Gespeichert in:
| Titel: | Half-linear differential equations |
|---|---|
| Von: |
Ondřej Došlý, Pavel Řehák
|
| Person: |
Došlý, Ondřej
Verfasser aut Řehák, Pavel |
| Hauptverfassende: | , |
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Amsterdam [u.a.]
Elsevier
2005
|
| Ausgabe: | 1. ed. |
| Schriftenreihe: | North-Holland mathematics studies
202 |
| Schlagworte: | |
| Online-Zugang: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013312604&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Beschreibung: | XIV, 517 S. graph. Darst. |
| ISBN: | 0444520392 |
Internformat
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Datensatz im Suchindex
| _version_ | 1819340155255783424 |
|---|---|
| adam_text | HALF-LINEAR DIFFERENTIAL EQUATIONS ONDREJ DOSLY DEPARTMENT OF
MATHEMATICS MASARYK UNIVERSITY BRNO, CZECH REPUBLIC PAVEL REHAK
MATHEMATICAL INSTITUTE ACADEMY OF SCIENCES OF THE CZECH REPUBLIC BRNO,
CZECH REPUBLIC ELSEVIER 2005 AMSTERDAM - BOSTON - HEIDELBERG - LONDON -
NEW YORK - OXFORD PARIS - SAN DIEGO - SAN FRANCISCO - SINGAPORE - SYDNEY
- TOKYO CONTENTS PREFACE BASIC THEORY 1 1.1 EXISTENCE AND UNIQUENESS 2
1.1.1 FIRST ORDER HALF-LINEAR SYSTEM AND OTHER FORMS OF HALF-LINEAR
EQUATIONS 2 1.1.2 HALF-LINEAR TRIGONOMETRIC FUNCTIONS 4 1.1.3
HALF-LINEAR PRIIFER TRANSFORMATION 7 1.1.4 HALF-LINEAR RICCATI
TRANSFORMATION 8 1.1.5 EXISTENCE AND UNIQUENESS THEOREM 8 1.1.6 AN
ALTERNATIVE APPROACH TO THE EXISTENCE THEORY 9 1.2 STURMIAN THEORY 13
1.2.1 PICONE S IDENTITY 13 1.2.2 ENERGY FUNCTIONAL 14 1.2.3 ROUNDABOUT
THEOREM 14 1.2.4 STURMIAN SEPARATION AND COMPARISON THEOREMS 16 1.2.5
MORE ON THE PROOF OF THE SEPARATION THEOREM 18 1.2.6 DISCONJUGACY ON
VARIOUS TYPES OF INTERVALS 20 1.2.7 TRANSFORMATION OF INDEPENDENT
VARIABLE 21 1.2.8 RECIPROCITY PRINCIPLE * 22 1.2.9 STURMIAN THEORY FOR
MIRZOV S SYSTEM 22 1.2.10 LEIGHTON-WINTNER OSCILLATION CRITERION 25 1.3
DIFFERENCES BETWEEN LINEAR AND HALF-LINEAR EQUATIONS 26 1.3.1 WRONSKIAN
IDENTITY 27 1.3.2 TRANSFORMATION FORMULA 30 1.3.3 FREDHOLM ALTERNATIVE
31 1.4 SOME ELEMENTARY HALF-LINEAR EQUATIONS 32 1.4.1 EQUATIONS WITH
CONSTANT COEFFICIENTS 32 CONTENTS 1.4.2 EULER TYPE HALF-LINEAR
DIFFERENTIAL EQUATION 37 1.4.3 KNESER TYPE OSCILLATION AND
NONOSCILLATION CRITERIA 43 1.5 NOTES AND REFERENCES 44 METHODS OF
