Nonlinear Analysis and Variational Problems: In Honor of George Isac / Edition 1

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Overview

Springer Optimization and Its Applications publishes undergraduate and graduate textbooks; monographs and .state-of-the-art expository works that focus on algorithms for solving optimization problems and applications involving such problems. Some of the topics include optimization, network flow problems, optimal control, multi-objective programming approximation and heuristic approaches.

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 I 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, networkanalysis, 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.

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Table of Contents

Preface vii

Biographical Sketch of George Isac xi

List of Contributors xxiii

PartI Nonlinear Analysis

1 Discrete Approximation Processes of King's Type Octavian Agratini Tudor Andrica 3

1.1 Introduction 3

1.2 Further Results on Vn Type Operators 4

1.3 A General Class in Study 7

References 11

2 Isometrics in Non-Archimedean Strictly Convex and Strictly 2-Convex 2-Normed Spaces Maryam Amyari Ghadir Sadeghi 13

2.1 Introduction and Preliminaries 13

2.2 Non-Archimedean Strictly Convex 2-Normed Spaces 15

2.3 Non-Archimedean Strictly 2-Convex 2-Normed Spaces 18

References 21

3 Fixed Points and Generalized Stability for ψ-Additive Mappings of Isac-Rassias Type Liviu Cădariu Viorel Radu 23

3.1 Introduction 23

3.2 Stability Properties for Cauchy Equation in β-Normed Spaces 25

3.3 Other Examples and Applications 31

References 35

4 A Remark on W*-Tensor Products of W*-Algebras Corneliu Constantinescu 37

4.1 Introduction 37

4.2 The Ordered Involutive Banach Space 39

4.3 The Multiplication 45

References 52

5 The Perturbed Median Principle for Integral Inequalities with Applications S.S. Dragomir 53

5.1 Introduction 53

5.2 A Perturbed Version of the Median Principle 56

5.3 Some Examples for 0th-Degree Inequalities 57

5.4 Inequalities of the 1st-Degree 62

References 63

6 Stability of a Mixed Type Additive, Quadratic, Cubic and Quartic Functional Equation M. Eshaghi-Gordji S. Kaboli-Gharelapeh M.S. Moslehian S. Zolfaghari 65

6.1 Introduction 66

6.2 General Solution 68

6.3 Stability 74

References 79

7 Ψ-Additive Mappings and Hyers-Ulam Stability P. Găvruţa L.Găvruţa 81

7.1 Introduction 81

7.2 Results 82

References 85

8 The Stability and Asymptotic Behavior of Quadratic Mappings on Restricted Domains Kil-Woung Jun Hark-Mahn Kim 87

8.1 Introduction 87

8.2 Approximately Quadratic Mappings 89

8.3 Quadratic Mappings on Restricted Domains 93

References 96

9 A Fixed Point Approach to the Stability of a Logarithmic Functional Equation Soon-Mo Jung Themistocles M. Rassias 99

9.1 Introduction 99

9.2 Preliminaries 101

9.3 Hyers-Ulam-Rassias Stability 102

9.4 Applications 106

References 108

10 Fixed Points and Stability of the Cauchy Functional Equation in Lie C*-Algebras Choonkil Park Jianlian Cuie C*-Algebras Choonkil Park Jianlian Cui 111

