for , or for g H in case (i). In fact, if A g H B = C in the sense (ii), then B = A − C and this does not imply A = B + C nor C = A − B unless B = B + C − C (i.e., C = { c}) or B = B + A − a }). Note also that, in general, A + (B g H A)  = A and A − (A g H B)  = B. A (i.e., A = {

4

1 Generalized Hukuhara Difference and Division for Interval and Fuzzy Arithmetic

For sets A, B ∈ K (X) over a normed space (X, · ) the Hausdorff distance is defined as usual by H (A, B) = max{d H (A, B), d H (B, A)}, where d H (A, B) = sup inf a − b and d H (B, A) = sup inf a − b , a∈A b∈B

b∈B a∈A

we denote A H = H (A, {0}) = sup a . a∈A

If X = Rn , n ≥ 1 is the real n-dimensional vector space with internal product x, y and √ corresponding norm x = x, x , we denote by K n and KCn the spaces of (nonempty) compact and compact convex sets of Rn , respectively. If A ⊆ Rn and S n−1 = { p| p ∈ Rn , p = 1} is the unit sphere, the support function associated to A is s A : Rn → R defined by s A ( p) = sup{ p, a |a ∈ A},

p ∈ Rn .

If A  = ∅ is compact, then s A ( p) ∈ R, ∀ p ∈ S n−1 . The following properties are well known (see e.g. [28] or [55]): • Any function s : Rn → R which is continuous (or, more generally, upper semicontinuous), positively homogeneous s(t p) = ts( p), ∀t ≥ 0, ∀ p ∈ Rn and subadditive s( p + p ) ≤ s( p ) + s( p ), ∀ p , p ∈ Rn is a support function of a compact convex set; the s of s to S n−1 is such that s( p/ p ) = (1/ p )s( p), ∀ p ∈ Rn , p  = 0 and we restriction n−1 can consider s restricted to S . It also follows that s : S n−1 → R is a convex function. • If A ∈ KCn is a compact convex set, then it is characterized by its support function and A = {x ∈ Rn | p, x ≤ s A ( p), ∀ p ∈ Rn } = {x ∈ Rn | p, x ≤ s A ( p), ∀ p ∈ S n−1 }. • For A, B ∈ KCn and ∀ p ∈ S n−1 we have s{0} ( p) = 0 and A ⊆ B =⇒ s A ( p) ≤ s B ( p);

A = B ⇐⇒ s A = s B ,

sk A ( p) = ks A ( p), ∀k ≥ 0; sk A+h B ( p) = sk A ( p) + sh B ( p), ∀k, h ≥ 0 and in particular s A+B ( p) = s A ( p) + s B ( p). • If s A is the support function of A ∈ KCn and s−A is the support function of −A ∈ KCn , then ∀ p ∈ S n−1 , s−A ( p) = s A (− p);  • If v is a measure on Rn such that v(S n−1 ) = S n−1 v(d p) = 1, a distance is defined by

1.1

Generalized Hukuhara Difference and Properties

  ρ2 (A, B) = s A − s B = n

5

2 S n−1

[s A ( p) − s B ( p)]2 v(d p)

,

the distance ρ2 (·, ·) induces the norm on KCn defined by A = ρ2 (A, {0}).  • The Steiner point of A ∈ KCn is defined by σ A = n S n−1 ps A ( p)v(d p) and σ A ∈ A. We can express the generalized Hukuhara difference of compact convex sets A, B ∈ KCn by the use of the support functions. Consider A, B, C ∈ KCn with C = A g H B as defined in (1.3); let s A , s B , sC and s(−1)C be the support functions of A, B, C, and (−1)C, respectively. In case (i) we have s A = s B + sC and in case (ii) we have s B = s A + s(−1)C . So, ∀ p ∈ S n−1 sC ( p) = i.e.,

 sC ( p) =

 s A (P) − s B (P) in cases (i), s B (−P) − s A (−P) in cases (ii), s A (P) − s B (P) in cases (i), s(−1)B (P) − s(−1)A (P) in cases (ii).

(1.4)

Now, sC in (1.4) is a correct support function if it is continuous (upper semicontinuous), positively homogeneous and subadditive and this requires that, in the corresponding cases (i) and (ii), s A − s B and/or s−B − s−A be support functions, assuming that s A and s B are support functions. Consider s1 = s A − s B and s2 = s B − s A . Continuity of s1 and s2 is obvious. To see their positive homogeneity let t ≥ 0; we have s1 (t p) = s A (t p) − s B (t p) = ts A ( p) − ts B ( p) = ts1 ( p) and similarly for s2 . But s1 and/or s2 may fail to be subadditive and the following four cases, related to the definition of g H -difference, are possible. Proposition 1.3 (see [75]) Let s A and s B be the support functions of A, B ∈ KCn and consider s1 = s A − s B and s2 = s B − s A ; the following four cases apply: 1 If s1 and s2 are both subadditive, then A g H B exists; (i) and (ii) are satisfied simultaneously and A g H B = {c}. 2 If s1 is subadditive and s2 is not, then C = A g H B exists, (i) is satisfied and sC = s A − sB . 3 If s1 is not subadditive and s2 is, then C = A g H B exists, (ii) is satisfied and sC = s−B − s−A . 4 If s1 and s2 are both not subadditive, then A g H B does not exist. Proof In case 1 subadditivity of s1 and s2 means that, ∀ p , p ∈ S n−1 s1 : s A ( p + p ) − s B ( p + p ) ≤ s A ( p ) + s A ( p ) − s B ( p ) − s B ( p ) s2 : s B ( p + p ) − s A ( p + p ) ≤ s B ( p ) + s B ( p ) − s A ( p ) − s A ( p ),

6

1 Generalized Hukuhara Difference and Division for Interval and Fuzzy Arithmetic

it follows that s A ( p + p ) − s A ( p ) − s A ( p ) ≤ s B ( p + p ) − s B ( p ) − s B ( p ) s B ( p + p ) − s B ( p ) − s B ( p ) ≤ s A ( p + p ) − s A ( p ) − s A ( p ) so that equality holds: s B ( p + p ) − s A ( p + p ) = s B ( p ) + s B ( p ) − s A ( p ) − s A ( p ) Taking p = − p = p produces, ∀ p ∈ S n−1 , s B ( p) + s B (− p) = s A ( p) + s A (− p) i.e., s B ( p) + s−B ( p) = s A ( p) + s−A ( p) i.e., s B−B ( p) = s A−A ( p) and B − B = A − A (A and B translate into each other); it follows that ∃c ∈ Rn such that A = B + {c} and B = A + {−c} so that A g H B = {c}. In case 2 we have that being s1 a support function it characterizes a nonempty set C ∈ KCn and sC ( p) = s1 ( p) = s A ( p) − s B ( p), ∀ p ∈ S n−1 ; then s A = s B + sC = s B+C and A = B + C from which (i) is satisfied. In case 3 we have that s2 the support function of a nonempty set D ∈ KCn and s D ( p) = s B ( p) − s A ( p), ∀ p ∈ S n−1 so that s B = s A + s D = s A+D and B = A + D. Defining C = (−1)D (or D = (−1)C) we obtain C ∈ KCn with sC ( p) = s−D ( p) = s D (− p) = s B (− p) − s A (− p) = s−B ( p) − s−A ( p) and (ii) is satisfied. In case 4 there is no C ∈ KCn such that A = B + C (otherwise s1 = s A − s B is a support function) and there is no D ∈ KCn such that B = A + D (otherwise s2 = s B − s A is a support function); it follows that (i) and (ii) cannot be satisfied and A g H B does not exist. This completes the proof.  Proposition 1.4 (see [75]) If C = A g H B exists, then A g H B = ρ2 (A, B); it follows that A g H B = 0 ⇐⇒ A = B. Proof In fact ρ2 (A, B) = s A − s B and, if A g H B exists, then either sC = s A − s B or sC = s−B − s−A ; but s A − s B = s−A − s−B as, changing variable p into −q and recalling that s−A ( p) = s A (− p), we have  

s−A − s−B = [s−A ( p) − s−B ( p)]2 v(d p) = [s A (− p) − s B (− p)]2 v(d p) n−1 n−1 S  S 2 [s A (q) − s B (q)] v(−dq) = s A − s B , S n−1

(1.5) the last property follows from the fact that A g H B = 0 implies ρ2 (A, B) = 0 so that A = B; on the other hand, for A = B, A g H A = {0}. This completes the proof. 

1.2 The Case of Compact Intervals in Rn

7

An interesting property relates the Steiner point of A g H B to the Steiner points of A and B. Proposition 1.5 (see [75]) If C = A g H B exists, let σ A , σ B and σC be the Steiner points of A, B and C, respectively; then σC = σ A − σ B . Proof For the Steiner points, we have   σA = n ps A ( p)v(d p), σ B = n S n−1

and

 S n−1

ps B ( p)v(d p) = −n

S n−1

qs B (q)v(−dq)

  n S n−1 p[s A ( p) − s B ( p)]v(d p) or σC =  n S n−1 q[s A (q) − s B (q)]v(−dq),

the result follows from the additivity of the integral. This completes the proof.

1.2

(1.6) (1.7) 

The Case of Compact Intervals in Rn

In this section, the g H -difference of compact intervals in Rn will be considered. For more details, one may consult [75]. If n = 1, i.e., for unidimensional compact intervals , the g H difference always exists. In fact, let A = [a − , a + ] and B = [b− , b+ ] be two intervals; the g H -difference is  ⎧ ⎪ a − = b− + c− , ⎪ ⎪ ⎪ ⎨ (1) a + = b+ + c+ ,  [a − , a + ] g H [b− , b+ ] = [c− , c+ ] ⇐⇒ ⎪ b− = a − − c+ , ⎪ ⎪ ⎪ or (2) ⎩ b+ = a + − c− , so that [a − , a + ] g H [b− , b+ ] = [c− , c+ ] is always defined by c− = min{a − − b− , a + − b+ }, c+ = max{a − − b− , a + − b+ }, i.e., [a, b] g H [c, d] = [min{a − c, b − d}, max{a − c, b − d}]. Conditions (i) and (ii) are satisfied simultaneously if and only if the two intervals have the same length and c− = c+ . Also, the result is {0} if and only if a − = b− and a + = b+ . Two simple examples on real compact intervals illustrate the generalization (from [28, p. 8]); [−1, 1]  [−1, 0] = [0, 1] as in fact (i) is [−1, 0] + [0, 1] = [−1, 1] but [0, 0] g H [0, 1] = [−1, 0] and [0, 1] g H [− 21 , 1] = [0, 21 ] satisfy (ii).

8

1 Generalized Hukuhara Difference and Division for Interval and Fuzzy Arithmetic

Of interest are the symmetric intervals A = [−a, a] and B = [−b, b] with a, b ≥ 0; it is well known that Minkowski operations with symmetric intervals are such that A − B = B − A = A + B and, in particular, A − A = A + A = 2 A. We have [−a, a] g H [−b, b] = [−|a − b|, |a + b|]. As S 0 = {−1, 1} and the support functions satisfy s A (−1) = −a − , s A (1) = a + , s B (−1) = −b− , s B (1) = b+ , the same results as before can be deduced from definition (1.4). Remark 1.5 An alternative representation of an interval A = [a − , a + ] is by the use of the a = a − + a + /2 and the (semi)width a¯ = a + − a − /2 and we can write A = ( a , a), ¯ midpoint − + ¯ ¯  a¯ ≥ 0, so that a =  a − a¯ and a =  a + a. ¯ If B = (b, b), b ≥ 0 is a second interval, ¯ and the g H -difference is obtained a + b, a¯ + b) the Minkowski addition is A + B = ( ¯ We see immediately that A g H A = {0}, A = B ⇐⇒ a − b, |a¯ − b|). by A g H B = ( ¯ A g H B = {0}, (A + B) g H B = A, but A + (B g H A) = B only if a¯ ≤ b. n A and B = ×n B where A = [a − , a + ], B = [b− , b+ ] are real Let now A = ×i=1 i i i i=1 i i i i i n denotes the cartesian product). compact intervals (×i=1 If A g H B exists, then the following equality holds: n A g H B = ×i=1 (Ai g H Bi ).

In fact, consider the support function of A (and similarly for B), defined by s A ( p) = max{ p, x |ai− ≤ xi ≤ ai+ },

p ∈ S n−1 ,

x

(1.8)

it can be obtained simply by s A ( p) = pi >0 pi ai+ + pi 0

and, being s−A ( p) = s−B ( p) = −



s−B ( p) − s−A ( p) =

pi >0

From the relations above, we deduce that

pi (ai− − bi− )

(1.9)

pi 0}. We will consider the case X = Rn with n ≥ 1. A particular class of fuzzy sets u is when the support is a convex set and the membership function is quasi-concave (i.e., μu ((1 − t)x + t x ) ≥ min{μu (x ), μu (x )} for every x , x ∈ supp(u) and t ∈ [0, 1]). Equivalently, μu is quasi-concave if the level sets [u]x are convex sets for all α ∈ [0, 1]. We will also require that the level-cuts [u]x are closed sets for all α ∈ [0, 1] and that the membership function is normal, i.e., the core [u]1 = {x|μu (x) = 1} is compact and nonempty. The following properties characterize the normal, convex and upper semicontinuous fuzzy sets (in terms of the level-cuts): (F1 ) [u]α ∈ KC (Rn ) for all α ∈ [0, 1]; (F2 ) [u]α ⊆ [u]β for α ≥ β (i.e., they are nested);  (F3 ) [u]α = ∞ k=1 [u]αk for all increasing sequences αk ↑ α converging to α. Furthermore, any family {Uα |α ∈ [0, 1]} satisfying conditions (F1) − (F3) represents the level-cuts of a fuzzy set u having [u]α = Uα . We will denote by F n the set of the fuzzy sets with the properties above (also called fuzzy quantities). The space F n of real fuzzy quantities is structured by an addition and a scalar multiplication, defined either by the level sets or, equivalently, by the Zadeh extension principle . Let u, v ∈ F n have membership functions μu , μv and α-cuts [u]α , [v]α , α ∈ [0, 1], + respectively. In the unidimensional case u ∈ F , we will denote by [u]α = [u − α , u α ] the compact intervals forming the α-cuts and the fuzzy quantities will be called fuzzy numbers.

1.3

g H -Difference of Fuzzy Numbers

11

The addition u + v ∈ F n and the scalar multiplication ku ∈ F n have level cuts [u + v]α = [u]α + [v]α = {x + y|x ∈ [u]α , y ∈ [v]α },

(1.12)

[ku]α = k[u]α = {kx|x ∈ [u]α },

(1.13)

In the fuzzy or in the interval arithmetic contexts, equation u = v + w is not equivalent to w = u − v = u + (−1)v or v = u − w = u + (−1)w and this has motivated the introduction of the following Hukuhara difference [28, 44, 55]. Definition 1.2 (see [75]) Given u, v ∈ F n , the H -difference is defined by u  v = w ⇐⇒ u = v + w. Clearly, u  v = {0}; if u  v exists, it is unique. In the unidimensional case (n = 1), the α-cuts of H -difference are [u  v]α = [u − α − + ] and [v] = [v − , v + ]. − vα+ ] where [u]α = [u − , u α α α α α The Hukuhara difference is also motivated by the problem of inverting the addition: if x, y are crisp numbers then (x + y) − y = x but this is not true if x, y are fuzzy. It is possible to see that (see [18]), if u and v are fuzzy numbers (and not in general fuzzy sets), then (u + v)  v = u i.e., the H -difference inverts the addition of fuzzy numbers. The g H -difference for fuzzy numbers can be defined as follows: vα− , u + α

Definition 1.3 Given u, v ∈ F n , the gH-difference is the fuzzy quantity w ∈ F n , if it exists, such that  (i) u = v + w, u g v ⇐⇒ (1.14) or (ii) v = u + (−1)w. If u g H v and u  v exist, u  v = u g H v; if (i) and (ii) are satisfied simultaneously, then w is a crisp quantity. Also, u g H u = u  u = {0}.

1.3.1

Support Functions and Fuzzy g H-Difference

An equivalent definition of w = u g H v for multidimensional fuzzy numbers can be obtained in terms of support functions in a way similar to Eq. (1.4)  in case(i), su ( p, α) − sv ( p, α) α ∈ [0, 1], sw ( p, α) = (1.15) s(−1)v ( p, α) − s(−1)u ( p, α) in case(ii),

12

1 Generalized Hukuhara Difference and Division for Interval and Fuzzy Arithmetic

where, for a fuzzy quantity u, the support functions are considered for each α-cut and defined to characterize the (compact) α-cuts [u]α : su : Sn−1 × [0, 1] → R defined by su ( p; α) = sup{ p, x |x ∈ [u]α } for each p ∈ Sn−1 , α ∈ [0, 1]. As a function of α, su ( p, ·) is nonincreasing for all p ∈ Sn−1 , due to the nesting property of the α-cuts. Proposition 1.8 (see [75]) Let su ( p, α) and sv ( p, α) be the support functions of two fuzzy quantities u, v ∈ F n . Consider s1 = su − sv , s2 = sv − su ; the following four cases apply:

1. If s1 and s2 are both subadditive in p for all α ∈ [0, 1] and are nonincreasing for all p, then u g H v exists; (i) and (ii) in (1.15) are satisfied simultaneously and u g H v is crisp. 2. If s1 is subadditive in p for all α ∈ [0, 1] and nonincreasing for all p and s2 is not, then w = u g H v exists, (i) is satisfied and sw = su − sv . 3. If s2 is subadditive in p for all α ∈ [0, 1] and nonincreasing for all p and s1 is not, then w = u g H v exists, (ii) is satisfied and sw = s−v − s−u . 4. If s1 and s2 are both not subadditive and nonincreasing for all p, then u g H v does not exist. Proof The proof is similar to the proof of Proposition 1.3. For α ∈]0, 1] consider the Wα = {x ∈ Rn | p, x ≤ su ( p; α) for all p ∈ Rn with p = α}, and W0 = sets  cl( α∈]0,1] Wα ). Property (F1) for {Wα } is ensured by the validity of the subadditive property for all α ∈ [0, 1]. The monotonicity condition ∀ p ensures the nesting property (F2)  of the α-cuts . It remains to show property (F3), i.e., Wα = ∞ all increasing k=1 Wαk for sequences αk ↑ α converging to α ∈]0, 1]. As Wα ⊆ Wαk we have Wα ⊆ ∞ k=1 Wαk ; let  W , for all p having

p

= α and all k = 1, 2, . . . we have (α /α) p, x ≤ now x ∈ ∞ α k k k=1 su ((αk /α) p; α) and taking the limit for k → ∞ we obtain, as sw is continuous (upper semicontinuous),  p, x = lim(αk /α) p, x ≤ lim sup su ((αk /α) p; α) = su ( p; α) and x ∈ Wα . The proof is completed.  It immediately follows a necessary and sufficient condition for u g H v to exist: Proposition 1.9 (see [75]) Let u, v ∈ F n be given with support functions su ( p, α) and sv ( p, α); then u g H v ∈ F n exists if and only if at least one of the two functions su − sv , s−v − s−u is a support function and is nonincreasing with α for all p.

1.3

g H -Difference of Fuzzy Numbers

13

In the unidimensional case, the conditions of the definition of w = u g H v are [w]α = [wα− , wα+ ] = [u]α g H [v]α and

 − + + wα− = min{u − α − vα , u α − vα }, − + + wα+ = max{u − α − vα , u α − vα }

(1.16)

provided that wα− is nondecreasing, wα+ is nonincreasing and w1− ≤ w1+ ; in particular, for α ∈ [0, 1],  − wα− = u − α − vα (1) if len([u]) ≤ len([v]), + wα+ = u + α − vα (1.17)  + wα− = u + α − vα (2) if len([u]) ≥ len([v]), − wα+ = u − α − vα − where len([u]α ) = u + α − u α is the length of the α-cuts of u (similarly len([v]α ) for v).

Proposition 1.10 (see [75]) Let u, v ∈ F be two fuzzy numbers with α-cuts given by [u]α and [v]α respectively; the g H -difference u g H v ∈ F exists if and only if one of the two conditions is satisfied: ⎧ ⎪ ⎪ ⎨len([u]α ) ≥ len([v]α ) for all α ∈ [0, 1], − (a) u − α − vα is increasing with respect to α, ⎪ ⎪ ⎩u + − v + is decreasing with respect to α α α

or

⎧ ⎪ ⎪ ⎨len([u]α ) ≤ len([v]α ) for all α ∈ [0, 1], + (b) u + α − vα is increasing with respect to α, ⎪ ⎪ ⎩u − − v − is decreasing with respect to α. α

α

Proof In fact, consider the support function of u(and similarly for v), obtained by + su ( p; α) = max{ px|u − α ≤ x ≤ u α }, x

p ∈ S 0 = {−1, 1},

(1.18)

α ∈ [0, 1], + i.e., simply by su (−1; α) = −u − α and su (1; α) = u α . Then

su ( p; α) − sv ( p; α) =

 − −(u − α − vα ) if p = −1, + u+ α − vα

if p = 1

(1.19)

14

1 Generalized Hukuhara Difference and Division for Interval and Fuzzy Arithmetic

and s−v ( p; α) − s−u ( p; α) =

 + −(u + α − vα ) if p = −1, − u− α − vα

if p = 1.

(1.20)

From the relations above and from Proposition 1.9. we deduce that (symbols  and  mean that the function is increasing or decreasing, respectively): ⎧ ⎧ − + + ⎪ ⎪ ⎪[w]α = [u − ⎪ α − vα , u α − vα ] ⎨ ⎪ ⎪ ⎪ − + + ⎪ (a) provided that u − ⎪ α − vα ≤ u α − vα , ∀α ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ − − + + ⎪ and u α − vα , u α − vα  ⎪ ⎨ u g H v = w ⇐⇒ or (1.21) ⎧ ⎪ ⎪ ⎪ + + − − ⎪[w]α = [u − v , u − v ] ⎪ ⎪ ⎪ α α α α ⎨ ⎪ ⎪ ⎪ − ≥ u + − v + , ∀α ⎪(b) provided that u − − v ⎪ α α α α ⎪ ⎪ ⎪ ⎩ ⎪ ⎩and u + − v + , u − − v −  α α α α 

and the proof is completed.

− + + The monotonicity of u − α − vα and u α − vα according to (a) or (b) in (1.21) is an important condition for the existence of u g H v and is to be verified explicitly as in fact it may not be satisfied. Consider [u]α = [5 + 4α, 11 − 2α] and [v]α = [12 + 3α, 19 − 4α]; then − + + + + − − − − u− α − vα = −6 + α, u α − vα = −8 + 2α and u α − vα < u α − vα but u α − vα is not decreasing as required by (1.21)(b).

Remark 1.6 Conditions (a) and (b) of the above proposition are both valid if len([u]α ) = len([v]α ) for all α ∈ [0, 1]; in this case, u g H v is a crisp quantity. Example 1.3.1 (Case of linear membership) If u and v are trapezoidal linear shaped fuzzy − + + − − + + numbers, denoted by u = u − 0 , u 1 , u 1 , u 0 with u 0 ≤ u 1 ≤ u 1 ≤ u 0 (similarly for v) and − − − + + + α-cuts [u]α = [u 0 + α(u 1 − u 0 ), u 0 + α(u 1 − u 0 )], then u g H v exists if and only if − − − + + + + (a) u − 0 − v0 ≤ u 1 − v1 ≤ u 1 − v1 ≤ u 0 − v0

or

− − − + + + + (b) u − 0 − v0 ≥ u 1 − v1 ≥ u 1 − v1 ≥ u 0 − v0 .

+ − + In particular, if u and v are (linear) triangular fuzzy numbers(i.e., u − 1 = u 1 and v1 = v1 ) − + − + u , u 0 and v = v0 , v , v0 then w = u g v exists if and only if denoted by u = u 0 ,  − + (a) u − u − v ≤ u+ 0 − v0 ≤  0 − v0

or

− + (b) u − u − v ≥ u+ 0 − v0 ≥  0 − v0 .

1.3

g H -Difference of Fuzzy Numbers

15

− + To illustrate: 12, 15, 19 g H 5, 9, 11 does not exist as u − α − vα = −6 + α and u α − + vα = −8 + 2α are both increasing with respect to α ∈ [0, 1] and conditions (1.21) are not satisfied; 12, 15, 19 g H 5, 7, 10 = 7, 8, 9 , according to (a); 12, 15, 19 g H 9, 13, 18 = 1, 2, 3 , according to (b). 

If u g H v is a fuzzy number, it has the same properties illustrated in Sect. 1.1 for intervals. In particular, the same properties as in Proposition 1.2 are also immediate. Proposition 1.11 (see [75]) Let u, v ∈ F . If u g H v exists, it is unique and has the following properties (0 denotes the crisp set {0}): (1) u g H v = 0. (2) (a) (u + v) g H v = u; (b) u g H (u − v) = v. (3) If u g H v exists then also (−v) g H (−u) does and 0 g H (u g H v) = (−v) g H (−u). (4) u g H v = v g H u = w if and only if w = −w(in particular w = 0 if and only if u = v). (5) If v g H u exists then either u + (v g H u) = u or v − (v g H u) = u and if both equalities hold then v g H u is a crisp set.

1.3.2

A Decomposition of Fuzzy Numbers and g H-Difference

In the literature [72] the authors proposed a decomposition of fuzzy numbers (or intervals) which is useful in the study of fuzzy arithmetic operations. The same decomposition can help in performing the g H -difference. Its basic elements are summarized as follows.   + Given u ∈ F with α-cuts [u]α = u − α , u α , define the following quantities:  − +  u=  u , u 

(1.22)

+

be the core u − 1 , u 1 , corresponding to the (α = 1)-cut;  uα =

+ u+ u−  u− +  α + ux − 2 2

(1.23)

be the symmetry profile (briefly the profile) of u; u¯ α =

− u− u+  u+ −  α − uα − 2 2

be the symmetric fuzzy component of u.

(1.24)

16

1 Generalized Hukuhara Difference and Division for Interval and Fuzzy Arithmetic

Clearly, we have u− u− +  u α − u¯ α , u + u+ +  u α + u¯ α α = α = and, in interval notation, [u]α =  u + { u α } + [−u¯ α , u¯ α ] , α ∈ [0, 1].

(1.25)

We then obtain a decomposition of u ∈ F in terms of three components:  − + 1.  u=  u , u ∈ I is a standard compact real interval; 2.  u : [0, 1] → R is a given function such that, for α = 1,  u 1 = 0; denote with P the set of all such (profile) functions; 3. u¯ is a symmetric fuzzy number with core given by the singleton {0}; denote by S0 the family of all such fuzzy numbers. By the use of the three previous elements, any fuzzy number or interval can be represented by a triplet (with a small abuse of notation we denote by u¯ both the function in (1.24) and the 0-symmetric fuzzy number obtained with it): u = ( u,  u , u) ¯ ∈ I × P × S0 , i.e.,  u ∈ I (crisp number or interval),  u ∈ P (crisp symmetry profile), u¯ ∈ S0 (0-symmetric fuzzy number). The profile function  u ∈ P and the symmetric fuzzy number u¯ ∈ S0 form what we call a valid pair: Definition 1.4 (see [75]) A pair of elements ( u , u) ¯ ∈ P × S0 is said to form a valid pair if it u α − u¯ α ,  u α + u¯ α ], i.e., if the following conditions represents a fuzzy number having α-cuts [ are satisfied:  u α − u¯ α ,  u α − u¯ α ≤  . α < α =⇒ u α + u¯ α ,  u α + u¯ α ≥  i.e., if  u α − u¯ α is a nondecreasing function and  u α + u¯ α is a nonincreasing function (note u and u¯ are zero). that for α = 1 both  By the definition above, we can define the following decomposition of the fuzzy numbers (intervals) u ∈ F :

1.3

g H -Difference of Fuzzy Numbers

17

Proposition 1.12 (see [75]) Any fuzzy number (interval) u ∈ F with α-cuts [u]α = + u,  u , u) ¯ ∈ I × P × S0 defined in (1.22)-(1.24) [u − α , u α ] can be represented in the form u = ( u , u) ¯ is a valid pair. Vice versa, valid triplet ( u,  u , u) ¯ ∈ I × P × S0 (i.e., ( u , u) ¯ is a and ( valid pair) represents a fuzzy number (interval) u ∈ F with α-cuts given by (1.25). 

Proof See [72]. Definition 1.5 (see [75]) We call u = ( u,  u , u) ¯ ∈ I × P × S0

(where we assume ( u , u) ¯ be a valid pair) as the CPS-decomposition of u ∈ F (C = Crisp, P = Profile, S = 0 − Symmetricfuzzy). We write this as

F = I × V(P, S0 ), where we assume V(P, S0 ) ⊂ P × S0 is the set of all valid pairs. Now, we can find the g H -difference u g H v of u, v ∈ F in terms of the CPS decomposition. Proposition 1.13 (see [75]) Given u = ( u,  u , u) ¯ ∈ F and v = ( v , v , v) ¯ ∈ F , the g H w,  w, w) ¯ ∈ F with difference w = u g H v is given by w = (  w = u g H  v,  wα =  uα −  vα ,  u¯ α − v¯α w¯ α = v¯α − u¯ α

if u¯ − v¯ ∈ S0 , if v¯ − u¯ ∈ S0

and u g H v exists if and only if one of the two conditions are satisfied: (1) u¯ − v¯ ∈ S0 ,  u g H  v exists in case (i) and ( u − v , u¯ − v) ¯ ∈ V(P, S0 ) or (2) v¯ − u¯ ∈ S0 ,  u g H  v exists in case (ii) and ( u − v , v¯ − u) ¯ ∈ V(P, S0 ). u,  u , u) ¯ ∈ F and v = ( v , v , v) ¯ ∈ F are fuzzy numbers, i.e.,  u− = In particular, if u = ( u and  v− =  v+ =  v , then u g H v exists (in this case  w = u − v is crisp) if and only  u+ =  if one of the two conditions are satisfied: (1 ) u¯ − v¯ ∈ S0 and ( u − v , u¯ − v) ¯ ∈ V(P, S0 ) or (2 ) v¯ − u¯ ∈ S0 and ( u − v , v¯ − u) ¯ ∈ V(P, S0 ).

18

1 Generalized Hukuhara Difference and Division for Interval and Fuzzy Arithmetic

Proof First, as  u , v ∈ I, then  w = u g H  v ∈ I always exists and if  u , v ∈ P then  w=  u − v ∈ P. If u g H v exists then clearly u¯ − v¯ ∈ S0 or v¯ − u¯ ∈ S0 (as in fact w¯ = u¯ − v¯ or w¯ = v¯ − u¯ from Eqs. (1.17)) and ( u − v , u¯ − v) ¯ or ( u − v , v¯ − u) ¯ is a valid pair. Vice u − v , u¯ − v) ¯ is a valid pair(or if v¯ − u¯ ∈ S0 and ( u − v , v¯ − u) ¯ is versa, if u¯ − v¯ ∈ S0 and ( u g H  v,  u − v , u¯ − v) ¯ (or the triplet ( u g H  v,  u − v , v¯ − u)) ¯ a valid pair), then the triplet ( u g H  v . The second part is immediate represents a fuzzy number (interval) and by (1.16) it is u g H  v is the standard crisp difference and  w is crisp. This completes as the g H -difference  the proof.  + Remark 1.7 If the functions u − u α and α and u α are differentiable with respect to α, then  u¯ α are differentiable and the condition for ( u , u) ¯ to be a valid pair is | u α | ≤ −u¯ α ∀α ∈]0, 1[ (recall that u¯ α is a decreasing function).

Example 1.3.2 (see [75]) If we have symmetric fuzzy numbers u, v ∈ F (i.e.,  u− =  u+ = − + v = v = v , and  uα =  vα = 0 ∀α ∈ [0, 1]) then (only) condition |u¯ − v| ¯ ∈ S0 i.e., (u¯ −  u , v¯ ∈ S0 or v¯ − u¯ ∈ S0 ) is necessary and sufficient for u g H v to exist and ( u , 0, u) ¯ g H ( v , 0, v) ¯ = ( u − v , 0, |u¯ − v|) ¯ i.e., with α-cuts [u g H v]α = [( u − v ) − |u¯ α − v¯α |, ( u − v ) + |u¯ α − v¯α |]. For example, (1, 0, 1 − α 2 ) g H (0, 0, 2 − 2α) = (1, 0, (1 − α)2 ), according to case (ii) of g H -difference, has α-cuts [1 − (1 − α)2 , 1 + (1 − α)2 ].  Example 1.3.3 (see [75]) If we have symmetric triangular (with linear membership) fuzzy numbers u =  u − Δu,  u,  u + Δu and v =  v − Δv, v , v + Δv then w = u g H v always exists and is the triangular symmetric fuzzy number w =  w − Δw,  w,  w + Δw where  w = u − v and Δw = |Δu − Δv|.

In fact, in this case, |u¯ α − v¯α | = |Δu − Δv|(1 − α), i.e., u¯ − v¯ ∈ S0 if |Δu − Δv| ≥ 0 or v¯ − u¯ ∈ S0 if |Δv − Δu| ≤ 0 so that [u g H v]α = [( u − v ) − |Δu − Δv|(1 − α), ( u − v ) + |Δu − Δv|(1 − α)]. In particular, if Δu = Δv then u g H v is a crisp number. To illustrate, 2, 4, 6 g H −2, 1, 4 = 2, 3, 4 , according to case (ii) of g H -difference, and in fact 2, 4, 6 + −2, −1, 4 g H 2, 4, 6 = (−1)2, 3, 4 = 2, 4, 6 + −4, −3, −2 = −2, −1, 4 ; −4, −3, −2 , according to case (i) of g H -difference, and in fact 2, 4, 6 + −4, −3, −2 = −2, −1, 4 .

1.3

g H -Difference of Fuzzy Numbers

1.3.3

19

Approximated Fuzzy g H-Difference

The approximated fuzzy g H -difference was proposed and studied [75]. If the g H -differences [u]α g H [v]α do not define a proper fuzzy number, we can use the nested property of the α-cuts and obtain a proper fuzzy number by    [u v]α := cl ([u]β g H [v]β ) for α ∈ [0, 1]. (1.26) β≥α

As each g H -difference [u]β g H [v]β exists for β ∈ [0, 1] and (1.26) defines a proper v can be considered as a generalization of Hukuhara fuzzy number, it follows that z = u  difference for fuzzy numbers, existing for any u, v. Example 1.3.4 (see [75]) 12, 15, 19 g H 5, 9, 11 does not exists; we obtain [u]β g H [v]β = [8 − 2β, 7 − β] v]α = [6, 7 − α]. and [u 



v can be obtained by choosing a partition 0 = α0 < α1 < A discretized version of z = u  · · · < α N = 1 of [0, 1] and from [wi− , wi+ ] = [u]αi g H [v]αi by the following backward iteration: − + + z− N = wN , z N = wN .

For k = N − 1, . . . , 0 :



− z k− = min{z k+1 , wk− },

+ , wk+ }. z k+ = max{z k+1

A third possibility for a g H -difference of fuzzy numbers may be obtained by defining z =  u g H v to be the fuzzy number whose α-cuts are as near as possible to the g H -differences [u]α g H [v]α , for example by minimizing the functional (wα ≥ 0 and γα ≥ 0 are weighting functions)  G(z|u, v) = 0

1

2 + + 2 (wα [z α− − (u g H v)− α ] + γα [z α − (u g H v)α ] )dα

(1.27)

such that z α− is increasing with α, z α+ is decreasing with α and z α− ≤ z α+ ∀α ∈ [0, 1]. A discretized version of G(z|u, v) can be obtained by choosing a partition 0 = α0 < α1 < · · · < α N = 1 of [0, 1] and defining the discretized G(z|u, v) as G N (z|u, v) =

N

i=0

wi [z i− − (u g H v)i− ]2 + γi [z i+ − (u g H v)i+ ]2 ,

20

1 Generalized Hukuhara Difference and Division for Interval and Fuzzy Arithmetic

− + + we minimize G N (z|u, v) with the given data (u g H v)i− = min{u − αi − vαi , u αi − vαi } + − − − − + + and (u g H v)i = max{u αi − vαi , u αi − vαi }, subject to the constraints z 0 ≤ z 1 ≤ · · · ≤ + + + z− N ≤ z N ≤ z N −1 ≤ · · · ≤ z 0 . We obtain a linearly constrained least squares minimization of the form

min (z − w)T D 2 (z − w) s.t.E z ≥ 0,

z∈R2N +2

(1.28)

+ + + − − + + where z = (z 0− , z 1− , . . . , z − N , z N , z N −1 , . . . , z 0 ), wi = (u g H v)i , wi = (u g H v)i , + + + w = (w0− , w1− , . . . , w− N , w N , w N −1 , . . . , w0 ),

√ √ √ √ D = diag{ w0 , . . . , w N , γ N , . . . , γ0 } and E is the (N , N + 1) matrix ⎡

−1 ⎢0 ⎢ E =⎢ . ⎣ .. 0

1 −1 .. . 0

0 1 .. . ···

··· 0 .. . ···

⎤ 0 0⎥ ⎥ .⎥ , · · · .. ⎦ −1 1 ··· ···

which can be solved by standard efficient procedures (see the classical book [56, Chap. 23]). If, at solution z ∗ , we have z ∗ = w, then we obtain the g H -difference as defined in (1.14).

1.4

Generalized Division

An ideal similar to the g H -difference can be used to introduce a division of real intervals and fuzzy numbers. One may consult [75] for more details. Consider first the case of real compact intervals A = [a − , a + ] and B = [b− , b+ ] with − b > 0 or b+ < 0 (i.e., 0 ∈ / B). The interval C = [c− , c+ ] defining the multiplication C = AB is given by c− = min{a − b− , a − b+ , a + b− , a + b+ }, c+ = max{a − b− , a − b+ , a + b− , a + b+ } and the multiplicative “inverse” (it is not the inverse in the algebraic sense) of an interval B is defined by B −1 = [1/b+ , 1/b− ]. Definition 1.6 (see [75]) For A = [a − , a + ] and B = [b− , b+ ] we define the generalized division (g-division) ÷g as follows:  A ÷g B = C ⇐⇒

(i) A = BC

or (ii) B = AC −1 .

(1.29)

1.4

Generalized Division

21

If both cases (i) and (ii) are valid, we have CC −1 = C −1 C = {1}, i.e., C = { c}, C −1 = c  = 0. It is immediate to see that A ÷g B always exists and is unique for given {1/ c} with  A = [a − , a + ] and B = [b− , b+ ] with 0 ∈ / B. It is easy to see that the following six cases are possible (see [75]), with the indicated rules: Case 1: If 0 < a − ≤ a + and b− ≤ b+ < 0 then a+ + a− , c = and (i) is satisfied, b− b+ a− a+ if a − b− ≤ a + b+ then c− = + , c+ = − and (ii) is satisfied. b b

if a − b− ≥ a + b+ then c− =

(1.30)

Case 2: If 0 < a − ≤ a + and 0 < b− ≤ b+ then a− + a+ , c = and (i) is satisfied, b− b+ + − a a if a − b+ ≥ a + b+ then c− = + , c+ = − and (ii) is satisfied. b b

if a − b+ ≤ a + b− then c− =

(1.31)

Case 3: If a − ≤ a + < 0 and b− ≤ b+ < 0 then a+ + a− , c = − and (i) is satisfied, + b b − + a a if a + b− ≥ a − b+ then c− = − , c+ = + and (ii) is satisfied. b b

if a + b− ≤ a − b+ then c− =

(1.32)

Case 4: If a − ≤ a + < 0 and 0 < b− ≤ b+ then a− + a+ , c = − and (i) is satisfied, + b b + − a a if a − b− ≥ a + b+ then c− = − , c+ = + and (ii) is satisfied. b b

if a − b− ≤ a + b+ then c− =

(1.33)

Case 5: If a − ≤ 0, a + ≥ 0 and b− ≤ b+ < 0 then the solution does not depend on b+ , c− =

a+ a− , c+ = − and (i) is satisfied. − b b

(1.34)

Case 6: If a − ≤ 0, a + ≥ 0 and 0 < b− ≤ b+ then the solution does not depend on b− , c− =

a− a+ + , c = and (i) is satisfied. b+ b+

(1.35)

Remark 1.8 If 0 ∈]b− , b+ [ the g-division is undefined: for intervals B = [0, b+ ] or B = [b− , 0] the division is possible but obtaining unbounded results C of the form C =] − ∞, c+ ] or C =]c− , +∞[: we work with B = [ε, b+ ] or B = [b− , −ε] and we obtain the result by

22

1 Generalized Hukuhara Difference and Division for Interval and Fuzzy Arithmetic

the limit for ε → 0+ . Example: for [−2, −1] ÷g [0, 3] we consider [−2, −1] ÷g [ε, 3] = −1 + [cε− , cε+ ] with (case 2) cε− = min{ −2 3 , −1/ε} and cε = max{−2/ε, 3 } and obtain the result 1 + C = [−∞, − 3 ] at the limit ε → 0 . The following properties are immediate. Proposition 1.14 (see [75]) For any A = [a − , a + ] and B = [b− , b+ ] with 0 ∈ / B, we have (here 1 is the same as {1}): 1. 2. 3. 4.

B ÷g B = 1, B ÷g B −1 = {b− b+ }(= { b2 } if b− = b+ =  b). (AB) ÷g B = A. 1 ÷g B = B −1 and 1 ÷g B −1 = B. At least one of the equality B(A ÷g B) = A or A(A ÷g B)−1 = B is valid and both hold if and only if A ÷g B is a singleton.

1.4.1

The Fuzzy Case

The search for alternative definitions of the division operator between fuzzy numbers has received some attention in recent literature, with the objective of inverting multiplication; some recent studies are in [14, 24, 48, 62, 63, 72]. In particular, Boukezzoula et al. [24] have analyzed the fuzzy division in a way similar to our setting (considering case (i) of the division operator). Using our notation, the generalized division introduced in [24] is z = u  v where the α-cuts [z α− , z α+ ] of z are defined by − + + + z α− = A− α /Bα , z α = Aα /Bα ,



with A− α

=

+ u+ α if vα < 0,



and A+ α

=

− u− α if vα > 0,

− u+ α if vα > 0,

+ u− α if vα < 0,

 Bα−

=

vα+ if A− α 1 the g H -difference may not exist. / A) as In a similar way, Proposition 1.14 is a motivation to consider X = B ÷ A (if 0 ∈ the solution to the interval equation AX = B.

(1.39)

Proposition 1.17 (see [75]) Let A, B ∈ KC (R); the g-division X = B ÷g A always exists / A) and (at least) one of the following equalities is satisfied: (if 0 ∈ 1. AX = B; 2. B/X = A. We say that X = B ÷g A is the solution of (1.39) in the multiplicative sense if AX = B and in the divisive sense if B X −1 = A. / A, consider the interval equation More generally, given A, B, C ∈ KC (R) with 0 ∈ AX + B = C.

(1.40)

Using g H -difference and g-division, Eq. (1.40) has the solution X = (C g H B) ÷g A and it always exists, i.e., (the proof is immediate): Proposition 1.18 (see [75]) Let A, B, C ∈ KC (R) with 0 ∈ / A; the interval X = (C g H B) ÷g A always exists and (at least) one of the four equalities is satisfied: 1. 2. 3. 4.

AX + B = C (X is additive and multiplicative); B = C − AX (X is subtractive and multiplicative); A ÷g X −1 + B = C (X is additive and divisive); B = C − A ÷g X −1 (X is subtractive and divisive).

The same results can be applied to Eqs. (1.38) and (1.40) in the case where A, B and C are fuzzy numbers, i.e., to equations ux = v, (1.41) ux + v = w.

(1.42)

1.5

Applications of g H -Difference and g-Division

27

We obtain the fuzzy solutions x = v ÷g u and x = (w g H v) ÷g u or, more generally, u using the approximated g H -difference and g-division of Sects. 1.3.3 and 1.4.2, x = v ÷ ˜ v)÷u. and x = (w We illustrate the application of g H -difference and g-division to Eqs. (1.41) and (1.42) with some examples. The data are triangular fuzzy numbers of the form a, b, c with linear left and right membership sides. The following examples can be found in [75]. Example 1.5.1 u = 0.2, 0.4, 0.6 , v = 0.1, 0.3, 0.5 ; the solution x = v ÷g u has α-cuts   0.1 + 0.2α 0.5 − 0.2α [x]α = , for α ∈ [0, 1] 0.2 + 0.2α 0.6 − 0.2α and satisfies case 1 of Proposition 1.17.



Example 1.5.2 u = 0.2, 0.4, 0.6 , v = 0.6, 0.8, 1 ; the solution x = v ÷g u has α-cuts   1 − 0.2α 0.6 + 0.2α for α ∈ [0, 1] [x]α = , 0.6 − 0.2α 0.2 − 0.2α and satisfies case 2 of Proposition 1.17.



Example 1.5.3 u = 0.5, 0.8, 1 , v = 0.3, 0.5, 0.6 ; the solution x = v ÷g u does not ˜ exists and its α-cuts are exist, but x = v ÷u   0.6 − 0.1α 5 for α ∈ [0, 1]. [x]α = , 1.0 − 0.2α 8 Example 1.5.4 (from [19, Chap.10]): u = 1, 2, 3 , v = −3, −2, −1 , w = 3, 4, 5 ; note that w = v + {6} but it is impossible to find x such that ux = {6}. Instead there exists x such that u = 6x −1 i.e., x = 6u −1 and cases 3 and 4 of Proposition 1.18 are both satisfied.  Example 1.5.5 (from [19, Chap.10]): u = 8, 9, 10 , v = −3, −2, −1 , w = 3, 5, 7 ; note that w = 2v + {9} and it is possible to find x such that ux = v + {9}. The solution x = (w g H v) ÷g u has α-cuts   6+α 8−α for α ∈ [0, 1] [x]α = , 8 + α 10 − α and satisfies cases 1 of Proposition 1.18.



