Handbook of Differential Equations: Ordinary Differential Equations

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ISBN: 9780444511287 | 0444511288
Cover: Hardcover
Copyright: 8/1/2004

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The book contains seven survey papers about ordinary differential equations. The common feature of all papers consists in the fact that nonlinear equations are focused on. This reflects the situation in modern mathematical modelling - nonlinear mathematical models are more realistic and describe the real world problems more accurately. The implications are that new methods and approaches have to be looked for, developed and adopted in order to understand and solve nonlinear ordinary differential equations. The purpose of this volume is to inform the mathematical community and also other scientists interested in and using the mathematical apparatus of ordinary differential equations, about some of these methods and possible applications.

Preface

List of Contributors

vii

1. A survey of recent results for initial and boundary value problems singular in the dependent variable

(68)

R.P. Agarwal and D. O'Regan

1. Introduction

(5)

2. Singular boundary value problems

(43)

2.1. Positone problems

(14)

2.2. Singular problems with sign changing nonlinearities

(29)

3. Singular initial value problems

(16)

References

(2)

2. The lower and upper solutions method for boundary value problems

(92)

C. De Coster and P. Habets

Introduction

(2)

1. Well ordered lower and upper solutions

(21)

1.1. A derivative independent periodic problem

(6)

1.2. A priori hounds on the derivatives

(4)

1.3. Derivative dependent periodic problems

(6)

1.4. Derivative dependent Dirichlet problem

(1)

1.5. A derivative independent Dirichlet problem

(2)

1.6. Historical and bibliographical notes

(2)

2. Relation with degree theory

(21)

2.1. The periodic problem

(7)

2.2. The Dirichlet problem

101

(7)

2.3. Non well-ordered lower and upper solutions

108

(6)

2.4. Historical and bibliographical notes

114

(1)

3. Variational methods

115

(20)

3.1. The minimization method

115

(6)

3.2. The minimax method

121

(13)

3.3. Historical and bibliographical notes

134

(1)

4. Monotone methods

135

(22)

4.1. Abstract results

135

(2)

4.2. Well-ordered lower and upper solutions

137

(8)

4.3. Lower and upper solutions in reversed order

145

(7)

4.4. A mixed approximation scheme

152

(2)

4.5. Historical and bibliographical notes

154

(3)

References

157

(4)

3. Half-linear differential equations

161

(198)

O. Došlý

Preface

165

(2)

3A. Basic Theory

167

(30)

1. Existence and uniqueness

167

(4)

1.1. First order half-linear system

167

(1)

1.2. Half-linear trigonometric functions

168

(1)

1.3. Half-linear Prefer transformation

169

(1)

1.4. Existence and uniqueness

170

(1)

2. Sturmian theory

171

(9)

2.1. Riccati equation

171

(1)

2.2. Picone's identity

172

(2)

2.3. Energy functional

174

(1)

2.4. Roundabout theorem

174

(2)

2.5. Sturmian separation and comparison theorems

176

(1)

2.6. Transformation of independent variable

177

(1)

2.7. Reciprocity principle

178

(1)

2.8. Leighton-Wintner oscillation criterion

178

(2)

3. Differences between linear and half-linear equations

180

(3)

3.1. Wronskian identity

180

(1)

3.2. Transformation formula

181

(1)

3.3. Fredholm alternative

182

(1)

4. Some elementary half-linear equations

183

(14)

4.1. Equations with constant coefficients

183

(5)

4.2. Euler-type half-linear differential equation

188

(6)

4.3. Kneser-type oscillation and nonoscillation criteria

194

(3)

3B. Nonoscillatory Equations

197

(58)

5. Nonoscillation criteria

197

(18)

5.1. Variational principle and Wirtinger's inequality

197

(1)

5.2. Nonoscillation criteria via Wirtinger inequality

198

(2)

5.3. Riccati inequality

200

(3)

5.4. Half-linear Hartman-Wintner theorem

203

(3)

5.5. Riccati integral equation and Hille-Wintner comparison theorem

206

(2)

5.6. Hille-Nehari criteria

208

(1)

5.7. Modified Hille-Nehari's criteria

209

(5)

5.8. Comparison theorem with respect to ρ

214

(1)

6. Asymptotic of nonoscillatory solutions

215

(14)

6.1. Integral conditions and classification of solutions

215

(3)

6.2. The case c negative

218

(6)

6.3. Uniqueness in M-

224

(2)

6.4. The case c positive

226

(3)

7. Principal solution

229

(12)

7.1. Principal solution of linear equations

229

(2)

7.2. Mirzov's construction of the principal solution

231

(1)

7.3. Construction of Elbert and Kusano

232

(2)

7.4. Comparison theorem for eventually minimal solutions of Riccati equations

234

(1)

7.5. Sturmian property of the principal solution

235

(1)

7.6. Integral characterization of the principal solution

236

(4)

7.7. Another integral characterization

240

(1)

7.8. Limit characterization of the principal solution

241

(1)

8. Conjugacy and disconjugacy of half-linear equations

241

(14)

8.1. Leighton's conjugacy criterion

241

(1)