OSCILLATION THEORY 47 2.1 VARIATIONAL PRINCIPLE 47 2.1.1 FORMULATION OF
VARIATIONAL PRINCIPLE 48 2.1.2 WIRTINGER INEQUALITY 48 2.1.3
APPLICATIONS 49 2.2 RICCATI TECHNIQUE 50 2.2.1 PRELIMINARIES 50 2.2.2
MORE GENERAL RICCATI TRANSFORMATION 52 2.2.3 RICCATI INEQUALITY 52 2.2.4
HALF-LINEAR HARTMAN-WINTNER THEOREM 54 2.2.5 POSITIVE SOLUTION OF
GENERALIZED RICCATI EQUATION 59 2.2.6 MODIFIED RICCATI INEQUALITY 63
2.2.7 APPLICATIONS 66 2.3 COMPARISON THEOREMS 69 2.3.1 HILLE-WINTNER
COMPARISON THEOREMS 69 2.3.2 LEIGHTON COMPARISON THEOREMS 73 2.3.3
MULTIPLIED COEFFICIENT COMPARISON 75 2.3.4 TELESCOPING PRINCIPLE 77
2.3.5 COMPARISON THEOREM WITH RESPECT TO P 80 2.4 NOTES AND REFERENCES
81 OSCILLATION AND NONOSCILLATION CRITERIA 83 3.1 CRITERIA OF CLASSICAL
TYPE 84 3.1.1 HILLE-NEHARI TYPE CRITERIA 85 3.1.2 OTHER CRITERIA 87
3.2 CRITERIA BY AVERAGING TECHNIQUE 91 3.2.1 COLES TYPE CRITERIA 91
3.2.2 GENERALIZED KAMENEV CRITERION 96 3.2.3 GENERALIZED H- FUNCTION
AVERAGING TECHNIQUE - PHILOS TYPE CRITERION 97 3.3 FURTHER EXTENSIONS OF
HILLE-NEHARI TYPE CRITERIA 99 3.3.1 Q,H TYPE CRITERIA 100 3.3.2
HILLE-NEHARI TYPE WEIGHTED CRITERIA AND EXTENSIONS 113 3.4 NOTES AND
REFERENCES 120 NONOSCILLATORY SOLUTIONS 123 4.1 ASYMPTOTIC OF
NONOSCILLATORY SOLUTIONS 123 4.1.1 INTEGRAL CONDITIONS AND
CLASSIFICATION OF SOLUTIONS 124 4.1.2 THE CASE C NEGATIVE 126 4.1.3
UNIQUENESS IN M~ 132 4.1.4 THE CASE C POSITIVE 134 4.1.5 GENERALIZED
FUBINI S THEOREM AND ITS APPLICATIONS 137 CONTENTS XI 4.2 PRINCIPAL
SOLUTION 140 4.2.1 PRINCIPAL SOLUTION OF LINEAR EQUATIONS 141 4.2.2
MIRZOV S CONSTRUCTION OF PRINCIPAL SOLUTION 142 4.2.3 CONSTRUCTION OF
ELBERT AND KUSANO 144 4.2.4 COMPARISON THEOREM FOR EVENTUALLY MINIMAL
SOLUTIONS OF RICCATI EQUATIONS 147 4.2.5 STURMIAN PROPERTY OF THE
PRINCIPAL SOLUTION 148 4.2.6 PRINCIPAL SOLUTION OF RECIPROCAL EQUATION
149 4.2.7 INTEGRALS ASSOCIATED WITH EVENTUALLY MINIMAL SOLUTION OF
RICCATI EQUATION 150 4.2.8 LIMIT CHARACTERIZATION OF THE PRINCIPAL
SOLUTION 154 4.2.9 INTEGRAL CHARACTERIZATION OF THE PRINCIPAL SOLUTION
158 4.2.10 ANOTHER INTEGRAL CHARACTERIZATION 164 4.2.11 OSCILLATION
CRITERIA AND (NON)PRINCIPAL SOLUTION 166 4.3 HALF-LINEAR DIFFERENTIAL
EQUATIONS AND KARAMATA FUNCTIONS 170 4.3.1 EXISTENCE OF REGULARLY
VARYING SOLUTIONS 171 4.3.2 EXISTENCE OF SLOWLY VARYING SOLUTIONS 183
