Nonlinear Analysis and Variational Problems

by Pardalos, Panos M.; Rassias, Themistocles M.; Khan, Akhtar A.

• ISBN13: 9781441901576
• ISBN10: 1441901574
• Format: Hardcover
• Copyright: 2009-11-10
• Publisher: Springer Verlag

Summary

The chapters in this volume, written by international experts from different fields of mathematics, are devoted to honoring George Isac, a renowned mathematician. These contributions focus on recent developments in complementarity theory, variational principles, stability theory of functional equations, nonsmooth optimization, and several other important topics at the forefront of nonlinear analysis and optimization. "Nonlinear Analysis and Variational Problems" is organized into two parts. Part I, Nonlinear Analysis, centers on stability issues for functional equations, fixed point theorems, critical point theorems, W*-algebras, the Brezis'Browder principle, and related topics. Part II, Variational Problems, addresses several important aspects of optimization and variational methods. This includes equilibrium problems, projected dynamical system, set-valued and set-semidefinite optimization, variational inequalities, variational principles, complementarity problems, and problems in optimal control. In the last few decades, the theory of complementarity, functional stability and variational principles have provided a unified framework for dealing with a wide range of problems in diverse branches of pure and applied mathematics, such as finance, operations research, economics, network analysis, control theory, biology, and others. This volume is well-suited to graduate students as well as researchers and practitioners in the fields of pure and applied mathematics, social sciences, economics, operations research, engineering, and related sciences.