8.2. Singular Leighton's theorem

242

(4)

8.3. Lyapunov inequality

246

(2)

8.4. Vallee Poussin-type inequality

248

(1)

8.5. Focal point criteria

249

(3)

8.6. Lyapunov-type focal points and conjugacy criteria

252

(3)

3C. Oscillatory Equations

255

(58)

9. Oscillation criteria

255

(20)

9.1. General observations

255

(1)

9.2. Coles-type criteria

256

(5)

9.3. Generalized Hartman-Wintner's criterion

261

(2)

9.4. Generalized Kamenev criterion

263

(1)

9.5. Another refinement of the Hartman-Wintner theorem

264

(2)

9.6. Half-linear Willet's criteria

266

(2)

9.7. Equations with periodic coefficient

268

(2)

9.8. Equations with almost periodic coefficient

270

(2)

9.9. Generalized H-function averaging technique

272

(3)

10. Various oscillation problems

275

(18)

10.1. Asymptotic formula for distance of zeros of oscillatory solutions

275

(4)

10.2. Half-linear Milloux and Armellini-Tonelli-Sansone theorems

279

(4)

10.3. Strongly and conditionally oscillatory equation

283

(1)

10.4. Oscillation of forced half-linear differential equations

284

(2)

10.5. Oscillation of retarded half-linear equations

286

(7)

11. Half-linear Sturm-Liouville problem

293

(9)

11.1. Basic Sturm-Liouville problem

293

(2)

11.2. Regular problem with indefinite weight

295

(5)

11.3. Singular Sturm-Liouville problem

300

(2)

12. Perturbation principle

302

(11)

12.1. General idea

302

(1)

12.2. Leighton-Wintner type oscillation criterion

303

(1)

12.3. Hille-Nehari-type oscillation criterion

304

(4)

12.4. Hille-Nehari-type nonoscillation criterion

308

(2)

12.5. Perturbed Eviler equation

310

(3)

3D. Related Equations and Problems

313

(48)

13. Half-linear boundary value problems

313

(9)

13.1. Basic boundary value problem

313

(1)

13.2. Variational characterization of eigenvalues

314

(1)

13.3. Nonresonance problem

315

(2)

13.4. Fredholm alternative for the scalar ρ-Laplacian

317

(2)

13.5. Homotopic deformation along ρ and Leray-Schauder degree

319

(2)

13.6. Resonance problem

321

(1)

14. Quasilinear and related differential equations

322

(7)

14.1. Equation (14.1) with constant coefficients

323

(3)

14.2. Emden-Fowler type equation

326

(1)

14.3. More about quasilinear equations

327

(2)

15. Partial differential equations with ρ-Laplacian

329

(8)

15.1. Dirichlet BVP with ρ-Laplacian

330

(3)

15.2. Picone's identity for equations with ρ-Laplacian

333

(2)

15.3. Second eigenvalue of ρ-Laplacian

335

(1)

15.4. Equations involving pseudolaplacian

336

(1)

16. Half-linear difference equations

337

(12)

16.1. Roundabout theorem for half-linear difference equations

338

(3)

16.2. Discrete Leighton-Wintner criterion

341

(2)

16.3. Riccati inequality

343

(2)

16.4. Hille-Nehari nonoscillation criterion

345

(3)

16.5. Half-linear dynamic equations on time scales

348

(1)

References

349

(10)

4. Radial solutions of quasilinear elliptic differential equations

359

(78)

J. Jacobsen and K. Schmitt

1. Introduction

361

(2)

2. Boundary value problems on a ball

363

(29)

2.1. Introduction

363

(1)

2.2. The case when f (0) = 0

364

(25)

2.3. The case when f (0) less than 0 387

2.4. The case when f (0) > 0

389

(3)

3. Problems on annular domains

392

(13)

3.1. Fixed point formulation

393

(3)

3.2. Existence results

396

(4)

3.3. Positone problems

400

(5)

4. The Liouville-Gelfand equation - A case study

405

(7)

4.1. Introduction

405

(1)

4.2. Solution set for the semilinear case

406

(3)

4.3. Solution set for the quasilinear case

409

(3)

5. Rellich-Pohozaev identities

412

(7)

5.1. More general equations

414

(4)

5.2. Critical dimensions

418

(1)

6. Problems linear at infinity

419

(5)

6.1. Landesman-Lazer results

421

(1)

6.2. Nonlinear terms which oscillate

422

(2)

7. Symmetry breaking

424

(3)

8. Whole space problems

427

(4)

References

431

(6)

5. Integrability of polynomial differential systems

437

(96)

J. Llibre

1. Basic notions

439

(94)

1.1. Polynomial systems

440

(1)

1.2. First integrals and invariants

441

(1)

1.3. Integrating factors

442

(2)

1.4. Invariant algebraic curves

444

(2)

1.5. Exponential factors

446

(2)

2. Darbouxian theory of integrability

448

(4)

3. Kapteyn-Bautin theorem

452

(2)

4. On the degree of the invariant algebraic curves

454

(6)

5. Darboux lemma

460

(6)

6. Applications of the Darboux lemma

466

(9)

