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E-raamat: Two-Point Boundary Value Problems: Lower and Upper Solutions

(Université du Littoral-Côte d'Opale, France), (Université Catholique de Louvain, Belgium)
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Two-Point Boundary Value Problems: Lower and Upper SolutionsSuurem pilt
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This book introduces the method of lower and upper solutions for ordinary differential equations. This method is known to be both easy and powerful to solve second order boundary value problems. Besides an extensive introduction to the method, the first half of the book describes some recent and more involved results on this subject. These concern the combined use of the method with degree theory, with variational methods and positive operators. The second half of the book concerns applications. This part exemplifies the method and provides the reader with a fairly large introduction to the problematic of boundary value problems. Although the book concerns mainly ordinary differential equations, some attention is given to other settings such as partial differential equations or functional differential equations. A detailed history of the problem is described in the introduction.



Key features:



- Presentation of the fundamental features of the method

- Actual construction of lower and upper solutions in problems

- Working applications

- Illustrate theorems by examples

- Description of the history of the method

- Bibliographical notes

Key features:


- Presentation of the fundamental features of the method

- Actual construction of lower and upper solutions in problems

- Working applications

- Illustrate theorems by examples

- Description of the history of the method

- Bibliographical notes

Arvustused

"In this excellent monograph the authors present a survey of classical and recent results in the theory of lower and upper solutions applied to two-point boundary value problems...essential for researchers in this field." --Zentralblatt MATH, Two-Point Boundary Value Problems: Lower and Upper Solutions

Contents v
Preface ix
Notations xi
Introduction -- The History 1(1)
The First Steps
1(4)
Other Boundary Value Problems
5(2)
Maximal and Minimal Solution
7(1)
The Nagumo Condition
8(3)
Degree Theory
11(1)
Non Well-ordered Lower and Upper Solutions
12(1)
Variational Methods
13(2)
Monotone Methods
15(4)
Applications
19(6)
The Periodic Problem
25(50)
A First Approach of Lower and Upper Solutions
25(6)
Existence of C2-Solutions
31(6)
Existence of W2,1-Solutions
37(5)
A Priori Bound on the Derivative
42(11)
BVP with Derivative Dependence: C2-Solutions
53(6)
BVP with Derivative Dependence: W2,1 Solutions
59(16)
The Separated BVP
75(60)
C2-Solutions
75(13)
W2,1-Solutions
88(9)
One-Sided Nagumo Condition
97(5)
Dirichlet Problem
102(12)
Other Boundary Value Problems
114(12)
PDE Problems
126(9)
Relation with Degree Theory
135(54)
The Periodic Problem
135(21)
The Dirichlet Problem
156(12)
Non Well-ordered Lower and Upper Solutions
168(21)
Variational Methods
189(52)
The Minimization Method
189(6)
Local Minimum of the Functional
195(7)
The Minimax Method
202(22)
A Reduction Method
224(17)
Monotone Iterative Methods
241(38)
An Abstract Formulation
241(2)
Well-ordered Lower and Upper Solutions
243(16)
Lower and Upper Solutions in Reversed Order
259(13)
A Mixed Approximation Scheme
272(7)
Parametric Multiplicity Problems
279(36)
Periodic Solutions of the Lienard Equation
279(6)
Periodic Solutions of the Rayleigh Equation
285(9)
A Two Parameters Dirichlet Problem
294(10)
Strong Resonance Problems
304(11)
Resonance and Non-resonance
315(30)
A Periodic Non-resonant Problem
315(4)
Resonance Conditions for the Periodic Problem
319(2)
A Non-resonant Dirichlet Problem
321(5)
Landesman-Lazer conditions for the Dirichlet problem
326(4)
Other Resonance Conditions
330(3)
Fredholm Alternative Results
333(4)
Derivative Dependent Dirichlet Problem
337(8)
Positive Solutions
345(30)
Existence of One Solution
345(13)
Some Multiplicity Results
358(7)
Parametric Problems
365(10)
Problems with Singular Forces
375(30)
A Periodic Problem
375(13)
A Sublinear Dirichlet Problem
388(8)
A Derivative Dependent Dirichlet Problem
396(3)
A Multiplicity Result
399(6)
Singular Perturbations
405(20)
Boundary Layers at One End Point
405(8)
Boundary Layers at Both End Points
413(2)
Angular Solutions
415(6)
Algebraic Boundary Layers
421(4)
Bibliographical Notes
425(10)
Notes on
Chapters 1 and 2
425(2)
Notes on
Chapter 3
427(1)
Notes on
Chapter 4
428(1)
Notes on
Chapter 5
429(1)
Notes on
Chapter 6
430(1)
Notes on
Chapter 7
431(1)
Notes on
Chapter 8
432(1)
Notes on
Chapter 9
432(1)
Notes on
Chapter 10
433(1)
Notes on the Appendix
433(2)
Appendix
435(28)
Degree Theory
435(3)
Variational Methods
438(3)
Spectral Results
441(11)
Inequalities
452(4)
Maximum Principle
456(3)
Anti-maximum Principle
459(4)
Bibliography 463(25)
Index 488


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