Table of Contents

Preface	p. vii
Biographical Sketch of George Isac	p. xi
List of Contributors	p. xxiii
Nonlinear Analysis
Discrete Approximation Processes of King's Type	p. 3
Introduction	p. 3
Further Results on Vn Type Operators	p. 4
A General Class in Study	p. 7
References	p. 11
Isometrics in Non-Archimedean Strictly Convex and Strictly 2-Convex 2-Normed Spaces	p. 13
Introduction and Preliminaries	p. 13
Non-Archimedean Strictly Convex 2-Normed Spaces	p. 15
Non-Archimedean Strictly 2-Convex 2-Normed Spaces	p. 18
References	p. 21
Fixed Points and Generalized Stability for ¿-Additive Mappings of Isac-Rassias Type	p. 23
Introduction	p. 23
Stability Properties for Cauchy Equation in ß-Normed Spaces	p. 25
Other Examples and Applications	p. 31
References	p. 35
A Remark on W*-Tensor Products of W*-Algebras	p. 37
Introduction	p. 37
The Ordered Involutive Banach Space	p. 39
The Multiplication	p. 45
References	p. 52
The Perturbed Median Principle for Integral Inequalities with Applications	p. 53
Introduction	p. 53
A Perturbed Version of the Median Principle	p. 56
Some Examples for 0th-Degree Inequalities	p. 57
Inequalities of the 1st-Degree	p. 62
References	p. 63
Stability of a Mixed Type Additive, Quadratic, Cubic and Quartic Functional Equation	p. 65
Introduction	p. 66
General Solution	p. 68
Stability	p. 74
References	p. 79
¿-Additive Mappings and Hyers-Ulam Stability	p. 81
Introduction	p. 81
Results	p. 82
References	p. 85
The Stability and Asymptotic Behavior of Quadratic Mappings on Restricted Domains	p. 87
Introduction	p. 87
Approximately Quadratic Mappings	p. 89
Quadratic Mappings on Restricted Domains	p. 93
References	p. 96
A Fixed Point Approach to the Stability of a Logarithmic Functional Equation	p. 99
Introduction	p. 99
Preliminaries	p. 101
Hyers-Ulam-Rassias Stability	p. 102
Applications	p. 106
References	p. 108
Fixed Points and Stability of the Cauchy Functional Equation in Lie C*-Algebras	p. 111
Introduction and Preliminaries	p. 111
Stability of Homomorphisms in C*-Algebras	p. 113
Stability of Derivations on C-Algebras	p. 117
Stability of Homomorphisms in Lie C*-Algebras	p. 119
Stability of Lie Derivations on C*-AIgebras	p. 121
References	p. 133
Fixed Points and Stability of Functional Equations	p. 125
Introduction and Preliminaries	p. 125
Fixed Points and Generalized Hyers-Ulam Stability of the Functional Equation (11.1): An Even Case	p. 127
Fixed Points and Generalized Hyers-Ulam Stability of the Functional Equation (11.1): An Odd Case	p. 130
References	p. 133
Compression-Expansion Critical Point Theorems in Conical Shells	p. 135
Introduction	p. 135
Main Results	p. 137
Proofs	p. 139
References	p. 145
Gronwall Lemma Approach to the Hyers-Ulam-Rassias Stability of an Integral Equation	p. 147
Introduction	p. 147
Gronwall Lemmas	p. 148
Stability of a Fixed Point Equation	p. 149
Stability of Volterra Integral Equations	p. 149
Stability of Fredholm Integral Equations	p. 150
References	p. 152
Brezis-Browder Principles and Applications	p. 153
Brezis-Browder Principles in General Separable Sets	p. 153
Introduction	p. 153
General Separable Sets	p. 154
Zorn-Bourbaki Principles	p. 161
Main Results	p. 163
Some Amorphous Versions	p. 167
Pseudometric Maximal Principles	p. 169
Introduction	p. 169
Logical Equivalents of Brezis-Browder's Principle	p. 170
Asymptotic Extensions	p. 171
Convergence and Uniform Versions	p. 173
Zorn Maximality Principles	p. 178
Relative KST Statements	p. 180
Introduction	p. 180
Maximal Principles	p. 181
Transitive (Pseudometric) Versions	p. 184
Main Results	p. 186
Extended KST Statements	p. 189
References	p. 193
Variational Problems
A Generalized Quasi-Equilibrium Problem	p. 201
Introduction	p. 201
Preliminaries	p. 202
Main Result	p. 203
Particular Cases of Theorem 15.8	p. 205
Applications	p. 209
References	p. 210
Double-Layer and Hybrid Dynamics of Equilibrium Problems: Applications to Markets of Environmental Products	p. 213
Introduction	p. 213
Dynamic Equilibrium Problems and Variational Inequalities	p. 215
General Formulation	p. 215
Double-Layer Dynamics and Hybrid Dynamical Systems	p. 220
DLD	p. 221
Tracking Equilibrium Dynamics: Hybrid Systems Approach	p. 222
Dynamics of Environmental Product Markets	p. 224
The Static Model	p. 224
Dynamic Equilibrium Model: EVI Formulation	p. 226
Example	p. 227
Dynamic Disequilibrium Model: DLD Formulation	p. 230
Conclusions and Acknowledgments	p. 232
References	p. 232
A Panoramic View on Projected Dynamical Systems	p. 235
Introduction	p. 235