28

1.5.2

1 Generalized Hukuhara Difference and Division for Interval and Fuzzy Arithmetic

Interval and Fuzzy Differential Equations

Equations with fuzzy numbers was discussed in [17]. In the literature [75], Stefanini summarized some results of some literatures related to interval and fuzzy differential equations based on Hukuhara difference. The generalized Hukuhara difference is (implicitly) used by Bede and Gal (see [10]) in their definition of generalized differentiability of a fuzzy-valued function. The g H -difference allows a new definition of the derivative for interval (or fuzzy) valued functions f : [a, b] → K C (R) (or f : [a, b] → F ). If x ∈]a, b[ and x + h ∈]a, b[, the g H -derivative of f at x is defined as the limit f (x + h) g f (x) lim = f (x) h→0 h provided that the g H -differences exist for small h. For interval and fuzzy-valued functions, using the results of the preceding sections, the g H -differences are very easy to be computed in terms of the lower and upper bounds of the intervals f (x) = [ f − (x), f + (x)] (or of the α-cuts [ f (x)]α = [ f α− (x), f α+ (x)], α ∈ [0, 1] in the fuzzy case) the interval derivative is   −   −  d f (x) d f + (x) d f (x) d f + (x) f (x) = min , , max , dx dx dx dx and the fuzzy derivative, in terms of level-cuts, is   −    − d f α (x) d f α+ (x) d f α (x) d f α+ (x) , max . [ f (x)]α = min , , dx dx dx dx This definition of the derivative allows to characterize a new solution concept in interval and fuzzy differential equations and, in particular, to design new computational procedures based on classical ODE methods (see [73]). A study by Bede and Stefanini (see [74]) presented a new approach to model interval uncertainty in dynamical systems and interval differential equations, related but more general than the differential inclusion model (see [45, 55]). In particular, the derivative of an intervalvalued function, based on the new gH-difference, allows the possibility of solutions without restrictions imposed by the classical Hukuhara derivative, i.e., the increasing uncertainty effect. See [29–32] for a critic of the use of classical Hukuhara derivative in fuzzy differential equations; see also [11] for an extended approach to fuzzy differential equations.

2

An Embedding Theorem and Multiplication of Fuzzy Vectors

In this chapter, an embedding theorem of fuzzy vectors is established. Furthermore, two new types of multiplication of fuzzy vectors are introduced and some of their operation regulations are established. Let x be a point in Rn and A a nonempty subset of Rn . The distance d(x, A) from x to A is defined by d(x, A) = inf{x − a : a ∈ A}. ¯ ≥ 0 and d(x, A) = 0 if and only if x ∈ A, ¯ the closure of A in Thus d(x, A) = d(x, A) n n R . We shall call the subset Sε (A) = {x ∈ R : d(x, A) < ε} an ε-neighborhood of A. Its closure is the subset S¯ε (A) = {x ∈ Rn : d(x, A) ≤ ε}. In particular, we shall denote the closed unit ball in Rn by S¯1n , that is, S¯1n = S¯1 ({0}). For more details of this knowledge, we  refer the readers to the book [28]. So if we let d H (B, A) = inf{ε > 0 : B ⊆ A + ε S¯1n }, then the Hausdorff distance between nonempty subsets A and B of Rn is defined by 





d H (A, B) = max{d H (A, B), d H (B, A)}.

(2.1)

Now, we consider a circumscribed cube V¯1n of S¯1n , i.e., n V1n = (−1, 1) × (−1, 1) × · · · × (−1, 1) := ×i=1 (−1, 1), n where ×i=1 denotes the Cartesian product. Now V¯1n is obviously a compact subset of Rn . We say Vε (A) := A + εV1n

is the ε-cube neighborhood of A. Obviously, Sε (A) ⊂ Vε (A). Let S n−1 and V n−1 be the surfaces of S¯1n and V¯1n , then the tangential points set between the surfaces of S¯1n (i.e., S n−1 ) and V¯1n (i.e., V n−1 ) is

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 C. Wang and R. P. Agarwal, Dynamic Equations and Almost Periodic Fuzzy Functions on Time Scales, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-11236-2_2

29

30

2 An Embedding Theorem and Multiplication of Fuzzy Vectors

S n−1 ∩ V n−1 = {P1 , P1∗ , P2 , P2∗ , . . . , Pn , Pn∗ }, where Pi = (0, . . . , 0,

1 

, 0, . . . , 0),

the ith element

Pi∗ = (0, . . . , 0,

−1 

, 0, . . . , 0),

the ith element

and S n−1 = { p ∈ Rn :  p = 1}. Now let A and B be nonempty subsets of Rn . We define the Hausdorff separation of B from A by ∗ dH (B, A) = inf{ε > 0 : B ⊆ A + ε V¯1n }, n [−ε, ε]. where ε V¯1n = [−ε, ε] × [−ε, ε] × · · · × [−ε, ε] := ×i=1 We define the Hausdorff distance between nonempty subsets A and B of Rn by

 ∗  ∗ d H (A, B) = max d H (A, B), d H (B, A) .

(2.2)

If we restrict our attention to nonempty closed subsets of Rn , we find that the distance (2.2) is a metric which is strongly equivalent to the Pompeiu-Hausdorff metric [40] when the ε-cube neighborhood is applied. Following Diamond and Kloeden (see [28]), denote by K (Rn ) and KC (Rn ) the spaces of nonempty compact and compact convex sets of Rn . For a vector a ∈ Rn and convex set C ⊆ Rn , denote by s(C, a) := sup{a, x , x ∈ C} the support function of C in direction a. In addition, for a convex function h : C → R on some convex set C ⊆ Rn and x ∈ C, we write   ∂h for the subdifferential of h in x and NC (x) := a ∈ Rn : s(C, a) = a, x the normal cone of C at x (for more knowledge of the normal cone, tangential cone and computational geometry, one may consult [15, 38, 54]). Theorem 2.1 Let A, B ∈ KC (Rn ). Then there exists a number Nn with 1 ≥ Nn ≥ 2√1 n such that d H (A, B) = Nn sup |s(A, u) − s(B, u)|. (2.3) u∈V n−1

Proof Let ε := d H (A, B) and u ∈ V¯1n . Since A ⊂ B + ε V¯1n , it follows that s(A, u) ≤ s(B + ε V¯1n , u) ≤ s(B, u) + εs(V¯1n , u) ≤ s(B, u) + Nn∗ ε, √ where Nn∗ is a positive number such that 2 n ≥ Nn∗ ≥ s(V¯1n , u). Similarly, we can also obtain s(B, u) ≤ s(A, u) + Nn∗ ε. Since u ∈ V¯1n is arbitrary, this yields that 1 sup |s(A, u) − s(B, u)| ≤ ε. Nn∗ u∈V¯ n 1

(2.4)

2 An Embedding Theorem and Multiplication of Fuzzy Vectors

31

For another inequality, let a ∗ ∈ A and b∗ ∈ B such that ε = a ∗ − b∗ . Let h(b) := b − a ∗ . Since h(b∗ ) ≤ h(b) for all b ∈ B, there exists    u ∈ ∂h(b∗ ) = u ∈ V¯1n : b∗ − a ∗ , u = h(b∗ ) 



such that −u ∈ N B (b∗ ). Thus, for u ∗ = −u , we have ε = h(b∗ ) = a ∗ − b∗ , u ∗ = a ∗ , u ∗ − maxb, u ∗ b∈B

≤ |s(A, u ∗ ) − s(B, u ∗ )| ≤ sup |s(A, u) − s(B, u)|. u∈V¯1n

(2.5)

Because of the homogeneity of s(A, ·), s(B, ·) and | · |, the supremum in (2.3) is attained for some vector u ∗ ∈ V n−1 . Hence, by (2.4) and (2.5), there exists 1 ≥ Nn ≥ 2√1 n such that  (2.3) holds. This completes the proof. Example 2.1 Consider A = [0, 1] × {1} and B = [−3, 1] × {0} two compact and convex sets of R2 . Then ∗ ∗ dH (A, B) = 2, d H (B, A) = 3, and so d H (A, B) = 3. Now for (q1 , q2 ) ∈ V 1 , we have s(q, A) = sup{q, a , a ∈ A} = sup(q1 , q2 ), (a1 , 1) : a1 ∈ [0, 1]} = sup{q1 a1 + q2 , a1 ∈ [0, 1]}  q1 + q2 if q1 ≥ 0, = q2 if q1 < 0 and s(q, B) = sup{q, b , b ∈ B} = sup(q1 , q2 ), (b1 , 0) : b1 ∈ [−3, −1]} = sup{q1 b1 , b1 ∈ [−3, −1]}  −q1 if q1 ≥ 0, = −3q1 if q1 < 0. Then we have |s(q, A) − s(q, B)| = =

 |q1 + q2 − (−q1 )|, if q1 ≥ 0, |q2 − (−3q1 )| if q1 < 0  |2q1 + q2 |, if q1 ≥ 0, |3q1 + q2 | if q1 < 0.

32

2 An Embedding Theorem and Multiplication of Fuzzy Vectors

Thus by Lemma 2.1, there exists Nn =

3 4

such that

3 3 sup |s(A, u) − s(B, u)| = · 4 = 3. 4 u∈V 1 4 Remark 2.1 As is well known that if the unit ball neighborhood is adopted (see [15, 28,  38, 54]), the Hausdorff distance via support functions can be represented by d H (A, B) = supu∈S n−1 |s(A, u) − s(B, u)|. Since V¯1n is the circumscribed cube of S¯1n , then from Theo rem 2.1, we can obtain d H (A, B) and d H (A, B) are strongly equivalent, i.e., there exist  ˜ H (A, B) ≤ positive constants α˜ and β˜ such that for any A, B ∈ KC (Rn ) we have αd  ˜ (A, B). Hence, the metric d H (A, B) is strongly equivalent to classical d H (A, B) ≤ βd H Pompeiu-Hausdorff metric defined by (2.1). In the sequel, we recall briefly the necessary knowledge of fuzzy sets. Definition 2.1 (see [13, 75]) Let X be a nonempty set. A fuzzy set u ∈ X is characterized by its membership function u : X → [0, 1]. Then u(x) is interpreted as the degree of membership of a element x in the fuzzy set u for each x ∈ X . Let us denote by RnF the class of fuzzy subsets u of Rn (i.e., u : Rn → [0, 1]), satisfying the following properties: (i) u is normal, i.e., there exists x0 ∈ Rn with u(x0 ) = 1; (ii) u is convex fuzzy set (i.e., u(t x + (1 − t)y) ≥ min{u(x), u(y)}, ∀t ∈ [0, 1], x, y ∈ Rn ); (iii) u is upper semicontinuous on Rn ; (iv) cl{x ∈ Rn , u(x) > 0} is compact, where cl denotes the closure of a subset. Then, RnF is called the space of fuzzy vectors . Obviously, Rn ⊂ RnF . Here Rn ⊂ RnF is understood as Rn = {χ{x} : x ∈ Rn is a usual real-vector}, where χ A = 1 if x ∈ A, χ A = 0 if x ∈ / A. For 0 < α ≤ 1, denote [u]α = {x ∈ Rn : u(x) ≥ α} (the α-level set) and [u]0 = cl{x ∈ Rn : u(x) > 0}. ˜ and the product λ · u are defined by For u, v ∈ RnF , and λ ∈ R, the sum u +v ˜ α = [u]α + [v]α , [λ · u]α = λ[u]α , ∀α ∈ [0, 1], where [u]α + [v]α = {x + y, [u +v] x ∈ [u]α , y ∈ [v]α } means the usual addition of two sets in Rn and [λ · u]α = {λx : x ∈ [u]α } means the usual product between a scalar and a subset of Rn . Definition 2.2 (see [13, 75]) The Hausdorff distance between two fuzzy vectors is the function D∞ : RnF × RnF → R+ ∪ {0}, defined in terms of the Hausdorff distance between their level sets, that is,

2 An Embedding Theorem and Multiplication of Fuzzy Vectors

33

  D∞ (u, v) = sup d H ([u]α , [v]α ) : α ∈ [0, 1] . 

Remark 2.2 Since d H ([u]α , [v]α ) and d H ([u]α , [v]α ) are strongly equivalent, then     D∞ (u, v) and D∞ (u, v) := sup d H ([u]α , [v]α ) : α ∈ [0, 1] are also strongly equivalent, which indicates that they only exist numerically different. ˜ defined for For convenience, let us denote by  · F the function uF = D∞ (u, 0) n ˜ ˜ ˜ ˜ ˜ all u ∈ RF , where 0 = (0, 0, . . . , 0) and 0 is a zero element of RF . The next lemma asserts that  · F has properties similar to the properties of a norm in the usual crisp sense, without being a norm. It is not a norm because RnF is not a linear space and, consequently, (RnF ,  · F ) is not a normed space.  Due to the strong equivalence of the metrics D∞ (u, v) and D∞ (u, v), the following lemma can be obtained immediately. Lemma 2.1 (see [13, 75]) The function  · F has the following properties: ˜ (i) uF = 0 if and only if u = 0; (ii) λ · uF = |λ| · uF for all u ∈ RnF and λ ∈ R; vF ≤ uF + vF for all u, v ∈ Rn . (iii) u + F Lemma 2.2 (see [13, 75]) , i.e., u + 0˜ = If we denote 0˜ = χ{0} then 0˜ ∈ RnF is neutral element with respect to + ˜0+ u = u, for all u ∈ Rn . F (ii) For any a, b ∈ R with a, b ≤ 0 or a, b ≥ 0 and any u ∈ RnF , we have (a + b) · u = b · u; for any general a, b ∈ R, the above property does not hold. a · u+ v) = λ · u + λ · v. (iii) For any λ ∈ R and any u, v ∈ RnF , we have λ · (u + n (iv) For any λ, μ ∈ R and any u ∈ RF , we have λ · (μ · u) = (λμ) · u. (i)

Definition 2.3 (see [75]) Given u, v ∈ RnF , the gH-difference is the fuzzy vector w, if it exists, such that  w or (I ) u = v + g H v = w ⇔ u− (−1) · w. (I I ) v = u + Remark 2.3 In the above definition, we also introduce the following notations: g H (I ) v = w ⇔ (I ) u = v + w u− g H (I I ) v = w ⇔ (I I ) v = u + (−1) · w. u−

34

2 An Embedding Theorem and Multiplication of Fuzzy Vectors

Remark 2.4 Note that it is possible that the gH-difference of two fuzzy vectors does not g H v exists, then v − g H u exists and v − g H u = −(u − g H v). exist. If u − Lemma 2.3 (see [75]) (i) If the gH-difference exists, it is unique; g H v = u − v or u − g H v = −(u − v) whenever the expressions on the right exist; in (ii) u − ˜ g H u = u − u = 0, where u − v denotes the Hukuhara difference of u, v; particular, u − g H v exists in the sense (I ), then v − g H u exists in the sense (I I ) and vice versa; (iii) if u − v)− g H v = u; (iv) (u + g H u = w if and only if w = −w ; furthermore, w = 0˜ if and only if (v) u −g H v = v − u = v. Definition 2.4 (see [13, 75]) Let KCn be the space of nonempty compact convex set of Rn , A, B ∈ KCn , we define the generalized Hukuhara difference of A and B as the set C ∈ KCn such that  (I ) A = B + C or A g H B = C ⇔ (2.6) (I I ) B = A + (−1) · C. Remark 2.5 In the above definition, we also introduce the following notations: A g H (I ) B = C ⇔ (I ) A = B + C A g H (I I ) B = C ⇔ (I I ) B = A + (−1) · C. Remark 2.6 In the Sect. 3 of the literature [75], Stefanini established some important basic results of g H -difference between compact intervals in Rn , including the special case of unidimensional and multidimensional intervals (boxes). For a compact subset A of Rn , we will consider its ε-cube neighborhood of A, then the g H -difference between multidimensional boxes can be applied. From [75], for any multidimensional boxes A, B of Rn , if − + n (A  n n A g H B exists, then A g H B = ×i=1 i g H Bi ), where A = ×i=1 Ai = ×i=1 [ai , ai ] − + n n and B = ×i=1 Bi = ×i=1 [bi , bi ]. Lemma 2.4 (see [75]) Let A, B be two multidimensional boxes of Rn . A necessary and sufficient condition for A g H B to exist is that either A contains a translate of B or B contains a translate of A. Remark 2.7 Given A = ([1, 2], [−2, 1]) and B = ([−2, 1], [1, 2]), let χ A , χ B : R2 → / A, and χ B (x) = 1 if [0, 1] be given, respectively, by χ A (x) = 1 if x ∈ A; χ A = 0 if x ∈ x ∈ B; χ B = 0 if x ∈ / B. Then χ A , χ B ∈ R2F and [χ A ]α = A, [χ B ]α = B for all α ∈ [0, 1].

2 An Embedding Theorem and Multiplication of Fuzzy Vectors

35

However, since A is not a translate of B and B is not a translate of A, we obtain that [χ A ]α g H [χ B ]α does not exist. Hence, Lemma 2.4 provides the significant condition to guarantee the existence of the g H -difference between two multidimensional boxes. Now, let u ∈ RnF and define su : [0, 1] × V n−1 → R by   su (α, p) = s( p, [u]α ) = sup  p, a : a ∈ [u]α

(2.7)

for (α, p) ∈ [0, 1] × V n−1 , where s(·, [u]α ) is the support function of [u]α . We shall call su the support function of the fuzzy set u. Note that the supremum in (2.7) is actually attained since the level set [u]α is compact and so can be replaced by the maximum. Lemma 2.5 The support function su has the following properties: (i) (ii) (iii) (iv)

u = v if and only if su = sv . uniformly bounded on [0, 1] × V n−1 . Lipschitz in p ∈ V n−1 uniformly on [0, 1].   For each α ∈ [0, 1], d H ([u]α , [v]α ) = √1n sup |su (α, p) − sv (α, p)| : p ∈ V n−1 .

(v) su (·, p) is nonincreasing and left continuous in α ∈ [0, 1] for each p ∈ V n−1 . (vi) A fuzzy set u ∈ RnF is called a Lipschitzian fuzzy set if it is a Lipschitz function of its membership grade in the sense that d H ([u]α , [u]β ) ≤ K |α − β| for all α, β ∈ [0, 1] and some fixed finite constant K . The support function su (·, p) is Lipschitz uniformly in p ∈ V n−1 if and only if u is a Lipschitzian fuzzy set. Proof (i). Since the support function on KC (Rn ) uniquely characterizes the elements of KC (Rn ), the result is obvious. √ (ii). Since [u]α ⊆ [u]0 , so |su (α, p)| = |s( p, [u]α )| ≤ [u]α  p ≤ n[u]0 , then the result follows. (iii). From the following inequality, |su (α, p) − su (α, q)| = |s( p, [u]α ) − s(q, [u]α )| ≤ [u]α  p − q ≤ [u]0  p − q. Then we can obtain the results. (iv). By Theorem 2.1, (iv) is the restatement of (2.3). (v). Since [u]β ⊆ [u]α for 0 ≤ α ≤ β, su (β, p) = s( p, [u]β ) ≤ s( p, [u]α ) = su (α, p) so su (·, p) is nonincreasing for each p ∈ V n−1 . Moreover, for a nondecreasing sequence αn˜ ↑ α in [0, 1], |su (αn˜ , p) − su (α, p)| ≤

√

nd H ([u]αn˜ , [u]α ) → 0 asn˜ → ∞.

36

2 An Embedding Theorem and Multiplication of Fuzzy Vectors

(vi). By (iii) and (iv), we can obtain the desired result immediately. This completes the proof.  From Definitions 2.2, 2.4 and Lemma 2.4, the following theorem follows immediately. g H v exists, then Theorem 2.2 For any u, v ∈ RnF and 0 ≤ α ≤ 1, if u − g H vF , D∞ (u, v) = sup {[u]α g H [v]α ∗ } = u − α∈[0,1]

(2.8)

where [u]α g H [v]α ∗ = d H ([u]α g H [v]α , {0}) and α

α

α

[u] g H [v] = [w] ⇔

 (I ) [u]α = [v]α + [w]α or (I I ) [v]α = [u]α + (−1) · [w]α .

g H v for u, v ∈ Rn via support funcRemark 2.8 The existence and nonexistence of u − F tions were established in Sect. 4.1 of the literature [75] (see pp. 1571-1572) by Stefanini. From the Proposition 20 in [75], Stefanini provided a necessary and sufficient condition to g H v. Hence, when we suppose that su − sv , s−v − s−u is a guarantee the existence of u − g H v ∈ Rn support function and is nonincreasing with α for all p in Theorem 2.2, then u − F exists. The monotonicity of su − sv , s−v − s−u is an important condition for the existence + g H v and it may not be satisfied. Consider u, v ∈ RF such that [u]α = [u − u− α , uα ] = α − + [5 + 4α, 14 − α] and [v] = [vα , vα ] = [9 + α, 12 − 2α] for all α ∈ [0, 1], through calculation, we have su (α, 1) = s−u (α, −1) = 14 − α and sv (α, 1) = s−v (α, −1) = 12 − 2α, then we obtain su (α, 1) − sv (α, 1) = s−u (α, −1) − s−v (α, −1) = 2 + α is increasing with − + + α. For the unidimensional case, the monotonicity of u − α − vα and u α − vα must be guaran g H v ∈ RF (see Proposition 21 in [75] that is a particular case teed for the existence of u − of Proposition 20).

2.1

An Embedding Theorem for Fuzzy Multidimensional Space

In this subsection, we will establish an embedding theorem for fuzzy multidimensional space. Definition 2.5 Let u i ∈ RF for each i = 1, 2, . . . , n. We say u = (u 1 , u 2 , . . . , u n ) ∈ n {R } := [Rn ] is a fuzzy (box) vector , where ×n RF × RF × · · · × RF = ×i=1 F i=1 F    n terms

denotes the Cartesian product .

2.1

An Embedding Theorem for Fuzzy Multidimensional Space

37

Remark 2.9 Let u = (u 1 , u 2 , . . . , u n ) ∈ [RnF ], then the α-level of u are multidimensional intervals (box) of Rn (see Sect. 3 from Stefanini [75]). In fact, a multidimensional intervals (box) of Rn can be regarded as a fuzzy (box) vector. Let u = (u 1 , u 2 , . . . , u n ) and v = (v1 , v2 , . . . , vn ) be two fuzzy vectors with (box) αlevels: + − + − + − + n [u]α = [u − 1,α , u 1,α ] × [u 2,α , u 2,α ] × · · · × [u n,α , u n,α ] := ×i=1 [u i,α , u i,α ],

− + − + − + − + n [v]α = [v1,α , v1,α ] × [v2,α , v2,α ] × · · · × [vn,α , vn,α ] := ×i=1 [vi,α , vi,α ].

The distance is defined by D∞ (u, v) = sup max

 n

α∈[0,1]

1 

1 n 2 2 ∗ ∗ 2 |su (α, Pi ) − sv (α, Pi )| , |su (α, Pi ) − sv (α, Pi )| 2

i=1

: α ∈ [0, 1],

Pi , Pi∗

i=1

∈S

n−1

∩V

n−1

 , i = 1, 2, . . . , n ,

(2.9)

˜ where 0˜ = the distance D∞ (·, ·) induces  · F on [RnF ] defined by uF = D∞ (u, 0), ˜ 0, ˜ . . . , 0) ˜ and 0˜ is a zero element of RF . In fact, because (0, 

 − + − su (α, Pi∗ ), su (α, Pi ) = [u i,α , u i,α ], i = 1, 2, . . . , n,



 − + − sv (α, Pi∗ ), sv (α, Pi ) = [vi,α , vi,α ], i = 1, 2, . . . , n,

then

=

g H v]α = [u]α g H [v]α [u −  n (i) ×i=1 [sv (α, Pi∗ ) − su (α, Pi∗ ), su (α, Pi ) − sv (α, Pi )] or n [su (α, Pi ) − sv (α, Pi ), sv (α, Pi∗ ) − su (α, Pi∗ )], (ii) ×i=1

so from (2.8), we have

38

2 An Embedding Theorem and Multiplication of Fuzzy Vectors

D∞ (u, v) g H vF = sup {[u]α g H [v]α ∗ } = u − α∈[0,1]

= sup max

 n

α∈[0,1]

|su (α, Pi ) − sv (α, Pi )|2

1 

1 n 2 2 , |su (α, Pi∗ ) − sv (α, Pi∗ )|2

i=1

i=1

 : α ∈ [0, 1], Pi , Pi∗ ∈ S n−1 ∩ V n−1 , i = 1, 2, . . . , n . Remark 2.10 For each i = 1, 2, . . . , n, if we introduce the distance  (i) D∞ (u i , vi ) = sup max |su (α, Pi ) − sv (α, Pi )|, |su (α, Pi∗ ) − sv (α, Pi∗ )| : α∈[0,1]

α ∈ [0, 1], Pi , Pi∗ ∈ S n−1 ∩ V n−1 }, (i) ˜ and then it the distance D∞ (·, ·) induces  · F0 on RF defined by u i F0 = D∞ (u i , 0), follows that

g H vF = D∞ (u, v) = u −

 n

(i) D∞ (u i , vi )

1 2

=

 n

i=1

1 u i − vi 2F0

2

.

i=1

Theorem 2.3 The metric space ([RnF ], D∞ ) is complete . (k)

(k)

(k)

Proof Let {u k } = {u 1 , u 2 , . . . , u n ) ∈ [RnF ] be a Cauchy sequence such that D∞ (u m , u n ) → 0 as m, n → ∞. For all α ∈ [0, 1], we obtain the following α-levels for each k ∈ Z+ : (k)−

(k)+

(k)−

(k)+

(k)+ n [u k ]α = [u 1,α , u 1,α ] × · · · × [u (k)− n,α , u n,α ] := ×i=1 [u i,α , u i,α ].

Then we obtain D∞ (u m , u n ) = u m − u n F = = sup max α∈[0,1]

 n

 n

(m)

u i

(n)

i=1



(m)−

u i,α

1

− u i F 0 (n)− 2

− u i,α

i=1

2

1 

1  n 2 2  (m)+ (n)+ 2 u i,α − u i,α , i=1

→ 0 as n, m → ∞.

(2.10)

So for each i = 1, 2, . . . , n, (2.10) yields that (m)−

|u i,α

(n)−

(m)+

− u i,α | → 0 and |u i,α

(n)+

− u i,α | → 0 as n, m → ∞.

2.1

An Embedding Theorem for Fuzzy Multidimensional Space (0)−

(0)+

(k)−

39

(0)−

(k)+

(0)+

Hence, there exist u i,α and u i,α such that u i,α → u i,α and u i,α → u i,α as k → ∞. Therefore, there exists some u 0 ∈ [RnF ] with the α-level which is also a multidimensional intervals (box): (0)−

(0)+

(0)−

(0)+

(0)+ n n [u 0 ]α = [u 1,α , u 1,α ] × · · · × [u (0)− n,α , u n,α ] = ×i=1 [u i,α , u i,α ] ∈ [RF ]

such that D∞ (u k , u 0 ) → 0 as k → ∞. The proof is completed.



¯ 1] on interval [0, 1] is the class of all real-valued bounded functions f on The space C[0, [0, 1] such that f is left continuous for any t ∈ (0, 1] and f has right limit for any t ∈ [0, 1), especially f is right continuous at 0. ¯ 1],  ·  ¯ ) is a Banach space with the norm  f  ¯ = Lemma 2.6 (see [16]) (C[0, C C supt∈[0,1] | f (t)|. In addition, it is easy to verify:   n ¯ 1] × C[0, ¯ 1] , with the norm defined by C[0, Theorem 2.4 ×i=1    ( f 1 , g1 ), ( f 2 , g2 ), . . . , ( f n , gn )  n ¯ ¯ × (C×C) = sup max x∈[0,1]

 n i=1

i=1

1  1  n 2 2 2 2 f i (x) , gi (x) i=1

is a Banach space.   n ¯ 1] × C[0, ¯ 1] , denote Proof For any Cauchy sequence {h n 0 }n 0 ∈N ⊂ ×i=1 C[0,  (n ) (n )  (n ) (n ) h n 0 = ( f 1 0 , g1 0 ), ( f 2 0 , g2 0 ), . . . , ( f n(n 0 ) , gn(n 0 ) ) , and then for any ε > 0, there exists N > 0 such that n 0 , m 0 > N h m 0 − h n 0 ×n (C× ¯ C) ¯ < ε, that is

implies

i=1

h m 0 − h n 0 ×n (C× ¯ C) ¯ i=1  (m 0 ) (m 0 )   (n ) (n )  =  ( f 1 , g1 ), . . . , ( f n(m 0 ) , gn(m 0 ) ) − ( f 1 0 , g1 0 ), . . . , ( f n(n 0 ) , gn(n 0 ) ) ×n (C× ¯ C) ¯ i=1  (m 0 )  (n ) (m ) (n ) =  ( f1 − f 1 0 , g1 0 − g1 0 ), . . . , ( f n(m 0 ) − f n(n 0 ) , gn(m 0 ) − gn(n 0 ) ) ×n (C× ¯ C) ¯ = sup max x∈[0,1]

 n i=1

i=1

1  1  n  (m 0 )  (m 0 ) 2 2 2 2 (n 0 ) (n 0 ) , fi gi (x) − gi (x) (x) − f i (x) , i=1

40

2 An Embedding Theorem and Multiplication of Fuzzy Vectors

which yields that (n 0 )

fi

(n 0 )

(x) → f i (x) and gi

(x) → gi (x) as n 0 → +∞

¯ 1] for each for each i = 1, 2, . . . , n. According to Lemma 2.6, we have f i (x), gi (x) ∈ C[0, i = 1, 2, . . . , n. Therefore, for any ε > 0, there is a N > 0 such that n 0 > N implies  (n ) (n )    h n 0 = ( f 1 0 , g1 0 ), . . . , ( f n(n 0 ) , gn(n 0 ) ) → ( f 1 , g1 ), . . . , ( f n , gn )   n ¯ 1] × C[0, ¯ 1] . C[0, := h ∈ ×i=1 

This completes the proof. The embedding theorem is established as follows.

Theorem 2.5 (Embedding theorem of fuzzy multidimensional space) For all u ∈ [RnF ],  − + n u i , u i . Then j([RnF ]) is a closed convex cone with vertex 0 in denote j(u) = ×i=1     n n ¯ ¯ ¯ 1] × C[0, ¯ 1] satisfies: C[0, ×i=1 C[0, 1] × C[0, 1] and j : [RnF ] → ×i=1 t · v) = sˆ j(u) + t j(v); (i) for all u, v ∈ [RnF ], sˆ , t ≥ 0, j(ˆs · u + (ii) D∞ (u, v) =  j(u) − j(v)×n (C× ¯ C) ¯ ; i=1

  n ¯ 1] × C[0, ¯ 1] isometrically and isomorphically. C[0, i.e., j embeds [RnF ] into ×i=1   n ¯ 1] × C[0, ¯ 1] . It is C[0, Proof From Lemma 2.5, j is meaningful and j([RnF ]) ⊂ ×i=1 easy to see that j is faithful (in fact, from (i) in Lemma 2.5, one can see j is absolutely a one-to-one mapping). In order to show that j([RnF ]) is a convex cone, it suffices to show (i), (ii) in Theorem 2.5. v]α = [u]α + [v]α , [k · u]α = k[u]α (k ≥ 0), for i = 1, 2, . . . , n, By the equations [u + − vi ) = u i− + vi− , (u i + vi )+ = u i+ + vi+ , (k · u i )− = ku i− , (k · u i )+ = ku i+ we have (u i + whenever k ≥ 0. This implies that   n t · vi )− , (ˆs · u i + t · vi )+ t · v) = ×i=1 (ˆs · u i + j(ˆs · u +   n (ˆs · u i )− + (t · vi )− , (ˆs · u i )+ + (t · vi )+ = ×i=1 n (ˆs u − + tv − , sˆ u + + tvi+ ) = ×i=1  n i − +i  i = sˆ ×i=1 (u i , u i )   n (vi− , vi+ ) +t ×i=1

= sˆ j(u) + t j(v) for any u, v ∈ [RnF ], sˆ ≥ 0, t ≥ 0. Hence we complete the proof of (i). Now, we prove (ii). Since

2.2

Multiplication of Fuzzy Vectors in Fuzzy Multidimensional Space

41

D∞ (u, v) = sup d H ([u]α , [v]α ) α∈[0,1]

= sup max α∈[0,1]

 n

|su (α, Pi ) − sv (α, Pi )|2

i=1

1 

1 n 2 2 , |su (α, Pi∗ ) − sv (α, Pi∗ )|2 i=1

 : α ∈ [0, 1], Pi , Pi∗ ∈ S n−1 ∩ V n−1 , i = 1, 2, . . . , n     n  − su (α, Pi∗ ), su (α, Pi ) − − sv (α, Pi∗ ), sv (α, Pi ) ×n (C× =  ×i=1 ¯ C) ¯ : i=1  ∗ n−1 n−1 α ∈ [0, 1], Pi , P ∈ S ∩V , i = 1, 2, . . . , n  n  − + i  − +  =  ×i=1 u i , u i − vi , vi ×n (C× ¯ C) ¯ i=1  n  − + − + n =  ×i=1 (u i , u i ) − ×i=1 (vi , vi )×n (C× ¯ C) ¯ i=1

=  j(u) − j(v)×n

¯ ¯ i=1 (C×C)

and noting that ([RnF ], D∞ ) is complete, we obtain that j([RnF ]) is a closed convex cone   n ¯ 1] × C[0, ¯ 1] . This completes the proof. C[0,  in ×i=1 Similar to Theorem 2.5, we introduce another embedding by   n ¯ 1] × C[0, ¯ 1] , C[0, j : [RnF ] → ×i=1 j(u) = j(−u), u ∈ [RnF ]. It is easy to prove the following properties: t · v) = sˆ (i) for all u, v ∈ [RnF ], sˆ , t ≥ 0, j(ˆs · u + j(u) + t j(v); n n (ii) D∞ (u, v) =  j(u) − j(v)×n (C× ¯ C) ¯ , j([RF ]) = j([RF ]). i=1

Remark 2.11 In Theorem 2.5, let n = 1, then j(u) = (u − , u + ), so one can obtain the Embedding Theorem from [119, 120] immediately. Remark 2.12 From the definitions of j and j, one can easily see that j(t · u) = t j(u) and j(t · u) = t j(u) for all t ∈ R.

2.2

Multiplication of Fuzzy Vectors in Fuzzy Multidimensional Space

In this section, we will introduce six new types of multiplication of two compact intervals. Let [u − , u + ] and [v − , v + ] be two compact intervals and ab denote the ordinary product of real numbers a, b. For convenience, we introduce the following notations:

42

2 An Embedding Theorem and Multiplication of Fuzzy Vectors

 − + u u  (I ) Iu,v =  − +  , v v (I V ) Iu,v

 + + u u  =  − +  , v v

 + − u u  (I I ) Iu,v =  − +  , v v (V ) Iu,v

 − − u u  (I I I ) Iu,v =  − +  , v v

 − + u u  =  − −  , v v

(V I ) Iu,v

 − + u u  =  + +  . v v

For any [a − , a + ] ⊆ [u − , u + ] and [b− , b+ ] ⊆ [v − , v + ], we define the following multiplications:   Type I. [a − , a + ] ◦ [b− , b+ ] = a  b : a ∈ [a − , a + ], b ∈ [b− , b+ ] ,

(2.11)

(I )

where if Iu,v ≤ 0, then

ab =

(I )

if Iu,v ≥ 0, then ab =

⎧ ⎪ ⎪ ⎨ab,

ab ∈ [u − v + , u + v − ],

⎧ ⎪ ⎪ ⎨ab,

ab ∈ [u + v − , u − v + ],

u − v + , ab < u − v + , ⎪ ⎪ ⎩u + v − , ab > u + v − ;

u + v − , ab < u + v − , ⎪ ⎪ ⎩u − v + , ab > u − v + .

  Type II. [a − , a + ]  [b− , b+ ] = a  b : a ∈ [a − , a + ], b ∈ [b− , b+ ] ,

(2.12)

(I I )

where if Iu,v ≤ 0, then

ab =

(I I )

if Iu,v ≥ 0, then

⎧ ⎪ ⎪ ⎨ab,

ab ∈ [u + v + , u − v − ],

u + v + , ab < u + v + , ⎪ ⎪ ⎩u − v − , ab > u − v − ;

⎧ ⎪ ab ∈ [u − v − , u + v + ], ⎪ ⎨ab, a  b = u − v − , ab < u − v − , ⎪ ⎪ ⎩u + v + , ab > u + v + .

  Type III. [a − , a + ]  [b− , b+ ] = a  b : a ∈ [a − , a + ], b ∈ [b− , b+ ] , (I I I )

where if Iu,v

≤ 0, then

(2.13)

2.2

Multiplication of Fuzzy Vectors in Fuzzy Multidimensional Space

ab =

(I I I )

if Iu,v

≥ 0, then

⎧ ⎪ ⎪ ⎨ab,

43

ab ∈ [u − v + , u − v − ],

u − v + , ab < u − v + , ⎪ ⎪ ⎩u − v − , ab > u − v − ;

⎧ ⎪ ab ∈ [u − v − , u − v + ], ⎪ ⎨ab, a  b = u − v − , ab < u − v − , ⎪ ⎪ ⎩u − v + , ab > u − v + .

  Type IV. [a − , a + ]  [b− , b+ ] = a  b : a ∈ [a − , a + ], b ∈ [b− , b+ ] ,

(2.14)

(I V )

where if Iu,v ≤ 0, then

ab =

(I V )

if Iu,v ≥ 0, then ab =

⎧ ⎪ ⎪ab, ⎨

ab ∈ [u + v − , u + v + ],

⎧ ⎪ ⎪ab, ⎨

ab ∈ [u + v + , u + v − ],

u +v−,

ab < u + v − , ⎪ ⎪ ⎩u + v + , ab > u + v + ;

u + v + , ab < u + v + , ⎪ ⎪ ⎩u + v − , ab > u + v − .

  Type V. [a − , a + ] ⊗ [b− , b+ ] = a  b : a ∈ [a − , a + ], b ∈ [b− , b+ ] , where if

(V ) Iu,v

(2.15)

≤ 0, then

ab =

(V )

if Iu,v ≥ 0, then ab =

⎧ ⎪ ⎪ ⎨ab,

ab ∈ [u − v − , u + v − ],

⎧ ⎪ ⎪ ⎨ab,

ab ∈ [u + v − , u − v − ],

u − v − , ab < u − v − , ⎪ ⎪ ⎩u + v − , ab > u + v − ;

u + v − , ab < u + v − , ⎪ ⎪ ⎩u − v − , ab > u − v − .

  Type VI. [a − , a + ]  [b− , b+ ] = a  b : a ∈ [a − , a + ], b ∈ [b− , b+ ] , (V I )

where if Iu,v ≤ 0, then

(2.16)

44

2 An Embedding Theorem and Multiplication of Fuzzy Vectors

ab =

(V I )

if Iu,v ≥ 0, then

⎧ ⎪ ⎪ ⎨ab,

ab ∈ [u − v + , u + v + ],

u − v + , ab < u − v + , ⎪ ⎪ ⎩u + v + , ab > u + v + ;

⎧ ⎪ ab ∈ [u + v + , u − v + ], ⎪ ⎨ab, a  b = u + v + , ab < u + v + , ⎪ ⎪ ⎩u − v + , ab > u − v + .

Remark 2.13 From the above definition of multiplications, for any [a − , a + ] ⊆ [u − , u + ] and [b− , b+ ] ⊆ [v − , v + ], we have [a − , a + ] ◦ [b− , b+ ] ⊆ [u − , u + ] ◦ [v − , v + ] and [a − , a + ]  [b− , b+ ] ⊆ [u − , u + ]  [v − , v + ], similarly, the multiplications , , ⊗,  between two compact intervals also have the same inclusion isotonicity . Now, we introduce six types of the multiplication of fuzzy vectors induced by the multiplications of compact intervals defined by (2.11)–(2.16). For any α ∈ [0, 1] and i = 1, 2, . . . , n, we introduce the notations:       u − u +  u + u −  u − u −   i,α i,α   i,α i,α   i,α i,α  α,(I ) α,(I I ) α,(I I I ) Iu i ,vi =  − +  , Iu i ,vi =  − +  , Iu i ,vi =  − +  ,  vi,α vi,α   vi,α vi,α   vi,α vi,α  V) Iuα,(I i ,vi

  u + u +   i,α i,α  =  − + ,  vi,α vi,α 

) Iuα,(V i ,vi

  u − u +   i,α i,α  =  − − ,  vi,α vi,α 

I) Iuα,(V i ,vi

  u − u +   i,α i,α  =  + + ,  vi,α vi,α 

then we define the following types I − V I with the (compact box) α-level set:  − + − +  n [u i,α , vi,α ] ◦ [u i,α Type I. [u ∗ v]α = ×i=1 , vi,α ] ,

where

− + − + [u i,α , u i,α ] ◦ [vi,α , vi,α ]

=

 α,(I ) − + + − vi,α , u i,α vi,α ] if Iu i ,vi ≤ 0, [u i,α α,(I )

+ − − + vi,α , u i,α vi,α ] if Iu i ,vi ≥ 0; [u i,α

(2.17)

 − + − +  n [u i,α , u i,α Type II. [u  v]α = ×i=1 ]  [vi,α , vi,α ] ,

where

− + − + [u i,α , u i,α ]  [vi,α , vi,α ]

=

 α,(I I ) + + − − vi,α , u i,α vi,α ] if Iu i ,vi ≤ 0, [u i,α α,(I I )

− − + + vi,α , u i,α vi,α ] if Iu i ,vi [u i,α

≥ 0;

 − + − +  n [u i,α , vi,α ]  [u i,α Type III. [u ∗ˆ v]α = ×i=1 , vi,α ] ,

(2.18)

2.2

Multiplication of Fuzzy Vectors in Fuzzy Multidimensional Space

where

− + − + [u i,α , u i,α ]  [vi,α , vi,α ]

=

45

 α,(I I I ) − + − − vi,α , u i,α vi,α ] if Iu i ,vi ≤ 0, [u i,α α,(I I I )

− − − + vi,α , u i,α vi,α ] if Iu i ,vi [u i,α

≥ 0;

(2.19)

 − + − +  n ˆ α = ×i=1 [u i,α , vi,α ]  [u i,α Type IV. [u v] , vi,α ] , − + − + where [u i,α , u i,α ]  [vi,α , vi,α ]=

 α,(I V ) + − + + vi,α , u i,α vi,α ] if Iu i ,vi ≤ 0, [u i,α α,(I V )

+ + + − vi,α , u i,α vi,α ] if Iu i ,vi [u i,α

≥ 0;

(2.20)

 − + − +  n [u i,α , vi,α ] ⊗ [u i,α Type V. [u ∗˜ v]α = ×i=1 , vi,α ] ,

where

− + − + [u i,α , u i,α ] ⊗ [vi,α , vi,α ]

=

 α,(V ) − − + − vi,α , u i,α vi,α ] if Iu i ,vi ≤ 0, [u i,α α,(V )

+ − − − vi,α , u i,α vi,α ] if Iu i ,vi ≥ 0; [u i,α

(2.21)

 − + − +  n ˜ α = ×i=1 [u i,α , vi,α ]  [u i,α Type VI. [u v] , vi,α ] ,

where

− + − + [u i,α , u i,α ]  [vi,α , vi,α ]

=

 α,(V I ) − + + + [u i,α vi,α , u i,α vi,α ] if Iu i ,vi ≤ 0, α,(V I )

+ + − + vi,α , u i,α vi,α ] if Iu i ,vi [u i,α

≥ 0.

(2.22)

α,(I )

− + Remark 2.14 For Iu i ,vi = 0 for all i = 1, 2, . . . , n, from (2.17), we have u i,α vi,α = + − u i,α vi,α , then − + − + − + + − n n n [u ∗ v]α = ×i=1 [u i,α , u i,α ] ◦ [vi,α , vi,α ] = ×i=1 {u i,α vi,α } = ×i=1 {u i,α vi,α }. α,(I I )

Similarly, For Iu i ,vi

= 0 for all i = 1, 2, . . . , n, from (2.18), we have

 − + − +  + + − − n n n [u i,α , u i,α [u  v]α = ×i=1 ]  [vi,α , vi,α ] = ×i=1 {u i,α vi,α } = ×i=1 {u i,α vi,α }, n [a , a ] = ×n {a } for any a ∈ R. For example, given u = χ noticing that ×i=1 i i i [−a,a] and i=1 i v = χ[−b,b] in RF , where a, b > 0, it follows that [u]α = [−a, a], [v]α = [−b, b] for all α,(I ) α,(I I ) α ∈ [0, 1]. Note that Iu,v = Iu,v = 0, it indicates that [u ∗ v]α = {−ab} and [u  v]α = {ab}, i.e. u ∗ v = χ{−ab} and u  v = χ{ab} . In fact, it is easy to see that if there exists some α,( Iˆ) Iˆ ∈ {I , I I , . . . , V I } such that Iu i ,vi = 0, then the corresponding product of α-levels defined by (2.17)–(2.22) is a one-point set for Type Iˆ.

Remark 2.15 Since the interval multiplications defined by (2.11) and (2.16) have a well inclusion isotonicity, then (2.17) and (2.22) also has well inclusion isotonicity naturally. α,(I ) For example, given u = χ[−1,0] and v = χ[−1,1] , then we have Iu,v < 0 for all α ∈ [0, 1]. Therefore, u ∗ v is given by + − + − + + − [u ∗ v]α = [u − α , u α ] ◦ [vα , vα ] = [u α vα , u α vα ] = [−1, 0]

46

2 An Embedding Theorem and Multiplication of Fuzzy Vectors

+ − + for all α ∈ [0, 1]. For any given a ∈ [−1, 0] = [u − α , u α ] and b ∈ [−1, 1] = [vα , vα ], it implies that ⎧ ⎪ ⎪ ⎨ab, ab ∈ [−1, 0], a  b = −1, ab < −1, ⎪ ⎪ ⎩0, ab > 0, + − + which indicates that for any [a, b] ⊆ [u − α , u α ], [c, d] ⊆ [vα , vα ], we obtain [a, b] ◦ [c, d] ⊆ − + − + [u α , u α ] ◦ [vα , vα ].

