Lakshmikantham Nonlinear Equations in Abstract Spaces

Proceedings of an International Symposium on Nonlinear Equations in Abstract Spaces, Held at the University of Texas at Arlington, Arlington, Texas, June 8-10, 1977
1. Auflage 2014
ISBN: 978-1-4832-7210-8
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 492 Seiten, Web PDF

Many problems in partial differential equations which arise from physical models can be considered as ordinary differential equations in appropriate infinite dimensional spaces, for which elegant theories and powerful techniques have recently been developed. This book gives a detailed account of the current state of the theory of nonlinear differential equations in a Banach space, and discusses existence theory for differential equations with continuous and discontinuous right-hand sides. Of special importance is the first systematic presentation of the very important and complex theory of multivalued discontinuous differential equations.

Autoren/Hrsg.

Lakshmikantham, V.

Inhaltsverzeichnis

1;Front Cover;1
2;Nonlinear Equations in Abstract Spaces;4
3;Copyright Page;5
4;Table of Contents;6
5;LIST OF CONTRIBUTORS;8
6;Preface;10
7;PART I: INVITED ADDRESSES AND RESEARCH REPORTS;14
7.1;CHAPTER 1. NEW RESULTS IN STOCHASTIC EQUATIONS THE NONLINEAR CASE;14
7.1.1;ACKNOWLEDGEMENT;32
7.1.2;REFERENCES;32
7.2;CHAPTER 2. POSITIVE OPERATORS AND STURMIAN THEORY OF NONSELFADJOINT SECOND-ORDER SYSTEMS;36
7.2.1;I. INTRODUCTION;36
7.2.2;II. SOME KNOWN RESULTS ABOUT POSITIVE LINEAR OPERATORS;37
7.2.3;III. STRICTLY POSITIVE LINEAR OPERATORS;38
7.2.4;IV. TWO BASIC EXAMPLES;40
7.2.5;VI. STURMIAN THEORY FOR x" + P(t)x = 0;47
7.2.6;V. MONOTONICITY AND CONTINUITY PROPERTIES OF .0 (a,b) AND µ0 (a,b);43
7.2.7;VII. REFERENCES;52
7.3;CHAPTER 3. NONLINEAR SUPERPOSITION FOR OPERATOR EQUATIONS;54
7.3.1;I. INTRODUCTION;54
7.3.2;II. CONSTANTS OF SUPERPOSITION;57
7.3.3;III. APPLICATIONS OF CONSTANTS OF SUPERPOSITION;59
7.3.4;IV. CONSTRUCTION OF A NONLINEAR SUPERPOSITION;64
7.3.5;V. A "MIXED" SUPERPOSITION FOR THE RICCATI EQUATION;67
7.3.6;VI. NONLINEAR DIFFUSION EQUATION;68
7.3.7;VII. BÄCKLUND TRANSFORMATIONS;70
7.3.8;VIII. REFERENCES;74
7.4;CHAPTER 4. RANDOM FIXED POINT THEOREMS;78
7.4.1;I. INTRODUCTION;78
7.4.2;II. CONTINUOUS RANDOM OPERATORS;80
7.4.3;III. UPPER SEMICONTINUOUS RANDOM OPERATORS;87
7.4.4;IV. REFERENCES;90
7.5;CHAPTER 5. DELAY EQUATIONS OF PARABOLIC TYPE IN BANACH SPACE;92
7.5.1;I. INTRODUCTION;92
7.5.2;II. THE QUASILINEAR CASE;93
7.5.3;III. THE SEMILINEAR CASE;96
7.5.4;IV. EXAMPLES;100
7.5.5;V. REFERENCES;102
7.6;CHAPTER 6. THE EXACT AMOUNT OF NONUNIQUENESS FOR SINGULAR ORDINARY DIFFERENTIAL EQUATIONS IN BANACH SPACES WITH AN APPLICATION TO THE EULER-POISSON-DARBOUX EQUATION;106
7.6.1;I. INTRODUCTION;106
7.6.2;II. THE ABSTRACT UNIQUENESS THEOREM;108
7.6.3;III. APPLICATION TO SINGULAR SECOND ORDER LINEAR EQUATIONS;110
7.6.4;IV. REFERENCES;114
7.7;CHAPTER 7. ON THE EQUATION Tx = y IN BANACH SPACES WITH WEAKLY CONTINUOUS DUALITY MAPS;116
7.7.1;I. INTRODUCTION;116
7.7.2;II. PRELIMINARIES;117
7.7.3;III. MAIN RESULTS;118
7.7.4;IV. DISCUSSION;121
7.7.5;V. REFERENCES;122
