Nonlinear differential equations in ordered spaces / S. Carl, S. Heikkilä.

Format:

Book

Author/Creator:

Carl, S. (Siegfried)

Contributor:

Heikkilä, Seppo.

James A. Crawford Memorial Fund.

Series:

Chapman & Hall/CRC monographs and surveys in pure and applied mathematics ; 111.

Chapman & Hall/CRC monographs and surveys in pure and applied mathematics ; 111

Language:

English

Subjects (All):

Differential equations, Nonlinear.

Ordered topological spaces.

Physical Description:

vi, 323 pages ; 24 cm.

Place of Publication:

Boca Raton : Chapman & Hall/CRC, [2000]

Summary:

Extremality results proved in this monograph for an abstract operator equation provide the theoretical framework for developing new methods that allow the treatment of a variety of discontinuous initial and boundary value problems for both ordinary and partial differential equations, in explicit and implicit forms. By mean of these extremality results, the authors prove the existence of extremal solutions between appropriate upper and lower solutions of first and second order discontinuous implicit and explicit ordinary and functional differential equations.

Contents:

1. Operator equations in ordered spaces and first applications 7

1.1. Operator equations in ordered spaces 7

1.2. Applications to differential equations 11

2. Extremality results for first order differential equations 21

2.1. Explicit initial value problems 22

2.2. Explicit boundary value problems 36

2.3. Explicit functional problems 47

2.4. Implicit functional problems 55

3. Uniqueness, comparison, and well-posedness results for quasilinear differential equations 67

3.1. First order boundary value problems 68

3.2. First order initial value problems 73

3.3. Well-posedness results 81

3.4. Second order problems 89

4. Second order functional differential equations 105

4.1. Explicit Sturm-Liouville boundary value problems 106

4.2. Implicit Sturm-Liouville boundary value problems 115

4.3. Convergence of successive approximations 121

4.4. Explicit phi-Laplacian problems 132

4.5. Implicit phi-Laplacian problems 142

5. Extremality results for quasilinear PDE 153

5.1. Quasilinear elliptic boundary value problems 154

5.2. Quasilinear parabolic problems 171

5.3. Discontinuous quasilinear problems 184

6. Differential inclusions of hemivariational inequality type 191

6.1. Quasilinear elliptic inclusions with state-dependent subdifferentials 192

6.2. State-dependent subdifferentials perturbed by discontinuous nonlinearities 207

6.3. Elliptic inclusions with generalized gradients 213

6.4. Quasilinear parabolic inclusions with state-dependent subdifferentials 229

7. Discontinuous implicit elliptic and parabolic problems 253

7.1. Statement of the problem and notations 253

7.2. Preliminaries 255

7.3. Main results 260

7.4. Implicit elliptic problems 270

A. Analysis in ordered spaces 277

B. Inequalities 282

C. Sobolev spaces 285

D. Pseudomonotone and quasilinear elliptic operators 289

E. Nonlinear first order evolution equations 292

F. Nonsmooth analysis 298.

Notes:

Includes bibliographical references (pages 303-318) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the James A. Crawford Memorial Fund.

ISBN:

1584880686

OCLC:

43903551