Ulam type stability

Preface......Page 5
Contents......Page 7
Contributors......Page 9
 1.1 Introduction......Page 11
 1.2 Stability of Maximum Preserving Functional Equations on Banach Lattices......Page 12
 1.3 Functional Equation Involving Both Lattice Operations......Page 21
  1.3.1 The Main Results of This Section......Page 23
 1.4 Schwaiger's Type Functional Equation......Page 29
 1.5 A More General Form of Problem 1.1 with Some Solutions......Page 36
  1.5.1 The Main Results of This Section......Page 37
  1.5.2 Proof of the Main Results......Page 38
 1.6 Lattice-Valued Maps Defined on Semigroups......Page 43
  1.6.1 Examples and Basic Properties of Solutions of (1.60), (1.61) and (1.62)......Page 45
  1.6.2 Separation Theorems in Lattice Environments......Page 48
  1.6.3 Stability Theorems......Page 52
 References......Page 55
2 A Purely Fixed Point Approach to the Ulam-Hyers Stability and Hyperstability of a General Functional Equation......Page 57
 2.1 Introduction......Page 58
 2.2 Stability of Eq. (2.1)......Page 59
 2.3 Hyperstability of Eq. (2.1)......Page 64
 References......Page 66
 3.1 Introduction......Page 67
 3.2 Birkhoff–James Orthogonality and Approximate Orthogonality......Page 68
  3.3.1 Exact Preservation or Reversal of the Orthogonality......Page 69
   3.3.2.1 AOP Operators: Review of Results......Page 71
   3.3.2.2 AOR Operators......Page 72
 3.4 Stability Problems......Page 74
 References......Page 80
4 Optimal Forward Contract Design for Inventory:A Value-of-Waiting Analysis......Page 82
 4.1 Problem Formulation and Model......Page 83
 4.2 Optimality: The Censor and Value of Waiting......Page 85
 4.3 Sensitivity and the Censor Functional Equation......Page 87
 4.4 Cobb-Douglas Revenue: Asymptotic Results......Page 88
 4.5 Cobb-Douglas Optimal Timing: Estimates......Page 90
 4.6 Asymptotic Analysis: The Proofs......Page 93
 4.7 Censor Comparative Statics: Reprise......Page 97
 4.8 Appendix......Page 100
 References......Page 105
 5.1 Quasi-Normed Spaces and Quasi-β-Normed Spaces......Page 106
 5.2 Ulam-Hyers Stability of Functional Equations in Quasi-Banach Spaces......Page 117
  5.2.1 Ulam-Hyers Stability of Functional Equations in p-Banach Spaces......Page 118
  5.2.2 Ulam-Hyers Stability of Functional Equations in Quasi-Banach Spaces Which Are Not Assumed to be p-Banach......Page 122
 5.3 Ulam-Hyers Stability of Functional Equations in Quasi-β-Banach Spaces......Page 128
  5.3.1 Ulam-Hyers Stability of Functional Equations in Quasi-β-Banach Spaces......Page 129
  5.3.2 Some Open Problems in Ulam-Hyers Stability of Functional Equations in Quasi-β-Normed Spaces......Page 135
 References......Page 136
 6.1 Introduction......Page 140
 6.2 Preliminaries......Page 141
 6.3 Fixed Point Theorem......Page 143
 6.4 Stability......Page 144
 6.5 An Observation on Superstability......Page 148
 References......Page 149
 7.1 Introduction......Page 151
 7.2 The Functional Equation......Page 152
  7.3.1 The Kernel as a Function in y......Page 153
  7.3.2 The Kernel as a Function in x......Page 154
 7.4 Solution of the Functional Equation......Page 156
 7.5 Remarks on Stability of the Functional Equation......Page 158
 References......Page 159
 8.1 Introduction......Page 161
 8.2 Approximation of Functions by Additive and by Quadratic Mappings......Page 163
 8.3 Approximation of Functions by Cubic Mappings......Page 165
 References......Page 171
9 Solutions and Stability of Some Functional Equationson Semigroups......Page 174
 9.1 Introduction......Page 175
 9.2 Stability of a Variant of the Jensen Functional Equation......Page 177
 9.3 Hyers-Ulam Stability of Eq. (9.1) on Amenable Semigroups......Page 182
 9.4 Application: Stability of the Symmetric Functional Equation (9.11)......Page 187
 9.5 μ-Jensen Functional Equation......Page 188
 9.6 Solutions of μ-Quadratic Functional Equation......Page 193
 9.7 Stability of the μ-Jensen Functional Equation......Page 197
 9.8 Stability of the μ-Quadratic Functional Equation......Page 200
 References......Page 204
10 Bi-additive s-Functional Inequalities and Quasi-*-Multipliers on Banach *-Algebras......Page 206
 10.1 Introduction and Preliminaries......Page 207
 10.2 Bi-additive s-Functional Inequality (10.1)......Page 208
 10.3 Bi-additive s-Functional Inequality (10.2)......Page 212
 10.4 Quasi-*-Multipliers in C*-Algebras......Page 214
 References......Page 221
 11.1 Introduction......Page 223
 11.2 The Main Result......Page 224
 11.3 Proof of Theorem 11.2......Page 226
 References......Page 233
 12.1 Introduction......Page 236
 12.2 Stability......Page 237
 12.3 Stability of System......Page 240
 12.5 Stability of Alternation......Page 241
 12.6 Stability of the Conditional Functional Equation......Page 242
 12.8 Stability of the Squares of Functional Equations......Page 247
 12.9 Inverse Stability......Page 248
 12.10 Inverse Stability of System......Page 249