Remark 2.16 Traditionally, the multiplication of compact intervals is induced by the ordinary multiplication of real numbers, i.e, for the real compact intervals U = [u − , u + ] and V = [v − , v + ], the interval C = [c− , c+ ] defining the multiplication C = U V is given by c− = min{u − v − , u − v + , u + v − , u + v + }, c+ = max{u − v − , u − v + , u + v − , u + v + }. In fact, C = U V = {ab : a ∈ U , b ∈ V }. However, note that such a multiplication of compact intervals induced by ordinary multiplication of real numbers is completely different from the multiplications of compact intervals induced by a  b above. In the example 1 1 1 1 + − + / of Remark 2.15, given − 21 ∈ [u − α , u α ], − 4 ∈ [vα , vα ], we have ab = (− 2 )(− 4 ) = 8 ∈ 1 1 [−1, 0] = [−1, 0] ◦ [−1, 1] but (− 2 )  (− 4 ) = 0 ∈ [−1, 0] = [−1, 0] ◦ [−1, 1]. Theorem 2.6 Let u, v, ω ∈ [RnF ], we have the following related properties with respect to g H(I ) v exists. the multiplications u ∗ v and u  v. For (i) − (xi) we assume that u − (i) (ii) (iii) (iv)

u ∗ v = v ∗ u; u  v = v  u. α,(I ) α,(I ) α,(I ) α,(I ) v) ∗ ω = u ∗ ω+ v ∗ ω. If Iu i ,ωi , Ivi ,ωi ≥ 0 or Iu i ,ωi , Ivi ,ωi ≤ 0, then (u + α,(I I ) α,(I I ) α,(I I ) α,(I I ) v)  ω = u  ω+ v  ω. If Iu i ,ωi , Ivi ,ωi ≥ 0 or Iu i ,ωi , Ivi ,ωi ≤ 0, then (u + α,(I ) α,(I ) α,(I ) If Iu − ≥ 0, I , I ≥ 0, then (u − v) ∗ ω = u ∗ ω − g H(I ) g H(I ) v ∗ ω. u i ,ωi vi ,ωi v ,ω

(v)

If Iu

g H(I ) v) ∗ ω = u ∗ ω− g H(I I ) v ∗ ω. ≥ 0, Iu i ,ωi , Ivi ,ωi ≤ 0, then (u −

(vi)

If

g H(I ) v) ∗ ω = u ∗ ω− g H(I I ) v ∗ ω. ≤ 0, Iu i ,ωi , Ivi ,ωi ≥ 0, then (u −

i

(vii) If

g H(I ) i

i

α,(I ) g H vi ,ωi i− (I ) α,(I ) Iu − i g H(I ) vi ,ωi α,(I ) Iu − i g H(I ) vi ,ωi α,(I I ) Iu − i g H(I ) vi ,ωi

(viii) If ω. α,(I I ) (i x) If Iu − i g H(I ) vi ,ωi ω. α,(I I ) (x) If Iu − i g H(I ) vi ,ωi ω. α,(I I ) (xi) If Iu − i g H(I ) vi ,ωi ω.

α,(I )

α,(I )

α,(I )

α,(I )

α,(I )

α,(I )

g H(I ) v) ∗ ω = u ∗ ω− g H(I ) v ∗ ω. ≤ 0, Iu i ,ωi , Ivi ,ωi ≤ 0, then (u − α,(I I ) α,(I I ) g H(I ) v)  ω = u  ω− g H(I ) v  ≥ 0, Iu i ,ωi , Ivi ,ωi ≥ 0, then (u − α,(I I )

α,(I I )

g H(I ) v)  ω = u  ω− g H(I I ) v  ≥ 0, Iu i ,ωi , Ivi ,ωi ≤ 0, then (u − α,(I I ) α,(I I ) g H(I ) v)  ω = u  ω− g H(I I ) v  ≤ 0, Iu i ,ωi , Ivi ,ωi ≥ 0, then (u − α,(I I )

α,(I I )

g H(I ) v)  ω = u  ω− g H(I ) v  ≤ 0, Iu i ,ωi , Ivi ,ωi ≤ 0, then (u −

2.2

Multiplication of Fuzzy Vectors in Fuzzy Multidimensional Space

Proof (i). From (2.17) and (2.18), the results are obvious. α,(I ) α,(I ) (ii). Case I. If Iu i ,ωi , Ivi ,ωi ≥ 0, we have + − − + + − − + n n [u ∗ ω]α = ×i=1 [u i,α ωi,α , u i,α ωi,α ], [v ∗ ω]α = ×i=1 [vi,α ωi,α , vi,α ωi,α ]. α,(I )

α,(I )

Case II. If Iu i ,ωi , Ivi ,ωi ≤ 0, we have − + + − − + + − n n [u ∗ ω]α = ×i=1 [u i,α ωi,α , u i,α ωi,α ], [v ∗ ω]α = ×i=1 [vi,α ωi,α , vi,α ωi,α ].

Then for Case I, we have α,(I ) Iu + i vi ,ωi

  u − + v − u + + v +   i,α  i,α i,α i,α =  − +  ωi,α  ωi,α − − + + + − = (u i,α + vi,α )ωi,α − (u i,α + vi,α )ωi,α ≥ 0,

and then we obtain + + − − − + n v) ∗ ω]α = ×i=1 [(u + [(u i,α + vi,α )ωi,α , (u i,α + vi,α )ωi,α ]

= [u ∗ ω]α + [v ∗ ω]α . Similarly, we can also have the same results for Case II. α,(I I ) α,(I I ) (iii). Case I. If Iu i ,ωi , Ivi ,ωi ≥ 0, we have − − + + − − + + n n [u  ω]α = ×i=1 [u i,α ωi,α , u i,α ωi,α ], [v  ω]α = ×i=1 [vi,α ωi,α , vi,α ωi,α ]. α,(I I )

α,(I I )

Case II. If Iu i ,ωi , Ivi ,ωi ≤ 0, we have + + − − + + − − n n [u  ω]α = ×i=1 [u i,α ωi,α , u i,α ωi,α ], [v  ω]α = ×i=1 [vi,α ωi,α , vi,α ωi,α ].

Then for Case I, we have α,(I I ) Iu + i vi ,ωi

  u + + v + u − + v −   i,α i,α i,α i,α  =  − +  ωi,α  ωi,α + + + − − − = (u i,α + vi,α )ωi,α − (u i,α + vi,α )ωi,α ≥ 0,

and then we obtain − − − + + + n v)  ω]α = ×i=1 [(u + [(u i,α + vi,α )ωi,α , (u i,α + vi,α )ωi,α ]

= [u  ω]α + [v  ω]α . Similarly, we can also have the same results for Case II. α,(I ) α,(I ) (iv). Since Iu i ,ωi , Ivi ,ωi ≥ 0, we have

47

48

2 An Embedding Theorem and Multiplication of Fuzzy Vectors + − − + + − − + n n [u ∗ ω]α = ×i=1 [u i,α ωi,α , u i,α ωi,α ], [v ∗ ω]α = ×i=1 [vi,α ωi,α , vi,α ωi,α ]. α,(I ) g H i−

Moreover, it follows from Iu α,(I ) Iu − i g H

v ,ω (I ) i i

v ,ω (I ) i i

≥ 0 that

  u − − v − u + − v +   i,α  i,α i,α i,α =  − +  ωi,α  ωi,α − − + + + − = (u i,α − vi,α )ωi,α − (u i,α − vi,α )ωi,α ≥ 0,

which indicates that [u ∗ ω]α g H(I ) [v ∗ ω]α exists, so we obtain + + − − − + n g H(I ) v) ∗ ω]α = ×i=1 [(u − [(u i,α − vi,α )ωi,α , (u i,α − vi,α )ωi,α ]

= [u ∗ ω]α g H(I ) [v ∗ ω]α .

α,(I )

α,(I )

(v). Since Iu i ,ωi , Ivi ,ωi ≤ 0, we have − + + − − + + − n n [u ∗ ω]α = ×i=1 [u i,α ωi,α , u i,α ωi,α ], [v ∗ ω]α = ×i=1 [vi,α ωi,α , vi,α ωi,α ]. α,(I ) g H i−

Moreover, it follows from Iu α,(I ) Iu − i g H

v ,ω (I ) i i

v ,ω (I ) i i

≥ 0 that

  u − − v − u + − v +   i,α  i,α i,α i,α =  − +  ωi,α  ωi,α − − + + + − = (u i,α − vi,α )ωi,α − (u i,α − vi,α )ωi,α ≥ 0,

which indicates that [u ∗ ω]α g H(I I ) [v ∗ ω]α exists, so we obtain + + + − − − n g H(I ) v) ∗ ω]α = ×i=1 [(u − [(u i,α − vi,α )ωi,α , (u i,α − vi,α )ωi,α ]

= [u ∗ ω]α g H(I I ) [v ∗ ω]α .

α,(I )

α,(I )

(vi). Since Iu i ,ωi , Ivi ,ωi ≥ 0, we have + − − + + − − + n n [u ∗ ω]α = ×i=1 [u i,α ωi,α , u i,α ωi,α ], [v ∗ ω]α = ×i=1 [vi,α ωi,α , vi,α ωi,α ]. α,(I ) g H i−

Moreover, it follows from Iu α,(I ) Iu − i g H

v ,ω (I ) i i

v ,ω (I ) i i

≤ 0 that

  u − − v − u + − v +   i,α  i,α i,α i,α =  − +  ωi,α  ωi,α − − + + + − = (u i,α − vi,α )ωi,α − (u i,α − vi,α )ωi,α ≤ 0,

which indicates that [u ∗ ω]α g H(I I ) [v ∗ ω]α exists, so we obtain

2.2

Multiplication of Fuzzy Vectors in Fuzzy Multidimensional Space − − + + + − n g H(I ) v) ∗ ω]α = ×i=1 [(u − [(u i,α − vi,α )ωi,α , (u i,α − vi,α )ωi,α ]

= [u ∗ ω]α g H(I I ) [v ∗ ω]α .

α,(I )

α,(I )

(vii). Since Iu i ,ωi , Ivi ,ωi ≤ 0, we have − + + − − + + − n n [u ∗ ω]α = ×i=1 [u i,α ωi,α , u i,α ωi,α ], [v ∗ ω]α = ×i=1 [vi,α ωi,α , vi,α ωi,α ]. α,(I ) g H i−

Moreover, it follows from Iu α,(I ) Iu − i g H

v ,ω (I ) i i

v ,ω (I ) i i

≤ 0 that

  u − − v − u + − v +   i,α  i,α i,α i,α =  − +  ωi,α  ωi,α − − + + + − = (u i,α − vi,α )ωi,α − (u i,α − vi,α )ωi,α ≤ 0,

which indicates that [u ∗ ω]α g H(I ) [v ∗ ω]α exists, so we obtain − − + + + − n g H(I ) v) ∗ ω]α = ×i=1 [(u − [(u i,α − vi,α )ωi,α , (u i,α − vi,α )ωi,α ]

= [u ∗ ω]α g H(I ) [v ∗ ω]α .

α,(I I )

α,(I I )

(viii). Since Iu i ,ωi , Ivi ,ωi ≥ 0, we have − − + + − − + + n n [u  ω]α = ×i=1 [u i,α ωi,α , u i,α ωi,α ], [v  ω]α = ×i=1 [vi,α ωi,α , vi,α ωi,α ]. α,(I I ) g H i−

Moreover, it follows from Iu α,(I I ) Iu − i g H

v ,ω (I ) i i

v ,ω (I ) i i

≥ 0 that

  u + − v + u − − v −   i,α  i,α i,α i,α =  − +  ωi,α  ωi,α + + + − − − = (u i,α − vi,α )ωi,α − (u i,α − vi,α )ωi,α ≥ 0,

which indicates that [u  ω]α g H(I ) [v  ω]α exists, so we obtain − − − + + + n g H(I ) v)  ω]α = ×i=1 [(u − [(u i,α − vi,α )ωi,α , (u i,α − vi,α )ωi,α ]

= [u  ω]α g H(I ) [v  ω]α .

α,(I I )

α,(I I )

(i x). Since Iu i ,ωi , Ivi ,ωi ≤ 0, we have + + − − + + − − n n [u  ω]α = ×i=1 [u i,α ωi,α , u i,α ωi,α ], [v  ω]α = ×i=1 [vi,α ωi,α , vi,α ωi,α ]. α,(I I ) g H i−

Moreover, it follows from Iu

v ,ω (I ) i i

≥ 0 that

49

50

2 An Embedding Theorem and Multiplication of Fuzzy Vectors α,(I I ) Iu − i g H(I ) vi ,ωi

  u + − v + u − − v −   i,α i,α i,α i,α  =  − +  ωi,α  ωi,α + + + − − − = (u i,α − vi,α )ωi,α − (u i,α − vi,α )ωi,α ≥ 0,

which indicates that [u  ω]α g H(I I ) [v  ω]α exists, so we obtain − − − + + + n g H(I ) v)  ω]α = ×i=1 [(u − [(u i,α − vi,α )ωi,α , (u i,α − vi,α )ωi,α ]

= [u  ω]α g H(I I ) [v  ω]α .

α,(I I )

α,(I I )

(x). Since Iu i ,ωi , Ivi ,ωi ≥ 0, we have − − + + − − + + n n [u  ω]α = ×i=1 [u i,α ωi,α , u i,α ωi,α ], [v  ω]α = ×i=1 [vi,α ωi,α , vi,α ωi,α ]. α,(I I ) g H vi ,ωi i− (I )

Moreover, it follows from Iu α,(I I ) Iu − i g H(I ) vi ,ωi

≤ 0 that

  u + − v + u − − v −   i,α i,α i,α i,α  =  − +  ωi,α  ωi,α + + + − − − = (u i,α − vi,α )ωi,α − (u i,α − vi,α )ωi,α ≤ 0,

which indicates that [u  ω]α g H(I I ) [v  ω]α exists, so we obtain + + + − − − n g H(I ) v)  ω]α = ×i=1 [(u − [(u i,α − vi,α )ωi,α , (u i,α − vi,α )ωi,α ]

= [u  ω]α g H(I I ) [v  ω]α .

α,(I I )

α,(I I )

(xi). Since Iu i ,ωi , Ivi ,ωi ≤ 0, we have + + − − + + − − n n [u  ω]α = ×i=1 [u i,α ωi,α , u i,α ωi,α ], [v  ω]α = ×i=1 [vi,α ωi,α , vi,α ωi,α ]. α,(I I ) g H vi ,ωi i− (I )

Moreover, it follows from Iu α,(I I ) Iu − i g H

v ,ω (I ) i i

≤ 0 that

  u + − v + u − − v −   i,α i,α i,α i,α  =  − +  ωi,α  ωi,α + + + − − − = (u i,α − vi,α )ωi,α − (u i,α − vi,α )ωi,α ≤ 0,

which indicates that [u  ω]α g H(I ) [v  ω]α exists, so we obtain + + + − − − n g H(I ) v)  ω]α = ×i=1 [(u − [(u i,α − vi,α )ωi,α , (u i,α − vi,α )ωi,α ]

= [u  ω]α g H(I ) [v  ω]α .

This completes the proof.



2.2

Multiplication of Fuzzy Vectors in Fuzzy Multidimensional Space

51

Theorem 2.7 Let u, v, ω ∈ [RnF ], we have the following related properties with respect to g H(I I ) v exists. the multiplications u ∗ v and u  v. For (i) − (viii) we assume that u − α,(I I ) g H v ,ω i− (I I ) i i α,(I I ) Iu − i g H(I I ) vi ,ωi α,(I I ) Iu − i g H(I I ) vi ,ωi α,(I I ) Iu − i g H(I I ) vi ,ωi α,(I ) Iu − i g H(I I ) vi ,ωi

α,(I )

α,(I )

α,(I )

α,(I )

(i)

If Iu

g H(I I ) v)  ω = u ∗ ω− g H(I ) v ∗ ω. ≥ 0, Iu i ,ωi , Ivi ,ωi ≥ 0, then (u −

(ii)

If

g H(I I ) v)  ω = u ∗ ω− g H(I ) v ∗ ω. ≥ 0, Iu i ,ωi , Ivi ,ωi ≤ 0, then (u −

(iii)

If

(iv)

If

(v)

If ω. α,(I ) (vi) If Iu − i g H(I I ) vi ,ωi ω. α,(I ) (vii) If Iu − i g H(I I ) vi ,ωi ω. α,(I ) (viii) If Iu − i g H(I I ) vi ,ωi ω. α,(I )

α,(I ) α,(I ) g H(I I ) v)  ω = u ∗ ω− g H(I I ) v ∗ ω. ≤ 0, Iu i ,ωi , Ivi ,ωi ≥ 0, then (u − α,(I )

α,(I )

g H(I I ) v)  ω = u ∗ ω− g H(I ) v ∗ ω. ≤ 0, Iu i ,ωi , Ivi ,ωi ≤ 0, then (u − α,(I I )

α,(I I )

g H(I I ) v) ∗ ω = u  ω− g H(I ) v  ≥ 0, Iu i ,ωi , Ivi ,ωi ≥ 0, then (u − α,(I I ) α,(I I ) g H(I I ) v) ∗ ω = u  ω− g H(I I ) v  ≥ 0, Iu i ,ωi , Ivi ,ωi ≤ 0, then (u − α,(I I )

α,(I I )

g H(I I ) v) ∗ ω = u  ω− g H(I ) v  ≤ 0, Iu i ,ωi , Ivi ,ωi ≥ 0, then (u − α,(I I ) α,(I I ) g H(I I ) v) ∗ ω = u  ω− g H(I ) v  ≤ 0, Iu i ,ωi , Ivi ,ωi ≤ 0, then (u −

α,(I )

Proof (i). Since Iu i ,ωi , Ivi ,ωi ≥ 0, we have + − − + + − − + n n [u ∗ ω]α = ×i=1 [u i,α ωi,α , u i,α ωi,α ], [v ∗ ω]α = ×i=1 [vi,α ωi,α , vi,α ωi,α ]. α,(I I ) g H i−

Moreover, it follows from Iu α,(I I ) Iu − i g H

v ,ω (I I ) i i

v ,ω (I I ) i i

≥ 0 that

  u − − v − u + − v +   i,α  i,α i,α i,α =  − +  ωi,α  ωi,α − − + + + − = (u i,α − vi,α )ωi,α − (u i,α − vi,α )ωi,α ≥ 0,

which indicates that [u ∗ ω]α g H(I ) [v ∗ ω]α exists, so we obtain + + − − − + n g H(I I ) v)  ω]α = ×i=1 [(u − [(u i,α − vi,α )ωi,α , (u i,α − vi,α )ωi,α ]

= [u ∗ ω]α g H(I ) [v ∗ ω]α .

α,(I )

α,(I )

(ii). Since Iu i ,ωi , Ivi ,ωi ≤ 0, we have − + + − − + + − n n [u ∗ ω]α = ×i=1 [u i,α ωi,α , u i,α ωi,α ], [v ∗ ω]α = ×i=1 [vi,α ωi,α , vi,α ωi,α ]. α,(I I ) g H i−

Moreover, it follows from Iu

v ,ω (I I ) i i

≥ 0 that

52

2 An Embedding Theorem and Multiplication of Fuzzy Vectors α,(I I ) Iu − i g H(I I ) vi ,ωi

  u − − v − u + − v +   i,α i,α i,α i,α  =  − +  ωi,α  ωi,α − − + + + − = (u i,α − vi,α )ωi,α − (u i,α − vi,α )ωi,α ≥ 0,

which indicates that [u ∗ ω]α g H(I I ) [v ∗ ω]α exists, so we obtain + + + − − − n g H(I I ) v)  ω]α = ×i=1 [(u − [(u i,α − vi,α )ωi,α , (u i,α − vi,α )ωi,α ]

= [u ∗ ω]α g H(I I ) [v ∗ ω]α .

α,(I )

α,(I )

(iii). Since Iu i ,ωi , Ivi ,ωi ≥ 0, we have + − − + + − − + n n [u ∗ ω]α = ×i=1 [u i,α ωi,α , u i,α ωi,α ], [v ∗ ω]α = ×i=1 [vi,α ωi,α , vi,α ωi,α ]. α,(I I ) g H v ,ω i− (I I ) i i

Moreover, it follows from Iu

α,(I I ) Iu − i g H(I I ) vi ,ωi

≤ 0 that

  u − − v − u + − v +   i,α i,α i,α i,α  =  − +  ωi,α  ωi,α − − + + + − = (u i,α − vi,α )ωi,α − (u i,α − vi,α )ωi,α ≤ 0,

which indicates that [u ∗ ω]α g H(I I ) [v ∗ ω]α exists, so we obtain − − + + + − n g H(I I ) v)  ω]α = ×i=1 [(u − [(u i,α − vi,α )ωi,α , (u i,α − vi,α )ωi,α ]

= [u ∗ ω]α g H(I I ) [v ∗ ω]α .

α,(I )

α,(I )

(iv). Since Iu i ,ωi , Ivi ,ωi ≤ 0, we have − + + − − + + − n n [u ∗ ω]α = ×i=1 [u i,α ωi,α , u i,α ωi,α ], [v ∗ ω]α = ×i=1 [vi,α ωi,α , vi,α ωi,α ]. α,(I I ) g H v ,ω i− (I I ) i i

Moreover, it follows from Iu α,(I I ) Iu − i g H

v ,ω (I I ) i i

≤ 0 that

  u − − v − u + − v +   i,α i,α i,α i,α  =  − +  ωi,α  ωi,α − − + + + − = (u i,α − vi,α )ωi,α − (u i,α − vi,α )ωi,α ≤ 0,

which indicates that [u ∗ ω]α g H(I ) [v ∗ ω]α exists, so we obtain − − + + + − n g H(I I ) v)  ω]α = ×i=1 [(u − [(u i,α − vi,α )ωi,α , (u i,α − vi,α )ωi,α ]

= [u ∗ ω]α g H(I ) [v ∗ ω]α .

α,(I I )

α,(I I )

(v). Since Iu i ,ωi , Ivi ,ωi ≥ 0, we have

2.2

Multiplication of Fuzzy Vectors in Fuzzy Multidimensional Space − − + + − − + + n n [u  ω]α = ×i=1 [u i,α ωi,α , u i,α ωi,α ], [v  ω]α = ×i=1 [vi,α ωi,α , vi,α ωi,α ]. α,(I ) g H i−

Moreover, it follows from Iu α,(I ) Iu − i g H

v ,ω (I I ) i i

v ,ω (I I ) i i

≥ 0 that

  u + − v + u − − v −   i,α  i,α i,α i,α =  − +  ωi,α  ωi,α + + + − − − = (u i,α − vi,α )ωi,α − (u i,α − vi,α )ωi,α ≥ 0,

which indicates that [u  ω]α g H(I ) [v  ω]α exists, so we obtain − − − + + + n g H(I I ) v) ∗ ω]α = ×i=1 [(u − [(u i,α − vi,α )ωi,α , (u i,α − vi,α )ωi,α ]

= [u  ω]α g H(I ) [v  ω]α .

α,(I I )

α,(I I )

(vi). Since Iu i ,ωi , Ivi ,ωi ≤ 0, we have + + − − + + − − n n [u  ω]α = ×i=1 [u i,α ωi,α , u i,α ωi,α ], [v  ω]α = ×i=1 [vi,α ωi,α , vi,α ωi,α ]. α,(I ) g H i−

Moreover, it follows from Iu α,(I ) Iu − i g H

v ,ω (I I ) i i

v ,ω (I I ) i i

≥ 0 that

  u + − v + u − − v −   i,α  i,α i,α i,α =  − +  ωi,α  ωi,α + + + − − − = (u i,α − vi,α )ωi,α − (u i,α − vi,α )ωi,α ≥ 0,

which indicates that [u  ω]α g H(I I ) [v  ω]α exists, so we obtain − − − + + + n g H(I I ) v) ∗ ω]α = ×i=1 [(u − [(u i,α − vi,α )ωi,α , (u i,α − vi,α )ωi,α ]

= [u  ω]α g H(I I ) [v  ω]α .

α,(I I )

α,(I I )

(vii). Since Iu i ,ωi , Ivi ,ωi ≥ 0, we have − − + + − − + + n n [u  ω]α = ×i=1 [u i,α ωi,α , u i,α ωi,α ], [v  ω]α = ×i=1 [vi,α ωi,α , vi,α ωi,α ]. α,(I ) g H i−

Moreover, it follows from Iu α,(I ) Iu − i g H

v ,ω (I I ) i i

v ,ω (I I ) i i

≤ 0 that

  u + − v + u − − v −   i,α  i,α i,α i,α =  − +  ωi,α  ωi,α + + + − − − = (u i,α − vi,α )ωi,α − (u i,α − vi,α )ωi,α ≤ 0,

which indicates that [u  ω]α g H(I I ) [v  ω]α exists, so we obtain

53

54

2 An Embedding Theorem and Multiplication of Fuzzy Vectors + + + − − − n g H(I I ) v) ∗ ω]α = ×i=1 [(u − [(u i,α − vi,α )ωi,α , (u i,α − vi,α )ωi,α ]

= [u  ω]α g H(I I ) [v  ω]α .

α,(I I )

α,(I I )

(viii). Since Iu i ,ωi , Ivi ,ωi ≤ 0, we have + + − − + + − − n n [u  ω]α = ×i=1 [u i,α ωi,α , u i,α ωi,α ], [v  ω]α = ×i=1 [vi,α ωi,α , vi,α ωi,α ]. α,(I ) g H i−

Moreover, it follows from Iu α,(I ) Iu − i g H

v ,ω (I I ) i i

v ,ω (I I ) i i

≤ 0 that

  u + − v + u − − v −   i,α  i,α i,α i,α =  − +  ωi,α  ωi,α + + + − − − = (u i,α − vi,α )ωi,α − (u i,α − vi,α )ωi,α ≤ 0,

which indicates that [u  ω]α g H(I ) [v  ω]α exists, so we obtain + + + − − − n g H(I I ) v) ∗ ω]α = ×i=1 [(u − [(u i,α − vi,α )ωi,α , (u i,α − vi,α )ωi,α ]

= [u  ω]α g H(I ) [v  ω]α .



This completes the proof.

Theorem 2.8 If u, v ∈ [RnF ], then u ∗ vF ≤ uF · vF and u  vF ≤ uF · vF . Proof From Theorem 2.5, we have u ∗ vF =  j(u ∗ v)×n

¯

¯

i=1 (C×C)

=

(u i− vi+ , u i+ vi− )×n (C× ¯ C) ¯ i=1

1  1 

 n n 2 2 − + 2 + − 2 = sup max (u i vi ) , (u i vi ) α∈[0,1]

≤ sup max

i=1

n

α∈[0,1]

≤ sup max ≤ sup max

i=1

i=1

 n

α∈[0,1]

× sup max α∈[0,1]

|u i− vi+ |,

 n

α∈[0,1]

i=1 n



|u i+ vi− |

i=1

1  1  1  1  n n n 2 2 2 2 − 2 + 2 + 2 − 2 |u i | · |vi | , |u i | · |vi | i=1

1  1  n 2 2 |u i− |2 , |u i+ |2

i=1

 n i=1

i=1

1  1  n 2 2 |vi− |2 , |vi+ |2 i=1

i=1

i=1

2.2

Multiplication of Fuzzy Vectors in Fuzzy Multidimensional Space n =  ×i=1 (u i− , u i+ )×n

=  j(u)×n

¯

¯

i=1 (C×C)

¯

n ·  ×i=1 (vi+ , vi− )×n

¯

i=1 (C×C)

·  j(v)×n

¯

i=1 (C×C)

¯

55

¯

¯

i=1 (C×C)

= uF · uF .

Similar to the above discussion, we also have u  vF =  j(u  v)×n

¯

¯

i=1 (C×C)

=

(u i+ vi+ , u i− vi− )×n (C× ¯ C) ¯ i=1

1  1 

 n n 2 2 + + 2 − − 2 = sup max (u i vi ) , (u i vi ) α∈[0,1]

i=1

≤ sup max

n

α∈[0,1]

|u i+ vi+ |,

i=1

≤ sup max

 n

α∈[0,1]

i=1

≤ sup max

 n

α∈[0,1]

i=1

× sup max =

i=1

i=1

1  1  n 2 2 |u i− |2 , |u i+ |2

¯

¯

i=1 (C×C)

= uF · vF .

i=1

i=1

i=1

1  1  n 2 2 |vi− |2 , |vi+ |2

i=1 − + (u i , u i )×n (C× ¯ C) ¯ i=1

=  j(u)×n



|u i− vi− |

1  1  1  1  n n n 2 2 2 2 + 2 + 2 − 2 − 2 |u i | · |vi | , |u i | · |vi |

 n

α∈[0,1]

n  ×i=1

i=1 n

·  j(v)×n

i=1 n ·  ×i=1 ¯

(vi+ , vi− )×n

¯

¯

i=1 (C×C)

¯

i=1 (C×C)

This completes the proof.



From Theorem 2.5 and the definition embedding j, we can prove the following properties easily. Theorem 2.9 For u, v, w ∈ [RnF ], if the g H -difference among them exist, then the following properties hold: (i) (ii) (iii) (iv)

g H w, v ± g H w) = D∞ (u, v); D∞ (u ± g H w, v ± g H e) ≤ D∞ (u, v) + D∞ (w, e); D∞ (u ± D∞ (μ · u, μ · v) = |μ|D∞ (u, v) for μ ∈ R; g H v) ∗ ω = u ∗ ω− g H v ∗ ω; D∞ (u ∗ w, v ∗ w) ≤ wF D∞ (u, v) if (u − g H v)  ω = u  ω− g H v  ω; D∞ (u  w, v  w) ≤ wF D∞ (u, v) if (u −

56

2 An Embedding Theorem and Multiplication of Fuzzy Vectors

(v)

D∞ (μ · u, ν · u) = |μ − ν|uF for μ, ν ≥ 0 or μ, ν ≤ 0.

Proof (i). For u, v, w ∈ [RnF ], we have g H w, v ± g H w) =  j(u ± g H w) − j(v ± g H w)×n D∞ (u ±

¯

¯

i=1 (C×C)

=  j(u) ± j(w) − j(v) ∓ j(w)×n

¯

¯

i=1 (C×C)

=  j(u) − j(v)×n

¯

¯

i=1 (C×C)

= D∞ (u, v).

(ii). For u, v, w, e ∈ [RnF ], we obtain g H w, v ± g H e) =  j(u ± g H w) − j(v ± g H e)×n D∞ (u ±

¯

¯

i=1 (C×C)

=  j(u) ± j(w) − j(v) ∓ j(e)×n

¯

¯

i=1 (C×C)

= ( j(u) − j(v)) ± ( j(w) − j(e))×n ≤  j(u) − j(v) +  j(w) − j(e)×n

¯

¯

¯

i=1 (C×C)

= D∞ (u, v) + D∞ (w, e). (iii). For u, v ∈ [RnF ] and μ ∈ R, we have    D∞ (μ · u, μ · v) = μ j(u) − j(v) 

¯

i=1 (C×C)

n (C× ¯ C) ¯ ×i=1

= |μ| jα, p (u) − jα, p (v)×n

¯

¯

i=1 (C×C)

= |μ|D∞ (u, v).

(iv). By Theorem 2.8, we have D∞ (u ∗ w, v ∗ w) =  j(u ∗ w) − j(v ∗ w)×n (C× ¯ C) ¯ i=1    g H v) ∗ w  n ¯ ¯ =  j (u − × (C×C) i=1

g H vF wF = wF D∞ (u, v). ≤ u − Similarly, we also have D∞ (u  w, v  w) =  j(u  w) − j(v  w)×n (C× ¯ C) ¯ i=1    g H v)  w  n ¯ ¯ =  j (u − × (C×C) i=1

g H vF wF = wF D∞ (u, v). ≤ u − (v). D∞ (μ · u, ν · u) = μ j(u) − ν j(u)×n

¯

¯

i=1 (C×C)

˜ = (μ − ν)( j(u) − j(0)) ×n

¯

¯

i=1 (C×C)

This completes the proof.

= |μ − ν|uF . 

Calculus of Fuzzy Vector-Valued Functions on Time Scales

In this chapter, we will establish some basic results of calculus of fuzzy vector-valued functions on time scales. For convenience, we introduce the following notations. Let f , g : T → [RnF ], where f = ( f 1 , f 2 , . . . , f n ), g = (g1 , g2 , . . . , gn ) with the box α-level sets (0 ≤ α < 1) as follows − + − + − + [ f (t)]α = [ f 1,α (t), f 1,α (t)] × [ f 2,α (t), f 2,α (t)] × · · · × [ f n,α (t), f n,α (t)] − + n = ×i=1 [ f i,α (t), f i,α (t)]

and − + − + − + [g(t)]α = [g1,α (t), g1,α (t)] × [g2,α (t), g2,α (t)] × · · · × [gn,α (t), gn,α (t)] − + n = ×i=1 [gi,α (t), gi,α (t)].

Next, we will introduce some knowledge of time scales which will be used in later chapters. A time scale T is a closed subset of R. It follows that the jump operators σ,  : T → T defined by σ (t) = inf{s ∈ T : s > t} and ρ(t) = sup{s ∈ T : s < t} (supplemented by inf φ := sup T and sup φ := inf T) are well defined. The point t ∈ T is left-dense, leftscattered, right-dense, right-scattered if ρ(t) = t, ρ(t) < t, σ (t) = t, σ (t) > t, respectively. If T has a right scattered minimum m, define Tk := T \ m; if T has a left scattered maximum M, define Tk = T\M. The notations [a, b]T , [a, b)T and so on, we will denote time scale intervals [a, b]T = {t ∈ T : a ≤ t ≤ b},

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 C. Wang and R. P. Agarwal, Dynamic Equations and Almost Periodic Fuzzy Functions on Time Scales, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-11236-2_3

57

3

58

3 Calculus of Fuzzy Vector-Valued Functions on Time Scales

where a, b ∈ T with a < (b). The graininess function is defined by μ : T → [0, ∞) : μ(t) := σ (t) − t, for all t ∈ T. For more details, we refer the reader to [4, 20, 41].

3.1

g H-Δ-Derivative of Fuzzy Vector-Valued Functions on Time Scales

In this subsection, we will introduce the definition of g H -Δ-derivative of fuzzy vector-valued functions on time scales. Definition 3.1 For f : T → [RnF ] and t ∈ Tκ , we define the g H -Δ-derivative of f (t), f Δ (t) = ( f 1Δ , f 2Δ , . . . , f nΔ ), to be the fuzzy vector (if it exists) with the property that for a given ε > 0, there exists a neighborhood U of t (i.e., U = (t − δ, t + δ)T for some δ > 0) such that   (i)  g H f i (s), f iΔ (t)(σ (t) − s) < ε|σ (t) − s|, i = 1, 2, . . . , n f i (σ (t))− D∞ for all s ∈ U . That is, the limit f iΔ (t)

   g H f i (s) f i σ (t) − = lim s→t σ (t) − s

exists for each i = 1, 2, . . . , n. The following definition is obviously equivalent to Definition 3.1. Definition 3.2 For f : T → RnF and t ∈ Tκ , we define the g H -Δ-derivative of f (t), f Δ (t) = ( f 1Δ , f 2Δ , . . . , f nΔ ), to be the fuzzy vector (if it exists) with the property that for a given ε > 0, there exists a δ > 0 such that |h| < δ implies   (i)  g H f i (t + h), f iΔ (t)(μ(t) − h) ≤ ε|μ(t) − h|, f i (σ (t))− D∞ i.e.    g H f i (t + h) f i σ (t) − lim = f iΔ (t) h→0 μ(t) − h exists for each i = 1, 2, . . . , n. A sufficient and necessary condition for g H -Δ-differentiability of functions is given by the following theorem.

3.1

g H -Δ-Derivative of Fuzzy Vector-Valued Functions on Time Scales

59

n [ f − (t), f + (t)], α ∈ Theorem 3.1 Let f : T → RnF be a function and [ f (t)]α = ×i=1 i,α i,α − + [0, 1]. The function f (t) is g H -Δ-differentiable if f i,α (t) and f i,α (t) are Δ-differentiable real-valued functions for each i = 1, 2, . . . , n. Furthermore

  − Δ + Δ − Δ + Δ n min{( f i,α [ f Δ (t)]α = ×i=1 ) (t), ( f i,α ) (t)}, max{( f i,α ) (t), ( f i,α ) (t)} . Proof In fact, for sufficiently small |h| > 0, we have      g H f (t + h) α f σ (t) − μ(t) − h      g H f i (t + h) α f i σ (t) − n = ×i=1 μ(t) − h       1 n  g H f i (t + h) α = ×i=1 · f i σ (t) − μ(t) − h   α    1 n g H [ f i (t + h)]α · f i σ (t) = ×i=1 μ(t) − h     − 1 + − + n (σ (t)), f i,α (σ (t))] g H [ f i,α (t + h), f i,α (t + h)] · [ f i,α = ×i=1 μ(t) − h   1 − − + + n (σ (t)) − f i,α (t + h), f i,α (σ (t)) − f i,α (t + h)}, · min{ f i,α = ×i=1 μ(t) − h   − − + + (σ (t)) − f i,α (t + h), f i,α (σ (t)) − f i,α (t + h)} max{ f i,α

− − + + (σ (t)) − f i,α (t + h) f i,α (σ (t)) − f i,α (t + h) f i,α , min = , μ(t) − h μ(t) − h

 − − + + (t + h) f i,α (σ (t)) − f i,α (t + h) f i,α (σ (t)) − f i,α . max , μ(t) − h μ(t) − h 



n ×i=1

Let h → 0, by Definition 3.2 we have  − Δ + Δ − Δ + Δ  n min{( f i,α [ f Δ (t)]α = ×i=1 ) , ( f i,α ) }, max{( f i,α ) , ( f i,α ) }. This completes the proof.



Remark 3.1 Note that the condition of Theorem 3.1 is sufficient but not necessary, that is, − + (t) and f i,α (t) may not be Δ-differentiable realif f (t) is g H -Δ-differentiable, then f i,α valued functions for each i = 1, 2, . . . , n and α ∈ [0, 1]. For example, consider the function f : [−1, 1] → RF , where [ f (t)]α = [ f α− (t), f α+ (t)] = [−|t|, |t|] for all α ∈ [0, 1]. Given t = 0, then t = 0 is the right and left dense point. It is obvious that f α− and f α+ are not Δ-differentiable at t = 0. However, f is g H -Δ-differentiable at t = 0. In fact, since

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3 Calculus of Fuzzy Vector-Valued Functions on Time Scales

 lim

s→0+

 g H f (s) f (0)− −s

α

 −1  [ f (0)]α g H [ f (s)]α s  −1  [0, 0] g H [−|s|, |s|] = lim + s s→0   −|s| |s| = lim , s s s→0+ = lim [−1, 1] = [−1, 1] for all α ∈ [0, 1], = lim

s→0+

s→0+



 g H f (s) f (0)− lim −s s→0−

α

 −1  [ f (0)]α g H [ f (s)]α s  −1  = lim [0, 0] g H [−|s|, |s|] s→0− s   |s| −|s| = lim , s s s→0− = lim [−1, 1] = [−1, 1] for all α ∈ [0, 1], = lim

s→0−

s→0−

it follows that f is g H -Δ-differentiable at t = 0 and its α-levels are given by [ f Δ (0)]α = [−1, 1] for all α ∈ [0, 1]. Now, we introduce the definition of continuous fuzzy vector-valued functions on time scales. Definition 3.3 A function f : T → RnF is said a fuzzy continuous function, if for arbitrary fixed t0 ∈ T and ε > 0, there exists a δ > 0 such that |t − t0 | < δ implies   D∞ f (t), f (t0 ) < ε. Throughout the paper, the set of all fuzzy continuous vector-valued functions is denoted by C(T, RnF ). From Definition 3.3, the following lemma is clear. Theorem 3.2 Let the fuzzy continuous function f : [a, b]T → RnF . Then f is bounded, i.e., there exists a positive number M such that  f (t)F ≤ M for all t ∈ [a, b]T . Theorem 3.3 Assume f : T → RnF is a function and let t ∈ Tk then we have the following: (i) If f is continuous at t and t is right-scattered, then f is g H -Δ-differentiable at t with f Δ (t) =

 g H f (t) f (σ (t))− . μ(t)

3.1

g H -Δ-Derivative of Fuzzy Vector-Valued Functions on Time Scales

61

(ii) If t is right-dense, then f is g H -Δ-differentiable at t if and only if the limits  g H f (t) f (t + h)− h

(3.1)

 g H f (t) f (t + h)− = f Δ (t). h

(3.2)

lim

h→0

exist and satisfy the following lim

h→0

Proof (i). If f is continuous at t and t is right-scattered, then we obtain      g H f i (t + h)  g H f i (t) f i σ (t) − f i σ (t) − lim = , h→0 μ(t) − h μ(t) then for any ε > 0, there exists δ > 0 such that |h| < δ implies        (i) f i σ (t) −g H f i (t + h) f i σ (t) −g H f i (t) < ε, D∞ , μ(t) − h μ(t) which yields that     g H f i (t)   f i σ (t) − 1 (i)  g H f i (t + h), f i σ (t) − D (μ(t) − h) < ε. |μ(t) − h| ∞ μ(t) Hence we obtain that f iΔ (t)

   g H f i (t) f i σ (t) − = , i = 1, 2, . . . , n. μ(t)

(ii). If t is right-dense, then σ (t) = t, i.e., μ(t) = 0. If f is g H -Δ-differentiable at t, then for a given ε > 0, there exists a δ > 0 such that |h| < δ implies   (i)  g H f i (t + h), f iΔ (t) · (−h) ≤ ε|h|, f i (t)− D∞ i.e.

 g H f i (t + h)  g H f i (t) f i (t)− f i (t + h)− = lim = f iΔ (t). h→0 h→0 −h h exists for i = 1, 2, . . . , n. Conversely, if (3.1) and (3.2) hold, then for a given ε > 0, there exists a δ > 0 such that |h| < δ implies lim

  (i)  g H f i (t + h), f iΔ (t)(μ(t) − h) f i (σ (t))− D∞   g H f i (t) f i (t + h)− (i)  = D∞ f i (t)−g H f i (t + h), (−h) h   (i)  g H f i (t + h), f i (t)−  g H f i (t + h) = 0. f i (t)− = D∞

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3 Calculus of Fuzzy Vector-Valued Functions on Time Scales

Hence, we can easily obtain the desired results. This completes the proof.



Theorem 3.4 Assume f : T → RnF is g H -Δ-differentiable at t ∈ Tk , then f is continuous at t. Proof Assume that f is g H -Δ-differentiable at t, let ε ∈ (0, 1) and define ε∗ =  ε  f iΔ (t)F0 + 1 + 2μ(t)]−1 . Obviously, ε∗ ∈ (0, 1). By Definition 3.1, for each i = 1, 2, . . . , n, we have   (i)  g H f i (t + h), f iΔ (t)(μ(t) − h) ≤ ε ∗ |μ(t) − h|. f i (σ (t))− D∞ Therefore, for t + h ∈ U ∩ (t − ε∗ , t + ε∗ ) with h ∈ [0, ε∗ ), since   Δ   (i) (i) (μ(t) − h) f iΔ (t), 0˜ = |μ(t) − h|D∞ f i (t), 0˜ D∞

= |μ(t) − h| f iΔ (t)F0   (i) μ(t) f iΔ (t), h f iΔ (t) , = D∞

we obtain

= ≤ ≤ = ≤

  (i) f i (t + h), f i (t) D∞   (i)  g H f i (t + h), f i (σ (t))−  g H f i (t) f i (σ (t))− D∞     (i) (i)  g H f i (t + h), 0˜ + D∞  g H f i (t), 0˜ D∞ f i (σ (t))− f i (σ (t))−   Δ   (i) (i)  g H f i (t + h), f iΔ (t)(μ(t) − h) + D∞ D∞ f i (σ (t))− f i (t)(μ(t) − h), 0˜   (i)  g H f i (t), 0˜ +D∞ f i (σ (t))−     (i) (i)  g H f i (t + h), f iΔ (t)(μ(t) − h) + D∞ f i (σ (t))− μ(t) f iΔ (t), h f iΔ (t) D∞   (i)  g H f i (t), 0˜ f i (σ (t))− +D∞   (i)  g H f i (t + h), f Δ (t)(μ(t) − h) D∞ f i (σ (t))−   Δ   (i) (i)  g H f i (t), f iΔ (t)μ(t) + h D∞ f i (σ (t))− f i (t), 0˜ +D∞

≤ ε∗ |μ(t) − h| + ε∗ μ(t) + h f iΔ (t)F0   = ε∗ 1 +  f iΔ (t)F0 + 2μ(t) = ε. Similarly, for h ∈ (−ε∗ , 0), since

  Δ   (i) (i) (μ(t) − h) f iΔ (t), 0˜ = |μ(t) − h|D∞ f i (t), 0˜ D∞

= |μ(t) − h| f iΔ (t)F0   (i) − μ(t) f iΔ (t), h f iΔ (t) = D∞   (i) μ(t) f iΔ (t), −h f iΔ (t) . = D∞

(3.3)

3.1

g H -Δ-Derivative of Fuzzy Vector-Valued Functions on Time Scales

63

 (i)  Similar to the proof of (3.3), we can also obtain D∞ f i (t + h), f i (t) < ε. This completes the proof.  By Theorem 3.1, for the definition of g H -Δ-differentiability, we distinguished two cases, corresponding to (I ) and (I I ) of (2.6). n [ f − (t), f + (t)], α ∈ Definition 3.4 Let f : T → RnF be a function and [ f (t)]α = ×i=1 i,α i,α − + [0, 1]. Let f i,α (t) and f i,α (t) be Δ-differentiable real-valued functions at t0 ∈ (a, b)T for each i = 1, 2, . . . , n and α ∈ [0, 1]. We say that f is (I )-g H -Δ-differentiable at t0 ∈ (a, b)T   if f Δ I (t) = f 1Δ I (t), f 2Δ I (t), . . . , f nΔ I (t) with α-level set − Δ + Δ n [ f Δ I (t)]α = ×i=1 [( f i,α ) (t), ( f i,α ) (t)],

(3.4)

and f is (I I )-g H -Δ-differentiable at t0 ∈ (a, b)T if   f Δ I I (t) = f 1Δ I I (t), f 2Δ I I (t), . . . , f nΔ I I (t) with α-level set

+ Δ − Δ n [ f Δ I I (t)]α = ×i=1 [( f i,α ) (t), ( f i,α ) (t)].

(3.5)

Similar to the literature [74], we will introduce and study the switch between the two cases (I ) and (I I ) in Definition 3.4. Definition 3.5 We say a point t0 ∈ (a, b)T is a switching point for the g H -Δ-differentiability of f , if in any neighborhood U of t0 there exists points t1 < t0 < t2 such that (i) (type-I switch ) at t1 (3.4) holds while (3.5) does not hold and at t2 (3.5) holds while (3.4) does not hold, or (ii) (type-II switch ) at t1 (3.5) holds while (3.4) does not hold and at t2 (3.4) holds while (3.5) does not hold. n [ f − (t), f + (t)], α ∈ Theorem 3.5 Let f : T → RnF be a function with [ f (t)]α = ×i=1 i,α i,α − + (t) and f i,α (t) are Δ-differentiable real-valued functions for each i = [0, 1], and f i,α   n f i+ (t) − f i− (t), 1, 2, . . . , n and α ∈ [0, 1]. Let the length function be len f (t) = i=1 if a point t0 ∈ (a, b)T is a switching point, then in any neighborhood U of t0 (i.e. U = (t0 −     δ, t0 + δ)T ) there exists points t1 , t2 ∈ U with t1 < t0 < t2 such that len f Δ (t1 ) len f Δ (t2 ) < 0.

