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E-raamat: Ordinary Differential Equations And Boundary Value Problems - Volume Ii: Boundary Value Problems

(Univ Of New Orleans, Usa), (Univ Of Tennessee At Chattanooga, Usa), (Univ Of Tennessee At Chattanooga, Usa), (Baylor Univ, Usa)
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Ordinary Differential Equations And Boundary Value Problems - Volume Ii: Boundary Value ProblemsSuurem pilt
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The authors give a systematic introduction to boundary value problems (BVPs) for ordinary differential equations. The book is a graduate level text and good to use for individual study. With the relaxed style of writing, the reader will find it to be an enticing invitation to join this important area of mathematical research. Starting with the basics of boundary value problems for ordinary differential equations, linear equations and the construction of Green's functions are presented clearly. A discussion of the important question of the existence of solutions to both linear and nonlinear problems plays a central role in this volume and this includes solution matching and the comparison of eigenvalues. The important and very active research area on existence and multiplicity of positive solutions is treated in detail. The last chapter is devoted to nodal solutions for BVPs with separated boundary conditions as well as for non-local problems. While this Volume II complements Volume I: Advanced Ordinary Differential Equations , it can be used as a stand-alone work.

Preface vii
1 Introduction to Boundary Value Problems
1(24)
1.1 Introduction
1(1)
1.2 Disconjugacy
2(16)
1.3 Disconjugacy for Equations of Arbitrary Order
18(7)
Bibliography
23(2)
2 Linear Problems and Green's Functions
25(24)
2.1 Introduction to Linear ODE's
25(6)
2.2 Linear BVP's
31(2)
2.3 Green's Functions for Linear BVP's
33(2)
2.4 Properties of the Green's Function
35(14)
2.4.1 Second method for constructing the Green's function
36(1)
2.4.2 Another approach using the properties
37(3)
2.4.3 Green's functions for general linear homogeneous BVP's
40(7)
Bibliography
47(2)
3 Existence of Solutions I
49(58)
3.1 Local Existence Theorems for Solutions of BVP's for the Nonlinear Equation, y" = f(x, y, y')
49(14)
3.2 Linear Second Order Inequalities
63(12)
3.3 Existence of Solutions of BVP's for Nonlinear ODE's
75(14)
3.4 Existence of Solutions of BVP's for Nonlinear Higher Order ODE's
89(18)
Bibliography
105(2)
4 Existence of Solutions II
107(28)
4.1 Existence of Solutions of BVP's Associated with y"' = f(x, y, y', y")
107(20)
4.2 A Converse of Lemma 4.3
127(8)
Bibliography
133(2)
5 Solution Matching
135(14)
5.1 Introduction
135(6)
5.1.1 Uniqueness and existence of solutions by matching
136(4)
5.1.2 Uniqueness and existence of solutions for a larger class of BVP's by matching
140(1)
5.2 Five-Point Nonlocal BVP's for nth Order ODE's by Solution Matching
141(8)
5.2.1 Uniqueness of solutions
142(2)
5.2.2 Existence of solutions
144(3)
Bibliography
147(2)
6 Comparison of Smallest Eigenvalues
149(38)
6.1 Comparison Theorems for 2-Point Conjugate Eigenvalue Problems
149(7)
6.2 Eigenvalue Comparisons for Nonlocal BVP's
156(4)
6.3 Solutions of Second Order Multi-Point BVP's
160(27)
6.3.1 Preliminary lemmas
161(7)
6.3.2 Existence results
168(17)
Bibliography
185(2)
7 BVP's for Functional Differential Equations
187(34)
7.1 Existence of Solutions for BVP's for Second Order Functional Differential Equations
187(6)
7.2 Periodic Solutions for First Order Functional Differential Equations
193(6)
7.3 Periodic Solutions for Functional Differential Equations with Sign-Changing Nonlinearities
199(22)
7.3.1 Preliminary results
200(8)
7.3.2 Existence results
208(3)
7.3.3 Proofs of the existence results
211(8)
Bibliography
219(2)
8 Positive Solutions
221(50)
8.1 Positive Solutions for Nonlinear Eigenvalue Problems
221(5)
8.1.1 Solutions in the cone
222(4)
8.2 Multiplicity of Positive Solutions
226(4)
8.3 Positive Solutions for Nonlinear Difference Equations
230(7)
8.4 Multiple Symmetric Positive Solutions
237(4)
8.5 Positive Solutions for a Singular Third Order BVP
241(8)
8.5.1 A priori, bounds on norms of solutions
245(2)
8.5.2 Existence of positive solutions
247(2)
8.6 Positive Solutions for Third Order BVP's with p-Laplacian
249(22)
8.6.1 Main results
250(2)
8.6.2 Proofs of the main results
252(15)
Bibliography
267(4)
9 Boundary Data Smoothness
271(12)
9.1 Differentiation of Solutions with Respect to Boundary Conditions
271(12)
Bibliography
281(2)
10 Nodal Solutions of BVP's for ODE's
283(46)
10.1 Nodal Solutions of Nonlinear BVP's with Separated BC's
283(19)
10.1.1 Main results
284(3)
10.1.2 Proofs of the main results
287(15)
10.2 Nodal Solutions of Nonlocal BVP's --- I
302(9)
10.2.1 Main results
303(4)
10.2.2 Proofs of the main results
307(4)
10.3 Nodal Solutions of Nonlocal BVP's --- II
311(18)
10.3.1 Existence and nonexistence of nodal solutions
313(6)
10.3.2 Dependence of nodal solutions on the problem
319(8)
Bibliography
327(2)
Index 329
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