Theory of Fuzzy Differential Equations and Inclusions (Series in Mathematicalanalysis and Applications)

This volume details the current state of the theory of fuzzy differential equations and inclusions and a systematic account of recent developments. tf1617_fm.pdf......Page 1 Theory of Fuzzy Differential Equations and Inclusions......Page 3 Table of Contents......Page 5 Preface......Page 7 1.1 Introduction......Page 10 Table of Contents......Page 0 1.2 Fuzzy Sets......Page 11 1.3 THe Hausdorff Metric......Page 14 Proposition 1.3.4.......Page 16 1.4 Support Functions......Page 17 1.5 The Space En......Page 19 Lemma 1.5.1.......Page 20 Proof.......Page 21 Theorem 1.5.2.......Page 22 Proof.......Page 23 Proof.......Page 24 Proof.......Page 25 1.6 Metric Space......Page 26 Example 1.6.1.......Page 27 Proof.......Page 28 1.7 Note and Comments......Page 30 2.2 Convergence of Fuzzy Sets......Page 31 Proof.......Page 32 Lemma 2.2.3.......Page 33 Proof.......Page 34 Example 2.2.1.......Page 35 Proof.......Page 36 Proof.......Page 38 Case 1.......Page 39 Case 2.......Page 40 Remark 2.2.1.......Page 41 Proof.......Page 42 Proof.......Page 43 Proof.......Page 44 Rernark 2.4.1.......Page 45 Proof.......Page 46 Theorem 2.4.3.......Page 47 Proof.......Page 48 Example 2.4.2.......Page 49 Theorem 2.5.1.......Page 50 Proof.......Page 51 Proof.......Page 52 Proof.......Page 53 Example 2.5.1.......Page 54 2.6 Notes and Comments......Page 55 3.1 Introduction......Page 56 Proof.......Page 57 Example 3.2.1.......Page 58 Proof.......Page 59 Theorem 3.4.1.......Page 60 Proof.......Page 61 Theorem 3.4.3.......Page 63 Corollary 3.4.1.......Page 64 3.5 Convergence of Successive Approximations......Page 65 Proof.......Page 66 Lemma 3.6.1.......Page 68 Proof of Theorem 3.6.1.......Page 69 Proof.......Page 70 Theorem 3.8.1.......Page 71 3.9 Stability Criteria......Page 72 Definition 3.9.1.......Page 74 Proof.......Page 75 3.10 Notes and Comments......Page 76 4.1 Introduction......Page 78 Proof.......Page 79 4.3 Stability Criteria......Page 81 Theorem 4.3.2.......Page 82 Proof.......Page 83 4.4 Nonuniform Stability Criteria......Page 84 Proof.......Page 85 Proof.......Page 87 Remark 4.4.2.......Page 88 4.5 Criteria for Boundedness......Page 89 Theorem 4.5.1.......Page 90 Theorem 4.5.3.......Page 91 Theorem 4.5.4.......Page 92 Proof.......Page 93 4.6 Fuzzy Differential Systems......Page 94 Proof.......Page 96 Corollary 4.6.1.......Page 97 Proof.......Page 98 Proof.......Page 100 4.8 Linear Variation of Parameters Formula......Page 101 4.9 Notes and Comments......Page 104 5.1 Introduction......Page 105 Proof.......Page 106 Theorem 5.2.2.......Page 107 Proof.......Page 108 Proof.......Page 109 Theorem 5.3.1.......Page 110 Theorem 5.3.2.......Page 111 Proof.......Page 112 Corollary 5.3.1.......Page 113 Proof.......Page 114 Proof.......Page 116 Theorem 5.4.2.......Page 117 Proof.......Page 118 Proof.......Page 119 5.5 Hybrid Fuzzy Differential Equations......Page 120 Proof.......Page 121 Proof.......Page 123 Proof.......Page 125 5.7 Boundary Value Problem......Page 126 Proof.......Page 127 Proof.......Page 128 Proof.......Page 129 Theorem 5.8.2.......Page 131 Theorem 5.9.2.......Page 132 Theorem 5.9.3.......Page 133 Definition 5.9.1.......Page 134 Proof.......Page 135 5.10 Notes and Comments......Page 138 6.1 Introduction......Page 139 6.2 Formulation of Fuzzy Diffenrential Inclusions......Page 142 Proof.......Page 143 Theorem 6.2.2.......Page 144 Proof.......Page 145 6.3 Differential Inclusions......Page 146 Example 6.3.1.......Page 148 Theorem 6.3.2.......Page 149 6.4 Fuzzy Differential......Page 150 Proof.......Page 151 Example 6.4.2.......Page 152 Example 6.4.3.......Page 154 Example 6.4.4.......Page 155 6.5 The Variation of Constants Formula......Page 156 Lemma 6.5.1.......Page 157 Example 6.5.1.......Page 158 Example 6.5.2.......Page 160 6.6 Fuzzy Volterra Integral Equations......Page 161 Remark 6.6.1.......Page 163 Remark 6.6.2.......Page 164 Lemma 6.6.1.......Page 165 Proof.......Page 166 Example 6.6.1.......Page 168 Example 6.6.2.......Page 169 Proof.......Page 170 6.7 Notes and Comments......Page 172 Bibliography......Page 173 "Fuzzy differential functions are applicable to real-world problems in engineering, computer science, and social science. That relevance makes for rapid development of new ideas and theories. This volume is a timely introduction to the subject that describes the current state of the theory of fuzzy differential equations and inclusions and provides a systematic account of recent developments. The chapters are presented in a clear and logical way and include the preliminary material for fuzzy set theory; a description of calculus for fuzzy functions, an investigation of the basic theory of fuzzy differential equations, and an introduction to fuzzy differential inclusions."--Provided by publisher

fuzzy Differential Functions Are Applicable To Real-world Problems In Engineering, Computer Science, And Social Science. That Relevance Makes For Rapid Development Of New Ideas And Theories. This Volume Is A Timely Introduction To The Subject That Describes The Current State Of The Theory Of Fuzzy Differential Equations And Inclusions And Provides A Systematic Account Of Recent Developments. The Chapters Are Presented In A Clear And Logical Way And Include The Preliminary Material For Fuzzy Set Theory; A Description Of Calculus For Fuzzy Functions, An Investigation Of The Basic Theory Of Fuzzy Differential Equations, And An Introduction To Fuzzy Differential Inclusions.

Detailing the theory of fuzzy differential equations and inclusions and a systematic account of recent developments, this text provides preliminary material of fuzzy set theory; description of calculus for fuzzy functions; an investigation of the basic theory of fuzzy differential equations; and an introduction to fuzzy differential inclusions. An exact description of any real world phenomenon is virtually impossible and one needs to accept this fact and adjust to it.