Conti Recent Advances in Differential Equations

1. Auflage 2014
ISBN: 978-1-4832-7391-4
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 462 Seiten, Web PDF

Recent Advances in Differential Equations contains the proceedings of a meeting held at the International Center for Theoretical Physics in Trieste, Italy, on August 24-28, 1978 under the auspices of the U.S. Army Research Office. The papers review the status of research in the field of differential equations (ordinary, partial, and functional). Both theoretical aspects (differential operators, periodic solutions, stability and bifurcation, asymptotic behavior of solutions, etc.) and problems arising from applications (reaction-diffusion equations, control problems, heat flow, etc.) are discussed. Comprised of 33 chapters, this book first examines non-cooperative trajectories of n-person dynamical games and stable non-cooperative equilibria, followed by a discussion on the determination and application of Vekua resolvents. The reader is then introduced to generalized Hopf bifurcation; some Cauchy problems arising in computational methods; and boundary value problems for pairs of ordinary differential operators. Subsequent chapters focus on degenerate evolution equations and singular optimal control; stability of neutral functional differential equations; local exact controllability of nonlinear evolution equations; and turbulence and higher order bifurcations. This monograph will be of interest to students and practitioners in the field of mathematics.

Autoren/Hrsg.

Conti, Roberto

Inhaltsverzeichnis

1;Front Cover;1
2;Recent Advances
in
Differential Equations;4
3;Copyright Page;5
4;Table of Contents;6
5;Contributors;10
6;Preface;12
7;CHAPTER 1. NONCOOPERATIVE TRAJECTORIES OF n-PERSON DYNAMICAL GAMES AND STABLE NONCOOPERATIVE EQUILIBRIA;16
7.1;I. INTRODUCTION;16
7.2;II. NONCOOPERATIVE TRAJECTORIES;17
7.3;III. STATEMENT OF EXISTENCE THEOREMS;20
7.4;IV. EXISTENCE OF DISCRETE NONCOOPERATIVE TRAJECTORIES;22
7.5;V. EXISTENCE OF CONTINUOUS NONCOOPERATIVE TRAJECTORIES;28
7.6;VI. STABLE NONCOOPERATIVE EQUILIBRIA;31
7.7;REFERENCES;35
8;CHAPTER 2. PROCESSUS DE CONTRÔLE AVEC CONTRÔLE INITIAL;38
8.1;I. INTRODUCTION;38
8.2;II. RESULTATS PRINCIPAUX;44
8.3;III. DEMONSTRATIONS;45
8.4;BIBLIOGRAPHIE;50
9;CHAPTER 3. DETERMINATION AND APPLICATION OF VEKUA RESOLVENTS;52
9.1;APPLICATIONS;55
9.2;REFERENCES;58
10;CHAPTER 4. GENERALIZED HOPF BIFURCATION;60
10.1;I. INTRODUCTION;60
10.2;II. RESULTS;62
10.3;III. OUTLINE OF PROOFS;65
10.4;REFERENCES;72
11;CHAPTER 5. PERTURBATION OF LINEAR DIFFERENTIAL EQUATIONS BY A HALF-LINEAR TERM DEPENDING ON A SMALL PARAMETER;74
12;CHAPTER 6. ON SOME CAUCHY PROBLEMS ARISING IN COMPUTATIONAL METHODS;80
12.1;REFERENCES;85
13;CHAPTER 7. COMPARISON RESULTS AND CRITICALITY IN SOME COMBUSTION PROBLEMS;86
13.1;I. INTRODUCTION;86
13.2;II. A MODEL PROBLEM;87