10.1 Introduction and Preliminaries 111

10.2 Stability of Homomorphisms in C*-Algebras 113

10.3 Stability of Derivations on C-Algebras 117

10.4 Stability of Homomorphisms in Lie C*-Algebras 119

10.5 Stability of Lie Derivations on C*-AIgebras 121

References 133

11 Fixed Points and Stability of Functional Equations Choonkil Park Themistocles M. Rassias 125

11.1 Introduction and Preliminaries 125

11.2 Fixed Points and Generalized Hyers-Ulam Stability of the Functional Equation (11.1): An Even Case 127

11.3 Fixed Points and Generalized Hyers-Ulam Stability of the Functional Equation (11.1): An Odd Case 130

References 133

12 Compression-Expansion Critical Point Theorems in Conical Shells Radu Precup 135

12.1 Introduction 135

12.2 Main Results 137

12.3 Proofs 139

References 145

13 Gronwall Lemma Approach to the Hyers-Ulam-Rassias Stability of an Integral Equation Ioan A. Rus 147

13.1 Introduction 147

13.2 Gronwall Lemmas 148

13.3 Stability of a Fixed Point Equation 149

13.4 Stability of Volterra Integral Equations 149

13.5 Stability of Fredholm Integral Equations 150

References 152

14 Brezis-Browder Principles and Applications Mihai Turinici 153

14.1 Brezis-Browder Principles in General Separable Sets 153

14.1.1 Introduction 153

14.1.2 General Separable Sets 154

14.1.3 Zorn-Bourbaki Principles 161

14.1.4 Main Results 163

14.1.5 Some Amorphous Versions 167

14.2 Pseudometric Maximal Principles 169

14.2.1 Introduction 169

14.2.2 Logical Equivalents of Brezis-Browder's Principle 170

14.2.3 Asymptotic Extensions 171

14.2.4 Convergence and Uniform Versions 173

14.2.5 Zorn Maximality Principles 178

14.3 Relative KST Statements 180

14.3.1 Introduction 180

14.3.2 Maximal Principles 181

14.3.3 Transitive (Pseudometric) Versions 184

14.3.4 Main Results 186

14.3.5 Extended KST Statements 189

References 193

Part II Variational Problems

15 A Generalized Quasi-Equilibrium Problem Mircea Balaj Donal O'Regan 201

15.1 Introduction 201

15.2 Preliminaries 202

15.3 Main Result 203

15.4 Particular Cases of Theorem 15.8 205

15.5 Applications 209

References 210

16 Double-Layer and Hybrid Dynamics of Equilibrium Problems: Applications to Markets of Environmental Products M. Cojocaru S. Hawkins H. Thille E. Thommes 213