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3 Calculus of Fuzzy Vector-Valued Functions on Time Scales

Proof Without loss of generality, we assume that t0 belongs to the switching point of type I in Definition 3.5 (the proof of the other case type II is similar). (i). If t0 is a switching point and right scattered, the α-level set of f (t) for t = t1 is that n [ f Δ (t1 )]α = ×i=1 [( f i− )Δ (t1 ), ( f i+ )Δ (t1 )]α ,

Δ  which implies that ( f i+ )Δ (t1 ) − ( f i− )Δ (t1 ) = f i+ − f i− (t1 ) > 0 for i = 1, 2, . . . , n. Also the α-level set of f (t) for t = t2 is that n [ f Δ (t2 )]α = ×i=1 [( f i+ )Δ (t2 ), ( f i− )Δ (t2 )]α ,

Δ  which implies that ( f i+ )Δ (t2 ) − ( f i− )Δ (t2 ) = f i+ − f i− (t2 ) < 0 for i = 1, 2, . . . , n. Hence we have     len f Δ (t1 ) len f Δ (t2 )    n n ( f i+ )Δ (t1 ) − ( f i− )Δ (t1 ) ( f i+ )Δ (t2 ) − ( f i− )Δ (t2 ) < 0. = i=1

i=1

(ii). By Definition 3.5, similar to the proof of (i), we can also easily obtain the desired result. This completes the proof.  n [ f − (t), f + (t)] , α ∈ Theorem 3.6 Let f : T → RnF be a function with [ f (t)]α = ×i=1 α i i [0, 1], and f i− (t) and f i+ (t) are Δ-differentiable real-valued functions for each i =   n f i+ (t) − f i− (t), if a point t0 ∈ 1, 2, . . . , n. Let the length function be len f (t) = i=1 (a, b)T is a switching point, then there exist a point c ∈ (t0 − δ, t0 + δ)T such that

  len f Δ (c) = 0 if c is right dense     or len f Δ (c) len f Δ (σ (c)) < 0 if c is right scattered. Proof According to Theorem 3.5, there exists points t1 , t2 ∈ U such that     len f Δ (t1 ) len f Δ (t2 ) < 0. Hence, by the Intermediate Value Theorem (Theorem 1.115 from [20]), we can obtain the desired result immediately. This completes the proof.  Theorem 3.7 Let f : (a, b)T → RnF be a fuzzy vector-valued function with the α-level n [ f − (t), f + (t)] and f − (t) and f + (t) are Δ-differentiable real-valued set [ f (t)]α = ×i=1 α i i i i functions for each i = 1, 2, . . . , n. Then the set of switching point is finite.

3.1

g H -Δ-Derivative of Fuzzy Vector-Valued Functions on Time Scales

65

Proof We argue by contradiction. Suppose that the set of switching point is infinite. Then this set has a cluster point t0 ∈ [a, b]T . Let tn → t0 be a convergent sequence of switching points. For δ > 0, according to the definition of a switching point there exist tn 1 , tn 2 ∈ (t0 − δ, t0 + δ)T such that tn 1 and tn 2 are in opposite differentiability case (I )-g H -Δ-differentiability  g H f (t0 ) or (I I )-g H -Δ-differentiability exclusively. Then the g H -differences f (t0 + h)− cannot exist for any h with 0 ≤ h < δ, and this is a contradiction because f i− (t) and f i+ (t) are Δ-differentiable real-valued functions for each i = 1, 2, . . . , n. This completes the proof.  Remark 3.2 Note that f is g H -Δ differentiable at t0 , the set of switching point may be infinite since that g H -Δ differentiable at t0 does not imply f i− (t) and f i+ (t) are Δdifferentiable real-valued functions for each i = 1, 2, . . . , n (see Remark 3.1). For example, given f : [−1, 1] → RF be the function whose α-levels are given by [ f (t)]α = g(t) · [−1, 1] for all α ∈ [0, 1], where  1 + t 2 sin 1t if t = 0, g(t) = 1 if t = 0. In fact, we have    g H f (s) α f (0)− lim −s s→0+  −1  = lim [ f (0)]α g H [ f (s)]α + s s→0    −1 1 1 2 2 [−1, 1] g H − 1 − s sin , 1 + s sin = lim s s s→0+ s 



 1 1 1 1 −1 2 2 2 2 = lim min s sin , −s sin , max s sin , −s sin s s s s s→0+ s



  1 1 1 1 , max s sin , −s sin = lim (−1) min s sin , −s sin s s s s s→0+



  1 1 1 1 = lim (−1) max s sin , −s sin , (−1) min s sin , −s sin s s s s s→0+



  1 1 1 1 , max − s sin , s sin = [0, 0] = lim min − s sin , s sin s s s s s→0+ and

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3 Calculus of Fuzzy Vector-Valued Functions on Time Scales

  g H f (s) α f (0)− −s s→0−  −1  [ f (0)]α g H [ f (s)]α lim − s s→0    1 1 lim [−1, 1] g H − 1 − s 2 sin , 1 + s 2 sin s s s→0− 



 −1 1 1 1 1 min s 2 sin , −s 2 sin , max s 2 sin , −s 2 sin lim s s s s s→0− s



  1 1 1 1 lim min − s sin , s sin , max − s sin , s sin = [0, 0], s s s s s→0− 

lim

= = = =

so f is g H -Δ-differentiable at t = 0 and [ f Δ (0)]α = [0, 0] for all α ∈ [0, 1]. Further, for any t ∈ (−1, 1)\{0}, it follows that    g H f (s) α f (t)− lim s→t t −s   1 = lim [ f (t)]α g H [ f (s)]α s→t t − s     1 1 1 1 1 2 2 2 2 − 1 − t sin , 1 + t sin g H − 1 − s sin , 1 + s sin = lim s→t t − s t t s s 

1 1 1 1 1 min − t 2 sin + s 2 sin , t 2 sin − s 2 sin , = lim s→t t − s t s t s

 1 1 1 1 max − t 2 sin + s 2 sin , t 2 sin − s 2 sin t s t s 

1 1 1 1 = min cos − 2t sin , − cos + 2t sin , t t t t

 1 1 1 1 max cos − 2t sin , − cos + 2t sin . t t t t Therefore, f is g H -Δ-differentiable for t ∈ (−1, 1)\{0}. However, there are infinite numbers of switching points. In fact, note that f α− (t) = −g(t) and f α+ (t) = g(t) are not Δdifferentiable at t = 0 which leads to the infinite numbers of switching points in the neighborhood of zero. Theorem 3.8 If f , g : T → RnF is g H -Δ-differentiable at t ∈ Tk , then (i)

μ(t) · f Δ (t) or f (t) = f (σ (t))+ (−1)μ(t) · f Δ (t), i.e., f (σ (t)) = f (t)+    g H f (t) = μ(t) · f Δ (t). f σ (t) −

(ii) Let f , g be (I )-g H -Δ-differentiable at t ∈ (a, b)T or (I I )-g H -Δ-differentiable at  g : T → Rn is g H -Δ-differentiable at t and t ∈ (a, b)T , then f + F

3.1

g H -Δ-Derivative of Fuzzy Vector-Valued Functions on Time Scales

 g)Δ = f Δ (t)+  g Δ (t). (f+

67

(3.6)

(iii) For any nonnegative constant λ ∈ R, λ · f : T → RnF is g H -Δ-differentiable at t with (λ · f )Δ (t) = λ · f Δ (t).    g H f (t) = μ(t) · Proof (i). If t is the right dense point, then σ (t) = t, we have f σ (t) − f Δ (t). If t is the right scattered point, then σ (t) > t, and it follows from Theorem 3.3 (i)    g H f (t) = μ(t) · f Δ (t). that f σ (t) − (ii). Case I. If f , g are (I )-g H -Δ-differentiable at t ∈ (a, b)T , then n n [ f Δ (t)]α = ×i=1 [( f i− )Δ , ( f i+ )Δ ]α , [g Δ (t)]α = ×i=1 [(gi− )Δ , (gi+ )Δ ]α ,

then one obtains n [ f Δ (t)]α + [g Δ (t)]α = ×i=1 [( f i− )Δ + (gi− )Δ , ( f i+ )Δ + (gi+ )Δ ]α .

Moreover, by Theorem 3.1, we have [( f + g)Δ ]α   n min{( f i− + g1− )Δ , ( f i+ + g1+ )Δ }, max{( f i− + g1− )Δ , ( f i+ + g1+ )Δ } α = ×i=1  n min{( f i− )Δ + (g1− )Δ , ( f i+ )Δ + (g1+ )Δ }, = ×i=1  max{( f i− )Δ + (g1− )Δ , ( f i+ )Δ + (g1+ )Δ } α n = = ×i=1 [( f i− )Δ + (gi− )Δ , ( f i+ )Δ + (gi+ )Δ ]α .

Hence, we have [ f Δ (t)]α + [g Δ (t)]α = [( f + g)Δ ]α for any 0 ≤ α ≤ 1. Case II. If f , g are (I I )-g H -Δ-differentiable at t ∈ (a, b)T , the proof of Case II is similar to Case I. (iii). For any nonnegative constant λ ∈ R and any α ∈ [0, 1], we have 

(λ · f )Δ (t)

α

 α n (λ · f i )Δ (t) . = ×i=1

Case I. If f is (I )-g H -Δ-differentiable at t ∈ (a, b)T , then 

(λ · f )Δ (t)

α

  n λ( f i− )Δ , λ( f i+ )Δ α = ×i=1

n = λ · ×i=1 [( f i− )Δ , ( f i+ )Δ ]α = λ · [ f Δ (t)]α .

Case II. If f is (I I )-g H -Δ-differentiable at t ∈ (a, b)T , then 

(λ · f )Δ (t)

α

  n λ( f i+ )Δ , λ( f i− )Δ α = ×i=1

n = λ · ×i=1 [( f i+ )Δ , ( f i− )Δ ]α = λ · [ f Δ (t)]α .

This completes the proof.



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3 Calculus of Fuzzy Vector-Valued Functions on Time Scales

Remark 3.3 In Theorem 3.8 (ii), the condition that f , g are (I )-g H -Δ-differentiable at t ∈ (a, b)T or (I I )-g H -Δ-differentiable at t ∈ (a, b)T simultaneously is essential, and without this condition, (3.6) will be invalid. For example, let f , g : [0, 5]T → RF with   [ f (t)]α = e−1 (t, 0) − t + 2, e−1 (t, 0) + t + 5 α ,  [g(t)]α = − e−1 (t, 0) + t − 10, −e−1 (t, 0) − t + 10]α , where f (t) is (I )-g H -Δ differentiable and g(t) is (I I )-g H -Δ differentiable at any t0 ∈ (0, 5)T . We have [( f + g)(t)]α = [−8, 15], so [( f + g)Δ (t)]α = {0}. However, we also obtain   [ f Δ (t) + g Δ (t)]α = − e−1 (t, 0) − 1, −e−1 (t, 0) + 1 α   + e−1 (t, 0) − 1, e−1 (t, 0) + 1 α = [−2, 2], which implies that f Δ + g Δ  ( f + g)Δ . Hence, this condition is necessary. Note that if we let T = R and Z, it turns into the examples of the continuous situation and discrete situation for fuzzy vector-valued functions. Now, we will provide some arithmetic properties of the g H -Δ-derivatives of the product of two fuzzy vector-valued functions on time scales. For convenience, we adopt the notation  f σ (t)) = f σ (t) in some statement. Theorem 3.9 Let f , g be (I )-g H -Δ-differentiable, then (i)

α,(I )

if I fi ,gi < 0, I

α,(I )

< 0, I

ΔI

f iσ ,gi

α,(I ) ΔI

fi

< 0 and f ∗ g is (I )-g H -Δ-differentiable, then

,gi

 f Δ I ∗ g. ( f ∗ g)Δ I = f σ ∗ g Δ I + (ii)

α,(I )

if I fi ,gi < 0, I

α,(I )

> 0, I

ΔI

f iσ ,gi

α,(I ) ΔI

fi

> 0 and f ∗ g is (I I )-g H -Δ-differentiable, then

,gi

 f Δ I ∗ g. ( f ∗ g)Δ I I = f σ ∗ g Δ I + (iii)

α,(I I )

if I fi ,gi

< 0, I

α,(I I )

ΔI

f iσ ,gi

< 0, I

α,(I I ) ΔI

fi

,gi

< 0 and f  g is (I )-g H -Δ-differentiable, then

g  f ΔI . ( f  g)Δ I = f σ  g Δ I + (iv)

α,(I I )

if I fi ,gi then

< 0, I

α,(I I )

ΔI

f iσ ,gi

> 0, I

α,(I I ) ΔI

fi

,gi

> 0 and f  g is (I I )-g H -Δ-differentiable,

 f Δ I  g. ( f  g)Δ I I = f σ  g Δ I +

3.1

g H -Δ-Derivative of Fuzzy Vector-Valued Functions on Time Scales α,(I )

(v)

if I fi ,gi > 0, I

α,(I )

> 0, I

ΔI

f iσ ,gi

α,(I ) ΔI

fi

69

> 0 and f ∗ g is (I )-g H -Δ-differentiable, then

,gi

 f Δ I ∗ g. ( f ∗ g)Δ I = f σ ∗ g Δ I + α,(I )

(vi)

if I fi ,gi > 0, I

α,(I )

< 0, I

ΔI

f iσ ,gi

α,(I ) ΔI

fi

< 0 and f ∗ g is (I I )-g H -Δ-differentiable, then

,gi

 f Δ I ∗ g. ( f ∗ g)Δ I I = f σ ∗ g Δ I + α,(I I )

(vii) if I fi ,gi

> 0, I

α,(I I )

> 0, I

ΔI

f iσ ,gi

α,(I I ) ΔI

fi

,gi

> 0 and f  g is (I )-g H -Δ-differentiable, then

g  f ΔI . ( f  g)Δ I = f σ  g Δ I + α,(I I )

(viii) if I fi ,gi

> 0, I

then

α,(I I )

ΔI

f iσ ,gi

< 0, I

α,(I I ) ΔI

fi

,gi

< 0 and f  g is (I I )-g H -Δ-differentiable,

 f Δ I  g. ( f  g)Δ I I = f σ  g Δ I +

Proof Since f , g be (I )-g H -Δ-differentiable, then  − Δ   − Δ + Δ n n ( f i ) , ( f i+ )Δ α , [g Δ I ]α = ×i=1 (gi ) , (gi ) α . [ f Δ I ]α = ×i=1 α,(I )

n [ f − g + , f + g − ] . Moreover, because f ∗ (i). Since I fi ,gi < 0, we have [ f ∗ g]α = ×i=1 i i i i α g is (I )-g H -Δ-differentiable, we obtain



( f ∗ g)Δ I

α

In addition, from I

 − + Δ  n ( f i gi ) , ( f i+ gi− )Δ α = ×i=1  − σ + Δ  n ( f i ) (gi ) + gi+ ( f i− )Δ , ( f i+ )σ (gi− )Δ + gi− ( f i+ )Δ α . = ×i=1 α,(I )

ΔI

f iσ ,gi



< 0, I

f σ ∗ gΔI

α

α,(I ) ΔI

fi

,gi

< 0, we have

 − σ + Δ n ( f i ) (gi ) , ( f i+ )σ (gi− )Δ ]α , = ×i=1



 + − Δ − + Δ α n f Δ I ∗ g = ×i=1 gi ( f i ) , gi ( f i ) α .  σ  α    α α Hence, we obtain f ∗ g Δ I + f Δ I ∗ g = ( f ∗ g)Δ I . (ii). Since f ∗ g is (I I )-g H -Δ-differentiable, through calculation, we have  α  + − Δ  n ( f ∗ g)Δ I I = ×i=1 ( f i gi ) , ( f i− gi+ )Δ α  + σ − Δ  n ( f i ) (gi ) + gi− ( f i+ )Δ , ( f i− )σ (gi+ )Δ + gi+ ( f i− )Δ α . = ×i=1 In addition, from I 

α,(I )

ΔI

f iσ ,gi

> 0, I

f σ ∗ gΔI

α

α,(I ) ΔI

fi

,gi

> 0, we have

 + σ − Δ n ( f i ) (gi ) , ( f i− )σ (gi+ )Δ ]α , = ×i=1

70

3 Calculus of Fuzzy Vector-Valued Functions on Time Scales



 − + Δ + − Δ α n f Δ I ∗ g = ×i=1 gi ( f i ) , gi ( f i ) α ,  σ  α  α  α and thus, we obtain f ∗ g Δ I + f Δ I ∗ g = ( f ∗ g)Δ I I . α,(I I ) n [ f + g + , f − g − ] . Moreover, because (iii). Since I fi ,gi < 0, we have [ f  g]α = ×i=1 i i i i α f  g is (I )-g H -Δ-differentiable, we obtain 

α

( f  g)Δ I

In addition, from I

 + + Δ  n ( f i gi ) , ( f i− gi− )Δ α = ×i=1  + σ + Δ  n ( f i ) (gi ) + gi+ ( f i+ )Δ , ( f i− )σ (gi− )Δ + gi− ( f i− )Δ α . = ×i=1

α,(I I )

ΔI

f iσ ,gi



< 0, I

f σ  gΔI

α

α,(I I ) ΔI

fi

,gi

< 0, we have

 + σ + Δ n ( f i ) (gi ) , ( f i− )σ (gi− )Δ ]α , = ×i=1



 + + Δ − − Δ α n f Δ I  g = ×i=1 gi ( f i ) , gi ( f i ) α ,  σ  α  α  α and thus, it follows that f  g Δ I + f Δ I  g = ( f  g)Δ I . (iv). Since f  g is (I I )-g H -Δ-differentiable, through calculation, we have 

( f  g)Δ I I

α

 − − Δ  n ( f i gi ) , ( f i+ gi+ )Δ α = ×i=1  − σ − Δ  n ( f i ) (gi ) + gi− ( f i− )Δ , ( f i+ )σ (gi+ )Δ + gi+ ( f i+ )Δ α . = ×i=1

In addition, from I 

α,(I I )

ΔI

f iσ ,gi

> 0, I

f σ  gΔI

α

α,(I I ) ΔI

fi

,gi

> 0, we have

 − σ − Δ n ( f i ) (gi ) , ( f i+ )σ (gi+ )Δ ]α , = ×i=1



 − − Δ + + Δ α n f Δ I  g = ×i=1 gi ( f i ) , gi ( f i ) α ,  σ  α  α  α and thus, we have f  g Δ I + f Δ I  g = ( f  g)Δ I I . α,(I ) n [ f + g − , f − g + ] . Moreover, because (v). Since I fi ,gi > 0, we have [ f ∗ g]α = ×i=1 i i i i α f ∗ g is (I )-g H -Δ-differentiable, we obtain 

( f ∗ g)Δ I

α

In addition, from I

 + − Δ  n ( f i gi ) , ( f i− gi+ )Δ α = ×i=1  + σ − Δ  n ( f i ) (gi ) + gi− ( f i+ )Δ , ( f i− )σ (gi+ )Δ + gi+ ( f i− )Δ α . = ×i=1 α,(I )

ΔI

f iσ ,gi



> 0, I

f σ ∗ gΔI 

α

α,(I ) ΔI

fi

,gi

> 0, we have

 + σ − Δ  n ( f i ) (gi ) , ( f i− )σ (gi+ )Δ α , = ×i=1

 − + Δ + − Δ α n f Δ I ∗ g = ×i=1 gi ( f i ) , gi ( f i ) α ,  σ  α  α  α and thus, we can obtain f ∗ g Δ I + f Δ I ∗ g = ( f ∗ g)Δ I . α,(I ) n [ f + g − , f − g + ] . Moreover, because (vi). Since I fi ,gi > 0, we have [ f ∗ g]α = ×i=1 i i i i α f ∗ g is (I I )-g H -Δ-differentiable, we obtain

3.1

g H -Δ-Derivative of Fuzzy Vector-Valued Functions on Time Scales



( f ∗ g)Δ I

α

In addition, from I

71

 − + Δ  n ( f i gi ) , ( f i+ gi− )Δ α = ×i=1  − σ + Δ  n ( f i ) (gi ) + gi+ ( f i− )Δ , ( f i+ )σ (gi− )Δ + gi− ( f i+ )Δ α . = ×i=1 α,(I )

ΔI

f iσ ,gi



< 0, I

f σ ∗ gΔI

α

α,(I ) ΔI

fi

,gi

< 0, we have

 − σ + Δ  n ( f i ) (gi ) , ( f i+ )σ (gi− )Δ α , = ×i=1



 + − Δ − + Δ α n f Δ I ∗ g = ×i=1 gi ( f i ) , gi ( f i ) α ,  σ  α  α  α and thus, we can obtain f ∗ g Δ I + f Δ I ∗ g = ( f ∗ g)Δ I I . α,(I I ) n [ f − g − , f + g + ] . Moreover, because (vii). Since I fi ,gi > 0, we have [ f  g]α = ×i=1 i i i i α f  g is (I )-g H -Δ-differentiable, we obtain 

( f  g)Δ I

α

In addition, from I

 − − Δ  n ( f i gi ) , ( f i+ gi+ )Δ α = ×i=1  − σ − Δ  n ( f i ) (gi ) + gi− ( f i− )Δ , ( f i+ )σ (gi+ )Δ + gi+ ( f i+ )Δ α . = ×i=1

α,(I I )

ΔI

f iσ ,gi



> 0, I

f σ  gΔI

α

α,(I I ) ΔI

fi

,gi

> 0, we have

 − σ − Δ  n ( f i ) (gi ) , ( f i+ )σ (gi+ )Δ α , = ×i=1



 − − Δ + + Δ α n f Δ I  g = ×i=1 gi ( f i ) , gi ( f i ) α ,  σ  α  α  α and thus, we can obtain f  g Δ I + f Δ I  g = ( f  g)Δ I . α,(I I ) n [ f − g − , f + g + ] . Moreover, because (viii). Since I fi ,gi > 0, we have [ f  g]α = ×i=1 i i i i α f  g is (I I )-g H -Δ-differentiable, we obtain 

( f  g)Δ I I

α

In addition, from I

 + + Δ  n ( f i gi ) , ( f i− gi− )Δ α = ×i=1  + σ + Δ  n ( f i ) (gi ) + gi+ ( f i+ )Δ , ( f i− )σ (gi− )Δ + gi− ( f i− )Δ α . = ×i=1

α,(I I )

ΔI

f iσ ,gi



< 0, I

f σ  gΔI 

α

α,(I I ) ΔI

fi

,gi

< 0, we have

 + σ + Δ  n ( f i ) (gi ) , ( f i− )σ (gi− )Δ α , = ×i=1

 + + Δ − − Δ α n f Δ I  g = ×i=1 gi ( f i ) , gi ( f i ) α ,  σ  α  α  α and thus, we can obtain f  g Δ I + f Δ I  g = ( f  g)Δ I I . This completes the proof. 

72

3 Calculus of Fuzzy Vector-Valued Functions on Time Scales

Theorem 3.10 Let f , g be (I I )-g H -Δ-differentiable, then α,(I I )

(i)

if I fi ,gi

< 0, I

then if I fi ,gi

< 0, I

α,(I )

α,(I I )

> 0, I

then

ΔI I

fi

,gi

< 0 and f  g is (I )-g H -Δ-differentiable,

α,(I ) ΔI I

fi

,gi

> 0 and f  g is (I I )-g H -Δ-differentiable,

α,(I )

ΔI I

f iσ ,gi

< 0, I

α,(I ) ΔI I

fi

,gi

> 0 and f  g is (I )-g H -Δ-differentiable,

g ∗ f ΔI I . ( f  g)Δ I = f σ ∗ g Δ I I +

α,(I I )

if I fi ,gi

> 0, I

then

α,(I )

ΔI I

f iσ ,gi

> 0, I

α,(I ) ΔI I

fi

,gi

< 0 and f  g is (I I )-g H -Δ-differentiable,

 f Δ I I ∗ g. ( f  g)Δ I I = f σ ∗ g Δ I I +

α,(I )

(v)

α,(I )

 f Δ I ∗ g. ( f  g)Δ I I = f σ ∗ g Δ I +

if I fi ,gi

(iv)

> 0, I

< 0, I

ΔI I

f iσ ,gi

then (iii)

ΔI I

f iσ ,gi

 f Δ I I ∗ g. ( f  g)Δ I = f σ ∗ g Δ I I +

α,(I I )

(ii)

α,(I )

if I fi ,gi < 0, I

α,(I I )

ΔI I

f iσ ,gi

> 0, I

α,(I I ) ΔI I

fi

,gi

< 0 and f ∗ g is (I )-g H -Δ-differentiable, then

 f Δ I I  g. ( f ∗ g)Δ I = f σ  g Δ I I + α,(I )

(vi)

if I fi ,gi < 0, I

α,(I I )

ΔI I

f iσ ,gi

< 0, I

α,(I I ) ΔI I

fi

,gi

> 0 and f ∗ g is (I I )-g H -Δ-differentiable, then

 f Δ I I  g. ( f ∗ g)Δ I I = f σ  g Δ I I + α,(I )

(vii) if I fi ,gi > 0, I

α,(I I )

ΔI I

f iσ ,gi

< 0, I

α,(I I ) ΔI I

fi

,gi

> 0 and f ∗ g is (I )-g H -Δ-differentiable, then

g  f ΔI I . ( f ∗ g)Δ I = f σ  g Δ I I + α,(I )

(viii) if I fi ,gi > 0, I

α,(I I )

ΔI I

f iσ ,gi

> 0, I

α,(I I ) ΔI I

fi

,gi

< 0 and f ∗ g is (I I )-g H -Δ-differentiable, then

 f Δ I I  g. ( f ∗ g)Δ I I = f σ  g Δ I I +

Proof Since f , g be (I I )-g H -Δ-differentiable, then  + Δ   + Δ − Δ n n ( f i ) , ( f i− )Δ α , [g Δ I I ]α = ×i=1 (gi ) , (gi ) α . [ f Δ I I ]α = ×i=1 α,(I I )

n [ f + g + , f − g − ] . Moreover, because (i). Since I fi ,gi < 0, we have [ f  g]α = ×i=1 i i i i α f  g is (I )-g H -Δ-differentiable, we obtain



( f  g)Δ I

α

 + + Δ  n ( f i gi ) , ( f i− gi− )Δ α = ×i=1  + σ + Δ  n ( f i ) (gi ) + gi+ ( f i+ )Δ , ( f i− )σ (gi− )Δ + gi− ( f i− )Δ α . = ×i=1

3.1

g H -Δ-Derivative of Fuzzy Vector-Valued Functions on Time Scales

In addition, from I

α,(I )

ΔI I

f iσ ,gi



> 0, I

f σ ∗ gΔI I

α

α,(I ) ΔI I

fi

,gi

73

< 0, we have

 + σ + Δ n ( f i ) (gi ) , ( f i− )σ (gi− )Δ ]α , = ×i=1



 + + Δ − − Δ α n f Δ I I ∗ g = ×i=1 gi ( f i ) , gi ( f i ) α ,  σ  α  α  α and thus, we can obtain f ∗ g Δ I I + f Δ I I ∗ g = ( f  g)Δ I . α,(I I ) n [ f + g + , f − g − ] . Moreover, because (ii). Since I fi ,gi < 0, we have [ f  g]α = ×i=1 i i i i α f  g is (I I )-g H -Δ-differentiable, we obtain 

( f  g)Δ I I

α

In addition, from I

 − − Δ  n ( f i gi ) , ( f i+ gi+ )Δ α = ×i=1   n ( f i− )σ (gi− )Δ + gi− ( f i− )Δ , ( f i+ )σ (gi+ )Δ + gi+ ( f i+ )Δ α . = ×i=1

α,(I )

ΔI I

f iσ ,gi



< 0, I

f σ ∗ gΔI I

α

α,(I ) ΔI I

fi

,gi

> 0, we have

 − σ − Δ n ( f i ) (gi ) , ( f i+ )σ (gi+ )Δ ]α , = ×i=1



 − − Δ + + Δ α n f Δ I I ∗ g = ×i=1 gi ( f i ) , gi ( f i ) α ,  σ  α  α  α and thus, we can obtain f ∗ g Δ I I + f Δ I I ∗ g = ( f  g)Δ I I . α,(I I ) n [ f − g − , f + g + ] . Moreover, because (iii). Since I fi ,gi > 0, we have [ f  g]α = ×i=1 i i i i α f  g is (I )-g H -Δ-differentiable, we obtain 

( f  g)Δ I

α

In addition, from I

 − − Δ  n ( f i gi ) , ( f i+ gi+ )Δ α = ×i=1  − σ − Δ  n ( f i ) (gi ) + gi− ( f i− )Δ , ( f i+ )σ (gi+ )Δ + gi+ ( f i+ )Δ α . = ×i=1

α,(I )

ΔI I

f iσ ,gi



< 0, I

f σ ∗ gΔI I

α

α,(I ) ΔI I

fi

,gi

> 0, we have

 − σ − Δ n ( f i ) (gi ) , ( f i+ )σ (gi+ )Δ ]α , = ×i=1



 − − Δ + + Δ α n f Δ I I ∗ g = ×i=1 gi ( f i ) , gi ( f i ) α ,  σ  α  α  α and thus, we can obtain f ∗ g Δ I I + f Δ I I ∗ g = ( f  g)Δ I . α,(I I ) n [ f − g − , f + g + ] . Moreover, because (iv). Since I fi ,gi > 0, we have [ f  g]α = ×i=1 i i i i α f  g is (I I )-g H -Δ-differentiable, we obtain 

( f  g)Δ I I

α

In addition, from I

 + + Δ  n ( f i gi ) , ( f i− gi− )Δ α = ×i=1  + σ + Δ  n ( f i ) (gi ) + gi+ ( f i+ )Δ , ( f i− )σ (gi− )Δ + gi− ( f i− )Δ α . = ×i=1

α,(I )

ΔI I

f iσ ,gi



> 0, I

f σ ∗ gΔI I

α

α,(I ) ΔI I

fi

,gi

< 0, we have

 + σ + Δ n ( f i ) (gi ) , ( f i− )σ (gi− )Δ ]α , = ×i=1

74

3 Calculus of Fuzzy Vector-Valued Functions on Time Scales



 + + Δ − − Δ α n f Δ I I ∗ g = ×i=1 gi ( f i ) , gi ( f i ) α ,  σ  α  α  α and thus, we can obtain f ∗ g Δ I I + f Δ I I ∗ g = ( f  g)Δ I I . α,(I ) n [ f − g + , f + g − ] . Moreover, because (v). Since I fi ,gi < 0, we have [ f ∗ g]α = ×i=1 i i i i α f ∗ g is (I )-g H -Δ-differentiable, we obtain 

( f ∗ g)Δ I

α

In addition, from I

 − + Δ  n ( f i gi ) , ( f i+ gi− )Δ α = ×i=1  − σ + Δ  n ( f i ) (gi ) + gi+ ( f i− )Δ , ( f i+ )σ (gi− )Δ + gi− ( f i+ )Δ α . = ×i=1 α,(I I )

ΔI I

f iσ ,gi



> 0, I

f σ  gΔI I

α

α,(I I ) ΔI I

fi

,gi

< 0, we have

 − σ + Δ n ( f i ) (gi ) , ( f i+ )σ (gi− )Δ ]α , = ×i=1



 + − Δ − + Δ α n f Δ I I  g = ×i=1 gi ( f i ) , gi ( f i ) α ,  σ  α  α  α and thus, we can obtain f  g Δ I I + f Δ I I  g = ( f ∗ g)Δ I . α,(I ) n [ f − g + , f + g − ] . Moreover, because (vi). Since I fi ,gi < 0, we have [ f ∗ g]α = ×i=1 i i i i α f ∗ g is (I I )-g H -Δ-differentiable, we obtain 

( f ∗ g)Δ I

α

In addition, from I

 + − Δ  n ( f i gi ) , ( f i− gi+ )Δ α = ×i=1  + σ − Δ  n ( f i ) (gi ) + gi− ( f i+ )Δ , ( f i− )σ (gi+ )Δ + gi+ ( f i− )Δ α . = ×i=1 α,(I I )

ΔI I

f iσ ,gi



< 0, I

f σ  gΔI I

α

α,(I I ) ΔI I

fi

,gi

> 0, we have

 + σ − Δ n ( f i ) (gi ) , ( f i− )σ (gi+ )Δ ]α , = ×i=1



 − + Δ + − Δ α n f Δ I I  g = ×i=1 gi ( f i ) , gi ( f i ) α ,  σ  α  α  α and thus, we can obtain f  g Δ I I + f Δ I I  g = ( f ∗ g)Δ I I . α,(I ) n [ f + g − , f − g + ] . Moreover, because (vii). Since I fi ,gi > 0, we have [ f ∗ g]α = ×i=1 i i i i α f ∗ g is (I )-g H -Δ-differentiable, we obtain 

( f ∗ g)Δ I

α

In addition, from I

 + − Δ  n ( f i gi ) , ( f i− gi+ )Δ α = ×i=1  + σ − Δ  n ( f i ) (gi ) + gi− ( f i+ )Δ , ( f i− )σ (gi+ )Δ + gi+ ( f i− )Δ α . = ×i=1 α,(I I )

ΔI I

f iσ ,gi



< 0, I

f σ  gΔI I 

α

α,(I I ) ΔI I

fi

,gi

> 0, we have

 + σ − Δ n ( f i ) (gi ) , ( f i− )σ (gi+ )Δ ]α , = ×i=1

 − + Δ + − Δ α n f Δ I I  g = ×i=1 gi ( f i ) , gi ( f i ) α , α  Δ α  σ α  Δ and thus, we can obtain f  g I I + f I I  g = ( f ∗ g)Δ I .

3.1

g H -Δ-Derivative of Fuzzy Vector-Valued Functions on Time Scales

75

α,(I )

n [ f + g − , f − g + ] . Moreover, because (viii). Since I fi ,gi > 0, we have [ f ∗ g]α = ×i=1 i i i i α f ∗ g is (I I )-g H -Δ-differentiable, we obtain

 α  − + Δ  n ( f ∗ g)Δ I I = ×i=1 ( f i gi ) , ( f i+ gi− )Δ α  − σ + Δ  n ( f i ) (gi ) + gi+ ( f i− )Δ , ( f i+ )σ (gi− )Δ + gi− ( f i+ )Δ α . = ×i=1 In addition, from I

α,(I I )

> 0, I

ΔI I

f iσ ,gi



f σ  gΔI I

α

α,(I I ) ΔI I

fi

,gi

< 0, we have

 − σ + Δ n ( f i ) (gi ) , ( f i+ )σ (gi− )Δ ]α , = ×i=1



 + − Δ − + Δ α n f Δ I I  g = ×i=1 gi ( f i ) , gi ( f i ) α ,  σ  α  α  α and thus, we can obtain f  g Δ I I + f Δ I I  g = ( f ∗ g)Δ I I . This completes the proof.  Example 3.1.1 We provide an example which satisfies the condition of Theorem 3.11 (i). Let T = [ 21 , 1]T , let f , g : T → RF be given such that α

[ f (t)] = [−2t, t]α ,



1 [g(t)] = t, t 2 α

 α

,

and then  −2t [ f (t) ∗ g(t)]α =  1 2t

   1 2 t  2 α 2 1 2 . = −2t − < 0, i.e. [ f (t) ∗ g(t)] = − 2t , t t t 2 2

Thus we have 

f (t) ∗ g(t)

Δ α

       1  1 2 Δ 2 Δ = (−2t ) , = − 2 t + σ (t) , t + σ (t) . t 2 2 α α

In addition, we also have 

i.e.

σ

f ∗g

 Δ α

  −2σ (t) σ (t)   = −2σ (t) − 1 σ (t) < 0, = 1  1 2 2 

 1 f ∗g = − 2σ (t), σ (t) , 2     α  Δ α −2 1 α  Δ 1 1 f ∗ g =  1  = −2t − t < 0, i.e. f ∗ g = − 2t, t , 2 2 α 2t t

so we have



σ

 Δ α

76

3 Calculus of Fuzzy Vector-Valued Functions on Time Scales



f σ ∗ gΔ

α

 α  f Δ ∗ g]α + f Δ ∗ g = [ f σ ∗ gΔ +     1  = − 2 t + σ (t) , t + σ (t) 2 α  Δ α = f (t) ∗ g(t) .

Remark 3.4 For the traditional multiplication of fuzzy functions, i.e., let f , g : T → RF with the α-level sets [ f ]α = [ f − , f + ]α and [g]α = [g − , g + ]α , the traditional multiplication of fuzzy functions f , g is defined by their α-level set: [ f g]α = [min{ f + g + , f − g + , f + g − , f − g − }, max{ f + g + , f − g + , f + g − , f − g − }]α , and we cannot obtain ( f g)Δ = f σ g Δ + f Δ g. For example, let T = R and f , g : [ 21 , 1] → RF be given such that [ f (t)]α = [ f − (t), f + (t)]α = [−2, 1], [g(t)]α = [g − (t), g + (t)]α = [t 2 , t]α ,





then [ f (t)]α = [0, 0], [g (t)]α = [1, 2t]α , [( f g) ]α = [−2, 1], where t ∈ ( 21 , 1), α ∈ [0, 1]. Consequently,

[ f (t)g (t)]α = [−2, 1][1, 2t]α = [−4t, 2t]α ,

[g(t) f (t)]α = [t 2 , t]α [0, 0] = [0, 0],



where t ∈ ( 21 , 1) and α ∈ [0, 1]. Thus, it follows that [ f (t)g (t)]α + [g(t) f (t)]α 





[( f g) (t)]α , i.e., ( f g) = f g + f g. Theorem 3.11 Let f , g : T → [RnF ] be (I )-g H -Δ-differentiable. Then (i)

α,(I )

if I fi ,gi ≤ 0, I

α,(I )

≤ 0, I

ΔI

f i ,gi

α,(I ) ΔI

,giσ

fi

≤ 0 and f ∗ g is (I )-g H -Δ-differentiable, then

 f ΔI ∗ gσ . ( f ∗ g)Δ I = f ∗ g Δ I + (ii)

α,(I )

if I fi ,gi ≤ 0, I

α,(I )

ΔI

f i ,gi

≥ 0, I

α,(I ) ΔI

fi

,giσ

≥ 0 and f ∗ g is (I I )-g H -Δ-differentiable, then

 f ΔI ∗ gσ . ( f ∗ g)Δ I I = f ∗ g Δ I + (iii)

α,(I I )

if I fi ,gi

≤ 0, I

α,(I I ) ΔI

f i ,gi

≤ 0, I

α,(I I ) ΔI

fi

,giσ

≤ 0 and f  g is (I )-g H -Δ-differentiable, then

gσ  f ΔI . ( f  g)Δ I = f  g Δ I + (iv)

α,(I I )

if I fi ,gi

≤ 0, I

α,(I I ) ΔI

f i ,gi

≥ 0, I

α,(I I ) ΔI

fi

,giσ

≥ 0 and f  g is (I I )-g H -Δ-differentiable, then

 f ΔI  gσ . ( f  g)Δ I I = f  g Δ I +

3.1

g H -Δ-Derivative of Fuzzy Vector-Valued Functions on Time Scales α,(I )

(v)

if I fi ,gi ≥ 0, I

α,(I )

≥ 0, I

ΔI

f i ,gi

α,(I ) ΔI

,giσ

fi

77

≥ 0 and f ∗ g is (I )-g H -Δ-differentiable, then

 f ΔI ∗ gσ . ( f ∗ g)Δ I = f ∗ g Δ I + (vi)

α,(I )

if I fi ,gi ≥ 0, I

α,(I )

≤ 0, I

ΔI

f i ,gi

α,(I ) ΔI

fi

,giσ

≤ 0 and f ∗ g is (I I )-g H -Δ-differentiable, then

 f ΔI ∗ gσ . ( f ∗ g)Δ I I = f ∗ g Δ I + α,(I I )

(vii) if I fi ,gi

α,(I I )

≥ 0, I

ΔI

f i ,gi

≥ 0, I

α,(I I ) ΔI

fi

≥ 0 and f  g is (I )-g H -Δ-differentiable, then

,giσ

gσ  f ΔI . ( f  g)Δ I = f  g Δ I + α,(I I )

(viii) if I fi ,gi

≥ 0, I

α,(I I ) ΔI

f i ,gi

≤ 0, I

α,(I I ) ΔI

fi

,giσ

≤ 0 and f  g is (I I )-g H -Δ-differentiable, then

 f ΔI  gσ . ( f  g)Δ I I = f  g Δ I +

Proof (i). Since f , g be (I )-g H -Δ-differentiable, then  − Δ   − Δ + Δ n n ( f i,α ) , ( f i,+α )Δ , [g Δ I ]α = ×i=1 (gi,α ) , (gi,α ) . [ f Δ I ]α = ×i=1 α,(I )

n [ f − g + , f + g − ]. Moreover, because f ∗ g From I fi ,gi ≤ 0, we have [ f ∗ g]α = ×i=1 i,α i,α i,α i,α is (I )-g H -Δ-differentiable, we obtain



( f ∗ g)Δ I

α

 − + Δ + − Δ n ( f i,α gi,α ) , ( f i,α = ×i=1 gi,α )  − + Δ + σ − Δ + − Δ − σ + Δ n = ×i=1 f i,α (gi,α ) + (gi,α ) ( f i,α ) , f i,α (gi,α ) + (gi,α ) ( f i,α ) .

In addition, from I

α,(I )

ΔI

f i ,gi



≤ 0, I

f ∗ gΔI

α,(I ) ΔI

fi

α

,giσ

≤ 0, we have

 − + Δ + − Δ n f i,α (gi,α ) , f i,α (gi,α ) ], = ×i=1



α  + σ − Δ − σ + Δ n f Δ I ∗ g σ = ×i=1 (gi,α ) ( f i,α ) , (gi,α ) ( f i,α ) ,   α  α  α and thus, we obtain f ∗ g Δ I + f Δ I ∗ g σ = ( f ∗ g)Δ I . By applying a similar discussion as in (i) to (ii) − (viii), one will obtain the results (ii) − (viii) immediately, so we will not repeat these similar proof steps here.  Theorem 3.12 Let f , g : T → [RnF ] be (I I )-g H -Δ-differentiable. Then (i)

α,(I I )

if I fi ,gi then

≤ 0, I

α,(I )

ΔI I

f iσ ,gi

≥ 0, I

α,(I ) ΔI I

fi

,gi

≤ 0 and f  g is (I )-g H -Δ-differentiable,

 f ΔI I ∗ gσ . ( f  g)Δ I = f ∗ g Δ I I +

78

3 Calculus of Fuzzy Vector-Valued Functions on Time Scales

(ii)

α,(I I )

if I fi ,gi

≤ 0, I

then (iii)

α,(I )

ΔI I

f i ,gi

≤ 0, I

α,(I ) ΔI I

fi

,giσ

≥ 0 and f  g is (I I )-g H -Δ-differentiable,

 f ΔI ∗ gσ . ( f  g)Δ I I = f ∗ g Δ I +

α,(I I )

if I fi ,gi

≥ 0, I

α,(I )

≤ 0, I

ΔI I

f i ,gi

α,(I ) ΔI I

fi

,giσ

≥ 0 and f  g is (I )-g H -Δ-differentiable, then

gσ ∗ f ΔI I . ( f  g)Δ I = f ∗ g Δ I I + (iv)

α,(I I )

if I fi ,gi

≥ 0, I

then (v)

α,(I )

ΔI I

f i ,gi

≥ 0, I

α,(I ) ΔI I

fi

,giσ

≤ 0 and f  g is (I I )-g H -Δ-differentiable,

 f ΔI I ∗ gσ . ( f  g)Δ I I = f ∗ g Δ I I +

α,(I )

if I fi ,gi ≤ 0, I

α,(I I )

ΔI I

f i ,gi

≥ 0, I

α,(I I ) ΔI I

fi

,giσ

≤ 0 and f ∗ g is (I )-g H -Δ-differentiable, then

 f ΔI I  gσ . ( f ∗ g)Δ I = f  g Δ I I + (vi)

α,(I )

if I fi ,gi ≤ 0, I

α,(I I )

ΔI I

f i ,gi

≤ 0, I

α,(I I ) ΔI I

fi

,giσ

≥ 0 and f ∗ g is (I I )-g H -Δ-differentiable, then

 f ΔI I  gσ . ( f ∗ g)Δ I I = f  g Δ I I + α,(I )

(vii) if I fi ,gi ≥ 0, I

α,(I I )

ΔI I

f i ,gi

≤ 0, I

α,(I I ) ΔI I

fi

,giσ

≥ 0 and f ∗ g is (I )-g H -Δ-differentiable, then

gσ  f ΔI I . ( f ∗ g)Δ I = f  g Δ I I + α,(I )

(viii) if I fi ,gi ≥ 0, I

α,(I I )

ΔI I

f i ,gi

≥ 0, I

α,(I I ) ΔI I

fi

,giσ

≤ 0 and f ∗ g is (I I )-g H -Δ-differentiable, then

 f ΔI I  gσ . ( f ∗ g)Δ I I = f  g Δ I I + Proof (i). Since f , g be (I I )-g H -Δ-differentiable, then  + Δ  + Δ − Δ − Δ n n ( f i,α ) , ( f i,α (gi,α ) , (gi,α ) . [ f Δ I I ]α = ×i=1 ) , [g Δ I I ]α = ×i=1 α,(I I )

n [ f + g + , f − g − ]. Moreover, because f  From I fi ,gi ≤ 0, we have [ f  g]α = ×i=1 i,α i,α i,α i,α g is (I )-g H -Δ-differentiable, we obtain



( f  g)Δ I

α

 + + Δ − − Δ n ( f i,α gi,α ) , ( f i,α = ×i=1 gi,α )  + + Δ + σ + Δ − − Δ − σ − Δ n = ×i=1 ( f i,α )(gi,α ) + (gi,α ) ( f i,α ) , f i,α (gi,α ) + (gi,α ) ( f i,α ) .