13.3;III. A COMPARISON RESULT;88
13.4;IV. SOME CRITICALITY CALCULATIONS;90
13.5;CAPTION FOR FIGURE;93
13.6;REFERENCES;95
14;CHAPTER 8. BOUNDARY VALUE PROBLEMS FOR PAIRS OF ORDINARY DIFFERENTIAL OPERATORS;96
14.1;I. INTRODUCTION;96
14.2;II. THE DIFFERENTIAL OPERATOR M;97
14.3;III. THE HILBERT
SPACE;99
14.4;IV. SUBSPACES A DETERMINED BY L,M;101
14.5;V. RESOLVENTS;103
15;CHAPTER 9. SEMILINEAR ELLIPTIC EQUATIONS AT RESONANCE: HIGHER EIGENVALUES AND UNBOUNDED NONLINEARITIES;104
15.1;REFERENCES;113
16;CHAPTER 10. COUNTABLE SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS;116
16.1;ACKNOWLEDGEMENT;122
16.2;REFERENCES;123
17;CHAPTER 11. THE ROLE OF THE STRUCTURAL OPERATOR AND THE QUOTIENT SPACE STRUCTURE IN THE THEORY OF HEREDITARY DIFFERENTIAL EQUATIONS;126
17.1;I. INTRODUCTION;126
17.2;II. REVIEW OF AUTONOMOUS LINEAR SYSTEMS;128
17.3;III. THE STRUCTURAL OPERATOR;134
17.4;IV. QUOTIENT SPACE AND QUOTIENT SEMIGROUP;140
17.5;REFERENCES;145
18;CHAPTER 12. DEGENERATE EVOLUTION EQUATIONS AND SINGULAR OPTIMAL CONTROL;150
18.1;REFERENCES;156
19;CHAPTER 13. COMMUTATIVE LINEAR DIFFERENTIAL OPERATORS;158
19.1;I. INTRODUCTION;158
19.2;II. NOTATIONS;159
19.3;III. THE RING OF OPERATORS;160
19.4;IV. COMMUTATIVE OPERATORS;162
20;CHAPTER 14. APPROXIMATIONS OF DELAYS BY ORDINARY
DIFFERENTIAL EQUATIONS;170
20.1;I. INTRODUCTION. STATE OF THE ART;170
20.2;PART I: COUPLED DIFFERENTIAL AND DIFFERENCE EQUATIONS;172
20.3;PART II: APPLICATIONS TO THE FOUNDATIONSOF THE METHOD OF
LINES;196
20.4;REFERENCES;211
21;CHAPTER 15. LINEAR STIELTJES INTEGRO-DIFFERENTIAL EQUATIONS;214
21.1;ABSTRACT;214
21.2;I. INTRODUCTION;215
21.3;II. NOTATIONS;215
21.4;III. THE MAIN THEOREM;219
21.5;REFERENCES;222
22;CHAPTER 16. A CRITICAL STUDY OF STABILITY OF NEUTRAL
FUNCTIONAL DIFFERENTIAL EQUATIONS;224
22.1;I. SOME STABILITY IMPLICATIONS;225
22.2;II. DISSIPATIVE PROCESS;232
22.3;III. TOTAL AND INTEGRAL STABILITY;233
22.4;IV. INTEGRAL STABILITY;237
22.5;REFERENCES;244
23;CHAPTER 17. NONLINEAR PERTURBATIONS OF LINEAR PROBLEMS
WITH INFINITE DIMENSIONAL KERNEL;246
23.1;I. INTRODUCTION;246
23.2;II. AN ABSTRACT EXISTENCE THEOREMFOR THE
CASE;247
23.3;III. PRELIMINARIES FOR THE
CASE;249
23.4;IV. AN ABSTRACT THEOREM FOR THE
CASE;250
23.5;V. TWO APPLICATIONS;252
23.6;VI. REMARKS;255
23.7;REFERENCES;255
24;CHAPTER 18. COMPARISON RESULTS FOR REACTION-DIFFUSION EQUATIONS;258
24.1;I. INTRODUCTION;258
24.2;II. MAIN RESULTS;260
24.3;III. APPLICATIONS;268
24.4;REFERENCES;272
25;CHAPTER 19. ON THE SYNTHESIS OF SOLUTIONS
OF INTEGRAL EQUATIONS;276
25.1;REFERENCES;285
26;CHAPTER 20. LOCAL EXACT CONTROLLABILITY
OF NONLINEAR EVOLUTION EQUATIONS;286
26.1;I. INTRODUCTION;286
26.2;II. THE LINEARIZED SYSTEM;289
26.3;III. EXACT CONTROLLABILITY;293
26.4;REFERENCES;295
27;CHAPTER 21. TOPOLOGICAL DEGREE AND THE STABILITY
OF A CLASS OF VOLTERRA INTEGRAL EQUATIONS;296
