Lakshmikantham Nonlinear Systems and Applications

An International Conference
1. Auflage 2014
ISBN: 978-1-4832-7224-5
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 716 Seiten, Web PDF

Nonlinear Systems and Applications: An International Conference contains the proceedings of an International Conference on Nonlinear Systems and Applications held at the University of Texas at Arlington, on July 19-23, 1976. The conference provided a forum for reviewing advances in nonlinear systems and their applications and tackled a wide array of topics ranging from abstract evolution equations and nonlinear semigroups to controllability and reachability. Various methods used in solving equations are also discussed, including approximation techniques for delay systems. Most of the applications are in the area of the life sciences. Comprised of 59 chapters, this book begins with a discussion on monotonically convergent upper and lower bounds for classes of conflicting populations, followed by an analysis of constrained problems. The reader is then introduced to approximation techniques for delay systems in biological models; differential inequalities for Liapunov functions; and stability or chaos in discrete epidemic models. Subsequent chapters deal with nonlinear boundary value problems for elliptic systems; bounds for solutions of reaction-diffusion equations; monotonicity and measurability; and periodic solutions of some integral equations from the theory of epidemics. This monograph will be helpful to students, practitioners, and researchers in the field of mathematics.

Autoren/Hrsg.

Lakshmikantham, V.

Inhaltsverzeichnis

1;Front Cover;1
2;Nonlinear Systems and Applications: An International Conference;4
3;Copyright Page;5
4;Table of Contents;6
5;List of Contributors;12
6;Preface;16
7;PART I: Contributed Papers;18
7.1;CHAPTER 1. MONOTONICALLY CONVERGENT UPPER AND LOWER BOUNDSFOR CLASSES OF CONFLICTING POPULATIONS;20
7.1.1;1. Introduction;20
7.1.2;2. Introductory Example;21
7.1.3;3. Remark on the Theory;23
7.1.4;4. Chemical Reaction Example;26
7.1.5;5. Bounds for Parabolic Equations;30
7.1.6;REFERENCES;30
7.2;CHAPTER 2. A NEW VIEW ON CONSTRAINED PROBLEMS;32
7.2.1;ACKNOWLEDGMENT;35
7.2.2;REFERENCES;36
7.3;CHAPTER 3. DELAY SYSTEMS IN BIOLOGICAL MODELS:APPROXIMATION TECHNIQUES;38
7.3.1;Introduction;38
7.3.2;I. delays in Biomodels;38
7.3.3;II. Approximation of Delay Systems;44
7.3.4;REFERENCES;53
7.4;CHAPTER 4. CONVERGENCE OF VOLTERRA SERIES ON INFINITE INTERVALSAND BILINEAR APPROXIMATIONS;56
7.4.1;1. Introduction;56
7.4.2;2. The Radius of Convergence r([0, 8));57
7.4.3;3. Some Nonlinear Phenomena;60
7.4.4;REFERENCES;63
7.5;CHAPTER 5. DIFFERENTIAL INEQUALITIES FOR LIAPUNOV FUNCTIONS;64
7.6;CHAPTER 6. NONLINEAR OSCILLATIONS;66
7.6.1;1. Introduction;66
7.6.2;2. The Alternative, or Bifurcation Process;67
7.6.3;3. Forces Oscillations of Liénard Systems;68
7.6.4;4. Nonlinear Eigenvalues;69
7.6.5;5. A Future Decomposition of the Spaces X and Y;72