In addition, from I

α,(I )

ΔI I

f i ,gi

 

≥ 0, I

f ∗ gΔI I

f ΔI I ∗ gσ

α

α,(I ) ΔI I

fi

α

,giσ

≤ 0, we have

 + + Δ − − Δ n f i,α (gi,α ) , f i,α (gi,α ) ], = ×i=1

 + σ + Δ − σ − Δ n (gi,α ) ( f i,α ) , (gi,α ) ( f i,α ) , = ×i=1

3.2

Δ-Integral of Fuzzy Vector-Valued Functions on Time Scales

79

 α  α  α and thus, we obtain f ∗ g Δ I I + f Δ I I ∗ g σ = ( f  g)Δ I . By applying a similar discussion as in (i) to (ii) − (viii), the results (ii) − (viii) may be checked immediately and we will not repeat these similar proof steps here. 

3.2

Δ-Integral of Fuzzy Vector-Valued Functions on Time Scales

In this section, we examine the relations between g H -Δ-differentiability and the integral of fuzzy vector-valued functions on time scales. Definition 3.6 The fuzzy Aumann Δ-integral (or Δ-integral for short) of f : [a, b]T → RnF is defined level-wise by 

b

f (t)Δt

α

 =

a

b



a

α n f (t) Δt = ×i=1 

n = ×i=1

b

f i− (t)Δt,

a



b

[ f i (t)]α Δt



a



 f i+ (t)Δt , α ∈ [0, 1].

b

a

n [ f − , f + ] . Then Theorem 3.13 Let f : [a, b]T → RnF be continuous with [ f (t)]α = ×i=1 i i α

t (i) the function F(t) = a f (s)Δs is g H -Δ-differentiable and F Δ (t) = f (t); b (ii) the function F(t) = t f (s)Δs is g H -Δ-differentiable and G Δ (t) = − f (t); Proof We have  t α n [F(t)]α = ×i=1 [Fi− , Fi+ ]α = f (s)Δs a  t   t n = ×i=1 f i− (s)Δs, f i+ (s)Δs , a

n [G(t)]α = ×i=1 [G i− , G i+ ]α =

 =

n ×i=1

t

α

a

b



f i− (s)Δs,



b

f (s)Δs

α

t b

f i+ (s)Δs

t

 α

.

Then 

F Δ (t)

α

  n = ×i=1

=

t

f i− (s)Δs

a − n ×i=1 [ f i , f i+ ]α

Δ   , a α

= [ f (t)] ,

t

f i+ (s)Δs

Δ  α

80

3 Calculus of Fuzzy Vector-Valued Functions on Time Scales



G Δ (t)

α

  n = ×i=1

b

t

f i− (s)Δs

Δ   ,

b

f i+ (s)Δs

t

Δ  α

n = ×i=1 [− f i+ , − f i− ]α = [− f (t)]α .



This completes the proof. Theorem 3.14 If f : [a, b]T → RnF is Δ-integrable and c ∈ [a, b]T . Then  b  c  b  f (t)Δt = f (t)Δt + f (t)Δt. a

a

c

Proof Clearly the integrability of f implies that f is integrable over any subinterval of  − n f i (t), f i+ (t)]α . [a, b]T . Now let α ∈ [0, 1] and f be a measurable selection for ×i=1 b + b − c + b + c − b Since a f i (t) = a f i (t)Δt + c f i (t)Δt and a f i (t) = a f i (t)Δt + c f i− (t)Δt for each i = 1, 2, . . . , n then we obtain  b  b α   b n f (t)Δt = ×i=1 f i− (t)Δt, f i+ (t)Δt a

a

 =

n ×i=1

=

f i− (t)Δt

a



 n ×i=1

c

n + ×i=1

 =

c

b c

f (t)Δt

a



α

b

+

f i− (t)Δt,

c

f i− (t)Δt,

a





α

a

c



+

b

f i+ (t)Δt

a

f i− (t)Δt, 

c

 c

b

 a

f (t)Δt

f i+ (t)Δt



 + c

b

f i+ (t)Δt

 α

α

f i+ (t)Δt α

c

 α

.

c



This completes the proof.

Theorem 3.15 If f is g H -Δ differentiable with no switching point in the interval [a, b]T , then  b

 g H f (a). f Δ (t)Δt = f (b)−

a

Proof If there is no switching point in the interval [a, b]T then f is (I ) or (I I ) g H -Δdifferentiable as in Definition 3.4. Without loss of generality, we assume that f is (I I )-g H Δ-differentiable (the proof for the (I )-g H -Δ-differentiability case being similar).

Δ-Integral of Fuzzy Vector-Valued Functions on Time Scales

3.2

81

We obtain 

b

f Δ (t)Δt

α

a

 n = ×i=1

a

 =

n ×i=1

b

b

 + Δ  ( f i ) (t), ( f i− )Δ (t) α Δt ( f i+ )Δ (t)Δt,



b

 α

( f i− )Δ (t)Δt



 +  α n f i (b) − f i+ (a), f i− (b) − f i− (a) α = ×i=1 a

a

= [ f (b)]α g H [ f (a)]α .



This completes the proof.

Theorem 3.16 Assume that function f is g H -Δ-differentiable with n switching points at ci , i = 1, 2 . . . , n, a = c0 < c1 < c2 < · · · < cn < cn+1 = b and exactly at these points. Then   ci+1 n   ci   g H f (a) =  g H (−1) f (b)− f Δ (t)Δt − f Δ (t)Δt . (3.1) i=1

Also,



b

ci−1

f Δ (t)Δt =

a

ci n+1  

  g H f (ci−1 ) , f (ci )−

(3.2)

i=1

where summation denotes standard fuzzy addition in this statement. Proof For simplicity we consider only one switch point, the case of a finite number of switch-points follows similarly. Suppose that f is (I )-g H -Δ-differentiable on [a, c]T and (I I )–g H -Δ-differentiable on [c, b]T . Then by calculation we obtain 

= = = = =

α  b g H f Δ (t)Δt − f Δ (t)Δt a c      g H f (a) −  g H (−1) f (b)−  g H f (c) α f (c)−      g H f (b) α  g H f (c)− f (c) − f (a) −  − n f (c) − f i− (a), f i+ (c) − f i+ (a)]α ×i=1  −i   g H f i (c) − f i− (b), f i+ (c) − f i+ (b) α  n min{ f i− (b) − f i− (a), f i+ (b) − f i+ (a)}, ×i=1  max{ f i− (b) − f i− (a), f i+ (b) − f i+ (a)} α  g H f (a)]α . [ f (b)− c

Also, one can easily check that

82

3 Calculus of Fuzzy Vector-Valued Functions on Time Scales



b

f Δ (t)Δt =



a

c

 f Δ (t)Δt +



a

     g H f (a) +  f (b)−  g H f (c) f Δ (t)Δt = f (c)−

b c



    g H f (a) +  g H f (a) if and only if f (c) ∈ Rn is a sin f (b)−  g H f (c) = f (b)− and f (c)− gleton. This completes the proof.  Theorem 3.17 If f , g : [a, b]T → RnF are Δ-integrable on [a, b]T , then α˜ · f + β˜ · g, ˜ β˜ ∈ R, is Δ-integrable on [a, b]T and where α, 

  α˜ · f (t) + β˜ · g(t) Δt = α˜ ·

b

a



b

β˜ · f (t)Δt +

a



b

g(t)Δt. a

Proof For any 0 ≤ α ≤ 1, we have 



b

α˜ ·  = α˜ ·

a



n ×i=1

+β˜ ·

a

n ×i=1

 n ×i=1



n + ×i=1

 n ×i=1

 =

α˜ ·

a



=

a

b

b

a



b

a b





α

b

g(t)Δt a

b





 =

β˜ · f (t)Δt +

f i− (t)Δt, b



f i+ (t)Δt

a



gi− (t)Δt,

f i− (t)Δt,

b

a b



b

β˜ · gi− (t)Δt,



b

a

 α˜ · f (t) + β˜ · g(t) Δt

 α

f i+ (t)Δt

 α

β˜ · gi+ (t)Δt

 α˜ · f i− (t) + β˜ · gi− (t) Δt, α

α

gi+ (t)Δt

α˜ ·

a



 a

b



 α

α˜ ·

f i+ (t) + β˜

·

 gi+ (t) Δt

 α

.

a



This completes the proof. Theorem 3.18 Assume f , g : [a, b]T → RnF are g H -Δ-differentiable, then a − b f (t)Δt. Proof In fact, for any α ∈ [0, 1], we can obtain

b a

f (t)Δt =

Δ-Integral of Fuzzy Vector-Valued Functions on Time Scales

3.2



b

α

f (t)Δt

a

 n = ×i=1



a

b

83

 f i+ (t)Δt a α   a − + f i (t)Δt, − f i (t)Δt

f i− (t)Δt,



b

 a n − = ×i=1 b b  a  n  −   ×i=1 f i (t), f i+ (t) α Δt =− b   a α  a =− [ f (t)]α Δt = − f (t)Δt . b

α

b



This completes the proof.

Theorem 3.19 Assume f , g : [a, b]T → RnF are (I )-g H -Δ-differentiable and f ∗ g is α,(I ) also (I )-g H -Δ-differentiable. If there is no switching point in [a, b]T and I fi ,gi > 0, I

α,(I )

ΔI

f iσ ,gi



α,(I ) ΔI

fi

,gi

> 0 for each i = 1, 2, . . . , n, then

   g H f (a) ∗ g(a) −  g HI f (t) ∗ g Δ I (t)Δt = f (b) ∗ g(b)−

b

a



> 0, I

b

f (t) ∗ g Δ I (t)Δt =

a



b

a



  f Δ I (t) ∗ g σ (t) Δt or

b

a

(3.3)

     g H f (b) ∗ g(b) .  g HI I f (a) ∗ g(a)− g σ (t) ∗ f Δ I (t)Δt − (3.4)

Proof First, let us prove (3.3). Since  g H f (a) ∗ g(a) f (b) ∗ g(b)−  b  Δ f (t) ∗ g(t) I Δt = a

 =

b



   f Δ I (t) ∗ g σ (t) Δt f (t) ∗ g Δ I (t)+

a

 =

b

f (t) ∗ g

a

ΔI

 (t)Δt +



b

  f Δ I (t) ∗ g σ (t) Δt,

a

which implies that  a

b

   g H f (a) ∗ g(a) −  g HI f (t) ∗ g Δ I (t)Δt = f (b) ∗ g(b)−

Now, we prove (3.4). Because

 a

b

  f Δ I (t) ∗ g σ (t) Δt.

84

3 Calculus of Fuzzy Vector-Valued Functions on Time Scales

 g H f (b) ∗ g(b) f (a) ∗ g(a)−  b  Δ f (t) ∗ g(t) I Δt = (−1) a

 = (−1)  =

b

f (t) ∗ g

ΔI

(−1) (t)Δt +

a b

  (−1) g σ (t) ∗ f Δ I (t)Δt +

a

This completes the proof.



a b

b

  f Δ I (t) ∗ g σ (t) Δt

f (t) ∗ g Δ I (t)Δt,

a

which indicates that  b  f (t) ∗ g Δ I (t)Δt = a



a

b

     g H f (b) ∗ g(b) .  g HI I f (a) ∗ g(a)− g σ (t) ∗ f Δ I (t)Δt − 

4

Shift Almost Periodic Fuzzy Vector-Valued Functions

Almost periodic functions and their generalized function theory on time scales have been developed in many literatures (see [57, 58, 60, 77–81, 83, 86–88, 90–93, 96–103, 106–114]) and these results have been applied to study various types of dynamic equations on time scales. In this chapter, we will develop a theory of shift almost periodic fuzzy vector-valued functions on time scales.

4.1

Shift Operators on Time Scales

In this section, the concept of shift operators on time scales and their basic properties will be presented which are necessary to study dynamic equations with shift operators on time scales (see [1, 2]).

4.1.1

Shift Operators

Definition 4.1 (see [1, 2]) Let T∗ be a non-empty subset of the time scale T including a fixed number t0 ∈ T∗ such that there exist operators δ± : [t0 , ∞)T × T∗ → T∗ satisfying the following properties: P.1 The functions δ± are strictly increasing with respect to their second arguments, i.e., if (T0 , t), (T0 , u) ∈ D± := {(s, t) ∈ [t0 , ∞)T × T∗ : δ± (s, t) ∈ T∗ }, then T0  t < u ⇒ δ± (T0 , t) < δ± (T0 , u). P.2 If (T1 u), (T2 , u) ∈ D− with T1 < T2 , then © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 C. Wang and R. P. Agarwal, Dynamic Equations and Almost Periodic Fuzzy Functions on Time Scales, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-11236-2_4

85

86

4 Shift Almost Periodic Fuzzy Vector-Valued Functions

δ− (T1 , u) > δ− (T2 , u), and if (T1 , u), (T2 , u) ∈ D+ with T1 < T2 , then δ+ (T1 , u) < δ+ (T2 , u). P.3 If t ∈ [t0 , ∞)T , then (t, t0 ) ∈ D+ and δ+ (t, t0 ) = t. Moreover, if t ∈ T∗ , then (t0 , t) ∈ D+ and δ+ (t0 , t) = t holds.     P.4 If (s, t) ∈ D± , then s, δ± (s, t) ∈ D and δ∓ s, δ± (s, t) = t, respectively.   P.5 If (s, t) ∈ D± and u, δ± (s, t) ∈ D∓ , then (s, δ∓ (u, t)) ∈ D± and

  δ∓ (u, δ± (s, t)) = δ± s, δ∓ (u, t) ,

respectively. Then the operators δ− and δ+ associated with t0 ∈ T∗ (called the initial point) are said to be backward and forward shift operators on the set T∗ , respectively. The variable s ∈ [t0 , ∞)T in δ± (s, t) is called the shift size. The values δ+ (s, t) and δ− (s, t) in T∗ indicate s units translation of the term t ∈ T∗ to the right and left, respectively. The sets D± are the domains of the shift operators δ± , respectively. Now we shall denote by T∗ the largest subset of the time scale T such that the shift operators δ± : [t0 , ∞)T × T∗ → T∗ exist. Example 4.1.1 Let T = R and t0 = 1. The operators  t/s if t  0, δ− (s, t) = for s ∈ [1, ∞) st if t < 0, and δ+ (s, t) =

 st

if t  0,

t/s if t < 0,

for s ∈ [1, ∞)

(4.1)

(4.2)

are backward and forward shift operators (on the set R∗ = R − {0}) associated with the initial point t0 = 1. In the table below, we state different time scales with their corresponding shift operators. 

4.1

Shift Operators on Time Scales T R Z q Z ∪ {0} N1/2

87 t0 T∗ δ− (s, t) δ+ (s, t) 0 R t −s t +s 0 Z t −s t +s t 1 qZ st  s  1/2 2 2 2 0 N t −s t + s2

The proof of the next lemma is a direct consequence of Definition 4.1. Lemma 4.1 (see [1, 2]) Let δ− and δ+ be the shift operators associated with the initial point t0 . We have δ− (t, t) = t0 for all t ∈ [t0 , ∞)T . δ− (t0 , t) = t for all t ∈ T∗ . If (s, t) ∈ D+ , then δ+ (s, t) = u implies δ− (s, u) = t. Conversely, If (s, u) ∈ D− , then δ− (s, u) = t implies δ+ (s, t) = u.   (iv) δ+ t, δ− (s, t0 ) = δ− (s, t) for all (s, t) ∈ D+ with t  t0 .     (v) δ+ u, t = δ+ (t, u) for all (u, t) ∈ [t0 , ∞)T × [t0 , ∞)T ∩ D+ . (vi) δ+ (s, t) ∈ [t0 , ∞)T for all (s, t) ∈ D+ with t  t0 .   (vii) δ− (s, t) ∈ [t0 , ∞)T for all (s, t) ∈ [t0 , ∞)T × [s, ∞)T ∩ D− . δt (s, ·) > 0. (viii) If δ+ (s, ·) is Δ-differentiable in its second variable, then δ+     (ix) δ+ δ− (u, s), δ− (s, v) = δ− (u, v) for all (s, v) ∈ [t0 , ∞)T × [s, ∞)T ∩ D− and   (u, s) ∈ [t0 , ∞)T × [u, ∞)T ∩ D− . (x) If (s, t) ∈ D− and δ− (s, t) = t0 , then s = t.

(i) (ii) (iii)

Proof (i). The result follows from P.3–P.5 and that   δ− (t, t) = δ− t, δ+ (t, t0 ) = t0 for all t ∈ [t0 , ∞)T . (ii). From P.3–P.4 and   δ− (t0 , t) = δ− t0 , δ+ (t0 , t) = t for all t ∈ T∗ . Let u := δ+ (s, t). By P.4 we have (s, u) ∈ D− for all (s, t) ∈ D+ , and hence,   δ− (s, u) = δ− s, δ+ (s, t) = t. The latter part of (iii) can be done in the similar way. We have (iv) since P.3 and P.5 yield     δ+ t, δ− (s, t0 ) = δ− (s, δ+ t, t0 ) = δ− (s, t). P.3 and P.5 guarantee that

88

4 Shift Almost Periodic Fuzzy Vector-Valued Functions

    t = δ+ (t, t0 ) = δ+ t, δ− (u, u) = δ− u, δ+ (t, u)   for all (u, t) ∈ [t0 , ∞)T × [t0 , ∞)T ∩ D+ . Using (iii) we have    δ+ (u, t) = δ+ u, δ− u, δ+ (t, u) = δ+ (t, u). This proves (v). To prove (vi) and (vii), we use P.1–P.2 to get δ+ (s, t)  δ+ (t0 , t) = t  t0   for all (s, t) ∈ [t0 , ∞)T × [t0 , ∞)T ∩ D+ and δ− (s, t)  δ− (s, s) = t0 for all (s, t) ∈ ([t0 , ∞)T × [s, ∞)T ) ∩ D− . Since δ+ (s, t) is strictly increasing in its second variable we have (viii) by (see [20]: Corollary 1.16). (ix) According to P.5 and (v) we have      δ+ δ− (u, s), δ− (s, v) = δ− s, δ+ v, δ− (u, s)   = δ− s, δ− (u, δ+ (v, s))   = δ− s, δ+ (s, δ− (u, v)) = δ− (u, v)

for all (s, v) ∈ ([t0 , ∞)T × [s, ∞)T ) ∩ D− and (u, s) ∈ ([t0 , ∞)T × [u, ∞)T ) ∩ D− . Suppose (s, t) ∈ D− = {(s, t) ∈ [t0 , ∞)T × T∗ : δ− (s, t) ∈ T∗ } and δ− (s, t) = t0 . Then by P.4 we have t = δ+ (s, δ− (s, t)) = δ+ (s, t0 ) = s. This is (x). The proof is completed.



Notice that the shift operators δ± are defined once the initial point t0 ∈ T is fixed. For instance, we choose the initial point t0 = 0 to define shift operators δ± (s, t) = t ± s on T = R. However, if we choose λ ∈ (0, ∞) as the initial point, then the new shift operators ∼

associated with λ are defined by δ ± (s, t) = t ∓ λ ± s. In terms of δ± the new shift operators

∼

δ ± can be defined as

∼

δ ± (s, t) = δ± (λ, δ± (s, t)).

Example 4.1.2 Some particular time scales with shift operators associated with different initial points are provided to show the change in the formula of shift operators as the initial √ point changes. where λ ∈ Z+ , N1/2 = { n : n ∈ N}, 2N = {2n : n ∈ N}, and hZ = {hn :  n ∈ Z}.

4.1

Shift Operators on Time Scales

t0 δ− (s, t) δ+ (s, t)

4.1.2

T = N1/2 0 λ  2 2 t − s 2 + λ2 − s 2 t  2 2 t + s t 2 − λ2 + s 2

89 T = hZ 0 hλ t −s t + hλ − s t +s t − hλ + s

T = 2N 1 2λ t/s 2λ ts −1 ts 2−λ ts

Periodicity of Time Scales

In this subsection, we will state a classical concept of periodic time scales which was introduced in [47]. Definition 4.2 (see [47]) A time scale T is said to be periodic if there exists a P > 0 such that t ± P ∈ T for all t ∈ T. (4.3) If T = R, the smallest positive P is called the period of the time scale. In the following, a new periodicity notion will be presented which does not require the time scale to be closed under the operation t ± P for a fixed P > 0 or to be unbounded. Definition 4.3 (see [1, 2], Periodicity in shifts) Let T be a time scale with the shift operators δ± associated with the initial point t0 ∈ T∗ . The time scale T is said to be periodic in shifts δ± if there exists a p ∈ (t0 , ∞)T∗ such that ( p, t) ∈ D∓ for all t ∈ T∗ . Furthermore, if P := inf{ p ∈ (t0 , ∞)T∗ : ( p, t) ∈ D∓ for all t ∈ T∗ } = t0 ,

(4.4)

then P is called the period of the time scale T. A time scale periodic in shifts may be bounded, the following example will be presented to show that a time scale, periodic in shifts, does not have to satisfy (4.3). Example 4.1.3 The following time scales are not periodic in the sense of Definition 4.2 but periodic with respect to the notion of shift operators given in Definition 4.3 (see [1, 2]). (1) T1 = {±n 2 : n ∈ Z}, ⎧√ √ ⎪ if t > 0, ⎪( t ± P)2 ⎨ δ± (P, t) =

±P if t = 0, ⎪ ⎪ ⎩−(√−t ± √ P)2 if t < 0

(2) T2 = q Z , δ± (P, t) = P ±1 t, P = q, t0 = 1,

P = 1, t0 = 0,

90

4 Shift Almost Periodic Fuzzy Vector-Valued Functions

(3) T3 = (4) T4 =



[22n , 22n+1 ], δ± (P, t) = P ±1 t, P = 4, t0 = 1, : q > 1 is constant and n ∈ Z ∪ {0, 1},

n∈Z  n q 1+q n

δ± (P, t) =

q

    t ±ln P ln 1−t 1−P ln q

1+q

    t ±ln P ln 1−t 1−P ln q

,

P=

q . 1+q

Note that the time scale T4 in Example 4.1.3 is bounded above and below and   qn T4 = : q > 1 is constant and n ∈ Z . 1 + qn  Remark 4.1 Let T be a time scale that is periodic in shifts with the period P. Thorough P.4 of Definition 4.1, it follows that the mapping δ+P : T∗ → T∗ defined by δ+P (t) = δ+ (P, t) is surjective. On the other hand, we know by P.1 of Definition 4.1 that shift operators δ± are strictly increasing in their second arguments. That is, the mapping δ+P (t) := δ+ (P, t) is injective. Hence, δ+P is an invertible mapping with the inverse (δ+P )−1 = δ−P defined by δ−P (t) := δ− (P, t). For next two results, we will present that if T is a periodic time scale in shifts δ± with period P, then the operators σ±P : T∗ → T∗ are commutative with the forward jump operator σ : T → T given by σ (t) := inf{s ∈ T : s > t}. That is,

 P    δ± ◦ σ (t) = σ ◦ δ±P (t) for all t ∈ T∗ .

(4.5)

T : T∗ → T∗ preserves the structure of the points Lemma 4.2 (see [1, 2]) The mapping δ+ ∗ in T . That is,   δ( t) =  t =⇒ σ δ+ (P,  t) = δ+ (P,  t).

  δ( t) >  t =⇒ σ δ+ (P,  t) > δ+ (P,  t). Proof From the definition, it follows that σ (t)  t for all t ∈ T∗ . Thus, by P.1 δ+ (P, σ (t))  δ+ (P, t). Since σ (δ+ (P, t)) is the smallest element satisfying

4.1

Shift Operators on Time Scales

91

  σ δ+ (P, t)  δ+ (P, t), we get

δ+ (P, σ (t))  σ (δ+ (P, t)) for all t ∈ T∗ .

(4.6)

t) =  t, the (4.6) implies If σ ( δ+ (P,  t) = δ+ (P, σ ( t))  σ (δ+ (P,  t)). That is, δ+ (P,  t) = σ (δ+ (P,  t)) if σ ( t) =  t. If σ ( t) >  t, then by definition of σ we have ( t, σ ( t))T∗ = ∅

(4.7)

and by P.1 δ+ (P, σ ( t)) > δ+ (P,  t). Suppose to the contrary that δ+ (P,  t) is right dense, i.e., σ (δ+ (P,  t)) = δ+ (P,  t). This along with (4.6) implies (δ+ (P,  t), δ+ (P, σ ( t)))T∗ = ∅. Pick one element s ∈ (δ+ (P,  t), δ+ (P, σ ( t)))T . Since δ+ (P, t) is strictly increasing in t and t, σ ( t))T such that δ+ (P, t) = s. This contradicts invertible there should be an element t ∈ ( t) must be right scattered, i.e., σ (δ+ (P,  t)) > δ+ (P,  t). The proof is (4.7). Hence, δ+ (P,  completed.  Corollary 4.1 (see [1, 2]) The following equalities hold: δ+ (P, σ (t)) = σ (δ+ (P, t)) for all t ∈ T∗ .

(4.8)

δ− (P, σ (t)) = σ (δ− (P, t)) for all t ∈ T∗ .

(4.9)

Proof The equality (4.8) can be obtained similar to the proof of preceding lemma. By (4.8) we have δ+ (P, σ (s)) = σ (δ+ (P, s)) for all s ∈ T∗ . Substituting s = δ− (P, t) we obtain δ+ (P, σ (δ− (P, t))) = σ (δ+ (P, δ− (P, t))) = σ (t). This and (iii) of Lemma 4.1 imply σ (δ− (P, t)) = δ− (P, σ (t)) for all t ∈ T∗ . This completes the proof.



92

4 Shift Almost Periodic Fuzzy Vector-Valued Functions

Note that (4.8) along with (4.9) yields (4.5). Definition 4.4 (see [1, 2], Periodic function in shifts δ± ) Let T be a time scale that is periodic in shifts δ± with the period P. We say that a real valued function f defined on T∗ is periodic in shifts δ± if there exists a T ∈ [P, ∞)T∗ such that T (T , t) ∈ D± and f (δ± (t)) = f (t) for all t ∈ T∗ ,

(4.10)

T (t) := δ (T , t). The smallest number T ∈ [P, ∞) ∗ such that (4.10) holds is where δ± ± T called the period of f .

Example 4.1.4 By Definition 4.3 we know that the real line R is periodic in shifts δ± defined by (4.1)–(4.2) associated with the initial point t0 = 1. The function   ln |t| , t ∈ R∗ := R − {0} f (t) = sin ln(1/2)π is periodic in shifts δ± defined by (4.1)–(4.2) with the period T = 4 since    f (t4±1 ) if t  0, ln |t| ± 2 ln(1/2) = sin f (δ± (T , t)) = π ln(1/2) f (t/4±1 ) if t < 0,   ln |t| π ± 2π = sin ln(1/2)   ln |t| π = sin ln(1/2) = f (t) for all t ∈ R∗ .



Example 4.1.5 The time scale q Z = {q n : n ∈ Z and q > 1} ∪ {0} is periodic in shifts δ± (P, t) = P ±1 t with the period P = q. The function f defined by ln t

f (t) = (−1) ln q , t ∈ q N

(4.11) ∗

is periodic in shifts δ± with the period T = q 2 since δ+ (q 2 , t) ∈ q Z = q Z and ln t ln t   ±2 f δ± (q 2 , t) = (−1) ln q = (−1) ln q = f (t)

for all t ∈ q Z . However, f is not periodic in the sense of Definition 4.2 since there is no any  positive number T such that f (t ± T ) = f (t) holds. Next, we introduce Δ-periodic function in shifts.

4.1

Shift Operators on Time Scales

93

Definition 4.5 (see [1, 2], Δ-periodic function in shifts δ± ) Let T be a time scale that is periodic in shifts δ± with period P. We say that a real valued function f defined on T∗ is Δ-periodic in shifts on T if there exists a T ∈ [P, ∞)T∗ such that (T , t) ∈ D± for all t ∈ T∗ ,

(4.12)

the shifts δ± are Δ-differentiable with rd-continuous derivatives, and

T ΔT f (δ± (t))δ± (t) = f (t)

(4.13) (4.14)

T (t) := δ (T , t). The smallest number T ∈ [P, ∞) ∗ such that for all t ∈ T∗ , where δ± ± T (4.12)–(4.14) hold is called the period of f .

Note that Definitions 4.4 and 4.5 give the classic periodicity definition (i.e. Definition T (t) = t ± T are the shifts satisfying the assumptions of 4.2) on time scales whenever δ± Definitions 4.4 and 4.5. Example 4.1.6 The real valued function g(t) = 1/t defined on 2Z = {2n : n ∈ Z} is Δperiodic in shifts δ± (T , t) = T ±1 t with the period T = 2 since 1

Δ f (δ± (2, t))δ± (2, t) =

2±1 t

2±1 =

1 = f (t). t 

To present the proof of next theorem, the following result is essential. Theorem 4.1 (Substitution, see [20] Theorem 1.98) Assume : T → R is strictly increasing ∼

and T := v(T) is a time scale. If f : T → R is an rd-continuous function and is differentiable with rd-continuous derivative, then for a, b ∈ T, 

b

g(s)v Δ (s)Δs =



v(b)

v(a)

a

∼

g(v −1 (s))Δs.

(4.15)

Theorem 4.2 (see [1, 2]) Let T be a time scale that is periodic in shifts δ± with period P ∈ [t0 , ∞)T∗ and f a Δ-periodic function in shifts δ± with the period T ∈ [P, ∞)T∗ . Suppose that f ∈ Cr d (T), then 

t

t0

 f (s)Δs =

T (t) δ±

T (t ) δ± 0

f (s)Δs.

T (s) and g(s) = f (δ T (s)) in (4.15) and taking (4.14) into Proof Substituting v(s) = δ+ + account we have

94

4 Shift Almost Periodic Fuzzy Vector-Valued Functions



T (t) δ+

T (t ) δ+ 0

 f (s)Δs = = =

v(t)

v(t0 )  t

g(v −1 (s))Δs

g(s)v Δ (s)Δs =

t0  t



t t0

T ΔT f (δ+ (s))δ+ (s)Δs

f (s)Δs.

t0

The equality



T (t) δ−

T (t) δ−

 f (s)Δs =

t

f (s)Δs

t0



can be obtained similarly. This completes the proof.

4.2

Complete-Closed Time Scales Under Non-translational Shifts

To introduce the concept of almost periodic fuzzy vector-valued functions on irregular time scales in the next section, in this part, we will introduce a concept of complete-closed time scales under shift operators. We assume that δ± are shift operators fulfilling Definition 3 from [2] and D˜ ± := {(s, t) ∈ [t0 , ∞)T × T∗ : δ± (s, t) ∈ T∗ }, where T∗ is the largest subset of the time scale T, i.e., Δ (s, t) denotes the T∗ = T. If δ± (s, t) is Δ-differentiable to its second argument, then δ± Δ-derivative of δ± (s, t) to its second argument. Definition 4.6 (see [2]) Let T be a time scale with the shift operators δ± associated with the initial point t0 ∈ T∗ . The time scale T is said to be periodic in shifts δ± if there exists a p ∈ (t0 , ∞)T∗ such that ( p, t) ∈ D˜ ∓ for all t ∈ T∗ . Furthermore, if   P := inf p ∈ (t0 , ∞)T∗ : ( p, t) ∈ D˜ ∓ for all t ∈ T∗ = t0 ,

(4.16)

 then P is called the period of the time scale T, where D˜ ± = (s, t) ∈ [t0 , ∞)T × T∗ :  δ± (s, t) ∈ T∗ . Now, we will provide an example to show Definition 4.6 needs a further extension. Example 4.1.1 Consider the following time scale:   T = −q Z ∪ {1} = − q n : q > 1, n ∈ Z ∪ {1}.

(4.17)

  For such a time scale, we obtain that T∗ = − q n : q > 1, n ∈ Z ∪ {1}. Take the initial  point t0 = 1 and the shift operators δ− (s, t) = −st, δ+ (s, t) = − st . Let Π − = − q n :  q > 1, n ∈ Z+ ⊂ T∗ , we obtain δ± (s, t) ∈ T∗ for any s ∈ Π − . Hence, according to

4.2

Complete-Closed Time Scales Under Non-translational Shifts

95

Definition 4.6, the time scale T cannot be regarded as a periodic time scale under shifts δ± since there is no number P ∈ (1, +∞)T∗ satisfying (4.16). However, this time scale is   the opposite number set of the time scale q Z = q n : q > 1, n ∈ Z ∪ {0}, and this time scale (4.17) also plays an important role in q-difference equations. In fact, from (4.17), it is / [1, +∞).  easy to observe that for any t ∈ T∗ , we have −(−q)t ∈ T∗ but −q ∈ For convenience, we introduce some notations. Let   D± = (s, t) ∈ T∗ × T∗ : δ± (s, t) ∈ T∗ . For any s ∈ T∗ , denote   δ− T∗s := δ− (s, T∗ ) := δ− (s, t) : (s, t) ∈ D− , ∀t ∈ T∗ ,

(4.18)

  δ+ T∗s := δ+ (s, T∗ ) := δ+ (s, t) : (s, t) ∈ D+ , ∀t ∈ T∗ .

(4.19)

Definition 4.7 Let T be a time scale with the shift operators δ± associated with the initial point t0 ∈ T∗ . The time scale T is said to be bi-direction shift complete-closed time scales (i.e. S-CCTS for short) in shifts δ± if     Π := p ∈ T∗ : ( p, t) ∈ D± for all t ∈ T∗ = Π ± ∈ / {t0 }, ∅ .

(4.20)

Remark 4.2 Note that from (4.18) and (4.19), it follows that (4.20) can be written into the     δ ± equivalent form Π = p ∈ T∗ : T∗p ⊆ T∗ ∈ / {t0 }, ∅ . Furthermore, from (4.20), we will refine the following the concept of S-CCTS attached with shift direction. For convenience, we will use the notations     δ − δ Π + := p ∈ T∗ : T∗p ⊆ T∗ , Π − := p ∈ T∗ : T∗p ⊆ T∗ . Definition 4.8 Let T be a S-CCTS, then (i) we say S-CCTS is with positive-direction if     Π+ ∈ / {t0 }, ∅ ; (ii) we say S-CCTS is with negative-direction if Π − ∈ / {t0 }, ∅ ; (iii) we   / {t0 }, ∅ . say S-CCTS is with bi-direction if Π ∈ Remark 4.3 From Definition 4.8, it follows that a bi-direction S-CCTS is also with a positive-direction and a negative-direction. Example 4.1.2 From Definitions 4.7 and 4.8, we provide the following examples of SCCTS.

96

4 Shift Almost Periodic Fuzzy Vector-Valued Functions

  (1) Let T = (−q)Z ∪ {1} = (−q)n : q > 1, n ∈ Z ∪ {0, 1}. Then we obtain that Π ± =   (−q)2n : q > 1, n ∈ Z+ . For any t ∈ T∗ , by taking t0 = 1, we attach the following shift operators   t , t > 0, st, t > 0, δ− (s, t) = s δ+ (s, t) = t st, t < 0. s , t < 0, Hence, there exists q 2 ∈ Π ± such that δ± (q 2 , t) ∈ T∗ for all t ∈ T∗ , i.e., Π ± ∈ /   {1}, ∅ . From Definition 4.8, it follows that T is a S-CCTS with bi-direction.     (2) Consider T = q n : q > 1, n ∈ Z ∪ − q n : q > 1, n ∈ Z ∪ {0}. For any t ∈ T∗ , by taking t0 = 1, we attach the shift operators   t , t > 0, st, t > 0, δ− (s, t) = s δ+ (s, t) = t st, t < 0. s , t < 0,   Then we obtain that Π ± = q n : q > 1, n ∈ Z+ . Hence, there exists q ∈ Π ± such   / {1}, ∅ . From Definition 4.8, it follows that δ± (q, t) ∈ T∗ for all t ∈ T∗ , i.e., Π ± ∈ that T is a S-CCTS with bi-direction. 1  √  (3) Consider N±2 = ± n, n ∈ N , For any t ∈ T∗ , we take t0 = 0 and attach the shift operators √ √ s 2 + t 2 , t > 0, t 2 − s 2 , t > 0, √ √ δ− (s, t) = δ+ (s, t) = − t 2 − s 2 , t < 0, − t 2 + s 2 , t < 0. √  1 Then it follows that Π ± = N 2 = n : n ∈ N . Hence, there exists 1 ∈ Π ± such that   δ± (1, t) ∈ T∗ for all t ∈ T∗ , i.e., Π ± ∈ / {0}, ∅ . From Definition 4.8, we obtain T is a S-CCTS with bi-direction.     (4) Let T1 = q n : q > 1, n ∈ Z+ ∪ {1} and T2 = q n : q > 1, n ∈ Z− ∪ {0, 1}. Now we take t0 = 1 and     Π1+ = q n : q > 1, n ∈ Z+ ⊆ T∗1 , Π2− = q n : q > 1, n ∈ Z− ⊆ T∗2 . Noting that for any s1 ∈ Π1+ , s2 ∈ Π2− , we have δ+ (s1 , t1 ) = s1 t1 ∈ T∗1 for all t1 ∈ T∗1 , δ− (s1 , t1 ) =

t1 ∈ / T∗1 for t1 = q, s1 = q 2 , s1

and δ− (s2 , t2 ) = s2 t2 ∈ T∗2 for all t2 ∈ T∗2 , δ+ (s2 , t2 ) =

t2 1 1 ∈ / T∗2 for t2 = , s2 = 2 . s2 q q

Hence, for the shift operator δ+ (s, t) = st, we obatin T1 is a positive-direction S-CCTS.  For the shift operator δ− (s, t) = st, T2 is a negative-direction S-CCTS.

4.3

Shift Almost Periodic Fuzzy Vector-Valued Functions

97

Remark 4.4 We attached the translation direction to time scales in [89] and introduced the concept of translation complete-closed time scales (i.e. T -CCTS). We also introduced the concepts of some special functions arising from differential and difference equations on T -CCTS including almost periodic functions and almost automorphic functions. However, these results will never cover some important and irregular time scales such as (−q)Z and q Z , etc. Hence, these results can be applied to fuzzy q-difference equations. Remark 4.5 Note that if T is a periodic time scales under translations and Π ⊆ T∗ , then the shift operators will fulfill δ± (τ, t) = t ± τ ∈ T with an initial point t0 = 0. Hence, if Π ⊆ T∗ , then T -CCTS is included in S-CCTS. If T is a bi-direction S-CCTS and t0 is an initial point, then for any s ∈ Π , we define a function A : Π → Π ,  δ+ (s, t0 ), s > t0 , A(s) = (4.21) δ− (s, t0 ), s < t0 , which will be used later. Note that A(s) > t0 and A(s) ≥ s. Remark 4.6 If T is a T -CCTS with the translation operators δ± (s, t) = t ± s and t0 = 0. Then (4.21) will become  s, s > 0, A(s) = −s, s < 0, that is, A(s) = |s| ≥ s.

4.3

Shift Almost Periodic Fuzzy Vector-Valued Functions

In this section, we assume that D ⊆ RnF is an open set. In what follows, we will introduce a new type of almost periodic fuzzy vector-valued functions called shift almost periodic fuzzy vector-valued functions (or S-almost periodic fuzzy vector-valued functions) on time scales. Definition 4.9 Let T be a bi-direction S-CCTS and f : T × D → RnF be continuous on T × D. (i)

A function f ∈ C(T × D, RnF ) is called shift almost periodic fuzzy vector-valued function in t ∈ T uniformly for x ∈ D with shift operators if the ε-shift number set of f     E{ε, f , S0 } = τ ∈ Π : D∞ f (δ± (τ, t), x), f (t, x) < ε, for all t ∈ T∗ and x ∈ S0

98

4 Shift Almost Periodic Fuzzy Vector-Valued Functions

is a relatively dense set with respect to the pair (Π , δ± ) for all ε > 0 and for each compact subset S0 of D; that is, for any given ε > 0 and each compact subset S0 of D, there exists a constant l(ε, S0 ) > 0 such that each interval of length l(ε, S0 ) contains a τ (ε, S0 ) ∈ E{ε, f , S0 } such that     D∞ f δ± (τ, t), x , f (t, x) < ε, for all t ∈ T∗ and x ∈ S0 . Now τ is called the ε-shift number of f and l(ε, S0 ) is called the inclusion length of E{ε, f , S0 }. (ii) A function f ∈ C(T × D, RnF ) is called shift normal function if for any sequence   Fn : T × D → RnF of the form Fn (t, x) = f δ+ (h n , t), x , n ∈ N, where (h n )n ⊂ Π is a sequence of real numbers, one can extract a subsequence of (Fn )n , converging uniformly on T × D (i.e., ∀(h n )n ⊂ Π , ∃(h n )k , ∃F : T → RnF which may depend on (h n )n ), such that   D∞ Fn k (t, x), F(t, x) → 0 as k → ∞ uniformly with respect to (t, x) ∈ T × D. (iii) Let δ± (s, t) be Δ-differentiable to its second argument. A function f ∈ C(T × D, RnF ) is called shift Δ-almost periodic fuzzy vector-valued function in t ∈ T uniformly for x ∈ D with shift operators if the ε-shift number set of f    Δ E{ε, f , S0 } = τ ∈ Π : D∞ f (δ± (τ, t), x)δ± (τ, t), f (t, x) < ε,  for all t ∈ T∗ and x ∈ S0 is a relatively dense set with respect to the pair (Π , δ± ) for all ε > 0 and for each compact subset S0 of D; that is, for any given ε > 0 and each compact subset S0 of D, there exists a constant l(ε, S0 ) > 0 such that each interval of length l(ε, S0 ) contains a τ (ε, S0 ) ∈ E{ε, f , S0 } such that    Δ  D∞ f δ± (τ, t), x δ± (τ, t), f (t, x) < ε, for all t ∈ T∗ and x ∈ S0 . Now τ is called the ε-shift number of f and l(ε, S0 ) is called the inclusion length of E{ε, f , S0 }. (iv) Let δ± (s, t) be Δ-differentiable to its second argument. A function f ∈ C(T × D, RnF ) is called shift Δ-normal function if for any sequence Fn : T × D → RnF of   Δ (h n , t), n ∈ N, where (h n )n ⊂ Π is a sequence the form Fn (t, x) = f δ+ (h n , t), x δ+ of real numbers, one can extract a subsequence of (Fn )n , converging uniformly on T × D (i.e., ∀(h n )n ⊂ Π , ∃(h n )k , ∃F : T → RnF which may depend on (h n )n ), such that   D∞ Fn k (t, x), F(t, x) → 0 as k → ∞ uniformly with respect to (t, x) ∈ T × D.

4.3

Shift Almost Periodic Fuzzy Vector-Valued Functions

99

For convenience, we denote A PS (T) the set of all shift almost periodic functions in shifts on T and we introduce some notations. Let α = {αn } ⊂ Π and β = {βn } ⊂ Π be two sequences. Then β ⊂ α means that β is a subsequence of α; δ± (α, β) =   {δ± (αn , βn )}; δ− (α, t0 ) = {δ− (αn , t0 )}, α and β are common subsequences of α and β ,   respectively, means that αn = αn(k) and βn = βn(k) for some given function n(k). We introduce the moving-operator T S , TαS f (t, x) = g(t, x) by g(t, x) = lim

n→+∞

  f δ+ (αn , t), x

and is written only when the limit exists. The mode of convergence, e.g. pointwise, uniform, etc., will be specified at each use of the symbol. In what follows, we will establish some basic properties of S-almost periodic fuzzy vector-valued functions. Theorem 4.3 Let T be a bi-direction S-CCTS with shifts δ± and f ∈ C(T × D, RnF ) be S-almost periodic in t uniformly for x ∈ D, where δ+ (τ, t) is continuous in t. Then it is uniformly continuous and bounded on T∗ × S0 . Proof For any given ε ≤ 1 and some compact set S0 ⊂ D, there exists a constant l(ε, S0 ) such that in any interval of length l(ε, S0 ), there exists τ ∈ E{ε, f , S0 } such that     D∞ f δ+ (τ, t), x , f (t, x) < ε ≤ 1, for all (t, x) ∈ T∗ × S0 .    Since f ∈ C(T × D, RnF ), for any (t, x) ∈ t0 , δ+ (l, t0 ) T × S0 , where t0 ∈ T∗ is the initial point, there exists an M > 0 such that  f (t, x) < M. For any given t ∈ T∗ , choose      τ ∈ E(ε, f , S0 ) ∩ δ− (t, t0 ), δ+ l, δ− (t, t0 ) T , then δ+ (τ, t) ∈ t0 , δ+ (l, t0 ) T . Hence, for x ∈ S0 , we can obtain         f δ+ (τ, t), x  < M and D∞ f δ+ (τ, t), x , f (t, x) < 1. F Thus for all (t, x) ∈ T∗ × S0 , we have f (t, x) F < M + 1.   ε ε Moreover, for any ε > 0, let l1 = l1 , S0 be an inclusion length of E , f , S0 . 3 3 ∗ We can seclect the initial point t0 ∈ T such that f (t, x) is uniformly on  continuous     ε t0 , δ+ (l1 , t0 ) T × S0 . Hence, there exists a positive constant δ ∗ = δ ∗ , S0 , for any 3   t1 , t2 ∈ t0 , δ+ (l1 , t0 ) T and |t1 − t2 | < δ ∗ ,   ε D∞ f (t1 , x), f (t2 , x) < for all x ∈ S0 . 3 Now, we choose an arbitrary υ, t ∈ T∗ , satisfying |t − υ| < δ ∗ , and we take

100

4 Shift Almost Periodic Fuzzy Vector-Valued Functions

    ε τ∈E , f , S0 ∩ δ− (t, t0 ), δ+ l, δ− (t, t0 ) T , 3   then δ+ (τ, t), δ+ (τ, υ) ∈ t0 , δ+ (l, t0 ) T . Since δ+ (τ, t) is continuous in t, then there exists δ ∗∗ > 0 such that |t − v| < δ ∗∗ implies 

|δ+ (τ, t) − δ+ (τ, υ)| < δ ∗ ,   Now, we take δ∗∗ = min δ ∗ , δ ∗∗ , when |t − v| < δ∗∗ implies      ε D∞ f δ+ (τ, t), x , f δ+ (τ, υ), x < for all x ∈ S0 . 3 Therefore, for (t, x) ∈ T∗ × S0 , we have    D∞ ( f (t, x), f (υ, x)) ≤ D∞ f (t, x), f δ+ (τ, t), x      +D∞ f δ+ (τ, t), x , f δ+ (τ, υ), x     +D∞ f δ+ (τ, υ), x , f (υ, x) < ε. 

The proof is completed.