 12.11 Inverse Stability for Alternation......Page 250
 12.13 Approximation of Approximation......Page 252
 12.15 Stability in the Class......Page 253
 12.16 Stability of the Translation Equation......Page 255
 12.17 Stability of the Dynamical System......Page 261
 12.18 b-Stability and Inverse b-Stability......Page 263
 12.19 Stability of Difference Equations......Page 266
 12.20 On the Stability of Idempotent Function with Constant Derivative......Page 267
 References......Page 275
 13.1 Introduction and Motivation......Page 277
  13.1.1 Continuous-Time Analogue......Page 283
  13.1.2 Control-Theory Analogies......Page 284
  13.1.3 The Accounting Context......Page 285
 13.2 Main Theorems and Auxiliary Propositions......Page 286
  13.2.1 Preliminaries......Page 288
 13.3 Eigenvalue Location: General Analysis......Page 294
  13.3.1 Root Locus for Two-Pole Case......Page 297
  13.3.2 Multi-Pole Case: A Distortion Estimate......Page 298
  13.3.3 Analysis of the Real Part via Taylor's Theorem......Page 299
  13.4.1 The Case n=2......Page 300
  13.4.2 General Case δ1ω1<0 with δjωj>0 for j=2,…,n......Page 302
  13.4.4 General n: Case when δkωk=0 for some k=2,…,n with δjωj>0 for Remaining Indices j......Page 303
  13.4.5 Effect of the Dividend-on-Dividend Multiplier: A Distortion Example......Page 304
   13.4.5.1 A Case Study: Distorted Circle......Page 305
 13.5 Differential Properties of Eigenvalues: Some Bifurcation Analysis......Page 306
 13.6 Proof of the Dominance Theorem......Page 311
 13.7 Obtaining Dividend Irrelevance......Page 313
 13.8 Derivation of Equivalences......Page 315
 13.9 Invertible Parametrization and Zero Placement......Page 316
 13.10 Asymptotics of the Unbounded Roots......Page 321
 13.11 Strip-and-Two-Circles Theorem......Page 324
 13.12 The Third-Derivative Test......Page 326
 References......Page 327
 14.1 Introduction......Page 329
 14.3 The Results......Page 330
 References......Page 334
 15.1 Introduction......Page 335
 15.3 Symmetry of Birkhoff-James Orthogonality in Operator Spaces......Page 339
 15.4 Future Directions for Research and Ulam Stability......Page 345
 References......Page 346
 16.1 Introduction......Page 349
 16.2 Preliminaries......Page 350
  16.2.1 Retraction-Displacement Condition in the Metric Fixed Point Theory......Page 352
  16.2.2 Operatorial Equations and Corresponding Ulam Inequations......Page 354
  16.2.3 Basic Notions in Ulam Stability......Page 356
 16.3 Ulam Stability of Zero Point Equations......Page 358
 16.4 Abstract Models of Ulam Stability: Examples......Page 360
  16.5.1 The Root Equation......Page 363
  16.5.4 More General Spaces......Page 364
 References......Page 365
 17.1 Introduction......Page 369
 17.2 Stability of the Cauchy Equation......Page 374
 17.3 Cauchy Difference as a Linear Operator......Page 382
 References......Page 384
 18.1 Introduction......Page 386
 18.2 Semi-inner Products......Page 387
 18.3 The Induced Seminorms......Page 389
 18.4 Some Further Properties of the Induced Seminorm......Page 393
 18.5 Parapreseminorms......Page 395
 18.6 Some Further Theorems on Parapreseminorms......Page 398
 18.7 Characterizations of Additive and Jensen Functions......Page 400
 18.8 Further Characterizations of Additive and Jensen Functions......Page 403
 References......Page 406
 19.1 Amenable Groups and G-Sets......Page 412
 19.2 Paradoxical Decompositions......Page 415
 19.3 Invariant Means......Page 416
 19.4 Elementary Groups......Page 420
 19.5 Growth of Groups......Page 427
 19.6 The Grigorchuk Group and the Burnside Groups......Page 429
 19.7 Ulam Stability......Page 431
 19.8 Invariant Means and Ulam-Stability......Page 434
 19.9 Elementary Operations and Stability......Page 437
 19.10 Stability of Linear Functional Equations......Page 441
 19.12 The Stability of the Sine and the Cosine Equations......Page 447
 19.13 A Functional Equation Related to Spherical Functions......Page 449
 19.14 Generalizations......Page 452
 References......Page 453
 20.1 Preliminaries......Page 455
  20.1.1 Function Modules......Page 456
 20.2 Multismoothness in Banach Function Modules......Page 457
 20.3 Norm Derivatives in Banach Function Module......Page 461
  20.3.2 Main Results......Page 462
 20.4 Stability of Orthogonality Equation in Normed Spaces......Page 468
 References......Page 470
 21.1 Introduction......Page 471
  21.2.1 ρ-Orthogonality......Page 475
  21.2.2 ρ*-Orthogonality......Page 476
  21.2.3 Some Generalized Norm Derivatives Orthogonality......Page 479
  21.2.4 The λ-Angularly Property of Norms......Page 482
  21.2.5 Approximate Norm Derivatives Orthogonalities......Page 487
  21.3.1 Linear Mappings Preserving Orthogonality......Page 491
  21.3.2 Mappings Which Exactly Preserve Norm Derivatives Orthogonality......Page 494
  21.3.3 Approximate Orthogonality Preserving Mapping......Page 499
 References......Page 505
Index......Page 510