7.8;CHAPTER 8. NONLINEAR EVOLUTION OPERATORS IN BANACH SPACES;124
7.8.1;REFERENCES;126
7.9;CHAPTER 9. ABSTRACT BOUNDARY VALUE PROBLEMS;128
7.9.1;I. INTRODUCTION;128
7.9.2;II. GENERAL COMPARISON RESULT;129
7.9.3;III. EXISTENCE RESULTS;130
7.9.4;IV. MONOTONE ITERATIVE METHOD;132
7.9.5;V. REFERENCES;133
7.10;CHAPTER 10. EXISTENCE THEORY OF DELAY DIFFERENTIAL EQUATIONS IN BANACH SPACES;136
7.10.1;REFERENCES;142
7.11;CHAPTER 11. INVARIANT SETS AND A MATHEMATICAL MODEL INVOLVING SEMILINEAR DIFFERENTIAL EQUATIONS;146
7.11.1;I. AN ABSTRACT SYSTEM;146
7.11.2;II. A MATHEMATICAL MODEL;151
7.11.3;III. REFERENCES;159
7.12;CHAPTER 12. TOTAL STABILITY AND CLASSICAL HAMILTONIAN THEORY;160
7.12.1;I. INTRODUCTION;160
7.12.2;II. PRELIMINARIES;161
7.12.3;III. CRITICAL ANALYSIS OF A CETAEV'S REMARK ON THE VALIDITY OF HAMILTONIAN SCHEME;163
7.12.4;IV. NON-OBSERVABLE MOTIONS;167
7.12.5;V. REFERENCES;169
7.13;CHAPTER 13. ON SOME MATHEMATICAL MODELS OF SOCIAL PHENOMENA;172
7.13.1;I. INTRODUCTION;172
7.13.2;II. THE FIRST AND SECOND "LAWS" OF SOCIAL DYNAMICS;173
7.13.3;III. THE LINEAR FORCE AND VERHULST'S LOGISTIC MODEL OF POPULATION GROWTH AND SATURATION;175
7.13.4;IV. SOME REMARKS ON OBSERVED POPULATION FIGURES;182
7.13.5;V. ON THE INTERACTION BETWEEN TWO SPECIES, THE LOTKA9-VOLTERRA 10,11 MODEL AND CERTAIN EXTENSIONS OF IT;191
7.13.6;VI. ON EVOLUTIONARY PATTERNS IN SYSTEMS OF SEVERAL VARIABLES;195
7.13.7;VII. INFLUENCE OF RANDOM FORCES;199
7.13.8;VIII. ON THE PRICES OF THINGS WITH SOME REMARKS ABOUT SEARS-ROEBUCK CATALOGUES;203
7.13.9;IX. THE PLANNER'S DILEMMA AND THE GUMBEL DISTRIBUTION OF EXTREME VALUES;217
7.13.10;X. ACKNOWLEDGEMENTS;225
7.13.11;XI. REFERENCES;225
7.14;CHAPTER 14. GENERALIZED INVERSE MAPPING THEOREMS AND RELATED APPLICATIONS OF GENERALIZED INVERSES IN NONLINEAR ANALYSIS;228
7.14.1;INTRODUCTION;228
7.14.2;I. A. Strong Differentiability in Normed Spaces;230
7.14.3;II. A. Generalized Inverses of Domain-Decomposable Linear Operators in Banach Spaces.;241
7.14.4;III. INVERSE MAPPING THEOREMS IN THE FRAMEWORK OF GENERALIZEDINVERSES;245
7.14.5;IV. REFERENCES;257
7.15;CHAPTER 15. ITERATION FOR SYSTEMS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS;264
7.15.1;I. STATEMENT OF MAIN LEMMA;264
7.15.2;II. A SIMPLE APPLICATION;264
7.15.3;III. APPLICATION TO CONSERVATION SYSTEMS;266
7.15.4;IV. NUMERICAL APPROXIMATIONS;268
7.15.5;V. PROOFS;270
7.15.6;VI. CONCLUSION;272
7.15.7;VII. REFERENCES;274
7.16;CHAPTER 16. EXISTENCE THEOREMS AND APPROXIMATIONS IN NONLINEAR ELASTICITY;276
7.16.1;I. INTRODUCTION;276
7.16.2;II. BOUNDARY-VALUE PROBLEMS IN ELASTOSTATICS;277
7.16.3;III. AN EXISTENCE THEOREM;279
7.16.4;IV. A MODEL PROBLEM;280
7.16.5;V. APPROXIMATIONS;282
7.16.6;VI. REFERENCES;284
7.17;CHAPTER 17. EXISTENCE THEOREMS FOR SEMILINEAR ABSTRACT AND DIFFERENTIAL EQUATIONS WITH NONINVERTIBLE LINEAR PARTS AND NONCOMPACT PERTURBATIONS;286
7.17.1;INTRODUCTION;286
7.17.2;SECTION 1;291
7.17.3;SECTION 2;306
7.17.4;SECTION 3;311
7.17.5;REFERENCES;324
7.18;CHAPTER 18. ITERATIVE METHODS FOR ACCRETIVE SETS;328
7.18.1;REFERENCES;336
7.19;CHAPTER 19. MODEL EQUATIONS FOR NONLINEAR DISPERSIVE SYSTEMS;338