In the following, we will establish a shift-convergence theorem of S-almost periodic fuzzy vector-valued functions. Theorem 4.4 Let f ∈ C(T × D, RnF ) be S-almost periodic in t uniformly for x ∈ D under   shifts δ± . Then for any given sequence α ⊂ Π , there exists a subsequence β ⊂ α and n S ∗ g ∈ C(T × D, RF ) such that Tβ f (t, x) = g(t, x) holds uniformly on T × S0 and g(t, x) is S-almost periodic in t uniformly for x ∈ D under shifts δ± .    ε ε Proof For any ε > 0 and S0 ⊂ D, let l = l , S0 be an inclusion length of E , f , S0 . 4 4     For any given subsequence α = {αn } ⊂ Π , there γn ∈ Π and t0 ≤ γn ≤ l, n =  exists  ε     1, 2, . . ., such that αn = δ+ (τn , γn ), where τn ∈ E , f , S0 . In fact, for any interval with 4   ε  length of l, there exists τn ∈ E , f , S0 , thus, we can choose a proper interval with length    4 of l such that t0 ≤ δ− τn , αn ≤ l, and from the definition of Π , it is easy to observe that       γn = δ− τn , αn ∈ Π . Therefore, there exists a subsequence γ = {γn } ⊂ γ = {γn } such that γn → s as n → ∞, where t0 ≤ s ≤ l. Also, it follows from Theorem 4.3 that f (t, x) is uniformly continuous on T∗ × S0 . Hence, there exists δ ∗ (ε, S0 ) > 0 so that |t1 − t2 | < δ ∗ , for x ∈ S0 , implies 

     ε D∞ f δ+ (τ, t1 ), x , f δ+ (τ, t2 ), x < . 2

4.3

Shift Almost Periodic Fuzzy Vector-Valued Functions

101

Since γ is a convergent sequence, there exists N = N (δ) so that p, m ≥ N implies    |γ p − γm | < δ ∗ . Now, one can take α ⊂ α , τ ⊂ τ = {τn } such that α, τ common with γ , then for any integers p, m ≥ N , we obtain       D∞ f δ+ δ− (τm , τ p ), t , x , f (t, x)            ≤ D∞ f δ− τm , δ+ (τ p , t) , x , f δ+ (τ p , t), x + D∞ f δ+ (τ p , t), x , f (t, x) ε ε ε < + = , 4 4 2     since δ− γn , αn = τn , and we obtain   δ+ δ− (αm , α p ), δ− (γ p , γm )       ε = δ+ δ− (γ p , α p ), δ− (αm , γm ) = δ+ τ p , δ− (τm , t0 ) = δ− (τm , τ p ) ∈ E , f , S0 . 2 Hence, we obtain      D∞ f δ+ (α p , t), x , f δ+ (αm , t), x      D∞ f δ+ (α p , t), x , f δ+ (αm , t), x ≤ sup (t,x)∈T∗ ×S0

≤ ≤

sup

    D∞ f δ+ (δ− (αm , α p ), t), x , f (t, x)

sup

     D∞ f δ+ (δ− (αm , α p ), t), x , f δ+ (δ− (γm , γ p ), t), x

(t,x)∈T∗ ×S0 (t,x)∈T∗ ×S0

+

sup

(t,x)∈T∗ ×S0

    ε ε D∞ f δ+ (δ− (γm , γ p ), t), x , f (t, x) < + = ε. 2 2 (k)

Thus, we can take sequences α (k) = {αn }, k = 1, 2, . . . , and α (k+1) ⊂ α (k) ⊂ α such that for any integers m, p, and all (t, x) ∈ T∗ × S0 , the following holds:      1 (k) D∞ f δ+ (α (k) < , k = 1, 2, . . . . p , t), x , f δ+ (αm , t), x k (n)

For all sequences α (k) , k = 1, 2, . . ., we can take a sequence β = {βn }, βn = αn , and then       it follows that f δ+ (βn , t), x ⊂ f δ+ (αn , t), x for any integers p, m with p < m and all (t, x) ∈ T∗ × S0 , the following holds:      1 D∞ f δ+ (β p , t), x , f δ+ (βm , t), x < . p    Therefore, f δ+ (βn , t), x converges uniformly on T∗ × S0 , i.e., TβS f (t, x) = g(t, x) holds uniformly on T∗ × S0 , where β = {βn } ⊂ α. Next, we show that g(t, x) is continuous on T∗ × D. If this is not true, there will exist (t0 , x0 ) ∈ T∗ × D such that g(t, x) is not continuous at this point. Then there exist ε0 > 0 and

102

4 Shift Almost Periodic Fuzzy Vector-Valued Functions

∗ }, {t }, {x }, where δ ∗ > 0, δ ∗ → 0 as m → +∞, |t − t | + |x − x | < sequences {δm m m 0 m 0 m m m ∗ δm and   D∞ g(t0 , x0 ), g(tm , xm ) ≥ ε0 . (4.22)

Let X = {xm } {x0 }, and obviously, X is a compact subset of D. Hence, there exists positive integer N = N (ε0 , X ) such that n > N implies

    ε0 D∞ f δ+ (βn , tm ), xm , g(tm , xm ) < for all m ∈ Z+ 3

(4.23)

and

    ε0 D∞ f δ+ (βn , t0 ), x0 , g(t0 , x0 ) < . (4.24) 3 According to the uniform continuity of f (t, x) on T∗ × D, for sufficiently large m, we have

     ε0 D∞ f δ+ (βn , t0 ), x0 , f δ+ (βn , tm ), xm < . (4.25) 3   From (4.23)–(4.25), we get D∞ g(t0 , x0 ), g(tm , xm ) < ε0 , which contradicts (4.22). Therefore, g(t, x) is continuous on T∗ × D. Finally, for any compact set S0 ⊂ D and given ε > 0, one can select τ ∈ E{ε, f , S0 }, and then for all (t, x) ∈ T∗ × S0 , we have      D∞ f δ+ (βn , δ+ (τ, t)), x , f δ+ (βn , t), x < ε. Let n → +∞, for all (t, x) ∈ T∗ × S0 , and we have     D∞ g δ+ (τ, t), x , g(t, x) ≤ ε, which implies that E{ε, g, S0 } is relatively dense. Therefore, g(t, x) is S-almost periodic in t uniformly for x ∈ D under shifts δ± . This completes the proof.  Next, we will give a sequentially compact criterion of S-almost periodic fuzzy vectorvalued functions through the shift operator T S . 



Theorem 4.5 Let f ∈ C(T × D, RnF ). If for any sequence α ⊂ Π , there exists α ⊂ α such that TαS f (t, x) exists uniformly on T∗ × S0 , then f (t, x) is S-almost periodic in t uniformly for x ∈ D under shifts δ± . Proof (By contradiction). If this is not true, then there exist ε0 > 0 and S0 ⊂ D such that for any sufficiently large l > 0, there is an interval with length of l such that there is no ε0 -shift numbers of f (t, x) in this interval, that is, every point in this interval is not in the set E{ε0 , f , S0 }.   One can select a number α1 ∈ Π to obtain an interval (a1 , b1 ) with b1 − a1 > 2|α1 |, where a1 , b1 ∈ Π such that there is no ε0 -shift numbers of f (t, x) in this interval. Next,

4.3

Shift Almost Periodic Fuzzy Vector-Valued Functions

103

           / taking α2 ∈ δ+ (α1 , a1 ), δ+ (α1 , b1 ) , obviously, δ− α1 , α2 ∈ (a1 , b1 ), so δ− α1 , α2 ∈   E{ε0 , f , S0 }; then one can obtain an interval (a2 , b2 ) with b2 − a2 > 2(|α1 | + |α2 |), where a2 , b2 ∈ Π such that there is no ε0 -shift numbers of f (t, x) in this interval.           Now, selecting α3 ∈ δ+ (α2 , a2 ), δ+ (α2 , b2 ) , α3 ∈ δ+ (α1 , a2 ), δ+ (α1 , b2 ) , obviously,       / E{ε0 , f , S0 }. One can repeat this process again to get the numδ− α2 , α3 , δ− α1 , α3 ∈      / E{ε0 , f , S0 }, i > j. Hence, for any i = j, i, j = bers α4 , α5 , . . . , such that δ− α j , αi ∈ 1, 2, . . . , without loss of generality, let i > j, for x ∈ S0 we obtain sup

       D∞ f δ+ (αi , t), x , f δ+ (α j , t), x

sup

      D∞ f δ+ (δ− (α j , αi ), t), x , f (t, x) ≥ ε0 .

(t,x)∈T∗ ×S0

=

(t,x)∈T∗ ×S0

    Therefore, there is no uniformly convergent subsequence of f δ+ (αn , t), x for (t, x) ∈ T∗ × S0 , and this is a contradiction. Thus, f (t, x) is S-almost periodic in t uniformly for x ∈ D under shifts δ± . This completes the proof.  From Theorems 4.4 and 4.5, we obtain the following equivalent definition of uniformly S-almost periodic fuzzy vector-valued functions. 

Definition 4.10 Let f ∈ C(T × D, RnF ). If for any given sequence α ⊂ Π , there exists a  subsequence α ⊂ α such that TαS f (t, x) exists uniformly on T∗ × S, then f (t, x) is called a S-almost periodic fuzzy vector-valued function in t uniformly for x ∈ D under shifts δ± . Theorem 4.6 If f ∈ C(T × D, RnF ) is S-almost periodic in t uniformly for x ∈ D and   ϕ(t) is S-almost periodic with {ϕ(t) : t ∈ T} ⊂ S0 , then f t, ϕ(t) is S-almost periodic. 



Proof For any given sequence α ⊂ Π , there exists α ⊂ α , ψ(t), g(t, x) such that TαS ϕ(t) = ψ(t) exists uniformly on T∗ and TαS f (t, x) = g(t, x) exists uniformly on T∗ × S0 , where ψ(t) is S-almost periodic and g(t, x) is S-almost periodic in t uniformly for x ∈ D. There∗ 0, there exists fore,  g(t,  x) is uniformly continuous on T × S0 , then for any given ε>  ε ε δ∗ implies > 0 such that for any x1 , x2 ∈ S0 and all t ∈ T∗ , |x1 − x2 | < δ ∗ 2 2   ε D∞ g(t, x1 ), g(t, x2 ) < . 2 For a sufficiently large N0 (ε) > 0 such that n ≥ N0 (ε), we obtain     ε D∞ f δ+ (αn , t), x , g(t, x) < , ∀(t, x) ∈ T∗ × S0 , 2

104

4 Shift Almost Periodic Fuzzy Vector-Valued Functions

      ∗ ε , ∀t ∈ T∗ , D∞ ϕ δ+ (αn , t) , ψ(t) < δ 2     where ϕ δ+ (αn , t) : t ∈ T∗ ⊂ S0 , {ψ(t) : t ∈ T∗ } ⊂ S0 . Therefore, when n ≥ N0 (ε), we have      D∞ TαS f t, ϕ(t) , g t, ψ(t)       = D∞ f δ+ (αn , t), ϕ δ+ (αn , t) , g t, ψ(t)       ≤ D∞ f δ+ (αn , t), ϕ δ+ (αn , t) , g t, ϕ(δ+ (αn , t))       +D∞ g t, ϕ δ+ (αn , t) , g t, ψ(t) < ε. Note that

      lim f δ+ (αn , t), ϕ δ+ (αn , t) = TαS f t, ϕ(t) ,       and thus, TαS f t, ϕ(t) = g t, ψ(t) exists uniformly on T∗ × S0 . Thus, f t, ϕ(t) is S almost periodic under shifts δ± . The proof is completed. n→∞

Definition 4.11 Let f ∈ C(T × D, RnF ). Then HS ( f ) = {g(t, x) : T × D → RnF | there exits α ∈ Π such that TαS f (t, x) = g(t, x) exists uniformly on T∗ × S0 } is called the S-hull of f (t, x) under shifts δ± . Theorem 4.7 HS ( f ) is compact if and only if f (t, x) is S-almost periodic in t uniformly for x ∈ D. 



Proof If HS ( f ) is compact, for any given α = {αn } ⊆ Π , there exists subsequence        f δ+ (αn k , t), x k∈N of the sequence f δ+ (αn , t), x n∈N such that   f δ+ (αn k , t), x → g(t, x) (k → ∞), ∀(t, x) ∈ T∗ × S0 . 

Note that α = {αn k }, obviously, α ⊂ α , so TαS f (t, x) exists uniformly on T∗ × S0 . Conversely, if f (t, x) is S-almost periodic in t uniformly for x ∈ D and {gn (t, x)} ⊂   HS ( f ), then we can choose α = {αn } such that     1  D∞ f δ+ (αn , t), x , gn (t, x) < , ∀(t, x) ∈ T∗ × S0 . n 

One can choose α ⊂ α so that TαS f (t, x) exists uniformly. Let β ⊂ γ = {n} such that β and α are common subsequences, then     D∞ f δ+ (αn , t), x , gβn (t, x) → 0 (n → ∞), ∀(t, x) ∈ T∗ × S0 , so that

lim gβn (t, x) = TαS f (t, x) ∈ HS ( f ), ∀(t, x) ∈ T∗ × S0 .

n→∞

4.3

Shift Almost Periodic Fuzzy Vector-Valued Functions

105

Thus, HS ( f ) is compact. The proof is completed.



Theorem 4.8 If f ∈ C(T × D, RnF ) is S-almost periodic in t uniformly for x ∈ D under shifts δ± , then for any g(t, x) ∈ HS ( f ), HS ( f ) = HS (g). 

Proof For any h(t, x) ∈ HS (g) there exists α ⊆ Π such that T S g(t, x) = h(t, x). Since α



f (t, x) is S-almost periodic in t uniformly for x ∈ D, from the sequence {αn } ⊆ Π , one   can extract a sequence {αn } such that TαS f (t, x) = lim f δ+ (αn , t), x exists uniformly n→∞

on T∗ × S0 . For g(t, x) ∈ HS ( f ), there exists α (1) ⊆ Π such that lim

n→+∞

  f δ+ (αn(1) , t), x = g(t, x), ∀(t, x) ∈ T∗ × S0 ,

so we have lim

n→+∞

    f δ+ (δ+ (αn(1) , αn ), t), x = g δ+ (αn , t), x , ∀(t, x) ∈ T∗ × S0 , (1)

and then we can take β = {βn } = {δ+ (αn , αn )} such that      1 D∞ f δ+ (βn , t), x , g δ+ (αn , t), x < , ∀(t, x) ∈ T∗ × S0 . n It follows that TβS f (t, x) = TαS g(t, x) = T S g(t, x) = h(t, x). Hence h(t, x) ∈ HS ( f ). α Thus, HS (g) ⊆ HS ( f ). On the other hand, for any g(t, x) ∈ HS ( f ), there exists α such that TαS f (t, x) = g(t, x), then     D∞ f δ+ (αn , t), x , g(t, x) → 0, n → ∞, ∀(t, x) ∈ T∗ × S0 , so making the change of variable δ+ (αn , t) = s, one has    D∞ f (s, x), g δ− (αn , s), x → 0, ∀(s, x) ∈ T∗ × S0 , that is

  lim g δ− (αn , t), x =: TαS−1 g(t, x) = f (t, x).

n→∞

Thus, f (t, x) ∈ HS (g), then by what was shown above, HS ( f ) ⊆ HS (g). According to the above, it follows that HS ( f ) = HS (g). The proof is completed.



From Definition 4.11 and Theorem 4.8, one can directly obtain the following theorem. Theorem 4.9 If f ∈ C(T × D, RnF ) is S-almost periodic in t uniformly for x ∈ D under shifts δ± , then for any g(t, x) ∈ HS ( f ), g(t, x) is S-almost periodic in t uniformly for x ∈ D under shifts δ± .

106

4 Shift Almost Periodic Fuzzy Vector-Valued Functions

Now, we establish the following theorems to guarantee four fundamental operations among S-almost periodic fuzzy vector-valued functions. Theorem 4.10 Let f ∈ C(T × D, RnF ) be S-almost periodic in t uniformly for x ∈ D under shifts δ± . Then for any ε > 0, there exists a positive constant L = L(ε, S), for any a ∈ Π , there exist a constant η > 0 and α ∈ Π such that [α, δ+ (α, η)]T∗ ⊂ [a, δ+ (a, L)]T∗ and [α, δ+ (α, η)]T∗ ⊂ E(ε, f , S). Proof Since f (t, x) is uniformly continuous on T∗ × S0 , given any ε > 0, there exists δ ∗ (ε1 , S0 ) > 0 so that |t1 − t2 | < δ ∗ (ε1 , S0 ) implies   D∞ f (t1 , x), f (t2 , x) < ε1 , ∀x ∈ S0 , where ε1 =

ε . 2

   ε , S0 = δ ∗ (ε1 , S0 ), and L = δ+ l(ε1 , S0 ), η , where l(ε1 , S0 ) is the 2 inclusion length of E(ε1 , f , S0 ).   For any a ∈ Π , consider an interval a, δ+ (a, L) T∗ , and take We take η = δ ∗



    τ ∈ E( f , ε1 , S) ∩ δ+ (η∗ , a), δ− η∗ , δ+ a, l(ε1 , S0 ) T∗ ,

where η∗ > t0 (t0 is the initial point) satisfying δ+ (η∗ , η∗ ) ≤ δ− (η∗ , t0 ) ≤ η∗ < δ+ (η∗ , η∗ ) ≤ η2 ), so we obtain

η 2

(in fact, we also have η∗ −

      δ+ τ, δ− (η∗ , t0 ) , δ+ (τ, η∗ ) T∗ ⊂ a, δ+ (a, L) T∗ .

    Hence, for all ξ ∈ δ+ τ, δ− (η∗ , t0 ) , δ+ (τ, η∗ ) T∗ , we can obtain

  |ξ − τ | ≤ |δ+ τ, δ− (η∗ , t0 ) − δ+ (τ, η∗ )| ≤ 2δ+ (η∗ , η∗ ) ≤ η.

Therefore, for any (t, x) ∈ T∗ × S0 ,          D∞ f δ+ (ξ, t), x , f (t, x) ≤ D∞ f δ+ (ξ, t), x , f δ+ (τ, t), x     +D∞ f δ+ (τ, t), x , f (t, x) ≤ ε.   We let α = δ− (η∗ , τ ), then α, δ+ (α, η) T∗ ⊂ E(ε, f , S0 ). This completes the proof.



+ In the following theorem, for η ∈ T∗ , we will use notations δη+ = δ+ (η, t0 ) = η, δ2η = + + + + + δ+ (η, δη ), δ3η = δ+ (η, δ2η ), . . . , δmη := δ+ (η, δ(m−1)η ) for simplicity.

Theorem 4.11 If f , g ∈ C(T × D, RnF ) are S-almost periodic in t uniformly for x ∈ D under shifts δ± , then for any ε > 0, E( f , ε, S0 ) ∩ E(g, ε, S0 ) is a nonempty relatively dense set in Π .

4.3

Shift Almost Periodic Fuzzy Vector-Valued Functions

107

Proof Since f , g are S-almost periodic in t uniformly for x ∈ D under shifts δ± , they  are ε ∗ ∗ ∗ uniformly continuous on T × S0 . For any given ε > 0, one can take δi = δi , S0 (i = 2     ε ε ε 1, 2); and l1 = l1 , S0 , l2 = l2 , S0 are inclusion lengths of E( f , , 2 2 2 ε S0 ), E(g, , S0 ), respectively. 2 According to Theorem 4.7, we choose η = η(ε, S) = min(δ1∗ , δ2∗ ) ∈ Π , L i = δ+ (li , η) (i = 1, 2), L = max(L 1 , L 2 ). ε + and τ = δ + , respecHence, one can select -shift numbers of f (t, x) and g(t, x): τ1 = δmη 2 nη 2  tively, where τ1 , τ2 ∈ a, δ+ (a, L) T∗ , m, n are integers. Since     δ− (τ2 , t0 ) ∈ δ− a, δ− (L, t0 ) , δ− (a, t0 ) T∗ ,

then it follows that        δ+ δ− a, δ− (L, t0 ) , a ≤ δ+ τ1 , δ− a, δ− (L, t0 )

  ≤ δ− (τ2 , τ1 ) ≤ δ− (a, τ1 ) ≤ δ− a, δ+ (a, L) ,

  thus, A δ− (τ2 , τ1 ) ≤ L. Let m − n = s, then s can only be selected from a finite number ε set {s1 , s2 , . . . , s p }. When m − n = s j , j = 1, 2, . . . , p, then the -shift numbers of f (t) 2  j j j j and g(t) can be denoted by τ1 , τ2 , respectively, i.e., δ− τ2 , τ1 = δs+j η , j = 1, 2, . . . , p, j

j

and we choose T = max{A(τ1 ), A(τ2 )}. j    ε For any a ∈ Π , on the interval δ+ (a, T ), δ+ δ+ (a, T ), L T∗ , we can take -shift num2 bers of f (t, x) and g(t, x) as τ1 and τ2 , respectively, and there exists some integer s j such that  j j δ− (τ2 , τ1 ) = δs+j η = δ− τ2 , τ1 .  j   j      Let τ (ε, S0 ) = δ− τ1 , τ1 = δ− τ2 , τ2 , then τ (ε, S0 ) ∈ a, δ+ a, δ+ L, δ+ (T , T ) T∗ , and for any (t, x) ∈ T∗ × S0 , we have          j j D∞ f δ+ (τ, t), x , f (t, x) ≤ D∞ f δ+ (δ− (τ1 , τ1 ), t), x , f δ− (τ1 , t), x     j +D∞ f δ− (τ1 , t), x , f (t, x) < ε and          j j D∞ g δ+ (τ, t), x , g(t, x) ≤ D∞ g δ+ (δ− (τ2 , τ2 ), t), x , g δ− (τ2 , t), x     j +D∞ g δ− (τ2 , t), x , g(t, x) < ε. Therefore, there exists at least a τ = τ (ε, S) on any interval

108

4 Shift Almost Periodic Fuzzy Vector-Valued Functions

    a, δ+ a, δ+ L, δ+ (T , T ) T∗

  with the length δ+ L, δ+ (T , T ) such that τ ∈ E( f , ε, S0 ) ∩ E(g, ε, S0 ). The proof is completed.  From Definition 4.9, one can easily prove the following theorem. Theorem 4.12 If f ∈ C(T × D, RnF ) is S-almost periodic in t uniformly for x ∈ D under   shifts δ± , then for any α ∈ R, b ∈ Π , the functions α · f (t, x), f δ± (b, t), x are S-almost periodic in t uniformly for x ∈ D under shifts δ± . Theorem 4.13 Let f , g ∈ C(T × D, RnF ) be S-almost periodic in t uniformly for x ∈ D  g H(I ) v exists and one of the following conditions is satisfied, under shifts δ± . If u − α,(I )

α,(I )

α,(I )

α,(I )

(i) If I fi ,ωi , Igi ,ωi > 0 or I fi ,ωi , Igi ,ωi < 0, α,(I I )

α,(I I )

α,(I I )

α,(I I )

(ii) If I fi ,ωi , Igi ,ωi > 0 or I fi ,ωi , Igi ,ωi < 0, then f ∗ g and f  g are S-almost periodic in t uniformly for x ∈ D. ε ε Proof From Theorem 4.11, for any ε > 0, E( f , , S0 ) ∩ E(g, , S0 ) is a nonempty rel2 2 atively dense set. Without loss of generality, we assume that condition (i) is satisfied (the proof of the other case when condition (ii) is satisfied is similar). According to Theorem 2.6 (ii), we have       g δ+ (τ, t), x ∗ f δ+ (τ, t), x + g δ+ (τ, t), x ∗ f (t, x)       = g δ+ (τ, t), x ∗ f δ+ (τ, t), x + f (t, x) ,   g(δ+ (τ, t), x) ∗ f (t, x) + g(t, x) ∗ f (t, x) = g(δ+ (τ, t), x) + g(t, x) ∗ f (t, x). Denote

sup

(t,x)∈T∗ ×S0

 f (t, x)F = M1 ,

sup

(t,x)∈T∗ ×S0

g(t, x)F = M2 , take

ε ε τ ∈ E( f , , S0 ) ∩ E(g, , S0 ), 2 2 then for all (t, x) ∈ T∗ × S0 we have       D∞ f δ+ (τ, t), x ∗ g δ+ (τ, t), x , f (t, x) ∗ g(t, x)        ≤ g δ+ (τ, t), x F D∞ f δ+ (τ, t), x , f (t, x)   + f (t, x)F D∞ g(δ+ (τ, t), x), g(t, x) ≤ (M1 + M2 )ε ≡ ε1 .

4.3

Shift Almost Periodic Fuzzy Vector-Valued Functions

109

Therefore, τ ∈ E( f ∗ g, ε1 , S0 ) and E( f ∗ g, ε1 , S0 ) is a relatively dense set, so f ∗ g is S-almost periodic in t uniformly for x ∈ D. This completes the proof.  Theorem 4.14 If f , g ∈ C(T × D, RnF ) are S-almost periodic in t uniformly for x ∈ D  g are S-almost periodic in t uniformly for x ∈ D. under shifts δ± , then f +   ε  ε f , , S0 ∩ E g, , S0 is a nonempty 2 2  ε   ε relatively dense set. It is easy to observe that if τ ∈ E f , , S0 ∩ E g, , S0 , then 2 2    g, ε, S0 . Hence τ ∈ E f+ Proof From Theorem 4.11, for any ε > 0, E





 ε ε  g, ε, S0 ). E( f , , S0 ) ∩ E(g, , S0 ) ⊂ E( f + 2 2

 g, ε, S0 ) is a relatively dense set, so f +  g is S-almost periodic in t uniTherefore, E( f +  formly for x ∈ D. The proof is completed. In what follows, we will establish a convergence theorem of S-almost periodic function sequences. Theorem 4.15 If f n ∈ C(T × D, RnF ), n = 1, 2, . . . are S-almost periodic in t for x ∈ D, and the sequence { f n (t, x)} uniformly converges to f (t, x) on T∗ × S0 , then f (t, x) is S-almost periodic in t uniformly for x ∈ D. Proof For any ε > 0, there exists sufficiently large n˜ 0 such that for all (t, x) ∈ T∗ × S0 , D∞ ( f (t, x), f n˜ 0 (t, x))
0, there exists a constant l(ε) > 0 such that each interval of length l(ε) contains a τ ∈ E{ε, f } such that   D∞ f (δ+ (τ, t)), f (t) < ε. From (4.26), one obtains   ( j ◦ f )(δ+ (τ, t)) − ( j ◦ f )(t) n ×

¯

¯

i=1 (C×C)

< ε,

which implies that j ◦ f is S-almost periodic. If j ◦ f is S-almost periodic, similarly, according to (4.26), one can obtain f is S-almost periodic immediately. Now we show (iii). If f is gH-Δ-differentiable, then there exists a neighborhood U of t such that

4.3

Shift Almost Periodic Fuzzy Vector-Valued Functions

111

  ( j ◦ f )(σ (t)) − ( j ◦ f )(t + h) − ( j ◦ f Δ )(t)(μ(t) − h) n ¯ ¯ ×i=1 (C×C)      Δ     = j ◦ f (σ (t))−g H f (t + h) −g H f (t) μ(t) − h ×n (C× ¯ C) ¯ i=1    g H f (t + h), f Δ (t)(μ(t) − h) < ε|μ(t) − h| ≤ D∞ f (σ (t))− for all t + h ∈ U with |h| < δ. The proof is completed.



Theorem 4.18 If f ∈ C(T × D, RnF ) is shift-Δ-almost periodic in t uniformly for x ∈ D under shifts δ± , denote  t

F(t, x) =

f (s, x)Δs, t0 ∈ T∗ ,

t0

then F(t, x) is S-almost periodic in t uniformly for x ∈ D under shifts δ± if and only if F(t, x) is bounded on T∗ × S0 , where S0 is any compact subset of D. Proof For any α ∈ [0, 1], we have 

F(t, x)

α

 −  n Fi (t, x), Fi+ (t, x) α = = ×i=1 

n = ×i=1

t t0

f i− (s, x)Δs,



t

t0



t

f (s, x)Δs

t0

f i+ (s, x)Δs

α

t



α f (s, x) Δs

t0

 α

 =

.

According to Definition 4.9 (iii), for i = 1, 2, . . . , n, we obtain     (i) Δ Δ  g H f i (t, x) < ε, f i (δ± (τ, t), x)δ± D∞ (τ, t), f i (t, x) =  f i (δ± (τ, t), x)δ± (τ, t)− F 0

which implies that for each i ∈ {1, 2, . . . , n}, we have     Δ Δ max  f i− (δ± (τ, t), x)δ± (τ, t) − f i− (t, x),  f i+ (δ± (τ, t), x)δ± (τ, t) − f i+ (t, x) < ε. Hence, f i+ , f i− ∈ C(T × D, R) is shift-Δ-almost periodic in t uniformly for x ∈ D under t shifts δ± . Without loss of generality, we will prove that Fi− (t, x) = t0 f i− (s, x)Δs is Salmost periodic in t uniformly for x ∈ D under shifts δ± if and only if Fi− (t, x) is bounded t on T∗ × S0 (the proof of the case Fi+ (t, x) = t0 f i+ (s, x)Δs is similar). First, if Fi− (t, x) is S-almost periodic in t uniformly for x ∈ D, it is easy to see that − Fi (t, x) is bounded on T∗ × S0 . Suppose Fi− (t, x) is bounded. Denote G − :=

sup

(t,x)∈T∗ ×S0

Fi− (t, x) > g − :=

inf

(t,x)∈T∗ ×S0

Fi− (t, x),

for any ε > 0, there exist t1 and t2 such that ε Fi− (t1 , x) < g − + , 6

ε Fi− (t2 , x) > G − − , ∀x ∈ S0 . 6

112

4 Shift Almost Periodic Fuzzy Vector-Valued Functions

Let l = l(ε1 , S0 ) be an inclusion length of E( f , ε1 , S0 ), where ε1 = any α ∈ Π , take τ ∈ E( f , ε1 , S0 ) such that

ε , d = |t1 − t2 |. For 6d

   δ+ (τ, t1 ) ∈ δ+ (α, t0 ), δ+ δ+ (α, l), t0 T . ˜ where d˜ ∈ Π and d˜ > d, so s1 , s2 ∈ Denote si = δ+ (τ, ti ), (i = 1, 2), L = δ+ (l, d),    δ+ (α, t0 ), δ+ δ+ (α, L), t0 T , for all x ∈ S0 , Fi− (s2 , x) − Fi− (s1 , x) = Fi− (t2 , x) − Fi− (t1 , x) − = Fi− (t2 , x) − Fi− (t1 , x) +



t2



t1 t2

f i− (t, x)Δt + 

t1



δ+ (τ,t2 ) δ+ (τ,t1 )

f i− (t, x)Δt

  Δ  f i− δ+ (τ, t), x δ+ (τ, t) − f i− (t, x) Δt

ε ε > G − g − − ε1 d = G − − g − − , 3 2 −

−

that is



Since

   ε Fi− (s1 , x) − g − + G − − Fi− (s2 , x) < . 2

Fi− (s1 , x) − g − ≥ 0,

G − − Fi− (s2 , x) ≥ 0,

in any interval with length L, there exist s1 , s2 such that ε Fi− (s1 , x) < g − + , 2

ε Fi− (s2 , x) > G − − . 2

Next, note that L = t0 , we have inf |δ L (t) − t| > q > 0 for all t ∈ T∗ , where q is ε , and we will show that if τ ∈ E( f , ε2 , S), then some positive constant. Let ε2 = 2q τ ∈ E(F, ε, S). In fact, for any (t, x) ∈ T∗ × S, we can choose s1 , s2 ∈ [t, δ+ (L, t)]T such that ε ε Fi− (s1 , x) < g − + , Fi− (s2 , x) > G − − . 2 2 Thus, for τ ∈ E( f , ε2 , S), one obtains   Fi− δ+ (τ, t), x − Fi− (t, x)

= Fi− (δ+ (τ, s1 ), x) − Fi− (s1 , x)  s1  δ+ (τ,s1 ) − + f i (t, x)Δt − f i− (t, x)Δt t

ε > g − − (g − + ) − 2 ε > − − ε2 q = −ε 2

δ+ (τ,t)



s1 t



  Δ  f i− δ+ (τ, t), x δ+ (τ, t) − f i− (t, x) Δt

4.3

Shift Almost Periodic Fuzzy Vector-Valued Functions

113

since |t − s1 | = |δ+ (τ, t1 ) − t| ≤ |δ+ (L, t) − t|. Moreover, we also have Fi− (δ+ (τ, t), x) − Fi− (t, x) = Fi− (δ+ (τ, s2 ), x) − Fi− (s2 , x) +  −

δ+ (τ,s2 )

δ+ (τ,t)

 t

s2

f i− (t, x)Δt

  ε f i− (t, x)Δt < G − − G − − + ε2 q = ε. 2

Similar to the proof above, we can also obtain Fi+ is S-almost periodic in t uniformly for x ∈ D under shifts δ± if and only if Fi+ (t, x) is bounded on T∗ × S0 . Therefore, for τ ∈ E( f , ε2 , S0 ), we have τ ∈ E(F, ε, S0 ), thus F(t, x) is S-almost periodic in t uniformly for x ∈ D. The proof is completed.  Finally, we will establish a sufficient and necessary criterion for S-almost periodic functions. Theorem 4.19 A function f ∈ C(T × D, RnF ) is S-almost periodic in t uniformly for x ∈   D under shifts δ± if and only if for every pair of sequences α , β ⊆ Π , there exist common   subsequences α ⊂ α , β ⊂ β such that TδS+ (α,β) f (t, x) = TαS TβS f (t, x).

(4.27)

Proof If f (t, x) is S–almost periodic in t uniformly for x ∈ D, for any two sequences     α , β ⊆ Π , there exists subsequence β ⊂ β such that T S f (t, x) = g(t, x) holds uniβ formly on T∗ × S0 and g(t, x) is S-almost periodic in t uniformly for x ∈ D.       Choose α ⊂ α and α , β are the common subsequences of α , β , respectively, then   there exists α ⊂ α such that T S g(t, x) = h(t, x) holds uniformly on T∗ × S0 . 



α









Similarly, take β ⊂ β such that β , α are the common subsequences of β , α ,   respectively, then there exist common subsequence α ⊂ α , β ⊂ β such that TδS+ (α,β) f (t, x) = k(t, x) holds uniformly on T∗ × S0 . From the above, one can easily observe that TβS f (t, x) = g(t, x), TαS g(t, x) = h(t, x) hold uniformly on T∗ × S0 . Thus, for all ε > 0, if n is sufficiently large, then for any (t, x) ∈ T∗ × S0 , we have     ε D∞ f δ+ (δ+ (αn , βn ), t), x , k(t, x) < , 3      ε  ε D∞ g(t, x), f δ+ (βn , t), x < , D∞ h(t, x), g δαn (t), x < . 3 3 Therefore

114

4 Shift Almost Periodic Fuzzy Vector-Valued Functions

   D∞ (h(t, x), k(t, x)) ≤ D∞ h(t, x), g δ+ (αn , t), x      +D∞ g δ+ (αn , t), x , f δ+ (δ+ (αn , βn ), t), x     +D∞ f δ+ (δ+ (αn , βn ), t), x , k(t, x) < ε holds for all (t, x) ∈ T∗ × S0 . Since ε > 0 is arbitrary, we have h(t, x) ≡ k(t, x), that is, TδS+ (α,β) f (t, x) = TαS TβS f (t, x) holds uniformly on T∗ × S0 . 

On the other hand, if (4.27) holds, then for any sequence γ ⊆ Π , there exists subsequence  γ ⊂ γ , such that TγS f (t, x) exists uniformly on T∗ × S0 . In the following, we will show that f (t, x) is S-almost periodic in t uniformly for x ∈ D. If this is not true, i.e., TγS f (t, x) does not converge uniformly on T∗ × S0 , then there exist        ε0 > 0 and t0 ∈ T, subsequences α ⊂ γ , β ⊂ γ , s ⊆ Π and α = {αn }, β = {βn }, s =  {sn } such that          D∞ f δ+ (δ+ (sn , αn ), t0 ), x , f δ+ (δ+ (sn , βn ), t0 ), x ≥ ε0 > 0. 





(4.28)



According to (4.27), there exist common subsequences α ⊂ α , s ⊂ s such that for all (t, x) ∈ T∗ × S0 , we have TδS (s  ,α  ) f (t, x) = TsS TαS f (t, x).

(4.29)

+

















Selecting β ⊂ β such that β , α , s are common subsequences of β , α , s , respectively, such that for all (t, x) ∈ T∗ × S0 , one has TδS+ (s,β) f (t, x) = TsS TβS f (t, x). 

(4.30) 





Similarly, taking α ⊂ α satisfying α, β, s are common subsequences of α , β , s , respectively, according to (4.29), for all (t, x) ∈ T∗ × S0 , we get TδS+ (s,α) f (t, x) = TsS TαS f (t, x).

(4.31)

Since TαS f (t, x) = TβS f (t, x) = TγS f (t, x), from (4.30) and (4.31), for all (t, x) ∈ T∗ × S0 , we obtain TδS+ (s,β) f (t, x) = TδS+ (s,α) f (t, x), that is, for all (t, x) ∈ T∗ × S0 , lim

n→+∞

  f δ+ (δ+ (sn , βn ), t), x = lim

n→+∞

  f δ+ (δ+ (sn , αn ), t), x .

Letting t = t0 , this contradicts (4.28). Therefore, f (t, x) is S-almost periodic in t uniformly  for x ∈ D under shifts δ± . The proof is completed.

5

Division of Fuzzy Vector-Valued Functions Depending on Determinant Algorithm

5.1

Basic Results of Fuzzy Multidimensional Spaces

This section is mainly concerned with an embedding theorem in a fuzzy multidimensional space. Now we introduce an equivalent norm  · L for any     n ¯ 1] × C[0, ¯ 1] ( f 1 , g1 ), ( f 2 , g2 ), . . . , ( f n , gn ) ∈ ×i=1 C[0, as follows:    ( f 1 , g1 ), ( f 2 , g2 ), . . . , ( f n , gn ) 

L

=

 n  i=1

0

1

f i2 (x) + gi2 (x)

1 2 dx .

In fact, there exists two positive constants c1 , c2 such that c1  · L ≤  · ×n (C× ¯ C) ¯ ≤ i=1 c 2  · L .  n    ¯ 1] × C[0, ¯ 1] , ·, · is an inner product space C[0, Theorem 5.1 The space ×i=1 equipped with the inner product x, y =

n   i=1

0

1

 (1) (1) (2) (2) xi (tˆ)yi (tˆ) + xi (tˆ)yi (tˆ) dtˆ,

where

 (1) (2)  (1) (2) x = (x1 , x1 ), (x2 , x2 ), . . . , (xn(1) , xn(2) )

and

 (1) (2)  (1) (2) y = (y1 , y1 ), (y2 , y2 ), . . . , (yn(1) , yn(2) ) .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 C. Wang and R. P. Agarwal, Dynamic Equations and Almost Periodic Fuzzy Functions on Time Scales, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-11236-2_5

115

116

5 Division of Fuzzy Vector-Valued Functions Depending …

  n ¯ 1] × C[0, ¯ 1] and α ∈ R, we can obtain (i). x, x ≥ 0 C[0, Proof For any x, y, z ∈ ×i=1 and x, x = 0 ⇔ x = 0; (ii). x, y = y, x; (iii). αx, y = αx, y for α ∈ R; (iv). we can also obtain

= = =

x + y, z n  1   i=1 0 n  1  i=1 0 n  1  i=1

+



0

 (1)  (2)  (2) (1) (1) (2) xi (tˆ) + yi (tˆ) z i (tˆ) + xi (tˆ) + yi (tˆ) z i (tˆ)

0

dtˆ

(1) (1) (1) (1) (2) (2) (2) (2) xi (tˆ)z i (tˆ) + yi (tˆ)z i (tˆ) + xi (tˆ)z i (tˆ) + yi (tˆ)z i (tˆ) dtˆ

 (1) (1) (2) (2) xi (tˆ)z i (tˆ) + xi (tˆ)z i (tˆ) dtˆ

n  1   i=1



 (1) (1) (2) (2) yi (tˆ)z i (tˆ) + yi (tˆ)z i (tˆ) dtˆ

= x, z + y, z.  n    ¯ 1] × C[0, ¯ 1] , ·, · is an inner product space. The proof is comC[0, Therefore, ×i=1 pleted.   n    ¯ 1] × C[0, ¯ 1] , ·, · , the C[0, Remark 5.1 Notice that in the inner product space ×i=1 inner product will generate the norm  · L , i.e.,  x, x = xL =

 n  i=1

which is equivalent to the norm  · ×n

1

(1) (xi (tˆ))2

0 ¯

¯

i=1 (C×C)

1 2

.

Remark 5.2 From Theorem 2.4, we obtain that Hilbert space.

5.2

 (2) + (xi (tˆ))2 d(tˆ)



   n ¯ 1] × C[0, ¯ 1] , ·, · is a ×i=1 C[0,

A New Division of Multidimensional Intervals and Fuzzy Vectors

In this subsection, using the algorithm determinant of the multiplication of Sect. 2.2, we introduce the corresponding division of multidimensional intervals and fuzzy vectors. / H . The “inverse” of H (it is not Definition 5.1 Let an interval be H = [h − , h + ] and 0 ∈ −1 the inverse in the algebraic sense) is defined by H = [1/h + , 1/h −1 ].

5.2

A New Division of Multidimensional Intervals and Fuzzy Vectors

117

Remark 5.3 If 0 ∈ (h − , h + ), the inverse of H is nonexistent. However, for intervals H = [0, h + ] or H = [h − , 0], the general inverse of H can be defined by letting H = [ε, H + ] or H = [h − , −ε] and we obtain the result by the limit as ε → 0+ . For example, consider H = [0, 3], and we have H −1 = [1/3, limε→0+ 1/ε] = [1/3, +∞). Similarly, for H = [−3, 0], we have H −1 = [limε→0+ 1/ − ε, −1/3] = (−∞, −1/3]. Notice that the concept of inverse and general inverse of an interval H was introduced in the literature [75] by Stefanini and it will be useful to establish the transformational relation between the multiplication and the division of fuzzy vectors under a determinant algorithm. For any [a − , a + ] ⊆ [u − , u + ] and [b− , b+ ] ⊆ [v − , v + ] and 0 ∈ / [v − , v + ], we introduce the following division by a determinant algorithm: 

Type I. [a − , a + ] ÷ I [b− , b+ ] = a  b : a ∈ [a − , a + ], b ∈ 1/b+ , 1/b− ,

(5.1)

(I )

where if Iu,v −1 ≤ 0, then

ab =

(I )

if Iu,v −1 ≥ 0, then ab =

⎧ ⎪ ⎪ ⎨ab,

ab ∈ [u − /v − , u + /v + ],

⎧ ⎪ ⎪ ⎨ab,

ab ∈ [u + /v + , u − /v − ],

u − /v − ,

ab < u − /v − , ⎪ ⎪ ⎩u + /v + , ab > u + /v + ;

u + /v + , ab < u + /v + , ⎪ ⎪ ⎩u − /v − , ab > u − /v − .

 Type II. [a − , a + ] ÷ I I [b− , b+ ] = a  b : a ∈ [a − , a + ], b ∈ [1/b+ , 1/b− ] , (I I )

where if Iu,v −1 ≤ 0, then ⎧ ⎪ ab ∈ [u + /v − , u − /v + ], ⎪ ⎨ab, a  b = u + /v − , ab < u + /v − , ⎪ ⎪ ⎩u − /v + , ab > u − /v + ; (I I )

if Iu,v −1 ≥ 0, then ab =

⎧ ⎪ ⎪ ⎨ab,

ab ∈ [u − /v + , u + /v − ],

u − /v + , ab < u − /v + , ⎪ ⎪ ⎩u + /v − , ab > u + /v − .

(5.2)

118

5 Division of Fuzzy Vector-Valued Functions Depending …

 Type III. [a − , a + ] ÷ I I I [b− , b+ ] = a  b : a ∈ [a − , a + ], b ∈ [1/b+ , 1/b− ] , (5.3) (I I I ) where if Iu,v −1 ≤ 0, then

ab =

(I I I )

if Iu,v −1 ≥ 0, then ab =

⎧ ⎪ ⎪ ⎨ab,

ab ∈ [u − /v − , u − /v + ],

⎧ ⎪ ⎪ ⎨ab,

ab ∈ [u − /v + , u − /v − ],

u − /v − , ab < u − /v − , ⎪ ⎪ ⎩u − /v + , ab > u − /v + ;

u − /v + , ab < u − /v + , ⎪ ⎪ ⎩u − /v − , ab > u − /v − .

 Type IV. [a − , a + ] ÷ I V [b− , b+ ] = a  b : a ∈ [a − , a + ], b ∈ [1/b+ , 1/b− ] , (5.4) (I V )

where if Iu,v −1 ≤ 0, then

ab =

(I V )

if Iu,v −1 ≥ 0, then ab =

⎧ ⎪ ⎪ ⎨ab,

ab ∈ [u + /v − , u + /v + ],

⎧ ⎪ ⎪ab, ⎨

ab ∈ [u + /v + , u + /v − ],

u + /v − , ab < u + /v − , ⎪ ⎪ ⎩u + /v + , ab > u + /v + ;

u + /v + , ab < u + /v + , ⎪ ⎪ ⎩u + /v − , ab > u + /v − .

 Type V. [a − , a + ] ÷V [b− , b+ ] = a  b : a ∈ [a − , a + ], b ∈ [1/b+ , 1/b− ] , where if

(V ) Iu,v −1

≤ 0, then

ab =

(V )

if Iu,v −1 ≥ 0, then ab =

⎧ ⎪ ⎪ab, ⎨

ab ∈ [u − /v + , u + /v + ],

⎧ ⎪ ⎪ ⎨ab,

ab ∈ [u + /v + , u − /v + ],

u − /v + , ab < u − /v + , ⎪ ⎪ ⎩u + /v + , ab > u + /v + ;

u + /v + , ab < u + /v + , ⎪ ⎪ ⎩u − /v + , ab > u − /v + .