7.6.6;6. The Use of Finite Elements;73
7.6.7;7. Boundary Value Problems for Schauder's Hyperbolic Systems;75
7.6.8;8. A Problem in Nonlinear Optics;78
7.6.9;REFERENCES;80
7.7;CHAPTER 7. A COMPARISON PRINCIPLE FOR STEADY-STATE DIFFUSION OPERATORS;84
7.7.1;REFERENCES;87
7.8;CHAPTER 8. STABILITY OR CHAOS IN DISCRETE EPIDEMIC MODELS;90
7.8.1;1. Introduction;90
7.8.2;2. The Model and Derivation of Equations;92
7.8.3;3. Analysis for One-dimensional Model;96
7.8.4;4. Higher dimensional Systems;99
7.8.5;REFERENCES;104
7.8.6;ACKNOWLEDGMENT;105
7.9;CHAPTER 9. ANALYTIC METHODS FOR APPROXIMATE SOLUTION OFELLIPTIC FREE BOUNDARY PROBLEMS;112
7.10;CHAPTER 10. NONLINEAR BOUNDARY VALUE PROBLEMS FOR ELLIPTIC SYSTEMS IN THE PLANE;114
7.10.1;Introduction;114
7.10.2;I. Nonlinear Cauchy-Riemann Problems;114
7.10.3;II. Elements of Hypercomplex Function Theory;119
7.10.4;III. Boundary Value Problems for Generalized Hyperanalytic Function (Linear Systems);124
7.10.5;IV. Boundary Value Problems for Nonlinear Systems in the Plane;132
7.10.6;REFERENCES;140
7.11;CHAPTER 11. ERROR PROPAGATION AND CATASTROPHES IN PROTEIN SYNTHESIZING MACHINERY;142
7.11.1;Introduction;142
7.11.2;2. Simple Basic Model;145
7.11.3;3. General Model;152
7.11.4;ACKNOWLEDGMENTS;159
7.11.5;REFERENCES;159
7.12;CHAPTER 12. SOME NEW APPLICATIONS OF POPOV'S FREQUENCY-DOMAIN METHOD;164
7.12.1;1. Frequency-Domain Criteria for Nuclear Reactor Stability;164
7.12.2;2. Absolute Stability and Forced Oscillations for Some Distributed Parmeter Systems;172
7.12.3;REFERENCES;178
7.13;CHAPTER 13. BOUNDS FOR SOLUTIONS OF REACTION-DIFFUSION EQUATIONS;180
7.13.1;1. Introduction;180
7.13.2;2. The Nonlinear Analysis Approximation Technique;180
7.13.3;3. Illustration of the Method;185
7.13.4;REFERENCES;187
7.14;CHAPTER 14. IMBEDDING METHODS FOR BIFURCATION PROBLEMS AND POST-BUCKLING BEHAVIOR OF NONLINEAR COLUMNS;190
7.14.1;1. Introduction;190
7.14.2;2. The Potential Energy Function (2);191
7.14.3;3. The Euler Equation;194
7.14.4;4. The Moment Function;195
7.14.5;3. Numerical Results;199
7.14.6;6. The Curvature and Free-End Angle Functions;202
7.14.7;7. Discussion;204
7.14.8;REFERENCES;204
7.15;CHAPTER 15. MONOTONICITY AND MEASURABILITY;206
7.15.1;REFERENCES;214
7.16;CHAPTER 16. CHAOTIC BEHAVIOR IN DYNAMICAL SYSTEMS;216
7.16.1;REFERENCES;226
7.17;CHAPTER 17. STABILITY TECHNIQUE AND THOUGHT PROVOCATIVE DYNAMICAL SYSTEMS;228
7.17.1;REFERENCES;234
7.18;CHAPTER 18. THE CURRENT STATUS OF ABSTRACT CAUCHY PROBLEM;236
7.18.1;V. Maximal Solution;241
7.18.2;VI. Delay Differential Equations;241
7.18.3;VII. Set-valued Differential Equations;241
7.18.4;REFERENCES;245
7.19;CHAPTER 19. ON THE METHOD OF CONE-VALUED LYAPUNOV FUNCTIONS;248
7.19.1;REFERENCES;251
7.20;CHAPTER 20. PERIODIC SOLUTIONS OF SOME INTEGRAL EQUATIONS FROM THE THEORY OF EPIDEMICS;252
7.20.1;1. Introduction;252
7.20.2;2. A Global Bifurcation Theorem;253
7.20.3;3. Existence and Uniqueness of Solutions to Some Nonlinear Integral Equations;254