7.19.1;REFERENCES;339
7.20;CHAPTER 20. SECOND ORDER DIFFERENTIAL EQUATIONS IN BANACH SPACE;342
7.20.1;I. INTRODUCTION;342
7.20.2;II. BASIC THEORY OF STRONGLY CONTINUOUS COSINE FAMILIES;344
7.20.3;III. THE PROBLEM OF CONVERSION TO A FIRST ORDER SYSTEM;351
7.20.4;IV. PERTURBATION AND APPROXIMATION RESULTS FOR STRONGLY CONTINUOUS COSINE FAMILIES;357
7.20.5;V. SPECIAL PROPERTIES OF STRONGLY CONTINUOUS COSINE FAMILIES: COMPACTNESS, UNIFORM CONTINUITY, INHOMOGENEOUS EQUATIONS;360
7.20.6;VI. ABSTRACT SECOND ORDER SEMILINEAR EQUATIONS;365
7.20.7;VII. REFERENCES;369
8;PART II: CONTRIBUTED PAPERS;376
8.1;CHAPTER 21. A CHARACTERIZATION OF THE RANGE OF A NONLINEAR VOLTERRA INTEGRAL OPERATOR;376
8.1.1;I. INTRODUCTION;376
8.1.2;II. STATEMENT AND DISCUSSION OF RESULTS;377
8.1.3;III. PROOF OF PROPOSITION 1;379
8.1.4;IV. PROOF OF THEOREM 1;380
8.1.5;V. PROOF OF THEOREM 2;381
8.1.6;VI. PROOF OF THEOREM 3;382
8.1.7;VII. EXAMPLES;384
8.1.8;VIII. REFERENCES;385
8.2;CHAPTER 22. DISCONTINUOUS PERTURBATIONS OF ELLIPTIC BOUNDARY VALUE PROBLEMS AT RESONANCE;386
8.2.1;I. DISCONTINUOUS PERTURBATIONS AT RESONANCE;386
8.2.2;II. THE ABSTRACT FRAMEWORK;387
8.2.3;III. APPLICATIONS TO SEMILINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS WITH KERNEL;391
8.2.4;IV. REFERENCES;397
8.3;CHAPTER 23. AN EXISTENCE THEOREM FOR WEAK SOLUTIONS OF DIFFERENTIAL EQUATIONS IN BANACH SPACES;398
8.3.1;I. PROPERTIES OF THE WEAK TOPOLOGY AND A MEASURE OF WEAK NONCOMPACTNESS;398
8.3.2;II. INTEGRATION AND DIFFERENTIATION;409
8.3.3;III. EXISTENCE OF WEAK SOLUTIONS TO THE ABSTRACT CAUCHY PROBLEM;411
8.3.4;IV. REFERENCES;414
8.4;CHAPTER 24. MONOTONICITY AND ALTERNATIVE METHODS;416
8.4.1;I. INTRODUCTION;416
8.4.2;II. REDUCTION TO AN ALTERNATIVE PROBLEM;418
8.4.3;III. SOLVING THE ALTERNATIVE PROBLEM;421
8.4.4;IV. DISCUSSION OF THE RESULTS;423
8.4.5;V. REFERENCES;425
8.5;CHAPTER 25. THE OLP1 METHOD OF NON-LINEAR STABILITY ANALYSIS OF TURBULENCE IN NEWTONIAN FLUIDS2;428
8.5.1;I. INTRODUCTION;428
8.5.2;II. DEFINITION OF TURBULENCE;429
8.5.3;III. PRIOR METHODS OF FLOW STABILITY ANALYSES;430
8.5.4;IV. BRIEF OVERVIEW OF THE OLP METHODOLOGY;431
8.5.5;V. FUNDAMENTAL PHYSICAL AND MATHEMATICAL QUESTIONS;433
8.5.6;VI. OLP DERIVATION;436
8.5.7;VII. PRIOR APPLICATIONS OF OLP;438
8.5.8;VIII. FORMULATION OF OLP FOR THE BOUNDARY LAYER;443
8.5.9;IX. SUMMARY AND RECAPITULATION:;445
8.5.10;X. REFERENCES CITED;446
8.6;CHAPTER 26. GENERALIZED CONTRACTIONS AND SEQUENCE OF ITERATES;450
8.6.1;I. INTRODUCTION;450
8.6.2;II. REFERENCES;470
8.7;CHAPTER 27. CRITERIA FOR THE EXISTENCE AND COMPARISON OF SOLUTIONS TO NONLINEAR VOLTERRA INTEGRAL EQUATIONS IN BANACH SPACE;474
8.7.1;I. INTRODUCTION;474
8.7.2;II. PRELIMINARIES;475
8.7.3;III. EXISTENCE CRITERIA;475
8.7.4;IV. MAXIMAL SOLUTIONS;477
8.7.5;V. REFERENCES;479
8.8;CHAPTER 28. SEMILINEAR BOUNDARY VALUE PROBLEMS IN BANACH SPACE;480
8.8.1;I. INTRODUCTION;480
8.8.2;II. MAIN RESULTS;482
8.8.3;III. EXAMPLES;485
8.8.4;IV. REFERENCES;488
8.9;CHAPTER 29. POLYNOMIAL PERTURBATIONS TO THE LAPLACIAN L.;490
8.9.1;I. INTRODUCTION;490
8.9.2;II. FIRST METHOD;490
8.9.3;III. SECOND METHOD;492
8.9.4;IV. REFERENCES;494