(5.5)

5.2

A New Division of Multidimensional Intervals and Fuzzy Vectors

119

 Type VI. [a − , a + ] ÷V I [b− , b+ ] = a  b : a ∈ [a − , a + ], b ∈ [1/b+ , 1/b− ] , (5.6) (V I )

where if Iu,v −1 ≤ 0, then

ab =

(V I )

if Iu,v −1 ≥ 0, then ab =

⎧ ⎪ ⎪ ⎨ab,

ab ∈ [u − /v − , u + /v − ],

⎧ ⎪ ⎪ab, ⎨

ab ∈ [u + /v − , u − /v − ],

u − /v − , ab < u − /v − , ⎪ ⎪ ⎩u + /v − , ab > u + /v − ;

u + /v − ,

ab < u + /v − , ⎪ ⎪ ⎩u − /v − , ab > u − /v − .

Next, we will introduce six types of the division of fuzzy vectors induced by the division of compact intervals defined by (5.1)–(5.6). For any α ∈ [0, 1] and i = 1, 2, . . . , n, we define the following types I − V I (see (5.7)–(5.12)) with the (compact box) α-level set:  α  − u + − +  n [u i,α , u i,α Type I. := [u ÷ I˚ v]α = ×i=1 ] ÷ I [vi,α , vi,α ] , v I˚ − + − + where [u i,α , u i,α ] ÷ I [vi,α , vi,α ]=

⎧ α,(I ) − − + + ⎨ [u i,α /vi,α , u i,α /vi,α ] if I −1 ≤ 0, u i ,vi

α,(I ) + + − − ⎩ [u i,α /vi,α , u i,α /vi,α ] if I −1 ≥ 0;

(5.7)

u i ,vi

 α  − u + − +  n [u i,α , u i,α Type II. := [u ÷ I˚I v]α = ×i=1 ] ÷ I I [vi,α , vi,α ] , v I˚I ⎧ α,(I I ) + − − + ⎨ [u i,α /vi,α , u i,α /vi,α ] if I −1 ≤ 0, u i ,vi − + − + where [u i,α , u i,α ] ÷ I I [vi,α , vi,α ] = α,(I I ) − + + − ⎩ [u i,α /vi,α , u i,α /vi,α ] if I −1 ≥ 0;

(5.8)

u i ,vi

Type III.

 α  − u + − +  n [u i,α , u i,α := [u ÷ I I˚ I v]α = ×i=1 ] ÷ I I I [vi,α , vi,α ] , v I I˚ I

− + − + where [u i,α , u i,α ] ÷ I I I [vi,α , vi,α ]=

⎧ α,(I I I ) − − − + ⎨ [u i,α /vi,α , u i,α /vi,α ] if I −1 ≤ 0, u i ,vi

α,(I I I ) − + − − ⎩ [u i,α /vi,α , u i,α /vi,α ] if I −1 ≥ 0; u i ,vi

Type IV.

 α  − u + − +  n [u i,α , u i,α := [u ÷ I ˚V v]α = ×i=1 ] ÷ I V [vi,α , vi,α ] , v I ˚V

(5.9)

120

5 Division of Fuzzy Vector-Valued Functions Depending …

− + − + where [u i,α , u i,α ] ÷ I V [vi,α , vi,α ]=

⎧ α,(I V ) + − + + ⎨ [u i,α /vi,α , u i,α /vi,α ] if I −1 ≤ 0, u i ,vi

(5.10)

α,(I V ) + + + − ⎩ [u i,α /vi,α , u i,α /vi,α ] if I −1 ≥ 0; u i ,vi

 α  − u + − +  n [u i,α , u i,α Type V. := [u ÷V˚ v]α = ×i=1 ] ÷V [vi,α , vi,α ] , v V˚ ⎧ α,(V ) − + + + ⎨ [u i,α /vi,α , u i,α /vi,α ] if I −1 ≤ 0, u i ,vi − + − + where [u i,α , u i,α ] ÷V [vi,α , vi,α ] = (5.11) α,(V ) + + − + ⎩ [u i,α /vi,α , u i,α /vi,α ] if I −1 ≥ 0; u i ,vi

 α  − u + − +  n [u i,α , u i,α := [u ÷V˚I v]α = ×i=1 ] ÷V I [vi,α , vi,α ] , v V˚I

Type VI.

− + − + where [u i,α , u i,α ] ÷V I [vi,α , vi,α ]=

⎧ α,(V I ) − − + − ⎨ [u i,α /vi,α , u i,α /vi,α ] if I −1 ≤ 0, u i ,vi

(5.12)

α,(V I ) + − − − ⎩ [u i,α /vi,α , u i,α /vi,α ] if I −1 ≥ 0. u i ,vi

For convenience, we introduce some notations. Let u ÷ I˚ v := I w; u ÷ I˚I v := u ÷ I I˚ I v := ˆ

ˆ

u ÷ I ˚V v := I V w; u ÷V˚ v := V w; u ÷V˚I v := V I w; and [ I w]α :=  ], where Iˆ ∈ I , I I , . . . , V I . The following properties are immediate.

I I I w; ˆ

n [ I w− , I w+ ×i=1 i,α i,α

Theorem 5.2 (i)

If I

α,(I ) u i ,vi−1

≤ 0, then − + − + − I +  n n [vi,α , vi,α ]  [ I wi,α ×i=1 [u i,α , u i,α ] = ×i=1 , wi,α ] ; − − + + − I + −1  n n [u i,α , u i,α ×i=1 [vi,α , vi,α ] = ×i=1 ] ◦ [ I wi,α , wi,α ] ;

(ii)

if I

α,(I ) u i ,vi−1

≥ 0, then − + − + − I +  n n [vi,α , vi,α ] ◦ [ I wi,α ×i=1 [u i,α , u i,α ] = ×i=1 , wi,α ] ; − − + + − I + −1  n n [u i,α , u i,α ×i=1 [vi,α , vi,α ] = ×i=1 ]  [ I wi,α , wi,α ] ;

(iii)

if I

α,(I I ) u i ,vi−1

I I w;

≤ 0, then − + − + − II +  n n [vi,α , vi,α ]  [ I I wi,α ×i=1 [u i,α , u i,α ] = ×i=1 , wi,α ] ; − − + + − I I + −1  n n [u i,α , u i,α ×i=1 [vi,α , vi,α ] = ×i=1 ]  [ I I wi,α , wi,α ] ;

5.2

(iv)

A New Division of Multidimensional Intervals and Fuzzy Vectors

if I

α,(I I ) u i ,vi−1

121

≥ 0, then − + − + − II +  n n [vi,α , vi,α ] ◦ [ I I wi,α ×i=1 [u i,α , u i,α ] = ×i=1 , wi,α ] ; − − + + − I I + −1  n n [u i,α , u i,α ×i=1 [vi,α , vi,α ] = ×i=1 ] ◦ [ I I wi,α , wi,α ] ;

(v)

if I

α,(I I I ) u i ,vi−1

≤ 0 or I

α,(I I I ) u i ,vi−1

≥ 0, then

− − + + − I I I + −1  n n [u i,α , u i,α ×i=1 [vi,α , vi,α ] = ×i=1 ]  [ I I I wi,α , wi,α ] ; (vi)

if I

α,(I V ) u i ,vi−1

≤ 0 or I

α,(I V ) u i ,vi−1

≥ 0, then

− − + + − I V + −1  n n [u i,α , u i,α ×i=1 [vi,α , vi,α ] = ×i=1 ]  [ I V wi,α , wi,α ] ; (vii) if I

α,(V ) u i ,vi−1

≤ 0 or I

α,(V ) u i ,vi−1

≥ 0, then

− + − + − V +  n n [vi,α , vi,α ]  [V wi,α ×i=1 [u i,α , u i,α ] = ×i=1 , wi,α ] ; (viii) if I

α,(V I ) u i ,vi−1

≤ 0 or I

α,(V I ) u i ,vi−1

≥ 0, then

− + − + − VI +  n n [vi,α , vi,α ]  [V I wi,α ×i=1 [u i,α , u i,α ] = ×i=1 , wi,α ] . Proof (i). From I

α,(I ) u i ,vi−1

≤ 0, we have

 −

− − + +  + − +  n n [u i,α , u i,α u i,α /vi,α , u i,α /vi,α , ×i=1 ] ÷ I [vi,α , vi,α ] = ×i=1

− − + +  n [ I w − , I w + ] = ×n so we obtain ×i=1 i=1 u i,α /vi,α , u i,α /vi,α . Then we have i,α i,α    v+  − v   + − − u i,α ≥ 0,  − i,α − + i,α +  = u i,α u i,α /vi,α u i,α /vi,α  which implies that − + − I +  − + n n [vi,α , vi,α ]  [ I wi,α ×i=1 , wi,α ] = ×i=1 [u i,α , u i,α ].

+ + − −  n [ I w − , I w + ]−1 = ×n On the other hand, we have ×i=1 i=1 vi,α /u i,α , vi,α /u i,α , so we have i,α i,α    u−  + u i,α   − + i,α − vi,α ≤ 0,  + + − −  = vi,α vi,α /u i,α vi,α /u i,α 

122

5 Division of Fuzzy Vector-Valued Functions Depending …

which implies that − + − I + −1  − + n n [u i,α , u i,α = ×i=1 ×i=1 ]  [ I wi,α , wi,α ] [vi,α , vi,α ]. By using a similar way, one can also check (ii) − (viii) and we will not repeat their proof process here.  Theorem 5.3 Let u, v ∈ [RnF ] (here 1 is the same as {1}). Then α,(I ) ≤ 0 and u ÷ I˚ v := I w, then u = v  ( I w) and v = u ∗ ( I w)−1 ; u i ,vi−1 α,(I ) I −1 ≥ 0 and u ÷ I˚ v := I w, then u = v ∗ ( I w) and v = u  ( I w)−1 ; u i ,vi α,(I I ) I −1 ≤ 0 and u ÷ I˚I v := I I w, then u = v  ( I I w) and v = u  ( I I w)−1 ; u i ,vi α,(I I ) I −1 ≥ 0 and u ÷ I˚I v := I I w, then u = v ∗ ( I I w); v = u ∗ (w I I )−1 ; u i ,vi α,(I I I ) α,(I I I ) I −1 ≤ 0 or I −1 ≥ 0 and u ÷ I I˚ I v := I I I w, then v = u ∗ˆ (w I I I )−1 ; u i ,vi u i ,vi α,(I V ) α,(I V ) ˆ I V w)−1 ; I −1 ≤ 0 or I −1 ≥ 0 and u ÷ I ˚V v := I V w, then v = u ( u i ,vi u i ,vi α,(V ) α,(V ) ˆ V w); I −1 ≤ 0 or I −1 ≥ 0 and u ÷V˚ v := V w, then u = v ( u i ,vi u i ,vi α,(V I ) α,(V I ) I −1 ≤ 0 or I −1 ≥ 0 and u ÷V˚I v := V I w, then u = v ∗ˆ (V I w). u i ,v u i ,v

(i)

if I

(ii)

if

(iii)

if

(iv)

if

(v)

if

(vi)

if

(vii) if (viii) if

i

i

Proof By Theorem 5.2, the results (i) − (viii) are immediate.

5.3



Basic Results of Calculus of Fuzzy Vector-Valued Functions on Time Scales

In this section, we will provide some determinant algorithm properties of the g H -Δderivatives of the product of two fuzzy vector-valued functions on time scales which belong to the transformations of Theorems 3.10 and 3.11 from the literature [104]. For convenience,  we adopt the notation f σ (t)) = f σ (t) in some statements. Using the following properties, we will deduce the determinant algorithm properties of the g H -Δ-derivatives of the division of two fuzzy vector-valued functions on time scales. Theorem 5.4 Let f , g −1 : T → [RnF ] be (I )-g H -Δ-differentiable and 0 ∈ / [g −1 ]α for α ∈ [0, 1]. Then (i)

if I

α,(I ) f i ,gi−1

≤ 0, I

differentiable, then

α,(I ) f i ,(gi−1 )Δ I

≤ 0, I

α,(I ) ΔI

fi

,(gi−1 )σ

≤ 0 and

f ÷ I g is (I )-g H -Δ-

 f Δ I ∗ (g −1 )σ . ( f ÷ I g)Δ I = ( f ∗ g −1 )Δ I = f ∗ (g −1 )Δ I +

5.3

(ii)

Basic Results of Calculus of Fuzzy Vector-Valued Functions on Time Scales α,(I ) f i ,gi−1

if I

α,(I ) f i ,(gi−1 )Δ I

≤ 0, I

α,(I )

≥ 0, I

ΔI

,(gi−1 )σ

fi

differentiable, then

123

≥ 0 and f ÷ I g is (I I )-g H -Δ-

 f Δ I ∗ (g −1 )σ . ( f ÷ I g)Δ I I = ( f ∗ g −1 )Δ I I = f ∗ (g −1 )Δ I + (iii)

α,(I I ) f i ,gi−1

if I

α,(I I ) f i ,(gi−1 )Δ I

≤ 0, I

≤ 0, I

α,(I I ) ΔI

,(gi−1 )σ

fi

differentiable, then

≤ 0 and

f ÷ I I g is (I )-g H -Δ-

(g −1 )σ  f Δ I . ( f ÷ I I g)Δ I = ( f  g −1 )Δ I = f  (g −1 )Δ I + (iv)

if I

α,(I I ) f i ,gi−1

≤ 0, I

α,(I I ) f i ,(gi−1 )Δ I

≥ 0, I

α,(I I ) ΔI

fi

differentiable, then

,(gi−1 )σ

≥ 0 and f ÷ I I g is (I I )-g H -Δ-

 f Δ I  (g −1 )σ . ( f ÷ I I g)Δ I I = ( f  g −1 )Δ I I = f  (g −1 )Δ I + (v)

if I

α,(I ) f i ,gi−1

α,(I ) f i ,(gi−1 )Δ I

≥ 0, I

≥ 0, I

differentiable, then

α,(I ) ΔI

fi

,(gi−1 )σ

≥ 0 and

f ÷ I g is (I )-g H -Δ-

 f Δ I ∗ (g −1 )σ . ( f ÷ I g)Δ I = ( f ∗ g −1 )Δ I = f ∗ (g −1 )Δ I + (vi)

if I

α,(I ) f i ,gi−1

α,(I ) f i ,(gi−1 )Δ I

≥ 0, I

α,(I )

≤ 0, I

ΔI

,(gi−1 )σ

fi

differentiable, then

≤ 0 and f ÷ I g is (I I )-g H -Δ-

 f Δ I ∗ (g −1 )σ . ( f ÷ I g)Δ I I = ( f ∗ g −1 )Δ I I = f ∗ (g −1 )Δ I + (vii) if I

α,(I I ) f i ,gi−1

≥ 0, I

α,(I I ) f i ,(gi−1 )Δ I

≥ 0, I

differentiable, then

α,(I I ) ΔI

fi

,(gi−1 )σ

≥ 0 and

f ÷ I I g is (I )-g H -Δ-

(g −1 )σ  f Δ I . ( f ÷ I I g)Δ I = ( f  g −1 )Δ I = f  (g −1 )Δ I + (viii) if I

α,(I I ) f i ,gi−1

≥ 0, I

α,(I I ) f i ,(gi−1 )Δ I

≤ 0, I

differentiable, then

α,(I I ) ΔI

fi

,(gi−1 )σ

≤ 0 and f ÷ I I g is (I I )-g H -Δ-

 f Δ I  (g −1 )σ . ( f ÷ I I g)Δ I I = ( f  g −1 )Δ I I = f  (g −1 )Δ I + n [ f − (t), f + (t)] and [g(t)]α = ×n [g − (t), g + (t)], then Proof Let [ f (t)]α = ×i=1 i=1 i,α i,α i,α i,α n [ 1 , 1 ]. [g −1 (t)]α = ([g(t)]α )−1 = ×i=1 + − g (t) g (t) i,α

i,α

124

Since I

5 Division of Fuzzy Vector-Valued Functions Depending … α,(I ) f i ,gi−1

≤ 0, then   − +  f− +  f i,α f i,α f i,α  i,α  = −   + − − + ≤ 0, 1/gi,α  gi,α 1/gi,α gi,α f−

n [ i,α , which implies that [ f ∗ g −1 ]α = ×i=1 g− i,α

+ f i,α

+ gi,α

].

Thus we obtain  −  − Δ  + Δ  + Δ I f i,α f i,α f i,α f i,α , = , − + − + gi,α gi,α gi,α gi,α  − Δ − − − Δ + Δ + + + Δ (gi,α ) ( f i,α ) gi,α − f i,α (gi,α ) ( f i,α ) gi,α − f i,α n . , = ×i=1 − − σ + + σ gi,α (gi,α ) gi,α (gi,α ) On the other hand, from I

α,(I ) f i ,(gi−1 )Δ I

≤ 0 and I

α,(I ) ΔI

fi

,(gi−1 )σ

≤ 0, we have

  − − Δ + + Δ  f−  + f i,α f i,α (gi,α ) (gi,α ) f i,α   i,α +  + Δ − Δ = − − − σ + + σ ≤ 0, (1/gi,α ) (1/gi,α )  gi,α (gi,α ) gi,α (gi,α )   − Δ + Δ  ( f − )Δ + Δ  ( f i,α ) ( f i,α )  ( f i,α )  i,α = − ≤ 0,  + −1 σ  − −1 σ − + σ ((gi,α ) ) ((gi,α ) )  (gi,α ) (gi,α )σ then

 − − Δ + + Δ f i,α f i,α (gi,α ) (gi,α ) n − − − σ ,− + + σ [ f ∗ (g −1 )Δ I ]α = ×i=1 gi,α (gi,α ) gi,α (gi,α )

and n [ f Δ I ∗ (g −1 )σ ]α = ×i=1



− Δ + Δ ) ( f i,α ) ( f i,α , − σ + σ , (gi,α ) (gi,α )

so we have [ f ∗ (g −1 )Δ I + f Δ I ∗ (g −1 )σ ]α  − Δ − − − Δ + Δ + + + Δ (gi,α ) ( f i,α ) gi,α − f i,α (gi,α ) ( f i,α ) gi,α − f i,α n , = ×i=1 , − − σ + + σ gi,α (gi,α ) gi,α (gi,α )  f Δ I ∗ (g −1 )σ . which implies that ( f ∗ g −1 )Δ I = f ∗ (g −1 )Δ I + By applying a similar analysis as in (i) to (ii) − (viii), one can obtain the results (ii) −  (viii) immediately. Theorem 5.5 Let f , g −1 : T → [RnF ] be (I I )-g H -Δ-differentiable and 0 ∈ / [g −1 ]α for α ∈ [0, 1]. Then

5.3

(i)

Basic Results of Calculus of Fuzzy Vector-Valued Functions on Time Scales α,(I I ) f i ,gi−1

if I

α,(I ) f i ,(gi−1 )Δ I I

≤ 0, I

≥ 0, I

α,(I ) ΔI I

,(gi−1 )σ

fi

differentiable, then

125

≤ 0 and f ÷ I I g is (I )-g H -Δ-

 f Δ I I ∗ (g −1 )σ . ( f ÷ I I g)Δ I = ( f  g −1 )Δ I = f ∗ (g −1 )Δ I I + (ii)

if I

α,(I I ) f i ,gi−1

≤ 0, I

α,(I ) f i ,(gi−1 )Δ I I

≤ 0, I

α,(I ) ΔI I

fi

differentiable, then

,(gi−1 )σ

≥ 0 and f ÷ I I g is (I I )-g H -Δ-

 f Δ I ∗ (g −1 )σ . ( f ÷ I I g)Δ I I = ( f  g −1 )Δ I I = f ∗ (g −1 )Δ I + (iii)

α,(I I ) f i ,gi−1

if I

α,(I ) f i ,(gi−1 )Δ I I

≥ 0, I

≤ 0, I

α,(I ) ΔI I

,(gi−1 )σ

fi

differentiable, then

≥ 0 and f ÷ I I g is (I )-g H -Δ-

(g −1 )σ ∗ f Δ I I . ( f ÷ I I g)Δ I = ( f  g −1 )Δ I = f ∗ (g −1 )Δ I I + (iv)

if I

α,(I I ) f i ,gi−1

≥ 0, I

α,(I ) f i ,(gi−1 )Δ I I

≥ 0, I

α,(I )

differentiable, then

ΔI I

fi

,(gi−1 )σ

≤ 0 and f ÷ I I g is (I I )-g H -Δ-

 f Δ I I ∗ (g −1 )σ . ( f ÷ I I g)Δ I I = ( f  g −1 )Δ I I = f ∗ (g −1 )Δ I I + (v)

if I

α,(I ) f i ,gi−1

≤ 0, I

α,(I I ) f i ,(gi−1 )Δ I I

≥ 0, I

differentiable, then

α,(I I ) ΔI I

fi

,(gi−1 )σ

≤ 0 and f ÷ I g is (I )-g H -Δ-

 f Δ I I  (g −1 )σ . ( f ÷ I g)Δ I = ( f ∗ g −1 )Δ I = f  (g −1 )Δ I I + (vi)

if I

α,(I ) f i ,gi−1

≤ 0, I

α,(I I ) f i ,(gi−1 )Δ I I

≤ 0, I

differentiable, then

α,(I I ) ΔI I

fi

,(gi−1 )σ

≥ 0 and f ÷ I g is (I I )-g H -Δ-

 f Δ I I  (g −1 )σ . ( f ÷ I g)Δ I I = ( f ∗ g −1 )Δ I I = f  (g −1 )Δ I I + (vii) if I

α,(I ) f i ,gi−1

≥ 0, I

α,(I I ) f i ,(gi−1 )Δ I I

≤ 0, I

differentiable, then

α,(I I ) ΔI I

fi

,(gi−1 )σ

≥ 0 and f ÷ I g is (I )-g H -Δ-

(g −1 )σ  f Δ I I . ( f ÷ I g)Δ I = ( f ∗ g −1 )Δ I = f  (g −1 )Δ I I + (viii) if I

α,(I ) f i ,gi−1

≥ 0, I

α,(I I ) f i ,(gi−1 )Δ I I

≥ 0, I

differentiable, then

α,(I I ) ΔI I

fi

,(gi−1 )σ

≤ 0 and f ÷ I g is (I I )-g H -Δ-

 f Δ I I  (g −1 )σ . ( f ÷ I g)Δ I I = ( f ∗ g −1 )Δ I I = f  (g −1 )Δ I I + n [ f − (t), f + (t)] and [g(t)]α = ×n [g − (t), g + (t)], then Proof Denote [ f (t)]α = ×i=1 i=1 i,α i,α i,α i,α n [ 1 , 1 ]. [g −1 (t)]α = ([g(t)]α )−1 = ×i=1 + − g (t) g (t) i,α

i,α

126

Since I

5 Division of Fuzzy Vector-Valued Functions Depending … α,(I I ) f i ,gi−1

≤ 0, then   + −  f+ −  f i,α f i,α f i,α  i,α  = −   + − − + ≤ 0, 1/gi,α  gi,α 1/gi,α gi,α f+

n [ i,α , which implies that [ f  g −1 ]α = ×i=1 g− i,α

− f i,α

+ gi,α

].

Thus we obtain  +  + Δ  − Δ  − Δ I f i,α f i,α f i,α f i,α , = , − + − + gi,α gi,α gi,α gi,α  + Δ − + − Δ − Δ + − + Δ (gi,α ) ( f i,α ) gi,α − f i,α (gi,α ) ( f i,α ) gi,α − f i,α . , = − − σ + + σ gi,α (gi,α ) gi,α (gi,α ) On the other hand, from I

α,(I ) f i ,(gi−1 )Δ I I

≥ 0, I

α,(I ) ΔI I

fi

,(gi−1 )σ

≤ 0, we have

  − + Δ + − Δ  f−  + f i,α f i,α (gi,α ) (gi,α ) f i,α   i,α +  − Δ + Δ = − + + σ − − σ ≥ 0, (1/gi,α ) (1/gi,α )  gi,α (gi,α ) gi,α (gi,α )   + Δ − Δ  ( f + )Δ − Δ  ( f i,α ) ( f i,α )  ( f i,α )  i,α = − ≤ 0,  + −1 σ  − −1 σ − + σ ((gi,α ) ) ((gi,α ) )  (gi,α ) (gi,α )σ then

 + − Δ − + Δ f i,α f i,α (gi,α ) (gi,α ) n − − − σ ,− + + σ [ f ∗ (g −1 )Δ I ]α = ×i=1 gi,α (gi,α ) gi,α (gi,α )

and n [ f Δ I ∗ (g −1 )σ ]α = ×i=1



+ Δ − Δ ) ( f i,α ) ( f i,α , − σ + σ , (gi,α ) (gi,α )

so we have [ f ∗ (g −1 )Δ I + f Δ I ∗ (g −1 )σ ]α  + Δ − + − Δ − Δ + − + Δ (gi,α ) ( f i,α ) gi,α − f i,α (gi,α ) ( f i,α ) gi,α − f i,α n , = ×i=1 , − − σ + + σ gi,α (gi,α ) gi,α (gi,α )  f Δ I I ∗ (g −1 )σ . which implies that ( f  g −1 )Δ I = f ∗ (g −1 )Δ I I + By applying a similar analysis as in (i) to (ii) − (viii), one can obtain the results (ii) −  (viii) immediately.

6

Almost Periodic Generalized Fuzzy Multidimensional Dynamic Equations and Applications

6.1

Almost Periodic Generalized Fuzzy Multidimensional Dynamic Equations

In the literature [116], the authors established a theory of fuzzy multidimensional dynamic equations on time scales. In this section, we introduce some new concepts related to fuzzy dynamic equations under shifts on time scales, and then we establish some basic results on almost periodic generalized fuzzy multidimensional dynamic equations . Consider the following nonlinear fuzzy multidimensional dynamic equation x Δ (t) = f (t, x),

(6.1)

where f ∈ C(T × [RnF ], [RnF ]), and let  = {x ∈ [RnF ] : x(t) is a bounded solution to (6.1)}. Definition 6.1 If   = ∅, then λ = inf xF exists, and λ is called the least-value of solux∈ tions to (6.1). If there exists ϕ(t) ∈  such that ϕ∞ = λ, then ϕ(t) is called a minimum ˜ and  · ∞ = supt∈T  · F . norm solution to (6.1), where  · F = D∞ (·, 0) A similar proof to that in Theorem 5.1 in [26] gives the following. Lemma 6.1 If f ∈ C(T × S, [RnF ]) is bounded on T∗ × S0 and (6.1) has a bounded solution ϕ(t) such that {ϕ(t), t ∈ T} ⊂ S0 and 0˜ ∈ S0 , then there is a minimum norm solution to (6.1). Lemma 6.2 If f ∈ C(T × [RnF ], [RnF ]) is S-almost periodic in t uniformly for x ∈ [RnF ] under shifts δ± , S0 = {ϕ(t) : t ≥ t0 } and (6.1) has a bounded solution ϕ(t) on [t0 , ∞)T , then (6.1) has a S-almost periodic solution ψ(t) satisfying {ψ(t), t ∈ T} ⊂ S0 . © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 C. Wang and R. P. Agarwal, Dynamic Equations and Almost Periodic Fuzzy Functions on Time Scales, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-11236-2_6

127

128

6 Almost Periodic Generalized Fuzzy Multidimensional …





Proof First, we may choose α = {αk } ⊂ such that lim αk = +∞ implies k→+∞



 TαS f (t, x) = lim f δ+ (αk , t), x = f (t, x)

k→∞

holds uniformly on T∗ × S0 . For any fixed a ∈ T∗ , consider the interval (a, ∞)T and ϕk (t) =  

ϕ δ+ (αk , t) . Now, we show that for k sufficiently large, {ϕk } is defined on (a, ∞)T and is a solution   to x Δ (t) = f (t, x). Because ϕ(t) is a solution to (6.1), then it follows that TαS ϕ Δ (t) =   TαS f (t, x) , i.e.,     Δ  Δ  



lim ϕ δ+ (αk , t) = lim ϕ δ+ (αk , t) = lim f δ+ (αk , t), x = f (t, x).

k→∞

k→∞

k→∞

One can easily observe that {ϕk (t)} is uniformly bounded and equicontinuous on (a, ∞)T . Then let α be a sequence which goes to +∞, and it follows from Corollary 3.4 of [85]  



that there will exist α = {αkn } ⊂ α such that TαS ϕ(t) = lim ϕ δ+ (αkn , t) = ψ(t) holds n→∞

uniformly on T∗ . Thus, for all t ∈ T∗ , one can obtain ψ(·) ∈ S0 . Since TαS f (t, x) =  

lim f δ+ (αkn , t), x = f (t, x), then ψ(t) is a S-almost periodic solution to (6.1). This n→∞ completes the proof.  Lemma 6.3 Let f ∈ C(T × [RnF ], [RnF ]) be S-almost periodic in t uniformly for x ∈ [RnF ] under shifts δ± . If (6.1) has a minimum norm solution, then for any g ∈ HS ( f ), the following equation x Δ (t) = g(t, x) (6.2) has the same least-value of solutions as that to (6.1). Proof Let ϕ(t) be the minimum norm solution to (6.1) and λ is the least-value. Since

g ∈ HS ( f ), then there exists a sequence α ∈ such that T S f (t, x) = g(t, x) holds uniα



formly on T∗ × S0 . From Corollary 3.4 in [85], there exists α ⊂ α such that TαS ϕ(t) = ψ(t) ˜ ≤ λ, one has holds uniformly on T∗ . Hence, ψ(t) is a solution to (6.2). For D∞ (ϕ(t), 0)

S ˜ ˜ ≤ λ , D∞ (ψ(t), 0) ≤ λ, thus, λ = ψ∞ ≤ λ. Since ϕ(t) = Tα −1 ψ(t) and D∞ (ψ(t), 0) ˜ ≤ λ , thus, λ = ϕ∞ ≤ λ . Therefore, λ = λ , where  · ∞ = it follows that D∞ (ϕ(t), 0) ˜ The proof is completed.  supt∈T D∞ (·, 0). From the proof process of Lemma 6.3, one can obtain the following lemma immediately.



Lemma 6.4 If ϕ(t) is a minimum norm solution to (6.1) then there exists a sequence α ⊆ such that T S f (t, x) = g(t, x) exists uniformly on T∗ × S0 . Moreover, if there exists α

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a subsequence α ⊂ α such that TαS ϕ(t) = ψ(t) holds uniformly on T∗ , then ψ(t) is a minimum norm solution to (6.2). Lemma 6.5 If f ∈ C(T × [RnF ], [RnF ]) is S-almost periodic in t uniformly for x ∈ [RnF ] and for every g ∈ HS ( f ), (6.2) has a unique minimum norm solution, then these minimum norm solutions are S-almost periodic on T∗ . Proof For a fixed g ∈ HS ( f ), we obtain that (6.2) has the unique minimum norm solution ψ(t). Because g(t, x) is S-almost periodic in t uniformly for x ∈ [RnF ], it follows that for





any sequences α , β ⊆ , there exist common subsequences α ⊂ α , β ⊂ β such that TδS+ (α,β) g(t, x) = TαS TβS g(t, x) holds uniformly on T∗ × S0 and TαS TβS ψ(t), TδS+ (α,β) ψ(t) exist uniformly on T∗ . It follows from Lemmas 6.3 and 6.4 that TαS TβS ψ(t) and TδS+ (α,β) ψ(t) are minimum norm solutions to the following equation: x Δ (t) = TδS+ (α,β) g(t, x). From the uniqueness of the minimum norm solution, one has TαS TβS ψ(t) = TδS+ (α,β) ψ(t). Therefore, ψ(t) is S-almost periodic. The proof is completed.



We will now discuss the linear S-almost periodic dynamic equation on T as follows:  f (t) x Δ (t) = A(t)x +

(6.3)

and its associated homogeneous equation x Δ (t) = A(t)x,

(6.4)

where A : T → Rn×n is an S-almost periodic matrix-valued function and f : T → [RnF ] is an S-almost periodic fuzzy vector-valued function. Definition 6.2 If B ∈ HS (A), we say that y Δ (t) = B(t)y is a homogeneous equation in the hull of (6.3).

(6.5)

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6 Almost Periodic Generalized Fuzzy Multidimensional …

Definition 6.3 If B ∈ HS (A) and g ∈ HS ( f ), we say that  g(t) y Δ (t) = B(t)y +

(6.6)

is an equation in the hull of (6.3). Definition 6.4 We say the vector functions x1 , x2 , . . . , xn ∈ C(T, [RnF ]) are linearly independent if there exist constants c1 , c2 , . . . , cn which are not complete zeroes such that ˜ c1 x1 (t) + c2 x2 (t) + · · · + cn xn (t) = 0. Otherwise, we say x1 , x2 , . . . , xn ∈ C(T, [RnF ]) are linearly dependent.   Definition 6.5 We say X (t) = x1 (t), x2 (t), . . . , xn (t) is a fundamental solution matrix for (6.4) if X Δ (t) = A(t)X (t) for all t ∈ T and x1 , x2 , . . . , xn ∈ C(T, RnF ) are linearly independent. In particular, for any initial point t0 ∈ T, we say X (t) = e A (t, t0 ) is a matrixvalued nontrivial solution for (6.4). Remark 6.1 We let [Rn×n [RnF ] × [RnF ] × · · · × [RnF ]. Notice that a matrix-valued F ] :=    n terms

nontrivial solution X (t) = e A (t, t0 ) satisfies X (t) ∈ Rn×n ⊂ [Rn×n F ], where





Rn×n := χ{x1 } , x1 ∈ Rn × χ{x2 } , x2 ∈ Rn × · · · × χ{xn } , xn ∈ Rn ,    n terms

where x1 , x2 , . . . , xn are real vectors which are linearly independent. Notice that if X (t) is invertible, then X −1 (t) ∈ Rn×n . By considering the embedding mapping from Theorem   2.5, if we let ( j ◦ X )(t) = ( j ◦ x1 )(t), ( j ◦ x2 )(t) . . . , ( j ◦ xn )(t) , from X (t) ∈ Rn×n , we have    ( j ◦ X )(t) = j ◦ X )(t) = ( j ◦ x1 )(t), ( j ◦ x2 )(t) . . . , ( j ◦ xn )(t)   = j ◦ χ{x1 (t)} , j ◦ χ{x2 (t)} . . . , j ◦ χ{xn (t)} ⎛ ⎞ x11 (t) x21 (t) · · · xn1 (t) ⎜ x12 (t) x22 (t) · · · xn2 (t)⎟ ⎜ ⎟ =⎜ . .. ⎟ = X (t) .. . . ⎝ .. . . ⎠ . x1n (t) x2n (t) · · · xnn (t)  T where xi (t) = xi1 (t), xi2 (t), . . . , xin (t) ∈ Rn , i = 1, 2, . . . , n. Hence, we can obtain that ( j ◦ X )−1 (t) = X −1 (t) = ( j ◦ X −1 )(t) for all invertible X (t) ∈ Rn×n . For the definin×n tion of [Rn×n F ] and A ∈ [RF ], where A = (xi j )n×n and xi j ∈ RF , we equip the space  n 1 n×n [RF ] with the function A Fˆ = ( nj=1 i=1 xi j 2F0 ) 2 .

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From Remark 6.1, one can obtain the following lemma directly. Lemma 6.6 ( j ◦ X )−1 (t) = X −1 (t) = ( j ◦ X −1 )(t) for all invertible X (t) ∈ Rn×n . In what follows, we introduce the following exponential dichotomy which will be used later. Definition 6.6 Let A(t) be an n × n rd-continuous matrix-valued function on T. The linear system x Δ (t) = A(t)x(t) (6.7) is said to admit an exponential dichotomy on T if there exist positive constants K , α, projection P and the fundamental solution matrix X (t) of (6.7), satisfying    X (t)P X −1 (s)Fˆ ≤ K eα t, ρ(s) , s, t ∈ T, t ≥ s,   X (t)(I − P)X −1 (s)Fˆ ≤ K eα s, ρ(t) , s, t ∈ T, t ≤ s. In the following, we give sufficient conditions to guarantee that (6.7) admits exponential dichotomy. Lemma 6.7 For any solution x(t, x0 ) = x(t), of (6.7) and t ≥ t0 , the following inequality holds 



t

exp t0

 ≤



t

exp t0

ξμ(s) ξμ(s)

 1



 2λ(s) + λ0 (s)μ(s) Δs



 2(s) + 0 (s)μ(s) Δs

2

x0 F ≤ x(t, x0 )F

 1 2

x0 F ,

(6.8)

ˆ = where λ(t) and (t) are the smallest and the largest eigenvalue of the matrix A(t)   1 T 2 A(t) + A (t) respectively, and λ0 (t) and 0 (t) are the smallest and the largest ˜ = A(t)A T (t), A T (t) is the transpose of the matrix A(t), eigenvalue of the matrix A(t) ˜ xF = D∞ (x, 0). Proof Let z(t) = ( j ◦ x)(t). Then (6.7) turns into ( j ◦ x)Δ (t) = A(t)( j ◦ x)(t), i.e., z Δ (t) = A(t)z(t).

(6.9)

˜ then (6.8) holds. Let z(t, t0 ) = z(t) be a nontrivial solution If z(t, t0 ) = 0, i.e., x(t, t0 ) = 0, of the fuzzy system (6.9), by Theorem 5.1, z2L = z, z. Then for t ∈ T, we have

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6 Almost Periodic Generalized Fuzzy Multidimensional …



z(t)2L

Δ

 Δ = z(t), z(t) = z Δ (t), z(t) + z σ (t), z Δ (t) = z Δ (t), z(t) + z Δ (t)μ(t) + z(t), z Δ (t) = z Δ (t), z(t) + μ(t)z Δ (t), z Δ (t) + z(t), z Δ (t) = 2A(t)z(t), z(t) + μ(t)A(t)z(t), A(t)z(t) ˆ = 2 A(t)z(t), z(t) + μ(t)A(t)A T (t)z(t), z(t).

ˆ ˜ are symmetric, then we have Since A(t), A(t) ˆ λ(t)z(t), z(t) ≤  A(t)z(t), z(t) ≤ (t)z(t), z(t),

(6.10)

ˆ where λ(t) and (t) are the smallest and the largest eigenvalue of the matrix A(t). We also have λ0 (t)z(t), z(t) ≤ A(t)A T (t)z(t), z(t) ≤ 0 (t)z(t), z(t). (6.11) Then from (6.10) and (6.11), we obtain 

Δ     2λ(t) + λ0 (t)μ(t) z(t), z(t) ≤ z(t)2L ≤ 2(t) + 0 (t)μ(t) z(t), z(t). (6.12) Thus, for t ≥ t0 , from (6.12) and by Lemmas 3.4 and 3.6 from [82], we have 



t

exp t0

 ≤



t

exp t0

  ξμ(s) 2λ(s) + λ0 (s)μ(s) Δs

 1

  ξμ(s) 2(s) + 0 (s)μ(s) Δs

2

z 0 L ≤ z(t, t0 )L

 1 2

z 0 L .

˜ the estimate (6.8) follows ˜ L = D∞ (x, 0), Thus, from zL =  j ◦ xL =  j ◦ x − j ◦ 0 from this inequality. This completes the proof.  ˆ Theorem 6.1 For (6.7), if  : T → R is the largest eigenvalue of matrix A(t) and λ0 : ˜ there exists a positive constant cμ > 0 T → R is the smallest eigenvalue of matrix A(t), and M > 0 such that (i) p(t) ≤ cμ m[ p] < 0 and cμ μ ≤ M for all t ∈ T; (ii) μν ≤ M for all right scattered points of T; (iii) Mm[ p] ∈ R + ;   ˆ = 1 A(t) + A T (t) and A(t) ˜ = A(t)A T (t), A T (t) is the transpose of the matrix where A(t) 2 A(t), i = 1, 2, . . . , n and 1 p(t) = (t) + 0 (t)μ(t), p ∈ R + . 2

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133

Assume |m[ p]| < +∞. Then the linear system (6.7) admits an exponential dichotomy on T, where  1 δ+ (T ,t) m[ p] = lim p(s)Δs T →∞ T t denotes the mean-value of the function p under the shift δ+ . Proof Let X (t, τ ) be the solution the matrix Cauchy problem X Δ = A(t)X , X (τ, τ ) = I , i.e., a matriciant of (6.4). Then any nontrivial matrix-valued solution X (t) of (6.4) can be represented as X (t) = X (t, s)X (s), i.e., X (t)X −1 (s) = X (t, s). The linear system (6.4) has a unique solution x(t) = X (t, s)x(s). Using Lemma 6.7, we obtain   x(t)F =  X (t)X −1 (s)x(s)F = X (t, s)x(s)F   t  1 2   ≤ exp ξμ(s) 2(s) + 0 (s)μ(s) Δs x(s)F s    t  t     ≤ exp ξμ(s) p(s) Δs x(s)F ≤ exp ξμ(s) cm[ p] Δs x(s)F s s     = ecm[ p] (t, s)x(s) = ecm[ p] t, ρ(s) ecm[ p] ρ(s), s x(s)   ≤ k0 ecm[ p] t, ρ(s) x(s)F ,   where k0 = ecm[ p] ρ(s), s ≥ 1. In fact, if ρ(s) is a right scatted point, then  ln(1 + cμ μ(τ )m[ p]) ecm[ p] ρ(s), s = exp − Δτ μ(τ ) ρ(s)   ln(1 + cμ μ(s)m[ p]) = exp − ν(s) · μ(s)     ν 1 1 = exp · ln ≤ exp M · ln := k0 , μ 1 + cμ μm[ p] 1 + Mm[ p]     if ρ(s) is a right dense point, then σ ρ(s) = ρ(s) = s, and then k0 = ecm[ p] ρ(s), s = 1. Thus, we have   X (t, x)x(s)F ≤ k0 ecm[ p] t, ρ(s) , x(s)F   which implies that X (t, s)Fˆ ≤ k0 ecm[ p] t, ρ(s) , i.e., (6.7) admits an exponential  dichotomy on T with the projection P = I . This completes the proof. 







σ (ρ(s))

Corollary 6.1 Let Rin ⊂ Rn be the vector subspace in which all the elements of the vector are equal to zero except the ith one and

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6 Almost Periodic Generalized Fuzzy Multidimensional …

  A(t) = χ{c1 (t)} , χ{c2 (t)} , . . . , χ{cn (t)} , ci (t) ∈ Rin , where ci ∈ C(T, R− ) are bounded functions and ci (t) ∈ R + , i = 1, 2, . . . , n. Assume there are positive constants cμ0 , M such that (i) cμ μ ≤ M for all t ∈ T; (ii) μν ≤ M for all right scattered points of T;  δ (T ,t)  M    2  2  c (t) + 21 cim (t) μ Δt < 0; (iii) m cM (t) + 21 cim (t) μ = lim T1 t + T →∞

(iv) |m[ciM + 21 (cim )2 μ]| < ∞ and Mm[ciM + 21 (cim )2 μ] ∈ R + ;     where cM (t) = maxi ci (t) , cm (t) = mini ci (t) . Then (6.7) admits an exponential dichotomy on T.  2 Proof From Corollary 6.1, one can easily obtain (t) = cM (t) and 0 = cim (t) . According to Theorem 6.1, (6.7) admits an exponential dichotomy. This completes the proof.  In the following, we will establish Favard’s theorem for homogeneous linear fuzzy dynamic equation. Lemma 6.8 If A(t) is an S-almost periodic fuzzy matrix-valued function and x(t) is an S-almost periodic solution of the homogeneous linear fuzzy dynamic equation x Δ (t) = A(t)x,   ˜ then inf t∈T x(t)F = inf t∈T D∞ x(t), 0˜ > 0 or x(t) ≡ 0. Proof If inf t∈T x(t)F = 0, then there exists {tn } ⊂ T such that x(tn )F → 0 as n →

∞. It follows from x ∈ HS (x) that there exists α ⊂ such that  

lim x δ+ (αn , t) = x(t)

n→+∞

for all t ∈ T∗ , which implies that for any ε > 0, there exists N > 0 so that n > N implies     ε x δ+ (α , tn ) −  g H x(tn ) < . Furthermore, since x(t) is S-almost periodic on T, it is n F 2 uniformly continuous on T, one can choose t0 ∈ T such that |t0 − tn | < δ ∗ implies      ε x δ (t0 ) −  g H x δα (tn )  < . αn F n 2 Therefore, for sufficiently large n ∈ N, we have

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135

    x δ+ (α , t0 ) −  g H x(tn )

    



 g H x δ+ (αn , tn )  ≤ x δ+ (αn , t0 ) − F     ε ε

   + x δ+ (αn , tn ) −g H x(tn ) F < + < ε. 2 2   

Hence, we can easily see that x δ+ (αn , t0 ) F → 0 as n → +∞. Since A(t) is S-almost

periodic on T, there exists sequence α ⊂ α such that F

n

TαS A(t) = B(t), TαS−1 B(t) = A(t),

TαS x(t) = y(t), TαS−1 y(t) = x(t)

hold uniformly on T∗ , then one obtains    Δ   Δ Δ y Δ (t) = TαS x(t) = lim x δ+ (αn , t) = lim x δ+ (αn , t) n→∞ n→∞         = lim A δ+ (αn , t) x δ+ (αn , t) = lim A δ+ (αn , t) · lim x δ+ (αn , t) , n→∞

n→∞

n→∞

that is, y(t) is a solution to the following equation: y Δ = B(t)y satisfying the initial condition: ˜ y(t0 ) = TαS x(t0 ) = lim x(δ+ (αn , t0 )) = 0. n→+∞

˜ and therefore, x(t) = T S−1 y(t) ≡ 0. ˜ The proof is comHence, y(t) = y(t0 )e B (t, t0 ) ≡ 0, α pleted.  Lemma 6.9 Suppose that (6.4) has an S-almost periodic solution x(t) and inf x(t)F > 0.

t∈T

If (6.3) has bounded solution on [t0 , ∞)T , then (6.3) has an S-almost periodic solution. Proof From Lemmas 6.1 and 6.2, we know that there is a minimum norm solution to (6.3) on T∗ and for every pair of TαS A(t) = B(t) and TαS f (t) = g(t), (6.6) has a minimum norm solution. Next, we show that the minimum norm solution to (6.6) is unique. For a fixed pair of B(t) and g(t), we consider (6.6). If x(t) is a minimum norm solution with least-value λ then

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6 Almost Periodic Generalized Fuzzy Multidimensional …

    ( j ◦ x)(t) = s( p, [x(t)]α ) ≤ [x(t)]α ∗  p ≤ sup [x(t)]α ∗ L α∈[0,1]

= x(t)F ≤ x∞ = λ. Now, let (6.6) have two different minimum norm solutions x1 (t) and x2 (t), and their least 1  g H x2 (t) is a bounded non-trivial solution to (6.5), values are equal to λ. Since x1 (t)− 2 from the condition of the lemma, there exists a real number ρ > 0 such that    1 1 x1 (t)−  g H x2 (t) = inf D∞ x1 (t), x2 (t) F t∈T 2 t∈T 2 1 = inf ( j ◦ x1 )(t) − ( j ◦ x2 )(t)L ≥ ρ > 0. t∈T 2 inf

Now by the parallelogram law, we have     1  1     2 2     (t) + j ◦ x (t) + )(t) − ( j ◦ x )(t) j ◦ x ( j ◦ x 1 2 1 2  2  2 L L  1 2 2 2 = ( j ◦ x1 )(t) + ( j ◦ x2 )(t)L ≤ λ , 2 and note that 1 2

then

1 2



 x2 (t) x1 (t)+

Δ

1 Δ 1 Δ  x x + 2 1 2 2  1  1  g(t) +  B(t)x2 +  g(t) = B(t)x1 + 2 2  1  x2 ) +  g(t), = B(t) (x1 + 2 =

  x2 (t) is a solution to (6.6). Hence, we obtain x1 (t)+        1      j ◦ 1 (x1 +  x2 ) (t) =  ( j ◦ x1 )(t) + ( j ◦ x2 )(t)    2 2 L L  2 2 < λ − ρ < λ.