7.20.4;4. Some Linear Spectral Theory;260
7.20.5;5. The Fixed Point Index of Some Cone Maps;262
7.20.6;REFERENCES;272
7.21;CHAPTER 21. SOME RECENT PROGRESS IN NEURO-MUSCULAR SYSTEMS;274
7.21.1;1. Introduction;274
7.21.2;2. Physiologically Significant Properties of the Neuro-Muscular Model;282
7.21.3;3. Effect of Multiple Pathways;294
7.21.4;4. Acceleration Sensitivity;298
7.21.5;5. The Response of the Neuro-Muscular System to Centrally Generated Oscillations;303
7.21.6;6. Final Remarks;305
7.21.7;REFERENCES;307
7.22;CHAPTER 22. APPLICATIONS OF THE SATURABILITY TECHNIQUE IN THE PROBLEMOF STABILITY OF NONLINEAR SYSTEMS;312
7.23;CHAPTER 23. NONLINEAR EVOLUTION EQUATIONS AND NONLINEAR ERGODIC THEOREMS;314
7.23.1;REFERENCES;315
7.24;CHAPTER 24. ON PURE STRUCTURE OF DYNAMIC SYSTEMS;316
7.24.1;REFERENCES;317
7.25;CHAPTER 25. OPTIMIZING AND EXTREMIZING NONLINEAR BOUNDARY VALUE PROBLEMS INLENTICULAR ANTENNAS IN OCEANOGRAPHY, MEDICINE & COMMUNICATIONS:SOME SOLUTIONS AND SOME QUESTIONS;318
7.25.1;1. Introduction: The Scanning Problem at Low F-Numbers;318
7.25.2;2. The Bifocal Boundary Value Problem;320
7.25.3;3. Solution by Polynomial Approximations and Infinite Series;326
7.25.4;4. Optimization for Azimuth vs. Elevation Scanning Trade Off;334
7.25.5;5. General Theory, Applications and Some Open Questions;337
7.25.6;ACKNOWLEDGMENTS;341
7.25.7;REFERENCES;342
7.26;CHAPTER 26. A MODEL FOR THE STATISTICAL ANALYSIS OF CELL RADIOSENSITIVITY DURING THE CELL CYCLE;344
7.26.1;1. Introduction;344
7.26.2;2. Radiosensitivity Model;346
7.26.3;3. statistical Characterization of the System;348
7.26.4;4. Results and Conclusions;353
7.26.5;Acknowledgments;363
7.26.6;REFERENCES;364
7.27;CHAPTER 27. VOLTERRA INTEGRAL EQUATIONS AND NONLINEAR SEMIGROUPS;366
8;PART II: Invited Addresses and Research Reports;368
8.1;CHAPTER 28. THE MATHEMATICAL ANALYSIS OF A FOUR-COMPARTMENT STOCHASTIC MODELOF ROSE BENGAL TRANSPORT THROUGH THE HEPATIC SYSTEM;370
8.1.1;Introduction;370
8.1.2;Development of the Mathematical Model;370
8.1.3;Analysis of the Mathematical Model;375
8.1.4;Estimation of the Model's Parameters;380
8.1.5;Observations and Conclusions;387
8.1.6;Acknowledgments;388
8.1.7;References;388
8.2;CHAPTER 29. A MORE EFFICIENT ALGORITHM FOR AN OPTIMAL TOUR OF A HEALTH CARE CONSUMER WITH MULTIPLEHEALTH CARE FACILITIES IN EACH COUNTY;390
8.2.1;1. Introduction;390
8.2.2;2. Bansal and Kumar's Algorithm [1];391
8.2.3;3. Modified Algorithm;392
8.2.4;4. Numerical Example;394
8.2.5;REFERENCES;397
8.3;CHAPTER 30. STABILITY OF NON-COMPACT SETS;398
8.3.1;1. Introduction;398
8.3.2;2. Basic Notions;398
8.3.3;3. Limit sets, Prolongations and Prolongational Limit Sets;399
8.3.4;4. Stability and Asymptotic Stability;402
8.3.5;REFERENCES;405
8.4;CHAPTER 31. PARAMETRIC EXCITATION AS THE MEANS OF ENERGY TRANSFERIN QUANTAL SYSTEMS WITH REFERENCE TO CARCINOGENESIS AND BIOENERGETICS;406
8.5;CHAPTER 32. QUANTUM STATISTICAL FOUNDATIONS FOR STRUCTURAL INFORMATION THEORY AND COMMUNICATION THEORY;408