General Background Material	p. 237
Spaces	p. 237
Cones and Properties	p. 241
Projectors	p. 242
Weighted Traffic Equilibrium Problem	p. 245
Time-Dependent Equilibria	p. 246
Projected Dynamical Systems in Hilbert Spaces	p. 247
Projected Dynamical Systems in Pivot Hilbert Spaces	p. 247
Projected Dynamical Systems in Non-pivot Hilbert Spaces	p. 248
Projected Dynamical Systems in Banach Spaces	p. 249
The Strictly Convex and Uniformly Smooth Case	p. 250
Projected Dynamical Systems and Unilateral Differential Inclusions	p. 251
Bridge with Variational Inequalities	p. 253
Conclusion	p. 256
References	p. 256
Foundations of Set-Semidefinite Optimization	p. 259
Introduction	p. 259
Applications of Set-Semidefinite Optimization	p. 261
Semidefinite Optimization	p. 261
Copositive Optimization	p. 262
Second-Order Optimality Conditions	p. 264
Semi-in finite Optimization	p. 265
Set-Semidefinite Cone	p. 267
Properties of the Set-Semidefinite Cone	p. 267
Dual and Interior of the Set-Semidefinite Cone	p. 271
Optimality Conditions	p. 274
Nonconvex Duality	p. 278
Future Research	p. 282
References	p. 283
On the Envelope of a Variational Inequality	p. 285
Introduction	p. 285
Auxiliary Variational Inequality	p. 287
A Particular Case	p. 290
References	p. 293
On the Nonlinear Generalized Ordered Complementarity Problem	p. 295
Introduction	p. 295
A Spectral Condition for the Generalized Ordered Complementarity Problem	p. 297
Existence and Uniqueness Results	p. 300
References	p. 303
Optimality Conditions for Several Types of Efficient Solutions of Set-Valued Optimization Problems	p. 305
Introduction	p. 305
Subdifferentials, Derivatives and Coderivatives	p. 307
Some Concepts of Efficient Points	p. 309
Optimality Conditions for Set-Valued Optimization Problem	p. 316
References	p. 323
Mean Value Theorems for the Scalar Derivative and Applications	p. 325
Introduction	p. 325
Preliminaries	p. 326
Scalar Derivatives and Scalar Differentiability	p. 327
Computational Formulae for the Scalar Derivatives	p. 328
Mean Value Theorems	p. 329
Applications to Complementarity Problems	p. 331
Comments	p. 340
References	p. 340
Application of a Vector-Valued Ekeland-Type Variational Principle for Deriving Optimality Conditions	p. 343
Introduction	p. 343
Properties of Cones	p. 345
An Ekeland-Type Variational Principle for Vector Optimization Problems	p. 349
Nonlinear Scalarization Scheme	p. 350
Differentiability Properties of Vector-Valued Functions	p. 353
Necessary Optimality Conditions for Vector Optimization Problems in General Spaces Based on Directional Derivatives	p. 357
Vector Optimization Problems with Finite-Dimensional Image Spaces	p. 363
References	p. 364
Nonlinear Variational Methods for Estimating Effective Properties of Multiscale Materials	p. 367
Introduction	p. 367
Preliminaries	p. 370
Some Nonlinear Bounds of Classical Type	p. 371
Some Useful Means of Power Type	p. 375
A Particular Power Type Mean	p. 376
Composition of Power Means	p. 380
Nonlinear Bounds	p. 386
Further Results for the Case p=2	p. 398
The Reiterated Cell Structure	p. 403
The Scalar Case	p. 404
The Vector-Valued Case	p. 405
Bounds Related to a Reynold-Type Equation	p. 407
Some Final Comments	p. 412
References	p. 412
On Common Linear/Quadratic Lyapunov Functions for Switched Linear Systems	p. 415
Introduction	p. 415
Preliminaries	p. 417
Matrix Theory Concepts	p. 417
Z-Transformations	p. 418
Complementarity Ideas	p. 421
Duality Ideas	p. 422
Positive Switched Linear Systems	p. 425
References	p. 428
Nonlinear Problems in Mathematical Programming and Optimal Control	p. 431
Introduction	p. 431
Main Result	p. 432
Proof of Theorem 26.1	p. 434
An Application	p. 436
References	p. 440
On Variational Inequalities Involving Mappings of Type (S)	p. 441
Main Results	p. 441
References	p. 448
Completely Generalized Co-complementarity Problems Involving p-Relaxed Accretive Operators with Fuzzy Mappings	p. 451
Introduction	p. 451
Background of Problem Formulation	p. 452
The Characterization of Problem and Solutions	p. 454
Iterative Algorithm and Pertinent Concepts	p. 455
Existence and Convergence Result for CGCCPFM	p. 458
References	p. 462
Generating Eigenvalue Bounds Using Optimization	p. 465
Introduction	p. 465
Outline	p. 467
Optimality Conditions	p. 467
Equality Constraints	p. 467
Equality and Inequality Constraints	p. 470
Sensitivity Analysis	p. 472
Generating Eigenvalue Bounds	p. 473
Fractional Programming	p. 484
Conclusion	p. 489
References	p. 490
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