4.4 NOTES AND REFERENCES 187 5 VARIOUS OSCILLATION PROBLEMS 189 5.1
CONJUGACY AND DISCONJUGACY 190 5.1.1 LYAPUNOV INEQUALITY 190 5.1.2
VALLEE-POUSSIN TYPE INEQUALITY 191 5.1.3 FOCAL POINT CRITERIA 193 5.1.4
LYAPUNOV-TYPE FOCAL POINTS AND CONJUGACY CRITERIA 196 5.1.5 FURTHER
RELATED RESULTS 200 5.2 PERTURBATION PRINCIPLE 201 5.2.1 GENERAL IDEA
201 5.2.2 SINGULAR LEIGHTON S THEOREM 202 5.2.3 LEIGHTON-WINTNER TYPE
OSCILLATION CRITERION 205 5.2.4 HILLE-NEHARI TYPE OSCILLATION CRITERION
207 5.2.5 HILLE-NEHARI TYPE NONOSCILLATION CRITERION 210 5.2.6 PERTURBED
EULER EQUATION 212 5.2.7 LINEARIZATION METHOD IN HALF-LINEAR OSCILLATION
THEORY . . . . 215 5.3 NONOSCILLATION DOMAINS AND (ALMOST) PERIODICITY
217 5.3.1 DISCONJUGACY DOMAIN AND NONOSCILLATION DOMAIN 217 5.3.2
EQUATIONS WITH PERIODIC COEFFICIENTS 229 5.3.3 EQUATIONS WITH ALMOST
PERIODIC COEFFICIENTS 230 5.4 STRONGLY AND CONDITIONALLY OSCILLATORY
EQUATION 237 5.4.1 STRONG (NON)OSCILLATION CRITERIA 238 5.4.2
OSCILLATION CONSTANT 240 5.5 FUNCTION SEQUENCE TECHNIQUE 243 5.5.1
FUNCTION SEQUENCES AND RICCATI INTEGRAL EQUATION 243 5.5.2 MODIFIED
APPROACHES 250 5.6 DISTANCE BETWEEN ZEROS OF OSCILLATORY SOLUTIONS 254
5.6.1 ASYMPTOTIC FORMULA FOR DISTRIBUTION OF ZEROS 255 XII CONTENTS
5.6.2 QUICKLY OSCILLATING SOLUTION 258 5.6.3 SLOWLY OSCILLATING SOLUTION
259 5.7 HALF-LINEAR STURM-LIOUVILLE PROBLEM 261 5.7.1 BASIC
STURM-LIOUVILLE PROBLEM 261 5.7.2 REGULAR PROBLEM WITH INDEFINITE WEIGHT
263 5.7.3 SINGULAR STURM-LIOUVILLE PROBLEM 267 5.7.4 SINGULAR EIGENVALUE
PROBLEM ASSOCIATED WITH RADIAL P-LAPLACIAN 269 5.7.5 ROTATION INDEX AND
PERIODIC POTENTIAL 270 5.8 ENERGY FUNCTIONAL AND VARIOUS BOUNDARY
CONDITIONS 282 5.8.1 DISFOCALITY 284 5.8.2 NONEXISTENCE OF COUPLED
POINTS 290 5.8.3 COMPARISON THEOREMS OF LEIGHTON-LEVIN TYPE 293 5.9
MISCELLANEOUS 298 5.9.1 EXTENDED HARTMAN-WINTNER CRITERION 298 5.9.2
HALF-LINEAR MILLOUX AND ARMELLINI-TONELLI-SANSONE THEOREMS 301 5.9.3
INTERVAL OSCILLATION CRITERIA 305 5.10 NOTES AND REFERENCES 309 6 BVP S
FOR HALF-LINEAR DIFFERENTIAL EQUATIONS 311 6.1 EIGENVALUES, EXISTENCE,
AND NONUNIQUENESS PROBLEMS 312 6.1.1 BASIC BOUNDARY VALUE PROBLEM 312
6.1.2 VARIATIONAL CHARACTERIZATION OF EIGENVALUES 312 6.1.3 EXISTENCE
AND (NON)UNIQUENESS BELOW THE FIRST EIGENVALUE . 315 6.1.4 HOMOTOPIC