This is a contradiction. The proof is completed.



Lemma 6.10 If every bounded solution of a homogeneous equation in the hull of (6.3) is S-almost periodic, then all bounded solutions of (6.3) are S-almost periodic. Proof From Lemma 6.8, we know that every non-trivial bounded solution of equations in the hull of (6.3) satisfies inf t∈T x(t)F > 0. From Lemma 6.9 it follows that if (6.3) has bounded solutions on T, then (6.3) will have an S-almost periodic solution ψ(t). If ϕ(t) is an  g H ϕ(t) is a bounded solution of its arbitrary bounded solution of (6.3), then η(t) = ψ(t)−

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associated homogeneous equation (6.4), and it is S-almost periodic. Thus, ϕ(t) is S-almost periodic. This completes the proof.  Lemma 6.11 If a homogeneous equation in the hull of (6.3) has the unique bounded solu˜ then (6.3) has a unique S-almost periodic solution . tion x(t) ≡ 0,  g H ψ(t) is Proof Let ψ(t), ϕ(t) be two bounded solutions to (6.3), and then x(t) = ϕ(t)− ˜ we have that a solution of a homogeneous equation in the hull of (6.3), since x(t) ≡ 0, ϕ(t) ≡ ψ(t). Thus, by Lemma 6.10, (6.3) has a unique S-almost periodic solution. This completes the proof.  Using a proof similar to that in Lemmas 7.4–7.5 in [36], one can easily prove Lemmas 6.12–6.13 (we omit their proofs). Lemma 6.12 Let P be a projection and X a differentiable invertible matrix such that X P X −1 is bounded on T. Then there exists a differentiable matrix S such that X P X −1 = S P S −1 for all t ∈ T and S, S −1 are bounded on T. In fact, there is an S of the form S = X Q −1 , where Q commutes with P. Lemma 6.13 If (6.4) has an exponential dichotomy and X (t) is the fundamental solution matrix of (6.4), C non-singular, then X (t)C has an exponential dichotomy with the same projection P if and only if C P = PC. Lemma 6.14 Suppose that A(t) is an S-almost periodic matrix function under shifts δ± and (6.4) has an exponential dichotomy, then for every B(·) ∈ HS (A), (6.5) has an exponential dichotomy with the same projection P and the same constants K , α. Proof Let X be the fundamental solution matrix satisfying (6.6). Let Q and S be given as in Lemma 6.12 and TαS A = B uniformly on T∗ . For any given t0 ∈ T∗ , let X n (t) =     X δ+ (αn , t) Q −1 δ+ (αn , t0 ) , it follows from (6.4) that X Δ (t) = A(t)X (t),

(6.13)

Now, replacing t with δ+ (αn , t) in (6.13), we can obtain       X Δ δ+ (αn , t) = A δ+ (αn , t) X δ+ (αn , t) ,   i.e., X nΔ (t) = A δ+ (αn , t) X n (t), then X n (t) is a fundamental solution matrix to x Δ (t) =   A δ+ (αn , t) x, by Lemma 6.13, it has an exponential dichotomy with the same projection P and the same constants. This is true since Q −1 commutes with P. One may select sub  sequences so that X n (t0 ) and X n−1 (t0 ) converge since their forms are S δ+ (αn , t0 ) and

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6 Almost Periodic Generalized Fuzzy Multidimensional …

  S −1 δ+ (αn , t0 ) and they are bounded. Without changing notation we may assume that X n (t0 ) → Y0 , then X n−1 (t0 ) → Z 0 , where Z 0 = Y0−1 . But now for a suitable subsequence X n (t) converges to a solution of y Δ = By uniformly on T∗ . Denote this solution by Y . Then Y (t0 ) = Y0 is non-singular and clearly Y satisfies (6.6) since X n does for all n. The proof is completed.  Lemma 6.15 If the homogeneous equation (6.4) has an exponential dichotomy , then (6.4) ˜ has only one bounded solution x(t) ≡ 0. Proof Let X (t) be the fundamental solution matrix to (6.4). For any sequence α ⊂ , we     have An = A δ+ (αn , t) , X n (t) = X δ+ (αn , t) . Since the homogeneous equation (6.4) has an exponential dichotomy, it is easy to observe that there exists a constant M such that ¯ X n (t)Fˆ ≤ M and X nΔ (t)Fˆ = An (t)X n (t)Fˆ ≤ AM, where A¯ = supt∈T A(t)Fˆ .

Therefore, from Corollary 3.4 in [85], there exists {αn k } := α ⊂ α such that {X n k } con  verges uniformly on T∗ and lim X δ+ (αn , t) exists uniformly on T∗ . Thus X (t) is n→+∞

S-almost periodic. Since the homogeneous equation (6.4) has an exponential dichotomy, ˜ This completes the proof.  then inf t∈T x(t)F = 0, from Lemma 6.8, x(t) ≡ 0. Lemma 6.16 If the homogeneous equation (6.4) has an exponential dichotomy, then all ˜ equations in the hull of (6.4) have only one bounded solution x(t) ≡ 0. Proof From Lemma 6.14, all equations in the hull of (6.4) have an exponential dichotomy, according to Lemma 6.15, all equations in the hull of (6.4) have only one bounded solution ˜ This completes the proof. x(t) ≡ 0.  Lemma 6.17 Let A ∈ R and xi : T → [RnF ]. Now (6.4) has a fundamental solution matrix     X (t) = x1 (t), x2 (t), . . . , xn (t) if and only if ( j ◦ x1 )(t), ( j ◦ x2 )(t) . . . , ( j ◦ xn )(t) is a fundamental solution for the dynamic equations: Z Δ (t) = A(t)Z (t),

(6.14)

  where Z (t) = ( j ◦ X )(t) = ( j ◦ x1 )(t), ( j ◦ x2 )(t) . . . , ( j ◦ xn )(t) . Proof If X (t) is the fundamental solution matrix for (6.4), one obtains that     j X Δ (t) = j A(t)X (t) . From Theorem 2.5 and Theorem 4.17 (iii), since A(t) is a matrix-valued function, we can obtain  ( j ◦ X )Δ (t) = A(t) j ◦ X )(t). (6.15)

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Almost Periodic Generalized Fuzzy Multidimensional Dynamic Equations

139

On the other hand, if Z (t) is the fundamental matrix solution for (6.14), because j is a bijective mapping, by employing the mapping j −1 on the both sides of (6.15), one gets the desired result immediately. This completes the proof.  Theorem 6.2 Let A : T → Rn×n be an S-almost periodic matrix-valued function and f : T → [RnF ] be an S-almost periodic fuzzy vector-valued function. If (6.4) admits an exponential dichotomy, then (6.3) has a unique S-almost periodic fuzzy vector-valued solution  +∞  t g H x(t) = X (t)P X −1 (σ (s)) f (s)Δs − X (t)(I − P)X −1 (σ (s)) f (s)Δs, −∞

t

where X (t) is the fundamental trivial solution matrix of (6.4). Proof First, by Lemma 6.17, let Z (t) = ( j ◦ X )(t), we prove that z(t) = ( j ◦ x)(t) is a bounded solution of the following vector-valued system: z(t) = A(t)z(t) + f˜(t), where f˜(t) = ( j ◦ f )(t).

(6.16)

In fact, z Δ (t) − A(t)z(t)  t Δ = Z (t) P Z −1 (σ (s)) f˜(s)Δs + Z (σ (t))P Z −1 (σ (t)) f˜(t) −∞  +∞

(I − P)Z −1 (σ (s)) f˜(s)Δs + Z (σ (t))(I − P)Z −1 (σ (t)) f˜(t)  +∞  t −1 ˜ P Z (σ (s)) f (s)Δs + A(t)Z (t) (I − P)Z −1 (σ (s)) f˜(s)Δs −A(t)Z (t)

−Z Δ (t)

t

−∞

t

= Z (σ (t))(P + I − P)Z −1 (σ (t)) f˜(t) = f˜(t), thus, we obtain (6.16) has a bounded solution z(t) satisfying  +∞  t z(t) + Z (t)(I − P)Z −1 (σ (s)) f˜(s)Δs = Z (t)P Z −1 (σ (s)) f˜(s)Δs, t

−∞

by employing the inverse mapping j −1 and Lemma 6.6, and one can obtain x(t) = ( j −1 ◦ z)(t) is a solution of (6.3). Moreover, we also have

140

6 Almost Periodic Generalized Fuzzy Multidimensional …

 t  xF = sup  X (t)P X −1 (σ (s)) f (s)Δs  −∞ t∈T   +∞  g H − X (t)(I − P)X −1 (σ (s)) f (s)Δs   t F    +∞    t     ≤ sup  eα (t, s) f (s)Δs  +  eα (σ (s), t) f (s)Δs  K  f F t∈T

−∞

t

2K ≤  f ∞ , α ˜ where  · ∞ = supt∈T D∞ (·, 0). Now, we show that the fuzzy vector-valued solution x(t) is S-almost periodic. According to Lemma 6.16, we obtain that all equations in the hull of (6.4) have only one bounded solu˜ Thus, for given α , β , one can choose common subsequences α ⊂ α , β ⊂ ¯ ≡ 0. tion x(t)

β such that TδS+ (α,β) A = TαS TβS A, Tδδ+ (α,β) f = TαS TβS f , y = TδS+ (α,β) x, and z = TαS TβS x  g H z = TδS (α,β) x −  g H TαS TβS x ≡ 0˜ because y −  g H z is exists uniformly on T∗ . However, y − +

the bounded solution to all equations in the hull of (6.4). Hence, TδS+ (α,β) x = TαS TβS x, then  it follows from Theorem 4.19 that x ∈ A PS (T). This completes the proof.

6.2

Applications on Fuzzy Dynamic Equations and Models on Time Scales

In this section, we provide several applications to demonstrate the S-almost periodic theory of fuzzy-vector-valued functions on irregular time scales. Application 6.2.1 As an application of our results obtained in the previous sections, in what follows, we consider the following S-almost periodic fuzzy delay vector-valued dynamic equation with variable delays under shifts δ± :  x Δ (t) = A(t)x(t)+

n     f t, x δ− (τi (t), t) ,

(6.1)

i=1

where A(t) is an S-almost periodic matrix-valued function on T, τi (t) : T∗ → is S-almost periodic on T for every i = 1, 2, . . . , n, f ∈ C(T × [RnF ], [RnF ]) is S-almost periodic uniformly in t for x ∈ [RnF ]. Theorem 6.3 Suppose that the following hold: (H1 ) x Δ (t) = A(t)x(t) admits an exponential dichotomy on T with positive constants K and α.

6.2

Applications on Fuzzy Dynamic Equations and Models on Time Scales

141

  α such that D∞ f (t, x), f (t, y) ≤ M D∞ (x, y) for t ∈ T, x, 2K n y ∈ [RnF ]. Then system (6.1) has a unique S-almost periodic fuzzy vector-valued solution.

(H2 ) There exists M
0, there exists at least a ω ∈ in an interval with a length of l such that ! ( f 1− (δ+ (ω, t), V ) − f 1− (t, V ))2 + ( f 2− (δ+ (ω, t), V ) − f 2− (t, V ))2 < ε, ! ( f 1− (δ+ (ω, t), V ) − f 1− (t, V ))2 + ( f 2+ (δ+ (ω, t), V ) − f 2+ (t, V ))2 < ε, ! ( f 1+ (δ+ (ω, t), V ) − f 1+ (t, V ))2 + ( f 2+ (δ+ (ω, t), V ) − f 2+ (t, V ))2 < ε, ! ( f 1+ (δ+ (ω, t), V ) − f 1+ (t, V ))2 + ( f 2− (δ+ (ω, t), V ) − f 2− (t, V ))2 < ε, which implies that D∞ ([F(δ+ (ω, t), V )]α , [F(t, V )]α ) < ε. Therefore, F : T × [R2F ] → [R2F ] is S-almost periodic in t uniformly for V ∈ [R2F ]. From Theorem 6.2, one can easily obtain that (6.3) has a S-almost periodic fuzzy solution:  x(t) = =

2      e A t, σ (s) F s, V δ− (τi (s), s) Δs

t −∞

2   i=1

i=1 t

−∞

     e A t, σ (s) F s, V δ− (τi (s), s) Δs.

Based on (6.4), we show the S-almost periodic fuzzy solutions for the following case.

(6.4)

144

6 Almost Periodic Generalized Fuzzy Multidimensional …

Suppose F(t, V ) = F(t) is a two dimensional trapezoidal fuzzy vector-valued function   and A(t) = χ{c1 (t)} , χ{c2 (t)} satisfies Corollary 6.1. Let a trapezoidal fuzzy number N = (l, m, n, r ) be ⎧ ⎪ 0, t ≤ l, ⎪ ⎪ ⎪ ⎪ t−l ⎪ ⎪ ⎨ m−l , l < t ≤ m, N (t) =

1, m < t ≤ n, ⎪ ⎪ r −t ⎪ ⎪ ⎪ r −n , n < t ≤ r , ⎪ ⎪ ⎩0, r < t,

where l < m < n < r and l, m, n, r ∈ R.

One can obtain that [N (t)]1 =[m, n], [N (t)]0 =[l, r ], [N (t)]α = [m − (1 − α)(m − l), n + (1 − α)(r − n)]. Now, we construct the two dimensional S-almost periodic trapezoidal fuzzy vector-valued function √ √  T F(t) = | sin 3t|N (t), | cos 2t|N (t) . We can obtain the α-level set of F(t) as √  √     [F(t)]α = | sin 3t| m − (1 − α)(m − l) , | sin 3t| n + (1 − α)(r − n) √  √     × | cos 2t| m − (1 − α)(m − l) , | cos 2t| n + (1 − α)(r − n) . Hence, in this case, we can obtain the α-level set of the S-almost periodic fuzzy solution for (6.3) as follows.  t α   e A t, σ (s) F(s)Δs −∞  t √     = ec1 t, σ (s) | sin 3s| m − (1 − α)(m − l) Δs, −∞  t √     ec1 t, σ (s) | sin 3s| n + (1 − α)(r − n) Δs −∞  t √     ec2 t, σ (s) | cos 2s| m − (1 − α)(m − l) Δs, × −∞  t √     ec2 t, σ (s) | cos 2s| n + (1 − α)(r − n) Δs −∞ & ' √   t   ec1 t, σ (s) | sin 3s|Δs α α −∞ √    := [N (t)] × [N (t)] · . t −∞ ec2 t, σ (s) | cos 2s|Δs Application 6.2.3 The theory of quantum calculus plays an important role in the real world, due to its applications in several physics fields such as black holes, cosmic strings, confor-

6.2

Applications on Fuzzy Dynamic Equations and Models on Time Scales

145

mal quantum mechanics, nuclear and high energy physics, fractional quantum Hall effect, high-Tc superconductors, etc. (see [46]). In applied dynamic equations, the q-difference equations which arises from quantum calculus are important dynamic systems which can be applied to model linear and nonlinear problems and play an important role in different fields of engineering and biological science (see [21]), but there is no result on almost periodic problems of fuzzy q-difference dynamic systems. With the developments of time scales, quantum time scales calculus can be extended to quantum-like time scales such as {(−q)n : q > 1, n ∈ Z}, {−q n :, q > 1, n ∈ Z}, etc. Now, based on the general quantum time scale {(−q)n : q > 1, n ∈ Z}, we consider the generalized two-dimensional q-difference fuzzy dynamic equations as follows: V Δ (t) = A(t)V (t) + F(t),

(6.5)

where A : T → R2×2 is a matrix-valued function. In fact, the general quantum time scale {(−q)n : q > 1, n ∈ Z} can be divided into two quantum-like time scales: T1 = {−q 2n+1 : q > 1, n ∈ Z}, T2 = {q 2n : q > 1, n ∈ Z}, T = T1 ∪ T2 . Therefore, we can introduce two derivatives on time scales T1 and T2 , respectively, and divide (6.5) into the following two q-difference fuzzy dynamic equations: ( )  g H f (t) f qt2 − ( ) V Δ (t) = Dq V (t) = (6.6) = A(t)V (t) + F(t), t ∈ T1 , 1 − 1 t 2 q    g H f (t) f q 2t − V (t) = Dq V (t) = = A(t)V (t) + F(t), t ∈ T2 . (q 2 − 1)t Δ

(6.7)

Hence, using time scales calculus, we can unify (6.6) and (6.7) by letting   1 t t ∈ T1 , 2 , t ∈ T1 , 2, q σ (t) = q μ(t) = 2 2 q , t ∈ T2 , q t, t ∈ T2 , and the following generalized fuzzy dynamic equations can be obtained:    g H f (t) f σ (t) − Δ V (t) = DG q V (t) = = A(t)V (t) + F(t), μ(t) where DG q denote the generalized q-difference operator for fuzzy vector-valued functions. Based on (6.5), we show the S-almost periodic fuzzy solutions for the following case. Suppose F(t) is a two dimensional triangular fuzzy vector-valued function and A(t) =    = (l, m, r ) be χ{c1 (t)} , χ{c2 (t)} satisfies Corollary 6.1. Let a triangular fuzzy number N

146

6 Almost Periodic Generalized Fuzzy Multidimensional …

⎧ ⎪ ⎪0, t ≤ l, ⎪ ⎪ ⎨ t−l , l < t ≤ m,  N (t) = m−l r −t ⎪ ⎪ r −n , m < t ≤ r , ⎪ ⎪ ⎩ 0, r < t,

where l < m < r and l, m, r ∈ R.

One can obtain that (t)]1 = {m}, [ N (t)]0 = [l, r ], [ N (t)]α = [m − (1 − α)(m − l), m + (1 − α)(r − m)]. [N Now, we construct the two dimensional S- almost periodic triangular fuzzy vector-valued function √ √   (t), | cos 7t| N (t) T . F(t) = | sin 5t| N We obtain the α-level set of F(t) as √  √     [F(t)]α = | sin 5t| m − (1 − α)(m − l) , | sin 5t| m + (1 − α)(r − m) √  √     × | cos 7t| m − (1 − α)(m − l) , | cos 7t| m + (1 − α)(r − m) . Hence, in this case, we obtain the α-level set of the S-almost periodic fuzzy solution for (6.5) as follows: 

t



−∞ t

= 

−∞



× 

α

√     ec1 t, σ (s) | sin 5s| m − (1 − α)(m − l) Δs,

√     ec1 t, σ (s) | sin 5s| m + (1 − α)(r − m) Δs

t

−∞

  e A t, σ (s) F(s)Δs

t

−∞

√     ec2 t, σ (s) | cos 7s| m − (1 − α)(m − l) Δs,

√     ec2 t, σ (s) | cos 7s| m + (1 − α)(r − m) Δs −∞ ' & √   t   ec1 t, σ (s) | sin 5s|Δs α α −∞   √   := [ N (t)] × [ N (t)] ·  t −∞ ec2 t, σ (s) | cos 7s|Δs ' & t  1  * √ ⎧ −2 − 1)c s ]| sin 5s|Δs   ⎪ 1 0 s0 ∈[s,t) [1 + (q −∞ 1+c1 q −2 ⎪ α α   ⎪ √ , [ N (t)] × [ N (t)] ·  t  1  * ⎪ −2 − 1)c s ]| cos 7s|Δs ⎪ ⎪ 2 0 ⎪ s0 ∈[s,t) [1 + (q −∞ 1+c2 q −2 ⎪ ⎪ ⎨ t ∈T , 1 &  +∞  1  * ' √ = ⎪ [1 + (q 2 − 1)c1 s0 ]| sin 5s|Δs   ⎪ 2 s ∈[t,s) t 0 1+c q ⎪ α α 1 (t)] × [ N (t)] ·  +∞  ⎪ √ * [N , ⎪ 1 2 ⎪ ⎪ s0 ∈[t,s) [1 + (q − 1)c2 s0 ]| cos 7s|Δs t ⎪ 1+c2 q 2 ⎪ ⎩ t ∈ T2 . t

Application 6.2.4 Consider the generalized two-dimensional difference fuzzy dynamic equations as follows: V Δ (t) = A(t)V (t) + F(t). (6.8)

6.2

Applications on Fuzzy Dynamic Equations and Models on Time Scales

147

For V (t) = [V1 (t), V2 (t)]T and   (t − 1)π t(t − 1)π A(t) = sin + 1 I , F(t) = cos I V (t − 1), t 2 where I is an identity matrix, t ∈ Z, a Fibonacci sequence with the background of the generalized two-dimensional difference fuzzy dynamic equations will be established as follows: + , + , t(t−1)π sin (t−1)π cos 0 0 t 2 V (t + 1) = V (t) + V (t − 1), 0 sin (t−1)π 0 cos t(t−1)π t 2 with the initial condition V (1) = V (2) = V0 , where t ∈ Z. For α ∈ [0, 1] and λ ∈ {1, 2}, 

V (t)

α

 −   −  + + = [V1 (t)]α × [V2 (t)]α = v1,α (t), v1,α (t) × v2,α (t), v2,α (t) , − − − vλ,α (t) = w(t, ηλ− , n − λ ) + (1 − α)w(t − 1, ηλ , n λ ), + + + vλ,α (t) = w(t, ηλ+ , n + λ ) + (1 − α)w(t − 1, ηλ , n λ ),

w(t + 1, ηλ− , n − λ) (t − 1)π t(t = sin w(t, ηλ− , n − λ ) + cos t w(t + 1, ηλ+ , n + λ) t(t (t − 1)π = sin w(t, ηλ+ , n + λ ) + cos t

− 1)π w(t − 1, ηλ− , n − λ ), 2 − 1)π w(t − 1, ηλ+ , n + λ) 2

(6.9)

− − − − + + + + + for w(2, ηλ− , n − λ ) = ηλ , w(1, ηλ , n λ ) = n λ , w(2, ηλ , n λ ) = ηλ , w(1, ηλ , n λ ) = + − + − + n λ , ηλ , ηλ , n λ , n λ ∈ N. In the real practice, it is difficult to obtain the exact numbers of the rabbits in the wild. Assume that the numbers of the rabbits continuously increase at the time t, the interval  −  −   + + v1,α (t), v1,α (t) is the range of the numbers of the male rabbits and v2,α (t), v2,α (t) is the range of the numbers of the female rabbits. Since the limits of the living conditions in the wild, it cannot be guaranteed that all of rabbits are survival. Hence, assume also that the numbers of the rabbits at time t + 1 is of survival and at the sum of the numbers of the rabbits at time t with the rate sin (t−1)π t t(t−1)π time t − 1 with the rate cos 2 of survival (maybe the rate of survival at time t − 1 is negative), i.e., (6.9) holds. + In particular, for η1− = η1+ = 6 and n − 1 = n 1 = 3, the status of the numbers range of the male rabbits see Fig. 6.1, which is an unique S-almost periodic fuzzy vector-valued solution

148

6 Almost Periodic Generalized Fuzzy Multidimensional …

Fig. 6.1 The status of the numbers range of the male rabbits

for the component V1 (t) of (6.8). Similarly, for any other cases, we can obtain the status of the numbers range of the rabbits.

Application 6.2.5 Consider the generalized two-dimensional homogeneous fuzzy dynamic equations as follows: V Δ (t) = A(t)V (t). (6.10) For T = R, V (t) = [V1 (t), V2 (t)]T , where 

V (t)

α

 α  α  −   −  + + = V1 (t) × V2 (t) = v1,α (t), v1,α (t) × v2,α (t), v2,α (t) ,    k sin(5tπ ) + cos t −a(cos t + sin 2t)V2 (t) , A(t) = −l(cos t + sin 2t) b(cos t + sin 2t)V1 (t) α,(I I )

which is the fuzzy predator-prey dynamic system . Since I V1 ,V2 ≥ 0, by Sect. 2.2, the fuzzy dynamic system (6.10) with the predator-prey application can be rewritten as  

  (−a)(cos t + sin 2t)V2 (t)  V1 (t) V1 (t) k sin(5tπ ) + cos t V1 (t)+ = .

b(cos t + sin 2t)V1 (t)  V2 (t) V2 (t) −l(cos t + sin 2t)V2 (t)+

6.2

Applications on Fuzzy Dynamic Equations and Models on Time Scales

149

In another real practice, assume that there are some carps and pikes in the same pond,  (t)  (t) [V1 (t)]α = v1− (α), v1+ (α) is the range of the numbers of the craps and [V2 (t)]α =   − + (t) is the range of the numbers of the pikes with 100(1 − α) percent chance v2,α (t), v2,α since the numbers of the carps and pikes cannot be measured exactly in general. In particular, for k = 4, a = 2, l = 2, b = 3, we have 

V2 (t)  V1 (t)

α

 α  α  −  − + + = V1 (t)  V2 (t) = v1,α (t)v2,α (t), v1,α (t)v2,α (t) .

− − Hence, for v1,α (t) and v2,α (t), we have

+

,  , +   − − − − (t) (t) − 2(cos t + sin 2t)v1,α (t)v2,α (t) v1,α 4 sin(5tπ ) + cos t v1,α  −  = . (6.11) − − − −2(cos t + sin 2t)v2,α (t) + 3(cos t + sin 2t)v1,α (t)v2,α (t) v2,α (t)

Hence, the status of the S-almost periodic solution of the system (6.11) see Fig. 6.2. Similarly, + + (t) and v2,α (t), we can obtain its status. for v1,α

Fig. 6.2 The status of the S-almost periodic solution of the system (6.11)

150

6 Almost Periodic Generalized Fuzzy Multidimensional …

Based on [104], we introduce a new division of fuzzy vectors induced by a determinant algorithm and obtain the relation between the division and the corresponding multiplication proposed in [104]. It is well known that interval analysis is a difficult task that plays a very important role in studying fuzzy calculus and dynamic equations (see [64]). In general, the classical fundamental formulas of calculus are not true and cannot be applied to the fuzzy situation directly. The proposed multiplication and division induced by a determinant algorithm is well defined and can be used with the g H -difference to form a traditional mixed operation, which lays a foundation for obtaining the basic formulas of calculus under a fuzzy background. Using the determinant algorithm, we suggest six types of division for fuzzy vectors in a fuzzy multidimensional space:             u u u u u u , , , , , , v I˚ v I˚I v I I˚ I v I ˚V v V˚ v V˚I and obtain the basic formulas of calculus such as  Δ I  Δ I I  Δ I  Δ I I f f f f , , , ,.... g I˚ g I˚ g I˚I g I˚I Moreover, based on the determinant algorithm and fuzzy calculus on time scales, we develop a theory of almost periodic fuzzy multidimensional dynamic systems on time scales and several applications are provided. Since we establish the theory on non-translational shifts of time scales, the results are valid for other fuzzy multidimensional dynamic systems on various hybrid domains and especially can cover the almost periodic fuzzy q-dynamic systems (i.e., fuzzy quantum dynamic systems which has important applications in quantum theory).

A

Almost Anti-periodic Discrete Oscillation

In 2017, M. Kosti´c introduced an interesting notion of almost anti-periodic functions in Banach space and studied the relationship between the types of anti-periodic functions and almost periodic functions in Banach spaces. After this, some related works were published (see [49–53]). It is natural to ask how to define the almost anti-periodic discrete process and explore what properties they will possess. The answer of this question will make it possible to study the almost anti-periodic discrete dynamic equation and contribute to establishing almost anti-periodic results on time scales. In the literature [117], a theory of almost anti-periodic discrete oscillation was developed. The main aim of this appendix is to introduce the notion of the almost anti-periodic discrete process of the N -dimensional vector-valued and N × N matrix-valued functions and to establish the stability of the almost anti-periodic discrete solutions to the general N -dimensional mechanical system and underactuated Euler–Lagrange system. For some relating the finite-dimensional vector spaces, one may see [39] and the underactuated Euler– Lagrange system with N degrees of freedom and m independent controls (see [118]), as well as the general N -dimensional mechanical system (see [25]).

A.1

Almost Anti-periodic Discrete Functions

In this section, we will introduce the notions of the almost anti-periodic discrete functions for the N × N matrix-valued function and the N -dimensional vector-valued function and establish some of their basic properties. Definition A.1 Let M(·) = [m i j (·)] N ×N : Z → R N ×N be an N × N matrix-valued discrete function and p(·) = [ p1 (·), . . . , p N (·)]T : Z → R N be a N -dimensional vector-valued discrete function, we define © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 C. Wang and R. P. Agarwal, Dynamic Equations and Almost Periodic Fuzzy Functions on Time Scales, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-11236-2

151

152

Appendix: Almost Anti-periodic Discrete Oscillation N N N 1  1  |m (n)|,  p(n) = | p j (n)|. i j N2 N

M(n) =

j=1 i=1

j=1

If for ∀ε > 0, there exists a positive integer l(ε) and τ ∈ [m, m + l]Z , ∀m ∈ Z, such that p(n) satisfies the following condition  p(n + τ ) + p(n) < ε, ∀n ∈ Z. Then, p(n) is called the almost anti-periodic discrete function, τ is called the ε-almost anti-period of p(n), and l is called an inclusion length. Similarly, M(n) is an almost antiperiodic N × N matrix-valued discrete function if M(n + τ ) + M(n) < ε, ∀n ∈ Z. The set of all almost anti-periodic N × N matrix-valued discrete functions is denoted by G. Definition A.2 ([53]) Let {q(n)} be a discrete sequence, if for ∀ε > 0, there exists a positive integer l(ε) such that q(n) satisfies the following condition |q(n + τ ) − q(n)| < ε, ∀n ∈ Z for some τ ∈ [m, m + l]Z , where m ∈ Z. Then, q(n) is called the almost periodic discrete function and τ is called the ε-almost period of q(n). Lemma A.1 Let M(n) be an almost anti-periodic discrete function; then, it is an almost periodic discrete function. Proof By Definition A.1, we have M(n + 2τ ) − M(n) ≤ |M(n + 2τ ) + M(n + τ ) + M(n + τ ) + M(n) < 2ε. The proof is completed.



Through Lemma A.1, the following lemma is immediate. Lemma A.2 ([105]) Let M(n) be an almost periodic discrete function (or an almost antiperiodic discrete function). Then, M(·) : Z → R is bounded. In what follows, some basic properties of the almost anti-periodic discrete functions will be established.

Appendix: Almost Anti-periodic Discrete Oscillation

153

Theorem A.1 Let M(·) ∈ G, then cM(·) ∈ G for all c ∈ R. Proof By Definition A.1, one has M(n + τ ) + M(n)
0, there exists N0 > 0 such that 

S(n) < ε.

|n|>N0

By Definition A.1 and Theorem A.2 , one has B(n + τ − m) + B(n − m)
0 and any n ∈ Z. Hence, we have

154

Appendix: Almost Anti-periodic Discrete Oscillation

       M(n + τ ) + M(n) =  S(m)B(n + τ − m) + S(m)B(n − m)  m∈Z  m∈Z       ≤ S(m)B(n + τ − m) + S(m)B(n − m)   |m|≤N0  |m|≤N0      + S(m)B(n + τ − m) + S(m)B(n − m)   |m|>N0 |m|>N0        ≤ S(m)   B(n + τ − m) + B(n − m)  |m|≤N0             + S(m)B(n + τ − m) +  S(m)B(n − m)  |m|>N0  |m|>N0           ≤ S0 Sε0 +  S(m) S(m)  λ +  λ ≤ ε + 2λ, |m|>N0

|m|>N0

which means M(·) ∈ G. The proof is completed.



Theorem A.4 Let M(·) ∈ G, then M(n) is an almost periodic discrete function. Proof Since M(·) ∈ G, we have M(n + τ ) + M(n) < ε, n ∈ Z. Hence,  N N      M(n + τ ) − M(n) =  12 |m i j (n + τ )| − N j=1 i=1

≤

1 N2

1 N2

N N   j=1 i=1

  |m i j (n)|

N  N    |m i j (n + τ )| − |m i j (n)| j=1 i=1 N  N  

 m i j (n + τ ) + m i j (n) j=1 i=1  = M(n + τ ) + M(n) < ε, ≤

1 N2

which means that {M(n)} is an almost periodic discrete function. The proof is completed.  Based on the theorems above, the following result can be proved. Proposition A.1 Let M(·) ∈ G, c > 0, if M(n) ≥ c > 0. Then, the following limit exists: lim

n→∞

 n j=1

1 M( j)

n

.

(A.1)

Appendix: Almost Anti-periodic Discrete Oscillation

 Proof Let h(n) =

n

155

1 M( j)

n

, we have

j=1

 n  1  ln [h(n)] = ln M( j) . n j=1

Step 1. We will prove that M(n) is bounded. Since M(n) is almost anti-periodic, by Theorem A.4, we obtain M(n), which is almost periodic. Moreover, by Lemma A.2, one has M(n), which is bounded; i.e., there exists λ ∈ R+ such that supn∈Z | ln M(n)| = λ. Step 2. We will prove that ln M(n) is an almost periodic discrete function. Since

 f (x) = ln x is uniformly continuous on c, eλ , i.e., for ∀ε > 0, there exists δ(ε) > 0 such that |x1 − x2 | < δ implies | f (x1 ) − f (x2 )| < ε,

λ for x1 , x2 ∈ c, e . On the other hand, M(·) ∈ G, i.e., M(τ + n) + M(n) < ε. By Theorem A.4, one has   M(τ + n) − M(n) < ε. 

By Step 1 and M(n) ≥ c > 0, we have M(n) ∈ c, eλ . Thus, one has | ln M(n + τ ) − ln M(n)| < ε for all n ∈ Z, which means that ln M(n) is almost periodic, i.e., for ∀ε > 0, there exists a positive integer l0 (ε) such that M(n) satisfies the following condition: | ln M(n + τ ) − ln M(n)| < ε, n ∈ Z, for some τ ∈ [a, a + l0 ]Z and a ∈ Z. Step 3. By Step 2, for τ ∈ [a, a + l0 ]Z and ∀ 4ε > 0, there exists an positive integer l0 (ε) such that M(n) satisfies the following condition: | ln M(n + τ ) − ln M(n)|
s, we have x(n + 1) = R n+1−s x(s) +

0 

R j b(n − j)

j=n−s 1 

= R n+1−s x(s) + R j b(n − j) + b(n) j=n−s   1  = R R n−s x(s) + R j−1 b(n − j) + b(n) = Rx(n) + b(n) j=n−s

and x(s) = x0 , which means that x(n) is a solution of Eq. (A.4). Step 2, we will prove the uniqueness of the solution of Eq. (A.5). Let x(n) and y(n) be two solutions of Eq. (A.5) with the initial value condition x(s) = y(s) = x0 , then we have x(n) = Rx(n − 1) + b(n − 1) and y(n) = Ry(n − 1) + b(n − 1). Assume that V (n) = x(n) − y(n), then V (n) = RV (n − 1). Hence, V (n) = R n−s V (s). On the other hand, x(s) = y(s) = x0 , i.e., V (s) = 0, which implies V (n) = 0, i.e., x(n) = y(n). The proof is completed. 

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161

Lemma A.3 Let τ be a ε-almost anti-period of f (n), if R + λB −1  < 1 − λ, R ≤ λ, S(q) + S( p) ≤ Bλ−1   p + q for p, q ∈ R N and 0 < λ < 1. Then,  b(n + τ ) + b(n) ≤ υq(n + τ ) + q(n) q(n + 1 + τ ) + q(n + 1) < ζ q(n + τ ) + q(n) where υ ∈ (λ, 1) and 0 < ζ =

R+λB −1  1−R

< 1.

 T Proof Since b(n) = 0, B −1 f (n) − B −1 S q(n) , we have b(n τ ) + b(n)  +      B −1 f (n + τ ) − B −1 S q(n + τ ) + B −1 f (n) − B −1 S q(n)  =       −1 S q(n)  + τ ) + B ≤  B −1 f (n + τ ) + B −1 f (n) +  B −1 S q(n  ≤ B −1 ε + B −1  Bλ−1  q(n + τ ) + q(n)   = B −1 ε + λq(n + τ ) + q(n). Hence, there exists υ ∈ (λ, 1) such that  b(n + τ ) + b(n) ≤ υq(n + τ ) + q(n). On the other hand, τ is a ε-almost anti-period of f (n) and x(n) = [ p(n), p(n + 1)]T , we have = = ≤ ≤ + ≤

x(n + 1 + τ ) + x(n + 1) q(n + 1 + τ ) + q(n + 1) + q(n + 2 + τ ) + q(n + 2) Rx(n + τ ) + b(n + τ ) + Rx(n) + b(n) Rx(n + τ ) + x(n) + b(n + τ ) + b(n)   R q(n + τ ) + q(n) + q(n + 1 + τ ) + q(n + 1)   B −1   f (n + τ ) + f (n) + S(q(n + τ )) + S(q(n)) (R + λB −1 )q(n + τ ) + q(n) + Rq(n + 1 + τ ) + q(n + 1) + B −1 ε,

i.e., (1 − R)q(n + 1 + τ ) + q(n + 1) + q(n + 2 + τ ) + q(n + 2) ≤ (R + λB −1 )q(n + τ ) + q(n) + B −1 ε. Hence, q(n + 1 + τ ) + q(n + 1) < The proof is completed.

R + λB −1  q(n + τ ) + q(n). 1 − R 

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Now, consider the underactuated Euler–Lagrange system with N degrees of freedom and m independent controls by a discrete dynamic equation as follows:    B q(n) q(n + 2) − 2q(n + 1) + q(n)   +C q(n), q(n + 1) − q(n) [q(n + 1) − q(n)] + G(q(n)) = Lω,

(A.6)

where B is a positive definite matrix denoting the inertia matrix, C is the Coriolis matrix, G is the gravity vector, L = [0, Im ], q(·) = [q1 (·), . . . , q N (·)] ∈ R N is the generalised coordinate vector, and ω ∈ Rm is the control. Assume that x(n) = [q(n), q(n + 1)]T , Eq. (A.6) can be rewritten as   q(n + 1) x(n + 1) = q(n + 2)     0 q(n + 1)  

  + = (A.7) B −1 q(n) Lω − G q(n) γ q(n) + γ1 q(n + 1)  0  0 I = x(n) + a(n) γ0 γ1 = H (n)x(n) + a(n), where

  γ0 = B −1 q(n) C(q(n), q(n + 1) − q(n)) − I ,   γ1 = 2I − B −1 q(n) C(q(n), q(n + 1) − q(n)),

and    0 0 I  

  . , a(n) = H (n) = B −1 q(n) Lω − G q(n) γ0 γ1 

Theorem A.6 Let s < n, the unique solution (A.7) with the initial value x(s) = x0 can be given as x(n) =

 n−1  j=s

  j+1) n−2  n−(   H ( j) x0 + H (n − i) a( j) + a(n − 1). j=s

i=1

Proof The proof process is similar to the proof process of Theorem A.5, we will not repeat it here.  Lemma A.4 Let τ ∈ Z+ ,  −1     B (q)C(q, p) − I  + 2I − B −1 (q)C(q, p) < η−λ − 1 < 1 − 1 , 1+η N N



B −1 (q) Lω − G(q)] + B −1 ( p) Lω − G( p)] ≤ λ p + q, H (n)a(n) + H (s)a(s) ≤ max{H (n), H (s)}a(n) + a(s)

Appendix: Almost Anti-periodic Discrete Oscillation

163

for any p, q ∈ R N , n, s ∈ Z and 0 < λ < η < 1. Then,  a(n + τ ) + a(n) ≤ λq(n + τ ) + q(n), q(n + 1 + τ ) + q(n + 1) < ηq(n + τ ) + q(n), H (n + τ )a(n + τ ) + H (n)a(n) ≤ max{H (n + τ ), H (n)}a(n + τ ) + a(n). Proof Since

 a(n) =

 0  

  , B −1 q(n) Lω − G q(n)

  

  a(n + τ ) + a(n) =  B −1 q(n + τ ) Lω − G q(n + τ )  

  −1 q(n) Lω − G q(n)  +B    ≤ λ q(n + τ ) + q(n)

we have

and x(n + 1 + τ ) + x(n + 1) q(n + 1 + τ ) + q(n + 1) + q(n + 2 + τ ) + q(n + 2) H (n)x(n + τ ) + a(n + τ ) + H (n)x(n) + a(n) H (n)x(n + τ ) + x(n) + a(n + τ ) + a(n)    H (n) q(n + τ ) + q(n) + q(n + 1 + τ ) + q(n + 1) + λq(n + τ ) + q(n) ,

= = ≤ ≤ i.e.,

(1 − H (n))q(n + 1 + τ ) + q(n + 1) + q(n + 2 + τ ) + q(n + 2) ≤ (H (n) + λ)q(n + τ ) + q(n). Hence, q(n + 1 + τ ) + q(n + 1) < and

q(n + 2 + τ ) + q(n + 2) < (H (n) + λB −1 )q(n + τ ) + q(n).

On the other hand, since

 H (n) =

where

H (n) + λ q(n + τ ) + q(n) 1 − H (n)

 0 I , γ0 γ1

  γ0 = B −1 q(n) C(q(n), q(n + 1) − q(n)) − I ,   γ1 = 2I − B −1 q(n) C(q(n), q(n + 1) − q(n)),

we have

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Appendix: Almost Anti-periodic Discrete Oscillation

 

  H (n) = N12 N +  B −1 q(n) C(q(n), q(n + 1) − q(n)) − I       + 2I − B −1 q(n) C(q(n), q(n + 1) − q(n)) η−λ 1 ≤ N1 + η−λ 1+η − N = 1+η < 1, i.e.,

H (n)+λ 1−H (n)

< η. Thus, q(n + 1 + τ ) + q(n + 1) < ηq(n + τ ) + q(n). Since

H (n)a(n) + H (s)a(s) ≤ max{H (n), H (s)}a(n) + a(s), we have H (n + τ )a(n + τ ) + H (n)a(n) ≤ max{H (n + τ ), H (n)}a(n + τ ) + a(n). 

The proof is completed.

A.2.2

Stability of the Almost Anti-periodic Solutions

In this section, we will establish the stability of the general N -dimensional mechanical system and the underactuated Euler–Lagrange system under the almost anti-periodic discrete process. Through Eqs. (A.5) and (A.7), the systems (A.4) and (A.6) can be turned into the following nonhomogeneous linear system x(n + 1) = M(n)x(n) + c(n), n ∈ Z,

(A.8)

which leads to the corresponding homogeneous linear system x(n + 1) = M(n)x(n), n ∈ Z, where



0 I M(n) = R = B −1 (C − B) B −1 (2B − C) and

in Eq. (A.5),

(A.9) 

 0   c(n) = b(n) = B −1 f (n) − B −1 S q(n) 

 0 I , γ γ   0 1 0  

  c(n) = a(n) = B −1 q(n) Lω − G q(n) 

M(n) = H (n) =

in Eq. (A.7), where   γ0 = B −1 q(n) C(q(n), q(n + 1) − q(n)) − I ,

Appendix: Almost Anti-periodic Discrete Oscillation

165

  γ1 = 2I − B −1 q(n) C(q(n), q(n + 1) − q(n)). Next, we will establish the stability of Eq. (A.8) to obtain the stability of Eqs. (A.4) and (A.6). For convenience, denote M∗ := inf{M(n) : n ∈ Z}, M ∗ := sup{M(n) : n ∈ Z}. Theorem A.7 ([33]) Let n, s ∈ Z,

Ψ (n, s) =

⎧n−s ⎪ ⎪ M(n − j), n > s, ⎪ ⎨ j=1 I , n = s, ⎪ ⎪ ⎪ ⎩ 0, n < s.

Then, the following results hold. (i) The zero solution of Eq. (A.9) is stable if and only if   Ψ (n, s) ≤ K (s), where K (s) is a positive constant dependent on s. (ii) The zero solution of Eq. (A.9) is uniformly stable if and only if   Ψ (n, s) ≤ K , where K is a positive constant independent of s. (iii) The zero solution of Eq. (A.9) is asymptotically stable if and only if   lim Ψ (n, s) = 0. n→∞

(iv) The zero solution of Eq. (A.9) is uniformly asymptotically stable if and only if   Ψ (n, s) ≤ K ηn−s for some constants K > 0 and η ∈ (0, 1), n ≥ s. Theorem A.8 If all the conditions of Lemma A.3 hold for M(n) = R and c(n) = b(n) (or all the conditions of Lemma A.4 hold for M(n) = H (n) and c(n) = a(n)) and λ0 = lim

n→∞

 n

1 M( j)

n

< 1.

(A.10)

j=1

Then, Eq. (A.8) has a unique uniformly asymptotically stable almost anti-periodic solution

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Appendix: Almost Anti-periodic Discrete Oscillation

x0 (n) =

j−1 n−2  n−   j=−∞

 M(n − i) c( j) + c(n − 1) n ∈ Z.

i=1

Moreover, Eq. (A.4) (or Eq. (A.6)) has an unique uniformly asymptotically stable almost anti-periodic solution. Proof Step 1. We will prove that x0 (n) is well-defined for all n ∈ Z. By Proposition A.2 (ii) and Eq. (A.10), we can obtain 1 lim n→∞ n

  −1

ln M( j) −

n 

j=−n

i.e., 1 lim n→∞ n

 ln M( j) = 0,

j=1

  −1

Hence, lim

 ln M( j) = ln λ0 .

j=−n

  −1

n→∞

1 M( j)

n

= λ0 ,

j=−n

which implies that for ∀ε > 0, there exists N2 > 0 such that  −1  1 n      M( j) − λ 0 < ε  j=−n

for n > N2 , i.e., λ0 − ε