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E-raamat: Higher Order Boundary Value Problems On Unbounded Domains: Types Of Solutions, Functional Problems And Applications

(Research Centre In Mathematics & Applications (Cima), Portugal), (Univ De Evora, Portugal)
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Higher Order Boundary Value Problems On Unbounded Domains: Types Of Solutions, Functional Problems And ApplicationsSuurem pilt
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This volume provides a comprehensive overview on different types of higher order boundary value problems defined on the half-line or on the real line (Sturm–Liouville and Lidstone types, impulsive, functional and problems defined by Hammerstein integral equations). It also includes classical and new methods and techniques to deal with the lack of compactness of the related operators.The reader will find a selection of original and recent results in this field, conditions to obtain solutions with particular qualitative properties, such as homoclinic and heteroclinic solutions and its relation with the solutions of Lidstone problems on all the real line.Each chapter contains applications to real phenomena, to classical equations or problems, with a common denominator: they are defined on unbounded intervals and the existing results in the literature are scarce or proven only numerically in discrete cases.The last part features some higher order functional problems, which generalize the classical two-point or multi-point boundary conditions, to more comprehensive data where an overall behavior of the unknown functions and their derivatives is involved.
Preface ix
Introduction xi
Part I Boundary Value Problems on the Half-Line
1(38)
Introduction
3(2)
1 Third-Order Boundary Value Problems
5(14)
1.1 Introduction
5(1)
1.2 Definitions and auxiliary results
6(5)
1.3 Existence and localization result
11(6)
1.4 Example
17(2)
2 General nth-Order Problems
19(6)
2.1 Introduction
19(1)
2.2 Preliminary results
20(2)
2.3 Existence and localization result
22(1)
2.4 Example
23(2)
3 Impulsive Problems on the Half-Line with Infinite Impulse Moments
25(14)
3.1 Introduction
25(1)
3.2 Definitions and preliminary results
26(2)
3.3 Main result
28(8)
3.4 Example
36(3)
Part II Homoclinic Solutions and Lidstone Problems
39(42)
Introduction
41(2)
4 Homoclinic Solutions for Second-Order Problems
43(14)
4.1 Introduction
43(1)
4.2 Preliminaries
44(3)
4.3 Existence and localization of homoclinics
47(5)
4.4 Example of a discontinuous BVP
52(1)
4.5 Duffing equation
53(1)
4.6 Forced cantilever beam equation with damping
54(3)
5 Homoclinic Solutions to Fourth-Order Problems
57(14)
5.1 Introduction
57(2)
5.2 Definitions and auxiliary results
59(2)
5.3 Existence results
61(4)
5.4 Example
65(1)
5.5 Bernoulli--Euler--v. Karman problem
66(1)
5.6 Extended Fisher--Kolmogorov and Swift--Hohenberg problems
67(4)
6 Lidstone Boundary Value Problems
71(10)
6.1 Introduction
71(2)
6.2 Auxiliary definitions and Green's functions
73(2)
6.3 Existence result
75(3)
6.4 An infinite beam resting on granular foundations
78(3)
Part III Heteroclinic Solutions and Hammerstein Equations
81(52)
Introduction
83(2)
7 Heteroclinic Solutions for Semi-linear Problems (i)
85(12)
7.1 Introduction
85(2)
7.2 Definitions and preliminary results
87(2)
7.3 Existence of heteroclinics
89(5)
7.4 Example
94(3)
8 Heteroclinic Solutions for Semi-linear Problems (ii)
97(14)
8.1 Introduction
97(2)
8.2 Auxiliary results
99(3)
8.3 Existence of heteroclinics solutions
102(6)
8.4 Examples
108(3)
9 Heteroclinic Solutions for Semi-linear Problems (iii)
111(16)
9.1 Introduction
111(2)
9.2 Existence results
113(11)
9.3 Example
124(1)
9.4 Singular Φ-Laplacian equations
125(2)
10 Hammerstein Integral Equations with Sign-Changing Kernels
127(6)
10.1 Introduction
127(2)
10.2 Main result
129(2)
10.3 Application to fourth-order BVPs and infinite beams
131(2)
Part IV Functional Boundary Value Problems
133(56)
Introduction
135(4)
11 Second-Order Functional Problems
139(14)
11.1 Introduction
139(1)
11.2 Definitions and auxiliary results
140(2)
11.3 Existence and localization results
142(7)
11.4 Example
149(1)
11.5 Emden-Fowler equation
149(4)
12 Third-Order Functional Problems
153(18)
12.1 Introduction
153(1)
12.2 Definitions and a priori bounds
154(5)
12.3 Existence and localization results
159(9)
12.4 Falkner-Skan equation
168(3)
13 Φ-Laplacian Equations with Functional Boundary Conditions
171(18)
13.1 Introduction
171(2)
13.2 Preliminary results
173(8)
13.3 Existence and localization result
181(5)
13.4 Examples
186(3)
Bibliography 189(10)
Index 199
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