7. Algebraic limit cycles for quadratic systems

475

(5)

8. Limit cycles and algebraic limit cycles

480

(5)

9. Darbouxian theory of integrability and centres

485

(2)

10. Non-existence of limit cycles

487

(4)

11. The inverse problem

491

(9)

12. Elementary and Liouvillian first integrals

500

(3)

13. Liouvillian first integrals for the planar Lotka-Volterra system

503

(11)

14. On the integrability of two-dimensional flows

514

(4)

15. Darbouxian theory of integrability for polynomial vector fields on surfaces

518

(10)

15.1. Invariant algebraic curves and exponential factors

519

(1)

15.2. The surfaces

520

(4)

15.3. Darbouxian theory

524

(4)

References

528

(5)

6. Global results for the forced pendulum equation

533

(58)

J. Mawhin

1. Introduction

535

(2)

2. Autonomous pendulum equations

537

(4)

2.1. The free conservative pendulum

537

(2)

2.2. The free damped pendulum

539

(1)

2.3. The pendulum with constant torque

540

(1)

3. Periodic solutions of the forced pendulum

541

(24)

3.1. The problems

541

(2)

3.2. The possibly dissipative case c > 0

543

(3)

3.3. The conservative case c = 0

546

(5)

3.4. The dissipative case c > 0

551

(3)

3.5. The degeneracy problem

554

(4)

3.6. Multiplicity and stability results under restrictions of the coefficients

558

(2)

3.7. Many Τ-periodic solutions for special forcings

560

(2)

3.8. Many Τ-periodic solutions for small length pendulum

562

(1)

3.9. Subharmonic solutions when c = 0

563

(2)

4. Rotating solutions and Mather sets

565

(6)

4.1. Periodic solutions of the second kind

565

(1)

4.2. Solutions with an arbitrary rotation number

566

(4)

4.3. Mather set

570

(1)

5. KAM theory and Lagrange stability

571

(6)

5.1. Twist mappings

571

(3)

5.2. Chaotic dynamics

574

(3)

6. Bounded forcing

577

(5)

6.1. Bounded functions and their averages

577

(1)

6.2. A necessary condition for the existence of bounded solutions

577

(1)

6.3. Sufficient conditions for the existence of bounded solutions

578

(2)

6.4. Local uniqueness of bounded solutions

580

(2)

7. Almost periodic forcings

582

(3)

7.1. Almost periodic functions

582

(1)

7.2. Almost periodic solutions

583

(2)

References

585

(6)

7. Waewski method and Conley index

591

(94)

R. Srzednicki

1. Introduction

593

(9)

1.1. Historical background of topological methods

593

(1)

1.2. An outline of the Waiewski method in a historical context

594

(6)

1.3. On the origin of the Conley index theory

600

(1)

1.4. Review of the current exposition

601

(1)

2. Waiewski method for local semiflows and flows

602

(8)

2.1. Local semiflows and Waiewski sets

602

(3)

2.2. Properties of Waiewski sets

605

(1)

2.3. Versions of the Waiewski theorem

606

(4)

3. Waiewski method in ordinary differential equations and inclusions

610

(10)

3.1. Polyfacial sets

610

(4)

3.2. Equations without the uniqueness property and differential inclusions

614

(2)

3.3. Weak inequalities in polyfacial sets

616

(1)

3.4. Generalized polyfacial sets

617

(3)

4. Waiewski method for retarded functional differential equations

620

(6)

4.1. Local semiflows generated by RFDEs

620

(1)

4.2. Waiewski-type theorems for RFDEs

621

(5)

5. Some topological concepts

626

(10)

5.1. Topological pairs, quotient spaces, and pointed spaces

626

(1)

5.2. Homotopy types

627

(2)

5.3. Absolute neighborhood retracts

629

(1)

5.4. Lusternik-Schnirelmann category

630

(2)

5.5. Cup-length

632

(2)

5.6. Lefschetz theorem and the fixed point index

634

(2)

6. Properties of the asymptotic parts of Waiewski sets

636

(2)

6.1. Estimates on the category of the asymptotic part

636

(1)

6.2. Stationary points of gradient-like local semiflows

637

(1)

6.3. Stationary points of local semiflows in the general case

637

(1)

7. Isolating blocks and segments

638

(20)

7.1. Isolating blocks and their structure

638

(4)

7.2. Estimates on the cohomology of the invariant part

642

(6)

7.3. On the number of stationary points of gradient-like flows

648

(3)

7.4. Fixed point index and the index of stationary points

651

(1)

7.5. Isolating segments

652

(6)

8. Selected applications of the Wazewski method

658

(9)

8.1. Asymptotic solutions

659

(3)

8.2. Two-point boundary value problems

662

(2)

8.3. Detection of chaotic dynamics

664

(3)

9. Conley index

667

(13)

9.1. Isolated invariant sets and the Conley index

667

(4)

9.2. Properties of the Conley index

671

(4)

9.3. Examples of Conley indices and an application

675

(5)

9.4. Concluding remarks

680

(1)

References

680

(5)

Author Index

685

(8)

Subject Index

693

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Handbook of Differential Equations: Ordinary Differential Equations

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