DEFORMATION ALONG P AND LERAY-SCHAUDER DEGREE 318 6.1.5 MULTIPLICITY
NONRESONANCE RESULTS 320 6.2 FREDHOLM ALTERNATIVE FOR ONE-DIMENSIONAL
P-LAPLACIAN 325 6.2.1 RESONANCE AT THE FIRST EIGENVALUE 325 6.2.2
RESONANCE AT HIGHER EIGENVALUES 335 6.3 BOUNDARY VALUE PROBLEMS AT
RESONANCE 337 6.3.1 AMBROSETTI-PRODI TYPE RESULT 337 6.3.2
LANDESMAN-LAZER SOLVABILITY CONDITION 342 6.3.3 FUCIK SPECTRUM 347 6.4
NOTES AND REFERENCES 350 7 PARTIAL DIFFERENTIAL EQUATIONS WITH
P-LAPLACIAN 353 7.1 EIGENVALUES AND COMPARISON PRINCIPLE 353 7.1.1
DIRICHLET BVP WITH P-LAPLACIAN 354 7.1.2 SECOND EIGENVALUE OF
P-LAPLACIAN 357 7.1.3 COMPARISON AND ANTIMAXIMUM PRINCIPLE FOR
P-LAPLACIAN . . 358 7.1.4 FUCIK SPECTRUM FOR P-LAPLACIAN 361 7.2
BOUNDARY VALUE PROBLEMS AT RESONANCE 364 7.2.1 RESONANCE AT THE FIRST
EIGENVALUE IN HIGHER DIMENSION . . . 364 7.2.2 RESONANCE AT THE FIRST
EIGENVALUE - MULTIPLICITY RESULTS . . . 367 CONTENTS XIII 7.2.3
LANDESMAN-LAZER RESULT IN HIGHER DIMENSION 368 7.3 OSCILLATION THEORY OF
PDE S WITH P-LAPLACIAN 371 7.3.1 PICONE S IDENTITY FOR EQUATIONS WITH
P-LAPLACIAN 372 7.3.2 NONEXISTENCE OF POSITIVE SOLUTIONS IN R N 373
7.3.3 OSCILLATION CRITERIA 378 7.3.4 EQUATIONS INVOLVING PSEUDOLAPLACIAN
380 .7.4 NOTES AND REFERENCES 381 8 HALF-LINEAR DIFFERENCE EQUATIONS 383
8.1 BASIC INFORMATION 383 8.1.1 LINEAR DIFFERENCE EQUATIONS 383 8.1.2
DISCRETIZATION, DIFFICULTIES VERSUS EASES 384 8.2 HALF-LINEAR DISCRETE
OSCILLATION THEORY 386 8.2.1 DISCRETE ROUNDABOUT THEOREM AND STURMIAN
THEORY 386 8.2.2 METHODS OF HALF-LINEAR DISCRETE OSCILLATION THEORY 390
8.2.3 REFINEMENTS OF RICCATI TECHNIQUE 391 8.2.4 DISCRETE OSCILLATION
CRITERIA 397 8.2.5 HILLE-NEHARI DISCRETE NONOSCILLATION CRITERIA 401
8.2.6 SOME DISCRETE COMPARISON THEOREMS 408 8.3 HALF-LINEAR DYNAMIC
EQUATIONS ON TIME SCALES 412 8.3.1 ESSENTIALS ON TIME SCALES, BASIC
PROPERTIES 412 8.3.2 OSCILLATION THEORY OF HALF-LINEAR DYNAMIC EQUATIONS
414 8.4 NOTES AND REFERENCES 416 9 RELATED DIFFERENTIAL EQUATIONS AND
INEQUALITIES 417 9.1 QUASILINEAR DIFFERENTIAL EQUATIONS 417 9.1.1
QUASILINEAR EQUATIONS WITH CONSTANT COEFFICIENTS 418 9.1.2 EXISTENCE,
UNIQUENESS AND SINGULAR SOLUTIONS 421 9.1.3 ASYMPTOTIC OF NONOSCILLATORY
SOLUTIONS 422 9.1.4 SUFFICIENT AND NECESSARY CONDITIONS FOR OSCILLATION
424 9.1.5 GENERALIZED RICCATI TRANSFORMATION AND APPLICATIONS . . . .
