Pardalos / Rassias / Khan Nonlinear Analysis and Variational Problems

In Honor of George Isac
1. Auflage 2009
ISBN: 978-1-4419-0158-3
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 35, 490 Seiten

Reihe: Springer Optimization and Its Applications

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.

Autoren/Hrsg.

Pardalos, Panos M.

Rassias, Themistocles M.

Khan, Akhtar A.

Inhaltsverzeichnis

1;Preface;7
2;Biographical Sketch of George Isac;11
3;Contents;15
4;List of Contributors;23
5;Part I Nonlinear Analysis;28
5.1;Discrete Approximation Processes of King's Type;29
5.1.1;Octavian Agratini and Tudor Andrica;29
5.1.1.1;Introduction;29
5.1.1.2;Further Results on Vn Type Operators;30
5.1.1.3;A General Class in Study;33
5.1.1.4;References;37
5.2;Isometrics in Non-Archimedean Strictly Convex and Strictly 2-Convex 2-Normed Spaces;39
5.2.1;Maryam Amyari and Ghadir Sadeghi;39
5.2.1.1;Introduction and Preliminaries;39
5.2.1.2;Non-Archimedean Strictly Convex 2-Normed Spaces;41
5.2.1.3;Non-Archimedean Strictly 2-Convex 2-Normed Spaces;44
5.2.1.4;References;47
5.3;Fixed Points and Generalized Stability for .-Additive Mappingsof Isac--Rassias Type;49
5.3.1;Liviu Cadariu and Viorel Radu;49
5.3.1.1;Introduction;49
5.3.1.2;Stability Properties for Cauchy Equationin -Normed Spaces;51
5.3.1.3;Other Examples and Applications;57
5.3.1.4;References;61
5.4;A Remark on W*-Tensor Products of W*-Algebras ;63
5.4.1;Corneliu Constantinescu;63
5.4.1.1;Introduction;63
5.4.1.2;The Ordered Involutive Banach Space ;65
5.4.1.3;The Multiplication;71
5.4.1.4;References;78
5.5;The Perturbed Median Principle for Integral Inequalitieswith Applications;79
5.5.1;S.S. Dragomir;79
5.5.1.1;Introduction;79
5.5.1.2;A Perturbed Version of the Median Principle;82
5.5.1.3;Some Examples for 0th-Degree Inequalities;83
5.5.1.4;Inequalities of the 1st-Degree;88
5.5.1.5;References;89
5.6;Stability of a Mixed Type Additive, Quadratic, Cubic and Quartic Functional Equation;90
5.6.1;M. Eshaghi-Gordji, S. Kaboli-Gharetapeh, M.S. Moslehian, and S. Zolfaghari;90
5.6.1.1;Introduction;91
5.6.1.2;General Solution;93
5.6.1.3;Stability;99
5.6.1.4;References;104
5.7;.-Additive Mappings and Hyers--Ulam Stability;106
5.7.1;P. Gavruta and L. Gavruta;106
5.7.1.1;Introduction;106
5.7.1.2;Results;107
5.7.1.3;References;110
5.8;The Stability and Asymptotic Behavior of Quadratic Mappingson Restricted Domains;112
5.8.1;Kil-Woung Jun and Hark-Mahn Kim;112
5.8.1.1;Introduction;112
5.8.1.2;Approximately Quadratic Mappings;114
5.8.1.3;Quadratic Mappings on Restricted Domains;118
5.8.1.4;References;121
5.9;A Fixed Point Approach to the Stability of a Logarithmic Functional Equation;123
5.9.1;Soon-Mo Jung and Themistocles M. Rassias;123
5.9.1.1;Introduction;123
5.9.1.2;Preliminaries;125
5.9.1.3;Hyers--Ulam--Rassias Stability;126
5.9.1.4;Applications;130
5.9.1.5;References;132
5.10;Fixed Points and Stability of the Cauchy Functional Equationin Lie C*-Algebras;134