8.5.1;REFERENCES;423
8.6;CHAPTER 33. BOUNDARY VALUE PROBLEMS FOR NONLINEAR DIFFERENTIAL EQUATIONS;428
8.6.1;REFERENCES;429
8.7;CHAPTER 34. GLOBAL SOLUTION FOR A PROBLEM OF NEUTRON TRANSPORT WITH TEMPERATURE FEEDBACK;430
8.7.1;1. Introduction;430
8.7.2;2. The Nonlinear Initial-Value Problem;431
8.7.3;3. The Auxiliary Linear Problems;433
8.7.4;4. A Priori Properties of u(t);434
8.7.5;5. The Nonlinear Operator F;436
8.7.6;6. Global Strong Solution of System (12);436
8.7.7;7. Concluding Remarks;437
8.7.8;Appendix A;438
8.7.9;Appendix B;439
8.7.10;Appendix C;440
8.7.11;REFERENCES;440
8.8;CHAPTER 35. OPTIMAL HARVESTING FOR THE LOGISTIC AND GOMPERTZ GROWTH CURVE;442
8.8.1;Introduction;442
8.8.2;Yield Formula;443
8.8.3;The Logistic Curve;444
8.8.4;Yield for the Logistic Curve;447
8.8.5;The Gompertz Curve;449
8.8.6;REFERENCES;452
8.9;CHAPTER 36. MAXIMUM AND MINIMUM DEGENERACY SET OF LINEAR TIME-INVARIANTDELAY-DIFFERENTIAL SYSTEMS OF THE NEUTRAL TYPE;454
8.9.1;1. Introduction;454
8.9.2;2. Preliminary Computations;455
8.9.3;3. Representation Formula and Necessary Conditions for Degeneracy;457
8.9.4;4. Proof of the Theorem;458
8.9.5;REFERENCES;460
8.10;CHAPTER 37. ON THE CONTROLLABILITY TO CLOSED SETS OF NONLINEAR AND RELATED LINEAR SYSTEMS;462
8.11;CHAPTER 38. A DISCRETE-TIME NONLINEAR (m + n)-PERSON LABOR MARKET SYSTEM WITH UNCERTAINTY;464
8.11.1;1. Introduction;464
8.11.2;2. The Model;466
8.11.3;3. The Main Results;473
8.11.4;4. Proofs;476
8.11.5;REFERENCES;484
8.12;CHAPTER 39. TIME DELAYS IN PREDATOR-PREY SYSTEMS;486
8.12.1;REFERENCE;489
8.13;CHAPTER 40. ON SELECTING A RESPONSE FUNCTION IN NONLINEAR REGRESSION;490
8.13.1;Generic Functions and Hierarchies;490
8.13.2;Statistical Methodology;492
8.13.3;Numerical Example;493
8.13.4;REFERENCES;495
8.14;CHAPTER 41. ON THE MOMENTS OF EXPONENTIAL DECAY;498
8.14.1;1. Introduction;498
8.14.2;2. Analysis;502
8.14.3;3. Problems for Future Research;505
8.14.4;REFERENCES;506
8.15;CHAPTER 42. ON THE STEADY STATE OF AN AGE DEPENDENT MODEL FOR MALARIA;508
8.15.1;0. Introduction;508
8.15.2;1. Existence and Non-uniqueness of Solutions;511
8.15.3;2. Properties of Solutions;520
8.15.4;3. Numerical Evidence and Further Questions;527
8.15.5;REFERENCES;529
8.16;CHAPTER 43. ABSTRACT VOLTERRA EQUATIONS WITH INFINITE DELAY;530
8.16.1;Introduction;530
8.16.2;2. The Evolution Operator;531
8.16.3;3. Stability;533
8.16.4;4. An Infinitesimal Generator;535
8.16.5;5. Representation and Approximation;537
8.16.6;REFERENCES;540
8.17;CHAPTER 44. THE INCIDENCE OF INFECTIOUS DISEASES UNDER THE INFLUENCEOF SEASONAL FLUCTUATIONS - ANALYTICAL APPROACH;542
8.17.1;1. Introduction;542
8.17.2;2. Passive Oscillations;545
8.17.3;3. Subharmonic Resonance;552
8.17.4;4. Subharmonic Resonance - the Amplitude;559
8.17.5;5. Conclusions;561
8.17.6;REFERENCES;563
8.18;CHAPTER 45. SUM OF RANGES OF OPERATORS AND APPLICATIONS;564
8.18.1;Section 1;564
8.18.2;Section 2;570