426 9.1.6 (HALF-)LINEARIZATION TECHNIQUE 428 9.1.7 SINGULAR SOLUTIONS OF
BLACK HOLE AND WHITE HOLE TYPE . . . . 431 9.1.8 CURIOUS DOUBLY SINGULAR
EQUATIONS 434 9.1.9 COUPLED QUASILINEAR SYSTEMS . 435 9.2 FORCED
HALF-LINEAR DIFFERENTIAL EQUATIONS 438 9.2.1 TWO OSCILLATORY CRITERIA
438 9.2.2 FORCED SUPER-HALF-LINEAR OSCILLATION 440 9.3 HALF-LINEAR
DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS 444 9.3.1 OSCILLATION OF
EQUATION WITH NONNEGATIVE SECOND COEFFICIENT . 444 9.3.2 OSCILLATION OF
EQUATION WITH NONPOSITIVE SECOND COEFFICIENT . 453 9.3.3 EXISTENCE AND
ASYMPTOTIC BEHAVIOR OF NONOSCILLATORY SOLUTIONS 457 9.4 HIGHER ORDER
HALF-LINEAR DIFFERENTIAL EQUATIONS 462 9.4.1 P-BIHARMONIC OPERATOR 462
9.4.2 HIGHER ORDER HALF-LINEAR EIGENVALUE PROBLEM 463 XIV CONTENTS 9.5
INEQUALITIES RELATED TO HALF-LINEAR DIFFERENTIAL EQUATIONS 466 9.5.1
INEQUALITIES OF WIRTINGER AND HARDY TYPE 466 9.5.2 INEQUALITIES OF OPIAL
TYPE 468 9.6 NOTES AND REFERENCES 469 BIBLIOGRAPHY 471 INDEX 497 AUTHOR
INDEX 515
|
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| id | DE-604.BV019990683 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T12:09:42Z |
| institution | BVB |
| isbn | 0444520392 |
| language | English |
| lccn | 2005049897 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-013312604 |
| oclc_num | 60371071 |
| open_access_boolean | |
| owner | DE-703 |
| owner_facet | DE-703 |
| physical | XIV, 517 S. graph. Darst. |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | Elsevier |
| record_format | marc |
| series | North-Holland mathematics studies |
| series2 | North-Holland mathematics studies |
| spellingShingle | Došlý, Ondřej Řehák, Pavel Half-linear differential equations North-Holland mathematics studies Differential equations Oscillation theory Differential equations, Linear Differential equations, Nonlinear Ordnung 2 (DE-588)4350619-7 gnd Differentialgleichung (DE-588)4012249-9 gnd |
| subject_GND | (DE-588)4350619-7 (DE-588)4012249-9 |
| title | Half-linear differential equations |
| title_auth | Half-linear differential equations |
| title_exact_search | Half-linear differential equations |
| title_full | Half-linear differential equations Ondřej Došlý, Pavel Řehák |
| title_fullStr | Half-linear differential equations Ondřej Došlý, Pavel Řehák |
| title_full_unstemmed | Half-linear differential equations Ondřej Došlý, Pavel Řehák |
| title_short | Half-linear differential equations |
| title_sort | half linear differential equations |
| topic | Differential equations Oscillation theory Differential equations, Linear Differential equations, Nonlinear Ordnung 2 (DE-588)4350619-7 gnd Differentialgleichung (DE-588)4012249-9 gnd |
| topic_facet | Differential equations Oscillation theory Differential equations, Linear Differential equations, Nonlinear Ordnung 2 Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013312604&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000003247 |
| work_keys_str_mv | AT doslyondrej halflineardifferentialequations AT rehakpavel halflineardifferentialequations |