5.10.1;Choonkil Park and Jianlian Cui;134
5.10.1.1;Introduction and Preliminaries;134
5.10.1.2;Stability of Homomorphisms in C*-Algebras;136
5.10.1.3;Stability of Derivations on C*-Algebras;140
5.10.1.4;Stability of Homomorphisms in Lie C*-Algebras;142
5.10.1.5;Stability of Lie Derivations on C*-Algebras;144
5.10.1.6;References;145
5.11;Fixed Points and Stability of Functional Equations;148
5.11.1;Choonkil Park and Themistocles M. Rassias;148
5.11.1.1;Introduction and Preliminaries;148
5.11.1.2;Fixed Points and Generalized Hyers--Ulam Stability of the Functional Equation (11.1): An Even Case;150
5.11.1.3;Fixed Points and Generalized Hyers--Ulam Stability of the Functional Equation (11.1): An Odd Case;153
5.11.1.4;References;156
5.12;Compression--Expansion Critical Point Theorems in ConicalShells;158
5.12.1;Radu Precup;158
5.12.1.1;Introduction;158
5.12.1.2;Main Results;160
5.12.1.3;Proofs;162
5.12.1.4;References;168
5.13;Gronwall Lemma Approach to the Hyers--Ulam--Rassias Stability of an Integral Equation;169
5.13.1;Ioan A. Rus;169
5.13.1.1;Introduction;169
5.13.1.2;Gronwall Lemmas;170
5.13.1.3;Stability of a Fixed Point Equation;171
5.13.1.4;Stability of Volterra Integral Equations;171
5.13.1.5;Stability of Fredholm Integral Equations;172
5.13.1.6;References;174
5.14;Brezis–Browder Principles and Applications;175
5.14.1;Mihai Turinici;175
5.14.2;Brezis–Browder Principles in General Separable Sets;175
5.14.2.1;Introduction;175
5.14.2.2;General Separable Sets;176
5.14.2.3;Zorn–Bourbaki Principles;183
5.14.2.4;Main Results;185
5.14.2.5;Some Amorphous Versions;189
5.14.3;Pseudometric Maximal Principles;191
5.14.3.1;Introduction;191
5.14.3.2;Logical Equivalents of Brezis–Browder’s Principle;192
5.14.3.3;Asymptotic Extensions;193
5.14.3.4;Convergence and Uniform Versions;195
5.14.3.5;Zorn Maximality Principles;200
5.14.4;Relative KST Statements;202
5.14.4.1;Introduction;202
5.14.4.2;Maximal Principles;203
5.14.4.3;Transitive (Pseudometric) Versions;206
5.14.4.4;Main Results;208
5.14.4.5;Extended KST Statements;211
5.14.5;References;215
6;Part II Variational Problems;220
6.1;A Generalized Quasi-Equilibrium Problem;221
6.1.1;Mircea Balaj and Donal O'Regan;221
6.1.1.1;Introduction;221
6.1.1.2;Preliminaries;222
6.1.1.3;Main Result;223
6.1.1.4;Particular Cases of Theorem15.8;225
6.1.1.5;Applications;229
6.1.1.6;References;230
6.2;Double-Layer and Hybrid Dynamics of Equilibrium Problems: Applications to Markets of Environmental Products;232
6.2.1;M. Cojocaru, S. Hawkins, H. Thille, and E. Thommes;232
6.2.1.1;Introduction;232
6.2.1.2;Dynamic Equilibrium Problems and Variational Inequalities;234
6.2.1.2.1;General Formulation;234
6.2.1.3;Double-Layer Dynamics and Hybrid Dynamical Systems;239
6.2.1.3.1;DLD;240
6.2.1.3.2;Tracking Equilibrium Dynamics: Hybrid Systems Approach;241
6.2.1.4;Dynamics of Environmental Product Markets;243
6.2.1.4.1;The Static Model;243