28;CHAPTER 22. PERIODIC SOLUTIONS OF SOME NONLINEAR SECOND
ORDER DIFFERENTIAL EQUATIONS IN HILBERT SPACES;302
28.1;I. INTRODUCTION;302
28.2;II. FORMULATION OF THE PROBLEM AND THE EXISTENCE
RESULT IN THE FINITE-DIMENSIONAL CASE;304
28.3;III. THE EXISTENCE THEOREM IN THE
INFINITE-DIMENSIONAL CASE;305
28.4;REFERENCES;308
29;CHAPTER 23. OPERATORS OF MONOTONE TYPE AND ALTERNATIVE PROBLEMS
WITH INFINITE DIMENSIONAL KERNEL;310
29.1;I. INTRODUCTION;310
29.2;II. MAPPINGS OF TYPE GT AND THEIR PROPERTIES;311
29.3;III. A CONTINUATION THEOREM FOR EQUATIONSINVOLVING SOME PERTURBATIONS OF
TYPE;312
29.4;IV. APPLICATIONS;317
29.5;REFERENCES;321
30;CHAPTER 24. STABILITY THEORY FOR COUNTABLY INFINITE SYSTEMS;324
30.1;I. INTRODUCTION;324
30.2;II. COUNTABLY INFINITE SYSTEMS"OF DIFFERENTIAL EQUATIONS;325
30.3;REFERENCES;331
31;CHAPTER 25. A NONLINEAR HYPERBOLIC VOLTERRA EQUATION
ARISING IN HEAT FLOW;332
31.1;I. INTRODUCTION;332
31.2;II. STATEMENT OF RESULTS;337
31.3;III. OUTLINE OF PROOF OF THEOREM 2.1;340
31.4;REFERENCES;350
32;CHAPTER 26. LINEARITY AND NONLINEARITYIN THE THEORY OF G-CONVERGENCE;352
32.1;0. INTRODUCTION;352
32.2;I. ABSTRACT DEFINITION OF G-CONVERGENCE;353
32.3;II. G-CONVERGENCE FOR ORDINARY DIFFERENTIAL EQUATIONS;354
32.4;III. MAIN PROBLEMS;357
32.5;IV. G-CONVERGENCE AND CONVERGENCE OF COEFFICIENTS;358
32.6;V. MORE ABOUT CAUCHY G-CONVERGENCE. NON-LINEARITY;367
32.7;VI. LIMITS OF THE STANDARD THEORY OF G-CONVERGENCE;374
32.8;VII. GENERAL THEORY OF G-CONVERGENCE FOR
PARAMETRIC DIFFERENTIAL EQUATIONS;376
32.9;VIII. SOME RESULTS IN GENERALIZED HOMOGENEIZATION;384
32.10;REFERENCES;386
33;CHAPTER 27. PATH INTEGRALS AND PARTIAL DIFFERENTIAL EQUATIONS;388
34;CHAPTER 28. ON PERIODIC SOLUTIONS OF HAMILTONIAN SYSTEMS
OF ORDINARY DIFFERENTIAL EQUATIONS;394
34.1;REFERENCES;399
35;CHAPTER 29. SOME RESULTS IN FUNCTIONAL INTEGRAL EQUATIONS
IN A BANACH SPACE;402
35.1;ACKNOWLEDGEMENT;407
35.2;REFERENCES;407
36;CHAPTER 30. TURBULENCE AND HIGHER ORDER BIFURCATIONS;408
36.1;I. SPECULATIONS ABOUT TURBULENCE;408
36.2;III. SECOND-ORDER BIFURCATION;412
36.3;IV. HIGHER-ORDER BIFURCATION;413
36.4;V. THE HOPF-LANDAU CONJECTURE;417
36.5;REFERENCES;418
37;CHAPTER 31. CONVERGENCE OF POWER SERIES SOLUTIONS OF p-ADIC
NONLINEAR DIFFERENTIAL EQUATION;420
37.1;I. INTRODUCTION;420
37.2;II. PROOF OF THEOREM 1;423
37.3;III. PROOF OF THEOREM 2: PART I;424
37.4;IV. PROOF OF THEOREM 2: PART II;426
37.5;V. PROOF OF THEOREM 2: PART III;428
37.6;VI. PROOF OF THEOREM 2: PART IV;433
37.7;REFERENCES;434
38;CHAPTER 32. UNIQUENESS OF PERIODIC SOLUTIONS
OF THE LIENARD EQUATION;436
38.1;I. INTRODUCTION;436
38.2;II. EXISTENCE OF PERIODIC SOLUTIONS;437
38.3;III. UNIQUENESS OF PERIODIC SOLUTIONS;438
38.4;REFERENCES;442
39;CHAPTER 33. BOUNDARY STABILIZABILITY FOR DIFFUSION PROCESSES;446
39.1;I. THE STABILIZABILITY PROBLEM;447
39.2;II. WELL-POSEDNESS OF EQ. (1.5);450
39.3;REFERENCES;461