16.1 Introduction 213

16.2 Dynamic Equilibrium Problems and Variational Inequalities 215

16.2.1 General Formulation 215

16.3 Double-Layer Dynamics and Hybrid Dynamical Systems 220

16.3.1 DLD 221

16.3.2 Tracking Equilibrium Dynamics: Hybrid Systems Approach 222

16.4 Dynamics of Environmental Product Markets 224

16.4.1 The Static Model 224

16.4.2 Dynamic Equilibrium Model: EVI Formulation 226

16.4.3 Example 227

16.4.4 Dynamic Disequilibrium Model: DLD Formulation 230

16.5 Conclusions and Acknowledgments 232

References 232

17 A Panoramic View on Projected Dynamical Systems Patrizia Daniele Sofia Giuffré Antonino Maugeri Stephane Pia 235

17.1 Introduction 235

17.2 General Background Material 237

17.2.1 Spaces 237

17.2.2 Cones and Properties 241

17.2.3 Projectors 242

17.2.4 Weighted Traffic Equilibrium Problem 245

17.2.5 Time-Dependent Equilibria 246

17.3 Projected Dynamical Systems in Hilbert Spaces 247

17.3.1 Projected Dynamical Systems in Pivot Hilbert Spaces 247

17.3.2 Projected Dynamical Systems in Non-pivot Hilbert Spaces 248

17.4 Projected Dynamical Systems in Banach Spaces 249

17.4.1 The Strictly Convex and Uniformly Smooth Case 250

17.4.2 Projected Dynamical Systems and Unilateral Differential Inclusions 251

17.5 Bridge with Variational Inequalities 253

17.6 Conclusion 256

References 256

18 Foundations of Set-Semidefinite Optimization Gabriele Eichfelder Johannes Jahn 259

18.1 Introduction 259

18.2 Applications of Set-Semidefinite Optimization 261

18.2.1 Semidefinite Optimization 261

18.2.2 Copositive Optimization 262

18.2.3 Second-Order Optimality Conditions 264

18.2.4 Semi-in finite Optimization 265

18.3 Set-Semidefinite Cone 267

18.3.1 Properties of the Set-Semidefinite Cone 267

18.3.2 Dual and Interior of the Set-Semidefinite Cone 271

18.4 Optimality Conditions 274

18.5 Nonconvex Duality 278

18.6 Future Research 282

References 283

19 On the Envelope of a Variational Inequality F. Giannessi A.A. Khan 285

19.1 Introduction 285

19.2 Auxiliary Variational Inequality 287

19.3 A Particular Case 290

References 293

20 On the Nonlinear Generalized Ordered Complementarity Problem D. Goeleven 295

20.1 Introduction 295

20.2 A Spectral Condition for the Generalized Ordered Complementarity Problem 297

20.3 Existence and Uniqueness Results 300

References 303

21 Optimality Conditions for Several Types of Efficient Solutions of Set-Valued Optimization Problems T.X.D. Ha 305

21.1 Introduction 305

21.2 Subdifferentials, Derivatives and Coderivatives 307

21.3 Some Concepts of Efficient Points 309

21.4 Optimality Conditions for Set-Valued Optimization Problem 316

References 323

22 Mean Value Theorems for the Scalar Derivative and Applications G. Isac S.Z. Nemeth 325

22.1 Introduction 325

22.2 Preliminaries 326

22.3 Scalar Derivatives and Scalar Differentiability 327

22.3.1 Computational Formulae for the Scalar Derivatives 328

22.4 Mean Value Theorems 329

22.5 Applications to Complementarity Problems 331

22.6 Comments 340

References 340

23 Application of a Vector-Valued Ekeland-Type Variational Principle for Deriving Optimality Conditions G. Isac C. Tammer 343

23.1 Introduction 343

23.2 Properties of Cones 345

23.3 An Ekeland-Type Variational Principle for Vector Optimization Problems 349

23.4 Nonlinear Scalarization Scheme 350

23.5 Differentiability Properties of Vector-Valued Functions 353

23.6 Necessary Optimality Conditions for Vector Optimization Problems in General Spaces Based on Directional Derivatives 357

23.7 Vector Optimization Problems with Finite-Dimensional Image Spaces 363

References 364

24 Nonlinear Variational Methods for Estimating Effective Properties of Multiscale Materials Dag Lukkassen Annette Meidell Lars-Erik Persson 367

24.1 Introduction 367

24.2 Preliminaries 370

24.3 Some Nonlinear Bounds of Classical Type 371

24.4 Some Useful Means of Power Type 375

24.4.1 A Particular Power Type Mean 376

24.4.2 Composition of Power Means 380

24.5 Nonlinear Bounds 386

24.6 Further Results for the Case p=2 398

24.7 The Reiterated Cell Structure 403

24.7.1 The Scalar Case 404

24.7.2 The Vector-Valued Case 405

24.8 Bounds Related to a Reynold-Type Equation 407

24.9 Some Final Comments 412

References 412

25 On Common Linear/Quadratic Lyapunov Functions for Switched Linear Systems Melania M. Moldovan M. Seetharama Gowda 415

25.1 Introduction 415

25.2 Preliminaries 417

25.2.1 Matrix Theory Concepts 417

25.2.2 Z-Transformations 418

25.3 Complementarity Ideas 421

25.4 Duality Ideas 422

25.5 Positive Switched Linear Systems 425

References 428

26 Nonlinear Problems in Mathematical Programming and Optimal Control Dumitru Motreanuxz 431

26.1 Introduction 431

26.2 Main Result 432

26.3 Proof of Theorem 26.1 434

26.4 An Application 436

References 440

27 On Variational Inequalities Involving Mappings of Type (S) Dan Pascali 441

27.1 Main Results 441

References 448

28 Completely Generalized Co-complementarity Problems Involving p-Relaxed Accretive Operators with Fuzzy Mappings Abul Hasan Siddiqi Syed Shakaib Irfan 451

28.1 Introduction 451

28.2 Background of Problem Formulation 452

28.3 The Characterization of Problem and Solutions 454

28.4 Iterative Algorithm and Pertinent Concepts 455

28.5 Existence and Convergence Result for CGCCPFM 458

References 462

29 Generating Eigenvalue Bounds Using Optimization Henry Wolkowicz 465

29.1 Introduction 465

29.1.1 Outline 467

29.2 Optimality Conditions 467

29.2.1 Equality Constraints 467

29.2.2 Equality and Inequality Constraints 470

29.2.3 Sensitivity Analysis 472

29.3 Generating Eigenvalue Bounds 473

29.4 Fractional Programming 484

29.5 Conclusion 489

References 490

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