6.2.1.4.2;Dynamic Equilibrium Model: EVI Formulation;245
6.2.1.4.3;Example;246
6.2.1.4.4;Dynamic Disequilibrium Model: DLD Formulation;249
6.2.1.5;Conclusions and Acknowledgments;251
6.2.1.6;References;251
6.3;A Panoramic View on Projected Dynamical Systems;254
6.3.1;Patrizia Daniele, Sofia Giuffré, Antonino Maugeri, and Stephane Pia;254
6.3.1.1;Introduction;254
6.3.1.2;General Background Material;256
6.3.1.2.1;Spaces;256
6.3.1.2.2;Cones and Properties;260
6.3.1.2.3;Projectors;261
6.3.1.2.4;Weighted Traffic Equilibrium Problem;264
6.3.1.2.5;Time-Dependent Equilibria;265
6.3.1.3;Projected Dynamical Systems in Hilbert Spaces;266
6.3.1.3.1;Projected Dynamical Systems in Pivot Hilbert Spaces;266
6.3.1.3.2;Projected Dynamical Systems in Non-pivot Hilbert Spaces;267
6.3.1.4;Projected Dynamical Systems in Banach Spaces;268
6.3.1.4.1;The Strictly Convex and Uniformly Smooth Case;269
6.3.1.4.2;Projected Dynamical Systems and Unilateral Differential Inclusions;270
6.3.1.5;Bridge with Variational Inequalities;272
6.3.1.6;Conclusion;275
6.3.1.7;References;275
6.4;Foundations of Set-Semidefinite Optimization;278
6.4.1;Gabriele Eichfelder and Johannes Jahn;278
6.4.1.1;Introduction;278
6.4.1.2;Applications of Set-Semidefinite Optimization;280
6.4.1.2.1;Semidefinite Optimization;280
6.4.1.2.2;Copositive Optimization;281
6.4.1.2.3;Second-Order Optimality Conditions;283
6.4.1.2.4;Semi-infinite Optimization;284
6.4.1.3;Set-Semidefinite Cone;286
6.4.1.3.1;Properties of the Set-Semidefinite Cone;286
6.4.1.3.2;Dual and Interior of the Set-Semidefinite Cone;290
6.4.1.4;Optimality Conditions;293
6.4.1.5;Nonconvex Duality;297
6.4.1.6;Future Research;301
6.4.1.7;References;302
6.5;On the Envelope of a Variational Inequality;304
6.5.1;F. Giannessi and A.A. Khan;304
6.5.1.1;Introduction;304
6.5.1.2;Auxiliary Variational Inequality;306
6.5.1.3;A Particular Case;309
6.5.1.4;References;312
6.6;On the Nonlinear Generalized Ordered ComplementarityProblem;313
6.6.1;D. Goeleven;313
6.6.1.1;Introduction;313
6.6.1.2;A Spectral Condition for the Generalized Ordered Complementarity Problem;315
6.6.1.3;Existence and Uniqueness Results;318
6.6.1.4;References;321
6.7;Optimality Conditions for Several Types of Efficient Solutionsof Set-Valued Optimization Problems;322
6.7.1;T.X.D. Ha;322
6.7.1.1;Introduction;322
6.7.1.2;Subdifferentials, Derivatives and Coderivatives;324
6.7.1.3;Some Concepts of Efficient Points ;326
6.7.1.4;Optimality Conditions for Set-Valued Optimization Problem;333
6.7.1.5;References;340
6.8;Mean Value Theorems for the Scalar Derivative and Applications;342
6.8.1;G. Isac and S.Z. Németh;342
6.8.1.1;Introduction;342
6.8.1.2;Preliminaries;343
6.8.1.3;Scalar Derivatives and Scalar Differentiability;344
6.8.1.3.1;Computational Formulae for the Scalar Derivatives;345
6.8.1.4;Mean Value Theorems;346
6.8.1.5;Applications to Complementarity Problems;348
6.8.1.6;Comments;357
6.8.1.7;References;357