8.18.3;REFERENCES;575
8.19;CHAPTER 46. A VARIATION OF RAZUMIKHIN'S METHOD FOR

RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS;578
8.19.1;1. Introduction;578
8.19.2;2. Notation;579
8.19.3;3. The Main Result;580
8.19.4;REFERENCES;583
8.20;CHAPTER 47. A DEGENERATE PROBLEM OF SINGULAR PERTURBATIONS;584
8.21;CHAPTER 48. BOCHER-OSGOOD TYPE THE OREMS FOR THIRD ORDER DIFFERENTIAL EQUATIONS;590
8.21.1;REFERENCES;592
8.22;CHAPTER 49. ON THE CHARACTERIZATION OF MACROMOLECULAR LENGTH DISTRIBUTIONSBY ANALYSIS OF ELECTRICAL BIREFRINGENCE DECAY;594
8.22.1;Introduction;594
8.22.2;Derivation of Equations;600
8.22.3;Discussion of the Equations;603
8.22.4;Plans for Future Work;605
8.22.5;REFERENCES;606
8.23;CHAPTER 50. PERIODIC SOLUTIONS AND PERTURBED SEMIGROUPS OF LINEAR OPERATORS;608
8.23.1;Section 1. Preliminaries and Existence;609
8.23.2;Section 2. Periodic Solutions;612
8.23.3;Section3. Example;616
8.23.4;Acknowledgements;619
8.23.5;REFERENCES;619
8.24;CHAPTER 51. SPECULATIVE DEMAND WITH SUPPLY RESPONSE LAG;620
8.24.1;1. Introduction;620
8.24.2;2. The Model;620
8.24.3;3. Mathematical Results;622
8.24.4;4. atypical Supply Curves;624
8.24.5;5. Concluding Remarks;624
8.24.6;REFERENCES;627
8.25;CHAPTER 52. A GEOMETRICAL STUDY OF B–CELL STIMULATION AND HUMORAL IMMUNE RESPONSE;628
8.25.1;Introduction;628
8.25.2;Section 1. Description of Humoral Immune Response;629
8.25.3;Section 2. A Model of B-Cell Stimulation;630
8.25.4;Section 3. Construction of a Model of Humoral Immune Response;637
8.25.5;REFERENCES;646
8.26;CHAPTER 53. GENERALIZED STABILITY OF MOTIONAND VECTOR LYAPUNOV FUNCTIONS;648
8.26.1;1. Introduction;648
8.26.2;2. Preliminaries;648
8.26.3;3. Main Results;651
8.26.4;4. Example;660
8.26.5;REFERENCES;662
8.27;CHAPTER 54. SIMPLE ANALOGS FOR NERVE MEMBRANE EQUATIONS;664
8.27.1;REFERENCES;671
8.28;CHAPTER 55. OSCILLATION RESULTS FOR A NONHOMOGENEOUS EQUATION;672
8.29;CHAPTER 56. THE COMPARISON OF A FOUR-COMPARTMENT AND A FIVE-COMPARTMENT MODEL OF ROSE BENGALTRANSPORT THROUGH THE HEPATIC SYSTEM;674
8.29.1;Introduction;674
8.29.2;Background;674
8.29.3;Bilirubin Transfer;675
8.29.4;Rose Bengal;676
8.29.5;A Four-Compartmental Model (Summary);676
8.29.6;A Five-Compartment Model;680
8.29.7;Procedure;682
8.29.8;Results Using Patient Data;683
8.29.9;Results;684
8.29.10;Conclusion;686
8.29.11;REFERENCES;687
8.30;CHAPTER 57. SOME PROPERTIES OF INCREASING DENSIFYING MAPPINGS;688
8.30.1;Introduction;688
8.30.2;REFERENCES;701
8.31;CHAPTER 58. OSCILLATION IN A NON-LINEAR PARABOLIC MODELOF SEPARATED, COOPERATIVELY COUPLED ENZYMES;704
8.31.1;I. Introduction;704
8.31.2;II. Model;705
8.31.3;III. Stability of Equilibrium State. Periodic Solutions;706
8.31.4;IV. Discussion;709
8.31.5;REFERENCES;710
8.32;CHAPTER 59. ITERATIVE TECHNIQUES FOR INVERSION OF THE NONLINEARPERRIN EQUATIONS FOR DIFFUSION OF SPHEROIDS;712
8.32.1;Inversion Procedure for R1 and R3;713
8.32.2;Inversion Procedure for R1 and D;714
8.32.3;Discussion;715
8.32.4;REFERENCES;716