6.9;Application of a Vector-Valued Ekeland-Type Variational Principle for Deriving Optimality Conditions;359
6.9.1;G. Isac and C. Tammer;359
6.9.1.1;Introduction;359
6.9.1.2;Properties of Cones;361
6.9.1.3;An Ekeland-Type Variational Principle for Vector Optimization Problems;365
6.9.1.4;Nonlinear Scalarization Scheme;366
6.9.1.5;Differentiability Properties of Vector-Valued Functions;369
6.9.1.6;Necessary Optimality Conditions for Vector Optimization Problems in General Spaces Based on Directional Derivatives;373
6.9.1.7;Vector Optimization Problems with Finite-Dimensional Image Spaces;379
6.9.1.8;References;380
6.10;Nonlinear Variational Methods for Estimating Effective Properties of Multiscale Materials;382
6.10.1;Dag Lukkassen, Annette Meidell, and Lars-Erik Persson;382
6.10.1.1;Introduction;382
6.10.1.2;Preliminaries;385
6.10.1.3;Some Nonlinear Bounds of Classical Type;386
6.10.1.4;Some Useful Means of Power Type;390
6.10.1.4.1;A Particular Power Type Mean;391
6.10.1.4.2;Composition of Power Means;395
6.10.1.5;Nonlinear Bounds;401
6.10.1.6;Further Results for the Case p=2;413
6.10.1.7;The Reiterated Cell Structure;418
6.10.1.7.1;The Scalar Case;419
6.10.1.7.2;The Vector-Valued Case;420
6.10.1.8;Bounds Related to a Reynold-Type Equation;422
6.10.1.9;Some Final Comments;427
6.10.1.10;References;427
6.11;On Common Linear/Quadratic Lyapunov Functions for Switched Linear Systems;430
6.11.1;Melania M. Moldovan and M. Seetharama Gowda;430
6.11.1.1;Introduction;430
6.11.1.2;Preliminaries;432
6.11.1.2.1;Matrix Theory Concepts;432
6.11.1.2.2;Z-Transformations;433
6.11.1.3;Complementarity Ideas;436
6.11.1.4;Duality Ideas;437
6.11.1.5;Positive Switched Linear Systems;440
6.11.1.6;References;443
6.12;Nonlinear Problems in Mathematical Programming and Optimal Control;445
6.12.1;Dumitru Motreanu;445
6.12.1.1;Introduction;445
6.12.1.2;Main Result;446
6.12.1.3;Proof of Theorem 26.1;448
6.12.1.4;An Application;450
6.12.1.5;References;454
6.13;On Variational Inequalities Involving Mappings of Type (S);455
6.13.1;Dan Pascali;455
6.13.1.1;Main Results;455
6.13.1.2;References;462
6.14;Completely Generalized Co-complementarity Problems Involving p-Relaxed Accretive Operators with Fuzzy Mappings;464
6.14.1;Abul Hasan Siddiqi and Syed Shakaib Irfan;464
6.14.1.1;Introduction;464
6.14.1.2;Background of Problem Formulation;465
6.14.1.3;The Characterization of Problem and Solutions;467
6.14.1.4;Iterative Algorithm and Pertinent Concepts;468
6.14.1.5;Existence and Convergence Result for CGCCPFM;471
6.14.1.6;References;475
6.15;Generating Eigenvalue Bounds Using Optimization;477
6.15.1;Henry Wolkowicz;477
6.15.1.1;Introduction;477
6.15.1.1.1;Outline;479
6.15.1.2;Optimality Conditions;479
6.15.1.2.1;Equality Constraints;479
6.15.1.2.2;Equality and Inequality Constraints;482
6.15.1.2.3;Sensitivity Analysis;484
6.15.1.3;Generating Eigenvalue Bounds;485
6.15.1.4;Fractional Programming;496
6.15.1.5;Conclusion;501
6.15.1.6;References;502